LMFDB - Embedded newform 324.5.k.a.17.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,5,Mod(17,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 11]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("324.17");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	324.k (of order $18$, degree $6$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$33.4918680392$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{18})$
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			17.4
Character	$\chi$	$=$	324.17
Dual form			324.5.k.a.305.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-12.7195 - 2.24280i) q^{5} +(-50.9819 + 42.7789i) q^{7} +O(q^{10})$	$q+(-12.7195 - 2.24280i) q^{5} +(-50.9819 + 42.7789i) q^{7} +(-53.1143 + 9.36548i) q^{11} +(-35.6301 + 12.9683i) q^{13} +(110.279 + 63.6697i) q^{17} +(32.3968 + 56.1128i) q^{19} +(552.034 - 657.888i) q^{23} +(-430.551 - 156.708i) q^{25} +(103.294 - 283.798i) q^{29} +(1049.54 + 880.673i) q^{31} +(744.411 - 429.786i) q^{35} +(554.706 - 960.778i) q^{37} +(-733.845 - 2016.22i) q^{41} +(-48.5881 - 275.557i) q^{43} +(-775.618 - 924.345i) q^{47} +(352.191 - 1997.37i) q^{49} +2045.36i q^{53} +696.594 q^{55} +(-664.820 - 117.226i) q^{59} +(3039.96 - 2550.83i) q^{61} +(482.284 - 85.0396i) q^{65} +(6936.65 - 2524.73i) q^{67} +(-7606.56 - 4391.65i) q^{71} +(323.073 + 559.578i) q^{73} +(2307.22 - 2749.64i) q^{77} +(8964.88 + 3262.95i) q^{79} +(1737.55 - 4773.87i) q^{83} +(-1259.90 - 1057.18i) q^{85} +(10674.4 - 6162.88i) q^{89} +(1261.72 - 2185.36i) q^{91} +(-286.222 - 786.389i) q^{95} +(1027.42 + 5826.76i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 9 q^{5}+O(q^{10}) $ 72 * q - 9 * q^5	$ 72 q - 9 q^{5} - 18 q^{11} + 1278 q^{23} + 441 q^{25} - 1854 q^{29} - 1665 q^{31} + 2673 q^{35} + 5472 q^{41} + 1260 q^{43} - 5103 q^{47} - 5904 q^{49} + 10944 q^{59} + 8352 q^{61} - 8757 q^{65} + 378 q^{67} + 19764 q^{71} + 6111 q^{73} + 5679 q^{77} - 5652 q^{79} + 20061 q^{83} + 26100 q^{85} - 15633 q^{89} - 6039 q^{91} - 48024 q^{95} - 37530 q^{97}+O(q^{100}) $ 72 * q - 9 * q^5 - 18 * q^11 + 1278 * q^23 + 441 * q^25 - 1854 * q^29 - 1665 * q^31 + 2673 * q^35 + 5472 * q^41 + 1260 * q^43 - 5103 * q^47 - 5904 * q^49 + 10944 * q^59 + 8352 * q^61 - 8757 * q^65 + 378 * q^67 + 19764 * q^71 + 6111 * q^73 + 5679 * q^77 - 5652 * q^79 + 20061 * q^83 + 26100 * q^85 - 15633 * q^89 - 6039 * q^91 - 48024 * q^95 - 37530 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{11}{18}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−12.7195	−	2.24280i	−0.508782	−	0.0897119i	−0.0866362	−	0.996240i	$-0.527612\pi$
$5$	−12.7195	−	2.24280i	−0.508782	−	0.0897119i	−0.422146	+	0.906528i	$0.638723\pi$
$6$		0			0
$7$	−50.9819	+	42.7789i	−1.04045	+	0.873039i	−0.992057	−	0.125790i	$-0.959853\pi$
$7$	−50.9819	+	42.7789i	−1.04045	+	0.873039i	−0.0483899	+	0.998829i	$0.515409\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−53.1143	+	9.36548i	−0.438961	+	0.0774007i	−0.388761	−	0.921339i	$-0.627097\pi$
$11$	−53.1143	+	9.36548i	−0.438961	+	0.0774007i	−0.0501997	+	0.998739i	$0.515986\pi$
$12$		0			0
$13$	−35.6301	+	12.9683i	−0.210829	+	0.0767355i	−0.445276	−	0.895393i	$-0.646894\pi$
$13$	−35.6301	+	12.9683i	−0.210829	+	0.0767355i	0.234447	+	0.972129i	$0.424672\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	110.279	+	63.6697i	0.381589	+	0.220310i	0.678509	−	0.734592i	$-0.262626\pi$
$17$	110.279	+	63.6697i	0.381589	+	0.220310i	−0.296921	+	0.954902i	$0.595960\pi$
$18$		0			0
$19$	32.3968	+	56.1128i	0.0897417	+	0.155437i	0.907402	−	0.420264i	$-0.138063\pi$
$19$	32.3968	+	56.1128i	0.0897417	+	0.155437i	−0.817660	+	0.575701i	$0.804729\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	552.034	−	657.888i	1.04354	−	1.24364i	0.0743772	−	0.997230i	$-0.476303\pi$
$23$	552.034	−	657.888i	1.04354	−	1.24364i	0.969165	−	0.246414i	$-0.0792524\pi$
$24$		0			0
$25$	−430.551	−	156.708i	−0.688882	−	0.250733i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	103.294	−	283.798i	0.122823	−	0.337453i	−0.863009	−	0.505188i	$-0.831423\pi$
$29$	103.294	−	283.798i	0.122823	−	0.337453i	0.985832	+	0.167735i	$0.0536453\pi$
$30$		0			0
$31$	1049.54	+	880.673i	1.09214	+	0.916413i	0.996872	−	0.0790392i	$-0.0251852\pi$
$31$	1049.54	+	880.673i	1.09214	+	0.916413i	0.0952665	+	0.995452i	$0.469630\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	744.411	−	429.786i	0.607682	−	0.350846i
$36$		0			0
$37$	554.706	−	960.778i	0.405190	−	0.701810i	−0.589153	−	0.808021i	$-0.700539\pi$
$37$	554.706	−	960.778i	0.405190	−	0.701810i	0.994344	+	0.106211i	$0.0338719\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−733.845	−	2016.22i	−0.436553	−	1.19942i	−0.941720	−	0.336398i	$-0.890791\pi$
$41$	−733.845	−	2016.22i	−0.436553	−	1.19942i	0.505167	−	0.863021i	$-0.331431\pi$
$42$		0			0
$43$	−48.5881	−	275.557i	−0.0262780	−	0.149030i	0.968846	−	0.247665i	$-0.0796634\pi$
$43$	−48.5881	−	275.557i	−0.0262780	−	0.149030i	−0.995124	+	0.0986354i	$0.968552\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−775.618	−	924.345i	−0.351117	−	0.418445i	0.561361	−	0.827571i	$-0.310278\pi$
$47$	−775.618	−	924.345i	−0.351117	−	0.418445i	−0.912478	+	0.409126i	$0.865834\pi$
$48$		0			0
$49$	352.191	−	1997.37i	0.146685	−	0.831893i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			2045.36i			0.728147i	0.931370	+	0.364073i	$0.118614\pi$
$53$			2045.36i			0.728147i	−0.931370	+	0.364073i	$0.881386\pi$
$54$		0			0
$55$	696.594			0.230279
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−664.820	−	117.226i	−0.190985	−	0.0336759i	0.0773372	−	0.997005i	$-0.475358\pi$
$59$	−664.820	−	117.226i	−0.190985	−	0.0336759i	−0.268323	+	0.963329i	$0.586469\pi$
$60$		0			0
$61$	3039.96	−	2550.83i	0.816973	−	0.685521i	−0.135288	−	0.990806i	$-0.543196\pi$
$61$	3039.96	−	2550.83i	0.816973	−	0.685521i	0.952261	+	0.305285i	$0.0987517\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	482.284	−	85.0396i	0.114150	−	0.0201277i
$66$		0			0
$67$	6936.65	−	2524.73i	1.54526	−	0.562427i	0.577957	−	0.816067i	$-0.303850\pi$
$67$	6936.65	−	2524.73i	1.54526	−	0.562427i	0.967299	+	0.253640i	$0.0816279\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7606.56	−	4391.65i	−1.50894	−	0.871186i	−0.999946	−	0.0104155i	$-0.996685\pi$
