LMFDB - Embedded newform 9464.2.a.y.1.3

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)

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);

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")

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 = [4,0,-1,0,5,0,4,0,5,0,-3,0,0,0,-18,0,-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:	$4$
Coefficient field:	4.4.122260.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 2x + 2 $ x^4 - x^3 - 8x^2 + 2x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.401251$ 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)

$f(q)$	$=$	$q+0.401251 q^{3} +0.598749 q^{5} +1.00000 q^{7} -2.83900 q^{9} -4.58315 q^{11} +0.240249 q^{15} -1.54666 q^{17} +5.58315 q^{19} +0.401251 q^{21} +1.98441 q^{23} -4.64150 q^{25} -2.34291 q^{27} +7.82340 q^{29} +0.546660 q^{31} -1.83900 q^{33} +0.598749 q^{35} -5.38566 q^{37} -0.854591 q^{41} +9.02090 q^{43} -1.69985 q^{45} +3.24025 q^{47} +1.00000 q^{49} -0.620600 q^{51} +0.327312 q^{53} -2.74416 q^{55} +2.24025 q^{57} -1.81810 q^{59} +2.82340 q^{61} -2.83900 q^{63} -2.05209 q^{67} +0.796246 q^{69} +6.24025 q^{71} -6.43775 q^{73} -1.86241 q^{75} -4.58315 q^{77} -11.3638 q^{79} +7.57690 q^{81} -4.67174 q^{83} -0.926061 q^{85} +3.13915 q^{87} +13.2611 q^{89} +0.219348 q^{93} +3.34291 q^{95} +8.11574 q^{97} +13.0116 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 5 q^{5} + 4 q^{7} + 5 q^{9} - 3 q^{11} - 18 q^{15} - 2 q^{17} + 7 q^{19} - q^{21} - 10 q^{23} + 3 q^{25} - 13 q^{27} - 3 q^{29} - 2 q^{31} + 9 q^{33} + 5 q^{35} - q^{37} - 5 q^{41} + 7 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) $ 4 * q - q^3 + 5 * q^5 + 4 * q^7 + 5 * q^9 - 3 * q^11 - 18 * q^15 - 2 * q^17 + 7 * q^19 - q^21 - 10 * q^23 + 3 * q^25 - 13 * q^27 - 3 * q^29 - 2 * q^31 + 9 * q^33 + 5 * q^35 - q^37 - 5 * q^41 + 7 * q^43 + 21 * q^45 - 6 * q^47 + 4 * q^49 - 10 * q^51 - 13 * q^53 - 12 * q^55 - 10 * q^57 - 20 * q^59 - 23 * q^61 + 5 * q^63 - 15 * q^67 + 11 * q^69 + 6 * q^71 - 12 * q^73 - 49 * q^75 - 3 * q^77 - 20 * q^79 + 28 * q^81 + q^83 + 8 * q^85 + 24 * q^87 + 5 * q^89 + 11 * q^93 + 17 * q^95 - 14 * q^97 - 8 * 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
$3$	0.401251	0.231663	0.115831	−	0.993269i	$-0.463047\pi$
$3$	0.401251	0.231663	0.115831	+	0.993269i	$0.463047\pi$
$4$	0	0
$5$	0.598749	0.267769	0.133884	−	0.990997i	$-0.457255\pi$
$5$	0.598749	0.267769	0.133884	+	0.990997i	$0.457255\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.83900	−0.946332
$10$	0	0
$11$	−4.58315	−1.38187	−0.690937	−	0.722915i	$-0.742802\pi$
$11$	−4.58315	−1.38187	−0.690937	+	0.722915i	$0.742802\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.240249	0.0620320
$16$	0	0
$17$	−1.54666	−0.375120	−0.187560	−	0.982253i	$-0.560058\pi$
$17$	−1.54666	−0.375120	−0.187560	+	0.982253i	$0.560058\pi$
$18$	0	0
$19$	5.58315	1.28086	0.640432	−	0.768015i	$-0.278755\pi$
$19$	5.58315	1.28086	0.640432	+	0.768015i	$0.278755\pi$
$20$	0	0
$21$	0.401251	0.0875602
$22$	0	0
$23$	1.98441	0.413777	0.206889	−	0.978364i	$-0.433666\pi$
$23$	1.98441	0.413777	0.206889	+	0.978364i	$0.433666\pi$
$24$	0	0
$25$	−4.64150	−0.928300
$26$	0	0
$27$	−2.34291	−0.450892
$28$	0	0
$29$	7.82340	1.45277	0.726385	−	0.687288i	$-0.241199\pi$
$29$	7.82340	1.45277	0.726385	+	0.687288i	$0.241199\pi$
$30$	0	0
$31$	0.546660	0.0981831	0.0490915	−	0.998794i	$-0.484367\pi$
$31$	0.546660	0.0981831	0.0490915	+	0.998794i	$0.484367\pi$
$32$	0	0
$33$	−1.83900	−0.320128
$34$	0	0
$35$	0.598749	0.101207
$36$	0	0
$37$	−5.38566	−0.885397	−0.442698	−	0.896671i	$-0.645979\pi$
$37$	−5.38566	−0.885397	−0.442698	+	0.896671i	$0.645979\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.854591	−0.133465	−0.0667324	−	0.997771i	$-0.521257\pi$
$41$	−0.854591	−0.133465	−0.0667324	+	0.997771i	$0.521257\pi$
$42$	0	0
$43$	9.02090	1.37567	0.687837	−	0.725865i	$-0.258560\pi$
$43$	9.02090	1.37567	0.687837	+	0.725865i	$0.258560\pi$
$44$	0	0
$45$	−1.69985	−0.253398
$46$	0	0
$47$	3.24025	0.472639	0.236319	−	0.971675i	$-0.424059\pi$
$47$	3.24025	0.472639	0.236319	+	0.971675i	$0.424059\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.620600	−0.0869013
$52$	0	0
$53$	0.327312	0.0449598	0.0224799	−	0.999747i	$-0.492844\pi$
$53$	0.327312	0.0449598	0.0224799	+	0.999747i	$0.492844\pi$
$54$	0	0
$55$	−2.74416	−0.370022
$56$	0	0
$57$	2.24025	0.296728
$58$	0	0
$59$	−1.81810	−0.236696	−0.118348	−	0.992972i	$-0.537760\pi$
$59$	−1.81810	−0.236696	−0.118348	+	0.992972i	$0.537760\pi$
$60$	0	0
$61$	2.82340	0.361500	0.180750	−	0.983529i	$-0.442148\pi$
$61$	2.82340	0.361500	0.180750	+	0.983529i	$0.442148\pi$
$62$	0	0
$63$	−2.83900	−0.357680
