LMFDB - Embedded newform 8470.2.a.dg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 16x^{6} + 69x^{4} - 10x^{3} - 70x^{2} + 10x + 5 $ x^8 - 16x^6 + 69x^4 - 10x^3 - 70x^2 + 10*x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.12455$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.12455 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.12455 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.73538 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.12455 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.12455 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.73538 q^{9} +1.00000 q^{10} -1.12455 q^{12} -7.06945 q^{13} -1.00000 q^{14} +1.12455 q^{15} +1.00000 q^{16} -3.44477 q^{17} +1.73538 q^{18} -3.43616 q^{19} -1.00000 q^{20} -1.12455 q^{21} +3.63913 q^{23} +1.12455 q^{24} +1.00000 q^{25} +7.06945 q^{26} +5.32519 q^{27} +1.00000 q^{28} +8.69298 q^{29} -1.12455 q^{30} -4.52145 q^{31} -1.00000 q^{32} +3.44477 q^{34} -1.00000 q^{35} -1.73538 q^{36} -0.780576 q^{37} +3.43616 q^{38} +7.94999 q^{39} +1.00000 q^{40} +5.01886 q^{41} +1.12455 q^{42} +5.39500 q^{43} +1.73538 q^{45} -3.63913 q^{46} +9.82775 q^{47} -1.12455 q^{48} +1.00000 q^{49} -1.00000 q^{50} +3.87383 q^{51} -7.06945 q^{52} +11.3718 q^{53} -5.32519 q^{54} -1.00000 q^{56} +3.86415 q^{57} -8.69298 q^{58} -10.5247 q^{59} +1.12455 q^{60} -0.456235 q^{61} +4.52145 q^{62} -1.73538 q^{63} +1.00000 q^{64} +7.06945 q^{65} +10.4937 q^{67} -3.44477 q^{68} -4.09241 q^{69} +1.00000 q^{70} -6.81376 q^{71} +1.73538 q^{72} +15.1151 q^{73} +0.780576 q^{74} -1.12455 q^{75} -3.43616 q^{76} -7.94999 q^{78} +16.2368 q^{79} -1.00000 q^{80} -0.782335 q^{81} -5.01886 q^{82} -7.40344 q^{83} -1.12455 q^{84} +3.44477 q^{85} -5.39500 q^{86} -9.77573 q^{87} -4.97143 q^{89} -1.73538 q^{90} -7.06945 q^{91} +3.63913 q^{92} +5.08461 q^{93} -9.82775 q^{94} +3.43616 q^{95} +1.12455 q^{96} +2.88993 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} + 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^5 + 8 * q^7 - 8 * q^8 + 8 * q^9	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} + 8 q^{7} - 8 q^{8} + 8 q^{9} + 8 q^{10} + q^{13} - 8 q^{14} + 8 q^{16} - 6 q^{17} - 8 q^{18} - 5 q^{19} - 8 q^{20} + 10 q^{23} + 8 q^{25} - q^{26} + 8 q^{28} - 3 q^{29} - 8 q^{31} - 8 q^{32} + 6 q^{34} - 8 q^{35} + 8 q^{36} - 6 q^{37} + 5 q^{38} - 35 q^{39} + 8 q^{40} - 11 q^{41} + 5 q^{43} - 8 q^{45} - 10 q^{46} - 15 q^{47} + 8 q^{49} - 8 q^{50} + 6 q^{51} + q^{52} - 16 q^{53} - 8 q^{56} - 38 q^{57} + 3 q^{58} - 9 q^{59} - 32 q^{61} + 8 q^{62} + 8 q^{63} + 8 q^{64} - q^{65} + 33 q^{67} - 6 q^{68} - 22 q^{69} + 8 q^{70} + 11 q^{71} - 8 q^{72} + 34 q^{73} + 6 q^{74} - 5 q^{76} + 35 q^{78} - 31 q^{79} - 8 q^{80} + 20 q^{81} + 11 q^{82} - 50 q^{83} + 6 q^{85} - 5 q^{86} + 12 q^{87} + q^{89} + 8 q^{90} + q^{91} + 10 q^{92} + 26 q^{93} + 15 q^{94} + 5 q^{95} - 4 q^{97} - 8 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^5 + 8 * q^7 - 8 * q^8 + 8 * q^9 + 8 * q^10 + q^13 - 8 * q^14 + 8 * q^16 - 6 * q^17 - 8 * q^18 - 5 * q^19 - 8 * q^20 + 10 * q^23 + 8 * q^25 - q^26 + 8 * q^28 - 3 * q^29 - 8 * q^31 - 8 * q^32 + 6 * q^34 - 8 * q^35 + 8 * q^36 - 6 * q^37 + 5 * q^38 - 35 * q^39 + 8 * q^40 - 11 * q^41 + 5 * q^43 - 8 * q^45 - 10 * q^46 - 15 * q^47 + 8 * q^49 - 8 * q^50 + 6 * q^51 + q^52 - 16 * q^53 - 8 * q^56 - 38 * q^57 + 3 * q^58 - 9 * q^59 - 32 * q^61 + 8 * q^62 + 8 * q^63 + 8 * q^64 - q^65 + 33 * q^67 - 6 * q^68 - 22 * q^69 + 8 * q^70 + 11 * q^71 - 8 * q^72 + 34 * q^73 + 6 * q^74 - 5 * q^76 + 35 * q^78 - 31 * q^79 - 8 * q^80 + 20 * q^81 + 11 * q^82 - 50 * q^83 + 6 * q^85 - 5 * q^86 + 12 * q^87 + q^89 + 8 * q^90 + q^91 + 10 * q^92 + 26 * q^93 + 15 * q^94 + 5 * q^95 - 4 * q^97 - 8 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.12455	−0.649262	−0.324631	−	0.945841i	$-0.605240\pi$
$3$	−1.12455	−0.649262	−0.324631	+	0.945841i	$0.605240\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.12455	0.459097
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	−1.73538	−0.578459
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	−1.12455	−0.324631
$13$	−7.06945	−1.96071	−0.980357	−	0.197232i	$-0.936805\pi$
$13$	−7.06945	−1.96071	−0.980357	+	0.197232i	$0.936805\pi$
$14$	−1.00000	−0.267261
$15$	1.12455	0.290359
$16$	1.00000	0.250000
$17$	−3.44477	−0.835479	−0.417740	−	0.908567i	$-0.637178\pi$
$17$	−3.44477	−0.835479	−0.417740	+	0.908567i	$0.637178\pi$
$18$	1.73538	0.409032
$19$	−3.43616	−0.788308	−0.394154	−	0.919044i	$-0.628962\pi$
$19$	−3.43616	−0.788308	−0.394154	+	0.919044i	$0.628962\pi$
$20$	−1.00000	−0.223607
$21$	−1.12455	−0.245398
$22$	0	0
$23$	3.63913	0.758812	0.379406	−	0.925230i	$-0.376128\pi$
$23$	3.63913	0.758812	0.379406	+	0.925230i	$0.376128\pi$
$24$	1.12455	0.229549
$25$	1.00000	0.200000
$26$	7.06945	1.38643
$27$	5.32519	1.02483
$28$	1.00000	0.188982
$29$	8.69298	1.61425	0.807123	−	0.590383i	$-0.201023\pi$
$29$	8.69298	1.61425	0.807123	+	0.590383i	$0.201023\pi$
$30$	−1.12455	−0.205315
$31$	−4.52145	−0.812076	−0.406038	−	0.913856i	$-0.633090\pi$
$31$	−4.52145	−0.812076	−0.406038	+	0.913856i	$0.633090\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.44477	0.590773
$35$	−1.00000	−0.169031
$36$	−1.73538	−0.289230
$37$	−0.780576	−0.128326	−0.0641630	−	0.997939i	$-0.520438\pi$
$37$	−0.780576	−0.128326	−0.0641630	+	0.997939i	$0.520438\pi$
$38$	3.43616	0.557418
$39$	7.94999	1.27302
$40$	1.00000	0.158114
$41$	5.01886	0.783814	0.391907	−	0.920005i	$-0.371815\pi$
$41$	5.01886	0.783814	0.391907	+	0.920005i	$0.371815\pi$
$42$	1.12455	0.173523
$43$	5.39500	0.822729	0.411365	−	0.911471i	$-0.365052\pi$
$43$	5.39500	0.822729	0.411365	+	0.911471i	$0.365052\pi$
$44$	0	0
$45$	1.73538	0.258695
$46$	−3.63913	−0.536561
$47$	9.82775	1.43352	0.716762	−	0.697318i	$-0.245623\pi$
$47$	9.82775	1.43352	0.716762	+	0.697318i	$0.245623\pi$
