LMFDB - Embedded newform 8008.2.a.o.1.1

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:	$9$
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} - 2x^{8} - 12x^{7} + 20x^{6} + 47x^{5} - 55x^{4} - 68x^{3} + 37x^{2} + 21x - 9 $ x^9 - 2x^8 - 12x^7 + 20x^6 + 47x^5 - 55x^4 - 68x^3 + 37x^2 + 21x - 9
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			1.1
Root			$2.76592$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.76592 q^{3} +1.47621 q^{5} +1.00000 q^{7} +4.65032 q^{9} +O(q^{10})$	$q-2.76592 q^{3} +1.47621 q^{5} +1.00000 q^{7} +4.65032 q^{9} -1.00000 q^{11} +1.00000 q^{13} -4.08309 q^{15} -6.18911 q^{17} +7.32587 q^{19} -2.76592 q^{21} +1.73852 q^{23} -2.82079 q^{25} -4.56466 q^{27} +1.32946 q^{29} +1.36215 q^{31} +2.76592 q^{33} +1.47621 q^{35} -10.4816 q^{37} -2.76592 q^{39} +4.59184 q^{41} -12.2461 q^{43} +6.86487 q^{45} +8.18515 q^{47} +1.00000 q^{49} +17.1186 q^{51} -1.78000 q^{53} -1.47621 q^{55} -20.2628 q^{57} +7.05916 q^{59} -8.50203 q^{61} +4.65032 q^{63} +1.47621 q^{65} +5.50989 q^{67} -4.80860 q^{69} -2.10620 q^{71} -0.373869 q^{73} +7.80209 q^{75} -1.00000 q^{77} -1.34237 q^{79} -1.32548 q^{81} -13.0259 q^{83} -9.13645 q^{85} -3.67719 q^{87} -15.2762 q^{89} +1.00000 q^{91} -3.76760 q^{93} +10.8146 q^{95} -9.75343 q^{97} -4.65032 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9}+O(q^{10}) $ 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9	$ 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9} - 9 q^{11} + 9 q^{13} - 4 q^{15} - 9 q^{17} + 9 q^{19} - 2 q^{21} - 10 q^{23} - q^{25} - 8 q^{27} - 13 q^{29} + 7 q^{31} + 2 q^{33} - 4 q^{35} - 15 q^{37} - 2 q^{39} - 18 q^{41} + 7 q^{43} + 9 q^{45} - 11 q^{47} + 9 q^{49} + q^{51} - 16 q^{53} + 4 q^{55} - 26 q^{57} - 2 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + 13 q^{67} - 3 q^{69} - q^{71} - 6 q^{73} - 4 q^{75} - 9 q^{77} - 21 q^{79} - 23 q^{81} - 30 q^{83} + q^{85} + q^{87} - 36 q^{89} + 9 q^{91} - 22 q^{93} - 23 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) $ 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9 - 9 * q^11 + 9 * q^13 - 4 * q^15 - 9 * q^17 + 9 * q^19 - 2 * q^21 - 10 * q^23 - q^25 - 8 * q^27 - 13 * q^29 + 7 * q^31 + 2 * q^33 - 4 * q^35 - 15 * q^37 - 2 * q^39 - 18 * q^41 + 7 * q^43 + 9 * q^45 - 11 * q^47 + 9 * q^49 + q^51 - 16 * q^53 + 4 * q^55 - 26 * q^57 - 2 * q^59 + 2 * q^61 + q^63 - 4 * q^65 + 13 * q^67 - 3 * q^69 - q^71 - 6 * q^73 - 4 * q^75 - 9 * q^77 - 21 * q^79 - 23 * q^81 - 30 * q^83 + q^85 + q^87 - 36 * q^89 + 9 * q^91 - 22 * q^93 - 23 * q^95 - 5 * q^97 - q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.76592	−1.59691	−0.798453	−	0.602058i	$-0.794348\pi$
$3$	−2.76592	−1.59691	−0.798453	+	0.602058i	$0.794348\pi$
$4$	0	0
$5$	1.47621	0.660183	0.330092	−	0.943949i	$-0.392920\pi$
$5$	1.47621	0.660183	0.330092	+	0.943949i	$0.392920\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	4.65032	1.55011
$10$	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
$15$	−4.08309	−1.05425
$16$	0	0
$17$	−6.18911	−1.50108	−0.750539	−	0.660826i	$-0.770206\pi$
$17$	−6.18911	−1.50108	−0.750539	+	0.660826i	$0.770206\pi$
$18$	0	0
$19$	7.32587	1.68067	0.840335	−	0.542067i	$-0.182358\pi$
$19$	7.32587	1.68067	0.840335	+	0.542067i	$0.182358\pi$
$20$	0	0
$21$	−2.76592	−0.603573
$22$	0	0
$23$	1.73852	0.362505	0.181253	−	0.983437i	$-0.441985\pi$
$23$	1.73852	0.362505	0.181253	+	0.983437i	$0.441985\pi$
$24$	0	0
$25$	−2.82079	−0.564158
$26$	0	0
$27$	−4.56466	−0.878468
$28$	0	0
$29$	1.32946	0.246875	0.123437	−	0.992352i	$-0.460608\pi$
$29$	1.32946	0.246875	0.123437	+	0.992352i	$0.460608\pi$
$30$	0	0
$31$	1.36215	0.244649	0.122325	−	0.992490i	$-0.460965\pi$
$31$	1.36215	0.244649	0.122325	+	0.992490i	$0.460965\pi$
$32$	0	0
$33$	2.76592	0.481485
$34$	0	0
$35$	1.47621	0.249526
$36$	0	0
$37$	−10.4816	−1.72317	−0.861584	−	0.507616i	$-0.830527\pi$
$37$	−10.4816	−1.72317	−0.861584	+	0.507616i	$0.830527\pi$
$38$	0	0
$39$	−2.76592	−0.442902
$40$	0	0
$41$	4.59184	0.717125	0.358563	−	0.933506i	$-0.383267\pi$
$41$	4.59184	0.717125	0.358563	+	0.933506i	$0.383267\pi$
$42$	0	0
$43$	−12.2461	−1.86751	−0.933755	−	0.357912i	$-0.883489\pi$
$43$	−12.2461	−1.86751	−0.933755	+	0.357912i	$0.883489\pi$
$44$	0	0
$45$	6.86487	1.02335
$46$	0	0
$47$	8.18515	1.19393	0.596963	−	0.802269i	$-0.296374\pi$
$47$	8.18515	1.19393	0.596963	+	0.802269i	$0.296374\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	17.1186	2.39708
$52$	0	0
$53$	−1.78000	−0.244502	−0.122251	−	0.992499i	$-0.539011\pi$
$53$	−1.78000	−0.244502	−0.122251	+	0.992499i	$0.539011\pi$
$54$	0	0
$55$	−1.47621	−0.199053
$56$	0	0
$57$	−20.2628	−2.68387
$58$	0	0
$59$	7.05916	0.919024	0.459512	−	0.888172i	$-0.348024\pi$
$59$	7.05916	0.919024	0.459512	+	0.888172i	$0.348024\pi$
