LMFDB - Embedded newform 1456.2.k.e.337.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1456,2,Mod(337,1456)] 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("1456.337"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1456, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$11.6262185343$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 17x^{6} + 72x^{4} + 36x^{2} + 4 $ x^8 + 17x^6 + 72x^4 + 36*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.4
Root			$0.401251i$ of defining polynomial
Character	$\chi$	$=$	1456.337
Dual form			1456.2.k.e.337.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.401251 q^{3} +0.598749i q^{5} +1.00000i q^{7} -2.83900 q^{9} -4.58315i q^{11} +(-0.401251 + 3.58315i) q^{13} -0.240249i q^{15} +1.54666 q^{17} -5.58315i q^{19} -0.401251i q^{21} +1.98441 q^{23} +4.64150 q^{25} +2.34291 q^{27} +7.82340 q^{29} -0.546660i q^{31} +1.83900i q^{33} -0.598749 q^{35} +5.38566i q^{37} +(0.161003 - 1.43775i) q^{39} -0.854591i q^{41} +9.02090 q^{43} -1.69985i q^{45} +3.24025i q^{47} -1.00000 q^{49} -0.620600 q^{51} +0.327312 q^{53} +2.74416 q^{55} +2.24025i q^{57} -1.81810i q^{59} +2.82340 q^{61} -2.83900i q^{63} +(-2.14541 - 0.240249i) q^{65} +2.05209i q^{67} -0.796246 q^{69} -6.24025i q^{71} +6.43775i q^{73} -1.86241 q^{75} +4.58315 q^{77} +11.3638 q^{79} +7.57690 q^{81} +4.67174i q^{83} +0.926061i q^{85} -3.13915 q^{87} -13.2611i q^{89} +(-3.58315 - 0.401251i) q^{91} +0.219348i q^{93} +3.34291 q^{95} +8.11574i q^{97} +13.0116i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{3} + 10 q^{9} + 2 q^{13} + 4 q^{17} - 20 q^{23} - 6 q^{25} + 26 q^{27} - 6 q^{29} - 10 q^{35} + 34 q^{39} + 14 q^{43} - 8 q^{49} - 20 q^{51} - 26 q^{53} + 24 q^{55} - 46 q^{61} - 14 q^{65} - 22 q^{69}+ \cdots + 34 q^{95}+O(q^{100}) $ 8 * q + 2 * q^3 + 10 * q^9 + 2 * q^13 + 4 * q^17 - 20 * q^23 - 6 * q^25 + 26 * q^27 - 6 * q^29 - 10 * q^35 + 34 * q^39 + 14 * q^43 - 8 * q^49 - 20 * q^51 - 26 * q^53 + 24 * q^55 - 46 * q^61 - 14 * q^65 - 22 * q^69 - 98 * q^75 + 6 * q^77 + 40 * q^79 + 56 * q^81 - 48 * q^87 + 2 * q^91 + 34 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$.

$n$	$561$	$911$	$1093$	$1249$
$\chi(n)$	$-1$	$1$	$1$	$1$

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	$-0.536953\pi$
$3$	−0.401251			−0.231663			−0.115831	+	0.993269i	$0.536953\pi$
$4$		0			0
$5$			0.598749i			0.267769i	0.990997	+	0.133884i	$0.0427450\pi$
$5$			0.598749i			0.267769i	−0.990997	+	0.133884i	$0.957255\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−2.83900			−0.946332
$10$		0			0
$11$		−	4.58315i		−	1.38187i	−0.722915	−	0.690937i	$-0.757198\pi$
$11$		−	4.58315i		−	1.38187i	0.722915	−	0.690937i	$-0.242802\pi$
$12$		0			0
$13$	−0.401251	+	3.58315i	−0.111287	+	0.993788i
$13$	−0.401251	+	3.58315i	−0.111287	+	0.993788i
$14$		0			0
$15$		−	0.240249i		−	0.0620320i
$16$		0			0
$17$	1.54666			0.375120			0.187560	−	0.982253i	$-0.439942\pi$
$17$	1.54666			0.375120			0.187560	+	0.982253i	$0.439942\pi$
$18$		0			0
$19$		−	5.58315i		−	1.28086i	−0.768015	−	0.640432i	$-0.778755\pi$
$19$		−	5.58315i		−	1.28086i	0.768015	−	0.640432i	$-0.221245\pi$
$20$		0			0
$21$		−	0.401251i		−	0.0875602i
$22$		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
$31$		−	0.546660i		−	0.0981831i	−0.998794	−	0.0490915i	$-0.984367\pi$
$31$		−	0.546660i		−	0.0981831i	0.998794	−	0.0490915i	$-0.0156326\pi$
$32$		0			0
$33$			1.83900i			0.320128i
$34$		0			0
$35$	−0.598749			−0.101207
$36$		0			0
$37$			5.38566i			0.885397i	0.896671	+	0.442698i	$0.145979\pi$
$37$			5.38566i			0.885397i	−0.896671	+	0.442698i	$0.854021\pi$
$38$		0			0
$39$	0.161003	−	1.43775i	0.0257811	−	0.230224i
$40$		0			0
$41$		−	0.854591i		−	0.133465i	−0.997771	−	0.0667324i	$-0.978743\pi$
$41$		−	0.854591i		−	0.133465i	0.997771	−	0.0667324i	$-0.0212574\pi$
$42$		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
$45$		−	1.69985i		−	0.253398i
$46$		0			0
$47$			3.24025i			0.472639i	0.971675	+	0.236319i	$0.0759412\pi$
$47$			3.24025i			0.472639i	−0.971675	+	0.236319i	$0.924059\pi$
$48$		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
$57$			2.24025i			0.296728i
$58$		0			0
$59$		−	1.81810i		−	0.236696i	−0.992972	−	0.118348i	$-0.962240\pi$
$59$		−	1.81810i		−	0.236696i	0.992972	−	0.118348i	$-0.0377598\pi$
$60$		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
$63$		−	2.83900i		−	0.357680i
$64$		0			0
$65$	−2.14541	−	0.240249i	−0.266105	−	0.0297992i
$66$		0			0
