LMFDB - Embedded newform 4004.2.e.a.3849.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(3849,4004)]

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, 1, 1, 0]))

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

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

chi := DirichletCharacter("4004.3849");

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.e (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:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3849.6
Character	$\chi$	$=$	4004.3849
Dual form			4004.2.e.a.3849.43

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73693i q^{3} -3.14756i q^{5} +(-1.11363 - 2.39996i) q^{7} -4.49077 q^{9} +O(q^{10})$	$q-2.73693i q^{3} -3.14756i q^{5} +(-1.11363 - 2.39996i) q^{7} -4.49077 q^{9} +(2.27935 + 2.40927i) q^{11} +1.00000 q^{13} -8.61465 q^{15} -3.90044 q^{17} -1.86130 q^{19} +(-6.56852 + 3.04793i) q^{21} -1.60142 q^{23} -4.90715 q^{25} +4.08011i q^{27} +6.06837i q^{29} -4.12973i q^{31} +(6.59399 - 6.23841i) q^{33} +(-7.55403 + 3.50523i) q^{35} +3.63858 q^{37} -2.73693i q^{39} +3.75880 q^{41} +11.3386i q^{43} +14.1350i q^{45} +11.1376i q^{47} +(-4.51964 + 5.34535i) q^{49} +10.6752i q^{51} -14.0057 q^{53} +(7.58333 - 7.17439i) q^{55} +5.09425i q^{57} -10.2976i q^{59} -4.42305 q^{61} +(5.00106 + 10.7777i) q^{63} -3.14756i q^{65} -12.0761 q^{67} +4.38297i q^{69} -4.22097 q^{71} -0.120130 q^{73} +13.4305i q^{75} +(3.24380 - 8.15339i) q^{77} +5.64789i q^{79} -2.30532 q^{81} +5.02539 q^{83} +12.2769i q^{85} +16.6087 q^{87} +14.7433i q^{89} +(-1.11363 - 2.39996i) q^{91} -11.3028 q^{93} +5.85857i q^{95} -16.7847i q^{97} +(-10.2360 - 10.8195i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 4 q^{7} - 48 q^{9}+O(q^{10}) $ 48 * q - 4 * q^7 - 48 * q^9	$ 48 q - 4 q^{7} - 48 q^{9} + 2 q^{11} + 48 q^{13} + 8 q^{15} + 4 q^{17} + 10 q^{21} + 4 q^{23} - 44 q^{25} + 10 q^{33} - 14 q^{35} - 16 q^{37} + 10 q^{49} - 8 q^{53} - 12 q^{55} + 16 q^{61} + 16 q^{63} + 4 q^{67} - 16 q^{73} + 2 q^{77} + 64 q^{81} - 4 q^{83} - 12 q^{87} - 4 q^{91} + 36 q^{93} - 40 q^{99}+O(q^{100}) $ 48 * q - 4 * q^7 - 48 * q^9 + 2 * q^11 + 48 * q^13 + 8 * q^15 + 4 * q^17 + 10 * q^21 + 4 * q^23 - 44 * q^25 + 10 * q^33 - 14 * q^35 - 16 * q^37 + 10 * q^49 - 8 * q^53 - 12 * q^55 + 16 * q^61 + 16 * q^63 + 4 * q^67 - 16 * q^73 + 2 * q^77 + 64 * q^81 - 4 * q^83 - 12 * q^87 - 4 * q^91 + 36 * q^93 - 40 * q^99

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
$3$		−	2.73693i		−	1.58017i	−0.613000	−	0.790083i	$-0.710038\pi$
$3$		−	2.73693i		−	1.58017i	0.613000	−	0.790083i	$-0.289962\pi$
$4$		0			0
$5$		−	3.14756i		−	1.40763i	−0.710382	−	0.703817i	$-0.751478\pi$
$5$		−	3.14756i		−	1.40763i	0.710382	−	0.703817i	$-0.248522\pi$
$6$		0			0
$7$	−1.11363	−	2.39996i	−0.420914	−	0.907101i
$7$	−1.11363	−	2.39996i	−0.420914	−	0.907101i
$8$		0			0
$9$	−4.49077			−1.49692
$10$		0			0
$11$	2.27935	+	2.40927i	0.687249	+	0.726422i
$11$	2.27935	+	2.40927i	0.687249	+	0.726422i
$12$		0			0
$13$	1.00000			0.277350
$13$	1.00000			0.277350
$14$		0			0
$15$	−8.61465			−2.22429
$16$		0			0
$17$	−3.90044			−0.945996			−0.472998	−	0.881064i	$-0.656828\pi$
$17$	−3.90044			−0.945996			−0.472998	+	0.881064i	$0.656828\pi$
$18$		0			0
$19$	−1.86130			−0.427012			−0.213506	−	0.976942i	$-0.568488\pi$
$19$	−1.86130			−0.427012			−0.213506	+	0.976942i	$0.568488\pi$
$20$		0			0
$21$	−6.56852	+	3.04793i	−1.43337	+	0.665113i
$22$		0			0
$23$	−1.60142			−0.333920			−0.166960	−	0.985964i	$-0.553395\pi$
$23$	−1.60142			−0.333920			−0.166960	+	0.985964i	$0.553395\pi$
$24$		0			0
$25$	−4.90715			−0.981431
$26$		0			0
$27$			4.08011i			0.785218i
$28$		0			0
$29$			6.06837i			1.12687i	0.826161	+	0.563434i	$0.190520\pi$
$29$			6.06837i			1.12687i	−0.826161	+	0.563434i	$0.809480\pi$
$30$		0			0
$31$		−	4.12973i		−	0.741722i	−0.928688	−	0.370861i	$-0.879063\pi$
$31$		−	4.12973i		−	0.741722i	0.928688	−	0.370861i	$-0.120937\pi$
$32$		0			0
$33$	6.59399	−	6.23841i	1.14787	−	1.08597i
$34$		0			0
$35$	−7.55403	+	3.50523i	−1.27686	+	0.592492i
$36$		0			0
$37$	3.63858			0.598180			0.299090	−	0.954225i	$-0.403317\pi$
$37$	3.63858			0.598180			0.299090	+	0.954225i	$0.403317\pi$
$38$		0			0
$39$		−	2.73693i		−	0.438259i
$40$		0			0
$41$	3.75880			0.587026			0.293513	−	0.955955i	$-0.405176\pi$
$41$	3.75880			0.587026			0.293513	+	0.955955i	$0.405176\pi$
$42$		0			0
$43$			11.3386i			1.72912i	0.502526	+	0.864562i	$0.332404\pi$
$43$			11.3386i			1.72912i	−0.502526	+	0.864562i	$0.667596\pi$
$44$		0			0
$45$			14.1350i			2.10712i
$46$		0			0
$47$			11.1376i			1.62459i	0.583248	+	0.812294i	$0.301782\pi$
$47$			11.1376i			1.62459i	−0.583248	+	0.812294i	$0.698218\pi$
$48$		0			0
$49$	−4.51964	+	5.34535i	−0.645663	+	0.763622i
$50$		0			0
$51$			10.6752i			1.49483i
$52$		0			0
$53$	−14.0057			−1.92383			−0.961917	−	0.273342i	$-0.911871\pi$
$53$	−14.0057			−1.92383			−0.961917	+	0.273342i	$0.911871\pi$
$54$		0			0
$55$	7.58333	−	7.17439i	1.02254	−	0.967395i
$56$		0			0
$57$			5.09425i			0.674750i
$58$		0			0
$59$		−	10.2976i		−	1.34063i	−0.742078	−	0.670314i	$-0.766160\pi$
$59$		−	10.2976i		−	1.34063i	0.742078	−	0.670314i	$-0.233840\pi$
$60$		0			0
$61$	−4.42305			−0.566314			−0.283157	−	0.959074i	$-0.591382\pi$
$61$	−4.42305			−0.566314			−0.283157	+	0.959074i	$0.591382\pi$
$62$		0			0
$63$	5.00106	+	10.7777i	0.630075	+	1.35786i
$64$		0			0
$65$		−	3.14756i		−	0.390407i
$66$		0			0
$67$	−12.0761			−1.47533			−0.737664	−	0.675168i	$-0.764071\pi$
$67$	−12.0761			−1.47533			−0.737664	+	0.675168i	$0.764071\pi$
$68$		0			0
