LMFDB - Embedded newform 755.2.a.i.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 = [5,0,-3] 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:	$1$
Dimension:	$5$
Coefficient field:	5.5.220669.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 7x^{3} - 2x^{2} + 11x + 5 $ x^5 - 7x^3 - 2x^2 + 11*x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.480943$ 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)

$f(q)$	$=$	\(q-0.480943 q^{2} -2.56539 q^{3} -1.76869 q^{4} -1.00000 q^{5} +1.23381 q^{6} +1.04633 q^{7} +1.81253 q^{8} +3.58122 q^{9} +0.480943 q^{10} +0.956168 q^{11} +4.53739 q^{12} +1.76869 q^{13} -0.503226 q^{14} +2.56539 q^{15} +2.66567 q^{16} +1.46511 q^{17} -1.72236 q^{18} -4.52156 q^{19} +1.76869 q^{20} -2.68425 q^{21} -0.459862 q^{22} +3.65555 q^{23} -4.64984 q^{24} +1.00000 q^{25} -0.850641 q^{26} -1.49106 q^{27} -1.85064 q^{28} -6.82264 q^{29} -1.23381 q^{30} +6.70628 q^{31} -4.90709 q^{32} -2.45294 q^{33} -0.704635 q^{34} -1.04633 q^{35} -6.33408 q^{36} -8.10028 q^{37} +2.17461 q^{38} -4.53739 q^{39} -1.81253 q^{40} -8.80931 q^{41} +1.29097 q^{42} -9.17066 q^{43} -1.69117 q^{44} -3.58122 q^{45} -1.75811 q^{46} +10.8251 q^{47} -6.83847 q^{48} -5.90519 q^{49} -0.480943 q^{50} -3.75858 q^{51} -3.12828 q^{52} +9.16684 q^{53} +0.717113 q^{54} -0.956168 q^{55} +1.89650 q^{56} +11.5996 q^{57} +3.28130 q^{58} -11.9096 q^{59} -4.53739 q^{60} -0.221193 q^{61} -3.22534 q^{62} +3.74714 q^{63} -2.97131 q^{64} -1.76869 q^{65} +1.17972 q^{66} +5.06931 q^{67} -2.59133 q^{68} -9.37792 q^{69} +0.503226 q^{70} -7.62183 q^{71} +6.49106 q^{72} -8.91530 q^{73} +3.89577 q^{74} -2.56539 q^{75} +7.99725 q^{76} +1.00047 q^{77} +2.18222 q^{78} +5.99975 q^{79} -2.66567 q^{80} -6.91852 q^{81} +4.23677 q^{82} -0.0357498 q^{83} +4.74761 q^{84} -1.46511 q^{85} +4.41056 q^{86} +17.5027 q^{87} +1.73308 q^{88} +2.00822 q^{89} +1.72236 q^{90} +1.85064 q^{91} -6.46555 q^{92} -17.2042 q^{93} -5.20627 q^{94} +4.52156 q^{95} +12.5886 q^{96} +4.17186 q^{97} +2.84006 q^{98} +3.42425 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} + 4 q^{4} - 5 q^{5} - 3 q^{6} - 7 q^{7} + 6 q^{8} + 2 q^{9} - 5 q^{11} - 3 q^{12} - 4 q^{13} - 11 q^{14} + 3 q^{15} - 10 q^{16} + 11 q^{17} - 8 q^{18} - 3 q^{19} - 4 q^{20} - 8 q^{21} - 4 q^{22}+ \cdots - 8 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 + 4 * q^4 - 5 * q^5 - 3 * q^6 - 7 * q^7 + 6 * q^8 + 2 * q^9 - 5 * q^11 - 3 * q^12 - 4 * q^13 - 11 * q^14 + 3 * q^15 - 10 * q^16 + 11 * q^17 - 8 * q^18 - 3 * q^19 - 4 * q^20 - 8 * q^21 - 4 * q^22 + 6 * q^23 - 6 * q^24 + 5 * q^25 - 6 * q^26 + 6 * q^27 - 11 * q^28 - 20 * q^29 + 3 * q^30 - 7 * q^31 - 3 * q^32 + 6 * q^33 - 3 * q^34 + 7 * q^35 - 9 * q^36 - 27 * q^37 + q^38 + 3 * q^39 - 6 * q^40 - 14 * q^41 + 23 * q^42 + 3 * q^43 - 23 * q^44 - 2 * q^45 - 4 * q^46 + 18 * q^47 - 14 * q^48 - 2 * q^49 - 17 * q^51 - 18 * q^52 - 2 * q^53 + q^54 + 5 * q^55 - 4 * q^56 + 14 * q^57 - 11 * q^58 - 16 * q^59 + 3 * q^60 - 25 * q^61 + 22 * q^62 + 7 * q^63 - 8 * q^64 + 4 * q^65 + 29 * q^66 + 15 * q^67 + 14 * q^68 - 34 * q^69 + 11 * q^70 - 5 * q^71 + 19 * q^72 - 6 * q^73 - 6 * q^74 - 3 * q^75 + 16 * q^76 + 8 * q^77 + 12 * q^78 - 16 * q^79 + 10 * q^80 - 23 * q^81 - 7 * q^82 + 4 * q^83 + 15 * q^84 - 11 * q^85 + 55 * q^86 + 4 * q^87 - 31 * q^88 - 7 * q^89 + 8 * q^90 + 11 * q^91 + 19 * q^92 - 31 * q^93 - 4 * q^94 + 3 * q^95 + 12 * q^96 - 13 * q^97 + 24 * q^98 - 8 * 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 $

$2$	−0.480943	−0.340078	−0.170039	−	0.985437i	$-0.554389\pi$
$2$	−0.480943	−0.340078	−0.170039	+	0.985437i	$0.554389\pi$
$3$	−2.56539	−1.48113	−0.740564	−	0.671986i	$-0.765442\pi$
$3$	−2.56539	−1.48113	−0.740564	+	0.671986i	$0.765442\pi$
$4$	−1.76869	−0.884347
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.23381	0.503699
$7$	1.04633	0.395476	0.197738	−	0.980255i	$-0.436640\pi$
$7$	1.04633	0.395476	0.197738	+	0.980255i	$0.436640\pi$
$8$	1.81253	0.640825
$9$	3.58122	1.19374
$10$	0.480943	0.152087
$11$	0.956168	0.288295	0.144148	−	0.989556i	$-0.453956\pi$
$11$	0.956168	0.288295	0.144148	+	0.989556i	$0.453956\pi$
$12$	4.53739	1.30983
$13$	1.76869	0.490547	0.245274	−	0.969454i	$-0.421122\pi$
$13$	1.76869	0.490547	0.245274	+	0.969454i	$0.421122\pi$
$14$	−0.503226	−0.134493
$15$	2.56539	0.662381
$16$	2.66567	0.666417
$17$	1.46511	0.355342	0.177671	−	0.984090i	$-0.443144\pi$
$17$	1.46511	0.355342	0.177671	+	0.984090i	$0.443144\pi$
$18$	−1.72236	−0.405965
$19$	−4.52156	−1.03732	−0.518658	−	0.854982i	$-0.673568\pi$
$19$	−4.52156	−1.03732	−0.518658	+	0.854982i	$0.673568\pi$
$20$	1.76869	0.395492
$21$	−2.68425	−0.585751
$22$	−0.459862	−0.0980429
$23$	3.65555	0.762235	0.381118	−	0.924527i	$-0.375539\pi$
$23$	3.65555	0.762235	0.381118	+	0.924527i	$0.375539\pi$
$24$	−4.64984	−0.949144
$25$	1.00000	0.200000
$26$	−0.850641	−0.166824
$27$	−1.49106	−0.286954
$28$	−1.85064	−0.349738
$29$	−6.82264	−1.26693	−0.633466	−	0.773770i	$-0.718368\pi$
$29$	−6.82264	−1.26693	−0.633466	+	0.773770i	$0.718368\pi$
$30$	−1.23381	−0.225261
$31$	6.70628	1.20448	0.602242	−	0.798314i	$-0.294274\pi$
$31$	6.70628	1.20448	0.602242	+	0.798314i	$0.294274\pi$
$32$	−4.90709	−0.867458
$33$	−2.45294	−0.427002
$34$	−0.704635	−0.120844
$35$	−1.04633	−0.176862
$36$	−6.33408	−1.05568
$37$	−8.10028	−1.33168	−0.665839	−	0.746096i	$-0.731926\pi$
$37$	−8.10028	−1.33168	−0.665839	+	0.746096i	$0.731926\pi$
$38$	2.17461	0.352768
$39$	−4.53739	−0.726564
$40$	−1.81253	−0.286586
$41$	−8.80931	−1.37578	−0.687891	−	0.725814i	$-0.741463\pi$
$41$	−8.80931	−1.37578	−0.687891	+	0.725814i	$0.741463\pi$
$42$	1.29097	0.199201
$43$	−9.17066	−1.39851	−0.699256	−	0.714871i	$-0.746485\pi$
$43$	−9.17066	−1.39851	−0.699256	+	0.714871i	$0.746485\pi$
$44$	−1.69117	−0.254953
$45$	−3.58122	−0.533857
$46$	−1.75811	−0.259219
$47$	10.8251	1.57901	0.789504	−	0.613745i	$-0.210338\pi$
$47$	10.8251	1.57901	0.789504	+	0.613745i	$0.210338\pi$
