LMFDB - Embedded newform 5796.2.k.d.5473.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5796,2,Mod(5473,5796)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5796, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5796.5473");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5796 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5796.k (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$46.2812930115$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 1932)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5473.16
Character	$\chi$	$=$	5796.5473
Dual form			5796.2.k.d.5473.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.248143 q^{5} +(0.652227 + 2.56410i) q^{7} +O(q^{10})$	$q-0.248143 q^{5} +(0.652227 + 2.56410i) q^{7} +4.05549i q^{11} -5.46658i q^{13} +0.349644 q^{17} -3.87946 q^{19} +(-0.300350 + 4.78642i) q^{23} -4.93843 q^{25} +3.50660 q^{29} -7.87222i q^{31} +(-0.161846 - 0.636263i) q^{35} -3.76156i q^{37} +5.40062i q^{41} +11.4582i q^{43} +1.86784i q^{47} +(-6.14920 + 3.34475i) q^{49} -9.71322i q^{53} -1.00634i q^{55} +12.5854i q^{59} -4.03300 q^{61} +1.35649i q^{65} +4.70293i q^{67} -13.9003 q^{71} -9.64759i q^{73} +(-10.3987 + 2.64510i) q^{77} -6.98013i q^{79} -12.2060 q^{83} -0.0867616 q^{85} +6.84062 q^{89} +(14.0169 - 3.56545i) q^{91} +0.962660 q^{95} -1.02306 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 12 q^{23} + 40 q^{25} + 16 q^{29} - 8 q^{35} - 32 q^{71} - 24 q^{77} + 8 q^{85} + 64 q^{95}+O(q^{100}) $ 32 * q - 12 * q^23 + 40 * q^25 + 16 * q^29 - 8 * q^35 - 32 * q^71 - 24 * q^77 + 8 * q^85 + 64 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5796\mathbb{Z}\right)^\times$.

$n$	$829$	$1289$	$2899$	$4789$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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			0
$3$		0			0
$4$		0			0
$5$	−0.248143			−0.110973			−0.0554865	−	0.998459i	$-0.517671\pi$
$5$	−0.248143			−0.110973			−0.0554865	+	0.998459i	$0.517671\pi$
$6$		0			0
$7$	0.652227	+	2.56410i	0.246519	+	0.969138i
$7$	0.652227	+	2.56410i	0.246519	+	0.969138i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			4.05549i			1.22278i	0.791331	+	0.611388i	$0.209388\pi$
$11$			4.05549i			1.22278i	−0.791331	+	0.611388i	$0.790612\pi$
$12$		0			0
$13$		−	5.46658i		−	1.51616i	−0.652163	−	0.758079i	$-0.726138\pi$
$13$		−	5.46658i		−	1.51616i	0.652163	−	0.758079i	$-0.273862\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.349644			0.0848011			0.0424005	−	0.999101i	$-0.486499\pi$
$17$	0.349644			0.0848011			0.0424005	+	0.999101i	$0.486499\pi$
$18$		0			0
$19$	−3.87946			−0.890008			−0.445004	−	0.895528i	$-0.646798\pi$
$19$	−3.87946			−0.890008			−0.445004	+	0.895528i	$0.646798\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.300350	+	4.78642i	−0.0626272	+	0.998037i
$23$	−0.300350	+	4.78642i	−0.0626272	+	0.998037i
$24$		0			0
$25$	−4.93843			−0.987685
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.50660			0.651158			0.325579	−	0.945515i	$-0.394441\pi$
$29$	3.50660			0.651158			0.325579	+	0.945515i	$0.394441\pi$
$30$		0			0
$31$		−	7.87222i		−	1.41389i	−0.707267	−	0.706947i	$-0.750072\pi$
$31$		−	7.87222i		−	1.41389i	0.707267	−	0.706947i	$-0.249928\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.161846	−	0.636263i	−0.0273569	−	0.107548i
$36$		0			0
$37$		−	3.76156i		−	0.618396i	−0.950998	−	0.309198i	$-0.899939\pi$
$37$		−	3.76156i		−	0.618396i	0.950998	−	0.309198i	$-0.100061\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			5.40062i			0.843435i	0.906727	+	0.421718i	$0.138573\pi$
$41$			5.40062i			0.843435i	−0.906727	+	0.421718i	$0.861427\pi$
$42$		0			0
$43$			11.4582i			1.74737i	0.486495	+	0.873683i	$0.338275\pi$
$43$			11.4582i			1.74737i	−0.486495	+	0.873683i	$0.661725\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			1.86784i			0.272452i	0.990678	+	0.136226i	$0.0434973\pi$
$47$			1.86784i			0.272452i	−0.990678	+	0.136226i	$0.956503\pi$
$48$		0			0
$49$	−6.14920	+	3.34475i	−0.878457	+	0.477821i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	9.71322i		−	1.33421i	−0.744962	−	0.667107i	$-0.767532\pi$
$53$		−	9.71322i		−	1.33421i	0.744962	−	0.667107i	$-0.232468\pi$
$54$		0			0
$55$		−	1.00634i		−	0.135695i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			12.5854i			1.63848i	0.573453	+	0.819239i	$0.305604\pi$
$59$			12.5854i			1.63848i	−0.573453	+	0.819239i	$0.694396\pi$
$60$		0			0
$61$	−4.03300			−0.516372			−0.258186	−	0.966095i	$-0.583125\pi$
$61$	−4.03300			−0.516372			−0.258186	+	0.966095i	$0.583125\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			1.35649i			0.168252i
$66$		0			0
$67$			4.70293i			0.574554i	0.957848	+	0.287277i	$0.0927500\pi$
$67$			4.70293i			0.574554i	−0.957848	+	0.287277i	$0.907250\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−13.9003			−1.64966			−0.824829	−	0.565382i	$-0.808729\pi$
$71$	−13.9003			−1.64966			−0.824829	+	0.565382i	$0.808729\pi$
$72$		0			0
$73$		−	9.64759i		−	1.12917i	−0.825377	−	0.564583i	$-0.809037\pi$
$73$		−	9.64759i		−	1.12917i	0.825377	−	0.564583i	$-0.190963\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.3987	+	2.64510i	−1.18504	+	0.301437i
