LMFDB - Embedded newform 810.3.j.e.269.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,3,Mod(269,810)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([1, 3]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("810.269");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	810.j (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.0709014132$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.2
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	810.269
Dual form			810.3.j.e.539.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(1.29410 - 4.82963i) q^{5} +(-4.33013 - 2.50000i) q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+(-0.707107 + 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(1.29410 - 4.82963i) q^{5} +(-4.33013 - 2.50000i) q^{7} +2.82843 q^{8} +(5.00000 + 5.00000i) q^{10} +(1.22474 + 0.707107i) q^{11} +(7.79423 - 4.50000i) q^{13} +(6.12372 - 3.53553i) q^{14} +(-2.00000 + 3.46410i) q^{16} -11.3137 q^{17} +21.0000 q^{19} +(-9.65926 + 2.58819i) q^{20} +(-1.73205 + 1.00000i) q^{22} +(0.707107 + 1.22474i) q^{23} +(-21.6506 - 12.5000i) q^{25} +12.7279i q^{26} +10.0000i q^{28} +(33.0681 + 19.0919i) q^{29} +(-20.0000 - 34.6410i) q^{31} +(-2.82843 - 4.89898i) q^{32} +(8.00000 - 13.8564i) q^{34} +(-17.6777 + 17.6777i) q^{35} -25.0000i q^{37} +(-14.8492 + 25.7196i) q^{38} +(3.66025 - 13.6603i) q^{40} +(-45.3156 + 26.1630i) q^{41} +(-55.4256 - 32.0000i) q^{43} -2.82843i q^{44} -2.00000 q^{46} +(11.3137 - 19.5959i) q^{47} +(-12.0000 - 20.7846i) q^{49} +(30.6186 - 17.6777i) q^{50} +(-15.5885 - 9.00000i) q^{52} -72.1249 q^{53} +(5.00000 - 5.00000i) q^{55} +(-12.2474 - 7.07107i) q^{56} +(-46.7654 + 27.0000i) q^{58} +(-78.3837 + 45.2548i) q^{59} +(48.5000 - 84.0045i) q^{61} +56.5685 q^{62} +8.00000 q^{64} +(-11.6469 - 43.4667i) q^{65} +(-113.449 + 65.5000i) q^{67} +(11.3137 + 19.5959i) q^{68} +(-9.15064 - 34.1506i) q^{70} +89.0955i q^{71} -17.0000i q^{73} +(30.6186 + 17.6777i) q^{74} +(-21.0000 - 36.3731i) q^{76} +(-3.53553 - 6.12372i) q^{77} +(58.5000 - 101.325i) q^{79} +(14.1421 + 14.1421i) q^{80} -74.0000i q^{82} +(-28.9914 + 50.2145i) q^{83} +(-14.6410 + 54.6410i) q^{85} +(78.3837 - 45.2548i) q^{86} +(3.46410 + 2.00000i) q^{88} -147.078i q^{89} -45.0000 q^{91} +(1.41421 - 2.44949i) q^{92} +(16.0000 + 27.7128i) q^{94} +(27.1760 - 101.422i) q^{95} +(35.5070 + 20.5000i) q^{97} +33.9411 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} + 40 q^{10} - 16 q^{16} + 168 q^{19} - 160 q^{31} + 64 q^{34} - 40 q^{40} - 16 q^{46} - 96 q^{49} + 40 q^{55} + 388 q^{61} + 64 q^{64} + 100 q^{70} - 168 q^{76} + 468 q^{79} + 160 q^{85} - 360 q^{91} + 128 q^{94}+O(q^{100}) $ 8 * q - 8 * q^4 + 40 * q^10 - 16 * q^16 + 168 * q^19 - 160 * q^31 + 64 * q^34 - 40 * q^40 - 16 * q^46 - 96 * q^49 + 40 * q^55 + 388 * q^61 + 64 * q^64 + 100 * q^70 - 168 * q^76 + 468 * q^79 + 160 * q^85 - 360 * q^91 + 128 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.707107	+	1.22474i	−0.353553	+	0.612372i
$2$	−0.707107	+	1.22474i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−1.00000	−	1.73205i	−0.250000	−	0.433013i
$5$	1.29410	−	4.82963i	0.258819	−	0.965926i
$5$	1.29410	−	4.82963i	0.258819	−	0.965926i
$6$		0			0
$7$	−4.33013	−	2.50000i	−0.618590	−	0.357143i	0.157730	−	0.987482i	$-0.449582\pi$
$7$	−4.33013	−	2.50000i	−0.618590	−	0.357143i	−0.776320	+	0.630339i	$0.782916\pi$
$8$	2.82843			0.353553
$9$		0			0
$10$	5.00000	+	5.00000i	0.500000	+	0.500000i
$11$	1.22474	+	0.707107i	0.111340	+	0.0642824i	0.554636	−	0.832093i	$-0.312857\pi$
$11$	1.22474	+	0.707107i	0.111340	+	0.0642824i	−0.443296	+	0.896375i	$0.646191\pi$
$12$		0			0
$13$	7.79423	−	4.50000i	0.599556	−	0.346154i	−0.169311	−	0.985563i	$-0.554154\pi$
$13$	7.79423	−	4.50000i	0.599556	−	0.346154i	0.768867	+	0.639409i	$0.220821\pi$
$14$	6.12372	−	3.53553i	0.437409	−	0.252538i
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.125000	+	0.216506i
$17$	−11.3137			−0.665512			−0.332756	−	0.943013i	$-0.607979\pi$
$17$	−11.3137			−0.665512			−0.332756	+	0.943013i	$0.607979\pi$
$18$		0			0
$19$	21.0000			1.10526			0.552632	−	0.833426i	$-0.313624\pi$
$19$	21.0000			1.10526			0.552632	+	0.833426i	$0.313624\pi$
$20$	−9.65926	+	2.58819i	−0.482963	+	0.129410i
$21$		0			0
$22$	−1.73205	+	1.00000i	−0.0787296	+	0.0454545i
$23$	0.707107	+	1.22474i	0.0307438	+	0.0532498i	0.880988	−	0.473139i	$-0.156879\pi$
$23$	0.707107	+	1.22474i	0.0307438	+	0.0532498i	−0.850244	+	0.526389i	$0.823546\pi$
$24$		0			0
$25$	−21.6506	−	12.5000i	−0.866025	−	0.500000i
$26$			12.7279i			0.489535i
$27$		0			0
$28$			10.0000i			0.357143i
$29$	33.0681	+	19.0919i	1.14028	+	0.658341i	0.946500	−	0.322704i	$-0.104592\pi$
$29$	33.0681	+	19.0919i	1.14028	+	0.658341i	0.193780	+	0.981045i	$0.437925\pi$
$30$		0			0
$31$	−20.0000	−	34.6410i	−0.645161	−	1.11745i	−0.984264	−	0.176703i	$-0.943457\pi$
$31$	−20.0000	−	34.6410i	−0.645161	−	1.11745i	0.339103	−	0.940749i	$-0.389876\pi$
$32$	−2.82843	−	4.89898i	−0.0883883	−	0.153093i
$33$		0			0
$34$	8.00000	−	13.8564i	0.235294	−	0.407541i
$35$	−17.6777	+	17.6777i	−0.505076	+	0.505076i
$36$		0			0
$37$		−	25.0000i		−	0.675676i	−0.941204	−	0.337838i	$-0.890304\pi$
$37$		−	25.0000i		−	0.675676i	0.941204	−	0.337838i	$-0.109696\pi$
$38$	−14.8492	+	25.7196i	−0.390770	+	0.676833i
$39$		0			0
$40$	3.66025	−	13.6603i	0.0915064	−	0.341506i
$41$	−45.3156	+	26.1630i	−1.10526	+	0.638121i	−0.937597	−	0.347724i	$-0.886955\pi$
$41$	−45.3156	+	26.1630i	−1.10526	+	0.638121i	−0.167661	+	0.985845i	$0.553621\pi$
$42$		0			0
$43$	−55.4256	−	32.0000i	−1.28897	−	0.744186i	−0.310498	−	0.950574i	$-0.600496\pi$
$43$	−55.4256	−	32.0000i	−1.28897	−	0.744186i	−0.978470	+	0.206388i	$0.933829\pi$
$44$		−	2.82843i		−	0.0642824i
$45$		0			0
$46$	−2.00000			−0.0434783
$47$	11.3137	−	19.5959i	0.240717	−	0.416934i	−0.720202	−	0.693765i	$-0.755951\pi$
$47$	11.3137	−	19.5959i	0.240717	−	0.416934i	0.960919	+	0.276830i	$0.0892840\pi$
$48$		0			0
$49$	−12.0000	−	20.7846i	−0.244898	−	0.424176i
$50$	30.6186	−	17.6777i	0.612372	−	0.353553i
$51$		0			0
$52$	−15.5885	−	9.00000i	−0.299778	−	0.173077i
$53$	−72.1249			−1.36085			−0.680424	−	0.732819i	$-0.738204\pi$
$53$	−72.1249			−1.36085			−0.680424	+	0.732819i	$0.738204\pi$
$54$		0			0
$55$	5.00000	−	5.00000i	0.0909091	−	0.0909091i
$56$	−12.2474	−	7.07107i	−0.218704	−	0.126269i
$57$		0			0
$58$	−46.7654	+	27.0000i	−0.806300	+	0.465517i
$59$	−78.3837	+	45.2548i	−1.32854	+	0.767031i	−0.985073	−	0.172135i	$-0.944933\pi$
$59$	−78.3837	+	45.2548i	−1.32854	+	0.767031i	−0.343463	+	0.939166i	$0.611600\pi$
$60$		0			0
$61$	48.5000	−	84.0045i	0.795082	−	1.37712i	−0.127705	−	0.991812i	$-0.540761\pi$
$61$	48.5000	−	84.0045i	0.795082	−	1.37712i	0.922787	−	0.385310i	$-0.125906\pi$
$62$	56.5685			0.912396
$63$		0			0
$64$	8.00000			0.125000
$65$	−11.6469	−	43.4667i	−0.179182	−	0.668718i
$66$		0			0
$67$	−113.449	+	65.5000i	−1.69327	+	0.977612i	−0.741429	+	0.671032i	$0.765851\pi$
$67$	−113.449	+	65.5000i	−1.69327	+	0.977612i	−0.951845	+	0.306580i	$0.900815\pi$
$68$	11.3137	+	19.5959i	0.166378	+	0.288175i
$69$		0			0
$70$	−9.15064	−	34.1506i	−0.130723	−	0.487866i
$71$			89.0955i			1.25487i	0.778671	+	0.627433i	$0.215894\pi$
$71$			89.0955i			1.25487i	−0.778671	+	0.627433i	$0.784106\pi$
$72$		0			0
$73$		−	17.0000i		−	0.232877i	−0.993198	−	0.116438i	$-0.962852\pi$