$69$			4.38297i			0.527648i
$70$		0			0
$71$	−4.22097			−0.500937			−0.250469	−	0.968125i	$-0.580585\pi$
$71$	−4.22097			−0.500937			−0.250469	+	0.968125i	$0.580585\pi$
$72$		0			0
$73$	−0.120130			−0.0140602			−0.00703009	−	0.999975i	$-0.502238\pi$
$73$	−0.120130			−0.0140602			−0.00703009	+	0.999975i	$0.502238\pi$
$74$		0			0
$75$			13.4305i			1.55082i
$76$		0			0
$77$	3.24380	−	8.15339i	0.369665	−	0.929165i
$78$		0			0
$79$			5.64789i			0.635437i	0.948185	+	0.317718i	$0.102917\pi$
$79$			5.64789i			0.635437i	−0.948185	+	0.317718i	$0.897083\pi$
$80$		0			0
$81$	−2.30532			−0.256147
$82$		0			0
$83$	5.02539			0.551608			0.275804	−	0.961214i	$-0.411056\pi$
$83$	5.02539			0.551608			0.275804	+	0.961214i	$0.411056\pi$
$84$		0			0
$85$			12.2769i			1.33162i
$86$		0			0
$87$	16.6087			1.78064
$88$		0			0
$89$			14.7433i			1.56279i	0.624039	+	0.781393i	$0.285491\pi$
$89$			14.7433i			1.56279i	−0.624039	+	0.781393i	$0.714509\pi$
$90$		0			0
$91$	−1.11363	−	2.39996i	−0.116740	−	0.251584i
$92$		0			0
$93$	−11.3028			−1.17204
$94$		0			0
$95$			5.85857i			0.601077i
$96$		0			0
$97$		−	16.7847i		−	1.70423i	−0.523352	−	0.852116i	$-0.675319\pi$
$97$		−	16.7847i		−	1.70423i	0.523352	−	0.852116i	$-0.324681\pi$
$98$		0			0
$99$	−10.2360	−	10.8195i	−1.02876	−	1.08740i
$100$		0			0
$101$	−14.2083			−1.41378			−0.706891	−	0.707323i	$-0.749903\pi$
$101$	−14.2083			−1.41378			−0.706891	+	0.707323i	$0.749903\pi$
$102$		0			0
$103$		−	9.65651i		−	0.951485i	−0.879585	−	0.475742i	$-0.842180\pi$
$103$		−	9.65651i		−	0.951485i	0.879585	−	0.475742i	$-0.157820\pi$
$104$		0			0
$105$	9.59355	+	20.6748i	0.936235	+	2.01766i
$106$		0			0
$107$		−	19.3026i		−	1.86606i	−0.359803	−	0.933028i	$-0.617156\pi$
$107$		−	19.3026i		−	1.86606i	0.359803	−	0.933028i	$-0.382844\pi$
$108$		0			0
$109$		−	15.8326i		−	1.51649i	−0.651969	−	0.758246i	$-0.726057\pi$
$109$		−	15.8326i		−	1.51649i	0.651969	−	0.758246i	$-0.273943\pi$
$110$		0			0
$111$		−	9.95854i		−	0.945223i
$112$		0			0
$113$	7.76604			0.730567			0.365284	−	0.930896i	$-0.380972\pi$
$113$	7.76604			0.730567			0.365284	+	0.930896i	$0.380972\pi$
$114$		0			0
$115$			5.04058i			0.470036i
$116$		0			0
$117$	−4.49077			−0.415171
$118$		0			0
$119$	4.34366	+	9.36091i	0.398183	+	0.858114i
$120$		0			0
$121$	−0.609151	+	10.9831i	−0.0553774	+	0.998465i
$122$		0			0
$123$		−	10.2876i		−	0.927598i
$124$		0			0
$125$		−	0.292239i		−	0.0261387i
$126$		0			0
$127$		−	8.27704i		−	0.734468i	−0.930129	−	0.367234i	$-0.880305\pi$
$127$		−	8.27704i		−	0.734468i	0.930129	−	0.367234i	$-0.119695\pi$
$128$		0			0
$129$	31.0330			2.73230
$130$		0			0
$131$	−2.47316			−0.216081			−0.108041	−	0.994146i	$-0.534458\pi$
$131$	−2.47316			−0.216081			−0.108041	+	0.994146i	$0.534458\pi$
$132$		0			0
$133$	2.07281	+	4.46706i	0.179735	+	0.387343i
$134$		0			0
$135$	12.8424			1.10530
$136$		0			0
$137$	−4.40553			−0.376390			−0.188195	−	0.982132i	$-0.560264\pi$
$137$	−4.40553			−0.376390			−0.188195	+	0.982132i	$0.560264\pi$
$138$		0			0
$139$	−2.55783			−0.216952			−0.108476	−	0.994099i	$-0.534597\pi$
$139$	−2.55783			−0.216952			−0.108476	+	0.994099i	$0.534597\pi$
$140$		0			0
$141$	30.4828			2.56712
$142$		0			0
$143$	2.27935	+	2.40927i	0.190609	+	0.201473i
$144$		0			0
$145$	19.1006			1.58622
$146$		0			0
$147$	14.6298	+	12.3699i	1.20665	+	1.02025i
$148$		0			0
$149$			0.0193873i			0.00158827i	1.00000		0.000794135i	$0.000252781\pi$
$149$			0.0193873i			0.00158827i	−1.00000		0.000794135i	$0.999747\pi$
$150$		0			0
$151$			6.47804i			0.527175i	0.964635	+	0.263588i	$0.0849058\pi$
$151$			6.47804i			0.527175i	−0.964635	+	0.263588i	$0.915094\pi$
$152$		0			0
$153$	17.5160			1.41608
$154$		0			0
$155$	−12.9986			−1.04407
$156$		0			0
$157$		−	1.78371i		−	0.142356i	−0.997464	−	0.0711779i	$-0.977324\pi$
$157$		−	1.78371i		−	0.142356i	0.997464	−	0.0711779i	$-0.0226758\pi$
$158$		0			0
$159$			38.3326i			3.03998i
$160$		0			0
$161$	1.78340	+	3.84335i	0.140551	+	0.302899i
$162$		0			0
$163$	7.32835			0.574001			0.287000	−	0.957930i	$-0.407342\pi$
$163$	7.32835			0.574001			0.287000	+	0.957930i	$0.407342\pi$
$164$		0			0
$165$	−19.6358	−	20.7550i	−1.52864	−	1.61577i
$166$		0			0
$167$	−5.49608			−0.425299			−0.212650	−	0.977129i	$-0.568209\pi$
$167$	−5.49608			−0.425299			−0.212650	+	0.977129i	$0.568209\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	8.35868			0.639204
$172$		0			0
$173$	−20.5935			−1.56570			−0.782849	−	0.622212i	$-0.786234\pi$
$173$	−20.5935			−1.56570			−0.782849	+	0.622212i	$0.786234\pi$
$174$		0			0
$175$	5.46477	+	11.7770i	0.413098	+	0.890257i
$176$		0			0
$177$	−28.1836			−2.11841
$178$		0			0
$179$	24.9131			1.86210			0.931048	−	0.364898i	$-0.118896\pi$
$179$	24.9131			1.86210			0.931048	+	0.364898i	$0.118896\pi$
$180$		0			0
$181$		−	6.64734i		−	0.494093i	−0.969004	−	0.247047i	$-0.920540\pi$
$181$		−	6.64734i		−	0.494093i	0.969004	−	0.247047i	$-0.0794601\pi$
$182$		0			0
$183$			12.1056i			0.894870i
$184$		0			0
$185$		−	11.4527i		−	0.842017i
$186$		0			0
$187$	−8.89046	−	9.39721i	−0.650135	−	0.687192i
$188$		0			0
$189$	9.79212	−	4.54375i	0.712272	−	0.330509i
$190$		0			0
$191$	0.750554			0.0543082			0.0271541	−	0.999631i	$-0.491356\pi$
$191$	0.750554			0.0543082			0.0271541	+	0.999631i	$0.491356\pi$
$192$		0			0
$193$			17.1123i			1.23177i	0.787836	+	0.615885i	$0.211201\pi$
$193$			17.1123i			1.23177i	−0.787836	+	0.615885i	$0.788799\pi$
$194$		0			0
$195$	−8.61465			−0.616908
$196$		0			0
$197$			8.77197i			0.624977i	0.949922	+	0.312489i	$0.101163\pi$
$197$			8.77197i			0.624977i	−0.949922	+	0.312489i	$0.898837\pi$
