LMFDB - Embedded newform 637.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(1,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("637.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 637 = 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	637.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$5.08647060876$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.4507648.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.20475$ of defining polynomial
Character	$\chi$	$=$	637.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.656184 q^{2} +0.204753 q^{3} -1.56942 q^{4} +1.35996 q^{5} -0.134356 q^{6} +2.34220 q^{8} -2.95808 q^{9} +O(q^{10})$	\(q-0.656184 q^{2} +0.204753 q^{3} -1.56942 q^{4} +1.35996 q^{5} -0.134356 q^{6} +2.34220 q^{8} -2.95808 q^{9} -0.892385 q^{10} -1.90551 q^{11} -0.321345 q^{12} -1.00000 q^{13} +0.278457 q^{15} +1.60193 q^{16} -3.56633 q^{17} +1.94104 q^{18} -0.985255 q^{19} -2.13436 q^{20} +1.25037 q^{22} +1.69605 q^{23} +0.479573 q^{24} -3.15050 q^{25} +0.656184 q^{26} -1.21994 q^{27} +6.54835 q^{29} -0.182719 q^{30} -7.69550 q^{31} -5.73556 q^{32} -0.390160 q^{33} +2.34017 q^{34} +4.64247 q^{36} -2.02005 q^{37} +0.646508 q^{38} -0.204753 q^{39} +3.18530 q^{40} -9.88779 q^{41} -3.16639 q^{43} +2.99055 q^{44} -4.02287 q^{45} -1.11292 q^{46} -7.76416 q^{47} +0.328001 q^{48} +2.06731 q^{50} -0.730219 q^{51} +1.56942 q^{52} +0.354194 q^{53} +0.800502 q^{54} -2.59143 q^{55} -0.201734 q^{57} -4.29692 q^{58} -2.16385 q^{59} -0.437016 q^{60} -12.2002 q^{61} +5.04967 q^{62} +0.559715 q^{64} -1.35996 q^{65} +0.256017 q^{66} +11.3134 q^{67} +5.59709 q^{68} +0.347272 q^{69} -9.05268 q^{71} -6.92840 q^{72} +7.13619 q^{73} +1.32553 q^{74} -0.645076 q^{75} +1.54628 q^{76} +0.134356 q^{78} -5.39629 q^{79} +2.17857 q^{80} +8.62444 q^{81} +6.48821 q^{82} +2.03494 q^{83} -4.85008 q^{85} +2.07773 q^{86} +1.34080 q^{87} -4.46309 q^{88} +6.89864 q^{89} +2.63974 q^{90} -2.66182 q^{92} -1.57568 q^{93} +5.09472 q^{94} -1.33991 q^{95} -1.17437 q^{96} +14.6223 q^{97} +5.63665 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{3} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 6 q^{9}+O(q^{10}) $ 6 * q - 8 * q^3 + 4 * q^4 - 6 * q^5 - 4 * q^6 + 6 * q^9	$ 6 q - 8 q^{3} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 6 q^{9} - 4 q^{10} + 4 q^{11} + 4 q^{12} - 6 q^{13} + 12 q^{15} - 16 q^{17} - 4 q^{18} - 2 q^{19} - 16 q^{20} - 12 q^{22} - 6 q^{23} - 12 q^{24} - 4 q^{25} - 20 q^{27} - 6 q^{29} - 6 q^{31} - 20 q^{32} - 4 q^{33} - 24 q^{36} - 8 q^{38} + 8 q^{39} - 4 q^{40} + 8 q^{41} + 2 q^{43} - 4 q^{44} - 14 q^{45} + 8 q^{46} - 30 q^{47} + 8 q^{48} + 8 q^{50} - 4 q^{51} - 4 q^{52} - 14 q^{53} + 48 q^{54} + 8 q^{55} + 4 q^{57} - 8 q^{58} - 24 q^{59} + 12 q^{60} - 28 q^{62} - 20 q^{64} + 6 q^{65} + 4 q^{66} + 16 q^{67} - 28 q^{68} + 20 q^{69} + 8 q^{71} + 28 q^{72} + 6 q^{73} - 12 q^{74} - 12 q^{75} + 16 q^{76} + 4 q^{78} - 22 q^{79} + 28 q^{80} + 46 q^{81} + 40 q^{82} - 50 q^{83} - 8 q^{85} - 16 q^{86} + 16 q^{87} - 44 q^{88} - 26 q^{89} + 40 q^{90} + 20 q^{92} + 16 q^{93} + 32 q^{94} - 6 q^{95} + 20 q^{96} + 14 q^{97} + 12 q^{99}+O(q^{100}) $ 6 * q - 8 * q^3 + 4 * q^4 - 6 * q^5 - 4 * q^6 + 6 * q^9 - 4 * q^10 + 4 * q^11 + 4 * q^12 - 6 * q^13 + 12 * q^15 - 16 * q^17 - 4 * q^18 - 2 * q^19 - 16 * q^20 - 12 * q^22 - 6 * q^23 - 12 * q^24 - 4 * q^25 - 20 * q^27 - 6 * q^29 - 6 * q^31 - 20 * q^32 - 4 * q^33 - 24 * q^36 - 8 * q^38 + 8 * q^39 - 4 * q^40 + 8 * q^41 + 2 * q^43 - 4 * q^44 - 14 * q^45 + 8 * q^46 - 30 * q^47 + 8 * q^48 + 8 * q^50 - 4 * q^51 - 4 * q^52 - 14 * q^53 + 48 * q^54 + 8 * q^55 + 4 * q^57 - 8 * q^58 - 24 * q^59 + 12 * q^60 - 28 * q^62 - 20 * q^64 + 6 * q^65 + 4 * q^66 + 16 * q^67 - 28 * q^68 + 20 * q^69 + 8 * q^71 + 28 * q^72 + 6 * q^73 - 12 * q^74 - 12 * q^75 + 16 * q^76 + 4 * q^78 - 22 * q^79 + 28 * q^80 + 46 * q^81 + 40 * q^82 - 50 * q^83 - 8 * q^85 - 16 * q^86 + 16 * q^87 - 44 * q^88 - 26 * q^89 + 40 * q^90 + 20 * q^92 + 16 * q^93 + 32 * q^94 - 6 * q^95 + 20 * q^96 + 14 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−0.656184	−0.463992	−0.231996	−	0.972717i	$-0.574526\pi$
$2$	−0.656184	−0.463992	−0.231996	+	0.972717i	$0.574526\pi$
$3$	0.204753	0.118214	0.0591072	−	0.998252i	$-0.481175\pi$
$3$	0.204753	0.118214	0.0591072	+	0.998252i	$0.481175\pi$
$4$	−1.56942	−0.784711
$5$	1.35996	0.608194	0.304097	−	0.952641i	$-0.401645\pi$
$5$	1.35996	0.608194	0.304097	+	0.952641i	$0.401645\pi$
$6$	−0.134356	−0.0548505
$7$	0	0
$7$	0	0
$8$	2.34220	0.828092
$9$	−2.95808	−0.986025
$10$	−0.892385	−0.282197
$11$	−1.90551	−0.574534	−0.287267	−	0.957851i	$-0.592747\pi$
$11$	−1.90551	−0.574534	−0.287267	+	0.957851i	$0.592747\pi$
$12$	−0.321345	−0.0927642
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0.278457	0.0718972
$16$	1.60193	0.400483
$17$	−3.56633	−0.864963	−0.432482	−	0.901643i	$-0.642362\pi$
$17$	−3.56633	−0.864963	−0.432482	+	0.901643i	$0.642362\pi$
$18$	1.94104	0.457508
$19$	−0.985255	−0.226033	−0.113016	−	0.993593i	$-0.536051\pi$
$19$	−0.985255	−0.226033	−0.113016	+	0.993593i	$0.536051\pi$
$20$	−2.13436	−0.477256
$21$	0	0
$22$	1.25037	0.266579
$23$	1.69605	0.353651	0.176826	−	0.984242i	$-0.443417\pi$
$23$	1.69605	0.353651	0.176826	+	0.984242i	$0.443417\pi$
$24$	0.479573	0.0978924
$25$	−3.15050	−0.630100
$26$	0.656184	0.128688
$27$	−1.21994	−0.234777
$28$	0	0
$29$	6.54835	1.21600	0.607999	−	0.793938i	$-0.291972\pi$
$29$	6.54835	1.21600	0.607999	+	0.793938i	$0.291972\pi$
$30$	−0.182719	−0.0333598
$31$	−7.69550	−1.38215	−0.691077	−	0.722781i	$-0.742863\pi$
$31$	−7.69550	−1.38215	−0.691077	+	0.722781i	$0.742863\pi$
$32$	−5.73556	−1.01391
$33$	−0.390160	−0.0679181
$34$	2.34017	0.401336
$35$	0	0
$36$	4.64247	0.773745
$37$	−2.02005	−0.332095	−0.166047	−	0.986118i	$-0.553100\pi$
$37$	−2.02005	−0.332095	−0.166047	+	0.986118i	$0.553100\pi$
$38$	0.646508	0.104877
