LMFDB - Embedded newform 8008.2.a.r.1.9

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 9x^{7} + 36x^{6} + 23x^{5} - 89x^{4} - 20x^{3} + 51x^{2} + 18x + 1 $ x^9 - 4x^8 - 9x^7 + 36x^6 + 23x^5 - 89x^4 - 20x^3 + 51x^2 + 18x + 1
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			1.9
Root			$-2.27563$ 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)

$f(q)$	$=$	$q+3.27563 q^{3} +1.56622 q^{5} -1.00000 q^{7} +7.72976 q^{9} +O(q^{10})$	$q+3.27563 q^{3} +1.56622 q^{5} -1.00000 q^{7} +7.72976 q^{9} -1.00000 q^{11} -1.00000 q^{13} +5.13038 q^{15} -3.96806 q^{17} +5.63967 q^{19} -3.27563 q^{21} -2.53405 q^{23} -2.54694 q^{25} +15.4929 q^{27} +8.75659 q^{29} -3.68835 q^{31} -3.27563 q^{33} -1.56622 q^{35} +6.52445 q^{37} -3.27563 q^{39} -2.06895 q^{41} +5.57197 q^{43} +12.1065 q^{45} +2.26128 q^{47} +1.00000 q^{49} -12.9979 q^{51} +10.8910 q^{53} -1.56622 q^{55} +18.4735 q^{57} +0.931751 q^{59} -5.40620 q^{61} -7.72976 q^{63} -1.56622 q^{65} +9.98281 q^{67} -8.30062 q^{69} +11.8688 q^{71} +6.84761 q^{73} -8.34283 q^{75} +1.00000 q^{77} -1.30713 q^{79} +27.5599 q^{81} -5.68837 q^{83} -6.21487 q^{85} +28.6834 q^{87} -7.44526 q^{89} +1.00000 q^{91} -12.0817 q^{93} +8.83299 q^{95} +8.98492 q^{97} -7.72976 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9}+O(q^{10}) $ 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9	$ 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9} - 9 q^{11} - 9 q^{13} - 3 q^{15} - 7 q^{17} + 13 q^{19} - 5 q^{21} + 9 q^{23} - 2 q^{25} + 5 q^{27} + q^{29} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 14 q^{37} - 5 q^{39} - 2 q^{41} + 5 q^{43} - 15 q^{45} + 13 q^{47} + 9 q^{49} - 3 q^{51} + 22 q^{53} - 3 q^{55} + 16 q^{57} + 43 q^{59} - 10 q^{61} - 8 q^{63} - 3 q^{65} + 26 q^{67} - 30 q^{69} + 18 q^{71} - 8 q^{73} + 28 q^{75} + 9 q^{77} - 9 q^{79} + 33 q^{81} + 4 q^{83} + 5 q^{85} + 33 q^{87} + 7 q^{89} + 9 q^{91} + 13 q^{93} + 7 q^{95} + 2 q^{97} - 8 q^{99}+O(q^{100}) $ 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9 - 9 * q^11 - 9 * q^13 - 3 * q^15 - 7 * q^17 + 13 * q^19 - 5 * q^21 + 9 * q^23 - 2 * q^25 + 5 * q^27 + q^29 + 10 * q^31 - 5 * q^33 - 3 * q^35 + 14 * q^37 - 5 * q^39 - 2 * q^41 + 5 * q^43 - 15 * q^45 + 13 * q^47 + 9 * q^49 - 3 * q^51 + 22 * q^53 - 3 * q^55 + 16 * q^57 + 43 * q^59 - 10 * q^61 - 8 * q^63 - 3 * q^65 + 26 * q^67 - 30 * q^69 + 18 * q^71 - 8 * q^73 + 28 * q^75 + 9 * q^77 - 9 * q^79 + 33 * q^81 + 4 * q^83 + 5 * q^85 + 33 * q^87 + 7 * q^89 + 9 * q^91 + 13 * q^93 + 7 * q^95 + 2 * q^97 - 8 * 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
$3$	3.27563	1.89119	0.945593	−	0.325351i	$-0.105482\pi$
$3$	3.27563	1.89119	0.945593	+	0.325351i	$0.105482\pi$
$4$	0	0
$5$	1.56622	0.700437	0.350219	−	0.936668i	$-0.386107\pi$
$5$	1.56622	0.700437	0.350219	+	0.936668i	$0.386107\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	7.72976	2.57659
$10$	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
$15$	5.13038	1.32466
$16$	0	0
$17$	−3.96806	−0.962396	−0.481198	−	0.876612i	$-0.659798\pi$
$17$	−3.96806	−0.962396	−0.481198	+	0.876612i	$0.659798\pi$
$18$	0	0
$19$	5.63967	1.29383	0.646914	−	0.762563i	$-0.276059\pi$
$19$	5.63967	1.29383	0.646914	+	0.762563i	$0.276059\pi$
$20$	0	0
$21$	−3.27563	−0.714801
$22$	0	0
$23$	−2.53405	−0.528386	−0.264193	−	0.964470i	$-0.585106\pi$
$23$	−2.53405	−0.528386	−0.264193	+	0.964470i	$0.585106\pi$
$24$	0	0
$25$	−2.54694	−0.509388
$26$	0	0
$27$	15.4929	2.98162
$28$	0	0
$29$	8.75659	1.62606	0.813029	−	0.582223i	$-0.197817\pi$
$29$	8.75659	1.62606	0.813029	+	0.582223i	$0.197817\pi$
$30$	0	0
$31$	−3.68835	−0.662446	−0.331223	−	0.943552i	$-0.607461\pi$
$31$	−3.68835	−0.662446	−0.331223	+	0.943552i	$0.607461\pi$
$32$	0	0
$33$	−3.27563	−0.570214
$34$	0	0
$35$	−1.56622	−0.264740
$36$	0	0
$37$	6.52445	1.07261	0.536307	−	0.844023i	$-0.319819\pi$
$37$	6.52445	1.07261	0.536307	+	0.844023i	$0.319819\pi$
$38$	0	0
$39$	−3.27563	−0.524521
$40$	0	0
$41$	−2.06895	−0.323115	−0.161558	−	0.986863i	$-0.551652\pi$
$41$	−2.06895	−0.323115	−0.161558	+	0.986863i	$0.551652\pi$
$42$	0	0
$43$	5.57197	0.849718	0.424859	−	0.905259i	$-0.360324\pi$
$43$	5.57197	0.849718	0.424859	+	0.905259i	$0.360324\pi$
$44$	0	0
$45$	12.1065	1.80474
$46$	0	0
$47$	2.26128	0.329842	0.164921	−	0.986307i	$-0.447263\pi$
$47$	2.26128	0.329842	0.164921	+	0.986307i	$0.447263\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−12.9979	−1.82007
$52$	0	0
$53$	10.8910	1.49600	0.747999	−	0.663700i	$-0.231015\pi$
$53$	10.8910	1.49600	0.747999	+	0.663700i	$0.231015\pi$
$54$	0	0
$55$	−1.56622	−0.211190
$56$	0	0
$57$	18.4735	2.44687
$58$	0	0
$59$	0.931751	0.121304	0.0606518	−	0.998159i	$-0.480682\pi$
$59$	0.931751	0.121304	0.0606518	+	0.998159i	$0.480682\pi$