$48$	−6.83847	−0.987048
$49$	−5.90519	−0.843599
$50$	−0.480943	−0.0680156
$51$	−3.75858	−0.526307
$52$	−3.12828	−0.433814
$53$	9.16684	1.25916	0.629581	−	0.776935i	$-0.283227\pi$
$53$	9.16684	1.25916	0.629581	+	0.776935i	$0.283227\pi$
$54$	0.717113	0.0975867
$55$	−0.956168	−0.128930
$56$	1.89650	0.253431
$57$	11.5996	1.53640
$58$	3.28130	0.430856
$59$	−11.9096	−1.55050	−0.775248	−	0.631657i	$-0.782375\pi$
$59$	−11.9096	−1.55050	−0.775248	+	0.631657i	$0.782375\pi$
$60$	−4.53739	−0.585774
$61$	−0.221193	−0.0283208	−0.0141604	−	0.999900i	$-0.504508\pi$
$61$	−0.221193	−0.0283208	−0.0141604	+	0.999900i	$0.504508\pi$
$62$	−3.22534	−0.409618
$63$	3.74714	0.472096
$64$	−2.97131	−0.371413
$65$	−1.76869	−0.219379
$66$	1.17972	0.145214
$67$	5.06931	0.619315	0.309657	−	0.950848i	$-0.399786\pi$
$67$	5.06931	0.619315	0.309657	+	0.950848i	$0.399786\pi$
$68$	−2.59133	−0.314245
$69$	−9.37792	−1.12897
$70$	0.503226	0.0601470
$71$	−7.62183	−0.904545	−0.452273	−	0.891880i	$-0.649387\pi$
$71$	−7.62183	−0.904545	−0.452273	+	0.891880i	$0.649387\pi$
$72$	6.49106	0.764978
$73$	−8.91530	−1.04346	−0.521729	−	0.853111i	$-0.674713\pi$
$73$	−8.91530	−1.04346	−0.521729	+	0.853111i	$0.674713\pi$
$74$	3.89577	0.452874
$75$	−2.56539	−0.296226
$76$	7.99725	0.917347
$77$	1.00047	0.114014
$78$	2.18222	0.247088
$79$	5.99975	0.675025	0.337512	−	0.941321i	$-0.390415\pi$
$79$	5.99975	0.675025	0.337512	+	0.941321i	$0.390415\pi$
$80$	−2.66567	−0.298031
$81$	−6.91852	−0.768725
$82$	4.23677	0.467873
$83$	−0.0357498	−0.00392405	−0.00196202	−	0.999998i	$-0.500625\pi$
$83$	−0.0357498	−0.00392405	−0.00196202	+	0.999998i	$0.500625\pi$
$84$	4.74761	0.518007
$85$	−1.46511	−0.158914
$86$	4.41056	0.475603
$87$	17.5027	1.87649
$88$	1.73308	0.184747
$89$	2.00822	0.212871	0.106435	−	0.994320i	$-0.466056\pi$
$89$	2.00822	0.212871	0.106435	+	0.994320i	$0.466056\pi$
$90$	1.72236	0.181553
$91$	1.85064	0.194000
$92$	−6.46555	−0.674081
$93$	−17.2042	−1.78399
$94$	−5.20627	−0.536986
$95$	4.52156	0.463902
$96$	12.5886	1.28482
$97$	4.17186	0.423588	0.211794	−	0.977314i	$-0.432069\pi$
$97$	4.17186	0.423588	0.211794	+	0.977314i	$0.432069\pi$
$98$	2.84006	0.286889
$99$	3.42425	0.344150
$100$	−1.76869	−0.176869
$101$	5.52595	0.549853	0.274926	−	0.961465i	$-0.411347\pi$
$101$	5.52595	0.549853	0.274926	+	0.961465i	$0.411347\pi$
$102$	1.80766	0.178985
$103$	2.58169	0.254381	0.127191	−	0.991878i	$-0.459404\pi$
$103$	2.58169	0.254381	0.127191	+	0.991878i	$0.459404\pi$
$104$	3.20580	0.314355
$105$	2.68425	0.261956
$106$	−4.40872	−0.428213
$107$	7.33705	0.709299	0.354650	−	0.934999i	$-0.384600\pi$
$107$	7.33705	0.709299	0.354650	+	0.934999i	$0.384600\pi$
$108$	2.63722	0.253767
$109$	−10.8766	−1.04179	−0.520894	−	0.853621i	$-0.674401\pi$
$109$	−10.8766	−1.04179	−0.520894	+	0.853621i	$0.674401\pi$
$110$	0.459862	0.0438461
$111$	20.7804	1.97238
$112$	2.78917	0.263552
$113$	−7.15031	−0.672645	−0.336322	−	0.941747i	$-0.609183\pi$
$113$	−7.15031	−0.672645	−0.336322	+	0.941747i	$0.609183\pi$
$114$	−5.57872	−0.522495
$115$	−3.65555	−0.340882
$116$	12.0672	1.12041
$117$	6.33408	0.585586
$118$	5.72783	0.527289
$119$	1.53299	0.140529
$120$	4.64984	0.424470
$121$	−10.0857	−0.916886
$122$	0.106381	0.00963128
$123$	22.5993	2.03771
$124$	−11.8614	−1.06518
$125$	−1.00000	−0.0894427
$126$	−1.80216	−0.160549
$127$	−5.61817	−0.498532	−0.249266	−	0.968435i	$-0.580189\pi$
$127$	−5.61817	−0.498532	−0.249266	+	0.968435i	$0.580189\pi$
$128$	11.2432	0.993768
$129$	23.5263	2.07138
$130$	0.850641	0.0746061
$131$	1.48009	0.129316	0.0646580	−	0.997907i	$-0.479404\pi$
$131$	1.48009	0.129316	0.0646580	+	0.997907i	$0.479404\pi$
$132$	4.33850	0.377618
$133$	−4.73105	−0.410234
$134$	−2.43805	−0.210615
$135$	1.49106	0.128330
$136$	2.65555	0.227712
$137$	−11.0949	−0.947902	−0.473951	−	0.880551i	$-0.657173\pi$
$137$	−11.0949	−0.947902	−0.473951	+	0.880551i	$0.657173\pi$
$138$	4.51024	0.383937
$139$	−15.1042	−1.28112	−0.640559	−	0.767909i	$-0.721297\pi$
$139$	−15.1042	−1.28112	−0.640559	+	0.767909i	$0.721297\pi$
$140$	1.85064	0.156408
$141$	−27.7707	−2.33871
$142$	3.66567	0.307616
$143$	1.69117	0.141423
$144$	9.54634	0.795528
$145$	6.82264	0.566589
$146$	4.28775	0.354857
$147$	15.1491	1.24948
$148$	14.3269	1.17766
$149$	13.3667	1.09504	0.547520	−	0.836793i	$-0.315572\pi$
$149$	13.3667	1.09504	0.547520	+	0.836793i	$0.315572\pi$
$150$	1.23381	0.100740
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	−8.19544	−0.664738
$153$	5.24689	0.424186
$154$	−0.481168	−0.0387736
$155$	−6.70628	−0.538661
$156$	8.02525	0.642534
$157$	−13.4184	−1.07090	−0.535451	−	0.844566i	$-0.679858\pi$
$157$	−13.4184	−1.07090	−0.535451	+	0.844566i	$0.679858\pi$
$158$	−2.88554	−0.229561
$159$	−23.5165	−1.86498
$160$	4.90709	0.387939
$161$	3.82492	0.301446
$162$	3.32741	0.261426
$163$	−20.7937	−1.62869	−0.814344	−	0.580383i	$-0.802903\pi$
$163$	−20.7937	−1.62869	−0.814344	+	0.580383i	$0.802903\pi$
$164$	15.5810	1.21667
$165$	2.45294	0.190961
$166$	0.0171936	0.00133448
$167$	2.88664	0.223375	0.111687	−	0.993743i	$-0.464374\pi$
$167$	2.88664	0.223375	0.111687	+	0.993743i	$0.464374\pi$
$168$	−4.86527	−0.375364
$169$	−9.87172	−0.759363
$170$	0.704635	0.0540430
$171$	−16.1927	−1.23829
$172$	16.2201	1.23677
$173$	3.85227	0.292883	0.146441	−	0.989219i	$-0.453218\pi$
$173$	3.85227	0.292883	0.146441	+	0.989219i	$0.453218\pi$
$174$	−8.41781	−0.638153
$175$	1.04633	0.0790952
$176$	2.54882	0.192125
$177$	30.5527	2.29648
$178$	−0.965838	−0.0723926
$179$	5.60027	0.418584	0.209292	−	0.977853i	$-0.432884\pi$
$179$	5.60027	0.418584	0.209292	+	0.977853i	$0.432884\pi$
$180$	6.33408	0.472115
$181$	−16.3286	−1.21369	−0.606847	−	0.794819i	$-0.707566\pi$
$181$	−16.3286	−1.21369	−0.606847	+	0.794819i	$0.707566\pi$
$182$	−0.890052	−0.0659751
$183$	0.567445	0.0419467
$184$	6.62579	0.488459
$185$	8.10028	0.595544
$186$	8.27424	0.606697
$187$	1.40089	0.102443
$188$	−19.1464	−1.39639
$189$	−1.56014	−0.113483
