LMFDB - Embedded newform 1375.2.b.f.749.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1375,2,Mod(749,1375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1375, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("1375.749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1375 = 5^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1375.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:	$10.9794302779$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 31 x^{18} + 401 x^{16} + 2814 x^{14} + 11674 x^{12} + 29278 x^{10} + 43479 x^{8} + 35324 x^{6} + \cdots + 1 $ x^20 + 31x^18 + 401x^16 + 2814x^14 + 11674x^12 + 29278x^10 + 43479x^8 + 35324x^6 + 12586x^4 + 636*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			749.16
Root			$1.50007i$ of defining polynomial
Character	$\chi$	$=$	1375.749
Dual form			1375.2.b.f.749.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.50007i q^{2} +3.20096i q^{3} -0.250195 q^{4} -4.80165 q^{6} +1.22398i q^{7} +2.62482i q^{8} -7.24615 q^{9} +O(q^{10})$	\(q+1.50007i q^{2} +3.20096i q^{3} -0.250195 q^{4} -4.80165 q^{6} +1.22398i q^{7} +2.62482i q^{8} -7.24615 q^{9} +1.00000 q^{11} -0.800866i q^{12} +0.288860i q^{13} -1.83605 q^{14} -4.43779 q^{16} -1.57860i q^{17} -10.8697i q^{18} -5.19542 q^{19} -3.91790 q^{21} +1.50007i q^{22} +1.84423i q^{23} -8.40195 q^{24} -0.433309 q^{26} -13.5918i q^{27} -0.306233i q^{28} +8.59916 q^{29} +3.94329 q^{31} -1.40734i q^{32} +3.20096i q^{33} +2.36800 q^{34} +1.81295 q^{36} +4.38869i q^{37} -7.79346i q^{38} -0.924631 q^{39} -12.1201 q^{41} -5.87711i q^{42} +5.82561i q^{43} -0.250195 q^{44} -2.76646 q^{46} -12.3196i q^{47} -14.2052i q^{48} +5.50188 q^{49} +5.05304 q^{51} -0.0722715i q^{52} +8.72294i q^{53} +20.3885 q^{54} -3.21272 q^{56} -16.6303i q^{57} +12.8993i q^{58} -12.4707 q^{59} +4.93306 q^{61} +5.91520i q^{62} -8.86913i q^{63} -6.76449 q^{64} -4.80165 q^{66} -10.1382i q^{67} +0.394959i q^{68} -5.90330 q^{69} -1.86054 q^{71} -19.0198i q^{72} -0.180227i q^{73} -6.58332 q^{74} +1.29987 q^{76} +1.22398i q^{77} -1.38701i q^{78} +9.66140 q^{79} +21.7683 q^{81} -18.1810i q^{82} +6.28419i q^{83} +0.980241 q^{84} -8.73880 q^{86} +27.5256i q^{87} +2.62482i q^{88} +8.79900 q^{89} -0.353559 q^{91} -0.461418i q^{92} +12.6223i q^{93} +18.4802 q^{94} +4.50483 q^{96} +11.0023i q^{97} +8.25318i q^{98} -7.24615 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 22 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10}) $ 20 * q - 22 * q^4 + 2 * q^6 - 16 * q^9	$ 20 q - 22 q^{4} + 2 q^{6} - 16 q^{9} + 20 q^{11} + 28 q^{14} + 26 q^{16} - 10 q^{19} - 6 q^{21} - 48 q^{24} + 4 q^{26} + 14 q^{29} - 6 q^{31} - 2 q^{34} + 12 q^{36} + 32 q^{39} - 8 q^{41} - 22 q^{44} + 72 q^{46} - 10 q^{49} + 50 q^{51} + 14 q^{54} - 88 q^{56} - 8 q^{59} - 30 q^{61} - 28 q^{64} + 2 q^{66} - 36 q^{69} + 44 q^{71} - 28 q^{74} - 10 q^{76} + 18 q^{79} - 60 q^{81} + 84 q^{84} - 54 q^{86} + 86 q^{89} - 52 q^{91} + 40 q^{94} + 182 q^{96} - 16 q^{99}+O(q^{100}) $ 20 * q - 22 * q^4 + 2 * q^6 - 16 * q^9 + 20 * q^11 + 28 * q^14 + 26 * q^16 - 10 * q^19 - 6 * q^21 - 48 * q^24 + 4 * q^26 + 14 * q^29 - 6 * q^31 - 2 * q^34 + 12 * q^36 + 32 * q^39 - 8 * q^41 - 22 * q^44 + 72 * q^46 - 10 * q^49 + 50 * q^51 + 14 * q^54 - 88 * q^56 - 8 * q^59 - 30 * q^61 - 28 * q^64 + 2 * q^66 - 36 * q^69 + 44 * q^71 - 28 * q^74 - 10 * q^76 + 18 * q^79 - 60 * q^81 + 84 * q^84 - 54 * q^86 + 86 * q^89 - 52 * q^91 + 40 * q^94 + 182 * q^96 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1375\mathbb{Z}\right)^\times$.

$n$	$376$	$1002$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.50007i	1.06071i	0.847777	+	0.530353i	$0.177941\pi$
$2$	1.50007i	1.06071i	−0.847777	+	0.530353i	$0.822059\pi$
$3$	3.20096i	1.84808i	0.382301	+	0.924038i	$0.375132\pi$
$3$	3.20096i	1.84808i	−0.382301	+	0.924038i	$0.624868\pi$
$4$	−0.250195	−0.125098
$5$	0	0
$5$	0	0
$6$	−4.80165	−1.96027
$7$	1.22398i	0.462620i	0.972880	+	0.231310i	$0.0743012\pi$
$7$	1.22398i	0.462620i	−0.972880	+	0.231310i	$0.925699\pi$
$8$	2.62482i	0.928014i
$9$	−7.24615	−2.41538
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	− 0.800866i	− 0.231190i
$13$	0.288860i	0.0801154i	0.999197	+	0.0400577i	$0.0127542\pi$
$13$	0.288860i	0.0801154i	−0.999197	+	0.0400577i	$0.987246\pi$
$14$	−1.83605	−0.490704
$15$	0	0
$16$	−4.43779	−1.10945
$17$	− 1.57860i	− 0.382867i	−0.981506	−	0.191433i	$-0.938686\pi$
$17$	− 1.57860i	− 0.382867i	0.981506	−	0.191433i	$-0.0613136\pi$
$18$	− 10.8697i	− 2.56201i
$19$	−5.19542	−1.19191	−0.595955	−	0.803018i	$-0.703226\pi$
$19$	−5.19542	−1.19191	−0.595955	+	0.803018i	$0.703226\pi$
$20$	0	0
$21$	−3.91790	−0.854957
$22$	1.50007i	0.319815i
$23$	1.84423i	0.384548i	0.981341	+	0.192274i	$0.0615863\pi$
$23$	1.84423i	0.384548i	−0.981341	+	0.192274i	$0.938414\pi$
$24$	−8.40195	−1.71504
$25$	0	0
$26$	−0.433309	−0.0849790
$27$	− 13.5918i	− 2.61574i
$28$	− 0.306233i	− 0.0578727i
$29$	8.59916	1.59682	0.798412	−	0.602112i	$-0.205674\pi$
$29$	8.59916	1.59682	0.798412	+	0.602112i	$0.205674\pi$
$30$	0	0
$31$	3.94329	0.708236	0.354118	−	0.935201i	$-0.384781\pi$
$31$	3.94329	0.708236	0.354118	+	0.935201i	$0.384781\pi$
$32$	− 1.40734i	− 0.248784i
$33$	3.20096i	0.557216i
$34$	2.36800	0.406109
$35$	0	0
$36$	1.81295	0.302159
$37$	4.38869i	0.721496i	0.932663	+	0.360748i	$0.117479\pi$
$37$	4.38869i	0.721496i	−0.932663	+	0.360748i	$0.882521\pi$
$38$	− 7.79346i	− 1.26427i
$39$	−0.924631	−0.148059
$40$	0	0
$41$	−12.1201	−1.89285	−0.946424	−	0.322928i	$-0.895333\pi$
$41$	−12.1201	−1.89285	−0.946424	+	0.322928i	$0.895333\pi$
$42$	− 5.87711i	− 0.906858i
$43$	5.82561i	0.888398i	0.895928	+	0.444199i	$0.146512\pi$
$43$	5.82561i	0.888398i	−0.895928	+	0.444199i	$0.853488\pi$
