LMFDB - Embedded newform 6422.2.a.bi.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, 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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 12x^{7} + 14x^{6} + 54x^{5} - 11x^{4} - 84x^{3} - 48x^{2} - 4x + 1 $ x^9 - 2x^8 - 12x^7 + 14x^6 + 54x^5 - 11x^4 - 84x^3 - 48x^2 - 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.785289$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.782742 q^{3} +1.00000 q^{4} -1.76453 q^{5} -0.782742 q^{6} +2.34025 q^{7} -1.00000 q^{8} -2.38731 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.782742 q^{3} +1.00000 q^{4} -1.76453 q^{5} -0.782742 q^{6} +2.34025 q^{7} -1.00000 q^{8} -2.38731 q^{9} +1.76453 q^{10} +5.03713 q^{11} +0.782742 q^{12} -2.34025 q^{14} -1.38117 q^{15} +1.00000 q^{16} +5.98944 q^{17} +2.38731 q^{18} -1.00000 q^{19} -1.76453 q^{20} +1.83181 q^{21} -5.03713 q^{22} +2.35701 q^{23} -0.782742 q^{24} -1.88644 q^{25} -4.21688 q^{27} +2.34025 q^{28} -4.79286 q^{29} +1.38117 q^{30} +0.870512 q^{31} -1.00000 q^{32} +3.94278 q^{33} -5.98944 q^{34} -4.12944 q^{35} -2.38731 q^{36} +6.18309 q^{37} +1.00000 q^{38} +1.76453 q^{40} +5.23838 q^{41} -1.83181 q^{42} +0.868371 q^{43} +5.03713 q^{44} +4.21249 q^{45} -2.35701 q^{46} +0.145776 q^{47} +0.782742 q^{48} -1.52324 q^{49} +1.88644 q^{50} +4.68819 q^{51} +4.67320 q^{53} +4.21688 q^{54} -8.88817 q^{55} -2.34025 q^{56} -0.782742 q^{57} +4.79286 q^{58} +3.48712 q^{59} -1.38117 q^{60} -1.64103 q^{61} -0.870512 q^{62} -5.58691 q^{63} +1.00000 q^{64} -3.94278 q^{66} +10.6347 q^{67} +5.98944 q^{68} +1.84493 q^{69} +4.12944 q^{70} +5.29239 q^{71} +2.38731 q^{72} +1.44995 q^{73} -6.18309 q^{74} -1.47659 q^{75} -1.00000 q^{76} +11.7881 q^{77} +6.95437 q^{79} -1.76453 q^{80} +3.86121 q^{81} -5.23838 q^{82} -2.22732 q^{83} +1.83181 q^{84} -10.5685 q^{85} -0.868371 q^{86} -3.75157 q^{87} -5.03713 q^{88} +0.111402 q^{89} -4.21249 q^{90} +2.35701 q^{92} +0.681386 q^{93} -0.145776 q^{94} +1.76453 q^{95} -0.782742 q^{96} -8.21775 q^{97} +1.52324 q^{98} -12.0252 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{2} - 5 q^{3} + 9 q^{4} - q^{5} + 5 q^{6} + 13 q^{7} - 9 q^{8} + 10 q^{9}+O(q^{10}) $ 9 * q - 9 * q^2 - 5 * q^3 + 9 * q^4 - q^5 + 5 * q^6 + 13 * q^7 - 9 * q^8 + 10 * q^9	\( 9 q - 9 q^{2} - 5 q^{3} + 9 q^{4} - q^{5} + 5 q^{6} + 13 q^{7} - 9 q^{8} + 10 q^{9} + q^{10} + 13 q^{11} - 5 q^{12} - 13 q^{14} + q^{15} + 9 q^{16} - 12 q^{17} - 10 q^{18} - 9 q^{19} - q^{20} - 18 q^{21} - 13 q^{22} - 22 q^{23} + 5 q^{24} + 4 q^{25} - 26 q^{27} + 13 q^{28} + 12 q^{29} - q^{30} - q^{31} - 9 q^{32} + 28 q^{33} + 12 q^{34} - 18 q^{35} + 10 q^{36} + 25 q^{37} + 9 q^{38} + q^{40} - 11 q^{41} + 18 q^{42} - 10 q^{43} + 13 q^{44} + q^{45} + 22 q^{46} + 12 q^{47} - 5 q^{48} + 2 q^{49} - 4 q^{50} + 35 q^{51} + 9 q^{53} + 26 q^{54} + 18 q^{55} - 13 q^{56} + 5 q^{57} - 12 q^{58} - 10 q^{59} + q^{60} + 32 q^{61} + q^{62} + 63 q^{63} + 9 q^{64} - 28 q^{66} + 73 q^{67} - 12 q^{68} + 2 q^{69} + 18 q^{70} + 51 q^{71} - 10 q^{72} + 14 q^{73} - 25 q^{74} - 49 q^{75} - 9 q^{76} + 18 q^{77} - 28 q^{79} - q^{80} + 29 q^{81} + 11 q^{82} + 22 q^{83} - 18 q^{84} + 51 q^{85} + 10 q^{86} - 20 q^{87} - 13 q^{88} - 3 q^{89} - q^{90} - 22 q^{92} + 59 q^{93} - 12 q^{94} + q^{95} + 5 q^{96} - 2 q^{98} - 25 q^{99}+O(q^{100}) \) 9 * q - 9 * q^2 - 5 * q^3 + 9 * q^4 - q^5 + 5 * q^6 + 13 * q^7 - 9 * q^8 + 10 * q^9 + q^10 + 13 * q^11 - 5 * q^12 - 13 * q^14 + q^15 + 9 * q^16 - 12 * q^17 - 10 * q^18 - 9 * q^19 - q^20 - 18 * q^21 - 13 * q^22 - 22 * q^23 + 5 * q^24 + 4 * q^25 - 26 * q^27 + 13 * q^28 + 12 * q^29 - q^30 - q^31 - 9 * q^32 + 28 * q^33 + 12 * q^34 - 18 * q^35 + 10 * q^36 + 25 * q^37 + 9 * q^38 + q^40 - 11 * q^41 + 18 * q^42 - 10 * q^43 + 13 * q^44 + q^45 + 22 * q^46 + 12 * q^47 - 5 * q^48 + 2 * q^49 - 4 * q^50 + 35 * q^51 + 9 * q^53 + 26 * q^54 + 18 * q^55 - 13 * q^56 + 5 * q^57 - 12 * q^58 - 10 * q^59 + q^60 + 32 * q^61 + q^62 + 63 * q^63 + 9 * q^64 - 28 * q^66 + 73 * q^67 - 12 * q^68 + 2 * q^69 + 18 * q^70 + 51 * q^71 - 10 * q^72 + 14 * q^73 - 25 * q^74 - 49 * q^75 - 9 * q^76 + 18 * q^77 - 28 * q^79 - q^80 + 29 * q^81 + 11 * q^82 + 22 * q^83 - 18 * q^84 + 51 * q^85 + 10 * q^86 - 20 * q^87 - 13 * q^88 - 3 * q^89 - q^90 - 22 * q^92 + 59 * q^93 - 12 * q^94 + q^95 + 5 * q^96 - 2 * q^98 - 25 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0.782742	0.451917	0.225958	−	0.974137i	$-0.427449\pi$
$3$	0.782742	0.451917	0.225958	+	0.974137i	$0.427449\pi$
$4$	1.00000	0.500000
$5$	−1.76453	−0.789122	−0.394561	−	0.918870i	$-0.629103\pi$
$5$	−1.76453	−0.789122	−0.394561	+	0.918870i	$0.629103\pi$
$6$	−0.782742	−0.319553
$7$	2.34025	0.884530	0.442265	−	0.896884i	$-0.354175\pi$
$7$	2.34025	0.884530	0.442265	+	0.896884i	$0.354175\pi$
$8$	−1.00000	−0.353553
$9$	−2.38731	−0.795771
$10$	1.76453	0.557993
$11$	5.03713	1.51875	0.759376	−	0.650652i	$-0.225504\pi$
$11$	5.03713	1.51875	0.759376	+	0.650652i	$0.225504\pi$
$12$	0.782742	0.225958
$13$	0	0
$13$	0	0
$14$	−2.34025	−0.625457
$15$	−1.38117	−0.356617
$16$	1.00000	0.250000
$17$	5.98944	1.45265	0.726326	−	0.687351i	$-0.241226\pi$
$17$	5.98944	1.45265	0.726326	+	0.687351i	$0.241226\pi$
$18$	2.38731	0.562695
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.76453	−0.394561
$21$	1.83181	0.399734
$22$	−5.03713	−1.07392
$23$	2.35701	0.491470	0.245735	−	0.969337i	$-0.420971\pi$
$23$	2.35701	0.491470	0.245735	+	0.969337i	$0.420971\pi$
$24$	−0.782742	−0.159777
$25$	−1.88644	−0.377287
$26$	0	0
$27$	−4.21688	−0.811539
$28$	2.34025	0.442265
$29$	−4.79286	−0.890011	−0.445006	−	0.895528i	$-0.646798\pi$
$29$	−4.79286	−0.890011	−0.445006	+	0.895528i	$0.646798\pi$
$30$	1.38117	0.252166
$31$	0.870512	0.156348	0.0781742	−	0.996940i	$-0.475091\pi$
$31$	0.870512	0.156348	0.0781742	+	0.996940i	$0.475091\pi$
$32$	−1.00000	−0.176777
$33$	3.94278	0.686349
$34$	−5.98944	−1.02718
$35$	−4.12944	−0.698002
$36$	−2.38731	−0.397886
$37$	6.18309	1.01649	0.508247	−	0.861212i	$-0.330294\pi$
