LMFDB - Embedded newform 2008.2.a.d.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2008,2,Mod(1,2008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2008 = 2^{3} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2008.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:	$16.0339607259$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	2008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.201914 q^{3} -1.79819 q^{5} +3.34181 q^{7} -2.95923 q^{9} +O(q^{10})$	$q-0.201914 q^{3} -1.79819 q^{5} +3.34181 q^{7} -2.95923 q^{9} -5.67355 q^{11} -3.71810 q^{13} +0.363080 q^{15} +3.29744 q^{17} -0.297273 q^{19} -0.674758 q^{21} +8.29149 q^{23} -1.76650 q^{25} +1.20325 q^{27} +7.61220 q^{29} +7.30347 q^{31} +1.14557 q^{33} -6.00922 q^{35} -5.21090 q^{37} +0.750736 q^{39} +7.48959 q^{41} -0.751052 q^{43} +5.32127 q^{45} +7.73856 q^{47} +4.16770 q^{49} -0.665799 q^{51} +8.79352 q^{53} +10.2021 q^{55} +0.0600236 q^{57} -11.6953 q^{59} +3.46801 q^{61} -9.88919 q^{63} +6.68586 q^{65} +14.0862 q^{67} -1.67417 q^{69} -1.05938 q^{71} -13.8342 q^{73} +0.356681 q^{75} -18.9599 q^{77} -0.746282 q^{79} +8.63474 q^{81} -2.81989 q^{83} -5.92943 q^{85} -1.53701 q^{87} +11.6951 q^{89} -12.4252 q^{91} -1.47467 q^{93} +0.534554 q^{95} +4.20944 q^{97} +16.7893 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) $ 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9	$ 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9} + 8 q^{11} + 8 q^{13} + 7 q^{15} + 19 q^{17} - 9 q^{19} + 9 q^{21} + 21 q^{23} + 65 q^{25} + 5 q^{27} + 10 q^{29} - 9 q^{31} + 34 q^{33} + 12 q^{35} + 11 q^{37} - 9 q^{39} + 35 q^{41} - 9 q^{43} + 29 q^{45} + 37 q^{47} + 77 q^{49} - 17 q^{51} + 38 q^{53} - 20 q^{55} + 51 q^{57} + 17 q^{59} + 22 q^{63} + 41 q^{65} + 9 q^{67} + 8 q^{69} + 13 q^{71} + 41 q^{73} + 25 q^{75} + 36 q^{77} - 36 q^{79} + 127 q^{81} + 29 q^{83} + 34 q^{85} + 10 q^{87} + 36 q^{89} - 6 q^{91} + 36 q^{93} + 25 q^{95} + 40 q^{97} + 19 q^{99}+O(q^{100}) $ 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9 + 8 * q^11 + 8 * q^13 + 7 * q^15 + 19 * q^17 - 9 * q^19 + 9 * q^21 + 21 * q^23 + 65 * q^25 + 5 * q^27 + 10 * q^29 - 9 * q^31 + 34 * q^33 + 12 * q^35 + 11 * q^37 - 9 * q^39 + 35 * q^41 - 9 * q^43 + 29 * q^45 + 37 * q^47 + 77 * q^49 - 17 * q^51 + 38 * q^53 - 20 * q^55 + 51 * q^57 + 17 * q^59 + 22 * q^63 + 41 * q^65 + 9 * q^67 + 8 * q^69 + 13 * q^71 + 41 * q^73 + 25 * q^75 + 36 * q^77 - 36 * q^79 + 127 * q^81 + 29 * q^83 + 34 * q^85 + 10 * q^87 + 36 * q^89 - 6 * q^91 + 36 * q^93 + 25 * q^95 + 40 * q^97 + 19 * 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
$3$	−0.201914	−0.116575	−0.0582875	−	0.998300i	$-0.518564\pi$
$3$	−0.201914	−0.116575	−0.0582875	+	0.998300i	$0.518564\pi$
$4$	0	0
$5$	−1.79819	−0.804176	−0.402088	−	0.915601i	$-0.631715\pi$
$5$	−1.79819	−0.804176	−0.402088	+	0.915601i	$0.631715\pi$
$6$	0	0
$7$	3.34181	1.26309	0.631543	−	0.775341i	$-0.282422\pi$
$7$	3.34181	1.26309	0.631543	+	0.775341i	$0.282422\pi$
$8$	0	0
$9$	−2.95923	−0.986410
$10$	0	0
$11$	−5.67355	−1.71064	−0.855319	−	0.518102i	$-0.826639\pi$
$11$	−5.67355	−1.71064	−0.855319	+	0.518102i	$0.826639\pi$
$12$	0	0
$13$	−3.71810	−1.03122	−0.515608	−	0.856825i	$-0.672434\pi$
$13$	−3.71810	−1.03122	−0.515608	+	0.856825i	$0.672434\pi$
$14$	0	0
$15$	0.363080	0.0937469
$16$	0	0
$17$	3.29744	0.799746	0.399873	−	0.916570i	$-0.369054\pi$
$17$	3.29744	0.799746	0.399873	+	0.916570i	$0.369054\pi$
$18$	0	0
$19$	−0.297273	−0.0681991	−0.0340996	−	0.999418i	$-0.510856\pi$
$19$	−0.297273	−0.0681991	−0.0340996	+	0.999418i	$0.510856\pi$
$20$	0	0
$21$	−0.674758	−0.147244
$22$	0	0
$23$	8.29149	1.72890	0.864448	−	0.502723i	$-0.167668\pi$
$23$	8.29149	1.72890	0.864448	+	0.502723i	$0.167668\pi$
$24$	0	0
$25$	−1.76650	−0.353300
$26$	0	0
$27$	1.20325	0.231566
$28$	0	0
$29$	7.61220	1.41355	0.706775	−	0.707438i	$-0.250149\pi$
$29$	7.61220	1.41355	0.706775	+	0.707438i	$0.250149\pi$
$30$	0	0
$31$	7.30347	1.31174	0.655871	−	0.754873i	$-0.272302\pi$
$31$	7.30347	1.31174	0.655871	+	0.754873i	$0.272302\pi$
$32$	0	0
$33$	1.14557	0.199418
$34$	0	0
$35$	−6.00922	−1.01574
$36$	0	0
$37$	−5.21090	−0.856666	−0.428333	−	0.903621i	$-0.640899\pi$
$37$	−5.21090	−0.856666	−0.428333	+	0.903621i	$0.640899\pi$
$38$	0	0
$39$	0.750736	0.120214
$40$	0	0
$41$	7.48959	1.16968	0.584838	−	0.811150i	$-0.301158\pi$
$41$	7.48959	1.16968	0.584838	+	0.811150i	$0.301158\pi$
$42$	0	0
$43$	−0.751052	−0.114534	−0.0572672	−	0.998359i	$-0.518239\pi$
$43$	−0.751052	−0.114534	−0.0572672	+	0.998359i	$0.518239\pi$
$44$	0	0
$45$	5.32127	0.793248
$46$	0	0
$47$	7.73856	1.12879	0.564393	−	0.825506i	$-0.309110\pi$
$47$	7.73856	1.12879	0.564393	+	0.825506i	$0.309110\pi$
$48$	0	0
$49$	4.16770	0.595385
$50$	0	0
$51$	−0.665799	−0.0932305
$52$	0	0
