LMFDB - Embedded newform 8008.2.a.z.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 35 x^{13} + 32 x^{12} + 477 x^{11} - 392 x^{10} - 3236 x^{9} + 2330 x^{8} + \cdots + 2560 $ x^15 - x^14 - 35x^13 + 32x^12 + 477x^11 - 392x^10 - 3236x^9 + 2330x^8 + 11690x^7 - 7119x^6 - 22246x^5 + 11137x^4 + 20034x^3 - 8392x^2 - 6016*x + 2560
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.40500$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.40500 q^{3} +3.56716 q^{5} -1.00000 q^{7} -1.02598 q^{9} +O(q^{10})$	$q-1.40500 q^{3} +3.56716 q^{5} -1.00000 q^{7} -1.02598 q^{9} -1.00000 q^{11} +1.00000 q^{13} -5.01185 q^{15} +6.72672 q^{17} -7.75471 q^{19} +1.40500 q^{21} -4.37653 q^{23} +7.72462 q^{25} +5.65649 q^{27} +5.83038 q^{29} +8.23976 q^{31} +1.40500 q^{33} -3.56716 q^{35} -11.3857 q^{37} -1.40500 q^{39} +2.41784 q^{41} -3.42064 q^{43} -3.65984 q^{45} +11.3028 q^{47} +1.00000 q^{49} -9.45103 q^{51} -5.39121 q^{53} -3.56716 q^{55} +10.8953 q^{57} -7.81386 q^{59} +0.208032 q^{61} +1.02598 q^{63} +3.56716 q^{65} -7.95876 q^{67} +6.14901 q^{69} -7.39527 q^{71} +13.7726 q^{73} -10.8531 q^{75} +1.00000 q^{77} +13.8757 q^{79} -4.86942 q^{81} -0.786323 q^{83} +23.9953 q^{85} -8.19167 q^{87} -4.96883 q^{89} -1.00000 q^{91} -11.5768 q^{93} -27.6623 q^{95} +9.57127 q^{97} +1.02598 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * 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
$3$	−1.40500	−0.811176	−0.405588	−	0.914056i	$-0.632933\pi$
$3$	−1.40500	−0.811176	−0.405588	+	0.914056i	$0.632933\pi$
$4$	0	0
$5$	3.56716	1.59528	0.797641	−	0.603133i	$-0.206081\pi$
$5$	3.56716	1.59528	0.797641	+	0.603133i	$0.206081\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.02598	−0.341994
$10$	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
$15$	−5.01185	−1.29405
$16$	0	0
$17$	6.72672	1.63147	0.815735	−	0.578426i	$-0.196333\pi$
$17$	6.72672	1.63147	0.815735	+	0.578426i	$0.196333\pi$
$18$	0	0
$19$	−7.75471	−1.77905	−0.889526	−	0.456885i	$-0.848965\pi$
$19$	−7.75471	−1.77905	−0.889526	+	0.456885i	$0.848965\pi$
$20$	0	0
$21$	1.40500	0.306596
$22$	0	0
$23$	−4.37653	−0.912569	−0.456284	−	0.889834i	$-0.650820\pi$
$23$	−4.37653	−0.912569	−0.456284	+	0.889834i	$0.650820\pi$
$24$	0	0
$25$	7.72462	1.54492
$26$	0	0
$27$	5.65649	1.08859
$28$	0	0
$29$	5.83038	1.08267	0.541337	−	0.840806i	$-0.317918\pi$
$29$	5.83038	1.08267	0.541337	+	0.840806i	$0.317918\pi$
$30$	0	0
$31$	8.23976	1.47990	0.739952	−	0.672659i	$-0.234848\pi$
$31$	8.23976	1.47990	0.739952	+	0.672659i	$0.234848\pi$
$32$	0	0
$33$	1.40500	0.244579
$34$	0	0
$35$	−3.56716	−0.602960
$36$	0	0
$37$	−11.3857	−1.87181	−0.935903	−	0.352258i	$-0.885414\pi$
$37$	−11.3857	−1.87181	−0.935903	+	0.352258i	$0.885414\pi$
$38$	0	0
$39$	−1.40500	−0.224980
$40$	0	0
$41$	2.41784	0.377603	0.188802	−	0.982015i	$-0.439540\pi$
$41$	2.41784	0.377603	0.188802	+	0.982015i	$0.439540\pi$
$42$	0	0
$43$	−3.42064	−0.521642	−0.260821	−	0.965387i	$-0.583993\pi$
$43$	−3.42064	−0.521642	−0.260821	+	0.965387i	$0.583993\pi$
$44$	0	0
$45$	−3.65984	−0.545577
$46$	0	0
$47$	11.3028	1.64869	0.824343	−	0.566090i	$-0.191545\pi$
$47$	11.3028	1.64869	0.824343	+	0.566090i	$0.191545\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−9.45103	−1.32341
$52$	0	0
$53$	−5.39121	−0.740540	−0.370270	−	0.928924i	$-0.620735\pi$
$53$	−5.39121	−0.740540	−0.370270	+	0.928924i	$0.620735\pi$
$54$	0	0
$55$	−3.56716	−0.480996
$56$	0	0
$57$	10.8953	1.44312
$58$	0	0
$59$	−7.81386	−1.01728	−0.508639	−	0.860980i	$-0.669851\pi$
$59$	−7.81386	−1.01728	−0.508639	+	0.860980i	$0.669851\pi$
$60$	0	0
$61$	0.208032	0.0266358	0.0133179	−	0.999911i	$-0.495761\pi$
$61$	0.208032	0.0266358	0.0133179	+	0.999911i	$0.495761\pi$
$62$	0	0
$63$	1.02598	0.129262
$64$	0	0
$65$	3.56716	0.442452
$66$	0	0
$67$	−7.95876	−0.972318	−0.486159	−	0.873871i	$-0.661602\pi$
$67$	−7.95876	−0.972318	−0.486159	+	0.873871i	$0.661602\pi$
$68$	0	0
$69$	6.14901	0.740254
$70$	0	0
$71$	−7.39527	−0.877657	−0.438828	−	0.898571i	$-0.644606\pi$
$71$	−7.39527	−0.877657	−0.438828	+	0.898571i	$0.644606\pi$
$72$	0	0
$73$	13.7726	1.61196	0.805982	−	0.591941i	$-0.201638\pi$
$73$	13.7726	1.61196	0.805982	+	0.591941i	$0.201638\pi$
$74$	0	0
$75$	−10.8531	−1.25321
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	13.8757	1.56114	0.780568	−	0.625070i	$-0.214930\pi$
$79$	13.8757	1.56114	0.780568	+	0.625070i	$0.214930\pi$
$80$	0	0
$81$	−4.86942	−0.541046
$82$	0	0
