LMFDB - Embedded newform 625.2.b.c.624.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(624,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("625.624");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 625 = 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	625.b (of order $2$, degree $1$, 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:	$4.99065012633$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.58140625.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 $ x^8 - 3x^7 + 4x^6 - 7x^5 + 11x^4 + 5x^3 - 10x^2 - 25*x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			624.4
Root			$1.17421 + 0.0566033i$ of defining polynomial
Character	$\chi$	$=$	625.624
Dual form			625.2.b.c.624.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.183172i q^{2} +1.47195i q^{3} +1.96645 q^{4} +0.269620 q^{6} -3.26086i q^{7} -0.726543i q^{8} +0.833366 q^{9} +O(q^{10})$	\(q-0.183172i q^{2} +1.47195i q^{3} +1.96645 q^{4} +0.269620 q^{6} -3.26086i q^{7} -0.726543i q^{8} +0.833366 q^{9} +2.00000 q^{11} +2.89451i q^{12} +0.296379i q^{13} -0.597298 q^{14} +3.79981 q^{16} -5.16297i q^{17} -0.152649i q^{18} -1.73038 q^{19} +4.79981 q^{21} -0.366344i q^{22} -0.879192i q^{23} +1.06943 q^{24} +0.0542883 q^{26} +5.64252i q^{27} -6.41230i q^{28} -5.91216 q^{29} +6.09953 q^{31} -2.14910i q^{32} +2.94390i q^{33} -0.945712 q^{34} +1.63877 q^{36} +8.08800i q^{37} +0.316957i q^{38} -0.436254 q^{39} -1.01515 q^{41} -0.879192i q^{42} +3.24199i q^{43} +3.93290 q^{44} -0.161043 q^{46} +4.21996i q^{47} +5.59313i q^{48} -3.63318 q^{49} +7.59963 q^{51} +0.582813i q^{52} +8.10072i q^{53} +1.03355 q^{54} -2.36915 q^{56} -2.54703i q^{57} +1.08294i q^{58} -5.93635 q^{59} +0.915615 q^{61} -1.11726i q^{62} -2.71749i q^{63} +7.20597 q^{64} +0.539240 q^{66} +6.88806i q^{67} -10.1527i q^{68} +1.29413 q^{69} -5.96878 q^{71} -0.605476i q^{72} -8.83341i q^{73} +1.48150 q^{74} -3.40270 q^{76} -6.52171i q^{77} +0.0799096i q^{78} +7.76067 q^{79} -5.80540 q^{81} +0.185946i q^{82} -14.5154i q^{83} +9.43858 q^{84} +0.593842 q^{86} -8.70240i q^{87} -1.45309i q^{88} -7.52642 q^{89} +0.966448 q^{91} -1.72889i q^{92} +8.97820i q^{93} +0.772978 q^{94} +3.16337 q^{96} -6.72649i q^{97} +0.665497i q^{98} +1.66673 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{4} + 6 q^{6} - 4 q^{9}+O(q^{10}) $ 8 * q - 6 * q^4 + 6 * q^6 - 4 * q^9	$ 8 q - 6 q^{4} + 6 q^{6} - 4 q^{9} + 16 q^{11} - 12 q^{14} - 2 q^{16} - 10 q^{19} + 6 q^{21} - 20 q^{24} + 6 q^{26} - 20 q^{29} + 16 q^{31} - 2 q^{34} - 12 q^{36} - 18 q^{39} + 26 q^{41} - 12 q^{44} + 6 q^{46} + 14 q^{49} - 4 q^{51} + 30 q^{54} + 10 q^{56} - 30 q^{59} + 6 q^{61} + 44 q^{64} + 12 q^{66} - 8 q^{69} + 46 q^{71} - 12 q^{74} - 20 q^{76} - 10 q^{79} - 32 q^{81} + 18 q^{84} - 14 q^{86} - 30 q^{89} - 14 q^{91} + 68 q^{94} - 54 q^{96} - 8 q^{99}+O(q^{100}) $ 8 * q - 6 * q^4 + 6 * q^6 - 4 * q^9 + 16 * q^11 - 12 * q^14 - 2 * q^16 - 10 * q^19 + 6 * q^21 - 20 * q^24 + 6 * q^26 - 20 * q^29 + 16 * q^31 - 2 * q^34 - 12 * q^36 - 18 * q^39 + 26 * q^41 - 12 * q^44 + 6 * q^46 + 14 * q^49 - 4 * q^51 + 30 * q^54 + 10 * q^56 - 30 * q^59 + 6 * q^61 + 44 * q^64 + 12 * q^66 - 8 * q^69 + 46 * q^71 - 12 * q^74 - 20 * q^76 - 10 * q^79 - 32 * q^81 + 18 * q^84 - 14 * q^86 - 30 * q^89 - 14 * q^91 + 68 * q^94 - 54 * q^96 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/625\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 0.183172i	− 0.129522i	−0.997901	−	0.0647611i	$-0.979371\pi$
$2$	− 0.183172i	− 0.129522i	0.997901	−	0.0647611i	$-0.0206285\pi$
$3$	1.47195i	0.849830i	0.905233	+	0.424915i	$0.139696\pi$
$3$	1.47195i	0.849830i	−0.905233	+	0.424915i	$0.860304\pi$
$4$	1.96645	0.983224
$5$	0	0
$5$	0	0
$6$	0.269620	0.110072
$7$	− 3.26086i	− 1.23249i	−0.787555	−	0.616244i	$-0.788654\pi$
$7$	− 3.26086i	− 1.23249i	0.787555	−	0.616244i	$-0.211346\pi$
$8$	− 0.726543i	− 0.256872i
$9$	0.833366	0.277789
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	2.89451i	0.835573i
$13$	0.296379i	0.0822006i	0.999155	+	0.0411003i	$0.0130863\pi$
$13$	0.296379i	0.0822006i	−0.999155	+	0.0411003i	$0.986914\pi$
$14$	−0.597298	−0.159635
$15$	0	0
$16$	3.79981	0.949953
$17$	− 5.16297i	− 1.25220i	−0.779741	−	0.626102i	$-0.784649\pi$
$17$	− 5.16297i	− 1.25220i	0.779741	−	0.626102i	$-0.215351\pi$
$18$	− 0.152649i	− 0.0359798i
$19$	−1.73038	−0.396976	−0.198488	−	0.980103i	$-0.563603\pi$
$19$	−1.73038	−0.396976	−0.198488	+	0.980103i	$0.563603\pi$
$20$	0	0
$21$	4.79981	1.04741
$22$	− 0.366344i	− 0.0781048i
$23$	− 0.879192i	− 0.183324i	−0.995790	−	0.0916621i	$-0.970782\pi$
$23$	− 0.879192i	− 0.183324i	0.995790	−	0.0916621i	$-0.0292180\pi$
$24$	1.06943	0.218297
$25$	0	0
$26$	0.0542883	0.0106468
$27$	5.64252i	1.08590i
$28$	− 6.41230i	− 1.21181i
$29$	−5.91216	−1.09786	−0.548930	−	0.835868i	$-0.684965\pi$
$29$	−5.91216	−1.09786	−0.548930	+	0.835868i	$0.684965\pi$
$30$	0	0
$31$	6.09953	1.09551	0.547754	−	0.836639i	$-0.315483\pi$
$31$	6.09953	1.09551	0.547754	+	0.836639i	$0.315483\pi$
$32$	− 2.14910i	− 0.379912i
$33$	2.94390i	0.512467i
$34$	−0.945712	−0.162188
$35$	0	0
$36$	1.63877	0.273128
$37$	8.08800i	1.32966i	0.746995	+	0.664830i	$0.231496\pi$
$37$	8.08800i	1.32966i	−0.746995	+	0.664830i	$0.768504\pi$
$38$	0.316957i	0.0514173i
$39$	−0.436254	−0.0698566
$40$	0	0
$41$	−1.01515	−0.158539	−0.0792695	−	0.996853i	$-0.525259\pi$
$41$	−1.01515	−0.158539	−0.0792695	+	0.996853i	$0.525259\pi$
$42$	− 0.879192i	− 0.135662i
$43$	3.24199i	0.494399i	0.968965	+	0.247200i	$0.0795103\pi$
$43$	3.24199i	0.494399i	−0.968965	+	0.247200i	$0.920490\pi$
$44$	3.93290	0.592906
$45$	0	0
$46$	−0.161043	−0.0237446
$47$	4.21996i	0.615544i	0.951460	+	0.307772i	$0.0995834\pi$
$47$	4.21996i	0.615544i	−0.951460	+	0.307772i	$0.900417\pi$
$48$	5.59313i	0.807299i
$49$	−3.63318	−0.519026
$50$	0	0
$51$	7.59963	1.06416
$52$	0.582813i	0.0808216i
$53$	8.10072i	1.11272i	0.830941	+	0.556360i	$0.187802\pi$