$60$	0	0
$61$	−5.40620	−0.692193	−0.346097	−	0.938199i	$-0.612493\pi$
$61$	−5.40620	−0.692193	−0.346097	+	0.938199i	$0.612493\pi$
$62$	0	0
$63$	−7.72976	−0.973858
$64$	0	0
$65$	−1.56622	−0.194266
$66$	0	0
$67$	9.98281	1.21959	0.609797	−	0.792557i	$-0.291251\pi$
$67$	9.98281	1.21959	0.609797	+	0.792557i	$0.291251\pi$
$68$	0	0
$69$	−8.30062	−0.999277
$70$	0	0
$71$	11.8688	1.40856	0.704282	−	0.709920i	$-0.251269\pi$
$71$	11.8688	1.40856	0.704282	+	0.709920i	$0.251269\pi$
$72$	0	0
$73$	6.84761	0.801452	0.400726	−	0.916198i	$-0.368758\pi$
$73$	6.84761	0.801452	0.400726	+	0.916198i	$0.368758\pi$
$74$	0	0
$75$	−8.34283	−0.963348
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−1.30713	−0.147063	−0.0735316	−	0.997293i	$-0.523427\pi$
$79$	−1.30713	−0.147063	−0.0735316	+	0.997293i	$0.523427\pi$
$80$	0	0
$81$	27.5599	3.06221
$82$	0	0
$83$	−5.68837	−0.624380	−0.312190	−	0.950020i	$-0.601063\pi$
$83$	−5.68837	−0.624380	−0.312190	+	0.950020i	$0.601063\pi$
$84$	0	0
$85$	−6.21487	−0.674098
$86$	0	0
$87$	28.6834	3.07518
$88$	0	0
$89$	−7.44526	−0.789196	−0.394598	−	0.918854i	$-0.629116\pi$
$89$	−7.44526	−0.789196	−0.394598	+	0.918854i	$0.629116\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−12.0817	−1.25281
$94$	0	0
$95$	8.83299	0.906245
$96$	0	0
$97$	8.98492	0.912280	0.456140	−	0.889908i	$-0.349232\pi$
$97$	8.98492	0.912280	0.456140	+	0.889908i	$0.349232\pi$
$98$	0	0
$99$	−7.72976	−0.776870
$100$	0	0
$101$	−2.53796	−0.252536	−0.126268	−	0.991996i	$-0.540300\pi$
$101$	−2.53796	−0.252536	−0.126268	+	0.991996i	$0.540300\pi$
$102$	0	0
$103$	8.53448	0.840927	0.420464	−	0.907309i	$-0.361867\pi$
$103$	8.53448	0.840927	0.420464	+	0.907309i	$0.361867\pi$
$104$	0	0
$105$	−5.13038	−0.500673
$106$	0	0
$107$	0.933711	0.0902653	0.0451326	−	0.998981i	$-0.485629\pi$
$107$	0.933711	0.0902653	0.0451326	+	0.998981i	$0.485629\pi$
$108$	0	0
$109$	−0.138088	−0.0132265	−0.00661323	−	0.999978i	$-0.502105\pi$
$109$	−0.138088	−0.0132265	−0.00661323	+	0.999978i	$0.502105\pi$
$110$	0	0
$111$	21.3717	2.02851
$112$	0	0
$113$	3.72160	0.350098	0.175049	−	0.984560i	$-0.443992\pi$
$113$	3.72160	0.350098	0.175049	+	0.984560i	$0.443992\pi$
$114$	0	0
$115$	−3.96889	−0.370101
$116$	0	0
$117$	−7.72976	−0.714617
$118$	0	0
$119$	3.96806	0.363751
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.77711	−0.611071
$124$	0	0
$125$	−11.8202	−1.05723
$126$	0	0
$127$	−22.2025	−1.97015	−0.985077	−	0.172117i	$-0.944939\pi$
$127$	−22.2025	−1.97015	−0.985077	+	0.172117i	$0.944939\pi$
$128$	0	0
$129$	18.2517	1.60698
$130$	0	0
$131$	19.4246	1.69713	0.848567	−	0.529088i	$-0.177466\pi$
$131$	19.4246	1.69713	0.848567	+	0.529088i	$0.177466\pi$
$132$	0	0
$133$	−5.63967	−0.489021
$134$	0	0
$135$	24.2654	2.08844
$136$	0	0
$137$	20.4570	1.74776	0.873882	−	0.486139i	$-0.161595\pi$
$137$	20.4570	1.74776	0.873882	+	0.486139i	$0.161595\pi$
$138$	0	0
$139$	−16.3622	−1.38783	−0.693914	−	0.720058i	$-0.744115\pi$
$139$	−16.3622	−1.38783	−0.693914	+	0.720058i	$0.744115\pi$
$140$	0	0
$141$	7.40712	0.623792
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	13.7148	1.13895
$146$	0	0
$147$	3.27563	0.270170
$148$	0	0
$149$	−12.0747	−0.989199	−0.494599	−	0.869121i	$-0.664685\pi$
$149$	−12.0747	−0.989199	−0.494599	+	0.869121i	$0.664685\pi$
$150$	0	0
$151$	16.8358	1.37008	0.685041	−	0.728505i	$-0.259784\pi$
$151$	16.8358	1.37008	0.685041	+	0.728505i	$0.259784\pi$
$152$	0	0
$153$	−30.6721	−2.47970
$154$	0	0
$155$	−5.77678	−0.464002
$156$	0	0
$157$	−13.8731	−1.10719	−0.553596	−	0.832786i	$-0.686745\pi$
$157$	−13.8731	−1.10719	−0.553596	+	0.832786i	$0.686745\pi$
$158$	0	0
$159$	35.6750	2.82921
$160$	0	0
$161$	2.53405	0.199711
$162$	0	0
$163$	4.46010	0.349342	0.174671	−	0.984627i	$-0.444114\pi$
$163$	4.46010	0.349342	0.174671	+	0.984627i	$0.444114\pi$
$164$	0	0
$165$	−5.13038	−0.399399
$166$	0	0
$167$	4.22153	0.326671	0.163336	−	0.986571i	$-0.447775\pi$
$167$	4.22153	0.326671	0.163336	+	0.986571i	$0.447775\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	43.5933	3.33366
$172$	0	0
$173$	−25.5142	−1.93981	−0.969906	−	0.243480i	$-0.921711\pi$
$173$	−25.5142	−1.93981	−0.969906	+	0.243480i	$0.921711\pi$
$174$	0	0
$175$	2.54694	0.192531
$176$	0	0
$177$	3.05207	0.229408
$178$	0	0
$179$	16.0519	1.19978	0.599888	−	0.800084i	$-0.295212\pi$
$179$	16.0519	1.19978	0.599888	+	0.800084i	$0.295212\pi$
$180$	0	0
$181$	−13.6846	−1.01717	−0.508585	−	0.861012i	$-0.669831\pi$
$181$	−13.6846	−1.01717	−0.508585	+	0.861012i	$0.669831\pi$
$182$	0	0
$183$	−17.7087	−1.30907
$184$	0	0
$185$	10.2188	0.751298
$186$	0	0
$187$	3.96806	0.290173
$188$	0	0
$189$	−15.4929	−1.12695
$190$	0	0
$191$	−13.5815	−0.982721	−0.491361	−	0.870956i	$-0.663500\pi$
$191$	−13.5815	−0.982721	−0.491361	+	0.870956i	$0.663500\pi$
