LMFDB - Embedded newform 4004.2.m.b.2157.15

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			2157.15
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.b.2157.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0840880 q^{3} -3.49341i q^{5} -1.00000i q^{7} -2.99293 q^{9} +O(q^{10})$	$q+0.0840880 q^{3} -3.49341i q^{5} -1.00000i q^{7} -2.99293 q^{9} -1.00000i q^{11} +(-2.83672 - 2.22554i) q^{13} -0.293754i q^{15} -0.970948 q^{17} -6.96670i q^{19} -0.0840880i q^{21} +2.68221 q^{23} -7.20395 q^{25} -0.503933 q^{27} +3.14113 q^{29} +1.13445i q^{31} -0.0840880i q^{33} -3.49341 q^{35} -1.41572i q^{37} +(-0.238534 - 0.187141i) q^{39} +2.80825i q^{41} +3.04376 q^{43} +10.4555i q^{45} -1.87989i q^{47} -1.00000 q^{49} -0.0816451 q^{51} -11.8444 q^{53} -3.49341 q^{55} -0.585815i q^{57} -0.275899i q^{59} +4.89604 q^{61} +2.99293i q^{63} +(-7.77473 + 9.90984i) q^{65} +0.948663i q^{67} +0.225542 q^{69} -4.35832i q^{71} +3.72601i q^{73} -0.605765 q^{75} -1.00000 q^{77} -11.7679 q^{79} +8.93641 q^{81} +11.4834i q^{83} +3.39193i q^{85} +0.264132 q^{87} -4.56347i q^{89} +(-2.22554 + 2.83672i) q^{91} +0.0953932i q^{93} -24.3376 q^{95} -10.4405i q^{97} +2.99293i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.0840880			0.0485482			0.0242741	−	0.999705i	$-0.492273\pi$
$3$	0.0840880			0.0485482			0.0242741	+	0.999705i	$0.492273\pi$
$4$		0			0
$5$		−	3.49341i		−	1.56230i	−0.624342	−	0.781151i	$-0.714633\pi$
$5$		−	3.49341i		−	1.56230i	0.624342	−	0.781151i	$-0.285367\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−2.99293			−0.997643
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	−2.83672	−	2.22554i	−0.786765	−	0.617253i
$13$	−2.83672	−	2.22554i	−0.786765	−	0.617253i
$14$		0			0
$15$		−	0.293754i		−	0.0758470i
$16$		0			0
$17$	−0.970948			−0.235490			−0.117745	−	0.993044i	$-0.537566\pi$
$17$	−0.970948			−0.235490			−0.117745	+	0.993044i	$0.537566\pi$
$18$		0			0
$19$		−	6.96670i		−	1.59827i	−0.601152	−	0.799135i	$-0.705291\pi$
$19$		−	6.96670i		−	1.59827i	0.601152	−	0.799135i	$-0.294709\pi$
$20$		0			0
$21$		−	0.0840880i		−	0.0183495i
$22$		0			0
$23$	2.68221			0.559279			0.279640	−	0.960105i	$-0.409785\pi$
$23$	2.68221			0.559279			0.279640	+	0.960105i	$0.409785\pi$
$24$		0			0
$25$	−7.20395			−1.44079
$26$		0			0
$27$	−0.503933			−0.0969820
$28$		0			0
$29$	3.14113			0.583294			0.291647	−	0.956526i	$-0.405797\pi$
$29$	3.14113			0.583294			0.291647	+	0.956526i	$0.405797\pi$
$30$		0			0
$31$			1.13445i			0.203752i	0.994797	+	0.101876i	$0.0324846\pi$
$31$			1.13445i			0.203752i	−0.994797	+	0.101876i	$0.967515\pi$
$32$		0			0
$33$		−	0.0840880i		−	0.0146378i
$34$		0			0
$35$	−3.49341			−0.590495
$36$		0			0
$37$		−	1.41572i		−	0.232743i	−0.993206	−	0.116371i	$-0.962874\pi$
$37$		−	1.41572i		−	0.232743i	0.993206	−	0.116371i	$-0.0371262\pi$
$38$		0			0
$39$	−0.238534	−	0.187141i	−0.0381960	−	0.0299665i
$40$		0			0
$41$			2.80825i			0.438575i	0.975660	+	0.219287i	$0.0703732\pi$
$41$			2.80825i			0.438575i	−0.975660	+	0.219287i	$0.929627\pi$
$42$		0			0
$43$	3.04376			0.464168			0.232084	−	0.972696i	$-0.425445\pi$
$43$	3.04376			0.464168			0.232084	+	0.972696i	$0.425445\pi$
$44$		0			0
$45$			10.4555i			1.55862i
$46$		0			0
$47$		−	1.87989i		−	0.274210i	−0.990557	−	0.137105i	$-0.956220\pi$
$47$		−	1.87989i		−	0.274210i	0.990557	−	0.137105i	$-0.0437798\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−0.0816451			−0.0114326
$52$		0			0
$53$	−11.8444			−1.62696			−0.813478	−	0.581596i	$-0.802428\pi$
$53$	−11.8444			−1.62696			−0.813478	+	0.581596i	$0.802428\pi$
$54$		0			0
$55$	−3.49341			−0.471052
$56$		0			0
$57$		−	0.585815i		−	0.0775932i
$58$		0			0
$59$		−	0.275899i		−	0.0359191i	−0.999839	−	0.0179595i	$-0.994283\pi$
$59$		−	0.275899i		−	0.0359191i	0.999839	−	0.0179595i	$-0.00571700\pi$
$60$		0			0
$61$	4.89604			0.626874			0.313437	−	0.949609i	$-0.398520\pi$
$61$	4.89604			0.626874			0.313437	+	0.949609i	$0.398520\pi$
$62$		0			0
$63$			2.99293i			0.377074i
$64$		0			0
$65$	−7.77473	+	9.90984i	−0.964336	+	1.22916i
$66$		0			0
$67$			0.948663i			0.115898i	0.998320	+	0.0579488i	$0.0184560\pi$
$67$			0.948663i			0.115898i	−0.998320	+	0.0579488i	$0.981544\pi$
$68$		0			0
$69$	0.225542			0.0271520
$70$		0			0
$71$		−	4.35832i		−	0.517237i	−0.965980	−	0.258619i	$-0.916733\pi$
$71$		−	4.35832i		−	0.517237i	0.965980	−	0.258619i	$-0.0832673\pi$
$72$		0			0
$73$			3.72601i			0.436097i	0.975938	+	0.218049i	$0.0699691\pi$
$73$			3.72601i			0.436097i	−0.975938	+	0.218049i	$0.930031\pi$
$74$		0			0
$75$	−0.605765			−0.0699478
$76$		0			0
$77$	−1.00000			−0.113961
$78$		0			0
$79$	−11.7679			−1.32400			−0.661998	−	0.749506i	$-0.730291\pi$
$79$	−11.7679			−1.32400			−0.661998	+	0.749506i	$0.730291\pi$
$80$		0			0
$81$	8.93641			0.992935
$82$		0			0
$83$			11.4834i			1.26046i	0.776407	+	0.630232i	$0.217040\pi$
$83$			11.4834i			1.26046i	−0.776407	+	0.630232i	$0.782960\pi$
$84$		0			0
$85$			3.39193i			0.367906i
$86$		0			0
$87$	0.264132			0.0283179
$88$		0			0
$89$		−	4.56347i		−	0.483726i	−0.970310	−	0.241863i	$-0.922241\pi$
$89$		−	4.56347i		−	0.483726i	0.970310	−	0.241863i	$-0.0777585\pi$
$90$		0			0
