LMFDB - Embedded newform 4004.2.a.j.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(1,4004)]

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, 0]))

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

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

chi := DirichletCharacter("4004.1");

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.a (trivial)

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:	yes
Analytic conductor:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 18x^{8} + 32x^{7} + 93x^{6} - 119x^{5} - 174x^{4} + 107x^{3} + 51x^{2} - 17x - 2 $ x^10 - 2x^9 - 18x^8 + 32x^7 + 93x^6 - 119x^5 - 174x^4 + 107x^3 + 51x^2 - 17*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.357256$ of defining polynomial
Character	$\chi$	$=$	4004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.357256 q^{3} +3.31742 q^{5} -1.00000 q^{7} -2.87237 q^{9} +O(q^{10})$	$q-0.357256 q^{3} +3.31742 q^{5} -1.00000 q^{7} -2.87237 q^{9} -1.00000 q^{11} -1.00000 q^{13} -1.18517 q^{15} +2.87740 q^{17} -0.480550 q^{19} +0.357256 q^{21} -8.87774 q^{23} +6.00528 q^{25} +2.09794 q^{27} +2.06825 q^{29} +8.14340 q^{31} +0.357256 q^{33} -3.31742 q^{35} +9.94343 q^{37} +0.357256 q^{39} +6.95677 q^{41} -5.10805 q^{43} -9.52885 q^{45} -3.30475 q^{47} +1.00000 q^{49} -1.02797 q^{51} -6.64805 q^{53} -3.31742 q^{55} +0.171679 q^{57} +8.21164 q^{59} +13.8367 q^{61} +2.87237 q^{63} -3.31742 q^{65} +10.8104 q^{67} +3.17163 q^{69} +8.48947 q^{71} +15.9546 q^{73} -2.14543 q^{75} +1.00000 q^{77} -1.62192 q^{79} +7.86760 q^{81} +10.8207 q^{83} +9.54555 q^{85} -0.738894 q^{87} +9.23781 q^{89} +1.00000 q^{91} -2.90928 q^{93} -1.59419 q^{95} -14.8240 q^{97} +2.87237 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 2 * q^3 - 2 * q^5 - 10 * q^7 + 10 * q^9	$ 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} + 9 q^{17} + 3 q^{19} + 2 q^{21} - 4 q^{23} + 22 q^{25} - 8 q^{27} + 13 q^{29} - 17 q^{31} + 2 q^{33} + 2 q^{35} + 11 q^{37} + 2 q^{39} + 8 q^{41} + 21 q^{43} - 15 q^{45} + q^{47} + 10 q^{49} + 19 q^{51} + 22 q^{53} + 2 q^{55} + 8 q^{57} - 24 q^{59} + 16 q^{61} - 10 q^{63} + 2 q^{65} + 19 q^{67} + 51 q^{69} - 25 q^{71} + 22 q^{73} + 28 q^{75} + 10 q^{77} + 41 q^{79} + 54 q^{81} + 8 q^{83} + 9 q^{85} - 5 q^{87} + 36 q^{89} + 10 q^{91} + 12 q^{93} + 13 q^{95} - 5 q^{97} - 10 q^{99}+O(q^{100}) $ 10 * q - 2 * q^3 - 2 * q^5 - 10 * q^7 + 10 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 + 9 * q^17 + 3 * q^19 + 2 * q^21 - 4 * q^23 + 22 * q^25 - 8 * q^27 + 13 * q^29 - 17 * q^31 + 2 * q^33 + 2 * q^35 + 11 * q^37 + 2 * q^39 + 8 * q^41 + 21 * q^43 - 15 * q^45 + q^47 + 10 * q^49 + 19 * q^51 + 22 * q^53 + 2 * q^55 + 8 * q^57 - 24 * q^59 + 16 * q^61 - 10 * q^63 + 2 * q^65 + 19 * q^67 + 51 * q^69 - 25 * q^71 + 22 * q^73 + 28 * q^75 + 10 * q^77 + 41 * q^79 + 54 * q^81 + 8 * q^83 + 9 * q^85 - 5 * q^87 + 36 * q^89 + 10 * q^91 + 12 * q^93 + 13 * q^95 - 5 * q^97 - 10 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−0.357256	−0.206262	−0.103131	−	0.994668i	$-0.532886\pi$
$3$	−0.357256	−0.206262	−0.103131	+	0.994668i	$0.532886\pi$
$4$	0	0
$5$	3.31742	1.48360	0.741798	−	0.670623i	$-0.233973\pi$
$5$	3.31742	1.48360	0.741798	+	0.670623i	$0.233973\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.87237	−0.957456
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.18517	−0.306010
$16$	0	0
$17$	2.87740	0.697873	0.348936	−	0.937146i	$-0.386543\pi$
$17$	2.87740	0.697873	0.348936	+	0.937146i	$0.386543\pi$
$18$	0	0
$19$	−0.480550	−0.110246	−0.0551228	−	0.998480i	$-0.517555\pi$
$19$	−0.480550	−0.110246	−0.0551228	+	0.998480i	$0.517555\pi$
$20$	0	0
$21$	0.357256	0.0779597
$22$	0	0
$23$	−8.87774	−1.85114	−0.925569	−	0.378580i	$-0.876412\pi$
$23$	−8.87774	−1.85114	−0.925569	+	0.378580i	$0.876412\pi$
$24$	0	0
$25$	6.00528	1.20106
$26$	0	0
$27$	2.09794	0.403749
$28$	0	0
$29$	2.06825	0.384064	0.192032	−	0.981389i	$-0.438492\pi$
$29$	2.06825	0.384064	0.192032	+	0.981389i	$0.438492\pi$
$30$	0	0
$31$	8.14340	1.46260	0.731299	−	0.682057i	$-0.238915\pi$
$31$	8.14340	1.46260	0.731299	+	0.682057i	$0.238915\pi$
$32$	0	0
$33$	0.357256	0.0621904
$34$	0	0
$35$	−3.31742	−0.560747
$36$	0	0
$37$	9.94343	1.63469	0.817345	−	0.576148i	$-0.195445\pi$
$37$	9.94343	1.63469	0.817345	+	0.576148i	$0.195445\pi$
$38$	0	0
$39$	0.357256	0.0572068
$40$	0	0
$41$	6.95677	1.08646	0.543232	−	0.839583i	$-0.317200\pi$
$41$	6.95677	1.08646	0.543232	+	0.839583i	$0.317200\pi$
$42$	0	0
$43$	−5.10805	−0.778970	−0.389485	−	0.921033i	$-0.627347\pi$
$43$	−5.10805	−0.778970	−0.389485	+	0.921033i	$0.627347\pi$
$44$	0	0
$45$	−9.52885	−1.42048
$46$	0	0
$47$	−3.30475	−0.482048	−0.241024	−	0.970519i	$-0.577483\pi$
$47$	−3.30475	−0.482048	−0.241024	+	0.970519i	$0.577483\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.02797	−0.143945