$192$	0	0
$193$	9.12990	0.657184	0.328592	−	0.944472i	$-0.393426\pi$
$193$	9.12990	0.657184	0.328592	+	0.944472i	$0.393426\pi$
$194$	0	0
$195$	−5.13038	−0.367394
$196$	0	0
$197$	−16.8756	−1.20234	−0.601168	−	0.799123i	$-0.705298\pi$
$197$	−16.8756	−1.20234	−0.601168	+	0.799123i	$0.705298\pi$
$198$	0	0
$199$	−27.2334	−1.93052	−0.965261	−	0.261289i	$-0.915853\pi$
$199$	−27.2334	−1.93052	−0.965261	+	0.261289i	$0.915853\pi$
$200$	0	0
$201$	32.7000	2.30648
$202$	0	0
$203$	−8.75659	−0.614592
$204$	0	0
$205$	−3.24044	−0.226322
$206$	0	0
$207$	−19.5876	−1.36143
$208$	0	0
$209$	−5.63967	−0.390104
$210$	0	0
$211$	11.8908	0.818600	0.409300	−	0.912400i	$-0.365773\pi$
$211$	11.8908	0.818600	0.409300	+	0.912400i	$0.365773\pi$
$212$	0	0
$213$	38.8777	2.66386
$214$	0	0
$215$	8.72697	0.595174
$216$	0	0
$217$	3.68835	0.250381
$218$	0	0
$219$	22.4302	1.51569
$220$	0	0
$221$	3.96806	0.266921
$222$	0	0
$223$	5.26041	0.352263	0.176132	−	0.984367i	$-0.443642\pi$
$223$	5.26041	0.352263	0.176132	+	0.984367i	$0.443642\pi$
$224$	0	0
$225$	−19.6872	−1.31248
$226$	0	0
$227$	−22.1531	−1.47035	−0.735176	−	0.677877i	$-0.762900\pi$
$227$	−22.1531	−1.47035	−0.735176	+	0.677877i	$0.762900\pi$
$228$	0	0
$229$	−21.7179	−1.43516	−0.717579	−	0.696477i	$-0.754750\pi$
$229$	−21.7179	−1.43516	−0.717579	+	0.696477i	$0.754750\pi$
$230$	0	0
$231$	3.27563	0.215521
$232$	0	0
$233$	−22.7482	−1.49029	−0.745144	−	0.666904i	$-0.767619\pi$
$233$	−22.7482	−1.49029	−0.745144	+	0.666904i	$0.767619\pi$
$234$	0	0
$235$	3.54167	0.231033
$236$	0	0
$237$	−4.28166	−0.278124
$238$	0	0
$239$	2.07295	0.134088	0.0670440	−	0.997750i	$-0.478643\pi$
$239$	2.07295	0.134088	0.0670440	+	0.997750i	$0.478643\pi$
$240$	0	0
$241$	−5.93032	−0.382005	−0.191003	−	0.981590i	$-0.561174\pi$
$241$	−5.93032	−0.382005	−0.191003	+	0.981590i	$0.561174\pi$
$242$	0	0
$243$	43.7972	2.80959
$244$	0	0
$245$	1.56622	0.100062
$246$	0	0
$247$	−5.63967	−0.358843
$248$	0	0
$249$	−18.6330	−1.18082
$250$	0	0
$251$	28.8241	1.81936	0.909681	−	0.415307i	$-0.136326\pi$
$251$	28.8241	1.81936	0.909681	+	0.415307i	$0.136326\pi$
$252$	0	0
$253$	2.53405	0.159314
$254$	0	0
$255$	−20.3576	−1.27484
$256$	0	0
$257$	−8.11683	−0.506314	−0.253157	−	0.967425i	$-0.581469\pi$
$257$	−8.11683	−0.506314	−0.253157	+	0.967425i	$0.581469\pi$
$258$	0	0
$259$	−6.52445	−0.405410
$260$	0	0
$261$	67.6864	4.18968
$262$	0	0
$263$	−11.6043	−0.715553	−0.357777	−	0.933807i	$-0.616465\pi$
$263$	−11.6043	−0.715553	−0.357777	+	0.933807i	$0.616465\pi$
$264$	0	0
$265$	17.0578	1.04785
$266$	0	0
$267$	−24.3879	−1.49252
$268$	0	0
$269$	10.6195	0.647480	0.323740	−	0.946146i	$-0.395060\pi$
$269$	10.6195	0.647480	0.323740	+	0.946146i	$0.395060\pi$
$270$	0	0
$271$	−26.1799	−1.59032	−0.795158	−	0.606402i	$-0.792612\pi$
$271$	−26.1799	−1.59032	−0.795158	+	0.606402i	$0.792612\pi$
$272$	0	0
$273$	3.27563	0.198250
$274$	0	0
$275$	2.54694	0.153586
$276$	0	0
$277$	14.6389	0.879566	0.439783	−	0.898104i	$-0.355055\pi$
$277$	14.6389	0.879566	0.439783	+	0.898104i	$0.355055\pi$
$278$	0	0
$279$	−28.5100	−1.70685
$280$	0	0
$281$	25.7437	1.53574	0.767870	−	0.640606i	$-0.221317\pi$
$281$	25.7437	1.53574	0.767870	+	0.640606i	$0.221317\pi$
$282$	0	0
$283$	−28.2792	−1.68102	−0.840512	−	0.541793i	$-0.817746\pi$
$283$	−28.2792	−1.68102	−0.840512	+	0.541793i	$0.817746\pi$
$284$	0	0
$285$	28.9336	1.71388
$286$	0	0
$287$	2.06895	0.122126
$288$	0	0
$289$	−1.25451	−0.0737946
$290$	0	0
$291$	29.4313	1.72529
$292$	0	0
$293$	−7.71732	−0.450851	−0.225425	−	0.974260i	$-0.572377\pi$
$293$	−7.71732	−0.450851	−0.225425	+	0.974260i	$0.572377\pi$
$294$	0	0
$295$	1.45933	0.0849656
$296$	0	0
$297$	−15.4929	−0.898992
$298$	0	0
$299$	2.53405	0.146548
$300$	0	0
$301$	−5.57197	−0.321163
$302$	0	0
$303$	−8.31341	−0.477593
$304$	0	0
$305$	−8.46733	−0.484838
$306$	0	0
$307$	−13.4923	−0.770047	−0.385024	−	0.922907i	$-0.625807\pi$
$307$	−13.4923	−0.770047	−0.385024	+	0.922907i	$0.625807\pi$
$308$	0	0
$309$	27.9558	1.59035
$310$	0	0
$311$	−4.16267	−0.236044	−0.118022	−	0.993011i	$-0.537655\pi$
$311$	−4.16267	−0.236044	−0.118022	+	0.993011i	$0.537655\pi$
$312$	0	0
$313$	6.43698	0.363840	0.181920	−	0.983313i	$-0.441769\pi$
$313$	6.43698	0.363840	0.181920	+	0.983313i	$0.441769\pi$
$314$	0	0
$315$	−12.1065	−0.682126
$316$	0	0
$317$	−19.5795	−1.09969	−0.549846	−	0.835266i	$-0.685314\pi$
$317$	−19.5795	−1.09969	−0.549846	+	0.835266i	$0.685314\pi$
$318$	0	0
$319$	−8.75659	−0.490275
$320$	0	0
$321$	3.05849	0.170709
$322$	0	0
$323$	−22.3785	−1.24517
$324$	0	0
$325$	2.54694	0.141279
$326$	0	0
$327$	−0.452326	−0.0250137
$328$	0	0
$329$	−2.26128	−0.124668
$330$	0	0
$331$	0.672975	0.0369901	0.0184950	−	0.999829i	$-0.494113\pi$
$331$	0.672975	0.0369901	0.0184950	+	0.999829i	$0.494113\pi$
