LMFDB - Embedded newform 9464.2.a.br.1.13

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			1.13
Root			$-2.32680$ 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)

$f(q)$	$=$	$q+2.32680 q^{3} +1.27247 q^{5} -1.00000 q^{7} +2.41400 q^{9} -6.42384 q^{11} +2.96078 q^{15} +3.33232 q^{17} +1.32253 q^{19} -2.32680 q^{21} +0.849071 q^{23} -3.38082 q^{25} -1.36350 q^{27} +8.55156 q^{29} -1.84234 q^{31} -14.9470 q^{33} -1.27247 q^{35} -9.91958 q^{37} -2.84770 q^{41} -6.03321 q^{43} +3.07174 q^{45} -4.70658 q^{47} +1.00000 q^{49} +7.75365 q^{51} -4.57771 q^{53} -8.17414 q^{55} +3.07727 q^{57} +1.56268 q^{59} -7.50272 q^{61} -2.41400 q^{63} -1.34362 q^{67} +1.97562 q^{69} -12.0208 q^{71} +13.1954 q^{73} -7.86650 q^{75} +6.42384 q^{77} -6.31729 q^{79} -10.4146 q^{81} -5.72464 q^{83} +4.24028 q^{85} +19.8978 q^{87} -12.5426 q^{89} -4.28676 q^{93} +1.68288 q^{95} +17.6875 q^{97} -15.5072 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.32680	1.34338	0.671690	−	0.740833i	$-0.265569\pi$
$3$	2.32680	1.34338	0.671690	+	0.740833i	$0.265569\pi$
$4$	0	0
$5$	1.27247	0.569066	0.284533	−	0.958666i	$-0.408162\pi$
$5$	1.27247	0.569066	0.284533	+	0.958666i	$0.408162\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.41400	0.804668
$10$	0	0
$11$	−6.42384	−1.93686	−0.968430	−	0.249284i	$-0.919805\pi$
$11$	−6.42384	−1.93686	−0.968430	+	0.249284i	$0.919805\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.96078	0.764471
$16$	0	0
$17$	3.33232	0.808206	0.404103	−	0.914713i	$-0.367584\pi$
$17$	3.33232	0.808206	0.404103	+	0.914713i	$0.367584\pi$
$18$	0	0
$19$	1.32253	0.303410	0.151705	−	0.988426i	$-0.451524\pi$
$19$	1.32253	0.303410	0.151705	+	0.988426i	$0.451524\pi$
$20$	0	0
$21$	−2.32680	−0.507750
$22$	0	0
$23$	0.849071	0.177044	0.0885218	−	0.996074i	$-0.471786\pi$
$23$	0.849071	0.177044	0.0885218	+	0.996074i	$0.471786\pi$
$24$	0	0
$25$	−3.38082	−0.676164
$26$	0	0
$27$	−1.36350	−0.262405
$28$	0	0
$29$	8.55156	1.58798	0.793992	−	0.607928i	$-0.207999\pi$
$29$	8.55156	1.58798	0.793992	+	0.607928i	$0.207999\pi$
$30$	0	0
$31$	−1.84234	−0.330894	−0.165447	−	0.986219i	$-0.552907\pi$
$31$	−1.84234	−0.330894	−0.165447	+	0.986219i	$0.552907\pi$
$32$	0	0
$33$	−14.9470	−2.60194
$34$	0	0
$35$	−1.27247	−0.215087
$36$	0	0
$37$	−9.91958	−1.63077	−0.815384	−	0.578920i	$-0.803474\pi$
$37$	−9.91958	−1.63077	−0.815384	+	0.578920i	$0.803474\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.84770	−0.444736	−0.222368	−	0.974963i	$-0.571379\pi$
$41$	−2.84770	−0.444736	−0.222368	+	0.974963i	$0.571379\pi$
$42$	0	0
$43$	−6.03321	−0.920055	−0.460028	−	0.887905i	$-0.652161\pi$
$43$	−6.03321	−0.920055	−0.460028	+	0.887905i	$0.652161\pi$
$44$	0	0
$45$	3.07174	0.457909
$46$	0	0
$47$	−4.70658	−0.686525	−0.343263	−	0.939239i	$-0.611532\pi$
$47$	−4.70658	−0.686525	−0.343263	+	0.939239i	$0.611532\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.75365	1.08573
$52$	0	0
$53$	−4.57771	−0.628797	−0.314399	−	0.949291i	$-0.601803\pi$
$53$	−4.57771	−0.628797	−0.314399	+	0.949291i	$0.601803\pi$
$54$	0	0
$55$	−8.17414	−1.10220
$56$	0	0
$57$	3.07727	0.407594
$58$	0	0
$59$	1.56268	0.203443	0.101722	−	0.994813i	$-0.467565\pi$
$59$	1.56268	0.203443	0.101722	+	0.994813i	$0.467565\pi$
$60$	0	0
$61$	−7.50272	−0.960625	−0.480313	−	0.877097i	$-0.659477\pi$
$61$	−7.50272	−0.960625	−0.480313	+	0.877097i	$0.659477\pi$
$62$	0	0
$63$	−2.41400	−0.304136
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.34362	−0.164149	−0.0820747	−	0.996626i	$-0.526155\pi$
$67$	−1.34362	−0.164149	−0.0820747	+	0.996626i	$0.526155\pi$
$68$	0	0
$69$	1.97562	0.237837
$70$	0	0
$71$	−12.0208	−1.42661	−0.713306	−	0.700853i	$-0.752803\pi$
$71$	−12.0208	−1.42661	−0.713306	+	0.700853i	$0.752803\pi$
$72$	0	0
$73$	13.1954	1.54440	0.772201	−	0.635378i	$-0.219156\pi$
$73$	13.1954	1.54440	0.772201	+	0.635378i	$0.219156\pi$
$74$	0	0
$75$	−7.86650	−0.908345
$76$	0	0
$77$	6.42384	0.732065
$78$	0	0
$79$	−6.31729	−0.710750	−0.355375	−	0.934724i	$-0.615647\pi$
$79$	−6.31729	−0.710750	−0.355375	+	0.934724i	$0.615647\pi$
$80$	0	0
$81$	−10.4146	−1.15718
$82$	0	0
$83$	−5.72464	−0.628360	−0.314180	−	0.949363i	$-0.601730\pi$
$83$	−5.72464	−0.628360	−0.314180	+	0.949363i	$0.601730\pi$
$84$	0	0
$85$	4.24028	0.459922
$86$	0	0
$87$	19.8978	2.13327
$88$	0	0
$89$	−12.5426	−1.32951	−0.664756	−	0.747061i	$-0.731464\pi$
$89$	−12.5426	−1.32951	−0.664756	+	0.747061i	$0.731464\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−4.28676	−0.444517
