LMFDB - Embedded newform 360.8.f.b.289.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [360,8,Mod(289,360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("360.289"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 360 = 2^{3} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	360.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$112.458609174$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 196x^{6} + 7674x^{4} + 75204x^{2} + 18225 $ x^8 + 196x^6 + 7674x^4 + 75204*x^2 + 18225
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{38}\cdot 3^{2}\cdot 5^{3} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.2
Root			$12.1404i$ of defining polynomial
Character	$\chi$	$=$	360.289
Dual form			360.8.f.b.289.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-61.1163 + 272.745i) q^{5} -1567.21i q^{7} -7334.60 q^{11} +6045.58i q^{13} +797.737i q^{17} -629.101 q^{19} +102919. i q^{23} +(-70654.6 - 33338.3i) q^{25} -134474. q^{29} +198533. q^{31} +(427448. + 95781.9i) q^{35} -111197. i q^{37} -10889.0 q^{41} -561169. i q^{43} +12720.3i q^{47} -1.63260e6 q^{49} -919664. i q^{53} +(448264. - 2.00048e6i) q^{55} -403069. q^{59} +230286. q^{61} +(-1.64890e6 - 369483. i) q^{65} +987745. i q^{67} +3.63077e6 q^{71} +376686. i q^{73} +1.14948e7i q^{77} +4.84319e6 q^{79} -3.05462e6i q^{83} +(-217579. - 48754.8i) q^{85} +4.54896e6 q^{89} +9.47467e6 q^{91} +(38448.3 - 171584. i) q^{95} -1.55682e6i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 744 q^{5} - 9120 q^{11} - 61600 q^{19} - 254424 q^{25} - 35760 q^{29} + 519168 q^{31} + 162144 q^{35} + 852336 q^{41} - 4310376 q^{49} + 829664 q^{55} - 1202976 q^{59} - 1290128 q^{61} - 1750848 q^{65}+ \cdots + 8588256 q^{95}+O(q^{100}) $ 8 * q + 744 * q^5 - 9120 * q^11 - 61600 * q^19 - 254424 * q^25 - 35760 * q^29 + 519168 * q^31 + 162144 * q^35 + 852336 * q^41 - 4310376 * q^49 + 829664 * q^55 - 1202976 * q^59 - 1290128 * q^61 - 1750848 * q^65 + 12982848 * q^71 - 14162816 * q^79 + 6461952 * q^85 - 16890960 * q^89 + 40041664 * q^91 + 8588256 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/360\mathbb{Z}\right)^\times$.

$n$	$181$	$217$	$271$	$281$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−61.1163	+	272.745i	−0.218656	+	0.975802i
$5$	−61.1163	+	272.745i	−0.218656	+	0.975802i
$6$		0			0
$7$		−	1567.21i		−	1.72696i	−0.504380	−	0.863482i	$-0.668279\pi$
$7$		−	1567.21i		−	1.72696i	0.504380	−	0.863482i	$-0.331721\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−7334.60			−1.66151			−0.830754	−	0.556640i	$-0.812090\pi$
$11$	−7334.60			−1.66151			−0.830754	+	0.556640i	$0.812090\pi$
$12$		0			0
$13$			6045.58i			0.763196i	0.924328	+	0.381598i	$0.124626\pi$
$13$			6045.58i			0.763196i	−0.924328	+	0.381598i	$0.875374\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			797.737i			0.0393812i	0.999806	+	0.0196906i	$0.00626811\pi$
$17$			797.737i			0.0393812i	−0.999806	+	0.0196906i	$0.993732\pi$
$18$		0			0
$19$	−629.101			−0.0210418			−0.0105209	−	0.999945i	$-0.503349\pi$
$19$	−629.101			−0.0210418			−0.0105209	+	0.999945i	$0.503349\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			102919.i			1.76380i	0.471434	+	0.881902i	$0.343737\pi$
$23$			102919.i			1.76380i	−0.471434	+	0.881902i	$0.656263\pi$
$24$		0			0
$25$	−70654.6	−	33338.3i	−0.904379	−	0.426731i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−134474.			−1.02387			−0.511937	−	0.859023i	$-0.671072\pi$
$29$	−134474.			−1.02387			−0.511937	+	0.859023i	$0.671072\pi$
$30$		0			0
$31$	198533.			1.19693			0.598463	−	0.801150i	$-0.295778\pi$
$31$	198533.			1.19693			0.598463	+	0.801150i	$0.295778\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	427448.	+	95781.9i	1.68517	+	0.377612i
$36$		0			0
$37$		−	111197.i		−	0.360901i	−0.983584	−	0.180450i	$-0.942244\pi$
$37$		−	111197.i		−	0.360901i	0.983584	−	0.180450i	$-0.0577555\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−10889.0			−0.0246743			−0.0123372	−	0.999924i	$-0.503927\pi$
$41$	−10889.0			−0.0246743			−0.0123372	+	0.999924i	$0.503927\pi$
$42$		0			0
$43$		−	561169.i		−	1.07635i	−0.842833	−	0.538176i	$-0.819114\pi$
$43$		−	561169.i		−	1.07635i	0.842833	−	0.538176i	$-0.180886\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			12720.3i			0.0178712i	0.999960	+	0.00893559i	$0.00284432\pi$
$47$			12720.3i			0.0178712i	−0.999960	+	0.00893559i	$0.997156\pi$
$48$		0			0
$49$	−1.63260e6			−1.98240
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	919664.i		−	0.848523i	−0.905540	−	0.424261i	$-0.860534\pi$
$53$		−	919664.i		−	0.848523i	0.905540	−	0.424261i	$-0.139466\pi$
$54$		0			0
$55$	448264.	−	2.00048e6i	0.363299	−	1.62130i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−403069.			−0.255504			−0.127752	−	0.991806i	$-0.540776\pi$
$59$	−403069.			−0.255504			−0.127752	+	0.991806i	$0.540776\pi$
$60$		0			0
$61$	230286.			0.129901			0.0649506	−	0.997888i	$-0.479311\pi$
$61$	230286.			0.129901			0.0649506	+	0.997888i	$0.479311\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.64890e6	−	369483.i	−0.744728	−	0.166878i
$66$		0			0
$67$			987745.i			0.401221i	0.979671	+	0.200610i	$0.0642925\pi$
$67$			987745.i			0.401221i	−0.979671	+	0.200610i	$0.935708\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.63077e6			1.20391			0.601956	−	0.798529i	$-0.294388\pi$
$71$	3.63077e6			1.20391			0.601956	+	0.798529i	$0.294388\pi$