$53$	8.10072i	1.11272i	−0.830941	+	0.556360i	$0.812198\pi$
$54$	1.03355	0.140649
$55$	0	0
$56$	−2.36915	−0.316591
$57$	− 2.54703i	− 0.337363i
$58$	1.08294i	0.142197i
$59$	−5.93635	−0.772847	−0.386424	−	0.922321i	$-0.626290\pi$
$59$	−5.93635	−0.772847	−0.386424	+	0.922321i	$0.626290\pi$
$60$	0	0
$61$	0.915615	0.117232	0.0586162	−	0.998281i	$-0.481331\pi$
$61$	0.915615	0.117232	0.0586162	+	0.998281i	$0.481331\pi$
$62$	− 1.11726i	− 0.141893i
$63$	− 2.71749i	− 0.342371i
$64$	7.20597	0.900746
$65$	0	0
$66$	0.539240	0.0663759
$67$	6.88806i	0.841510i	0.907174	+	0.420755i	$0.138235\pi$
$67$	6.88806i	0.841510i	−0.907174	+	0.420755i	$0.861765\pi$
$68$	− 10.1527i	− 1.23120i
$69$	1.29413	0.155794
$70$	0	0
$71$	−5.96878	−0.708364	−0.354182	−	0.935177i	$-0.615241\pi$
$71$	−5.96878	−0.708364	−0.354182	+	0.935177i	$0.615241\pi$
$72$	− 0.605476i	− 0.0713560i
$73$	− 8.83341i	− 1.03387i	−0.856024	−	0.516936i	$-0.827072\pi$
$73$	− 8.83341i	− 1.03387i	0.856024	−	0.516936i	$-0.172928\pi$
$74$	1.48150	0.172220
$75$	0	0
$76$	−3.40270	−0.390317
$77$	− 6.52171i	− 0.743218i
$78$	0.0799096i	0.00904798i
$79$	7.76067	0.873144	0.436572	−	0.899669i	$-0.356192\pi$
$79$	7.76067	0.873144	0.436572	+	0.899669i	$0.356192\pi$
$80$	0	0
$81$	−5.80540	−0.645045
$82$	0.185946i	0.0205343i
$83$	− 14.5154i	− 1.59327i	−0.604461	−	0.796635i	$-0.706612\pi$
$83$	− 14.5154i	− 1.59327i	0.604461	−	0.796635i	$-0.293388\pi$
$84$	9.43858	1.02983
$85$	0	0
$86$	0.593842	0.0640357
$87$	− 8.70240i	− 0.932995i
$88$	− 1.45309i	− 0.154899i
$89$	−7.52642	−0.797799	−0.398900	−	0.916995i	$-0.630608\pi$
$89$	−7.52642	−0.797799	−0.398900	+	0.916995i	$0.630608\pi$
$90$	0	0
$91$	0.966448	0.101311
$92$	− 1.72889i	− 0.180249i
$93$	8.97820i	0.930996i
$94$	0.772978	0.0797266
$95$	0	0
$96$	3.16337	0.322860
$97$	− 6.72649i	− 0.682971i	−0.939887	−	0.341486i	$-0.889070\pi$
$97$	− 6.72649i	− 0.682971i	0.939887	−	0.341486i	$-0.110930\pi$
$98$	0.665497i	0.0672254i
$99$	1.66673	0.167513
$100$	0	0
$101$	−12.1955	−1.21350	−0.606748	−	0.794894i	$-0.707526\pi$
$101$	−12.1955	−1.21350	−0.606748	+	0.794894i	$0.707526\pi$
$102$	− 1.39204i	− 0.137832i
$103$	− 1.38140i	− 0.136114i	−0.997681	−	0.0680569i	$-0.978320\pi$
$103$	− 1.38140i	− 0.136114i	0.997681	−	0.0680569i	$-0.0216800\pi$
$104$	0.215332	0.0211150
$105$	0	0
$106$	1.48383	0.144122
$107$	15.8285i	1.53020i	0.643911	+	0.765101i	$0.277311\pi$
$107$	15.8285i	1.53020i	−0.643911	+	0.765101i	$0.722689\pi$
$108$	11.0957i	1.06769i
$109$	2.00377	0.191926	0.0959632	−	0.995385i	$-0.469407\pi$
$109$	2.00377	0.191926	0.0959632	+	0.995385i	$0.469407\pi$
$110$	0	0
$111$	−11.9051	−1.12998
$112$	− 12.3906i	− 1.17081i
$113$	10.4275i	0.980935i	0.871459	+	0.490468i	$0.163174\pi$
$113$	10.4275i	0.980935i	−0.871459	+	0.490468i	$0.836826\pi$
$114$	−0.466545	−0.0436959
$115$	0	0
$116$	−11.6260	−1.07944
$117$	0.246992i	0.0228344i
$118$	1.08737i	0.100101i
$119$	−16.8357	−1.54333
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 0.167715i	− 0.0151842i
$123$	− 1.49424i	− 0.134731i
$124$	11.9944	1.07713
$125$	0	0
$126$	−0.497767	−0.0443446
$127$	− 5.85007i	− 0.519110i	−0.965728	−	0.259555i	$-0.916424\pi$
$127$	− 5.85007i	− 0.519110i	0.965728	−	0.259555i	$-0.0835758\pi$
$128$	− 5.61814i	− 0.496578i
$129$	−4.77205	−0.420155
$130$	0	0
$131$	−1.49664	−0.130762	−0.0653811	−	0.997860i	$-0.520826\pi$
$131$	−1.49664	−0.130762	−0.0653811	+	0.997860i	$0.520826\pi$
$132$	5.78902i	0.503870i
$133$	5.64252i	0.489268i
$134$	1.26170	0.108994
$135$	0	0
$136$	−3.75112	−0.321656
$137$	− 7.84887i	− 0.670574i	−0.942116	−	0.335287i	$-0.891167\pi$
$137$	− 7.84887i	− 0.670574i	0.942116	−	0.335287i	$-0.108833\pi$
$138$	− 0.237048i	− 0.0201788i
$139$	−5.39152	−0.457303	−0.228651	−	0.973508i	$-0.573432\pi$
$139$	−5.39152	−0.457303	−0.228651	+	0.973508i	$0.573432\pi$
$140$	0	0
$141$	−6.21156	−0.523108
$142$	1.09331i	0.0917488i
$143$	0.592757i	0.0495689i
$144$	3.16663	0.263886
$145$	0	0
$146$	−1.61803	−0.133909
$147$	− 5.34786i	− 0.441084i
$148$	15.9046i	1.30735i
$149$	−18.8229	−1.54203	−0.771015	−	0.636817i	$-0.780251\pi$
$149$	−18.8229	−1.54203	−0.771015	+	0.636817i	$0.780251\pi$
$150$	0	0
$151$	−3.88797	−0.316398	−0.158199	−	0.987407i	$-0.550569\pi$
$151$	−3.88797	−0.316398	−0.158199	+	0.987407i	$0.550569\pi$
$152$	1.25719i	0.101972i
$153$	− 4.30264i	− 0.347848i
$154$	−1.19460	−0.0962632
$155$	0	0
$156$	−0.857871	−0.0686847
$157$	4.28378i	0.341883i	0.985281	+	0.170941i	$0.0546808\pi$
$157$	4.28378i	0.341883i	−0.985281	+	0.170941i	$0.945319\pi$
$158$	− 1.42154i	− 0.113092i
$159$	−11.9238	−0.945623
$160$	0	0
$161$	−2.86692	−0.225945
$162$	1.06339i	0.0835477i
$163$	− 15.7263i	− 1.23178i	−0.787832	−	0.615890i	$-0.788796\pi$
$163$	− 15.7263i	− 1.23178i	0.787832	−	0.615890i	$-0.211204\pi$
$164$	−1.99623	−0.155879
$165$	0	0
$166$	−2.65881	−0.206364
$167$	21.0330i	1.62758i	0.581156	+	0.813792i	$0.302601\pi$
$167$	21.0330i	1.62758i	−0.581156	+	0.813792i	$0.697399\pi$
$168$	− 3.48727i	− 0.269049i
$169$	12.9122	0.993243
$170$	0	0
$171$	−1.44204	−0.110276
$172$	6.37521i	0.486105i
$173$	− 7.16663i	− 0.544869i	−0.962174	−	0.272434i	$-0.912171\pi$
$173$	− 7.16663i	− 0.544869i	0.962174	−	0.272434i	$-0.0878288\pi$
$174$	−1.59404	−0.120844
$175$	0	0
$176$	7.59963	0.572843
$177$	− 8.73801i	− 0.656789i
$178$	1.37863i	0.103333i
$179$	−8.03934	−0.600888	−0.300444	−	0.953799i	$-0.597135\pi$
$179$	−8.03934	−0.600888	−0.300444	+	0.953799i	$0.597135\pi$
$180$	0	0
$181$	−20.6171	−1.53246	−0.766229	−	0.642568i	$-0.777869\pi$
$181$	−20.6171	−1.53246	−0.766229	+	0.642568i	$0.777869\pi$
$182$	− 0.177026i	− 0.0131221i
$183$	1.34774i	0.0996277i
$184$	−0.638770	−0.0470908
$185$	0	0
$186$	1.64456	0.120585
$187$	− 10.3259i	− 0.755107i
$188$	8.29833i	0.605218i
$189$	18.3994	1.33836
$190$	0	0
$191$	18.0303	1.30463	0.652313	−	0.757950i	$-0.273799\pi$
$191$	18.0303	1.30463	0.652313	+	0.757950i	$0.273799\pi$
$192$	10.6068i	0.765482i
