LMFDB - Embedded newform 731.2.s.a.558.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [731,2,Mod(214,731)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(731, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([3, 8]))

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

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

chi := DirichletCharacter("731.214");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 731 = 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	731.s (of order $16$, degree $8$, 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:	$5.83706438776$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{16})$
Coefficient field:	16.0.3289935900927224469054816256.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 10319x^{8} + 214358881 $ x^16 + 10319*x^8 + 214358881
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{16}]$

Embedding invariants

Embedding label			558.1
Root			$3.22048 - 0.792772i$ of defining polynomial
Character	$\chi$	$=$	731.558
Dual form			731.2.s.a.300.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.41421 - 1.41421i) q^{4} +(-2.77164 - 1.14805i) q^{9} +O(q^{10})$	$q+(1.41421 - 1.41421i) q^{4} +(-2.77164 - 1.14805i) q^{9} +(-4.38861 + 0.872949i) q^{11} +(-4.77872 - 4.77872i) q^{13} -4.00000i q^{16} +(-2.07243 - 3.56441i) q^{17} +(1.52113 + 7.64725i) q^{23} +(1.91342 - 4.61940i) q^{25} +(10.2516 + 2.03917i) q^{31} +(-5.54328 + 2.29610i) q^{36} +(-10.0341 - 6.70458i) q^{41} +(-6.05828 - 2.50942i) q^{43} +(-4.97189 + 7.44097i) q^{44} +(6.16199 + 6.16199i) q^{47} +(2.67878 + 6.46716i) q^{49} -13.5163 q^{52} +(10.1933 - 4.22221i) q^{53} +(3.46524 - 8.36582i) q^{59} +(-5.65685 - 5.65685i) q^{64} -11.3488i q^{67} +(-7.97170 - 2.10997i) q^{68} +(9.03385 - 1.79695i) q^{79} +(6.36396 + 6.36396i) q^{81} +(-2.50942 - 6.05828i) q^{83} +(12.9661 + 8.66364i) q^{92} +(1.58732 + 2.37559i) q^{97} +(13.1658 + 2.61885i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 24 q^{13} + 56 q^{23} - 64 q^{59} + 96 q^{79}+O(q^{100}) $ 16 * q - 24 * q^13 + 56 * q^23 - 64 * q^59 + 96 * q^79

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/731\mathbb{Z}\right)^\times$.

$n$	$173$	$562$
$\chi(n)$	$e\left(\frac{9}{16}\right)$	$-1$

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		0.923880	−	0.382683i	$-0.125000\pi$
$2$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$3$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$3$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$4$	1.41421	−	1.41421i	0.707107	−	0.707107i
$5$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$5$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$6$		0			0
$7$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$7$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$8$		0			0
$9$	−2.77164	−	1.14805i	−0.923880	−	0.382683i
$10$		0			0
$11$	−4.38861	+	0.872949i	−1.32322	+	0.263204i	−0.805626	−	0.592425i	$-0.798171\pi$
$11$	−4.38861	+	0.872949i	−1.32322	+	0.263204i	−0.517590	+	0.855629i	$0.673171\pi$
$12$		0			0
$13$	−4.77872	−	4.77872i	−1.32538	−	1.32538i	−0.909353	−	0.416025i	$-0.863423\pi$
$13$	−4.77872	−	4.77872i	−1.32538	−	1.32538i	−0.416025	−	0.909353i	$-0.636577\pi$
$14$		0			0
$15$		0			0
$16$		−	4.00000i		−	1.00000i
$17$	−2.07243	−	3.56441i	−0.502639	−	0.864496i
$17$	−2.07243	−	3.56441i	−0.502639	−	0.864496i
$18$		0			0
$19$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$19$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.52113	+	7.64725i	0.317178	+	1.59456i	0.729800	+	0.683660i	$0.239613\pi$
$23$	1.52113	+	7.64725i	0.317178	+	1.59456i	−0.412622	+	0.910902i	$0.635387\pi$
$24$		0			0
$25$	1.91342	−	4.61940i	0.382683	−	0.923880i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$29$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$30$		0			0
$31$	10.2516	+	2.03917i	1.84124	+	0.366246i	0.987898	−	0.155103i	$-0.0495709\pi$
$31$	10.2516	+	2.03917i	1.84124	+	0.366246i	0.853344	+	0.521349i	$0.174571\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−5.54328	+	2.29610i	−0.923880	+	0.382683i
$37$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$37$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−10.0341	−	6.70458i	−1.56707	−	1.04708i	−0.969447	−	0.245299i	$-0.921114\pi$
$41$	−10.0341	−	6.70458i	−1.56707	−	1.04708i	−0.597619	−	0.801781i	$-0.703886\pi$
$42$		0			0
$43$	−6.05828	−	2.50942i	−0.923880	−	0.382683i
$43$	−6.05828	−	2.50942i	−0.923880	−	0.382683i
$44$	−4.97189	+	7.44097i	−0.749541	+	1.12177i
