LMFDB - Embedded newform 9264.2.a.bp.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9264,2,Mod(1,9264)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9264, 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("9264.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9264 = 2^{4} \cdot 3 \cdot 193 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9264.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:	$73.9734124325$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 20 x^{11} + 39 x^{10} + 148 x^{9} - 275 x^{8} - 508 x^{7} + 865 x^{6} + 823 x^{5} + \cdots - 48 $ x^13 - 2x^12 - 20x^11 + 39x^10 + 148x^9 - 275x^8 - 508x^7 + 865x^6 + 823x^5 - 1187x^4 - 556x^3 + 576x^2 + 64x - 48
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 579)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-1.05223$ of defining polynomial
Character	$\chi$	$=$	9264.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.01396 q^{5} -3.77896 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.01396 q^{5} -3.77896 q^{7} +1.00000 q^{9} -4.93419 q^{11} -5.50864 q^{13} -3.01396 q^{15} -7.70179 q^{17} -5.52057 q^{19} +3.77896 q^{21} +7.92401 q^{23} +4.08397 q^{25} -1.00000 q^{27} +4.62451 q^{29} -3.53869 q^{31} +4.93419 q^{33} -11.3896 q^{35} +5.57165 q^{37} +5.50864 q^{39} -4.18854 q^{41} +0.0489147 q^{43} +3.01396 q^{45} -8.14508 q^{47} +7.28054 q^{49} +7.70179 q^{51} -12.7647 q^{53} -14.8715 q^{55} +5.52057 q^{57} +2.57352 q^{59} -6.68322 q^{61} -3.77896 q^{63} -16.6028 q^{65} +6.27347 q^{67} -7.92401 q^{69} -0.729787 q^{71} +3.26717 q^{73} -4.08397 q^{75} +18.6461 q^{77} -13.5577 q^{79} +1.00000 q^{81} +9.98298 q^{83} -23.2129 q^{85} -4.62451 q^{87} +3.67690 q^{89} +20.8169 q^{91} +3.53869 q^{93} -16.6388 q^{95} +0.726031 q^{97} -4.93419 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{3} + 6 q^{5} - 15 q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q - 13 * q^3 + 6 * q^5 - 15 * q^7 + 13 * q^9	$ 13 q - 13 q^{3} + 6 q^{5} - 15 q^{7} + 13 q^{9} - 3 q^{11} + 11 q^{13} - 6 q^{15} - 2 q^{17} - 9 q^{19} + 15 q^{21} + 8 q^{23} + 21 q^{25} - 13 q^{27} + 5 q^{29} - 25 q^{31} + 3 q^{33} + 10 q^{35} + 29 q^{37} - 11 q^{39} - 11 q^{41} - 8 q^{43} + 6 q^{45} + 12 q^{47} + 20 q^{49} + 2 q^{51} + 14 q^{53} - 12 q^{55} + 9 q^{57} - 10 q^{59} + 6 q^{61} - 15 q^{63} - 15 q^{65} - 25 q^{67} - 8 q^{69} + 8 q^{73} - 21 q^{75} - 25 q^{77} - 7 q^{79} + 13 q^{81} + 28 q^{83} - 3 q^{85} - 5 q^{87} + 7 q^{89} - 7 q^{91} + 25 q^{93} + 26 q^{95} + 26 q^{97} - 3 q^{99}+O(q^{100}) $ 13 * q - 13 * q^3 + 6 * q^5 - 15 * q^7 + 13 * q^9 - 3 * q^11 + 11 * q^13 - 6 * q^15 - 2 * q^17 - 9 * q^19 + 15 * q^21 + 8 * q^23 + 21 * q^25 - 13 * q^27 + 5 * q^29 - 25 * q^31 + 3 * q^33 + 10 * q^35 + 29 * q^37 - 11 * q^39 - 11 * q^41 - 8 * q^43 + 6 * q^45 + 12 * q^47 + 20 * q^49 + 2 * q^51 + 14 * q^53 - 12 * q^55 + 9 * q^57 - 10 * q^59 + 6 * q^61 - 15 * q^63 - 15 * q^65 - 25 * q^67 - 8 * q^69 + 8 * q^73 - 21 * q^75 - 25 * q^77 - 7 * q^79 + 13 * q^81 + 28 * q^83 - 3 * q^85 - 5 * q^87 + 7 * q^89 - 7 * q^91 + 25 * q^93 + 26 * q^95 + 26 * q^97 - 3 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.01396	1.34789	0.673943	−	0.738784i	$-0.264600\pi$
$5$	3.01396	1.34789	0.673943	+	0.738784i	$0.264600\pi$
$6$	0	0
$7$	−3.77896	−1.42831	−0.714156	−	0.699986i	$-0.753189\pi$
$7$	−3.77896	−1.42831	−0.714156	+	0.699986i	$0.753189\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.93419	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$11$	−4.93419	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$12$	0	0
$13$	−5.50864	−1.52782	−0.763911	−	0.645321i	$-0.776723\pi$
$13$	−5.50864	−1.52782	−0.763911	+	0.645321i	$0.776723\pi$
$14$	0	0
$15$	−3.01396	−0.778202
$16$	0	0
$17$	−7.70179	−1.86796	−0.933979	−	0.357329i	$-0.883688\pi$
$17$	−7.70179	−1.86796	−0.933979	+	0.357329i	$0.883688\pi$
$18$	0	0
$19$	−5.52057	−1.26650	−0.633252	−	0.773945i	$-0.718280\pi$
$19$	−5.52057	−1.26650	−0.633252	+	0.773945i	$0.718280\pi$
$20$	0	0
$21$	3.77896	0.824637
$22$	0	0
$23$	7.92401	1.65227	0.826136	−	0.563471i	$-0.190534\pi$
$23$	7.92401	1.65227	0.826136	+	0.563471i	$0.190534\pi$
$24$	0	0
$25$	4.08397	0.816794
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.62451	0.858750	0.429375	−	0.903126i	$-0.358734\pi$
$29$	4.62451	0.858750	0.429375	+	0.903126i	$0.358734\pi$
$30$	0	0
$31$	−3.53869	−0.635568	−0.317784	−	0.948163i	$-0.602939\pi$
$31$	−3.53869	−0.635568	−0.317784	+	0.948163i	$0.602939\pi$
$32$	0	0
$33$	4.93419	0.858933
$34$	0	0
$35$	−11.3896	−1.92520
$36$	0	0
$37$	5.57165	0.915974	0.457987	−	0.888959i	$-0.348571\pi$
$37$	5.57165	0.915974	0.457987	+	0.888959i	$0.348571\pi$
$38$	0	0
$39$	5.50864	0.882089
$40$	0	0
$41$	−4.18854	−0.654140	−0.327070	−	0.945000i	$-0.606061\pi$
$41$	−4.18854	−0.654140	−0.327070	+	0.945000i	$0.606061\pi$
$42$	0	0