$71$	−7606.56	−	4391.65i	−1.50894	−	0.871186i	−0.508993	−	0.860771i	$-0.669982\pi$
$72$		0			0
$73$	323.073	+	559.578i	0.0606254	+	0.105006i	0.894745	−	0.446577i	$-0.147357\pi$
$73$	323.073	+	559.578i	0.0606254	+	0.105006i	−0.834120	+	0.551583i	$0.814024\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2307.22	−	2749.64i	0.389142	−	0.463761i
$78$		0			0
$79$	8964.88	+	3262.95i	1.43645	+	0.522825i	0.938772	−	0.344538i	$-0.111964\pi$
$79$	8964.88	+	3262.95i	1.43645	+	0.522825i	0.497676	+	0.867363i	$0.334187\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1737.55	−	4773.87i	0.252220	−	0.692970i	−0.747372	−	0.664406i	$-0.768684\pi$
$83$	1737.55	−	4773.87i	0.252220	−	0.692970i	0.999592	−	0.0285636i	$-0.00909331\pi$
$84$		0			0
$85$	−1259.90	−	1057.18i	−0.174381	−	0.146323i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10674.4	−	6162.88i	1.34761	−	0.778043i	0.359700	−	0.933068i	$-0.382879\pi$
$89$	10674.4	−	6162.88i	1.34761	−	0.778043i	0.987911	+	0.155024i	$0.0495456\pi$
$90$		0			0
$91$	1261.72	−	2185.36i	0.152363	−	0.263901i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−286.222	−	786.389i	−0.0317144	−	0.0871345i
$96$		0			0
$97$	1027.42	+	5826.76i	0.109195	+	0.619275i	0.989462	+	0.144796i	$0.0462526\pi$
$97$	1027.42	+	5826.76i	0.109195	+	0.619275i	−0.880267	+	0.474479i	$0.842636\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6861.65	−	8177.40i	−0.672645	−	0.801627i	0.316497	−	0.948594i	$-0.397493\pi$
$101$	−6861.65	−	8177.40i	−0.672645	−	0.801627i	−0.989141	+	0.146967i	$0.953049\pi$
$102$		0			0
$103$	2452.40	−	13908.3i	0.231162	−	1.31099i	−0.619385	−	0.785088i	$-0.712618\pi$
$103$	2452.40	−	13908.3i	0.231162	−	1.31099i	0.850547	−	0.525899i	$-0.176271\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			5869.63i			0.512676i	0.966587	+	0.256338i	$0.0825161\pi$
$107$			5869.63i			0.512676i	−0.966587	+	0.256338i	$0.917484\pi$
$108$		0			0
$109$	−7570.66			−0.637207			−0.318604	−	0.947888i	$-0.603214\pi$
$109$	−7570.66			−0.637207			−0.318604	+	0.947888i	$0.603214\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7266.11	+	1281.21i	0.569043	+	0.100338i	0.450764	−	0.892643i	$-0.351151\pi$
$113$	7266.11	+	1281.21i	0.569043	+	0.100338i	0.118278	+	0.992980i	$0.462263\pi$
$114$		0			0
$115$	−8497.12	+	7129.93i	−0.642505	+	0.539125i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−8345.95	+	1471.62i	−0.589362	+	0.103920i
$120$		0			0
$121$	−11024.6	+	4012.64i	−0.752997	+	0.274068i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12115.8	+	6995.07i	0.775412	+	0.447684i
$126$		0			0
$127$	−9299.46	−	16107.1i	−0.576567	−	0.998644i	−0.995869	−	0.0907972i	$-0.971058\pi$
$127$	−9299.46	−	16107.1i	−0.576567	−	0.998644i	0.419302	−	0.907847i	$-0.362275\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	13911.9	−	16579.6i	0.810671	−	0.966120i	−0.189204	−	0.981938i	$-0.560591\pi$
$131$	13911.9	−	16579.6i	0.810671	−	0.966120i	0.999875	+	0.0158182i	$0.00503530\pi$
$132$		0			0
$133$	−4052.09	−	1474.84i	−0.229074	−	0.0833762i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6938.93	+	19064.6i	−0.369702	+	1.01575i	0.605774	+	0.795637i	$0.292864\pi$
$137$	−6938.93	+	19064.6i	−0.369702	+	1.01575i	−0.975475	+	0.220110i	$0.929358\pi$
$138$		0			0
$139$	25354.8	+	21275.2i	1.31229	+	1.10115i	0.987878	+	0.155231i	$0.0496121\pi$
$139$	25354.8	+	21275.2i	1.31229	+	1.10115i	0.324416	+	0.945915i	$0.394832\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1771.01	−	1022.49i	0.0866063	−	0.0500022i
$144$		0			0
$145$	−1950.36	+	3378.11i	−0.0927636	+	0.160671i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8127.74	−	22330.8i	−0.366098	−	1.00585i	−0.976832	−	0.214010i	$-0.931348\pi$
$149$	−8127.74	−	22330.8i	−0.366098	−	1.00585i	0.610733	−	0.791836i	$-0.290875\pi$
$150$		0			0
$151$	4804.82	+	27249.5i	0.210728	+	1.19510i	0.888167	+	0.459521i	$0.151979\pi$
$151$	4804.82	+	27249.5i	0.210728	+	1.19510i	−0.677438	+	0.735579i	$0.736910\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−11374.6	−	13555.7i	−0.473447	−	0.564232i
$156$		0			0
$157$	1360.25	−	7714.37i	0.0551848	−	0.312969i	−0.944703	−	0.327926i	$-0.893650\pi$
$157$	1360.25	−	7714.37i	0.0551848	−	0.312969i	0.999888	+	0.0149576i	$0.00476134\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			57155.8i			2.20500i
$162$		0			0
$163$	−15688.8			−0.590491			−0.295246	−	0.955421i	$-0.595401\pi$
$163$	−15688.8			−0.590491			−0.295246	+	0.955421i	$0.595401\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	14825.7	+	2614.16i	0.531595	+	0.0937345i	0.433001	−	0.901393i	$-0.357455\pi$
$167$	14825.7	+	2614.16i	0.531595	+	0.0937345i	0.0985936	+	0.995128i	$0.468566\pi$
$168$		0			0
$169$	−20777.7	+	17434.5i	−0.727484	+	0.610432i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−47917.4	+	8449.13i	−1.60104	+	0.282306i	−0.901659	−	0.432447i	$-0.857650\pi$
$173$	−47917.4	+	8449.13i	−1.60104	+	0.282306i	−0.699377	+	0.714753i	$0.746539\pi$
$174$		0			0
$175$	28654.1	−	10429.2i	0.935644	−	0.340547i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	39279.8	+	22678.2i	1.22592	+	0.707786i	0.966174	−	0.257891i	$-0.0830274\pi$
$179$	39279.8	+	22678.2i	1.22592	+	0.707786i	0.259747	+	0.965677i	$0.416361\pi$
$180$		0			0
$181$	−18860.8	−	32667.9i	−0.575710	−	0.997158i	−0.995964	−	0.0897520i	$-0.971393\pi$
$181$	−18860.8	−	32667.9i	−0.575710	−	0.997158i	0.420255	−	0.907406i	$-0.361941\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−9210.43	+	10976.6i	−0.269114	+	0.320718i
$186$		0			0
$187$	−6453.69	−	2348.95i	−0.184555	−	0.0671724i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−14692.1	+	40366.3i	−0.402734	+	1.10650i	0.558195	+	0.829710i	$0.311494\pi$
$191$	−14692.1	+	40366.3i	−0.402734	+	1.10650i	−0.960929	+	0.276793i	$0.910728\pi$
$192$		0			0
$193$	−22734.0	−	19076.1i	−0.610324	−	0.512122i	0.284421	−	0.958699i	$-0.408199\pi$
$193$	−22734.0	−	19076.1i	−0.610324	−	0.512122i	−0.894745	+	0.446577i	$0.852643\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−55365.5	+	31965.3i	−1.42661	+	0.823656i	−0.996852	−	0.0792854i	$-0.974736\pi$