$45$		0			0
$46$		0			0
$47$	6.16199	+	6.16199i	0.898818	+	0.898818i	0.995332	−	0.0965136i	$-0.0307691\pi$
$47$	6.16199	+	6.16199i	0.898818	+	0.898818i	−0.0965136	+	0.995332i	$0.530769\pi$
$48$		0			0
$49$	2.67878	+	6.46716i	0.382683	+	0.923880i
$50$		0			0
$51$		0			0
$52$	−13.5163			−1.87437
$53$	10.1933	−	4.22221i	1.40016	−	0.579965i	0.450367	−	0.892844i	$-0.351293\pi$
$53$	10.1933	−	4.22221i	1.40016	−	0.579965i	0.949793	+	0.312878i	$0.101293\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.46524	−	8.36582i	0.451135	−	1.08914i	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	3.46524	−	8.36582i	0.451135	−	1.08914i	0.971891	−	0.235431i	$-0.0756503\pi$
$60$		0			0
$61$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$61$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$62$		0			0
$63$		0			0
$64$	−5.65685	−	5.65685i	−0.707107	−	0.707107i
$65$		0			0
$66$		0			0
$67$		−	11.3488i		−	1.38647i	−0.720710	−	0.693236i	$-0.756184\pi$
$67$		−	11.3488i		−	1.38647i	0.720710	−	0.693236i	$-0.243816\pi$
$68$	−7.97170	−	2.10997i	−0.966711	−	0.255872i
$69$		0			0
$70$		0			0
$71$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$71$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$72$		0			0
$73$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$73$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	9.03385	−	1.79695i	1.01639	−	0.202172i	0.341335	−	0.939942i	$-0.389121\pi$
$79$	9.03385	−	1.79695i	1.01639	−	0.202172i	0.675053	+	0.737769i	$0.264121\pi$
$80$		0			0
$81$	6.36396	+	6.36396i	0.707107	+	0.707107i
$82$		0			0
$83$	−2.50942	−	6.05828i	−0.275445	−	0.664983i	0.724254	−	0.689534i	$-0.242184\pi$
$83$	−2.50942	−	6.05828i	−0.275445	−	0.664983i	−0.999699	+	0.0245507i	$0.992184\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$89$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$90$		0			0
$91$		0			0
$92$	12.9661	+	8.66364i	1.35180	+	0.903247i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.58732	+	2.37559i	0.161168	+	0.241204i	0.903260	−	0.429093i	$-0.141167\pi$
$97$	1.58732	+	2.37559i	0.161168	+	0.241204i	−0.742093	+	0.670297i	$0.766167\pi$
$98$		0			0
$99$	13.1658	+	2.61885i	1.32322	+	0.263204i
$100$	−3.82683	−	9.23880i	−0.382683	−	0.923880i
$101$			20.0631i			1.99636i	0.0603342	+	0.998178i	$0.480783\pi$
$101$			20.0631i			1.99636i	−0.0603342	+	0.998178i	$0.519217\pi$
$102$		0			0
$103$	−20.0883			−1.97936			−0.989678	−	0.143310i	$-0.954225\pi$
$103$	−20.0883			−1.97936			−0.989678	+	0.143310i	$0.954225\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.67583	−	1.11975i	0.162008	−	0.108250i	−0.471923	−	0.881640i	$-0.656440\pi$
$107$	1.67583	−	1.11975i	0.162008	−	0.108250i	0.633932	+	0.773389i	$0.281440\pi$
$108$		0			0
$109$	−10.3667	−	6.92682i	−0.992951	−	0.663469i	−0.0508181	−	0.998708i	$-0.516183\pi$
$109$	−10.3667	−	6.92682i	−0.992951	−	0.663469i	−0.942133	+	0.335239i	$0.891183\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$113$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	7.75867	+	18.7311i	0.717290	+	1.73169i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.33518	−	3.45254i	0.757743	−	0.313868i
$122$		0			0
$123$		0			0
$124$	17.3818	−	11.6141i	1.56093	−	1.04298i
$125$		0			0
$126$		0			0
$127$	−1.00814	+	2.43387i	−0.0894583	+	0.215971i	−0.962276	−	0.272075i	$-0.912290\pi$
$127$	−1.00814	+	2.43387i	−0.0894583	+	0.215971i	0.872818	+	0.488046i	$0.162290\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$131$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$	3.26935	−	16.4361i	0.277303	−	1.39410i	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	3.26935	−	16.4361i	0.277303	−	1.39410i	0.828626	−	0.559803i	$-0.189123\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	25.1435	+	16.8004i	2.10261	+	1.40492i
$144$	−4.59220	+	11.0866i	−0.382683	+	0.923880i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$		0			0
$151$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$151$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$152$		0			0
$153$	1.65191	+	12.2585i	0.133549	+	0.991042i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$163$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$164$	−23.6721	+	4.70867i	−1.84848	+	0.367686i
$165$		0			0
$166$		0			0
$167$	−8.27483	−	1.64597i	−0.640326	−	0.127369i	−0.135760	−	0.990742i	$-0.543347\pi$
$167$	−8.27483	−	1.64597i	−0.640326	−	0.127369i	−0.504566	+	0.863373i	$0.668347\pi$
$168$		0			0
$169$			32.6723i			2.51326i
$170$		0			0
$171$		0			0