$193$	− 6.78859i	− 0.488653i	−0.969693	−	0.244327i	$-0.921433\pi$
$193$	− 6.78859i	− 0.488653i	0.969693	−	0.244327i	$-0.0785669\pi$
$194$	−1.23210	−0.0884599
$195$	0	0
$196$	−7.14446	−0.510318
$197$	7.99537i	0.569647i	0.958580	+	0.284823i	$0.0919350\pi$
$197$	7.99537i	0.569647i	−0.958580	+	0.284823i	$0.908065\pi$
$198$	− 0.305299i	− 0.0216966i
$199$	5.20485	0.368962	0.184481	−	0.982836i	$-0.440940\pi$
$199$	5.20485	0.368962	0.184481	+	0.982836i	$0.440940\pi$
$200$	0	0
$201$	−10.1389	−0.715141
$202$	2.23387i	0.157175i
$203$	19.2787i	1.35310i
$204$	14.9443	1.04631
$205$	0	0
$206$	−0.253035	−0.0176298
$207$	− 0.732688i	− 0.0509254i
$208$	1.12618i	0.0780868i
$209$	−3.46076	−0.239386
$210$	0	0
$211$	16.6020	1.14293	0.571463	−	0.820628i	$-0.306376\pi$
$211$	16.6020	1.14293	0.571463	+	0.820628i	$0.306376\pi$
$212$	15.9296i	1.09405i
$213$	− 8.78574i	− 0.601989i
$214$	2.89934	0.198195
$215$	0	0
$216$	4.09953	0.278938
$217$	− 19.8897i	− 1.35020i
$218$	− 0.367035i	− 0.0248587i
$219$	13.0023	0.878616
$220$	0	0
$221$	1.53019	0.102932
$222$	2.18069i	0.146358i
$223$	− 6.62808i	− 0.443849i	−0.975064	−	0.221925i	$-0.928766\pi$
$223$	− 6.62808i	− 0.443849i	0.975064	−	0.221925i	$-0.0712339\pi$
$224$	−7.00792	−0.468236
$225$	0	0
$226$	1.91002	0.127053
$227$	13.3919i	0.888853i	0.895815	+	0.444427i	$0.146593\pi$
$227$	13.3919i	0.888853i	−0.895815	+	0.444427i	$0.853407\pi$
$228$	− 5.00860i	− 0.331703i
$229$	−10.0867	−0.666549	−0.333274	−	0.942830i	$-0.608154\pi$
$229$	−10.0867	−0.666549	−0.333274	+	0.942830i	$0.608154\pi$
$230$	0	0
$231$	9.59963	0.631609
$232$	4.29544i	0.282009i
$233$	− 22.0055i	− 1.44163i	−0.693129	−	0.720814i	$-0.743768\pi$
$233$	− 22.0055i	− 1.44163i	0.693129	−	0.720814i	$-0.256232\pi$
$234$	0.0452420	0.00295756
$235$	0	0
$236$	−11.6735	−0.759882
$237$	11.4233i	0.742024i
$238$	3.08383i	0.199895i
$239$	−7.56029	−0.489035	−0.244517	−	0.969645i	$-0.578629\pi$
$239$	−7.56029	−0.489035	−0.244517	+	0.969645i	$0.578629\pi$
$240$	0	0
$241$	−20.3945	−1.31373	−0.656864	−	0.754009i	$-0.728117\pi$
$241$	−20.3945	−1.31373	−0.656864	+	0.754009i	$0.728117\pi$
$242$	1.28220i	0.0824232i
$243$	8.38230i	0.537725i
$244$	1.80051	0.115266
$245$	0	0
$246$	−0.273703	−0.0174507
$247$	− 0.512848i	− 0.0326317i
$248$	− 4.43157i	− 0.281405i
$249$	21.3659	1.35401
$250$	0	0
$251$	10.5717	0.667278	0.333639	−	0.942701i	$-0.391723\pi$
$251$	10.5717	0.667278	0.333639	+	0.942701i	$0.391723\pi$
$252$	− 5.34379i	− 0.336627i
$253$	− 1.75838i	− 0.110549i
$254$	−1.07157	−0.0672362
$255$	0	0
$256$	13.3829	0.836428
$257$	− 20.2700i	− 1.26441i	−0.774801	−	0.632205i	$-0.782150\pi$
$257$	− 20.2700i	− 1.26441i	0.774801	−	0.632205i	$-0.217850\pi$
$258$	0.874106i	0.0544194i
$259$	26.3738	1.63879
$260$	0	0
$261$	−4.92699	−0.304973
$262$	0.274143i	0.0169366i
$263$	28.0909i	1.73216i	0.499906	+	0.866079i	$0.333368\pi$
$263$	28.0909i	1.73216i	−0.499906	+	0.866079i	$0.666632\pi$
$264$	2.13887	0.131638
$265$	0	0
$266$	1.03355	0.0633711
$267$	− 11.0785i	− 0.677994i
$268$	13.5450i	0.827393i
$269$	−20.3230	−1.23911	−0.619557	−	0.784952i	$-0.712688\pi$
$269$	−20.3230	−1.23911	−0.619557	+	0.784952i	$0.712688\pi$
$270$	0	0
$271$	31.4467	1.91025	0.955125	−	0.296202i	$-0.0957200\pi$
$271$	31.4467	1.91025	0.955125	+	0.296202i	$0.0957200\pi$
$272$	− 19.6183i	− 1.18954i
$273$	1.42256i	0.0860974i
$274$	−1.43769	−0.0868543
$275$	0	0
$276$	2.54483	0.153181
$277$	13.9305i	0.837001i	0.908217	+	0.418501i	$0.137444\pi$
$277$	13.9305i	0.837001i	−0.908217	+	0.418501i	$0.862556\pi$
$278$	0.987576i	0.0592309i
$279$	5.08314	0.304320
$280$	0	0
$281$	25.8777	1.54373	0.771866	−	0.635785i	$-0.219323\pi$
$281$	25.8777	1.54373	0.771866	+	0.635785i	$0.219323\pi$
$282$	1.13778i	0.0677541i
$283$	23.7316i	1.41070i	0.708859	+	0.705350i	$0.249210\pi$
$283$	23.7316i	1.41070i	−0.708859	+	0.705350i	$0.750790\pi$
$284$	−11.7373	−0.696480
$285$	0	0
$286$	0.108577	0.00642027
$287$	3.31024i	0.195397i
$288$	− 1.79099i	− 0.105535i
$289$	−9.65625	−0.568014
$290$	0	0
$291$	9.90105	0.580410
$292$	− 17.3704i	− 1.01653i
$293$	− 12.3029i	− 0.718742i	−0.933195	−	0.359371i	$-0.882991\pi$
$293$	− 12.3029i	− 0.718742i	0.933195	−	0.359371i	$-0.117009\pi$
$294$	−0.979578	−0.0571301
$295$	0	0
$296$	5.87628	0.341552
$297$	11.2850i	0.654824i
$298$	3.44783i	0.199727i
$299$	0.260574	0.0150694
$300$	0	0
$301$	10.5717	0.609341
$302$	0.712167i	0.0409806i
$303$	− 17.9511i	− 1.03127i
$304$	−6.57512	−0.377109
$305$	0	0
$306$	−0.788124	−0.0450540
$307$	− 4.28249i	− 0.244415i	−0.992505	−	0.122207i	$-0.961003\pi$
$307$	− 4.28249i	− 0.244415i	0.992505	−	0.122207i	$-0.0389973\pi$
$308$	− 12.8246i	− 0.730750i
$309$	2.03336	0.115674
$310$	0	0
$311$	25.6467	1.45429	0.727145	−	0.686484i	$-0.240847\pi$
$311$	25.6467	1.45429	0.727145	+	0.686484i	$0.240847\pi$
$312$	0.316957i	0.0179442i
$313$	− 22.3513i	− 1.26337i	−0.775226	−	0.631684i	$-0.782364\pi$
$313$	− 22.3513i	− 1.26337i	0.775226	−	0.631684i	$-0.217636\pi$
$314$	0.784668	0.0442814
$315$	0	0
$316$	15.2610	0.858496
$317$	− 21.9228i	− 1.23131i	−0.788016	−	0.615655i	$-0.788892\pi$
$317$	− 21.9228i	− 1.23131i	0.788016	−	0.615655i	$-0.211108\pi$
$318$	2.18412i	0.122479i
$319$	−11.8243	−0.662035
$320$	0	0
$321$	−23.2988	−1.30041
$322$	0.525139i	0.0292649i
$323$	8.93390i	0.497095i
$324$	−11.4160	−0.634224
$325$	0	0
$326$	−2.88062	−0.159543
$327$	2.94945i	0.163105i
$328$	0.737546i	0.0407242i
$329$	13.7607	0.758650
$330$	0	0
$331$	−8.96299	−0.492651	−0.246325	−	0.969187i	$-0.579223\pi$
$331$	−8.96299	−0.492651	−0.246325	+	0.969187i	$0.579223\pi$
$332$	− 28.5437i	− 1.56654i
$333$	6.74026i	0.369364i
$334$	3.85266	0.210808
$335$	0	0
$336$	18.2384	0.994986
$337$	29.0836i	1.58428i	0.610338	+	0.792141i	$0.291034\pi$
$337$	29.0836i	1.58428i	−0.610338	+	0.792141i	$0.708966\pi$
$338$	− 2.36515i	− 0.128647i
$339$	−15.3487	−0.833628
$340$	0	0
$341$	12.1991	0.660616
$342$	0.264141i	0.0142831i
$343$	− 10.9787i	− 0.592795i
$344$	2.35544	0.126997
