LMFDB - Embedded newform 9464.2.a.br.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9464,2,Mod(1,9464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9464 = 2^{3} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,0,-3,0,-4,0,-15,0,16,0,-15,0,0,0,-8,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$75.5704204729$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 26 x^{13} + 78 x^{12} + 253 x^{11} - 782 x^{10} - 1087 x^{9} + 3776 x^{8} + \cdots - 344 $ x^15 - 3x^14 - 26x^13 + 78x^12 + 253x^11 - 782x^10 - 1087x^9 + 3776x^8 + 1576x^7 - 8849x^6 + 1791x^5 + 8496x^4 - 5353x^3 - 1252x^2 + 1724x - 344
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.05916$ of defining polynomial
Character	$\chi$	$=$	9464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.05916 q^{3} +1.23771 q^{5} -1.00000 q^{7} -1.87817 q^{9} +1.37211 q^{11} -1.31094 q^{15} +1.41900 q^{17} +6.61988 q^{19} +1.05916 q^{21} -5.86724 q^{23} -3.46807 q^{25} +5.16678 q^{27} -5.41201 q^{29} +3.88551 q^{31} -1.45329 q^{33} -1.23771 q^{35} -0.373505 q^{37} +1.42901 q^{41} -5.83736 q^{43} -2.32464 q^{45} +0.595667 q^{47} +1.00000 q^{49} -1.50295 q^{51} +4.15776 q^{53} +1.69828 q^{55} -7.01153 q^{57} -0.154867 q^{59} +2.86393 q^{61} +1.87817 q^{63} -13.3682 q^{67} +6.21436 q^{69} -2.63494 q^{71} -3.56409 q^{73} +3.67325 q^{75} -1.37211 q^{77} -6.55447 q^{79} +0.162065 q^{81} -0.649696 q^{83} +1.75631 q^{85} +5.73220 q^{87} +6.18665 q^{89} -4.11539 q^{93} +8.19350 q^{95} +18.1559 q^{97} -2.57707 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 3 q^{3} - 4 q^{5} - 15 q^{7} + 16 q^{9} - 15 q^{11} - 8 q^{15} + 2 q^{17} - 13 q^{19} + 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} - 19 q^{31} + 24 q^{33} + 4 q^{35} + 2 q^{37} - 30 q^{41}+ \cdots - 70 q^{99}+O(q^{100}) $ 15 * q - 3 * q^3 - 4 * q^5 - 15 * q^7 + 16 * q^9 - 15 * q^11 - 8 * q^15 + 2 * q^17 - 13 * q^19 + 3 * q^21 - 10 * q^23 + 23 * q^25 - 9 * q^27 + 25 * q^29 - 19 * q^31 + 24 * q^33 + 4 * q^35 + 2 * q^37 - 30 * q^41 + 15 * q^43 + 2 * q^45 - 13 * q^47 + 15 * q^49 + q^51 + 62 * q^53 - 3 * q^55 + 2 * q^57 - 22 * q^59 - q^61 - 16 * q^63 - 29 * q^67 + 32 * q^69 - 20 * q^71 - 21 * q^73 - 46 * q^75 + 15 * q^77 + 29 * q^79 + 7 * q^81 - 23 * q^83 - 8 * q^85 - 22 * q^87 - 9 * q^89 - 10 * q^93 + 28 * q^95 - 11 * q^97 - 70 * q^99

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
$3$	−1.05916	−0.611508	−0.305754	−	0.952111i	$-0.598908\pi$
$3$	−1.05916	−0.611508	−0.305754	+	0.952111i	$0.598908\pi$
$4$	0	0
$5$	1.23771	0.553521	0.276760	−	0.960939i	$-0.410739\pi$
$5$	1.23771	0.553521	0.276760	+	0.960939i	$0.410739\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.87817	−0.626058
$10$	0	0
$11$	1.37211	0.413707	0.206854	−	0.978372i	$-0.433678\pi$
$11$	1.37211	0.413707	0.206854	+	0.978372i	$0.433678\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−1.31094	−0.338482
$16$	0	0
$17$	1.41900	0.344159	0.172079	−	0.985083i	$-0.444951\pi$
$17$	1.41900	0.344159	0.172079	+	0.985083i	$0.444951\pi$
$18$	0	0
$19$	6.61988	1.51871	0.759353	−	0.650679i	$-0.225516\pi$
$19$	6.61988	1.51871	0.759353	+	0.650679i	$0.225516\pi$
$20$	0	0
$21$	1.05916	0.231128
$22$	0	0
$23$	−5.86724	−1.22340	−0.611702	−	0.791089i	$-0.709515\pi$
$23$	−5.86724	−1.22340	−0.611702	+	0.791089i	$0.709515\pi$
$24$	0	0
$25$	−3.46807	−0.693615
$26$	0	0
$27$	5.16678	0.994347
$28$	0	0
$29$	−5.41201	−1.00498	−0.502492	−	0.864582i	$-0.667584\pi$
$29$	−5.41201	−1.00498	−0.502492	+	0.864582i	$0.667584\pi$
$30$	0	0
$31$	3.88551	0.697859	0.348929	−	0.937149i	$-0.386545\pi$
$31$	3.88551	0.697859	0.348929	+	0.937149i	$0.386545\pi$
$32$	0	0
$33$	−1.45329	−0.252985
$34$	0	0
$35$	−1.23771	−0.209211
$36$	0	0
$37$	−0.373505	−0.0614038	−0.0307019	−	0.999529i	$-0.509774\pi$
$37$	−0.373505	−0.0614038	−0.0307019	+	0.999529i	$0.509774\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.42901	0.223174	0.111587	−	0.993755i	$-0.464407\pi$
$41$	1.42901	0.223174	0.111587	+	0.993755i	$0.464407\pi$
$42$	0	0
$43$	−5.83736	−0.890189	−0.445095	−	0.895484i	$-0.646830\pi$
$43$	−5.83736	−0.890189	−0.445095	+	0.895484i	$0.646830\pi$
$44$	0	0
$45$	−2.32464	−0.346536
$46$	0	0
$47$	0.595667	0.0868869	0.0434435	−	0.999056i	$-0.486167\pi$
$47$	0.595667	0.0868869	0.0434435	+	0.999056i	$0.486167\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.50295	−0.210456
$52$	0	0
$53$	4.15776	0.571112	0.285556	−	0.958362i	$-0.407822\pi$
$53$	4.15776	0.571112	0.285556	+	0.958362i	$0.407822\pi$
$54$	0	0
$55$	1.69828	0.228996
$56$	0	0
$57$	−7.01153	−0.928700
$58$	0	0
$59$	−0.154867	−0.0201619	−0.0100810	−	0.999949i	$-0.503209\pi$
$59$	−0.154867	−0.0201619	−0.0100810	+	0.999949i	$0.503209\pi$
$60$	0	0