$48$	−1.12455	−0.162315
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	3.87383	0.542445
$52$	−7.06945	−0.980357
$53$	11.3718	1.56204	0.781020	−	0.624507i	$-0.214700\pi$
$53$	11.3718	1.56204	0.781020	+	0.624507i	$0.214700\pi$
$54$	−5.32519	−0.724667
$55$	0	0
$56$	−1.00000	−0.133631
$57$	3.86415	0.511819
$58$	−8.69298	−1.14144
$59$	−10.5247	−1.37020	−0.685099	−	0.728450i	$-0.740241\pi$
$59$	−10.5247	−1.37020	−0.685099	+	0.728450i	$0.740241\pi$
$60$	1.12455	0.145179
$61$	−0.456235	−0.0584149	−0.0292074	−	0.999573i	$-0.509298\pi$
$61$	−0.456235	−0.0584149	−0.0292074	+	0.999573i	$0.509298\pi$
$62$	4.52145	0.574224
$63$	−1.73538	−0.218637
$64$	1.00000	0.125000
$65$	7.06945	0.876858
$66$	0	0
$67$	10.4937	1.28201	0.641006	−	0.767536i	$-0.278517\pi$
$67$	10.4937	1.28201	0.641006	+	0.767536i	$0.278517\pi$
$68$	−3.44477	−0.417740
$69$	−4.09241	−0.492668
$70$	1.00000	0.119523
$71$	−6.81376	−0.808645	−0.404322	−	0.914617i	$-0.632493\pi$
$71$	−6.81376	−0.808645	−0.404322	+	0.914617i	$0.632493\pi$
$72$	1.73538	0.204516
$73$	15.1151	1.76909	0.884547	−	0.466452i	$-0.154468\pi$
$73$	15.1151	1.76909	0.884547	+	0.466452i	$0.154468\pi$
$74$	0.780576	0.0907401
$75$	−1.12455	−0.129852
$76$	−3.43616	−0.394154
$77$	0	0
$78$	−7.94999	−0.900159
$79$	16.2368	1.82678	0.913390	−	0.407087i	$-0.133455\pi$
$79$	16.2368	1.82678	0.913390	+	0.407087i	$0.133455\pi$
$80$	−1.00000	−0.111803
$81$	−0.782335	−0.0869261
$82$	−5.01886	−0.554240
$83$	−7.40344	−0.812633	−0.406317	−	0.913732i	$-0.633187\pi$
$83$	−7.40344	−0.812633	−0.406317	+	0.913732i	$0.633187\pi$
$84$	−1.12455	−0.122699
$85$	3.44477	0.373638
$86$	−5.39500	−0.581757
$87$	−9.77573	−1.04807
$88$	0	0
$89$	−4.97143	−0.526970	−0.263485	−	0.964663i	$-0.584872\pi$
$89$	−4.97143	−0.526970	−0.263485	+	0.964663i	$0.584872\pi$
$90$	−1.73538	−0.182925
$91$	−7.06945	−0.741080
$92$	3.63913	0.379406
$93$	5.08461	0.527250
$94$	−9.82775	−1.01365
$95$	3.43616	0.352542
$96$	1.12455	0.114774
$97$	2.88993	0.293427	0.146714	−	0.989179i	$-0.453130\pi$
$97$	2.88993	0.293427	0.146714	+	0.989179i	$0.453130\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−12.2266	−1.21659	−0.608295	−	0.793711i	$-0.708146\pi$
$101$	−12.2266	−1.21659	−0.608295	+	0.793711i	$0.708146\pi$
$102$	−3.87383	−0.383566
$103$	3.70633	0.365196	0.182598	−	0.983188i	$-0.441549\pi$
$103$	3.70633	0.365196	0.182598	+	0.983188i	$0.441549\pi$
$104$	7.06945	0.693217
$105$	1.12455	0.109745
$106$	−11.3718	−1.10453
$107$	2.85107	0.275623	0.137812	−	0.990458i	$-0.455993\pi$
$107$	2.85107	0.275623	0.137812	+	0.990458i	$0.455993\pi$
$108$	5.32519	0.512417
$109$	−9.80121	−0.938786	−0.469393	−	0.882989i	$-0.655527\pi$
$109$	−9.80121	−0.938786	−0.469393	+	0.882989i	$0.655527\pi$
$110$	0	0
$111$	0.877800	0.0833171
$112$	1.00000	0.0944911
$113$	10.8032	1.01628	0.508140	−	0.861275i	$-0.330333\pi$
$113$	10.8032	1.01628	0.508140	+	0.861275i	$0.330333\pi$
$114$	−3.86415	−0.361910
$115$	−3.63913	−0.339351
$116$	8.69298	0.807123
$117$	12.2682	1.13419
$118$	10.5247	0.968877
$119$	−3.44477	−0.315781
$120$	−1.12455	−0.102657
$121$	0	0
$122$	0.456235	0.0413055
$123$	−5.64398	−0.508901
$124$	−4.52145	−0.406038
$125$	−1.00000	−0.0894427
$126$	1.73538	0.154600
$127$	−21.2610	−1.88661	−0.943303	−	0.331934i	$-0.892299\pi$
$127$	−21.2610	−1.88661	−0.943303	+	0.331934i	$0.892299\pi$
$128$	−1.00000	−0.0883883
$129$	−6.06697	−0.534167
$130$	−7.06945	−0.620032
$131$	−10.2637	−0.896747	−0.448373	−	0.893846i	$-0.647997\pi$
$131$	−10.2637	−0.896747	−0.448373	+	0.893846i	$0.647997\pi$
$132$	0	0
$133$	−3.43616	−0.297953
$134$	−10.4937	−0.906519
$135$	−5.32519	−0.458319
$136$	3.44477	0.295386
$137$	−0.760911	−0.0650090	−0.0325045	−	0.999472i	$-0.510348\pi$
$137$	−0.760911	−0.0650090	−0.0325045	+	0.999472i	$0.510348\pi$
$138$	4.09241	0.348369
$139$	−17.6377	−1.49601	−0.748007	−	0.663691i	$-0.768989\pi$
$139$	−17.6377	−1.49601	−0.748007	+	0.663691i	$0.768989\pi$
$140$	−1.00000	−0.0845154
$141$	−11.0518	−0.930733
$142$	6.81376	0.571798
$143$	0	0
$144$	−1.73538	−0.144615
$145$	−8.69298	−0.721913
$146$	−15.1151	−1.25094
$147$	−1.12455	−0.0927517
$148$	−0.780576	−0.0641630
$149$	−0.778180	−0.0637510	−0.0318755	−	0.999492i	$-0.510148\pi$
$149$	−0.778180	−0.0637510	−0.0318755	+	0.999492i	$0.510148\pi$
$150$	1.12455	0.0918195
$151$	−13.4154	−1.09173	−0.545864	−	0.837874i	$-0.683798\pi$
$151$	−13.4154	−1.09173	−0.545864	+	0.837874i	$0.683798\pi$
$152$	3.43616	0.278709
$153$	5.97797	0.483290
$154$	0	0
$155$	4.52145	0.363171
$156$	7.94999	0.636508
$157$	−24.0361	−1.91829	−0.959146	−	0.282912i	$-0.908700\pi$
$157$	−24.0361	−1.91829	−0.959146	+	0.282912i	$0.908700\pi$
$158$	−16.2368	−1.29173
$159$	−12.7882	−1.01417
$160$	1.00000	0.0790569
$161$	3.63913	0.286804
$162$	0.782335	0.0614660
$163$	0.905391	0.0709157	0.0354579	−	0.999371i	$-0.488711\pi$
$163$	0.905391	0.0709157	0.0354579	+	0.999371i	$0.488711\pi$
$164$	5.01886	0.391907
$165$	0	0
$166$	7.40344	0.574618
$167$	3.64082	0.281735	0.140867	−	0.990028i	$-0.455011\pi$
$167$	3.64082	0.281735	0.140867	+	0.990028i	$0.455011\pi$
$168$	1.12455	0.0867613
$169$	36.9772	2.84440
$170$	−3.44477	−0.264202
$171$	5.96303	0.456004
$172$	5.39500	0.411365
$173$	3.17934	0.241721	0.120860	−	0.992670i	$-0.461435\pi$
$173$	3.17934	0.241721	0.120860	+	0.992670i	$0.461435\pi$
$174$	9.77573	0.741096
$175$	1.00000	0.0755929
$176$	0	0
$177$	11.8356	0.889618
$178$	4.97143	0.372624
$179$	4.48137	0.334953	0.167477	−	0.985876i	$-0.446438\pi$
$179$	4.48137	0.334953	0.167477	+	0.985876i	$0.446438\pi$
$180$	1.73538	0.129347
$181$	−9.02937	−0.671148	−0.335574	−	0.942014i	$-0.608930\pi$
$181$	−9.02937	−0.671148	−0.335574	+	0.942014i	$0.608930\pi$
$182$	7.06945	0.524023
$183$	0.513061	0.0379265
$184$	−3.63913	−0.268281
$185$	0.780576	0.0573891
$186$	−5.08461	−0.372822
$187$	0	0