$91$	−2.22554	+	2.83672i	−0.233300	+	0.297369i
$92$		0			0
$93$			0.0953932i			0.00989182i
$94$		0			0
$95$	−24.3376			−2.49698
$96$		0			0
$97$		−	10.4405i		−	1.06007i	−0.847976	−	0.530034i	$-0.822179\pi$
$97$		−	10.4405i		−	1.06007i	0.847976	−	0.530034i	$-0.177821\pi$
$98$		0			0
$99$			2.99293i			0.300801i
$100$		0			0
$101$	−1.61623			−0.160821			−0.0804104	−	0.996762i	$-0.525623\pi$
$101$	−1.61623			−0.160821			−0.0804104	+	0.996762i	$0.525623\pi$
$102$		0			0
$103$	5.03537			0.496150			0.248075	−	0.968741i	$-0.420202\pi$
$103$	5.03537			0.496150			0.248075	+	0.968741i	$0.420202\pi$
$104$		0			0
$105$	−0.293754			−0.0286675
$106$		0			0
$107$	4.92787			0.476396			0.238198	−	0.971217i	$-0.423443\pi$
$107$	4.92787			0.476396			0.238198	+	0.971217i	$0.423443\pi$
$108$		0			0
$109$			18.9632i			1.81634i	0.418601	+	0.908170i	$0.362521\pi$
$109$			18.9632i			1.81634i	−0.418601	+	0.908170i	$0.637479\pi$
$110$		0			0
$111$		−	0.119045i		−	0.0112992i
$112$		0			0
$113$	2.93498			0.276100			0.138050	−	0.990425i	$-0.455917\pi$
$113$	2.93498			0.276100			0.138050	+	0.990425i	$0.455917\pi$
$114$		0			0
$115$		−	9.37007i		−	0.873764i
$116$		0			0
$117$	8.49010	+	6.66088i	0.784910	+	0.615798i
$118$		0			0
$119$			0.970948i			0.0890067i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			0.236140i			0.0212920i
$124$		0			0
$125$			7.69930i			0.688646i
$126$		0			0
$127$	−18.3032			−1.62415			−0.812073	−	0.583556i	$-0.801661\pi$
$127$	−18.3032			−1.62415			−0.812073	+	0.583556i	$0.801661\pi$
$128$		0			0
$129$	0.255943			0.0225346
$130$		0			0
$131$	−6.50099			−0.567994			−0.283997	−	0.958825i	$-0.591661\pi$
$131$	−6.50099			−0.567994			−0.283997	+	0.958825i	$0.591661\pi$
$132$		0			0
$133$	−6.96670			−0.604089
$134$		0			0
$135$			1.76045i			0.151515i
$136$		0			0
$137$			4.68047i			0.399879i	0.979808	+	0.199940i	$0.0640746\pi$
$137$			4.68047i			0.399879i	−0.979808	+	0.199940i	$0.935925\pi$
$138$		0			0
$139$	17.8209			1.51155			0.755774	−	0.654832i	$-0.227261\pi$
$139$	17.8209			1.51155			0.755774	+	0.654832i	$0.227261\pi$
$140$		0			0
$141$		−	0.158076i		−	0.0133124i
$142$		0			0
$143$	−2.22554	+	2.83672i	−0.186109	+	0.237218i
$144$		0			0
$145$		−	10.9733i		−	0.911282i
$146$		0			0
$147$	−0.0840880			−0.00693546
$148$		0			0
$149$			20.1548i			1.65114i	0.564296	+	0.825572i	$0.309148\pi$
$149$			20.1548i			1.65114i	−0.564296	+	0.825572i	$0.690852\pi$
$150$		0			0
$151$		−	14.2431i		−	1.15909i	−0.814941	−	0.579543i	$-0.803231\pi$
$151$		−	14.2431i		−	1.15909i	0.814941	−	0.579543i	$-0.196769\pi$
$152$		0			0
$153$	2.90598			0.234935
$154$		0			0
$155$	3.96309			0.318323
$156$		0			0
$157$	−7.88136			−0.629001			−0.314500	−	0.949257i	$-0.601837\pi$
$157$	−7.88136			−0.629001			−0.314500	+	0.949257i	$0.601837\pi$
$158$		0			0
$159$	−0.995973			−0.0789858
$160$		0			0
$161$		−	2.68221i		−	0.211388i
$162$		0			0
$163$			22.7014i			1.77811i	0.457800	+	0.889055i	$0.348637\pi$
$163$			22.7014i			1.77811i	−0.457800	+	0.889055i	$0.651363\pi$
$164$		0			0
$165$	−0.293754			−0.0228687
$166$		0			0
$167$		−	2.40137i		−	0.185824i	−0.995674	−	0.0929119i	$-0.970383\pi$
$167$		−	2.40137i		−	0.185824i	0.995674	−	0.0929119i	$-0.0296175\pi$
$168$		0			0
$169$	3.09396	+	12.6265i	0.237997	+	0.971266i
$170$		0			0
$171$			20.8508i			1.59450i
$172$		0			0
$173$	6.96838			0.529796			0.264898	−	0.964276i	$-0.414662\pi$
$173$	6.96838			0.529796			0.264898	+	0.964276i	$0.414662\pi$
$174$		0			0
$175$			7.20395i			0.544567i
$176$		0			0
$177$		−	0.0231998i		−	0.00174381i
$178$		0			0
$179$	−7.43029			−0.555366			−0.277683	−	0.960673i	$-0.589566\pi$
$179$	−7.43029			−0.555366			−0.277683	+	0.960673i	$0.589566\pi$
$180$		0			0
$181$	7.50290			0.557686			0.278843	−	0.960337i	$-0.410049\pi$
$181$	7.50290			0.557686			0.278843	+	0.960337i	$0.410049\pi$
$182$		0			0
$183$	0.411698			0.0304336
$184$		0			0
$185$	−4.94569			−0.363614
$186$		0			0
$187$			0.970948i			0.0710028i
$188$		0			0
$189$			0.503933i			0.0366558i
$190$		0			0
$191$	−0.747810			−0.0541097			−0.0270548	−	0.999634i	$-0.508613\pi$
$191$	−0.747810			−0.0541097			−0.0270548	+	0.999634i	$0.508613\pi$
$192$		0			0
$193$		−	8.17119i		−	0.588175i	−0.955779	−	0.294087i	$-0.904984\pi$
$193$		−	8.17119i		−	0.588175i	0.955779	−	0.294087i	$-0.0950157\pi$
$194$		0			0
$195$	−0.653761	+	0.833298i	−0.0468168	+	0.0596737i
$196$		0			0
$197$		−	9.22368i		−	0.657160i	−0.944476	−	0.328580i	$-0.893430\pi$
$197$		−	9.22368i		−	0.657160i	0.944476	−	0.328580i	$-0.106570\pi$
$198$		0			0
$199$	10.0824			0.714723			0.357362	−	0.933966i	$-0.383676\pi$
$199$	10.0824			0.714723			0.357362	+	0.933966i	$0.383676\pi$
$200$		0			0
$201$			0.0797711i			0.00562662i
$202$		0			0
$203$		−	3.14113i		−	0.220464i
$204$		0			0
$205$	9.81038			0.685187
$206$		0			0
$207$	−8.02766			−0.557961
$208$		0			0
$209$	−6.96670			−0.481896
$210$		0			0
$211$	−3.54869			−0.244302			−0.122151	−	0.992512i	$-0.538979\pi$
$211$	−3.54869			−0.244302			−0.122151	+	0.992512i	$0.538979\pi$
$212$		0			0
$213$		−	0.366482i		−	0.0251109i
$214$		0			0
$215$		−	10.6331i		−	0.725172i
$216$		0			0
$217$	1.13445			0.0770112
$218$		0			0
