LMFDB - Embedded newform 2576.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2576,2,Mod(1,2576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2576 = 2^{4} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2576.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:	$20.5694635607$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	2576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{3} +2.73205 q^{5} -1.00000 q^{7} -2.46410 q^{9} +O(q^{10})$	$q-0.732051 q^{3} +2.73205 q^{5} -1.00000 q^{7} -2.46410 q^{9} +3.46410 q^{11} -1.46410 q^{13} -2.00000 q^{15} -0.732051 q^{17} +2.00000 q^{19} +0.732051 q^{21} -1.00000 q^{23} +2.46410 q^{25} +4.00000 q^{27} +8.00000 q^{29} -0.196152 q^{31} -2.53590 q^{33} -2.73205 q^{35} +0.535898 q^{37} +1.07180 q^{39} -2.00000 q^{41} -6.73205 q^{45} +8.19615 q^{47} +1.00000 q^{49} +0.535898 q^{51} +3.46410 q^{53} +9.46410 q^{55} -1.46410 q^{57} +11.6603 q^{59} +8.19615 q^{61} +2.46410 q^{63} -4.00000 q^{65} +4.53590 q^{67} +0.732051 q^{69} -8.39230 q^{71} -7.46410 q^{73} -1.80385 q^{75} -3.46410 q^{77} -0.928203 q^{79} +4.46410 q^{81} +6.39230 q^{83} -2.00000 q^{85} -5.85641 q^{87} +14.1962 q^{89} +1.46410 q^{91} +0.143594 q^{93} +5.46410 q^{95} -8.73205 q^{97} -8.53590 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{13} - 4 q^{15} + 2 q^{17} + 4 q^{19} - 2 q^{21} - 2 q^{23} - 2 q^{25} + 8 q^{27} + 16 q^{29} + 10 q^{31} - 12 q^{33} - 2 q^{35} + 8 q^{37} + 16 q^{39} - 4 q^{41} - 10 q^{45} + 6 q^{47} + 2 q^{49} + 8 q^{51} + 12 q^{55} + 4 q^{57} + 6 q^{59} + 6 q^{61} - 2 q^{63} - 8 q^{65} + 16 q^{67} - 2 q^{69} + 4 q^{71} - 8 q^{73} - 14 q^{75} + 12 q^{79} + 2 q^{81} - 8 q^{83} - 4 q^{85} + 16 q^{87} + 18 q^{89} - 4 q^{91} + 28 q^{93} + 4 q^{95} - 14 q^{97} - 24 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^13 - 4 * q^15 + 2 * q^17 + 4 * q^19 - 2 * q^21 - 2 * q^23 - 2 * q^25 + 8 * q^27 + 16 * q^29 + 10 * q^31 - 12 * q^33 - 2 * q^35 + 8 * q^37 + 16 * q^39 - 4 * q^41 - 10 * q^45 + 6 * q^47 + 2 * q^49 + 8 * q^51 + 12 * q^55 + 4 * q^57 + 6 * q^59 + 6 * q^61 - 2 * q^63 - 8 * q^65 + 16 * q^67 - 2 * q^69 + 4 * q^71 - 8 * q^73 - 14 * q^75 + 12 * q^79 + 2 * q^81 - 8 * q^83 - 4 * q^85 + 16 * q^87 + 18 * q^89 - 4 * q^91 + 28 * q^93 + 4 * q^95 - 14 * q^97 - 24 * 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
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	0	0
$5$	2.73205	1.22181	0.610905	−	0.791704i	$-0.290806\pi$
$5$	2.73205	1.22181	0.610905	+	0.791704i	$0.290806\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	−1.46410	−0.406069	−0.203034	−	0.979172i	$-0.565080\pi$
$13$	−1.46410	−0.406069	−0.203034	+	0.979172i	$0.565080\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	−0.732051	−0.177548	−0.0887742	−	0.996052i	$-0.528295\pi$
$17$	−0.732051	−0.177548	−0.0887742	+	0.996052i	$0.528295\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0.732051	0.159747
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	2.46410	0.492820
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	−0.196152	−0.0352300	−0.0176150	−	0.999845i	$-0.505607\pi$
$31$	−0.196152	−0.0352300	−0.0176150	+	0.999845i	$0.505607\pi$
$32$	0	0
$33$	−2.53590	−0.441443
$34$	0	0
$35$	−2.73205	−0.461801
$36$	0	0
$37$	0.535898	0.0881012	0.0440506	−	0.999029i	$-0.485974\pi$
$37$	0.535898	0.0881012	0.0440506	+	0.999029i	$0.485974\pi$
$38$	0	0
$39$	1.07180	0.171625
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−6.73205	−1.00355
$46$	0	0
$47$	8.19615	1.19553	0.597766	−	0.801671i	$-0.296055\pi$
$47$	8.19615	1.19553	0.597766	+	0.801671i	$0.296055\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.535898	0.0750408
$52$	0	0
$53$	3.46410	0.475831	0.237915	−	0.971286i	$-0.423536\pi$
$53$	3.46410	0.475831	0.237915	+	0.971286i	$0.423536\pi$
$54$	0	0
$55$	9.46410	1.27614
$56$	0	0
$57$	−1.46410	−0.193925
$58$	0	0
$59$	11.6603	1.51804	0.759018	−	0.651070i	$-0.225679\pi$
$59$	11.6603	1.51804	0.759018	+	0.651070i	$0.225679\pi$
$60$	0	0
$61$	8.19615	1.04941	0.524705	−	0.851284i	$-0.324176\pi$
$61$	8.19615	1.04941	0.524705	+	0.851284i	$0.324176\pi$
$62$	0	0
$63$	2.46410	0.310448
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	4.53590	0.554148	0.277074	−	0.960849i	$-0.410635\pi$
$67$	4.53590	0.554148	0.277074	+	0.960849i	$0.410635\pi$
$68$	0	0
$69$	0.732051	0.0881286
$70$	0	0
$71$	−8.39230	−0.995983	−0.497992	−	0.867182i	$-0.665929\pi$
$71$	−8.39230	−0.995983	−0.497992	+	0.867182i	$0.665929\pi$
$72$	0	0
$73$	−7.46410	−0.873607	−0.436804	−	0.899557i	$-0.643889\pi$
$73$	−7.46410	−0.873607	−0.436804	+	0.899557i	$0.643889\pi$
$74$	0	0
$75$	−1.80385	−0.208290
