LMFDB - Embedded newform 450.5.g.f.343.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [450,5,Mod(307,450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("450.307"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 5, names="a")

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

Newform invariants

comment:select newform

sage:traces = [4,-8,0,0,0,0,28,64,0,0,-464,0,336,0,0,-256,392] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$46.5164833877$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			343.2
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	450.343
Dual form			450.5.g.f.307.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.00000 + 2.00000i) q^{2} -8.00000i q^{4} +(58.4393 - 58.4393i) q^{7} +(16.0000 + 16.0000i) q^{8} -64.5607 q^{11} +(-55.6209 - 55.6209i) q^{13} +233.757i q^{14} -64.0000 q^{16} +(267.015 - 267.015i) q^{17} -580.211i q^{19} +(129.121 - 129.121i) q^{22} +(396.318 + 396.318i) q^{23} +222.484 q^{26} +(-467.514 - 467.514i) q^{28} +291.832i q^{29} -683.787 q^{31} +(128.000 - 128.000i) q^{32} +1068.06i q^{34} +(-1335.59 + 1335.59i) q^{37} +(1160.42 + 1160.42i) q^{38} -1582.39 q^{41} +(460.393 + 460.393i) q^{43} +516.486i q^{44} -1585.27 q^{46} +(-1893.23 + 1893.23i) q^{47} -4429.30i q^{49} +(-444.967 + 444.967i) q^{52} +(-761.364 - 761.364i) q^{53} +1870.06 q^{56} +(-583.664 - 583.664i) q^{58} -2752.50i q^{59} +1937.66 q^{61} +(1367.57 - 1367.57i) q^{62} +512.000i q^{64} +(3956.99 - 3956.99i) q^{67} +(-2136.12 - 2136.12i) q^{68} -3692.97 q^{71} +(-3396.76 - 3396.76i) q^{73} -5342.36i q^{74} -4641.69 q^{76} +(-3772.88 + 3772.88i) q^{77} -757.307i q^{79} +(3164.78 - 3164.78i) q^{82} +(-4279.98 - 4279.98i) q^{83} -1841.57 q^{86} +(-1032.97 - 1032.97i) q^{88} +7195.62i q^{89} -6500.89 q^{91} +(3170.54 - 3170.54i) q^{92} -7572.91i q^{94} +(9513.12 - 9513.12i) q^{97} +(8858.60 + 8858.60i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 28 q^{7} + 64 q^{8} - 464 q^{11} + 336 q^{13} - 256 q^{16} + 392 q^{17} + 928 q^{22} + 968 q^{23} - 1344 q^{26} - 224 q^{28} - 560 q^{31} + 512 q^{32} - 2256 q^{37} + 1232 q^{38} - 392 q^{41}+ \cdots + 23912 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 + 28 * q^7 + 64 * q^8 - 464 * q^11 + 336 * q^13 - 256 * q^16 + 392 * q^17 + 928 * q^22 + 968 * q^23 - 1344 * q^26 - 224 * q^28 - 560 * q^31 + 512 * q^32 - 2256 * q^37 + 1232 * q^38 - 392 * q^41 - 216 * q^43 - 3872 * q^46 - 9072 * q^47 + 2688 * q^52 - 4280 * q^53 + 896 * q^56 + 2192 * q^58 - 4536 * q^61 + 1120 * q^62 + 2248 * q^67 - 3136 * q^68 - 18064 * q^71 - 20524 * q^73 - 4928 * q^76 + 7336 * q^77 + 784 * q^82 - 336 * q^83 + 864 * q^86 - 7424 * q^88 - 52752 * q^91 + 7744 * q^92 + 40404 * q^97 + 23912 * q^98

Character values

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

$n$	$101$	$127$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	−2.00000	+	2.00000i	−0.500000	+	0.500000i
$2$	−2.00000	+	2.00000i	−0.500000	+	0.500000i
$3$		0			0
$3$		0			0
$4$		−	8.00000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	58.4393	−	58.4393i	1.19264	−	1.19264i	0.216315	−	0.976324i	$-0.430596\pi$
$7$	58.4393	−	58.4393i	1.19264	−	1.19264i	0.976324	−	0.216315i	$-0.0694037\pi$
$8$	16.0000	+	16.0000i	0.250000	+	0.250000i
$9$		0			0
$10$		0			0
$11$	−64.5607			−0.533560			−0.266780	−	0.963758i	$-0.585960\pi$
$11$	−64.5607			−0.533560			−0.266780	+	0.963758i	$0.585960\pi$
$12$		0			0
$13$	−55.6209	−	55.6209i	−0.329118	−	0.329118i	0.523133	−	0.852251i	$-0.324763\pi$
$13$	−55.6209	−	55.6209i	−0.329118	−	0.329118i	−0.852251	+	0.523133i	$0.824763\pi$
$14$			233.757i			1.19264i
$15$		0			0
$16$	−64.0000			−0.250000
$17$	267.015	−	267.015i	0.923927	−	0.923927i	−0.0733776	−	0.997304i	$-0.523378\pi$
$17$	267.015	−	267.015i	0.923927	−	0.923927i	0.997304	+	0.0733776i	$0.0233778\pi$
$18$		0			0
$19$		−	580.211i		−	1.60723i	−0.595147	−	0.803617i	$-0.702906\pi$
$19$		−	580.211i		−	1.60723i	0.595147	−	0.803617i	$-0.297094\pi$
$20$		0			0
$21$		0			0
$22$	129.121	−	129.121i	0.266780	−	0.266780i
$23$	396.318	+	396.318i	0.749183	+	0.749183i	0.974326	−	0.225143i	$-0.0722848\pi$
$23$	396.318	+	396.318i	0.749183	+	0.749183i	−0.225143	+	0.974326i	$0.572285\pi$
$24$		0			0
$25$		0			0
$26$	222.484			0.329118
$27$		0			0
$28$	−467.514	−	467.514i	−0.596319	−	0.596319i
$29$			291.832i			0.347006i	0.984833	+	0.173503i	$0.0555086\pi$
$29$			291.832i			0.347006i	−0.984833	+	0.173503i	$0.944491\pi$
$30$		0			0
$31$	−683.787			−0.711537			−0.355768	−	0.934574i	$-0.615781\pi$
$31$	−683.787			−0.711537			−0.355768	+	0.934574i	$0.615781\pi$
$32$	128.000	−	128.000i	0.125000	−	0.125000i
$33$		0			0
$34$			1068.06i			0.923927i
$35$		0			0
$36$		0			0
$37$	−1335.59	+	1335.59i	−0.975595	+	0.975595i	−0.999709	−	0.0241144i	$-0.992323\pi$
$37$	−1335.59	+	1335.59i	−0.975595	+	0.975595i	0.0241144	+	0.999709i	$0.492323\pi$
$38$	1160.42	+	1160.42i	0.803617	+	0.803617i
$39$		0			0
$40$		0			0
$41$	−1582.39			−0.941339			−0.470669	−	0.882310i	$-0.655988\pi$
$41$	−1582.39			−0.941339			−0.470669	+	0.882310i	$0.655988\pi$
$42$		0			0
$43$	460.393	+	460.393i	0.248996	+	0.248996i	0.820558	−	0.571563i	$-0.193663\pi$
$43$	460.393	+	460.393i	0.248996	+	0.248996i	−0.571563	+	0.820558i	$0.693663\pi$
$44$			516.486i			0.266780i
$45$		0			0
$46$	−1585.27			−0.749183
$47$	−1893.23	+	1893.23i	−0.857052	+	0.857052i	−0.990990	−	0.133938i	$-0.957238\pi$
$47$	−1893.23	+	1893.23i	−0.857052	+	0.857052i	0.133938	+	0.990990i	$0.457238\pi$
$48$		0			0
$49$		−	4429.30i		−	1.84477i
$50$		0			0
$51$		0			0
$52$	−444.967	+	444.967i	−0.164559	+	0.164559i
$53$	−761.364	−	761.364i	−0.271045	−	0.271045i	0.558476	−	0.829521i	$-0.311386\pi$
$53$	−761.364	−	761.364i	−0.271045	−	0.271045i	−0.829521	+	0.558476i	$0.811386\pi$
$54$		0			0
$55$		0			0
$56$	1870.06			0.596319
$57$		0			0
$58$	−583.664	−	583.664i	−0.173503	−	0.173503i
$59$		−	2752.50i		−	0.790720i	−0.918526	−	0.395360i	$-0.870620\pi$
$59$		−	2752.50i		−	0.790720i	0.918526	−	0.395360i	$-0.129380\pi$
$60$		0			0
$61$	1937.66			0.520736			0.260368	−	0.965509i	$-0.416156\pi$
$61$	1937.66			0.520736			0.260368	+	0.965509i	$0.416156\pi$
$62$	1367.57	−	1367.57i	0.355768	−	0.355768i
$63$		0			0
$64$			512.000i			0.125000i
$65$		0			0
$66$		0			0
$67$	3956.99	−	3956.99i	0.881486	−	0.881486i	−0.112199	−	0.993686i	$-0.535789\pi$