$345$	0	0
$346$	−1.31273	−0.0705726
$347$	− 14.2972i	− 0.767513i	−0.923434	−	0.383757i	$-0.874630\pi$
$347$	− 14.2972i	− 0.767513i	0.923434	−	0.383757i	$-0.125370\pi$
$348$	− 17.1128i	− 0.917343i
$349$	−5.62382	−0.301036	−0.150518	−	0.988607i	$-0.548094\pi$
$349$	−5.62382	−0.301036	−0.150518	+	0.988607i	$0.548094\pi$
$350$	0	0
$351$	−1.67232	−0.0892620
$352$	− 4.29821i	− 0.229095i
$353$	1.90997i	0.101658i	0.998707	+	0.0508288i	$0.0161863\pi$
$353$	1.90997i	0.101658i	−0.998707	+	0.0508288i	$0.983814\pi$
$354$	−1.60056	−0.0850688
$355$	0	0
$356$	−14.8003	−0.784415
$357$	− 24.7813i	− 1.31156i
$358$	1.47258i	0.0778284i
$359$	−22.2047	−1.17192	−0.585958	−	0.810341i	$-0.699282\pi$
$359$	−22.2047	−1.17192	−0.585958	+	0.810341i	$0.699282\pi$
$360$	0	0
$361$	−16.0058	−0.842410
$362$	3.77648i	0.198487i
$363$	− 10.3036i	− 0.540801i
$364$	1.90047	0.0996117
$365$	0	0
$366$	0.246868	0.0129040
$367$	21.4450i	1.11942i	0.828689	+	0.559709i	$0.189087\pi$
$367$	21.4450i	1.11942i	−0.828689	+	0.559709i	$0.810913\pi$
$368$	− 3.34077i	− 0.174149i
$369$	−0.845987	−0.0440403
$370$	0	0
$371$	26.4153	1.37141
$372$	17.6552i	0.915377i
$373$	23.1399i	1.19814i	0.800697	+	0.599070i	$0.204463\pi$
$373$	23.1399i	1.19814i	−0.800697	+	0.599070i	$0.795537\pi$
$374$	−1.89142	−0.0978032
$375$	0	0
$376$	3.06598	0.158116
$377$	− 1.75224i	− 0.0902448i
$378$	− 3.37026i	− 0.173348i
$379$	21.6501	1.11209	0.556047	−	0.831151i	$-0.312318\pi$
$379$	21.6501	1.11209	0.556047	+	0.831151i	$0.312318\pi$
$380$	0	0
$381$	8.61100	0.441155
$382$	− 3.30265i	− 0.168978i
$383$	5.65481i	0.288947i	0.989509	+	0.144474i	$0.0461489\pi$
$383$	5.65481i	0.288947i	−0.989509	+	0.144474i	$0.953851\pi$
$384$	8.26962	0.422007
$385$	0	0
$386$	−1.24348	−0.0632915
$387$	2.70176i	0.137338i
$388$	− 13.2273i	− 0.671514i
$389$	8.12605	0.412007	0.206004	−	0.978551i	$-0.433954\pi$
$389$	8.12605	0.412007	0.206004	+	0.978551i	$0.433954\pi$
$390$	0	0
$391$	−4.53924	−0.229559
$392$	2.63966i	0.133323i
$393$	− 2.20298i	− 0.111126i
$394$	1.46453	0.0737819
$395$	0	0
$396$	3.27754	0.164703
$397$	20.7689i	1.04236i	0.853447	+	0.521180i	$0.174508\pi$
$397$	20.7689i	1.04236i	−0.853447	+	0.521180i	$0.825492\pi$
$398$	− 0.953382i	− 0.0477887i
$399$	−8.30550	−0.415795
$400$	0	0
$401$	30.1195	1.50410	0.752049	−	0.659107i	$-0.229066\pi$
$401$	30.1195	1.50410	0.752049	+	0.659107i	$0.229066\pi$
$402$	1.85716i	0.0926266i
$403$	1.80777i	0.0900515i
$404$	−23.9818	−1.19314
$405$	0	0
$406$	3.53132	0.175256
$407$	16.1760i	0.801815i
$408$	− 5.52145i	− 0.273353i
$409$	−11.0452	−0.546152	−0.273076	−	0.961992i	$-0.588041\pi$
$409$	−11.0452	−0.546152	−0.273076	+	0.961992i	$0.588041\pi$
$410$	0	0
$411$	11.5531	0.569874
$412$	− 2.71646i	− 0.133830i
$413$	19.3576i	0.952524i
$414$	−0.134208	−0.00659597
$415$	0	0
$416$	0.636949	0.0312290
$417$	− 7.93604i	− 0.388630i
$418$	0.633915i	0.0310058i
$419$	32.9212	1.60830	0.804152	−	0.594424i	$-0.202620\pi$
$419$	32.9212	1.60830	0.804152	+	0.594424i	$0.202620\pi$
$420$	0	0
$421$	−20.2647	−0.987642	−0.493821	−	0.869564i	$-0.664400\pi$
$421$	−20.2647	−0.987642	−0.493821	+	0.869564i	$0.664400\pi$
$422$	− 3.04101i	− 0.148034i
$423$	3.51677i	0.170991i
$424$	5.88552	0.285826
$425$	0	0
$426$	−1.60930	−0.0779709
$427$	− 2.98569i	− 0.144488i
$428$	31.1260i	1.50453i
$429$	−0.872509	−0.0421251
$430$	0	0
$431$	7.15526	0.344657	0.172328	−	0.985040i	$-0.444871\pi$
$431$	7.15526	0.344657	0.172328	+	0.985040i	$0.444871\pi$
$432$	21.4405i	1.03156i
$433$	− 5.48264i	− 0.263479i	−0.991284	−	0.131739i	$-0.957944\pi$
$433$	− 5.48264i	− 0.263479i	0.991284	−	0.131739i	$-0.0420562\pi$
$434$	−3.64324	−0.174881
$435$	0	0
$436$	3.94031	0.188707
$437$	1.52134i	0.0727754i
$438$	− 2.38166i	− 0.113800i
$439$	30.7561	1.46791	0.733954	−	0.679199i	$-0.237673\pi$
$439$	30.7561	1.46791	0.733954	+	0.679199i	$0.237673\pi$
$440$	0	0
$441$	−3.02777	−0.144179
$442$	− 0.280289i	− 0.0133320i
$443$	11.3527i	0.539381i	0.962947	+	0.269691i	$0.0869214\pi$
$443$	11.3527i	0.539381i	−0.962947	+	0.269691i	$0.913079\pi$
$444$	−23.4108	−1.11103
$445$	0	0
$446$	−1.21408	−0.0574883
$447$	− 27.7063i	− 1.31046i
$448$	− 23.4976i	− 1.11016i
$449$	15.7661	0.744050	0.372025	−	0.928223i	$-0.378664\pi$
$449$	15.7661	0.744050	0.372025	+	0.928223i	$0.378664\pi$
$450$	0	0
$451$	−2.03029	−0.0956027
$452$	20.5051i	0.964479i
$453$	− 5.72289i	− 0.268885i
$454$	2.45303	0.115126
$455$	0	0
$456$	−1.85053	−0.0866588
$457$	4.16714i	0.194931i	0.995239	+	0.0974653i	$0.0310735\pi$
$457$	4.16714i	0.194931i	−0.995239	+	0.0974653i	$0.968926\pi$
$458$	1.84760i	0.0863329i
$459$	29.1322	1.35977
$460$	0	0
$461$	−23.9714	−1.11646	−0.558230	−	0.829686i	$-0.688519\pi$
$461$	−23.9714	−1.11646	−0.558230	+	0.829686i	$0.688519\pi$
$462$	− 1.75838i	− 0.0818074i
$463$	41.3247i	1.92052i	0.279107	+	0.960260i	$0.409962\pi$
$463$	41.3247i	1.92052i	−0.279107	+	0.960260i	$0.590038\pi$
$464$	−22.4651	−1.04292
$465$	0	0
$466$	−4.03079	−0.186723
$467$	10.3966i	0.481097i	0.970637	+	0.240548i	$0.0773273\pi$
$467$	10.3966i	0.481097i	−0.970637	+	0.240548i	$0.922673\pi$
$468$	0.485697i	0.0224513i
$469$	22.4610	1.03715
$470$	0	0
$471$	−6.30550	−0.290542
$472$	4.31301i	0.198522i
$473$	6.48398i	0.298134i
$474$	2.09243	0.0961086
$475$	0	0
$476$	−33.1065	−1.51743
$477$	6.75086i	0.309101i
$478$	1.38483i	0.0633408i
$479$	−36.0081	−1.64525	−0.822626	−	0.568582i	$-0.807492\pi$
$479$	−36.0081	−1.64525	−0.822626	+	0.568582i	$0.807492\pi$
$480$	0	0
$481$	−2.39711	−0.109299
$482$	3.73571i	0.170157i
$483$	− 4.21996i	− 0.192015i
$484$	−13.7651	−0.625688
$485$	0	0
$486$	1.53540	0.0696473
$487$	− 10.6378i	− 0.482045i	−0.970520	−	0.241023i	$-0.922517\pi$
$487$	− 10.6378i	− 0.482045i	0.970520	−	0.241023i	$-0.0774828\pi$
$488$	− 0.665233i	− 0.0301137i
$489$	23.1483	1.04680
$490$	0	0
$491$	−17.6693	−0.797402	−0.398701	−	0.917081i	$-0.630539\pi$
$491$	−17.6693	−0.797402	−0.398701	+	0.917081i	$0.630539\pi$