$52$	0	0
$53$	−6.64805	−0.913180	−0.456590	−	0.889677i	$-0.650929\pi$
$53$	−6.64805	−0.913180	−0.456590	+	0.889677i	$0.650929\pi$
$54$	0	0
$55$	−3.31742	−0.447321
$56$	0	0
$57$	0.171679	0.0227395
$58$	0	0
$59$	8.21164	1.06907	0.534533	−	0.845148i	$-0.320488\pi$
$59$	8.21164	1.06907	0.534533	+	0.845148i	$0.320488\pi$
$60$	0	0
$61$	13.8367	1.77160	0.885801	−	0.464065i	$-0.153610\pi$
$61$	13.8367	1.77160	0.885801	+	0.464065i	$0.153610\pi$
$62$	0	0
$63$	2.87237	0.361884
$64$	0	0
$65$	−3.31742	−0.411475
$66$	0	0
$67$	10.8104	1.32070	0.660349	−	0.750959i	$-0.270408\pi$
$67$	10.8104	1.32070	0.660349	+	0.750959i	$0.270408\pi$
$68$	0	0
$69$	3.17163	0.381819
$70$	0	0
$71$	8.48947	1.00752	0.503758	−	0.863845i	$-0.331951\pi$
$71$	8.48947	1.00752	0.503758	+	0.863845i	$0.331951\pi$
$72$	0	0
$73$	15.9546	1.86735	0.933675	−	0.358122i	$-0.116583\pi$
$73$	15.9546	1.86735	0.933675	+	0.358122i	$0.116583\pi$
$74$	0	0
$75$	−2.14543	−0.247732
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−1.62192	−0.182481	−0.0912403	−	0.995829i	$-0.529083\pi$
$79$	−1.62192	−0.182481	−0.0912403	+	0.995829i	$0.529083\pi$
$80$	0	0
$81$	7.86760	0.874178
$82$	0	0
$83$	10.8207	1.18773	0.593865	−	0.804564i	$-0.297601\pi$
$83$	10.8207	1.18773	0.593865	+	0.804564i	$0.297601\pi$
$84$	0	0
$85$	9.54555	1.03536
$86$	0	0
$87$	−0.738894	−0.0792178
$88$	0	0
$89$	9.23781	0.979206	0.489603	−	0.871946i	$-0.337142\pi$
$89$	9.23781	0.979206	0.489603	+	0.871946i	$0.337142\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−2.90928	−0.301678
$94$	0	0
$95$	−1.59419	−0.163560
$96$	0	0
$97$	−14.8240	−1.50515	−0.752574	−	0.658508i	$-0.771188\pi$
$97$	−14.8240	−1.50515	−0.752574	+	0.658508i	$0.771188\pi$
$98$	0	0
$99$	2.87237	0.288684
$100$	0	0
$101$	6.81473	0.678091	0.339045	−	0.940770i	$-0.389896\pi$
$101$	6.81473	0.678091	0.339045	+	0.940770i	$0.389896\pi$
$102$	0	0
$103$	5.56915	0.548745	0.274372	−	0.961624i	$-0.411530\pi$
$103$	5.56915	0.548745	0.274372	+	0.961624i	$0.411530\pi$
$104$	0	0
$105$	1.18517	0.115661
$106$	0	0
$107$	−12.1594	−1.17550	−0.587748	−	0.809044i	$-0.699985\pi$
$107$	−12.1594	−1.17550	−0.587748	+	0.809044i	$0.699985\pi$
$108$	0	0
$109$	−7.42599	−0.711281	−0.355640	−	0.934623i	$-0.615737\pi$
$109$	−7.42599	−0.711281	−0.355640	+	0.934623i	$0.615737\pi$
$110$	0	0
$111$	−3.55236	−0.337175
$112$	0	0
$113$	−11.9437	−1.12357	−0.561783	−	0.827284i	$-0.689885\pi$
$113$	−11.9437	−1.12357	−0.561783	+	0.827284i	$0.689885\pi$
$114$	0	0
$115$	−29.4512	−2.74634
$116$	0	0
$117$	2.87237	0.265551
$118$	0	0
$119$	−2.87740	−0.263771
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.48535	−0.224096
$124$	0	0
$125$	3.33495	0.298287
$126$	0	0
$127$	−4.19697	−0.372421	−0.186210	−	0.982510i	$-0.559621\pi$
$127$	−4.19697	−0.372421	−0.186210	+	0.982510i	$0.559621\pi$
$128$	0	0
$129$	1.82488	0.160672
$130$	0	0
$131$	18.3104	1.59978	0.799892	−	0.600143i	$-0.204890\pi$
$131$	18.3104	1.59978	0.799892	+	0.600143i	$0.204890\pi$
$132$	0	0
$133$	0.480550	0.0416689
$134$	0	0
$135$	6.95975	0.599000
$136$	0	0
$137$	8.81368	0.753003	0.376502	−	0.926416i	$-0.377127\pi$
$137$	8.81368	0.753003	0.376502	+	0.926416i	$0.377127\pi$
$138$	0	0
$139$	−5.52322	−0.468473	−0.234237	−	0.972180i	$-0.575259\pi$
$139$	−5.52322	−0.468473	−0.234237	+	0.972180i	$0.575259\pi$
$140$	0	0
$141$	1.18064	0.0994281
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	6.86124	0.569795
$146$	0	0
$147$	−0.357256	−0.0294660
$148$	0	0
$149$	−22.7718	−1.86554	−0.932770	−	0.360473i	$-0.882615\pi$
$149$	−22.7718	−1.86554	−0.932770	+	0.360473i	$0.882615\pi$
$150$	0	0
$151$	11.1108	0.904184	0.452092	−	0.891971i	$-0.350678\pi$
$151$	11.1108	0.904184	0.452092	+	0.891971i	$0.350678\pi$
$152$	0	0
$153$	−8.26496	−0.668182
$154$	0	0
$155$	27.0151	2.16990
$156$	0	0
$157$	−17.9105	−1.42941	−0.714705	−	0.699426i	$-0.753439\pi$
$157$	−17.9105	−1.42941	−0.714705	+	0.699426i	$0.753439\pi$
$158$	0	0
$159$	2.37506	0.188354
$160$	0	0
$161$	8.87774	0.699664
$162$	0	0
$163$	−3.88780	−0.304516	−0.152258	−	0.988341i	$-0.548654\pi$
$163$	−3.88780	−0.304516	−0.152258	+	0.988341i	$0.548654\pi$
$164$	0	0
$165$	1.18517	0.0922654
$166$	0	0
$167$	9.13044	0.706534	0.353267	−	0.935522i	$-0.385071\pi$
$167$	9.13044	0.706534	0.353267	+	0.935522i	$0.385071\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.38032	0.105555
$172$	0	0
$173$	22.1924	1.68726	0.843629	−	0.536927i	$-0.180415\pi$
$173$	22.1924	1.68726	0.843629	+	0.536927i	$0.180415\pi$
$174$	0	0
$175$	−6.00528	−0.453957
$176$	0	0
$177$	−2.93366	−0.220508
$178$	0	0
$179$	−7.70749	−0.576085	−0.288043	−	0.957618i	$-0.593004\pi$
$179$	−7.70749	−0.576085	−0.288043	+	0.957618i	$0.593004\pi$
$180$	0	0
$181$	8.39961	0.624338	0.312169	−	0.950027i	$-0.398944\pi$
$181$	8.39961	0.624338	0.312169	+	0.950027i	$0.398944\pi$