$73$		−	17.0000i		−	0.232877i	0.993198	−	0.116438i	$-0.0371477\pi$
$74$	30.6186	+	17.6777i	0.413765	+	0.238887i
$75$		0			0
$76$	−21.0000	−	36.3731i	−0.276316	−	0.478593i
$77$	−3.53553	−	6.12372i	−0.0459160	−	0.0795289i
$78$		0			0
$79$	58.5000	−	101.325i	0.740506	−	1.28259i	−0.211759	−	0.977322i	$-0.567919\pi$
$79$	58.5000	−	101.325i	0.740506	−	1.28259i	0.952265	−	0.305273i	$-0.0987476\pi$
$80$	14.1421	+	14.1421i	0.176777	+	0.176777i
$81$		0			0
$82$		−	74.0000i		−	0.902439i
$83$	−28.9914	+	50.2145i	−0.349294	+	0.604994i	−0.986124	−	0.166009i	$-0.946912\pi$
$83$	−28.9914	+	50.2145i	−0.349294	+	0.604994i	0.636830	+	0.771004i	$0.280245\pi$
$84$		0			0
$85$	−14.6410	+	54.6410i	−0.172247	+	0.642835i
$86$	78.3837	−	45.2548i	0.911438	−	0.526219i
$87$		0			0
$88$	3.46410	+	2.00000i	0.0393648	+	0.0227273i
$89$		−	147.078i		−	1.65256i	−0.563257	−	0.826282i	$-0.690452\pi$
$89$		−	147.078i		−	1.65256i	0.563257	−	0.826282i	$-0.309548\pi$
$90$		0			0
$91$	−45.0000			−0.494505
$92$	1.41421	−	2.44949i	0.0153719	−	0.0266249i
$93$		0			0
$94$	16.0000	+	27.7128i	0.170213	+	0.294817i
$95$	27.1760	−	101.422i	0.286063	−	1.06760i
$96$		0			0
$97$	35.5070	+	20.5000i	0.366052	+	0.211340i	0.671732	−	0.740794i	$-0.265551\pi$
$97$	35.5070	+	20.5000i	0.366052	+	0.211340i	−0.305680	+	0.952134i	$0.598884\pi$
$98$	33.9411			0.346338
$99$		0			0
$100$			50.0000i			0.500000i
$101$	−78.3837	−	45.2548i	−0.776076	−	0.448068i	0.0589618	−	0.998260i	$-0.481221\pi$
$101$	−78.3837	−	45.2548i	−0.776076	−	0.448068i	−0.835038	+	0.550193i	$0.814554\pi$
$102$		0			0
$103$	−11.2583	+	6.50000i	−0.109304	+	0.0631068i	−0.553655	−	0.832746i	$-0.686768\pi$
$103$	−11.2583	+	6.50000i	−0.109304	+	0.0631068i	0.444351	+	0.895853i	$0.353434\pi$
$104$	22.0454	−	12.7279i	0.211975	−	0.122384i
$105$		0			0
$106$	51.0000	−	88.3346i	0.481132	−	0.833345i
$107$	123.037			1.14987			0.574937	−	0.818197i	$-0.305026\pi$
$107$	123.037			1.14987			0.574937	+	0.818197i	$0.305026\pi$
$108$		0			0
$109$	8.00000			0.0733945			0.0366972	−	0.999326i	$-0.488316\pi$
$109$	8.00000			0.0733945			0.0366972	+	0.999326i	$0.488316\pi$
$110$	2.58819	+	9.65926i	0.0235290	+	0.0878114i
$111$		0			0
$112$	17.3205	−	10.0000i	0.154647	−	0.0892857i
$113$	−19.0919	−	33.0681i	−0.168955	−	0.292638i	0.769098	−	0.639131i	$-0.220706\pi$
$113$	−19.0919	−	33.0681i	−0.168955	−	0.292638i	−0.938053	+	0.346493i	$0.887372\pi$
$114$		0			0
$115$	6.83013	−	1.83013i	0.0593924	−	0.0159141i
$116$		−	76.3675i		−	0.658341i
$117$		0			0
$118$		−	128.000i		−	1.08475i
$119$	48.9898	+	28.2843i	0.411679	+	0.237683i
$120$		0			0
$121$	−59.5000	−	103.057i	−0.491736	−	0.851711i
$122$	68.5894	+	118.800i	0.562208	+	0.973773i
$123$		0			0
$124$	−40.0000	+	69.2820i	−0.322581	+	0.558726i
$125$	−88.3883	+	88.3883i	−0.707107	+	0.707107i
$126$		0			0
$127$			8.00000i			0.0629921i	0.999504	+	0.0314961i	$0.0100272\pi$
$127$			8.00000i			0.0629921i	−0.999504	+	0.0314961i	$0.989973\pi$
$128$	−5.65685	+	9.79796i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	61.4711	+	16.4711i	0.472855	+	0.126701i
$131$	117.576	−	67.8823i	0.897523	−	0.518185i	0.0211272	−	0.999777i	$-0.493274\pi$
$131$	117.576	−	67.8823i	0.897523	−	0.518185i	0.876396	+	0.481592i	$0.159941\pi$
$132$		0			0
$133$	−90.9327	−	52.5000i	−0.683704	−	0.394737i
$134$		−	185.262i		−	1.38255i
$135$		0			0
$136$	−32.0000			−0.235294
$137$	133.643	−	231.477i	0.975498	−	1.68961i	0.297215	−	0.954811i	$-0.403942\pi$
$137$	133.643	−	231.477i	0.975498	−	1.68961i	0.678283	−	0.734801i	$-0.262724\pi$
$138$		0			0
$139$	18.5000	+	32.0429i	0.133094	+	0.230525i	0.924868	−	0.380289i	$-0.124176\pi$
$139$	18.5000	+	32.0429i	0.133094	+	0.230525i	−0.791774	+	0.610814i	$0.790842\pi$
$140$	48.2963	+	12.9410i	0.344974	+	0.0924354i
$141$		0			0
$142$	−109.119	−	63.0000i	−0.768445	−	0.443662i
$143$	12.7279			0.0890064
$144$		0			0
$145$	135.000	−	135.000i	0.931034	−	0.931034i
$146$	20.8207	+	12.0208i	0.142607	+	0.0823344i
$147$		0			0
$148$	−43.3013	+	25.0000i	−0.292576	+	0.168919i
$149$	−225.353	+	130.108i	−1.51244	+	0.873206i	−0.512542	+	0.858662i	$0.671296\pi$
$149$	−225.353	+	130.108i	−1.51244	+	0.873206i	−0.999894	+	0.0145438i	$0.995370\pi$
$150$		0			0
$151$	−54.5000	+	94.3968i	−0.360927	+	0.625144i	−0.988114	−	0.153725i	$-0.950873\pi$
$151$	−54.5000	+	94.3968i	−0.360927	+	0.625144i	0.627187	+	0.778869i	$0.284206\pi$
$152$	59.3970			0.390770
$153$		0			0
$154$	10.0000			0.0649351
$155$	−193.185	+	51.7638i	−1.24636	+	0.333960i
$156$		0			0
$157$	102.191	−	59.0000i	0.650898	−	0.375796i	−0.137902	−	0.990446i	$-0.544036\pi$
$157$	102.191	−	59.0000i	0.650898	−	0.375796i	0.788800	+	0.614650i	$0.210703\pi$
$158$	82.7315	+	143.295i	0.523617	+	0.906931i
$159$		0			0
$160$	−27.3205	+	7.32051i	−0.170753	+	0.0457532i
$161$		−	7.07107i		−	0.0439197i
$162$		0			0
$163$			203.000i			1.24540i	0.782461	+	0.622699i	$0.213964\pi$
$163$			203.000i			1.24540i	−0.782461	+	0.622699i	$0.786036\pi$
$164$	90.6311	+	52.3259i	0.552629	+	0.319060i
$165$		0			0
$166$	−41.0000	−	71.0141i	−0.246988	−	0.427796i
$167$	−50.9117	−	88.1816i	−0.304860	−	0.528034i	0.672370	−	0.740215i	$-0.265276\pi$
$167$	−50.9117	−	88.1816i	−0.304860	−	0.528034i	−0.977230	+	0.212182i	$0.931943\pi$
$168$		0			0
$169$	−44.0000	+	76.2102i	−0.260355	+	0.450948i
$170$	−56.5685	−	56.5685i	−0.332756	−	0.332756i
$171$		0			0
$172$			128.000i			0.744186i
$173$	5.65685	−	9.79796i	0.0326986	−	0.0566356i	−0.849213	−	0.528050i	$-0.822923\pi$
$173$	5.65685	−	9.79796i	0.0326986	−	0.0566356i	0.881912	+	0.471415i	$0.156257\pi$
$174$		0			0
$175$	62.5000	+	108.253i	0.357143	+	0.618590i
$176$	−4.89898	+	2.82843i	−0.0278351	+	0.0160706i
$177$		0			0
$178$	180.133	+	104.000i	1.01198	+	0.584270i
$179$			125.865i			0.703156i	0.936159	+	0.351578i	$0.114355\pi$
$179$			125.865i			0.703156i	−0.936159	+	0.351578i	$0.885645\pi$
$180$		0			0
$181$	−127.000			−0.701657			−0.350829	−	0.936440i	$-0.614100\pi$
$181$	−127.000			−0.701657			−0.350829	+	0.936440i	$0.614100\pi$
$182$	31.8198	−	55.1135i	0.174834	−	0.302822i
$183$		0			0
$184$	2.00000	+	3.46410i	0.0108696	+	0.0188266i
$185$	−120.741	−	32.3524i	−0.652653	−	0.174878i
$186$		0			0
$187$	−13.8564	−	8.00000i	−0.0740984	−	0.0427807i
$188$	−45.2548			−0.240717
$189$		0			0
$190$	105.000	+	105.000i	0.552632	+	0.552632i
$191$	−88.1816	−	50.9117i	−0.461684	−	0.266553i	0.251068	−	0.967969i	$-0.419218\pi$
$191$	−88.1816	−	50.9117i	−0.461684	−	0.266553i	−0.712752	+	0.701416i	$0.752552\pi$
$192$		0			0
$193$	234.693	−	135.500i	1.21603	−	0.702073i	0.251960	−	0.967738i	$-0.418925\pi$
$193$	234.693	−	135.500i	1.21603	−	0.702073i	0.964065	+	0.265665i	$0.0855915\pi$
$194$	−50.2145	+	28.9914i	−0.258838	+	0.149440i
$195$		0			0
$196$	−24.0000	+	41.5692i	−0.122449	+	0.212088i
$197$	316.784			1.60804			0.804020	−	0.594602i	$-0.202691\pi$
$197$	316.784			1.60804			0.804020	+	0.594602i	$0.202691\pi$
$198$		0			0
$199$	147.000			0.738693			0.369347	−	0.929292i	$-0.379581\pi$
$199$	147.000			0.738693			0.369347	+	0.929292i	$0.379581\pi$
$200$	−61.2372	−	35.3553i	−0.306186	−	0.176777i
$201$		0			0
$202$	110.851	−	64.0000i	0.548769	−	0.316832i
$203$	−95.4594	−	165.341i	−0.470243	−	0.814486i
$204$		0			0
$205$	67.7147	+	252.715i	0.330316	+	1.23275i
$206$		−	18.3848i		−	0.0892465i
$207$		0			0
$208$			36.0000i			0.173077i