$172$	−12.1166	+	5.01885i	−0.923880	+	0.382683i
$173$	4.47155	−	22.4800i	0.339965	−	1.70912i	−0.311335	−	0.950300i	$-0.600776\pi$
$173$	4.47155	−	22.4800i	0.339965	−	1.70912i	0.651300	−	0.758820i	$-0.274224\pi$
$174$		0			0
$175$		0			0
$176$	3.49179	+	17.5544i	0.263204	+	1.32322i
$177$		0			0
$178$		0			0
$179$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$179$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$180$		0			0
$181$	24.0834	−	4.79048i	1.79010	−	0.356074i	0.815283	−	0.579062i	$-0.196581\pi$
$181$	24.0834	−	4.79048i	1.79010	−	0.356074i	0.974821	+	0.222988i	$0.0715812\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	12.2066	+	13.8337i	0.892638	+	1.01162i
$188$	17.4287			1.27112
$189$		0			0
$190$		0			0
$191$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$191$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$192$		0			0
$193$	0.227085	+	1.14163i	0.0163460	+	0.0821767i	0.988097	−	0.153831i	$-0.0491610\pi$
$193$	0.227085	+	1.14163i	0.0163460	+	0.0821767i	−0.971751	+	0.236007i	$0.924161\pi$
$194$		0			0
$195$		0			0
$196$	12.9343	+	5.35757i	0.923880	+	0.382683i
$197$	−8.29190	+	12.4097i	−0.590773	+	0.884155i	−0.999595	−	0.0284595i	$-0.990940\pi$
$197$	−8.29190	+	12.4097i	−0.590773	+	0.884155i	0.408822	+	0.912614i	$0.365940\pi$
$198$		0			0
$199$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$199$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	4.56340	−	22.9418i	0.317178	−	1.59456i
$208$	−19.1149	+	19.1149i	−1.32538	+	1.32538i
$209$		0			0
$210$		0			0
$211$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$211$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$212$	8.44442	−	20.3866i	0.579965	−	1.40016i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−7.12974	+	26.9369i	−0.479598	+	1.81197i
$222$		0			0
$223$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$223$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$224$		0			0
$225$	−10.6066	+	10.6066i	−0.707107	+	0.707107i
$226$		0			0
$227$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$227$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$228$		0			0
$229$	11.0127	−	26.5869i	0.727738	−	1.75691i	0.0777462	−	0.996973i	$-0.475228\pi$
$229$	11.0127	−	26.5869i	0.727738	−	1.75691i	0.649992	−	0.759941i	$-0.274772\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$233$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$234$		0			0
$235$		0			0
$236$	−6.93047	−	16.7316i	−0.451135	−	1.08914i
$237$		0			0
$238$		0			0
$239$	10.5254			0.680830			0.340415	−	0.940275i	$-0.389432\pi$
$239$	10.5254			0.680830			0.340415	+	0.940275i	$0.389432\pi$
$240$		0			0
$241$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$241$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.2213	+	12.2213i	0.771400	+	0.771400i	0.978351	−	0.206951i	$-0.0663540\pi$
$251$	12.2213	+	12.2213i	0.771400	+	0.771400i	−0.206951	+	0.978351i	$0.566354\pi$
$252$		0			0
$253$	−13.3513	−	32.2329i	−0.839390	−	2.02647i
$254$		0			0
$255$		0			0
$256$	−16.0000			−1.00000
$257$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$257$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$263$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−16.0496	−	16.0496i	−0.980384	−	0.980384i
$269$	31.7306	+	6.31162i	1.93465	+	0.384826i	0.999535	+	0.0304855i	$0.00970535\pi$
$269$	31.7306	+	6.31162i	1.93465	+	0.384826i	0.935116	+	0.354341i	$0.115295\pi$
$270$		0			0
$271$		−	29.1434i		−	1.77033i	−0.465274	−	0.885167i	$-0.654044\pi$
$271$		−	29.1434i		−	1.77033i	0.465274	−	0.885167i	$-0.345956\pi$
$272$	−14.2576	+	8.28973i	−0.864496	+	0.502639i
$273$		0			0
$274$		0			0
$275$	−4.36474	+	21.9430i	−0.263204	+	1.32322i
$276$		0			0
$277$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$277$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$278$		0			0
$279$	−26.0727	−	17.4212i	−1.56093	−	1.04298i
$280$		0			0
$281$	5.61715	+	2.32670i	0.335091	+	0.138799i	0.543884	−	0.839161i	$-0.316953\pi$
$281$	5.61715	+	2.32670i	0.335091	+	0.138799i	−0.208792	+	0.977960i	$0.566953\pi$
$282$		0			0
$283$	23.3382	−	4.64225i	1.38731	−	0.275953i	0.555732	−	0.831362i	$-0.312438\pi$
$283$	23.3382	−	4.64225i	1.38731	−	0.275953i	0.831578	+	0.555409i	$0.187438\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.41004	+	14.7740i	−0.494708	+	0.869059i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.11488	+	2.11488i	−0.123552	+	0.123552i	−0.766179	−	0.642627i	$-0.777845\pi$