$76$	0	0
$77$	−3.46410	−0.394771
$78$	0	0
$79$	−0.928203	−0.104431	−0.0522155	−	0.998636i	$-0.516628\pi$
$79$	−0.928203	−0.104431	−0.0522155	+	0.998636i	$0.516628\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	6.39230	0.701647	0.350823	−	0.936442i	$-0.385902\pi$
$83$	6.39230	0.701647	0.350823	+	0.936442i	$0.385902\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−5.85641	−0.627873
$88$	0	0
$89$	14.1962	1.50479	0.752395	−	0.658713i	$-0.228899\pi$
$89$	14.1962	1.50479	0.752395	+	0.658713i	$0.228899\pi$
$90$	0	0
$91$	1.46410	0.153480
$92$	0	0
$93$	0.143594	0.0148900
$94$	0	0
$95$	5.46410	0.560605
$96$	0	0
$97$	−8.73205	−0.886605	−0.443303	−	0.896372i	$-0.646193\pi$
$97$	−8.73205	−0.886605	−0.443303	+	0.896372i	$0.646193\pi$
$98$	0	0
$99$	−8.53590	−0.857890
$100$	0	0
$101$	−1.46410	−0.145684	−0.0728418	−	0.997344i	$-0.523207\pi$
$101$	−1.46410	−0.145684	−0.0728418	+	0.997344i	$0.523207\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	−6.92820	−0.669775	−0.334887	−	0.942258i	$-0.608698\pi$
$107$	−6.92820	−0.669775	−0.334887	+	0.942258i	$0.608698\pi$
$108$	0	0
$109$	−8.92820	−0.855167	−0.427583	−	0.903976i	$-0.640635\pi$
$109$	−8.92820	−0.855167	−0.427583	+	0.903976i	$0.640635\pi$
$110$	0	0
$111$	−0.392305	−0.0372359
$112$	0	0
$113$	19.4641	1.83103	0.915514	−	0.402285i	$-0.131784\pi$
$113$	19.4641	1.83103	0.915514	+	0.402285i	$0.131784\pi$
$114$	0	0
$115$	−2.73205	−0.254765
$116$	0	0
$117$	3.60770	0.333532
$118$	0	0
$119$	0.732051	0.0671070
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.46410	0.132014
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	15.3205	1.35948	0.679738	−	0.733455i	$-0.262094\pi$
$127$	15.3205	1.35948	0.679738	+	0.733455i	$0.262094\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.26795	0.285522	0.142761	−	0.989757i	$-0.454402\pi$
$131$	3.26795	0.285522	0.142761	+	0.989757i	$0.454402\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	0	0
$135$	10.9282	0.940550
$136$	0	0
$137$	17.3205	1.47979	0.739895	−	0.672722i	$-0.234875\pi$
$137$	17.3205	1.47979	0.739895	+	0.672722i	$0.234875\pi$
$138$	0	0
$139$	1.80385	0.153000	0.0765002	−	0.997070i	$-0.475625\pi$
$139$	1.80385	0.153000	0.0765002	+	0.997070i	$0.475625\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	−5.07180	−0.424125
$144$	0	0
$145$	21.8564	1.81508
$146$	0	0
$147$	−0.732051	−0.0603785
$148$	0	0
$149$	22.7846	1.86659	0.933294	−	0.359113i	$-0.116921\pi$
$149$	22.7846	1.86659	0.933294	+	0.359113i	$0.116921\pi$
$150$	0	0
$151$	17.4641	1.42121	0.710604	−	0.703592i	$-0.248422\pi$
$151$	17.4641	1.42121	0.710604	+	0.703592i	$0.248422\pi$
$152$	0	0
$153$	1.80385	0.145832
$154$	0	0
$155$	−0.535898	−0.0430444
$156$	0	0
$157$	−8.19615	−0.654124	−0.327062	−	0.945003i	$-0.606059\pi$
$157$	−8.19615	−0.654124	−0.327062	+	0.945003i	$0.606059\pi$
$158$	0	0
$159$	−2.53590	−0.201110
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−12.3923	−0.970640	−0.485320	−	0.874337i	$-0.661297\pi$
$163$	−12.3923	−0.970640	−0.485320	+	0.874337i	$0.661297\pi$
$164$	0	0
$165$	−6.92820	−0.539360
$166$	0	0
$167$	−22.0526	−1.70648	−0.853239	−	0.521520i	$-0.825365\pi$
$167$	−22.0526	−1.70648	−0.853239	+	0.521520i	$0.825365\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	−4.92820	−0.376869
$172$	0	0
$173$	6.92820	0.526742	0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820	0.526742	0.263371	+	0.964695i	$0.415166\pi$
$174$	0	0
$175$	−2.46410	−0.186269
$176$	0	0
$177$	−8.53590	−0.641597
$178$	0	0
$179$	13.8564	1.03568	0.517838	−	0.855479i	$-0.326737\pi$
$179$	13.8564	1.03568	0.517838	+	0.855479i	$0.326737\pi$
$180$	0	0
$181$	14.0526	1.04452	0.522259	−	0.852787i	$-0.325089\pi$
$181$	14.0526	1.04452	0.522259	+	0.852787i	$0.325089\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	1.46410	0.107643
$186$	0	0
$187$	−2.53590	−0.185443
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	4.92820	0.354740	0.177370	−	0.984144i	$-0.443241\pi$
$193$	4.92820	0.354740	0.177370	+	0.984144i	$0.443241\pi$
$194$	0	0
$195$	2.92820	0.209693
$196$	0	0
$197$	8.92820	0.636108	0.318054	−	0.948073i	$-0.396971\pi$
$197$	8.92820	0.636108	0.318054	+	0.948073i	$0.396971\pi$
$198$	0	0
$199$	24.7846	1.75693	0.878467	−	0.477803i	$-0.158567\pi$
$199$	24.7846	1.75693	0.878467	+	0.477803i	$0.158567\pi$
$200$	0	0
$201$	−3.32051	−0.234211
$202$	0	0
$203$	−8.00000	−0.561490
$204$	0	0
$205$	−5.46410	−0.381629
$206$	0	0
$207$	2.46410	0.171267
$208$	0	0
$209$	6.92820	0.479234
$210$	0	0
$211$	−27.7128	−1.90783	−0.953914	−	0.300079i	$-0.902987\pi$
$211$	−27.7128	−1.90783	−0.953914	+	0.300079i	$0.902987\pi$
$212$	0	0
$213$	6.14359	0.420952
