LMFDB - Embedded newform 755.2.a.j.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [755,2,Mod(1,755)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(755, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("755.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 755 = 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	755.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,-2,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$6.02870535261$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 2 x^{14} - 22 x^{13} + 48 x^{12} + 171 x^{11} - 423 x^{10} - 527 x^{9} + 1641 x^{8} + 400 x^{7} + \cdots + 8 $ x^15 - 2x^14 - 22x^13 + 48x^12 + 171x^11 - 423x^10 - 527x^9 + 1641x^8 + 400x^7 - 2651x^6 + 421x^5 + 1591x^4 - 396x^3 - 288x^2 + 60x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.12744$ of defining polynomial
Character	$\chi$	$=$	755.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.12744 q^{2} +2.55334 q^{3} +2.52599 q^{4} +1.00000 q^{5} -5.43206 q^{6} +2.60553 q^{7} -1.11901 q^{8} +3.51953 q^{9} -2.12744 q^{10} +3.86021 q^{11} +6.44970 q^{12} +2.63130 q^{13} -5.54309 q^{14} +2.55334 q^{15} -2.67136 q^{16} -6.79915 q^{17} -7.48758 q^{18} +3.02807 q^{19} +2.52599 q^{20} +6.65279 q^{21} -8.21236 q^{22} +2.56609 q^{23} -2.85720 q^{24} +1.00000 q^{25} -5.59792 q^{26} +1.32654 q^{27} +6.58153 q^{28} -3.99265 q^{29} -5.43206 q^{30} -2.40837 q^{31} +7.92116 q^{32} +9.85642 q^{33} +14.4648 q^{34} +2.60553 q^{35} +8.89030 q^{36} -1.36589 q^{37} -6.44202 q^{38} +6.71859 q^{39} -1.11901 q^{40} -8.52346 q^{41} -14.1534 q^{42} +4.38895 q^{43} +9.75085 q^{44} +3.51953 q^{45} -5.45920 q^{46} -7.24681 q^{47} -6.82088 q^{48} -0.211234 q^{49} -2.12744 q^{50} -17.3605 q^{51} +6.64663 q^{52} -3.11675 q^{53} -2.82213 q^{54} +3.86021 q^{55} -2.91560 q^{56} +7.73168 q^{57} +8.49410 q^{58} -1.36025 q^{59} +6.44970 q^{60} -12.1363 q^{61} +5.12365 q^{62} +9.17023 q^{63} -11.5091 q^{64} +2.63130 q^{65} -20.9689 q^{66} +11.4763 q^{67} -17.1746 q^{68} +6.55210 q^{69} -5.54309 q^{70} +11.9053 q^{71} -3.93838 q^{72} +4.34482 q^{73} +2.90584 q^{74} +2.55334 q^{75} +7.64887 q^{76} +10.0579 q^{77} -14.2934 q^{78} -0.434182 q^{79} -2.67136 q^{80} -7.17149 q^{81} +18.1331 q^{82} -3.33674 q^{83} +16.8049 q^{84} -6.79915 q^{85} -9.33722 q^{86} -10.1946 q^{87} -4.31961 q^{88} -9.70616 q^{89} -7.48758 q^{90} +6.85592 q^{91} +6.48192 q^{92} -6.14938 q^{93} +15.4171 q^{94} +3.02807 q^{95} +20.2254 q^{96} +14.0904 q^{97} +0.449387 q^{98} +13.5861 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 2 q^{2} + 5 q^{3} + 18 q^{4} + 15 q^{5} - 4 q^{6} + 11 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{10} + 2 q^{11} + 4 q^{12} + 11 q^{13} - 9 q^{14} + 5 q^{15} + 40 q^{16} + 25 q^{17} - 15 q^{18} - 3 q^{19}+ \cdots - 11 q^{99}+O(q^{100}) $ 15 * q - 2 * q^2 + 5 * q^3 + 18 * q^4 + 15 * q^5 - 4 * q^6 + 11 * q^7 + 12 * q^8 + 10 * q^9 - 2 * q^10 + 2 * q^11 + 4 * q^12 + 11 * q^13 - 9 * q^14 + 5 * q^15 + 40 * q^16 + 25 * q^17 - 15 * q^18 - 3 * q^19 + 18 * q^20 - 11 * q^21 + 13 * q^22 + 8 * q^23 - 21 * q^24 + 15 * q^25 - 45 * q^26 + 20 * q^27 + 29 * q^28 + 24 * q^29 - 4 * q^30 - 11 * q^31 + 27 * q^32 + 26 * q^33 - 11 * q^34 + 11 * q^35 - 22 * q^36 + 35 * q^37 + 2 * q^38 + 11 * q^39 + 12 * q^40 - 20 * q^41 - 17 * q^42 - 13 * q^43 - 32 * q^44 + 10 * q^45 - 5 * q^46 + 6 * q^47 - 11 * q^48 + 40 * q^49 - 2 * q^50 + 23 * q^51 + 27 * q^52 + 38 * q^53 - 30 * q^54 + 2 * q^55 - 36 * q^56 + 4 * q^57 + 6 * q^58 - 15 * q^59 + 4 * q^60 + 33 * q^61 - 13 * q^62 + 2 * q^63 + 74 * q^64 + 11 * q^65 + 8 * q^66 + 24 * q^67 + 8 * q^68 + 13 * q^69 - 9 * q^70 + 19 * q^71 + 56 * q^72 + 4 * q^73 - 28 * q^74 + 5 * q^75 + 19 * q^76 + 14 * q^77 - 15 * q^78 - 4 * q^79 + 40 * q^80 - 13 * q^81 + 33 * q^82 + 19 * q^83 - 71 * q^84 + 25 * q^85 - 59 * q^86 + 49 * q^87 - 48 * q^88 - 35 * q^89 - 15 * q^90 - q^91 + 18 * q^92 + 2 * q^93 - 70 * q^94 - 3 * q^95 - 35 * q^96 + 55 * q^97 - 64 * q^98 - 11 * q^99

Coefficient data

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

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

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

$2$	−2.12744	−1.50433	−0.752163	−	0.658978i	$-0.770989\pi$
$2$	−2.12744	−1.50433	−0.752163	+	0.658978i	$0.770989\pi$
$3$	2.55334	1.47417	0.737085	−	0.675800i	$-0.236202\pi$
$3$	2.55334	1.47417	0.737085	+	0.675800i	$0.236202\pi$
$4$	2.52599	1.26299
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−5.43206	−2.21763
$7$	2.60553	0.984796	0.492398	−	0.870370i	$-0.336120\pi$
$7$	2.60553	0.984796	0.492398	+	0.870370i	$0.336120\pi$
$8$	−1.11901	−0.395629
$9$	3.51953	1.17318
$10$	−2.12744	−0.672755
$11$	3.86021	1.16390	0.581949	−	0.813225i	$-0.302290\pi$
$11$	3.86021	1.16390	0.581949	+	0.813225i	$0.302290\pi$
$12$	6.44970	1.86187
$13$	2.63130	0.729791	0.364895	−	0.931048i	$-0.381105\pi$
$13$	2.63130	0.729791	0.364895	+	0.931048i	$0.381105\pi$
$14$	−5.54309	−1.48145
$15$	2.55334	0.659269
$16$	−2.67136	−0.667840
$17$	−6.79915	−1.64904	−0.824518	−	0.565836i	$-0.808554\pi$
$17$	−6.79915	−1.64904	−0.824518	+	0.565836i	$0.808554\pi$
$18$	−7.48758	−1.76484
$19$	3.02807	0.694686	0.347343	−	0.937738i	$-0.387084\pi$
$19$	3.02807	0.694686	0.347343	+	0.937738i	$0.387084\pi$
$20$	2.52599	0.564828
$21$	6.65279	1.45176
$22$	−8.21236	−1.75088
$23$	2.56609	0.535067	0.267534	−	0.963549i	$-0.413791\pi$
$23$	2.56609	0.535067	0.267534	+	0.963549i	$0.413791\pi$
$24$	−2.85720	−0.583224
$25$	1.00000	0.200000
$26$	−5.59792	−1.09784
$27$	1.32654	0.255293
$28$	6.58153	1.24379
$29$	−3.99265	−0.741416	−0.370708	−	0.928750i	$-0.620885\pi$
$29$	−3.99265	−0.741416	−0.370708	+	0.928750i	$0.620885\pi$
$30$	−5.43206	−0.991755
$31$	−2.40837	−0.432556	−0.216278	−	0.976332i	$-0.569392\pi$
$31$	−2.40837	−0.432556	−0.216278	+	0.976332i	$0.569392\pi$
$32$	7.92116	1.40028
$33$	9.85642	1.71578
$34$	14.4648	2.48069
$35$	2.60553	0.440414
$36$	8.89030	1.48172
$37$	−1.36589	−0.224551	−0.112275	−	0.993677i	$-0.535814\pi$
