LMFDB - Embedded newform 4004.2.m.b.2157.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(2157,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, 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("4004.2157");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.m (of order $2$, degree $1$, 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:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2157.14
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.b.2157.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0797053 q^{3} -0.579061i q^{5} +1.00000i q^{7} -2.99365 q^{9} +O(q^{10})$	$q-0.0797053 q^{3} -0.579061i q^{5} +1.00000i q^{7} -2.99365 q^{9} +1.00000i q^{11} +(-1.19435 - 3.40199i) q^{13} +0.0461542i q^{15} +6.63982 q^{17} +3.33399i q^{19} -0.0797053i q^{21} -8.63162 q^{23} +4.66469 q^{25} +0.477726 q^{27} +0.664989 q^{29} -0.646758i q^{31} -0.0797053i q^{33} +0.579061 q^{35} -0.0692840i q^{37} +(0.0951964 + 0.271157i) q^{39} +0.306795i q^{41} -5.52303 q^{43} +1.73350i q^{45} -2.26906i q^{47} -1.00000 q^{49} -0.529229 q^{51} +4.29017 q^{53} +0.579061 q^{55} -0.265737i q^{57} +11.0513i q^{59} -2.98125 q^{61} -2.99365i q^{63} +(-1.96996 + 0.691604i) q^{65} +2.33089i q^{67} +0.687986 q^{69} -3.49278i q^{71} +9.99030i q^{73} -0.371801 q^{75} -1.00000 q^{77} -12.0953 q^{79} +8.94286 q^{81} +9.43919i q^{83} -3.84486i q^{85} -0.0530032 q^{87} +6.81854i q^{89} +(3.40199 - 1.19435i) q^{91} +0.0515501i q^{93} +1.93058 q^{95} +14.5484i q^{97} -2.99365i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 18 q^{9}+O(q^{10}) $ 30 * q + 18 * q^9	$ 30 q + 18 q^{9} + 4 q^{13} + 12 q^{17} + 6 q^{23} - 6 q^{29} - 2 q^{35} + 8 q^{39} - 22 q^{43} - 30 q^{49} + 60 q^{51} - 38 q^{53} - 2 q^{55} - 36 q^{61} + 10 q^{65} + 36 q^{69} - 20 q^{75} - 30 q^{77} - 10 q^{79} - 42 q^{81} + 36 q^{87} + 6 q^{91} - 2 q^{95}+O(q^{100}) $ 30 * q + 18 * q^9 + 4 * q^13 + 12 * q^17 + 6 * q^23 - 6 * q^29 - 2 * q^35 + 8 * q^39 - 22 * q^43 - 30 * q^49 + 60 * q^51 - 38 * q^53 - 2 * q^55 - 36 * q^61 + 10 * q^65 + 36 * q^69 - 20 * q^75 - 30 * q^77 - 10 * q^79 - 42 * q^81 + 36 * q^87 + 6 * q^91 - 2 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times$.

$n$	$365$	$925$	$2003$	$3433$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	−0.0797053			−0.0460179			−0.0230090	−	0.999735i	$-0.507325\pi$
$3$	−0.0797053			−0.0460179			−0.0230090	+	0.999735i	$0.507325\pi$
$4$		0			0
$5$		−	0.579061i		−	0.258964i	−0.991582	−	0.129482i	$-0.958669\pi$
$5$		−	0.579061i		−	0.258964i	0.991582	−	0.129482i	$-0.0413314\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−2.99365			−0.997882
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	−1.19435	−	3.40199i	−0.331254	−	0.943542i
$13$	−1.19435	−	3.40199i	−0.331254	−	0.943542i
$14$		0			0
$15$			0.0461542i			0.0119170i
$16$		0			0
$17$	6.63982			1.61039			0.805196	−	0.593008i	$-0.202060\pi$
$17$	6.63982			1.61039			0.805196	+	0.593008i	$0.202060\pi$
$18$		0			0
$19$			3.33399i			0.764870i	0.923982	+	0.382435i	$0.124914\pi$
$19$			3.33399i			0.764870i	−0.923982	+	0.382435i	$0.875086\pi$
$20$		0			0
$21$		−	0.0797053i		−	0.0173931i
$22$		0			0
$23$	−8.63162			−1.79982			−0.899909	−	0.436079i	$-0.856367\pi$
$23$	−8.63162			−1.79982			−0.899909	+	0.436079i	$0.856367\pi$
$24$		0			0
$25$	4.66469			0.932938
$26$		0			0
$27$	0.477726			0.0919384
$28$		0			0
$29$	0.664989			0.123485			0.0617427	−	0.998092i	$-0.480334\pi$
$29$	0.664989			0.123485			0.0617427	+	0.998092i	$0.480334\pi$
$30$		0			0
$31$		−	0.646758i		−	0.116161i	−0.998312	−	0.0580806i	$-0.981502\pi$
$31$		−	0.646758i		−	0.116161i	0.998312	−	0.0580806i	$-0.0184980\pi$
$32$		0			0
$33$		−	0.0797053i		−	0.0138749i
$34$		0			0
$35$	0.579061			0.0978791
$36$		0			0
$37$		−	0.0692840i		−	0.0113902i	−0.999984	−	0.00569511i	$-0.998187\pi$
$37$		−	0.0692840i		−	0.0113902i	0.999984	−	0.00569511i	$-0.00181282\pi$
$38$		0			0
$39$	0.0951964	+	0.271157i	0.0152436	+	0.0434198i
$40$		0			0
$41$			0.306795i			0.0479133i	0.999713	+	0.0239567i	$0.00762637\pi$
$41$			0.306795i			0.0479133i	−0.999713	+	0.0239567i	$0.992374\pi$
$42$		0			0
$43$	−5.52303			−0.842254			−0.421127	−	0.907002i	$-0.638365\pi$
$43$	−5.52303			−0.842254			−0.421127	+	0.907002i	$0.638365\pi$
$44$		0			0
$45$			1.73350i			0.258415i
$46$		0			0
$47$		−	2.26906i		−	0.330977i	−0.986212	−	0.165489i	$-0.947080\pi$
$47$		−	2.26906i		−	0.330977i	0.986212	−	0.165489i	$-0.0529201\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−0.529229			−0.0741069
$52$		0			0
$53$	4.29017			0.589300			0.294650	−	0.955605i	$-0.404797\pi$
$53$	4.29017			0.589300			0.294650	+	0.955605i	$0.404797\pi$
$54$		0			0
$55$	0.579061			0.0780805
$56$		0			0
$57$		−	0.265737i		−	0.0351977i
$58$		0			0
$59$			11.0513i			1.43876i	0.694615	+	0.719381i	$0.255575\pi$
$59$			11.0513i			1.43876i	−0.694615	+	0.719381i	$0.744425\pi$
$60$		0			0
$61$	−2.98125			−0.381710			−0.190855	−	0.981618i	$-0.561126\pi$
$61$	−2.98125			−0.381710			−0.190855	+	0.981618i	$0.561126\pi$
$62$		0			0
$63$		−	2.99365i		−	0.377164i
$64$		0			0
$65$	−1.96996	+	0.691604i	−0.244343	+	0.0857829i
$66$		0			0
$67$			2.33089i			0.284764i	0.989812	+	0.142382i	$0.0454761\pi$
$67$			2.33089i			0.284764i	−0.989812	+	0.142382i	$0.954524\pi$
$68$		0			0
$69$	0.687986			0.0828238
$70$		0			0
$71$		−	3.49278i		−	0.414517i	−0.978286	−	0.207259i	$-0.933546\pi$
