LMFDB - Embedded newform 111.2.q.a.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [111,2,Mod(2,111)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(111, base_ring=CyclotomicField(36))

chi = DirichletCharacter(H, H._module([18, 1]))

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

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

chi := DirichletCharacter("111.2");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 111 = 3 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	111.q (of order $36$, degree $12$, 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:	$0.886339462436$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{36})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{6} + 1 $ x^12 - x^6 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{36}]$

Embedding invariants

Embedding label			17.1
Root			$0.342020 + 0.939693i$ of defining polynomial
Character	$\chi$	$=$	111.17
Dual form			111.2.q.a.98.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.70574 - 0.300767i) q^{3} +(0.684040 + 1.87939i) q^{4} +(3.54163 + 2.97178i) q^{7} +(2.81908 + 1.02606i) q^{9} +O(q^{10})$	$q+(-1.70574 - 0.300767i) q^{3} +(0.684040 + 1.87939i) q^{4} +(3.54163 + 2.97178i) q^{7} +(2.81908 + 1.02606i) q^{9} +(-0.601535 - 3.41147i) q^{12} +(-4.92474 - 2.29644i) q^{13} +(-3.06418 + 2.57115i) q^{16} +(5.59549 - 3.91800i) q^{19} +(-5.14728 - 6.13429i) q^{21} +(-4.92404 + 0.868241i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(-3.16250 + 8.68891i) q^{28} +(7.07399 - 7.07399i) q^{31} +6.00000i q^{36} +(0.500000 - 6.06218i) q^{37} +(7.70961 + 5.39833i) q^{39} +(-0.418915 - 0.418915i) q^{43} +(6.00000 - 3.46410i) q^{48} +(2.49613 + 14.1563i) q^{49} +(0.947182 - 10.8263i) q^{52} +(-10.7228 + 5.00014i) q^{57} +(-2.11331 + 4.53201i) q^{61} +(6.93491 + 12.0116i) q^{63} +(-6.92820 - 4.00000i) q^{64} +(-0.406951 + 0.484985i) q^{67} -15.3824i q^{73} +8.66025 q^{75} +(11.1910 + 7.83601i) q^{76} +(1.03662 + 11.8486i) q^{79} +(6.89440 + 5.78509i) q^{81} +(8.00774 - 13.8698i) q^{84} +(-10.6171 - 22.7684i) q^{91} +(-14.1940 + 9.93874i) q^{93} +(-18.7543 + 5.02519i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 54 q^{27} - 48 q^{28} + 6 q^{37} + 18 q^{39} + 72 q^{48} + 78 q^{49} - 114 q^{91}+O(q^{100}) $ 12 * q - 54 * q^27 - 48 * q^28 + 6 * q^37 + 18 * q^39 + 72 * q^48 + 78 * q^49 - 114 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/111\mathbb{Z}\right)^\times$.

$n$	$38$	$76$
$\chi(n)$	$-1$	$e\left(\frac{7}{36}\right)$

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		−0.819152	−	0.573576i	$-0.805556\pi$
$2$		0			0		0.819152	+	0.573576i	$0.194444\pi$
$3$	−1.70574	−	0.300767i	−0.984808	−	0.173648i
$3$	−1.70574	−	0.300767i	−0.984808	−	0.173648i
$4$	0.684040	+	1.87939i	0.342020	+	0.939693i
$5$		0			0		−0.0871557	−	0.996195i	$-0.527778\pi$
$5$		0			0		0.0871557	+	0.996195i	$0.472222\pi$
$6$		0			0
$7$	3.54163	+	2.97178i	1.33861	+	1.12323i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	3.54163	+	2.97178i	1.33861	+	1.12323i	0.356630	+	0.934246i	$0.383926\pi$
$8$		0			0
$9$	2.81908	+	1.02606i	0.939693	+	0.342020i
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$	−0.601535	−	3.41147i	−0.173648	−	0.984808i
$13$	−4.92474	−	2.29644i	−1.36588	−	0.636919i	−0.405108	−	0.914269i	$-0.632766\pi$
$13$	−4.92474	−	2.29644i	−1.36588	−	0.636919i	−0.960769	+	0.277350i	$0.910544\pi$
$14$		0			0
$15$		0			0
$16$	−3.06418	+	2.57115i	−0.766044	+	0.642788i
$17$		0			0		−0.422618	−	0.906308i	$-0.638889\pi$
