LMFDB - Embedded newform 450.6.f.f.107.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(107,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("450.107");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.f (of order $4$, degree $2$, 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:	$72.1727189158$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 252 x^{14} + 27174 x^{12} - 1635700 x^{10} + 60061815 x^{8} - 1376564028 x^{6} + \cdots + 498214340649 $ x^16 - 252x^14 + 27174x^12 - 1635700x^10 + 60061815x^8 - 1376564028x^6 + 19220200150x^4 - 149540021784*x^2 + 498214340649
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{8}\cdot 5^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			107.8
Root			$7.20885 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	450.107
Dual form			450.6.f.f.143.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(117.170 + 117.170i) q^{7} +(-45.2548 - 45.2548i) q^{8} +O(q^{10})$	\(q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(117.170 + 117.170i) q^{7} +(-45.2548 - 45.2548i) q^{8} +28.2771i q^{11} +(283.237 - 283.237i) q^{13} +662.815 q^{14} -256.000 q^{16} +(-137.842 + 137.842i) q^{17} +2809.99i q^{19} +(79.9797 + 79.9797i) q^{22} +(-902.809 - 902.809i) q^{23} -1602.23i q^{26} +(1874.73 - 1874.73i) q^{28} -827.573 q^{29} +2049.64 q^{31} +(-724.077 + 724.077i) q^{32} +779.754i q^{34} +(-1320.83 - 1320.83i) q^{37} +(7947.85 + 7947.85i) q^{38} +10791.6i q^{41} +(-7831.78 + 7831.78i) q^{43} +452.433 q^{44} -5107.06 q^{46} +(-11749.4 + 11749.4i) q^{47} +10650.8i q^{49} +(-4531.80 - 4531.80i) q^{52} +(22576.3 + 22576.3i) q^{53} -10605.0i q^{56} +(-2340.73 + 2340.73i) q^{58} -33192.4 q^{59} +32011.7 q^{61} +(5797.27 - 5797.27i) q^{62} +4096.00i q^{64} +(27419.3 + 27419.3i) q^{67} +(2205.48 + 2205.48i) q^{68} -54641.5i q^{71} +(14250.4 - 14250.4i) q^{73} -7471.75 q^{74} +44959.8 q^{76} +(-3313.23 + 3313.23i) q^{77} +79274.9i q^{79} +(30523.2 + 30523.2i) q^{82} +(36508.8 + 36508.8i) q^{83} +44303.2i q^{86} +(1279.67 - 1279.67i) q^{88} +69924.7 q^{89} +66374.0 q^{91} +(-14444.9 + 14444.9i) q^{92} +66464.6i q^{94} +(69381.8 + 69381.8i) q^{97} +(30124.9 + 30124.9i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 528 q^{7}+O(q^{10}) $ 16 * q - 528 * q^7	$ 16 q - 528 q^{7} + 192 q^{13} - 4096 q^{16} - 2688 q^{22} - 8448 q^{28} + 13024 q^{31} - 47328 q^{37} - 55440 q^{43} + 44544 q^{46} - 3072 q^{52} - 101184 q^{58} + 28400 q^{61} + 242256 q^{67} - 430944 q^{73} - 7168 q^{76} + 158208 q^{82} - 43008 q^{88} - 185472 q^{91} + 457152 q^{97}+O(q^{100}) $ 16 * q - 528 * q^7 + 192 * q^13 - 4096 * q^16 - 2688 * q^22 - 8448 * q^28 + 13024 * q^31 - 47328 * q^37 - 55440 * q^43 + 44544 * q^46 - 3072 * q^52 - 101184 * q^58 + 28400 * q^61 + 242256 * q^67 - 430944 * q^73 - 7168 * q^76 + 158208 * q^82 - 43008 * q^88 - 185472 * q^91 + 457152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	2.82843	−	2.82843i	0.500000	−	0.500000i
$2$	2.82843	−	2.82843i	0.500000	−	0.500000i
$3$		0			0
$3$		0			0
$4$		−	16.0000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	117.170	+	117.170i	0.903800	+	0.903800i	0.995763	−	0.0919622i	$-0.0293139\pi$
$7$	117.170	+	117.170i	0.903800	+	0.903800i	−0.0919622	+	0.995763i	$0.529314\pi$
$8$	−45.2548	−	45.2548i	−0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$			28.2771i			0.0704616i	0.999379	+	0.0352308i	$0.0112166\pi$
$11$			28.2771i			0.0704616i	−0.999379	+	0.0352308i	$0.988783\pi$
$12$		0			0
$13$	283.237	−	283.237i	0.464828	−	0.464828i	−0.435406	−	0.900234i	$-0.643395\pi$
$13$	283.237	−	283.237i	0.464828	−	0.464828i	0.900234	+	0.435406i	$0.143395\pi$
$14$	662.815			0.903800
$15$		0			0
$16$	−256.000			−0.250000
$17$	−137.842	+	137.842i	−0.115681	+	0.115681i	−0.762577	−	0.646897i	$-0.776066\pi$
$17$	−137.842	+	137.842i	−0.115681	+	0.115681i	0.646897	+	0.762577i	$0.276066\pi$
$18$		0			0
$19$			2809.99i			1.78575i	0.450305	+	0.892875i	$0.351315\pi$
$19$			2809.99i			1.78575i	−0.450305	+	0.892875i	$0.648685\pi$
$20$		0			0
$21$		0			0
$22$	79.9797	+	79.9797i	0.0352308	+	0.0352308i
$23$	−902.809	−	902.809i	−0.355858	−	0.355858i	0.506426	−	0.862284i	$-0.330966\pi$
$23$	−902.809	−	902.809i	−0.355858	−	0.355858i	−0.862284	+	0.506426i	$0.830966\pi$
$24$		0			0
$25$		0			0
$26$		−	1602.23i		−	0.464828i
$27$		0			0
$28$	1874.73	−	1874.73i	0.451900	−	0.451900i
$29$	−827.573			−0.182731			−0.0913653	−	0.995817i	$-0.529123\pi$
$29$	−827.573			−0.182731			−0.0913653	+	0.995817i	$0.529123\pi$
$30$		0			0
$31$	2049.64			0.383067			0.191533	−	0.981486i	$-0.438654\pi$
$31$	2049.64			0.383067			0.191533	+	0.981486i	$0.438654\pi$
$32$	−724.077	+	724.077i	−0.125000	+	0.125000i
$33$		0			0
$34$			779.754i			0.115681i
$35$		0			0
$36$		0			0
$37$	−1320.83	−	1320.83i	−0.158615	−	0.158615i	0.623338	−	0.781953i	$-0.285776\pi$
$37$	−1320.83	−	1320.83i	−0.158615	−	0.158615i	−0.781953	+	0.623338i	$0.785776\pi$
$38$	7947.85	+	7947.85i	0.892875	+	0.892875i
$39$		0			0
$40$		0			0
$41$			10791.6i			1.00259i	0.865275	+	0.501297i	$0.167144\pi$
$41$			10791.6i			1.00259i	−0.865275	+	0.501297i	$0.832856\pi$
$42$		0			0
$43$	−7831.78	+	7831.78i	−0.645936	+	0.645936i	−0.952008	−	0.306073i	$-0.900985\pi$
$43$	−7831.78	+	7831.78i	−0.645936	+	0.645936i	0.306073	+	0.952008i	$0.400985\pi$
$44$	452.433			0.0352308
$45$		0			0
$46$	−5107.06			−0.355858
$47$	−11749.4	+	11749.4i	−0.775838	+	0.775838i	−0.979120	−	0.203282i	$-0.934839\pi$
$47$	−11749.4	+	11749.4i	−0.775838	+	0.775838i	0.203282	+	0.979120i	$0.434839\pi$
$48$		0			0
$49$			10650.8i			0.633710i
$50$		0			0
$51$		0			0
$52$	−4531.80	−	4531.80i	−0.232414	−	0.232414i
$53$	22576.3	+	22576.3i	1.10398	+	1.10398i	0.993925	+	0.110059i	$0.0351038\pi$
$53$	22576.3	+	22576.3i	1.10398	+	1.10398i	0.110059	+	0.993925i	$0.464896\pi$
$54$		0			0
$55$		0			0
$56$		−	10605.0i		−	0.451900i
$57$		0			0
$58$	−2340.73	+	2340.73i	−0.0913653	+	0.0913653i
$59$	−33192.4			−1.24139			−0.620695	−	0.784052i	$-0.713150\pi$
$59$	−33192.4			−1.24139			−0.620695	+	0.784052i	$0.713150\pi$
$60$		0			0
$61$	32011.7			1.10150			0.550749	−	0.834671i	$-0.314342\pi$
$61$	32011.7			1.10150			0.550749	+	0.834671i	$0.314342\pi$
$62$	5797.27	−	5797.27i	0.191533	−	0.191533i
$63$		0			0
$64$			4096.00i			0.125000i
$65$		0			0
$66$		0			0
$67$	27419.3	+	27419.3i	0.746223	+	0.746223i	0.973768	−	0.227544i	$-0.0730696\pi$
$67$	27419.3	+	27419.3i	0.746223	+	0.746223i	−0.227544	+	0.973768i	$0.573070\pi$
$68$	2205.48	+	2205.48i	0.0578403	+	0.0578403i