$44$	−0.250195	−0.0377184
$45$	0	0
$46$	−2.76646	−0.407893
$47$	− 12.3196i	− 1.79700i	−0.438971	−	0.898501i	$-0.644657\pi$
$47$	− 12.3196i	− 1.79700i	0.438971	−	0.898501i	$-0.355343\pi$
$48$	− 14.2052i	− 2.05034i
$49$	5.50188	0.785983
$50$	0	0
$51$	5.05304	0.707567
$52$	− 0.0722715i	− 0.0100223i
$53$	8.72294i	1.19819i	0.800679	+	0.599094i	$0.204472\pi$
$53$	8.72294i	1.19819i	−0.800679	+	0.599094i	$0.795528\pi$
$54$	20.3885	2.77453
$55$	0	0
$56$	−3.21272	−0.429318
$57$	− 16.6303i	− 2.20274i
$58$	12.8993i	1.69376i
$59$	−12.4707	−1.62354	−0.811772	−	0.583974i	$-0.801497\pi$
$59$	−12.4707	−1.62354	−0.811772	+	0.583974i	$0.801497\pi$
$60$	0	0
$61$	4.93306	0.631614	0.315807	−	0.948823i	$-0.397725\pi$
$61$	4.93306	0.631614	0.315807	+	0.948823i	$0.397725\pi$
$62$	5.91520i	0.751231i
$63$	− 8.86913i	− 1.11740i
$64$	−6.76449	−0.845561
$65$	0	0
$66$	−4.80165	−0.591042
$67$	− 10.1382i	− 1.23858i	−0.785162	−	0.619290i	$-0.787421\pi$
$67$	− 10.1382i	− 1.23858i	0.785162	−	0.619290i	$-0.212579\pi$
$68$	0.394959i	0.0478958i
$69$	−5.90330	−0.710674
$70$	0	0
$71$	−1.86054	−0.220806	−0.110403	−	0.993887i	$-0.535214\pi$
$71$	−1.86054	−0.220806	−0.110403	+	0.993887i	$0.535214\pi$
$72$	− 19.0198i	− 2.24151i
$73$	− 0.180227i	− 0.0210940i	−0.999944	−	0.0105470i	$-0.996643\pi$
$73$	− 0.180227i	− 0.0210940i	0.999944	−	0.0105470i	$-0.00335727\pi$
$74$	−6.58332	−0.765296
$75$	0	0
$76$	1.29987	0.149105
$77$	1.22398i	0.139485i
$78$	− 1.38701i	− 0.157048i
$79$	9.66140	1.08699	0.543496	−	0.839412i	$-0.317100\pi$
$79$	9.66140	1.08699	0.543496	+	0.839412i	$0.317100\pi$
$80$	0	0
$81$	21.7683	2.41870
$82$	− 18.1810i	− 2.00775i
$83$	6.28419i	0.689780i	0.938643	+	0.344890i	$0.112084\pi$
$83$	6.28419i	0.689780i	−0.938643	+	0.344890i	$0.887916\pi$
$84$	0.980241	0.106953
$85$	0	0
$86$	−8.73880	−0.942329
$87$	27.5256i	2.95105i
$88$	2.62482i	0.279807i
$89$	8.79900	0.932692	0.466346	−	0.884602i	$-0.345570\pi$
$89$	8.79900	0.932692	0.466346	+	0.884602i	$0.345570\pi$
$90$	0	0
$91$	−0.353559	−0.0370630
$92$	− 0.461418i	− 0.0481061i
$93$	12.6223i	1.30887i
$94$	18.4802	1.90609
$95$	0	0
$96$	4.50483	0.459772
$97$	11.0023i	1.11711i	0.829468	+	0.558555i	$0.188644\pi$
$97$	11.0023i	1.11711i	−0.829468	+	0.558555i	$0.811356\pi$
$98$	8.25318i	0.833697i
$99$	−7.24615	−0.728266
$100$	0	0
$101$	3.98763	0.396784	0.198392	−	0.980123i	$-0.436428\pi$
$101$	3.98763	0.396784	0.198392	+	0.980123i	$0.436428\pi$
$102$	7.57989i	0.750521i
$103$	3.96230i	0.390417i	0.980762	+	0.195209i	$0.0625384\pi$
$103$	3.96230i	0.390417i	−0.980762	+	0.195209i	$0.937462\pi$
$104$	−0.758207	−0.0743483
$105$	0	0
$106$	−13.0850	−1.27093
$107$	10.5764i	1.02246i	0.859444	+	0.511231i	$0.170810\pi$
$107$	10.5764i	1.02246i	−0.859444	+	0.511231i	$0.829190\pi$
$108$	3.40060i	0.327223i
$109$	15.1940	1.45532	0.727659	−	0.685939i	$-0.240608\pi$
$109$	15.1940	1.45532	0.727659	+	0.685939i	$0.240608\pi$
$110$	0	0
$111$	−14.0480	−1.33338
$112$	− 5.43176i	− 0.513253i
$113$	18.0620i	1.69913i	0.527481	+	0.849567i	$0.323137\pi$
$113$	18.0620i	1.69913i	−0.527481	+	0.849567i	$0.676863\pi$
$114$	24.9466	2.33646
$115$	0	0
$116$	−2.15147	−0.199759
$117$	− 2.09313i	− 0.193510i
$118$	− 18.7068i	− 1.72210i
$119$	1.93217	0.177122
$120$	0	0
$121$	1.00000	0.0909091
$122$	7.39991i	0.669957i
$123$	− 38.7961i	− 3.49812i
$124$	−0.986594	−0.0885987
$125$	0	0
$126$	13.3043	1.18524
$127$	8.03182i	0.712709i	0.934351	+	0.356354i	$0.115980\pi$
$127$	8.03182i	0.712709i	−0.934351	+	0.356354i	$0.884020\pi$
$128$	− 12.9618i	− 1.14568i
$129$	−18.6476	−1.64183
$130$	0	0
$131$	11.8660	1.03673	0.518367	−	0.855158i	$-0.326540\pi$
$131$	11.8660	1.03673	0.518367	+	0.855158i	$0.326540\pi$
$132$	− 0.800866i	− 0.0697064i
$133$	− 6.35907i	− 0.551401i
$134$	15.2080	1.31377
$135$	0	0
$136$	4.14354	0.355306
$137$	− 8.43692i	− 0.720815i	−0.932795	−	0.360407i	$-0.882638\pi$
$137$	− 8.43692i	− 0.720815i	0.932795	−	0.360407i	$-0.117362\pi$
$138$	− 8.85534i	− 0.753817i
$139$	−14.0788	−1.19415	−0.597076	−	0.802185i	$-0.703671\pi$
$139$	−14.0788	−1.19415	−0.597076	+	0.802185i	$0.703671\pi$
$140$	0	0
$141$	39.4347	3.32100
$142$	− 2.79093i	− 0.234210i
$143$	0.288860i	0.0241557i
$144$	32.1569	2.67974
$145$	0	0
$146$	0.270352	0.0223745
$147$	17.6113i	1.45256i
$148$	− 1.09803i	− 0.0902575i
$149$	0.594780	0.0487263	0.0243631	−	0.999703i	$-0.492244\pi$
$149$	0.594780	0.0487263	0.0243631	+	0.999703i	$0.492244\pi$
$150$	0	0
$151$	−1.38071	−0.112361	−0.0561804	−	0.998421i	$-0.517892\pi$
$151$	−1.38071	−0.112361	−0.0561804	+	0.998421i	$0.517892\pi$
$152$	− 13.6370i	− 1.10611i
$153$	11.4388i	0.924770i
$154$	−1.83605	−0.147953
$155$	0	0
$156$	0.231338	0.0185219
$157$	0.300286i	0.0239654i	0.999928	+	0.0119827i	$0.00381430\pi$
$157$	0.300286i	0.0239654i	−0.999928	+	0.0119827i	$0.996186\pi$
$158$	14.4927i	1.15298i
$159$	−27.9218	−2.21434
$160$	0	0
$161$	−2.25729	−0.177900
$162$	32.6538i	2.56552i
$163$	− 17.5768i	− 1.37672i	−0.725367	−	0.688362i	$-0.758330\pi$
$163$	− 17.5768i	− 1.37672i	0.725367	−	0.688362i	$-0.241670\pi$
$164$	3.03240	0.236791
$165$	0	0
$166$	−9.42670	−0.731654
$167$	7.06410i	0.546636i	0.961924	+	0.273318i	$0.0881212\pi$
$167$	7.06410i	0.546636i	−0.961924	+	0.273318i	$0.911879\pi$
$168$	− 10.2838i	− 0.793412i
$169$	12.9166	0.993582
$170$	0	0
$171$	37.6468	2.87892
$172$	− 1.45754i	− 0.111137i
$173$	8.99187i	0.683639i	0.939766	+	0.341820i	$0.111043\pi$
$173$	8.99187i	0.683639i	−0.939766	+	0.341820i	$0.888957\pi$
$174$	−41.2901	−3.13020
$175$	0	0
$176$	−4.43779	−0.334511
$177$	− 39.9182i	− 3.00043i
$178$	13.1991i	0.989312i
$179$	−13.3813	−1.00017	−0.500084	−	0.865977i	$-0.666697\pi$
$179$	−13.3813	−1.00017	−0.500084	+	0.865977i	$0.666697\pi$
$180$	0	0