$219$			0.313313i			0.0211717i
$220$		0			0
$221$	2.75431	+	2.16088i	0.185275	+	0.145357i
$222$		0			0
$223$			24.2913i			1.62667i	0.581797	+	0.813334i	$0.302350\pi$
$223$			24.2913i			1.62667i	−0.581797	+	0.813334i	$0.697650\pi$
$224$		0			0
$225$	21.5609			1.43739
$226$		0			0
$227$		−	13.7076i		−	0.909804i	−0.890542	−	0.454902i	$-0.849674\pi$
$227$		−	13.7076i		−	0.909804i	0.890542	−	0.454902i	$-0.150326\pi$
$228$		0			0
$229$			2.28258i			0.150837i	0.997152	+	0.0754186i	$0.0240293\pi$
$229$			2.28258i			0.150837i	−0.997152	+	0.0754186i	$0.975971\pi$
$230$		0			0
$231$	−0.0840880			−0.00553258
$232$		0			0
$233$	20.9248			1.37083			0.685415	−	0.728153i	$-0.259621\pi$
$233$	20.9248			1.37083			0.685415	+	0.728153i	$0.259621\pi$
$234$		0			0
$235$	−6.56724			−0.428399
$236$		0			0
$237$	−0.989542			−0.0642776
$238$		0			0
$239$		−	2.11444i		−	0.136772i	−0.997659	−	0.0683860i	$-0.978215\pi$
$239$		−	2.11444i		−	0.136772i	0.997659	−	0.0683860i	$-0.0217849\pi$
$240$		0			0
$241$		−	6.41437i		−	0.413186i	−0.978427	−	0.206593i	$-0.933762\pi$
$241$		−	6.41437i		−	0.413186i	0.978427	−	0.206593i	$-0.0662375\pi$
$242$		0			0
$243$	2.26324			0.145187
$244$		0			0
$245$			3.49341i			0.223186i
$246$		0			0
$247$	−15.5046	+	19.7626i	−0.986537	+	1.25746i
$248$		0			0
$249$			0.965614i			0.0611933i
$250$		0			0
$251$	−7.32369			−0.462267			−0.231134	−	0.972922i	$-0.574243\pi$
$251$	−7.32369			−0.462267			−0.231134	+	0.972922i	$0.574243\pi$
$252$		0			0
$253$		−	2.68221i		−	0.168629i
$254$		0			0
$255$			0.285220i			0.0178612i
$256$		0			0
$257$	11.1913			0.698097			0.349049	−	0.937105i	$-0.386505\pi$
$257$	11.1913			0.698097			0.349049	+	0.937105i	$0.386505\pi$
$258$		0			0
$259$	−1.41572			−0.0879684
$260$		0			0
$261$	−9.40119			−0.581919
$262$		0			0
$263$	−26.7906			−1.65198			−0.825990	−	0.563684i	$-0.809384\pi$
$263$	−26.7906			−1.65198			−0.825990	+	0.563684i	$0.809384\pi$
$264$		0			0
$265$			41.3774i			2.54180i
$266$		0			0
$267$		−	0.383733i		−	0.0234841i
$268$		0			0
$269$	10.6662			0.650330			0.325165	−	0.945657i	$-0.394580\pi$
$269$	10.6662			0.650330			0.325165	+	0.945657i	$0.394580\pi$
$270$		0			0
$271$		−	7.39856i		−	0.449431i	−0.974424	−	0.224715i	$-0.927855\pi$
$271$		−	7.39856i		−	0.449431i	0.974424	−	0.224715i	$-0.0721452\pi$
$272$		0			0
$273$	−0.187141	+	0.238534i	−0.0113263	+	0.0144367i
$274$		0			0
$275$			7.20395i			0.434414i
$276$		0			0
$277$	14.8678			0.893317			0.446658	−	0.894705i	$-0.352614\pi$
$277$	14.8678			0.893317			0.446658	+	0.894705i	$0.352614\pi$
$278$		0			0
$279$		−	3.39532i		−	0.203272i
$280$		0			0
$281$		−	5.23586i		−	0.312345i	−0.987730	−	0.156173i	$-0.950084\pi$
$281$		−	5.23586i		−	0.312345i	0.987730	−	0.156173i	$-0.0499157\pi$
$282$		0			0
$283$	−28.8552			−1.71526			−0.857632	−	0.514265i	$-0.828065\pi$
$283$	−28.8552			−1.71526			−0.857632	+	0.514265i	$0.828065\pi$
$284$		0			0
$285$	−2.04650			−0.121224
$286$		0			0
$287$	2.80825			0.165766
$288$		0			0
$289$	−16.0573			−0.944545
$290$		0			0
$291$		−	0.877918i		−	0.0514644i
$292$		0			0
$293$		−	18.7478i		−	1.09526i	−0.836721	−	0.547630i	$-0.815530\pi$
$293$		−	18.7478i		−	1.09526i	0.836721	−	0.547630i	$-0.184470\pi$
$294$		0			0
$295$	−0.963831			−0.0561164
$296$		0			0
$297$			0.503933i			0.0292412i
$298$		0			0
$299$	−7.60868	−	5.96936i	−0.440021	−	0.345217i
$300$		0			0
$301$		−	3.04376i		−	0.175439i
$302$		0			0
$303$	−0.135905			−0.00780756
$304$		0			0
$305$		−	17.1039i		−	0.979366i
$306$		0			0
$307$			8.77504i			0.500818i	0.968140	+	0.250409i	$0.0805651\pi$
$307$			8.77504i			0.500818i	−0.968140	+	0.250409i	$0.919435\pi$
$308$		0			0
$309$	0.423414			0.0240872
$310$		0			0
$311$	−28.7015			−1.62751			−0.813757	−	0.581205i	$-0.802582\pi$
$311$	−28.7015			−1.62751			−0.813757	+	0.581205i	$0.802582\pi$
$312$		0			0
$313$	−8.59052			−0.485565			−0.242782	−	0.970081i	$-0.578060\pi$
$313$	−8.59052			−0.485565			−0.242782	+	0.970081i	$0.578060\pi$
$314$		0			0
$315$	10.4555			0.589103
$316$		0			0
$317$		−	13.2942i		−	0.746676i	−0.927695	−	0.373338i	$-0.878213\pi$
$317$		−	13.2942i		−	0.746676i	0.927695	−	0.373338i	$-0.121787\pi$
$318$		0			0
$319$		−	3.14113i		−	0.175870i
$320$		0			0
$321$	0.414375			0.0231282
$322$		0			0
$323$			6.76430i			0.376376i
$324$		0			0
$325$	20.4356	+	16.0327i	1.13356	+	0.889332i
$326$		0			0
$327$			1.59457i			0.0881801i
$328$		0			0
$329$	−1.87989			−0.103642
$330$		0			0
$331$			0.728416i			0.0400374i	0.999800	+	0.0200187i	$0.00637257\pi$
$331$			0.728416i			0.0400374i	−0.999800	+	0.0200187i	$0.993627\pi$
$332$		0			0
$333$			4.23714i			0.232194i
$334$		0			0
$335$	3.31407			0.181067
$336$		0			0
$337$	−16.0620			−0.874955			−0.437477	−	0.899229i	$-0.644128\pi$
$337$	−16.0620			−0.874955			−0.437477	+	0.899229i	$0.644128\pi$
$338$		0			0
$339$	0.246797			0.0134042
$340$		0			0
$341$	1.13445			0.0614337
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	0.787910i		−	0.0424197i
$346$		0			0
$347$	6.77393			0.363644			0.181822	−	0.983331i	$-0.441801\pi$
$347$	6.77393			0.363644			0.181822	+	0.983331i	$0.441801\pi$
$348$		0			0
$349$		−	23.2267i		−	1.24330i	−0.783295	−	0.621650i	$-0.786463\pi$