$53$	8.79352	1.20788	0.603941	−	0.797029i	$-0.293596\pi$
$53$	8.79352	1.20788	0.603941	+	0.797029i	$0.293596\pi$
$54$	0	0
$55$	10.2021	1.37565
$56$	0	0
$57$	0.0600236	0.00795032
$58$	0	0
$59$	−11.6953	−1.52259	−0.761297	−	0.648403i	$-0.775437\pi$
$59$	−11.6953	−1.52259	−0.761297	+	0.648403i	$0.775437\pi$
$60$	0	0
$61$	3.46801	0.444033	0.222016	−	0.975043i	$-0.428736\pi$
$61$	3.46801	0.444033	0.222016	+	0.975043i	$0.428736\pi$
$62$	0	0
$63$	−9.88919	−1.24592
$64$	0	0
$65$	6.68586	0.829279
$66$	0	0
$67$	14.0862	1.72090	0.860452	−	0.509532i	$-0.170181\pi$
$67$	14.0862	1.72090	0.860452	+	0.509532i	$0.170181\pi$
$68$	0	0
$69$	−1.67417	−0.201546
$70$	0	0
$71$	−1.05938	−0.125726	−0.0628629	−	0.998022i	$-0.520023\pi$
$71$	−1.05938	−0.125726	−0.0628629	+	0.998022i	$0.520023\pi$
$72$	0	0
$73$	−13.8342	−1.61917	−0.809587	−	0.587000i	$-0.800309\pi$
$73$	−13.8342	−1.61917	−0.809587	+	0.587000i	$0.800309\pi$
$74$	0	0
$75$	0.356681	0.0411860
$76$	0	0
$77$	−18.9599	−2.16068
$78$	0	0
$79$	−0.746282	−0.0839633	−0.0419816	−	0.999118i	$-0.513367\pi$
$79$	−0.746282	−0.0839633	−0.0419816	+	0.999118i	$0.513367\pi$
$80$	0	0
$81$	8.63474	0.959415
$82$	0	0
$83$	−2.81989	−0.309523	−0.154761	−	0.987952i	$-0.549461\pi$
$83$	−2.81989	−0.309523	−0.154761	+	0.987952i	$0.549461\pi$
$84$	0	0
$85$	−5.92943	−0.643137
$86$	0	0
$87$	−1.53701	−0.164785
$88$	0	0
$89$	11.6951	1.23968	0.619838	−	0.784730i	$-0.287198\pi$
$89$	11.6951	1.23968	0.619838	+	0.784730i	$0.287198\pi$
$90$	0	0
$91$	−12.4252	−1.30251
$92$	0	0
$93$	−1.47467	−0.152916
$94$	0	0
$95$	0.534554	0.0548441
$96$	0	0
$97$	4.20944	0.427404	0.213702	−	0.976899i	$-0.431448\pi$
$97$	4.20944	0.427404	0.213702	+	0.976899i	$0.431448\pi$
$98$	0	0
$99$	16.7893	1.68739
$100$	0	0
$101$	5.63117	0.560323	0.280161	−	0.959953i	$-0.409612\pi$
$101$	5.63117	0.560323	0.280161	+	0.959953i	$0.409612\pi$
$102$	0	0
$103$	−19.3367	−1.90531	−0.952653	−	0.304060i	$-0.901657\pi$
$103$	−19.3367	−1.90531	−0.952653	+	0.304060i	$0.901657\pi$
$104$	0	0
$105$	1.21335	0.118410
$106$	0	0
$107$	0.533744	0.0515990	0.0257995	−	0.999667i	$-0.491787\pi$
$107$	0.533744	0.0515990	0.0257995	+	0.999667i	$0.491787\pi$
$108$	0	0
$109$	−0.151594	−0.0145200	−0.00726001	−	0.999974i	$-0.502311\pi$
$109$	−0.151594	−0.0145200	−0.00726001	+	0.999974i	$0.502311\pi$
$110$	0	0
$111$	1.05215	0.0998659
$112$	0	0
$113$	18.6063	1.75033	0.875164	−	0.483825i	$-0.160753\pi$
$113$	18.6063	1.75033	0.875164	+	0.483825i	$0.160753\pi$
$114$	0	0
$115$	−14.9097	−1.39034
$116$	0	0
$117$	11.0027	1.01720
$118$	0	0
$119$	11.0194	1.01015
$120$	0	0
$121$	21.1891	1.92628
$122$	0	0
$123$	−1.51225	−0.136355
$124$	0	0
$125$	12.1675	1.08829
$126$	0	0
$127$	−10.9350	−0.970328	−0.485164	−	0.874423i	$-0.661240\pi$
$127$	−10.9350	−0.970328	−0.485164	+	0.874423i	$0.661240\pi$
$128$	0	0
$129$	0.151648	0.0133518
$130$	0	0
$131$	8.51858	0.744272	0.372136	−	0.928178i	$-0.378626\pi$
$131$	8.51858	0.744272	0.372136	+	0.928178i	$0.378626\pi$
$132$	0	0
$133$	−0.993430	−0.0861413
$134$	0	0
$135$	−2.16368	−0.186220
$136$	0	0
$137$	2.57273	0.219804	0.109902	−	0.993942i	$-0.464946\pi$
$137$	2.57273	0.219804	0.109902	+	0.993942i	$0.464946\pi$
$138$	0	0
$139$	−1.48109	−0.125624	−0.0628120	−	0.998025i	$-0.520007\pi$
$139$	−1.48109	−0.125624	−0.0628120	+	0.998025i	$0.520007\pi$
$140$	0	0
$141$	−1.56252	−0.131588
$142$	0	0
$143$	21.0948	1.76404
$144$	0	0
$145$	−13.6882	−1.13674
$146$	0	0
$147$	−0.841516	−0.0694071
$148$	0	0
$149$	18.8091	1.54090	0.770450	−	0.637500i	$-0.220031\pi$
$149$	18.8091	1.54090	0.770450	+	0.637500i	$0.220031\pi$
$150$	0	0
$151$	−12.1457	−0.988407	−0.494203	−	0.869346i	$-0.664540\pi$
$151$	−12.1457	−0.988407	−0.494203	+	0.869346i	$0.664540\pi$
$152$	0	0
$153$	−9.75788	−0.788878
$154$	0	0
$155$	−13.1330	−1.05487
$156$	0	0
$157$	6.19537	0.494444	0.247222	−	0.968959i	$-0.420482\pi$
$157$	6.19537	0.494444	0.247222	+	0.968959i	$0.420482\pi$
$158$	0	0
$159$	−1.77553	−0.140809
$160$	0	0
$161$	27.7086	2.18374
$162$	0	0
$163$	7.57212	0.593094	0.296547	−	0.955018i	$-0.404165\pi$
$163$	7.57212	0.593094	0.296547	+	0.955018i	$0.404165\pi$
$164$	0	0
$165$	−2.05995	−0.160367
$166$	0	0
$167$	9.61169	0.743775	0.371888	−	0.928278i	$-0.378711\pi$
$167$	9.61169	0.743775	0.371888	+	0.928278i	$0.378711\pi$
$168$	0	0
$169$	0.824263	0.0634049
$170$	0	0
$171$	0.879700	0.0672723
$172$	0	0
$173$	6.48244	0.492851	0.246426	−	0.969162i	$-0.420744\pi$
$173$	6.48244	0.492851	0.246426	+	0.969162i	$0.420744\pi$
$174$	0	0
$175$	−5.90331	−0.446249
$176$	0	0
$177$	2.36144	0.177496
$178$	0	0
$179$	−20.9179	−1.56348	−0.781740	−	0.623605i	$-0.785667\pi$
$179$	−20.9179	−1.56348	−0.781740	+	0.623605i	$0.785667\pi$
$180$	0	0
$181$	14.2301	1.05771	0.528857	−	0.848711i	$-0.322621\pi$
$181$	14.2301	1.05771	0.528857	+	0.848711i	$0.322621\pi$
$182$	0	0
$183$	−0.700239	−0.0517632
