LMFDB - Embedded newform 4004.2.e.b.3849.17

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			3849.17
Character	$\chi$	$=$	4004.3849
Dual form			4004.2.e.b.3849.32

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00653i q^{3} +1.31275i q^{5} +(1.40205 + 2.24371i) q^{7} +1.98690 q^{9} +O(q^{10})$	$q-1.00653i q^{3} +1.31275i q^{5} +(1.40205 + 2.24371i) q^{7} +1.98690 q^{9} +(-2.38054 - 2.30934i) q^{11} -1.00000 q^{13} +1.32132 q^{15} +1.70089 q^{17} -7.19891 q^{19} +(2.25836 - 1.41121i) q^{21} -3.72958 q^{23} +3.27668 q^{25} -5.01946i q^{27} -7.88526i q^{29} -5.82116i q^{31} +(-2.32441 + 2.39608i) q^{33} +(-2.94544 + 1.84055i) q^{35} -8.84246 q^{37} +1.00653i q^{39} -7.03447 q^{41} -7.07868i q^{43} +2.60831i q^{45} -3.62470i q^{47} +(-3.06849 + 6.29161i) q^{49} -1.71200i q^{51} +6.34029 q^{53} +(3.03159 - 3.12506i) q^{55} +7.24591i q^{57} +1.65669i q^{59} +3.42210 q^{61} +(2.78574 + 4.45803i) q^{63} -1.31275i q^{65} +13.5872 q^{67} +3.75393i q^{69} +7.55149 q^{71} +11.0352 q^{73} -3.29807i q^{75} +(1.84384 - 8.57906i) q^{77} -12.8141i q^{79} +0.908469 q^{81} -15.2426 q^{83} +2.23285i q^{85} -7.93675 q^{87} -10.7277i q^{89} +(-1.40205 - 2.24371i) q^{91} -5.85916 q^{93} -9.45039i q^{95} +1.83241i q^{97} +(-4.72989 - 4.58842i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 22 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 + 22 * 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
$3$		−	1.00653i		−	0.581120i	−0.956857	−	0.290560i	$-0.906158\pi$
$3$		−	1.00653i		−	0.581120i	0.956857	−	0.290560i	$-0.0938415\pi$
$4$		0			0
$5$			1.31275i			0.587081i	0.955947	+	0.293541i	$0.0948336\pi$
$5$			1.31275i			0.587081i	−0.955947	+	0.293541i	$0.905166\pi$
$6$		0			0
$7$	1.40205	+	2.24371i	0.529927	+	0.848043i
$7$	1.40205	+	2.24371i	0.529927	+	0.848043i
$8$		0			0
$9$	1.98690			0.662300
$10$		0			0
$11$	−2.38054	−	2.30934i	−0.717760	−	0.696291i
$11$	−2.38054	−	2.30934i	−0.717760	−	0.696291i
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$	1.32132			0.341165
$16$		0			0
$17$	1.70089			0.412527			0.206264	−	0.978496i	$-0.433870\pi$
$17$	1.70089			0.412527			0.206264	+	0.978496i	$0.433870\pi$
$18$		0			0
$19$	−7.19891			−1.65154			−0.825771	−	0.564005i	$-0.809260\pi$
$19$	−7.19891			−1.65154			−0.825771	+	0.564005i	$0.809260\pi$
$20$		0			0
$21$	2.25836	−	1.41121i	0.492815	−	0.307951i
$22$		0			0
$23$	−3.72958			−0.777671			−0.388836	−	0.921307i	$-0.627123\pi$
$23$	−3.72958			−0.777671			−0.388836	+	0.921307i	$0.627123\pi$
$24$		0			0
$25$	3.27668			0.655335
$26$		0			0
$27$		−	5.01946i		−	0.965995i
$28$		0			0
$29$		−	7.88526i		−	1.46426i	−0.681167	−	0.732128i	$-0.738527\pi$
$29$		−	7.88526i		−	1.46426i	0.681167	−	0.732128i	$-0.261473\pi$
$30$		0			0
$31$		−	5.82116i		−	1.04551i	−0.852483	−	0.522755i	$-0.824904\pi$
$31$		−	5.82116i		−	1.04551i	0.852483	−	0.522755i	$-0.175096\pi$
$32$		0			0
$33$	−2.32441	+	2.39608i	−0.404628	+	0.417104i
$34$		0			0
$35$	−2.94544	+	1.84055i	−0.497871	+	0.311110i
$36$		0			0
$37$	−8.84246			−1.45369			−0.726845	−	0.686801i	$-0.759014\pi$
$37$	−8.84246			−1.45369			−0.726845	+	0.686801i	$0.759014\pi$
$38$		0			0
$39$			1.00653i			0.161174i
$40$		0			0
$41$	−7.03447			−1.09860			−0.549300	−	0.835625i	$-0.685105\pi$
$41$	−7.03447			−1.09860			−0.549300	+	0.835625i	$0.685105\pi$
$42$		0			0
$43$		−	7.07868i		−	1.07949i	−0.841829	−	0.539744i	$-0.818521\pi$
$43$		−	7.07868i		−	1.07949i	0.841829	−	0.539744i	$-0.181479\pi$
$44$		0			0
$45$			2.60831i			0.388824i
$46$		0			0
$47$		−	3.62470i		−	0.528717i	−0.964424	−	0.264358i	$-0.914840\pi$
$47$		−	3.62470i		−	0.528717i	0.964424	−	0.264358i	$-0.0851602\pi$
$48$		0			0
$49$	−3.06849	+	6.29161i	−0.438355	+	0.898802i
$50$		0			0
$51$		−	1.71200i		−	0.239728i
$52$		0			0
$53$	6.34029			0.870906			0.435453	−	0.900211i	$-0.356588\pi$
$53$	6.34029			0.870906			0.435453	+	0.900211i	$0.356588\pi$
$54$		0			0
$55$	3.03159	−	3.12506i	0.408779	−	0.421383i
$56$		0			0
$57$			7.24591i			0.959744i
$58$		0			0
$59$			1.65669i			0.215682i	0.994168	+	0.107841i	$0.0343938\pi$
$59$			1.65669i			0.215682i	−0.994168	+	0.107841i	$0.965606\pi$
$60$		0			0
$61$	3.42210			0.438154			0.219077	−	0.975708i	$-0.429695\pi$
$61$	3.42210			0.438154			0.219077	+	0.975708i	$0.429695\pi$
$62$		0			0
$63$	2.78574	+	4.45803i	0.350970	+	0.561659i
$64$		0			0
$65$		−	1.31275i		−	0.162827i
$66$		0			0
$67$	13.5872			1.65995			0.829973	−	0.557803i	$-0.188356\pi$
$67$	13.5872			1.65995			0.829973	+	0.557803i	$0.188356\pi$
$68$		0			0
$69$			3.75393i			0.451920i
$70$		0			0
$71$	7.55149			0.896197			0.448099	−	0.893984i	$-0.352101\pi$
$71$	7.55149			0.896197			0.448099	+	0.893984i	$0.352101\pi$
$72$		0			0
$73$	11.0352			1.29157			0.645785	−	0.763520i	$-0.276530\pi$
$73$	11.0352			1.29157			0.645785	+	0.763520i	$0.276530\pi$
$74$		0			0
$75$		−	3.29807i		−	0.380828i
$76$		0			0
$77$	1.84384	−	8.57906i	0.210125	−	0.977675i
$78$		0			0
$79$		−	12.8141i		−	1.44170i	−0.693091	−	0.720850i	$-0.743752\pi$
$79$		−	12.8141i		−	1.44170i	0.693091	−	0.720850i	$-0.256248\pi$
$80$		0			0
$81$	0.908469			0.100941
$82$		0			0
$83$	−15.2426			−1.67310			−0.836548	−	0.547894i	$-0.815430\pi$
$83$	−15.2426			−1.67310			−0.836548	+	0.547894i	$0.815430\pi$
$84$		0			0
$85$			2.23285i			0.242187i
$86$		0			0
$87$	−7.93675			−0.850909
$88$		0			0
$89$		−	10.7277i		−	1.13713i	−0.822639	−	0.568564i	$-0.807499\pi$