$349$		−	23.2267i		−	1.24330i	0.783295	−	0.621650i	$-0.213537\pi$
$350$		0			0
$351$	1.42952	+	1.12152i	0.0763020	+	0.0598625i
$352$		0			0
$353$		−	9.10827i		−	0.484784i	−0.970178	−	0.242392i	$-0.922068\pi$
$353$		−	9.10827i		−	0.484784i	0.970178	−	0.242392i	$-0.0779321\pi$
$354$		0			0
$355$	−15.2254			−0.808081
$356$		0			0
$357$			0.0816451i			0.00432112i
$358$		0			0
$359$		−	33.4593i		−	1.76592i	−0.469453	−	0.882958i	$-0.655549\pi$
$359$		−	33.4593i		−	1.76592i	0.469453	−	0.882958i	$-0.344451\pi$
$360$		0			0
$361$	−29.5349			−1.55447
$362$		0			0
$363$	−0.0840880			−0.00441347
$364$		0			0
$365$	13.0165			0.681315
$366$		0			0
$367$	−9.45135			−0.493356			−0.246678	−	0.969097i	$-0.579339\pi$
$367$	−9.45135			−0.493356			−0.246678	+	0.969097i	$0.579339\pi$
$368$		0			0
$369$		−	8.40489i		−	0.437541i
$370$		0			0
$371$			11.8444i			0.614931i
$372$		0			0
$373$	−17.3470			−0.898192			−0.449096	−	0.893483i	$-0.648254\pi$
$373$	−17.3470			−0.898192			−0.449096	+	0.893483i	$0.648254\pi$
$374$		0			0
$375$			0.647418i			0.0334325i
$376$		0			0
$377$	−8.91052	−	6.99071i	−0.458915	−	0.360040i
$378$		0			0
$379$			13.4938i			0.693131i	0.938026	+	0.346565i	$0.112652\pi$
$379$			13.4938i			0.693131i	−0.938026	+	0.346565i	$0.887348\pi$
$380$		0			0
$381$	−1.53908			−0.0788494
$382$		0			0
$383$		−	10.0603i		−	0.514059i	−0.966404	−	0.257030i	$-0.917256\pi$
$383$		−	10.0603i		−	0.514059i	0.966404	−	0.257030i	$-0.0827438\pi$
$384$		0			0
$385$			3.49341i			0.178041i
$386$		0			0
$387$	−9.10975			−0.463074
$388$		0			0
$389$	−18.9985			−0.963261			−0.481631	−	0.876374i	$-0.659955\pi$
$389$	−18.9985			−0.963261			−0.481631	+	0.876374i	$0.659955\pi$
$390$		0			0
$391$	−2.60429			−0.131704
$392$		0			0
$393$	−0.546655			−0.0275751
$394$		0			0
$395$			41.1103i			2.06848i
$396$		0			0
$397$			8.99876i			0.451635i	0.974170	+	0.225817i	$0.0725053\pi$
$397$			8.99876i			0.451635i	−0.974170	+	0.225817i	$0.927495\pi$
$398$		0			0
$399$	−0.585815			−0.0293275
$400$		0			0
$401$		−	26.7224i		−	1.33445i	−0.744855	−	0.667227i	$-0.767481\pi$
$401$		−	26.7224i		−	1.33445i	0.744855	−	0.667227i	$-0.232519\pi$
$402$		0			0
$403$	2.52475	−	3.21810i	0.125767	−	0.160305i
$404$		0			0
$405$		−	31.2186i		−	1.55126i
$406$		0			0
$407$	−1.41572			−0.0701745
$408$		0			0
$409$			17.6632i			0.873390i	0.899610	+	0.436695i	$0.143851\pi$
$409$			17.6632i			0.873390i	−0.899610	+	0.436695i	$0.856149\pi$
$410$		0			0
$411$			0.393571i			0.0194134i
$412$		0			0
$413$	−0.275899			−0.0135761
$414$		0			0
$415$	40.1162			1.96923
$416$		0			0
$417$	1.49852			0.0733830
$418$		0			0
$419$	22.4274			1.09565			0.547824	−	0.836593i	$-0.315456\pi$
$419$	22.4274			1.09565			0.547824	+	0.836593i	$0.315456\pi$
$420$		0			0
$421$			9.68618i			0.472076i	0.971744	+	0.236038i	$0.0758489\pi$
$421$			9.68618i			0.472076i	−0.971744	+	0.236038i	$0.924151\pi$
$422$		0			0
$423$			5.62638i			0.273564i
$424$		0			0
$425$	6.99466			0.339291
$426$		0			0
$427$		−	4.89604i		−	0.236936i
$428$		0			0
$429$	−0.187141	+	0.238534i	−0.00903525	+	0.0115165i
$430$		0			0
$431$		−	26.6309i		−	1.28276i	−0.767222	−	0.641382i	$-0.778362\pi$
$431$		−	26.6309i		−	1.28276i	0.767222	−	0.641382i	$-0.221638\pi$
$432$		0			0
$433$	17.3201			0.832349			0.416175	−	0.909285i	$-0.363370\pi$
$433$	17.3201			0.832349			0.416175	+	0.909285i	$0.363370\pi$
$434$		0			0
$435$		−	0.922721i		−	0.0442411i
$436$		0			0
$437$		−	18.6861i		−	0.893879i
$438$		0			0
$439$	−18.9835			−0.906032			−0.453016	−	0.891502i	$-0.649652\pi$
$439$	−18.9835			−0.906032			−0.453016	+	0.891502i	$0.649652\pi$
$440$		0			0
$441$	2.99293			0.142520
$442$		0			0
$443$	−18.1450			−0.862096			−0.431048	−	0.902329i	$-0.641856\pi$
$443$	−18.1450			−0.862096			−0.431048	+	0.902329i	$0.641856\pi$
$444$		0			0
$445$	−15.9421			−0.755727
$446$		0			0
$447$			1.69478i			0.0801601i
$448$		0			0
$449$			22.9902i			1.08498i	0.840063	+	0.542488i	$0.182518\pi$
$449$			22.9902i			1.08498i	−0.840063	+	0.542488i	$0.817482\pi$
$450$		0			0
$451$	2.80825			0.132235
$452$		0			0
$453$		−	1.19767i		−	0.0562716i
$454$		0			0
$455$	9.90984	+	7.77473i	0.464580	+	0.364485i
$456$		0			0
$457$		−	26.2036i		−	1.22575i	−0.790179	−	0.612876i	$-0.790013\pi$
$457$		−	26.2036i		−	1.22575i	0.790179	−	0.612876i	$-0.209987\pi$
$458$		0			0
$459$	0.489293			0.0228383
$460$		0			0
$461$			5.72612i			0.266692i	0.991070	+	0.133346i	$0.0425722\pi$
$461$			5.72612i			0.266692i	−0.991070	+	0.133346i	$0.957428\pi$
$462$		0			0
$463$		−	33.0086i		−	1.53404i	−0.641622	−	0.767021i	$-0.721738\pi$
$463$		−	33.0086i		−	1.53404i	0.641622	−	0.767021i	$-0.278262\pi$
$464$		0			0
$465$	0.333248			0.0154540
$466$		0			0
$467$	−3.64897			−0.168854			−0.0844270	−	0.996430i	$-0.526906\pi$
$467$	−3.64897			−0.168854			−0.0844270	+	0.996430i	$0.526906\pi$
$468$		0			0
$469$	0.948663			0.0438052
$470$		0			0
$471$	−0.662727			−0.0305369
$472$		0			0
$473$		−	3.04376i		−	0.139952i
$474$		0			0
$475$			50.1877i			2.30277i
$476$		0			0
$477$	35.4495			1.62312
$478$		0			0
$479$		−	5.91128i		−	0.270093i	−0.990839	−	0.135047i	$-0.956882\pi$
$479$		−	5.91128i		−	0.270093i	0.990839	−	0.135047i	$-0.0431184\pi$