$181$	−1.28865	−0.0957848	−0.0478924	−	0.998853i	$-0.515250\pi$
$181$	−1.28865	−0.0957848	−0.0478924	+	0.998853i	$0.515250\pi$
$182$	− 0.530361i	− 0.0393130i
$183$	15.7905i	1.16727i
$184$	−4.84077	−0.356866
$185$	0	0
$186$	−18.9343	−1.38833
$187$	− 1.57860i	− 0.115439i
$188$	3.08231i	0.224801i
$189$	16.6360	1.21009
$190$	0	0
$191$	−23.3257	−1.68779	−0.843896	−	0.536507i	$-0.819743\pi$
$191$	−23.3257	−1.68779	−0.843896	+	0.536507i	$0.819743\pi$
$192$	− 21.6529i	− 1.56266i
$193$	− 0.327637i	− 0.0235838i	−0.999930	−	0.0117919i	$-0.996246\pi$
$193$	− 0.327637i	− 0.0235838i	0.999930	−	0.0117919i	$-0.00375357\pi$
$194$	−16.5041	−1.18492
$195$	0	0
$196$	−1.37654	−0.0983246
$197$	18.1023i	1.28973i	0.764295	+	0.644867i	$0.223087\pi$
$197$	18.1023i	1.28973i	−0.764295	+	0.644867i	$0.776913\pi$
$198$	− 10.8697i	− 0.772476i
$199$	18.2554	1.29409	0.647045	−	0.762451i	$-0.276004\pi$
$199$	18.2554	1.29409	0.647045	+	0.762451i	$0.276004\pi$
$200$	0	0
$201$	32.4520	2.28899
$202$	5.98170i	0.420871i
$203$	10.5252i	0.738722i
$204$	−1.26425	−0.0885150
$205$	0	0
$206$	−5.94371	−0.414118
$207$	− 13.3636i	− 0.928832i
$208$	− 1.28190i	− 0.0888839i
$209$	−5.19542	−0.359374
$210$	0	0
$211$	−20.5222	−1.41280	−0.706402	−	0.707811i	$-0.749683\pi$
$211$	−20.5222	−1.41280	−0.706402	+	0.707811i	$0.749683\pi$
$212$	− 2.18244i	− 0.149891i
$213$	− 5.95552i	− 0.408065i
$214$	−15.8653	−1.08453
$215$	0	0
$216$	35.6759	2.42744
$217$	4.82650i	0.327644i
$218$	22.7919i	1.54367i
$219$	0.576900	0.0389833
$220$	0	0
$221$	0.455995	0.0306736
$222$	− 21.0730i	− 1.41432i
$223$	− 22.0220i	− 1.47471i	−0.675508	−	0.737353i	$-0.736076\pi$
$223$	− 22.0220i	− 1.47471i	0.675508	−	0.737353i	$-0.263924\pi$
$224$	1.72255	0.115093
$225$	0	0
$226$	−27.0942	−1.80228
$227$	− 8.36860i	− 0.555443i	−0.960662	−	0.277722i	$-0.910421\pi$
$227$	− 8.36860i	− 0.555443i	0.960662	−	0.277722i	$-0.0895793\pi$
$228$	4.16083i	0.275558i
$229$	7.58253	0.501068	0.250534	−	0.968108i	$-0.419394\pi$
$229$	7.58253	0.501068	0.250534	+	0.968108i	$0.419394\pi$
$230$	0	0
$231$	−3.91790	−0.257779
$232$	22.5712i	1.48187i
$233$	20.9119i	1.36999i	0.728550	+	0.684993i	$0.240195\pi$
$233$	20.9119i	1.36999i	−0.728550	+	0.684993i	$0.759805\pi$
$234$	3.13983	0.205257
$235$	0	0
$236$	3.12011	0.203102
$237$	30.9258i	2.00884i
$238$	2.89838i	0.187874i
$239$	−24.4959	−1.58451	−0.792254	−	0.610191i	$-0.791093\pi$
$239$	−24.4959	−1.58451	−0.792254	+	0.610191i	$0.791093\pi$
$240$	0	0
$241$	1.43188	0.0922353	0.0461177	−	0.998936i	$-0.485315\pi$
$241$	1.43188	0.0922353	0.0461177	+	0.998936i	$0.485315\pi$
$242$	1.50007i	0.0964278i
$243$	28.9040i	1.85420i
$244$	−1.23423	−0.0790134
$245$	0	0
$246$	58.1966	3.71048
$247$	− 1.50075i	− 0.0954904i
$248$	10.3504i	0.657253i
$249$	−20.1155	−1.27477
$250$	0	0
$251$	−2.85477	−0.180192	−0.0900958	−	0.995933i	$-0.528717\pi$
$251$	−2.85477	−0.180192	−0.0900958	+	0.995933i	$0.528717\pi$
$252$	2.21901i	0.139785i
$253$	1.84423i	0.115946i
$254$	−12.0483	−0.755975
$255$	0	0
$256$	5.91464	0.369665
$257$	− 24.3844i	− 1.52106i	−0.649303	−	0.760530i	$-0.724939\pi$
$257$	− 24.3844i	− 1.52106i	0.649303	−	0.760530i	$-0.275061\pi$
$258$	− 27.9726i	− 1.74150i
$259$	−5.37166	−0.333779
$260$	0	0
$261$	−62.3108	−3.85694
$262$	17.7997i	1.09967i
$263$	− 20.4099i	− 1.25853i	−0.777192	−	0.629263i	$-0.783357\pi$
$263$	− 20.4099i	− 1.25853i	0.777192	−	0.629263i	$-0.216643\pi$
$264$	−8.40195	−0.517104
$265$	0	0
$266$	9.53902	0.584875
$267$	28.1653i	1.72369i
$268$	2.53653i	0.154943i
$269$	−16.5189	−1.00718	−0.503588	−	0.863944i	$-0.667987\pi$
$269$	−16.5189	−1.00718	−0.503588	+	0.863944i	$0.667987\pi$
$270$	0	0
$271$	14.6001	0.886892	0.443446	−	0.896301i	$-0.353756\pi$
$271$	14.6001	0.886892	0.443446	+	0.896301i	$0.353756\pi$
$272$	7.00550i	0.424771i
$273$	− 1.13173i	− 0.0684952i
$274$	12.6559	0.764573
$275$	0	0
$276$	1.47698	0.0889037
$277$	4.93008i	0.296220i	0.988971	+	0.148110i	$0.0473190\pi$
$277$	4.93008i	0.296220i	−0.988971	+	0.148110i	$0.952681\pi$
$278$	− 21.1192i	− 1.26664i
$279$	−28.5737	−1.71066
$280$	0	0
$281$	−31.6194	−1.88626	−0.943128	−	0.332431i	$-0.892131\pi$
$281$	−31.6194	−1.88626	−0.943128	+	0.332431i	$0.892131\pi$
$282$	59.1545i	3.52260i
$283$	10.1494i	0.603318i	0.953416	+	0.301659i	$0.0975403\pi$
$283$	10.1494i	0.603318i	−0.953416	+	0.301659i	$0.902460\pi$
$284$	0.465499	0.0276223
$285$	0	0
$286$	−0.433309	−0.0256221
$287$	− 14.8348i	− 0.875669i
$288$	10.1978i	0.600910i
$289$	14.5080	0.853413
$290$	0	0
$291$	−35.2178	−2.06450
$292$	0.0450920i	0.00263881i
$293$	18.8640i	1.10204i	0.834490	+	0.551022i	$0.185762\pi$
$293$	18.8640i	1.10204i	−0.834490	+	0.551022i	$0.814238\pi$
$294$	−26.4181	−1.54073
$295$	0	0
$296$	−11.5195	−0.669559
$297$	− 13.5918i	− 0.788674i
$298$	0.892209i	0.0516843i
$299$	−0.532725	−0.0308083
$300$	0	0
$301$	−7.13042	−0.410991
$302$	− 2.07116i	− 0.119182i
$303$	12.7642i	0.733286i
$304$	23.0562	1.32236
$305$	0	0
$306$	−17.1589	−0.980910
$307$	− 3.75619i	− 0.214377i	−0.994239	−	0.107188i	$-0.965815\pi$
$307$	− 3.75619i	− 0.214377i	0.994239	−	0.107188i	$-0.0341848\pi$
$308$	− 0.306233i	− 0.0174493i
$309$	−12.6832	−0.721521
$310$	0	0
$311$	10.4251	0.591155	0.295578	−	0.955319i	$-0.404488\pi$
$311$	10.4251	0.591155	0.295578	+	0.955319i	$0.404488\pi$
$312$	− 2.42699i	− 0.137401i
$313$	− 9.73050i	− 0.550001i	−0.961444	−	0.275000i	$-0.911322\pi$
$313$	− 9.73050i	− 0.550001i	0.961444	−	0.275000i	$-0.0886780\pi$
$314$	−0.450448	−0.0254203
$315$	0	0
$316$	−2.41724	−0.135980
$317$	18.1884i	1.02156i	0.859711	+	0.510781i	$0.170644\pi$
$317$	18.1884i	1.02156i	−0.859711	+	0.510781i	$0.829356\pi$
$318$	− 41.8845i	− 2.34877i
$319$	8.59916	0.481460
$320$	0	0
$321$	−33.8547	−1.88959
$322$	− 3.38609i	− 0.188699i
$323$	8.20149i	0.456343i