$184$	0	0
$185$	9.37020	0.688911
$186$	0	0
$187$	−18.7082	−1.36808
$188$	0	0
$189$	4.02104	0.292488
$190$	0	0
$191$	9.18371	0.664510	0.332255	−	0.943190i	$-0.392191\pi$
$191$	9.18371	0.664510	0.332255	+	0.943190i	$0.392191\pi$
$192$	0	0
$193$	−2.62839	−0.189195	−0.0945977	−	0.995516i	$-0.530156\pi$
$193$	−2.62839	−0.189195	−0.0945977	+	0.995516i	$0.530156\pi$
$194$	0	0
$195$	−1.34997	−0.0966732
$196$	0	0
$197$	−1.75695	−0.125178	−0.0625889	−	0.998039i	$-0.519936\pi$
$197$	−1.75695	−0.125178	−0.0625889	+	0.998039i	$0.519936\pi$
$198$	0	0
$199$	17.2691	1.22417	0.612087	−	0.790790i	$-0.290330\pi$
$199$	17.2691	1.22417	0.612087	+	0.790790i	$0.290330\pi$
$200$	0	0
$201$	−2.84420	−0.200614
$202$	0	0
$203$	25.4385	1.78544
$204$	0	0
$205$	−13.4677	−0.940627
$206$	0	0
$207$	−24.5364	−1.70540
$208$	0	0
$209$	1.68659	0.116664
$210$	0	0
$211$	0.0218577	0.00150475	0.000752373	−	1.00000i	$-0.499761\pi$
$211$	0.0218577	0.00150475	0.000752373		1.00000i	$0.499761\pi$
$212$	0	0
$213$	0.213904	0.0146565
$214$	0	0
$215$	1.35054	0.0921058
$216$	0	0
$217$	24.4068	1.65684
$218$	0	0
$219$	2.79332	0.188755
$220$	0	0
$221$	−12.2602	−0.824711
$222$	0	0
$223$	−13.3927	−0.896842	−0.448421	−	0.893823i	$-0.648013\pi$
$223$	−13.3927	−0.896842	−0.448421	+	0.893823i	$0.648013\pi$
$224$	0	0
$225$	5.22749	0.348499
$226$	0	0
$227$	17.8476	1.18459	0.592294	−	0.805722i	$-0.298222\pi$
$227$	17.8476	1.18459	0.592294	+	0.805722i	$0.298222\pi$
$228$	0	0
$229$	5.13644	0.339426	0.169713	−	0.985494i	$-0.445716\pi$
$229$	5.13644	0.339426	0.169713	+	0.985494i	$0.445716\pi$
$230$	0	0
$231$	3.82827	0.251882
$232$	0	0
$233$	−25.5471	−1.67365	−0.836823	−	0.547474i	$-0.815590\pi$
$233$	−25.5471	−1.67365	−0.836823	+	0.547474i	$0.815590\pi$
$234$	0	0
$235$	−13.9154	−0.907743
$236$	0	0
$237$	0.150685	0.00978802
$238$	0	0
$239$	−21.8897	−1.41593	−0.707964	−	0.706249i	$-0.750386\pi$
$239$	−21.8897	−1.41593	−0.707964	+	0.706249i	$0.750386\pi$
$240$	0	0
$241$	17.8301	1.14854	0.574269	−	0.818667i	$-0.305286\pi$
$241$	17.8301	1.14854	0.574269	+	0.818667i	$0.305286\pi$
$242$	0	0
$243$	−5.35323	−0.343410
$244$	0	0
$245$	−7.49433	−0.478795
$246$	0	0
$247$	1.10529	0.0703280
$248$	0	0
$249$	0.569374	0.0360826
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−47.0422	−2.95751
$254$	0	0
$255$	1.19723	0.0749737
$256$	0	0
$257$	−23.4810	−1.46471	−0.732353	−	0.680925i	$-0.761578\pi$
$257$	−23.4810	−1.46471	−0.732353	+	0.680925i	$0.761578\pi$
$258$	0	0
$259$	−17.4138	−1.08204
$260$	0	0
$261$	−22.5263	−1.39434
$262$	0	0
$263$	−26.9639	−1.66267	−0.831333	−	0.555775i	$-0.812422\pi$
$263$	−26.9639	−1.66267	−0.831333	+	0.555775i	$0.812422\pi$
$264$	0	0
$265$	−15.8124	−0.971351
$266$	0	0
$267$	−2.36140	−0.144515
$268$	0	0
$269$	12.5841	0.767264	0.383632	−	0.923486i	$-0.374673\pi$
$269$	12.5841	0.767264	0.383632	+	0.923486i	$0.374673\pi$
$270$	0	0
$271$	23.7129	1.44046	0.720228	−	0.693737i	$-0.244037\pi$
$271$	23.7129	1.44046	0.720228	+	0.693737i	$0.244037\pi$
$272$	0	0
$273$	2.50882	0.151841
$274$	0	0
$275$	10.0223	0.604369
$276$	0	0
$277$	−23.1666	−1.39194	−0.695972	−	0.718069i	$-0.745026\pi$
$277$	−23.1666	−1.39194	−0.695972	+	0.718069i	$0.745026\pi$
$278$	0	0
$279$	−21.6126	−1.29392
$280$	0	0
$281$	−11.1605	−0.665780	−0.332890	−	0.942966i	$-0.608024\pi$
$281$	−11.1605	−0.665780	−0.332890	+	0.942966i	$0.608024\pi$
$282$	0	0
$283$	−13.6610	−0.812064	−0.406032	−	0.913859i	$-0.633088\pi$
$283$	−13.6610	−0.812064	−0.406032	+	0.913859i	$0.633088\pi$
$284$	0	0
$285$	−0.107934	−0.00639346
$286$	0	0
$287$	25.0288	1.47740
$288$	0	0
$289$	−6.12690	−0.360406
$290$	0	0
$291$	−0.849944	−0.0498246
$292$	0	0
$293$	−6.77614	−0.395866	−0.197933	−	0.980216i	$-0.563423\pi$
$293$	−6.77614	−0.395866	−0.197933	+	0.980216i	$0.563423\pi$
$294$	0	0
$295$	21.0303	1.22443
$296$	0	0
$297$	−6.82670	−0.396125
$298$	0	0
$299$	−30.8286	−1.78286
$300$	0	0
$301$	−2.50987	−0.144667
$302$	0	0
$303$	−1.13701	−0.0653196
$304$	0	0
$305$	−6.23615	−0.357081
$306$	0	0
$307$	27.0975	1.54653	0.773267	−	0.634081i	$-0.218621\pi$
$307$	27.0975	1.54653	0.773267	+	0.634081i	$0.218621\pi$
$308$	0	0
$309$	3.90436	0.222111
$310$	0	0
$311$	−1.85540	−0.105210	−0.0526051	−	0.998615i	$-0.516752\pi$
$311$	−1.85540	−0.105210	−0.0526051	+	0.998615i	$0.516752\pi$
$312$	0	0
$313$	0.494028	0.0279241	0.0139621	−	0.999903i	$-0.495556\pi$
$313$	0.494028	0.0279241	0.0139621	+	0.999903i	$0.495556\pi$
$314$	0	0
$315$	17.7827	1.00194
$316$	0	0
$317$	22.0357	1.23765	0.618824	−	0.785530i	$-0.287610\pi$
$317$	22.0357	1.23765	0.618824	+	0.785530i	$0.287610\pi$
$318$	0	0
$319$	−43.1882	−2.41807
$320$	0	0
$321$	−0.107770	−0.00601515
$322$	0	0
$323$	−0.980240	−0.0545420
$324$	0	0
$325$	6.56803	0.364329
$326$	0	0
$327$	0.0306088	0.00169267
$328$	0	0
$329$	25.8608	1.42575
$330$	0	0