$190$	−2.17461	−0.157763
$191$	−17.8639	−1.29259	−0.646295	−	0.763088i	$-0.723682\pi$
$191$	−17.8639	−1.29259	−0.646295	+	0.763088i	$0.723682\pi$
$192$	7.62255	0.550110
$193$	12.8070	0.921870	0.460935	−	0.887434i	$-0.347514\pi$
$193$	12.8070	0.921870	0.460935	+	0.887434i	$0.347514\pi$
$194$	−2.00643	−0.144053
$195$	4.53739	0.324929
$196$	10.4445	0.746034
$197$	−20.0190	−1.42630	−0.713148	−	0.701013i	$-0.752731\pi$
$197$	−20.0190	−1.42630	−0.713148	+	0.701013i	$0.752731\pi$
$198$	−1.64687	−0.117038
$199$	17.3920	1.23289	0.616445	−	0.787398i	$-0.288572\pi$
$199$	17.3920	1.23289	0.616445	+	0.787398i	$0.288572\pi$
$200$	1.81253	0.128165
$201$	−13.0047	−0.917284
$202$	−2.65767	−0.186993
$203$	−7.13874	−0.501042
$204$	6.64778	0.465438
$205$	8.80931	0.615269
$206$	−1.24164	−0.0865095
$207$	13.0913	0.909911
$208$	4.71475	0.326909
$209$	−4.32337	−0.299053
$210$	−1.29097	−0.0890854
$211$	−2.47071	−0.170091	−0.0850453	−	0.996377i	$-0.527104\pi$
$211$	−2.47071	−0.170091	−0.0850453	+	0.996377i	$0.527104\pi$
$212$	−16.2133	−1.11354
$213$	19.5530	1.33975
$214$	−3.52870	−0.241217
$215$	9.17066	0.625434
$216$	−2.70258	−0.183887
$217$	7.01699	0.476345
$218$	5.23102	0.354289
$219$	22.8712	1.54549
$220$	1.69117	0.114019
$221$	2.59133	0.174312
$222$	−9.99417	−0.670765
$223$	−15.3953	−1.03094	−0.515471	−	0.856907i	$-0.672383\pi$
$223$	−15.3953	−1.03094	−0.515471	+	0.856907i	$0.672383\pi$
$224$	−5.13444	−0.343059
$225$	3.58122	0.238748
$226$	3.43889	0.228752
$227$	7.07697	0.469715	0.234857	−	0.972030i	$-0.424538\pi$
$227$	7.07697	0.469715	0.234857	+	0.972030i	$0.424538\pi$
$228$	−20.5161	−1.35871
$229$	−7.45630	−0.492726	−0.246363	−	0.969178i	$-0.579236\pi$
$229$	−7.45630	−0.492726	−0.246363	+	0.969178i	$0.579236\pi$
$230$	1.75811	0.115926
$231$	−2.56659	−0.168869
$232$	−12.3662	−0.811882
$233$	−16.2506	−1.06461	−0.532306	−	0.846552i	$-0.678674\pi$
$233$	−16.2506	−1.06461	−0.532306	+	0.846552i	$0.678674\pi$
$234$	−3.04633	−0.199145
$235$	−10.8251	−0.706154
$236$	21.0644	1.37118
$237$	−15.3917	−0.999798
$238$	−0.737282	−0.0477909
$239$	−8.89255	−0.575211	−0.287606	−	0.957749i	$-0.592859\pi$
$239$	−8.89255	−0.575211	−0.287606	+	0.957749i	$0.592859\pi$
$240$	6.83847	0.441421
$241$	−13.8522	−0.892298	−0.446149	−	0.894959i	$-0.647205\pi$
$241$	−13.8522	−0.892298	−0.446149	+	0.894959i	$0.647205\pi$
$242$	4.85067	0.311813
$243$	22.2219	1.42553
$244$	0.391222	0.0250454
$245$	5.90519	0.377269
$246$	−10.8690	−0.692980
$247$	−7.99725	−0.508853
$248$	12.1553	0.771863
$249$	0.0917121	0.00581202
$250$	0.480943	0.0304175
$251$	−14.0265	−0.885348	−0.442674	−	0.896683i	$-0.645970\pi$
$251$	−14.0265	−0.885348	−0.442674	+	0.896683i	$0.645970\pi$
$252$	−6.62755	−0.417497
$253$	3.49532	0.219749
$254$	2.70202	0.169540
$255$	3.75858	0.235371
$256$	0.535274	0.0334546
$257$	19.8708	1.23950	0.619752	−	0.784798i	$-0.287233\pi$
$257$	19.8708	1.23950	0.619752	+	0.784798i	$0.287233\pi$
$258$	−11.3148	−0.704429
$259$	−8.47558	−0.526647
$260$	3.12828	0.194008
$261$	−24.4334	−1.51239
$262$	−0.711838	−0.0439775
$263$	−3.97177	−0.244910	−0.122455	−	0.992474i	$-0.539077\pi$
$263$	−3.97177	−0.244910	−0.122455	+	0.992474i	$0.539077\pi$
$264$	−4.44602	−0.273634
$265$	−9.16684	−0.563114
$266$	2.27536	0.139512
$267$	−5.15186	−0.315289
$268$	−8.96606	−0.547689
$269$	21.5351	1.31302	0.656510	−	0.754318i	$-0.272032\pi$
$269$	21.5351	1.31302	0.656510	+	0.754318i	$0.272032\pi$
$270$	−0.717113	−0.0436421
$271$	26.2764	1.59618	0.798089	−	0.602540i	$-0.205844\pi$
$271$	26.2764	1.59618	0.798089	+	0.602540i	$0.205844\pi$
$272$	3.90550	0.236806
$273$	−4.74761	−0.287339
$274$	5.33602	0.322361
$275$	0.956168	0.0576591
$276$	16.5867	0.998400
$277$	−22.9919	−1.38145	−0.690725	−	0.723117i	$-0.742709\pi$
$277$	−22.9919	−1.38145	−0.690725	+	0.723117i	$0.742709\pi$
$278$	7.26424	0.435680
$279$	24.0167	1.43784
$280$	−1.89650	−0.113338
$281$	−2.43664	−0.145358	−0.0726789	−	0.997355i	$-0.523155\pi$
$281$	−2.43664	−0.145358	−0.0726789	+	0.997355i	$0.523155\pi$
$282$	13.3561	0.795345
$283$	4.88829	0.290578	0.145289	−	0.989389i	$-0.453589\pi$
$283$	4.88829	0.290578	0.145289	+	0.989389i	$0.453589\pi$
$284$	13.4807	0.799932
$285$	−11.5996	−0.687098
$286$	−0.813355	−0.0480947
$287$	−9.21746	−0.544089
$288$	−17.5734	−1.03552
$289$	−14.8534	−0.873732
$290$	−3.28130	−0.192685
$291$	−10.7024	−0.627388
$292$	15.7684	0.922778
$293$	18.1387	1.05967	0.529837	−	0.848099i	$-0.322253\pi$
$293$	18.1387	1.05967	0.529837	+	0.848099i	$0.322253\pi$
$294$	−7.28586	−0.424920
$295$	11.9096	0.693403
$296$	−14.6820	−0.853372
$297$	−1.42570	−0.0827275
$298$	−6.42860	−0.372399
$299$	6.46555	0.373913
$300$	4.53739	0.261966
$301$	−9.59555	−0.553078
$302$	0.480943	0.0276751
$303$	−14.1762	−0.814402
$304$	−12.0530	−0.691285
$305$	0.221193	0.0126655
$306$	−2.52345	−0.144256
$307$	10.3810	0.592473	0.296237	−	0.955115i	$-0.404268\pi$
$307$	10.3810	0.592473	0.296237	+	0.955115i	$0.404268\pi$
$308$	−1.76952	−0.100828
$309$	−6.62304	−0.376771
$310$	3.22534	0.183187
$311$	16.5272	0.937169	0.468585	−	0.883419i	$-0.344764\pi$
$311$	16.5272	0.937169	0.468585	+	0.883419i	$0.344764\pi$
$312$	−8.22414	−0.465600
$313$	−11.8263	−0.668460	−0.334230	−	0.942492i	$-0.608476\pi$
$313$	−11.8263	−0.668460	−0.334230	+	0.942492i	$0.608476\pi$
$314$	6.45347	0.364190
$315$	−3.74714	−0.211128
$316$	−10.6117	−0.596956
$317$	8.02337	0.450637	0.225319	−	0.974285i	$-0.427658\pi$
$317$	8.02337	0.450637	0.225319	+	0.974285i	$0.427658\pi$
$318$	11.3101	0.634238
$319$	−6.52359	−0.365251
$320$	2.97131	0.166101
$321$	−18.8224	−1.05056
$322$	−1.83957	−0.102515
$323$	−6.62458	−0.368602
$324$	12.2367	0.679819
$325$	1.76869	0.0981095
$326$	10.0006	0.553881
$327$	27.9027	1.54302
$328$	−15.9671	−0.881636
$329$	11.3267	0.624460
$330$	−1.17972	−0.0649417
$331$	−5.93410	−0.326168	−0.163084	−	0.986612i	$-0.552144\pi$
$331$	−5.93410	−0.326168	−0.163084	+	0.986612i	$0.552144\pi$