$72$		0			0
$73$			376686.i			0.113331i	0.998393	+	0.0566657i	$0.0180469\pi$
$73$			376686.i			0.113331i	−0.998393	+	0.0566657i	$0.981953\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			1.14948e7i			2.86936i
$78$		0			0
$79$	4.84319e6			1.10519			0.552595	−	0.833450i	$-0.313638\pi$
$79$	4.84319e6			1.10519			0.552595	+	0.833450i	$0.313638\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	3.05462e6i		−	0.586386i	−0.956053	−	0.293193i	$-0.905282\pi$
$83$		−	3.05462e6i		−	0.586386i	0.956053	−	0.293193i	$-0.0947179\pi$
$84$		0			0
$85$	−217579.	−	48754.8i	−0.0384282	−	0.00861094i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.54896e6			0.683986			0.341993	−	0.939703i	$-0.388898\pi$
$89$	4.54896e6			0.683986			0.341993	+	0.939703i	$0.388898\pi$
$90$		0			0
$91$	9.47467e6			1.31801
$92$		0			0
$93$		0			0
$94$		0			0
$95$	38448.3	−	171584.i	0.00460092	−	0.0205326i
$96$		0			0
$97$		−	1.55682e6i		−	0.173195i	−0.996243	−	0.0865977i	$-0.972401\pi$
$97$		−	1.55682e6i		−	0.173195i	0.996243	−	0.0865977i	$-0.0275995\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.42005e7			1.37145			0.685723	−	0.727862i	$-0.259486\pi$
$101$	1.42005e7			1.37145			0.685723	+	0.727862i	$0.259486\pi$
$102$		0			0
$103$			1.51725e7i			1.36813i	0.729421	+	0.684065i	$0.239789\pi$
$103$			1.51725e7i			1.36813i	−0.729421	+	0.684065i	$0.760211\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			8.20296e6i			0.647332i	0.946171	+	0.323666i	$0.104915\pi$
$107$			8.20296e6i			0.647332i	−0.946171	+	0.323666i	$0.895085\pi$
$108$		0			0
$109$	−1.66069e7			−1.22828			−0.614139	−	0.789198i	$-0.710497\pi$
$109$	−1.66069e7			−1.22828			−0.614139	+	0.789198i	$0.710497\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1.90200e7i		−	1.24004i	−0.784587	−	0.620019i	$-0.787125\pi$
$113$		−	1.90200e7i		−	1.24004i	0.784587	−	0.620019i	$-0.212875\pi$
$114$		0			0
$115$	−2.80708e7	−	6.29006e6i	−1.72112	−	0.385667i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.25022e6			0.0680099
$120$		0			0
$121$	3.43092e7			1.76061
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.34110e7	−	1.72332e7i	0.614153	−	0.789187i
$126$		0			0
$127$		−	1.50382e7i		−	0.651451i	−0.945464	−	0.325725i	$-0.894392\pi$
$127$		−	1.50382e7i		−	0.651451i	0.945464	−	0.325725i	$-0.105608\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.27882e6			−0.205158			−0.102579	−	0.994725i	$-0.532709\pi$
$131$	−5.27882e6			−0.205158			−0.102579	+	0.994725i	$0.532709\pi$
$132$		0			0
$133$			985932.i			0.0363384i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	5.33664e7i		−	1.77315i	−0.462583	−	0.886576i	$-0.653077\pi$
$137$		−	5.33664e7i		−	1.77315i	0.462583	−	0.886576i	$-0.346923\pi$
$138$		0			0
$139$	2.80350e7			0.885418			0.442709	−	0.896665i	$-0.354017\pi$
$139$	2.80350e7			0.885418			0.442709	+	0.896665i	$0.354017\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	4.43419e7i		−	1.26806i
$144$		0			0
$145$	8.21858e6	−	3.66772e7i	0.223877	−	0.999098i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.41075e6			−0.208297			−0.104148	−	0.994562i	$-0.533212\pi$
$149$	−8.41075e6			−0.208297			−0.104148	+	0.994562i	$0.533212\pi$
$150$		0			0
$151$	7.66401e7			1.81149			0.905747	−	0.423820i	$-0.139311\pi$
$151$	7.66401e7			1.81149			0.905747	+	0.423820i	$0.139311\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.21336e7	+	5.41489e7i	−0.261716	+	1.16796i
$156$		0			0
$157$			4.32322e7i			0.891577i	0.895138	+	0.445788i	$0.147077\pi$
$157$			4.32322e7i			0.891577i	−0.895138	+	0.445788i	$0.852923\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.61296e8			3.04602
$162$		0			0
$163$		−	2.78860e7i		−	0.504347i	−0.967682	−	0.252173i	$-0.918855\pi$
$163$		−	2.78860e7i		−	0.504347i	0.967682	−	0.252173i	$-0.0811454\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4.92856e7i		−	0.818865i	−0.912340	−	0.409433i	$-0.865727\pi$
$167$		−	4.92856e7i		−	0.818865i	0.912340	−	0.409433i	$-0.134273\pi$
$168$		0			0
$169$	2.61995e7			0.417532
$170$		0			0
$171$		0			0
$172$		0			0
$173$			6.89940e7i			1.01309i	0.862212	+	0.506547i	$0.169078\pi$
$173$			6.89940e7i			1.01309i	−0.862212	+	0.506547i	$0.830922\pi$
$174$		0			0
$175$	−5.22481e7	+	1.10730e8i	−0.736948	+	1.56183i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.96347e7			1.29845			0.649225	−	0.760596i	$-0.275093\pi$
$179$	9.96347e7			1.29845			0.649225	+	0.760596i	$0.275093\pi$
$180$		0			0
$181$	−6.79192e7			−0.851369			−0.425684	−	0.904872i	$-0.639967\pi$
$181$	−6.79192e7			−0.851369			−0.425684	+	0.904872i	$0.639967\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.03285e7	+	6.79596e6i	0.352168	+	0.0789133i
$186$		0			0
$187$		−	5.85109e6i		−	0.0654321i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.12927e7			0.948024			0.474012	−	0.880518i	$-0.342805\pi$
$191$	9.12927e7			0.948024			0.474012	+	0.880518i	$0.342805\pi$
$192$		0			0
$193$			3.18801e7i			0.319204i	0.987181	+	0.159602i	$0.0510211\pi$
$193$			3.18801e7i			0.319204i	−0.987181	+	0.159602i	$0.948979\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4.50081e7i			0.419429i	0.977763	+	0.209715i	$0.0672535\pi$
$197$			4.50081e7i			0.419429i	−0.977763	+	0.209715i	$0.932747\pi$
$198$		0			0
$199$	1.79160e8			1.61159			0.805797	−	0.592192i	$-0.201737\pi$