$61$	2.86393	0.366688	0.183344	−	0.983049i	$-0.441308\pi$
$61$	2.86393	0.366688	0.183344	+	0.983049i	$0.441308\pi$
$62$	0	0
$63$	1.87817	0.236628
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−13.3682	−1.63319	−0.816595	−	0.577211i	$-0.804141\pi$
$67$	−13.3682	−1.63319	−0.816595	+	0.577211i	$0.804141\pi$
$68$	0	0
$69$	6.21436	0.748121
$70$	0	0
$71$	−2.63494	−0.312710	−0.156355	−	0.987701i	$-0.549974\pi$
$71$	−2.63494	−0.312710	−0.156355	+	0.987701i	$0.549974\pi$
$72$	0	0
$73$	−3.56409	−0.417145	−0.208572	−	0.978007i	$-0.566882\pi$
$73$	−3.56409	−0.417145	−0.208572	+	0.978007i	$0.566882\pi$
$74$	0	0
$75$	3.67325	0.424151
$76$	0	0
$77$	−1.37211	−0.156367
$78$	0	0
$79$	−6.55447	−0.737436	−0.368718	−	0.929541i	$-0.620203\pi$
$79$	−6.55447	−0.737436	−0.368718	+	0.929541i	$0.620203\pi$
$80$	0	0
$81$	0.162065	0.0180072
$82$	0	0
$83$	−0.649696	−0.0713134	−0.0356567	−	0.999364i	$-0.511352\pi$
$83$	−0.649696	−0.0713134	−0.0356567	+	0.999364i	$0.511352\pi$
$84$	0	0
$85$	1.75631	0.190499
$86$	0	0
$87$	5.73220	0.614556
$88$	0	0
$89$	6.18665	0.655783	0.327892	−	0.944715i	$-0.393662\pi$
$89$	6.18665	0.655783	0.327892	+	0.944715i	$0.393662\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−4.11539	−0.426746
$94$	0	0
$95$	8.19350	0.840635
$96$	0	0
$97$	18.1559	1.84346	0.921728	−	0.387836i	$-0.126777\pi$
$97$	18.1559	1.84346	0.921728	+	0.387836i	$0.126777\pi$
$98$	0	0
$99$	−2.57707	−0.259005
$100$	0	0
$101$	−7.85848	−0.781948	−0.390974	−	0.920402i	$-0.627862\pi$
$101$	−7.85848	−0.781948	−0.390974	+	0.920402i	$0.627862\pi$
$102$	0	0
$103$	15.9947	1.57601	0.788004	−	0.615670i	$-0.211115\pi$
$103$	15.9947	1.57601	0.788004	+	0.615670i	$0.211115\pi$
$104$	0	0
$105$	1.31094	0.127934
$106$	0	0
$107$	0.184176	0.0178050	0.00890248	−	0.999960i	$-0.497166\pi$
$107$	0.184176	0.0178050	0.00890248	+	0.999960i	$0.497166\pi$
$108$	0	0
$109$	0.0168999	0.00161872	0.000809360	−	1.00000i	$-0.499742\pi$
$109$	0.0168999	0.00161872	0.000809360		1.00000i	$0.499742\pi$
$110$	0	0
$111$	0.395602	0.0375489
$112$	0	0
$113$	15.2006	1.42996	0.714978	−	0.699147i	$-0.246437\pi$
$113$	15.2006	1.42996	0.714978	+	0.699147i	$0.246437\pi$
$114$	0	0
$115$	−7.26194	−0.677179
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.41900	−0.130080
$120$	0	0
$121$	−9.11731	−0.828846
$122$	0	0
$123$	−1.51355	−0.136472
$124$	0	0
$125$	−10.4810	−0.937451
$126$	0	0
$127$	−18.1868	−1.61382	−0.806910	−	0.590675i	$-0.798862\pi$
$127$	−18.1868	−1.61382	−0.806910	+	0.590675i	$0.798862\pi$
$128$	0	0
$129$	6.18271	0.544358
$130$	0	0
$131$	−12.4168	−1.08486	−0.542430	−	0.840101i	$-0.682495\pi$
$131$	−12.4168	−1.08486	−0.542430	+	0.840101i	$0.682495\pi$
$132$	0	0
$133$	−6.61988	−0.574017
$134$	0	0
$135$	6.39498	0.550392
$136$	0	0
$137$	3.95141	0.337592	0.168796	−	0.985651i	$-0.446012\pi$
$137$	3.95141	0.337592	0.168796	+	0.985651i	$0.446012\pi$
$138$	0	0
$139$	−12.7364	−1.08028	−0.540142	−	0.841574i	$-0.681630\pi$
$139$	−12.7364	−1.08028	−0.540142	+	0.841574i	$0.681630\pi$
$140$	0	0
$141$	−0.630908	−0.0531320
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−6.69850	−0.556280
$146$	0	0
$147$	−1.05916	−0.0873583
$148$	0	0
$149$	15.2204	1.24691	0.623453	−	0.781861i	$-0.285729\pi$
$149$	15.2204	1.24691	0.623453	+	0.781861i	$0.285729\pi$
$150$	0	0
$151$	−2.99459	−0.243697	−0.121848	−	0.992549i	$-0.538882\pi$
$151$	−2.99459	−0.243697	−0.121848	+	0.992549i	$0.538882\pi$
$152$	0	0
$153$	−2.66514	−0.215463
$154$	0	0
$155$	4.80914	0.386279
$156$	0	0
$157$	23.6841	1.89020	0.945098	−	0.326786i	$-0.105966\pi$
$157$	23.6841	1.89020	0.945098	+	0.326786i	$0.105966\pi$
$158$	0	0
$159$	−4.40374	−0.349239
$160$	0	0
$161$	5.86724	0.462403
$162$	0	0
$163$	−22.3927	−1.75393	−0.876965	−	0.480555i	$-0.840435\pi$
$163$	−22.3927	−1.75393	−0.876965	+	0.480555i	$0.840435\pi$
$164$	0	0
$165$	−1.79875	−0.140033
$166$	0	0
$167$	6.87488	0.531994	0.265997	−	0.963974i	$-0.414299\pi$
$167$	6.87488	0.531994	0.265997	+	0.963974i	$0.414299\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−12.4333	−0.950798
$172$	0	0
$173$	1.08791	0.0827123	0.0413562	−	0.999144i	$-0.486832\pi$
$173$	1.08791	0.0827123	0.0413562	+	0.999144i	$0.486832\pi$
$174$	0	0
$175$	3.46807	0.262162
$176$	0	0
$177$	0.164029	0.0123292
$178$	0	0
$179$	−9.78052	−0.731030	−0.365515	−	0.930805i	$-0.619107\pi$
$179$	−9.78052	−0.731030	−0.365515	+	0.930805i	$0.619107\pi$
$180$	0	0
$181$	3.60262	0.267781	0.133890	−	0.990996i	$-0.457253\pi$
$181$	3.60262	0.267781	0.133890	+	0.990996i	$0.457253\pi$
$182$	0	0
$183$	−3.03336	−0.224233
$184$	0	0
$185$	−0.462291	−0.0339883
$186$	0	0
$187$	1.94703	0.142381
$188$	0	0
$189$	−5.16678	−0.375828
$190$	0	0
$191$	−12.2511	−0.886457	−0.443228	−	0.896409i	$-0.646167\pi$