$331$	34.2083	1.88026	0.940129	−	0.340819i	$-0.110704\pi$
$331$	34.2083	1.88026	0.940129	+	0.340819i	$0.110704\pi$
$332$	0	0
$333$	15.4202	0.845024
$334$	0	0
$335$	−25.3297	−1.38391
$336$	0	0
$337$	20.0437	1.09185	0.545925	−	0.837834i	$-0.316178\pi$
$337$	20.0437	1.09185	0.545925	+	0.837834i	$0.316178\pi$
$338$	0	0
$339$	−3.75686	−0.204045
$340$	0	0
$341$	−41.4365	−2.24391
$342$	0	0
$343$	−9.46502	−0.511063
$344$	0	0
$345$	3.01048	0.162079
$346$	0	0
$347$	−12.0886	−0.648948	−0.324474	−	0.945895i	$-0.605187\pi$
$347$	−12.0886	−0.648948	−0.324474	+	0.945895i	$0.605187\pi$
$348$	0	0
$349$	−9.53974	−0.510650	−0.255325	−	0.966855i	$-0.582183\pi$
$349$	−9.53974	−0.510650	−0.255325	+	0.966855i	$0.582183\pi$
$350$	0	0
$351$	−4.47381	−0.238794
$352$	0	0
$353$	32.6432	1.73742	0.868712	−	0.495317i	$-0.164948\pi$
$353$	32.6432	1.73742	0.868712	+	0.495317i	$0.164948\pi$
$354$	0	0
$355$	1.90498	0.101106
$356$	0	0
$357$	−2.22497	−0.117758
$358$	0	0
$359$	15.4908	0.817574	0.408787	−	0.912630i	$-0.365952\pi$
$359$	15.4908	0.817574	0.408787	+	0.912630i	$0.365952\pi$
$360$	0	0
$361$	−18.9116	−0.995349
$362$	0	0
$363$	−4.27838	−0.224557
$364$	0	0
$365$	24.8766	1.30210
$366$	0	0
$367$	−5.91353	−0.308684	−0.154342	−	0.988018i	$-0.549326\pi$
$367$	−5.91353	−0.308684	−0.154342	+	0.988018i	$0.549326\pi$
$368$	0	0
$369$	−22.1634	−1.15378
$370$	0	0
$371$	29.3863	1.52566
$372$	0	0
$373$	−18.5340	−0.959652	−0.479826	−	0.877364i	$-0.659300\pi$
$373$	−18.5340	−0.959652	−0.479826	+	0.877364i	$0.659300\pi$
$374$	0	0
$375$	−2.45678	−0.126868
$376$	0	0
$377$	−28.3029	−1.45768
$378$	0	0
$379$	12.6334	0.648933	0.324467	−	0.945897i	$-0.394815\pi$
$379$	12.6334	0.648933	0.324467	+	0.945897i	$0.394815\pi$
$380$	0	0
$381$	2.20794	0.113116
$382$	0	0
$383$	−19.7360	−1.00846	−0.504231	−	0.863569i	$-0.668224\pi$
$383$	−19.7360	−1.00846	−0.504231	+	0.863569i	$0.668224\pi$
$384$	0	0
$385$	34.0936	1.73757
$386$	0	0
$387$	2.22254	0.112978
$388$	0	0
$389$	29.9438	1.51821	0.759106	−	0.650967i	$-0.225637\pi$
$389$	29.9438	1.51821	0.759106	+	0.650967i	$0.225637\pi$
$390$	0	0
$391$	27.3407	1.38268
$392$	0	0
$393$	−1.72002	−0.0867635
$394$	0	0
$395$	1.34196	0.0675213
$396$	0	0
$397$	9.08019	0.455721	0.227861	−	0.973694i	$-0.426827\pi$
$397$	9.08019	0.455721	0.227861	+	0.973694i	$0.426827\pi$
$398$	0	0
$399$	0.200587	0.0100419
$400$	0	0
$401$	−5.18938	−0.259145	−0.129573	−	0.991570i	$-0.541361\pi$
$401$	−5.18938	−0.259145	−0.129573	+	0.991570i	$0.541361\pi$
$402$	0	0
$403$	−27.1550	−1.35269
$404$	0	0
$405$	−15.5269	−0.771539
$406$	0	0
$407$	29.5643	1.46545
$408$	0	0
$409$	−16.7709	−0.829269	−0.414634	−	0.909988i	$-0.636091\pi$
$409$	−16.7709	−0.829269	−0.414634	+	0.909988i	$0.636091\pi$
$410$	0	0
$411$	−0.519471	−0.0256236
$412$	0	0
$413$	−39.0834	−1.92317
$414$	0	0
$415$	5.07070	0.248911
$416$	0	0
$417$	0.299052	0.0146446
$418$	0	0
$419$	35.9968	1.75856	0.879279	−	0.476307i	$-0.158025\pi$
$419$	35.9968	1.75856	0.879279	+	0.476307i	$0.158025\pi$
$420$	0	0
$421$	12.6445	0.616256	0.308128	−	0.951345i	$-0.400298\pi$
$421$	12.6445	0.616256	0.308128	+	0.951345i	$0.400298\pi$
$422$	0	0
$423$	−22.9002	−1.11345
$424$	0	0
$425$	−5.82493	−0.282551
$426$	0	0
$427$	11.5894	0.560852
$428$	0	0
$429$	−4.25933	−0.205643
$430$	0	0
$431$	−8.66730	−0.417489	−0.208744	−	0.977970i	$-0.566938\pi$
$431$	−8.66730	−0.417489	−0.208744	+	0.977970i	$0.566938\pi$
$432$	0	0
$433$	12.3222	0.592168	0.296084	−	0.955162i	$-0.404319\pi$
$433$	12.3222	0.592168	0.296084	+	0.955162i	$0.404319\pi$
$434$	0	0
$435$	2.76384	0.132516
$436$	0	0
$437$	−2.46484	−0.117909
$438$	0	0
$439$	6.56535	0.313347	0.156674	−	0.987650i	$-0.449923\pi$
$439$	6.56535	0.313347	0.156674	+	0.987650i	$0.449923\pi$
$440$	0	0
$441$	−12.3332	−0.587294
$442$	0	0
$443$	7.06939	0.335877	0.167938	−	0.985797i	$-0.446289\pi$
$443$	7.06939	0.335877	0.167938	+	0.985797i	$0.446289\pi$
$444$	0	0
$445$	−21.0300	−0.996918
$446$	0	0
$447$	−3.79782	−0.179631
$448$	0	0
$449$	17.0058	0.802554	0.401277	−	0.915957i	$-0.368566\pi$
$449$	17.0058	0.802554	0.401277	+	0.915957i	$0.368566\pi$
$450$	0	0
$451$	−42.4925	−2.00089
$452$	0	0
$453$	2.45239	0.115224
$454$	0	0
$455$	22.3429	1.04745
$456$	0	0
$457$	30.9648	1.44847	0.724237	−	0.689551i	$-0.242192\pi$
$457$	30.9648	1.44847	0.724237	+	0.689551i	$0.242192\pi$
$458$	0	0
$459$	3.96765	0.185194
$460$	0	0
$461$	−15.9416	−0.742473	−0.371237	−	0.928538i	$-0.621066\pi$
$461$	−15.9416	−0.742473	−0.371237	+	0.928538i	$0.621066\pi$
$462$	0	0
$463$	−18.7896	−0.873227	−0.436613	−	0.899649i	$-0.643822\pi$
$463$	−18.7896	−0.873227	−0.436613	+	0.899649i	$0.643822\pi$
$464$	0	0
$465$	2.65174	0.122972
$466$	0	0
$467$	−33.5619	−1.55306	−0.776531	−	0.630079i	$-0.783022\pi$
$467$	−33.5619	−1.55306	−0.776531	+	0.630079i	$0.783022\pi$
$468$	0	0
$469$	47.0734	2.17365
$470$	0	0