$332$	0.0632304	0.00347022
$333$	−29.0089	−1.58968
$334$	−1.38831	−0.0759649
$335$	−5.06931	−0.276966
$336$	−7.15531	−0.390354
$337$	−20.7197	−1.12868	−0.564338	−	0.825544i	$-0.690869\pi$
$337$	−20.7197	−1.12868	−0.564338	+	0.825544i	$0.690869\pi$
$338$	4.74773	0.258243
$339$	18.3433	0.996273
$340$	2.59133	0.140535
$341$	6.41233	0.347247
$342$	7.78776	0.421114
$343$	−13.5031	−0.729099
$344$	−16.6221	−0.896201
$345$	9.37792	0.504890
$346$	−1.85272	−0.0996030
$347$	10.1583	0.545325	0.272662	−	0.962110i	$-0.412096\pi$
$347$	10.1583	0.545325	0.272662	+	0.962110i	$0.412096\pi$
$348$	−30.9570	−1.65947
$349$	−19.8519	−1.06265	−0.531323	−	0.847169i	$-0.678305\pi$
$349$	−19.8519	−1.06265	−0.531323	+	0.847169i	$0.678305\pi$
$350$	−0.503226	−0.0268985
$351$	−2.63722	−0.140765
$352$	−4.69200	−0.250084
$353$	15.0182	0.799338	0.399669	−	0.916660i	$-0.369125\pi$
$353$	15.0182	0.799338	0.399669	+	0.916660i	$0.369125\pi$
$354$	−14.6941	−0.780983
$355$	7.62183	0.404525
$356$	−3.55192	−0.188251
$357$	−3.93272	−0.208142
$358$	−2.69341	−0.142351
$359$	17.2763	0.911807	0.455903	−	0.890029i	$-0.349316\pi$
$359$	17.2763	0.911807	0.455903	+	0.890029i	$0.349316\pi$
$360$	−6.49106	−0.342109
$361$	1.44447	0.0760249
$362$	7.85311	0.412751
$363$	25.8739	1.35803
$364$	−3.27322	−0.171563
$365$	8.91530	0.466648
$366$	−0.272909	−0.0142652
$367$	0.406316	0.0212095	0.0106048	−	0.999944i	$-0.496624\pi$
$367$	0.406316	0.0212095	0.0106048	+	0.999944i	$0.496624\pi$
$368$	9.74449	0.507966
$369$	−31.5481	−1.64233
$370$	−3.89577	−0.202531
$371$	9.59155	0.497969
$372$	30.4290	1.57767
$373$	−23.7379	−1.22910	−0.614551	−	0.788877i	$-0.710663\pi$
$373$	−23.7379	−1.22910	−0.614551	+	0.788877i	$0.710663\pi$
$374$	−0.673749	−0.0348387
$375$	2.56539	0.132476
$376$	19.6208	1.01187
$377$	−12.0672	−0.621490
$378$	0.750338	0.0385932
$379$	−1.17066	−0.0601327	−0.0300663	−	0.999548i	$-0.509572\pi$
$379$	−1.17066	−0.0601327	−0.0300663	+	0.999548i	$0.509572\pi$
$380$	−7.99725	−0.410250
$381$	14.4128	0.738390
$382$	8.59154	0.439581
$383$	6.62759	0.338654	0.169327	−	0.985560i	$-0.445841\pi$
$383$	6.62759	0.338654	0.169327	+	0.985560i	$0.445841\pi$
$384$	−28.8432	−1.47190
$385$	−1.00047	−0.0509886
$386$	−6.15945	−0.313508
$387$	−32.8421	−1.66946
$388$	−7.37874	−0.374599
$389$	−14.9726	−0.759139	−0.379569	−	0.925163i	$-0.623928\pi$
$389$	−14.9726	−0.759139	−0.379569	+	0.925163i	$0.623928\pi$
$390$	−2.18222	−0.110501
$391$	5.35579	0.270854
$392$	−10.7033	−0.540599
$393$	−3.79700	−0.191533
$394$	9.62801	0.485052
$395$	−5.99975	−0.301880
$396$	−6.05645	−0.304348
$397$	−0.0925144	−0.00464316	−0.00232158	−	0.999997i	$-0.500739\pi$
$397$	−0.0925144	−0.00464316	−0.00232158	+	0.999997i	$0.500739\pi$
$398$	−8.36458	−0.419278
$399$	12.1370	0.607609
$400$	2.66567	0.133283
$401$	28.8917	1.44278	0.721391	−	0.692528i	$-0.243503\pi$
$401$	28.8917	1.44278	0.721391	+	0.692528i	$0.243503\pi$
$402$	6.25454	0.311948
$403$	11.8614	0.590856
$404$	−9.77372	−0.486261
$405$	6.91852	0.343784
$406$	3.43333	0.170393
$407$	−7.74522	−0.383916
$408$	−6.81253	−0.337270
$409$	18.0864	0.894314	0.447157	−	0.894455i	$-0.352436\pi$
$409$	18.0864	0.894314	0.447157	+	0.894455i	$0.352436\pi$
$410$	−4.23677	−0.209239
$411$	28.4628	1.40396
$412$	−4.56622	−0.224961
$413$	−12.4614	−0.613184
$414$	−6.29619	−0.309441
$415$	0.0357498	0.00175489
$416$	−8.67913	−0.425530
$417$	38.7480	1.89750
$418$	2.07929	0.101701
$419$	−7.88849	−0.385378	−0.192689	−	0.981260i	$-0.561721\pi$
$419$	−7.88849	−0.385378	−0.192689	+	0.981260i	$0.561721\pi$
$420$	−4.74761	−0.231660
$421$	20.2146	0.985197	0.492599	−	0.870257i	$-0.336047\pi$
$421$	20.2146	0.985197	0.492599	+	0.870257i	$0.336047\pi$
$422$	1.18827	0.0578441
$423$	38.7672	1.88493
$424$	16.6151	0.806902
$425$	1.46511	0.0710683
$426$	−9.40386	−0.455619
$427$	−0.231441	−0.0112002
$428$	−12.9770	−0.627267
$429$	−4.33850	−0.209465
$430$	−4.41056	−0.212696
$431$	−24.8208	−1.19557	−0.597787	−	0.801655i	$-0.703953\pi$
$431$	−24.8208	−1.19557	−0.597787	+	0.801655i	$0.703953\pi$
$432$	−3.97466	−0.191231
$433$	−38.9868	−1.87358	−0.936792	−	0.349886i	$-0.886220\pi$
$433$	−38.9868	−1.87358	−0.936792	+	0.349886i	$0.886220\pi$
$434$	−3.37477	−0.161994
$435$	−17.5027	−0.839191
$436$	19.2374	0.921302
$437$	−16.5288	−0.790679
$438$	−10.9997	−0.525588
$439$	−8.14148	−0.388572	−0.194286	−	0.980945i	$-0.562239\pi$
$439$	−8.14148	−0.388572	−0.194286	+	0.980945i	$0.562239\pi$
$440$	−1.73308	−0.0826213
$441$	−21.1478	−1.00704
$442$	−1.24628	−0.0592797
$443$	−16.9706	−0.806297	−0.403149	−	0.915135i	$-0.632084\pi$
$443$	−16.9706	−0.806297	−0.403149	+	0.915135i	$0.632084\pi$
$444$	−36.7541	−1.74427
$445$	−2.00822	−0.0951986
$446$	7.40424	0.350601
$447$	−34.2907	−1.62189
$448$	−3.10897	−0.146885
$449$	18.0433	0.851517	0.425759	−	0.904837i	$-0.360007\pi$
$449$	18.0433	0.851517	0.425759	+	0.904837i	$0.360007\pi$
$450$	−1.72236	−0.0811929
$451$	−8.42318	−0.396632
$452$	12.6467	0.594851
$453$	2.56539	0.120532
$454$	−3.40362	−0.159740
$455$	−1.85064	−0.0867594
$456$	21.0245	0.984562
$457$	−7.05558	−0.330046	−0.165023	−	0.986290i	$-0.552770\pi$
$457$	−7.05558	−0.330046	−0.165023	+	0.986290i	$0.552770\pi$
$458$	3.58605	0.167565
$459$	−2.18456	−0.101967
$460$	6.46555	0.301458
$461$	−7.89631	−0.367768	−0.183884	−	0.982948i	$-0.558867\pi$
$461$	−7.89631	−0.367768	−0.183884	+	0.982948i	$0.558867\pi$
$462$	1.23438	0.0574287
$463$	20.4539	0.950574	0.475287	−	0.879831i	$-0.342344\pi$
$463$	20.4539	0.950574	0.475287	+	0.879831i	$0.342344\pi$
$464$	−18.1869	−0.844305
$465$	17.2042	0.797826
$466$	7.81560	0.362051
$467$	−36.6137	−1.69428	−0.847140	−	0.531370i	$-0.821677\pi$
$467$	−36.6137	−1.69428	−0.847140	+	0.531370i	$0.821677\pi$
$468$	−11.2031	−0.517861
$469$	5.30418	0.244924
$470$	5.20627	0.240147
$471$	34.4233	1.58614
$472$	−21.5864	−0.993596
$473$	−8.76869	−0.403185
$474$	7.40252	0.340009
$475$	−4.52156	−0.207463
$476$	−2.71139	−0.124277
