LMFDB - Embedded newform 8008.2.a.n.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$9$
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} - 3x^{8} - 15x^{7} + 45x^{6} + 64x^{5} - 201x^{4} - 63x^{3} + 282x^{2} + 3x - 116 $ x^9 - 3x^8 - 15x^7 + 45x^6 + 64x^5 - 201x^4 - 63x^3 + 282x^2 + 3x - 116
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.763644$ 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)

$f(q)$	$=$	$q+0.763644 q^{3} +1.55421 q^{5} -1.00000 q^{7} -2.41685 q^{9} +O(q^{10})$	$q+0.763644 q^{3} +1.55421 q^{5} -1.00000 q^{7} -2.41685 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.18686 q^{15} +3.61102 q^{17} +1.02809 q^{19} -0.763644 q^{21} +4.55145 q^{23} -2.58443 q^{25} -4.13654 q^{27} -4.78553 q^{29} +2.84863 q^{31} +0.763644 q^{33} -1.55421 q^{35} +1.27046 q^{37} -0.763644 q^{39} +3.11068 q^{41} +8.35432 q^{43} -3.75629 q^{45} +9.30529 q^{47} +1.00000 q^{49} +2.75753 q^{51} +1.24483 q^{53} +1.55421 q^{55} +0.785092 q^{57} -2.95515 q^{59} -6.44819 q^{61} +2.41685 q^{63} -1.55421 q^{65} -6.51850 q^{67} +3.47569 q^{69} -10.4003 q^{71} +6.95921 q^{73} -1.97358 q^{75} -1.00000 q^{77} +2.23287 q^{79} +4.09170 q^{81} -10.5278 q^{83} +5.61229 q^{85} -3.65444 q^{87} +12.4084 q^{89} +1.00000 q^{91} +2.17534 q^{93} +1.59786 q^{95} +1.06194 q^{97} -2.41685 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9	$ 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} + 11 q^{15} + 7 q^{17} - 17 q^{19} + 3 q^{21} + 11 q^{23} + 18 q^{25} - 9 q^{27} + 9 q^{29} - 8 q^{31} - 3 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} + 18 q^{41} + 7 q^{43} + 5 q^{45} + 15 q^{47} + 9 q^{49} - 7 q^{51} - 4 q^{53} + q^{55} + 22 q^{57} - 23 q^{59} + 12 q^{61} - 12 q^{63} - q^{65} - 16 q^{67} - 32 q^{69} - 6 q^{71} + 4 q^{73} - 14 q^{75} - 9 q^{77} + 21 q^{79} + 5 q^{81} - 16 q^{83} + 53 q^{85} + 41 q^{87} + 5 q^{89} + 9 q^{91} + 29 q^{93} + 19 q^{95} + 18 q^{97} + 12 q^{99}+O(q^{100}) $ 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 + 11 * q^15 + 7 * q^17 - 17 * q^19 + 3 * q^21 + 11 * q^23 + 18 * q^25 - 9 * q^27 + 9 * q^29 - 8 * q^31 - 3 * q^33 - q^35 + 2 * q^37 + 3 * q^39 + 18 * q^41 + 7 * q^43 + 5 * q^45 + 15 * q^47 + 9 * q^49 - 7 * q^51 - 4 * q^53 + q^55 + 22 * q^57 - 23 * q^59 + 12 * q^61 - 12 * q^63 - q^65 - 16 * q^67 - 32 * q^69 - 6 * q^71 + 4 * q^73 - 14 * q^75 - 9 * q^77 + 21 * q^79 + 5 * q^81 - 16 * q^83 + 53 * q^85 + 41 * q^87 + 5 * q^89 + 9 * q^91 + 29 * q^93 + 19 * q^95 + 18 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.763644	0.440890	0.220445	−	0.975399i	$-0.429249\pi$
$3$	0.763644	0.440890	0.220445	+	0.975399i	$0.429249\pi$
$4$	0	0
$5$	1.55421	0.695064	0.347532	−	0.937668i	$-0.387020\pi$
$5$	1.55421	0.695064	0.347532	+	0.937668i	$0.387020\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.41685	−0.805616
$10$	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
$15$	1.18686	0.306447
$16$	0	0
$17$	3.61102	0.875801	0.437900	−	0.899024i	$-0.355722\pi$
$17$	3.61102	0.875801	0.437900	+	0.899024i	$0.355722\pi$
$18$	0	0
$19$	1.02809	0.235859	0.117930	−	0.993022i	$-0.462374\pi$
$19$	1.02809	0.235859	0.117930	+	0.993022i	$0.462374\pi$
$20$	0	0
$21$	−0.763644	−0.166641
$22$	0	0
$23$	4.55145	0.949043	0.474522	−	0.880244i	$-0.342621\pi$
$23$	4.55145	0.949043	0.474522	+	0.880244i	$0.342621\pi$
$24$	0	0
$25$	−2.58443	−0.516885
$26$	0	0
$27$	−4.13654	−0.796078
$28$	0	0
$29$	−4.78553	−0.888651	−0.444325	−	0.895865i	$-0.646557\pi$
$29$	−4.78553	−0.888651	−0.444325	+	0.895865i	$0.646557\pi$
$30$	0	0
$31$	2.84863	0.511630	0.255815	−	0.966726i	$-0.417656\pi$
$31$	2.84863	0.511630	0.255815	+	0.966726i	$0.417656\pi$
$32$	0	0
$33$	0.763644	0.132933
$34$	0	0
$35$	−1.55421	−0.262710
$36$	0	0
$37$	1.27046	0.208862	0.104431	−	0.994532i	$-0.466698\pi$
$37$	1.27046	0.208862	0.104431	+	0.994532i	$0.466698\pi$
$38$	0	0
$39$	−0.763644	−0.122281
$40$	0	0
$41$	3.11068	0.485806	0.242903	−	0.970051i	$-0.421900\pi$
$41$	3.11068	0.485806	0.242903	+	0.970051i	$0.421900\pi$
$42$	0	0
$43$	8.35432	1.27402	0.637011	−	0.770855i	$-0.280171\pi$
$43$	8.35432	1.27402	0.637011	+	0.770855i	$0.280171\pi$
$44$	0	0
$45$	−3.75629	−0.559955
$46$	0	0
$47$	9.30529	1.35732	0.678658	−	0.734455i	$-0.262562\pi$
$47$	9.30529	1.35732	0.678658	+	0.734455i	$0.262562\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.75753	0.386132
$52$	0	0
$53$	1.24483	0.170990	0.0854950	−	0.996339i	$-0.472753\pi$
$53$	1.24483	0.170990	0.0854950	+	0.996339i	$0.472753\pi$
$54$	0	0
$55$	1.55421	0.209570
$56$	0	0
$57$	0.785092	0.103988
$58$	0	0
$59$	−2.95515	−0.384728	−0.192364	−	0.981324i	$-0.561615\pi$
$59$	−2.95515	−0.384728	−0.192364	+	0.981324i	$0.561615\pi$
$60$	0	0
$61$	−6.44819	−0.825606	−0.412803	−	0.910820i	$-0.635450\pi$