$332$	0	0
$333$	50.4324	2.76368
$334$	0	0
$335$	15.6353	0.854249
$336$	0	0
$337$	3.75966	0.204802	0.102401	−	0.994743i	$-0.467348\pi$
$337$	3.75966	0.204802	0.102401	+	0.994743i	$0.467348\pi$
$338$	0	0
$339$	12.1906	0.662101
$340$	0	0
$341$	3.68835	0.199735
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−13.0006	−0.699931
$346$	0	0
$347$	−18.9429	−1.01691	−0.508453	−	0.861090i	$-0.669782\pi$
$347$	−18.9429	−1.01691	−0.508453	+	0.861090i	$0.669782\pi$
$348$	0	0
$349$	−11.1404	−0.596329	−0.298165	−	0.954514i	$-0.596374\pi$
$349$	−11.1404	−0.596329	−0.298165	+	0.954514i	$0.596374\pi$
$350$	0	0
$351$	−15.4929	−0.826952
$352$	0	0
$353$	15.6080	0.830732	0.415366	−	0.909654i	$-0.363654\pi$
$353$	15.6080	0.830732	0.415366	+	0.909654i	$0.363654\pi$
$354$	0	0
$355$	18.5892	0.986611
$356$	0	0
$357$	12.9979	0.687922
$358$	0	0
$359$	−9.22663	−0.486963	−0.243482	−	0.969906i	$-0.578290\pi$
$359$	−9.22663	−0.486963	−0.243482	+	0.969906i	$0.578290\pi$
$360$	0	0
$361$	12.8058	0.673991
$362$	0	0
$363$	3.27563	0.171926
$364$	0	0
$365$	10.7249	0.561366
$366$	0	0
$367$	−6.32095	−0.329951	−0.164975	−	0.986298i	$-0.552754\pi$
$367$	−6.32095	−0.329951	−0.164975	+	0.986298i	$0.552754\pi$
$368$	0	0
$369$	−15.9925	−0.832534
$370$	0	0
$371$	−10.8910	−0.565434
$372$	0	0
$373$	−5.06328	−0.262167	−0.131083	−	0.991371i	$-0.541846\pi$
$373$	−5.06328	−0.262167	−0.131083	+	0.991371i	$0.541846\pi$
$374$	0	0
$375$	−38.7186	−1.99942
$376$	0	0
$377$	−8.75659	−0.450988
$378$	0	0
$379$	1.66915	0.0857386	0.0428693	−	0.999081i	$-0.486350\pi$
$379$	1.66915	0.0857386	0.0428693	+	0.999081i	$0.486350\pi$
$380$	0	0
$381$	−72.7272	−3.72593
$382$	0	0
$383$	8.37176	0.427777	0.213889	−	0.976858i	$-0.431387\pi$
$383$	8.37176	0.427777	0.213889	+	0.976858i	$0.431387\pi$
$384$	0	0
$385$	1.56622	0.0798222
$386$	0	0
$387$	43.0700	2.18937
$388$	0	0
$389$	12.1034	0.613665	0.306833	−	0.951763i	$-0.400731\pi$
$389$	12.1034	0.613665	0.306833	+	0.951763i	$0.400731\pi$
$390$	0	0
$391$	10.0553	0.508517
$392$	0	0
$393$	63.6278	3.20960
$394$	0	0
$395$	−2.04725	−0.103009
$396$	0	0
$397$	−2.42971	−0.121944	−0.0609718	−	0.998139i	$-0.519420\pi$
$397$	−2.42971	−0.121944	−0.0609718	+	0.998139i	$0.519420\pi$
$398$	0	0
$399$	−18.4735	−0.924830
$400$	0	0
$401$	−16.3887	−0.818411	−0.409205	−	0.912442i	$-0.634194\pi$
$401$	−16.3887	−0.818411	−0.409205	+	0.912442i	$0.634194\pi$
$402$	0	0
$403$	3.68835	0.183730
$404$	0	0
$405$	43.1650	2.14489
$406$	0	0
$407$	−6.52445	−0.323405
$408$	0	0
$409$	18.2487	0.902342	0.451171	−	0.892438i	$-0.351007\pi$
$409$	18.2487	0.902342	0.451171	+	0.892438i	$0.351007\pi$
$410$	0	0
$411$	67.0097	3.30535
$412$	0	0
$413$	−0.931751	−0.0458485
$414$	0	0
$415$	−8.90927	−0.437339
$416$	0	0
$417$	−53.5967	−2.62464
$418$	0	0
$419$	−32.0834	−1.56738	−0.783688	−	0.621155i	$-0.786664\pi$
$419$	−32.0834	−1.56738	−0.783688	+	0.621155i	$0.786664\pi$
$420$	0	0
$421$	−12.7199	−0.619932	−0.309966	−	0.950748i	$-0.600318\pi$
$421$	−12.7199	−0.619932	−0.309966	+	0.950748i	$0.600318\pi$
$422$	0	0
$423$	17.4792	0.849865
$424$	0	0
$425$	10.1064	0.490233
$426$	0	0
$427$	5.40620	0.261624
$428$	0	0
$429$	3.27563	0.158149
$430$	0	0
$431$	7.49675	0.361106	0.180553	−	0.983565i	$-0.442211\pi$
$431$	7.49675	0.361106	0.180553	+	0.983565i	$0.442211\pi$
$432$	0	0
$433$	5.36675	0.257909	0.128955	−	0.991650i	$-0.458838\pi$
$433$	5.36675	0.257909	0.128955	+	0.991650i	$0.458838\pi$
$434$	0	0
$435$	44.9246	2.15397
$436$	0	0
$437$	−14.2912	−0.683641
$438$	0	0
$439$	−15.6745	−0.748101	−0.374051	−	0.927408i	$-0.622031\pi$
$439$	−15.6745	−0.748101	−0.374051	+	0.927408i	$0.622031\pi$
$440$	0	0
$441$	7.72976	0.368084
$442$	0	0
$443$	−1.20664	−0.0573291	−0.0286645	−	0.999589i	$-0.509125\pi$
$443$	−1.20664	−0.0573291	−0.0286645	+	0.999589i	$0.509125\pi$
$444$	0	0
$445$	−11.6609	−0.552782
$446$	0	0
$447$	−39.5523	−1.87076
$448$	0	0
$449$	10.7102	0.505445	0.252722	−	0.967539i	$-0.418674\pi$
$449$	10.7102	0.505445	0.252722	+	0.967539i	$0.418674\pi$
$450$	0	0
$451$	2.06895	0.0974229
$452$	0	0
$453$	55.1480	2.59108
$454$	0	0
$455$	1.56622	0.0734258
$456$	0	0
$457$	37.1928	1.73980	0.869902	−	0.493224i	$-0.164182\pi$
$457$	37.1928	1.73980	0.869902	+	0.493224i	$0.164182\pi$
$458$	0	0
$459$	−61.4769	−2.86950
$460$	0	0
$461$	−1.34512	−0.0626485	−0.0313243	−	0.999509i	$-0.509972\pi$
$461$	−1.34512	−0.0626485	−0.0313243	+	0.999509i	$0.509972\pi$
$462$	0	0
$463$	38.2347	1.77692	0.888460	−	0.458955i	$-0.151776\pi$
$463$	38.2347	1.77692	0.888460	+	0.458955i	$0.151776\pi$
$464$	0	0
$465$	−18.9226	−0.877515
$466$	0	0
$467$	−7.07386	−0.327339	−0.163669	−	0.986515i	$-0.552333\pi$
$467$	−7.07386	−0.327339	−0.163669	+	0.986515i	$0.552333\pi$
$468$	0	0
$469$	−9.98281	−0.460963
$470$	0	0
$471$	−45.4430	−2.09390
