LMFDB - Embedded newform 4004.2.a.j.1.6

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			1.6
Root			$-0.0962745$ 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)

$f(q)$	$=$	$q+0.0962745 q^{3} -1.48698 q^{5} -1.00000 q^{7} -2.99073 q^{9} +O(q^{10})$	$q+0.0962745 q^{3} -1.48698 q^{5} -1.00000 q^{7} -2.99073 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.143158 q^{15} -7.53478 q^{17} +7.59969 q^{19} -0.0962745 q^{21} -3.71789 q^{23} -2.78889 q^{25} -0.576755 q^{27} -9.29580 q^{29} +6.27772 q^{31} -0.0962745 q^{33} +1.48698 q^{35} +0.976345 q^{37} -0.0962745 q^{39} +4.24087 q^{41} +11.8980 q^{43} +4.44716 q^{45} +12.9010 q^{47} +1.00000 q^{49} -0.725407 q^{51} -4.71919 q^{53} +1.48698 q^{55} +0.731656 q^{57} -5.01808 q^{59} -3.65793 q^{61} +2.99073 q^{63} +1.48698 q^{65} +5.47832 q^{67} -0.357938 q^{69} -9.99423 q^{71} +8.02335 q^{73} -0.268499 q^{75} +1.00000 q^{77} +2.16743 q^{79} +8.91667 q^{81} -12.0745 q^{83} +11.2041 q^{85} -0.894948 q^{87} +12.1364 q^{89} +1.00000 q^{91} +0.604385 q^{93} -11.3006 q^{95} -9.20899 q^{97} +2.99073 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.0962745	0.0555841	0.0277920	−	0.999614i	$-0.491152\pi$
$3$	0.0962745	0.0555841	0.0277920	+	0.999614i	$0.491152\pi$
$4$	0	0
$5$	−1.48698	−0.664998	−0.332499	−	0.943104i	$-0.607892\pi$
$5$	−1.48698	−0.664998	−0.332499	+	0.943104i	$0.607892\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.99073	−0.996910
$10$	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
$15$	−0.143158	−0.0369633
$16$	0	0
$17$	−7.53478	−1.82745	−0.913726	−	0.406331i	$-0.866808\pi$
$17$	−7.53478	−1.82745	−0.913726	+	0.406331i	$0.866808\pi$
$18$	0	0
$19$	7.59969	1.74349	0.871744	−	0.489961i	$-0.162989\pi$
$19$	7.59969	1.74349	0.871744	+	0.489961i	$0.162989\pi$
$20$	0	0
$21$	−0.0962745	−0.0210088
$22$	0	0
$23$	−3.71789	−0.775233	−0.387616	−	0.921821i	$-0.626701\pi$
$23$	−3.71789	−0.775233	−0.387616	+	0.921821i	$0.626701\pi$
$24$	0	0
$25$	−2.78889	−0.557778
$26$	0	0
$27$	−0.576755	−0.110996
$28$	0	0
$29$	−9.29580	−1.72619	−0.863093	−	0.505044i	$-0.831476\pi$
$29$	−9.29580	−1.72619	−0.863093	+	0.505044i	$0.831476\pi$
$30$	0	0
$31$	6.27772	1.12751	0.563756	−	0.825941i	$-0.309356\pi$
$31$	6.27772	1.12751	0.563756	+	0.825941i	$0.309356\pi$
$32$	0	0
$33$	−0.0962745	−0.0167592
$34$	0	0
$35$	1.48698	0.251345
$36$	0	0
$37$	0.976345	0.160510	0.0802551	−	0.996774i	$-0.474427\pi$
$37$	0.976345	0.160510	0.0802551	+	0.996774i	$0.474427\pi$
$38$	0	0
$39$	−0.0962745	−0.0154163
$40$	0	0
$41$	4.24087	0.662312	0.331156	−	0.943576i	$-0.392561\pi$
$41$	4.24087	0.662312	0.331156	+	0.943576i	$0.392561\pi$
$42$	0	0
$43$	11.8980	1.81443	0.907217	−	0.420663i	$-0.138202\pi$
$43$	11.8980	1.81443	0.907217	+	0.420663i	$0.138202\pi$
$44$	0	0
$45$	4.44716	0.662943
$46$	0	0
$47$	12.9010	1.88180	0.940901	−	0.338681i	$-0.109981\pi$
$47$	12.9010	1.88180	0.940901	+	0.338681i	$0.109981\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.725407	−0.101577
$52$	0	0
$53$	−4.71919	−0.648231	−0.324116	−	0.946017i	$-0.605067\pi$
$53$	−4.71919	−0.648231	−0.324116	+	0.946017i	$0.605067\pi$
$54$	0	0
$55$	1.48698	0.200504
$56$	0	0
$57$	0.731656	0.0969102
$58$	0	0
$59$	−5.01808	−0.653298	−0.326649	−	0.945146i	$-0.605919\pi$
$59$	−5.01808	−0.653298	−0.326649	+	0.945146i	$0.605919\pi$
$60$	0	0
$61$	−3.65793	−0.468350	−0.234175	−	0.972194i	$-0.575239\pi$
$61$	−3.65793	−0.468350	−0.234175	+	0.972194i	$0.575239\pi$
$62$	0	0
$63$	2.99073	0.376797
$64$	0	0
$65$	1.48698	0.184437
$66$	0	0
$67$	5.47832	0.669283	0.334641	−	0.942346i	$-0.391385\pi$
$67$	5.47832	0.669283	0.334641	+	0.942346i	$0.391385\pi$
$68$	0	0
$69$	−0.357938	−0.0430906
$70$	0	0
$71$	−9.99423	−1.18610	−0.593049	−	0.805167i	$-0.702076\pi$
$71$	−9.99423	−1.18610	−0.593049	+	0.805167i	$0.702076\pi$
$72$	0	0
$73$	8.02335	0.939062	0.469531	−	0.882916i	$-0.344423\pi$
$73$	8.02335	0.939062	0.469531	+	0.882916i	$0.344423\pi$
$74$	0	0
$75$	−0.268499	−0.0310036
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	2.16743	0.243855	0.121927	−	0.992539i	$-0.461093\pi$
$79$	2.16743	0.243855	0.121927	+	0.992539i	$0.461093\pi$
$80$	0	0
$81$	8.91667	0.990741
$82$	0	0
$83$	−12.0745	−1.32535	−0.662673	−	0.748909i	$-0.730578\pi$
$83$	−12.0745	−1.32535	−0.662673	+	0.748909i	$0.730578\pi$
$84$	0	0
$85$	11.2041	1.21525
$86$	0	0
$87$	−0.894948	−0.0959485
$88$	0	0
$89$	12.1364	1.28645	0.643226	−	0.765676i	$-0.277596\pi$