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.05209	−0.250702	−0.125351	−	0.992112i	$-0.540006\pi$
$67$	−2.05209	−0.250702	−0.125351	+	0.992112i	$0.540006\pi$
$68$	0	0
$69$	0.796246	0.0958567
$70$	0	0
$71$	6.24025	0.740581	0.370291	−	0.928916i	$-0.379258\pi$
$71$	6.24025	0.740581	0.370291	+	0.928916i	$0.379258\pi$
$72$	0	0
$73$	−6.43775	−0.753481	−0.376741	−	0.926319i	$-0.622955\pi$
$73$	−6.43775	−0.753481	−0.376741	+	0.926319i	$0.622955\pi$
$74$	0	0
$75$	−1.86241	−0.215052
$76$	0	0
$77$	−4.58315	−0.522299
$78$	0	0
$79$	−11.3638	−1.27853	−0.639264	−	0.768987i	$-0.720761\pi$
$79$	−11.3638	−1.27853	−0.639264	+	0.768987i	$0.720761\pi$
$80$	0	0
$81$	7.57690	0.841878
$82$	0	0
$83$	−4.67174	−0.512790	−0.256395	−	0.966572i	$-0.582535\pi$
$83$	−4.67174	−0.512790	−0.256395	+	0.966572i	$0.582535\pi$
$84$	0	0
$85$	−0.926061	−0.100445
$86$	0	0
$87$	3.13915	0.336552
$88$	0	0
$89$	13.2611	1.40568	0.702840	−	0.711348i	$-0.251915\pi$
$89$	13.2611	1.40568	0.702840	+	0.711348i	$0.251915\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.219348	0.0227453
$94$	0	0
$95$	3.34291	0.342975
$96$	0	0
$97$	8.11574	0.824029	0.412014	−	0.911177i	$-0.364825\pi$
$97$	8.11574	0.824029	0.412014	+	0.911177i	$0.364825\pi$
$98$	0	0
$99$	13.0116	1.30771
$100$	0	0
$101$	−18.7339	−1.86409	−0.932045	−	0.362343i	$-0.881977\pi$
$101$	−18.7339	−1.86409	−0.932045	+	0.362343i	$0.881977\pi$
$102$	0	0
$103$	−15.5738	−1.53453	−0.767267	−	0.641328i	$-0.778384\pi$
$103$	−15.5738	−1.53453	−0.767267	+	0.641328i	$0.778384\pi$
$104$	0	0
$105$	0.240249	0.0234459
$106$	0	0
$107$	−13.8287	−1.33687	−0.668436	−	0.743770i	$-0.733036\pi$
$107$	−13.8287	−1.33687	−0.668436	+	0.743770i	$0.733036\pi$
$108$	0	0
$109$	−11.3336	−1.08556	−0.542780	−	0.839875i	$-0.682628\pi$
$109$	−11.3336	−1.08556	−0.542780	+	0.839875i	$0.682628\pi$
$110$	0	0
$111$	−2.16100	−0.205113
$112$	0	0
$113$	−11.8802	−1.11760	−0.558799	−	0.829303i	$-0.688738\pi$
$113$	−11.8802	−1.11760	−0.558799	+	0.829303i	$0.688738\pi$
$114$	0	0
$115$	1.18816	0.110797
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.54666	−0.141782
$120$	0	0
$121$	10.0053	0.909573
$122$	0	0
$123$	−0.342906	−0.0309188
$124$	0	0
$125$	−5.77283	−0.516338
$126$	0	0
$127$	−1.06768	−0.0947415	−0.0473707	−	0.998877i	$-0.515084\pi$
$127$	−1.06768	−0.0947415	−0.0473707	+	0.998877i	$0.515084\pi$
$128$	0	0
$129$	3.61965	0.318692
$130$	0	0
$131$	−4.90668	−0.428699	−0.214349	−	0.976757i	$-0.568763\pi$
$131$	−4.90668	−0.428699	−0.214349	+	0.976757i	$0.568763\pi$
$132$	0	0
$133$	5.58315	0.484121
$134$	0	0
$135$	−1.40281	−0.120735
$136$	0	0
$137$	12.4999	1.06794	0.533968	−	0.845504i	$-0.320700\pi$
$137$	12.4999	1.06794	0.533968	+	0.845504i	$0.320700\pi$
$138$	0	0
$139$	−6.34916	−0.538529	−0.269264	−	0.963066i	$-0.586781\pi$
$139$	−6.34916	−0.538529	−0.269264	+	0.963066i	$0.586781\pi$
$140$	0	0
$141$	1.30015	0.109493
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4.68425	0.389006
$146$	0	0
$147$	0.401251	0.0330947
$148$	0	0
$149$	−21.8225	−1.78777	−0.893883	−	0.448301i	$-0.852029\pi$
$149$	−21.8225	−1.78777	−0.893883	+	0.448301i	$0.852029\pi$
$150$	0	0
$151$	−10.0924	−0.821305	−0.410653	−	0.911792i	$-0.634699\pi$
$151$	−10.0924	−0.821305	−0.410653	+	0.911792i	$0.634699\pi$
$152$	0	0
$153$	4.39096	0.354988
$154$	0	0
$155$	0.327312	0.0262903
$156$	0	0
$157$	3.56756	0.284722	0.142361	−	0.989815i	$-0.454531\pi$
$157$	3.56756	0.284722	0.142361	+	0.989815i	$0.454531\pi$
$158$	0	0
$159$	0.131334	0.0104155
$160$	0	0
$161$	1.98441	0.156393
$162$	0	0
$163$	−11.0933	−0.868896	−0.434448	−	0.900697i	$-0.643057\pi$
$163$	−11.0933	−0.868896	−0.434448	+	0.900697i	$0.643057\pi$
$164$	0	0
$165$	−1.10110	−0.0857203
$166$	0	0
$167$	15.9158	1.23160	0.615800	−	0.787903i	$-0.288833\pi$
$167$	15.9158	1.23160	0.615800	+	0.787903i	$0.288833\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−15.8506	−1.21212
$172$	0	0
$173$	−13.2665	−1.00863	−0.504315	−	0.863520i	$-0.668255\pi$
$173$	−13.2665	−1.00863	−0.504315	+	0.863520i	$0.668255\pi$
$174$	0	0
$175$	−4.64150	−0.350864
$176$	0	0
$177$	−0.729514	−0.0548336
$178$	0	0
$179$	−14.6989	−1.09865	−0.549324	−	0.835610i	$-0.685115\pi$
$179$	−14.6989	−1.09865	−0.549324	+	0.835610i	$0.685115\pi$
$180$	0	0
$181$	−15.2038	−1.13009	−0.565043	−	0.825061i	$-0.691140\pi$
$181$	−15.2038	−1.13009	−0.565043	+	0.825061i	$0.691140\pi$
$182$	0	0
$183$	1.13289	0.0837460
$184$	0	0
$185$	−3.22465	−0.237081
$186$	0	0
$187$	7.08858	0.518369
$188$	0	0
$189$	−2.34291	−0.170421
$190$	0	0
$191$	−7.75572	−0.561184	−0.280592	−	0.959827i	$-0.590531\pi$
$191$	−7.75572	−0.561184	−0.280592	+	0.959827i	$0.590531\pi$
$192$	0	0
$193$	6.87549	0.494909	0.247454	−	0.968900i	$-0.420406\pi$