$37$	6.18309	1.01649	0.508247	+	0.861212i	$0.330294\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	1.76453	0.278997
$41$	5.23838	0.818097	0.409049	−	0.912513i	$-0.365861\pi$
$41$	5.23838	0.818097	0.409049	+	0.912513i	$0.365861\pi$
$42$	−1.83181	−0.282655
$43$	0.868371	0.132425	0.0662127	−	0.997806i	$-0.478908\pi$
$43$	0.868371	0.132425	0.0662127	+	0.997806i	$0.478908\pi$
$44$	5.03713	0.759376
$45$	4.21249	0.627960
$46$	−2.35701	−0.347522
$47$	0.145776	0.0212637	0.0106318	−	0.999943i	$-0.496616\pi$
$47$	0.145776	0.0212637	0.0106318	+	0.999943i	$0.496616\pi$
$48$	0.782742	0.112979
$49$	−1.52324	−0.217606
$50$	1.88644	0.266782
$51$	4.68819	0.656477
$52$	0	0
$53$	4.67320	0.641914	0.320957	−	0.947094i	$-0.395996\pi$
$53$	4.67320	0.641914	0.320957	+	0.947094i	$0.395996\pi$
$54$	4.21688	0.573845
$55$	−8.88817	−1.19848
$56$	−2.34025	−0.312729
$57$	−0.782742	−0.103677
$58$	4.79286	0.629333
$59$	3.48712	0.453984	0.226992	−	0.973897i	$-0.427111\pi$
$59$	3.48712	0.453984	0.226992	+	0.973897i	$0.427111\pi$
$60$	−1.38117	−0.178309
$61$	−1.64103	−0.210113	−0.105056	−	0.994466i	$-0.533502\pi$
$61$	−1.64103	−0.210113	−0.105056	+	0.994466i	$0.533502\pi$
$62$	−0.870512	−0.110555
$63$	−5.58691	−0.703884
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.94278	−0.485322
$67$	10.6347	1.29924	0.649619	−	0.760260i	$-0.274928\pi$
$67$	10.6347	1.29924	0.649619	+	0.760260i	$0.274928\pi$
$68$	5.98944	0.726326
$69$	1.84493	0.222103
$70$	4.12944	0.493562
$71$	5.29239	0.628091	0.314046	−	0.949408i	$-0.398316\pi$
$71$	5.29239	0.628091	0.314046	+	0.949408i	$0.398316\pi$
$72$	2.38731	0.281348
$73$	1.44995	0.169703	0.0848517	−	0.996394i	$-0.472958\pi$
$73$	1.44995	0.169703	0.0848517	+	0.996394i	$0.472958\pi$
$74$	−6.18309	−0.718769
$75$	−1.47659	−0.170502
$76$	−1.00000	−0.114708
$77$	11.7881	1.34338
$78$	0	0
$79$	6.95437	0.782427	0.391214	−	0.920300i	$-0.372055\pi$
$79$	6.95437	0.782427	0.391214	+	0.920300i	$0.372055\pi$
$80$	−1.76453	−0.197280
$81$	3.86121	0.429024
$82$	−5.23838	−0.578482
$83$	−2.22732	−0.244481	−0.122240	−	0.992501i	$-0.539008\pi$
$83$	−2.22732	−0.244481	−0.122240	+	0.992501i	$0.539008\pi$
$84$	1.83181	0.199867
$85$	−10.5685	−1.14632
$86$	−0.868371	−0.0936389
$87$	−3.75157	−0.402211
$88$	−5.03713	−0.536960
$89$	0.111402	0.0118086	0.00590430	−	0.999983i	$-0.498121\pi$
$89$	0.111402	0.0118086	0.00590430	+	0.999983i	$0.498121\pi$
$90$	−4.21249	−0.444035
$91$	0	0
$92$	2.35701	0.245735
$93$	0.681386	0.0706565
$94$	−0.145776	−0.0150357
$95$	1.76453	0.181037
$96$	−0.782742	−0.0798883
$97$	−8.21775	−0.834386	−0.417193	−	0.908818i	$-0.636986\pi$
$97$	−8.21775	−0.834386	−0.417193	+	0.908818i	$0.636986\pi$
$98$	1.52324	0.153871
$99$	−12.0252	−1.20858
$100$	−1.88644	−0.188644
$101$	−17.2558	−1.71702	−0.858509	−	0.512799i	$-0.828608\pi$
$101$	−17.2558	−1.71702	−0.858509	+	0.512799i	$0.828608\pi$
$102$	−4.68819	−0.464200
$103$	−5.40241	−0.532315	−0.266158	−	0.963930i	$-0.585754\pi$
$103$	−5.40241	−0.532315	−0.266158	+	0.963930i	$0.585754\pi$
$104$	0	0
$105$	−3.23228	−0.315439
$106$	−4.67320	−0.453902
$107$	−17.9901	−1.73916	−0.869582	−	0.493788i	$-0.835612\pi$
$107$	−17.9901	−1.73916	−0.869582	+	0.493788i	$0.835612\pi$
$108$	−4.21688	−0.405769
$109$	11.7518	1.12562	0.562811	−	0.826586i	$-0.309720\pi$
$109$	11.7518	1.12562	0.562811	+	0.826586i	$0.309720\pi$
$110$	8.88817	0.847453
$111$	4.83977	0.459370
$112$	2.34025	0.221133
$113$	−16.7496	−1.57567	−0.787834	−	0.615888i	$-0.788798\pi$
$113$	−16.7496	−1.57567	−0.787834	+	0.615888i	$0.788798\pi$
$114$	0.782742	0.0733105
$115$	−4.15901	−0.387829
$116$	−4.79286	−0.445006
$117$	0	0
$118$	−3.48712	−0.321015
$119$	14.0168	1.28491
$120$	1.38117	0.126083
$121$	14.3727	1.30661
$122$	1.64103	0.148572
$123$	4.10030	0.369712
$124$	0.870512	0.0781742
$125$	12.1513	1.08685
$126$	5.58691	0.497721
$127$	0.742321	0.0658703	0.0329352	−	0.999457i	$-0.489515\pi$
$127$	0.742321	0.0658703	0.0329352	+	0.999457i	$0.489515\pi$
$128$	−1.00000	−0.0883883
$129$	0.679711	0.0598452
$130$	0	0
$131$	8.11281	0.708820	0.354410	−	0.935090i	$-0.384682\pi$
$131$	8.11281	0.708820	0.354410	+	0.935090i	$0.384682\pi$
$132$	3.94278	0.343175
$133$	−2.34025	−0.202925
$134$	−10.6347	−0.918700
$135$	7.44081	0.640403
$136$	−5.98944	−0.513590
$137$	−2.36989	−0.202474	−0.101237	−	0.994862i	$-0.532280\pi$
$137$	−2.36989	−0.202474	−0.101237	+	0.994862i	$0.532280\pi$
$138$	−1.84493	−0.157051
$139$	18.5642	1.57459	0.787296	−	0.616576i	$-0.211481\pi$
$139$	18.5642	1.57459	0.787296	+	0.616576i	$0.211481\pi$
$140$	−4.12944	−0.349001
$141$	0.114105	0.00960940
$142$	−5.29239	−0.444128
$143$	0	0
$144$	−2.38731	−0.198943
$145$	8.45714	0.702327
$146$	−1.44995	−0.119998
$147$	−1.19231	−0.0983397
$148$	6.18309	0.508247
$149$	−5.59030	−0.457975	−0.228988	−	0.973429i	$-0.573542\pi$
$149$	−5.59030	−0.457975	−0.228988	+	0.973429i	$0.573542\pi$
$150$	1.47659	0.120563
$151$	11.5195	0.937442	0.468721	−	0.883346i	$-0.344715\pi$
$151$	11.5195	0.937442	0.468721	+	0.883346i	$0.344715\pi$
$152$	1.00000	0.0811107
$153$	−14.2987	−1.15598
$154$	−11.7881	−0.949915
$155$	−1.53604	−0.123378
$156$	0	0
$157$	−3.17622	−0.253490	−0.126745	−	0.991935i	$-0.540453\pi$
$157$	−3.17622	−0.253490	−0.126745	+	0.991935i	$0.540453\pi$
$158$	−6.95437	−0.553260
$159$	3.65791	0.290092
$160$	1.76453	0.139498
$161$	5.51598	0.434720
$162$	−3.86121	−0.303365
$163$	13.5142	1.05851	0.529256	−	0.848462i	$-0.322471\pi$
$163$	13.5142	1.05851	0.529256	+	0.848462i	$0.322471\pi$
$164$	5.23838	0.409049
$165$	−6.95714	−0.541613
$166$	2.22732	0.172874
$167$	0.462218	0.0357675	0.0178837	−	0.999840i	$-0.494307\pi$
$167$	0.462218	0.0357675	0.0178837	+	0.999840i	$0.494307\pi$
$168$	−1.83181	−0.141327
$169$	0	0
$170$	10.5685	0.810570
$171$	2.38731	0.182562
$172$	0.868371	0.0662127
$173$	13.4733	1.02436	0.512179	−	0.858879i	$-0.328838\pi$
$173$	13.4733	1.02436	0.512179	+	0.858879i	$0.328838\pi$
$174$	3.75157	0.284406
$175$	−4.41473	−0.333722
$176$	5.03713	0.379688