$477$	32.8285	1.50311
$478$	4.27681	0.195617
$479$	27.5049	1.25673	0.628365	−	0.777919i	$-0.283725\pi$
$479$	27.5049	1.25673	0.628365	+	0.777919i	$0.283725\pi$
$480$	−12.5886	−0.574588
$481$	−14.3269	−0.653251
$482$	6.66211	0.303451
$483$	−9.81241	−0.446480
$484$	17.8386	0.810845
$485$	−4.17186	−0.189434
$486$	−10.6874	−0.484792
$487$	−39.9722	−1.81131	−0.905656	−	0.424012i	$-0.860621\pi$
$487$	−39.9722	−1.81131	−0.905656	+	0.424012i	$0.860621\pi$
$488$	−0.400917	−0.0181487
$489$	53.3439	2.41230
$490$	−2.84006	−0.128301
$491$	35.6452	1.60865	0.804323	−	0.594192i	$-0.202528\pi$
$491$	35.6452	1.60865	0.804323	+	0.594192i	$0.202528\pi$
$492$	−39.9712	−1.80204
$493$	−9.99593	−0.450194
$494$	3.84622	0.173050
$495$	−3.42425	−0.153908
$496$	17.8767	0.802688
$497$	−7.97497	−0.357726
$498$	−0.0441083	−0.00197654
$499$	5.95255	0.266473	0.133236	−	0.991084i	$-0.457463\pi$
$499$	5.95255	0.266473	0.133236	+	0.991084i	$0.457463\pi$
$500$	1.76869	0.0790984
$501$	−7.40536	−0.330847
$502$	6.74597	0.301087
$503$	15.9458	0.710988	0.355494	−	0.934679i	$-0.384313\pi$
$503$	15.9458	0.710988	0.355494	+	0.934679i	$0.384313\pi$
$504$	6.79180	0.302531
$505$	−5.52595	−0.245902
$506$	−1.68105	−0.0747318
$507$	25.3248	1.12471
$508$	9.93683	0.440875
$509$	−9.78955	−0.433914	−0.216957	−	0.976181i	$-0.569613\pi$
$509$	−9.78955	−0.433914	−0.216957	+	0.976181i	$0.569613\pi$
$510$	−1.80766	−0.0800446
$511$	−9.32836	−0.412663
$512$	−22.7438	−1.00515
$513$	6.74190	0.297662
$514$	−9.55670	−0.421528
$515$	−2.58169	−0.113763
$516$	−41.6108	−1.83181
$517$	10.3506	0.455221
$518$	4.07627	0.179101
$519$	−9.88258	−0.433797
$520$	−3.20580	−0.140584
$521$	−36.4879	−1.59856	−0.799282	−	0.600956i	$-0.794787\pi$
$521$	−36.4879	−1.59856	−0.799282	+	0.600956i	$0.794787\pi$
$522$	11.7511	0.514330
$523$	20.0867	0.878331	0.439166	−	0.898406i	$-0.355274\pi$
$523$	20.0867	0.878331	0.439166	+	0.898406i	$0.355274\pi$
$524$	−2.61782	−0.114360
$525$	−2.68425	−0.117150
$526$	1.91020	0.0832885
$527$	9.82545	0.428003
$528$	−6.53872	−0.284561
$529$	−9.63693	−0.418997
$530$	4.40872	0.191503
$531$	−42.6508	−1.85089
$532$	8.36778	0.362789
$533$	−15.5810	−0.674887
$534$	2.47775	0.107223
$535$	−7.33705	−0.317208
$536$	9.18826	0.396872
$537$	−14.3669	−0.619976
$538$	−10.3572	−0.446529
$539$	−5.64635	−0.243206
$540$	−2.63722	−0.113488
$541$	17.8861	0.768984	0.384492	−	0.923128i	$-0.374377\pi$
$541$	17.8861	0.768984	0.384492	+	0.923128i	$0.374377\pi$
$542$	−12.6374	−0.542825
$543$	41.8892	1.79764
$544$	−7.18943	−0.308244
$545$	10.8766	0.465902
$546$	2.28333	0.0977175
$547$	29.8421	1.27596	0.637978	−	0.770054i	$-0.279771\pi$
$547$	29.8421	1.27596	0.637978	+	0.770054i	$0.279771\pi$
$548$	19.6235	0.838275
$549$	−0.792139	−0.0338077
$550$	−0.459862	−0.0196086
$551$	30.8490	1.31421
$552$	−16.9977	−0.723471
$553$	6.27773	0.266956
$554$	11.0578	0.469801
$555$	−20.7804	−0.882077
$556$	26.7146	1.13295
$557$	20.7798	0.880467	0.440234	−	0.897883i	$-0.354896\pi$
$557$	20.7798	0.880467	0.440234	+	0.897883i	$0.354896\pi$
$558$	−11.5506	−0.488978
$559$	−16.2201	−0.686037
$560$	−2.78917	−0.117864
$561$	−3.59383	−0.151732
$562$	1.17189	0.0494330
$563$	20.9064	0.881098	0.440549	−	0.897728i	$-0.354784\pi$
$563$	20.9064	0.881098	0.440549	+	0.897728i	$0.354784\pi$
$564$	49.1179	2.06823
$565$	7.15031	0.300816
$566$	−2.35099	−0.0988193
$567$	−7.23907	−0.304012
$568$	−13.8148	−0.579655
$569$	13.5182	0.566711	0.283356	−	0.959015i	$-0.408552\pi$
$569$	13.5182	0.566711	0.283356	+	0.959015i	$0.408552\pi$
$570$	5.57872	0.233667
$571$	45.2293	1.89279	0.946393	−	0.323017i	$-0.104697\pi$
$571$	45.2293	1.89279	0.946393	+	0.323017i	$0.104697\pi$
$572$	−2.99116	−0.125067
$573$	45.8280	1.91449
$574$	4.43307	0.185033
$575$	3.65555	0.152447
$576$	−10.6409	−0.443371
$577$	22.3699	0.931272	0.465636	−	0.884976i	$-0.345826\pi$
$577$	22.3699	0.931272	0.465636	+	0.884976i	$0.345826\pi$
$578$	7.14366	0.297137
$579$	−32.8550	−1.36541
$580$	−12.0672	−0.501062
$581$	−0.0374061	−0.00155187
$582$	5.14726	0.213361
$583$	8.76503	0.363011
$584$	−16.1592	−0.668673
$585$	−6.33408	−0.261882
$586$	−8.72368	−0.360372
$587$	−18.6107	−0.768144	−0.384072	−	0.923303i	$-0.625479\pi$
$587$	−18.6107	−0.768144	−0.384072	+	0.923303i	$0.625479\pi$
$588$	−26.7941	−1.10497
$589$	−30.3228	−1.24943
$590$	−5.72783	−0.235811
$591$	51.3566	2.11253
$592$	−21.5926	−0.887452
$593$	14.0814	0.578254	0.289127	−	0.957291i	$-0.406635\pi$
$593$	14.0814	0.578254	0.289127	+	0.957291i	$0.406635\pi$
$594$	0.685680	0.0281338
$595$	−1.53299	−0.0628466
$596$	−23.6415	−0.968395
$597$	−44.6174	−1.82607
$598$	−3.10956	−0.127159
$599$	−2.18003	−0.0890738	−0.0445369	−	0.999008i	$-0.514181\pi$
$599$	−2.18003	−0.0890738	−0.0445369	+	0.999008i	$0.514181\pi$
$600$	−4.64984	−0.189829
$601$	1.89273	0.0772059	0.0386030	−	0.999255i	$-0.487709\pi$
$601$	1.89273	0.0772059	0.0386030	+	0.999255i	$0.487709\pi$
$602$	4.61491	0.188090
$603$	18.1543	0.739301
$604$	1.76869	0.0719671
$605$	10.0857	0.410044
$606$	6.81795	0.276960
$607$	13.8032	0.560254	0.280127	−	0.959963i	$-0.409623\pi$
$607$	13.8032	0.560254	0.280127	+	0.959963i	$0.409623\pi$
$608$	22.1877	0.899829
$609$	18.3137	0.742107
$610$	−0.106381	−0.00430724
$611$	19.1464	0.774579
$612$	−9.28014	−0.375127
$613$	−27.0975	−1.09446	−0.547229	−	0.836983i	$-0.684317\pi$
$613$	−27.0975	−1.09446	−0.547229	+	0.836983i	$0.684317\pi$
$614$	−4.99265	−0.201487
$615$	−22.5993	−0.911292
$616$	1.81338	0.0730630
$617$	25.4622	1.02507	0.512534	−	0.858667i	$-0.328707\pi$
$617$	25.4622	1.02507	0.512534	+	0.858667i	$0.328707\pi$
$618$	3.18530	0.128132
$619$	−41.4042	−1.66418	−0.832088	−	0.554644i	$-0.812854\pi$
$619$	−41.4042	−1.66418	−0.832088	+	0.554644i	$0.812854\pi$
$620$	11.8614	0.476364
$621$	−5.45064	−0.218726
$622$	−7.94862	−0.318711
$623$	2.10126	0.0841853
$624$	−12.0952	−0.484194
$625$	1.00000	0.0400000
$626$	5.68775	0.227328
$627$	11.0911	0.442936
$628$	23.7330	0.947049