$324$	−5.44632	−0.302573
$325$	0	0
$326$	26.3664	1.46030
$327$	48.6353i	2.68954i
$328$	− 31.8132i	− 1.75659i
$329$	15.0789	0.831329
$330$	0	0
$331$	−12.3936	−0.681214	−0.340607	−	0.940206i	$-0.610633\pi$
$331$	−12.3936	−0.681214	−0.340607	+	0.940206i	$0.610633\pi$
$332$	− 1.57228i	− 0.0862899i
$333$	− 31.8011i	− 1.74269i
$334$	−10.5966	−0.579821
$335$	0	0
$336$	17.3868	0.948530
$337$	− 11.3540i	− 0.618494i	−0.950982	−	0.309247i	$-0.899923\pi$
$337$	− 11.3540i	− 0.618494i	0.950982	−	0.309247i	$-0.100077\pi$
$338$	19.3757i	1.05390i
$339$	−57.8159	−3.14013
$340$	0	0
$341$	3.94329	0.213541
$342$	56.4726i	3.05369i
$343$	15.3020i	0.826231i
$344$	−15.2912	−0.824446
$345$	0	0
$346$	−13.4884	−0.725141
$347$	25.7332i	1.38143i	0.723126	+	0.690716i	$0.242705\pi$
$347$	25.7332i	1.38143i	−0.723126	+	0.690716i	$0.757295\pi$
$348$	− 6.88677i	− 0.369170i
$349$	22.9327	1.22756	0.613779	−	0.789478i	$-0.289649\pi$
$349$	22.9327	1.22756	0.613779	+	0.789478i	$0.289649\pi$
$350$	0	0
$351$	3.92612	0.209561
$352$	− 1.40734i	− 0.0750113i
$353$	− 35.4667i	− 1.88770i	−0.330369	−	0.943852i	$-0.607173\pi$
$353$	− 35.4667i	− 1.88770i	0.330369	−	0.943852i	$-0.392827\pi$
$354$	59.8798	3.18258
$355$	0	0
$356$	−2.20147	−0.116678
$357$	6.18481i	0.327335i
$358$	− 20.0729i	− 1.06088i
$359$	22.5576	1.19054	0.595272	−	0.803524i	$-0.297044\pi$
$359$	22.5576	1.19054	0.595272	+	0.803524i	$0.297044\pi$
$360$	0	0
$361$	7.99235	0.420650
$362$	− 1.93306i	− 0.101600i
$363$	3.20096i	0.168007i
$364$	0.0884587	0.00463650
$365$	0	0
$366$	−23.6868	−1.23813
$367$	5.09524i	0.265969i	0.991118	+	0.132985i	$0.0424561\pi$
$367$	5.09524i	0.265969i	−0.991118	+	0.132985i	$0.957544\pi$
$368$	− 8.18431i	− 0.426636i
$369$	87.8243	4.57195
$370$	0	0
$371$	−10.6767	−0.554306
$372$	− 3.15805i	− 0.163737i
$373$	1.43608i	0.0743574i	0.999309	+	0.0371787i	$0.0118371\pi$
$373$	1.43608i	0.0743574i	−0.999309	+	0.0371787i	$0.988163\pi$
$374$	2.36800	0.122447
$375$	0	0
$376$	32.3368	1.66764
$377$	2.48396i	0.127930i
$378$	24.9551i	1.28355i
$379$	20.0927	1.03209	0.516046	−	0.856561i	$-0.327403\pi$
$379$	20.0927	1.03209	0.516046	+	0.856561i	$0.327403\pi$
$380$	0	0
$381$	−25.7095	−1.31714
$382$	− 34.9901i	− 1.79025i
$383$	− 34.2841i	− 1.75184i	−0.482458	−	0.875919i	$-0.660256\pi$
$383$	− 34.2841i	− 1.75184i	0.482458	−	0.875919i	$-0.339744\pi$
$384$	41.4904	2.11730
$385$	0	0
$386$	0.491477	0.0250155
$387$	− 42.2133i	− 2.14582i
$388$	− 2.75271i	− 0.139748i
$389$	20.1409	1.02119	0.510593	−	0.859822i	$-0.329426\pi$
$389$	20.1409	1.02119	0.510593	+	0.859822i	$0.329426\pi$
$390$	0	0
$391$	2.91130	0.147231
$392$	14.4414i	0.729403i
$393$	37.9825i	1.91596i
$394$	−27.1546	−1.36803
$395$	0	0
$396$	1.81295	0.0911043
$397$	24.7335i	1.24134i	0.784073	+	0.620669i	$0.213139\pi$
$397$	24.7335i	1.24134i	−0.784073	+	0.620669i	$0.786861\pi$
$398$	27.3843i	1.37265i
$399$	20.3551	1.01903
$400$	0	0
$401$	17.5472	0.876266	0.438133	−	0.898910i	$-0.355640\pi$
$401$	17.5472	0.876266	0.438133	+	0.898910i	$0.355640\pi$
$402$	48.6801i	2.42795i
$403$	1.13906i	0.0567407i
$404$	−0.997686	−0.0496367
$405$	0	0
$406$	−15.7884	−0.783567
$407$	4.38869i	0.217539i
$408$	13.2633i	0.656632i
$409$	3.59423	0.177723	0.0888616	−	0.996044i	$-0.471677\pi$
$409$	3.59423	0.177723	0.0888616	+	0.996044i	$0.471677\pi$
$410$	0	0
$411$	27.0063	1.33212
$412$	− 0.991350i	− 0.0488403i
$413$	− 15.2638i	− 0.751084i
$414$	20.0462	0.985218
$415$	0	0
$416$	0.406524	0.0199315
$417$	− 45.0658i	− 2.20688i
$418$	− 7.79346i	− 0.381191i
$419$	11.3269	0.553357	0.276679	−	0.960963i	$-0.410766\pi$
$419$	11.3269	0.553357	0.276679	+	0.960963i	$0.410766\pi$
$420$	0	0
$421$	−10.5985	−0.516537	−0.258269	−	0.966073i	$-0.583152\pi$
$421$	−10.5985	−0.516537	−0.258269	+	0.966073i	$0.583152\pi$
$422$	− 30.7846i	− 1.49857i
$423$	89.2699i	4.34045i
$424$	−22.8961	−1.11194
$425$	0	0
$426$	8.93367	0.432838
$427$	6.03796i	0.292197i
$428$	− 2.64617i	− 0.127908i
$429$	−0.924631	−0.0446416
$430$	0	0
$431$	−36.2599	−1.74658	−0.873290	−	0.487201i	$-0.838018\pi$
$431$	−36.2599	−1.74658	−0.873290	+	0.487201i	$0.838018\pi$
$432$	60.3174i	2.90202i
$433$	− 9.23138i	− 0.443632i	−0.975089	−	0.221816i	$-0.928802\pi$
$433$	− 9.23138i	− 0.443632i	0.975089	−	0.221816i	$-0.0711984\pi$
$434$	−7.24007	−0.347534
$435$	0	0
$436$	−3.80146	−0.182057
$437$	− 9.58154i	− 0.458347i
$438$	0.865387i	0.0413498i
$439$	−11.4383	−0.545918	−0.272959	−	0.962026i	$-0.588002\pi$
$439$	−11.4383	−0.545918	−0.272959	+	0.962026i	$0.588002\pi$
$440$	0	0
$441$	−39.8675	−1.89845
$442$	0.684022i	0.0325356i
$443$	25.3909i	1.20636i	0.797606	+	0.603179i	$0.206100\pi$
$443$	25.3909i	1.20636i	−0.797606	+	0.603179i	$0.793900\pi$
$444$	3.51475	0.166803
$445$	0	0
$446$	33.0345	1.56423
$447$	1.90387i	0.0900499i
$448$	− 8.27958i	− 0.391173i
$449$	−27.6895	−1.30675	−0.653375	−	0.757035i	$-0.726647\pi$
$449$	−27.6895	−1.30675	−0.653375	+	0.757035i	$0.726647\pi$
$450$	0	0
$451$	−12.1201	−0.570715
$452$	− 4.51904i	− 0.212558i
$453$	− 4.41961i	− 0.207651i
$454$	12.5534	0.589162
$455$	0	0
$456$	43.6516	2.04417
$457$	− 9.54606i	− 0.446546i	−0.974756	−	0.223273i	$-0.928326\pi$
$457$	− 9.54606i	− 0.446546i	0.974756	−	0.223273i	$-0.0716741\pi$
$458$	11.3743i	0.531486i
$459$	−21.4560	−1.00148
$460$	0	0
$461$	15.7565	0.733854	0.366927	−	0.930250i	$-0.380410\pi$
$461$	15.7565	0.733854	0.366927	+	0.930250i	$0.380410\pi$
$462$	− 5.87711i	− 0.273428i
$463$	− 3.06197i	− 0.142302i	−0.997466	−	0.0711510i	$-0.977333\pi$
$463$	− 3.06197i	− 0.142302i	0.997466	−	0.0711510i	$-0.0226672\pi$
$464$	−38.1613	−1.77159
$465$	0	0
$466$	−31.3692	−1.45315
$467$	11.0959i	0.513458i	0.966483	+	0.256729i	$0.0826448\pi$
$467$	11.0959i	0.513458i	−0.966483	+	0.256729i	$0.917355\pi$