$94$	0	0
$95$	1.68288	0.172660
$96$	0	0
$97$	17.6875	1.79589	0.897946	−	0.440105i	$-0.145059\pi$
$97$	17.6875	1.79589	0.897946	+	0.440105i	$0.145059\pi$
$98$	0	0
$99$	−15.5072	−1.55853
$100$	0	0
$101$	−0.199640	−0.0198649	−0.00993244	−	0.999951i	$-0.503162\pi$
$101$	−0.199640	−0.0198649	−0.00993244	+	0.999951i	$0.503162\pi$
$102$	0	0
$103$	−12.5235	−1.23398	−0.616989	−	0.786972i	$-0.711648\pi$
$103$	−12.5235	−1.23398	−0.616989	+	0.786972i	$0.711648\pi$
$104$	0	0
$105$	−2.96078	−0.288943
$106$	0	0
$107$	−8.67551	−0.838693	−0.419347	−	0.907826i	$-0.637741\pi$
$107$	−8.67551	−0.838693	−0.419347	+	0.907826i	$0.637741\pi$
$108$	0	0
$109$	3.03474	0.290675	0.145338	−	0.989382i	$-0.453573\pi$
$109$	3.03474	0.290675	0.145338	+	0.989382i	$0.453573\pi$
$110$	0	0
$111$	−23.0809	−2.19074
$112$	0	0
$113$	−4.77391	−0.449091	−0.224546	−	0.974464i	$-0.572090\pi$
$113$	−4.77391	−0.449091	−0.224546	+	0.974464i	$0.572090\pi$
$114$	0	0
$115$	1.08042	0.100749
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.33232	−0.305473
$120$	0	0
$121$	30.2657	2.75143
$122$	0	0
$123$	−6.62603	−0.597449
$124$	0	0
$125$	−10.6643	−0.953847
$126$	0	0
$127$	22.0644	1.95790	0.978948	−	0.204110i	$-0.0654299\pi$
$127$	22.0644	1.95790	0.978948	+	0.204110i	$0.0654299\pi$
$128$	0	0
$129$	−14.0381	−1.23598
$130$	0	0
$131$	12.8975	1.12686	0.563430	−	0.826164i	$-0.309481\pi$
$131$	12.8975	1.12686	0.563430	+	0.826164i	$0.309481\pi$
$132$	0	0
$133$	−1.32253	−0.114678
$134$	0	0
$135$	−1.73501	−0.149326
$136$	0	0
$137$	5.13983	0.439126	0.219563	−	0.975598i	$-0.429537\pi$
$137$	5.13983	0.439126	0.219563	+	0.975598i	$0.429537\pi$
$138$	0	0
$139$	10.5030	0.890854	0.445427	−	0.895318i	$-0.353052\pi$
$139$	10.5030	0.890854	0.445427	+	0.895318i	$0.353052\pi$
$140$	0	0
$141$	−10.9513	−0.922264
$142$	0	0
$143$	0	0
$144$	0	0
$145$	10.8816	0.903668
$146$	0	0
$147$	2.32680	0.191911
$148$	0	0
$149$	11.5611	0.947122	0.473561	−	0.880761i	$-0.342968\pi$
$149$	11.5611	0.947122	0.473561	+	0.880761i	$0.342968\pi$
$150$	0	0
$151$	−9.23704	−0.751700	−0.375850	−	0.926681i	$-0.622649\pi$
$151$	−9.23704	−0.751700	−0.375850	+	0.926681i	$0.622649\pi$
$152$	0	0
$153$	8.04423	0.650338
$154$	0	0
$155$	−2.34432	−0.188301
$156$	0	0
$157$	−6.14929	−0.490767	−0.245383	−	0.969426i	$-0.578914\pi$
$157$	−6.14929	−0.490767	−0.245383	+	0.969426i	$0.578914\pi$
$158$	0	0
$159$	−10.6514	−0.844713
$160$	0	0
$161$	−0.849071	−0.0669162
$162$	0	0
$163$	−14.5021	−1.13589	−0.567945	−	0.823067i	$-0.692261\pi$
$163$	−14.5021	−1.13589	−0.567945	+	0.823067i	$0.692261\pi$
$164$	0	0
$165$	−19.0196	−1.48067
$166$	0	0
$167$	−15.2120	−1.17714	−0.588571	−	0.808446i	$-0.700309\pi$
$167$	−15.2120	−1.17714	−0.588571	+	0.808446i	$0.700309\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	3.19259	0.244144
$172$	0	0
$173$	17.3939	1.32244	0.661218	−	0.750194i	$-0.270040\pi$
$173$	17.3939	1.32244	0.661218	+	0.750194i	$0.270040\pi$
$174$	0	0
$175$	3.38082	0.255566
$176$	0	0
$177$	3.63603	0.273301
$178$	0	0
$179$	2.60245	0.194516	0.0972579	−	0.995259i	$-0.468993\pi$
$179$	2.60245	0.194516	0.0972579	+	0.995259i	$0.468993\pi$
$180$	0	0
$181$	−3.77181	−0.280357	−0.140178	−	0.990126i	$-0.544768\pi$
$181$	−3.77181	−0.280357	−0.140178	+	0.990126i	$0.544768\pi$
$182$	0	0
$183$	−17.4573	−1.29048
$184$	0	0
$185$	−12.6224	−0.928014
$186$	0	0
$187$	−21.4063	−1.56538
$188$	0	0
$189$	1.36350	0.0991799
$190$	0	0
$191$	−5.15435	−0.372956	−0.186478	−	0.982459i	$-0.559707\pi$
$191$	−5.15435	−0.372956	−0.186478	+	0.982459i	$0.559707\pi$
$192$	0	0
$193$	−2.38435	−0.171629	−0.0858146	−	0.996311i	$-0.527349\pi$
$193$	−2.38435	−0.171629	−0.0858146	+	0.996311i	$0.527349\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.9851	−0.925153	−0.462577	−	0.886579i	$-0.653075\pi$
$197$	−12.9851	−0.925153	−0.462577	+	0.886579i	$0.653075\pi$
$198$	0	0
$199$	15.3134	1.08554	0.542768	−	0.839883i	$-0.317376\pi$
$199$	15.3134	1.08554	0.542768	+	0.839883i	$0.317376\pi$
$200$	0	0
$201$	−3.12634	−0.220515
$202$	0	0
$203$	−8.55156	−0.600202
$204$	0	0
$205$	−3.62361	−0.253084
$206$	0	0
$207$	2.04966	0.142461
$208$	0	0
$209$	−8.49573	−0.587662
$210$	0	0
$211$	−15.8677	−1.09237	−0.546187	−	0.837663i	$-0.683921\pi$
$211$	−15.8677	−1.09237	−0.546187	+	0.837663i	$0.683921\pi$
$212$	0	0
$213$	−27.9701	−1.91648
$214$	0	0
$215$	−7.67707	−0.523572
$216$	0	0
$217$	1.84234	0.125066
$218$	0	0
$219$	30.7030	2.07472