$37$	−1.36589	−0.224551	−0.112275	+	0.993677i	$0.535814\pi$
$38$	−6.44202	−1.04503
$39$	6.71859	1.07584
$40$	−1.11901	−0.176931
$41$	−8.52346	−1.33114	−0.665570	−	0.746335i	$-0.731812\pi$
$41$	−8.52346	−1.33114	−0.665570	+	0.746335i	$0.731812\pi$
$42$	−14.1534	−2.18391
$43$	4.38895	0.669309	0.334655	−	0.942341i	$-0.391380\pi$
$43$	4.38895	0.669309	0.334655	+	0.942341i	$0.391380\pi$
$44$	9.75085	1.47000
$45$	3.51953	0.524661
$46$	−5.45920	−0.804915
$47$	−7.24681	−1.05706	−0.528528	−	0.848916i	$-0.677256\pi$
$47$	−7.24681	−1.05706	−0.528528	+	0.848916i	$0.677256\pi$
$48$	−6.82088	−0.984509
$49$	−0.211234	−0.0301763
$50$	−2.12744	−0.300865
$51$	−17.3605	−2.43096
$52$	6.64663	0.921722
$53$	−3.11675	−0.428119	−0.214059	−	0.976821i	$-0.568669\pi$
$53$	−3.11675	−0.428119	−0.214059	+	0.976821i	$0.568669\pi$
$54$	−2.82213	−0.384044
$55$	3.86021	0.520511
$56$	−2.91560	−0.389614
$57$	7.73168	1.02409
$58$	8.49410	1.11533
$59$	−1.36025	−0.177089	−0.0885447	−	0.996072i	$-0.528222\pi$
$59$	−1.36025	−0.177089	−0.0885447	+	0.996072i	$0.528222\pi$
$60$	6.44970	0.832653
$61$	−12.1363	−1.55389	−0.776944	−	0.629570i	$-0.783231\pi$
$61$	−12.1363	−1.55389	−0.776944	+	0.629570i	$0.783231\pi$
$62$	5.12365	0.650704
$63$	9.17023	1.15534
$64$	−11.5091	−1.43863
$65$	2.63130	0.326372
$66$	−20.9689	−2.58110
$67$	11.4763	1.40205	0.701026	−	0.713136i	$-0.252726\pi$
$67$	11.4763	1.40205	0.701026	+	0.713136i	$0.252726\pi$
$68$	−17.1746	−2.08272
$69$	6.55210	0.788780
$70$	−5.54309	−0.662526
$71$	11.9053	1.41290	0.706449	−	0.707764i	$-0.250296\pi$
$71$	11.9053	1.41290	0.706449	+	0.707764i	$0.250296\pi$
$72$	−3.93838	−0.464143
$73$	4.34482	0.508522	0.254261	−	0.967136i	$-0.418168\pi$
$73$	4.34482	0.508522	0.254261	+	0.967136i	$0.418168\pi$
$74$	2.90584	0.337797
$75$	2.55334	0.294834
$76$	7.64887	0.877385
$77$	10.0579	1.14620
$78$	−14.2934	−1.61841
$79$	−0.434182	−0.0488493	−0.0244247	−	0.999702i	$-0.507775\pi$
$79$	−0.434182	−0.0488493	−0.0244247	+	0.999702i	$0.507775\pi$
$80$	−2.67136	−0.298667
$81$	−7.17149	−0.796832
$82$	18.1331	2.00247
$83$	−3.33674	−0.366255	−0.183128	−	0.983089i	$-0.558622\pi$
$83$	−3.33674	−0.366255	−0.183128	+	0.983089i	$0.558622\pi$
$84$	16.8049	1.83356
$85$	−6.79915	−0.737471
$86$	−9.33722	−1.00686
$87$	−10.1946	−1.09297
$88$	−4.31961	−0.460471
$89$	−9.70616	−1.02885	−0.514425	−	0.857535i	$-0.671995\pi$
$89$	−9.70616	−1.02885	−0.514425	+	0.857535i	$0.671995\pi$
$90$	−7.48758	−0.789261
$91$	6.85592	0.718695
$92$	6.48192	0.675787
$93$	−6.14938	−0.637660
$94$	15.4171	1.59015
$95$	3.02807	0.310673
$96$	20.2254	2.06425
$97$	14.0904	1.43066	0.715331	−	0.698786i	$-0.246276\pi$
$97$	14.0904	1.43066	0.715331	+	0.698786i	$0.246276\pi$
$98$	0.449387	0.0453949
$99$	13.5861	1.36546
$100$	2.52599	0.252599
$101$	−6.96453	−0.692996	−0.346498	−	0.938051i	$-0.612629\pi$
$101$	−6.96453	−0.692996	−0.346498	+	0.938051i	$0.612629\pi$
$102$	36.9334	3.65695
$103$	20.2580	1.99608	0.998042	−	0.0625543i	$-0.0199247\pi$
$103$	20.2580	1.99608	0.998042	+	0.0625543i	$0.0199247\pi$
$104$	−2.94444	−0.288726
$105$	6.65279	0.649246
$106$	6.63069	0.644030
$107$	13.6211	1.31680	0.658399	−	0.752669i	$-0.271234\pi$
$107$	13.6211	1.31680	0.658399	+	0.752669i	$0.271234\pi$
$108$	3.35083	0.322433
$109$	15.4421	1.47909	0.739543	−	0.673109i	$-0.235042\pi$
$109$	15.4421	1.47909	0.739543	+	0.673109i	$0.235042\pi$
$110$	−8.21236	−0.783017
$111$	−3.48758	−0.331026
$112$	−6.96030	−0.657686
$113$	−0.332750	−0.0313024	−0.0156512	−	0.999878i	$-0.504982\pi$
$113$	−0.332750	−0.0313024	−0.0156512	+	0.999878i	$0.504982\pi$
$114$	−16.4487	−1.54056
$115$	2.56609	0.239289
$116$	−10.0854	−0.936404
$117$	9.26094	0.856174
$118$	2.89384	0.266400
$119$	−17.7154	−1.62396
$120$	−2.85720	−0.260826
$121$	3.90123	0.354657
$122$	25.8191	2.33755
$123$	−21.7633	−1.96233
$124$	−6.08351	−0.546315
$125$	1.00000	0.0894427
$126$	−19.5091	−1.73801
$127$	−5.62503	−0.499141	−0.249570	−	0.968357i	$-0.580289\pi$
$127$	−5.62503	−0.499141	−0.249570	+	0.968357i	$0.580289\pi$
$128$	8.64247	0.763894
$129$	11.2065	0.986676
$130$	−5.59792	−0.490970
$131$	−13.3688	−1.16803	−0.584017	−	0.811741i	$-0.698520\pi$
$131$	−13.3688	−1.16803	−0.584017	+	0.811741i	$0.698520\pi$
$132$	24.8972	2.16702
$133$	7.88971	0.684125
$134$	−24.4151	−2.10914
$135$	1.32654	0.114170
$136$	7.60830	0.652406
$137$	9.04633	0.772880	0.386440	−	0.922314i	$-0.373705\pi$
$137$	9.04633	0.772880	0.386440	+	0.922314i	$0.373705\pi$
$138$	−13.9392	−1.18658
$139$	−7.83821	−0.664828	−0.332414	−	0.943134i	$-0.607863\pi$
$139$	−7.83821	−0.664828	−0.332414	+	0.943134i	$0.607863\pi$
$140$	6.58153	0.556241
$141$	−18.5035	−1.55828
$142$	−25.3278	−2.12546
$143$	10.1574	0.849402
$144$	−9.40193	−0.783494
$145$	−3.99265	−0.331571
$146$	−9.24332	−0.764983
$147$	−0.539351	−0.0444849
$148$	−3.45022	−0.283606
$149$	−8.35930	−0.684821	−0.342410	−	0.939550i	$-0.611243\pi$
$149$	−8.35930	−0.684821	−0.342410	+	0.939550i	$0.611243\pi$
$150$	−5.43206	−0.443526
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	−3.38843	−0.274838
$153$	−23.9298	−1.93461
$154$	−21.3975	−1.72426
$155$	−2.40837	−0.193445
$156$	16.9711	1.35877
$157$	5.71733	0.456292	0.228146	−	0.973627i	$-0.426734\pi$
$157$	5.71733	0.456292	0.228146	+	0.973627i	$0.426734\pi$
$158$	0.923696	0.0734853
$159$	−7.95812	−0.631120
$160$	7.92116	0.626223
$161$	6.68602	0.526932
$162$	15.2569	1.19869
$163$	−10.7609	−0.842855	−0.421428	−	0.906862i	$-0.638471\pi$
$163$	−10.7609	−0.842855	−0.421428	+	0.906862i	$0.638471\pi$
$164$	−21.5302	−1.68122
$165$	9.85642	0.767321
$166$	7.09871	0.550967
$167$	−8.29351	−0.641771	−0.320885	−	0.947118i	$-0.603980\pi$
$167$	−8.29351	−0.641771	−0.320885	+	0.947118i	$0.603980\pi$
$168$	−7.44452	−0.574357
$169$	−6.07627	−0.467405
$170$	14.4648	1.10940
$171$	10.6574	0.814990
$172$	11.0864	0.845334
$173$	−24.0069	−1.82521	−0.912605	−	0.408842i	$-0.865933\pi$
$173$	−24.0069	−1.82521	−0.912605	+	0.408842i	$0.865933\pi$
$174$	21.6883	1.64419
$175$	2.60553	0.196959