$43$	0.0489147	0.00745942	0.00372971	−	0.999993i	$-0.498813\pi$
$43$	0.0489147	0.00745942	0.00372971	+	0.999993i	$0.498813\pi$
$44$	0	0
$45$	3.01396	0.449295
$46$	0	0
$47$	−8.14508	−1.18808	−0.594041	−	0.804435i	$-0.702468\pi$
$47$	−8.14508	−1.18808	−0.594041	+	0.804435i	$0.702468\pi$
$48$	0	0
$49$	7.28054	1.04008
$50$	0	0
$51$	7.70179	1.07847
$52$	0	0
$53$	−12.7647	−1.75336	−0.876681	−	0.481072i	$-0.840248\pi$
$53$	−12.7647	−1.75336	−0.876681	+	0.481072i	$0.840248\pi$
$54$	0	0
$55$	−14.8715	−2.00527
$56$	0	0
$57$	5.52057	0.731217
$58$	0	0
$59$	2.57352	0.335044	0.167522	−	0.985868i	$-0.446424\pi$
$59$	2.57352	0.335044	0.167522	+	0.985868i	$0.446424\pi$
$60$	0	0
$61$	−6.68322	−0.855698	−0.427849	−	0.903850i	$-0.640729\pi$
$61$	−6.68322	−0.855698	−0.427849	+	0.903850i	$0.640729\pi$
$62$	0	0
$63$	−3.77896	−0.476104
$64$	0	0
$65$	−16.6028	−2.05933
$66$	0	0
$67$	6.27347	0.766426	0.383213	−	0.923660i	$-0.374818\pi$
$67$	6.27347	0.766426	0.383213	+	0.923660i	$0.374818\pi$
$68$	0	0
$69$	−7.92401	−0.953939
$70$	0	0
$71$	−0.729787	−0.0866097	−0.0433049	−	0.999062i	$-0.513789\pi$
$71$	−0.729787	−0.0866097	−0.0433049	+	0.999062i	$0.513789\pi$
$72$	0	0
$73$	3.26717	0.382393	0.191197	−	0.981552i	$-0.438763\pi$
$73$	3.26717	0.382393	0.191197	+	0.981552i	$0.438763\pi$
$74$	0	0
$75$	−4.08397	−0.471576
$76$	0	0
$77$	18.6461	2.12492
$78$	0	0
$79$	−13.5577	−1.52536	−0.762679	−	0.646777i	$-0.776116\pi$
$79$	−13.5577	−1.52536	−0.762679	+	0.646777i	$0.776116\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.98298	1.09577	0.547887	−	0.836552i	$-0.315432\pi$
$83$	9.98298	1.09577	0.547887	+	0.836552i	$0.315432\pi$
$84$	0	0
$85$	−23.2129	−2.51779
$86$	0	0
$87$	−4.62451	−0.495800
$88$	0	0
$89$	3.67690	0.389750	0.194875	−	0.980828i	$-0.437570\pi$
$89$	3.67690	0.389750	0.194875	+	0.980828i	$0.437570\pi$
$90$	0	0
$91$	20.8169	2.18221
$92$	0	0
$93$	3.53869	0.366945
$94$	0	0
$95$	−16.6388	−1.70710
$96$	0	0
$97$	0.726031	0.0737172	0.0368586	−	0.999320i	$-0.488265\pi$
$97$	0.726031	0.0737172	0.0368586	+	0.999320i	$0.488265\pi$
$98$	0	0
$99$	−4.93419	−0.495905
$100$	0	0
$101$	−6.63669	−0.660375	−0.330188	−	0.943915i	$-0.607112\pi$
$101$	−6.63669	−0.660375	−0.330188	+	0.943915i	$0.607112\pi$
$102$	0	0
$103$	6.08147	0.599225	0.299613	−	0.954061i	$-0.403143\pi$
$103$	6.08147	0.599225	0.299613	+	0.954061i	$0.403143\pi$
$104$	0	0
$105$	11.3896	1.11152
$106$	0	0
$107$	4.39098	0.424492	0.212246	−	0.977216i	$-0.431922\pi$
$107$	4.39098	0.424492	0.212246	+	0.977216i	$0.431922\pi$
$108$	0	0
$109$	13.9009	1.33146	0.665731	−	0.746192i	$-0.268120\pi$
$109$	13.9009	1.33146	0.665731	+	0.746192i	$0.268120\pi$
$110$	0	0
$111$	−5.57165	−0.528838
$112$	0	0
$113$	17.9146	1.68526	0.842632	−	0.538489i	$-0.181005\pi$
$113$	17.9146	1.68526	0.842632	+	0.538489i	$0.181005\pi$
$114$	0	0
$115$	23.8827	2.22707
$116$	0	0
$117$	−5.50864	−0.509274
$118$	0	0
$119$	29.1047	2.66803
$120$	0	0
$121$	13.3463	1.21330
$122$	0	0
$123$	4.18854	0.377668
$124$	0	0
$125$	−2.76087	−0.246940
$126$	0	0
$127$	4.10722	0.364457	0.182228	−	0.983256i	$-0.441669\pi$
$127$	4.10722	0.364457	0.182228	+	0.983256i	$0.441669\pi$
$128$	0	0
$129$	−0.0489147	−0.00430670
$130$	0	0
$131$	8.72840	0.762604	0.381302	−	0.924451i	$-0.375476\pi$
$131$	8.72840	0.762604	0.381302	+	0.924451i	$0.375476\pi$
$132$	0	0
$133$	20.8620	1.80896
$134$	0	0
$135$	−3.01396	−0.259401
$136$	0	0
$137$	−9.86915	−0.843178	−0.421589	−	0.906787i	$-0.638527\pi$
$137$	−9.86915	−0.843178	−0.421589	+	0.906787i	$0.638527\pi$
$138$	0	0
$139$	11.1281	0.943873	0.471937	−	0.881632i	$-0.343555\pi$
$139$	11.1281	0.943873	0.471937	+	0.881632i	$0.343555\pi$
$140$	0	0
$141$	8.14508	0.685939
$142$	0	0
$143$	27.1807	2.27297
$144$	0	0
$145$	13.9381	1.15750
$146$	0	0
$147$	−7.28054	−0.600489
$148$	0	0
$149$	−4.12963	−0.338312	−0.169156	−	0.985589i	$-0.554104\pi$
$149$	−4.12963	−0.338312	−0.169156	+	0.985589i	$0.554104\pi$
$150$	0	0
$151$	0.708420	0.0576504	0.0288252	−	0.999584i	$-0.490823\pi$
$151$	0.708420	0.0576504	0.0288252	+	0.999584i	$0.490823\pi$
$152$	0	0
$153$	−7.70179	−0.622652
$154$	0	0
$155$	−10.6655	−0.856672
$156$	0	0
$157$	7.67216	0.612305	0.306152	−	0.951983i	$-0.400958\pi$
$157$	7.67216	0.612305	0.306152	+	0.951983i	$0.400958\pi$
$158$	0	0
$159$	12.7647	1.01230
$160$	0	0
$161$	−29.9445	−2.35996
$162$	0	0
$163$	3.90736	0.306048	0.153024	−	0.988222i	$-0.451099\pi$
$163$	3.90736	0.306048	0.153024	+	0.988222i	$0.451099\pi$
$164$	0	0
$165$	14.8715	1.15774
$166$	0	0
$167$	−8.60125	−0.665585	−0.332792	−	0.943000i	$-0.607991\pi$
$167$	−8.60125	−0.665585	−0.332792	+	0.943000i	$0.607991\pi$
$168$	0	0
$169$	17.3451	1.33424
$170$	0	0
$171$	−5.52057	−0.422168
$172$	0	0
$173$	−11.2628	−0.856291	−0.428146	−	0.903710i	$-0.640833\pi$
$173$	−11.2628	−0.856291	−0.428146	+	0.903710i	$0.640833\pi$
$174$	0	0
$175$	−15.4332	−1.16664
$176$	0	0
$177$	−2.57352	−0.193437
$178$	0	0