$193$	6.87549	0.494909	0.247454	+	0.968900i	$0.420406\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.24959	0.516512	0.258256	−	0.966077i	$-0.416852\pi$
$197$	7.24959	0.516512	0.258256	+	0.966077i	$0.416852\pi$
$198$	0	0
$199$	−26.7236	−1.89438	−0.947192	−	0.320666i	$-0.896093\pi$
$199$	−26.7236	−1.89438	−0.947192	+	0.320666i	$0.896093\pi$
$200$	0	0
$201$	−0.823403	−0.0580784
$202$	0	0
$203$	7.82340	0.549095
$204$	0	0
$205$	−0.511685	−0.0357377
$206$	0	0
$207$	−5.63372	−0.391571
$208$	0	0
$209$	−25.5885	−1.76999
$210$	0	0
$211$	13.9011	0.956993	0.478497	−	0.878089i	$-0.341182\pi$
$211$	13.9011	0.956993	0.478497	+	0.878089i	$0.341182\pi$
$212$	0	0
$213$	2.50391	0.171565
$214$	0	0
$215$	5.40125	0.368362
$216$	0	0
$217$	0.546660	0.0371097
$218$	0	0
$219$	−2.58315	−0.174553
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−19.9026	−1.33278	−0.666390	−	0.745603i	$-0.732161\pi$
$223$	−19.9026	−1.33278	−0.666390	+	0.745603i	$0.732161\pi$
$224$	0	0
$225$	13.1772	0.878480
$226$	0	0
$227$	−20.2596	−1.34468	−0.672339	−	0.740243i	$-0.734710\pi$
$227$	−20.2596	−1.34468	−0.672339	+	0.740243i	$0.734710\pi$
$228$	0	0
$229$	−9.30641	−0.614985	−0.307492	−	0.951551i	$-0.599490\pi$
$229$	−9.30641	−0.614985	−0.307492	+	0.951551i	$0.599490\pi$
$230$	0	0
$231$	−1.83900	−0.120997
$232$	0	0
$233$	13.3326	0.873449	0.436724	−	0.899595i	$-0.356138\pi$
$233$	13.3326	0.873449	0.436724	+	0.899595i	$0.356138\pi$
$234$	0	0
$235$	1.94009	0.126558
$236$	0	0
$237$	−4.55974	−0.296187
$238$	0	0
$239$	0.729514	0.0471883	0.0235942	−	0.999722i	$-0.492489\pi$
$239$	0.729514	0.0471883	0.0235942	+	0.999722i	$0.492489\pi$
$240$	0	0
$241$	−1.44305	−0.0929552	−0.0464776	−	0.998919i	$-0.514800\pi$
$241$	−1.44305	−0.0929552	−0.0464776	+	0.998919i	$0.514800\pi$
$242$	0	0
$243$	10.0690	0.645924
$244$	0	0
$245$	0.598749	0.0382526
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1.87454	−0.118794
$250$	0	0
$251$	20.3389	1.28378	0.641889	−	0.766797i	$-0.278151\pi$
$251$	20.3389	1.28378	0.641889	+	0.766797i	$0.278151\pi$
$252$	0	0
$253$	−9.09484	−0.571788
$254$	0	0
$255$	−0.371583	−0.0232694
$256$	0	0
$257$	−24.4951	−1.52796	−0.763982	−	0.645237i	$-0.776759\pi$
$257$	−24.4951	−1.52796	−0.763982	+	0.645237i	$0.776759\pi$
$258$	0	0
$259$	−5.38566	−0.334648
$260$	0	0
$261$	−22.2106	−1.37480
$262$	0	0
$263$	18.0100	1.11055	0.555273	−	0.831668i	$-0.312614\pi$
$263$	18.0100	1.11055	0.555273	+	0.831668i	$0.312614\pi$
$264$	0	0
$265$	0.195978	0.0120388
$266$	0	0
$267$	5.32105	0.325643
$268$	0	0
$269$	13.5822	0.828122	0.414061	−	0.910249i	$-0.364110\pi$
$269$	13.5822	0.828122	0.414061	+	0.910249i	$0.364110\pi$
$270$	0	0
$271$	−23.9460	−1.45462	−0.727308	−	0.686311i	$-0.759229\pi$
$271$	−23.9460	−1.45462	−0.727308	+	0.686311i	$0.759229\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	21.2727	1.28279
$276$	0	0
$277$	16.6989	1.00334	0.501670	−	0.865059i	$-0.332719\pi$
$277$	16.6989	1.00334	0.501670	+	0.865059i	$0.332719\pi$
$278$	0	0
$279$	−1.55197	−0.0929138
$280$	0	0
$281$	−16.4269	−0.979946	−0.489973	−	0.871738i	$-0.662993\pi$
$281$	−16.4269	−0.979946	−0.489973	+	0.871738i	$0.662993\pi$
$282$	0	0
$283$	−32.0627	−1.90593	−0.952965	−	0.303081i	$-0.901985\pi$
$283$	−32.0627	−1.90593	−0.952965	+	0.303081i	$0.901985\pi$
$284$	0	0
$285$	1.34135	0.0794545
$286$	0	0
$287$	−0.854591	−0.0504449
$288$	0	0
$289$	−14.6078	−0.859285
$290$	0	0
$291$	3.25645	0.190897
$292$	0	0
$293$	17.1882	1.00414	0.502072	−	0.864826i	$-0.332571\pi$
$293$	17.1882	1.00414	0.502072	+	0.864826i	$0.332571\pi$
$294$	0	0
$295$	−1.08858	−0.0633797
$296$	0	0
$297$	10.7379	0.623076
$298$	0	0
$299$	0	0
$300$	0	0
$301$	9.02090	0.519956
$302$	0	0
$303$	−7.51699	−0.431840
$304$	0	0
$305$	1.69051	0.0967983
$306$	0	0
$307$	−17.5160	−0.999693	−0.499847	−	0.866114i	$-0.666610\pi$
$307$	−17.5160	−0.999693	−0.499847	+	0.866114i	$0.666610\pi$
$308$	0	0
$309$	−6.24902	−0.355494
$310$	0	0
$311$	−12.9338	−0.733411	−0.366705	−	0.930337i	$-0.619514\pi$
$311$	−12.9338	−0.733411	−0.366705	+	0.930337i	$0.619514\pi$
$312$	0	0
$313$	−20.3910	−1.15257	−0.576283	−	0.817250i	$-0.695497\pi$
$313$	−20.3910	−1.15257	−0.576283	+	0.817250i	$0.695497\pi$
$314$	0	0
$315$	−1.69985	−0.0957755
$316$	0	0
$317$	11.6134	0.652273	0.326137	−	0.945323i	$-0.394253\pi$
$317$	11.6134	0.652273	0.326137	+	0.945323i	$0.394253\pi$
$318$	0	0
$319$	−35.8559	−2.00754
$320$	0	0
$321$	−5.54879	−0.309703
$322$	0	0
$323$	−8.63524	−0.480478
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−4.54761	−0.251483
$328$	0	0
$329$	3.24025	0.178641
$330$	0	0
$331$	27.4468	1.50861	0.754307	−	0.656521i	$-0.227973\pi$
$331$	27.4468	1.50861	0.754307	+	0.656521i	$0.227973\pi$
$332$	0	0
$333$	15.2899	0.837880
$334$	0	0
$335$	−1.22869	−0.0671302
$336$	0	0