$89$	12.1364	1.28645	0.643226	+	0.765676i	$0.277596\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0.604385	0.0626718
$94$	0	0
$95$	−11.3006	−1.15942
$96$	0	0
$97$	−9.20899	−0.935031	−0.467515	−	0.883985i	$-0.654851\pi$
$97$	−9.20899	−0.935031	−0.467515	+	0.883985i	$0.654851\pi$
$98$	0	0
$99$	2.99073	0.300580
$100$	0	0
$101$	4.95809	0.493348	0.246674	−	0.969098i	$-0.420662\pi$
$101$	4.95809	0.493348	0.246674	+	0.969098i	$0.420662\pi$
$102$	0	0
$103$	−0.232419	−0.0229009	−0.0114504	−	0.999934i	$-0.503645\pi$
$103$	−0.232419	−0.0229009	−0.0114504	+	0.999934i	$0.503645\pi$
$104$	0	0
$105$	0.143158	0.0139708
$106$	0	0
$107$	2.52905	0.244492	0.122246	−	0.992500i	$-0.460990\pi$
$107$	2.52905	0.244492	0.122246	+	0.992500i	$0.460990\pi$
$108$	0	0
$109$	1.23676	0.118460	0.0592301	−	0.998244i	$-0.481135\pi$
$109$	1.23676	0.118460	0.0592301	+	0.998244i	$0.481135\pi$
$110$	0	0
$111$	0.0939972	0.00892181
$112$	0	0
$113$	−4.71296	−0.443358	−0.221679	−	0.975120i	$-0.571154\pi$
$113$	−4.71296	−0.443358	−0.221679	+	0.975120i	$0.571154\pi$
$114$	0	0
$115$	5.52842	0.515528
$116$	0	0
$117$	2.99073	0.276493
$118$	0	0
$119$	7.53478	0.690712
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.408287	0.0368140
$124$	0	0
$125$	11.5819	1.03592
$126$	0	0
$127$	20.6384	1.83137	0.915683	−	0.401902i	$-0.131651\pi$
$127$	20.6384	1.83137	0.915683	+	0.401902i	$0.131651\pi$
$128$	0	0
$129$	1.14548	0.100854
$130$	0	0
$131$	−1.43721	−0.125570	−0.0627850	−	0.998027i	$-0.519998\pi$
$131$	−1.43721	−0.125570	−0.0627850	+	0.998027i	$0.519998\pi$
$132$	0	0
$133$	−7.59969	−0.658977
$134$	0	0
$135$	0.857622	0.0738124
$136$	0	0
$137$	21.9061	1.87156	0.935782	−	0.352580i	$-0.114695\pi$
$137$	21.9061	1.87156	0.935782	+	0.352580i	$0.114695\pi$
$138$	0	0
$139$	−4.33207	−0.367441	−0.183721	−	0.982978i	$-0.558814\pi$
$139$	−4.33207	−0.367441	−0.183721	+	0.982978i	$0.558814\pi$
$140$	0	0
$141$	1.24204	0.104598
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	13.8227	1.14791
$146$	0	0
$147$	0.0962745	0.00794059
$148$	0	0
$149$	12.4447	1.01951	0.509755	−	0.860320i	$-0.329736\pi$
$149$	12.4447	1.01951	0.509755	+	0.860320i	$0.329736\pi$
$150$	0	0
$151$	20.1067	1.63626	0.818132	−	0.575031i	$-0.195010\pi$
$151$	20.1067	1.63626	0.818132	+	0.575031i	$0.195010\pi$
$152$	0	0
$153$	22.5345	1.82181
$154$	0	0
$155$	−9.33485	−0.749793
$156$	0	0
$157$	5.44888	0.434868	0.217434	−	0.976075i	$-0.430231\pi$
$157$	5.44888	0.434868	0.217434	+	0.976075i	$0.430231\pi$
$158$	0	0
$159$	−0.454338	−0.0360313
$160$	0	0
$161$	3.71789	0.293011
$162$	0	0
$163$	−3.33546	−0.261253	−0.130627	−	0.991432i	$-0.541699\pi$
$163$	−3.33546	−0.261253	−0.130627	+	0.991432i	$0.541699\pi$
$164$	0	0
$165$	0.143158	0.0111449
$166$	0	0
$167$	−17.6360	−1.36471	−0.682356	−	0.731020i	$-0.739045\pi$
$167$	−17.6360	−1.36471	−0.682356	+	0.731020i	$0.739045\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−22.7286	−1.73810
$172$	0	0
$173$	−17.9723	−1.36641	−0.683204	−	0.730227i	$-0.739414\pi$
$173$	−17.9723	−1.36641	−0.683204	+	0.730227i	$0.739414\pi$
$174$	0	0
$175$	2.78889	0.210820
$176$	0	0
$177$	−0.483113	−0.0363130
$178$	0	0
$179$	10.5233	0.786547	0.393273	−	0.919422i	$-0.371343\pi$
$179$	10.5233	0.786547	0.393273	+	0.919422i	$0.371343\pi$
$180$	0	0
$181$	−5.25277	−0.390436	−0.195218	−	0.980760i	$-0.562541\pi$
$181$	−5.25277	−0.390436	−0.195218	+	0.980760i	$0.562541\pi$
$182$	0	0
$183$	−0.352165	−0.0260328
$184$	0	0
$185$	−1.45181	−0.106739
$186$	0	0
$187$	7.53478	0.550997
$188$	0	0
$189$	0.576755	0.0419527
$190$	0	0
$191$	11.5412	0.835091	0.417545	−	0.908656i	$-0.362890\pi$
$191$	11.5412	0.835091	0.417545	+	0.908656i	$0.362890\pi$
$192$	0	0
$193$	0.170184	0.0122501	0.00612505	−	0.999981i	$-0.498050\pi$
$193$	0.170184	0.0122501	0.00612505	+	0.999981i	$0.498050\pi$
$194$	0	0
$195$	0.143158	0.0102518
$196$	0	0
$197$	−27.4451	−1.95538	−0.977691	−	0.210047i	$-0.932638\pi$
$197$	−27.4451	−1.95538	−0.977691	+	0.210047i	$0.932638\pi$
$198$	0	0
$199$	−13.2838	−0.941664	−0.470832	−	0.882223i	$-0.656046\pi$
$199$	−13.2838	−0.941664	−0.470832	+	0.882223i	$0.656046\pi$
$200$	0	0
$201$	0.527422	0.0372015
$202$	0	0
$203$	9.29580	0.652437
$204$	0	0
$205$	−6.30609	−0.440436
$206$	0	0
$207$	11.1192	0.772838
$208$	0	0
$209$	−7.59969	−0.525682
$210$	0	0
$211$	4.47275	0.307917	0.153958	−	0.988077i	$-0.450798\pi$
$211$	4.47275	0.307917	0.153958	+	0.988077i	$0.450798\pi$
$212$	0	0
$213$	−0.962190	−0.0659281
$214$	0	0
$215$	−17.6921	−1.20659
$216$	0	0