$177$	2.72951	0.205163
$178$	−0.111402	−0.00834994
$179$	−17.6427	−1.31868	−0.659338	−	0.751847i	$-0.729163\pi$
$179$	−17.6427	−1.31868	−0.659338	+	0.751847i	$0.729163\pi$
$180$	4.21249	0.313980
$181$	5.49870	0.408715	0.204358	−	0.978896i	$-0.434489\pi$
$181$	5.49870	0.408715	0.204358	+	0.978896i	$0.434489\pi$
$182$	0	0
$183$	−1.28451	−0.0949534
$184$	−2.35701	−0.173761
$185$	−10.9102	−0.802137
$186$	−0.681386	−0.0499617
$187$	30.1696	2.20622
$188$	0.145776	0.0106318
$189$	−9.86854	−0.717831
$190$	−1.76453	−0.128012
$191$	−7.72016	−0.558611	−0.279306	−	0.960202i	$-0.590104\pi$
$191$	−7.72016	−0.558611	−0.279306	+	0.960202i	$0.590104\pi$
$192$	0.782742	0.0564896
$193$	−18.9326	−1.36280	−0.681399	−	0.731913i	$-0.738628\pi$
$193$	−18.9326	−1.36280	−0.681399	+	0.731913i	$0.738628\pi$
$194$	8.21775	0.590000
$195$	0	0
$196$	−1.52324	−0.108803
$197$	−18.2720	−1.30183	−0.650914	−	0.759151i	$-0.725614\pi$
$197$	−18.2720	−1.30183	−0.650914	+	0.759151i	$0.725614\pi$
$198$	12.0252	0.854595
$199$	−11.6266	−0.824190	−0.412095	−	0.911141i	$-0.635203\pi$
$199$	−11.6266	−0.824190	−0.412095	+	0.911141i	$0.635203\pi$
$200$	1.88644	0.133391
$201$	8.32425	0.587147
$202$	17.2558	1.21411
$203$	−11.2165	−0.787242
$204$	4.68819	0.328239
$205$	−9.24327	−0.645578
$206$	5.40241	0.376404
$207$	−5.62691	−0.391098
$208$	0	0
$209$	−5.03713	−0.348426
$210$	3.23228	0.223049
$211$	5.87305	0.404318	0.202159	−	0.979353i	$-0.435204\pi$
$211$	5.87305	0.404318	0.202159	+	0.979353i	$0.435204\pi$
$212$	4.67320	0.320957
$213$	4.14258	0.283845
$214$	17.9901	1.22977
$215$	−1.53227	−0.104500
$216$	4.21688	0.286922
$217$	2.03721	0.138295
$218$	−11.7518	−0.795935
$219$	1.13493	0.0766918
$220$	−8.88817	−0.599240
$221$	0	0
$222$	−4.83977	−0.324824
$223$	8.94247	0.598832	0.299416	−	0.954123i	$-0.403208\pi$
$223$	8.94247	0.598832	0.299416	+	0.954123i	$0.403208\pi$
$224$	−2.34025	−0.156364
$225$	4.50352	0.300234
$226$	16.7496	1.11417
$227$	14.0792	0.934472	0.467236	−	0.884133i	$-0.345250\pi$
$227$	14.0792	0.934472	0.467236	+	0.884133i	$0.345250\pi$
$228$	−0.782742	−0.0518384
$229$	−9.30882	−0.615144	−0.307572	−	0.951525i	$-0.599517\pi$
$229$	−9.30882	−0.615144	−0.307572	+	0.951525i	$0.599517\pi$
$230$	4.15901	0.274237
$231$	9.22707	0.607097
$232$	4.79286	0.314667
$233$	18.3417	1.20161	0.600803	−	0.799397i	$-0.294848\pi$
$233$	18.3417	1.20161	0.600803	+	0.799397i	$0.294848\pi$
$234$	0	0
$235$	−0.257227	−0.0167796
$236$	3.48712	0.226992
$237$	5.44348	0.353592
$238$	−14.0168	−0.908572
$239$	26.8693	1.73803	0.869016	−	0.494783i	$-0.164753\pi$
$239$	26.8693	1.73803	0.869016	+	0.494783i	$0.164753\pi$
$240$	−1.38117	−0.0891543
$241$	27.7878	1.78997	0.894983	−	0.446100i	$-0.147187\pi$
$241$	27.7878	1.78997	0.894983	+	0.446100i	$0.147187\pi$
$242$	−14.3727	−0.923912
$243$	15.6730	1.00542
$244$	−1.64103	−0.105056
$245$	2.68780	0.171717
$246$	−4.10030	−0.261426
$247$	0	0
$248$	−0.870512	−0.0552775
$249$	−1.74342	−0.110485
$250$	−12.1513	−0.768517
$251$	23.8510	1.50546	0.752730	−	0.658329i	$-0.228737\pi$
$251$	23.8510	1.50546	0.752730	+	0.658329i	$0.228737\pi$
$252$	−5.58691	−0.351942
$253$	11.8725	0.746421
$254$	−0.742321	−0.0465773
$255$	−8.27244	−0.518040
$256$	1.00000	0.0625000
$257$	−8.28542	−0.516830	−0.258415	−	0.966034i	$-0.583200\pi$
$257$	−8.28542	−0.516830	−0.258415	+	0.966034i	$0.583200\pi$
$258$	−0.679711	−0.0423169
$259$	14.4700	0.899119
$260$	0	0
$261$	11.4421	0.708246
$262$	−8.11281	−0.501211
$263$	18.0106	1.11058	0.555291	−	0.831656i	$-0.312607\pi$
$263$	18.0106	1.11058	0.555291	+	0.831656i	$0.312607\pi$
$264$	−3.94278	−0.242661
$265$	−8.24601	−0.506548
$266$	2.34025	0.143490
$267$	0.0871992	0.00533650
$268$	10.6347	0.649619
$269$	−14.7201	−0.897501	−0.448751	−	0.893657i	$-0.648131\pi$
$269$	−14.7201	−0.897501	−0.448751	+	0.893657i	$0.648131\pi$
$270$	−7.44081	−0.452833
$271$	27.3591	1.66194	0.830972	−	0.556314i	$-0.187785\pi$
$271$	27.3591	1.66194	0.830972	+	0.556314i	$0.187785\pi$
$272$	5.98944	0.363163
$273$	0	0
$274$	2.36989	0.143170
$275$	−9.50223	−0.573006
$276$	1.84493	0.111052
$277$	30.3660	1.82451	0.912257	−	0.409618i	$-0.134338\pi$
$277$	30.3660	1.82451	0.912257	+	0.409618i	$0.134338\pi$
$278$	−18.5642	−1.11340
$279$	−2.07818	−0.124418
$280$	4.12944	0.246781
$281$	−1.85596	−0.110717	−0.0553585	−	0.998467i	$-0.517630\pi$
$281$	−1.85596	−0.110717	−0.0553585	+	0.998467i	$0.517630\pi$
$282$	−0.114105	−0.00679487
$283$	10.7759	0.640562	0.320281	−	0.947323i	$-0.396223\pi$
$283$	10.7759	0.640562	0.320281	+	0.947323i	$0.396223\pi$
$284$	5.29239	0.314046
$285$	1.38117	0.0818136
$286$	0	0
$287$	12.2591	0.723632
$288$	2.38731	0.140674
$289$	18.8733	1.11020
$290$	−8.45714	−0.496620
$291$	−6.43238	−0.377073
$292$	1.44995	0.0848517
$293$	−30.4684	−1.77998	−0.889992	−	0.455977i	$-0.849290\pi$
$293$	−30.4684	−1.77998	−0.889992	+	0.455977i	$0.849290\pi$
$294$	1.19231	0.0695367
$295$	−6.15312	−0.358248
$296$	−6.18309	−0.359385
$297$	−21.2410	−1.23253
$298$	5.59030	0.323838
$299$	0	0
$300$	−1.47659	−0.0852512
$301$	2.03220	0.117134
$302$	−11.5195	−0.662872
$303$	−13.5069	−0.775948
$304$	−1.00000	−0.0573539
$305$	2.89565	0.165804
$306$	14.2987	0.817400
$307$	−19.1868	−1.09505	−0.547523	−	0.836791i	$-0.684429\pi$
$307$	−19.1868	−1.09505	−0.547523	+	0.836791i	$0.684429\pi$
$308$	11.7881	0.671691
$309$	−4.22869	−0.240562
$310$	1.53604	0.0872414
$311$	17.7139	1.00446	0.502230	−	0.864734i	$-0.332513\pi$
$311$	17.7139	1.00446	0.502230	+	0.864734i	$0.332513\pi$
$312$	0	0
$313$	4.68889	0.265032	0.132516	−	0.991181i	$-0.457694\pi$
$313$	4.68889	0.265032	0.132516	+	0.991181i	$0.457694\pi$
$314$	3.17622	0.179245
$315$	9.85826	0.555450
$316$	6.95437	0.391214
$317$	−20.8847	−1.17300	−0.586502	−	0.809948i	$-0.699495\pi$
$317$	−20.8847	−1.17300	−0.586502	+	0.809948i	$0.699495\pi$
$318$	−3.65791	−0.205126
$319$	−24.1423	−1.35171
$320$	−1.76453	−0.0986402
$321$	−14.0816	−0.785957
$322$	−5.51598	−0.307393
$323$	−5.98944	−0.333261
$324$	3.86121	0.214512