$176$	−10.3120	−0.777297
$177$	−3.47318	−0.261060
$178$	20.6492	1.54773
$179$	11.8924	0.888881	0.444440	−	0.895808i	$-0.353403\pi$
$179$	11.8924	0.888881	0.444440	+	0.895808i	$0.353403\pi$
$180$	8.89030	0.662644
$181$	−0.175932	−0.0130769	−0.00653846	−	0.999979i	$-0.502081\pi$
$181$	−0.175932	−0.0130769	−0.00653846	+	0.999979i	$0.502081\pi$
$182$	−14.5855	−1.08115
$183$	−30.9879	−2.29070
$184$	−2.87148	−0.211688
$185$	−1.36589	−0.100422
$186$	13.0824	0.959249
$187$	−26.2461	−1.91931
$188$	−18.3053	−1.33505
$189$	3.45634	0.251412
$190$	−6.44202	−0.467354
$191$	27.2502	1.97175	0.985876	−	0.167475i	$-0.0535613\pi$
$191$	27.2502	1.97175	0.985876	+	0.167475i	$0.0535613\pi$
$192$	−29.3865	−2.12079
$193$	−4.82078	−0.347008	−0.173504	−	0.984833i	$-0.555509\pi$
$193$	−4.82078	−0.347008	−0.173504	+	0.984833i	$0.555509\pi$
$194$	−29.9764	−2.15218
$195$	6.71859	0.481128
$196$	−0.533574	−0.0381124
$197$	−10.8355	−0.771996	−0.385998	−	0.922500i	$-0.626143\pi$
$197$	−10.8355	−0.771996	−0.385998	+	0.922500i	$0.626143\pi$
$198$	−28.9037	−2.05409
$199$	5.11188	0.362371	0.181186	−	0.983449i	$-0.442007\pi$
$199$	5.11188	0.362371	0.181186	+	0.983449i	$0.442007\pi$
$200$	−1.11901	−0.0791258
$201$	29.3028	2.06686
$202$	14.8166	1.04249
$203$	−10.4029	−0.730143
$204$	−43.8525	−3.07029
$205$	−8.52346	−0.595304
$206$	−43.0977	−3.00276
$207$	9.03145	0.627729
$208$	−7.02914	−0.487383
$209$	11.6890	0.808544
$210$	−14.1534	−0.976676
$211$	−6.01660	−0.414200	−0.207100	−	0.978320i	$-0.566403\pi$
$211$	−6.01660	−0.414200	−0.207100	+	0.978320i	$0.566403\pi$
$212$	−7.87288	−0.540712
$213$	30.3982	2.08285
$214$	−28.9780	−1.98089
$215$	4.38895	0.299324
$216$	−1.48441	−0.101001
$217$	−6.27506	−0.425979
$218$	−32.8521	−2.22503
$219$	11.0938	0.749648
$220$	9.75085	0.657402
$221$	−17.8906	−1.20345
$222$	7.41960	0.497971
$223$	16.7656	1.12271	0.561353	−	0.827577i	$-0.310281\pi$
$223$	16.7656	1.12271	0.561353	+	0.827577i	$0.310281\pi$
$224$	20.6388	1.37899
$225$	3.51953	0.234635
$226$	0.707904	0.0470891
$227$	−22.0615	−1.46427	−0.732137	−	0.681157i	$-0.761477\pi$
$227$	−22.0615	−1.46427	−0.732137	+	0.681157i	$0.761477\pi$
$228$	19.5301	1.29341
$229$	−16.9196	−1.11808	−0.559039	−	0.829141i	$-0.688830\pi$
$229$	−16.9196	−1.11808	−0.559039	+	0.829141i	$0.688830\pi$
$230$	−5.45920	−0.359969
$231$	25.6812	1.68970
$232$	4.46780	0.293325
$233$	−0.0701274	−0.00459420	−0.00229710	−	0.999997i	$-0.500731\pi$
$233$	−0.0701274	−0.00459420	−0.00229710	+	0.999997i	$0.500731\pi$
$234$	−19.7021	−1.28796
$235$	−7.24681	−0.472729
$236$	−3.43597	−0.223663
$237$	−1.10861	−0.0720122
$238$	37.6883	2.44297
$239$	17.8971	1.15767	0.578833	−	0.815446i	$-0.303508\pi$
$239$	17.8971	1.15767	0.578833	+	0.815446i	$0.303508\pi$
$240$	−6.82088	−0.440286
$241$	−30.4880	−1.96390	−0.981951	−	0.189135i	$-0.939432\pi$
$241$	−30.4880	−1.96390	−0.981951	+	0.189135i	$0.939432\pi$
$242$	−8.29962	−0.533520
$243$	−22.2909	−1.42996
$244$	−30.6560	−1.96255
$245$	−0.211234	−0.0134952
$246$	46.3000	2.95198
$247$	7.96775	0.506976
$248$	2.69498	0.171131
$249$	−8.51983	−0.539923
$250$	−2.12744	−0.134551
$251$	0.407597	0.0257273	0.0128636	−	0.999917i	$-0.495905\pi$
$251$	0.407597	0.0257273	0.0128636	+	0.999917i	$0.495905\pi$
$252$	23.1639	1.45919
$253$	9.90566	0.622764
$254$	11.9669	0.750870
$255$	−17.3605	−1.08716
$256$	4.63180	0.289488
$257$	−9.60743	−0.599295	−0.299648	−	0.954050i	$-0.596869\pi$
$257$	−9.60743	−0.599295	−0.299648	+	0.954050i	$0.596869\pi$
$258$	−23.8411	−1.48428
$259$	−3.55886	−0.221137
$260$	6.64663	0.412206
$261$	−14.0522	−0.869812
$262$	28.4412	1.75710
$263$	11.3293	0.698592	0.349296	−	0.937012i	$-0.386421\pi$
$263$	11.3293	0.698592	0.349296	+	0.937012i	$0.386421\pi$
$264$	−11.0294	−0.678813
$265$	−3.11675	−0.191461
$266$	−16.7849	−1.02915
$267$	−24.7831	−1.51670
$268$	28.9890	1.77078
$269$	−14.1573	−0.863185	−0.431593	−	0.902069i	$-0.642048\pi$
$269$	−14.1573	−0.863185	−0.431593	+	0.902069i	$0.642048\pi$
$270$	−2.82213	−0.171749
$271$	−24.5420	−1.49082	−0.745411	−	0.666605i	$-0.767747\pi$
$271$	−24.5420	−1.49082	−0.745411	+	0.666605i	$0.767747\pi$
$272$	18.1630	1.10129
$273$	17.5055	1.05948
$274$	−19.2455	−1.16266
$275$	3.86021	0.232779
$276$	16.5505	0.996225
$277$	19.0519	1.14472	0.572358	−	0.820004i	$-0.306029\pi$
$277$	19.0519	1.14472	0.572358	+	0.820004i	$0.306029\pi$
$278$	16.6753	1.00012
$279$	−8.47633	−0.507464
$280$	−2.91560	−0.174241
$281$	−11.1117	−0.662867	−0.331434	−	0.943479i	$-0.607532\pi$
$281$	−11.1117	−0.662867	−0.331434	+	0.943479i	$0.607532\pi$
$282$	39.3651	2.34416
$283$	27.0500	1.60796	0.803978	−	0.594660i	$-0.202713\pi$
$283$	27.0500	1.60796	0.803978	+	0.594660i	$0.202713\pi$
$284$	30.0726	1.78448
$285$	7.73168	0.457985
$286$	−21.6092	−1.27778
$287$	−22.2081	−1.31090
$288$	27.8788	1.64277
$289$	29.2284	1.71932
$290$	8.49410	0.498791
$291$	35.9775	2.10904
$292$	10.9750	0.642261
$293$	15.8783	0.927620	0.463810	−	0.885935i	$-0.346482\pi$
$293$	15.8783	0.927620	0.463810	+	0.885935i	$0.346482\pi$
$294$	1.14744	0.0669198
$295$	−1.36025	−0.0791968
$296$	1.52844	0.0888388
$297$	5.12073	0.297135
$298$	17.7839	1.03019
$299$	6.75216	0.390487
$300$	6.44970	0.372374
$301$	11.4355	0.659133
$302$	2.12744	0.122420
$303$	−17.7828	−1.02159
$304$	−8.08906	−0.463939
$305$	−12.1363	−0.694920
$306$	50.9092	2.91028
$307$	−32.1198	−1.83317	−0.916586	−	0.399837i	$-0.869067\pi$
$307$	−32.1198	−1.83317	−0.916586	+	0.399837i	$0.869067\pi$
$308$	25.4061	1.44765
$309$	51.7256	2.94257
$310$	5.12365	0.291004
$311$	−8.50777	−0.482431	−0.241216	−	0.970472i	$-0.577546\pi$
$311$	−8.50777	−0.482431	−0.241216	+	0.970472i	$0.577546\pi$
$312$	−7.51816	−0.425632
$313$	13.6919	0.773910	0.386955	−	0.922099i	$-0.373527\pi$
$313$	13.6919	0.773910	0.386955	+	0.922099i	$0.373527\pi$
$314$	−12.1633	−0.686412
$315$	9.17023	0.516684
$316$	−1.09674	−0.0616964
$317$	−22.1162	−1.24217	−0.621084	−	0.783744i	$-0.713307\pi$
$317$	−22.1162	−1.24217	−0.621084	+	0.783744i	$0.713307\pi$
$318$	16.9304	0.949410