$69$		0			0
$70$		0			0
$71$		−	54641.5i		−	1.28640i	−0.765698	−	0.643200i	$-0.777606\pi$
$71$		−	54641.5i		−	1.28640i	0.765698	−	0.643200i	$-0.222394\pi$
$72$		0			0
$73$	14250.4	−	14250.4i	0.312982	−	0.312982i	−0.533081	−	0.846064i	$-0.678966\pi$
$73$	14250.4	−	14250.4i	0.312982	−	0.312982i	0.846064	+	0.533081i	$0.178966\pi$
$74$	−7471.75			−0.158615
$75$		0			0
$76$	44959.8			0.892875
$77$	−3313.23	+	3313.23i	−0.0636833	+	0.0636833i
$78$		0			0
$79$			79274.9i			1.42912i	0.699576	+	0.714559i	$0.253372\pi$
$79$			79274.9i			1.42912i	−0.699576	+	0.714559i	$0.746628\pi$
$80$		0			0
$81$		0			0
$82$	30523.2	+	30523.2i	0.501297	+	0.501297i
$83$	36508.8	+	36508.8i	0.581704	+	0.581704i	0.935371	−	0.353667i	$-0.115065\pi$
$83$	36508.8	+	36508.8i	0.581704	+	0.581704i	−0.353667	+	0.935371i	$0.615065\pi$
$84$		0			0
$85$		0			0
$86$			44303.2i			0.645936i
$87$		0			0
$88$	1279.67	−	1279.67i	0.0176154	−	0.0176154i
$89$	69924.7			0.935741			0.467871	−	0.883797i	$-0.345021\pi$
$89$	69924.7			0.935741			0.467871	+	0.883797i	$0.345021\pi$
$90$		0			0
$91$	66374.0			0.840223
$92$	−14444.9	+	14444.9i	−0.177929	+	0.177929i
$93$		0			0
$94$			66464.6i			0.775838i
$95$		0			0
$96$		0			0
$97$	69381.8	+	69381.8i	0.748715	+	0.748715i	0.974238	−	0.225523i	$-0.0724091\pi$
$97$	69381.8	+	69381.8i	0.748715	+	0.748715i	−0.225523	+	0.974238i	$0.572409\pi$
$98$	30124.9	+	30124.9i	0.316855	+	0.316855i
$99$		0			0
$100$		0			0
$101$		−	163404.i		−	1.59389i	−0.604049	−	0.796947i	$-0.706447\pi$
$101$		−	163404.i		−	1.59389i	0.604049	−	0.796947i	$-0.293553\pi$
$102$		0			0
$103$	−78791.9	+	78791.9i	−0.731793	+	0.731793i	−0.970975	−	0.239181i	$-0.923121\pi$
$103$	−78791.9	+	78791.9i	−0.731793	+	0.731793i	0.239181	+	0.970975i	$0.423121\pi$
$104$	−25635.7			−0.232414
$105$		0			0
$106$	127711.			1.10398
$107$	59824.6	−	59824.6i	0.505150	−	0.505150i	−0.407884	−	0.913034i	$-0.633733\pi$
$107$	59824.6	−	59824.6i	0.505150	−	0.505150i	0.913034	+	0.407884i	$0.133733\pi$
$108$		0			0
$109$			3640.35i			0.0293479i	0.999892	+	0.0146739i	$0.00467103\pi$
$109$			3640.35i			0.0293479i	−0.999892	+	0.0146739i	$0.995329\pi$
$110$		0			0
$111$		0			0
$112$	−29995.6	−	29995.6i	−0.225950	−	0.225950i
$113$	141328.	+	141328.i	1.04120	+	1.04120i	0.999114	+	0.0420834i	$0.0133995\pi$
$113$	141328.	+	141328.i	1.04120	+	1.04120i	0.0420834	+	0.999114i	$0.486600\pi$
$114$		0			0
$115$		0			0
$116$			13241.2i			0.0913653i
$117$		0			0
$118$	−93882.3	+	93882.3i	−0.620695	+	0.620695i
$119$	−32302.1			−0.209104
$120$		0			0
$121$	160251.			0.995035
$122$	90542.7	−	90542.7i	0.550749	−	0.550749i
$123$		0			0
$124$		−	32794.3i		−	0.191533i
$125$		0			0
$126$		0			0
$127$	−99907.0	−	99907.0i	−0.549650	−	0.549650i	0.376689	−	0.926340i	$-0.377062\pi$
$127$	−99907.0	−	99907.0i	−0.549650	−	0.549650i	−0.926340	+	0.376689i	$0.877062\pi$
$128$	11585.2	+	11585.2i	0.0625000	+	0.0625000i
$129$		0			0
$130$		0			0
$131$		−	189944.i		−	0.967047i	−0.875331	−	0.483524i	$-0.839357\pi$
$131$		−	189944.i		−	0.967047i	0.875331	−	0.483524i	$-0.160643\pi$
$132$		0			0
$133$	−329247.	+	329247.i	−1.61396	+	1.61396i
$134$	155107.			0.746223
$135$		0			0
$136$	12476.1			0.0578403
$137$	−266688.	+	266688.i	−1.21395	+	1.21395i	−0.244236	+	0.969716i	$0.578537\pi$
$137$	−266688.	+	266688.i	−1.21395	+	1.21395i	−0.969716	+	0.244236i	$0.921463\pi$
$138$		0			0
$139$			242739.i			1.06562i	0.846234	+	0.532811i	$0.178864\pi$
$139$			242739.i			1.06562i	−0.846234	+	0.532811i	$0.821136\pi$
$140$		0			0
$141$		0			0
$142$	−154549.	−	154549.i	−0.643200	−	0.643200i
$143$	8009.12	+	8009.12i	0.0327525	+	0.0327525i
$144$		0			0
$145$		0			0
$146$		−	80612.4i		−	0.312982i
$147$		0			0
$148$	−21133.3	+	21133.3i	−0.0793073	+	0.0793073i
$149$	479131.			1.76803			0.884013	−	0.467462i	$-0.154832\pi$
$149$	479131.			1.76803			0.884013	+	0.467462i	$0.154832\pi$
$150$		0			0
$151$	181321.			0.647151			0.323576	−	0.946202i	$-0.395115\pi$
$151$	181321.			0.647151			0.323576	+	0.946202i	$0.395115\pi$
$152$	127166.	−	127166.i	0.446437	−	0.446437i
$153$		0			0
$154$			18742.5i			0.0636833i
$155$		0			0
$156$		0			0
$157$	261002.	+	261002.i	0.845075	+	0.845075i	0.989514	−	0.144439i	$-0.0461377\pi$
$157$	261002.	+	261002.i	0.845075	+	0.845075i	−0.144439	+	0.989514i	$0.546138\pi$
$158$	224223.	+	224223.i	0.714559	+	0.714559i
$159$		0			0
$160$		0			0
$161$		−	211565.i		−	0.643249i
$162$		0			0
$163$	−414824.	+	414824.i	−1.22291	+	1.22291i	−0.256317	+	0.966593i	$0.582509\pi$
$163$	−414824.	+	414824.i	−1.22291	+	1.22291i	−0.966593	+	0.256317i	$0.917491\pi$
$164$	172665.			0.501297
$165$		0			0
$166$	206525.			0.581704
$167$	282143.	−	282143.i	0.782850	−	0.782850i	−0.197461	−	0.980311i	$-0.563269\pi$
$167$	282143.	−	282143.i	0.782850	−	0.782850i	0.980311	+	0.197461i	$0.0632694\pi$
$168$		0			0
$169$			210846.i			0.567870i
$170$		0			0
$171$		0			0
$172$	125308.	+	125308.i	0.322968	+	0.322968i
$173$	−516050.	−	516050.i	−1.31092	−	1.31092i	−0.920737	−	0.390183i	$-0.872412\pi$
$173$	−516050.	−	516050.i	−1.31092	−	1.31092i	−0.390183	−	0.920737i	$-0.627588\pi$
$174$		0			0
$175$		0			0
$176$		−	7238.93i		−	0.0176154i
$177$		0			0
$178$	197777.	−	197777.i	0.467871	−	0.467871i
$179$	183417.			0.427865			0.213933	−	0.976848i	$-0.431373\pi$
$179$	183417.			0.427865			0.213933	+	0.976848i	$0.431373\pi$
$180$		0			0
$181$	333015.			0.755557			0.377779	−	0.925896i	$-0.376688\pi$
$181$	333015.			0.755557			0.377779	+	0.925896i	$0.376688\pi$
$182$	187734.	−	187734.i	0.420112	−	0.420112i
$183$		0			0
$184$			81712.9i			0.177929i
$185$		0			0
$186$		0			0
$187$	−3897.78	−	3897.78i	−0.00815105	−	0.00815105i
$188$	187990.	+	187990.i	0.387919	+	0.387919i
$189$		0			0
$190$		0			0
$191$		−	872731.i		−	1.73100i	−0.500910	−	0.865500i	$-0.667001\pi$
$191$		−	872731.i		−	1.73100i	0.500910	−	0.865500i	$-0.332999\pi$
$192$		0			0
$193$	−198341.	+	198341.i	−0.383283	+	0.383283i	−0.872284	−	0.489000i	$-0.837362\pi$
$193$	−198341.	+	198341.i	−0.383283	+	0.383283i	0.489000	+	0.872284i	$0.337362\pi$
$194$	392483.			0.748715
$195$		0			0
$196$	170412.			0.316855
$197$	−63308.1	+	63308.1i	−0.116223	+	0.116223i	−0.762827	−	0.646603i	$-0.776189\pi$
$197$	−63308.1	+	63308.1i	−0.116223	+	0.116223i	0.646603	+	0.762827i	$0.276189\pi$
$198$		0			0
$199$			354056.i			0.633782i	0.948462	+	0.316891i	$0.102639\pi$
$199$			354056.i			0.633782i	−0.948462	+	0.316891i	$0.897361\pi$
$200$		0			0
$201$		0			0
$202$	−462177.	−	462177.i	−0.796947	−	0.796947i