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−7.21823	−0.483369	−0.241684	−	0.970355i	$-0.577700\pi$
$223$	−7.21823	−0.483369	−0.241684	+	0.970355i	$0.577700\pi$
$224$	0	0
$225$	−8.16131	−0.544088
$226$	0	0
$227$	27.8725	1.84996	0.924981	−	0.380014i	$-0.124081\pi$
$227$	27.8725	1.84996	0.924981	+	0.380014i	$0.124081\pi$
$228$	0	0
$229$	7.66707	0.506654	0.253327	−	0.967381i	$-0.418475\pi$
$229$	7.66707	0.506654	0.253327	+	0.967381i	$0.418475\pi$
$230$	0	0
$231$	14.9470	0.983440
$232$	0	0
$233$	−21.4770	−1.40701	−0.703503	−	0.710693i	$-0.748382\pi$
$233$	−21.4770	−1.40701	−0.703503	+	0.710693i	$0.748382\pi$
$234$	0	0
$235$	−5.98898	−0.390678
$236$	0	0
$237$	−14.6991	−0.954807
$238$	0	0
$239$	−28.8767	−1.86788	−0.933940	−	0.357430i	$-0.883653\pi$
$239$	−28.8767	−1.86788	−0.933940	+	0.357430i	$0.883653\pi$
$240$	0	0
$241$	−15.3763	−0.990474	−0.495237	−	0.868758i	$-0.664919\pi$
$241$	−15.3763	−0.990474	−0.495237	+	0.868758i	$0.664919\pi$
$242$	0	0
$243$	−20.1422	−1.29212
$244$	0	0
$245$	1.27247	0.0812951
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−13.3201	−0.844126
$250$	0	0
$251$	−16.1428	−1.01892	−0.509462	−	0.860493i	$-0.670155\pi$
$251$	−16.1428	−1.01892	−0.509462	+	0.860493i	$0.670155\pi$
$252$	0	0
$253$	−5.45430	−0.342909
$254$	0	0
$255$	9.86628	0.617850
$256$	0	0
$257$	−23.7811	−1.48342	−0.741712	−	0.670719i	$-0.765986\pi$
$257$	−23.7811	−1.48342	−0.741712	+	0.670719i	$0.765986\pi$
$258$	0	0
$259$	9.91958	0.616372
$260$	0	0
$261$	20.6435	1.27780
$262$	0	0
$263$	−7.98942	−0.492649	−0.246324	−	0.969187i	$-0.579223\pi$
$263$	−7.98942	−0.492649	−0.246324	+	0.969187i	$0.579223\pi$
$264$	0	0
$265$	−5.82500	−0.357827
$266$	0	0
$267$	−29.1841	−1.78604
$268$	0	0
$269$	24.4005	1.48772	0.743862	−	0.668333i	$-0.232992\pi$
$269$	24.4005	1.48772	0.743862	+	0.668333i	$0.232992\pi$
$270$	0	0
$271$	20.4146	1.24010	0.620048	−	0.784564i	$-0.287113\pi$
$271$	20.4146	1.24010	0.620048	+	0.784564i	$0.287113\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	21.7179	1.30964
$276$	0	0
$277$	7.24043	0.435035	0.217518	−	0.976056i	$-0.430204\pi$
$277$	7.24043	0.435035	0.217518	+	0.976056i	$0.430204\pi$
$278$	0	0
$279$	−4.44742	−0.266260
$280$	0	0
$281$	−21.8164	−1.30146	−0.650730	−	0.759309i	$-0.725537\pi$
$281$	−21.8164	−1.30146	−0.650730	+	0.759309i	$0.725537\pi$
$282$	0	0
$283$	−16.2625	−0.966705	−0.483353	−	0.875426i	$-0.660581\pi$
$283$	−16.2625	−0.966705	−0.483353	+	0.875426i	$0.660581\pi$
$284$	0	0
$285$	3.91573	0.231948
$286$	0	0
$287$	2.84770	0.168094
$288$	0	0
$289$	−5.89564	−0.346802
$290$	0	0
$291$	41.1553	2.41256
$292$	0	0
$293$	19.7217	1.15215	0.576077	−	0.817395i	$-0.304583\pi$
$293$	19.7217	1.15215	0.576077	+	0.817395i	$0.304583\pi$
$294$	0	0
$295$	1.98846	0.115772
$296$	0	0
$297$	8.75890	0.508243
$298$	0	0
$299$	0	0
$300$	0	0
$301$	6.03321	0.347748
$302$	0	0
$303$	−0.464521	−0.0266861
$304$	0	0
$305$	−9.54699	−0.546659
$306$	0	0
$307$	4.26868	0.243627	0.121813	−	0.992553i	$-0.461129\pi$
$307$	4.26868	0.243627	0.121813	+	0.992553i	$0.461129\pi$
$308$	0	0
$309$	−29.1397	−1.65770
$310$	0	0
$311$	7.67686	0.435315	0.217657	−	0.976025i	$-0.430158\pi$
$311$	7.67686	0.435315	0.217657	+	0.976025i	$0.430158\pi$
$312$	0	0
$313$	−5.96026	−0.336894	−0.168447	−	0.985711i	$-0.553875\pi$
$313$	−5.96026	−0.336894	−0.168447	+	0.985711i	$0.553875\pi$
$314$	0	0
$315$	−3.07174	−0.173073
$316$	0	0
$317$	−5.50647	−0.309274	−0.154637	−	0.987971i	$-0.549421\pi$
$317$	−5.50647	−0.309274	−0.154637	+	0.987971i	$0.549421\pi$
$318$	0	0
$319$	−54.9339	−3.07571
$320$	0	0
$321$	−20.1862	−1.12668
$322$	0	0
$323$	4.40710	0.245218
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.06123	0.390487
$328$	0	0
$329$	4.70658	0.259482
$330$	0	0
$331$	−18.6624	−1.02578	−0.512889	−	0.858455i	$-0.671425\pi$
$331$	−18.6624	−1.02578	−0.512889	+	0.858455i	$0.671425\pi$
$332$	0	0
$333$	−23.9459	−1.31223
$334$	0	0
$335$	−1.70972	−0.0934118
$336$	0	0
$337$	33.9443	1.84907	0.924533	−	0.381102i	$-0.124455\pi$
$337$	33.9443	1.84907	0.924533	+	0.381102i	$0.124455\pi$
$338$	0	0
$339$	−11.1079	−0.603300
$340$	0	0
$341$	11.8349	0.640897
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	2.51391	0.135345
$346$	0	0
$347$	−3.50971	−0.188411	−0.0942056	−	0.995553i	$-0.530031\pi$
$347$	−3.50971	−0.188411	−0.0942056	+	0.995553i	$0.530031\pi$
$348$	0	0
$349$	−26.0776	−1.39590	−0.697950	−	0.716147i	$-0.745904\pi$