$319$	−15.4125	−0.862932
$320$	−11.5091	−0.643376
$321$	34.7792	1.94119
$322$	−14.2241	−0.792678
$323$	−20.5883	−1.14556
$324$	−18.1151	−1.00639
$325$	2.63130	0.145958
$326$	22.8930	1.26793
$327$	39.4289	2.18042
$328$	9.53781	0.526638
$329$	−18.8817	−1.04098
$330$	−20.9689	−1.15430
$331$	5.17282	0.284324	0.142162	−	0.989843i	$-0.454595\pi$
$331$	5.17282	0.284324	0.142162	+	0.989843i	$0.454595\pi$
$332$	−8.42858	−0.462578
$333$	−4.80729	−0.263438
$334$	17.6439	0.965432
$335$	11.4763	0.627017
$336$	−17.7720	−0.969541
$337$	−28.5827	−1.55700	−0.778499	−	0.627646i	$-0.784018\pi$
$337$	−28.5827	−1.55700	−0.778499	+	0.627646i	$0.784018\pi$
$338$	12.9269	0.703129
$339$	−0.849622	−0.0461451
$340$	−17.1746	−0.931422
$341$	−9.29681	−0.503450
$342$	−22.6729	−1.22601
$343$	−18.7891	−1.01451
$344$	−4.91127	−0.264798
$345$	6.55210	0.352753
$346$	51.0732	2.74571
$347$	20.4069	1.09550	0.547749	−	0.836643i	$-0.315485\pi$
$347$	20.4069	1.09550	0.547749	+	0.836643i	$0.315485\pi$
$348$	−25.7514	−1.38042
$349$	−18.8070	−1.00671	−0.503357	−	0.864079i	$-0.667902\pi$
$349$	−18.8070	−1.00671	−0.503357	+	0.864079i	$0.667902\pi$
$350$	−5.54309	−0.296291
$351$	3.49052	0.186310
$352$	30.5774	1.62978
$353$	27.7640	1.47773	0.738864	−	0.673855i	$-0.235363\pi$
$353$	27.7640	1.47773	0.738864	+	0.673855i	$0.235363\pi$
$354$	7.38896	0.392719
$355$	11.9053	0.631867
$356$	−24.5176	−1.29943
$357$	−45.2333	−2.39400
$358$	−25.3004	−1.33717
$359$	21.5754	1.13871	0.569353	−	0.822093i	$-0.307194\pi$
$359$	21.5754	1.13871	0.569353	+	0.822093i	$0.307194\pi$
$360$	−3.93838	−0.207571
$361$	−9.83080	−0.517411
$362$	0.374284	0.0196719
$363$	9.96116	0.522825
$364$	17.3180	0.907708
$365$	4.34482	0.227418
$366$	65.9249	3.44595
$367$	0.913369	0.0476775	0.0238387	−	0.999716i	$-0.492411\pi$
$367$	0.913369	0.0476775	0.0238387	+	0.999716i	$0.492411\pi$
$368$	−6.85496	−0.357339
$369$	−29.9986	−1.56166
$370$	2.90584	0.151068
$371$	−8.12078	−0.421610
$372$	−15.5333	−0.805362
$373$	35.6167	1.84416	0.922080	−	0.386998i	$-0.126488\pi$
$373$	35.6167	1.84416	0.922080	+	0.386998i	$0.126488\pi$
$374$	55.8370	2.88726
$375$	2.55334	0.131854
$376$	8.10923	0.418202
$377$	−10.5058	−0.541078
$378$	−7.35314	−0.378205
$379$	−7.14559	−0.367044	−0.183522	−	0.983016i	$-0.558750\pi$
$379$	−7.14559	−0.367044	−0.183522	+	0.983016i	$0.558750\pi$
$380$	7.64887	0.392379
$381$	−14.3626	−0.735818
$382$	−57.9730	−2.96616
$383$	−4.18015	−0.213596	−0.106798	−	0.994281i	$-0.534060\pi$
$383$	−4.18015	−0.213596	−0.106798	+	0.994281i	$0.534060\pi$
$384$	22.0671	1.12611
$385$	10.0579	0.512597
$386$	10.2559	0.522012
$387$	15.4471	0.785218
$388$	35.5922	1.80692
$389$	22.2373	1.12748	0.563738	−	0.825954i	$-0.309363\pi$
$389$	22.2373	1.12748	0.563738	+	0.825954i	$0.309363\pi$
$390$	−14.2934	−0.723774
$391$	−17.4472	−0.882345
$392$	0.236372	0.0119386
$393$	−34.1350	−1.72188
$394$	23.0518	1.16133
$395$	−0.434182	−0.0218461
$396$	34.3184	1.72457
$397$	18.9109	0.949110	0.474555	−	0.880226i	$-0.342609\pi$
$397$	18.9109	0.949110	0.474555	+	0.880226i	$0.342609\pi$
$398$	−10.8752	−0.545124
$399$	20.1451	1.00852
$400$	−2.67136	−0.133568
$401$	39.2798	1.96154	0.980770	−	0.195165i	$-0.0625241\pi$
$401$	39.2798	1.96154	0.980770	+	0.195165i	$0.0625241\pi$
$402$	−62.3400	−3.10923
$403$	−6.33713	−0.315675
$404$	−17.5923	−0.875250
$405$	−7.17149	−0.356354
$406$	22.1316	1.09837
$407$	−5.27262	−0.261354
$408$	19.4266	0.961758
$409$	−23.0528	−1.13989	−0.569943	−	0.821684i	$-0.693035\pi$
$409$	−23.0528	−1.13989	−0.569943	+	0.821684i	$0.693035\pi$
$410$	18.1331	0.895531
$411$	23.0983	1.13936
$412$	51.1716	2.52104
$413$	−3.54416	−0.174397
$414$	−19.2138	−0.944308
$415$	−3.33674	−0.163794
$416$	20.8429	1.02191
$417$	−20.0136	−0.980070
$418$	−24.8676	−1.21631
$419$	−14.8416	−0.725061	−0.362530	−	0.931972i	$-0.618087\pi$
$419$	−14.8416	−0.725061	−0.362530	+	0.931972i	$0.618087\pi$
$420$	16.8049	0.819993
$421$	9.08416	0.442735	0.221367	−	0.975190i	$-0.428948\pi$
$421$	9.08416	0.442735	0.221367	+	0.975190i	$0.428948\pi$
$422$	12.7999	0.623092
$423$	−25.5054	−1.24011
$424$	3.48767	0.169376
$425$	−6.79915	−0.329807
$426$	−64.6703	−3.13329
$427$	−31.6213	−1.53026
$428$	34.4067	1.66311
$429$	25.9352	1.25216
$430$	−9.33722	−0.450281
$431$	0.460039	0.0221593	0.0110797	−	0.999939i	$-0.496473\pi$
$431$	0.460039	0.0221593	0.0110797	+	0.999939i	$0.496473\pi$
$432$	−3.54367	−0.170495
$433$	−39.3903	−1.89298	−0.946488	−	0.322738i	$-0.895397\pi$
$433$	−39.3903	−1.89298	−0.946488	+	0.322738i	$0.895397\pi$
$434$	13.3498	0.640811
$435$	−10.1946	−0.488792
$436$	39.0066	1.86808
$437$	7.77030	0.371704
$438$	−23.6013	−1.12771
$439$	10.5825	0.505075	0.252538	−	0.967587i	$-0.418735\pi$
$439$	10.5825	0.505075	0.252538	+	0.967587i	$0.418735\pi$
$440$	−4.31961	−0.205929
$441$	−0.743444	−0.0354021
$442$	38.0611	1.81038
$443$	−0.268262	−0.0127455	−0.00637275	−	0.999980i	$-0.502029\pi$
$443$	−0.268262	−0.0127455	−0.00637275	+	0.999980i	$0.502029\pi$
$444$	−8.80958	−0.418084
$445$	−9.70616	−0.460116
$446$	−35.6677	−1.68891
$447$	−21.3441	−1.00954
$448$	−29.9872	−1.41676
$449$	−9.88369	−0.466440	−0.233220	−	0.972424i	$-0.574926\pi$
$449$	−9.88369	−0.466440	−0.233220	+	0.972424i	$0.574926\pi$
$450$	−7.48758	−0.352968
$451$	−32.9023	−1.54931
$452$	−0.840522	−0.0395348
$453$	−2.55334	−0.119966
$454$	46.9345	2.20274
$455$	6.85592	0.321410
$456$	−8.65181	−0.405158
$457$	32.8158	1.53506	0.767529	−	0.641014i	$-0.221486\pi$
$457$	32.8158	1.53506	0.767529	+	0.641014i	$0.221486\pi$
$458$	35.9954	1.68195
$459$	−9.01935	−0.420987
$460$	6.48192	0.302221
$461$	19.1507	0.891938	0.445969	−	0.895048i	$-0.352859\pi$
$461$	19.1507	0.891938	0.445969	+	0.895048i	$0.352859\pi$
$462$	−54.6351	−2.54185
$463$	−27.6639	−1.28565	−0.642824	−	0.766014i	$-0.722237\pi$
$463$	−27.6639	−1.28565	−0.642824	+	0.766014i	$0.722237\pi$
$464$	10.6658	0.495147
$465$	−6.14938	−0.285170
$466$	0.149192	0.00691117
$467$	6.87738	0.318247	0.159124	−	0.987259i	$-0.449133\pi$