$203$	−96967.0	−	96967.0i	−0.165152	−	0.165152i
$204$		0			0
$205$		0			0
$206$			445714.i			0.731793i
$207$		0			0
$208$	−72508.8	+	72508.8i	−0.116207	+	0.116207i
$209$	−79458.3			−0.125827
$210$		0			0
$211$	−295633.			−0.457138			−0.228569	−	0.973528i	$-0.573405\pi$
$211$	−295633.			−0.457138			−0.228569	+	0.973528i	$0.573405\pi$
$212$	361220.	−	361220.i	0.551992	−	0.551992i
$213$		0			0
$214$		−	338419.i		−	0.505150i
$215$		0			0
$216$		0			0
$217$	240158.	+	240158.i	0.346216	+	0.346216i
$218$	10296.5	+	10296.5i	0.0146739	+	0.0146739i
$219$		0			0
$220$		0			0
$221$			78084.2i			0.107543i
$222$		0			0
$223$	−72165.1	+	72165.1i	−0.0971773	+	0.0971773i	−0.754024	−	0.656847i	$-0.771890\pi$
$223$	−72165.1	+	72165.1i	−0.0971773	+	0.0971773i	0.656847	+	0.754024i	$0.271890\pi$
$224$	−169681.			−0.225950
$225$		0			0
$226$	799474.			1.04120
$227$	−674789.	+	674789.i	−0.869168	+	0.869168i	−0.992380	−	0.123213i	$-0.960680\pi$
$227$	−674789.	+	674789.i	−0.869168	+	0.869168i	0.123213	+	0.992380i	$0.460680\pi$
$228$		0			0
$229$		−	1.07253e6i		−	1.35152i	−0.737122	−	0.675760i	$-0.763816\pi$
$229$		−	1.07253e6i		−	1.35152i	0.737122	−	0.675760i	$-0.236184\pi$
$230$		0			0
$231$		0			0
$232$	37451.7	+	37451.7i	0.0456827	+	0.0456827i
$233$	668430.	+	668430.i	0.806615	+	0.806615i	0.984120	−	0.177505i	$-0.0568025\pi$
$233$	668430.	+	668430.i	0.806615	+	0.806615i	−0.177505	+	0.984120i	$0.556803\pi$
$234$		0			0
$235$		0			0
$236$			531078.i			0.620695i
$237$		0			0
$238$	−91364.1	+	91364.1i	−0.104552	+	0.104552i
$239$	−1.02490e6			−1.16061			−0.580306	−	0.814399i	$-0.697067\pi$
$239$	−1.02490e6			−1.16061			−0.580306	+	0.814399i	$0.697067\pi$
$240$		0			0
$241$	−465821.			−0.516626			−0.258313	−	0.966061i	$-0.583167\pi$
$241$	−465821.			−0.516626			−0.258313	+	0.966061i	$0.583167\pi$
$242$	453259.	−	453259.i	0.497518	−	0.497518i
$243$		0			0
$244$		−	512187.i		−	0.550749i
$245$		0			0
$246$		0			0
$247$	795894.	+	795894.i	0.830066	+	0.830066i
$248$	−92756.3	−	92756.3i	−0.0957667	−	0.0957667i
$249$		0			0
$250$		0			0
$251$		−	1.02118e6i		−	1.02310i	−0.859254	−	0.511550i	$-0.829072\pi$
$251$		−	1.02118e6i		−	1.02310i	0.859254	−	0.511550i	$-0.170928\pi$
$252$		0			0
$253$	25528.8	−	25528.8i	0.0250743	−	0.0250743i
$254$	−565159.			−0.549650
$255$		0			0
$256$	65536.0			0.0625000
$257$	−510335.	+	510335.i	−0.481973	+	0.481973i	−0.905761	−	0.423789i	$-0.860700\pi$
$257$	−510335.	+	510335.i	−0.481973	+	0.481973i	0.423789	+	0.905761i	$0.360700\pi$
$258$		0			0
$259$		−	309525.i		−	0.286712i
$260$		0			0
$261$		0			0
$262$	−537243.	−	537243.i	−0.483524	−	0.483524i
$263$	−320756.	−	320756.i	−0.285947	−	0.285947i	0.549528	−	0.835475i	$-0.314807\pi$
$263$	−320756.	−	320756.i	−0.285947	−	0.285947i	−0.835475	+	0.549528i	$0.814807\pi$
$264$		0			0
$265$		0			0
$266$			1.86250e6i			1.61396i
$267$		0			0
$268$	438708.	−	438708.i	0.373112	−	0.373112i
$269$	−85841.8			−0.0723300			−0.0361650	−	0.999346i	$-0.511514\pi$
$269$	−85841.8			−0.0723300			−0.0361650	+	0.999346i	$0.511514\pi$
$270$		0			0
$271$	−2.28993e6			−1.89409			−0.947043	−	0.321107i	$-0.895945\pi$
$271$	−2.28993e6			−1.89409			−0.947043	+	0.321107i	$0.895945\pi$
$272$	35287.7	−	35287.7i	0.0289202	−	0.0289202i
$273$		0			0
$274$			1.50861e6i			1.21395i
$275$		0			0
$276$		0			0
$277$	−431331.	−	431331.i	−0.337762	−	0.337762i	0.517762	−	0.855525i	$-0.326765\pi$
$277$	−431331.	−	431331.i	−0.337762	−	0.337762i	−0.855525	+	0.517762i	$0.826765\pi$
$278$	686570.	+	686570.i	0.532811	+	0.532811i
$279$		0			0
$280$		0			0
$281$		−	226762.i		−	0.171319i	−0.996324	−	0.0856594i	$-0.972700\pi$
$281$		−	226762.i		−	0.171319i	0.996324	−	0.0856594i	$-0.0272997\pi$
$282$		0			0
$283$	1.26414e6	−	1.26414e6i	0.938271	−	0.938271i	−0.0599317	−	0.998202i	$-0.519088\pi$
$283$	1.26414e6	−	1.26414e6i	0.938271	−	0.938271i	0.998202	+	0.0599317i	$0.0190883\pi$
$284$	−874263.			−0.643200
$285$		0			0
$286$	45306.4			0.0327525
$287$	−1.26445e6	+	1.26445e6i	−0.906145	+	0.906145i
$288$		0			0
$289$			1.38186e6i			0.973236i
$290$		0			0
$291$		0			0
$292$	−228006.	−	228006.i	−0.156491	−	0.156491i
$293$	−1.45042e6	−	1.45042e6i	−0.987015	−	0.987015i	0.0129015	−	0.999917i	$-0.495893\pi$
$293$	−1.45042e6	−	1.45042e6i	−0.987015	−	0.987015i	−0.999917	+	0.0129015i	$0.995893\pi$
$294$		0			0
$295$		0			0
$296$			119548.i			0.0793073i
$297$		0			0
$298$	1.35519e6	−	1.35519e6i	0.884013	−	0.884013i
$299$	−511418.			−0.330825
$300$		0			0
$301$	−1.83530e6			−1.16759
$302$	512853.	−	512853.i	0.323576	−	0.323576i
$303$		0			0
$304$		−	719357.i		−	0.446437i
$305$		0			0
$306$		0			0
$307$	800295.	+	800295.i	0.484623	+	0.484623i	0.906604	−	0.421982i	$-0.138665\pi$
$307$	800295.	+	800295.i	0.484623	+	0.484623i	−0.421982	+	0.906604i	$0.638665\pi$
$308$	53011.7	+	53011.7i	0.0318416	+	0.0318416i
$309$		0			0
$310$		0			0
$311$			1.63357e6i			0.957714i	0.877893	+	0.478857i	$0.158949\pi$
$311$			1.63357e6i			0.957714i	−0.877893	+	0.478857i	$0.841051\pi$
$312$		0			0
$313$	1.99538e6	−	1.99538e6i	1.15124	−	1.15124i	0.164933	−	0.986305i	$-0.447259\pi$
$313$	1.99538e6	−	1.99538e6i	1.15124	−	1.15124i	0.986305	−	0.164933i	$-0.0527407\pi$
$314$	1.47645e6			0.845075
$315$		0			0
$316$	1.26840e6			0.714559
$317$	−537671.	+	537671.i	−0.300517	+	0.300517i	−0.841216	−	0.540699i	$-0.818160\pi$
$317$	−537671.	+	537671.i	−0.300517	+	0.300517i	0.540699	+	0.841216i	$0.318160\pi$
$318$		0			0
$319$		−	23401.4i		−	0.0128755i
$320$		0			0
$321$		0			0
$322$	−598396.	−	598396.i	−0.321624	−	0.321624i
$323$	−387336.	−	387336.i	−0.206577	−	0.206577i
$324$		0			0
$325$		0			0
$326$			2.34660e6i			1.22291i
$327$		0			0
$328$	488371.	−	488371.i	0.250649	−	0.250649i
$329$	−2.75336e6			−1.40240
$330$		0			0
$331$	−31025.8			−0.0155651			−0.00778257	−	0.999970i	$-0.502477\pi$
$331$	−31025.8			−0.0155651			−0.00778257	+	0.999970i	$0.502477\pi$
$332$	584141.	−	584141.i	0.290852	−	0.290852i
$333$		0			0
$334$		−	1.59604e6i		−	0.782850i
$335$		0			0
$336$		0			0
$337$	702444.	+	702444.i	0.336928	+	0.336928i	0.855210	−	0.518282i	$-0.173428\pi$
$337$	702444.	+	702444.i	0.336928	+	0.336928i	−0.518282	+	0.855210i	$0.673428\pi$
$338$	596363.	+	596363.i	0.283935	+	0.283935i
$339$		0			0
$340$		0			0
$341$			57958.0i			0.0269915i
$342$		0			0
$343$	721328.	−	721328.i	0.331053	−	0.331053i
$344$	708852.			0.322968
$345$		0			0
$346$	−2.91922e6			−1.31092
$347$	−1.84881e6	+	1.84881e6i	−0.824266	+	0.824266i	−0.986717	−	0.162451i	$-0.948060\pi$