$349$	−26.0776	−1.39590	−0.697950	+	0.716147i	$0.745904\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−26.5545	−1.41335	−0.706677	−	0.707537i	$-0.749806\pi$
$353$	−26.5545	−1.41335	−0.706677	+	0.707537i	$0.749806\pi$
$354$	0	0
$355$	−15.2962	−0.811836
$356$	0	0
$357$	−7.75365	−0.410367
$358$	0	0
$359$	13.0339	0.687901	0.343950	−	0.938988i	$-0.388235\pi$
$359$	13.0339	0.687901	0.343950	+	0.938988i	$0.388235\pi$
$360$	0	0
$361$	−17.2509	−0.907943
$362$	0	0
$363$	70.4223	3.69621
$364$	0	0
$365$	16.7907	0.878866
$366$	0	0
$367$	−1.99562	−0.104171	−0.0520854	−	0.998643i	$-0.516587\pi$
$367$	−1.99562	−0.104171	−0.0520854	+	0.998643i	$0.516587\pi$
$368$	0	0
$369$	−6.87436	−0.357865
$370$	0	0
$371$	4.57771	0.237663
$372$	0	0
$373$	10.9350	0.566194	0.283097	−	0.959091i	$-0.408638\pi$
$373$	10.9350	0.566194	0.283097	+	0.959091i	$0.408638\pi$
$374$	0	0
$375$	−24.8138	−1.28138
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−13.6921	−0.703314	−0.351657	−	0.936129i	$-0.614382\pi$
$379$	−13.6921	−0.703314	−0.351657	+	0.936129i	$0.614382\pi$
$380$	0	0
$381$	51.3394	2.63020
$382$	0	0
$383$	32.9325	1.68277	0.841387	−	0.540433i	$-0.181739\pi$
$383$	32.9325	1.68277	0.841387	+	0.540433i	$0.181739\pi$
$384$	0	0
$385$	8.17414	0.416593
$386$	0	0
$387$	−14.5642	−0.740339
$388$	0	0
$389$	−29.6277	−1.50218	−0.751092	−	0.660197i	$-0.770473\pi$
$389$	−29.6277	−1.50218	−0.751092	+	0.660197i	$0.770473\pi$
$390$	0	0
$391$	2.82938	0.143088
$392$	0	0
$393$	30.0099	1.51380
$394$	0	0
$395$	−8.03855	−0.404463
$396$	0	0
$397$	19.4546	0.976397	0.488198	−	0.872733i	$-0.337654\pi$
$397$	19.4546	0.976397	0.488198	+	0.872733i	$0.337654\pi$
$398$	0	0
$399$	−3.07727	−0.154056
$400$	0	0
$401$	−14.3380	−0.716008	−0.358004	−	0.933720i	$-0.616543\pi$
$401$	−14.3380	−0.716008	−0.358004	+	0.933720i	$0.616543\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−13.2523	−0.658510
$406$	0	0
$407$	63.7218	3.15857
$408$	0	0
$409$	20.0449	0.991156	0.495578	−	0.868564i	$-0.334956\pi$
$409$	20.0449	0.991156	0.495578	+	0.868564i	$0.334956\pi$
$410$	0	0
$411$	11.9594	0.589912
$412$	0	0
$413$	−1.56268	−0.0768942
$414$	0	0
$415$	−7.28442	−0.357578
$416$	0	0
$417$	24.4384	1.19676
$418$	0	0
$419$	37.8811	1.85061	0.925307	−	0.379219i	$-0.123807\pi$
$419$	37.8811	1.85061	0.925307	+	0.379219i	$0.123807\pi$
$420$	0	0
$421$	−6.72479	−0.327746	−0.163873	−	0.986481i	$-0.552399\pi$
$421$	−6.72479	−0.327746	−0.163873	+	0.986481i	$0.552399\pi$
$422$	0	0
$423$	−11.3617	−0.552425
$424$	0	0
$425$	−11.2660	−0.546480
$426$	0	0
$427$	7.50272	0.363082
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.12684	0.246951	0.123476	−	0.992348i	$-0.460596\pi$
$431$	5.12684	0.246951	0.123476	+	0.992348i	$0.460596\pi$
$432$	0	0
$433$	−21.0031	−1.00935	−0.504673	−	0.863310i	$-0.668387\pi$
$433$	−21.0031	−1.00935	−0.504673	+	0.863310i	$0.668387\pi$
$434$	0	0
$435$	25.3193	1.21397
$436$	0	0
$437$	1.12292	0.0537167
$438$	0	0
$439$	−12.2500	−0.584660	−0.292330	−	0.956317i	$-0.594431\pi$
$439$	−12.2500	−0.584660	−0.292330	+	0.956317i	$0.594431\pi$
$440$	0	0
$441$	2.41400	0.114953
$442$	0	0
$443$	13.1819	0.626290	0.313145	−	0.949705i	$-0.398617\pi$
$443$	13.1819	0.626290	0.313145	+	0.949705i	$0.398617\pi$
$444$	0	0
$445$	−15.9601	−0.756579
$446$	0	0
$447$	26.9004	1.27234
$448$	0	0
$449$	−8.58174	−0.404997	−0.202499	−	0.979283i	$-0.564906\pi$
$449$	−8.58174	−0.404997	−0.202499	+	0.979283i	$0.564906\pi$
$450$	0	0
$451$	18.2932	0.861392
$452$	0	0
$453$	−21.4928	−1.00982
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.89657	0.322608	0.161304	−	0.986905i	$-0.448430\pi$
$457$	6.89657	0.322608	0.161304	+	0.986905i	$0.448430\pi$
$458$	0	0
$459$	−4.54361	−0.212078
$460$	0	0
$461$	12.8190	0.597041	0.298520	−	0.954403i	$-0.403507\pi$
$461$	12.8190	0.597041	0.298520	+	0.954403i	$0.403507\pi$
$462$	0	0
$463$	−19.1638	−0.890618	−0.445309	−	0.895377i	$-0.646906\pi$
$463$	−19.1638	−0.890618	−0.445309	+	0.895377i	$0.646906\pi$
$464$	0	0
$465$	−5.45478	−0.252959
$466$	0	0
$467$	6.80660	0.314972	0.157486	−	0.987521i	$-0.449661\pi$
$467$	6.80660	0.314972	0.157486	+	0.987521i	$0.449661\pi$
$468$	0	0
$469$	1.34362	0.0620427
$470$	0	0
$471$	−14.3082	−0.659286
$472$	0	0
$473$	38.7564	1.78202
$474$	0	0
$475$	−4.47124	−0.205155
$476$	0	0
$477$	−11.0506	−0.505973
$478$	0	0
$479$	18.1357	0.828641	0.414320	−	0.910131i	$-0.364019\pi$
$479$	18.1357	0.828641	0.414320	+	0.910131i	$0.364019\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−1.97562	−0.0898938