$467$	6.87738	0.318247	0.159124	+	0.987259i	$0.449133\pi$
$468$	23.3930	1.08134
$469$	29.9018	1.38074
$470$	15.4171	0.711139
$471$	14.5983	0.672653
$472$	1.52213	0.0700617
$473$	16.9423	0.779007
$474$	2.35851	0.108330
$475$	3.02807	0.138937
$476$	−44.7488	−2.05106
$477$	−10.9695	−0.502259
$478$	−38.0749	−1.74151
$479$	30.7468	1.40486	0.702429	−	0.711754i	$-0.252099\pi$
$479$	30.7468	1.40486	0.702429	+	0.711754i	$0.252099\pi$
$480$	20.2254	0.923159
$481$	−3.59406	−0.163875
$482$	64.8612	2.95435
$483$	17.0717	0.776788
$484$	9.85446	0.447930
$485$	14.0904	0.639812
$486$	47.4224	2.15112
$487$	17.9966	0.815504	0.407752	−	0.913093i	$-0.366313\pi$
$487$	17.9966	0.815504	0.407752	+	0.913093i	$0.366313\pi$
$488$	13.5806	0.614763
$489$	−27.4761	−1.24251
$490$	0.449387	0.0203012
$491$	20.5368	0.926812	0.463406	−	0.886146i	$-0.346627\pi$
$491$	20.5368	0.926812	0.463406	+	0.886146i	$0.346627\pi$
$492$	−54.9737	−2.47841
$493$	27.1466	1.22262
$494$	−16.9509	−0.762657
$495$	13.5861	0.610651
$496$	6.43361	0.288878
$497$	31.0196	1.39142
$498$	18.1254	0.812219
$499$	−29.1826	−1.30639	−0.653196	−	0.757189i	$-0.726572\pi$
$499$	−29.1826	−1.30639	−0.653196	+	0.757189i	$0.726572\pi$
$500$	2.52599	0.112966
$501$	−21.1761	−0.946080
$502$	−0.867136	−0.0387022
$503$	−35.2746	−1.57282	−0.786409	−	0.617706i	$-0.788062\pi$
$503$	−35.2746	−1.57282	−0.786409	+	0.617706i	$0.788062\pi$
$504$	−10.2616	−0.457086
$505$	−6.96453	−0.309917
$506$	−21.0737	−0.936839
$507$	−15.5148	−0.689035
$508$	−14.2088	−0.630412
$509$	−5.84464	−0.259059	−0.129529	−	0.991576i	$-0.541347\pi$
$509$	−5.84464	−0.259059	−0.129529	+	0.991576i	$0.541347\pi$
$510$	36.9334	1.63544
$511$	11.3205	0.500791
$512$	−27.1388	−1.19938
$513$	4.01686	0.177349
$514$	20.4392	0.901535
$515$	20.2580	0.892676
$516$	28.3074	1.24617
$517$	−27.9742	−1.23030
$518$	7.57125	0.332662
$519$	−61.2977	−2.69067
$520$	−2.94444	−0.129122
$521$	5.44758	0.238663	0.119331	−	0.992854i	$-0.461925\pi$
$521$	5.44758	0.238663	0.119331	+	0.992854i	$0.461925\pi$
$522$	29.8953	1.30848
$523$	−7.53899	−0.329657	−0.164829	−	0.986322i	$-0.552707\pi$
$523$	−7.53899	−0.329657	−0.164829	+	0.986322i	$0.552707\pi$
$524$	−33.7694	−1.47522
$525$	6.65279	0.290351
$526$	−24.1023	−1.05091
$527$	16.3748	0.713300
$528$	−26.3300	−1.14587
$529$	−16.4152	−0.713703
$530$	6.63069	0.288019
$531$	−4.78744	−0.207757
$532$	19.9293	0.864046
$533$	−22.4278	−0.971454
$534$	52.7245	2.28161
$535$	13.6211	0.588890
$536$	−12.8421	−0.554692
$537$	30.3653	1.31036
$538$	30.1188	1.29851
$539$	−0.815407	−0.0351221
$540$	3.35083	0.144197
$541$	40.1043	1.72422	0.862110	−	0.506722i	$-0.169143\pi$
$541$	40.1043	1.72422	0.862110	+	0.506722i	$0.169143\pi$
$542$	52.2116	2.24268
$543$	−0.449214	−0.0192776
$544$	−53.8572	−2.30911
$545$	15.4421	0.661467
$546$	−37.2418	−1.59380
$547$	−24.7785	−1.05945	−0.529727	−	0.848169i	$-0.677705\pi$
$547$	−24.7785	−1.05945	−0.529727	+	0.848169i	$0.677705\pi$
$548$	22.8509	0.976143
$549$	−42.7139	−1.82299
$550$	−8.21236	−0.350176
$551$	−12.0900	−0.515051
$552$	−7.33185	−0.312064
$553$	−1.13127	−0.0481067
$554$	−40.5316	−1.72202
$555$	−3.48758	−0.148039
$556$	−19.7992	−0.839674
$557$	−15.6487	−0.663056	−0.331528	−	0.943445i	$-0.607564\pi$
$557$	−15.6487	−0.663056	−0.331528	+	0.943445i	$0.607564\pi$
$558$	18.0329	0.763392
$559$	11.5486	0.488456
$560$	−6.96030	−0.294126
$561$	−67.0153	−2.82939
$562$	23.6394	0.997168
$563$	−8.37689	−0.353044	−0.176522	−	0.984297i	$-0.556485\pi$
$563$	−8.37689	−0.353044	−0.176522	+	0.984297i	$0.556485\pi$
$564$	−46.7397	−1.96810
$565$	−0.332750	−0.0139989
$566$	−57.5472	−2.41889
$567$	−18.6855	−0.784717
$568$	−13.3221	−0.558984
$569$	−27.5893	−1.15661	−0.578303	−	0.815822i	$-0.696285\pi$
$569$	−27.5893	−1.15661	−0.578303	+	0.815822i	$0.696285\pi$
$570$	−16.4487	−0.688959
$571$	23.2913	0.974709	0.487355	−	0.873204i	$-0.337962\pi$
$571$	23.2913	0.974709	0.487355	+	0.873204i	$0.337962\pi$
$572$	25.6574	1.07279
$573$	69.5789	2.90670
$574$	47.2463	1.97202
$575$	2.56609	0.107013
$576$	−40.5065	−1.68777
$577$	−37.7795	−1.57278	−0.786390	−	0.617731i	$-0.788052\pi$
$577$	−37.7795	−1.57278	−0.786390	+	0.617731i	$0.788052\pi$
$578$	−62.1816	−2.58641
$579$	−12.3091	−0.511548
$580$	−10.0854	−0.418772
$581$	−8.69397	−0.360687
$582$	−76.5399	−3.17268
$583$	−12.0313	−0.498286
$584$	−4.86188	−0.201186
$585$	9.26094	0.382893
$586$	−33.7801	−1.39544
$587$	−15.4233	−0.636586	−0.318293	−	0.947992i	$-0.603110\pi$
$587$	−15.4233	−0.636586	−0.318293	+	0.947992i	$0.603110\pi$
$588$	−1.36240	−0.0561842
$589$	−7.29270	−0.300491
$590$	2.89384	0.119138
$591$	−27.6666	−1.13805
$592$	3.64878	0.149964
$593$	27.1534	1.11506	0.557529	−	0.830158i	$-0.311750\pi$
$593$	27.1534	1.11506	0.557529	+	0.830158i	$0.311750\pi$
$594$	−10.8940	−0.446987
$595$	−17.7154	−0.726259
$596$	−21.1155	−0.864925
$597$	13.0523	0.534197
$598$	−14.3648	−0.587420
$599$	36.2963	1.48303	0.741513	−	0.670938i	$-0.234109\pi$
$599$	36.2963	1.48303	0.741513	+	0.670938i	$0.234109\pi$
$600$	−2.85720	−0.116645
$601$	34.1202	1.39179	0.695895	−	0.718143i	$-0.255008\pi$
$601$	34.1202	1.39179	0.695895	+	0.718143i	$0.255008\pi$
$602$	−24.3284	−0.991551
$603$	40.3912	1.64486
$604$	−2.52599	−0.102781
$605$	3.90123	0.158608
$606$	37.8318	1.53681
$607$	23.0592	0.935946	0.467973	−	0.883743i	$-0.344985\pi$
$607$	23.0592	0.935946	0.467973	+	0.883743i	$0.344985\pi$
$608$	23.9858	0.972754
$609$	−26.5622	−1.07636
$610$	25.8191	1.04539
$611$	−19.0685	−0.771429
$612$	−60.4465	−2.44340
$613$	−3.40514	−0.137532	−0.0687662	−	0.997633i	$-0.521906\pi$
$613$	−3.40514	−0.137532	−0.0687662	+	0.997633i	$0.521906\pi$
$614$	68.3328	2.75769
$615$	−21.7633	−0.877579
$616$	−11.2548	−0.453471
$617$	3.61313	0.145459	0.0727295	−	0.997352i	$-0.476829\pi$
$617$	3.61313	0.145459	0.0727295	+	0.997352i	$0.476829\pi$
$618$	−110.043	−4.42658
$619$	10.8369	0.435574	0.217787	−	0.975996i	$-0.430116\pi$
$619$	10.8369	0.435574	0.217787	+	0.975996i	$0.430116\pi$
$620$	−6.08351	−0.244320
$621$	3.40403	0.136599