$209$	25.7196	+	14.8492i	0.123060	+	0.0710490i
$210$		0			0
$211$	−70.5000	−	122.110i	−0.334123	−	0.578718i	0.649193	−	0.760624i	$-0.275107\pi$
$211$	−70.5000	−	122.110i	−0.334123	−	0.578718i	−0.983316	+	0.181906i	$0.941774\pi$
$212$	72.1249	+	124.924i	0.340212	+	0.589264i
$213$		0			0
$214$	−87.0000	+	150.688i	−0.406542	+	0.704151i
$215$	−226.274	+	226.274i	−1.05244	+	1.05244i
$216$		0			0
$217$			200.000i			0.921659i
$218$	−5.65685	+	9.79796i	−0.0259489	+	0.0449448i
$219$		0			0
$220$	−13.6603	−	3.66025i	−0.0620921	−	0.0166375i
$221$	−88.1816	+	50.9117i	−0.399012	+	0.230370i
$222$		0			0
$223$	6.92820	+	4.00000i	0.0310682	+	0.0179372i	0.515454	−	0.856917i	$-0.327623\pi$
$223$	6.92820	+	4.00000i	0.0310682	+	0.0179372i	−0.484385	+	0.874855i	$0.660957\pi$
$224$			28.2843i			0.126269i
$225$		0			0
$226$	54.0000			0.238938
$227$	−34.6482	+	60.0125i	−0.152635	+	0.264372i	−0.932196	−	0.361955i	$-0.882109\pi$
$227$	−34.6482	+	60.0125i	−0.152635	+	0.264372i	0.779560	+	0.626327i	$0.215443\pi$
$228$		0			0
$229$	4.00000	+	6.92820i	0.0174672	+	0.0302542i	0.874627	−	0.484797i	$-0.161106\pi$
$229$	4.00000	+	6.92820i	0.0174672	+	0.0302542i	−0.857160	+	0.515051i	$0.827773\pi$
$230$	−2.58819	+	9.65926i	−0.0112530	+	0.0419968i
$231$		0			0
$232$	93.5307	+	54.0000i	0.403150	+	0.232759i
$233$	−316.784			−1.35959			−0.679794	−	0.733403i	$-0.737931\pi$
$233$	−316.784			−1.35959			−0.679794	+	0.733403i	$0.737931\pi$
$234$		0			0
$235$	−80.0000	−	80.0000i	−0.340426	−	0.340426i
$236$	156.767	+	90.5097i	0.664268	+	0.383516i
$237$		0			0
$238$	−69.2820	+	40.0000i	−0.291101	+	0.168067i
$239$	177.588	−	102.530i	0.743046	−	0.428998i	−0.0801297	−	0.996784i	$-0.525533\pi$
$239$	177.588	−	102.530i	0.743046	−	0.428998i	0.823176	+	0.567787i	$0.192200\pi$
$240$		0			0
$241$	−39.5000	+	68.4160i	−0.163900	+	0.283884i	−0.936264	−	0.351297i	$-0.885741\pi$
$241$	−39.5000	+	68.4160i	−0.163900	+	0.283884i	0.772364	+	0.635180i	$0.219074\pi$
$242$	168.291			0.695419
$243$		0			0
$244$	−194.000			−0.795082
$245$	−115.911	+	31.0583i	−0.473107	+	0.126769i
$246$		0			0
$247$	163.679	−	94.5000i	0.662667	−	0.382591i
$248$	−56.5685	−	97.9796i	−0.228099	−	0.395079i
$249$		0			0
$250$	−45.7532	−	170.753i	−0.183013	−	0.683013i
$251$			46.6690i			0.185932i	0.995669	+	0.0929662i	$0.0296349\pi$
$251$			46.6690i			0.185932i	−0.995669	+	0.0929662i	$0.970365\pi$
$252$		0			0
$253$			2.00000i			0.00790514i
$254$	−9.79796	−	5.65685i	−0.0385746	−	0.0222711i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.0312500	−	0.0541266i
$257$	84.8528	+	146.969i	0.330167	+	0.571865i	0.982544	−	0.186029i	$-0.0595617\pi$
$257$	84.8528	+	146.969i	0.330167	+	0.571865i	−0.652378	+	0.757894i	$0.726228\pi$
$258$		0			0
$259$	−62.5000	+	108.253i	−0.241313	+	0.417966i
$260$	−63.6396	+	63.6396i	−0.244768	+	0.244768i
$261$		0			0
$262$			192.000i			0.732824i
$263$	−141.421	+	244.949i	−0.537724	+	0.931365i	0.461302	+	0.887243i	$0.347382\pi$
$263$	−141.421	+	244.949i	−0.537724	+	0.931365i	−0.999026	+	0.0441219i	$0.985951\pi$
$264$		0			0
$265$	−93.3365	+	348.336i	−0.352213	+	1.31448i
$266$	128.598	−	74.2462i	0.483452	−	0.279121i
$267$		0			0
$268$	226.899	+	131.000i	0.846637	+	0.488806i
$269$			101.823i			0.378526i	0.981926	+	0.189263i	$0.0606098\pi$
$269$			101.823i			0.378526i	−0.981926	+	0.189263i	$0.939390\pi$
$270$		0			0
$271$	−221.000			−0.815498			−0.407749	−	0.913094i	$-0.633686\pi$
$271$	−221.000			−0.815498			−0.407749	+	0.913094i	$0.633686\pi$
$272$	22.6274	−	39.1918i	0.0831890	−	0.144088i
$273$		0			0
$274$	189.000	+	327.358i	0.689781	+	1.19474i
$275$	−17.6777	−	30.6186i	−0.0642824	−	0.111340i
$276$		0			0
$277$	76.2102	+	44.0000i	0.275127	+	0.158845i	0.631215	−	0.775608i	$-0.282556\pi$
$277$	76.2102	+	44.0000i	0.275127	+	0.158845i	−0.356088	+	0.934452i	$0.615890\pi$
$278$	−52.3259			−0.188223
$279$		0			0
$280$	−50.0000	+	50.0000i	−0.178571	+	0.178571i
$281$	176.363	+	101.823i	0.627627	+	0.362361i	0.779833	−	0.625988i	$-0.215304\pi$
$281$	176.363	+	101.823i	0.627627	+	0.362361i	−0.152205	+	0.988349i	$0.548638\pi$
$282$		0			0
$283$	27.7128	−	16.0000i	0.0979251	−	0.0565371i	−0.450238	−	0.892909i	$-0.648661\pi$
$283$	27.7128	−	16.0000i	0.0979251	−	0.0565371i	0.548163	+	0.836372i	$0.315327\pi$
$284$	154.318	−	89.0955i	0.543373	−	0.313716i
$285$		0			0
$286$	−9.00000	+	15.5885i	−0.0314685	+	0.0545051i
$287$	261.630			0.911601
$288$		0			0
$289$	−161.000			−0.557093
$290$	69.8811	+	260.800i	0.240969	+	0.899310i
$291$		0			0
$292$	−29.4449	+	17.0000i	−0.100839	+	0.0582192i
$293$	−86.9741	−	150.644i	−0.296840	−	0.514142i	0.678571	−	0.734535i	$-0.262599\pi$
$293$	−86.9741	−	150.644i	−0.296840	−	0.514142i	−0.975411	+	0.220393i	$0.929266\pi$
$294$		0			0
$295$	117.128	+	437.128i	0.397045	+	1.48179i
$296$		−	70.7107i		−	0.238887i
$297$		0			0
$298$		−	368.000i		−	1.23490i
$299$	11.0227	+	6.36396i	0.0368652	+	0.0212842i
$300$		0			0
$301$	160.000	+	277.128i	0.531561	+	0.920691i
$302$	−77.0746	−	133.497i	−0.255214	−	0.442044i
$303$		0			0
$304$	−42.0000	+	72.7461i	−0.138158	+	0.239296i
$305$	−342.947	−	342.947i	−1.12442	−	1.12442i
$306$		0			0
$307$		−	486.000i		−	1.58306i	−0.611129	−	0.791531i	$-0.709284\pi$
$307$		−	486.000i		−	1.58306i	0.611129	−	0.791531i	$-0.290716\pi$
$308$	−7.07107	+	12.2474i	−0.0229580	+	0.0397644i
$309$		0			0
$310$	73.2051	−	273.205i	0.236145	−	0.881307i
$311$	−58.7878	+	33.9411i	−0.189028	+	0.109135i	−0.591528	−	0.806285i	$-0.701475\pi$
$311$	−58.7878	+	33.9411i	−0.189028	+	0.109135i	0.402499	+	0.915420i	$0.368142\pi$
$312$		0			0
$313$	243.353	+	140.500i	0.777486	+	0.448882i	0.835539	−	0.549432i	$-0.185156\pi$
$313$	243.353	+	140.500i	0.777486	+	0.448882i	−0.0580525	+	0.998314i	$0.518489\pi$
$314$			166.877i			0.531456i
$315$		0			0
$316$	−234.000			−0.740506
$317$	260.215	−	450.706i	0.820868	−	1.42179i	−0.0841679	−	0.996452i	$-0.526823\pi$
$317$	260.215	−	450.706i	0.820868	−	1.42179i	0.905036	−	0.425334i	$-0.139843\pi$
$318$		0			0
$319$	27.0000	+	46.7654i	0.0846395	+	0.146600i
$320$	10.3528	−	38.6370i	0.0323524	−	0.120741i
$321$		0			0
$322$	8.66025	+	5.00000i	0.0268952	+	0.0155280i
$323$	−237.588			−0.735566
$324$		0			0
$325$	−225.000			−0.692308
$326$	−248.623	−	143.543i	−0.762648	−	0.440315i
$327$		0			0
$328$	−128.172	+	74.0000i	−0.390768	+	0.225610i
$329$	−97.9796	+	56.5685i	−0.297810	+	0.171941i
$330$		0			0
$331$	29.5000	−	51.0955i	0.0891239	−	0.154367i	−0.818017	−	0.575194i	$-0.804927\pi$
$331$	29.5000	−	51.0955i	0.0891239	−	0.154367i	0.907141	+	0.420827i	$0.138260\pi$
$332$	115.966			0.349294
$333$		0			0
$334$	144.000			0.431138
$335$	169.526	+	632.681i	0.506049	+	1.88860i
$336$		0			0
$337$	−47.6314	+	27.5000i	−0.141339	+	0.0816024i	−0.569002	−	0.822336i	$-0.692670\pi$
$337$	−47.6314	+	27.5000i	−0.141339	+	0.0816024i	0.427663	+	0.903938i	$0.359337\pi$
$338$	−62.2254	−	107.778i	−0.184099	−	0.318868i
$339$		0			0
$340$	109.282	−	29.2820i	0.321418	−	0.0861236i
$341$		−	56.5685i		−	0.165890i
$342$		0			0
$343$			365.000i			1.06414i
$344$	−156.767	−	90.5097i	−0.455719	−	0.263109i
$345$		0			0
$346$	8.00000	+	13.8564i	0.0231214	+	0.0400474i
$347$	299.106	+	518.067i	0.861977	+	1.49299i	0.870017	+	0.493021i	$0.164107\pi$
$347$	299.106	+	518.067i	0.861977	+	1.49299i	−0.00803996	+	0.999968i	$0.502559\pi$
$348$		0			0