$60$	0	0
$61$	−8.50203	−1.08857	−0.544287	−	0.838899i	$-0.683200\pi$
$61$	−8.50203	−1.08857	−0.544287	+	0.838899i	$0.683200\pi$
$62$	0	0
$63$	4.65032	0.585885
$64$	0	0
$65$	1.47621	0.183102
$66$	0	0
$67$	5.50989	0.673140	0.336570	−	0.941659i	$-0.390733\pi$
$67$	5.50989	0.673140	0.336570	+	0.941659i	$0.390733\pi$
$68$	0	0
$69$	−4.80860	−0.578887
$70$	0	0
$71$	−2.10620	−0.249960	−0.124980	−	0.992159i	$-0.539887\pi$
$71$	−2.10620	−0.249960	−0.124980	+	0.992159i	$0.539887\pi$
$72$	0	0
$73$	−0.373869	−0.0437580	−0.0218790	−	0.999761i	$-0.506965\pi$
$73$	−0.373869	−0.0437580	−0.0218790	+	0.999761i	$0.506965\pi$
$74$	0	0
$75$	7.80209	0.900907
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−1.34237	−0.151029	−0.0755144	−	0.997145i	$-0.524060\pi$
$79$	−1.34237	−0.151029	−0.0755144	+	0.997145i	$0.524060\pi$
$80$	0	0
$81$	−1.32548	−0.147276
$82$	0	0
$83$	−13.0259	−1.42978	−0.714889	−	0.699238i	$-0.753523\pi$
$83$	−13.0259	−1.42978	−0.714889	+	0.699238i	$0.753523\pi$
$84$	0	0
$85$	−9.13645	−0.990987
$86$	0	0
$87$	−3.67719	−0.394236
$88$	0	0
$89$	−15.2762	−1.61928	−0.809639	−	0.586928i	$-0.800337\pi$
$89$	−15.2762	−1.61928	−0.809639	+	0.586928i	$0.800337\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−3.76760	−0.390682
$94$	0	0
$95$	10.8146	1.10955
$96$	0	0
$97$	−9.75343	−0.990311	−0.495155	−	0.868804i	$-0.664889\pi$
$97$	−9.75343	−0.990311	−0.495155	+	0.868804i	$0.664889\pi$
$98$	0	0
$99$	−4.65032	−0.467375
$100$	0	0
$101$	−0.100179	−0.00996814	−0.00498407	−	0.999988i	$-0.501586\pi$
$101$	−0.100179	−0.00996814	−0.00498407	+	0.999988i	$0.501586\pi$
$102$	0	0
$103$	−18.7556	−1.84804	−0.924022	−	0.382339i	$-0.875119\pi$
$103$	−18.7556	−1.84804	−0.924022	+	0.382339i	$0.875119\pi$
$104$	0	0
$105$	−4.08309	−0.398469
$106$	0	0
$107$	4.74472	0.458690	0.229345	−	0.973345i	$-0.426342\pi$
$107$	4.74472	0.458690	0.229345	+	0.973345i	$0.426342\pi$
$108$	0	0
$109$	13.2282	1.26704	0.633518	−	0.773728i	$-0.281610\pi$
$109$	13.2282	1.26704	0.633518	+	0.773728i	$0.281610\pi$
$110$	0	0
$111$	28.9913	2.75174
$112$	0	0
$113$	1.32703	0.124836	0.0624181	−	0.998050i	$-0.480119\pi$
$113$	1.32703	0.124836	0.0624181	+	0.998050i	$0.480119\pi$
$114$	0	0
$115$	2.56642	0.239320
$116$	0	0
$117$	4.65032	0.429922
$118$	0	0
$119$	−6.18911	−0.567355
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−12.7007	−1.14518
$124$	0	0
$125$	−11.5452	−1.03263
$126$	0	0
$127$	11.7341	1.04124	0.520618	−	0.853790i	$-0.325702\pi$
$127$	11.7341	1.04124	0.520618	+	0.853790i	$0.325702\pi$
$128$	0	0
$129$	33.8717	2.98224
$130$	0	0
$131$	14.5479	1.27105	0.635526	−	0.772079i	$-0.280783\pi$
$131$	14.5479	1.27105	0.635526	+	0.772079i	$0.280783\pi$
$132$	0	0
$133$	7.32587	0.635234
$134$	0	0
$135$	−6.73841	−0.579950
$136$	0	0
$137$	13.8362	1.18211	0.591053	−	0.806633i	$-0.298712\pi$
$137$	13.8362	1.18211	0.591053	+	0.806633i	$0.298712\pi$
$138$	0	0
$139$	0.523614	0.0444124	0.0222062	−	0.999753i	$-0.492931\pi$
$139$	0.523614	0.0444124	0.0222062	+	0.999753i	$0.492931\pi$
$140$	0	0
$141$	−22.6395	−1.90659
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	1.96257	0.162983
$146$	0	0
$147$	−2.76592	−0.228129
$148$	0	0
$149$	6.96098	0.570265	0.285133	−	0.958488i	$-0.407962\pi$
$149$	6.96098	0.570265	0.285133	+	0.958488i	$0.407962\pi$
$150$	0	0
$151$	3.40564	0.277147	0.138574	−	0.990352i	$-0.455748\pi$
$151$	3.40564	0.277147	0.138574	+	0.990352i	$0.455748\pi$
$152$	0	0
$153$	−28.7813	−2.32683
$154$	0	0
$155$	2.01083	0.161513
$156$	0	0
$157$	3.75167	0.299416	0.149708	−	0.988730i	$-0.452167\pi$
$157$	3.75167	0.299416	0.149708	+	0.988730i	$0.452167\pi$
$158$	0	0
$159$	4.92335	0.390447
$160$	0	0
$161$	1.73852	0.137014
$162$	0	0
$163$	21.7917	1.70686	0.853430	−	0.521207i	$-0.174518\pi$
$163$	21.7917	1.70686	0.853430	+	0.521207i	$0.174518\pi$
$164$	0	0
$165$	4.08309	0.317868
$166$	0	0
$167$	−12.5350	−0.969986	−0.484993	−	0.874518i	$-0.661178\pi$
$167$	−12.5350	−0.969986	−0.484993	+	0.874518i	$0.661178\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	34.0677	2.60522
$172$	0	0
$173$	−2.12209	−0.161339	−0.0806696	−	0.996741i	$-0.525706\pi$
$173$	−2.12209	−0.161339	−0.0806696	+	0.996741i	$0.525706\pi$
$174$	0	0
$175$	−2.82079	−0.213232
$176$	0	0
$177$	−19.5251	−1.46759
$178$	0	0
$179$	−11.0068	−0.822686	−0.411343	−	0.911481i	$-0.634940\pi$
$179$	−11.0068	−0.822686	−0.411343	+	0.911481i	$0.634940\pi$
$180$	0	0
$181$	−2.09372	−0.155625	−0.0778127	−	0.996968i	$-0.524794\pi$
$181$	−2.09372	−0.155625	−0.0778127	+	0.996968i	$0.524794\pi$
$182$	0	0
$183$	23.5159	1.73835
$184$	0	0
$185$	−15.4731	−1.13761
$186$	0	0
$187$	6.18911	0.452592
$188$	0	0
$189$	−4.56466	−0.332030
$190$	0	0
$191$	21.0597	1.52382	0.761912	−	0.647681i	$-0.224261\pi$
