LMFDB - Embedded newform 63.10.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,10,Mod(1,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("63.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	63.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:	$32.4472576783$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 426x + 2016 $ x^3 - x^2 - 426*x + 2016
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$18.2745$ of defining polynomial
Character	$\chi$	$=$	63.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+34.1627 q^{2} +655.088 q^{4} -1423.70 q^{5} +2401.00 q^{7} +4888.28 q^{8} -48637.4 q^{10} -69354.4 q^{11} +105959. q^{13} +82024.6 q^{14} -168409. q^{16} -568267. q^{17} -396405. q^{19} -932649. q^{20} -2.36933e6 q^{22} +620765. q^{23} +73796.7 q^{25} +3.61984e6 q^{26} +1.57287e6 q^{28} -4.87652e6 q^{29} -1.42482e6 q^{31} -8.25609e6 q^{32} -1.94135e7 q^{34} -3.41830e6 q^{35} +1.31092e7 q^{37} -1.35423e7 q^{38} -6.95944e6 q^{40} +2.03049e7 q^{41} -1.11768e7 q^{43} -4.54332e7 q^{44} +2.12070e7 q^{46} +1.99352e7 q^{47} +5.76480e6 q^{49} +2.52109e6 q^{50} +6.94125e7 q^{52} -5.65007e7 q^{53} +9.87398e7 q^{55} +1.17367e7 q^{56} -1.66595e8 q^{58} +1.09340e8 q^{59} +3.20008e7 q^{61} -4.86755e7 q^{62} -1.95825e8 q^{64} -1.50854e8 q^{65} +8.02869e7 q^{67} -3.72265e8 q^{68} -1.16778e8 q^{70} -2.07893e8 q^{71} -2.70274e8 q^{73} +4.47844e8 q^{74} -2.59680e8 q^{76} -1.66520e8 q^{77} -5.16196e8 q^{79} +2.39763e8 q^{80} +6.93671e8 q^{82} +6.82693e8 q^{83} +8.09042e8 q^{85} -3.81830e8 q^{86} -3.39023e8 q^{88} +1.47150e8 q^{89} +2.54408e8 q^{91} +4.06656e8 q^{92} +6.81040e8 q^{94} +5.64362e8 q^{95} +1.09643e9 q^{97} +1.96941e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 21 q^{2} + 1557 q^{4} - 1554 q^{5} + 7203 q^{7} - 14055 q^{8} - 97860 q^{10} + 3444 q^{11} - 19782 q^{13} - 50421 q^{14} + 482961 q^{16} - 1016694 q^{17} + 222852 q^{19} + 1922088 q^{20} - 2847048 q^{22}+ \cdots - 121060821 q^{98}+O(q^{100}) $ 3 * q - 21 * q^2 + 1557 * q^4 - 1554 * q^5 + 7203 * q^7 - 14055 * q^8 - 97860 * q^10 + 3444 * q^11 - 19782 * q^13 - 50421 * q^14 + 482961 * q^16 - 1016694 * q^17 + 222852 * q^19 + 1922088 * q^20 - 2847048 * q^22 - 1885632 * q^23 + 3073221 * q^25 + 8785056 * q^26 + 3738357 * q^28 - 4081818 * q^29 + 2869440 * q^31 - 25221951 * q^32 - 3981642 * q^34 - 3731154 * q^35 + 1395618 * q^37 - 43479870 * q^38 - 82859280 * q^40 + 14420658 * q^41 - 61631172 * q^43 - 97011984 * q^44 + 89747664 * q^46 + 10368960 * q^47 + 17294403 * q^49 - 73325055 * q^50 - 80908044 * q^52 - 67502610 * q^53 - 105823032 * q^55 - 33746055 * q^56 - 159163830 * q^58 + 42590100 * q^59 + 191746842 * q^61 + 46983468 * q^62 + 7852161 * q^64 - 364283220 * q^65 - 255175788 * q^67 - 743485806 * q^68 - 234961860 * q^70 - 296514504 * q^71 + 344213310 * q^73 + 690696462 * q^74 + 728839986 * q^76 + 8269044 * q^77 - 960412656 * q^79 + 1333333344 * q^80 + 562675302 * q^82 + 1100517180 * q^83 + 438179412 * q^85 + 880982256 * q^86 + 1206124800 * q^88 - 506816478 * q^89 - 47496582 * q^91 - 691123488 * q^92 + 1388004828 * q^94 + 2203071072 * q^95 - 647498250 * q^97 - 121060821 * q^98

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 $

$2$	34.1627	1.50979	0.754896	−	0.655845i	$-0.227688\pi$
$2$	34.1627	1.50979	0.754896	+	0.655845i	$0.227688\pi$
$3$	0	0
$3$	0	0
$4$	655.088	1.27947
$5$	−1423.70	−1.01872	−0.509358	−	0.860554i	$-0.670117\pi$
$5$	−1423.70	−1.01872	−0.509358	+	0.860554i	$0.670117\pi$
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	4888.28	0.421940
$9$	0	0
$10$	−48637.4	−1.53805
$11$	−69354.4	−1.42826	−0.714129	−	0.700014i	$-0.753177\pi$
$11$	−69354.4	−1.42826	−0.714129	+	0.700014i	$0.753177\pi$
$12$	0	0
$13$	105959.	1.02895	0.514473	−	0.857506i	$-0.327987\pi$
$13$	105959.	1.02895	0.514473	+	0.857506i	$0.327987\pi$
$14$	82024.6	0.570647
$15$	0	0
$16$	−168409.	−0.642428
$17$	−568267.	−1.65018	−0.825092	−	0.564998i	$-0.808877\pi$
$17$	−568267.	−1.65018	−0.825092	+	0.564998i	$0.808877\pi$
$18$	0	0
$19$	−396405.	−0.697828	−0.348914	−	0.937155i	$-0.613449\pi$
$19$	−396405.	−0.697828	−0.348914	+	0.937155i	$0.613449\pi$
$20$	−932649.	−1.30342
$21$	0	0
$22$	−2.36933e6	−2.15637
$23$	620765.	0.462543	0.231271	−	0.972889i	$-0.425712\pi$
$23$	620765.	0.462543	0.231271	+	0.972889i	$0.425712\pi$
$24$	0	0
$25$	73796.7	0.0377839
$26$	3.61984e6	1.55349
$27$	0	0
$28$	1.57287e6	0.483594
$29$	−4.87652e6	−1.28032	−0.640161	−	0.768241i	$-0.721132\pi$
$29$	−4.87652e6	−1.28032	−0.640161	+	0.768241i	$0.721132\pi$
$30$	0	0
$31$	−1.42482e6	−0.277096	−0.138548	−	0.990356i	$-0.544244\pi$
$31$	−1.42482e6	−0.277096	−0.138548	+	0.990356i	$0.544244\pi$
$32$	−8.25609e6	−1.39187
$33$	0	0
$34$	−1.94135e7	−2.49143
$35$	−3.41830e6	−0.385039
$36$	0	0
$37$	1.31092e7	1.14992	0.574960	−	0.818182i	$-0.305018\pi$
$37$	1.31092e7	1.14992	0.574960	+	0.818182i	$0.305018\pi$
$38$	−1.35423e7	−1.05357
$39$	0	0
$40$	−6.95944e6	−0.429837
$41$	2.03049e7	1.12221	0.561105	−	0.827744i	$-0.310376\pi$
$41$	2.03049e7	1.12221	0.561105	+	0.827744i	$0.310376\pi$
$42$	0	0
$43$	−1.11768e7	−0.498551	−0.249276	−	0.968433i	$-0.580193\pi$
$43$	−1.11768e7	−0.498551	−0.249276	+	0.968433i	$0.580193\pi$
$44$	−4.54332e7	−1.82741
$45$	0	0
$46$	2.12070e7	0.698343
$47$	1.99352e7	0.595909	0.297955	−	0.954580i	$-0.403696\pi$
$47$	1.99352e7	0.595909	0.297955	+	0.954580i	$0.403696\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	2.52109e6	0.0570458
$51$	0	0
$52$	6.94125e7	1.31651
$53$	−5.65007e7	−0.983586	−0.491793	−	0.870712i	$-0.663658\pi$
$53$	−5.65007e7	−0.983586	−0.491793	+	0.870712i	$0.663658\pi$