$217$	−6.27772	−0.426160
$218$	0	0
$219$	0.772444	0.0521969
$220$	0	0
$221$	7.53478	0.506844
$222$	0	0
$223$	−16.6613	−1.11573	−0.557863	−	0.829933i	$-0.688378\pi$
$223$	−16.6613	−1.11573	−0.557863	+	0.829933i	$0.688378\pi$
$224$	0	0
$225$	8.34082	0.556055
$226$	0	0
$227$	27.9568	1.85556	0.927779	−	0.373130i	$-0.121716\pi$
$227$	27.9568	1.85556	0.927779	+	0.373130i	$0.121716\pi$
$228$	0	0
$229$	9.91041	0.654898	0.327449	−	0.944869i	$-0.393811\pi$
$229$	9.91041	0.654898	0.327449	+	0.944869i	$0.393811\pi$
$230$	0	0
$231$	0.0962745	0.00633440
$232$	0	0
$233$	−13.0561	−0.855334	−0.427667	−	0.903936i	$-0.640664\pi$
$233$	−13.0561	−0.855334	−0.427667	+	0.903936i	$0.640664\pi$
$234$	0	0
$235$	−19.1835	−1.25139
$236$	0	0
$237$	0.208668	0.0135544
$238$	0	0
$239$	14.8509	0.960623	0.480311	−	0.877098i	$-0.340524\pi$
$239$	14.8509	0.960623	0.480311	+	0.877098i	$0.340524\pi$
$240$	0	0
$241$	5.60632	0.361135	0.180567	−	0.983563i	$-0.442207\pi$
$241$	5.60632	0.361135	0.180567	+	0.983563i	$0.442207\pi$
$242$	0	0
$243$	2.58871	0.166066
$244$	0	0
$245$	−1.48698	−0.0949997
$246$	0	0
$247$	−7.59969	−0.483557
$248$	0	0
$249$	−1.16246	−0.0736682
$250$	0	0
$251$	6.74971	0.426038	0.213019	−	0.977048i	$-0.431670\pi$
$251$	6.74971	0.426038	0.213019	+	0.977048i	$0.431670\pi$
$252$	0	0
$253$	3.71789	0.233742
$254$	0	0
$255$	1.07867	0.0675486
$256$	0	0
$257$	−15.1654	−0.945992	−0.472996	−	0.881065i	$-0.656828\pi$
$257$	−15.1654	−0.945992	−0.472996	+	0.881065i	$0.656828\pi$
$258$	0	0
$259$	−0.976345	−0.0606672
$260$	0	0
$261$	27.8012	1.72085
$262$	0	0
$263$	18.9930	1.17116	0.585578	−	0.810616i	$-0.300867\pi$
$263$	18.9930	1.17116	0.585578	+	0.810616i	$0.300867\pi$
$264$	0	0
$265$	7.01735	0.431072
$266$	0	0
$267$	1.16842	0.0715063
$268$	0	0
$269$	26.4136	1.61046	0.805231	−	0.592961i	$-0.202041\pi$
$269$	26.4136	1.61046	0.805231	+	0.592961i	$0.202041\pi$
$270$	0	0
$271$	−13.3654	−0.811888	−0.405944	−	0.913898i	$-0.633057\pi$
$271$	−13.3654	−0.811888	−0.405944	+	0.913898i	$0.633057\pi$
$272$	0	0
$273$	0.0962745	0.00582680
$274$	0	0
$275$	2.78889	0.168176
$276$	0	0
$277$	31.7062	1.90504	0.952521	−	0.304473i	$-0.0984802\pi$
$277$	31.7062	1.90504	0.952521	+	0.304473i	$0.0984802\pi$
$278$	0	0
$279$	−18.7750	−1.12403
$280$	0	0
$281$	−14.7863	−0.882074	−0.441037	−	0.897489i	$-0.645389\pi$
$281$	−14.7863	−0.882074	−0.441037	+	0.897489i	$0.645389\pi$
$282$	0	0
$283$	−27.6881	−1.64589	−0.822943	−	0.568124i	$-0.807669\pi$
$283$	−27.6881	−1.64589	−0.822943	+	0.568124i	$0.807669\pi$
$284$	0	0
$285$	−1.08796	−0.0644451
$286$	0	0
$287$	−4.24087	−0.250331
$288$	0	0
$289$	39.7729	2.33958
$290$	0	0
$291$	−0.886590	−0.0519728
$292$	0	0
$293$	2.36339	0.138071	0.0690355	−	0.997614i	$-0.478008\pi$
$293$	2.36339	0.138071	0.0690355	+	0.997614i	$0.478008\pi$
$294$	0	0
$295$	7.46178	0.434442
$296$	0	0
$297$	0.576755	0.0334667
$298$	0	0
$299$	3.71789	0.215011
$300$	0	0
$301$	−11.8980	−0.685792
$302$	0	0
$303$	0.477337	0.0274223
$304$	0	0
$305$	5.43927	0.311452
$306$	0	0
$307$	11.1239	0.634875	0.317438	−	0.948279i	$-0.397178\pi$
$307$	11.1239	0.634875	0.317438	+	0.948279i	$0.397178\pi$
$308$	0	0
$309$	−0.0223760	−0.00127293
$310$	0	0
$311$	7.39075	0.419091	0.209546	−	0.977799i	$-0.432802\pi$
$311$	7.39075	0.419091	0.209546	+	0.977799i	$0.432802\pi$
$312$	0	0
$313$	18.1279	1.02465	0.512325	−	0.858792i	$-0.328784\pi$
$313$	18.1279	1.02465	0.512325	+	0.858792i	$0.328784\pi$
$314$	0	0
$315$	−4.44716	−0.250569
$316$	0	0
$317$	11.9342	0.670293	0.335147	−	0.942166i	$-0.391214\pi$
$317$	11.9342	0.670293	0.335147	+	0.942166i	$0.391214\pi$
$318$	0	0
$319$	9.29580	0.520465
$320$	0	0
$321$	0.243483	0.0135899
$322$	0	0
$323$	−57.2620	−3.18614
$324$	0	0
$325$	2.78889	0.154700
$326$	0	0
$327$	0.119068	0.00658450
$328$	0	0
$329$	−12.9010	−0.711254
$330$	0	0
$331$	−27.7216	−1.52372	−0.761858	−	0.647744i	$-0.775713\pi$
$331$	−27.7216	−1.52372	−0.761858	+	0.647744i	$0.775713\pi$
$332$	0	0
$333$	−2.91999	−0.160014
$334$	0	0
$335$	−8.14614	−0.445071
$336$	0	0
$337$	−14.3079	−0.779400	−0.389700	−	0.920942i	$-0.627421\pi$
$337$	−14.3079	−0.779400	−0.389700	+	0.920942i	$0.627421\pi$
$338$	0	0
$339$	−0.453738	−0.0246437
$340$	0	0
$341$	−6.27772	−0.339958
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0.532246	0.0286552
$346$	0	0
$347$	19.2795	1.03498	0.517489	−	0.855690i	$-0.326867\pi$
$347$	19.2795	1.03498	0.517489	+	0.855690i	$0.326867\pi$
$348$	0	0
$349$	−9.55297	−0.511359	−0.255679	−	0.966762i	$-0.582299\pi$