$39$	−0.204753	−0.0327868
$40$	3.18530	0.503640
$41$	−9.88779	−1.54421	−0.772107	−	0.635493i	$-0.780797\pi$
$41$	−9.88779	−1.54421	−0.772107	+	0.635493i	$0.780797\pi$
$42$	0	0
$43$	−3.16639	−0.482869	−0.241435	−	0.970417i	$-0.577618\pi$
$43$	−3.16639	−0.482869	−0.241435	+	0.970417i	$0.577618\pi$
$44$	2.99055	0.450843
$45$	−4.02287	−0.599694
$46$	−1.11292	−0.164091
$47$	−7.76416	−1.13252	−0.566260	−	0.824227i	$-0.691610\pi$
$47$	−7.76416	−1.13252	−0.566260	+	0.824227i	$0.691610\pi$
$48$	0.328001	0.0473429
$49$	0	0
$50$	2.06731	0.292362
$51$	−0.730219	−0.102251
$52$	1.56942	0.217640
$53$	0.354194	0.0486522	0.0243261	−	0.999704i	$-0.492256\pi$
$53$	0.354194	0.0486522	0.0243261	+	0.999704i	$0.492256\pi$
$54$	0.800502	0.108935
$55$	−2.59143	−0.349428
$56$	0	0
$57$	−0.201734	−0.0267203
$58$	−4.29692	−0.564214
$59$	−2.16385	−0.281709	−0.140854	−	0.990030i	$-0.544985\pi$
$59$	−2.16385	−0.281709	−0.140854	+	0.990030i	$0.544985\pi$
$60$	−0.437016	−0.0564186
$61$	−12.2002	−1.56207	−0.781037	−	0.624484i	$-0.785309\pi$
$61$	−12.2002	−1.56207	−0.781037	+	0.624484i	$0.785309\pi$
$62$	5.04967	0.641308
$63$	0	0
$64$	0.559715	0.0699644
$65$	−1.35996	−0.168683
$66$	0.256017	0.0315135
$67$	11.3134	1.38215	0.691076	−	0.722782i	$-0.257137\pi$
$67$	11.3134	1.38215	0.691076	+	0.722782i	$0.257137\pi$
$68$	5.59709	0.678746
$69$	0.347272	0.0418067
$70$	0	0
$71$	−9.05268	−1.07436	−0.537178	−	0.843469i	$-0.680510\pi$
$71$	−9.05268	−1.07436	−0.537178	+	0.843469i	$0.680510\pi$
$72$	−6.92840	−0.816520
$73$	7.13619	0.835228	0.417614	−	0.908625i	$-0.362866\pi$
$73$	7.13619	0.835228	0.417614	+	0.908625i	$0.362866\pi$
$74$	1.32553	0.154089
$75$	−0.645076	−0.0744869
$76$	1.54628	0.177371
$77$	0	0
$78$	0.134356	0.0152128
$79$	−5.39629	−0.607130	−0.303565	−	0.952811i	$-0.598177\pi$
$79$	−5.39629	−0.607130	−0.303565	+	0.952811i	$0.598177\pi$
$80$	2.17857	0.243571
$81$	8.62444	0.958271
$82$	6.48821	0.716503
$83$	2.03494	0.223364	0.111682	−	0.993744i	$-0.464376\pi$
$83$	2.03494	0.223364	0.111682	+	0.993744i	$0.464376\pi$
$84$	0	0
$85$	−4.85008	−0.526065
$86$	2.07773	0.224048
$87$	1.34080	0.143749
$88$	−4.46309	−0.475767
$89$	6.89864	0.731254	0.365627	−	0.930761i	$-0.380855\pi$
$89$	6.89864	0.731254	0.365627	+	0.930761i	$0.380855\pi$
$90$	2.63974	0.278253
$91$	0	0
$92$	−2.66182	−0.277514
$93$	−1.57568	−0.163390
$94$	5.09472	0.525480
$95$	−1.33991	−0.137472
$96$	−1.17437	−0.119859
$97$	14.6223	1.48467	0.742334	−	0.670030i	$-0.233719\pi$
$97$	14.6223	1.48467	0.742334	+	0.670030i	$0.233719\pi$
$98$	0	0
$99$	5.63665	0.566505
$100$	4.94447	0.494447
$101$	−2.81368	−0.279972	−0.139986	−	0.990153i	$-0.544706\pi$
$101$	−2.81368	−0.279972	−0.139986	+	0.990153i	$0.544706\pi$
$102$	0.479158	0.0474437
$103$	−6.04988	−0.596112	−0.298056	−	0.954548i	$-0.596338\pi$
$103$	−6.04988	−0.596112	−0.298056	+	0.954548i	$0.596338\pi$
$104$	−2.34220	−0.229671
$105$	0	0
$106$	−0.232416	−0.0225743
$107$	17.3204	1.67442	0.837211	−	0.546880i	$-0.184185\pi$
$107$	17.3204	1.67442	0.837211	+	0.546880i	$0.184185\pi$
$108$	1.91460	0.184232
$109$	10.5762	1.01302	0.506509	−	0.862234i	$-0.330936\pi$
$109$	10.5762	1.01302	0.506509	+	0.862234i	$0.330936\pi$
$110$	1.70045	0.162132
$111$	−0.413613	−0.0392584
$112$	0	0
$113$	−2.34665	−0.220755	−0.110377	−	0.993890i	$-0.535206\pi$
$113$	−2.34665	−0.220755	−0.110377	+	0.993890i	$0.535206\pi$
$114$	0.132375	0.0123980
$115$	2.30657	0.215089
$116$	−10.2771	−0.954208
$117$	2.95808	0.273474
$118$	1.41988	0.130711
$119$	0	0
$120$	0.652201	0.0595375
$121$	−7.36902	−0.669911
$122$	8.00557	0.724790
$123$	−2.02456	−0.182548
$124$	12.0775	1.08459
$125$	−11.0844	−0.991417
$126$	0	0
$127$	−18.1639	−1.61179	−0.805894	−	0.592060i	$-0.798315\pi$
$127$	−18.1639	−1.61179	−0.805894	+	0.592060i	$0.798315\pi$
$128$	11.1038	0.981450
$129$	−0.648328	−0.0570821
$130$	0.892385	0.0782674
$131$	−10.1014	−0.882560	−0.441280	−	0.897369i	$-0.645476\pi$
$131$	−10.1014	−0.882560	−0.441280	+	0.897369i	$0.645476\pi$
$132$	0.612326	0.0532961
$133$	0	0
$134$	−7.42367	−0.641308
$135$	−1.65907	−0.142790
$136$	−8.35306	−0.716269
$137$	6.55601	0.560118	0.280059	−	0.959983i	$-0.409646\pi$
$137$	6.55601	0.560118	0.280059	+	0.959983i	$0.409646\pi$
$138$	−0.227875	−0.0193980
$139$	13.8243	1.17256	0.586281	−	0.810108i	$-0.300591\pi$
$139$	13.8243	1.17256	0.586281	+	0.810108i	$0.300591\pi$
$140$	0	0
$141$	−1.58974	−0.133880
$142$	5.94023	0.498493
$143$	1.90551	0.159347
$144$	−4.73864	−0.394887
$145$	8.90551	0.739563
$146$	−4.68265	−0.387539
$147$	0	0
$148$	3.17032	0.260598
$149$	8.09389	0.663078	0.331539	−	0.943442i	$-0.392432\pi$
$149$	8.09389	0.663078	0.331539	+	0.943442i	$0.392432\pi$
$150$	0.423288	0.0345614
$151$	12.0365	0.979520	0.489760	−	0.871857i	$-0.337084\pi$
$151$	12.0365	0.979520	0.489760	+	0.871857i	$0.337084\pi$
$152$	−2.30766	−0.187176
$153$	10.5495	0.852876
$154$	0	0
$155$	−10.4656	−0.840617
$156$	0.321345	0.0257282
$157$	18.3935	1.46796	0.733982	−	0.679169i	$-0.237660\pi$
$157$	18.3935	1.46796	0.733982	+	0.679169i	$0.237660\pi$
$158$	3.54096	0.281704
$159$	0.0725223	0.00575139
$160$	−7.80014	−0.616655
$161$	0	0
$162$	−5.65922	−0.444630
$163$	2.51286	0.196823	0.0984114	−	0.995146i	$-0.468624\pi$
$163$	2.51286	0.196823	0.0984114	+	0.995146i	$0.468624\pi$
$164$	15.5181	1.21176
$165$	−0.530603	−0.0413074
$166$	−1.33530	−0.103639
$167$	−17.9095	−1.38588	−0.692941	−	0.720995i	$-0.743685\pi$
$167$	−17.9095	−1.38588	−0.692941	+	0.720995i	$0.743685\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	3.18254	0.244090
$171$	2.91446	0.222874
$172$	4.96940	0.378913
$173$	−6.51981	−0.495692	−0.247846	−	0.968799i	$-0.579723\pi$
$173$	−6.51981	−0.495692	−0.247846	+	0.968799i	$0.579723\pi$
$174$	−0.879809	−0.0666982
$175$	0	0
$176$	−3.05250	−0.230091
$177$	−0.443055	−0.0333020
$178$	−4.52678	−0.339296
$179$	−26.6014	−1.98828	−0.994141	−	0.108091i	$-0.965526\pi$
