LMFDB - Embedded newform 448.2.ba.b.81.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,2,Mod(81,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 9, 8]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("448.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	448.ba (of order $12$, degree $4$, 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:	$3.57729801055$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			81.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	448.81
Dual form			448.2.ba.b.177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.86603 - 0.500000i) q^{3} +(-3.23205 - 0.866025i) q^{5} +(-1.73205 - 2.00000i) q^{7} +(0.633975 - 0.366025i) q^{9} +O(q^{10})$	\(q+(1.86603 - 0.500000i) q^{3} +(-3.23205 - 0.866025i) q^{5} +(-1.73205 - 2.00000i) q^{7} +(0.633975 - 0.366025i) q^{9} +(-1.13397 - 4.23205i) q^{11} +(0.267949 + 0.267949i) q^{13} -6.46410 q^{15} +(0.232051 - 0.401924i) q^{17} +(1.13397 - 4.23205i) q^{19} +(-4.23205 - 2.86603i) q^{21} +(2.13397 - 1.23205i) q^{23} +(5.36603 + 3.09808i) q^{25} +(-3.09808 + 3.09808i) q^{27} +(-3.73205 - 3.73205i) q^{29} +(0.133975 - 0.232051i) q^{31} +(-4.23205 - 7.33013i) q^{33} +(3.86603 + 7.96410i) q^{35} +(10.6962 + 2.86603i) q^{37} +(0.633975 + 0.366025i) q^{39} +8.92820i q^{41} +(0.464102 - 0.464102i) q^{43} +(-2.36603 + 0.633975i) q^{45} +(3.86603 + 6.69615i) q^{47} +(-1.00000 + 6.92820i) q^{49} +(0.232051 - 0.866025i) q^{51} +(-2.96410 - 11.0622i) q^{53} +14.6603i q^{55} -8.46410i q^{57} +(-2.66987 - 9.96410i) q^{59} +(-0.0358984 + 0.133975i) q^{61} +(-1.83013 - 0.633975i) q^{63} +(-0.633975 - 1.09808i) q^{65} +(7.33013 - 1.96410i) q^{67} +(3.36603 - 3.36603i) q^{69} -7.46410i q^{71} +(-2.76795 - 1.59808i) q^{73} +(11.5622 + 3.09808i) q^{75} +(-6.50000 + 9.59808i) q^{77} +(0.330127 + 0.571797i) q^{79} +(-5.33013 + 9.23205i) q^{81} +(-8.46410 - 8.46410i) q^{83} +(-1.09808 + 1.09808i) q^{85} +(-8.83013 - 5.09808i) q^{87} +(4.50000 - 2.59808i) q^{89} +(0.0717968 - 1.00000i) q^{91} +(0.133975 - 0.500000i) q^{93} +(-7.33013 + 12.6962i) q^{95} +10.9282 q^{97} +(-2.26795 - 2.26795i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 6 q^{5} + 6 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 6 * q^5 + 6 * q^9	$ 4 q + 4 q^{3} - 6 q^{5} + 6 q^{9} - 8 q^{11} + 8 q^{13} - 12 q^{15} - 6 q^{17} + 8 q^{19} - 10 q^{21} + 12 q^{23} + 18 q^{25} - 2 q^{27} - 8 q^{29} + 4 q^{31} - 10 q^{33} + 12 q^{35} + 22 q^{37} + 6 q^{39} - 12 q^{43} - 6 q^{45} + 12 q^{47} - 4 q^{49} - 6 q^{51} + 2 q^{53} - 28 q^{59} - 14 q^{61} + 10 q^{63} - 6 q^{65} + 12 q^{67} + 10 q^{69} - 18 q^{73} + 22 q^{75} - 26 q^{77} - 16 q^{79} - 4 q^{81} - 20 q^{83} + 6 q^{85} - 18 q^{87} + 18 q^{89} + 28 q^{91} + 4 q^{93} - 12 q^{95} + 16 q^{97} - 16 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 6 * q^5 + 6 * q^9 - 8 * q^11 + 8 * q^13 - 12 * q^15 - 6 * q^17 + 8 * q^19 - 10 * q^21 + 12 * q^23 + 18 * q^25 - 2 * q^27 - 8 * q^29 + 4 * q^31 - 10 * q^33 + 12 * q^35 + 22 * q^37 + 6 * q^39 - 12 * q^43 - 6 * q^45 + 12 * q^47 - 4 * q^49 - 6 * q^51 + 2 * q^53 - 28 * q^59 - 14 * q^61 + 10 * q^63 - 6 * q^65 + 12 * q^67 + 10 * q^69 - 18 * q^73 + 22 * q^75 - 26 * q^77 - 16 * q^79 - 4 * q^81 - 20 * q^83 + 6 * q^85 - 18 * q^87 + 18 * q^89 + 28 * q^91 + 4 * q^93 - 12 * q^95 + 16 * q^97 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{3}{4}\right)$

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			0
$2$		0			0
$3$	1.86603	−	0.500000i	1.07735	−	0.288675i	0.323840	−	0.946112i	$-0.395026\pi$
$3$	1.86603	−	0.500000i	1.07735	−	0.288675i	0.753510	+	0.657437i	$0.228359\pi$
$4$		0			0
$5$	−3.23205	−	0.866025i	−1.44542	−	0.387298i	−0.550990	−	0.834512i	$-0.685750\pi$
$5$	−3.23205	−	0.866025i	−1.44542	−	0.387298i	−0.894427	+	0.447214i	$0.852416\pi$
$6$		0			0
$7$	−1.73205	−	2.00000i	−0.654654	−	0.755929i
$7$	−1.73205	−	2.00000i	−0.654654	−	0.755929i
$8$		0			0
$9$	0.633975	−	0.366025i	0.211325	−	0.122008i
$10$		0			0
$11$	−1.13397	−	4.23205i	−0.341906	−	1.27601i	−0.896185	−	0.443680i	$-0.853673\pi$
$11$	−1.13397	−	4.23205i	−0.341906	−	1.27601i	0.554279	−	0.832331i	$-0.312994\pi$
$12$		0			0
$13$	0.267949	+	0.267949i	0.0743157	+	0.0743157i	0.743288	−	0.668972i	$-0.233265\pi$
$13$	0.267949	+	0.267949i	0.0743157	+	0.0743157i	−0.668972	+	0.743288i	$0.733265\pi$
$14$		0			0
$15$	−6.46410			−1.66902
$16$		0			0
$17$	0.232051	−	0.401924i	0.0562806	−	0.0974808i	−0.836512	−	0.547948i	$-0.815409\pi$
$17$	0.232051	−	0.401924i	0.0562806	−	0.0974808i	0.892793	+	0.450467i	$0.148743\pi$
$18$		0			0
$19$	1.13397	−	4.23205i	0.260152	−	0.970899i	−0.705000	−	0.709207i	$-0.749053\pi$
$19$	1.13397	−	4.23205i	0.260152	−	0.970899i	0.965152	−	0.261692i	$-0.0842803\pi$
$20$		0			0
$21$	−4.23205	−	2.86603i	−0.923509	−	0.625418i
$22$		0			0
$23$	2.13397	−	1.23205i	0.444964	−	0.256900i	−0.260737	−	0.965410i	$-0.583965\pi$
$23$	2.13397	−	1.23205i	0.444964	−	0.256900i	0.705701	+	0.708510i	$0.250632\pi$
$24$		0			0
$25$	5.36603	+	3.09808i	1.07321	+	0.619615i
$26$		0			0
$27$	−3.09808	+	3.09808i	−0.596225	+	0.596225i
$28$		0			0
$29$	−3.73205	−	3.73205i	−0.693024	−	0.693024i	0.269872	−	0.962896i	$-0.413019\pi$
$29$	−3.73205	−	3.73205i	−0.693024	−	0.693024i	−0.962896	+	0.269872i	$0.913019\pi$
$30$		0			0
$31$	0.133975	−	0.232051i	0.0240625	−	0.0416776i	−0.853743	−	0.520694i	$-0.825673\pi$
$31$	0.133975	−	0.232051i	0.0240625	−	0.0416776i	0.877806	+	0.479016i	$0.159007\pi$
$32$		0			0
$33$	−4.23205	−	7.33013i	−0.736705	−	1.27601i
$34$		0			0
$35$	3.86603	+	7.96410i	0.653478	+	1.34618i
$36$		0			0
$37$	10.6962	+	2.86603i	1.75844	+	0.471172i	0.986394	−	0.164399i	$-0.0525685\pi$
$37$	10.6962	+	2.86603i	1.75844	+	0.471172i	0.772043	+	0.635571i	$0.219235\pi$
$38$		0			0
$39$	0.633975	+	0.366025i	0.101517	+	0.0586110i
$40$		0			0
$41$			8.92820i			1.39435i	0.716900	+	0.697176i	$0.245560\pi$
$41$			8.92820i			1.39435i	−0.716900	+	0.697176i	$0.754440\pi$
$42$		0			0
$43$	0.464102	−	0.464102i	0.0707748	−	0.0707748i	−0.670833	−	0.741608i	$-0.734063\pi$
$43$	0.464102	−	0.464102i	0.0707748	−	0.0707748i	0.741608	+	0.670833i	$0.234063\pi$
$44$		0			0
$45$	−2.36603	+	0.633975i	−0.352706	+	0.0945074i
$46$		0			0
$47$	3.86603	+	6.69615i	0.563918	+	0.976734i	0.997149	+	0.0754516i	$0.0240398\pi$
$47$	3.86603	+	6.69615i	0.563918	+	0.976734i	−0.433232	+	0.901283i	$0.642627\pi$
$48$		0			0
$49$	−1.00000	+	6.92820i	−0.142857	+	0.989743i
$50$		0			0
$51$	0.232051	−	0.866025i	0.0324936	−	0.121268i
$52$		0			0
$53$	−2.96410	−	11.0622i	−0.407151	−	1.51951i	−0.800054	−	0.599927i	$-0.795196\pi$
$53$	−2.96410	−	11.0622i	−0.407151	−	1.51951i	0.392904	−	0.919580i	$-0.371471\pi$
$54$		0			0
$55$			14.6603i			1.97679i
$56$		0			0
$57$		−	8.46410i		−	1.12110i
$58$		0			0
$59$	−2.66987	−	9.96410i	−0.347588	−	1.29722i	−0.889560	−	0.456819i	$-0.848989\pi$
$59$	−2.66987	−	9.96410i	−0.347588	−	1.29722i	0.541972	−	0.840397i	$-0.317678\pi$
$60$		0			0
$61$	−0.0358984	+	0.133975i	−0.00459632	+	0.0171537i	−0.968186	−	0.250232i	$-0.919493\pi$
$61$	−0.0358984	+	0.133975i	−0.00459632	+	0.0171537i	0.963590	+	0.267386i	$0.0861598\pi$
$62$		0			0
