LMFDB - Embedded newform 8008.2.a.x.1.2

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 17 x^{10} + 79 x^{9} + 80 x^{8} - 536 x^{7} - 4 x^{6} + 1484 x^{5} - 682 x^{4} + \cdots - 64 $ x^12 - 4x^11 - 17x^10 + 79x^9 + 80x^8 - 536x^7 - 4x^6 + 1484x^5 - 682x^4 - 1431x^3 + 1069x^2 - 64
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			1.2
Root			$-2.33021$ 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)

$f(q)$	$=$	$q-2.33021 q^{3} -0.450718 q^{5} +1.00000 q^{7} +2.42987 q^{9} +O(q^{10})$	$q-2.33021 q^{3} -0.450718 q^{5} +1.00000 q^{7} +2.42987 q^{9} +1.00000 q^{11} +1.00000 q^{13} +1.05027 q^{15} -4.58105 q^{17} -1.00053 q^{19} -2.33021 q^{21} -2.87448 q^{23} -4.79685 q^{25} +1.32852 q^{27} +5.30382 q^{29} -0.0259529 q^{31} -2.33021 q^{33} -0.450718 q^{35} +6.00386 q^{37} -2.33021 q^{39} -0.530599 q^{41} -6.30811 q^{43} -1.09519 q^{45} +1.34854 q^{47} +1.00000 q^{49} +10.6748 q^{51} -5.06192 q^{53} -0.450718 q^{55} +2.33145 q^{57} +0.893254 q^{59} +9.35972 q^{61} +2.42987 q^{63} -0.450718 q^{65} -10.2223 q^{67} +6.69815 q^{69} +4.88228 q^{71} +9.71338 q^{73} +11.1777 q^{75} +1.00000 q^{77} -7.26545 q^{79} -10.3853 q^{81} +10.3646 q^{83} +2.06476 q^{85} -12.3590 q^{87} -2.17830 q^{89} +1.00000 q^{91} +0.0604757 q^{93} +0.450959 q^{95} +17.3624 q^{97} +2.42987 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) $ 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9	$ 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{13} - 3 q^{15} + 16 q^{17} - 2 q^{19} + 4 q^{21} + 9 q^{23} + 14 q^{25} + 7 q^{27} + 15 q^{29} + 10 q^{31} + 4 q^{33} + 6 q^{35} + 18 q^{37} + 4 q^{39} + 24 q^{41} + 15 q^{45} + 5 q^{47} + 12 q^{49} + 4 q^{51} + 15 q^{53} + 6 q^{55} - 4 q^{57} + 15 q^{59} + 17 q^{61} + 14 q^{63} + 6 q^{65} - 7 q^{67} + 9 q^{71} + 32 q^{73} - 8 q^{75} + 12 q^{77} + 20 q^{79} - 4 q^{81} - 5 q^{83} + 25 q^{85} + 19 q^{87} + 16 q^{89} + 12 q^{91} + 21 q^{93} + 8 q^{95} + 10 q^{97} + 14 q^{99}+O(q^{100}) $ 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9 + 12 * q^11 + 12 * q^13 - 3 * q^15 + 16 * q^17 - 2 * q^19 + 4 * q^21 + 9 * q^23 + 14 * q^25 + 7 * q^27 + 15 * q^29 + 10 * q^31 + 4 * q^33 + 6 * q^35 + 18 * q^37 + 4 * q^39 + 24 * q^41 + 15 * q^45 + 5 * q^47 + 12 * q^49 + 4 * q^51 + 15 * q^53 + 6 * q^55 - 4 * q^57 + 15 * q^59 + 17 * q^61 + 14 * q^63 + 6 * q^65 - 7 * q^67 + 9 * q^71 + 32 * q^73 - 8 * q^75 + 12 * q^77 + 20 * q^79 - 4 * q^81 - 5 * q^83 + 25 * q^85 + 19 * q^87 + 16 * q^89 + 12 * q^91 + 21 * q^93 + 8 * q^95 + 10 * q^97 + 14 * 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
$3$	−2.33021	−1.34535	−0.672673	−	0.739940i	$-0.734854\pi$
$3$	−2.33021	−1.34535	−0.672673	+	0.739940i	$0.734854\pi$
$4$	0	0
$5$	−0.450718	−0.201567	−0.100784	−	0.994908i	$-0.532135\pi$
$5$	−0.450718	−0.201567	−0.100784	+	0.994908i	$0.532135\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.42987	0.809957
$10$	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
$15$	1.05027	0.271178
$16$	0	0
$17$	−4.58105	−1.11107	−0.555534	−	0.831494i	$-0.687486\pi$
$17$	−4.58105	−1.11107	−0.555534	+	0.831494i	$0.687486\pi$
$18$	0	0
$19$	−1.00053	−0.229538	−0.114769	−	0.993392i	$-0.536613\pi$
$19$	−1.00053	−0.229538	−0.114769	+	0.993392i	$0.536613\pi$
$20$	0	0
$21$	−2.33021	−0.508493
$22$	0	0
$23$	−2.87448	−0.599371	−0.299686	−	0.954038i	$-0.596882\pi$
$23$	−2.87448	−0.599371	−0.299686	+	0.954038i	$0.596882\pi$
$24$	0	0
$25$	−4.79685	−0.959371
$26$	0	0
$27$	1.32852	0.255673
$28$	0	0
$29$	5.30382	0.984895	0.492447	−	0.870342i	$-0.336102\pi$
$29$	5.30382	0.984895	0.492447	+	0.870342i	$0.336102\pi$
$30$	0	0
$31$	−0.0259529	−0.00466128	−0.00233064	−	0.999997i	$-0.500742\pi$
$31$	−0.0259529	−0.00466128	−0.00233064	+	0.999997i	$0.500742\pi$
$32$	0	0
$33$	−2.33021	−0.405637
$34$	0	0
$35$	−0.450718	−0.0761852
$36$	0	0
$37$	6.00386	0.987028	0.493514	−	0.869738i	$-0.335712\pi$
$37$	6.00386	0.987028	0.493514	+	0.869738i	$0.335712\pi$
$38$	0	0
$39$	−2.33021	−0.373132
$40$	0	0
$41$	−0.530599	−0.0828657	−0.0414329	−	0.999141i	$-0.513192\pi$
$41$	−0.530599	−0.0828657	−0.0414329	+	0.999141i	$0.513192\pi$
$42$	0	0
$43$	−6.30811	−0.961978	−0.480989	−	0.876727i	$-0.659722\pi$
$43$	−6.30811	−0.961978	−0.480989	+	0.876727i	$0.659722\pi$
$44$	0	0
$45$	−1.09519	−0.163261
$46$	0	0
$47$	1.34854	0.196705	0.0983526	−	0.995152i	$-0.468643\pi$
$47$	1.34854	0.196705	0.0983526	+	0.995152i	$0.468643\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	10.6748	1.49477
$52$	0	0
$53$	−5.06192	−0.695308	−0.347654	−	0.937623i	$-0.613022\pi$
$53$	−5.06192	−0.695308	−0.347654	+	0.937623i	$0.613022\pi$
$54$	0	0
$55$	−0.450718	−0.0607748
$56$	0	0
$57$	2.33145	0.308809
$58$	0	0
$59$	0.893254	0.116292	0.0581459	−	0.998308i	$-0.481481\pi$