$179$	−4.73076	−0.353594	−0.176797	−	0.984247i	$-0.556574\pi$
$179$	−4.73076	−0.353594	−0.176797	+	0.984247i	$0.556574\pi$
$180$	0	0
$181$	−6.26039	−0.465331	−0.232666	−	0.972557i	$-0.574745\pi$
$181$	−6.26039	−0.465331	−0.232666	+	0.972557i	$0.574745\pi$
$182$	0	0
$183$	6.68322	0.494038
$184$	0	0
$185$	16.7928	1.23463
$186$	0	0
$187$	38.0021	2.77899
$188$	0	0
$189$	3.77896	0.274879
$190$	0	0
$191$	5.20719	0.376779	0.188390	−	0.982094i	$-0.439673\pi$
$191$	5.20719	0.376779	0.188390	+	0.982094i	$0.439673\pi$
$192$	0	0
$193$	−1.00000	−0.0719816
$193$	−1.00000	−0.0719816
$194$	0	0
$195$	16.6028	1.18895
$196$	0	0
$197$	−13.3487	−0.951057	−0.475528	−	0.879700i	$-0.657743\pi$
$197$	−13.3487	−0.951057	−0.475528	+	0.879700i	$0.657743\pi$
$198$	0	0
$199$	3.36005	0.238188	0.119094	−	0.992883i	$-0.462001\pi$
$199$	3.36005	0.238188	0.119094	+	0.992883i	$0.462001\pi$
$200$	0	0
$201$	−6.27347	−0.442496
$202$	0	0
$203$	−17.4758	−1.22656
$204$	0	0
$205$	−12.6241	−0.881705
$206$	0	0
$207$	7.92401	0.550757
$208$	0	0
$209$	27.2395	1.88420
$210$	0	0
$211$	13.9904	0.963139	0.481570	−	0.876408i	$-0.340067\pi$
$211$	13.9904	0.963139	0.481570	+	0.876408i	$0.340067\pi$
$212$	0	0
$213$	0.729787	0.0500042
$214$	0	0
$215$	0.147427	0.0100544
$216$	0	0
$217$	13.3726	0.907789
$218$	0	0
$219$	−3.26717	−0.220775
$220$	0	0
$221$	42.4264	2.85391
$222$	0	0
$223$	−1.11740	−0.0748264	−0.0374132	−	0.999300i	$-0.511912\pi$
$223$	−1.11740	−0.0748264	−0.0374132	+	0.999300i	$0.511912\pi$
$224$	0	0
$225$	4.08397	0.272265
$226$	0	0
$227$	−8.01883	−0.532229	−0.266114	−	0.963941i	$-0.585740\pi$
$227$	−8.01883	−0.532229	−0.266114	+	0.963941i	$0.585740\pi$
$228$	0	0
$229$	28.2834	1.86902	0.934511	−	0.355934i	$-0.115837\pi$
$229$	28.2834	1.86902	0.934511	+	0.355934i	$0.115837\pi$
$230$	0	0
$231$	−18.6461	−1.22682
$232$	0	0
$233$	0.459141	0.0300793	0.0150397	−	0.999887i	$-0.495213\pi$
$233$	0.459141	0.0300793	0.0150397	+	0.999887i	$0.495213\pi$
$234$	0	0
$235$	−24.5490	−1.60140
$236$	0	0
$237$	13.5577	0.880666
$238$	0	0
$239$	19.0197	1.23028	0.615140	−	0.788418i	$-0.289100\pi$
$239$	19.0197	1.23028	0.615140	+	0.788418i	$0.289100\pi$
$240$	0	0
$241$	−21.9125	−1.41151	−0.705755	−	0.708456i	$-0.749392\pi$
$241$	−21.9125	−1.41151	−0.705755	+	0.708456i	$0.749392\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	21.9433	1.40190
$246$	0	0
$247$	30.4108	1.93499
$248$	0	0
$249$	−9.98298	−0.632645
$250$	0	0
$251$	−16.0018	−1.01003	−0.505013	−	0.863111i	$-0.668512\pi$
$251$	−16.0018	−1.01003	−0.505013	+	0.863111i	$0.668512\pi$
$252$	0	0
$253$	−39.0986	−2.45811
$254$	0	0
$255$	23.2129	1.45365
$256$	0	0
$257$	7.13264	0.444922	0.222461	−	0.974942i	$-0.428591\pi$
$257$	7.13264	0.444922	0.222461	+	0.974942i	$0.428591\pi$
$258$	0	0
$259$	−21.0551	−1.30830
$260$	0	0
$261$	4.62451	0.286250
$262$	0	0
$263$	−16.1941	−0.998572	−0.499286	−	0.866437i	$-0.666404\pi$
$263$	−16.1941	−0.998572	−0.499286	+	0.866437i	$0.666404\pi$
$264$	0	0
$265$	−38.4722	−2.36333
$266$	0	0
$267$	−3.67690	−0.225022
$268$	0	0
$269$	−29.5483	−1.80159	−0.900795	−	0.434245i	$-0.857015\pi$
$269$	−29.5483	−1.80159	−0.900795	+	0.434245i	$0.857015\pi$
$270$	0	0
$271$	11.9367	0.725101	0.362550	−	0.931964i	$-0.381906\pi$
$271$	11.9367	0.725101	0.362550	+	0.931964i	$0.381906\pi$
$272$	0	0
$273$	−20.8169	−1.25990
$274$	0	0
$275$	−20.1511	−1.21516
$276$	0	0
$277$	−11.9932	−0.720602	−0.360301	−	0.932836i	$-0.617326\pi$
$277$	−11.9932	−0.720602	−0.360301	+	0.932836i	$0.617326\pi$
$278$	0	0
$279$	−3.53869	−0.211856
$280$	0	0
$281$	−0.120341	−0.00717896	−0.00358948	−	0.999994i	$-0.501143\pi$
$281$	−0.120341	−0.00717896	−0.00358948	+	0.999994i	$0.501143\pi$
$282$	0	0
$283$	−20.3583	−1.21018	−0.605088	−	0.796159i	$-0.706862\pi$
$283$	−20.3583	−1.21018	−0.605088	+	0.796159i	$0.706862\pi$
$284$	0	0
$285$	16.6388	0.985596
$286$	0	0
$287$	15.8283	0.934316
$288$	0	0
$289$	42.3175	2.48926
$290$	0	0
$291$	−0.726031	−0.0425607
$292$	0	0
$293$	11.0146	0.643480	0.321740	−	0.946828i	$-0.395732\pi$
$293$	11.0146	0.643480	0.321740	+	0.946828i	$0.395732\pi$
$294$	0	0
$295$	7.75649	0.451600
$296$	0	0
$297$	4.93419	0.286311
$298$	0	0
$299$	−43.6506	−2.52438
$300$	0	0
$301$	−0.184847	−0.0106544
$302$	0	0
$303$	6.63669	0.381268
$304$	0	0
$305$	−20.1430	−1.15338
$306$	0	0
$307$	−2.08748	−0.119139	−0.0595694	−	0.998224i	$-0.518973\pi$
$307$	−2.08748	−0.119139	−0.0595694	+	0.998224i	$0.518973\pi$
$308$	0	0
$309$	−6.08147	−0.345963
$310$	0	0
$311$	7.25318	0.411290	0.205645	−	0.978627i	$-0.434071\pi$
$311$	7.25318	0.411290	0.205645	+	0.978627i	$0.434071\pi$
$312$	0	0
$313$	−6.58271	−0.372077	−0.186038	−	0.982542i	$-0.559565\pi$
$313$	−6.58271	−0.372077	−0.186038	+	0.982542i	$0.559565\pi$
$314$	0	0
$315$	−11.3896	−0.641734
$316$	0	0
$317$	22.9791	1.29063	0.645317	−	0.763915i	$-0.276725\pi$
$317$	22.9791	1.29063	0.645317	+	0.763915i	$0.276725\pi$
