LMFDB - Embedded newform 2320.2.g.i.1681.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2320,2,Mod(1681,2320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2320.1681");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2320.g (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$18.5252932689$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.16516096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 13x^{2} + 1 $ x^6 + 9x^4 + 13x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 145)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1681.2
Root			$-2.68667i$ of defining polynomial
Character	$\chi$	$=$	2320.1681
Dual form			2320.2.g.i.1681.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.31446i q^{3} +1.00000 q^{5} -4.21819 q^{7} -2.35673 q^{9} +O(q^{10})$	$q-2.31446i q^{3} +1.00000 q^{5} -4.21819 q^{7} -2.35673 q^{9} +2.31446i q^{11} -0.643274 q^{13} -2.31446i q^{15} +0.586195i q^{17} +5.95953i q^{19} +9.76282i q^{21} +5.57491 q^{23} +1.00000 q^{25} -1.48883i q^{27} +(-5.21819 + 1.33061i) q^{29} +0.158221i q^{31} +5.35673 q^{33} -4.21819 q^{35} -4.04272i q^{37} +1.48883i q^{39} +9.76282i q^{41} +4.97568i q^{43} -2.35673 q^{45} +2.31446i q^{47} +10.7931 q^{49} +1.35673 q^{51} +13.0796 q^{53} +2.31446i q^{55} +13.7931 q^{57} +2.64327 q^{59} +0.983848i q^{61} +9.94111 q^{63} -0.643274 q^{65} -1.57491 q^{67} -12.9029i q^{69} +9.35673 q^{71} +9.17663i q^{73} -2.31446i q^{75} -9.76282i q^{77} +4.97568i q^{79} -10.5160 q^{81} +8.21819 q^{83} +0.586195i q^{85} +(3.07965 + 12.0773i) q^{87} -7.29014i q^{89} +2.71345 q^{91} +0.366196 q^{93} +5.95953i q^{95} -3.05888i q^{97} -5.45455i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{5} - 14 q^{9}+O(q^{10}) $ 6 * q + 6 * q^5 - 14 * q^9	$ 6 q + 6 q^{5} - 14 q^{9} - 4 q^{13} + 8 q^{23} + 6 q^{25} - 6 q^{29} + 32 q^{33} - 14 q^{45} + 14 q^{49} + 8 q^{51} + 28 q^{53} + 32 q^{57} + 16 q^{59} - 16 q^{63} - 4 q^{65} + 16 q^{67} + 56 q^{71} + 38 q^{81} + 24 q^{83} - 32 q^{87} + 16 q^{91} - 48 q^{93}+O(q^{100}) $ 6 * q + 6 * q^5 - 14 * q^9 - 4 * q^13 + 8 * q^23 + 6 * q^25 - 6 * q^29 + 32 * q^33 - 14 * q^45 + 14 * q^49 + 8 * q^51 + 28 * q^53 + 32 * q^57 + 16 * q^59 - 16 * q^63 - 4 * q^65 + 16 * q^67 + 56 * q^71 + 38 * q^81 + 24 * q^83 - 32 * q^87 + 16 * q^91 - 48 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times$.

$n$	$321$	$581$	$1857$	$2031$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		−	2.31446i		−	1.33625i	−0.744047	−	0.668127i	$-0.767096\pi$
$3$		−	2.31446i		−	1.33625i	0.744047	−	0.668127i	$-0.232904\pi$
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	−4.21819			−1.59432			−0.797162	−	0.603765i	$-0.793667\pi$
$7$	−4.21819			−1.59432			−0.797162	+	0.603765i	$0.793667\pi$
$8$		0			0
$9$	−2.35673			−0.785575
$10$		0			0
$11$			2.31446i			0.697836i	0.937153	+	0.348918i	$0.113451\pi$
$11$			2.31446i			0.697836i	−0.937153	+	0.348918i	$0.886549\pi$
$12$		0			0
$13$	−0.643274			−0.178412			−0.0892061	−	0.996013i	$-0.528433\pi$
$13$	−0.643274			−0.178412			−0.0892061	+	0.996013i	$0.528433\pi$
$14$		0			0
$15$		−	2.31446i		−	0.597591i
$16$		0			0
$17$			0.586195i			0.142173i	0.997470	+	0.0710866i	$0.0226467\pi$
$17$			0.586195i			0.142173i	−0.997470	+	0.0710866i	$0.977353\pi$
$18$		0			0
$19$			5.95953i			1.36721i	0.729852	+	0.683605i	$0.239589\pi$
$19$			5.95953i			1.36721i	−0.729852	+	0.683605i	$0.760411\pi$
$20$		0			0
$21$			9.76282i			2.13042i
$22$		0			0
$23$	5.57491			1.16245			0.581225	−	0.813743i	$-0.302574\pi$
$23$	5.57491			1.16245			0.581225	+	0.813743i	$0.302574\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		−	1.48883i		−	0.286526i
$28$		0			0
$29$	−5.21819	+	1.33061i	−0.968993	+	0.247088i
$29$	−5.21819	+	1.33061i	−0.968993	+	0.247088i
$30$		0			0
$31$			0.158221i			0.0284174i	0.999899	+	0.0142087i	$0.00452291\pi$
$31$			0.158221i			0.0284174i	−0.999899	+	0.0142087i	$0.995477\pi$
$32$		0			0
$33$	5.35673			0.932486
$34$		0			0
$35$	−4.21819			−0.713004
$36$		0			0
$37$		−	4.04272i		−	0.664620i	−0.943170	−	0.332310i	$-0.892172\pi$
$37$		−	4.04272i		−	0.664620i	0.943170	−	0.332310i	$-0.107828\pi$
$38$		0			0
$39$			1.48883i			0.238404i
$40$		0			0
$41$			9.76282i			1.52470i	0.647167	+	0.762349i	$0.275954\pi$
$41$			9.76282i			1.52470i	−0.647167	+	0.762349i	$0.724046\pi$
$42$		0			0
$43$			4.97568i			0.758785i	0.925236	+	0.379392i	$0.123867\pi$
$43$			4.97568i			0.758785i	−0.925236	+	0.379392i	$0.876133\pi$
$44$		0			0
$45$	−2.35673			−0.351320
$46$		0			0
$47$			2.31446i			0.337599i	0.985650	+	0.168799i	$0.0539890\pi$
$47$			2.31446i			0.337599i	−0.985650	+	0.168799i	$0.946011\pi$
$48$		0			0
$49$	10.7931			1.54187
$50$		0			0
$51$	1.35673			0.189980
$52$		0			0
$53$	13.0796			1.79663			0.898314	−	0.439354i	$-0.144793\pi$
$53$	13.0796			1.79663			0.898314	+	0.439354i	$0.144793\pi$
$54$		0			0
$55$			2.31446i			0.312082i
$56$		0			0
$57$	13.7931			1.82694
$58$		0			0
$59$	2.64327			0.344125			0.172063	−	0.985086i	$-0.444957\pi$
$59$	2.64327			0.344125			0.172063	+	0.985086i	$0.444957\pi$
$60$		0			0
$61$			0.983848i			0.125969i	0.998015	+	0.0629844i	$0.0200618\pi$
$61$			0.983848i			0.125969i	−0.998015	+	0.0629844i	$0.979938\pi$
$62$		0			0
$63$	9.94111			1.25246
$64$		0			0
$65$	−0.643274			−0.0797884
$66$		0			0
$67$	−1.57491			−0.192406			−0.0962031	−	0.995362i	$-0.530670\pi$
$67$	−1.57491			−0.192406			−0.0962031	+	0.995362i	$0.530670\pi$