$59$	0.893254	0.116292	0.0581459	+	0.998308i	$0.481481\pi$
$60$	0	0
$61$	9.35972	1.19839	0.599195	−	0.800603i	$-0.295487\pi$
$61$	9.35972	1.19839	0.599195	+	0.800603i	$0.295487\pi$
$62$	0	0
$63$	2.42987	0.306135
$64$	0	0
$65$	−0.450718	−0.0559046
$66$	0	0
$67$	−10.2223	−1.24885	−0.624427	−	0.781083i	$-0.714667\pi$
$67$	−10.2223	−1.24885	−0.624427	+	0.781083i	$0.714667\pi$
$68$	0	0
$69$	6.69815	0.806362
$70$	0	0
$71$	4.88228	0.579420	0.289710	−	0.957114i	$-0.406441\pi$
$71$	4.88228	0.579420	0.289710	+	0.957114i	$0.406441\pi$
$72$	0	0
$73$	9.71338	1.13686	0.568432	−	0.822730i	$-0.307550\pi$
$73$	9.71338	1.13686	0.568432	+	0.822730i	$0.307550\pi$
$74$	0	0
$75$	11.1777	1.29069
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−7.26545	−0.817427	−0.408713	−	0.912663i	$-0.634022\pi$
$79$	−7.26545	−0.817427	−0.408713	+	0.912663i	$0.634022\pi$
$80$	0	0
$81$	−10.3853	−1.15393
$82$	0	0
$83$	10.3646	1.13767	0.568833	−	0.822453i	$-0.307395\pi$
$83$	10.3646	1.13767	0.568833	+	0.822453i	$0.307395\pi$
$84$	0	0
$85$	2.06476	0.223955
$86$	0	0
$87$	−12.3590	−1.32502
$88$	0	0
$89$	−2.17830	−0.230899	−0.115449	−	0.993313i	$-0.536831\pi$
$89$	−2.17830	−0.230899	−0.115449	+	0.993313i	$0.536831\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0.0604757	0.00627104
$94$	0	0
$95$	0.450959	0.0462674
$96$	0	0
$97$	17.3624	1.76289	0.881443	−	0.472290i	$-0.156572\pi$
$97$	17.3624	1.76289	0.881443	+	0.472290i	$0.156572\pi$
$98$	0	0
$99$	2.42987	0.244211
$100$	0	0
$101$	−11.9323	−1.18730	−0.593652	−	0.804722i	$-0.702314\pi$
$101$	−11.9323	−1.18730	−0.593652	+	0.804722i	$0.702314\pi$
$102$	0	0
$103$	−8.95618	−0.882478	−0.441239	−	0.897390i	$-0.645461\pi$
$103$	−8.95618	−0.882478	−0.441239	+	0.897390i	$0.645461\pi$
$104$	0	0
$105$	1.05027	0.102495
$106$	0	0
$107$	2.05546	0.198709	0.0993545	−	0.995052i	$-0.468322\pi$
$107$	2.05546	0.198709	0.0993545	+	0.995052i	$0.468322\pi$
$108$	0	0
$109$	−17.9187	−1.71630	−0.858151	−	0.513397i	$-0.828387\pi$
$109$	−17.9187	−1.71630	−0.858151	+	0.513397i	$0.828387\pi$
$110$	0	0
$111$	−13.9902	−1.32789
$112$	0	0
$113$	1.54475	0.145318	0.0726588	−	0.997357i	$-0.476852\pi$
$113$	1.54475	0.145318	0.0726588	+	0.997357i	$0.476852\pi$
$114$	0	0
$115$	1.29558	0.120814
$116$	0	0
$117$	2.42987	0.224642
$118$	0	0
$119$	−4.58105	−0.419944
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.23641	0.111483
$124$	0	0
$125$	4.41562	0.394945
$126$	0	0
$127$	−10.0917	−0.895491	−0.447746	−	0.894161i	$-0.647773\pi$
$127$	−10.0917	−0.895491	−0.447746	+	0.894161i	$0.647773\pi$
$128$	0	0
$129$	14.6992	1.29419
$130$	0	0
$131$	3.00573	0.262612	0.131306	−	0.991342i	$-0.458083\pi$
$131$	3.00573	0.262612	0.131306	+	0.991342i	$0.458083\pi$
$132$	0	0
$133$	−1.00053	−0.0867574
$134$	0	0
$135$	−0.598787	−0.0515354
$136$	0	0
$137$	17.4517	1.49100	0.745499	−	0.666506i	$-0.232211\pi$
$137$	17.4517	1.49100	0.745499	+	0.666506i	$0.232211\pi$
$138$	0	0
$139$	−12.8482	−1.08977	−0.544885	−	0.838511i	$-0.683427\pi$
$139$	−12.8482	−1.08977	−0.544885	+	0.838511i	$0.683427\pi$
$140$	0	0
$141$	−3.14239	−0.264637
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−2.39053	−0.198522
$146$	0	0
$147$	−2.33021	−0.192192
$148$	0	0
$149$	−13.3756	−1.09577	−0.547884	−	0.836554i	$-0.684567\pi$
$149$	−13.3756	−1.09577	−0.547884	+	0.836554i	$0.684567\pi$
$150$	0	0
$151$	−2.03284	−0.165430	−0.0827152	−	0.996573i	$-0.526359\pi$
$151$	−2.03284	−0.165430	−0.0827152	+	0.996573i	$0.526359\pi$
$152$	0	0
$153$	−11.1314	−0.899918
$154$	0	0
$155$	0.0116974	0.000939561 0
$156$	0	0
$157$	9.57967	0.764541	0.382271	−	0.924050i	$-0.375142\pi$
$157$	9.57967	0.764541	0.382271	+	0.924050i	$0.375142\pi$
$158$	0	0
$159$	11.7953	0.935430
$160$	0	0
$161$	−2.87448	−0.226541
$162$	0	0
$163$	−19.6300	−1.53754	−0.768772	−	0.639523i	$-0.779132\pi$
$163$	−19.6300	−1.53754	−0.768772	+	0.639523i	$0.779132\pi$
$164$	0	0
$165$	1.05027	0.0817631
$166$	0	0
$167$	13.1028	1.01392	0.506962	−	0.861968i	$-0.330768\pi$
$167$	13.1028	1.01392	0.506962	+	0.861968i	$0.330768\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.43117	−0.185916
$172$	0	0
$173$	12.4125	0.943704	0.471852	−	0.881678i	$-0.343586\pi$
$173$	12.4125	0.943704	0.471852	+	0.881678i	$0.343586\pi$
$174$	0	0
$175$	−4.79685	−0.362608
$176$	0	0
$177$	−2.08147	−0.156453
$178$	0	0
$179$	−2.47029	−0.184638	−0.0923190	−	0.995729i	$-0.529428\pi$
$179$	−2.47029	−0.184638	−0.0923190	+	0.995729i	$0.529428\pi$
$180$	0	0
$181$	−15.1195	−1.12382	−0.561912	−	0.827197i	$-0.689934\pi$
$181$	−15.1195	−1.12382	−0.561912	+	0.827197i	$0.689934\pi$
$182$	0	0
$183$	−21.8101	−1.61225
$184$	0	0
$185$	−2.70604	−0.198952
$186$	0	0
$187$	−4.58105	−0.335000
$188$	0	0
$189$	1.32852	0.0966355
$190$	0	0
$191$	−9.65521	−0.698626	−0.349313	−	0.937006i	$-0.613585\pi$