$83$	−0.786323	−0.0863101	−0.0431551	−	0.999068i	$-0.513741\pi$
$83$	−0.786323	−0.0863101	−0.0431551	+	0.999068i	$0.513741\pi$
$84$	0	0
$85$	23.9953	2.60266
$86$	0	0
$87$	−8.19167	−0.878239
$88$	0	0
$89$	−4.96883	−0.526694	−0.263347	−	0.964701i	$-0.584826\pi$
$89$	−4.96883	−0.526694	−0.263347	+	0.964701i	$0.584826\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−11.5768	−1.20046
$94$	0	0
$95$	−27.6623	−2.83809
$96$	0	0
$97$	9.57127	0.971815	0.485908	−	0.874010i	$-0.338489\pi$
$97$	9.57127	0.971815	0.485908	+	0.874010i	$0.338489\pi$
$98$	0	0
$99$	1.02598	0.103115
$100$	0	0
$101$	16.0120	1.59325	0.796625	−	0.604474i	$-0.206617\pi$
$101$	16.0120	1.59325	0.796625	+	0.604474i	$0.206617\pi$
$102$	0	0
$103$	10.1927	1.00432	0.502159	−	0.864775i	$-0.332539\pi$
$103$	10.1927	1.00432	0.502159	+	0.864775i	$0.332539\pi$
$104$	0	0
$105$	5.01185	0.489106
$106$	0	0
$107$	18.0949	1.74930	0.874649	−	0.484757i	$-0.161092\pi$
$107$	18.0949	1.74930	0.874649	+	0.484757i	$0.161092\pi$
$108$	0	0
$109$	−7.42996	−0.711661	−0.355831	−	0.934550i	$-0.615802\pi$
$109$	−7.42996	−0.711661	−0.355831	+	0.934550i	$0.615802\pi$
$110$	0	0
$111$	15.9970	1.51836
$112$	0	0
$113$	−0.584345	−0.0549706	−0.0274853	−	0.999622i	$-0.508750\pi$
$113$	−0.584345	−0.0549706	−0.0274853	+	0.999622i	$0.508750\pi$
$114$	0	0
$115$	−15.6118	−1.45580
$116$	0	0
$117$	−1.02598	−0.0948520
$118$	0	0
$119$	−6.72672	−0.616638
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−3.39706	−0.306302
$124$	0	0
$125$	9.71916	0.869308
$126$	0	0
$127$	−11.5471	−1.02464	−0.512320	−	0.858794i	$-0.671214\pi$
$127$	−11.5471	−1.02464	−0.512320	+	0.858794i	$0.671214\pi$
$128$	0	0
$129$	4.80599	0.423144
$130$	0	0
$131$	−15.1756	−1.32590	−0.662948	−	0.748665i	$-0.730695\pi$
$131$	−15.1756	−1.32590	−0.662948	+	0.748665i	$0.730695\pi$
$132$	0	0
$133$	7.75471	0.672418
$134$	0	0
$135$	20.1776	1.73661
$136$	0	0
$137$	6.55768	0.560261	0.280130	−	0.959962i	$-0.409622\pi$
$137$	6.55768	0.560261	0.280130	+	0.959962i	$0.409622\pi$
$138$	0	0
$139$	20.4130	1.73141	0.865703	−	0.500558i	$-0.166872\pi$
$139$	20.4130	1.73141	0.865703	+	0.500558i	$0.166872\pi$
$140$	0	0
$141$	−15.8804	−1.33737
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	20.7979	1.72717
$146$	0	0
$147$	−1.40500	−0.115882
$148$	0	0
$149$	5.46204	0.447468	0.223734	−	0.974650i	$-0.428175\pi$
$149$	5.46204	0.447468	0.223734	+	0.974650i	$0.428175\pi$
$150$	0	0
$151$	5.25418	0.427579	0.213789	−	0.976880i	$-0.431419\pi$
$151$	5.25418	0.427579	0.213789	+	0.976880i	$0.431419\pi$
$152$	0	0
$153$	−6.90150	−0.557953
$154$	0	0
$155$	29.3925	2.36087
$156$	0	0
$157$	6.57383	0.524649	0.262325	−	0.964980i	$-0.415511\pi$
$157$	6.57383	0.524649	0.262325	+	0.964980i	$0.415511\pi$
$158$	0	0
$159$	7.57464	0.600708
$160$	0	0
$161$	4.37653	0.344919
$162$	0	0
$163$	−1.41792	−0.111060	−0.0555300	−	0.998457i	$-0.517685\pi$
$163$	−1.41792	−0.111060	−0.0555300	+	0.998457i	$0.517685\pi$
$164$	0	0
$165$	5.01185	0.390172
$166$	0	0
$167$	−6.62175	−0.512406	−0.256203	−	0.966623i	$-0.582472\pi$
$167$	−6.62175	−0.512406	−0.256203	+	0.966623i	$0.582472\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	7.95618	0.608425
$172$	0	0
$173$	8.52908	0.648454	0.324227	−	0.945979i	$-0.394896\pi$
$173$	8.52908	0.648454	0.324227	+	0.945979i	$0.394896\pi$
$174$	0	0
$175$	−7.72462	−0.583927
$176$	0	0
$177$	10.9785	0.825191
$178$	0	0
$179$	6.19123	0.462754	0.231377	−	0.972864i	$-0.425677\pi$
$179$	6.19123	0.462754	0.231377	+	0.972864i	$0.425677\pi$
$180$	0	0
$181$	7.83046	0.582033	0.291017	−	0.956718i	$-0.406006\pi$
$181$	7.83046	0.582033	0.291017	+	0.956718i	$0.406006\pi$
$182$	0	0
$183$	−0.292285	−0.0216063
$184$	0	0
$185$	−40.6148	−2.98606
$186$	0	0
$187$	−6.72672	−0.491907
$188$	0	0
$189$	−5.65649	−0.411449
$190$	0	0
$191$	8.18273	0.592081	0.296041	−	0.955175i	$-0.404334\pi$
$191$	8.18273	0.592081	0.296041	+	0.955175i	$0.404334\pi$
$192$	0	0
$193$	−21.0011	−1.51169	−0.755845	−	0.654751i	$-0.772774\pi$
$193$	−21.0011	−1.51169	−0.755845	+	0.654751i	$0.772774\pi$
$194$	0	0
$195$	−5.01185	−0.358906
$196$	0	0
$197$	−12.7264	−0.906719	−0.453360	−	0.891328i	$-0.649775\pi$
$197$	−12.7264	−0.906719	−0.453360	+	0.891328i	$0.649775\pi$
$198$	0	0
$199$	21.7881	1.54451	0.772257	−	0.635310i	$-0.219128\pi$
$199$	21.7881	1.54451	0.772257	+	0.635310i	$0.219128\pi$
$200$	0	0
$201$	11.1820	0.788720
$202$	0	0
$203$	−5.83038	−0.409212
$204$	0	0
$205$	8.62482	0.602383
$206$	0	0