$337$	−17.9114	−0.975697	−0.487849	−	0.872928i	$-0.662218\pi$
$337$	−17.9114	−0.975697	−0.487849	+	0.872928i	$0.662218\pi$
$338$	0	0
$339$	−4.76696	−0.258906
$340$	0	0
$341$	−2.50543	−0.135677
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.476751	0.0256674
$346$	0	0
$347$	−0.480497	−0.0257945	−0.0128972	−	0.999917i	$-0.504105\pi$
$347$	−0.480497	−0.0257945	−0.0128972	+	0.999917i	$0.504105\pi$
$348$	0	0
$349$	−21.1258	−1.13084	−0.565419	−	0.824804i	$-0.691285\pi$
$349$	−21.1258	−1.13084	−0.565419	+	0.824804i	$0.691285\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.2446	1.29041	0.645205	−	0.764009i	$-0.276772\pi$
$353$	24.2446	1.29041	0.645205	+	0.764009i	$0.276772\pi$
$354$	0	0
$355$	3.73634	0.198304
$356$	0	0
$357$	−0.620600	−0.0328456
$358$	0	0
$359$	27.0846	1.42947	0.714734	−	0.699396i	$-0.246548\pi$
$359$	27.0846	1.42947	0.714734	+	0.699396i	$0.246548\pi$
$360$	0	0
$361$	12.1716	0.640611
$362$	0	0
$363$	4.01464	0.210714
$364$	0	0
$365$	−3.85459	−0.201759
$366$	0	0
$367$	−2.13133	−0.111255	−0.0556274	−	0.998452i	$-0.517716\pi$
$367$	−2.13133	−0.111255	−0.0556274	+	0.998452i	$0.517716\pi$
$368$	0	0
$369$	2.42618	0.126302
$370$	0	0
$371$	0.327312	0.0169932
$372$	0	0
$373$	−4.57690	−0.236983	−0.118491	−	0.992955i	$-0.537806\pi$
$373$	−4.57690	−0.236983	−0.118491	+	0.992955i	$0.537806\pi$
$374$	0	0
$375$	−2.31636	−0.119616
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−2.88426	−0.148154	−0.0740772	−	0.997253i	$-0.523601\pi$
$379$	−2.88426	−0.148154	−0.0740772	+	0.997253i	$0.523601\pi$
$380$	0	0
$381$	−0.428409	−0.0219481
$382$	0	0
$383$	8.35542	0.426942	0.213471	−	0.976949i	$-0.431523\pi$
$383$	8.35542	0.426942	0.213471	+	0.976949i	$0.431523\pi$
$384$	0	0
$385$	−2.74416	−0.139855
$386$	0	0
$387$	−25.6103	−1.30185
$388$	0	0
$389$	−33.4212	−1.69452	−0.847261	−	0.531177i	$-0.821750\pi$
$389$	−33.4212	−1.69452	−0.847261	+	0.531177i	$0.821750\pi$
$390$	0	0
$391$	−3.06920	−0.155216
$392$	0	0
$393$	−1.96881	−0.0993134
$394$	0	0
$395$	−6.80406	−0.342350
$396$	0	0
$397$	37.2611	1.87008	0.935042	−	0.354538i	$-0.115362\pi$
$397$	37.2611	1.87008	0.935042	+	0.354538i	$0.115362\pi$
$398$	0	0
$399$	2.24025	0.112153
$400$	0	0
$401$	10.6858	0.533624	0.266812	−	0.963749i	$-0.414030\pi$
$401$	10.6858	0.533624	0.266812	+	0.963749i	$0.414030\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	4.53666	0.225428
$406$	0	0
$407$	24.6833	1.22351
$408$	0	0
$409$	32.8643	1.62503	0.812516	−	0.582938i	$-0.198097\pi$
$409$	32.8643	1.62503	0.812516	+	0.582938i	$0.198097\pi$
$410$	0	0
$411$	5.01559	0.247401
$412$	0	0
$413$	−1.81810	−0.0894627
$414$	0	0
$415$	−2.79720	−0.137309
$416$	0	0
$417$	−2.54761	−0.124757
$418$	0	0
$419$	27.8302	1.35960	0.679798	−	0.733400i	$-0.262068\pi$
$419$	27.8302	1.35960	0.679798	+	0.733400i	$0.262068\pi$
$420$	0	0
$421$	35.5004	1.73019	0.865093	−	0.501612i	$-0.167259\pi$
$421$	35.5004	1.73019	0.865093	+	0.501612i	$0.167259\pi$
$422$	0	0
$423$	−9.19906	−0.447273
$424$	0	0
$425$	7.17882	0.348224
$426$	0	0
$427$	2.82340	0.136634
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.6546	0.609552	0.304776	−	0.952424i	$-0.401418\pi$
$431$	12.6546	0.609552	0.304776	+	0.952424i	$0.401418\pi$
$432$	0	0
$433$	16.6671	0.800972	0.400486	−	0.916303i	$-0.368841\pi$
$433$	16.6671	0.800972	0.400486	+	0.916303i	$0.368841\pi$
$434$	0	0
$435$	1.87956	0.0901181
$436$	0	0
$437$	11.0792	0.529992
$438$	0	0
$439$	−19.3841	−0.925154	−0.462577	−	0.886579i	$-0.653075\pi$
$439$	−19.3841	−0.925154	−0.462577	+	0.886579i	$0.653075\pi$
$440$	0	0
$441$	−2.83900	−0.135190
$442$	0	0
$443$	−20.9120	−0.993558	−0.496779	−	0.867877i	$-0.665484\pi$
$443$	−20.9120	−0.993558	−0.496779	+	0.867877i	$0.665484\pi$
$444$	0	0
$445$	7.94009	0.376397
$446$	0	0
$447$	−8.75629	−0.414158
$448$	0	0
$449$	16.1585	0.762566	0.381283	−	0.924458i	$-0.375482\pi$
$449$	16.1585	0.762566	0.381283	+	0.924458i	$0.375482\pi$
$450$	0	0
$451$	3.91672	0.184431
$452$	0	0
$453$	−4.04958	−0.190266
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.6274	0.918132	0.459066	−	0.888402i	$-0.348184\pi$
$457$	19.6274	0.918132	0.459066	+	0.888402i	$0.348184\pi$
$458$	0	0
$459$	3.62368	0.169139
$460$	0	0
$461$	17.8627	0.831950	0.415975	−	0.909376i	$-0.363440\pi$
$461$	17.8627	0.831950	0.415975	+	0.909376i	$0.363440\pi$
$462$	0	0
$463$	10.0924	0.469032	0.234516	−	0.972112i	$-0.424649\pi$
$463$	10.0924	0.469032	0.234516	+	0.972112i	$0.424649\pi$
$464$	0	0
$465$	0.131334	0.00609049
$466$	0	0
$467$	−10.6118	−0.491057	−0.245529	−	0.969389i	$-0.578961\pi$
$467$	−10.6118	−0.491057	−0.245529	+	0.969389i	$0.578961\pi$
$468$	0	0
$469$	−2.05209	−0.0947566
$470$	0	0
$471$	1.43149	0.0659595
$472$	0	0
$473$	−41.3442	−1.90101
$474$	0	0