$349$	219.500	−	380.185i	0.628940	−	1.08936i	−0.358825	−	0.933405i	$-0.616823\pi$
$349$	219.500	−	380.185i	0.628940	−	1.08936i	0.987765	−	0.155951i	$-0.0498442\pi$
$350$	−176.777			−0.505076
$351$		0			0
$352$		−	8.00000i		−	0.0227273i
$353$	260.215	−	450.706i	0.737154	−	1.27679i	−0.216618	−	0.976256i	$-0.569503\pi$
$353$	260.215	−	450.706i	0.737154	−	1.27679i	0.953772	−	0.300531i	$-0.0971640\pi$
$354$		0			0
$355$	430.298	+	115.298i	1.21211	+	0.324783i
$356$	−254.747	+	147.078i	−0.715581	+	0.413141i
$357$		0			0
$358$	−154.153	−	89.0000i	−0.430594	−	0.248603i
$359$			55.1543i			0.153633i	0.997045	+	0.0768166i	$0.0244756\pi$
$359$			55.1543i			0.153633i	−0.997045	+	0.0768166i	$0.975524\pi$
$360$		0			0
$361$	80.0000			0.221607
$362$	89.8026	−	155.543i	0.248073	−	0.429676i
$363$		0			0
$364$	45.0000	+	77.9423i	0.123626	+	0.214127i
$365$	−82.1037	−	21.9996i	−0.224942	−	0.0602729i
$366$		0			0
$367$	510.089	+	294.500i	1.38989	+	0.802452i	0.993302	−	0.115545i	$-0.0368615\pi$
$367$	510.089	+	294.500i	1.38989	+	0.802452i	0.396586	+	0.917998i	$0.370195\pi$
$368$	−5.65685			−0.0153719
$369$		0			0
$370$	125.000	−	125.000i	0.337838	−	0.337838i
$371$	312.310	+	180.312i	0.841806	+	0.486017i
$372$		0			0
$373$	7.79423	−	4.50000i	0.0208961	−	0.0120643i	−0.489516	−	0.871995i	$-0.662826\pi$
$373$	7.79423	−	4.50000i	0.0208961	−	0.0120643i	0.510412	+	0.859930i	$0.329493\pi$
$374$	19.5959	−	11.3137i	0.0523955	−	0.0302506i
$375$		0			0
$376$	32.0000	−	55.4256i	0.0851064	−	0.147409i
$377$	343.654			0.911549
$378$		0			0
$379$	157.000			0.414248			0.207124	−	0.978315i	$-0.433590\pi$
$379$	157.000			0.414248			0.207124	+	0.978315i	$0.433590\pi$
$380$	−202.844	+	54.3520i	−0.533801	+	0.143032i
$381$		0			0
$382$	124.708	−	72.0000i	0.326460	−	0.188482i
$383$	141.421	+	244.949i	0.369246	+	0.639553i	0.989448	−	0.144889i	$-0.0462825\pi$
$383$	141.421	+	244.949i	0.369246	+	0.639553i	−0.620202	+	0.784443i	$0.712949\pi$
$384$		0			0
$385$	−34.1506	+	9.15064i	−0.0887029	+	0.0237679i
$386$			383.252i			0.992881i
$387$		0			0
$388$		−	82.0000i		−	0.211340i
$389$	−519.292	−	299.813i	−1.33494	−	0.770728i	−0.348888	−	0.937164i	$-0.613441\pi$
$389$	−519.292	−	299.813i	−1.33494	−	0.770728i	−0.986052	+	0.166436i	$0.946774\pi$
$390$		0			0
$391$	−8.00000	−	13.8564i	−0.0204604	−	0.0354384i
$392$	−33.9411	−	58.7878i	−0.0865845	−	0.149969i
$393$		0			0
$394$	−224.000	+	387.979i	−0.568528	+	0.984719i
$395$	−413.657	−	413.657i	−1.04723	−	1.04723i
$396$		0			0
$397$			296.000i			0.745592i	0.927913	+	0.372796i	$0.121601\pi$
$397$			296.000i			0.745592i	−0.927913	+	0.372796i	$0.878399\pi$
$398$	−103.945	+	180.037i	−0.261168	+	0.452356i
$399$		0			0
$400$	86.6025	−	50.0000i	0.216506	−	0.125000i
$401$	−336.805	+	194.454i	−0.839912	+	0.484924i	−0.857234	−	0.514926i	$-0.827819\pi$
$401$	−336.805	+	194.454i	−0.839912	+	0.484924i	0.0173221	+	0.999850i	$0.494486\pi$
$402$		0			0
$403$	−311.769	−	180.000i	−0.773621	−	0.446650i
$404$			181.019i			0.448068i
$405$		0			0
$406$	270.000			0.665025
$407$	17.6777	−	30.6186i	0.0434341	−	0.0752300i
$408$		0			0
$409$	−72.5000	−	125.574i	−0.177262	−	0.307026i	0.763680	−	0.645595i	$-0.223391\pi$
$409$	−72.5000	−	125.574i	−0.177262	−	0.307026i	−0.940942	+	0.338569i	$0.890057\pi$
$410$	−357.393	−	95.7630i	−0.871689	−	0.233568i
$411$		0			0
$412$	22.5167	+	13.0000i	0.0546521	+	0.0315534i
$413$	452.548			1.09576
$414$		0			0
$415$	205.000	+	205.000i	0.493976	+	0.493976i
$416$	−44.0908	−	25.4558i	−0.105988	−	0.0611919i
$417$		0			0
$418$	−36.3731	+	21.0000i	−0.0870169	+	0.0502392i
$419$	607.473	−	350.725i	1.44982	−	0.837052i	0.451347	−	0.892348i	$-0.350944\pi$
$419$	607.473	−	350.725i	1.44982	−	0.837052i	0.998470	+	0.0552959i	$0.0176102\pi$
$420$		0			0
$421$	−252.500	+	437.343i	−0.599762	+	1.03882i	0.393093	+	0.919499i	$0.371405\pi$
$421$	−252.500	+	437.343i	−0.599762	+	1.03882i	−0.992856	+	0.119320i	$0.961928\pi$
$422$	199.404			0.472522
$423$		0			0
$424$	−204.000			−0.481132
$425$	244.949	+	141.421i	0.576351	+	0.332756i
$426$		0			0
$427$	−420.022	+	242.500i	−0.983659	+	0.567916i
$428$	−123.037	−	213.106i	−0.287469	−	0.497910i
$429$		0			0
$430$	−117.128	−	437.128i	−0.272391	−	1.01658i
$431$		−	43.8406i		−	0.101718i	−0.998706	−	0.0508592i	$-0.983804\pi$
$431$		−	43.8406i		−	0.101718i	0.998706	−	0.0508592i	$-0.0161960\pi$
$432$		0			0
$433$		−	32.0000i		−	0.0739030i	−0.999317	−	0.0369515i	$-0.988235\pi$
$433$		−	32.0000i		−	0.0739030i	0.999317	−	0.0369515i	$-0.0117647\pi$
$434$	−244.949	−	141.421i	−0.564399	−	0.325856i
$435$		0			0
$436$	−8.00000	−	13.8564i	−0.0183486	−	0.0317807i
$437$	14.8492	+	25.7196i	0.0339800	+	0.0588550i
$438$		0			0
$439$	252.000	−	436.477i	0.574032	−	0.994252i	−0.422114	−	0.906543i	$-0.638712\pi$
$439$	252.000	−	436.477i	0.574032	−	0.994252i	0.996146	−	0.0877097i	$-0.0279548\pi$
$440$	14.1421	−	14.1421i	0.0321412	−	0.0321412i
$441$		0			0
$442$		−	144.000i		−	0.325792i
$443$	−118.794	+	205.757i	−0.268158	+	0.464463i	−0.968386	−	0.249456i	$-0.919748\pi$
$443$	−118.794	+	205.757i	−0.268158	+	0.464463i	0.700228	+	0.713919i	$0.253082\pi$
$444$		0			0
$445$	−710.333	−	190.333i	−1.59625	−	0.427715i
$446$	−9.79796	+	5.65685i	−0.0219685	+	0.0126835i
$447$		0			0
$448$	−34.6410	−	20.0000i	−0.0773237	−	0.0446429i
$449$			67.8823i			0.151185i	0.997139	+	0.0755927i	$0.0240849\pi$
$449$			67.8823i			0.151185i	−0.997139	+	0.0755927i	$0.975915\pi$
$450$		0			0
$451$	−74.0000			−0.164080
$452$	−38.1838	+	66.1362i	−0.0844774	+	0.146319i
$453$		0			0
$454$	−49.0000	−	84.8705i	−0.107930	−	0.186939i
$455$	−58.2343	+	217.333i	−0.127987	+	0.477656i
$456$		0			0
$457$	−651.251	−	376.000i	−1.42506	−	0.822757i	−0.428332	−	0.903621i	$-0.640899\pi$
$457$	−651.251	−	376.000i	−1.42506	−	0.822757i	−0.996725	+	0.0808643i	$0.974232\pi$
$458$	−11.3137			−0.0247024
$459$		0			0
$460$	−10.0000	−	10.0000i	−0.0217391	−	0.0217391i
$461$	−529.090	−	305.470i	−1.14770	−	0.662625i	−0.199374	−	0.979923i	$-0.563891\pi$
$461$	−529.090	−	305.470i	−1.14770	−	0.662625i	−0.948326	+	0.317298i	$0.897224\pi$
$462$		0			0
$463$	517.017	−	298.500i	1.11667	−	0.644708i	0.176120	−	0.984369i	$-0.443645\pi$
$463$	517.017	−	298.500i	1.11667	−	0.644708i	0.940548	+	0.339660i	$0.110312\pi$
$464$	−132.272	+	76.3675i	−0.285070	+	0.164585i
$465$		0			0
$466$	224.000	−	387.979i	0.480687	−	0.832574i
$467$	848.528			1.81698			0.908488	−	0.417910i	$-0.137237\pi$
$467$	848.528			1.81698			0.908488	+	0.417910i	$0.137237\pi$
$468$		0			0
$469$	655.000			1.39659
$470$	154.548	−	41.4110i	0.328826	−	0.0881086i
$471$		0			0
$472$	−221.703	+	128.000i	−0.469709	+	0.271186i
$473$	−45.2548	−	78.3837i	−0.0956762	−	0.165716i
$474$		0			0
$475$	−454.663	−	262.500i	−0.957186	−	0.552632i
$476$		−	113.137i		−	0.237683i
$477$		0			0
$478$			290.000i			0.606695i
$479$	145.745	+	84.1457i	0.304269	+	0.175670i	0.644359	−	0.764723i	$-0.277124\pi$
$479$	145.745	+	84.1457i	0.304269	+	0.175670i	−0.340090	+	0.940393i	$0.610458\pi$
$480$		0			0
$481$	−112.500	−	194.856i	−0.233888	−	0.405105i
$482$	−55.8614	−	96.7548i	−0.115895	−	0.200736i
$483$		0			0
$484$	−119.000	+	206.114i	−0.245868	+	0.425855i
$485$	144.957	−	144.957i	0.298880	−	0.298880i
$486$		0			0
$487$		−	507.000i		−	1.04107i	−0.853841	−	0.520534i	$-0.825733\pi$
$487$		−	507.000i		−	1.04107i	0.853841	−	0.520534i	$-0.174267\pi$