$347$	−1.84881e6	+	1.84881e6i	−0.824266	+	0.824266i	0.162451	+	0.986717i	$0.448060\pi$
$348$		0			0
$349$			285458.i			0.125452i	0.998031	+	0.0627262i	$0.0199795\pi$
$349$			285458.i			0.125452i	−0.998031	+	0.0627262i	$0.980021\pi$
$350$		0			0
$351$		0			0
$352$	−20474.8	−	20474.8i	−0.00880771	−	0.00880771i
$353$	−1.11482e6	−	1.11482e6i	−0.476178	−	0.476178i	0.427729	−	0.903907i	$-0.359314\pi$
$353$	−1.11482e6	−	1.11482e6i	−0.476178	−	0.476178i	−0.903907	+	0.427729i	$0.859314\pi$
$354$		0			0
$355$		0			0
$356$		−	1.11880e6i		−	0.467871i
$357$		0			0
$358$	518782.	−	518782.i	0.213933	−	0.213933i
$359$	−1.09213e6			−0.447237			−0.223618	−	0.974677i	$-0.571787\pi$
$359$	−1.09213e6			−0.447237			−0.223618	+	0.974677i	$0.571787\pi$
$360$		0			0
$361$	−5.41994e6			−2.18890
$362$	941908.	−	941908.i	0.377779	−	0.377779i
$363$		0			0
$364$		−	1.06198e6i		−	0.420112i
$365$		0			0
$366$		0			0
$367$	−1.38502e6	−	1.38502e6i	−0.536774	−	0.536774i	0.385806	−	0.922580i	$-0.373923\pi$
$367$	−1.38502e6	−	1.38502e6i	−0.536774	−	0.536774i	−0.922580	+	0.385806i	$0.873923\pi$
$368$	231119.	+	231119.i	0.0889644	+	0.0889644i
$369$		0			0
$370$		0			0
$371$			5.29054e6i			1.99556i
$372$		0			0
$373$	−2.32237e6	+	2.32237e6i	−0.864291	+	0.864291i	−0.991833	−	0.127542i	$-0.959291\pi$
$373$	−2.32237e6	+	2.32237e6i	−0.864291	+	0.864291i	0.127542	+	0.991833i	$0.459291\pi$
$374$	−22049.2			−0.00815105
$375$		0			0
$376$	1.06343e6			0.387919
$377$	−234400.	+	234400.i	−0.0849383	+	0.0849383i
$378$		0			0
$379$		−	2.91676e6i		−	1.04304i	−0.853238	−	0.521522i	$-0.825364\pi$
$379$		−	2.91676e6i		−	1.04304i	0.853238	−	0.521522i	$-0.174636\pi$
$380$		0			0
$381$		0			0
$382$	−2.46846e6	−	2.46846e6i	−0.865500	−	0.865500i
$383$	−1.46470e6	−	1.46470e6i	−0.510212	−	0.510212i	0.404379	−	0.914591i	$-0.367488\pi$
$383$	−1.46470e6	−	1.46470e6i	−0.510212	−	0.510212i	−0.914591	+	0.404379i	$0.867488\pi$
$384$		0			0
$385$		0			0
$386$			1.12199e6i			0.383283i
$387$		0			0
$388$	1.11011e6	−	1.11011e6i	0.374357	−	0.374357i
$389$	−2.43062e6			−0.814410			−0.407205	−	0.913337i	$-0.633497\pi$
$389$	−2.43062e6			−0.814410			−0.407205	+	0.913337i	$0.633497\pi$
$390$		0			0
$391$	248891.			0.0823317
$392$	481999.	−	481999.i	0.158428	−	0.158428i
$393$		0			0
$394$			358125.i			0.116223i
$395$		0			0
$396$		0			0
$397$	−499831.	−	499831.i	−0.159165	−	0.159165i	0.623032	−	0.782197i	$-0.285901\pi$
$397$	−499831.	−	499831.i	−0.159165	−	0.159165i	−0.782197	+	0.623032i	$0.785901\pi$
$398$	1.00142e6	+	1.00142e6i	0.316891	+	0.316891i
$399$		0			0
$400$		0			0
$401$		−	5.80023e6i		−	1.80129i	−0.434554	−	0.900646i	$-0.643094\pi$
$401$		−	5.80023e6i		−	1.80129i	0.434554	−	0.900646i	$-0.356906\pi$
$402$		0			0
$403$	580536.	−	580536.i	0.178060	−	0.178060i
$404$	−2.61447e6			−0.796947
$405$		0			0
$406$	−548528.			−0.165152
$407$	37349.3	−	37349.3i	0.0111762	−	0.0111762i
$408$		0			0
$409$		−	6.25836e6i		−	1.84992i	−0.380067	−	0.924959i	$-0.624099\pi$
$409$		−	6.25836e6i		−	1.84992i	0.380067	−	0.924959i	$-0.375901\pi$
$410$		0			0
$411$		0			0
$412$	1.26067e6	+	1.26067e6i	0.365897	+	0.365897i
$413$	−3.88916e6	−	3.88916e6i	−1.12197	−	1.12197i
$414$		0			0
$415$		0			0
$416$			410171.i			0.116207i
$417$		0			0
$418$	−224742.	+	224742.i	−0.0629134	+	0.0629134i
$419$	1.90211e6			0.529298			0.264649	−	0.964345i	$-0.414744\pi$
$419$	1.90211e6			0.529298			0.264649	+	0.964345i	$0.414744\pi$
$420$		0			0
$421$	396767.			0.109101			0.0545507	−	0.998511i	$-0.482627\pi$
$421$	396767.			0.109101			0.0545507	+	0.998511i	$0.482627\pi$
$422$	−836178.	+	836178.i	−0.228569	+	0.228569i
$423$		0			0
$424$		−	2.04337e6i		−	0.551992i
$425$		0			0
$426$		0			0
$427$	3.75082e6	+	3.75082e6i	0.995535	+	0.995535i
$428$	−957194.	−	957194.i	−0.252575	−	0.252575i
$429$		0			0
$430$		0			0
$431$		−	1.37490e6i		−	0.356514i	−0.983984	−	0.178257i	$-0.942954\pi$
$431$		−	1.37490e6i		−	0.356514i	0.983984	−	0.178257i	$-0.0570459\pi$
$432$		0			0
$433$	361016.	−	361016.i	0.0925351	−	0.0925351i	−0.659324	−	0.751859i	$-0.729157\pi$
$433$	361016.	−	361016.i	0.0925351	−	0.0925351i	0.751859	+	0.659324i	$0.229157\pi$
$434$	1.35854e6			0.346216
$435$		0			0
$436$	58245.6			0.0146739
$437$	2.53688e6	−	2.53688e6i	0.635473	−	0.635473i
$438$		0			0
$439$		−	844631.i		−	0.209173i	−0.994516	−	0.104586i	$-0.966648\pi$
$439$		−	844631.i		−	0.209173i	0.994516	−	0.104586i	$-0.0333519\pi$
$440$		0			0
$441$		0			0
$442$	220856.	+	220856.i	0.0537716	+	0.0537716i
$443$	904722.	+	904722.i	0.219031	+	0.219031i	0.808090	−	0.589059i	$-0.200501\pi$
$443$	904722.	+	904722.i	0.219031	+	0.219031i	−0.589059	+	0.808090i	$0.700501\pi$
$444$		0			0
$445$		0			0
$446$			408227.i			0.0971773i
$447$		0			0
$448$	−479930.	+	479930.i	−0.112975	+	0.112975i
$449$	7.62319e6			1.78452			0.892259	−	0.451525i	$-0.149120\pi$
$449$	7.62319e6			1.78452			0.892259	+	0.451525i	$0.149120\pi$
$450$		0			0
$451$	−305154.			−0.0706445
$452$	2.26125e6	−	2.26125e6i	0.520599	−	0.520599i
$453$		0			0
$454$			3.81718e6i			0.869168i
$455$		0			0
$456$		0			0
$457$	−3.38843e6	−	3.38843e6i	−0.758941	−	0.758941i	0.217188	−	0.976130i	$-0.430311\pi$
$457$	−3.38843e6	−	3.38843e6i	−0.758941	−	0.758941i	−0.976130	+	0.217188i	$0.930311\pi$
$458$	−3.03358e6	−	3.03358e6i	−0.675760	−	0.675760i
$459$		0			0
$460$		0			0
$461$			3.37767e6i			0.740226i	0.928987	+	0.370113i	$0.120681\pi$
$461$			3.37767e6i			0.740226i	−0.928987	+	0.370113i	$0.879319\pi$
$462$		0			0
$463$	841979.	−	841979.i	0.182536	−	0.182536i	−0.609924	−	0.792460i	$-0.708800\pi$
$463$	841979.	−	841979.i	0.182536	−	0.182536i	0.792460	+	0.609924i	$0.208800\pi$
$464$	211859.			0.0456827
$465$		0			0
$466$	3.78121e6			0.806615
$467$	−774792.	+	774792.i	−0.164397	+	0.164397i	−0.784511	−	0.620115i	$-0.787086\pi$
$467$	−774792.	+	774792.i	−0.164397	+	0.164397i	0.620115	+	0.784511i	$0.287086\pi$
$468$		0			0
$469$			6.42545e6i			1.34887i
$470$		0			0
$471$		0			0
$472$	1.50212e6	+	1.50212e6i	0.310348	+	0.310348i
$473$	−221460.	−	221460.i	−0.0455137	−	0.0455137i
$474$		0			0
$475$		0			0
$476$			516833.i			0.104552i
$477$		0			0
$478$	−2.89886e6	+	2.89886e6i	−0.580306	+	0.580306i
$479$	2.64210e6			0.526150			0.263075	−	0.964775i	$-0.415263\pi$
$479$	2.64210e6			0.526150			0.263075	+	0.964775i	$0.415263\pi$
$480$		0			0
$481$	−748218.			−0.147457
$482$	−1.31754e6	+	1.31754e6i	−0.258313	+	0.258313i
$483$		0			0
$484$		−	2.56402e6i		−	0.497518i
$485$		0			0
$486$		0			0
$487$	190754.	+	190754.i	0.0364461	+	0.0364461i	0.725095	−	0.688649i	$-0.241796\pi$