$207$	4.49023	0.312093
$208$	0	0
$209$	7.75471	0.536404
$210$	0	0
$211$	14.8631	1.02322	0.511608	−	0.859219i	$-0.329050\pi$
$211$	14.8631	1.02322	0.511608	+	0.859219i	$0.329050\pi$
$212$	0	0
$213$	10.3903	0.711934
$214$	0	0
$215$	−12.2020	−0.832167
$216$	0	0
$217$	−8.23976	−0.559351
$218$	0	0
$219$	−19.3505	−1.30759
$220$	0	0
$221$	6.72672	0.452488
$222$	0	0
$223$	17.4924	1.17138	0.585689	−	0.810536i	$-0.300824\pi$
$223$	17.4924	1.17138	0.585689	+	0.810536i	$0.300824\pi$
$224$	0	0
$225$	−7.92532	−0.528355
$226$	0	0
$227$	4.95157	0.328647	0.164324	−	0.986406i	$-0.447456\pi$
$227$	4.95157	0.328647	0.164324	+	0.986406i	$0.447456\pi$
$228$	0	0
$229$	−17.7646	−1.17392	−0.586958	−	0.809617i	$-0.699675\pi$
$229$	−17.7646	−1.17392	−0.586958	+	0.809617i	$0.699675\pi$
$230$	0	0
$231$	−1.40500	−0.0924421
$232$	0	0
$233$	−25.9750	−1.70168	−0.850839	−	0.525427i	$-0.823906\pi$
$233$	−25.9750	−1.70168	−0.850839	+	0.525427i	$0.823906\pi$
$234$	0	0
$235$	40.3190	2.63012
$236$	0	0
$237$	−19.4953	−1.26636
$238$	0	0
$239$	−26.7117	−1.72783	−0.863917	−	0.503634i	$-0.831996\pi$
$239$	−26.7117	−1.72783	−0.863917	+	0.503634i	$0.831996\pi$
$240$	0	0
$241$	2.25030	0.144954	0.0724771	−	0.997370i	$-0.476910\pi$
$241$	2.25030	0.144954	0.0724771	+	0.997370i	$0.476910\pi$
$242$	0	0
$243$	−10.1280	−0.649709
$244$	0	0
$245$	3.56716	0.227897
$246$	0	0
$247$	−7.75471	−0.493420
$248$	0	0
$249$	1.10478	0.0700127
$250$	0	0
$251$	−2.17576	−0.137333	−0.0686664	−	0.997640i	$-0.521874\pi$
$251$	−2.17576	−0.137333	−0.0686664	+	0.997640i	$0.521874\pi$
$252$	0	0
$253$	4.37653	0.275150
$254$	0	0
$255$	−33.7133	−2.11121
$256$	0	0
$257$	25.2191	1.57312	0.786562	−	0.617511i	$-0.211859\pi$
$257$	25.2191	1.57312	0.786562	+	0.617511i	$0.211859\pi$
$258$	0	0
$259$	11.3857	0.707476
$260$	0	0
$261$	−5.98186	−0.370268
$262$	0	0
$263$	11.9413	0.736331	0.368165	−	0.929760i	$-0.379986\pi$
$263$	11.9413	0.736331	0.368165	+	0.929760i	$0.379986\pi$
$264$	0	0
$265$	−19.2313	−1.18137
$266$	0	0
$267$	6.98119	0.427242
$268$	0	0
$269$	14.1379	0.862003	0.431002	−	0.902351i	$-0.358160\pi$
$269$	14.1379	0.862003	0.431002	+	0.902351i	$0.358160\pi$
$270$	0	0
$271$	6.98997	0.424610	0.212305	−	0.977203i	$-0.431903\pi$
$271$	6.98997	0.424610	0.212305	+	0.977203i	$0.431903\pi$
$272$	0	0
$273$	1.40500	0.0850343
$274$	0	0
$275$	−7.72462	−0.465812
$276$	0	0
$277$	−13.2602	−0.796727	−0.398363	−	0.917228i	$-0.630422\pi$
$277$	−13.2602	−0.796727	−0.398363	+	0.917228i	$0.630422\pi$
$278$	0	0
$279$	−8.45384	−0.506118
$280$	0	0
$281$	−10.3451	−0.617137	−0.308568	−	0.951202i	$-0.599850\pi$
$281$	−10.3451	−0.617137	−0.308568	+	0.951202i	$0.599850\pi$
$282$	0	0
$283$	−8.03242	−0.477478	−0.238739	−	0.971084i	$-0.576734\pi$
$283$	−8.03242	−0.477478	−0.238739	+	0.971084i	$0.576734\pi$
$284$	0	0
$285$	38.8654	2.30219
$286$	0	0
$287$	−2.41784	−0.142721
$288$	0	0
$289$	28.2488	1.66170
$290$	0	0
$291$	−13.4476	−0.788313
$292$	0	0
$293$	−11.7768	−0.688010	−0.344005	−	0.938968i	$-0.611784\pi$
$293$	−11.7768	−0.688010	−0.344005	+	0.938968i	$0.611784\pi$
$294$	0	0
$295$	−27.8733	−1.62284
$296$	0	0
$297$	−5.65649	−0.328223
$298$	0	0
$299$	−4.37653	−0.253101
$300$	0	0
$301$	3.42064	0.197162
$302$	0	0
$303$	−22.4968	−1.29241
$304$	0	0
$305$	0.742083	0.0424916
$306$	0	0
$307$	29.4916	1.68318	0.841588	−	0.540120i	$-0.181621\pi$
$307$	29.4916	1.68318	0.841588	+	0.540120i	$0.181621\pi$
$308$	0	0
$309$	−14.3207	−0.814679
$310$	0	0
$311$	23.1107	1.31049	0.655245	−	0.755417i	$-0.272565\pi$
$311$	23.1107	1.31049	0.655245	+	0.755417i	$0.272565\pi$
$312$	0	0
$313$	−14.6421	−0.827622	−0.413811	−	0.910363i	$-0.635803\pi$
$313$	−14.6421	−0.827622	−0.413811	+	0.910363i	$0.635803\pi$
$314$	0	0
$315$	3.65984	0.206209
$316$	0	0
$317$	33.8690	1.90227	0.951137	−	0.308770i	$-0.0999174\pi$
$317$	33.8690	1.90227	0.951137	+	0.308770i	$0.0999174\pi$
$318$	0	0
$319$	−5.83038	−0.326438
$320$	0	0
$321$	−25.4233	−1.41899
$322$	0	0
$323$	−52.1638	−2.90247
$324$	0	0
$325$	7.72462	0.428485
$326$	0	0
$327$	10.4391	0.577282
$328$	0	0
$329$	−11.3028	−0.623145
$330$	0	0
$331$	−11.6763	−0.641787	−0.320894	−	0.947115i	$-0.603983\pi$
$331$	−11.6763	−0.641787	−0.320894	+	0.947115i	$0.603983\pi$
$332$	0	0
$333$	11.6816	0.640146
$334$	0	0
$335$	−28.3902	−1.55112
$336$	0	0
$337$	13.7562	0.749349	0.374675	−	0.927156i	$-0.377754\pi$
$337$	13.7562	0.749349	0.374675	+	0.927156i	$0.377754\pi$
$338$	0	0
$339$	0.821004	0.0445908
$340$	0	0
