LMFDB - Embedded newform 8008.2.a.s.1.8

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:	$1$
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} - 3x^{9} - 15x^{8} + 43x^{7} + 66x^{6} - 173x^{5} - 127x^{4} + 246x^{3} + 99x^{2} - 82x + 8 $ x^10 - 3x^9 - 15x^8 + 43x^7 + 66x^6 - 173x^5 - 127x^4 + 246x^3 + 99x^2 - 82*x + 8
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			1.8
Root			$-1.48379$ 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)

$f(q)$	$=$	$q+1.48379 q^{3} +1.24336 q^{5} +1.00000 q^{7} -0.798370 q^{9} +O(q^{10})$	$q+1.48379 q^{3} +1.24336 q^{5} +1.00000 q^{7} -0.798370 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.84489 q^{15} -6.51837 q^{17} +0.227934 q^{19} +1.48379 q^{21} -6.29398 q^{23} -3.45405 q^{25} -5.63598 q^{27} +5.55506 q^{29} +0.926360 q^{31} +1.48379 q^{33} +1.24336 q^{35} +7.65797 q^{37} -1.48379 q^{39} -6.02168 q^{41} +2.92471 q^{43} -0.992665 q^{45} -9.23385 q^{47} +1.00000 q^{49} -9.67189 q^{51} -9.80380 q^{53} +1.24336 q^{55} +0.338206 q^{57} +0.662663 q^{59} -3.66560 q^{61} -0.798370 q^{63} -1.24336 q^{65} -3.65499 q^{67} -9.33893 q^{69} +7.73384 q^{71} +12.6127 q^{73} -5.12508 q^{75} +1.00000 q^{77} -4.56202 q^{79} -5.96749 q^{81} -16.2276 q^{83} -8.10471 q^{85} +8.24254 q^{87} +7.78829 q^{89} -1.00000 q^{91} +1.37452 q^{93} +0.283405 q^{95} -9.75868 q^{97} -0.798370 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9}+O(q^{10}) $ 10 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 9 * q^9	$ 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9} + 10 q^{11} - 10 q^{13} - 5 q^{15} - 11 q^{17} + 2 q^{19} - 3 q^{21} - 8 q^{23} + 2 q^{25} - 15 q^{27} - 8 q^{29} - 23 q^{31} - 3 q^{33} - 4 q^{35} + 10 q^{37} + 3 q^{39} - 18 q^{41} + 12 q^{43} - 10 q^{45} - 36 q^{47} + 10 q^{49} + 9 q^{51} - 21 q^{53} - 4 q^{55} - 30 q^{57} - 13 q^{59} - 2 q^{61} + 9 q^{63} + 4 q^{65} - 4 q^{67} - 26 q^{69} - 24 q^{71} - 23 q^{73} - 28 q^{75} + 10 q^{77} + 14 q^{79} + 30 q^{81} - 9 q^{83} - 17 q^{85} + 7 q^{87} - 18 q^{89} - 10 q^{91} + q^{93} - 4 q^{95} - 9 q^{97} + 9 q^{99}+O(q^{100}) $ 10 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 9 * q^9 + 10 * q^11 - 10 * q^13 - 5 * q^15 - 11 * q^17 + 2 * q^19 - 3 * q^21 - 8 * q^23 + 2 * q^25 - 15 * q^27 - 8 * q^29 - 23 * q^31 - 3 * q^33 - 4 * q^35 + 10 * q^37 + 3 * q^39 - 18 * q^41 + 12 * q^43 - 10 * q^45 - 36 * q^47 + 10 * q^49 + 9 * q^51 - 21 * q^53 - 4 * q^55 - 30 * q^57 - 13 * q^59 - 2 * q^61 + 9 * q^63 + 4 * q^65 - 4 * q^67 - 26 * q^69 - 24 * q^71 - 23 * q^73 - 28 * q^75 + 10 * q^77 + 14 * q^79 + 30 * q^81 - 9 * q^83 - 17 * q^85 + 7 * q^87 - 18 * q^89 - 10 * q^91 + q^93 - 4 * q^95 - 9 * q^97 + 9 * 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
$3$	1.48379	0.856666	0.428333	−	0.903621i	$-0.359101\pi$
$3$	1.48379	0.856666	0.428333	+	0.903621i	$0.359101\pi$
$4$	0	0
$5$	1.24336	0.556049	0.278025	−	0.960574i	$-0.410320\pi$
$5$	1.24336	0.556049	0.278025	+	0.960574i	$0.410320\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.798370	−0.266123
$10$	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
$15$	1.84489	0.476348
$16$	0	0
$17$	−6.51837	−1.58094	−0.790469	−	0.612503i	$-0.790163\pi$
$17$	−6.51837	−1.58094	−0.790469	+	0.612503i	$0.790163\pi$
$18$	0	0
$19$	0.227934	0.0522917	0.0261459	−	0.999658i	$-0.491677\pi$
$19$	0.227934	0.0522917	0.0261459	+	0.999658i	$0.491677\pi$
$20$	0	0
$21$	1.48379	0.323789
$22$	0	0
$23$	−6.29398	−1.31238	−0.656192	−	0.754594i	$-0.727834\pi$
$23$	−6.29398	−1.31238	−0.656192	+	0.754594i	$0.727834\pi$
$24$	0	0
$25$	−3.45405	−0.690809
$26$	0	0
$27$	−5.63598	−1.08464
$28$	0	0
$29$	5.55506	1.03155	0.515775	−	0.856724i	$-0.327504\pi$
$29$	5.55506	1.03155	0.515775	+	0.856724i	$0.327504\pi$
$30$	0	0
$31$	0.926360	0.166379	0.0831896	−	0.996534i	$-0.473489\pi$
$31$	0.926360	0.166379	0.0831896	+	0.996534i	$0.473489\pi$
$32$	0	0
$33$	1.48379	0.258295
$34$	0	0
$35$	1.24336	0.210167
$36$	0	0
$37$	7.65797	1.25896	0.629481	−	0.777016i	$-0.283267\pi$
$37$	7.65797	1.25896	0.629481	+	0.777016i	$0.283267\pi$
$38$	0	0
$39$	−1.48379	−0.237596
$40$	0	0
$41$	−6.02168	−0.940428	−0.470214	−	0.882552i	$-0.655823\pi$
$41$	−6.02168	−0.940428	−0.470214	+	0.882552i	$0.655823\pi$
$42$	0	0
$43$	2.92471	0.446014	0.223007	−	0.974817i	$-0.428413\pi$
$43$	2.92471	0.446014	0.223007	+	0.974817i	$0.428413\pi$
$44$	0	0
$45$	−0.992665	−0.147978
$46$	0	0
$47$	−9.23385	−1.34690	−0.673448	−	0.739235i	$-0.735187\pi$
$47$	−9.23385	−1.34690	−0.673448	+	0.739235i	$0.735187\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−9.67189	−1.35433
$52$	0	0
$53$	−9.80380	−1.34665	−0.673327	−	0.739344i	$-0.735136\pi$
$53$	−9.80380	−1.34665	−0.673327	+	0.739344i	$0.735136\pi$
$54$	0	0
$55$	1.24336	0.167655
$56$	0	0
$57$	0.338206	0.0447965
$58$	0	0