$63$	−1.83013	−	0.633975i	−0.230574	−	0.0798733i
$64$		0			0
$65$	−0.633975	−	1.09808i	−0.0786349	−	0.136200i
$66$		0			0
$67$	7.33013	−	1.96410i	0.895518	−	0.239953i	0.218427	−	0.975853i	$-0.429907\pi$
$67$	7.33013	−	1.96410i	0.895518	−	0.239953i	0.677090	+	0.735900i	$0.263241\pi$
$68$		0			0
$69$	3.36603	−	3.36603i	0.405222	−	0.405222i
$70$		0			0
$71$		−	7.46410i		−	0.885826i	−0.896565	−	0.442913i	$-0.853945\pi$
$71$		−	7.46410i		−	0.885826i	0.896565	−	0.442913i	$-0.146055\pi$
$72$		0			0
$73$	−2.76795	−	1.59808i	−0.323964	−	0.187041i	0.329194	−	0.944262i	$-0.393223\pi$
$73$	−2.76795	−	1.59808i	−0.323964	−	0.187041i	−0.653158	+	0.757222i	$0.726556\pi$
$74$		0			0
$75$	11.5622	+	3.09808i	1.33509	+	0.357735i
$76$		0			0
$77$	−6.50000	+	9.59808i	−0.740744	+	1.09380i
$78$		0			0
$79$	0.330127	+	0.571797i	0.0371422	+	0.0643322i	0.883999	−	0.467489i	$-0.154841\pi$
$79$	0.330127	+	0.571797i	0.0371422	+	0.0643322i	−0.846857	+	0.531821i	$0.821508\pi$
$80$		0			0
$81$	−5.33013	+	9.23205i	−0.592236	+	1.02578i
$82$		0			0
$83$	−8.46410	−	8.46410i	−0.929056	−	0.929056i	0.0685891	−	0.997645i	$-0.478150\pi$
$83$	−8.46410	−	8.46410i	−0.929056	−	0.929056i	−0.997645	+	0.0685891i	$0.978150\pi$
$84$		0			0
$85$	−1.09808	+	1.09808i	−0.119103	+	0.119103i
$86$		0			0
$87$	−8.83013	−	5.09808i	−0.946689	−	0.546571i
$88$		0			0
$89$	4.50000	−	2.59808i	0.476999	−	0.275396i	−0.242166	−	0.970235i	$-0.577858\pi$
$89$	4.50000	−	2.59808i	0.476999	−	0.275396i	0.719165	+	0.694839i	$0.244525\pi$
$90$		0			0
$91$	0.0717968	−	1.00000i	0.00752635	−	0.104828i
$92$		0			0
$93$	0.133975	−	0.500000i	0.0138925	−	0.0518476i
$94$		0			0
$95$	−7.33013	+	12.6962i	−0.752055	+	1.30260i
$96$		0			0
$97$	10.9282			1.10959			0.554795	−	0.831987i	$-0.312797\pi$
$97$	10.9282			1.10959			0.554795	+	0.831987i	$0.312797\pi$
$98$		0			0
$99$	−2.26795	−	2.26795i	−0.227937	−	0.227937i
$100$		0			0
$101$	1.89230	+	7.06218i	0.188291	+	0.702713i	0.993902	+	0.110268i	$0.0351708\pi$
$101$	1.89230	+	7.06218i	0.188291	+	0.702713i	−0.805611	+	0.592445i	$0.798163\pi$
$102$		0			0
$103$	−0.401924	+	0.232051i	−0.0396027	+	0.0228646i	−0.519671	−	0.854367i	$-0.673945\pi$
$103$	−0.401924	+	0.232051i	−0.0396027	+	0.0228646i	0.480068	+	0.877231i	$0.340612\pi$
$104$		0			0
$105$	11.1962	+	12.9282i	1.09263	+	1.26166i
$106$		0			0
$107$	−0.866025	−	0.232051i	−0.0837218	−	0.0224332i	0.216715	−	0.976235i	$-0.430466\pi$
$107$	−0.866025	−	0.232051i	−0.0837218	−	0.0224332i	−0.300437	+	0.953802i	$0.597132\pi$
$108$		0			0
$109$	14.4282	−	3.86603i	1.38197	−	0.370298i	0.510134	−	0.860095i	$-0.329596\pi$
$109$	14.4282	−	3.86603i	1.38197	−	0.370298i	0.871837	+	0.489797i	$0.162929\pi$
$110$		0			0
$111$	21.3923			2.03047
$112$		0			0
$113$	5.46410			0.514019			0.257010	−	0.966409i	$-0.417263\pi$
$113$	5.46410			0.514019			0.257010	+	0.966409i	$0.417263\pi$
$114$		0			0
$115$	−7.96410	+	2.13397i	−0.742656	+	0.198994i
$116$		0			0
$117$	0.267949	+	0.0717968i	0.0247719	+	0.00663761i
$118$		0			0
$119$	−1.20577	+	0.232051i	−0.110533	+	0.0212721i
$120$		0			0
$121$	−7.09808	+	4.09808i	−0.645280	+	0.372552i
$122$		0			0
$123$	4.46410	+	16.6603i	0.402514	+	1.50220i
$124$		0			0
$125$	−2.83013	−	2.83013i	−0.253134	−	0.253134i
$126$		0			0
$127$	−2.53590			−0.225025			−0.112512	−	0.993650i	$-0.535890\pi$
$127$	−2.53590			−0.225025			−0.112512	+	0.993650i	$0.535890\pi$
$128$		0			0
$129$	0.633975	−	1.09808i	0.0558184	−	0.0966802i
$130$		0			0
$131$	−2.40192	+	8.96410i	−0.209857	+	0.783197i	0.778057	+	0.628194i	$0.216206\pi$
$131$	−2.40192	+	8.96410i	−0.209857	+	0.783197i	−0.987914	+	0.155003i	$0.950461\pi$
$132$		0			0
$133$	−10.4282	+	5.06218i	−0.904240	+	0.438946i
$134$		0			0
$135$	12.6962	−	7.33013i	1.09271	−	0.630877i
$136$		0			0
$137$	−11.7679	−	6.79423i	−1.00540	−	0.580470i	−0.0955611	−	0.995424i	$-0.530465\pi$
$137$	−11.7679	−	6.79423i	−1.00540	−	0.580470i	−0.909843	+	0.414953i	$0.863798\pi$
$138$		0			0
$139$	−1.92820	+	1.92820i	−0.163548	+	0.163548i	−0.784136	−	0.620588i	$-0.786894\pi$
$139$	−1.92820	+	1.92820i	−0.163548	+	0.163548i	0.620588	+	0.784136i	$0.286894\pi$
$140$		0			0
$141$	10.5622	+	10.5622i	0.889496	+	0.889496i
$142$		0			0
$143$	0.830127	−	1.43782i	0.0694187	−	0.120237i
$144$		0			0
$145$	8.83013	+	15.2942i	0.733302	+	1.27012i
$146$		0			0
$147$	1.59808	+	13.4282i	0.131807	+	1.10754i
$148$		0			0
$149$	8.96410	+	2.40192i	0.734368	+	0.196773i	0.606573	−	0.795027i	$-0.292544\pi$
$149$	8.96410	+	2.40192i	0.734368	+	0.196773i	0.127794	+	0.991801i	$0.459210\pi$
$150$		0			0
$151$	8.13397	+	4.69615i	0.661933	+	0.382167i	0.793013	−	0.609204i	$-0.208511\pi$
$151$	8.13397	+	4.69615i	0.661933	+	0.382167i	−0.131080	+	0.991372i	$0.541844\pi$
$152$		0			0
$153$		−	0.339746i		−	0.0274668i
$154$		0			0
$155$	−0.633975	+	0.633975i	−0.0509221	+	0.0509221i
$156$		0			0
$157$	−15.8923	+	4.25833i	−1.26834	+	0.339852i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	−15.8923	+	4.25833i	−1.26834	+	0.339852i	−0.438948	+	0.898513i	$0.644649\pi$
$158$		0			0
$159$	−11.0622	−	19.1603i	−0.877288	−	1.51951i
$160$		0			0
$161$	−6.16025	−	2.13397i	−0.485496	−	0.168181i
$162$		0			0
$163$	−0.0621778	+	0.232051i	−0.00487014	+	0.0181756i	−0.968318	−	0.249720i	$-0.919661\pi$
$163$	−0.0621778	+	0.232051i	−0.00487014	+	0.0181756i	0.963448	+	0.267895i	$0.0863282\pi$
$164$		0			0
$165$	7.33013	+	27.3564i	0.570650	+	2.12969i
$166$		0			0
$167$		−	5.85641i		−	0.453182i	−0.973990	−	0.226591i	$-0.927242\pi$
$167$		−	5.85641i		−	0.453182i	0.973990	−	0.226591i	$-0.0727581\pi$
$168$		0			0
$169$		−	12.8564i		−	0.988954i
$170$		0			0
$171$	−0.830127	−	3.09808i	−0.0634814	−	0.236916i
$172$		0			0
$173$	1.23205	−	4.59808i	0.0936711	−	0.349585i	−0.903144	−	0.429339i	$-0.858747\pi$
$173$	1.23205	−	4.59808i	0.0936711	−	0.349585i	0.996815	+	0.0797535i	$0.0254133\pi$
$174$		0			0
$175$	−3.09808	−	16.0981i	−0.234193	−	1.21690i
$176$		0			0
$177$	−9.96410	−	17.2583i	−0.748948	−	1.29722i
$178$		0			0
$179$	7.59808	−	2.03590i	0.567907	−	0.152170i	0.0365704	−	0.999331i	$-0.488357\pi$
$179$	7.59808	−	2.03590i	0.567907	−	0.152170i	0.531336	+	0.847161i	$0.321690\pi$
$180$		0			0
$181$	−7.39230	+	7.39230i	−0.549466	+	0.549466i	−0.926286	−	0.376821i	$-0.877017\pi$
$181$	−7.39230	+	7.39230i	−0.549466	+	0.549466i	0.376821	+	0.926286i	$0.377017\pi$
$182$		0			0
$183$			0.267949i			0.0198074i
$184$		0			0
$185$	−32.0885	−	18.5263i	−2.35919	−	1.36208i
$186$		0			0
$187$	−1.96410	−	0.526279i	−0.143629	−	0.0384854i
$188$		0			0
$189$	11.5622	+	0.830127i	0.841025	+	0.0603829i
$190$		0			0
$191$	−4.33013	−	7.50000i	−0.313317	−	0.542681i	0.665761	−	0.746165i	$-0.268107\pi$
$191$	−4.33013	−	7.50000i	−0.313317	−	0.542681i	−0.979078	+	0.203484i	$0.934774\pi$
$192$		0			0
$193$	11.5000	−	19.9186i	0.827788	−	1.43377i	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	11.5000	−	19.9186i	0.827788	−	1.43377i	0.899770	−	0.436365i	$-0.143734\pi$
$194$		0			0
$195$	−1.73205	−	1.73205i	−0.124035	−	0.124035i
$196$		0			0
$197$	0.660254	−	0.660254i	0.0470412	−	0.0470412i	−0.683195	−	0.730236i	$-0.739410\pi$