$199$	1.79160e8			1.61159			0.805797	+	0.592192i	$0.201737\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			2.10749e8i			1.76819i
$204$		0			0
$205$	665497.	−	2.96993e6i	0.00539520	−	0.0240773i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.61421e6			0.0349611
$210$		0			0
$211$	8.85380e7			0.648845			0.324422	−	0.945912i	$-0.394830\pi$
$211$	8.85380e7			0.648845			0.324422	+	0.945912i	$0.394830\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.53056e8	+	3.42966e7i	1.05031	+	0.235351i
$216$		0			0
$217$		−	3.11143e8i		−	2.06705i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.82278e6			−0.0300556
$222$		0			0
$223$		−	3.22465e6i		−	0.0194722i	−0.999953	−	0.00973612i	$-0.996901\pi$
$223$		−	3.22465e6i		−	0.0194722i	0.999953	−	0.00973612i	$-0.00309915\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	2.90542e8i		−	1.64861i	−0.566143	−	0.824307i	$-0.691565\pi$
$227$		−	2.90542e8i		−	1.64861i	0.566143	−	0.824307i	$-0.308435\pi$
$228$		0			0
$229$	8.89116e7			0.489254			0.244627	−	0.969617i	$-0.421335\pi$
$229$	8.89116e7			0.489254			0.244627	+	0.969617i	$0.421335\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	1.87337e8i		−	0.970235i	−0.874449	−	0.485117i	$-0.838777\pi$
$233$		−	1.87337e8i		−	0.970235i	0.874449	−	0.485117i	$-0.161223\pi$
$234$		0			0
$235$	−3.46938e6	−	777415.i	−0.0174387	−	0.00390765i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.14791e8			1.01771			0.508854	−	0.860853i	$-0.330069\pi$
$239$	2.14791e8			1.01771			0.508854	+	0.860853i	$0.330069\pi$
$240$		0			0
$241$	1.00588e8			0.462898			0.231449	−	0.972847i	$-0.425653\pi$
$241$	1.00588e8			0.462898			0.231449	+	0.972847i	$0.425653\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.97782e7	−	4.45282e8i	0.433465	−	1.93443i
$246$		0			0
$247$		−	3.80328e6i		−	0.0160590i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.40407e8			−0.560443			−0.280222	−	0.959935i	$-0.590408\pi$
$251$	−1.40407e8			−0.560443			−0.280222	+	0.959935i	$0.590408\pi$
$252$		0			0
$253$		−	7.54873e8i		−	2.93057i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			4.07786e8i			1.49853i	0.662268	+	0.749267i	$0.269594\pi$
$257$			4.07786e8i			1.49853i	−0.662268	+	0.749267i	$0.730406\pi$
$258$		0			0
$259$	−1.74269e8			−0.623263
$260$		0			0
$261$		0			0
$262$		0			0
$263$			5.38653e7i			0.182585i	0.995824	+	0.0912923i	$0.0290997\pi$
$263$			5.38653e7i			0.182585i	−0.995824	+	0.0912923i	$0.970900\pi$
$264$		0			0
$265$	2.50834e8	+	5.62065e7i	0.827990	+	0.185535i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.40416e8			−1.37953			−0.689764	−	0.724034i	$-0.742286\pi$
$269$	−4.40416e8			−1.37953			−0.689764	+	0.724034i	$0.742286\pi$
$270$		0			0
$271$	−5.80619e8			−1.77214			−0.886072	−	0.463548i	$-0.846576\pi$
$271$	−5.80619e8			−1.77214			−0.886072	+	0.463548i	$0.846576\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.18223e8	+	2.44523e8i	1.50263	+	0.709016i
$276$		0			0
$277$			5.99169e8i			1.69383i	0.531728	+	0.846915i	$0.321543\pi$
$277$			5.99169e8i			1.69383i	−0.531728	+	0.846915i	$0.678457\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.15072e8			0.578246			0.289123	−	0.957292i	$-0.406636\pi$
$281$	2.15072e8			0.578246			0.289123	+	0.957292i	$0.406636\pi$
$282$		0			0
$283$			6.33158e8i			1.66058i	0.557333	+	0.830289i	$0.311825\pi$
$283$			6.33158e8i			1.66058i	−0.557333	+	0.830289i	$0.688175\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			1.70654e7i			0.0426117i
$288$		0			0
$289$	4.09702e8			0.998449
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	1.98465e8i		−	0.460942i	−0.973079	−	0.230471i	$-0.925973\pi$
$293$		−	1.98465e8i		−	0.460942i	0.973079	−	0.230471i	$-0.0740267\pi$
$294$		0			0
$295$	2.46341e7	−	1.09935e8i	0.0558676	−	0.249321i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.22208e8			−1.34613
$300$		0			0
$301$	−8.79468e8			−1.85882
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.40742e7	+	6.28094e7i	−0.0284037	+	0.126758i
$306$		0			0
$307$			3.12274e8i			0.615958i	0.951393	+	0.307979i	$0.0996526\pi$
$307$			3.12274e8i			0.615958i	−0.951393	+	0.307979i	$0.900347\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.87905e8			1.10827			0.554136	−	0.832426i	$-0.313049\pi$
$311$	5.87905e8			1.10827			0.554136	+	0.832426i	$0.313049\pi$
$312$		0			0
$313$			2.65789e8i			0.489928i	0.969532	+	0.244964i	$0.0787762\pi$
$313$			2.65789e8i			0.489928i	−0.969532	+	0.244964i	$0.921224\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			5.69257e7i			0.100369i	0.998740	+	0.0501846i	$0.0159810\pi$
$317$			5.69257e7i			0.100369i	−0.998740	+	0.0501846i	$0.984019\pi$
$318$		0			0
$319$	9.86316e8			1.70117
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	501857.i		−	0.000828651i
$324$		0			0
$325$	2.01549e8	−	4.27148e8i	0.325679	−	0.690218i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.99353e7			0.0308629
$330$		0			0
$331$	2.23025e8			0.338030			0.169015	−	0.985614i	$-0.445941\pi$
$331$	2.23025e8			0.338030			0.169015	+	0.985614i	$0.445941\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.69403e8	−	6.03674e7i	−0.391512	−	0.0877294i
$336$		0			0
$337$		−	3.52001e8i		−	0.501002i	−0.968116	−	0.250501i	$-0.919405\pi$
$337$		−	3.52001e8i		−	0.501002i	0.968116	−	0.250501i	$-0.0805953\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.45616e9			−1.98870
$342$		0			0