$89$		−	10.7277i		−	1.13713i	0.822639	−	0.568564i	$-0.192501\pi$
$90$		0			0
$91$	−1.40205	−	2.24371i	−0.146975	−	0.235205i
$92$		0			0
$93$	−5.85916			−0.607567
$94$		0			0
$95$		−	9.45039i		−	0.969590i
$96$		0			0
$97$			1.83241i			0.186053i	0.995664	+	0.0930267i	$0.0296542\pi$
$97$			1.83241i			0.186053i	−0.995664	+	0.0930267i	$0.970346\pi$
$98$		0			0
$99$	−4.72989	−	4.58842i	−0.475372	−	0.461153i
$100$		0			0
$101$	−3.20666			−0.319075			−0.159537	−	0.987192i	$-0.551000\pi$
$101$	−3.20666			−0.319075			−0.159537	+	0.987192i	$0.551000\pi$
$102$		0			0
$103$		−	13.8025i		−	1.36000i	−0.733211	−	0.680001i	$-0.761979\pi$
$103$		−	13.8025i		−	1.36000i	0.733211	−	0.680001i	$-0.238021\pi$
$104$		0			0
$105$	1.85257	+	2.96467i	0.180792	+	0.289322i
$106$		0			0
$107$			0.571531i			0.0552520i	0.999618	+	0.0276260i	$0.00879475\pi$
$107$			0.571531i			0.0552520i	−0.999618	+	0.0276260i	$0.991205\pi$
$108$		0			0
$109$		−	8.30624i		−	0.795593i	−0.917474	−	0.397797i	$-0.869775\pi$
$109$		−	8.30624i		−	0.795593i	0.917474	−	0.397797i	$-0.130225\pi$
$110$		0			0
$111$			8.90019i			0.844768i
$112$		0			0
$113$	19.5589			1.83994			0.919972	−	0.391983i	$-0.128211\pi$
$113$	19.5589			1.83994			0.919972	+	0.391983i	$0.128211\pi$
$114$		0			0
$115$		−	4.89602i		−	0.456557i
$116$		0			0
$117$	−1.98690			−0.183689
$118$		0			0
$119$	2.38474	+	3.81631i	0.218609	+	0.349841i
$120$		0			0
$121$	0.333938	+	10.9949i	0.0303580	+	0.999539i
$122$		0			0
$123$			7.08039i			0.638418i
$124$		0			0
$125$			10.8652i			0.971817i
$126$		0			0
$127$			10.5090i			0.932523i	0.884647	+	0.466261i	$0.154399\pi$
$127$			10.5090i			0.932523i	−0.884647	+	0.466261i	$0.845601\pi$
$128$		0			0
$129$	−7.12490			−0.627312
$130$		0			0
$131$	−2.88594			−0.252146			−0.126073	−	0.992021i	$-0.540237\pi$
$131$	−2.88594			−0.252146			−0.126073	+	0.992021i	$0.540237\pi$
$132$		0			0
$133$	−10.0933	−	16.1523i	−0.875197	−	1.40058i
$134$		0			0
$135$	6.58931			0.567118
$136$		0			0
$137$	10.6971			0.913918			0.456959	−	0.889488i	$-0.348939\pi$
$137$	10.6971			0.913918			0.456959	+	0.889488i	$0.348939\pi$
$138$		0			0
$139$	2.61088			0.221452			0.110726	−	0.993851i	$-0.464682\pi$
$139$	2.61088			0.221452			0.110726	+	0.993851i	$0.464682\pi$
$140$		0			0
$141$	−3.64837			−0.307248
$142$		0			0
$143$	2.38054	+	2.30934i	0.199071	+	0.193116i
$144$		0			0
$145$	10.3514			0.859638
$146$		0			0
$147$	6.33269	+	3.08852i	0.522311	+	0.254737i
$148$		0			0
$149$		−	5.94047i		−	0.486662i	−0.969943	−	0.243331i	$-0.921760\pi$
$149$		−	5.94047i		−	0.486662i	0.969943	−	0.243331i	$-0.0782401\pi$
$150$		0			0
$151$			14.1424i			1.15089i	0.817839	+	0.575447i	$0.195172\pi$
$151$			14.1424i			1.15089i	−0.817839	+	0.575447i	$0.804828\pi$
$152$		0			0
$153$	3.37950			0.273217
$154$		0			0
$155$	7.64175			0.613800
$156$		0			0
$157$		−	8.04414i		−	0.641992i	−0.947081	−	0.320996i	$-0.895982\pi$
$157$		−	8.04414i		−	0.641992i	0.947081	−	0.320996i	$-0.104018\pi$
$158$		0			0
$159$		−	6.38169i		−	0.506101i
$160$		0			0
$161$	−5.22908	−	8.36811i	−0.412109	−	0.659499i
$162$		0			0
$163$	13.5458			1.06099			0.530495	−	0.847688i	$-0.322006\pi$
$163$	13.5458			1.06099			0.530495	+	0.847688i	$0.322006\pi$
$164$		0			0
$165$	−3.14547	−	3.05138i	−0.244874	−	0.237550i
$166$		0			0
$167$	−16.3839			−1.26782			−0.633912	−	0.773405i	$-0.718552\pi$
$167$	−16.3839			−1.26782			−0.633912	+	0.773405i	$0.718552\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−14.3035			−1.09382
$172$		0			0
$173$	−3.82489			−0.290801			−0.145400	−	0.989373i	$-0.546447\pi$
$173$	−3.82489			−0.290801			−0.145400	+	0.989373i	$0.546447\pi$
$174$		0			0
$175$	4.59408	+	7.35192i	0.347280	+	0.555753i
$176$		0			0
$177$	1.66750			0.125337
$178$		0			0
$179$	−8.24268			−0.616087			−0.308043	−	0.951372i	$-0.599674\pi$
$179$	−8.24268			−0.616087			−0.308043	+	0.951372i	$0.599674\pi$
$180$		0			0
$181$			2.14905i			0.159737i	0.996805	+	0.0798686i	$0.0254501\pi$
$181$			2.14905i			0.159737i	−0.996805	+	0.0798686i	$0.974550\pi$
$182$		0			0
$183$		−	3.44444i		−	0.254620i
$184$		0			0
$185$		−	11.6080i		−	0.853435i
$186$		0			0
$187$	−4.04904	−	3.92793i	−0.296095	−	0.287239i
$188$		0			0
$189$	11.2622	−	7.03756i	0.819206	−	0.511907i
$190$		0			0
$191$	−6.10931			−0.442054			−0.221027	−	0.975268i	$-0.570941\pi$
$191$	−6.10931			−0.442054			−0.221027	+	0.975268i	$0.570941\pi$
$192$		0			0
$193$		−	18.9207i		−	1.36194i	−0.732310	−	0.680971i	$-0.761558\pi$
$193$		−	18.9207i		−	1.36194i	0.732310	−	0.680971i	$-0.238442\pi$
$194$		0			0
$195$	−1.32132			−0.0946220
$196$		0			0
$197$		−	13.1734i		−	0.938566i	−0.883048	−	0.469283i	$-0.844512\pi$
$197$		−	13.1734i		−	0.938566i	0.883048	−	0.469283i	$-0.155488\pi$
$198$		0			0
$199$		−	6.35522i		−	0.450509i	−0.974300	−	0.225255i	$-0.927679\pi$
$199$		−	6.35522i		−	0.450509i	0.974300	−	0.225255i	$-0.0723214\pi$
$200$		0			0
$201$		−	13.6760i		−	0.964628i
$202$		0			0
$203$	17.6923	−	11.0556i	1.24175	−	0.775949i
$204$		0			0
$205$		−	9.23453i		−	0.644967i
$206$		0			0
$207$	−7.41030			−0.515052
$208$		0			0
$209$	17.1373	+	16.6247i	1.18541	+	1.14995i
$210$		0			0
$211$			12.8201i			0.882570i	0.897367	+	0.441285i	$0.145477\pi$
$211$			12.8201i			0.882570i	−0.897367	+	0.441285i	$0.854523\pi$
$212$		0			0
$213$		−	7.60080i		−	0.520798i
$214$		0			0
$215$	9.29257			0.633748
$216$		0			0
$217$	13.0610	−	8.16158i	0.886638	−	0.554044i
$218$		0			0