$318$	0	0
$319$	−22.8182	−1.27758
$320$	0	0
$321$	−4.39098	−0.245080
$322$	0	0
$323$	42.5182	2.36578
$324$	0	0
$325$	−22.4971	−1.24792
$326$	0	0
$327$	−13.9009	−0.768719
$328$	0	0
$329$	30.7799	1.69695
$330$	0	0
$331$	−27.2963	−1.50034	−0.750171	−	0.661244i	$-0.770029\pi$
$331$	−27.2963	−1.50034	−0.750171	+	0.661244i	$0.770029\pi$
$332$	0	0
$333$	5.57165	0.305325
$334$	0	0
$335$	18.9080	1.03305
$336$	0	0
$337$	7.93185	0.432075	0.216038	−	0.976385i	$-0.430687\pi$
$337$	7.93185	0.432075	0.216038	+	0.976385i	$0.430687\pi$
$338$	0	0
$339$	−17.9146	−0.972988
$340$	0	0
$341$	17.4606	0.945544
$342$	0	0
$343$	−1.06015	−0.0572427
$344$	0	0
$345$	−23.8827	−1.28580
$346$	0	0
$347$	−4.00227	−0.214853	−0.107427	−	0.994213i	$-0.534261\pi$
$347$	−4.00227	−0.214853	−0.107427	+	0.994213i	$0.534261\pi$
$348$	0	0
$349$	7.14623	0.382529	0.191265	−	0.981539i	$-0.438741\pi$
$349$	7.14623	0.382529	0.191265	+	0.981539i	$0.438741\pi$
$350$	0	0
$351$	5.50864	0.294030
$352$	0	0
$353$	−24.4226	−1.29988	−0.649941	−	0.759985i	$-0.725206\pi$
$353$	−24.4226	−1.29988	−0.649941	+	0.759985i	$0.725206\pi$
$354$	0	0
$355$	−2.19955	−0.116740
$356$	0	0
$357$	−29.1047	−1.54039
$358$	0	0
$359$	−21.3702	−1.12788	−0.563939	−	0.825817i	$-0.690715\pi$
$359$	−21.3702	−1.12788	−0.563939	+	0.825817i	$0.690715\pi$
$360$	0	0
$361$	11.4766	0.604034
$362$	0	0
$363$	−13.3463	−0.700498
$364$	0	0
$365$	9.84713	0.515422
$366$	0	0
$367$	−11.7128	−0.611404	−0.305702	−	0.952127i	$-0.598891\pi$
$367$	−11.7128	−0.611404	−0.305702	+	0.952127i	$0.598891\pi$
$368$	0	0
$369$	−4.18854	−0.218047
$370$	0	0
$371$	48.2372	2.50435
$372$	0	0
$373$	−9.30022	−0.481547	−0.240774	−	0.970581i	$-0.577401\pi$
$373$	−9.30022	−0.481547	−0.240774	+	0.970581i	$0.577401\pi$
$374$	0	0
$375$	2.76087	0.142571
$376$	0	0
$377$	−25.4748	−1.31202
$378$	0	0
$379$	21.6015	1.10959	0.554797	−	0.831986i	$-0.312796\pi$
$379$	21.6015	1.10959	0.554797	+	0.831986i	$0.312796\pi$
$380$	0	0
$381$	−4.10722	−0.210419
$382$	0	0
$383$	35.1084	1.79395	0.896977	−	0.442077i	$-0.145758\pi$
$383$	35.1084	1.79395	0.896977	+	0.442077i	$0.145758\pi$
$384$	0	0
$385$	56.1987	2.86415
$386$	0	0
$387$	0.0489147	0.00248647
$388$	0	0
$389$	12.8340	0.650709	0.325355	−	0.945592i	$-0.394516\pi$
$389$	12.8340	0.650709	0.325355	+	0.945592i	$0.394516\pi$
$390$	0	0
$391$	−61.0291	−3.08637
$392$	0	0
$393$	−8.72840	−0.440290
$394$	0	0
$395$	−40.8623	−2.05601
$396$	0	0
$397$	−12.6635	−0.635565	−0.317783	−	0.948164i	$-0.602938\pi$
$397$	−12.6635	−0.635565	−0.317783	+	0.948164i	$0.602938\pi$
$398$	0	0
$399$	−20.8620	−1.04441
$400$	0	0
$401$	4.24500	0.211985	0.105993	−	0.994367i	$-0.466198\pi$
$401$	4.24500	0.211985	0.105993	+	0.994367i	$0.466198\pi$
$402$	0	0
$403$	19.4934	0.971034
$404$	0	0
$405$	3.01396	0.149765
$406$	0	0
$407$	−27.4916	−1.36271
$408$	0	0
$409$	13.3699	0.661099	0.330550	−	0.943789i	$-0.392766\pi$
$409$	13.3699	0.661099	0.330550	+	0.943789i	$0.392766\pi$
$410$	0	0
$411$	9.86915	0.486809
$412$	0	0
$413$	−9.72522	−0.478547
$414$	0	0
$415$	30.0883	1.47698
$416$	0	0
$417$	−11.1281	−0.544945
$418$	0	0
$419$	24.9650	1.21962	0.609810	−	0.792547i	$-0.291246\pi$
$419$	24.9650	1.21962	0.609810	+	0.792547i	$0.291246\pi$
$420$	0	0
$421$	23.3482	1.13792	0.568962	−	0.822364i	$-0.307345\pi$
$421$	23.3482	1.13792	0.568962	+	0.822364i	$0.307345\pi$
$422$	0	0
$423$	−8.14508	−0.396027
$424$	0	0
$425$	−31.4539	−1.52574
$426$	0	0
$427$	25.2556	1.22220
$428$	0	0
$429$	−27.1807	−1.31230
$430$	0	0
$431$	22.4189	1.07988	0.539940	−	0.841703i	$-0.318447\pi$
$431$	22.4189	1.07988	0.539940	+	0.841703i	$0.318447\pi$
$432$	0	0
$433$	−30.0547	−1.44433	−0.722167	−	0.691718i	$-0.756854\pi$
$433$	−30.0547	−1.44433	−0.722167	+	0.691718i	$0.756854\pi$
$434$	0	0
$435$	−13.9381	−0.668281
$436$	0	0
$437$	−43.7450	−2.09261
$438$	0	0
$439$	15.8831	0.758058	0.379029	−	0.925385i	$-0.376258\pi$
$439$	15.8831	0.758058	0.379029	+	0.925385i	$0.376258\pi$
$440$	0	0
$441$	7.28054	0.346692
$442$	0	0
$443$	14.2038	0.674845	0.337422	−	0.941353i	$-0.390445\pi$
$443$	14.2038	0.674845	0.337422	+	0.941353i	$0.390445\pi$
$444$	0	0
$445$	11.0820	0.525339
$446$	0	0
$447$	4.12963	0.195325
$448$	0	0
$449$	−19.6860	−0.929040	−0.464520	−	0.885563i	$-0.653773\pi$
$449$	−19.6860	−0.929040	−0.464520	+	0.885563i	$0.653773\pi$
$450$	0	0
$451$	20.6671	0.973174
$452$	0	0
$453$	−0.708420	−0.0332845
$454$	0	0
$455$	62.7415	2.94137
$456$	0	0
$457$	−18.6845	−0.874024	−0.437012	−	0.899456i	$-0.643963\pi$
$457$	−18.6845	−0.874024	−0.437012	+	0.899456i	$0.643963\pi$
$458$	0	0
$459$	7.70179	0.359489
$460$	0	0
$461$	−16.1678	−0.753012	−0.376506	−	0.926414i	$-0.622875\pi$
$461$	−16.1678	−0.753012	−0.376506	+	0.926414i	$0.622875\pi$
$462$	0	0
$463$	6.67465	0.310197	0.155099	−	0.987899i	$-0.450430\pi$
$463$	6.67465	0.310197	0.155099	+	0.987899i	$0.450430\pi$
$464$	0	0
$465$	10.6655	0.494600