$468$	0.523690i	0.0242076i
$469$	12.4089	0.572992
$470$	0	0
$471$	−0.961203	−0.0442899
$472$	− 32.7333i	− 1.50667i
$473$	5.82561i	0.267862i
$474$	−46.3907	−2.13079
$475$	0	0
$476$	−0.483420	−0.0221575
$477$	− 63.2077i	− 2.89408i
$478$	− 36.7455i	− 1.68070i
$479$	27.3993	1.25190	0.625952	−	0.779861i	$-0.284710\pi$
$479$	27.3993	1.25190	0.625952	+	0.779861i	$0.284710\pi$
$480$	0	0
$481$	−1.26772	−0.0578030
$482$	2.14791i	0.0978346i
$483$	− 7.22551i	− 0.328772i
$484$	−0.250195	−0.0113725
$485$	0	0
$486$	−43.3579	−1.96676
$487$	− 3.69858i	− 0.167598i	−0.996483	−	0.0837992i	$-0.973295\pi$
$487$	− 3.69858i	− 0.167598i	0.996483	−	0.0837992i	$-0.0267054\pi$
$488$	12.9484i	0.586147i
$489$	56.2628	2.54429
$490$	0	0
$491$	16.3266	0.736809	0.368404	−	0.929666i	$-0.379904\pi$
$491$	16.3266	0.736809	0.368404	+	0.929666i	$0.379904\pi$
$492$	9.70660i	0.437607i
$493$	− 13.5746i	− 0.611371i
$494$	2.25122	0.101287
$495$	0	0
$496$	−17.4995	−0.785752
$497$	− 2.27726i	− 0.102149i
$498$	− 30.1745i	− 1.35215i
$499$	16.7245	0.748693	0.374347	−	0.927289i	$-0.377867\pi$
$499$	16.7245	0.748693	0.374347	+	0.927289i	$0.377867\pi$
$500$	0	0
$501$	−22.6119	−1.01023
$502$	− 4.28234i	− 0.191130i
$503$	21.0625i	0.939132i	0.882897	+	0.469566i	$0.155590\pi$
$503$	21.0625i	0.939132i	−0.882897	+	0.469566i	$0.844410\pi$
$504$	23.2799	1.03697
$505$	0	0
$506$	−2.76646	−0.122984
$507$	41.3454i	1.83621i
$508$	− 2.00952i	− 0.0891582i
$509$	5.09722	0.225930	0.112965	−	0.993599i	$-0.463965\pi$
$509$	5.09722	0.225930	0.112965	+	0.993599i	$0.463965\pi$
$510$	0	0
$511$	0.220594	0.00975850
$512$	− 17.0514i	− 0.753571i
$513$	70.6149i	3.11772i
$514$	36.5783	1.61340
$515$	0	0
$516$	4.66553	0.205389
$517$	− 12.3196i	− 0.541817i
$518$	− 8.05784i	− 0.354041i
$519$	−28.7826	−1.26342
$520$	0	0
$521$	8.12234	0.355846	0.177923	−	0.984044i	$-0.443062\pi$
$521$	8.12234	0.355846	0.177923	+	0.984044i	$0.443062\pi$
$522$	− 93.4702i	− 4.09108i
$523$	40.8582i	1.78660i	0.449456	+	0.893302i	$0.351618\pi$
$523$	40.8582i	1.78660i	−0.449456	+	0.893302i	$0.648382\pi$
$524$	−2.96881	−0.129693
$525$	0	0
$526$	30.6161	1.33493
$527$	− 6.22488i	− 0.271160i
$528$	− 14.2052i	− 0.618202i
$529$	19.5988	0.852123
$530$	0	0
$531$	90.3644	3.92148
$532$	1.59101i	0.0689790i
$533$	− 3.50103i	− 0.151646i
$534$	−42.2497	−1.82832
$535$	0	0
$536$	26.6110	1.14942
$537$	− 42.8331i	− 1.84839i
$538$	− 24.7794i	− 1.06832i
$539$	5.50188	0.236983
$540$	0	0
$541$	34.9937	1.50450	0.752248	−	0.658880i	$-0.228970\pi$
$541$	34.9937	1.50450	0.752248	+	0.658880i	$0.228970\pi$
$542$	21.9011i	0.940732i
$543$	− 4.12493i	− 0.177018i
$544$	−2.22162	−0.0952513
$545$	0	0
$546$	1.69766	0.0726533
$547$	25.9477i	1.10944i	0.832035	+	0.554722i	$0.187176\pi$
$547$	25.9477i	1.10944i	−0.832035	+	0.554722i	$0.812824\pi$
$548$	2.11088i	0.0901723i
$549$	−35.7457	−1.52559
$550$	0	0
$551$	−44.6762	−1.90327
$552$	− 15.4951i	− 0.659516i
$553$	11.8253i	0.502864i
$554$	−7.39545	−0.314202
$555$	0	0
$556$	3.52246	0.149386
$557$	14.1413i	0.599187i	0.954067	+	0.299593i	$0.0968510\pi$
$557$	14.1413i	0.599187i	−0.954067	+	0.299593i	$0.903149\pi$
$558$	− 42.8624i	− 1.81451i
$559$	−1.68279	−0.0711744
$560$	0	0
$561$	5.05304	0.213339
$562$	− 47.4312i	− 2.00076i
$563$	− 22.7762i	− 0.959901i	−0.877296	−	0.479951i	$-0.840655\pi$
$563$	− 22.7762i	− 0.959901i	0.877296	−	0.479951i	$-0.159345\pi$
$564$	−9.86637	−0.415449
$565$	0	0
$566$	−15.2247	−0.639943
$567$	26.6439i	1.11894i
$568$	− 4.88359i	− 0.204911i
$569$	−14.0904	−0.590700	−0.295350	−	0.955389i	$-0.595436\pi$
$569$	−14.0904	−0.590700	−0.295350	+	0.955389i	$0.595436\pi$
$570$	0	0
$571$	21.3286	0.892576	0.446288	−	0.894889i	$-0.352746\pi$
$571$	21.3286	0.892576	0.446288	+	0.894889i	$0.352746\pi$
$572$	− 0.0722715i	− 0.00302182i
$573$	− 74.6648i	− 3.11917i
$574$	22.2531	0.928827
$575$	0	0
$576$	49.0165	2.04235
$577$	12.5674i	0.523187i	0.965178	+	0.261594i	$0.0842480\pi$
$577$	12.5674i	0.523187i	−0.965178	+	0.261594i	$0.915752\pi$
$578$	21.7630i	0.905220i
$579$	1.04875	0.0435847
$580$	0	0
$581$	−7.69171	−0.319106
$582$	− 52.8290i	− 2.18983i
$583$	8.72294i	0.361267i
$584$	0.473064	0.0195755
$585$	0	0
$586$	−28.2972	−1.16895
$587$	− 32.0871i	− 1.32438i	−0.749337	−	0.662189i	$-0.769628\pi$
$587$	− 32.0871i	− 1.32438i	0.749337	−	0.662189i	$-0.230372\pi$
$588$	− 4.40627i	− 0.181711i
$589$	−20.4870	−0.844154
$590$	0	0
$591$	−57.9447	−2.38353
$592$	− 19.4761i	− 0.800463i
$593$	7.14177i	0.293277i	0.989190	+	0.146639i	$0.0468455\pi$
$593$	7.14177i	0.293277i	−0.989190	+	0.146639i	$0.953155\pi$
$594$	20.3885	0.836552
$595$	0	0
$596$	−0.148811	−0.00609555
$597$	58.4348i	2.39158i
$598$	− 0.799122i	− 0.0326785i
$599$	1.40611	0.0574519	0.0287260	−	0.999587i	$-0.490855\pi$
$599$	1.40611	0.0574519	0.0287260	+	0.999587i	$0.490855\pi$
$600$	0	0
$601$	−4.29464	−0.175182	−0.0875910	−	0.996157i	$-0.527917\pi$
$601$	−4.29464	−0.175182	−0.0875910	+	0.996157i	$0.527917\pi$
$602$	− 10.6961i	− 0.435940i
$603$	73.4630i	2.99165i
$604$	0.345448	0.0140561
$605$	0	0
$606$	−19.1472	−0.777802
$607$	16.6781i	0.676943i	0.940977	+	0.338472i	$0.109910\pi$
$607$	16.6781i	0.676943i	−0.940977	+	0.338472i	$0.890090\pi$
$608$	7.31170i	0.296529i
$609$	−33.6907	−1.36521
$610$	0	0
$611$	3.55865	0.143968
$612$	− 2.86193i	− 0.115687i
$613$	− 48.2654i	− 1.94942i	−0.223468	−	0.974711i	$-0.571738\pi$
$613$	− 48.2654i	− 1.94942i	0.223468	−	0.974711i	$-0.428262\pi$
$614$	5.63452	0.227391
$615$	0	0
$616$	−3.21272	−0.129444
$617$	27.6863i	1.11461i	0.830309	+	0.557303i	$0.188164\pi$
$617$	27.6863i	1.11461i	−0.830309	+	0.557303i	$0.811836\pi$
$618$	− 19.0256i	− 0.765321i
$619$	12.3773	0.497485	0.248743	−	0.968570i	$-0.419983\pi$
$619$	12.3773	0.497485	0.248743	+	0.968570i	$0.419983\pi$