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.196152	0.0133157
$218$	0	0
$219$	5.46410	0.369230
$220$	0	0
$221$	1.07180	0.0720969
$222$	0	0
$223$	21.2679	1.42421	0.712104	−	0.702074i	$-0.247743\pi$
$223$	21.2679	1.42421	0.712104	+	0.702074i	$0.247743\pi$
$224$	0	0
$225$	−6.07180	−0.404786
$226$	0	0
$227$	0.928203	0.0616070	0.0308035	−	0.999525i	$-0.490193\pi$
$227$	0.928203	0.0616070	0.0308035	+	0.999525i	$0.490193\pi$
$228$	0	0
$229$	0.196152	0.0129621	0.00648106	−	0.999979i	$-0.497937\pi$
$229$	0.196152	0.0129621	0.00648106	+	0.999979i	$0.497937\pi$
$230$	0	0
$231$	2.53590	0.166850
$232$	0	0
$233$	16.3923	1.07390	0.536948	−	0.843615i	$-0.319577\pi$
$233$	16.3923	1.07390	0.536948	+	0.843615i	$0.319577\pi$
$234$	0	0
$235$	22.3923	1.46071
$236$	0	0
$237$	0.679492	0.0441377
$238$	0	0
$239$	4.39230	0.284115	0.142057	−	0.989858i	$-0.454628\pi$
$239$	4.39230	0.284115	0.142057	+	0.989858i	$0.454628\pi$
$240$	0	0
$241$	21.1244	1.36074	0.680370	−	0.732869i	$-0.261819\pi$
$241$	21.1244	1.36074	0.680370	+	0.732869i	$0.261819\pi$
$242$	0	0
$243$	−15.2679	−0.979439
$244$	0	0
$245$	2.73205	0.174544
$246$	0	0
$247$	−2.92820	−0.186317
$248$	0	0
$249$	−4.67949	−0.296551
$250$	0	0
$251$	7.85641	0.495892	0.247946	−	0.968774i	$-0.420244\pi$
$251$	7.85641	0.495892	0.247946	+	0.968774i	$0.420244\pi$
$252$	0	0
$253$	−3.46410	−0.217786
$254$	0	0
$255$	1.46410	0.0916856
$256$	0	0
$257$	−23.8564	−1.48812	−0.744061	−	0.668112i	$-0.767103\pi$
$257$	−23.8564	−1.48812	−0.744061	+	0.668112i	$0.767103\pi$
$258$	0	0
$259$	−0.535898	−0.0332991
$260$	0	0
$261$	−19.7128	−1.22019
$262$	0	0
$263$	8.92820	0.550537	0.275268	−	0.961367i	$-0.411233\pi$
$263$	8.92820	0.550537	0.275268	+	0.961367i	$0.411233\pi$
$264$	0	0
$265$	9.46410	0.581375
$266$	0	0
$267$	−10.3923	−0.635999
$268$	0	0
$269$	1.85641	0.113187	0.0565935	−	0.998397i	$-0.481976\pi$
$269$	1.85641	0.113187	0.0565935	+	0.998397i	$0.481976\pi$
$270$	0	0
$271$	10.7321	0.651926	0.325963	−	0.945383i	$-0.394312\pi$
$271$	10.7321	0.651926	0.325963	+	0.945383i	$0.394312\pi$
$272$	0	0
$273$	−1.07180	−0.0648681
$274$	0	0
$275$	8.53590	0.514734
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	0	0
$279$	0.483340	0.0289368
$280$	0	0
$281$	−19.4641	−1.16113	−0.580565	−	0.814214i	$-0.697168\pi$
$281$	−19.4641	−1.16113	−0.580565	+	0.814214i	$0.697168\pi$
$282$	0	0
$283$	−11.0718	−0.658150	−0.329075	−	0.944304i	$-0.606737\pi$
$283$	−11.0718	−0.658150	−0.329075	+	0.944304i	$0.606737\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	2.00000	0.118056
$288$	0	0
$289$	−16.4641	−0.968477
$290$	0	0
$291$	6.39230	0.374724
$292$	0	0
$293$	26.7321	1.56170	0.780852	−	0.624717i	$-0.214786\pi$
$293$	26.7321	1.56170	0.780852	+	0.624717i	$0.214786\pi$
$294$	0	0
$295$	31.8564	1.85475
$296$	0	0
$297$	13.8564	0.804030
$298$	0	0
$299$	1.46410	0.0846712
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.07180	0.0615731
$304$	0	0
$305$	22.3923	1.28218
$306$	0	0
$307$	−8.05256	−0.459584	−0.229792	−	0.973240i	$-0.573805\pi$
$307$	−8.05256	−0.459584	−0.229792	+	0.973240i	$0.573805\pi$
$308$	0	0
$309$	11.7128	0.666319
$310$	0	0
$311$	21.2679	1.20599	0.602997	−	0.797743i	$-0.293973\pi$
$311$	21.2679	1.20599	0.602997	+	0.797743i	$0.293973\pi$
$312$	0	0
$313$	5.12436	0.289646	0.144823	−	0.989458i	$-0.453739\pi$
$313$	5.12436	0.289646	0.144823	+	0.989458i	$0.453739\pi$
$314$	0	0
$315$	6.73205	0.379308
$316$	0	0
$317$	−29.8564	−1.67690	−0.838451	−	0.544976i	$-0.816539\pi$
$317$	−29.8564	−1.67690	−0.838451	+	0.544976i	$0.816539\pi$
$318$	0	0
$319$	27.7128	1.55162
$320$	0	0
$321$	5.07180	0.283080
$322$	0	0
$323$	−1.46410	−0.0814648
$324$	0	0
$325$	−3.60770	−0.200119
$326$	0	0
$327$	6.53590	0.361436
$328$	0	0
$329$	−8.19615	−0.451869
$330$	0	0
$331$	0.392305	0.0215630	0.0107815	−	0.999942i	$-0.496568\pi$
$331$	0.392305	0.0215630	0.0107815	+	0.999942i	$0.496568\pi$
$332$	0	0
$333$	−1.32051	−0.0723634
$334$	0	0
$335$	12.3923	0.677064
$336$	0	0
$337$	−31.8564	−1.73533	−0.867665	−	0.497150i	$-0.834380\pi$
$337$	−31.8564	−1.73533	−0.867665	+	0.497150i	$0.834380\pi$
$338$	0	0
$339$	−14.2487	−0.773884
$340$	0	0
$341$	−0.679492	−0.0367966
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	2.00000	0.107676
$346$	0	0
$347$	−33.4641	−1.79645	−0.898224	−	0.439539i	$-0.855142\pi$
$347$	−33.4641	−1.79645	−0.898224	+	0.439539i	$0.855142\pi$
$348$	0	0
$349$	−28.7846	−1.54080	−0.770402	−	0.637558i	$-0.779945\pi$
$349$	−28.7846	−1.54080	−0.770402	+	0.637558i	$0.779945\pi$
$350$	0	0
$351$	−5.85641	−0.312592
$352$	0	0
$353$	26.3923	1.40472	0.702360	−	0.711822i	$-0.252130\pi$