$182$	0	0
$183$	−4.94323	−0.365414
$184$	0	0
$185$	32.9866	2.42522
$186$	0	0
$187$	−2.87740	−0.210416
$188$	0	0
$189$	−2.09794	−0.152603
$190$	0	0
$191$	−12.8806	−0.932005	−0.466002	−	0.884784i	$-0.654306\pi$
$191$	−12.8806	−0.932005	−0.466002	+	0.884784i	$0.654306\pi$
$192$	0	0
$193$	−9.88160	−0.711293	−0.355646	−	0.934621i	$-0.615739\pi$
$193$	−9.88160	−0.711293	−0.355646	+	0.934621i	$0.615739\pi$
$194$	0	0
$195$	1.18517	0.0848718
$196$	0	0
$197$	−6.78543	−0.483442	−0.241721	−	0.970346i	$-0.577712\pi$
$197$	−6.78543	−0.483442	−0.241721	+	0.970346i	$0.577712\pi$
$198$	0	0
$199$	−22.4228	−1.58951	−0.794755	−	0.606930i	$-0.792401\pi$
$199$	−22.4228	−1.58951	−0.794755	+	0.606930i	$0.792401\pi$
$200$	0	0
$201$	−3.86208	−0.272410
$202$	0	0
$203$	−2.06825	−0.145162
$204$	0	0
$205$	23.0785	1.61187
$206$	0	0
$207$	25.5001	1.77238
$208$	0	0
$209$	0.480550	0.0332403
$210$	0	0
$211$	0.726378	0.0500059	0.0250030	−	0.999687i	$-0.492040\pi$
$211$	0.726378	0.0500059	0.0250030	+	0.999687i	$0.492040\pi$
$212$	0	0
$213$	−3.03292	−0.207812
$214$	0	0
$215$	−16.9455	−1.15568
$216$	0	0
$217$	−8.14340	−0.552810
$218$	0	0
$219$	−5.69990	−0.385163
$220$	0	0
$221$	−2.87740	−0.193555
$222$	0	0
$223$	16.9176	1.13289	0.566444	−	0.824100i	$-0.308319\pi$
$223$	16.9176	1.13289	0.566444	+	0.824100i	$0.308319\pi$
$224$	0	0
$225$	−17.2494	−1.14996
$226$	0	0
$227$	−3.43324	−0.227872	−0.113936	−	0.993488i	$-0.536346\pi$
$227$	−3.43324	−0.227872	−0.113936	+	0.993488i	$0.536346\pi$
$228$	0	0
$229$	21.7443	1.43691	0.718453	−	0.695576i	$-0.244851\pi$
$229$	21.7443	1.43691	0.718453	+	0.695576i	$0.244851\pi$
$230$	0	0
$231$	−0.357256	−0.0235057
$232$	0	0
$233$	20.7972	1.36247	0.681236	−	0.732064i	$-0.261443\pi$
$233$	20.7972	1.36247	0.681236	+	0.732064i	$0.261443\pi$
$234$	0	0
$235$	−10.9633	−0.715164
$236$	0	0
$237$	0.579442	0.0376388
$238$	0	0
$239$	−15.5909	−1.00849	−0.504245	−	0.863561i	$-0.668229\pi$
$239$	−15.5909	−1.00849	−0.504245	+	0.863561i	$0.668229\pi$
$240$	0	0
$241$	−13.2478	−0.853368	−0.426684	−	0.904401i	$-0.640318\pi$
$241$	−13.2478	−0.853368	−0.426684	+	0.904401i	$0.640318\pi$
$242$	0	0
$243$	−9.10457	−0.584059
$244$	0	0
$245$	3.31742	0.211942
$246$	0	0
$247$	0.480550	0.0305766
$248$	0	0
$249$	−3.86578	−0.244984
$250$	0	0
$251$	−24.0058	−1.51524	−0.757618	−	0.652699i	$-0.773637\pi$
$251$	−24.0058	−1.51524	−0.757618	+	0.652699i	$0.773637\pi$
$252$	0	0
$253$	8.87774	0.558139
$254$	0	0
$255$	−3.41021	−0.213556
$256$	0	0
$257$	23.3774	1.45824	0.729121	−	0.684385i	$-0.239929\pi$
$257$	23.3774	1.45824	0.729121	+	0.684385i	$0.239929\pi$
$258$	0	0
$259$	−9.94343	−0.617855
$260$	0	0
$261$	−5.94076	−0.367724
$262$	0	0
$263$	−8.87124	−0.547024	−0.273512	−	0.961869i	$-0.588185\pi$
$263$	−8.87124	−0.547024	−0.273512	+	0.961869i	$0.588185\pi$
$264$	0	0
$265$	−22.0544	−1.35479
$266$	0	0
$267$	−3.30027	−0.201973
$268$	0	0
$269$	−3.59668	−0.219293	−0.109647	−	0.993971i	$-0.534972\pi$
$269$	−3.59668	−0.219293	−0.109647	+	0.993971i	$0.534972\pi$
$270$	0	0
$271$	25.2641	1.53469	0.767343	−	0.641237i	$-0.221578\pi$
$271$	25.2641	1.53469	0.767343	+	0.641237i	$0.221578\pi$
$272$	0	0
$273$	−0.357256	−0.0216221
$274$	0	0
$275$	−6.00528	−0.362132
$276$	0	0
$277$	−19.5662	−1.17562	−0.587810	−	0.808999i	$-0.700010\pi$
$277$	−19.5662	−1.17562	−0.587810	+	0.808999i	$0.700010\pi$
$278$	0	0
$279$	−23.3908	−1.40037
$280$	0	0
$281$	19.0672	1.13746	0.568728	−	0.822526i	$-0.307436\pi$
$281$	19.0672	1.13746	0.568728	+	0.822526i	$0.307436\pi$
$282$	0	0
$283$	−10.8890	−0.647281	−0.323641	−	0.946180i	$-0.604907\pi$
$283$	−10.8890	−0.647281	−0.323641	+	0.946180i	$0.604907\pi$
$284$	0	0
$285$	0.569533	0.0337362
$286$	0	0
$287$	−6.95677	−0.410645
$288$	0	0
$289$	−8.72056	−0.512974
$290$	0	0
$291$	5.29596	0.310455
$292$	0	0
$293$	20.6171	1.20446	0.602231	−	0.798322i	$-0.294279\pi$
$293$	20.6171	1.20446	0.602231	+	0.798322i	$0.294279\pi$
$294$	0	0
$295$	27.2415	1.58606
$296$	0	0
$297$	−2.09794	−0.121735
$298$	0	0
$299$	8.87774	0.513413
$300$	0	0
$301$	5.10805	0.294423
$302$	0	0
$303$	−2.43461	−0.139864
$304$	0	0
$305$	45.9020	2.62834
$306$	0	0
$307$	1.96417	0.112101	0.0560506	−	0.998428i	$-0.482149\pi$
$307$	1.96417	0.112101	0.0560506	+	0.998428i	$0.482149\pi$
$308$	0	0
$309$	−1.98961	−0.113185
$310$	0	0
$311$	−7.78177	−0.441264	−0.220632	−	0.975357i	$-0.570812\pi$
$311$	−7.78177	−0.441264	−0.220632	+	0.975357i	$0.570812\pi$
$312$	0	0
$313$	20.2610	1.14522	0.572609	−	0.819829i	$-0.305931\pi$
$313$	20.2610	1.14522	0.572609	+	0.819829i	$0.305931\pi$
$314$	0	0
$315$	9.52885	0.536890
$316$	0	0
$317$	24.6451	1.38421	0.692104	−	0.721798i	$-0.256684\pi$
$317$	24.6451	1.38421	0.692104	+	0.721798i	$0.256684\pi$
$318$	0	0
$319$	−2.06825	−0.115800
$320$	0	0
$321$	4.34403	0.242460
$322$	0	0
$323$	−1.38273	−0.0769374