$629$	−11.8678	−0.473201
$630$	1.80216	0.0717999
$631$	−21.9555	−0.874036	−0.437018	−	0.899453i	$-0.643965\pi$
$631$	−21.9555	−0.874036	−0.437018	+	0.899453i	$0.643965\pi$
$632$	10.8747	0.432572
$633$	6.33833	0.251926
$634$	−3.85878	−0.153252
$635$	5.61817	0.222950
$636$	41.5935	1.64929
$637$	−10.4445	−0.413825
$638$	3.13747	0.124214
$639$	−27.2955	−1.07979
$640$	−11.2432	−0.444426
$641$	−50.3786	−1.98983	−0.994917	−	0.100694i	$-0.967894\pi$
$641$	−50.3786	−1.98983	−0.994917	+	0.100694i	$0.967894\pi$
$642$	9.05249	0.357273
$643$	4.88892	0.192800	0.0964001	−	0.995343i	$-0.469267\pi$
$643$	4.88892	0.192800	0.0964001	+	0.995343i	$0.469267\pi$
$644$	−6.76512	−0.266583
$645$	−23.5263	−0.926347
$646$	3.18605	0.125353
$647$	0.810735	0.0318733	0.0159366	−	0.999873i	$-0.494927\pi$
$647$	0.810735	0.0318733	0.0159366	+	0.999873i	$0.494927\pi$
$648$	−12.5400	−0.492618
$649$	−11.3876	−0.447001
$650$	−0.850641	−0.0333649
$651$	−18.0013	−0.705527
$652$	36.7777	1.44033
$653$	47.4414	1.85653	0.928263	−	0.371925i	$-0.121302\pi$
$653$	47.4414	1.85653	0.928263	+	0.371925i	$0.121302\pi$
$654$	−13.4196	−0.524747
$655$	−1.48009	−0.0578319
$656$	−23.4827	−0.916844
$657$	−31.9277	−1.24562
$658$	−5.44749	−0.212365
$659$	12.3148	0.479717	0.239859	−	0.970808i	$-0.422899\pi$
$659$	12.3148	0.479717	0.239859	+	0.970808i	$0.422899\pi$
$660$	−4.33850	−0.168876
$661$	−1.96257	−0.0763353	−0.0381677	−	0.999271i	$-0.512152\pi$
$661$	−1.96257	−0.0763353	−0.0381677	+	0.999271i	$0.512152\pi$
$662$	2.85396	0.110922
$663$	−6.64778	−0.258178
$664$	−0.0647974	−0.00251463
$665$	4.73105	0.183462
$666$	13.9516	0.540614
$667$	−24.9405	−0.965701
$668$	−5.10558	−0.197541
$669$	39.4948	1.52696
$670$	2.43805	0.0941900
$671$	−0.211497	−0.00816476
$672$	13.1718	0.508115
$673$	37.3451	1.43955	0.719774	−	0.694208i	$-0.244245\pi$
$673$	37.3451	1.43955	0.719774	+	0.694208i	$0.244245\pi$
$674$	9.96501	0.383838
$675$	−1.49106	−0.0573908
$676$	17.4601	0.671541
$677$	−46.1601	−1.77408	−0.887039	−	0.461695i	$-0.847241\pi$
$677$	−46.1601	−1.77408	−0.887039	+	0.461695i	$0.847241\pi$
$678$	−8.82209	−0.338810
$679$	4.36515	0.167519
$680$	−2.65555	−0.101836
$681$	−18.1552	−0.695708
$682$	−3.08396	−0.118091
$683$	27.1238	1.03786	0.518932	−	0.854815i	$-0.326330\pi$
$683$	27.1238	1.03786	0.518932	+	0.854815i	$0.326330\pi$
$684$	28.6399	1.09507
$685$	11.0949	0.423915
$686$	6.49422	0.247951
$687$	19.1283	0.729790
$688$	−24.4459	−0.931992
$689$	16.2133	0.617679
$690$	−4.51024	−0.171702
$691$	−5.77816	−0.219812	−0.109906	−	0.993942i	$-0.535055\pi$
$691$	−5.77816	−0.219812	−0.109906	+	0.993942i	$0.535055\pi$
$692$	−6.81349	−0.259010
$693$	3.58290	0.136103
$694$	−4.88555	−0.185453
$695$	15.1042	0.572933
$696$	31.7241	1.20250
$697$	−12.9066	−0.488873
$698$	9.54761	0.361382
$699$	41.6891	1.57683
$700$	−1.85064	−0.0699476
$701$	−11.4867	−0.433846	−0.216923	−	0.976189i	$-0.569602\pi$
$701$	−11.4867	−0.433846	−0.216923	+	0.976189i	$0.569602\pi$
$702$	1.26835	0.0478709
$703$	36.6259	1.38137
$704$	−2.84107	−0.107077
$705$	27.7707	1.04590
$706$	−7.22289	−0.271837
$707$	5.78198	0.217454
$708$	−54.0384	−2.03089
$709$	29.1067	1.09312	0.546562	−	0.837419i	$-0.315936\pi$
$709$	29.1067	1.09312	0.546562	+	0.837419i	$0.315936\pi$
$710$	−3.66567	−0.137570
$711$	21.4864	0.805804
$712$	3.63995	0.136413
$713$	24.5152	0.918100
$714$	1.89141	0.0707844
$715$	−1.69117	−0.0632461
$716$	−9.90516	−0.370173
$717$	22.8129	0.851962
$718$	−8.30890	−0.310085
$719$	11.0870	0.413477	0.206738	−	0.978396i	$-0.433715\pi$
$719$	11.0870	0.413477	0.206738	+	0.978396i	$0.433715\pi$
$720$	−9.54634	−0.355771
$721$	2.70130	0.100602
$722$	−0.694709	−0.0258544
$723$	35.5363	1.32161
$724$	28.8803	1.07333
$725$	−6.82264	−0.253386
$726$	−12.4438	−0.461834
$727$	18.3052	0.678902	0.339451	−	0.940624i	$-0.389759\pi$
$727$	18.3052	0.678902	0.339451	+	0.940624i	$0.389759\pi$
$728$	3.35434	0.124320
$729$	−36.2522	−1.34267
$730$	−4.28775	−0.158697
$731$	−13.4360	−0.496950
$732$	−1.00364	−0.0370955
$733$	38.9889	1.44009	0.720045	−	0.693928i	$-0.244121\pi$
$733$	38.9889	1.44009	0.720045	+	0.693928i	$0.244121\pi$
$734$	−0.195415	−0.00721289
$735$	−15.1491	−0.558783
$736$	−17.9381	−0.661208
$737$	4.84711	0.178546
$738$	15.1728	0.558519
$739$	−3.63358	−0.133663	−0.0668317	−	0.997764i	$-0.521289\pi$
$739$	−3.63358	−0.133663	−0.0668317	+	0.997764i	$0.521289\pi$
$740$	−14.3269	−0.526668
$741$	20.5161	0.753676
$742$	−4.61299	−0.169348
$743$	−2.60444	−0.0955476	−0.0477738	−	0.998858i	$-0.515213\pi$
$743$	−2.60444	−0.0955476	−0.0477738	+	0.998858i	$0.515213\pi$
$744$	−31.1831	−1.14323
$745$	−13.3667	−0.489716
$746$	11.4166	0.417991
$747$	−0.128028	−0.00468429
$748$	−2.47775	−0.0905955
$749$	7.67699	0.280511
$750$	−1.23381	−0.0450522
$751$	53.9790	1.96972	0.984861	−	0.173347i	$-0.0554581\pi$
$751$	53.9790	1.96972	0.984861	+	0.173347i	$0.0554581\pi$
$752$	28.8562	1.05228
$753$	35.9836	1.31131
$754$	5.80362	0.211355
$755$	1.00000	0.0363937
$756$	2.75941	0.100359
$757$	−1.75196	−0.0636762	−0.0318381	−	0.999493i	$-0.510136\pi$
$757$	−1.75196	−0.0636762	−0.0318381	+	0.999493i	$0.510136\pi$
$758$	0.563020	0.0204498
$759$	−8.96686	−0.325476
$760$	8.19544	0.297280
$761$	48.1562	1.74566	0.872830	−	0.488023i	$-0.162282\pi$
$761$	48.1562	1.74566	0.872830	+	0.488023i	$0.162282\pi$
$762$	−6.93173	−0.251110
$763$	−11.3805	−0.412002
$764$	31.5959	1.14310
$765$	−5.24689	−0.189702
$766$	−3.18749	−0.115169
$767$	−21.0644	−0.760592
$768$	−1.37319	−0.0495506
$769$	33.6508	1.21348	0.606739	−	0.794901i	$-0.292477\pi$
$769$	33.6508	1.21348	0.606739	+	0.794901i	$0.292477\pi$
$770$	0.481168	0.0173401
$771$	−50.9762	−1.83586
$772$	−22.6517	−0.815253
$773$	−8.05893	−0.289860	−0.144930	−	0.989442i	$-0.546296\pi$
$773$	−8.05893	−0.289860	−0.144930	+	0.989442i	$0.546296\pi$
$774$	15.7952	0.567747
$775$	6.70628	0.240897
$776$	7.56161	0.271446
$777$	21.7432	0.780031
$778$	7.20094	0.258166
$779$	39.8318	1.42712