$78$		0			0
$79$		−	6.98013i		−	0.785326i	−0.919682	−	0.392663i	$-0.871554\pi$
$79$		−	6.98013i		−	0.785326i	0.919682	−	0.392663i	$-0.128446\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−12.2060			−1.33978			−0.669890	−	0.742461i	$-0.733659\pi$
$83$	−12.2060			−1.33978			−0.669890	+	0.742461i	$0.733659\pi$
$84$		0			0
$85$	−0.0867616			−0.00941062
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.84062			0.725104			0.362552	−	0.931964i	$-0.381906\pi$
$89$	6.84062			0.725104			0.362552	+	0.931964i	$0.381906\pi$
$90$		0			0
$91$	14.0169	−	3.56545i	1.46937	−	0.373761i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.962660			0.0987668
$96$		0			0
$97$	−1.02306			−0.103876			−0.0519379	−	0.998650i	$-0.516540\pi$
$97$	−1.02306			−0.103876			−0.0519379	+	0.998650i	$0.516540\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	9.90088i		−	0.985174i	−0.870263	−	0.492587i	$-0.836051\pi$
$101$		−	9.90088i		−	0.985174i	0.870263	−	0.492587i	$-0.163949\pi$
$102$		0			0
$103$	−6.54945			−0.645337			−0.322668	−	0.946512i	$-0.604580\pi$
$103$	−6.54945			−0.645337			−0.322668	+	0.946512i	$0.604580\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	5.20163i		−	0.502860i	−0.967875	−	0.251430i	$-0.919099\pi$
$107$		−	5.20163i		−	0.502860i	0.967875	−	0.251430i	$-0.0809009\pi$
$108$		0			0
$109$		−	6.98642i		−	0.669177i	−0.942364	−	0.334589i	$-0.891403\pi$
$109$		−	6.98642i		−	0.669177i	0.942364	−	0.334589i	$-0.108597\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			5.89960i			0.554987i	0.960727	+	0.277494i	$0.0895038\pi$
$113$			5.89960i			0.554987i	−0.960727	+	0.277494i	$0.910496\pi$
$114$		0			0
$115$	0.0745296	−	1.18772i	0.00694992	−	0.110755i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.228047	+	0.896521i	0.0209051	+	0.0821839i
$120$		0			0
$121$	−5.44697			−0.495179
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.46615			0.220579
$126$		0			0
$127$	−9.08378			−0.806055			−0.403028	−	0.915188i	$-0.632042\pi$
$127$	−9.08378			−0.806055			−0.403028	+	0.915188i	$0.632042\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			16.8087i			1.46858i	0.678835	+	0.734291i	$0.262485\pi$
$131$			16.8087i			1.46858i	−0.678835	+	0.734291i	$0.737515\pi$
$132$		0			0
$133$	−2.53029	−	9.94731i	−0.219404	−	0.862541i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	14.9328i		−	1.27579i	−0.770122	−	0.637897i	$-0.779805\pi$
$137$		−	14.9328i		−	1.27579i	0.770122	−	0.637897i	$-0.220195\pi$
$138$		0			0
$139$			0.0692691i			0.00587533i	0.999996	+	0.00293766i	$0.000935089\pi$
$139$			0.0692691i			0.00587533i	−0.999996	+	0.00293766i	$0.999065\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	22.1697			1.85392
$144$		0			0
$145$	−0.870137			−0.0722609
$146$		0			0
$147$		0			0
$148$		0			0
$149$			5.35388i			0.438607i	0.975657	+	0.219303i	$0.0703785\pi$
$149$			5.35388i			0.438607i	−0.975657	+	0.219303i	$0.929622\pi$
$150$		0			0
$151$	7.79280			0.634169			0.317085	−	0.948397i	$-0.397296\pi$
$151$	7.79280			0.634169			0.317085	+	0.948397i	$0.397296\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			1.95344i			0.156904i
$156$		0			0
$157$	8.35019			0.666418			0.333209	−	0.942853i	$-0.391869\pi$
$157$	8.35019			0.666418			0.333209	+	0.942853i	$0.391869\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−12.4687	+	2.35171i	−0.982674	+	0.185340i
$162$		0			0
$163$	−3.50912			−0.274856			−0.137428	−	0.990512i	$-0.543884\pi$
$163$	−3.50912			−0.274856			−0.137428	+	0.990512i	$0.543884\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			7.07693i			0.547629i	0.961782	+	0.273815i	$0.0882855\pi$
$167$			7.07693i			0.547629i	−0.961782	+	0.273815i	$0.911715\pi$
$168$		0			0
$169$	−16.8835			−1.29873
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	10.8180i		−	0.822477i	−0.911528	−	0.411238i	$-0.865096\pi$
$173$		−	10.8180i		−	0.822477i	0.911528	−	0.411238i	$-0.134904\pi$
$174$		0			0
$175$	−3.22098	−	12.6626i	−0.243483	−	0.957203i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.95147			−0.743808			−0.371904	−	0.928271i	$-0.621295\pi$
$179$	−9.95147			−0.743808			−0.371904	+	0.928271i	$0.621295\pi$
$180$		0			0
$181$	−6.14027			−0.456402			−0.228201	−	0.973614i	$-0.573284\pi$
$181$	−6.14027			−0.456402			−0.228201	+	0.973614i	$0.573284\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			0.933404i			0.0686252i
$186$		0			0
$187$			1.41798i			0.103693i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	0.686980i		−	0.0497081i	−0.999691	−	0.0248541i	$-0.992088\pi$
$191$		−	0.686980i		−	0.0497081i	0.999691	−	0.0248541i	$-0.00791211\pi$
$192$		0			0
$193$	−2.09247			−0.150619			−0.0753096	−	0.997160i	$-0.523995\pi$
$193$	−2.09247			−0.150619			−0.0753096	+	0.997160i	$0.523995\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−9.87120			−0.703294			−0.351647	−	0.936133i	$-0.614378\pi$
$197$	−9.87120			−0.703294			−0.351647	+	0.936133i	$0.614378\pi$
$198$		0			0
$199$	−14.7198			−1.04346			−0.521731	−	0.853110i	$-0.674714\pi$