$67$	3956.99	−	3956.99i	0.881486	−	0.881486i	0.993686	+	0.112199i	$0.0357895\pi$
$68$	−2136.12	−	2136.12i	−0.461963	−	0.461963i
$69$		0			0
$70$		0			0
$71$	−3692.97			−0.732587			−0.366294	−	0.930499i	$-0.619373\pi$
$71$	−3692.97			−0.732587			−0.366294	+	0.930499i	$0.619373\pi$
$72$		0			0
$73$	−3396.76	−	3396.76i	−0.637411	−	0.637411i	0.312505	−	0.949916i	$-0.398832\pi$
$73$	−3396.76	−	3396.76i	−0.637411	−	0.637411i	−0.949916	+	0.312505i	$0.898832\pi$
$74$		−	5342.36i		−	0.975595i
$75$		0			0
$76$	−4641.69			−0.803617
$77$	−3772.88	+	3772.88i	−0.636344	+	0.636344i
$78$		0			0
$79$		−	757.307i		−	0.121344i	−0.998158	−	0.0606719i	$-0.980676\pi$
$79$		−	757.307i		−	0.121344i	0.998158	−	0.0606719i	$-0.0193243\pi$
$80$		0			0
$81$		0			0
$82$	3164.78	−	3164.78i	0.470669	−	0.470669i
$83$	−4279.98	−	4279.98i	−0.621277	−	0.621277i	0.324581	−	0.945858i	$-0.394777\pi$
$83$	−4279.98	−	4279.98i	−0.621277	−	0.621277i	−0.945858	+	0.324581i	$0.894777\pi$
$84$		0			0
$85$		0			0
$86$	−1841.57			−0.248996
$87$		0			0
$88$	−1032.97	−	1032.97i	−0.133390	−	0.133390i
$89$			7195.62i			0.908423i	0.890894	+	0.454211i	$0.150079\pi$
$89$			7195.62i			0.908423i	−0.890894	+	0.454211i	$0.849921\pi$
$90$		0			0
$91$	−6500.89			−0.785037
$92$	3170.54	−	3170.54i	0.374592	−	0.374592i
$93$		0			0
$94$		−	7572.91i		−	0.857052i
$95$		0			0
$96$		0			0
$97$	9513.12	−	9513.12i	1.01107	−	1.01107i	0.0111282	−	0.999938i	$-0.496458\pi$
$97$	9513.12	−	9513.12i	1.01107	−	1.01107i	0.999938	−	0.0111282i	$-0.00354228\pi$
$98$	8858.60	+	8858.60i	0.922386	+	0.922386i
$99$		0			0
$100$		0			0
$101$	−10303.1			−1.01001			−0.505004	−	0.863117i	$-0.668509\pi$
$101$	−10303.1			−1.01001			−0.505004	+	0.863117i	$0.668509\pi$
$102$		0			0
$103$	12187.7	+	12187.7i	1.14881	+	1.14881i	0.986787	+	0.162024i	$0.0518021\pi$
$103$	12187.7	+	12187.7i	1.14881	+	1.14881i	0.162024	+	0.986787i	$0.448198\pi$
$104$		−	1779.87i		−	0.164559i
$105$		0			0
$106$	3045.46			0.271045
$107$	13641.0	−	13641.0i	1.19146	−	1.19146i	0.214797	−	0.976659i	$-0.431091\pi$
$107$	13641.0	−	13641.0i	1.19146	−	1.19146i	0.976659	−	0.214797i	$-0.0689089\pi$
$108$		0			0
$109$		−	2560.46i		−	0.215509i	−0.994178	−	0.107755i	$-0.965634\pi$
$109$		−	2560.46i		−	0.215509i	0.994178	−	0.107755i	$-0.0343661\pi$
$110$		0			0
$111$		0			0
$112$	−3740.11	+	3740.11i	−0.298160	+	0.298160i
$113$	−15148.6	−	15148.6i	−1.18636	−	1.18636i	−0.978067	−	0.208291i	$-0.933210\pi$
$113$	−15148.6	−	15148.6i	−1.18636	−	1.18636i	−0.208291	−	0.978067i	$-0.566790\pi$
$114$		0			0
$115$		0			0
$116$	2334.66			0.173503
$117$		0			0
$118$	5504.99	+	5504.99i	0.395360	+	0.395360i
$119$		−	31208.3i		−	2.20382i
$120$		0			0
$121$	−10472.9			−0.715314
$122$	−3875.32	+	3875.32i	−0.260368	+	0.260368i
$123$		0			0
$124$			5470.29i			0.355768i
$125$		0			0
$126$		0			0
$127$	3033.54	−	3033.54i	0.188080	−	0.188080i	−0.606786	−	0.794865i	$-0.707541\pi$
$127$	3033.54	−	3033.54i	0.188080	−	0.188080i	0.794865	+	0.606786i	$0.207541\pi$
$128$	−1024.00	−	1024.00i	−0.0625000	−	0.0625000i
$129$		0			0
$130$		0			0
$131$	−1726.38			−0.100599			−0.0502994	−	0.998734i	$-0.516018\pi$
$131$	−1726.38			−0.100599			−0.0502994	+	0.998734i	$0.516018\pi$
$132$		0			0
$133$	−33907.1	−	33907.1i	−1.91685	−	1.91685i
$134$			15828.0i			0.881486i
$135$		0			0
$136$	8544.47			0.461963
$137$	476.561	−	476.561i	0.0253908	−	0.0253908i	−0.694297	−	0.719688i	$-0.744285\pi$
$137$	476.561	−	476.561i	0.0253908	−	0.0253908i	0.719688	+	0.694297i	$0.244285\pi$
$138$		0			0
$139$		−	26869.8i		−	1.39070i	−0.718670	−	0.695351i	$-0.755249\pi$
$139$		−	26869.8i		−	1.39070i	0.718670	−	0.695351i	$-0.244751\pi$
$140$		0			0
$141$		0			0
$142$	7385.94	−	7385.94i	0.366294	−	0.366294i
$143$	3590.93	+	3590.93i	0.175604	+	0.175604i
$144$		0			0
$145$		0			0
$146$	13587.0			0.637411
$147$		0			0
$148$	10684.7	+	10684.7i	0.487797	+	0.487797i
$149$		−	16444.1i		−	0.740690i	−0.928894	−	0.370345i	$-0.879239\pi$
$149$		−	16444.1i		−	0.740690i	0.928894	−	0.370345i	$-0.120761\pi$
$150$		0			0
$151$	10400.6			0.456146			0.228073	−	0.973644i	$-0.426758\pi$
$151$	10400.6			0.456146			0.228073	+	0.973644i	$0.426758\pi$
$152$	9283.38	−	9283.38i	0.401808	−	0.401808i
$153$		0			0
$154$		−	15091.5i		−	0.636344i
$155$		0			0
$156$		0			0
$157$	−9435.15	+	9435.15i	−0.382780	+	0.382780i	−0.872103	−	0.489323i	$-0.837244\pi$
$157$	−9435.15	+	9435.15i	−0.382780	+	0.382780i	0.489323	+	0.872103i	$0.337244\pi$
$158$	1514.61	+	1514.61i	0.0606719	+	0.0606719i
$159$		0			0
$160$		0			0
$161$	46321.1			1.78701
$162$		0			0
$163$	−21947.0	−	21947.0i	−0.826038	−	0.826038i	0.160928	−	0.986966i	$-0.448551\pi$
$163$	−21947.0	−	21947.0i	−0.826038	−	0.826038i	−0.986966	+	0.160928i	$0.948551\pi$
$164$			12659.1i			0.470669i
$165$		0			0
$166$	17119.9			0.621277
$167$	1963.25	−	1963.25i	0.0703950	−	0.0703950i	−0.671033	−	0.741428i	$-0.734149\pi$
$167$	1963.25	−	1963.25i	0.0703950	−	0.0703950i	0.741428	+	0.671033i	$0.234149\pi$
$168$		0			0
$169$		−	22373.6i		−	0.783363i
$170$		0			0
$171$		0			0
$172$	3683.14	−	3683.14i	0.124498	−	0.124498i
$173$	3854.25	+	3854.25i	0.128780	+	0.128780i	0.768559	−	0.639779i	$-0.220974\pi$
$173$	3854.25	+	3854.25i	0.128780	+	0.128780i	−0.639779	+	0.768559i	$0.720974\pi$
$174$		0			0
$175$		0			0
$176$	4131.89			0.133390
$177$		0			0
$178$	−14391.2	−	14391.2i	−0.454211	−	0.454211i
$179$			908.189i			0.0283446i	0.999900	+	0.0141723i	$0.00451133\pi$
$179$			908.189i			0.0283446i	−0.999900	+	0.0141723i	$0.995489\pi$
$180$		0			0
$181$	−31189.9			−0.952045			−0.476022	−	0.879433i	$-0.657922\pi$
$181$	−31189.9			−0.952045			−0.476022	+	0.879433i	$0.657922\pi$
$182$	13001.8	−	13001.8i	0.392519	−	0.392519i
$183$		0			0
$184$			12682.2i			0.374592i
$185$		0			0
$186$		0			0
$187$	−17238.7	+	17238.7i	−0.492970	+	0.492970i
$188$	15145.8	+	15145.8i	0.428526	+	0.428526i
$189$		0			0
$190$		0			0
$191$	−19579.0			−0.536691			−0.268345	−	0.963323i	$-0.586477\pi$
$191$	−19579.0			−0.536691			−0.268345	+	0.963323i	$0.586477\pi$
$192$		0			0
$193$	17164.9	+	17164.9i	0.460815	+	0.460815i	0.898922	−	0.438108i	$-0.144351\pi$
$193$	17164.9	+	17164.9i	0.460815	+	0.460815i	−0.438108	+	0.898922i	$0.644351\pi$
$194$			38052.5i			1.01107i
$195$		0			0
$196$	−35434.4			−0.922386
$197$	−33382.9	+	33382.9i	−0.860185	+	0.860185i	−0.991359	−	0.131175i	$-0.958125\pi$