$475$	−25.9142	−1.18903
$476$	0	0
$477$	−0.929238	−0.0425469
$478$	0	0
$479$	−11.0063	−0.502889	−0.251444	−	0.967872i	$-0.580906\pi$
$479$	−11.0063	−0.502889	−0.251444	+	0.967872i	$0.580906\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0.796246	0.0362304
$484$	0	0
$485$	4.85929	0.220649
$486$	0	0
$487$	9.87715	0.447576	0.223788	−	0.974638i	$-0.428158\pi$
$487$	9.87715	0.447576	0.223788	+	0.974638i	$0.428158\pi$
$488$	0	0
$489$	−4.45121	−0.201291
$490$	0	0
$491$	23.0992	1.04245	0.521226	−	0.853419i	$-0.325475\pi$
$491$	23.0992	1.04245	0.521226	+	0.853419i	$0.325475\pi$
$492$	0	0
$493$	−12.1001	−0.544963
$494$	0	0
$495$	7.79066	0.350164
$496$	0	0
$497$	6.24025	0.279913
$498$	0	0
$499$	−12.7310	−0.569919	−0.284960	−	0.958540i	$-0.591980\pi$
$499$	−12.7310	−0.569919	−0.284960	+	0.958540i	$0.591980\pi$
$500$	0	0
$501$	6.38623	0.285316
$502$	0	0
$503$	−11.9688	−0.533663	−0.266831	−	0.963743i	$-0.585977\pi$
$503$	−11.9688	−0.533663	−0.266831	+	0.963743i	$0.585977\pi$
$504$	0	0
$505$	−11.2169	−0.499145
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4.13664	−0.183353	−0.0916767	−	0.995789i	$-0.529223\pi$
$509$	−4.13664	−0.183353	−0.0916767	+	0.995789i	$0.529223\pi$
$510$	0	0
$511$	−6.43775	−0.284789
$512$	0	0
$513$	−13.0808	−0.577532
$514$	0	0
$515$	−9.32480	−0.410900
$516$	0	0
$517$	−14.8506	−0.653127
$518$	0	0
$519$	−5.32318	−0.233662
$520$	0	0
$521$	−1.96881	−0.0862552	−0.0431276	−	0.999070i	$-0.513732\pi$
$521$	−1.96881	−0.0862552	−0.0431276	+	0.999070i	$0.513732\pi$
$522$	0	0
$523$	30.6056	1.33829	0.669144	−	0.743133i	$-0.266661\pi$
$523$	30.6056	1.33829	0.669144	+	0.743133i	$0.266661\pi$
$524$	0	0
$525$	−1.86241	−0.0812822
$526$	0	0
$527$	−0.845498	−0.0368305
$528$	0	0
$529$	−19.0621	−0.828788
$530$	0	0
$531$	5.16157	0.223993
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−8.27992	−0.357972
$536$	0	0
$537$	−5.89795	−0.254515
$538$	0	0
$539$	−4.58315	−0.197410
$540$	0	0
$541$	0.375613	0.0161489	0.00807444	−	0.999967i	$-0.497430\pi$
$541$	0.375613	0.0161489	0.00807444	+	0.999967i	$0.497430\pi$
$542$	0	0
$543$	−6.10053	−0.261799
$544$	0	0
$545$	−6.78596	−0.290679
$546$	0	0
$547$	−13.4331	−0.574360	−0.287180	−	0.957877i	$-0.592718\pi$
$547$	−13.4331	−0.574360	−0.287180	+	0.957877i	$0.592718\pi$
$548$	0	0
$549$	−8.01563	−0.342099
$550$	0	0
$551$	43.6793	1.86080
$552$	0	0
$553$	−11.3638	−0.483238
$554$	0	0
$555$	−1.29390	−0.0549229
$556$	0	0
$557$	39.8669	1.68921	0.844607	−	0.535387i	$-0.179834\pi$
$557$	39.8669	1.68921	0.844607	+	0.535387i	$0.179834\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	2.84430	0.120087
$562$	0	0
$563$	31.5217	1.32848	0.664241	−	0.747518i	$-0.268755\pi$
$563$	31.5217	1.32848	0.664241	+	0.747518i	$0.268755\pi$
$564$	0	0
$565$	−7.11327	−0.299258
$566$	0	0
$567$	7.57690	0.318200
$568$	0	0
$569$	1.55349	0.0651255	0.0325628	−	0.999470i	$-0.489633\pi$
$569$	1.55349	0.0651255	0.0325628	+	0.999470i	$0.489633\pi$
$570$	0	0
$571$	7.06744	0.295763	0.147882	−	0.989005i	$-0.452755\pi$
$571$	7.06744	0.295763	0.147882	+	0.989005i	$0.452755\pi$
$572$	0	0
$573$	−3.11199	−0.130005
$574$	0	0
$575$	−9.21062	−0.384109
$576$	0	0
$577$	−39.3248	−1.63711	−0.818556	−	0.574426i	$-0.805225\pi$
$577$	−39.3248	−1.63711	−0.818556	+	0.574426i	$0.805225\pi$
$578$	0	0
$579$	2.75880	0.114652
$580$	0	0
$581$	−4.67174	−0.193816
$582$	0	0
$583$	−1.50012	−0.0621287
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−38.2144	−1.57728	−0.788638	−	0.614858i	$-0.789213\pi$
$587$	−38.2144	−1.57728	−0.788638	+	0.614858i	$0.789213\pi$
$588$	0	0
$589$	3.05209	0.125759
$590$	0	0
$591$	2.90891	0.119656
$592$	0	0
$593$	−0.734113	−0.0301464	−0.0150732	−	0.999886i	$-0.504798\pi$
$593$	−0.734113	−0.0301464	−0.0150732	+	0.999886i	$0.504798\pi$
$594$	0	0
$595$	−0.926061	−0.0379648
$596$	0	0
$597$	−10.7229	−0.438858
$598$	0	0
$599$	−24.6593	−1.00755	−0.503776	−	0.863834i	$-0.668056\pi$
$599$	−24.6593	−1.00755	−0.503776	+	0.863834i	$0.668056\pi$
$600$	0	0
$601$	20.4181	0.832873	0.416436	−	0.909165i	$-0.363279\pi$
$601$	20.4181	0.832873	0.416436	+	0.909165i	$0.363279\pi$
$602$	0	0
$603$	5.82587	0.237248
$604$	0	0
$605$	5.99066	0.243555
$606$	0	0
$607$	−46.8821	−1.90288	−0.951442	−	0.307827i	$-0.900398\pi$
$607$	−46.8821	−1.90288	−0.951442	+	0.307827i	$0.900398\pi$
$608$	0	0
$609$	3.13915	0.127205
$610$	0	0
$611$	0	0
$612$	0	0
$613$	19.6696	0.794448	0.397224	−	0.917722i	$-0.369974\pi$
$613$	19.6696	0.794448	0.397224	+	0.917722i	$0.369974\pi$
$614$	0	0
$615$	−0.205314	−0.00827908
$616$	0	0
$617$	−14.9143	−0.600425	−0.300213	−	0.953872i	$-0.597058\pi$
$617$	−14.9143	−0.600425	−0.300213	+	0.953872i	$0.597058\pi$
$618$	0	0
$619$	35.7198	1.43570	0.717850	−	0.696198i	$-0.245126\pi$