$219$		−	11.1072i		−	0.750556i
$220$		0			0
$221$	−1.70089			−0.114414
$222$		0			0
$223$			13.6672i			0.915224i	0.889152	+	0.457612i	$0.151295\pi$
$223$			13.6672i			0.915224i	−0.889152	+	0.457612i	$0.848705\pi$
$224$		0			0
$225$	6.51043			0.434028
$226$		0			0
$227$	−23.7831			−1.57854			−0.789271	−	0.614045i	$-0.789541\pi$
$227$	−23.7831			−1.57854			−0.789271	+	0.614045i	$0.789541\pi$
$228$		0			0
$229$			21.2904i			1.40691i	0.710742	+	0.703453i	$0.248359\pi$
$229$			21.2904i			1.40691i	−0.710742	+	0.703453i	$0.751641\pi$
$230$		0			0
$231$	−8.63507	−	1.85588i	−0.568146	−	0.122108i
$232$		0			0
$233$			5.13110i			0.336149i	0.985774	+	0.168075i	$0.0537550\pi$
$233$			5.13110i			0.336149i	−0.985774	+	0.168075i	$0.946245\pi$
$234$		0			0
$235$	4.75834			0.310400
$236$		0			0
$237$	−12.8978			−0.837800
$238$		0			0
$239$			4.19004i			0.271031i	0.990775	+	0.135516i	$0.0432691\pi$
$239$			4.19004i			0.271031i	−0.990775	+	0.135516i	$0.956731\pi$
$240$		0			0
$241$	−12.8107			−0.825211			−0.412605	−	0.910910i	$-0.635381\pi$
$241$	−12.8107			−0.825211			−0.412605	+	0.910910i	$0.635381\pi$
$242$		0			0
$243$		−	15.9728i		−	1.02465i
$244$		0			0
$245$	−8.25934	−	4.02817i	−0.527670	−	0.257350i
$246$		0			0
$247$	7.19891			0.458055
$248$		0			0
$249$			15.3421i			0.972269i
$250$		0			0
$251$		−	29.9578i		−	1.89092i	−0.325743	−	0.945458i	$-0.605614\pi$
$251$		−	29.9578i		−	1.89092i	0.325743	−	0.945458i	$-0.394386\pi$
$252$		0			0
$253$	8.87842	+	8.61286i	0.558181	+	0.541486i
$254$		0			0
$255$	2.24743			0.140740
$256$		0			0
$257$		−	8.47185i		−	0.528460i	−0.964460	−	0.264230i	$-0.914882\pi$
$257$		−	8.47185i		−	0.528460i	0.964460	−	0.264230i	$-0.0851177\pi$
$258$		0			0
$259$	−12.3976	−	19.8399i	−0.770350	−	1.23279i
$260$		0			0
$261$		−	15.6672i		−	0.969777i
$262$		0			0
$263$			8.46777i			0.522145i	0.965319	+	0.261072i	$0.0840761\pi$
$263$			8.46777i			0.522145i	−0.965319	+	0.261072i	$0.915924\pi$
$264$		0			0
$265$			8.32324i			0.511293i
$266$		0			0
$267$	−10.7977			−0.660808
$268$		0			0
$269$		−	9.80321i		−	0.597712i	−0.954298	−	0.298856i	$-0.903395\pi$
$269$		−	9.80321i		−	0.597712i	0.954298	−	0.298856i	$-0.0966051\pi$
$270$		0			0
$271$	−0.371106			−0.0225431			−0.0112716	−	0.999936i	$-0.503588\pi$
$271$	−0.371106			−0.0225431			−0.0112716	+	0.999936i	$0.503588\pi$
$272$		0			0
$273$	−2.25836	+	1.41121i	−0.136682	+	0.0854102i
$274$		0			0
$275$	−7.80026	−	7.56695i	−0.470373	−	0.456304i
$276$		0			0
$277$			12.8240i			0.770522i	0.922808	+	0.385261i	$0.125889\pi$
$277$			12.8240i			0.770522i	−0.922808	+	0.385261i	$0.874111\pi$
$278$		0			0
$279$		−	11.5661i		−	0.692442i
$280$		0			0
$281$		−	19.3006i		−	1.15138i	−0.817668	−	0.575690i	$-0.804734\pi$
$281$		−	19.3006i		−	1.15138i	0.817668	−	0.575690i	$-0.195266\pi$
$282$		0			0
$283$	23.8527			1.41790			0.708948	−	0.705261i	$-0.249170\pi$
$283$	23.8527			1.41790			0.708948	+	0.705261i	$0.249170\pi$
$284$		0			0
$285$	−9.51209			−0.563448
$286$		0			0
$287$	−9.86271	−	15.7833i	−0.582177	−	0.931660i
$288$		0			0
$289$	−14.1070			−0.829821
$290$		0			0
$291$	1.84438			0.108119
$292$		0			0
$293$	−21.8364			−1.27570			−0.637848	−	0.770162i	$-0.720175\pi$
$293$	−21.8364			−1.27570			−0.637848	+	0.770162i	$0.720175\pi$
$294$		0			0
$295$	−2.17482			−0.126623
$296$		0			0
$297$	−11.5916	+	11.9490i	−0.672614	+	0.693352i
$298$		0			0
$299$	3.72958			0.215687
$300$		0			0
$301$	15.8825	−	9.92470i	0.915453	−	0.572050i
$302$		0			0
$303$			3.22760i			0.185421i
$304$		0			0
$305$			4.49237i			0.257232i
$306$		0			0
$307$	−17.6196			−1.00560			−0.502802	−	0.864401i	$-0.667698\pi$
$307$	−17.6196			−1.00560			−0.502802	+	0.864401i	$0.667698\pi$
$308$		0			0
$309$	−13.8926			−0.790324
$310$		0			0
$311$			33.4804i			1.89850i	0.314527	+	0.949249i	$0.398154\pi$
$311$			33.4804i			1.89850i	−0.314527	+	0.949249i	$0.601846\pi$
$312$		0			0
$313$			27.8369i			1.57344i	0.617312	+	0.786718i	$0.288222\pi$
$313$			27.8369i			1.57344i	−0.617312	+	0.786718i	$0.711778\pi$
$314$		0			0
$315$	−5.85230	+	3.65699i	−0.329740	+	0.206048i
$316$		0			0
$317$	10.9180			0.613217			0.306608	−	0.951836i	$-0.400806\pi$
$317$	10.9180			0.613217			0.306608	+	0.951836i	$0.400806\pi$
$318$		0			0
$319$	−18.2097	+	18.7712i	−1.01955	+	1.05098i
$320$		0			0
$321$	0.575263			0.0321080
$322$		0			0
$323$	−12.2446			−0.681306
$324$		0			0
$325$	−3.27668			−0.181757
$326$		0			0
$327$	−8.36047			−0.462335
$328$		0			0
$329$	8.13278	−	5.08203i	0.448375	−	0.280181i
$330$		0			0
$331$	−4.63203			−0.254600			−0.127300	−	0.991864i	$-0.540631\pi$
$331$	−4.63203			−0.254600			−0.127300	+	0.991864i	$0.540631\pi$
$332$		0			0
$333$	−17.5691			−0.962779
$334$		0			0
$335$			17.8367i			0.974524i
$336$		0			0
$337$			24.0028i			1.30752i	0.756703	+	0.653759i	$0.226809\pi$
$337$			24.0028i			1.30752i	−0.756703	+	0.653759i	$0.773191\pi$
$338$		0			0
$339$		−	19.6866i		−	1.06923i
$340$		0			0
$341$	−13.4430	+	13.8575i	−0.727980	+	0.750425i
$342$		0			0
$343$	−18.4188	+	1.93639i	−0.994519	+	0.104555i
$344$		0			0
$345$	−4.92799			−0.265314
$346$		0			0
$347$		−	35.2322i		−	1.89136i	−0.325093	−	0.945682i	$-0.605396\pi$
$347$		−	35.2322i		−	1.89136i	0.325093	−	0.945682i	$-0.394604\pi$
$348$		0			0
$349$	−24.3959			−1.30588			−0.652942	−	0.757408i	$-0.726465\pi$
$349$	−24.3959			−1.30588			−0.652942	+	0.757408i	$0.726465\pi$
$350$		0			0
$351$			5.01946i			0.267919i