$61$	−6.44819	−0.825606	−0.412803	+	0.910820i	$0.635450\pi$
$62$	0	0
$63$	2.41685	0.304494
$64$	0	0
$65$	−1.55421	−0.192776
$66$	0	0
$67$	−6.51850	−0.796362	−0.398181	−	0.917307i	$-0.630358\pi$
$67$	−6.51850	−0.796362	−0.398181	+	0.917307i	$0.630358\pi$
$68$	0	0
$69$	3.47569	0.418424
$70$	0	0
$71$	−10.4003	−1.23429	−0.617146	−	0.786849i	$-0.711711\pi$
$71$	−10.4003	−1.23429	−0.617146	+	0.786849i	$0.711711\pi$
$72$	0	0
$73$	6.95921	0.814514	0.407257	−	0.913314i	$-0.366485\pi$
$73$	6.95921	0.814514	0.407257	+	0.913314i	$0.366485\pi$
$74$	0	0
$75$	−1.97358	−0.227890
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	2.23287	0.251217	0.125609	−	0.992080i	$-0.459912\pi$
$79$	2.23287	0.251217	0.125609	+	0.992080i	$0.459912\pi$
$80$	0	0
$81$	4.09170	0.454633
$82$	0	0
$83$	−10.5278	−1.15558	−0.577788	−	0.816187i	$-0.696084\pi$
$83$	−10.5278	−1.15558	−0.577788	+	0.816187i	$0.696084\pi$
$84$	0	0
$85$	5.61229	0.608738
$86$	0	0
$87$	−3.65444	−0.391797
$88$	0	0
$89$	12.4084	1.31529	0.657646	−	0.753327i	$-0.271552\pi$
$89$	12.4084	1.31529	0.657646	+	0.753327i	$0.271552\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	2.17534	0.225572
$94$	0	0
$95$	1.59786	0.163937
$96$	0	0
$97$	1.06194	0.107824	0.0539120	−	0.998546i	$-0.482831\pi$
$97$	1.06194	0.107824	0.0539120	+	0.998546i	$0.482831\pi$
$98$	0	0
$99$	−2.41685	−0.242902
$100$	0	0
$101$	9.07986	0.903480	0.451740	−	0.892150i	$-0.350804\pi$
$101$	9.07986	0.903480	0.451740	+	0.892150i	$0.350804\pi$
$102$	0	0
$103$	14.1645	1.39567	0.697837	−	0.716257i	$-0.254146\pi$
$103$	14.1645	1.39567	0.697837	+	0.716257i	$0.254146\pi$
$104$	0	0
$105$	−1.18686	−0.115826
$106$	0	0
$107$	0.191694	0.0185317	0.00926586	−	0.999957i	$-0.497051\pi$
$107$	0.191694	0.0185317	0.00926586	+	0.999957i	$0.497051\pi$
$108$	0	0
$109$	15.3709	1.47226	0.736131	−	0.676839i	$-0.236651\pi$
$109$	15.3709	1.47226	0.736131	+	0.676839i	$0.236651\pi$
$110$	0	0
$111$	0.970177	0.0920851
$112$	0	0
$113$	16.4325	1.54584	0.772921	−	0.634503i	$-0.218795\pi$
$113$	16.4325	1.54584	0.772921	+	0.634503i	$0.218795\pi$
$114$	0	0
$115$	7.07392	0.659646
$116$	0	0
$117$	2.41685	0.223438
$118$	0	0
$119$	−3.61102	−0.331022
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.37545	0.214187
$124$	0	0
$125$	−11.7878	−1.05433
$126$	0	0
$127$	−4.82092	−0.427788	−0.213894	−	0.976857i	$-0.568615\pi$
$127$	−4.82092	−0.427788	−0.213894	+	0.976857i	$0.568615\pi$
$128$	0	0
$129$	6.37973	0.561704
$130$	0	0
$131$	3.95281	0.345358	0.172679	−	0.984978i	$-0.444758\pi$
$131$	3.95281	0.345358	0.172679	+	0.984978i	$0.444758\pi$
$132$	0	0
$133$	−1.02809	−0.0891464
$134$	0	0
$135$	−6.42906	−0.553326
$136$	0	0
$137$	−13.4629	−1.15021	−0.575105	−	0.818080i	$-0.695039\pi$
$137$	−13.4629	−1.15021	−0.575105	+	0.818080i	$0.695039\pi$
$138$	0	0
$139$	8.51502	0.722234	0.361117	−	0.932520i	$-0.382395\pi$
$139$	8.51502	0.722234	0.361117	+	0.932520i	$0.382395\pi$
$140$	0	0
$141$	7.10593	0.598427
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−7.43773	−0.617669
$146$	0	0
$147$	0.763644	0.0629843
$148$	0	0
$149$	12.1535	0.995653	0.497826	−	0.867277i	$-0.334132\pi$
$149$	12.1535	0.995653	0.497826	+	0.867277i	$0.334132\pi$
$150$	0	0
$151$	−8.43997	−0.686835	−0.343417	−	0.939183i	$-0.611585\pi$
$151$	−8.43997	−0.686835	−0.343417	+	0.939183i	$0.611585\pi$
$152$	0	0
$153$	−8.72728	−0.705559
$154$	0	0
$155$	4.42738	0.355616
$156$	0	0
$157$	−11.7574	−0.938346	−0.469173	−	0.883106i	$-0.655448\pi$
$157$	−11.7574	−0.938346	−0.469173	+	0.883106i	$0.655448\pi$
$158$	0	0
$159$	0.950604	0.0753878
$160$	0	0
$161$	−4.55145	−0.358705
$162$	0	0
$163$	−12.4942	−0.978620	−0.489310	−	0.872110i	$-0.662751\pi$
$163$	−12.4942	−0.978620	−0.489310	+	0.872110i	$0.662751\pi$
$164$	0	0
$165$	1.18686	0.0923973
$166$	0	0
$167$	24.4830	1.89455	0.947277	−	0.320415i	$-0.103822\pi$
$167$	24.4830	1.89455	0.947277	+	0.320415i	$0.103822\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.48473	−0.190012
$172$	0	0
$173$	−19.8952	−1.51261	−0.756303	−	0.654221i	$-0.772997\pi$
$173$	−19.8952	−1.51261	−0.756303	+	0.654221i	$0.772997\pi$
$174$	0	0
$175$	2.58443	0.195364
$176$	0	0
$177$	−2.25668	−0.169623
$178$	0	0
$179$	−8.89979	−0.665202	−0.332601	−	0.943068i	$-0.607926\pi$
$179$	−8.89979	−0.665202	−0.332601	+	0.943068i	$0.607926\pi$
$180$	0	0
$181$	18.7538	1.39396	0.696980	−	0.717090i	$-0.254526\pi$
$181$	18.7538	1.39396	0.696980	+	0.717090i	$0.254526\pi$
$182$	0	0
$183$	−4.92412	−0.364002
$184$	0	0
$185$	1.97456	0.145172
$186$	0	0
$187$	3.61102	0.264064
$188$	0	0
$189$	4.13654	0.300889
$190$	0	0
$191$	−12.1866	−0.881794	−0.440897	−	0.897558i	$-0.645340\pi$
$191$	−12.1866	−0.881794	−0.440897	+	0.897558i	$0.645340\pi$
$192$	0	0
$193$	14.7549	1.06208	0.531042	−	0.847346i	$-0.321801\pi$