$325$	0	0
$326$	−13.5142	−0.748480
$327$	9.19866	0.508687
$328$	−5.23838	−0.289241
$329$	0.341153	0.0188084
$330$	6.95714	0.382978
$331$	14.1797	0.779389	0.389695	−	0.920944i	$-0.372581\pi$
$331$	14.1797	0.779389	0.389695	+	0.920944i	$0.372581\pi$
$332$	−2.22732	−0.122240
$333$	−14.7610	−0.808896
$334$	−0.462218	−0.0252914
$335$	−18.7653	−1.02526
$336$	1.83181	0.0999335
$337$	6.92799	0.377392	0.188696	−	0.982036i	$-0.439574\pi$
$337$	6.92799	0.377392	0.188696	+	0.982036i	$0.439574\pi$
$338$	0	0
$339$	−13.1106	−0.712071
$340$	−10.5685	−0.573159
$341$	4.38488	0.237455
$342$	−2.38731	−0.129091
$343$	−19.9465	−1.07701
$344$	−0.868371	−0.0468194
$345$	−3.25543	−0.175266
$346$	−13.4733	−0.724331
$347$	−14.8567	−0.797548	−0.398774	−	0.917049i	$-0.630564\pi$
$347$	−14.8567	−0.797548	−0.398774	+	0.917049i	$0.630564\pi$
$348$	−3.75157	−0.201105
$349$	−19.3683	−1.03676	−0.518381	−	0.855150i	$-0.673465\pi$
$349$	−19.3683	−1.03676	−0.518381	+	0.855150i	$0.673465\pi$
$350$	4.41473	0.235977
$351$	0	0
$352$	−5.03713	−0.268480
$353$	−5.27186	−0.280593	−0.140296	−	0.990110i	$-0.544806\pi$
$353$	−5.27186	−0.280593	−0.140296	+	0.990110i	$0.544806\pi$
$354$	−2.72951	−0.145072
$355$	−9.33858	−0.495640
$356$	0.111402	0.00590430
$357$	10.9715	0.580674
$358$	17.6427	0.932444
$359$	30.5453	1.61212	0.806061	−	0.591833i	$-0.201595\pi$
$359$	30.5453	1.61212	0.806061	+	0.591833i	$0.201595\pi$
$360$	−4.21249	−0.222018
$361$	1.00000	0.0526316
$362$	−5.49870	−0.289005
$363$	11.2501	0.590478
$364$	0	0
$365$	−2.55847	−0.133917
$366$	1.28451	0.0671422
$367$	5.94574	0.310365	0.155182	−	0.987886i	$-0.450403\pi$
$367$	5.94574	0.310365	0.155182	+	0.987886i	$0.450403\pi$
$368$	2.35701	0.122867
$369$	−12.5057	−0.651018
$370$	10.9102	0.567196
$371$	10.9365	0.567792
$372$	0.681386	0.0353282
$373$	32.0382	1.65887	0.829436	−	0.558601i	$-0.188662\pi$
$373$	32.0382	1.65887	0.829436	+	0.558601i	$0.188662\pi$
$374$	−30.1696	−1.56003
$375$	9.51135	0.491164
$376$	−0.145776	−0.00751784
$377$	0	0
$378$	9.86854	0.507583
$379$	32.0536	1.64648	0.823240	−	0.567693i	$-0.192164\pi$
$379$	32.0536	1.64648	0.823240	+	0.567693i	$0.192164\pi$
$380$	1.76453	0.0905184
$381$	0.581046	0.0297679
$382$	7.72016	0.394998
$383$	25.9231	1.32461	0.662304	−	0.749235i	$-0.269579\pi$
$383$	25.9231	1.32461	0.662304	+	0.749235i	$0.269579\pi$
$384$	−0.782742	−0.0399442
$385$	−20.8005	−1.06009
$386$	18.9326	0.963643
$387$	−2.07307	−0.105380
$388$	−8.21775	−0.417193
$389$	25.9537	1.31591	0.657953	−	0.753059i	$-0.271423\pi$
$389$	25.9537	1.31591	0.657953	+	0.753059i	$0.271423\pi$
$390$	0	0
$391$	14.1171	0.713934
$392$	1.52324	0.0769353
$393$	6.35024	0.320327
$394$	18.2720	0.920532
$395$	−12.2712	−0.617430
$396$	−12.0252	−0.604290
$397$	29.9370	1.50250	0.751248	−	0.660019i	$-0.229452\pi$
$397$	29.9370	1.50250	0.751248	+	0.660019i	$0.229452\pi$
$398$	11.6266	0.582790
$399$	−1.83181	−0.0917053
$400$	−1.88644	−0.0943218
$401$	−20.3776	−1.01761	−0.508805	−	0.860882i	$-0.669913\pi$
$401$	−20.3776	−1.01761	−0.508805	+	0.860882i	$0.669913\pi$
$402$	−8.32425	−0.415176
$403$	0	0
$404$	−17.2558	−0.858509
$405$	−6.81322	−0.338552
$406$	11.2165	0.556664
$407$	31.1450	1.54380
$408$	−4.68819	−0.232100
$409$	−37.8118	−1.86968	−0.934838	−	0.355076i	$-0.884455\pi$
$409$	−37.8118	−1.86968	−0.934838	+	0.355076i	$0.884455\pi$
$410$	9.24327	0.456493
$411$	−1.85502	−0.0915012
$412$	−5.40241	−0.266158
$413$	8.16071	0.401562
$414$	5.62691	0.276548
$415$	3.93018	0.192925
$416$	0	0
$417$	14.5310	0.711584
$418$	5.03713	0.246374
$419$	−28.0383	−1.36976	−0.684881	−	0.728655i	$-0.740146\pi$
$419$	−28.0383	−1.36976	−0.684881	+	0.728655i	$0.740146\pi$
$420$	−3.23228	−0.157719
$421$	−3.91159	−0.190639	−0.0953197	−	0.995447i	$-0.530387\pi$
$421$	−3.91159	−0.190639	−0.0953197	+	0.995447i	$0.530387\pi$
$422$	−5.87305	−0.285896
$423$	−0.348014	−0.0169210
$424$	−4.67320	−0.226951
$425$	−11.2987	−0.548067
$426$	−4.14258	−0.200709
$427$	−3.84042	−0.185851
$428$	−17.9901	−0.869582
$429$	0	0
$430$	1.53227	0.0738924
$431$	13.0490	0.628550	0.314275	−	0.949332i	$-0.398239\pi$
$431$	13.0490	0.628550	0.314275	+	0.949332i	$0.398239\pi$
$432$	−4.21688	−0.202885
$433$	29.2972	1.40793	0.703966	−	0.710233i	$-0.251411\pi$
$433$	29.2972	1.40793	0.703966	+	0.710233i	$0.251411\pi$
$434$	−2.03721	−0.0977893
$435$	6.61976	0.317393
$436$	11.7518	0.562811
$437$	−2.35701	−0.112751
$438$	−1.13493	−0.0542293
$439$	−3.91843	−0.187017	−0.0935083	−	0.995618i	$-0.529808\pi$
$439$	−3.91843	−0.187017	−0.0935083	+	0.995618i	$0.529808\pi$
$440$	8.88817	0.423727
$441$	3.63646	0.173165
$442$	0	0
$443$	30.3206	1.44057	0.720287	−	0.693676i	$-0.244010\pi$
$443$	30.3206	1.44057	0.720287	+	0.693676i	$0.244010\pi$
$444$	4.83977	0.229685
$445$	−0.196572	−0.00931842
$446$	−8.94247	−0.423438
$447$	−4.37577	−0.206967
$448$	2.34025	0.110566
$449$	−36.2384	−1.71020	−0.855098	−	0.518466i	$-0.826503\pi$
$449$	−36.2384	−1.71020	−0.855098	+	0.518466i	$0.826503\pi$
$450$	−4.50352	−0.212298
$451$	26.3864	1.24249
$452$	−16.7496	−0.787834
$453$	9.01679	0.423646
$454$	−14.0792	−0.660772
$455$	0	0
$456$	0.782742	0.0366553
$457$	−18.0369	−0.843732	−0.421866	−	0.906658i	$-0.638625\pi$
$457$	−18.0369	−0.843732	−0.421866	+	0.906658i	$0.638625\pi$
$458$	9.30882	0.434973
$459$	−25.2567	−1.17888
$460$	−4.15901	−0.193915
$461$	26.3160	1.22566	0.612828	−	0.790216i	$-0.290032\pi$
$461$	26.3160	1.22566	0.612828	+	0.790216i	$0.290032\pi$
$462$	−9.22707	−0.429282
$463$	18.4306	0.856543	0.428272	−	0.903650i	$-0.359123\pi$
$463$	18.4306	0.856543	0.428272	+	0.903650i	$0.359123\pi$
$464$	−4.79286	−0.222503
$465$	−1.20233	−0.0557565
$466$	−18.3417	−0.849664
$467$	10.5772	0.489456	0.244728	−	0.969592i	$-0.421301\pi$
$467$	10.5772	0.489456	0.244728	+	0.969592i	$0.421301\pi$
$468$	0	0
$469$	24.8879	1.14922
$470$	0.257227	0.0118650
$471$	−2.48616	−0.114556
$472$	−3.48712	−0.160507
$473$	4.37410	0.201121
$474$	−5.44348	−0.250027
$475$	1.88644	0.0865556
$476$	14.0168	0.642457