$67$			2.05209i			0.250702i	0.992112	+	0.125351i	$0.0400058\pi$
$67$			2.05209i			0.250702i	−0.992112	+	0.125351i	$0.959994\pi$
$68$		0			0
$69$	−0.796246			−0.0958567
$70$		0			0
$71$		−	6.24025i		−	0.740581i	−0.928916	−	0.370291i	$-0.879258\pi$
$71$		−	6.24025i		−	0.740581i	0.928916	−	0.370291i	$-0.120742\pi$
$72$		0			0
$73$			6.43775i			0.753481i	0.926319	+	0.376741i	$0.122955\pi$
$73$			6.43775i			0.753481i	−0.926319	+	0.376741i	$0.877045\pi$
$74$		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	$-0.279239\pi$
$79$	11.3638			1.27853			0.639264	+	0.768987i	$0.279239\pi$
$80$		0			0
$81$	7.57690			0.841878
$82$		0			0
$83$			4.67174i			0.512790i	0.966572	+	0.256395i	$0.0825348\pi$
$83$			4.67174i			0.512790i	−0.966572	+	0.256395i	$0.917465\pi$
$84$		0			0
$85$			0.926061i			0.100445i
$86$		0			0
$87$	−3.13915			−0.336552
$88$		0			0
$89$		−	13.2611i		−	1.40568i	−0.711348	−	0.702840i	$-0.751915\pi$
$89$		−	13.2611i		−	1.40568i	0.711348	−	0.702840i	$-0.248085\pi$
$90$		0			0
$91$	−3.58315	−	0.401251i	−0.375617	−	0.0420626i
$92$		0			0
$93$			0.219348i			0.0227453i
$94$		0			0
$95$	3.34291			0.342975
$96$		0			0
$97$			8.11574i			0.824029i	0.911177	+	0.412014i	$0.135175\pi$
$97$			8.11574i			0.824029i	−0.911177	+	0.412014i	$0.864825\pi$
$98$		0			0
$99$			13.0116i			1.30771i
$100$		0			0
$101$	18.7339			1.86409			0.932045	−	0.362343i	$-0.118023\pi$
$101$	18.7339			1.86409			0.932045	+	0.362343i	$0.118023\pi$
$102$		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	$-0.266964\pi$
$107$	13.8287			1.33687			0.668436	+	0.743770i	$0.266964\pi$
$108$		0			0
$109$		−	11.3336i		−	1.08556i	−0.839875	−	0.542780i	$-0.817372\pi$
$109$		−	11.3336i		−	1.08556i	0.839875	−	0.542780i	$-0.182628\pi$
$110$		0			0
$111$		−	2.16100i		−	0.205113i
$112$		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
$115$			1.18816i			0.110797i
$116$		0			0
$117$	1.13915	−	10.1726i	0.105315	−	0.940454i
$118$		0			0
$119$			1.54666i			0.141782i
$120$		0			0
$121$	−10.0053			−0.909573
$122$		0			0
$123$			0.342906i			0.0309188i
$124$		0			0
$125$			5.77283i			0.516338i
$126$		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	$-0.431237\pi$
$131$	4.90668			0.428699			0.214349	+	0.976757i	$0.431237\pi$
$132$		0			0
$133$	5.58315			0.484121
$134$		0			0
$135$			1.40281i			0.120735i
$136$		0			0
$137$		−	12.4999i		−	1.06794i	−0.845504	−	0.533968i	$-0.820700\pi$
$137$		−	12.4999i		−	1.06794i	0.845504	−	0.533968i	$-0.179300\pi$
$138$		0			0
$139$	6.34916			0.538529			0.269264	−	0.963066i	$-0.413219\pi$
$139$	6.34916			0.538529			0.269264	+	0.963066i	$0.413219\pi$
$140$		0			0
$141$		−	1.30015i		−	0.109493i
$142$		0			0
$143$	16.4222	+	1.83900i	1.37329	+	0.153785i
$144$		0			0
$145$			4.68425i			0.389006i
$146$		0			0
$147$	0.401251			0.0330947
$148$		0			0
$149$		−	21.8225i		−	1.78777i	−0.448301	−	0.893883i	$-0.647971\pi$
$149$		−	21.8225i		−	1.78777i	0.448301	−	0.893883i	$-0.352029\pi$
$150$		0			0
$151$		−	10.0924i		−	0.821305i	−0.911792	−	0.410653i	$-0.865301\pi$
$151$		−	10.0924i		−	0.821305i	0.911792	−	0.410653i	$-0.134699\pi$
$152$		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
$161$			1.98441i			0.156393i
$162$		0			0
$163$		−	11.0933i		−	0.868896i	−0.900697	−	0.434448i	$-0.856943\pi$
$163$		−	11.0933i		−	0.868896i	0.900697	−	0.434448i	$-0.143057\pi$
$164$		0			0
$165$	−1.10110			−0.0857203
$166$		0			0
$167$			15.9158i			1.23160i	0.787903	+	0.615800i	$0.211167\pi$
$167$			15.9158i			1.23160i	−0.787903	+	0.615800i	$0.788833\pi$
$168$		0			0
$169$	−12.6780	−	2.87549i	−0.975230	−	0.221192i
$170$		0			0
$171$			15.8506i			1.21212i
$172$		0			0
$173$	13.2665			1.00863			0.504315	−	0.863520i	$-0.331745\pi$
$173$	13.2665			1.00863			0.504315	+	0.863520i	$0.331745\pi$
$174$		0			0
$175$			4.64150i			0.350864i
$176$		0			0
$177$			0.729514i			0.0548336i
$178$		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	$-0.308860\pi$
$181$	15.2038			1.13009			0.565043	+	0.825061i	$0.308860\pi$
$182$		0			0
$183$	−1.13289			−0.0837460
$184$		0			0
$185$	−3.22465			−0.237081
$186$		0			0
$187$		−	7.08858i		−	0.518369i
$188$		0			0
$189$			2.34291i			0.170421i
$190$		0			0
$191$	7.75572			0.561184			0.280592	−	0.959827i	$-0.409469\pi$
$191$	7.75572			0.561184			0.280592	+	0.959827i	$0.409469\pi$
$192$		0			0
$193$		−	6.87549i		−	0.494909i	−0.968900	−	0.247454i	$-0.920406\pi$
$193$		−	6.87549i		−	0.494909i	0.968900	−	0.247454i	$-0.0795940\pi$
$194$		0			0
$195$	0.860848	+	0.0964001i	0.0616466	+	0.00690336i