$341$	−8.23976	−0.446208
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	21.9345	1.18091
$346$	0	0
$347$	18.3705	0.986179	0.493090	−	0.869979i	$-0.335867\pi$
$347$	18.3705	0.986179	0.493090	+	0.869979i	$0.335867\pi$
$348$	0	0
$349$	6.23371	0.333683	0.166842	−	0.985984i	$-0.446643\pi$
$349$	6.23371	0.333683	0.166842	+	0.985984i	$0.446643\pi$
$350$	0	0
$351$	5.65649	0.301921
$352$	0	0
$353$	−10.4692	−0.557220	−0.278610	−	0.960404i	$-0.589874\pi$
$353$	−10.4692	−0.557220	−0.278610	+	0.960404i	$0.589874\pi$
$354$	0	0
$355$	−26.3801	−1.40011
$356$	0	0
$357$	9.45103	0.500202
$358$	0	0
$359$	−27.9031	−1.47267	−0.736334	−	0.676619i	$-0.763445\pi$
$359$	−27.9031	−1.47267	−0.736334	+	0.676619i	$0.763445\pi$
$360$	0	0
$361$	41.1355	2.16502
$362$	0	0
$363$	−1.40500	−0.0737433
$364$	0	0
$365$	49.1291	2.57154
$366$	0	0
$367$	8.70128	0.454203	0.227101	−	0.973871i	$-0.427075\pi$
$367$	8.70128	0.454203	0.227101	+	0.973871i	$0.427075\pi$
$368$	0	0
$369$	−2.48066	−0.129138
$370$	0	0
$371$	5.39121	0.279898
$372$	0	0
$373$	−17.3731	−0.899544	−0.449772	−	0.893143i	$-0.648495\pi$
$373$	−17.3731	−0.899544	−0.449772	+	0.893143i	$0.648495\pi$
$374$	0	0
$375$	−13.6554	−0.705162
$376$	0	0
$377$	5.83038	0.300280
$378$	0	0
$379$	28.1508	1.44601	0.723004	−	0.690843i	$-0.242761\pi$
$379$	28.1508	1.44601	0.723004	+	0.690843i	$0.242761\pi$
$380$	0	0
$381$	16.2237	0.831164
$382$	0	0
$383$	3.42193	0.174852	0.0874261	−	0.996171i	$-0.472136\pi$
$383$	3.42193	0.174852	0.0874261	+	0.996171i	$0.472136\pi$
$384$	0	0
$385$	3.56716	0.181799
$386$	0	0
$387$	3.50951	0.178398
$388$	0	0
$389$	22.8008	1.15605	0.578023	−	0.816021i	$-0.303824\pi$
$389$	22.8008	1.15605	0.578023	+	0.816021i	$0.303824\pi$
$390$	0	0
$391$	−29.4397	−1.48883
$392$	0	0
$393$	21.3216	1.07553
$394$	0	0
$395$	49.4968	2.49045
$396$	0	0
$397$	−17.9193	−0.899344	−0.449672	−	0.893194i	$-0.648459\pi$
$397$	−17.9193	−0.899344	−0.449672	+	0.893194i	$0.648459\pi$
$398$	0	0
$399$	−10.8953	−0.545449
$400$	0	0
$401$	7.38688	0.368883	0.184442	−	0.982843i	$-0.440952\pi$
$401$	7.38688	0.368883	0.184442	+	0.982843i	$0.440952\pi$
$402$	0	0
$403$	8.23976	0.410452
$404$	0	0
$405$	−17.3700	−0.863121
$406$	0	0
$407$	11.3857	0.564371
$408$	0	0
$409$	34.4252	1.70222	0.851108	−	0.524990i	$-0.175931\pi$
$409$	34.4252	1.70222	0.851108	+	0.524990i	$0.175931\pi$
$410$	0	0
$411$	−9.21353	−0.454470
$412$	0	0
$413$	7.81386	0.384495
$414$	0	0
$415$	−2.80494	−0.137689
$416$	0	0
$417$	−28.6802	−1.40447
$418$	0	0
$419$	13.6112	0.664950	0.332475	−	0.943112i	$-0.392116\pi$
$419$	13.6112	0.664950	0.332475	+	0.943112i	$0.392116\pi$
$420$	0	0
$421$	−0.793681	−0.0386817	−0.0193408	−	0.999813i	$-0.506157\pi$
$421$	−0.793681	−0.0386817	−0.0193408	+	0.999813i	$0.506157\pi$
$422$	0	0
$423$	−11.5965	−0.563841
$424$	0	0
$425$	51.9614	2.52050
$426$	0	0
$427$	−0.208032	−0.0100674
$428$	0	0
$429$	1.40500	0.0678339
$430$	0	0
$431$	−22.1104	−1.06502	−0.532510	−	0.846424i	$-0.678751\pi$
$431$	−22.1104	−1.06502	−0.532510	+	0.846424i	$0.678751\pi$
$432$	0	0
$433$	15.2633	0.733508	0.366754	−	0.930318i	$-0.380469\pi$
$433$	15.2633	0.733508	0.366754	+	0.930318i	$0.380469\pi$
$434$	0	0
$435$	−29.2210	−1.40104
$436$	0	0
$437$	33.9387	1.62351
$438$	0	0
$439$	−6.01567	−0.287113	−0.143556	−	0.989642i	$-0.545854\pi$
$439$	−6.01567	−0.287113	−0.143556	+	0.989642i	$0.545854\pi$
$440$	0	0
$441$	−1.02598	−0.0488563
$442$	0	0
$443$	14.9202	0.708878	0.354439	−	0.935079i	$-0.384672\pi$
$443$	14.9202	0.708878	0.354439	+	0.935079i	$0.384672\pi$
$444$	0	0
$445$	−17.7246	−0.840226
$446$	0	0
$447$	−7.67415	−0.362975
$448$	0	0
$449$	2.12064	0.100079	0.0500395	−	0.998747i	$-0.484065\pi$
$449$	2.12064	0.100079	0.0500395	+	0.998747i	$0.484065\pi$
$450$	0	0
$451$	−2.41784	−0.113852
$452$	0	0
$453$	−7.38211	−0.346842
$454$	0	0
$455$	−3.56716	−0.167231
$456$	0	0
$457$	15.8377	0.740856	0.370428	−	0.928861i	$-0.379211\pi$
$457$	15.8377	0.740856	0.370428	+	0.928861i	$0.379211\pi$
$458$	0	0
$459$	38.0497	1.77601
$460$	0	0
$461$	−27.9586	−1.30216	−0.651080	−	0.759009i	$-0.725684\pi$
$461$	−27.9586	−1.30216	−0.651080	+	0.759009i	$0.725684\pi$
$462$	0	0
$463$	−6.10637	−0.283787	−0.141893	−	0.989882i	$-0.545319\pi$
$463$	−6.10637	−0.283787	−0.141893	+	0.989882i	$0.545319\pi$
$464$	0	0
$465$	−41.2964	−1.91508
$466$	0	0
$467$	2.00502	0.0927811	0.0463906	−	0.998923i	$-0.485228\pi$
$467$	2.00502	0.0927811	0.0463906	+	0.998923i	$0.485228\pi$
$468$	0	0