$197$	−33382.9	+	33382.9i	−0.860185	+	0.860185i	0.131175	+	0.991359i	$0.458125\pi$
$198$		0			0
$199$			33728.5i			0.851708i	0.904792	+	0.425854i	$0.140026\pi$
$199$			33728.5i			0.851708i	−0.904792	+	0.425854i	$0.859974\pi$
$200$		0			0
$201$		0			0
$202$	20606.2	−	20606.2i	0.505004	−	0.505004i
$203$	17054.5	+	17054.5i	0.413853	+	0.413853i
$204$		0			0
$205$		0			0
$206$	−48750.9			−1.14881
$207$		0			0
$208$	3559.74	+	3559.74i	0.0822795	+	0.0822795i
$209$			37458.9i			0.857555i
$210$		0			0
$211$	−52018.2			−1.16840			−0.584198	−	0.811611i	$-0.698591\pi$
$211$	−52018.2			−1.16840			−0.584198	+	0.811611i	$0.698591\pi$
$212$	−6090.91	+	6090.91i	−0.135522	+	0.135522i
$213$		0			0
$214$			54563.9i			1.19146i
$215$		0			0
$216$		0			0
$217$	−39960.0	+	39960.0i	−0.848606	+	0.848606i
$218$	5120.93	+	5120.93i	0.107755	+	0.107755i
$219$		0			0
$220$		0			0
$221$	−29703.2			−0.608161
$222$		0			0
$223$	47003.8	+	47003.8i	0.945198	+	0.945198i	0.998574	−	0.0533764i	$-0.0169983\pi$
$223$	47003.8	+	47003.8i	0.945198	+	0.945198i	−0.0533764	+	0.998574i	$0.516998\pi$
$224$		−	14960.5i		−	0.298160i
$225$		0			0
$226$	60594.4			1.18636
$227$	−4524.24	+	4524.24i	−0.0877998	+	0.0877998i	−0.749643	−	0.661843i	$-0.769775\pi$
$227$	−4524.24	+	4524.24i	−0.0877998	+	0.0877998i	0.661843	+	0.749643i	$0.269775\pi$
$228$		0			0
$229$			25303.6i			0.482516i	0.970461	+	0.241258i	$0.0775600\pi$
$229$			25303.6i			0.482516i	−0.970461	+	0.241258i	$0.922440\pi$
$230$		0			0
$231$		0			0
$232$	−4669.31	+	4669.31i	−0.0867515	+	0.0867515i
$233$	−35292.2	−	35292.2i	−0.650081	−	0.650081i	0.302931	−	0.953012i	$-0.402035\pi$
$233$	−35292.2	−	35292.2i	−0.650081	−	0.650081i	−0.953012	+	0.302931i	$0.902035\pi$
$234$		0			0
$235$		0			0
$236$	−22020.0			−0.395360
$237$		0			0
$238$	62416.6	+	62416.6i	1.10191	+	1.10191i
$239$		−	9835.76i		−	0.172192i	−0.996287	−	0.0860958i	$-0.972561\pi$
$239$		−	9835.76i		−	0.172192i	0.996287	−	0.0860958i	$-0.0274391\pi$
$240$		0			0
$241$	115888.			1.99529			0.997643	−	0.0686212i	$-0.0218600\pi$
$241$	115888.			1.99529			0.997643	+	0.0686212i	$0.0218600\pi$
$242$	20945.8	−	20945.8i	0.357657	−	0.357657i
$243$		0			0
$244$		−	15501.3i		−	0.260368i
$245$		0			0
$246$		0			0
$247$	−32271.9	+	32271.9i	−0.528969	+	0.528969i
$248$	−10940.6	−	10940.6i	−0.177884	−	0.177884i
$249$		0			0
$250$		0			0
$251$	−58915.8			−0.935156			−0.467578	−	0.883952i	$-0.654873\pi$
$251$	−58915.8			−0.935156			−0.467578	+	0.883952i	$0.654873\pi$
$252$		0			0
$253$	−25586.6	−	25586.6i	−0.399734	−	0.399734i
$254$			12134.2i			0.188080i
$255$		0			0
$256$	4096.00			0.0625000
$257$	−42257.6	+	42257.6i	−0.639791	+	0.639791i	−0.950504	−	0.310713i	$-0.899432\pi$
$257$	−42257.6	+	42257.6i	−0.639791	+	0.639791i	0.310713	+	0.950504i	$0.399432\pi$
$258$		0			0
$259$			156102.i			2.32706i
$260$		0			0
$261$		0			0
$262$	3452.75	−	3452.75i	0.0502994	−	0.0502994i
$263$	−33247.6	−	33247.6i	−0.480673	−	0.480673i	0.424674	−	0.905346i	$-0.360389\pi$
$263$	−33247.6	−	33247.6i	−0.480673	−	0.480673i	−0.905346	+	0.424674i	$0.860389\pi$
$264$		0			0
$265$		0			0
$266$	135629.			1.91685
$267$		0			0
$268$	−31655.9	−	31655.9i	−0.440743	−	0.440743i
$269$		−	53427.3i		−	0.738343i	−0.929361	−	0.369172i	$-0.879641\pi$
$269$		−	53427.3i		−	0.738343i	0.929361	−	0.369172i	$-0.120359\pi$
$270$		0			0
$271$	34292.5			0.466939			0.233470	−	0.972364i	$-0.424992\pi$
$271$	34292.5			0.466939			0.233470	+	0.972364i	$0.424992\pi$
$272$	−17088.9	+	17088.9i	−0.230982	+	0.230982i
$273$		0			0
$274$			1906.24i			0.0253908i
$275$		0			0
$276$		0			0
$277$	64926.7	−	64926.7i	0.846182	−	0.846182i	−0.143472	−	0.989654i	$-0.545827\pi$
$277$	64926.7	−	64926.7i	0.846182	−	0.846182i	0.989654	+	0.143472i	$0.0458268\pi$
$278$	53739.5	+	53739.5i	0.695351	+	0.695351i
$279$		0			0
$280$		0			0
$281$	97350.7			1.23290			0.616448	−	0.787396i	$-0.288571\pi$
$281$	97350.7			1.23290			0.616448	+	0.787396i	$0.288571\pi$
$282$		0			0
$283$	39561.0	+	39561.0i	0.493962	+	0.493962i	0.909552	−	0.415590i	$-0.136425\pi$
$283$	39561.0	+	39561.0i	0.493962	+	0.493962i	−0.415590	+	0.909552i	$0.636425\pi$
$284$			29543.8i			0.366294i
$285$		0			0
$286$	−14363.7			−0.175604
$287$	−92473.8	+	92473.8i	−1.12268	+	1.12268i
$288$		0			0
$289$		−	59072.8i		−	0.707281i
$290$		0			0
$291$		0			0
$292$	−27174.1	+	27174.1i	−0.318705	+	0.318705i
$293$	41217.2	+	41217.2i	0.480112	+	0.480112i	0.905167	−	0.425055i	$-0.139745\pi$
$293$	41217.2	+	41217.2i	0.480112	+	0.480112i	−0.425055	+	0.905167i	$0.639745\pi$
$294$		0			0
$295$		0			0
$296$	−42738.9			−0.487797
$297$		0			0
$298$	32888.1	+	32888.1i	0.370345	+	0.370345i
$299$		−	44087.1i		−	0.493139i
$300$		0			0
$301$	53810.1			0.593923
$302$	−20801.2	+	20801.2i	−0.228073	+	0.228073i
$303$		0			0
$304$			37133.5i			0.401808i
$305$		0			0
$306$		0			0
$307$	−47762.9	+	47762.9i	−0.506773	+	0.506773i	−0.913534	−	0.406761i	$-0.866658\pi$
$307$	−47762.9	+	47762.9i	−0.506773	+	0.506773i	0.406761	+	0.913534i	$0.366658\pi$
$308$	30183.1	+	30183.1i	0.318172	+	0.318172i
$309$		0			0
$310$		0			0
$311$	−72396.5			−0.748508			−0.374254	−	0.927326i	$-0.622101\pi$
$311$	−72396.5			−0.748508			−0.374254	+	0.927326i	$0.622101\pi$
$312$		0			0
$313$	54859.6	+	54859.6i	0.559969	+	0.559969i	0.929298	−	0.369330i	$-0.120413\pi$
$313$	54859.6	+	54859.6i	0.559969	+	0.559969i	−0.369330	+	0.929298i	$0.620413\pi$
$314$		−	37740.6i		−	0.382780i
$315$		0			0
$316$	−6058.46			−0.0606719
$317$	75699.9	−	75699.9i	0.753315	−	0.753315i	−0.221781	−	0.975096i	$-0.571187\pi$
$317$	75699.9	−	75699.9i	0.753315	−	0.753315i	0.975096	+	0.221781i	$0.0711872\pi$
$318$		0			0
$319$		−	18840.9i		−	0.185148i
$320$		0			0
$321$		0			0
$322$	−92642.1	+	92642.1i	−0.893505	+	0.893505i
$323$	−154925.	−	154925.i	−1.48497	−	1.48497i
$324$		0			0
$325$		0			0
$326$	87788.0			0.826038
$327$		0			0
$328$	−25318.3	−	25318.3i	−0.235335	−	0.235335i
$329$			221278.i			2.04431i
$330$		0			0
$331$	25992.7			0.237244			0.118622	−	0.992939i	$-0.462152\pi$
$331$	25992.7			0.237244			0.118622	+	0.992939i	$0.462152\pi$
$332$	−34239.8	+	34239.8i	−0.310638	+	0.310638i
$333$		0			0
$334$			7852.98i			0.0703950i
$335$		0			0
$336$		0			0
$337$	80677.7	−	80677.7i	0.710385	−	0.710385i	−0.256231	−	0.966616i	$-0.582481\pi$
$337$	80677.7	−	80677.7i	0.710385	−	0.710385i	0.966616	+	0.256231i	$0.0824808\pi$