$71$		−	3.49278i		−	0.414517i	0.978286	−	0.207259i	$-0.0664542\pi$
$72$		0			0
$73$			9.99030i			1.16928i	0.811294	+	0.584638i	$0.198764\pi$
$73$			9.99030i			1.16928i	−0.811294	+	0.584638i	$0.801236\pi$
$74$		0			0
$75$	−0.371801			−0.0429318
$76$		0			0
$77$	−1.00000			−0.113961
$78$		0			0
$79$	−12.0953			−1.36082			−0.680412	−	0.732830i	$-0.738199\pi$
$79$	−12.0953			−1.36082			−0.680412	+	0.732830i	$0.738199\pi$
$80$		0			0
$81$	8.94286			0.993652
$82$		0			0
$83$			9.43919i			1.03609i	0.855355	+	0.518043i	$0.173339\pi$
$83$			9.43919i			1.03609i	−0.855355	+	0.518043i	$0.826661\pi$
$84$		0			0
$85$		−	3.84486i		−	0.417033i
$86$		0			0
$87$	−0.0530032			−0.00568254
$88$		0			0
$89$			6.81854i			0.722764i	0.932418	+	0.361382i	$0.117695\pi$
$89$			6.81854i			0.722764i	−0.932418	+	0.361382i	$0.882305\pi$
$90$		0			0
$91$	3.40199	−	1.19435i	0.356625	−	0.125202i
$92$		0			0
$93$			0.0515501i			0.00534549i
$94$		0			0
$95$	1.93058			0.198074
$96$		0			0
$97$			14.5484i			1.47716i	0.674165	+	0.738581i	$0.264504\pi$
$97$			14.5484i			1.47716i	−0.674165	+	0.738581i	$0.735496\pi$
$98$		0			0
$99$		−	2.99365i		−	0.300873i
$100$		0			0
$101$	−14.3153			−1.42442			−0.712211	−	0.701966i	$-0.752306\pi$
$101$	−14.3153			−1.42442			−0.712211	+	0.701966i	$0.752306\pi$
$102$		0			0
$103$	10.0201			0.987311			0.493655	−	0.869658i	$-0.335660\pi$
$103$	10.0201			0.987311			0.493655	+	0.869658i	$0.335660\pi$
$104$		0			0
$105$	−0.0461542			−0.00450419
$106$		0			0
$107$	−8.94064			−0.864325			−0.432162	−	0.901796i	$-0.642249\pi$
$107$	−8.94064			−0.864325			−0.432162	+	0.901796i	$0.642249\pi$
$108$		0			0
$109$			7.49245i			0.717646i	0.933406	+	0.358823i	$0.116822\pi$
$109$			7.49245i			0.717646i	−0.933406	+	0.358823i	$0.883178\pi$
$110$		0			0
$111$			0.00552231i			0.000524154i
$112$		0			0
$113$	2.14873			0.202135			0.101068	−	0.994880i	$-0.467774\pi$
$113$	2.14873			0.202135			0.101068	+	0.994880i	$0.467774\pi$
$114$		0			0
$115$			4.99823i			0.466087i
$116$		0			0
$117$	3.57548	+	10.1843i	0.330553	+	0.941543i
$118$		0			0
$119$			6.63982i			0.608671i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	0.0244532i		−	0.00220487i
$124$		0			0
$125$		−	5.59644i		−	0.500561i
$126$		0			0
$127$	−9.92608			−0.880797			−0.440398	−	0.897802i	$-0.645163\pi$
$127$	−9.92608			−0.880797			−0.440398	+	0.897802i	$0.645163\pi$
$128$		0			0
$129$	0.440215			0.0387588
$130$		0			0
$131$	−5.26373			−0.459895			−0.229947	−	0.973203i	$-0.573855\pi$
$131$	−5.26373			−0.459895			−0.229947	+	0.973203i	$0.573855\pi$
$132$		0			0
$133$	−3.33399			−0.289094
$134$		0			0
$135$		−	0.276632i		−	0.0238087i
$136$		0			0
$137$			11.6164i			0.992456i	0.868192	+	0.496228i	$0.165282\pi$
$137$			11.6164i			0.992456i	−0.868192	+	0.496228i	$0.834718\pi$
$138$		0			0
$139$	−0.106286			−0.00901503			−0.00450751	−	0.999990i	$-0.501435\pi$
$139$	−0.106286			−0.00901503			−0.00450751	+	0.999990i	$0.501435\pi$
$140$		0			0
$141$			0.180857i			0.0152309i
$142$		0			0
$143$	3.40199	−	1.19435i	0.284488	−	0.0998769i
$144$		0			0
$145$		−	0.385069i		−	0.0319782i
$146$		0			0
$147$	0.0797053			0.00657399
$148$		0			0
$149$			7.90346i			0.647477i	0.946147	+	0.323738i	$0.104940\pi$
$149$			7.90346i			0.647477i	−0.946147	+	0.323738i	$0.895060\pi$
$150$		0			0
$151$			4.41831i			0.359557i	0.983707	+	0.179779i	$0.0575381\pi$
$151$			4.41831i			0.359557i	−0.983707	+	0.179779i	$0.942462\pi$
$152$		0			0
$153$	−19.8773			−1.60698
$154$		0			0
$155$	−0.374512			−0.0300815
$156$		0			0
$157$	10.4175			0.831408			0.415704	−	0.909500i	$-0.363535\pi$
$157$	10.4175			0.831408			0.415704	+	0.909500i	$0.363535\pi$
$158$		0			0
$159$	−0.341949			−0.0271183
$160$		0			0
$161$		−	8.63162i		−	0.680267i
$162$		0			0
$163$			8.54204i			0.669064i	0.942384	+	0.334532i	$0.108578\pi$
$163$			8.54204i			0.669064i	−0.942384	+	0.334532i	$0.891422\pi$
$164$		0			0
$165$	−0.0461542			−0.00359310
$166$		0			0
$167$		−	6.88449i		−	0.532738i	−0.963871	−	0.266369i	$-0.914176\pi$
$167$		−	6.88449i		−	0.532738i	0.963871	−	0.266369i	$-0.0858239\pi$
$168$		0			0
$169$	−10.1470	+	8.12636i	−0.780541	+	0.625104i
$170$		0			0
$171$		−	9.98079i		−	0.763250i
$172$		0			0
$173$	10.1150			0.769031			0.384516	−	0.923118i	$-0.374368\pi$
$173$	10.1150			0.769031			0.384516	+	0.923118i	$0.374368\pi$
$174$		0			0
$175$			4.66469i			0.352617i
$176$		0			0
$177$		−	0.880851i		−	0.0662088i
$178$		0			0
$179$	−0.992942			−0.0742160			−0.0371080	−	0.999311i	$-0.511815\pi$
$179$	−0.992942			−0.0742160			−0.0371080	+	0.999311i	$0.511815\pi$
$180$		0			0
$181$	−3.14771			−0.233968			−0.116984	−	0.993134i	$-0.537323\pi$
$181$	−3.14771			−0.233968			−0.116984	+	0.993134i	$0.537323\pi$
$182$		0			0
$183$	0.237621			0.0175655
$184$		0			0
$185$	−0.0401196			−0.00294965
$186$		0			0
$187$			6.63982i			0.485552i
$188$		0			0
$189$			0.477726i			0.0347494i
$190$		0			0
$191$	−0.813925			−0.0588935			−0.0294468	−	0.999566i	$-0.509375\pi$
$191$	−0.813925			−0.0588935			−0.0294468	+	0.999566i	$0.509375\pi$
$192$		0			0
$193$			4.02413i			0.289663i	0.989456	+	0.144832i	$0.0462640\pi$
$193$			4.02413i			0.289663i	−0.989456	+	0.144832i	$0.953736\pi$
$194$		0			0
$195$	0.157016	−	0.0551245i	0.0112442	−	0.00394755i
$196$		0			0
$197$			5.65712i			0.403053i	0.979483	+	0.201526i	$0.0645902\pi$
$197$			5.65712i			0.403053i	−0.979483	+	0.201526i	$0.935410\pi$