$469$	7.95876	0.367501
$470$	0	0
$471$	−9.23622	−0.425583
$472$	0	0
$473$	3.42064	0.157281
$474$	0	0
$475$	−59.9022	−2.74850
$476$	0	0
$477$	5.53128	0.253260
$478$	0	0
$479$	17.6575	0.806790	0.403395	−	0.915026i	$-0.367830\pi$
$479$	17.6575	0.806790	0.403395	+	0.915026i	$0.367830\pi$
$480$	0	0
$481$	−11.3857	−0.519145
$482$	0	0
$483$	−6.14901	−0.279790
$484$	0	0
$485$	34.1422	1.55032
$486$	0	0
$487$	19.5956	0.887962	0.443981	−	0.896036i	$-0.353566\pi$
$487$	19.5956	0.887962	0.443981	+	0.896036i	$0.353566\pi$
$488$	0	0
$489$	1.99217	0.0900892
$490$	0	0
$491$	−35.2125	−1.58912	−0.794559	−	0.607187i	$-0.792298\pi$
$491$	−35.2125	−1.58912	−0.794559	+	0.607187i	$0.792298\pi$
$492$	0	0
$493$	39.2193	1.76635
$494$	0	0
$495$	3.65984	0.164498
$496$	0	0
$497$	7.39527	0.331723
$498$	0	0
$499$	26.5395	1.18807	0.594036	−	0.804438i	$-0.297534\pi$
$499$	26.5395	1.18807	0.594036	+	0.804438i	$0.297534\pi$
$500$	0	0
$501$	9.30354	0.415652
$502$	0	0
$503$	6.22621	0.277613	0.138807	−	0.990320i	$-0.455673\pi$
$503$	6.22621	0.277613	0.138807	+	0.990320i	$0.455673\pi$
$504$	0	0
$505$	57.1172	2.54168
$506$	0	0
$507$	−1.40500	−0.0623981
$508$	0	0
$509$	−28.2049	−1.25016	−0.625080	−	0.780561i	$-0.714934\pi$
$509$	−28.2049	−1.25016	−0.625080	+	0.780561i	$0.714934\pi$
$510$	0	0
$511$	−13.7726	−0.609265
$512$	0	0
$513$	−43.8644	−1.93666
$514$	0	0
$515$	36.3590	1.60217
$516$	0	0
$517$	−11.3028	−0.497098
$518$	0	0
$519$	−11.9833	−0.526010
$520$	0	0
$521$	22.6514	0.992376	0.496188	−	0.868215i	$-0.334733\pi$
$521$	22.6514	0.992376	0.496188	+	0.868215i	$0.334733\pi$
$522$	0	0
$523$	18.5694	0.811983	0.405992	−	0.913877i	$-0.366926\pi$
$523$	18.5694	0.811983	0.405992	+	0.913877i	$0.366926\pi$
$524$	0	0
$525$	10.8531	0.473667
$526$	0	0
$527$	55.4266	2.41442
$528$	0	0
$529$	−3.84602	−0.167218
$530$	0	0
$531$	8.01687	0.347903
$532$	0	0
$533$	2.41784	0.104728
$534$	0	0
$535$	64.5473	2.79062
$536$	0	0
$537$	−8.69866	−0.375375
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	12.5219	0.538360	0.269180	−	0.963090i	$-0.413247\pi$
$541$	12.5219	0.538360	0.269180	+	0.963090i	$0.413247\pi$
$542$	0	0
$543$	−11.0018	−0.472131
$544$	0	0
$545$	−26.5039	−1.13530
$546$	0	0
$547$	−43.3551	−1.85373	−0.926866	−	0.375393i	$-0.877508\pi$
$547$	−43.3551	−1.85373	−0.926866	+	0.375393i	$0.877508\pi$
$548$	0	0
$549$	−0.213437	−0.00910927
$550$	0	0
$551$	−45.2129	−1.92613
$552$	0	0
$553$	−13.8757	−0.590054
$554$	0	0
$555$	57.0637	2.42222
$556$	0	0
$557$	−7.53513	−0.319274	−0.159637	−	0.987176i	$-0.551032\pi$
$557$	−7.53513	−0.319274	−0.159637	+	0.987176i	$0.551032\pi$
$558$	0	0
$559$	−3.42064	−0.144678
$560$	0	0
$561$	9.45103	0.399023
$562$	0	0
$563$	4.73585	0.199592	0.0997960	−	0.995008i	$-0.468181\pi$
$563$	4.73585	0.199592	0.0997960	+	0.995008i	$0.468181\pi$
$564$	0	0
$565$	−2.08445	−0.0876936
$566$	0	0
$567$	4.86942	0.204496
$568$	0	0
$569$	35.9503	1.50711	0.753557	−	0.657382i	$-0.228336\pi$
$569$	35.9503	1.50711	0.753557	+	0.657382i	$0.228336\pi$
$570$	0	0
$571$	−22.2471	−0.931013	−0.465507	−	0.885044i	$-0.654128\pi$
$571$	−22.2471	−0.931013	−0.465507	+	0.885044i	$0.654128\pi$
$572$	0	0
$573$	−11.4967	−0.480282
$574$	0	0
$575$	−33.8070	−1.40985
$576$	0	0
$577$	20.2789	0.844220	0.422110	−	0.906545i	$-0.361290\pi$
$577$	20.2789	0.844220	0.422110	+	0.906545i	$0.361290\pi$
$578$	0	0
$579$	29.5064	1.22625
$580$	0	0
$581$	0.786323	0.0326222
$582$	0	0
$583$	5.39121	0.223281
$584$	0	0
$585$	−3.65984	−0.151316
$586$	0	0
$587$	−22.6493	−0.934836	−0.467418	−	0.884036i	$-0.654816\pi$
$587$	−22.6493	−0.934836	−0.467418	+	0.884036i	$0.654816\pi$
$588$	0	0
$589$	−63.8969	−2.63283
$590$	0	0
$591$	17.8806	0.735509
$592$	0	0
$593$	29.2892	1.20276	0.601381	−	0.798962i	$-0.294617\pi$
$593$	29.2892	1.20276	0.601381	+	0.798962i	$0.294617\pi$
$594$	0	0
$595$	−23.9953	−0.983711
$596$	0	0
$597$	−30.6122	−1.25287
$598$	0	0
$599$	−25.8476	−1.05610	−0.528052	−	0.849212i	$-0.677077\pi$
$599$	−25.8476	−1.05610	−0.528052	+	0.849212i	$0.677077\pi$
$600$	0	0
$601$	40.0689	1.63444	0.817222	−	0.576323i	$-0.195513\pi$
$601$	40.0689	1.63444	0.817222	+	0.576323i	$0.195513\pi$
$602$	0	0
$603$	8.16554	0.332527
$604$	0	0
$605$	3.56716	0.145026
$606$	0	0
$607$	−7.46851	−0.303137	−0.151569	−	0.988447i	$-0.548432\pi$
$607$	−7.46851	−0.303137	−0.151569	+	0.988447i	$0.548432\pi$
$608$	0	0
$609$	8.19167	0.331943
$610$	0	0
$611$	11.3028	0.457263
$612$	0	0
$613$	10.5986	0.428074	0.214037	−	0.976826i	$-0.431339\pi$