$492$	− 2.93835i	− 0.132471i
$493$	30.5243i	1.37475i
$494$	−0.0939394	−0.00422653
$495$	0	0
$496$	23.1771	1.04068
$497$	19.4633i	0.873049i
$498$	− 3.91363i	− 0.175374i
$499$	9.41734	0.421578	0.210789	−	0.977532i	$-0.432397\pi$
$499$	9.41734	0.421578	0.210789	+	0.977532i	$0.432397\pi$
$500$	0	0
$501$	−30.9595	−1.38317
$502$	− 1.93643i	− 0.0864273i
$503$	− 18.0133i	− 0.803172i	−0.915821	−	0.401586i	$-0.868459\pi$
$503$	− 18.0133i	− 0.803172i	0.915821	−	0.401586i	$-0.131541\pi$
$504$	−1.97437	−0.0879454
$505$	0	0
$506$	−0.322087	−0.0143185
$507$	19.0060i	0.844088i
$508$	− 11.5039i	− 0.510401i
$509$	16.0485	0.711337	0.355669	−	0.934612i	$-0.384253\pi$
$509$	16.0485	0.711337	0.355669	+	0.934612i	$0.384253\pi$
$510$	0	0
$511$	−28.8045	−1.27423
$512$	− 13.6877i	− 0.604914i
$513$	− 9.76370i	− 0.431078i
$514$	−3.71291	−0.163769
$515$	0	0
$516$	−9.38398	−0.413107
$517$	8.43991i	0.371187i
$518$	− 4.83095i	− 0.212260i
$519$	10.5489	0.463046
$520$	0	0
$521$	−2.20069	−0.0964142	−0.0482071	−	0.998837i	$-0.515351\pi$
$521$	−2.20069	−0.0964142	−0.0482071	+	0.998837i	$0.515351\pi$
$522$	0.902487i	0.0395008i
$523$	− 7.43588i	− 0.325148i	−0.986696	−	0.162574i	$-0.948020\pi$
$523$	− 7.43588i	− 0.325148i	0.986696	−	0.162574i	$-0.0519797\pi$
$524$	−2.94307	−0.128569
$525$	0	0
$526$	5.14547	0.224353
$527$	− 31.4917i	− 1.37180i
$528$	11.1863i	0.486820i
$529$	22.2270	0.966392
$530$	0	0
$531$	−4.94715	−0.214688
$532$	11.0957i	0.481061i
$533$	− 0.300867i	− 0.0130320i
$534$	−2.02927	−0.0878153
$535$	0	0
$536$	5.00447	0.216160
$537$	− 11.8335i	− 0.510653i
$538$	3.72260i	0.160493i
$539$	−7.26636	−0.312984
$540$	0	0
$541$	20.6474	0.887701	0.443850	−	0.896101i	$-0.353612\pi$
$541$	20.6474	0.887701	0.443850	+	0.896101i	$0.353612\pi$
$542$	− 5.76016i	− 0.247420i
$543$	− 30.3473i	− 1.30233i
$544$	−11.0958	−0.475727
$545$	0	0
$546$	0.260574	0.0111515
$547$	− 31.3762i	− 1.34155i	−0.741662	−	0.670774i	$-0.765962\pi$
$547$	− 31.3762i	− 1.34155i	0.741662	−	0.670774i	$-0.234038\pi$
$548$	− 15.4344i	− 0.659325i
$549$	0.763042	0.0325658
$550$	0	0
$551$	10.2303	0.435825
$552$	− 0.940237i	− 0.0400192i
$553$	− 25.3064i	− 1.07614i
$554$	2.55167	0.108410
$555$	0	0
$556$	−10.6021	−0.449631
$557$	22.3515i	0.947064i	0.880776	+	0.473532i	$0.157021\pi$
$557$	22.3515i	0.947064i	−0.880776	+	0.473532i	$0.842979\pi$
$558$	− 0.931089i	− 0.0394161i
$559$	−0.960857	−0.0406399
$560$	0	0
$561$	15.1993	0.641713
$562$	− 4.74007i	− 0.199948i
$563$	− 34.3663i	− 1.44837i	−0.689607	−	0.724184i	$-0.742217\pi$
$563$	− 34.3663i	− 1.44837i	0.689607	−	0.724184i	$-0.257783\pi$
$564$	−12.2147	−0.514332
$565$	0	0
$566$	4.34697	0.182717
$567$	18.9306i	0.795010i
$568$	4.33657i	0.181958i
$569$	−32.1662	−1.34848	−0.674239	−	0.738513i	$-0.735528\pi$
$569$	−32.1662	−1.34848	−0.674239	+	0.738513i	$0.735528\pi$
$570$	0	0
$571$	−27.1668	−1.13690	−0.568448	−	0.822719i	$-0.692456\pi$
$571$	−27.1668	−1.13690	−0.568448	+	0.822719i	$0.692456\pi$
$572$	1.16563i	0.0487373i
$573$	26.5397i	1.10871i
$574$	0.606344	0.0253083
$575$	0	0
$576$	6.00521	0.250217
$577$	− 13.8503i	− 0.576596i	−0.957541	−	0.288298i	$-0.906911\pi$
$577$	− 13.8503i	− 0.576596i	0.957541	−	0.288298i	$-0.0930893\pi$
$578$	1.76875i	0.0735705i
$579$	9.99246	0.415273
$580$	0	0
$581$	−47.3325	−1.96368
$582$	− 1.81360i	− 0.0751759i
$583$	16.2014i	0.670995i
$584$	−6.41785	−0.265572
$585$	0	0
$586$	−2.25354	−0.0930930
$587$	− 44.1499i	− 1.82226i	−0.412118	−	0.911131i	$-0.635211\pi$
$587$	− 44.1499i	− 1.82226i	0.412118	−	0.911131i	$-0.364789\pi$
$588$	− 10.5163i	− 0.433684i
$589$	−10.5545	−0.434891
$590$	0	0
$591$	−11.7688	−0.484103
$592$	30.7329i	1.26311i
$593$	− 16.2531i	− 0.667437i	−0.942673	−	0.333718i	$-0.891697\pi$
$593$	− 16.2531i	− 0.667437i	0.942673	−	0.333718i	$-0.108303\pi$
$594$	2.06710	0.0848143
$595$	0	0
$596$	−37.0142	−1.51616
$597$	7.66127i	0.313555i
$598$	− 0.0477298i	− 0.00195182i
$599$	−30.4822	−1.24547	−0.622734	−	0.782433i	$-0.713978\pi$
$599$	−30.4822	−1.24547	−0.622734	+	0.782433i	$0.713978\pi$
$600$	0	0
$601$	−28.9162	−1.17952	−0.589758	−	0.807580i	$-0.700777\pi$
$601$	−28.9162	−1.17952	−0.589758	+	0.807580i	$0.700777\pi$
$602$	− 1.93643i	− 0.0789232i
$603$	5.74027i	0.233762i
$604$	−7.64549	−0.311090
$605$	0	0
$606$	−3.28815	−0.133572
$607$	− 8.23276i	− 0.334157i	−0.985944	−	0.167079i	$-0.946567\pi$
$607$	− 8.23276i	− 0.334157i	0.985944	−	0.167079i	$-0.0534334\pi$
$608$	3.71877i	0.150816i
$609$	−28.3773	−1.14990
$610$	0	0
$611$	−1.25071	−0.0505981
$612$	− 8.46092i	− 0.342012i
$613$	4.79811i	0.193794i	0.995294	+	0.0968969i	$0.0308917\pi$
$613$	4.79811i	0.193794i	−0.995294	+	0.0968969i	$0.969108\pi$
$614$	−0.784432	−0.0316571
$615$	0	0
$616$	−4.73830	−0.190912
$617$	2.03184i	0.0817986i	0.999163	+	0.0408993i	$0.0130223\pi$
$617$	2.03184i	0.0817986i	−0.999163	+	0.0408993i	$0.986978\pi$
$618$	− 0.372454i	− 0.0149823i
$619$	−8.14100	−0.327215	−0.163607	−	0.986526i	$-0.552313\pi$
$619$	−8.14100	−0.327215	−0.163607	+	0.986526i	$0.552313\pi$
$620$	0	0
$621$	4.96086	0.199072
$622$	− 4.69776i	− 0.188363i
$623$	24.5426i	0.983278i
$624$	−1.65769	−0.0663605
$625$	0	0
$626$	−4.09413	−0.163634
$627$	− 5.09406i	− 0.203437i
$628$	8.42382i	0.336147i
$629$	41.7581	1.66500
$630$	0	0
$631$	33.2847	1.32504	0.662522	−	0.749042i	$-0.269486\pi$
$631$	33.2847	1.32504	0.662522	+	0.749042i	$0.269486\pi$
$632$	− 5.63846i	− 0.224286i
$633$	24.4372i	0.971293i
$634$	−4.01565	−0.159482
$635$	0	0
$636$	−23.4476	−0.929759
$637$	− 1.07680i	− 0.0426642i
$638$	2.16589i	0.0857482i
$639$	−4.97417	−0.196775
$640$	0	0
$641$	−40.0481	−1.58180	−0.790902	−	0.611943i	$-0.790388\pi$
$641$	−40.0481	−1.58180	−0.790902	+	0.611943i	$0.790388\pi$
$642$	4.26769i	0.168432i
$643$	− 11.6870i	− 0.460890i	−0.973085	−	0.230445i	$-0.925982\pi$
$643$	− 11.6870i	− 0.460890i	0.973085	−	0.230445i	$-0.0740182\pi$
$644$	−5.63764	−0.222154
$645$	0	0
$646$	1.63644	0.0643849
$647$	7.94936i	0.312522i	0.987716	+	0.156261i	$0.0499440\pi$