$198$		0			0
$199$	−24.3745			−1.72786			−0.863931	−	0.503610i	$-0.832005\pi$
$199$	−24.3745			−1.72786			−0.863931	+	0.503610i	$0.832005\pi$
$200$		0			0
$201$		−	0.185784i		−	0.0131042i
$202$		0			0
$203$			0.664989i			0.0466731i
$204$		0			0
$205$	0.177653			0.0124078
$206$		0			0
$207$	25.8400			1.79601
$208$		0			0
$209$	−3.33399			−0.230617
$210$		0			0
$211$	14.6501			1.00856			0.504279	−	0.863541i	$-0.331758\pi$
$211$	14.6501			1.00856			0.504279	+	0.863541i	$0.331758\pi$
$212$		0			0
$213$			0.278394i			0.0190752i
$214$		0			0
$215$			3.19817i			0.218113i
$216$		0			0
$217$	0.646758			0.0439048
$218$		0			0
$219$		−	0.796281i		−	0.0538077i
$220$		0			0
$221$	−7.93030	−	22.5886i	−0.533450	−	1.51947i
$222$		0			0
$223$		−	11.3616i		−	0.760826i	−0.924817	−	0.380413i	$-0.875782\pi$
$223$		−	11.3616i		−	0.760826i	0.924817	−	0.380413i	$-0.124218\pi$
$224$		0			0
$225$	−13.9644			−0.930962
$226$		0			0
$227$			6.28464i			0.417126i	0.978009	+	0.208563i	$0.0668787\pi$
$227$			6.28464i			0.417126i	−0.978009	+	0.208563i	$0.933121\pi$
$228$		0			0
$229$			11.9984i			0.792876i	0.918061	+	0.396438i	$0.129754\pi$
$229$			11.9984i			0.792876i	−0.918061	+	0.396438i	$0.870246\pi$
$230$		0			0
$231$	0.0797053			0.00524423
$232$		0			0
$233$	15.6933			1.02810			0.514050	−	0.857760i	$-0.328145\pi$
$233$	15.6933			1.02810			0.514050	+	0.857760i	$0.328145\pi$
$234$		0			0
$235$	−1.31393			−0.0857111
$236$		0			0
$237$	0.964058			0.0626223
$238$		0			0
$239$			23.1855i			1.49975i	0.661582	+	0.749873i	$0.269885\pi$
$239$			23.1855i			1.49975i	−0.661582	+	0.749873i	$0.730115\pi$
$240$		0			0
$241$		−	11.2466i		−	0.724455i	−0.932090	−	0.362228i	$-0.882016\pi$
$241$		−	11.2466i		−	0.724455i	0.932090	−	0.362228i	$-0.117984\pi$
$242$		0			0
$243$	−2.14597			−0.137664
$244$		0			0
$245$			0.579061i			0.0369948i
$246$		0			0
$247$	11.3422	−	3.98197i	0.721686	−	0.253366i
$248$		0			0
$249$		−	0.752354i		−	0.0476785i
$250$		0			0
$251$	8.75861			0.552838			0.276419	−	0.961037i	$-0.410852\pi$
$251$	8.75861			0.552838			0.276419	+	0.961037i	$0.410852\pi$
$252$		0			0
$253$		−	8.63162i		−	0.542665i
$254$		0			0
$255$			0.306456i			0.0191910i
$256$		0			0
$257$	−14.7351			−0.919149			−0.459575	−	0.888139i	$-0.651998\pi$
$257$	−14.7351			−0.919149			−0.459575	+	0.888139i	$0.651998\pi$
$258$		0			0
$259$	0.0692840			0.00430510
$260$		0			0
$261$	−1.99074			−0.123224
$262$		0			0
$263$	−0.712037			−0.0439061			−0.0219531	−	0.999759i	$-0.506988\pi$
$263$	−0.712037			−0.0439061			−0.0219531	+	0.999759i	$0.506988\pi$
$264$		0			0
$265$		−	2.48427i		−	0.152607i
$266$		0			0
$267$		−	0.543474i		−	0.0332601i
$268$		0			0
$269$	10.2597			0.625547			0.312773	−	0.949828i	$-0.398742\pi$
$269$	10.2597			0.625547			0.312773	+	0.949828i	$0.398742\pi$
$270$		0			0
$271$			1.00483i			0.0610388i	0.999534	+	0.0305194i	$0.00971614\pi$
$271$			1.00483i			0.0610388i	−0.999534	+	0.0305194i	$0.990284\pi$
$272$		0			0
$273$	−0.271157	+	0.0951964i	−0.0164111	+	0.00576155i
$274$		0			0
$275$			4.66469i			0.281291i
$276$		0			0
$277$	7.50150			0.450721			0.225361	−	0.974275i	$-0.427644\pi$
$277$	7.50150			0.450721			0.225361	+	0.974275i	$0.427644\pi$
$278$		0			0
$279$			1.93617i			0.115915i
$280$		0			0
$281$		−	22.1260i		−	1.31993i	−0.751298	−	0.659963i	$-0.770572\pi$
$281$		−	22.1260i		−	1.31993i	0.751298	−	0.659963i	$-0.229428\pi$
$282$		0			0
$283$	−3.29148			−0.195658			−0.0978291	−	0.995203i	$-0.531190\pi$
$283$	−3.29148			−0.195658			−0.0978291	+	0.995203i	$0.531190\pi$
$284$		0			0
$285$	−0.153878			−0.00911493
$286$		0			0
$287$	−0.306795			−0.0181095
$288$		0			0
$289$	27.0872			1.59337
$290$		0			0
$291$		−	1.15958i		−	0.0679759i
$292$		0			0
$293$		−	1.03351i		−	0.0603782i	−0.999544	−	0.0301891i	$-0.990389\pi$
$293$		−	1.03351i		−	0.0603782i	0.999544	−	0.0301891i	$-0.00961094\pi$
$294$		0			0
$295$	6.39940			0.372587
$296$		0			0
$297$			0.477726i			0.0277205i
$298$		0			0
$299$	10.3092	+	29.3647i	0.596197	+	1.69820i
$300$		0			0
$301$		−	5.52303i		−	0.318342i
$302$		0			0
$303$	1.14100			0.0655489
$304$		0			0
$305$			1.72632i			0.0988489i
$306$		0			0
$307$			18.3599i			1.04785i	0.851763	+	0.523927i	$0.175533\pi$
$307$			18.3599i			1.04785i	−0.851763	+	0.523927i	$0.824467\pi$
$308$		0			0
$309$	−0.798656			−0.0454340
$310$		0			0
$311$	−19.6862			−1.11630			−0.558150	−	0.829740i	$-0.688489\pi$
$311$	−19.6862			−1.11630			−0.558150	+	0.829740i	$0.688489\pi$
$312$		0			0
$313$	−2.65738			−0.150204			−0.0751020	−	0.997176i	$-0.523928\pi$
$313$	−2.65738			−0.150204			−0.0751020	+	0.997176i	$0.523928\pi$
$314$		0			0
$315$	−1.73350			−0.0976718
$316$		0			0
$317$		−	5.83514i		−	0.327734i	−0.986482	−	0.163867i	$-0.947603\pi$
$317$		−	5.83514i		−	0.327734i	0.986482	−	0.163867i	$-0.0523968\pi$
$318$		0			0
$319$			0.664989i			0.0372322i
$320$		0			0
$321$	0.712617			0.0397744
$322$		0			0
$323$			22.1371i			1.23174i
$324$		0			0
$325$	−5.57129	−	15.8692i	−0.309040	−	0.880266i
$326$		0			0
$327$		−	0.597188i		−	0.0330246i
$328$		0			0
$329$	2.26906			0.125098
$330$		0			0
$331$			19.6492i			1.08002i	0.841660	+	0.540008i	$0.181579\pi$
$331$			19.6492i			1.08002i	−0.841660	+	0.540008i	$0.818421\pi$
$332$		0			0
$333$			0.207412i			0.0113661i
$334$		0			0
$335$	1.34973			0.0737434
$336$		0			0
$337$	−14.8862			−0.810902			−0.405451	−	0.914117i	$-0.632885\pi$