$353$	26.3923	1.40472	0.702360	+	0.711822i	$0.252130\pi$
$354$	0	0
$355$	−22.9282	−1.21690
$356$	0	0
$357$	−0.535898	−0.0283628
$358$	0	0
$359$	0.928203	0.0489887	0.0244943	−	0.999700i	$-0.492202\pi$
$359$	0.928203	0.0489887	0.0244943	+	0.999700i	$0.492202\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−0.732051	−0.0384227
$364$	0	0
$365$	−20.3923	−1.06738
$366$	0	0
$367$	26.2487	1.37017	0.685086	−	0.728462i	$-0.259765\pi$
$367$	26.2487	1.37017	0.685086	+	0.728462i	$0.259765\pi$
$368$	0	0
$369$	4.92820	0.256552
$370$	0	0
$371$	−3.46410	−0.179847
$372$	0	0
$373$	8.92820	0.462285	0.231142	−	0.972920i	$-0.425754\pi$
$373$	8.92820	0.462285	0.231142	+	0.972920i	$0.425754\pi$
$374$	0	0
$375$	5.07180	0.261906
$376$	0	0
$377$	−11.7128	−0.603241
$378$	0	0
$379$	0.535898	0.0275273	0.0137636	−	0.999905i	$-0.495619\pi$
$379$	0.535898	0.0275273	0.0137636	+	0.999905i	$0.495619\pi$
$380$	0	0
$381$	−11.2154	−0.574582
$382$	0	0
$383$	18.5359	0.947140	0.473570	−	0.880756i	$-0.342965\pi$
$383$	18.5359	0.947140	0.473570	+	0.880756i	$0.342965\pi$
$384$	0	0
$385$	−9.46410	−0.482335
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.07180	0.358554	0.179277	−	0.983799i	$-0.442624\pi$
$389$	7.07180	0.358554	0.179277	+	0.983799i	$0.442624\pi$
$390$	0	0
$391$	0.732051	0.0370214
$392$	0	0
$393$	−2.39230	−0.120676
$394$	0	0
$395$	−2.53590	−0.127595
$396$	0	0
$397$	−31.3205	−1.57193	−0.785966	−	0.618270i	$-0.787834\pi$
$397$	−31.3205	−1.57193	−0.785966	+	0.618270i	$0.787834\pi$
$398$	0	0
$399$	1.46410	0.0732968
$400$	0	0
$401$	−29.7128	−1.48379	−0.741894	−	0.670518i	$-0.766072\pi$
$401$	−29.7128	−1.48379	−0.741894	+	0.670518i	$0.766072\pi$
$402$	0	0
$403$	0.287187	0.0143058
$404$	0	0
$405$	12.1962	0.606032
$406$	0	0
$407$	1.85641	0.0920187
$408$	0	0
$409$	7.46410	0.369076	0.184538	−	0.982825i	$-0.440921\pi$
$409$	7.46410	0.369076	0.184538	+	0.982825i	$0.440921\pi$
$410$	0	0
$411$	−12.6795	−0.625433
$412$	0	0
$413$	−11.6603	−0.573764
$414$	0	0
$415$	17.4641	0.857279
$416$	0	0
$417$	−1.32051	−0.0646656
$418$	0	0
$419$	11.0718	0.540893	0.270446	−	0.962735i	$-0.412829\pi$
$419$	11.0718	0.540893	0.270446	+	0.962735i	$0.412829\pi$
$420$	0	0
$421$	32.5359	1.58570	0.792851	−	0.609415i	$-0.208596\pi$
$421$	32.5359	1.58570	0.792851	+	0.609415i	$0.208596\pi$
$422$	0	0
$423$	−20.1962	−0.981971
$424$	0	0
$425$	−1.80385	−0.0874995
$426$	0	0
$427$	−8.19615	−0.396640
$428$	0	0
$429$	3.71281	0.179256
$430$	0	0
$431$	−27.7128	−1.33488	−0.667440	−	0.744664i	$-0.732610\pi$
$431$	−27.7128	−1.33488	−0.667440	+	0.744664i	$0.732610\pi$
$432$	0	0
$433$	1.41154	0.0678344	0.0339172	−	0.999425i	$-0.489202\pi$
$433$	1.41154	0.0678344	0.0339172	+	0.999425i	$0.489202\pi$
$434$	0	0
$435$	−16.0000	−0.767141
$436$	0	0
$437$	−2.00000	−0.0956730
$438$	0	0
$439$	−18.7321	−0.894032	−0.447016	−	0.894526i	$-0.647513\pi$
$439$	−18.7321	−0.894032	−0.447016	+	0.894526i	$0.647513\pi$
$440$	0	0
$441$	−2.46410	−0.117338
$442$	0	0
$443$	−34.9282	−1.65949	−0.829745	−	0.558143i	$-0.811514\pi$
$443$	−34.9282	−1.65949	−0.829745	+	0.558143i	$0.811514\pi$
$444$	0	0
$445$	38.7846	1.83857
$446$	0	0
$447$	−16.6795	−0.788913
$448$	0	0
$449$	−23.3205	−1.10056	−0.550281	−	0.834979i	$-0.685480\pi$
$449$	−23.3205	−1.10056	−0.550281	+	0.834979i	$0.685480\pi$
$450$	0	0
$451$	−6.92820	−0.326236
$452$	0	0
$453$	−12.7846	−0.600673
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	−6.39230	−0.299019	−0.149510	−	0.988760i	$-0.547770\pi$
$457$	−6.39230	−0.299019	−0.149510	+	0.988760i	$0.547770\pi$
$458$	0	0
$459$	−2.92820	−0.136677
$460$	0	0
$461$	35.7128	1.66331	0.831656	−	0.555292i	$-0.187393\pi$
$461$	35.7128	1.66331	0.831656	+	0.555292i	$0.187393\pi$
$462$	0	0
$463$	−10.9282	−0.507877	−0.253938	−	0.967220i	$-0.581726\pi$
$463$	−10.9282	−0.507877	−0.253938	+	0.967220i	$0.581726\pi$
$464$	0	0
$465$	0.392305	0.0181927
$466$	0	0
$467$	−23.8564	−1.10394	−0.551971	−	0.833863i	$-0.686124\pi$
$467$	−23.8564	−1.10394	−0.551971	+	0.833863i	$0.686124\pi$
$468$	0	0
$469$	−4.53590	−0.209448
$470$	0	0
$471$	6.00000	0.276465
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.92820	0.226121
$476$	0	0
$477$	−8.53590	−0.390832
$478$	0	0
$479$	−9.07180	−0.414501	−0.207250	−	0.978288i	$-0.566452\pi$
$479$	−9.07180	−0.414501	−0.207250	+	0.978288i	$0.566452\pi$
$480$	0	0
$481$	−0.784610	−0.0357751
$482$	0	0
$483$	−0.732051	−0.0333095
$484$	0	0
$485$	−23.8564	−1.08326
$486$	0	0
$487$	−8.78461	−0.398069	−0.199034	−	0.979993i	$-0.563781\pi$
$487$	−8.78461	−0.398069	−0.199034	+	0.979993i	$0.563781\pi$
$488$	0	0
$489$	9.07180	0.410241
$490$	0	0
$491$	−12.7846	−0.576961	−0.288481	−	0.957486i	$-0.593150\pi$