$193$	14.7549	1.06208	0.531042	+	0.847346i	$0.321801\pi$
$194$	0	0
$195$	−1.18686	−0.0849931
$196$	0	0
$197$	11.1653	0.795496	0.397748	−	0.917495i	$-0.369792\pi$
$197$	11.1653	0.795496	0.397748	+	0.917495i	$0.369792\pi$
$198$	0	0
$199$	26.7758	1.89808	0.949042	−	0.315149i	$-0.102055\pi$
$199$	26.7758	1.89808	0.949042	+	0.315149i	$0.102055\pi$
$200$	0	0
$201$	−4.97782	−0.351108
$202$	0	0
$203$	4.78553	0.335878
$204$	0	0
$205$	4.83465	0.337666
$206$	0	0
$207$	−11.0002	−0.764565
$208$	0	0
$209$	1.02809	0.0711142
$210$	0	0
$211$	20.7307	1.42716	0.713582	−	0.700572i	$-0.247072\pi$
$211$	20.7307	1.42716	0.713582	+	0.700572i	$0.247072\pi$
$212$	0	0
$213$	−7.94214	−0.544187
$214$	0	0
$215$	12.9844	0.885528
$216$	0	0
$217$	−2.84863	−0.193378
$218$	0	0
$219$	5.31436	0.359111
$220$	0	0
$221$	−3.61102	−0.242903
$222$	0	0
$223$	−5.70228	−0.381853	−0.190926	−	0.981604i	$-0.561149\pi$
$223$	−5.70228	−0.381853	−0.190926	+	0.981604i	$0.561149\pi$
$224$	0	0
$225$	6.24617	0.416411
$226$	0	0
$227$	−9.36092	−0.621306	−0.310653	−	0.950523i	$-0.600548\pi$
$227$	−9.36092	−0.621306	−0.310653	+	0.950523i	$0.600548\pi$
$228$	0	0
$229$	21.4426	1.41697	0.708483	−	0.705728i	$-0.249380\pi$
$229$	21.4426	1.41697	0.708483	+	0.705728i	$0.249380\pi$
$230$	0	0
$231$	−0.763644	−0.0502441
$232$	0	0
$233$	19.6958	1.29031	0.645157	−	0.764050i	$-0.276792\pi$
$233$	19.6958	1.29031	0.645157	+	0.764050i	$0.276792\pi$
$234$	0	0
$235$	14.4624	0.943422
$236$	0	0
$237$	1.70511	0.110759
$238$	0	0
$239$	−1.59704	−0.103304	−0.0516520	−	0.998665i	$-0.516449\pi$
$239$	−1.59704	−0.103304	−0.0516520	+	0.998665i	$0.516449\pi$
$240$	0	0
$241$	19.0106	1.22458	0.612290	−	0.790634i	$-0.290249\pi$
$241$	19.0106	1.22458	0.612290	+	0.790634i	$0.290249\pi$
$242$	0	0
$243$	15.5342	0.996521
$244$	0	0
$245$	1.55421	0.0992949
$246$	0	0
$247$	−1.02809	−0.0654155
$248$	0	0
$249$	−8.03949	−0.509482
$250$	0	0
$251$	5.42958	0.342712	0.171356	−	0.985209i	$-0.445185\pi$
$251$	5.42958	0.342712	0.171356	+	0.985209i	$0.445185\pi$
$252$	0	0
$253$	4.55145	0.286147
$254$	0	0
$255$	4.28579	0.268387
$256$	0	0
$257$	14.7935	0.922791	0.461396	−	0.887194i	$-0.347349\pi$
$257$	14.7935	0.922791	0.461396	+	0.887194i	$0.347349\pi$
$258$	0	0
$259$	−1.27046	−0.0789424
$260$	0	0
$261$	11.5659	0.715911
$262$	0	0
$263$	−7.19227	−0.443494	−0.221747	−	0.975104i	$-0.571176\pi$
$263$	−7.19227	−0.443494	−0.221747	+	0.975104i	$0.571176\pi$
$264$	0	0
$265$	1.93472	0.118849
$266$	0	0
$267$	9.47563	0.579899
$268$	0	0
$269$	23.6556	1.44231	0.721155	−	0.692774i	$-0.243612\pi$
$269$	23.6556	1.44231	0.721155	+	0.692774i	$0.243612\pi$
$270$	0	0
$271$	−16.2942	−0.989805	−0.494902	−	0.868949i	$-0.664796\pi$
$271$	−16.2942	−0.989805	−0.494902	+	0.868949i	$0.664796\pi$
$272$	0	0
$273$	0.763644	0.0462178
$274$	0	0
$275$	−2.58443	−0.155847
$276$	0	0
$277$	4.65531	0.279710	0.139855	−	0.990172i	$-0.455336\pi$
$277$	4.65531	0.279710	0.139855	+	0.990172i	$0.455336\pi$
$278$	0	0
$279$	−6.88471	−0.412177
$280$	0	0
$281$	11.1294	0.663922	0.331961	−	0.943293i	$-0.392290\pi$
$281$	11.1294	0.663922	0.331961	+	0.943293i	$0.392290\pi$
$282$	0	0
$283$	3.69259	0.219502	0.109751	−	0.993959i	$-0.464995\pi$
$283$	3.69259	0.219502	0.109751	+	0.993959i	$0.464995\pi$
$284$	0	0
$285$	1.22020	0.0722783
$286$	0	0
$287$	−3.11068	−0.183617
$288$	0	0
$289$	−3.96054	−0.232973
$290$	0	0
$291$	0.810946	0.0475385
$292$	0	0
$293$	−13.8153	−0.807098	−0.403549	−	0.914958i	$-0.632223\pi$
$293$	−13.8153	−0.807098	−0.403549	+	0.914958i	$0.632223\pi$
$294$	0	0
$295$	−4.59293	−0.267411
$296$	0	0
$297$	−4.13654	−0.240027
$298$	0	0
$299$	−4.55145	−0.263217
$300$	0	0
$301$	−8.35432	−0.481535
$302$	0	0
$303$	6.93378	0.398335
$304$	0	0
$305$	−10.0218	−0.573849
$306$	0	0
$307$	−15.6569	−0.893589	−0.446794	−	0.894637i	$-0.647434\pi$
$307$	−15.6569	−0.893589	−0.446794	+	0.894637i	$0.647434\pi$
$308$	0	0
$309$	10.8167	0.615339
$310$	0	0
$311$	−16.1559	−0.916119	−0.458059	−	0.888922i	$-0.651455\pi$
$311$	−16.1559	−0.916119	−0.458059	+	0.888922i	$0.651455\pi$
$312$	0	0
$313$	16.5609	0.936077	0.468038	−	0.883708i	$-0.344961\pi$
$313$	16.5609	0.936077	0.468038	+	0.883708i	$0.344961\pi$
$314$	0	0
$315$	3.75629	0.211643
$316$	0	0
$317$	27.6460	1.55275	0.776376	−	0.630270i	$-0.217056\pi$
$317$	27.6460	1.55275	0.776376	+	0.630270i	$0.217056\pi$
$318$	0	0
$319$	−4.78553	−0.267938
$320$	0	0
$321$	0.146386	0.00817046
$322$	0	0
$323$	3.71244	0.206566
$324$	0	0
$325$	2.58443	0.143358
$326$	0	0
$327$	11.7379	0.649106
$328$	0	0
$329$	−9.30529	−0.513017
$330$	0	0
$331$	10.7209	0.589274	0.294637	−	0.955609i	$-0.404801\pi$
$331$	10.7209	0.589274	0.294637	+	0.955609i	$0.404801\pi$
$332$	0	0
$333$	−3.07050	−0.168262
$334$	0	0
$335$	−10.1311	−0.553523