$488$	137.179	−	237.601i	0.281104	−	0.486886i
$489$		0			0
$490$	43.9230	−	163.923i	0.0896389	−	0.334537i
$491$	371.098	−	214.253i	0.755800	−	0.436361i	−0.0719859	−	0.997406i	$-0.522934\pi$
$491$	371.098	−	214.253i	0.755800	−	0.436361i	0.827786	+	0.561044i	$0.189600\pi$
$492$		0			0
$493$	−374.123	−	216.000i	−0.758870	−	0.438134i
$494$			267.286i			0.541066i
$495$		0			0
$496$	160.000			0.322581
$497$	222.739	−	385.795i	0.448166	−	0.776247i
$498$		0			0
$499$	−435.000	−	753.442i	−0.871743	−	1.50990i	−0.860192	−	0.509971i	$-0.829656\pi$
$499$	−435.000	−	753.442i	−0.871743	−	1.50990i	−0.0115517	−	0.999933i	$-0.503677\pi$
$500$	241.481	+	64.7048i	0.482963	+	0.129410i
$501$		0			0
$502$	−57.1577	−	33.0000i	−0.113860	−	0.0657371i
$503$	−462.448			−0.919379			−0.459690	−	0.888080i	$-0.652039\pi$
$503$	−462.448			−0.919379			−0.459690	+	0.888080i	$0.652039\pi$
$504$		0			0
$505$	−320.000	+	320.000i	−0.633663	+	0.633663i
$506$	−2.44949	−	1.41421i	−0.00484089	−	0.00279489i
$507$		0			0
$508$	13.8564	−	8.00000i	0.0272764	−	0.0157480i
$509$	709.127	−	409.415i	1.39318	−	0.804351i	0.399512	−	0.916728i	$-0.369180\pi$
$509$	709.127	−	409.415i	1.39318	−	0.804351i	0.993666	+	0.112377i	$0.0358464\pi$
$510$		0			0
$511$	−42.5000	+	73.6122i	−0.0831703	+	0.144055i
$512$	22.6274			0.0441942
$513$		0			0
$514$	−240.000			−0.466926
$515$	16.8232	+	62.7852i	0.0326665	+	0.121913i
$516$		0			0
$517$	27.7128	−	16.0000i	0.0536031	−	0.0309478i
$518$	−88.3883	−	153.093i	−0.170634	−	0.295547i
$519$		0			0
$520$	−32.9423	−	122.942i	−0.0633506	−	0.236427i
$521$		−	864.084i		−	1.65851i	−0.558869	−	0.829256i	$-0.688765\pi$
$521$		−	864.084i		−	1.65851i	0.558869	−	0.829256i	$-0.311235\pi$
$522$		0			0
$523$		−	163.000i		−	0.311663i	−0.987784	−	0.155832i	$-0.950194\pi$
$523$		−	163.000i		−	0.311663i	0.987784	−	0.155832i	$-0.0498058\pi$
$524$	−235.151	−	135.765i	−0.448761	−	0.259093i
$525$		0			0
$526$	−200.000	−	346.410i	−0.380228	−	0.658574i
$527$	226.274	+	391.918i	0.429363	+	0.743678i
$528$		0			0
$529$	263.500	−	456.395i	0.498110	−	0.862751i
$530$	−360.624	−	360.624i	−0.680424	−	0.680424i
$531$		0			0
$532$			210.000i			0.394737i
$533$	−235.467	+	407.840i	−0.441776	+	0.765178i
$534$		0			0
$535$	159.221	−	594.221i	0.297609	−	1.11069i
$536$	−320.883	+	185.262i	−0.598663	+	0.345638i
$537$		0			0
$538$	−124.708	−	72.0000i	−0.231799	−	0.133829i
$539$		−	33.9411i		−	0.0629705i
$540$		0			0
$541$	697.000			1.28835			0.644177	−	0.764876i	$-0.277200\pi$
$541$	697.000			1.28835			0.644177	+	0.764876i	$0.277200\pi$
$542$	156.271	−	270.669i	0.288322	−	0.499389i
$543$		0			0
$544$	32.0000	+	55.4256i	0.0588235	+	0.101885i
$545$	10.3528	−	38.6370i	0.0189959	−	0.0708936i
$546$		0			0
$547$	−336.884	−	194.500i	−0.615875	−	0.355576i	0.159386	−	0.987216i	$-0.449049\pi$
$547$	−336.884	−	194.500i	−0.615875	−	0.355576i	−0.775261	+	0.631640i	$0.782382\pi$
$548$	−534.573			−0.975498
$549$		0			0
$550$	50.0000			0.0909091
$551$	694.430	+	400.930i	1.26031	+	0.727640i
$552$		0			0
$553$	−506.625	+	292.500i	−0.916139	+	0.528933i
$554$	−107.778	+	62.2254i	−0.194544	+	0.112320i
$555$		0			0
$556$	37.0000	−	64.0859i	0.0665468	−	0.115262i
$557$	1086.12			1.94994			0.974969	−	0.222339i	$-0.0713691\pi$
$557$	1086.12			1.94994			0.974969	+	0.222339i	$0.0713691\pi$
$558$		0			0
$559$	−576.000			−1.03041
$560$	−25.8819	−	96.5926i	−0.0462177	−	0.172487i
$561$		0			0
$562$	−249.415	+	144.000i	−0.443799	+	0.256228i
$563$	311.127	+	538.888i	0.552623	+	0.957172i	0.998084	+	0.0618704i	$0.0197065\pi$
$563$	311.127	+	538.888i	0.552623	+	0.957172i	−0.445461	+	0.895301i	$0.646960\pi$
$564$		0			0
$565$	−184.413	+	49.4134i	−0.326395	+	0.0874574i
$566$			45.2548i			0.0799555i
$567$		0			0
$568$			252.000i			0.443662i
$569$	−401.716	−	231.931i	−0.706004	−	0.407612i	0.103576	−	0.994622i	$-0.466972\pi$
$569$	−401.716	−	231.931i	−0.706004	−	0.407612i	−0.809580	+	0.587010i	$0.800305\pi$
$570$		0			0
$571$	461.500	+	799.341i	0.808231	+	1.39990i	0.914088	+	0.405516i	$0.132908\pi$
$571$	461.500	+	799.341i	0.808231	+	1.39990i	−0.105857	+	0.994381i	$0.533758\pi$
$572$	−12.7279	−	22.0454i	−0.0222516	−	0.0385409i
$573$		0			0
$574$	−185.000	+	320.429i	−0.322300	+	0.558239i
$575$		−	35.3553i		−	0.0614875i
$576$		0			0
$577$		−	247.000i		−	0.428076i	−0.976825	−	0.214038i	$-0.931338\pi$
$577$		−	247.000i		−	0.428076i	0.976825	−	0.214038i	$-0.0686617\pi$
$578$	113.844	−	197.184i	0.196962	−	0.341149i
$579$		0			0
$580$	−368.827	−	98.8269i	−0.635908	−	0.170391i
$581$	251.073	−	144.957i	0.432139	−	0.249496i
$582$		0			0
$583$	−88.3346	−	51.0000i	−0.151517	−	0.0874786i
$584$		−	48.0833i		−	0.0823344i
$585$		0			0
$586$	246.000			0.419795
$587$	226.981	−	393.143i	0.386680	−	0.669750i	−0.605321	−	0.795982i	$-0.706955\pi$
$587$	226.981	−	393.143i	0.386680	−	0.669750i	0.992001	+	0.126232i	$0.0402884\pi$
$588$		0			0
$589$	−420.000	−	727.461i	−0.713073	−	1.23508i
$590$	−618.193	−	165.644i	−1.04778	−	0.280753i
$591$		0			0
$592$	86.6025	+	50.0000i	0.146288	+	0.0844595i
$593$	−4.24264			−0.00715454			−0.00357727	−	0.999994i	$-0.501139\pi$
$593$	−4.24264			−0.00715454			−0.00357727	+	0.999994i	$0.501139\pi$
$594$		0			0
$595$	200.000	−	200.000i	0.336134	−	0.336134i
$596$	450.706	+	260.215i	0.756218	+	0.436603i
$597$		0			0
$598$	−15.5885	+	9.00000i	−0.0260677	+	0.0150502i
$599$	194.734	−	112.430i	0.325099	−	0.187696i	−0.328564	−	0.944482i	$-0.606565\pi$
$599$	194.734	−	112.430i	0.325099	−	0.187696i	0.653663	+	0.756786i	$0.273231\pi$
$600$		0			0
$601$	368.000	−	637.395i	0.612313	−	1.06056i	−0.378537	−	0.925586i	$-0.623573\pi$
$601$	368.000	−	637.395i	0.612313	−	1.06056i	0.990850	−	0.134971i	$-0.0430940\pi$
$602$	−452.548			−0.751741
$603$		0			0
$604$	218.000			0.360927
$605$	−574.726	+	153.997i	−0.949960	+	0.254541i
$606$		0			0
$607$	−378.453	+	218.500i	−0.623481	+	0.359967i	−0.778223	−	0.627988i	$-0.783879\pi$
$607$	−378.453	+	218.500i	−0.623481	+	0.359967i	0.154742	+	0.987955i	$0.450545\pi$
$608$	−59.3970	−	102.879i	−0.0976924	−	0.169208i
$609$		0			0
$610$	662.522	−	177.522i	1.08610	−	0.291020i
$611$		−	203.647i		−	0.333301i
$612$		0			0
$613$		−	335.000i		−	0.546493i	−0.961944	−	0.273246i	$-0.911903\pi$
$613$		−	335.000i		−	0.546493i	0.961944	−	0.273246i	$-0.0880974\pi$
$614$	595.226	+	343.654i	0.969423	+	0.559697i
$615$		0			0
$616$	−10.0000	−	17.3205i	−0.0162338	−	0.0281177i
$617$	−127.986	−	221.679i	−0.207433	−	0.359285i	0.743472	−	0.668767i	$-0.233178\pi$
$617$	−127.986	−	221.679i	−0.207433	−	0.359285i	−0.950905	+	0.309482i	$0.899844\pi$
$618$		0			0
$619$	482.500	−	835.715i	0.779483	−	1.35010i	−0.152757	−	0.988264i	$-0.548815\pi$
$619$	482.500	−	835.715i	0.779483	−	1.35010i	0.932240	−	0.361840i	$-0.117851\pi$
$620$	282.843	+	282.843i	0.456198	+	0.456198i
$621$		0			0
$622$		−	96.0000i		−	0.154341i
$623$	−367.696	+	636.867i	−0.590201	+	1.02226i
$624$		0			0
$625$	312.500	+	541.266i	0.500000	+	0.866025i
$626$	−344.153	+	198.697i	−0.549766	+	0.317407i
$627$		0			0
$628$	−204.382	−	118.000i	−0.325449	−	0.187898i
$629$			282.843i			0.449670i
$630$		0			0
$631$	275.000			0.435816			0.217908	−	0.975969i	$-0.430077\pi$
$631$	275.000			0.435816			0.217908	+	0.975969i	$0.430077\pi$
$632$	165.463	−	286.590i	0.261809	−	0.453466i
$633$		0			0
$634$	368.000	+	637.395i	0.580442	+	1.00535i
$635$	38.6370	+	10.3528i	0.0608457	+	0.0163036i