$338$	44747.3	+	44747.3i	0.391681	+	0.391681i
$339$		0			0
$340$		0			0
$341$	44145.8			0.379647
$342$		0			0
$343$	−118532.	−	118532.i	−1.00751	−	1.00751i
$344$			14732.6i			0.124498i
$345$		0			0
$346$	−15417.0			−0.128780
$347$	47528.3	−	47528.3i	0.394724	−	0.394724i	−0.481643	−	0.876367i	$-0.659960\pi$
$347$	47528.3	−	47528.3i	0.394724	−	0.394724i	0.876367	+	0.481643i	$0.159960\pi$
$348$		0			0
$349$		−	122431.i		−	1.00517i	−0.864527	−	0.502587i	$-0.832382\pi$
$349$		−	122431.i		−	1.00517i	0.864527	−	0.502587i	$-0.167618\pi$
$350$		0			0
$351$		0			0
$352$	−8263.77	+	8263.77i	−0.0666950	+	0.0666950i
$353$	124879.	+	124879.i	1.00217	+	1.00217i	0.999998	+	0.00217111i	$0.000691086\pi$
$353$	124879.	+	124879.i	1.00217	+	1.00217i	0.00217111	+	0.999998i	$0.499309\pi$
$354$		0			0
$355$		0			0
$356$	57564.9			0.454211
$357$		0			0
$358$	−1816.38	−	1816.38i	−0.0141723	−	0.0141723i
$359$			105425.i			0.818006i	0.912533	+	0.409003i	$0.134123\pi$
$359$			105425.i			0.818006i	−0.912533	+	0.409003i	$0.865877\pi$
$360$		0			0
$361$	−206324.			−1.58320
$362$	62379.9	−	62379.9i	0.476022	−	0.476022i
$363$		0			0
$364$			52007.1i			0.392519i
$365$		0			0
$366$		0			0
$367$	−113172.	+	113172.i	−0.840247	+	0.840247i	−0.988891	−	0.148643i	$-0.952509\pi$
$367$	−113172.	+	113172.i	−0.840247	+	0.840247i	0.148643	+	0.988891i	$0.452509\pi$
$368$	−25364.3	−	25364.3i	−0.187296	−	0.187296i
$369$		0			0
$370$		0			0
$371$	−88987.2			−0.646516
$372$		0			0
$373$	−49863.2	−	49863.2i	−0.358395	−	0.358395i	0.504826	−	0.863221i	$-0.331557\pi$
$373$	−49863.2	−	49863.2i	−0.358395	−	0.358395i	−0.863221	+	0.504826i	$0.831557\pi$
$374$		−	68954.7i		−	0.492970i
$375$		0			0
$376$	−60583.3			−0.428526
$377$	16232.0	−	16232.0i	0.114206	−	0.114206i
$378$		0			0
$379$			70095.0i			0.487988i	0.969777	+	0.243994i	$0.0784577\pi$
$379$			70095.0i			0.487988i	−0.969777	+	0.243994i	$0.921542\pi$
$380$		0			0
$381$		0			0
$382$	39158.0	−	39158.0i	0.268345	−	0.268345i
$383$	136097.	+	136097.i	0.927789	+	0.927789i	0.997563	−	0.0697735i	$-0.0222276\pi$
$383$	136097.	+	136097.i	0.927789	+	0.927789i	−0.0697735	+	0.997563i	$0.522228\pi$
$384$		0			0
$385$		0			0
$386$	−68659.5			−0.460815
$387$		0			0
$388$	−76105.0	−	76105.0i	−0.505533	−	0.505533i
$389$		−	192464.i		−	1.27189i	−0.771733	−	0.635947i	$-0.780610\pi$
$389$		−	192464.i		−	1.27189i	0.771733	−	0.635947i	$-0.219390\pi$
$390$		0			0
$391$	211645.			1.38438
$392$	70868.8	−	70868.8i	0.461193	−	0.461193i
$393$		0			0
$394$		−	133532.i		−	0.860185i
$395$		0			0
$396$		0			0
$397$	31478.6	−	31478.6i	0.199726	−	0.199726i	−0.600157	−	0.799882i	$-0.704895\pi$
$397$	31478.6	−	31478.6i	0.199726	−	0.199726i	0.799882	+	0.600157i	$0.204895\pi$
$398$	−67457.0	−	67457.0i	−0.425854	−	0.425854i
$399$		0			0
$400$		0			0
$401$	−7702.89			−0.0479032			−0.0239516	−	0.999713i	$-0.507625\pi$
$401$	−7702.89			−0.0479032			−0.0239516	+	0.999713i	$0.507625\pi$
$402$		0			0
$403$	38032.8	+	38032.8i	0.234179	+	0.234179i
$404$			82424.7i			0.505004i
$405$		0			0
$406$	−68217.8			−0.413853
$407$	86226.6	−	86226.6i	0.520538	−	0.520538i
$408$		0			0
$409$			174162.i			1.04113i	0.853821	+	0.520566i	$0.174279\pi$
$409$			174162.i			1.04113i	−0.853821	+	0.520566i	$0.825721\pi$
$410$		0			0
$411$		0			0
$412$	97501.9	−	97501.9i	0.574405	−	0.574405i
$413$	−160854.	−	160854.i	−0.943043	−	0.943043i
$414$		0			0
$415$		0			0
$416$	−14239.0			−0.0822795
$417$		0			0
$418$	−74917.7	−	74917.7i	−0.428777	−	0.428777i
$419$		−	9181.39i		−	0.0522974i	−0.999658	−	0.0261487i	$-0.991676\pi$
$419$		−	9181.39i		−	0.0522974i	0.999658	−	0.0261487i	$-0.00832434\pi$
$420$		0			0
$421$	−197015.			−1.11157			−0.555783	−	0.831327i	$-0.687581\pi$
$421$	−197015.			−1.11157			−0.555783	+	0.831327i	$0.687581\pi$
$422$	104036.	−	104036.i	0.584198	−	0.584198i
$423$		0			0
$424$		−	24363.7i		−	0.135522i
$425$		0			0
$426$		0			0
$427$	113235.	−	113235.i	0.621050	−	0.621050i
$428$	−109128.	−	109128.i	−0.595728	−	0.595728i
$429$		0			0
$430$		0			0
$431$	55446.0			0.298480			0.149240	−	0.988801i	$-0.452317\pi$
$431$	55446.0			0.298480			0.149240	+	0.988801i	$0.452317\pi$
$432$		0			0
$433$	21947.4	+	21947.4i	0.117059	+	0.117059i	0.763210	−	0.646151i	$-0.223622\pi$
$433$	21947.4	+	21947.4i	0.117059	+	0.117059i	−0.646151	+	0.763210i	$0.723622\pi$
$434$		−	159840.i		−	0.848606i
$435$		0			0
$436$	−20483.7			−0.107755
$437$	229948.	−	229948.i	1.20411	−	1.20411i
$438$		0			0
$439$		−	335784.i		−	1.74233i	−0.490988	−	0.871166i	$-0.663364\pi$
$439$		−	335784.i		−	1.74233i	0.490988	−	0.871166i	$-0.336636\pi$
$440$		0			0
$441$		0			0
$442$	59406.4	−	59406.4i	0.304081	−	0.304081i
$443$	96887.9	+	96887.9i	0.493699	+	0.493699i	0.909470	−	0.415771i	$-0.136488\pi$
$443$	96887.9	+	96887.9i	0.493699	+	0.493699i	−0.415771	+	0.909470i	$0.636488\pi$
$444$		0			0
$445$		0			0
$446$	−188015.			−0.945198
$447$		0			0
$448$	29920.9	+	29920.9i	0.149080	+	0.149080i
$449$		−	7802.43i		−	0.0387024i	−0.999813	−	0.0193512i	$-0.993840\pi$
$449$		−	7802.43i		−	0.0387024i	0.999813	−	0.0193512i	$-0.00616006\pi$
$450$		0			0
$451$	102160.			0.502260
$452$	−121189.	+	121189.i	−0.593179	+	0.593179i
$453$		0			0
$454$		−	18096.9i		−	0.0877998i
$455$		0			0
$456$		0			0
$457$	72251.3	−	72251.3i	0.345950	−	0.345950i	−0.512649	−	0.858599i	$-0.671336\pi$
$457$	72251.3	−	72251.3i	0.345950	−	0.345950i	0.858599	+	0.512649i	$0.171336\pi$
$458$	−50607.2	−	50607.2i	−0.241258	−	0.241258i
$459$		0			0
$460$		0			0
$461$	397196.			1.86897			0.934485	−	0.356001i	$-0.115860\pi$
$461$	397196.			1.86897			0.934485	+	0.356001i	$0.115860\pi$
$462$		0			0
$463$	56717.3	+	56717.3i	0.264578	+	0.264578i	0.826911	−	0.562333i	$-0.190096\pi$
$463$	56717.3	+	56717.3i	0.264578	+	0.264578i	−0.562333	+	0.826911i	$0.690096\pi$
$464$		−	18677.3i		−	0.0867515i
$465$		0			0
$466$	141169.			0.650081
$467$	−188196.	+	188196.i	−0.862933	+	0.862933i	−0.991678	−	0.128744i	$-0.958905\pi$
$467$	−188196.	+	188196.i	−0.862933	+	0.862933i	0.128744	+	0.991678i	$0.458905\pi$
$468$		0			0
$469$		−	462488.i		−	2.10259i
$470$		0			0
$471$		0			0
$472$	44040.0	−	44040.0i	0.197680	−	0.197680i
$473$	−29723.3	−	29723.3i	−0.132854	−	0.132854i
$474$		0			0
$475$		0			0
$476$	−249666.			−1.10191
$477$		0			0
$478$	19671.5	+	19671.5i	0.0860958	+	0.0860958i