$199$	−14.7198			−1.04346			−0.521731	+	0.853110i	$0.674714\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.28710	+	8.99126i	0.160523	+	0.631062i
$204$		0			0
$205$		−	1.34013i		−	0.0935985i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	15.7331i		−	1.08828i
$210$		0			0
$211$	−5.39018			−0.371075			−0.185538	−	0.982637i	$-0.559403\pi$
$211$	−5.39018			−0.371075			−0.185538	+	0.982637i	$0.559403\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	2.84328i		−	0.193910i
$216$		0			0
$217$	20.1852	−	5.13448i	1.37026	−	0.348551i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	1.91136i		−	0.128572i
$222$		0			0
$223$		−	20.0807i		−	1.34470i	−0.740233	−	0.672351i	$-0.765285\pi$
$223$		−	20.0807i		−	1.34470i	0.740233	−	0.672351i	$-0.234715\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−15.5820			−1.03421			−0.517107	−	0.855921i	$-0.672991\pi$
$227$	−15.5820			−1.03421			−0.517107	+	0.855921i	$0.672991\pi$
$228$		0			0
$229$	−7.89135			−0.521475			−0.260738	−	0.965410i	$-0.583966\pi$
$229$	−7.89135			−0.521475			−0.260738	+	0.965410i	$0.583966\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	15.5621			1.01951			0.509753	−	0.860321i	$-0.329737\pi$
$233$	15.5621			1.01951			0.509753	+	0.860321i	$0.329737\pi$
$234$		0			0
$235$		−	0.463491i		−	0.0302348i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−13.6315			−0.881748			−0.440874	−	0.897569i	$-0.645331\pi$
$239$	−13.6315			−0.881748			−0.440874	+	0.897569i	$0.645331\pi$
$240$		0			0
$241$	27.6041			1.77814			0.889069	−	0.457773i	$-0.151353\pi$
$241$	27.6041			1.77814			0.889069	+	0.457773i	$0.151353\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.52588	−	0.829976i	0.0974849	−	0.0530252i
$246$		0			0
$247$			21.2074i			1.34939i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−19.8899			−1.25544			−0.627721	−	0.778439i	$-0.716012\pi$
$251$	−19.8899			−1.25544			−0.627721	+	0.778439i	$0.716012\pi$
$252$		0			0
$253$	−19.4113	−	1.21806i	−1.22037	−	0.0765790i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	5.75374i		−	0.358909i	−0.983766	−	0.179454i	$-0.942567\pi$
$257$		−	5.75374i		−	0.358909i	0.983766	−	0.179454i	$-0.0574332\pi$
$258$		0			0
$259$	9.64500	−	2.45339i	0.599311	−	0.152446i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			11.0037i			0.678516i	0.940693	+	0.339258i	$0.110176\pi$
$263$			11.0037i			0.678516i	−0.940693	+	0.339258i	$0.889824\pi$
$264$		0			0
$265$			2.41027i			0.148062i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	18.3564i		−	1.11921i	−0.828759	−	0.559606i	$-0.810952\pi$
$269$		−	18.3564i		−	1.11921i	0.828759	−	0.559606i	$-0.189048\pi$
$270$		0			0
$271$		−	29.9547i		−	1.81962i	−0.415027	−	0.909809i	$-0.636228\pi$
$271$		−	29.9547i		−	1.81962i	0.415027	−	0.909809i	$-0.363772\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	20.0277i		−	1.20772i
$276$		0			0
$277$	−18.6594			−1.12113			−0.560567	−	0.828109i	$-0.689417\pi$
$277$	−18.6594			−1.12113			−0.560567	+	0.828109i	$0.689417\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			0.0173097i			0.00103261i	1.00000		0.000516305i	$0.000164345\pi$
$281$			0.0173097i			0.00103261i	−1.00000		0.000516305i	$0.999836\pi$
$282$		0			0
$283$	−0.120744			−0.00717752			−0.00358876	−	0.999994i	$-0.501142\pi$
$283$	−0.120744			−0.00717752			−0.00358876	+	0.999994i	$0.501142\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−13.8477	+	3.52243i	−0.817405	+	0.207923i
$288$		0			0
$289$	−16.8777			−0.992809
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−28.5499			−1.66790			−0.833950	−	0.551839i	$-0.813926\pi$
$293$	−28.5499			−1.66790			−0.833950	+	0.551839i	$0.813926\pi$
$294$		0			0
$295$		−	3.12297i		−	0.181827i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	26.1653	+	1.64189i	1.51318	+	0.0949527i
$300$		0			0
$301$	−29.3801	+	7.47338i	−1.69344	+	0.430759i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.00076			0.0573033
$306$		0			0
$307$			9.73639i			0.555685i	0.960627	+	0.277843i	$0.0896194\pi$
$307$			9.73639i			0.555685i	−0.960627	+	0.277843i	$0.910381\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	24.8161i		−	1.40719i	−0.710601	−	0.703596i	$-0.751577\pi$
$311$		−	24.8161i		−	1.40719i	0.710601	−	0.703596i	$-0.248423\pi$
$312$		0			0
$313$	−24.1571			−1.36544			−0.682719	−	0.730681i	$-0.739203\pi$
$313$	−24.1571			−1.36544			−0.682719	+	0.730681i	$0.739203\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.882476			−0.0495648			−0.0247824	−	0.999693i	$-0.507889\pi$
$317$	−0.882476			−0.0495648			−0.0247824	+	0.999693i	$0.507889\pi$
$318$		0			0
$319$			14.2210i			0.796220i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.35643			−0.0754736
$324$		0			0
$325$			26.9963i			1.49749i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.78932	+	1.21825i	−0.264044	+	0.0671645i
$330$		0			0
$331$	4.48030			0.246259			0.123130	−	0.992391i	$-0.460707\pi$
$331$	4.48030			0.246259			0.123130	+	0.992391i	$0.460707\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	1.16700i		−	0.0637599i
$336$		0			0
$337$			7.99001i			0.435243i	0.976033	+	0.217622i	$0.0698299\pi$