$622$	18.0997	0.725733
$623$	−25.2897	−1.01321
$624$	−17.9478	−0.718486
$625$	1.00000	0.0400000
$626$	−29.1286	−1.16421
$627$	29.8459	1.19193
$628$	14.4419	0.576295
$629$	9.28688	0.370292
$630$	−19.5091	−0.777261
$631$	−31.4947	−1.25379	−0.626893	−	0.779106i	$-0.715674\pi$
$631$	−31.4947	−1.25379	−0.626893	+	0.779106i	$0.715674\pi$
$632$	0.485853	0.0193262
$633$	−15.3624	−0.610601
$634$	47.0507	1.86862
$635$	−5.62503	−0.223222
$636$	−20.1021	−0.797101
$637$	−0.555819	−0.0220224
$638$	32.7890	1.29813
$639$	41.9011	1.65758
$640$	8.64247	0.341624
$641$	4.00732	0.158280	0.0791398	−	0.996864i	$-0.474783\pi$
$641$	4.00732	0.158280	0.0791398	+	0.996864i	$0.474783\pi$
$642$	−73.9906	−2.92017
$643$	11.1296	0.438908	0.219454	−	0.975623i	$-0.429572\pi$
$643$	11.1296	0.438908	0.219454	+	0.975623i	$0.429572\pi$
$644$	16.8888	0.665513
$645$	11.2065	0.441255
$646$	43.8003	1.72330
$647$	24.4197	0.960038	0.480019	−	0.877258i	$-0.340630\pi$
$647$	24.4197	0.960038	0.480019	+	0.877258i	$0.340630\pi$
$648$	8.02495	0.315250
$649$	−5.25085	−0.206114
$650$	−5.59792	−0.219569
$651$	−16.0224	−0.627966
$652$	−27.1818	−1.06452
$653$	28.0387	1.09724	0.548620	−	0.836072i	$-0.315153\pi$
$653$	28.0387	1.09724	0.548620	+	0.836072i	$0.315153\pi$
$654$	−83.8825	−3.28007
$655$	−13.3688	−0.522361
$656$	22.7692	0.888988
$657$	15.2917	0.596587
$658$	40.1697	1.56598
$659$	−28.3497	−1.10435	−0.552174	−	0.833729i	$-0.686202\pi$
$659$	−28.3497	−1.10435	−0.552174	+	0.833729i	$0.686202\pi$
$660$	24.8972	0.969122
$661$	5.24727	0.204095	0.102048	−	0.994780i	$-0.467461\pi$
$661$	5.24727	0.204095	0.102048	+	0.994780i	$0.467461\pi$
$662$	−11.0048	−0.427715
$663$	−45.6807	−1.77409
$664$	3.73384	0.144901
$665$	7.88971	0.305950
$666$	10.2272	0.396296
$667$	−10.2455	−0.396707
$668$	−20.9493	−0.810553
$669$	42.8081	1.65506
$670$	−24.4151	−0.943237
$671$	−46.8485	−1.80857
$672$	52.6978	2.03286
$673$	35.1032	1.35313	0.676565	−	0.736383i	$-0.263468\pi$
$673$	35.1032	1.35313	0.676565	+	0.736383i	$0.263468\pi$
$674$	60.8078	2.34223
$675$	1.32654	0.0510586
$676$	−15.3486	−0.590330
$677$	−24.0150	−0.922972	−0.461486	−	0.887148i	$-0.652683\pi$
$677$	−24.0150	−0.922972	−0.461486	+	0.887148i	$0.652683\pi$
$678$	1.80752	0.0694173
$679$	36.7129	1.40891
$680$	7.60830	0.291765
$681$	−56.3305	−2.15859
$682$	19.7784	0.757353
$683$	−21.9473	−0.839791	−0.419896	−	0.907572i	$-0.637933\pi$
$683$	−21.9473	−0.839791	−0.419896	+	0.907572i	$0.637933\pi$
$684$	26.9204	1.02933
$685$	9.04633	0.345643
$686$	39.9725	1.52616
$687$	−43.2015	−1.64824
$688$	−11.7245	−0.446991
$689$	−8.20111	−0.312437
$690$	−13.9392	−0.530656
$691$	22.1705	0.843406	0.421703	−	0.906734i	$-0.361432\pi$
$691$	22.1705	0.843406	0.421703	+	0.906734i	$0.361432\pi$
$692$	−60.6411	−2.30523
$693$	35.3990	1.34470
$694$	−43.4143	−1.64798
$695$	−7.83821	−0.297320
$696$	11.4078	0.432412
$697$	57.9522	2.19510
$698$	40.0106	1.51442
$699$	−0.179059	−0.00677263
$700$	6.58153	0.248758
$701$	41.5650	1.56989	0.784945	−	0.619565i	$-0.212691\pi$
$701$	41.5650	1.56989	0.784945	+	0.619565i	$0.212691\pi$
$702$	−7.42587	−0.280271
$703$	−4.13601	−0.155992
$704$	−44.4274	−1.67442
$705$	−18.5035	−0.696884
$706$	−59.0661	−2.22298
$707$	−18.1463	−0.682460
$708$	−8.77320	−0.329717
$709$	30.1633	1.13281	0.566403	−	0.824128i	$-0.308334\pi$
$709$	30.1633	1.13281	0.566403	+	0.824128i	$0.308334\pi$
$710$	−25.3278	−0.950534
$711$	−1.52812	−0.0573089
$712$	10.8613	0.407043
$713$	−6.18009	−0.231446
$714$	96.2310	3.60135
$715$	10.1574	0.379864
$716$	30.0401	1.12265
$717$	45.6973	1.70660
$718$	−45.9003	−1.71299
$719$	0.118341	0.00441339	0.00220669	−	0.999998i	$-0.499298\pi$
$719$	0.118341	0.00441339	0.00220669	+	0.999998i	$0.499298\pi$
$720$	−9.40193	−0.350389
$721$	52.7828	1.96574
$722$	20.9144	0.778354
$723$	−77.8461	−2.89513
$724$	−0.444402	−0.0165161
$725$	−3.99265	−0.148283
$726$	−21.1917	−0.786499
$727$	31.1824	1.15649	0.578246	−	0.815862i	$-0.303737\pi$
$727$	31.1824	1.15649	0.578246	+	0.815862i	$0.303737\pi$
$728$	−7.67182	−0.284337
$729$	−35.4016	−1.31117
$730$	−9.24332	−0.342111
$731$	−29.8411	−1.10371
$732$	−78.2752	−2.89313
$733$	−7.16079	−0.264490	−0.132245	−	0.991217i	$-0.542219\pi$
$733$	−7.16079	−0.264490	−0.132245	+	0.991217i	$0.542219\pi$
$734$	−1.94314	−0.0717225
$735$	−0.539351	−0.0198943
$736$	20.3264	0.749243
$737$	44.3009	1.63184
$738$	63.8201	2.34925
$739$	−31.8925	−1.17319	−0.586593	−	0.809882i	$-0.699531\pi$
$739$	−31.8925	−1.17319	−0.586593	+	0.809882i	$0.699531\pi$
$740$	−3.45022	−0.126833
$741$	20.3444	0.747369
$742$	17.2764	0.634238
$743$	46.8778	1.71978	0.859890	−	0.510480i	$-0.170532\pi$
$743$	46.8778	1.71978	0.859890	+	0.510480i	$0.170532\pi$
$744$	6.88120	0.252277
$745$	−8.35930	−0.306261
$746$	−75.7722	−2.77422
$747$	−11.7438	−0.429682
$748$	−66.2975	−2.42408
$749$	35.4901	1.29678
$750$	−5.43206	−0.198351
$751$	1.76497	0.0644049	0.0322024	−	0.999481i	$-0.489748\pi$
$751$	1.76497	0.0644049	0.0322024	+	0.999481i	$0.489748\pi$
$752$	19.3588	0.705944
$753$	1.04073	0.0379264
$754$	22.3505	0.813958
$755$	−1.00000	−0.0363937
$756$	8.73067	0.317531
$757$	−18.6205	−0.676775	−0.338387	−	0.941007i	$-0.609881\pi$
$757$	−18.6205	−0.676775	−0.338387	+	0.941007i	$0.609881\pi$
$758$	15.2018	0.552154
$759$	25.2925	0.918059
$760$	−3.38843	−0.122911
$761$	23.2657	0.843381	0.421690	−	0.906740i	$-0.361437\pi$
$761$	23.2657	0.843381	0.421690	+	0.906740i	$0.361437\pi$
$762$	30.5555	1.10691
$763$	40.2348	1.45660
$764$	68.8336	2.49031
$765$	−23.9298	−0.865185
$766$	8.89301	0.321317
$767$	−3.57922	−0.129238
$768$	11.8266	0.426754
$769$	−12.3090	−0.443874	−0.221937	−	0.975061i	$-0.571238\pi$
$769$	−12.3090	−0.443874	−0.221937	+	0.975061i	$0.571238\pi$
$770$	−21.3975	−0.771113
$771$	−24.5310	−0.883463
$772$	−12.1772	−0.438268
$773$	32.1410	1.15603	0.578016	−	0.816025i	$-0.303827\pi$
$773$	32.1410	1.15603	0.578016	+	0.816025i	$0.303827\pi$
$774$	−32.8627	−1.18122
$775$	−2.40837	−0.0865111
$776$	−15.7672	−0.566011
$777$	−9.08697	−0.325993