$343$			1.26795e9i			1.69658i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			2.38775e8i			0.306786i	0.988165	+	0.153393i	$0.0490200\pi$
$347$			2.38775e8i			0.306786i	−0.988165	+	0.153393i	$0.950980\pi$
$348$		0			0
$349$	8.46269e8			1.06566			0.532831	−	0.846222i	$-0.321128\pi$
$349$	8.46269e8			1.06566			0.532831	+	0.846222i	$0.321128\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	8.30033e8i		−	1.00435i	−0.864767	−	0.502174i	$-0.832534\pi$
$353$		−	8.30033e8i		−	1.00435i	0.864767	−	0.502174i	$-0.167466\pi$
$354$		0			0
$355$	−2.21900e8	+	9.90275e8i	−0.263243	+	1.17478i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−5.83160e8			−0.665207			−0.332604	−	0.943067i	$-0.607927\pi$
$359$	−5.83160e8			−0.665207			−0.332604	+	0.943067i	$0.607927\pi$
$360$		0			0
$361$	−8.93476e8			−0.999557
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.02739e8	−	2.30217e7i	−0.110589	−	0.0247806i
$366$		0			0
$367$		−	5.96738e8i		−	0.630163i	−0.949065	−	0.315081i	$-0.897968\pi$
$367$		−	5.96738e8i		−	0.630163i	0.949065	−	0.315081i	$-0.102032\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.44130e9			−1.46537
$372$		0			0
$373$		−	1.15850e9i		−	1.15588i	−0.816078	−	0.577942i	$-0.803856\pi$
$373$		−	1.15850e9i		−	1.15588i	0.816078	−	0.577942i	$-0.196144\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	8.12975e8i		−	0.781416i
$378$		0			0
$379$	−7.41506e8			−0.699644			−0.349822	−	0.936816i	$-0.613758\pi$
$379$	−7.41506e8			−0.699644			−0.349822	+	0.936816i	$0.613758\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			4.06298e8i			0.369529i	0.982783	+	0.184765i	$0.0591523\pi$
$383$			4.06298e8i			0.369529i	−0.982783	+	0.184765i	$0.940848\pi$
$384$		0			0
$385$	−3.13516e9	−	7.02522e8i	−2.79993	−	0.627405i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.78517e8			0.153764			0.0768822	−	0.997040i	$-0.475503\pi$
$389$	1.78517e8			0.153764			0.0768822	+	0.997040i	$0.475503\pi$
$390$		0			0
$391$	−8.21027e7			−0.0694606
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.95998e8	+	1.32096e9i	−0.241657	+	1.07845i
$396$		0			0
$397$		−	1.79226e9i		−	1.43758i	−0.695225	−	0.718792i	$-0.744695\pi$
$397$		−	1.79226e9i		−	1.43758i	0.695225	−	0.718792i	$-0.255305\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.77831e8			0.137722			0.0688610	−	0.997626i	$-0.478064\pi$
$401$	1.77831e8			0.137722			0.0688610	+	0.997626i	$0.478064\pi$
$402$		0			0
$403$			1.20025e9i			0.913489i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			8.15587e8i			0.599639i
$408$		0			0
$409$	−1.79387e9			−1.29646			−0.648231	−	0.761443i	$-0.724491\pi$
$409$	−1.79387e9			−1.29646			−0.648231	+	0.761443i	$0.724491\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			6.31693e8i			0.441246i
$414$		0			0
$415$	8.33132e8	+	1.86687e8i	0.572197	+	0.128217i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.19067e9			−1.45488			−0.727442	−	0.686169i	$-0.759291\pi$
$419$	−2.19067e9			−1.45488			−0.727442	+	0.686169i	$0.759291\pi$
$420$		0			0
$421$	1.65657e9			1.08199			0.540994	−	0.841027i	$-0.318048\pi$
$421$	1.65657e9			1.08199			0.540994	+	0.841027i	$0.318048\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.65952e7	−	5.63638e7i	0.0168052	−	0.0356155i
$426$		0			0
$427$		−	3.60906e8i		−	0.224335i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.03672e9			0.623722			0.311861	−	0.950128i	$-0.399048\pi$
$431$	1.03672e9			0.623722			0.311861	+	0.950128i	$0.399048\pi$
$432$		0			0
$433$			3.80495e8i			0.225238i	0.993638	+	0.112619i	$0.0359240\pi$
$433$			3.80495e8i			0.225238i	−0.993638	+	0.112619i	$0.964076\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	6.47468e7i		−	0.0371136i
$438$		0			0
$439$	−1.14225e9			−0.644371			−0.322185	−	0.946677i	$-0.604417\pi$
$439$	−1.14225e9			−0.644371			−0.322185	+	0.946677i	$0.604417\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			5.15916e8i			0.281946i	0.990013	+	0.140973i	$0.0450231\pi$
$443$			5.15916e8i			0.281946i	−0.990013	+	0.140973i	$0.954977\pi$
$444$		0			0
$445$	−2.78016e8	+	1.24071e9i	−0.149558	+	0.667434i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.43839e9			−1.27128			−0.635639	−	0.771987i	$-0.719263\pi$
$449$	−2.43839e9			−1.27128			−0.635639	+	0.771987i	$0.719263\pi$
$450$		0			0
$451$	7.98667e7			0.0409966
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5.79057e8	+	2.58417e9i	−0.288192	+	1.28612i
$456$		0			0
$457$		−	2.74663e9i		−	1.34615i	−0.739575	−	0.673074i	$-0.764974\pi$
$457$		−	2.74663e9i		−	1.34615i	0.739575	−	0.673074i	$-0.235026\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.24337e9			0.591082			0.295541	−	0.955330i	$-0.404500\pi$
$461$	1.24337e9			0.591082			0.295541	+	0.955330i	$0.404500\pi$
$462$		0			0
$463$		−	1.94862e9i		−	0.912416i	−0.889873	−	0.456208i	$-0.849207\pi$
$463$		−	1.94862e9i		−	0.912416i	0.889873	−	0.456208i	$-0.150793\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	5.39865e8i		−	0.245288i	−0.992451	−	0.122644i	$-0.960863\pi$
$467$		−	5.39865e8i		−	0.245288i	0.992451	−	0.122644i	$-0.0391373\pi$
$468$		0			0
$469$	1.54800e9			0.692894
$470$		0			0
$471$		0			0
$472$		0			0
$473$			4.11595e9i			1.78837i
$474$		0			0
$475$	4.44489e7	+	2.09732e7i	0.0190298	+	0.00897918i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.05001e9			−1.26802			−0.634011	−	0.773324i	$-0.718592\pi$
$479$	−3.05001e9			−1.26802			−0.634011	+	0.773324i	$0.718592\pi$