$472$	0	0
$473$	−5.57197	−0.256200
$474$	0	0
$475$	−14.3639	−0.659060
$476$	0	0
$477$	84.1850	3.85457
$478$	0	0
$479$	16.2079	0.740556	0.370278	−	0.928921i	$-0.379262\pi$
$479$	16.2079	0.740556	0.370278	+	0.928921i	$0.379262\pi$
$480$	0	0
$481$	−6.52445	−0.297489
$482$	0	0
$483$	8.30062	0.377691
$484$	0	0
$485$	14.0724	0.638995
$486$	0	0
$487$	−15.2066	−0.689078	−0.344539	−	0.938772i	$-0.611965\pi$
$487$	−15.2066	−0.689078	−0.344539	+	0.938772i	$0.611965\pi$
$488$	0	0
$489$	14.6096	0.660670
$490$	0	0
$491$	26.7314	1.20637	0.603187	−	0.797600i	$-0.293897\pi$
$491$	26.7314	1.20637	0.603187	+	0.797600i	$0.293897\pi$
$492$	0	0
$493$	−34.7467	−1.56491
$494$	0	0
$495$	−12.1065	−0.544149
$496$	0	0
$497$	−11.8688	−0.532387
$498$	0	0
$499$	9.35422	0.418752	0.209376	−	0.977835i	$-0.432857\pi$
$499$	9.35422	0.418752	0.209376	+	0.977835i	$0.432857\pi$
$500$	0	0
$501$	13.8282	0.617797
$502$	0	0
$503$	−5.40092	−0.240815	−0.120408	−	0.992725i	$-0.538420\pi$
$503$	−5.40092	−0.240815	−0.120408	+	0.992725i	$0.538420\pi$
$504$	0	0
$505$	−3.97501	−0.176886
$506$	0	0
$507$	3.27563	0.145476
$508$	0	0
$509$	−8.21205	−0.363993	−0.181996	−	0.983299i	$-0.558256\pi$
$509$	−8.21205	−0.363993	−0.181996	+	0.983299i	$0.558256\pi$
$510$	0	0
$511$	−6.84761	−0.302920
$512$	0	0
$513$	87.3751	3.85770
$514$	0	0
$515$	13.3669	0.589017
$516$	0	0
$517$	−2.26128	−0.0994510
$518$	0	0
$519$	−83.5753	−3.66855
$520$	0	0
$521$	−9.32959	−0.408737	−0.204368	−	0.978894i	$-0.565514\pi$
$521$	−9.32959	−0.408737	−0.204368	+	0.978894i	$0.565514\pi$
$522$	0	0
$523$	−26.2813	−1.14920	−0.574601	−	0.818434i	$-0.694843\pi$
$523$	−26.2813	−1.14920	−0.574601	+	0.818434i	$0.694843\pi$
$524$	0	0
$525$	8.34283	0.364111
$526$	0	0
$527$	14.6356	0.637536
$528$	0	0
$529$	−16.5786	−0.720808
$530$	0	0
$531$	7.20221	0.312549
$532$	0	0
$533$	2.06895	0.0896160
$534$	0	0
$535$	1.46240	0.0632252
$536$	0	0
$537$	52.5802	2.26900
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−36.9986	−1.59069	−0.795347	−	0.606154i	$-0.792712\pi$
$541$	−36.9986	−1.59069	−0.795347	+	0.606154i	$0.792712\pi$
$542$	0	0
$543$	−44.8258	−1.92366
$544$	0	0
$545$	−0.216277	−0.00926430
$546$	0	0
$547$	22.3363	0.955031	0.477516	−	0.878623i	$-0.341537\pi$
$547$	22.3363	0.955031	0.477516	+	0.878623i	$0.341537\pi$
$548$	0	0
$549$	−41.7886	−1.78350
$550$	0	0
$551$	49.3843	2.10384
$552$	0	0
$553$	1.30713	0.0555847
$554$	0	0
$555$	33.4729	1.42084
$556$	0	0
$557$	7.60292	0.322146	0.161073	−	0.986942i	$-0.448505\pi$
$557$	7.60292	0.322146	0.161073	+	0.986942i	$0.448505\pi$
$558$	0	0
$559$	−5.57197	−0.235669
$560$	0	0
$561$	12.9979	0.548772
$562$	0	0
$563$	34.7912	1.46628	0.733138	−	0.680080i	$-0.238055\pi$
$563$	34.7912	1.46628	0.733138	+	0.680080i	$0.238055\pi$
$564$	0	0
$565$	5.82886	0.245222
$566$	0	0
$567$	−27.5599	−1.15741
$568$	0	0
$569$	−28.1580	−1.18044	−0.590222	−	0.807241i	$-0.700960\pi$
$569$	−28.1580	−1.18044	−0.590222	+	0.807241i	$0.700960\pi$
$570$	0	0
$571$	9.19433	0.384771	0.192385	−	0.981319i	$-0.438378\pi$
$571$	9.19433	0.384771	0.192385	+	0.981319i	$0.438378\pi$
$572$	0	0
$573$	−44.4879	−1.85851
$574$	0	0
$575$	6.45408	0.269154
$576$	0	0
$577$	−29.8932	−1.24447	−0.622235	−	0.782831i	$-0.713775\pi$
$577$	−29.8932	−1.24447	−0.622235	+	0.782831i	$0.713775\pi$
$578$	0	0
$579$	29.9062	1.24286
$580$	0	0
$581$	5.68837	0.235993
$582$	0	0
$583$	−10.8910	−0.451060
$584$	0	0
$585$	−12.1065	−0.500544
$586$	0	0
$587$	−40.5591	−1.67405	−0.837027	−	0.547161i	$-0.815709\pi$
$587$	−40.5591	−1.67405	−0.837027	+	0.547161i	$0.815709\pi$
$588$	0	0
$589$	−20.8010	−0.857092
$590$	0	0
$591$	−55.2782	−2.27384
$592$	0	0
$593$	−27.9552	−1.14798	−0.573992	−	0.818861i	$-0.694606\pi$
$593$	−27.9552	−1.14798	−0.573992	+	0.818861i	$0.694606\pi$
$594$	0	0
$595$	6.21487	0.254785
$596$	0	0
$597$	−89.2064	−3.65098
$598$	0	0
$599$	13.5478	0.553547	0.276774	−	0.960935i	$-0.410735\pi$
$599$	13.5478	0.553547	0.276774	+	0.960935i	$0.410735\pi$
$600$	0	0
$601$	−42.4972	−1.73350	−0.866748	−	0.498747i	$-0.833794\pi$
$601$	−42.4972	−1.73350	−0.866748	+	0.498747i	$0.833794\pi$
$602$	0	0
$603$	77.1647	3.14239
$604$	0	0
$605$	1.56622	0.0636761
$606$	0	0
$607$	16.4657	0.668323	0.334162	−	0.942516i	$-0.391547\pi$
$607$	16.4657	0.668323	0.334162	+	0.942516i	$0.391547\pi$
$608$	0	0
$609$	−28.6834	−1.16231
$610$	0	0
$611$	−2.26128	−0.0914816
$612$	0	0
$613$	30.7359	1.24141	0.620707	−	0.784043i	$-0.286846\pi$
$613$	30.7359	1.24141	0.620707	+	0.784043i	$0.286846\pi$
$614$	0	0
$615$	−10.6145	−0.428017
$616$	0	0
$617$	−9.16784	−0.369083	−0.184542	−	0.982825i	$-0.559080\pi$
$617$	−9.16784	−0.369083	−0.184542	+	0.982825i	$0.559080\pi$
$618$	0	0
$619$	10.5551	0.424247	0.212124	−	0.977243i	$-0.431962\pi$