$17$		0			0		0.422618	+	0.906308i	$0.361111\pi$
$18$		0			0
$19$	5.59549	−	3.91800i	1.28369	−	0.898852i	0.285467	−	0.958388i	$-0.407851\pi$
$19$	5.59549	−	3.91800i	1.28369	−	0.898852i	0.998226	+	0.0595368i	$0.0189624\pi$
$20$		0			0
$21$	−5.14728	−	6.13429i	−1.12323	−	1.33861i
$22$		0			0
$23$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$23$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$24$		0			0
$25$	−4.92404	+	0.868241i	−0.984808	+	0.173648i
$26$		0			0
$27$	−4.50000	−	2.59808i	−0.866025	−	0.500000i
$28$	−3.16250	+	8.68891i	−0.597657	+	1.64205i
$29$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$29$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$30$		0			0
$31$	7.07399	−	7.07399i	1.27053	−	1.27053i	0.324714	−	0.945812i	$-0.394732\pi$
$31$	7.07399	−	7.07399i	1.27053	−	1.27053i	0.945812	−	0.324714i	$-0.105268\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$			6.00000i			1.00000i
$37$	0.500000	−	6.06218i	0.0821995	−	0.996616i
$37$	0.500000	−	6.06218i	0.0821995	−	0.996616i
$38$		0			0
$39$	7.70961	+	5.39833i	1.23453	+	0.864425i
$40$		0			0
$41$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$41$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$42$		0			0
$43$	−0.418915	−	0.418915i	−0.0638839	−	0.0638839i	0.674443	−	0.738327i	$-0.264384\pi$
$43$	−0.418915	−	0.418915i	−0.0638839	−	0.0638839i	−0.738327	+	0.674443i	$0.764384\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	6.00000	−	3.46410i	0.866025	−	0.500000i
$49$	2.49613	+	14.1563i	0.356590	+	2.02232i
$50$		0			0
$51$		0			0
$52$	0.947182	−	10.8263i	0.131351	−	1.50134i
$53$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$53$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−10.7228	+	5.00014i	−1.42028	+	0.662285i
$58$		0			0
$59$		0			0		−0.996195	−	0.0871557i	$-0.972222\pi$
$59$		0			0		0.996195	+	0.0871557i	$0.0277778\pi$
$60$		0			0
$61$	−2.11331	+	4.53201i	−0.270582	+	0.580264i	−0.993904	−	0.110246i	$-0.964836\pi$
$61$	−2.11331	+	4.53201i	−0.270582	+	0.580264i	0.723323	+	0.690510i	$0.242614\pi$
$62$		0			0
$63$	6.93491	+	12.0116i	0.873716	+	1.51332i
$64$	−6.92820	−	4.00000i	−0.866025	−	0.500000i
$65$		0			0
$66$		0			0
$67$	−0.406951	+	0.484985i	−0.0497170	+	0.0592504i	−0.790330	−	0.612682i	$-0.790091\pi$
$67$	−0.406951	+	0.484985i	−0.0497170	+	0.0592504i	0.740613	+	0.671932i	$0.234535\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$71$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$72$		0			0
$73$		−	15.3824i		−	1.80037i	−0.435506	−	0.900186i	$-0.643431\pi$
$73$		−	15.3824i		−	1.80037i	0.435506	−	0.900186i	$-0.356569\pi$
$74$		0			0
$75$	8.66025			1.00000
$76$	11.1910	+	7.83601i	1.28369	+	0.898852i
$77$		0			0
$78$		0			0
$79$	1.03662	+	11.8486i	0.116628	+	1.33307i	0.798677	+	0.601760i	$0.205534\pi$
$79$	1.03662	+	11.8486i	0.116628	+	1.33307i	−0.682048	+	0.731307i	$0.738911\pi$
$80$		0			0
$81$	6.89440	+	5.78509i	0.766044	+	0.642788i
$82$		0			0
$83$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$83$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$84$	8.00774	−	13.8698i	0.873716	−	1.51332i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.0871557	−	0.996195i	$-0.472222\pi$
$89$		0			0		−0.0871557	+	0.996195i	$0.527778\pi$
$90$		0			0
$91$	−10.6171	−	22.7684i	−1.11297	−	2.38678i
$92$		0			0
$93$	−14.1940	+	9.93874i	−1.47185	+	1.03060i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−18.7543	+	5.02519i	−1.90421	+	0.510231i	−0.908474	+	0.417941i	$0.862752\pi$