$337$			7.99001i			0.435243i	−0.976033	+	0.217622i	$0.930170\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	31.9257			1.72887
$342$		0			0
$343$	−12.5869	−	13.5856i	−0.679631	−	0.733554i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.18595			0.439445			0.219723	−	0.975562i	$-0.429485\pi$
$347$	8.18595			0.439445			0.219723	+	0.975562i	$0.429485\pi$
$348$		0			0
$349$			23.3084i			1.24767i	0.781557	+	0.623834i	$0.214426\pi$
$349$			23.3084i			1.24767i	−0.781557	+	0.623834i	$0.785574\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			20.6822i			1.10080i	0.834901	+	0.550400i	$0.185525\pi$
$353$			20.6822i			1.10080i	−0.834901	+	0.550400i	$0.814475\pi$
$354$		0			0
$355$	3.44925			0.183067
$356$		0			0
$357$		0			0
$358$		0			0
$359$			17.3509i			0.915748i	0.889017	+	0.457874i	$0.151389\pi$
$359$			17.3509i			0.915748i	−0.889017	+	0.457874i	$0.848611\pi$
$360$		0			0
$361$	−3.94982			−0.207885
$362$		0			0
$363$		0			0
$364$		0			0
$365$			2.39398i			0.125307i
$366$		0			0
$367$	−15.2971			−0.798501			−0.399250	−	0.916842i	$-0.630730\pi$
$367$	−15.2971			−0.798501			−0.399250	+	0.916842i	$0.630730\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	24.9057	−	6.33523i	1.29304	−	0.328909i
$372$		0			0
$373$			19.8672i			1.02869i	0.857584	+	0.514343i	$0.171964\pi$
$373$			19.8672i			1.02869i	−0.857584	+	0.514343i	$0.828036\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	19.1691i		−	0.987258i
$378$		0			0
$379$		−	2.96163i		−	0.152129i	−0.997103	−	0.0760644i	$-0.975765\pi$
$379$		−	2.96163i		−	0.152129i	0.997103	−	0.0760644i	$-0.0242355\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6.82827			0.348908			0.174454	−	0.984665i	$-0.444184\pi$
$383$	6.82827			0.348908			0.174454	+	0.984665i	$0.444184\pi$
$384$		0			0
$385$	2.58036	−	0.656363i	0.131507	−	0.0334513i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			13.7609i			0.697706i	0.937177	+	0.348853i	$0.113429\pi$
$389$			13.7609i			0.697706i	−0.937177	+	0.348853i	$0.886571\pi$
$390$		0			0
$391$	−0.105015	+	1.67354i	−0.00531085	+	0.0846346i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			1.73207i			0.0871499i
$396$		0			0
$397$			26.0290i			1.30636i	0.757203	+	0.653180i	$0.226565\pi$
$397$			26.0290i			1.30636i	−0.757203	+	0.653180i	$0.773435\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			33.8292i			1.68935i	0.535279	+	0.844675i	$0.320207\pi$
$401$			33.8292i			1.68935i	−0.535279	+	0.844675i	$0.679793\pi$
$402$		0			0
$403$	−43.0342			−2.14368
$404$		0			0
$405$		0			0
$406$		0			0
$407$	15.2549			0.756160
$408$		0			0
$409$			24.2940i			1.20126i	0.799527	+	0.600630i	$0.205083\pi$
$409$			24.2940i			1.20126i	−0.799527	+	0.600630i	$0.794917\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−32.2702	+	8.20853i	−1.58791	+	0.403915i
$414$		0			0
$415$	3.02883			0.148679
$416$		0			0
$417$		0			0
$418$		0			0
$419$	21.5498			1.05277			0.526387	−	0.850245i	$-0.323546\pi$
$419$	21.5498			1.05277			0.526387	+	0.850245i	$0.323546\pi$
$420$		0			0
$421$			37.1329i			1.80975i	0.425682	+	0.904873i	$0.360034\pi$
$421$			37.1329i			1.80975i	−0.425682	+	0.904873i	$0.639966\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.72669			−0.0837567
$426$		0			0
$427$	−2.63043	−	10.3410i	−0.127295	−	0.500436i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	35.6918i		−	1.71921i	−0.510958	−	0.859606i	$-0.670709\pi$
$431$		−	35.6918i		−	1.71921i	0.510958	−	0.859606i	$-0.329291\pi$
$432$		0			0
$433$	17.2866			0.830743			0.415372	−	0.909652i	$-0.363652\pi$
$433$	17.2866			0.830743			0.415372	+	0.909652i	$0.363652\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.16519	−	18.5687i	0.0557387	−	0.888261i
$438$		0			0
$439$			28.6110i			1.36553i	0.730638	+	0.682765i	$0.239223\pi$
$439$			28.6110i			1.36553i	−0.730638	+	0.682765i	$0.760777\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−21.0921			−1.00212			−0.501058	−	0.865414i	$-0.667056\pi$
$443$	−21.0921			−1.00212			−0.501058	+	0.865414i	$0.667056\pi$
$444$		0			0
$445$	−1.69745			−0.0804669
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−23.5513			−1.11146			−0.555728	−	0.831364i	$-0.687560\pi$
$449$	−23.5513			−1.11146			−0.555728	+	0.831364i	$0.687560\pi$
$450$		0			0
$451$	−21.9021			−1.03133
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.47818	+	0.884742i	−0.163060	+	0.0414774i
$456$		0			0
$457$		−	26.9541i		−	1.26086i	−0.776247	−	0.630429i	$-0.782879\pi$
$457$		−	26.9541i		−	1.26086i	0.776247	−	0.630429i	$-0.217121\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			18.4634i			0.859924i	0.902847	+	0.429962i	$0.141473\pi$
$461$			18.4634i			0.859924i	−0.902847	+	0.429962i	$0.858527\pi$
$462$		0			0
$463$	35.3889			1.64466			0.822331	−	0.569009i	$-0.192673\pi$
$463$	35.3889			1.64466			0.822331	+	0.569009i	$0.192673\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	14.9569			0.692123			0.346062	−	0.938212i	$-0.387519\pi$
$467$	14.9569			0.692123			0.346062	+	0.938212i	$0.387519\pi$
$468$		0			0
$469$	−12.0588	+	3.06738i	−0.556822	+	0.141638i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−46.4688			−2.13664
$474$		0			0