$780$	−8.02525	−0.287350
$781$	−7.28775	−0.260776
$782$	−2.57583	−0.0921115
$783$	10.1729	0.363551
$784$	−15.7413	−0.562188
$785$	13.4184	0.478922
$786$	1.82614	0.0651363
$787$	2.90863	0.103681	0.0518407	−	0.998655i	$-0.483491\pi$
$787$	2.90863	0.103681	0.0518407	+	0.998655i	$0.483491\pi$
$788$	35.4075	1.26134
$789$	10.1891	0.362743
$790$	2.88554	0.102663
$791$	−7.48160	−0.266015
$792$	6.20654	0.220540
$793$	−0.391222	−0.0138927
$794$	0.0444941	0.00157904
$795$	23.5165	0.834044
$796$	−30.7612	−1.09030
$797$	0.451798	0.0160035	0.00800175	−	0.999968i	$-0.497453\pi$
$797$	0.451798	0.0160035	0.00800175	+	0.999968i	$0.497453\pi$
$798$	−5.83719	−0.206634
$799$	15.8600	0.561088
$800$	−4.90709	−0.173492
$801$	7.19187	0.254112
$802$	−13.8953	−0.490659
$803$	−8.52452	−0.300824
$804$	23.0014	0.811198
$805$	−3.82492	−0.134811
$806$	−5.70463	−0.200937
$807$	−55.2460	−1.94475
$808$	10.0159	0.352359
$809$	−1.16162	−0.0408405	−0.0204203	−	0.999791i	$-0.506500\pi$
$809$	−1.16162	−0.0408405	−0.0204203	+	0.999791i	$0.506500\pi$
$810$	−3.32741	−0.116913
$811$	−40.5281	−1.42313	−0.711567	−	0.702618i	$-0.752014\pi$
$811$	−40.5281	−1.42313	−0.711567	+	0.702618i	$0.752014\pi$
$812$	12.6263	0.443095
$813$	−67.4092	−2.36414
$814$	3.72501	0.130562
$815$	20.7937	0.728371
$816$	−10.0191	−0.350739
$817$	41.4656	1.45070
$818$	−8.69852	−0.304137
$819$	6.62755	0.231585
$820$	−15.5810	−0.544111
$821$	−50.2158	−1.75254	−0.876272	−	0.481816i	$-0.839977\pi$
$821$	−50.2158	−1.75254	−0.876272	+	0.481816i	$0.839977\pi$
$822$	−13.6890	−0.477457
$823$	54.3282	1.89376	0.946882	−	0.321583i	$-0.104215\pi$
$823$	54.3282	1.89376	0.946882	+	0.321583i	$0.104215\pi$
$824$	4.67938	0.163014
$825$	−2.45294	−0.0854005
$826$	5.99321	0.208530
$827$	−43.6955	−1.51944	−0.759720	−	0.650250i	$-0.774664\pi$
$827$	−43.6955	−1.51944	−0.759720	+	0.650250i	$0.774664\pi$
$828$	−23.1546	−0.804677
$829$	49.2585	1.71082	0.855410	−	0.517952i	$-0.173305\pi$
$829$	49.2585	1.71082	0.855410	+	0.517952i	$0.173305\pi$
$830$	−0.0171936	−0.000596798 0
$831$	58.9832	2.04610
$832$	−5.25533	−0.182196
$833$	−8.65176	−0.299766
$834$	−18.6356	−0.645298
$835$	−2.88664	−0.0998963
$836$	7.64671	0.264467
$837$	−9.99944	−0.345631
$838$	3.79391	0.131059
$839$	−3.82552	−0.132072	−0.0660358	−	0.997817i	$-0.521035\pi$
$839$	−3.82552	−0.132072	−0.0660358	+	0.997817i	$0.521035\pi$
$840$	4.86527	0.167868
$841$	17.5484	0.605118
$842$	−9.72205	−0.335044
$843$	6.25093	0.215294
$844$	4.36993	0.150419
$845$	9.87172	0.339598
$846$	−18.6448	−0.641022
$847$	−10.5530	−0.362607
$848$	24.4357	0.839126
$849$	−12.5404	−0.430384
$850$	−0.704635	−0.0241688
$851$	−29.6110	−1.01505
$852$	−34.5832	−1.18480
$853$	−31.4864	−1.07807	−0.539037	−	0.842282i	$-0.681212\pi$
$853$	−31.4864	−1.07807	−0.539037	+	0.842282i	$0.681212\pi$
$854$	0.111310	0.00380894
$855$	16.1927	0.553778
$856$	13.2986	0.454537
$857$	0.0162565	0.000555310 0	0.000277655	−	1.00000i	$-0.499912\pi$
$857$	0.0162565	0.000555310 0	0.000277655		1.00000i	$0.499912\pi$
$858$	2.08657	0.0712344
$859$	−55.3186	−1.88745	−0.943723	−	0.330737i	$-0.892703\pi$
$859$	−55.3186	−1.88745	−0.943723	+	0.330737i	$0.892703\pi$
$860$	−16.2201	−0.553100
$861$	23.6464	0.805866
$862$	11.9374	0.406589
$863$	3.69078	0.125636	0.0628178	−	0.998025i	$-0.479991\pi$
$863$	3.69078	0.125636	0.0628178	+	0.998025i	$0.479991\pi$
$864$	7.31674	0.248921
$865$	−3.85227	−0.130981
$866$	18.7504	0.637165
$867$	38.1049	1.29411
$868$	−12.4109	−0.421254
$869$	5.73677	0.194606
$870$	8.41781	0.285390
$871$	8.96606	0.303803
$872$	−19.7141	−0.667604
$873$	14.9404	0.505654
$874$	7.94940	0.268893
$875$	−1.04633	−0.0353725
$876$	−40.4522	−1.36675
$877$	1.76591	0.0596306	0.0298153	−	0.999555i	$-0.490508\pi$
$877$	1.76591	0.0596306	0.0298153	+	0.999555i	$0.490508\pi$
$878$	3.91559	0.132145
$879$	−46.5328	−1.56951
$880$	−2.54882	−0.0859208
$881$	28.3124	0.953868	0.476934	−	0.878939i	$-0.341748\pi$
$881$	28.3124	0.953868	0.476934	+	0.878939i	$0.341748\pi$
$882$	10.1709	0.342471
$883$	−11.9216	−0.401193	−0.200596	−	0.979674i	$-0.564288\pi$
$883$	−11.9216	−0.401193	−0.200596	+	0.979674i	$0.564288\pi$
$884$	−4.58328	−0.154152
$885$	−30.5527	−1.02702
$886$	8.16188	0.274204
$887$	−48.6486	−1.63346	−0.816730	−	0.577020i	$-0.804215\pi$
$887$	−48.6486	−1.63346	−0.816730	+	0.577020i	$0.804215\pi$
$888$	37.6650	1.26395
$889$	−5.87847	−0.197158
$890$	0.965838	0.0323750
$891$	−6.61527	−0.221620
$892$	27.2295	0.911711
$893$	−48.9465	−1.63793
$894$	16.4919	0.551570
$895$	−5.60027	−0.187196
$896$	11.7641	0.393012
$897$	−16.5867	−0.553813
$898$	−8.67781	−0.289582
$899$	−45.7545	−1.52600
$900$	−6.33408	−0.211136
$901$	13.4304	0.447433
$902$	4.05107	0.134886
$903$	24.6163	0.819180
$904$	−12.9601	−0.431047
$905$	16.3286	0.542780
$906$	−1.23381	−0.0409904
$907$	−28.8087	−0.956576	−0.478288	−	0.878203i	$-0.658742\pi$
$907$	−28.8087	−0.956576	−0.478288	+	0.878203i	$0.658742\pi$
$908$	−12.5170	−0.415391
$909$	19.7897	0.656381
$910$	0.890052	0.0295049
$911$	−27.1064	−0.898075	−0.449037	−	0.893513i	$-0.648233\pi$
$911$	−27.1064	−0.898075	−0.449037	+	0.893513i	$0.648233\pi$
$912$	30.9205	1.02388
$913$	−0.0341828	−0.00113128
$914$	3.39333	0.112241
$915$	−0.567445	−0.0187592
$916$	13.1879	0.435741
$917$	1.54866	0.0511414
$918$	1.05065	0.0346766
$919$	−7.39369	−0.243895	−0.121948	−	0.992537i	$-0.538914\pi$
$919$	−7.39369	−0.243895	−0.121948	+	0.992537i	$0.538914\pi$
$920$	−6.62579	−0.218446
$921$	−26.6312	−0.877529
$922$	3.79767	0.125070
$923$	−13.4807	−0.443722
$924$	4.53951	0.149339
$925$	−8.10028	−0.266335
$926$	−9.83717	−0.323269
$927$	9.24560	0.303665
$928$	33.4793	1.09901
$929$	−54.7297	−1.79562	−0.897811	−	0.440381i	$-0.854843\pi$
$929$	−54.7297	−1.79562	−0.897811	+	0.440381i	$0.854843\pi$
$930$	−8.27424	−0.271323
$931$	26.7007	0.875078
$932$	28.7423	0.941486
$933$	−42.3986	−1.38807
$934$	17.6091	0.576187
$935$	−1.40089	−0.0458141
$936$	11.4807	0.375258