$97$	−18.7543	+	5.02519i	−1.90421	+	0.510231i	−0.995732	+	0.0922897i	$0.970581\pi$
$98$		0			0
$99$		0			0
$100$	−5.00000	−	8.66025i	−0.500000	−	0.866025i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−1.76795	−	0.473721i	−0.174201	−	0.0466771i	0.170664	−	0.985329i	$-0.445409\pi$
$103$	−1.76795	−	0.473721i	−0.174201	−	0.0466771i	−0.344865	+	0.938652i	$0.612075\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$107$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$108$	1.80460	−	10.2344i	0.173648	−	0.984808i
$109$	6.07794	−	8.68020i	0.582161	−	0.831413i	−0.414751	−	0.909935i	$-0.636131\pi$
$109$	6.07794	−	8.68020i	0.582161	−	0.831413i	0.996912	+	0.0785223i	$0.0250202\pi$
$110$		0			0
$111$	−2.67617	+	10.1901i	−0.254011	+	0.967201i
$112$	−18.4931			−1.74743
$113$		0			0		−0.819152	−	0.573576i	$-0.805556\pi$
$113$		0			0		0.819152	+	0.573576i	$0.194444\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−11.5269	−	11.5269i	−1.06567	−	1.06567i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	18.1337	+	8.45586i	1.62845	+	0.759359i
$125$		0			0
$126$		0			0
$127$	−6.61927	+	5.55423i	−0.587365	+	0.492858i	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−6.61927	+	5.55423i	−0.587365	+	0.492858i	0.299991	+	0.953942i	$0.403016\pi$
$128$		0			0
$129$	0.588563	+	0.840555i	0.0518201	+	0.0740067i
$130$		0			0
$131$		0			0		0.906308	−	0.422618i	$-0.138889\pi$
$131$		0			0		−0.906308	+	0.422618i	$0.861111\pi$
$132$		0			0
$133$	31.4606	+	2.75245i	2.72798	+	0.238667i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$	−7.99989	+	21.9795i	−0.678542	+	1.86428i	−0.220366	+	0.975417i	$0.570725\pi$
$139$	−7.99989	+	21.9795i	−0.678542	+	1.86428i	−0.458176	+	0.888861i	$0.651497\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−11.2763	+	4.10424i	−0.939693	+	0.342020i
$145$		0			0
$146$		0			0
$147$		−	24.8976i		−	2.05352i
$148$	11.7352	−	3.20708i	0.964626	−	0.263620i
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	8.02616	+	1.41523i	0.653160	+	0.115170i	0.490402	−	0.871496i	$-0.336850\pi$
$151$	8.02616	+	1.41523i	0.653160	+	0.115170i	0.162758	+	0.986666i	$0.447961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−4.87185	+	18.1820i	−0.390061	+	1.45573i
$157$	16.3474	+	5.94996i	1.30466	+	0.474858i	0.898513	−	0.438948i	$-0.144649\pi$
$157$	16.3474	+	5.94996i	1.30466	+	0.474858i	0.406150	+	0.913806i	$0.366871\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−2.22186	+	25.3960i	−0.174030	+	1.98917i	−0.0303908	+	0.999538i	$0.509675\pi$
$163$	−2.22186	+	25.3960i	−0.174030	+	1.98917i	−0.143639	+	0.989630i	$0.545880\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.819152	−	0.573576i	$-0.194444\pi$
$167$		0			0		−0.819152	+	0.573576i	$0.805556\pi$
$168$		0			0
$169$	10.6232	+	12.6602i	0.817166	+	0.973861i
$170$		0			0
$171$	19.7942	−	5.30385i	1.51370	−	0.405595i
$172$	0.500748	−	1.07386i	0.0381817	−	0.0818808i
$173$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$173$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$174$		0			0
$175$	−20.0194	−	11.5582i	−1.51332	−	0.873716i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$179$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$180$		0			0
$181$	−20.6918	+	7.53121i	−1.53801	+	0.559791i	−0.965567	−	0.260153i	$-0.916227\pi$
$181$	−20.6918	+	7.53121i	−1.53801	+	0.559791i	−0.572444	+	0.819943i	$0.694005\pi$
$182$		0			0
$183$	4.96783	−	7.09480i	0.367233	−	0.524462i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−8.21643	−	22.5744i	−0.597657	−	1.64205i