$324$	0	0
$325$	−6.00528	−0.333113
$326$	0	0
$327$	2.65298	0.146710
$328$	0	0
$329$	3.30475	0.182197
$330$	0	0
$331$	25.7539	1.41556	0.707781	−	0.706431i	$-0.249696\pi$
$331$	25.7539	1.41556	0.707781	+	0.706431i	$0.249696\pi$
$332$	0	0
$333$	−28.5612	−1.56514
$334$	0	0
$335$	35.8626	1.95938
$336$	0	0
$337$	14.0916	0.767618	0.383809	−	0.923413i	$-0.374612\pi$
$337$	14.0916	0.767618	0.383809	+	0.923413i	$0.374612\pi$
$338$	0	0
$339$	4.26696	0.231749
$340$	0	0
$341$	−8.14340	−0.440990
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	10.5216	0.566466
$346$	0	0
$347$	−29.7026	−1.59452	−0.797259	−	0.603638i	$-0.793717\pi$
$347$	−29.7026	−1.59452	−0.797259	+	0.603638i	$0.793717\pi$
$348$	0	0
$349$	−3.32866	−0.178179	−0.0890896	−	0.996024i	$-0.528396\pi$
$349$	−3.32866	−0.178179	−0.0890896	+	0.996024i	$0.528396\pi$
$350$	0	0
$351$	−2.09794	−0.111980
$352$	0	0
$353$	−14.5088	−0.772223	−0.386111	−	0.922452i	$-0.626182\pi$
$353$	−14.5088	−0.772223	−0.386111	+	0.922452i	$0.626182\pi$
$354$	0	0
$355$	28.1632	1.49475
$356$	0	0
$357$	1.02797	0.0544060
$358$	0	0
$359$	30.6298	1.61658	0.808289	−	0.588786i	$-0.200394\pi$
$359$	30.6298	1.61658	0.808289	+	0.588786i	$0.200394\pi$
$360$	0	0
$361$	−18.7691	−0.987846
$362$	0	0
$363$	−0.357256	−0.0187511
$364$	0	0
$365$	52.9283	2.77039
$366$	0	0
$367$	−9.79391	−0.511238	−0.255619	−	0.966778i	$-0.582279\pi$
$367$	−9.79391	−0.511238	−0.255619	+	0.966778i	$0.582279\pi$
$368$	0	0
$369$	−19.9824	−1.04024
$370$	0	0
$371$	6.64805	0.345150
$372$	0	0
$373$	8.92267	0.461998	0.230999	−	0.972954i	$-0.425801\pi$
$373$	8.92267	0.461998	0.230999	+	0.972954i	$0.425801\pi$
$374$	0	0
$375$	−1.19143	−0.0615252
$376$	0	0
$377$	−2.06825	−0.106520
$378$	0	0
$379$	−25.5806	−1.31399	−0.656993	−	0.753897i	$-0.728172\pi$
$379$	−25.5806	−1.31399	−0.656993	+	0.753897i	$0.728172\pi$
$380$	0	0
$381$	1.49939	0.0768163
$382$	0	0
$383$	−21.9910	−1.12369	−0.561843	−	0.827244i	$-0.689907\pi$
$383$	−21.9910	−1.12369	−0.561843	+	0.827244i	$0.689907\pi$
$384$	0	0
$385$	3.31742	0.169071
$386$	0	0
$387$	14.6722	0.745830
$388$	0	0
$389$	4.27389	0.216695	0.108347	−	0.994113i	$-0.465444\pi$
$389$	4.27389	0.216695	0.108347	+	0.994113i	$0.465444\pi$
$390$	0	0
$391$	−25.5448	−1.29186
$392$	0	0
$393$	−6.54150	−0.329975
$394$	0	0
$395$	−5.38060	−0.270727
$396$	0	0
$397$	14.7076	0.738155	0.369077	−	0.929399i	$-0.379674\pi$
$397$	14.7076	0.738155	0.369077	+	0.929399i	$0.379674\pi$
$398$	0	0
$399$	−0.171679	−0.00859472
$400$	0	0
$401$	32.2978	1.61288	0.806438	−	0.591318i	$-0.201392\pi$
$401$	32.2978	1.61288	0.806438	+	0.591318i	$0.201392\pi$
$402$	0	0
$403$	−8.14340	−0.405652
$404$	0	0
$405$	26.1001	1.29693
$406$	0	0
$407$	−9.94343	−0.492878
$408$	0	0
$409$	27.1068	1.34035	0.670173	−	0.742205i	$-0.266220\pi$
$409$	27.1068	1.34035	0.670173	+	0.742205i	$0.266220\pi$
$410$	0	0
$411$	−3.14874	−0.155316
$412$	0	0
$413$	−8.21164	−0.404069
$414$	0	0
$415$	35.8970	1.76211
$416$	0	0
$417$	1.97321	0.0966283
$418$	0	0
$419$	−26.3010	−1.28489	−0.642443	−	0.766333i	$-0.722079\pi$
$419$	−26.3010	−1.28489	−0.642443	+	0.766333i	$0.722079\pi$
$420$	0	0
$421$	1.33347	0.0649892	0.0324946	−	0.999472i	$-0.489655\pi$
$421$	1.33347	0.0649892	0.0324946	+	0.999472i	$0.489655\pi$
$422$	0	0
$423$	9.49246	0.461539
$424$	0	0
$425$	17.2796	0.838184
$426$	0	0
$427$	−13.8367	−0.669603
$428$	0	0
$429$	−0.357256	−0.0172485
$430$	0	0
$431$	37.8787	1.82455	0.912277	−	0.409574i	$-0.134323\pi$
$431$	37.8787	1.82455	0.912277	+	0.409574i	$0.134323\pi$
$432$	0	0
$433$	8.39699	0.403534	0.201767	−	0.979434i	$-0.435332\pi$
$433$	8.39699	0.403534	0.201767	+	0.979434i	$0.435332\pi$
$434$	0	0
$435$	−2.45122	−0.117527
$436$	0	0
$437$	4.26620	0.204080
$438$	0	0
$439$	15.1229	0.721777	0.360889	−	0.932609i	$-0.382473\pi$
$439$	15.1229	0.721777	0.360889	+	0.932609i	$0.382473\pi$
$440$	0	0
$441$	−2.87237	−0.136779
$442$	0	0
$443$	−35.0430	−1.66494	−0.832471	−	0.554068i	$-0.813075\pi$
$443$	−35.0430	−1.66494	−0.832471	+	0.554068i	$0.813075\pi$
$444$	0	0
$445$	30.6457	1.45275
$446$	0	0
$447$	8.13537	0.384790
$448$	0	0
$449$	−3.97901	−0.187781	−0.0938906	−	0.995583i	$-0.529930\pi$
$449$	−3.97901	−0.187781	−0.0938906	+	0.995583i	$0.529930\pi$
$450$	0	0
$451$	−6.95677	−0.327581
$452$	0	0
$453$	−3.96940	−0.186499
$454$	0	0
$455$	3.31742	0.155523
$456$	0	0
$457$	−21.8963	−1.02426	−0.512132	−	0.858907i	$-0.671144\pi$
$457$	−21.8963	−1.02426	−0.512132	+	0.858907i	$0.671144\pi$
$458$	0	0
$459$	6.03662	0.281765
$460$	0	0
$461$	−21.6825	−1.00985	−0.504926	−	0.863162i	$-0.668480\pi$
$461$	−21.6825	−1.00985	−0.504926	+	0.863162i	$0.668480\pi$
$462$	0	0
$463$	−23.4072	−1.08782	−0.543912	−	0.839142i	$-0.683058\pi$
$463$	−23.4072	−1.08782	−0.543912	+	0.839142i	$0.683058\pi$
$464$	0	0
$465$	−9.65131	−0.447569
$466$	0	0