$197$	0.660254	−	0.660254i	0.0470412	−	0.0470412i	0.730236	+	0.683195i	$0.239410\pi$
$198$		0			0
$199$	−1.66987	−	0.964102i	−0.118374	−	0.0683434i	0.439644	−	0.898172i	$-0.355105\pi$
$199$	−1.66987	−	0.964102i	−0.118374	−	0.0683434i	−0.558018	+	0.829829i	$0.688438\pi$
$200$		0			0
$201$	12.6962	−	7.33013i	0.895518	−	0.517027i
$202$		0			0
$203$	−1.00000	+	13.9282i	−0.0701862	+	0.977568i
$204$		0			0
$205$	7.73205	−	28.8564i	0.540030	−	2.01542i
$206$		0			0
$207$	0.901924	−	1.56218i	0.0626880	−	0.108579i
$208$		0			0
$209$	−19.1962			−1.32783
$210$		0			0
$211$	2.07180	+	2.07180i	0.142628	+	0.142628i	0.774816	−	0.632187i	$-0.217843\pi$
$211$	2.07180	+	2.07180i	0.142628	+	0.142628i	−0.632187	+	0.774816i	$0.717843\pi$
$212$		0			0
$213$	−3.73205	−	13.9282i	−0.255716	−	0.954345i
$214$		0			0
$215$	−1.90192	+	1.09808i	−0.129710	+	0.0748882i
$216$		0			0
$217$	−0.696152	+	0.133975i	−0.0472579	+	0.00909479i
$218$		0			0
$219$	−5.96410	−	1.59808i	−0.403017	−	0.107988i
$220$		0			0
$221$	0.169873	−	0.0455173i	0.0114269	−	0.00306183i
$222$		0			0
$223$	17.8564			1.19575			0.597877	−	0.801588i	$-0.296011\pi$
$223$	17.8564			1.19575			0.597877	+	0.801588i	$0.296011\pi$
$224$		0			0
$225$	4.53590			0.302393
$226$		0			0
$227$	−22.9904	+	6.16025i	−1.52593	+	0.408870i	−0.921687	−	0.387934i	$-0.873189\pi$
$227$	−22.9904	+	6.16025i	−1.52593	+	0.408870i	−0.604238	+	0.796804i	$0.706522\pi$
$228$		0			0
$229$	−17.4282	−	4.66987i	−1.15169	−	0.308594i	−0.368047	−	0.929807i	$-0.619973\pi$
$229$	−17.4282	−	4.66987i	−1.15169	−	0.308594i	−0.783641	+	0.621213i	$0.786640\pi$
$230$		0			0
$231$	−7.33013	+	21.1603i	−0.482287	+	1.39224i
$232$		0			0
$233$	9.69615	−	5.59808i	0.635216	−	0.366742i	−0.147553	−	0.989054i	$-0.547140\pi$
$233$	9.69615	−	5.59808i	0.635216	−	0.366742i	0.782769	+	0.622312i	$0.213806\pi$
$234$		0			0
$235$	−6.69615	−	24.9904i	−0.436809	−	1.63019i
$236$		0			0
$237$	0.901924	+	0.901924i	0.0585862	+	0.0585862i
$238$		0			0
$239$	15.4641			1.00029			0.500145	−	0.865942i	$-0.333280\pi$
$239$	15.4641			1.00029			0.500145	+	0.865942i	$0.333280\pi$
$240$		0			0
$241$	−0.0358984	+	0.0621778i	−0.00231242	+	0.00400523i	−0.867179	−	0.497996i	$-0.834069\pi$
$241$	−0.0358984	+	0.0621778i	−0.00231242	+	0.00400523i	0.864867	+	0.502001i	$0.167403\pi$
$242$		0			0
$243$	−1.92820	+	7.19615i	−0.123694	+	0.461633i
$244$		0			0
$245$	9.23205	−	21.5263i	0.589814	−	1.37526i
$246$		0			0
$247$	1.43782	−	0.830127i	0.0914864	−	0.0528197i
$248$		0			0
$249$	−20.0263	−	11.5622i	−1.26911	−	0.732723i
$250$		0			0
$251$	−13.5885	+	13.5885i	−0.857696	+	0.857696i	−0.991066	−	0.133370i	$-0.957420\pi$
$251$	−13.5885	+	13.5885i	−0.857696	+	0.857696i	0.133370	+	0.991066i	$0.457420\pi$
$252$		0			0
$253$	−7.63397	−	7.63397i	−0.479944	−	0.479944i
$254$		0			0
$255$	−1.50000	+	2.59808i	−0.0939336	+	0.162698i
$256$		0			0
$257$	−0.696152	−	1.20577i	−0.0434248	−	0.0752140i	0.843496	−	0.537135i	$-0.180494\pi$
$257$	−0.696152	−	1.20577i	−0.0434248	−	0.0752140i	−0.886921	+	0.461921i	$0.847160\pi$
$258$		0			0
$259$	−12.7942	−	26.3564i	−0.794995	−	1.63771i
$260$		0			0
$261$	−3.73205	−	1.00000i	−0.231008	−	0.0618984i
$262$		0			0
$263$	−21.9904	−	12.6962i	−1.35598	−	0.782878i	−0.366905	−	0.930258i	$-0.619583\pi$
$263$	−21.9904	−	12.6962i	−1.35598	−	0.782878i	−0.989080	+	0.147380i	$0.952916\pi$
$264$		0			0
$265$			38.3205i			2.35401i
$266$		0			0
$267$	7.09808	−	7.09808i	0.434395	−	0.434395i
$268$		0			0
$269$	21.6244	−	5.79423i	1.31846	−	0.353280i	0.470059	−	0.882635i	$-0.344232\pi$
$269$	21.6244	−	5.79423i	1.31846	−	0.353280i	0.848401	+	0.529354i	$0.177566\pi$
$270$		0			0
$271$	−6.06218	−	10.5000i	−0.368251	−	0.637830i	0.621041	−	0.783778i	$-0.286710\pi$
$271$	−6.06218	−	10.5000i	−0.368251	−	0.637830i	−0.989292	+	0.145948i	$0.953377\pi$
$272$		0			0
$273$	−0.366025	−	1.90192i	−0.0221529	−	0.115110i
$274$		0			0
$275$	7.02628	−	26.2224i	0.423701	−	1.58127i
$276$		0			0
$277$	4.50000	+	16.7942i	0.270379	+	1.00907i	0.958875	+	0.283828i	$0.0916044\pi$
$277$	4.50000	+	16.7942i	0.270379	+	1.00907i	−0.688496	+	0.725240i	$0.741729\pi$
$278$		0			0
$279$		−	0.196152i		−	0.0117433i
$280$		0			0
$281$			12.9282i			0.771232i	0.922659	+	0.385616i	$0.126011\pi$
$281$			12.9282i			0.771232i	−0.922659	+	0.385616i	$0.873989\pi$
$282$		0			0
$283$	4.52628	+	16.8923i	0.269059	+	1.00414i	0.959719	+	0.280963i	$0.0906538\pi$
$283$	4.52628	+	16.8923i	0.269059	+	1.00414i	−0.690659	+	0.723180i	$0.742680\pi$
$284$		0			0
$285$	−7.33013	+	27.3564i	−0.434199	+	1.62045i
$286$		0			0
$287$	17.8564	−	15.4641i	1.05403	−	0.912817i
$288$		0			0
$289$	8.39230	+	14.5359i	0.493665	+	0.855053i
$290$		0			0
$291$	20.3923	−	5.46410i	1.19542	−	0.320311i
$292$		0			0
$293$	5.92820	−	5.92820i	0.346329	−	0.346329i	−0.512411	−	0.858740i	$-0.671248\pi$
$293$	5.92820	−	5.92820i	0.346329	−	0.346329i	0.858740	+	0.512411i	$0.171248\pi$
$294$		0			0
$295$			34.5167i			2.00964i
$296$		0			0
$297$	16.6244	+	9.59808i	0.964643	+	0.556937i
$298$		0			0
$299$	0.901924	+	0.241670i	0.0521596	+	0.0139761i
$300$		0			0
$301$	−1.73205	−	0.124356i	−0.0998337	−	0.00716774i
$302$		0			0
$303$	7.06218	+	12.2321i	0.405712	+	0.702713i
$304$		0			0
$305$	0.232051	−	0.401924i	0.0132872	−	0.0230141i
$306$		0			0
$307$	−9.00000	−	9.00000i	−0.513657	−	0.513657i	0.401988	−	0.915645i	$-0.368319\pi$
$307$	−9.00000	−	9.00000i	−0.513657	−	0.513657i	−0.915645	+	0.401988i	$0.868319\pi$
$308$		0			0
$309$	−0.633975	+	0.633975i	−0.0360656	+	0.0360656i
$310$		0			0
$311$	0.401924	+	0.232051i	0.0227910	+	0.0131584i	0.511352	−	0.859371i	$-0.329145\pi$
$311$	0.401924	+	0.232051i	0.0227910	+	0.0131584i	−0.488561	+	0.872530i	$0.662478\pi$
$312$		0			0
$313$	8.30385	−	4.79423i	0.469361	−	0.270986i	−0.246611	−	0.969115i	$-0.579317\pi$
$313$	8.30385	−	4.79423i	0.469361	−	0.270986i	0.715972	+	0.698129i	$0.245984\pi$
$314$		0			0
$315$	5.36603	+	3.63397i	0.302341	+	0.204751i
$316$		0			0
$317$	−3.16025	+	11.7942i	−0.177498	+	0.662430i	0.818615	+	0.574342i	$0.194742\pi$
$317$	−3.16025	+	11.7942i	−0.177498	+	0.662430i	−0.996113	+	0.0880875i	$0.971924\pi$
$318$		0			0
$319$	−11.5622	+	20.0263i	−0.647358	+	1.12126i
$320$		0			0
$321$	−1.73205			−0.0966736
$322$		0			0
$323$	−1.43782	−	1.43782i	−0.0800026	−	0.0800026i
$324$		0			0
$325$	0.607695	+	2.26795i	0.0337089	+	0.125803i
$326$		0			0
$327$	24.9904	−	14.4282i	1.38197	−	0.797881i
$328$		0			0
$329$	6.69615	−	19.3301i	0.369171	−	1.06570i
$330$		0			0
$331$	17.2583	+	4.62436i	0.948604	+	0.254178i	0.699770	−	0.714369i	$-0.253286\pi$
$331$	17.2583	+	4.62436i	0.948604	+	0.254178i	0.248834	+	0.968546i	$0.419953\pi$
$332$		0			0
$333$	7.83013	−	2.09808i	0.429088	−	0.114974i
$334$		0			0
$335$	−25.3923			−1.38733
$336$		0			0
$337$	−33.8564			−1.84428			−0.922138	−	0.386861i	$-0.873559\pi$
$337$	−33.8564			−1.84428			−0.922138	+	0.386861i	$0.873559\pi$
$338$		0			0
$339$	10.1962	−	2.73205i	0.553779	−	0.148385i
$340$		0			0
$341$	−1.13397	−	0.303848i	−0.0614082	−	0.0164543i
$342$		0			0
$343$	15.5885	−	10.0000i	0.841698	−	0.539949i
$344$		0			0
$345$	−13.7942	+	7.96410i	−0.742656	+	0.428773i
$346$		0			0