$191$	21.0597	1.52382	0.761912	+	0.647681i	$0.224261\pi$
$192$	0	0
$193$	2.91064	0.209513	0.104756	−	0.994498i	$-0.466594\pi$
$193$	2.91064	0.209513	0.104756	+	0.994498i	$0.466594\pi$
$194$	0	0
$195$	−4.08309	−0.292396
$196$	0	0
$197$	−19.1934	−1.36748	−0.683738	−	0.729728i	$-0.739647\pi$
$197$	−19.1934	−1.36748	−0.683738	+	0.729728i	$0.739647\pi$
$198$	0	0
$199$	3.70949	0.262959	0.131480	−	0.991319i	$-0.458027\pi$
$199$	3.70949	0.262959	0.131480	+	0.991319i	$0.458027\pi$
$200$	0	0
$201$	−15.2399	−1.07494
$202$	0	0
$203$	1.32946	0.0933099
$204$	0	0
$205$	6.77854	0.473434
$206$	0	0
$207$	8.08465	0.561922
$208$	0	0
$209$	−7.32587	−0.506741
$210$	0	0
$211$	3.89600	0.268212	0.134106	−	0.990967i	$-0.457184\pi$
$211$	3.89600	0.268212	0.134106	+	0.990967i	$0.457184\pi$
$212$	0	0
$213$	5.82559	0.399163
$214$	0	0
$215$	−18.0778	−1.23290
$216$	0	0
$217$	1.36215	0.0924688
$218$	0	0
$219$	1.03409	0.0698774
$220$	0	0
$221$	−6.18911	−0.416324
$222$	0	0
$223$	10.1156	0.677389	0.338695	−	0.940896i	$-0.390015\pi$
$223$	10.1156	0.677389	0.338695	+	0.940896i	$0.390015\pi$
$224$	0	0
$225$	−13.1176	−0.874506
$226$	0	0
$227$	−9.44653	−0.626988	−0.313494	−	0.949590i	$-0.601500\pi$
$227$	−9.44653	−0.626988	−0.313494	+	0.949590i	$0.601500\pi$
$228$	0	0
$229$	−19.7921	−1.30790	−0.653951	−	0.756537i	$-0.726890\pi$
$229$	−19.7921	−1.30790	−0.653951	+	0.756537i	$0.726890\pi$
$230$	0	0
$231$	2.76592	0.181984
$232$	0	0
$233$	−21.5715	−1.41320	−0.706599	−	0.707615i	$-0.749771\pi$
$233$	−21.5715	−1.41320	−0.706599	+	0.707615i	$0.749771\pi$
$234$	0	0
$235$	12.0830	0.788210
$236$	0	0
$237$	3.71290	0.241179
$238$	0	0
$239$	−17.6948	−1.14458	−0.572291	−	0.820051i	$-0.693945\pi$
$239$	−17.6948	−1.14458	−0.572291	+	0.820051i	$0.693945\pi$
$240$	0	0
$241$	4.53565	0.292167	0.146083	−	0.989272i	$-0.453333\pi$
$241$	4.53565	0.292167	0.146083	+	0.989272i	$0.453333\pi$
$242$	0	0
$243$	17.3601	1.11365
$244$	0	0
$245$	1.47621	0.0943119
$246$	0	0
$247$	7.32587	0.466134
$248$	0	0
$249$	36.0286	2.28322
$250$	0	0
$251$	−7.12131	−0.449493	−0.224747	−	0.974417i	$-0.572155\pi$
$251$	−7.12131	−0.449493	−0.224747	+	0.974417i	$0.572155\pi$
$252$	0	0
$253$	−1.73852	−0.109300
$254$	0	0
$255$	25.2707	1.58251
$256$	0	0
$257$	−4.31400	−0.269100	−0.134550	−	0.990907i	$-0.542959\pi$
$257$	−4.31400	−0.269100	−0.134550	+	0.990907i	$0.542959\pi$
$258$	0	0
$259$	−10.4816	−0.651296
$260$	0	0
$261$	6.18242	0.382682
$262$	0	0
$263$	2.23625	0.137893	0.0689467	−	0.997620i	$-0.478036\pi$
$263$	2.23625	0.137893	0.0689467	+	0.997620i	$0.478036\pi$
$264$	0	0
$265$	−2.62766	−0.161416
$266$	0	0
$267$	42.2529	2.58583
$268$	0	0
$269$	29.5497	1.80167	0.900837	−	0.434157i	$-0.142954\pi$
$269$	29.5497	1.80167	0.900837	+	0.434157i	$0.142954\pi$
$270$	0	0
$271$	−14.7101	−0.893572	−0.446786	−	0.894641i	$-0.647432\pi$
$271$	−14.7101	−0.893572	−0.446786	+	0.894641i	$0.647432\pi$
$272$	0	0
$273$	−2.76592	−0.167401
$274$	0	0
$275$	2.82079	0.170100
$276$	0	0
$277$	16.5344	0.993457	0.496728	−	0.867906i	$-0.334535\pi$
$277$	16.5344	0.993457	0.496728	+	0.867906i	$0.334535\pi$
$278$	0	0
$279$	6.33444	0.379233
$280$	0	0
$281$	−14.9836	−0.893846	−0.446923	−	0.894572i	$-0.647480\pi$
$281$	−14.9836	−0.893846	−0.446923	+	0.894572i	$0.647480\pi$
$282$	0	0
$283$	−30.7403	−1.82732	−0.913660	−	0.406479i	$-0.866756\pi$
$283$	−30.7403	−1.82732	−0.913660	+	0.406479i	$0.866756\pi$
$284$	0	0
$285$	−29.9122	−1.77185
$286$	0	0
$287$	4.59184	0.271048
$288$	0	0
$289$	21.3050	1.25324
$290$	0	0
$291$	26.9772	1.58143
$292$	0	0
$293$	−13.2293	−0.772863	−0.386431	−	0.922318i	$-0.626292\pi$
$293$	−13.2293	−0.772863	−0.386431	+	0.922318i	$0.626292\pi$
$294$	0	0
$295$	10.4208	0.606724
$296$	0	0
$297$	4.56466	0.264868
$298$	0	0
$299$	1.73852	0.100541
$300$	0	0
$301$	−12.2461	−0.705853
$302$	0	0
$303$	0.277086	0.0159182
$304$	0	0
$305$	−12.5508	−0.718658
$306$	0	0
$307$	−12.4182	−0.708743	−0.354372	−	0.935105i	$-0.615305\pi$
$307$	−12.4182	−0.708743	−0.354372	+	0.935105i	$0.615305\pi$
$308$	0	0
$309$	51.8765	2.95115
$310$	0	0
$311$	−26.9454	−1.52793	−0.763966	−	0.645257i	$-0.776750\pi$
$311$	−26.9454	−1.52793	−0.763966	+	0.645257i	$0.776750\pi$
$312$	0	0
$313$	−27.0454	−1.52870	−0.764348	−	0.644804i	$-0.776939\pi$
$313$	−27.0454	−1.52870	−0.764348	+	0.644804i	$0.776939\pi$
$314$	0	0
$315$	6.86487	0.386792
$316$	0	0
$317$	−12.1600	−0.682973	−0.341487	−	0.939887i	$-0.610930\pi$
$317$	−12.1600	−0.682973	−0.341487	+	0.939887i	$0.610930\pi$
$318$	0	0
$319$	−1.32946	−0.0744355
$320$	0	0
$321$	−13.1235	−0.732484
$322$	0	0
$323$	−45.3406	−2.52282
$324$	0	0
$325$	−2.82079	−0.156469
$326$	0	0
$327$	−36.5883	−2.02334
$328$	0	0
$329$	8.18515	0.451262
$330$	0	0
$331$	25.5998	1.40709	0.703547	−	0.710649i	$-0.251598\pi$