$491$	−12.7846	−0.576961	−0.288481	+	0.957486i	$0.593150\pi$
$492$	0	0
$493$	−5.85641	−0.263759
$494$	0	0
$495$	−23.3205	−1.04818
$496$	0	0
$497$	8.39230	0.376446
$498$	0	0
$499$	1.85641	0.0831042	0.0415521	−	0.999136i	$-0.486770\pi$
$499$	1.85641	0.0831042	0.0415521	+	0.999136i	$0.486770\pi$
$500$	0	0
$501$	16.1436	0.721243
$502$	0	0
$503$	−42.6410	−1.90127	−0.950634	−	0.310313i	$-0.899566\pi$
$503$	−42.6410	−1.90127	−0.950634	+	0.310313i	$0.899566\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	0	0
$507$	7.94744	0.352958
$508$	0	0
$509$	−40.0000	−1.77297	−0.886484	−	0.462758i	$-0.846860\pi$
$509$	−40.0000	−1.77297	−0.886484	+	0.462758i	$0.846860\pi$
$510$	0	0
$511$	7.46410	0.330192
$512$	0	0
$513$	8.00000	0.353209
$514$	0	0
$515$	−43.7128	−1.92622
$516$	0	0
$517$	28.3923	1.24869
$518$	0	0
$519$	−5.07180	−0.222627
$520$	0	0
$521$	−11.6603	−0.510845	−0.255423	−	0.966830i	$-0.582215\pi$
$521$	−11.6603	−0.510845	−0.255423	+	0.966830i	$0.582215\pi$
$522$	0	0
$523$	12.9282	0.565311	0.282655	−	0.959222i	$-0.408785\pi$
$523$	12.9282	0.565311	0.282655	+	0.959222i	$0.408785\pi$
$524$	0	0
$525$	1.80385	0.0787264
$526$	0	0
$527$	0.143594	0.00625503
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−28.7321	−1.24686
$532$	0	0
$533$	2.92820	0.126835
$534$	0	0
$535$	−18.9282	−0.818338
$536$	0	0
$537$	−10.1436	−0.437728
$538$	0	0
$539$	3.46410	0.149209
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	0	0
$543$	−10.2872	−0.441465
$544$	0	0
$545$	−24.3923	−1.04485
$546$	0	0
$547$	−44.3923	−1.89808	−0.949039	−	0.315159i	$-0.897942\pi$
$547$	−44.3923	−1.89808	−0.949039	+	0.315159i	$0.897942\pi$
$548$	0	0
$549$	−20.1962	−0.861951
$550$	0	0
$551$	16.0000	0.681623
$552$	0	0
$553$	0.928203	0.0394712
$554$	0	0
$555$	−1.07180	−0.0454952
$556$	0	0
$557$	−31.8564	−1.34980	−0.674900	−	0.737910i	$-0.735813\pi$
$557$	−31.8564	−1.34980	−0.674900	+	0.737910i	$0.735813\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.85641	0.0783775
$562$	0	0
$563$	21.7128	0.915086	0.457543	−	0.889188i	$-0.348730\pi$
$563$	21.7128	0.915086	0.457543	+	0.889188i	$0.348730\pi$
$564$	0	0
$565$	53.1769	2.23717
$566$	0	0
$567$	−4.46410	−0.187475
$568$	0	0
$569$	−19.8564	−0.832424	−0.416212	−	0.909268i	$-0.636643\pi$
$569$	−19.8564	−0.832424	−0.416212	+	0.909268i	$0.636643\pi$
$570$	0	0
$571$	−13.0718	−0.547038	−0.273519	−	0.961867i	$-0.588188\pi$
$571$	−13.0718	−0.547038	−0.273519	+	0.961867i	$0.588188\pi$
$572$	0	0
$573$	11.7128	0.489310
$574$	0	0
$575$	−2.46410	−0.102760
$576$	0	0
$577$	17.3205	0.721062	0.360531	−	0.932747i	$-0.382595\pi$
$577$	17.3205	0.721062	0.360531	+	0.932747i	$0.382595\pi$
$578$	0	0
$579$	−3.60770	−0.149931
$580$	0	0
$581$	−6.39230	−0.265197
$582$	0	0
$583$	12.0000	0.496989
$584$	0	0
$585$	9.85641	0.407512
$586$	0	0
$587$	−19.2679	−0.795273	−0.397637	−	0.917543i	$-0.630170\pi$
$587$	−19.2679	−0.795273	−0.397637	+	0.917543i	$0.630170\pi$
$588$	0	0
$589$	−0.392305	−0.0161646
$590$	0	0
$591$	−6.53590	−0.268851
$592$	0	0
$593$	−26.7846	−1.09991	−0.549956	−	0.835194i	$-0.685356\pi$
$593$	−26.7846	−1.09991	−0.549956	+	0.835194i	$0.685356\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	0	0
$597$	−18.1436	−0.742568
$598$	0	0
$599$	20.7846	0.849236	0.424618	−	0.905373i	$-0.360408\pi$
$599$	20.7846	0.849236	0.424618	+	0.905373i	$0.360408\pi$
$600$	0	0
$601$	−7.46410	−0.304467	−0.152234	−	0.988345i	$-0.548647\pi$
$601$	−7.46410	−0.304467	−0.152234	+	0.988345i	$0.548647\pi$
$602$	0	0
$603$	−11.1769	−0.455159
$604$	0	0
$605$	2.73205	0.111074
$606$	0	0
$607$	20.1962	0.819737	0.409868	−	0.912145i	$-0.365575\pi$
$607$	20.1962	0.819737	0.409868	+	0.912145i	$0.365575\pi$
$608$	0	0
$609$	5.85641	0.237314
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	2.78461	0.112469	0.0562347	−	0.998418i	$-0.482091\pi$
$613$	2.78461	0.112469	0.0562347	+	0.998418i	$0.482091\pi$
$614$	0	0
$615$	4.00000	0.161296
$616$	0	0
$617$	−33.7128	−1.35723	−0.678613	−	0.734496i	$-0.737419\pi$
$617$	−33.7128	−1.35723	−0.678613	+	0.734496i	$0.737419\pi$
$618$	0	0
$619$	16.2487	0.653091	0.326545	−	0.945182i	$-0.394115\pi$
$619$	16.2487	0.653091	0.326545	+	0.945182i	$0.394115\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	−14.1962	−0.568757
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	−5.07180	−0.202548
$628$	0	0
$629$	−0.392305	−0.0156422
$630$	0	0
$631$	10.7846	0.429329	0.214664	−	0.976688i	$-0.431134\pi$
$631$	10.7846	0.429329	0.214664	+	0.976688i	$0.431134\pi$
$632$	0	0
$633$	20.2872	0.806343
$634$	0	0
$635$	41.8564	1.66102
$636$	0	0
$637$	−1.46410	−0.0580098
$638$	0	0
$639$	20.6795	0.818068
$640$	0	0