$68$		0			0
$69$		−	12.9029i		−	1.55333i
$70$		0			0
$71$	9.35673			1.11044			0.555220	−	0.831704i	$-0.312634\pi$
$71$	9.35673			1.11044			0.555220	+	0.831704i	$0.312634\pi$
$72$		0			0
$73$			9.17663i			1.07404i	0.843568	+	0.537022i	$0.180451\pi$
$73$			9.17663i			1.07404i	−0.843568	+	0.537022i	$0.819549\pi$
$74$		0			0
$75$		−	2.31446i		−	0.267251i
$76$		0			0
$77$		−	9.76282i		−	1.11258i
$78$		0			0
$79$			4.97568i			0.559808i	0.960028	+	0.279904i	$0.0903027\pi$
$79$			4.97568i			0.559808i	−0.960028	+	0.279904i	$0.909697\pi$
$80$		0			0
$81$	−10.5160			−1.16845
$82$		0			0
$83$	8.21819			0.902063			0.451032	−	0.892508i	$-0.351056\pi$
$83$	8.21819			0.902063			0.451032	+	0.892508i	$0.351056\pi$
$84$		0			0
$85$			0.586195i			0.0635818i
$86$		0			0
$87$	3.07965	+	12.0773i	0.330173	+	1.29482i
$88$		0			0
$89$		−	7.29014i		−	0.772754i	−0.922341	−	0.386377i	$-0.873726\pi$
$89$		−	7.29014i		−	0.772754i	0.922341	−	0.386377i	$-0.126274\pi$
$90$		0			0
$91$	2.71345			0.284447
$92$		0			0
$93$	0.366196			0.0379728
$94$		0			0
$95$			5.95953i			0.611435i
$96$		0			0
$97$		−	3.05888i		−	0.310582i	−0.987869	−	0.155291i	$-0.950369\pi$
$97$		−	3.05888i		−	0.310582i	0.987869	−	0.155291i	$-0.0496315\pi$
$98$		0			0
$99$		−	5.45455i		−	0.548203i
$100$		0			0
$101$		−	1.48883i		−	0.148144i	−0.997253	−	0.0740722i	$-0.976400\pi$
$101$		−	1.48883i		−	0.148144i	0.997253	−	0.0740722i	$-0.0235995\pi$
$102$		0			0
$103$	−11.2978			−1.11321			−0.556604	−	0.830778i	$-0.687896\pi$
$103$	−11.2978			−1.11321			−0.556604	+	0.830778i	$0.687896\pi$
$104$		0			0
$105$			9.76282i			0.952754i
$106$		0			0
$107$	2.93164			0.283412			0.141706	−	0.989909i	$-0.454741\pi$
$107$	2.93164			0.283412			0.141706	+	0.989909i	$0.454741\pi$
$108$		0			0
$109$	−17.0796			−1.63593			−0.817967	−	0.575265i	$-0.804899\pi$
$109$	−17.0796			−1.63593			−0.817967	+	0.575265i	$0.804899\pi$
$110$		0			0
$111$	−9.35673			−0.888101
$112$		0			0
$113$			13.4891i			1.26895i	0.772944	+	0.634474i	$0.218783\pi$
$113$			13.4891i			1.26895i	−0.772944	+	0.634474i	$0.781217\pi$
$114$		0			0
$115$	5.57491			0.519863
$116$		0			0
$117$	1.51602			0.140156
$118$		0			0
$119$		−	2.47268i		−	0.226670i
$120$		0			0
$121$	5.64327			0.513025
$122$		0			0
$123$	22.5957			2.03738
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$		−	11.0934i		−	0.984383i	−0.870487	−	0.492192i	$-0.836196\pi$
$127$		−	11.0934i		−	0.984383i	0.870487	−	0.492192i	$-0.163804\pi$
$128$		0			0
$129$	11.5160			1.01393
$130$		0			0
$131$			7.13192i			0.623119i	0.950227	+	0.311559i	$0.100851\pi$
$131$			7.13192i			0.623119i	−0.950227	+	0.311559i	$0.899149\pi$
$132$		0			0
$133$		−	25.1384i		−	2.17978i
$134$		0			0
$135$		−	1.48883i		−	0.128138i
$136$		0			0
$137$			14.7894i			1.26354i	0.775154	+	0.631772i	$0.217672\pi$
$137$			14.7894i			1.26354i	−0.775154	+	0.631772i	$0.782328\pi$
$138$		0			0
$139$	−3.56363			−0.302263			−0.151131	−	0.988514i	$-0.548292\pi$
$139$	−3.56363			−0.302263			−0.151131	+	0.988514i	$0.548292\pi$
$140$		0			0
$141$	5.35673			0.451118
$142$		0			0
$143$		−	1.48883i		−	0.124502i
$144$		0			0
$145$	−5.21819	+	1.33061i	−0.433347	+	0.110501i
$146$		0			0
$147$		−	24.9802i		−	2.06033i
$148$		0			0
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$		0			0
$151$	−4.50655			−0.366738			−0.183369	−	0.983044i	$-0.558700\pi$
$151$	−4.50655			−0.366738			−0.183369	+	0.983044i	$0.558700\pi$
$152$		0			0
$153$		−	1.38150i		−	0.111688i
$154$		0			0
$155$			0.158221i			0.0127086i
$156$		0			0
$157$		−	16.9456i		−	1.35241i	−0.736714	−	0.676205i	$-0.763624\pi$
$157$		−	16.9456i		−	1.35241i	0.736714	−	0.676205i	$-0.236376\pi$
$158$		0			0
$159$		−	30.2723i		−	2.40075i
$160$		0			0
$161$	−23.5160			−1.85332
$162$		0			0
$163$			11.0934i			0.868905i	0.900695	+	0.434453i	$0.143058\pi$
$163$			11.0934i			0.868905i	−0.900695	+	0.434453i	$0.856942\pi$
$164$		0			0
$165$	5.35673			0.417021
$166$		0			0
$167$	5.57491			0.431400			0.215700	−	0.976460i	$-0.430797\pi$
$167$	5.57491			0.431400			0.215700	+	0.976460i	$0.430797\pi$
$168$		0			0
$169$	−12.5862			−0.968169
$170$		0			0
$171$		−	14.0450i		−	1.07405i
$172$		0			0
$173$	7.79310			0.592498			0.296249	−	0.955111i	$-0.404264\pi$
$173$	7.79310			0.592498			0.296249	+	0.955111i	$0.404264\pi$
$174$		0			0
$175$	−4.21819			−0.318865
$176$		0			0
$177$		−	6.11775i		−	0.459838i
$178$		0			0
$179$	4.43637			0.331590			0.165795	−	0.986160i	$-0.446981\pi$
$179$	4.43637			0.331590			0.165795	+	0.986160i	$0.446981\pi$
$180$		0			0
$181$	13.5160			1.00464			0.502319	−	0.864682i	$-0.332480\pi$
$181$	13.5160			1.00464			0.502319	+	0.864682i	$0.332480\pi$
$182$		0			0
$183$	2.27708			0.168326
$184$		0			0
$185$		−	4.04272i		−	0.297227i
$186$		0			0
$187$	−1.35673			−0.0992136
$188$		0			0
$189$			6.28017i			0.456816i
$190$		0			0
$191$		−	11.5723i		−	0.837342i	−0.908138	−	0.418671i	$-0.862496\pi$
$191$		−	11.5723i		−	0.837342i	0.908138	−	0.418671i	$-0.137504\pi$
$192$		0			0
$193$			9.84404i			0.708589i	0.935134	+	0.354295i	$0.115279\pi$
$193$			9.84404i			0.708589i	−0.935134	+	0.354295i	$0.884721\pi$
$194$		0			0
$195$			1.48883i			0.106618i
$196$		0			0
$197$	−5.14982			−0.366910			−0.183455	−	0.983028i	$-0.558728\pi$
$197$	−5.14982			−0.366910			−0.183455	+	0.983028i	$0.558728\pi$
$198$		0			0
$199$	17.7931			1.26132			0.630660	−	0.776060i	$-0.282784\pi$