$331$	25.5998	1.40709	0.703547	+	0.710649i	$0.251598\pi$
$332$	0	0
$333$	−48.7429	−2.67109
$334$	0	0
$335$	8.13377	0.444395
$336$	0	0
$337$	−5.95728	−0.324514	−0.162257	−	0.986749i	$-0.551877\pi$
$337$	−5.95728	−0.324514	−0.162257	+	0.986749i	$0.551877\pi$
$338$	0	0
$339$	−3.67045	−0.199352
$340$	0	0
$341$	−1.36215	−0.0737646
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−7.09852	−0.382171
$346$	0	0
$347$	27.3599	1.46876	0.734379	−	0.678739i	$-0.237473\pi$
$347$	27.3599	1.46876	0.734379	+	0.678739i	$0.237473\pi$
$348$	0	0
$349$	23.5801	1.26221	0.631107	−	0.775696i	$-0.282601\pi$
$349$	23.5801	1.26221	0.631107	+	0.775696i	$0.282601\pi$
$350$	0	0
$351$	−4.56466	−0.243643
$352$	0	0
$353$	−30.4400	−1.62016	−0.810078	−	0.586323i	$-0.800575\pi$
$353$	−30.4400	−1.62016	−0.810078	+	0.586323i	$0.800575\pi$
$354$	0	0
$355$	−3.10920	−0.165019
$356$	0	0
$357$	17.1186	0.906011
$358$	0	0
$359$	7.66794	0.404698	0.202349	−	0.979313i	$-0.435142\pi$
$359$	7.66794	0.404698	0.202349	+	0.979313i	$0.435142\pi$
$360$	0	0
$361$	34.6684	1.82465
$362$	0	0
$363$	−2.76592	−0.145173
$364$	0	0
$365$	−0.551910	−0.0288883
$366$	0	0
$367$	−14.5933	−0.761763	−0.380882	−	0.924624i	$-0.624379\pi$
$367$	−14.5933	−0.761763	−0.380882	+	0.924624i	$0.624379\pi$
$368$	0	0
$369$	21.3535	1.11162
$370$	0	0
$371$	−1.78000	−0.0924131
$372$	0	0
$373$	23.6749	1.22584	0.612920	−	0.790145i	$-0.289995\pi$
$373$	23.6749	1.22584	0.612920	+	0.790145i	$0.289995\pi$
$374$	0	0
$375$	31.9330	1.64901
$376$	0	0
$377$	1.32946	0.0684707
$378$	0	0
$379$	3.69695	0.189900	0.0949498	−	0.995482i	$-0.469731\pi$
$379$	3.69695	0.189900	0.0949498	+	0.995482i	$0.469731\pi$
$380$	0	0
$381$	−32.4557	−1.66276
$382$	0	0
$383$	−9.98658	−0.510290	−0.255145	−	0.966903i	$-0.582123\pi$
$383$	−9.98658	−0.510290	−0.255145	+	0.966903i	$0.582123\pi$
$384$	0	0
$385$	−1.47621	−0.0752348
$386$	0	0
$387$	−56.9482	−2.89484
$388$	0	0
$389$	−10.8335	−0.549282	−0.274641	−	0.961547i	$-0.588559\pi$
$389$	−10.8335	−0.549282	−0.274641	+	0.961547i	$0.588559\pi$
$390$	0	0
$391$	−10.7599	−0.544149
$392$	0	0
$393$	−40.2382	−2.02975
$394$	0	0
$395$	−1.98163	−0.0997066
$396$	0	0
$397$	38.6542	1.94000	0.970000	−	0.243107i	$-0.0781665\pi$
$397$	38.6542	1.94000	0.970000	+	0.243107i	$0.0781665\pi$
$398$	0	0
$399$	−20.2628	−1.01441
$400$	0	0
$401$	7.90733	0.394873	0.197437	−	0.980316i	$-0.436738\pi$
$401$	7.90733	0.394873	0.197437	+	0.980316i	$0.436738\pi$
$402$	0	0
$403$	1.36215	0.0678536
$404$	0	0
$405$	−1.95670	−0.0972290
$406$	0	0
$407$	10.4816	0.519555
$408$	0	0
$409$	−3.90710	−0.193194	−0.0965969	−	0.995324i	$-0.530796\pi$
$409$	−3.90710	−0.193194	−0.0965969	+	0.995324i	$0.530796\pi$
$410$	0	0
$411$	−38.2698	−1.88771
$412$	0	0
$413$	7.05916	0.347358
$414$	0	0
$415$	−19.2290	−0.943915
$416$	0	0
$417$	−1.44828	−0.0709224
$418$	0	0
$419$	−20.1947	−0.986574	−0.493287	−	0.869867i	$-0.664205\pi$
$419$	−20.1947	−0.986574	−0.493287	+	0.869867i	$0.664205\pi$
$420$	0	0
$421$	−3.70159	−0.180404	−0.0902021	−	0.995923i	$-0.528751\pi$
$421$	−3.70159	−0.180404	−0.0902021	+	0.995923i	$0.528751\pi$
$422$	0	0
$423$	38.0636	1.85071
$424$	0	0
$425$	17.4582	0.846846
$426$	0	0
$427$	−8.50203	−0.411442
$428$	0	0
$429$	2.76592	0.133540
$430$	0	0
$431$	−20.3524	−0.980341	−0.490171	−	0.871627i	$-0.663065\pi$
$431$	−20.3524	−0.980341	−0.490171	+	0.871627i	$0.663065\pi$
$432$	0	0
$433$	−0.902581	−0.0433753	−0.0216876	−	0.999765i	$-0.506904\pi$
$433$	−0.902581	−0.0433753	−0.0216876	+	0.999765i	$0.506904\pi$
$434$	0	0
$435$	−5.42831	−0.260268
$436$	0	0
$437$	12.7361	0.609252
$438$	0	0
$439$	−4.57423	−0.218316	−0.109158	−	0.994024i	$-0.534815\pi$
$439$	−4.57423	−0.218316	−0.109158	+	0.994024i	$0.534815\pi$
$440$	0	0
$441$	4.65032	0.221444
$442$	0	0
$443$	38.8592	1.84626	0.923129	−	0.384491i	$-0.125623\pi$
$443$	38.8592	1.84626	0.923129	+	0.384491i	$0.125623\pi$
$444$	0	0
$445$	−22.5510	−1.06902
$446$	0	0
$447$	−19.2535	−0.910660
$448$	0	0
$449$	−38.5639	−1.81994	−0.909971	−	0.414671i	$-0.863897\pi$
$449$	−38.5639	−1.81994	−0.909971	+	0.414671i	$0.863897\pi$
$450$	0	0
$451$	−4.59184	−0.216221
$452$	0	0
$453$	−9.41974	−0.442578
$454$	0	0
$455$	1.47621	0.0692060
$456$	0	0
$457$	29.4361	1.37696	0.688480	−	0.725255i	$-0.258278\pi$
$457$	29.4361	1.37696	0.688480	+	0.725255i	$0.258278\pi$
$458$	0	0
$459$	28.2511	1.31865
$460$	0	0
$461$	0.133567	0.00622082	0.00311041	−	0.999995i	$-0.499010\pi$
$461$	0.133567	0.00622082	0.00311041	+	0.999995i	$0.499010\pi$
$462$	0	0
$463$	−14.9059	−0.692736	−0.346368	−	0.938099i	$-0.612585\pi$
$463$	−14.9059	−0.692736	−0.346368	+	0.938099i	$0.612585\pi$
$464$	0	0
$465$	−5.56179	−0.257922
$466$	0	0
$467$	−39.3924	−1.82286	−0.911430	−	0.411454i	$-0.865021\pi$
$467$	−39.3924	−1.82286	−0.911430	+	0.411454i	$0.865021\pi$