$337$	−14.8862			−0.810902			−0.405451	+	0.914117i	$0.632885\pi$
$338$		0			0
$339$	−0.171265			−0.00930184
$340$		0			0
$341$	0.646758			0.0350239
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		−	0.398386i		−	0.0214484i
$346$		0			0
$347$	−5.74011			−0.308145			−0.154073	−	0.988060i	$-0.549239\pi$
$347$	−5.74011			−0.308145			−0.154073	+	0.988060i	$0.549239\pi$
$348$		0			0
$349$			14.6144i			0.782291i	0.920329	+	0.391146i	$0.127921\pi$
$349$			14.6144i			0.782291i	−0.920329	+	0.391146i	$0.872079\pi$
$350$		0			0
$351$	−0.570574	−	1.62522i	−0.0304550	−	0.0867477i
$352$		0			0
$353$			21.4683i			1.14264i	0.820727	+	0.571321i	$0.193569\pi$
$353$			21.4683i			1.14264i	−0.820727	+	0.571321i	$0.806431\pi$
$354$		0			0
$355$	−2.02253			−0.107345
$356$		0			0
$357$		−	0.529229i		−	0.0280098i
$358$		0			0
$359$			29.5066i			1.55730i	0.627461	+	0.778648i	$0.284094\pi$
$359$			29.5066i			1.55730i	−0.627461	+	0.778648i	$0.715906\pi$
$360$		0			0
$361$	7.88451			0.414974
$362$		0			0
$363$	0.0797053			0.00418345
$364$		0			0
$365$	5.78499			0.302800
$366$		0			0
$367$	−16.1014			−0.840488			−0.420244	−	0.907411i	$-0.638055\pi$
$367$	−16.1014			−0.840488			−0.420244	+	0.907411i	$0.638055\pi$
$368$		0			0
$369$		−	0.918436i		−	0.0478118i
$370$		0			0
$371$			4.29017i			0.222734i
$372$		0			0
$373$	30.6332			1.58613			0.793063	−	0.609139i	$-0.208485\pi$
$373$	30.6332			1.58613			0.793063	+	0.609139i	$0.208485\pi$
$374$		0			0
$375$			0.446066i			0.0230348i
$376$		0			0
$377$	−0.794232	−	2.26228i	−0.0409051	−	0.116514i
$378$		0			0
$379$			34.5219i			1.77327i	0.462469	+	0.886636i	$0.346964\pi$
$379$			34.5219i			1.77327i	−0.462469	+	0.886636i	$0.653036\pi$
$380$		0			0
$381$	0.791161			0.0405324
$382$		0			0
$383$		−	1.07605i		−	0.0549835i	−0.999622	−	0.0274918i	$-0.991248\pi$
$383$		−	1.07605i		−	0.0549835i	0.999622	−	0.0274918i	$-0.00875200\pi$
$384$		0			0
$385$			0.579061i			0.0295117i
$386$		0			0
$387$	16.5340			0.840470
$388$		0			0
$389$	−32.3547			−1.64045			−0.820224	−	0.572043i	$-0.806151\pi$
$389$	−32.3547			−1.64045			−0.820224	+	0.572043i	$0.806151\pi$
$390$		0			0
$391$	−57.3124			−2.89841
$392$		0			0
$393$	0.419548			0.0211634
$394$		0			0
$395$			7.00389i			0.352404i
$396$		0			0
$397$		−	1.00679i		−	0.0505295i	−0.999681	−	0.0252648i	$-0.991957\pi$
$397$		−	1.00679i		−	0.0505295i	0.999681	−	0.0252648i	$-0.00804288\pi$
$398$		0			0
$399$	0.265737			0.0133035
$400$		0			0
$401$		−	38.0916i		−	1.90220i	−0.308881	−	0.951101i	$-0.599954\pi$
$401$		−	38.0916i		−	1.90220i	0.308881	−	0.951101i	$-0.400046\pi$
$402$		0			0
$403$	−2.20026	+	0.772458i	−0.109603	+	0.0384789i
$404$		0			0
$405$		−	5.17846i		−	0.257320i
$406$		0			0
$407$	0.0692840			0.00343428
$408$		0			0
$409$		−	19.0243i		−	0.940693i	−0.882482	−	0.470347i	$-0.844129\pi$
$409$		−	19.0243i		−	0.940693i	0.882482	−	0.470347i	$-0.155871\pi$
$410$		0			0
$411$		−	0.925889i		−	0.0456707i
$412$		0			0
$413$	−11.0513			−0.543801
$414$		0			0
$415$	5.46586			0.268309
$416$		0			0
$417$	0.00847153			0.000414853
$418$		0			0
$419$	−23.5893			−1.15241			−0.576207	−	0.817304i	$-0.695468\pi$
$419$	−23.5893			−1.15241			−0.576207	+	0.817304i	$0.695468\pi$
$420$		0			0
$421$			24.2343i			1.18111i	0.806998	+	0.590554i	$0.201091\pi$
$421$			24.2343i			1.18111i	−0.806998	+	0.590554i	$0.798909\pi$
$422$		0			0
$423$			6.79278i			0.330276i
$424$		0			0
$425$	30.9727			1.50240
$426$		0			0
$427$		−	2.98125i		−	0.144273i
$428$		0			0
$429$	−0.271157	+	0.0951964i	−0.0130916	+	0.00459613i
$430$		0			0
$431$			26.4305i			1.27311i	0.771231	+	0.636555i	$0.219641\pi$
$431$			26.4305i			1.27311i	−0.771231	+	0.636555i	$0.780359\pi$
$432$		0			0
$433$	−32.7615			−1.57442			−0.787209	−	0.616687i	$-0.788475\pi$
$433$	−32.7615			−1.57442			−0.787209	+	0.616687i	$0.788475\pi$
$434$		0			0
$435$			0.0306920i			0.00147157i
$436$		0			0
$437$		−	28.7777i		−	1.37663i
$438$		0			0
$439$	8.62244			0.411527			0.205763	−	0.978602i	$-0.434032\pi$
$439$	8.62244			0.411527			0.205763	+	0.978602i	$0.434032\pi$
$440$		0			0
$441$	2.99365			0.142555
$442$		0			0
$443$	−30.6381			−1.45566			−0.727830	−	0.685757i	$-0.759471\pi$
$443$	−30.6381			−1.45566			−0.727830	+	0.685757i	$0.759471\pi$
$444$		0			0
$445$	3.94835			0.187170
$446$		0			0
$447$		−	0.629948i		−	0.0297955i
$448$		0			0
$449$		−	13.9360i		−	0.657681i	−0.944385	−	0.328841i	$-0.893342\pi$
$449$		−	13.9360i		−	0.657681i	0.944385	−	0.328841i	$-0.106658\pi$
$450$		0			0
$451$	−0.306795			−0.0144464
$452$		0			0
$453$		−	0.352163i		−	0.0165461i
$454$		0			0
$455$	−0.691604	−	1.96996i	−0.0324229	−	0.0923530i
$456$		0			0
$457$		−	1.64962i		−	0.0771660i	−0.999255	−	0.0385830i	$-0.987716\pi$
$457$		−	1.64962i		−	0.0771660i	0.999255	−	0.0385830i	$-0.0122844\pi$
$458$		0			0
$459$	3.17201			0.148057
$460$		0			0
$461$		−	11.8520i		−	0.552004i	−0.961157	−	0.276002i	$-0.910990\pi$
$461$		−	11.8520i		−	0.552004i	0.961157	−	0.276002i	$-0.0890097\pi$
$462$		0			0
$463$		−	30.3489i		−	1.41043i	−0.708993	−	0.705215i	$-0.750850\pi$
$463$		−	30.3489i		−	1.41043i	0.708993	−	0.705215i	$-0.249150\pi$
$464$		0			0
$465$	0.0298506			0.00138429
$466$		0			0
$467$	20.8659			0.965560			0.482780	−	0.875742i	$-0.339627\pi$
$467$	20.8659			0.965560			0.482780	+	0.875742i	$0.339627\pi$
$468$		0			0
$469$	−2.33089			−0.107631
$470$		0			0