$475$	19.1584			0.879048
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.1485			0.555078			0.277539	−	0.960714i	$-0.410481\pi$
$479$	12.1485			0.555078			0.277539	+	0.960714i	$0.410481\pi$
$480$		0			0
$481$	−20.5629			−0.937586
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.253865			0.0115274
$486$		0			0
$487$	19.1745			0.868879			0.434440	−	0.900701i	$-0.356946\pi$
$487$	19.1745			0.868879			0.434440	+	0.900701i	$0.356946\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	13.0026			0.586800			0.293400	−	0.955990i	$-0.405213\pi$
$491$	13.0026			0.586800			0.293400	+	0.955990i	$0.405213\pi$
$492$		0			0
$493$	1.22606			0.0552189
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.06614	−	35.6417i	−0.406672	−	1.59875i
$498$		0			0
$499$	−37.0374			−1.65802			−0.829011	−	0.559232i	$-0.811096\pi$
$499$	−37.0374			−1.65802			−0.829011	+	0.559232i	$0.811096\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−13.8200			−0.616206			−0.308103	−	0.951353i	$-0.599694\pi$
$503$	−13.8200			−0.616206			−0.308103	+	0.951353i	$0.599694\pi$
$504$		0			0
$505$			2.45683i			0.109328i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	33.4182i		−	1.48123i	−0.671927	−	0.740617i	$-0.734533\pi$
$509$		−	33.4182i		−	1.48123i	0.671927	−	0.740617i	$-0.265467\pi$
$510$		0			0
$511$	24.7374	−	6.29242i	1.09432	−	0.278360i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.62520			0.0716149
$516$		0			0
$517$	−7.57499			−0.333148
$518$		0			0
$519$		0			0
$520$		0			0
$521$	33.6964			1.47627			0.738133	−	0.674655i	$-0.235708\pi$
$521$	33.6964			1.47627			0.738133	+	0.674655i	$0.235708\pi$
$522$		0			0
$523$	3.03787			0.132837			0.0664185	−	0.997792i	$-0.478843\pi$
$523$	3.03787			0.132837			0.0664185	+	0.997792i	$0.478843\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	2.75247i		−	0.119900i
$528$		0			0
$529$	−22.8196	−	2.87520i	−0.992156	−	0.125009i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	29.5229			1.27878
$534$		0			0
$535$			1.29075i			0.0558039i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−13.5646	−	24.9380i	−0.584268	−	1.07416i
$540$		0			0
$541$	−23.9327			−1.02895			−0.514474	−	0.857506i	$-0.672013\pi$
$541$	−23.9327			−1.02895			−0.514474	+	0.857506i	$0.672013\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			1.73363i			0.0742605i
$546$		0			0
$547$	−9.70243			−0.414846			−0.207423	−	0.978251i	$-0.566508\pi$
$547$	−9.70243			−0.414846			−0.207423	+	0.978251i	$0.566508\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−13.6037			−0.579536
$552$		0			0
$553$	17.8977	−	4.55263i	0.761090	−	0.193598i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	12.9311i		−	0.547907i	−0.961743	−	0.273953i	$-0.911669\pi$
$557$		−	12.9311i		−	0.547907i	0.961743	−	0.273953i	$-0.0883314\pi$
$558$		0			0
$559$	62.6375			2.64928
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−28.3780			−1.19599			−0.597996	−	0.801499i	$-0.704036\pi$
$563$	−28.3780			−1.19599			−0.597996	+	0.801499i	$0.704036\pi$
$564$		0			0
$565$		−	1.46394i		−	0.0615886i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	3.41585i		−	0.143200i	−0.997433	−	0.0716000i	$-0.977190\pi$
$569$		−	3.41585i		−	0.143200i	0.997433	−	0.0716000i	$-0.0228105\pi$
$570$		0			0
$571$		−	43.5325i		−	1.82178i	−0.412651	−	0.910889i	$-0.635397\pi$
$571$		−	43.5325i		−	1.82178i	0.412651	−	0.910889i	$-0.364603\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.48325	−	23.6374i	0.0618560	−	0.985746i
$576$		0			0
$577$		−	39.0578i		−	1.62600i	−0.582267	−	0.812998i	$-0.697834\pi$
$577$		−	39.0578i		−	1.62600i	0.582267	−	0.812998i	$-0.302166\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.96107	−	31.2973i	−0.330281	−	1.29843i
$582$		0			0
$583$	39.3919			1.63144
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	10.8977i		−	0.449798i	−0.974382	−	0.224899i	$-0.927795\pi$
$587$		−	10.8977i		−	0.449798i	0.974382	−	0.224899i	$-0.0722052\pi$
$588$		0			0
$589$			30.5399i			1.25838i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	34.7874i		−	1.42855i	−0.699866	−	0.714275i	$-0.746757\pi$
$593$		−	34.7874i		−	1.42855i	0.699866	−	0.714275i	$-0.253243\pi$
$594$		0			0
$595$	−0.0565883	−	0.222465i	−0.00231989	−	0.00912019i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	18.1144			0.740134			0.370067	−	0.929005i	$-0.379335\pi$
$599$	18.1144			0.740134			0.370067	+	0.929005i	$0.379335\pi$
$600$		0			0
$601$			0.729850i			0.0297712i	0.999889	+	0.0148856i	$0.00473841\pi$
$601$			0.729850i			0.0297712i	−0.999889	+	0.0148856i	$0.995262\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.35163			0.0549515
$606$		0			0
$607$		−	20.2777i		−	0.823047i	−0.911399	−	0.411523i	$-0.864997\pi$
$607$		−	20.2777i		−	0.823047i	0.911399	−	0.411523i	$-0.135003\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	10.2107			0.413080
$612$		0			0
$613$		−	29.7029i		−	1.19969i	−0.800117	−	0.599844i	$-0.795229\pi$
$613$		−	29.7029i		−	1.19969i	0.800117	−	0.599844i	$-0.204771\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			32.0102i			1.28868i	0.764739	+	0.644340i	$0.222868\pi$