$198$		0			0
$199$			0.964316i			0.0683585i	0.999416	+	0.0341793i	$0.0108817\pi$
$199$			0.964316i			0.0683585i	−0.999416	+	0.0341793i	$0.989118\pi$
$200$		0			0
$201$			33.0513i			2.33126i
$202$		0			0
$203$	14.5639	−	6.75794i	1.02218	−	0.474314i
$204$		0			0
$205$		−	11.8311i		−	0.826318i
$206$		0			0
$207$	7.19161			0.499852
$208$		0			0
$209$	−4.24256	−	4.48438i	−0.293464	−	0.310191i
$210$		0			0
$211$			6.85981i			0.472249i	0.971723	+	0.236124i	$0.0758773\pi$
$211$			6.85981i			0.472249i	−0.971723	+	0.236124i	$0.924123\pi$
$212$		0			0
$213$			11.5525i			0.791563i
$214$		0			0
$215$	35.6890			2.43397
$216$		0			0
$217$	−9.91120	+	4.59900i	−0.672816	+	0.312201i
$218$		0			0
$219$			0.328788i			0.0222174i
$220$		0			0
$221$	−3.90044			−0.262372
$222$		0			0
$223$			8.74386i			0.585532i	0.956184	+	0.292766i	$0.0945757\pi$
$223$			8.74386i			0.585532i	−0.956184	+	0.292766i	$0.905424\pi$
$224$		0			0
$225$	22.0369			1.46913
$226$		0			0
$227$	15.8234			1.05023			0.525117	−	0.851030i	$-0.324022\pi$
$227$	15.8234			1.05023			0.525117	+	0.851030i	$0.324022\pi$
$228$		0			0
$229$		−	21.9838i		−	1.45273i	−0.687308	−	0.726366i	$-0.741208\pi$
$229$		−	21.9838i		−	1.45273i	0.687308	−	0.726366i	$-0.258792\pi$
$230$		0			0
$231$	−22.3152	−	8.87804i	−1.46823	−	0.584132i
$232$		0			0
$233$		−	15.0836i		−	0.988161i	−0.869416	−	0.494081i	$-0.835505\pi$
$233$		−	15.0836i		−	0.988161i	0.869416	−	0.494081i	$-0.164495\pi$
$234$		0			0
$235$	35.0563			2.28682
$236$		0			0
$237$	15.4578			1.00410
$238$		0			0
$239$		−	26.2329i		−	1.69686i	−0.529305	−	0.848432i	$-0.677547\pi$
$239$		−	26.2329i		−	1.69686i	0.529305	−	0.848432i	$-0.322453\pi$
$240$		0			0
$241$	−3.69058			−0.237731			−0.118866	−	0.992910i	$-0.537926\pi$
$241$	−3.69058			−0.237731			−0.118866	+	0.992910i	$0.537926\pi$
$242$		0			0
$243$			18.5498i			1.18997i
$244$		0			0
$245$	16.8248	+	14.2259i	1.07490	+	0.908857i
$246$		0			0
$247$	−1.86130			−0.118432
$248$		0			0
$249$		−	13.7541i		−	0.871632i
$250$		0			0
$251$			18.8493i			1.18975i	0.803816	+	0.594877i	$0.202799\pi$
$251$			18.8493i			1.18975i	−0.803816	+	0.594877i	$0.797201\pi$
$252$		0			0
$253$	−3.65020	−	3.85826i	−0.229486	−	0.242566i
$254$		0			0
$255$	33.6009			2.10417
$256$		0			0
$257$		−	12.0766i		−	0.753319i	−0.926352	−	0.376659i	$-0.877073\pi$
$257$		−	12.0766i		−	0.753319i	0.926352	−	0.376659i	$-0.122927\pi$
$258$		0			0
$259$	−4.05205	−	8.73247i	−0.251782	−	0.542609i
$260$		0			0
$261$		−	27.2516i		−	1.68683i
$262$		0			0
$263$		−	27.6745i		−	1.70648i	−0.521517	−	0.853241i	$-0.674634\pi$
$263$		−	27.6745i		−	1.70648i	0.521517	−	0.853241i	$-0.325366\pi$
$264$		0			0
$265$			44.0839i			2.70805i
$266$		0			0
$267$	40.3513			2.46946
$268$		0			0
$269$		−	2.67178i		−	0.162901i	−0.996677	−	0.0814507i	$-0.974045\pi$
$269$		−	2.67178i		−	0.162901i	0.996677	−	0.0814507i	$-0.0259553\pi$
$270$		0			0
$271$	−5.87211			−0.356705			−0.178353	−	0.983967i	$-0.557077\pi$
$271$	−5.87211			−0.356705			−0.178353	+	0.983967i	$0.557077\pi$
$272$		0			0
$273$	−6.56852	+	3.04793i	−0.397545	+	0.184469i
$274$		0			0
$275$	−11.1851	−	11.8227i	−0.674487	−	0.712933i
$276$		0			0
$277$		−	26.4655i		−	1.59016i	−0.606507	−	0.795078i	$-0.707430\pi$
$277$		−	26.4655i		−	1.59016i	0.606507	−	0.795078i	$-0.292570\pi$
$278$		0			0
$279$			18.5457i			1.11030i
$280$		0			0
$281$		−	7.84478i		−	0.467980i	−0.972239	−	0.233990i	$-0.924822\pi$
$281$		−	7.84478i		−	0.467980i	0.972239	−	0.233990i	$-0.0751784\pi$
$282$		0			0
$283$	18.7144			1.11246			0.556228	−	0.831030i	$-0.312248\pi$
$283$	18.7144			1.11246			0.556228	+	0.831030i	$0.312248\pi$
$284$		0			0
$285$	16.0345			0.949801
$286$		0			0
$287$	−4.18592	−	9.02099i	−0.247087	−	0.532492i
$288$		0			0
$289$	−1.78656			−0.105092
$290$		0			0
$291$	−45.9386			−2.69297
$292$		0			0
$293$	0.530045			0.0309656			0.0154828	−	0.999880i	$-0.495071\pi$
$293$	0.530045			0.0309656			0.0154828	+	0.999880i	$0.495071\pi$
$294$		0			0
$295$	−32.4122			−1.88711
$296$		0			0
$297$	−9.83009	+	9.30000i	−0.570400	+	0.539641i
$298$		0			0
$299$	−1.60142			−0.0926126
$300$		0			0
$301$	27.2123	−	12.6271i	1.56849	−	0.727812i
$302$		0			0
$303$			38.8871i			2.23401i
$304$		0			0
$305$			13.9218i			0.797162i
$306$		0			0
$307$	−5.42819			−0.309803			−0.154901	−	0.987930i	$-0.549506\pi$
$307$	−5.42819			−0.309803			−0.154901	+	0.987930i	$0.549506\pi$
$308$		0			0
$309$	−26.4292			−1.50350
$310$		0			0
$311$			3.21768i			0.182458i	0.995830	+	0.0912291i	$0.0290796\pi$
$311$			3.21768i			0.182458i	−0.995830	+	0.0912291i	$0.970920\pi$
$312$		0			0
$313$		−	20.0100i		−	1.13103i	−0.824737	−	0.565517i	$-0.808677\pi$
$313$		−	20.0100i		−	1.13103i	0.824737	−	0.565517i	$-0.191323\pi$
$314$		0			0
$315$	33.9234	−	15.7412i	1.91137	−	0.886914i
$316$		0			0
$317$	−5.35916			−0.301001			−0.150500	−	0.988610i	$-0.548088\pi$
$317$	−5.35916			−0.301001			−0.150500	+	0.988610i	$0.548088\pi$
$318$		0			0
$319$	−14.6203	+	13.8319i	−0.818582	+	0.774439i
$320$		0			0
$321$	−52.8299			−2.94868
$322$		0			0
$323$	7.25991			0.403952
$324$		0			0
$325$	−4.90715			−0.272200
$326$		0			0
$327$	−43.3327			−2.39631
$328$		0			0
$329$	26.7299	−	12.4032i	1.47367	−	0.683811i
$330$		0			0
$331$	−21.3483			−1.17341			−0.586706	−	0.809800i	$-0.699575\pi$
$331$	−21.3483			−1.17341			−0.586706	+	0.809800i	$0.699575\pi$
$332$		0			0
$333$	−16.3400			−0.895428
$334$		0			0
$335$			38.0102i			2.07672i
$336$		0			0
$337$			6.17992i			0.336642i	0.985732	+	0.168321i	$0.0538344\pi$