$484$	0	0
$485$	22.5068	1.02198
$486$	0	0
$487$	−42.2938	−1.91651	−0.958257	−	0.285907i	$-0.907705\pi$
$487$	−42.2938	−1.91651	−0.958257	+	0.285907i	$0.907705\pi$
$488$	0	0
$489$	−33.7434	−1.52593
$490$	0	0
$491$	−23.0597	−1.04067	−0.520334	−	0.853963i	$-0.674193\pi$
$491$	−23.0597	−1.04067	−0.520334	+	0.853963i	$0.674193\pi$
$492$	0	0
$493$	28.4965	1.28342
$494$	0	0
$495$	−19.7324	−0.886905
$496$	0	0
$497$	12.0208	0.539209
$498$	0	0
$499$	−11.4377	−0.512024	−0.256012	−	0.966674i	$-0.582409\pi$
$499$	−11.4377	−0.512024	−0.256012	+	0.966674i	$0.582409\pi$
$500$	0	0
$501$	−35.3954	−1.58135
$502$	0	0
$503$	21.6016	0.963167	0.481584	−	0.876400i	$-0.340062\pi$
$503$	21.6016	0.963167	0.481584	+	0.876400i	$0.340062\pi$
$504$	0	0
$505$	−0.254035	−0.0113044
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.68536	0.429296	0.214648	−	0.976691i	$-0.431140\pi$
$509$	9.68536	0.429296	0.214648	+	0.976691i	$0.431140\pi$
$510$	0	0
$511$	−13.1954	−0.583729
$512$	0	0
$513$	−1.80327	−0.0796163
$514$	0	0
$515$	−15.9358	−0.702215
$516$	0	0
$517$	30.2343	1.32970
$518$	0	0
$519$	40.4722	1.77653
$520$	0	0
$521$	37.8117	1.65656	0.828281	−	0.560314i	$-0.189319\pi$
$521$	37.8117	1.65656	0.828281	+	0.560314i	$0.189319\pi$
$522$	0	0
$523$	16.8832	0.738252	0.369126	−	0.929379i	$-0.379657\pi$
$523$	16.8832	0.738252	0.369126	+	0.929379i	$0.379657\pi$
$524$	0	0
$525$	7.86650	0.343322
$526$	0	0
$527$	−6.13928	−0.267431
$528$	0	0
$529$	−22.2791	−0.968656
$530$	0	0
$531$	3.77230	0.163704
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−11.0393	−0.477271
$536$	0	0
$537$	6.05537	0.261309
$538$	0	0
$539$	−6.42384	−0.276694
$540$	0	0
$541$	22.0672	0.948743	0.474372	−	0.880325i	$-0.342675\pi$
$541$	22.0672	0.948743	0.474372	+	0.880325i	$0.342675\pi$
$542$	0	0
$543$	−8.77626	−0.376625
$544$	0	0
$545$	3.86161	0.165413
$546$	0	0
$547$	−36.0808	−1.54270	−0.771351	−	0.636410i	$-0.780419\pi$
$547$	−36.0808	−1.54270	−0.771351	+	0.636410i	$0.780419\pi$
$548$	0	0
$549$	−18.1116	−0.772984
$550$	0	0
$551$	11.3097	0.481810
$552$	0	0
$553$	6.31729	0.268638
$554$	0	0
$555$	−29.3697	−1.24667
$556$	0	0
$557$	36.7874	1.55873	0.779366	−	0.626569i	$-0.215542\pi$
$557$	36.7874	1.55873	0.779366	+	0.626569i	$0.215542\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−49.8082	−2.10290
$562$	0	0
$563$	−30.4095	−1.28161	−0.640803	−	0.767705i	$-0.721398\pi$
$563$	−30.4095	−1.28161	−0.640803	+	0.767705i	$0.721398\pi$
$564$	0	0
$565$	−6.07465	−0.255562
$566$	0	0
$567$	10.4146	0.437372
$568$	0	0
$569$	−20.4066	−0.855490	−0.427745	−	0.903899i	$-0.640692\pi$
$569$	−20.4066	−0.855490	−0.427745	+	0.903899i	$0.640692\pi$
$570$	0	0
$571$	−4.86680	−0.203669	−0.101835	−	0.994801i	$-0.532471\pi$
$571$	−4.86680	−0.203669	−0.101835	+	0.994801i	$0.532471\pi$
$572$	0	0
$573$	−11.9931	−0.501021
$574$	0	0
$575$	−2.87056	−0.119711
$576$	0	0
$577$	15.1891	0.632332	0.316166	−	0.948704i	$-0.397604\pi$
$577$	15.1891	0.632332	0.316166	+	0.948704i	$0.397604\pi$
$578$	0	0
$579$	−5.54791	−0.230563
$580$	0	0
$581$	5.72464	0.237498
$582$	0	0
$583$	29.4065	1.21789
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.24477	−0.175200	−0.0876001	−	0.996156i	$-0.527920\pi$
$587$	−4.24477	−0.175200	−0.0876001	+	0.996156i	$0.527920\pi$
$588$	0	0
$589$	−2.43656	−0.100397
$590$	0	0
$591$	−30.2138	−1.24283
$592$	0	0
$593$	−42.1424	−1.73058	−0.865291	−	0.501270i	$-0.832866\pi$
$593$	−42.1424	−1.73058	−0.865291	+	0.501270i	$0.832866\pi$
$594$	0	0
$595$	−4.24028	−0.173834
$596$	0	0
$597$	35.6312	1.45829
$598$	0	0
$599$	42.2108	1.72469	0.862343	−	0.506324i	$-0.168996\pi$
$599$	42.2108	1.72469	0.862343	+	0.506324i	$0.168996\pi$
$600$	0	0
$601$	−33.2585	−1.35664	−0.678320	−	0.734766i	$-0.737292\pi$
$601$	−33.2585	−1.35664	−0.678320	+	0.734766i	$0.737292\pi$
$602$	0	0
$603$	−3.24351	−0.132086
$604$	0	0
$605$	38.5122	1.56574
$606$	0	0
$607$	27.7681	1.12707	0.563536	−	0.826091i	$-0.309441\pi$
$607$	27.7681	1.12707	0.563536	+	0.826091i	$0.309441\pi$
$608$	0	0
$609$	−19.8978	−0.806299
$610$	0	0
$611$	0	0
$612$	0	0
$613$	2.02464	0.0817746	0.0408873	−	0.999164i	$-0.486982\pi$
$613$	2.02464	0.0817746	0.0408873	+	0.999164i	$0.486982\pi$
$614$	0	0
$615$	−8.43142	−0.339988
$616$	0	0
$617$	30.8567	1.24224	0.621122	−	0.783714i	$-0.286677\pi$
$617$	30.8567	1.24224	0.621122	+	0.783714i	$0.286677\pi$
$618$	0	0
$619$	−1.33848	−0.0537982	−0.0268991	−	0.999638i	$-0.508563\pi$