$352$		0			0
$353$			3.36472i			0.179086i	0.995983	+	0.0895430i	$0.0285406\pi$
$353$			3.36472i			0.179086i	−0.995983	+	0.0895430i	$0.971459\pi$
$354$		0			0
$355$			9.91325i			0.526141i
$356$		0			0
$357$	3.84123	−	2.40031i	0.203299	−	0.127038i
$358$		0			0
$359$		−	25.7941i		−	1.36136i	−0.732581	−	0.680680i	$-0.761684\pi$
$359$		−	25.7941i		−	1.36136i	0.732581	−	0.680680i	$-0.238316\pi$
$360$		0			0
$361$	32.8242			1.72759
$362$		0			0
$363$	11.0667	−	0.336118i	0.580852	−	0.0176416i
$364$		0			0
$365$			14.4865i			0.758256i
$366$		0			0
$367$		−	33.5294i		−	1.75022i	−0.483922	−	0.875111i	$-0.660788\pi$
$367$		−	33.5294i		−	1.75022i	0.483922	−	0.875111i	$-0.339212\pi$
$368$		0			0
$369$	−13.9768			−0.727602
$370$		0			0
$371$	8.88944	+	14.2258i	0.461516	+	0.738566i
$372$		0			0
$373$			0.497932i			0.0257819i	0.999917	+	0.0128910i	$0.00410344\pi$
$373$			0.497932i			0.0257819i	−0.999917	+	0.0128910i	$0.995897\pi$
$374$		0			0
$375$	10.9362			0.564742
$376$		0			0
$377$			7.88526i			0.406112i
$378$		0			0
$379$	6.01654			0.309049			0.154524	−	0.987989i	$-0.450615\pi$
$379$	6.01654			0.309049			0.154524	+	0.987989i	$0.450615\pi$
$380$		0			0
$381$	10.5776			0.541907
$382$		0			0
$383$		−	32.5070i		−	1.66103i	−0.556995	−	0.830516i	$-0.688046\pi$
$383$		−	32.5070i		−	1.66103i	0.556995	−	0.830516i	$-0.311954\pi$
$384$		0			0
$385$	11.2622	+	2.42050i	0.573975	+	0.123360i
$386$		0			0
$387$		−	14.0646i		−	0.714945i
$388$		0			0
$389$	−37.2662			−1.88947			−0.944737	−	0.327830i	$-0.893683\pi$
$389$	−37.2662			−1.88947			−0.944737	+	0.327830i	$0.893683\pi$
$390$		0			0
$391$	−6.34362			−0.320811
$392$		0			0
$393$			2.90478i			0.146527i
$394$		0			0
$395$	16.8218			0.846395
$396$		0			0
$397$		−	14.5983i		−	0.732666i	−0.930484	−	0.366333i	$-0.880613\pi$
$397$		−	14.5983i		−	0.732666i	0.930484	−	0.366333i	$-0.119387\pi$
$398$		0			0
$399$	−16.2577	+	10.1592i	−0.813904	+	0.508594i
$400$		0			0
$401$	−7.28955			−0.364023			−0.182011	−	0.983296i	$-0.558261\pi$
$401$	−7.28955			−0.364023			−0.182011	+	0.983296i	$0.558261\pi$
$402$		0			0
$403$			5.82116i			0.289972i
$404$		0			0
$405$			1.19260i			0.0592606i
$406$		0			0
$407$	21.0498	+	20.4202i	1.04340	+	1.01219i
$408$		0			0
$409$	18.4061			0.910123			0.455062	−	0.890460i	$-0.349617\pi$
$409$	18.4061			0.910123			0.455062	+	0.890460i	$0.349617\pi$
$410$		0			0
$411$		−	10.7670i		−	0.531096i
$412$		0			0
$413$	−3.71713	+	2.32277i	−0.182908	+	0.114296i
$414$		0			0
$415$		−	20.0098i		−	0.982244i
$416$		0			0
$417$		−	2.62793i		−	0.128690i
$418$		0			0
$419$		−	31.0776i		−	1.51824i	−0.650952	−	0.759119i	$-0.725630\pi$
$419$		−	31.0776i		−	1.51824i	0.650952	−	0.759119i	$-0.274370\pi$
$420$		0			0
$421$	1.55924			0.0759929			0.0379964	−	0.999278i	$-0.487902\pi$
$421$	1.55924			0.0759929			0.0379964	+	0.999278i	$0.487902\pi$
$422$		0			0
$423$		−	7.20192i		−	0.350169i
$424$		0			0
$425$	5.57328			0.270344
$426$		0			0
$427$	4.79796	+	7.67820i	0.232190	+	0.371574i
$428$		0			0
$429$	2.32441	−	2.39608i	0.112224	−	0.115684i
$430$		0			0
$431$			6.40646i			0.308588i	0.988025	+	0.154294i	$0.0493103\pi$
$431$			6.40646i			0.308588i	−0.988025	+	0.154294i	$0.950690\pi$
$432$		0			0
$433$		−	5.12752i		−	0.246413i	−0.992381	−	0.123206i	$-0.960682\pi$
$433$		−	5.12752i		−	0.246413i	0.992381	−	0.123206i	$-0.0393177\pi$
$434$		0			0
$435$		−	10.4190i		−	0.499553i
$436$		0			0
$437$	26.8489			1.28436
$438$		0			0
$439$	−37.6222			−1.79561			−0.897805	−	0.440392i	$-0.854839\pi$
$439$	−37.6222			−1.79561			−0.897805	+	0.440392i	$0.854839\pi$
$440$		0			0
$441$	−6.09677	+	12.5008i	−0.290323	+	0.595276i
$442$		0			0
$443$	7.30278			0.346966			0.173483	−	0.984837i	$-0.444498\pi$
$443$	7.30278			0.346966			0.173483	+	0.984837i	$0.444498\pi$
$444$		0			0
$445$	14.0828			0.667587
$446$		0			0
$447$	−5.97925			−0.282809
$448$		0			0
$449$	19.2253			0.907300			0.453650	−	0.891180i	$-0.350122\pi$
$449$	19.2253			0.907300			0.453650	+	0.891180i	$0.350122\pi$
$450$		0			0
$451$	16.7458	+	16.2449i	0.788530	+	0.764945i
$452$		0			0
$453$	14.2348			0.668807
$454$		0			0
$455$	2.94544	−	1.84055i	0.138084	−	0.0862865i
$456$		0			0
$457$			11.1292i			0.520603i	0.965527	+	0.260301i	$0.0838219\pi$
$457$			11.1292i			0.520603i	−0.965527	+	0.260301i	$0.916178\pi$
$458$		0			0
$459$		−	8.53756i		−	0.398499i
$460$		0			0
$461$	3.21050			0.149528			0.0747639	−	0.997201i	$-0.476180\pi$
$461$	3.21050			0.149528			0.0747639	+	0.997201i	$0.476180\pi$
$462$		0			0
$463$	−33.3513			−1.54996			−0.774982	−	0.631983i	$-0.782241\pi$
$463$	−33.3513			−1.54996			−0.774982	+	0.631983i	$0.782241\pi$
$464$		0			0
$465$		−	7.69164i		−	0.356691i
$466$		0			0
$467$			25.8696i			1.19710i	0.801084	+	0.598551i	$0.204257\pi$
$467$			25.8696i			1.19710i	−0.801084	+	0.598551i	$0.795743\pi$
$468$		0			0
$469$	19.0501	+	30.4859i	0.879650	+	1.40771i
$470$		0			0
$471$	−8.09666			−0.373074
$472$		0			0
$473$	−16.3470	+	16.8511i	−0.751638	+	0.774813i
$474$		0			0
$475$	−23.5885			−1.08231
$476$		0			0
$477$	12.5975			0.576801
$478$		0			0
$479$	6.47795			0.295985			0.147993	−	0.988988i	$-0.452719\pi$
$479$	6.47795			0.295985			0.147993	+	0.988988i	$0.452719\pi$
$480$		0			0
$481$	8.84246			0.403181
$482$		0			0
$483$	−8.42274	+	5.26322i	−0.383248	+	0.239485i
$484$		0			0
$485$	−2.40551			−0.109228
$486$		0			0
$487$	2.34987			0.106483			0.0532413	−	0.998582i	$-0.483045\pi$