$337$			6.17992i			0.336642i	−0.985732	+	0.168321i	$0.946166\pi$
$338$		0			0
$339$		−	21.2551i		−	1.15442i
$340$		0			0
$341$	9.94963	−	9.41309i	0.538803	−	0.509748i
$342$		0			0
$343$	17.8619	+	4.89422i	0.964451	+	0.264263i
$344$		0			0
$345$	13.7957			0.742735
$346$		0			0
$347$		−	10.7497i		−	0.577075i	−0.957469	−	0.288537i	$-0.906831\pi$
$347$		−	10.7497i		−	0.577075i	0.957469	−	0.288537i	$-0.0931689\pi$
$348$		0			0
$349$	−31.6426			−1.69379			−0.846894	−	0.531762i	$-0.821530\pi$
$349$	−31.6426			−1.69379			−0.846894	+	0.531762i	$0.821530\pi$
$350$		0			0
$351$			4.08011i			0.217780i
$352$		0			0
$353$			4.87946i			0.259707i	0.991533	+	0.129854i	$0.0414508\pi$
$353$			4.87946i			0.259707i	−0.991533	+	0.129854i	$0.958549\pi$
$354$		0			0
$355$			13.2858i			0.705136i
$356$		0			0
$357$	25.6201	−	11.8883i	1.35596	−	0.629194i
$358$		0			0
$359$			8.02193i			0.423381i	0.977337	+	0.211691i	$0.0678969\pi$
$359$			8.02193i			0.423381i	−0.977337	+	0.211691i	$0.932103\pi$
$360$		0			0
$361$	−15.5355			−0.817660
$362$		0			0
$363$	30.0600	+	1.66720i	1.57774	+	0.0875054i
$364$		0			0
$365$			0.378118i			0.0197916i
$366$		0			0
$367$		−	17.7288i		−	0.925438i	−0.886505	−	0.462719i	$-0.846874\pi$
$367$		−	17.7288i		−	0.925438i	0.886505	−	0.462719i	$-0.153126\pi$
$368$		0			0
$369$	−16.8799			−0.878732
$370$		0			0
$371$	15.5972	+	33.6132i	0.809768	+	1.74511i
$372$		0			0
$373$			31.0939i			1.60998i	0.593288	+	0.804990i	$0.297829\pi$
$373$			31.0939i			1.60998i	−0.593288	+	0.804990i	$0.702171\pi$
$374$		0			0
$375$	−0.799837			−0.0413034
$376$		0			0
$377$			6.06837i			0.312537i
$378$		0			0
$379$	−11.8604			−0.609226			−0.304613	−	0.952476i	$-0.598527\pi$
$379$	−11.8604			−0.609226			−0.304613	+	0.952476i	$0.598527\pi$
$380$		0			0
$381$	−22.6536			−1.16058
$382$		0			0
$383$			31.6769i			1.61861i	0.587386	+	0.809307i	$0.300157\pi$
$383$			31.6769i			1.61861i	−0.587386	+	0.809307i	$0.699843\pi$
$384$		0			0
$385$	−25.6633	−	10.2101i	−1.30792	−	0.520353i
$386$		0			0
$387$		−	50.9191i		−	2.58836i
$388$		0			0
$389$	−30.2305			−1.53275			−0.766375	−	0.642394i	$-0.777941\pi$
$389$	−30.2305			−1.53275			−0.766375	+	0.642394i	$0.777941\pi$
$390$		0			0
$391$	6.24625			0.315887
$392$		0			0
$393$			6.76886i			0.341444i
$394$		0			0
$395$	17.7771			0.894462
$396$		0			0
$397$		−	13.5078i		−	0.677937i	−0.940798	−	0.338968i	$-0.889922\pi$
$397$		−	13.5078i		−	0.677937i	0.940798	−	0.338968i	$-0.110078\pi$
$398$		0			0
$399$	12.2260	−	5.67313i	0.612066	−	0.284012i
$400$		0			0
$401$	15.3629			0.767185			0.383592	−	0.923502i	$-0.374687\pi$
$401$	15.3629			0.767185			0.383592	+	0.923502i	$0.374687\pi$
$402$		0			0
$403$		−	4.12973i		−	0.205717i
$404$		0			0
$405$			7.25615i			0.360561i
$406$		0			0
$407$	8.29360	+	8.76633i	0.411098	+	0.434531i
$408$		0			0
$409$	−21.5197			−1.06408			−0.532041	−	0.846719i	$-0.678575\pi$
$409$	−21.5197			−1.06408			−0.532041	+	0.846719i	$0.678575\pi$
$410$		0			0
$411$			12.0576i			0.594758i
$412$		0			0
$413$	−24.7137	+	11.4677i	−1.21608	+	0.564288i
$414$		0			0
$415$		−	15.8177i		−	0.776462i
$416$		0			0
$417$			7.00060i			0.342821i
$418$		0			0
$419$			25.5313i			1.24728i	0.781710	+	0.623642i	$0.214348\pi$
$419$			25.5313i			1.24728i	−0.781710	+	0.623642i	$0.785652\pi$
$420$		0			0
$421$	12.8524			0.626386			0.313193	−	0.949689i	$-0.398601\pi$
$421$	12.8524			0.626386			0.313193	+	0.949689i	$0.398601\pi$
$422$		0			0
$423$		−	50.0164i		−	2.43188i
$424$		0			0
$425$	19.1401			0.928429
$426$		0			0
$427$	4.92566	+	10.6152i	0.238369	+	0.513704i
$428$		0			0
$429$	6.59399	−	6.23841i	0.318361	−	0.301193i
$430$		0			0
$431$			1.86201i			0.0896900i	0.998994	+	0.0448450i	$0.0142794\pi$
$431$			1.86201i			0.0896900i	−0.998994	+	0.0448450i	$0.985721\pi$
$432$		0			0
$433$		−	1.62280i		−	0.0779867i	−0.999239	−	0.0389933i	$-0.987585\pi$
$433$		−	1.62280i		−	0.0779867i	0.999239	−	0.0389933i	$-0.0124151\pi$
$434$		0			0
$435$		−	52.2769i		−	2.50648i
$436$		0			0
$437$	2.98073			0.142588
$438$		0			0
$439$	−22.3837			−1.06831			−0.534157	−	0.845386i	$-0.679371\pi$
$439$	−22.3837			−1.06831			−0.534157	+	0.845386i	$0.679371\pi$
$440$		0			0
$441$	20.2967	−	24.0047i	0.966508	−	1.14308i
$442$		0			0
$443$	32.0511			1.52279			0.761396	−	0.648287i	$-0.224514\pi$
$443$	32.0511			1.52279			0.761396	+	0.648287i	$0.224514\pi$
$444$		0			0
$445$	46.4055			2.19983
$446$		0			0
$447$	0.0530616			0.00250973
$448$		0			0
$449$	−2.24122			−0.105770			−0.0528848	−	0.998601i	$-0.516842\pi$
$449$	−2.24122			−0.105770			−0.0528848	+	0.998601i	$0.516842\pi$
$450$		0			0
$451$	8.56762	+	9.05596i	0.403433	+	0.426429i
$452$		0			0
$453$	17.7299			0.833024
$454$		0			0
$455$	−7.55403	+	3.50523i	−0.354139	+	0.164328i
$456$		0			0
$457$			24.6556i			1.15334i	0.816977	+	0.576670i	$0.195648\pi$
$457$			24.6556i			1.15334i	−0.816977	+	0.576670i	$0.804352\pi$
$458$		0			0
$459$		−	15.9142i		−	0.742813i
$460$		0			0
$461$	−36.4822			−1.69915			−0.849574	−	0.527470i	$-0.823141\pi$
$461$	−36.4822			−1.69915			−0.849574	+	0.527470i	$0.823141\pi$
$462$		0			0
$463$	4.81549			0.223795			0.111897	−	0.993720i	$-0.464307\pi$
$463$	4.81549			0.223795			0.111897	+	0.993720i	$0.464307\pi$
$464$		0			0
$465$			35.5762i			1.64981i
$466$		0			0
$467$		−	28.6090i		−	1.32387i	−0.749562	−	0.661934i	$-0.769736\pi$
$467$		−	28.6090i		−	1.32387i	0.749562	−	0.661934i	$-0.230264\pi$
$468$		0			0
$469$	13.4483	+	28.9821i	0.620986	+	1.33827i
$470$		0			0
$471$	−4.88189			−0.224946
$472$		0			0
$473$	−27.3178	+	25.8447i	−1.25607	+	1.18834i
$474$		0			0
$475$	9.13371			0.419083