$197$	−55365.5	+	31965.3i	−1.42661	+	0.823656i	−0.429763	+	0.902942i	$0.641403\pi$
$198$		0			0
$199$	−16884.5	+	29244.8i	−0.426365	+	0.738486i	−0.996547	−	0.0830327i	$-0.973539\pi$
$199$	−16884.5	+	29244.8i	−0.426365	+	0.738486i	0.570182	+	0.821519i	$0.306873\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6874.44	+	18887.4i	0.166819	+	0.458331i
$204$		0			0
$205$	4812.19	+	27291.3i	0.114508	+	0.649406i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2246.25	−	2676.98i	−0.0514241	−	0.0612848i
$210$		0			0
$211$	5066.91	−	28735.9i	0.113809	−	0.645445i	−0.873523	−	0.486782i	$-0.838171\pi$
$211$	5066.91	−	28735.9i	0.113809	−	0.645445i	0.987333	−	0.158663i	$-0.0507183\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			3613.93i			0.0781812i
$216$		0			0
$217$	−91182.0			−1.93638
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4754.94	−	838.424i	−0.0973555	−	0.0171664i
$222$		0			0
$223$	−16956.2	+	14227.9i	−0.340972	+	0.286109i	−0.797153	−	0.603778i	$-0.793661\pi$
$223$	−16956.2	+	14227.9i	−0.340972	+	0.286109i	0.456181	+	0.889887i	$0.349217\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	39043.0	−	6884.34i	0.757690	−	0.133601i	0.218560	−	0.975824i	$-0.429864\pi$
$227$	39043.0	−	6884.34i	0.757690	−	0.133601i	0.539130	+	0.842222i	$0.318753\pi$
$228$		0			0
$229$	38414.7	−	13981.8i	0.732532	−	0.266620i	0.0512959	−	0.998684i	$-0.483665\pi$
$229$	38414.7	−	13981.8i	0.732532	−	0.266620i	0.681236	+	0.732064i	$0.261443\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	31898.3	+	18416.5i	0.587565	+	0.339231i	0.764134	−	0.645058i	$-0.223167\pi$
$233$	31898.3	+	18416.5i	0.587565	+	0.339231i	−0.176569	+	0.984288i	$0.556500\pi$
$234$		0			0
$235$	7792.38	+	13496.8i	0.141102	+	0.244397i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	70607.6	−	84146.8i	1.23611	−	1.47313i	0.407591	−	0.913164i	$-0.366369\pi$
$239$	70607.6	−	84146.8i	1.23611	−	1.47313i	0.828514	−	0.559969i	$-0.189187\pi$
$240$		0			0
$241$	40453.6	+	14723.9i	0.696504	+	0.253507i	0.665917	−	0.746026i	$-0.268040\pi$
$241$	40453.6	+	14723.9i	0.696504	+	0.253507i	0.0305863	+	0.999532i	$0.490263\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−8959.42	+	24615.8i	−0.149261	+	0.410092i
$246$		0			0
$247$	−1881.99	−	1579.17i	−0.0308477	−	0.0258843i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−11903.2	+	6872.34i	−0.188937	+	0.109083i	−0.591485	−	0.806316i	$-0.701458\pi$
$251$	−11903.2	+	6872.34i	−0.188937	+	0.109083i	0.402548	+	0.915399i	$0.368125\pi$
$252$		0			0
$253$	−23159.4	+	40113.3i	−0.361815	+	0.626682i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1233.88	−	3390.05i	−0.0186812	−	0.0513263i	0.930002	−	0.367556i	$-0.119805\pi$
$257$	−1233.88	−	3390.05i	−0.0186812	−	0.0513263i	−0.948683	+	0.316229i	$0.897583\pi$
$258$		0			0
$259$	12821.1	+	72712.0i	0.191128	+	1.08394i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−58225.4	−	69390.4i	−0.841785	−	1.00320i	−0.999876	−	0.0157758i	$-0.994978\pi$
$263$	−58225.4	−	69390.4i	−0.841785	−	1.00320i	0.158090	−	0.987425i	$-0.449466\pi$
$264$		0			0
$265$	4587.34	−	26016.1i	0.0653235	−	0.370468i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	42707.4i		−	0.590199i	−0.955466	−	0.295100i	$-0.904647\pi$
$269$		−	42707.4i		−	0.590199i	0.955466	−	0.295100i	$-0.0953528\pi$
$270$		0			0
$271$	−32308.2			−0.439920			−0.219960	−	0.975509i	$-0.570593\pi$
$271$	−32308.2			−0.439920			−0.219960	+	0.975509i	$0.570593\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	24336.1	+	4291.10i	0.321799	+	0.0567419i
$276$		0			0
$277$	−37675.3	+	31613.3i	−0.491017	+	0.412012i	−0.854391	−	0.519631i	$-0.826069\pi$
$277$	−37675.3	+	31613.3i	−0.491017	+	0.412012i	0.363373	+	0.931644i	$0.381625\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	15371.6	−	2710.43i	0.194673	−	0.0343262i	−0.0754612	−	0.997149i	$-0.524043\pi$
$281$	15371.6	−	2710.43i	0.194673	−	0.0343262i	0.270135	+	0.962823i	$0.412932\pi$
$282$		0			0
$283$	19523.3	−	7105.88i	0.243769	−	0.0887248i	−0.217246	−	0.976117i	$-0.569707\pi$
$283$	19523.3	−	7105.88i	0.243769	−	0.0887248i	0.461016	+	0.887392i	$0.347485\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	123665.	+	71397.8i	1.50135	+	0.866804i
$288$		0			0
$289$	−33652.8	−	58288.4i	−0.402927	−	0.697890i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−38793.1	+	46231.8i	−0.451876	+	0.538525i	−0.943100	−	0.332508i	$-0.892105\pi$
$293$	−38793.1	+	46231.8i	−0.451876	+	0.538525i	0.491225	+	0.871033i	$0.336549\pi$
$294$		0			0
$295$	8193.30	+	2982.12i	0.0941488	+	0.0342673i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−11137.3	+	30599.5i	−0.124577	+	0.342273i
$300$		0			0
$301$	14265.1	+	11969.9i	0.157450	+	0.132116i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−44387.8	+	25627.3i	−0.477160	+	0.275489i
$306$		0			0
$307$	66235.6	−	114723.i	0.702773	−	1.21724i	−0.264717	−	0.964326i	$-0.585279\pi$
$307$	66235.6	−	114723.i	0.702773	−	1.21724i	0.967489	−	0.252912i	$-0.0813882\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−49779.2	−	136767.i	−0.514668	−	1.41404i	−0.876322	−	0.481726i	$-0.840010\pi$
$311$	−49779.2	−	136767.i	−0.514668	−	1.41404i	0.361654	−	0.932312i	$-0.382212\pi$
$312$		0			0
$313$	−2180.14	−	12364.2i	−0.0222534	−	0.126205i	0.971657	−	0.236394i	$-0.0759656\pi$
$313$	−2180.14	−	12364.2i	−0.0222534	−	0.126205i	−0.993911	+	0.110189i	$0.964854\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−93196.9	−	111068.i	−0.927433	−	1.10527i	−0.994205	−	0.107504i	$-0.965714\pi$
$317$	−93196.9	−	111068.i	−0.927433	−	1.10527i	0.0667713	−	0.997768i	$-0.478730\pi$
$318$		0			0
$319$	−2828.48	+	16041.1i	−0.0277954	+	0.157635i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			8250.76i			0.0790841i
$324$		0			0
$325$	17372.8			0.164476
$326$		0			0
$327$		0			0
$328$		0			0
$329$	79084.9	+	13944.8i	0.730637	+	0.128831i
$330$		0			0
$331$	70836.2	−	59438.6i	0.646546	−	0.542516i	−0.259475	−	0.965750i	$-0.583549\pi$
$331$	70836.2	−	59438.6i	0.646546	−	0.542516i	0.906021	+	0.423234i	$0.139105\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−93893.5	+	16556.0i	−0.836654	+	0.147525i
$336$		0			0