$477$	−11.1564	−0.510817
$478$	−26.8693	−1.22897
$479$	−36.4162	−1.66390	−0.831950	−	0.554851i	$-0.812775\pi$
$479$	−36.4162	−1.66390	−0.831950	+	0.554851i	$0.812775\pi$
$480$	1.38117	0.0630416
$481$	0	0
$482$	−27.7878	−1.26570
$483$	4.31759	0.196457
$484$	14.3727	0.653304
$485$	14.5005	0.658432
$486$	−15.6730	−0.710941
$487$	19.4971	0.883499	0.441749	−	0.897138i	$-0.354358\pi$
$487$	19.4971	0.883499	0.441749	+	0.897138i	$0.354358\pi$
$488$	1.64103	0.0742860
$489$	10.5781	0.478359
$490$	−2.68780	−0.121423
$491$	21.8767	0.987283	0.493642	−	0.869665i	$-0.335665\pi$
$491$	21.8767	0.987283	0.493642	+	0.869665i	$0.335665\pi$
$492$	4.10030	0.184856
$493$	−28.7065	−1.29288
$494$	0	0
$495$	21.2188	0.953716
$496$	0.870512	0.0390871
$497$	12.3855	0.555566
$498$	1.74342	0.0781246
$499$	1.39352	0.0623823	0.0311912	−	0.999513i	$-0.490070\pi$
$499$	1.39352	0.0623823	0.0311912	+	0.999513i	$0.490070\pi$
$500$	12.1513	0.543423
$501$	0.361798	0.0161639
$502$	−23.8510	−1.06452
$503$	−33.0184	−1.47222	−0.736109	−	0.676863i	$-0.763339\pi$
$503$	−33.0184	−1.47222	−0.736109	+	0.676863i	$0.763339\pi$
$504$	5.58691	0.248861
$505$	30.4484	1.35494
$506$	−11.8725	−0.527799
$507$	0	0
$508$	0.742321	0.0329352
$509$	26.3255	1.16686	0.583429	−	0.812164i	$-0.301711\pi$
$509$	26.3255	1.16686	0.583429	+	0.812164i	$0.301711\pi$
$510$	8.27244	0.366310
$511$	3.39323	0.150108
$512$	−1.00000	−0.0441942
$513$	4.21688	0.186180
$514$	8.28542	0.365454
$515$	9.53271	0.420061
$516$	0.679711	0.0299226
$517$	0.734294	0.0322942
$518$	−14.4700	−0.635773
$519$	10.5462	0.462925
$520$	0	0
$521$	3.72720	0.163292	0.0816458	−	0.996661i	$-0.473982\pi$
$521$	3.72720	0.163292	0.0816458	+	0.996661i	$0.473982\pi$
$522$	−11.4421	−0.500805
$523$	−18.4801	−0.808080	−0.404040	−	0.914741i	$-0.632394\pi$
$523$	−18.4801	−0.808080	−0.404040	+	0.914741i	$0.632394\pi$
$524$	8.11281	0.354410
$525$	−3.45559	−0.150815
$526$	−18.0106	−0.785300
$527$	5.21387	0.227120
$528$	3.94278	0.171587
$529$	−17.4445	−0.758458
$530$	8.24601	0.358184
$531$	−8.32484	−0.361267
$532$	−2.34025	−0.101463
$533$	0	0
$534$	−0.0871992	−0.00377348
$535$	31.7440	1.37241
$536$	−10.6347	−0.459350
$537$	−13.8097	−0.595931
$538$	14.7201	0.634629
$539$	−7.67277	−0.330489
$540$	7.44081	0.320201
$541$	24.5629	1.05604	0.528021	−	0.849232i	$-0.322934\pi$
$541$	24.5629	1.05604	0.528021	+	0.849232i	$0.322934\pi$
$542$	−27.3591	−1.17517
$543$	4.30407	0.184705
$544$	−5.98944	−0.256795
$545$	−20.7365	−0.888252
$546$	0	0
$547$	26.7202	1.14247	0.571237	−	0.820785i	$-0.306464\pi$
$547$	26.7202	1.14247	0.571237	+	0.820785i	$0.306464\pi$
$548$	−2.36989	−0.101237
$549$	3.91766	0.167202
$550$	9.50223	0.405176
$551$	4.79286	0.204183
$552$	−1.84493	−0.0785254
$553$	16.2749	0.692081
$554$	−30.3660	−1.29013
$555$	−8.53991	−0.362499
$556$	18.5642	0.787296
$557$	−9.18596	−0.389221	−0.194611	−	0.980881i	$-0.562344\pi$
$557$	−9.18596	−0.389221	−0.194611	+	0.980881i	$0.562344\pi$
$558$	2.07818	0.0879766
$559$	0	0
$560$	−4.12944	−0.174500
$561$	23.6150	0.997026
$562$	1.85596	0.0782888
$563$	−30.9192	−1.30309	−0.651543	−	0.758611i	$-0.725878\pi$
$563$	−30.9192	−1.30309	−0.651543	+	0.758611i	$0.725878\pi$
$564$	0.114105	0.00480470
$565$	29.5551	1.24339
$566$	−10.7759	−0.452946
$567$	9.03619	0.379484
$568$	−5.29239	−0.222064
$569$	−16.4304	−0.688798	−0.344399	−	0.938823i	$-0.611917\pi$
$569$	−16.4304	−0.688798	−0.344399	+	0.938823i	$0.611917\pi$
$570$	−1.38117	−0.0578509
$571$	42.2387	1.76763	0.883817	−	0.467834i	$-0.154965\pi$
$571$	42.2387	1.76763	0.883817	+	0.467834i	$0.154965\pi$
$572$	0	0
$573$	−6.04290	−0.252446
$574$	−12.2591	−0.511685
$575$	−4.44634	−0.185425
$576$	−2.38731	−0.0994714
$577$	−15.1645	−0.631305	−0.315653	−	0.948875i	$-0.602223\pi$
$577$	−15.1645	−0.631305	−0.315653	+	0.948875i	$0.602223\pi$
$578$	−18.8733	−0.785028
$579$	−14.8193	−0.615871
$580$	8.45714	0.351164
$581$	−5.21249	−0.216251
$582$	6.43238	0.266631
$583$	23.5395	0.974908
$584$	−1.44995	−0.0599992
$585$	0	0
$586$	30.4684	1.25864
$587$	−1.22383	−0.0505128	−0.0252564	−	0.999681i	$-0.508040\pi$
$587$	−1.22383	−0.0505128	−0.0252564	+	0.999681i	$0.508040\pi$
$588$	−1.19231	−0.0491699
$589$	−0.870512	−0.0358688
$590$	6.15312	0.253320
$591$	−14.3023	−0.588318
$592$	6.18309	0.254123
$593$	27.2021	1.11706	0.558529	−	0.829485i	$-0.311366\pi$
$593$	27.2021	1.11706	0.558529	+	0.829485i	$0.311366\pi$
$594$	21.2410	0.871528
$595$	−24.7330	−1.01395
$596$	−5.59030	−0.228988
$597$	−9.10066	−0.372465
$598$	0	0
$599$	−7.31560	−0.298907	−0.149454	−	0.988769i	$-0.547752\pi$
$599$	−7.31560	−0.298907	−0.149454	+	0.988769i	$0.547752\pi$
$600$	1.47659	0.0602817
$601$	−38.6006	−1.57455	−0.787276	−	0.616601i	$-0.788509\pi$
$601$	−38.6006	−1.57455	−0.787276	+	0.616601i	$0.788509\pi$
$602$	−2.03220	−0.0828264
$603$	−25.3884	−1.03390
$604$	11.5195	0.468721
$605$	−25.3610	−1.03107
$606$	13.5069	0.548678
$607$	−19.5391	−0.793069	−0.396534	−	0.918020i	$-0.629787\pi$
$607$	−19.5391	−0.793069	−0.396534	+	0.918020i	$0.629787\pi$
$608$	1.00000	0.0405554
$609$	−8.77961	−0.355768
$610$	−2.89565	−0.117241
$611$	0	0
$612$	−14.2987	−0.577989
$613$	−37.1023	−1.49855	−0.749275	−	0.662259i	$-0.769598\pi$
$613$	−37.1023	−1.49855	−0.749275	+	0.662259i	$0.769598\pi$
$614$	19.1868	0.774315
$615$	−7.23510	−0.291747
$616$	−11.7881	−0.474957
$617$	1.19861	0.0482544	0.0241272	−	0.999709i	$-0.492319\pi$
$617$	1.19861	0.0482544	0.0241272	+	0.999709i	$0.492319\pi$
$618$	4.22869	0.170103
$619$	20.4675	0.822659	0.411329	−	0.911487i	$-0.365065\pi$
$619$	20.4675	0.822659	0.411329	+	0.911487i	$0.365065\pi$
$620$	−1.53604	−0.0616890
$621$	−9.93921	−0.398847
$622$	−17.7139	−0.710261
$623$	0.260709	0.0104451
$624$	0	0
$625$	−12.0092	−0.480367
$626$	−4.68889	−0.187406
$627$	−3.94278	−0.157459
$628$	−3.17622	−0.126745
$629$	37.0332	1.47661
$630$	−9.85826	−0.392762
$631$	7.67083	0.305371	0.152686	−	0.988275i	$-0.451208\pi$
$631$	7.67083	0.305371	0.152686	+	0.988275i	$0.451208\pi$
$632$	−6.95437	−0.276630