$617$			32.0102i			1.28868i	−0.764739	+	0.644340i	$0.777132\pi$
$618$		0			0
$619$	−47.0226			−1.89000			−0.944999	−	0.327073i	$-0.893938\pi$
$619$	−47.0226			−1.89000			−0.944999	+	0.327073i	$0.893938\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4.46164	+	17.5400i	0.178752	+	0.702726i
$624$		0			0
$625$	24.0802			0.963207
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.31521i		−	0.0524407i
$630$		0			0
$631$			34.8593i			1.38772i	0.720108	+	0.693862i	$0.244092\pi$
$631$			34.8593i			1.38772i	−0.720108	+	0.693862i	$0.755908\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.25408			0.0894503
$636$		0			0
$637$	18.2844	+	33.6151i	0.724452	+	1.33188i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			23.7753i			0.939066i	0.882915	+	0.469533i	$0.155578\pi$
$641$			23.7753i			0.939066i	−0.882915	+	0.469533i	$0.844422\pi$
$642$		0			0
$643$	25.3414			0.999367			0.499684	−	0.866208i	$-0.333450\pi$
$643$	25.3414			0.999367			0.499684	+	0.866208i	$0.333450\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			47.5590i			1.86974i	0.354994	+	0.934868i	$0.384483\pi$
$647$			47.5590i			1.86974i	−0.354994	+	0.934868i	$0.615517\pi$
$648$		0			0
$649$	−51.0399			−2.00349
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8.23220			−0.322151			−0.161075	−	0.986942i	$-0.551496\pi$
$653$	−8.23220			−0.322151			−0.161075	+	0.986942i	$0.551496\pi$
$654$		0			0
$655$		−	4.17096i		−	0.162973i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			41.6169i			1.62117i	0.585624	+	0.810583i	$0.300849\pi$
$659$			41.6169i			1.62117i	−0.585624	+	0.810583i	$0.699151\pi$
$660$		0			0
$661$	−4.58613			−0.178380			−0.0891898	−	0.996015i	$-0.528428\pi$
$661$	−4.58613			−0.178380			−0.0891898	+	0.996015i	$0.528428\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.627873	+	2.46835i	0.0243479	+	0.0957187i
$666$		0			0
$667$	−1.05320	+	16.7840i	−0.0407802	+	0.649880i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	16.3558i		−	0.631407i
$672$		0			0
$673$	19.5644			0.754151			0.377076	−	0.926182i	$-0.376930\pi$
$673$	19.5644			0.754151			0.377076	+	0.926182i	$0.376930\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8.22886			0.316261			0.158130	−	0.987418i	$-0.449453\pi$
$677$	8.22886			0.316261			0.158130	+	0.987418i	$0.449453\pi$
$678$		0			0
$679$	−0.667266	−	2.62322i	−0.0256073	−	0.100670i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−45.1557			−1.72783			−0.863917	−	0.503634i	$-0.831996\pi$
$683$	−45.1557			−1.72783			−0.863917	+	0.503634i	$0.831996\pi$
$684$		0			0
$685$			3.70546i			0.141578i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−53.0981			−2.02288
$690$		0			0
$691$		−	18.1471i		−	0.690349i	−0.938539	−	0.345174i	$-0.887820\pi$
$691$		−	18.1471i		−	0.690349i	0.938539	−	0.345174i	$-0.112180\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	0.0171886i		−	0.000652002i
$696$		0			0
$697$			1.88829i			0.0715242i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	23.7085i		−	0.895458i	−0.894169	−	0.447729i	$-0.852233\pi$
$701$		−	23.7085i		−	0.895458i	0.894169	−	0.447729i	$-0.147767\pi$
$702$		0			0
$703$			14.5928i			0.550378i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	25.3868	−	6.45762i	0.954769	−	0.242864i
$708$		0			0
$709$			25.0512i			0.940819i	0.882448	+	0.470409i	$0.155894\pi$
$709$			25.0512i			0.940819i	−0.882448	+	0.470409i	$0.844106\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	37.6797	+	2.36442i	1.41112	+	0.0885482i
$714$		0			0
$715$	−5.50124			−0.205735
$716$		0			0
$717$		0			0
$718$		0			0
$719$			11.1554i			0.416028i	0.978126	+	0.208014i	$0.0667000\pi$
$719$			11.1554i			0.416028i	−0.978126	+	0.208014i	$0.933300\pi$
$720$		0			0
$721$	−4.27173	−	16.7934i	−0.159088	−	0.625420i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−17.3171			−0.643139
$726$		0			0
$727$	23.8412			0.884221			0.442110	−	0.896961i	$-0.354230\pi$
$727$	23.8412			0.884221			0.442110	+	0.896961i	$0.354230\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			4.00630i			0.148179i
$732$		0			0
$733$	−21.7749			−0.804273			−0.402137	−	0.915580i	$-0.631732\pi$
$733$	−21.7749			−0.804273			−0.402137	+	0.915580i	$0.631732\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−19.0727			−0.702550
$738$		0			0
$739$	37.8156			1.39107			0.695534	−	0.718493i	$-0.255168\pi$
$739$	37.8156			1.39107			0.695534	+	0.718493i	$0.255168\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			32.1362i			1.17896i	0.807782	+	0.589482i	$0.200668\pi$
$743$			32.1362i			1.17896i	−0.807782	+	0.589482i	$0.799332\pi$
$744$		0			0
$745$		−	1.32853i		−	0.0486735i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	13.3375	−	3.39264i	0.487341	−	0.123965i
$750$		0			0
$751$		−	53.2188i		−	1.94198i	−0.239116	−	0.970991i	$-0.576857\pi$
$751$		−	53.2188i		−	1.94198i	0.239116	−	0.970991i	$-0.423143\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.93373			−0.0703756
$756$		0			0
$757$		−	0.976339i		−	0.0354857i	−0.999843	−	0.0177428i	$-0.994352\pi$
$757$		−	0.976339i		−	0.0354857i	0.999843	−	0.0177428i	$-0.00564801\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			42.1794i			1.52900i	0.644623	+	0.764500i	$0.277014\pi$