$620$	0	0
$621$	25.0663	1.00588
$622$	15.6384i	0.627042i
$623$	10.7698i	0.431482i
$624$	4.10332	0.164264
$625$	0	0
$626$	14.5964	0.583389
$627$	− 16.6303i	− 0.664151i
$628$	− 0.0751301i	− 0.00299802i
$629$	6.92799	0.276237
$630$	0	0
$631$	3.96878	0.157995	0.0789974	−	0.996875i	$-0.474828\pi$
$631$	3.96878	0.157995	0.0789974	+	0.996875i	$0.474828\pi$
$632$	25.3594i	1.00874i
$633$	− 65.6906i	− 2.61097i
$634$	−27.2838	−1.08358
$635$	0	0
$636$	6.98590	0.277009
$637$	1.58927i	0.0629694i
$638$	12.8993i	0.510688i
$639$	13.4818	0.533330
$640$	0	0
$641$	−11.9847	−0.473367	−0.236683	−	0.971587i	$-0.576060\pi$
$641$	−11.9847	−0.473367	−0.236683	+	0.971587i	$0.576060\pi$
$642$	− 50.7843i	− 2.00430i
$643$	− 43.0681i	− 1.69844i	−0.528038	−	0.849221i	$-0.677072\pi$
$643$	− 43.0681i	− 1.69844i	0.528038	−	0.849221i	$-0.322928\pi$
$644$	0.564765	0.0222548
$645$	0	0
$646$	−12.3028	−0.484046
$647$	− 0.396319i	− 0.0155809i	−0.999970	−	0.00779045i	$-0.997520\pi$
$647$	− 0.396319i	− 0.0155809i	0.999970	−	0.00779045i	$-0.00247980\pi$
$648$	57.1378i	2.24458i
$649$	−12.4707	−0.489517
$650$	0	0
$651$	−15.4494	−0.605511
$652$	4.39765i	0.172225i
$653$	− 36.1248i	− 1.41367i	−0.707378	−	0.706835i	$-0.750122\pi$
$653$	− 36.1248i	− 1.41367i	0.707378	−	0.706835i	$-0.249878\pi$
$654$	−72.9561	−2.85281
$655$	0	0
$656$	53.7867	2.10002
$657$	1.30595i	0.0509501i
$658$	22.6194i	0.881796i
$659$	−37.7433	−1.47027	−0.735135	−	0.677921i	$-0.762881\pi$
$659$	−37.7433	−1.47027	−0.735135	+	0.677921i	$0.762881\pi$
$660$	0	0
$661$	35.8915	1.39602	0.698008	−	0.716090i	$-0.254070\pi$
$661$	35.8915	1.39602	0.698008	+	0.716090i	$0.254070\pi$
$662$	− 18.5912i	− 0.722568i
$663$	1.45962i	0.0566870i
$664$	−16.4949	−0.640126
$665$	0	0
$666$	47.7037	1.84848
$667$	15.8588i	0.614056i
$668$	− 1.76741i	− 0.0683830i
$669$	70.4917	2.72537
$670$	0	0
$671$	4.93306	0.190439
$672$	5.51381i	0.212700i
$673$	14.6746i	0.565665i	0.959169	+	0.282832i	$0.0912740\pi$
$673$	14.6746i	0.565665i	−0.959169	+	0.282832i	$0.908726\pi$
$674$	17.0318	0.656040
$675$	0	0
$676$	−3.23166	−0.124295
$677$	17.1886i	0.660612i	0.943874	+	0.330306i	$0.107152\pi$
$677$	17.1886i	0.660612i	−0.943874	+	0.330306i	$0.892848\pi$
$678$	− 86.7276i	− 3.33075i
$679$	−13.4665	−0.516797
$680$	0	0
$681$	26.7875	1.02650
$682$	5.91520i	0.226505i
$683$	18.5912i	0.711374i	0.934605	+	0.355687i	$0.115753\pi$
$683$	18.5912i	0.711374i	−0.934605	+	0.355687i	$0.884247\pi$
$684$	−9.41905	−0.360146
$685$	0	0
$686$	−22.9540	−0.876389
$687$	24.2714i	0.926012i
$688$	− 25.8529i	− 0.985631i
$689$	−2.51971	−0.0959933
$690$	0	0
$691$	28.5832	1.08736	0.543678	−	0.839294i	$-0.317031\pi$
$691$	28.5832	1.08736	0.543678	+	0.839294i	$0.317031\pi$
$692$	− 2.24973i	− 0.0855217i
$693$	− 8.86913i	− 0.336910i
$694$	−38.6015	−1.46529
$695$	0	0
$696$	−72.2497	−2.73862
$697$	19.1329i	0.724709i
$698$	34.4005i	1.30208i
$699$	−66.9382	−2.53184
$700$	0	0
$701$	14.3823	0.543212	0.271606	−	0.962408i	$-0.412445\pi$
$701$	14.3823	0.543212	0.271606	+	0.962408i	$0.412445\pi$
$702$	5.88944i	0.222283i
$703$	− 22.8011i	− 0.859959i
$704$	−6.76449	−0.254946
$705$	0	0
$706$	53.2024	2.00230
$707$	4.88077i	0.183560i
$708$	9.98734i	0.375347i
$709$	4.40230	0.165332	0.0826659	−	0.996577i	$-0.473657\pi$
$709$	4.40230	0.165332	0.0826659	+	0.996577i	$0.473657\pi$
$710$	0	0
$711$	−70.0080	−2.62550
$712$	23.0958i	0.865552i
$713$	7.27233i	0.272351i
$714$	−9.27761	−0.347206
$715$	0	0
$716$	3.34795	0.125119
$717$	− 78.4105i	− 2.92829i
$718$	33.8379i	1.26282i
$719$	1.10294	0.0411326	0.0205663	−	0.999788i	$-0.493453\pi$
$719$	1.10294	0.0411326	0.0205663	+	0.999788i	$0.493453\pi$
$720$	0	0
$721$	−4.84977	−0.180615
$722$	11.9890i	0.446186i
$723$	4.58338i	0.170458i
$724$	0.322415	0.0119825
$725$	0	0
$726$	−4.80165	−0.178206
$727$	20.8390i	0.772877i	0.922315	+	0.386438i	$0.126295\pi$
$727$	20.8390i	0.772877i	−0.922315	+	0.386438i	$0.873705\pi$
$728$	− 0.928028i	− 0.0343950i
$729$	−27.2159	−1.00800
$730$	0	0
$731$	9.19632	0.340138
$732$	− 3.95072i	− 0.146023i
$733$	5.74125i	0.212058i	0.994363	+	0.106029i	$0.0338136\pi$
$733$	5.74125i	0.212058i	−0.994363	+	0.106029i	$0.966186\pi$
$734$	−7.64319	−0.282115
$735$	0	0
$736$	2.59545	0.0956696
$737$	− 10.1382i	− 0.373446i
$738$	131.742i	4.84950i
$739$	33.3538	1.22694	0.613470	−	0.789718i	$-0.289773\pi$
$739$	33.3538	1.22694	0.613470	+	0.789718i	$0.289773\pi$
$740$	0	0
$741$	4.80384	0.176474
$742$	− 16.0157i	− 0.587955i
$743$	34.0596i	1.24953i	0.780815	+	0.624763i	$0.214804\pi$
$743$	34.0596i	1.24953i	−0.780815	+	0.624763i	$0.785196\pi$
$744$	−33.1313	−1.21465
$745$	0	0
$746$	−2.15421	−0.0788713
$747$	− 45.5362i	− 1.66608i
$748$	0.394959i	0.0144411i
$749$	−12.9453	−0.473011
$750$	0	0
$751$	52.3853	1.91157	0.955784	−	0.294070i	$-0.0950098\pi$
$751$	52.3853	1.91157	0.955784	+	0.294070i	$0.0950098\pi$
$752$	54.6720i	1.99368i
$753$	− 9.13802i	− 0.333008i
$754$	−3.72609	−0.135696
$755$	0	0
$756$	−4.16225	−0.151380
$757$	− 46.3336i	− 1.68402i	−0.539460	−	0.842011i	$-0.681371\pi$
$757$	− 46.3336i	− 1.68402i	0.539460	−	0.842011i	$-0.318629\pi$
$758$	30.1403i	1.09475i
$759$	−5.90330	−0.214276
$760$	0	0
$761$	28.5647	1.03547	0.517734	−	0.855542i	$-0.326776\pi$
$761$	28.5647	1.03547	0.517734	+	0.855542i	$0.326776\pi$
$762$	− 38.5660i	− 1.39710i
$763$	18.5971i	0.673259i
$764$	5.83599	0.211139
$765$	0	0
$766$	51.4284	1.85819
$767$	− 3.60229i	− 0.130071i
$768$	18.9325i	0.683169i
$769$	−9.38477	−0.338424	−0.169212	−	0.985580i	$-0.554122\pi$
$769$	−9.38477	−0.338424	−0.169212	+	0.985580i	$0.554122\pi$
$770$	0	0
$771$	78.0537	2.81103
$772$	0.0819733i	0.00295028i
$773$	50.4277i	1.81376i	0.421391	+	0.906879i	$0.361542\pi$
$773$	50.4277i	1.81376i	−0.421391	+	0.906879i	$0.638458\pi$