$191$	−9.65521	−0.698626	−0.349313	+	0.937006i	$0.613585\pi$
$192$	0	0
$193$	22.8775	1.64676	0.823378	−	0.567493i	$-0.192087\pi$
$193$	22.8775	1.64676	0.823378	+	0.567493i	$0.192087\pi$
$194$	0	0
$195$	1.05027	0.0752111
$196$	0	0
$197$	16.4437	1.17156	0.585782	−	0.810469i	$-0.300787\pi$
$197$	16.4437	1.17156	0.585782	+	0.810469i	$0.300787\pi$
$198$	0	0
$199$	24.7124	1.75181	0.875907	−	0.482479i	$-0.160264\pi$
$199$	24.7124	1.75181	0.875907	+	0.482479i	$0.160264\pi$
$200$	0	0
$201$	23.8201	1.68014
$202$	0	0
$203$	5.30382	0.372255
$204$	0	0
$205$	0.239151	0.0167030
$206$	0	0
$207$	−6.98463	−0.485465
$208$	0	0
$209$	−1.00053	−0.0692084
$210$	0	0
$211$	−9.25844	−0.637377	−0.318689	−	0.947859i	$-0.603242\pi$
$211$	−9.25844	−0.637377	−0.318689	+	0.947859i	$0.603242\pi$
$212$	0	0
$213$	−11.3767	−0.779521
$214$	0	0
$215$	2.84318	0.193903
$216$	0	0
$217$	−0.0259529	−0.00176180
$218$	0	0
$219$	−22.6342	−1.52948
$220$	0	0
$221$	−4.58105	−0.308155
$222$	0	0
$223$	−3.75137	−0.251210	−0.125605	−	0.992080i	$-0.540087\pi$
$223$	−3.75137	−0.251210	−0.125605	+	0.992080i	$0.540087\pi$
$224$	0	0
$225$	−11.6557	−0.777049
$226$	0	0
$227$	−24.7262	−1.64114	−0.820568	−	0.571549i	$-0.806343\pi$
$227$	−24.7262	−1.64114	−0.820568	+	0.571549i	$0.806343\pi$
$228$	0	0
$229$	−15.6933	−1.03704	−0.518520	−	0.855066i	$-0.673517\pi$
$229$	−15.6933	−1.03704	−0.518520	+	0.855066i	$0.673517\pi$
$230$	0	0
$231$	−2.33021	−0.153316
$232$	0	0
$233$	12.4394	0.814931	0.407465	−	0.913221i	$-0.366413\pi$
$233$	12.4394	0.814931	0.407465	+	0.913221i	$0.366413\pi$
$234$	0	0
$235$	−0.607812	−0.0396493
$236$	0	0
$237$	16.9300	1.09972
$238$	0	0
$239$	−26.7912	−1.73298	−0.866489	−	0.499196i	$-0.833629\pi$
$239$	−26.7912	−1.73298	−0.866489	+	0.499196i	$0.833629\pi$
$240$	0	0
$241$	17.6366	1.13608	0.568038	−	0.823003i	$-0.307703\pi$
$241$	17.6366	1.13608	0.568038	+	0.823003i	$0.307703\pi$
$242$	0	0
$243$	20.2144	1.29676
$244$	0	0
$245$	−0.450718	−0.0287953
$246$	0	0
$247$	−1.00053	−0.0636625
$248$	0	0
$249$	−24.1517	−1.53055
$250$	0	0
$251$	−3.25129	−0.205220	−0.102610	−	0.994722i	$-0.532719\pi$
$251$	−3.25129	−0.205220	−0.102610	+	0.994722i	$0.532719\pi$
$252$	0	0
$253$	−2.87448	−0.180717
$254$	0	0
$255$	−4.81132	−0.301297
$256$	0	0
$257$	0.379665	0.0236828	0.0118414	−	0.999930i	$-0.496231\pi$
$257$	0.379665	0.0236828	0.0118414	+	0.999930i	$0.496231\pi$
$258$	0	0
$259$	6.00386	0.373062
$260$	0	0
$261$	12.8876	0.797722
$262$	0	0
$263$	−0.135924	−0.00838146	−0.00419073	−	0.999991i	$-0.501334\pi$
$263$	−0.135924	−0.00838146	−0.00419073	+	0.999991i	$0.501334\pi$
$264$	0	0
$265$	2.28150	0.140151
$266$	0	0
$267$	5.07588	0.310639
$268$	0	0
$269$	−10.2112	−0.622588	−0.311294	−	0.950314i	$-0.600762\pi$
$269$	−10.2112	−0.622588	−0.311294	+	0.950314i	$0.600762\pi$
$270$	0	0
$271$	25.5262	1.55060	0.775302	−	0.631590i	$-0.217598\pi$
$271$	25.5262	1.55060	0.775302	+	0.631590i	$0.217598\pi$
$272$	0	0
$273$	−2.33021	−0.141031
$274$	0	0
$275$	−4.79685	−0.289261
$276$	0	0
$277$	−4.03154	−0.242232	−0.121116	−	0.992638i	$-0.538647\pi$
$277$	−4.03154	−0.242232	−0.121116	+	0.992638i	$0.538647\pi$
$278$	0	0
$279$	−0.0630623	−0.00377544
$280$	0	0
$281$	25.8208	1.54034	0.770169	−	0.637840i	$-0.220172\pi$
$281$	25.8208	1.54034	0.770169	+	0.637840i	$0.220172\pi$
$282$	0	0
$283$	10.5421	0.626661	0.313331	−	0.949644i	$-0.398555\pi$
$283$	10.5421	0.626661	0.313331	+	0.949644i	$0.398555\pi$
$284$	0	0
$285$	−1.05083	−0.0622457
$286$	0	0
$287$	−0.530599	−0.0313203
$288$	0	0
$289$	3.98603	0.234472
$290$	0	0
$291$	−40.4581	−2.37169
$292$	0	0
$293$	23.6892	1.38394	0.691969	−	0.721927i	$-0.256743\pi$
$293$	23.6892	1.38394	0.691969	+	0.721927i	$0.256743\pi$
$294$	0	0
$295$	−0.402605	−0.0234406
$296$	0	0
$297$	1.32852	0.0770885
$298$	0	0
$299$	−2.87448	−0.166236
$300$	0	0
$301$	−6.30811	−0.363594
$302$	0	0
$303$	27.8046	1.59734
$304$	0	0
$305$	−4.21859	−0.241556
$306$	0	0
$307$	−1.72141	−0.0982462	−0.0491231	−	0.998793i	$-0.515643\pi$
$307$	−1.72141	−0.0982462	−0.0491231	+	0.998793i	$0.515643\pi$
$308$	0	0
$309$	20.8698	1.18724
$310$	0	0
$311$	−3.60043	−0.204162	−0.102081	−	0.994776i	$-0.532550\pi$
$311$	−3.60043	−0.204162	−0.102081	+	0.994776i	$0.532550\pi$
$312$	0	0
$313$	14.1492	0.799760	0.399880	−	0.916568i	$-0.369052\pi$
$313$	14.1492	0.799760	0.399880	+	0.916568i	$0.369052\pi$
$314$	0	0
$315$	−1.09519	−0.0617067
$316$	0	0
$317$	20.6807	1.16154	0.580772	−	0.814067i	$-0.302751\pi$
$317$	20.6807	1.16154	0.580772	+	0.814067i	$0.302751\pi$
$318$	0	0
$319$	5.30382	0.296957
$320$	0	0
$321$	−4.78965	−0.267332
$322$	0	0
$323$	4.58350	0.255033
$324$	0	0
$325$	−4.79685	−0.266082
$326$	0	0
$327$	41.7544	2.30902
$328$	0	0
$329$	1.34854	0.0743476
$330$	0	0
$331$	4.98348	0.273917	0.136959	−	0.990577i	$-0.456267\pi$