$349$	−9.55297	−0.511359	−0.255679	+	0.966762i	$0.582299\pi$
$350$	0	0
$351$	0.576755	0.0307849
$352$	0	0
$353$	13.4085	0.713664	0.356832	−	0.934169i	$-0.383857\pi$
$353$	13.4085	0.713664	0.356832	+	0.934169i	$0.383857\pi$
$354$	0	0
$355$	14.8612	0.788752
$356$	0	0
$357$	0.725407	0.0383926
$358$	0	0
$359$	30.0564	1.58632	0.793159	−	0.609014i	$-0.208435\pi$
$359$	30.0564	1.58632	0.793159	+	0.609014i	$0.208435\pi$
$360$	0	0
$361$	38.7553	2.03975
$362$	0	0
$363$	0.0962745	0.00505310
$364$	0	0
$365$	−11.9306	−0.624474
$366$	0	0
$367$	−1.13021	−0.0589963	−0.0294982	−	0.999565i	$-0.509391\pi$
$367$	−1.13021	−0.0589963	−0.0294982	+	0.999565i	$0.509391\pi$
$368$	0	0
$369$	−12.6833	−0.660266
$370$	0	0
$371$	4.71919	0.245008
$372$	0	0
$373$	−24.2307	−1.25462	−0.627309	−	0.778770i	$-0.715844\pi$
$373$	−24.2307	−1.25462	−0.627309	+	0.778770i	$0.715844\pi$
$374$	0	0
$375$	1.11504	0.0575806
$376$	0	0
$377$	9.29580	0.478758
$378$	0	0
$379$	−14.5563	−0.747705	−0.373852	−	0.927488i	$-0.621963\pi$
$379$	−14.5563	−0.747705	−0.373852	+	0.927488i	$0.621963\pi$
$380$	0	0
$381$	1.98696	0.101795
$382$	0	0
$383$	10.1806	0.520202	0.260101	−	0.965581i	$-0.416244\pi$
$383$	10.1806	0.520202	0.260101	+	0.965581i	$0.416244\pi$
$384$	0	0
$385$	−1.48698	−0.0757835
$386$	0	0
$387$	−35.5838	−1.80883
$388$	0	0
$389$	0.101396	0.00514100	0.00257050	−	0.999997i	$-0.499182\pi$
$389$	0.101396	0.00514100	0.00257050	+	0.999997i	$0.499182\pi$
$390$	0	0
$391$	28.0134	1.41670
$392$	0	0
$393$	−0.138367	−0.00697970
$394$	0	0
$395$	−3.22292	−0.162163
$396$	0	0
$397$	−32.8536	−1.64888	−0.824438	−	0.565953i	$-0.808508\pi$
$397$	−32.8536	−1.64888	−0.824438	+	0.565953i	$0.808508\pi$
$398$	0	0
$399$	−0.731656	−0.0366286
$400$	0	0
$401$	−22.2263	−1.10993	−0.554964	−	0.831874i	$-0.687268\pi$
$401$	−22.2263	−1.10993	−0.554964	+	0.831874i	$0.687268\pi$
$402$	0	0
$403$	−6.27772	−0.312716
$404$	0	0
$405$	−13.2589	−0.658840
$406$	0	0
$407$	−0.976345	−0.0483956
$408$	0	0
$409$	7.74445	0.382939	0.191469	−	0.981499i	$-0.438675\pi$
$409$	7.74445	0.382939	0.191469	+	0.981499i	$0.438675\pi$
$410$	0	0
$411$	2.10900	0.104029
$412$	0	0
$413$	5.01808	0.246923
$414$	0	0
$415$	17.9545	0.881352
$416$	0	0
$417$	−0.417068	−0.0204239
$418$	0	0
$419$	32.4417	1.58488	0.792440	−	0.609950i	$-0.208811\pi$
$419$	32.4417	1.58488	0.792440	+	0.609950i	$0.208811\pi$
$420$	0	0
$421$	−27.2537	−1.32827	−0.664133	−	0.747614i	$-0.731199\pi$
$421$	−27.2537	−1.32827	−0.664133	+	0.747614i	$0.731199\pi$
$422$	0	0
$423$	−38.5834	−1.87599
$424$	0	0
$425$	21.0137	1.01931
$426$	0	0
$427$	3.65793	0.177020
$428$	0	0
$429$	0.0962745	0.00464818
$430$	0	0
$431$	21.3245	1.02716	0.513582	−	0.858040i	$-0.328318\pi$
$431$	21.3245	1.02716	0.513582	+	0.858040i	$0.328318\pi$
$432$	0	0
$433$	−1.57252	−0.0755706	−0.0377853	−	0.999286i	$-0.512030\pi$
$433$	−1.57252	−0.0755706	−0.0377853	+	0.999286i	$0.512030\pi$
$434$	0	0
$435$	1.33077	0.0638055
$436$	0	0
$437$	−28.2548	−1.35161
$438$	0	0
$439$	−25.4666	−1.21546	−0.607728	−	0.794145i	$-0.707919\pi$
$439$	−25.4666	−1.21546	−0.607728	+	0.794145i	$0.707919\pi$
$440$	0	0
$441$	−2.99073	−0.142416
$442$	0	0
$443$	34.9443	1.66025	0.830126	−	0.557576i	$-0.188268\pi$
$443$	34.9443	1.66025	0.830126	+	0.557576i	$0.188268\pi$
$444$	0	0
$445$	−18.0465	−0.855488
$446$	0	0
$447$	1.19811	0.0566685
$448$	0	0
$449$	21.6772	1.02301	0.511505	−	0.859281i	$-0.329088\pi$
$449$	21.6772	1.02301	0.511505	+	0.859281i	$0.329088\pi$
$450$	0	0
$451$	−4.24087	−0.199695
$452$	0	0
$453$	1.93577	0.0909502
$454$	0	0
$455$	−1.48698	−0.0697107
$456$	0	0
$457$	−13.2417	−0.619418	−0.309709	−	0.950831i	$-0.600232\pi$
$457$	−13.2417	−0.619418	−0.309709	+	0.950831i	$0.600232\pi$
$458$	0	0
$459$	4.34572	0.202841
$460$	0	0
$461$	37.1588	1.73066	0.865328	−	0.501206i	$-0.167110\pi$
$461$	37.1588	1.73066	0.865328	+	0.501206i	$0.167110\pi$
$462$	0	0
$463$	−0.624236	−0.0290107	−0.0145054	−	0.999895i	$-0.504617\pi$
$463$	−0.624236	−0.0290107	−0.0145054	+	0.999895i	$0.504617\pi$
$464$	0	0
$465$	−0.898708	−0.0416766
$466$	0	0
$467$	−4.53216	−0.209723	−0.104862	−	0.994487i	$-0.533440\pi$
$467$	−4.53216	−0.209723	−0.104862	+	0.994487i	$0.533440\pi$
$468$	0	0
$469$	−5.47832	−0.252965
$470$	0	0
$471$	0.524588	0.0241717
$472$	0	0
$473$	−11.8980	−0.547072
$474$	0	0
$475$	−21.1947	−0.972480
$476$	0	0
$477$	14.1138	0.646228
$478$	0	0
$479$	−22.2480	−1.01654	−0.508269	−	0.861199i	$-0.669714\pi$