$480$		0			0
$481$	−3.15073	+	4.01599i	−0.143661	+	0.183114i
$482$		0			0
$483$		−	0.225542i		−	0.0102625i
$484$		0			0
$485$	−36.4729			−1.65615
$486$		0			0
$487$			25.9540i			1.17609i	0.808828	+	0.588045i	$0.200102\pi$
$487$			25.9540i			1.17609i	−0.808828	+	0.588045i	$0.799898\pi$
$488$		0			0
$489$			1.90891i			0.0863241i
$490$		0			0
$491$	6.21985			0.280698			0.140349	−	0.990102i	$-0.455178\pi$
$491$	6.21985			0.280698			0.140349	+	0.990102i	$0.455178\pi$
$492$		0			0
$493$	−3.04988			−0.137360
$494$		0			0
$495$	10.4555			0.469942
$496$		0			0
$497$	−4.35832			−0.195497
$498$		0			0
$499$			23.0550i			1.03208i	0.856564	+	0.516041i	$0.172595\pi$
$499$			23.0550i			1.03208i	−0.856564	+	0.516041i	$0.827405\pi$
$500$		0			0
$501$		−	0.201927i		−	0.00902142i
$502$		0			0
$503$	−34.3228			−1.53038			−0.765190	−	0.643804i	$-0.777355\pi$
$503$	−34.3228			−1.53038			−0.765190	+	0.643804i	$0.777355\pi$
$504$		0			0
$505$			5.64616i			0.251251i
$506$		0			0
$507$	0.260165	+	1.06173i	0.0115543	+	0.0471532i
$508$		0			0
$509$		−	31.2406i		−	1.38471i	−0.721555	−	0.692357i	$-0.756572\pi$
$509$		−	31.2406i		−	1.38471i	0.721555	−	0.692357i	$-0.243428\pi$
$510$		0			0
$511$	3.72601			0.164829
$512$		0			0
$513$			3.51075i			0.155003i
$514$		0			0
$515$		−	17.5906i		−	0.775136i
$516$		0			0
$517$	−1.87989			−0.0826775
$518$		0			0
$519$	0.585957			0.0257207
$520$		0			0
$521$	30.7806			1.34852			0.674261	−	0.738493i	$-0.264462\pi$
$521$	30.7806			1.34852			0.674261	+	0.738493i	$0.264462\pi$
$522$		0			0
$523$	13.8857			0.607178			0.303589	−	0.952803i	$-0.401815\pi$
$523$	13.8857			0.607178			0.303589	+	0.952803i	$0.401815\pi$
$524$		0			0
$525$			0.605765i			0.0264378i
$526$		0			0
$527$		−	1.10149i		−	0.0479816i
$528$		0			0
$529$	−15.8058			−0.687207
$530$		0			0
$531$			0.825748i			0.0358344i
$532$		0			0
$533$	6.24987	−	7.96622i	0.270712	−	0.345055i
$534$		0			0
$535$		−	17.2151i		−	0.744274i
$536$		0			0
$537$	−0.624798			−0.0269620
$538$		0			0
$539$			1.00000i			0.0430730i
$540$		0			0
$541$		−	22.9182i		−	0.985330i	−0.870219	−	0.492665i	$-0.836023\pi$
$541$		−	22.9182i		−	0.985330i	0.870219	−	0.492665i	$-0.163977\pi$
$542$		0			0
$543$	0.630904			0.0270747
$544$		0			0
$545$	66.2462			2.83767
$546$		0			0
$547$	21.0358			0.899425			0.449713	−	0.893173i	$-0.351526\pi$
$547$	21.0358			0.899425			0.449713	+	0.893173i	$0.351526\pi$
$548$		0			0
$549$	−14.6535			−0.625396
$550$		0			0
$551$		−	21.8833i		−	0.932261i
$552$		0			0
$553$			11.7679i			0.500423i
$554$		0			0
$555$	−0.415873			−0.0176528
$556$		0			0
$557$		−	30.6323i		−	1.29793i	−0.760818	−	0.648966i	$-0.775202\pi$
$557$		−	30.6323i		−	1.29793i	0.760818	−	0.648966i	$-0.224798\pi$
$558$		0			0
$559$	−8.63428	−	6.77399i	−0.365191	−	0.286509i
$560$		0			0
$561$			0.0816451i			0.00344706i
$562$		0			0
$563$	−4.58350			−0.193172			−0.0965858	−	0.995325i	$-0.530792\pi$
$563$	−4.58350			−0.193172			−0.0965858	+	0.995325i	$0.530792\pi$
$564$		0			0
$565$		−	10.2531i		−	0.431352i
$566$		0			0
$567$		−	8.93641i		−	0.375294i
$568$		0			0
$569$	−32.7235			−1.37184			−0.685920	−	0.727677i	$-0.740600\pi$
$569$	−32.7235			−1.37184			−0.685920	+	0.727677i	$0.740600\pi$
$570$		0			0
$571$	−5.33161			−0.223121			−0.111560	−	0.993758i	$-0.535585\pi$
$571$	−5.33161			−0.223121			−0.111560	+	0.993758i	$0.535585\pi$
$572$		0			0
$573$	−0.0628819			−0.00262693
$574$		0			0
$575$	−19.3225			−0.805804
$576$		0			0
$577$			7.85337i			0.326940i	0.986548	+	0.163470i	$0.0522687\pi$
$577$			7.85337i			0.326940i	−0.986548	+	0.163470i	$0.947731\pi$
$578$		0			0
$579$		−	0.687099i		−	0.0285548i
$580$		0			0
$581$	11.4834			0.476411
$582$		0			0
$583$			11.8444i			0.490545i
$584$		0			0
$585$	23.2692	−	29.6594i	0.962063	−	1.22627i
$586$		0			0
$587$		−	32.8177i		−	1.35453i	−0.735739	−	0.677265i	$-0.763165\pi$
$587$		−	32.8177i		−	1.35453i	0.735739	−	0.677265i	$-0.236835\pi$
$588$		0			0
$589$	7.90334			0.325651
$590$		0			0
$591$		−	0.775601i		−	0.0319040i
$592$		0			0
$593$		−	16.0222i		−	0.657953i	−0.944338	−	0.328977i	$-0.893296\pi$
$593$		−	16.0222i		−	0.657953i	0.944338	−	0.328977i	$-0.106704\pi$
$594$		0			0
$595$	3.39193			0.139055
$596$		0			0
$597$	0.847810			0.0346985
$598$		0			0
$599$	13.2161			0.539996			0.269998	−	0.962861i	$-0.412977\pi$
$599$	13.2161			0.539996			0.269998	+	0.962861i	$0.412977\pi$
$600$		0			0
$601$	14.2322			0.580544			0.290272	−	0.956944i	$-0.406254\pi$
$601$	14.2322			0.580544			0.290272	+	0.956944i	$0.406254\pi$
$602$		0			0
$603$		−	2.83928i		−	0.115624i
$604$		0			0
$605$			3.49341i			0.142028i
$606$		0			0
$607$	20.4629			0.830565			0.415282	−	0.909693i	$-0.363683\pi$
$607$	20.4629			0.830565			0.415282	+	0.909693i	$0.363683\pi$
$608$		0			0
$609$		−	0.264132i		−	0.0107032i
$610$		0			0
$611$	−4.18377	+	5.33272i	−0.169257	+	0.215739i
$612$		0			0
$613$		−	26.5811i		−	1.07360i	−0.843710	−	0.536800i	$-0.819633\pi$
$613$		−	26.5811i		−	1.07360i	0.843710	−	0.536800i	$-0.180367\pi$
$614$		0			0
$615$	0.824935			0.0332646
$616$		0			0
$617$			6.97272i			0.280711i	0.990101	+	0.140356i	$0.0448245\pi$
$617$			6.97272i			0.280711i	−0.990101	+	0.140356i	$0.955175\pi$