$466$	0	0
$467$	−6.01221	−0.278212	−0.139106	−	0.990277i	$-0.544423\pi$
$467$	−6.01221	−0.278212	−0.139106	+	0.990277i	$0.544423\pi$
$468$	0	0
$469$	−23.7072	−1.09470
$470$	0	0
$471$	−7.67216	−0.353514
$472$	0	0
$473$	−0.241355	−0.0110975
$474$	0	0
$475$	−22.5458	−1.03447
$476$	0	0
$477$	−12.7647	−0.584454
$478$	0	0
$479$	7.74813	0.354021	0.177011	−	0.984209i	$-0.443357\pi$
$479$	7.74813	0.354021	0.177011	+	0.984209i	$0.443357\pi$
$480$	0	0
$481$	−30.6922	−1.39945
$482$	0	0
$483$	29.9445	1.36252
$484$	0	0
$485$	2.18823	0.0993624
$486$	0	0
$487$	−23.7671	−1.07699	−0.538495	−	0.842629i	$-0.681007\pi$
$487$	−23.7671	−1.07699	−0.538495	+	0.842629i	$0.681007\pi$
$488$	0	0
$489$	−3.90736	−0.176697
$490$	0	0
$491$	18.9106	0.853426	0.426713	−	0.904387i	$-0.359672\pi$
$491$	18.9106	0.853426	0.426713	+	0.904387i	$0.359672\pi$
$492$	0	0
$493$	−35.6170	−1.60411
$494$	0	0
$495$	−14.8715	−0.668423
$496$	0	0
$497$	2.75783	0.123706
$498$	0	0
$499$	27.2736	1.22093	0.610466	−	0.792042i	$-0.290982\pi$
$499$	27.2736	1.22093	0.610466	+	0.792042i	$0.290982\pi$
$500$	0	0
$501$	8.60125	0.384275
$502$	0	0
$503$	1.64508	0.0733507	0.0366754	−	0.999327i	$-0.488323\pi$
$503$	1.64508	0.0733507	0.0366754	+	0.999327i	$0.488323\pi$
$504$	0	0
$505$	−20.0027	−0.890110
$506$	0	0
$507$	−17.3451	−0.770324
$508$	0	0
$509$	31.0713	1.37721	0.688607	−	0.725135i	$-0.258223\pi$
$509$	31.0713	1.37721	0.688607	+	0.725135i	$0.258223\pi$
$510$	0	0
$511$	−12.3465	−0.546177
$512$	0	0
$513$	5.52057	0.243739
$514$	0	0
$515$	18.3293	0.807687
$516$	0	0
$517$	40.1894	1.76753
$518$	0	0
$519$	11.2628	0.494380
$520$	0	0
$521$	11.0807	0.485454	0.242727	−	0.970095i	$-0.421958\pi$
$521$	11.0807	0.485454	0.242727	+	0.970095i	$0.421958\pi$
$522$	0	0
$523$	−20.4180	−0.892816	−0.446408	−	0.894830i	$-0.647297\pi$
$523$	−20.4180	−0.892816	−0.446408	+	0.894830i	$0.647297\pi$
$524$	0	0
$525$	15.4332	0.673559
$526$	0	0
$527$	27.2542	1.18721
$528$	0	0
$529$	39.7900	1.73000
$530$	0	0
$531$	2.57352	0.111681
$532$	0	0
$533$	23.0732	0.999409
$534$	0	0
$535$	13.2342	0.572166
$536$	0	0
$537$	4.73076	0.204147
$538$	0	0
$539$	−35.9236	−1.54734
$540$	0	0
$541$	25.6550	1.10299	0.551496	−	0.834177i	$-0.314057\pi$
$541$	25.6550	1.10299	0.551496	+	0.834177i	$0.314057\pi$
$542$	0	0
$543$	6.26039	0.268659
$544$	0	0
$545$	41.8967	1.79466
$546$	0	0
$547$	−21.9198	−0.937222	−0.468611	−	0.883405i	$-0.655245\pi$
$547$	−21.9198	−0.937222	−0.468611	+	0.883405i	$0.655245\pi$
$548$	0	0
$549$	−6.68322	−0.285233
$550$	0	0
$551$	−25.5299	−1.08761
$552$	0	0
$553$	51.2339	2.17869
$554$	0	0
$555$	−16.7928	−0.712813
$556$	0	0
$557$	40.8767	1.73200	0.866001	−	0.500042i	$-0.166682\pi$
$557$	40.8767	1.73200	0.866001	+	0.500042i	$0.166682\pi$
$558$	0	0
$559$	−0.269454	−0.0113967
$560$	0	0
$561$	−38.0021	−1.60445
$562$	0	0
$563$	−26.8758	−1.13268	−0.566341	−	0.824171i	$-0.691641\pi$
$563$	−26.8758	−1.13268	−0.566341	+	0.824171i	$0.691641\pi$
$564$	0	0
$565$	53.9940	2.27154
$566$	0	0
$567$	−3.77896	−0.158701
$568$	0	0
$569$	−23.3458	−0.978709	−0.489354	−	0.872085i	$-0.662768\pi$
$569$	−23.3458	−0.978709	−0.489354	+	0.872085i	$0.662768\pi$
$570$	0	0
$571$	36.1861	1.51434	0.757170	−	0.653218i	$-0.226581\pi$
$571$	36.1861	1.51434	0.757170	+	0.653218i	$0.226581\pi$
$572$	0	0
$573$	−5.20719	−0.217534
$574$	0	0
$575$	32.3615	1.34957
$576$	0	0
$577$	35.2043	1.46557	0.732786	−	0.680459i	$-0.238219\pi$
$577$	35.2043	1.46557	0.732786	+	0.680459i	$0.238219\pi$
$578$	0	0
$579$	1.00000	0.0415586
$580$	0	0
$581$	−37.7253	−1.56511
$582$	0	0
$583$	62.9834	2.60850
$584$	0	0
$585$	−16.6028	−0.686443
$586$	0	0
$587$	−13.9848	−0.577213	−0.288607	−	0.957448i	$-0.593192\pi$
$587$	−13.9848	−0.577213	−0.288607	+	0.957448i	$0.593192\pi$
$588$	0	0
$589$	19.5356	0.804949
$590$	0	0
$591$	13.3487	0.549093
$592$	0	0
$593$	−35.0102	−1.43769	−0.718847	−	0.695168i	$-0.755330\pi$
$593$	−35.0102	−1.43769	−0.718847	+	0.695168i	$0.755330\pi$
$594$	0	0
$595$	87.7206	3.59619
$596$	0	0
$597$	−3.36005	−0.137518
$598$	0	0
$599$	−27.3102	−1.11587	−0.557933	−	0.829886i	$-0.688405\pi$
$599$	−27.3102	−1.11587	−0.557933	+	0.829886i	$0.688405\pi$
$600$	0	0
$601$	18.6952	0.762593	0.381297	−	0.924453i	$-0.375478\pi$
$601$	18.6952	0.762593	0.381297	+	0.924453i	$0.375478\pi$
$602$	0	0
$603$	6.27347	0.255475
$604$	0	0
$605$	40.2252	1.63539
$606$	0	0
$607$	−26.2091	−1.06379	−0.531897	−	0.846809i	$-0.678521\pi$
$607$	−26.2091	−1.06379	−0.531897	+	0.846809i	$0.678521\pi$
$608$	0	0
$609$	17.4758	0.708157
$610$	0	0
$611$	44.8683	1.81518
$612$	0	0
$613$	39.7830	1.60682	0.803409	−	0.595427i	$-0.203017\pi$
$613$	39.7830	1.60682	0.803409	+	0.595427i	$0.203017\pi$
$614$	0	0
$615$	12.6241	0.509053
$616$	0	0
$617$	4.60878	0.185543	0.0927713	−	0.995687i	$-0.470427\pi$
$617$	4.60878	0.185543	0.0927713	+	0.995687i	$0.470427\pi$
$618$	0	0
$619$	5.75041	0.231129	0.115564	−	0.993300i	$-0.463132\pi$