$59$	0.662663	0.0862714	0.0431357	−	0.999069i	$-0.486265\pi$
$59$	0.662663	0.0862714	0.0431357	+	0.999069i	$0.486265\pi$
$60$	0	0
$61$	−3.66560	−0.469332	−0.234666	−	0.972076i	$-0.575400\pi$
$61$	−3.66560	−0.469332	−0.234666	+	0.972076i	$0.575400\pi$
$62$	0	0
$63$	−0.798370	−0.100585
$64$	0	0
$65$	−1.24336	−0.154220
$66$	0	0
$67$	−3.65499	−0.446529	−0.223264	−	0.974758i	$-0.571671\pi$
$67$	−3.65499	−0.446529	−0.223264	+	0.974758i	$0.571671\pi$
$68$	0	0
$69$	−9.33893	−1.12428
$70$	0	0
$71$	7.73384	0.917838	0.458919	−	0.888478i	$-0.348237\pi$
$71$	7.73384	0.917838	0.458919	+	0.888478i	$0.348237\pi$
$72$	0	0
$73$	12.6127	1.47621	0.738103	−	0.674688i	$-0.235722\pi$
$73$	12.6127	1.47621	0.738103	+	0.674688i	$0.235722\pi$
$74$	0	0
$75$	−5.12508	−0.591793
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−4.56202	−0.513267	−0.256634	−	0.966509i	$-0.582613\pi$
$79$	−4.56202	−0.513267	−0.256634	+	0.966509i	$0.582613\pi$
$80$	0	0
$81$	−5.96749	−0.663055
$82$	0	0
$83$	−16.2276	−1.78121	−0.890603	−	0.454781i	$-0.849717\pi$
$83$	−16.2276	−1.78121	−0.890603	+	0.454781i	$0.849717\pi$
$84$	0	0
$85$	−8.10471	−0.879079
$86$	0	0
$87$	8.24254	0.883693
$88$	0	0
$89$	7.78829	0.825557	0.412778	−	0.910831i	$-0.364558\pi$
$89$	7.78829	0.825557	0.412778	+	0.910831i	$0.364558\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	1.37452	0.142531
$94$	0	0
$95$	0.283405	0.0290768
$96$	0	0
$97$	−9.75868	−0.990844	−0.495422	−	0.868653i	$-0.664986\pi$
$97$	−9.75868	−0.990844	−0.495422	+	0.868653i	$0.664986\pi$
$98$	0	0
$99$	−0.798370	−0.0802392
$100$	0	0
$101$	−0.974919	−0.0970081	−0.0485040	−	0.998823i	$-0.515445\pi$
$101$	−0.974919	−0.0970081	−0.0485040	+	0.998823i	$0.515445\pi$
$102$	0	0
$103$	−9.84556	−0.970112	−0.485056	−	0.874483i	$-0.661201\pi$
$103$	−9.84556	−0.970112	−0.485056	+	0.874483i	$0.661201\pi$
$104$	0	0
$105$	1.84489	0.180043
$106$	0	0
$107$	−13.5285	−1.30785	−0.653924	−	0.756560i	$-0.726879\pi$
$107$	−13.5285	−1.30785	−0.653924	+	0.756560i	$0.726879\pi$
$108$	0	0
$109$	5.32818	0.510347	0.255174	−	0.966895i	$-0.417867\pi$
$109$	5.32818	0.510347	0.255174	+	0.966895i	$0.417867\pi$
$110$	0	0
$111$	11.3628	1.07851
$112$	0	0
$113$	−7.46315	−0.702074	−0.351037	−	0.936362i	$-0.614171\pi$
$113$	−7.46315	−0.702074	−0.351037	+	0.936362i	$0.614171\pi$
$114$	0	0
$115$	−7.82570	−0.729750
$116$	0	0
$117$	0.798370	0.0738093
$118$	0	0
$119$	−6.51837	−0.597538
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−8.93490	−0.805633
$124$	0	0
$125$	−10.5115	−0.940173
$126$	0	0
$127$	7.87179	0.698508	0.349254	−	0.937028i	$-0.386435\pi$
$127$	7.87179	0.698508	0.349254	+	0.937028i	$0.386435\pi$
$128$	0	0
$129$	4.33965	0.382085
$130$	0	0
$131$	−18.7472	−1.63795	−0.818973	−	0.573831i	$-0.805456\pi$
$131$	−18.7472	−1.63795	−0.818973	+	0.573831i	$0.805456\pi$
$132$	0	0
$133$	0.227934	0.0197644
$134$	0	0
$135$	−7.00757	−0.603116
$136$	0	0
$137$	7.09757	0.606386	0.303193	−	0.952929i	$-0.401947\pi$
$137$	7.09757	0.606386	0.303193	+	0.952929i	$0.401947\pi$
$138$	0	0
$139$	−17.9843	−1.52541	−0.762704	−	0.646747i	$-0.776129\pi$
$139$	−17.9843	−1.52541	−0.762704	+	0.646747i	$0.776129\pi$
$140$	0	0
$141$	−13.7011	−1.15384
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	6.90696	0.573592
$146$	0	0
$147$	1.48379	0.122381
$148$	0	0
$149$	0.852519	0.0698411	0.0349205	−	0.999390i	$-0.488882\pi$
$149$	0.852519	0.0698411	0.0349205	+	0.999390i	$0.488882\pi$
$150$	0	0
$151$	10.7824	0.877462	0.438731	−	0.898618i	$-0.355428\pi$
$151$	10.7824	0.877462	0.438731	+	0.898618i	$0.355428\pi$
$152$	0	0
$153$	5.20407	0.420724
$154$	0	0
$155$	1.15180	0.0925150
$156$	0	0
$157$	−23.8569	−1.90399	−0.951993	−	0.306120i	$-0.900969\pi$
$157$	−23.8569	−1.90399	−0.951993	+	0.306120i	$0.900969\pi$
$158$	0	0
$159$	−14.5468	−1.15363
$160$	0	0
$161$	−6.29398	−0.496035
$162$	0	0
$163$	20.6016	1.61364	0.806820	−	0.590797i	$-0.201187\pi$
$163$	20.6016	1.61364	0.806820	+	0.590797i	$0.201187\pi$
$164$	0	0
$165$	1.84489	0.143624
$166$	0	0
$167$	19.7279	1.52659	0.763294	−	0.646051i	$-0.223581\pi$
$167$	19.7279	1.52659	0.763294	+	0.646051i	$0.223581\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.181976	−0.0139160
$172$	0	0
$173$	−17.0751	−1.29820	−0.649100	−	0.760704i	$-0.724854\pi$
$173$	−17.0751	−1.29820	−0.649100	+	0.760704i	$0.724854\pi$
$174$	0	0
$175$	−3.45405	−0.261101
$176$	0	0
$177$	0.983252	0.0739058
$178$	0	0
$179$	−0.850231	−0.0635492	−0.0317746	−	0.999495i	$-0.510116\pi$
$179$	−0.850231	−0.0635492	−0.0317746	+	0.999495i	$0.510116\pi$
$180$	0	0
$181$	22.8421	1.69784	0.848920	−	0.528521i	$-0.177253\pi$
$181$	22.8421	1.69784	0.848920	+	0.528521i	$0.177253\pi$
$182$	0	0
$183$	−5.43898	−0.402061
$184$	0	0
$185$	9.52164	0.700045
$186$	0	0
$187$	−6.51837	−0.476670
$188$	0	0
$189$	−5.63598	−0.409957
$190$	0	0