$613$	10.5986	0.428074	0.214037	+	0.976826i	$0.431339\pi$
$614$	0	0
$615$	−12.1178	−0.488639
$616$	0	0
$617$	0.191306	0.00770167	0.00385084	−	0.999993i	$-0.498774\pi$
$617$	0.191306	0.00770167	0.00385084	+	0.999993i	$0.498774\pi$
$618$	0	0
$619$	30.9994	1.24597	0.622985	−	0.782234i	$-0.285920\pi$
$619$	30.9994	1.24597	0.622985	+	0.782234i	$0.285920\pi$
$620$	0	0
$621$	−24.7558	−0.993416
$622$	0	0
$623$	4.96883	0.199072
$624$	0	0
$625$	−3.95332	−0.158133
$626$	0	0
$627$	−10.8953	−0.435118
$628$	0	0
$629$	−76.5888	−3.05380
$630$	0	0
$631$	−42.6507	−1.69790	−0.848950	−	0.528474i	$-0.822765\pi$
$631$	−42.6507	−1.69790	−0.848950	+	0.528474i	$0.822765\pi$
$632$	0	0
$633$	−20.8826	−0.830008
$634$	0	0
$635$	−41.1904	−1.63459
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	7.58741	0.300153
$640$	0	0
$641$	35.9646	1.42051	0.710257	−	0.703942i	$-0.248579\pi$
$641$	35.9646	1.42051	0.710257	+	0.703942i	$0.248579\pi$
$642$	0	0
$643$	28.3182	1.11676	0.558380	−	0.829585i	$-0.311423\pi$
$643$	28.3182	1.11676	0.558380	+	0.829585i	$0.311423\pi$
$644$	0	0
$645$	17.1437	0.675033
$646$	0	0
$647$	28.0120	1.10126	0.550632	−	0.834748i	$-0.314387\pi$
$647$	28.0120	1.10126	0.550632	+	0.834748i	$0.314387\pi$
$648$	0	0
$649$	7.81386	0.306721
$650$	0	0
$651$	11.5768	0.453732
$652$	0	0
$653$	−12.3213	−0.482168	−0.241084	−	0.970504i	$-0.577503\pi$
$653$	−12.3213	−0.482168	−0.241084	+	0.970504i	$0.577503\pi$
$654$	0	0
$655$	−54.1337	−2.11518
$656$	0	0
$657$	−14.1305	−0.551281
$658$	0	0
$659$	5.73997	0.223598	0.111799	−	0.993731i	$-0.464339\pi$
$659$	5.73997	0.223598	0.111799	+	0.993731i	$0.464339\pi$
$660$	0	0
$661$	−7.68336	−0.298848	−0.149424	−	0.988773i	$-0.547742\pi$
$661$	−7.68336	−0.298848	−0.149424	+	0.988773i	$0.547742\pi$
$662$	0	0
$663$	−9.45103	−0.367048
$664$	0	0
$665$	27.6623	1.07270
$666$	0	0
$667$	−25.5168	−0.988014
$668$	0	0
$669$	−24.5768	−0.950193
$670$	0	0
$671$	−0.208032	−0.00803099
$672$	0	0
$673$	−26.5872	−1.02486	−0.512430	−	0.858729i	$-0.671254\pi$
$673$	−26.5872	−1.02486	−0.512430	+	0.858729i	$0.671254\pi$
$674$	0	0
$675$	43.6943	1.68179
$676$	0	0
$677$	−1.58113	−0.0607676	−0.0303838	−	0.999538i	$-0.509673\pi$
$677$	−1.58113	−0.0607676	−0.0303838	+	0.999538i	$0.509673\pi$
$678$	0	0
$679$	−9.57127	−0.367312
$680$	0	0
$681$	−6.95695	−0.266591
$682$	0	0
$683$	−28.9895	−1.10925	−0.554627	−	0.832099i	$-0.687139\pi$
$683$	−28.9895	−1.10925	−0.554627	+	0.832099i	$0.687139\pi$
$684$	0	0
$685$	23.3923	0.893774
$686$	0	0
$687$	24.9592	0.952252
$688$	0	0
$689$	−5.39121	−0.205389
$690$	0	0
$691$	−15.6121	−0.593911	−0.296955	−	0.954891i	$-0.595971\pi$
$691$	−15.6121	−0.593911	−0.296955	+	0.954891i	$0.595971\pi$
$692$	0	0
$693$	−1.02598	−0.0389738
$694$	0	0
$695$	72.8163	2.76208
$696$	0	0
$697$	16.2641	0.616048
$698$	0	0
$699$	36.4948	1.38036
$700$	0	0
$701$	8.46008	0.319533	0.159766	−	0.987155i	$-0.448926\pi$
$701$	8.46008	0.319533	0.159766	+	0.987155i	$0.448926\pi$
$702$	0	0
$703$	88.2931	3.33004
$704$	0	0
$705$	−56.6481	−2.13349
$706$	0	0
$707$	−16.0120	−0.602192
$708$	0	0
$709$	−6.59201	−0.247568	−0.123784	−	0.992309i	$-0.539503\pi$
$709$	−6.59201	−0.247568	−0.123784	+	0.992309i	$0.539503\pi$
$710$	0	0
$711$	−14.2362	−0.533899
$712$	0	0
$713$	−36.0615	−1.35051
$714$	0	0
$715$	−3.56716	−0.133404
$716$	0	0
$717$	37.5298	1.40158
$718$	0	0
$719$	−3.82640	−0.142701	−0.0713503	−	0.997451i	$-0.522731\pi$
$719$	−3.82640	−0.142701	−0.0713503	+	0.997451i	$0.522731\pi$
$720$	0	0
$721$	−10.1927	−0.379597
$722$	0	0
$723$	−3.16166	−0.117583
$724$	0	0
$725$	45.0375	1.67265
$726$	0	0
$727$	20.6518	0.765933	0.382967	−	0.923762i	$-0.374902\pi$
$727$	20.6518	0.765933	0.382967	+	0.923762i	$0.374902\pi$
$728$	0	0
$729$	28.8380	1.06807
$730$	0	0
$731$	−23.0097	−0.851044
$732$	0	0
$733$	−38.7441	−1.43105	−0.715523	−	0.698589i	$-0.753812\pi$
$733$	−38.7441	−1.43105	−0.715523	+	0.698589i	$0.753812\pi$
$734$	0	0
$735$	−5.01185	−0.184865
$736$	0	0
$737$	7.95876	0.293165
$738$	0	0
$739$	−7.44966	−0.274040	−0.137020	−	0.990568i	$-0.543753\pi$
$739$	−7.44966	−0.274040	−0.137020	+	0.990568i	$0.543753\pi$
$740$	0	0
$741$	10.8953	0.400250
$742$	0	0
$743$	11.5827	0.424926	0.212463	−	0.977169i	$-0.431851\pi$
$743$	11.5827	0.424926	0.212463	+	0.977169i	$0.431851\pi$
$744$	0	0
$745$	19.4840	0.713837
$746$	0	0
$747$	0.806752	0.0295175
$748$	0	0
$749$	−18.0949	−0.661173
$750$	0	0
$751$	39.9852	1.45908	0.729540	−	0.683938i	$-0.239734\pi$