$471$	−0.830332			−0.0382597
$472$		0			0
$473$		−	5.52303i		−	0.253949i
$474$		0			0
$475$			15.5520i			0.713576i
$476$		0			0
$477$	−12.8433			−0.588052
$478$		0			0
$479$		−	35.9280i		−	1.64159i	−0.571221	−	0.820796i	$-0.693530\pi$
$479$		−	35.9280i		−	1.64159i	0.571221	−	0.820796i	$-0.306470\pi$
$480$		0			0
$481$	−0.235703	+	0.0827496i	−0.0107471	+	0.00377306i
$482$		0			0
$483$			0.687986i			0.0313045i
$484$		0			0
$485$	8.42438			0.382531
$486$		0			0
$487$			40.7037i			1.84446i	0.386643	+	0.922229i	$0.373635\pi$
$487$			40.7037i			1.84446i	−0.386643	+	0.922229i	$0.626365\pi$
$488$		0			0
$489$		−	0.680846i		−	0.0307889i
$490$		0			0
$491$	24.6372			1.11186			0.555930	−	0.831229i	$-0.312362\pi$
$491$	24.6372			1.11186			0.555930	+	0.831229i	$0.312362\pi$
$492$		0			0
$493$	4.41541			0.198860
$494$		0			0
$495$	−1.73350			−0.0779152
$496$		0			0
$497$	3.49278			0.156673
$498$		0			0
$499$		−	36.5698i		−	1.63709i	−0.574444	−	0.818544i	$-0.694782\pi$
$499$		−	36.5698i		−	1.63709i	0.574444	−	0.818544i	$-0.305218\pi$
$500$		0			0
$501$			0.548730i			0.0245155i
$502$		0			0
$503$	9.91394			0.442041			0.221020	−	0.975269i	$-0.429061\pi$
$503$	9.91394			0.442041			0.221020	+	0.975269i	$0.429061\pi$
$504$		0			0
$505$			8.28940i			0.368874i
$506$		0			0
$507$	0.808773	−	0.647714i	0.0359189	−	0.0287660i
$508$		0			0
$509$		−	36.6195i		−	1.62313i	−0.584263	−	0.811564i	$-0.698616\pi$
$509$		−	36.6195i		−	1.62313i	0.584263	−	0.811564i	$-0.301384\pi$
$510$		0			0
$511$	−9.99030			−0.441945
$512$		0			0
$513$			1.59273i			0.0703209i
$514$		0			0
$515$		−	5.80225i		−	0.255678i
$516$		0			0
$517$	2.26906			0.0997933
$518$		0			0
$519$	−0.806222			−0.0353892
$520$		0			0
$521$	−30.6635			−1.34339			−0.671696	−	0.740827i	$-0.734434\pi$
$521$	−30.6635			−1.34339			−0.671696	+	0.740827i	$0.734434\pi$
$522$		0			0
$523$	−26.3525			−1.15231			−0.576157	−	0.817339i	$-0.695448\pi$
$523$	−26.3525			−1.15231			−0.576157	+	0.817339i	$0.695448\pi$
$524$		0			0
$525$		−	0.371801i		−	0.0162267i
$526$		0			0
$527$		−	4.29436i		−	0.187065i
$528$		0			0
$529$	51.5049			2.23934
$530$		0			0
$531$		−	33.0838i		−	1.43572i
$532$		0			0
$533$	1.04371	−	0.366422i	0.0452082	−	0.0158715i
$534$		0			0
$535$			5.17717i			0.223829i
$536$		0			0
$537$	0.0791428			0.00341526
$538$		0			0
$539$		−	1.00000i		−	0.0430730i
$540$		0			0
$541$		−	7.03422i		−	0.302425i	−0.988501	−	0.151212i	$-0.951682\pi$
$541$		−	7.03422i		−	0.302425i	0.988501	−	0.151212i	$-0.0483177\pi$
$542$		0			0
$543$	0.250890			0.0107667
$544$		0			0
$545$	4.33858			0.185844
$546$		0			0
$547$	−14.4911			−0.619593			−0.309797	−	0.950803i	$-0.600261\pi$
$547$	−14.4911			−0.619593			−0.309797	+	0.950803i	$0.600261\pi$
$548$		0			0
$549$	8.92480			0.380901
$550$		0			0
$551$			2.21707i			0.0944502i
$552$		0			0
$553$		−	12.0953i		−	0.514343i
$554$		0			0
$555$	0.00319775			0.000135737
$556$		0			0
$557$		−	38.0933i		−	1.61407i	−0.590507	−	0.807033i	$-0.701072\pi$
$557$		−	38.0933i		−	1.61407i	0.590507	−	0.807033i	$-0.298928\pi$
$558$		0			0
$559$	6.59645	+	18.7893i	0.279000	+	0.794702i
$560$		0			0
$561$		−	0.529229i		−	0.0223441i
$562$		0			0
$563$	27.3241			1.15157			0.575786	−	0.817600i	$-0.304696\pi$
$563$	27.3241			1.15157			0.575786	+	0.817600i	$0.304696\pi$
$564$		0			0
$565$		−	1.24424i		−	0.0523457i
$566$		0			0
$567$			8.94286i			0.375565i
$568$		0			0
$569$	13.1103			0.549613			0.274807	−	0.961499i	$-0.411386\pi$
$569$	13.1103			0.549613			0.274807	+	0.961499i	$0.411386\pi$
$570$		0			0
$571$	31.3561			1.31221			0.656105	−	0.754670i	$-0.272203\pi$
$571$	31.3561			1.31221			0.656105	+	0.754670i	$0.272203\pi$
$572$		0			0
$573$	0.0648742			0.00271016
$574$		0			0
$575$	−40.2638			−1.67912
$576$		0			0
$577$		−	22.2865i		−	0.927801i	−0.885887	−	0.463901i	$-0.846449\pi$
$577$		−	22.2865i		−	0.927801i	0.885887	−	0.463901i	$-0.153551\pi$
$578$		0			0
$579$		−	0.320744i		−	0.0133297i
$580$		0			0
$581$	−9.43919			−0.391603
$582$		0			0
$583$			4.29017i			0.177681i
$584$		0			0
$585$	5.89735	−	2.07042i	0.243826	−	0.0856012i
$586$		0			0
$587$		−	35.9582i		−	1.48415i	−0.670315	−	0.742076i	$-0.733841\pi$
$587$		−	35.9582i		−	1.48415i	0.670315	−	0.742076i	$-0.266159\pi$
$588$		0			0
$589$	2.15628			0.0888482
$590$		0			0
$591$		−	0.450903i		−	0.0185477i
$592$		0			0
$593$		−	1.04896i		−	0.0430757i	−0.999768	−	0.0215378i	$-0.993144\pi$
$593$		−	1.04896i		−	0.0430757i	0.999768	−	0.0215378i	$-0.00685623\pi$
$594$		0			0
$595$	3.84486			0.157624
$596$		0			0
$597$	1.94278			0.0795126
$598$		0			0
$599$	11.7314			0.479334			0.239667	−	0.970855i	$-0.422962\pi$
$599$	11.7314			0.479334			0.239667	+	0.970855i	$0.422962\pi$
$600$		0			0
$601$	−13.6460			−0.556634			−0.278317	−	0.960489i	$-0.589777\pi$
$601$	−13.6460			−0.556634			−0.278317	+	0.960489i	$0.589777\pi$
$602$		0			0
$603$		−	6.97786i		−	0.284161i
$604$		0			0
$605$			0.579061i			0.0235422i
$606$		0			0
$607$	16.4832			0.669031			0.334516	−	0.942390i	$-0.391427\pi$
$607$	16.4832			0.669031			0.334516	+	0.942390i	$0.391427\pi$
$608$		0			0
$609$		−	0.0530032i		−	0.00214780i
$610$		0			0
$611$	−7.71933	+	2.71007i	−0.312291	+	0.109638i
$612$		0			0
$613$			44.4322i			1.79460i	0.441420	+	0.897301i	$0.354475\pi$
$613$			44.4322i			1.79460i	−0.441420	+	0.897301i	$0.645525\pi$
$614$		0			0
$615$	−0.0141599			−0.000570981
$616$		0			0