$54$	0	0
$55$	9.87398e7	1.45499
$56$	1.17367e7	0.159478
$57$	0	0
$58$	−1.66595e8	−1.93302
$59$	1.09340e8	1.17475	0.587375	−	0.809315i	$-0.300161\pi$
$59$	1.09340e8	1.17475	0.587375	+	0.809315i	$0.300161\pi$
$60$	0	0
$61$	3.20008e7	0.295921	0.147961	−	0.988993i	$-0.452729\pi$
$61$	3.20008e7	0.295921	0.147961	+	0.988993i	$0.452729\pi$
$62$	−4.86755e7	−0.418358
$63$	0	0
$64$	−1.95825e8	−1.45901
$65$	−1.50854e8	−1.04821
$66$	0	0
$67$	8.02869e7	0.486752	0.243376	−	0.969932i	$-0.421745\pi$
$67$	8.02869e7	0.486752	0.243376	+	0.969932i	$0.421745\pi$
$68$	−3.72265e8	−2.11136
$69$	0	0
$70$	−1.16778e8	−0.581328
$71$	−2.07893e8	−0.970906	−0.485453	−	0.874263i	$-0.661345\pi$
$71$	−2.07893e8	−0.970906	−0.485453	+	0.874263i	$0.661345\pi$
$72$	0	0
$73$	−2.70274e8	−1.11391	−0.556957	−	0.830541i	$-0.688031\pi$
$73$	−2.70274e8	−1.11391	−0.556957	+	0.830541i	$0.688031\pi$
$74$	4.47844e8	1.73614
$75$	0	0
$76$	−2.59680e8	−0.892849
$77$	−1.66520e8	−0.539831
$78$	0	0
$79$	−5.16196e8	−1.49105	−0.745526	−	0.666477i	$-0.767802\pi$
$79$	−5.16196e8	−1.49105	−0.745526	+	0.666477i	$0.767802\pi$
$80$	2.39763e8	0.654452
$81$	0	0
$82$	6.93671e8	1.69430
$83$	6.82693e8	1.57897	0.789486	−	0.613769i	$-0.210347\pi$
$83$	6.82693e8	1.57897	0.789486	+	0.613769i	$0.210347\pi$
$84$	0	0
$85$	8.09042e8	1.68107
$86$	−3.81830e8	−0.752708
$87$	0	0
$88$	−3.39023e8	−0.602639
$89$	1.47150e8	0.248602	0.124301	−	0.992245i	$-0.460331\pi$
$89$	1.47150e8	0.248602	0.124301	+	0.992245i	$0.460331\pi$
$90$	0	0
$91$	2.54408e8	0.388905
$92$	4.06656e8	0.591809
$93$	0	0
$94$	6.81040e8	0.899699
$95$	5.64362e8	0.710889
$96$	0	0
$97$	1.09643e9	1.25750	0.628750	−	0.777608i	$-0.283567\pi$
$97$	1.09643e9	1.25750	0.628750	+	0.777608i	$0.283567\pi$
$98$	1.96941e8	0.215684
$99$	0	0
$100$	4.83433e7	0.0483433
$101$	−2.08683e8	−0.199545	−0.0997727	−	0.995010i	$-0.531812\pi$
$101$	−2.08683e8	−0.199545	−0.0997727	+	0.995010i	$0.531812\pi$
$102$	0	0
$103$	6.78194e8	0.593727	0.296863	−	0.954920i	$-0.404059\pi$
$103$	6.78194e8	0.593727	0.296863	+	0.954920i	$0.404059\pi$
$104$	5.17957e8	0.434154
$105$	0	0
$106$	−1.93021e9	−1.48501
$107$	4.59542e8	0.338921	0.169461	−	0.985537i	$-0.445797\pi$
$107$	4.59542e8	0.338921	0.169461	+	0.985537i	$0.445797\pi$
$108$	0	0
$109$	5.21086e8	0.353582	0.176791	−	0.984248i	$-0.443428\pi$
$109$	5.21086e8	0.353582	0.176791	+	0.984248i	$0.443428\pi$
$110$	3.37322e9	2.19673
$111$	0	0
$112$	−4.04349e8	−0.242815
$113$	−4.45612e8	−0.257101	−0.128551	−	0.991703i	$-0.541032\pi$
$113$	−4.45612e8	−0.257101	−0.128551	+	0.991703i	$0.541032\pi$
$114$	0	0
$115$	−8.83783e8	−0.471200
$116$	−3.19455e9	−1.63813
$117$	0	0
$118$	3.73535e9	1.77363
$119$	−1.36441e9	−0.623711
$120$	0	0
$121$	2.45208e9	1.03992
$122$	1.09323e9	0.446779
$123$	0	0
$124$	−9.33380e8	−0.354536
$125$	2.67560e9	0.980226
$126$	0	0
$127$	9.28626e8	0.316755	0.158378	−	0.987379i	$-0.449374\pi$
$127$	9.28626e8	0.316755	0.158378	+	0.987379i	$0.449374\pi$
$128$	−2.46278e9	−0.810925
$129$	0	0
$130$	−5.15357e9	−1.58257
$131$	−2.57694e9	−0.764509	−0.382255	−	0.924057i	$-0.624852\pi$
$131$	−2.57694e9	−0.764509	−0.382255	+	0.924057i	$0.624852\pi$
$132$	0	0
$133$	−9.51769e8	−0.263754
$134$	2.74281e9	0.734894
$135$	0	0
$136$	−2.77785e9	−0.696279
$137$	4.44116e9	1.07710	0.538548	−	0.842595i	$-0.318973\pi$
$137$	4.44116e9	1.07710	0.538548	+	0.842595i	$0.318973\pi$
$138$	0	0
$139$	−7.28389e9	−1.65499	−0.827497	−	0.561470i	$-0.810236\pi$
$139$	−7.28389e9	−1.65499	−0.827497	+	0.561470i	$0.810236\pi$
$140$	−2.23929e9	−0.492645
$141$	0	0
$142$	−7.10218e9	−1.46587
$143$	−7.34872e9	−1.46960
$144$	0	0
$145$	6.94270e9	1.30428
$146$	−9.23328e9	−1.68178
$147$	0	0
$148$	8.58766e9	1.47129
$149$	−4.87355e9	−0.810042	−0.405021	−	0.914307i	$-0.632736\pi$
$149$	−4.87355e9	−0.810042	−0.405021	+	0.914307i	$0.632736\pi$
$150$	0	0
$151$	−8.63776e9	−1.35209	−0.676044	−	0.736861i	$-0.736307\pi$
$151$	−8.63776e9	−1.35209	−0.676044	+	0.736861i	$0.736307\pi$
$152$	−1.93774e9	−0.294442
$153$	0	0
$154$	−5.68876e9	−0.815032
$155$	2.02851e9	0.282283
$156$	0	0
$157$	−9.34170e9	−1.22709	−0.613546	−	0.789659i	$-0.710258\pi$
$157$	−9.34170e9	−1.22709	−0.613546	+	0.789659i	$0.710258\pi$
$158$	−1.76346e10	−2.25118
$159$	0	0
$160$	1.17542e10	1.41792
$161$	1.49046e9	0.174825
$162$	0	0
$163$	−2.10680e9	−0.233765	−0.116883	−	0.993146i	$-0.537290\pi$
$163$	−2.10680e9	−0.233765	−0.116883	+	0.993146i	$0.537290\pi$
$164$	1.33015e10	1.43583
$165$	0	0
$166$	2.33226e10	2.38392
$167$	−1.39800e10	−1.39086	−0.695431	−	0.718593i	$-0.744787\pi$
$167$	−1.39800e10	−1.39086	−0.695431	+	0.718593i	$0.744787\pi$
$168$	0	0
$169$	6.22820e8	0.0587316
$170$	2.76390e10	2.53806
$171$	0	0
$172$	−7.32180e9	−0.637881
$173$	1.24435e10	1.05618	0.528088	−	0.849190i	$-0.322909\pi$
$173$	1.24435e10	1.05618	0.528088	+	0.849190i	$0.322909\pi$
$174$	0	0
$175$	1.77186e8	0.0142810
$176$	1.16799e10	0.917553
$177$	0	0
$178$	5.02704e9	0.375338
$179$	−7.30178e9	−0.531606	−0.265803	−	0.964027i	$-0.585637\pi$
$179$	−7.30178e9	−0.531606	−0.265803	+	0.964027i	$0.585637\pi$
$180$	0	0
$181$	−1.27074e10	−0.880038	−0.440019	−	0.897988i	$-0.645028\pi$
$181$	−1.27074e10	−0.880038	−0.440019	+	0.897988i	$0.645028\pi$
$182$	8.69125e9	0.587166
$183$	0	0
$184$	3.03447e9	0.195165
$185$	−1.86635e10	−1.17144
$186$	0	0
$187$	3.94118e10	2.35689
$188$	1.30593e10	0.762448
$189$	0	0
$190$	1.92801e10	1.07329
$191$	−1.61547e10	−0.878311	−0.439155	−	0.898411i	$-0.644722\pi$
$191$	−1.61547e10	−0.878311	−0.439155	+	0.898411i	$0.644722\pi$
$192$	0	0
$193$	−1.52841e10	−0.792924	−0.396462	−	0.918051i	$-0.629762\pi$