$196$		0			0
$197$			7.24959i			0.516512i	0.966077	+	0.258256i	$0.0831477\pi$
$197$			7.24959i			0.516512i	−0.966077	+	0.258256i	$0.916852\pi$
$198$		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
$201$		−	0.823403i		−	0.0580784i
$202$		0			0
$203$			7.82340i			0.549095i
$204$		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	$-0.658818\pi$
$211$	−13.9011			−0.956993			−0.478497	+	0.878089i	$0.658818\pi$
$212$		0			0
$213$			2.50391i			0.171565i
$214$		0			0
$215$			5.40125i			0.368362i
$216$		0			0
$217$	0.546660			0.0371097
$218$		0			0
$219$		−	2.58315i		−	0.174553i
$220$		0			0
$221$	−0.620600	+	5.54192i	−0.0417460	+	0.372790i
$222$		0			0
$223$			19.9026i			1.33278i	0.745603	+	0.666390i	$0.232161\pi$
$223$			19.9026i			1.33278i	−0.745603	+	0.666390i	$0.767839\pi$
$224$		0			0
$225$	−13.1772			−0.878480
$226$		0			0
$227$			20.2596i			1.34468i	0.740243	+	0.672339i	$0.234710\pi$
$227$			20.2596i			1.34468i	−0.740243	+	0.672339i	$0.765290\pi$
$228$		0			0
$229$			9.30641i			0.614985i	0.951551	+	0.307492i	$0.0994899\pi$
$229$			9.30641i			0.614985i	−0.951551	+	0.307492i	$0.900510\pi$
$230$		0			0
$231$	−1.83900			−0.120997
$232$		0			0
$233$	−13.3326			−0.873449			−0.436724	−	0.899595i	$-0.643862\pi$
$233$	−13.3326			−0.873449			−0.436724	+	0.899595i	$0.643862\pi$
$234$		0			0
$235$	−1.94009			−0.126558
$236$		0			0
$237$	−4.55974			−0.296187
$238$		0			0
$239$		−	0.729514i		−	0.0471883i	−0.999722	−	0.0235942i	$-0.992489\pi$
$239$		−	0.729514i		−	0.0471883i	0.999722	−	0.0235942i	$-0.00751095\pi$
$240$		0			0
$241$			1.44305i			0.0929552i	0.998919	+	0.0464776i	$0.0147996\pi$
$241$			1.44305i			0.0929552i	−0.998919	+	0.0464776i	$0.985200\pi$
$242$		0			0
$243$	−10.0690			−0.645924
$244$		0			0
$245$		−	0.598749i		−	0.0382526i
$246$		0			0
$247$	20.0053	+	2.24025i	1.27291	+	0.142544i
$248$		0			0
$249$		−	1.87454i		−	0.118794i
$250$		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
$253$		−	9.09484i		−	0.571788i
$254$		0			0
$255$		−	0.371583i		−	0.0232694i
$256$		0			0
$257$	24.4951			1.52796			0.763982	−	0.645237i	$-0.223241\pi$
$257$	24.4951			1.52796			0.763982	+	0.645237i	$0.223241\pi$
$258$		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	$-0.687386\pi$
$263$	−18.0100			−1.11055			−0.555273	+	0.831668i	$0.687386\pi$
$264$		0			0
$265$			0.195978i			0.0120388i
$266$		0			0
$267$			5.32105i			0.325643i
$268$		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
$271$		−	23.9460i		−	1.45462i	−0.686311	−	0.727308i	$-0.740771\pi$
$271$		−	23.9460i		−	1.45462i	0.686311	−	0.727308i	$-0.259229\pi$
$272$		0			0
$273$	1.43775	+	0.161003i	0.0870163	+	0.00974433i
$274$		0			0
$275$		−	21.2727i		−	1.28279i
$276$		0			0
$277$	−16.6989			−1.00334			−0.501670	−	0.865059i	$-0.667281\pi$
$277$	−16.6989			−1.00334			−0.501670	+	0.865059i	$0.667281\pi$
$278$		0			0
$279$			1.55197i			0.0929138i
$280$		0			0
$281$			16.4269i			0.979946i	0.871738	+	0.489973i	$0.162993\pi$
$281$			16.4269i			0.979946i	−0.871738	+	0.489973i	$0.837007\pi$
$282$		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
$291$		−	3.25645i		−	0.190897i
$292$		0			0
$293$		−	17.1882i		−	1.00414i	−0.864826	−	0.502072i	$-0.832571\pi$
$293$		−	17.1882i		−	1.00414i	0.864826	−	0.502072i	$-0.167429\pi$
$294$		0			0
$295$	1.08858			0.0633797
$296$		0			0
$297$		−	10.7379i		−	0.623076i
$298$		0			0
$299$	−0.796246	+	7.11043i	−0.0460481	+	0.411207i
$300$		0			0
$301$			9.02090i			0.519956i
$302$		0			0
$303$	−7.51699			−0.431840
$304$		0			0
$305$			1.69051i			0.0967983i
$306$		0			0
$307$		−	17.5160i		−	0.999693i	−0.866114	−	0.499847i	$-0.833390\pi$
$307$		−	17.5160i		−	0.999693i	0.866114	−	0.499847i	$-0.166610\pi$
$308$		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
$317$			11.6134i			0.652273i	0.945323	+	0.326137i	$0.105747\pi$
$317$			11.6134i			0.652273i	−0.945323	+	0.326137i	$0.894253\pi$
$318$		0			0
$319$		−	35.8559i		−	2.00754i
$320$		0			0
$321$	−5.54879			−0.309703
$322$		0			0
$323$		−	8.63524i		−	0.480478i
$324$		0			0
$325$	−1.86241	+	16.6312i	−0.103308	+	0.922534i
$326$		0			0
$327$			4.54761i			0.251483i
$328$		0			0
$329$	−3.24025			−0.178641
$330$		0			0
$331$		−	27.4468i		−	1.50861i	−0.656521	−	0.754307i	$-0.727973\pi$
$331$		−	27.4468i		−	1.50861i	0.656521	−	0.754307i	$-0.272027\pi$
$332$		0			0
$333$		−	15.2899i		−	0.837880i
$334$		0			0
$335$	−1.22869			−0.0671302
$336$		0			0
$337$	17.9114			0.975697			0.487849	−	0.872928i	$-0.337782\pi$