$751$	39.9852	1.45908	0.729540	+	0.683938i	$0.239734\pi$
$752$	0	0
$753$	3.05694	0.111401
$754$	0	0
$755$	18.7425	0.682109
$756$	0	0
$757$	35.2045	1.27953	0.639764	−	0.768571i	$-0.279032\pi$
$757$	35.2045	1.27953	0.639764	+	0.768571i	$0.279032\pi$
$758$	0	0
$759$	−6.14901	−0.223195
$760$	0	0
$761$	−4.66198	−0.168997	−0.0844983	−	0.996424i	$-0.526929\pi$
$761$	−4.66198	−0.168997	−0.0844983	+	0.996424i	$0.526929\pi$
$762$	0	0
$763$	7.42996	0.268983
$764$	0	0
$765$	−24.6187	−0.890092
$766$	0	0
$767$	−7.81386	−0.282142
$768$	0	0
$769$	29.1921	1.05270	0.526348	−	0.850269i	$-0.323561\pi$
$769$	29.1921	1.05270	0.526348	+	0.850269i	$0.323561\pi$
$770$	0	0
$771$	−35.4328	−1.27608
$772$	0	0
$773$	18.1029	0.651115	0.325558	−	0.945522i	$-0.394448\pi$
$773$	18.1029	0.651115	0.325558	+	0.945522i	$0.394448\pi$
$774$	0	0
$775$	63.6490	2.28634
$776$	0	0
$777$	−15.9970	−0.573887
$778$	0	0
$779$	−18.7496	−0.671775
$780$	0	0
$781$	7.39527	0.264623
$782$	0	0
$783$	32.9795	1.17859
$784$	0	0
$785$	23.4499	0.836963
$786$	0	0
$787$	12.7346	0.453939	0.226969	−	0.973902i	$-0.427118\pi$
$787$	12.7346	0.453939	0.226969	+	0.973902i	$0.427118\pi$
$788$	0	0
$789$	−16.7775	−0.597294
$790$	0	0
$791$	0.584345	0.0207769
$792$	0	0
$793$	0.208032	0.00738743
$794$	0	0
$795$	27.0199	0.958299
$796$	0	0
$797$	−27.8950	−0.988092	−0.494046	−	0.869436i	$-0.664483\pi$
$797$	−27.8950	−0.988092	−0.494046	+	0.869436i	$0.664483\pi$
$798$	0	0
$799$	76.0310	2.68978
$800$	0	0
$801$	5.09792	0.180126
$802$	0	0
$803$	−13.7726	−0.486025
$804$	0	0
$805$	15.6118	0.550242
$806$	0	0
$807$	−19.8637	−0.699236
$808$	0	0
$809$	−8.90818	−0.313195	−0.156597	−	0.987663i	$-0.550053\pi$
$809$	−8.90818	−0.313195	−0.156597	+	0.987663i	$0.550053\pi$
$810$	0	0
$811$	48.5905	1.70624	0.853122	−	0.521711i	$-0.174706\pi$
$811$	48.5905	1.70624	0.853122	+	0.521711i	$0.174706\pi$
$812$	0	0
$813$	−9.82089	−0.344434
$814$	0	0
$815$	−5.05795	−0.177172
$816$	0	0
$817$	26.5260	0.928028
$818$	0	0
$819$	1.02598	0.0358507
$820$	0	0
$821$	26.3366	0.919154	0.459577	−	0.888138i	$-0.348001\pi$
$821$	26.3366	0.919154	0.459577	+	0.888138i	$0.348001\pi$
$822$	0	0
$823$	17.9901	0.627096	0.313548	−	0.949572i	$-0.398482\pi$
$823$	17.9901	0.627096	0.313548	+	0.949572i	$0.398482\pi$
$824$	0	0
$825$	10.8531	0.377856
$826$	0	0
$827$	49.3514	1.71612	0.858058	−	0.513553i	$-0.171671\pi$
$827$	49.3514	1.71612	0.858058	+	0.513553i	$0.171671\pi$
$828$	0	0
$829$	31.6445	1.09906	0.549529	−	0.835475i	$-0.314807\pi$
$829$	31.6445	1.09906	0.549529	+	0.835475i	$0.314807\pi$
$830$	0	0
$831$	18.6305	0.646286
$832$	0	0
$833$	6.72672	0.233067
$834$	0	0
$835$	−23.6208	−0.817433
$836$	0	0
$837$	46.6082	1.61101
$838$	0	0
$839$	−44.0270	−1.51998	−0.759991	−	0.649934i	$-0.774797\pi$
$839$	−44.0270	−1.51998	−0.759991	+	0.649934i	$0.774797\pi$
$840$	0	0
$841$	4.99330	0.172183
$842$	0	0
$843$	14.5348	0.500607
$844$	0	0
$845$	3.56716	0.122714
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	11.2855	0.387319
$850$	0	0
$851$	49.8300	1.70815
$852$	0	0
$853$	21.2571	0.727829	0.363915	−	0.931432i	$-0.381440\pi$
$853$	21.2571	0.727829	0.363915	+	0.931432i	$0.381440\pi$
$854$	0	0
$855$	28.3810	0.970609
$856$	0	0
$857$	−49.5701	−1.69328	−0.846641	−	0.532164i	$-0.821379\pi$
$857$	−49.5701	−1.69328	−0.846641	+	0.532164i	$0.821379\pi$
$858$	0	0
$859$	−21.9034	−0.747333	−0.373666	−	0.927563i	$-0.621899\pi$
$859$	−21.9034	−0.747333	−0.373666	+	0.927563i	$0.621899\pi$
$860$	0	0
$861$	3.39706	0.115771
$862$	0	0
$863$	−4.51320	−0.153631	−0.0768156	−	0.997045i	$-0.524475\pi$
$863$	−4.51320	−0.153631	−0.0768156	+	0.997045i	$0.524475\pi$
$864$	0	0
$865$	30.4246	1.03447
$866$	0	0
$867$	−39.6895	−1.34793
$868$	0	0
$869$	−13.8757	−0.470701
$870$	0	0
$871$	−7.95876	−0.269672
$872$	0	0
$873$	−9.81994	−0.332355
$874$	0	0
$875$	−9.71916	−0.328568
$876$	0	0
$877$	18.2175	0.615162	0.307581	−	0.951522i	$-0.400481\pi$
$877$	18.2175	0.615162	0.307581	+	0.951522i	$0.400481\pi$
$878$	0	0
$879$	16.5464	0.558097
$880$	0	0
$881$	−12.0454	−0.405820	−0.202910	−	0.979197i	$-0.565040\pi$
$881$	−12.0454	−0.405820	−0.202910	+	0.979197i	$0.565040\pi$
$882$	0	0
$883$	−9.66843	−0.325369	−0.162684	−	0.986678i	$-0.552015\pi$
$883$	−9.66843	−0.325369	−0.162684	+	0.986678i	$0.552015\pi$
$884$	0	0
$885$	39.1619	1.31641
$886$	0	0
$887$	−3.46704	−0.116412	−0.0582060	−	0.998305i	$-0.518538\pi$
$887$	−3.46704	−0.116412	−0.0582060	+	0.998305i	$0.518538\pi$