$778$	−47.3084	−1.69609
$779$	−25.8096	−0.924725
$780$	16.9711	0.607662
$781$	45.9570	1.64447
$782$	37.1179	1.32733
$783$	−5.29641	−0.189278
$784$	0.564281	0.0201529
$785$	5.71733	0.204060
$786$	72.6200	2.59027
$787$	−28.4217	−1.01312	−0.506561	−	0.862204i	$-0.669084\pi$
$787$	−28.4217	−1.01312	−0.506561	+	0.862204i	$0.669084\pi$
$788$	−27.3703	−0.975027
$789$	28.9274	1.02984
$790$	0.923696	0.0328636
$791$	−0.866988	−0.0308265
$792$	−15.2030	−0.540215
$793$	−31.9341	−1.13401
$794$	−40.2317	−1.42777
$795$	−7.95812	−0.282245
$796$	12.9125	0.457673
$797$	35.1465	1.24495	0.622477	−	0.782638i	$-0.286127\pi$
$797$	35.1465	1.24495	0.622477	+	0.782638i	$0.286127\pi$
$798$	−42.8574	−1.51714
$799$	49.2721	1.74312
$800$	7.92116	0.280055
$801$	−34.1611	−1.20702
$802$	−83.5654	−2.95080
$803$	16.7719	0.591868
$804$	74.0187	2.61044
$805$	6.68602	0.235651
$806$	13.4819	0.474878
$807$	−36.1483	−1.27248
$808$	7.79336	0.274169
$809$	18.8754	0.663624	0.331812	−	0.943345i	$-0.392340\pi$
$809$	18.8754	0.663624	0.331812	+	0.943345i	$0.392340\pi$
$810$	15.2569	0.536073
$811$	−48.6416	−1.70804	−0.854018	−	0.520243i	$-0.825841\pi$
$811$	−48.6416	−1.70804	−0.854018	+	0.520243i	$0.825841\pi$
$812$	−26.2777	−0.922167
$813$	−62.6641	−2.19772
$814$	11.2172	0.393162
$815$	−10.7609	−0.376936
$816$	46.3762	1.62349
$817$	13.2901	0.464960
$818$	49.0433	1.71476
$819$	24.1296	0.843157
$820$	−21.5302	−0.751866
$821$	−1.02623	−0.0358156	−0.0179078	−	0.999840i	$-0.505701\pi$
$821$	−1.02623	−0.0358156	−0.0179078	+	0.999840i	$0.505701\pi$
$822$	−49.1403	−1.71396
$823$	42.4724	1.48049	0.740247	−	0.672335i	$-0.234709\pi$
$823$	42.4724	1.48049	0.740247	+	0.672335i	$0.234709\pi$
$824$	−22.6689	−0.789708
$825$	9.85642	0.343157
$826$	7.53999	0.262350
$827$	−29.2421	−1.01685	−0.508424	−	0.861107i	$-0.669772\pi$
$827$	−29.2421	−1.01685	−0.508424	+	0.861107i	$0.669772\pi$
$828$	22.8133	0.792818
$829$	−27.7155	−0.962598	−0.481299	−	0.876556i	$-0.659835\pi$
$829$	−27.7155	−0.962598	−0.481299	+	0.876556i	$0.659835\pi$
$830$	7.09871	0.246400
$831$	48.6458	1.68750
$832$	−30.2838	−1.04990
$833$	1.43621	0.0497617
$834$	42.5777	1.47434
$835$	−8.29351	−0.287009
$836$	29.5262	1.02119
$837$	−3.19480	−0.110428
$838$	31.5746	1.09073
$839$	34.3966	1.18750	0.593751	−	0.804649i	$-0.297646\pi$
$839$	34.3966	1.18750	0.593751	+	0.804649i	$0.297646\pi$
$840$	−7.44452	−0.256860
$841$	−13.0588	−0.450303
$842$	−19.3260	−0.666017
$843$	−28.3719	−0.977179
$844$	−15.1979	−0.523132
$845$	−6.07627	−0.209030
$846$	54.2611	1.86553
$847$	10.1648	0.349265
$848$	8.32596	0.285915
$849$	69.0678	2.37040
$850$	14.4648	0.496137
$851$	−3.50500	−0.120150
$852$	76.7856	2.63063
$853$	−19.1054	−0.654157	−0.327079	−	0.944997i	$-0.606064\pi$
$853$	−19.1054	−0.654157	−0.327079	+	0.944997i	$0.606064\pi$
$854$	67.2724	2.30201
$855$	10.6574	0.364475
$856$	−15.2421	−0.520964
$857$	22.4004	0.765182	0.382591	−	0.923918i	$-0.375032\pi$
$857$	22.4004	0.765182	0.382591	+	0.923918i	$0.375032\pi$
$858$	−55.1755	−1.88366
$859$	6.67865	0.227873	0.113936	−	0.993488i	$-0.463654\pi$
$859$	6.67865	0.227873	0.113936	+	0.993488i	$0.463654\pi$
$860$	11.0864	0.378045
$861$	−56.7047	−1.93249
$862$	−0.978705	−0.0333348
$863$	−22.9271	−0.780447	−0.390224	−	0.920720i	$-0.627602\pi$
$863$	−22.9271	−0.780447	−0.390224	+	0.920720i	$0.627602\pi$
$864$	10.5077	0.357481
$865$	−24.0069	−0.816259
$866$	83.8004	2.84765
$867$	74.6300	2.53457
$868$	−15.8507	−0.538009
$869$	−1.67604	−0.0568556
$870$	21.6883	0.735302
$871$	30.1975	1.02320
$872$	−17.2798	−0.585169
$873$	49.5916	1.67842
$874$	−16.5308	−0.559164
$875$	2.60553	0.0880829
$876$	28.0228	0.946801
$877$	44.4195	1.49994	0.749969	−	0.661472i	$-0.230068\pi$
$877$	44.4195	1.49994	0.749969	+	0.661472i	$0.230068\pi$
$878$	−22.5136	−0.759797
$879$	40.5427	1.36747
$880$	−10.3120	−0.347618
$881$	−15.7568	−0.530862	−0.265431	−	0.964130i	$-0.585514\pi$
$881$	−15.7568	−0.530862	−0.265431	+	0.964130i	$0.585514\pi$
$882$	1.58163	0.0532563
$883$	6.57822	0.221375	0.110687	−	0.993855i	$-0.464695\pi$
$883$	6.57822	0.221375	0.110687	+	0.993855i	$0.464695\pi$
$884$	−45.1914	−1.51995
$885$	−3.47318	−0.116750
$886$	0.570710	0.0191734
$887$	−7.59189	−0.254911	−0.127455	−	0.991844i	$-0.540681\pi$
$887$	−7.59189	−0.254911	−0.127455	+	0.991844i	$0.540681\pi$
$888$	3.90262	0.130963
$889$	−14.6562	−0.491552
$890$	20.6492	0.692164
$891$	−27.6835	−0.927431
$892$	42.3496	1.41797
$893$	−21.9438	−0.734322
$894$	45.4083	1.51868
$895$	11.8924	0.397520
$896$	22.5182	0.752280
$897$	17.2405	0.575645
$898$	21.0269	0.701678
$899$	9.61576	0.320703
$900$	8.89030	0.296343
$901$	21.1913	0.705983
$902$	69.9977	2.33067
$903$	29.1988	0.971674
$904$	0.372349	0.0123842
$905$	−0.175932	−0.00584818
$906$	5.43206	0.180468
$907$	−29.7885	−0.989110	−0.494555	−	0.869146i	$-0.664669\pi$
$907$	−29.7885	−0.989110	−0.494555	+	0.869146i	$0.664669\pi$
$908$	−55.7271	−1.84937
$909$	−24.5119	−0.813008
$910$	−14.5855	−0.483506
$911$	−14.4593	−0.479058	−0.239529	−	0.970889i	$-0.576993\pi$
$911$	−14.4593	−0.479058	−0.239529	+	0.970889i	$0.576993\pi$
$912$	−20.6541	−0.683925
$913$	−12.8805	−0.426284
$914$	−69.8135	−2.30923
$915$	−30.9879	−1.02443
$916$	−42.7387	−1.41213
$917$	−34.8327	−1.15028
$918$	19.1881	0.633302
$919$	−50.2535	−1.65771	−0.828854	−	0.559464i	$-0.811007\pi$
$919$	−50.2535	−1.65771	−0.828854	+	0.559464i	$0.811007\pi$
$920$	−2.87148	−0.0946698
$921$	−82.0126	−2.70241
$922$	−40.7420	−1.34177
$923$	31.3264	1.03112
$924$	64.8703	2.13408
$925$	−1.36589	−0.0449102
$926$	58.8531	1.93403
$927$	71.2988	2.34176
$928$	−31.6264	−1.03819
$929$	24.5042	0.803955	0.401978	−	0.915650i	$-0.368323\pi$
$929$	24.5042	0.803955	0.401978	+	0.915650i	$0.368323\pi$
$930$	13.0824	0.428989
$931$	−0.639630	−0.0209630
$932$	−0.177141	−0.00580245
$933$	−21.7232	−0.711186
$934$	−14.6312	−0.478747
$935$	−26.2461	−0.858341
$936$	−10.3631	−0.338727
$937$	−15.1200	−0.493948	−0.246974	−	0.969022i	$-0.579436\pi$
$937$	−15.1200	−0.493948	−0.246974	+	0.969022i	$0.579436\pi$