$487$	190754.	+	190754.i	0.0364461	+	0.0364461i	−0.688649	+	0.725095i	$0.741796\pi$
$488$	−1.44868e6	−	1.44868e6i	−0.275375	−	0.275375i
$489$		0			0
$490$		0			0
$491$		−	6.69770e6i		−	1.25378i	−0.779107	−	0.626891i	$-0.784327\pi$
$491$		−	6.69770e6i		−	1.25378i	0.779107	−	0.626891i	$-0.215673\pi$
$492$		0			0
$493$	114075.	−	114075.i	0.0211384	−	0.0211384i
$494$	4.50226e6			0.830066
$495$		0			0
$496$	−524709.			−0.0957667
$497$	6.40236e6	−	6.40236e6i	1.16265	−	1.16265i
$498$		0			0
$499$			4.04738e6i			0.727650i	0.931467	+	0.363825i	$0.118529\pi$
$499$			4.04738e6i			0.727650i	−0.931467	+	0.363825i	$0.881471\pi$
$500$		0			0
$501$		0			0
$502$	−2.88833e6	−	2.88833e6i	−0.511550	−	0.511550i
$503$	4.85945e6	+	4.85945e6i	0.856381	+	0.856381i	0.990910	−	0.134528i	$-0.0429519\pi$
$503$	4.85945e6	+	4.85945e6i	0.856381	+	0.856381i	−0.134528	+	0.990910i	$0.542952\pi$
$504$		0			0
$505$		0			0
$506$		−	144413.i		−	0.0250743i
$507$		0			0
$508$	−1.59851e6	+	1.59851e6i	−0.274825	+	0.274825i
$509$	−3.04228e6			−0.520481			−0.260240	−	0.965544i	$-0.583802\pi$
$509$	−3.04228e6			−0.520481			−0.260240	+	0.965544i	$0.583802\pi$
$510$		0			0
$511$	3.33945e6			0.565747
$512$	185364.	−	185364.i	0.0312500	−	0.0312500i
$513$		0			0
$514$			2.88689e6i			0.481973i
$515$		0			0
$516$		0			0
$517$	−332239.	−	332239.i	−0.0546668	−	0.0546668i
$518$	−875468.	−	875468.i	−0.143356	−	0.143356i
$519$		0			0
$520$		0			0
$521$		−	1.34709e6i		−	0.217422i	−0.994073	−	0.108711i	$-0.965328\pi$
$521$		−	1.34709e6i		−	0.217422i	0.994073	−	0.108711i	$-0.0346723\pi$
$522$		0			0
$523$	−3.71364e6	+	3.71364e6i	−0.593671	+	0.593671i	−0.938621	−	0.344950i	$-0.887896\pi$
$523$	−3.71364e6	+	3.71364e6i	−0.593671	+	0.593671i	0.344950	+	0.938621i	$0.387896\pi$
$524$	−3.03911e6			−0.483524
$525$		0			0
$526$	−1.81447e6			−0.285947
$527$	−282528.	+	282528.i	−0.0443134	+	0.0443134i
$528$		0			0
$529$		−	4.80621e6i		−	0.746731i
$530$		0			0
$531$		0			0
$532$	5.26796e6	+	5.26796e6i	0.806981	+	0.806981i
$533$	3.05658e6	+	3.05658e6i	0.466034	+	0.466034i
$534$		0			0
$535$		0			0
$536$		−	2.48171e6i		−	0.373112i
$537$		0			0
$538$	−242797.	+	242797.i	−0.0361650	+	0.0361650i
$539$	−301173.			−0.0446523
$540$		0			0
$541$	1.31604e7			1.93320			0.966598	−	0.256298i	$-0.0825027\pi$
$541$	1.31604e7			1.93320			0.966598	+	0.256298i	$0.0825027\pi$
$542$	−6.47691e6	+	6.47691e6i	−0.947043	+	0.947043i
$543$		0			0
$544$		−	199617.i		−	0.0289202i
$545$		0			0
$546$		0			0
$547$	−5.13609e6	−	5.13609e6i	−0.733946	−	0.733946i	0.237453	−	0.971399i	$-0.423687\pi$
$547$	−5.13609e6	−	5.13609e6i	−0.733946	−	0.733946i	−0.971399	+	0.237453i	$0.923687\pi$
$548$	4.26700e6	+	4.26700e6i	0.606976	+	0.606976i
$549$		0			0
$550$		0			0
$551$		−	2.32547e6i		−	0.326311i
$552$		0			0
$553$	−9.28866e6	+	9.28866e6i	−1.29164	+	1.29164i
$554$	−2.43998e6			−0.337762
$555$		0			0
$556$	3.88383e6			0.532811
$557$	−4.26355e6	+	4.26355e6i	−0.582282	+	0.582282i	−0.935530	−	0.353248i	$-0.885077\pi$
$557$	−4.26355e6	+	4.26355e6i	−0.582282	+	0.582282i	0.353248	+	0.935530i	$0.385077\pi$
$558$		0			0
$559$			4.43650e6i			0.600498i
$560$		0			0
$561$		0			0
$562$	−641381.	−	641381.i	−0.0856594	−	0.0856594i
$563$	4.98183e6	+	4.98183e6i	0.662396	+	0.662396i	0.955944	−	0.293548i	$-0.0948361\pi$
$563$	4.98183e6	+	4.98183e6i	0.662396	+	0.662396i	−0.293548	+	0.955944i	$0.594836\pi$
$564$		0			0
$565$		0			0
$566$		−	7.15104e6i		−	0.938271i
$567$		0			0
$568$	−2.47279e6	+	2.47279e6i	−0.321600	+	0.321600i
$569$	1.53062e7			1.98193			0.990963	−	0.134136i	$-0.0428260\pi$
$569$	1.53062e7			1.98193			0.990963	+	0.134136i	$0.0428260\pi$
$570$		0			0
$571$	834423.			0.107102			0.0535508	−	0.998565i	$-0.482946\pi$
$571$	834423.			0.107102			0.0535508	+	0.998565i	$0.482946\pi$
$572$	128146.	−	128146.i	0.0163763	−	0.0163763i
$573$		0			0
$574$			7.15282e6i			0.906145i
$575$		0			0
$576$		0			0
$577$	−1.12011e6	−	1.12011e6i	−0.140062	−	0.140062i	0.633599	−	0.773661i	$-0.281577\pi$
$577$	−1.12011e6	−	1.12011e6i	−0.140062	−	0.140062i	−0.773661	+	0.633599i	$0.781577\pi$
$578$	3.90848e6	+	3.90848e6i	0.486618	+	0.486618i
$579$		0			0
$580$		0			0
$581$			8.55549e6i			1.05149i
$582$		0			0
$583$	−638391.	+	638391.i	−0.0777885	+	0.0777885i
$584$	−1.28980e6			−0.156491
$585$		0			0
$586$	−8.20480e6			−0.987015
$587$	−1.49924e6	+	1.49924e6i	−0.179587	+	0.179587i	−0.791176	−	0.611589i	$-0.790531\pi$
$587$	−1.49924e6	+	1.49924e6i	−0.179587	+	0.179587i	0.611589	+	0.791176i	$0.290531\pi$
$588$		0			0
$589$			5.75948e6i			0.684061i
$590$		0			0
$591$		0			0
$592$	338133.	+	338133.i	0.0396536	+	0.0396536i
$593$	5.77287e6	+	5.77287e6i	0.674147	+	0.674147i	0.958669	−	0.284522i	$-0.0918349\pi$
$593$	5.77287e6	+	5.77287e6i	0.674147	+	0.674147i	−0.284522	+	0.958669i	$0.591835\pi$
$594$		0			0
$595$		0			0
$596$		−	7.66610e6i		−	0.884013i
$597$		0			0
$598$	−1.44651e6	+	1.44651e6i	−0.165413	+	0.165413i
$599$	−1.36531e7			−1.55476			−0.777379	−	0.629032i	$-0.783451\pi$
$599$	−1.36531e7			−1.55476			−0.777379	+	0.629032i	$0.783451\pi$
$600$		0			0
$601$	1.24801e7			1.40939			0.704696	−	0.709509i	$-0.251083\pi$
$601$	1.24801e7			1.40939			0.704696	+	0.709509i	$0.251083\pi$
$602$	−5.19102e6	+	5.19102e6i	−0.583797	+	0.583797i
$603$		0			0
$604$		−	2.90114e6i		−	0.323576i
$605$		0			0
$606$		0			0
$607$	7.28190e6	+	7.28190e6i	0.802182	+	0.802182i	0.983436	−	0.181255i	$-0.0580158\pi$
$607$	7.28190e6	+	7.28190e6i	0.802182	+	0.802182i	−0.181255	+	0.983436i	$0.558016\pi$
$608$	−2.03465e6	−	2.03465e6i	−0.223219	−	0.223219i
$609$		0			0
$610$		0			0
$611$			6.65574e6i			0.721262i
$612$		0			0
$613$	−4.08795e6	+	4.08795e6i	−0.439394	+	0.439394i	−0.891808	−	0.452414i	$-0.850563\pi$
$613$	−4.08795e6	+	4.08795e6i	−0.439394	+	0.439394i	0.452414	+	0.891808i	$0.350563\pi$
$614$	4.52715e6			0.484623
$615$		0			0
$616$	299880.			0.0318416
$617$	8.93170e6	−	8.93170e6i	0.944542	−	0.944542i	−0.0539990	−	0.998541i	$-0.517197\pi$
$617$	8.93170e6	−	8.93170e6i	0.944542	−	0.944542i	0.998541	+	0.0539990i	$0.0171968\pi$
$618$		0			0
$619$			1.35977e7i			1.42639i	0.700966	+	0.713195i	$0.252752\pi$
$619$			1.35977e7i			1.42639i	−0.700966	+	0.713195i	$0.747248\pi$
$620$		0			0
$621$		0			0
$622$	4.62042e6	+	4.62042e6i	0.478857	+	0.478857i
$623$	8.19310e6	+	8.19310e6i	0.845723	+	0.845723i
$624$		0			0
$625$		0			0
$626$		−	1.12876e7i		−	1.15124i
$627$		0			0
$628$	4.17604e6	−	4.17604e6i	0.422537	−	0.422537i
$629$	364133.			0.0366973