$193$	−1.52841e10	−0.792924	−0.396462	+	0.918051i	$0.629762\pi$
$194$	3.74570e10	1.89856
$195$	0	0
$196$	3.77645e9	0.182781
$197$	−2.33886e10	−1.10639	−0.553193	−	0.833053i	$-0.686591\pi$
$197$	−2.33886e10	−1.10639	−0.553193	+	0.833053i	$0.686591\pi$
$198$	0	0
$199$	−2.53610e10	−1.14638	−0.573189	−	0.819423i	$-0.694294\pi$
$199$	−2.53610e10	−1.14638	−0.573189	+	0.819423i	$0.694294\pi$
$200$	3.60738e8	0.0159425
$201$	0	0
$202$	−7.12918e9	−0.301272
$203$	−1.17085e10	−0.483916
$204$	0	0
$205$	−2.89082e10	−1.14322
$206$	2.31689e10	0.896403
$207$	0	0
$208$	−1.78444e10	−0.661024
$209$	2.74924e10	0.996678
$210$	0	0
$211$	−9.23757e8	−0.0320838	−0.0160419	−	0.999871i	$-0.505107\pi$
$211$	−9.23757e8	−0.0320838	−0.0160419	+	0.999871i	$0.505107\pi$
$212$	−3.70129e10	−1.25847
$213$	0	0
$214$	1.56992e10	0.511700
$215$	1.59124e10	0.507883
$216$	0	0
$217$	−3.42098e9	−0.104733
$218$	1.78017e10	0.533835
$219$	0	0
$220$	6.46833e10	1.86162
$221$	−6.02130e10	−1.69795
$222$	0	0
$223$	6.68635e9	0.181058	0.0905290	−	0.995894i	$-0.471144\pi$
$223$	6.68635e9	0.181058	0.0905290	+	0.995894i	$0.471144\pi$
$224$	−1.98229e10	−0.526078
$225$	0	0
$226$	−1.52233e10	−0.388169
$227$	4.82876e10	1.20703	0.603516	−	0.797351i	$-0.293766\pi$
$227$	4.82876e10	1.20703	0.603516	+	0.797351i	$0.293766\pi$
$228$	0	0
$229$	−2.34264e10	−0.562918	−0.281459	−	0.959573i	$-0.590818\pi$
$229$	−2.34264e10	−0.562918	−0.281459	+	0.959573i	$0.590818\pi$
$230$	−3.01924e10	−0.711413
$231$	0	0
$232$	−2.38378e10	−0.540219
$233$	3.29140e10	0.731609	0.365805	−	0.930692i	$-0.380794\pi$
$233$	3.29140e10	0.731609	0.365805	+	0.930692i	$0.380794\pi$
$234$	0	0
$235$	−2.83817e10	−0.607063
$236$	7.16275e10	1.50306
$237$	0	0
$238$	−4.66119e10	−0.941673
$239$	−2.51973e10	−0.499533	−0.249766	−	0.968306i	$-0.580354\pi$
$239$	−2.51973e10	−0.499533	−0.249766	+	0.968306i	$0.580354\pi$
$240$	0	0
$241$	7.00815e10	1.33822	0.669109	−	0.743165i	$-0.266676\pi$
$241$	7.00815e10	1.33822	0.669109	+	0.743165i	$0.266676\pi$
$242$	8.37696e10	1.57006
$243$	0	0
$244$	2.09633e10	0.378622
$245$	−8.20735e9	−0.145531
$246$	0	0
$247$	−4.20027e10	−0.718028
$248$	−6.96489e9	−0.116918
$249$	0	0
$250$	9.14056e10	1.47994
$251$	−4.19681e10	−0.667401	−0.333701	−	0.942679i	$-0.608297\pi$
$251$	−4.19681e10	−0.667401	−0.333701	+	0.942679i	$0.608297\pi$
$252$	0	0
$253$	−4.30527e10	−0.660630
$254$	3.17243e10	0.478235
$255$	0	0
$256$	1.61271e10	0.234680
$257$	−8.37595e10	−1.19766	−0.598832	−	0.800875i	$-0.704368\pi$
$257$	−8.37595e10	−1.19766	−0.598832	+	0.800875i	$0.704368\pi$
$258$	0	0
$259$	3.14751e10	0.434629
$260$	−9.88226e10	−1.34115
$261$	0	0
$262$	−8.80350e10	−1.15425
$263$	2.34604e9	0.0302367	0.0151184	−	0.999886i	$-0.495187\pi$
$263$	2.34604e9	0.0302367	0.0151184	+	0.999886i	$0.495187\pi$
$264$	0	0
$265$	8.04400e10	1.00200
$266$	−3.25150e10	−0.398214
$267$	0	0
$268$	5.25950e10	0.622784
$269$	6.44659e10	0.750663	0.375332	−	0.926891i	$-0.377529\pi$
$269$	6.44659e10	0.750663	0.375332	+	0.926891i	$0.377529\pi$
$270$	0	0
$271$	−1.40530e10	−0.158273	−0.0791365	−	0.996864i	$-0.525216\pi$
$271$	−1.40530e10	−0.158273	−0.0791365	+	0.996864i	$0.525216\pi$
$272$	9.57011e10	1.06012
$273$	0	0
$274$	1.51722e11	1.62619
$275$	−5.11812e9	−0.0539652
$276$	0	0
$277$	−7.52768e10	−0.768249	−0.384124	−	0.923281i	$-0.625497\pi$
$277$	−7.52768e10	−0.768249	−0.384124	+	0.923281i	$0.625497\pi$
$278$	−2.48837e11	−2.49870
$279$	0	0
$280$	−1.67096e10	−0.162463
$281$	−9.56085e10	−0.914783	−0.457391	−	0.889266i	$-0.651216\pi$
$281$	−9.56085e10	−0.914783	−0.457391	+	0.889266i	$0.651216\pi$
$282$	0	0
$283$	4.82806e10	0.447439	0.223719	−	0.974654i	$-0.428180\pi$
$283$	4.82806e10	0.447439	0.223719	+	0.974654i	$0.428180\pi$
$284$	−1.36188e11	−1.24224
$285$	0	0
$286$	−2.51052e11	−2.21879
$287$	4.87522e10	0.424156
$288$	0	0
$289$	2.04340e11	1.72311
$290$	2.37181e11	1.96920
$291$	0	0
$292$	−1.77053e11	−1.42522
$293$	7.07439e10	0.560770	0.280385	−	0.959888i	$-0.409538\pi$
$293$	7.07439e10	0.560770	0.280385	+	0.959888i	$0.409538\pi$
$294$	0	0
$295$	−1.55668e11	−1.19674
$296$	6.40812e10	0.485197
$297$	0	0
$298$	−1.66494e11	−1.22299
$299$	6.57756e10	0.475932
$300$	0	0
$301$	−2.68355e10	−0.188435
$302$	−2.95089e11	−2.04137
$303$	0	0
$304$	6.67581e10	0.448304
$305$	−4.55595e10	−0.301460
$306$	0	0
$307$	1.30493e11	0.838429	0.419214	−	0.907887i	$-0.362306\pi$
$307$	1.30493e11	0.838429	0.419214	+	0.907887i	$0.362306\pi$
$308$	−1.09085e11	−0.690697
$309$	0	0
$310$	6.92993e10	0.426188
$311$	−8.51715e10	−0.516265	−0.258132	−	0.966110i	$-0.583107\pi$
$311$	−8.51715e10	−0.516265	−0.258132	+	0.966110i	$0.583107\pi$
$312$	0	0
$313$	−1.21745e11	−0.716969	−0.358484	−	0.933536i	$-0.616706\pi$
$313$	−1.21745e11	−0.716969	−0.358484	+	0.933536i	$0.616706\pi$
$314$	−3.19137e11	−1.85265
$315$	0	0
$316$	−3.38154e11	−1.90775
$317$	2.14595e11	1.19358	0.596791	−	0.802397i	$-0.296442\pi$
$317$	2.14595e11	1.19358	0.596791	+	0.802397i	$0.296442\pi$
$318$	0	0
$319$	3.38208e11	1.82863
$320$	2.78796e11	1.48632
$321$	0	0
$322$	5.09180e10	0.263949
$323$	2.25264e11	1.15154
$324$	0	0
$325$	7.81943e9	0.0388776
$326$	−7.19740e10	−0.352936
$327$	0	0
$328$	9.92562e10	0.473506
$329$	4.78644e10	0.225233
$330$	0	0
$331$	−5.48000e10	−0.250931	−0.125466	−	0.992098i	$-0.540042\pi$
$331$	−5.48000e10	−0.250931	−0.125466	+	0.992098i	$0.540042\pi$
$332$	4.47224e11	2.02025
$333$	0	0
$334$	−4.77595e11	−2.09991
$335$	−1.14304e11	−0.495863
$336$	0	0
$337$	−2.34297e11	−0.989538	−0.494769	−	0.869025i	$-0.664747\pi$
$337$	−2.34297e11	−0.989538	−0.494769	+	0.869025i	$0.664747\pi$