$619$	−1.33848	−0.0537982	−0.0268991	+	0.999638i	$0.508563\pi$
$620$	0	0
$621$	−1.15771	−0.0464572
$622$	0	0
$623$	12.5426	0.502508
$624$	0	0
$625$	3.33407	0.133363
$626$	0	0
$627$	−19.7679	−0.789453
$628$	0	0
$629$	−33.0552	−1.31800
$630$	0	0
$631$	−0.610406	−0.0242999	−0.0121499	−	0.999926i	$-0.503868\pi$
$631$	−0.610406	−0.0242999	−0.0121499	+	0.999926i	$0.503868\pi$
$632$	0	0
$633$	−36.9209	−1.46747
$634$	0	0
$635$	28.0762	1.11417
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−29.0184	−1.14795
$640$	0	0
$641$	−0.890388	−0.0351682	−0.0175841	−	0.999845i	$-0.505597\pi$
$641$	−0.890388	−0.0351682	−0.0175841	+	0.999845i	$0.505597\pi$
$642$	0	0
$643$	50.2105	1.98011	0.990055	−	0.140682i	$-0.0449295\pi$
$643$	50.2105	1.98011	0.990055	+	0.140682i	$0.0449295\pi$
$644$	0	0
$645$	−17.8630	−0.703356
$646$	0	0
$647$	17.9782	0.706794	0.353397	−	0.935473i	$-0.385026\pi$
$647$	17.9782	0.706794	0.353397	+	0.935473i	$0.385026\pi$
$648$	0	0
$649$	−10.0384	−0.394041
$650$	0	0
$651$	4.28676	0.168012
$652$	0	0
$653$	0.614495	0.0240471	0.0120235	−	0.999928i	$-0.496173\pi$
$653$	0.614495	0.0240471	0.0120235	+	0.999928i	$0.496173\pi$
$654$	0	0
$655$	16.4117	0.641258
$656$	0	0
$657$	31.8537	1.24273
$658$	0	0
$659$	0.679615	0.0264740	0.0132370	−	0.999912i	$-0.495786\pi$
$659$	0.679615	0.0264740	0.0132370	+	0.999912i	$0.495786\pi$
$660$	0	0
$661$	29.0473	1.12981	0.564905	−	0.825156i	$-0.308913\pi$
$661$	29.0473	1.12981	0.564905	+	0.825156i	$0.308913\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.68288	−0.0652593
$666$	0	0
$667$	7.26088	0.281142
$668$	0	0
$669$	−16.7954	−0.649347
$670$	0	0
$671$	48.1963	1.86060
$672$	0	0
$673$	−3.15331	−0.121551	−0.0607755	−	0.998151i	$-0.519357\pi$
$673$	−3.15331	−0.121551	−0.0607755	+	0.998151i	$0.519357\pi$
$674$	0	0
$675$	4.60975	0.177429
$676$	0	0
$677$	27.4370	1.05449	0.527245	−	0.849713i	$-0.323225\pi$
$677$	27.4370	1.05449	0.527245	+	0.849713i	$0.323225\pi$
$678$	0	0
$679$	−17.6875	−0.678783
$680$	0	0
$681$	64.8537	2.48520
$682$	0	0
$683$	−20.2967	−0.776633	−0.388316	−	0.921526i	$-0.626943\pi$
$683$	−20.2967	−0.776633	−0.388316	+	0.921526i	$0.626943\pi$
$684$	0	0
$685$	6.54028	0.249891
$686$	0	0
$687$	17.8397	0.680629
$688$	0	0
$689$	0	0
$690$	0	0
$691$	43.0810	1.63888	0.819439	−	0.573167i	$-0.194285\pi$
$691$	43.0810	1.63888	0.819439	+	0.573167i	$0.194285\pi$
$692$	0	0
$693$	15.5072	0.589069
$694$	0	0
$695$	13.3648	0.506955
$696$	0	0
$697$	−9.48945	−0.359439
$698$	0	0
$699$	−49.9727	−1.89014
$700$	0	0
$701$	27.6972	1.04611	0.523055	−	0.852299i	$-0.324792\pi$
$701$	27.6972	1.04611	0.523055	+	0.852299i	$0.324792\pi$
$702$	0	0
$703$	−13.1190	−0.494791
$704$	0	0
$705$	−13.9352	−0.524828
$706$	0	0
$707$	0.199640	0.00750822
$708$	0	0
$709$	−42.7243	−1.60454	−0.802272	−	0.596959i	$-0.796375\pi$
$709$	−42.7243	−1.60454	−0.802272	+	0.596959i	$0.796375\pi$
$710$	0	0
$711$	−15.2499	−0.571918
$712$	0	0
$713$	−1.56428	−0.0585827
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−67.1904	−2.50927
$718$	0	0
$719$	−23.7582	−0.886032	−0.443016	−	0.896514i	$-0.646091\pi$
$719$	−23.7582	−0.886032	−0.443016	+	0.896514i	$0.646091\pi$
$720$	0	0
$721$	12.5235	0.466400
$722$	0	0
$723$	−35.7776	−1.33058
$724$	0	0
$725$	−28.9113	−1.07374
$726$	0	0
$727$	0.0456124	0.00169167	0.000845835	−	1.00000i	$-0.499731\pi$
$727$	0.0456124	0.00169167	0.000845835		1.00000i	$0.499731\pi$
$728$	0	0
$729$	−15.6231	−0.578633
$730$	0	0
$731$	−20.1046	−0.743595
$732$	0	0
$733$	1.40512	0.0518992	0.0259496	−	0.999663i	$-0.491739\pi$
$733$	1.40512	0.0518992	0.0259496	+	0.999663i	$0.491739\pi$
$734$	0	0
$735$	2.96078	0.109210
$736$	0	0
$737$	8.63121	0.317935
$738$	0	0
$739$	38.0191	1.39856	0.699278	−	0.714850i	$-0.253505\pi$
$739$	38.0191	1.39856	0.699278	+	0.714850i	$0.253505\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.37149	0.270434	0.135217	−	0.990816i	$-0.456827\pi$
$743$	7.37149	0.270434	0.135217	+	0.990816i	$0.456827\pi$
$744$	0	0
$745$	14.7111	0.538975
$746$	0	0
$747$	−13.8193	−0.505621
$748$	0	0
$749$	8.67551	0.316996
$750$	0	0
$751$	38.3548	1.39959	0.699794	−	0.714345i	$-0.253275\pi$
$751$	38.3548	1.39959	0.699794	+	0.714345i	$0.253275\pi$
$752$	0	0
$753$	−37.5610	−1.36880
$754$	0	0
$755$	−11.7539	−0.427767
$756$	0	0
$757$	46.1074	1.67580	0.837900	−	0.545823i	$-0.183783\pi$
$757$	46.1074	1.67580	0.837900	+	0.545823i	$0.183783\pi$
$758$	0	0
$759$	−12.6911	−0.460656
$760$	0	0
$761$	−24.1313	−0.874760	−0.437380	−	0.899277i	$-0.644093\pi$