$190$		0			0
$191$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$191$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$192$	10.6146	+	8.90673i	0.766044	+	0.642788i
$193$	−1.14080	+	4.25753i	−0.0821167	+	0.306464i	−0.994753	−	0.102310i	$-0.967377\pi$
$193$	−1.14080	+	4.25753i	−0.0821167	+	0.306464i	0.912636	+	0.408773i	$0.134043\pi$
$194$		0			0
$195$		0			0
$196$	−24.8976	+	14.3746i	−1.77840	+	1.02676i
$197$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$197$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$198$		0			0
$199$	−7.29750	−	27.2346i	−0.517306	−	1.93061i	−0.293251	−	0.956036i	$-0.594737\pi$
$199$	−7.29750	−	27.2346i	−0.517306	−	1.93061i	−0.224055	−	0.974576i	$-0.571930\pi$
$200$		0			0
$201$	0.840019	−	0.704860i	0.0592504	−	0.0497170i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	20.9948	−	5.62553i	1.45573	−	0.390061i
$209$		0			0
$210$		0			0
$211$	3.37077	+	5.83834i	0.232053	+	0.401928i	0.958412	−	0.285388i	$-0.0921223\pi$
$211$	3.37077	+	5.83834i	0.232053	+	0.401928i	−0.726359	+	0.687315i	$0.758789\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	46.0758	−	4.03111i	3.12783	−	0.273650i
$218$		0			0
$219$	−4.62652	+	26.2383i	−0.312631	+	1.77302i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	22.7570			1.52392			0.761961	−	0.647623i	$-0.224237\pi$
$223$	22.7570			1.52392			0.761961	+	0.647623i	$0.224237\pi$
$224$		0			0
$225$	−14.7721	−	2.60472i	−0.984808	−	0.173648i
$226$		0			0
$227$		0			0		−0.0871557	−	0.996195i	$-0.527778\pi$
$227$		0			0		0.0871557	+	0.996195i	$0.472222\pi$
$228$	−16.7321	−	16.7321i	−1.10811	−	1.10811i
$229$	−12.7536	−	10.7015i	−0.842779	−	0.707175i	0.115408	−	0.993318i	$-0.463182\pi$
$229$	−12.7536	−	10.7015i	−0.842779	−	0.707175i	−0.958187	+	0.286143i	$0.907627\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$233$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.79547	−	20.5223i	0.116628	−	1.33307i
$238$		0			0
$239$		0			0		−0.422618	−	0.906308i	$-0.638889\pi$
$239$		0			0		0.422618	+	0.906308i	$0.361111\pi$
$240$		0			0
$241$	−23.6141	+	16.5348i	−1.52112	+	1.06510i	−0.547533	+	0.836784i	$0.684433\pi$
$241$	−23.6141	+	16.5348i	−1.52112	+	1.06510i	−0.973587	+	0.228316i	$0.926678\pi$
$242$		0			0
$243$	−10.0201	−	11.9415i	−0.642788	−	0.766044i
$244$	−9.96297	−	0.871647i	−0.637814	−	0.0558015i
$245$		0			0
$246$		0			0
$247$	−36.5538	+	6.44542i	−2.32586	+	0.410112i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$251$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$252$	−17.8307	+	21.2498i	−1.12323	+	1.33861i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	2.77837	−	15.7569i	0.173648	−	0.984808i
$257$		0			0		0.573576	−	0.819152i	$-0.305556\pi$
$257$		0			0		−0.573576	+	0.819152i	$0.694444\pi$
$258$		0			0
$259$	19.7863	−	19.9841i	1.22946	−	1.24175i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$263$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.18984	−	0.433068i	−0.0726813	−	0.0264538i
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	−5.42934	−	30.7913i	−0.329809	−	1.87044i	−0.473466	−	0.880812i	$-0.656997\pi$
$271$	−5.42934	−	30.7913i	−0.329809	−	1.87044i	0.143657	−	0.989628i	$-0.454114\pi$
$272$		0			0
$273$	11.2620	+	42.0302i	0.681604	+	2.54378i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−17.4904	−	24.9788i	−1.05089	−	1.50083i	−0.853906	−	0.520427i	$-0.825773\pi$
$277$	−17.4904	−	24.9788i	−1.05089	−	1.50083i	−0.196988	−	0.980406i	$-0.563116\pi$
$278$		0			0