$338$	2.12772e10	0.0886725
$339$	0	0
$340$	5.29994e11	2.15088
$341$	9.88172e10	0.395765
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	−5.46353e10	−0.210359
$345$	0	0
$346$	4.25104e11	1.59460
$347$	−3.43449e10	−0.127169	−0.0635843	−	0.997976i	$-0.520253\pi$
$347$	−3.43449e10	−0.127169	−0.0635843	+	0.997976i	$0.520253\pi$
$348$	0	0
$349$	−2.13485e11	−0.770288	−0.385144	−	0.922856i	$-0.625848\pi$
$349$	−2.13485e11	−0.770288	−0.385144	+	0.922856i	$0.625848\pi$
$350$	6.05314e9	0.0215613
$351$	0	0
$352$	5.72595e11	1.98795
$353$	−2.75882e11	−0.945664	−0.472832	−	0.881153i	$-0.656768\pi$
$353$	−2.75882e11	−0.945664	−0.472832	+	0.881153i	$0.656768\pi$
$354$	0	0
$355$	2.95977e11	0.989079
$356$	9.63962e10	0.318079
$357$	0	0
$358$	−2.49448e11	−0.802614
$359$	3.46238e11	1.10015	0.550073	−	0.835117i	$-0.314600\pi$
$359$	3.46238e11	1.10015	0.550073	+	0.835117i	$0.314600\pi$
$360$	0	0
$361$	−1.65550e11	−0.513036
$362$	−4.34117e11	−1.32867
$363$	0	0
$364$	1.66659e11	0.497592
$365$	3.84789e11	1.13476
$366$	0	0
$367$	3.56842e11	1.02678	0.513391	−	0.858155i	$-0.328389\pi$
$367$	3.56842e11	1.02678	0.513391	+	0.858155i	$0.328389\pi$
$368$	−1.04542e11	−0.297150
$369$	0	0
$370$	−6.37596e11	−1.76863
$371$	−1.35658e11	−0.371760
$372$	0	0
$373$	6.73833e11	1.80245	0.901224	−	0.433354i	$-0.142670\pi$
$373$	6.73833e11	1.80245	0.901224	+	0.433354i	$0.142670\pi$
$374$	1.34641e12	3.55841
$375$	0	0
$376$	9.74487e10	0.251438
$377$	−5.16711e11	−1.31738
$378$	0	0
$379$	5.90163e11	1.46925	0.734625	−	0.678473i	$-0.237358\pi$
$379$	5.90163e11	1.46925	0.734625	+	0.678473i	$0.237358\pi$
$380$	3.69707e11	0.909561
$381$	0	0
$382$	−5.51887e11	−1.32607
$383$	1.58931e11	0.377412	0.188706	−	0.982034i	$-0.439571\pi$
$383$	1.58931e11	0.377412	0.188706	+	0.982034i	$0.439571\pi$
$384$	0	0
$385$	2.37074e11	0.549935
$386$	−5.22145e11	−1.19715
$387$	0	0
$388$	7.18258e11	1.60893
$389$	3.75434e11	0.831304	0.415652	−	0.909524i	$-0.363553\pi$
$389$	3.75434e11	0.831304	0.415652	+	0.909524i	$0.363553\pi$
$390$	0	0
$391$	−3.52760e11	−0.763280
$392$	2.81799e10	0.0602771
$393$	0	0
$394$	−7.99018e11	−1.67041
$395$	7.34908e11	1.51896
$396$	0	0
$397$	−4.33507e11	−0.875869	−0.437935	−	0.899007i	$-0.644290\pi$
$397$	−4.33507e11	−0.875869	−0.437935	+	0.899007i	$0.644290\pi$
$398$	−8.66400e11	−1.73079
$399$	0	0
$400$	−1.24280e10	−0.0242734
$401$	1.77805e11	0.343395	0.171698	−	0.985150i	$-0.445075\pi$
$401$	1.77805e11	0.343395	0.171698	+	0.985150i	$0.445075\pi$
$402$	0	0
$403$	−1.50972e11	−0.285118
$404$	−1.36706e11	−0.255312
$405$	0	0
$406$	−3.99994e11	−0.730612
$407$	−9.09178e11	−1.64238
$408$	0	0
$409$	−7.67870e11	−1.35685	−0.678427	−	0.734668i	$-0.737338\pi$
$409$	−7.67870e11	−1.35685	−0.678427	+	0.734668i	$0.737338\pi$
$410$	−9.87580e11	−1.72602
$411$	0	0
$412$	4.44277e11	0.759655
$413$	2.62526e11	0.444014
$414$	0	0
$415$	−9.71951e11	−1.60852
$416$	−8.74807e11	−1.43216
$417$	0	0
$418$	9.39215e11	1.50478
$419$	−4.96552e11	−0.787048	−0.393524	−	0.919314i	$-0.628744\pi$
$419$	−4.96552e11	−0.787048	−0.393524	+	0.919314i	$0.628744\pi$
$420$	0	0
$421$	−4.48514e11	−0.695835	−0.347917	−	0.937525i	$-0.613111\pi$
$421$	−4.48514e11	−0.695835	−0.347917	+	0.937525i	$0.613111\pi$
$422$	−3.15580e10	−0.0484399
$423$	0	0
$424$	−2.76191e11	−0.415014
$425$	−4.19362e10	−0.0623504
$426$	0	0
$427$	7.68338e10	0.111848
$428$	3.01041e11	0.433639
$429$	0	0
$430$	5.43611e11	0.766797
$431$	1.65940e11	0.231635	0.115817	−	0.993271i	$-0.463051\pi$
$431$	1.65940e11	0.231635	0.115817	+	0.993271i	$0.463051\pi$
$432$	0	0
$433$	−4.10674e11	−0.561438	−0.280719	−	0.959790i	$-0.590573\pi$
$433$	−4.10674e11	−0.561438	−0.280719	+	0.959790i	$0.590573\pi$
$434$	−1.16870e11	−0.158124
$435$	0	0
$436$	3.41358e11	0.452398
$437$	−2.46074e11	−0.322775
$438$	0	0
$439$	−3.29593e11	−0.423534	−0.211767	−	0.977320i	$-0.567922\pi$
$439$	−3.29593e11	−0.423534	−0.211767	+	0.977320i	$0.567922\pi$
$440$	4.82667e11	0.613919
$441$	0	0
$442$	−2.05704e12	−2.56355
$443$	−5.09091e11	−0.628027	−0.314014	−	0.949418i	$-0.601674\pi$
$443$	−5.09091e11	−0.628027	−0.314014	+	0.949418i	$0.601674\pi$
$444$	0	0
$445$	−2.09497e11	−0.253255
$446$	2.28424e11	0.273360
$447$	0	0
$448$	−4.70175e11	−0.551453
$449$	−1.49596e12	−1.73705	−0.868525	−	0.495646i	$-0.834931\pi$
$449$	−1.49596e12	−1.73705	−0.868525	+	0.495646i	$0.834931\pi$
$450$	0	0
$451$	−1.40824e12	−1.60281
$452$	−2.91915e11	−0.328953
$453$	0	0
$454$	1.64963e12	1.82237
$455$	−3.62200e11	−0.396184
$456$	0	0
$457$	1.43920e12	1.54347	0.771735	−	0.635944i	$-0.219389\pi$
$457$	1.43920e12	1.54347	0.771735	+	0.635944i	$0.219389\pi$
$458$	−8.00307e11	−0.849889
$459$	0	0
$460$	−5.78956e11	−0.602886
$461$	1.37741e12	1.42039	0.710195	−	0.704005i	$-0.248607\pi$
$461$	1.37741e12	1.42039	0.710195	+	0.704005i	$0.248607\pi$
$462$	0	0
$463$	1.76612e12	1.78610	0.893049	−	0.449960i	$-0.148562\pi$
$463$	1.76612e12	1.78610	0.893049	+	0.449960i	$0.148562\pi$
$464$	8.21248e11	0.822514
$465$	0	0
$466$	1.12443e12	1.10458
$467$	1.17323e12	1.14145	0.570727	−	0.821140i	$-0.306661\pi$
$467$	1.17323e12	1.14145	0.570727	+	0.821140i	$0.306661\pi$
$468$	0	0
$469$	1.92769e11	0.183975
$470$	−9.69596e11	−0.916538
$471$	0	0
$472$	5.34485e11	0.495674
$473$	7.75161e11	0.712060
$474$	0	0
$475$	−2.92534e10	−0.0263667
$476$	−8.93808e11	−0.798019
$477$	0	0
$478$	−8.60808e11	−0.754190
$479$	1.98723e12	1.72480	0.862401	−	0.506225i	$-0.168960\pi$
$479$	1.98723e12	1.72480	0.862401	+	0.506225i	$0.168960\pi$
$480$	0	0
$481$	1.38903e12	1.18321
$482$	2.39417e12	2.02043
$483$	0	0
$484$	1.60633e12	1.33055
$485$	−1.56099e12	−1.28104