$617$		−	36.3782i		−	1.46453i	−0.681020	−	0.732265i	$-0.738463\pi$
$617$		−	36.3782i		−	1.46453i	0.681020	−	0.732265i	$-0.261537\pi$
$618$		0			0
$619$			37.6940i			1.51505i	0.652805	+	0.757526i	$0.273592\pi$
$619$			37.6940i			1.51505i	−0.652805	+	0.757526i	$0.726408\pi$
$620$		0			0
$621$	−4.12355			−0.165472
$622$		0			0
$623$	−6.81854			−0.273179
$624$		0			0
$625$	20.0828			0.803311
$626$		0			0
$627$	0.265737			0.0106125
$628$		0			0
$629$		−	0.460033i		−	0.0183427i
$630$		0			0
$631$			15.6916i			0.624672i	0.949972	+	0.312336i	$0.101112\pi$
$631$			15.6916i			0.624672i	−0.949972	+	0.312336i	$0.898888\pi$
$632$		0			0
$633$	−1.16769			−0.0464117
$634$		0			0
$635$			5.74780i			0.228094i
$636$		0			0
$637$	1.19435	+	3.40199i	0.0473220	+	0.134792i
$638$		0			0
$639$			10.4562i			0.413639i
$640$		0			0
$641$	27.4218			1.08310			0.541549	−	0.840669i	$-0.317838\pi$
$641$	27.4218			1.08310			0.541549	+	0.840669i	$0.317838\pi$
$642$		0			0
$643$		−	6.69536i		−	0.264039i	−0.991247	−	0.132020i	$-0.957854\pi$
$643$		−	6.69536i		−	0.264039i	0.991247	−	0.132020i	$-0.0421462\pi$
$644$		0			0
$645$		−	0.254911i		−	0.0100371i
$646$		0			0
$647$	10.5045			0.412975			0.206487	−	0.978449i	$-0.433797\pi$
$647$	10.5045			0.412975			0.206487	+	0.978449i	$0.433797\pi$
$648$		0			0
$649$	−11.0513			−0.433803
$650$		0			0
$651$	−0.0515501			−0.00202041
$652$		0			0
$653$	−28.0593			−1.09805			−0.549023	−	0.835807i	$-0.685000\pi$
$653$	−28.0593			−1.09805			−0.549023	+	0.835807i	$0.685000\pi$
$654$		0			0
$655$			3.04802i			0.119096i
$656$		0			0
$657$		−	29.9074i		−	1.16680i
$658$		0			0
$659$	18.1233			0.705984			0.352992	−	0.935626i	$-0.385164\pi$
$659$	18.1233			0.705984			0.352992	+	0.935626i	$0.385164\pi$
$660$		0			0
$661$		−	1.71799i		−	0.0668222i	−0.999442	−	0.0334111i	$-0.989363\pi$
$661$		−	1.71799i		−	0.0668222i	0.999442	−	0.0334111i	$-0.0106371\pi$
$662$		0			0
$663$	0.632087	+	1.80043i	0.0245482	+	0.0699229i
$664$		0			0
$665$			1.93058i			0.0748648i
$666$		0			0
$667$	−5.73993			−0.222251
$668$		0			0
$669$			0.905577i			0.0350116i
$670$		0			0
$671$		−	2.98125i		−	0.115090i
$672$		0			0
$673$	14.0413			0.541251			0.270626	−	0.962685i	$-0.412769\pi$
$673$	14.0413			0.541251			0.270626	+	0.962685i	$0.412769\pi$
$674$		0			0
$675$	2.22844			0.0857728
$676$		0			0
$677$	38.7943			1.49099			0.745493	−	0.666514i	$-0.232214\pi$
$677$	38.7943			1.49099			0.745493	+	0.666514i	$0.232214\pi$
$678$		0			0
$679$	−14.5484			−0.558315
$680$		0			0
$681$		−	0.500920i		−	0.0191953i
$682$		0			0
$683$		−	31.6219i		−	1.20998i	−0.796233	−	0.604990i	$-0.793177\pi$
$683$		−	31.6219i		−	1.20998i	0.796233	−	0.604990i	$-0.206823\pi$
$684$		0			0
$685$	6.72660			0.257010
$686$		0			0
$687$		−	0.956336i		−	0.0364865i
$688$		0			0
$689$	−5.12398	−	14.5951i	−0.195208	−	0.556029i
$690$		0			0
$691$		−	46.8544i		−	1.78243i	−0.453585	−	0.891213i	$-0.649855\pi$
$691$		−	46.8544i		−	1.78243i	0.453585	−	0.891213i	$-0.350145\pi$
$692$		0			0
$693$	2.99365			0.113719
$694$		0			0
$695$			0.0615458i			0.00233456i
$696$		0			0
$697$			2.03706i			0.0771593i
$698$		0			0
$699$	−1.25084			−0.0473110
$700$		0			0
$701$	9.20641			0.347721			0.173861	−	0.984770i	$-0.444376\pi$
$701$	9.20641			0.347721			0.173861	+	0.984770i	$0.444376\pi$
$702$		0			0
$703$	0.230992			0.00871203
$704$		0			0
$705$	0.104727			0.00394424
$706$		0			0
$707$		−	14.3153i		−	0.538381i
$708$		0			0
$709$			34.5805i			1.29870i	0.760491	+	0.649349i	$0.224958\pi$
$709$			34.5805i			1.29870i	−0.760491	+	0.649349i	$0.775042\pi$
$710$		0			0
$711$	36.2090			1.35794
$712$		0			0
$713$			5.58257i			0.209069i
$714$		0			0
$715$	−0.691604	−	1.96996i	−0.0258645	−	0.0736722i
$716$		0			0
$717$		−	1.84801i		−	0.0690151i
$718$		0			0
$719$	22.6023			0.842924			0.421462	−	0.906846i	$-0.361517\pi$
$719$	22.6023			0.842924			0.421462	+	0.906846i	$0.361517\pi$
$720$		0			0
$721$			10.0201i			0.373168i
$722$		0			0
$723$			0.896411i			0.0333379i
$724$		0			0
$725$	3.10197			0.115204
$726$		0			0
$727$	−14.5453			−0.539454			−0.269727	−	0.962937i	$-0.586933\pi$
$727$	−14.5453			−0.539454			−0.269727	+	0.962937i	$0.586933\pi$
$728$		0			0
$729$	−26.6575			−0.987317
$730$		0			0
$731$	−36.6719			−1.35636
$732$		0			0
$733$			27.9364i			1.03185i	0.856632	+	0.515927i	$0.172552\pi$
$733$			27.9364i			1.03185i	−0.856632	+	0.515927i	$0.827448\pi$
$734$		0			0
$735$		−	0.0461542i		−	0.00170242i
$736$		0			0
$737$	−2.33089			−0.0858594
$738$		0			0
$739$			28.5046i			1.04856i	0.851547	+	0.524279i	$0.175665\pi$
$739$			28.5046i			1.04856i	−0.851547	+	0.524279i	$0.824335\pi$
$740$		0			0
$741$	−0.904033	+	0.317384i	−0.0332105	+	0.0116594i
$742$		0			0
$743$			4.03999i			0.148213i	0.997250	+	0.0741063i	$0.0236104\pi$
$743$			4.03999i			0.148213i	−0.997250	+	0.0741063i	$0.976390\pi$
$744$		0			0
$745$	4.57658			0.167673
$746$		0			0
$747$		−	28.2576i		−	1.03389i
$748$		0			0
$749$		−	8.94064i		−	0.326684i
$750$		0			0
$751$	33.1909			1.21115			0.605577	−	0.795787i	$-0.292942\pi$
$751$	33.1909			1.21115			0.605577	+	0.795787i	$0.292942\pi$
$752$		0			0
$753$	−0.698108			−0.0254405
$754$		0			0
$755$	2.55847			0.0931123
$756$		0			0
$757$	−10.8799			−0.395436			−0.197718	−	0.980259i	$-0.563353\pi$
$757$	−10.8799			−0.395436			−0.197718	+	0.980259i	$0.563353\pi$
$758$		0			0
$759$			0.687986i			0.0249723i
$760$		0			0