$761$	−24.1313	−0.874760	−0.437380	+	0.899277i	$0.644093\pi$
$762$	0	0
$763$	−3.03474	−0.109865
$764$	0	0
$765$	10.2360	0.370085
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−28.6587	−1.03346	−0.516729	−	0.856149i	$-0.672850\pi$
$769$	−28.6587	−1.03346	−0.516729	+	0.856149i	$0.672850\pi$
$770$	0	0
$771$	−55.3339	−1.99280
$772$	0	0
$773$	40.3274	1.45047	0.725237	−	0.688499i	$-0.241730\pi$
$773$	40.3274	1.45047	0.725237	+	0.688499i	$0.241730\pi$
$774$	0	0
$775$	6.22863	0.223739
$776$	0	0
$777$	23.0809	0.828022
$778$	0	0
$779$	−3.76617	−0.134937
$780$	0	0
$781$	77.2200	2.76315
$782$	0	0
$783$	−11.6600	−0.416696
$784$	0	0
$785$	−7.82478	−0.279278
$786$	0	0
$787$	−15.1690	−0.540717	−0.270359	−	0.962760i	$-0.587142\pi$
$787$	−15.1690	−0.540717	−0.270359	+	0.962760i	$0.587142\pi$
$788$	0	0
$789$	−18.5898	−0.661814
$790$	0	0
$791$	4.77391	0.169741
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−13.5536	−0.480697
$796$	0	0
$797$	16.4333	0.582096	0.291048	−	0.956708i	$-0.405996\pi$
$797$	16.4333	0.582096	0.291048	+	0.956708i	$0.405996\pi$
$798$	0	0
$799$	−15.6838	−0.554854
$800$	0	0
$801$	−30.2778	−1.06981
$802$	0	0
$803$	−84.7650	−2.99129
$804$	0	0
$805$	−1.08042	−0.0380797
$806$	0	0
$807$	56.7751	1.99858
$808$	0	0
$809$	27.9907	0.984102	0.492051	−	0.870566i	$-0.336247\pi$
$809$	27.9907	0.984102	0.492051	+	0.870566i	$0.336247\pi$
$810$	0	0
$811$	−39.5128	−1.38748	−0.693742	−	0.720224i	$-0.744039\pi$
$811$	−39.5128	−1.38748	−0.693742	+	0.720224i	$0.744039\pi$
$812$	0	0
$813$	47.5006	1.66592
$814$	0	0
$815$	−18.4534	−0.646396
$816$	0	0
$817$	−7.97911	−0.279154
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−24.0197	−0.838294	−0.419147	−	0.907918i	$-0.637671\pi$
$821$	−24.0197	−0.838294	−0.419147	+	0.907918i	$0.637671\pi$
$822$	0	0
$823$	15.4832	0.539710	0.269855	−	0.962901i	$-0.413024\pi$
$823$	15.4832	0.539710	0.269855	+	0.962901i	$0.413024\pi$
$824$	0	0
$825$	50.5331	1.75934
$826$	0	0
$827$	−5.51389	−0.191737	−0.0958683	−	0.995394i	$-0.530563\pi$
$827$	−5.51389	−0.191737	−0.0958683	+	0.995394i	$0.530563\pi$
$828$	0	0
$829$	7.72297	0.268230	0.134115	−	0.990966i	$-0.457181\pi$
$829$	7.72297	0.268230	0.134115	+	0.990966i	$0.457181\pi$
$830$	0	0
$831$	16.8470	0.584417
$832$	0	0
$833$	3.33232	0.115458
$834$	0	0
$835$	−19.3568	−0.669871
$836$	0	0
$837$	2.51203	0.0868285
$838$	0	0
$839$	48.5972	1.67776	0.838881	−	0.544314i	$-0.183210\pi$
$839$	48.5972	1.67776	0.838881	+	0.544314i	$0.183210\pi$
$840$	0	0
$841$	44.1292	1.52170
$842$	0	0
$843$	−50.7625	−1.74835
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−30.2657	−1.03994
$848$	0	0
$849$	−37.8396	−1.29865
$850$	0	0
$851$	−8.42242	−0.288717
$852$	0	0
$853$	−16.8464	−0.576811	−0.288406	−	0.957508i	$-0.593125\pi$
$853$	−16.8464	−0.576811	−0.288406	+	0.957508i	$0.593125\pi$
$854$	0	0
$855$	4.06248	0.138934
$856$	0	0
$857$	−3.05743	−0.104440	−0.0522199	−	0.998636i	$-0.516630\pi$
$857$	−3.05743	−0.104440	−0.0522199	+	0.998636i	$0.516630\pi$
$858$	0	0
$859$	23.9404	0.816836	0.408418	−	0.912795i	$-0.366081\pi$
$859$	23.9404	0.816836	0.408418	+	0.912795i	$0.366081\pi$
$860$	0	0
$861$	6.62603	0.225815
$862$	0	0
$863$	−3.11424	−0.106010	−0.0530049	−	0.998594i	$-0.516880\pi$
$863$	−3.11424	−0.106010	−0.0530049	+	0.998594i	$0.516880\pi$
$864$	0	0
$865$	22.1333	0.752553
$866$	0	0
$867$	−13.7180	−0.465887
$868$	0	0
$869$	40.5812	1.37662
$870$	0	0
$871$	0	0
$872$	0	0
$873$	42.6976	1.44510
$874$	0	0
$875$	10.6643	0.360520
$876$	0	0
$877$	−18.2711	−0.616971	−0.308486	−	0.951229i	$-0.599822\pi$
$877$	−18.2711	−0.616971	−0.308486	+	0.951229i	$0.599822\pi$
$878$	0	0
$879$	45.8885	1.54778
$880$	0	0
$881$	8.36565	0.281846	0.140923	−	0.990021i	$-0.454993\pi$
$881$	8.36565	0.281846	0.140923	+	0.990021i	$0.454993\pi$
$882$	0	0
$883$	−52.4830	−1.76619	−0.883096	−	0.469191i	$-0.844545\pi$
$883$	−52.4830	−1.76619	−0.883096	+	0.469191i	$0.844545\pi$
$884$	0	0
$885$	4.62674	0.155526
$886$	0	0
$887$	−53.7876	−1.80601	−0.903006	−	0.429628i	$-0.858645\pi$
$887$	−53.7876	−1.80601	−0.903006	+	0.429628i	$0.858645\pi$
$888$	0	0
$889$	−22.0644	−0.740015
$890$	0	0
$891$	66.9017	2.24129
$892$	0	0
$893$	−6.22460	−0.208298
$894$	0	0
$895$	3.31153	0.110692
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−15.7549	−0.525455
$900$	0	0
$901$	−15.2544	−0.508198
$902$	0	0
$903$	14.0381	0.467158
$904$	0	0
$905$	−4.79952	−0.159541
$906$	0	0