$486$	0	0
$487$	−1.23240e12	−0.992825	−0.496413	−	0.868087i	$-0.665350\pi$
$487$	−1.23240e12	−0.992825	−0.496413	+	0.868087i	$0.665350\pi$
$488$	1.56428e11	0.124861
$489$	0	0
$490$	−2.80385e11	−0.219721
$491$	2.03763e12	1.58219	0.791095	−	0.611693i	$-0.209511\pi$
$491$	2.03763e12	1.58219	0.791095	+	0.611693i	$0.209511\pi$
$492$	0	0
$493$	2.77117e12	2.11277
$494$	−1.43493e12	−1.08407
$495$	0	0
$496$	2.39951e11	0.178014
$497$	−4.99151e11	−0.366968
$498$	0	0
$499$	−3.26299e11	−0.235594	−0.117797	−	0.993038i	$-0.537583\pi$
$499$	−3.26299e11	−0.235594	−0.117797	+	0.993038i	$0.537583\pi$
$500$	1.75275e12	1.25417
$501$	0	0
$502$	−1.43374e12	−1.00764
$503$	4.46869e11	0.311261	0.155630	−	0.987815i	$-0.450259\pi$
$503$	4.46869e11	0.311261	0.155630	+	0.987815i	$0.450259\pi$
$504$	0	0
$505$	2.97103e11	0.203280
$506$	−1.47080e12	−0.997413
$507$	0	0
$508$	6.08332e11	0.405279
$509$	−1.34100e12	−0.885523	−0.442761	−	0.896639i	$-0.646001\pi$
$509$	−1.34100e12	−0.885523	−0.442761	+	0.896639i	$0.646001\pi$
$510$	0	0
$511$	−6.48928e11	−0.421020
$512$	1.81189e12	1.16524
$513$	0	0
$514$	−2.86145e12	−1.80822
$515$	−9.65545e11	−0.604839
$516$	0	0
$517$	−1.38259e12	−0.851112
$518$	1.07527e12	0.656199
$519$	0	0
$520$	−7.37415e11	−0.442280
$521$	−2.98523e12	−1.77504	−0.887520	−	0.460769i	$-0.847574\pi$
$521$	−2.98523e12	−1.77504	−0.887520	+	0.460769i	$0.847574\pi$
$522$	0	0
$523$	1.64651e12	0.962289	0.481145	−	0.876641i	$-0.340221\pi$
$523$	1.64651e12	0.962289	0.481145	+	0.876641i	$0.340221\pi$
$524$	−1.68812e12	−0.978166
$525$	0	0
$526$	8.01470e10	0.0456511
$527$	8.09676e11	0.457260
$528$	0	0
$529$	−1.41580e12	−0.786054
$530$	2.74805e12	1.51280
$531$	0	0
$532$	−6.23493e11	−0.337465
$533$	2.15149e12	1.15470
$534$	0	0
$535$	−6.54250e11	−0.345265
$536$	3.92464e11	0.205380
$537$	0	0
$538$	2.20233e12	1.13334
$539$	−3.99814e11	−0.204037
$540$	0	0
$541$	−4.57968e11	−0.229852	−0.114926	−	0.993374i	$-0.536663\pi$
$541$	−4.57968e11	−0.229852	−0.114926	+	0.993374i	$0.536663\pi$
$542$	−4.80087e11	−0.238959
$543$	0	0
$544$	4.69166e12	2.29684
$545$	−7.41871e11	−0.360200
$546$	0	0
$547$	2.39624e12	1.14443	0.572213	−	0.820105i	$-0.306085\pi$
$547$	2.39624e12	1.14443	0.572213	+	0.820105i	$0.306085\pi$
$548$	2.90935e12	1.37811
$549$	0	0
$550$	−1.74849e11	−0.0814761
$551$	1.93308e12	0.893444
$552$	0	0
$553$	−1.23939e12	−0.563565
$554$	−2.57166e12	−1.15990
$555$	0	0
$556$	−4.77159e12	−2.11751
$557$	−1.07863e12	−0.474814	−0.237407	−	0.971410i	$-0.576298\pi$
$557$	−1.07863e12	−0.474814	−0.237407	+	0.971410i	$0.576298\pi$
$558$	0	0
$559$	−1.18428e12	−0.512983
$560$	5.75672e11	0.247360
$561$	0	0
$562$	−3.26624e12	−1.38113
$563$	−2.33140e11	−0.0977976	−0.0488988	−	0.998804i	$-0.515571\pi$
$563$	−2.33140e11	−0.0977976	−0.0488988	+	0.998804i	$0.515571\pi$
$564$	0	0
$565$	6.34418e11	0.261913
$566$	1.64939e12	0.675539
$567$	0	0
$568$	−1.01624e12	−0.409664
$569$	−4.50535e11	−0.180187	−0.0900934	−	0.995933i	$-0.528717\pi$
$569$	−4.50535e11	−0.180187	−0.0900934	+	0.995933i	$0.528717\pi$
$570$	0	0
$571$	−4.38839e12	−1.72760	−0.863800	−	0.503835i	$-0.831922\pi$
$571$	−4.38839e12	−1.72760	−0.863800	+	0.503835i	$0.831922\pi$
$572$	−4.81406e12	−1.88031
$573$	0	0
$574$	1.66550e12	0.640387
$575$	4.58104e10	0.0174767
$576$	0	0
$577$	3.13994e12	1.17932	0.589658	−	0.807653i	$-0.299263\pi$
$577$	3.13994e12	1.17932	0.589658	+	0.807653i	$0.299263\pi$
$578$	6.98079e12	2.60153
$579$	0	0
$580$	4.54808e12	1.66879
$581$	1.63915e12	0.596795
$582$	0	0
$583$	3.91857e12	1.40481
$584$	−1.32117e12	−0.470005
$585$	0	0
$586$	2.41680e12	0.846645
$587$	−2.52512e12	−0.877829	−0.438915	−	0.898529i	$-0.644637\pi$
$587$	−2.52512e12	−0.877829	−0.438915	+	0.898529i	$0.644637\pi$
$588$	0	0
$589$	5.64805e11	0.193366
$590$	−5.31802e12	−1.80682
$591$	0	0
$592$	−2.20770e12	−0.738740
$593$	9.35417e11	0.310641	0.155321	−	0.987864i	$-0.450359\pi$
$593$	9.35417e11	0.310641	0.155321	+	0.987864i	$0.450359\pi$
$594$	0	0
$595$	1.94251e12	0.635385
$596$	−3.19261e12	−1.03642
$597$	0	0
$598$	2.24707e12	0.718557
$599$	−4.73586e12	−1.50307	−0.751534	−	0.659694i	$-0.770686\pi$
$599$	−4.73586e12	−1.50307	−0.751534	+	0.659694i	$0.770686\pi$
$600$	0	0
$601$	−5.99855e12	−1.87548	−0.937738	−	0.347344i	$-0.887084\pi$
$601$	−5.99855e12	−1.87548	−0.937738	+	0.347344i	$0.887084\pi$
$602$	−9.16773e11	−0.284497
$603$	0	0
$604$	−5.65850e12	−1.72995
$605$	−3.49103e12	−1.05938
$606$	0	0
$607$	3.43715e12	1.02766	0.513830	−	0.857892i	$-0.328226\pi$
$607$	3.43715e12	1.02766	0.513830	+	0.857892i	$0.328226\pi$
$608$	3.27276e12	0.971287
$609$	0	0
$610$	−1.55643e12	−0.455141
$611$	2.11231e12	0.613159
$612$	0	0
$613$	4.15273e12	1.18785	0.593925	−	0.804520i	$-0.297577\pi$
$613$	4.15273e12	1.18785	0.593925	+	0.804520i	$0.297577\pi$
$614$	4.45801e12	1.26585
$615$	0	0
$616$	−8.13995e11	−0.227776
$617$	1.12196e12	0.311669	0.155834	−	0.987783i	$-0.450193\pi$
$617$	1.12196e12	0.311669	0.155834	+	0.987783i	$0.450193\pi$
$618$	0	0
$619$	−5.98940e12	−1.63974	−0.819871	−	0.572549i	$-0.805955\pi$
$619$	−5.98940e12	−1.63974	−0.819871	+	0.572549i	$0.805955\pi$
$620$	1.32885e12	0.361172
$621$	0	0
$622$	−2.90969e12	−0.779452
$623$	3.53307e11	0.0939628
$624$	0	0
$625$	−3.95339e12	−1.03636
$626$	−4.15912e12	−1.08247
$627$	0	0
$628$	−6.11963e12	−1.57003
$629$	−7.44951e12	−1.89758
$630$	0	0
$631$	4.97135e12	1.24837	0.624184	−	0.781277i	$-0.285432\pi$
$631$	4.97135e12	1.24837	0.624184	+	0.781277i	$0.285432\pi$
$632$	−2.52331e12	−0.629134
$633$	0	0
$634$	7.33113e12	1.80206
$635$	−1.32208e12	−0.322684
$636$	0	0
$637$	6.10833e11	0.146992
$638$	1.15541e13	2.76085