$471$	−1.25093	−0.0576399
$472$	0	0
$473$	4.26113	0.195927
$474$	0	0
$475$	0.525133	0.0240948
$476$	0	0
$477$	−26.0221	−1.19147
$478$	0	0
$479$	2.55330	0.116663	0.0583316	−	0.998297i	$-0.481422\pi$
$479$	2.55330	0.116663	0.0583316	+	0.998297i	$0.481422\pi$
$480$	0	0
$481$	19.3746	0.883407
$482$	0	0
$483$	−5.59475	−0.254570
$484$	0	0
$485$	−7.56938	−0.343708
$486$	0	0
$487$	3.10886	0.140876	0.0704380	−	0.997516i	$-0.477560\pi$
$487$	3.10886	0.140876	0.0704380	+	0.997516i	$0.477560\pi$
$488$	0	0
$489$	−1.52892	−0.0691399
$490$	0	0
$491$	−17.6594	−0.796956	−0.398478	−	0.917178i	$-0.630461\pi$
$491$	−17.6594	−0.796956	−0.398478	+	0.917178i	$0.630461\pi$
$492$	0	0
$493$	25.1008	1.13048
$494$	0	0
$495$	−30.1905	−1.35696
$496$	0	0
$497$	−3.54026	−0.158802
$498$	0	0
$499$	19.2196	0.860388	0.430194	−	0.902736i	$-0.358445\pi$
$499$	19.2196	0.860388	0.430194	+	0.902736i	$0.358445\pi$
$500$	0	0
$501$	−1.94073	−0.0867056
$502$	0	0
$503$	−0.483311	−0.0215498	−0.0107749	−	0.999942i	$-0.503430\pi$
$503$	−0.483311	−0.0215498	−0.0107749	+	0.999942i	$0.503430\pi$
$504$	0	0
$505$	−10.1259	−0.450598
$506$	0	0
$507$	−0.166430	−0.00739143
$508$	0	0
$509$	−36.2312	−1.60592	−0.802961	−	0.596032i	$-0.796743\pi$
$509$	−36.2312	−1.60592	−0.802961	+	0.596032i	$0.796743\pi$
$510$	0	0
$511$	−46.2314	−2.04515
$512$	0	0
$513$	−0.357694	−0.0157926
$514$	0	0
$515$	34.7712	1.53220
$516$	0	0
$517$	−43.9051	−1.93094
$518$	0	0
$519$	−1.30890	−0.0574541
$520$	0	0
$521$	−11.2503	−0.492882	−0.246441	−	0.969158i	$-0.579261\pi$
$521$	−11.2503	−0.492882	−0.246441	+	0.969158i	$0.579261\pi$
$522$	0	0
$523$	2.69662	0.117915	0.0589574	−	0.998261i	$-0.481222\pi$
$523$	2.69662	0.117915	0.0589574	+	0.998261i	$0.481222\pi$
$524$	0	0
$525$	1.19196	0.0520214
$526$	0	0
$527$	24.0827	1.04906
$528$	0	0
$529$	45.7488	1.98908
$530$	0	0
$531$	34.6090	1.50190
$532$	0	0
$533$	−27.8470	−1.20619
$534$	0	0
$535$	−0.959775	−0.0414947
$536$	0	0
$537$	4.22362	0.182263
$538$	0	0
$539$	−23.6456	−1.01849
$540$	0	0
$541$	4.54198	0.195275	0.0976375	−	0.995222i	$-0.468871\pi$
$541$	4.54198	0.195275	0.0976375	+	0.995222i	$0.468871\pi$
$542$	0	0
$543$	−2.87325	−0.123303
$544$	0	0
$545$	0.272594	0.0116767
$546$	0	0
$547$	23.4436	1.00238	0.501188	−	0.865339i	$-0.332897\pi$
$547$	23.4436	1.00238	0.501188	+	0.865339i	$0.332897\pi$
$548$	0	0
$549$	−10.2626	−0.437999
$550$	0	0
$551$	−2.26290	−0.0964029
$552$	0	0
$553$	−2.49393	−0.106053
$554$	0	0
$555$	−1.89197	−0.0803098
$556$	0	0
$557$	−13.4645	−0.570511	−0.285255	−	0.958452i	$-0.592078\pi$
$557$	−13.4645	−0.570511	−0.285255	+	0.958452i	$0.592078\pi$
$558$	0	0
$559$	2.79249	0.118110
$560$	0	0
$561$	3.77744	0.159484
$562$	0	0
$563$	−3.30829	−0.139428	−0.0697140	−	0.997567i	$-0.522209\pi$
$563$	−3.30829	−0.139428	−0.0697140	+	0.997567i	$0.522209\pi$
$564$	0	0
$565$	−33.4576	−1.40757
$566$	0	0
$567$	28.8557	1.21182
$568$	0	0
$569$	−22.3573	−0.937268	−0.468634	−	0.883392i	$-0.655254\pi$
$569$	−22.3573	−0.937268	−0.468634	+	0.883392i	$0.655254\pi$
$570$	0	0
$571$	−25.1726	−1.05344	−0.526721	−	0.850038i	$-0.676579\pi$
$571$	−25.1726	−1.05344	−0.526721	+	0.850038i	$0.676579\pi$
$572$	0	0
$573$	−1.85432	−0.0774653
$574$	0	0
$575$	−14.6469	−0.610819
$576$	0	0
$577$	−3.46194	−0.144122	−0.0720612	−	0.997400i	$-0.522958\pi$
$577$	−3.46194	−0.144122	−0.0720612	+	0.997400i	$0.522958\pi$
$578$	0	0
$579$	0.530708	0.0220555
$580$	0	0
$581$	−9.42352	−0.390954
$582$	0	0
$583$	−49.8904	−2.06625
$584$	0	0
$585$	−19.7850	−0.818009
$586$	0	0
$587$	−18.6098	−0.768107	−0.384053	−	0.923311i	$-0.625472\pi$
$587$	−18.6098	−0.768107	−0.384053	+	0.923311i	$0.625472\pi$
$588$	0	0
$589$	−2.17112	−0.0894596
$590$	0	0
$591$	0.354754	0.0145926
$592$	0	0
$593$	−4.00728	−0.164559	−0.0822797	−	0.996609i	$-0.526220\pi$
$593$	−4.00728	−0.164559	−0.0822797	+	0.996609i	$0.526220\pi$
$594$	0	0
$595$	−19.8150	−0.812337
$596$	0	0
$597$	−3.48687	−0.142708
$598$	0	0
$599$	−46.1066	−1.88386	−0.941932	−	0.335805i	$-0.890992\pi$
$599$	−46.1066	−1.88386	−0.941932	+	0.335805i	$0.890992\pi$
$600$	0	0
$601$	−15.4501	−0.630222	−0.315111	−	0.949055i	$-0.602042\pi$
$601$	−15.4501	−0.630222	−0.315111	+	0.949055i	$0.602042\pi$
$602$	0	0
$603$	−41.6843	−1.69752
$604$	0	0
$605$	−38.1021	−1.54907
$606$	0	0
$607$	36.9285	1.49888	0.749441	−	0.662071i	$-0.230322\pi$
$607$	36.9285	1.49888	0.749441	+	0.662071i	$0.230322\pi$
$608$	0	0
$609$	−5.13640	−0.208137
$610$	0	0
$611$	−28.7727	−1.16402
$612$	0	0
$613$	30.6685	1.23869	0.619345	−	0.785119i	$-0.287398\pi$
$613$	30.6685	1.23869	0.619345	+	0.785119i	$0.287398\pi$
$614$	0	0
$615$	2.71932	0.109654
$616$	0	0
$617$	30.4887	1.22743	0.613714	−	0.789528i	$-0.289675\pi$
$617$	30.4887	1.22743	0.613714	+	0.789528i	$0.289675\pi$
$618$	0	0
$619$	−11.4511	−0.460258	−0.230129	−	0.973160i	$-0.573915\pi$