$619$	35.7198	1.43570	0.717850	+	0.696198i	$0.245126\pi$
$620$	0	0
$621$	−4.64928	−0.186569
$622$	0	0
$623$	13.2611	0.531297
$624$	0	0
$625$	19.7510	0.790041
$626$	0	0
$627$	−10.2674	−0.410041
$628$	0	0
$629$	8.32978	0.332130
$630$	0	0
$631$	13.4696	0.536218	0.268109	−	0.963389i	$-0.413601\pi$
$631$	13.4696	0.536218	0.268109	+	0.963389i	$0.413601\pi$
$632$	0	0
$633$	5.57785	0.221700
$634$	0	0
$635$	−0.639273	−0.0253688
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−17.7160	−0.700836
$640$	0	0
$641$	23.8880	0.943520	0.471760	−	0.881727i	$-0.343619\pi$
$641$	23.8880	0.943520	0.471760	+	0.881727i	$0.343619\pi$
$642$	0	0
$643$	−20.1680	−0.795347	−0.397673	−	0.917527i	$-0.630182\pi$
$643$	−20.1680	−0.795347	−0.397673	+	0.917527i	$0.630182\pi$
$644$	0	0
$645$	2.16726	0.0853358
$646$	0	0
$647$	−14.0895	−0.553917	−0.276958	−	0.960882i	$-0.589326\pi$
$647$	−14.0895	−0.553917	−0.276958	+	0.960882i	$0.589326\pi$
$648$	0	0
$649$	8.33262	0.327084
$650$	0	0
$651$	0.219348	0.00859693
$652$	0	0
$653$	37.0727	1.45077	0.725384	−	0.688344i	$-0.241662\pi$
$653$	37.0727	1.45077	0.725384	+	0.688344i	$0.241662\pi$
$654$	0	0
$655$	−2.93787	−0.114792
$656$	0	0
$657$	18.2767	0.713044
$658$	0	0
$659$	27.0986	1.05561	0.527806	−	0.849365i	$-0.323015\pi$
$659$	27.0986	1.05561	0.527806	+	0.849365i	$0.323015\pi$
$660$	0	0
$661$	−22.0372	−0.857148	−0.428574	−	0.903507i	$-0.640984\pi$
$661$	−22.0372	−0.857148	−0.428574	+	0.903507i	$0.640984\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.34291	0.129632
$666$	0	0
$667$	15.5248	0.601123
$668$	0	0
$669$	−7.98597	−0.308755
$670$	0	0
$671$	−12.9401	−0.499547
$672$	0	0
$673$	−34.5797	−1.33295	−0.666475	−	0.745528i	$-0.732198\pi$
$673$	−34.5797	−1.33295	−0.666475	+	0.745528i	$0.732198\pi$
$674$	0	0
$675$	10.8746	0.418563
$676$	0	0
$677$	10.2952	0.395676	0.197838	−	0.980235i	$-0.436608\pi$
$677$	10.2952	0.395676	0.197838	+	0.980235i	$0.436608\pi$
$678$	0	0
$679$	8.11574	0.311454
$680$	0	0
$681$	−8.12921	−0.311512
$682$	0	0
$683$	−41.4905	−1.58759	−0.793796	−	0.608184i	$-0.791898\pi$
$683$	−41.4905	−1.58759	−0.793796	+	0.608184i	$0.791898\pi$
$684$	0	0
$685$	7.48428	0.285960
$686$	0	0
$687$	−3.73421	−0.142469
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−28.4461	−1.08214	−0.541071	−	0.840977i	$-0.681981\pi$
$691$	−28.4461	−1.08214	−0.541071	+	0.840977i	$0.681981\pi$
$692$	0	0
$693$	13.0116	0.494268
$694$	0	0
$695$	−3.80155	−0.144201
$696$	0	0
$697$	1.32176	0.0500653
$698$	0	0
$699$	5.34973	0.202345
$700$	0	0
$701$	−13.2699	−0.501198	−0.250599	−	0.968091i	$-0.580628\pi$
$701$	−13.2699	−0.501198	−0.250599	+	0.968091i	$0.580628\pi$
$702$	0	0
$703$	−30.0690	−1.13407
$704$	0	0
$705$	0.778466	0.0293187
$706$	0	0
$707$	−18.7339	−0.704560
$708$	0	0
$709$	12.7416	0.478523	0.239261	−	0.970955i	$-0.423095\pi$
$709$	12.7416	0.478523	0.239261	+	0.970955i	$0.423095\pi$
$710$	0	0
$711$	32.2618	1.20991
$712$	0	0
$713$	1.08480	0.0406259
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.292718	0.0109318
$718$	0	0
$719$	35.7470	1.33314	0.666568	−	0.745444i	$-0.267763\pi$
$719$	35.7470	1.33314	0.666568	+	0.745444i	$0.267763\pi$
$720$	0	0
$721$	−15.5738	−0.579999
$722$	0	0
$723$	−0.579027	−0.0215342
$724$	0	0
$725$	−36.3123	−1.34861
$726$	0	0
$727$	−19.3435	−0.717410	−0.358705	−	0.933451i	$-0.616782\pi$
$727$	−19.3435	−0.717410	−0.358705	+	0.933451i	$0.616782\pi$
$728$	0	0
$729$	−18.6905	−0.692241
$730$	0	0
$731$	−13.9523	−0.516043
$732$	0	0
$733$	26.7154	0.986757	0.493379	−	0.869815i	$-0.335762\pi$
$733$	26.7154	0.986757	0.493379	+	0.869815i	$0.335762\pi$
$734$	0	0
$735$	0.240249	0.00886171
$736$	0	0
$737$	9.40504	0.346439
$738$	0	0
$739$	−12.6071	−0.463759	−0.231880	−	0.972744i	$-0.574488\pi$
$739$	−12.6071	−0.463759	−0.231880	+	0.972744i	$0.574488\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−42.5348	−1.56045	−0.780225	−	0.625498i	$-0.784896\pi$
$743$	−42.5348	−1.56045	−0.780225	+	0.625498i	$0.784896\pi$
$744$	0	0
$745$	−13.0662	−0.478707
$746$	0	0
$747$	13.2631	0.485270
$748$	0	0
$749$	−13.8287	−0.505290
$750$	0	0
$751$	−16.9563	−0.618744	−0.309372	−	0.950941i	$-0.600119\pi$
$751$	−16.9563	−0.618744	−0.309372	+	0.950941i	$0.600119\pi$
$752$	0	0
$753$	8.16100	0.297403
$754$	0	0
$755$	−6.04279	−0.219920
$756$	0	0
$757$	−29.5619	−1.07444	−0.537222	−	0.843441i	$-0.680526\pi$
$757$	−29.5619	−1.07444	−0.537222	+	0.843441i	$0.680526\pi$
$758$	0	0
$759$	−3.64932	−0.132462
$760$	0	0
$761$	13.9775	0.506685	0.253343	−	0.967377i	$-0.418470\pi$
$761$	13.9775	0.506685	0.253343	+	0.967377i	$0.418470\pi$
$762$	0	0
$763$	−11.3336	−0.410303
$764$	0	0
$765$	2.62908	0.0950547
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−18.5280	−0.668136	−0.334068	−	0.942549i	$-0.608422\pi$