$336$	0	0
$337$	−9.30889	−0.507088	−0.253544	−	0.967324i	$-0.581596\pi$
$337$	−9.30889	−0.507088	−0.253544	+	0.967324i	$0.581596\pi$
$338$	0	0
$339$	12.5486	0.681546
$340$	0	0
$341$	2.84863	0.154262
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	5.40196	0.290832
$346$	0	0
$347$	−28.1924	−1.51345	−0.756724	−	0.653734i	$-0.773201\pi$
$347$	−28.1924	−1.51345	−0.756724	+	0.653734i	$0.773201\pi$
$348$	0	0
$349$	19.8050	1.06014	0.530068	−	0.847955i	$-0.322166\pi$
$349$	19.8050	1.06014	0.530068	+	0.847955i	$0.322166\pi$
$350$	0	0
$351$	4.13654	0.220792
$352$	0	0
$353$	−21.9753	−1.16963	−0.584813	−	0.811168i	$-0.698832\pi$
$353$	−21.9753	−1.16963	−0.584813	+	0.811168i	$0.698832\pi$
$354$	0	0
$355$	−16.1643	−0.857912
$356$	0	0
$357$	−2.75753	−0.145944
$358$	0	0
$359$	20.8772	1.10186	0.550928	−	0.834553i	$-0.314274\pi$
$359$	20.8772	1.10186	0.550928	+	0.834553i	$0.314274\pi$
$360$	0	0
$361$	−17.9430	−0.944370
$362$	0	0
$363$	0.763644	0.0400809
$364$	0	0
$365$	10.8161	0.566140
$366$	0	0
$367$	7.25726	0.378826	0.189413	−	0.981898i	$-0.439342\pi$
$367$	7.25726	0.378826	0.189413	+	0.981898i	$0.439342\pi$
$368$	0	0
$369$	−7.51803	−0.391373
$370$	0	0
$371$	−1.24483	−0.0646281
$372$	0	0
$373$	−9.05017	−0.468600	−0.234300	−	0.972164i	$-0.575280\pi$
$373$	−9.05017	−0.468600	−0.234300	+	0.972164i	$0.575280\pi$
$374$	0	0
$375$	−9.00169	−0.464845
$376$	0	0
$377$	4.78553	0.246467
$378$	0	0
$379$	−7.92065	−0.406856	−0.203428	−	0.979090i	$-0.565208\pi$
$379$	−7.92065	−0.406856	−0.203428	+	0.979090i	$0.565208\pi$
$380$	0	0
$381$	−3.68147	−0.188607
$382$	0	0
$383$	−33.8448	−1.72939	−0.864695	−	0.502298i	$-0.832488\pi$
$383$	−33.8448	−1.72939	−0.864695	+	0.502298i	$0.832488\pi$
$384$	0	0
$385$	−1.55421	−0.0792099
$386$	0	0
$387$	−20.1911	−1.02637
$388$	0	0
$389$	−23.1083	−1.17164	−0.585818	−	0.810443i	$-0.699227\pi$
$389$	−23.1083	−1.17164	−0.585818	+	0.810443i	$0.699227\pi$
$390$	0	0
$391$	16.4354	0.831173
$392$	0	0
$393$	3.01854	0.152265
$394$	0	0
$395$	3.47035	0.174612
$396$	0	0
$397$	13.7326	0.689217	0.344609	−	0.938746i	$-0.388012\pi$
$397$	13.7326	0.689217	0.344609	+	0.938746i	$0.388012\pi$
$398$	0	0
$399$	−0.785092	−0.0393038
$400$	0	0
$401$	−38.2794	−1.91158	−0.955790	−	0.294051i	$-0.904997\pi$
$401$	−38.2794	−1.91158	−0.955790	+	0.294051i	$0.904997\pi$
$402$	0	0
$403$	−2.84863	−0.141901
$404$	0	0
$405$	6.35936	0.315999
$406$	0	0
$407$	1.27046	0.0629742
$408$	0	0
$409$	37.5213	1.85531	0.927655	−	0.373439i	$-0.121821\pi$
$409$	37.5213	1.85531	0.927655	+	0.373439i	$0.121821\pi$
$410$	0	0
$411$	−10.2808	−0.507116
$412$	0	0
$413$	2.95515	0.145414
$414$	0	0
$415$	−16.3624	−0.803200
$416$	0	0
$417$	6.50244	0.318426
$418$	0	0
$419$	26.8698	1.31267	0.656337	−	0.754468i	$-0.272105\pi$
$419$	26.8698	1.31267	0.656337	+	0.754468i	$0.272105\pi$
$420$	0	0
$421$	−15.8231	−0.771170	−0.385585	−	0.922672i	$-0.626000\pi$
$421$	−15.8231	−0.771170	−0.385585	+	0.922672i	$0.626000\pi$
$422$	0	0
$423$	−22.4895	−1.09347
$424$	0	0
$425$	−9.33242	−0.452689
$426$	0	0
$427$	6.44819	0.312050
$428$	0	0
$429$	−0.763644	−0.0368691
$430$	0	0
$431$	30.3780	1.46325	0.731627	−	0.681705i	$-0.238761\pi$
$431$	30.3780	1.46325	0.731627	+	0.681705i	$0.238761\pi$
$432$	0	0
$433$	−10.2984	−0.494911	−0.247456	−	0.968899i	$-0.579594\pi$
$433$	−10.2984	−0.494911	−0.247456	+	0.968899i	$0.579594\pi$
$434$	0	0
$435$	−5.67978	−0.272324
$436$	0	0
$437$	4.67928	0.223841
$438$	0	0
$439$	−0.227077	−0.0108378	−0.00541889	−	0.999985i	$-0.501725\pi$
$439$	−0.227077	−0.0108378	−0.00541889	+	0.999985i	$0.501725\pi$
$440$	0	0
$441$	−2.41685	−0.115088
$442$	0	0
$443$	2.57932	0.122547	0.0612737	−	0.998121i	$-0.480484\pi$
$443$	2.57932	0.122547	0.0612737	+	0.998121i	$0.480484\pi$
$444$	0	0
$445$	19.2853	0.914212
$446$	0	0
$447$	9.28094	0.438973
$448$	0	0
$449$	21.8040	1.02899	0.514497	−	0.857492i	$-0.327979\pi$
$449$	21.8040	1.02899	0.514497	+	0.857492i	$0.327979\pi$
$450$	0	0
$451$	3.11068	0.146476
$452$	0	0
$453$	−6.44513	−0.302819
$454$	0	0
$455$	1.55421	0.0728625
$456$	0	0
$457$	−29.0371	−1.35830	−0.679149	−	0.734000i	$-0.737651\pi$
$457$	−29.0371	−1.35830	−0.679149	+	0.734000i	$0.737651\pi$
$458$	0	0
$459$	−14.9371	−0.697206
$460$	0	0
$461$	−34.2576	−1.59553	−0.797767	−	0.602966i	$-0.793985\pi$
$461$	−34.2576	−1.59553	−0.797767	+	0.602966i	$0.793985\pi$
$462$	0	0
$463$	−25.4791	−1.18412	−0.592058	−	0.805895i	$-0.701684\pi$
$463$	−25.4791	−1.18412	−0.592058	+	0.805895i	$0.701684\pi$
$464$	0	0
$465$	3.38094	0.156787
$466$	0	0
$467$	−26.5467	−1.22843	−0.614216	−	0.789138i	$-0.710528\pi$
$467$	−26.5467	−1.22843	−0.614216	+	0.789138i	$0.710528\pi$
$468$	0	0
$469$	6.51850	0.300997
$470$	0	0
$471$	−8.97850	−0.413707
$472$	0	0
$473$	8.35432	0.384132