$337$	114875.	−	41811.2i	1.01150	−	0.368157i	0.217493	−	0.976062i	$-0.430212\pi$
$337$	114875.	−	41811.2i	1.01150	−	0.368157i	0.794010	+	0.607905i	$0.207990\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−63993.7	−	36946.8i	−0.550337	−	0.317737i
$342$		0			0
$343$	−12405.8	−	21487.4i	−0.105447	−	0.182640i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	143847.	−	171431.i	1.19466	−	1.42374i	0.314367	−	0.949302i	$-0.398208\pi$
$347$	143847.	−	171431.i	1.19466	−	1.42374i	0.880290	−	0.474435i	$-0.157348\pi$
$348$		0			0
$349$	87358.3	+	31795.8i	0.717221	+	0.261047i	0.674746	−	0.738050i	$-0.264253\pi$
$349$	87358.3	+	31795.8i	0.717221	+	0.261047i	0.0424757	+	0.999098i	$0.486476\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−28610.0	+	78605.2i	−0.229598	+	0.630815i	−0.999977	−	0.00677099i	$-0.997845\pi$
$353$	−28610.0	+	78605.2i	−0.229598	+	0.630815i	0.770379	+	0.637586i	$0.220067\pi$
$354$		0			0
$355$	86902.4	+	72919.8i	0.689565	+	0.578613i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	175614.	−	101391.i	1.36260	−	0.786699i	0.372633	−	0.927979i	$-0.378455\pi$
$359$	175614.	−	101391.i	1.36260	−	0.786699i	0.989970	+	0.141279i	$0.0451215\pi$
$360$		0			0
$361$	63061.4	−	109226.i	0.483893	−	0.838127i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2854.31	−	7842.16i	−0.0214248	−	0.0588641i
$366$		0			0
$367$	−6653.31	−	37732.8i	−0.0493976	−	0.280148i	0.950096	−	0.311957i	$-0.100984\pi$
$367$	−6653.31	−	37732.8i	−0.0493976	−	0.280148i	−0.999494	+	0.0318088i	$0.989873\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−87498.4	−	104277.i	−0.635700	−	0.757598i
$372$		0			0
$373$	14463.8	−	82028.1i	0.103959	−	0.589583i	−0.887671	−	0.460477i	$-0.847678\pi$
$373$	14463.8	−	82028.1i	0.103959	−	0.589583i	0.991631	−	0.129106i	$-0.0412107\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			11451.3i			0.0805698i
$378$		0			0
$379$	272691.			1.89842			0.949211	−	0.314641i	$-0.101884\pi$
$379$	272691.			1.89842			0.949211	+	0.314641i	$0.101884\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−148837.	−	26243.9i	−1.01464	−	0.178909i	−0.358486	−	0.933535i	$-0.616707\pi$
$383$	−148837.	−	26243.9i	−1.01464	−	0.178909i	−0.656155	+	0.754626i	$0.727818\pi$
$384$		0			0
$385$	−35513.7	+	29799.5i	−0.239593	+	0.201043i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	219772.	−	38751.7i	1.45235	−	0.256089i	0.608881	−	0.793262i	$-0.291619\pi$
$389$	219772.	−	38751.7i	1.45235	−	0.256089i	0.843472	+	0.537173i	$0.180508\pi$
$390$		0			0
$391$	102765.	−	37403.5i	0.672191	−	0.244658i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−106711.	−	61609.6i	−0.683935	−	0.394870i
$396$		0			0
$397$	23756.9	+	41148.1i	0.150733	+	0.261077i	0.931497	−	0.363749i	$-0.118503\pi$
$397$	23756.9	+	41148.1i	0.150733	+	0.261077i	−0.780764	+	0.624826i	$0.785170\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−121108.	+	144331.i	−0.753152	+	0.897572i	−0.997394	−	0.0721422i	$-0.977016\pi$
$401$	−121108.	+	144331.i	−0.753152	+	0.897572i	0.244242	+	0.969714i	$0.421461\pi$
$402$		0			0
$403$	−48816.2	−	17767.6i	−0.300576	−	0.109401i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−20464.6	+	56226.1i	−0.123542	+	0.339429i
$408$		0			0
$409$	−211385.	−	177373.i	−1.26365	−	1.06033i	−0.995283	−	0.0970133i	$-0.969071\pi$
$409$	−211385.	−	177373.i	−1.26365	−	1.06033i	−0.268369	−	0.963316i	$-0.586484\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	38908.6	−	22463.9i	0.228111	−	0.131700i
$414$		0			0
$415$	−32807.6	+	56824.5i	−0.190493	+	0.329943i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	44328.9	+	121793.i	0.252499	+	0.693734i	0.999579	+	0.0290010i	$0.00923260\pi$
$419$	44328.9	+	121793.i	0.252499	+	0.693734i	−0.747081	+	0.664733i	$0.768545\pi$
$420$		0			0
$421$	−2327.75	−	13201.3i	−0.0131332	−	0.0744823i	0.977537	−	0.210765i	$-0.0675955\pi$
$421$	−2327.75	−	13201.3i	−0.0131332	−	0.0744823i	−0.990670	+	0.136283i	$0.956484\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−37503.3	−	44694.7i	−0.207631	−	0.247444i
$426$		0			0
$427$	−45861.2	+	260092.i	−0.251530	+	1.42650i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			46492.4i			0.250281i	0.992139	+	0.125140i	$0.0399381\pi$
$431$			46492.4i			0.250281i	−0.992139	+	0.125140i	$0.960062\pi$
$432$		0			0
$433$	−185099.			−0.987252			−0.493626	−	0.869674i	$-0.664329\pi$
$433$	−185099.			−0.987252			−0.493626	+	0.869674i	$0.664329\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	54800.1	+	9662.73i	0.286958	+	0.0505984i
$438$		0			0
$439$	179417.	−	150549.i	0.930969	−	0.781176i	−0.0450221	−	0.998986i	$-0.514336\pi$
$439$	179417.	−	150549.i	0.930969	−	0.781176i	0.975991	+	0.217810i	$0.0698914\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	371286.	−	65467.7i	1.89191	−	0.333595i	0.897661	−	0.440686i	$-0.145265\pi$
$443$	371286.	−	65467.7i	1.89191	−	0.333595i	0.994249	+	0.107091i	$0.0341536\pi$
$444$		0			0
$445$	−149596.	+	54448.5i	−0.755440	+	0.274957i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−316831.	−	182922.i	−1.57157	−	0.907348i	−0.995977	−	0.0896134i	$-0.971437\pi$
$449$	−316831.	−	182922.i	−1.57157	−	0.907348i	−0.575596	−	0.817734i	$-0.695230\pi$
$450$		0			0
$451$	57860.6	+	100217.i	0.284466	+	0.492709i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−20949.8	+	24967.0i	−0.101195	+	0.120599i
$456$		0			0
$457$	−121964.	−	44391.3i	−0.583983	−	0.212552i	0.0330982	−	0.999452i	$-0.489463\pi$
$457$	−121964.	−	44391.3i	−0.583983	−	0.212552i	−0.617081	+	0.786900i	$0.711685\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−42641.6	+	117157.i	−0.200647	+	0.551272i	−0.998681	−	0.0513395i	$-0.983651\pi$
$461$	−42641.6	+	117157.i	−0.200647	+	0.551272i	0.798035	+	0.602612i	$0.205873\pi$
$462$		0			0
$463$	−20676.2	−	17349.4i	−0.0964513	−	0.0809322i	0.593287	−	0.804991i	$-0.297830\pi$
$463$	−20676.2	−	17349.4i	−0.0964513	−	0.0809322i	−0.689738	+	0.724059i	$0.742274\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6504.77	+	3755.53i	−0.0298262	+	0.0172202i	−0.514839	−	0.857287i	$-0.672148\pi$
$467$	−6504.77	+	3755.53i	−0.0298262	+	0.0172202i	0.485013	+	0.874507i	$0.338815\pi$
$468$		0			0