$293$	−2.11488	+	2.11488i	−0.123552	+	0.123552i	0.642627	+	0.766179i	$0.277845\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	29.2750	−	43.8131i	1.69302	−	2.53378i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−7.07703			−0.403907			−0.201954	−	0.979395i	$-0.564729\pi$
$307$	−7.07703			−0.403907			−0.201954	+	0.979395i	$0.564729\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.28295	−	3.52995i	0.299568	−	0.200165i	−0.396697	−	0.917950i	$-0.629844\pi$
$311$	5.28295	−	3.52995i	0.299568	−	0.200165i	0.696265	+	0.717784i	$0.254844\pi$
$312$		0			0
$313$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$313$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$314$		0			0
$315$		0			0
$316$	10.2345	−	15.3171i	0.575738	−	0.861652i
$317$	−32.1154	+	6.38815i	−1.80378	+	0.358794i	−0.978551	−	0.206005i	$-0.933954\pi$
$317$	−32.1154	+	6.38815i	−1.80378	+	0.358794i	−0.825228	+	0.564799i	$0.808954\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	18.0000			1.00000
$325$	−31.2185	+	12.9311i	−1.73169	+	0.717290i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$331$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$332$	−12.1166	−	5.01885i	−0.664983	−	0.275445i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−14.5609	−	2.89635i	−0.793185	−	0.157774i	−0.218163	−	0.975912i	$-0.570006\pi$
$337$	−14.5609	−	2.89635i	−0.793185	−	0.157774i	−0.575022	+	0.818138i	$0.695006\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−46.7704			−2.53276
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$347$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$348$		0			0
$349$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$349$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−21.7787	−	21.7787i	−1.15916	−	1.15916i	−0.984656	−	0.174509i	$-0.944166\pi$
$353$	−21.7787	−	21.7787i	−1.15916	−	1.15916i	−0.174509	−	0.984656i	$-0.555834\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.00692	+	3.73079i	−0.475367	+	0.196903i	−0.607486	−	0.794330i	$-0.707822\pi$
$359$	−9.00692	+	3.73079i	−0.475367	+	0.196903i	0.132119	+	0.991234i	$0.457822\pi$
$360$		0			0
$361$	13.4350	−	13.4350i	0.707107	−	0.707107i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−20.6778	+	30.9466i	−1.07937	+	1.61540i	−0.342296	+	0.939592i	$0.611204\pi$
$367$	−20.6778	+	30.9466i	−1.07937	+	1.61540i	−0.737079	+	0.675806i	$0.763796\pi$
$368$	30.5890	−	6.08453i	1.59456	−	0.317178i
$369$	20.1137	+	30.1024i	1.04708	+	1.56707i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−22.4438	+	14.9965i	−1.15286	+	0.770318i	−0.976819	−	0.214065i	$-0.931330\pi$
$379$	−22.4438	+	14.9965i	−1.15286	+	0.770318i	−0.176042	+	0.984383i	$0.556330\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$383$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	13.9104	+	13.9104i	0.707107	+	0.707107i
$388$	5.60439	+	1.11478i	0.284520	+	0.0565945i
$389$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$389$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$390$		0			0
$391$	24.1055	−	21.2704i	1.21907	−	1.07569i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	22.3229	−	14.9157i	1.12177	−	0.749541i
$397$	−2.02340	−	10.1723i	−0.101552	−	0.510535i	−0.997760	−	0.0669005i	$-0.978689\pi$
$397$	−2.02340	−	10.1723i	−0.101552	−	0.510535i	0.896208	−	0.443634i	$-0.146311\pi$
$398$		0			0
$399$		0			0
$400$	−18.4776	−	7.65367i	−0.923880	−	0.382683i
$401$	8.04657	−	12.0425i	0.401826	−	0.601376i	−0.574283	−	0.818657i	$-0.694719\pi$
$401$	8.04657	−	12.0425i	0.401826	−	0.601376i	0.976109	+	0.217281i	$0.0697189\pi$
$402$		0			0
$403$	−39.2449	−	58.7342i	−1.95493	−	2.92576i
$404$	28.3736	+	28.3736i	1.41164	+	1.41164i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	−28.4091	+	28.4091i	−1.39962	+	1.39962i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$419$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$420$		0			0
$421$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$421$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$422$		0			0
$423$	−10.0045	−	24.1531i	−0.486437	−	1.17436i
$424$		0			0
$425$	−20.4309	+	2.75319i	−0.991042	+	0.133549i
$426$		0			0
$427$		0			0
$428$	0.786408	−	3.95354i	0.0380125	−	0.191102i
$429$		0			0
$430$		0			0
$431$	2.81518	+	14.1529i	0.135603	+	0.681720i	0.987450	+	0.157930i	$0.0504821\pi$
$431$	2.81518	+	14.1529i	0.135603	+	0.681720i	−0.851848	+	0.523789i	$0.824518\pi$
$432$		0			0
$433$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$433$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$434$		0			0
