LMFDB - Embedded newform 56.10.a.c.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [56,10,Mod(1,56)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("56.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(56, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 56 = 2^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	56.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,70] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$28.8420068252$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5576x^{2} - 170673x - 607824 $ x^4 - 2x^3 - 5576x^2 - 170673*x - 607824
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{11}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-38.9670$ of defining polynomial
Character	$\chi$	$=$	56.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+128.448 q^{3} -1333.35 q^{5} +2401.00 q^{7} -3184.23 q^{9} -22287.0 q^{11} +185165. q^{13} -171266. q^{15} +351739. q^{17} +1.10798e6 q^{19} +308403. q^{21} -1.99049e6 q^{23} -175297. q^{25} -2.93724e6 q^{27} +3.18384e6 q^{29} +7.42986e6 q^{31} -2.86270e6 q^{33} -3.20138e6 q^{35} +1.11534e7 q^{37} +2.37840e7 q^{39} +2.22681e7 q^{41} +1.98693e7 q^{43} +4.24571e6 q^{45} -569943. q^{47} +5.76480e6 q^{49} +4.51801e7 q^{51} -6.76141e7 q^{53} +2.97164e7 q^{55} +1.42317e8 q^{57} +6.89400e7 q^{59} -9.40859e7 q^{61} -7.64534e6 q^{63} -2.46890e8 q^{65} +7.82459e7 q^{67} -2.55673e8 q^{69} +1.61299e8 q^{71} +1.68908e8 q^{73} -2.25164e7 q^{75} -5.35110e7 q^{77} -3.52815e8 q^{79} -3.14606e8 q^{81} +1.80933e8 q^{83} -4.68993e8 q^{85} +4.08957e8 q^{87} -8.90321e8 q^{89} +4.44581e8 q^{91} +9.54347e8 q^{93} -1.47732e9 q^{95} +5.34984e8 q^{97} +7.09669e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 70 q^{3} + 1022 q^{5} + 9604 q^{7} + 12176 q^{9} - 31476 q^{11} + 11466 q^{13} + 98504 q^{15} + 397012 q^{17} + 246610 q^{19} + 168070 q^{21} + 1976552 q^{23} + 1086512 q^{25} + 3139108 q^{27} + 3718940 q^{29}+ \cdots + 1279453212 q^{99}+O(q^{100}) $ 4 * q + 70 * q^3 + 1022 * q^5 + 9604 * q^7 + 12176 * q^9 - 31476 * q^11 + 11466 * q^13 + 98504 * q^15 + 397012 * q^17 + 246610 * q^19 + 168070 * q^21 + 1976552 * q^23 + 1086512 * q^25 + 3139108 * q^27 + 3718940 * q^29 + 11118436 * q^31 + 16891056 * q^33 + 2453822 * q^35 + 30221340 * q^37 + 37296384 * q^39 + 27162604 * q^41 + 64096964 * q^43 + 35015302 * q^45 + 104378204 * q^47 + 23059204 * q^49 + 111871540 * q^51 + 20587304 * q^53 + 251355720 * q^55 - 49920188 * q^57 + 130065474 * q^59 - 134490482 * q^61 + 29234576 * q^63 - 227739180 * q^65 + 104031144 * q^67 - 408620800 * q^69 - 71827896 * q^71 - 586388432 * q^73 + 134360114 * q^75 - 75573876 * q^77 + 164282440 * q^79 - 1324238632 * q^81 + 130538394 * q^83 - 1710305284 * q^85 + 142567628 * q^87 - 2036960408 * q^89 + 27529866 * q^91 - 2396678120 * q^93 - 498649144 * q^95 - 2049365612 * q^97 + 1279453212 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	128.448	0.915546	0.457773	−	0.889069i	$-0.348647\pi$
$3$	128.448	0.915546	0.457773	+	0.889069i	$0.348647\pi$
$4$	0	0
$5$	−1333.35	−0.954069	−0.477035	−	0.878884i	$-0.658288\pi$
$5$	−1333.35	−0.954069	−0.477035	+	0.878884i	$0.658288\pi$
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	0	0
$9$	−3184.23	−0.161776
$10$	0	0
$11$	−22287.0	−0.458969	−0.229485	−	0.973312i	$-0.573704\pi$
$11$	−22287.0	−0.458969	−0.229485	+	0.973312i	$0.573704\pi$
$12$	0	0
$13$	185165.	1.79810	0.899049	−	0.437849i	$-0.144259\pi$
$13$	185165.	1.79810	0.899049	+	0.437849i	$0.144259\pi$
$14$	0	0
$15$	−171266.	−0.873494
$16$	0	0
$17$	351739.	1.02141	0.510706	−	0.859755i	$-0.329384\pi$
$17$	351739.	1.02141	0.510706	+	0.859755i	$0.329384\pi$
$18$	0	0
$19$	1.10798e6	1.95047	0.975235	−	0.221169i	$-0.0709873\pi$
$19$	1.10798e6	1.95047	0.975235	+	0.221169i	$0.0709873\pi$
$20$	0	0
$21$	308403.	0.346044
$22$	0	0
$23$	−1.99049e6	−1.48315	−0.741573	−	0.670872i	$-0.765920\pi$
$23$	−1.99049e6	−1.48315	−0.741573	+	0.670872i	$0.765920\pi$
$24$	0	0
$25$	−175297.	−0.0897518
$26$	0	0
$27$	−2.93724e6	−1.06366
$28$	0	0
$29$	3.18384e6	0.835912	0.417956	−	0.908467i	$-0.362747\pi$
$29$	3.18384e6	0.835912	0.417956	+	0.908467i	$0.362747\pi$
$30$	0	0
$31$	7.42986e6	1.44495	0.722476	−	0.691396i	$-0.243004\pi$
$31$	7.42986e6	1.44495	0.722476	+	0.691396i	$0.243004\pi$
$32$	0	0
$33$	−2.86270e6	−0.420207
$34$	0	0
$35$	−3.20138e6	−0.360604
$36$	0	0
$37$	1.11534e7	0.978360	0.489180	−	0.872183i	$-0.337296\pi$
$37$	1.11534e7	0.978360	0.489180	+	0.872183i	$0.337296\pi$
$38$	0	0
$39$	2.37840e7	1.64624
$40$	0	0
$41$	2.22681e7	1.23071	0.615354	−	0.788251i	$-0.289013\pi$
$41$	2.22681e7	1.23071	0.615354	+	0.788251i	$0.289013\pi$
$42$	0	0
$43$	1.98693e7	0.886287	0.443144	−	0.896451i	$-0.353863\pi$
$43$	1.98693e7	0.886287	0.443144	+	0.896451i	$0.353863\pi$
$44$	0	0
$45$	4.24571e6	0.154345
$46$	0	0
$47$	−569943.	−0.0170369	−0.00851847	−	0.999964i	$-0.502712\pi$
$47$	−569943.	−0.0170369	−0.00851847	+	0.999964i	$0.502712\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	0	0
$51$	4.51801e7	0.935149
$52$	0	0