$636$		0			0
$637$	−187.061	−	108.000i	−0.293660	−	0.169545i
$638$	−76.3675			−0.119698
$639$		0			0
$640$	40.0000	+	40.0000i	0.0625000	+	0.0625000i
$641$	−421.312	−	243.245i	−0.657273	−	0.379477i	0.133964	−	0.990986i	$-0.457229\pi$
$641$	−421.312	−	243.245i	−0.657273	−	0.379477i	−0.791237	+	0.611509i	$0.790563\pi$
$642$		0			0
$643$	−997.661	+	576.000i	−1.55157	+	0.895801i	−0.553559	+	0.832810i	$0.686731\pi$
$643$	−997.661	+	576.000i	−1.55157	+	0.895801i	−0.998014	+	0.0629907i	$0.979936\pi$
$644$	−12.2474	+	7.07107i	−0.0190178	+	0.0109799i
$645$		0			0
$646$	168.000	−	290.985i	0.260062	−	0.450440i
$647$	−691.550			−1.06886			−0.534428	−	0.845214i	$-0.679473\pi$
$647$	−691.550			−1.06886			−0.534428	+	0.845214i	$0.679473\pi$
$648$		0			0
$649$	−128.000			−0.197227
$650$	159.099	−	275.568i	0.244768	−	0.423950i
$651$		0			0
$652$	351.606	−	203.000i	0.539273	−	0.311350i
$653$	−175.362	−	303.737i	−0.268549	−	0.465140i	0.699938	−	0.714203i	$-0.253211\pi$
$653$	−175.362	−	303.737i	−0.268549	−	0.465140i	−0.968487	+	0.249063i	$0.919877\pi$
$654$		0			0
$655$	−175.692	−	655.692i	−0.268232	−	1.00106i
$656$		−	209.304i		−	0.319060i
$657$		0			0
$658$		−	160.000i		−	0.243161i
$659$	431.110	+	248.902i	0.654188	+	0.377696i	0.790059	−	0.613031i	$-0.210050\pi$
$659$	431.110	+	248.902i	0.654188	+	0.377696i	−0.135871	+	0.990727i	$0.543383\pi$
$660$		0			0
$661$	288.500	+	499.697i	0.436460	+	0.755971i	0.997414	−	0.0718765i	$-0.0228987\pi$
$661$	288.500	+	499.697i	0.436460	+	0.755971i	−0.560954	+	0.827847i	$0.689565\pi$
$662$	41.7193	+	72.2599i	0.0630201	+	0.109154i
$663$		0			0
$664$	−82.0000	+	142.028i	−0.123494	+	0.213898i
$665$	−371.231	+	371.231i	−0.558242	+	0.558242i
$666$		0			0
$667$			54.0000i			0.0809595i
$668$	−101.823	+	176.363i	−0.152430	+	0.264017i
$669$		0			0
$670$	−894.747	−	239.747i	−1.33544	−	0.357831i
$671$	118.800	−	68.5894i	0.177050	−	0.102220i
$672$		0			0
$673$	423.486	+	244.500i	0.629252	+	0.363299i	0.780462	−	0.625203i	$-0.214984\pi$
$673$	423.486	+	244.500i	0.629252	+	0.363299i	−0.151210	+	0.988502i	$0.548317\pi$
$674$		−	77.7817i		−	0.115403i
$675$		0			0
$676$	176.000			0.260355
$677$	−299.813	+	519.292i	−0.442856	+	0.767048i	−0.997900	−	0.0647722i	$-0.979368\pi$
$677$	−299.813	+	519.292i	−0.442856	+	0.767048i	0.555044	+	0.831821i	$0.312701\pi$
$678$		0			0
$679$	−102.500	−	177.535i	−0.150957	−	0.261466i
$680$	−41.4110	+	154.548i	−0.0608986	+	0.227277i
$681$		0			0
$682$	69.2820	+	40.0000i	0.101587	+	0.0586510i
$683$	236.174			0.345789			0.172894	−	0.984940i	$-0.444688\pi$
$683$	236.174			0.345789			0.172894	+	0.984940i	$0.444688\pi$
$684$		0			0
$685$	−945.000	−	945.000i	−1.37956	−	1.37956i
$686$	−447.032	−	258.094i	−0.651650	−	0.376230i
$687$		0			0
$688$	221.703	−	128.000i	0.322242	−	0.186047i
$689$	−562.158	+	324.562i	−0.815904	+	0.471062i
$690$		0			0
$691$	−320.000	+	554.256i	−0.463097	+	0.802107i	−0.999113	−	0.0421001i	$-0.986595\pi$
$691$	−320.000	+	554.256i	−0.463097	+	0.802107i	0.536016	+	0.844208i	$0.319928\pi$
$692$	−22.6274			−0.0326986
$693$		0			0
$694$	−846.000			−1.21902
$695$	178.696	−	47.8815i	0.257117	−	0.0688943i
$696$		0			0
$697$	512.687	−	296.000i	0.735562	−	0.424677i
$698$	310.420	+	537.663i	0.444728	+	0.770291i
$699$		0			0
$700$	125.000	−	216.506i	0.178571	−	0.309295i
$701$			1093.19i			1.55947i	0.626111	+	0.779734i	$0.284646\pi$
$701$			1093.19i			1.55947i	−0.626111	+	0.779734i	$0.715354\pi$
$702$		0			0
$703$		−	525.000i		−	0.746799i
$704$	9.79796	+	5.65685i	0.0139176	+	0.00803530i
$705$		0			0
$706$	368.000	+	637.395i	0.521246	+	0.902825i
$707$	226.274	+	391.918i	0.320048	+	0.554340i
$708$		0			0
$709$	244.500	−	423.486i	0.344852	−	0.597301i	−0.640475	−	0.767979i	$-0.721262\pi$
$709$	244.500	−	423.486i	0.344852	−	0.597301i	0.985327	+	0.170678i	$0.0545958\pi$
$710$	−445.477	+	445.477i	−0.627433	+	0.627433i
$711$		0			0
$712$		−	416.000i		−	0.584270i
$713$	28.2843	−	48.9898i	0.0396694	−	0.0687094i
$714$		0			0
$715$	16.4711	−	61.4711i	0.0230366	−	0.0859736i
$716$	218.005	−	125.865i	0.304476	−	0.175789i
$717$		0			0
$718$	−67.5500	−	39.0000i	−0.0940808	−	0.0543175i
$719$			620.840i			0.863477i	0.901999	+	0.431738i	$0.142100\pi$
$719$			620.840i			0.863477i	−0.901999	+	0.431738i	$0.857900\pi$
$720$		0			0
$721$	65.0000			0.0901526
$722$	−56.5685	+	97.9796i	−0.0783498	+	0.135706i
$723$		0			0
$724$	127.000	+	219.970i	0.175414	+	0.303827i
$725$	−477.297	−	826.703i	−0.658341	−	1.14028i
$726$		0			0
$727$	935.307	+	540.000i	1.28653	+	0.742779i	0.978034	−	0.208447i	$-0.0668409\pi$
$727$	935.307	+	540.000i	1.28653	+	0.742779i	0.308496	+	0.951225i	$0.400174\pi$
$728$	−127.279			−0.174834
$729$		0			0
$730$	85.0000	−	85.0000i	0.116438	−	0.116438i
$731$	627.069	+	362.039i	0.857824	+	0.495265i
$732$		0			0
$733$	214.774	−	124.000i	0.293007	−	0.169168i	−0.346290	−	0.938128i	$-0.612559\pi$
$733$	214.774	−	124.000i	0.293007	−	0.169168i	0.639297	+	0.768960i	$0.279225\pi$
$734$	−721.375	+	416.486i	−0.982799	+	0.567419i
$735$		0			0
$736$	4.00000	−	6.92820i	0.00543478	−	0.00941332i
$737$	−185.262			−0.251373
$738$		0			0
$739$	848.000			1.14750			0.573748	−	0.819032i	$-0.305489\pi$
$739$	848.000			1.14750			0.573748	+	0.819032i	$0.305489\pi$
$740$	64.7048	+	241.481i	0.0874389	+	0.326326i
$741$		0			0
$742$	−441.673	+	255.000i	−0.595247	+	0.343666i
$743$	−16.9706	−	29.3939i	−0.0228406	−	0.0395611i	0.854379	−	0.519650i	$-0.173938\pi$
$743$	−16.9706	−	29.3939i	−0.0228406	−	0.0395611i	−0.877220	+	0.480089i	$0.840604\pi$
$744$		0			0
$745$	336.743	+	1256.74i	0.452005	+	1.68690i
$746$			12.7279i			0.0170616i
$747$		0			0
$748$			32.0000i			0.0427807i
$749$	−532.764	−	307.591i	−0.711300	−	0.410669i
$750$		0			0
$751$	66.5000	+	115.181i	0.0885486	+	0.153371i	0.906898	−	0.421351i	$-0.138444\pi$
$751$	66.5000	+	115.181i	0.0885486	+	0.153371i	−0.818349	+	0.574721i	$0.805110\pi$
$752$	45.2548	+	78.3837i	0.0601793	+	0.104234i
$753$		0			0
$754$	−243.000	+	420.888i	−0.322281	+	0.558207i
$755$	385.373	+	385.373i	0.510428	+	0.510428i
$756$		0			0
$757$			1271.00i			1.67900i	0.543363	+	0.839498i	$0.317151\pi$
$757$			1271.00i			1.67900i	−0.543363	+	0.839498i	$0.682849\pi$
$758$	−111.016	+	192.285i	−0.146459	+	0.253674i
$759$		0			0
$760$	76.8653	−	286.865i	0.101139	−	0.377454i
$761$	466.628	−	269.408i	0.613177	−	0.354018i	−0.161031	−	0.986949i	$-0.551482\pi$
$761$	466.628	−	269.408i	0.613177	−	0.354018i	0.774208	+	0.632931i	$0.218148\pi$
$762$		0			0
$763$	−34.6410	−	20.0000i	−0.0454011	−	0.0262123i
$764$			203.647i			0.266553i
$765$		0			0
$766$	−400.000			−0.522193
$767$	−407.294	+	705.453i	−0.531022	+	0.919756i
$768$		0			0
$769$	40.5000	+	70.1481i	0.0526658	+	0.0912198i	0.891156	−	0.453696i	$-0.149895\pi$
$769$	40.5000	+	70.1481i	0.0526658	+	0.0912198i	−0.838491	+	0.544916i	$0.816562\pi$
$770$	12.9410	−	48.2963i	0.0168064	−	0.0627225i
$771$		0			0
$772$	−469.386	−	271.000i	−0.608013	−	0.351036i
$773$	445.477			0.576297			0.288148	−	0.957586i	$-0.406960\pi$
$773$	445.477			0.576297			0.288148	+	0.957586i	$0.406960\pi$
$774$		0			0
$775$			1000.00i			1.29032i
$776$	100.429	+	57.9828i	0.129419	+	0.0747200i
$777$		0			0
$778$	734.390	−	424.000i	0.943945	−	0.544987i
$779$	−951.627	+	549.422i	−1.22160	+	0.705291i
$780$		0			0
$781$	−63.0000	+	109.119i	−0.0806658	+	0.139717i