$633$	4.59709	0.182718
$634$	20.8847	0.829439
$635$	−1.30985	−0.0519797
$636$	3.65791	0.145046
$637$	0	0
$638$	24.1423	0.955801
$639$	−12.6346	−0.499817
$640$	1.76453	0.0697491
$641$	−49.1615	−1.94176	−0.970881	−	0.239563i	$-0.922996\pi$
$641$	−49.1615	−1.94176	−0.970881	+	0.239563i	$0.922996\pi$
$642$	14.0816	0.555756
$643$	26.6857	1.05238	0.526190	−	0.850367i	$-0.323620\pi$
$643$	26.6857	1.05238	0.526190	+	0.850367i	$0.323620\pi$
$644$	5.51598	0.217360
$645$	−1.19937	−0.0472251
$646$	5.98944	0.235651
$647$	47.0769	1.85078	0.925392	−	0.379012i	$-0.123736\pi$
$647$	47.0769	1.85078	0.925392	+	0.379012i	$0.123736\pi$
$648$	−3.86121	−0.151683
$649$	17.5651	0.689489
$650$	0	0
$651$	1.59461	0.0624978
$652$	13.5142	0.529256
$653$	34.6956	1.35774	0.678872	−	0.734257i	$-0.262469\pi$
$653$	34.6956	1.35774	0.678872	+	0.734257i	$0.262469\pi$
$654$	−9.19866	−0.359696
$655$	−14.3153	−0.559345
$656$	5.23838	0.204524
$657$	−3.46148	−0.135045
$658$	−0.341153	−0.0132995
$659$	−24.9592	−0.972273	−0.486136	−	0.873883i	$-0.661594\pi$
$659$	−24.9592	−0.972273	−0.486136	+	0.873883i	$0.661594\pi$
$660$	−6.95714	−0.270806
$661$	13.9916	0.544210	0.272105	−	0.962268i	$-0.412280\pi$
$661$	13.9916	0.544210	0.272105	+	0.962268i	$0.412280\pi$
$662$	−14.1797	−0.551111
$663$	0	0
$664$	2.22732	0.0864370
$665$	4.12944	0.160133
$666$	14.7610	0.571976
$667$	−11.2968	−0.437414
$668$	0.462218	0.0178837
$669$	6.99965	0.270622
$670$	18.7653	0.724966
$671$	−8.26609	−0.319109
$672$	−1.83181	−0.0706636
$673$	26.9743	1.03978	0.519892	−	0.854232i	$-0.325972\pi$
$673$	26.9743	1.03978	0.519892	+	0.854232i	$0.325972\pi$
$674$	−6.92799	−0.266856
$675$	7.95487	0.306183
$676$	0	0
$677$	−6.92926	−0.266313	−0.133157	−	0.991095i	$-0.542511\pi$
$677$	−6.92926	−0.266313	−0.133157	+	0.991095i	$0.542511\pi$
$678$	13.1106	0.503510
$679$	−19.2316	−0.738039
$680$	10.5685	0.405285
$681$	11.0204	0.422303
$682$	−4.38488	−0.167906
$683$	−14.7573	−0.564672	−0.282336	−	0.959316i	$-0.591109\pi$
$683$	−14.7573	−0.564672	−0.282336	+	0.959316i	$0.591109\pi$
$684$	2.38731	0.0912812
$685$	4.18174	0.159776
$686$	19.9465	0.761561
$687$	−7.28641	−0.277994
$688$	0.868371	0.0331063
$689$	0	0
$690$	3.25543	0.123932
$691$	17.5201	0.666494	0.333247	−	0.942840i	$-0.391856\pi$
$691$	17.5201	0.666494	0.333247	+	0.942840i	$0.391856\pi$
$692$	13.4733	0.512179
$693$	−28.1420	−1.06903
$694$	14.8567	0.563952
$695$	−32.7570	−1.24254
$696$	3.75157	0.142203
$697$	31.3749	1.18841
$698$	19.3683	0.733101
$699$	14.3568	0.543026
$700$	−4.41473	−0.166861
$701$	−41.8360	−1.58012	−0.790061	−	0.613028i	$-0.789951\pi$
$701$	−41.8360	−1.58012	−0.790061	+	0.613028i	$0.789951\pi$
$702$	0	0
$703$	−6.18309	−0.233200
$704$	5.03713	0.189844
$705$	−0.201342	−0.00758298
$706$	5.27186	0.198409
$707$	−40.3829	−1.51875
$708$	2.72951	0.102581
$709$	−42.3097	−1.58897	−0.794487	−	0.607281i	$-0.792260\pi$
$709$	−42.3097	−1.58897	−0.794487	+	0.607281i	$0.792260\pi$
$710$	9.33858	0.350471
$711$	−16.6023	−0.622633
$712$	−0.111402	−0.00417497
$713$	2.05180	0.0768405
$714$	−10.9715	−0.410599
$715$	0	0
$716$	−17.6427	−0.659338
$717$	21.0318	0.785446
$718$	−30.5453	−1.13994
$719$	7.10100	0.264823	0.132411	−	0.991195i	$-0.457728\pi$
$719$	7.10100	0.264823	0.132411	+	0.991195i	$0.457728\pi$
$720$	4.21249	0.156990
$721$	−12.6430	−0.470849
$722$	−1.00000	−0.0372161
$723$	21.7507	0.808916
$724$	5.49870	0.204358
$725$	9.04142	0.335790
$726$	−11.2501	−0.417531
$727$	−15.4980	−0.574790	−0.287395	−	0.957812i	$-0.592789\pi$
$727$	−15.4980	−0.574790	−0.287395	+	0.957812i	$0.592789\pi$
$728$	0	0
$729$	0.684264	0.0253431
$730$	2.55847	0.0946933
$731$	5.20105	0.192368
$732$	−1.28451	−0.0474767
$733$	28.7469	1.06179	0.530895	−	0.847438i	$-0.321856\pi$
$733$	28.7469	1.06179	0.530895	+	0.847438i	$0.321856\pi$
$734$	−5.94574	−0.219461
$735$	2.10386	0.0776020
$736$	−2.35701	−0.0868804
$737$	53.5685	1.97322
$738$	12.5057	0.460340
$739$	18.4927	0.680263	0.340132	−	0.940378i	$-0.389528\pi$
$739$	18.4927	0.680263	0.340132	+	0.940378i	$0.389528\pi$
$740$	−10.9102	−0.401068
$741$	0	0
$742$	−10.9365	−0.401490
$743$	−9.62790	−0.353213	−0.176607	−	0.984281i	$-0.556512\pi$
$743$	−9.62790	−0.353213	−0.176607	+	0.984281i	$0.556512\pi$
$744$	−0.681386	−0.0249808
$745$	9.86425	0.361398
$746$	−32.0382	−1.17300
$747$	5.31732	0.194551
$748$	30.1696	1.10311
$749$	−42.1012	−1.53834
$750$	−9.51135	−0.347305
$751$	30.5592	1.11512	0.557560	−	0.830137i	$-0.311738\pi$
$751$	30.5592	1.11512	0.557560	+	0.830137i	$0.311738\pi$
$752$	0.145776	0.00531591
$753$	18.6692	0.680342
$754$	0	0
$755$	−20.3265	−0.739756
$756$	−9.86854	−0.358915
$757$	37.3681	1.35817	0.679084	−	0.734061i	$-0.262377\pi$
$757$	37.3681	1.35817	0.679084	+	0.734061i	$0.262377\pi$
$758$	−32.0536	−1.16424
$759$	9.29315	0.337320
$760$	−1.76453	−0.0640062
$761$	37.7326	1.36781	0.683903	−	0.729573i	$-0.260281\pi$
$761$	37.7326	1.36781	0.683903	+	0.729573i	$0.260281\pi$
$762$	−0.581046	−0.0210491
$763$	27.5022	0.995647
$764$	−7.72016	−0.279306
$765$	25.2304	0.912208
$766$	−25.9231	−0.936639
$767$	0	0
$768$	0.782742	0.0282448
$769$	16.4011	0.591439	0.295720	−	0.955275i	$-0.404441\pi$
$769$	16.4011	0.591439	0.295720	+	0.955275i	$0.404441\pi$
$770$	20.8005	0.749598
$771$	−6.48535	−0.233564
$772$	−18.9326	−0.681399
$773$	−32.4644	−1.16766	−0.583831	−	0.811875i	$-0.698447\pi$
$773$	−32.4644	−1.16766	−0.583831	+	0.811875i	$0.698447\pi$
$774$	2.07307	0.0745151
$775$	−1.64216	−0.0589883
$776$	8.21775	0.295000
$777$	11.3262	0.406327
$778$	−25.9537	−0.930486
$779$	−5.23838	−0.187684
$780$	0	0
$781$	26.6585	0.953915
$782$	−14.1171	−0.504828
$783$	20.2109	0.722279
$784$	−1.52324	−0.0544015
$785$	5.60454	0.200034
$786$	−6.35024	−0.226506
$787$	0.645004	0.0229919	0.0114960	−	0.999934i	$-0.496341\pi$
$787$	0.645004	0.0229919	0.0114960	+	0.999934i	$0.496341\pi$
$788$	−18.2720	−0.650914
$789$	14.0977	0.501890
$790$	12.2712	0.436589
$791$	−39.1982	−1.39373
$792$	12.0252	0.427297
$793$	0	0
$794$	−29.9370	−1.06243