$199$	17.7931			1.26132			0.630660	+	0.776060i	$0.282784\pi$
$200$		0			0
$201$			3.64507i			0.257104i
$202$		0			0
$203$	22.0113	−	5.61277i	1.54489	−	0.393939i
$204$		0			0
$205$			9.76282i			0.681865i
$206$		0			0
$207$	−13.1385			−0.913192
$208$		0			0
$209$	−13.7931			−0.954089
$210$		0			0
$211$			18.8624i			1.29854i	0.760556	+	0.649272i	$0.224926\pi$
$211$			18.8624i			1.29854i	−0.760556	+	0.649272i	$0.775074\pi$
$212$		0			0
$213$		−	21.6558i		−	1.48383i
$214$		0			0
$215$			4.97568i			0.339339i
$216$		0			0
$217$		−	0.667406i		−	0.0453065i
$218$		0			0
$219$	21.2389			1.43519
$220$		0			0
$221$		−	0.377084i		−	0.0253654i
$222$		0			0
$223$	−13.5749			−0.909043			−0.454522	−	0.890736i	$-0.650190\pi$
$223$	−13.5749			−0.909043			−0.454522	+	0.890736i	$0.650190\pi$
$224$		0			0
$225$	−2.35673			−0.157115
$226$		0			0
$227$	−7.00181			−0.464727			−0.232363	−	0.972629i	$-0.574646\pi$
$227$	−7.00181			−0.464727			−0.232363	+	0.972629i	$0.574646\pi$
$228$		0			0
$229$			24.1546i			1.59618i	0.602539	+	0.798089i	$0.294156\pi$
$229$			24.1546i			1.59618i	−0.602539	+	0.798089i	$0.705844\pi$
$230$		0			0
$231$	−22.5957			−1.48669
$232$		0			0
$233$	5.14982			0.337376			0.168688	−	0.985669i	$-0.446047\pi$
$233$	5.14982			0.337376			0.168688	+	0.985669i	$0.446047\pi$
$234$		0			0
$235$			2.31446i			0.150979i
$236$		0			0
$237$	11.5160			0.748046
$238$		0			0
$239$	−14.6433			−0.947195			−0.473597	−	0.880741i	$-0.657045\pi$
$239$	−14.6433			−0.947195			−0.473597	+	0.880741i	$0.657045\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$		0			0
$243$			19.8724i			1.27482i
$244$		0			0
$245$	10.7931			0.689546
$246$		0			0
$247$		−	3.83361i		−	0.243927i
$248$		0			0
$249$		−	19.0207i		−	1.20539i
$250$		0			0
$251$			22.6354i			1.42873i	0.699771	+	0.714367i	$0.253285\pi$
$251$			22.6354i			1.42873i	−0.699771	+	0.714367i	$0.746715\pi$
$252$		0			0
$253$			12.9029i			0.811199i
$254$		0			0
$255$	1.35673			0.0849615
$256$		0			0
$257$	10.9204			0.681193			0.340596	−	0.940210i	$-0.389371\pi$
$257$	10.9204			0.681193			0.340596	+	0.940210i	$0.389371\pi$
$258$		0			0
$259$			17.0530i			1.05962i
$260$		0			0
$261$	12.2978	−	3.13589i	0.761217	−	0.194107i
$262$		0			0
$263$		−	9.92105i		−	0.611758i	−0.952070	−	0.305879i	$-0.901050\pi$
$263$		−	9.92105i		−	0.611758i	0.952070	−	0.305879i	$-0.0989503\pi$
$264$		0			0
$265$	13.0796			0.803476
$266$		0			0
$267$	−16.8727			−1.03260
$268$		0			0
$269$		−	12.4240i		−	0.757508i	−0.925497	−	0.378754i	$-0.876353\pi$
$269$		−	12.4240i		−	0.757508i	0.925497	−	0.378754i	$-0.123647\pi$
$270$		0			0
$271$			27.2643i			1.65619i	0.560587	+	0.828095i	$0.310575\pi$
$271$			27.2643i			1.65619i	−0.560587	+	0.828095i	$0.689425\pi$
$272$		0			0
$273$		−	6.28017i		−	0.380093i
$274$		0			0
$275$			2.31446i			0.139567i
$276$		0			0
$277$	−2.92035			−0.175467			−0.0877335	−	0.996144i	$-0.527962\pi$
$277$	−2.92035			−0.175467			−0.0877335	+	0.996144i	$0.527962\pi$
$278$		0			0
$279$		−	0.372884i		−	0.0223240i
$280$		0			0
$281$	−18.3662			−1.09564			−0.547818	−	0.836598i	$-0.684541\pi$
$281$	−18.3662			−1.09564			−0.547818	+	0.836598i	$0.684541\pi$
$282$		0			0
$283$	14.8615			0.883422			0.441711	−	0.897157i	$-0.354372\pi$
$283$	14.8615			0.883422			0.441711	+	0.897157i	$0.354372\pi$
$284$		0			0
$285$	13.7931			0.817033
$286$		0			0
$287$		−	41.1814i		−	2.43086i
$288$		0			0
$289$	16.6564			0.979787
$290$		0			0
$291$	−7.07965			−0.415016
$292$		0			0
$293$			1.57004i			0.0917229i	0.998948	+	0.0458615i	$0.0146033\pi$
$293$			1.57004i			0.0917229i	−0.998948	+	0.0458615i	$0.985397\pi$
$294$		0			0
$295$	2.64327			0.153897
$296$		0			0
$297$	3.44584			0.199948
$298$		0			0
$299$	−3.58620			−0.207395
$300$		0			0
$301$		−	20.9884i		−	1.20975i
$302$		0			0
$303$	−3.44584			−0.197959
$304$		0			0
$305$			0.983848i			0.0563350i
$306$		0			0
$307$		−	4.65924i		−	0.265917i	−0.991122	−	0.132958i	$-0.957552\pi$
$307$		−	4.65924i		−	0.265917i	0.991122	−	0.132958i	$-0.0424477\pi$
$308$		0			0
$309$			26.1484i			1.48753i
$310$		0			0
$311$			21.6516i			1.22775i	0.789404	+	0.613874i	$0.210390\pi$
$311$			21.6516i			1.22775i	−0.789404	+	0.613874i	$0.789610\pi$
$312$		0			0
$313$	11.3567			0.641920			0.320960	−	0.947093i	$-0.395994\pi$
$313$	11.3567			0.641920			0.320960	+	0.947093i	$0.395994\pi$
$314$		0			0
$315$	9.94111			0.560118
$316$		0			0
$317$		−	9.36517i		−	0.526000i	−0.964796	−	0.263000i	$-0.915288\pi$
$317$		−	9.36517i		−	0.526000i	0.964796	−	0.263000i	$-0.0847120\pi$
$318$		0			0
$319$	−3.07965	−	12.0773i	−0.172427	−	0.676198i
$320$		0			0
$321$		−	6.78516i		−	0.378711i
$322$		0			0
$323$	−3.49345			−0.194381
$324$		0			0
$325$	−0.643274			−0.0356824
$326$		0			0
$327$			39.5302i			2.18602i
$328$		0			0
$329$		−	9.76282i		−	0.538242i
$330$		0			0
$331$		−	18.5115i		−	1.01748i	−0.860919	−	0.508741i	$-0.830111\pi$
$331$		−	18.5115i		−	1.01748i	0.860919	−	0.508741i	$-0.169889\pi$
$332$		0			0
$333$			9.52759i			0.522109i
$334$		0			0
$335$	−1.57491			−0.0860467
$336$		0			0
$337$			18.8116i			1.02473i	0.858768	+	0.512365i	$0.171231\pi$
$337$			18.8116i			1.02473i	−0.858768	+	0.512365i	$0.828769\pi$
$338$		0			0
$339$	31.2200			1.69564
$340$		0			0
$341$	−0.366196			−0.0198306
$342$		0			0
$343$	−16.0000			−0.863919
$344$		0			0
$345$		−	12.9029i		−	0.694669i
$346$		0			0
$347$	25.2313			1.35449			0.677243	−	0.735759i	$-0.263174\pi$