$782$	22.6274			0.0289353
$783$		0			0
$784$	96.0000			0.122449
$785$	−152.703	−	569.896i	−0.194526	−	0.725982i
$786$		0			0
$787$	342.080	−	197.500i	0.434663	−	0.250953i	−0.266668	−	0.963788i	$-0.585923\pi$
$787$	342.080	−	197.500i	0.434663	−	0.250953i	0.701331	+	0.712835i	$0.252589\pi$
$788$	−316.784	−	548.686i	−0.402010	−	0.696302i
$789$		0			0
$790$	799.125	−	214.125i	1.01155	−	0.271044i
$791$			190.919i			0.241364i
$792$		0			0
$793$		−	873.000i		−	1.10088i
$794$	−362.524	−	209.304i	−0.456580	−	0.263607i
$795$		0			0
$796$	−147.000	−	254.611i	−0.184673	−	0.319864i
$797$	−537.401	−	930.806i	−0.674280	−	1.16789i	−0.976679	−	0.214706i	$-0.931121\pi$
$797$	−537.401	−	930.806i	−0.674280	−	1.16789i	0.302399	−	0.953181i	$-0.402213\pi$
$798$		0			0
$799$	−128.000	+	221.703i	−0.160200	+	0.277475i
$800$			141.421i			0.176777i
$801$		0			0
$802$		−	550.000i		−	0.685786i
$803$	12.0208	−	20.8207i	0.0149699	−	0.0259286i
$804$		0			0
$805$	−34.1506	−	9.15064i	−0.0424231	−	0.0113672i
$806$	440.908	−	254.558i	0.547032	−	0.315829i
$807$		0			0
$808$	−221.703	−	128.000i	−0.274384	−	0.158416i
$809$		−	1255.82i		−	1.55231i	−0.630540	−	0.776157i	$-0.717167\pi$
$809$		−	1255.82i		−	1.55231i	0.630540	−	0.776157i	$-0.282833\pi$
$810$		0			0
$811$	−752.000			−0.927250			−0.463625	−	0.886031i	$-0.653452\pi$
$811$	−752.000			−0.927250			−0.463625	+	0.886031i	$0.653452\pi$
$812$	−190.919	+	330.681i	−0.235122	+	0.407243i
$813$		0			0
$814$	25.0000	+	43.3013i	0.0307125	+	0.0531957i
$815$	980.415	+	262.701i	1.20296	+	0.322333i
$816$		0			0
$817$	−1163.94	−	672.000i	−1.42465	−	0.822521i
$818$	205.061			0.250686
$819$		0			0
$820$	370.000	−	370.000i	0.451220	−	0.451220i
$821$	−431.110	−	248.902i	−0.525104	−	0.303169i	0.213917	−	0.976852i	$-0.431378\pi$
$821$	−431.110	−	248.902i	−0.525104	−	0.303169i	−0.739020	+	0.673683i	$0.764711\pi$
$822$		0			0
$823$	−459.859	+	265.500i	−0.558760	+	0.322600i	−0.752648	−	0.658423i	$-0.771224\pi$
$823$	−459.859	+	265.500i	−0.558760	+	0.322600i	0.193888	+	0.981024i	$0.437890\pi$
$824$	−31.8434	+	18.3848i	−0.0386449	+	0.0223116i
$825$		0			0
$826$	−320.000	+	554.256i	−0.387409	+	0.671012i
$827$	−350.725			−0.424093			−0.212047	−	0.977260i	$-0.568013\pi$
$827$	−350.725			−0.424093			−0.212047	+	0.977260i	$0.568013\pi$
$828$		0			0
$829$	−705.000			−0.850422			−0.425211	−	0.905094i	$-0.639800\pi$
$829$	−705.000			−0.850422			−0.425211	+	0.905094i	$0.639800\pi$
$830$	−396.030	+	106.116i	−0.477144	+	0.127850i
$831$		0			0
$832$	62.3538	−	36.0000i	0.0749445	−	0.0432692i
$833$	135.765	+	235.151i	0.162983	+	0.282294i
$834$		0			0
$835$	−491.769	+	131.769i	−0.588945	+	0.157807i
$836$		−	59.3970i		−	0.0710490i
$837$		0			0
$838$			992.000i			1.18377i
$839$	1401.11	+	808.930i	1.66997	+	0.964160i	0.967648	+	0.252304i	$0.0811885\pi$
$839$	1401.11	+	808.930i	1.66997	+	0.964160i	0.702326	+	0.711855i	$0.252145\pi$
$840$		0			0
$841$	308.500	+	534.338i	0.366825	+	0.635360i
$842$	−357.089	−	618.496i	−0.424096	−	0.734556i
$843$		0			0
$844$	−141.000	+	244.219i	−0.167062	+	0.289359i
$845$	311.127	+	311.127i	0.368198	+	0.368198i
$846$		0			0
$847$			595.000i			0.702479i
$848$	144.250	−	249.848i	0.170106	−	0.294632i
$849$		0			0
$850$	−346.410	+	200.000i	−0.407541	+	0.235294i
$851$	30.6186	−	17.6777i	0.0359796	−	0.0207728i
$852$		0			0
$853$	1419.42	+	819.500i	1.66403	+	0.960727i	0.970762	+	0.240045i	$0.0771622\pi$
$853$	1419.42	+	819.500i	1.66403	+	0.960727i	0.693266	+	0.720682i	$0.256171\pi$
$854$		−	685.894i		−	0.803154i
$855$		0			0
$856$	348.000			0.406542
$857$	494.268	−	856.097i	0.576742	−	0.998946i	−0.419108	−	0.907936i	$-0.637657\pi$
$857$	494.268	−	856.097i	0.576742	−	0.998946i	0.995850	−	0.0910097i	$-0.0290094\pi$
$858$		0			0
$859$	−177.500	−	307.439i	−0.206636	−	0.357903i	0.744017	−	0.668161i	$-0.232918\pi$
$859$	−177.500	−	307.439i	−0.206636	−	0.357903i	−0.950653	+	0.310257i	$0.899585\pi$
$860$	618.193	+	165.644i	0.718829	+	0.192610i
$861$		0			0
$862$	53.6936	+	31.0000i	0.0622895	+	0.0359629i
$863$	−1459.47			−1.69116			−0.845578	−	0.533851i	$-0.820744\pi$
$863$	−1459.47			−1.69116			−0.845578	+	0.533851i	$0.820744\pi$
$864$		0			0
$865$	−40.0000	−	40.0000i	−0.0462428	−	0.0462428i
$866$	39.1918	+	22.6274i	0.0452562	+	0.0261287i
$867$		0			0
$868$	346.410	−	200.000i	0.399090	−	0.230415i
$869$	143.295	−	82.7315i	0.164897	−	0.0952031i
$870$		0			0
$871$	−589.500	+	1021.04i	−0.676808	+	1.17227i
$872$	22.6274			0.0259489
$873$		0			0
$874$	−42.0000			−0.0480549
$875$	603.704	−	161.762i	0.689947	−	0.184871i
$876$		0			0
$877$	977.743	−	564.500i	1.11487	−	0.643672i	0.174785	−	0.984607i	$-0.444077\pi$
$877$	977.743	−	564.500i	1.11487	−	0.643672i	0.940087	+	0.340935i	$0.110744\pi$
$878$	356.382	+	617.271i	0.405902	+	0.703043i
$879$		0			0
$880$	7.32051	+	27.3205i	0.00831876	+	0.0310460i
$881$			165.463i			0.187813i	0.995581	+	0.0939063i	$0.0299354\pi$
$881$			165.463i			0.187813i	−0.995581	+	0.0939063i	$0.970065\pi$
$882$		0			0
$883$			1227.00i			1.38958i	0.719212	+	0.694790i	$0.244503\pi$
$883$			1227.00i			1.38958i	−0.719212	+	0.694790i	$0.755497\pi$
$884$	176.363	+	101.823i	0.199506	+	0.115185i
$885$		0			0
$886$	−168.000	−	290.985i	−0.189616	−	0.328425i
$887$	−118.794	−	205.757i	−0.133928	−	0.231970i	0.791260	−	0.611480i	$-0.209426\pi$
$887$	−118.794	−	205.757i	−0.133928	−	0.231970i	−0.925187	+	0.379511i	$0.876092\pi$
$888$		0			0
$889$	20.0000	−	34.6410i	0.0224972	−	0.0389663i
$890$	735.391	−	735.391i	0.826282	−	0.826282i
$891$		0			0
$892$		−	16.0000i		−	0.0179372i
$893$	237.588	−	411.514i	0.266056	−	0.460822i
$894$		0			0
$895$	607.881	+	162.881i	0.679197	+	0.181990i
$896$	48.9898	−	28.2843i	0.0546761	−	0.0315673i
$897$		0			0
$898$	−83.1384	−	48.0000i	−0.0925818	−	0.0534521i
$899$		−	1527.35i		−	1.69894i
$900$		0			0
$901$	816.000			0.905660
$902$	52.3259	−	90.6311i	0.0580110	−	0.100478i
$903$		0			0
$904$	−54.0000	−	93.5307i	−0.0597345	−	0.103463i
$905$	−164.350	+	613.363i	−0.181602	+	0.677749i
$906$		0			0
$907$	870.356	+	502.500i	0.959598	+	0.554024i	0.896049	−	0.443954i	$-0.146425\pi$
$907$	870.356	+	502.500i	0.959598	+	0.554024i	0.0635488	+	0.997979i	$0.479758\pi$
$908$	138.593			0.152635
$909$		0			0
$910$	−225.000	−	225.000i	−0.247253	−	0.247253i
$911$	−676.059	−	390.323i	−0.742107	−	0.428455i	0.0807281	−	0.996736i	$-0.474275\pi$
$911$	−676.059	−	390.323i	−0.742107	−	0.428455i	−0.822835	+	0.568281i	$0.807609\pi$
$912$		0			0
$913$	−71.0141	+	41.0000i	−0.0777810	+	0.0449069i
$914$	921.008	−	531.744i	1.00767	−	0.581777i
$915$		0			0
$916$	8.00000	−	13.8564i	0.00873362	−	0.0151271i
$917$	−678.823			−0.740264
$918$		0			0
$919$	600.000			0.652884			0.326442	−	0.945217i	$-0.394150\pi$
$919$	600.000			0.652884			0.326442	+	0.945217i	$0.394150\pi$
$920$	19.3185	−	5.17638i	0.0209984	−	0.00562650i
$921$		0			0
$922$	748.246	−	432.000i	0.811547	−	0.468547i
$923$	400.930	+	694.430i	0.434377	+	0.752362i
$924$		0			0
$925$	−312.500	+	541.266i	−0.337838	+	0.585152i
$926$			844.285i			0.911755i
$927$		0			0
$928$		−	216.000i		−	0.232759i
$929$	1091.25	+	630.032i	1.17465	+	0.678183i	0.954770	−	0.297344i	$-0.0961009\pi$
$929$	1091.25	+	630.032i	1.17465	+	0.678183i	0.219877	+	0.975528i	$0.429434\pi$
$930$		0			0
$931$	−252.000	−	436.477i	−0.270677	−	0.468826i
$932$	316.784	+	548.686i	0.339897	+	0.588719i