$191$	10.7448	0.777466	0.388733	−	0.921350i	$-0.372913\pi$
$191$	10.7448	0.777466	0.388733	+	0.921350i	$0.372913\pi$
$192$	0	0
$193$	−8.66158	−0.623474	−0.311737	−	0.950168i	$-0.600911\pi$
$193$	−8.66158	−0.623474	−0.311737	+	0.950168i	$0.600911\pi$
$194$	0	0
$195$	−1.84489	−0.132115
$196$	0	0
$197$	−14.2408	−1.01462	−0.507308	−	0.861765i	$-0.669359\pi$
$197$	−14.2408	−1.01462	−0.507308	+	0.861765i	$0.669359\pi$
$198$	0	0
$199$	10.5751	0.749646	0.374823	−	0.927096i	$-0.377703\pi$
$199$	10.5751	0.749646	0.374823	+	0.927096i	$0.377703\pi$
$200$	0	0
$201$	−5.42324	−0.382526
$202$	0	0
$203$	5.55506	0.389889
$204$	0	0
$205$	−7.48713	−0.522924
$206$	0	0
$207$	5.02492	0.349256
$208$	0	0
$209$	0.227934	0.0157665
$210$	0	0
$211$	−0.550447	−0.0378943	−0.0189472	−	0.999820i	$-0.506031\pi$
$211$	−0.550447	−0.0378943	−0.0189472	+	0.999820i	$0.506031\pi$
$212$	0	0
$213$	11.4754	0.786281
$214$	0	0
$215$	3.63648	0.248006
$216$	0	0
$217$	0.926360	0.0628854
$218$	0	0
$219$	18.7146	1.26461
$220$	0	0
$221$	6.51837	0.438473
$222$	0	0
$223$	−3.82378	−0.256059	−0.128030	−	0.991770i	$-0.540865\pi$
$223$	−3.82378	−0.256059	−0.128030	+	0.991770i	$0.540865\pi$
$224$	0	0
$225$	2.75761	0.183840
$226$	0	0
$227$	8.32135	0.552308	0.276154	−	0.961113i	$-0.410940\pi$
$227$	8.32135	0.552308	0.276154	+	0.961113i	$0.410940\pi$
$228$	0	0
$229$	5.57186	0.368199	0.184100	−	0.982908i	$-0.441063\pi$
$229$	5.57186	0.368199	0.184100	+	0.982908i	$0.441063\pi$
$230$	0	0
$231$	1.48379	0.0976262
$232$	0	0
$233$	−4.41004	−0.288911	−0.144456	−	0.989511i	$-0.546143\pi$
$233$	−4.41004	−0.288911	−0.144456	+	0.989511i	$0.546143\pi$
$234$	0	0
$235$	−11.4810	−0.748940
$236$	0	0
$237$	−6.76908	−0.439699
$238$	0	0
$239$	22.2076	1.43649	0.718245	−	0.695790i	$-0.244946\pi$
$239$	22.2076	1.43649	0.718245	+	0.695790i	$0.244946\pi$
$240$	0	0
$241$	1.62710	0.104811	0.0524055	−	0.998626i	$-0.483311\pi$
$241$	1.62710	0.104811	0.0524055	+	0.998626i	$0.483311\pi$
$242$	0	0
$243$	8.05344	0.516628
$244$	0	0
$245$	1.24336	0.0794356
$246$	0	0
$247$	−0.227934	−0.0145031
$248$	0	0
$249$	−24.0783	−1.52590
$250$	0	0
$251$	6.12331	0.386500	0.193250	−	0.981150i	$-0.438097\pi$
$251$	6.12331	0.386500	0.193250	+	0.981150i	$0.438097\pi$
$252$	0	0
$253$	−6.29398	−0.395699
$254$	0	0
$255$	−12.0257	−0.753077
$256$	0	0
$257$	−13.6927	−0.854129	−0.427065	−	0.904221i	$-0.640452\pi$
$257$	−13.6927	−0.854129	−0.427065	+	0.904221i	$0.640452\pi$
$258$	0	0
$259$	7.65797	0.475843
$260$	0	0
$261$	−4.43500	−0.274519
$262$	0	0
$263$	−6.05416	−0.373315	−0.186658	−	0.982425i	$-0.559766\pi$
$263$	−6.05416	−0.373315	−0.186658	+	0.982425i	$0.559766\pi$
$264$	0	0
$265$	−12.1897	−0.748806
$266$	0	0
$267$	11.5562	0.707226
$268$	0	0
$269$	17.7668	1.08326	0.541629	−	0.840617i	$-0.317808\pi$
$269$	17.7668	1.08326	0.541629	+	0.840617i	$0.317808\pi$
$270$	0	0
$271$	10.3285	0.627410	0.313705	−	0.949521i	$-0.398430\pi$
$271$	10.3285	0.627410	0.313705	+	0.949521i	$0.398430\pi$
$272$	0	0
$273$	−1.48379	−0.0898030
$274$	0	0
$275$	−3.45405	−0.208287
$276$	0	0
$277$	0.712780	0.0428268	0.0214134	−	0.999771i	$-0.493183\pi$
$277$	0.712780	0.0428268	0.0214134	+	0.999771i	$0.493183\pi$
$278$	0	0
$279$	−0.739578	−0.0442774
$280$	0	0
$281$	−8.33921	−0.497475	−0.248738	−	0.968571i	$-0.580016\pi$
$281$	−8.33921	−0.497475	−0.248738	+	0.968571i	$0.580016\pi$
$282$	0	0
$283$	−10.8406	−0.644404	−0.322202	−	0.946671i	$-0.604423\pi$
$283$	−10.8406	−0.644404	−0.322202	+	0.946671i	$0.604423\pi$
$284$	0	0
$285$	0.420514	0.0249091
$286$	0	0
$287$	−6.02168	−0.355448
$288$	0	0
$289$	25.4892	1.49936
$290$	0	0
$291$	−14.4798	−0.848822
$292$	0	0
$293$	−11.6009	−0.677730	−0.338865	−	0.940835i	$-0.610043\pi$
$293$	−11.6009	−0.677730	−0.338865	+	0.940835i	$0.610043\pi$
$294$	0	0
$295$	0.823932	0.0479712
$296$	0	0
$297$	−5.63598	−0.327033
$298$	0	0
$299$	6.29398	0.363990
$300$	0	0
$301$	2.92471	0.168577
$302$	0	0
$303$	−1.44657	−0.0831035
$304$	0	0
$305$	−4.55768	−0.260972
$306$	0	0
$307$	25.3208	1.44513	0.722566	−	0.691302i	$-0.242963\pi$
$307$	25.3208	1.44513	0.722566	+	0.691302i	$0.242963\pi$
$308$	0	0
$309$	−14.6087	−0.831062
$310$	0	0
$311$	−13.1153	−0.743703	−0.371851	−	0.928292i	$-0.621277\pi$
$311$	−13.1153	−0.743703	−0.371851	+	0.928292i	$0.621277\pi$
$312$	0	0
$313$	−14.6322	−0.827060	−0.413530	−	0.910490i	$-0.635704\pi$
$313$	−14.6322	−0.827060	−0.413530	+	0.910490i	$0.635704\pi$
$314$	0	0
$315$	−0.992665	−0.0559303
$316$	0	0
$317$	8.70649	0.489005	0.244502	−	0.969649i	$-0.421375\pi$
$317$	8.70649	0.489005	0.244502	+	0.969649i	$0.421375\pi$
$318$	0	0
$319$	5.55506	0.311024
$320$	0	0
$321$	−20.0734	−1.12039
$322$	0	0
$323$	−1.48576	−0.0826699
$324$	0	0
$325$	3.45405	0.191596
$326$	0	0
$327$	7.90590	0.437197
$328$	0	0
$329$	−9.23385	−0.509079
$330$	0	0
$331$	−13.1570	−0.723173	−0.361586	−	0.932339i	$-0.617765\pi$