$435$		0			0
$436$	−24.4567	+	4.86475i	−1.17127	+	0.232979i
$437$		0			0
$438$		0			0
$439$	40.8926	+	8.13404i	1.95170	+	0.388217i	0.995398	+	0.0958262i	$0.0305493\pi$
$439$	40.8926	+	8.13404i	1.95170	+	0.388217i	0.956299	+	0.292391i	$0.0944507\pi$
$440$		0			0
$441$		−	21.0000i		−	1.00000i
$442$		0			0
$443$	−37.5579			−1.78443			−0.892215	−	0.451612i	$-0.850849\pi$
$443$	−37.5579			−1.78443			−0.892215	+	0.451612i	$0.850849\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$449$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$450$		0			0
$451$	49.8886	+	20.6645i	2.34916	+	0.973055i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$457$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−24.2331	+	10.0377i	−1.12865	+	0.467502i	−0.867322	−	0.497748i	$-0.834160\pi$
$461$	−24.2331	+	10.0377i	−1.12865	+	0.467502i	−0.261328	+	0.965250i	$0.584160\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$467$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$468$	37.4622	+	15.5173i	1.73169	+	0.717290i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	28.7780	+	5.72431i	1.32322	+	0.263204i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−33.0995			−1.51552
$478$		0			0
$479$	1.15045	−	5.78372i	0.0525656	−	0.264265i	−0.945561	−	0.325444i	$-0.894486\pi$
$479$	1.15045	−	5.78372i	0.0525656	−	0.264265i	0.998127	+	0.0611793i	$0.0194862\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	6.90509	−	16.6704i	0.313868	−	0.757743i
$485$		0			0
$486$		0			0
$487$	−42.9832	+	8.54989i	−1.94775	+	0.387432i	−0.950762	+	0.309921i	$0.899697\pi$
$487$	−42.9832	+	8.54989i	−1.94775	+	0.387432i	−0.996991	+	0.0775113i	$0.975303\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$491$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	8.15668	−	41.0064i	0.366246	−	1.84124i
$497$		0			0
$498$		0			0
$499$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$499$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$503$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	2.01629	+	4.86775i	0.0894583	+	0.215971i
$509$		−	42.1546i		−	1.86847i	−0.356658	−	0.934235i	$-0.616084\pi$
$509$		−	42.1546i		−	1.86847i	0.356658	−	0.934235i	$-0.383916\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−32.4216	−	21.6635i	−1.42590	−	0.952758i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$521$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−13.9773	−	40.7670i	−0.608862	−	1.77584i
$528$		0			0
$529$	−34.9174	+	14.4633i	−1.51815	+	0.628837i
$530$		0			0
$531$	−19.2088	+	19.2088i	−0.833589	+	0.833589i
$532$		0			0
$533$	15.9109	+	79.9896i	0.689178	+	3.46473i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−17.4016	−	26.0434i	−0.749541	−	1.12177i
$540$		0			0
$541$	25.2250	+	5.01757i	1.08451	+	0.215722i	0.704816	−	0.709390i	$-0.251030\pi$
$541$	25.2250	+	5.01757i	1.08451	+	0.215722i	0.379693	+	0.925113i	$0.376030\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	3.46912	−	17.4404i	0.148329	−	0.745699i	−0.832987	−	0.553293i	$-0.813371\pi$
$547$	3.46912	−	17.4404i	0.148329	−	0.745699i	0.981315	−	0.192406i	$-0.0616291\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−18.6207	−	27.8678i	−0.789692	−	1.18186i
$557$	27.0104	+	27.0104i	1.14447	+	1.14447i	0.987623	+	0.156844i	$0.0501319\pi$
$557$	27.0104	+	27.0104i	1.14447	+	1.14447i	0.156844	+	0.987623i	$0.449868\pi$
$558$		0			0
$559$	16.9590	+	40.9427i	0.717290	+	1.73169i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	22.2141	−	9.20140i	0.936214	−	0.387793i	0.138182	−	0.990407i	$-0.455874\pi$
$563$	22.2141	−	9.20140i	0.936214	−	0.387793i	0.798033	+	0.602614i	$0.205874\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8.90324	−	21.4943i	0.373243	−	0.901088i	−0.619953	−	0.784639i	$-0.712849\pi$
$569$	8.90324	−	21.4943i	0.373243	−	0.901088i	0.993197	−	0.116450i	$-0.0371515\pi$
$570$		0			0
$571$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$571$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$572$	59.3176	−	11.7990i	2.48019	−	0.493341i
$573$		0			0
$574$		0			0
$575$	38.2363	+	7.60567i	1.59456	+	0.317178i
$576$	9.18440	+	22.1731i	0.382683	+	0.923880i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−41.0487	+	27.4279i	−1.70006	+	1.13595i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$587$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$593$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	32.4577	−	32.4577i	1.32618	−	1.32618i	0.417514	−	0.908671i	$-0.362902\pi$