$468$	0	0
$469$	5.50989	0.254423
$470$	0	0
$471$	−10.3768	−0.478138
$472$	0	0
$473$	12.2461	0.563076
$474$	0	0
$475$	−20.6648	−0.948164
$476$	0	0
$477$	−8.27758	−0.379004
$478$	0	0
$479$	−6.74289	−0.308091	−0.154045	−	0.988064i	$-0.549230\pi$
$479$	−6.74289	−0.308091	−0.154045	+	0.988064i	$0.549230\pi$
$480$	0	0
$481$	−10.4816	−0.477921
$482$	0	0
$483$	−4.80860	−0.218799
$484$	0	0
$485$	−14.3982	−0.653786
$486$	0	0
$487$	−21.1468	−0.958253	−0.479127	−	0.877746i	$-0.659047\pi$
$487$	−21.1468	−0.958253	−0.479127	+	0.877746i	$0.659047\pi$
$488$	0	0
$489$	−60.2742	−2.72569
$490$	0	0
$491$	−35.3244	−1.59417	−0.797083	−	0.603870i	$-0.793625\pi$
$491$	−35.3244	−1.59417	−0.797083	+	0.603870i	$0.793625\pi$
$492$	0	0
$493$	−8.22818	−0.370579
$494$	0	0
$495$	−6.86487	−0.308553
$496$	0	0
$497$	−2.10620	−0.0944760
$498$	0	0
$499$	−3.17067	−0.141939	−0.0709694	−	0.997478i	$-0.522609\pi$
$499$	−3.17067	−0.141939	−0.0709694	+	0.997478i	$0.522609\pi$
$500$	0	0
$501$	34.6708	1.54898
$502$	0	0
$503$	−26.8560	−1.19745	−0.598725	−	0.800955i	$-0.704326\pi$
$503$	−26.8560	−1.19745	−0.598725	+	0.800955i	$0.704326\pi$
$504$	0	0
$505$	−0.147885	−0.00658080
$506$	0	0
$507$	−2.76592	−0.122839
$508$	0	0
$509$	1.72269	0.0763567	0.0381783	−	0.999271i	$-0.487845\pi$
$509$	1.72269	0.0763567	0.0381783	+	0.999271i	$0.487845\pi$
$510$	0	0
$511$	−0.373869	−0.0165390
$512$	0	0
$513$	−33.4401	−1.47642
$514$	0	0
$515$	−27.6873	−1.22005
$516$	0	0
$517$	−8.18515	−0.359982
$518$	0	0
$519$	5.86952	0.257643
$520$	0	0
$521$	−34.4306	−1.50843	−0.754217	−	0.656625i	$-0.771983\pi$
$521$	−34.4306	−1.50843	−0.754217	+	0.656625i	$0.771983\pi$
$522$	0	0
$523$	−28.6072	−1.25090	−0.625452	−	0.780262i	$-0.715086\pi$
$523$	−28.6072	−1.25090	−0.625452	+	0.780262i	$0.715086\pi$
$524$	0	0
$525$	7.80209	0.340511
$526$	0	0
$527$	−8.43050	−0.367238
$528$	0	0
$529$	−19.9776	−0.868590
$530$	0	0
$531$	32.8273	1.42459
$532$	0	0
$533$	4.59184	0.198895
$534$	0	0
$535$	7.00423	0.302819
$536$	0	0
$537$	30.4439	1.31375
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−40.5888	−1.74505	−0.872524	−	0.488571i	$-0.837518\pi$
$541$	−40.5888	−1.74505	−0.872524	+	0.488571i	$0.837518\pi$
$542$	0	0
$543$	5.79108	0.248519
$544$	0	0
$545$	19.5277	0.836475
$546$	0	0
$547$	30.4530	1.30208	0.651038	−	0.759045i	$-0.274334\pi$
$547$	30.4530	1.30208	0.651038	+	0.759045i	$0.274334\pi$
$548$	0	0
$549$	−39.5372	−1.68740
$550$	0	0
$551$	9.73946	0.414915
$552$	0	0
$553$	−1.34237	−0.0570835
$554$	0	0
$555$	42.7974	1.81665
$556$	0	0
$557$	−5.89381	−0.249729	−0.124864	−	0.992174i	$-0.539850\pi$
$557$	−5.89381	−0.249729	−0.124864	+	0.992174i	$0.539850\pi$
$558$	0	0
$559$	−12.2461	−0.517954
$560$	0	0
$561$	−17.1186	−0.722747
$562$	0	0
$563$	−15.9747	−0.673252	−0.336626	−	0.941638i	$-0.609286\pi$
$563$	−15.9747	−0.673252	−0.336626	+	0.941638i	$0.609286\pi$
$564$	0	0
$565$	1.95898	0.0824147
$566$	0	0
$567$	−1.32548	−0.0556650
$568$	0	0
$569$	−33.9162	−1.42184	−0.710921	−	0.703272i	$-0.751722\pi$
$569$	−33.9162	−1.42184	−0.710921	+	0.703272i	$0.751722\pi$
$570$	0	0
$571$	12.3194	0.515549	0.257775	−	0.966205i	$-0.417011\pi$
$571$	12.3194	0.515549	0.257775	+	0.966205i	$0.417011\pi$
$572$	0	0
$573$	−58.2494	−2.43340
$574$	0	0
$575$	−4.90399	−0.204510
$576$	0	0
$577$	−21.5539	−0.897302	−0.448651	−	0.893707i	$-0.648095\pi$
$577$	−21.5539	−0.897302	−0.448651	+	0.893707i	$0.648095\pi$
$578$	0	0
$579$	−8.05061	−0.334572
$580$	0	0
$581$	−13.0259	−0.540405
$582$	0	0
$583$	1.78000	0.0737202
$584$	0	0
$585$	6.86487	0.283827
$586$	0	0
$587$	−33.8982	−1.39913	−0.699563	−	0.714571i	$-0.746622\pi$
$587$	−33.8982	−1.39913	−0.699563	+	0.714571i	$0.746622\pi$
$588$	0	0
$589$	9.97894	0.411175
$590$	0	0
$591$	53.0875	2.18373
$592$	0	0
$593$	9.19508	0.377597	0.188798	−	0.982016i	$-0.439541\pi$
$593$	9.19508	0.377597	0.188798	+	0.982016i	$0.439541\pi$
$594$	0	0
$595$	−9.13645	−0.374558
$596$	0	0
$597$	−10.2602	−0.419921
$598$	0	0
$599$	−18.5289	−0.757069	−0.378534	−	0.925587i	$-0.623572\pi$
$599$	−18.5289	−0.757069	−0.378534	+	0.925587i	$0.623572\pi$
$600$	0	0
$601$	20.8048	0.848645	0.424323	−	0.905511i	$-0.360512\pi$
$601$	20.8048	0.848645	0.424323	+	0.905511i	$0.360512\pi$
$602$	0	0
$603$	25.6227	1.04344
$604$	0	0
$605$	1.47621	0.0600166
$606$	0	0
$607$	24.9475	1.01259	0.506294	−	0.862361i	$-0.331015\pi$
$607$	24.9475	1.01259	0.506294	+	0.862361i	$0.331015\pi$
$608$	0	0
$609$	−3.67719	−0.149007
$610$	0	0
$611$	8.18515	0.331136
$612$	0	0
$613$	−40.3591	−1.63009	−0.815045	−	0.579398i	$-0.803288\pi$
$613$	−40.3591	−1.63009	−0.815045	+	0.579398i	$0.803288\pi$
$614$	0	0
$615$	−18.7489	−0.756029
$616$	0	0
$617$	−42.4416	−1.70864	−0.854318	−	0.519751i	$-0.826025\pi$
$617$	−42.4416	−1.70864	−0.854318	+	0.519751i	$0.826025\pi$