$619$	10.5551	0.424247	0.212124	+	0.977243i	$0.431962\pi$
$620$	0	0
$621$	−39.2599	−1.57545
$622$	0	0
$623$	7.44526	0.298288
$624$	0	0
$625$	−5.77840	−0.231136
$626$	0	0
$627$	−18.4735	−0.737759
$628$	0	0
$629$	−25.8894	−1.03228
$630$	0	0
$631$	−8.76459	−0.348913	−0.174456	−	0.984665i	$-0.555817\pi$
$631$	−8.76459	−0.348913	−0.174456	+	0.984665i	$0.555817\pi$
$632$	0	0
$633$	38.9500	1.54812
$634$	0	0
$635$	−34.7741	−1.37997
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	91.7428	3.62929
$640$	0	0
$641$	−31.0110	−1.22486	−0.612430	−	0.790525i	$-0.709808\pi$
$641$	−31.0110	−1.22486	−0.612430	+	0.790525i	$0.709808\pi$
$642$	0	0
$643$	−6.10606	−0.240799	−0.120400	−	0.992725i	$-0.538418\pi$
$643$	−6.10606	−0.240799	−0.120400	+	0.992725i	$0.538418\pi$
$644$	0	0
$645$	28.5863	1.12559
$646$	0	0
$647$	39.9066	1.56889	0.784445	−	0.620199i	$-0.212948\pi$
$647$	39.9066	1.56889	0.784445	+	0.620199i	$0.212948\pi$
$648$	0	0
$649$	−0.931751	−0.0365744
$650$	0	0
$651$	12.0817	0.473518
$652$	0	0
$653$	−31.4127	−1.22928	−0.614638	−	0.788810i	$-0.710698\pi$
$653$	−31.4127	−1.22928	−0.614638	+	0.788810i	$0.710698\pi$
$654$	0	0
$655$	30.4233	1.18874
$656$	0	0
$657$	52.9303	2.06501
$658$	0	0
$659$	−38.3063	−1.49220	−0.746101	−	0.665833i	$-0.768076\pi$
$659$	−38.3063	−1.49220	−0.746101	+	0.665833i	$0.768076\pi$
$660$	0	0
$661$	33.0041	1.28371	0.641855	−	0.766826i	$-0.278165\pi$
$661$	33.0041	1.28371	0.641855	+	0.766826i	$0.278165\pi$
$662$	0	0
$663$	12.9979	0.504797
$664$	0	0
$665$	−8.83299	−0.342529
$666$	0	0
$667$	−22.1897	−0.859187
$668$	0	0
$669$	17.2312	0.666195
$670$	0	0
$671$	5.40620	0.208704
$672$	0	0
$673$	−34.0220	−1.31145	−0.655727	−	0.754998i	$-0.727638\pi$
$673$	−34.0220	−1.31145	−0.655727	+	0.754998i	$0.727638\pi$
$674$	0	0
$675$	−39.4596	−1.51880
$676$	0	0
$677$	38.2233	1.46904	0.734521	−	0.678586i	$-0.237407\pi$
$677$	38.2233	1.46904	0.734521	+	0.678586i	$0.237407\pi$
$678$	0	0
$679$	−8.98492	−0.344809
$680$	0	0
$681$	−72.5653	−2.78071
$682$	0	0
$683$	−38.2188	−1.46240	−0.731201	−	0.682162i	$-0.761040\pi$
$683$	−38.2188	−1.46240	−0.731201	+	0.682162i	$0.761040\pi$
$684$	0	0
$685$	32.0403	1.22420
$686$	0	0
$687$	−71.1397	−2.71415
$688$	0	0
$689$	−10.8910	−0.414915
$690$	0	0
$691$	47.2799	1.79861	0.899306	−	0.437320i	$-0.144072\pi$
$691$	47.2799	1.79861	0.899306	+	0.437320i	$0.144072\pi$
$692$	0	0
$693$	7.72976	0.293629
$694$	0	0
$695$	−25.6270	−0.972086
$696$	0	0
$697$	8.20970	0.310965
$698$	0	0
$699$	−74.5149	−2.81841
$700$	0	0
$701$	17.7256	0.669485	0.334742	−	0.942310i	$-0.391351\pi$
$701$	17.7256	0.669485	0.334742	+	0.942310i	$0.391351\pi$
$702$	0	0
$703$	36.7957	1.38778
$704$	0	0
$705$	11.6012	0.436927
$706$	0	0
$707$	2.53796	0.0954497
$708$	0	0
$709$	−46.2174	−1.73573	−0.867865	−	0.496801i	$-0.834508\pi$
$709$	−46.2174	−1.73573	−0.867865	+	0.496801i	$0.834508\pi$
$710$	0	0
$711$	−10.1038	−0.378921
$712$	0	0
$713$	9.34646	0.350028
$714$	0	0
$715$	1.56622	0.0585735
$716$	0	0
$717$	6.79022	0.253585
$718$	0	0
$719$	11.3486	0.423232	0.211616	−	0.977353i	$-0.432127\pi$
$719$	11.3486	0.423232	0.211616	+	0.977353i	$0.432127\pi$
$720$	0	0
$721$	−8.53448	−0.317841
$722$	0	0
$723$	−19.4255	−0.722443
$724$	0	0
$725$	−22.3025	−0.828295
$726$	0	0
$727$	10.7671	0.399331	0.199666	−	0.979864i	$-0.436014\pi$
$727$	10.7671	0.399331	0.199666	+	0.979864i	$0.436014\pi$
$728$	0	0
$729$	60.7839	2.25126
$730$	0	0
$731$	−22.1099	−0.817765
$732$	0	0
$733$	−9.45026	−0.349053	−0.174527	−	0.984652i	$-0.555840\pi$
$733$	−9.45026	−0.349053	−0.174527	+	0.984652i	$0.555840\pi$
$734$	0	0
$735$	5.13038	0.189237
$736$	0	0
$737$	−9.98281	−0.367722
$738$	0	0
$739$	−3.44157	−0.126600	−0.0633000	−	0.997995i	$-0.520163\pi$
$739$	−3.44157	−0.126600	−0.0633000	+	0.997995i	$0.520163\pi$
$740$	0	0
$741$	−18.4735	−0.678640
$742$	0	0
$743$	−44.7146	−1.64042	−0.820210	−	0.572062i	$-0.806144\pi$
$743$	−44.7146	−1.64042	−0.820210	+	0.572062i	$0.806144\pi$
$744$	0	0
$745$	−18.9117	−0.692872
$746$	0	0
$747$	−43.9697	−1.60877
$748$	0	0
$749$	−0.933711	−0.0341171
$750$	0	0
$751$	39.3070	1.43433	0.717166	−	0.696902i	$-0.245439\pi$
$751$	39.3070	1.43433	0.717166	+	0.696902i	$0.245439\pi$
$752$	0	0
$753$	94.4172	3.44075
$754$	0	0
$755$	26.3687	0.959656
$756$	0	0
$757$	−22.2679	−0.809341	−0.404671	−	0.914463i	$-0.632614\pi$
$757$	−22.2679	−0.809341	−0.404671	+	0.914463i	$0.632614\pi$
$758$	0	0
$759$	8.30062	0.301293
$760$	0	0
$761$	28.8274	1.04499	0.522495	−	0.852642i	$-0.325001\pi$
$761$	28.8274	1.04499	0.522495	+	0.852642i	$0.325001\pi$
$762$	0	0
$763$	0.138088	0.00499913
$764$	0	0
$765$	−48.0395	−1.73687
$766$	0	0
$767$	−0.931751	−0.0336436
$768$	0	0
$769$	22.7504	0.820400	0.410200	−	0.911996i	$-0.365459\pi$
$769$	22.7504	0.820400	0.410200	+	0.911996i	$0.365459\pi$