$53$	−6.76141e7	−1.17705	−0.588527	−	0.808478i	$-0.700292\pi$
$53$	−6.76141e7	−1.17705	−0.588527	+	0.808478i	$0.700292\pi$
$54$	0	0
$55$	2.97164e7	0.437889
$56$	0	0
$57$	1.42317e8	1.78575
$58$	0	0
$59$	6.89400e7	0.740691	0.370345	−	0.928894i	$-0.379239\pi$
$59$	6.89400e7	0.740691	0.370345	+	0.928894i	$0.379239\pi$
$60$	0	0
$61$	−9.40859e7	−0.870042	−0.435021	−	0.900420i	$-0.643259\pi$
$61$	−9.40859e7	−0.870042	−0.435021	+	0.900420i	$0.643259\pi$
$62$	0	0
$63$	−7.64534e6	−0.0611455
$64$	0	0
$65$	−2.46890e8	−1.71551
$66$	0	0
$67$	7.82459e7	0.474378	0.237189	−	0.971463i	$-0.423774\pi$
$67$	7.82459e7	0.474378	0.237189	+	0.971463i	$0.423774\pi$
$68$	0	0
$69$	−2.55673e8	−1.35789
$70$	0	0
$71$	1.61299e8	0.753302	0.376651	−	0.926355i	$-0.377076\pi$
$71$	1.61299e8	0.753302	0.376651	+	0.926355i	$0.377076\pi$
$72$	0	0
$73$	1.68908e8	0.696142	0.348071	−	0.937468i	$-0.386837\pi$
$73$	1.68908e8	0.696142	0.348071	+	0.937468i	$0.386837\pi$
$74$	0	0
$75$	−2.25164e7	−0.0821719
$76$	0	0
$77$	−5.35110e7	−0.173474
$78$	0	0
$79$	−3.52815e8	−1.01912	−0.509559	−	0.860436i	$-0.670192\pi$
$79$	−3.52815e8	−1.01912	−0.509559	+	0.860436i	$0.670192\pi$
$80$	0	0
$81$	−3.14606e8	−0.812053
$82$	0	0
$83$	1.80933e8	0.418471	0.209235	−	0.977865i	$-0.432902\pi$
$83$	1.80933e8	0.418471	0.209235	+	0.977865i	$0.432902\pi$
$84$	0	0
$85$	−4.68993e8	−0.974498
$86$	0	0
$87$	4.08957e8	0.765316
$88$	0	0
$89$	−8.90321e8	−1.50415	−0.752076	−	0.659076i	$-0.770947\pi$
$89$	−8.90321e8	−1.50415	−0.752076	+	0.659076i	$0.770947\pi$
$90$	0	0
$91$	4.44581e8	0.679617
$92$	0	0
$93$	9.54347e8	1.32292
$94$	0	0
$95$	−1.47732e9	−1.86088
$96$	0	0
$97$	5.34984e8	0.613576	0.306788	−	0.951778i	$-0.400746\pi$
$97$	5.34984e8	0.613576	0.306788	+	0.951778i	$0.400746\pi$
$98$	0	0
$99$	7.09669e7	0.0742501
$100$	0	0
$101$	−1.68385e9	−1.61012	−0.805059	−	0.593194i	$-0.797867\pi$
$101$	−1.68385e9	−1.61012	−0.805059	+	0.593194i	$0.797867\pi$
$102$	0	0
$103$	−3.93045e8	−0.344092	−0.172046	−	0.985089i	$-0.555038\pi$
$103$	−3.93045e8	−0.344092	−0.172046	+	0.985089i	$0.555038\pi$
$104$	0	0
$105$	−4.11209e8	−0.330150
$106$	0	0
$107$	−5.83255e8	−0.430161	−0.215081	−	0.976596i	$-0.569001\pi$
$107$	−5.83255e8	−0.430161	−0.215081	+	0.976596i	$0.569001\pi$
$108$	0	0
$109$	−1.78621e9	−1.21203	−0.606013	−	0.795455i	$-0.707232\pi$
$109$	−1.78621e9	−1.21203	−0.606013	+	0.795455i	$0.707232\pi$
$110$	0	0
$111$	1.43262e9	0.895733
$112$	0	0
$113$	9.58596e8	0.553073	0.276537	−	0.961003i	$-0.410813\pi$
$113$	9.58596e8	0.553073	0.276537	+	0.961003i	$0.410813\pi$
$114$	0	0
$115$	2.65402e9	1.41502
$116$	0	0
$117$	−5.89608e8	−0.290889
$118$	0	0
$119$	8.44526e8	0.386057
$120$	0	0
$121$	−1.86124e9	−0.789347
$122$	0	0
$123$	2.86028e9	1.12677
$124$	0	0
$125$	2.83794e9	1.03970
$126$	0	0
$127$	−1.76411e9	−0.601741	−0.300870	−	0.953665i	$-0.597277\pi$
$127$	−1.76411e9	−0.601741	−0.300870	+	0.953665i	$0.597277\pi$
$128$	0	0
$129$	2.55216e9	0.811437
$130$	0	0
$131$	6.26391e9	1.85834	0.929169	−	0.369656i	$-0.120524\pi$
$131$	6.26391e9	1.85834	0.929169	+	0.369656i	$0.120524\pi$
$132$	0	0
$133$	2.66025e9	0.737209
$134$	0	0
$135$	3.91638e9	1.01480
$136$	0	0
$137$	7.81567e8	0.189550	0.0947750	−	0.995499i	$-0.469787\pi$
$137$	7.81567e8	0.189550	0.0947750	+	0.995499i	$0.469787\pi$
$138$	0	0
$139$	−1.55133e9	−0.352484	−0.176242	−	0.984347i	$-0.556394\pi$
$139$	−1.55133e9	−0.352484	−0.176242	+	0.984347i	$0.556394\pi$
$140$	0	0
$141$	−7.32078e7	−0.0155981
$142$	0	0
$143$	−4.12676e9	−0.825271
$144$	0	0
$145$	−4.24518e9	−0.797518
$146$	0	0
$147$	7.40474e8	0.130792
$148$	0	0
$149$	7.91062e9	1.31484	0.657419	−	0.753525i	$-0.271648\pi$
$149$	7.91062e9	1.31484	0.657419	+	0.753525i	$0.271648\pi$
$150$	0	0
$151$	−4.28928e9	−0.671411	−0.335705	−	0.941967i	$-0.608975\pi$
$151$	−4.28928e9	−0.671411	−0.335705	+	0.941967i	$0.608975\pi$
$152$	0	0
$153$	−1.12002e9	−0.165240
$154$	0	0
$155$	−9.90662e9	−1.37858
$156$	0	0
$157$	9.14889e9	1.20177	0.600883	−	0.799337i	$-0.294816\pi$
$157$	9.14889e9	1.20177	0.600883	+	0.799337i	$0.294816\pi$
$158$	0	0
$159$	−8.68487e9	−1.07765
$160$	0	0
$161$	−4.77916e9	−0.560577
$162$	0	0
$163$	−1.15239e10	−1.27866	−0.639329	−	0.768933i	$-0.720788\pi$
$163$	−1.15239e10	−1.27866	−0.639329	+	0.768933i	$0.720788\pi$
$164$	0	0
$165$	3.81699e9	0.400907
$166$	0	0
$167$	1.41989e9	0.141264	0.0706321	−	0.997502i	$-0.477498\pi$
$167$	1.41989e9	0.141264	0.0706321	+	0.997502i	$0.477498\pi$
$168$	0	0
$169$	2.36815e10	2.23315
$170$	0	0
$171$	−3.52806e9	−0.315539
$172$	0	0
$173$	−1.29140e10	−1.09611	−0.548055	−	0.836442i	$-0.684631\pi$
$173$	−1.29140e10	−1.09611	−0.548055	+	0.836442i	$0.684631\pi$
$174$	0	0
$175$	−4.20887e8	−0.0339230
$176$	0	0
$177$	8.85517e9	0.678136
$178$	0	0
$179$	1.18543e10	0.863050	0.431525	−	0.902101i	$-0.357976\pi$
$179$	1.18543e10	0.863050	0.431525	+	0.902101i	$0.357976\pi$
$180$	0	0
$181$	−1.65050e10	−1.14304	−0.571519	−	0.820589i	$-0.693646\pi$
$181$	−1.65050e10	−1.14304	−0.571519	+	0.820589i	$0.693646\pi$
$182$	0	0
$183$	−1.20851e10	−0.796564
$184$	0	0
$185$	−1.48714e10	−0.933423
$186$	0	0