$469$	−245638.	+	425458.i	−1.11674	+	1.93424i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5161.44	+	14180.9i	0.0230700	+	0.0633844i
$474$		0			0
$475$	−5155.15	−	29236.3i	−0.0228483	−	0.129579i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	18692.0	+	22276.3i	0.0814677	+	0.0970895i	0.805239	−	0.592951i	$-0.202037\pi$
$479$	18692.0	+	22276.3i	0.0814677	+	0.0970895i	−0.723771	+	0.690040i	$0.757593\pi$
$480$		0			0
$481$	−7304.56	+	41426.2i	−0.0315721	+	0.179054i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	76418.0i		−	0.324872i
$486$		0			0
$487$	−49764.8			−0.209828			−0.104914	−	0.994481i	$-0.533457\pi$
$487$	−49764.8			−0.209828			−0.104914	+	0.994481i	$0.533457\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−90254.7	−	15914.3i	−0.374375	−	0.0660124i	−0.0167050	−	0.999860i	$-0.505318\pi$
$491$	−90254.7	−	15914.3i	−0.374375	−	0.0660124i	−0.357670	+	0.933848i	$0.616429\pi$
$492$		0			0
$493$	29460.5	−	24720.3i	0.121212	−	0.101709i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	575667.	−	101506.i	2.33055	−	0.410939i
$498$		0			0
$499$	−191913.	+	69850.7i	−0.770733	+	0.280524i	−0.697303	−	0.716776i	$-0.745617\pi$
$499$	−191913.	+	69850.7i	−0.770733	+	0.280524i	−0.0734301	+	0.997300i	$0.523395\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	13803.7	+	7969.57i	0.0545582	+	0.0314992i	0.527031	−	0.849846i	$-0.323305\pi$
$503$	13803.7	+	7969.57i	0.0545582	+	0.0314992i	−0.472473	+	0.881345i	$0.656639\pi$
$504$		0			0
$505$	68936.8	+	119402.i	0.270314	+	0.468197i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−244551.	+	291444.i	−0.943916	+	1.12492i	0.0481046	+	0.998842i	$0.484682\pi$
$509$	−244551.	+	291444.i	−0.943916	+	1.12492i	−0.992021	+	0.126073i	$0.959763\pi$
$510$		0			0
$511$	−40409.0	−	14707.7i	−0.154752	−	0.0563251i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−62386.8	+	171406.i	−0.235222	+	0.646268i
$516$		0			0
$517$	49853.3	+	41831.9i	0.186515	+	0.156504i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	111957.	−	64638.2i	0.412453	−	0.238130i	−0.279390	−	0.960178i	$-0.590132\pi$
$521$	111957.	−	64638.2i	0.412453	−	0.238130i	0.691843	+	0.722048i	$0.256799\pi$
$522$		0			0
$523$	165029.	−	285838.i	0.603332	−	1.04500i	−0.388981	−	0.921246i	$-0.627173\pi$
$523$	165029.	−	285838.i	0.603332	−	1.04500i	0.992313	−	0.123756i	$-0.0394939\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	59670.7	+	163944.i	0.214852	+	0.590302i
$528$		0			0
$529$	−79481.7	−	450763.i	−0.284024	−	1.61078i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	52294.0	+	62321.5i	0.184076	+	0.219373i
$534$		0			0
$535$	13164.4	−	74659.0i	0.0459932	−	0.260840i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			109388.i			0.376522i
$540$		0			0
$541$	398933.			1.36303			0.681515	−	0.731805i	$-0.261322\pi$
$541$	398933.			1.36303			0.681515	+	0.731805i	$0.261322\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	96295.3	+	16979.5i	0.324199	+	0.0571651i
$546$		0			0
$547$	−401021.	+	336496.i	−1.34027	+	1.12462i	−0.358712	+	0.933448i	$0.616784\pi$
$547$	−401021.	+	336496.i	−1.34027	+	1.12462i	−0.981557	+	0.191171i	$0.938771\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	19271.1	−	3398.02i	0.0634751	−	0.0111924i
$552$		0			0
$553$	−596632.	+	217156.i	−1.95099	+	0.710104i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−85866.2	−	49574.8i	−0.276765	−	0.159791i	0.355193	−	0.934793i	$-0.384415\pi$
$557$	−85866.2	−	49574.8i	−0.276765	−	0.159791i	−0.631958	+	0.775003i	$0.717749\pi$
$558$		0			0
$559$	5304.69	+	9188.00i	0.0169761	+	0.0294034i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−165576.	+	197325.i	−0.522372	+	0.622538i	−0.961140	−	0.276062i	$-0.910970\pi$
$563$	−165576.	+	197325.i	−0.522372	+	0.622538i	0.438768	+	0.898600i	$0.355415\pi$
$564$		0			0
$565$	−89548.0	−	32592.8i	−0.280517	−	0.102100i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−181684.	+	499173.i	−0.561168	+	1.54180i	0.256760	+	0.966475i	$0.417345\pi$
$569$	−181684.	+	499173.i	−0.561168	+	1.54180i	−0.817928	+	0.575321i	$0.804877\pi$
$570$		0			0
$571$	149431.	+	125387.i	0.458319	+	0.384575i	0.842512	−	0.538677i	$-0.181076\pi$
$571$	149431.	+	125387.i	0.458319	+	0.384575i	−0.384193	+	0.923253i	$0.625520\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−340775.	+	196747.i	−1.03070	+	0.595075i
$576$		0			0
$577$	−36785.1	+	63713.7i	−0.110489	+	0.191373i	−0.915968	−	0.401252i	$-0.868575\pi$
$577$	−36785.1	+	63713.7i	−0.110489	+	0.191373i	0.805478	+	0.592625i	$0.201908\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	115637.	+	317711.i	0.342568	+	0.941196i
$582$		0			0
$583$	−19155.8	−	108638.i	−0.0563591	−	0.319628i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−267913.	−	319286.i	−0.777531	−	0.926625i	0.221289	−	0.975208i	$-0.428974\pi$
$587$	−267913.	−	319286.i	−0.777531	−	0.926625i	−0.998819	+	0.0485835i	$0.984529\pi$
$588$		0			0
$589$	−15415.2	+	87423.9i	−0.0444343	+	0.251999i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	66415.7i		−	0.188869i	−0.995531	−	0.0944347i	$-0.969896\pi$
$593$		−	66415.7i		−	0.188869i	0.995531	−	0.0944347i	$-0.0301044\pi$
$594$		0			0
$595$	109457.			0.309179
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−453163.	−	79904.9i	−1.26299	−	0.222700i	−0.498249	−	0.867034i	$-0.666024\pi$
$599$	−453163.	−	79904.9i	−1.26299	−	0.222700i	−0.764744	+	0.644334i	$0.777135\pi$
$600$		0			0
$601$	423835.	−	355640.i	1.17340	−	0.984603i	0.173404	−	0.984851i	$-0.444523\pi$
$601$	423835.	−	355640.i	1.17340	−	0.984603i	1.00000		0.000247732i	$7.88556e-5\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	149228.	−	26312.9i	0.407698	−	0.0718882i
$606$		0			0
$607$	−369178.	+	134370.i	−1.00198	+	0.364690i	−0.790347	−	0.612660i	$-0.790100\pi$
$607$	−369178.	+	134370.i	−1.00198	+	0.364690i	−0.211631	+	0.977350i	$0.567877\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	39622.5	+	22876.1i	0.106135	+	0.0612772i
$612$		0			0
$613$	179948.	+	311679.i	0.478879	+	0.829443i	0.999707	−	0.0242190i	$-0.00770991\pi$
$613$	179948.	+	311679.i	0.478879	+	0.829443i	−0.520828	+	0.853662i	$0.674377\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4613.78	−	5498.49i	0.0121196	−	0.0144435i	−0.759951	−	0.649981i	$-0.774777\pi$