$179$	−26.6014	−1.98828	−0.994141	+	0.108091i	$0.965526\pi$
$180$	6.31359	0.470587
$181$	24.6402	1.83149	0.915746	−	0.401758i	$-0.131601\pi$
$181$	24.6402	1.83149	0.915746	+	0.401758i	$0.131601\pi$
$182$	0	0
$183$	−2.49803	−0.184660
$184$	3.97249	0.292856
$185$	−2.74720	−0.201978
$186$	1.03394	0.0758119
$187$	6.79569	0.496950
$188$	12.1853	0.888701
$189$	0	0
$190$	0.879227	0.0637858
$191$	3.02099	0.218591	0.109295	−	0.994009i	$-0.465141\pi$
$191$	3.02099	0.218591	0.109295	+	0.994009i	$0.465141\pi$
$192$	0.114604	0.00827080
$193$	21.2522	1.52977	0.764884	−	0.644168i	$-0.222796\pi$
$193$	21.2522	1.52977	0.764884	+	0.644168i	$0.222796\pi$
$194$	−9.59491	−0.688874
$195$	−0.278457	−0.0199407
$196$	0	0
$197$	−2.72191	−0.193928	−0.0969639	−	0.995288i	$-0.530913\pi$
$197$	−2.72191	−0.193928	−0.0969639	+	0.995288i	$0.530913\pi$
$198$	−3.69868	−0.262854
$199$	21.6670	1.53593	0.767967	−	0.640490i	$-0.221269\pi$
$199$	21.6670	1.53593	0.767967	+	0.640490i	$0.221269\pi$
$200$	−7.37910	−0.521781
$201$	2.31646	0.163390
$202$	1.84629	0.129905
$203$	0	0
$204$	1.14602	0.0802376
$205$	−13.4470	−0.939181
$206$	3.96983	0.276591
$207$	−5.01705	−0.348709
$208$	−1.60193	−0.111074
$209$	1.87741	0.129864
$210$	0	0
$211$	18.3885	1.26592	0.632959	−	0.774186i	$-0.281840\pi$
$211$	18.3885	1.26592	0.632959	+	0.774186i	$0.281840\pi$
$212$	−0.555879	−0.0381780
$213$	−1.85357	−0.127004
$214$	−11.3653	−0.776918
$215$	−4.30617	−0.293678
$216$	−2.85733	−0.194417
$217$	0	0
$218$	−6.93995	−0.470033
$219$	1.46116	0.0987359
$220$	4.06704	0.274200
$221$	3.56633	0.239898
$222$	0.271406	0.0182156
$223$	−22.2216	−1.48807	−0.744035	−	0.668141i	$-0.767090\pi$
$223$	−22.2216	−1.48807	−0.744035	+	0.668141i	$0.767090\pi$
$224$	0	0
$225$	9.31943	0.621295
$226$	1.53984	0.102428
$227$	−26.7374	−1.77463	−0.887313	−	0.461167i	$-0.847431\pi$
$227$	−26.7374	−1.77463	−0.887313	+	0.461167i	$0.847431\pi$
$228$	0.316606	0.0209678
$229$	16.7008	1.10362	0.551809	−	0.833970i	$-0.313938\pi$
$229$	16.7008	1.10362	0.551809	+	0.833970i	$0.313938\pi$
$230$	−1.51353	−0.0997994
$231$	0	0
$232$	15.3375	1.00696
$233$	−13.8776	−0.909149	−0.454575	−	0.890709i	$-0.650209\pi$
$233$	−13.8776	−0.909149	−0.454575	+	0.890709i	$0.650209\pi$
$234$	−1.94104	−0.126890
$235$	−10.5590	−0.688791
$236$	3.39599	0.221060
$237$	−1.10491	−0.0717715
$238$	0	0
$239$	0.465845	0.0301330	0.0150665	−	0.999886i	$-0.495204\pi$
$239$	0.465845	0.0301330	0.0150665	+	0.999886i	$0.495204\pi$
$240$	0.446069	0.0287936
$241$	−5.69314	−0.366727	−0.183364	−	0.983045i	$-0.558699\pi$
$241$	−5.69314	−0.366727	−0.183364	+	0.983045i	$0.558699\pi$
$242$	4.83543	0.310833
$243$	5.42569	0.348058
$244$	19.1473	1.22578
$245$	0	0
$246$	1.32848	0.0847010
$247$	0.985255	0.0626902
$248$	−18.0244	−1.14455
$249$	0.416662	0.0264049
$250$	7.27339	0.460010
$251$	−17.9066	−1.13025	−0.565126	−	0.825005i	$-0.691172\pi$
$251$	−17.9066	−1.13025	−0.565126	+	0.825005i	$0.691172\pi$
$252$	0	0
$253$	−3.23185	−0.203185
$254$	11.9189	0.747857
$255$	−0.993070	−0.0621885
$256$	−8.40559	−0.525350
$257$	−25.9756	−1.62031	−0.810156	−	0.586214i	$-0.800618\pi$
$257$	−25.9756	−1.62031	−0.810156	+	0.586214i	$0.800618\pi$
$258$	0.425423	0.0264857
$259$	0	0
$260$	2.13436	0.132367
$261$	−19.3705	−1.19901
$262$	6.62835	0.409501
$263$	27.8402	1.71670	0.858349	−	0.513067i	$-0.171491\pi$
$263$	27.8402	1.71670	0.858349	+	0.513067i	$0.171491\pi$
$264$	−0.913832	−0.0562425
$265$	0.481690	0.0295900
$266$	0	0
$267$	1.41252	0.0864448
$268$	−17.7555	−1.08459
$269$	−5.90486	−0.360025	−0.180013	−	0.983664i	$-0.557614\pi$
$269$	−5.90486	−0.360025	−0.180013	+	0.983664i	$0.557614\pi$
$270$	1.08865	0.0662533
$271$	4.34932	0.264203	0.132101	−	0.991236i	$-0.457828\pi$
$271$	4.34932	0.264203	0.132101	+	0.991236i	$0.457828\pi$
$272$	−5.71303	−0.346403
$273$	0	0
$274$	−4.30195	−0.259890
$275$	6.00332	0.362014
$276$	−0.545017	−0.0328062
$277$	1.08051	0.0649217	0.0324609	−	0.999473i	$-0.489666\pi$
$277$	1.08051	0.0649217	0.0324609	+	0.999473i	$0.489666\pi$
$278$	−9.07129	−0.544060
$279$	22.7639	1.36284
$280$	0	0
$281$	14.4912	0.864473	0.432236	−	0.901760i	$-0.357725\pi$
$281$	14.4912	0.864473	0.432236	+	0.901760i	$0.357725\pi$
$282$	1.04316	0.0621193
$283$	−24.4979	−1.45625	−0.728123	−	0.685446i	$-0.759607\pi$
$283$	−24.4979	−1.45625	−0.728123	+	0.685446i	$0.759607\pi$
$284$	14.2075	0.843059
$285$	−0.274351	−0.0162511
$286$	−1.25037	−0.0739357
$287$	0	0
$288$	16.9662	0.999744
$289$	−4.28126	−0.251839
$290$	−5.84365	−0.343151
$291$	2.99396	0.175509
$292$	−11.1997	−0.655413
$293$	−1.16105	−0.0678296	−0.0339148	−	0.999425i	$-0.510797\pi$
$293$	−1.16105	−0.0678296	−0.0339148	+	0.999425i	$0.510797\pi$
$294$	0	0
$295$	−2.94275	−0.171334
$296$	−4.73136	−0.275005
$297$	2.32460	0.134887
$298$	−5.31108	−0.307663
$299$	−1.69605	−0.0980852
$300$	1.01240	0.0584507
$301$	0	0
$302$	−7.89818	−0.454489
$303$	−0.576111	−0.0330967
$304$	−1.57831	−0.0905224
$305$	−16.5918	−0.950044
$306$	−6.92240	−0.395728
$307$	−25.3731	−1.44812	−0.724060	−	0.689737i	$-0.757726\pi$
$307$	−25.3731	−1.44812	−0.724060	+	0.689737i	$0.757726\pi$
$308$	0	0
$309$	−1.23873	−0.0704690
$310$	6.86736	0.390040
$311$	−24.4645	−1.38726	−0.693628	−	0.720333i	$-0.743989\pi$
$311$	−24.4645	−1.38726	−0.693628	+	0.720333i	$0.743989\pi$
$312$	−0.479573	−0.0271505
$313$	−6.61665	−0.373995	−0.186997	−	0.982360i	$-0.559876\pi$
$313$	−6.61665	−0.373995	−0.186997	+	0.982360i	$0.559876\pi$
$314$	−12.0695	−0.681124
$315$	0	0
$316$	8.46906	0.476422
$317$	8.19803	0.460447	0.230224	−	0.973138i	$-0.426054\pi$
$317$	8.19803	0.460447	0.230224	+	0.973138i	$0.426054\pi$
$318$	−0.0475880	−0.00266860
$319$	−12.4780	−0.698632
$320$	0.761192	0.0425519
$321$	3.54640	0.197941
$322$	0	0
$323$	3.51375	0.195510
$324$	−13.5354	−0.751966
$325$	3.15050	0.174758
$326$	−1.64890	−0.0913242
$327$	2.16552	0.119753
$328$	−23.1592	−1.27875
$329$	0	0
$330$	0.348173	0.0191663