$795$	−6.45450	−0.228917
$796$	−11.6266	−0.412095
$797$	39.4882	1.39874	0.699372	−	0.714758i	$-0.253463\pi$
$797$	39.4882	1.39874	0.699372	+	0.714758i	$0.253463\pi$
$798$	1.83181	0.0648454
$799$	0.873118	0.0308887
$800$	1.88644	0.0666956
$801$	−0.265952	−0.00939695
$802$	20.3776	0.719559
$803$	7.30357	0.257737
$804$	8.32425	0.293574
$805$	−9.73310	−0.343047
$806$	0	0
$807$	−11.5221	−0.405596
$808$	17.2558	0.607057
$809$	−34.6882	−1.21957	−0.609787	−	0.792566i	$-0.708745\pi$
$809$	−34.6882	−1.21957	−0.609787	+	0.792566i	$0.708745\pi$
$810$	6.81322	0.239392
$811$	−47.9079	−1.68227	−0.841137	−	0.540821i	$-0.818114\pi$
$811$	−47.9079	−1.68227	−0.841137	+	0.540821i	$0.818114\pi$
$812$	−11.2165	−0.393621
$813$	21.4151	0.751060
$814$	−31.1450	−1.09163
$815$	−23.8461	−0.835294
$816$	4.68819	0.164119
$817$	−0.868371	−0.0303805
$818$	37.8118	1.32206
$819$	0	0
$820$	−9.24327	−0.322789
$821$	−25.8271	−0.901372	−0.450686	−	0.892683i	$-0.648821\pi$
$821$	−25.8271	−0.901372	−0.450686	+	0.892683i	$0.648821\pi$
$822$	1.85502	0.0647011
$823$	−35.1210	−1.22424	−0.612120	−	0.790765i	$-0.709683\pi$
$823$	−35.1210	−1.22424	−0.612120	+	0.790765i	$0.709683\pi$
$824$	5.40241	0.188202
$825$	−7.43780	−0.258951
$826$	−8.16071	−0.283948
$827$	−16.1634	−0.562056	−0.281028	−	0.959700i	$-0.590675\pi$
$827$	−16.1634	−0.562056	−0.281028	+	0.959700i	$0.590675\pi$
$828$	−5.62691	−0.195549
$829$	−30.4427	−1.05732	−0.528660	−	0.848834i	$-0.677305\pi$
$829$	−30.4427	−1.05732	−0.528660	+	0.848834i	$0.677305\pi$
$830$	−3.93018	−0.136419
$831$	23.7687	0.824528
$832$	0	0
$833$	−9.12336	−0.316106
$834$	−14.5310	−0.503166
$835$	−0.815597	−0.0282249
$836$	−5.03713	−0.174213
$837$	−3.67084	−0.126883
$838$	28.0383	0.968568
$839$	19.1260	0.660303	0.330152	−	0.943928i	$-0.392900\pi$
$839$	19.1260	0.660303	0.330152	+	0.943928i	$0.392900\pi$
$840$	3.23228	0.111524
$841$	−6.02851	−0.207880
$842$	3.91159	0.134802
$843$	−1.45274	−0.0500349
$844$	5.87305	0.202159
$845$	0	0
$846$	0.348014	0.0119650
$847$	33.6357	1.15573
$848$	4.67320	0.160478
$849$	8.43477	0.289481
$850$	11.2987	0.387542
$851$	14.5736	0.499576
$852$	4.14258	0.141922
$853$	−20.4540	−0.700331	−0.350166	−	0.936688i	$-0.613875\pi$
$853$	−20.4540	−0.700331	−0.350166	+	0.936688i	$0.613875\pi$
$854$	3.84042	0.131417
$855$	−4.21249	−0.144064
$856$	17.9901	0.614887
$857$	3.63453	0.124153	0.0620765	−	0.998071i	$-0.480228\pi$
$857$	3.63453	0.124153	0.0620765	+	0.998071i	$0.480228\pi$
$858$	0	0
$859$	−7.84999	−0.267838	−0.133919	−	0.990992i	$-0.542756\pi$
$859$	−7.84999	−0.267838	−0.133919	+	0.990992i	$0.542756\pi$
$860$	−1.53227	−0.0522498
$861$	9.59572	0.327021
$862$	−13.0490	−0.444452
$863$	−13.2290	−0.450322	−0.225161	−	0.974322i	$-0.572291\pi$
$863$	−13.2290	−0.450322	−0.225161	+	0.974322i	$0.572291\pi$
$864$	4.21688	0.143461
$865$	−23.7741	−0.808343
$866$	−29.2972	−0.995559
$867$	14.7730	0.501716
$868$	2.03721	0.0691475
$869$	35.0301	1.18831
$870$	−6.61976	−0.224431
$871$	0	0
$872$	−11.7518	−0.397967
$873$	19.6183	0.663980
$874$	2.35701	0.0797269
$875$	28.4371	0.961349
$876$	1.13493	0.0383459
$877$	29.2693	0.988355	0.494177	−	0.869361i	$-0.335469\pi$
$877$	29.2693	0.988355	0.494177	+	0.869361i	$0.335469\pi$
$878$	3.91843	0.132241
$879$	−23.8489	−0.804404
$880$	−8.88817	−0.299620
$881$	−26.6284	−0.897135	−0.448567	−	0.893749i	$-0.648066\pi$
$881$	−26.6284	−0.897135	−0.448567	+	0.893749i	$0.648066\pi$
$882$	−3.63646	−0.122446
$883$	−0.321650	−0.0108244	−0.00541220	−	0.999985i	$-0.501723\pi$
$883$	−0.321650	−0.0108244	−0.00541220	+	0.999985i	$0.501723\pi$
$884$	0	0
$885$	−4.81631	−0.161898
$886$	−30.3206	−1.01864
$887$	−15.1500	−0.508688	−0.254344	−	0.967114i	$-0.581860\pi$
$887$	−15.1500	−0.508688	−0.254344	+	0.967114i	$0.581860\pi$
$888$	−4.83977	−0.162412
$889$	1.73721	0.0582643
$890$	0.196572	0.00658912
$891$	19.4494	0.651581
$892$	8.94247	0.299416
$893$	−0.145776	−0.00487822
$894$	4.37577	0.146348
$895$	31.1310	1.04060
$896$	−2.34025	−0.0781822
$897$	0	0
$898$	36.2384	1.20929
$899$	−4.17224	−0.139152
$900$	4.50352	0.150117
$901$	27.9899	0.932477
$902$	−26.3864	−0.878571
$903$	1.59069	0.0529349
$904$	16.7496	0.557083
$905$	−9.70262	−0.322526
$906$	−9.01679	−0.299563
$907$	43.2174	1.43501	0.717506	−	0.696552i	$-0.245284\pi$
$907$	43.2174	1.43501	0.717506	+	0.696552i	$0.245284\pi$
$908$	14.0792	0.467236
$909$	41.1950	1.36635
$910$	0	0
$911$	−15.4723	−0.512621	−0.256310	−	0.966595i	$-0.582507\pi$
$911$	−15.4723	−0.512621	−0.256310	+	0.966595i	$0.582507\pi$
$912$	−0.782742	−0.0259192
$913$	−11.2193	−0.371306
$914$	18.0369	0.596608
$915$	2.26655	0.0749297
$916$	−9.30882	−0.307572
$917$	18.9860	0.626973
$918$	25.2567	0.833596
$919$	−21.4089	−0.706214	−0.353107	−	0.935583i	$-0.614875\pi$
$919$	−21.4089	−0.706214	−0.353107	+	0.935583i	$0.614875\pi$
$920$	4.15901	0.137118
$921$	−15.0183	−0.494870
$922$	−26.3160	−0.866670
$923$	0	0
$924$	9.22707	0.303548
$925$	−11.6640	−0.383510
$926$	−18.4306	−0.605668
$927$	12.8972	0.423601
$928$	4.79286	0.157333
$929$	−9.56755	−0.313901	−0.156951	−	0.987606i	$-0.550166\pi$
$929$	−9.56755	−0.313901	−0.156951	+	0.987606i	$0.550166\pi$
$930$	1.20233	0.0394258
$931$	1.52324	0.0499222
$932$	18.3417	0.600803
$933$	13.8654	0.453932
$934$	−10.5772	−0.346098
$935$	−53.2351	−1.74097
$936$	0	0
$937$	−29.4433	−0.961872	−0.480936	−	0.876756i	$-0.659703\pi$
$937$	−29.4433	−0.961872	−0.480936	+	0.876756i	$0.659703\pi$
$938$	−24.8879	−0.812618
$939$	3.67019	0.119772
$940$	−0.257227	−0.00838980
$941$	26.1539	0.852591	0.426296	−	0.904584i	$-0.359818\pi$
$941$	26.1539	0.852591	0.426296	+	0.904584i	$0.359818\pi$
$942$	2.48616	0.0810036
$943$	12.3469	0.402070
$944$	3.48712	0.113496
$945$	17.4133	0.566456
$946$	−4.37410	−0.142214
$947$	−1.08053	−0.0351124	−0.0175562	−	0.999846i	$-0.505589\pi$
$947$	−1.08053	−0.0351124	−0.0175562	+	0.999846i	$0.505589\pi$
$948$	5.44348	0.176796
$949$	0	0
$950$	−1.88644	−0.0612041
$951$	−16.3474	−0.530100
$952$	−14.0168	−0.454286