$191$	−12.2511	−0.886457	−0.443228	+	0.896409i	$0.646167\pi$
$192$	0	0
$193$	−1.07798	−0.0775948	−0.0387974	−	0.999247i	$-0.512353\pi$
$193$	−1.07798	−0.0775948	−0.0387974	+	0.999247i	$0.512353\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.50815	−0.677427	−0.338714	−	0.940890i	$-0.609992\pi$
$197$	−9.50815	−0.677427	−0.338714	+	0.940890i	$0.609992\pi$
$198$	0	0
$199$	−0.893041	−0.0633060	−0.0316530	−	0.999499i	$-0.510077\pi$
$199$	−0.893041	−0.0633060	−0.0316530	+	0.999499i	$0.510077\pi$
$200$	0	0
$201$	14.1591	0.998708
$202$	0	0
$203$	5.41201	0.379849
$204$	0	0
$205$	1.76870	0.123531
$206$	0	0
$207$	11.0197	0.765922
$208$	0	0
$209$	9.08322	0.628299
$210$	0	0
$211$	−13.7160	−0.944248	−0.472124	−	0.881532i	$-0.656513\pi$
$211$	−13.7160	−0.944248	−0.472124	+	0.881532i	$0.656513\pi$
$212$	0	0
$213$	2.79083	0.191224
$214$	0	0
$215$	−7.22496	−0.492738
$216$	0	0
$217$	−3.88551	−0.263766
$218$	0	0
$219$	3.77495	0.255087
$220$	0	0
$221$	0	0
$222$	0	0
$223$	12.6297	0.845744	0.422872	−	0.906189i	$-0.361022\pi$
$223$	12.6297	0.845744	0.422872	+	0.906189i	$0.361022\pi$
$224$	0	0
$225$	6.51365	0.434243
$226$	0	0
$227$	−16.1615	−1.07268	−0.536339	−	0.844003i	$-0.680193\pi$
$227$	−16.1615	−1.07268	−0.536339	+	0.844003i	$0.680193\pi$
$228$	0	0
$229$	−5.98521	−0.395514	−0.197757	−	0.980251i	$-0.563366\pi$
$229$	−5.98521	−0.395514	−0.197757	+	0.980251i	$0.563366\pi$
$230$	0	0
$231$	1.45329	0.0956194
$232$	0	0
$233$	22.5399	1.47664	0.738319	−	0.674451i	$-0.235620\pi$
$233$	22.5399	1.47664	0.738319	+	0.674451i	$0.235620\pi$
$234$	0	0
$235$	0.737263	0.0480937
$236$	0	0
$237$	6.94225	0.450948
$238$	0	0
$239$	3.97115	0.256873	0.128436	−	0.991718i	$-0.459004\pi$
$239$	3.97115	0.256873	0.128436	+	0.991718i	$0.459004\pi$
$240$	0	0
$241$	−24.2510	−1.56214	−0.781071	−	0.624442i	$-0.785327\pi$
$241$	−24.2510	−1.56214	−0.781071	+	0.624442i	$0.785327\pi$
$242$	0	0
$243$	−15.6720	−1.00536
$244$	0	0
$245$	1.23771	0.0790744
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.688134	0.0436087
$250$	0	0
$251$	9.40345	0.593541	0.296770	−	0.954949i	$-0.404090\pi$
$251$	9.40345	0.593541	0.296770	+	0.954949i	$0.404090\pi$
$252$	0	0
$253$	−8.05050	−0.506131
$254$	0	0
$255$	−1.86022	−0.116492
$256$	0	0
$257$	−20.1983	−1.25993	−0.629967	−	0.776622i	$-0.716932\pi$
$257$	−20.1983	−1.25993	−0.629967	+	0.776622i	$0.716932\pi$
$258$	0	0
$259$	0.373505	0.0232085
$260$	0	0
$261$	10.1647	0.629179
$262$	0	0
$263$	−12.6854	−0.782213	−0.391107	−	0.920345i	$-0.627908\pi$
$263$	−12.6854	−0.782213	−0.391107	+	0.920345i	$0.627908\pi$
$264$	0	0
$265$	5.14610	0.316122
$266$	0	0
$267$	−6.55266	−0.401017
$268$	0	0
$269$	−17.7217	−1.08051	−0.540255	−	0.841502i	$-0.681672\pi$
$269$	−17.7217	−1.08051	−0.540255	+	0.841502i	$0.681672\pi$
$270$	0	0
$271$	−18.5887	−1.12918	−0.564591	−	0.825371i	$-0.690966\pi$
$271$	−18.5887	−1.12918	−0.564591	+	0.825371i	$0.690966\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.75858	−0.286953
$276$	0	0
$277$	17.0438	1.02406	0.512032	−	0.858966i	$-0.328893\pi$
$277$	17.0438	1.02406	0.512032	+	0.858966i	$0.328893\pi$
$278$	0	0
$279$	−7.29767	−0.436900
$280$	0	0
$281$	−21.0737	−1.25715	−0.628575	−	0.777749i	$-0.716361\pi$
$281$	−21.0737	−1.25715	−0.628575	+	0.777749i	$0.716361\pi$
$282$	0	0
$283$	7.01726	0.417133	0.208566	−	0.978008i	$-0.433120\pi$
$283$	7.01726	0.417133	0.208566	+	0.978008i	$0.433120\pi$
$284$	0	0
$285$	−8.67825	−0.514055
$286$	0	0
$287$	−1.42901	−0.0843517
$288$	0	0
$289$	−14.9864	−0.881555
$290$	0	0
$291$	−19.2301	−1.12729
$292$	0	0
$293$	−14.2645	−0.833342	−0.416671	−	0.909057i	$-0.636803\pi$
$293$	−14.2645	−0.833342	−0.416671	+	0.909057i	$0.636803\pi$
$294$	0	0
$295$	−0.191680	−0.0111600
$296$	0	0
$297$	7.08940	0.411369
$298$	0	0
$299$	0	0
$300$	0	0
$301$	5.83736	0.336460
$302$	0	0
$303$	8.32341	0.478168
$304$	0	0
$305$	3.54471	0.202970
$306$	0	0
$307$	−9.89226	−0.564581	−0.282291	−	0.959329i	$-0.591094\pi$
$307$	−9.89226	−0.564581	−0.282291	+	0.959329i	$0.591094\pi$
$308$	0	0
$309$	−16.9410	−0.963741
$310$	0	0
$311$	6.47324	0.367064	0.183532	−	0.983014i	$-0.441247\pi$
$311$	6.47324	0.367064	0.183532	+	0.983014i	$0.441247\pi$
$312$	0	0
$313$	−3.66440	−0.207124	−0.103562	−	0.994623i	$-0.533024\pi$
$313$	−3.66440	−0.207124	−0.103562	+	0.994623i	$0.533024\pi$
$314$	0	0
$315$	2.32464	0.130978
$316$	0	0
$317$	−3.91107	−0.219668	−0.109834	−	0.993950i	$-0.535032\pi$
$317$	−3.91107	−0.219668	−0.109834	+	0.993950i	$0.535032\pi$
$318$	0	0
$319$	−7.42588	−0.415770
$320$	0	0
$321$	−0.195072	−0.0108879
$322$	0	0
$323$	9.39363	0.522676
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.0178998	−0.000989860 0
$328$	0	0
$329$	−0.595667	−0.0328402