$337$	17.9114			0.975697			0.487849	+	0.872928i	$0.337782\pi$
$338$		0			0
$339$	4.76696			0.258906
$340$		0			0
$341$	−2.50543			−0.135677
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		−	0.476751i		−	0.0256674i
$346$		0			0
$347$	0.480497			0.0257945			0.0128972	−	0.999917i	$-0.495895\pi$
$347$	0.480497			0.0257945			0.0128972	+	0.999917i	$0.495895\pi$
$348$		0			0
$349$			21.1258i			1.13084i	0.824804	+	0.565419i	$0.191285\pi$
$349$			21.1258i			1.13084i	−0.824804	+	0.565419i	$0.808715\pi$
$350$		0			0
$351$	−0.940094	+	8.39499i	−0.0501785	+	0.448092i
$352$		0			0
$353$			24.2446i			1.29041i	0.764009	+	0.645205i	$0.223228\pi$
$353$			24.2446i			1.29041i	−0.764009	+	0.645205i	$0.776772\pi$
$354$		0			0
$355$	3.73634			0.198304
$356$		0			0
$357$		−	0.620600i		−	0.0328456i
$358$		0			0
$359$			27.0846i			1.42947i	0.699396	+	0.714734i	$0.253452\pi$
$359$			27.0846i			1.42947i	−0.699396	+	0.714734i	$0.746548\pi$
$360$		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	$-0.482284\pi$
$367$	2.13133			0.111255			0.0556274	+	0.998452i	$0.482284\pi$
$368$		0			0
$369$			2.42618i			0.126302i
$370$		0			0
$371$			0.327312i			0.0169932i
$372$		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
$375$		−	2.31636i		−	0.119616i
$376$		0			0
$377$	−3.13915	+	28.0325i	−0.161675	+	1.44375i
$378$		0			0
$379$			2.88426i			0.148154i	0.997253	+	0.0740772i	$0.0236011\pi$
$379$			2.88426i			0.148154i	−0.997253	+	0.0740772i	$0.976399\pi$
$380$		0			0
$381$	0.428409			0.0219481
$382$		0			0
$383$		−	8.35542i		−	0.426942i	−0.976949	−	0.213471i	$-0.931523\pi$
$383$		−	8.35542i		−	0.426942i	0.976949	−	0.213471i	$-0.0684769\pi$
$384$		0			0
$385$			2.74416i			0.139855i
$386$		0			0
$387$	−25.6103			−1.30185
$388$		0			0
$389$	33.4212			1.69452			0.847261	−	0.531177i	$-0.178250\pi$
$389$	33.4212			1.69452			0.847261	+	0.531177i	$0.178250\pi$
$390$		0			0
$391$	3.06920			0.155216
$392$		0			0
$393$	−1.96881			−0.0993134
$394$		0			0
$395$			6.80406i			0.342350i
$396$		0			0
$397$		−	37.2611i		−	1.87008i	−0.354538	−	0.935042i	$-0.615362\pi$
$397$		−	37.2611i		−	1.87008i	0.354538	−	0.935042i	$-0.384638\pi$
$398$		0			0
$399$	−2.24025			−0.112153
$400$		0			0
$401$		−	10.6858i		−	0.533624i	−0.963749	−	0.266812i	$-0.914030\pi$
$401$		−	10.6858i		−	0.533624i	0.963749	−	0.266812i	$-0.0859703\pi$
$402$		0			0
$403$	1.95877	+	0.219348i	0.0975732	+	0.0109265i
$404$		0			0
$405$			4.53666i			0.225428i
$406$		0			0
$407$	24.6833			1.22351
$408$		0			0
$409$			32.8643i			1.62503i	0.582938	+	0.812516i	$0.301903\pi$
$409$			32.8643i			1.62503i	−0.582938	+	0.812516i	$0.698097\pi$
$410$		0			0
$411$			5.01559i			0.247401i
$412$		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	$-0.737932\pi$
$419$	−27.8302			−1.35960			−0.679798	+	0.733400i	$0.737932\pi$
$420$		0			0
$421$			35.5004i			1.73019i	0.501612	+	0.865093i	$0.332741\pi$
$421$			35.5004i			1.73019i	−0.501612	+	0.865093i	$0.667259\pi$
$422$		0			0
$423$		−	9.19906i		−	0.447273i
$424$		0			0
$425$	7.17882			0.348224
$426$		0			0
$427$			2.82340i			0.136634i
$428$		0			0
$429$	−6.58941	−	0.737900i	−0.318140	−	0.0356262i
$430$		0			0
$431$		−	12.6546i		−	0.609552i	−0.952424	−	0.304776i	$-0.901418\pi$
$431$		−	12.6546i		−	0.609552i	0.952424	−	0.304776i	$-0.0985816\pi$
$432$		0			0
$433$	−16.6671			−0.800972			−0.400486	−	0.916303i	$-0.631159\pi$
$433$	−16.6671			−0.800972			−0.400486	+	0.916303i	$0.631159\pi$
$434$		0			0
$435$		−	1.87956i		−	0.0901181i
$436$		0			0
$437$		−	11.0792i		−	0.529992i
$438$		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	$-0.334516\pi$
$443$	20.9120			0.993558			0.496779	+	0.867877i	$0.334516\pi$
$444$		0			0
$445$	7.94009			0.376397
$446$		0			0
$447$			8.75629i			0.414158i
$448$		0			0
$449$		−	16.1585i		−	0.762566i	−0.924458	−	0.381283i	$-0.875482\pi$
$449$		−	16.1585i		−	0.762566i	0.924458	−	0.381283i	$-0.124518\pi$
$450$		0			0
$451$	−3.91672			−0.184431
$452$		0			0
$453$			4.04958i			0.190266i
$454$		0			0
$455$	0.240249	−	2.14541i	0.0112630	−	0.100578i
$456$		0			0
$457$			19.6274i			0.918132i	0.888402	+	0.459066i	$0.151816\pi$
$457$			19.6274i			0.918132i	−0.888402	+	0.459066i	$0.848184\pi$
$458$		0			0
$459$	3.62368			0.169139
$460$		0			0
$461$			17.8627i			0.831950i	0.909376	+	0.415975i	$0.136560\pi$
$461$			17.8627i			0.831950i	−0.909376	+	0.415975i	$0.863440\pi$
$462$		0			0
$463$			10.0924i			0.469032i	0.972112	+	0.234516i	$0.0753506\pi$