$774$	63.3227	2.27609
$775$	0	0
$776$	−28.8789	−1.03669
$777$	− 17.1945i	− 0.616848i
$778$	30.2127i	1.08318i
$779$	62.9691	2.25610
$780$	0	0
$781$	−1.86054	−0.0665754
$782$	4.36714i	0.156169i
$783$	− 116.878i	− 4.17687i
$784$	−24.4162	−0.872007
$785$	0	0
$786$	−56.9762	−2.03227
$787$	32.2280i	1.14880i	0.818573	+	0.574402i	$0.194765\pi$
$787$	32.2280i	1.14880i	−0.818573	+	0.574402i	$0.805235\pi$
$788$	− 4.52911i	− 0.161343i
$789$	65.3312	2.32585
$790$	0	0
$791$	−22.1075	−0.786053
$792$	− 19.0198i	− 0.675841i
$793$	1.42497i	0.0506020i
$794$	−37.1018	−1.31669
$795$	0	0
$796$	−4.56742	−0.161888
$797$	− 43.8497i	− 1.55324i	−0.629971	−	0.776618i	$-0.716933\pi$
$797$	− 43.8497i	− 1.55324i	0.629971	−	0.776618i	$-0.283067\pi$
$798$	30.5340i	1.08089i
$799$	−19.4478	−0.688013
$800$	0	0
$801$	−63.7589	−2.25281
$802$	26.3219i	0.929460i
$803$	− 0.180227i	− 0.00636008i
$804$	−8.11935	−0.286347
$805$	0	0
$806$	−1.70867	−0.0601852
$807$	− 52.8764i	− 1.86134i
$808$	10.4668i	0.368221i
$809$	41.6426	1.46407	0.732037	−	0.681264i	$-0.238570\pi$
$809$	41.6426	1.46407	0.732037	+	0.681264i	$0.238570\pi$
$810$	0	0
$811$	12.6463	0.444071	0.222035	−	0.975039i	$-0.428730\pi$
$811$	12.6463	0.444071	0.222035	+	0.975039i	$0.428730\pi$
$812$	− 2.63335i	− 0.0924125i
$813$	46.7343i	1.63904i
$814$	−6.58332	−0.230745
$815$	0	0
$816$	−22.4243	−0.785009
$817$	− 30.2665i	− 1.05889i
$818$	5.39158i	0.188512i
$819$	2.56194	0.0895214
$820$	0	0
$821$	11.5890	0.404458	0.202229	−	0.979338i	$-0.435181\pi$
$821$	11.5890	0.404458	0.202229	+	0.979338i	$0.435181\pi$
$822$	40.5112i	1.41299i
$823$	− 27.6578i	− 0.964089i	−0.876147	−	0.482045i	$-0.839894\pi$
$823$	− 27.6578i	− 0.964089i	0.876147	−	0.482045i	$-0.160106\pi$
$824$	−10.4003	−0.362313
$825$	0	0
$826$	22.8967	0.796680
$827$	− 9.28121i	− 0.322739i	−0.986894	−	0.161370i	$-0.948409\pi$
$827$	− 9.28121i	− 0.322739i	0.986894	−	0.161370i	$-0.0515911\pi$
$828$	3.34350i	0.116195i
$829$	−24.6674	−0.856734	−0.428367	−	0.903605i	$-0.640911\pi$
$829$	−24.6674	−0.856734	−0.428367	+	0.903605i	$0.640911\pi$
$830$	0	0
$831$	−15.7810	−0.547437
$832$	− 1.95399i	− 0.0677425i
$833$	− 8.68527i	− 0.300927i
$834$	67.6017	2.34085
$835$	0	0
$836$	1.29987	0.0449569
$837$	− 53.5963i	− 1.85256i
$838$	16.9911i	0.586949i
$839$	−29.4086	−1.01530	−0.507649	−	0.861564i	$-0.669485\pi$
$839$	−29.4086	−1.01530	−0.507649	+	0.861564i	$0.669485\pi$
$840$	0	0
$841$	44.9455	1.54984
$842$	− 15.8984i	− 0.547894i
$843$	− 101.212i	− 3.48594i
$844$	5.13455	0.176738
$845$	0	0
$846$	−133.911	−4.60394
$847$	1.22398i	0.0420564i
$848$	− 38.7106i	− 1.32933i
$849$	−32.4877	−1.11498
$850$	0	0
$851$	−8.09375	−0.277450
$852$	1.49004i	0.0510480i
$853$	− 25.9758i	− 0.889394i	−0.895681	−	0.444697i	$-0.853311\pi$
$853$	− 25.9758i	− 0.889394i	0.895681	−	0.444697i	$-0.146689\pi$
$854$	−9.05733	−0.309935
$855$	0	0
$856$	−27.7612	−0.948859
$857$	− 16.5357i	− 0.564849i	−0.959290	−	0.282424i	$-0.908861\pi$
$857$	− 16.5357i	− 0.564849i	0.959290	−	0.282424i	$-0.0911386\pi$
$858$	− 1.38701i	− 0.0473516i
$859$	34.2100	1.16723	0.583616	−	0.812030i	$-0.301637\pi$
$859$	34.2100	1.16723	0.583616	+	0.812030i	$0.301637\pi$
$860$	0	0
$861$	47.4855	1.61830
$862$	− 54.3923i	− 1.85261i
$863$	14.1618i	0.482072i	0.970516	+	0.241036i	$0.0774871\pi$
$863$	14.1618i	0.482072i	−0.970516	+	0.241036i	$0.922513\pi$
$864$	−19.1282	−0.650754
$865$	0	0
$866$	13.8477	0.470563
$867$	46.4396i	1.57717i
$868$	− 1.20757i	− 0.0409875i
$869$	9.66140	0.327741
$870$	0	0
$871$	2.92853	0.0992294
$872$	39.8815i	1.35056i
$873$	− 79.7240i	− 2.69825i
$874$	14.3729	0.486172
$875$	0	0
$876$	−0.144338	−0.00487672
$877$	7.64867i	0.258277i	0.991627	+	0.129139i	$0.0412212\pi$
$877$	7.64867i	0.258277i	−0.991627	+	0.129139i	$0.958779\pi$
$878$	− 17.1581i	− 0.579059i
$879$	−60.3828	−2.03666
$880$	0	0
$881$	14.9148	0.502492	0.251246	−	0.967923i	$-0.419160\pi$
$881$	14.9148	0.502492	0.251246	+	0.967923i	$0.419160\pi$
$882$	− 59.8038i	− 2.01370i
$883$	14.2982i	0.481173i	0.970628	+	0.240587i	$0.0773398\pi$
$883$	14.2982i	0.481173i	−0.970628	+	0.240587i	$0.922660\pi$
$884$	−0.114088	−0.00383719
$885$	0	0
$886$	−38.0880	−1.27959
$887$	− 27.9589i	− 0.938767i	−0.882994	−	0.469384i	$-0.844476\pi$
$887$	− 27.9589i	− 0.938767i	0.882994	−	0.469384i	$-0.155524\pi$
$888$	− 36.8736i	− 1.23740i
$889$	−9.83077	−0.329713
$890$	0	0
$891$	21.7683	0.729264
$892$	5.50981i	0.184482i
$893$	64.0056i	2.14187i
$894$	−2.85593	−0.0955165
$895$	0	0
$896$	15.8650	0.530013
$897$	− 1.70523i	− 0.0569360i
$898$	− 41.5361i	− 1.38608i
$899$	33.9090	1.13093
$900$	0	0
$901$	13.7700	0.458746
$902$	− 18.1810i	− 0.605361i
$903$	− 22.8242i	− 0.759542i
$904$	−47.4096	−1.57682
$905$	0	0
$906$	6.62970	0.220257
$907$	− 35.0205i	− 1.16284i	−0.813605	−	0.581419i	$-0.802498\pi$
$907$	− 35.0205i	− 1.16284i	0.813605	−	0.581419i	$-0.197502\pi$
$908$	2.09378i	0.0694847i
$909$	−28.8950	−0.958385
$910$	0	0
$911$	33.1195	1.09730	0.548648	−	0.836053i	$-0.315143\pi$
$911$	33.1195	1.09730	0.548648	+	0.836053i	$0.315143\pi$
$912$	73.8019i	2.44383i
$913$	6.28419i	0.207976i
$914$	14.3197	0.473654
$915$	0	0
$916$	−1.89712	−0.0626824
$917$	14.5237i	0.479614i
$918$	− 32.1853i	− 1.06227i
$919$	−4.61314	−0.152173	−0.0760867	−	0.997101i	$-0.524243\pi$
$919$	−4.61314	−0.152173	−0.0760867	+	0.997101i	$0.524243\pi$
$920$	0	0
$921$	12.0234	0.396185
$922$	23.6358i	0.778404i
$923$	− 0.537437i	− 0.0176899i
$924$	0.980241	0.0322476
$925$	0	0
$926$	4.59316	0.150941
$927$	− 28.7114i	− 0.943008i
$928$	− 12.1019i	− 0.397265i
$929$	12.1376	0.398222	0.199111	−	0.979977i	$-0.436195\pi$
$929$	12.1376	0.398222	0.199111	+	0.979977i	$0.436195\pi$
$930$	0	0
$931$	−28.5846	−0.936821
$932$	− 5.23207i	− 0.171382i