$188$	9.82775	0.716762
$189$	5.32519	0.387351
$190$	−3.43616	−0.249285
$191$	2.90870	0.210466	0.105233	−	0.994448i	$-0.466441\pi$
$191$	2.90870	0.210466	0.105233	+	0.994448i	$0.466441\pi$
$192$	−1.12455	−0.0811577
$193$	7.77320	0.559527	0.279764	−	0.960069i	$-0.409744\pi$
$193$	7.77320	0.559527	0.279764	+	0.960069i	$0.409744\pi$
$194$	−2.88993	−0.207485
$195$	−7.94999	−0.569310
$196$	1.00000	0.0714286
$197$	−26.2257	−1.86851	−0.934254	−	0.356609i	$-0.883933\pi$
$197$	−26.2257	−1.86851	−0.934254	+	0.356609i	$0.883933\pi$
$198$	0	0
$199$	5.64963	0.400491	0.200246	−	0.979746i	$-0.435826\pi$
$199$	5.64963	0.400491	0.200246	+	0.979746i	$0.435826\pi$
$200$	−1.00000	−0.0707107
$201$	−11.8008	−0.832361
$202$	12.2266	0.860260
$203$	8.69298	0.610128
$204$	3.87383	0.271222
$205$	−5.01886	−0.350532
$206$	−3.70633	−0.258233
$207$	−6.31527	−0.438942
$208$	−7.06945	−0.490178
$209$	0	0
$210$	−1.12455	−0.0776016
$211$	−7.42973	−0.511484	−0.255742	−	0.966745i	$-0.582320\pi$
$211$	−7.42973	−0.511484	−0.255742	+	0.966745i	$0.582320\pi$
$212$	11.3718	0.781020
$213$	7.66245	0.525022
$214$	−2.85107	−0.194895
$215$	−5.39500	−0.367936
$216$	−5.32519	−0.362333
$217$	−4.52145	−0.306936
$218$	9.80121	0.663822
$219$	−16.9978	−1.14860
$220$	0	0
$221$	24.3526	1.63814
$222$	−0.877800	−0.0589141
$223$	1.51167	0.101229	0.0506145	−	0.998718i	$-0.483882\pi$
$223$	1.51167	0.101229	0.0506145	+	0.998718i	$0.483882\pi$
$224$	−1.00000	−0.0668153
$225$	−1.73538	−0.115692
$226$	−10.8032	−0.718618
$227$	−13.8558	−0.919640	−0.459820	−	0.888012i	$-0.652086\pi$
$227$	−13.8558	−0.919640	−0.459820	+	0.888012i	$0.652086\pi$
$228$	3.86415	0.255909
$229$	20.5463	1.35774	0.678870	−	0.734258i	$-0.262470\pi$
$229$	20.5463	1.35774	0.678870	+	0.734258i	$0.262470\pi$
$230$	3.63913	0.239957
$231$	0	0
$232$	−8.69298	−0.570722
$233$	−16.2345	−1.06356	−0.531780	−	0.846883i	$-0.678477\pi$
$233$	−16.2345	−1.06356	−0.531780	+	0.846883i	$0.678477\pi$
$234$	−12.2682	−0.801995
$235$	−9.82775	−0.641092
$236$	−10.5247	−0.685099
$237$	−18.2591	−1.18606
$238$	3.44477	0.223291
$239$	0.392921	0.0254160	0.0127080	−	0.999919i	$-0.495955\pi$
$239$	0.392921	0.0254160	0.0127080	+	0.999919i	$0.495955\pi$
$240$	1.12455	0.0725897
$241$	3.32681	0.214298	0.107149	−	0.994243i	$-0.465828\pi$
$241$	3.32681	0.214298	0.107149	+	0.994243i	$0.465828\pi$
$242$	0	0
$243$	−15.0958	−0.968395
$244$	−0.456235	−0.0292074
$245$	−1.00000	−0.0638877
$246$	5.64398	0.359847
$247$	24.2918	1.54565
$248$	4.52145	0.287112
$249$	8.32557	0.527612
$250$	1.00000	0.0632456
$251$	17.2755	1.09042	0.545211	−	0.838299i	$-0.316450\pi$
$251$	17.2755	1.09042	0.545211	+	0.838299i	$0.316450\pi$
$252$	−1.73538	−0.109318
$253$	0	0
$254$	21.2610	1.33403
$255$	−3.87383	−0.242589
$256$	1.00000	0.0625000
$257$	−0.448545	−0.0279795	−0.0139897	−	0.999902i	$-0.504453\pi$
$257$	−0.448545	−0.0279795	−0.0139897	+	0.999902i	$0.504453\pi$
$258$	6.06697	0.377713
$259$	−0.780576	−0.0485026
$260$	7.06945	0.438429
$261$	−15.0856	−0.933775
$262$	10.2637	0.634096
$263$	−26.1440	−1.61211	−0.806053	−	0.591843i	$-0.798400\pi$
$263$	−26.1440	−1.61211	−0.806053	+	0.591843i	$0.798400\pi$
$264$	0	0
$265$	−11.3718	−0.698565
$266$	3.43616	0.210684
$267$	5.59064	0.342142
$268$	10.4937	0.641006
$269$	14.1617	0.863453	0.431727	−	0.902004i	$-0.357904\pi$
$269$	14.1617	0.863453	0.431727	+	0.902004i	$0.357904\pi$
$270$	5.32519	0.324081
$271$	0.934211	0.0567493	0.0283746	−	0.999597i	$-0.490967\pi$
$271$	0.934211	0.0567493	0.0283746	+	0.999597i	$0.490967\pi$
$272$	−3.44477	−0.208870
$273$	7.94999	0.481155
$274$	0.760911	0.0459683
$275$	0	0
$276$	−4.09241	−0.246334
$277$	25.3249	1.52162	0.760812	−	0.648973i	$-0.224801\pi$
$277$	25.3249	1.52162	0.760812	+	0.648973i	$0.224801\pi$
$278$	17.6377	1.05784
$279$	7.84641	0.469752
$280$	1.00000	0.0597614
$281$	−27.7779	−1.65709	−0.828545	−	0.559922i	$-0.810831\pi$
$281$	−27.7779	−1.65709	−0.828545	+	0.559922i	$0.810831\pi$
$282$	11.0518	0.658127
$283$	19.5153	1.16006	0.580032	−	0.814594i	$-0.303040\pi$
$283$	19.5153	1.16006	0.580032	+	0.814594i	$0.303040\pi$
$284$	−6.81376	−0.404322
$285$	−3.86415	−0.228892
$286$	0	0
$287$	5.01886	0.296254
$288$	1.73538	0.102258
$289$	−5.13357	−0.301975
$290$	8.69298	0.510469
$291$	−3.24988	−0.190511
$292$	15.1151	0.884547
$293$	13.3505	0.779945	0.389972	−	0.920827i	$-0.372485\pi$
$293$	13.3505	0.779945	0.389972	+	0.920827i	$0.372485\pi$
$294$	1.12455	0.0655854
$295$	10.5247	0.612771
$296$	0.780576	0.0453701
$297$	0	0
$298$	0.778180	0.0450788
$299$	−25.7267	−1.48781
$300$	−1.12455	−0.0649262
$301$	5.39500	0.310962
$302$	13.4154	0.771968
$303$	13.7495	0.789886
$304$	−3.43616	−0.197077
$305$	0.456235	0.0261239
$306$	−5.97797	−0.341738
$307$	−25.7848	−1.47162	−0.735809	−	0.677189i	$-0.763198\pi$
$307$	−25.7848	−1.47162	−0.735809	+	0.677189i	$0.763198\pi$
$308$	0	0
$309$	−4.16798	−0.237108
$310$	−4.52145	−0.256801
$311$	5.11540	0.290068	0.145034	−	0.989427i	$-0.453671\pi$
$311$	5.11540	0.290068	0.145034	+	0.989427i	$0.453671\pi$
$312$	−7.94999	−0.450079
$313$	16.5571	0.935861	0.467930	−	0.883765i	$-0.345000\pi$
$313$	16.5571	0.935861	0.467930	+	0.883765i	$0.345000\pi$
$314$	24.0361	1.35644
$315$	1.73538	0.0977774
$316$	16.2368	0.913390
$317$	15.3500	0.862143	0.431072	−	0.902318i	$-0.358136\pi$
$317$	15.3500	0.862143	0.431072	+	0.902318i	$0.358136\pi$
$318$	12.7882	0.717128
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−3.20618	−0.178952
$322$	−3.63913	−0.202801
$323$	11.8368	0.658615
$324$	−0.782335	−0.0434630
$325$	−7.06945	−0.392143
$326$	−0.905391	−0.0501450
$327$	11.0220	0.609518
$328$	−5.01886	−0.277120
$329$	9.82775	0.541821
$330$	0	0
$331$	−6.59151	−0.362302	−0.181151	−	0.983455i	$-0.557982\pi$
$331$	−6.59151	−0.362302	−0.181151	+	0.983455i	$0.557982\pi$
$332$	−7.40344	−0.406317
$333$	1.35459	0.0742313
$334$	−3.64082	−0.199217