$618$		0			0
$619$			20.4333i			0.821283i	0.911797	+	0.410641i	$0.134695\pi$
$619$			20.4333i			0.821283i	−0.911797	+	0.410641i	$0.865305\pi$
$620$		0			0
$621$	−1.35165			−0.0542400
$622$		0			0
$623$	−4.56347			−0.182831
$624$		0			0
$625$	−9.12289			−0.364916
$626$		0			0
$627$	−0.585815			−0.0233952
$628$		0			0
$629$			1.37459i			0.0548084i
$630$		0			0
$631$			23.8699i			0.950247i	0.879919	+	0.475123i	$0.157597\pi$
$631$			23.8699i			0.950247i	−0.879919	+	0.475123i	$0.842403\pi$
$632$		0			0
$633$	−0.298402			−0.0118604
$634$		0			0
$635$			63.9406i			2.53741i
$636$		0			0
$637$	2.83672	+	2.22554i	0.112395	+	0.0881790i
$638$		0			0
$639$			13.0441i			0.516018i
$640$		0			0
$641$	12.0536			0.476089			0.238045	−	0.971254i	$-0.423494\pi$
$641$	12.0536			0.476089			0.238045	+	0.971254i	$0.423494\pi$
$642$		0			0
$643$			30.1980i			1.19089i	0.803394	+	0.595447i	$0.203025\pi$
$643$			30.1980i			1.19089i	−0.803394	+	0.595447i	$0.796975\pi$
$644$		0			0
$645$		−	0.894116i		−	0.0352058i
$646$		0			0
$647$	9.78337			0.384624			0.192312	−	0.981334i	$-0.438401\pi$
$647$	9.78337			0.384624			0.192312	+	0.981334i	$0.438401\pi$
$648$		0			0
$649$	−0.275899			−0.0108300
$650$		0			0
$651$	0.0953932			0.00373876
$652$		0			0
$653$	−2.05721			−0.0805048			−0.0402524	−	0.999190i	$-0.512816\pi$
$653$	−2.05721			−0.0805048			−0.0402524	+	0.999190i	$0.512816\pi$
$654$		0			0
$655$			22.7107i			0.887379i
$656$		0			0
$657$		−	11.1517i		−	0.435069i
$658$		0			0
$659$	−13.0143			−0.506964			−0.253482	−	0.967340i	$-0.581576\pi$
$659$	−13.0143			−0.506964			−0.253482	+	0.967340i	$0.581576\pi$
$660$		0			0
$661$		−	0.322348i		−	0.0125379i	−0.999980	−	0.00626894i	$-0.998005\pi$
$661$		−	0.322348i		−	0.0125379i	0.999980	−	0.00626894i	$-0.00199548\pi$
$662$		0			0
$663$	0.231604	+	0.181704i	0.00899477	+	0.00705681i
$664$		0			0
$665$			24.3376i			0.943770i
$666$		0			0
$667$	8.42518			0.326224
$668$		0			0
$669$			2.04261i			0.0789718i
$670$		0			0
$671$		−	4.89604i		−	0.189010i
$672$		0			0
$673$	−13.6248			−0.525198			−0.262599	−	0.964905i	$-0.584580\pi$
$673$	−13.6248			−0.525198			−0.262599	+	0.964905i	$0.584580\pi$
$674$		0			0
$675$	3.63031			0.139731
$676$		0			0
$677$	20.7816			0.798703			0.399351	−	0.916798i	$-0.369235\pi$
$677$	20.7816			0.798703			0.399351	+	0.916798i	$0.369235\pi$
$678$		0			0
$679$	−10.4405			−0.400668
$680$		0			0
$681$		−	1.15264i		−	0.0441693i
$682$		0			0
$683$			42.5659i			1.62874i	0.580347	+	0.814369i	$0.302917\pi$
$683$			42.5659i			1.62874i	−0.580347	+	0.814369i	$0.697083\pi$
$684$		0			0
$685$	16.3508			0.624733
$686$		0			0
$687$			0.191938i			0.00732288i
$688$		0			0
$689$	33.5993	+	26.3602i	1.28003	+	1.00424i
$690$		0			0
$691$		−	4.05706i		−	0.154338i	−0.997018	−	0.0771689i	$-0.975412\pi$
$691$		−	4.05706i		−	0.154338i	0.997018	−	0.0771689i	$-0.0245881\pi$
$692$		0			0
$693$	2.99293			0.113692
$694$		0			0
$695$		−	62.2558i		−	2.36150i
$696$		0			0
$697$		−	2.72667i		−	0.103280i
$698$		0			0
$699$	1.75953			0.0665514
$700$		0			0
$701$	42.6393			1.61046			0.805232	−	0.592960i	$-0.202041\pi$
$701$	42.6393			1.61046			0.805232	+	0.592960i	$0.202041\pi$
$702$		0			0
$703$	−9.86287			−0.371985
$704$		0			0
$705$	−0.552226			−0.0207980
$706$		0			0
$707$			1.61623i			0.0607845i
$708$		0			0
$709$		−	8.34238i		−	0.313305i	−0.987654	−	0.156652i	$-0.949930\pi$
$709$		−	8.34238i		−	0.313305i	0.987654	−	0.156652i	$-0.0500702\pi$
$710$		0			0
$711$	35.2206			1.32088
$712$		0			0
$713$			3.04282i			0.113955i
$714$		0			0
$715$	9.90984	+	7.77473i	0.370607	+	0.290758i
$716$		0			0
$717$		−	0.177799i		−	0.00664004i
$718$		0			0
$719$	−4.36188			−0.162671			−0.0813353	−	0.996687i	$-0.525918\pi$
$719$	−4.36188			−0.162671			−0.0813353	+	0.996687i	$0.525918\pi$
$720$		0			0
$721$		−	5.03537i		−	0.187527i
$722$		0			0
$723$		−	0.539371i		−	0.0200594i
$724$		0			0
$725$	−22.6286			−0.840404
$726$		0			0
$727$	28.8662			1.07059			0.535295	−	0.844665i	$-0.320201\pi$
$727$	28.8662			1.07059			0.535295	+	0.844665i	$0.320201\pi$
$728$		0			0
$729$	−26.6189			−0.985886
$730$		0			0
$731$	−2.95533			−0.109307
$732$		0			0
$733$		−	15.3204i		−	0.565872i	−0.959139	−	0.282936i	$-0.908692\pi$
$733$		−	15.3204i		−	0.565872i	0.959139	−	0.282936i	$-0.0913083\pi$
$734$		0			0
$735$			0.293754i			0.0108353i
$736$		0			0
$737$	0.948663			0.0349444
$738$		0			0
$739$		−	37.5122i		−	1.37991i	−0.723854	−	0.689953i	$-0.757631\pi$
$739$		−	37.5122i		−	1.37991i	0.723854	−	0.689953i	$-0.242369\pi$
$740$		0			0
$741$	−1.30375	+	1.66179i	−0.0478946	+	0.0610475i
$742$		0			0
$743$			18.4641i			0.677382i	0.940898	+	0.338691i	$0.109984\pi$
$743$			18.4641i			0.677382i	−0.940898	+	0.338691i	$0.890016\pi$
$744$		0			0
$745$	70.4090			2.57959
$746$		0			0
$747$		−	34.3689i		−	1.25749i
$748$		0			0
$749$		−	4.92787i		−	0.180061i
$750$		0			0
$751$	42.3735			1.54623			0.773115	−	0.634266i	$-0.218698\pi$
$751$	42.3735			1.54623			0.773115	+	0.634266i	$0.218698\pi$
$752$		0			0
$753$	−0.615835			−0.0224423
$754$		0			0
$755$	−49.7570			−1.81084
$756$		0			0
$757$	7.62517			0.277141			0.138571	−	0.990353i	$-0.455749\pi$