$480$		0			0
$481$	6.72251e8			0.275438
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.24614e8	+	9.51469e7i	0.169004	+	0.0378703i
$486$		0			0
$487$			3.23990e9i			1.27110i	0.772060	+	0.635550i	$0.219227\pi$
$487$			3.23990e9i			1.27110i	−0.772060	+	0.635550i	$0.780773\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.52405e9			0.581052			0.290526	−	0.956867i	$-0.406170\pi$
$491$	1.52405e9			0.581052			0.290526	+	0.956867i	$0.406170\pi$
$492$		0			0
$493$		−	1.07275e8i		−	0.0403214i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	5.69018e9i		−	2.07911i
$498$		0			0
$499$	2.42876e8			0.0875050			0.0437525	−	0.999042i	$-0.486069\pi$
$499$	2.42876e8			0.0875050			0.0437525	+	0.999042i	$0.486069\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	1.44474e9i		−	0.506178i	−0.967443	−	0.253089i	$-0.918553\pi$
$503$		−	1.44474e9i		−	0.506178i	0.967443	−	0.253089i	$-0.0814465\pi$
$504$		0			0
$505$	−8.67882e8	+	3.87311e9i	−0.299876	+	1.33826i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.96881e9			−0.661747			−0.330873	−	0.943675i	$-0.607343\pi$
$509$	−1.96881e9			−0.661747			−0.330873	+	0.943675i	$0.607343\pi$
$510$		0			0
$511$	5.90346e8			0.195719
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.13822e9	−	9.27288e8i	−1.33502	−	0.299150i
$516$		0			0
$517$		−	9.32980e7i		−	0.0296931i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.45071e9			−0.449416			−0.224708	−	0.974426i	$-0.572143\pi$
$521$	−1.45071e9			−0.449416			−0.224708	+	0.974426i	$0.572143\pi$
$522$		0			0
$523$			4.57337e9i			1.39791i	0.715164	+	0.698957i	$0.246352\pi$
$523$			4.57337e9i			1.39791i	−0.715164	+	0.698957i	$0.753648\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			1.58377e8i			0.0471364i
$528$		0			0
$529$	−7.18759e9			−2.11100
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	6.58305e7i		−	0.0188314i
$534$		0			0
$535$	−2.23731e9	−	5.01334e8i	−0.631668	−	0.141543i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.19744e10			3.29378
$540$		0			0
$541$	2.84589e9			0.772731			0.386365	−	0.922346i	$-0.373730\pi$
$541$	2.84589e9			0.772731			0.386365	+	0.922346i	$0.373730\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.01495e9	−	4.52946e9i	0.268571	−	1.19856i
$546$		0			0
$547$		−	1.68110e9i		−	0.439175i	−0.975593	−	0.219587i	$-0.929529\pi$
$547$		−	1.68110e9i		−	0.439175i	0.975593	−	0.219587i	$-0.0704711\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8.45980e7			0.0215442
$552$		0			0
$553$		−	7.59028e9i		−	1.90862i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	1.60972e9i		−	0.394692i	−0.980334	−	0.197346i	$-0.936768\pi$
$557$		−	1.60972e9i		−	0.394692i	0.980334	−	0.197346i	$-0.0632322\pi$
$558$		0			0
$559$	3.39259e9			0.821467
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	2.60363e9i		−	0.614894i	−0.951565	−	0.307447i	$-0.900525\pi$
$563$		−	2.60363e9i		−	0.614894i	0.951565	−	0.307447i	$-0.0994748\pi$
$564$		0			0
$565$	5.18760e9	+	1.16243e9i	1.21003	+	0.271142i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.44326e9			1.46627			0.733133	−	0.680085i	$-0.238057\pi$
$569$	6.44326e9			1.46627			0.733133	+	0.680085i	$0.238057\pi$
$570$		0			0
$571$	4.63353e9			1.04156			0.520781	−	0.853690i	$-0.325641\pi$
$571$	4.63353e9			1.04156			0.520781	+	0.853690i	$0.325641\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.43116e9	−	7.27173e9i	0.752669	−	1.59515i
$576$		0			0
$577$			1.84995e9i			0.400908i	0.979703	+	0.200454i	$0.0642418\pi$
$577$			1.84995e9i			0.400908i	−0.979703	+	0.200454i	$0.935758\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.78722e9			−1.01267
$582$		0			0
$583$			6.74537e9i			1.40983i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			8.24371e9i			1.68225i	0.540844	+	0.841123i	$0.318105\pi$
$587$			8.24371e9i			1.68225i	−0.540844	+	0.841123i	$0.681895\pi$
$588$		0			0
$589$	−1.24897e8			−0.0251855
$590$		0			0
$591$		0			0
$592$		0			0
$593$			6.14399e9i			1.20993i	0.796253	+	0.604963i	$0.206812\pi$
$593$			6.14399e9i			1.20993i	−0.796253	+	0.604963i	$0.793188\pi$
$594$		0			0
$595$	−7.64088e7	+	3.40991e8i	−0.0148708	+	0.0663642i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.96742e9			−0.374028			−0.187014	−	0.982357i	$-0.559881\pi$
$599$	−1.96742e9			−0.374028			−0.187014	+	0.982357i	$0.559881\pi$
$600$		0			0
$601$	1.40986e9			0.264920			0.132460	−	0.991188i	$-0.457712\pi$
$601$	1.40986e9			0.264920			0.132460	+	0.991188i	$0.457712\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.09685e9	+	9.35767e9i	−0.384968	+	1.71800i
$606$		0			0
$607$		−	3.90414e9i		−	0.708541i	−0.935143	−	0.354270i	$-0.884729\pi$
$607$		−	3.90414e9i		−	0.708541i	0.935143	−	0.354270i	$-0.115271\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.69013e7			−0.0136392
$612$		0			0
$613$			1.75471e9i			0.307677i	0.988096	+	0.153838i	$0.0491635\pi$
$613$			1.75471e9i			0.307677i	−0.988096	+	0.153838i	$0.950836\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.96725e9i		−	0.337180i	−0.985686	−	0.168590i	$-0.946079\pi$
$617$		−	1.96725e9i		−	0.337180i	0.985686	−	0.168590i	$-0.0539214\pi$
$618$		0			0
$619$	5.83043e9			0.988060			0.494030	−	0.869445i	$-0.335523\pi$
$619$	5.83043e9			0.988060			0.494030	+	0.869445i	$0.335523\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	7.12916e9i		−	1.18122i
$624$		0			0