$479$		−	79286.0i		−	0.345562i	−0.984960	−	0.172781i	$-0.944725\pi$
$479$		−	79286.0i		−	0.345562i	0.984960	−	0.172781i	$-0.0552752\pi$
$480$		0			0
$481$	148573.			0.642171
$482$	−231776.	+	231776.i	−0.997643	+	0.997643i
$483$		0			0
$484$			83783.3i			0.357657i
$485$		0			0
$486$		0			0
$487$	−109632.	+	109632.i	−0.462254	+	0.462254i	−0.899394	−	0.437139i	$-0.855992\pi$
$487$	−109632.	+	109632.i	−0.462254	+	0.462254i	0.437139	+	0.899394i	$0.355992\pi$
$488$	31002.6	+	31002.6i	0.130184	+	0.130184i
$489$		0			0
$490$		0			0
$491$	48561.1			0.201430			0.100715	−	0.994915i	$-0.467887\pi$
$491$	48561.1			0.201430			0.100715	+	0.994915i	$0.467887\pi$
$492$		0			0
$493$	77923.5	+	77923.5i	0.320608	+	0.320608i
$494$		−	129088.i		−	0.528969i
$495$		0			0
$496$	43762.4			0.177884
$497$	−215815.	+	215815.i	−0.873712	+	0.873712i
$498$		0			0
$499$			409828.i			1.64589i	0.568120	+	0.822945i	$0.307671\pi$
$499$			409828.i			1.64589i	−0.568120	+	0.822945i	$0.692329\pi$
$500$		0			0
$501$		0			0
$502$	117832.	−	117832.i	0.467578	−	0.467578i
$503$	73740.9	+	73740.9i	0.291456	+	0.291456i	0.837655	−	0.546199i	$-0.183926\pi$
$503$	73740.9	+	73740.9i	0.291456	+	0.291456i	−0.546199	+	0.837655i	$0.683926\pi$
$504$		0			0
$505$		0			0
$506$	102346.			0.399734
$507$		0			0
$508$	−24268.3	−	24268.3i	−0.0940399	−	0.0940399i
$509$		−	374277.i		−	1.44463i	−0.691562	−	0.722317i	$-0.743077\pi$
$509$		−	374277.i		−	1.44463i	0.691562	−	0.722317i	$-0.256923\pi$
$510$		0			0
$511$	−397009.			−1.52040
$512$	−8192.00	+	8192.00i	−0.0312500	+	0.0312500i
$513$		0			0
$514$		−	169030.i		−	0.639791i
$515$		0			0
$516$		0			0
$517$	122228.	−	122228.i	0.457288	−	0.457288i
$518$	−312204.	−	312204.i	−1.16353	−	1.16353i
$519$		0			0
$520$		0			0
$521$	−496201.			−1.82803			−0.914013	−	0.405685i	$-0.867033\pi$
$521$	−496201.			−1.82803			−0.914013	+	0.405685i	$0.867033\pi$
$522$		0			0
$523$	−49228.5	−	49228.5i	−0.179976	−	0.179976i	0.611370	−	0.791345i	$-0.290619\pi$
$523$	−49228.5	−	49228.5i	−0.179976	−	0.179976i	−0.791345	+	0.611370i	$0.790619\pi$
$524$			13811.0i			0.0502994i
$525$		0			0
$526$	132991.			0.480673
$527$	−182581.	+	182581.i	−0.657408	+	0.657408i
$528$		0			0
$529$			34294.7i			0.122551i
$530$		0			0
$531$		0			0
$532$	−271257.	+	271257.i	−0.958424	+	0.958424i
$533$	88014.0	+	88014.0i	0.309811	+	0.309811i
$534$		0			0
$535$		0			0
$536$	126624.			0.440743
$537$		0			0
$538$	106855.	+	106855.i	0.369172	+	0.369172i
$539$			285959.i			0.984296i
$540$		0			0
$541$	−349181.			−1.19304			−0.596521	−	0.802597i	$-0.703451\pi$
$541$	−349181.			−1.19304			−0.596521	+	0.802597i	$0.703451\pi$
$542$	−68585.0	+	68585.0i	−0.233470	+	0.233470i
$543$		0			0
$544$		−	68355.8i		−	0.230982i
$545$		0			0
$546$		0			0
$547$	110716.	−	110716.i	0.370031	−	0.370031i	−0.497458	−	0.867488i	$-0.665733\pi$
$547$	110716.	−	110716.i	0.370031	−	0.370031i	0.867488	+	0.497458i	$0.165733\pi$
$548$	−3812.49	−	3812.49i	−0.0126954	−	0.0126954i
$549$		0			0
$550$		0			0
$551$	169324.			0.557720
$552$		0			0
$553$	−44256.5	−	44256.5i	−0.144719	−	0.144719i
$554$			259707.i			0.846182i
$555$		0			0
$556$	−214958.			−0.695351
$557$	420140.	−	420140.i	1.35420	−	1.35420i	0.473303	−	0.880900i	$-0.343062\pi$
$557$	420140.	−	420140.i	1.35420	−	1.35420i	0.880900	−	0.473303i	$-0.156938\pi$
$558$		0			0
$559$		−	51214.9i		−	0.163898i
$560$		0			0
$561$		0			0
$562$	−194701.	+	194701.i	−0.616448	+	0.616448i
$563$	24524.2	+	24524.2i	0.0773711	+	0.0773711i	0.744733	−	0.667362i	$-0.232577\pi$
$563$	24524.2	+	24524.2i	0.0773711	+	0.0773711i	−0.667362	+	0.744733i	$0.732577\pi$
$564$		0			0
$565$		0			0
$566$	−158244.			−0.493962
$567$		0			0
$568$	−59087.5	−	59087.5i	−0.183147	−	0.183147i
$569$			110083.i			0.340014i	0.985443	+	0.170007i	$0.0543790\pi$
$569$			110083.i			0.340014i	−0.985443	+	0.170007i	$0.945621\pi$
$570$		0			0
$571$	−96373.5			−0.295587			−0.147794	−	0.989018i	$-0.547217\pi$
$571$	−96373.5			−0.295587			−0.147794	+	0.989018i	$0.547217\pi$
$572$	28727.4	−	28727.4i	0.0878020	−	0.0878020i
$573$		0			0
$574$		−	369895.i		−	1.12268i
$575$		0			0
$576$		0			0
$577$	84333.2	−	84333.2i	0.253307	−	0.253307i	−0.569018	−	0.822325i	$-0.692677\pi$
$577$	84333.2	−	84333.2i	0.253307	−	0.253307i	0.822325	+	0.569018i	$0.192677\pi$
$578$	118146.	+	118146.i	0.353640	+	0.353640i
$579$		0			0
$580$		0			0
$581$	−500237.			−1.48192
$582$		0			0
$583$	49154.2	+	49154.2i	0.144618	+	0.144618i
$584$		−	108696.i		−	0.318705i
$585$		0			0
$586$	−164869.			−0.480112
$587$	380824.	−	380824.i	1.10522	−	1.10522i	0.111447	−	0.993770i	$-0.464451\pi$
$587$	380824.	−	380824.i	1.10522	−	1.10522i	0.993770	−	0.111447i	$-0.0355485\pi$
$588$		0			0
$589$			396741.i			1.14361i
$590$		0			0
$591$		0			0
$592$	85477.7	−	85477.7i	0.243899	−	0.243899i
$593$	128314.	+	128314.i	0.364893	+	0.364893i	0.865611	−	0.500717i	$-0.166930\pi$
$593$	128314.	+	128314.i	0.364893	+	0.364893i	−0.500717	+	0.865611i	$0.666930\pi$
$594$		0			0
$595$		0			0
$596$	−131552.			−0.370345
$597$		0			0
$598$	88174.2	+	88174.2i	0.246570	+	0.246570i
$599$		−	289021.i		−	0.805519i	−0.915306	−	0.402759i	$-0.868051\pi$
$599$		−	289021.i		−	0.805519i	0.915306	−	0.402759i	$-0.131949\pi$
$600$		0			0
$601$	429037.			1.18781			0.593904	−	0.804536i	$-0.297586\pi$
$601$	429037.			1.18781			0.593904	+	0.804536i	$0.297586\pi$
$602$	−107620.	+	107620.i	−0.296962	+	0.296962i
$603$		0			0
$604$		−	83204.7i		−	0.228073i
$605$		0			0
$606$		0			0
$607$	259307.	−	259307.i	0.703779	−	0.703779i	−0.261441	−	0.965220i	$-0.584198\pi$
$607$	259307.	−	259307.i	0.703779	−	0.703779i	0.965220	+	0.261441i	$0.0841975\pi$
$608$	−74267.0	−	74267.0i	−0.200904	−	0.200904i
$609$		0			0
$610$		0			0
$611$	210606.			0.564142
$612$		0			0
$613$	−228930.	−	228930.i	−0.609230	−	0.609230i	0.333515	−	0.942745i	$-0.391765\pi$
$613$	−228930.	−	228930.i	−0.609230	−	0.609230i	−0.942745	+	0.333515i	$0.891765\pi$
$614$		−	191051.i		−	0.506773i
$615$		0			0
$616$	−120732.			−0.318172
$617$	324502.	−	324502.i	0.852406	−	0.852406i	−0.138023	−	0.990429i	$-0.544075\pi$
$617$	324502.	−	324502.i	0.852406	−	0.852406i	0.990429	+	0.138023i	$0.0440748\pi$
$618$		0			0
$619$			56923.5i			0.148563i	0.997237	+	0.0742814i	$0.0236663\pi$
$619$			56923.5i			0.148563i	−0.997237	+	0.0742814i	$0.976334\pi$
$620$		0			0
$621$		0			0