$187$	−7.83920e9	−0.468797
$188$	0	0
$189$	−7.05231e9	−0.402025
$190$	0	0
$191$	1.10136e10	0.598795	0.299398	−	0.954128i	$-0.403214\pi$
$191$	1.10136e10	0.598795	0.299398	+	0.954128i	$0.403214\pi$
$192$	0	0
$193$	2.86468e10	1.48617	0.743083	−	0.669199i	$-0.233363\pi$
$193$	2.86468e10	1.48617	0.743083	+	0.669199i	$0.233363\pi$
$194$	0	0
$195$	−3.17124e10	−1.57063
$196$	0	0
$197$	−2.40217e10	−1.13633	−0.568166	−	0.822914i	$-0.692347\pi$
$197$	−2.40217e10	−1.13633	−0.568166	+	0.822914i	$0.692347\pi$
$198$	0	0
$199$	−3.60955e9	−0.163160	−0.0815801	−	0.996667i	$-0.525997\pi$
$199$	−3.60955e9	−0.163160	−0.0815801	+	0.996667i	$0.525997\pi$
$200$	0	0
$201$	1.00505e10	0.434315
$202$	0	0
$203$	7.64441e9	0.315945
$204$	0	0
$205$	−2.96912e10	−1.17418
$206$	0	0
$207$	6.33818e9	0.239937
$208$	0	0
$209$	−2.46934e10	−0.895206
$210$	0	0
$211$	−1.67653e10	−0.582292	−0.291146	−	0.956679i	$-0.594037\pi$
$211$	−1.67653e10	−0.582292	−0.291146	+	0.956679i	$0.594037\pi$
$212$	0	0
$213$	2.07185e10	0.689682
$214$	0	0
$215$	−2.64928e10	−0.845579
$216$	0	0
$217$	1.78391e10	0.546140
$218$	0	0
$219$	2.16958e10	0.637350
$220$	0	0
$221$	6.51297e10	1.83660
$222$	0	0
$223$	4.85947e10	1.31588	0.657942	−	0.753069i	$-0.271427\pi$
$223$	4.85947e10	1.31588	0.657942	+	0.753069i	$0.271427\pi$
$224$	0	0
$225$	5.58185e8	0.0145197
$226$	0	0
$227$	1.54430e10	0.386025	0.193013	−	0.981196i	$-0.438174\pi$
$227$	1.54430e10	0.386025	0.193013	+	0.981196i	$0.438174\pi$
$228$	0	0
$229$	−6.16550e10	−1.48152	−0.740761	−	0.671768i	$-0.765535\pi$
$229$	−6.16550e10	−1.48152	−0.740761	+	0.671768i	$0.765535\pi$
$230$	0	0
$231$	−6.87335e9	−0.158823
$232$	0	0
$233$	1.58392e10	0.352073	0.176037	−	0.984384i	$-0.443672\pi$
$233$	1.58392e10	0.352073	0.176037	+	0.984384i	$0.443672\pi$
$234$	0	0
$235$	7.59935e8	0.0162544
$236$	0	0
$237$	−4.53182e10	−0.933050
$238$	0	0
$239$	−3.09957e10	−0.614484	−0.307242	−	0.951631i	$-0.599406\pi$
$239$	−3.09957e10	−0.614484	−0.307242	+	0.951631i	$0.599406\pi$
$240$	0	0
$241$	5.28203e10	1.00861	0.504306	−	0.863525i	$-0.331748\pi$
$241$	5.28203e10	1.00861	0.504306	+	0.863525i	$0.331748\pi$
$242$	0	0
$243$	1.74033e10	0.320187
$244$	0	0
$245$	−7.68651e9	−0.136296
$246$	0	0
$247$	2.05158e11	3.50714
$248$	0	0
$249$	2.32403e10	0.383129
$250$	0	0
$251$	1.86620e10	0.296775	0.148387	−	0.988929i	$-0.452592\pi$
$251$	1.86620e10	0.296775	0.148387	+	0.988929i	$0.452592\pi$
$252$	0	0
$253$	4.43619e10	0.680719
$254$	0	0
$255$	−6.02409e10	−0.892197
$256$	0	0
$257$	−3.71525e10	−0.531237	−0.265619	−	0.964078i	$-0.585576\pi$
$257$	−3.71525e10	−0.531237	−0.265619	+	0.964078i	$0.585576\pi$
$258$	0	0
$259$	2.67793e10	0.369785
$260$	0	0
$261$	−1.01381e10	−0.135230
$262$	0	0
$263$	7.63904e10	0.984550	0.492275	−	0.870440i	$-0.336165\pi$
$263$	7.63904e10	0.984550	0.492275	+	0.870440i	$0.336165\pi$
$264$	0	0
$265$	9.01535e10	1.12299
$266$	0	0
$267$	−1.14360e11	−1.37712
$268$	0	0
$269$	−1.97181e10	−0.229604	−0.114802	−	0.993388i	$-0.536623\pi$
$269$	−1.97181e10	−0.229604	−0.114802	+	0.993388i	$0.536623\pi$
$270$	0	0
$271$	4.51685e10	0.508714	0.254357	−	0.967110i	$-0.418136\pi$
$271$	4.51685e10	0.508714	0.254357	+	0.967110i	$0.418136\pi$
$272$	0	0
$273$	5.71053e10	0.622220
$274$	0	0
$275$	3.90683e9	0.0411933
$276$	0	0
$277$	−1.19051e11	−1.21500	−0.607498	−	0.794321i	$-0.707827\pi$
$277$	−1.19051e11	−1.21500	−0.607498	+	0.794321i	$0.707827\pi$
$278$	0	0
$279$	−2.36584e10	−0.233758
$280$	0	0
$281$	−1.45380e11	−1.39099	−0.695496	−	0.718530i	$-0.744815\pi$
$281$	−1.45380e11	−1.39099	−0.695496	+	0.718530i	$0.744815\pi$
$282$	0	0
$283$	1.18157e11	1.09502	0.547508	−	0.836800i	$-0.315576\pi$
$283$	1.18157e11	1.09502	0.547508	+	0.836800i	$0.315576\pi$
$284$	0	0
$285$	−1.89759e11	−1.70372
$286$	0	0
$287$	5.34656e10	0.465164
$288$	0	0
$289$	5.13275e9	0.0432822
$290$	0	0
$291$	6.87174e10	0.561757
$292$	0	0
$293$	−1.09887e11	−0.871050	−0.435525	−	0.900177i	$-0.643437\pi$
$293$	−1.09887e11	−0.871050	−0.435525	+	0.900177i	$0.643437\pi$
$294$	0	0
$295$	−9.19212e10	−0.706670
$296$	0	0
$297$	6.54621e10	0.488187
$298$	0	0
$299$	−3.68568e11	−2.66684
$300$	0	0
$301$	4.77062e10	0.334985
$302$	0	0
$303$	−2.16287e11	−1.47414
$304$	0	0
$305$	1.25450e11	0.830081
$306$	0	0
$307$	3.01613e10	0.193788	0.0968940	−	0.995295i	$-0.469109\pi$
$307$	3.01613e10	0.193788	0.0968940	+	0.995295i	$0.469109\pi$
$308$	0	0
$309$	−5.04857e10	−0.315032
$310$	0	0
$311$	2.23160e11	1.35268	0.676339	−	0.736590i	$-0.263565\pi$
$311$	2.23160e11	1.35268	0.676339	+	0.736590i	$0.263565\pi$
$312$	0	0
$313$	−2.17968e11	−1.28364	−0.641821	−	0.766854i	$-0.721821\pi$
$313$	−2.17968e11	−1.28364	−0.641821	+	0.766854i	$0.721821\pi$
$314$	0	0
$315$	1.01939e10	0.0583371
$316$	0	0
$317$	7.47610e9	0.0415823	0.0207912	−	0.999784i	$-0.493381\pi$
$317$	7.47610e9	0.0415823	0.0207912	+	0.999784i	$0.493381\pi$
$318$	0	0
$319$	−7.09582e10	−0.383658
$320$	0	0
$321$	−7.49176e10	−0.393832
$322$	0	0
$323$	3.89719e11	1.99223
$324$	0	0
$325$	−3.24587e10	−0.161383
$326$	0	0
$327$	−2.29434e11	−1.10967
$328$	0	0
$329$	−1.36843e9	−0.00643935