$487$	2.34987			0.106483			0.0532413	+	0.998582i	$0.483045\pi$
$488$		0			0
$489$		−	13.6342i		−	0.616562i
$490$		0			0
$491$		−	32.6434i		−	1.47318i	−0.676342	−	0.736588i	$-0.736436\pi$
$491$		−	32.6434i		−	1.47318i	0.676342	−	0.736588i	$-0.263564\pi$
$492$		0			0
$493$		−	13.4120i		−	0.604046i
$494$		0			0
$495$	6.02346	−	6.20919i	0.270735	−	0.279082i
$496$		0			0
$497$	10.5876	+	16.9434i	0.474919	+	0.760014i
$498$		0			0
$499$	−30.6207			−1.37077			−0.685385	−	0.728181i	$-0.740366\pi$
$499$	−30.6207			−1.37077			−0.685385	+	0.728181i	$0.740366\pi$
$500$		0			0
$501$			16.4909i			0.736758i
$502$		0			0
$503$	−0.244600			−0.0109062			−0.00545309	−	0.999985i	$-0.501736\pi$
$503$	−0.244600			−0.0109062			−0.00545309	+	0.999985i	$0.501736\pi$
$504$		0			0
$505$		−	4.20956i		−	0.187323i
$506$		0			0
$507$		−	1.00653i		−	0.0447015i
$508$		0			0
$509$			31.9726i			1.41716i	0.705629	+	0.708581i	$0.250664\pi$
$509$			31.9726i			1.41716i	−0.705629	+	0.708581i	$0.749336\pi$
$510$		0			0
$511$	15.4719	+	24.7597i	0.684437	+	1.09531i
$512$		0			0
$513$			36.1346i			1.59538i
$514$		0			0
$515$	18.1193			0.798432
$516$		0			0
$517$	−8.37065	+	8.62874i	−0.368141	+	0.379492i
$518$		0			0
$519$			3.84986i			0.168990i
$520$		0			0
$521$			3.34864i			0.146707i	0.997306	+	0.0733533i	$0.0233701\pi$
$521$			3.34864i			0.146707i	−0.997306	+	0.0733533i	$0.976630\pi$
$522$		0			0
$523$	28.0406			1.22613			0.613065	−	0.790032i	$-0.289936\pi$
$523$	28.0406			1.22613			0.613065	+	0.790032i	$0.289936\pi$
$524$		0			0
$525$	7.39992	−	4.62407i	0.322959	−	0.201811i
$526$		0			0
$527$		−	9.90116i		−	0.431301i
$528$		0			0
$529$	−9.09022			−0.395227
$530$		0			0
$531$			3.29167i			0.142846i
$532$		0			0
$533$	7.03447			0.304697
$534$		0			0
$535$	−0.750280			−0.0324374
$536$		0			0
$537$			8.29649i			0.358020i
$538$		0			0
$539$	21.8341	−	7.89127i	0.940461	−	0.339901i
$540$		0			0
$541$		−	2.94304i		−	0.126531i	−0.997997	−	0.0632656i	$-0.979848\pi$
$541$		−	2.94304i		−	0.126531i	0.997997	−	0.0632656i	$-0.0201515\pi$
$542$		0			0
$543$	2.16308			0.0928265
$544$		0			0
$545$	10.9040			0.467078
$546$		0			0
$547$			13.0601i			0.558409i	0.960232	+	0.279205i	$0.0900708\pi$
$547$			13.0601i			0.558409i	−0.960232	+	0.279205i	$0.909929\pi$
$548$		0			0
$549$	6.79936			0.290190
$550$		0			0
$551$			56.7653i			2.41828i
$552$		0			0
$553$	28.7512	−	17.9661i	1.22262	−	0.763995i
$554$		0			0
$555$	−11.6838			−0.495948
$556$		0			0
$557$			11.4167i			0.483743i	0.970308	+	0.241871i	$0.0777612\pi$
$557$			11.4167i			0.483743i	−0.970308	+	0.241871i	$0.922239\pi$
$558$		0			0
$559$			7.07868i			0.299396i
$560$		0			0
$561$	−3.95358	+	4.07548i	−0.166920	+	0.172067i
$562$		0			0
$563$	36.5969			1.54238			0.771188	−	0.636608i	$-0.219663\pi$
$563$	36.5969			1.54238			0.771188	+	0.636608i	$0.219663\pi$
$564$		0			0
$565$			25.6760i			1.08020i
$566$		0			0
$567$	1.27372	+	2.03834i	0.0534913	+	0.0856023i
$568$		0			0
$569$		−	13.7107i		−	0.574782i	−0.957813	−	0.287391i	$-0.907212\pi$
$569$		−	13.7107i		−	0.574782i	0.957813	−	0.287391i	$-0.0927880\pi$
$570$		0			0
$571$			10.9035i			0.456296i	0.973627	+	0.228148i	$0.0732669\pi$
$571$			10.9035i			0.456296i	−0.973627	+	0.228148i	$0.926733\pi$
$572$		0			0
$573$			6.14920i			0.256886i
$574$		0			0
$575$	−12.2206			−0.509636
$576$		0			0
$577$		−	12.0240i		−	0.500565i	−0.968173	−	0.250282i	$-0.919477\pi$
$577$		−	12.0240i		−	0.500565i	0.968173	−	0.250282i	$-0.0805235\pi$
$578$		0			0
$579$	−19.0442			−0.791452
$580$		0			0
$581$	−21.3710	−	34.2001i	−0.886618	−	1.41886i
$582$		0			0
$583$	−15.0933	−	14.6419i	−0.625101	−	0.606404i
$584$		0			0
$585$		−	2.60831i		−	0.107840i
$586$		0			0
$587$		−	5.32031i		−	0.219593i	−0.993954	−	0.109796i	$-0.964980\pi$
$587$		−	5.32031i		−	0.219593i	0.993954	−	0.109796i	$-0.0350199\pi$
$588$		0			0
$589$			41.9060i			1.72670i
$590$		0			0
$591$	−13.2594			−0.545419
$592$		0			0
$593$	−9.39434			−0.385779			−0.192890	−	0.981220i	$-0.561786\pi$
$593$	−9.39434			−0.385779			−0.192890	+	0.981220i	$0.561786\pi$
$594$		0			0
$595$	−5.00988	+	3.13058i	−0.205385	+	0.128341i
$596$		0			0
$597$	−6.39671			−0.261800
$598$		0			0
$599$	1.90235			0.0777280			0.0388640	−	0.999245i	$-0.487626\pi$
$599$	1.90235			0.0777280			0.0388640	+	0.999245i	$0.487626\pi$
$600$		0			0
$601$	35.6770			1.45529			0.727647	−	0.685952i	$-0.240614\pi$
$601$	35.6770			1.45529			0.727647	+	0.685952i	$0.240614\pi$
$602$		0			0
$603$	26.9965			1.09938
$604$		0			0
$605$	−14.4336	+	0.438378i	−0.586811	+	0.0178226i
$606$		0			0
$607$	9.19712			0.373300			0.186650	−	0.982427i	$-0.440237\pi$
$607$	9.19712			0.373300			0.186650	+	0.982427i	$0.440237\pi$
$608$		0			0
$609$	−11.1278	−	17.8078i	−0.450919	−	0.721607i
$610$		0			0
$611$			3.62470i			0.146640i
$612$		0			0
$613$			2.92417i			0.118106i	0.998255	+	0.0590531i	$0.0188081\pi$
$613$			2.92417i			0.118106i	−0.998255	+	0.0590531i	$0.981192\pi$
$614$		0			0
$615$	−9.29482			−0.374803
$616$		0			0
$617$	−30.4042			−1.22403			−0.612014	−	0.790847i	$-0.709641\pi$
$617$	−30.4042			−1.22403			−0.612014	+	0.790847i	$0.709641\pi$
$618$		0			0
$619$			17.0031i			0.683412i	0.939807	+	0.341706i	$0.111005\pi$
$619$			17.0031i			0.683412i	−0.939807	+	0.341706i	$0.888995\pi$
$620$		0			0
$621$			18.7205i			0.751227i
$622$		0			0
$623$	24.0698	−	15.0408i	0.964334	−	0.602595i
$624$		0			0
$625$	2.11999			0.0847997