$757$	7.62517			0.277141			0.138571	+	0.990353i	$0.455749\pi$
$758$		0			0
$759$		−	0.225542i		−	0.00818664i
$760$		0			0
$761$		−	15.5163i		−	0.562466i	−0.959639	−	0.281233i	$-0.909257\pi$
$761$		−	15.5163i		−	0.562466i	0.959639	−	0.281233i	$-0.0907434\pi$
$762$		0			0
$763$	18.9632			0.686512
$764$		0			0
$765$		−	10.1518i		−	0.367039i
$766$		0			0
$767$	−0.614025	+	0.782650i	−0.0221712	+	0.0282598i
$768$		0			0
$769$		−	46.5468i		−	1.67852i	−0.543729	−	0.839261i	$-0.682988\pi$
$769$		−	46.5468i		−	1.67852i	0.543729	−	0.839261i	$-0.317012\pi$
$770$		0			0
$771$	0.941058			0.0338914
$772$		0			0
$773$			36.6789i			1.31925i	0.751596	+	0.659624i	$0.229284\pi$
$773$			36.6789i			1.31925i	−0.751596	+	0.659624i	$0.770716\pi$
$774$		0			0
$775$		−	8.17248i		−	0.293564i
$776$		0			0
$777$	−0.119045			−0.00427071
$778$		0			0
$779$	19.5642			0.700961
$780$		0			0
$781$	−4.35832			−0.155953
$782$		0			0
$783$	−1.58292			−0.0565690
$784$		0			0
$785$			27.5328i			0.982689i
$786$		0			0
$787$			20.4239i			0.728032i	0.931393	+	0.364016i	$0.118595\pi$
$787$			20.4239i			0.728032i	−0.931393	+	0.364016i	$0.881405\pi$
$788$		0			0
$789$	−2.25277			−0.0802007
$790$		0			0
$791$		−	2.93498i		−	0.104356i
$792$		0			0
$793$	−13.8887	−	10.8963i	−0.493202	−	0.386940i
$794$		0			0
$795$			3.47935i			0.123400i
$796$		0			0
$797$	42.2141			1.49530			0.747650	−	0.664093i	$-0.231182\pi$
$797$	42.2141			1.49530			0.747650	+	0.664093i	$0.231182\pi$
$798$		0			0
$799$			1.82528i			0.0645736i
$800$		0			0
$801$			13.6581i			0.482586i
$802$		0			0
$803$	3.72601			0.131488
$804$		0			0
$805$	−9.37007			−0.330252
$806$		0			0
$807$	0.896900			0.0315724
$808$		0			0
$809$	18.8253			0.661862			0.330931	−	0.943655i	$-0.392637\pi$
$809$	18.8253			0.661862			0.330931	+	0.943655i	$0.392637\pi$
$810$		0			0
$811$			5.92132i			0.207926i	0.994581	+	0.103963i	$0.0331523\pi$
$811$			5.92132i			0.207926i	−0.994581	+	0.103963i	$0.966848\pi$
$812$		0			0
$813$		−	0.622130i		−	0.0218191i
$814$		0			0
$815$	79.3054			2.77795
$816$		0			0
$817$		−	21.2049i		−	0.741866i
$818$		0			0
$819$	6.66088	−	8.49010i	0.232750	−	0.296668i
$820$		0			0
$821$			8.01842i			0.279845i	0.990162	+	0.139922i	$0.0446853\pi$
$821$			8.01842i			0.279845i	−0.990162	+	0.139922i	$0.955315\pi$
$822$		0			0
$823$	−25.5217			−0.889631			−0.444816	−	0.895622i	$-0.646731\pi$
$823$	−25.5217			−0.889631			−0.444816	+	0.895622i	$0.646731\pi$
$824$		0			0
$825$			0.605765i			0.0210900i
$826$		0			0
$827$		−	13.1240i		−	0.456367i	−0.973618	−	0.228183i	$-0.926721\pi$
$827$		−	13.1240i		−	0.456367i	0.973618	−	0.228183i	$-0.0732786\pi$
$828$		0			0
$829$	14.4207			0.500853			0.250426	−	0.968136i	$-0.419429\pi$
$829$	14.4207			0.500853			0.250426	+	0.968136i	$0.419429\pi$
$830$		0			0
$831$	1.25020			0.0433689
$832$		0			0
$833$	0.970948			0.0336414
$834$		0			0
$835$	−8.38899			−0.290313
$836$		0			0
$837$		−	0.571685i		−	0.0197603i
$838$		0			0
$839$			1.58974i			0.0548839i	0.999623	+	0.0274420i	$0.00873614\pi$
$839$			1.58974i			0.0548839i	−0.999623	+	0.0274420i	$0.991264\pi$
$840$		0			0
$841$	−19.1333			−0.659768
$842$		0			0
$843$		−	0.440273i		−	0.0151638i
$844$		0			0
$845$	44.1094	−	10.8085i	1.51741	−	0.371823i
$846$		0			0
$847$			1.00000i			0.0343604i
$848$		0			0
$849$	−2.42638			−0.0832730
$850$		0			0
$851$		−	3.79725i		−	0.130168i
$852$		0			0
$853$			22.9029i			0.784180i	0.919927	+	0.392090i	$0.128248\pi$
$853$			22.9029i			0.784180i	−0.919927	+	0.392090i	$0.871752\pi$
$854$		0			0
$855$	72.8406			2.49110
$856$		0			0
$857$	5.83442			0.199300			0.0996499	−	0.995023i	$-0.468228\pi$
$857$	5.83442			0.199300			0.0996499	+	0.995023i	$0.468228\pi$
$858$		0			0
$859$	−46.2903			−1.57940			−0.789702	−	0.613491i	$-0.789765\pi$
$859$	−46.2903			−1.57940			−0.789702	+	0.613491i	$0.789765\pi$
$860$		0			0
$861$	0.236140			0.00804763
$862$		0			0
$863$		−	31.6335i		−	1.07682i	−0.842684	−	0.538408i	$-0.819026\pi$
$863$		−	31.6335i		−	1.07682i	0.842684	−	0.538408i	$-0.180974\pi$
$864$		0			0
$865$		−	24.3434i		−	0.827702i
$866$		0			0
$867$	−1.35022			−0.0458560
$868$		0			0
$869$			11.7679i			0.399200i
$870$		0			0
$871$	2.11128	−	2.69109i	0.0715381	−	0.0911841i
$872$		0			0
$873$			31.2476i			1.05757i
$874$		0			0
$875$	7.69930			0.260284
$876$		0			0
$877$			15.2080i			0.513537i	0.966473	+	0.256768i	$0.0826577\pi$
$877$			15.2080i			0.513537i	−0.966473	+	0.256768i	$0.917342\pi$
$878$		0			0
$879$		−	1.57647i		−	0.0531729i
$880$		0			0
$881$	6.71576			0.226260			0.113130	−	0.993580i	$-0.463912\pi$
$881$	6.71576			0.226260			0.113130	+	0.993580i	$0.463912\pi$
$882$		0			0
$883$	25.0425			0.842746			0.421373	−	0.906887i	$-0.361548\pi$
$883$	25.0425			0.842746			0.421373	+	0.906887i	$0.361548\pi$
$884$		0			0
$885$	−0.0810466			−0.00272435
$886$		0			0
$887$	−21.0342			−0.706261			−0.353130	−	0.935574i	$-0.614883\pi$
$887$	−21.0342			−0.706261			−0.353130	+	0.935574i	$0.614883\pi$
$888$		0			0
$889$			18.3032i			0.613869i
$890$		0			0
$891$		−	8.93641i		−	0.299381i
$892$		0			0
$893$	−13.0966			−0.438262
$894$		0			0
$895$			25.9571i			0.867649i
$896$		0			0