$476$		0			0
$477$	62.8964			2.87983
$478$		0			0
$479$	28.2301			1.28987			0.644934	−	0.764239i	$-0.276885\pi$
$479$	28.2301			1.28987			0.644934	+	0.764239i	$0.276885\pi$
$480$		0			0
$481$	3.63858			0.165905
$482$		0			0
$483$	10.5190	−	4.88102i	0.478630	−	0.222094i
$484$		0			0
$485$	−52.8310			−2.39893
$486$		0			0
$487$	−1.92644			−0.0872953			−0.0436476	−	0.999047i	$-0.513898\pi$
$487$	−1.92644			−0.0872953			−0.0436476	+	0.999047i	$0.513898\pi$
$488$		0			0
$489$		−	20.0572i		−	0.907016i
$490$		0			0
$491$			34.9755i			1.57842i	0.614121	+	0.789212i	$0.289511\pi$
$491$			34.9755i			1.57842i	−0.614121	+	0.789212i	$0.710489\pi$
$492$		0			0
$493$		−	23.6693i		−	1.06601i
$494$		0			0
$495$	−34.0549	+	32.2185i	−1.53066	+	1.44811i
$496$		0			0
$497$	4.70061	+	10.1302i	0.210851	+	0.454400i
$498$		0			0
$499$	−24.1798			−1.08244			−0.541218	−	0.840882i	$-0.682037\pi$
$499$	−24.1798			−1.08244			−0.541218	+	0.840882i	$0.682037\pi$
$500$		0			0
$501$			15.0424i			0.672043i
$502$		0			0
$503$	40.5145			1.80645			0.903225	−	0.429167i	$-0.141193\pi$
$503$	40.5145			1.80645			0.903225	+	0.429167i	$0.141193\pi$
$504$		0			0
$505$			44.7216i			1.99009i
$506$		0			0
$507$		−	2.73693i		−	0.121551i
$508$		0			0
$509$			11.4921i			0.509379i	0.967023	+	0.254690i	$0.0819733\pi$
$509$			11.4921i			0.509379i	−0.967023	+	0.254690i	$0.918027\pi$
$510$		0			0
$511$	0.133781	+	0.288308i	0.00591812	+	0.0127540i
$512$		0			0
$513$		−	7.59433i		−	0.335298i
$514$		0			0
$515$	−30.3945			−1.33934
$516$		0			0
$517$	−26.8335	+	25.3865i	−1.18014	+	1.11650i
$518$		0			0
$519$			56.3630i			2.47406i
$520$		0			0
$521$		−	41.1078i		−	1.80097i	−0.434890	−	0.900483i	$-0.643213\pi$
$521$		−	41.1078i		−	1.80097i	0.434890	−	0.900483i	$-0.356787\pi$
$522$		0			0
$523$	38.6399			1.68961			0.844803	−	0.535077i	$-0.179717\pi$
$523$	38.6399			1.68961			0.844803	+	0.535077i	$0.179717\pi$
$524$		0			0
$525$	32.2327	−	14.9567i	1.40675	−	0.652762i
$526$		0			0
$527$			16.1078i			0.701666i
$528$		0			0
$529$	−20.4354			−0.888498
$530$		0			0
$531$			46.2439i			2.00681i
$532$		0			0
$533$	3.75880			0.162812
$534$		0			0
$535$	−60.7563			−2.62672
$536$		0			0
$537$		−	68.1854i		−	2.94242i
$538$		0			0
$539$	−23.1802	+	1.29488i	−0.998443	+	0.0557745i
$540$		0			0
$541$		−	35.8170i		−	1.53990i	−0.638107	−	0.769948i	$-0.720282\pi$
$541$		−	35.8170i		−	1.53990i	0.638107	−	0.769948i	$-0.279718\pi$
$542$		0			0
$543$	−18.1933			−0.780749
$544$		0			0
$545$	−49.8342			−2.13466
$546$		0			0
$547$			0.993160i			0.0424644i	0.999775	+	0.0212322i	$0.00675893\pi$
$547$			0.993160i			0.0424644i	−0.999775	+	0.0212322i	$0.993241\pi$
$548$		0			0
$549$	19.8629			0.847728
$550$		0			0
$551$		−	11.2951i		−	0.481187i
$552$		0			0
$553$	13.5547	−	6.28967i	0.576405	−	0.267464i
$554$		0			0
$555$	−31.3451			−1.33053
$556$		0			0
$557$		−	27.8814i		−	1.18137i	−0.806902	−	0.590686i	$-0.798857\pi$
$557$		−	27.8814i		−	1.18137i	0.806902	−	0.590686i	$-0.201143\pi$
$558$		0			0
$559$			11.3386i			0.479573i
$560$		0			0
$561$	−25.7195	+	24.3325i	−1.08588	+	1.02732i
$562$		0			0
$563$	−17.9338			−0.755820			−0.377910	−	0.925842i	$-0.623357\pi$
$563$	−17.9338			−0.755820			−0.377910	+	0.925842i	$0.623357\pi$
$564$		0			0
$565$		−	24.4441i		−	1.02837i
$566$		0			0
$567$	2.56728	+	5.53269i	0.107816	+	0.232351i
$568$		0			0
$569$			29.2554i			1.22645i	0.789908	+	0.613226i	$0.210128\pi$
$569$			29.2554i			1.22645i	−0.789908	+	0.613226i	$0.789872\pi$
$570$		0			0
$571$			5.83082i			0.244012i	0.992529	+	0.122006i	$0.0389328\pi$
$571$			5.83082i			0.244012i	−0.992529	+	0.122006i	$0.961067\pi$
$572$		0			0
$573$		−	2.05421i		−	0.0858159i
$574$		0			0
$575$	7.85842			0.327719
$576$		0			0
$577$		−	25.5527i		−	1.06377i	−0.846816	−	0.531886i	$-0.821483\pi$
$577$		−	25.5527i		−	1.06377i	0.846816	−	0.531886i	$-0.178517\pi$
$578$		0			0
$579$	46.8351			1.94640
$580$		0			0
$581$	−5.59644	−	12.0607i	−0.232179	−	0.500364i
$582$		0			0
$583$	−31.9239	−	33.7435i	−1.32215	−	1.39751i
$584$		0			0
$585$			14.1350i			0.584409i
$586$		0			0
$587$		−	1.67119i		−	0.0689775i	−0.999405	−	0.0344888i	$-0.989020\pi$
$587$		−	1.67119i		−	0.0689775i	0.999405	−	0.0344888i	$-0.0109803\pi$
$588$		0			0
$589$			7.68669i			0.316724i
$590$		0			0
$591$	24.0082			0.987567
$592$		0			0
$593$	−37.1191			−1.52430			−0.762149	−	0.647402i	$-0.775856\pi$
$593$	−37.1191			−1.52430			−0.762149	+	0.647402i	$0.775856\pi$
$594$		0			0
$595$	29.4641	−	13.6719i	1.20791	−	0.560495i
$596$		0			0
$597$	2.63926			0.108018
$598$		0			0
$599$	7.62652			0.311611			0.155806	−	0.987788i	$-0.450203\pi$
$599$	7.62652			0.311611			0.155806	+	0.987788i	$0.450203\pi$
$600$		0			0
$601$	−25.8465			−1.05430			−0.527151	−	0.849771i	$-0.676740\pi$
$601$	−25.8465			−1.05430			−0.527151	+	0.849771i	$0.676740\pi$
$602$		0			0
$603$	54.2308			2.20845
$604$		0			0
$605$	34.5701	+	1.91734i	1.40547	+	0.0779511i
$606$		0			0
$607$	8.09391			0.328522			0.164261	−	0.986417i	$-0.447476\pi$
$607$	8.09391			0.328522			0.164261	+	0.986417i	$0.447476\pi$
$608$		0			0
$609$	−18.4960	−	39.8602i	−0.749495	−	1.61522i
$610$		0			0
$611$			11.1376i			0.450580i
$612$		0			0
$613$		−	26.0344i		−	1.05152i	−0.850633	−	0.525760i	$-0.823781\pi$
$613$		−	26.0344i		−	1.05152i	0.850633	−	0.525760i	$-0.176219\pi$
$614$		0			0
$615$	−32.3808			−1.30572
$616$		0			0
$617$	27.1587			1.09337			0.546685	−	0.837338i	$-0.315890\pi$
$617$	27.1587			1.09337			0.546685	+	0.837338i	$0.315890\pi$
$618$		0			0
$619$			15.4093i			0.619350i	0.950842	+	0.309675i	$0.100220\pi$