$331$	4.98348	0.273917	0.136959	+	0.990577i	$0.456267\pi$
$332$	0	0
$333$	14.5886	0.799450
$334$	0	0
$335$	4.60738	0.251728
$336$	0	0
$337$	25.7649	1.40350	0.701752	−	0.712421i	$-0.252401\pi$
$337$	25.7649	1.40350	0.701752	+	0.712421i	$0.252401\pi$
$338$	0	0
$339$	−3.59958	−0.195503
$340$	0	0
$341$	−0.0259529	−0.00140543
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−3.01897	−0.162536
$346$	0	0
$347$	−2.98884	−0.160449	−0.0802247	−	0.996777i	$-0.525564\pi$
$347$	−2.98884	−0.160449	−0.0802247	+	0.996777i	$0.525564\pi$
$348$	0	0
$349$	4.76464	0.255045	0.127523	−	0.991836i	$-0.459297\pi$
$349$	4.76464	0.255045	0.127523	+	0.991836i	$0.459297\pi$
$350$	0	0
$351$	1.32852	0.0709111
$352$	0	0
$353$	5.68291	0.302471	0.151235	−	0.988498i	$-0.451675\pi$
$353$	5.68291	0.302471	0.151235	+	0.988498i	$0.451675\pi$
$354$	0	0
$355$	−2.20053	−0.116792
$356$	0	0
$357$	10.6748	0.564971
$358$	0	0
$359$	18.6108	0.982238	0.491119	−	0.871092i	$-0.336588\pi$
$359$	18.6108	0.982238	0.491119	+	0.871092i	$0.336588\pi$
$360$	0	0
$361$	−17.9989	−0.947312
$362$	0	0
$363$	−2.33021	−0.122304
$364$	0	0
$365$	−4.37799	−0.229154
$366$	0	0
$367$	−26.9528	−1.40692	−0.703462	−	0.710733i	$-0.748363\pi$
$367$	−26.9528	−1.40692	−0.703462	+	0.710733i	$0.748363\pi$
$368$	0	0
$369$	−1.28929	−0.0671177
$370$	0	0
$371$	−5.06192	−0.262802
$372$	0	0
$373$	−29.3689	−1.52067	−0.760333	−	0.649533i	$-0.774964\pi$
$373$	−29.3689	−1.52067	−0.760333	+	0.649533i	$0.774964\pi$
$374$	0	0
$375$	−10.2893	−0.531337
$376$	0	0
$377$	5.30382	0.273161
$378$	0	0
$379$	31.1019	1.59760	0.798799	−	0.601597i	$-0.205469\pi$
$379$	31.1019	1.59760	0.798799	+	0.601597i	$0.205469\pi$
$380$	0	0
$381$	23.5157	1.20475
$382$	0	0
$383$	18.4474	0.942621	0.471310	−	0.881967i	$-0.343781\pi$
$383$	18.4474	0.942621	0.471310	+	0.881967i	$0.343781\pi$
$384$	0	0
$385$	−0.450718	−0.0229707
$386$	0	0
$387$	−15.3279	−0.779161
$388$	0	0
$389$	38.0797	1.93072	0.965358	−	0.260929i	$-0.0840289\pi$
$389$	38.0797	1.93072	0.965358	+	0.260929i	$0.0840289\pi$
$390$	0	0
$391$	13.1682	0.665942
$392$	0	0
$393$	−7.00397	−0.353304
$394$	0	0
$395$	3.27467	0.164766
$396$	0	0
$397$	31.0979	1.56076	0.780380	−	0.625306i	$-0.215026\pi$
$397$	31.0979	1.56076	0.780380	+	0.625306i	$0.215026\pi$
$398$	0	0
$399$	2.33145	0.116719
$400$	0	0
$401$	30.1397	1.50510	0.752552	−	0.658533i	$-0.228823\pi$
$401$	30.1397	1.50510	0.752552	+	0.658533i	$0.228823\pi$
$402$	0	0
$403$	−0.0259529	−0.00129281
$404$	0	0
$405$	4.68086	0.232594
$406$	0	0
$407$	6.00386	0.297600
$408$	0	0
$409$	2.23995	0.110759	0.0553793	−	0.998465i	$-0.482363\pi$
$409$	2.23995	0.110759	0.0553793	+	0.998465i	$0.482363\pi$
$410$	0	0
$411$	−40.6661	−2.00591
$412$	0	0
$413$	0.893254	0.0439541
$414$	0	0
$415$	−4.67152	−0.229316
$416$	0	0
$417$	29.9390	1.46612
$418$	0	0
$419$	−1.18453	−0.0578680	−0.0289340	−	0.999581i	$-0.509211\pi$
$419$	−1.18453	−0.0578680	−0.0289340	+	0.999581i	$0.509211\pi$
$420$	0	0
$421$	11.5501	0.562917	0.281458	−	0.959573i	$-0.409182\pi$
$421$	11.5501	0.562917	0.281458	+	0.959573i	$0.409182\pi$
$422$	0	0
$423$	3.27679	0.159323
$424$	0	0
$425$	21.9746	1.06593
$426$	0	0
$427$	9.35972	0.452949
$428$	0	0
$429$	−2.33021	−0.112504
$430$	0	0
$431$	35.3887	1.70461	0.852307	−	0.523042i	$-0.175203\pi$
$431$	35.3887	1.70461	0.852307	+	0.523042i	$0.175203\pi$
$432$	0	0
$433$	−9.11186	−0.437888	−0.218944	−	0.975737i	$-0.570261\pi$
$433$	−9.11186	−0.437888	−0.218944	+	0.975737i	$0.570261\pi$
$434$	0	0
$435$	5.57042	0.267081
$436$	0	0
$437$	2.87602	0.137579
$438$	0	0
$439$	19.0021	0.906923	0.453462	−	0.891276i	$-0.350189\pi$
$439$	19.0021	0.906923	0.453462	+	0.891276i	$0.350189\pi$
$440$	0	0
$441$	2.42987	0.115708
$442$	0	0
$443$	29.2548	1.38994	0.694968	−	0.719041i	$-0.255419\pi$
$443$	29.2548	1.38994	0.694968	+	0.719041i	$0.255419\pi$
$444$	0	0
$445$	0.981796	0.0465416
$446$	0	0
$447$	31.1678	1.47419
$448$	0	0
$449$	0.232309	0.0109633	0.00548166	−	0.999985i	$-0.498255\pi$
$449$	0.232309	0.0109633	0.00548166	+	0.999985i	$0.498255\pi$
$450$	0	0
$451$	−0.530599	−0.0249850
$452$	0	0
$453$	4.73695	0.222561
$454$	0	0
$455$	−0.450718	−0.0211300
$456$	0	0
$457$	−10.1754	−0.475986	−0.237993	−	0.971267i	$-0.576490\pi$
$457$	−10.1754	−0.475986	−0.237993	+	0.971267i	$0.576490\pi$
$458$	0	0
$459$	−6.08601	−0.284071
$460$	0	0
$461$	−19.8112	−0.922698	−0.461349	−	0.887219i	$-0.652634\pi$
$461$	−19.8112	−0.922698	−0.461349	+	0.887219i	$0.652634\pi$
$462$	0	0
$463$	10.4491	0.485613	0.242806	−	0.970075i	$-0.421932\pi$
$463$	10.4491	0.485613	0.242806	+	0.970075i	$0.421932\pi$
$464$	0	0
$465$	−0.0272575	−0.00126404
$466$	0	0
$467$	26.3079	1.21739	0.608693	−	0.793406i	$-0.291694\pi$
$467$	26.3079	1.21739	0.608693	+	0.793406i	$0.291694\pi$
$468$	0	0
$469$	−10.2223	−0.472022
$470$	0	0