$331$	−0.724715	−0.0398339	−0.0199170	−	0.999802i	$-0.506340\pi$
$331$	−0.724715	−0.0398339	−0.0199170	+	0.999802i	$0.506340\pi$
$332$	−3.19369	−0.175276
$333$	5.97547	0.327454
$334$	11.7519	0.643038
$335$	15.3858	0.840616
$336$	0	0
$337$	−8.58299	−0.467545	−0.233773	−	0.972291i	$-0.575107\pi$
$337$	−8.58299	−0.467545	−0.233773	+	0.972291i	$0.575107\pi$
$338$	−0.656184	−0.0356917
$339$	−0.480485	−0.0260964
$340$	7.61183	0.412809
$341$	14.6639	0.794094
$342$	−1.91242	−0.103412
$343$	0	0
$344$	−7.41630	−0.399860
$345$	0.472277	0.0254266
$346$	4.27820	0.229997
$347$	−25.9250	−1.39173	−0.695864	−	0.718173i	$-0.744979\pi$
$347$	−25.9250	−1.39173	−0.695864	+	0.718173i	$0.744979\pi$
$348$	−2.10428	−0.112801
$349$	−17.1403	−0.917501	−0.458750	−	0.888565i	$-0.651703\pi$
$349$	−17.1403	−0.917501	−0.458750	+	0.888565i	$0.651703\pi$
$350$	0	0
$351$	1.21994	0.0651154
$352$	10.9292	0.582527
$353$	18.9101	1.00648	0.503242	−	0.864146i	$-0.332140\pi$
$353$	18.9101	1.00648	0.503242	+	0.864146i	$0.332140\pi$
$354$	0.290725	0.0154519
$355$	−12.3113	−0.653417
$356$	−10.8269	−0.573823
$357$	0	0
$358$	17.4554	0.922547
$359$	14.1049	0.744431	0.372215	−	0.928146i	$-0.378598\pi$
$359$	14.1049	0.744431	0.372215	+	0.928146i	$0.378598\pi$
$360$	−9.42236	−0.496602
$361$	−18.0293	−0.948909
$362$	−16.1685	−0.849798
$363$	−1.50883	−0.0791931
$364$	0	0
$365$	9.70495	0.507980
$366$	1.63917	0.0856806
$367$	−0.360258	−0.0188053	−0.00940266	−	0.999956i	$-0.502993\pi$
$367$	−0.360258	−0.0188053	−0.00940266	+	0.999956i	$0.502993\pi$
$368$	2.71696	0.141631
$369$	29.2488	1.52263
$370$	1.80267	0.0937162
$371$	0	0
$372$	2.47291	0.128214
$373$	−25.1680	−1.30315	−0.651575	−	0.758584i	$-0.725892\pi$
$373$	−25.1680	−1.30315	−0.651575	+	0.758584i	$0.725892\pi$
$374$	−4.45923	−0.230581
$375$	−2.26956	−0.117200
$376$	−18.1852	−0.937830
$377$	−6.54835	−0.337257
$378$	0	0
$379$	31.8947	1.63832	0.819161	−	0.573564i	$-0.194440\pi$
$379$	31.8947	1.63832	0.819161	+	0.573564i	$0.194440\pi$
$380$	2.10288	0.107876
$381$	−3.71913	−0.190537
$382$	−1.98232	−0.101424
$383$	−32.8172	−1.67688	−0.838440	−	0.544994i	$-0.816532\pi$
$383$	−32.8172	−1.67688	−0.838440	+	0.544994i	$0.816532\pi$
$384$	2.27355	0.116022
$385$	0	0
$386$	−13.9454	−0.709800
$387$	9.36641	0.476122
$388$	−22.9486	−1.16504
$389$	−25.3087	−1.28320	−0.641600	−	0.767040i	$-0.721729\pi$
$389$	−25.3087	−1.28320	−0.641600	+	0.767040i	$0.721729\pi$
$390$	0.182719	0.00925233
$391$	−6.04869	−0.305895
$392$	0	0
$393$	−2.06829	−0.104331
$394$	1.78607	0.0899810
$395$	−7.33875	−0.369253
$396$	−8.84629	−0.444543
$397$	5.29395	0.265696	0.132848	−	0.991136i	$-0.457588\pi$
$397$	5.29395	0.265696	0.132848	+	0.991136i	$0.457588\pi$
$398$	−14.2175	−0.712661
$399$	0	0
$400$	−5.04689	−0.252345
$401$	−24.6335	−1.23014	−0.615070	−	0.788473i	$-0.710872\pi$
$401$	−24.6335	−1.23014	−0.615070	+	0.788473i	$0.710872\pi$
$402$	−1.52002	−0.0758118
$403$	7.69550	0.383340
$404$	4.41586	0.219697
$405$	11.7289	0.582815
$406$	0	0
$407$	3.84924	0.190800
$408$	−1.71032	−0.0846733
$409$	−10.2223	−0.505458	−0.252729	−	0.967537i	$-0.581328\pi$
$409$	−10.2223	−0.505458	−0.252729	+	0.967537i	$0.581328\pi$
$410$	8.82372	0.435773
$411$	1.34236	0.0662140
$412$	9.49481	0.467776
$413$	0	0
$414$	3.29211	0.161798
$415$	2.76745	0.135849
$416$	5.73556	0.281209
$417$	2.83057	0.138614
$418$	−1.23193	−0.0602556
$419$	−30.6503	−1.49736	−0.748682	−	0.662929i	$-0.769313\pi$
$419$	−30.6503	−1.49736	−0.748682	+	0.662929i	$0.769313\pi$
$420$	0	0
$421$	25.9764	1.26601	0.633007	−	0.774146i	$-0.281820\pi$
$421$	25.9764	1.26601	0.633007	+	0.774146i	$0.281820\pi$
$422$	−12.0662	−0.587376
$423$	22.9670	1.11669
$424$	0.829592	0.0402885
$425$	11.2357	0.545014
$426$	1.21628	0.0589290
$427$	0	0
$428$	−27.1830	−1.31394
$429$	0.390160	0.0188371
$430$	2.82564	0.136264
$431$	14.8818	0.716829	0.358415	−	0.933563i	$-0.383317\pi$
$431$	14.8818	0.716829	0.358415	+	0.933563i	$0.383317\pi$
$432$	−1.95426	−0.0940242
$433$	40.0871	1.92647	0.963233	−	0.268669i	$-0.0865837\pi$
$433$	40.0871	1.92647	0.963233	+	0.268669i	$0.0865837\pi$
$434$	0	0
$435$	1.82343	0.0874269
$436$	−16.5986	−0.794927
$437$	−1.67104	−0.0799368
$438$	−0.958789	−0.0458127
$439$	3.00243	0.143298	0.0716491	−	0.997430i	$-0.477174\pi$
$439$	3.00243	0.143298	0.0716491	+	0.997430i	$0.477174\pi$
$440$	−6.06963	−0.289358
$441$	0	0
$442$	−2.34017	−0.111311
$443$	1.43566	0.0682101	0.0341051	−	0.999418i	$-0.489142\pi$
$443$	1.43566	0.0682101	0.0341051	+	0.999418i	$0.489142\pi$
$444$	0.649133	0.0308065
$445$	9.38189	0.444744
$446$	14.5815	0.690453
$447$	1.65725	0.0783853
$448$	0	0
$449$	19.6313	0.926460	0.463230	−	0.886238i	$-0.346690\pi$
$449$	19.6313	0.926460	0.463230	+	0.886238i	$0.346690\pi$
$450$	−6.11526	−0.288276
$451$	18.8413	0.887203
$452$	3.68289	0.173229
$453$	2.46452	0.115793
$454$	17.5447	0.823413
$455$	0	0
$456$	−0.472501	−0.0221269
$457$	10.6421	0.497815	0.248907	−	0.968527i	$-0.419929\pi$
$457$	10.6421	0.497815	0.248907	+	0.968527i	$0.419929\pi$
$458$	−10.9588	−0.512070
$459$	4.35070	0.203073
$460$	−3.61998	−0.168782
$461$	38.5930	1.79745	0.898727	−	0.438509i	$-0.144493\pi$
$461$	38.5930	1.79745	0.898727	+	0.438509i	$0.144493\pi$
$462$	0	0
$463$	−27.9993	−1.30124	−0.650618	−	0.759405i	$-0.725490\pi$
$463$	−27.9993	−1.30124	−0.650618	+	0.759405i	$0.725490\pi$
$464$	10.4900	0.486987
$465$	−2.14287	−0.0993730
$466$	9.10623	0.421838
$467$	25.8199	1.19480	0.597401	−	0.801943i	$-0.296200\pi$
$467$	25.8199	1.19480	0.597401	+	0.801943i	$0.296200\pi$
$468$	−4.64247	−0.214598
$469$	0	0
$470$	6.92862	0.319594
$471$	3.76614	0.173535
$472$	−5.06816	−0.233281
$473$	6.03359	0.277425
$474$	0.725023	0.0333014
$475$	3.10405	0.142423
$476$	0	0
$477$	−1.04773	−0.0479723
$478$	−0.305680	−0.0139815
$479$	39.4071	1.80055	0.900277	−	0.435317i	$-0.143364\pi$
$479$	39.4071	1.80055	0.900277	+	0.435317i	$0.143364\pi$