$331$	−13.1570	−0.723173	−0.361586	+	0.932339i	$0.617765\pi$
$332$	0	0
$333$	−6.11390	−0.335039
$334$	0	0
$335$	−4.54449	−0.248292
$336$	0	0
$337$	−1.98382	−0.108066	−0.0540329	−	0.998539i	$-0.517208\pi$
$337$	−1.98382	−0.108066	−0.0540329	+	0.998539i	$0.517208\pi$
$338$	0	0
$339$	−11.0737	−0.601443
$340$	0	0
$341$	0.926360	0.0501652
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−11.6117	−0.625152
$346$	0	0
$347$	−11.2606	−0.604503	−0.302252	−	0.953228i	$-0.597738\pi$
$347$	−11.2606	−0.604503	−0.302252	+	0.953228i	$0.597738\pi$
$348$	0	0
$349$	−30.9040	−1.65425	−0.827127	−	0.562015i	$-0.810026\pi$
$349$	−30.9040	−1.65425	−0.827127	+	0.562015i	$0.810026\pi$
$350$	0	0
$351$	5.63598	0.300826
$352$	0	0
$353$	29.2323	1.55588	0.777940	−	0.628338i	$-0.216265\pi$
$353$	29.2323	1.55588	0.777940	+	0.628338i	$0.216265\pi$
$354$	0	0
$355$	9.61598	0.510363
$356$	0	0
$357$	−9.67189	−0.511891
$358$	0	0
$359$	−8.63342	−0.455654	−0.227827	−	0.973702i	$-0.573162\pi$
$359$	−8.63342	−0.455654	−0.227827	+	0.973702i	$0.573162\pi$
$360$	0	0
$361$	−18.9480	−0.997266
$362$	0	0
$363$	1.48379	0.0778787
$364$	0	0
$365$	15.6822	0.820843
$366$	0	0
$367$	−14.6463	−0.764531	−0.382266	−	0.924052i	$-0.624856\pi$
$367$	−14.6463	−0.764531	−0.382266	+	0.924052i	$0.624856\pi$
$368$	0	0
$369$	4.80753	0.250270
$370$	0	0
$371$	−9.80380	−0.508988
$372$	0	0
$373$	−32.9080	−1.70391	−0.851956	−	0.523613i	$-0.824584\pi$
$373$	−32.9080	−1.70391	−0.851956	+	0.523613i	$0.824584\pi$
$374$	0	0
$375$	−15.5968	−0.805414
$376$	0	0
$377$	−5.55506	−0.286100
$378$	0	0
$379$	0.661396	0.0339736	0.0169868	−	0.999856i	$-0.494593\pi$
$379$	0.661396	0.0339736	0.0169868	+	0.999856i	$0.494593\pi$
$380$	0	0
$381$	11.6801	0.598388
$382$	0	0
$383$	−1.54297	−0.0788420	−0.0394210	−	0.999223i	$-0.512551\pi$
$383$	−1.54297	−0.0788420	−0.0394210	+	0.999223i	$0.512551\pi$
$384$	0	0
$385$	1.24336	0.0633677
$386$	0	0
$387$	−2.33500	−0.118695
$388$	0	0
$389$	−12.9405	−0.656107	−0.328054	−	0.944659i	$-0.606393\pi$
$389$	−12.9405	−0.656107	−0.328054	+	0.944659i	$0.606393\pi$
$390$	0	0
$391$	41.0265	2.07480
$392$	0	0
$393$	−27.8168	−1.40317
$394$	0	0
$395$	−5.67225	−0.285402
$396$	0	0
$397$	12.8697	0.645913	0.322956	−	0.946414i	$-0.395323\pi$
$397$	12.8697	0.645913	0.322956	+	0.946414i	$0.395323\pi$
$398$	0	0
$399$	0.338206	0.0169315
$400$	0	0
$401$	−3.07712	−0.153664	−0.0768320	−	0.997044i	$-0.524481\pi$
$401$	−3.07712	−0.153664	−0.0768320	+	0.997044i	$0.524481\pi$
$402$	0	0
$403$	−0.926360	−0.0461453
$404$	0	0
$405$	−7.41977	−0.368691
$406$	0	0
$407$	7.65797	0.379592
$408$	0	0
$409$	−0.718152	−0.0355103	−0.0177552	−	0.999842i	$-0.505652\pi$
$409$	−0.718152	−0.0355103	−0.0177552	+	0.999842i	$0.505652\pi$
$410$	0	0
$411$	10.5313	0.519470
$412$	0	0
$413$	0.662663	0.0326075
$414$	0	0
$415$	−20.1768	−0.990438
$416$	0	0
$417$	−26.6849	−1.30677
$418$	0	0
$419$	−6.18050	−0.301937	−0.150969	−	0.988539i	$-0.548239\pi$
$419$	−6.18050	−0.301937	−0.150969	+	0.988539i	$0.548239\pi$
$420$	0	0
$421$	28.5164	1.38981	0.694903	−	0.719104i	$-0.255447\pi$
$421$	28.5164	1.38981	0.694903	+	0.719104i	$0.255447\pi$
$422$	0	0
$423$	7.37203	0.358440
$424$	0	0
$425$	22.5148	1.09213
$426$	0	0
$427$	−3.66560	−0.177391
$428$	0	0
$429$	−1.48379	−0.0716380
$430$	0	0
$431$	9.11154	0.438888	0.219444	−	0.975625i	$-0.429576\pi$
$431$	9.11154	0.438888	0.219444	+	0.975625i	$0.429576\pi$
$432$	0	0
$433$	31.7503	1.52582	0.762910	−	0.646505i	$-0.223770\pi$
$433$	31.7503	1.52582	0.762910	+	0.646505i	$0.223770\pi$
$434$	0	0
$435$	10.2485	0.491377
$436$	0	0
$437$	−1.43461	−0.0686268
$438$	0	0
$439$	−17.1777	−0.819845	−0.409922	−	0.912120i	$-0.634444\pi$
$439$	−17.1777	−0.819845	−0.409922	+	0.912120i	$0.634444\pi$
$440$	0	0
$441$	−0.798370	−0.0380176
$442$	0	0
$443$	17.4669	0.829876	0.414938	−	0.909850i	$-0.363803\pi$
$443$	17.4669	0.829876	0.414938	+	0.909850i	$0.363803\pi$
$444$	0	0
$445$	9.68367	0.459050
$446$	0	0
$447$	1.26496	0.0598305
$448$	0	0
$449$	22.6622	1.06949	0.534747	−	0.845012i	$-0.320407\pi$
$449$	22.6622	1.06949	0.534747	+	0.845012i	$0.320407\pi$
$450$	0	0
$451$	−6.02168	−0.283550
$452$	0	0
$453$	15.9989	0.751692
$454$	0	0
$455$	−1.24336	−0.0582898
$456$	0	0
$457$	1.09697	0.0513140	0.0256570	−	0.999671i	$-0.491832\pi$
$457$	1.09697	0.0513140	0.0256570	+	0.999671i	$0.491832\pi$
$458$	0	0
$459$	36.7374	1.71476
$460$	0	0
$461$	−2.82511	−0.131578	−0.0657892	−	0.997834i	$-0.520956\pi$
$461$	−2.82511	−0.131578	−0.0657892	+	0.997834i	$0.520956\pi$
$462$	0	0
$463$	−36.6237	−1.70205	−0.851024	−	0.525127i	$-0.824018\pi$
$463$	−36.6237	−1.70205	−0.851024	+	0.525127i	$0.824018\pi$
$464$	0	0
$465$	1.70903	0.0792544
$466$	0	0
$467$	40.1125	1.85618	0.928092	−	0.372350i	$-0.121448\pi$
$467$	40.1125	1.85618	0.928092	+	0.372350i	$0.121448\pi$