$933$	33.3705i	1.09250i
$934$	−16.6446	−0.544628
$935$	0	0
$936$	5.49408	0.179580
$937$	19.4908i	0.636738i	0.947967	+	0.318369i	$0.103135\pi$
$937$	19.4908i	0.636738i	−0.947967	+	0.318369i	$0.896865\pi$
$938$	18.6142i	0.607776i
$939$	31.1470	1.01644
$940$	0	0
$941$	−18.4539	−0.601580	−0.300790	−	0.953690i	$-0.597250\pi$
$941$	−18.4539	−0.601580	−0.300790	+	0.953690i	$0.597250\pi$
$942$	− 1.44187i	− 0.0469786i
$943$	− 22.3523i	− 0.727891i
$944$	55.3423	1.80124
$945$	0	0
$946$	−8.73880	−0.284123
$947$	− 17.9344i	− 0.582789i	−0.956603	−	0.291395i	$-0.905881\pi$
$947$	− 17.9344i	− 0.582789i	0.956603	−	0.291395i	$-0.0941193\pi$
$948$	− 7.73748i	− 0.251302i
$949$	0.0520605	0.00168995
$950$	0	0
$951$	−58.2203	−1.88792
$952$	5.07160i	0.164372i
$953$	− 12.9446i	− 0.419316i	−0.977775	−	0.209658i	$-0.932765\pi$
$953$	− 12.9446i	− 0.419316i	0.977775	−	0.209658i	$-0.0672350\pi$
$954$	94.8157	3.06977
$955$	0	0
$956$	6.12876	0.198218
$957$	27.5256i	0.889775i
$958$	41.1007i	1.32790i
$959$	10.3266	0.333463
$960$	0	0
$961$	−15.4504	−0.498401
$962$	− 1.90166i	− 0.0613120i
$963$	− 76.6384i	− 2.46964i
$964$	−0.358249	−0.0115384
$965$	0	0
$966$	10.8387	0.348731
$967$	0.410829i	0.0132114i	0.999978	+	0.00660568i	$0.00210267\pi$
$967$	0.410829i	0.0132114i	−0.999978	+	0.00660568i	$0.997897\pi$
$968$	2.62482i	0.0843649i
$969$	−26.2526	−0.843356
$970$	0	0
$971$	18.8694	0.605549	0.302774	−	0.953062i	$-0.402087\pi$
$971$	18.8694	0.605549	0.302774	+	0.953062i	$0.402087\pi$
$972$	− 7.23166i	− 0.231956i
$973$	− 17.2322i	− 0.552438i
$974$	5.54810	0.177773
$975$	0	0
$976$	−21.8919	−0.700743
$977$	9.23029i	0.295303i	0.989039	+	0.147652i	$0.0471714\pi$
$977$	9.23029i	0.295303i	−0.989039	+	0.147652i	$0.952829\pi$
$978$	84.3979i	2.69875i
$979$	8.79900	0.281217
$980$	0	0
$981$	−110.098	−3.51515
$982$	24.4910i	0.781538i
$983$	− 5.10972i	− 0.162975i	−0.996674	−	0.0814874i	$-0.974033\pi$
$983$	− 5.10972i	− 0.162975i	0.996674	−	0.0814874i	$-0.0259670\pi$
$984$	101.833	3.24631
$985$	0	0
$986$	20.3628	0.648485
$987$	48.2671i	1.53636i
$988$	0.375481i	0.0119456i
$989$	−10.7438	−0.341632
$990$	0	0
$991$	−3.71545	−0.118025	−0.0590126	−	0.998257i	$-0.518795\pi$
$991$	−3.71545	−0.118025	−0.0590126	+	0.998257i	$0.518795\pi$
$992$	− 5.54954i	− 0.176198i
$993$	− 39.6714i	− 1.25893i
$994$	3.41604	0.108350
$995$	0	0
$996$	5.03280	0.159470
$997$	− 19.8706i	− 0.629307i	−0.949207	−	0.314654i	$-0.898112\pi$
$997$	− 19.8706i	− 0.629307i	0.949207	−	0.314654i	$-0.101888\pi$
$998$	25.0879i	0.794144i
$999$	59.6500	1.88724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1375.2.b.f.749.16		20
5.2	odd	4		1375.2.a.f.1.2	yes	10
5.3	odd	4		1375.2.a.e.1.9	✓	10
5.4	even	2	inner	1375.2.b.f.749.5		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1375.2.a.e.1.9	✓	10	5.3	odd	4
1375.2.a.f.1.2	yes	10	5.2	odd	4
1375.2.b.f.749.5		20	5.4	even	2	inner
1375.2.b.f.749.16		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1375 = 5^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1375.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.50007i q^{2} +3.20096i q^{3} -0.250195 q^{4} -4.80165 q^{6} +1.22398i q^{7} +2.62482i q^{8} -7.24615 q^{9} +O(q^{10})\)	\(q+1.50007i q^{2} +3.20096i q^{3} -0.250195 q^{4} -4.80165 q^{6} +1.22398i q^{7} +2.62482i q^{8} -7.24615 q^{9} +1.00000 q^{11} -0.800866i q^{12} +0.288860i q^{13} -1.83605 q^{14} -4.43779 q^{16} -1.57860i q^{17} -10.8697i q^{18} -5.19542 q^{19} -3.91790 q^{21} +1.50007i q^{22} +1.84423i q^{23} -8.40195 q^{24} -0.433309 q^{26} -13.5918i q^{27} -0.306233i q^{28} +8.59916 q^{29} +3.94329 q^{31} -1.40734i q^{32} +3.20096i q^{33} +2.36800 q^{34} +1.81295 q^{36} +4.38869i q^{37} -7.79346i q^{38} -0.924631 q^{39} -12.1201 q^{41} -5.87711i q^{42} +5.82561i q^{43} -0.250195 q^{44} -2.76646 q^{46} -12.3196i q^{47} -14.2052i q^{48} +5.50188 q^{49} +5.05304 q^{51} -0.0722715i q^{52} +8.72294i q^{53} +20.3885 q^{54} -3.21272 q^{56} -16.6303i q^{57} +12.8993i q^{58} -12.4707 q^{59} +4.93306 q^{61} +5.91520i q^{62} -8.86913i q^{63} -6.76449 q^{64} -4.80165 q^{66} -10.1382i q^{67} +0.394959i q^{68} -5.90330 q^{69} -1.86054 q^{71} -19.0198i q^{72} -0.180227i q^{73} -6.58332 q^{74} +1.29987 q^{76} +1.22398i q^{77} -1.38701i q^{78} +9.66140 q^{79} +21.7683 q^{81} -18.1810i q^{82} +6.28419i q^{83} +0.980241 q^{84} -8.73880 q^{86} +27.5256i q^{87} +2.62482i q^{88} +8.79900 q^{89} -0.353559 q^{91} -0.461418i q^{92} +12.6223i q^{93} +18.4802 q^{94} +4.50483 q^{96} +11.0023i q^{97} +8.25318i q^{98} -7.24615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 22 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10}) \) 20 * q - 22 * q^4 + 2 * q^6 - 16 * q^9	\( 20 q - 22 q^{4} + 2 q^{6} - 16 q^{9} + 20 q^{11} + 28 q^{14} + 26 q^{16} - 10 q^{19} - 6 q^{21} - 48 q^{24} + 4 q^{26} + 14 q^{29} - 6 q^{31} - 2 q^{34} + 12 q^{36} + 32 q^{39} - 8 q^{41} - 22 q^{44} + 72 q^{46} - 10 q^{49} + 50 q^{51} + 14 q^{54} - 88 q^{56} - 8 q^{59} - 30 q^{61} - 28 q^{64} + 2 q^{66} - 36 q^{69} + 44 q^{71} - 28 q^{74} - 10 q^{76} + 18 q^{79} - 60 q^{81} + 84 q^{84} - 54 q^{86} + 86 q^{89} - 52 q^{91} + 40 q^{94} + 182 q^{96} - 16 q^{99}+O(q^{100}) \) 20 * q - 22 * q^4 + 2 * q^6 - 16 * q^9 + 20 * q^11 + 28 * q^14 + 26 * q^16 - 10 * q^19 - 6 * q^21 - 48 * q^24 + 4 * q^26 + 14 * q^29 - 6 * q^31 - 2 * q^34 + 12 * q^36 + 32 * q^39 - 8 * q^41 - 22 * q^44 + 72 * q^46 - 10 * q^49 + 50 * q^51 + 14 * q^54 - 88 * q^56 - 8 * q^59 - 30 * q^61 - 28 * q^64 + 2 * q^66 - 36 * q^69 + 44 * q^71 - 28 * q^74 - 10 * q^76 + 18 * q^79 - 60 * q^81 + 84 * q^84 - 54 * q^86 + 86 * q^89 - 52 * q^91 + 40 * q^94 + 182 * q^96 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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