$347$	25.2313			1.35449			0.677243	+	0.735759i	$0.263174\pi$
$348$		0			0
$349$	2.85018			0.152566			0.0762832	−	0.997086i	$-0.475695\pi$
$349$	2.85018			0.152566			0.0762832	+	0.997086i	$0.475695\pi$
$350$		0			0
$351$			0.957728i			0.0511197i
$352$		0			0
$353$	−30.0927			−1.60168			−0.800838	−	0.598881i	$-0.795612\pi$
$353$	−30.0927			−1.60168			−0.800838	+	0.598881i	$0.795612\pi$
$354$		0			0
$355$	9.35673			0.496603
$356$		0			0
$357$	−5.72292			−0.302889
$358$		0			0
$359$			23.6193i			1.24658i	0.781992	+	0.623289i	$0.214204\pi$
$359$			23.6193i			1.24658i	−0.781992	+	0.623289i	$0.785796\pi$
$360$		0			0
$361$	−16.5160			−0.869264
$362$		0			0
$363$		−	13.0611i		−	0.685532i
$364$		0			0
$365$			9.17663i			0.480327i
$366$		0			0
$367$		−	17.2112i		−	0.898417i	−0.893427	−	0.449208i	$-0.851706\pi$
$367$		−	17.2112i		−	0.898417i	0.893427	−	0.449208i	$-0.148294\pi$
$368$		0			0
$369$		−	23.0083i		−	1.19776i
$370$		0			0
$371$	−55.1724			−2.86441
$372$		0			0
$373$	22.8727			1.18431			0.592153	−	0.805826i	$-0.298278\pi$
$373$	22.8727			1.18431			0.592153	+	0.805826i	$0.298278\pi$
$374$		0			0
$375$		−	2.31446i		−	0.119518i
$376$		0			0
$377$	3.35673	−	0.855948i	0.172880	−	0.0440836i
$378$		0			0
$379$		−	27.2643i		−	1.40047i	−0.713910	−	0.700237i	$-0.753077\pi$
$379$		−	27.2643i		−	1.40047i	0.713910	−	0.700237i	$-0.246923\pi$
$380$		0			0
$381$	−25.6753			−1.31539
$382$		0			0
$383$	−37.2313			−1.90243			−0.951215	−	0.308529i	$-0.900163\pi$
$383$	−37.2313			−1.90243			−0.951215	+	0.308529i	$0.900163\pi$
$384$		0			0
$385$		−	9.76282i		−	0.497560i
$386$		0			0
$387$		−	11.7263i		−	0.596082i
$388$		0			0
$389$			23.6496i			1.19908i	0.800344	+	0.599541i	$0.204650\pi$
$389$			23.6496i			1.19908i	−0.800344	+	0.599541i	$0.795350\pi$
$390$		0			0
$391$			3.26799i			0.165269i
$392$		0			0
$393$	16.5066			0.832645
$394$		0			0
$395$			4.97568i			0.250354i
$396$		0			0
$397$	−30.0226			−1.50679			−0.753395	−	0.657568i	$-0.771585\pi$
$397$	−30.0226			−1.50679			−0.753395	+	0.657568i	$0.771585\pi$
$398$		0			0
$399$	−58.1819			−2.91274
$400$		0			0
$401$	−32.3698			−1.61647			−0.808236	−	0.588859i	$-0.799577\pi$
$401$	−32.3698			−1.61647			−0.808236	+	0.588859i	$0.799577\pi$
$402$		0			0
$403$		−	0.101780i		−	0.00507000i
$404$		0			0
$405$	−10.5160			−0.522545
$406$		0			0
$407$	9.35673			0.463796
$408$		0			0
$409$			24.5055i			1.21172i	0.795571	+	0.605860i	$0.207171\pi$
$409$			24.5055i			1.21172i	−0.795571	+	0.605860i	$0.792829\pi$
$410$		0			0
$411$	34.2295			1.68842
$412$		0			0
$413$	−11.1498			−0.548647
$414$		0			0
$415$	8.21819			0.403415
$416$		0			0
$417$			8.24787i			0.403900i
$418$		0			0
$419$	−6.22947			−0.304330			−0.152165	−	0.988355i	$-0.548624\pi$
$419$	−6.22947			−0.304330			−0.152165	+	0.988355i	$0.548624\pi$
$420$		0			0
$421$			32.9074i			1.60381i	0.597452	+	0.801905i	$0.296180\pi$
$421$			32.9074i			1.60381i	−0.597452	+	0.801905i	$0.703820\pi$
$422$		0			0
$423$		−	5.45455i		−	0.265209i
$424$		0			0
$425$			0.586195i			0.0284346i
$426$		0			0
$427$		−	4.15006i		−	0.200835i
$428$		0			0
$429$	−3.44584			−0.166367
$430$		0			0
$431$	−3.00947			−0.144961			−0.0724806	−	0.997370i	$-0.523092\pi$
$431$	−3.00947			−0.144961			−0.0724806	+	0.997370i	$0.523092\pi$
$432$		0			0
$433$			34.3150i			1.64908i	0.565807	+	0.824538i	$0.308565\pi$
$433$			34.3150i			1.64908i	−0.565807	+	0.824538i	$0.691435\pi$
$434$		0			0
$435$	3.07965	+	12.0773i	0.147658	+	0.579061i
$436$		0			0
$437$			33.2239i			1.58931i
$438$		0			0
$439$	−29.8157			−1.42302			−0.711512	−	0.702674i	$-0.751989\pi$
$439$	−29.8157			−1.42302			−0.711512	+	0.702674i	$0.751989\pi$
$440$		0			0
$441$	−25.4364			−1.21126
$442$		0			0
$443$		−	27.9579i		−	1.32832i	−0.747591	−	0.664159i	$-0.768790\pi$
$443$		−	27.9579i		−	1.32832i	0.747591	−	0.664159i	$-0.231210\pi$
$444$		0			0
$445$		−	7.29014i		−	0.345586i
$446$		0			0
$447$		−	13.8868i		−	0.656821i
$448$		0			0
$449$			4.27796i			0.201889i	0.994892	+	0.100945i	$0.0321865\pi$
$449$			4.27796i			0.201889i	−0.994892	+	0.100945i	$0.967814\pi$
$450$		0			0
$451$	−22.5957			−1.06399
$452$		0			0
$453$			10.4302i			0.490055i
$454$		0			0
$455$	2.71345			0.127209
$456$		0			0
$457$	−10.3662			−0.484910			−0.242455	−	0.970163i	$-0.577953\pi$
$457$	−10.3662			−0.484910			−0.242455	+	0.970163i	$0.577953\pi$
$458$		0			0
$459$	0.872747			0.0407363
$460$		0			0
$461$		−	13.0915i		−	0.609730i	−0.952396	−	0.304865i	$-0.901389\pi$
$461$		−	13.0915i		−	0.609730i	0.952396	−	0.304865i	$-0.0986113\pi$
$462$		0			0
$463$	33.4571			1.55488			0.777442	−	0.628954i	$-0.216517\pi$
$463$	33.4571			1.55488			0.777442	+	0.628954i	$0.216517\pi$
$464$		0			0
$465$	0.366196			0.0169820
$466$		0			0
$467$			32.5868i			1.50794i	0.656911	+	0.753968i	$0.271863\pi$
$467$			32.5868i			1.50794i	−0.656911	+	0.753968i	$0.728137\pi$
$468$		0			0
$469$	6.64327			0.306758
$470$		0			0
$471$	−39.2200			−1.80716
$472$		0			0
$473$	−11.5160			−0.529507
$474$		0			0
$475$			5.95953i			0.273442i
$476$		0			0
$477$	−30.8251			−1.41139
$478$		0			0
$479$			36.5222i			1.66874i	0.551204	+	0.834370i	$0.314169\pi$
$479$			36.5222i			1.66874i	−0.551204	+	0.834370i	$0.685831\pi$
$480$		0			0
$481$			2.60058i			0.118576i
$482$		0			0
$483$			54.4269i			2.47651i
$484$		0			0
$485$		−	3.05888i		−	0.138896i
$486$		0			0