$770$	0	0
$771$	−26.5877	−0.957534
$772$	0	0
$773$	22.1718	0.797464	0.398732	−	0.917068i	$-0.369450\pi$
$773$	22.1718	0.797464	0.398732	+	0.917068i	$0.369450\pi$
$774$	0	0
$775$	9.39399	0.337442
$776$	0	0
$777$	−21.3717	−0.766705
$778$	0	0
$779$	−11.6682	−0.418056
$780$	0	0
$781$	−11.8688	−0.424698
$782$	0	0
$783$	135.665	4.84829
$784$	0	0
$785$	−21.7283	−0.775518
$786$	0	0
$787$	16.6919	0.595002	0.297501	−	0.954721i	$-0.403847\pi$
$787$	16.6919	0.595002	0.297501	+	0.954721i	$0.403847\pi$
$788$	0	0
$789$	−38.0115	−1.35324
$790$	0	0
$791$	−3.72160	−0.132325
$792$	0	0
$793$	5.40620	0.191980
$794$	0	0
$795$	55.8751	1.98168
$796$	0	0
$797$	5.67891	0.201157	0.100579	−	0.994929i	$-0.467931\pi$
$797$	5.67891	0.201157	0.100579	+	0.994929i	$0.467931\pi$
$798$	0	0
$799$	−8.97289	−0.317438
$800$	0	0
$801$	−57.5501	−2.03343
$802$	0	0
$803$	−6.84761	−0.241647
$804$	0	0
$805$	3.96889	0.139885
$806$	0	0
$807$	34.7854	1.22451
$808$	0	0
$809$	−47.1300	−1.65700	−0.828502	−	0.559987i	$-0.810806\pi$
$809$	−47.1300	−1.65700	−0.828502	+	0.559987i	$0.810806\pi$
$810$	0	0
$811$	13.0566	0.458478	0.229239	−	0.973370i	$-0.426376\pi$
$811$	13.0566	0.458478	0.229239	+	0.973370i	$0.426376\pi$
$812$	0	0
$813$	−85.7558	−3.00759
$814$	0	0
$815$	6.98551	0.244692
$816$	0	0
$817$	31.4241	1.09939
$818$	0	0
$819$	7.72976	0.270100
$820$	0	0
$821$	18.3452	0.640254	0.320127	−	0.947375i	$-0.396274\pi$
$821$	18.3452	0.640254	0.320127	+	0.947375i	$0.396274\pi$
$822$	0	0
$823$	−4.78669	−0.166854	−0.0834268	−	0.996514i	$-0.526586\pi$
$823$	−4.78669	−0.166854	−0.0834268	+	0.996514i	$0.526586\pi$
$824$	0	0
$825$	8.34283	0.290460
$826$	0	0
$827$	41.0739	1.42828	0.714140	−	0.700003i	$-0.246818\pi$
$827$	41.0739	1.42828	0.714140	+	0.700003i	$0.246818\pi$
$828$	0	0
$829$	−53.4556	−1.85659	−0.928294	−	0.371846i	$-0.878725\pi$
$829$	−53.4556	−1.85659	−0.928294	+	0.371846i	$0.878725\pi$
$830$	0	0
$831$	47.9516	1.66342
$832$	0	0
$833$	−3.96806	−0.137485
$834$	0	0
$835$	6.61186	0.228813
$836$	0	0
$837$	−57.1434	−1.97516
$838$	0	0
$839$	52.3671	1.80791	0.903956	−	0.427626i	$-0.140650\pi$
$839$	52.3671	1.80791	0.903956	+	0.427626i	$0.140650\pi$
$840$	0	0
$841$	47.6780	1.64407
$842$	0	0
$843$	84.3268	2.90437
$844$	0	0
$845$	1.56622	0.0538798
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−92.6323	−3.17913
$850$	0	0
$851$	−16.5333	−0.566754
$852$	0	0
$853$	18.3321	0.627680	0.313840	−	0.949476i	$-0.398384\pi$
$853$	18.3321	0.627680	0.313840	+	0.949476i	$0.398384\pi$
$854$	0	0
$855$	68.2769	2.33502
$856$	0	0
$857$	−2.29105	−0.0782607	−0.0391304	−	0.999234i	$-0.512459\pi$
$857$	−2.29105	−0.0782607	−0.0391304	+	0.999234i	$0.512459\pi$
$858$	0	0
$859$	29.2336	0.997438	0.498719	−	0.866764i	$-0.333804\pi$
$859$	29.2336	0.997438	0.498719	+	0.866764i	$0.333804\pi$
$860$	0	0
$861$	6.77711	0.230963
$862$	0	0
$863$	−22.3267	−0.760010	−0.380005	−	0.924984i	$-0.624078\pi$
$863$	−22.3267	−0.760010	−0.380005	+	0.924984i	$0.624078\pi$
$864$	0	0
$865$	−39.9610	−1.35872
$866$	0	0
$867$	−4.10931	−0.139559
$868$	0	0
$869$	1.30713	0.0443412
$870$	0	0
$871$	−9.98281	−0.338255
$872$	0	0
$873$	69.4512	2.35057
$874$	0	0
$875$	11.8202	0.399596
$876$	0	0
$877$	−44.2994	−1.49588	−0.747942	−	0.663764i	$-0.768958\pi$
$877$	−44.2994	−1.49588	−0.747942	+	0.663764i	$0.768958\pi$
$878$	0	0
$879$	−25.2791	−0.852643
$880$	0	0
$881$	−11.4592	−0.386069	−0.193035	−	0.981192i	$-0.561833\pi$
$881$	−11.4592	−0.386069	−0.193035	+	0.981192i	$0.561833\pi$
$882$	0	0
$883$	43.5627	1.46600	0.733001	−	0.680227i	$-0.238119\pi$
$883$	43.5627	1.46600	0.733001	+	0.680227i	$0.238119\pi$
$884$	0	0
$885$	4.78023	0.160686
$886$	0	0
$887$	−34.2945	−1.15150	−0.575748	−	0.817627i	$-0.695289\pi$
$887$	−34.2945	−1.15150	−0.575748	+	0.817627i	$0.695289\pi$
$888$	0	0
$889$	22.2025	0.744648
$890$	0	0
$891$	−27.5599	−0.923292
$892$	0	0
$893$	12.7529	0.426758
$894$	0	0
$895$	25.1409	0.840368
$896$	0	0
$897$	8.30062	0.277150
$898$	0	0
$899$	−32.2974	−1.07718
$900$	0	0
$901$	−43.2162	−1.43974
$902$	0	0
$903$	−18.2517	−0.607380
$904$	0	0
$905$	−21.4332	−0.712464
$906$	0	0
$907$	−4.13667	−0.137356	−0.0686780	−	0.997639i	$-0.521878\pi$
$907$	−4.13667	−0.137356	−0.0686780	+	0.997639i	$0.521878\pi$
$908$	0	0
$909$	−19.6178	−0.650682
$910$	0	0
$911$	33.3136	1.10373	0.551865	−	0.833933i	$-0.313916\pi$
$911$	33.3136	1.10373	0.551865	+	0.833933i	$0.313916\pi$
$912$	0	0
$913$	5.68837	0.188258
$914$	0	0
$915$	−27.7358	−0.916919
$916$	0	0
$917$	−19.4246	−0.641456
$918$	0	0
$919$	−23.9039	−0.788516	−0.394258	−	0.919000i	$-0.628998\pi$
$919$	−23.9039	−0.788516	−0.394258	+	0.919000i	$0.628998\pi$
$920$	0	0
$921$	−44.1958	−1.45630
$922$	0	0
$923$	−11.8688	−0.390666
$924$	0	0
$925$	−16.6174	−0.546376
$926$	0	0
$927$	65.9695	2.16672