$619$	−11.4511	−0.460258	−0.230129	+	0.973160i	$0.573915\pi$
$620$	0	0
$621$	9.97675	0.400353
$622$	0	0
$623$	39.0827	1.56582
$624$	0	0
$625$	−13.0470	−0.521879
$626$	0	0
$627$	−0.340546	−0.0136001
$628$	0	0
$629$	−17.1826	−0.685116
$630$	0	0
$631$	−15.2408	−0.606727	−0.303363	−	0.952875i	$-0.598110\pi$
$631$	−15.2408	−0.606727	−0.303363	+	0.952875i	$0.598110\pi$
$632$	0	0
$633$	−0.00441337	−0.000175416 0
$634$	0	0
$635$	19.6633	0.780315
$636$	0	0
$637$	−15.4959	−0.613970
$638$	0	0
$639$	3.13496	0.124017
$640$	0	0
$641$	44.4574	1.75596	0.877981	−	0.478695i	$-0.158890\pi$
$641$	44.4574	1.75596	0.877981	+	0.478695i	$0.158890\pi$
$642$	0	0
$643$	23.7397	0.936204	0.468102	−	0.883675i	$-0.344938\pi$
$643$	23.7397	0.936204	0.468102	+	0.883675i	$0.344938\pi$
$644$	0	0
$645$	−0.272692	−0.0107372
$646$	0	0
$647$	4.14223	0.162848	0.0814239	−	0.996680i	$-0.474053\pi$
$647$	4.14223	0.162848	0.0814239	+	0.996680i	$0.474053\pi$
$648$	0	0
$649$	66.3536	2.60461
$650$	0	0
$651$	−4.92807	−0.193146
$652$	0	0
$653$	21.5920	0.844959	0.422479	−	0.906373i	$-0.361160\pi$
$653$	21.5920	0.844959	0.422479	+	0.906373i	$0.361160\pi$
$654$	0	0
$655$	−15.3180	−0.598526
$656$	0	0
$657$	40.9387	1.59717
$658$	0	0
$659$	0.00140660	5.47932e−5 0	2.73966e−5	−	1.00000i	$-0.499991\pi$
$659$	0.00140660	5.47932e−5 0	2.73966e−5		1.00000i	$0.499991\pi$
$660$	0	0
$661$	−40.7801	−1.58616	−0.793081	−	0.609116i	$-0.791525\pi$
$661$	−40.7801	−1.58616	−0.793081	+	0.609116i	$0.791525\pi$
$662$	0	0
$663$	2.47551	0.0961407
$664$	0	0
$665$	1.78638	0.0692728
$666$	0	0
$667$	63.1165	2.44388
$668$	0	0
$669$	2.70417	0.104549
$670$	0	0
$671$	−19.6759	−0.759580
$672$	0	0
$673$	15.3615	0.592141	0.296071	−	0.955166i	$-0.404324\pi$
$673$	15.3615	0.592141	0.296071	+	0.955166i	$0.404324\pi$
$674$	0	0
$675$	−2.12555	−0.0818123
$676$	0	0
$677$	−27.9057	−1.07250	−0.536251	−	0.844059i	$-0.680160\pi$
$677$	−27.9057	−1.07250	−0.536251	+	0.844059i	$0.680160\pi$
$678$	0	0
$679$	14.0671	0.539848
$680$	0	0
$681$	−3.60369	−0.138094
$682$	0	0
$683$	17.6691	0.676091	0.338045	−	0.941130i	$-0.390234\pi$
$683$	17.6691	0.676091	0.338045	+	0.941130i	$0.390234\pi$
$684$	0	0
$685$	−4.62627	−0.176761
$686$	0	0
$687$	−1.03712	−0.0395686
$688$	0	0
$689$	−32.6952	−1.24559
$690$	0	0
$691$	−21.6744	−0.824532	−0.412266	−	0.911063i	$-0.635263\pi$
$691$	−21.6744	−0.824532	−0.412266	+	0.911063i	$0.635263\pi$
$692$	0	0
$693$	56.1068	2.13132
$694$	0	0
$695$	2.66328	0.101024
$696$	0	0
$697$	24.6965	0.935445
$698$	0	0
$699$	5.15831	0.195105
$700$	0	0
$701$	15.6555	0.591299	0.295649	−	0.955296i	$-0.404464\pi$
$701$	15.6555	0.591299	0.295649	+	0.955296i	$0.404464\pi$
$702$	0	0
$703$	1.54906	0.0584239
$704$	0	0
$705$	2.80972	0.105820
$706$	0	0
$707$	18.8183	0.707735
$708$	0	0
$709$	25.4117	0.954356	0.477178	−	0.878807i	$-0.341660\pi$
$709$	25.4117	0.954356	0.477178	+	0.878807i	$0.341660\pi$
$710$	0	0
$711$	2.20842	0.0828222
$712$	0	0
$713$	60.5566	2.26786
$714$	0	0
$715$	−37.9325	−1.41860
$716$	0	0
$717$	4.41984	0.165062
$718$	0	0
$719$	−10.0811	−0.375960	−0.187980	−	0.982173i	$-0.560194\pi$
$719$	−10.0811	−0.375960	−0.187980	+	0.982173i	$0.560194\pi$
$720$	0	0
$721$	−64.6197	−2.40656
$722$	0	0
$723$	−3.60015	−0.133891
$724$	0	0
$725$	−13.4470	−0.499408
$726$	0	0
$727$	−41.3278	−1.53276	−0.766381	−	0.642386i	$-0.777944\pi$
$727$	−41.3278	−1.53276	−0.766381	+	0.642386i	$0.777944\pi$
$728$	0	0
$729$	−24.8233	−0.919382
$730$	0	0
$731$	−2.47655	−0.0915984
$732$	0	0
$733$	−7.71188	−0.284845	−0.142422	−	0.989806i	$-0.545489\pi$
$733$	−7.71188	−0.284845	−0.142422	+	0.989806i	$0.545489\pi$
$734$	0	0
$735$	1.51321	0.0558155
$736$	0	0
$737$	−79.9187	−2.94384
$738$	0	0
$739$	8.81195	0.324153	0.162076	−	0.986778i	$-0.448181\pi$
$739$	8.81195	0.324153	0.162076	+	0.986778i	$0.448181\pi$
$740$	0	0
$741$	−0.223174	−0.00819849
$742$	0	0
$743$	10.2517	0.376100	0.188050	−	0.982159i	$-0.439783\pi$
$743$	10.2517	0.376100	0.188050	+	0.982159i	$0.439783\pi$
$744$	0	0
$745$	−33.8224	−1.23916
$746$	0	0
$747$	8.34469	0.305316
$748$	0	0
$749$	1.78367	0.0651739
$750$	0	0
$751$	14.9259	0.544653	0.272326	−	0.962205i	$-0.412207\pi$
$751$	14.9259	0.544653	0.272326	+	0.962205i	$0.412207\pi$
$752$	0	0
$753$	0.201914	0.00735815
$754$	0	0
$755$	21.8404	0.794853
$756$	0	0
$757$	−22.3520	−0.812396	−0.406198	−	0.913785i	$-0.633146\pi$
$757$	−22.3520	−0.812396	−0.406198	+	0.913785i	$0.633146\pi$
$758$	0	0
$759$	9.49846	0.344772
$760$	0	0
$761$	−0.735230	−0.0266521	−0.0133260	−	0.999911i	$-0.504242\pi$
$761$	−0.735230	−0.0266521	−0.0133260	+	0.999911i	$0.504242\pi$
$762$	0	0
$763$	−0.506597	−0.0183400
$764$	0	0
$765$	17.5466	0.634397
$766$	0	0
$767$	43.4842	1.57012
$768$	0	0
$769$	9.13226	0.329318	0.164659	−	0.986351i	$-0.447348\pi$
$769$	9.13226	0.329318	0.164659	+	0.986351i	$0.447348\pi$
$770$	0	0