$335$	−10.4937	−0.573333
$336$	−1.12455	−0.0613495
$337$	19.1547	1.04342	0.521711	−	0.853123i	$-0.325294\pi$
$337$	19.1547	1.04342	0.521711	+	0.853123i	$0.325294\pi$
$338$	−36.9772	−2.01129
$339$	−12.1488	−0.659832
$340$	3.44477	0.186819
$341$	0	0
$342$	−5.96303	−0.322444
$343$	1.00000	0.0539949
$344$	−5.39500	−0.290879
$345$	4.09241	0.220328
$346$	−3.17934	−0.170922
$347$	9.00265	0.483288	0.241644	−	0.970365i	$-0.422313\pi$
$347$	9.00265	0.483288	0.241644	+	0.970365i	$0.422313\pi$
$348$	−9.77573	−0.524034
$349$	6.72881	0.360185	0.180092	−	0.983650i	$-0.442360\pi$
$349$	6.72881	0.360185	0.180092	+	0.983650i	$0.442360\pi$
$350$	−1.00000	−0.0534522
$351$	−37.6462	−2.00940
$352$	0	0
$353$	−29.9182	−1.59238	−0.796192	−	0.605044i	$-0.793156\pi$
$353$	−29.9182	−1.59238	−0.796192	+	0.605044i	$0.793156\pi$
$354$	−11.8356	−0.629055
$355$	6.81376	0.361637
$356$	−4.97143	−0.263485
$357$	3.87383	0.205025
$358$	−4.48137	−0.236848
$359$	23.5622	1.24356	0.621782	−	0.783190i	$-0.286409\pi$
$359$	23.5622	1.24356	0.621782	+	0.783190i	$0.286409\pi$
$360$	−1.73538	−0.0914624
$361$	−7.19282	−0.378570
$362$	9.02937	0.474573
$363$	0	0
$364$	−7.06945	−0.370540
$365$	−15.1151	−0.791163
$366$	−0.513061	−0.0268181
$367$	−10.1699	−0.530862	−0.265431	−	0.964130i	$-0.585514\pi$
$367$	−10.1699	−0.530862	−0.265431	+	0.964130i	$0.585514\pi$
$368$	3.63913	0.189703
$369$	−8.70961	−0.453404
$370$	−0.780576	−0.0405802
$371$	11.3718	0.590395
$372$	5.08461	0.263625
$373$	−10.7767	−0.557996	−0.278998	−	0.960292i	$-0.590002\pi$
$373$	−10.7767	−0.557996	−0.278998	+	0.960292i	$0.590002\pi$
$374$	0	0
$375$	1.12455	0.0580717
$376$	−9.82775	−0.506827
$377$	−61.4546	−3.16507
$378$	−5.32519	−0.273898
$379$	−5.39945	−0.277351	−0.138676	−	0.990338i	$-0.544285\pi$
$379$	−5.39945	−0.277351	−0.138676	+	0.990338i	$0.544285\pi$
$380$	3.43616	0.176271
$381$	23.9091	1.22490
$382$	−2.90870	−0.148822
$383$	7.17505	0.366628	0.183314	−	0.983054i	$-0.441318\pi$
$383$	7.17505	0.366628	0.183314	+	0.983054i	$0.441318\pi$
$384$	1.12455	0.0573872
$385$	0	0
$386$	−7.77320	−0.395646
$387$	−9.36235	−0.475915
$388$	2.88993	0.146714
$389$	16.5388	0.838549	0.419274	−	0.907860i	$-0.362285\pi$
$389$	16.5388	0.838549	0.419274	+	0.907860i	$0.362285\pi$
$390$	7.94999	0.402563
$391$	−12.5360	−0.633972
$392$	−1.00000	−0.0505076
$393$	11.5421	0.582224
$394$	26.2257	1.32123
$395$	−16.2368	−0.816960
$396$	0	0
$397$	16.7413	0.840221	0.420111	−	0.907473i	$-0.361991\pi$
$397$	16.7413	0.840221	0.420111	+	0.907473i	$0.361991\pi$
$398$	−5.64963	−0.283190
$399$	3.86415	0.193449
$400$	1.00000	0.0500000
$401$	−20.3412	−1.01579	−0.507896	−	0.861419i	$-0.669576\pi$
$401$	−20.3412	−1.01579	−0.507896	+	0.861419i	$0.669576\pi$
$402$	11.8008	0.588568
$403$	31.9641	1.59225
$404$	−12.2266	−0.608295
$405$	0.782335	0.0388745
$406$	−8.69298	−0.431425
$407$	0	0
$408$	−3.87383	−0.191783
$409$	0.443589	0.0219341	0.0109670	−	0.999940i	$-0.496509\pi$
$409$	0.443589	0.0219341	0.0109670	+	0.999940i	$0.496509\pi$
$410$	5.01886	0.247864
$411$	0.855686	0.0422079
$412$	3.70633	0.182598
$413$	−10.5247	−0.517886
$414$	6.31527	0.310379
$415$	7.40344	0.363421
$416$	7.06945	0.346608
$417$	19.8346	0.971305
$418$	0	0
$419$	−18.6545	−0.911330	−0.455665	−	0.890151i	$-0.650599\pi$
$419$	−18.6545	−0.911330	−0.455665	+	0.890151i	$0.650599\pi$
$420$	1.12455	0.0548726
$421$	12.2716	0.598081	0.299040	−	0.954240i	$-0.403333\pi$
$421$	12.2716	0.598081	0.299040	+	0.954240i	$0.403333\pi$
$422$	7.42973	0.361674
$423$	−17.0549	−0.829235
$424$	−11.3718	−0.552264
$425$	−3.44477	−0.167096
$426$	−7.66245	−0.371247
$427$	−0.456235	−0.0220787
$428$	2.85107	0.137812
$429$	0	0
$430$	5.39500	0.260170
$431$	−21.9795	−1.05872	−0.529358	−	0.848398i	$-0.677567\pi$
$431$	−21.9795	−1.05872	−0.529358	+	0.848398i	$0.677567\pi$
$432$	5.32519	0.256208
$433$	22.2345	1.06852	0.534262	−	0.845319i	$-0.320590\pi$
$433$	22.2345	1.06852	0.534262	+	0.845319i	$0.320590\pi$
$434$	4.52145	0.217036
$435$	9.77573	0.468710
$436$	−9.80121	−0.469393
$437$	−12.5046	−0.598178
$438$	16.9978	0.812186
$439$	11.0302	0.526444	0.263222	−	0.964735i	$-0.415215\pi$
$439$	11.0302	0.526444	0.263222	+	0.964735i	$0.415215\pi$
$440$	0	0
$441$	−1.73538	−0.0826370
$442$	−24.3526	−1.15834
$443$	30.8702	1.46669	0.733345	−	0.679857i	$-0.237958\pi$
$443$	30.8702	1.46669	0.733345	+	0.679857i	$0.237958\pi$
$444$	0.877800	0.0416586
$445$	4.97143	0.235668
$446$	−1.51167	−0.0715797
$447$	0.875106	0.0413911
$448$	1.00000	0.0472456
$449$	−13.9952	−0.660475	−0.330238	−	0.943898i	$-0.607129\pi$
$449$	−13.9952	−0.660475	−0.330238	+	0.943898i	$0.607129\pi$
$450$	1.73538	0.0818065
$451$	0	0
$452$	10.8032	0.508140
$453$	15.0863	0.708817
$454$	13.8558	0.650284
$455$	7.06945	0.331421
$456$	−3.86415	−0.180955
$457$	7.06533	0.330502	0.165251	−	0.986252i	$-0.447157\pi$
$457$	7.06533	0.330502	0.165251	+	0.986252i	$0.447157\pi$
$458$	−20.5463	−0.960067
$459$	−18.3440	−0.856227
$460$	−3.63913	−0.169676
$461$	−10.6011	−0.493744	−0.246872	−	0.969048i	$-0.579403\pi$
$461$	−10.6011	−0.493744	−0.246872	+	0.969048i	$0.579403\pi$
$462$	0	0
$463$	4.59137	0.213379	0.106690	−	0.994292i	$-0.465975\pi$
$463$	4.59137	0.213379	0.106690	+	0.994292i	$0.465975\pi$
$464$	8.69298	0.403561
$465$	−5.08461	−0.235793
$466$	16.2345	0.752050
$467$	−30.9287	−1.43121	−0.715606	−	0.698504i	$-0.753849\pi$
$467$	−30.9287	−1.43121	−0.715606	+	0.698504i	$0.753849\pi$
$468$	12.2682	0.567096
$469$	10.4937	0.484555
$470$	9.82775	0.453320
$471$	27.0299	1.24547
$472$	10.5247	0.484438
$473$	0	0
$474$	18.2591	0.838670
$475$	−3.43616	−0.157662
$476$	−3.44477	−0.157891
$477$	−19.7344	−0.903576
$478$	−0.392921	−0.0179718
$479$	−3.01362	−0.137696	−0.0688480	−	0.997627i	$-0.521932\pi$
$479$	−3.01362	−0.137696	−0.0688480	+	0.997627i	$0.521932\pi$
$480$	−1.12455	−0.0513287
$481$	5.51825	0.251610