$622$	144793.	−	144793.i	0.374254	−	0.374254i
$623$	420507.	+	420507.i	1.08342	+	1.08342i
$624$		0			0
$625$		0			0
$626$	−219438.			−0.559969
$627$		0			0
$628$	75481.2	+	75481.2i	0.191390	+	0.191390i
$629$			713244.i			1.80276i
$630$		0			0
$631$	−89685.8			−0.225250			−0.112625	−	0.993638i	$-0.535926\pi$
$631$	−89685.8			−0.225250			−0.112625	+	0.993638i	$0.535926\pi$
$632$	12116.9	−	12116.9i	0.0303360	−	0.0303360i
$633$		0			0
$634$			302799.i			0.753315i
$635$		0			0
$636$		0			0
$637$	−246362.	+	246362.i	−0.607148	+	0.607148i
$638$	37681.8	+	37681.8i	0.0925742	+	0.0925742i
$639$		0			0
$640$		0			0
$641$	388065.			0.944471			0.472235	−	0.881473i	$-0.343447\pi$
$641$	388065.			0.944471			0.472235	+	0.881473i	$0.343447\pi$
$642$		0			0
$643$	−311459.	−	311459.i	−0.753319	−	0.753319i	0.221778	−	0.975097i	$-0.428814\pi$
$643$	−311459.	−	311459.i	−0.753319	−	0.753319i	−0.975097	+	0.221778i	$0.928814\pi$
$644$		−	370569.i		−	0.893505i
$645$		0			0
$646$	619700.			1.48497
$647$	−412897.	+	412897.i	−0.986355	+	0.986355i	−0.999908	−	0.0135532i	$-0.995686\pi$
$647$	−412897.	+	412897.i	−0.986355	+	0.986355i	0.0135532	+	0.999908i	$0.495686\pi$
$648$		0			0
$649$			177703.i			0.421896i
$650$		0			0
$651$		0			0
$652$	−175576.	+	175576.i	−0.413019	+	0.413019i
$653$	−424771.	−	424771.i	−0.996158	−	0.996158i	0.00383509	−	0.999993i	$-0.498779\pi$
$653$	−424771.	−	424771.i	−0.996158	−	0.996158i	−0.999993	+	0.00383509i	$0.998779\pi$
$654$		0			0
$655$		0			0
$656$	101273.			0.235335
$657$		0			0
$658$	−442556.	−	442556.i	−1.02215	−	1.02215i
$659$			538552.i			1.24010i	0.784562	+	0.620050i	$0.212888\pi$
$659$			538552.i			1.24010i	−0.784562	+	0.620050i	$0.787112\pi$
$660$		0			0
$661$	57725.6			0.132119			0.0660595	−	0.997816i	$-0.478957\pi$
$661$	57725.6			0.132119			0.0660595	+	0.997816i	$0.478957\pi$
$662$	−51985.4	+	51985.4i	−0.118622	+	0.118622i
$663$		0			0
$664$		−	136959.i		−	0.310638i
$665$		0			0
$666$		0			0
$667$	−115658.	+	115658.i	−0.259971	+	0.259971i
$668$	−15706.0	−	15706.0i	−0.0351975	−	0.0351975i
$669$		0			0
$670$		0			0
$671$	−125097.			−0.277844
$672$		0			0
$673$	61056.8	+	61056.8i	0.134804	+	0.134804i	0.771289	−	0.636485i	$-0.219612\pi$
$673$	61056.8	+	61056.8i	0.134804	+	0.134804i	−0.636485	+	0.771289i	$0.719612\pi$
$674$			322711.i			0.710385i
$675$		0			0
$676$	−178989.			−0.391681
$677$	163948.	−	163948.i	0.357708	−	0.357708i	−0.505259	−	0.862968i	$-0.668603\pi$
$677$	163948.	−	163948.i	0.357708	−	0.357708i	0.862968	+	0.505259i	$0.168603\pi$
$678$		0			0
$679$		−	1.11188e6i		−	2.41167i
$680$		0			0
$681$		0			0
$682$	−88291.5	+	88291.5i	−0.189824	+	0.189824i
$683$	240846.	+	240846.i	0.516294	+	0.516294i	0.916448	−	0.400154i	$-0.131043\pi$
$683$	240846.	+	240846.i	0.516294	+	0.516294i	−0.400154	+	0.916448i	$0.631043\pi$
$684$		0			0
$685$		0			0
$686$	474130.			1.00751
$687$		0			0
$688$	−29465.1	−	29465.1i	−0.0622489	−	0.0622489i
$689$			84695.6i			0.178411i
$690$		0			0
$691$	−517319.			−1.08343			−0.541717	−	0.840561i	$-0.682225\pi$
$691$	−517319.			−1.08343			−0.541717	+	0.840561i	$0.682225\pi$
$692$	30834.0	−	30834.0i	0.0643899	−	0.0643899i
$693$		0			0
$694$			190113.i			0.394724i
$695$		0			0
$696$		0			0
$697$	−422522.	+	422522.i	−0.869728	+	0.869728i
$698$	244862.	+	244862.i	0.502587	+	0.502587i
$699$		0			0
$700$		0			0
$701$	517565.			1.05324			0.526621	−	0.850100i	$-0.323459\pi$
$701$	517565.			1.05324			0.526621	+	0.850100i	$0.323459\pi$
$702$		0			0
$703$	774924.	+	774924.i	1.56801	+	1.56801i
$704$		−	33055.1i		−	0.0666950i
$705$		0			0
$706$	−499517.			−1.00217
$707$	−602105.	+	602105.i	−1.20457	+	1.20457i
$708$		0			0
$709$			959761.i			1.90928i	0.297756	+	0.954642i	$0.403762\pi$
$709$			959761.i			1.90928i	−0.297756	+	0.954642i	$0.596238\pi$
$710$		0			0
$711$		0			0
$712$	−115130.	+	115130.i	−0.227106	+	0.227106i
$713$	−270997.	−	270997.i	−0.533071	−	0.533071i
$714$		0			0
$715$		0			0
$716$	7265.51			0.0141723
$717$		0			0
$718$	−210851.	−	210851.i	−0.409003	−	0.409003i
$719$			477318.i			0.923315i	0.887058	+	0.461657i	$0.152745\pi$
$719$			477318.i			0.923315i	−0.887058	+	0.461657i	$0.847255\pi$
$720$		0			0
$721$	1.42448e6			2.74023
$722$	412648.	−	412648.i	0.791599	−	0.791599i
$723$		0			0
$724$			249520.i			0.476022i
$725$		0			0
$726$		0			0
$727$	−441668.	+	441668.i	−0.835656	+	0.835656i	−0.988284	−	0.152628i	$-0.951226\pi$
$727$	−441668.	+	441668.i	−0.835656	+	0.835656i	0.152628	+	0.988284i	$0.451226\pi$
$728$	−104014.	−	104014.i	−0.196259	−	0.196259i
$729$		0			0
$730$		0			0
$731$	245863.			0.460107
$732$		0			0
$733$	361096.	+	361096.i	0.672071	+	0.672071i	0.958193	−	0.286122i	$-0.0923664\pi$
$733$	361096.	+	361096.i	0.672071	+	0.672071i	−0.286122	+	0.958193i	$0.592366\pi$
$734$		−	452688.i		−	0.840247i
$735$		0			0
$736$	101457.			0.187296
$737$	−255466.	+	255466.i	−0.470326	+	0.470326i
$738$		0			0
$739$			45841.2i			0.0839396i	0.999119	+	0.0419698i	$0.0133633\pi$
$739$			45841.2i			0.0839396i	−0.999119	+	0.0419698i	$0.986637\pi$
$740$		0			0
$741$		0			0
$742$	177974.	−	177974.i	0.323258	−	0.323258i
$743$	241293.	+	241293.i	0.437086	+	0.437086i	0.891030	−	0.453944i	$-0.149983\pi$
$743$	241293.	+	241293.i	0.437086	+	0.437086i	−0.453944	+	0.891030i	$0.649983\pi$
$744$		0			0
$745$		0			0
$746$	199453.			0.358395
$747$		0			0
$748$	137909.	+	137909.i	0.246485	+	0.246485i
$749$		−	1.59434e6i		−	2.84195i
$750$		0			0
$751$	448347.			0.794940			0.397470	−	0.917615i	$-0.369888\pi$
$751$	448347.			0.794940			0.397470	+	0.917615i	$0.369888\pi$
$752$	121167.	−	121167.i	0.214263	−	0.214263i
$753$		0			0
$754$			64927.9i			0.114206i
$755$		0			0
$756$		0			0
$757$	156632.	−	156632.i	0.273331	−	0.273331i	−0.557109	−	0.830440i	$-0.688089\pi$
$757$	156632.	−	156632.i	0.273331	−	0.273331i	0.830440	+	0.557109i	$0.188089\pi$
$758$	−140190.	−	140190.i	−0.243994	−	0.243994i
$759$		0			0
$760$		0			0
$761$	720605.			1.24431			0.622154	−	0.782895i	$-0.286258\pi$
$761$	720605.			1.24431			0.622154	+	0.782895i	$0.286258\pi$
$762$		0			0
$763$	−149632.	−	149632.i	−0.257025	−	0.257025i
$764$			156632.i			0.268345i
$765$		0			0
$766$	−544386.			−0.927789
$767$	−153096.	+	153096.i	−0.260240	+	0.260240i
$768$		0			0
$769$		−	340366.i		−	0.575564i	−0.957696	−	0.287782i	$-0.907082\pi$
$769$		−	340366.i		−	0.575564i	0.957696	−	0.287782i	$-0.0929179\pi$