$468$	0	0
$469$	−3.65499	−0.168772
$470$	0	0
$471$	−35.3986	−1.63108
$472$	0	0
$473$	2.92471	0.134478
$474$	0	0
$475$	−0.787296	−0.0361236
$476$	0	0
$477$	7.82706	0.358376
$478$	0	0
$479$	7.68707	0.351231	0.175616	−	0.984459i	$-0.443808\pi$
$479$	7.68707	0.351231	0.175616	+	0.984459i	$0.443808\pi$
$480$	0	0
$481$	−7.65797	−0.349173
$482$	0	0
$483$	−9.33893	−0.424936
$484$	0	0
$485$	−12.1336	−0.550958
$486$	0	0
$487$	−4.57378	−0.207258	−0.103629	−	0.994616i	$-0.533045\pi$
$487$	−4.57378	−0.207258	−0.103629	+	0.994616i	$0.533045\pi$
$488$	0	0
$489$	30.5684	1.38235
$490$	0	0
$491$	15.2434	0.687924	0.343962	−	0.938984i	$-0.388231\pi$
$491$	15.2434	0.687924	0.343962	+	0.938984i	$0.388231\pi$
$492$	0	0
$493$	−36.2100	−1.63081
$494$	0	0
$495$	−0.992665	−0.0446170
$496$	0	0
$497$	7.73384	0.346910
$498$	0	0
$499$	13.1183	0.587257	0.293629	−	0.955920i	$-0.405137\pi$
$499$	13.1183	0.587257	0.293629	+	0.955920i	$0.405137\pi$
$500$	0	0
$501$	29.2720	1.30778
$502$	0	0
$503$	−8.03016	−0.358047	−0.179024	−	0.983845i	$-0.557294\pi$
$503$	−8.03016	−0.358047	−0.179024	+	0.983845i	$0.557294\pi$
$504$	0	0
$505$	−1.21218	−0.0539413
$506$	0	0
$507$	1.48379	0.0658974
$508$	0	0
$509$	14.6499	0.649345	0.324672	−	0.945827i	$-0.394746\pi$
$509$	14.6499	0.649345	0.324672	+	0.945827i	$0.394746\pi$
$510$	0	0
$511$	12.6127	0.557953
$512$	0	0
$513$	−1.28463	−0.0567179
$514$	0	0
$515$	−12.2416	−0.539430
$516$	0	0
$517$	−9.23385	−0.406104
$518$	0	0
$519$	−25.3359	−1.11212
$520$	0	0
$521$	7.13690	0.312673	0.156337	−	0.987704i	$-0.450032\pi$
$521$	7.13690	0.312673	0.156337	+	0.987704i	$0.450032\pi$
$522$	0	0
$523$	−31.8586	−1.39308	−0.696540	−	0.717518i	$-0.745278\pi$
$523$	−31.8586	−1.39308	−0.696540	+	0.717518i	$0.745278\pi$
$524$	0	0
$525$	−5.12508	−0.223677
$526$	0	0
$527$	−6.03836	−0.263035
$528$	0	0
$529$	16.6141	0.722353
$530$	0	0
$531$	−0.529051	−0.0229588
$532$	0	0
$533$	6.02168	0.260828
$534$	0	0
$535$	−16.8208	−0.727228
$536$	0	0
$537$	−1.26156	−0.0544405
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	33.1526	1.42534	0.712672	−	0.701498i	$-0.247485\pi$
$541$	33.1526	1.42534	0.712672	+	0.701498i	$0.247485\pi$
$542$	0	0
$543$	33.8929	1.45448
$544$	0	0
$545$	6.62487	0.283778
$546$	0	0
$547$	31.9248	1.36501	0.682503	−	0.730883i	$-0.260891\pi$
$547$	31.9248	1.36501	0.682503	+	0.730883i	$0.260891\pi$
$548$	0	0
$549$	2.92651	0.124900
$550$	0	0
$551$	1.26619	0.0539415
$552$	0	0
$553$	−4.56202	−0.193997
$554$	0	0
$555$	14.1281	0.599705
$556$	0	0
$557$	−19.7937	−0.838686	−0.419343	−	0.907828i	$-0.637740\pi$
$557$	−19.7937	−0.838686	−0.419343	+	0.907828i	$0.637740\pi$
$558$	0	0
$559$	−2.92471	−0.123702
$560$	0	0
$561$	−9.67189	−0.408347
$562$	0	0
$563$	5.52518	0.232858	0.116429	−	0.993199i	$-0.462855\pi$
$563$	5.52518	0.232858	0.116429	+	0.993199i	$0.462855\pi$
$564$	0	0
$565$	−9.27941	−0.390388
$566$	0	0
$567$	−5.96749	−0.250611
$568$	0	0
$569$	−28.9002	−1.21156	−0.605780	−	0.795632i	$-0.707139\pi$
$569$	−28.9002	−1.21156	−0.605780	+	0.795632i	$0.707139\pi$
$570$	0	0
$571$	36.0758	1.50972	0.754862	−	0.655884i	$-0.227704\pi$
$571$	36.0758	1.50972	0.754862	+	0.655884i	$0.227704\pi$
$572$	0	0
$573$	15.9430	0.666029
$574$	0	0
$575$	21.7397	0.906607
$576$	0	0
$577$	−19.0013	−0.791036	−0.395518	−	0.918458i	$-0.629435\pi$
$577$	−19.0013	−0.791036	−0.395518	+	0.918458i	$0.629435\pi$
$578$	0	0
$579$	−12.8520	−0.534109
$580$	0	0
$581$	−16.2276	−0.673233
$582$	0	0
$583$	−9.80380	−0.406032
$584$	0	0
$585$	0.992665	0.0410416
$586$	0	0
$587$	−7.16053	−0.295547	−0.147773	−	0.989021i	$-0.547211\pi$
$587$	−7.16053	−0.295547	−0.147773	+	0.989021i	$0.547211\pi$
$588$	0	0
$589$	0.211149	0.00870025
$590$	0	0
$591$	−21.1304	−0.869187
$592$	0	0
$593$	−10.9413	−0.449305	−0.224652	−	0.974439i	$-0.572125\pi$
$593$	−10.9413	−0.449305	−0.224652	+	0.974439i	$0.572125\pi$
$594$	0	0
$595$	−8.10471	−0.332261
$596$	0	0
$597$	15.6912	0.642197
$598$	0	0
$599$	−13.5274	−0.552715	−0.276357	−	0.961055i	$-0.589127\pi$
$599$	−13.5274	−0.552715	−0.276357	+	0.961055i	$0.589127\pi$
$600$	0	0
$601$	10.2181	0.416806	0.208403	−	0.978043i	$-0.433173\pi$
$601$	10.2181	0.416806	0.208403	+	0.978043i	$0.433173\pi$
$602$	0	0
$603$	2.91804	0.118832
$604$	0	0
$605$	1.24336	0.0505499
$606$	0	0
$607$	−14.3365	−0.581900	−0.290950	−	0.956738i	$-0.593971\pi$
$607$	−14.3365	−0.581900	−0.290950	+	0.956738i	$0.593971\pi$
$608$	0	0
$609$	8.24254	0.334005
$610$	0	0
$611$	9.23385	0.373562
$612$	0	0
$613$	−42.8929	−1.73243	−0.866214	−	0.499674i	$-0.833453\pi$
$613$	−42.8929	−1.73243	−0.866214	+	0.499674i	$0.833453\pi$
$614$	0	0
$615$	−11.1093	−0.447971
$616$	0	0
$617$	1.08510	0.0436843	0.0218422	−	0.999761i	$-0.493047\pi$
$617$	1.08510	0.0436843	0.0218422	+	0.999761i	$0.493047\pi$
$618$	0	0