$330$	0	0
$331$	3.15127e11	1.44298	0.721489	−	0.692426i	$-0.243458\pi$
$331$	3.15127e11	1.44298	0.721489	+	0.692426i	$0.243458\pi$
$332$	0	0
$333$	−3.55150e10	−0.158275
$334$	0	0
$335$	−1.04329e11	−0.452590
$336$	0	0
$337$	1.49538e11	0.631565	0.315782	−	0.948832i	$-0.397733\pi$
$337$	1.49538e11	0.631565	0.315782	+	0.948832i	$0.397733\pi$
$338$	0	0
$339$	1.23129e11	0.506364
$340$	0	0
$341$	−1.65589e11	−0.663188
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	0	0
$345$	3.40902e11	1.29552
$346$	0	0
$347$	−4.08977e11	−1.51431	−0.757157	−	0.653233i	$-0.773412\pi$
$347$	−4.08977e11	−1.51431	−0.757157	+	0.653233i	$0.773412\pi$
$348$	0	0
$349$	3.94310e10	0.142273	0.0711366	−	0.997467i	$-0.477337\pi$
$349$	3.94310e10	0.142273	0.0711366	+	0.997467i	$0.477337\pi$
$350$	0	0
$351$	−5.43873e11	−1.91256
$352$	0	0
$353$	−9.27700e10	−0.317996	−0.158998	−	0.987279i	$-0.550826\pi$
$353$	−9.27700e10	−0.317996	−0.158998	+	0.987279i	$0.550826\pi$
$354$	0	0
$355$	−2.15068e11	−0.718702
$356$	0	0
$357$	1.08477e11	0.353453
$358$	0	0
$359$	−6.16203e10	−0.195794	−0.0978969	−	0.995197i	$-0.531212\pi$
$359$	−6.16203e10	−0.195794	−0.0978969	+	0.995197i	$0.531212\pi$
$360$	0	0
$361$	9.04925e11	2.80434
$362$	0	0
$363$	−2.39072e11	−0.722684
$364$	0	0
$365$	−2.25214e11	−0.664168
$366$	0	0
$367$	3.96168e11	1.13994	0.569969	−	0.821666i	$-0.306955\pi$
$367$	3.96168e11	1.13994	0.569969	+	0.821666i	$0.306955\pi$
$368$	0	0
$369$	−7.09067e10	−0.199099
$370$	0	0
$371$	−1.62342e11	−0.444884
$372$	0	0
$373$	1.29472e11	0.346327	0.173164	−	0.984893i	$-0.444601\pi$
$373$	1.29472e11	0.346327	0.173164	+	0.984893i	$0.444601\pi$
$374$	0	0
$375$	3.64526e11	0.951892
$376$	0	0
$377$	5.89535e11	1.50305
$378$	0	0
$379$	6.61542e10	0.164695	0.0823476	−	0.996604i	$-0.473758\pi$
$379$	6.61542e10	0.164695	0.0823476	+	0.996604i	$0.473758\pi$
$380$	0	0
$381$	−2.26596e11	−0.550921
$382$	0	0
$383$	−2.57103e11	−0.610538	−0.305269	−	0.952266i	$-0.598746\pi$
$383$	−2.57103e11	−0.610538	−0.305269	+	0.952266i	$0.598746\pi$
$384$	0	0
$385$	7.13490e10	0.165506
$386$	0	0
$387$	−6.32685e10	−0.143380
$388$	0	0
$389$	−3.51966e11	−0.779341	−0.389670	−	0.920954i	$-0.627411\pi$
$389$	−3.51966e11	−0.779341	−0.389670	+	0.920954i	$0.627411\pi$
$390$	0	0
$391$	−7.00133e11	−1.51490
$392$	0	0
$393$	8.04584e11	1.70139
$394$	0	0
$395$	4.70426e11	0.972310
$396$	0	0
$397$	6.13801e10	0.124014	0.0620070	−	0.998076i	$-0.480250\pi$
$397$	6.13801e10	0.124014	0.0620070	+	0.998076i	$0.480250\pi$
$398$	0	0
$399$	3.41703e11	0.674948
$400$	0	0
$401$	6.84492e11	1.32196	0.660981	−	0.750403i	$-0.270141\pi$
$401$	6.84492e11	1.32196	0.660981	+	0.750403i	$0.270141\pi$
$402$	0	0
$403$	1.37575e12	2.59816
$404$	0	0
$405$	4.19480e11	0.774755
$406$	0	0
$407$	−2.48575e11	−0.449037
$408$	0	0
$409$	5.35688e11	0.946580	0.473290	−	0.880907i	$-0.343066\pi$
$409$	5.35688e11	0.946580	0.473290	+	0.880907i	$0.343066\pi$
$410$	0	0
$411$	1.00390e11	0.173542
$412$	0	0
$413$	1.65525e11	0.279955
$414$	0	0
$415$	−2.41247e11	−0.399250
$416$	0	0
$417$	−1.99265e11	−0.322715
$418$	0	0
$419$	−8.57177e11	−1.35865	−0.679324	−	0.733838i	$-0.737727\pi$
$419$	−8.57177e11	−1.35865	−0.679324	+	0.733838i	$0.737727\pi$
$420$	0	0
$421$	−1.06825e12	−1.65731	−0.828655	−	0.559759i	$-0.810894\pi$
$421$	−1.06825e12	−1.65731	−0.828655	+	0.559759i	$0.810894\pi$
$422$	0	0
$423$	1.81483e9	0.00275616
$424$	0	0
$425$	−6.16587e10	−0.0916736
$426$	0	0
$427$	−2.25900e11	−0.328845
$428$	0	0
$429$	−5.30072e11	−0.755574
$430$	0	0
$431$	−1.82289e11	−0.254455	−0.127228	−	0.991874i	$-0.540608\pi$
$431$	−1.82289e11	−0.254455	−0.127228	+	0.991874i	$0.540608\pi$
$432$	0	0
$433$	−3.16598e11	−0.432826	−0.216413	−	0.976302i	$-0.569436\pi$
$433$	−3.16598e11	−0.432826	−0.216413	+	0.976302i	$0.569436\pi$
$434$	0	0
$435$	−5.45283e11	−0.730164
$436$	0	0
$437$	−2.20541e12	−2.89283
$438$	0	0
$439$	1.35181e12	1.73710	0.868552	−	0.495597i	$-0.165051\pi$
$439$	1.35181e12	1.73710	0.868552	+	0.495597i	$0.165051\pi$
$440$	0	0
$441$	−1.83565e10	−0.0231108
$442$	0	0
$443$	8.80722e11	1.08648	0.543240	−	0.839577i	$-0.317197\pi$
$443$	8.80722e11	1.08648	0.543240	+	0.839577i	$0.317197\pi$
$444$	0	0
$445$	1.18711e12	1.43507
$446$	0	0
$447$	1.01610e12	1.20379
$448$	0	0
$449$	−1.53710e12	−1.78481	−0.892406	−	0.451233i	$-0.850984\pi$
$449$	−1.53710e12	−1.78481	−0.892406	+	0.451233i	$0.850984\pi$
$450$	0	0
$451$	−4.96287e11	−0.564857
$452$	0	0
$453$	−5.50947e11	−0.614707
$454$	0	0
$455$	−5.92782e11	−0.648402
$456$	0	0
$457$	−6.06859e9	−0.00650826	−0.00325413	−	0.999995i	$-0.501036\pi$
$457$	−6.06859e9	−0.00650826	−0.00325413	+	0.999995i	$0.501036\pi$
$458$	0	0
$459$	−1.03314e12	−1.08643
$460$	0	0
$461$	1.63752e11	0.168862	0.0844309	−	0.996429i	$-0.473093\pi$
$461$	1.63752e11	0.168862	0.0844309	+	0.996429i	$0.473093\pi$
$462$	0	0
$463$	3.53914e10	0.0357918	0.0178959	−	0.999840i	$-0.494303\pi$
$463$	3.53914e10	0.0357918	0.0178959	+	0.999840i	$0.494303\pi$
$464$	0	0
$465$	−1.27248e12	−1.26216
$466$	0	0
$467$	−1.34951e12	−1.31295	−0.656476	−	0.754347i	$-0.727954\pi$