$626$		0			0
$627$	16.7332	−	17.2492i	0.668261	−	0.688865i
$628$		0			0
$629$	−15.0401			−0.599687
$630$		0			0
$631$	24.7626			0.985785			0.492893	−	0.870090i	$-0.335940\pi$
$631$	24.7626			0.985785			0.492893	+	0.870090i	$0.335940\pi$
$632$		0			0
$633$	12.9038			0.512879
$634$		0			0
$635$	−13.7957			−0.547467
$636$		0			0
$637$	3.06849	−	6.29161i	0.121578	−	0.249283i
$638$		0			0
$639$	15.0041			0.593551
$640$		0			0
$641$	47.1920			1.86397			0.931985	−	0.362496i	$-0.118075\pi$
$641$	47.1920			1.86397			0.931985	+	0.362496i	$0.118075\pi$
$642$		0			0
$643$			39.6814i			1.56488i	0.622724	+	0.782441i	$0.286026\pi$
$643$			39.6814i			1.56488i	−0.622724	+	0.782441i	$0.713974\pi$
$644$		0			0
$645$		−	9.35324i		−	0.368283i
$646$		0			0
$647$		−	40.1001i		−	1.57650i	−0.615356	−	0.788249i	$-0.710988\pi$
$647$		−	40.1001i		−	1.57650i	0.615356	−	0.788249i	$-0.289012\pi$
$648$		0			0
$649$	3.82585	−	3.94381i	0.150178	−	0.154808i
$650$		0			0
$651$	−8.21487	−	13.1463i	−0.321966	−	0.515243i
$652$		0			0
$653$	−6.44816			−0.252336			−0.126168	−	0.992009i	$-0.540268\pi$
$653$	−6.44816			−0.252336			−0.126168	+	0.992009i	$0.540268\pi$
$654$		0			0
$655$		−	3.78853i		−	0.148030i
$656$		0			0
$657$	21.9258			0.855406
$658$		0			0
$659$			20.0549i			0.781228i	0.920555	+	0.390614i	$0.127737\pi$
$659$			20.0549i			0.781228i	−0.920555	+	0.390614i	$0.872263\pi$
$660$		0			0
$661$			30.0749i			1.16978i	0.811113	+	0.584889i	$0.198862\pi$
$661$			30.0749i			1.16978i	−0.811113	+	0.584889i	$0.801138\pi$
$662$		0			0
$663$			1.71200i			0.0664885i
$664$		0			0
$665$	21.2040	−	13.2500i	0.822254	−	0.513812i
$666$		0			0
$667$			29.4087i			1.13871i
$668$		0			0
$669$	13.7564			0.531855
$670$		0			0
$671$	−8.14643	−	7.90277i	−0.314490	−	0.305083i
$672$		0			0
$673$			12.1638i			0.468879i	0.972131	+	0.234439i	$0.0753255\pi$
$673$			12.1638i			0.468879i	−0.972131	+	0.234439i	$0.924675\pi$
$674$		0			0
$675$		−	16.4471i		−	0.633051i
$676$		0			0
$677$	14.7567			0.567145			0.283572	−	0.958951i	$-0.408480\pi$
$677$	14.7567			0.567145			0.283572	+	0.958951i	$0.408480\pi$
$678$		0			0
$679$	−4.11141	+	2.56914i	−0.157781	+	0.0985947i
$680$		0			0
$681$			23.9384i			0.917322i
$682$		0			0
$683$	19.8283			0.758708			0.379354	−	0.925252i	$-0.376146\pi$
$683$	19.8283			0.758708			0.379354	+	0.925252i	$0.376146\pi$
$684$		0			0
$685$			14.0427i			0.536545i
$686$		0			0
$687$	21.4294			0.817581
$688$		0			0
$689$	−6.34029			−0.241546
$690$		0			0
$691$			17.3888i			0.661501i	0.943718	+	0.330750i	$0.107302\pi$
$691$			17.3888i			0.661501i	−0.943718	+	0.330750i	$0.892698\pi$
$692$		0			0
$693$	3.66352	−	17.0457i	0.139166	−	0.647514i
$694$		0			0
$695$			3.42744i			0.130010i
$696$		0			0
$697$	−11.9649			−0.453202
$698$		0			0
$699$	5.16460			0.195343
$700$		0			0
$701$		−	29.7580i		−	1.12394i	−0.827156	−	0.561972i	$-0.810043\pi$
$701$		−	29.7580i		−	1.12394i	0.827156	−	0.561972i	$-0.189957\pi$
$702$		0			0
$703$	63.6560			2.40083
$704$		0			0
$705$		−	4.78941i		−	0.180380i
$706$		0			0
$707$	−4.49592	−	7.19483i	−0.169086	−	0.270589i
$708$		0			0
$709$	3.93603			0.147821			0.0739104	−	0.997265i	$-0.476452\pi$
$709$	3.93603			0.147821			0.0739104	+	0.997265i	$0.476452\pi$
$710$		0			0
$711$		−	25.4603i		−	0.954837i
$712$		0			0
$713$			21.7105i			0.813064i
$714$		0			0
$715$	−3.03159	+	3.12506i	−0.113375	+	0.116871i
$716$		0			0
$717$	4.21740			0.157502
$718$		0			0
$719$			27.3877i			1.02139i	0.859762	+	0.510695i	$0.170612\pi$
$719$			27.3877i			1.02139i	−0.859762	+	0.510695i	$0.829388\pi$
$720$		0			0
$721$	30.9689	−	19.3519i	1.15334	−	0.720702i
$722$		0			0
$723$			12.8944i			0.479546i
$724$		0			0
$725$		−	25.8375i		−	0.959579i
$726$		0			0
$727$			31.4031i			1.16468i	0.812947	+	0.582338i	$0.197862\pi$
$727$			31.4031i			1.16468i	−0.812947	+	0.582338i	$0.802138\pi$
$728$		0			0
$729$	−13.3517			−0.494506
$730$		0			0
$731$		−	12.0401i		−	0.445318i
$732$		0			0
$733$	−17.8121			−0.657905			−0.328953	−	0.944346i	$-0.606696\pi$
$733$	−17.8121			−0.657905			−0.328953	+	0.944346i	$0.606696\pi$
$734$		0			0
$735$	−4.05447	+	8.31327i	−0.149551	+	0.306639i
$736$		0			0
$737$	−32.3450	−	31.3775i	−1.19144	−	1.15581i
$738$		0			0
$739$		−	30.2753i		−	1.11369i	−0.830615	−	0.556847i	$-0.812011\pi$
$739$		−	30.2753i		−	1.11369i	0.830615	−	0.556847i	$-0.187989\pi$
$740$		0			0
$741$		−	7.24591i		−	0.266185i
$742$		0			0
$743$		−	24.5190i		−	0.899516i	−0.893151	−	0.449758i	$-0.851510\pi$
$743$		−	24.5190i		−	0.899516i	0.893151	−	0.449758i	$-0.148490\pi$
$744$		0			0
$745$	7.79837			0.285710
$746$		0			0
$747$	−30.2856			−1.10809
$748$		0			0
$749$	−1.28235	+	0.801318i	−0.0468561	+	0.0292795i
$750$		0			0
$751$	−21.5572			−0.786633			−0.393317	−	0.919403i	$-0.628672\pi$
$751$	−21.5572			−0.786633			−0.393317	+	0.919403i	$0.628672\pi$
$752$		0			0
$753$	−30.1533			−1.09885
$754$		0			0
$755$	−18.5655			−0.675668
$756$		0			0
$757$	−21.7238			−0.789566			−0.394783	−	0.918774i	$-0.629180\pi$
$757$	−21.7238			−0.789566			−0.394783	+	0.918774i	$0.629180\pi$
$758$		0			0
$759$	8.66909	−	8.93638i	0.314668	−	0.324370i
$760$		0			0
$761$	6.90527			0.250316			0.125158	−	0.992137i	$-0.460056\pi$
$761$	6.90527			0.250316			0.125158	+	0.992137i	$0.460056\pi$
$762$		0			0
$763$	18.6368	−	11.6458i	0.674698	−	0.421606i
$764$		0			0
$765$			4.43646i			0.160400i
$766$		0			0
$767$		−	1.65669i		−	0.0598196i