$619$			15.4093i			0.619350i	−0.950842	+	0.309675i	$0.899780\pi$
$620$		0			0
$621$		−	6.53399i		−	0.262200i
$622$		0			0
$623$	35.3834	−	16.4186i	1.41760	−	0.657798i
$624$		0			0
$625$	−25.4556			−1.01822
$626$		0			0
$627$	−12.2734	+	11.6116i	−0.490153	+	0.463721i
$628$		0			0
$629$	−14.1921			−0.565876
$630$		0			0
$631$	13.1607			0.523920			0.261960	−	0.965079i	$-0.415631\pi$
$631$	13.1607			0.523920			0.261960	+	0.965079i	$0.415631\pi$
$632$		0			0
$633$	18.7748			0.746231
$634$		0			0
$635$	−26.0525			−1.03386
$636$		0			0
$637$	−4.51964	+	5.34535i	−0.179075	+	0.211791i
$638$		0			0
$639$	18.9554			0.749864
$640$		0			0
$641$	−15.4401			−0.609847			−0.304923	−	0.952377i	$-0.598631\pi$
$641$	−15.4401			−0.609847			−0.304923	+	0.952377i	$0.598631\pi$
$642$		0			0
$643$		−	3.04987i		−	0.120275i	−0.998190	−	0.0601377i	$-0.980846\pi$
$643$		−	3.04987i		−	0.120275i	0.998190	−	0.0601377i	$-0.0191540\pi$
$644$		0			0
$645$		−	97.6782i		−	3.84608i
$646$		0			0
$647$			39.9576i			1.57090i	0.618927	+	0.785449i	$0.287568\pi$
$647$			39.9576i			1.57090i	−0.618927	+	0.785449i	$0.712432\pi$
$648$		0			0
$649$	24.8096	−	23.4717i	0.973861	−	0.921345i
$650$		0			0
$651$	12.5871	+	27.1262i	0.493329	+	1.06316i
$652$		0			0
$653$	11.2157			0.438906			0.219453	−	0.975623i	$-0.429573\pi$
$653$	11.2157			0.438906			0.219453	+	0.975623i	$0.429573\pi$
$654$		0			0
$655$			7.78444i			0.304163i
$656$		0			0
$657$	0.539477			0.0210470
$658$		0			0
$659$			29.9908i			1.16827i	0.811655	+	0.584137i	$0.198567\pi$
$659$			29.9908i			1.16827i	−0.811655	+	0.584137i	$0.801433\pi$
$660$		0			0
$661$		−	8.07350i		−	0.314023i	−0.987597	−	0.157011i	$-0.949814\pi$
$661$		−	8.07350i		−	0.314023i	0.987597	−	0.157011i	$-0.0501860\pi$
$662$		0			0
$663$			10.6752i			0.414591i
$664$		0			0
$665$	14.0604	−	6.52430i	0.545237	−	0.253001i
$666$		0			0
$667$		−	9.71802i		−	0.376283i
$668$		0			0
$669$	23.9313			0.925238
$670$		0			0
$671$	−10.0817	−	10.6563i	−0.389199	−	0.411383i
$672$		0			0
$673$			14.2728i			0.550174i	0.961419	+	0.275087i	$0.0887067\pi$
$673$			14.2728i			0.550174i	−0.961419	+	0.275087i	$0.911293\pi$
$674$		0			0
$675$		−	20.0217i		−	0.770637i
$676$		0			0
$677$	23.1024			0.887898			0.443949	−	0.896052i	$-0.353577\pi$
$677$	23.1024			0.887898			0.443949	+	0.896052i	$0.353577\pi$
$678$		0			0
$679$	−40.2828	+	18.6920i	−1.54591	+	0.717335i
$680$		0			0
$681$		−	43.3074i		−	1.65954i
$682$		0			0
$683$	6.85995			0.262489			0.131244	−	0.991350i	$-0.458103\pi$
$683$	6.85995			0.262489			0.131244	+	0.991350i	$0.458103\pi$
$684$		0			0
$685$			13.8667i			0.529819i
$686$		0			0
$687$	−60.1681			−2.29556
$688$		0			0
$689$	−14.0057			−0.533575
$690$		0			0
$691$			5.14073i			0.195563i	0.995208	+	0.0977813i	$0.0311746\pi$
$691$			5.14073i			0.195563i	−0.995208	+	0.0977813i	$0.968825\pi$
$692$		0			0
$693$	−14.5671	+	36.6150i	−0.553360	+	1.39089i
$694$		0			0
$695$			8.05094i			0.305389i
$696$		0			0
$697$	−14.6610			−0.555324
$698$		0			0
$699$	−41.2828			−1.56146
$700$		0			0
$701$			25.6739i			0.969688i	0.874601	+	0.484844i	$0.161124\pi$
$701$			25.6739i			0.969688i	−0.874601	+	0.484844i	$0.838876\pi$
$702$		0			0
$703$	−6.77251			−0.255430
$704$		0			0
$705$		−	95.9466i		−	3.61356i
$706$		0			0
$707$	15.8229	+	34.0995i	0.595080	+	1.28244i
$708$		0			0
$709$	45.6876			1.71583			0.857917	−	0.513788i	$-0.171758\pi$
$709$	45.6876			1.71583			0.857917	+	0.513788i	$0.171758\pi$
$710$		0			0
$711$		−	25.3633i		−	0.951199i
$712$		0			0
$713$			6.61344i			0.247675i
$714$		0			0
$715$	7.58333	−	7.17439i	0.283600	−	0.268307i
$716$		0			0
$717$	−71.7974			−2.68132
$718$		0			0
$719$		−	6.45916i		−	0.240886i	−0.992720	−	0.120443i	$-0.961568\pi$
$719$		−	6.45916i		−	0.240886i	0.992720	−	0.120443i	$-0.0384315\pi$
$720$		0			0
$721$	−23.1753	+	10.7538i	−0.863092	+	0.400493i
$722$		0			0
$723$			10.1008i			0.375654i
$724$		0			0
$725$		−	29.7784i		−	1.10594i
$726$		0			0
$727$		−	19.8947i		−	0.737852i	−0.929459	−	0.368926i	$-0.879726\pi$
$727$		−	19.8947i		−	0.737852i	0.929459	−	0.368926i	$-0.120274\pi$
$728$		0			0
$729$	43.8536			1.62421
$730$		0			0
$731$		−	44.2256i		−	1.63574i
$732$		0			0
$733$	−27.1610			−1.00322			−0.501608	−	0.865095i	$-0.667258\pi$
$733$	−27.1610			−1.00322			−0.501608	+	0.865095i	$0.667258\pi$
$734$		0			0
$735$	38.9351	−	46.0483i	1.43614	−	1.69852i
$736$		0			0
$737$	−27.5256	−	29.0945i	−1.01392	−	1.07171i
$738$		0			0
$739$		−	2.03226i		−	0.0747578i	−0.999301	−	0.0373789i	$-0.988099\pi$
$739$		−	2.03226i		−	0.0747578i	0.999301	−	0.0373789i	$-0.0119008\pi$
$740$		0			0
$741$			5.09425i			0.187142i
$742$		0			0
$743$			5.41192i			0.198544i	0.995060	+	0.0992721i	$0.0316514\pi$
$743$			5.41192i			0.198544i	−0.995060	+	0.0992721i	$0.968349\pi$
$744$		0			0
$745$	0.0610228			0.00223570
$746$		0			0
$747$	−22.5678			−0.825714
$748$		0			0
$749$	−46.3256	+	21.4960i	−1.69270	+	0.785449i
$750$		0			0
$751$	−37.6496			−1.37385			−0.686926	−	0.726727i	$-0.741041\pi$
$751$	−37.6496			−1.37385			−0.686926	+	0.726727i	$0.741041\pi$
$752$		0			0
$753$	51.5890			1.88001
$754$		0			0
$755$	20.3900			0.742069
$756$		0			0
$757$	−5.74126			−0.208670			−0.104335	−	0.994542i	$-0.533271\pi$
$757$	−5.74126			−0.208670			−0.104335	+	0.994542i	$0.533271\pi$
$758$		0			0
$759$	−10.5598	+	9.99032i	−0.383295	+	0.362626i
$760$		0			0
$761$	25.5350			0.925643			0.462821	−	0.886452i	$-0.346837\pi$
$761$	25.5350			0.925643			0.462821	+	0.886452i	$0.346837\pi$
$762$		0			0
$763$	−37.9977	+	17.6317i	−1.37561	+	0.638312i
$764$		0			0