$641$	−9.71281	−0.383633	−0.191817	−	0.981431i	$-0.561438\pi$
$641$	−9.71281	−0.383633	−0.191817	+	0.981431i	$0.561438\pi$
$642$	0	0
$643$	−1.21539	−0.0479303	−0.0239652	−	0.999713i	$-0.507629\pi$
$643$	−1.21539	−0.0479303	−0.0239652	+	0.999713i	$0.507629\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	35.5167	1.39630	0.698152	−	0.715950i	$-0.254006\pi$
$647$	35.5167	1.39630	0.698152	+	0.715950i	$0.254006\pi$
$648$	0	0
$649$	40.3923	1.58554
$650$	0	0
$651$	−0.143594	−0.00562787
$652$	0	0
$653$	29.7128	1.16275	0.581376	−	0.813635i	$-0.302515\pi$
$653$	29.7128	1.16275	0.581376	+	0.813635i	$0.302515\pi$
$654$	0	0
$655$	8.92820	0.348854
$656$	0	0
$657$	18.3923	0.717552
$658$	0	0
$659$	35.7128	1.39117	0.695587	−	0.718442i	$-0.255144\pi$
$659$	35.7128	1.39117	0.695587	+	0.718442i	$0.255144\pi$
$660$	0	0
$661$	−30.4449	−1.18417	−0.592084	−	0.805876i	$-0.701695\pi$
$661$	−30.4449	−1.18417	−0.592084	+	0.805876i	$0.701695\pi$
$662$	0	0
$663$	−0.784610	−0.0304717
$664$	0	0
$665$	−5.46410	−0.211889
$666$	0	0
$667$	−8.00000	−0.309761
$668$	0	0
$669$	−15.5692	−0.601941
$670$	0	0
$671$	28.3923	1.09607
$672$	0	0
$673$	3.32051	0.127996	0.0639981	−	0.997950i	$-0.479615\pi$
$673$	3.32051	0.127996	0.0639981	+	0.997950i	$0.479615\pi$
$674$	0	0
$675$	9.85641	0.379373
$676$	0	0
$677$	−35.8038	−1.37605	−0.688027	−	0.725685i	$-0.741523\pi$
$677$	−35.8038	−1.37605	−0.688027	+	0.725685i	$0.741523\pi$
$678$	0	0
$679$	8.73205	0.335105
$680$	0	0
$681$	−0.679492	−0.0260382
$682$	0	0
$683$	−10.9282	−0.418156	−0.209078	−	0.977899i	$-0.567046\pi$
$683$	−10.9282	−0.418156	−0.209078	+	0.977899i	$0.567046\pi$
$684$	0	0
$685$	47.3205	1.80802
$686$	0	0
$687$	−0.143594	−0.00547844
$688$	0	0
$689$	−5.07180	−0.193220
$690$	0	0
$691$	8.73205	0.332183	0.166091	−	0.986110i	$-0.446885\pi$
$691$	8.73205	0.332183	0.166091	+	0.986110i	$0.446885\pi$
$692$	0	0
$693$	8.53590	0.324252
$694$	0	0
$695$	4.92820	0.186937
$696$	0	0
$697$	1.46410	0.0554568
$698$	0	0
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−3.75129	−0.141684	−0.0708421	−	0.997488i	$-0.522569\pi$
$701$	−3.75129	−0.141684	−0.0708421	+	0.997488i	$0.522569\pi$
$702$	0	0
$703$	1.07180	0.0404236
$704$	0	0
$705$	−16.3923	−0.617370
$706$	0	0
$707$	1.46410	0.0550632
$708$	0	0
$709$	−0.535898	−0.0201261	−0.0100630	−	0.999949i	$-0.503203\pi$
$709$	−0.535898	−0.0201261	−0.0100630	+	0.999949i	$0.503203\pi$
$710$	0	0
$711$	2.28719	0.0857762
$712$	0	0
$713$	0.196152	0.00734597
$714$	0	0
$715$	−13.8564	−0.518200
$716$	0	0
$717$	−3.21539	−0.120081
$718$	0	0
$719$	15.8038	0.589384	0.294692	−	0.955592i	$-0.404783\pi$
$719$	15.8038	0.589384	0.294692	+	0.955592i	$0.404783\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	−15.4641	−0.575116
$724$	0	0
$725$	19.7128	0.732115
$726$	0	0
$727$	33.0718	1.22657	0.613283	−	0.789864i	$-0.289849\pi$
$727$	33.0718	1.22657	0.613283	+	0.789864i	$0.289849\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	0	0
$732$	0	0
$733$	13.6603	0.504553	0.252276	−	0.967655i	$-0.418821\pi$
$733$	13.6603	0.504553	0.252276	+	0.967655i	$0.418821\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	15.7128	0.578789
$738$	0	0
$739$	19.3205	0.710716	0.355358	−	0.934730i	$-0.384359\pi$
$739$	19.3205	0.710716	0.355358	+	0.934730i	$0.384359\pi$
$740$	0	0
$741$	2.14359	0.0787469
$742$	0	0
$743$	12.9282	0.474290	0.237145	−	0.971474i	$-0.423788\pi$
$743$	12.9282	0.474290	0.237145	+	0.971474i	$0.423788\pi$
$744$	0	0
$745$	62.2487	2.28062
$746$	0	0
$747$	−15.7513	−0.576310
$748$	0	0
$749$	6.92820	0.253151
$750$	0	0
$751$	−48.6410	−1.77494	−0.887468	−	0.460869i	$-0.847538\pi$
$751$	−48.6410	−1.77494	−0.887468	+	0.460869i	$0.847538\pi$
$752$	0	0
$753$	−5.75129	−0.209589
$754$	0	0
$755$	47.7128	1.73645
$756$	0	0
$757$	5.21539	0.189557	0.0947783	−	0.995498i	$-0.469786\pi$
$757$	5.21539	0.189557	0.0947783	+	0.995498i	$0.469786\pi$
$758$	0	0
$759$	2.53590	0.0920473
$760$	0	0
$761$	−41.3205	−1.49787	−0.748934	−	0.662645i	$-0.769434\pi$
$761$	−41.3205	−1.49787	−0.748934	+	0.662645i	$0.769434\pi$
$762$	0	0
$763$	8.92820	0.323223
$764$	0	0
$765$	4.92820	0.178180
$766$	0	0
$767$	−17.0718	−0.616427
$768$	0	0
$769$	21.1244	0.761764	0.380882	−	0.924624i	$-0.375620\pi$
$769$	21.1244	0.761764	0.380882	+	0.924624i	$0.375620\pi$
$770$	0	0
$771$	17.4641	0.628954
$772$	0	0
$773$	9.66025	0.347455	0.173728	−	0.984794i	$-0.444419\pi$
$773$	9.66025	0.347455	0.173728	+	0.984794i	$0.444419\pi$
$774$	0	0
$775$	−0.483340	−0.0173621
$776$	0	0
$777$	0.392305	0.0140739
$778$	0	0
$779$	−4.00000	−0.143315
$780$	0	0
$781$	−29.0718	−1.04027
$782$	0	0
$783$	32.0000	1.14359
$784$	0	0