$639$	0	0
$640$	3.50626e12	0.826103
$641$	2.41181e12	0.564263	0.282131	−	0.959376i	$-0.408959\pi$
$641$	2.41181e12	0.564263	0.282131	+	0.959376i	$0.408959\pi$
$642$	0	0
$643$	6.86804e12	1.58447	0.792234	−	0.610217i	$-0.208918\pi$
$643$	6.86804e12	1.58447	0.792234	+	0.610217i	$0.208918\pi$
$644$	9.76380e11	0.223683
$645$	0	0
$646$	7.69562e12	1.73859
$647$	1.73394e12	0.389013	0.194506	−	0.980901i	$-0.437690\pi$
$647$	1.73394e12	0.389013	0.194506	+	0.980901i	$0.437690\pi$
$648$	0	0
$649$	−7.58322e12	−1.67785
$650$	2.67132e11	0.0586971
$651$	0	0
$652$	−1.38014e12	−0.299095
$653$	−2.86440e10	−0.00616488	−0.00308244	−	0.999995i	$-0.500981\pi$
$653$	−2.86440e10	−0.00616488	−0.00308244	+	0.999995i	$0.500981\pi$
$654$	0	0
$655$	3.66878e12	0.778818
$656$	−3.41953e12	−0.720940
$657$	0	0
$658$	1.63518e12	0.340054
$659$	6.31728e12	1.30481	0.652403	−	0.757872i	$-0.273761\pi$
$659$	6.31728e12	1.30481	0.652403	+	0.757872i	$0.273761\pi$
$660$	0	0
$661$	−3.49558e12	−0.712217	−0.356109	−	0.934445i	$-0.615897\pi$
$661$	−3.49558e12	−0.712217	−0.356109	+	0.934445i	$0.615897\pi$
$662$	−1.87212e12	−0.378854
$663$	0	0
$664$	3.33719e12	0.666231
$665$	1.35503e12	0.268691
$666$	0	0
$667$	−3.02717e12	−0.592203
$668$	−9.15816e12	−1.77957
$669$	0	0
$670$	−3.90494e12	−0.748649
$671$	−2.21939e12	−0.422652
$672$	0	0
$673$	−8.02535e12	−1.50798	−0.753991	−	0.656885i	$-0.771874\pi$
$673$	−8.02535e12	−1.50798	−0.753991	+	0.656885i	$0.771874\pi$
$674$	−8.00422e12	−1.49400
$675$	0	0
$676$	4.08002e11	0.0751453
$677$	−1.17163e12	−0.214358	−0.107179	−	0.994240i	$-0.534182\pi$
$677$	−1.17163e12	−0.214358	−0.107179	+	0.994240i	$0.534182\pi$
$678$	0	0
$679$	2.63253e12	0.475290
$680$	3.95482e12	0.709311
$681$	0	0
$682$	3.37586e12	0.597523
$683$	−4.69754e12	−0.825995	−0.412997	−	0.910732i	$-0.635518\pi$
$683$	−4.69754e12	−0.825995	−0.412997	+	0.910732i	$0.635518\pi$
$684$	0	0
$685$	−6.32288e12	−1.09726
$686$	4.72855e11	0.0815211
$687$	0	0
$688$	1.88227e12	0.320283
$689$	−5.98676e12	−1.01206
$690$	0	0
$691$	−7.83193e12	−1.30683	−0.653413	−	0.757001i	$-0.726664\pi$
$691$	−7.83193e12	−1.30683	−0.653413	+	0.757001i	$0.726664\pi$
$692$	8.15161e12	1.35134
$693$	0	0
$694$	−1.17332e12	−0.191998
$695$	1.03701e13	1.68597
$696$	0	0
$697$	−1.15386e13	−1.85185
$698$	−7.29322e12	−1.16297
$699$	0	0
$700$	1.16072e11	0.0182721
$701$	−7.13243e12	−1.11559	−0.557797	−	0.829977i	$-0.688353\pi$
$701$	−7.13243e12	−1.11559	−0.557797	+	0.829977i	$0.688353\pi$
$702$	0	0
$703$	−5.19655e12	−0.802446
$704$	1.35813e13	2.08384
$705$	0	0
$706$	−9.42486e12	−1.42776
$707$	−5.01049e11	−0.0754211
$708$	0	0
$709$	8.65274e12	1.28601	0.643007	−	0.765861i	$-0.277687\pi$
$709$	8.65274e12	1.28601	0.643007	+	0.765861i	$0.277687\pi$
$710$	1.01114e13	1.49330
$711$	0	0
$712$	7.19310e11	0.104895
$713$	−8.84475e11	−0.128169
$714$	0	0
$715$	1.04624e13	1.49711
$716$	−4.78331e12	−0.680174
$717$	0	0
$718$	1.18284e13	1.66099
$719$	−4.58446e11	−0.0639747	−0.0319873	−	0.999488i	$-0.510184\pi$
$719$	−4.58446e11	−0.0639747	−0.0319873	+	0.999488i	$0.510184\pi$
$720$	0	0
$721$	1.62834e12	0.224408
$722$	−5.65565e12	−0.774577
$723$	0	0
$724$	−8.32444e12	−1.12598
$725$	−3.59871e11	−0.0483755
$726$	0	0
$727$	−2.53850e12	−0.337033	−0.168517	−	0.985699i	$-0.553898\pi$
$727$	−2.53850e12	−0.337033	−0.168517	+	0.985699i	$0.553898\pi$
$728$	1.24361e12	0.164095
$729$	0	0
$730$	1.31454e13	1.71325
$731$	6.35141e12	0.822701
$732$	0	0
$733$	1.09361e13	1.39925	0.699624	−	0.714511i	$-0.253351\pi$
$733$	1.09361e13	1.39925	0.699624	+	0.714511i	$0.253351\pi$
$734$	1.21907e13	1.55023
$735$	0	0
$736$	−5.12509e12	−0.643800
$737$	−5.56824e12	−0.695208
$738$	0	0
$739$	7.34996e12	0.906536	0.453268	−	0.891374i	$-0.350258\pi$
$739$	7.34996e12	0.906536	0.453268	+	0.891374i	$0.350258\pi$
$740$	−1.22263e13	−1.49882
$741$	0	0
$742$	−4.63444e12	−0.561281
$743$	1.60813e12	0.193584	0.0967922	−	0.995305i	$-0.469142\pi$
$743$	1.60813e12	0.193584	0.0967922	+	0.995305i	$0.469142\pi$
$744$	0	0
$745$	6.93848e12	0.825203
$746$	2.30199e13	2.72132
$747$	0	0
$748$	2.58182e13	3.01557
$749$	1.10336e12	0.128100
$750$	0	0
$751$	−2.95903e12	−0.339446	−0.169723	−	0.985492i	$-0.554287\pi$
$751$	−2.95903e12	−0.339446	−0.169723	+	0.985492i	$0.554287\pi$
$752$	−3.35726e12	−0.382829
$753$	0	0
$754$	−1.76522e13	−1.98897
$755$	1.22976e13	1.37739
$756$	0	0
$757$	−4.83223e12	−0.534830	−0.267415	−	0.963581i	$-0.586170\pi$
$757$	−4.83223e12	−0.534830	−0.267415	+	0.963581i	$0.586170\pi$
$758$	2.01616e13	2.21826
$759$	0	0
$760$	2.75876e12	0.299953
$761$	−3.15213e12	−0.340701	−0.170350	−	0.985384i	$-0.554490\pi$
$761$	−3.15213e12	−0.340701	−0.170350	+	0.985384i	$0.554490\pi$
$762$	0	0
$763$	1.25113e12	0.133642
$764$	−1.05827e13	−1.12377
$765$	0	0
$766$	5.42952e12	0.569813
$767$	1.15856e13	1.20876
$768$	0	0
$769$	−6.87650e12	−0.709086	−0.354543	−	0.935040i	$-0.615364\pi$
$769$	−6.87650e12	−0.709086	−0.354543	+	0.935040i	$0.615364\pi$
$770$	8.09909e12	0.830286
$771$	0	0
$772$	−1.00124e13	−1.01452
$773$	−1.16083e13	−1.16939	−0.584696	−	0.811253i	$-0.698786\pi$
$773$	−1.16083e13	−1.16939	−0.584696	+	0.811253i	$0.698786\pi$
$774$	0	0
$775$	−1.05147e11	−0.0104698
$776$	5.35965e12	0.530590
$777$	0	0
$778$	1.28258e13	1.25510
$779$	−8.04899e12	−0.783110
$780$	0	0
$781$	1.44183e13	1.38670
$782$	−1.20512e13	−1.15239
$783$	0	0
$784$	−9.70842e11	−0.0917754
$785$	1.32998e13	1.25006
$786$	0	0
$787$	−2.50005e12	−0.232307	−0.116154	−	0.993231i	$-0.537057\pi$
$787$	−2.50005e12	−0.232307	−0.116154	+	0.993231i	$0.537057\pi$
$788$	−1.53216e13	−1.41559
$789$	0	0
$790$	2.51064e13	2.29331