$474$	0	0
$475$	−2.65701	−0.121912
$476$	0	0
$477$	−3.00855	−0.137752
$478$	0	0
$479$	22.3747	1.02233	0.511164	−	0.859483i	$-0.329215\pi$
$479$	22.3747	1.02233	0.511164	+	0.859483i	$0.329215\pi$
$480$	0	0
$481$	−1.27046	−0.0579279
$482$	0	0
$483$	−3.47569	−0.158149
$484$	0	0
$485$	1.65048	0.0749446
$486$	0	0
$487$	−3.34573	−0.151610	−0.0758048	−	0.997123i	$-0.524153\pi$
$487$	−3.34573	−0.151610	−0.0758048	+	0.997123i	$0.524153\pi$
$488$	0	0
$489$	−9.54111	−0.431464
$490$	0	0
$491$	9.58951	0.432768	0.216384	−	0.976308i	$-0.430574\pi$
$491$	9.58951	0.432768	0.216384	+	0.976308i	$0.430574\pi$
$492$	0	0
$493$	−17.2806	−0.778281
$494$	0	0
$495$	−3.75629	−0.168833
$496$	0	0
$497$	10.4003	0.466518
$498$	0	0
$499$	4.05910	0.181710	0.0908552	−	0.995864i	$-0.471040\pi$
$499$	4.05910	0.181710	0.0908552	+	0.995864i	$0.471040\pi$
$500$	0	0
$501$	18.6963	0.835290
$502$	0	0
$503$	25.3910	1.13213	0.566065	−	0.824361i	$-0.308465\pi$
$503$	25.3910	1.13213	0.566065	+	0.824361i	$0.308465\pi$
$504$	0	0
$505$	14.1120	0.627977
$506$	0	0
$507$	0.763644	0.0339146
$508$	0	0
$509$	−10.4880	−0.464873	−0.232436	−	0.972612i	$-0.574670\pi$
$509$	−10.4880	−0.464873	−0.232436	+	0.972612i	$0.574670\pi$
$510$	0	0
$511$	−6.95921	−0.307858
$512$	0	0
$513$	−4.25272	−0.187762
$514$	0	0
$515$	22.0147	0.970083
$516$	0	0
$517$	9.30529	0.409246
$518$	0	0
$519$	−15.1929	−0.666893
$520$	0	0
$521$	−3.86958	−0.169529	−0.0847646	−	0.996401i	$-0.527014\pi$
$521$	−3.86958	−0.169529	−0.0847646	+	0.996401i	$0.527014\pi$
$522$	0	0
$523$	−13.4579	−0.588472	−0.294236	−	0.955733i	$-0.595065\pi$
$523$	−13.4579	−0.588472	−0.294236	+	0.955733i	$0.595065\pi$
$524$	0	0
$525$	1.97358	0.0861342
$526$	0	0
$527$	10.2865	0.448086
$528$	0	0
$529$	−2.28428	−0.0993165
$530$	0	0
$531$	7.14215	0.309943
$532$	0	0
$533$	−3.11068	−0.134738
$534$	0	0
$535$	0.297933	0.0128807
$536$	0	0
$537$	−6.79627	−0.293281
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−12.4446	−0.535037	−0.267518	−	0.963553i	$-0.586204\pi$
$541$	−12.4446	−0.535037	−0.267518	+	0.963553i	$0.586204\pi$
$542$	0	0
$543$	14.3212	0.614583
$544$	0	0
$545$	23.8896	1.02332
$546$	0	0
$547$	21.0611	0.900507	0.450254	−	0.892901i	$-0.351334\pi$
$547$	21.0611	0.900507	0.450254	+	0.892901i	$0.351334\pi$
$548$	0	0
$549$	15.5843	0.665121
$550$	0	0
$551$	−4.91994	−0.209596
$552$	0	0
$553$	−2.23287	−0.0949511
$554$	0	0
$555$	1.50786	0.0640051
$556$	0	0
$557$	3.47725	0.147336	0.0736679	−	0.997283i	$-0.476530\pi$
$557$	3.47725	0.147336	0.0736679	+	0.997283i	$0.476530\pi$
$558$	0	0
$559$	−8.35432	−0.353350
$560$	0	0
$561$	2.75753	0.116423
$562$	0	0
$563$	10.5980	0.446652	0.223326	−	0.974744i	$-0.428309\pi$
$563$	10.5980	0.446652	0.223326	+	0.974744i	$0.428309\pi$
$564$	0	0
$565$	25.5396	1.07446
$566$	0	0
$567$	−4.09170	−0.171835
$568$	0	0
$569$	−2.94541	−0.123478	−0.0617390	−	0.998092i	$-0.519665\pi$
$569$	−2.94541	−0.123478	−0.0617390	+	0.998092i	$0.519665\pi$
$570$	0	0
$571$	10.7824	0.451230	0.225615	−	0.974217i	$-0.427561\pi$
$571$	10.7824	0.451230	0.225615	+	0.974217i	$0.427561\pi$
$572$	0	0
$573$	−9.30626	−0.388774
$574$	0	0
$575$	−11.7629	−0.490547
$576$	0	0
$577$	14.0827	0.586271	0.293135	−	0.956071i	$-0.405301\pi$
$577$	14.0827	0.586271	0.293135	+	0.956071i	$0.405301\pi$
$578$	0	0
$579$	11.2675	0.468262
$580$	0	0
$581$	10.5278	0.436767
$582$	0	0
$583$	1.24483	0.0515554
$584$	0	0
$585$	3.75629	0.155304
$586$	0	0
$587$	2.60754	0.107625	0.0538123	−	0.998551i	$-0.482863\pi$
$587$	2.60754	0.107625	0.0538123	+	0.998551i	$0.482863\pi$
$588$	0	0
$589$	2.92864	0.120673
$590$	0	0
$591$	8.52633	0.350726
$592$	0	0
$593$	−9.05565	−0.371871	−0.185936	−	0.982562i	$-0.559532\pi$
$593$	−9.05565	−0.371871	−0.185936	+	0.982562i	$0.559532\pi$
$594$	0	0
$595$	−5.61229	−0.230081
$596$	0	0
$597$	20.4472	0.836847
$598$	0	0
$599$	−44.1995	−1.80594	−0.902971	−	0.429702i	$-0.858619\pi$
$599$	−44.1995	−1.80594	−0.902971	+	0.429702i	$0.858619\pi$
$600$	0	0
$601$	−36.4066	−1.48506	−0.742528	−	0.669815i	$-0.766373\pi$
$601$	−36.4066	−1.48506	−0.742528	+	0.669815i	$0.766373\pi$
$602$	0	0
$603$	15.7542	0.641562
$604$	0	0
$605$	1.55421	0.0631877
$606$	0	0
$607$	−27.3187	−1.10883	−0.554417	−	0.832239i	$-0.687059\pi$
$607$	−27.3187	−1.10883	−0.554417	+	0.832239i	$0.687059\pi$
$608$	0	0
$609$	3.65444	0.148085
$610$	0	0
$611$	−9.30529	−0.376452
$612$	0	0
$613$	28.6930	1.15890	0.579449	−	0.815008i	$-0.303268\pi$
$613$	28.6930	1.15890	0.579449	+	0.815008i	$0.303268\pi$
$614$	0	0
$615$	3.69195	0.148874
$616$	0	0
$617$	46.4942	1.87178	0.935892	−	0.352286i	$-0.114596\pi$
$617$	46.4942	1.87178	0.935892	+	0.352286i	$0.114596\pi$
$618$	0	0
$619$	−22.6265	−0.909434	−0.454717	−	0.890636i	$-0.650260\pi$
$619$	−22.6265	−0.909434	−0.454717	+	0.890636i	$0.650260\pi$