$771$	4.74115	0.170748
$772$	0	0
$773$	20.2905	0.729797	0.364899	−	0.931047i	$-0.381104\pi$
$773$	20.2905	0.729797	0.364899	+	0.931047i	$0.381104\pi$
$774$	0	0
$775$	−12.9016	−0.463439
$776$	0	0
$777$	3.51609	0.126139
$778$	0	0
$779$	−2.22645	−0.0797710
$780$	0	0
$781$	6.01046	0.215071
$782$	0	0
$783$	9.15940	0.327330
$784$	0	0
$785$	−11.1405	−0.397620
$786$	0	0
$787$	−43.6653	−1.55650	−0.778250	−	0.627954i	$-0.783892\pi$
$787$	−43.6653	−1.55650	−0.778250	+	0.627954i	$0.783892\pi$
$788$	0	0
$789$	5.44439	0.193825
$790$	0	0
$791$	62.1786	2.21082
$792$	0	0
$793$	−12.8944	−0.457894
$794$	0	0
$795$	3.19275	0.113235
$796$	0	0
$797$	−43.9528	−1.55689	−0.778443	−	0.627715i	$-0.783990\pi$
$797$	−43.9528	−1.55689	−0.778443	+	0.627715i	$0.783990\pi$
$798$	0	0
$799$	25.5174	0.902742
$800$	0	0
$801$	−34.6084	−1.22283
$802$	0	0
$803$	78.4891	2.76982
$804$	0	0
$805$	−49.8254	−1.75611
$806$	0	0
$807$	−2.54090	−0.0894439
$808$	0	0
$809$	−23.2769	−0.818371	−0.409185	−	0.912451i	$-0.634187\pi$
$809$	−23.2769	−0.818371	−0.409185	+	0.912451i	$0.634187\pi$
$810$	0	0
$811$	15.8360	0.556076	0.278038	−	0.960570i	$-0.410316\pi$
$811$	15.8360	0.556076	0.278038	+	0.960570i	$0.410316\pi$
$812$	0	0
$813$	−4.78796	−0.167921
$814$	0	0
$815$	−13.6161	−0.476952
$816$	0	0
$817$	0.223268	0.00781114
$818$	0	0
$819$	36.7690	1.28481
$820$	0	0
$821$	54.6129	1.90600	0.953001	−	0.302967i	$-0.0979770\pi$
$821$	54.6129	1.90600	0.953001	+	0.302967i	$0.0979770\pi$
$822$	0	0
$823$	−9.89849	−0.345040	−0.172520	−	0.985006i	$-0.555191\pi$
$823$	−9.89849	−0.345040	−0.172520	+	0.985006i	$0.555191\pi$
$824$	0	0
$825$	−2.02365	−0.0704543
$826$	0	0
$827$	43.8078	1.52335	0.761673	−	0.647962i	$-0.224378\pi$
$827$	43.8078	1.52335	0.761673	+	0.647962i	$0.224378\pi$
$828$	0	0
$829$	29.4488	1.02280	0.511400	−	0.859343i	$-0.329127\pi$
$829$	29.4488	1.02280	0.511400	+	0.859343i	$0.329127\pi$
$830$	0	0
$831$	4.67765	0.162266
$832$	0	0
$833$	13.7427	0.476157
$834$	0	0
$835$	−17.2837	−0.598126
$836$	0	0
$837$	8.78791	0.303755
$838$	0	0
$839$	21.9325	0.757194	0.378597	−	0.925562i	$-0.376407\pi$
$839$	21.9325	0.757194	0.378597	+	0.925562i	$0.376407\pi$
$840$	0	0
$841$	28.9456	0.998126
$842$	0	0
$843$	2.25346	0.0776133
$844$	0	0
$845$	−1.48218	−0.0509887
$846$	0	0
$847$	70.8100	2.43306
$848$	0	0
$849$	2.75835	0.0946664
$850$	0	0
$851$	−43.2061	−1.48109
$852$	0	0
$853$	28.6166	0.979813	0.489907	−	0.871775i	$-0.337031\pi$
$853$	28.6166	0.979813	0.489907	+	0.871775i	$0.337031\pi$
$854$	0	0
$855$	−1.58187	−0.0540988
$856$	0	0
$857$	48.8480	1.66862	0.834308	−	0.551299i	$-0.185868\pi$
$857$	48.8480	1.66862	0.834308	+	0.551299i	$0.185868\pi$
$858$	0	0
$859$	−50.7577	−1.73183	−0.865914	−	0.500192i	$-0.833263\pi$
$859$	−50.7577	−1.73183	−0.865914	+	0.500192i	$0.833263\pi$
$860$	0	0
$861$	−5.05366	−0.172228
$862$	0	0
$863$	45.3849	1.54492	0.772460	−	0.635064i	$-0.219026\pi$
$863$	45.3849	1.54492	0.772460	+	0.635064i	$0.219026\pi$
$864$	0	0
$865$	−11.6567	−0.396339
$866$	0	0
$867$	1.23711	0.0420143
$868$	0	0
$869$	4.23406	0.143631
$870$	0	0
$871$	−52.3739	−1.77462
$872$	0	0
$873$	−12.4567	−0.421596
$874$	0	0
$875$	40.6614	1.37461
$876$	0	0
$877$	24.2677	0.819462	0.409731	−	0.912206i	$-0.365623\pi$
$877$	24.2677	0.819462	0.409731	+	0.912206i	$0.365623\pi$
$878$	0	0
$879$	1.36820	0.0461481
$880$	0	0
$881$	−48.7992	−1.64409	−0.822044	−	0.569424i	$-0.807166\pi$
$881$	−48.7992	−1.64409	−0.822044	+	0.569424i	$0.807166\pi$
$882$	0	0
$883$	27.5261	0.926327	0.463164	−	0.886273i	$-0.346714\pi$
$883$	27.5261	0.926327	0.463164	+	0.886273i	$0.346714\pi$
$884$	0	0
$885$	−4.24632	−0.142738
$886$	0	0
$887$	51.0022	1.71249	0.856244	−	0.516572i	$-0.172792\pi$
$887$	51.0022	1.71249	0.856244	+	0.516572i	$0.172792\pi$
$888$	0	0
$889$	−36.5428	−1.22561
$890$	0	0
$891$	−48.9896	−1.64121
$892$	0	0
$893$	−2.30047	−0.0769822
$894$	0	0
$895$	37.6145	1.25731
$896$	0	0
$897$	6.22472	0.207837
$898$	0	0
$899$	55.5955	1.85421
$900$	0	0
$901$	28.9961	0.966000
$902$	0	0
$903$	0.506778	0.0168645
$904$	0	0
$905$	−25.5884	−0.850588
$906$	0	0
$907$	−6.17559	−0.205057	−0.102528	−	0.994730i	$-0.532693\pi$
$907$	−6.17559	−0.205057	−0.102528	+	0.994730i	$0.532693\pi$
$908$	0	0
$909$	−16.6639	−0.552708
$910$	0	0
$911$	−17.6971	−0.586331	−0.293165	−	0.956062i	$-0.594709\pi$
$911$	−17.6971	−0.586331	−0.293165	+	0.956062i	$0.594709\pi$
$912$	0	0
$913$	15.9987	0.529481
$914$	0	0
$915$	1.25917	0.0416267
$916$	0	0
$917$	28.4675	0.940079
$918$	0	0
$919$	44.6548	1.47303	0.736514	−	0.676423i	$-0.236471\pi$
$919$	44.6548	1.47303	0.736514	+	0.676423i	$0.236471\pi$
$920$	0	0
$921$	−5.47135	−0.180287
$922$	0	0
$923$	3.93889	0.129650
$924$	0	0
$925$	9.20506	0.302660
$926$	0	0
$927$	57.2219	1.87941
$928$	0	0
$929$	−27.5676	−0.904462	−0.452231	−	0.891901i	$-0.649372\pi$