$791$	−1.06991e12	−0.0971751
$792$	0	0
$793$	3.39077e12	0.304487
$794$	−1.48098e13	−1.32238
$795$	0	0
$796$	−1.66137e13	−1.46675
$797$	−2.77023e12	−0.243194	−0.121597	−	0.992580i	$-0.538802\pi$
$797$	−2.77023e12	−0.243194	−0.121597	+	0.992580i	$0.538802\pi$
$798$	0	0
$799$	−1.13285e13	−0.983360
$800$	−6.09272e11	−0.0525904
$801$	0	0
$802$	6.07429e12	0.518455
$803$	1.87447e13	1.59096
$804$	0	0
$805$	−2.12196e12	−0.178097
$806$	−5.15761e12	−0.430468
$807$	0	0
$808$	−1.02010e12	−0.0841962
$809$	2.96640e12	0.243479	0.121739	−	0.992562i	$-0.461153\pi$
$809$	2.96640e12	0.243479	0.121739	+	0.992562i	$0.461153\pi$
$810$	0	0
$811$	−9.01447e12	−0.731722	−0.365861	−	0.930669i	$-0.619225\pi$
$811$	−9.01447e12	−0.731722	−0.365861	+	0.930669i	$0.619225\pi$
$812$	−7.67012e12	−0.619156
$813$	0	0
$814$	−3.10599e13	−2.47965
$815$	2.99946e12	0.238140
$816$	0	0
$817$	4.43055e12	0.347903
$818$	−2.62325e13	−2.04857
$819$	0	0
$820$	−1.89374e13	−1.46271
$821$	−1.25176e13	−0.961563	−0.480782	−	0.876840i	$-0.659647\pi$
$821$	−1.25176e13	−0.961563	−0.480782	+	0.876840i	$0.659647\pi$
$822$	0	0
$823$	−1.50348e13	−1.14234	−0.571172	−	0.820830i	$-0.693511\pi$
$823$	−1.50348e13	−1.14234	−0.571172	+	0.820830i	$0.693511\pi$
$824$	3.31520e12	0.250517
$825$	0	0
$826$	8.96858e12	0.670369
$827$	−2.31522e13	−1.72115	−0.860574	−	0.509325i	$-0.829895\pi$
$827$	−2.31522e13	−1.72115	−0.860574	+	0.509325i	$0.829895\pi$
$828$	0	0
$829$	4.88071e12	0.358912	0.179456	−	0.983766i	$-0.442566\pi$
$829$	4.88071e12	0.358912	0.179456	+	0.983766i	$0.442566\pi$
$830$	−3.32044e13	−2.42854
$831$	0	0
$832$	−2.07494e13	−1.50124
$833$	−3.27595e12	−0.235741
$834$	0	0
$835$	1.99034e13	1.41690
$836$	1.80100e13	1.27522
$837$	0	0
$838$	−1.69635e13	−1.18828
$839$	1.18015e13	0.822258	0.411129	−	0.911577i	$-0.365135\pi$
$839$	1.18015e13	0.822258	0.411129	+	0.911577i	$0.365135\pi$
$840$	0	0
$841$	9.27330e12	0.639223
$842$	−1.53224e13	−1.05057
$843$	0	0
$844$	−6.05142e11	−0.0410503
$845$	−8.86708e11	−0.0598309
$846$	0	0
$847$	5.88744e12	0.393053
$848$	9.51520e12	0.631883
$849$	0	0
$850$	−1.43265e12	−0.0941361
$851$	8.13771e12	0.531887
$852$	0	0
$853$	−2.54707e13	−1.64729	−0.823646	−	0.567104i	$-0.808064\pi$
$853$	−2.54707e13	−1.64729	−0.823646	+	0.567104i	$0.808064\pi$
$854$	2.62485e12	0.168867
$855$	0	0
$856$	2.24637e12	0.143004
$857$	−1.26584e13	−0.801615	−0.400807	−	0.916162i	$-0.631270\pi$
$857$	−1.26584e13	−0.801615	−0.400807	+	0.916162i	$0.631270\pi$
$858$	0	0
$859$	2.01387e13	1.26201	0.631004	−	0.775779i	$-0.282643\pi$
$859$	2.01387e13	1.26201	0.631004	+	0.775779i	$0.282643\pi$
$860$	1.04240e13	0.649820
$861$	0	0
$862$	5.66896e12	0.349720
$863$	1.44977e13	0.889716	0.444858	−	0.895601i	$-0.353254\pi$
$863$	1.44977e13	0.889716	0.444858	+	0.895601i	$0.353254\pi$
$864$	0	0
$865$	−1.77159e13	−1.07594
$866$	−1.40297e13	−0.847655
$867$	0	0
$868$	−2.24104e12	−0.134002
$869$	3.58004e13	2.12961
$870$	0	0
$871$	8.50712e12	0.500842
$872$	2.54721e12	0.149191
$873$	0	0
$874$	−8.40656e12	−0.487323
$875$	6.42411e12	0.370490
$876$	0	0
$877$	1.83297e13	1.04630	0.523150	−	0.852241i	$-0.324757\pi$
$877$	1.83297e13	1.04630	0.523150	+	0.852241i	$0.324757\pi$
$878$	−1.12598e13	−0.639448
$879$	0	0
$880$	−1.66286e13	−0.934726
$881$	−4.57369e12	−0.255785	−0.127892	−	0.991788i	$-0.540821\pi$
$881$	−4.57369e12	−0.255785	−0.127892	+	0.991788i	$0.540821\pi$
$882$	0	0
$883$	3.07854e13	1.70420	0.852101	−	0.523377i	$-0.175328\pi$
$883$	3.07854e13	1.70420	0.852101	+	0.523377i	$0.175328\pi$
$884$	−3.94448e13	−2.17248
$885$	0	0
$886$	−1.73919e13	−0.948190
$887$	2.58803e13	1.40382	0.701911	−	0.712264i	$-0.252330\pi$
$887$	2.58803e13	1.40382	0.701911	+	0.712264i	$0.252330\pi$
$888$	0	0
$889$	2.22963e12	0.119722
$890$	−7.15699e12	−0.382363
$891$	0	0
$892$	4.38015e12	0.231658
$893$	−7.90242e12	−0.415842
$894$	0	0
$895$	1.03955e13	0.541556
$896$	−5.91314e12	−0.306501
$897$	0	0
$898$	−5.11061e13	−2.62258
$899$	6.94814e12	0.354773
$900$	0	0
$901$	3.21075e13	1.62310
$902$	−4.81091e13	−2.41990
$903$	0	0
$904$	−2.17827e12	−0.108481
$905$	1.80915e13	0.896510
$906$	0	0
$907$	9.64118e12	0.473040	0.236520	−	0.971627i	$-0.423993\pi$
$907$	9.64118e12	0.473040	0.236520	+	0.971627i	$0.423993\pi$
$908$	3.16326e13	1.54436
$909$	0	0
$910$	−1.23737e13	−0.598156
$911$	−7.28746e12	−0.350545	−0.175272	−	0.984520i	$-0.556081\pi$
$911$	−7.28746e12	−0.350545	−0.175272	+	0.984520i	$0.556081\pi$
$912$	0	0
$913$	−4.73478e13	−2.25518
$914$	4.91669e13	2.33032
$915$	0	0
$916$	−1.53463e13	−0.720236
$917$	−6.18722e12	−0.288957
$918$	0	0
$919$	2.73276e13	1.26381	0.631906	−	0.775045i	$-0.282273\pi$
$919$	2.73276e13	1.26381	0.631906	+	0.775045i	$0.282273\pi$
$920$	−4.32017e12	−0.198818
$921$	0	0
$922$	4.70559e13	2.14449
$923$	−2.20281e13	−0.999011
$924$	0	0
$925$	9.67413e11	0.0434484
$926$	6.03353e13	2.69663
$927$	0	0
$928$	4.02610e13	1.78204
$929$	7.21805e12	0.317943	0.158971	−	0.987283i	$-0.449182\pi$
$929$	7.21805e12	0.317943	0.158971	+	0.987283i	$0.449182\pi$
$930$	0	0
$931$	−2.28520e12	−0.0996897
$932$	2.15616e13	0.936071
$933$	0	0
$934$	4.00808e13	1.72336
$935$	−5.61106e13	−2.40100
$936$	0	0
$937$	1.41988e13	0.601761	0.300880	−	0.953662i	$-0.402719\pi$
$937$	1.41988e13	0.601761	0.300880	+	0.953662i	$0.402719\pi$
$938$	6.58550e12	0.277764
$939$	0	0
$940$	−1.85925e13	−0.776718
$941$	2.53242e13	1.05289	0.526445	−	0.850209i	$-0.323525\pi$
$941$	2.53242e13	1.05289	0.526445	+	0.850209i	$0.323525\pi$
$942$	0	0
$943$	1.26046e13	0.519070
$944$	−1.84138e13	−0.754693
$945$	0	0
$946$	2.64816e13	1.07506
$947$	1.37160e13	0.554182	0.277091	−	0.960844i	$-0.410630\pi$