$630$		0			0
$631$	1.43251e7			1.43227			0.716133	−	0.697963i	$-0.245910\pi$
$631$	1.43251e7			1.43227			0.716133	+	0.697963i	$0.245910\pi$
$632$	3.58757e6	−	3.58757e6i	0.357279	−	0.357279i
$633$		0			0
$634$			3.04153e6i			0.300517i
$635$		0			0
$636$		0			0
$637$	3.01669e6	+	3.01669e6i	0.294566	+	0.294566i
$638$	−66189.0	−	66189.0i	−0.00643775	−	0.00643775i
$639$		0			0
$640$		0			0
$641$		−	8.36174e6i		−	0.803807i	−0.915682	−	0.401903i	$-0.868349\pi$
$641$		−	8.36174e6i		−	0.803807i	0.915682	−	0.401903i	$-0.131651\pi$
$642$		0			0
$643$	9.10919e6	−	9.10919e6i	0.868865	−	0.868865i	−0.123482	−	0.992347i	$-0.539406\pi$
$643$	9.10919e6	−	9.10919e6i	0.868865	−	0.868865i	0.992347	+	0.123482i	$0.0394061\pi$
$644$	−3.38504e6			−0.321624
$645$		0			0
$646$	−2.19110e6			−0.206577
$647$	6.96716e6	−	6.96716e6i	0.654328	−	0.654328i	−0.299704	−	0.954032i	$-0.596888\pi$
$647$	6.96716e6	−	6.96716e6i	0.654328	−	0.654328i	0.954032	+	0.299704i	$0.0968881\pi$
$648$		0			0
$649$		−	938584.i		−	0.0874704i
$650$		0			0
$651$		0			0
$652$	6.63718e6	+	6.63718e6i	0.611455	+	0.611455i
$653$	−2.40446e6	−	2.40446e6i	−0.220666	−	0.220666i	0.588113	−	0.808779i	$-0.299871\pi$
$653$	−2.40446e6	−	2.40446e6i	−0.220666	−	0.220666i	−0.808779	+	0.588113i	$0.799871\pi$
$654$		0			0
$655$		0			0
$656$		−	2.76264e6i		−	0.250649i
$657$		0			0
$658$	−7.78768e6	+	7.78768e6i	−0.701202	+	0.701202i
$659$	−1.11541e7			−1.00051			−0.500257	−	0.865877i	$-0.666761\pi$
$659$	−1.11541e7			−1.00051			−0.500257	+	0.865877i	$0.666761\pi$
$660$		0			0
$661$	8.37133e6			0.745231			0.372615	−	0.927986i	$-0.378461\pi$
$661$	8.37133e6			0.745231			0.372615	+	0.927986i	$0.378461\pi$
$662$	−87754.2	+	87754.2i	−0.00778257	+	0.00778257i
$663$		0			0
$664$		−	3.30440e6i		−	0.290852i
$665$		0			0
$666$		0			0
$667$	747141.	+	747141.i	0.0650261	+	0.0650261i
$668$	−4.51429e6	−	4.51429e6i	−0.391425	−	0.391425i
$669$		0			0
$670$		0			0
$671$			905197.i			0.0776134i
$672$		0			0
$673$	7.04166e6	−	7.04166e6i	0.599291	−	0.599291i	−0.340833	−	0.940124i	$-0.610709\pi$
$673$	7.04166e6	−	7.04166e6i	0.599291	−	0.599291i	0.940124	+	0.340833i	$0.110709\pi$
$674$	3.97362e6			0.336928
$675$		0			0
$676$	3.37354e6			0.283935
$677$	−7.32224e6	+	7.32224e6i	−0.614005	+	0.614005i	−0.943987	−	0.329982i	$-0.892957\pi$
$677$	−7.32224e6	+	7.32224e6i	−0.614005	+	0.614005i	0.329982	+	0.943987i	$0.392957\pi$
$678$		0			0
$679$			1.62590e7i			1.35338i
$680$		0			0
$681$		0			0
$682$	163930.	+	163930.i	0.0134958	+	0.0134958i
$683$	9.77498e6	+	9.77498e6i	0.801796	+	0.801796i	0.983376	−	0.181580i	$-0.0581211\pi$
$683$	9.77498e6	+	9.77498e6i	0.801796	+	0.801796i	−0.181580	+	0.983376i	$0.558121\pi$
$684$		0			0
$685$		0			0
$686$		−	4.08045e6i		−	0.331053i
$687$		0			0
$688$	2.00494e6	−	2.00494e6i	0.161484	−	0.161484i
$689$	1.27889e7			1.02632
$690$		0			0
$691$	2.14497e7			1.70894			0.854470	−	0.519501i	$-0.173882\pi$
$691$	2.14497e7			1.70894			0.854470	+	0.519501i	$0.173882\pi$
$692$	−8.25680e6	+	8.25680e6i	−0.655460	+	0.655460i
$693$		0			0
$694$			1.04584e7i			0.824266i
$695$		0			0
$696$		0			0
$697$	−1.48754e6	−	1.48754e6i	−0.115981	−	0.115981i
$698$	807398.	+	807398.i	0.0627262	+	0.0627262i
$699$		0			0
$700$		0			0
$701$			1.45950e7i			1.12179i	0.827888	+	0.560893i	$0.189542\pi$
$701$			1.45950e7i			1.12179i	−0.827888	+	0.560893i	$0.810458\pi$
$702$		0			0
$703$	3.71152e6	−	3.71152e6i	0.283246	−	0.283246i
$704$	−115823.			−0.00880771
$705$		0			0
$706$	−6.30639e6			−0.476178
$707$	1.91461e7	−	1.91461e7i	1.44056	−	1.44056i
$708$		0			0
$709$		−	1.88701e7i		−	1.40980i	−0.709306	−	0.704900i	$-0.750992\pi$
$709$		−	1.88701e7i		−	1.40980i	0.709306	−	0.704900i	$-0.249008\pi$
$710$		0			0
$711$		0			0
$712$	−3.16443e6	−	3.16443e6i	−0.233935	−	0.233935i
$713$	−1.85044e6	−	1.85044e6i	−0.136317	−	0.136317i
$714$		0			0
$715$		0			0
$716$		−	2.93467e6i		−	0.213933i
$717$		0			0
$718$	−3.08900e6	+	3.08900e6i	−0.223618	+	0.223618i
$719$	3.25833e6			0.235057			0.117528	−	0.993070i	$-0.462503\pi$
$719$	3.25833e6			0.235057			0.117528	+	0.993070i	$0.462503\pi$
$720$		0			0
$721$	−1.84641e7			−1.32279
$722$	−1.53299e7	+	1.53299e7i	−1.09445	+	1.09445i
$723$		0			0
$724$		−	5.32824e6i		−	0.377779i
$725$		0			0
$726$		0			0
$727$	−1.56667e7	−	1.56667e7i	−1.09936	−	1.09936i	−0.994485	−	0.104879i	$-0.966554\pi$
$727$	−1.56667e7	−	1.56667e7i	−1.09936	−	1.09936i	−0.104879	−	0.994485i	$-0.533446\pi$
$728$	−3.00375e6	−	3.00375e6i	−0.210056	−	0.210056i
$729$		0			0
$730$		0			0
$731$		−	2.15910e6i		−	0.149444i
$732$		0			0
$733$	−1.29191e7	+	1.29191e7i	−0.888121	+	0.888121i	−0.994343	−	0.106221i	$-0.966125\pi$
$733$	−1.29191e7	+	1.29191e7i	−0.888121	+	0.888121i	0.106221	+	0.994343i	$0.466125\pi$
$734$	−7.83487e6			−0.536774
$735$		0			0
$736$	1.30741e6			0.0889644
$737$	−775337.	+	775337.i	−0.0525801	+	0.0525801i
$738$		0			0
$739$		−	8.83812e6i		−	0.595318i	−0.954672	−	0.297659i	$-0.903794\pi$
$739$		−	8.83812e6i		−	0.595318i	0.954672	−	0.297659i	$-0.0962058\pi$
$740$		0			0
$741$		0			0
$742$	1.49639e7	+	1.49639e7i	0.997781	+	0.997781i
$743$	1.16046e7	+	1.16046e7i	0.771187	+	0.771187i	0.978314	−	0.207127i	$-0.0664115\pi$
$743$	1.16046e7	+	1.16046e7i	0.771187	+	0.771187i	−0.207127	+	0.978314i	$0.566411\pi$
$744$		0			0
$745$		0			0
$746$			1.31373e7i			0.864291i
$747$		0			0
$748$	−62364.5	+	62364.5i	−0.00407552	+	0.00407552i
$749$	1.40193e7			0.913110
$750$		0			0
$751$	−1.83362e7			−1.18634			−0.593171	−	0.805076i	$-0.702124\pi$
$751$	−1.83362e7			−1.18634			−0.593171	+	0.805076i	$0.702124\pi$
$752$	3.00785e6	−	3.00785e6i	0.193959	−	0.193959i
$753$		0			0
$754$			1.32596e6i			0.0849383i
$755$		0			0
$756$		0			0
$757$	2.21679e6	+	2.21679e6i	0.140600	+	0.140600i	0.773904	−	0.633303i	$-0.218301\pi$
$757$	2.21679e6	+	2.21679e6i	0.140600	+	0.140600i	−0.633303	+	0.773904i	$0.718301\pi$
$758$	−8.24984e6	−	8.24984e6i	−0.521522	−	0.521522i
$759$		0			0
$760$		0			0
$761$		−	1.85062e7i		−	1.15839i	−0.815188	−	0.579197i	$-0.803366\pi$
$761$		−	1.85062e7i		−	1.15839i	0.815188	−	0.579197i	$-0.196634\pi$
$762$		0			0
$763$	−426541.	+	426541.i	−0.0265246	+	0.0265246i
$764$	−1.39637e7			−0.865500
$765$		0			0
$766$	−8.28557e6			−0.510212
$767$	−9.40132e6	+	9.40132e6i	−0.577033	+	0.577033i
$768$		0			0
$769$		−	2.16451e7i		−	1.31991i	−0.751307	−	0.659953i	$-0.770576\pi$
$769$		−	2.16451e7i		−	1.31991i	0.751307	−	0.659953i	$-0.229424\pi$
$770$		0			0
$771$		0			0
$772$	3.17346e6	+	3.17346e6i	0.191642	+	0.191642i