$928$	0	0
$929$	19.8590	0.651553	0.325776	−	0.945447i	$-0.394374\pi$
$929$	19.8590	0.651553	0.325776	+	0.945447i	$0.394374\pi$
$930$	0	0
$931$	5.63967	0.184833
$932$	0	0
$933$	−13.6354	−0.446402
$934$	0	0
$935$	6.21487	0.203248
$936$	0	0
$937$	56.1483	1.83428	0.917142	−	0.398561i	$-0.130490\pi$
$937$	56.1483	1.83428	0.917142	+	0.398561i	$0.130490\pi$
$938$	0	0
$939$	21.0852	0.688088
$940$	0	0
$941$	−47.1656	−1.53755	−0.768777	−	0.639517i	$-0.779134\pi$
$941$	−47.1656	−1.53755	−0.768777	+	0.639517i	$0.779134\pi$
$942$	0	0
$943$	5.24282	0.170730
$944$	0	0
$945$	−24.2654	−0.789355
$946$	0	0
$947$	46.8759	1.52326	0.761631	−	0.648011i	$-0.224399\pi$
$947$	46.8759	1.52326	0.761631	+	0.648011i	$0.224399\pi$
$948$	0	0
$949$	−6.84761	−0.222283
$950$	0	0
$951$	−64.1351	−2.07972
$952$	0	0
$953$	11.5525	0.374222	0.187111	−	0.982339i	$-0.440088\pi$
$953$	11.5525	0.374222	0.187111	+	0.982339i	$0.440088\pi$
$954$	0	0
$955$	−21.2717	−0.688334
$956$	0	0
$957$	−28.6834	−0.927202
$958$	0	0
$959$	−20.4570	−0.660593
$960$	0	0
$961$	−17.3961	−0.561165
$962$	0	0
$963$	7.21736	0.232576
$964$	0	0
$965$	14.2995	0.460316
$966$	0	0
$967$	−3.44499	−0.110783	−0.0553917	−	0.998465i	$-0.517641\pi$
$967$	−3.44499	−0.110783	−0.0553917	+	0.998465i	$0.517641\pi$
$968$	0	0
$969$	−73.3038	−2.35486
$970$	0	0
$971$	−12.5481	−0.402689	−0.201344	−	0.979521i	$-0.564531\pi$
$971$	−12.5481	−0.402689	−0.201344	+	0.979521i	$0.564531\pi$
$972$	0	0
$973$	16.3622	0.524550
$974$	0	0
$975$	8.34283	0.267185
$976$	0	0
$977$	36.4720	1.16684	0.583422	−	0.812169i	$-0.301714\pi$
$977$	36.4720	1.16684	0.583422	+	0.812169i	$0.301714\pi$
$978$	0	0
$979$	7.44526	0.237951
$980$	0	0
$981$	−1.06739	−0.0340791
$982$	0	0
$983$	18.7249	0.597233	0.298616	−	0.954373i	$-0.403475\pi$
$983$	18.7249	0.597233	0.298616	+	0.954373i	$0.403475\pi$
$984$	0	0
$985$	−26.4310	−0.842161
$986$	0	0
$987$	−7.40712	−0.235771
$988$	0	0
$989$	−14.1197	−0.448979
$990$	0	0
$991$	−55.0043	−1.74727	−0.873635	−	0.486582i	$-0.838243\pi$
$991$	−55.0043	−1.74727	−0.873635	+	0.486582i	$0.838243\pi$
$992$	0	0
$993$	2.20442	0.0699551
$994$	0	0
$995$	−42.6536	−1.35221
$996$	0	0
$997$	35.5096	1.12460	0.562300	−	0.826933i	$-0.309917\pi$
$997$	35.5096	1.12460	0.562300	+	0.826933i	$0.309917\pi$
$998$	0	0
$999$	101.083	3.19812

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.r.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.r.1.9	✓	9	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)

\(f(q)\)	\(=\)	\(q+3.27563 q^{3} +1.56622 q^{5} -1.00000 q^{7} +7.72976 q^{9} +O(q^{10})\)	\(q+3.27563 q^{3} +1.56622 q^{5} -1.00000 q^{7} +7.72976 q^{9} -1.00000 q^{11} -1.00000 q^{13} +5.13038 q^{15} -3.96806 q^{17} +5.63967 q^{19} -3.27563 q^{21} -2.53405 q^{23} -2.54694 q^{25} +15.4929 q^{27} +8.75659 q^{29} -3.68835 q^{31} -3.27563 q^{33} -1.56622 q^{35} +6.52445 q^{37} -3.27563 q^{39} -2.06895 q^{41} +5.57197 q^{43} +12.1065 q^{45} +2.26128 q^{47} +1.00000 q^{49} -12.9979 q^{51} +10.8910 q^{53} -1.56622 q^{55} +18.4735 q^{57} +0.931751 q^{59} -5.40620 q^{61} -7.72976 q^{63} -1.56622 q^{65} +9.98281 q^{67} -8.30062 q^{69} +11.8688 q^{71} +6.84761 q^{73} -8.34283 q^{75} +1.00000 q^{77} -1.30713 q^{79} +27.5599 q^{81} -5.68837 q^{83} -6.21487 q^{85} +28.6834 q^{87} -7.44526 q^{89} +1.00000 q^{91} -12.0817 q^{93} +8.83299 q^{95} +8.98492 q^{97} -7.72976 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9}+O(q^{10}) \) 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9	\( 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9} - 9 q^{11} - 9 q^{13} - 3 q^{15} - 7 q^{17} + 13 q^{19} - 5 q^{21} + 9 q^{23} - 2 q^{25} + 5 q^{27} + q^{29} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 14 q^{37} - 5 q^{39} - 2 q^{41} + 5 q^{43} - 15 q^{45} + 13 q^{47} + 9 q^{49} - 3 q^{51} + 22 q^{53} - 3 q^{55} + 16 q^{57} + 43 q^{59} - 10 q^{61} - 8 q^{63} - 3 q^{65} + 26 q^{67} - 30 q^{69} + 18 q^{71} - 8 q^{73} + 28 q^{75} + 9 q^{77} - 9 q^{79} + 33 q^{81} + 4 q^{83} + 5 q^{85} + 33 q^{87} + 7 q^{89} + 9 q^{91} + 13 q^{93} + 7 q^{95} + 2 q^{97} - 8 q^{99}+O(q^{100}) \) 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9 - 9 * q^11 - 9 * q^13 - 3 * q^15 - 7 * q^17 + 13 * q^19 - 5 * q^21 + 9 * q^23 - 2 * q^25 + 5 * q^27 + q^29 + 10 * q^31 - 5 * q^33 - 3 * q^35 + 14 * q^37 - 5 * q^39 - 2 * q^41 + 5 * q^43 - 15 * q^45 + 13 * q^47 + 9 * q^49 - 3 * q^51 + 22 * q^53 - 3 * q^55 + 16 * q^57 + 43 * q^59 - 10 * q^61 - 8 * q^63 - 3 * q^65 + 26 * q^67 - 30 * q^69 + 18 * q^71 - 8 * q^73 + 28 * q^75 + 9 * q^77 - 9 * q^79 + 33 * q^81 + 4 * q^83 + 5 * q^85 + 33 * q^87 + 7 * q^89 + 9 * q^91 + 13 * q^93 + 7 * q^95 + 2 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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