$953$	14.0462	0.455001	0.227501	−	0.973778i	$-0.426945\pi$
$953$	14.0462	0.455001	0.227501	+	0.973778i	$0.426945\pi$
$954$	11.1564	0.361202
$955$	13.6225	0.440812
$956$	26.8693	0.869016
$957$	−18.8972	−0.610859
$958$	36.4162	1.17655
$959$	−5.54613	−0.179094
$960$	−1.38117	−0.0445771
$961$	−30.2422	−0.975555
$962$	0	0
$963$	42.9479	1.38398
$964$	27.7878	0.894983
$965$	33.4071	1.07541
$966$	−4.31759	−0.138916
$967$	−30.5706	−0.983084	−0.491542	−	0.870854i	$-0.663567\pi$
$967$	−30.5706	−0.983084	−0.491542	+	0.870854i	$0.663567\pi$
$968$	−14.3727	−0.461956
$969$	−4.68819	−0.150606
$970$	−14.5005	−0.465581
$971$	−33.4166	−1.07239	−0.536195	−	0.844094i	$-0.680139\pi$
$971$	−33.4166	−1.07239	−0.536195	+	0.844094i	$0.680139\pi$
$972$	15.6730	0.502711
$973$	43.4447	1.39277
$974$	−19.4971	−0.624728
$975$	0	0
$976$	−1.64103	−0.0525282
$977$	−15.6919	−0.502027	−0.251013	−	0.967984i	$-0.580764\pi$
$977$	−15.6919	−0.502027	−0.251013	+	0.967984i	$0.580764\pi$
$978$	−10.5781	−0.338251
$979$	0.561147	0.0179343
$980$	2.68780	0.0858587
$981$	−28.0553	−0.895737
$982$	−21.8767	−0.698115
$983$	15.6480	0.499095	0.249547	−	0.968363i	$-0.419718\pi$
$983$	15.6480	0.499095	0.249547	+	0.968363i	$0.419718\pi$
$984$	−4.10030	−0.130713
$985$	32.2416	1.02730
$986$	28.7065	0.914202
$987$	0.267035	0.00849980
$988$	0	0
$989$	2.04676	0.0650830
$990$	−21.2188	−0.674379
$991$	55.6043	1.76633	0.883165	−	0.469063i	$-0.155408\pi$
$991$	55.6043	1.76633	0.883165	+	0.469063i	$0.155408\pi$
$992$	−0.870512	−0.0276388
$993$	11.0991	0.352219
$994$	−12.3855	−0.392844
$995$	20.5155	0.650386
$996$	−1.74342	−0.0552424
$997$	26.5671	0.841388	0.420694	−	0.907203i	$-0.361787\pi$
$997$	26.5671	0.841388	0.420694	+	0.907203i	$0.361787\pi$
$998$	−1.39352	−0.0441110
$999$	−26.0733	−0.824924

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bi.1.7	✓	9
13.12	even	2		6422.2.a.bk.1.7	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bi.1.7	✓	9	1.1	even	1	trivial
6422.2.a.bk.1.7	yes	9	13.12	even	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.782742 q^{3} +1.00000 q^{4} -1.76453 q^{5} -0.782742 q^{6} +2.34025 q^{7} -1.00000 q^{8} -2.38731 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.782742 q^{3} +1.00000 q^{4} -1.76453 q^{5} -0.782742 q^{6} +2.34025 q^{7} -1.00000 q^{8} -2.38731 q^{9} +1.76453 q^{10} +5.03713 q^{11} +0.782742 q^{12} -2.34025 q^{14} -1.38117 q^{15} +1.00000 q^{16} +5.98944 q^{17} +2.38731 q^{18} -1.00000 q^{19} -1.76453 q^{20} +1.83181 q^{21} -5.03713 q^{22} +2.35701 q^{23} -0.782742 q^{24} -1.88644 q^{25} -4.21688 q^{27} +2.34025 q^{28} -4.79286 q^{29} +1.38117 q^{30} +0.870512 q^{31} -1.00000 q^{32} +3.94278 q^{33} -5.98944 q^{34} -4.12944 q^{35} -2.38731 q^{36} +6.18309 q^{37} +1.00000 q^{38} +1.76453 q^{40} +5.23838 q^{41} -1.83181 q^{42} +0.868371 q^{43} +5.03713 q^{44} +4.21249 q^{45} -2.35701 q^{46} +0.145776 q^{47} +0.782742 q^{48} -1.52324 q^{49} +1.88644 q^{50} +4.68819 q^{51} +4.67320 q^{53} +4.21688 q^{54} -8.88817 q^{55} -2.34025 q^{56} -0.782742 q^{57} +4.79286 q^{58} +3.48712 q^{59} -1.38117 q^{60} -1.64103 q^{61} -0.870512 q^{62} -5.58691 q^{63} +1.00000 q^{64} -3.94278 q^{66} +10.6347 q^{67} +5.98944 q^{68} +1.84493 q^{69} +4.12944 q^{70} +5.29239 q^{71} +2.38731 q^{72} +1.44995 q^{73} -6.18309 q^{74} -1.47659 q^{75} -1.00000 q^{76} +11.7881 q^{77} +6.95437 q^{79} -1.76453 q^{80} +3.86121 q^{81} -5.23838 q^{82} -2.22732 q^{83} +1.83181 q^{84} -10.5685 q^{85} -0.868371 q^{86} -3.75157 q^{87} -5.03713 q^{88} +0.111402 q^{89} -4.21249 q^{90} +2.35701 q^{92} +0.681386 q^{93} -0.145776 q^{94} +1.76453 q^{95} -0.782742 q^{96} -8.21775 q^{97} +1.52324 q^{98} -12.0252 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{2} - 5 q^{3} + 9 q^{4} - q^{5} + 5 q^{6} + 13 q^{7} - 9 q^{8} + 10 q^{9}+O(q^{10}) \) 9 * q - 9 * q^2 - 5 * q^3 + 9 * q^4 - q^5 + 5 * q^6 + 13 * q^7 - 9 * q^8 + 10 * q^9	\( 9 q - 9 q^{2} - 5 q^{3} + 9 q^{4} - q^{5} + 5 q^{6} + 13 q^{7} - 9 q^{8} + 10 q^{9} + q^{10} + 13 q^{11} - 5 q^{12} - 13 q^{14} + q^{15} + 9 q^{16} - 12 q^{17} - 10 q^{18} - 9 q^{19} - q^{20} - 18 q^{21} - 13 q^{22} - 22 q^{23} + 5 q^{24} + 4 q^{25} - 26 q^{27} + 13 q^{28} + 12 q^{29} - q^{30} - q^{31} - 9 q^{32} + 28 q^{33} + 12 q^{34} - 18 q^{35} + 10 q^{36} + 25 q^{37} + 9 q^{38} + q^{40} - 11 q^{41} + 18 q^{42} - 10 q^{43} + 13 q^{44} + q^{45} + 22 q^{46} + 12 q^{47} - 5 q^{48} + 2 q^{49} - 4 q^{50} + 35 q^{51} + 9 q^{53} + 26 q^{54} + 18 q^{55} - 13 q^{56} + 5 q^{57} - 12 q^{58} - 10 q^{59} + q^{60} + 32 q^{61} + q^{62} + 63 q^{63} + 9 q^{64} - 28 q^{66} + 73 q^{67} - 12 q^{68} + 2 q^{69} + 18 q^{70} + 51 q^{71} - 10 q^{72} + 14 q^{73} - 25 q^{74} - 49 q^{75} - 9 q^{76} + 18 q^{77} - 28 q^{79} - q^{80} + 29 q^{81} + 11 q^{82} + 22 q^{83} - 18 q^{84} + 51 q^{85} + 10 q^{86} - 20 q^{87} - 13 q^{88} - 3 q^{89} - q^{90} - 22 q^{92} + 59 q^{93} - 12 q^{94} + q^{95} + 5 q^{96} - 2 q^{98} - 25 q^{99}+O(q^{100}) \) 9 * q - 9 * q^2 - 5 * q^3 + 9 * q^4 - q^5 + 5 * q^6 + 13 * q^7 - 9 * q^8 + 10 * q^9 + q^10 + 13 * q^11 - 5 * q^12 - 13 * q^14 + q^15 + 9 * q^16 - 12 * q^17 - 10 * q^18 - 9 * q^19 - q^20 - 18 * q^21 - 13 * q^22 - 22 * q^23 + 5 * q^24 + 4 * q^25 - 26 * q^27 + 13 * q^28 + 12 * q^29 - q^30 - q^31 - 9 * q^32 + 28 * q^33 + 12 * q^34 - 18 * q^35 + 10 * q^36 + 25 * q^37 + 9 * q^38 + q^40 - 11 * q^41 + 18 * q^42 - 10 * q^43 + 13 * q^44 + q^45 + 22 * q^46 + 12 * q^47 - 5 * q^48 + 2 * q^49 - 4 * q^50 + 35 * q^51 + 9 * q^53 + 26 * q^54 + 18 * q^55 - 13 * q^56 + 5 * q^57 - 12 * q^58 - 10 * q^59 + q^60 + 32 * q^61 + q^62 + 63 * q^63 + 9 * q^64 - 28 * q^66 + 73 * q^67 - 12 * q^68 + 2 * q^69 + 18 * q^70 + 51 * q^71 - 10 * q^72 + 14 * q^73 - 25 * q^74 - 49 * q^75 - 9 * q^76 + 18 * q^77 - 28 * q^79 - q^80 + 29 * q^81 + 11 * q^82 + 22 * q^83 - 18 * q^84 + 51 * q^85 + 10 * q^86 - 20 * q^87 - 13 * q^88 - 3 * q^89 - q^90 - 22 * q^92 + 59 * q^93 - 12 * q^94 + q^95 + 5 * q^96 - 2 * q^98 - 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more