$785$	−22.3923	−0.799216
$786$	0	0
$787$	−16.2487	−0.579204	−0.289602	−	0.957147i	$-0.593523\pi$
$787$	−16.2487	−0.579204	−0.289602	+	0.957147i	$0.593523\pi$
$788$	0	0
$789$	−6.53590	−0.232684
$790$	0	0
$791$	−19.4641	−0.692064
$792$	0	0
$793$	−12.0000	−0.426132
$794$	0	0
$795$	−6.92820	−0.245718
$796$	0	0
$797$	49.7654	1.76278	0.881390	−	0.472389i	$-0.156608\pi$
$797$	49.7654	1.76278	0.881390	+	0.472389i	$0.156608\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	0	0
$801$	−34.9808	−1.23598
$802$	0	0
$803$	−25.8564	−0.912453
$804$	0	0
$805$	2.73205	0.0962921
$806$	0	0
$807$	−1.35898	−0.0478385
$808$	0	0
$809$	−30.7846	−1.08233	−0.541165	−	0.840917i	$-0.682016\pi$
$809$	−30.7846	−1.08233	−0.541165	+	0.840917i	$0.682016\pi$
$810$	0	0
$811$	49.9090	1.75254	0.876270	−	0.481820i	$-0.160024\pi$
$811$	49.9090	1.75254	0.876270	+	0.481820i	$0.160024\pi$
$812$	0	0
$813$	−7.85641	−0.275536
$814$	0	0
$815$	−33.8564	−1.18594
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−3.60770	−0.126063
$820$	0	0
$821$	−0.287187	−0.0100229	−0.00501145	−	0.999987i	$-0.501595\pi$
$821$	−0.287187	−0.0100229	−0.00501145	+	0.999987i	$0.501595\pi$
$822$	0	0
$823$	−25.4641	−0.887623	−0.443811	−	0.896120i	$-0.646374\pi$
$823$	−25.4641	−0.887623	−0.443811	+	0.896120i	$0.646374\pi$
$824$	0	0
$825$	−6.24871	−0.217552
$826$	0	0
$827$	−9.85641	−0.342741	−0.171370	−	0.985207i	$-0.554819\pi$
$827$	−9.85641	−0.342741	−0.171370	+	0.985207i	$0.554819\pi$
$828$	0	0
$829$	−14.9282	−0.518478	−0.259239	−	0.965813i	$-0.583472\pi$
$829$	−14.9282	−0.518478	−0.259239	+	0.965813i	$0.583472\pi$
$830$	0	0
$831$	5.85641	0.203156
$832$	0	0
$833$	−0.732051	−0.0253641
$834$	0	0
$835$	−60.2487	−2.08499
$836$	0	0
$837$	−0.784610	−0.0271201
$838$	0	0
$839$	−15.3205	−0.528923	−0.264461	−	0.964396i	$-0.585194\pi$
$839$	−15.3205	−0.528923	−0.264461	+	0.964396i	$0.585194\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	14.2487	0.490752
$844$	0	0
$845$	−29.6603	−1.02034
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	8.10512	0.278167
$850$	0	0
$851$	−0.535898	−0.0183704
$852$	0	0
$853$	2.53590	0.0868275	0.0434138	−	0.999057i	$-0.486177\pi$
$853$	2.53590	0.0868275	0.0434138	+	0.999057i	$0.486177\pi$
$854$	0	0
$855$	−13.4641	−0.460463
$856$	0	0
$857$	52.9282	1.80799	0.903996	−	0.427540i	$-0.140620\pi$
$857$	52.9282	1.80799	0.903996	+	0.427540i	$0.140620\pi$
$858$	0	0
$859$	−1.12436	−0.0383625	−0.0191813	−	0.999816i	$-0.506106\pi$
$859$	−1.12436	−0.0383625	−0.0191813	+	0.999816i	$0.506106\pi$
$860$	0	0
$861$	−1.46410	−0.0498964
$862$	0	0
$863$	−21.4641	−0.730647	−0.365323	−	0.930881i	$-0.619042\pi$
$863$	−21.4641	−0.730647	−0.365323	+	0.930881i	$0.619042\pi$
$864$	0	0
$865$	18.9282	0.643578
$866$	0	0
$867$	12.0526	0.409326
$868$	0	0
$869$	−3.21539	−0.109075
$870$	0	0
$871$	−6.64102	−0.225022
$872$	0	0
$873$	21.5167	0.728229
$874$	0	0
$875$	6.92820	0.234216
$876$	0	0
$877$	50.9282	1.71972	0.859862	−	0.510527i	$-0.170550\pi$
$877$	50.9282	1.71972	0.859862	+	0.510527i	$0.170550\pi$
$878$	0	0
$879$	−19.5692	−0.660053
$880$	0	0
$881$	−36.3397	−1.22432	−0.612159	−	0.790735i	$-0.709699\pi$
$881$	−36.3397	−1.22432	−0.612159	+	0.790735i	$0.709699\pi$
$882$	0	0
$883$	40.0000	1.34611	0.673054	−	0.739594i	$-0.264982\pi$
$883$	40.0000	1.34611	0.673054	+	0.739594i	$0.264982\pi$
$884$	0	0
$885$	−23.3205	−0.783910
$886$	0	0
$887$	−31.5167	−1.05823	−0.529113	−	0.848551i	$-0.677475\pi$
$887$	−31.5167	−1.05823	−0.529113	+	0.848551i	$0.677475\pi$
$888$	0	0
$889$	−15.3205	−0.513833
$890$	0	0
$891$	15.4641	0.518067
$892$	0	0
$893$	16.3923	0.548548
$894$	0	0
$895$	37.8564	1.26540
$896$	0	0
$897$	−1.07180	−0.0357863
$898$	0	0
$899$	−1.56922	−0.0523364
$900$	0	0
$901$	−2.53590	−0.0844830
$902$	0	0
$903$	0	0
$904$	0	0
$905$	38.3923	1.27620
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	3.60770	0.119660
$910$	0	0
$911$	20.9282	0.693382	0.346691	−	0.937979i	$-0.387305\pi$
$911$	20.9282	0.693382	0.346691	+	0.937979i	$0.387305\pi$
$912$	0	0
$913$	22.1436	0.732846
$914$	0	0
$915$	−16.3923	−0.541913
$916$	0	0
$917$	−3.26795	−0.107917
$918$	0	0
$919$	8.14359	0.268632	0.134316	−	0.990939i	$-0.457116\pi$
$919$	8.14359	0.268632	0.134316	+	0.990939i	$0.457116\pi$
$920$	0	0
$921$	5.89488	0.194243
$922$	0	0
$923$	12.2872	0.404438
$924$	0	0
$925$	1.32051	0.0434180
$926$	0	0
$927$	39.4256	1.29491
$928$	0	0
$929$	−11.8564	−0.388996	−0.194498	−	0.980903i	$-0.562308\pi$
$929$	−11.8564	−0.388996	−0.194498	+	0.980903i	$0.562308\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	−15.5692	−0.509713
$934$	0	0
$935$	−6.92820	−0.226576
$936$	0	0