$487$	−25.9411			−1.17550			−0.587752	−	0.809041i	$-0.699987\pi$
$487$	−25.9411			−1.17550			−0.587752	+	0.809041i	$0.699987\pi$
$488$		0			0
$489$	25.6753			1.16108
$490$		0			0
$491$		−	26.3411i		−	1.18876i	−0.804185	−	0.594379i	$-0.797398\pi$
$491$		−	26.3411i		−	1.18876i	0.804185	−	0.594379i	$-0.202602\pi$
$492$		0			0
$493$	−0.779998	−	3.05888i	−0.0351294	−	0.137765i
$494$		0			0
$495$		−	5.45455i		−	0.245164i
$496$		0			0
$497$	−39.4684			−1.77040
$498$		0			0
$499$	21.0131			0.940676			0.470338	−	0.882486i	$-0.344132\pi$
$499$	21.0131			0.940676			0.470338	+	0.882486i	$0.344132\pi$
$500$		0			0
$501$		−	12.9029i		−	0.576460i
$502$		0			0
$503$			14.5500i			0.648751i	0.945928	+	0.324375i	$0.105154\pi$
$503$			14.5500i			0.648751i	−0.945928	+	0.324375i	$0.894846\pi$
$504$		0			0
$505$		−	1.48883i		−	0.0662522i
$506$		0			0
$507$			29.1303i			1.29372i
$508$		0			0
$509$	−12.2295			−0.542062			−0.271031	−	0.962571i	$-0.587365\pi$
$509$	−12.2295			−0.542062			−0.271031	+	0.962571i	$0.587365\pi$
$510$		0			0
$511$		−	38.7087i		−	1.71237i
$512$		0			0
$513$	8.87275			0.391741
$514$		0			0
$515$	−11.2978			−0.497842
$516$		0			0
$517$	−5.35673			−0.235589
$518$		0			0
$519$		−	18.0368i		−	0.791728i
$520$		0			0
$521$	17.5160			0.767391			0.383695	−	0.923460i	$-0.374651\pi$
$521$	17.5160			0.767391			0.383695	+	0.923460i	$0.374651\pi$
$522$		0			0
$523$	10.8840			0.475926			0.237963	−	0.971274i	$-0.423520\pi$
$523$	10.8840			0.475926			0.237963	+	0.971274i	$0.423520\pi$
$524$		0			0
$525$			9.76282i			0.426085i
$526$		0			0
$527$	−0.0927485			−0.00404019
$528$		0			0
$529$	8.07965			0.351289
$530$		0			0
$531$	−6.22947			−0.270336
$532$		0			0
$533$		−	6.28017i		−	0.272025i
$534$		0			0
$535$	2.93164			0.126746
$536$		0			0
$537$		−	10.2678i		−	0.443089i
$538$		0			0
$539$			24.9802i			1.07597i
$540$		0			0
$541$		−	35.7572i		−	1.53732i	−0.639657	−	0.768661i	$-0.720923\pi$
$541$		−	35.7572i		−	1.53732i	0.639657	−	0.768661i	$-0.279077\pi$
$542$		0			0
$543$		−	31.2823i		−	1.34245i
$544$		0			0
$545$	−17.0796			−0.731612
$546$		0			0
$547$	−12.6320			−0.540105			−0.270052	−	0.962846i	$-0.587041\pi$
$547$	−12.6320			−0.540105			−0.270052	+	0.962846i	$0.587041\pi$
$548$		0			0
$549$		−	2.31866i		−	0.0989580i
$550$		0			0
$551$	−7.92982	−	31.0980i	−0.337822	−	1.32482i
$552$		0			0
$553$		−	20.9884i		−	0.892516i
$554$		0			0
$555$	−9.35673			−0.397171
$556$		0			0
$557$	−21.0095			−0.890200			−0.445100	−	0.895481i	$-0.646832\pi$
$557$	−21.0095			−0.890200			−0.445100	+	0.895481i	$0.646832\pi$
$558$		0			0
$559$		−	3.20073i		−	0.135376i
$560$		0			0
$561$			3.14009i			0.132575i
$562$		0			0
$563$		−	45.6782i		−	1.92511i	−0.271090	−	0.962554i	$-0.587384\pi$
$563$		−	45.6782i		−	1.92511i	0.271090	−	0.962554i	$-0.412616\pi$
$564$		0			0
$565$			13.4891i			0.567491i
$566$		0			0
$567$	44.3585			1.86288
$568$		0			0
$569$		−	38.5463i		−	1.61595i	−0.589220	−	0.807973i	$-0.700565\pi$
$569$		−	38.5463i		−	1.61595i	0.589220	−	0.807973i	$-0.299435\pi$
$570$		0			0
$571$	25.7229			1.07647			0.538235	−	0.842795i	$-0.319091\pi$
$571$	25.7229			1.07647			0.538235	+	0.842795i	$0.319091\pi$
$572$		0			0
$573$	−26.7836			−1.11890
$574$		0			0
$575$	5.57491			0.232490
$576$		0			0
$577$		−	31.7145i		−	1.32029i	−0.751138	−	0.660145i	$-0.770495\pi$
$577$		−	31.7145i		−	1.32029i	0.751138	−	0.660145i	$-0.229505\pi$
$578$		0			0
$579$	22.7836			0.946855
$580$		0			0
$581$	−34.6658			−1.43818
$582$		0			0
$583$			30.2723i			1.25375i
$584$		0			0
$585$	1.51602			0.0626798
$586$		0			0
$587$	−37.5975			−1.55181			−0.775907	−	0.630847i	$-0.782707\pi$
$587$	−37.5975			−1.55181			−0.775907	+	0.630847i	$0.782707\pi$
$588$		0			0
$589$	−0.942924			−0.0388525
$590$		0			0
$591$			11.9191i			0.490285i
$592$		0			0
$593$	36.2295			1.48777			0.743883	−	0.668310i	$-0.232982\pi$
$593$	36.2295			1.48777			0.743883	+	0.668310i	$0.232982\pi$
$594$		0			0
$595$		−	2.47268i		−	0.101370i
$596$		0			0
$597$		−	41.1814i		−	1.68544i
$598$		0			0
$599$		−	3.48685i		−	0.142469i	−0.997460	−	0.0712344i	$-0.977306\pi$
$599$		−	3.48685i		−	0.142469i	0.997460	−	0.0712344i	$-0.0226938\pi$
$600$		0			0
$601$			4.81746i			0.196508i	0.995161	+	0.0982542i	$0.0313258\pi$
$601$			4.81746i			0.196508i	−0.995161	+	0.0982542i	$0.968674\pi$
$602$		0			0
$603$	3.71164			0.151150
$604$		0			0
$605$	5.64327			0.229432
$606$		0			0
$607$		−	7.57626i		−	0.307511i	−0.988109	−	0.153756i	$-0.950863\pi$
$607$		−	7.57626i		−	0.307511i	0.988109	−	0.153756i	$-0.0491368\pi$
$608$		0			0
$609$	−12.9905	−	50.9442i	−0.526403	−	2.06436i
$610$		0			0
$611$		−	1.48883i		−	0.0602317i
$612$		0			0
$613$	4.15930			0.167992			0.0839962	−	0.996466i	$-0.473232\pi$
$613$	4.15930			0.167992			0.0839962	+	0.996466i	$0.473232\pi$
$614$		0			0
$615$	22.5957			0.911145
$616$		0			0
$617$			25.7246i			1.03563i	0.855491	+	0.517817i	$0.173255\pi$
$617$			25.7246i			1.03563i	−0.855491	+	0.517817i	$0.826745\pi$
$618$		0			0
$619$		−	31.1997i		−	1.25402i	−0.779010	−	0.627012i	$-0.784278\pi$
$619$		−	31.1997i		−	1.25402i	0.779010	−	0.627012i	$-0.215722\pi$
$620$		0			0
$621$		−	8.30011i		−	0.333072i
$622$		0			0
$623$			30.7512i			1.23202i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			31.9236i			1.27490i
$628$		0			0
$629$	2.36983			0.0944912
$630$		0			0
$631$	−22.1593			−0.882148			−0.441074	−	0.897471i	$-0.645402\pi$