$467$	−1.34951e12	−1.31295	−0.656476	+	0.754347i	$0.727954\pi$
$468$	0	0
$469$	1.87868e11	0.179298
$470$	0	0
$471$	1.17515e12	1.10027
$472$	0	0
$473$	−4.42826e11	−0.406779
$474$	0	0
$475$	−1.94225e11	−0.175058
$476$	0	0
$477$	2.15299e11	0.190419
$478$	0	0
$479$	5.48236e11	0.475837	0.237918	−	0.971285i	$-0.423535\pi$
$479$	5.48236e11	0.475837	0.237918	+	0.971285i	$0.423535\pi$
$480$	0	0
$481$	2.06521e12	1.75919
$482$	0	0
$483$	−6.13871e11	−0.513234
$484$	0	0
$485$	−7.13323e11	−0.585394
$486$	0	0
$487$	−1.53461e12	−1.23628	−0.618139	−	0.786069i	$-0.712113\pi$
$487$	−1.53461e12	−1.23628	−0.618139	+	0.786069i	$0.712113\pi$
$488$	0	0
$489$	−1.48021e12	−1.17067
$490$	0	0
$491$	1.44127e12	1.11913	0.559563	−	0.828788i	$-0.310969\pi$
$491$	1.44127e12	1.11913	0.559563	+	0.828788i	$0.310969\pi$
$492$	0	0
$493$	1.11988e12	0.853811
$494$	0	0
$495$	−9.46238e10	−0.0708398
$496$	0	0
$497$	3.87279e11	0.284721
$498$	0	0
$499$	−1.37787e12	−0.994849	−0.497425	−	0.867507i	$-0.665721\pi$
$499$	−1.37787e12	−0.994849	−0.497425	+	0.867507i	$0.665721\pi$
$500$	0	0
$501$	1.82382e11	0.129334
$502$	0	0
$503$	−6.19613e10	−0.0431583	−0.0215792	−	0.999767i	$-0.506869\pi$
$503$	−6.19613e10	−0.0431583	−0.0215792	+	0.999767i	$0.506869\pi$
$504$	0	0
$505$	2.24517e12	1.53616
$506$	0	0
$507$	3.04183e12	2.04455
$508$	0	0
$509$	−1.28353e12	−0.847568	−0.423784	−	0.905763i	$-0.639298\pi$
$509$	−1.28353e12	−0.847568	−0.423784	+	0.905763i	$0.639298\pi$
$510$	0	0
$511$	4.05548e11	0.263117
$512$	0	0
$513$	−3.25439e12	−2.07464
$514$	0	0
$515$	5.24068e11	0.328288
$516$	0	0
$517$	1.27023e10	0.00781943
$518$	0	0
$519$	−1.65877e12	−1.00354
$520$	0	0
$521$	−2.47356e12	−1.47080	−0.735398	−	0.677636i	$-0.763005\pi$
$521$	−2.47356e12	−1.47080	−0.735398	+	0.677636i	$0.763005\pi$
$522$	0	0
$523$	1.16606e12	0.681499	0.340749	−	0.940154i	$-0.389319\pi$
$523$	1.16606e12	0.681499	0.340749	+	0.940154i	$0.389319\pi$
$524$	0	0
$525$	−5.40619e10	−0.0310581
$526$	0	0
$527$	2.61338e12	1.47589
$528$	0	0
$529$	2.16089e12	1.19972
$530$	0	0
$531$	−2.19521e11	−0.119826
$532$	0	0
$533$	4.12326e12	2.21293
$534$	0	0
$535$	7.77684e11	0.410403
$536$	0	0
$537$	1.52265e12	0.790162
$538$	0	0
$539$	−1.28480e11	−0.0655670
$540$	0	0
$541$	−1.92562e12	−0.966457	−0.483229	−	0.875494i	$-0.660536\pi$
$541$	−1.92562e12	−0.966457	−0.483229	+	0.875494i	$0.660536\pi$
$542$	0	0
$543$	−2.12002e12	−1.04650
$544$	0	0
$545$	2.38164e12	1.15636
$546$	0	0
$547$	7.68308e11	0.366938	0.183469	−	0.983026i	$-0.441267\pi$
$547$	7.68308e11	0.366938	0.183469	+	0.983026i	$0.441267\pi$
$548$	0	0
$549$	2.99592e11	0.140752
$550$	0	0
$551$	3.52762e12	1.63042
$552$	0	0
$553$	−8.47108e11	−0.385191
$554$	0	0
$555$	−1.91019e12	−0.854592
$556$	0	0
$557$	−3.12571e12	−1.37594	−0.687971	−	0.725739i	$-0.741498\pi$
$557$	−3.12571e12	−1.37594	−0.687971	+	0.725739i	$0.741498\pi$
$558$	0	0
$559$	3.67909e12	1.59363
$560$	0	0
$561$	−1.00693e12	−0.429205
$562$	0	0
$563$	9.23093e11	0.387220	0.193610	−	0.981079i	$-0.437980\pi$
$563$	9.23093e11	0.387220	0.193610	+	0.981079i	$0.437980\pi$
$564$	0	0
$565$	−1.27815e12	−0.527670
$566$	0	0
$567$	−7.55369e11	−0.306927
$568$	0	0
$569$	−2.57947e11	−0.103163	−0.0515816	−	0.998669i	$-0.516426\pi$
$569$	−2.57947e11	−0.103163	−0.0515816	+	0.998669i	$0.516426\pi$
$570$	0	0
$571$	−4.59120e12	−1.80744	−0.903720	−	0.428124i	$-0.859175\pi$
$571$	−4.59120e12	−1.80744	−0.903720	+	0.428124i	$0.859175\pi$
$572$	0	0
$573$	1.41467e12	0.548225
$574$	0	0
$575$	3.48926e11	0.133115
$576$	0	0
$577$	−2.30955e12	−0.867434	−0.433717	−	0.901049i	$-0.642798\pi$
$577$	−2.30955e12	−0.867434	−0.433717	+	0.901049i	$0.642798\pi$
$578$	0	0
$579$	3.67961e12	1.36065
$580$	0	0
$581$	4.34419e11	0.158167
$582$	0	0
$583$	1.50691e12	0.540231
$584$	0	0
$585$	7.86155e11	0.277528
$586$	0	0
$587$	−9.87038e11	−0.343133	−0.171566	−	0.985173i	$-0.554883\pi$
$587$	−9.87038e11	−0.343133	−0.171566	+	0.985173i	$0.554883\pi$
$588$	0	0
$589$	8.23212e12	2.81834
$590$	0	0
$591$	−3.08552e12	−1.04036
$592$	0	0
$593$	−7.39847e11	−0.245695	−0.122847	−	0.992426i	$-0.539203\pi$
$593$	−7.39847e11	−0.245695	−0.122847	+	0.992426i	$0.539203\pi$
$594$	0	0
$595$	−1.12605e12	−0.368326
$596$	0	0
$597$	−4.63638e11	−0.149381
$598$	0	0
$599$	−7.64932e11	−0.242774	−0.121387	−	0.992605i	$-0.538734\pi$
$599$	−7.64932e11	−0.242774	−0.121387	+	0.992605i	$0.538734\pi$
$600$	0	0
$601$	2.33332e12	0.729524	0.364762	−	0.931101i	$-0.381150\pi$
$601$	2.33332e12	0.729524	0.364762	+	0.931101i	$0.381150\pi$
$602$	0	0
$603$	−2.49153e11	−0.0767430
$604$	0	0
$605$	2.48169e12	0.753092
$606$	0	0
$607$	−2.30933e12	−0.690459	−0.345229	−	0.938518i	$-0.612199\pi$
$607$	−2.30933e12	−0.690459	−0.345229	+	0.938518i	$0.612199\pi$
$608$	0	0
$609$	9.81905e11	0.289262
$610$	0	0
$611$	−1.05533e11	−0.0306341
$612$	0	0
$613$	−4.10547e12	−1.17433	−0.587166	−	0.809467i	$-0.699756\pi$
$613$	−4.10547e12	−1.17433	−0.587166	+	0.809467i	$0.699756\pi$
$614$	0	0
$615$	−3.81376e12	−1.07502
$616$	0	0
$617$	2.03654e12	0.565730	0.282865	−	0.959160i	$-0.408715\pi$
$617$	2.03654e12	0.565730	0.282865	+	0.959160i	$0.408715\pi$