$599$	32.4577	−	32.4577i	1.32618	−	1.32618i	0.908671	−	0.417514i	$-0.137098\pi$
$600$		0			0
$601$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$601$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$602$		0			0
$603$	−13.0290	+	31.4547i	−0.530580	+	1.28093i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$607$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	58.8928i		−	2.38255i
$612$	19.6723	+	15.0000i	0.795206	+	0.606339i
$613$	48.8406			1.97265			0.986327	−	0.164798i	$-0.0526973\pi$
$613$	48.8406			1.97265			0.986327	+	0.164798i	$0.0526973\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−34.3508	+	22.9525i	−1.38291	+	0.924032i	−0.382911	+	0.923785i	$0.625078\pi$
$617$	−34.3508	+	22.9525i	−1.38291	+	0.924032i	−1.00000		0.000246592i	$0.999922\pi$
$618$		0			0
$619$	18.4813	+	12.3488i	0.742824	+	0.496339i	0.868474	−	0.495735i	$-0.165101\pi$
$619$	18.4813	+	12.3488i	0.742824	+	0.496339i	−0.125649	+	0.992075i	$0.540101\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−17.6777	−	17.6777i	−0.707107	−	0.707107i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$631$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	18.1036	−	43.7059i	0.717290	−	1.73169i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$641$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$642$		0			0
$643$	38.3103	+	7.62040i	1.51081	+	0.300519i	0.879834	−	0.475281i	$-0.157654\pi$
$643$	38.3103	+	7.62040i	1.51081	+	0.300519i	0.630978	+	0.775800i	$0.282654\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	−7.90464	+	39.7393i	−0.310284	+	1.55990i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$653$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$654$		0			0
$655$		0			0
$656$	−26.8183	+	40.1365i	−1.04708	+	1.56707i
$657$		0			0
$658$		0			0
$659$	36.2107	+	36.2107i	1.41057	+	1.41057i	0.756013	+	0.654557i	$0.227145\pi$
$659$	36.2107	+	36.2107i	1.41057	+	1.41057i	0.654557	+	0.756013i	$0.272855\pi$
$660$		0			0
$661$	−2.50942	−	6.05828i	−0.0976052	−	0.235640i	0.867533	−	0.497379i	$-0.165704\pi$
$661$	−2.50942	−	6.05828i	−0.0976052	−	0.235640i	−0.965139	+	0.261739i	$0.915704\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−14.0301	+	9.37463i	−0.542842	+	0.362715i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$673$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$674$		0			0
$675$		0			0
$676$	46.2056	+	46.2056i	1.77714	+	1.77714i
$677$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$677$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3.03650	+	15.2655i	−0.116188	+	0.584119i	0.878197	+	0.478299i	$0.158747\pi$
$683$	−3.03650	+	15.2655i	−0.116188	+	0.584119i	−0.994385	+	0.105819i	$0.966253\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−10.0377	+	24.2331i	−0.382683	+	0.923880i
$689$	−68.8878	−	28.5342i	−2.62441	−	1.08707i
$690$		0			0
$691$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$691$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$692$	−25.4678	−	38.1152i	−0.968139	−	1.44892i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−3.10285	+	49.6605i	−0.117529	+	1.88103i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−36.1149	+	36.1149i	−1.36404	+	1.36404i	−0.495342	+	0.868698i	$0.664957\pi$
$701$	−36.1149	+	36.1149i	−1.36404	+	1.36404i	−0.868698	+	0.495342i	$0.835043\pi$
$702$		0			0
$703$		0			0
$704$	29.7639	+	19.8876i	1.12177	+	0.749541i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.69568	−	2.53776i	−0.0636826	−	0.0953078i	0.798262	−	0.602311i	$-0.205753\pi$
$709$	−1.69568	−	2.53776i	−0.0636826	−	0.0953078i	−0.861944	+	0.507003i	$0.830753\pi$
$710$		0			0
$711$	−27.1016	−	5.39084i	−1.01639	−	0.202172i
$712$		0			0
$713$			81.4984i			3.05214i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	33.0877	−	22.1085i	1.23396	−	0.824508i	0.244551	−	0.969636i	$-0.421359\pi$
$719$	33.0877	−	22.1085i	1.23396	−	0.824508i	0.989413	+	0.145128i	$0.0463594\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$	27.2843	−	40.8338i	1.01401	−	1.51758i
$725$		0			0
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$	−10.3325	−	24.9447i	−0.382683	−	0.923880i
$730$		0			0
$731$	3.61077	+	26.7948i	0.133549	+	0.991042i
$732$		0			0
$733$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$733$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9.90689	+	49.8053i	0.364925	+	1.83460i
$738$		0			0
$739$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$739$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$743$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			19.6723i			0.719772i