$479$	−22.2480	−1.01654	−0.508269	+	0.861199i	$0.669714\pi$
$480$	0	0
$481$	−0.976345	−0.0445175
$482$	0	0
$483$	0.357938	0.0162867
$484$	0	0
$485$	13.6936	0.621793
$486$	0	0
$487$	−9.46077	−0.428709	−0.214354	−	0.976756i	$-0.568765\pi$
$487$	−9.46077	−0.428709	−0.214354	+	0.976756i	$0.568765\pi$
$488$	0	0
$489$	−0.321119	−0.0145215
$490$	0	0
$491$	24.2146	1.09279	0.546394	−	0.837528i	$-0.316000\pi$
$491$	24.2146	1.09279	0.546394	+	0.837528i	$0.316000\pi$
$492$	0	0
$493$	70.0418	3.15452
$494$	0	0
$495$	−4.44716	−0.199885
$496$	0	0
$497$	9.99423	0.448303
$498$	0	0
$499$	31.3417	1.40305	0.701523	−	0.712646i	$-0.252504\pi$
$499$	31.3417	1.40305	0.701523	+	0.712646i	$0.252504\pi$
$500$	0	0
$501$	−1.69789	−0.0758563
$502$	0	0
$503$	−33.5662	−1.49664	−0.748322	−	0.663335i	$-0.769140\pi$
$503$	−33.5662	−1.49664	−0.748322	+	0.663335i	$0.769140\pi$
$504$	0	0
$505$	−7.37258	−0.328075
$506$	0	0
$507$	0.0962745	0.00427570
$508$	0	0
$509$	−15.5318	−0.688436	−0.344218	−	0.938890i	$-0.611856\pi$
$509$	−15.5318	−0.688436	−0.344218	+	0.938890i	$0.611856\pi$
$510$	0	0
$511$	−8.02335	−0.354932
$512$	0	0
$513$	−4.38316	−0.193521
$514$	0	0
$515$	0.345602	0.0152290
$516$	0	0
$517$	−12.9010	−0.567385
$518$	0	0
$519$	−1.73027	−0.0759506
$520$	0	0
$521$	−27.3683	−1.19903	−0.599514	−	0.800364i	$-0.704640\pi$
$521$	−27.3683	−1.19903	−0.599514	+	0.800364i	$0.704640\pi$
$522$	0	0
$523$	−7.82671	−0.342238	−0.171119	−	0.985250i	$-0.554738\pi$
$523$	−7.82671	−0.342238	−0.171119	+	0.985250i	$0.554738\pi$
$524$	0	0
$525$	0.268499	0.0117183
$526$	0	0
$527$	−47.3013	−2.06047
$528$	0	0
$529$	−9.17732	−0.399014
$530$	0	0
$531$	15.0077	0.651280
$532$	0	0
$533$	−4.24087	−0.183692
$534$	0	0
$535$	−3.76064	−0.162587
$536$	0	0
$537$	1.01312	0.0437195
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−6.75618	−0.290471	−0.145235	−	0.989397i	$-0.546394\pi$
$541$	−6.75618	−0.290471	−0.145235	+	0.989397i	$0.546394\pi$
$542$	0	0
$543$	−0.505708	−0.0217020
$544$	0	0
$545$	−1.83904	−0.0787757
$546$	0	0
$547$	−24.9014	−1.06471	−0.532354	−	0.846522i	$-0.678692\pi$
$547$	−24.9014	−1.06471	−0.532354	+	0.846522i	$0.678692\pi$
$548$	0	0
$549$	10.9399	0.466903
$550$	0	0
$551$	−70.6452	−3.00959
$552$	0	0
$553$	−2.16743	−0.0921684
$554$	0	0
$555$	−0.139772	−0.00593299
$556$	0	0
$557$	30.6165	1.29726	0.648632	−	0.761102i	$-0.275341\pi$
$557$	30.6165	1.29726	0.648632	+	0.761102i	$0.275341\pi$
$558$	0	0
$559$	−11.8980	−0.503233
$560$	0	0
$561$	0.725407	0.0306267
$562$	0	0
$563$	3.43712	0.144857	0.0724286	−	0.997374i	$-0.476925\pi$
$563$	3.43712	0.144857	0.0724286	+	0.997374i	$0.476925\pi$
$564$	0	0
$565$	7.00808	0.294832
$566$	0	0
$567$	−8.91667	−0.374465
$568$	0	0
$569$	−7.77965	−0.326140	−0.163070	−	0.986615i	$-0.552140\pi$
$569$	−7.77965	−0.326140	−0.163070	+	0.986615i	$0.552140\pi$
$570$	0	0
$571$	44.9038	1.87916	0.939582	−	0.342324i	$-0.111214\pi$
$571$	44.9038	1.87916	0.939582	+	0.342324i	$0.111214\pi$
$572$	0	0
$573$	1.11112	0.0464178
$574$	0	0
$575$	10.3688	0.432408
$576$	0	0
$577$	4.01254	0.167044	0.0835221	−	0.996506i	$-0.473383\pi$
$577$	4.01254	0.167044	0.0835221	+	0.996506i	$0.473383\pi$
$578$	0	0
$579$	0.0163844	0.000680911 0
$580$	0	0
$581$	12.0745	0.500934
$582$	0	0
$583$	4.71919	0.195449
$584$	0	0
$585$	−4.44716	−0.183867
$586$	0	0
$587$	37.6248	1.55294	0.776471	−	0.630152i	$-0.217008\pi$
$587$	37.6248	1.55294	0.776471	+	0.630152i	$0.217008\pi$
$588$	0	0
$589$	47.7088	1.96581
$590$	0	0
$591$	−2.64226	−0.108688
$592$	0	0
$593$	35.3853	1.45310	0.726551	−	0.687113i	$-0.241122\pi$
$593$	35.3853	1.45310	0.726551	+	0.687113i	$0.241122\pi$
$594$	0	0
$595$	−11.2041	−0.459322
$596$	0	0
$597$	−1.27889	−0.0523415
$598$	0	0
$599$	−19.7877	−0.808503	−0.404252	−	0.914648i	$-0.632468\pi$
$599$	−19.7877	−0.808503	−0.404252	+	0.914648i	$0.632468\pi$
$600$	0	0
$601$	3.93876	0.160666	0.0803328	−	0.996768i	$-0.474402\pi$
$601$	3.93876	0.160666	0.0803328	+	0.996768i	$0.474402\pi$
$602$	0	0
$603$	−16.3842	−0.667215
$604$	0	0
$605$	−1.48698	−0.0604543
$606$	0	0
$607$	46.0332	1.86843	0.934215	−	0.356711i	$-0.116102\pi$
$607$	46.0332	1.86843	0.934215	+	0.356711i	$0.116102\pi$
$608$	0	0
$609$	0.894948	0.0362651
$610$	0	0
$611$	−12.9010	−0.521918
$612$	0	0
$613$	−4.74471	−0.191637	−0.0958186	−	0.995399i	$-0.530547\pi$
$613$	−4.74471	−0.191637	−0.0958186	+	0.995399i	$0.530547\pi$
$614$	0	0
$615$	−0.607115	−0.0244812
$616$	0	0
$617$	−12.5014	−0.503289	−0.251645	−	0.967820i	$-0.580971\pi$