$463$			10.0924i			0.469032i	−0.972112	+	0.234516i	$0.924649\pi$
$464$		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
$473$		−	41.3442i		−	1.90101i
$474$		0			0
$475$		−	25.9142i		−	1.18903i
$476$		0			0
$477$	−0.929238			−0.0425469
$478$		0			0
$479$		−	11.0063i		−	0.502889i	−0.967872	−	0.251444i	$-0.919094\pi$
$479$		−	11.0063i		−	0.502889i	0.967872	−	0.251444i	$-0.0809055\pi$
$480$		0			0
$481$	−19.2976	−	2.16100i	−0.879897	−	0.0985332i
$482$		0			0
$483$		−	0.796246i		−	0.0362304i
$484$		0			0
$485$	−4.85929			−0.220649
$486$		0			0
$487$		−	9.87715i		−	0.447576i	−0.974638	−	0.223788i	$-0.928158\pi$
$487$		−	9.87715i		−	0.447576i	0.974638	−	0.223788i	$-0.0718424\pi$
$488$		0			0
$489$			4.45121i			0.201291i
$490$		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
$499$			12.7310i			0.569919i	0.958540	+	0.284960i	$0.0919802\pi$
$499$			12.7310i			0.569919i	−0.958540	+	0.284960i	$0.908020\pi$
$500$		0			0
$501$		−	6.38623i		−	0.285316i
$502$		0			0
$503$	11.9688			0.533663			0.266831	−	0.963743i	$-0.414023\pi$
$503$	11.9688			0.533663			0.266831	+	0.963743i	$0.414023\pi$
$504$		0			0
$505$			11.2169i			0.499145i
$506$		0			0
$507$	5.08706	+	1.15380i	0.225924	+	0.0512418i
$508$		0			0
$509$		−	4.13664i		−	0.183353i	−0.995789	−	0.0916767i	$-0.970777\pi$
$509$		−	4.13664i		−	0.183353i	0.995789	−	0.0916767i	$-0.0292226\pi$
$510$		0			0
$511$	−6.43775			−0.284789
$512$		0			0
$513$		−	13.0808i		−	0.577532i
$514$		0			0
$515$		−	9.32480i		−	0.410900i
$516$		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	$-0.733339\pi$
$523$	−30.6056			−1.33829			−0.669144	+	0.743133i	$0.733339\pi$
$524$		0			0
$525$		−	1.86241i		−	0.0812822i
$526$		0			0
$527$		−	0.845498i		−	0.0368305i
$528$		0			0
$529$	−19.0621			−0.828788
$530$		0			0
$531$			5.16157i			0.223993i
$532$		0			0
$533$	3.06213	+	0.342906i	0.132636	+	0.0148529i
$534$		0			0
$535$			8.27992i			0.357972i
$536$		0			0
$537$	5.89795			0.254515
$538$		0			0
$539$			4.58315i			0.197410i
$540$		0			0
$541$		−	0.375613i		−	0.0161489i	−0.999967	−	0.00807444i	$-0.997430\pi$
$541$		−	0.375613i		−	0.0161489i	0.999967	−	0.00807444i	$-0.00257020\pi$
$542$		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	$-0.407282\pi$
$547$	13.4331			0.574360			0.287180	+	0.957877i	$0.407282\pi$
$548$		0			0
$549$	−8.01563			−0.342099
$550$		0			0
$551$		−	43.6793i		−	1.86080i
$552$		0			0
$553$			11.3638i			0.483238i
$554$		0			0
$555$	1.29390			0.0549229
$556$		0			0
$557$		−	39.8669i		−	1.68921i	−0.535387	−	0.844607i	$-0.679834\pi$
$557$		−	39.8669i		−	1.68921i	0.535387	−	0.844607i	$-0.320166\pi$
$558$		0			0
$559$	−3.61965	+	32.3233i	−0.153095	+	1.36713i
$560$		0			0
$561$			2.84430i			0.120087i
$562$		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
$565$		−	7.11327i		−	0.299258i
$566$		0			0
$567$			7.57690i			0.318200i
$568$		0			0
$569$	−1.55349			−0.0651255			−0.0325628	−	0.999470i	$-0.510367\pi$
$569$	−1.55349			−0.0651255			−0.0325628	+	0.999470i	$0.510367\pi$
$570$		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
$577$		−	39.3248i		−	1.63711i	−0.574426	−	0.818556i	$-0.694775\pi$
$577$		−	39.3248i		−	1.63711i	0.574426	−	0.818556i	$-0.305225\pi$
$578$		0			0
$579$			2.75880i			0.114652i
$580$		0			0
$581$	−4.67174			−0.193816
$582$		0			0
$583$		−	1.50012i		−	0.0621287i
$584$		0			0
$585$	6.09081	+	0.682065i	0.251824	+	0.0281999i
$586$		0			0
$587$			38.2144i			1.57728i	0.614858	+	0.788638i	$0.289213\pi$
$587$			38.2144i			1.57728i	−0.614858	+	0.788638i	$0.710787\pi$
$588$		0			0
$589$	−3.05209			−0.125759
$590$		0			0
$591$		−	2.90891i		−	0.119656i
$592$		0			0
$593$			0.734113i			0.0301464i	0.999886	+	0.0150732i	$0.00479813\pi$
$593$			0.734113i			0.0301464i	−0.999886	+	0.0150732i	$0.995202\pi$
$594$		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	$-0.331944\pi$
$599$	24.6593			1.00755			0.503776	+	0.863834i	$0.331944\pi$
$600$		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
$603$		−	5.82587i		−	0.237248i
$604$		0			0
$605$		−	5.99066i		−	0.243555i
$606$		0			0
$607$	46.8821			1.90288			0.951442	−	0.307827i	$-0.0996018\pi$
$607$	46.8821			1.90288			0.951442	+	0.307827i	$0.0996018\pi$
$608$		0			0
$609$		−	3.13915i		−	0.127205i
$610$		0			0
$611$	−11.6103	−	1.30015i	−0.469703	−	0.0525986i
$612$		0			0
$613$			19.6696i			0.794448i	0.917722	+	0.397224i	$0.130026\pi$
$613$			19.6696i			0.794448i	−0.917722	+	0.397224i	$0.869974\pi$
$614$		0			0