$773$	−1.55356e7	−	1.55356e7i	−0.935143	−	0.935143i	0.0628778	−	0.998021i	$-0.479972\pi$
$773$	−1.55356e7	−	1.55356e7i	−0.935143	−	0.935143i	−0.998021	+	0.0628778i	$0.979972\pi$
$774$		0			0
$775$		0			0
$776$		−	6.27973e6i		−	0.374357i
$777$		0			0
$778$	−6.87484e6	+	6.87484e6i	−0.407205	+	0.407205i
$779$	−3.03242e7			−1.79038
$780$		0			0
$781$	1.54510e6			0.0906419
$782$	703969.	−	703969.i	0.0411658	−	0.0411658i
$783$		0			0
$784$		−	2.72660e6i		−	0.158428i
$785$		0			0
$786$		0			0
$787$	−9.00535e6	−	9.00535e6i	−0.518279	−	0.518279i	0.398771	−	0.917050i	$-0.369437\pi$
$787$	−9.00535e6	−	9.00535e6i	−0.518279	−	0.518279i	−0.917050	+	0.398771i	$0.869437\pi$
$788$	1.01293e6	+	1.01293e6i	0.0581117	+	0.0581117i
$789$		0			0
$790$		0			0
$791$			3.31190e7i			1.88207i
$792$		0			0
$793$	9.06691e6	−	9.06691e6i	0.512007	−	0.512007i
$794$	−2.82747e6			−0.159165
$795$		0			0
$796$	5.66490e6			0.316891
$797$	2.44170e7	−	2.44170e7i	1.36159	−	1.36159i	0.489697	−	0.871893i	$-0.337107\pi$
$797$	2.44170e7	−	2.44170e7i	1.36159	−	1.36159i	0.871893	−	0.489697i	$-0.162893\pi$
$798$		0			0
$799$		−	3.23913e6i		−	0.179499i
$800$		0			0
$801$		0			0
$802$	−1.64055e7	−	1.64055e7i	−0.900646	−	0.900646i
$803$	402960.	+	402960.i	0.0220533	+	0.0220533i
$804$		0			0
$805$		0			0
$806$		−	3.28401e6i		−	0.178060i
$807$		0			0
$808$	−7.39483e6	+	7.39483e6i	−0.398474	+	0.398474i
$809$	2.80955e7			1.50926			0.754632	−	0.656149i	$-0.227816\pi$
$809$	2.80955e7			1.50926			0.754632	+	0.656149i	$0.227816\pi$
$810$		0			0
$811$	−2.35077e7			−1.25504			−0.627522	−	0.778599i	$-0.715931\pi$
$811$	−2.35077e7			−1.25504			−0.627522	+	0.778599i	$0.715931\pi$
$812$	−1.55147e6	+	1.55147e6i	−0.0825760	+	0.0825760i
$813$		0			0
$814$		−	211279.i		−	0.0111762i
$815$		0			0
$816$		0			0
$817$	−2.20072e7	−	2.20072e7i	−1.15348	−	1.15348i
$818$	−1.77013e7	−	1.77013e7i	−0.924959	−	0.924959i
$819$		0			0
$820$		0			0
$821$			1.08516e7i			0.561870i	0.959727	+	0.280935i	$0.0906446\pi$
$821$			1.08516e7i			0.561870i	−0.959727	+	0.280935i	$0.909355\pi$
$822$		0			0
$823$	1.20680e7	−	1.20680e7i	0.621061	−	0.621061i	−0.324742	−	0.945803i	$-0.605277\pi$
$823$	1.20680e7	−	1.20680e7i	0.621061	−	0.621061i	0.945803	+	0.324742i	$0.105277\pi$
$824$	7.13143e6			0.365897
$825$		0			0
$826$	−2.20004e7			−1.12197
$827$	−2.12279e7	+	2.12279e7i	−1.07930	+	1.07930i	−0.0827289	+	0.996572i	$0.526364\pi$
$827$	−2.12279e7	+	2.12279e7i	−1.07930	+	1.07930i	−0.996572	+	0.0827289i	$0.973636\pi$
$828$		0			0
$829$		−	6.04829e6i		−	0.305665i	−0.988252	−	0.152833i	$-0.951160\pi$
$829$		−	6.04829e6i		−	0.305665i	0.988252	−	0.152833i	$-0.0488396\pi$
$830$		0			0
$831$		0			0
$832$	1.16014e6	+	1.16014e6i	0.0581035	+	0.0581035i
$833$	−1.46813e6	−	1.46813e6i	−0.0733080	−	0.0733080i
$834$		0			0
$835$		0			0
$836$			1.27133e6i			0.0629134i
$837$		0			0
$838$	5.37998e6	−	5.37998e6i	0.264649	−	0.264649i
$839$	1.15583e7			0.566879			0.283440	−	0.958990i	$-0.408524\pi$
$839$	1.15583e7			0.566879			0.283440	+	0.958990i	$0.408524\pi$
$840$		0			0
$841$	−1.98263e7			−0.966610
$842$	1.12223e6	−	1.12223e6i	0.0545507	−	0.0545507i
$843$		0			0
$844$			4.73013e6i			0.228569i
$845$		0			0
$846$		0			0
$847$	1.87767e7	+	1.87767e7i	0.899313	+	0.899313i
$848$	−5.77953e6	−	5.77953e6i	−0.275996	−	0.275996i
$849$		0			0
$850$		0			0
$851$			2.38492e6i			0.112888i
$852$		0			0
$853$	6.22871e6	−	6.22871e6i	0.293107	−	0.293107i	−0.545200	−	0.838306i	$-0.683546\pi$
$853$	6.22871e6	−	6.22871e6i	0.293107	−	0.293107i	0.838306	+	0.545200i	$0.183546\pi$
$854$	2.12178e7			0.995535
$855$		0			0
$856$	−5.41471e6			−0.252575
$857$	2.14608e6	−	2.14608e6i	0.0998145	−	0.0998145i	−0.655436	−	0.755251i	$-0.727515\pi$
$857$	2.14608e6	−	2.14608e6i	0.0998145	−	0.0998145i	0.755251	+	0.655436i	$0.227515\pi$
$858$		0			0
$859$			1.59281e7i			0.736514i	0.929724	+	0.368257i	$0.120045\pi$
$859$			1.59281e7i			0.736514i	−0.929724	+	0.368257i	$0.879955\pi$
$860$		0			0
$861$		0			0
$862$	−3.88879e6	−	3.88879e6i	−0.178257	−	0.178257i
$863$	−1.87460e7	−	1.87460e7i	−0.856806	−	0.856806i	0.134155	−	0.990960i	$-0.457168\pi$
$863$	−1.87460e7	−	1.87460e7i	−0.856806	−	0.856806i	−0.990960	+	0.134155i	$0.957168\pi$
$864$		0			0
$865$		0			0
$866$		−	2.04221e6i		−	0.0925351i
$867$		0			0
$868$	3.84252e6	−	3.84252e6i	0.173108	−	0.173108i
$869$	−2.24166e6			−0.100698
$870$		0			0
$871$	1.55323e7			0.693731
$872$	164743.	−	164743.i	0.00733697	−	0.00733697i
$873$		0			0
$874$		−	1.43508e7i		−	0.635473i
$875$		0			0
$876$		0			0
$877$	−1.91128e7	−	1.91128e7i	−0.839125	−	0.839125i	0.149619	−	0.988744i	$-0.452195\pi$
$877$	−1.91128e7	−	1.91128e7i	−0.839125	−	0.839125i	−0.988744	+	0.149619i	$0.952195\pi$
$878$	−2.38898e6	−	2.38898e6i	−0.104586	−	0.104586i
$879$		0			0
$880$		0			0
$881$			3.02404e7i			1.31265i	0.754479	+	0.656324i	$0.227890\pi$
$881$			3.02404e7i			1.31265i	−0.754479	+	0.656324i	$0.772110\pi$
$882$		0			0
$883$	2.28275e7	−	2.28275e7i	0.985272	−	0.985272i	−0.0146211	−	0.999893i	$-0.504654\pi$
$883$	2.28275e7	−	2.28275e7i	0.985272	−	0.985272i	0.999893	+	0.0146211i	$0.00465420\pi$
$884$	1.24935e6			0.0537716
$885$		0			0
$886$	5.11788e6			0.219031
$887$	4.94314e6	−	4.94314e6i	0.210957	−	0.210957i	−0.593717	−	0.804674i	$-0.702340\pi$
$887$	4.94314e6	−	4.94314e6i	0.210957	−	0.210957i	0.804674	+	0.593717i	$0.202340\pi$
$888$		0			0
$889$		−	2.34123e7i		−	0.993548i
$890$		0			0
$891$		0			0
$892$	1.15464e6	+	1.15464e6i	0.0485886	+	0.0485886i
$893$	−3.30157e7	−	3.30157e7i	−1.38545	−	1.38545i
$894$		0			0
$895$		0			0
$896$			2.71489e6i			0.112975i
$897$		0			0
$898$	2.15616e7	−	2.15616e7i	0.892259	−	0.892259i
$899$	−1.69623e6			−0.0699980
$900$		0			0
$901$	−6.22394e6			−0.255419
$902$	−863107.	+	863107.i	−0.0353222	+	0.0353222i
$903$		0			0
$904$		−	1.27916e7i		−	0.520599i
$905$		0			0
$906$		0			0
$907$	1.85920e7	+	1.85920e7i	0.750428	+	0.750428i	0.974559	−	0.224131i	$-0.0719544\pi$
$907$	1.85920e7	+	1.85920e7i	0.750428	+	0.750428i	−0.224131	+	0.974559i	$0.571954\pi$
$908$	1.07966e7	+	1.07966e7i	0.434584	+	0.434584i
$909$		0			0
$910$		0			0
$911$			6.12298e6i			0.244437i	0.992503	+	0.122219i	$0.0390009\pi$
$911$			6.12298e6i			0.244437i	−0.992503	+	0.122219i	$0.960999\pi$
$912$		0			0
$913$	−1.03236e6	+	1.03236e6i	−0.0409878	+	0.0409878i
$914$	−1.91679e7			−0.758941
$915$		0			0
$916$	−1.71605e7			−0.675760
$917$	2.22558e7	−	2.22558e7i	0.874018	−	0.874018i
$918$		0			0
$919$		−	2.02334e7i		−	0.790279i	−0.918621	−	0.395140i	$-0.870696\pi$