$617$	−12.5014	−0.503289	−0.251645	+	0.967820i	$0.580971\pi$
$618$	0	0
$619$	1.53170	0.0615644	0.0307822	−	0.999526i	$-0.490200\pi$
$619$	1.53170	0.0615644	0.0307822	+	0.999526i	$0.490200\pi$
$620$	0	0
$621$	2.14431	0.0860481
$622$	0	0
$623$	−12.1364	−0.486233
$624$	0	0
$625$	−3.27763	−0.131105
$626$	0	0
$627$	−0.731656	−0.0292195
$628$	0	0
$629$	−7.35655	−0.293325
$630$	0	0
$631$	−17.9627	−0.715083	−0.357542	−	0.933897i	$-0.616385\pi$
$631$	−17.9627	−0.715083	−0.357542	+	0.933897i	$0.616385\pi$
$632$	0	0
$633$	0.430612	0.0171153
$634$	0	0
$635$	−30.6890	−1.21785
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	29.8901	1.18243
$640$	0	0
$641$	−22.6689	−0.895367	−0.447683	−	0.894192i	$-0.647751\pi$
$641$	−22.6689	−0.895367	−0.447683	+	0.894192i	$0.647751\pi$
$642$	0	0
$643$	−9.22386	−0.363753	−0.181877	−	0.983321i	$-0.558217\pi$
$643$	−9.22386	−0.363753	−0.181877	+	0.983321i	$0.558217\pi$
$644$	0	0
$645$	−1.70330	−0.0670675
$646$	0	0
$647$	−32.0275	−1.25913	−0.629566	−	0.776947i	$-0.716767\pi$
$647$	−32.0275	−1.25913	−0.629566	+	0.776947i	$0.716767\pi$
$648$	0	0
$649$	5.01808	0.196977
$650$	0	0
$651$	−0.604385	−0.0236877
$652$	0	0
$653$	−22.9334	−0.897452	−0.448726	−	0.893669i	$-0.648122\pi$
$653$	−22.9334	−0.897452	−0.448726	+	0.893669i	$0.648122\pi$
$654$	0	0
$655$	2.13711	0.0835037
$656$	0	0
$657$	−23.9957	−0.936161
$658$	0	0
$659$	27.1761	1.05863	0.529316	−	0.848425i	$-0.322449\pi$
$659$	27.1761	1.05863	0.529316	+	0.848425i	$0.322449\pi$
$660$	0	0
$661$	30.0555	1.16902	0.584512	−	0.811385i	$-0.301286\pi$
$661$	30.0555	1.16902	0.584512	+	0.811385i	$0.301286\pi$
$662$	0	0
$663$	0.725407	0.0281725
$664$	0	0
$665$	11.3006	0.438218
$666$	0	0
$667$	34.5607	1.33820
$668$	0	0
$669$	−1.60406	−0.0620166
$670$	0	0
$671$	3.65793	0.141213
$672$	0	0
$673$	12.1199	0.467186	0.233593	−	0.972334i	$-0.424952\pi$
$673$	12.1199	0.467186	0.233593	+	0.972334i	$0.424952\pi$
$674$	0	0
$675$	1.60851	0.0619114
$676$	0	0
$677$	2.61418	0.100471	0.0502356	−	0.998737i	$-0.484003\pi$
$677$	2.61418	0.100471	0.0502356	+	0.998737i	$0.484003\pi$
$678$	0	0
$679$	9.20899	0.353408
$680$	0	0
$681$	2.69153	0.103140
$682$	0	0
$683$	−34.1398	−1.30632	−0.653161	−	0.757219i	$-0.726558\pi$
$683$	−34.1398	−1.30632	−0.653161	+	0.757219i	$0.726558\pi$
$684$	0	0
$685$	−32.5739	−1.24459
$686$	0	0
$687$	0.954119	0.0364019
$688$	0	0
$689$	4.71919	0.179787
$690$	0	0
$691$	36.7273	1.39717	0.698586	−	0.715526i	$-0.253813\pi$
$691$	36.7273	1.39717	0.698586	+	0.715526i	$0.253813\pi$
$692$	0	0
$693$	−2.99073	−0.113608
$694$	0	0
$695$	6.44170	0.244348
$696$	0	0
$697$	−31.9540	−1.21034
$698$	0	0
$699$	−1.25697	−0.0475429
$700$	0	0
$701$	30.5869	1.15525	0.577626	−	0.816301i	$-0.303979\pi$
$701$	30.5869	1.15525	0.577626	+	0.816301i	$0.303979\pi$
$702$	0	0
$703$	7.41992	0.279848
$704$	0	0
$705$	−1.84688	−0.0695576
$706$	0	0
$707$	−4.95809	−0.186468
$708$	0	0
$709$	22.2292	0.834836	0.417418	−	0.908715i	$-0.362935\pi$
$709$	22.2292	0.834836	0.417418	+	0.908715i	$0.362935\pi$
$710$	0	0
$711$	−6.48219	−0.243101
$712$	0	0
$713$	−23.3399	−0.874085
$714$	0	0
$715$	−1.48698	−0.0556099
$716$	0	0
$717$	1.42976	0.0533953
$718$	0	0
$719$	−23.0582	−0.859925	−0.429963	−	0.902847i	$-0.641473\pi$
$719$	−23.0582	−0.859925	−0.429963	+	0.902847i	$0.641473\pi$
$720$	0	0
$721$	0.232419	0.00865572
$722$	0	0
$723$	0.539745	0.0200733
$724$	0	0
$725$	25.9250	0.962829
$726$	0	0
$727$	−37.1745	−1.37873	−0.689364	−	0.724415i	$-0.742110\pi$
$727$	−37.1745	−1.37873	−0.689364	+	0.724415i	$0.742110\pi$
$728$	0	0
$729$	−26.5008	−0.981510
$730$	0	0
$731$	−89.6491	−3.31579
$732$	0	0
$733$	22.7524	0.840380	0.420190	−	0.907436i	$-0.361963\pi$
$733$	22.7524	0.840380	0.420190	+	0.907436i	$0.361963\pi$
$734$	0	0
$735$	−0.143158	−0.00528047
$736$	0	0
$737$	−5.47832	−0.201796
$738$	0	0
$739$	−21.0686	−0.775020	−0.387510	−	0.921866i	$-0.626665\pi$
$739$	−21.0686	−0.775020	−0.387510	+	0.921866i	$0.626665\pi$
$740$	0	0
$741$	−0.731656	−0.0268781
$742$	0	0
$743$	−41.6490	−1.52795	−0.763976	−	0.645244i	$-0.776756\pi$
$743$	−41.6490	−1.52795	−0.763976	+	0.645244i	$0.776756\pi$
$744$	0	0
$745$	−18.5050	−0.677972
$746$	0	0
$747$	36.1115	1.32125
$748$	0	0
$749$	−2.52905	−0.0924093
$750$	0	0
$751$	−0.399650	−0.0145834	−0.00729171	−	0.999973i	$-0.502321\pi$
$751$	−0.399650	−0.0145834	−0.00729171	+	0.999973i	$0.502321\pi$
$752$	0	0
$753$	0.649825	0.0236809
$754$	0	0
$755$	−29.8983	−1.08811
$756$	0	0