$480$	−1.59711	−0.0728976
$481$	2.02005	0.0921065
$482$	3.73575	0.170159
$483$	0	0
$484$	11.5651	0.525687
$485$	19.8858	0.902966
$486$	−3.56025	−0.161496
$487$	−34.0959	−1.54503	−0.772517	−	0.634994i	$-0.781003\pi$
$487$	−34.0959	−1.54503	−0.772517	+	0.634994i	$0.781003\pi$
$488$	−28.5753	−1.29354
$489$	0.514517	0.0232673
$490$	0	0
$491$	−9.70414	−0.437942	−0.218971	−	0.975731i	$-0.570270\pi$
$491$	−9.70414	−0.437942	−0.218971	+	0.975731i	$0.570270\pi$
$492$	3.17739	0.143248
$493$	−23.3536	−1.05179
$494$	−0.646508	−0.0290878
$495$	7.66563	0.344545
$496$	−12.3277	−0.553529
$497$	0	0
$498$	−0.273407	−0.0122516
$499$	18.0463	0.807862	0.403931	−	0.914789i	$-0.367644\pi$
$499$	18.0463	0.807862	0.403931	+	0.914789i	$0.367644\pi$
$500$	17.3961	0.777976
$501$	−3.66704	−0.163831
$502$	11.7500	0.524428
$503$	−16.4922	−0.735352	−0.367676	−	0.929954i	$-0.619846\pi$
$503$	−16.4922	−0.735352	−0.367676	+	0.929954i	$0.619846\pi$
$504$	0	0
$505$	−3.82650	−0.170277
$506$	2.12069	0.0942761
$507$	0.204753	0.00909341
$508$	28.5069	1.26479
$509$	19.1977	0.850921	0.425460	−	0.904977i	$-0.360112\pi$
$509$	19.1977	0.850921	0.425460	+	0.904977i	$0.360112\pi$
$510$	0.651637	0.0288550
$511$	0	0
$512$	−16.6921	−0.737692
$513$	1.20195	0.0530673
$514$	17.0448	0.751812
$515$	−8.22760	−0.362552
$516$	1.01750	0.0447930
$517$	14.7947	0.650670
$518$	0	0
$519$	−1.33495	−0.0585979
$520$	−3.18530	−0.139685
$521$	8.76034	0.383797	0.191899	−	0.981415i	$-0.438535\pi$
$521$	8.76034	0.383797	0.191899	+	0.981415i	$0.438535\pi$
$522$	12.7106	0.556329
$523$	11.7406	0.513381	0.256690	−	0.966494i	$-0.417368\pi$
$523$	11.7406	0.513381	0.256690	+	0.966494i	$0.417368\pi$
$524$	15.8533	0.692555
$525$	0	0
$526$	−18.2683	−0.796534
$527$	27.4447	1.19551
$528$	−0.625010	−0.0272001
$529$	−20.1234	−0.874931
$530$	−0.316077	−0.0137295
$531$	6.40082	0.277772
$532$	0	0
$533$	9.88779	0.428288
$534$	−0.926872	−0.0401097
$535$	23.5550	1.01837
$536$	26.4982	1.14455
$537$	−5.44673	−0.235044
$538$	3.87467	0.167049
$539$	0	0
$540$	2.60378	0.112049
$541$	16.4587	0.707614	0.353807	−	0.935318i	$-0.384887\pi$
$541$	16.4587	0.707614	0.353807	+	0.935318i	$0.384887\pi$
$542$	−2.85396	−0.122588
$543$	5.04516	0.216509
$544$	20.4549	0.876997
$545$	14.3833	0.616112
$546$	0	0
$547$	−8.19375	−0.350339	−0.175170	−	0.984538i	$-0.556047\pi$
$547$	−8.19375	−0.350339	−0.175170	+	0.984538i	$0.556047\pi$
$548$	−10.2891	−0.439531
$549$	36.0891	1.54025
$550$	−3.93928	−0.167972
$551$	−6.45179	−0.274856
$552$	0.813381	0.0346198
$553$	0	0
$554$	−0.709015	−0.0301232
$555$	−0.562498	−0.0238767
$556$	−21.6962	−0.920123
$557$	−13.8144	−0.585335	−0.292667	−	0.956214i	$-0.594543\pi$
$557$	−13.8144	−0.585335	−0.292667	+	0.956214i	$0.594543\pi$
$558$	−14.9373	−0.632346
$559$	3.16639	0.133924
$560$	0	0
$561$	1.39144	0.0587467
$562$	−9.50889	−0.401108
$563$	−9.75559	−0.411149	−0.205575	−	0.978641i	$-0.565906\pi$
$563$	−9.75559	−0.411149	−0.205575	+	0.978641i	$0.565906\pi$
$564$	2.49497	0.105057
$565$	−3.19136	−0.134262
$566$	16.0751	0.675687
$567$	0	0
$568$	−21.2032	−0.889666
$569$	−1.43264	−0.0600593	−0.0300296	−	0.999549i	$-0.509560\pi$
$569$	−1.43264	−0.0600593	−0.0300296	+	0.999549i	$0.509560\pi$
$570$	0.180025	0.00754040
$571$	16.2872	0.681598	0.340799	−	0.940136i	$-0.389303\pi$
$571$	16.2872	0.681598	0.340799	+	0.940136i	$0.389303\pi$
$572$	−2.99055	−0.125041
$573$	0.618557	0.0258406
$574$	0	0
$575$	−5.34342	−0.222836
$576$	−1.65568	−0.0689867
$577$	4.81846	0.200595	0.100297	−	0.994957i	$-0.468021\pi$
$577$	4.81846	0.200595	0.100297	+	0.994957i	$0.468021\pi$
$578$	2.80929	0.116851
$579$	4.35146	0.180841
$580$	−13.9765	−0.580343
$581$	0	0
$582$	−1.96459	−0.0814349
$583$	−0.674920	−0.0279523
$584$	16.7144	0.691645
$585$	4.02287	0.166325
$586$	0.761866	0.0314724
$587$	−36.6215	−1.51153	−0.755765	−	0.654843i	$-0.772735\pi$
$587$	−36.6215	−1.51153	−0.755765	+	0.654843i	$0.772735\pi$
$588$	0	0
$589$	7.58203	0.312412
$590$	1.93099	0.0794974
$591$	−0.557319	−0.0229251
$592$	−3.23599	−0.132998
$593$	18.7081	0.768251	0.384125	−	0.923281i	$-0.374503\pi$
$593$	18.7081	0.768251	0.384125	+	0.923281i	$0.374503\pi$
$594$	−1.52537	−0.0625866
$595$	0	0
$596$	−12.7027	−0.520325
$597$	4.43639	0.181569
$598$	1.11292	0.0455108
$599$	−25.4194	−1.03861	−0.519305	−	0.854589i	$-0.673809\pi$
$599$	−25.4194	−1.03861	−0.519305	+	0.854589i	$0.673809\pi$
$600$	−1.51090	−0.0616820
$601$	−17.7790	−0.725219	−0.362609	−	0.931941i	$-0.618114\pi$
$601$	−17.7790	−0.725219	−0.362609	+	0.931941i	$0.618114\pi$
$602$	0	0
$603$	−33.4659	−1.36284
$604$	−18.8904	−0.768640
$605$	−10.0216	−0.407436
$606$	0.378035	0.0153566
$607$	−26.0047	−1.05550	−0.527750	−	0.849400i	$-0.676964\pi$
$607$	−26.0047	−1.05550	−0.527750	+	0.849400i	$0.676964\pi$
$608$	5.65098	0.229178
$609$	0	0
$610$	10.8873	0.440813
$611$	7.76416	0.314104
$612$	−16.5566	−0.669261
$613$	26.8230	1.08337	0.541685	−	0.840582i	$-0.317787\pi$
$613$	26.8230	1.08337	0.541685	+	0.840582i	$0.317787\pi$
$614$	16.6494	0.671916
$615$	−2.75332	−0.111025
$616$	0	0
$617$	−18.6491	−0.750784	−0.375392	−	0.926866i	$-0.622492\pi$
$617$	−18.6491	−0.750784	−0.375392	+	0.926866i	$0.622492\pi$
$618$	0.812836	0.0326971
$619$	−13.0685	−0.525269	−0.262634	−	0.964895i	$-0.584591\pi$
$619$	−13.0685	−0.525269	−0.262634	+	0.964895i	$0.584591\pi$
$620$	16.4249	0.659642
$621$	−2.06908	−0.0830291
$622$	16.0532	0.643676
$623$	0	0
$624$	−0.328001	−0.0131306
$625$	0.678175	0.0271270
$626$	4.34174	0.173531
$627$	0.384407	0.0153517
$628$	−28.8672	−1.15193
$629$	7.20418	0.287250
$630$	0	0
$631$	−49.1745	−1.95761	−0.978804	−	0.204801i	$-0.934345\pi$
$631$	−49.1745	−1.95761	−0.978804	+	0.204801i	$0.934345\pi$
$632$	−12.6392	−0.502760
$633$	3.76511	0.149650
$634$	−5.37941	−0.213644
$635$	−24.7023	−0.980279
$636$	−0.113818	−0.00451318
$637$	0	0
$638$	8.18784	0.324160
$639$	26.7785	1.05934
$640$	15.1008	0.596912