$467$	−2.99684	−0.138677	−0.0693385	−	0.997593i	$-0.522089\pi$
$467$	−2.99684	−0.138677	−0.0693385	+	0.997593i	$0.522089\pi$
$468$	0	0
$469$	−10.8104	−0.499177
$470$	0	0
$471$	6.39863	0.294833
$472$	0	0
$473$	5.10805	0.234868
$474$	0	0
$475$	−2.88584	−0.132411
$476$	0	0
$477$	19.0957	0.874330
$478$	0	0
$479$	38.2005	1.74542	0.872712	−	0.488235i	$-0.162359\pi$
$479$	38.2005	1.74542	0.872712	+	0.488235i	$0.162359\pi$
$480$	0	0
$481$	−9.94343	−0.453382
$482$	0	0
$483$	−3.17163	−0.144314
$484$	0	0
$485$	−49.1774	−2.23303
$486$	0	0
$487$	8.58420	0.388987	0.194494	−	0.980904i	$-0.437694\pi$
$487$	8.58420	0.388987	0.194494	+	0.980904i	$0.437694\pi$
$488$	0	0
$489$	1.38894	0.0628101
$490$	0	0
$491$	−13.6705	−0.616941	−0.308470	−	0.951234i	$-0.599817\pi$
$491$	−13.6705	−0.616941	−0.308470	+	0.951234i	$0.599817\pi$
$492$	0	0
$493$	5.95118	0.268028
$494$	0	0
$495$	9.52885	0.428290
$496$	0	0
$497$	−8.48947	−0.380805
$498$	0	0
$499$	2.10973	0.0944443	0.0472222	−	0.998884i	$-0.484963\pi$
$499$	2.10973	0.0944443	0.0472222	+	0.998884i	$0.484963\pi$
$500$	0	0
$501$	−3.26191	−0.145731
$502$	0	0
$503$	−1.80469	−0.0804671	−0.0402336	−	0.999190i	$-0.512810\pi$
$503$	−1.80469	−0.0804671	−0.0402336	+	0.999190i	$0.512810\pi$
$504$	0	0
$505$	22.6073	1.00601
$506$	0	0
$507$	−0.357256	−0.0158663
$508$	0	0
$509$	3.18204	0.141042	0.0705208	−	0.997510i	$-0.477534\pi$
$509$	3.18204	0.141042	0.0705208	+	0.997510i	$0.477534\pi$
$510$	0	0
$511$	−15.9546	−0.705792
$512$	0	0
$513$	−1.00816	−0.0445116
$514$	0	0
$515$	18.4752	0.814115
$516$	0	0
$517$	3.30475	0.145343
$518$	0	0
$519$	−7.92838	−0.348017
$520$	0	0
$521$	−13.1784	−0.577355	−0.288677	−	0.957426i	$-0.593215\pi$
$521$	−13.1784	−0.577355	−0.288677	+	0.957426i	$0.593215\pi$
$522$	0	0
$523$	11.9880	0.524199	0.262099	−	0.965041i	$-0.415585\pi$
$523$	11.9880	0.524199	0.262099	+	0.965041i	$0.415585\pi$
$524$	0	0
$525$	2.14543	0.0936341
$526$	0	0
$527$	23.4318	1.02071
$528$	0	0
$529$	55.8143	2.42671
$530$	0	0
$531$	−23.5869	−1.02358
$532$	0	0
$533$	−6.95677	−0.301331
$534$	0	0
$535$	−40.3379	−1.74396
$536$	0	0
$537$	2.75355	0.118824
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−38.0987	−1.63799	−0.818995	−	0.573800i	$-0.805469\pi$
$541$	−38.0987	−1.63799	−0.818995	+	0.573800i	$0.805469\pi$
$542$	0	0
$543$	−3.00082	−0.128777
$544$	0	0
$545$	−24.6351	−1.05525
$546$	0	0
$547$	33.8679	1.44809	0.724044	−	0.689754i	$-0.242281\pi$
$547$	33.8679	1.44809	0.724044	+	0.689754i	$0.242281\pi$
$548$	0	0
$549$	−39.7440	−1.69623
$550$	0	0
$551$	−0.993895	−0.0423414
$552$	0	0
$553$	1.62192	0.0689712
$554$	0	0
$555$	−11.7847	−0.500231
$556$	0	0
$557$	20.1538	0.853943	0.426972	−	0.904265i	$-0.359580\pi$
$557$	20.1538	0.853943	0.426972	+	0.904265i	$0.359580\pi$
$558$	0	0
$559$	5.10805	0.216047
$560$	0	0
$561$	1.02797	0.0434009
$562$	0	0
$563$	10.0802	0.424828	0.212414	−	0.977180i	$-0.431867\pi$
$563$	10.0802	0.424828	0.212414	+	0.977180i	$0.431867\pi$
$564$	0	0
$565$	−39.6222	−1.66692
$566$	0	0
$567$	−7.86760	−0.330408
$568$	0	0
$569$	−14.3423	−0.601259	−0.300630	−	0.953741i	$-0.597197\pi$
$569$	−14.3423	−0.601259	−0.300630	+	0.953741i	$0.597197\pi$
$570$	0	0
$571$	28.3207	1.18518	0.592592	−	0.805502i	$-0.298104\pi$
$571$	28.3207	1.18518	0.592592	+	0.805502i	$0.298104\pi$
$572$	0	0
$573$	4.60166	0.192237
$574$	0	0
$575$	−53.3133	−2.22332
$576$	0	0
$577$	7.62104	0.317268	0.158634	−	0.987337i	$-0.449291\pi$
$577$	7.62104	0.317268	0.158634	+	0.987337i	$0.449291\pi$
$578$	0	0
$579$	3.53026	0.146713
$580$	0	0
$581$	−10.8207	−0.448920
$582$	0	0
$583$	6.64805	0.275334
$584$	0	0
$585$	9.52885	0.393970
$586$	0	0
$587$	−27.5017	−1.13511	−0.567557	−	0.823334i	$-0.692111\pi$
$587$	−27.5017	−1.13511	−0.567557	+	0.823334i	$0.692111\pi$
$588$	0	0
$589$	−3.91331	−0.161245
$590$	0	0
$591$	2.42414	0.0997157
$592$	0	0
$593$	34.7880	1.42857	0.714285	−	0.699855i	$-0.246752\pi$
$593$	34.7880	1.42857	0.714285	+	0.699855i	$0.246752\pi$
$594$	0	0
$595$	−9.54555	−0.391330
$596$	0	0
$597$	8.01069	0.327856
$598$	0	0
$599$	−35.6803	−1.45786	−0.728929	−	0.684590i	$-0.759981\pi$
$599$	−35.6803	−1.45786	−0.728929	+	0.684590i	$0.759981\pi$
$600$	0	0
$601$	−32.2088	−1.31383	−0.656913	−	0.753966i	$-0.728138\pi$
$601$	−32.2088	−1.31383	−0.656913	+	0.753966i	$0.728138\pi$
$602$	0	0
$603$	−31.0514	−1.26451
$604$	0	0
$605$	3.31742	0.134872
$606$	0	0
$607$	−22.3174	−0.905836	−0.452918	−	0.891552i	$-0.649617\pi$
$607$	−22.3174	−0.905836	−0.452918	+	0.891552i	$0.649617\pi$
$608$	0	0
$609$	0.738894	0.0299415
$610$	0	0
$611$	3.30475	0.133696
$612$	0	0
$613$	−12.6487	−0.510875	−0.255437	−	0.966826i	$-0.582219\pi$
$613$	−12.6487	−0.510875	−0.255437	+	0.966826i	$0.582219\pi$
$614$	0	0
$615$	−8.24495	−0.332468
$616$	0	0
$617$	18.7287	0.753989	0.376994	−	0.926216i	$-0.376958\pi$