$482$	−3.32681	−0.151532
$483$	−4.09241	−0.186211
$484$	0	0
$485$	−2.88993	−0.131225
$486$	15.0958	0.684759
$487$	3.00551	0.136193	0.0680963	−	0.997679i	$-0.478307\pi$
$487$	3.00551	0.136193	0.0680963	+	0.997679i	$0.478307\pi$
$488$	0.456235	0.0206528
$489$	−1.01816	−0.0460429
$490$	1.00000	0.0451754
$491$	−34.6723	−1.56474	−0.782369	−	0.622815i	$-0.785989\pi$
$491$	−34.6723	−1.56474	−0.782369	+	0.622815i	$0.785989\pi$
$492$	−5.64398	−0.254450
$493$	−29.9453	−1.34867
$494$	−24.2918	−1.09294
$495$	0	0
$496$	−4.52145	−0.203019
$497$	−6.81376	−0.305639
$498$	−8.32557	−0.373078
$499$	−5.25216	−0.235119	−0.117559	−	0.993066i	$-0.537507\pi$
$499$	−5.25216	−0.235119	−0.117559	+	0.993066i	$0.537507\pi$
$500$	−1.00000	−0.0447214
$501$	−4.09430	−0.182920
$502$	−17.2755	−0.771044
$503$	−33.6525	−1.50049	−0.750245	−	0.661160i	$-0.770064\pi$
$503$	−33.6525	−1.50049	−0.750245	+	0.661160i	$0.770064\pi$
$504$	1.73538	0.0772998
$505$	12.2266	0.544076
$506$	0	0
$507$	−41.5828	−1.84676
$508$	−21.2610	−0.943303
$509$	−20.1633	−0.893724	−0.446862	−	0.894603i	$-0.647458\pi$
$509$	−20.1633	−0.893724	−0.446862	+	0.894603i	$0.647458\pi$
$510$	3.87383	0.171536
$511$	15.1151	0.668654
$512$	−1.00000	−0.0441942
$513$	−18.2982	−0.807885
$514$	0.448545	0.0197845
$515$	−3.70633	−0.163321
$516$	−6.06697	−0.267083
$517$	0	0
$518$	0.780576	0.0342965
$519$	−3.57534	−0.156940
$520$	−7.06945	−0.310016
$521$	−34.1002	−1.49396	−0.746979	−	0.664848i	$-0.768496\pi$
$521$	−34.1002	−1.49396	−0.746979	+	0.664848i	$0.768496\pi$
$522$	15.0856	0.660279
$523$	37.5694	1.64280	0.821398	−	0.570355i	$-0.193194\pi$
$523$	37.5694	1.64280	0.821398	+	0.570355i	$0.193194\pi$
$524$	−10.2637	−0.448373
$525$	−1.12455	−0.0490796
$526$	26.1440	1.13993
$527$	15.5753	0.678472
$528$	0	0
$529$	−9.75670	−0.424204
$530$	11.3718	0.493960
$531$	18.2643	0.792604
$532$	−3.43616	−0.148976
$533$	−35.4806	−1.53683
$534$	−5.59064	−0.241931
$535$	−2.85107	−0.123263
$536$	−10.4937	−0.453260
$537$	−5.03954	−0.217472
$538$	−14.1617	−0.610554
$539$	0	0
$540$	−5.32519	−0.229160
$541$	22.6441	0.973547	0.486773	−	0.873528i	$-0.338174\pi$
$541$	22.6441	0.973547	0.486773	+	0.873528i	$0.338174\pi$
$542$	−0.934211	−0.0401278
$543$	10.1540	0.435751
$544$	3.44477	0.147693
$545$	9.80121	0.419838
$546$	−7.94999	−0.340228
$547$	−24.3655	−1.04179	−0.520896	−	0.853620i	$-0.674402\pi$
$547$	−24.3655	−1.04179	−0.520896	+	0.853620i	$0.674402\pi$
$548$	−0.760911	−0.0325045
$549$	0.791739	0.0337906
$550$	0	0
$551$	−29.8704	−1.27252
$552$	4.09241	0.174184
$553$	16.2368	0.690458
$554$	−25.3249	−1.07595
$555$	−0.877800	−0.0372605
$556$	−17.6377	−0.748007
$557$	−25.1506	−1.06567	−0.532833	−	0.846220i	$-0.678873\pi$
$557$	−25.1506	−1.06567	−0.532833	+	0.846220i	$0.678873\pi$
$558$	−7.84641	−0.332165
$559$	−38.1397	−1.61314
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	27.7779	1.17174
$563$	20.1972	0.851212	0.425606	−	0.904909i	$-0.360061\pi$
$563$	20.1972	0.851212	0.425606	+	0.904909i	$0.360061\pi$
$564$	−11.0518	−0.465366
$565$	−10.8032	−0.454494
$566$	−19.5153	−0.820289
$567$	−0.782335	−0.0328550
$568$	6.81376	0.285899
$569$	−25.8942	−1.08554	−0.542771	−	0.839881i	$-0.682625\pi$
$569$	−25.8942	−1.08554	−0.542771	+	0.839881i	$0.682625\pi$
$570$	3.86415	0.161851
$571$	−22.4564	−0.939771	−0.469885	−	0.882727i	$-0.655705\pi$
$571$	−22.4564	−0.939771	−0.469885	+	0.882727i	$0.655705\pi$
$572$	0	0
$573$	−3.27099	−0.136648
$574$	−5.01886	−0.209483
$575$	3.63913	0.151762
$576$	−1.73538	−0.0723074
$577$	−33.8577	−1.40952	−0.704758	−	0.709448i	$-0.748945\pi$
$577$	−33.8577	−1.40952	−0.704758	+	0.709448i	$0.748945\pi$
$578$	5.13357	0.213528
$579$	−8.74139	−0.363280
$580$	−8.69298	−0.360956
$581$	−7.40344	−0.307146
$582$	3.24988	0.134712
$583$	0	0
$584$	−15.1151	−0.625469
$585$	−12.2682	−0.507226
$586$	−13.3505	−0.551504
$587$	28.5545	1.17857	0.589285	−	0.807925i	$-0.299409\pi$
$587$	28.5545	1.17857	0.589285	+	0.807925i	$0.299409\pi$
$588$	−1.12455	−0.0463758
$589$	15.5364	0.640166
$590$	−10.5247	−0.433295
$591$	29.4923	1.21315
$592$	−0.780576	−0.0320815
$593$	−34.9417	−1.43488	−0.717442	−	0.696618i	$-0.754687\pi$
$593$	−34.9417	−1.43488	−0.717442	+	0.696618i	$0.754687\pi$
$594$	0	0
$595$	3.44477	0.141222
$596$	−0.778180	−0.0318755
$597$	−6.35331	−0.260024
$598$	25.7267	1.05204
$599$	−31.0639	−1.26924	−0.634618	−	0.772826i	$-0.718843\pi$
$599$	−31.0639	−1.26924	−0.634618	+	0.772826i	$0.718843\pi$
$600$	1.12455	0.0459097
$601$	43.2602	1.76462	0.882310	−	0.470668i	$-0.155987\pi$
$601$	43.2602	1.76462	0.882310	+	0.470668i	$0.155987\pi$
$602$	−5.39500	−0.219884
$603$	−18.2106	−0.741591
$604$	−13.4154	−0.545864
$605$	0	0
$606$	−13.7495	−0.558534
$607$	33.1013	1.34354	0.671771	−	0.740759i	$-0.265534\pi$
$607$	33.1013	1.34354	0.671771	+	0.740759i	$0.265534\pi$
$608$	3.43616	0.139355
$609$	−9.77573	−0.396133
$610$	−0.456235	−0.0184724
$611$	−69.4768	−2.81073
$612$	5.97797	0.241645
$613$	17.0269	0.687708	0.343854	−	0.939023i	$-0.388267\pi$
$613$	17.0269	0.687708	0.343854	+	0.939023i	$0.388267\pi$
$614$	25.7848	1.04059
$615$	5.64398	0.227587
$616$	0	0
$617$	3.82168	0.153855	0.0769276	−	0.997037i	$-0.475489\pi$
$617$	3.82168	0.153855	0.0769276	+	0.997037i	$0.475489\pi$
$618$	4.16798	0.167661
$619$	−28.4586	−1.14385	−0.571924	−	0.820307i	$-0.693803\pi$
$619$	−28.4586	−1.14385	−0.571924	+	0.820307i	$0.693803\pi$
$620$	4.52145	0.181586
$621$	19.3791	0.777656
$622$	−5.11540	−0.205109
$623$	−4.97143	−0.199176
$624$	7.94999	0.318254
$625$	1.00000	0.0400000
$626$	−16.5571	−0.661753
$627$	0	0
$628$	−24.0361	−0.959146
$629$	2.68890	0.107214
$630$	−1.73538	−0.0691391
$631$	8.90784	0.354615	0.177308	−	0.984155i	$-0.443261\pi$
$631$	8.90784	0.354615	0.177308	+	0.984155i	$0.443261\pi$
$632$	−16.2368	−0.645864
$633$	8.35514	0.332087
$634$	−15.3500	−0.609627
$635$	21.2610	0.843716