$641$	4.37503	0.172803	0.0864016	−	0.996260i	$-0.472463\pi$
$641$	4.37503	0.172803	0.0864016	+	0.996260i	$0.472463\pi$
$642$	−2.32709	−0.0918429
$643$	5.34651	0.210846	0.105423	−	0.994427i	$-0.466380\pi$
$643$	5.34651	0.210846	0.105423	+	0.994427i	$0.466380\pi$
$644$	0	0
$645$	−0.881702	−0.0347170
$646$	−2.30566	−0.0907151
$647$	29.7346	1.16899	0.584493	−	0.811398i	$-0.301293\pi$
$647$	29.7346	1.16899	0.584493	+	0.811398i	$0.301293\pi$
$648$	20.2001	0.793537
$649$	4.12324	0.161851
$650$	−2.06731	−0.0810865
$651$	0	0
$652$	−3.94375	−0.154449
$653$	−2.54262	−0.0995005	−0.0497503	−	0.998762i	$-0.515843\pi$
$653$	−2.54262	−0.0995005	−0.0497503	+	0.998762i	$0.515843\pi$
$654$	−1.42098	−0.0555646
$655$	−13.7375	−0.536768
$656$	−15.8396	−0.618432
$657$	−21.1094	−0.823556
$658$	0	0
$659$	1.87019	0.0728523	0.0364261	−	0.999336i	$-0.488403\pi$
$659$	1.87019	0.0728523	0.0364261	+	0.999336i	$0.488403\pi$
$660$	0.832740	0.0324144
$661$	12.0763	0.469714	0.234857	−	0.972030i	$-0.424538\pi$
$661$	12.0763	0.469714	0.234857	+	0.972030i	$0.424538\pi$
$662$	0.475546	0.0184826
$663$	0.730219	0.0283593
$664$	4.76624	0.184966
$665$	0	0
$666$	−3.92101	−0.151936
$667$	11.1063	0.430040
$668$	28.1076	1.08752
$669$	−4.54995	−0.175911
$670$	−10.0959	−0.390039
$671$	23.2476	0.897464
$672$	0	0
$673$	−7.81691	−0.301320	−0.150660	−	0.988586i	$-0.548140\pi$
$673$	−7.81691	−0.301320	−0.150660	+	0.988586i	$0.548140\pi$
$674$	5.63202	0.216937
$675$	3.84341	0.147933
$676$	−1.56942	−0.0603624
$677$	−6.25628	−0.240448	−0.120224	−	0.992747i	$-0.538361\pi$
$677$	−6.25628	−0.240448	−0.120224	+	0.992747i	$0.538361\pi$
$678$	0.315287	0.0121085
$679$	0	0
$680$	−11.3598	−0.435630
$681$	−5.47458	−0.209786
$682$	−9.62220	−0.368453
$683$	22.3656	0.855795	0.427898	−	0.903827i	$-0.359254\pi$
$683$	22.3656	0.855795	0.427898	+	0.903827i	$0.359254\pi$
$684$	−4.57402	−0.174892
$685$	8.91593	0.340660
$686$	0	0
$687$	3.41954	0.130464
$688$	−5.07234	−0.193381
$689$	−0.354194	−0.0134937
$690$	−0.309901	−0.0117977
$691$	3.80771	0.144852	0.0724260	−	0.997374i	$-0.476926\pi$
$691$	3.80771	0.144852	0.0724260	+	0.997374i	$0.476926\pi$
$692$	10.2323	0.388975
$693$	0	0
$694$	17.0116	0.645751
$695$	18.8005	0.713145
$696$	3.14041	0.119037
$697$	35.2632	1.33569
$698$	11.2472	0.425713
$699$	−2.84148	−0.107475
$700$	0	0
$701$	−8.14218	−0.307526	−0.153763	−	0.988108i	$-0.549139\pi$
$701$	−8.14218	−0.307526	−0.153763	+	0.988108i	$0.549139\pi$
$702$	−0.800502	−0.0302130
$703$	1.99027	0.0750643
$704$	−1.06654	−0.0401969
$705$	−2.16198	−0.0814250
$706$	−12.4085	−0.467001
$707$	0	0
$708$	0.695340	0.0261325
$709$	−24.0319	−0.902536	−0.451268	−	0.892388i	$-0.649028\pi$
$709$	−24.0319	−0.902536	−0.451268	+	0.892388i	$0.649028\pi$
$710$	8.07848	0.303180
$711$	15.9626	0.598646
$712$	16.1580	0.605546
$713$	−13.0520	−0.488800
$714$	0	0
$715$	2.59143	0.0969138
$716$	41.7488	1.56023
$717$	0.0953833	0.00356215
$718$	−9.25544	−0.345410
$719$	32.6960	1.21936	0.609678	−	0.792649i	$-0.291299\pi$
$719$	32.6960	1.21936	0.609678	+	0.792649i	$0.291299\pi$
$720$	−6.44437	−0.240168
$721$	0	0
$722$	11.8305	0.440286
$723$	−1.16569	−0.0433525
$724$	−38.6709	−1.43719
$725$	−20.6306	−0.766201
$726$	0.990071	0.0367450
$727$	15.7712	0.584921	0.292460	−	0.956278i	$-0.405526\pi$
$727$	15.7712	0.584921	0.292460	+	0.956278i	$0.405526\pi$
$728$	0	0
$729$	−24.7624	−0.917126
$730$	−6.36823	−0.235699
$731$	11.2924	0.417664
$732$	3.92046	0.144905
$733$	−23.5094	−0.868340	−0.434170	−	0.900831i	$-0.642958\pi$
$733$	−23.5094	−0.868340	−0.434170	+	0.900831i	$0.642958\pi$
$734$	0.236396	0.00872552
$735$	0	0
$736$	−9.72781	−0.358572
$737$	−21.5578	−0.794093
$738$	−19.1926	−0.706490
$739$	−22.5267	−0.828658	−0.414329	−	0.910127i	$-0.635984\pi$
$739$	−22.5267	−0.828658	−0.414329	+	0.910127i	$0.635984\pi$
$740$	4.31151	0.158494
$741$	0.201734	0.00741089
$742$	0	0
$743$	−13.9795	−0.512857	−0.256429	−	0.966563i	$-0.582546\pi$
$743$	−13.9795	−0.512857	−0.256429	+	0.966563i	$0.582546\pi$
$744$	−3.69055	−0.135302
$745$	11.0074	0.403280
$746$	16.5149	0.604652
$747$	−6.01952	−0.220243
$748$	−10.6653	−0.389963
$749$	0	0
$750$	1.48925	0.0543798
$751$	2.54591	0.0929017	0.0464509	−	0.998921i	$-0.485209\pi$
$751$	2.54591	0.0929017	0.0464509	+	0.998921i	$0.485209\pi$
$752$	−12.4377	−0.453555
$753$	−3.66643	−0.133612
$754$	4.29692	0.156485
$755$	16.3692	0.595738
$756$	0	0
$757$	19.6921	0.715721	0.357861	−	0.933775i	$-0.383506\pi$
$757$	19.6921	0.715721	0.357861	+	0.933775i	$0.383506\pi$
$758$	−20.9288	−0.760168
$759$	−0.661732	−0.0240193
$760$	−3.13833	−0.113839
$761$	39.0905	1.41703	0.708515	−	0.705696i	$-0.249365\pi$
$761$	39.0905	1.41703	0.708515	+	0.705696i	$0.249365\pi$
$762$	2.44043	0.0884075
$763$	0	0
$764$	−4.74120	−0.171531
$765$	14.3469	0.518714
$766$	21.5341	0.778059
$767$	2.16385	0.0781320
$768$	−1.72107	−0.0621039
$769$	21.7389	0.783926	0.391963	−	0.919981i	$-0.371796\pi$
$769$	21.7389	0.783926	0.391963	+	0.919981i	$0.371796\pi$
$770$	0	0
$771$	−5.31859	−0.191544
$772$	−33.3537	−1.20043
$773$	−4.12806	−0.148476	−0.0742381	−	0.997241i	$-0.523652\pi$
$773$	−4.12806	−0.148476	−0.0742381	+	0.997241i	$0.523652\pi$
$774$	−6.14609	−0.220917
$775$	24.2447	0.870895
$776$	34.2483	1.22944
$777$	0	0
$778$	16.6071	0.595395
$779$	9.74199	0.349043
$780$	0.437016	0.0156477
$781$	17.2500	0.617254
$782$	3.96905	0.141933
$783$	−7.98857	−0.285488
$784$	0	0
$785$	25.0145	0.892807
$786$	1.35718	0.0484089
$787$	−38.7863	−1.38258	−0.691291	−	0.722576i	$-0.742958\pi$
$787$	−38.7863	−1.38258	−0.691291	+	0.722576i	$0.742958\pi$
$788$	4.27182	0.152177
$789$	5.70037	0.202938
$790$	4.81557	0.171330
$791$	0	0
$792$	13.2022	0.469118
$793$	12.2002	0.433242
$794$	−3.47381	−0.123281
$795$	0.0986276	0.00349796
$796$	−34.0047	−1.20526
$797$	−19.9651	−0.707201	−0.353601	−	0.935397i	$-0.615043\pi$
$797$	−19.9651	−0.707201	−0.353601	+	0.935397i	$0.615043\pi$