$768$		0			0
$769$	26.3034			0.948527			0.474263	−	0.880383i	$-0.342714\pi$
$769$	26.3034			0.948527			0.474263	+	0.880383i	$0.342714\pi$
$770$		0			0
$771$	−8.52716			−0.307098
$772$		0			0
$773$			35.8072i			1.28790i	0.765069	+	0.643948i	$0.222705\pi$
$773$			35.8072i			1.28790i	−0.765069	+	0.643948i	$0.777295\pi$
$774$		0			0
$775$		−	19.0740i		−	0.685160i
$776$		0			0
$777$	−19.9695	+	12.4785i	−0.716400	+	0.447665i
$778$		0			0
$779$	50.6405			1.81438
$780$		0			0
$781$	−17.9766	−	17.4389i	−0.643254	−	0.624014i
$782$		0			0
$783$	−39.5798			−1.41447
$784$		0			0
$785$	10.5600			0.376902
$786$		0			0
$787$	−4.28684			−0.152809			−0.0764046	−	0.997077i	$-0.524344\pi$
$787$	−4.28684			−0.152809			−0.0764046	+	0.997077i	$0.524344\pi$
$788$		0			0
$789$	8.52305			0.303429
$790$		0			0
$791$	27.4226	+	43.8845i	0.975036	+	1.56035i
$792$		0			0
$793$	−3.42210			−0.121522
$794$		0			0
$795$	8.37759			0.297122
$796$		0			0
$797$			35.8045i			1.26826i	0.773226	+	0.634130i	$0.218642\pi$
$797$			35.8045i			1.26826i	−0.773226	+	0.634130i	$0.781358\pi$
$798$		0			0
$799$		−	6.16523i		−	0.218110i
$800$		0			0
$801$		−	21.3148i		−	0.753120i
$802$		0			0
$803$	−26.2697	−	25.4839i	−0.927036	−	0.899308i
$804$		0			0
$805$	10.9853	−	6.86449i	0.387180	−	0.241942i
$806$		0			0
$807$	−9.86721			−0.347342
$808$		0			0
$809$		−	44.5501i		−	1.56630i	−0.621834	−	0.783149i	$-0.713612\pi$
$809$		−	44.5501i		−	1.56630i	0.621834	−	0.783149i	$-0.286388\pi$
$810$		0			0
$811$	6.38928			0.224358			0.112179	−	0.993688i	$-0.464217\pi$
$811$	6.38928			0.224358			0.112179	+	0.993688i	$0.464217\pi$
$812$		0			0
$813$			0.373529i			0.0131002i
$814$		0			0
$815$			17.7823i			0.622887i
$816$		0			0
$817$			50.9588i			1.78282i
$818$		0			0
$819$	−2.78574	−	4.45803i	−0.0973417	−	0.155776i
$820$		0			0
$821$			19.6789i			0.686797i	0.939190	+	0.343399i	$0.111578\pi$
$821$			19.6789i			0.686797i	−0.939190	+	0.343399i	$0.888422\pi$
$822$		0			0
$823$	−17.5959			−0.613354			−0.306677	−	0.951814i	$-0.599217\pi$
$823$	−17.5959			−0.613354			−0.306677	+	0.951814i	$0.599217\pi$
$824$		0			0
$825$	−7.61635	+	7.85119i	−0.265167	+	0.273343i
$826$		0			0
$827$		−	37.5703i		−	1.30645i	−0.757164	−	0.653224i	$-0.773416\pi$
$827$		−	37.5703i		−	1.30645i	0.757164	−	0.653224i	$-0.226584\pi$
$828$		0			0
$829$		−	14.8038i		−	0.514157i	−0.966390	−	0.257079i	$-0.917240\pi$
$829$		−	14.8038i		−	0.514157i	0.966390	−	0.257079i	$-0.0827599\pi$
$830$		0			0
$831$	12.9078			0.447766
$832$		0			0
$833$	−5.21917	+	10.7014i	−0.180833	+	0.370780i
$834$		0			0
$835$		−	21.5080i		−	0.744316i
$836$		0			0
$837$	−29.2191			−1.00996
$838$		0			0
$839$			37.2935i			1.28751i	0.765230	+	0.643757i	$0.222625\pi$
$839$			37.2935i			1.28751i	−0.765230	+	0.643757i	$0.777375\pi$
$840$		0			0
$841$	−33.1774			−1.14405
$842$		0			0
$843$	−19.4267			−0.669089
$844$		0			0
$845$			1.31275i			0.0451601i
$846$		0			0
$847$	−24.2013	+	16.1648i	−0.831565	+	0.555427i
$848$		0			0
$849$		−	24.0084i		−	0.823967i
$850$		0			0
$851$	32.9787			1.13049
$852$		0			0
$853$	3.73241			0.127795			0.0638976	−	0.997956i	$-0.479647\pi$
$853$	3.73241			0.127795			0.0638976	+	0.997956i	$0.479647\pi$
$854$		0			0
$855$		−	18.7770i		−	0.642159i
$856$		0			0
$857$	38.9323			1.32990			0.664950	−	0.746887i	$-0.268453\pi$
$857$	38.9323			1.32990			0.664950	+	0.746887i	$0.268453\pi$
$858$		0			0
$859$			9.70132i			0.331005i	0.986209	+	0.165502i	$0.0529245\pi$
$859$			9.70132i			0.331005i	−0.986209	+	0.165502i	$0.947075\pi$
$860$		0			0
$861$	−15.8864	+	9.92710i	−0.541406	+	0.338315i
$862$		0			0
$863$	−11.9587			−0.407078			−0.203539	−	0.979067i	$-0.565244\pi$
$863$	−11.9587			−0.407078			−0.203539	+	0.979067i	$0.565244\pi$
$864$		0			0
$865$		−	5.02114i		−	0.170724i
$866$		0			0
$867$			14.1991i			0.482226i
$868$		0			0
$869$	−29.5921	+	30.5045i	−1.00384	+	1.03479i
$870$		0			0
$871$	−13.5872			−0.460386
$872$		0			0
$873$			3.64082i			0.123223i
$874$		0			0
$875$	−24.3785	+	15.2337i	−0.824143	+	0.514992i
$876$		0			0
$877$			43.0738i			1.45450i	0.686374	+	0.727249i	$0.259201\pi$
$877$			43.0738i			1.45450i	−0.686374	+	0.727249i	$0.740799\pi$
$878$		0			0
$879$			21.9790i			0.741332i
$880$		0			0
$881$			51.9360i			1.74977i	0.484333	+	0.874884i	$0.339062\pi$
$881$			51.9360i			1.74977i	−0.484333	+	0.874884i	$0.660938\pi$
$882$		0			0
$883$	−9.86162			−0.331870			−0.165935	−	0.986137i	$-0.553064\pi$
$883$	−9.86162			−0.331870			−0.165935	+	0.986137i	$0.553064\pi$
$884$		0			0
$885$			2.18902i			0.0735832i
$886$		0			0
$887$	32.6581			1.09655			0.548276	−	0.836298i	$-0.315285\pi$
$887$	32.6581			1.09655			0.548276	+	0.836298i	$0.315285\pi$
$888$		0			0
$889$	−23.5792	+	14.7342i	−0.790820	+	0.494169i
$890$		0			0
$891$	−2.16265	−	2.09796i	−0.0724514	−	0.0702843i
$892$		0			0
$893$			26.0939i			0.873198i
$894$		0			0
$895$		−	10.8206i		−	0.361693i
$896$		0			0
$897$		−	3.75393i		−	0.125340i
$898$		0			0
$899$	−45.9014			−1.53090
$900$		0			0
$901$	10.7842			0.359272
$902$		0			0
$903$	−9.98949	−	15.9862i	−0.332430	−	0.531988i
$904$		0			0
$905$	−2.82117			−0.0937788
$906$		0			0
$907$	−36.3870			−1.20821			−0.604105	−	0.796905i	$-0.706469\pi$
$907$	−36.3870			−1.20821			−0.604105	+	0.796905i	$0.706469\pi$
$908$		0			0
$909$	−6.37132			−0.211323
$910$		0			0
$911$	31.4048			1.04049			0.520244	−	0.854018i	$-0.325841\pi$
$911$	31.4048			1.04049			0.520244	+	0.854018i	$0.325841\pi$