$937$	−2.98076	−0.0973773	−0.0486886	−	0.998814i	$-0.515504\pi$
$937$	−2.98076	−0.0973773	−0.0486886	+	0.998814i	$0.515504\pi$
$938$	0	0
$939$	−3.75129	−0.122419
$940$	0	0
$941$	−45.3731	−1.47912	−0.739560	−	0.673091i	$-0.764966\pi$
$941$	−45.3731	−1.47912	−0.739560	+	0.673091i	$0.764966\pi$
$942$	0	0
$943$	2.00000	0.0651290
$944$	0	0
$945$	−10.9282	−0.355494
$946$	0	0
$947$	−12.6795	−0.412028	−0.206014	−	0.978549i	$-0.566049\pi$
$947$	−12.6795	−0.412028	−0.206014	+	0.978549i	$0.566049\pi$
$948$	0	0
$949$	10.9282	0.354744
$950$	0	0
$951$	21.8564	0.708743
$952$	0	0
$953$	−44.9282	−1.45537	−0.727684	−	0.685913i	$-0.759403\pi$
$953$	−44.9282	−1.45537	−0.727684	+	0.685913i	$0.759403\pi$
$954$	0	0
$955$	−43.7128	−1.41451
$956$	0	0
$957$	−20.2872	−0.655792
$958$	0	0
$959$	−17.3205	−0.559308
$960$	0	0
$961$	−30.9615	−0.998759
$962$	0	0
$963$	17.0718	0.550131
$964$	0	0
$965$	13.4641	0.433425
$966$	0	0
$967$	17.8564	0.574223	0.287112	−	0.957897i	$-0.407305\pi$
$967$	17.8564	0.574223	0.287112	+	0.957897i	$0.407305\pi$
$968$	0	0
$969$	1.07180	0.0344311
$970$	0	0
$971$	34.1051	1.09449	0.547243	−	0.836974i	$-0.315677\pi$
$971$	34.1051	1.09449	0.547243	+	0.836974i	$0.315677\pi$
$972$	0	0
$973$	−1.80385	−0.0578287
$974$	0	0
$975$	2.64102	0.0845802
$976$	0	0
$977$	−27.4641	−0.878654	−0.439327	−	0.898327i	$-0.644783\pi$
$977$	−27.4641	−0.878654	−0.439327	+	0.898327i	$0.644783\pi$
$978$	0	0
$979$	49.1769	1.57170
$980$	0	0
$981$	22.0000	0.702406
$982$	0	0
$983$	3.60770	0.115068	0.0575338	−	0.998344i	$-0.481676\pi$
$983$	3.60770	0.115068	0.0575338	+	0.998344i	$0.481676\pi$
$984$	0	0
$985$	24.3923	0.777203
$986$	0	0
$987$	6.00000	0.190982
$988$	0	0
$989$	0	0
$990$	0	0
$991$	26.5359	0.842941	0.421470	−	0.906842i	$-0.361514\pi$
$991$	26.5359	0.842941	0.421470	+	0.906842i	$0.361514\pi$
$992$	0	0
$993$	−0.287187	−0.00911361
$994$	0	0
$995$	67.7128	2.14664
$996$	0	0
$997$	3.21539	0.101832	0.0509162	−	0.998703i	$-0.483786\pi$
$997$	3.21539	0.101832	0.0509162	+	0.998703i	$0.483786\pi$
$998$	0	0
$999$	2.14359	0.0678203

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.a.u.1.1		2
4.3	odd	2		322.2.a.f.1.2	✓	2
12.11	even	2		2898.2.a.w.1.1		2
20.19	odd	2		8050.2.a.x.1.1		2
28.27	even	2		2254.2.a.o.1.1		2
92.91	even	2		7406.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.a.f.1.2	✓	2	4.3	odd	2
2254.2.a.o.1.1		2	28.27	even	2
2576.2.a.u.1.1		2	1.1	even	1	trivial
2898.2.a.w.1.1		2	12.11	even	2
7406.2.a.s.1.2		2	92.91	even	2
8050.2.a.x.1.1		2	20.19	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:	\( N \)	\(=\)	\( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2576.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{3} +2.73205 q^{5} -1.00000 q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q-0.732051 q^{3} +2.73205 q^{5} -1.00000 q^{7} -2.46410 q^{9} +3.46410 q^{11} -1.46410 q^{13} -2.00000 q^{15} -0.732051 q^{17} +2.00000 q^{19} +0.732051 q^{21} -1.00000 q^{23} +2.46410 q^{25} +4.00000 q^{27} +8.00000 q^{29} -0.196152 q^{31} -2.53590 q^{33} -2.73205 q^{35} +0.535898 q^{37} +1.07180 q^{39} -2.00000 q^{41} -6.73205 q^{45} +8.19615 q^{47} +1.00000 q^{49} +0.535898 q^{51} +3.46410 q^{53} +9.46410 q^{55} -1.46410 q^{57} +11.6603 q^{59} +8.19615 q^{61} +2.46410 q^{63} -4.00000 q^{65} +4.53590 q^{67} +0.732051 q^{69} -8.39230 q^{71} -7.46410 q^{73} -1.80385 q^{75} -3.46410 q^{77} -0.928203 q^{79} +4.46410 q^{81} +6.39230 q^{83} -2.00000 q^{85} -5.85641 q^{87} +14.1962 q^{89} +1.46410 q^{91} +0.143594 q^{93} +5.46410 q^{95} -8.73205 q^{97} -8.53590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{13} - 4 q^{15} + 2 q^{17} + 4 q^{19} - 2 q^{21} - 2 q^{23} - 2 q^{25} + 8 q^{27} + 16 q^{29} + 10 q^{31} - 12 q^{33} - 2 q^{35} + 8 q^{37} + 16 q^{39} - 4 q^{41} - 10 q^{45} + 6 q^{47} + 2 q^{49} + 8 q^{51} + 12 q^{55} + 4 q^{57} + 6 q^{59} + 6 q^{61} - 2 q^{63} - 8 q^{65} + 16 q^{67} - 2 q^{69} + 4 q^{71} - 8 q^{73} - 14 q^{75} + 12 q^{79} + 2 q^{81} - 8 q^{83} - 4 q^{85} + 16 q^{87} + 18 q^{89} - 4 q^{91} + 28 q^{93} + 4 q^{95} - 14 q^{97} - 24 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^13 - 4 * q^15 + 2 * q^17 + 4 * q^19 - 2 * q^21 - 2 * q^23 - 2 * q^25 + 8 * q^27 + 16 * q^29 + 10 * q^31 - 12 * q^33 - 2 * q^35 + 8 * q^37 + 16 * q^39 - 4 * q^41 - 10 * q^45 + 6 * q^47 + 2 * q^49 + 8 * q^51 + 12 * q^55 + 4 * q^57 + 6 * q^59 + 6 * q^61 - 2 * q^63 - 8 * q^65 + 16 * q^67 - 2 * q^69 + 4 * q^71 - 8 * q^73 - 14 * q^75 + 12 * q^79 + 2 * q^81 - 8 * q^83 - 4 * q^85 + 16 * q^87 + 18 * q^89 - 4 * q^91 + 28 * q^93 + 4 * q^95 - 14 * q^97 - 24 * q^99

Introduction

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