$619$	36.3887	1.46259	0.731293	−	0.682063i	$-0.238917\pi$
$619$	36.3887	1.46259	0.731293	+	0.682063i	$0.238917\pi$
$620$	0	0
$621$	35.4727	1.42347
$622$	0	0
$623$	7.78829	0.312031
$624$	0	0
$625$	4.20067	0.168027
$626$	0	0
$627$	0.338206	0.0135067
$628$	0	0
$629$	−49.9175	−1.99034
$630$	0	0
$631$	−20.3738	−0.811066	−0.405533	−	0.914080i	$-0.632914\pi$
$631$	−20.3738	−0.811066	−0.405533	+	0.914080i	$0.632914\pi$
$632$	0	0
$633$	−0.816747	−0.0324628
$634$	0	0
$635$	9.78749	0.388405
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−6.17447	−0.244258
$640$	0	0
$641$	−1.31921	−0.0521056	−0.0260528	−	0.999661i	$-0.508294\pi$
$641$	−1.31921	−0.0521056	−0.0260528	+	0.999661i	$0.508294\pi$
$642$	0	0
$643$	23.1191	0.911730	0.455865	−	0.890049i	$-0.349330\pi$
$643$	23.1191	0.911730	0.455865	+	0.890049i	$0.349330\pi$
$644$	0	0
$645$	5.39576	0.212458
$646$	0	0
$647$	−19.2565	−0.757053	−0.378526	−	0.925590i	$-0.623569\pi$
$647$	−19.2565	−0.757053	−0.378526	+	0.925590i	$0.623569\pi$
$648$	0	0
$649$	0.662663	0.0260118
$650$	0	0
$651$	1.37452	0.0538718
$652$	0	0
$653$	−30.5734	−1.19643	−0.598215	−	0.801336i	$-0.704123\pi$
$653$	−30.5734	−1.19643	−0.598215	+	0.801336i	$0.704123\pi$
$654$	0	0
$655$	−23.3095	−0.910779
$656$	0	0
$657$	−10.0696	−0.392853
$658$	0	0
$659$	−17.1129	−0.666623	−0.333311	−	0.942817i	$-0.608166\pi$
$659$	−17.1129	−0.666623	−0.333311	+	0.942817i	$0.608166\pi$
$660$	0	0
$661$	5.54480	0.215668	0.107834	−	0.994169i	$-0.465609\pi$
$661$	5.54480	0.215668	0.107834	+	0.994169i	$0.465609\pi$
$662$	0	0
$663$	9.67189	0.375625
$664$	0	0
$665$	0.283405	0.0109900
$666$	0	0
$667$	−34.9634	−1.35379
$668$	0	0
$669$	−5.67368	−0.219357
$670$	0	0
$671$	−3.66560	−0.141509
$672$	0	0
$673$	16.4987	0.635979	0.317990	−	0.948094i	$-0.396992\pi$
$673$	16.4987	0.635979	0.317990	+	0.948094i	$0.396992\pi$
$674$	0	0
$675$	19.4669	0.749283
$676$	0	0
$677$	−47.7528	−1.83529	−0.917645	−	0.397400i	$-0.869912\pi$
$677$	−47.7528	−1.83529	−0.917645	+	0.397400i	$0.869912\pi$
$678$	0	0
$679$	−9.75868	−0.374504
$680$	0	0
$681$	12.3471	0.473143
$682$	0	0
$683$	7.13651	0.273071	0.136536	−	0.990635i	$-0.456403\pi$
$683$	7.13651	0.273071	0.136536	+	0.990635i	$0.456403\pi$
$684$	0	0
$685$	8.82486	0.337181
$686$	0	0
$687$	8.26747	0.315424
$688$	0	0
$689$	9.80380	0.373495
$690$	0	0
$691$	27.9592	1.06362	0.531809	−	0.846864i	$-0.321512\pi$
$691$	27.9592	1.06362	0.531809	+	0.846864i	$0.321512\pi$
$692$	0	0
$693$	−0.798370	−0.0303276
$694$	0	0
$695$	−22.3610	−0.848202
$696$	0	0
$697$	39.2515	1.48676
$698$	0	0
$699$	−6.54357	−0.247501
$700$	0	0
$701$	6.46299	0.244104	0.122052	−	0.992524i	$-0.461053\pi$
$701$	6.46299	0.244104	0.122052	+	0.992524i	$0.461053\pi$
$702$	0	0
$703$	1.74551	0.0658333
$704$	0	0
$705$	−17.0354	−0.641592
$706$	0	0
$707$	−0.974919	−0.0366656
$708$	0	0
$709$	−37.7675	−1.41839	−0.709194	−	0.705013i	$-0.750941\pi$
$709$	−37.7675	−1.41839	−0.709194	+	0.705013i	$0.750941\pi$
$710$	0	0
$711$	3.64218	0.136592
$712$	0	0
$713$	−5.83049	−0.218353
$714$	0	0
$715$	−1.24336	−0.0464992
$716$	0	0
$717$	32.9514	1.23059
$718$	0	0
$719$	26.7807	0.998752	0.499376	−	0.866385i	$-0.333563\pi$
$719$	26.7807	0.998752	0.499376	+	0.866385i	$0.333563\pi$
$720$	0	0
$721$	−9.84556	−0.366668
$722$	0	0
$723$	2.41428	0.0897880
$724$	0	0
$725$	−19.1874	−0.712604
$726$	0	0
$727$	−19.1445	−0.710030	−0.355015	−	0.934861i	$-0.615524\pi$
$727$	−19.1445	−0.710030	−0.355015	+	0.934861i	$0.615524\pi$
$728$	0	0
$729$	29.8521	1.10563
$730$	0	0
$731$	−19.0643	−0.705120
$732$	0	0
$733$	12.3655	0.456731	0.228366	−	0.973575i	$-0.426662\pi$
$733$	12.3655	0.456731	0.228366	+	0.973575i	$0.426662\pi$
$734$	0	0
$735$	1.84489	0.0680498
$736$	0	0
$737$	−3.65499	−0.134633
$738$	0	0
$739$	−26.8212	−0.986633	−0.493316	−	0.869850i	$-0.664215\pi$
$739$	−26.8212	−0.986633	−0.493316	+	0.869850i	$0.664215\pi$
$740$	0	0
$741$	−0.338206	−0.0124243
$742$	0	0
$743$	18.3865	0.674535	0.337267	−	0.941409i	$-0.390497\pi$
$743$	18.3865	0.674535	0.337267	+	0.941409i	$0.390497\pi$
$744$	0	0
$745$	1.05999	0.0388351
$746$	0	0
$747$	12.9556	0.474021
$748$	0	0
$749$	−13.5285	−0.494320
$750$	0	0
$751$	11.8401	0.432050	0.216025	−	0.976388i	$-0.430691\pi$
$751$	11.8401	0.432050	0.216025	+	0.976388i	$0.430691\pi$
$752$	0	0
$753$	9.08570	0.331101
$754$	0	0
$755$	13.4065	0.487912
$756$	0	0
$757$	17.4786	0.635269	0.317635	−	0.948213i	$-0.397111\pi$
$757$	17.4786	0.635269	0.317635	+	0.948213i	$0.397111\pi$
$758$	0	0
$759$	−9.33893	−0.338982
$760$	0	0
$761$	31.0674	1.12619	0.563096	−	0.826391i	$-0.309610\pi$
$761$	31.0674	1.12619	0.563096	+	0.826391i	$0.309610\pi$
$762$	0	0
$763$	5.32818	0.192893
$764$	0	0
$765$	6.47056	0.233943
$766$	0	0
$767$	−0.662663	−0.0239274
$768$	0	0
$769$	−45.7034	−1.64811	−0.824053	−	0.566512i	$-0.808292\pi$
$769$	−45.7034	−1.64811	−0.824053	+	0.566512i	$0.808292\pi$