$631$	−22.1593			−0.882148			−0.441074	+	0.897471i	$0.645402\pi$
$632$		0			0
$633$	43.6564			1.73519
$634$		0			0
$635$		−	11.0934i		−	0.440230i
$636$		0			0
$637$	−6.94292			−0.275089
$638$		0			0
$639$	−22.0512			−0.872333
$640$		0			0
$641$		−	22.3754i		−	0.883776i	−0.897070	−	0.441888i	$-0.854309\pi$
$641$		−	22.3754i		−	0.883776i	0.897070	−	0.441888i	$-0.145691\pi$
$642$		0			0
$643$	20.1480			0.794560			0.397280	−	0.917697i	$-0.369954\pi$
$643$	20.1480			0.794560			0.397280	+	0.917697i	$0.369954\pi$
$644$		0			0
$645$	11.5160			0.453443
$646$		0			0
$647$	−34.9542			−1.37419			−0.687096	−	0.726567i	$-0.741115\pi$
$647$	−34.9542			−1.37419			−0.687096	+	0.726567i	$0.741115\pi$
$648$		0			0
$649$			6.11775i			0.240143i
$650$		0			0
$651$	−1.54468			−0.0605410
$652$		0			0
$653$			17.3227i			0.677890i	0.940806	+	0.338945i	$0.110070\pi$
$653$			17.3227i			0.677890i	−0.940806	+	0.338945i	$0.889930\pi$
$654$		0			0
$655$			7.13192i			0.278667i
$656$		0			0
$657$		−	21.6268i		−	0.843742i
$658$		0			0
$659$			17.1851i			0.669435i	0.942318	+	0.334718i	$0.108641\pi$
$659$			17.1851i			0.669435i	−0.942318	+	0.334718i	$0.891359\pi$
$660$		0			0
$661$	24.0891			0.936958			0.468479	−	0.883475i	$-0.344802\pi$
$661$	24.0891			0.936958			0.468479	+	0.883475i	$0.344802\pi$
$662$		0			0
$663$	−0.872747			−0.0338947
$664$		0			0
$665$		−	25.1384i		−	0.974826i
$666$		0			0
$667$	−29.0909	+	7.41804i	−1.12641	+	0.287228i
$668$		0			0
$669$			31.4186i			1.21471i
$670$		0			0
$671$	−2.27708			−0.0879056
$672$		0			0
$673$	−31.4495			−1.21229			−0.606144	−	0.795355i	$-0.707285\pi$
$673$	−31.4495			−1.21229			−0.606144	+	0.795355i	$0.707285\pi$
$674$		0			0
$675$		−	1.48883i		−	0.0573052i
$676$		0			0
$677$		−	29.0187i		−	1.11528i	−0.830083	−	0.557640i	$-0.811707\pi$
$677$		−	29.0187i		−	1.11528i	0.830083	−	0.557640i	$-0.188293\pi$
$678$		0			0
$679$			12.9029i			0.495168i
$680$		0			0
$681$			16.2054i			0.620993i
$682$		0			0
$683$	−12.2884			−0.470201			−0.235101	−	0.971971i	$-0.575542\pi$
$683$	−12.2884			−0.470201			−0.235101	+	0.971971i	$0.575542\pi$
$684$		0			0
$685$			14.7894i			0.565074i
$686$		0			0
$687$	55.9048			2.13290
$688$		0			0
$689$	−8.41380			−0.320540
$690$		0			0
$691$	41.6753			1.58540			0.792702	−	0.609609i	$-0.208674\pi$
$691$	41.6753			1.58540			0.792702	+	0.609609i	$0.208674\pi$
$692$		0			0
$693$			23.0083i			0.874013i
$694$		0			0
$695$	−3.56363			−0.135176
$696$		0			0
$697$	−5.72292			−0.216771
$698$		0			0
$699$		−	11.9191i		−	0.450820i
$700$		0			0
$701$	27.8157			1.05058			0.525292	−	0.850922i	$-0.323956\pi$
$701$	27.8157			1.05058			0.525292	+	0.850922i	$0.323956\pi$
$702$		0			0
$703$	24.0927			0.908675
$704$		0			0
$705$	5.35673			0.201746
$706$		0			0
$707$			6.28017i			0.236190i
$708$		0			0
$709$	−0.643274			−0.0241587			−0.0120793	−	0.999927i	$-0.503845\pi$
$709$	−0.643274			−0.0241587			−0.0120793	+	0.999927i	$0.503845\pi$
$710$		0			0
$711$		−	11.7263i		−	0.439771i
$712$		0			0
$713$			0.882069i			0.0330337i
$714$		0			0
$715$		−	1.48883i		−	0.0556792i
$716$		0			0
$717$			33.8913i			1.26569i
$718$		0			0
$719$	−21.2389			−0.792079			−0.396039	−	0.918233i	$-0.629616\pi$
$719$	−21.2389			−0.792079			−0.396039	+	0.918233i	$0.629616\pi$
$720$		0			0
$721$	47.6564			1.77482
$722$		0			0
$723$		−	4.62892i		−	0.172151i
$724$		0			0
$725$	−5.21819	+	1.33061i	−0.193799	+	0.0494177i
$726$		0			0
$727$			19.0771i			0.707531i	0.935334	+	0.353765i	$0.115099\pi$
$727$			19.0771i			0.707531i	−0.935334	+	0.353765i	$0.884901\pi$
$728$		0			0
$729$	14.4458			0.535031
$730$		0			0
$731$	−2.91672			−0.107879
$732$		0			0
$733$		−	39.4835i		−	1.45836i	−0.684324	−	0.729178i	$-0.739903\pi$
$733$		−	39.4835i		−	1.45836i	0.684324	−	0.729178i	$-0.260097\pi$
$734$		0			0
$735$		−	24.9802i		−	0.921408i
$736$		0			0
$737$		−	3.64507i		−	0.134268i
$738$		0			0
$739$			5.95953i			0.219225i	0.993974	+	0.109612i	$0.0349610\pi$
$739$			5.95953i			0.219225i	−0.993974	+	0.109612i	$0.965039\pi$
$740$		0			0
$741$	−8.87275			−0.325948
$742$		0			0
$743$		−	18.0065i		−	0.660594i	−0.943877	−	0.330297i	$-0.892851\pi$
$743$		−	18.0065i		−	0.660594i	0.943877	−	0.330297i	$-0.107149\pi$
$744$		0			0
$745$	6.00000			0.219823
$746$		0			0
$747$	−19.3680			−0.708638
$748$		0			0
$749$	−12.3662			−0.451851
$750$		0			0
$751$		−	25.8623i		−	0.943727i	−0.881671	−	0.471864i	$-0.843581\pi$
$751$		−	25.8623i		−	0.943727i	0.881671	−	0.471864i	$-0.156419\pi$
$752$		0			0
$753$	52.3888			1.90915
$754$		0			0
$755$	−4.50655			−0.164010
$756$		0			0
$757$			53.0193i			1.92702i	0.267676	+	0.963509i	$0.413744\pi$
$757$			53.0193i			1.92702i	−0.267676	+	0.963509i	$0.586256\pi$
$758$		0			0
$759$	29.8633			1.08397
$760$		0			0
$761$	−34.8953			−1.26495			−0.632477	−	0.774579i	$-0.717962\pi$
$761$	−34.8953			−1.26495			−0.632477	+	0.774579i	$0.717962\pi$
$762$		0			0
$763$	72.0451			2.60821
$764$		0			0
$765$		−	1.38150i		−	0.0499483i
$766$		0			0
$767$	−1.70035			−0.0613961
$768$		0			0
$769$			30.5888i			1.10306i	0.834155	+	0.551530i	$0.185956\pi$
$769$			30.5888i			1.10306i	−0.834155	+	0.551530i	$0.814044\pi$
$770$		0			0
$771$		−	25.2747i		−	0.910247i
$772$		0			0
$773$		−	11.9658i		−	0.430378i	−0.976572	−	0.215189i	$-0.930963\pi$
$773$		−	11.9658i		−	0.430378i	0.976572	−	0.215189i	$-0.0690368\pi$
$774$		0			0
$775$			0.158221i			0.00568347i
$776$		0			0