$761$			32.4200i			1.17523i	0.809142	+	0.587613i	$0.199932\pi$
$761$			32.4200i			1.17523i	−0.809142	+	0.587613i	$0.800068\pi$
$762$		0			0
$763$	−7.49245			−0.271245
$764$		0			0
$765$			11.5101i			0.416150i
$766$		0			0
$767$	37.5965	−	13.1992i	1.35753	−	0.476596i
$768$		0			0
$769$		−	28.4458i		−	1.02578i	−0.858454	−	0.512891i	$-0.828574\pi$
$769$		−	28.4458i		−	1.02578i	0.858454	−	0.512891i	$-0.171426\pi$
$770$		0			0
$771$	1.17446			0.0422973
$772$		0			0
$773$			38.8523i			1.39742i	0.715405	+	0.698710i	$0.246242\pi$
$773$			38.8523i			1.39742i	−0.715405	+	0.698710i	$0.753758\pi$
$774$		0			0
$775$		−	3.01693i		−	0.108371i
$776$		0			0
$777$	−0.00552231			−0.000198112
$778$		0			0
$779$	−1.02285			−0.0366474
$780$		0			0
$781$	3.49278			0.124982
$782$		0			0
$783$	0.317682			0.0113530
$784$		0			0
$785$		−	6.03237i		−	0.215305i
$786$		0			0
$787$		−	38.6429i		−	1.37747i	−0.725013	−	0.688735i	$-0.758166\pi$
$787$		−	38.6429i		−	1.37747i	0.725013	−	0.688735i	$-0.241834\pi$
$788$		0			0
$789$	0.0567532			0.00202047
$790$		0			0
$791$			2.14873i			0.0763999i
$792$		0			0
$793$	3.56067	+	10.1422i	0.126443	+	0.360159i
$794$		0			0
$795$			0.198009i			0.00702267i
$796$		0			0
$797$	30.9629			1.09676			0.548382	−	0.836228i	$-0.315244\pi$
$797$	30.9629			1.09676			0.548382	+	0.836228i	$0.315244\pi$
$798$		0			0
$799$		−	15.0662i		−	0.533003i
$800$		0			0
$801$		−	20.4123i		−	0.721234i
$802$		0			0
$803$	−9.99030			−0.352550
$804$		0			0
$805$	−4.99823			−0.176164
$806$		0			0
$807$	−0.817755			−0.0287863
$808$		0			0
$809$	−30.0419			−1.05622			−0.528109	−	0.849177i	$-0.677099\pi$
$809$	−30.0419			−1.05622			−0.528109	+	0.849177i	$0.677099\pi$
$810$		0			0
$811$			39.6029i			1.39065i	0.718698	+	0.695323i	$0.244739\pi$
$811$			39.6029i			1.39065i	−0.718698	+	0.695323i	$0.755261\pi$
$812$		0			0
$813$		−	0.0800900i		−	0.00280888i
$814$		0			0
$815$	4.94636			0.173263
$816$		0			0
$817$		−	18.4137i		−	0.644215i
$818$		0			0
$819$	−10.1843	+	3.57548i	−0.355870	+	0.124937i
$820$		0			0
$821$		−	27.4490i		−	0.957977i	−0.877821	−	0.478989i	$-0.841003\pi$
$821$		−	27.4490i		−	0.957977i	0.877821	−	0.478989i	$-0.158997\pi$
$822$		0			0
$823$	2.85219			0.0994210			0.0497105	−	0.998764i	$-0.484170\pi$
$823$	2.85219			0.0994210			0.0497105	+	0.998764i	$0.484170\pi$
$824$		0			0
$825$		−	0.371801i		−	0.0129444i
$826$		0			0
$827$		−	37.3734i		−	1.29960i	−0.760105	−	0.649800i	$-0.774853\pi$
$827$		−	37.3734i		−	1.29960i	0.760105	−	0.649800i	$-0.225147\pi$
$828$		0			0
$829$	−34.6517			−1.20350			−0.601752	−	0.798683i	$-0.705530\pi$
$829$	−34.6517			−1.20350			−0.601752	+	0.798683i	$0.705530\pi$
$830$		0			0
$831$	−0.597909			−0.0207412
$832$		0			0
$833$	−6.63982			−0.230056
$834$		0			0
$835$	−3.98654			−0.137960
$836$		0			0
$837$		−	0.308973i		−	0.0106797i
$838$		0			0
$839$		−	46.6245i		−	1.60966i	−0.593507	−	0.804829i	$-0.702257\pi$
$839$		−	46.6245i		−	1.60966i	0.593507	−	0.804829i	$-0.297743\pi$
$840$		0			0
$841$	−28.5578			−0.984751
$842$		0			0
$843$			1.76356i			0.0607402i
$844$		0			0
$845$	4.70565	+	5.87575i	0.161879	+	0.202132i
$846$		0			0
$847$		−	1.00000i		−	0.0343604i
$848$		0			0
$849$	0.262349			0.00900378
$850$		0			0
$851$			0.598033i			0.0205003i
$852$		0			0
$853$			21.4187i			0.733364i	0.930346	+	0.366682i	$0.119506\pi$
$853$			21.4187i			0.733364i	−0.930346	+	0.366682i	$0.880494\pi$
$854$		0			0
$855$	−5.77948			−0.197654
$856$		0			0
$857$	−1.58945			−0.0542945			−0.0271472	−	0.999631i	$-0.508642\pi$
$857$	−1.58945			−0.0542945			−0.0271472	+	0.999631i	$0.508642\pi$
$858$		0			0
$859$	−10.3518			−0.353198			−0.176599	−	0.984283i	$-0.556510\pi$
$859$	−10.3518			−0.353198			−0.176599	+	0.984283i	$0.556510\pi$
$860$		0			0
$861$	0.0244532			0.000833363
$862$		0			0
$863$		−	57.0306i		−	1.94134i	−0.240406	−	0.970672i	$-0.577281\pi$
$863$		−	57.0306i		−	1.94134i	0.240406	−	0.970672i	$-0.422719\pi$
$864$		0			0
$865$		−	5.85721i		−	0.199151i
$866$		0			0
$867$	−2.15900			−0.0733233
$868$		0			0
$869$		−	12.0953i		−	0.410304i
$870$		0			0
$871$	7.92966	−	2.78391i	0.268686	−	0.0943292i
$872$		0			0
$873$		−	43.5526i		−	1.47403i
$874$		0			0
$875$	5.59644			0.189194
$876$		0			0
$877$		−	36.9301i		−	1.24704i	−0.781807	−	0.623521i	$-0.785702\pi$
$877$		−	36.9301i		−	1.24704i	0.781807	−	0.623521i	$-0.214298\pi$
$878$		0			0
$879$			0.0823761i			0.00277848i
$880$		0			0
$881$	20.3444			0.685419			0.342710	−	0.939441i	$-0.388655\pi$
$881$	20.3444			0.685419			0.342710	+	0.939441i	$0.388655\pi$
$882$		0			0
$883$	33.4838			1.12682			0.563410	−	0.826177i	$-0.309489\pi$
$883$	33.4838			1.12682			0.563410	+	0.826177i	$0.309489\pi$
$884$		0			0
$885$	−0.510066			−0.0171457
$886$		0			0
$887$	11.1375			0.373961			0.186980	−	0.982364i	$-0.440130\pi$
$887$	11.1375			0.373961			0.186980	+	0.982364i	$0.440130\pi$
$888$		0			0
$889$		−	9.92608i		−	0.332910i
$890$		0			0
$891$			8.94286i			0.299597i
$892$		0			0
$893$	7.56504			0.253154
$894$		0			0
$895$			0.574974i			0.0192192i
$896$		0			0
$897$	−0.821699	−	2.34052i	−0.0274357	−	0.0781477i
$898$		0			0
$899$		−	0.430087i		−	0.0143442i
$900$		0			0
$901$	28.4859			0.949005
$902$		0			0
$903$			0.440215i			0.0146494i
$904$		0			0
$905$			1.82272i			0.0605891i
$906$		0			0
$907$	8.30368			0.275719			0.137860	−	0.990452i	$-0.455978\pi$
$907$	8.30368			0.275719			0.137860	+	0.990452i	$0.455978\pi$