$330$	0	0
$331$	−15.1164	−0.830874	−0.415437	−	0.909622i	$-0.636371\pi$
$331$	−15.1164	−0.830874	−0.415437	+	0.909622i	$0.636371\pi$
$332$	0	0
$333$	0.701507	0.0384424
$334$	0	0
$335$	−16.5460	−0.904005
$336$	0	0
$337$	−18.6325	−1.01498	−0.507488	−	0.861659i	$-0.669426\pi$
$337$	−18.6325	−1.01498	−0.507488	+	0.861659i	$0.669426\pi$
$338$	0	0
$339$	−16.1000	−0.874429
$340$	0	0
$341$	5.33136	0.288709
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	7.69157	0.414100
$346$	0	0
$347$	8.93281	0.479539	0.239769	−	0.970830i	$-0.422928\pi$
$347$	8.93281	0.479539	0.239769	+	0.970830i	$0.422928\pi$
$348$	0	0
$349$	9.08280	0.486191	0.243096	−	0.970002i	$-0.421837\pi$
$349$	9.08280	0.486191	0.243096	+	0.970002i	$0.421837\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.80414	−0.362148	−0.181074	−	0.983469i	$-0.557957\pi$
$353$	−6.80414	−0.362148	−0.181074	+	0.983469i	$0.557957\pi$
$354$	0	0
$355$	−3.26129	−0.173091
$356$	0	0
$357$	1.50295	0.0795448
$358$	0	0
$359$	12.1227	0.639813	0.319906	−	0.947449i	$-0.396349\pi$
$359$	12.1227	0.639813	0.319906	+	0.947449i	$0.396349\pi$
$360$	0	0
$361$	24.8228	1.30647
$362$	0	0
$363$	9.65671	0.506846
$364$	0	0
$365$	−4.41131	−0.230898
$366$	0	0
$367$	−7.97685	−0.416388	−0.208194	−	0.978088i	$-0.566759\pi$
$367$	−7.97685	−0.416388	−0.208194	+	0.978088i	$0.566759\pi$
$368$	0	0
$369$	−2.68393	−0.139720
$370$	0	0
$371$	−4.15776	−0.215860
$372$	0	0
$373$	23.3848	1.21082	0.605409	−	0.795915i	$-0.293010\pi$
$373$	23.3848	1.21082	0.605409	+	0.795915i	$0.293010\pi$
$374$	0	0
$375$	11.1011	0.573259
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−13.4075	−0.688700	−0.344350	−	0.938841i	$-0.611901\pi$
$379$	−13.4075	−0.688700	−0.344350	+	0.938841i	$0.611901\pi$
$380$	0	0
$381$	19.2628	0.986863
$382$	0	0
$383$	15.6316	0.798737	0.399369	−	0.916790i	$-0.369229\pi$
$383$	15.6316	0.798737	0.399369	+	0.916790i	$0.369229\pi$
$384$	0	0
$385$	−1.69828	−0.0865522
$386$	0	0
$387$	10.9636	0.557310
$388$	0	0
$389$	28.0188	1.42061	0.710306	−	0.703893i	$-0.248557\pi$
$389$	28.0188	1.42061	0.710306	+	0.703893i	$0.248557\pi$
$390$	0	0
$391$	−8.32562	−0.421045
$392$	0	0
$393$	13.1514	0.663400
$394$	0	0
$395$	−8.11254	−0.408186
$396$	0	0
$397$	2.48649	0.124793	0.0623966	−	0.998051i	$-0.480126\pi$
$397$	2.48649	0.124793	0.0623966	+	0.998051i	$0.480126\pi$
$398$	0	0
$399$	7.01153	0.351016
$400$	0	0
$401$	−21.1731	−1.05734	−0.528668	−	0.848829i	$-0.677308\pi$
$401$	−21.1731	−1.05734	−0.528668	+	0.848829i	$0.677308\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0.200589	0.00996734
$406$	0	0
$407$	−0.512490	−0.0254032
$408$	0	0
$409$	3.01891	0.149276	0.0746378	−	0.997211i	$-0.476220\pi$
$409$	3.01891	0.149276	0.0746378	+	0.997211i	$0.476220\pi$
$410$	0	0
$411$	−4.18519	−0.206440
$412$	0	0
$413$	0.154867	0.00762049
$414$	0	0
$415$	−0.804136	−0.0394735
$416$	0	0
$417$	13.4899	0.660603
$418$	0	0
$419$	−25.5288	−1.24716	−0.623582	−	0.781758i	$-0.714323\pi$
$419$	−25.5288	−1.24716	−0.623582	+	0.781758i	$0.714323\pi$
$420$	0	0
$421$	−17.7108	−0.863169	−0.431585	−	0.902072i	$-0.642045\pi$
$421$	−17.7108	−0.863169	−0.431585	+	0.902072i	$0.642045\pi$
$422$	0	0
$423$	−1.11877	−0.0543963
$424$	0	0
$425$	−4.92121	−0.238714
$426$	0	0
$427$	−2.86393	−0.138595
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.4492	−1.17768	−0.588838	−	0.808251i	$-0.700414\pi$
$431$	−24.4492	−1.17768	−0.588838	+	0.808251i	$0.700414\pi$
$432$	0	0
$433$	27.0243	1.29870	0.649351	−	0.760489i	$-0.275040\pi$
$433$	27.0243	1.29870	0.649351	+	0.760489i	$0.275040\pi$
$434$	0	0
$435$	7.09480	0.340170
$436$	0	0
$437$	−38.8404	−1.85799
$438$	0	0
$439$	−18.7592	−0.895326	−0.447663	−	0.894202i	$-0.647744\pi$
$439$	−18.7592	−0.895326	−0.447663	+	0.894202i	$0.647744\pi$
$440$	0	0
$441$	−1.87817	−0.0894369
$442$	0	0
$443$	11.7308	0.557346	0.278673	−	0.960386i	$-0.410105\pi$
$443$	11.7308	0.557346	0.278673	+	0.960386i	$0.410105\pi$
$444$	0	0
$445$	7.65728	0.362990
$446$	0	0
$447$	−16.1209	−0.762493
$448$	0	0
$449$	−28.1773	−1.32977	−0.664884	−	0.746947i	$-0.731519\pi$
$449$	−28.1773	−1.32977	−0.664884	+	0.746947i	$0.731519\pi$
$450$	0	0
$451$	1.96076	0.0923285
$452$	0	0
$453$	3.17176	0.149022
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−30.8561	−1.44339	−0.721694	−	0.692213i	$-0.756636\pi$
$457$	−30.8561	−1.44339	−0.721694	+	0.692213i	$0.756636\pi$
$458$	0	0
$459$	7.33168	0.342213
$460$	0	0
$461$	−14.3631	−0.668954	−0.334477	−	0.942404i	$-0.608560\pi$
$461$	−14.3631	−0.668954	−0.334477	+	0.942404i	$0.608560\pi$
$462$	0	0
$463$	−7.95490	−0.369696	−0.184848	−	0.982767i	$-0.559179\pi$
$463$	−7.95490	−0.369696	−0.184848	+	0.982767i	$0.559179\pi$
$464$	0	0
$465$	−5.09366	−0.236213