$647$	7.94936i	0.312522i	−0.987716	+	0.156261i	$0.950056\pi$
$648$	4.21787i	0.165694i
$649$	−11.8727	−0.466044
$650$	0	0
$651$	29.2766	1.14744
$652$	− 30.9250i	− 1.21112i
$653$	3.33736i	0.130601i	0.997866	+	0.0653005i	$0.0208006\pi$
$653$	3.33736i	0.130601i	−0.997866	+	0.0653005i	$0.979199\pi$
$654$	0.540256	0.0211257
$655$	0	0
$656$	−3.85736	−0.150605
$657$	− 7.36146i	− 0.287198i
$658$	− 2.52057i	− 0.0982621i
$659$	30.0508	1.17061	0.585306	−	0.810812i	$-0.300974\pi$
$659$	30.0508	1.17061	0.585306	+	0.810812i	$0.300974\pi$
$660$	0	0
$661$	−6.58417	−0.256094	−0.128047	−	0.991768i	$-0.540871\pi$
$661$	−6.58417	−0.256094	−0.128047	+	0.991768i	$0.540871\pi$
$662$	1.64177i	0.0638092i
$663$	2.25237i	0.0874747i
$664$	−10.5460	−0.409266
$665$	0	0
$666$	1.23463	0.0478409
$667$	5.19792i	0.201264i
$668$	41.3603i	1.60028i
$669$	9.75620	0.377196
$670$	0	0
$671$	1.83123	0.0706939
$672$	− 10.3153i	− 0.397921i
$673$	6.73140i	0.259476i	0.991548	+	0.129738i	$0.0414137\pi$
$673$	6.73140i	0.259476i	−0.991548	+	0.129738i	$0.958586\pi$
$674$	5.32730	0.205200
$675$	0	0
$676$	25.3911	0.976580
$677$	− 13.6478i	− 0.524529i	−0.964996	−	0.262265i	$-0.915531\pi$
$677$	− 13.6478i	− 0.524529i	0.964996	−	0.262265i	$-0.0844693\pi$
$678$	2.81146i	0.107973i
$679$	−21.9341	−0.841753
$680$	0	0
$681$	−19.7122	−0.755374
$682$	− 2.23453i	− 0.0855645i
$683$	1.17220i	0.0448529i	0.999748	+	0.0224265i	$0.00713917\pi$
$683$	1.17220i	0.0448529i	−0.999748	+	0.0224265i	$0.992861\pi$
$684$	−2.83570	−0.108426
$685$	0	0
$686$	−2.01099	−0.0767801
$687$	− 14.8471i	− 0.566453i
$688$	12.3190i	0.469656i
$689$	−2.40088	−0.0914663
$690$	0	0
$691$	12.2674	0.466673	0.233336	−	0.972396i	$-0.425036\pi$
$691$	12.2674	0.466673	0.233336	+	0.972396i	$0.425036\pi$
$692$	− 14.0928i	− 0.535728i
$693$	− 5.43497i	− 0.206457i
$694$	−2.61885	−0.0994100
$695$	0	0
$696$	−6.32266	−0.239660
$697$	5.24116i	0.198523i
$698$	1.03013i	0.0389909i
$699$	32.3910	1.22514
$700$	0	0
$701$	−20.0271	−0.756415	−0.378207	−	0.925721i	$-0.623459\pi$
$701$	−20.0271	−0.756415	−0.378207	+	0.925721i	$0.623459\pi$
$702$	0.306323i	0.0115614i
$703$	− 13.9953i	− 0.527843i
$704$	14.4119	0.543171
$705$	0	0
$706$	0.349854	0.0131669
$707$	39.7677i	1.49562i
$708$	− 17.1828i	− 0.645771i
$709$	4.39510	0.165061	0.0825306	−	0.996589i	$-0.473700\pi$
$709$	4.39510	0.165061	0.0825306	+	0.996589i	$0.473700\pi$
$710$	0	0
$711$	6.46748	0.242549
$712$	5.46827i	0.204932i
$713$	− 5.36266i	− 0.200833i
$714$	−4.53924	−0.169877
$715$	0	0
$716$	−15.8089	−0.590808
$717$	− 11.1284i	− 0.415596i
$718$	4.06727i	0.151789i
$719$	10.9283	0.407557	0.203779	−	0.979017i	$-0.434678\pi$
$719$	10.9283	0.407557	0.203779	+	0.979017i	$0.434678\pi$
$720$	0	0
$721$	−4.50456	−0.167759
$722$	2.93181i	0.109111i
$723$	− 30.0197i	− 1.11645i
$724$	−40.5425	−1.50675
$725$	0	0
$726$	−1.88734	−0.0700458
$727$	32.9032i	1.22031i	0.792281	+	0.610156i	$0.208893\pi$
$727$	32.9032i	1.22031i	−0.792281	+	0.610156i	$0.791107\pi$
$728$	− 0.702166i	− 0.0260240i
$729$	−29.7545	−1.10202
$730$	0	0
$731$	16.7383	0.619088
$732$	2.65026i	0.0979564i
$733$	− 13.7498i	− 0.507859i	−0.967223	−	0.253929i	$-0.918277\pi$
$733$	− 13.7498i	− 0.507859i	0.967223	−	0.253929i	$-0.0817231\pi$
$734$	3.92812	0.144990
$735$	0	0
$736$	−1.88948	−0.0696470
$737$	13.7761i	0.507450i
$738$	0.154961i	0.00570420i
$739$	43.1893	1.58874	0.794372	−	0.607432i	$-0.207800\pi$
$739$	43.1893	1.58874	0.794372	+	0.607432i	$0.207800\pi$
$740$	0	0
$741$	0.754886	0.0277314
$742$	− 4.83854i	− 0.177628i
$743$	− 31.8479i	− 1.16838i	−0.811615	−	0.584192i	$-0.801411\pi$
$743$	− 31.8479i	− 1.16838i	0.811615	−	0.584192i	$-0.198589\pi$
$744$	6.52304	0.239146
$745$	0	0
$746$	4.23859	0.155186
$747$	− 12.0966i	− 0.442592i
$748$	− 20.3054i	− 0.742440i
$749$	51.6145	1.88595
$750$	0	0
$751$	−29.5952	−1.07995	−0.539973	−	0.841682i	$-0.681565\pi$
$751$	−29.5952	−1.07995	−0.539973	+	0.841682i	$0.681565\pi$
$752$	16.0351i	0.584738i
$753$	15.5610i	0.567073i
$754$	−0.320961	−0.0116887
$755$	0	0
$756$	36.1815	1.31591
$757$	− 0.0984401i	− 0.00357786i	−0.999998	−	0.00178893i	$-0.999431\pi$
$757$	− 0.0984401i	− 0.00357786i	0.999998	−	0.00178893i	$-0.000569435\pi$
$758$	− 3.96570i	− 0.144041i
$759$	2.58825	0.0939476
$760$	0	0
$761$	−3.54402	−0.128471	−0.0642353	−	0.997935i	$-0.520461\pi$
$761$	−3.54402	−0.128471	−0.0642353	+	0.997935i	$0.520461\pi$
$762$	− 1.57730i	− 0.0571394i
$763$	− 6.53400i	− 0.236547i
$764$	35.4556	1.28274
$765$	0	0
$766$	1.03580	0.0374251
$767$	− 1.75941i	− 0.0635285i
$768$	19.6989i	0.710822i
$769$	−1.42759	−0.0514802	−0.0257401	−	0.999669i	$-0.508194\pi$
$769$	−1.42759	−0.0514802	−0.0257401	+	0.999669i	$0.508194\pi$
$770$	0	0
$771$	29.8365	1.07453
$772$	− 13.3494i	− 0.480456i
$773$	− 33.7807i	− 1.21501i	−0.794317	−	0.607504i	$-0.792171\pi$
$773$	− 33.7807i	− 1.21501i	0.794317	−	0.607504i	$-0.207829\pi$
$774$	0.494888	0.0177884
$775$	0	0
$776$	−4.88708	−0.175436
$777$	38.8209i	1.39269i
$778$	− 1.48847i	− 0.0533641i
$779$	1.75659	0.0629363
$780$	0	0
$781$	−11.9376	−0.427159
$782$	0.831462i	0.0297330i
$783$	− 33.3595i	− 1.19217i
$784$	−13.8054	−0.493050
$785$	0	0
$786$	−0.403525	−0.0143932
$787$	2.18030i	0.0777192i	0.999245	+	0.0388596i	$0.0123725\pi$
$787$	2.18030i	0.0777192i	−0.999245	+	0.0388596i	$0.987627\pi$
$788$	15.7225i	0.560090i
$789$	−41.3484	−1.47204
$790$	0	0
$791$	34.0025	1.20899
$792$	− 1.21095i	− 0.0430293i
$793$	0.271369i	0.00963659i
$794$	3.80428	0.135009
$795$	0	0
$796$	10.2351	0.362772
$797$	23.4919i	0.832124i	0.909336	+	0.416062i	$0.136590\pi$
$797$	23.4919i	0.832124i	−0.909336	+	0.416062i	$0.863410\pi$
$798$	1.52134i	0.0538547i
$799$	21.7875	0.770787
$800$	0	0
$801$	−6.27226	−0.221620
$802$	− 5.51706i	− 0.194814i
$803$	− 17.6668i	− 0.623449i
$804$	−19.9376	−0.703143
$805$	0	0
$806$	0.331133	0.0116637
$807$	− 29.9144i	− 1.05304i
$808$	8.86054i	0.311713i
$809$	38.1509	1.34132	0.670658	−	0.741767i	$-0.266012\pi$