$636$	−12.7882	−0.507086
$637$	−7.06945	−0.280102
$638$	0	0
$639$	11.8244	0.467768
$640$	1.00000	0.0395285
$641$	28.2250	1.11482	0.557410	−	0.830237i	$-0.311795\pi$
$641$	28.2250	1.11482	0.557410	+	0.830237i	$0.311795\pi$
$642$	3.20618	0.126538
$643$	27.7470	1.09423	0.547117	−	0.837056i	$-0.315725\pi$
$643$	27.7470	1.09423	0.547117	+	0.837056i	$0.315725\pi$
$644$	3.63913	0.143402
$645$	6.06697	0.238887
$646$	−11.8368	−0.465711
$647$	26.0679	1.02484	0.512418	−	0.858736i	$-0.328750\pi$
$647$	26.0679	1.02484	0.512418	+	0.858736i	$0.328750\pi$
$648$	0.782335	0.0307330
$649$	0	0
$650$	7.06945	0.277287
$651$	5.08461	0.199282
$652$	0.905391	0.0354579
$653$	10.8574	0.424884	0.212442	−	0.977174i	$-0.431858\pi$
$653$	10.8574	0.424884	0.212442	+	0.977174i	$0.431858\pi$
$654$	−11.0220	−0.430994
$655$	10.2637	0.401037
$656$	5.01886	0.195954
$657$	−26.2305	−1.02335
$658$	−9.82775	−0.383126
$659$	8.76914	0.341597	0.170799	−	0.985306i	$-0.445365\pi$
$659$	8.76914	0.341597	0.170799	+	0.985306i	$0.445365\pi$
$660$	0	0
$661$	−18.7665	−0.729934	−0.364967	−	0.931020i	$-0.618920\pi$
$661$	−18.7665	−0.729934	−0.364967	+	0.931020i	$0.618920\pi$
$662$	6.59151	0.256186
$663$	−27.3859	−1.06358
$664$	7.40344	0.287309
$665$	3.43616	0.133248
$666$	−1.35459	−0.0524894
$667$	31.6349	1.22491
$668$	3.64082	0.140867
$669$	−1.69996	−0.0657241
$670$	10.4937	0.405408
$671$	0	0
$672$	1.12455	0.0433806
$673$	−13.0246	−0.502061	−0.251031	−	0.967979i	$-0.580770\pi$
$673$	−13.0246	−0.502061	−0.251031	+	0.967979i	$0.580770\pi$
$674$	−19.1547	−0.737810
$675$	5.32519	0.204967
$676$	36.9772	1.42220
$677$	10.0804	0.387422	0.193711	−	0.981059i	$-0.437948\pi$
$677$	10.0804	0.387422	0.193711	+	0.981059i	$0.437948\pi$
$678$	12.1488	0.466571
$679$	2.88993	0.110905
$680$	−3.44477	−0.132101
$681$	15.5816	0.597087
$682$	0	0
$683$	−8.98900	−0.343955	−0.171977	−	0.985101i	$-0.555016\pi$
$683$	−8.98900	−0.343955	−0.171977	+	0.985101i	$0.555016\pi$
$684$	5.96303	0.228002
$685$	0.760911	0.0290729
$686$	−1.00000	−0.0381802
$687$	−23.1055	−0.881529
$688$	5.39500	0.205682
$689$	−80.3925	−3.06271
$690$	−4.09241	−0.155795
$691$	45.5777	1.73386	0.866929	−	0.498432i	$-0.166090\pi$
$691$	45.5777	1.73386	0.866929	+	0.498432i	$0.166090\pi$
$692$	3.17934	0.120860
$693$	0	0
$694$	−9.00265	−0.341736
$695$	17.6377	0.669038
$696$	9.77573	0.370548
$697$	−17.2888	−0.654860
$698$	−6.72881	−0.254689
$699$	18.2566	0.690529
$700$	1.00000	0.0377964
$701$	−11.8996	−0.449442	−0.224721	−	0.974423i	$-0.572147\pi$
$701$	−11.8996	−0.449442	−0.224721	+	0.974423i	$0.572147\pi$
$702$	37.6462	1.42086
$703$	2.68218	0.101160
$704$	0	0
$705$	11.0518	0.416236
$706$	29.9182	1.12599
$707$	−12.2266	−0.459828
$708$	11.8356	0.444809
$709$	−51.2048	−1.92304	−0.961518	−	0.274740i	$-0.911408\pi$
$709$	−51.2048	−1.92304	−0.961518	+	0.274740i	$0.911408\pi$
$710$	−6.81376	−0.255716
$711$	−28.1769	−1.05672
$712$	4.97143	0.186312
$713$	−16.4541	−0.616213
$714$	−3.87383	−0.144974
$715$	0	0
$716$	4.48137	0.167477
$717$	−0.441861	−0.0165016
$718$	−23.5622	−0.879333
$719$	−27.8658	−1.03922	−0.519610	−	0.854403i	$-0.673923\pi$
$719$	−27.8658	−1.03922	−0.519610	+	0.854403i	$0.673923\pi$
$720$	1.73538	0.0646737
$721$	3.70633	0.138031
$722$	7.19282	0.267689
$723$	−3.74118	−0.139136
$724$	−9.02937	−0.335574
$725$	8.69298	0.322849
$726$	0	0
$727$	41.3933	1.53519	0.767597	−	0.640933i	$-0.221453\pi$
$727$	41.3933	1.53519	0.767597	+	0.640933i	$0.221453\pi$
$728$	7.06945	0.262011
$729$	19.3230	0.715668
$730$	15.1151	0.559436
$731$	−18.5845	−0.687373
$732$	0.513061	0.0189633
$733$	5.84935	0.216051	0.108025	−	0.994148i	$-0.465547\pi$
$733$	5.84935	0.216051	0.108025	+	0.994148i	$0.465547\pi$
$734$	10.1699	0.375376
$735$	1.12455	0.0414798
$736$	−3.63913	−0.134140
$737$	0	0
$738$	8.70961	0.320605
$739$	−48.5521	−1.78602	−0.893009	−	0.450039i	$-0.851410\pi$
$739$	−48.5521	−1.78602	−0.893009	+	0.450039i	$0.851410\pi$
$740$	0.780576	0.0286945
$741$	−27.3174	−1.00353
$742$	−11.3718	−0.417473
$743$	−33.8585	−1.24215	−0.621073	−	0.783752i	$-0.713303\pi$
$743$	−33.8585	−1.24215	−0.621073	+	0.783752i	$0.713303\pi$
$744$	−5.08461	−0.186411
$745$	0.778180	0.0285103
$746$	10.7767	0.394562
$747$	12.8478	0.470075
$748$	0	0
$749$	2.85107	0.104176
$750$	−1.12455	−0.0410629
$751$	34.9061	1.27374	0.636870	−	0.770971i	$-0.280229\pi$
$751$	34.9061	1.27374	0.636870	+	0.770971i	$0.280229\pi$
$752$	9.82775	0.358381
$753$	−19.4273	−0.707969
$754$	61.4546	2.23804
$755$	13.4154	0.488235
$756$	5.32519	0.193675
$757$	−18.7354	−0.680948	−0.340474	−	0.940254i	$-0.610588\pi$
$757$	−18.7354	−0.680948	−0.340474	+	0.940254i	$0.610588\pi$
$758$	5.39945	0.196117
$759$	0	0
$760$	−3.43616	−0.124643
$761$	26.6630	0.966531	0.483266	−	0.875474i	$-0.339451\pi$
$761$	26.6630	0.966531	0.483266	+	0.875474i	$0.339451\pi$
$762$	−23.9091	−0.866136
$763$	−9.80121	−0.354828
$764$	2.90870	0.105233
$765$	−5.97797	−0.216134
$766$	−7.17505	−0.259245
$767$	74.4038	2.68657
$768$	−1.12455	−0.0405789
$769$	38.2777	1.38033	0.690165	−	0.723652i	$-0.257538\pi$
$769$	38.2777	1.38033	0.690165	+	0.723652i	$0.257538\pi$
$770$	0	0
$771$	0.504413	0.0181660
$772$	7.77320	0.279764
$773$	7.52232	0.270559	0.135280	−	0.990807i	$-0.456807\pi$
$773$	7.52232	0.270559	0.135280	+	0.990807i	$0.456807\pi$
$774$	9.36235	0.336523
$775$	−4.52145	−0.162415
$776$	−2.88993	−0.103742
$777$	0.877800	0.0314909
$778$	−16.5388	−0.592943
$779$	−17.2456	−0.617887
$780$	−7.94999	−0.284655
$781$	0	0
$782$	12.5360	0.448286
$783$	46.2918	1.65433
$784$	1.00000	0.0357143
$785$	24.0361	0.857886
$786$	−11.5421	−0.411694
$787$	−0.215236	−0.00767235	−0.00383618	−	0.999993i	$-0.501221\pi$
$787$	−0.215236	−0.00767235	−0.00383618	+	0.999993i	$0.501221\pi$
$788$	−26.2257	−0.934254
$789$	29.4003	1.04668
$790$	16.2368	0.577678
$791$	10.8032	0.384118
$792$	0	0