$933$		0			0
$934$	−600.000	+	1039.23i	−0.642398	+	1.11267i
$935$	−56.5685	+	56.5685i	−0.0605011	+	0.0605011i
$936$		0			0
$937$			1465.00i			1.56350i	0.623592	+	0.781750i	$0.285673\pi$
$937$			1465.00i			1.56350i	−0.623592	+	0.781750i	$0.714327\pi$
$938$	−463.155	+	802.208i	−0.493769	+	0.855232i
$939$		0			0
$940$	−58.5641	+	218.564i	−0.0623022	+	0.232515i
$941$	−221.679	+	127.986i	−0.235578	+	0.136011i	−0.613143	−	0.789972i	$-0.710095\pi$
$941$	−221.679	+	127.986i	−0.235578	+	0.136011i	0.377565	+	0.925983i	$0.376762\pi$
$942$		0			0
$943$	−64.0859	−	37.0000i	−0.0679596	−	0.0392365i
$944$		−	362.039i		−	0.383516i
$945$		0			0
$946$	128.000			0.135307
$947$	−203.647	+	352.727i	−0.215044	+	0.372467i	−0.953286	−	0.302069i	$-0.902323\pi$
$947$	−203.647	+	352.727i	−0.215044	+	0.372467i	0.738242	+	0.674536i	$0.235656\pi$
$948$		0			0
$949$	−76.5000	−	132.502i	−0.0806112	−	0.139623i
$950$	642.991	−	371.231i	0.676833	−	0.390770i
$951$		0			0
$952$	138.564	+	80.0000i	0.145550	+	0.0840336i
$953$	497.803			0.522354			0.261177	−	0.965291i	$-0.415889\pi$
$953$	497.803			0.522354			0.261177	+	0.965291i	$0.415889\pi$
$954$		0			0
$955$	−360.000	+	360.000i	−0.376963	+	0.376963i
$956$	−355.176	−	205.061i	−0.371523	−	0.214499i
$957$		0			0
$958$	−206.114	+	119.000i	−0.215150	+	0.124217i
$959$	−1157.38	+	668.216i	−1.20687	+	0.696784i
$960$		0			0
$961$	−319.500	+	553.390i	−0.332466	+	0.575848i
$962$	318.198			0.330767
$963$		0			0
$964$	158.000			0.163900
$965$	−350.700	−	1308.83i	−0.363419	−	1.35630i
$966$		0			0
$967$	−425.218	+	245.500i	−0.439730	+	0.253878i	−0.703483	−	0.710712i	$-0.748373\pi$
$967$	−425.218	+	245.500i	−0.439730	+	0.253878i	0.263753	+	0.964590i	$0.415039\pi$
$968$	−168.291	−	291.489i	−0.173855	−	0.301125i
$969$		0			0
$970$	75.0352	+	280.035i	0.0773559	+	0.288696i
$971$		−	509.117i		−	0.524322i	−0.965024	−	0.262161i	$-0.915565\pi$
$971$		−	509.117i		−	0.524322i	0.965024	−	0.262161i	$-0.0844352\pi$
$972$		0			0
$973$		−	185.000i		−	0.190134i
$974$	620.946	+	358.503i	0.637521	+	0.368073i
$975$		0			0
$976$	194.000	+	336.018i	0.198770	+	0.344281i
$977$	235.467	+	407.840i	0.241010	+	0.417441i	0.961002	−	0.276541i	$-0.0891881\pi$
$977$	235.467	+	407.840i	0.241010	+	0.417441i	−0.719992	+	0.693982i	$0.755855\pi$
$978$		0			0
$979$	104.000	−	180.133i	0.106231	−	0.183997i
$980$	169.706	+	169.706i	0.173169	+	0.173169i
$981$		0			0
$982$			606.000i			0.617108i
$983$	300.520	−	520.517i	0.305718	−	0.529518i	−0.671703	−	0.740820i	$-0.734437\pi$
$983$	300.520	−	520.517i	0.305718	−	0.529518i	0.977421	+	0.211302i	$0.0677703\pi$
$984$		0			0
$985$	409.948	−	1529.95i	0.416191	−	1.55325i
$986$	529.090	−	305.470i	0.536602	−	0.309807i
$987$		0			0
$988$	−327.358	−	189.000i	−0.331334	−	0.191296i
$989$		−	90.5097i		−	0.0915163i
$990$		0			0
$991$	−755.000			−0.761857			−0.380928	−	0.924605i	$-0.624396\pi$
$991$	−755.000			−0.761857			−0.380928	+	0.924605i	$0.624396\pi$
$992$	−113.137	+	195.959i	−0.114049	+	0.197539i
$993$		0			0
$994$	315.000	+	545.596i	0.316901	+	0.548889i
$995$	190.232	−	709.955i	0.191188	−	0.713523i
$996$		0			0
$997$	−20.7846	−	12.0000i	−0.0208472	−	0.0120361i	0.489540	−	0.871981i	$-0.337165\pi$
$997$	−20.7846	−	12.0000i	−0.0208472	−	0.0120361i	−0.510387	+	0.859945i	$0.670498\pi$
$998$	1230.37			1.23283
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.3.j.e.269.2		8
3.2	odd	2	inner	810.3.j.e.269.3		8
5.4	even	2	inner	810.3.j.e.269.4		8
9.2	odd	6		270.3.b.c.269.1	✓	4
9.4	even	3	inner	810.3.j.e.539.1		8
9.5	odd	6	inner	810.3.j.e.539.4		8
9.7	even	3		270.3.b.c.269.4	yes	4
15.14	odd	2	inner	810.3.j.e.269.1		8
36.7	odd	6		2160.3.c.i.1889.4		4
36.11	even	6		2160.3.c.i.1889.1		4
45.2	even	12		1350.3.d.f.701.1		2
45.4	even	6	inner	810.3.j.e.539.3		8
45.7	odd	12		1350.3.d.f.701.2		2
45.14	odd	6	inner	810.3.j.e.539.2		8
45.29	odd	6		270.3.b.c.269.3	yes	4
45.34	even	6		270.3.b.c.269.2	yes	4
45.38	even	12		1350.3.d.g.701.2		2
45.43	odd	12		1350.3.d.g.701.1		2
180.79	odd	6		2160.3.c.i.1889.2		4
180.119	even	6		2160.3.c.i.1889.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.3.b.c.269.1	✓	4	9.2	odd	6
270.3.b.c.269.2	yes	4	45.34	even	6
270.3.b.c.269.3	yes	4	45.29	odd	6
270.3.b.c.269.4	yes	4	9.7	even	3
810.3.j.e.269.1		8	15.14	odd	2	inner
810.3.j.e.269.2		8	1.1	even	1	trivial
810.3.j.e.269.3		8	3.2	odd	2	inner
810.3.j.e.269.4		8	5.4	even	2	inner
810.3.j.e.539.1		8	9.4	even	3	inner
810.3.j.e.539.2		8	45.14	odd	6	inner
810.3.j.e.539.3		8	45.4	even	6	inner
810.3.j.e.539.4		8	9.5	odd	6	inner
1350.3.d.f.701.1		2	45.2	even	12
1350.3.d.f.701.2		2	45.7	odd	12
1350.3.d.g.701.1		2	45.43	odd	12
1350.3.d.g.701.2		2	45.38	even	12
2160.3.c.i.1889.1		4	36.11	even	6
2160.3.c.i.1889.2		4	180.79	odd	6
2160.3.c.i.1889.3		4	180.119	even	6
2160.3.c.i.1889.4		4	36.7	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	810.j (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(1.29410 - 4.82963i) q^{5} +(-4.33013 - 2.50000i) q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+(-0.707107 + 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(1.29410 - 4.82963i) q^{5} +(-4.33013 - 2.50000i) q^{7} +2.82843 q^{8} +(5.00000 + 5.00000i) q^{10} +(1.22474 + 0.707107i) q^{11} +(7.79423 - 4.50000i) q^{13} +(6.12372 - 3.53553i) q^{14} +(-2.00000 + 3.46410i) q^{16} -11.3137 q^{17} +21.0000 q^{19} +(-9.65926 + 2.58819i) q^{20} +(-1.73205 + 1.00000i) q^{22} +(0.707107 + 1.22474i) q^{23} +(-21.6506 - 12.5000i) q^{25} +12.7279i q^{26} +10.0000i q^{28} +(33.0681 + 19.0919i) q^{29} +(-20.0000 - 34.6410i) q^{31} +(-2.82843 - 4.89898i) q^{32} +(8.00000 - 13.8564i) q^{34} +(-17.6777 + 17.6777i) q^{35} -25.0000i q^{37} +(-14.8492 + 25.7196i) q^{38} +(3.66025 - 13.6603i) q^{40} +(-45.3156 + 26.1630i) q^{41} +(-55.4256 - 32.0000i) q^{43} -2.82843i q^{44} -2.00000 q^{46} +(11.3137 - 19.5959i) q^{47} +(-12.0000 - 20.7846i) q^{49} +(30.6186 - 17.6777i) q^{50} +(-15.5885 - 9.00000i) q^{52} -72.1249 q^{53} +(5.00000 - 5.00000i) q^{55} +(-12.2474 - 7.07107i) q^{56} +(-46.7654 + 27.0000i) q^{58} +(-78.3837 + 45.2548i) q^{59} +(48.5000 - 84.0045i) q^{61} +56.5685 q^{62} +8.00000 q^{64} +(-11.6469 - 43.4667i) q^{65} +(-113.449 + 65.5000i) q^{67} +(11.3137 + 19.5959i) q^{68} +(-9.15064 - 34.1506i) q^{70} +89.0955i q^{71} -17.0000i q^{73} +(30.6186 + 17.6777i) q^{74} +(-21.0000 - 36.3731i) q^{76} +(-3.53553 - 6.12372i) q^{77} +(58.5000 - 101.325i) q^{79} +(14.1421 + 14.1421i) q^{80} -74.0000i q^{82} +(-28.9914 + 50.2145i) q^{83} +(-14.6410 + 54.6410i) q^{85} +(78.3837 - 45.2548i) q^{86} +(3.46410 + 2.00000i) q^{88} -147.078i q^{89} -45.0000 q^{91} +(1.41421 - 2.44949i) q^{92} +(16.0000 + 27.7128i) q^{94} +(27.1760 - 101.422i) q^{95} +(35.5070 + 20.5000i) q^{97} +33.9411 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} + 40 q^{10} - 16 q^{16} + 168 q^{19} - 160 q^{31} + 64 q^{34} - 40 q^{40} - 16 q^{46} - 96 q^{49} + 40 q^{55} + 388 q^{61} + 64 q^{64} + 100 q^{70} - 168 q^{76} + 468 q^{79} + 160 q^{85} - 360 q^{91} + 128 q^{94}+O(q^{100}) \) 8 * q - 8 * q^4 + 40 * q^10 - 16 * q^16 + 168 * q^19 - 160 * q^31 + 64 * q^34 - 40 * q^40 - 16 * q^46 - 96 * q^49 + 40 * q^55 + 388 * q^61 + 64 * q^64 + 100 * q^70 - 168 * q^76 + 468 * q^79 + 160 * q^85 - 360 * q^91 + 128 * q^94

Introduction

L-functions

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