$888$	0	0
$889$	11.5471	0.387278
$890$	0	0
$891$	4.86942	0.163132
$892$	0	0
$893$	−87.6501	−2.93310
$894$	0	0
$895$	22.0851	0.738223
$896$	0	0
$897$	6.14901	0.205309
$898$	0	0
$899$	48.0409	1.60225
$900$	0	0
$901$	−36.2652	−1.20817
$902$	0	0
$903$	−4.80599	−0.159933
$904$	0	0
$905$	27.9325	0.928507
$906$	0	0
$907$	36.1170	1.19925	0.599623	−	0.800283i	$-0.295317\pi$
$907$	36.1170	1.19925	0.599623	+	0.800283i	$0.295317\pi$
$908$	0	0
$909$	−16.4280	−0.544882
$910$	0	0
$911$	−8.53483	−0.282772	−0.141386	−	0.989955i	$-0.545156\pi$
$911$	−8.53483	−0.282772	−0.141386	+	0.989955i	$0.545156\pi$
$912$	0	0
$913$	0.786323	0.0260235
$914$	0	0
$915$	−1.04263	−0.0344681
$916$	0	0
$917$	15.1756	0.501142
$918$	0	0
$919$	−23.1619	−0.764042	−0.382021	−	0.924154i	$-0.624772\pi$
$919$	−23.1619	−0.764042	−0.382021	+	0.924154i	$0.624772\pi$
$920$	0	0
$921$	−41.4357	−1.36535
$922$	0	0
$923$	−7.39527	−0.243418
$924$	0	0
$925$	−87.9506	−2.89180
$926$	0	0
$927$	−10.4575	−0.343471
$928$	0	0
$929$	41.2484	1.35332	0.676658	−	0.736297i	$-0.263427\pi$
$929$	41.2484	1.35332	0.676658	+	0.736297i	$0.263427\pi$
$930$	0	0
$931$	−7.75471	−0.254150
$932$	0	0
$933$	−32.4705	−1.06304
$934$	0	0
$935$	−23.9953	−0.784730
$936$	0	0
$937$	−26.8171	−0.876076	−0.438038	−	0.898956i	$-0.644326\pi$
$937$	−26.8171	−0.876076	−0.438038	+	0.898956i	$0.644326\pi$
$938$	0	0
$939$	20.5721	0.671347
$940$	0	0
$941$	35.5742	1.15969	0.579843	−	0.814728i	$-0.303114\pi$
$941$	35.5742	1.15969	0.579843	+	0.814728i	$0.303114\pi$
$942$	0	0
$943$	−10.5817	−0.344589
$944$	0	0
$945$	−20.1776	−0.656378
$946$	0	0
$947$	−14.5280	−0.472096	−0.236048	−	0.971741i	$-0.575852\pi$
$947$	−14.5280	−0.472096	−0.236048	+	0.971741i	$0.575852\pi$
$948$	0	0
$949$	13.7726	0.447078
$950$	0	0
$951$	−47.5859	−1.54308
$952$	0	0
$953$	20.6535	0.669033	0.334517	−	0.942390i	$-0.391427\pi$
$953$	20.6535	0.669033	0.334517	+	0.942390i	$0.391427\pi$
$954$	0	0
$955$	29.1891	0.944537
$956$	0	0
$957$	8.19167	0.264799
$958$	0	0
$959$	−6.55768	−0.211759
$960$	0	0
$961$	36.8936	1.19012
$962$	0	0
$963$	−18.5650	−0.598249
$964$	0	0
$965$	−74.9141	−2.41157
$966$	0	0
$967$	−3.68620	−0.118540	−0.0592701	−	0.998242i	$-0.518877\pi$
$967$	−3.68620	−0.118540	−0.0592701	+	0.998242i	$0.518877\pi$
$968$	0	0
$969$	73.2900	2.35441
$970$	0	0
$971$	21.3480	0.685090	0.342545	−	0.939501i	$-0.388711\pi$
$971$	21.3480	0.685090	0.342545	+	0.939501i	$0.388711\pi$
$972$	0	0
$973$	−20.4130	−0.654410
$974$	0	0
$975$	−10.8531	−0.347577
$976$	0	0
$977$	40.9622	1.31050	0.655249	−	0.755413i	$-0.272564\pi$
$977$	40.9622	1.31050	0.655249	+	0.755413i	$0.272564\pi$
$978$	0	0
$979$	4.96883	0.158804
$980$	0	0
$981$	7.62300	0.243384
$982$	0	0
$983$	−15.3459	−0.489460	−0.244730	−	0.969591i	$-0.578699\pi$
$983$	−15.3459	−0.489460	−0.244730	+	0.969591i	$0.578699\pi$
$984$	0	0
$985$	−45.3971	−1.44647
$986$	0	0
$987$	15.8804	0.505480
$988$	0	0
$989$	14.9705	0.476034
$990$	0	0
$991$	−8.34847	−0.265198	−0.132599	−	0.991170i	$-0.542332\pi$
$991$	−8.34847	−0.265198	−0.132599	+	0.991170i	$0.542332\pi$
$992$	0	0
$993$	16.4052	0.520602
$994$	0	0
$995$	77.7215	2.46394
$996$	0	0
$997$	−44.1777	−1.39912	−0.699561	−	0.714573i	$-0.746621\pi$
$997$	−44.1777	−1.39912	−0.699561	+	0.714573i	$0.746621\pi$
$998$	0	0
$999$	−64.4034	−2.03763

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.z.1.6	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.z.1.6	✓	15	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.40500 q^{3} +3.56716 q^{5} -1.00000 q^{7} -1.02598 q^{9} +O(q^{10})\)	\(q-1.40500 q^{3} +3.56716 q^{5} -1.00000 q^{7} -1.02598 q^{9} -1.00000 q^{11} +1.00000 q^{13} -5.01185 q^{15} +6.72672 q^{17} -7.75471 q^{19} +1.40500 q^{21} -4.37653 q^{23} +7.72462 q^{25} +5.65649 q^{27} +5.83038 q^{29} +8.23976 q^{31} +1.40500 q^{33} -3.56716 q^{35} -11.3857 q^{37} -1.40500 q^{39} +2.41784 q^{41} -3.42064 q^{43} -3.65984 q^{45} +11.3028 q^{47} +1.00000 q^{49} -9.45103 q^{51} -5.39121 q^{53} -3.56716 q^{55} +10.8953 q^{57} -7.81386 q^{59} +0.208032 q^{61} +1.02598 q^{63} +3.56716 q^{65} -7.95876 q^{67} +6.14901 q^{69} -7.39527 q^{71} +13.7726 q^{73} -10.8531 q^{75} +1.00000 q^{77} +13.8757 q^{79} -4.86942 q^{81} -0.786323 q^{83} +23.9953 q^{85} -8.19167 q^{87} -4.96883 q^{89} -1.00000 q^{91} -11.5768 q^{93} -27.6623 q^{95} +9.57127 q^{97} +1.02598 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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