$809$	38.1509	1.34132	0.670658	+	0.741767i	$0.266012\pi$
$810$	0	0
$811$	47.9069	1.68224	0.841120	−	0.540849i	$-0.181897\pi$
$811$	47.9069	1.68224	0.841120	+	0.540849i	$0.181897\pi$
$812$	37.9106i	1.33040i
$813$	46.2879i	1.62339i
$814$	2.96299	0.103853
$815$	0	0
$816$	28.8772	1.01090
$817$	− 5.60988i	− 0.196265i
$818$	2.02318i	0.0707388i
$819$	0.805405	0.0281431
$820$	0	0
$821$	18.9808	0.662434	0.331217	−	0.943555i	$-0.392541\pi$
$821$	18.9808	0.662434	0.331217	+	0.943555i	$0.392541\pi$
$822$	− 2.11621i	− 0.0738114i
$823$	22.1681i	0.772732i	0.922345	+	0.386366i	$0.126270\pi$
$823$	22.1681i	0.772732i	−0.922345	+	0.386366i	$0.873730\pi$
$824$	−1.00365	−0.0349638
$825$	0	0
$826$	3.54577	0.123373
$827$	− 4.72938i	− 0.164457i	−0.996614	−	0.0822283i	$-0.973796\pi$
$827$	− 4.72938i	− 0.164457i	0.996614	−	0.0822283i	$-0.0262037\pi$
$828$	− 1.44079i	− 0.0500710i
$829$	−16.4400	−0.570986	−0.285493	−	0.958381i	$-0.592157\pi$
$829$	−16.4400	−0.570986	−0.285493	+	0.958381i	$0.592157\pi$
$830$	0	0
$831$	−20.5050	−0.711309
$832$	2.13570i	0.0740419i
$833$	18.7580i	0.649926i
$834$	−1.45366	−0.0503362
$835$	0	0
$836$	−6.80540	−0.235370
$837$	34.4167i	1.18962i
$838$	− 6.03024i	− 0.208311i
$839$	5.60083	0.193362	0.0966811	−	0.995315i	$-0.469177\pi$
$839$	5.60083	0.193362	0.0966811	+	0.995315i	$0.469177\pi$
$840$	0	0
$841$	5.95363	0.205298
$842$	3.71193i	0.127922i
$843$	38.0906i	1.31191i
$844$	32.6469	1.12375
$845$	0	0
$846$	0.644174	0.0221471
$847$	22.8260i	0.784310i
$848$	30.7812i	1.05703i
$849$	−34.9318	−1.19886
$850$	0	0
$851$	7.11091	0.243759
$852$	− 17.2767i	− 0.591890i
$853$	17.8905i	0.612559i	0.951942	+	0.306279i	$0.0990842\pi$
$853$	17.8905i	0.612559i	−0.951942	+	0.306279i	$0.900916\pi$
$854$	−0.546895	−0.0187144
$855$	0	0
$856$	11.5001	0.393065
$857$	3.19536i	0.109151i	0.998510	+	0.0545757i	$0.0173806\pi$
$857$	3.19536i	0.109151i	−0.998510	+	0.0545757i	$0.982619\pi$
$858$	0.159819i	0.00545614i
$859$	−43.4196	−1.48146	−0.740728	−	0.671805i	$-0.765519\pi$
$859$	−43.4196	−1.48146	−0.740728	+	0.671805i	$0.765519\pi$
$860$	0	0
$861$	−4.87251	−0.166055
$862$	− 1.31064i	− 0.0446407i
$863$	− 43.2724i	− 1.47301i	−0.676433	−	0.736504i	$-0.736475\pi$
$863$	− 43.2724i	− 1.47301i	0.676433	−	0.736504i	$-0.263525\pi$
$864$	12.1264	0.412547
$865$	0	0
$866$	−1.00427	−0.0341264
$867$	− 14.2135i	− 0.482716i
$868$	− 39.1120i	− 1.32755i
$869$	15.5213	0.526525
$870$	0	0
$871$	−2.04147	−0.0691727
$872$	− 1.45582i	− 0.0493004i
$873$	− 5.60562i	− 0.189722i
$874$	0.278666	0.00942603
$875$	0	0
$876$	25.5684	0.863876
$877$	34.2339i	1.15600i	0.816038	+	0.577999i	$0.196166\pi$
$877$	34.2339i	1.15600i	−0.816038	+	0.577999i	$0.803834\pi$
$878$	− 5.63366i	− 0.190127i
$879$	18.1092	0.610808
$880$	0	0
$881$	27.6270	0.930779	0.465389	−	0.885106i	$-0.345914\pi$
$881$	27.6270	0.930779	0.465389	+	0.885106i	$0.345914\pi$
$882$	0.554602i	0.0186744i
$883$	− 25.8990i	− 0.871571i	−0.900051	−	0.435786i	$-0.856471\pi$
$883$	− 25.8990i	− 0.871571i	0.900051	−	0.435786i	$-0.143529\pi$
$884$	3.00905	0.101205
$885$	0	0
$886$	2.07949	0.0698618
$887$	17.2449i	0.579027i	0.957174	+	0.289514i	$0.0934936\pi$
$887$	17.2449i	0.579027i	−0.957174	+	0.289514i	$0.906506\pi$
$888$	8.64958i	0.290261i
$889$	−19.0762	−0.639796
$890$	0	0
$891$	−11.6108	−0.388977
$892$	− 13.0338i	− 0.436403i
$893$	− 7.30213i	− 0.244356i
$894$	−5.07502	−0.169734
$895$	0	0
$896$	−18.3200	−0.612027
$897$	0.383551i	0.0128064i
$898$	− 2.88792i	− 0.0963710i
$899$	−36.0614	−1.20271
$900$	0	0
$901$	41.8238	1.39335
$902$	0.371893i	0.0123827i
$903$	15.5610i	0.517836i
$904$	7.57601	0.251974
$905$	0	0
$906$	−1.04827	−0.0348266
$907$	− 57.0465i	− 1.89420i	−0.320940	−	0.947099i	$-0.603999\pi$
$907$	− 57.0465i	− 1.89420i	0.320940	−	0.947099i	$-0.396001\pi$
$908$	26.3345i	0.873942i
$909$	−10.1633	−0.337095
$910$	0	0
$911$	19.8683	0.658267	0.329133	−	0.944283i	$-0.393243\pi$
$911$	19.8683	0.658267	0.329133	+	0.944283i	$0.393243\pi$
$912$	− 9.67824i	− 0.320479i
$913$	− 29.0307i	− 0.960777i
$914$	0.763304	0.0252478
$915$	0	0
$916$	−19.8350	−0.655367
$917$	4.88033i	0.161163i
$918$	− 5.33620i	− 0.176121i
$919$	−27.9785	−0.922924	−0.461462	−	0.887160i	$-0.652675\pi$
$919$	−27.9785	−0.922924	−0.461462	+	0.887160i	$0.652675\pi$
$920$	0	0
$921$	6.30361	0.207711
$922$	4.39090i	0.144606i
$923$	− 1.76902i	− 0.0582279i
$924$	18.8772	0.621013
$925$	0	0
$926$	7.56952	0.248750
$927$	− 1.15122i	− 0.0378109i
$928$	12.7059i	0.417090i
$929$	14.1249	0.463424	0.231712	−	0.972784i	$-0.425567\pi$
$929$	14.1249	0.463424	0.231712	+	0.972784i	$0.425567\pi$
$930$	0	0
$931$	6.28678	0.206041
$932$	− 43.2727i	− 1.41744i
$933$	37.7506i	1.23590i
$934$	1.90437	0.0623127
$935$	0	0
$936$	0.179450	0.00586551
$937$	35.9546i	1.17459i	0.809374	+	0.587293i	$0.199806\pi$
$937$	35.9546i	1.17459i	−0.809374	+	0.587293i	$0.800194\pi$
$938$	− 4.11422i	− 0.134334i
$939$	32.8999	1.07365
$940$	0	0
$941$	26.6979	0.870327	0.435164	−	0.900351i	$-0.356691\pi$
$941$	26.6979	0.870327	0.435164	+	0.900351i	$0.356691\pi$
$942$	1.15499i	0.0376317i
$943$	0.892508i	0.0290640i
$944$	−22.5570	−0.734169
$945$	0	0
$946$	1.18768	0.0386150
$947$	17.3102i	0.562506i	0.959634	+	0.281253i	$0.0907499\pi$
$947$	17.3102i	0.562506i	−0.959634	+	0.281253i	$0.909250\pi$
$948$	22.4634i	0.729576i
$949$	2.61803	0.0849850
$950$	0	0
$951$	32.2693	1.04640
$952$	12.2318i	0.396436i
$953$	− 24.2622i	− 0.785931i	−0.919553	−	0.392965i	$-0.871449\pi$
$953$	− 24.2622i	− 0.785931i	0.919553	−	0.392965i	$-0.128551\pi$
$954$	1.23657	0.0400354
$955$	0	0
$956$	−14.8669	−0.480830
$957$	− 17.4048i	− 0.562617i
$958$	6.59568i	0.213097i
$959$	−25.5940	−0.826475
$960$	0	0
$961$	6.20427	0.200138
$962$	0.439084i	0.0141566i
$963$	13.1910i	0.425072i
$964$	−40.1048	−1.29169
$965$	0	0
$966$	−0.772978	−0.0248702
$967$	− 31.1143i	− 1.00057i	−0.865861	−	0.500284i	$-0.833229\pi$
$967$	− 31.1143i	− 1.00057i	0.865861	−	0.500284i	$-0.166771\pi$