$347$	−4.47372	−	16.6962i	−0.240162	−	0.896296i	−0.975754	−	0.218871i	$-0.929762\pi$
$347$	−4.47372	−	16.6962i	−0.240162	−	0.896296i	0.735592	−	0.677425i	$-0.236904\pi$
$348$		0			0
$349$	−18.1244	−	18.1244i	−0.970175	−	0.970175i	0.0293934	−	0.999568i	$-0.490642\pi$
$349$	−18.1244	−	18.1244i	−0.970175	−	0.970175i	−0.999568	+	0.0293934i	$0.990642\pi$
$350$		0			0
$351$	−1.66025			−0.0886178
$352$		0			0
$353$	−6.89230	+	11.9378i	−0.366840	+	0.635386i	−0.989070	−	0.147449i	$-0.952894\pi$
$353$	−6.89230	+	11.9378i	−0.366840	+	0.635386i	0.622229	+	0.782835i	$0.286227\pi$
$354$		0			0
$355$	−6.46410	+	24.1244i	−0.343079	+	1.28039i
$356$		0			0
$357$	−2.13397	+	1.03590i	−0.112942	+	0.0548256i
$358$		0			0
$359$	−2.13397	+	1.23205i	−0.112627	+	0.0650252i	−0.555255	−	0.831680i	$-0.687379\pi$
$359$	−2.13397	+	1.23205i	−0.112627	+	0.0650252i	0.442628	+	0.896705i	$0.354046\pi$
$360$		0			0
$361$	−0.169873	−	0.0980762i	−0.00894068	−	0.00516191i
$362$		0			0
$363$	−11.1962	+	11.1962i	−0.587646	+	0.587646i
$364$		0			0
$365$	7.56218	+	7.56218i	0.395822	+	0.395822i
$366$		0			0
$367$	−9.06218	+	15.6962i	−0.473042	+	0.819332i	−0.999524	−	0.0308537i	$-0.990177\pi$
$367$	−9.06218	+	15.6962i	−0.473042	+	0.819332i	0.526482	+	0.850186i	$0.323511\pi$
$368$		0			0
$369$	3.26795	+	5.66025i	0.170123	+	0.294661i
$370$		0			0
$371$	−16.9904	+	25.0885i	−0.882097	+	1.30253i
$372$		0			0
$373$	3.23205	+	0.866025i	0.167349	+	0.0448411i	0.341521	−	0.939874i	$-0.389058\pi$
$373$	3.23205	+	0.866025i	0.167349	+	0.0448411i	−0.174171	+	0.984715i	$0.555725\pi$
$374$		0			0
$375$	−6.69615	−	3.86603i	−0.345788	−	0.199641i
$376$		0			0
$377$		−	2.00000i		−	0.103005i
$378$		0			0
$379$	24.1244	−	24.1244i	1.23918	−	1.23918i	0.278850	−	0.960335i	$-0.410047\pi$
$379$	24.1244	−	24.1244i	1.23918	−	1.23918i	0.960335	−	0.278850i	$-0.0899533\pi$
$380$		0			0
$381$	−4.73205	+	1.26795i	−0.242430	+	0.0649590i
$382$		0			0
$383$	−7.40192	−	12.8205i	−0.378221	−	0.655097i	0.612583	−	0.790406i	$-0.290131\pi$
$383$	−7.40192	−	12.8205i	−0.378221	−	0.655097i	−0.990803	+	0.135309i	$0.956797\pi$
$384$		0			0
$385$	29.3205	−	25.3923i	1.49431	−	1.29411i
$386$		0			0
$387$	0.124356	−	0.464102i	0.00632135	−	0.0235916i
$388$		0			0
$389$	5.35641	+	19.9904i	0.271581	+	1.01355i	0.958098	+	0.286440i	$0.0924717\pi$
$389$	5.35641	+	19.9904i	0.271581	+	1.01355i	−0.686518	+	0.727113i	$0.740862\pi$
$390$		0			0
$391$		−	1.14359i		−	0.0578340i
$392$		0			0
$393$			17.9282i			0.904358i
$394$		0			0
$395$	−0.571797	−	2.13397i	−0.0287702	−	0.107372i
$396$		0			0
$397$	−1.37564	+	5.13397i	−0.0690416	+	0.257667i	−0.991816	−	0.127673i	$-0.959249\pi$
$397$	−1.37564	+	5.13397i	−0.0690416	+	0.257667i	0.922775	+	0.385340i	$0.125916\pi$
$398$		0			0
$399$	−16.9282	+	14.6603i	−0.847470	+	0.733931i
$400$		0			0
$401$	7.50000	+	12.9904i	0.374532	+	0.648709i	0.990257	−	0.139253i	$-0.0444700\pi$
$401$	7.50000	+	12.9904i	0.374532	+	0.648709i	−0.615725	+	0.787961i	$0.711137\pi$
$402$		0			0
$403$	0.0980762	−	0.0262794i	0.00488552	−	0.00130907i
$404$		0			0
$405$	25.2224	−	25.2224i	1.25331	−	1.25331i
$406$		0			0
$407$		−	48.5167i		−	2.40488i
$408$		0			0
$409$	7.50000	+	4.33013i	0.370851	+	0.214111i	0.673830	−	0.738886i	$-0.264648\pi$
$409$	7.50000	+	4.33013i	0.370851	+	0.214111i	−0.302979	+	0.952997i	$0.597981\pi$
$410$		0			0
$411$	−25.3564	−	6.79423i	−1.25074	−	0.335135i
$412$		0			0
$413$	−15.3038	+	22.5981i	−0.753053	+	1.11198i
$414$		0			0
$415$	20.0263	+	34.6865i	0.983051	+	1.70269i
$416$		0			0
$417$	−2.63397	+	4.56218i	−0.128986	+	0.223411i
$418$		0			0
$419$	19.0000	+	19.0000i	0.928211	+	0.928211i	0.997590	−	0.0693796i	$-0.0221020\pi$
$419$	19.0000	+	19.0000i	0.928211	+	0.928211i	−0.0693796	+	0.997590i	$0.522102\pi$
$420$		0			0
$421$	−8.66025	+	8.66025i	−0.422075	+	0.422075i	−0.885918	−	0.463843i	$-0.846470\pi$
$421$	−8.66025	+	8.66025i	−0.422075	+	0.422075i	0.463843	+	0.885918i	$0.346470\pi$
$422$		0			0
$423$	4.90192	+	2.83013i	0.238340	+	0.137605i
$424$		0			0
$425$	2.49038	−	1.43782i	0.120801	−	0.0697446i
$426$		0			0
$427$	0.330127	−	0.160254i	0.0159760	−	0.00775524i
$428$		0			0
$429$	0.830127	−	3.09808i	0.0400789	−	0.149577i
$430$		0			0
$431$	6.66987	−	11.5526i	0.321276	−	0.556467i	−0.659475	−	0.751726i	$-0.729221\pi$
$431$	6.66987	−	11.5526i	0.321276	−	0.556467i	0.980752	+	0.195259i	$0.0625548\pi$
$432$		0			0
$433$	29.1769			1.40215			0.701077	−	0.713086i	$-0.252703\pi$
$433$	29.1769			1.40215			0.701077	+	0.713086i	$0.252703\pi$
$434$		0			0
$435$	24.1244	+	24.1244i	1.15667	+	1.15667i
$436$		0			0
$437$	−2.79423	−	10.4282i	−0.133666	−	0.498849i
$438$		0			0
$439$	−12.5263	+	7.23205i	−0.597847	+	0.345167i	−0.768194	−	0.640217i	$-0.778844\pi$
$439$	−12.5263	+	7.23205i	−0.597847	+	0.345167i	0.170347	+	0.985384i	$0.445511\pi$
$440$		0			0
$441$	1.90192	+	4.75833i	0.0905678	+	0.226587i
$442$		0			0
$443$	12.7942	+	3.42820i	0.607872	+	0.162879i	0.549608	−	0.835422i	$-0.314777\pi$
$443$	12.7942	+	3.42820i	0.607872	+	0.162879i	0.0582637	+	0.998301i	$0.481444\pi$
$444$		0			0
$445$	−16.7942	+	4.50000i	−0.796123	+	0.213320i
$446$		0			0
$447$	17.9282			0.847975
$448$		0			0
$449$	11.3205			0.534248			0.267124	−	0.963662i	$-0.413927\pi$
$449$	11.3205			0.534248			0.267124	+	0.963662i	$0.413927\pi$
$450$		0			0
$451$	37.7846	−	10.1244i	1.77921	−	0.476737i
$452$		0			0
$453$	17.5263	+	4.69615i	0.823456	+	0.220644i
$454$		0			0
$455$	−1.09808	+	3.16987i	−0.0514786	+	0.148606i
$456$		0			0
$457$	25.2846	−	14.5981i	1.18276	−	0.682869i	0.226112	−	0.974101i	$-0.427399\pi$
$457$	25.2846	−	14.5981i	1.18276	−	0.682869i	0.956652	+	0.291232i	$0.0940652\pi$
$458$		0			0
$459$	0.526279	+	1.96410i	0.0245646	+	0.0916764i
$460$		0			0
$461$	1.33975	+	1.33975i	0.0623982	+	0.0623982i	0.737617	−	0.675219i	$-0.235951\pi$
$461$	1.33975	+	1.33975i	0.0623982	+	0.0623982i	−0.675219	+	0.737617i	$0.735951\pi$
$462$		0			0
$463$	−29.8564			−1.38754			−0.693772	−	0.720194i	$-0.744053\pi$
$463$	−29.8564			−1.38754			−0.693772	+	0.720194i	$0.744053\pi$
$464$		0			0
$465$	−0.866025	+	1.50000i	−0.0401610	+	0.0695608i
$466$		0			0
$467$	−0.00961894	+	0.0358984i	−0.000445112	+	0.00166118i	−0.966148	−	0.257988i	$-0.916940\pi$
$467$	−0.00961894	+	0.0358984i	−0.000445112	+	0.00166118i	0.965703	+	0.259650i	$0.0836070\pi$
$468$		0			0
$469$	−16.6244	−	11.2583i	−0.767641	−	0.519861i
$470$		0			0
$471$	−27.5263	+	15.8923i	−1.26834	+	0.732279i
$472$		0			0
$473$	−2.49038	−	1.43782i	−0.114508	−	0.0661111i
$474$		0			0
$475$	19.1962	−	19.1962i	0.880780	−	0.880780i
$476$		0			0
$477$	−5.92820	−	5.92820i	−0.271434	−	0.271434i
$478$		0			0
$479$	−7.79423	+	13.5000i	−0.356127	+	0.616831i	−0.987310	−	0.158803i	$-0.949236\pi$
$479$	−7.79423	+	13.5000i	−0.356127	+	0.616831i	0.631183	+	0.775634i	$0.282570\pi$
$480$		0			0
$481$	2.09808	+	3.63397i	0.0956640	+	0.165695i
$482$		0			0
$483$	−12.5622	−	0.901924i	−0.571599	−	0.0410390i
$484$		0			0
$485$	−35.3205	−	9.46410i	−1.60382	−	0.429743i
$486$		0			0
$487$	19.3301	+	11.1603i	0.875932	+	0.505719i	0.869315	−	0.494259i	$-0.164560\pi$
$487$	19.3301	+	11.1603i	0.875932	+	0.505719i	0.00661681	+	0.999978i	$0.497894\pi$