$897$	−0.639798	−	0.501951i	−0.0213622	−	0.0167597i
$898$		0			0
$899$			3.56345i			0.118848i
$900$		0			0
$901$	11.5003			0.383131
$902$		0			0
$903$		−	0.255943i		−	0.00851726i
$904$		0			0
$905$		−	26.2107i		−	0.871275i
$906$		0			0
$907$	−30.2671			−1.00500			−0.502502	−	0.864576i	$-0.667587\pi$
$907$	−30.2671			−1.00500			−0.502502	+	0.864576i	$0.667587\pi$
$908$		0			0
$909$	4.83726			0.160442
$910$		0			0
$911$	−17.1423			−0.567951			−0.283975	−	0.958832i	$-0.591653\pi$
$911$	−17.1423			−0.567951			−0.283975	+	0.958832i	$0.591653\pi$
$912$		0			0
$913$	11.4834			0.380044
$914$		0			0
$915$		−	1.43823i		−	0.0475465i
$916$		0			0
$917$			6.50099i			0.214682i
$918$		0			0
$919$	−50.2743			−1.65839			−0.829197	−	0.558956i	$-0.811202\pi$
$919$	−50.2743			−1.65839			−0.829197	+	0.558956i	$0.811202\pi$
$920$		0			0
$921$			0.737875i			0.0243138i
$922$		0			0
$923$	−9.69960	+	12.3633i	−0.319266	+	0.406944i
$924$		0			0
$925$			10.1988i			0.335333i
$926$		0			0
$927$	−15.0705			−0.494980
$928$		0			0
$929$			17.7293i			0.581680i	0.956772	+	0.290840i	$0.0939347\pi$
$929$			17.7293i			0.581680i	−0.956772	+	0.290840i	$0.906065\pi$
$930$		0			0
$931$			6.96670i			0.228324i
$932$		0			0
$933$	−2.41345			−0.0790129
$934$		0			0
$935$	3.39193			0.110928
$936$		0			0
$937$	5.70667			0.186429			0.0932144	−	0.995646i	$-0.470286\pi$
$937$	5.70667			0.186429			0.0932144	+	0.995646i	$0.470286\pi$
$938$		0			0
$939$	−0.722359			−0.0235733
$940$		0			0
$941$		−	13.0143i		−	0.424254i	−0.977242	−	0.212127i	$-0.931961\pi$
$941$		−	13.0143i		−	0.424254i	0.977242	−	0.212127i	$-0.0680390\pi$
$942$		0			0
$943$			7.53231i			0.245286i
$944$		0			0
$945$	1.76045			0.0572674
$946$		0			0
$947$		−	32.1312i		−	1.04412i	−0.852908	−	0.522061i	$-0.825163\pi$
$947$		−	32.1312i		−	1.04412i	0.852908	−	0.522061i	$-0.174837\pi$
$948$		0			0
$949$	8.29239	−	10.5697i	0.269182	−	0.343106i
$950$		0			0
$951$		−	1.11788i		−	0.0362498i
$952$		0			0
$953$	−23.0769			−0.747534			−0.373767	−	0.927523i	$-0.621934\pi$
$953$	−23.0769			−0.747534			−0.373767	+	0.927523i	$0.621934\pi$
$954$		0			0
$955$			2.61241i			0.0845357i
$956$		0			0
$957$		−	0.264132i		−	0.00853816i
$958$		0			0
$959$	4.68047			0.151140
$960$		0			0
$961$	29.7130			0.958485
$962$		0			0
$963$	−14.7488			−0.475273
$964$		0			0
$965$	−28.5453			−0.918907
$966$		0			0
$967$		−	27.2700i		−	0.876943i	−0.898745	−	0.438471i	$-0.855520\pi$
$967$		−	27.2700i		−	0.876943i	0.898745	−	0.438471i	$-0.144480\pi$
$968$		0			0
$969$			0.568797i			0.0182724i
$970$		0			0
$971$	40.4164			1.29702			0.648512	−	0.761205i	$-0.275392\pi$
$971$	40.4164			1.29702			0.648512	+	0.761205i	$0.275392\pi$
$972$		0			0
$973$		−	17.8209i		−	0.571311i
$974$		0			0
$975$	1.71839	+	1.34815i	0.0550324	+	0.0431755i
$976$		0			0
$977$		−	10.6283i		−	0.340029i	−0.985442	−	0.170015i	$-0.945618\pi$
$977$		−	10.6283i		−	0.340029i	0.985442	−	0.170015i	$-0.0543815\pi$
$978$		0			0
$979$	−4.56347			−0.145849
$980$		0			0
$981$		−	56.7554i		−	1.81206i
$982$		0			0
$983$			0.470200i			0.0149970i	0.999972	+	0.00749852i	$0.00238688\pi$
$983$			0.470200i			0.0149970i	−0.999972	+	0.00749852i	$0.997613\pi$
$984$		0			0
$985$	−32.2222			−1.02668
$986$		0			0
$987$	−0.158076			−0.00503162
$988$		0			0
$989$	8.16399			0.259600
$990$		0			0
$991$	−37.3201			−1.18551			−0.592756	−	0.805382i	$-0.701960\pi$
$991$	−37.3201			−1.18551			−0.592756	+	0.805382i	$0.701960\pi$
$992$		0			0
$993$			0.0612510i			0.00194374i
$994$		0			0
$995$		−	35.2220i		−	1.11661i
$996$		0			0
$997$	−16.0431			−0.508091			−0.254045	−	0.967192i	$-0.581761\pi$
$997$	−16.0431			−0.508091			−0.254045	+	0.967192i	$0.581761\pi$
$998$		0			0
$999$			0.713427i			0.0225718i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.b.2157.15	✓	30
13.12	even	2	inner	4004.2.m.b.2157.16	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.b.2157.15	✓	30	1.1	even	1	trivial
4004.2.m.b.2157.16	yes	30	13.12	even	2	inner

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)

\(f(q)\)	\(=\)	\(q+0.0840880 q^{3} -3.49341i q^{5} -1.00000i q^{7} -2.99293 q^{9} +O(q^{10})\)	\(q+0.0840880 q^{3} -3.49341i q^{5} -1.00000i q^{7} -2.99293 q^{9} -1.00000i q^{11} +(-2.83672 - 2.22554i) q^{13} -0.293754i q^{15} -0.970948 q^{17} -6.96670i q^{19} -0.0840880i q^{21} +2.68221 q^{23} -7.20395 q^{25} -0.503933 q^{27} +3.14113 q^{29} +1.13445i q^{31} -0.0840880i q^{33} -3.49341 q^{35} -1.41572i q^{37} +(-0.238534 - 0.187141i) q^{39} +2.80825i q^{41} +3.04376 q^{43} +10.4555i q^{45} -1.87989i q^{47} -1.00000 q^{49} -0.0816451 q^{51} -11.8444 q^{53} -3.49341 q^{55} -0.585815i q^{57} -0.275899i q^{59} +4.89604 q^{61} +2.99293i q^{63} +(-7.77473 + 9.90984i) q^{65} +0.948663i q^{67} +0.225542 q^{69} -4.35832i q^{71} +3.72601i q^{73} -0.605765 q^{75} -1.00000 q^{77} -11.7679 q^{79} +8.93641 q^{81} +11.4834i q^{83} +3.39193i q^{85} +0.264132 q^{87} -4.56347i q^{89} +(-2.22554 + 2.83672i) q^{91} +0.0953932i q^{93} -24.3376 q^{95} -10.4405i q^{97} +2.99293i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 18 q^{9}+O(q^{10}) \) 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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