$777$	39.4684			1.41592
$778$		0			0
$779$	−58.1819			−2.08458
$780$		0			0
$781$			21.6558i			0.774904i
$782$		0			0
$783$	1.98106	+	7.76901i	0.0707973	+	0.277642i
$784$		0			0
$785$		−	16.9456i		−	0.604816i
$786$		0			0
$787$	21.2087			0.756009			0.378005	−	0.925804i	$-0.376610\pi$
$787$	21.2087			0.756009			0.378005	+	0.925804i	$0.376610\pi$
$788$		0			0
$789$	−22.9619			−0.817464
$790$		0			0
$791$		−	56.8996i		−	2.02312i
$792$		0			0
$793$		−	0.632884i		−	0.0224744i
$794$		0			0
$795$		−	30.2723i		−	1.07365i
$796$		0			0
$797$			8.32068i			0.294734i	0.989082	+	0.147367i	$0.0470798\pi$
$797$			8.32068i			0.294734i	−0.989082	+	0.147367i	$0.952920\pi$
$798$		0			0
$799$	−1.35673			−0.0479975
$800$		0			0
$801$			17.1809i			0.607056i
$802$		0			0
$803$	−21.2389			−0.749506
$804$		0			0
$805$	−23.5160			−0.828831
$806$		0			0
$807$	−28.7550			−1.01222
$808$		0			0
$809$		−	23.4610i		−	0.824846i	−0.910992	−	0.412423i	$-0.864683\pi$
$809$		−	23.4610i		−	0.824846i	0.910992	−	0.412423i	$-0.135317\pi$
$810$		0			0
$811$	7.00947			0.246136			0.123068	−	0.992398i	$-0.460727\pi$
$811$	7.00947			0.246136			0.123068	+	0.992398i	$0.460727\pi$
$812$		0			0
$813$	63.1022			2.21309
$814$		0			0
$815$			11.0934i			0.388586i
$816$		0			0
$817$	−29.6527			−1.03742
$818$		0			0
$819$	−6.39486			−0.223455
$820$		0			0
$821$	1.44584			0.0504603			0.0252302	−	0.999682i	$-0.491968\pi$
$821$	1.44584			0.0504603			0.0252302	+	0.999682i	$0.491968\pi$
$822$		0			0
$823$		−	47.1671i		−	1.64414i	−0.569386	−	0.822070i	$-0.692819\pi$
$823$		−	47.1671i		−	1.64414i	0.569386	−	0.822070i	$-0.307181\pi$
$824$		0			0
$825$	5.35673			0.186497
$826$		0			0
$827$			20.1366i			0.700219i	0.936709	+	0.350109i	$0.113856\pi$
$827$			20.1366i			0.700219i	−0.936709	+	0.350109i	$0.886144\pi$
$828$		0			0
$829$			12.8011i			0.444602i	0.974978	+	0.222301i	$0.0713567\pi$
$829$			12.8011i			0.444602i	−0.974978	+	0.222301i	$0.928643\pi$
$830$		0			0
$831$			6.75904i			0.234468i
$832$		0			0
$833$			6.32686i			0.219213i
$834$		0			0
$835$	5.57491			0.192928
$836$		0			0
$837$	0.235565			0.00814231
$838$		0			0
$839$			16.8341i			0.581178i	0.956848	+	0.290589i	$0.0938512\pi$
$839$			16.8341i			0.581178i	−0.956848	+	0.290589i	$0.906149\pi$
$840$		0			0
$841$	25.4589	−	13.8868i	0.877895	−	0.478854i
$842$		0			0
$843$			42.5078i			1.46405i
$844$		0			0
$845$	−12.5862			−0.432978
$846$		0			0
$847$	−23.8044			−0.817928
$848$		0			0
$849$		−	34.3963i		−	1.18048i
$850$		0			0
$851$		−	22.5378i		−	0.772587i
$852$		0			0
$853$		−	40.1164i		−	1.37356i	−0.726866	−	0.686779i	$-0.759024\pi$
$853$		−	40.1164i		−	1.37356i	0.726866	−	0.686779i	$-0.240976\pi$
$854$		0			0
$855$		−	14.0450i		−	0.480328i
$856$		0			0
$857$	15.2389			0.520552			0.260276	−	0.965534i	$-0.416186\pi$
$857$	15.2389			0.520552			0.260276	+	0.965534i	$0.416186\pi$
$858$		0			0
$859$			10.7770i			0.367706i	0.982954	+	0.183853i	$0.0588571\pi$
$859$			10.7770i			0.367706i	−0.982954	+	0.183853i	$0.941143\pi$
$860$		0			0
$861$	−95.3128			−3.24825
$862$		0			0
$863$	32.1004			1.09271			0.546355	−	0.837554i	$-0.316015\pi$
$863$	32.1004			1.09271			0.546355	+	0.837554i	$0.316015\pi$
$864$		0			0
$865$	7.79310			0.264973
$866$		0			0
$867$		−	38.5505i		−	1.30924i
$868$		0			0
$869$	−11.5160			−0.390654
$870$		0			0
$871$	1.01310			0.0343276
$872$		0			0
$873$			7.20893i			0.243985i
$874$		0			0
$875$	−4.21819			−0.142601
$876$		0			0
$877$	28.2520			0.954004			0.477002	−	0.878902i	$-0.341723\pi$
$877$	28.2520			0.954004			0.477002	+	0.878902i	$0.341723\pi$
$878$		0			0
$879$	3.63380			0.122565
$880$		0			0
$881$		−	8.59043i		−	0.289419i	−0.989474	−	0.144710i	$-0.953775\pi$
$881$		−	8.59043i		−	0.289419i	0.989474	−	0.144710i	$-0.0462248\pi$
$882$		0			0
$883$	34.7913			1.17082			0.585410	−	0.810737i	$-0.300934\pi$
$883$	34.7913			1.17082			0.585410	+	0.810737i	$0.300934\pi$
$884$		0			0
$885$		−	6.11775i		−	0.205646i
$886$		0			0
$887$			40.3558i			1.35501i	0.735516	+	0.677507i	$0.236940\pi$
$887$			40.3558i			1.35501i	−0.735516	+	0.677507i	$0.763060\pi$
$888$		0			0
$889$			46.7942i			1.56943i
$890$		0			0
$891$		−	24.3389i		−	0.815384i
$892$		0			0
$893$	−13.7931			−0.461568
$894$		0			0
$895$	4.43637			0.148292
$896$		0			0
$897$			8.30011i			0.277133i
$898$		0			0
$899$	−0.210531	−	0.825627i	−0.00702160	−	0.0275362i
$900$		0			0
$901$			7.66723i			0.255432i
$902$		0			0
$903$	−48.5767			−1.61653
$904$		0			0
$905$	13.5160			0.449288
$906$		0			0
$907$		−	8.59463i		−	0.285380i	−0.989767	−	0.142690i	$-0.954425\pi$
$907$		−	8.59463i		−	0.285380i	0.989767	−	0.142690i	$-0.0455752\pi$
$908$		0			0
$909$			3.50877i			0.116379i
$910$		0			0
$911$			12.9332i			0.428497i	0.976779	+	0.214249i	$0.0687303\pi$
$911$			12.9332i			0.428497i	−0.976779	+	0.214249i	$0.931270\pi$
$912$		0			0
$913$			19.0207i			0.629492i
$914$		0			0
$915$	2.27708			0.0752779
$916$		0			0
$917$		−	30.0838i		−	0.993454i
$918$		0			0
$919$	−52.1153			−1.71913			−0.859563	−	0.511030i	$-0.829264\pi$
$919$	−52.1153			−1.71913			−0.859563	+	0.511030i	$0.829264\pi$
$920$		0			0
$921$	−10.7836			−0.355333
$922$		0			0
$923$	−6.01894			−0.198116
$924$		0			0
$925$		−	4.04272i		−	0.132924i
$926$		0			0
$927$	26.6259			0.874509
$928$		0			0
$929$	−26.5993			−0.872695			−0.436347	−	0.899778i	$-0.643728\pi$
$929$	−26.5993			−0.872695			−0.436347	+	0.899778i	$0.643728\pi$