$471$	−22.3226	−1.02857
$472$	0	0
$473$	−6.30811	−0.290047
$474$	0	0
$475$	4.79942	0.220212
$476$	0	0
$477$	−12.2998	−0.563170
$478$	0	0
$479$	6.95042	0.317573	0.158786	−	0.987313i	$-0.449242\pi$
$479$	6.95042	0.317573	0.158786	+	0.987313i	$0.449242\pi$
$480$	0	0
$481$	6.00386	0.273752
$482$	0	0
$483$	6.69815	0.304776
$484$	0	0
$485$	−7.82555	−0.355340
$486$	0	0
$487$	3.58146	0.162291	0.0811457	−	0.996702i	$-0.474142\pi$
$487$	3.58146	0.162291	0.0811457	+	0.996702i	$0.474142\pi$
$488$	0	0
$489$	45.7421	2.06853
$490$	0	0
$491$	−3.47121	−0.156654	−0.0783268	−	0.996928i	$-0.524958\pi$
$491$	−3.47121	−0.156654	−0.0783268	+	0.996928i	$0.524958\pi$
$492$	0	0
$493$	−24.2971	−1.09429
$494$	0	0
$495$	−1.09519	−0.0492250
$496$	0	0
$497$	4.88228	0.219000
$498$	0	0
$499$	−16.6368	−0.744766	−0.372383	−	0.928079i	$-0.621459\pi$
$499$	−16.6368	−0.744766	−0.372383	+	0.928079i	$0.621459\pi$
$500$	0	0
$501$	−30.5322	−1.36408
$502$	0	0
$503$	4.93936	0.220235	0.110118	−	0.993919i	$-0.464877\pi$
$503$	4.93936	0.220235	0.110118	+	0.993919i	$0.464877\pi$
$504$	0	0
$505$	5.37808	0.239321
$506$	0	0
$507$	−2.33021	−0.103488
$508$	0	0
$509$	−13.9000	−0.616106	−0.308053	−	0.951369i	$-0.599677\pi$
$509$	−13.9000	−0.616106	−0.308053	+	0.951369i	$0.599677\pi$
$510$	0	0
$511$	9.71338	0.429694
$512$	0	0
$513$	−1.32923	−0.0586869
$514$	0	0
$515$	4.03671	0.177879
$516$	0	0
$517$	1.34854	0.0593089
$518$	0	0
$519$	−28.9237	−1.26961
$520$	0	0
$521$	14.7891	0.647921	0.323960	−	0.946071i	$-0.394986\pi$
$521$	14.7891	0.647921	0.323960	+	0.946071i	$0.394986\pi$
$522$	0	0
$523$	−18.9123	−0.826978	−0.413489	−	0.910509i	$-0.635690\pi$
$523$	−18.9123	−0.826978	−0.413489	+	0.910509i	$0.635690\pi$
$524$	0	0
$525$	11.1777	0.487833
$526$	0	0
$527$	0.118892	0.00517900
$528$	0	0
$529$	−14.7373	−0.640754
$530$	0	0
$531$	2.17049	0.0941913
$532$	0	0
$533$	−0.530599	−0.0229828
$534$	0	0
$535$	−0.926433	−0.0400532
$536$	0	0
$537$	5.75628	0.248402
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	27.4396	1.17972	0.589860	−	0.807505i	$-0.299183\pi$
$541$	27.4396	1.17972	0.589860	+	0.807505i	$0.299183\pi$
$542$	0	0
$543$	35.2316	1.51193
$544$	0	0
$545$	8.07629	0.345950
$546$	0	0
$547$	17.5157	0.748916	0.374458	−	0.927244i	$-0.377829\pi$
$547$	17.5157	0.748916	0.374458	+	0.927244i	$0.377829\pi$
$548$	0	0
$549$	22.7429	0.970644
$550$	0	0
$551$	−5.30666	−0.226071
$552$	0	0
$553$	−7.26545	−0.308958
$554$	0	0
$555$	6.30565	0.267660
$556$	0	0
$557$	−3.54300	−0.150122	−0.0750609	−	0.997179i	$-0.523915\pi$
$557$	−3.54300	−0.150122	−0.0750609	+	0.997179i	$0.523915\pi$
$558$	0	0
$559$	−6.30811	−0.266805
$560$	0	0
$561$	10.6748	0.450691
$562$	0	0
$563$	3.26335	0.137534	0.0687669	−	0.997633i	$-0.478094\pi$
$563$	3.26335	0.137534	0.0687669	+	0.997633i	$0.478094\pi$
$564$	0	0
$565$	−0.696245	−0.0292912
$566$	0	0
$567$	−10.3853	−0.436143
$568$	0	0
$569$	−5.63675	−0.236305	−0.118152	−	0.992995i	$-0.537697\pi$
$569$	−5.63675	−0.236305	−0.118152	+	0.992995i	$0.537697\pi$
$570$	0	0
$571$	4.98849	0.208762	0.104381	−	0.994537i	$-0.466714\pi$
$571$	4.98849	0.208762	0.104381	+	0.994537i	$0.466714\pi$
$572$	0	0
$573$	22.4986	0.939894
$574$	0	0
$575$	13.7885	0.575019
$576$	0	0
$577$	20.0910	0.836401	0.418201	−	0.908355i	$-0.362661\pi$
$577$	20.0910	0.836401	0.418201	+	0.908355i	$0.362661\pi$
$578$	0	0
$579$	−53.3093	−2.21546
$580$	0	0
$581$	10.3646	0.429997
$582$	0	0
$583$	−5.06192	−0.209643
$584$	0	0
$585$	−1.09519	−0.0452804
$586$	0	0
$587$	7.37830	0.304535	0.152268	−	0.988339i	$-0.451342\pi$
$587$	7.37830	0.304535	0.152268	+	0.988339i	$0.451342\pi$
$588$	0	0
$589$	0.0259668	0.00106994
$590$	0	0
$591$	−38.3172	−1.57616
$592$	0	0
$593$	32.3134	1.32695	0.663476	−	0.748197i	$-0.269080\pi$
$593$	32.3134	1.32695	0.663476	+	0.748197i	$0.269080\pi$
$594$	0	0
$595$	2.06476	0.0846469
$596$	0	0
$597$	−57.5850	−2.35680
$598$	0	0
$599$	22.1865	0.906516	0.453258	−	0.891379i	$-0.350262\pi$
$599$	22.1865	0.906516	0.453258	+	0.891379i	$0.350262\pi$
$600$	0	0
$601$	31.4896	1.28449	0.642243	−	0.766501i	$-0.278004\pi$
$601$	31.4896	1.28449	0.642243	+	0.766501i	$0.278004\pi$
$602$	0	0
$603$	−24.8389	−1.01152
$604$	0	0
$605$	−0.450718	−0.0183243
$606$	0	0
$607$	−23.9960	−0.973967	−0.486984	−	0.873411i	$-0.661903\pi$
$607$	−23.9960	−0.973967	−0.486984	+	0.873411i	$0.661903\pi$
$608$	0	0
$609$	−12.3590	−0.500812
$610$	0	0
$611$	1.34854	0.0545562
$612$	0	0
$613$	−34.6706	−1.40033	−0.700167	−	0.713980i	$-0.746891\pi$
$613$	−34.6706	−1.40033	−0.700167	+	0.713980i	$0.746891\pi$
$614$	0	0
$615$	−0.557271	−0.0224713
$616$	0	0
$617$	28.2335	1.13664	0.568319	−	0.822809i	$-0.307594\pi$
$617$	28.2335	1.13664	0.568319	+	0.822809i	$0.307594\pi$
$618$	0	0
$619$	16.4763	0.662240	0.331120	−	0.943589i	$-0.392574\pi$