$488$		0			0
$489$			0.464102i			0.0209874i
$490$		0			0
$491$	−27.5885	+	27.5885i	−1.24505	+	1.24505i	−0.287170	+	0.957880i	$0.592714\pi$
$491$	−27.5885	+	27.5885i	−1.24505	+	1.24505i	−0.957880	+	0.287170i	$0.907286\pi$
$492$		0			0
$493$	−2.36603	+	0.633975i	−0.106560	+	0.0285528i
$494$		0			0
$495$	5.36603	+	9.29423i	0.241185	+	0.417745i
$496$		0			0
$497$	−14.9282	+	12.9282i	−0.669621	+	0.579909i
$498$		0			0
$499$	−0.990381	+	3.69615i	−0.0443355	+	0.165463i	−0.984544	−	0.175137i	$-0.943963\pi$
$499$	−0.990381	+	3.69615i	−0.0443355	+	0.165463i	0.940209	+	0.340599i	$0.110630\pi$
$500$		0			0
$501$	−2.92820	−	10.9282i	−0.130822	−	0.488236i
$502$		0			0
$503$			4.14359i			0.184754i	0.995724	+	0.0923769i	$0.0294464\pi$
$503$			4.14359i			0.184754i	−0.995724	+	0.0923769i	$0.970554\pi$
$504$		0			0
$505$		−	24.4641i		−	1.08864i
$506$		0			0
$507$	−6.42820	−	23.9904i	−0.285487	−	1.06545i
$508$		0			0
$509$	4.42820	−	16.5263i	0.196277	−	0.732514i	−0.795656	−	0.605749i	$-0.792874\pi$
$509$	4.42820	−	16.5263i	0.196277	−	0.732514i	0.991933	−	0.126766i	$-0.0404597\pi$
$510$		0			0
$511$	1.59808	+	8.30385i	0.0706947	+	0.367341i
$512$		0			0
$513$	9.59808	+	16.6244i	0.423765	+	0.733983i
$514$		0			0
$515$	1.50000	−	0.401924i	0.0660979	−	0.0177109i
$516$		0			0
$517$	23.9545	−	23.9545i	1.05352	−	1.05352i
$518$		0			0
$519$		−	9.19615i		−	0.403666i
$520$		0			0
$521$	37.6244	+	21.7224i	1.64835	+	0.951677i	0.977727	+	0.209881i	$0.0673078\pi$
$521$	37.6244	+	21.7224i	1.64835	+	0.951677i	0.670626	+	0.741796i	$0.266026\pi$
$522$		0			0
$523$	34.6506	+	9.28461i	1.51517	+	0.405988i	0.918147	−	0.396239i	$-0.129685\pi$
$523$	34.6506	+	9.28461i	1.51517	+	0.405988i	0.597019	+	0.802227i	$0.296352\pi$
$524$		0			0
$525$	−13.8301	−	28.4904i	−0.603596	−	1.24342i
$526$		0			0
$527$	−0.0621778	−	0.107695i	−0.00270851	−	0.00469127i
$528$		0			0
$529$	−8.46410	+	14.6603i	−0.368004	+	0.637402i
$530$		0			0
$531$	−5.33975	−	5.33975i	−0.231725	−	0.231725i
$532$		0			0
$533$	−2.39230	+	2.39230i	−0.103622	+	0.103622i
$534$		0			0
$535$	2.59808	+	1.50000i	0.112325	+	0.0648507i
$536$		0			0
$537$	13.1603	−	7.59808i	0.567907	−	0.327881i
$538$		0			0
$539$	30.4545	−	3.62436i	1.31177	−	0.156112i
$540$		0			0
$541$	−5.76795	+	21.5263i	−0.247984	+	0.925487i	0.723877	+	0.689929i	$0.242358\pi$
$541$	−5.76795	+	21.5263i	−0.247984	+	0.925487i	−0.971860	+	0.235558i	$0.924308\pi$
$542$		0			0
$543$	−10.0981	+	17.4904i	−0.433350	+	0.750584i
$544$		0			0
$545$	−49.9808			−2.14094
$546$		0			0
$547$	−15.0526	−	15.0526i	−0.643601	−	0.643601i	0.307838	−	0.951439i	$-0.400395\pi$
$547$	−15.0526	−	15.0526i	−0.643601	−	0.643601i	−0.951439	+	0.307838i	$0.900395\pi$
$548$		0			0
$549$	0.0262794	+	0.0980762i	0.00112158	+	0.00418579i
$550$		0			0
$551$	−20.0263	+	11.5622i	−0.853148	+	0.492565i
$552$		0			0
$553$	0.571797	−	1.65064i	0.0243153	−	0.0701921i
$554$		0			0
$555$	−69.1410	−	18.5263i	−2.93487	−	0.786397i
$556$		0			0
$557$	−5.23205	+	1.40192i	−0.221689	+	0.0594014i	−0.367954	−	0.929844i	$-0.619942\pi$
$557$	−5.23205	+	1.40192i	−0.221689	+	0.0594014i	0.146265	+	0.989245i	$0.453275\pi$
$558$		0			0
$559$	0.248711			0.0105194
$560$		0			0
$561$	−3.92820			−0.165849
$562$		0			0
$563$	42.6506	−	11.4282i	1.79751	−	0.481641i	0.803925	−	0.594731i	$-0.202741\pi$
$563$	42.6506	−	11.4282i	1.79751	−	0.481641i	0.993585	+	0.113089i	$0.0360746\pi$
$564$		0			0
$565$	−17.6603	−	4.73205i	−0.742972	−	0.199079i
$566$		0			0
$567$	27.6962	−	5.33013i	1.16313	−	0.223844i
$568$		0			0
$569$	−26.0885	+	15.0622i	−1.09369	+	0.631439i	−0.934555	−	0.355818i	$-0.884202\pi$
$569$	−26.0885	+	15.0622i	−1.09369	+	0.631439i	−0.159130	+	0.987258i	$0.550869\pi$
$570$		0			0
$571$	2.72243	+	10.1603i	0.113930	+	0.425193i	0.999205	−	0.0398756i	$-0.0126962\pi$
$571$	2.72243	+	10.1603i	0.113930	+	0.425193i	−0.885274	+	0.465069i	$0.846030\pi$
$572$		0			0
$573$	−11.8301	−	11.8301i	−0.494211	−	0.494211i
$574$		0			0
$575$	15.2679			0.636717
$576$		0			0
$577$	17.6244	−	30.5263i	0.733712	−	1.27083i	−0.221575	−	0.975143i	$-0.571120\pi$
$577$	17.6244	−	30.5263i	0.733712	−	1.27083i	0.955286	−	0.295682i	$-0.0955470\pi$
$578$		0			0
$579$	11.5000	−	42.9186i	0.477924	−	1.78364i
$580$		0			0
$581$	−2.26795	+	31.5885i	−0.0940904	+	1.31051i
$582$		0			0
$583$	−43.4545	+	25.0885i	−1.79970	+	1.03906i
$584$		0			0
$585$	−0.803848	−	0.464102i	−0.0332350	−	0.0191882i
$586$		0			0
$587$	−8.07180	+	8.07180i	−0.333159	+	0.333159i	−0.853785	−	0.520626i	$-0.825699\pi$
$587$	−8.07180	+	8.07180i	−0.333159	+	0.333159i	0.520626	+	0.853785i	$0.325699\pi$
$588$		0			0
$589$	−0.830127	−	0.830127i	−0.0342048	−	0.0342048i
$590$		0			0
$591$	0.901924	−	1.56218i	0.0371002	−	0.0642594i
$592$		0			0
$593$	−4.69615	−	8.13397i	−0.192848	−	0.334022i	0.753345	−	0.657625i	$-0.228439\pi$
$593$	−4.69615	−	8.13397i	−0.192848	−	0.334022i	−0.946193	+	0.323603i	$0.895106\pi$
$594$		0			0
$595$	4.09808	+	0.294229i	0.168005	+	0.0120622i
$596$		0			0
$597$	−3.59808	−	0.964102i	−0.147259	−	0.0394581i
$598$		0			0
$599$	25.3301	+	14.6244i	1.03496	+	0.597535i	0.918402	−	0.395649i	$-0.129480\pi$
$599$	25.3301	+	14.6244i	1.03496	+	0.597535i	0.116559	+	0.993184i	$0.462814\pi$
$600$		0			0
$601$			10.0000i			0.407909i	0.978980	+	0.203954i	$0.0653794\pi$
$601$			10.0000i			0.407909i	−0.978980	+	0.203954i	$0.934621\pi$
$602$		0			0
$603$	3.92820	−	3.92820i	0.159969	−	0.159969i
$604$		0			0
$605$	26.4904	−	7.09808i	1.07699	−	0.288578i
$606$		0			0
$607$	−10.5263	−	18.2321i	−0.427249	−	0.740016i	0.569379	−	0.822075i	$-0.307184\pi$
$607$	−10.5263	−	18.2321i	−0.427249	−	0.740016i	−0.996627	+	0.0820591i	$0.973850\pi$
$608$		0			0
$609$	5.09808	+	26.4904i	0.206584	+	1.07344i
$610$		0			0
$611$	−0.758330	+	2.83013i	−0.0306788	+	0.114495i
$612$		0			0
$613$	−5.76795	−	21.5263i	−0.232965	−	0.869438i	−0.979056	−	0.203593i	$-0.934738\pi$
$613$	−5.76795	−	21.5263i	−0.232965	−	0.869438i	0.746090	−	0.665845i	$-0.231929\pi$
$614$		0			0
$615$		−	57.7128i		−	2.32721i
$616$		0			0
$617$			7.46410i			0.300493i	0.988649	+	0.150247i	$0.0480068\pi$
$617$			7.46410i			0.300493i	−0.988649	+	0.150247i	$0.951993\pi$
$618$		0			0
$619$	6.93782	+	25.8923i	0.278855	+	1.04070i	0.953214	+	0.302296i	$0.0977532\pi$
$619$	6.93782	+	25.8923i	0.278855	+	1.04070i	−0.674359	+	0.738403i	$0.735580\pi$
$620$		0			0
$621$	−2.79423	+	10.4282i	−0.112129	+	0.418469i
$622$		0			0
$623$	−12.9904	−	4.50000i	−0.520449	−	0.180289i
$624$		0			0
$625$	−8.79423	−	15.2321i	−0.351769	−	0.609282i
$626$		0			0
$627$	−35.8205	+	9.59808i	−1.43053	+	0.383310i
$628$		0			0
$629$	3.63397	−	3.63397i	0.144896	−	0.144896i
$630$		0			0
$631$			16.2487i			0.646851i	0.946254	+	0.323425i	$0.104835\pi$
$631$			16.2487i			0.646851i	−0.946254	+	0.323425i	$0.895165\pi$
$632$		0			0
$633$	4.90192	+	2.83013i	0.194834	+	0.112487i
$634$		0			0
$635$	8.19615	+	2.19615i	0.325254	+	0.0871517i
$636$		0			0
$637$	−2.12436	+	1.58846i	−0.0841700	+	0.0629370i
$638$		0			0
$639$	−2.73205	−	4.73205i	−0.108078	−	0.187197i
$640$		0			0
$641$	19.4282	−	33.6506i	0.767368	−	1.32912i	−0.171618	−	0.985164i	$-0.554899\pi$