$757$	32.1194	1.16740	0.583699	−	0.811970i	$-0.301605\pi$
$757$	32.1194	1.16740	0.583699	+	0.811970i	$0.301605\pi$
$758$	0	0
$759$	0.357938	0.0129923
$760$	0	0
$761$	−28.5139	−1.03363	−0.516815	−	0.856097i	$-0.672882\pi$
$761$	−28.5139	−1.03363	−0.516815	+	0.856097i	$0.672882\pi$
$762$	0	0
$763$	−1.23676	−0.0447737
$764$	0	0
$765$	−33.5083	−1.21150
$766$	0	0
$767$	5.01808	0.181192
$768$	0	0
$769$	0.500827	0.0180603	0.00903014	−	0.999959i	$-0.497126\pi$
$769$	0.500827	0.0180603	0.00903014	+	0.999959i	$0.497126\pi$
$770$	0	0
$771$	−1.46004	−0.0525821
$772$	0	0
$773$	−25.6976	−0.924277	−0.462138	−	0.886808i	$-0.652918\pi$
$773$	−25.6976	−0.924277	−0.462138	+	0.886808i	$0.652918\pi$
$774$	0	0
$775$	−17.5079	−0.628902
$776$	0	0
$777$	−0.0939972	−0.00337213
$778$	0	0
$779$	32.2293	1.15473
$780$	0	0
$781$	9.99423	0.357622
$782$	0	0
$783$	5.36140	0.191601
$784$	0	0
$785$	−8.10237	−0.289186
$786$	0	0
$787$	−26.7939	−0.955100	−0.477550	−	0.878605i	$-0.658475\pi$
$787$	−26.7939	−0.955100	−0.477550	+	0.878605i	$0.658475\pi$
$788$	0	0
$789$	1.82854	0.0650977
$790$	0	0
$791$	4.71296	0.167574
$792$	0	0
$793$	3.65793	0.129897
$794$	0	0
$795$	0.675591	0.0239608
$796$	0	0
$797$	−24.6618	−0.873567	−0.436784	−	0.899567i	$-0.643882\pi$
$797$	−24.6618	−0.873567	−0.436784	+	0.899567i	$0.643882\pi$
$798$	0	0
$799$	−97.2061	−3.43890
$800$	0	0
$801$	−36.2966	−1.28248
$802$	0	0
$803$	−8.02335	−0.283138
$804$	0	0
$805$	−5.52842	−0.194851
$806$	0	0
$807$	2.54295	0.0895161
$808$	0	0
$809$	13.7534	0.483542	0.241771	−	0.970333i	$-0.422272\pi$
$809$	13.7534	0.483542	0.241771	+	0.970333i	$0.422272\pi$
$810$	0	0
$811$	−7.54296	−0.264869	−0.132435	−	0.991192i	$-0.542279\pi$
$811$	−7.54296	−0.264869	−0.132435	+	0.991192i	$0.542279\pi$
$812$	0	0
$813$	−1.28674	−0.0451280
$814$	0	0
$815$	4.95975	0.173733
$816$	0	0
$817$	90.4214	3.16345
$818$	0	0
$819$	−2.99073	−0.104505
$820$	0	0
$821$	−26.8612	−0.937461	−0.468730	−	0.883341i	$-0.655288\pi$
$821$	−26.8612	−0.937461	−0.468730	+	0.883341i	$0.655288\pi$
$822$	0	0
$823$	−49.5461	−1.72707	−0.863534	−	0.504290i	$-0.831754\pi$
$823$	−49.5461	−1.72707	−0.863534	+	0.504290i	$0.831754\pi$
$824$	0	0
$825$	0.268499	0.00934794
$826$	0	0
$827$	17.0075	0.591407	0.295704	−	0.955280i	$-0.404446\pi$
$827$	17.0075	0.591407	0.295704	+	0.955280i	$0.404446\pi$
$828$	0	0
$829$	−4.12070	−0.143118	−0.0715589	−	0.997436i	$-0.522797\pi$
$829$	−4.12070	−0.143118	−0.0715589	+	0.997436i	$0.522797\pi$
$830$	0	0
$831$	3.05250	0.105890
$832$	0	0
$833$	−7.53478	−0.261065
$834$	0	0
$835$	26.2243	0.907530
$836$	0	0
$837$	−3.62071	−0.125150
$838$	0	0
$839$	−24.9638	−0.861846	−0.430923	−	0.902389i	$-0.641812\pi$
$839$	−24.9638	−0.861846	−0.430923	+	0.902389i	$0.641812\pi$
$840$	0	0
$841$	57.4119	1.97972
$842$	0	0
$843$	−1.42354	−0.0490293
$844$	0	0
$845$	−1.48698	−0.0511537
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−2.66566	−0.0914851
$850$	0	0
$851$	−3.62994	−0.124433
$852$	0	0
$853$	8.42469	0.288456	0.144228	−	0.989544i	$-0.453930\pi$
$853$	8.42469	0.288456	0.144228	+	0.989544i	$0.453930\pi$
$854$	0	0
$855$	33.7970	1.15583
$856$	0	0
$857$	−36.9461	−1.26205	−0.631027	−	0.775761i	$-0.717366\pi$
$857$	−36.9461	−1.26205	−0.631027	+	0.775761i	$0.717366\pi$
$858$	0	0
$859$	52.5551	1.79316	0.896579	−	0.442884i	$-0.146045\pi$
$859$	52.5551	1.79316	0.896579	+	0.442884i	$0.146045\pi$
$860$	0	0
$861$	−0.408287	−0.0139144
$862$	0	0
$863$	53.4110	1.81813	0.909066	−	0.416653i	$-0.136797\pi$
$863$	53.4110	1.81813	0.909066	+	0.416653i	$0.136797\pi$
$864$	0	0
$865$	26.7244	0.908658
$866$	0	0
$867$	3.82911	0.130043
$868$	0	0
$869$	−2.16743	−0.0735249
$870$	0	0
$871$	−5.47832	−0.185626
$872$	0	0
$873$	27.5416	0.932142
$874$	0	0
$875$	−11.5819	−0.391540
$876$	0	0
$877$	−45.8164	−1.54711	−0.773556	−	0.633728i	$-0.781524\pi$
$877$	−45.8164	−1.54711	−0.773556	+	0.633728i	$0.781524\pi$
$878$	0	0
$879$	0.227535	0.00767455
$880$	0	0
$881$	−45.1659	−1.52168	−0.760839	−	0.648941i	$-0.775212\pi$
$881$	−45.1659	−1.52168	−0.760839	+	0.648941i	$0.775212\pi$
$882$	0	0
$883$	−4.93907	−0.166213	−0.0831064	−	0.996541i	$-0.526484\pi$
$883$	−4.93907	−0.166213	−0.0831064	+	0.996541i	$0.526484\pi$
$884$	0	0
$885$	0.718379	0.0241480
$886$	0	0
$887$	56.1806	1.88636	0.943179	−	0.332284i	$-0.107819\pi$
$887$	56.1806	1.88636	0.943179	+	0.332284i	$0.107819\pi$
$888$	0	0
$889$	−20.6384	−0.692191
$890$	0	0
$891$	−8.91667	−0.298720
$892$	0	0
$893$	98.0435	3.28090
$894$	0	0
$895$	−15.6479	−0.523052