$619$	16.4763	0.662240	0.331120	+	0.943589i	$0.392574\pi$
$620$	0	0
$621$	−3.81880	−0.153243
$622$	0	0
$623$	−2.17830	−0.0872716
$624$	0	0
$625$	21.9941	0.879763
$626$	0	0
$627$	2.33145	0.0931093
$628$	0	0
$629$	−27.5040	−1.09666
$630$	0	0
$631$	−19.0049	−0.756574	−0.378287	−	0.925688i	$-0.623487\pi$
$631$	−19.0049	−0.756574	−0.378287	+	0.925688i	$0.623487\pi$
$632$	0	0
$633$	21.5741	0.857494
$634$	0	0
$635$	4.54850	0.180502
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	11.8633	0.469305
$640$	0	0
$641$	−15.6486	−0.618084	−0.309042	−	0.951048i	$-0.600008\pi$
$641$	−15.6486	−0.618084	−0.309042	+	0.951048i	$0.600008\pi$
$642$	0	0
$643$	34.1378	1.34626	0.673131	−	0.739523i	$-0.264949\pi$
$643$	34.1378	1.34626	0.673131	+	0.739523i	$0.264949\pi$
$644$	0	0
$645$	−6.62520	−0.260867
$646$	0	0
$647$	0.228445	0.00898111	0.00449056	−	0.999990i	$-0.498571\pi$
$647$	0.228445	0.00898111	0.00449056	+	0.999990i	$0.498571\pi$
$648$	0	0
$649$	0.893254	0.0350633
$650$	0	0
$651$	0.0604757	0.00237023
$652$	0	0
$653$	−17.3097	−0.677380	−0.338690	−	0.940898i	$-0.609984\pi$
$653$	−17.3097	−0.677380	−0.338690	+	0.940898i	$0.609984\pi$
$654$	0	0
$655$	−1.35474	−0.0529339
$656$	0	0
$657$	23.6023	0.920812
$658$	0	0
$659$	−42.9142	−1.67170	−0.835850	−	0.548958i	$-0.815025\pi$
$659$	−42.9142	−1.67170	−0.835850	+	0.548958i	$0.815025\pi$
$660$	0	0
$661$	16.6341	0.646990	0.323495	−	0.946230i	$-0.395142\pi$
$661$	16.6341	0.646990	0.323495	+	0.946230i	$0.395142\pi$
$662$	0	0
$663$	10.6748	0.414575
$664$	0	0
$665$	0.450959	0.0174874
$666$	0	0
$667$	−15.2457	−0.590318
$668$	0	0
$669$	8.74147	0.337965
$670$	0	0
$671$	9.35972	0.361328
$672$	0	0
$673$	18.1582	0.699947	0.349974	−	0.936760i	$-0.386191\pi$
$673$	18.1582	0.699947	0.349974	+	0.936760i	$0.386191\pi$
$674$	0	0
$675$	−6.37271	−0.245286
$676$	0	0
$677$	24.4935	0.941363	0.470681	−	0.882303i	$-0.344008\pi$
$677$	24.4935	0.941363	0.470681	+	0.882303i	$0.344008\pi$
$678$	0	0
$679$	17.3624	0.666309
$680$	0	0
$681$	57.6172	2.20790
$682$	0	0
$683$	−30.3030	−1.15951	−0.579755	−	0.814791i	$-0.696852\pi$
$683$	−30.3030	−1.15951	−0.579755	+	0.814791i	$0.696852\pi$
$684$	0	0
$685$	−7.86579	−0.300536
$686$	0	0
$687$	36.5686	1.39518
$688$	0	0
$689$	−5.06192	−0.192844
$690$	0	0
$691$	12.9858	0.494002	0.247001	−	0.969015i	$-0.420555\pi$
$691$	12.9858	0.494002	0.247001	+	0.969015i	$0.420555\pi$
$692$	0	0
$693$	2.42987	0.0923032
$694$	0	0
$695$	5.79091	0.219662
$696$	0	0
$697$	2.43070	0.0920694
$698$	0	0
$699$	−28.9863	−1.09636
$700$	0	0
$701$	46.3447	1.75041	0.875207	−	0.483749i	$-0.160725\pi$
$701$	46.3447	1.75041	0.875207	+	0.483749i	$0.160725\pi$
$702$	0	0
$703$	−6.00707	−0.226561
$704$	0	0
$705$	1.41633	0.0533420
$706$	0	0
$707$	−11.9323	−0.448759
$708$	0	0
$709$	−46.0285	−1.72864	−0.864318	−	0.502947i	$-0.832249\pi$
$709$	−46.0285	−1.72864	−0.864318	+	0.502947i	$0.832249\pi$
$710$	0	0
$711$	−17.6541	−0.662080
$712$	0	0
$713$	0.0746013	0.00279384
$714$	0	0
$715$	−0.450718	−0.0168559
$716$	0	0
$717$	62.4291	2.33146
$718$	0	0
$719$	−1.88854	−0.0704307	−0.0352154	−	0.999380i	$-0.511212\pi$
$719$	−1.88854	−0.0704307	−0.0352154	+	0.999380i	$0.511212\pi$
$720$	0	0
$721$	−8.95618	−0.333546
$722$	0	0
$723$	−41.0970	−1.52841
$724$	0	0
$725$	−25.4416	−0.944879
$726$	0	0
$727$	18.4788	0.685340	0.342670	−	0.939456i	$-0.388669\pi$
$727$	18.4788	0.685340	0.342670	+	0.939456i	$0.388669\pi$
$728$	0	0
$729$	−15.9479	−0.590662
$730$	0	0
$731$	28.8978	1.06882
$732$	0	0
$733$	4.47512	0.165292	0.0826462	−	0.996579i	$-0.473663\pi$
$733$	4.47512	0.165292	0.0826462	+	0.996579i	$0.473663\pi$
$734$	0	0
$735$	1.05027	0.0387397
$736$	0	0
$737$	−10.2223	−0.376544
$738$	0	0
$739$	−21.4364	−0.788549	−0.394275	−	0.918993i	$-0.629004\pi$
$739$	−21.4364	−0.788549	−0.394275	+	0.918993i	$0.629004\pi$
$740$	0	0
$741$	2.33145	0.0856481
$742$	0	0
$743$	−20.7360	−0.760730	−0.380365	−	0.924836i	$-0.624202\pi$
$743$	−20.7360	−0.760730	−0.380365	+	0.924836i	$0.624202\pi$
$744$	0	0
$745$	6.02860	0.220871
$746$	0	0
$747$	25.1847	0.921460
$748$	0	0
$749$	2.05546	0.0751049
$750$	0	0
$751$	−1.12271	−0.0409682	−0.0204841	−	0.999790i	$-0.506521\pi$
$751$	−1.12271	−0.0409682	−0.0204841	+	0.999790i	$0.506521\pi$
$752$	0	0
$753$	7.57619	0.276092
$754$	0	0
$755$	0.916238	0.0333453
$756$	0	0
$757$	−13.6527	−0.496217	−0.248109	−	0.968732i	$-0.579809\pi$
$757$	−13.6527	−0.496217	−0.248109	+	0.968732i	$0.579809\pi$
$758$	0	0
$759$	6.69815	0.243127
$760$	0	0
$761$	−4.34747	−0.157596	−0.0787979	−	0.996891i	$-0.525108\pi$
$761$	−4.34747	−0.157596	−0.0787979	+	0.996891i	$0.525108\pi$
$762$	0	0
$763$	−17.9187	−0.648701
$764$	0	0
$765$	5.01710	0.181394
$766$	0	0
$767$	0.893254	0.0322535
$768$	0	0
$769$	1.96385	0.0708183	0.0354092	−	0.999373i	$-0.488727\pi$