$641$	19.4282	−	33.6506i	0.767368	−	1.32912i	0.938986	−	0.343957i	$-0.111767\pi$
$642$		0			0
$643$	−15.3923	−	15.3923i	−0.607013	−	0.607013i	0.335151	−	0.942164i	$-0.391213\pi$
$643$	−15.3923	−	15.3923i	−0.607013	−	0.607013i	−0.942164	+	0.335151i	$0.891213\pi$
$644$		0			0
$645$	−3.00000	+	3.00000i	−0.118125	+	0.118125i
$646$		0			0
$647$	29.1340	+	16.8205i	1.14537	+	0.661282i	0.947756	−	0.318997i	$-0.103346\pi$
$647$	29.1340	+	16.8205i	1.14537	+	0.661282i	0.197619	+	0.980279i	$0.436679\pi$
$648$		0			0
$649$	−39.1410	+	22.5981i	−1.53642	+	0.887052i
$650$		0			0
$651$	−1.23205	+	0.598076i	−0.0482879	+	0.0234405i
$652$		0			0
$653$	−5.55256	+	20.7224i	−0.217288	+	0.810931i	0.768060	+	0.640378i	$0.221222\pi$
$653$	−5.55256	+	20.7224i	−0.217288	+	0.810931i	−0.985348	+	0.170554i	$0.945444\pi$
$654$		0			0
$655$	15.5263	−	26.8923i	0.606662	−	1.05077i
$656$		0			0
$657$	−2.33975			−0.0912822
$658$		0			0
$659$	8.85641	+	8.85641i	0.344997	+	0.344997i	0.858242	−	0.513245i	$-0.171557\pi$
$659$	8.85641	+	8.85641i	0.344997	+	0.344997i	−0.513245	+	0.858242i	$0.671557\pi$
$660$		0			0
$661$	4.83975	+	18.0622i	0.188244	+	0.702537i	0.993913	+	0.110170i	$0.0351397\pi$
$661$	4.83975	+	18.0622i	0.188244	+	0.702537i	−0.805668	+	0.592367i	$0.798194\pi$
$662$		0			0
$663$	0.294229	−	0.169873i	0.0114269	−	0.00659732i
$664$		0			0
$665$	38.0885	−	7.33013i	1.47701	−	0.284250i
$666$		0			0
$667$	−12.5622	−	3.36603i	−0.486409	−	0.130333i
$668$		0			0
$669$	33.3205	−	8.92820i	1.28825	−	0.345184i
$670$		0			0
$671$	0.607695			0.0234598
$672$		0			0
$673$	−0.784610			−0.0302445			−0.0151222	−	0.999886i	$-0.504814\pi$
$673$	−0.784610			−0.0302445			−0.0151222	+	0.999886i	$0.504814\pi$
$674$		0			0
$675$	−26.2224	+	7.02628i	−1.00930	+	0.270442i
$676$		0			0
$677$	49.5526	+	13.2776i	1.90446	+	0.510298i	0.995660	+	0.0930654i	$0.0296666\pi$
$677$	49.5526	+	13.2776i	1.90446	+	0.510298i	0.908800	+	0.417233i	$0.137000\pi$
$678$		0			0
$679$	−18.9282	−	21.8564i	−0.726398	−	0.838772i
$680$		0			0
$681$	−39.8205	+	22.9904i	−1.52593	+	0.880993i
$682$		0			0
$683$	−11.5788	−	43.2128i	−0.443052	−	1.65349i	−0.721029	−	0.692905i	$-0.756331\pi$
$683$	−11.5788	−	43.2128i	−0.443052	−	1.65349i	0.277977	−	0.960588i	$-0.410336\pi$
$684$		0			0
$685$	32.1506	+	32.1506i	1.22841	+	1.22841i
$686$		0			0
$687$	−34.8564			−1.32985
$688$		0			0
$689$	2.16987	−	3.75833i	0.0826656	−	0.143181i
$690$		0			0
$691$	−2.99038	+	11.1603i	−0.113759	+	0.424556i	−0.999191	−	0.0402135i	$-0.987196\pi$
$691$	−2.99038	+	11.1603i	−0.113759	+	0.424556i	0.885432	+	0.464770i	$0.153863\pi$
$692$		0			0
$693$	−0.607695	+	8.46410i	−0.0230844	+	0.321525i
$694$		0			0
$695$	7.90192	−	4.56218i	0.299737	−	0.173053i
$696$		0			0
$697$	3.58846	+	2.07180i	0.135923	+	0.0784749i
$698$		0			0
$699$	15.2942	−	15.2942i	0.578481	−	0.578481i
$700$		0			0
$701$	7.39230	+	7.39230i	0.279204	+	0.279204i	0.832791	−	0.553588i	$-0.186742\pi$
$701$	7.39230	+	7.39230i	0.279204	+	0.279204i	−0.553588	+	0.832791i	$0.686742\pi$
$702$		0			0
$703$	24.2583	−	42.0167i	0.914920	−	1.58469i
$704$		0			0
$705$	−24.9904	−	43.2846i	−0.941192	−	1.63019i
$706$		0			0
$707$	10.8468	−	16.0167i	0.407935	−	0.602369i
$708$		0			0
$709$	−8.76795	−	2.34936i	−0.329287	−	0.0882323i	0.0903879	−	0.995907i	$-0.471189\pi$
$709$	−8.76795	−	2.34936i	−0.329287	−	0.0882323i	−0.419675	+	0.907674i	$0.637856\pi$
$710$		0			0
$711$	0.418584	+	0.241670i	0.0156981	+	0.00906332i
$712$		0			0
$713$		−	0.660254i		−	0.0247267i
$714$		0			0
$715$	−3.92820	+	3.92820i	−0.146906	+	0.146906i
$716$		0			0
$717$	28.8564	−	7.73205i	1.07766	−	0.288759i
$718$		0			0
$719$	15.7942	+	27.3564i	0.589025	+	1.02022i	0.994360	+	0.106054i	$0.0338215\pi$
$719$	15.7942	+	27.3564i	0.589025	+	1.02022i	−0.405335	+	0.914168i	$0.632845\pi$
$720$		0			0
$721$	1.16025	+	0.401924i	0.0432101	+	0.0149684i
$722$		0			0
$723$	−0.0358984	+	0.133975i	−0.00133508	+	0.00498257i
$724$		0			0
$725$	−8.46410	−	31.5885i	−0.314349	−	1.17317i
$726$		0			0
$727$			6.67949i			0.247729i	0.992299	+	0.123864i	$0.0395288\pi$
$727$			6.67949i			0.247729i	−0.992299	+	0.123864i	$0.960471\pi$
$728$		0			0
$729$		−	17.5885i		−	0.651424i
$730$		0			0
$731$	−0.0788383	−	0.294229i	−0.00291594	−	0.0108824i
$732$		0			0
$733$	12.1077	−	45.1865i	0.447208	−	1.66900i	−0.262832	−	0.964842i	$-0.584657\pi$
$733$	12.1077	−	45.1865i	0.447208	−	1.66900i	0.710040	−	0.704161i	$-0.248677\pi$
$734$		0			0
$735$	6.46410	−	44.7846i	0.238432	−	1.65191i
$736$		0			0
$737$	−16.6244	−	28.7942i	−0.612366	−	1.06065i
$738$		0			0
$739$	−50.3109	+	13.4808i	−1.85072	+	0.495898i	−0.999578	−	0.0290444i	$-0.990754\pi$
$739$	−50.3109	+	13.4808i	−1.85072	+	0.495898i	−0.851138	+	0.524942i	$0.824087\pi$
$740$		0			0
$741$	2.26795	−	2.26795i	0.0833152	−	0.0833152i
$742$		0			0
$743$			24.9282i			0.914527i	0.889331	+	0.457264i	$0.151170\pi$
$743$			24.9282i			0.914527i	−0.889331	+	0.457264i	$0.848830\pi$
$744$		0			0
$745$	−26.8923	−	15.5263i	−0.985258	−	0.568839i
$746$		0			0
$747$	−8.46410	−	2.26795i	−0.309685	−	0.0829799i
$748$		0			0
$749$	1.03590	+	2.13397i	0.0378509	+	0.0779737i
$750$		0			0
$751$	−12.5263	−	21.6962i	−0.457090	−	0.791704i	0.541715	−	0.840562i	$-0.317775\pi$
$751$	−12.5263	−	21.6962i	−0.457090	−	0.791704i	−0.998806	+	0.0488582i	$0.984442\pi$
$752$		0			0
$753$	−18.5622	+	32.1506i	−0.676443	+	1.17163i
$754$		0			0
$755$	−22.2224	−	22.2224i	−0.808757	−	0.808757i
$756$		0			0
$757$	−1.33975	+	1.33975i	−0.0486939	+	0.0486939i	−0.731034	−	0.682341i	$-0.760962\pi$
$757$	−1.33975	+	1.33975i	−0.0486939	+	0.0486939i	0.682341	+	0.731034i	$0.260962\pi$
$758$		0			0
$759$	−18.0622	−	10.4282i	−0.655616	−	0.378520i
$760$		0			0
$761$	−15.2321	+	8.79423i	−0.552161	+	0.318791i	−0.749993	−	0.661445i	$-0.769943\pi$
$761$	−15.2321	+	8.79423i	−0.552161	+	0.318791i	0.197832	+	0.980236i	$0.436610\pi$
$762$		0			0
$763$	−32.7224	−	22.1603i	−1.18463	−	0.802255i
$764$		0			0
$765$	−0.294229	+	1.09808i	−0.0106379	+	0.0397010i
$766$		0			0
$767$	1.95448	−	3.38526i	0.0705723	−	0.122235i
$768$		0			0
$769$	−17.8564			−0.643918			−0.321959	−	0.946754i	$-0.604341\pi$
$769$	−17.8564			−0.643918			−0.321959	+	0.946754i	$0.604341\pi$
$770$		0			0
$771$	−1.90192	−	1.90192i	−0.0684961	−	0.0684961i
$772$		0			0
$773$	4.03590	+	15.0622i	0.145161	+	0.541749i	0.999748	+	0.0224406i	$0.00714368\pi$
$773$	4.03590	+	15.0622i	0.145161	+	0.541749i	−0.854587	+	0.519308i	$0.826190\pi$
$774$		0			0
$775$	1.43782	−	0.830127i	0.0516481	−	0.0298190i
$776$		0			0
$777$	−37.0526	−	42.7846i	−1.32925	−	1.53489i
$778$		0			0
$779$	37.7846	+	10.1244i	1.35377	+	0.362743i
$780$		0			0
$781$	−31.5885	+	8.46410i	−1.13032	+	0.302869i
$782$		0			0
$783$	23.1244			0.826397
$784$		0			0
$785$	55.0526			1.96491
$786$		0			0
$787$	15.5263	−	4.16025i	0.553452	−	0.148297i	0.0287557	−	0.999586i	$-0.490846\pi$
$787$	15.5263	−	4.16025i	0.553452	−	0.148297i	0.524696	+	0.851289i	$0.324179\pi$
$788$		0			0
$789$	−47.3827	−	12.6962i	−1.68687	−	0.451995i
$790$		0			0
$791$	−9.46410	−	10.9282i	−0.336505	−	0.388562i
$792$		0			0
$793$	−0.0455173	+	0.0262794i	−0.00161637	+	0.000933210i