$761$			42.1794i			1.52900i	−0.644623	+	0.764500i	$0.722986\pi$
$762$		0			0
$763$	17.9139	−	4.55673i	0.648525	−	0.164965i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	68.7990			2.48419
$768$		0			0
$769$	29.3735			1.05924			0.529618	−	0.848236i	$-0.322335\pi$
$769$	29.3735			1.05924			0.529618	+	0.848236i	$0.322335\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	21.7233			0.781332			0.390666	−	0.920532i	$-0.372245\pi$
$773$	21.7233			0.781332			0.390666	+	0.920532i	$0.372245\pi$
$774$		0			0
$775$			38.8764i			1.39648i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	20.9515i		−	0.750664i
$780$		0			0
$781$		−	56.3724i		−	2.01716i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.07204			−0.0739543
$786$		0			0
$787$	−13.8818			−0.494834			−0.247417	−	0.968909i	$-0.579582\pi$
$787$	−13.8818			−0.494834			−0.247417	+	0.968909i	$0.579582\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−15.1271	+	3.84788i	−0.537859	+	0.136815i
$792$		0			0
$793$			22.0467i			0.782902i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	15.2781			0.541177			0.270588	−	0.962695i	$-0.412782\pi$
$797$	15.2781			0.541177			0.270588	+	0.962695i	$0.412782\pi$
$798$		0			0
$799$			0.653077i			0.0231042i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	39.1257			1.38072
$804$		0			0
$805$	3.09403	−	0.583559i	0.109050	−	0.0205678i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	22.7917			0.801313			0.400656	−	0.916228i	$-0.368782\pi$
$809$	22.7917			0.801313			0.400656	+	0.916228i	$0.368782\pi$
$810$		0			0
$811$		−	38.2968i		−	1.34478i	−0.740196	−	0.672391i	$-0.765268\pi$
$811$		−	38.2968i		−	1.34478i	0.740196	−	0.672391i	$-0.234732\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.870764			0.0305015
$816$		0			0
$817$		−	44.4518i		−	1.55517i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	10.3811			0.362304			0.181152	−	0.983455i	$-0.442017\pi$
$821$	10.3811			0.362304			0.181152	+	0.983455i	$0.442017\pi$
$822$		0			0
$823$	40.1306			1.39886			0.699432	−	0.714699i	$-0.253436\pi$
$823$	40.1306			1.39886			0.699432	+	0.714699i	$0.253436\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			17.4365i			0.606326i	0.952939	+	0.303163i	$0.0980426\pi$
$827$			17.4365i			0.606326i	−0.952939	+	0.303163i	$0.901957\pi$
$828$		0			0
$829$			2.92638i			0.101637i	0.998708	+	0.0508187i	$0.0161831\pi$
$829$			2.92638i			0.101637i	−0.998708	+	0.0508187i	$0.983817\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.15003	+	1.16947i	−0.0744941	+	0.0405198i
$834$		0			0
$835$		−	1.75609i		−	0.0607720i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−53.2071			−1.83691			−0.918456	−	0.395524i	$-0.870563\pi$
$839$	−53.2071			−1.83691			−0.918456	+	0.395524i	$0.870563\pi$
$840$		0			0
$841$	−16.7038			−0.575993
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.18953			0.144124
$846$		0			0
$847$	−3.55267	−	13.9666i	−0.122071	−	0.479897i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	18.0044	+	1.12978i	0.617182	+	0.0387284i
$852$		0			0
$853$		−	33.3053i		−	1.14035i	−0.821522	−	0.570176i	$-0.806875\pi$
$853$		−	33.3053i		−	1.14035i	0.821522	−	0.570176i	$-0.193125\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			28.3676i			0.969018i	0.874786	+	0.484509i	$0.161002\pi$
$857$			28.3676i			0.969018i	−0.874786	+	0.484509i	$0.838998\pi$
$858$		0			0
$859$			40.7811i			1.39143i	0.718316	+	0.695717i	$0.244913\pi$
$859$			40.7811i			1.39143i	−0.718316	+	0.695717i	$0.755087\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−37.0714			−1.26193			−0.630963	−	0.775813i	$-0.717340\pi$
$863$	−37.0714			−1.26193			−0.630963	+	0.775813i	$0.717340\pi$
$864$		0			0
$865$			2.68441i			0.0912726i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	28.3078			0.960278
$870$		0			0
$871$	25.7089			0.871114
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.60849	+	6.32345i	0.0543769	+	0.213772i
$876$		0			0
$877$	−0.932860			−0.0315004			−0.0157502	−	0.999876i	$-0.505014\pi$
$877$	−0.932860			−0.0315004			−0.0157502	+	0.999876i	$0.505014\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−38.0199			−1.28092			−0.640461	−	0.767991i	$-0.721257\pi$
$881$	−38.0199			−1.28092			−0.640461	+	0.767991i	$0.721257\pi$
$882$		0			0
$883$	19.9372			0.670941			0.335470	−	0.942051i	$-0.391105\pi$
$883$	19.9372			0.670941			0.335470	+	0.942051i	$0.391105\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	15.8042i		−	0.530654i	−0.964159	−	0.265327i	$-0.914520\pi$
$887$		−	15.8042i		−	0.530654i	0.964159	−	0.265327i	$-0.0854799\pi$
$888$		0			0
$889$	−5.92469	−	23.2917i	−0.198708	−	0.781179i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	7.24619i		−	0.242485i
$894$		0			0
$895$	2.46939			0.0825425
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	27.6047i		−	0.920668i
$900$		0			0
$901$		−	3.39617i		−	0.113143i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.52366			0.0506483
$906$		0			0
$907$			20.0631i			0.666185i	0.942894	+	0.333093i	$0.108092\pi$