$907$	−30.5595	−1.01471	−0.507357	−	0.861736i	$-0.669377\pi$
$907$	−30.5595	−1.01471	−0.507357	+	0.861736i	$0.669377\pi$
$908$	0	0
$909$	−0.481930	−0.0159846
$910$	0	0
$911$	−17.6465	−0.584655	−0.292327	−	0.956318i	$-0.594430\pi$
$911$	−17.6465	−0.584655	−0.292327	+	0.956318i	$0.594430\pi$
$912$	0	0
$913$	36.7742	1.21705
$914$	0	0
$915$	−22.2139	−0.734370
$916$	0	0
$917$	−12.8975	−0.425913
$918$	0	0
$919$	−28.1817	−0.929627	−0.464814	−	0.885409i	$-0.653879\pi$
$919$	−28.1817	−0.929627	−0.464814	+	0.885409i	$0.653879\pi$
$920$	0	0
$921$	9.93238	0.327283
$922$	0	0
$923$	0	0
$924$	0	0
$925$	33.5363	1.10267
$926$	0	0
$927$	−30.2318	−0.992943
$928$	0	0
$929$	38.6355	1.26759	0.633795	−	0.773501i	$-0.281496\pi$
$929$	38.6355	1.26759	0.633795	+	0.773501i	$0.281496\pi$
$930$	0	0
$931$	1.32253	0.0433442
$932$	0	0
$933$	17.8625	0.584792
$934$	0	0
$935$	−27.2389	−0.890806
$936$	0	0
$937$	−16.6889	−0.545203	−0.272601	−	0.962127i	$-0.587884\pi$
$937$	−16.6889	−0.545203	−0.272601	+	0.962127i	$0.587884\pi$
$938$	0	0
$939$	−13.8683	−0.452576
$940$	0	0
$941$	14.0185	0.456991	0.228495	−	0.973545i	$-0.426619\pi$
$941$	14.0185	0.456991	0.228495	+	0.973545i	$0.426619\pi$
$942$	0	0
$943$	−2.41790	−0.0787377
$944$	0	0
$945$	1.73501	0.0564399
$946$	0	0
$947$	−23.4628	−0.762440	−0.381220	−	0.924484i	$-0.624496\pi$
$947$	−23.4628	−0.762440	−0.381220	+	0.924484i	$0.624496\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−12.8125	−0.415473
$952$	0	0
$953$	−27.7415	−0.898634	−0.449317	−	0.893372i	$-0.648333\pi$
$953$	−27.7415	−0.898634	−0.449317	+	0.893372i	$0.648333\pi$
$954$	0	0
$955$	−6.55875	−0.212236
$956$	0	0
$957$	−127.820	−4.13184
$958$	0	0
$959$	−5.13983	−0.165974
$960$	0	0
$961$	−27.6058	−0.890509
$962$	0	0
$963$	−20.9427	−0.674869
$964$	0	0
$965$	−3.03401	−0.0976683
$966$	0	0
$967$	−12.6723	−0.407514	−0.203757	−	0.979022i	$-0.565315\pi$
$967$	−12.6723	−0.407514	−0.203757	+	0.979022i	$0.565315\pi$
$968$	0	0
$969$	10.2544	0.329420
$970$	0	0
$971$	−29.7405	−0.954417	−0.477208	−	0.878790i	$-0.658351\pi$
$971$	−29.7405	−0.954417	−0.477208	+	0.878790i	$0.658351\pi$
$972$	0	0
$973$	−10.5030	−0.336711
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.6607	0.916938	0.458469	−	0.888710i	$-0.348398\pi$
$977$	28.6607	0.916938	0.458469	+	0.888710i	$0.348398\pi$
$978$	0	0
$979$	80.5716	2.57508
$980$	0	0
$981$	7.32586	0.233897
$982$	0	0
$983$	57.3163	1.82811	0.914054	−	0.405593i	$-0.132935\pi$
$983$	57.3163	1.82811	0.914054	+	0.405593i	$0.132935\pi$
$984$	0	0
$985$	−16.5232	−0.526473
$986$	0	0
$987$	10.9513	0.348583
$988$	0	0
$989$	−5.12262	−0.162890
$990$	0	0
$991$	−57.6221	−1.83043	−0.915213	−	0.402970i	$-0.867978\pi$
$991$	−57.6221	−1.83043	−0.915213	+	0.402970i	$0.867978\pi$
$992$	0	0
$993$	−43.4237	−1.37801
$994$	0	0
$995$	19.4858	0.617741
$996$	0	0
$997$	8.54191	0.270525	0.135262	−	0.990810i	$-0.456812\pi$
$997$	8.54191	0.270525	0.135262	+	0.990810i	$0.456812\pi$
$998$	0	0
$999$	13.5253	0.427922

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.br.1.13	✓	15
13.12	even	2		9464.2.a.bs.1.13	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9464.2.a.br.1.13	✓	15	1.1	even	1	trivial
9464.2.a.bs.1.13	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)

\(f(q)\)	\(=\)	\(q+2.32680 q^{3} +1.27247 q^{5} -1.00000 q^{7} +2.41400 q^{9} -6.42384 q^{11} +2.96078 q^{15} +3.33232 q^{17} +1.32253 q^{19} -2.32680 q^{21} +0.849071 q^{23} -3.38082 q^{25} -1.36350 q^{27} +8.55156 q^{29} -1.84234 q^{31} -14.9470 q^{33} -1.27247 q^{35} -9.91958 q^{37} -2.84770 q^{41} -6.03321 q^{43} +3.07174 q^{45} -4.70658 q^{47} +1.00000 q^{49} +7.75365 q^{51} -4.57771 q^{53} -8.17414 q^{55} +3.07727 q^{57} +1.56268 q^{59} -7.50272 q^{61} -2.41400 q^{63} -1.34362 q^{67} +1.97562 q^{69} -12.0208 q^{71} +13.1954 q^{73} -7.86650 q^{75} +6.42384 q^{77} -6.31729 q^{79} -10.4146 q^{81} -5.72464 q^{83} +4.24028 q^{85} +19.8978 q^{87} -12.5426 q^{89} -4.28676 q^{93} +1.68288 q^{95} +17.6875 q^{97} -15.5072 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 3 q^{3} - 4 q^{5} - 15 q^{7} + 16 q^{9} - 15 q^{11} - 8 q^{15} + 2 q^{17} - 13 q^{19} + 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} - 19 q^{31} + 24 q^{33} + 4 q^{35} + 2 q^{37} - 30 q^{41}+ \cdots - 70 q^{99}+O(q^{100}) \) 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

Varieties

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Database

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