$615$	−0.205314			−0.00827908
$616$		0			0
$617$		−	14.9143i		−	0.600425i	−0.953872	−	0.300213i	$-0.902942\pi$
$617$		−	14.9143i		−	0.600425i	0.953872	−	0.300213i	$-0.0970576\pi$
$618$		0			0
$619$			35.7198i			1.43570i	0.696198	+	0.717850i	$0.254874\pi$
$619$			35.7198i			1.43570i	−0.696198	+	0.717850i	$0.745126\pi$
$620$		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
$629$			8.32978i			0.332130i
$630$		0			0
$631$			13.4696i			0.536218i	0.963389	+	0.268109i	$0.0863987\pi$
$631$			13.4696i			0.536218i	−0.963389	+	0.268109i	$0.913601\pi$
$632$		0			0
$633$	5.57785			0.221700
$634$		0			0
$635$		−	0.639273i		−	0.0253688i
$636$		0			0
$637$	0.401251	−	3.58315i	0.0158982	−	0.141970i
$638$		0			0
$639$			17.7160i			0.700836i
$640$		0			0
$641$	−23.8880			−0.943520			−0.471760	−	0.881727i	$-0.656381\pi$
$641$	−23.8880			−0.943520			−0.471760	+	0.881727i	$0.656381\pi$
$642$		0			0
$643$			20.1680i			0.795347i	0.917527	+	0.397673i	$0.130182\pi$
$643$			20.1680i			0.795347i	−0.917527	+	0.397673i	$0.869818\pi$
$644$		0			0
$645$		−	2.16726i		−	0.0853358i
$646$		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
$655$			2.93787i			0.114792i
$656$		0			0
$657$		−	18.2767i		−	0.713044i
$658$		0			0
$659$	−27.0986			−1.05561			−0.527806	−	0.849365i	$-0.676985\pi$
$659$	−27.0986			−1.05561			−0.527806	+	0.849365i	$0.676985\pi$
$660$		0			0
$661$			22.0372i			0.857148i	0.903507	+	0.428574i	$0.140984\pi$
$661$			22.0372i			0.857148i	−0.903507	+	0.428574i	$0.859016\pi$
$662$		0			0
$663$	0.249016	−	2.22370i	0.00967100	−	0.0863615i
$664$		0			0
$665$			3.34291i			0.129632i
$666$		0			0
$667$	15.5248			0.601123
$668$		0			0
$669$		−	7.98597i		−	0.308755i
$670$		0			0
$671$		−	12.9401i		−	0.499547i
$672$		0			0
$673$	34.5797			1.33295			0.666475	−	0.745528i	$-0.267802\pi$
$673$	34.5797			1.33295			0.666475	+	0.745528i	$0.267802\pi$
$674$		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
$681$		−	8.12921i		−	0.311512i
$682$		0			0
$683$		−	41.4905i		−	1.58759i	−0.608184	−	0.793796i	$-0.708102\pi$
$683$		−	41.4905i		−	1.58759i	0.608184	−	0.793796i	$-0.291898\pi$
$684$		0			0
$685$	7.48428			0.285960
$686$		0			0
$687$		−	3.73421i		−	0.142469i
$688$		0			0
$689$	−0.131334	+	1.17281i	−0.00500344	+	0.0446805i
$690$		0			0
$691$			28.4461i			1.08214i	0.840977	+	0.541071i	$0.181981\pi$
$691$			28.4461i			1.08214i	−0.840977	+	0.541071i	$0.818019\pi$
$692$		0			0
$693$	−13.0116			−0.494268
$694$		0			0
$695$			3.80155i			0.144201i
$696$		0			0
$697$		−	1.32176i		−	0.0500653i
$698$		0			0
$699$	5.34973			0.202345
$700$		0			0
$701$	13.2699			0.501198			0.250599	−	0.968091i	$-0.419372\pi$
$701$	13.2699			0.501198			0.250599	+	0.968091i	$0.419372\pi$
$702$		0			0
$703$	30.0690			1.13407
$704$		0			0
$705$	0.778466			0.0293187
$706$		0			0
$707$			18.7339i			0.704560i
$708$		0			0
$709$		−	12.7416i		−	0.478523i	−0.970955	−	0.239261i	$-0.923095\pi$
$709$		−	12.7416i		−	0.478523i	0.970955	−	0.239261i	$-0.0769053\pi$
$710$		0			0
$711$	−32.2618			−1.20991
$712$		0			0
$713$		−	1.08480i		−	0.0406259i
$714$		0			0
$715$	−1.10110	+	9.83274i	−0.0411787	+	0.367724i
$716$		0			0
$717$			0.292718i			0.0109318i
$718$		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
$721$		−	15.5738i		−	0.579999i
$722$		0			0
$723$		−	0.579027i		−	0.0215342i
$724$		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
$733$			26.7154i			0.986757i	0.869815	+	0.493379i	$0.164238\pi$
$733$			26.7154i			0.986757i	−0.869815	+	0.493379i	$0.835762\pi$
$734$		0			0
$735$			0.240249i			0.00886171i
$736$		0			0
$737$	9.40504			0.346439
$738$		0			0
$739$		−	12.6071i		−	0.463759i	−0.972744	−	0.231880i	$-0.925512\pi$
$739$		−	12.6071i		−	0.463759i	0.972744	−	0.231880i	$-0.0744876\pi$
$740$		0			0
$741$	−8.02716	−	0.898903i	−0.294885	−	0.0330220i
$742$		0			0
$743$			42.5348i			1.56045i	0.625498	+	0.780225i	$0.284896\pi$
$743$			42.5348i			1.56045i	−0.625498	+	0.780225i	$0.715104\pi$
$744$		0			0
$745$	13.0662			0.478707
$746$		0			0
$747$		−	13.2631i		−	0.485270i
$748$		0			0
$749$			13.8287i			0.505290i
$750$		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
$759$			3.64932i			0.132462i
$760$		0			0
$761$		−	13.9775i		−	0.506685i	−0.967377	−	0.253343i	$-0.918470\pi$
$761$		−	13.9775i		−	0.506685i	0.967377	−	0.253343i	$-0.0815300\pi$
$762$		0			0
$763$	11.3336			0.410303
$764$		0			0
$765$		−	2.62908i		−	0.0950547i
$766$		0			0
$767$	6.51452	+	0.729514i	0.235226	+	0.0263412i
$768$		0			0