$919$		−	2.02334e7i		−	0.790279i	0.918621	−	0.395140i	$-0.129304\pi$
$920$		0			0
$921$		0			0
$922$	9.55349e6	+	9.55349e6i	0.370113	+	0.370113i
$923$	−1.54765e7	−	1.54765e7i	−0.597955	−	0.597955i
$924$		0			0
$925$		0			0
$926$		−	4.76295e6i		−	0.182536i
$927$		0			0
$928$	599227.	−	599227.i	0.0228413	−	0.0228413i
$929$	−1.16624e7			−0.443353			−0.221676	−	0.975120i	$-0.571153\pi$
$929$	−1.16624e7			−0.443353			−0.221676	+	0.975120i	$0.571153\pi$
$930$		0			0
$931$	−2.99285e7			−1.13165
$932$	1.06949e7	−	1.06949e7i	0.403308	−	0.403308i
$933$		0			0
$934$			4.38289e6i			0.164397i
$935$		0			0
$936$		0			0
$937$	3.00134e6	+	3.00134e6i	0.111678	+	0.111678i	0.760737	−	0.649060i	$-0.224837\pi$
$937$	3.00134e6	+	3.00134e6i	0.111678	+	0.111678i	−0.649060	+	0.760737i	$0.724837\pi$
$938$	1.81739e7	+	1.81739e7i	0.674437	+	0.674437i
$939$		0			0
$940$		0			0
$941$		−	3.96549e6i		−	0.145990i	−0.997332	−	0.0729949i	$-0.976744\pi$
$941$		−	3.96549e6i		−	0.145990i	0.997332	−	0.0729949i	$-0.0232557\pi$
$942$		0			0
$943$	9.74274e6	−	9.74274e6i	0.356781	−	0.356781i
$944$	8.49725e6			0.310348
$945$		0			0
$946$	−1.25277e6			−0.0455137
$947$	−1.84772e7	+	1.84772e7i	−0.669518	+	0.669518i	−0.957604	−	0.288087i	$-0.906981\pi$
$947$	−1.84772e7	+	1.84772e7i	−0.669518	+	0.669518i	0.288087	+	0.957604i	$0.406981\pi$
$948$		0			0
$949$		−	8.07249e6i		−	0.290966i
$950$		0			0
$951$		0			0
$952$	1.46183e6	+	1.46183e6i	0.0522761	+	0.0522761i
$953$	−1.18785e7	−	1.18785e7i	−0.423672	−	0.423672i	0.462794	−	0.886466i	$-0.346847\pi$
$953$	−1.18785e7	−	1.18785e7i	−0.423672	−	0.423672i	−0.886466	+	0.462794i	$0.846847\pi$
$954$		0			0
$955$		0			0
$956$			1.63984e7i			0.580306i
$957$		0			0
$958$	7.47297e6	−	7.47297e6i	0.263075	−	0.263075i
$959$	−6.24957e7			−2.19434
$960$		0			0
$961$	−2.44281e7			−0.853260
$962$	−2.11628e6	+	2.11628e6i	−0.0737285	+	0.0737285i
$963$		0			0
$964$			7.45313e6i			0.258313i
$965$		0			0
$966$		0			0
$967$	−3.78721e6	−	3.78721e6i	−0.130243	−	0.130243i	0.638980	−	0.769223i	$-0.279356\pi$
$967$	−3.78721e6	−	3.78721e6i	−0.130243	−	0.130243i	−0.769223	+	0.638980i	$0.779356\pi$
$968$	−7.25215e6	−	7.25215e6i	−0.248759	−	0.248759i
$969$		0			0
$970$		0			0
$971$			4.75911e6i			0.161986i	0.996715	+	0.0809930i	$0.0258092\pi$
$971$			4.75911e6i			0.161986i	−0.996715	+	0.0809930i	$0.974191\pi$
$972$		0			0
$973$	−2.84418e7	+	2.84418e7i	−0.963109	+	0.963109i
$974$	1.07907e6			0.0364461
$975$		0			0
$976$	−8.19499e6			−0.275375
$977$	1.13234e7	−	1.13234e7i	0.379525	−	0.379525i	−0.491406	−	0.870931i	$-0.663517\pi$
$977$	1.13234e7	−	1.13234e7i	0.379525	−	0.379525i	0.870931	+	0.491406i	$0.163517\pi$
$978$		0			0
$979$			1.97727e6i			0.0659339i
$980$		0			0
$981$		0			0
$982$	−1.89440e7	−	1.89440e7i	−0.626891	−	0.626891i
$983$	3.71041e7	+	3.71041e7i	1.22472	+	1.22472i	0.965933	+	0.258790i	$0.0833239\pi$
$983$	3.71041e7	+	3.71041e7i	1.22472	+	1.22472i	0.258790	+	0.965933i	$0.416676\pi$
$984$		0			0
$985$		0			0
$986$		−	645304.i		−	0.0211384i
$987$		0			0
$988$	1.27343e7	−	1.27343e7i	0.415033	−	0.415033i
$989$	1.41412e7			0.459722
$990$		0			0
$991$	−1.18623e6			−0.0383693			−0.0191847	−	0.999816i	$-0.506107\pi$
$991$	−1.18623e6			−0.0383693			−0.0191847	+	0.999816i	$0.506107\pi$
$992$	−1.48410e6	+	1.48410e6i	−0.0478833	+	0.0478833i
$993$		0			0
$994$		−	3.62172e7i		−	1.16265i
$995$		0			0
$996$		0			0
$997$	−1.48509e7	−	1.48509e7i	−0.473167	−	0.473167i	0.429771	−	0.902938i	$-0.358594\pi$
$997$	−1.48509e7	−	1.48509e7i	−0.473167	−	0.473167i	−0.902938	+	0.429771i	$0.858594\pi$
$998$	1.14477e7	+	1.14477e7i	0.363825	+	0.363825i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.f.f.107.8	yes	16
3.2	odd	2	inner	450.6.f.f.107.4	✓	16
5.2	odd	4		450.6.f.g.143.5	yes	16
5.3	odd	4	inner	450.6.f.f.143.4	yes	16
5.4	even	2		450.6.f.g.107.1	yes	16
15.2	even	4		450.6.f.g.143.1	yes	16
15.8	even	4	inner	450.6.f.f.143.8	yes	16
15.14	odd	2		450.6.f.g.107.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.6.f.f.107.4	✓	16	3.2	odd	2	inner
450.6.f.f.107.8	yes	16	1.1	even	1	trivial
450.6.f.f.143.4	yes	16	5.3	odd	4	inner
450.6.f.f.143.8	yes	16	15.8	even	4	inner
450.6.f.g.107.1	yes	16	5.4	even	2
450.6.f.g.107.5	yes	16	15.14	odd	2
450.6.f.g.143.1	yes	16	15.2	even	4
450.6.f.g.143.5	yes	16	5.2	odd	4

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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(117.170 + 117.170i) q^{7} +(-45.2548 - 45.2548i) q^{8} +O(q^{10})\)	\(q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(117.170 + 117.170i) q^{7} +(-45.2548 - 45.2548i) q^{8} +28.2771i q^{11} +(283.237 - 283.237i) q^{13} +662.815 q^{14} -256.000 q^{16} +(-137.842 + 137.842i) q^{17} +2809.99i q^{19} +(79.9797 + 79.9797i) q^{22} +(-902.809 - 902.809i) q^{23} -1602.23i q^{26} +(1874.73 - 1874.73i) q^{28} -827.573 q^{29} +2049.64 q^{31} +(-724.077 + 724.077i) q^{32} +779.754i q^{34} +(-1320.83 - 1320.83i) q^{37} +(7947.85 + 7947.85i) q^{38} +10791.6i q^{41} +(-7831.78 + 7831.78i) q^{43} +452.433 q^{44} -5107.06 q^{46} +(-11749.4 + 11749.4i) q^{47} +10650.8i q^{49} +(-4531.80 - 4531.80i) q^{52} +(22576.3 + 22576.3i) q^{53} -10605.0i q^{56} +(-2340.73 + 2340.73i) q^{58} -33192.4 q^{59} +32011.7 q^{61} +(5797.27 - 5797.27i) q^{62} +4096.00i q^{64} +(27419.3 + 27419.3i) q^{67} +(2205.48 + 2205.48i) q^{68} -54641.5i q^{71} +(14250.4 - 14250.4i) q^{73} -7471.75 q^{74} +44959.8 q^{76} +(-3313.23 + 3313.23i) q^{77} +79274.9i q^{79} +(30523.2 + 30523.2i) q^{82} +(36508.8 + 36508.8i) q^{83} +44303.2i q^{86} +(1279.67 - 1279.67i) q^{88} +69924.7 q^{89} +66374.0 q^{91} +(-14444.9 + 14444.9i) q^{92} +66464.6i q^{94} +(69381.8 + 69381.8i) q^{97} +(30124.9 + 30124.9i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 528 q^{7}+O(q^{10}) \) 16 * q - 528 * q^7	\( 16 q - 528 q^{7} + 192 q^{13} - 4096 q^{16} - 2688 q^{22} - 8448 q^{28} + 13024 q^{31} - 47328 q^{37} - 55440 q^{43} + 44544 q^{46} - 3072 q^{52} - 101184 q^{58} + 28400 q^{61} + 242256 q^{67} - 430944 q^{73} - 7168 q^{76} + 158208 q^{82} - 43008 q^{88} - 185472 q^{91} + 457152 q^{97}+O(q^{100}) \) 16 * q - 528 * q^7 + 192 * q^13 - 4096 * q^16 - 2688 * q^22 - 8448 * q^28 + 13024 * q^31 - 47328 * q^37 - 55440 * q^43 + 44544 * q^46 - 3072 * q^52 - 101184 * q^58 + 28400 * q^61 + 242256 * q^67 - 430944 * q^73 - 7168 * q^76 + 158208 * q^82 - 43008 * q^88 - 185472 * q^91 + 457152 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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