$770$	0	0
$771$	−20.3171	−0.731704
$772$	0	0
$773$	−4.05766	−0.145944	−0.0729720	−	0.997334i	$-0.523248\pi$
$773$	−4.05766	−0.145944	−0.0729720	+	0.997334i	$0.523248\pi$
$774$	0	0
$775$	−3.19969	−0.114936
$776$	0	0
$777$	11.3628	0.407639
$778$	0	0
$779$	−1.37255	−0.0491766
$780$	0	0
$781$	7.73384	0.276739
$782$	0	0
$783$	−31.3082	−1.11886
$784$	0	0
$785$	−29.6628	−1.05871
$786$	0	0
$787$	−28.1849	−1.00468	−0.502341	−	0.864669i	$-0.667528\pi$
$787$	−28.1849	−1.00468	−0.502341	+	0.864669i	$0.667528\pi$
$788$	0	0
$789$	−8.98309	−0.319807
$790$	0	0
$791$	−7.46315	−0.265359
$792$	0	0
$793$	3.66560	0.130169
$794$	0	0
$795$	−18.0869	−0.641477
$796$	0	0
$797$	3.78169	0.133954	0.0669771	−	0.997755i	$-0.478665\pi$
$797$	3.78169	0.133954	0.0669771	+	0.997755i	$0.478665\pi$
$798$	0	0
$799$	60.1897	2.12936
$800$	0	0
$801$	−6.21794	−0.219700
$802$	0	0
$803$	12.6127	0.445093
$804$	0	0
$805$	−7.82570	−0.275820
$806$	0	0
$807$	26.3621	0.927991
$808$	0	0
$809$	−17.9864	−0.632369	−0.316185	−	0.948698i	$-0.602402\pi$
$809$	−17.9864	−0.632369	−0.316185	+	0.948698i	$0.602402\pi$
$810$	0	0
$811$	50.2984	1.76621	0.883107	−	0.469171i	$-0.155447\pi$
$811$	50.2984	1.76621	0.883107	+	0.469171i	$0.155447\pi$
$812$	0	0
$813$	15.3253	0.537481
$814$	0	0
$815$	25.6153	0.897263
$816$	0	0
$817$	0.666641	0.0233228
$818$	0	0
$819$	0.798370	0.0278973
$820$	0	0
$821$	−34.3262	−1.19799	−0.598997	−	0.800751i	$-0.704434\pi$
$821$	−34.3262	−1.19799	−0.598997	+	0.800751i	$0.704434\pi$
$822$	0	0
$823$	17.4286	0.607524	0.303762	−	0.952748i	$-0.401757\pi$
$823$	17.4286	0.607524	0.303762	+	0.952748i	$0.401757\pi$
$824$	0	0
$825$	−5.12508	−0.178432
$826$	0	0
$827$	−20.4744	−0.711965	−0.355983	−	0.934493i	$-0.615854\pi$
$827$	−20.4744	−0.711965	−0.355983	+	0.934493i	$0.615854\pi$
$828$	0	0
$829$	−14.4773	−0.502817	−0.251408	−	0.967881i	$-0.580894\pi$
$829$	−14.4773	−0.502817	−0.251408	+	0.967881i	$0.580894\pi$
$830$	0	0
$831$	1.05762	0.0366883
$832$	0	0
$833$	−6.51837	−0.225848
$834$	0	0
$835$	24.5289	0.848858
$836$	0	0
$837$	−5.22094	−0.180462
$838$	0	0
$839$	14.4244	0.497986	0.248993	−	0.968505i	$-0.419900\pi$
$839$	14.4244	0.497986	0.248993	+	0.968505i	$0.419900\pi$
$840$	0	0
$841$	1.85872	0.0640937
$842$	0	0
$843$	−12.3736	−0.426170
$844$	0	0
$845$	1.24336	0.0427730
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−16.0851	−0.552039
$850$	0	0
$851$	−48.1991	−1.65224
$852$	0	0
$853$	5.98544	0.204938	0.102469	−	0.994736i	$-0.467326\pi$
$853$	5.98544	0.204938	0.102469	+	0.994736i	$0.467326\pi$
$854$	0	0
$855$	−0.226262	−0.00773801
$856$	0	0
$857$	10.7103	0.365858	0.182929	−	0.983126i	$-0.441442\pi$
$857$	10.7103	0.365858	0.182929	+	0.983126i	$0.441442\pi$
$858$	0	0
$859$	5.64085	0.192463	0.0962316	−	0.995359i	$-0.469321\pi$
$859$	5.64085	0.192463	0.0962316	+	0.995359i	$0.469321\pi$
$860$	0	0
$861$	−8.93490	−0.304500
$862$	0	0
$863$	27.5782	0.938771	0.469386	−	0.882993i	$-0.344475\pi$
$863$	27.5782	0.938771	0.469386	+	0.882993i	$0.344475\pi$
$864$	0	0
$865$	−21.2306	−0.721863
$866$	0	0
$867$	37.8205	1.28445
$868$	0	0
$869$	−4.56202	−0.154756
$870$	0	0
$871$	3.65499	0.123845
$872$	0	0
$873$	7.79104	0.263687
$874$	0	0
$875$	−10.5115	−0.355352
$876$	0	0
$877$	−5.02453	−0.169666	−0.0848331	−	0.996395i	$-0.527036\pi$
$877$	−5.02453	−0.169666	−0.0848331	+	0.996395i	$0.527036\pi$
$878$	0	0
$879$	−17.2132	−0.580588
$880$	0	0
$881$	−18.4158	−0.620444	−0.310222	−	0.950664i	$-0.600403\pi$
$881$	−18.4158	−0.620444	−0.310222	+	0.950664i	$0.600403\pi$
$882$	0	0
$883$	2.52834	0.0850852	0.0425426	−	0.999095i	$-0.486454\pi$
$883$	2.52834	0.0850852	0.0425426	+	0.999095i	$0.486454\pi$
$884$	0	0
$885$	1.22254	0.0410953
$886$	0	0
$887$	7.44532	0.249989	0.124995	−	0.992157i	$-0.460109\pi$
$887$	7.44532	0.249989	0.124995	+	0.992157i	$0.460109\pi$
$888$	0	0
$889$	7.87179	0.264011
$890$	0	0
$891$	−5.96749	−0.199919
$892$	0	0
$893$	−2.10471	−0.0704315
$894$	0	0
$895$	−1.05715	−0.0353365
$896$	0	0
$897$	9.33893	0.311818
$898$	0	0
$899$	5.14599	0.171628
$900$	0	0
$901$	63.9048	2.12898
$902$	0	0
$903$	4.33965	0.144414
$904$	0	0
$905$	28.4011	0.944083
$906$	0	0
$907$	50.8747	1.68927	0.844633	−	0.535345i	$-0.179818\pi$
$907$	50.8747	1.68927	0.844633	+	0.535345i	$0.179818\pi$
$908$	0	0
$909$	0.778346	0.0258161
$910$	0	0
$911$	0.406731	0.0134756	0.00673780	−	0.999977i	$-0.497855\pi$
$911$	0.406731	0.0134756	0.00673780	+	0.999977i	$0.497855\pi$
$912$	0	0
$913$	−16.2276	−0.537054
$914$	0	0
$915$	−6.76263	−0.223566
$916$	0	0
$917$	−18.7472	−0.619086
$918$	0	0
$919$	25.8262	0.851929	0.425965	−	0.904740i	$-0.359935\pi$
$919$	25.8262	0.851929	0.425965	+	0.904740i	$0.359935\pi$
$920$	0	0
$921$	37.5707	1.23800
$922$	0	0
$923$	−7.73384	−0.254563
$924$	0	0
$925$	−26.4510	−0.869703
$926$	0	0
$927$	7.86040	0.258169