$798$	0	0
$799$	27.6896	0.979587
$800$	18.0699	0.638867
$801$	−20.4067	−0.721035
$802$	16.1641	0.570775
$803$	−13.5981	−0.479866
$804$	−3.63550	−0.128214
$805$	0	0
$806$	−5.04967	−0.177867
$807$	−1.20904	−0.0425602
$808$	−6.59020	−0.231842
$809$	−30.2441	−1.06332	−0.531662	−	0.846956i	$-0.678432\pi$
$809$	−30.2441	−1.06332	−0.531662	+	0.846956i	$0.678432\pi$
$810$	−7.69633	−0.270421
$811$	−23.4099	−0.822034	−0.411017	−	0.911628i	$-0.634826\pi$
$811$	−23.4099	−0.822034	−0.411017	+	0.911628i	$0.634826\pi$
$812$	0	0
$813$	0.890538	0.0312325
$814$	−2.52581	−0.0885295
$815$	3.41740	0.119706
$816$	−1.16976	−0.0409498
$817$	3.11970	0.109144
$818$	6.70768	0.234528
$819$	0	0
$820$	21.1041	0.736986
$821$	−46.5536	−1.62473	−0.812366	−	0.583147i	$-0.801821\pi$
$821$	−46.5536	−1.62473	−0.812366	+	0.583147i	$0.801821\pi$
$822$	−0.880838	−0.0307228
$823$	−5.68031	−0.198003	−0.0990017	−	0.995087i	$-0.531565\pi$
$823$	−5.68031	−0.198003	−0.0990017	+	0.995087i	$0.531565\pi$
$824$	−14.1700	−0.493636
$825$	1.22920	0.0427953
$826$	0	0
$827$	48.3198	1.68024	0.840122	−	0.542398i	$-0.182483\pi$
$827$	48.3198	1.68024	0.840122	+	0.542398i	$0.182483\pi$
$828$	7.87388	0.273636
$829$	11.8545	0.411722	0.205861	−	0.978581i	$-0.434000\pi$
$829$	11.8545	0.411722	0.205861	+	0.978581i	$0.434000\pi$
$830$	−1.81595	−0.0630327
$831$	0.221239	0.00767468
$832$	−0.559715	−0.0194046
$833$	0	0
$834$	−1.85738	−0.0643157
$835$	−24.3563	−0.842884
$836$	−2.94646	−0.101905
$837$	9.38802	0.324497
$838$	20.1122	0.694765
$839$	2.69386	0.0930025	0.0465013	−	0.998918i	$-0.485193\pi$
$839$	2.69386	0.0930025	0.0465013	+	0.998918i	$0.485193\pi$
$840$	0	0
$841$	13.8809	0.478652
$842$	−17.0453	−0.587420
$843$	2.96712	0.102193
$844$	−28.8593	−0.993380
$845$	1.35996	0.0467841
$846$	−15.0706	−0.518137
$847$	0	0
$848$	0.567394	0.0194844
$849$	−5.01602	−0.172149
$850$	−7.37271	−0.252882
$851$	−3.42612	−0.117446
$852$	2.90903	0.0996617
$853$	2.48965	0.0852440	0.0426220	−	0.999091i	$-0.486429\pi$
$853$	2.48965	0.0852440	0.0426220	+	0.999091i	$0.486429\pi$
$854$	0	0
$855$	3.96355	0.135551
$856$	40.5677	1.38658
$857$	−46.2110	−1.57854	−0.789270	−	0.614047i	$-0.789541\pi$
$857$	−46.2110	−1.57854	−0.789270	+	0.614047i	$0.789541\pi$
$858$	−0.256017	−0.00874027
$859$	10.9170	0.372482	0.186241	−	0.982504i	$-0.440369\pi$
$859$	10.9170	0.372482	0.186241	+	0.982504i	$0.440369\pi$
$860$	6.75820	0.230453
$861$	0	0
$862$	−9.76517	−0.332603
$863$	−29.2330	−0.995104	−0.497552	−	0.867434i	$-0.665768\pi$
$863$	−29.2330	−0.995104	−0.497552	+	0.867434i	$0.665768\pi$
$864$	6.99701	0.238043
$865$	−8.86670	−0.301477
$866$	−26.3045	−0.893865
$867$	−0.876603	−0.0297710
$868$	0	0
$869$	10.2827	0.348817
$870$	−1.19651	−0.0405654
$871$	−11.3134	−0.383340
$872$	24.7716	0.838873
$873$	−43.2538	−1.46392
$874$	1.09651	0.0370901
$875$	0	0
$876$	−2.29318	−0.0774792
$877$	−7.38110	−0.249242	−0.124621	−	0.992204i	$-0.539772\pi$
$877$	−7.38110	−0.249242	−0.124621	+	0.992204i	$0.539772\pi$
$878$	−1.97015	−0.0664892
$879$	−0.237730	−0.00801843
$880$	−4.15129	−0.139940
$881$	−16.4854	−0.555407	−0.277703	−	0.960667i	$-0.589573\pi$
$881$	−16.4854	−0.555407	−0.277703	+	0.960667i	$0.589573\pi$
$882$	0	0
$883$	12.4427	0.418732	0.209366	−	0.977837i	$-0.432860\pi$
$883$	12.4427	0.418732	0.209366	+	0.977837i	$0.432860\pi$
$884$	−5.59709	−0.188250
$885$	−0.602538	−0.0202541
$886$	−0.942055	−0.0316490
$887$	8.11331	0.272418	0.136209	−	0.990680i	$-0.456508\pi$
$887$	8.11331	0.272418	0.136209	+	0.990680i	$0.456508\pi$
$888$	−0.968763	−0.0325095
$889$	0	0
$890$	−6.15624	−0.206358
$891$	−16.4340	−0.550559
$892$	34.8751	1.16771
$893$	7.64968	0.255987
$894$	−1.08746	−0.0363702
$895$	−36.1769	−1.20926
$896$	0	0
$897$	−0.347272	−0.0115951
$898$	−12.8818	−0.429870
$899$	−50.3929	−1.68070
$900$	−14.6261	−0.487537
$901$	−1.26317	−0.0420824
$902$	−12.3634	−0.411655
$903$	0	0
$904$	−5.49633	−0.182805
$905$	33.5098	1.11390
$906$	−1.61718	−0.0537272
$907$	−18.4796	−0.613605	−0.306803	−	0.951773i	$-0.599259\pi$
$907$	−18.4796	−0.613605	−0.306803	+	0.951773i	$0.599259\pi$
$908$	41.9623	1.39257
$909$	8.32308	0.276059
$910$	0	0
$911$	15.8660	0.525663	0.262831	−	0.964842i	$-0.415344\pi$
$911$	15.8660	0.525663	0.262831	+	0.964842i	$0.415344\pi$
$912$	−0.323165	−0.0107010
$913$	−3.87761	−0.128330
$914$	−6.98315	−0.230982
$915$	−3.39723	−0.112309
$916$	−26.2106	−0.866022
$917$	0	0
$918$	−2.85486	−0.0942244
$919$	43.4568	1.43351	0.716753	−	0.697327i	$-0.245627\pi$
$919$	43.4568	1.43351	0.716753	+	0.697327i	$0.245627\pi$
$920$	5.40244	0.178113
$921$	−5.19523	−0.171189
$922$	−25.3241	−0.834004
$923$	9.05268	0.297973
$924$	0	0
$925$	6.36418	0.209253
$926$	18.3727	0.603763
$927$	17.8960	0.587782
$928$	−37.5585	−1.23292
$929$	25.6312	0.840931	0.420465	−	0.907309i	$-0.361867\pi$
$929$	25.6312	0.840931	0.420465	+	0.907309i	$0.361867\pi$
$930$	1.40611	0.0461083
$931$	0	0
$932$	21.7798	0.713420
$933$	−5.00920	−0.163994
$934$	−16.9426	−0.554379
$935$	9.24189	0.302242
$936$	6.92840	0.226462
$937$	−23.9639	−0.782867	−0.391433	−	0.920206i	$-0.628021\pi$
$937$	−23.9639	−0.782867	−0.391433	+	0.920206i	$0.628021\pi$
$938$	0	0
$939$	−1.35478	−0.0442116
$940$	16.5715	0.540502
$941$	42.2934	1.37872	0.689362	−	0.724417i	$-0.257891\pi$
$941$	42.2934	1.37872	0.689362	+	0.724417i	$0.257891\pi$
$942$	−2.47128	−0.0805186
$943$	−16.7702	−0.546113
$944$	−3.46634	−0.112820
$945$	0	0
$946$	−3.95914	−0.128723
$947$	−49.5957	−1.61165	−0.805823	−	0.592157i	$-0.798277\pi$
$947$	−49.5957	−1.61165	−0.805823	+	0.592157i	$0.798277\pi$
$948$	1.73407	0.0563199
$949$	−7.13619	−0.231650
$950$	−2.03683	−0.0660833
$951$	1.67857	0.0544315
$952$	0	0
$953$	18.7156	0.606259	0.303129	−	0.952949i	$-0.401969\pi$
$953$	18.7156	0.606259	0.303129	+	0.952949i	$0.401969\pi$
$954$	0.687505	0.0222588
$955$	4.10843	0.132946
$956$	−0.731107	−0.0236457
$957$	−2.55491	−0.0825884