$769$	1.96385	0.0708183	0.0354092	+	0.999373i	$0.488727\pi$
$770$	0	0
$771$	−0.884698	−0.0318616
$772$	0	0
$773$	24.4006	0.877629	0.438815	−	0.898578i	$-0.355398\pi$
$773$	24.4006	0.877629	0.438815	+	0.898578i	$0.355398\pi$
$774$	0	0
$775$	0.124492	0.00447190
$776$	0	0
$777$	−13.9902	−0.501897
$778$	0	0
$779$	0.530883	0.0190209
$780$	0	0
$781$	4.88228	0.174702
$782$	0	0
$783$	7.04622	0.251811
$784$	0	0
$785$	−4.31773	−0.154106
$786$	0	0
$787$	−24.0394	−0.856913	−0.428456	−	0.903562i	$-0.640942\pi$
$787$	−24.0394	−0.856913	−0.428456	+	0.903562i	$0.640942\pi$
$788$	0	0
$789$	0.316732	0.0112760
$790$	0	0
$791$	1.54475	0.0549249
$792$	0	0
$793$	9.35972	0.332374
$794$	0	0
$795$	−5.31636	−0.188552
$796$	0	0
$797$	−47.0838	−1.66780	−0.833898	−	0.551919i	$-0.813896\pi$
$797$	−47.0838	−1.66780	−0.833898	+	0.551919i	$0.813896\pi$
$798$	0	0
$799$	−6.17775	−0.218553
$800$	0	0
$801$	−5.29298	−0.187018
$802$	0	0
$803$	9.71338	0.342778
$804$	0	0
$805$	1.29558	0.0456632
$806$	0	0
$807$	23.7942	0.837596
$808$	0	0
$809$	6.29892	0.221458	0.110729	−	0.993851i	$-0.464681\pi$
$809$	6.29892	0.221458	0.110729	+	0.993851i	$0.464681\pi$
$810$	0	0
$811$	−11.1529	−0.391632	−0.195816	−	0.980641i	$-0.562736\pi$
$811$	−11.1529	−0.391632	−0.195816	+	0.980641i	$0.562736\pi$
$812$	0	0
$813$	−59.4813	−2.08610
$814$	0	0
$815$	8.84761	0.309918
$816$	0	0
$817$	6.31149	0.220811
$818$	0	0
$819$	2.42987	0.0849066
$820$	0	0
$821$	54.9812	1.91886	0.959429	−	0.281952i	$-0.0909819\pi$
$821$	54.9812	1.91886	0.959429	+	0.281952i	$0.0909819\pi$
$822$	0	0
$823$	10.5938	0.369277	0.184639	−	0.982806i	$-0.440889\pi$
$823$	10.5938	0.369277	0.184639	+	0.982806i	$0.440889\pi$
$824$	0	0
$825$	11.1777	0.389156
$826$	0	0
$827$	37.0066	1.28684	0.643422	−	0.765512i	$-0.277514\pi$
$827$	37.0066	1.28684	0.643422	+	0.765512i	$0.277514\pi$
$828$	0	0
$829$	−0.879127	−0.0305333	−0.0152667	−	0.999883i	$-0.504860\pi$
$829$	−0.879127	−0.0305333	−0.0152667	+	0.999883i	$0.504860\pi$
$830$	0	0
$831$	9.39433	0.325886
$832$	0	0
$833$	−4.58105	−0.158724
$834$	0	0
$835$	−5.90566	−0.204374
$836$	0	0
$837$	−0.0344789	−0.00119177
$838$	0	0
$839$	54.6764	1.88764	0.943819	−	0.330462i	$-0.107205\pi$
$839$	54.6764	1.88764	0.943819	+	0.330462i	$0.107205\pi$
$840$	0	0
$841$	−0.869492	−0.0299825
$842$	0	0
$843$	−60.1677	−2.07229
$844$	0	0
$845$	−0.450718	−0.0155052
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−24.5652	−0.843077
$850$	0	0
$851$	−17.2580	−0.591596
$852$	0	0
$853$	44.1183	1.51058	0.755290	−	0.655391i	$-0.227496\pi$
$853$	44.1183	1.51058	0.755290	+	0.655391i	$0.227496\pi$
$854$	0	0
$855$	1.09577	0.0374746
$856$	0	0
$857$	−50.1995	−1.71478	−0.857391	−	0.514665i	$-0.827916\pi$
$857$	−50.1995	−1.71478	−0.857391	+	0.514665i	$0.827916\pi$
$858$	0	0
$859$	41.7995	1.42618	0.713090	−	0.701073i	$-0.247295\pi$
$859$	41.7995	1.42618	0.713090	+	0.701073i	$0.247295\pi$
$860$	0	0
$861$	1.23641	0.0421366
$862$	0	0
$863$	7.55918	0.257317	0.128659	−	0.991689i	$-0.458933\pi$
$863$	7.55918	0.257317	0.128659	+	0.991689i	$0.458933\pi$
$864$	0	0
$865$	−5.59453	−0.190220
$866$	0	0
$867$	−9.28828	−0.315447
$868$	0	0
$869$	−7.26545	−0.246463
$870$	0	0
$871$	−10.2223	−0.346370
$872$	0	0
$873$	42.1884	1.42786
$874$	0	0
$875$	4.41562	0.149275
$876$	0	0
$877$	52.7925	1.78268	0.891338	−	0.453340i	$-0.149768\pi$
$877$	52.7925	1.78268	0.891338	+	0.453340i	$0.149768\pi$
$878$	0	0
$879$	−55.2008	−1.86188
$880$	0	0
$881$	24.9687	0.841217	0.420609	−	0.907242i	$-0.361817\pi$
$881$	24.9687	0.841217	0.420609	+	0.907242i	$0.361817\pi$
$882$	0	0
$883$	25.5900	0.861171	0.430586	−	0.902550i	$-0.358307\pi$
$883$	25.5900	0.861171	0.430586	+	0.902550i	$0.358307\pi$
$884$	0	0
$885$	0.938154	0.0315357
$886$	0	0
$887$	9.28699	0.311826	0.155913	−	0.987771i	$-0.450168\pi$
$887$	9.28699	0.311826	0.155913	+	0.987771i	$0.450168\pi$
$888$	0	0
$889$	−10.0917	−0.338464
$890$	0	0
$891$	−10.3853	−0.347922
$892$	0	0
$893$	−1.34926	−0.0451514
$894$	0	0
$895$	1.11340	0.0372169
$896$	0	0
$897$	6.69815	0.223645
$898$	0	0
$899$	−0.137650	−0.00459087
$900$	0	0
$901$	23.1889	0.772535
$902$	0	0
$903$	14.6992	0.489159
$904$	0	0
$905$	6.81462	0.226526
$906$	0	0
$907$	−26.2010	−0.869989	−0.434994	−	0.900433i	$-0.643250\pi$
$907$	−26.2010	−0.869989	−0.434994	+	0.900433i	$0.643250\pi$
$908$	0	0
$909$	−28.9938	−0.961665
$910$	0	0
$911$	49.4666	1.63890	0.819451	−	0.573149i	$-0.194279\pi$
$911$	49.4666	1.63890	0.819451	+	0.573149i	$0.194279\pi$
$912$	0	0
$913$	10.3646	0.343019
$914$	0	0
$915$	9.83020	0.324976
$916$	0	0
$917$	3.00573	0.0992579
$918$	0	0
$919$	−27.7549	−0.915550	−0.457775	−	0.889068i	$-0.651353\pi$
$919$	−27.7549	−0.915550	−0.457775	+	0.889068i	$0.651353\pi$
$920$	0	0
$921$	4.01125	0.132175
$922$	0	0
$923$	4.88228	0.160702
$924$	0	0
$925$	−28.7996	−0.946926