$617$	4613.78	−	5498.49i	0.0121196	−	0.0144435i	0.772070	+	0.635537i	$0.219221\pi$
$618$		0			0
$619$	−644677.	−	234643.i	−1.68252	−	0.612388i	−0.688870	−	0.724884i	$-0.741893\pi$
$619$	−644677.	−	234643.i	−1.68252	−	0.612388i	−0.993652	+	0.112496i	$0.964115\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−280561.	+	770835.i	−0.722855	+	1.98603i
$624$		0			0
$625$	80948.8	+	67924.1i	0.207229	+	0.173886i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	122345.	−	70635.8i	0.309232	−	0.178535i
$630$		0			0
$631$	171922.	−	297778.i	0.431790	−	0.747883i	−0.565237	−	0.824928i	$-0.691215\pi$
$631$	171922.	−	297778.i	0.431790	−	0.747883i	0.997028	+	0.0770456i	$0.0245487\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	82159.8	+	225732.i	0.203757	+	0.559817i
$636$		0			0
$637$	13353.9	+	75733.9i	0.0329102	+	0.186643i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−40211.0	−	47921.6i	−0.0978654	−	0.116631i	0.714889	−	0.699238i	$-0.246477\pi$
$641$	−40211.0	−	47921.6i	−0.0978654	−	0.116631i	−0.812754	+	0.582607i	$0.802033\pi$
$642$		0			0
$643$	−120245.	+	681942.i	−0.290834	+	1.64940i	0.392837	+	0.919608i	$0.371494\pi$
$643$	−120245.	+	681942.i	−0.290834	+	1.64940i	−0.683671	+	0.729791i	$0.739617\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	396881.i		−	0.948094i	−0.880499	−	0.474047i	$-0.842793\pi$
$647$		−	396881.i		−	0.948094i	0.880499	−	0.474047i	$-0.157207\pi$
$648$		0			0
$649$	36409.3			0.0864417
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−87977.7	−	15512.8i	−0.206322	−	0.0363802i	0.0695316	−	0.997580i	$-0.477850\pi$
$653$	−87977.7	−	15512.8i	−0.206322	−	0.0363802i	−0.275854	+	0.961200i	$0.588961\pi$
$654$		0			0
$655$	−214138.	+	179683.i	−0.499127	+	0.418817i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	603844.	−	106474.i	1.39044	−	0.245173i	0.572233	−	0.820091i	$-0.306077\pi$
$659$	603844.	−	106474.i	1.39044	−	0.245173i	0.818211	+	0.574918i	$0.194966\pi$
$660$		0			0
$661$	213765.	−	77804.0i	0.489252	−	0.178073i	−0.0856014	−	0.996329i	$-0.527281\pi$
$661$	213765.	−	77804.0i	0.489252	−	0.178073i	0.574854	+	0.818256i	$0.305059\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	48233.0	+	27847.3i	0.109069	+	0.0629710i
$666$		0			0
$667$	−129686.	−	224622.i	−0.291501	−	0.504895i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−137575.	+	163956.i	−0.305559	+	0.364151i
$672$		0			0
$673$	783882.	+	285310.i	1.73070	+	0.629922i	0.998680	−	0.0513653i	$-0.0163573\pi$
$673$	783882.	+	285310.i	1.73070	+	0.629922i	0.732016	+	0.681287i	$0.238580\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	200241.	−	550158.i	0.436894	−	1.20036i	−0.504608	−	0.863349i	$-0.668363\pi$
$677$	200241.	−	550158.i	0.436894	−	1.20036i	0.941502	−	0.337008i	$-0.109415\pi$
$678$		0			0
$679$	−301642.	−	253108.i	−0.654263	−	0.548992i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	72722.4	−	41986.3i	0.155893	−	0.0900050i	−0.420024	−	0.907513i	$-0.637978\pi$
$683$	72722.4	−	41986.3i	0.155893	−	0.0900050i	0.575917	+	0.817508i	$0.304645\pi$
$684$		0			0
$685$	131018.	−	226930.i	0.279222	−	0.483627i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−26524.9	−	72876.5i	−0.0558747	−	0.153514i
$690$		0			0
$691$	−60391.2	−	342495.i	−0.126479	−	0.717297i	−0.980419	−	0.196925i	$-0.936905\pi$
$691$	−60391.2	−	342495.i	−0.126479	−	0.717297i	0.853940	−	0.520372i	$-0.174207\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−274786.	−	327477.i	−0.568885	−	0.677971i
$696$		0			0
$697$	47444.5	−	269071.i	0.0976607	−	0.553862i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	878363.i		−	1.78747i	−0.448599	−	0.893733i	$-0.648077\pi$
$701$		−	878363.i		−	1.78747i	0.448599	−	0.893733i	$-0.351923\pi$
$702$		0			0
$703$	71882.7			0.145450
$704$		0			0
$705$		0			0
$706$		0			0
$707$	699640.	+	123365.i	1.39970	+	0.246805i
$708$		0			0
$709$	366127.	−	307217.i	0.728348	−	0.611156i	−0.201333	−	0.979523i	$-0.564527\pi$
$709$	366127.	−	307217.i	0.728348	−	0.611156i	0.929681	+	0.368366i	$0.120083\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.15877e6	−	204322.i	2.27938	−	0.401917i
$714$		0			0
$715$	−24819.7	+	9033.64i	−0.0485495	+	0.0176706i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	308819.	+	178297.i	0.597373	+	0.344894i	0.768008	−	0.640441i	$-0.221248\pi$
$719$	308819.	+	178297.i	0.597373	+	0.344894i	−0.170634	+	0.985334i	$0.554582\pi$
$720$		0			0
$721$	469952.	+	813980.i	0.904030	+	1.56583i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−88946.8	+	106003.i	−0.169221	+	0.201670i
$726$		0			0
$727$	−374717.	−	136386.i	−0.708982	−	0.258048i	−0.0377409	−	0.999288i	$-0.512016\pi$
$727$	−374717.	−	136386.i	−0.708982	−	0.258048i	−0.671241	+	0.741239i	$0.734238\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	12186.3	−	33481.7i	0.0228055	−	0.0626575i
$732$		0			0
$733$	−478278.	−	401323.i	−0.890169	−	0.746940i	0.0780751	−	0.996947i	$-0.475123\pi$
$733$	−478278.	−	401323.i	−0.890169	−	0.746940i	−0.968244	+	0.250007i	$0.919567\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−344790.	+	199065.i	−0.634775	+	0.366487i
$738$		0			0
$739$	−305145.	+	528526.i	−0.558749	+	0.967782i	0.438852	+	0.898559i	$0.355385\pi$
$739$	−305145.	+	528526.i	−0.558749	+	0.967782i	−0.997601	+	0.0692229i	$0.977948\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−195090.	−	536006.i	−0.353393	−	0.970940i	−0.981272	−	0.192629i	$-0.938299\pi$
$743$	−195090.	−	536006.i	−0.353393	−	0.970940i	0.627878	−	0.778311i	$-0.283923\pi$
$744$		0			0
$745$	53297.7	+	302266.i	0.0960276	+	0.544599i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−251096.	−	299245.i	−0.447586	−	0.533412i
$750$		0			0
$751$	42940.4	−	243527.i	0.0761353	−	0.431784i	−0.922785	−	0.385316i	$-0.874092\pi$
$751$	42940.4	−	243527.i	0.0761353	−	0.431784i	0.998920	−	0.0464681i	$-0.0147966\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	357377.i		−	0.626950i
$756$		0			0
$757$	−40518.7			−0.0707073			−0.0353536	−	0.999375i	$-0.511256\pi$
$757$	−40518.7			−0.0707073			−0.0353536	+	0.999375i	$0.511256\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−492161.	−	86781.2i	−0.849841	−	0.149850i	−0.268268	−	0.963344i	$-0.586451\pi$