$769$	−18.5280	−0.668136	−0.334068	+	0.942549i	$0.608422\pi$
$770$	0	0
$771$	−9.82871	−0.353972
$772$	0	0
$773$	−45.2936	−1.62910	−0.814549	−	0.580095i	$-0.803016\pi$
$773$	−45.2936	−1.62910	−0.814549	+	0.580095i	$0.803016\pi$
$774$	0	0
$775$	−2.53732	−0.0911433
$776$	0	0
$777$	−2.16100	−0.0775255
$778$	0	0
$779$	−4.77131	−0.170950
$780$	0	0
$781$	−28.6000	−1.02339
$782$	0	0
$783$	−18.3295	−0.655043
$784$	0	0
$785$	2.13607	0.0762397
$786$	0	0
$787$	−2.70483	−0.0964166	−0.0482083	−	0.998837i	$-0.515351\pi$
$787$	−2.70483	−0.0964166	−0.0482083	+	0.998837i	$0.515351\pi$
$788$	0	0
$789$	7.22656	0.257272
$790$	0	0
$791$	−11.8802	−0.422412
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0.0786363	0.00278894
$796$	0	0
$797$	−11.1435	−0.394723	−0.197362	−	0.980331i	$-0.563237\pi$
$797$	−11.1435	−0.394723	−0.197362	+	0.980331i	$0.563237\pi$
$798$	0	0
$799$	−5.01156	−0.177296
$800$	0	0
$801$	−37.6484	−1.33024
$802$	0	0
$803$	29.5052	1.04122
$804$	0	0
$805$	1.18816	0.0418771
$806$	0	0
$807$	5.44988	0.191845
$808$	0	0
$809$	−0.906111	−0.0318571	−0.0159286	−	0.999873i	$-0.505070\pi$
$809$	−0.906111	−0.0318571	−0.0159286	+	0.999873i	$0.505070\pi$
$810$	0	0
$811$	26.5953	0.933887	0.466943	−	0.884287i	$-0.345355\pi$
$811$	26.5953	0.933887	0.466943	+	0.884287i	$0.345355\pi$
$812$	0	0
$813$	−9.60837	−0.336980
$814$	0	0
$815$	−6.64211	−0.232663
$816$	0	0
$817$	50.3651	1.76205
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25.6655	0.895731	0.447866	−	0.894101i	$-0.352184\pi$
$821$	25.6655	0.895731	0.447866	+	0.894101i	$0.352184\pi$
$822$	0	0
$823$	−27.2992	−0.951589	−0.475795	−	0.879556i	$-0.657839\pi$
$823$	−27.2992	−0.951589	−0.475795	+	0.879556i	$0.657839\pi$
$824$	0	0
$825$	8.53571	0.297175
$826$	0	0
$827$	−6.88497	−0.239414	−0.119707	−	0.992809i	$-0.538195\pi$
$827$	−6.88497	−0.239414	−0.119707	+	0.992809i	$0.538195\pi$
$828$	0	0
$829$	50.8278	1.76532	0.882661	−	0.470011i	$-0.155750\pi$
$829$	50.8278	1.76532	0.882661	+	0.470011i	$0.155750\pi$
$830$	0	0
$831$	6.70046	0.232436
$832$	0	0
$833$	−1.54666	−0.0535886
$834$	0	0
$835$	9.52955	0.329784
$836$	0	0
$837$	−1.28077	−0.0442700
$838$	0	0
$839$	−0.915066	−0.0315916	−0.0157958	−	0.999875i	$-0.505028\pi$
$839$	−0.915066	−0.0315916	−0.0157958	+	0.999875i	$0.505028\pi$
$840$	0	0
$841$	32.2056	1.11054
$842$	0	0
$843$	−6.59131	−0.227017
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.0053	0.343786
$848$	0	0
$849$	−12.8652	−0.441533
$850$	0	0
$851$	−10.6873	−0.366357
$852$	0	0
$853$	14.3622	0.491754	0.245877	−	0.969301i	$-0.420924\pi$
$853$	14.3622	0.491754	0.245877	+	0.969301i	$0.420924\pi$
$854$	0	0
$855$	−9.49050	−0.324568
$856$	0	0
$857$	1.62368	0.0554638	0.0277319	−	0.999615i	$-0.491172\pi$
$857$	1.62368	0.0554638	0.0277319	+	0.999615i	$0.491172\pi$
$858$	0	0
$859$	−48.1045	−1.64130	−0.820652	−	0.571428i	$-0.806390\pi$
$859$	−48.1045	−1.64130	−0.820652	+	0.571428i	$0.806390\pi$
$860$	0	0
$861$	−0.342906	−0.0116862
$862$	0	0
$863$	−42.5828	−1.44953	−0.724767	−	0.688994i	$-0.758053\pi$
$863$	−42.5828	−1.44953	−0.724767	+	0.688994i	$0.758053\pi$
$864$	0	0
$865$	−7.94327	−0.270079
$866$	0	0
$867$	−5.86142	−0.199064
$868$	0	0
$869$	52.0821	1.76676
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−23.0406	−0.779805
$874$	0	0
$875$	−5.77283	−0.195157
$876$	0	0
$877$	−37.9080	−1.28006	−0.640030	−	0.768350i	$-0.721078\pi$
$877$	−37.9080	−1.28006	−0.640030	+	0.768350i	$0.721078\pi$
$878$	0	0
$879$	6.89677	0.232622
$880$	0	0
$881$	−55.8471	−1.88154	−0.940768	−	0.339050i	$-0.889894\pi$
$881$	−55.8471	−1.88154	−0.940768	+	0.339050i	$0.889894\pi$
$882$	0	0
$883$	−41.4193	−1.39387	−0.696935	−	0.717134i	$-0.745454\pi$
$883$	−41.4193	−1.39387	−0.696935	+	0.717134i	$0.745454\pi$
$884$	0	0
$885$	−0.436795	−0.0146827
$886$	0	0
$887$	1.12451	0.0377573	0.0188786	−	0.999822i	$-0.493990\pi$
$887$	1.12451	0.0377573	0.0188786	+	0.999822i	$0.493990\pi$
$888$	0	0
$889$	−1.06768	−0.0358089
$890$	0	0
$891$	−34.7261	−1.16337
$892$	0	0
$893$	18.0908	0.605386
$894$	0	0
$895$	−8.80094	−0.294183
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.27674	0.142637
$900$	0	0
$901$	−0.506240	−0.0168653
$902$	0	0
$903$	3.61965	0.120454
$904$	0	0
$905$	−9.10323	−0.302601
$906$	0	0
$907$	35.9711	1.19440	0.597200	−	0.802092i	$-0.296280\pi$
$907$	35.9711	1.19440	0.597200	+	0.802092i	$0.296280\pi$
$908$	0	0
$909$	53.1854	1.76405
$910$	0	0
$911$	13.2106	0.437686	0.218843	−	0.975760i	$-0.429772\pi$
$911$	13.2106	0.437686	0.218843	+	0.975760i	$0.429772\pi$
$912$	0	0
$913$	21.4113	0.708610
$914$	0	0
$915$	0.678319	0.0224245
$916$	0	0
$917$	−4.90668	−0.162033
$918$	0	0
$919$	4.00033	0.131959	0.0659793	−	0.997821i	$-0.478983\pi$