$929$	−27.5676	−0.904462	−0.452231	+	0.891901i	$0.649372\pi$
$930$	0	0
$931$	−1.23894	−0.0406048
$932$	0	0
$933$	0.374631	0.0122649
$934$	0	0
$935$	33.6409	1.10017
$936$	0	0
$937$	−1.22196	−0.0399196	−0.0199598	−	0.999801i	$-0.506354\pi$
$937$	−1.22196	−0.0399196	−0.0199598	+	0.999801i	$0.506354\pi$
$938$	0	0
$939$	−0.0997511	−0.00325526
$940$	0	0
$941$	−42.9102	−1.39883	−0.699416	−	0.714715i	$-0.746556\pi$
$941$	−42.9102	−1.39883	−0.699416	+	0.714715i	$0.746556\pi$
$942$	0	0
$943$	62.0998	2.02225
$944$	0	0
$945$	−7.23060	−0.235212
$946$	0	0
$947$	25.6450	0.833352	0.416676	−	0.909055i	$-0.363195\pi$
$947$	25.6450	0.833352	0.416676	+	0.909055i	$0.363195\pi$
$948$	0	0
$949$	51.4370	1.66972
$950$	0	0
$951$	−4.44931	−0.144279
$952$	0	0
$953$	37.8088	1.22475	0.612374	−	0.790568i	$-0.290215\pi$
$953$	37.8088	1.22475	0.612374	+	0.790568i	$0.290215\pi$
$954$	0	0
$955$	−16.5141	−0.534383
$956$	0	0
$957$	8.72029	0.281887
$958$	0	0
$959$	8.59759	0.277631
$960$	0	0
$961$	22.3406	0.720665
$962$	0	0
$963$	−1.57947	−0.0508978
$964$	0	0
$965$	4.72635	0.152146
$966$	0	0
$967$	4.36445	0.140351	0.0701757	−	0.997535i	$-0.477644\pi$
$967$	4.36445	0.140351	0.0701757	+	0.997535i	$0.477644\pi$
$968$	0	0
$969$	0.197924	0.00635824
$970$	0	0
$971$	−3.58064	−0.114908	−0.0574541	−	0.998348i	$-0.518298\pi$
$971$	−3.58064	−0.114908	−0.0574541	+	0.998348i	$0.518298\pi$
$972$	0	0
$973$	−4.94951	−0.158674
$974$	0	0
$975$	−1.32618	−0.0424716
$976$	0	0
$977$	29.3579	0.939244	0.469622	−	0.882868i	$-0.344390\pi$
$977$	29.3579	0.939244	0.469622	+	0.882868i	$0.344390\pi$
$978$	0	0
$979$	−66.3526	−2.12064
$980$	0	0
$981$	0.448600	0.0143227
$982$	0	0
$983$	−17.1708	−0.547663	−0.273832	−	0.961778i	$-0.588291\pi$
$983$	−17.1708	−0.547663	−0.273832	+	0.961778i	$0.588291\pi$
$984$	0	0
$985$	3.15934	0.100665
$986$	0	0
$987$	−5.22166	−0.166207
$988$	0	0
$989$	−6.22734	−0.198018
$990$	0	0
$991$	22.9166	0.727971	0.363985	−	0.931405i	$-0.381416\pi$
$991$	22.9166	0.727971	0.363985	+	0.931405i	$0.381416\pi$
$992$	0	0
$993$	−6.90713	−0.219191
$994$	0	0
$995$	−31.0532	−0.984452
$996$	0	0
$997$	−32.3122	−1.02334	−0.511669	−	0.859183i	$-0.670973\pi$
$997$	−32.3122	−1.02334	−0.511669	+	0.859183i	$0.670973\pi$
$998$	0	0
$999$	−6.27002	−0.198375

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2008.2.a.d.1.11	✓	23
4.3	odd	2		4016.2.a.m.1.13		23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.11	✓	23	1.1	even	1	trivial
4016.2.a.m.1.13		23	4.3	odd	2

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:	\( N \)	\(=\)	\( 2008 = 2^{3} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2008.a (trivial)

\(f(q)\)	\(=\)	\(q-0.201914 q^{3} -1.79819 q^{5} +3.34181 q^{7} -2.95923 q^{9} +O(q^{10})\)	\(q-0.201914 q^{3} -1.79819 q^{5} +3.34181 q^{7} -2.95923 q^{9} -5.67355 q^{11} -3.71810 q^{13} +0.363080 q^{15} +3.29744 q^{17} -0.297273 q^{19} -0.674758 q^{21} +8.29149 q^{23} -1.76650 q^{25} +1.20325 q^{27} +7.61220 q^{29} +7.30347 q^{31} +1.14557 q^{33} -6.00922 q^{35} -5.21090 q^{37} +0.750736 q^{39} +7.48959 q^{41} -0.751052 q^{43} +5.32127 q^{45} +7.73856 q^{47} +4.16770 q^{49} -0.665799 q^{51} +8.79352 q^{53} +10.2021 q^{55} +0.0600236 q^{57} -11.6953 q^{59} +3.46801 q^{61} -9.88919 q^{63} +6.68586 q^{65} +14.0862 q^{67} -1.67417 q^{69} -1.05938 q^{71} -13.8342 q^{73} +0.356681 q^{75} -18.9599 q^{77} -0.746282 q^{79} +8.63474 q^{81} -2.81989 q^{83} -5.92943 q^{85} -1.53701 q^{87} +11.6951 q^{89} -12.4252 q^{91} -1.47467 q^{93} +0.534554 q^{95} +4.20944 q^{97} +16.7893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) \) 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9} + 8 q^{11} + 8 q^{13} + 7 q^{15} + 19 q^{17} - 9 q^{19} + 9 q^{21} + 21 q^{23} + 65 q^{25} + 5 q^{27} + 10 q^{29} - 9 q^{31} + 34 q^{33} + 12 q^{35} + 11 q^{37} - 9 q^{39} + 35 q^{41} - 9 q^{43} + 29 q^{45} + 37 q^{47} + 77 q^{49} - 17 q^{51} + 38 q^{53} - 20 q^{55} + 51 q^{57} + 17 q^{59} + 22 q^{63} + 41 q^{65} + 9 q^{67} + 8 q^{69} + 13 q^{71} + 41 q^{73} + 25 q^{75} + 36 q^{77} - 36 q^{79} + 127 q^{81} + 29 q^{83} + 34 q^{85} + 10 q^{87} + 36 q^{89} - 6 q^{91} + 36 q^{93} + 25 q^{95} + 40 q^{97} + 19 q^{99}+O(q^{100}) \) 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9 + 8 * q^11 + 8 * q^13 + 7 * q^15 + 19 * q^17 - 9 * q^19 + 9 * q^21 + 21 * q^23 + 65 * q^25 + 5 * q^27 + 10 * q^29 - 9 * q^31 + 34 * q^33 + 12 * q^35 + 11 * q^37 - 9 * q^39 + 35 * q^41 - 9 * q^43 + 29 * q^45 + 37 * q^47 + 77 * q^49 - 17 * q^51 + 38 * q^53 - 20 * q^55 + 51 * q^57 + 17 * q^59 + 22 * q^63 + 41 * q^65 + 9 * q^67 + 8 * q^69 + 13 * q^71 + 41 * q^73 + 25 * q^75 + 36 * q^77 - 36 * q^79 + 127 * q^81 + 29 * q^83 + 34 * q^85 + 10 * q^87 + 36 * q^89 - 6 * q^91 + 36 * q^93 + 25 * q^95 + 40 * q^97 + 19 * q^99

Introduction

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