$619$	5.75041	0.231129	0.115564	+	0.993300i	$0.463132\pi$
$620$	0	0
$621$	−7.92401	−0.317980
$622$	0	0
$623$	−13.8948	−0.556685
$624$	0	0
$625$	−28.7410	−1.14964
$626$	0	0
$627$	−27.2395	−1.08784
$628$	0	0
$629$	−42.9117	−1.71100
$630$	0	0
$631$	−21.8440	−0.869595	−0.434798	−	0.900528i	$-0.643180\pi$
$631$	−21.8440	−0.869595	−0.434798	+	0.900528i	$0.643180\pi$
$632$	0	0
$633$	−13.9904	−0.556069
$634$	0	0
$635$	12.3790	0.491246
$636$	0	0
$637$	−40.1059	−1.58905
$638$	0	0
$639$	−0.729787	−0.0288699
$640$	0	0
$641$	38.2227	1.50970	0.754852	−	0.655895i	$-0.227709\pi$
$641$	38.2227	1.50970	0.754852	+	0.655895i	$0.227709\pi$
$642$	0	0
$643$	−17.4577	−0.688463	−0.344232	−	0.938885i	$-0.611861\pi$
$643$	−17.4577	−0.688463	−0.344232	+	0.938885i	$0.611861\pi$
$644$	0	0
$645$	−0.147427	−0.00580494
$646$	0	0
$647$	5.94129	0.233576	0.116788	−	0.993157i	$-0.462740\pi$
$647$	5.94129	0.233576	0.116788	+	0.993157i	$0.462740\pi$
$648$	0	0
$649$	−12.6982	−0.498450
$650$	0	0
$651$	−13.3726	−0.524112
$652$	0	0
$653$	28.2240	1.10449	0.552245	−	0.833682i	$-0.313771\pi$
$653$	28.2240	1.10449	0.552245	+	0.833682i	$0.313771\pi$
$654$	0	0
$655$	26.3071	1.02790
$656$	0	0
$657$	3.26717	0.127464
$658$	0	0
$659$	16.1919	0.630748	0.315374	−	0.948967i	$-0.397870\pi$
$659$	16.1919	0.630748	0.315374	+	0.948967i	$0.397870\pi$
$660$	0	0
$661$	28.8422	1.12183	0.560916	−	0.827872i	$-0.310449\pi$
$661$	28.8422	1.12183	0.560916	+	0.827872i	$0.310449\pi$
$662$	0	0
$663$	−42.4264	−1.64770
$664$	0	0
$665$	62.8773	2.43828
$666$	0	0
$667$	36.6447	1.41889
$668$	0	0
$669$	1.11740	0.0432010
$670$	0	0
$671$	32.9763	1.27304
$672$	0	0
$673$	−28.7029	−1.10642	−0.553208	−	0.833043i	$-0.686597\pi$
$673$	−28.7029	−1.10642	−0.553208	+	0.833043i	$0.686597\pi$
$674$	0	0
$675$	−4.08397	−0.157192
$676$	0	0
$677$	−3.50728	−0.134796	−0.0673978	−	0.997726i	$-0.521470\pi$
$677$	−3.50728	−0.134796	−0.0673978	+	0.997726i	$0.521470\pi$
$678$	0	0
$679$	−2.74364	−0.105291
$680$	0	0
$681$	8.01883	0.307282
$682$	0	0
$683$	−1.68618	−0.0645199	−0.0322600	−	0.999480i	$-0.510270\pi$
$683$	−1.68618	−0.0645199	−0.0322600	+	0.999480i	$0.510270\pi$
$684$	0	0
$685$	−29.7452	−1.13651
$686$	0	0
$687$	−28.2834	−1.07908
$688$	0	0
$689$	70.3160	2.67883
$690$	0	0
$691$	−44.3789	−1.68825	−0.844126	−	0.536145i	$-0.819880\pi$
$691$	−44.3789	−1.68825	−0.844126	+	0.536145i	$0.819880\pi$
$692$	0	0
$693$	18.6461	0.708308
$694$	0	0
$695$	33.5397	1.27223
$696$	0	0
$697$	32.2592	1.22190
$698$	0	0
$699$	−0.459141	−0.0173663
$700$	0	0
$701$	−40.8579	−1.54318	−0.771591	−	0.636118i	$-0.780539\pi$
$701$	−40.8579	−1.54318	−0.771591	+	0.636118i	$0.780539\pi$
$702$	0	0
$703$	−30.7587	−1.16009
$704$	0	0
$705$	24.5490	0.924567
$706$	0	0
$707$	25.0798	0.943222
$708$	0	0
$709$	5.05933	0.190007	0.0950036	−	0.995477i	$-0.469714\pi$
$709$	5.05933	0.190007	0.0950036	+	0.995477i	$0.469714\pi$
$710$	0	0
$711$	−13.5577	−0.508453
$712$	0	0
$713$	−28.0406	−1.05013
$714$	0	0
$715$	81.9216	3.06370
$716$	0	0
$717$	−19.0197	−0.710302
$718$	0	0
$719$	17.6286	0.657437	0.328719	−	0.944428i	$-0.393383\pi$
$719$	17.6286	0.657437	0.328719	+	0.944428i	$0.393383\pi$
$720$	0	0
$721$	−22.9816	−0.855881
$722$	0	0
$723$	21.9125	0.814935
$724$	0	0
$725$	18.8864	0.701422
$726$	0	0
$727$	−50.8085	−1.88438	−0.942192	−	0.335073i	$-0.891239\pi$
$727$	−50.8085	−1.88438	−0.942192	+	0.335073i	$0.891239\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.376731	−0.0139339
$732$	0	0
$733$	−22.3958	−0.827208	−0.413604	−	0.910457i	$-0.635730\pi$
$733$	−22.3958	−0.827208	−0.413604	+	0.910457i	$0.635730\pi$
$734$	0	0
$735$	−21.9433	−0.809390
$736$	0	0
$737$	−30.9545	−1.14022
$738$	0	0
$739$	−3.41760	−0.125719	−0.0628593	−	0.998022i	$-0.520022\pi$
$739$	−3.41760	−0.125719	−0.0628593	+	0.998022i	$0.520022\pi$
$740$	0	0
$741$	−30.4108	−1.11717
$742$	0	0
$743$	8.83892	0.324268	0.162134	−	0.986769i	$-0.448162\pi$
$743$	8.83892	0.324268	0.162134	+	0.986769i	$0.448162\pi$
$744$	0	0
$745$	−12.4465	−0.456006
$746$	0	0
$747$	9.98298	0.365258
$748$	0	0
$749$	−16.5933	−0.606307
$750$	0	0
$751$	−39.3748	−1.43681	−0.718403	−	0.695627i	$-0.755127\pi$
$751$	−39.3748	−1.43681	−0.718403	+	0.695627i	$0.755127\pi$
$752$	0	0
$753$	16.0018	0.583139
$754$	0	0
$755$	2.13515	0.0777061
$756$	0	0
$757$	−43.8895	−1.59519	−0.797596	−	0.603192i	$-0.793895\pi$
$757$	−43.8895	−1.59519	−0.797596	+	0.603192i	$0.793895\pi$
$758$	0	0
$759$	39.0986	1.41919
$760$	0	0
$761$	−54.3535	−1.97031	−0.985156	−	0.171660i	$-0.945087\pi$
$761$	−54.3535	−1.97031	−0.985156	+	0.171660i	$0.945087\pi$
$762$	0	0
$763$	−52.5308	−1.90174
$764$	0	0
$765$	−23.2129	−0.839264
$766$	0	0
$767$	−14.1766	−0.511887
$768$	0	0
$769$	−25.6784	−0.925987	−0.462994	−	0.886362i	$-0.653225\pi$
$769$	−25.6784	−0.925987	−0.462994	+	0.886362i	$0.653225\pi$
$770$	0	0
$771$	−7.13264	−0.256876
$772$	0	0