$928$	0	0
$929$	−44.0774	−1.44613	−0.723066	−	0.690779i	$-0.757268\pi$
$929$	−44.0774	−1.44613	−0.723066	+	0.690779i	$0.757268\pi$
$930$	0	0
$931$	0.227934	0.00747025
$932$	0	0
$933$	−19.4604	−0.637105
$934$	0	0
$935$	−8.10471	−0.265052
$936$	0	0
$937$	−60.9030	−1.98961	−0.994807	−	0.101775i	$-0.967548\pi$
$937$	−60.9030	−1.98961	−0.994807	+	0.101775i	$0.967548\pi$
$938$	0	0
$939$	−21.7111	−0.708514
$940$	0	0
$941$	−0.914238	−0.0298033	−0.0149017	−	0.999889i	$-0.504744\pi$
$941$	−0.914238	−0.0298033	−0.0149017	+	0.999889i	$0.504744\pi$
$942$	0	0
$943$	37.9003	1.23420
$944$	0	0
$945$	−7.00757	−0.227956
$946$	0	0
$947$	−49.5737	−1.61093	−0.805465	−	0.592644i	$-0.798084\pi$
$947$	−49.5737	−1.61093	−0.805465	+	0.592644i	$0.798084\pi$
$948$	0	0
$949$	−12.6127	−0.409426
$950$	0	0
$951$	12.9186	0.418914
$952$	0	0
$953$	33.6169	1.08896	0.544479	−	0.838775i	$-0.316727\pi$
$953$	33.6169	1.08896	0.544479	+	0.838775i	$0.316727\pi$
$954$	0	0
$955$	13.3597	0.432310
$956$	0	0
$957$	8.24254	0.266443
$958$	0	0
$959$	7.09757	0.229192
$960$	0	0
$961$	−30.1419	−0.972318
$962$	0	0
$963$	10.8007	0.348049
$964$	0	0
$965$	−10.7695	−0.346682
$966$	0	0
$967$	−16.6979	−0.536967	−0.268483	−	0.963284i	$-0.586522\pi$
$967$	−16.6979	−0.536967	−0.268483	+	0.963284i	$0.586522\pi$
$968$	0	0
$969$	−2.20455	−0.0708205
$970$	0	0
$971$	55.1454	1.76970	0.884850	−	0.465876i	$-0.154261\pi$
$971$	55.1454	1.76970	0.884850	+	0.465876i	$0.154261\pi$
$972$	0	0
$973$	−17.9843	−0.576550
$974$	0	0
$975$	5.12508	0.164134
$976$	0	0
$977$	50.9811	1.63103	0.815514	−	0.578737i	$-0.196454\pi$
$977$	50.9811	1.63103	0.815514	+	0.578737i	$0.196454\pi$
$978$	0	0
$979$	7.78829	0.248915
$980$	0	0
$981$	−4.25386	−0.135815
$982$	0	0
$983$	12.6080	0.402133	0.201066	−	0.979578i	$-0.435559\pi$
$983$	12.6080	0.402133	0.201066	+	0.979578i	$0.435559\pi$
$984$	0	0
$985$	−17.7065	−0.564176
$986$	0	0
$987$	−13.7011	−0.436110
$988$	0	0
$989$	−18.4080	−0.585342
$990$	0	0
$991$	49.7674	1.58091	0.790457	−	0.612518i	$-0.209843\pi$
$991$	49.7674	1.58091	0.790457	+	0.612518i	$0.209843\pi$
$992$	0	0
$993$	−19.5222	−0.619518
$994$	0	0
$995$	13.1487	0.416840
$996$	0	0
$997$	−42.9478	−1.36017	−0.680085	−	0.733133i	$-0.738057\pi$
$997$	−42.9478	−1.36017	−0.680085	+	0.733133i	$0.738057\pi$
$998$	0	0
$999$	−43.1602	−1.36553

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.s.1.8	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.s.1.8	✓	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.48379 q^{3} +1.24336 q^{5} +1.00000 q^{7} -0.798370 q^{9} +O(q^{10})\)	\(q+1.48379 q^{3} +1.24336 q^{5} +1.00000 q^{7} -0.798370 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.84489 q^{15} -6.51837 q^{17} +0.227934 q^{19} +1.48379 q^{21} -6.29398 q^{23} -3.45405 q^{25} -5.63598 q^{27} +5.55506 q^{29} +0.926360 q^{31} +1.48379 q^{33} +1.24336 q^{35} +7.65797 q^{37} -1.48379 q^{39} -6.02168 q^{41} +2.92471 q^{43} -0.992665 q^{45} -9.23385 q^{47} +1.00000 q^{49} -9.67189 q^{51} -9.80380 q^{53} +1.24336 q^{55} +0.338206 q^{57} +0.662663 q^{59} -3.66560 q^{61} -0.798370 q^{63} -1.24336 q^{65} -3.65499 q^{67} -9.33893 q^{69} +7.73384 q^{71} +12.6127 q^{73} -5.12508 q^{75} +1.00000 q^{77} -4.56202 q^{79} -5.96749 q^{81} -16.2276 q^{83} -8.10471 q^{85} +8.24254 q^{87} +7.78829 q^{89} -1.00000 q^{91} +1.37452 q^{93} +0.283405 q^{95} -9.75868 q^{97} -0.798370 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9}+O(q^{10}) \) 10 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 9 * q^9	\( 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9} + 10 q^{11} - 10 q^{13} - 5 q^{15} - 11 q^{17} + 2 q^{19} - 3 q^{21} - 8 q^{23} + 2 q^{25} - 15 q^{27} - 8 q^{29} - 23 q^{31} - 3 q^{33} - 4 q^{35} + 10 q^{37} + 3 q^{39} - 18 q^{41} + 12 q^{43} - 10 q^{45} - 36 q^{47} + 10 q^{49} + 9 q^{51} - 21 q^{53} - 4 q^{55} - 30 q^{57} - 13 q^{59} - 2 q^{61} + 9 q^{63} + 4 q^{65} - 4 q^{67} - 26 q^{69} - 24 q^{71} - 23 q^{73} - 28 q^{75} + 10 q^{77} + 14 q^{79} + 30 q^{81} - 9 q^{83} - 17 q^{85} + 7 q^{87} - 18 q^{89} - 10 q^{91} + q^{93} - 4 q^{95} - 9 q^{97} + 9 q^{99}+O(q^{100}) \) 10 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 9 * q^9 + 10 * q^11 - 10 * q^13 - 5 * q^15 - 11 * q^17 + 2 * q^19 - 3 * q^21 - 8 * q^23 + 2 * q^25 - 15 * q^27 - 8 * q^29 - 23 * q^31 - 3 * q^33 - 4 * q^35 + 10 * q^37 + 3 * q^39 - 18 * q^41 + 12 * q^43 - 10 * q^45 - 36 * q^47 + 10 * q^49 + 9 * q^51 - 21 * q^53 - 4 * q^55 - 30 * q^57 - 13 * q^59 - 2 * q^61 + 9 * q^63 + 4 * q^65 - 4 * q^67 - 26 * q^69 - 24 * q^71 - 23 * q^73 - 28 * q^75 + 10 * q^77 + 14 * q^79 + 30 * q^81 - 9 * q^83 - 17 * q^85 + 7 * q^87 - 18 * q^89 - 10 * q^91 + q^93 - 4 * q^95 - 9 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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