$618$	0	0
$619$	37.4256	1.50426	0.752131	−	0.659013i	$-0.229026\pi$
$619$	37.4256	1.50426	0.752131	+	0.659013i	$0.229026\pi$
$620$	0	0
$621$	−7.93572	−0.318450
$622$	0	0
$623$	−15.2762	−0.612030
$624$	0	0
$625$	−2.93918	−0.117567
$626$	0	0
$627$	20.2628	0.809218
$628$	0	0
$629$	64.8719	2.58661
$630$	0	0
$631$	−40.8889	−1.62776	−0.813880	−	0.581033i	$-0.802649\pi$
$631$	−40.8889	−1.62776	−0.813880	+	0.581033i	$0.802649\pi$
$632$	0	0
$633$	−10.7760	−0.428309
$634$	0	0
$635$	17.3221	0.687406
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−9.79451	−0.387465
$640$	0	0
$641$	8.21329	0.324406	0.162203	−	0.986757i	$-0.448140\pi$
$641$	8.21329	0.324406	0.162203	+	0.986757i	$0.448140\pi$
$642$	0	0
$643$	−1.95429	−0.0770698	−0.0385349	−	0.999257i	$-0.512269\pi$
$643$	−1.95429	−0.0770698	−0.0385349	+	0.999257i	$0.512269\pi$
$644$	0	0
$645$	50.0019	1.96882
$646$	0	0
$647$	21.7588	0.855425	0.427712	−	0.903915i	$-0.359320\pi$
$647$	21.7588	0.855425	0.427712	+	0.903915i	$0.359320\pi$
$648$	0	0
$649$	−7.05916	−0.277096
$650$	0	0
$651$	−3.76760	−0.147664
$652$	0	0
$653$	−16.1187	−0.630775	−0.315387	−	0.948963i	$-0.602134\pi$
$653$	−16.1187	−0.630775	−0.315387	+	0.948963i	$0.602134\pi$
$654$	0	0
$655$	21.4758	0.839127
$656$	0	0
$657$	−1.73861	−0.0678296
$658$	0	0
$659$	9.70353	0.377996	0.188998	−	0.981977i	$-0.439476\pi$
$659$	9.70353	0.377996	0.188998	+	0.981977i	$0.439476\pi$
$660$	0	0
$661$	−28.3256	−1.10174	−0.550870	−	0.834591i	$-0.685704\pi$
$661$	−28.3256	−1.10174	−0.550870	+	0.834591i	$0.685704\pi$
$662$	0	0
$663$	17.1186	0.664831
$664$	0	0
$665$	10.8146	0.419371
$666$	0	0
$667$	2.31129	0.0894935
$668$	0	0
$669$	−27.9789	−1.08173
$670$	0	0
$671$	8.50203	0.328217
$672$	0	0
$673$	−16.2319	−0.625693	−0.312846	−	0.949804i	$-0.601282\pi$
$673$	−16.2319	−0.625693	−0.312846	+	0.949804i	$0.601282\pi$
$674$	0	0
$675$	12.8759	0.495595
$676$	0	0
$677$	5.38387	0.206919	0.103459	−	0.994634i	$-0.467009\pi$
$677$	5.38387	0.206919	0.103459	+	0.994634i	$0.467009\pi$
$678$	0	0
$679$	−9.75343	−0.374302
$680$	0	0
$681$	26.1284	1.00124
$682$	0	0
$683$	−17.5890	−0.673026	−0.336513	−	0.941679i	$-0.609248\pi$
$683$	−17.5890	−0.673026	−0.336513	+	0.941679i	$0.609248\pi$
$684$	0	0
$685$	20.4252	0.780406
$686$	0	0
$687$	54.7435	2.08860
$688$	0	0
$689$	−1.78000	−0.0678127
$690$	0	0
$691$	−8.53303	−0.324612	−0.162306	−	0.986741i	$-0.551893\pi$
$691$	−8.53303	−0.324612	−0.162306	+	0.986741i	$0.551893\pi$
$692$	0	0
$693$	−4.65032	−0.176651
$694$	0	0
$695$	0.772967	0.0293203
$696$	0	0
$697$	−28.4194	−1.07646
$698$	0	0
$699$	59.6651	2.25674
$700$	0	0
$701$	−35.3195	−1.33400	−0.666999	−	0.745058i	$-0.732422\pi$
$701$	−35.3195	−1.33400	−0.666999	+	0.745058i	$0.732422\pi$
$702$	0	0
$703$	−76.7870	−2.89608
$704$	0	0
$705$	−33.4207	−1.25870
$706$	0	0
$707$	−0.100179	−0.00376760
$708$	0	0
$709$	11.3433	0.426005	0.213003	−	0.977052i	$-0.431676\pi$
$709$	11.3433	0.426005	0.213003	+	0.977052i	$0.431676\pi$
$710$	0	0
$711$	−6.24246	−0.234111
$712$	0	0
$713$	2.36812	0.0886868
$714$	0	0
$715$	−1.47621	−0.0552073
$716$	0	0
$717$	48.9425	1.82779
$718$	0	0
$719$	−26.2863	−0.980312	−0.490156	−	0.871635i	$-0.663060\pi$
$719$	−26.2863	−0.980312	−0.490156	+	0.871635i	$0.663060\pi$
$720$	0	0
$721$	−18.7556	−0.698495
$722$	0	0
$723$	−12.5452	−0.466563
$724$	0	0
$725$	−3.75013	−0.139276
$726$	0	0
$727$	10.8359	0.401883	0.200941	−	0.979603i	$-0.435600\pi$
$727$	10.8359	0.401883	0.200941	+	0.979603i	$0.435600\pi$
$728$	0	0
$729$	−44.0404	−1.63112
$730$	0	0
$731$	75.7923	2.80328
$732$	0	0
$733$	−0.108179	−0.00399567	−0.00199784	−	0.999998i	$-0.500636\pi$
$733$	−0.108179	−0.00399567	−0.00199784	+	0.999998i	$0.500636\pi$
$734$	0	0
$735$	−4.08309	−0.150607
$736$	0	0
$737$	−5.50989	−0.202959
$738$	0	0
$739$	−6.24529	−0.229737	−0.114868	−	0.993381i	$-0.536645\pi$
$739$	−6.24529	−0.229737	−0.114868	+	0.993381i	$0.536645\pi$
$740$	0	0
$741$	−20.2628	−0.744372
$742$	0	0
$743$	8.16345	0.299488	0.149744	−	0.988725i	$-0.452155\pi$
$743$	8.16345	0.299488	0.149744	+	0.988725i	$0.452155\pi$
$744$	0	0
$745$	10.2759	0.376480
$746$	0	0
$747$	−60.5746	−2.21631
$748$	0	0
$749$	4.74472	0.173368
$750$	0	0
$751$	−24.4724	−0.893010	−0.446505	−	0.894781i	$-0.647332\pi$
$751$	−24.4724	−0.893010	−0.446505	+	0.894781i	$0.647332\pi$
$752$	0	0
$753$	19.6970	0.717798
$754$	0	0
$755$	5.02746	0.182968
$756$	0	0
$757$	51.9607	1.88854	0.944272	−	0.329165i	$-0.106767\pi$
$757$	51.9607	1.88854	0.944272	+	0.329165i	$0.106767\pi$
$758$	0	0
$759$	4.80860	0.174541
$760$	0	0
$761$	−13.7933	−0.500006	−0.250003	−	0.968245i	$-0.580432\pi$
$761$	−13.7933	−0.500006	−0.250003	+	0.968245i	$0.580432\pi$
$762$	0	0
$763$	13.2282	0.478894
$764$	0	0
$765$	−42.4874	−1.53614
$766$	0	0
$767$	7.05916	0.254891