$907$			20.0631i			0.666185i	−0.942894	+	0.333093i	$0.891908\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	24.2258i		−	0.802637i	−0.915939	−	0.401318i	$-0.868552\pi$
$911$		−	24.2258i		−	0.802637i	0.915939	−	0.401318i	$-0.131448\pi$
$912$		0			0
$913$		−	49.5011i		−	1.63825i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−43.0991	+	10.9631i	−1.42326	+	0.362033i
$918$		0			0
$919$			28.8911i			0.953030i	0.879166	+	0.476515i	$0.158100\pi$
$919$			28.8911i			0.953030i	−0.879166	+	0.476515i	$0.841900\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			75.9870i			2.50114i
$924$		0			0
$925$			18.5762i			0.610781i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			40.6003i			1.33205i	0.745929	+	0.666026i	$0.232006\pi$
$929$			40.6003i			1.33205i	−0.745929	+	0.666026i	$0.767994\pi$
$930$		0			0
$931$	23.8556	−	12.9758i	0.781834	−	0.425265i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	0.351861i		−	0.0115071i
$936$		0			0
$937$	33.1559			1.08316			0.541579	−	0.840650i	$-0.317827\pi$
$937$	33.1559			1.08316			0.541579	+	0.840650i	$0.317827\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	20.1569			0.657095			0.328548	−	0.944487i	$-0.393441\pi$
$941$	20.1569			0.657095			0.328548	+	0.944487i	$0.393441\pi$
$942$		0			0
$943$	−25.8496	−	1.62207i	−0.841780	−	0.0528220i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	13.5374			0.439907			0.219954	−	0.975510i	$-0.429409\pi$
$947$	13.5374			0.439907			0.219954	+	0.975510i	$0.429409\pi$
$948$		0			0
$949$	−52.7394			−1.71199
$950$		0			0
$951$		0			0
$952$		0			0
$953$			29.0790i			0.941962i	0.882143	+	0.470981i	$0.156100\pi$
$953$			29.0790i			0.941962i	−0.882143	+	0.470981i	$0.843900\pi$
$954$		0			0
$955$			0.170469i			0.00551626i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	38.2891	−	9.73956i	1.23642	−	0.314507i
$960$		0			0
$961$	−30.9719			−0.999094
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.519231			0.0167146
$966$		0			0
$967$	8.24367			0.265099			0.132549	−	0.991176i	$-0.457684\pi$
$967$	8.24367			0.265099			0.132549	+	0.991176i	$0.457684\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	44.4456			1.42633			0.713163	−	0.700998i	$-0.247262\pi$
$971$	44.4456			1.42633			0.713163	+	0.700998i	$0.247262\pi$
$972$		0			0
$973$	−0.177613	+	0.0451792i	−0.00569400	+	0.00144838i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	27.6030i		−	0.883098i	−0.897237	−	0.441549i	$-0.854429\pi$
$977$		−	27.6030i		−	0.883098i	0.897237	−	0.441549i	$-0.145571\pi$
$978$		0			0
$979$			27.7420i			0.886639i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	57.9436			1.84811			0.924057	−	0.382255i	$-0.124852\pi$
$983$	57.9436			1.84811			0.924057	+	0.382255i	$0.124852\pi$
$984$		0			0
$985$	2.44947			0.0780465
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−54.8440	−	3.44148i	−1.74394	−	0.109433i
$990$		0			0
$991$	59.2749			1.88293			0.941464	−	0.337113i	$-0.109450\pi$
$991$	59.2749			1.88293			0.941464	+	0.337113i	$0.109450\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	3.65263			0.115796
$996$		0			0
$997$			0.648259i			0.0205306i	0.999947	+	0.0102653i	$0.00326760\pi$
$997$			0.648259i			0.0205306i	−0.999947	+	0.0102653i	$0.996732\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5796.2.k.d.5473.16		32
3.2	odd	2		1932.2.k.a.1609.24	yes	32
7.6	odd	2	inner	5796.2.k.d.5473.18		32
21.20	even	2		1932.2.k.a.1609.21	✓	32
23.22	odd	2	inner	5796.2.k.d.5473.17		32
69.68	even	2		1932.2.k.a.1609.23	yes	32
161.160	even	2	inner	5796.2.k.d.5473.15		32
483.482	odd	2		1932.2.k.a.1609.22	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.k.a.1609.21	✓	32	21.20	even	2
1932.2.k.a.1609.22	yes	32	483.482	odd	2
1932.2.k.a.1609.23	yes	32	69.68	even	2
1932.2.k.a.1609.24	yes	32	3.2	odd	2
5796.2.k.d.5473.15		32	161.160	even	2	inner
5796.2.k.d.5473.16		32	1.1	even	1	trivial
5796.2.k.d.5473.17		32	23.22	odd	2	inner
5796.2.k.d.5473.18		32	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5796 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5796.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.248143 q^{5} +(0.652227 + 2.56410i) q^{7} +O(q^{10})\)	\(q-0.248143 q^{5} +(0.652227 + 2.56410i) q^{7} +4.05549i q^{11} -5.46658i q^{13} +0.349644 q^{17} -3.87946 q^{19} +(-0.300350 + 4.78642i) q^{23} -4.93843 q^{25} +3.50660 q^{29} -7.87222i q^{31} +(-0.161846 - 0.636263i) q^{35} -3.76156i q^{37} +5.40062i q^{41} +11.4582i q^{43} +1.86784i q^{47} +(-6.14920 + 3.34475i) q^{49} -9.71322i q^{53} -1.00634i q^{55} +12.5854i q^{59} -4.03300 q^{61} +1.35649i q^{65} +4.70293i q^{67} -13.9003 q^{71} -9.64759i q^{73} +(-10.3987 + 2.64510i) q^{77} -6.98013i q^{79} -12.2060 q^{83} -0.0867616 q^{85} +6.84062 q^{89} +(14.0169 - 3.56545i) q^{91} +0.962660 q^{95} -1.02306 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 12 q^{23} + 40 q^{25} + 16 q^{29} - 8 q^{35} - 32 q^{71} - 24 q^{77} + 8 q^{85} + 64 q^{95}+O(q^{100}) \) 32 * q - 12 * q^23 + 40 * q^25 + 16 * q^29 - 8 * q^35 - 32 * q^71 - 24 * q^77 + 8 * q^85 + 64 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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