$466$	0	0
$467$	20.4167	0.944770	0.472385	−	0.881392i	$-0.343393\pi$
$467$	20.4167	0.944770	0.472385	+	0.881392i	$0.343393\pi$
$468$	0	0
$469$	13.3682	0.617288
$470$	0	0
$471$	−25.0853	−1.15587
$472$	0	0
$473$	−8.00951	−0.368278
$474$	0	0
$475$	−22.9582	−1.05340
$476$	0	0
$477$	−7.80900	−0.357549
$478$	0	0
$479$	27.6412	1.26296	0.631480	−	0.775393i	$-0.282448\pi$
$479$	27.6412	1.26296	0.631480	+	0.775393i	$0.282448\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−6.21436	−0.282763
$484$	0	0
$485$	22.4718	1.02039
$486$	0	0
$487$	22.5413	1.02144	0.510722	−	0.859746i	$-0.329378\pi$
$487$	22.5413	1.02144	0.510722	+	0.859746i	$0.329378\pi$
$488$	0	0
$489$	23.7175	1.07254
$490$	0	0
$491$	−20.2460	−0.913690	−0.456845	−	0.889546i	$-0.651021\pi$
$491$	−20.2460	−0.913690	−0.456845	+	0.889546i	$0.651021\pi$
$492$	0	0
$493$	−7.67966	−0.345874
$494$	0	0
$495$	−3.18966	−0.143365
$496$	0	0
$497$	2.63494	0.118193
$498$	0	0
$499$	2.39238	0.107098	0.0535489	−	0.998565i	$-0.482947\pi$
$499$	2.39238	0.107098	0.0535489	+	0.998565i	$0.482947\pi$
$500$	0	0
$501$	−7.28161	−0.325319
$502$	0	0
$503$	−26.6591	−1.18867	−0.594335	−	0.804217i	$-0.702585\pi$
$503$	−26.6591	−1.18867	−0.594335	+	0.804217i	$0.702585\pi$
$504$	0	0
$505$	−9.72653	−0.432825
$506$	0	0
$507$	0	0
$508$	0	0
$509$	41.2970	1.83046	0.915228	−	0.402937i	$-0.132011\pi$
$509$	41.2970	1.83046	0.915228	+	0.402937i	$0.132011\pi$
$510$	0	0
$511$	3.56409	0.157666
$512$	0	0
$513$	34.2035	1.51012
$514$	0	0
$515$	19.7969	0.872354
$516$	0	0
$517$	0.817321	0.0359458
$518$	0	0
$519$	−1.15227	−0.0505792
$520$	0	0
$521$	−12.4060	−0.543516	−0.271758	−	0.962366i	$-0.587605\pi$
$521$	−12.4060	−0.543516	−0.271758	+	0.962366i	$0.587605\pi$
$522$	0	0
$523$	−22.2391	−0.972448	−0.486224	−	0.873834i	$-0.661626\pi$
$523$	−22.2391	−0.972448	−0.486224	+	0.873834i	$0.661626\pi$
$524$	0	0
$525$	−3.67325	−0.160314
$526$	0	0
$527$	5.51355	0.240174
$528$	0	0
$529$	11.4245	0.496715
$530$	0	0
$531$	0.290867	0.0126225
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0.227957	0.00985542
$536$	0	0
$537$	10.3592	0.447031
$538$	0	0
$539$	1.37211	0.0591010
$540$	0	0
$541$	−26.9699	−1.15953	−0.579764	−	0.814785i	$-0.696855\pi$
$541$	−26.9699	−1.15953	−0.579764	+	0.814785i	$0.696855\pi$
$542$	0	0
$543$	−3.81576	−0.163750
$544$	0	0
$545$	0.0209172	0.000895996 0
$546$	0	0
$547$	−13.6736	−0.584642	−0.292321	−	0.956320i	$-0.594428\pi$
$547$	−13.6736	−0.584642	−0.292321	+	0.956320i	$0.594428\pi$
$548$	0	0
$549$	−5.37895	−0.229568
$550$	0	0
$551$	−35.8269	−1.52628
$552$	0	0
$553$	6.55447	0.278725
$554$	0	0
$555$	0.489641	0.0207841
$556$	0	0
$557$	−15.3542	−0.650577	−0.325288	−	0.945615i	$-0.605461\pi$
$557$	−15.3542	−0.650577	−0.325288	+	0.945615i	$0.605461\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−2.06222	−0.0870671
$562$	0	0
$563$	6.00693	0.253162	0.126581	−	0.991956i	$-0.459600\pi$
$563$	6.00693	0.253162	0.126581	+	0.991956i	$0.459600\pi$
$564$	0	0
$565$	18.8140	0.791511
$566$	0	0
$567$	−0.162065	−0.00680607
$568$	0	0
$569$	27.8052	1.16565	0.582827	−	0.812596i	$-0.301947\pi$
$569$	27.8052	1.16565	0.582827	+	0.812596i	$0.301947\pi$
$570$	0	0
$571$	35.0285	1.46590	0.732948	−	0.680285i	$-0.238144\pi$
$571$	35.0285	1.46590	0.732948	+	0.680285i	$0.238144\pi$
$572$	0	0
$573$	12.9759	0.542075
$574$	0	0
$575$	20.3480	0.848570
$576$	0	0
$577$	5.03316	0.209533	0.104767	−	0.994497i	$-0.466590\pi$
$577$	5.03316	0.209533	0.104767	+	0.994497i	$0.466590\pi$
$578$	0	0
$579$	1.14176	0.0474498
$580$	0	0
$581$	0.649696	0.0269539
$582$	0	0
$583$	5.70491	0.236273
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15.7257	0.649067	0.324534	−	0.945874i	$-0.394793\pi$
$587$	15.7257	0.649067	0.324534	+	0.945874i	$0.394793\pi$
$588$	0	0
$589$	25.7216	1.05984
$590$	0	0
$591$	10.0707	0.414252
$592$	0	0
$593$	−42.8984	−1.76162	−0.880812	−	0.473466i	$-0.843003\pi$
$593$	−42.8984	−1.76162	−0.880812	+	0.473466i	$0.843003\pi$
$594$	0	0
$595$	−1.75631	−0.0720019
$596$	0	0
$597$	0.945876	0.0387121
$598$	0	0
$599$	−38.6509	−1.57923	−0.789617	−	0.613600i	$-0.789721\pi$
$599$	−38.6509	−1.57923	−0.789617	+	0.613600i	$0.789721\pi$
$600$	0	0
$601$	0.845721	0.0344977	0.0172488	−	0.999851i	$-0.494509\pi$
$601$	0.845721	0.0344977	0.0172488	+	0.999851i	$0.494509\pi$
$602$	0	0
$603$	25.1079	1.02247
$604$	0	0
$605$	−11.2846	−0.458784
$606$	0	0
$607$	−8.66279	−0.351612	−0.175806	−	0.984425i	$-0.556253\pi$
$607$	−8.66279	−0.351612	−0.175806	+	0.984425i	$0.556253\pi$
$608$	0	0
$609$	−5.73220	−0.232280
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−8.03586	−0.324565	−0.162283	−	0.986744i	$-0.551886\pi$
$613$	−8.03586	−0.324565	−0.162283	+	0.986744i	$0.551886\pi$
$614$	0	0