$773$	47.4057	1.70507	0.852533	−	0.522674i	$-0.175065\pi$
$773$	47.4057	1.70507	0.852533	+	0.522674i	$0.175065\pi$
$774$	0	0
$775$	−14.4519	−0.519128
$776$	0	0
$777$	21.0551	0.755346
$778$	0	0
$779$	23.1231	0.828471
$780$	0	0
$781$	3.60091	0.128851
$782$	0	0
$783$	−4.62451	−0.165267
$784$	0	0
$785$	23.1236	0.825317
$786$	0	0
$787$	29.5503	1.05335	0.526677	−	0.850065i	$-0.323438\pi$
$787$	29.5503	1.05335	0.526677	+	0.850065i	$0.323438\pi$
$788$	0	0
$789$	16.1941	0.576526
$790$	0	0
$791$	−67.6986	−2.40708
$792$	0	0
$793$	36.8154	1.30735
$794$	0	0
$795$	38.4722	1.36447
$796$	0	0
$797$	−43.6617	−1.54658	−0.773289	−	0.634054i	$-0.781390\pi$
$797$	−43.6617	−1.54658	−0.773289	+	0.634054i	$0.781390\pi$
$798$	0	0
$799$	62.7316	2.21929
$800$	0	0
$801$	3.67690	0.129917
$802$	0	0
$803$	−16.1208	−0.568892
$804$	0	0
$805$	−90.2517	−3.18096
$806$	0	0
$807$	29.5483	1.04015
$808$	0	0
$809$	38.5615	1.35575	0.677875	−	0.735177i	$-0.262901\pi$
$809$	38.5615	1.35575	0.677875	+	0.735177i	$0.262901\pi$
$810$	0	0
$811$	−44.9627	−1.57885	−0.789427	−	0.613844i	$-0.789622\pi$
$811$	−44.9627	−1.57885	−0.789427	+	0.613844i	$0.789622\pi$
$812$	0	0
$813$	−11.9367	−0.418637
$814$	0	0
$815$	11.7766	0.412518
$816$	0	0
$817$	−0.270037	−0.00944739
$818$	0	0
$819$	20.8169	0.727403
$820$	0	0
$821$	30.6399	1.06934	0.534671	−	0.845061i	$-0.320436\pi$
$821$	30.6399	1.06934	0.534671	+	0.845061i	$0.320436\pi$
$822$	0	0
$823$	−42.9964	−1.49876	−0.749380	−	0.662140i	$-0.769648\pi$
$823$	−42.9964	−1.49876	−0.749380	+	0.662140i	$0.769648\pi$
$824$	0	0
$825$	20.1511	0.701572
$826$	0	0
$827$	−53.4632	−1.85910	−0.929548	−	0.368700i	$-0.879803\pi$
$827$	−53.4632	−1.85910	−0.929548	+	0.368700i	$0.879803\pi$
$828$	0	0
$829$	44.7236	1.55332	0.776658	−	0.629923i	$-0.216913\pi$
$829$	44.7236	1.55332	0.776658	+	0.629923i	$0.216913\pi$
$830$	0	0
$831$	11.9932	0.416040
$832$	0	0
$833$	−56.0732	−1.94282
$834$	0	0
$835$	−25.9238	−0.897132
$836$	0	0
$837$	3.53869	0.122315
$838$	0	0
$839$	48.7683	1.68367	0.841834	−	0.539737i	$-0.181476\pi$
$839$	48.7683	1.68367	0.841834	+	0.539737i	$0.181476\pi$
$840$	0	0
$841$	−7.61391	−0.262548
$842$	0	0
$843$	0.120341	0.00414477
$844$	0	0
$845$	52.2776	1.79840
$846$	0	0
$847$	−50.4350	−1.73297
$848$	0	0
$849$	20.3583	0.698695
$850$	0	0
$851$	44.1499	1.51344
$852$	0	0
$853$	−53.3269	−1.82588	−0.912940	−	0.408095i	$-0.866193\pi$
$853$	−53.3269	−1.82588	−0.912940	+	0.408095i	$0.866193\pi$
$854$	0	0
$855$	−16.6388	−0.569034
$856$	0	0
$857$	43.4856	1.48544	0.742721	−	0.669602i	$-0.233535\pi$
$857$	43.4856	1.48544	0.742721	+	0.669602i	$0.233535\pi$
$858$	0	0
$859$	21.9762	0.749818	0.374909	−	0.927062i	$-0.377674\pi$
$859$	21.9762	0.749818	0.374909	+	0.927062i	$0.377674\pi$
$860$	0	0
$861$	−15.8283	−0.539428
$862$	0	0
$863$	−50.2611	−1.71091	−0.855454	−	0.517878i	$-0.826722\pi$
$863$	−50.2611	−1.71091	−0.855454	+	0.517878i	$0.826722\pi$
$864$	0	0
$865$	−33.9455	−1.15418
$866$	0	0
$867$	−42.3175	−1.43718
$868$	0	0
$869$	66.8962	2.26930
$870$	0	0
$871$	−34.5583	−1.17096
$872$	0	0
$873$	0.726031	0.0245724
$874$	0	0
$875$	10.4332	0.352708
$876$	0	0
$877$	3.36641	0.113676	0.0568378	−	0.998383i	$-0.481898\pi$
$877$	3.36641	0.113676	0.0568378	+	0.998383i	$0.481898\pi$
$878$	0	0
$879$	−11.0146	−0.371513
$880$	0	0
$881$	−1.50473	−0.0506956	−0.0253478	−	0.999679i	$-0.508069\pi$
$881$	−1.50473	−0.0506956	−0.0253478	+	0.999679i	$0.508069\pi$
$882$	0	0
$883$	42.1862	1.41968	0.709839	−	0.704364i	$-0.248768\pi$
$883$	42.1862	1.41968	0.709839	+	0.704364i	$0.248768\pi$
$884$	0	0
$885$	−7.75649	−0.260732
$886$	0	0
$887$	16.5406	0.555378	0.277689	−	0.960671i	$-0.410432\pi$
$887$	16.5406	0.555378	0.277689	+	0.960671i	$0.410432\pi$
$888$	0	0
$889$	−15.5210	−0.520558
$890$	0	0
$891$	−4.93419	−0.165302
$892$	0	0
$893$	44.9654	1.50471
$894$	0	0
$895$	−14.2583	−0.476603
$896$	0	0
$897$	43.6506	1.45745
$898$	0	0
$899$	−16.3647	−0.545794
$900$	0	0
$901$	98.3108	3.27521
$902$	0	0
$903$	0.184847	0.00615131
$904$	0	0
$905$	−18.8686	−0.627213
$906$	0	0
$907$	16.8652	0.560001	0.280000	−	0.960000i	$-0.409665\pi$
$907$	16.8652	0.560001	0.280000	+	0.960000i	$0.409665\pi$
$908$	0	0
$909$	−6.63669	−0.220125
$910$	0	0
$911$	−28.7345	−0.952018	−0.476009	−	0.879440i	$-0.657917\pi$
$911$	−28.7345	−0.952018	−0.476009	+	0.879440i	$0.657917\pi$
$912$	0	0
$913$	−49.2579	−1.63020
$914$	0	0
$915$	20.1430	0.665906
$916$	0	0
$917$	−32.9843	−1.08924
$918$	0	0
$919$	37.6678	1.24255	0.621273	−	0.783594i	$-0.286616\pi$
$919$	37.6678	1.24255	0.621273	+	0.783594i	$0.286616\pi$
$920$	0	0
$921$	2.08748	0.0687848
$922$	0	0
$923$	4.02013	0.132324
$924$	0	0
$925$	22.7545	0.748162
$926$	0	0
$927$	6.08147	0.199742
$928$	0	0
$929$	−15.8367	−0.519586	−0.259793	−	0.965664i	$-0.583654\pi$
$929$	−15.8367	−0.519586	−0.259793	+	0.965664i	$0.583654\pi$
$930$	0	0
$931$	−40.1927	−1.31726
$932$	0	0