$908$		0			0
$909$	42.8548			1.42141
$910$		0			0
$911$	−14.8309			−0.491371			−0.245685	−	0.969350i	$-0.579013\pi$
$911$	−14.8309			−0.491371			−0.245685	+	0.969350i	$0.579013\pi$
$912$		0			0
$913$	−9.43919			−0.312391
$914$		0			0
$915$		−	0.137597i		−	0.00454882i
$916$		0			0
$917$		−	5.26373i		−	0.173824i
$918$		0			0
$919$	−15.8910			−0.524194			−0.262097	−	0.965042i	$-0.584414\pi$
$919$	−15.8910			−0.524194			−0.262097	+	0.965042i	$0.584414\pi$
$920$		0			0
$921$		−	1.46338i		−	0.0482200i
$922$		0			0
$923$	−11.8824	+	4.17162i	−0.391114	+	0.137311i
$924$		0			0
$925$		−	0.323188i		−	0.0106264i
$926$		0			0
$927$	−29.9967			−0.985220
$928$		0			0
$929$		−	52.4001i		−	1.71919i	−0.510976	−	0.859595i	$-0.670716\pi$
$929$		−	52.4001i		−	1.71919i	0.510976	−	0.859595i	$-0.329284\pi$
$930$		0			0
$931$		−	3.33399i		−	0.109267i
$932$		0			0
$933$	1.56909			0.0513698
$934$		0			0
$935$	3.84486			0.125740
$936$		0			0
$937$	−7.60786			−0.248538			−0.124269	−	0.992249i	$-0.539659\pi$
$937$	−7.60786			−0.248538			−0.124269	+	0.992249i	$0.539659\pi$
$938$		0			0
$939$	0.211807			0.00691208
$940$		0			0
$941$		−	28.6954i		−	0.935445i	−0.883875	−	0.467722i	$-0.845075\pi$
$941$		−	28.6954i		−	0.935445i	0.883875	−	0.467722i	$-0.154925\pi$
$942$		0			0
$943$		−	2.64814i		−	0.0862352i
$944$		0			0
$945$	0.276632			0.00899884
$946$		0			0
$947$			2.62856i			0.0854167i	0.999088	+	0.0427083i	$0.0135986\pi$
$947$			2.62856i			0.0854167i	−0.999088	+	0.0427083i	$0.986401\pi$
$948$		0			0
$949$	33.9869	−	11.9320i	1.10326	−	0.387328i
$950$		0			0
$951$			0.465092i			0.0150816i
$952$		0			0
$953$	12.1477			0.393503			0.196752	−	0.980453i	$-0.436961\pi$
$953$	12.1477			0.393503			0.196752	+	0.980453i	$0.436961\pi$
$954$		0			0
$955$			0.471312i			0.0152513i
$956$		0			0
$957$		−	0.0530032i		−	0.00171335i
$958$		0			0
$959$	−11.6164			−0.375113
$960$		0			0
$961$	30.5817			0.986507
$962$		0			0
$963$	26.7651			0.862494
$964$		0			0
$965$	2.33021			0.0750122
$966$		0			0
$967$			17.0783i			0.549202i	0.961558	+	0.274601i	$0.0885457\pi$
$967$			17.0783i			0.549202i	−0.961558	+	0.274601i	$0.911454\pi$
$968$		0			0
$969$		−	1.76444i		−	0.0566821i
$970$		0			0
$971$	−26.0583			−0.836251			−0.418125	−	0.908389i	$-0.637313\pi$
$971$	−26.0583			−0.836251			−0.418125	+	0.908389i	$0.637313\pi$
$972$		0			0
$973$		−	0.106286i		−	0.00340736i
$974$		0			0
$975$	0.444062	+	1.26486i	0.0142214	+	0.0405080i
$976$		0			0
$977$			10.9210i			0.349393i	0.984622	+	0.174697i	$0.0558945\pi$
$977$			10.9210i			0.349393i	−0.984622	+	0.174697i	$0.944106\pi$
$978$		0			0
$979$	−6.81854			−0.217922
$980$		0			0
$981$		−	22.4297i		−	0.716127i
$982$		0			0
$983$		−	0.882988i		−	0.0281630i	−0.999901	−	0.0140815i	$-0.995518\pi$
$983$		−	0.882988i		−	0.0281630i	0.999901	−	0.0140815i	$-0.00448242\pi$
$984$		0			0
$985$	3.27581			0.104376
$986$		0			0
$987$	−0.180857			−0.00575673
$988$		0			0
$989$	47.6727			1.51590
$990$		0			0
$991$	24.0996			0.765548			0.382774	−	0.923842i	$-0.374969\pi$
$991$	24.0996			0.765548			0.382774	+	0.923842i	$0.374969\pi$
$992$		0			0
$993$		−	1.56615i		−	0.0497001i
$994$		0			0
$995$			14.1143i			0.447454i
$996$		0			0
$997$	−52.2959			−1.65623			−0.828113	−	0.560560i	$-0.810586\pi$
$997$	−52.2959			−1.65623			−0.828113	+	0.560560i	$0.810586\pi$
$998$		0			0
$999$		−	0.0330987i		−	0.00104720i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.b.2157.14	yes	30
13.12	even	2	inner	4004.2.m.b.2157.13	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.b.2157.13	✓	30	13.12	even	2	inner
4004.2.m.b.2157.14	yes	30	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.0797053 q^{3} -0.579061i q^{5} +1.00000i q^{7} -2.99365 q^{9} +O(q^{10})\)	\(q-0.0797053 q^{3} -0.579061i q^{5} +1.00000i q^{7} -2.99365 q^{9} +1.00000i q^{11} +(-1.19435 - 3.40199i) q^{13} +0.0461542i q^{15} +6.63982 q^{17} +3.33399i q^{19} -0.0797053i q^{21} -8.63162 q^{23} +4.66469 q^{25} +0.477726 q^{27} +0.664989 q^{29} -0.646758i q^{31} -0.0797053i q^{33} +0.579061 q^{35} -0.0692840i q^{37} +(0.0951964 + 0.271157i) q^{39} +0.306795i q^{41} -5.52303 q^{43} +1.73350i q^{45} -2.26906i q^{47} -1.00000 q^{49} -0.529229 q^{51} +4.29017 q^{53} +0.579061 q^{55} -0.265737i q^{57} +11.0513i q^{59} -2.98125 q^{61} -2.99365i q^{63} +(-1.96996 + 0.691604i) q^{65} +2.33089i q^{67} +0.687986 q^{69} -3.49278i q^{71} +9.99030i q^{73} -0.371801 q^{75} -1.00000 q^{77} -12.0953 q^{79} +8.94286 q^{81} +9.43919i q^{83} -3.84486i q^{85} -0.0530032 q^{87} +6.81854i q^{89} +(3.40199 - 1.19435i) q^{91} +0.0515501i q^{93} +1.93058 q^{95} +14.5484i q^{97} -2.99365i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 18 q^{9}+O(q^{10}) \) 30 * q + 18 * q^9	\( 30 q + 18 q^{9} + 4 q^{13} + 12 q^{17} + 6 q^{23} - 6 q^{29} - 2 q^{35} + 8 q^{39} - 22 q^{43} - 30 q^{49} + 60 q^{51} - 38 q^{53} - 2 q^{55} - 36 q^{61} + 10 q^{65} + 36 q^{69} - 20 q^{75} - 30 q^{77} - 10 q^{79} - 42 q^{81} + 36 q^{87} + 6 q^{91} - 2 q^{95}+O(q^{100}) \) 30 * q + 18 * q^9 + 4 * q^13 + 12 * q^17 + 6 * q^23 - 6 * q^29 - 2 * q^35 + 8 * q^39 - 22 * q^43 - 30 * q^49 + 60 * q^51 - 38 * q^53 - 2 * q^55 - 36 * q^61 + 10 * q^65 + 36 * q^69 - 20 * q^75 - 30 * q^77 - 10 * q^79 - 42 * q^81 + 36 * q^87 + 6 * q^91 - 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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