$937$	25.5089	0.833339	0.416670	−	0.909058i	$-0.363197\pi$
$937$	25.5089	0.833339	0.416670	+	0.909058i	$0.363197\pi$
$938$	−2.55101	−0.0832933
$939$	30.3390	0.990074
$940$	19.1464	0.624485
$941$	10.9381	0.356571	0.178285	−	0.983979i	$-0.442945\pi$
$941$	10.9381	0.356571	0.178285	+	0.983979i	$0.442945\pi$
$942$	−16.5557	−0.539412
$943$	−32.2029	−1.04867
$944$	−31.7470	−1.03328
$945$	1.56014	0.0507513
$946$	4.21724	0.137114
$947$	−37.7577	−1.22696	−0.613481	−	0.789710i	$-0.710231\pi$
$947$	−37.7577	−1.22696	−0.613481	+	0.789710i	$0.710231\pi$
$948$	27.2232	0.884168
$949$	−15.7684	−0.511865
$950$	2.17461	0.0705537
$951$	−20.5831	−0.667452
$952$	2.77859	0.0900546
$953$	−31.3497	−1.01552	−0.507758	−	0.861500i	$-0.669525\pi$
$953$	−31.3497	−1.01552	−0.507758	+	0.861500i	$0.669525\pi$
$954$	−15.7886	−0.511175
$955$	17.8639	0.578064
$956$	15.7282	0.508686
$957$	16.7355	0.540983
$958$	−13.2283	−0.427386
$959$	−11.6090	−0.374873
$960$	−7.62255	−0.246017
$961$	13.9742	0.450780
$962$	6.89043	0.222156
$963$	26.2756	0.846719
$964$	24.5003	0.789101
$965$	−12.8070	−0.412273
$966$	4.71921	0.151838
$967$	8.53921	0.274603	0.137301	−	0.990529i	$-0.456157\pi$
$967$	8.53921	0.274603	0.137301	+	0.990529i	$0.456157\pi$
$968$	−18.2807	−0.587563
$969$	16.9946	0.545946
$970$	2.00643	0.0644225
$971$	16.1186	0.517271	0.258636	−	0.965975i	$-0.416727\pi$
$971$	16.1186	0.517271	0.258636	+	0.965975i	$0.416727\pi$
$972$	−39.3037	−1.26067
$973$	−15.8040	−0.506652
$974$	19.2243	0.615988
$975$	−4.53739	−0.145313
$976$	−0.589626	−0.0188735
$977$	−52.7230	−1.68676	−0.843378	−	0.537320i	$-0.819437\pi$
$977$	−52.7230	−1.68676	−0.843378	+	0.537320i	$0.819437\pi$
$978$	−25.6554	−0.820368
$979$	1.92019	0.0613696
$980$	−10.4445	−0.333636
$981$	−38.9514	−1.24362
$982$	−17.1433	−0.547065
$983$	41.3603	1.31919	0.659594	−	0.751622i	$-0.270728\pi$
$983$	41.3603	1.31919	0.659594	+	0.751622i	$0.270728\pi$
$984$	40.9618	1.30582
$985$	20.0190	0.637859
$986$	4.80747	0.153101
$987$	−29.0574	−0.924906
$988$	14.1447	0.450002
$989$	−33.5238	−1.06600
$990$	1.64687	0.0523409
$991$	−34.7877	−1.10507	−0.552534	−	0.833491i	$-0.686339\pi$
$991$	−34.7877	−1.10507	−0.552534	+	0.833491i	$0.686339\pi$
$992$	−32.9083	−1.04484
$993$	15.2233	0.483096
$994$	3.83550	0.121655
$995$	−17.3920	−0.551365
$996$	−0.162211	−0.00513984
$997$	−1.47976	−0.0468643	−0.0234322	−	0.999725i	$-0.507459\pi$
$997$	−1.47976	−0.0468643	−0.0234322	+	0.999725i	$0.507459\pi$
$998$	−2.86284	−0.0906215
$999$	12.0780	0.382130

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	755.2.a.i.1.3	✓	5
3.2	odd	2		6795.2.a.y.1.3		5
5.4	even	2		3775.2.a.n.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
755.2.a.i.1.3	✓	5	1.1	even	1	trivial
3775.2.a.n.1.3		5	5.4	even	2
6795.2.a.y.1.3		5	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)

\(f(q)\)	\(=\)	\(q-0.480943 q^{2} -2.56539 q^{3} -1.76869 q^{4} -1.00000 q^{5} +1.23381 q^{6} +1.04633 q^{7} +1.81253 q^{8} +3.58122 q^{9} +0.480943 q^{10} +0.956168 q^{11} +4.53739 q^{12} +1.76869 q^{13} -0.503226 q^{14} +2.56539 q^{15} +2.66567 q^{16} +1.46511 q^{17} -1.72236 q^{18} -4.52156 q^{19} +1.76869 q^{20} -2.68425 q^{21} -0.459862 q^{22} +3.65555 q^{23} -4.64984 q^{24} +1.00000 q^{25} -0.850641 q^{26} -1.49106 q^{27} -1.85064 q^{28} -6.82264 q^{29} -1.23381 q^{30} +6.70628 q^{31} -4.90709 q^{32} -2.45294 q^{33} -0.704635 q^{34} -1.04633 q^{35} -6.33408 q^{36} -8.10028 q^{37} +2.17461 q^{38} -4.53739 q^{39} -1.81253 q^{40} -8.80931 q^{41} +1.29097 q^{42} -9.17066 q^{43} -1.69117 q^{44} -3.58122 q^{45} -1.75811 q^{46} +10.8251 q^{47} -6.83847 q^{48} -5.90519 q^{49} -0.480943 q^{50} -3.75858 q^{51} -3.12828 q^{52} +9.16684 q^{53} +0.717113 q^{54} -0.956168 q^{55} +1.89650 q^{56} +11.5996 q^{57} +3.28130 q^{58} -11.9096 q^{59} -4.53739 q^{60} -0.221193 q^{61} -3.22534 q^{62} +3.74714 q^{63} -2.97131 q^{64} -1.76869 q^{65} +1.17972 q^{66} +5.06931 q^{67} -2.59133 q^{68} -9.37792 q^{69} +0.503226 q^{70} -7.62183 q^{71} +6.49106 q^{72} -8.91530 q^{73} +3.89577 q^{74} -2.56539 q^{75} +7.99725 q^{76} +1.00047 q^{77} +2.18222 q^{78} +5.99975 q^{79} -2.66567 q^{80} -6.91852 q^{81} +4.23677 q^{82} -0.0357498 q^{83} +4.74761 q^{84} -1.46511 q^{85} +4.41056 q^{86} +17.5027 q^{87} +1.73308 q^{88} +2.00822 q^{89} +1.72236 q^{90} +1.85064 q^{91} -6.46555 q^{92} -17.2042 q^{93} -5.20627 q^{94} +4.52156 q^{95} +12.5886 q^{96} +4.17186 q^{97} +2.84006 q^{98} +3.42425 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} + 4 q^{4} - 5 q^{5} - 3 q^{6} - 7 q^{7} + 6 q^{8} + 2 q^{9} - 5 q^{11} - 3 q^{12} - 4 q^{13} - 11 q^{14} + 3 q^{15} - 10 q^{16} + 11 q^{17} - 8 q^{18} - 3 q^{19} - 4 q^{20} - 8 q^{21} - 4 q^{22}+ \cdots - 8 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 + 4 * q^4 - 5 * q^5 - 3 * q^6 - 7 * q^7 + 6 * q^8 + 2 * q^9 - 5 * q^11 - 3 * q^12 - 4 * q^13 - 11 * q^14 + 3 * q^15 - 10 * q^16 + 11 * q^17 - 8 * q^18 - 3 * q^19 - 4 * q^20 - 8 * q^21 - 4 * q^22 + 6 * q^23 - 6 * q^24 + 5 * q^25 - 6 * q^26 + 6 * q^27 - 11 * q^28 - 20 * q^29 + 3 * q^30 - 7 * q^31 - 3 * q^32 + 6 * q^33 - 3 * q^34 + 7 * q^35 - 9 * q^36 - 27 * q^37 + q^38 + 3 * q^39 - 6 * q^40 - 14 * q^41 + 23 * q^42 + 3 * q^43 - 23 * q^44 - 2 * q^45 - 4 * q^46 + 18 * q^47 - 14 * q^48 - 2 * q^49 - 17 * q^51 - 18 * q^52 - 2 * q^53 + q^54 + 5 * q^55 - 4 * q^56 + 14 * q^57 - 11 * q^58 - 16 * q^59 + 3 * q^60 - 25 * q^61 + 22 * q^62 + 7 * q^63 - 8 * q^64 + 4 * q^65 + 29 * q^66 + 15 * q^67 + 14 * q^68 - 34 * q^69 + 11 * q^70 - 5 * q^71 + 19 * q^72 - 6 * q^73 - 6 * q^74 - 3 * q^75 + 16 * q^76 + 8 * q^77 + 12 * q^78 - 16 * q^79 + 10 * q^80 - 23 * q^81 - 7 * q^82 + 4 * q^83 + 15 * q^84 - 11 * q^85 + 55 * q^86 + 4 * q^87 - 31 * q^88 - 7 * q^89 + 8 * q^90 + 11 * q^91 + 19 * q^92 - 31 * q^93 - 4 * q^94 + 3 * q^95 + 12 * q^96 - 13 * q^97 + 24 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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