$947$	1.37160e13	0.554182	0.277091	+	0.960844i	$0.410630\pi$
$948$	0	0
$949$	−2.86380e13	−1.14616
$950$	−9.99374e11	−0.0398082
$951$	0	0
$952$	−6.66961e12	−0.263169
$953$	−2.44096e13	−0.958610	−0.479305	−	0.877648i	$-0.659111\pi$
$953$	−2.44096e13	−0.958610	−0.479305	+	0.877648i	$0.659111\pi$
$954$	0	0
$955$	2.29994e13	0.894750
$956$	−1.65065e13	−0.639137
$957$	0	0
$958$	6.78892e13	2.60409
$959$	1.06632e13	0.407104
$960$	0	0
$961$	−2.44095e13	−0.923218
$962$	4.74531e13	1.78639
$963$	0	0
$964$	4.59096e13	1.71221
$965$	2.17600e13	0.807765
$966$	0	0
$967$	−4.19850e13	−1.54410	−0.772049	−	0.635563i	$-0.780768\pi$
$967$	−4.19850e13	−1.54410	−0.772049	+	0.635563i	$0.780768\pi$
$968$	1.19864e13	0.438784
$969$	0	0
$970$	−5.33275e13	−1.93410
$971$	−3.13254e12	−0.113086	−0.0565432	−	0.998400i	$-0.518008\pi$
$971$	−3.13254e12	−0.113086	−0.0565432	+	0.998400i	$0.518008\pi$
$972$	0	0
$973$	−1.74886e13	−0.625529
$974$	−4.21022e13	−1.49896
$975$	0	0
$976$	−5.38920e12	−0.190108
$977$	−4.53124e13	−1.59108	−0.795539	−	0.605902i	$-0.792812\pi$
$977$	−4.53124e13	−1.59108	−0.795539	+	0.605902i	$0.792812\pi$
$978$	0	0
$979$	−1.02055e13	−0.355068
$980$	−5.37654e12	−0.186202
$981$	0	0
$982$	6.96109e13	2.38878
$983$	2.54481e13	0.869290	0.434645	−	0.900602i	$-0.356874\pi$
$983$	2.54481e13	0.869290	0.434645	+	0.900602i	$0.356874\pi$
$984$	0	0
$985$	3.32984e13	1.12709
$986$	9.46704e13	3.18983
$987$	0	0
$988$	−2.75155e13	−0.918694
$989$	−6.93817e12	−0.230601
$990$	0	0
$991$	−5.41691e13	−1.78410	−0.892052	−	0.451933i	$-0.850734\pi$
$991$	−5.41691e13	−1.78410	−0.892052	+	0.451933i	$0.850734\pi$
$992$	1.17634e13	0.385683
$993$	0	0
$994$	−1.70523e13	−0.554045
$995$	3.61065e13	1.16783
$996$	0	0
$997$	−4.29329e13	−1.37614	−0.688069	−	0.725646i	$-0.741541\pi$
$997$	−4.29329e13	−1.37614	−0.688069	+	0.725646i	$0.741541\pi$
$998$	−1.11473e13	−0.355697
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.10.a.e.1.3		3
3.2	odd	2		7.10.a.b.1.1	✓	3
12.11	even	2		112.10.a.h.1.3		3
15.2	even	4		175.10.b.d.99.2		6
15.8	even	4		175.10.b.d.99.5		6
15.14	odd	2		175.10.a.d.1.3		3
21.2	odd	6		49.10.c.d.18.3		6
21.5	even	6		49.10.c.e.18.3		6
21.11	odd	6		49.10.c.d.30.3		6
21.17	even	6		49.10.c.e.30.3		6
21.20	even	2		49.10.a.c.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.10.a.b.1.1	✓	3	3.2	odd	2
49.10.a.c.1.1		3	21.20	even	2
49.10.c.d.18.3		6	21.2	odd	6
49.10.c.d.30.3		6	21.11	odd	6
49.10.c.e.18.3		6	21.5	even	6
49.10.c.e.30.3		6	21.17	even	6
63.10.a.e.1.3		3	1.1	even	1	trivial
112.10.a.h.1.3		3	12.11	even	2
175.10.a.d.1.3		3	15.14	odd	2
175.10.b.d.99.2		6	15.2	even	4
175.10.b.d.99.5		6	15.8	even	4

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:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	63.a (trivial)

\(f(q)\)	\(=\)	\(q+34.1627 q^{2} +655.088 q^{4} -1423.70 q^{5} +2401.00 q^{7} +4888.28 q^{8} -48637.4 q^{10} -69354.4 q^{11} +105959. q^{13} +82024.6 q^{14} -168409. q^{16} -568267. q^{17} -396405. q^{19} -932649. q^{20} -2.36933e6 q^{22} +620765. q^{23} +73796.7 q^{25} +3.61984e6 q^{26} +1.57287e6 q^{28} -4.87652e6 q^{29} -1.42482e6 q^{31} -8.25609e6 q^{32} -1.94135e7 q^{34} -3.41830e6 q^{35} +1.31092e7 q^{37} -1.35423e7 q^{38} -6.95944e6 q^{40} +2.03049e7 q^{41} -1.11768e7 q^{43} -4.54332e7 q^{44} +2.12070e7 q^{46} +1.99352e7 q^{47} +5.76480e6 q^{49} +2.52109e6 q^{50} +6.94125e7 q^{52} -5.65007e7 q^{53} +9.87398e7 q^{55} +1.17367e7 q^{56} -1.66595e8 q^{58} +1.09340e8 q^{59} +3.20008e7 q^{61} -4.86755e7 q^{62} -1.95825e8 q^{64} -1.50854e8 q^{65} +8.02869e7 q^{67} -3.72265e8 q^{68} -1.16778e8 q^{70} -2.07893e8 q^{71} -2.70274e8 q^{73} +4.47844e8 q^{74} -2.59680e8 q^{76} -1.66520e8 q^{77} -5.16196e8 q^{79} +2.39763e8 q^{80} +6.93671e8 q^{82} +6.82693e8 q^{83} +8.09042e8 q^{85} -3.81830e8 q^{86} -3.39023e8 q^{88} +1.47150e8 q^{89} +2.54408e8 q^{91} +4.06656e8 q^{92} +6.81040e8 q^{94} +5.64362e8 q^{95} +1.09643e9 q^{97} +1.96941e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 21 q^{2} + 1557 q^{4} - 1554 q^{5} + 7203 q^{7} - 14055 q^{8} - 97860 q^{10} + 3444 q^{11} - 19782 q^{13} - 50421 q^{14} + 482961 q^{16} - 1016694 q^{17} + 222852 q^{19} + 1922088 q^{20} - 2847048 q^{22}+ \cdots - 121060821 q^{98}+O(q^{100}) \) 3 * q - 21 * q^2 + 1557 * q^4 - 1554 * q^5 + 7203 * q^7 - 14055 * q^8 - 97860 * q^10 + 3444 * q^11 - 19782 * q^13 - 50421 * q^14 + 482961 * q^16 - 1016694 * q^17 + 222852 * q^19 + 1922088 * q^20 - 2847048 * q^22 - 1885632 * q^23 + 3073221 * q^25 + 8785056 * q^26 + 3738357 * q^28 - 4081818 * q^29 + 2869440 * q^31 - 25221951 * q^32 - 3981642 * q^34 - 3731154 * q^35 + 1395618 * q^37 - 43479870 * q^38 - 82859280 * q^40 + 14420658 * q^41 - 61631172 * q^43 - 97011984 * q^44 + 89747664 * q^46 + 10368960 * q^47 + 17294403 * q^49 - 73325055 * q^50 - 80908044 * q^52 - 67502610 * q^53 - 105823032 * q^55 - 33746055 * q^56 - 159163830 * q^58 + 42590100 * q^59 + 191746842 * q^61 + 46983468 * q^62 + 7852161 * q^64 - 364283220 * q^65 - 255175788 * q^67 - 743485806 * q^68 - 234961860 * q^70 - 296514504 * q^71 + 344213310 * q^73 + 690696462 * q^74 + 728839986 * q^76 + 8269044 * q^77 - 960412656 * q^79 + 1333333344 * q^80 + 562675302 * q^82 + 1100517180 * q^83 + 438179412 * q^85 + 880982256 * q^86 + 1206124800 * q^88 - 506816478 * q^89 - 47496582 * q^91 - 691123488 * q^92 + 1388004828 * q^94 + 2203071072 * q^95 - 647498250 * q^97 - 121060821 * q^98

Introduction

L-functions

Modular forms

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Database

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