$765$		−	55.1326i		−	1.99332i
$766$		0			0
$767$		−	10.2976i		−	0.371823i
$768$		0			0
$769$	−28.5698			−1.03025			−0.515126	−	0.857114i	$-0.672255\pi$
$769$	−28.5698			−1.03025			−0.515126	+	0.857114i	$0.672255\pi$
$770$		0			0
$771$	−33.0528			−1.19037
$772$		0			0
$773$			30.5108i			1.09740i	0.836020	+	0.548699i	$0.184877\pi$
$773$			30.5108i			1.09740i	−0.836020	+	0.548699i	$0.815123\pi$
$774$		0			0
$775$			20.2652i			0.727948i
$776$		0			0
$777$	−23.9001	+	11.0902i	−0.857412	+	0.397857i
$778$		0			0
$779$	−6.99627			−0.250668
$780$		0			0
$781$	−9.62106	−	10.1695i	−0.344269	−	0.363892i
$782$		0			0
$783$	−24.7596			−0.884838
$784$		0			0
$785$	−5.61435			−0.200385
$786$		0			0
$787$	−19.0442			−0.678853			−0.339427	−	0.940633i	$-0.610233\pi$
$787$	−19.0442			−0.678853			−0.339427	+	0.940633i	$0.610233\pi$
$788$		0			0
$789$	−75.7430			−2.69652
$790$		0			0
$791$	−8.64851	−	18.6382i	−0.307506	−	0.662698i
$792$		0			0
$793$	−4.42305			−0.157067
$794$		0			0
$795$	120.654			4.27917
$796$		0			0
$797$		−	19.8172i		−	0.701961i	−0.936383	−	0.350980i	$-0.885848\pi$
$797$		−	19.8172i		−	0.701961i	0.936383	−	0.350980i	$-0.114152\pi$
$798$		0			0
$799$		−	43.4416i		−	1.53685i
$800$		0			0
$801$		−	66.2087i		−	2.33937i
$802$		0			0
$803$	−0.273819	−	0.289426i	−0.00966285	−	0.0102136i
$804$		0			0
$805$	12.0972	−	5.61335i	0.426370	−	0.197845i
$806$		0			0
$807$	−7.31247			−0.257411
$808$		0			0
$809$		−	22.6524i		−	0.796415i	−0.917295	−	0.398208i	$-0.869632\pi$
$809$		−	22.6524i		−	0.796415i	0.917295	−	0.398208i	$-0.130368\pi$
$810$		0			0
$811$	−13.8363			−0.485858			−0.242929	−	0.970044i	$-0.578108\pi$
$811$	−13.8363			−0.485858			−0.242929	+	0.970044i	$0.578108\pi$
$812$		0			0
$813$			16.0715i			0.563654i
$814$		0			0
$815$		−	23.0665i		−	0.807983i
$816$		0			0
$817$		−	21.1046i		−	0.738357i
$818$		0			0
$819$	5.00106	+	10.7777i	0.174751	+	0.376602i
$820$		0			0
$821$			32.4497i			1.13250i	0.824233	+	0.566250i	$0.191606\pi$
$821$			32.4497i			1.13250i	−0.824233	+	0.566250i	$0.808394\pi$
$822$		0			0
$823$	43.3744			1.51194			0.755969	−	0.654607i	$-0.227166\pi$
$823$	43.3744			1.51194			0.755969	+	0.654607i	$0.227166\pi$
$824$		0			0
$825$	−32.3577	+	30.6128i	−1.12655	+	1.06580i
$826$		0			0
$827$		−	19.5875i		−	0.681124i	−0.940222	−	0.340562i	$-0.889383\pi$
$827$		−	19.5875i		−	0.681124i	0.940222	−	0.340562i	$-0.110617\pi$
$828$		0			0
$829$			2.44840i			0.0850365i	0.999096	+	0.0425183i	$0.0135381\pi$
$829$			2.44840i			0.0850365i	−0.999096	+	0.0425183i	$0.986462\pi$
$830$		0			0
$831$	−72.4340			−2.51271
$832$		0			0
$833$	17.6286	−	20.8492i	0.610795	−	0.722383i
$834$		0			0
$835$			17.2993i			0.598665i
$836$		0			0
$837$	16.8498			0.582413
$838$		0			0
$839$		−	1.25631i		−	0.0433727i	−0.999765	−	0.0216863i	$-0.993096\pi$
$839$		−	1.25631i		−	0.0433727i	0.999765	−	0.0216863i	$-0.00690352\pi$
$840$		0			0
$841$	−7.82512			−0.269832
$842$		0			0
$843$	−21.4706			−0.739486
$844$		0			0
$845$		−	3.14756i		−	0.108279i
$846$		0			0
$847$	27.0375	−	10.7692i	0.929018	−	0.370035i
$848$		0			0
$849$		−	51.2199i		−	1.75786i
$850$		0			0
$851$	−5.82691			−0.199744
$852$		0			0
$853$	2.78063			0.0952071			0.0476036	−	0.998866i	$-0.484842\pi$
$853$	2.78063			0.0952071			0.0476036	+	0.998866i	$0.484842\pi$
$854$		0			0
$855$		−	26.3095i		−	0.899765i
$856$		0			0
$857$	12.0829			0.412744			0.206372	−	0.978474i	$-0.433834\pi$
$857$	12.0829			0.412744			0.206372	+	0.978474i	$0.433834\pi$
$858$		0			0
$859$		−	43.9975i		−	1.50118i	−0.660771	−	0.750588i	$-0.729770\pi$
$859$		−	43.9975i		−	1.50118i	0.660771	−	0.750588i	$-0.270230\pi$
$860$		0			0
$861$	−24.6898	+	11.4566i	−0.841425	+	0.390439i
$862$		0			0
$863$	44.5919			1.51793			0.758964	−	0.651133i	$-0.225706\pi$
$863$	44.5919			1.51793			0.758964	+	0.651133i	$0.225706\pi$
$864$		0			0
$865$			64.8194i			2.20393i
$866$		0			0
$867$			4.88968i			0.166062i
$868$		0			0
$869$	−13.6073	+	12.8735i	−0.461595	+	0.436703i
$870$		0			0
$871$	−12.0761			−0.409182
$872$		0			0
$873$			75.3763i			2.55110i
$874$		0			0
$875$	−0.701364	+	0.325447i	−0.0237104	+	0.0110021i
$876$		0			0
$877$			9.21406i			0.311137i	0.987825	+	0.155568i	$0.0497209\pi$
$877$			9.21406i			0.311137i	−0.987825	+	0.155568i	$0.950279\pi$
$878$		0			0
$879$		−	1.45069i		−	0.0489307i
$880$		0			0
$881$		−	24.1413i		−	0.813341i	−0.913575	−	0.406671i	$-0.866690\pi$
$881$		−	24.1413i		−	0.813341i	0.913575	−	0.406671i	$-0.133310\pi$
$882$		0			0
$883$	−33.4535			−1.12580			−0.562900	−	0.826525i	$-0.690314\pi$
$883$	−33.4535			−1.12580			−0.562900	+	0.826525i	$0.690314\pi$
$884$		0			0
$885$			88.7098i			2.98195i
$886$		0			0
$887$	−23.3757			−0.784880			−0.392440	−	0.919778i	$-0.628369\pi$
$887$	−23.3757			−0.784880			−0.392440	+	0.919778i	$0.628369\pi$
$888$		0			0
$889$	−19.8646	+	9.21758i	−0.666237	+	0.309148i
$890$		0			0
$891$	−5.25463	−	5.55414i	−0.176037	−	0.186071i
$892$		0			0
$893$		−	20.7305i		−	0.693719i
$894$		0			0
$895$		−	78.4157i		−	2.62115i
$896$		0			0
$897$			4.38297i			0.146343i
$898$		0			0
$899$	25.0607			0.835822
$900$		0			0
$901$	54.6285			1.81994
$902$		0			0
$903$	−34.5593	−	74.4780i	−1.15006	−	2.47847i
$904$		0			0
$905$	−20.9229			−0.695502
$906$		0			0
$907$	7.72648			0.256554			0.128277	−	0.991738i	$-0.459055\pi$
$907$	7.72648			0.256554			0.128277	+	0.991738i	$0.459055\pi$
$908$		0			0
$909$	63.8063			2.11632
$910$		0			0
$911$	27.3887			0.907428			0.453714	−	0.891147i	$-0.350099\pi$
$911$	27.3887			0.907428			0.453714	+	0.891147i	$0.350099\pi$
$912$		0			0