$896$	0	0
$897$	0.357938	0.0119512
$898$	0	0
$899$	−58.3565	−1.94630
$900$	0	0
$901$	35.5581	1.18461
$902$	0	0
$903$	−1.14548	−0.0381191
$904$	0	0
$905$	7.81077	0.259639
$906$	0	0
$907$	−0.175012	−0.00581117	−0.00290559	−	0.999996i	$-0.500925\pi$
$907$	−0.175012	−0.00581117	−0.00290559	+	0.999996i	$0.500925\pi$
$908$	0	0
$909$	−14.8283	−0.491824
$910$	0	0
$911$	−28.3521	−0.939348	−0.469674	−	0.882840i	$-0.655628\pi$
$911$	−28.3521	−0.939348	−0.469674	+	0.882840i	$0.655628\pi$
$912$	0	0
$913$	12.0745	0.399607
$914$	0	0
$915$	0.523663	0.0173118
$916$	0	0
$917$	1.43721	0.0474610
$918$	0	0
$919$	−11.6918	−0.385676	−0.192838	−	0.981231i	$-0.561769\pi$
$919$	−11.6918	−0.385676	−0.192838	+	0.981231i	$0.561769\pi$
$920$	0	0
$921$	1.07095	0.0352890
$922$	0	0
$923$	9.99423	0.328964
$924$	0	0
$925$	−2.72292	−0.0895291
$926$	0	0
$927$	0.695102	0.0228301
$928$	0	0
$929$	17.3318	0.568639	0.284320	−	0.958730i	$-0.408232\pi$
$929$	17.3318	0.568639	0.284320	+	0.958730i	$0.408232\pi$
$930$	0	0
$931$	7.59969	0.249070
$932$	0	0
$933$	0.711541	0.0232948
$934$	0	0
$935$	−11.2041	−0.366412
$936$	0	0
$937$	31.4827	1.02849	0.514247	−	0.857642i	$-0.328072\pi$
$937$	31.4827	1.02849	0.514247	+	0.857642i	$0.328072\pi$
$938$	0	0
$939$	1.74526	0.0569543
$940$	0	0
$941$	−33.6213	−1.09602	−0.548011	−	0.836471i	$-0.684615\pi$
$941$	−33.6213	−1.09602	−0.548011	+	0.836471i	$0.684615\pi$
$942$	0	0
$943$	−15.7671	−0.513446
$944$	0	0
$945$	−0.857622	−0.0278985
$946$	0	0
$947$	24.1792	0.785720	0.392860	−	0.919598i	$-0.371486\pi$
$947$	24.1792	0.785720	0.392860	+	0.919598i	$0.371486\pi$
$948$	0	0
$949$	−8.02335	−0.260449
$950$	0	0
$951$	1.14896	0.0372576
$952$	0	0
$953$	−18.3155	−0.593297	−0.296649	−	0.954987i	$-0.595869\pi$
$953$	−18.3155	−0.593297	−0.296649	+	0.954987i	$0.595869\pi$
$954$	0	0
$955$	−17.1615	−0.555333
$956$	0	0
$957$	0.894948	0.0289296
$958$	0	0
$959$	−21.9061	−0.707385
$960$	0	0
$961$	8.40982	0.271284
$962$	0	0
$963$	−7.56370	−0.243737
$964$	0	0
$965$	−0.253060	−0.00814629
$966$	0	0
$967$	25.0244	0.804731	0.402365	−	0.915479i	$-0.368188\pi$
$967$	25.0244	0.804731	0.402365	+	0.915479i	$0.368188\pi$
$968$	0	0
$969$	−5.51287	−0.177099
$970$	0	0
$971$	8.74207	0.280547	0.140273	−	0.990113i	$-0.455202\pi$
$971$	8.74207	0.280547	0.140273	+	0.990113i	$0.455202\pi$
$972$	0	0
$973$	4.33207	0.138880
$974$	0	0
$975$	0.268499	0.00859885
$976$	0	0
$977$	−4.82118	−0.154243	−0.0771216	−	0.997022i	$-0.524573\pi$
$977$	−4.82118	−0.154243	−0.0771216	+	0.997022i	$0.524573\pi$
$978$	0	0
$979$	−12.1364	−0.387880
$980$	0	0
$981$	−3.69882	−0.118094
$982$	0	0
$983$	1.55814	0.0496968	0.0248484	−	0.999691i	$-0.492090\pi$
$983$	1.55814	0.0496968	0.0248484	+	0.999691i	$0.492090\pi$
$984$	0	0
$985$	40.8103	1.30032
$986$	0	0
$987$	−1.24204	−0.0395344
$988$	0	0
$989$	−44.2356	−1.40661
$990$	0	0
$991$	23.5716	0.748776	0.374388	−	0.927272i	$-0.377853\pi$
$991$	23.5716	0.748776	0.374388	+	0.927272i	$0.377853\pi$
$992$	0	0
$993$	−2.66888	−0.0846944
$994$	0	0
$995$	19.7527	0.626204
$996$	0	0
$997$	38.9566	1.23377	0.616884	−	0.787054i	$-0.288395\pi$
$997$	38.9566	1.23377	0.616884	+	0.787054i	$0.288395\pi$
$998$	0	0
$999$	−0.563112	−0.0178161

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.j.1.6	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.j.1.6	✓	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)

\(f(q)\)	\(=\)	\(q+0.0962745 q^{3} -1.48698 q^{5} -1.00000 q^{7} -2.99073 q^{9} +O(q^{10})\)	\(q+0.0962745 q^{3} -1.48698 q^{5} -1.00000 q^{7} -2.99073 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.143158 q^{15} -7.53478 q^{17} +7.59969 q^{19} -0.0962745 q^{21} -3.71789 q^{23} -2.78889 q^{25} -0.576755 q^{27} -9.29580 q^{29} +6.27772 q^{31} -0.0962745 q^{33} +1.48698 q^{35} +0.976345 q^{37} -0.0962745 q^{39} +4.24087 q^{41} +11.8980 q^{43} +4.44716 q^{45} +12.9010 q^{47} +1.00000 q^{49} -0.725407 q^{51} -4.71919 q^{53} +1.48698 q^{55} +0.731656 q^{57} -5.01808 q^{59} -3.65793 q^{61} +2.99073 q^{63} +1.48698 q^{65} +5.47832 q^{67} -0.357938 q^{69} -9.99423 q^{71} +8.02335 q^{73} -0.268499 q^{75} +1.00000 q^{77} +2.16743 q^{79} +8.91667 q^{81} -12.0745 q^{83} +11.2041 q^{85} -0.894948 q^{87} +12.1364 q^{89} +1.00000 q^{91} +0.604385 q^{93} -11.3006 q^{95} -9.20899 q^{97} +2.99073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9}+O(q^{10}) \) 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

Properties

Related objects

Downloads

Learn more