$761$	−492161.	−	86781.2i	−0.849841	−	0.149850i	−0.581573	+	0.813494i	$0.697562\pi$
$762$		0			0
$763$	385967.	−	323864.i	0.662980	−	0.556307i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	25207.8	−	4444.82i	0.0428494	−	0.00755550i
$768$		0			0
$769$	−125460.	+	45663.7i	−0.212155	+	0.0772180i	−0.445911	−	0.895077i	$-0.647120\pi$
$769$	−125460.	+	45663.7i	−0.212155	+	0.0772180i	0.233757	+	0.972295i	$0.424898\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−685790.	−	395941.i	−1.14771	−	0.662630i	−0.199381	−	0.979922i	$-0.563893\pi$
$773$	−685790.	−	395941.i	−1.14771	−	0.662630i	−0.948328	+	0.317292i	$0.897227\pi$
$774$		0			0
$775$	−313874.	−	543647.i	−0.522580	−	0.905135i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	89361.8	−	106497.i	0.147257	−	0.175495i
$780$		0			0
$781$	445147.	+	162020.i	0.729796	+	0.265624i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−34603.5	+	95072.4i	−0.0561541	+	0.154282i
$786$		0			0
$787$	855952.	+	718229.i	1.38197	+	1.15961i	0.968476	+	0.249109i	$0.0801377\pi$
$787$	855952.	+	718229.i	1.38197	+	1.15961i	0.413498	+	0.910505i	$0.364307\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−425249.	+	245517.i	−0.679657	+	0.392400i
$792$		0			0
$793$	−75234.0	+	130309.i	−0.119638	+	0.207219i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	104497.	+	287102.i	0.164508	+	0.451981i	0.994367	−	0.105992i	$-0.0338017\pi$
$797$	104497.	+	287102.i	0.164508	+	0.451981i	−0.829859	+	0.557972i	$0.811579\pi$
$798$		0			0
$799$	−26681.7	−	151319.i	−0.0417945	−	0.237029i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−22400.5	−	26695.9i	−0.0347397	−	0.0414012i
$804$		0			0
$805$	128189.	−	726995.i	0.197815	−	1.12186i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	670428.i		−	1.02437i	−0.858876	−	0.512183i	$-0.828837\pi$
$809$		−	670428.i		−	1.02437i	0.858876	−	0.512183i	$-0.171163\pi$
$810$		0			0
$811$	228060.			0.346743			0.173371	−	0.984857i	$-0.444534\pi$
$811$	228060.			0.346743			0.173371	+	0.984857i	$0.444534\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	199554.	+	35186.7i	0.300431	+	0.0529741i
$816$		0			0
$817$	13888.2	−	11653.6i	0.0208066	−	0.0174588i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	679845.	−	119875.i	1.00861	−	0.177845i	0.355153	−	0.934808i	$-0.384429\pi$
$821$	679845.	−	119875.i	1.00861	−	0.177845i	0.653458	+	0.756963i	$0.273318\pi$
$822$		0			0
$823$	446735.	−	162598.i	0.659554	−	0.240058i	0.00950993	−	0.999955i	$-0.496973\pi$
$823$	446735.	−	162598.i	0.659554	−	0.240058i	0.650044	+	0.759897i	$0.274751\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	689188.	+	397903.i	1.00769	+	0.581790i	0.910514	−	0.413478i	$-0.135686\pi$
$827$	689188.	+	397903.i	1.00769	+	0.581790i	0.0971749	+	0.995267i	$0.469019\pi$
$828$		0			0
$829$	−209011.	−	362018.i	−0.304131	−	0.526771i	0.672936	−	0.739700i	$-0.265033\pi$
$829$	−209011.	−	362018.i	−0.304131	−	0.526771i	−0.977068	+	0.212930i	$0.931699\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	166011.	−	197845.i	0.239248	−	0.285124i
$834$		0			0
$835$	−182712.	−	66501.9i	−0.262057	−	0.0953808i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−394033.	+	1.08260e6i	−0.559769	+	1.53795i	0.260206	+	0.965553i	$0.416210\pi$
$839$	−394033.	+	1.08260e6i	−0.559769	+	1.53795i	−0.819975	+	0.572400i	$0.806013\pi$
$840$		0			0
$841$	471937.	+	396002.i	0.667255	+	0.559894i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	303385.	−	175159.i	0.424894	−	0.245312i
$846$		0			0
$847$	390400.	−	676193.i	0.544181	−	0.942549i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−325868.	−	895316.i	−0.449969	−	1.23628i
$852$		0			0
$853$	−59641.9	−	338246.i	−0.0819697	−	0.464873i	−0.997969	−	0.0636945i	$-0.979712\pi$
$853$	−59641.9	−	338246.i	−0.0819697	−	0.464873i	0.916000	−	0.401179i	$-0.131399\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−764467.	−	911057.i	−1.04087	−	1.24046i	−0.970033	−	0.242975i	$-0.921877\pi$
$857$	−764467.	−	911057.i	−1.04087	−	1.24046i	−0.0708392	−	0.997488i	$-0.522568\pi$
$858$		0			0
$859$	62428.0	−	354047.i	0.0846045	−	0.479816i	−0.912837	−	0.408325i	$-0.866113\pi$
$859$	62428.0	−	354047.i	0.0846045	−	0.479816i	0.997441	−	0.0714912i	$-0.0227758\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1.15231e6i			1.54721i	0.633671	+	0.773603i	$0.281547\pi$
$863$			1.15231e6i			1.54721i	−0.633671	+	0.773603i	$0.718453\pi$
$864$		0			0
$865$	628437.			0.839904
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−506722.	−	89348.8i	−0.671012	−	0.118318i
$870$		0			0
$871$	−214412.	+	179913.i	−0.282626	+	0.237152i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−916928.	+	161679.i	−1.19762	+	0.211173i
$876$		0			0
$877$	1.07476e6	−	391179.i	1.39737	−	0.508600i	0.469972	−	0.882682i	$-0.344264\pi$
$877$	1.07476e6	−	391179.i	1.39737	−	0.508600i	0.927396	+	0.374081i	$0.122042\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−102028.	−	58906.1i	−0.131453	−	0.0758942i	0.432832	−	0.901475i	$-0.357514\pi$
$881$	−102028.	−	58906.1i	−0.131453	−	0.0758942i	−0.564284	+	0.825581i	$0.690848\pi$
$882$		0			0
$883$	288165.	+	499117.i	0.369590	+	0.640149i	0.989501	−	0.144523i	$-0.0461648\pi$
$883$	288165.	+	499117.i	0.369590	+	0.640149i	−0.619911	+	0.784672i	$0.712831\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	659602.	−	786083.i	0.838368	−	0.999128i	−0.161557	−	0.986863i	$-0.551652\pi$
$887$	659602.	−	786083.i	0.838368	−	0.999128i	0.999925	−	0.0122644i	$-0.00390399\pi$
$888$		0			0
$889$	1.16315e6	+	423352.i	1.47174	+	0.535670i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	26740.1	−	73467.9i	0.0335321	−	0.0921287i
$894$		0			0
$895$	−448758.	−	376553.i	−0.560230	−	0.470088i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	358345.	−	206891.i	0.443386	−	0.255989i
$900$		0			0
$901$	−130228.	+	225561.i	−0.160418	+	0.277853i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	166633.	+	457822.i	0.203453	+	0.558984i
$906$		0			0
$907$	100331.	+	569005.i	0.121961	+	0.691674i	0.983067	+	0.183249i	$0.0586613\pi$