$618$	0	0
$619$	3.96655e12	1.08594	0.542970	−	0.839752i	$-0.317300\pi$
$619$	3.96655e12	1.08594	0.542970	+	0.839752i	$0.317300\pi$
$620$	0	0
$621$	5.84654e12	1.57756
$622$	0	0
$623$	−2.13766e12	−0.568516
$624$	0	0
$625$	−3.44159e12	−0.902193
$626$	0	0
$627$	−3.17181e12	−0.819602
$628$	0	0
$629$	3.92308e12	0.999308
$630$	0	0
$631$	−7.21178e12	−1.81097	−0.905483	−	0.424382i	$-0.860492\pi$
$631$	−7.21178e12	−1.81097	−0.905483	+	0.424382i	$0.860492\pi$
$632$	0	0
$633$	−2.15346e12	−0.533115
$634$	0	0
$635$	2.35218e12	0.574102
$636$	0	0
$637$	1.06744e12	0.256871
$638$	0	0
$639$	−5.13614e11	−0.121866
$640$	0	0
$641$	−5.26172e11	−0.123102	−0.0615512	−	0.998104i	$-0.519605\pi$
$641$	−5.26172e11	−0.123102	−0.0615512	+	0.998104i	$0.519605\pi$
$642$	0	0
$643$	−1.12420e12	−0.259355	−0.129677	−	0.991556i	$-0.541394\pi$
$643$	−1.12420e12	−0.259355	−0.129677	+	0.991556i	$0.541394\pi$
$644$	0	0
$645$	−3.40293e12	−0.774167
$646$	0	0
$647$	−4.10610e12	−0.921214	−0.460607	−	0.887604i	$-0.652368\pi$
$647$	−4.10610e12	−0.921214	−0.460607	+	0.887604i	$0.652368\pi$
$648$	0	0
$649$	−1.53646e12	−0.339954
$650$	0	0
$651$	2.29139e12	0.500016
$652$	0	0
$653$	−6.06598e12	−1.30554	−0.652772	−	0.757555i	$-0.726394\pi$
$653$	−6.06598e12	−1.30554	−0.652772	+	0.757555i	$0.726394\pi$
$654$	0	0
$655$	−8.35200e12	−1.77298
$656$	0	0
$657$	−5.37843e11	−0.112619
$658$	0	0
$659$	−5.94402e12	−1.22771	−0.613856	−	0.789418i	$-0.710382\pi$
$659$	−5.94402e12	−1.22771	−0.613856	+	0.789418i	$0.710382\pi$
$660$	0	0
$661$	−8.08693e12	−1.64770	−0.823848	−	0.566811i	$-0.808177\pi$
$661$	−8.08693e12	−1.64770	−0.823848	+	0.566811i	$0.808177\pi$
$662$	0	0
$663$	8.36575e12	1.68149
$664$	0	0
$665$	−3.54705e12	−0.703348
$666$	0	0
$667$	−6.33740e12	−1.23978
$668$	0	0
$669$	6.24187e12	1.20475
$670$	0	0
$671$	2.09689e12	0.399323
$672$	0	0
$673$	2.55058e12	0.479260	0.239630	−	0.970864i	$-0.422974\pi$
$673$	2.55058e12	0.479260	0.239630	+	0.970864i	$0.422974\pi$
$674$	0	0
$675$	5.14888e11	0.0954654
$676$	0	0
$677$	5.64109e12	1.03208	0.516041	−	0.856564i	$-0.327405\pi$
$677$	5.64109e12	1.03208	0.516041	+	0.856564i	$0.327405\pi$
$678$	0	0
$679$	1.28450e12	0.231910
$680$	0	0
$681$	1.98362e12	0.353424
$682$	0	0
$683$	1.20725e12	0.212277	0.106138	−	0.994351i	$-0.466151\pi$
$683$	1.20725e12	0.212277	0.106138	+	0.994351i	$0.466151\pi$
$684$	0	0
$685$	−1.04210e12	−0.180844
$686$	0	0
$687$	−7.91943e12	−1.35640
$688$	0	0
$689$	−1.25198e13	−2.11646
$690$	0	0
$691$	5.21659e12	0.870433	0.435216	−	0.900326i	$-0.356672\pi$
$691$	5.21659e12	0.870433	0.435216	+	0.900326i	$0.356672\pi$
$692$	0	0
$693$	1.70391e11	0.0280639
$694$	0	0
$695$	2.06848e12	0.336294
$696$	0	0
$697$	7.83256e12	1.25706
$698$	0	0
$699$	2.03451e12	0.322339
$700$	0	0
$701$	1.25403e13	1.96144	0.980722	−	0.195407i	$-0.0626027\pi$
$701$	1.25403e13	1.96144	0.980722	+	0.195407i	$0.0626027\pi$
$702$	0	0
$703$	1.23577e13	1.90826
$704$	0	0
$705$	9.76118e10	0.0148817
$706$	0	0
$707$	−4.04293e12	−0.608568
$708$	0	0
$709$	8.32940e12	1.23796	0.618979	−	0.785408i	$-0.287547\pi$
$709$	8.32940e12	1.23796	0.618979	+	0.785408i	$0.287547\pi$
$710$	0	0
$711$	1.12344e12	0.164869
$712$	0	0
$713$	−1.47890e13	−2.14307
$714$	0	0
$715$	5.50242e12	0.787366
$716$	0	0
$717$	−3.98132e12	−0.562589
$718$	0	0
$719$	6.67548e12	0.931542	0.465771	−	0.884905i	$-0.345777\pi$
$719$	6.67548e12	0.931542	0.465771	+	0.884905i	$0.345777\pi$
$720$	0	0
$721$	−9.43702e11	−0.130055
$722$	0	0
$723$	6.78464e12	0.923431
$724$	0	0
$725$	−5.58117e11	−0.0750247
$726$	0	0
$727$	−5.59131e12	−0.742350	−0.371175	−	0.928563i	$-0.621045\pi$
$727$	−5.59131e12	−0.742350	−0.371175	+	0.928563i	$0.621045\pi$
$728$	0	0
$729$	8.42780e12	1.10520
$730$	0	0
$731$	6.98882e12	0.905264
$732$	0	0
$733$	−7.00621e12	−0.896427	−0.448214	−	0.893927i	$-0.647940\pi$
$733$	−7.00621e12	−0.896427	−0.448214	+	0.893927i	$0.647940\pi$
$734$	0	0
$735$	−9.87313e11	−0.124785
$736$	0	0
$737$	−1.74386e12	−0.217725
$738$	0	0
$739$	−1.44169e12	−0.177817	−0.0889085	−	0.996040i	$-0.528338\pi$
$739$	−1.44169e12	−0.177817	−0.0889085	+	0.996040i	$0.528338\pi$
$740$	0	0
$741$	2.63521e13	3.21094
$742$	0	0
$743$	1.25229e13	1.50749	0.753746	−	0.657166i	$-0.228245\pi$
$743$	1.25229e13	1.50749	0.753746	+	0.657166i	$0.228245\pi$
$744$	0	0
$745$	−1.05476e13	−1.25445
$746$	0	0
$747$	−5.76132e11	−0.0676985
$748$	0	0
$749$	−1.40039e12	−0.162586
$750$	0	0
$751$	−1.02604e13	−1.17702	−0.588509	−	0.808491i	$-0.700285\pi$
$751$	−1.02604e13	−1.17702	−0.588509	+	0.808491i	$0.700285\pi$
$752$	0	0
$753$	2.39709e12	0.271711
$754$	0	0
$755$	5.71912e12	0.640572
$756$	0	0
$757$	−1.27370e13	−1.40973	−0.704865	−	0.709342i	$-0.748992\pi$
$757$	−1.27370e13	−1.40973	−0.704865	+	0.709342i	$0.748992\pi$
$758$	0	0
$759$	5.69818e12	0.623229
$760$	0	0
$761$	−1.58031e13	−1.70810	−0.854048	−	0.520194i	$-0.825860\pi$
$761$	−1.58031e13	−1.70810	−0.854048	+	0.520194i	$0.825860\pi$
$762$	0	0
$763$	−4.28868e12	−0.458103
$764$	0	0
$765$	1.49338e12	0.157650
$766$	0	0
$767$	1.27652e13	1.33183
$768$	0	0