$930$		0			0
$931$			64.3218i			2.10806i
$932$		0			0
$933$	50.1117			1.64058
$934$		0			0
$935$	−1.35673			−0.0443697
$936$		0			0
$937$	−14.5291			−0.474646			−0.237323	−	0.971431i	$-0.576270\pi$
$937$	−14.5291			−0.474646			−0.237323	+	0.971431i	$0.576270\pi$
$938$		0			0
$939$		−	26.2847i		−	0.857768i
$940$		0			0
$941$	3.14619			0.102563			0.0512815	−	0.998684i	$-0.483669\pi$
$941$	3.14619			0.102563			0.0512815	+	0.998684i	$0.483669\pi$
$942$		0			0
$943$			54.4269i			1.77238i
$944$		0			0
$945$			6.28017i			0.204294i
$946$		0			0
$947$		−	51.7960i		−	1.68314i	−0.540145	−	0.841572i	$-0.681631\pi$
$947$		−	51.7960i		−	1.68314i	0.540145	−	0.841572i	$-0.318369\pi$
$948$		0			0
$949$		−	5.90309i		−	0.191622i
$950$		0			0
$951$	−21.6753			−0.702870
$952$		0			0
$953$	−10.5542			−0.341883			−0.170941	−	0.985281i	$-0.554681\pi$
$953$	−10.5542			−0.341883			−0.170941	+	0.985281i	$0.554681\pi$
$954$		0			0
$955$		−	11.5723i		−	0.374471i
$956$		0			0
$957$	−27.9524	+	7.12772i	−0.903573	+	0.230407i
$958$		0			0
$959$		−	62.3844i		−	2.01450i
$960$		0			0
$961$	30.9750			0.999192
$962$		0			0
$963$	−6.90907			−0.222642
$964$		0			0
$965$			9.84404i			0.316891i
$966$		0			0
$967$			21.0448i			0.676755i	0.941010	+	0.338378i	$0.109878\pi$
$967$			21.0448i			0.676755i	−0.941010	+	0.338378i	$0.890122\pi$
$968$		0			0
$969$			8.08545i			0.259742i
$970$		0			0
$971$			1.01417i			0.0325462i	0.999868	+	0.0162731i	$0.00518012\pi$
$971$			1.01417i			0.0325462i	−0.999868	+	0.0162731i	$0.994820\pi$
$972$		0			0
$973$	15.0320			0.481905
$974$		0			0
$975$			1.48883i			0.0476808i
$976$		0			0
$977$	9.95239			0.318405			0.159203	−	0.987246i	$-0.449108\pi$
$977$	9.95239			0.318405			0.159203	+	0.987246i	$0.449108\pi$
$978$		0			0
$979$	16.8727			0.539255
$980$		0			0
$981$	40.2520			1.28515
$982$		0			0
$983$		−	15.4059i		−	0.491372i	−0.969349	−	0.245686i	$-0.920987\pi$
$983$		−	15.4059i		−	0.491372i	0.969349	−	0.245686i	$-0.0790133\pi$
$984$		0			0
$985$	−5.14982			−0.164087
$986$		0			0
$987$	−22.5957			−0.719228
$988$		0			0
$989$			27.7390i			0.882049i
$990$		0			0
$991$	8.92035			0.283364			0.141682	−	0.989912i	$-0.454749\pi$
$991$	8.92035			0.283364			0.141682	+	0.989912i	$0.454749\pi$
$992$		0			0
$993$	−42.8441			−1.35962
$994$		0			0
$995$	17.7931			0.564079
$996$		0			0
$997$			1.19296i			0.0377814i	0.999822	+	0.0188907i	$0.00601345\pi$
$997$			1.19296i			0.0377814i	−0.999822	+	0.0188907i	$0.993987\pi$
$998$		0			0
$999$	−6.01894			−0.190431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.g.i.1681.2		6
4.3	odd	2		145.2.c.b.86.6	yes	6
12.11	even	2		1305.2.d.b.811.1		6
20.3	even	4		725.2.d.c.724.11		12
20.7	even	4		725.2.d.c.724.2		12
20.19	odd	2		725.2.c.e.376.1		6
29.28	even	2	inner	2320.2.g.i.1681.5		6
116.75	even	4		4205.2.a.m.1.6		6
116.99	even	4		4205.2.a.m.1.1		6
116.115	odd	2		145.2.c.b.86.1	✓	6
348.347	even	2		1305.2.d.b.811.6		6
580.347	even	4		725.2.d.c.724.12		12
580.463	even	4		725.2.d.c.724.1		12
580.579	odd	2		725.2.c.e.376.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.c.b.86.1	✓	6	116.115	odd	2
145.2.c.b.86.6	yes	6	4.3	odd	2
725.2.c.e.376.1		6	20.19	odd	2
725.2.c.e.376.6		6	580.579	odd	2
725.2.d.c.724.1		12	580.463	even	4
725.2.d.c.724.2		12	20.7	even	4
725.2.d.c.724.11		12	20.3	even	4
725.2.d.c.724.12		12	580.347	even	4
1305.2.d.b.811.1		6	12.11	even	2
1305.2.d.b.811.6		6	348.347	even	2
2320.2.g.i.1681.2		6	1.1	even	1	trivial
2320.2.g.i.1681.5		6	29.28	even	2	inner
4205.2.a.m.1.1		6	116.99	even	4
4205.2.a.m.1.6		6	116.75	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:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2320.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.31446i q^{3} +1.00000 q^{5} -4.21819 q^{7} -2.35673 q^{9} +O(q^{10})\)	\(q-2.31446i q^{3} +1.00000 q^{5} -4.21819 q^{7} -2.35673 q^{9} +2.31446i q^{11} -0.643274 q^{13} -2.31446i q^{15} +0.586195i q^{17} +5.95953i q^{19} +9.76282i q^{21} +5.57491 q^{23} +1.00000 q^{25} -1.48883i q^{27} +(-5.21819 + 1.33061i) q^{29} +0.158221i q^{31} +5.35673 q^{33} -4.21819 q^{35} -4.04272i q^{37} +1.48883i q^{39} +9.76282i q^{41} +4.97568i q^{43} -2.35673 q^{45} +2.31446i q^{47} +10.7931 q^{49} +1.35673 q^{51} +13.0796 q^{53} +2.31446i q^{55} +13.7931 q^{57} +2.64327 q^{59} +0.983848i q^{61} +9.94111 q^{63} -0.643274 q^{65} -1.57491 q^{67} -12.9029i q^{69} +9.35673 q^{71} +9.17663i q^{73} -2.31446i q^{75} -9.76282i q^{77} +4.97568i q^{79} -10.5160 q^{81} +8.21819 q^{83} +0.586195i q^{85} +(3.07965 + 12.0773i) q^{87} -7.29014i q^{89} +2.71345 q^{91} +0.366196 q^{93} +5.95953i q^{95} -3.05888i q^{97} -5.45455i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{5} - 14 q^{9}+O(q^{10}) \) 6 * q + 6 * q^5 - 14 * q^9	\( 6 q + 6 q^{5} - 14 q^{9} - 4 q^{13} + 8 q^{23} + 6 q^{25} - 6 q^{29} + 32 q^{33} - 14 q^{45} + 14 q^{49} + 8 q^{51} + 28 q^{53} + 32 q^{57} + 16 q^{59} - 16 q^{63} - 4 q^{65} + 16 q^{67} + 56 q^{71} + 38 q^{81} + 24 q^{83} - 32 q^{87} + 16 q^{91} - 48 q^{93}+O(q^{100}) \) 6 * q + 6 * q^5 - 14 * q^9 - 4 * q^13 + 8 * q^23 + 6 * q^25 - 6 * q^29 + 32 * q^33 - 14 * q^45 + 14 * q^49 + 8 * q^51 + 28 * q^53 + 32 * q^57 + 16 * q^59 - 16 * q^63 - 4 * q^65 + 16 * q^67 + 56 * q^71 + 38 * q^81 + 24 * q^83 - 32 * q^87 + 16 * q^91 - 48 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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