$958$	−25.8583	−0.835443
$959$	0	0
$960$	0.155857	0.00503025
$961$	28.2208	0.910348
$962$	−1.32553	−0.0427367
$963$	−51.2349	−1.65102
$964$	8.93494	0.287775
$965$	28.9022	0.930395
$966$	0	0
$967$	−34.3284	−1.10393	−0.551964	−	0.833868i	$-0.686121\pi$
$967$	−34.3284	−1.10393	−0.551964	+	0.833868i	$0.686121\pi$
$968$	−17.2597	−0.554748
$969$	0.719451	0.0231121
$970$	−13.0487	−0.418969
$971$	−56.7118	−1.81997	−0.909984	−	0.414642i	$-0.863907\pi$
$971$	−56.7118	−1.81997	−0.909984	+	0.414642i	$0.863907\pi$
$972$	−8.51520	−0.273125
$973$	0	0
$974$	22.3732	0.716884
$975$	0.645076	0.0206590
$976$	−19.5439	−0.625585
$977$	−39.8809	−1.27590	−0.637951	−	0.770077i	$-0.720218\pi$
$977$	−39.8809	−1.27590	−0.637951	+	0.770077i	$0.720218\pi$
$978$	−0.337618	−0.0107958
$979$	−13.1454	−0.420130
$980$	0	0
$981$	−31.2853	−0.998862
$982$	6.36770	0.203201
$983$	−41.2678	−1.31624	−0.658120	−	0.752913i	$-0.728648\pi$
$983$	−41.2678	−1.31624	−0.658120	+	0.752913i	$0.728648\pi$
$984$	−4.74192	−0.151167
$985$	−3.70169	−0.117946
$986$	15.3243	0.488024
$987$	0	0
$988$	−1.54628	−0.0491937
$989$	−5.37036	−0.170767
$990$	−5.03007	−0.159866
$991$	−26.1784	−0.831583	−0.415792	−	0.909460i	$-0.636495\pi$
$991$	−26.1784	−0.831583	−0.415792	+	0.909460i	$0.636495\pi$
$992$	44.1380	1.40138
$993$	−0.148388	−0.00470894
$994$	0	0
$995$	29.4663	0.934145
$996$	−0.653918	−0.0207202
$997$	−8.62009	−0.273001	−0.136501	−	0.990640i	$-0.543586\pi$
$997$	−8.62009	−0.273001	−0.136501	+	0.990640i	$0.543586\pi$
$998$	−11.8417	−0.374842
$999$	2.46434	0.0779681

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	637.2.a.m.1.3	✓	6
3.2	odd	2		5733.2.a.bu.1.4		6
7.2	even	3		637.2.e.o.508.4		12
7.3	odd	6		637.2.e.n.79.4		12
7.4	even	3		637.2.e.o.79.4		12
7.5	odd	6		637.2.e.n.508.4		12
7.6	odd	2		637.2.a.n.1.3	yes	6
13.12	even	2		8281.2.a.cc.1.4		6
21.20	even	2		5733.2.a.br.1.4		6
91.90	odd	2		8281.2.a.cd.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.a.m.1.3	✓	6	1.1	even	1	trivial
637.2.a.n.1.3	yes	6	7.6	odd	2
637.2.e.n.79.4		12	7.3	odd	6
637.2.e.n.508.4		12	7.5	odd	6
637.2.e.o.79.4		12	7.4	even	3
637.2.e.o.508.4		12	7.2	even	3
5733.2.a.br.1.4		6	21.20	even	2
5733.2.a.bu.1.4		6	3.2	odd	2
8281.2.a.cc.1.4		6	13.12	even	2
8281.2.a.cd.1.4		6	91.90	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.a (trivial)

\(f(q)\)	\(=\)	\(q-0.656184 q^{2} +0.204753 q^{3} -1.56942 q^{4} +1.35996 q^{5} -0.134356 q^{6} +2.34220 q^{8} -2.95808 q^{9} +O(q^{10})\)	\(q-0.656184 q^{2} +0.204753 q^{3} -1.56942 q^{4} +1.35996 q^{5} -0.134356 q^{6} +2.34220 q^{8} -2.95808 q^{9} -0.892385 q^{10} -1.90551 q^{11} -0.321345 q^{12} -1.00000 q^{13} +0.278457 q^{15} +1.60193 q^{16} -3.56633 q^{17} +1.94104 q^{18} -0.985255 q^{19} -2.13436 q^{20} +1.25037 q^{22} +1.69605 q^{23} +0.479573 q^{24} -3.15050 q^{25} +0.656184 q^{26} -1.21994 q^{27} +6.54835 q^{29} -0.182719 q^{30} -7.69550 q^{31} -5.73556 q^{32} -0.390160 q^{33} +2.34017 q^{34} +4.64247 q^{36} -2.02005 q^{37} +0.646508 q^{38} -0.204753 q^{39} +3.18530 q^{40} -9.88779 q^{41} -3.16639 q^{43} +2.99055 q^{44} -4.02287 q^{45} -1.11292 q^{46} -7.76416 q^{47} +0.328001 q^{48} +2.06731 q^{50} -0.730219 q^{51} +1.56942 q^{52} +0.354194 q^{53} +0.800502 q^{54} -2.59143 q^{55} -0.201734 q^{57} -4.29692 q^{58} -2.16385 q^{59} -0.437016 q^{60} -12.2002 q^{61} +5.04967 q^{62} +0.559715 q^{64} -1.35996 q^{65} +0.256017 q^{66} +11.3134 q^{67} +5.59709 q^{68} +0.347272 q^{69} -9.05268 q^{71} -6.92840 q^{72} +7.13619 q^{73} +1.32553 q^{74} -0.645076 q^{75} +1.54628 q^{76} +0.134356 q^{78} -5.39629 q^{79} +2.17857 q^{80} +8.62444 q^{81} +6.48821 q^{82} +2.03494 q^{83} -4.85008 q^{85} +2.07773 q^{86} +1.34080 q^{87} -4.46309 q^{88} +6.89864 q^{89} +2.63974 q^{90} -2.66182 q^{92} -1.57568 q^{93} +5.09472 q^{94} -1.33991 q^{95} -1.17437 q^{96} +14.6223 q^{97} +5.63665 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{3} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 6 q^{9}+O(q^{10}) \) 6 * q - 8 * q^3 + 4 * q^4 - 6 * q^5 - 4 * q^6 + 6 * q^9	\( 6 q - 8 q^{3} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 6 q^{9} - 4 q^{10} + 4 q^{11} + 4 q^{12} - 6 q^{13} + 12 q^{15} - 16 q^{17} - 4 q^{18} - 2 q^{19} - 16 q^{20} - 12 q^{22} - 6 q^{23} - 12 q^{24} - 4 q^{25} - 20 q^{27} - 6 q^{29} - 6 q^{31} - 20 q^{32} - 4 q^{33} - 24 q^{36} - 8 q^{38} + 8 q^{39} - 4 q^{40} + 8 q^{41} + 2 q^{43} - 4 q^{44} - 14 q^{45} + 8 q^{46} - 30 q^{47} + 8 q^{48} + 8 q^{50} - 4 q^{51} - 4 q^{52} - 14 q^{53} + 48 q^{54} + 8 q^{55} + 4 q^{57} - 8 q^{58} - 24 q^{59} + 12 q^{60} - 28 q^{62} - 20 q^{64} + 6 q^{65} + 4 q^{66} + 16 q^{67} - 28 q^{68} + 20 q^{69} + 8 q^{71} + 28 q^{72} + 6 q^{73} - 12 q^{74} - 12 q^{75} + 16 q^{76} + 4 q^{78} - 22 q^{79} + 28 q^{80} + 46 q^{81} + 40 q^{82} - 50 q^{83} - 8 q^{85} - 16 q^{86} + 16 q^{87} - 44 q^{88} - 26 q^{89} + 40 q^{90} + 20 q^{92} + 16 q^{93} + 32 q^{94} - 6 q^{95} + 20 q^{96} + 14 q^{97} + 12 q^{99}+O(q^{100}) \) 6 * q - 8 * q^3 + 4 * q^4 - 6 * q^5 - 4 * q^6 + 6 * q^9 - 4 * q^10 + 4 * q^11 + 4 * q^12 - 6 * q^13 + 12 * q^15 - 16 * q^17 - 4 * q^18 - 2 * q^19 - 16 * q^20 - 12 * q^22 - 6 * q^23 - 12 * q^24 - 4 * q^25 - 20 * q^27 - 6 * q^29 - 6 * q^31 - 20 * q^32 - 4 * q^33 - 24 * q^36 - 8 * q^38 + 8 * q^39 - 4 * q^40 + 8 * q^41 + 2 * q^43 - 4 * q^44 - 14 * q^45 + 8 * q^46 - 30 * q^47 + 8 * q^48 + 8 * q^50 - 4 * q^51 - 4 * q^52 - 14 * q^53 + 48 * q^54 + 8 * q^55 + 4 * q^57 - 8 * q^58 - 24 * q^59 + 12 * q^60 - 28 * q^62 - 20 * q^64 + 6 * q^65 + 4 * q^66 + 16 * q^67 - 28 * q^68 + 20 * q^69 + 8 * q^71 + 28 * q^72 + 6 * q^73 - 12 * q^74 - 12 * q^75 + 16 * q^76 + 4 * q^78 - 22 * q^79 + 28 * q^80 + 46 * q^81 + 40 * q^82 - 50 * q^83 - 8 * q^85 - 16 * q^86 + 16 * q^87 - 44 * q^88 - 26 * q^89 + 40 * q^90 + 20 * q^92 + 16 * q^93 + 32 * q^94 - 6 * q^95 + 20 * q^96 + 14 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more