$769$		−	18.5280i		−	0.668136i	−0.942549	−	0.334068i	$-0.891578\pi$
$769$		−	18.5280i		−	0.668136i	0.942549	−	0.334068i	$-0.108422\pi$
$770$		0			0
$771$	−9.82871			−0.353972
$772$		0			0
$773$		−	45.2936i		−	1.62910i	−0.580095	−	0.814549i	$-0.696984\pi$
$773$		−	45.2936i		−	1.62910i	0.580095	−	0.814549i	$-0.303016\pi$
$774$		0			0
$775$		−	2.53732i		−	0.0911433i
$776$		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
$785$			2.13607i			0.0762397i
$786$		0			0
$787$		−	2.70483i		−	0.0964166i	−0.998837	−	0.0482083i	$-0.984649\pi$
$787$		−	2.70483i		−	0.0964166i	0.998837	−	0.0482083i	$-0.0153511\pi$
$788$		0			0
$789$	7.22656			0.257272
$790$		0			0
$791$		−	11.8802i		−	0.422412i
$792$		0			0
$793$	−1.13289	+	10.1167i	−0.0402303	+	0.359254i
$794$		0			0
$795$		−	0.0786363i		−	0.00278894i
$796$		0			0
$797$	11.1435			0.394723			0.197362	−	0.980331i	$-0.436763\pi$
$797$	11.1435			0.394723			0.197362	+	0.980331i	$0.436763\pi$
$798$		0			0
$799$			5.01156i			0.177296i
$800$		0			0
$801$			37.6484i			1.33024i
$802$		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
$811$		−	26.5953i		−	0.933887i	−0.884287	−	0.466943i	$-0.845355\pi$
$811$		−	26.5953i		−	0.933887i	0.884287	−	0.466943i	$-0.154645\pi$
$812$		0			0
$813$			9.60837i			0.336980i
$814$		0			0
$815$	6.64211			0.232663
$816$		0			0
$817$		−	50.3651i		−	1.76205i
$818$		0			0
$819$	10.1726	+	1.13915i	0.355458	+	0.0398052i
$820$		0			0
$821$			25.6655i			0.895731i	0.894101	+	0.447866i	$0.147816\pi$
$821$			25.6655i			0.895731i	−0.894101	+	0.447866i	$0.852184\pi$
$822$		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
$825$			8.53571i			0.297175i
$826$		0			0
$827$		−	6.88497i		−	0.239414i	−0.992809	−	0.119707i	$-0.961805\pi$
$827$		−	6.88497i		−	0.239414i	0.992809	−	0.119707i	$-0.0381955\pi$
$828$		0			0
$829$	−50.8278			−1.76532			−0.882661	−	0.470011i	$-0.844250\pi$
$829$	−50.8278			−1.76532			−0.882661	+	0.470011i	$0.844250\pi$
$830$		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
$837$		−	1.28077i		−	0.0442700i
$838$		0			0
$839$		−	0.915066i		−	0.0315916i	−0.999875	−	0.0157958i	$-0.994972\pi$
$839$		−	0.915066i		−	0.0315916i	0.999875	−	0.0157958i	$-0.00502817\pi$
$840$		0			0
$841$	32.2056			1.11054
$842$		0			0
$843$		−	6.59131i		−	0.227017i
$844$		0			0
$845$	1.72170	−	7.59093i	0.0592282	−	0.261136i
$846$		0			0
$847$		−	10.0053i		−	0.343786i
$848$		0			0
$849$	12.8652			0.441533
$850$		0			0
$851$			10.6873i			0.366357i
$852$		0			0
$853$		−	14.3622i		−	0.491754i	−0.969301	−	0.245877i	$-0.920924\pi$
$853$		−	14.3622i		−	0.491754i	0.969301	−	0.245877i	$-0.0790759\pi$
$854$		0			0
$855$	−9.49050			−0.324568
$856$		0			0
$857$	−1.62368			−0.0554638			−0.0277319	−	0.999615i	$-0.508828\pi$
$857$	−1.62368			−0.0554638			−0.0277319	+	0.999615i	$0.508828\pi$
$858$		0			0
$859$	48.1045			1.64130			0.820652	−	0.571428i	$-0.193610\pi$
$859$	48.1045			1.64130			0.820652	+	0.571428i	$0.193610\pi$
$860$		0			0
$861$	−0.342906			−0.0116862
$862$		0			0
$863$			42.5828i			1.44953i	0.688994	+	0.724767i	$0.258053\pi$
$863$			42.5828i			1.44953i	−0.688994	+	0.724767i	$0.741947\pi$
$864$		0			0
$865$			7.94327i			0.270079i
$866$		0			0
$867$	5.86142			0.199064
$868$		0			0
$869$		−	52.0821i		−	1.76676i
$870$		0			0
$871$	−7.35295	−	0.823403i	−0.249145	−	0.0279000i
$872$		0			0
$873$		−	23.0406i		−	0.779805i
$874$		0			0
$875$	−5.77283			−0.195157
$876$		0			0
$877$		−	37.9080i		−	1.28006i	−0.768350	−	0.640030i	$-0.778922\pi$
$877$		−	37.9080i		−	1.28006i	0.768350	−	0.640030i	$-0.221078\pi$
$878$		0			0
$879$			6.89677i			0.232622i
$880$		0			0
$881$	55.8471			1.88154			0.940768	−	0.339050i	$-0.110106\pi$
$881$	55.8471			1.88154			0.940768	+	0.339050i	$0.110106\pi$
$882$		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	$-0.506010\pi$
$887$	−1.12451			−0.0377573			−0.0188786	+	0.999822i	$0.506010\pi$
$888$		0			0
$889$		−	1.06768i		−	0.0358089i
$890$		0			0
$891$		−	34.7261i		−	1.16337i
$892$		0			0
$893$	18.0908			0.605386
$894$		0			0
$895$		−	8.80094i		−	0.294183i
$896$		0			0
$897$	0.319495	−	2.85307i	0.0106676	−	0.0952613i
$898$		0			0
$899$		−	4.27674i		−	0.142637i
$900$		0			0
$901$	0.506240			0.0168653
$902$		0			0
$903$		−	3.61965i		−	0.120454i
$904$		0			0
$905$			9.10323i			0.302601i
$906$		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	$-0.570228\pi$
$911$	−13.2106			−0.437686			−0.218843	+	0.975760i	$0.570228\pi$
$912$		0			0
$913$	21.4113			0.708610
$914$		0			0