$617$	18.7287	0.753989	0.376994	+	0.926216i	$0.376958\pi$
$618$	0	0
$619$	19.6059	0.788028	0.394014	−	0.919104i	$-0.371086\pi$
$619$	19.6059	0.788028	0.394014	+	0.919104i	$0.371086\pi$
$620$	0	0
$621$	−18.6250	−0.747395
$622$	0	0
$623$	−9.23781	−0.370105
$624$	0	0
$625$	−18.9630	−0.758520
$626$	0	0
$627$	−0.171679	−0.00685622
$628$	0	0
$629$	28.6113	1.14081
$630$	0	0
$631$	−34.6374	−1.37889	−0.689446	−	0.724337i	$-0.742146\pi$
$631$	−34.6374	−1.37889	−0.689446	+	0.724337i	$0.742146\pi$
$632$	0	0
$633$	−0.259503	−0.0103143
$634$	0	0
$635$	−13.9231	−0.552522
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−24.3849	−0.964651
$640$	0	0
$641$	−36.6600	−1.44798	−0.723991	−	0.689809i	$-0.757694\pi$
$641$	−36.6600	−1.44798	−0.723991	+	0.689809i	$0.757694\pi$
$642$	0	0
$643$	24.3192	0.959056	0.479528	−	0.877526i	$-0.340808\pi$
$643$	24.3192	0.959056	0.479528	+	0.877526i	$0.340808\pi$
$644$	0	0
$645$	6.05391	0.238372
$646$	0	0
$647$	2.91761	0.114703	0.0573515	−	0.998354i	$-0.481734\pi$
$647$	2.91761	0.114703	0.0573515	+	0.998354i	$0.481734\pi$
$648$	0	0
$649$	−8.21164	−0.322335
$650$	0	0
$651$	2.90928	0.114024
$652$	0	0
$653$	7.76774	0.303975	0.151988	−	0.988382i	$-0.451433\pi$
$653$	7.76774	0.303975	0.151988	+	0.988382i	$0.451433\pi$
$654$	0	0
$655$	60.7432	2.37343
$656$	0	0
$657$	−45.8276	−1.78790
$658$	0	0
$659$	18.9750	0.739161	0.369581	−	0.929199i	$-0.379501\pi$
$659$	18.9750	0.739161	0.369581	+	0.929199i	$0.379501\pi$
$660$	0	0
$661$	6.42026	0.249719	0.124860	−	0.992174i	$-0.460152\pi$
$661$	6.42026	0.249719	0.124860	+	0.992174i	$0.460152\pi$
$662$	0	0
$663$	1.02797	0.0399231
$664$	0	0
$665$	1.59419	0.0618199
$666$	0	0
$667$	−18.3614	−0.710955
$668$	0	0
$669$	−6.04393	−0.233672
$670$	0	0
$671$	−13.8367	−0.534158
$672$	0	0
$673$	35.6734	1.37511	0.687554	−	0.726133i	$-0.258684\pi$
$673$	35.6734	1.37511	0.687554	+	0.726133i	$0.258684\pi$
$674$	0	0
$675$	12.5987	0.484925
$676$	0	0
$677$	−6.68403	−0.256888	−0.128444	−	0.991717i	$-0.540998\pi$
$677$	−6.68403	−0.256888	−0.128444	+	0.991717i	$0.540998\pi$
$678$	0	0
$679$	14.8240	0.568892
$680$	0	0
$681$	1.22655	0.0470014
$682$	0	0
$683$	−39.3534	−1.50581	−0.752907	−	0.658127i	$-0.771349\pi$
$683$	−39.3534	−1.50581	−0.752907	+	0.658127i	$0.771349\pi$
$684$	0	0
$685$	29.2387	1.11715
$686$	0	0
$687$	−7.76830	−0.296379
$688$	0	0
$689$	6.64805	0.253271
$690$	0	0
$691$	−28.0281	−1.06624	−0.533119	−	0.846040i	$-0.678980\pi$
$691$	−28.0281	−1.06624	−0.533119	+	0.846040i	$0.678980\pi$
$692$	0	0
$693$	−2.87237	−0.109112
$694$	0	0
$695$	−18.3228	−0.695025
$696$	0	0
$697$	20.0174	0.758214
$698$	0	0
$699$	−7.42995	−0.281026
$700$	0	0
$701$	31.9693	1.20746	0.603732	−	0.797187i	$-0.293680\pi$
$701$	31.9693	1.20746	0.603732	+	0.797187i	$0.293680\pi$
$702$	0	0
$703$	−4.77831	−0.180217
$704$	0	0
$705$	3.91669	0.147511
$706$	0	0
$707$	−6.81473	−0.256294
$708$	0	0
$709$	14.2914	0.536726	0.268363	−	0.963318i	$-0.413517\pi$
$709$	14.2914	0.536726	0.268363	+	0.963318i	$0.413517\pi$
$710$	0	0
$711$	4.65876	0.174717
$712$	0	0
$713$	−72.2950	−2.70747
$714$	0	0
$715$	3.31742	0.124065
$716$	0	0
$717$	5.56994	0.208013
$718$	0	0
$719$	25.5202	0.951744	0.475872	−	0.879515i	$-0.342133\pi$
$719$	25.5202	0.951744	0.475872	+	0.879515i	$0.342133\pi$
$720$	0	0
$721$	−5.56915	−0.207406
$722$	0	0
$723$	4.73287	0.176017
$724$	0	0
$725$	12.4204	0.461282
$726$	0	0
$727$	27.6213	1.02442	0.512208	−	0.858862i	$-0.328828\pi$
$727$	27.6213	1.02442	0.512208	+	0.858862i	$0.328828\pi$
$728$	0	0
$729$	−20.3501	−0.753709
$730$	0	0
$731$	−14.6979	−0.543622
$732$	0	0
$733$	2.95299	0.109071	0.0545356	−	0.998512i	$-0.482632\pi$
$733$	2.95299	0.109071	0.0545356	+	0.998512i	$0.482632\pi$
$734$	0	0
$735$	−1.18517	−0.0437157
$736$	0	0
$737$	−10.8104	−0.398205
$738$	0	0
$739$	14.7567	0.542835	0.271417	−	0.962462i	$-0.412508\pi$
$739$	14.7567	0.542835	0.271417	+	0.962462i	$0.412508\pi$
$740$	0	0
$741$	−0.171679	−0.00630680
$742$	0	0
$743$	−46.2320	−1.69609	−0.848044	−	0.529926i	$-0.822220\pi$
$743$	−46.2320	−1.69609	−0.848044	+	0.529926i	$0.822220\pi$
$744$	0	0
$745$	−75.5437	−2.76771
$746$	0	0
$747$	−31.0812	−1.13720
$748$	0	0
$749$	12.1594	0.444295
$750$	0	0
$751$	8.10505	0.295757	0.147879	−	0.989006i	$-0.452755\pi$
$751$	8.10505	0.295757	0.147879	+	0.989006i	$0.452755\pi$
$752$	0	0
$753$	8.57624	0.312536
$754$	0	0
$755$	36.8592	1.34144
$756$	0	0
$757$	−2.55960	−0.0930303	−0.0465151	−	0.998918i	$-0.514812\pi$
$757$	−2.55960	−0.0930303	−0.0465151	+	0.998918i	$0.514812\pi$
$758$	0	0
$759$	−3.17163	−0.115123
$760$	0	0
$761$	3.44077	0.124728	0.0623639	−	0.998053i	$-0.480136\pi$
$761$	3.44077	0.124728	0.0623639	+	0.998053i	$0.480136\pi$
$762$	0	0
$763$	7.42599	0.268839
$764$	0	0
$765$	−27.4183	−0.991312
$766$	0	0
$767$	−8.21164	−0.296505
$768$	0	0
$769$	−33.0363	−1.19132	−0.595659	−	0.803238i	$-0.703109\pi$