$768$	0	0
$769$	−8.73799	−0.315100	−0.157550	−	0.987511i	$-0.550360\pi$
$769$	−8.73799	−0.315100	−0.157550	+	0.987511i	$0.550360\pi$
$770$	0	0
$771$	11.9322	0.429727
$772$	0	0
$773$	38.2474	1.37566	0.687831	−	0.725871i	$-0.258563\pi$
$773$	38.2474	1.37566	0.687831	+	0.725871i	$0.258563\pi$
$774$	0	0
$775$	−3.84234	−0.138021
$776$	0	0
$777$	28.9913	1.04006
$778$	0	0
$779$	33.6393	1.20525
$780$	0	0
$781$	2.10620	0.0753658
$782$	0	0
$783$	−6.06853	−0.216872
$784$	0	0
$785$	5.53826	0.197669
$786$	0	0
$787$	6.55994	0.233837	0.116918	−	0.993142i	$-0.462698\pi$
$787$	6.55994	0.233837	0.116918	+	0.993142i	$0.462698\pi$
$788$	0	0
$789$	−6.18530	−0.220203
$790$	0	0
$791$	1.32703	0.0471836
$792$	0	0
$793$	−8.50203	−0.301916
$794$	0	0
$795$	7.26791	0.257766
$796$	0	0
$797$	47.7228	1.69043	0.845214	−	0.534428i	$-0.179473\pi$
$797$	47.7228	1.69043	0.845214	+	0.534428i	$0.179473\pi$
$798$	0	0
$799$	−50.6588	−1.79218
$800$	0	0
$801$	−71.0394	−2.51005
$802$	0	0
$803$	0.373869	0.0131935
$804$	0	0
$805$	2.56642	0.0904544
$806$	0	0
$807$	−81.7320	−2.87710
$808$	0	0
$809$	38.5727	1.35614	0.678072	−	0.734995i	$-0.262816\pi$
$809$	38.5727	1.35614	0.678072	+	0.734995i	$0.262816\pi$
$810$	0	0
$811$	−4.65450	−0.163442	−0.0817208	−	0.996655i	$-0.526042\pi$
$811$	−4.65450	−0.163442	−0.0817208	+	0.996655i	$0.526042\pi$
$812$	0	0
$813$	40.6869	1.42695
$814$	0	0
$815$	32.1693	1.12684
$816$	0	0
$817$	−89.7133	−3.13867
$818$	0	0
$819$	4.65032	0.162495
$820$	0	0
$821$	14.7527	0.514874	0.257437	−	0.966295i	$-0.417122\pi$
$821$	14.7527	0.514874	0.257437	+	0.966295i	$0.417122\pi$
$822$	0	0
$823$	−15.5174	−0.540904	−0.270452	−	0.962733i	$-0.587173\pi$
$823$	−15.5174	−0.540904	−0.270452	+	0.962733i	$0.587173\pi$
$824$	0	0
$825$	−7.80209	−0.271634
$826$	0	0
$827$	8.60204	0.299122	0.149561	−	0.988752i	$-0.452214\pi$
$827$	8.60204	0.299122	0.149561	+	0.988752i	$0.452214\pi$
$828$	0	0
$829$	36.1283	1.25479	0.627393	−	0.778702i	$-0.284122\pi$
$829$	36.1283	1.25479	0.627393	+	0.778702i	$0.284122\pi$
$830$	0	0
$831$	−45.7329	−1.58646
$832$	0	0
$833$	−6.18911	−0.214440
$834$	0	0
$835$	−18.5043	−0.640368
$836$	0	0
$837$	−6.21775	−0.214917
$838$	0	0
$839$	25.4500	0.878631	0.439316	−	0.898333i	$-0.355221\pi$
$839$	25.4500	0.878631	0.439316	+	0.898333i	$0.355221\pi$
$840$	0	0
$841$	−27.2325	−0.939053
$842$	0	0
$843$	41.4434	1.42739
$844$	0	0
$845$	1.47621	0.0507833
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	85.0252	2.91806
$850$	0	0
$851$	−18.2225	−0.624658
$852$	0	0
$853$	6.60035	0.225992	0.112996	−	0.993595i	$-0.463955\pi$
$853$	6.60035	0.225992	0.112996	+	0.993595i	$0.463955\pi$
$854$	0	0
$855$	50.2912	1.71992
$856$	0	0
$857$	−46.6193	−1.59249	−0.796243	−	0.604977i	$-0.793182\pi$
$857$	−46.6193	−1.59249	−0.796243	+	0.604977i	$0.793182\pi$
$858$	0	0
$859$	−15.7583	−0.537667	−0.268834	−	0.963187i	$-0.586638\pi$
$859$	−15.7583	−0.537667	−0.268834	+	0.963187i	$0.586638\pi$
$860$	0	0
$861$	−12.7007	−0.432838
$862$	0	0
$863$	−36.7352	−1.25048	−0.625240	−	0.780433i	$-0.714999\pi$
$863$	−36.7352	−1.25048	−0.625240	+	0.780433i	$0.714999\pi$
$864$	0	0
$865$	−3.13265	−0.106513
$866$	0	0
$867$	−58.9281	−2.00130
$868$	0	0
$869$	1.34237	0.0455369
$870$	0	0
$871$	5.50989	0.186695
$872$	0	0
$873$	−45.3566	−1.53509
$874$	0	0
$875$	−11.5452	−0.390298
$876$	0	0
$877$	−41.4742	−1.40049	−0.700243	−	0.713905i	$-0.746925\pi$
$877$	−41.4742	−1.40049	−0.700243	+	0.713905i	$0.746925\pi$
$878$	0	0
$879$	36.5911	1.23419
$880$	0	0
$881$	−51.1180	−1.72221	−0.861105	−	0.508427i	$-0.830227\pi$
$881$	−51.1180	−1.72221	−0.861105	+	0.508427i	$0.830227\pi$
$882$	0	0
$883$	17.4683	0.587855	0.293928	−	0.955828i	$-0.405038\pi$
$883$	17.4683	0.587855	0.293928	+	0.955828i	$0.405038\pi$
$884$	0	0
$885$	−28.8232	−0.968881
$886$	0	0
$887$	−14.2768	−0.479369	−0.239684	−	0.970851i	$-0.577044\pi$
$887$	−14.2768	−0.479369	−0.239684	+	0.970851i	$0.577044\pi$
$888$	0	0
$889$	11.7341	0.393550
$890$	0	0
$891$	1.32548	0.0444053
$892$	0	0
$893$	59.9634	2.00660
$894$	0	0
$895$	−16.2484	−0.543123
$896$	0	0
$897$	−4.80860	−0.160554
$898$	0	0
$899$	1.81093	0.0603978
$900$	0	0
$901$	11.0166	0.367017
$902$	0	0
$903$	33.8717	1.12718
$904$	0	0
$905$	−3.09079	−0.102741
$906$	0	0
$907$	6.41086	0.212869	0.106435	−	0.994320i	$-0.466057\pi$
$907$	6.41086	0.212869	0.106435	+	0.994320i	$0.466057\pi$
$908$	0	0
$909$	−0.465863	−0.0154517
$910$	0	0
$911$	41.6729	1.38069	0.690343	−	0.723482i	$-0.257460\pi$
$911$	41.6729	1.38069	0.690343	+	0.723482i	$0.257460\pi$
$912$	0	0
$913$	13.0259	0.431094
$914$	0	0
$915$	34.7146	1.14763
$916$	0	0
$917$	14.5479	0.480413
$918$	0	0
$919$	37.1015	1.22386	0.611932	−	0.790910i	$-0.290393\pi$
$919$	37.1015	1.22386	0.611932	+	0.790910i	$0.290393\pi$
$920$	0	0