$915$		−	0.678319i		−	0.0224245i
$916$		0			0
$917$			4.90668i			0.162033i
$918$		0			0
$919$	−4.00033			−0.131959			−0.0659793	−	0.997821i	$-0.521017\pi$
$919$	−4.00033			−0.131959			−0.0659793	+	0.997821i	$0.521017\pi$
$920$		0			0
$921$			7.02834i			0.231592i
$922$		0			0
$923$	22.3598	+	2.50391i	0.735981	+	0.0824172i
$924$		0			0
$925$			24.9975i			0.821914i
$926$		0			0
$927$	44.2140			1.45218
$928$		0			0
$929$			54.3801i			1.78415i	0.451884	+	0.892077i	$0.350752\pi$
$929$			54.3801i			1.78415i	−0.451884	+	0.892077i	$0.649248\pi$
$930$		0			0
$931$			5.58315i			0.182981i
$932$		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
$941$			38.4447i			1.25326i	0.779316	+	0.626631i	$0.215567\pi$
$941$			38.4447i			1.25326i	−0.779316	+	0.626631i	$0.784433\pi$
$942$		0			0
$943$		−	1.69586i		−	0.0552247i
$944$		0			0
$945$	−1.40281			−0.0456335
$946$		0			0
$947$		−	45.4366i		−	1.47649i	−0.674533	−	0.738245i	$-0.735655\pi$
$947$		−	45.4366i		−	1.47649i	0.674533	−	0.738245i	$-0.264345\pi$
$948$		0			0
$949$	−23.0674	−	2.58315i	−0.748801	−	0.0838527i
$950$		0			0
$951$		−	4.65989i		−	0.151107i
$952$		0			0
$953$	12.0830			0.391408			0.195704	−	0.980663i	$-0.437301\pi$
$953$	12.0830			0.391408			0.195704	+	0.980663i	$0.437301\pi$
$954$		0			0
$955$			4.64373i			0.150267i
$956$		0			0
$957$			14.3872i			0.465073i
$958$		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
$967$		−	7.50012i		−	0.241188i	−0.992702	−	0.120594i	$-0.961520\pi$
$967$		−	7.50012i		−	0.241188i	0.992702	−	0.120594i	$-0.0384799\pi$
$968$		0			0
$969$			3.46490i			0.111309i
$970$		0			0
$971$	−10.6649			−0.342253			−0.171127	−	0.985249i	$-0.554741\pi$
$971$	−10.6649			−0.342253			−0.171127	+	0.985249i	$0.554741\pi$
$972$		0			0
$973$			6.34916i			0.203545i
$974$		0			0
$975$	0.747294	−	6.67330i	0.0239326	−	0.213717i
$976$		0			0
$977$		−	8.56035i		−	0.273870i	−0.990580	−	0.136935i	$-0.956275\pi$
$977$		−	8.56035i		−	0.273870i	0.990580	−	0.136935i	$-0.0437251\pi$
$978$		0			0
$979$	−60.7779			−1.94247
$980$		0			0
$981$			32.1760i			1.02730i
$982$		0			0
$983$		−	15.7511i		−	0.502383i	−0.967937	−	0.251191i	$-0.919178\pi$
$983$		−	15.7511i		−	0.502383i	0.967937	−	0.251191i	$-0.0808224\pi$
$984$		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	$-0.115841\pi$
$991$	58.8369			1.86901			0.934507	+	0.355944i	$0.115841\pi$
$992$		0			0
$993$			11.0131i			0.349490i
$994$		0			0
$995$		−	16.0007i		−	0.507257i
$996$		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
$999$			12.6181i			0.399219i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.k.a.337.5	✓	8	52.51	odd	2
728.2.k.a.337.6	yes	8	4.3	odd	2
1456.2.k.e.337.3		8	13.12	even	2	inner
1456.2.k.e.337.4		8	1.1	even	1	trivial
9464.2.a.x.1.3		4	52.47	even	4
9464.2.a.y.1.3		4	52.31	even	4

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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.401251 q^{3} +0.598749i q^{5} +1.00000i q^{7} -2.83900 q^{9} -4.58315i q^{11} +(-0.401251 + 3.58315i) q^{13} -0.240249i q^{15} +1.54666 q^{17} -5.58315i q^{19} -0.401251i q^{21} +1.98441 q^{23} +4.64150 q^{25} +2.34291 q^{27} +7.82340 q^{29} -0.546660i q^{31} +1.83900i q^{33} -0.598749 q^{35} +5.38566i q^{37} +(0.161003 - 1.43775i) q^{39} -0.854591i q^{41} +9.02090 q^{43} -1.69985i q^{45} +3.24025i q^{47} -1.00000 q^{49} -0.620600 q^{51} +0.327312 q^{53} +2.74416 q^{55} +2.24025i q^{57} -1.81810i q^{59} +2.82340 q^{61} -2.83900i q^{63} +(-2.14541 - 0.240249i) q^{65} +2.05209i q^{67} -0.796246 q^{69} -6.24025i q^{71} +6.43775i q^{73} -1.86241 q^{75} +4.58315 q^{77} +11.3638 q^{79} +7.57690 q^{81} +4.67174i q^{83} +0.926061i q^{85} -3.13915 q^{87} -13.2611i q^{89} +(-3.58315 - 0.401251i) q^{91} +0.219348i q^{93} +3.34291 q^{95} +8.11574i q^{97} +13.0116i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{3} + 10 q^{9} + 2 q^{13} + 4 q^{17} - 20 q^{23} - 6 q^{25} + 26 q^{27} - 6 q^{29} - 10 q^{35} + 34 q^{39} + 14 q^{43} - 8 q^{49} - 20 q^{51} - 26 q^{53} + 24 q^{55} - 46 q^{61} - 14 q^{65} - 22 q^{69}+ \cdots + 34 q^{95}+O(q^{100}) \) 8 * q + 2 * q^3 + 10 * q^9 + 2 * q^13 + 4 * q^17 - 20 * q^23 - 6 * q^25 + 26 * q^27 - 6 * q^29 - 10 * q^35 + 34 * q^39 + 14 * q^43 - 8 * q^49 - 20 * q^51 - 26 * q^53 + 24 * q^55 - 46 * q^61 - 14 * q^65 - 22 * q^69 - 98 * q^75 + 6 * q^77 + 40 * q^79 + 56 * q^81 - 48 * q^87 + 2 * q^91 + 34 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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