$770$		0			0
$771$		0			0
$772$	137319.	−	137319.i	0.230407	−	0.230407i
$773$	−236854.	−	236854.i	−0.396390	−	0.396390i	0.480568	−	0.876958i	$-0.340431\pi$
$773$	−236854.	−	236854.i	−0.396390	−	0.396390i	−0.876958	+	0.480568i	$0.840431\pi$
$774$		0			0
$775$		0			0
$776$	304420.			0.505533
$777$		0			0
$778$	384928.	+	384928.i	0.635947	+	0.635947i
$779$			918121.i			1.51295i
$780$		0			0
$781$	238421.			0.390879
$782$	−423291.	+	423291.i	−0.692190	+	0.692190i
$783$		0			0
$784$			283475.i			0.461193i
$785$		0			0
$786$		0			0
$787$	638474.	−	638474.i	1.03085	−	1.03085i	0.0313370	−	0.999509i	$-0.490023\pi$
$787$	638474.	−	638474.i	1.03085	−	1.03085i	0.999509	−	0.0313370i	$-0.00997651\pi$
$788$	267063.	+	267063.i	0.430092	+	0.430092i
$789$		0			0
$790$		0			0
$791$	−1.77055e6			−2.82979
$792$		0			0
$793$	−107774.	−	107774.i	−0.171384	−	0.171384i
$794$			125914.i			0.199726i
$795$		0			0
$796$	269828.			0.425854
$797$	−37438.2	+	37438.2i	−0.0589384	+	0.0589384i	−0.735962	−	0.677023i	$-0.763270\pi$
$797$	−37438.2	+	37438.2i	−0.0589384	+	0.0589384i	0.677023	+	0.735962i	$0.263270\pi$
$798$		0			0
$799$			1.01104e6i			1.58371i
$800$		0			0
$801$		0			0
$802$	15405.8	−	15405.8i	0.0239516	−	0.0239516i
$803$	219297.	+	219297.i	0.340097	+	0.340097i
$804$		0			0
$805$		0			0
$806$	−152131.			−0.234179
$807$		0			0
$808$	−164849.	−	164849.i	−0.252502	−	0.252502i
$809$		−	941647.i		−	1.43877i	−0.694612	−	0.719384i	$-0.744424\pi$
$809$		−	941647.i		−	1.43877i	0.694612	−	0.719384i	$-0.255576\pi$
$810$		0			0
$811$	223386.			0.339636			0.169818	−	0.985475i	$-0.445682\pi$
$811$	223386.			0.339636			0.169818	+	0.985475i	$0.445682\pi$
$812$	136436.	−	136436.i	0.206926	−	0.206926i
$813$		0			0
$814$			344906.i			0.520538i
$815$		0			0
$816$		0			0
$817$	267125.	−	267125.i	0.400194	−	0.400194i
$818$	−348323.	−	348323.i	−0.520566	−	0.520566i
$819$		0			0
$820$		0			0
$821$	774777.			1.14945			0.574726	−	0.818346i	$-0.305109\pi$
$821$	774777.			1.14945			0.574726	+	0.818346i	$0.305109\pi$
$822$		0			0
$823$	468343.	+	468343.i	0.691455	+	0.691455i	0.962552	−	0.271097i	$-0.0873864\pi$
$823$	468343.	+	468343.i	0.691455	+	0.691455i	−0.271097	+	0.962552i	$0.587386\pi$
$824$			390007.i			0.574405i
$825$		0			0
$826$	643416.			0.943043
$827$	339034.	−	339034.i	0.495715	−	0.495715i	−0.414386	−	0.910101i	$-0.636004\pi$
$827$	339034.	−	339034.i	0.495715	−	0.495715i	0.910101	+	0.414386i	$0.136004\pi$
$828$		0			0
$829$		−	78978.6i		−	0.114921i	−0.998348	−	0.0574606i	$-0.981700\pi$
$829$		−	78978.6i		−	0.114921i	0.998348	−	0.0574606i	$-0.0183004\pi$
$830$		0			0
$831$		0			0
$832$	28477.9	−	28477.9i	0.0411397	−	0.0411397i
$833$	−1.18269e6	−	1.18269e6i	−1.70443	−	1.70443i
$834$		0			0
$835$		0			0
$836$	299671.			0.428777
$837$		0			0
$838$	18362.8	+	18362.8i	0.0261487	+	0.0261487i
$839$			230443.i			0.327370i	0.986513	+	0.163685i	$0.0523381\pi$
$839$			230443.i			0.327370i	−0.986513	+	0.163685i	$0.947662\pi$
$840$		0			0
$841$	622115.			0.879587
$842$	394030.	−	394030.i	0.555783	−	0.555783i
$843$		0			0
$844$			416145.i			0.584198i
$845$		0			0
$846$		0			0
$847$	−612030.	+	612030.i	−0.853111	+	0.853111i
$848$	48727.3	+	48727.3i	0.0677612	+	0.0677612i
$849$		0			0
$850$		0			0
$851$	−1.05864e6			−1.46180
$852$		0			0
$853$	−321295.	−	321295.i	−0.441576	−	0.441576i	0.450965	−	0.892541i	$-0.351080\pi$
$853$	−321295.	−	321295.i	−0.441576	−	0.441576i	−0.892541	+	0.450965i	$0.851080\pi$
$854$			452942.i			0.621050i
$855$		0			0
$856$	436511.			0.595728
$857$	628471.	−	628471.i	0.855704	−	0.855704i	−0.135124	−	0.990829i	$-0.543143\pi$
$857$	628471.	−	628471.i	0.855704	−	0.855704i	0.990829	+	0.135124i	$0.0431434\pi$
$858$		0			0
$859$		−	281374.i		−	0.381327i	−0.981655	−	0.190664i	$-0.938936\pi$
$859$		−	281374.i		−	0.381327i	0.981655	−	0.190664i	$-0.0610640\pi$
$860$		0			0
$861$		0			0
$862$	−110892.	+	110892.i	−0.149240	+	0.149240i
$863$	−734342.	−	734342.i	−0.986000	−	0.986000i	0.0139036	−	0.999903i	$-0.495574\pi$
$863$	−734342.	−	734342.i	−0.986000	−	0.986000i	−0.999903	+	0.0139036i	$0.995574\pi$
$864$		0			0
$865$		0			0
$866$	−87789.4			−0.117059
$867$		0			0
$868$	319680.	+	319680.i	0.424303	+	0.424303i
$869$			48892.3i			0.0647442i
$870$		0			0
$871$	−440183.			−0.580226
$872$	40967.4	−	40967.4i	0.0538773	−	0.0538773i
$873$		0			0
$874$			919792.i			1.20411i
$875$		0			0
$876$		0			0
$877$	−277654.	+	277654.i	−0.360998	+	0.360998i	−0.864180	−	0.503182i	$-0.832163\pi$
$877$	−277654.	+	277654.i	−0.360998	+	0.360998i	0.503182	+	0.864180i	$0.332163\pi$
$878$	671568.	+	671568.i	0.871166	+	0.871166i
$879$		0			0
$880$		0			0
$881$	−1.36621e6			−1.76022			−0.880108	−	0.474774i	$-0.842530\pi$
$881$	−1.36621e6			−1.76022			−0.880108	+	0.474774i	$0.842530\pi$
$882$		0			0
$883$	197156.	+	197156.i	0.252865	+	0.252865i	0.822144	−	0.569279i	$-0.192778\pi$
$883$	197156.	+	197156.i	0.252865	+	0.252865i	−0.569279	+	0.822144i	$0.692778\pi$
$884$			237626.i			0.304081i
$885$		0			0
$886$	−387552.			−0.493699
$887$	−506908.	+	506908.i	−0.644291	+	0.644291i	−0.951607	−	0.307316i	$-0.900569\pi$
$887$	−506908.	+	506908.i	−0.644291	+	0.644291i	0.307316	+	0.951607i	$0.400569\pi$
$888$		0			0
$889$		−	354556.i		−	0.448622i
$890$		0			0
$891$		0			0
$892$	376030.	−	376030.i	0.472599	−	0.472599i
$893$	1.09847e6	+	1.09847e6i	1.37748	+	1.37748i
$894$		0			0
$895$		0			0
$896$	−119684.			−0.149080
$897$		0			0
$898$	15604.9	+	15604.9i	0.0193512	+	0.0193512i
$899$		−	199551.i		−	0.246908i
$900$		0			0
$901$	−406591.			−0.500851
$902$	−204321.	+	204321.i	−0.251130	+	0.251130i
$903$		0			0
$904$		−	484755.i		−	0.593179i
$905$		0			0
$906$		0			0
$907$	−136926.	+	136926.i	−0.166445	+	0.166445i	−0.785415	−	0.618970i	$-0.787550\pi$
$907$	−136926.	+	136926.i	−0.166445	+	0.166445i	0.618970	+	0.785415i	$0.287550\pi$
$908$	36193.9	+	36193.9i	0.0438999	+	0.0438999i
$909$		0			0
$910$		0			0
$911$	1.61508e6			1.94607			0.973035	−	0.230658i	$-0.0740878\pi$
$911$	1.61508e6			1.94607			0.973035	+	0.230658i	$0.0740878\pi$
$912$		0			0
$913$	276318.	+	276318.i	0.331488	+	0.331488i
$914$			289005.i			0.345950i
$915$		0			0
$916$	202429.			0.241258
$917$	−100888.	+	100888.i	−0.119978	+	0.119978i
$918$		0			0
$919$			976616.i			1.15636i	0.815910	+	0.578179i	$0.196237\pi$
$919$			976616.i			1.15636i	−0.815910	+	0.578179i	$0.803763\pi$