$620$	0	0
$621$	−18.8273	−0.755513
$622$	0	0
$623$	−12.4084	−0.497133
$624$	0	0
$625$	−5.39860	−0.215944
$626$	0	0
$627$	0.785092	0.0313535
$628$	0	0
$629$	4.58765	0.182921
$630$	0	0
$631$	−1.70304	−0.0677970	−0.0338985	−	0.999425i	$-0.510792\pi$
$631$	−1.70304	−0.0677970	−0.0338985	+	0.999425i	$0.510792\pi$
$632$	0	0
$633$	15.8309	0.629222
$634$	0	0
$635$	−7.49273	−0.297340
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	25.1360	0.994365
$640$	0	0
$641$	−18.3537	−0.724928	−0.362464	−	0.931998i	$-0.618064\pi$
$641$	−18.3537	−0.724928	−0.362464	+	0.931998i	$0.618064\pi$
$642$	0	0
$643$	11.0360	0.435219	0.217609	−	0.976036i	$-0.430174\pi$
$643$	11.0360	0.435219	0.217609	+	0.976036i	$0.430174\pi$
$644$	0	0
$645$	9.91545	0.390420
$646$	0	0
$647$	18.3081	0.719767	0.359884	−	0.932997i	$-0.382816\pi$
$647$	18.3081	0.719767	0.359884	+	0.932997i	$0.382816\pi$
$648$	0	0
$649$	−2.95515	−0.116000
$650$	0	0
$651$	−2.17534	−0.0852584
$652$	0	0
$653$	−1.10285	−0.0431577	−0.0215789	−	0.999767i	$-0.506869\pi$
$653$	−1.10285	−0.0431577	−0.0215789	+	0.999767i	$0.506869\pi$
$654$	0	0
$655$	6.14349	0.240046
$656$	0	0
$657$	−16.8194	−0.656186
$658$	0	0
$659$	−33.9239	−1.32149	−0.660743	−	0.750612i	$-0.729759\pi$
$659$	−33.9239	−1.32149	−0.660743	+	0.750612i	$0.729759\pi$
$660$	0	0
$661$	24.5526	0.954986	0.477493	−	0.878636i	$-0.341546\pi$
$661$	24.5526	0.954986	0.477493	+	0.878636i	$0.341546\pi$
$662$	0	0
$663$	−2.75753	−0.107094
$664$	0	0
$665$	−1.59786	−0.0619625
$666$	0	0
$667$	−21.7811	−0.843368
$668$	0	0
$669$	−4.35451	−0.168355
$670$	0	0
$671$	−6.44819	−0.248930
$672$	0	0
$673$	−14.1038	−0.543660	−0.271830	−	0.962345i	$-0.587629\pi$
$673$	−14.1038	−0.543660	−0.271830	+	0.962345i	$0.587629\pi$
$674$	0	0
$675$	10.6906	0.411481
$676$	0	0
$677$	31.2490	1.20100	0.600498	−	0.799626i	$-0.294969\pi$
$677$	31.2490	1.20100	0.600498	+	0.799626i	$0.294969\pi$
$678$	0	0
$679$	−1.06194	−0.0407536
$680$	0	0
$681$	−7.14841	−0.273928
$682$	0	0
$683$	−50.8580	−1.94603	−0.973013	−	0.230749i	$-0.925882\pi$
$683$	−50.8580	−1.94603	−0.973013	+	0.230749i	$0.925882\pi$
$684$	0	0
$685$	−20.9241	−0.799469
$686$	0	0
$687$	16.3745	0.624726
$688$	0	0
$689$	−1.24483	−0.0474241
$690$	0	0
$691$	8.29206	0.315445	0.157722	−	0.987484i	$-0.449585\pi$
$691$	8.29206	0.315445	0.157722	+	0.987484i	$0.449585\pi$
$692$	0	0
$693$	2.41685	0.0918085
$694$	0	0
$695$	13.2341	0.501999
$696$	0	0
$697$	11.2327	0.425469
$698$	0	0
$699$	15.0406	0.568886
$700$	0	0
$701$	22.1036	0.834840	0.417420	−	0.908714i	$-0.362934\pi$
$701$	22.1036	0.834840	0.417420	+	0.908714i	$0.362934\pi$
$702$	0	0
$703$	1.30614	0.0492620
$704$	0	0
$705$	11.0441	0.415945
$706$	0	0
$707$	−9.07986	−0.341483
$708$	0	0
$709$	−0.389720	−0.0146362	−0.00731811	−	0.999973i	$-0.502329\pi$
$709$	−0.389720	−0.0146362	−0.00731811	+	0.999973i	$0.502329\pi$
$710$	0	0
$711$	−5.39650	−0.202384
$712$	0	0
$713$	12.9654	0.485559
$714$	0	0
$715$	−1.55421	−0.0581242
$716$	0	0
$717$	−1.21957	−0.0455457
$718$	0	0
$719$	−12.7073	−0.473904	−0.236952	−	0.971521i	$-0.576148\pi$
$719$	−12.7073	−0.473904	−0.236952	+	0.971521i	$0.576148\pi$
$720$	0	0
$721$	−14.1645	−0.527515
$722$	0	0
$723$	14.5173	0.539905
$724$	0	0
$725$	12.3679	0.459331
$726$	0	0
$727$	−4.87754	−0.180898	−0.0904489	−	0.995901i	$-0.528830\pi$
$727$	−4.87754	−0.180898	−0.0904489	+	0.995901i	$0.528830\pi$
$728$	0	0
$729$	−0.412464	−0.0152765
$730$	0	0
$731$	30.1676	1.11579
$732$	0	0
$733$	21.9663	0.811345	0.405672	−	0.914019i	$-0.367037\pi$
$733$	21.9663	0.811345	0.405672	+	0.914019i	$0.367037\pi$
$734$	0	0
$735$	1.18686	0.0437781
$736$	0	0
$737$	−6.51850	−0.240112
$738$	0	0
$739$	24.9920	0.919346	0.459673	−	0.888088i	$-0.347967\pi$
$739$	24.9920	0.919346	0.459673	+	0.888088i	$0.347967\pi$
$740$	0	0
$741$	−0.785092	−0.0288411
$742$	0	0
$743$	−8.97156	−0.329135	−0.164567	−	0.986366i	$-0.552623\pi$
$743$	−8.97156	−0.329135	−0.164567	+	0.986366i	$0.552623\pi$
$744$	0	0
$745$	18.8891	0.692043
$746$	0	0
$747$	25.4441	0.930951
$748$	0	0
$749$	−0.191694	−0.00700434
$750$	0	0
$751$	9.35304	0.341297	0.170649	−	0.985332i	$-0.445414\pi$
$751$	9.35304	0.341297	0.170649	+	0.985332i	$0.445414\pi$
$752$	0	0
$753$	4.14626	0.151098
$754$	0	0
$755$	−13.1175	−0.477394
$756$	0	0
$757$	42.1416	1.53166	0.765832	−	0.643041i	$-0.222328\pi$
$757$	42.1416	1.53166	0.765832	+	0.643041i	$0.222328\pi$
$758$	0	0
$759$	3.47569	0.126160
$760$	0	0
$761$	−39.7772	−1.44192	−0.720962	−	0.692975i	$-0.756300\pi$
$761$	−39.7772	−1.44192	−0.720962	+	0.692975i	$0.756300\pi$
$762$	0	0
$763$	−15.3709	−0.556463
$764$	0	0
$765$	−13.5640	−0.490409
$766$	0	0
$767$	2.95515	0.106704
$768$	0	0
$769$	−5.61322	−0.202418	−0.101209	−	0.994865i	$-0.532271\pi$