$625$	3.88063e9	+	4.71101e9i	0.635802	+	0.771852i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	8.87062e7			0.0142127
$630$		0			0
$631$	−8.13487e9			−1.28899			−0.644493	−	0.764611i	$-0.722931\pi$
$631$	−8.13487e9			−1.28899			−0.644493	+	0.764611i	$0.722931\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.10158e9	+	9.19077e8i	0.635687	+	0.142444i
$636$		0			0
$637$		−	9.86998e9i		−	1.51296i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.22668e10			1.83961			0.919807	−	0.392371i	$-0.128345\pi$
$641$	1.22668e10			1.83961			0.919807	+	0.392371i	$0.128345\pi$
$642$		0			0
$643$		−	8.60241e9i		−	1.27609i	−0.769999	−	0.638045i	$-0.779743\pi$
$643$		−	8.60241e9i		−	1.27609i	0.769999	−	0.638045i	$-0.220257\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			5.28250e9i			0.766787i	0.923585	+	0.383393i	$0.125245\pi$
$647$			5.28250e9i			0.766787i	−0.923585	+	0.383393i	$0.874755\pi$
$648$		0			0
$649$	2.95635e9			0.424522
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	9.02641e9i		−	1.26858i	−0.773094	−	0.634291i	$-0.781292\pi$
$653$		−	9.02641e9i		−	1.26858i	0.773094	−	0.634291i	$-0.218708\pi$
$654$		0			0
$655$	3.22622e8	−	1.43977e9i	0.0448590	−	0.200193i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.61503e9			1.30873			0.654367	−	0.756177i	$-0.272935\pi$
$659$	9.61503e9			1.30873			0.654367	+	0.756177i	$0.272935\pi$
$660$		0			0
$661$	−1.20472e10			−1.62248			−0.811242	−	0.584710i	$-0.801208\pi$
$661$	−1.20472e10			−1.62248			−0.811242	+	0.584710i	$0.801208\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.68908e8	−	6.02565e7i	−0.0354591	−	0.00794563i
$666$		0			0
$667$		−	1.38400e10i		−	1.80591i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.68906e9			−0.215832
$672$		0			0
$673$			1.21089e10i			1.53127i	0.643278	+	0.765633i	$0.277574\pi$
$673$			1.21089e10i			1.53127i	−0.643278	+	0.765633i	$0.722426\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	8.88163e9i		−	1.10010i	−0.835132	−	0.550050i	$-0.814609\pi$
$677$		−	8.88163e9i		−	1.10010i	0.835132	−	0.550050i	$-0.185391\pi$
$678$		0			0
$679$	−2.43985e9			−0.299102
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1.11480e7i		−	0.00133883i	−1.00000		0.000669414i	$-0.999787\pi$
$683$		−	1.11480e7i		−	0.00133883i	1.00000		0.000669414i	$-0.000213081\pi$
$684$		0			0
$685$	1.45554e10	+	3.26156e9i	1.73024	+	0.387711i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	5.55990e9			0.647589
$690$		0			0
$691$	−1.27585e10			−1.47105			−0.735526	−	0.677497i	$-0.763065\pi$
$691$	−1.27585e10			−1.47105			−0.735526	+	0.677497i	$0.763065\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.71339e9	+	7.64640e9i	−0.193602	+	0.863993i
$696$		0			0
$697$		−	8.68658e6i		−	0.000971705i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−5.93252e8			−0.0650468			−0.0325234	−	0.999471i	$-0.510354\pi$
$701$	−5.93252e8			−0.0650468			−0.0325234	+	0.999471i	$0.510354\pi$
$702$		0			0
$703$			6.99543e7i			0.00759400i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	2.22551e10i		−	2.36844i
$708$		0			0
$709$	4.17030e9			0.439446			0.219723	−	0.975562i	$-0.429485\pi$
$709$	4.17030e9			0.439446			0.219723	+	0.975562i	$0.429485\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			2.04329e10i			2.11114i
$714$		0			0
$715$	1.20940e10	+	2.71001e9i	1.23737	+	0.277268i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	8.56642e9			0.859505			0.429752	−	0.902947i	$-0.358601\pi$
$719$	8.56642e9			0.859505			0.429752	+	0.902947i	$0.358601\pi$
$720$		0			0
$721$	2.37785e10			2.36271
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.50123e9	+	4.48315e9i	0.925970	+	0.436918i
$726$		0			0
$727$		−	1.45545e10i		−	1.40484i	−0.711765	−	0.702418i	$-0.752104\pi$
$727$		−	1.45545e10i		−	1.40484i	0.711765	−	0.702418i	$-0.247896\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.47665e8			0.0423880
$732$		0			0
$733$			3.87389e9i			0.363315i	0.983362	+	0.181657i	$0.0581462\pi$
$733$			3.87389e9i			0.363315i	−0.983362	+	0.181657i	$0.941854\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	7.24472e9i		−	0.666631i
$738$		0			0
$739$	−1.89101e10			−1.72361			−0.861804	−	0.507242i	$-0.830665\pi$
$739$	−1.89101e10			−1.72361			−0.861804	+	0.507242i	$0.830665\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	7.66965e8i		−	0.0685985i	−0.999412	−	0.0342993i	$-0.989080\pi$
$743$		−	7.66965e8i		−	0.0685985i	0.999412	−	0.0342993i	$-0.0109199\pi$
$744$		0			0
$745$	5.14034e8	−	2.29399e9i	0.0455454	−	0.203256i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.28557e10			1.11792
$750$		0			0
$751$	1.65279e10			1.42390			0.711948	−	0.702232i	$-0.247813\pi$
$751$	1.65279e10			1.42390			0.711948	+	0.702232i	$0.247813\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−4.68396e9	+	2.09032e10i	−0.396094	+	1.76766i
$756$		0			0
$757$		−	1.29897e10i		−	1.08834i	−0.838976	−	0.544168i	$-0.816845\pi$
$757$		−	1.29897e10i		−	1.08834i	0.838976	−	0.544168i	$-0.183155\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.70701e9			0.633927			0.316964	−	0.948438i	$-0.397337\pi$
$761$	7.70701e9			0.633927			0.316964	+	0.948438i	$0.397337\pi$
$762$		0			0
$763$			2.60265e10i			2.12119i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	2.43679e9i		−	0.195000i
$768$		0			0
$769$	5.65613e9			0.448516			0.224258	−	0.974530i	$-0.428004\pi$