$615$	−1.87334	−0.0755403
$616$	0	0
$617$	40.7197	1.63931	0.819657	−	0.572855i	$-0.194164\pi$
$617$	40.7197	1.63931	0.819657	+	0.572855i	$0.194164\pi$
$618$	0	0
$619$	15.6247	0.628010	0.314005	−	0.949421i	$-0.398329\pi$
$619$	15.6247	0.628010	0.314005	+	0.949421i	$0.398329\pi$
$620$	0	0
$621$	−30.3147	−1.21649
$622$	0	0
$623$	−6.18665	−0.247863
$624$	0	0
$625$	4.36789	0.174716
$626$	0	0
$627$	−9.62060	−0.384210
$628$	0	0
$629$	−0.530005	−0.0211327
$630$	0	0
$631$	−38.9641	−1.55114	−0.775568	−	0.631265i	$-0.782536\pi$
$631$	−38.9641	−1.55114	−0.775568	+	0.631265i	$0.782536\pi$
$632$	0	0
$633$	14.5275	0.577415
$634$	0	0
$635$	−22.5100	−0.893283
$636$	0	0
$637$	0	0
$638$	0	0
$639$	4.94887	0.195774
$640$	0	0
$641$	12.2964	0.485680	0.242840	−	0.970066i	$-0.421921\pi$
$641$	12.2964	0.485680	0.242840	+	0.970066i	$0.421921\pi$
$642$	0	0
$643$	−32.1534	−1.26801	−0.634003	−	0.773330i	$-0.718589\pi$
$643$	−32.1534	−1.26801	−0.634003	+	0.773330i	$0.718589\pi$
$644$	0	0
$645$	7.65241	0.301313
$646$	0	0
$647$	−10.8447	−0.426348	−0.213174	−	0.977014i	$-0.568380\pi$
$647$	−10.8447	−0.426348	−0.213174	+	0.977014i	$0.568380\pi$
$648$	0	0
$649$	−0.212494	−0.00834113
$650$	0	0
$651$	4.11539	0.161295
$652$	0	0
$653$	−12.4228	−0.486143	−0.243071	−	0.970008i	$-0.578155\pi$
$653$	−12.4228	−0.486143	−0.243071	+	0.970008i	$0.578155\pi$
$654$	0	0
$655$	−15.3684	−0.600492
$656$	0	0
$657$	6.69398	0.261157
$658$	0	0
$659$	−16.0704	−0.626016	−0.313008	−	0.949751i	$-0.601337\pi$
$659$	−16.0704	−0.626016	−0.313008	+	0.949751i	$0.601337\pi$
$660$	0	0
$661$	−26.2977	−1.02286	−0.511431	−	0.859324i	$-0.670884\pi$
$661$	−26.2977	−1.02286	−0.511431	+	0.859324i	$0.670884\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.19350	−0.317730
$666$	0	0
$667$	31.7535	1.22950
$668$	0	0
$669$	−13.3769	−0.517179
$670$	0	0
$671$	3.92963	0.151702
$672$	0	0
$673$	26.6022	1.02544	0.512719	−	0.858556i	$-0.328638\pi$
$673$	26.6022	1.02544	0.512719	+	0.858556i	$0.328638\pi$
$674$	0	0
$675$	−17.9188	−0.689694
$676$	0	0
$677$	33.9152	1.30347	0.651734	−	0.758447i	$-0.274042\pi$
$677$	33.9152	1.30347	0.651734	+	0.758447i	$0.274042\pi$
$678$	0	0
$679$	−18.1559	−0.696761
$680$	0	0
$681$	17.1177	0.655951
$682$	0	0
$683$	−9.67363	−0.370151	−0.185076	−	0.982724i	$-0.559253\pi$
$683$	−9.67363	−0.370151	−0.185076	+	0.982724i	$0.559253\pi$
$684$	0	0
$685$	4.89070	0.186864
$686$	0	0
$687$	6.33931	0.241860
$688$	0	0
$689$	0	0
$690$	0	0
$691$	41.8284	1.59123	0.795614	−	0.605804i	$-0.207149\pi$
$691$	41.8284	1.59123	0.795614	+	0.605804i	$0.207149\pi$
$692$	0	0
$693$	2.57707	0.0978946
$694$	0	0
$695$	−15.7639	−0.597960
$696$	0	0
$697$	2.02777	0.0768071
$698$	0	0
$699$	−23.8734	−0.902976
$700$	0	0
$701$	−49.1577	−1.85666	−0.928329	−	0.371759i	$-0.878755\pi$
$701$	−49.1577	−1.85666	−0.928329	+	0.371759i	$0.878755\pi$
$702$	0	0
$703$	−2.47256	−0.0932543
$704$	0	0
$705$	−0.780881	−0.0294097
$706$	0	0
$707$	7.85848	0.295549
$708$	0	0
$709$	−9.67548	−0.363370	−0.181685	−	0.983357i	$-0.558155\pi$
$709$	−9.67548	−0.363370	−0.181685	+	0.983357i	$0.558155\pi$
$710$	0	0
$711$	12.3104	0.461678
$712$	0	0
$713$	−22.7972	−0.853763
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−4.20610	−0.157080
$718$	0	0
$719$	−47.0901	−1.75617	−0.878083	−	0.478509i	$-0.841177\pi$
$719$	−47.0901	−1.75617	−0.878083	+	0.478509i	$0.841177\pi$
$720$	0	0
$721$	−15.9947	−0.595675
$722$	0	0
$723$	25.6857	0.955263
$724$	0	0
$725$	18.7692	0.697072
$726$	0	0
$727$	−10.5312	−0.390581	−0.195291	−	0.980745i	$-0.562565\pi$
$727$	−10.5312	−0.390581	−0.195291	+	0.980745i	$0.562565\pi$
$728$	0	0
$729$	16.1130	0.596778
$730$	0	0
$731$	−8.28323	−0.306366
$732$	0	0
$733$	−39.5113	−1.45938	−0.729692	−	0.683776i	$-0.760336\pi$
$733$	−39.5113	−1.45938	−0.729692	+	0.683776i	$0.760336\pi$
$734$	0	0
$735$	−1.31094	−0.0483546
$736$	0	0
$737$	−18.3427	−0.675662
$738$	0	0
$739$	7.50842	0.276202	0.138101	−	0.990418i	$-0.455900\pi$
$739$	7.50842	0.276202	0.138101	+	0.990418i	$0.455900\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.15873	0.152569	0.0762845	−	0.997086i	$-0.475694\pi$
$743$	4.15873	0.152569	0.0762845	+	0.997086i	$0.475694\pi$
$744$	0	0
$745$	18.8385	0.690189
$746$	0	0
$747$	1.22024	0.0446464
$748$	0	0
$749$	−0.184176	−0.00672964
$750$	0	0
$751$	9.26056	0.337923	0.168961	−	0.985623i	$-0.445959\pi$
$751$	9.26056	0.337923	0.168961	+	0.985623i	$0.445959\pi$
$752$	0	0
$753$	−9.95978	−0.362955
$754$	0	0
$755$	−3.70644	−0.134891
$756$	0	0
$757$	28.8481	1.04850	0.524251	−	0.851564i	$-0.324345\pi$
$757$	28.8481	1.04850	0.524251	+	0.851564i	$0.324345\pi$
$758$	0	0
$759$	8.52679	0.309503
$760$	0	0
$761$	8.91399	0.323132	0.161566	−	0.986862i	$-0.448346\pi$