$794$		0			0
$795$	19.1603	+	71.5070i	0.679544	+	2.53609i
$796$		0			0
$797$	−30.6603	−	30.6603i	−1.08604	−	1.08604i	−0.995932	−	0.0901101i	$-0.971278\pi$
$797$	−30.6603	−	30.6603i	−1.08604	−	1.08604i	−0.0901101	−	0.995932i	$-0.528722\pi$
$798$		0			0
$799$	3.58846			0.126950
$800$		0			0
$801$	1.90192	−	3.29423i	0.0672012	−	0.116396i
$802$		0			0
$803$	−3.62436	+	13.5263i	−0.127901	+	0.477332i
$804$		0			0
$805$	18.0622	+	12.2321i	0.636608	+	0.431123i
$806$		0			0
$807$	37.4545	−	21.6244i	1.31846	−	0.761213i
$808$		0			0
$809$	15.5718	+	8.99038i	0.547475	+	0.316085i	0.748103	−	0.663583i	$-0.230965\pi$
$809$	15.5718	+	8.99038i	0.547475	+	0.316085i	−0.200628	+	0.979668i	$0.564298\pi$
$810$		0			0
$811$	10.3205	−	10.3205i	0.362402	−	0.362402i	−0.502295	−	0.864697i	$-0.667511\pi$
$811$	10.3205	−	10.3205i	0.362402	−	0.362402i	0.864697	+	0.502295i	$0.167511\pi$
$812$		0			0
$813$	−16.5622	−	16.5622i	−0.580861	−	0.580861i
$814$		0			0
$815$	0.401924	−	0.696152i	0.0140788	−	0.0243852i
$816$		0			0
$817$	−1.43782	−	2.49038i	−0.0503030	−	0.0871274i
$818$		0			0
$819$	−0.320508	−	0.660254i	−0.0111995	−	0.0230711i
$820$		0			0
$821$	17.1603	+	4.59808i	0.598897	+	0.160474i	0.545516	−	0.838100i	$-0.316334\pi$
$821$	17.1603	+	4.59808i	0.598897	+	0.160474i	0.0533808	+	0.998574i	$0.483000\pi$
$822$		0			0
$823$	−27.0622	−	15.6244i	−0.943328	−	0.544631i	−0.0523262	−	0.998630i	$-0.516664\pi$
$823$	−27.0622	−	15.6244i	−0.943328	−	0.544631i	−0.891002	+	0.453999i	$0.849997\pi$
$824$		0			0
$825$		−	52.4449i		−	1.82590i
$826$		0			0
$827$	−37.7846	+	37.7846i	−1.31390	+	1.31390i	−0.395383	+	0.918516i	$0.629388\pi$
$827$	−37.7846	+	37.7846i	−1.31390	+	1.31390i	−0.918516	+	0.395383i	$0.870612\pi$
$828$		0			0
$829$	−33.8205	+	9.06218i	−1.17463	+	0.314742i	−0.792796	−	0.609487i	$-0.791376\pi$
$829$	−33.8205	+	9.06218i	−1.17463	+	0.314742i	−0.381838	+	0.924229i	$0.624709\pi$
$830$		0			0
$831$	16.7942	+	29.0885i	0.582585	+	1.00907i
$832$		0			0
$833$	2.55256	+	2.00962i	0.0884409	+	0.0696292i
$834$		0			0
$835$	−5.07180	+	18.9282i	−0.175517	+	0.655037i
$836$		0			0
$837$	0.303848	+	1.13397i	0.0105025	+	0.0391959i
$838$		0			0
$839$			37.7128i			1.30199i	0.759082	+	0.650995i	$0.225648\pi$
$839$			37.7128i			1.30199i	−0.759082	+	0.650995i	$0.774352\pi$
$840$		0			0
$841$		−	1.14359i		−	0.0394343i
$842$		0			0
$843$	6.46410	+	24.1244i	0.222635	+	0.830887i
$844$		0			0
$845$	−11.1340	+	41.5526i	−0.383020	+	1.42945i
$846$		0			0
$847$	20.4904	+	7.09808i	0.704058	+	0.243893i
$848$		0			0
$849$	16.8923	+	29.2583i	0.579742	+	1.00414i
$850$		0			0
$851$	26.3564	−	7.06218i	0.903486	−	0.242088i
$852$		0			0
$853$	−36.1244	+	36.1244i	−1.23687	+	1.23687i	−0.275603	+	0.961272i	$0.588877\pi$
$853$	−36.1244	+	36.1244i	−1.23687	+	1.23687i	−0.961272	+	0.275603i	$0.911123\pi$
$854$		0			0
$855$			10.7321i			0.367028i
$856$		0			0
$857$	8.89230	+	5.13397i	0.303755	+	0.175373i	0.644129	−	0.764917i	$-0.277220\pi$
$857$	8.89230	+	5.13397i	0.303755	+	0.175373i	−0.340373	+	0.940290i	$0.610553\pi$
$858$		0			0
$859$	−39.6506	−	10.6244i	−1.35286	−	0.362498i	−0.491672	−	0.870781i	$-0.663614\pi$
$859$	−39.6506	−	10.6244i	−1.35286	−	0.362498i	−0.861191	+	0.508282i	$0.830281\pi$
$860$		0			0
$861$	25.5885	−	37.7846i	0.872052	−	1.28770i
$862$		0			0
$863$	−7.66987	−	13.2846i	−0.261086	−	0.452213i	0.705445	−	0.708765i	$-0.250747\pi$
$863$	−7.66987	−	13.2846i	−0.261086	−	0.452213i	−0.966531	+	0.256551i	$0.917414\pi$
$864$		0			0
$865$	−7.96410	+	13.7942i	−0.270788	+	0.469018i
$866$		0			0
$867$	22.9282	+	22.9282i	0.778683	+	0.778683i
$868$		0			0
$869$	2.04552	−	2.04552i	0.0693894	−	0.0693894i
$870$		0			0
$871$	2.49038	+	1.43782i	0.0843833	+	0.0487187i
$872$		0			0
$873$	6.92820	−	4.00000i	0.234484	−	0.135379i
$874$		0			0
$875$	−0.758330	+	10.5622i	−0.0256362	+	0.357067i
$876$		0			0
$877$	7.55256	−	28.1865i	0.255032	−	0.951792i	−0.713041	−	0.701122i	$-0.752683\pi$
$877$	7.55256	−	28.1865i	0.255032	−	0.951792i	0.968073	−	0.250669i	$-0.0806507\pi$
$878$		0			0
$879$	8.09808	−	14.0263i	0.273141	−	0.473095i
$880$		0			0
$881$	50.0000			1.68454			0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000			1.68454			0.842271	+	0.539054i	$0.181218\pi$
$882$		0			0
$883$	5.00000	+	5.00000i	0.168263	+	0.168263i	0.786216	−	0.617952i	$-0.212037\pi$
$883$	5.00000	+	5.00000i	0.168263	+	0.168263i	−0.617952	+	0.786216i	$0.712037\pi$
$884$		0			0
$885$	17.2583	+	64.4090i	0.580132	+	2.16508i
$886$		0			0
$887$	27.7750	−	16.0359i	0.932593	−	0.538433i	0.0449622	−	0.998989i	$-0.485683\pi$
$887$	27.7750	−	16.0359i	0.932593	−	0.538433i	0.887631	+	0.460556i	$0.152350\pi$
$888$		0			0
$889$	4.39230	+	5.07180i	0.147313	+	0.170103i
$890$		0			0
$891$	45.1147	+	12.0885i	1.51140	+	0.404979i
$892$		0			0
$893$	32.7224	−	8.76795i	1.09501	−	0.293408i
$894$		0			0
$895$	−26.3205			−0.879798
$896$		0			0
$897$	1.80385			0.0602287
$898$		0			0
$899$	−1.36603	+	0.366025i	−0.0455595	+	0.0122076i
$900$		0			0
$901$	−5.13397	−	1.37564i	−0.171037	−	0.0458294i
$902$		0			0
$903$	−3.29423	+	0.633975i	−0.109625	+	0.0210974i
$904$		0			0
$905$	30.2942	−	17.4904i	1.00701	−	0.581400i
$906$		0			0
$907$	−2.72243	−	10.1603i	−0.0903969	−	0.337366i	0.905885	−	0.423525i	$-0.139207\pi$
$907$	−2.72243	−	10.1603i	−0.0903969	−	0.337366i	−0.996281	+	0.0861591i	$0.972541\pi$
$908$		0			0
$909$	3.78461	+	3.78461i	0.125528	+	0.125528i
$910$		0			0
$911$	27.3205			0.905169			0.452584	−	0.891722i	$-0.350502\pi$
$911$	27.3205			0.905169			0.452584	+	0.891722i	$0.350502\pi$
$912$		0			0
$913$	−26.2224	+	45.4186i	−0.867836	+	1.50314i
$914$		0			0
$915$	0.232051	−	0.866025i	0.00767136	−	0.0286299i
$916$		0			0
$917$	22.0885	−	10.7224i	0.729425	−	0.354086i
$918$		0			0
$919$	−14.1340	+	8.16025i	−0.466237	+	0.269182i	−0.714663	−	0.699469i	$-0.753420\pi$
$919$	−14.1340	+	8.16025i	−0.466237	+	0.269182i	0.248426	+	0.968651i	$0.420087\pi$
$920$		0			0
$921$	−21.2942	−	12.2942i	−0.701669	−	0.405109i
$922$		0			0
$923$	2.00000	−	2.00000i	0.0658308	−	0.0658308i
$924$		0			0
$925$	48.5167	+	48.5167i	1.59522	+	1.59522i
$926$		0			0
$927$	−0.169873	+	0.294229i	−0.00557936	+	0.00966374i
$928$		0			0
$929$	20.0167	+	34.6699i	0.656725	+	1.13748i	0.981458	+	0.191676i	$0.0613922\pi$
$929$	20.0167	+	34.6699i	0.656725	+	1.13748i	−0.324733	+	0.945806i	$0.605274\pi$
$930$		0			0
$931$	28.1865	+	12.0885i	0.923776	+	0.396183i
$932$		0			0
$933$	0.866025	+	0.232051i	0.0283524	+	0.00759700i
$934$		0			0
$935$	5.89230	+	3.40192i	0.192699	+	0.111255i
$936$		0			0
$937$		−	42.9282i		−	1.40240i	−0.712963	−	0.701202i	$-0.752647\pi$
$937$		−	42.9282i		−	1.40240i	0.712963	−	0.701202i	$-0.247353\pi$
$938$		0			0
$939$	13.0981	−	13.0981i	0.427440	−	0.427440i
$940$		0			0
$941$	16.6962	−	4.47372i	0.544279	−	0.145839i	0.0238050	−	0.999717i	$-0.492422\pi$
$941$	16.6962	−	4.47372i	0.544279	−	0.145839i	0.520474	+	0.853877i	$0.325755\pi$
$942$		0			0
$943$	11.0000	+	19.0526i	0.358209	+	0.620437i
$944$		0			0
$945$	−36.6506	−	12.6962i	−1.19225	−	0.413006i
$946$		0			0
$947$	−10.5981	+	39.5526i	−0.344391	+	1.28529i	0.548931	+	0.835868i	$0.315035\pi$