$907$	100331.	+	569005.i	0.121961	+	0.691674i	−0.861106	+	0.508426i	$0.830228\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−708200.	−	844000.i	−0.853334	−	1.01696i	−0.999616	−	0.0277210i	$-0.991175\pi$
$911$	−708200.	−	844000.i	−0.853334	−	1.01696i	0.146282	−	0.989243i	$-0.453269\pi$
$912$		0			0
$913$	−47578.9	+	269834.i	−0.0570786	+	0.323709i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1.44039e6i			1.71294i
$918$		0			0
$919$	−104165.			−0.123336			−0.0616679	−	0.998097i	$-0.519642\pi$
$919$	−104165.			−0.123336			−0.0616679	+	0.998097i	$0.519642\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	327975.	+	57830.8i	0.384979	+	0.0678822i
$924$		0			0
$925$	−389391.	+	326738.i	−0.455095	+	0.381870i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	440494.	−	77670.9i	0.510397	−	0.0899968i	0.0874820	−	0.996166i	$-0.472118\pi$
$929$	440494.	−	77670.9i	0.510397	−	0.0899968i	0.422915	+	0.906169i	$0.361007\pi$
$930$		0			0
$931$	123488.	−	44946.0i	0.142471	−	0.0518552i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	76819.8	+	44351.9i	0.0878719	+	0.0507328i
$936$		0			0
$937$	−578929.	−	1.00274e6i	−0.659396	−	1.14211i	−0.980772	−	0.195156i	$-0.937479\pi$
$937$	−578929.	−	1.00274e6i	−0.659396	−	1.14211i	0.321376	−	0.946952i	$-0.395855\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	307394.	−	366338.i	0.347149	−	0.413716i	−0.564012	−	0.825767i	$-0.690743\pi$
$941$	307394.	−	366338.i	0.347149	−	0.413716i	0.911161	+	0.412050i	$0.135187\pi$
$942$		0			0
$943$	−1.73156e6	−	630235.i	−1.94721	−	0.708727i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−334459.	+	918920.i	−0.372944	+	1.02465i	0.601273	+	0.799043i	$0.294660\pi$
$947$	−334459.	+	918920.i	−0.372944	+	1.02465i	−0.974217	+	0.225612i	$0.927562\pi$
$948$		0			0
$949$	−18767.9	−	15748.1i	−0.0208393	−	0.0174862i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	600191.	−	346520.i	0.660851	−	0.381542i	−0.131750	−	0.991283i	$-0.542060\pi$
$953$	600191.	−	346520.i	0.660851	−	0.381542i	0.792601	+	0.609741i	$0.208726\pi$
$954$		0			0
$955$	277411.	−	480490.i	0.304170	−	0.526838i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−461801.	−	1.26879e6i	−0.502131	−	1.37959i
$960$		0			0
$961$	165592.	+	939120.i	0.179305	+	1.01689i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	246382.	+	293626.i	0.264578	+	0.315312i
$966$		0			0
$967$	−237165.	+	1.34503e6i	−0.253628	+	1.43839i	0.545943	+	0.837822i	$0.316172\pi$
$967$	−237165.	+	1.34503e6i	−0.253628	+	1.43839i	−0.799571	+	0.600572i	$0.794940\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1.03468e6i			1.09741i	0.836016	+	0.548705i	$0.184879\pi$
$971$			1.03468e6i			1.09741i	−0.836016	+	0.548705i	$0.815121\pi$
$972$		0			0
$973$	−2.20277e6			−2.32671
$974$		0			0
$975$		0			0
$976$		0			0
$977$	594628.	+	104849.i	0.622954	+	0.109844i	0.476211	−	0.879331i	$-0.342010\pi$
$977$	594628.	+	104849.i	0.622954	+	0.109844i	0.146743	+	0.989175i	$0.453121\pi$
$978$		0			0
$979$	−509246.	+	427308.i	−0.531328	+	0.445837i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.59582e6	+	281387.i	−1.65150	+	0.291203i	−0.920373	−	0.391042i	$-0.872115\pi$
$983$	−1.59582e6	+	281387.i	−1.65150	+	0.291203i	−0.731123	+	0.682245i	$0.761004\pi$
$984$		0			0
$985$	775915.	−	282410.i	0.799727	−	0.291077i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−208108.	−	120151.i	−0.212763	−	0.122839i
$990$		0			0
$991$	−561943.	−	973314.i	−0.572196	−	0.991073i	−0.996340	−	0.0854783i	$-0.972758\pi$
$991$	−561943.	−	973314.i	−0.572196	−	0.991073i	0.424144	−	0.905595i	$-0.360575\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	280353.	−	334112.i	0.283178	−	0.337478i
$996$		0			0
$997$	261108.	+	95035.6i	0.262682	+	0.0956084i	0.470004	−	0.882664i	$-0.344253\pi$
$997$	261108.	+	95035.6i	0.262682	+	0.0956084i	−0.207322	+	0.978273i	$0.566475\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.5.k.a.17.4		72
3.2	odd	2		108.5.k.a.77.8	✓	72
27.7	even	9		108.5.k.a.101.8	yes	72
27.20	odd	18	inner	324.5.k.a.305.4		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.77.8	✓	72	3.2	odd	2
108.5.k.a.101.8	yes	72	27.7	even	9
324.5.k.a.17.4		72	1.1	even	1	trivial
324.5.k.a.305.4		72	27.20	odd	18	inner

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	324.k (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-12.7195 - 2.24280i) q^{5} +(-50.9819 + 42.7789i) q^{7} +O(q^{10})\)	\(q+(-12.7195 - 2.24280i) q^{5} +(-50.9819 + 42.7789i) q^{7} +(-53.1143 + 9.36548i) q^{11} +(-35.6301 + 12.9683i) q^{13} +(110.279 + 63.6697i) q^{17} +(32.3968 + 56.1128i) q^{19} +(552.034 - 657.888i) q^{23} +(-430.551 - 156.708i) q^{25} +(103.294 - 283.798i) q^{29} +(1049.54 + 880.673i) q^{31} +(744.411 - 429.786i) q^{35} +(554.706 - 960.778i) q^{37} +(-733.845 - 2016.22i) q^{41} +(-48.5881 - 275.557i) q^{43} +(-775.618 - 924.345i) q^{47} +(352.191 - 1997.37i) q^{49} +2045.36i q^{53} +696.594 q^{55} +(-664.820 - 117.226i) q^{59} +(3039.96 - 2550.83i) q^{61} +(482.284 - 85.0396i) q^{65} +(6936.65 - 2524.73i) q^{67} +(-7606.56 - 4391.65i) q^{71} +(323.073 + 559.578i) q^{73} +(2307.22 - 2749.64i) q^{77} +(8964.88 + 3262.95i) q^{79} +(1737.55 - 4773.87i) q^{83} +(-1259.90 - 1057.18i) q^{85} +(10674.4 - 6162.88i) q^{89} +(1261.72 - 2185.36i) q^{91} +(-286.222 - 786.389i) q^{95} +(1027.42 + 5826.76i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{5}+O(q^{10}) \) 72 * q - 9 * q^5	\( 72 q - 9 q^{5} - 18 q^{11} + 1278 q^{23} + 441 q^{25} - 1854 q^{29} - 1665 q^{31} + 2673 q^{35} + 5472 q^{41} + 1260 q^{43} - 5103 q^{47} - 5904 q^{49} + 10944 q^{59} + 8352 q^{61} - 8757 q^{65} + 378 q^{67} + 19764 q^{71} + 6111 q^{73} + 5679 q^{77} - 5652 q^{79} + 20061 q^{83} + 26100 q^{85} - 15633 q^{89} - 6039 q^{91} - 48024 q^{95} - 37530 q^{97}+O(q^{100}) \) 72 * q - 9 * q^5 - 18 * q^11 + 1278 * q^23 + 441 * q^25 - 1854 * q^29 - 1665 * q^31 + 2673 * q^35 + 5472 * q^41 + 1260 * q^43 - 5103 * q^47 - 5904 * q^49 + 10944 * q^59 + 8352 * q^61 - 8757 * q^65 + 378 * q^67 + 19764 * q^71 + 6111 * q^73 + 5679 * q^77 - 5652 * q^79 + 20061 * q^83 + 26100 * q^85 - 15633 * q^89 - 6039 * q^91 - 48024 * q^95 - 37530 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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