$769$	5.65613e9			0.448516			0.224258	+	0.974530i	$0.428004\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			9.82561e8i			0.0765123i	0.999268	+	0.0382561i	$0.0121803\pi$
$773$			9.82561e8i			0.0765123i	−0.999268	+	0.0382561i	$0.987820\pi$
$774$		0			0
$775$	−1.40273e10	−	6.61877e9i	−1.08247	−	0.510765i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.85030e6			0.000519193
$780$		0			0
$781$	−2.66303e10			−2.00031
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.17914e10	−	2.64219e9i	−0.870002	−	0.194949i
$786$		0			0
$787$			7.84313e9i			0.573559i	0.957997	+	0.286779i	$0.0925846\pi$
$787$			7.84313e9i			0.573559i	−0.957997	+	0.286779i	$0.907415\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.98082e10			−2.14150
$792$		0			0
$793$			1.39221e9i			0.0991401i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1.44628e10i			1.01193i	0.862555	+	0.505964i	$0.168863\pi$
$797$			1.44628e10i			1.01193i	−0.862555	+	0.505964i	$0.831137\pi$
$798$		0			0
$799$	−1.01474e7			−0.000703788
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	2.76285e9i		−	0.188301i
$804$		0			0
$805$	−9.85782e9	+	4.39927e10i	−0.666033	+	2.97232i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.35190e10			−0.897684			−0.448842	−	0.893611i	$-0.648163\pi$
$809$	−1.35190e10			−0.897684			−0.448842	+	0.893611i	$0.648163\pi$
$810$		0			0
$811$	2.47720e10			1.63075			0.815377	−	0.578930i	$-0.196530\pi$
$811$	2.47720e10			1.63075			0.815377	+	0.578930i	$0.196530\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	7.60576e9	+	1.70429e9i	0.492143	+	0.110279i
$816$		0			0
$817$			3.53032e8i			0.0226484i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.31400e10			−0.828692			−0.414346	−	0.910119i	$-0.635990\pi$
$821$	−1.31400e10			−0.828692			−0.414346	+	0.910119i	$0.635990\pi$
$822$		0			0
$823$		−	2.13367e10i		−	1.33422i	−0.744959	−	0.667111i	$-0.767531\pi$
$823$		−	2.13367e10i		−	1.33422i	0.744959	−	0.667111i	$-0.232469\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	2.27212e10i		−	1.39689i	−0.715663	−	0.698446i	$-0.753875\pi$
$827$		−	2.27212e10i		−	1.39689i	0.715663	−	0.698446i	$-0.246125\pi$
$828$		0			0
$829$	−5.03801e9			−0.307127			−0.153564	−	0.988139i	$-0.549075\pi$
$829$	−5.03801e9			−0.307127			−0.153564	+	0.988139i	$0.549075\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	1.30238e9i		−	0.0780694i
$834$		0			0
$835$	1.34424e10	+	3.01215e9i	0.799050	+	0.179050i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.18094e10			0.690334			0.345167	−	0.938541i	$-0.387822\pi$
$839$	1.18094e10			0.690334			0.345167	+	0.938541i	$0.387822\pi$
$840$		0			0
$841$	8.33482e8			0.0483182
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.60122e9	+	7.14579e9i	−0.0912960	+	0.407429i
$846$		0			0
$847$		−	5.37697e10i		−	3.04050i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.14444e10			0.636558
$852$		0			0
$853$			7.15162e9i			0.394532i	0.980350	+	0.197266i	$0.0632063\pi$
$853$			7.15162e9i			0.394532i	−0.980350	+	0.197266i	$0.936794\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1.74665e9i			0.0947921i	0.998876	+	0.0473960i	$0.0150923\pi$
$857$			1.74665e9i			0.0947921i	−0.998876	+	0.0473960i	$0.984908\pi$
$858$		0			0
$859$	−8.22608e9			−0.442810			−0.221405	−	0.975182i	$-0.571064\pi$
$859$	−8.22608e9			−0.442810			−0.221405	+	0.975182i	$0.571064\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	7.20533e9i		−	0.381607i	−0.981628	−	0.190803i	$-0.938891\pi$
$863$		−	7.20533e9i		−	0.381607i	0.981628	−	0.190803i	$-0.0611093\pi$
$864$		0			0
$865$	−1.88178e10	−	4.21666e9i	−0.988579	−	0.221519i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.55229e10			−1.83628
$870$		0			0
$871$	−5.97149e9			−0.306210
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.70079e10	−	2.10178e10i	−1.36290	−	1.06062i
$876$		0			0
$877$			1.52548e10i			0.763676i	0.924229	+	0.381838i	$0.124709\pi$
$877$			1.52548e10i			0.763676i	−0.924229	+	0.381838i	$0.875291\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.17539e10			1.07182			0.535911	−	0.844275i	$-0.319968\pi$
$881$	2.17539e10			1.07182			0.535911	+	0.844275i	$0.319968\pi$
$882$		0			0
$883$			8.10982e9i			0.396414i	0.980160	+	0.198207i	$0.0635118\pi$
$883$			8.10982e9i			0.396414i	−0.980160	+	0.198207i	$0.936488\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	3.27974e10i		−	1.57800i	−0.614393	−	0.789000i	$-0.710599\pi$
$887$		−	3.27974e10i		−	1.57800i	0.614393	−	0.789000i	$-0.289401\pi$
$888$		0			0
$889$	−2.35679e10			−1.12503
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	8.00233e6i		−	0.000376042i
$894$		0			0
$895$	−6.08931e9	+	2.71749e10i	−0.283914	+	1.26703i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.66976e10			−1.22550
$900$		0			0
$901$	7.33650e8			0.0334158
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4.15097e9	−	1.85246e10i	0.186157	−	0.830767i
$906$		0			0
$907$		−	3.51778e10i		−	1.56546i	−0.622360	−	0.782731i	$-0.713826\pi$
$907$		−	3.51778e10i		−	1.56546i	0.622360	−	0.782731i	$-0.286174\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.69402e10			1.18056			0.590278	−	0.807200i	$-0.299018\pi$
$911$	2.69402e10			1.18056			0.590278	+	0.807200i	$0.299018\pi$
$912$		0			0
$913$			2.24044e10i			0.974285i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			8.27301e9i			0.354300i
$918$		0			0
$919$	1.53332e10			0.651671			0.325836	−	0.945426i	$-0.394354\pi$