$913$	11.4546	+	12.1075i	0.379092	+	0.400700i
$914$		0			0
$915$	38.1030			1.25965
$916$		0			0
$917$	2.75419	+	5.93550i	0.0909515	+	0.196007i
$918$		0			0
$919$		−	48.4424i		−	1.59797i	−0.601352	−	0.798984i	$-0.705371\pi$
$919$		−	48.4424i		−	1.59797i	0.601352	−	0.798984i	$-0.294629\pi$
$920$		0			0
$921$			14.8565i			0.489540i
$922$		0			0
$923$	−4.22097			−0.138935
$924$		0			0
$925$	−17.8551			−0.587072
$926$		0			0
$927$			43.3651i			1.42430i
$928$		0			0
$929$		−	10.1484i		−	0.332960i	−0.986045	−	0.166480i	$-0.946760\pi$
$929$		−	10.1484i		−	0.332960i	0.986045	−	0.166480i	$-0.0532401\pi$
$930$		0			0
$931$	8.41243	−	9.94933i	0.275706	−	0.326076i
$932$		0			0
$933$	8.80657			0.288314
$934$		0			0
$935$	−29.5783	+	27.9833i	−0.967314	+	0.915151i
$936$		0			0
$937$	−25.7280			−0.840496			−0.420248	−	0.907409i	$-0.638057\pi$
$937$	−25.7280			−0.840496			−0.420248	+	0.907409i	$0.638057\pi$
$938$		0			0
$939$	−54.7660			−1.78722
$940$		0			0
$941$	−30.4661			−0.993167			−0.496583	−	0.867989i	$-0.665412\pi$
$941$	−30.4661			−0.993167			−0.496583	+	0.867989i	$0.665412\pi$
$942$		0			0
$943$	−6.01943			−0.196020
$944$		0			0
$945$	−14.3017	−	30.8213i	−0.465236	−	1.00262i
$946$		0			0
$947$	10.6722			0.346801			0.173401	−	0.984851i	$-0.444524\pi$
$947$	10.6722			0.346801			0.173401	+	0.984851i	$0.444524\pi$
$948$		0			0
$949$	−0.120130			−0.00389959
$950$		0			0
$951$			14.6676i			0.475630i
$952$		0			0
$953$			33.0552i			1.07076i	0.844610	+	0.535382i	$0.179832\pi$
$953$			33.0552i			1.07076i	−0.844610	+	0.535382i	$0.820168\pi$
$954$		0			0
$955$		−	2.36242i		−	0.0764460i
$956$		0			0
$957$	37.8570	+	40.0148i	1.22374	+	1.29349i
$958$		0			0
$959$	4.90614	+	10.5731i	0.158428	+	0.341423i
$960$		0			0
$961$	13.9453			0.449849
$962$		0			0
$963$			86.6836i			2.79334i
$964$		0			0
$965$	53.8620			1.73388
$966$		0			0
$967$		−	50.9875i		−	1.63965i	−0.572616	−	0.819824i	$-0.694071\pi$
$967$		−	50.9875i		−	1.63965i	0.572616	−	0.819824i	$-0.305929\pi$
$968$		0			0
$969$		−	19.8698i		−	0.638311i
$970$		0			0
$971$		−	12.1147i		−	0.388778i	−0.980925	−	0.194389i	$-0.937728\pi$
$971$		−	12.1147i		−	0.388778i	0.980925	−	0.194389i	$-0.0622724\pi$
$972$		0			0
$973$	2.84848	+	6.13870i	0.0913182	+	0.196798i
$974$		0			0
$975$			13.4305i			0.430121i
$976$		0			0
$977$	8.75497			0.280096			0.140048	−	0.990145i	$-0.455274\pi$
$977$	8.75497			0.280096			0.140048	+	0.990145i	$0.455274\pi$
$978$		0			0
$979$	−35.5206	+	33.6051i	−1.13524	+	1.07402i
$980$		0			0
$981$			71.1006i			2.27007i
$982$		0			0
$983$		−	11.2475i		−	0.358741i	−0.983782	−	0.179370i	$-0.942594\pi$
$983$		−	11.2475i		−	0.358741i	0.983782	−	0.179370i	$-0.0574060\pi$
$984$		0			0
$985$	27.6103			0.879738
$986$		0			0
$987$	−33.9467	−	73.1577i	−1.08053	−	2.32863i
$988$		0			0
$989$		−	18.1579i		−	0.577388i
$990$		0			0
$991$	−1.41802			−0.0450448			−0.0225224	−	0.999746i	$-0.507170\pi$
$991$	−1.41802			−0.0450448			−0.0225224	+	0.999746i	$0.507170\pi$
$992$		0			0
$993$			58.4289i			1.85418i
$994$		0			0
$995$	3.03525			0.0962237
$996$		0			0
$997$	31.1994			0.988094			0.494047	−	0.869435i	$-0.335517\pi$
$997$	31.1994			0.988094			0.494047	+	0.869435i	$0.335517\pi$
$998$		0			0
$999$			14.8458i			0.469702i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.e.a.3849.6	✓	48
7.6	odd	2		4004.2.e.b.3849.43	yes	48
11.10	odd	2		4004.2.e.b.3849.6	yes	48
77.76	even	2	inner	4004.2.e.a.3849.43	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.e.a.3849.6	✓	48	1.1	even	1	trivial
4004.2.e.a.3849.43	yes	48	77.76	even	2	inner
4004.2.e.b.3849.6	yes	48	11.10	odd	2
4004.2.e.b.3849.43	yes	48	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.73693i q^{3} -3.14756i q^{5} +(-1.11363 - 2.39996i) q^{7} -4.49077 q^{9} +O(q^{10})\)	\(q-2.73693i q^{3} -3.14756i q^{5} +(-1.11363 - 2.39996i) q^{7} -4.49077 q^{9} +(2.27935 + 2.40927i) q^{11} +1.00000 q^{13} -8.61465 q^{15} -3.90044 q^{17} -1.86130 q^{19} +(-6.56852 + 3.04793i) q^{21} -1.60142 q^{23} -4.90715 q^{25} +4.08011i q^{27} +6.06837i q^{29} -4.12973i q^{31} +(6.59399 - 6.23841i) q^{33} +(-7.55403 + 3.50523i) q^{35} +3.63858 q^{37} -2.73693i q^{39} +3.75880 q^{41} +11.3386i q^{43} +14.1350i q^{45} +11.1376i q^{47} +(-4.51964 + 5.34535i) q^{49} +10.6752i q^{51} -14.0057 q^{53} +(7.58333 - 7.17439i) q^{55} +5.09425i q^{57} -10.2976i q^{59} -4.42305 q^{61} +(5.00106 + 10.7777i) q^{63} -3.14756i q^{65} -12.0761 q^{67} +4.38297i q^{69} -4.22097 q^{71} -0.120130 q^{73} +13.4305i q^{75} +(3.24380 - 8.15339i) q^{77} +5.64789i q^{79} -2.30532 q^{81} +5.02539 q^{83} +12.2769i q^{85} +16.6087 q^{87} +14.7433i q^{89} +(-1.11363 - 2.39996i) q^{91} -11.3028 q^{93} +5.85857i q^{95} -16.7847i q^{97} +(-10.2360 - 10.8195i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 4 q^{7} - 48 q^{9}+O(q^{10}) \) 48 * q - 4 * q^7 - 48 * q^9	\( 48 q - 4 q^{7} - 48 q^{9} + 2 q^{11} + 48 q^{13} + 8 q^{15} + 4 q^{17} + 10 q^{21} + 4 q^{23} - 44 q^{25} + 10 q^{33} - 14 q^{35} - 16 q^{37} + 10 q^{49} - 8 q^{53} - 12 q^{55} + 16 q^{61} + 16 q^{63} + 4 q^{67} - 16 q^{73} + 2 q^{77} + 64 q^{81} - 4 q^{83} - 12 q^{87} - 4 q^{91} + 36 q^{93} - 40 q^{99}+O(q^{100}) \) 48 * q - 4 * q^7 - 48 * q^9 + 2 * q^11 + 48 * q^13 + 8 * q^15 + 4 * q^17 + 10 * q^21 + 4 * q^23 - 44 * q^25 + 10 * q^33 - 14 * q^35 - 16 * q^37 + 10 * q^49 - 8 * q^53 - 12 * q^55 + 16 * q^61 + 16 * q^63 + 4 * q^67 - 16 * q^73 + 2 * q^77 + 64 * q^81 - 4 * q^83 - 12 * q^87 - 4 * q^91 + 36 * q^93 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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