$933$	−7.25318	−0.237458
$934$	0	0
$935$	114.537	3.74576
$936$	0	0
$937$	−9.19075	−0.300249	−0.150124	−	0.988667i	$-0.547967\pi$
$937$	−9.19075	−0.300249	−0.150124	+	0.988667i	$0.547967\pi$
$938$	0	0
$939$	6.58271	0.214819
$940$	0	0
$941$	21.3247	0.695164	0.347582	−	0.937650i	$-0.387003\pi$
$941$	21.3247	0.695164	0.347582	+	0.937650i	$0.387003\pi$
$942$	0	0
$943$	−33.1900	−1.08082
$944$	0	0
$945$	11.3896	0.370505
$946$	0	0
$947$	−34.7747	−1.13003	−0.565013	−	0.825082i	$-0.691129\pi$
$947$	−34.7747	−1.13003	−0.565013	+	0.825082i	$0.691129\pi$
$948$	0	0
$949$	−17.9977	−0.584229
$950$	0	0
$951$	−22.9791	−0.745148
$952$	0	0
$953$	−0.0530140	−0.00171729	−0.000858646	−	1.00000i	$-0.500273\pi$
$953$	−0.0530140	−0.00171729	−0.000858646		1.00000i	$0.500273\pi$
$954$	0	0
$955$	15.6943	0.507855
$956$	0	0
$957$	22.8182	0.737609
$958$	0	0
$959$	37.2951	1.20432
$960$	0	0
$961$	−18.4777	−0.596054
$962$	0	0
$963$	4.39098	0.141497
$964$	0	0
$965$	−3.01396	−0.0970229
$966$	0	0
$967$	−25.2732	−0.812733	−0.406366	−	0.913710i	$-0.633204\pi$
$967$	−25.2732	−0.812733	−0.406366	+	0.913710i	$0.633204\pi$
$968$	0	0
$969$	−42.5182	−1.36588
$970$	0	0
$971$	−22.8028	−0.731775	−0.365888	−	0.930659i	$-0.619235\pi$
$971$	−22.8028	−0.731775	−0.365888	+	0.930659i	$0.619235\pi$
$972$	0	0
$973$	−42.0527	−1.34815
$974$	0	0
$975$	22.4971	0.720485
$976$	0	0
$977$	−13.7179	−0.438876	−0.219438	−	0.975626i	$-0.570422\pi$
$977$	−13.7179	−0.438876	−0.219438	+	0.975626i	$0.570422\pi$
$978$	0	0
$979$	−18.1425	−0.579838
$980$	0	0
$981$	13.9009	0.443820
$982$	0	0
$983$	−5.25861	−0.167724	−0.0838618	−	0.996477i	$-0.526725\pi$
$983$	−5.25861	−0.167724	−0.0838618	+	0.996477i	$0.526725\pi$
$984$	0	0
$985$	−40.2325	−1.28192
$986$	0	0
$987$	−30.7799	−0.979736
$988$	0	0
$989$	0.387601	0.0123250
$990$	0	0
$991$	29.5793	0.939618	0.469809	−	0.882768i	$-0.344323\pi$
$991$	29.5793	0.939618	0.469809	+	0.882768i	$0.344323\pi$
$992$	0	0
$993$	27.2963	0.866223
$994$	0	0
$995$	10.1271	0.321050
$996$	0	0
$997$	40.8901	1.29500	0.647502	−	0.762064i	$-0.275814\pi$
$997$	40.8901	1.29500	0.647502	+	0.762064i	$0.275814\pi$
$998$	0	0
$999$	−5.57165	−0.176279

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9264.2.a.bp.1.10		13
4.3	odd	2		579.2.a.g.1.5	✓	13
12.11	even	2		1737.2.a.j.1.9		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
579.2.a.g.1.5	✓	13	4.3	odd	2
1737.2.a.j.1.9		13	12.11	even	2
9264.2.a.bp.1.10		13	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:	\( N \)	\(=\)	\( 9264 = 2^{4} \cdot 3 \cdot 193 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9264.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.01396 q^{5} -3.77896 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.01396 q^{5} -3.77896 q^{7} +1.00000 q^{9} -4.93419 q^{11} -5.50864 q^{13} -3.01396 q^{15} -7.70179 q^{17} -5.52057 q^{19} +3.77896 q^{21} +7.92401 q^{23} +4.08397 q^{25} -1.00000 q^{27} +4.62451 q^{29} -3.53869 q^{31} +4.93419 q^{33} -11.3896 q^{35} +5.57165 q^{37} +5.50864 q^{39} -4.18854 q^{41} +0.0489147 q^{43} +3.01396 q^{45} -8.14508 q^{47} +7.28054 q^{49} +7.70179 q^{51} -12.7647 q^{53} -14.8715 q^{55} +5.52057 q^{57} +2.57352 q^{59} -6.68322 q^{61} -3.77896 q^{63} -16.6028 q^{65} +6.27347 q^{67} -7.92401 q^{69} -0.729787 q^{71} +3.26717 q^{73} -4.08397 q^{75} +18.6461 q^{77} -13.5577 q^{79} +1.00000 q^{81} +9.98298 q^{83} -23.2129 q^{85} -4.62451 q^{87} +3.67690 q^{89} +20.8169 q^{91} +3.53869 q^{93} -16.6388 q^{95} +0.726031 q^{97} -4.93419 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{3} + 6 q^{5} - 15 q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q - 13 * q^3 + 6 * q^5 - 15 * q^7 + 13 * q^9	\( 13 q - 13 q^{3} + 6 q^{5} - 15 q^{7} + 13 q^{9} - 3 q^{11} + 11 q^{13} - 6 q^{15} - 2 q^{17} - 9 q^{19} + 15 q^{21} + 8 q^{23} + 21 q^{25} - 13 q^{27} + 5 q^{29} - 25 q^{31} + 3 q^{33} + 10 q^{35} + 29 q^{37} - 11 q^{39} - 11 q^{41} - 8 q^{43} + 6 q^{45} + 12 q^{47} + 20 q^{49} + 2 q^{51} + 14 q^{53} - 12 q^{55} + 9 q^{57} - 10 q^{59} + 6 q^{61} - 15 q^{63} - 15 q^{65} - 25 q^{67} - 8 q^{69} + 8 q^{73} - 21 q^{75} - 25 q^{77} - 7 q^{79} + 13 q^{81} + 28 q^{83} - 3 q^{85} - 5 q^{87} + 7 q^{89} - 7 q^{91} + 25 q^{93} + 26 q^{95} + 26 q^{97} - 3 q^{99}+O(q^{100}) \) 13 * q - 13 * q^3 + 6 * q^5 - 15 * q^7 + 13 * q^9 - 3 * q^11 + 11 * q^13 - 6 * q^15 - 2 * q^17 - 9 * q^19 + 15 * q^21 + 8 * q^23 + 21 * q^25 - 13 * q^27 + 5 * q^29 - 25 * q^31 + 3 * q^33 + 10 * q^35 + 29 * q^37 - 11 * q^39 - 11 * q^41 - 8 * q^43 + 6 * q^45 + 12 * q^47 + 20 * q^49 + 2 * q^51 + 14 * q^53 - 12 * q^55 + 9 * q^57 - 10 * q^59 + 6 * q^61 - 15 * q^63 - 15 * q^65 - 25 * q^67 - 8 * q^69 + 8 * q^73 - 21 * q^75 - 25 * q^77 - 7 * q^79 + 13 * q^81 + 28 * q^83 - 3 * q^85 - 5 * q^87 + 7 * q^89 - 7 * q^91 + 25 * q^93 + 26 * q^95 + 26 * q^97 - 3 * q^99

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