$793$	3.22533	0.114535
$794$	−16.7413	−0.594126
$795$	12.7882	0.453552
$796$	5.64963	0.200246
$797$	28.2639	1.00116	0.500580	−	0.865690i	$-0.333120\pi$
$797$	28.2639	1.00116	0.500580	+	0.865690i	$0.333120\pi$
$798$	−3.86415	−0.136789
$799$	−33.8543	−1.19768
$800$	−1.00000	−0.0353553
$801$	8.62730	0.304831
$802$	20.3412	0.718273
$803$	0	0
$804$	−11.8008	−0.416181
$805$	−3.63913	−0.128263
$806$	−31.9641	−1.12589
$807$	−15.9256	−0.560607
$808$	12.2266	0.430130
$809$	4.57788	0.160950	0.0804748	−	0.996757i	$-0.474356\pi$
$809$	4.57788	0.160950	0.0804748	+	0.996757i	$0.474356\pi$
$810$	−0.782335	−0.0274884
$811$	−44.4278	−1.56007	−0.780035	−	0.625736i	$-0.784799\pi$
$811$	−44.4278	−1.56007	−0.780035	+	0.625736i	$0.784799\pi$
$812$	8.69298	0.305064
$813$	−1.05057	−0.0368451
$814$	0	0
$815$	−0.905391	−0.0317145
$816$	3.87383	0.135611
$817$	−18.5381	−0.648564
$818$	−0.443589	−0.0155097
$819$	12.2682	0.428684
$820$	−5.01886	−0.175266
$821$	−51.5288	−1.79837	−0.899184	−	0.437571i	$-0.855839\pi$
$821$	−51.5288	−1.79837	−0.899184	+	0.437571i	$0.855839\pi$
$822$	−0.855686	−0.0298455
$823$	3.01093	0.104954	0.0524772	−	0.998622i	$-0.483288\pi$
$823$	3.01093	0.104954	0.0524772	+	0.998622i	$0.483288\pi$
$824$	−3.70633	−0.129116
$825$	0	0
$826$	10.5247	0.366201
$827$	17.5581	0.610555	0.305277	−	0.952263i	$-0.401251\pi$
$827$	17.5581	0.610555	0.305277	+	0.952263i	$0.401251\pi$
$828$	−6.31527	−0.219471
$829$	−54.9773	−1.90944	−0.954720	−	0.297507i	$-0.903845\pi$
$829$	−54.9773	−1.90944	−0.954720	+	0.297507i	$0.903845\pi$
$830$	−7.40344	−0.256977
$831$	−28.4792	−0.987932
$832$	−7.06945	−0.245089
$833$	−3.44477	−0.119354
$834$	−19.8346	−0.686816
$835$	−3.64082	−0.125996
$836$	0	0
$837$	−24.0776	−0.832242
$838$	18.6545	0.644408
$839$	8.14300	0.281127	0.140564	−	0.990072i	$-0.455109\pi$
$839$	8.14300	0.281127	0.140564	+	0.990072i	$0.455109\pi$
$840$	−1.12455	−0.0388008
$841$	46.5679	1.60579
$842$	−12.2716	−0.422907
$843$	31.2378	1.07589
$844$	−7.42973	−0.255742
$845$	−36.9772	−1.27205
$846$	17.0549	0.586358
$847$	0	0
$848$	11.3718	0.390510
$849$	−21.9460	−0.753185
$850$	3.44477	0.118155
$851$	−2.84062	−0.0973753
$852$	7.66245	0.262511
$853$	18.4464	0.631593	0.315796	−	0.948827i	$-0.397728\pi$
$853$	18.4464	0.631593	0.315796	+	0.948827i	$0.397728\pi$
$854$	0.456235	0.0156120
$855$	−5.96303	−0.203931
$856$	−2.85107	−0.0974476
$857$	−36.0108	−1.23011	−0.615054	−	0.788485i	$-0.710866\pi$
$857$	−36.0108	−1.23011	−0.615054	+	0.788485i	$0.710866\pi$
$858$	0	0
$859$	25.6553	0.875349	0.437674	−	0.899134i	$-0.355802\pi$
$859$	25.6553	0.875349	0.437674	+	0.899134i	$0.355802\pi$
$860$	−5.39500	−0.183968
$861$	−5.64398	−0.192346
$862$	21.9795	0.748626
$863$	15.4519	0.525990	0.262995	−	0.964797i	$-0.415290\pi$
$863$	15.4519	0.525990	0.262995	+	0.964797i	$0.415290\pi$
$864$	−5.32519	−0.181167
$865$	−3.17934	−0.108101
$866$	−22.2345	−0.755561
$867$	5.77298	0.196061
$868$	−4.52145	−0.153468
$869$	0	0
$870$	−9.77573	−0.331428
$871$	−74.1848	−2.51366
$872$	9.80121	0.331911
$873$	−5.01511	−0.169736
$874$	12.5046	0.422976
$875$	−1.00000	−0.0338062
$876$	−16.9978	−0.574302
$877$	−27.9195	−0.942774	−0.471387	−	0.881926i	$-0.656247\pi$
$877$	−27.9195	−0.942774	−0.471387	+	0.881926i	$0.656247\pi$
$878$	−11.0302	−0.372252
$879$	−15.0134	−0.506388
$880$	0	0
$881$	23.1798	0.780946	0.390473	−	0.920614i	$-0.372311\pi$
$881$	23.1798	0.780946	0.390473	+	0.920614i	$0.372311\pi$
$882$	1.73538	0.0584332
$883$	24.4908	0.824181	0.412090	−	0.911143i	$-0.364799\pi$
$883$	24.4908	0.824181	0.412090	+	0.911143i	$0.364799\pi$
$884$	24.3526	0.819068
$885$	−11.8356	−0.397849
$886$	−30.8702	−1.03711
$887$	22.6184	0.759453	0.379726	−	0.925099i	$-0.376018\pi$
$887$	22.6184	0.759453	0.379726	+	0.925099i	$0.376018\pi$
$888$	−0.877800	−0.0294570
$889$	−21.2610	−0.713070
$890$	−4.97143	−0.166643
$891$	0	0
$892$	1.51167	0.0506145
$893$	−33.7697	−1.13006
$894$	−0.875106	−0.0292679
$895$	−4.48137	−0.149796
$896$	−1.00000	−0.0334077
$897$	28.9311	0.965980
$898$	13.9952	0.467027
$899$	−39.3048	−1.31089
$900$	−1.73538	−0.0578459
$901$	−39.1733	−1.30505
$902$	0	0
$903$	−6.06697	−0.201896
$904$	−10.8032	−0.359309
$905$	9.02937	0.300146
$906$	−15.0863	−0.501209
$907$	29.1264	0.967127	0.483563	−	0.875309i	$-0.339342\pi$
$907$	29.1264	0.967127	0.483563	+	0.875309i	$0.339342\pi$
$908$	−13.8558	−0.459820
$909$	21.2177	0.703748
$910$	−7.06945	−0.234350
$911$	−0.781467	−0.0258912	−0.0129456	−	0.999916i	$-0.504121\pi$
$911$	−0.781467	−0.0258912	−0.0129456	+	0.999916i	$0.504121\pi$
$912$	3.86415	0.127955
$913$	0	0
$914$	−7.06533	−0.233700
$915$	−0.513061	−0.0169613
$916$	20.5463	0.678870
$917$	−10.2637	−0.338938
$918$	18.3440	0.605444
$919$	−0.902346	−0.0297656	−0.0148828	−	0.999889i	$-0.504738\pi$
$919$	−0.902346	−0.0297656	−0.0148828	+	0.999889i	$0.504738\pi$
$920$	3.63913	0.119979
$921$	28.9965	0.955466
$922$	10.6011	0.349130
$923$	48.1696	1.58552
$924$	0	0
$925$	−0.780576	−0.0256652
$926$	−4.59137	−0.150882
$927$	−6.43189	−0.211251
$928$	−8.69298	−0.285361
$929$	56.9745	1.86927	0.934637	−	0.355603i	$-0.115724\pi$
$929$	56.9745	1.86927	0.934637	+	0.355603i	$0.115724\pi$
$930$	5.08461	0.166731
$931$	−3.43616	−0.112615
$932$	−16.2345	−0.531780
$933$	−5.75254	−0.188330
$934$	30.9287	1.01202
$935$	0	0
$936$	−12.2682	−0.400998
$937$	−35.7088	−1.16656	−0.583278	−	0.812273i	$-0.698230\pi$
$937$	−35.7088	−1.16656	−0.583278	+	0.812273i	$0.698230\pi$
$938$	−10.4937	−0.342632
$939$	−18.6193	−0.607619
$940$	−9.82775	−0.320546
$941$	−9.23591	−0.301082	−0.150541	−	0.988604i	$-0.548102\pi$
$941$	−9.23591	−0.301082	−0.150541	+	0.988604i	$0.548102\pi$
$942$	−27.0299	−0.880683
$943$	18.2643	0.594768
$944$	−10.5247	−0.342550
$945$	−5.32519	−0.173228
$946$	0	0
$947$	−26.7339	−0.868736	−0.434368	−	0.900735i	$-0.643028\pi$
$947$	−26.7339	−0.868736	−0.434368	+	0.900735i	$0.643028\pi$