$968$	5.08580i	0.163464i
$969$	−13.1502	−0.422447
$970$	0	0
$971$	−17.3721	−0.557497	−0.278749	−	0.960364i	$-0.589920\pi$
$971$	−17.3721	−0.557497	−0.278749	+	0.960364i	$0.589920\pi$
$972$	16.4834i	0.528704i
$973$	17.5810i	0.563620i
$974$	−1.94855	−0.0624356
$975$	0	0
$976$	3.47917	0.111365
$977$	− 42.8929i	− 1.37227i	−0.727476	−	0.686133i	$-0.759307\pi$
$977$	− 42.8929i	− 1.37227i	0.727476	−	0.686133i	$-0.240693\pi$
$978$	− 4.24013i	− 0.135584i
$979$	−15.0528	−0.481091
$980$	0	0
$981$	1.66987	0.0533149
$982$	3.23651i	0.103281i
$983$	37.4533i	1.19457i	0.802028	+	0.597287i	$0.203755\pi$
$983$	37.4533i	1.19457i	−0.802028	+	0.597287i	$0.796245\pi$
$984$	−1.08563	−0.0346086
$985$	0	0
$986$	5.59120	0.178060
$987$	20.2550i	0.644724i
$988$	− 1.00849i	− 0.0320843i
$989$	2.85033	0.0906353
$990$	0	0
$991$	16.7624	0.532476	0.266238	−	0.963907i	$-0.414219\pi$
$991$	16.7624	0.532476	0.266238	+	0.963907i	$0.414219\pi$
$992$	− 13.1085i	− 0.416196i
$993$	− 13.1931i	− 0.418669i
$994$	3.56514	0.113079
$995$	0	0
$996$	42.0149	1.33129
$997$	− 59.8595i	− 1.89577i	−0.318615	−	0.947884i	$-0.603218\pi$
$997$	− 59.8595i	− 1.89577i	0.318615	−	0.947884i	$-0.396782\pi$
$998$	− 1.72499i	− 0.0546037i
$999$	−45.6367	−1.44388

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.2.b.c.624.4		8
5.2	odd	4		625.2.a.f.1.5		8
5.3	odd	4		625.2.a.f.1.4		8
5.4	even	2	inner	625.2.b.c.624.5		8
15.2	even	4		5625.2.a.x.1.4		8
15.8	even	4		5625.2.a.x.1.5		8
20.3	even	4		10000.2.a.bj.1.6		8
20.7	even	4		10000.2.a.bj.1.3		8
25.2	odd	20		125.2.d.b.101.3		16
25.3	odd	20		625.2.d.o.376.3		16
25.4	even	10		625.2.e.a.249.2		8
25.6	even	5		625.2.e.a.374.2		8
25.8	odd	20		625.2.d.o.251.3		16
25.9	even	10		125.2.e.b.99.1		8
25.11	even	5		125.2.e.b.24.1		8
25.12	odd	20		125.2.d.b.26.3		16
25.13	odd	20		125.2.d.b.26.2		16
25.14	even	10		25.2.e.a.4.2	✓	8
25.16	even	5		25.2.e.a.19.2	yes	8
25.17	odd	20		625.2.d.o.251.2		16
25.19	even	10		625.2.e.i.374.1		8
25.21	even	5		625.2.e.i.249.1		8
25.22	odd	20		625.2.d.o.376.2		16
25.23	odd	20		125.2.d.b.101.2		16
75.14	odd	10		225.2.m.a.154.1		8
75.41	odd	10		225.2.m.a.19.1		8
100.39	odd	10		400.2.y.c.129.2		8
100.91	odd	10		400.2.y.c.369.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.e.a.4.2	✓	8	25.14	even	10
25.2.e.a.19.2	yes	8	25.16	even	5
125.2.d.b.26.2		16	25.13	odd	20
125.2.d.b.26.3		16	25.12	odd	20
125.2.d.b.101.2		16	25.23	odd	20
125.2.d.b.101.3		16	25.2	odd	20
125.2.e.b.24.1		8	25.11	even	5
125.2.e.b.99.1		8	25.9	even	10
225.2.m.a.19.1		8	75.41	odd	10
225.2.m.a.154.1		8	75.14	odd	10
400.2.y.c.129.2		8	100.39	odd	10
400.2.y.c.369.2		8	100.91	odd	10
625.2.a.f.1.4		8	5.3	odd	4
625.2.a.f.1.5		8	5.2	odd	4
625.2.b.c.624.4		8	1.1	even	1	trivial
625.2.b.c.624.5		8	5.4	even	2	inner
625.2.d.o.251.2		16	25.17	odd	20
625.2.d.o.251.3		16	25.8	odd	20
625.2.d.o.376.2		16	25.22	odd	20
625.2.d.o.376.3		16	25.3	odd	20
625.2.e.a.249.2		8	25.4	even	10
625.2.e.a.374.2		8	25.6	even	5
625.2.e.i.249.1		8	25.21	even	5
625.2.e.i.374.1		8	25.19	even	10
5625.2.a.x.1.4		8	15.2	even	4
5625.2.a.x.1.5		8	15.8	even	4
10000.2.a.bj.1.3		8	20.7	even	4
10000.2.a.bj.1.6		8	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 625 = 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	625.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.183172i q^{2} +1.47195i q^{3} +1.96645 q^{4} +0.269620 q^{6} -3.26086i q^{7} -0.726543i q^{8} +0.833366 q^{9} +O(q^{10})\)	\(q-0.183172i q^{2} +1.47195i q^{3} +1.96645 q^{4} +0.269620 q^{6} -3.26086i q^{7} -0.726543i q^{8} +0.833366 q^{9} +2.00000 q^{11} +2.89451i q^{12} +0.296379i q^{13} -0.597298 q^{14} +3.79981 q^{16} -5.16297i q^{17} -0.152649i q^{18} -1.73038 q^{19} +4.79981 q^{21} -0.366344i q^{22} -0.879192i q^{23} +1.06943 q^{24} +0.0542883 q^{26} +5.64252i q^{27} -6.41230i q^{28} -5.91216 q^{29} +6.09953 q^{31} -2.14910i q^{32} +2.94390i q^{33} -0.945712 q^{34} +1.63877 q^{36} +8.08800i q^{37} +0.316957i q^{38} -0.436254 q^{39} -1.01515 q^{41} -0.879192i q^{42} +3.24199i q^{43} +3.93290 q^{44} -0.161043 q^{46} +4.21996i q^{47} +5.59313i q^{48} -3.63318 q^{49} +7.59963 q^{51} +0.582813i q^{52} +8.10072i q^{53} +1.03355 q^{54} -2.36915 q^{56} -2.54703i q^{57} +1.08294i q^{58} -5.93635 q^{59} +0.915615 q^{61} -1.11726i q^{62} -2.71749i q^{63} +7.20597 q^{64} +0.539240 q^{66} +6.88806i q^{67} -10.1527i q^{68} +1.29413 q^{69} -5.96878 q^{71} -0.605476i q^{72} -8.83341i q^{73} +1.48150 q^{74} -3.40270 q^{76} -6.52171i q^{77} +0.0799096i q^{78} +7.76067 q^{79} -5.80540 q^{81} +0.185946i q^{82} -14.5154i q^{83} +9.43858 q^{84} +0.593842 q^{86} -8.70240i q^{87} -1.45309i q^{88} -7.52642 q^{89} +0.966448 q^{91} -1.72889i q^{92} +8.97820i q^{93} +0.772978 q^{94} +3.16337 q^{96} -6.72649i q^{97} +0.665497i q^{98} +1.66673 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{4} + 6 q^{6} - 4 q^{9}+O(q^{10}) \) 8 * q - 6 * q^4 + 6 * q^6 - 4 * q^9	\( 8 q - 6 q^{4} + 6 q^{6} - 4 q^{9} + 16 q^{11} - 12 q^{14} - 2 q^{16} - 10 q^{19} + 6 q^{21} - 20 q^{24} + 6 q^{26} - 20 q^{29} + 16 q^{31} - 2 q^{34} - 12 q^{36} - 18 q^{39} + 26 q^{41} - 12 q^{44} + 6 q^{46} + 14 q^{49} - 4 q^{51} + 30 q^{54} + 10 q^{56} - 30 q^{59} + 6 q^{61} + 44 q^{64} + 12 q^{66} - 8 q^{69} + 46 q^{71} - 12 q^{74} - 20 q^{76} - 10 q^{79} - 32 q^{81} + 18 q^{84} - 14 q^{86} - 30 q^{89} - 14 q^{91} + 68 q^{94} - 54 q^{96} - 8 q^{99}+O(q^{100}) \) 8 * q - 6 * q^4 + 6 * q^6 - 4 * q^9 + 16 * q^11 - 12 * q^14 - 2 * q^16 - 10 * q^19 + 6 * q^21 - 20 * q^24 + 6 * q^26 - 20 * q^29 + 16 * q^31 - 2 * q^34 - 12 * q^36 - 18 * q^39 + 26 * q^41 - 12 * q^44 + 6 * q^46 + 14 * q^49 - 4 * q^51 + 30 * q^54 + 10 * q^56 - 30 * q^59 + 6 * q^61 + 44 * q^64 + 12 * q^66 - 8 * q^69 + 46 * q^71 - 12 * q^74 - 20 * q^76 - 10 * q^79 - 32 * q^81 + 18 * q^84 - 14 * q^86 - 30 * q^89 - 14 * q^91 + 68 * q^94 - 54 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more