$926$	0	0
$927$	−21.7624	−0.714770
$928$	0	0
$929$	19.8034	0.649728	0.324864	−	0.945761i	$-0.394681\pi$
$929$	19.8034	0.649728	0.324864	+	0.945761i	$0.394681\pi$
$930$	0	0
$931$	−1.00053	−0.0327912
$932$	0	0
$933$	8.38975	0.274668
$934$	0	0
$935$	2.06476	0.0675249
$936$	0	0
$937$	−5.18988	−0.169546	−0.0847729	−	0.996400i	$-0.527016\pi$
$937$	−5.18988	−0.169546	−0.0847729	+	0.996400i	$0.527016\pi$
$938$	0	0
$939$	−32.9706	−1.07595
$940$	0	0
$941$	−7.43677	−0.242432	−0.121216	−	0.992626i	$-0.538679\pi$
$941$	−7.43677	−0.242432	−0.121216	+	0.992626i	$0.538679\pi$
$942$	0	0
$943$	1.52520	0.0496673
$944$	0	0
$945$	−0.598787	−0.0194785
$946$	0	0
$947$	−10.8128	−0.351370	−0.175685	−	0.984446i	$-0.556214\pi$
$947$	−10.8128	−0.351370	−0.175685	+	0.984446i	$0.556214\pi$
$948$	0	0
$949$	9.71338	0.315310
$950$	0	0
$951$	−48.1903	−1.56268
$952$	0	0
$953$	−40.1509	−1.30061	−0.650307	−	0.759671i	$-0.725360\pi$
$953$	−40.1509	−1.30061	−0.650307	+	0.759671i	$0.725360\pi$
$954$	0	0
$955$	4.35177	0.140820
$956$	0	0
$957$	−12.3590	−0.399510
$958$	0	0
$959$	17.4517	0.563545
$960$	0	0
$961$	−30.9993	−0.999978
$962$	0	0
$963$	4.99451	0.160946
$964$	0	0
$965$	−10.3113	−0.331932
$966$	0	0
$967$	−2.36045	−0.0759068	−0.0379534	−	0.999280i	$-0.512084\pi$
$967$	−2.36045	−0.0759068	−0.0379534	+	0.999280i	$0.512084\pi$
$968$	0	0
$969$	−10.6805	−0.343108
$970$	0	0
$971$	3.20956	0.103000	0.0514998	−	0.998673i	$-0.483600\pi$
$971$	3.20956	0.103000	0.0514998	+	0.998673i	$0.483600\pi$
$972$	0	0
$973$	−12.8482	−0.411894
$974$	0	0
$975$	11.1777	0.357972
$976$	0	0
$977$	−54.2369	−1.73519	−0.867596	−	0.497270i	$-0.834336\pi$
$977$	−54.2369	−1.73519	−0.867596	+	0.497270i	$0.834336\pi$
$978$	0	0
$979$	−2.17830	−0.0696186
$980$	0	0
$981$	−43.5402	−1.39013
$982$	0	0
$983$	−37.4556	−1.19465	−0.597325	−	0.801999i	$-0.703770\pi$
$983$	−37.4556	−1.19465	−0.597325	+	0.801999i	$0.703770\pi$
$984$	0	0
$985$	−7.41146	−0.236149
$986$	0	0
$987$	−3.14239	−0.100023
$988$	0	0
$989$	18.1326	0.576582
$990$	0	0
$991$	21.9300	0.696629	0.348314	−	0.937378i	$-0.386754\pi$
$991$	21.9300	0.696629	0.348314	+	0.937378i	$0.386754\pi$
$992$	0	0
$993$	−11.6126	−0.368513
$994$	0	0
$995$	−11.1383	−0.353108
$996$	0	0
$997$	−51.1721	−1.62064	−0.810319	−	0.585989i	$-0.800706\pi$
$997$	−51.1721	−1.62064	−0.810319	+	0.585989i	$0.800706\pi$
$998$	0	0
$999$	7.97624	0.252357

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.x.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.x.1.2	✓	12	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)

\(f(q)\)	\(=\)	\(q-2.33021 q^{3} -0.450718 q^{5} +1.00000 q^{7} +2.42987 q^{9} +O(q^{10})\)	\(q-2.33021 q^{3} -0.450718 q^{5} +1.00000 q^{7} +2.42987 q^{9} +1.00000 q^{11} +1.00000 q^{13} +1.05027 q^{15} -4.58105 q^{17} -1.00053 q^{19} -2.33021 q^{21} -2.87448 q^{23} -4.79685 q^{25} +1.32852 q^{27} +5.30382 q^{29} -0.0259529 q^{31} -2.33021 q^{33} -0.450718 q^{35} +6.00386 q^{37} -2.33021 q^{39} -0.530599 q^{41} -6.30811 q^{43} -1.09519 q^{45} +1.34854 q^{47} +1.00000 q^{49} +10.6748 q^{51} -5.06192 q^{53} -0.450718 q^{55} +2.33145 q^{57} +0.893254 q^{59} +9.35972 q^{61} +2.42987 q^{63} -0.450718 q^{65} -10.2223 q^{67} +6.69815 q^{69} +4.88228 q^{71} +9.71338 q^{73} +11.1777 q^{75} +1.00000 q^{77} -7.26545 q^{79} -10.3853 q^{81} +10.3646 q^{83} +2.06476 q^{85} -12.3590 q^{87} -2.17830 q^{89} +1.00000 q^{91} +0.0604757 q^{93} +0.450959 q^{95} +17.3624 q^{97} +2.42987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) \) 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9	\( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{13} - 3 q^{15} + 16 q^{17} - 2 q^{19} + 4 q^{21} + 9 q^{23} + 14 q^{25} + 7 q^{27} + 15 q^{29} + 10 q^{31} + 4 q^{33} + 6 q^{35} + 18 q^{37} + 4 q^{39} + 24 q^{41} + 15 q^{45} + 5 q^{47} + 12 q^{49} + 4 q^{51} + 15 q^{53} + 6 q^{55} - 4 q^{57} + 15 q^{59} + 17 q^{61} + 14 q^{63} + 6 q^{65} - 7 q^{67} + 9 q^{71} + 32 q^{73} - 8 q^{75} + 12 q^{77} + 20 q^{79} - 4 q^{81} - 5 q^{83} + 25 q^{85} + 19 q^{87} + 16 q^{89} + 12 q^{91} + 21 q^{93} + 8 q^{95} + 10 q^{97} + 14 q^{99}+O(q^{100}) \) 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9 + 12 * q^11 + 12 * q^13 - 3 * q^15 + 16 * q^17 - 2 * q^19 + 4 * q^21 + 9 * q^23 + 14 * q^25 + 7 * q^27 + 15 * q^29 + 10 * q^31 + 4 * q^33 + 6 * q^35 + 18 * q^37 + 4 * q^39 + 24 * q^41 + 15 * q^45 + 5 * q^47 + 12 * q^49 + 4 * q^51 + 15 * q^53 + 6 * q^55 - 4 * q^57 + 15 * q^59 + 17 * q^61 + 14 * q^63 + 6 * q^65 - 7 * q^67 + 9 * q^71 + 32 * q^73 - 8 * q^75 + 12 * q^77 + 20 * q^79 - 4 * q^81 - 5 * q^83 + 25 * q^85 + 19 * q^87 + 16 * q^89 + 12 * q^91 + 21 * q^93 + 8 * q^95 + 10 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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