$938$	−63.6141	−2.07708
$939$	34.9599	1.14087
$940$	−18.3053	−0.597055
$941$	−34.4769	−1.12391	−0.561957	−	0.827167i	$-0.689951\pi$
$941$	−34.4769	−1.12391	−0.561957	+	0.827167i	$0.689951\pi$
$942$	−31.0569	−1.01189
$943$	−21.8720	−0.712250
$944$	3.63371	0.118267
$945$	3.45634	0.112435
$946$	−36.0437	−1.17188
$947$	−34.9014	−1.13414	−0.567072	−	0.823668i	$-0.691924\pi$
$947$	−34.9014	−1.13414	−0.567072	+	0.823668i	$0.691924\pi$
$948$	−2.80035	−0.0909510
$949$	11.4325	0.371115
$950$	−6.44202	−0.209007
$951$	−56.4700	−1.83117
$952$	19.8236	0.642487
$953$	−13.7273	−0.444672	−0.222336	−	0.974970i	$-0.571368\pi$
$953$	−13.7273	−0.444672	−0.222336	+	0.974970i	$0.571368\pi$
$954$	23.3369	0.755561
$955$	27.2502	0.881795
$956$	45.2078	1.46213
$957$	−39.3532	−1.27211
$958$	−65.4119	−2.11336
$959$	23.5705	0.761130
$960$	−29.3865	−0.948446
$961$	−25.1998	−0.812896
$962$	7.64614	0.246521
$963$	47.9398	1.54484
$964$	−77.0122	−2.48040
$965$	−4.82078	−0.155186
$966$	−36.3189	−1.16854
$967$	−27.1956	−0.874552	−0.437276	−	0.899327i	$-0.644057\pi$
$967$	−27.1956	−0.874552	−0.437276	+	0.899327i	$0.644057\pi$
$968$	−4.36551	−0.140313
$969$	−52.5688	−1.68875
$970$	−29.9764	−0.962485
$971$	0.904766	0.0290353	0.0145177	−	0.999895i	$-0.495379\pi$
$971$	0.904766	0.0290353	0.0145177	+	0.999895i	$0.495379\pi$
$972$	−56.3064	−1.80603
$973$	−20.4227	−0.654720
$974$	−38.2866	−1.22678
$975$	6.71859	0.215167
$976$	32.4203	1.03775
$977$	−33.4445	−1.06998	−0.534992	−	0.844857i	$-0.679685\pi$
$977$	−33.4445	−1.06998	−0.534992	+	0.844857i	$0.679685\pi$
$978$	58.4537	1.86914
$979$	−37.4678	−1.19748
$980$	−0.533574	−0.0170444
$981$	54.3490	1.73523
$982$	−43.6907	−1.39423
$983$	13.3334	0.425270	0.212635	−	0.977132i	$-0.431796\pi$
$983$	13.3334	0.425270	0.212635	+	0.977132i	$0.431796\pi$
$984$	24.3533	0.776353
$985$	−10.8355	−0.345247
$986$	−57.7527	−1.83922
$987$	−48.2115	−1.53459
$988$	20.1264	0.640308
$989$	11.2625	0.358125
$990$	−28.9037	−0.918618
$991$	15.4382	0.490412	0.245206	−	0.969471i	$-0.421144\pi$
$991$	15.4382	0.490412	0.245206	+	0.969471i	$0.421144\pi$
$992$	−19.0771	−0.605698
$993$	13.2079	0.419141
$994$	−65.9922	−2.09314
$995$	5.11188	0.162057
$996$	−21.5210	−0.681919
$997$	−7.80480	−0.247181	−0.123590	−	0.992333i	$-0.539441\pi$
$997$	−7.80480	−0.247181	−0.123590	+	0.992333i	$0.539441\pi$
$998$	62.0841	1.96524
$999$	−1.81191	−0.0573262

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	755.2.a.j.1.3	✓	15
3.2	odd	2		6795.2.a.bh.1.13		15
5.4	even	2		3775.2.a.q.1.13		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
755.2.a.j.1.3	✓	15	1.1	even	1	trivial
3775.2.a.q.1.13		15	5.4	even	2
6795.2.a.bh.1.13		15	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 755 = 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	755.a (trivial)

\(f(q)\)	\(=\)	\(q-2.12744 q^{2} +2.55334 q^{3} +2.52599 q^{4} +1.00000 q^{5} -5.43206 q^{6} +2.60553 q^{7} -1.11901 q^{8} +3.51953 q^{9} -2.12744 q^{10} +3.86021 q^{11} +6.44970 q^{12} +2.63130 q^{13} -5.54309 q^{14} +2.55334 q^{15} -2.67136 q^{16} -6.79915 q^{17} -7.48758 q^{18} +3.02807 q^{19} +2.52599 q^{20} +6.65279 q^{21} -8.21236 q^{22} +2.56609 q^{23} -2.85720 q^{24} +1.00000 q^{25} -5.59792 q^{26} +1.32654 q^{27} +6.58153 q^{28} -3.99265 q^{29} -5.43206 q^{30} -2.40837 q^{31} +7.92116 q^{32} +9.85642 q^{33} +14.4648 q^{34} +2.60553 q^{35} +8.89030 q^{36} -1.36589 q^{37} -6.44202 q^{38} +6.71859 q^{39} -1.11901 q^{40} -8.52346 q^{41} -14.1534 q^{42} +4.38895 q^{43} +9.75085 q^{44} +3.51953 q^{45} -5.45920 q^{46} -7.24681 q^{47} -6.82088 q^{48} -0.211234 q^{49} -2.12744 q^{50} -17.3605 q^{51} +6.64663 q^{52} -3.11675 q^{53} -2.82213 q^{54} +3.86021 q^{55} -2.91560 q^{56} +7.73168 q^{57} +8.49410 q^{58} -1.36025 q^{59} +6.44970 q^{60} -12.1363 q^{61} +5.12365 q^{62} +9.17023 q^{63} -11.5091 q^{64} +2.63130 q^{65} -20.9689 q^{66} +11.4763 q^{67} -17.1746 q^{68} +6.55210 q^{69} -5.54309 q^{70} +11.9053 q^{71} -3.93838 q^{72} +4.34482 q^{73} +2.90584 q^{74} +2.55334 q^{75} +7.64887 q^{76} +10.0579 q^{77} -14.2934 q^{78} -0.434182 q^{79} -2.67136 q^{80} -7.17149 q^{81} +18.1331 q^{82} -3.33674 q^{83} +16.8049 q^{84} -6.79915 q^{85} -9.33722 q^{86} -10.1946 q^{87} -4.31961 q^{88} -9.70616 q^{89} -7.48758 q^{90} +6.85592 q^{91} +6.48192 q^{92} -6.14938 q^{93} +15.4171 q^{94} +3.02807 q^{95} +20.2254 q^{96} +14.0904 q^{97} +0.449387 q^{98} +13.5861 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 2 q^{2} + 5 q^{3} + 18 q^{4} + 15 q^{5} - 4 q^{6} + 11 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{10} + 2 q^{11} + 4 q^{12} + 11 q^{13} - 9 q^{14} + 5 q^{15} + 40 q^{16} + 25 q^{17} - 15 q^{18} - 3 q^{19}+ \cdots - 11 q^{99}+O(q^{100}) \) 15 * q - 2 * q^2 + 5 * q^3 + 18 * q^4 + 15 * q^5 - 4 * q^6 + 11 * q^7 + 12 * q^8 + 10 * q^9 - 2 * q^10 + 2 * q^11 + 4 * q^12 + 11 * q^13 - 9 * q^14 + 5 * q^15 + 40 * q^16 + 25 * q^17 - 15 * q^18 - 3 * q^19 + 18 * q^20 - 11 * q^21 + 13 * q^22 + 8 * q^23 - 21 * q^24 + 15 * q^25 - 45 * q^26 + 20 * q^27 + 29 * q^28 + 24 * q^29 - 4 * q^30 - 11 * q^31 + 27 * q^32 + 26 * q^33 - 11 * q^34 + 11 * q^35 - 22 * q^36 + 35 * q^37 + 2 * q^38 + 11 * q^39 + 12 * q^40 - 20 * q^41 - 17 * q^42 - 13 * q^43 - 32 * q^44 + 10 * q^45 - 5 * q^46 + 6 * q^47 - 11 * q^48 + 40 * q^49 - 2 * q^50 + 23 * q^51 + 27 * q^52 + 38 * q^53 - 30 * q^54 + 2 * q^55 - 36 * q^56 + 4 * q^57 + 6 * q^58 - 15 * q^59 + 4 * q^60 + 33 * q^61 - 13 * q^62 + 2 * q^63 + 74 * q^64 + 11 * q^65 + 8 * q^66 + 24 * q^67 + 8 * q^68 + 13 * q^69 - 9 * q^70 + 19 * q^71 + 56 * q^72 + 4 * q^73 - 28 * q^74 + 5 * q^75 + 19 * q^76 + 14 * q^77 - 15 * q^78 - 4 * q^79 + 40 * q^80 - 13 * q^81 + 33 * q^82 + 19 * q^83 - 71 * q^84 + 25 * q^85 - 59 * q^86 + 49 * q^87 - 48 * q^88 - 35 * q^89 - 15 * q^90 - q^91 + 18 * q^92 + 2 * q^93 - 70 * q^94 - 3 * q^95 - 35 * q^96 + 55 * q^97 - 64 * q^98 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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