$769$	−5.61322	−0.202418	−0.101209	+	0.994865i	$0.532271\pi$
$770$	0	0
$771$	11.2969	0.406850
$772$	0	0
$773$	41.2364	1.48317	0.741584	−	0.670860i	$-0.234075\pi$
$773$	41.2364	1.48317	0.741584	+	0.670860i	$0.234075\pi$
$774$	0	0
$775$	−7.36209	−0.264454
$776$	0	0
$777$	−0.970177	−0.0348049
$778$	0	0
$779$	3.19804	0.114582
$780$	0	0
$781$	−10.4003	−0.372153
$782$	0	0
$783$	19.7956	0.707435
$784$	0	0
$785$	−18.2735	−0.652211
$786$	0	0
$787$	−9.08239	−0.323752	−0.161876	−	0.986811i	$-0.551755\pi$
$787$	−9.08239	−0.323752	−0.161876	+	0.986811i	$0.551755\pi$
$788$	0	0
$789$	−5.49233	−0.195532
$790$	0	0
$791$	−16.4325	−0.584273
$792$	0	0
$793$	6.44819	0.228982
$794$	0	0
$795$	1.47744	0.0523994
$796$	0	0
$797$	−37.5216	−1.32908	−0.664542	−	0.747251i	$-0.731373\pi$
$797$	−37.5216	−1.32908	−0.664542	+	0.747251i	$0.731373\pi$
$798$	0	0
$799$	33.6016	1.18874
$800$	0	0
$801$	−29.9893	−1.05962
$802$	0	0
$803$	6.95921	0.245585
$804$	0	0
$805$	−7.07392	−0.249323
$806$	0	0
$807$	18.0645	0.635900
$808$	0	0
$809$	−2.22629	−0.0782721	−0.0391360	−	0.999234i	$-0.512461\pi$
$809$	−2.22629	−0.0782721	−0.0391360	+	0.999234i	$0.512461\pi$
$810$	0	0
$811$	−9.05440	−0.317943	−0.158971	−	0.987283i	$-0.550818\pi$
$811$	−9.05440	−0.317943	−0.158971	+	0.987283i	$0.550818\pi$
$812$	0	0
$813$	−12.4430	−0.436395
$814$	0	0
$815$	−19.4186	−0.680204
$816$	0	0
$817$	8.58896	0.300490
$818$	0	0
$819$	−2.41685	−0.0844515
$820$	0	0
$821$	29.2651	1.02136	0.510679	−	0.859771i	$-0.329394\pi$
$821$	29.2651	1.02136	0.510679	+	0.859771i	$0.329394\pi$
$822$	0	0
$823$	51.6067	1.79890	0.899448	−	0.437027i	$-0.143969\pi$
$823$	51.6067	1.79890	0.899448	+	0.437027i	$0.143969\pi$
$824$	0	0
$825$	−1.97358	−0.0687113
$826$	0	0
$827$	−17.6269	−0.612947	−0.306474	−	0.951879i	$-0.599149\pi$
$827$	−17.6269	−0.612947	−0.306474	+	0.951879i	$0.599149\pi$
$828$	0	0
$829$	35.5997	1.23643	0.618215	−	0.786009i	$-0.287856\pi$
$829$	35.5997	1.23643	0.618215	+	0.786009i	$0.287856\pi$
$830$	0	0
$831$	3.55500	0.123321
$832$	0	0
$833$	3.61102	0.125114
$834$	0	0
$835$	38.0518	1.31684
$836$	0	0
$837$	−11.7835	−0.407297
$838$	0	0
$839$	−35.6882	−1.23209	−0.616047	−	0.787709i	$-0.711267\pi$
$839$	−35.6882	−1.23209	−0.616047	+	0.787709i	$0.711267\pi$
$840$	0	0
$841$	−6.09870	−0.210300
$842$	0	0
$843$	8.49887	0.292717
$844$	0	0
$845$	1.55421	0.0534665
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	2.81982	0.0967761
$850$	0	0
$851$	5.78243	0.198219
$852$	0	0
$853$	22.2507	0.761849	0.380925	−	0.924606i	$-0.375606\pi$
$853$	22.2507	0.761849	0.380925	+	0.924606i	$0.375606\pi$
$854$	0	0
$855$	−3.86179	−0.132070
$856$	0	0
$857$	18.0586	0.616870	0.308435	−	0.951245i	$-0.400195\pi$
$857$	18.0586	0.616870	0.308435	+	0.951245i	$0.400195\pi$
$858$	0	0
$859$	−36.3958	−1.24181	−0.620905	−	0.783886i	$-0.713235\pi$
$859$	−36.3958	−1.24181	−0.620905	+	0.783886i	$0.713235\pi$
$860$	0	0
$861$	−2.37545	−0.0809551
$862$	0	0
$863$	−30.2661	−1.03027	−0.515136	−	0.857109i	$-0.672258\pi$
$863$	−30.2661	−1.03027	−0.515136	+	0.857109i	$0.672258\pi$
$864$	0	0
$865$	−30.9214	−1.05136
$866$	0	0
$867$	−3.02444	−0.102716
$868$	0	0
$869$	2.23287	0.0757448
$870$	0	0
$871$	6.51850	0.220871
$872$	0	0
$873$	−2.56655	−0.0868647
$874$	0	0
$875$	11.7878	0.398500
$876$	0	0
$877$	−36.1400	−1.22036	−0.610181	−	0.792262i	$-0.708903\pi$
$877$	−36.1400	−1.22036	−0.610181	+	0.792262i	$0.708903\pi$
$878$	0	0
$879$	−10.5500	−0.355841
$880$	0	0
$881$	0.936792	0.0315613	0.0157807	−	0.999875i	$-0.494977\pi$
$881$	0.936792	0.0315613	0.0157807	+	0.999875i	$0.494977\pi$
$882$	0	0
$883$	−37.9589	−1.27742	−0.638710	−	0.769447i	$-0.720532\pi$
$883$	−37.9589	−1.27742	−0.638710	+	0.769447i	$0.720532\pi$
$884$	0	0
$885$	−3.50736	−0.117899
$886$	0	0
$887$	51.3329	1.72359	0.861796	−	0.507256i	$-0.169340\pi$
$887$	51.3329	1.72359	0.861796	+	0.507256i	$0.169340\pi$
$888$	0	0
$889$	4.82092	0.161689
$890$	0	0
$891$	4.09170	0.137077
$892$	0	0
$893$	9.56664	0.320135
$894$	0	0
$895$	−13.8322	−0.462358
$896$	0	0
$897$	−3.47569	−0.116050
$898$	0	0
$899$	−13.6322	−0.454660
$900$	0	0
$901$	4.49509	0.149753
$902$	0	0
$903$	−6.37973	−0.212304
$904$	0	0
$905$	29.1474	0.968892
$906$	0	0
$907$	−27.3418	−0.907870	−0.453935	−	0.891035i	$-0.649980\pi$
$907$	−27.3418	−0.907870	−0.453935	+	0.891035i	$0.649980\pi$
$908$	0	0
$909$	−21.9446	−0.727858
$910$	0	0
$911$	47.8164	1.58423	0.792114	−	0.610373i	$-0.208980\pi$
$911$	47.8164	1.58423	0.792114	+	0.610373i	$0.208980\pi$
$912$	0	0
$913$	−10.5278	−0.348419
$914$	0	0
$915$	−7.65313	−0.253005
$916$	0	0
$917$	−3.95281	−0.130533
$918$	0	0
$919$	−42.2920	−1.39509	−0.697543	−	0.716543i	$-0.745723\pi$
$919$	−42.2920	−1.39509	−0.697543	+	0.716543i	$0.745723\pi$
$920$	0	0
$921$	−11.9563	−0.393974
$922$	0	0
$923$	10.4003	0.342331