$920$		0			0
$921$		0			0
$922$	−794391.	+	794391.i	−0.934485	+	0.934485i
$923$	205406.	+	205406.i	0.241107	+	0.241107i
$924$		0			0
$925$		0			0
$926$	−226869.			−0.264578
$927$		0			0
$928$	37354.5	+	37354.5i	0.0433758	+	0.0433758i
$929$		−	23445.3i		−	0.0271659i	−0.999908	−	0.0135830i	$-0.995676\pi$
$929$		−	23445.3i		−	0.0271659i	0.999908	−	0.0135830i	$-0.00432372\pi$
$930$		0			0
$931$	−2.56993e6			−2.96498
$932$	−282338.	+	282338.i	−0.325040	+	0.325040i
$933$		0			0
$934$		−	752785.i		−	0.862933i
$935$		0			0
$936$		0			0
$937$	782178.	−	782178.i	0.890895	−	0.890895i	−0.103712	−	0.994607i	$-0.533072\pi$
$937$	782178.	−	782178.i	0.890895	−	0.890895i	0.994607	+	0.103712i	$0.0330721\pi$
$938$	924975.	+	924975.i	1.05129	+	1.05129i
$939$		0			0
$940$		0			0
$941$	1.12116e6			1.26616			0.633082	−	0.774084i	$-0.281789\pi$
$941$	1.12116e6			1.26616			0.633082	+	0.774084i	$0.281789\pi$
$942$		0			0
$943$	−627130.	−	627130.i	−0.705235	−	0.705235i
$944$			176160.i			0.197680i
$945$		0			0
$946$	118893.			0.132854
$947$	−1.14123e6	+	1.14123e6i	−1.27254	+	1.27254i	−0.327789	+	0.944751i	$0.606304\pi$
$947$	−1.14123e6	+	1.14123e6i	−1.27254	+	1.27254i	−0.944751	+	0.327789i	$0.893696\pi$
$948$		0			0
$949$			377862.i			0.419566i
$950$		0			0
$951$		0			0
$952$	499333.	−	499333.i	0.550955	−	0.550955i
$953$	−390289.	−	390289.i	−0.429735	−	0.429735i	0.458803	−	0.888538i	$-0.348278\pi$
$953$	−390289.	−	390289.i	−0.429735	−	0.429735i	−0.888538	+	0.458803i	$0.848278\pi$
$954$		0			0
$955$		0			0
$956$	−78686.1			−0.0860958
$957$		0			0
$958$	158572.	+	158572.i	0.172781	+	0.172781i
$959$		−	55699.7i		−	0.0605642i
$960$		0			0
$961$	−455957.			−0.493716
$962$	−297147.	+	297147.i	−0.321086	+	0.321086i
$963$		0			0
$964$		−	927105.i		−	0.997643i
$965$		0			0
$966$		0			0
$967$	−749909.	+	749909.i	−0.801966	+	0.801966i	−0.983403	−	0.181437i	$-0.941925\pi$
$967$	−749909.	+	749909.i	−0.801966	+	0.801966i	0.181437	+	0.983403i	$0.441925\pi$
$968$	−167567.	−	167567.i	−0.178829	−	0.178829i
$969$		0			0
$970$		0			0
$971$	−1.83268e6			−1.94378			−0.971892	−	0.235426i	$-0.924352\pi$
$971$	−1.83268e6			−1.94378			−0.971892	+	0.235426i	$0.924352\pi$
$972$		0			0
$973$	−1.57025e6	−	1.57025e6i	−1.65861	−	1.65861i
$974$		−	438529.i		−	0.462254i
$975$		0			0
$976$	−124010.			−0.130184
$977$	726921.	−	726921.i	0.761549	−	0.761549i	−0.215053	−	0.976602i	$-0.568992\pi$
$977$	726921.	−	726921.i	0.761549	−	0.761549i	0.976602	+	0.215053i	$0.0689925\pi$
$978$		0			0
$979$		−	464554.i		−	0.484698i
$980$		0			0
$981$		0			0
$982$	−97122.1	+	97122.1i	−0.100715	+	0.100715i
$983$	310860.	+	310860.i	0.321705	+	0.321705i	0.849421	−	0.527716i	$-0.176951\pi$
$983$	310860.	+	310860.i	0.321705	+	0.321705i	−0.527716	+	0.849421i	$0.676951\pi$
$984$		0			0
$985$		0			0
$986$	−311694.			−0.320608
$987$		0			0
$988$	258175.	+	258175.i	0.264485	+	0.264485i
$989$			364924.i			0.373087i
$990$		0			0
$991$	−1.01850e6			−1.03708			−0.518541	−	0.855053i	$-0.673525\pi$
$991$	−1.01850e6			−1.03708			−0.518541	+	0.855053i	$0.673525\pi$
$992$	−87524.7	+	87524.7i	−0.0889421	+	0.0889421i
$993$		0			0
$994$		−	863258.i		−	0.873712i
$995$		0			0
$996$		0			0
$997$	−1.02194e6	+	1.02194e6i	−1.02810	+	1.02810i	−0.0285086	+	0.999594i	$0.509076\pi$
$997$	−1.02194e6	+	1.02194e6i	−1.02810	+	1.02810i	−0.999594	+	0.0285086i	$0.990924\pi$
$998$	−819657.	−	819657.i	−0.822945	−	0.822945i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.5.g.f.343.2		4
3.2	odd	2		150.5.f.e.43.1		4
5.2	odd	4	inner	450.5.g.f.307.2		4
5.3	odd	4		90.5.g.e.37.2		4
5.4	even	2		90.5.g.e.73.2		4
15.2	even	4		150.5.f.e.7.1		4
15.8	even	4		30.5.f.a.7.2	✓	4
15.14	odd	2		30.5.f.a.13.2	yes	4
60.23	odd	4		240.5.bg.b.97.1		4
60.59	even	2		240.5.bg.b.193.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.5.f.a.7.2	✓	4	15.8	even	4
30.5.f.a.13.2	yes	4	15.14	odd	2
90.5.g.e.37.2		4	5.3	odd	4
90.5.g.e.73.2		4	5.4	even	2
150.5.f.e.7.1		4	15.2	even	4
150.5.f.e.43.1		4	3.2	odd	2
240.5.bg.b.97.1		4	60.23	odd	4
240.5.bg.b.193.1		4	60.59	even	2
450.5.g.f.307.2		4	5.2	odd	4	inner
450.5.g.f.343.2		4	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-2.00000 + 2.00000i) q^{2} -8.00000i q^{4} +(58.4393 - 58.4393i) q^{7} +(16.0000 + 16.0000i) q^{8} -64.5607 q^{11} +(-55.6209 - 55.6209i) q^{13} +233.757i q^{14} -64.0000 q^{16} +(267.015 - 267.015i) q^{17} -580.211i q^{19} +(129.121 - 129.121i) q^{22} +(396.318 + 396.318i) q^{23} +222.484 q^{26} +(-467.514 - 467.514i) q^{28} +291.832i q^{29} -683.787 q^{31} +(128.000 - 128.000i) q^{32} +1068.06i q^{34} +(-1335.59 + 1335.59i) q^{37} +(1160.42 + 1160.42i) q^{38} -1582.39 q^{41} +(460.393 + 460.393i) q^{43} +516.486i q^{44} -1585.27 q^{46} +(-1893.23 + 1893.23i) q^{47} -4429.30i q^{49} +(-444.967 + 444.967i) q^{52} +(-761.364 - 761.364i) q^{53} +1870.06 q^{56} +(-583.664 - 583.664i) q^{58} -2752.50i q^{59} +1937.66 q^{61} +(1367.57 - 1367.57i) q^{62} +512.000i q^{64} +(3956.99 - 3956.99i) q^{67} +(-2136.12 - 2136.12i) q^{68} -3692.97 q^{71} +(-3396.76 - 3396.76i) q^{73} -5342.36i q^{74} -4641.69 q^{76} +(-3772.88 + 3772.88i) q^{77} -757.307i q^{79} +(3164.78 - 3164.78i) q^{82} +(-4279.98 - 4279.98i) q^{83} -1841.57 q^{86} +(-1032.97 - 1032.97i) q^{88} +7195.62i q^{89} -6500.89 q^{91} +(3170.54 - 3170.54i) q^{92} -7572.91i q^{94} +(9513.12 - 9513.12i) q^{97} +(8858.60 + 8858.60i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 28 q^{7} + 64 q^{8} - 464 q^{11} + 336 q^{13} - 256 q^{16} + 392 q^{17} + 928 q^{22} + 968 q^{23} - 1344 q^{26} - 224 q^{28} - 560 q^{31} + 512 q^{32} - 2256 q^{37} + 1232 q^{38} - 392 q^{41}+ \cdots + 23912 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 28 * q^7 + 64 * q^8 - 464 * q^11 + 336 * q^13 - 256 * q^16 + 392 * q^17 + 928 * q^22 + 968 * q^23 - 1344 * q^26 - 224 * q^28 - 560 * q^31 + 512 * q^32 - 2256 * q^37 + 1232 * q^38 - 392 * q^41 - 216 * q^43 - 3872 * q^46 - 9072 * q^47 + 2688 * q^52 - 4280 * q^53 + 896 * q^56 + 2192 * q^58 - 4536 * q^61 + 1120 * q^62 + 2248 * q^67 - 3136 * q^68 - 18064 * q^71 - 20524 * q^73 - 4928 * q^76 + 7336 * q^77 + 784 * q^82 - 336 * q^83 + 864 * q^86 - 7424 * q^88 - 52752 * q^91 + 7744 * q^92 + 40404 * q^97 + 23912 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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