$948$	−18.2591	−0.593029
$949$	−106.856	−3.46869
$950$	3.43616	0.111484
$951$	−17.2619	−0.559757
$952$	3.44477	0.111646
$953$	41.1619	1.33337	0.666683	−	0.745341i	$-0.267713\pi$
$953$	41.1619	1.33337	0.666683	+	0.745341i	$0.267713\pi$
$954$	19.7344	0.638924
$955$	−2.90870	−0.0941234
$956$	0.392921	0.0127080
$957$	0	0
$958$	3.01362	0.0973658
$959$	−0.760911	−0.0245711
$960$	1.12455	0.0362948
$961$	−10.5565	−0.340533
$962$	−5.51825	−0.177915
$963$	−4.94768	−0.159437
$964$	3.32681	0.107149
$965$	−7.77320	−0.250228
$966$	4.09241	0.131671
$967$	−31.2558	−1.00512	−0.502559	−	0.864543i	$-0.667608\pi$
$967$	−31.2558	−1.00512	−0.502559	+	0.864543i	$0.667608\pi$
$968$	0	0
$969$	−13.3111	−0.427614
$970$	2.88993	0.0927899
$971$	−10.2903	−0.330232	−0.165116	−	0.986274i	$-0.552800\pi$
$971$	−10.2903	−0.330232	−0.165116	+	0.986274i	$0.552800\pi$
$972$	−15.0958	−0.484198
$973$	−17.6377	−0.565440
$974$	−3.00551	−0.0963028
$975$	7.94999	0.254603
$976$	−0.456235	−0.0146037
$977$	−44.9080	−1.43673	−0.718367	−	0.695665i	$-0.755110\pi$
$977$	−44.9080	−1.43673	−0.718367	+	0.695665i	$0.755110\pi$
$978$	1.01816	0.0325572
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	17.0088	0.543049
$982$	34.6723	1.10644
$983$	11.9639	0.381588	0.190794	−	0.981630i	$-0.438894\pi$
$983$	11.9639	0.381588	0.190794	+	0.981630i	$0.438894\pi$
$984$	5.64398	0.179924
$985$	26.2257	0.835622
$986$	29.9453	0.953653
$987$	−11.0518	−0.351784
$988$	24.2918	0.772824
$989$	19.6331	0.624297
$990$	0	0
$991$	25.7714	0.818656	0.409328	−	0.912387i	$-0.365763\pi$
$991$	25.7714	0.818656	0.409328	+	0.912387i	$0.365763\pi$
$992$	4.52145	0.143556
$993$	7.41252	0.235229
$994$	6.81376	0.216119
$995$	−5.64963	−0.179105
$996$	8.32557	0.263806
$997$	−9.16694	−0.290320	−0.145160	−	0.989408i	$-0.546370\pi$
$997$	−9.16694	−0.290320	−0.145160	+	0.989408i	$0.546370\pi$
$998$	5.25216	0.166254
$999$	−4.15672	−0.131513

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.dg.1.3		8
11.2	odd	10		770.2.n.k.631.2	yes	16
11.6	odd	10		770.2.n.k.421.2	✓	16
11.10	odd	2		8470.2.a.dh.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.k.421.2	✓	16	11.6	odd	10
770.2.n.k.631.2	yes	16	11.2	odd	10
8470.2.a.dg.1.3		8	1.1	even	1	trivial
8470.2.a.dh.1.3		8	11.10	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:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.12455 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.12455 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.73538 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.12455 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.12455 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.73538 q^{9} +1.00000 q^{10} -1.12455 q^{12} -7.06945 q^{13} -1.00000 q^{14} +1.12455 q^{15} +1.00000 q^{16} -3.44477 q^{17} +1.73538 q^{18} -3.43616 q^{19} -1.00000 q^{20} -1.12455 q^{21} +3.63913 q^{23} +1.12455 q^{24} +1.00000 q^{25} +7.06945 q^{26} +5.32519 q^{27} +1.00000 q^{28} +8.69298 q^{29} -1.12455 q^{30} -4.52145 q^{31} -1.00000 q^{32} +3.44477 q^{34} -1.00000 q^{35} -1.73538 q^{36} -0.780576 q^{37} +3.43616 q^{38} +7.94999 q^{39} +1.00000 q^{40} +5.01886 q^{41} +1.12455 q^{42} +5.39500 q^{43} +1.73538 q^{45} -3.63913 q^{46} +9.82775 q^{47} -1.12455 q^{48} +1.00000 q^{49} -1.00000 q^{50} +3.87383 q^{51} -7.06945 q^{52} +11.3718 q^{53} -5.32519 q^{54} -1.00000 q^{56} +3.86415 q^{57} -8.69298 q^{58} -10.5247 q^{59} +1.12455 q^{60} -0.456235 q^{61} +4.52145 q^{62} -1.73538 q^{63} +1.00000 q^{64} +7.06945 q^{65} +10.4937 q^{67} -3.44477 q^{68} -4.09241 q^{69} +1.00000 q^{70} -6.81376 q^{71} +1.73538 q^{72} +15.1151 q^{73} +0.780576 q^{74} -1.12455 q^{75} -3.43616 q^{76} -7.94999 q^{78} +16.2368 q^{79} -1.00000 q^{80} -0.782335 q^{81} -5.01886 q^{82} -7.40344 q^{83} -1.12455 q^{84} +3.44477 q^{85} -5.39500 q^{86} -9.77573 q^{87} -4.97143 q^{89} -1.73538 q^{90} -7.06945 q^{91} +3.63913 q^{92} +5.08461 q^{93} -9.82775 q^{94} +3.43616 q^{95} +1.12455 q^{96} +2.88993 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} + 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^5 + 8 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} + 8 q^{7} - 8 q^{8} + 8 q^{9} + 8 q^{10} + q^{13} - 8 q^{14} + 8 q^{16} - 6 q^{17} - 8 q^{18} - 5 q^{19} - 8 q^{20} + 10 q^{23} + 8 q^{25} - q^{26} + 8 q^{28} - 3 q^{29} - 8 q^{31} - 8 q^{32} + 6 q^{34} - 8 q^{35} + 8 q^{36} - 6 q^{37} + 5 q^{38} - 35 q^{39} + 8 q^{40} - 11 q^{41} + 5 q^{43} - 8 q^{45} - 10 q^{46} - 15 q^{47} + 8 q^{49} - 8 q^{50} + 6 q^{51} + q^{52} - 16 q^{53} - 8 q^{56} - 38 q^{57} + 3 q^{58} - 9 q^{59} - 32 q^{61} + 8 q^{62} + 8 q^{63} + 8 q^{64} - q^{65} + 33 q^{67} - 6 q^{68} - 22 q^{69} + 8 q^{70} + 11 q^{71} - 8 q^{72} + 34 q^{73} + 6 q^{74} - 5 q^{76} + 35 q^{78} - 31 q^{79} - 8 q^{80} + 20 q^{81} + 11 q^{82} - 50 q^{83} + 6 q^{85} - 5 q^{86} + 12 q^{87} + q^{89} + 8 q^{90} + q^{91} + 10 q^{92} + 26 q^{93} + 15 q^{94} + 5 q^{95} - 4 q^{97} - 8 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^5 + 8 * q^7 - 8 * q^8 + 8 * q^9 + 8 * q^10 + q^13 - 8 * q^14 + 8 * q^16 - 6 * q^17 - 8 * q^18 - 5 * q^19 - 8 * q^20 + 10 * q^23 + 8 * q^25 - q^26 + 8 * q^28 - 3 * q^29 - 8 * q^31 - 8 * q^32 + 6 * q^34 - 8 * q^35 + 8 * q^36 - 6 * q^37 + 5 * q^38 - 35 * q^39 + 8 * q^40 - 11 * q^41 + 5 * q^43 - 8 * q^45 - 10 * q^46 - 15 * q^47 + 8 * q^49 - 8 * q^50 + 6 * q^51 + q^52 - 16 * q^53 - 8 * q^56 - 38 * q^57 + 3 * q^58 - 9 * q^59 - 32 * q^61 + 8 * q^62 + 8 * q^63 + 8 * q^64 - q^65 + 33 * q^67 - 6 * q^68 - 22 * q^69 + 8 * q^70 + 11 * q^71 - 8 * q^72 + 34 * q^73 + 6 * q^74 - 5 * q^76 + 35 * q^78 - 31 * q^79 - 8 * q^80 + 20 * q^81 + 11 * q^82 - 50 * q^83 + 6 * q^85 - 5 * q^86 + 12 * q^87 + q^89 + 8 * q^90 + q^91 + 10 * q^92 + 26 * q^93 + 15 * q^94 + 5 * q^95 - 4 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more