$761$	8.91399	0.323132	0.161566	+	0.986862i	$0.448346\pi$
$762$	0	0
$763$	−0.0168999	−0.000611819 0
$764$	0	0
$765$	−3.29867	−0.119264
$766$	0	0
$767$	0	0
$768$	0	0
$769$	9.64102	0.347664	0.173832	−	0.984775i	$-0.444385\pi$
$769$	9.64102	0.347664	0.173832	+	0.984775i	$0.444385\pi$
$770$	0	0
$771$	21.3933	0.770460
$772$	0	0
$773$	−6.24691	−0.224686	−0.112343	−	0.993670i	$-0.535835\pi$
$773$	−6.24691	−0.224686	−0.112343	+	0.993670i	$0.535835\pi$
$774$	0	0
$775$	−13.4752	−0.484045
$776$	0	0
$777$	−0.395602	−0.0141922
$778$	0	0
$779$	9.45986	0.338935
$780$	0	0
$781$	−3.61543	−0.129370
$782$	0	0
$783$	−27.9627	−0.999304
$784$	0	0
$785$	29.3141	1.04626
$786$	0	0
$787$	−23.4253	−0.835020	−0.417510	−	0.908672i	$-0.637097\pi$
$787$	−23.4253	−0.835020	−0.417510	+	0.908672i	$0.637097\pi$
$788$	0	0
$789$	13.4359	0.478330
$790$	0	0
$791$	−15.2006	−0.540473
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−5.45056	−0.193311
$796$	0	0
$797$	−44.9075	−1.59070	−0.795352	−	0.606147i	$-0.792714\pi$
$797$	−44.9075	−1.59070	−0.795352	+	0.606147i	$0.792714\pi$
$798$	0	0
$799$	0.845253	0.0299029
$800$	0	0
$801$	−11.6196	−0.410558
$802$	0	0
$803$	−4.89033	−0.172576
$804$	0	0
$805$	7.26194	0.255950
$806$	0	0
$807$	18.7701	0.660740
$808$	0	0
$809$	10.2072	0.358865	0.179433	−	0.983770i	$-0.442574\pi$
$809$	10.2072	0.358865	0.179433	+	0.983770i	$0.442574\pi$
$810$	0	0
$811$	−4.52825	−0.159008	−0.0795041	−	0.996835i	$-0.525334\pi$
$811$	−4.52825	−0.159008	−0.0795041	+	0.996835i	$0.525334\pi$
$812$	0	0
$813$	19.6884	0.690503
$814$	0	0
$815$	−27.7156	−0.970836
$816$	0	0
$817$	−38.6426	−1.35193
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−25.0809	−0.875331	−0.437665	−	0.899138i	$-0.644195\pi$
$821$	−25.0809	−0.875331	−0.437665	+	0.899138i	$0.644195\pi$
$822$	0	0
$823$	25.3258	0.882801	0.441400	−	0.897310i	$-0.354482\pi$
$823$	25.3258	0.882801	0.441400	+	0.897310i	$0.354482\pi$
$824$	0	0
$825$	5.04011	0.175474
$826$	0	0
$827$	−28.5573	−0.993035	−0.496517	−	0.868027i	$-0.665388\pi$
$827$	−28.5573	−0.993035	−0.496517	+	0.868027i	$0.665388\pi$
$828$	0	0
$829$	9.91533	0.344374	0.172187	−	0.985064i	$-0.444917\pi$
$829$	9.91533	0.344374	0.172187	+	0.985064i	$0.444917\pi$
$830$	0	0
$831$	−18.0522	−0.626223
$832$	0	0
$833$	1.41900	0.0491655
$834$	0	0
$835$	8.50911	0.294470
$836$	0	0
$837$	20.0756	0.693914
$838$	0	0
$839$	−49.2813	−1.70138	−0.850690	−	0.525668i	$-0.823816\pi$
$839$	−49.2813	−1.70138	−0.850690	+	0.525668i	$0.823816\pi$
$840$	0	0
$841$	0.289845	0.00999466
$842$	0	0
$843$	22.3204	0.768757
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9.11731	0.313274
$848$	0	0
$849$	−7.43241	−0.255080
$850$	0	0
$851$	2.19144	0.0751216
$852$	0	0
$853$	−37.8435	−1.29574	−0.647869	−	0.761752i	$-0.724340\pi$
$853$	−37.8435	−1.29574	−0.647869	+	0.761752i	$0.724340\pi$
$854$	0	0
$855$	−15.3888	−0.526287
$856$	0	0
$857$	−19.8967	−0.679659	−0.339830	−	0.940487i	$-0.610369\pi$
$857$	−19.8967	−0.679659	−0.339830	+	0.940487i	$0.610369\pi$
$858$	0	0
$859$	−34.4152	−1.17423	−0.587115	−	0.809503i	$-0.699736\pi$
$859$	−34.4152	−1.17423	−0.587115	+	0.809503i	$0.699736\pi$
$860$	0	0
$861$	1.51355	0.0515817
$862$	0	0
$863$	−23.8186	−0.810794	−0.405397	−	0.914141i	$-0.632867\pi$
$863$	−23.8186	−0.810794	−0.405397	+	0.914141i	$0.632867\pi$
$864$	0	0
$865$	1.34652	0.0457830
$866$	0	0
$867$	15.8731	0.539078
$868$	0	0
$869$	−8.99347	−0.305083
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−34.1000	−1.15411
$874$	0	0
$875$	10.4810	0.354323
$876$	0	0
$877$	−27.3355	−0.923054	−0.461527	−	0.887126i	$-0.652698\pi$
$877$	−27.3355	−0.923054	−0.461527	+	0.887126i	$0.652698\pi$
$878$	0	0
$879$	15.1084	0.509595
$880$	0	0
$881$	−50.6952	−1.70796	−0.853982	−	0.520303i	$-0.825819\pi$
$881$	−50.6952	−1.70796	−0.853982	+	0.520303i	$0.825819\pi$
$882$	0	0
$883$	12.1465	0.408762	0.204381	−	0.978891i	$-0.434482\pi$
$883$	12.1465	0.408762	0.204381	+	0.978891i	$0.434482\pi$
$884$	0	0
$885$	0.203020	0.00682445
$886$	0	0
$887$	36.0249	1.20960	0.604799	−	0.796378i	$-0.293253\pi$
$887$	36.0249	1.20960	0.604799	+	0.796378i	$0.293253\pi$
$888$	0	0
$889$	18.1868	0.609966
$890$	0	0
$891$	0.222371	0.00744970
$892$	0	0
$893$	3.94324	0.131956
$894$	0	0
$895$	−12.1054	−0.404640
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−21.0284	−0.701337
$900$	0	0
$901$	5.89987	0.196553
$902$	0	0
$903$	−6.18271	−0.205748
$904$	0	0
$905$	4.45900	0.148222
$906$	0	0
$907$	58.5463	1.94400	0.971999	−	0.234984i	$-0.0755038\pi$
$907$	58.5463	1.94400	0.971999	+	0.234984i	$0.0755038\pi$
$908$	0	0
$909$	14.7596	0.489545
$910$	0	0
$911$	−25.2015	−0.834962	−0.417481	−	0.908686i	$-0.637087\pi$
$911$	−25.2015	−0.834962	−0.417481	+	0.908686i	$0.637087\pi$