$947$	−10.5981	+	39.5526i	−0.344391	+	1.28529i	−0.893322	+	0.449418i	$0.851632\pi$
$948$		0			0
$949$	−0.313467	−	1.16987i	−0.0101756	−	0.0379757i
$950$		0			0
$951$			23.5885i			0.764908i
$952$		0			0
$953$			23.4641i			0.760077i	0.924971	+	0.380038i	$0.124089\pi$
$953$			23.4641i			0.760077i	−0.924971	+	0.380038i	$0.875911\pi$
$954$		0			0
$955$	7.50000	+	27.9904i	0.242694	+	0.905747i
$956$		0			0
$957$	−11.5622	+	43.1506i	−0.373752	+	1.39486i
$958$		0			0
$959$	6.79423	+	35.3038i	0.219397	+	1.14002i
$960$		0			0
$961$	15.4641	+	26.7846i	0.498842	+	0.864020i
$962$		0			0
$963$	−0.633975	+	0.169873i	−0.0204295	+	0.00547408i
$964$		0			0
$965$	−54.4186	+	54.4186i	−1.75180	+	1.75180i
$966$		0			0
$967$			11.7513i			0.377896i	0.981987	+	0.188948i	$0.0605078\pi$
$967$			11.7513i			0.377896i	−0.981987	+	0.188948i	$0.939492\pi$
$968$		0			0
$969$	−3.40192	−	1.96410i	−0.109286	−	0.0630960i
$970$		0			0
$971$	50.0429	+	13.4090i	1.60595	+	0.430314i	0.946834	−	0.321723i	$-0.104262\pi$
$971$	50.0429	+	13.4090i	1.60595	+	0.430314i	0.659121	+	0.752037i	$0.270929\pi$
$972$		0			0
$973$	7.19615	+	0.516660i	0.230698	+	0.0165634i
$974$		0			0
$975$	2.26795	+	3.92820i	0.0726325	+	0.125803i
$976$		0			0
$977$	22.4282	−	38.8468i	0.717542	−	1.24282i	−0.244429	−	0.969667i	$-0.578601\pi$
$977$	22.4282	−	38.8468i	0.717542	−	1.24282i	0.961971	−	0.273152i	$-0.0880661\pi$
$978$		0			0
$979$	−16.0981	−	16.0981i	−0.514497	−	0.514497i
$980$		0			0
$981$	7.73205	−	7.73205i	0.246865	−	0.246865i
$982$		0			0
$983$	18.8660	+	10.8923i	0.601733	+	0.347411i	0.769723	−	0.638378i	$-0.220394\pi$
$983$	18.8660	+	10.8923i	0.601733	+	0.347411i	−0.167990	+	0.985789i	$0.553728\pi$
$984$		0			0
$985$	−2.70577	+	1.56218i	−0.0862130	+	0.0497751i
$986$		0			0
$987$	2.83013	−	39.4186i	0.0900839	−	1.25471i
$988$		0			0
$989$	0.418584	−	1.56218i	0.0133102	−	0.0496744i
$990$		0			0
$991$	−19.7942	+	34.2846i	−0.628784	+	1.08909i	0.359012	+	0.933333i	$0.383114\pi$
$991$	−19.7942	+	34.2846i	−0.628784	+	1.08909i	−0.987796	+	0.155753i	$0.950219\pi$
$992$		0			0
$993$	34.5167			1.09535
$994$		0			0
$995$	4.56218	+	4.56218i	0.144631	+	0.144631i
$996$		0			0
$997$	−1.50000	−	5.59808i	−0.0475055	−	0.177293i	0.938097	−	0.346373i	$-0.112587\pi$
$997$	−1.50000	−	5.59808i	−0.0475055	−	0.177293i	−0.985602	+	0.169080i	$0.945920\pi$
$998$		0			0
$999$	−42.0167	+	24.2583i	−1.32935	+	0.767500i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.ba.b.81.1		4
4.3	odd	2		112.2.w.a.109.1	yes	4
7.2	even	3		448.2.ba.a.401.1		4
8.3	odd	2		896.2.ba.d.417.1		4
8.5	even	2		896.2.ba.a.417.1		4
16.3	odd	4		896.2.ba.b.865.1		4
16.5	even	4		448.2.ba.a.305.1		4
16.11	odd	4		112.2.w.b.53.1	yes	4
16.13	even	4		896.2.ba.c.865.1		4
28.3	even	6		784.2.m.d.589.1		4
28.11	odd	6		784.2.m.e.589.1		4
28.19	even	6		784.2.x.h.765.1		4
28.23	odd	6		112.2.w.b.93.1	yes	4
28.27	even	2		784.2.x.a.557.1		4
56.37	even	6		896.2.ba.c.289.1		4
56.51	odd	6		896.2.ba.b.289.1		4
112.11	odd	12		784.2.m.e.197.1		4
112.27	even	4		784.2.x.h.165.1		4
112.37	even	12	inner	448.2.ba.b.177.1		4
112.51	odd	12		896.2.ba.d.737.1		4
112.59	even	12		784.2.m.d.197.1		4
112.75	even	12		784.2.x.a.373.1		4
112.93	even	12		896.2.ba.a.737.1		4
112.107	odd	12		112.2.w.a.37.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.w.a.37.1	✓	4	112.107	odd	12
112.2.w.a.109.1	yes	4	4.3	odd	2
112.2.w.b.53.1	yes	4	16.11	odd	4
112.2.w.b.93.1	yes	4	28.23	odd	6
448.2.ba.a.305.1		4	16.5	even	4
448.2.ba.a.401.1		4	7.2	even	3
448.2.ba.b.81.1		4	1.1	even	1	trivial
448.2.ba.b.177.1		4	112.37	even	12	inner
784.2.m.d.197.1		4	112.59	even	12
784.2.m.d.589.1		4	28.3	even	6
784.2.m.e.197.1		4	112.11	odd	12
784.2.m.e.589.1		4	28.11	odd	6
784.2.x.a.373.1		4	112.75	even	12
784.2.x.a.557.1		4	28.27	even	2
784.2.x.h.165.1		4	112.27	even	4
784.2.x.h.765.1		4	28.19	even	6
896.2.ba.a.417.1		4	8.5	even	2
896.2.ba.a.737.1		4	112.93	even	12
896.2.ba.b.289.1		4	56.51	odd	6
896.2.ba.b.865.1		4	16.3	odd	4
896.2.ba.c.289.1		4	56.37	even	6
896.2.ba.c.865.1		4	16.13	even	4
896.2.ba.d.417.1		4	8.3	odd	2
896.2.ba.d.737.1		4	112.51	odd	12

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.ba (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.86603 - 0.500000i) q^{3} +(-3.23205 - 0.866025i) q^{5} +(-1.73205 - 2.00000i) q^{7} +(0.633975 - 0.366025i) q^{9} +O(q^{10})\)	\(q+(1.86603 - 0.500000i) q^{3} +(-3.23205 - 0.866025i) q^{5} +(-1.73205 - 2.00000i) q^{7} +(0.633975 - 0.366025i) q^{9} +(-1.13397 - 4.23205i) q^{11} +(0.267949 + 0.267949i) q^{13} -6.46410 q^{15} +(0.232051 - 0.401924i) q^{17} +(1.13397 - 4.23205i) q^{19} +(-4.23205 - 2.86603i) q^{21} +(2.13397 - 1.23205i) q^{23} +(5.36603 + 3.09808i) q^{25} +(-3.09808 + 3.09808i) q^{27} +(-3.73205 - 3.73205i) q^{29} +(0.133975 - 0.232051i) q^{31} +(-4.23205 - 7.33013i) q^{33} +(3.86603 + 7.96410i) q^{35} +(10.6962 + 2.86603i) q^{37} +(0.633975 + 0.366025i) q^{39} +8.92820i q^{41} +(0.464102 - 0.464102i) q^{43} +(-2.36603 + 0.633975i) q^{45} +(3.86603 + 6.69615i) q^{47} +(-1.00000 + 6.92820i) q^{49} +(0.232051 - 0.866025i) q^{51} +(-2.96410 - 11.0622i) q^{53} +14.6603i q^{55} -8.46410i q^{57} +(-2.66987 - 9.96410i) q^{59} +(-0.0358984 + 0.133975i) q^{61} +(-1.83013 - 0.633975i) q^{63} +(-0.633975 - 1.09808i) q^{65} +(7.33013 - 1.96410i) q^{67} +(3.36603 - 3.36603i) q^{69} -7.46410i q^{71} +(-2.76795 - 1.59808i) q^{73} +(11.5622 + 3.09808i) q^{75} +(-6.50000 + 9.59808i) q^{77} +(0.330127 + 0.571797i) q^{79} +(-5.33013 + 9.23205i) q^{81} +(-8.46410 - 8.46410i) q^{83} +(-1.09808 + 1.09808i) q^{85} +(-8.83013 - 5.09808i) q^{87} +(4.50000 - 2.59808i) q^{89} +(0.0717968 - 1.00000i) q^{91} +(0.133975 - 0.500000i) q^{93} +(-7.33013 + 12.6962i) q^{95} +10.9282 q^{97} +(-2.26795 - 2.26795i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 6 q^{5} + 6 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 6 * q^5 + 6 * q^9	\( 4 q + 4 q^{3} - 6 q^{5} + 6 q^{9} - 8 q^{11} + 8 q^{13} - 12 q^{15} - 6 q^{17} + 8 q^{19} - 10 q^{21} + 12 q^{23} + 18 q^{25} - 2 q^{27} - 8 q^{29} + 4 q^{31} - 10 q^{33} + 12 q^{35} + 22 q^{37} + 6 q^{39} - 12 q^{43} - 6 q^{45} + 12 q^{47} - 4 q^{49} - 6 q^{51} + 2 q^{53} - 28 q^{59} - 14 q^{61} + 10 q^{63} - 6 q^{65} + 12 q^{67} + 10 q^{69} - 18 q^{73} + 22 q^{75} - 26 q^{77} - 16 q^{79} - 4 q^{81} - 20 q^{83} + 6 q^{85} - 18 q^{87} + 18 q^{89} + 28 q^{91} + 4 q^{93} - 12 q^{95} + 16 q^{97} - 16 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 6 * q^5 + 6 * q^9 - 8 * q^11 + 8 * q^13 - 12 * q^15 - 6 * q^17 + 8 * q^19 - 10 * q^21 + 12 * q^23 + 18 * q^25 - 2 * q^27 - 8 * q^29 + 4 * q^31 - 10 * q^33 + 12 * q^35 + 22 * q^37 + 6 * q^39 - 12 * q^43 - 6 * q^45 + 12 * q^47 - 4 * q^49 - 6 * q^51 + 2 * q^53 - 28 * q^59 - 14 * q^61 + 10 * q^63 - 6 * q^65 + 12 * q^67 + 10 * q^69 - 18 * q^73 + 22 * q^75 - 26 * q^77 - 16 * q^79 - 4 * q^81 - 20 * q^83 + 6 * q^85 - 18 * q^87 + 18 * q^89 + 28 * q^91 + 4 * q^93 - 12 * q^95 + 16 * q^97 - 16 * q^99

Introduction

L-functions

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Database

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