$769$	−1.03153e13	−1.06369	−0.531844	−	0.846843i	$-0.678501\pi$
$769$	−1.03153e13	−1.06369	−0.531844	+	0.846843i	$0.678501\pi$
$770$	0	0
$771$	−4.77214e12	−0.486372
$772$	0	0
$773$	1.24848e13	1.25769	0.628843	−	0.777532i	$-0.283529\pi$
$773$	1.24848e13	1.25769	0.628843	+	0.777532i	$0.283529\pi$
$774$	0	0
$775$	−1.30243e12	−0.129687
$776$	0	0
$777$	3.43973e12	0.338555
$778$	0	0
$779$	2.46725e13	2.40046
$780$	0	0
$781$	−3.59486e12	−0.345742
$782$	0	0
$783$	−9.35171e12	−0.889126
$784$	0	0
$785$	−1.21987e13	−1.14657
$786$	0	0
$787$	−6.89443e12	−0.640638	−0.320319	−	0.947310i	$-0.603790\pi$
$787$	−6.89443e12	−0.640638	−0.320319	+	0.947310i	$0.603790\pi$
$788$	0	0
$789$	9.81215e12	0.901400
$790$	0	0
$791$	2.30159e12	0.209042
$792$	0	0
$793$	−1.74214e13	−1.56442
$794$	0	0
$795$	1.15800e13	1.02815
$796$	0	0
$797$	−8.25529e12	−0.724719	−0.362360	−	0.932038i	$-0.618029\pi$
$797$	−8.25529e12	−0.724719	−0.362360	+	0.932038i	$0.618029\pi$
$798$	0	0
$799$	−2.00472e11	−0.0174017
$800$	0	0
$801$	2.83499e12	0.243335
$802$	0	0
$803$	−3.76445e12	−0.319508
$804$	0	0
$805$	6.37230e12	0.534829
$806$	0	0
$807$	−2.53274e12	−0.210213
$808$	0	0
$809$	1.13656e13	0.932875	0.466438	−	0.884554i	$-0.345537\pi$
$809$	1.13656e13	0.932875	0.466438	+	0.884554i	$0.345537\pi$
$810$	0	0
$811$	−2.28367e13	−1.85370	−0.926849	−	0.375434i	$-0.877494\pi$
$811$	−2.28367e13	−1.85370	−0.926849	+	0.375434i	$0.877494\pi$
$812$	0	0
$813$	5.80178e12	0.465751
$814$	0	0
$815$	1.53654e13	1.21993
$816$	0	0
$817$	2.20147e13	1.72868
$818$	0	0
$819$	−1.41565e12	−0.109946
$820$	0	0
$821$	−1.17716e13	−0.904260	−0.452130	−	0.891952i	$-0.649336\pi$
$821$	−1.17716e13	−0.904260	−0.452130	+	0.891952i	$0.649336\pi$
$822$	0	0
$823$	2.41292e12	0.183334	0.0916671	−	0.995790i	$-0.470780\pi$
$823$	2.41292e12	0.183334	0.0916671	+	0.995790i	$0.470780\pi$
$824$	0	0
$825$	5.01822e11	0.0377144
$826$	0	0
$827$	8.35379e11	0.0621024	0.0310512	−	0.999518i	$-0.490115\pi$
$827$	8.35379e11	0.0621024	0.0310512	+	0.999518i	$0.490115\pi$
$828$	0	0
$829$	6.81892e12	0.501441	0.250721	−	0.968059i	$-0.419332\pi$
$829$	6.81892e12	0.501441	0.250721	+	0.968059i	$0.419332\pi$
$830$	0	0
$831$	−1.52918e13	−1.11238
$832$	0	0
$833$	2.02771e12	0.145916
$834$	0	0
$835$	−1.89322e12	−0.134776
$836$	0	0
$837$	−2.18233e13	−1.53694
$838$	0	0
$839$	1.60099e13	1.11548	0.557738	−	0.830017i	$-0.311669\pi$
$839$	1.60099e13	1.11548	0.557738	+	0.830017i	$0.311669\pi$
$840$	0	0
$841$	−4.37029e12	−0.301251
$842$	0	0
$843$	−1.86736e13	−1.27352
$844$	0	0
$845$	−3.15758e13	−2.13058
$846$	0	0
$847$	−4.46884e12	−0.298345
$848$	0	0
$849$	1.51770e13	1.00254
$850$	0	0
$851$	−2.22007e13	−1.45105
$852$	0	0
$853$	7.40758e12	0.479078	0.239539	−	0.970887i	$-0.423004\pi$
$853$	7.40758e12	0.479078	0.239539	+	0.970887i	$0.423004\pi$
$854$	0	0
$855$	4.70414e12	0.301046
$856$	0	0
$857$	−1.13540e13	−0.719008	−0.359504	−	0.933144i	$-0.617054\pi$
$857$	−1.13540e13	−0.719008	−0.359504	+	0.933144i	$0.617054\pi$
$858$	0	0
$859$	−3.09711e13	−1.94083	−0.970416	−	0.241440i	$-0.922380\pi$
$859$	−3.09711e13	−1.94083	−0.970416	+	0.241440i	$0.922380\pi$
$860$	0	0
$861$	6.86753e12	0.425879
$862$	0	0
$863$	5.56922e12	0.341780	0.170890	−	0.985290i	$-0.445336\pi$
$863$	5.56922e12	0.341780	0.170890	+	0.985290i	$0.445336\pi$
$864$	0	0
$865$	1.72189e13	1.04576
$866$	0	0
$867$	6.59289e11	0.0396269
$868$	0	0
$869$	7.86317e12	0.467744
$870$	0	0
$871$	1.44884e13	0.852979
$872$	0	0
$873$	−1.70352e12	−0.0992617
$874$	0	0
$875$	6.81388e12	0.392969
$876$	0	0
$877$	−5.51008e12	−0.314528	−0.157264	−	0.987557i	$-0.550267\pi$
$877$	−5.51008e12	−0.314528	−0.157264	+	0.987557i	$0.550267\pi$
$878$	0	0
$879$	−1.41147e13	−0.797486
$880$	0	0
$881$	1.96782e13	1.10051	0.550256	−	0.834996i	$-0.314530\pi$
$881$	1.96782e13	1.10051	0.550256	+	0.834996i	$0.314530\pi$
$882$	0	0
$883$	−7.12786e12	−0.394581	−0.197290	−	0.980345i	$-0.563214\pi$
$883$	−7.12786e12	−0.394581	−0.197290	+	0.980345i	$0.563214\pi$
$884$	0	0
$885$	−1.18071e13	−0.646989
$886$	0	0
$887$	−2.04067e13	−1.10692	−0.553461	−	0.832875i	$-0.686693\pi$
$887$	−2.04067e13	−1.10692	−0.553461	+	0.832875i	$0.686693\pi$
$888$	0	0
$889$	−4.23563e12	−0.227437
$890$	0	0
$891$	7.01161e12	0.372707
$892$	0	0
$893$	−6.31484e11	−0.0332300
$894$	0	0
$895$	−1.58059e13	−0.823410
$896$	0	0
$897$	−4.73417e13	−2.44162
$898$	0	0
$899$	2.36555e13	1.20785
$900$	0	0
$901$	−2.37826e13	−1.20226
$902$	0	0
$903$	6.12774e12	0.306694
$904$	0	0
$905$	2.20069e13	1.09054
$906$	0	0
$907$	−5.97935e12	−0.293374	−0.146687	−	0.989183i	$-0.546861\pi$
$907$	−5.97935e12	−0.293374	−0.146687	+	0.989183i	$0.546861\pi$
$908$	0	0
$909$	5.36178e12	0.260478
$910$	0	0
$911$	−1.21617e13	−0.585006	−0.292503	−	0.956265i	$-0.594488\pi$
$911$	−1.21617e13	−0.585006	−0.292503	+	0.956265i	$0.594488\pi$
$912$	0	0
$913$	−4.03244e12	−0.192065
$914$	0	0
$915$	1.61137e13	0.759977
$916$	0	0
$917$	1.50396e13	0.702386
$918$	0	0
$919$	1.76206e13	0.814895	0.407448	−	0.913229i	$-0.366419\pi$
$919$	1.76206e13	0.814895	0.407448	+	0.913229i	$0.366419\pi$
$920$	0	0
$921$	3.87414e12	0.177422