$919$	1.53332e10			0.651671			0.325836	+	0.945426i	$0.394354\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			2.19501e10i			0.918821i
$924$		0			0
$925$	−3.70713e9	+	7.85659e9i	−0.154007	+	0.326391i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.27999e10			−0.523783			−0.261891	−	0.965097i	$-0.584346\pi$
$929$	−1.27999e10			−0.523783			−0.261891	+	0.965097i	$0.584346\pi$
$930$		0			0
$931$	1.02707e9			0.0417134
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.59585e9	+	3.57597e8i	0.0638488	+	0.0143071i
$936$		0			0
$937$			4.53222e9i			0.179979i	0.995943	+	0.0899896i	$0.0286834\pi$
$937$			4.53222e9i			0.179979i	−0.995943	+	0.0899896i	$0.971317\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.40485e9			−0.250579			−0.125290	−	0.992120i	$-0.539986\pi$
$941$	−6.40485e9			−0.250579			−0.125290	+	0.992120i	$0.539986\pi$
$942$		0			0
$943$		−	1.12069e9i		−	0.0435207i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			2.55218e10i			0.976533i	0.872695	+	0.488266i	$0.162371\pi$
$947$			2.55218e10i			0.976533i	−0.872695	+	0.488266i	$0.837629\pi$
$948$		0			0
$949$	−2.27729e9			−0.0864940
$950$		0			0
$951$		0			0
$952$		0			0
$953$			2.24447e10i			0.840020i	0.907520	+	0.420010i	$0.137973\pi$
$953$			2.24447e10i			0.840020i	−0.907520	+	0.420010i	$0.862027\pi$
$954$		0			0
$955$	−5.57947e9	+	2.48996e10i	−0.207291	+	0.925083i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.36362e10			−3.06217
$960$		0			0
$961$	1.19028e10			0.432632
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8.69513e9	−	1.94839e9i	−0.311480	−	0.0697960i
$966$		0			0
$967$		−	1.19222e10i		−	0.423998i	−0.977270	−	0.211999i	$-0.932003\pi$
$967$		−	1.19222e10i		−	0.423998i	0.977270	−	0.211999i	$-0.0679973\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.70027e10			−1.29708			−0.648540	−	0.761181i	$-0.724620\pi$
$971$	−3.70027e10			−1.29708			−0.648540	+	0.761181i	$0.724620\pi$
$972$		0			0
$973$		−	4.39366e10i		−	1.52908i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			2.23468e9i			0.0766627i	0.999265	+	0.0383314i	$0.0122042\pi$
$977$			2.23468e9i			0.0766627i	−0.999265	+	0.0383314i	$0.987796\pi$
$978$		0			0
$979$	−3.33648e10			−1.13645
$980$		0			0
$981$		0			0
$982$		0			0
$983$			2.50064e10i			0.839679i	0.907598	+	0.419839i	$0.137914\pi$
$983$			2.50064e10i			0.839679i	−0.907598	+	0.419839i	$0.862086\pi$
$984$		0			0
$985$	−1.22757e10	−	2.75073e9i	−0.409280	−	0.0917109i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5.77552e10			1.89847
$990$		0			0
$991$	−1.33087e10			−0.434388			−0.217194	−	0.976128i	$-0.569690\pi$
$991$	−1.33087e10			−0.434388			−0.217194	+	0.976128i	$0.569690\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.09496e10	+	4.88650e10i	−0.352385	+	1.57260i
$996$		0			0
$997$			3.56881e10i			1.14049i	0.821476	+	0.570244i	$0.193151\pi$
$997$			3.56881e10i			1.14049i	−0.821476	+	0.570244i	$0.806849\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.8.f.b.289.2		8
3.2	odd	2		40.8.c.b.9.7	yes	8
5.4	even	2	inner	360.8.f.b.289.1		8
12.11	even	2		80.8.c.e.49.2		8
15.2	even	4		200.8.a.r.1.4		4
15.8	even	4		200.8.a.q.1.1		4
15.14	odd	2		40.8.c.b.9.2	✓	8
24.5	odd	2		320.8.c.k.129.2		8
24.11	even	2		320.8.c.l.129.7		8
60.23	odd	4		400.8.a.bl.1.4		4
60.47	odd	4		400.8.a.bj.1.1		4
60.59	even	2		80.8.c.e.49.7		8
120.29	odd	2		320.8.c.k.129.7		8
120.59	even	2		320.8.c.l.129.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.8.c.b.9.2	✓	8	15.14	odd	2
40.8.c.b.9.7	yes	8	3.2	odd	2
80.8.c.e.49.2		8	12.11	even	2
80.8.c.e.49.7		8	60.59	even	2
200.8.a.q.1.1		4	15.8	even	4
200.8.a.r.1.4		4	15.2	even	4
320.8.c.k.129.2		8	24.5	odd	2
320.8.c.k.129.7		8	120.29	odd	2
320.8.c.l.129.2		8	120.59	even	2
320.8.c.l.129.7		8	24.11	even	2
360.8.f.b.289.1		8	5.4	even	2	inner
360.8.f.b.289.2		8	1.1	even	1	trivial
400.8.a.bj.1.1		4	60.47	odd	4
400.8.a.bl.1.4		4	60.23	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	360.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-61.1163 + 272.745i) q^{5} -1567.21i q^{7} -7334.60 q^{11} +6045.58i q^{13} +797.737i q^{17} -629.101 q^{19} +102919. i q^{23} +(-70654.6 - 33338.3i) q^{25} -134474. q^{29} +198533. q^{31} +(427448. + 95781.9i) q^{35} -111197. i q^{37} -10889.0 q^{41} -561169. i q^{43} +12720.3i q^{47} -1.63260e6 q^{49} -919664. i q^{53} +(448264. - 2.00048e6i) q^{55} -403069. q^{59} +230286. q^{61} +(-1.64890e6 - 369483. i) q^{65} +987745. i q^{67} +3.63077e6 q^{71} +376686. i q^{73} +1.14948e7i q^{77} +4.84319e6 q^{79} -3.05462e6i q^{83} +(-217579. - 48754.8i) q^{85} +4.54896e6 q^{89} +9.47467e6 q^{91} +(38448.3 - 171584. i) q^{95} -1.55682e6i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 744 q^{5} - 9120 q^{11} - 61600 q^{19} - 254424 q^{25} - 35760 q^{29} + 519168 q^{31} + 162144 q^{35} + 852336 q^{41} - 4310376 q^{49} + 829664 q^{55} - 1202976 q^{59} - 1290128 q^{61} - 1750848 q^{65}+ \cdots + 8588256 q^{95}+O(q^{100}) \) 8 * q + 744 * q^5 - 9120 * q^11 - 61600 * q^19 - 254424 * q^25 - 35760 * q^29 + 519168 * q^31 + 162144 * q^35 + 852336 * q^41 - 4310376 * q^49 + 829664 * q^55 - 1202976 * q^59 - 1290128 * q^61 - 1750848 * q^65 + 12982848 * q^71 - 14162816 * q^79 + 6461952 * q^85 - 16890960 * q^89 + 40041664 * q^91 + 8588256 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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