$748$	36.8266	+	2.30097i	1.34651	+	0.0841318i
$749$		0			0
$750$		0			0
$751$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$751$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$752$	24.6479	−	24.6479i	0.898818	−	0.898818i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$757$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$761$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−56.5373	+	23.4185i	−2.04144	+	0.845594i
$768$		0			0
$769$	2.43637	−	2.43637i	0.0878579	−	0.0878579i	−0.661812	−	0.749670i	$-0.730212\pi$
$769$	2.43637	−	2.43637i	0.0878579	−	0.0878579i	0.749670	+	0.661812i	$0.230212\pi$
$770$		0			0
$771$		0			0
$772$	1.93566	+	1.29337i	0.0696660	+	0.0465493i
$773$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$773$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$774$		0			0
$775$	29.0353	−	43.4544i	1.04298	−	1.56093i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	25.8686	−	10.7151i	0.923880	−	0.382683i
$785$		0			0
$786$		0			0
$787$	45.8285	−	30.6216i	1.63361	−	1.09154i	0.712923	−	0.701242i	$-0.247371\pi$
$787$	45.8285	−	30.6216i	1.63361	−	1.09154i	0.920687	−	0.390301i	$-0.127629\pi$
$788$	5.82346	+	29.2765i	0.207452	+	1.04293i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.95129	−	4.71083i	−0.0691182	−	0.166866i	0.885545	−	0.464553i	$-0.153785\pi$
$797$	−1.95129	−	4.71083i	−0.0691182	−	0.166866i	−0.954664	+	0.297687i	$0.903785\pi$
$798$		0			0
$799$	9.19354	−	34.7342i	0.325244	−	1.22881i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	30.9694	−	46.3490i	1.08883	−	1.62955i	0.378620	−	0.925552i	$-0.376399\pi$
$809$	30.9694	−	46.3490i	1.08883	−	1.62955i	0.710207	−	0.703993i	$-0.248601\pi$
$810$		0			0
$811$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$811$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	33.0097	−	22.0564i	1.15205	−	0.769773i	0.175373	−	0.984502i	$-0.443887\pi$
$821$	33.0097	−	22.0564i	1.15205	−	0.769773i	0.976674	+	0.214729i	$0.0688868\pi$
$822$		0			0
$823$	45.4194	+	30.3483i	1.58322	+	1.05787i	0.961562	+	0.274587i	$0.0885410\pi$
$823$	45.4194	+	30.3483i	1.58322	+	1.05787i	0.621659	+	0.783288i	$0.286459\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−56.3780	+	11.2143i	−1.96046	+	0.389959i	−0.973655	+	0.228024i	$0.926773\pi$
$827$	−56.3780	+	11.2143i	−1.96046	+	0.389959i	−0.986801	+	0.161935i	$0.948227\pi$
$828$	−25.9909	−	38.8982i	−0.903247	−	1.35180i
$829$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$829$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$830$		0			0
$831$		0			0
$832$			54.0650i			1.87437i
$833$	17.5000	−	22.9510i	0.606339	−	0.795206i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$839$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$840$		0			0
$841$	−11.0978	+	26.7925i	−0.382683	+	0.923880i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−16.8888	−	40.7733i	−0.579965	−	1.40016i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	5.85745	−	29.4474i	0.200555	−	1.00826i	−0.741027	−	0.671475i	$-0.765661\pi$
$853$	5.85745	−	29.4474i	0.200555	−	1.00826i	0.941582	−	0.336784i	$-0.109339\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.25825	+	4.18163i	0.213778	+	0.142842i	0.657849	−	0.753150i	$-0.271466\pi$
$857$	6.25825	+	4.18163i	0.213778	+	0.142842i	−0.444072	+	0.895991i	$0.646466\pi$
$858$		0			0
$859$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$859$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−38.0774	+	15.7722i	−1.29169	+	0.535035i
$870$		0			0
$871$	−54.2326	+	54.2326i	−1.83760	+	1.83760i
$872$		0			0
$873$	−1.67217	−	8.40658i	−0.0565945	−	0.284520i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−31.6161	+	47.3169i	−1.06760	+	1.59778i	−0.303324	+	0.952887i	$0.598097\pi$
$877$	−31.6161	+	47.3169i	−1.06760	+	1.59778i	−0.764276	+	0.644889i	$0.776903\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−26.8013	−	5.33110i	−0.902958	−	0.179609i	−0.278287	−	0.960498i	$-0.589767\pi$
$881$	−26.8013	−	5.33110i	−0.902958	−	0.179609i	−0.624670	+	0.780889i	$0.714767\pi$
$882$		0			0
$883$		−	36.7815i		−	1.23780i	−0.785471	−	0.618899i	$-0.787579\pi$
$883$		−	36.7815i		−	1.23780i	0.785471	−	0.618899i	$-0.212421\pi$
$884$	28.0115	+	48.1775i	0.942130	+	1.62038i
$885$		0			0
$886$		0			0
$887$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$887$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−33.4844	−	22.3735i	−1.12177	−	0.749541i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$			30.0000i			1.00000i