$769$	−33.0363	−1.19132	−0.595659	+	0.803238i	$0.703109\pi$
$770$	0	0
$771$	−8.35172	−0.300780
$772$	0	0
$773$	−41.4235	−1.48990	−0.744949	−	0.667121i	$-0.767526\pi$
$773$	−41.4235	−1.48990	−0.744949	+	0.667121i	$0.767526\pi$
$774$	0	0
$775$	48.9034	1.75666
$776$	0	0
$777$	3.55236	0.127440
$778$	0	0
$779$	−3.34307	−0.119778
$780$	0	0
$781$	−8.48947	−0.303777
$782$	0	0
$783$	4.33906	0.155065
$784$	0	0
$785$	−59.4165	−2.12067
$786$	0	0
$787$	39.0531	1.39209	0.696047	−	0.717997i	$-0.254941\pi$
$787$	39.0531	1.39209	0.696047	+	0.717997i	$0.254941\pi$
$788$	0	0
$789$	3.16931	0.112830
$790$	0	0
$791$	11.9437	0.424668
$792$	0	0
$793$	−13.8367	−0.491354
$794$	0	0
$795$	7.87907	0.279442
$796$	0	0
$797$	−44.3846	−1.57218	−0.786091	−	0.618111i	$-0.787898\pi$
$797$	−44.3846	−1.57218	−0.786091	+	0.618111i	$0.787898\pi$
$798$	0	0
$799$	−9.50910	−0.336408
$800$	0	0
$801$	−26.5344	−0.937547
$802$	0	0
$803$	−15.9546	−0.563027
$804$	0	0
$805$	29.4512	1.03802
$806$	0	0
$807$	1.28494	0.0452319
$808$	0	0
$809$	−42.4134	−1.49117	−0.745587	−	0.666408i	$-0.767831\pi$
$809$	−42.4134	−1.49117	−0.745587	+	0.666408i	$0.767831\pi$
$810$	0	0
$811$	21.4979	0.754893	0.377446	−	0.926031i	$-0.376802\pi$
$811$	21.4979	0.754893	0.377446	+	0.926031i	$0.376802\pi$
$812$	0	0
$813$	−9.02577	−0.316548
$814$	0	0
$815$	−12.8975	−0.451779
$816$	0	0
$817$	2.45467	0.0858781
$818$	0	0
$819$	−2.87237	−0.100369
$820$	0	0
$821$	−20.9958	−0.732759	−0.366380	−	0.930465i	$-0.619403\pi$
$821$	−20.9958	−0.732759	−0.366380	+	0.930465i	$0.619403\pi$
$822$	0	0
$823$	38.0899	1.32773	0.663866	−	0.747852i	$-0.268915\pi$
$823$	38.0899	1.32773	0.663866	+	0.747852i	$0.268915\pi$
$824$	0	0
$825$	2.14543	0.0746941
$826$	0	0
$827$	2.36403	0.0822055	0.0411027	−	0.999155i	$-0.486913\pi$
$827$	2.36403	0.0822055	0.0411027	+	0.999155i	$0.486913\pi$
$828$	0	0
$829$	−40.0774	−1.39195	−0.695973	−	0.718068i	$-0.745027\pi$
$829$	−40.0774	−1.39195	−0.695973	+	0.718068i	$0.745027\pi$
$830$	0	0
$831$	6.99016	0.242486
$832$	0	0
$833$	2.87740	0.0996961
$834$	0	0
$835$	30.2895	1.04821
$836$	0	0
$837$	17.0844	0.590522
$838$	0	0
$839$	27.4305	0.947008	0.473504	−	0.880792i	$-0.342989\pi$
$839$	27.4305	0.947008	0.473504	+	0.880792i	$0.342989\pi$
$840$	0	0
$841$	−24.7224	−0.852495
$842$	0	0
$843$	−6.81190	−0.234614
$844$	0	0
$845$	3.31742	0.114123
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	3.89015	0.133510
$850$	0	0
$851$	−88.2752	−3.02604
$852$	0	0
$853$	−32.7588	−1.12164	−0.560820	−	0.827937i	$-0.689514\pi$
$853$	−32.7588	−1.12164	−0.560820	+	0.827937i	$0.689514\pi$
$854$	0	0
$855$	4.57909	0.156601
$856$	0	0
$857$	−14.4748	−0.494451	−0.247225	−	0.968958i	$-0.579519\pi$
$857$	−14.4748	−0.494451	−0.247225	+	0.968958i	$0.579519\pi$
$858$	0	0
$859$	−34.9700	−1.19316	−0.596580	−	0.802554i	$-0.703474\pi$
$859$	−34.9700	−1.19316	−0.596580	+	0.802554i	$0.703474\pi$
$860$	0	0
$861$	2.48535	0.0847005
$862$	0	0
$863$	3.62392	0.123360	0.0616798	−	0.998096i	$-0.480354\pi$
$863$	3.62392	0.123360	0.0616798	+	0.998096i	$0.480354\pi$
$864$	0	0
$865$	73.6216	2.50321
$866$	0	0
$867$	3.11547	0.105807
$868$	0	0
$869$	1.62192	0.0550200
$870$	0	0
$871$	−10.8104	−0.366296
$872$	0	0
$873$	42.5799	1.44111
$874$	0	0
$875$	−3.33495	−0.112742
$876$	0	0
$877$	39.0327	1.31804	0.659020	−	0.752125i	$-0.270971\pi$
$877$	39.0327	1.31804	0.659020	+	0.752125i	$0.270971\pi$
$878$	0	0
$879$	−7.36558	−0.248435
$880$	0	0
$881$	−42.9275	−1.44627	−0.723133	−	0.690709i	$-0.757299\pi$
$881$	−42.9275	−1.44627	−0.723133	+	0.690709i	$0.757299\pi$
$882$	0	0
$883$	−49.6472	−1.67076	−0.835380	−	0.549673i	$-0.814752\pi$
$883$	−49.6472	−1.67076	−0.835380	+	0.549673i	$0.814752\pi$
$884$	0	0
$885$	−9.73219	−0.327144
$886$	0	0
$887$	−8.28643	−0.278231	−0.139116	−	0.990276i	$-0.544426\pi$
$887$	−8.28643	−0.278231	−0.139116	+	0.990276i	$0.544426\pi$
$888$	0	0
$889$	4.19697	0.140762
$890$	0	0
$891$	−7.86760	−0.263575
$892$	0	0
$893$	1.58810	0.0531436
$894$	0	0
$895$	−25.5690	−0.854677
$896$	0	0
$897$	−3.17163	−0.105898
$898$	0	0
$899$	16.8426	0.561731
$900$	0	0
$901$	−19.1291	−0.637283
$902$	0	0
$903$	−1.82488	−0.0607283
$904$	0	0
$905$	27.8651	0.926266
$906$	0	0
$907$	−23.4630	−0.779076	−0.389538	−	0.921010i	$-0.627365\pi$
$907$	−23.4630	−0.779076	−0.389538	+	0.921010i	$0.627365\pi$
$908$	0	0
$909$	−19.5744	−0.649242
$910$	0	0
$911$	11.8253	0.391790	0.195895	−	0.980625i	$-0.437239\pi$
$911$	11.8253	0.391790	0.195895	+	0.980625i	$0.437239\pi$
$912$	0	0
$913$	−10.8207	−0.358114
$914$	0	0
$915$	−16.3988	−0.542127
$916$	0	0
$917$	−18.3104	−0.604662
$918$	0	0
$919$	0.351322	0.0115890	0.00579452	−	0.999983i	$-0.498156\pi$
$919$	0.351322	0.0115890	0.00579452	+	0.999983i	$0.498156\pi$
$920$	0	0
$921$	−0.701712	−0.0231222
$922$	0	0
$923$	−8.48947	−0.279434