$912$		0			0
$913$	36.2857	+	35.2003i	1.20088	+	1.16496i
$914$		0			0
$915$	4.52170			0.149483
$916$		0			0
$917$	−4.04624	−	6.47522i	−0.133619	−	0.213830i
$918$		0			0
$919$			31.8609i			1.05100i	0.850795	+	0.525498i	$0.176121\pi$
$919$			31.8609i			1.05100i	−0.850795	+	0.525498i	$0.823879\pi$
$920$		0			0
$921$			17.7347i			0.584377i
$922$		0			0
$923$	−7.55149			−0.248560
$924$		0			0
$925$	−28.9739			−0.952655
$926$		0			0
$927$		−	27.4242i		−	0.900730i
$928$		0			0
$929$		−	21.4956i		−	0.705248i	−0.935765	−	0.352624i	$-0.885289\pi$
$929$		−	21.4956i		−	0.705248i	0.935765	−	0.352624i	$-0.114711\pi$
$930$		0			0
$931$	22.0897	−	45.2927i	0.723962	−	1.48441i
$932$		0			0
$933$	33.6990			1.10325
$934$		0			0
$935$	5.15641	−	5.31540i	0.168633	−	0.173832i
$936$		0			0
$937$	4.27190			0.139557			0.0697785	−	0.997563i	$-0.477771\pi$
$937$	4.27190			0.139557			0.0697785	+	0.997563i	$0.477771\pi$
$938$		0			0
$939$	28.0187			0.914355
$940$		0			0
$941$	47.5457			1.54995			0.774973	−	0.631995i	$-0.217764\pi$
$941$	47.5457			1.54995			0.774973	+	0.631995i	$0.217764\pi$
$942$		0			0
$943$	26.2356			0.854349
$944$		0			0
$945$	9.23858	+	14.7845i	0.300531	+	0.480941i
$946$		0			0
$947$	35.0760			1.13982			0.569908	−	0.821708i	$-0.306979\pi$
$947$	35.0760			1.13982			0.569908	+	0.821708i	$0.306979\pi$
$948$		0			0
$949$	−11.0352			−0.358217
$950$		0			0
$951$		−	10.9893i		−	0.356352i
$952$		0			0
$953$			32.0160i			1.03710i	0.855048	+	0.518549i	$0.173528\pi$
$953$			32.0160i			1.03710i	−0.855048	+	0.518549i	$0.826472\pi$
$954$		0			0
$955$		−	8.02002i		−	0.259522i
$956$		0			0
$957$	18.8937	+	18.3286i	0.610748	+	0.592480i
$958$		0			0
$959$	14.9980	+	24.0013i	0.484310	+	0.775042i
$960$		0			0
$961$	−2.88587			−0.0930924
$962$		0			0
$963$			1.13558i			0.0365934i
$964$		0			0
$965$	24.8382			0.799571
$966$		0			0
$967$		−	39.2428i		−	1.26196i	−0.775797	−	0.630982i	$-0.782652\pi$
$967$		−	39.2428i		−	1.26196i	0.775797	−	0.630982i	$-0.217348\pi$
$968$		0			0
$969$			12.3245i			0.395920i
$970$		0			0
$971$		−	26.7848i		−	0.859565i	−0.902933	−	0.429782i	$-0.858590\pi$
$971$		−	26.7848i		−	0.859565i	0.902933	−	0.429782i	$-0.141410\pi$
$972$		0			0
$973$	3.66060	+	5.85806i	0.117353	+	0.187801i
$974$		0			0
$975$			3.29807i			0.105623i
$976$		0			0
$977$	−41.4895			−1.32737			−0.663683	−	0.748014i	$-0.731007\pi$
$977$	−41.4895			−1.32737			−0.663683	+	0.748014i	$0.731007\pi$
$978$		0			0
$979$	−24.7737	+	25.5376i	−0.791772	+	0.816185i
$980$		0			0
$981$		−	16.5037i		−	0.526921i
$982$		0			0
$983$		−	46.4682i		−	1.48211i	−0.671447	−	0.741053i	$-0.734327\pi$
$983$		−	46.4682i		−	1.48211i	0.671447	−	0.741053i	$-0.265673\pi$
$984$		0			0
$985$	17.2934			0.551015
$986$		0			0
$987$	−5.11521	−	8.18588i	−0.162819	−	0.260560i
$988$		0			0
$989$			26.4005i			0.839487i
$990$		0			0
$991$	−27.8370			−0.884272			−0.442136	−	0.896948i	$-0.645779\pi$
$991$	−27.8370			−0.884272			−0.442136	+	0.896948i	$0.645779\pi$
$992$		0			0
$993$			4.66228i			0.147953i
$994$		0			0
$995$	8.34283			0.264486
$996$		0			0
$997$	36.2802			1.14900			0.574502	−	0.818503i	$-0.305196\pi$
$997$	36.2802			1.14900			0.574502	+	0.818503i	$0.305196\pi$
$998$		0			0
$999$			44.3843i			1.40426i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.e.b.3849.17	yes	48
7.6	odd	2		4004.2.e.a.3849.32	yes	48
11.10	odd	2		4004.2.e.a.3849.17	✓	48
77.76	even	2	inner	4004.2.e.b.3849.32	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.e.a.3849.17	✓	48	11.10	odd	2
4004.2.e.a.3849.32	yes	48	7.6	odd	2
4004.2.e.b.3849.17	yes	48	1.1	even	1	trivial
4004.2.e.b.3849.32	yes	48	77.76	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.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00653i q^{3} +1.31275i q^{5} +(1.40205 + 2.24371i) q^{7} +1.98690 q^{9} +O(q^{10})\)	\(q-1.00653i q^{3} +1.31275i q^{5} +(1.40205 + 2.24371i) q^{7} +1.98690 q^{9} +(-2.38054 - 2.30934i) q^{11} -1.00000 q^{13} +1.32132 q^{15} +1.70089 q^{17} -7.19891 q^{19} +(2.25836 - 1.41121i) q^{21} -3.72958 q^{23} +3.27668 q^{25} -5.01946i q^{27} -7.88526i q^{29} -5.82116i q^{31} +(-2.32441 + 2.39608i) q^{33} +(-2.94544 + 1.84055i) q^{35} -8.84246 q^{37} +1.00653i q^{39} -7.03447 q^{41} -7.07868i q^{43} +2.60831i q^{45} -3.62470i q^{47} +(-3.06849 + 6.29161i) q^{49} -1.71200i q^{51} +6.34029 q^{53} +(3.03159 - 3.12506i) q^{55} +7.24591i q^{57} +1.65669i q^{59} +3.42210 q^{61} +(2.78574 + 4.45803i) q^{63} -1.31275i q^{65} +13.5872 q^{67} +3.75393i q^{69} +7.55149 q^{71} +11.0352 q^{73} -3.29807i q^{75} +(1.84384 - 8.57906i) q^{77} -12.8141i q^{79} +0.908469 q^{81} -15.2426 q^{83} +2.23285i q^{85} -7.93675 q^{87} -10.7277i q^{89} +(-1.40205 - 2.24371i) q^{91} -5.85916 q^{93} -9.45039i q^{95} +1.83241i q^{97} +(-4.72989 - 4.58842i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{7} - 48 q^{9}+O(q^{10}) \) 48 * q + 4 * q^7 - 48 * q^9	\( 48 q + 4 q^{7} - 48 q^{9} + 2 q^{11} - 48 q^{13} + 8 q^{15} - 4 q^{17} - 10 q^{21} + 4 q^{23} - 44 q^{25} - 10 q^{33} + 14 q^{35} - 16 q^{37} + 10 q^{49} - 8 q^{53} + 12 q^{55} - 16 q^{61} - 16 q^{63} + 4 q^{67} + 16 q^{73} + 22 q^{77} + 64 q^{81} + 4 q^{83} + 12 q^{87} - 4 q^{91} + 36 q^{93} - 40 q^{99}+O(q^{100}) \) 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 + 22 * 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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