$921$	34.3477	1.13180
$922$	0	0
$923$	−2.10620	−0.0693264
$924$	0	0
$925$	29.5665	0.972139
$926$	0	0
$927$	−87.2196	−2.86467
$928$	0	0
$929$	20.4293	0.670263	0.335132	−	0.942171i	$-0.391219\pi$
$929$	20.4293	0.670263	0.335132	+	0.942171i	$0.391219\pi$
$930$	0	0
$931$	7.32587	0.240096
$932$	0	0
$933$	74.5287	2.43996
$934$	0	0
$935$	9.13645	0.298794
$936$	0	0
$937$	−21.2774	−0.695103	−0.347551	−	0.937661i	$-0.612987\pi$
$937$	−21.2774	−0.695103	−0.347551	+	0.937661i	$0.612987\pi$
$938$	0	0
$939$	74.8055	2.44118
$940$	0	0
$941$	35.4256	1.15484	0.577420	−	0.816447i	$-0.304060\pi$
$941$	35.4256	1.15484	0.577420	+	0.816447i	$0.304060\pi$
$942$	0	0
$943$	7.98299	0.259962
$944$	0	0
$945$	−6.73841	−0.219200
$946$	0	0
$947$	6.13515	0.199366	0.0996828	−	0.995019i	$-0.468217\pi$
$947$	6.13515	0.199366	0.0996828	+	0.995019i	$0.468217\pi$
$948$	0	0
$949$	−0.373869	−0.0121363
$950$	0	0
$951$	33.6336	1.09064
$952$	0	0
$953$	16.8497	0.545816	0.272908	−	0.962040i	$-0.412015\pi$
$953$	16.8497	0.545816	0.272908	+	0.962040i	$0.412015\pi$
$954$	0	0
$955$	31.0886	1.00600
$956$	0	0
$957$	3.67719	0.118867
$958$	0	0
$959$	13.8362	0.446794
$960$	0	0
$961$	−29.1445	−0.940147
$962$	0	0
$963$	22.0645	0.711018
$964$	0	0
$965$	4.29673	0.138317
$966$	0	0
$967$	19.0552	0.612775	0.306388	−	0.951907i	$-0.400880\pi$
$967$	19.0552	0.612775	0.306388	+	0.951907i	$0.400880\pi$
$968$	0	0
$969$	125.409	4.02870
$970$	0	0
$971$	−24.0815	−0.772812	−0.386406	−	0.922329i	$-0.626284\pi$
$971$	−24.0815	−0.772812	−0.386406	+	0.922329i	$0.626284\pi$
$972$	0	0
$973$	0.523614	0.0167863
$974$	0	0
$975$	7.80209	0.249867
$976$	0	0
$977$	−24.0794	−0.770368	−0.385184	−	0.922840i	$-0.625862\pi$
$977$	−24.0794	−0.770368	−0.385184	+	0.922840i	$0.625862\pi$
$978$	0	0
$979$	15.2762	0.488231
$980$	0	0
$981$	61.5155	1.96404
$982$	0	0
$983$	39.4934	1.25964	0.629821	−	0.776740i	$-0.283128\pi$
$983$	39.4934	1.25964	0.629821	+	0.776740i	$0.283128\pi$
$984$	0	0
$985$	−28.3336	−0.902784
$986$	0	0
$987$	−22.6395	−0.720623
$988$	0	0
$989$	−21.2900	−0.676983
$990$	0	0
$991$	−39.9313	−1.26846	−0.634229	−	0.773145i	$-0.718683\pi$
$991$	−39.9313	−1.26846	−0.634229	+	0.773145i	$0.718683\pi$
$992$	0	0
$993$	−70.8071	−2.24700
$994$	0	0
$995$	5.47601	0.173601
$996$	0	0
$997$	7.93666	0.251357	0.125678	−	0.992071i	$-0.459889\pi$
$997$	7.93666	0.251357	0.125678	+	0.992071i	$0.459889\pi$
$998$	0	0
$999$	47.8450	1.51375

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.o.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-2.76592 q^{3} +1.47621 q^{5} +1.00000 q^{7} +4.65032 q^{9} +O(q^{10})\)	\(q-2.76592 q^{3} +1.47621 q^{5} +1.00000 q^{7} +4.65032 q^{9} -1.00000 q^{11} +1.00000 q^{13} -4.08309 q^{15} -6.18911 q^{17} +7.32587 q^{19} -2.76592 q^{21} +1.73852 q^{23} -2.82079 q^{25} -4.56466 q^{27} +1.32946 q^{29} +1.36215 q^{31} +2.76592 q^{33} +1.47621 q^{35} -10.4816 q^{37} -2.76592 q^{39} +4.59184 q^{41} -12.2461 q^{43} +6.86487 q^{45} +8.18515 q^{47} +1.00000 q^{49} +17.1186 q^{51} -1.78000 q^{53} -1.47621 q^{55} -20.2628 q^{57} +7.05916 q^{59} -8.50203 q^{61} +4.65032 q^{63} +1.47621 q^{65} +5.50989 q^{67} -4.80860 q^{69} -2.10620 q^{71} -0.373869 q^{73} +7.80209 q^{75} -1.00000 q^{77} -1.34237 q^{79} -1.32548 q^{81} -13.0259 q^{83} -9.13645 q^{85} -3.67719 q^{87} -15.2762 q^{89} +1.00000 q^{91} -3.76760 q^{93} +10.8146 q^{95} -9.75343 q^{97} -4.65032 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9}+O(q^{10}) \) 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9	\( 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9} - 9 q^{11} + 9 q^{13} - 4 q^{15} - 9 q^{17} + 9 q^{19} - 2 q^{21} - 10 q^{23} - q^{25} - 8 q^{27} - 13 q^{29} + 7 q^{31} + 2 q^{33} - 4 q^{35} - 15 q^{37} - 2 q^{39} - 18 q^{41} + 7 q^{43} + 9 q^{45} - 11 q^{47} + 9 q^{49} + q^{51} - 16 q^{53} + 4 q^{55} - 26 q^{57} - 2 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + 13 q^{67} - 3 q^{69} - q^{71} - 6 q^{73} - 4 q^{75} - 9 q^{77} - 21 q^{79} - 23 q^{81} - 30 q^{83} + q^{85} + q^{87} - 36 q^{89} + 9 q^{91} - 22 q^{93} - 23 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) \) 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9 - 9 * q^11 + 9 * q^13 - 4 * q^15 - 9 * q^17 + 9 * q^19 - 2 * q^21 - 10 * q^23 - q^25 - 8 * q^27 - 13 * q^29 + 7 * q^31 + 2 * q^33 - 4 * q^35 - 15 * q^37 - 2 * q^39 - 18 * q^41 + 7 * q^43 + 9 * q^45 - 11 * q^47 + 9 * q^49 + q^51 - 16 * q^53 + 4 * q^55 - 26 * q^57 - 2 * q^59 + 2 * q^61 + q^63 - 4 * q^65 + 13 * q^67 - 3 * q^69 - q^71 - 6 * q^73 - 4 * q^75 - 9 * q^77 - 21 * q^79 - 23 * q^81 - 30 * q^83 + q^85 + q^87 - 36 * q^89 + 9 * q^91 - 22 * q^93 - 23 * q^95 - 5 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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