$924$	0	0
$925$	−3.28340	−0.107958
$926$	0	0
$927$	−34.2335	−1.12438
$928$	0	0
$929$	25.2483	0.828369	0.414184	−	0.910193i	$-0.364067\pi$
$929$	25.2483	0.828369	0.414184	+	0.910193i	$0.364067\pi$
$930$	0	0
$931$	1.02809	0.0336942
$932$	0	0
$933$	−12.3374	−0.403908
$934$	0	0
$935$	5.61229	0.183541
$936$	0	0
$937$	−17.5207	−0.572377	−0.286189	−	0.958173i	$-0.592388\pi$
$937$	−17.5207	−0.572377	−0.286189	+	0.958173i	$0.592388\pi$
$938$	0	0
$939$	12.6466	0.412707
$940$	0	0
$941$	16.8160	0.548186	0.274093	−	0.961703i	$-0.411622\pi$
$941$	16.8160	0.548186	0.274093	+	0.961703i	$0.411622\pi$
$942$	0	0
$943$	14.1581	0.461051
$944$	0	0
$945$	6.42906	0.209137
$946$	0	0
$947$	56.9736	1.85139	0.925697	−	0.378266i	$-0.123479\pi$
$947$	56.9736	1.85139	0.925697	+	0.378266i	$0.123479\pi$
$948$	0	0
$949$	−6.95921	−0.225906
$950$	0	0
$951$	21.1117	0.684593
$952$	0	0
$953$	27.2538	0.882836	0.441418	−	0.897302i	$-0.354476\pi$
$953$	27.2538	0.882836	0.441418	+	0.897302i	$0.354476\pi$
$954$	0	0
$955$	−18.9406	−0.612904
$956$	0	0
$957$	−3.65444	−0.118131
$958$	0	0
$959$	13.4629	0.434738
$960$	0	0
$961$	−22.8853	−0.738235
$962$	0	0
$963$	−0.463294	−0.0149295
$964$	0	0
$965$	22.9323	0.738216
$966$	0	0
$967$	−3.65503	−0.117538	−0.0587689	−	0.998272i	$-0.518717\pi$
$967$	−3.65503	−0.117538	−0.0587689	+	0.998272i	$0.518717\pi$
$968$	0	0
$969$	2.83498	0.0910727
$970$	0	0
$971$	−19.8780	−0.637917	−0.318958	−	0.947769i	$-0.603333\pi$
$971$	−19.8780	−0.637917	−0.318958	+	0.947769i	$0.603333\pi$
$972$	0	0
$973$	−8.51502	−0.272979
$974$	0	0
$975$	1.97358	0.0632052
$976$	0	0
$977$	23.8678	0.763599	0.381799	−	0.924245i	$-0.375304\pi$
$977$	23.8678	0.763599	0.381799	+	0.924245i	$0.375304\pi$
$978$	0	0
$979$	12.4084	0.396575
$980$	0	0
$981$	−37.1491	−1.18608
$982$	0	0
$983$	−28.0856	−0.895791	−0.447895	−	0.894086i	$-0.647826\pi$
$983$	−28.0856	−0.895791	−0.447895	+	0.894086i	$0.647826\pi$
$984$	0	0
$985$	17.3533	0.552921
$986$	0	0
$987$	−7.10593	−0.226184
$988$	0	0
$989$	38.0243	1.20910
$990$	0	0
$991$	5.18594	0.164737	0.0823684	−	0.996602i	$-0.473752\pi$
$991$	5.18594	0.164737	0.0823684	+	0.996602i	$0.473752\pi$
$992$	0	0
$993$	8.18696	0.259805
$994$	0	0
$995$	41.6152	1.31929
$996$	0	0
$997$	−18.3344	−0.580656	−0.290328	−	0.956927i	$-0.593764\pi$
$997$	−18.3344	−0.580656	−0.290328	+	0.956927i	$0.593764\pi$
$998$	0	0
$999$	−5.25530	−0.166270

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.n.1.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.n.1.6	✓	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)

\(f(q)\)	\(=\)	\(q+0.763644 q^{3} +1.55421 q^{5} -1.00000 q^{7} -2.41685 q^{9} +O(q^{10})\)	\(q+0.763644 q^{3} +1.55421 q^{5} -1.00000 q^{7} -2.41685 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.18686 q^{15} +3.61102 q^{17} +1.02809 q^{19} -0.763644 q^{21} +4.55145 q^{23} -2.58443 q^{25} -4.13654 q^{27} -4.78553 q^{29} +2.84863 q^{31} +0.763644 q^{33} -1.55421 q^{35} +1.27046 q^{37} -0.763644 q^{39} +3.11068 q^{41} +8.35432 q^{43} -3.75629 q^{45} +9.30529 q^{47} +1.00000 q^{49} +2.75753 q^{51} +1.24483 q^{53} +1.55421 q^{55} +0.785092 q^{57} -2.95515 q^{59} -6.44819 q^{61} +2.41685 q^{63} -1.55421 q^{65} -6.51850 q^{67} +3.47569 q^{69} -10.4003 q^{71} +6.95921 q^{73} -1.97358 q^{75} -1.00000 q^{77} +2.23287 q^{79} +4.09170 q^{81} -10.5278 q^{83} +5.61229 q^{85} -3.65444 q^{87} +12.4084 q^{89} +1.00000 q^{91} +2.17534 q^{93} +1.59786 q^{95} +1.06194 q^{97} -2.41685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9	\( 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} + 11 q^{15} + 7 q^{17} - 17 q^{19} + 3 q^{21} + 11 q^{23} + 18 q^{25} - 9 q^{27} + 9 q^{29} - 8 q^{31} - 3 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} + 18 q^{41} + 7 q^{43} + 5 q^{45} + 15 q^{47} + 9 q^{49} - 7 q^{51} - 4 q^{53} + q^{55} + 22 q^{57} - 23 q^{59} + 12 q^{61} - 12 q^{63} - q^{65} - 16 q^{67} - 32 q^{69} - 6 q^{71} + 4 q^{73} - 14 q^{75} - 9 q^{77} + 21 q^{79} + 5 q^{81} - 16 q^{83} + 53 q^{85} + 41 q^{87} + 5 q^{89} + 9 q^{91} + 29 q^{93} + 19 q^{95} + 18 q^{97} + 12 q^{99}+O(q^{100}) \) 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 + 11 * q^15 + 7 * q^17 - 17 * q^19 + 3 * q^21 + 11 * q^23 + 18 * q^25 - 9 * q^27 + 9 * q^29 - 8 * q^31 - 3 * q^33 - q^35 + 2 * q^37 + 3 * q^39 + 18 * q^41 + 7 * q^43 + 5 * q^45 + 15 * q^47 + 9 * q^49 - 7 * q^51 - 4 * q^53 + q^55 + 22 * q^57 - 23 * q^59 + 12 * q^61 - 12 * q^63 - q^65 - 16 * q^67 - 32 * q^69 - 6 * q^71 + 4 * q^73 - 14 * q^75 - 9 * q^77 + 21 * q^79 + 5 * q^81 - 16 * q^83 + 53 * q^85 + 41 * q^87 + 5 * q^89 + 9 * q^91 + 29 * q^93 + 19 * q^95 + 18 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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