$912$	0	0
$913$	−0.891456	−0.0295029
$914$	0	0
$915$	−3.75442	−0.124117
$916$	0	0
$917$	12.4168	0.410038
$918$	0	0
$919$	41.0864	1.35531	0.677657	−	0.735378i	$-0.262995\pi$
$919$	41.0864	1.35531	0.677657	+	0.735378i	$0.262995\pi$
$920$	0	0
$921$	10.4775	0.345246
$922$	0	0
$923$	0	0
$924$	0	0
$925$	1.29534	0.0425906
$926$	0	0
$927$	−30.0409	−0.986673
$928$	0	0
$929$	38.9384	1.27753	0.638764	−	0.769402i	$-0.279446\pi$
$929$	38.9384	1.27753	0.638764	+	0.769402i	$0.279446\pi$
$930$	0	0
$931$	6.61988	0.216958
$932$	0	0
$933$	−6.85622	−0.224463
$934$	0	0
$935$	2.40986	0.0788109
$936$	0	0
$937$	3.23744	0.105763	0.0528813	−	0.998601i	$-0.483160\pi$
$937$	3.23744	0.105763	0.0528813	+	0.998601i	$0.483160\pi$
$938$	0	0
$939$	3.88120	0.126658
$940$	0	0
$941$	2.10528	0.0686303	0.0343151	−	0.999411i	$-0.489075\pi$
$941$	2.10528	0.0686303	0.0343151	+	0.999411i	$0.489075\pi$
$942$	0	0
$943$	−8.38432	−0.273031
$944$	0	0
$945$	−6.39498	−0.208029
$946$	0	0
$947$	−15.0716	−0.489760	−0.244880	−	0.969553i	$-0.578749\pi$
$947$	−15.0716	−0.489760	−0.244880	+	0.969553i	$0.578749\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	4.14246	0.134329
$952$	0	0
$953$	−59.5826	−1.93007	−0.965035	−	0.262121i	$-0.915578\pi$
$953$	−59.5826	−1.93007	−0.965035	+	0.262121i	$0.915578\pi$
$954$	0	0
$955$	−15.1633	−0.490672
$956$	0	0
$957$	7.86522	0.254246
$958$	0	0
$959$	−3.95141	−0.127598
$960$	0	0
$961$	−15.9028	−0.512993
$962$	0	0
$963$	−0.345915	−0.0111469
$964$	0	0
$965$	−1.33423	−0.0429504
$966$	0	0
$967$	−37.0692	−1.19207	−0.596033	−	0.802960i	$-0.703257\pi$
$967$	−37.0692	−1.19207	−0.596033	+	0.802960i	$0.703257\pi$
$968$	0	0
$969$	−9.94938	−0.319620
$970$	0	0
$971$	51.7412	1.66045	0.830227	−	0.557426i	$-0.188211\pi$
$971$	51.7412	1.66045	0.830227	+	0.557426i	$0.188211\pi$
$972$	0	0
$973$	12.7364	0.408309
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−58.0716	−1.85787	−0.928937	−	0.370237i	$-0.879277\pi$
$977$	−58.0716	−1.85787	−0.928937	+	0.370237i	$0.879277\pi$
$978$	0	0
$979$	8.48877	0.271302
$980$	0	0
$981$	−0.0317410	−0.00101341
$982$	0	0
$983$	48.4068	1.54394	0.771969	−	0.635661i	$-0.219272\pi$
$983$	48.4068	1.54394	0.771969	+	0.635661i	$0.219272\pi$
$984$	0	0
$985$	−11.7683	−0.374970
$986$	0	0
$987$	0.630908	0.0200820
$988$	0	0
$989$	34.2492	1.08906
$990$	0	0
$991$	−25.7527	−0.818060	−0.409030	−	0.912521i	$-0.634133\pi$
$991$	−25.7527	−0.818060	−0.409030	+	0.912521i	$0.634133\pi$
$992$	0	0
$993$	16.0108	0.508086
$994$	0	0
$995$	−1.10533	−0.0350412
$996$	0	0
$997$	−41.9505	−1.32859	−0.664293	−	0.747472i	$-0.731267\pi$
$997$	−41.9505	−1.32859	−0.664293	+	0.747472i	$0.731267\pi$
$998$	0	0
$999$	−1.92982	−0.0610567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.br.1.5	✓	15
13.12	even	2		9464.2.a.bs.1.5	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9464.2.a.br.1.5	✓	15	1.1	even	1	trivial
9464.2.a.bs.1.5	yes	15	13.12	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9464.a (trivial)

\(f(q)\)	\(=\)	\(q-1.05916 q^{3} +1.23771 q^{5} -1.00000 q^{7} -1.87817 q^{9} +1.37211 q^{11} -1.31094 q^{15} +1.41900 q^{17} +6.61988 q^{19} +1.05916 q^{21} -5.86724 q^{23} -3.46807 q^{25} +5.16678 q^{27} -5.41201 q^{29} +3.88551 q^{31} -1.45329 q^{33} -1.23771 q^{35} -0.373505 q^{37} +1.42901 q^{41} -5.83736 q^{43} -2.32464 q^{45} +0.595667 q^{47} +1.00000 q^{49} -1.50295 q^{51} +4.15776 q^{53} +1.69828 q^{55} -7.01153 q^{57} -0.154867 q^{59} +2.86393 q^{61} +1.87817 q^{63} -13.3682 q^{67} +6.21436 q^{69} -2.63494 q^{71} -3.56409 q^{73} +3.67325 q^{75} -1.37211 q^{77} -6.55447 q^{79} +0.162065 q^{81} -0.649696 q^{83} +1.75631 q^{85} +5.73220 q^{87} +6.18665 q^{89} -4.11539 q^{93} +8.19350 q^{95} +18.1559 q^{97} -2.57707 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 3 q^{3} - 4 q^{5} - 15 q^{7} + 16 q^{9} - 15 q^{11} - 8 q^{15} + 2 q^{17} - 13 q^{19} + 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} - 19 q^{31} + 24 q^{33} + 4 q^{35} + 2 q^{37} - 30 q^{41}+ \cdots - 70 q^{99}+O(q^{100}) \) 15 * q - 3 * q^3 - 4 * q^5 - 15 * q^7 + 16 * q^9 - 15 * q^11 - 8 * q^15 + 2 * q^17 - 13 * q^19 + 3 * q^21 - 10 * q^23 + 23 * q^25 - 9 * q^27 + 25 * q^29 - 19 * q^31 + 24 * q^33 + 4 * q^35 + 2 * q^37 - 30 * q^41 + 15 * q^43 + 2 * q^45 - 13 * q^47 + 15 * q^49 + q^51 + 62 * q^53 - 3 * q^55 + 2 * q^57 - 22 * q^59 - q^61 - 16 * q^63 - 29 * q^67 + 32 * q^69 - 20 * q^71 - 21 * q^73 - 46 * q^75 + 15 * q^77 + 29 * q^79 + 7 * q^81 - 23 * q^83 - 8 * q^85 - 22 * q^87 - 9 * q^89 - 10 * q^93 + 28 * q^95 - 11 * q^97 - 70 * q^99

Introduction

L-functions

Modular forms

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Database

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