$922$	0	0
$923$	2.98669e13	1.35451
$924$	0	0
$925$	−1.95515e12	−0.0878096
$926$	0	0
$927$	1.25155e12	0.0556658
$928$	0	0
$929$	1.42719e13	0.628651	0.314326	−	0.949315i	$-0.398222\pi$
$929$	1.42719e13	0.628651	0.314326	+	0.949315i	$0.398222\pi$
$930$	0	0
$931$	6.38727e12	0.278639
$932$	0	0
$933$	2.86644e13	1.23844
$934$	0	0
$935$	1.04524e13	0.447265
$936$	0	0
$937$	−1.24922e13	−0.529432	−0.264716	−	0.964326i	$-0.585278\pi$
$937$	−1.24922e13	−0.529432	−0.264716	+	0.964326i	$0.585278\pi$
$938$	0	0
$939$	−2.79975e13	−1.17523
$940$	0	0
$941$	2.56181e13	1.06511	0.532553	−	0.846396i	$-0.321233\pi$
$941$	2.56181e13	1.06511	0.532553	+	0.846396i	$0.321233\pi$
$942$	0	0
$943$	−4.43243e13	−1.82532
$944$	0	0
$945$	9.40322e12	0.383560
$946$	0	0
$947$	−2.53905e13	−1.02588	−0.512939	−	0.858425i	$-0.671443\pi$
$947$	−2.53905e13	−1.02588	−0.512939	+	0.858425i	$0.671443\pi$
$948$	0	0
$949$	3.12758e13	1.25173
$950$	0	0
$951$	9.60287e11	0.0380705
$952$	0	0
$953$	3.84720e11	0.0151087	0.00755435	−	0.999971i	$-0.497595\pi$
$953$	3.84720e11	0.0151087	0.00755435	+	0.999971i	$0.497595\pi$
$954$	0	0
$955$	−1.46850e13	−0.571292
$956$	0	0
$957$	−9.11440e12	−0.351257
$958$	0	0
$959$	1.87654e12	0.0716431
$960$	0	0
$961$	2.87632e13	1.08788
$962$	0	0
$963$	1.85722e12	0.0695897
$964$	0	0
$965$	−3.81962e13	−1.41791
$966$	0	0
$967$	4.35217e13	1.60062	0.800308	−	0.599589i	$-0.204669\pi$
$967$	4.35217e13	1.60062	0.800308	+	0.599589i	$0.204669\pi$
$968$	0	0
$969$	5.00585e13	1.82398
$970$	0	0
$971$	2.35851e13	0.851435	0.425717	−	0.904856i	$-0.360022\pi$
$971$	2.35851e13	0.851435	0.425717	+	0.904856i	$0.360022\pi$
$972$	0	0
$973$	−3.72475e12	−0.133226
$974$	0	0
$975$	−4.16925e12	−0.147753
$976$	0	0
$977$	7.85117e12	0.275682	0.137841	−	0.990454i	$-0.455984\pi$
$977$	7.85117e12	0.275682	0.137841	+	0.990454i	$0.455984\pi$
$978$	0	0
$979$	1.98426e13	0.690360
$980$	0	0
$981$	5.68769e12	0.196077
$982$	0	0
$983$	3.32046e12	0.113425	0.0567123	−	0.998391i	$-0.481938\pi$
$983$	3.32046e12	0.113425	0.0567123	+	0.998391i	$0.481938\pi$
$984$	0	0
$985$	3.20293e13	1.08414
$986$	0	0
$987$	−1.75772e11	−0.00589552
$988$	0	0
$989$	−3.95496e13	−1.31449
$990$	0	0
$991$	4.06670e13	1.33940	0.669701	−	0.742631i	$-0.266422\pi$
$991$	4.06670e13	1.33940	0.669701	+	0.742631i	$0.266422\pi$
$992$	0	0
$993$	4.04773e13	1.32111
$994$	0	0
$995$	4.81280e12	0.155666
$996$	0	0
$997$	3.65436e13	1.17134	0.585670	−	0.810550i	$-0.300832\pi$
$997$	3.65436e13	1.17134	0.585670	+	0.810550i	$0.300832\pi$
$998$	0	0
$999$	−3.27601e13	−1.04064

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	56.10.a.c.1.3	✓	4
4.3	odd	2		112.10.a.k.1.2		4
7.6	odd	2		392.10.a.f.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.10.a.c.1.3	✓	4	1.1	even	1	trivial
112.10.a.k.1.2		4	4.3	odd	2
392.10.a.f.1.2		4	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	56.a (trivial)

\(f(q)\)	\(=\)	\(q+128.448 q^{3} -1333.35 q^{5} +2401.00 q^{7} -3184.23 q^{9} -22287.0 q^{11} +185165. q^{13} -171266. q^{15} +351739. q^{17} +1.10798e6 q^{19} +308403. q^{21} -1.99049e6 q^{23} -175297. q^{25} -2.93724e6 q^{27} +3.18384e6 q^{29} +7.42986e6 q^{31} -2.86270e6 q^{33} -3.20138e6 q^{35} +1.11534e7 q^{37} +2.37840e7 q^{39} +2.22681e7 q^{41} +1.98693e7 q^{43} +4.24571e6 q^{45} -569943. q^{47} +5.76480e6 q^{49} +4.51801e7 q^{51} -6.76141e7 q^{53} +2.97164e7 q^{55} +1.42317e8 q^{57} +6.89400e7 q^{59} -9.40859e7 q^{61} -7.64534e6 q^{63} -2.46890e8 q^{65} +7.82459e7 q^{67} -2.55673e8 q^{69} +1.61299e8 q^{71} +1.68908e8 q^{73} -2.25164e7 q^{75} -5.35110e7 q^{77} -3.52815e8 q^{79} -3.14606e8 q^{81} +1.80933e8 q^{83} -4.68993e8 q^{85} +4.08957e8 q^{87} -8.90321e8 q^{89} +4.44581e8 q^{91} +9.54347e8 q^{93} -1.47732e9 q^{95} +5.34984e8 q^{97} +7.09669e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 70 q^{3} + 1022 q^{5} + 9604 q^{7} + 12176 q^{9} - 31476 q^{11} + 11466 q^{13} + 98504 q^{15} + 397012 q^{17} + 246610 q^{19} + 168070 q^{21} + 1976552 q^{23} + 1086512 q^{25} + 3139108 q^{27} + 3718940 q^{29}+ \cdots + 1279453212 q^{99}+O(q^{100}) \) 4 * q + 70 * q^3 + 1022 * q^5 + 9604 * q^7 + 12176 * q^9 - 31476 * q^11 + 11466 * q^13 + 98504 * q^15 + 397012 * q^17 + 246610 * q^19 + 168070 * q^21 + 1976552 * q^23 + 1086512 * q^25 + 3139108 * q^27 + 3718940 * q^29 + 11118436 * q^31 + 16891056 * q^33 + 2453822 * q^35 + 30221340 * q^37 + 37296384 * q^39 + 27162604 * q^41 + 64096964 * q^43 + 35015302 * q^45 + 104378204 * q^47 + 23059204 * q^49 + 111871540 * q^51 + 20587304 * q^53 + 251355720 * q^55 - 49920188 * q^57 + 130065474 * q^59 - 134490482 * q^61 + 29234576 * q^63 - 227739180 * q^65 + 104031144 * q^67 - 408620800 * q^69 - 71827896 * q^71 - 586388432 * q^73 + 134360114 * q^75 - 75573876 * q^77 + 164282440 * q^79 - 1324238632 * q^81 + 130538394 * q^83 - 1710305284 * q^85 + 142567628 * q^87 - 2036960408 * q^89 + 27529866 * q^91 - 2396678120 * q^93 - 498649144 * q^95 - 2049365612 * q^97 + 1279453212 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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