$924$	0	0
$925$	59.7131	1.96336
$926$	0	0
$927$	−15.9967	−0.525399
$928$	0	0
$929$	−47.8775	−1.57081	−0.785404	−	0.618983i	$-0.787545\pi$
$929$	−47.8775	−1.57081	−0.785404	+	0.618983i	$0.787545\pi$
$930$	0	0
$931$	−0.480550	−0.0157494
$932$	0	0
$933$	2.78009	0.0910160
$934$	0	0
$935$	−9.54555	−0.312173
$936$	0	0
$937$	−29.7007	−0.970281	−0.485141	−	0.874436i	$-0.661232\pi$
$937$	−29.7007	−0.970281	−0.485141	+	0.874436i	$0.661232\pi$
$938$	0	0
$939$	−7.23836	−0.236215
$940$	0	0
$941$	54.8926	1.78945	0.894724	−	0.446619i	$-0.147372\pi$
$941$	54.8926	1.78945	0.894724	+	0.446619i	$0.147372\pi$
$942$	0	0
$943$	−61.7604	−2.01119
$944$	0	0
$945$	−6.95975	−0.226401
$946$	0	0
$947$	0.691988	0.0224866	0.0112433	−	0.999937i	$-0.496421\pi$
$947$	0.691988	0.0224866	0.0112433	+	0.999937i	$0.496421\pi$
$948$	0	0
$949$	−15.9546	−0.517910
$950$	0	0
$951$	−8.80462	−0.285510
$952$	0	0
$953$	−11.3735	−0.368425	−0.184212	−	0.982886i	$-0.558973\pi$
$953$	−11.3735	−0.368425	−0.184212	+	0.982886i	$0.558973\pi$
$954$	0	0
$955$	−42.7302	−1.38272
$956$	0	0
$957$	0.738894	0.0238851
$958$	0	0
$959$	−8.81368	−0.284609
$960$	0	0
$961$	35.3149	1.13919
$962$	0	0
$963$	34.9263	1.12548
$964$	0	0
$965$	−32.7814	−1.05527
$966$	0	0
$967$	4.60866	0.148205	0.0741023	−	0.997251i	$-0.476391\pi$
$967$	4.60866	0.148205	0.0741023	+	0.997251i	$0.476391\pi$
$968$	0	0
$969$	0.493991	0.0158693
$970$	0	0
$971$	11.8954	0.381740	0.190870	−	0.981615i	$-0.438869\pi$
$971$	11.8954	0.381740	0.190870	+	0.981615i	$0.438869\pi$
$972$	0	0
$973$	5.52322	0.177066
$974$	0	0
$975$	2.14543	0.0687086
$976$	0	0
$977$	7.89304	0.252521	0.126260	−	0.991997i	$-0.459703\pi$
$977$	7.89304	0.252521	0.126260	+	0.991997i	$0.459703\pi$
$978$	0	0
$979$	−9.23781	−0.295242
$980$	0	0
$981$	21.3302	0.681020
$982$	0	0
$983$	−29.6468	−0.945585	−0.472792	−	0.881174i	$-0.656754\pi$
$983$	−29.6468	−0.945585	−0.472792	+	0.881174i	$0.656754\pi$
$984$	0	0
$985$	−22.5101	−0.717232
$986$	0	0
$987$	−1.18064	−0.0375803
$988$	0	0
$989$	45.3479	1.44198
$990$	0	0
$991$	56.9475	1.80900	0.904498	−	0.426477i	$-0.140246\pi$
$991$	56.9475	1.80900	0.904498	+	0.426477i	$0.140246\pi$
$992$	0	0
$993$	−9.20075	−0.291977
$994$	0	0
$995$	−74.3859	−2.35819
$996$	0	0
$997$	31.2545	0.989839	0.494920	−	0.868939i	$-0.335198\pi$
$997$	31.2545	0.989839	0.494920	+	0.868939i	$0.335198\pi$
$998$	0	0
$999$	20.8607	0.660004

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.j.1.5	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.j.1.5	✓	10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.a (trivial)

\(f(q)\)	\(=\)	\(q-0.357256 q^{3} +3.31742 q^{5} -1.00000 q^{7} -2.87237 q^{9} +O(q^{10})\)	\(q-0.357256 q^{3} +3.31742 q^{5} -1.00000 q^{7} -2.87237 q^{9} -1.00000 q^{11} -1.00000 q^{13} -1.18517 q^{15} +2.87740 q^{17} -0.480550 q^{19} +0.357256 q^{21} -8.87774 q^{23} +6.00528 q^{25} +2.09794 q^{27} +2.06825 q^{29} +8.14340 q^{31} +0.357256 q^{33} -3.31742 q^{35} +9.94343 q^{37} +0.357256 q^{39} +6.95677 q^{41} -5.10805 q^{43} -9.52885 q^{45} -3.30475 q^{47} +1.00000 q^{49} -1.02797 q^{51} -6.64805 q^{53} -3.31742 q^{55} +0.171679 q^{57} +8.21164 q^{59} +13.8367 q^{61} +2.87237 q^{63} -3.31742 q^{65} +10.8104 q^{67} +3.17163 q^{69} +8.48947 q^{71} +15.9546 q^{73} -2.14543 q^{75} +1.00000 q^{77} -1.62192 q^{79} +7.86760 q^{81} +10.8207 q^{83} +9.54555 q^{85} -0.738894 q^{87} +9.23781 q^{89} +1.00000 q^{91} -2.90928 q^{93} -1.59419 q^{95} -14.8240 q^{97} +2.87237 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 2 * q^3 - 2 * q^5 - 10 * q^7 + 10 * q^9	\( 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} + 9 q^{17} + 3 q^{19} + 2 q^{21} - 4 q^{23} + 22 q^{25} - 8 q^{27} + 13 q^{29} - 17 q^{31} + 2 q^{33} + 2 q^{35} + 11 q^{37} + 2 q^{39} + 8 q^{41} + 21 q^{43} - 15 q^{45} + q^{47} + 10 q^{49} + 19 q^{51} + 22 q^{53} + 2 q^{55} + 8 q^{57} - 24 q^{59} + 16 q^{61} - 10 q^{63} + 2 q^{65} + 19 q^{67} + 51 q^{69} - 25 q^{71} + 22 q^{73} + 28 q^{75} + 10 q^{77} + 41 q^{79} + 54 q^{81} + 8 q^{83} + 9 q^{85} - 5 q^{87} + 36 q^{89} + 10 q^{91} + 12 q^{93} + 13 q^{95} - 5 q^{97} - 10 q^{99}+O(q^{100}) \) 10 * q - 2 * q^3 - 2 * q^5 - 10 * q^7 + 10 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 + 9 * q^17 + 3 * q^19 + 2 * q^21 - 4 * q^23 + 22 * q^25 - 8 * q^27 + 13 * q^29 - 17 * q^31 + 2 * q^33 + 2 * q^35 + 11 * q^37 + 2 * q^39 + 8 * q^41 + 21 * q^43 - 15 * q^45 + q^47 + 10 * q^49 + 19 * q^51 + 22 * q^53 + 2 * q^55 + 8 * q^57 - 24 * q^59 + 16 * q^61 - 10 * q^63 + 2 * q^65 + 19 * q^67 + 51 * q^69 - 25 * q^71 + 22 * q^73 + 28 * q^75 + 10 * q^77 + 41 * q^79 + 54 * q^81 + 8 * q^83 + 9 * q^85 - 5 * q^87 + 36 * q^89 + 10 * q^91 + 12 * q^93 + 13 * q^95 - 5 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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