$901$	−36.1747	−	27.5829i	−1.20515	−	0.918920i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	3.21878	+	16.1819i	0.106878	+	0.537312i	0.996712	+	0.0810272i	$0.0258201\pi$
$907$	3.21878	+	16.1819i	0.106878	+	0.537312i	−0.889834	+	0.456284i	$0.849180\pi$
$908$		0			0
$909$	23.0335	−	55.6078i	0.763973	−	1.84439i
$910$		0			0
$911$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$911$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$912$		0			0
$913$	16.3014	+	24.3968i	0.539499	+	0.807418i
$914$		0			0
$915$		0			0
$916$	−22.0253	−	53.1739i	−0.727738	−	1.75691i
$917$		0			0
$918$		0			0
$919$	−32.7872			−1.08155			−0.540775	−	0.841167i	$-0.681869\pi$
$919$	−32.7872			−1.08155			−0.540775	+	0.841167i	$0.681869\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	55.6774	+	23.0623i	1.82869	+	0.757467i
$928$		0			0
$929$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$929$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$937$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.06696	−	5.36398i	−0.0347820	−	0.174861i	0.959490	−	0.281744i	$-0.0909129\pi$
$941$	−1.06696	−	5.36398i	−0.0347820	−	0.174861i	−0.994272	+	0.106883i	$0.965913\pi$
$942$		0			0
$943$	36.0084	−	86.9320i	1.17260	−	2.83090i
$944$	−33.4633	−	13.8609i	−1.08914	−	0.451135i
$945$		0			0
$946$		0			0
$947$	−33.1076	−	49.5490i	−1.07585	−	1.61013i	−0.745808	−	0.666160i	$-0.767937\pi$
$947$	−33.1076	−	49.5490i	−1.07585	−	1.61013i	−0.330044	−	0.943966i	$-0.607063\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	14.8851	−	14.8851i	0.481419	−	0.481419i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	72.2969	+	29.9463i	2.33216	+	0.966011i
$962$		0			0
$963$	−5.93031	+	1.17961i	−0.191102	+	0.0380125i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−10.6420	−	25.6920i	−0.342223	−	0.826201i	−0.997490	−	0.0708031i	$-0.977444\pi$
$967$	−10.6420	−	25.6920i	−0.342223	−	0.826201i	0.655267	−	0.755397i	$-0.272556\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	13.2047	−	5.46955i	0.423757	−	0.175526i	−0.160605	−	0.987019i	$-0.551345\pi$
$971$	13.2047	−	5.46955i	0.423757	−	0.175526i	0.584363	+	0.811493i	$0.301345\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−20.0754	+	48.4663i	−0.642268	+	1.55057i	0.181344	+	0.983420i	$0.441955\pi$
$977$	−20.0754	+	48.4663i	−0.642268	+	1.55057i	−0.823612	+	0.567153i	$0.808045\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	20.7805	+	31.1001i	0.663469	+	0.992951i
$982$		0			0
$983$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$983$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	9.97474	−	50.1464i	0.317178	−	1.59456i
$990$		0			0
$991$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$991$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$997$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.s.a.558.1	yes	16
17.11	odd	16	inner	731.2.s.a.300.1	✓	16
43.42	odd	2	CM	731.2.s.a.558.1	yes	16
731.300	even	16	inner	731.2.s.a.300.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.a.300.1	✓	16	17.11	odd	16	inner
731.2.s.a.300.1	✓	16	731.300	even	16	inner
731.2.s.a.558.1	yes	16	1.1	even	1	trivial
731.2.s.a.558.1	yes	16	43.42	odd	2	CM

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:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.s (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(1.41421 - 1.41421i) q^{4} +(-2.77164 - 1.14805i) q^{9} +O(q^{10})\)	\(q+(1.41421 - 1.41421i) q^{4} +(-2.77164 - 1.14805i) q^{9} +(-4.38861 + 0.872949i) q^{11} +(-4.77872 - 4.77872i) q^{13} -4.00000i q^{16} +(-2.07243 - 3.56441i) q^{17} +(1.52113 + 7.64725i) q^{23} +(1.91342 - 4.61940i) q^{25} +(10.2516 + 2.03917i) q^{31} +(-5.54328 + 2.29610i) q^{36} +(-10.0341 - 6.70458i) q^{41} +(-6.05828 - 2.50942i) q^{43} +(-4.97189 + 7.44097i) q^{44} +(6.16199 + 6.16199i) q^{47} +(2.67878 + 6.46716i) q^{49} -13.5163 q^{52} +(10.1933 - 4.22221i) q^{53} +(3.46524 - 8.36582i) q^{59} +(-5.65685 - 5.65685i) q^{64} -11.3488i q^{67} +(-7.97170 - 2.10997i) q^{68} +(9.03385 - 1.79695i) q^{79} +(6.36396 + 6.36396i) q^{81} +(-2.50942 - 6.05828i) q^{83} +(12.9661 + 8.66364i) q^{92} +(1.58732 + 2.37559i) q^{97} +(13.1658 + 2.61885i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 24 q^{13} + 56 q^{23} - 64 q^{59} + 96 q^{79}+O(q^{100}) \) 16 * q - 24 * q^13 + 56 * q^23 - 64 * q^59 + 96 * q^79

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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