$919$	4.00033	0.131959	0.0659793	+	0.997821i	$0.478983\pi$
$920$	0	0
$921$	−7.02834	−0.231592
$922$	0	0
$923$	0	0
$924$	0	0
$925$	24.9975	0.821914
$926$	0	0
$927$	44.2140	1.45218
$928$	0	0
$929$	54.3801	1.78415	0.892077	−	0.451884i	$-0.149248\pi$
$929$	54.3801	1.78415	0.892077	+	0.451884i	$0.149248\pi$
$930$	0	0
$931$	5.58315	0.182981
$932$	0	0
$933$	−5.18972	−0.169904
$934$	0	0
$935$	4.24428	0.138803
$936$	0	0
$937$	46.7217	1.52633	0.763165	−	0.646203i	$-0.223644\pi$
$937$	46.7217	1.52633	0.763165	+	0.646203i	$0.223644\pi$
$938$	0	0
$939$	−8.18190	−0.267006
$940$	0	0
$941$	38.4447	1.25326	0.626631	−	0.779316i	$-0.284433\pi$
$941$	38.4447	1.25326	0.626631	+	0.779316i	$0.284433\pi$
$942$	0	0
$943$	−1.69586	−0.0552247
$944$	0	0
$945$	−1.40281	−0.0456335
$946$	0	0
$947$	−45.4366	−1.47649	−0.738245	−	0.674533i	$-0.764345\pi$
$947$	−45.4366	−1.47649	−0.738245	+	0.674533i	$0.764345\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	4.65989	0.151107
$952$	0	0
$953$	−12.0830	−0.391408	−0.195704	−	0.980663i	$-0.562699\pi$
$953$	−12.0830	−0.391408	−0.195704	+	0.980663i	$0.562699\pi$
$954$	0	0
$955$	−4.64373	−0.150267
$956$	0	0
$957$	−14.3872	−0.465073
$958$	0	0
$959$	12.4999	0.403642
$960$	0	0
$961$	−30.7012	−0.990360
$962$	0	0
$963$	39.2597	1.26513
$964$	0	0
$965$	4.11669	0.132521
$966$	0	0
$967$	7.50012	0.241188	0.120594	−	0.992702i	$-0.461520\pi$
$967$	7.50012	0.241188	0.120594	+	0.992702i	$0.461520\pi$
$968$	0	0
$969$	−3.46490	−0.111309
$970$	0	0
$971$	10.6649	0.342253	0.171127	−	0.985249i	$-0.445259\pi$
$971$	10.6649	0.342253	0.171127	+	0.985249i	$0.445259\pi$
$972$	0	0
$973$	−6.34916	−0.203545
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−8.56035	−0.273870	−0.136935	−	0.990580i	$-0.543725\pi$
$977$	−8.56035	−0.273870	−0.136935	+	0.990580i	$0.543725\pi$
$978$	0	0
$979$	−60.7779	−1.94247
$980$	0	0
$981$	32.1760	1.02730
$982$	0	0
$983$	−15.7511	−0.502383	−0.251191	−	0.967937i	$-0.580822\pi$
$983$	−15.7511	−0.502383	−0.251191	+	0.967937i	$0.580822\pi$
$984$	0	0
$985$	4.34068	0.138306
$986$	0	0
$987$	1.30015	0.0413844
$988$	0	0
$989$	17.9011	0.569223
$990$	0	0
$991$	−58.8369	−1.86901	−0.934507	−	0.355944i	$-0.884159\pi$
$991$	−58.8369	−1.86901	−0.934507	+	0.355944i	$0.884159\pi$
$992$	0	0
$993$	11.0131	0.349490
$994$	0	0
$995$	−16.0007	−0.507257
$996$	0	0
$997$	−39.0644	−1.23718	−0.618590	−	0.785714i	$-0.712296\pi$
$997$	−39.0644	−1.23718	−0.618590	+	0.785714i	$0.712296\pi$
$998$	0	0
$999$	12.6181	0.399219

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.y.1.3		4
13.5	odd	4		728.2.k.a.337.5	✓	8
13.8	odd	4		728.2.k.a.337.6	yes	8
13.12	even	2		9464.2.a.x.1.3		4
52.31	even	4		1456.2.k.e.337.3		8
52.47	even	4		1456.2.k.e.337.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.k.a.337.5	✓	8	13.5	odd	4
728.2.k.a.337.6	yes	8	13.8	odd	4
1456.2.k.e.337.3		8	52.31	even	4
1456.2.k.e.337.4		8	52.47	even	4
9464.2.a.x.1.3		4	13.12	even	2
9464.2.a.y.1.3		4	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+0.401251 q^{3} +0.598749 q^{5} +1.00000 q^{7} -2.83900 q^{9} -4.58315 q^{11} +0.240249 q^{15} -1.54666 q^{17} +5.58315 q^{19} +0.401251 q^{21} +1.98441 q^{23} -4.64150 q^{25} -2.34291 q^{27} +7.82340 q^{29} +0.546660 q^{31} -1.83900 q^{33} +0.598749 q^{35} -5.38566 q^{37} -0.854591 q^{41} +9.02090 q^{43} -1.69985 q^{45} +3.24025 q^{47} +1.00000 q^{49} -0.620600 q^{51} +0.327312 q^{53} -2.74416 q^{55} +2.24025 q^{57} -1.81810 q^{59} +2.82340 q^{61} -2.83900 q^{63} -2.05209 q^{67} +0.796246 q^{69} +6.24025 q^{71} -6.43775 q^{73} -1.86241 q^{75} -4.58315 q^{77} -11.3638 q^{79} +7.57690 q^{81} -4.67174 q^{83} -0.926061 q^{85} +3.13915 q^{87} +13.2611 q^{89} +0.219348 q^{93} +3.34291 q^{95} +8.11574 q^{97} +13.0116 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 5 q^{5} + 4 q^{7} + 5 q^{9} - 3 q^{11} - 18 q^{15} - 2 q^{17} + 7 q^{19} - q^{21} - 10 q^{23} + 3 q^{25} - 13 q^{27} - 3 q^{29} - 2 q^{31} + 9 q^{33} + 5 q^{35} - q^{37} - 5 q^{41} + 7 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) \) 4 * q - q^3 + 5 * q^5 + 4 * q^7 + 5 * q^9 - 3 * q^11 - 18 * q^15 - 2 * q^17 + 7 * q^19 - q^21 - 10 * q^23 + 3 * q^25 - 13 * q^27 - 3 * q^29 - 2 * q^31 + 9 * q^33 + 5 * q^35 - q^37 - 5 * q^41 + 7 * q^43 + 21 * q^45 - 6 * q^47 + 4 * q^49 - 10 * q^51 - 13 * q^53 - 12 * q^55 - 10 * q^57 - 20 * q^59 - 23 * q^61 + 5 * q^63 - 15 * q^67 + 11 * q^69 + 6 * q^71 - 12 * q^73 - 49 * q^75 - 3 * q^77 - 20 * q^79 + 28 * q^81 + q^83 + 8 * q^85 + 24 * q^87 + 5 * q^89 + 11 * q^93 + 17 * q^95 - 14 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

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Database

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