$279$	27.2005	−	12.6838i	1.62845	−	0.759359i
$280$		0			0
$281$		0			0		−0.996195	−	0.0871557i	$-0.972222\pi$
$281$		0			0		0.996195	+	0.0871557i	$0.0277778\pi$
$282$		0			0
$283$	−4.15923	+	8.91949i	−0.247240	+	0.530209i	−0.990288	−	0.139030i	$-0.955602\pi$
$283$	−4.15923	+	8.91949i	−0.247240	+	0.530209i	0.743048	+	0.669238i	$0.233379\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−10.9274	+	13.0228i	−0.642788	+	0.766044i
$290$		0			0
$291$	33.5012	−	2.93098i	1.96388	−	0.171817i
$292$	28.9094	−	10.5222i	1.69180	−	0.615763i
$293$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$293$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	5.92396	+	16.2760i	0.342020	+	0.939693i
$301$	−0.238719	−	2.72857i	−0.0137595	−	0.157272i
$302$		0			0
$303$		0			0
$304$	−7.07180	+	26.3923i	−0.405595	+	1.51370i
$305$		0			0
$306$		0			0
$307$	−8.95740	+	5.17155i	−0.511226	+	0.295156i	−0.733337	−	0.679865i	$-0.762038\pi$
$307$	−8.95740	+	5.17155i	−0.511226	+	0.295156i	0.222112	+	0.975021i	$0.428705\pi$
$308$		0			0
$309$	2.87318	+	1.33978i	0.163449	+	0.0762177i
$310$		0			0
$311$		0			0		0.0871557	−	0.996195i	$-0.472222\pi$
$311$		0			0		−0.0871557	+	0.996195i	$0.527778\pi$
$312$		0			0
$313$	−3.47681	−	7.45604i	−0.196521	−	0.421440i	0.783210	−	0.621757i	$-0.213581\pi$
$313$	−3.47681	−	7.45604i	−0.196521	−	0.421440i	−0.979731	+	0.200316i	$0.935803\pi$
$314$		0			0
$315$		0			0
$316$	−21.5589	+	10.0531i	−1.21278	+	0.565531i
$317$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$317$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−6.15636	+	16.9145i	−0.342020	+	0.939693i
$325$	26.2435	+	7.03192i	1.45573	+	0.390061i
$326$		0			0
$327$	−12.9781	+	12.9781i	−0.717690	+	0.717690i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	16.7608	−	23.9368i	0.921255	−	1.31569i	−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	16.7608	−	23.9368i	0.921255	−	1.31569i	0.948737	−	0.316066i	$-0.102362\pi$
$332$		0			0
$333$	7.62970	−	16.5767i	0.418105	−	0.908399i
$334$		0			0
$335$		0			0
$336$	31.5443	+	5.56212i	1.72088	+	0.303438i
$337$	12.2750	+	33.7252i	0.668660	+	1.83713i	0.532476	+	0.846445i	$0.321262\pi$
$337$	12.2750	+	33.7252i	0.668660	+	1.83713i	0.136184	+	0.990684i	$0.456516\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−17.0475	+	29.5271i	−0.920477	+	1.59431i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$347$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$348$		0			0
$349$	−3.98048	+	3.34002i	−0.213070	+	0.178787i	−0.743076	−	0.669207i	$-0.766634\pi$
$349$	−3.98048	+	3.34002i	−0.213070	+	0.178787i	0.530006	+	0.847994i	$0.322190\pi$
$350$		0			0
$351$	16.1950	+	23.1288i	0.864425	+	1.23453i
$352$		0			0
$353$		0			0		0.906308	−	0.422618i	$-0.138889\pi$
$353$		0			0		−0.906308	+	0.422618i	$0.861111\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	9.46036	−	25.9921i	0.497914	−	1.36801i
$362$		0			0
$363$	−12.2467	+	14.5951i	−0.642788	+	0.766044i
$364$	35.5281	−	35.5281i	1.86218	−	1.86218i
$365$		0			0
$366$		0			0
$367$	−1.61040	+	9.13306i	−0.0840624	+	0.476742i	0.913493	+	0.406855i	$0.133375\pi$
$367$	−1.61040	+	9.13306i	−0.0840624	+	0.476742i	−0.997555	+	0.0698862i	$0.977736\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−28.3880	−	19.8775i	−1.47185	−	1.03060i
$373$	−24.2817	−	4.28153i	−1.25726	−	0.221689i	−0.494962	−	0.868914i	$-0.664818\pi$
$373$	−24.2817	−	4.28153i	−1.25726	−	0.221689i	−0.762299	+	0.647225i	$0.775929\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−11.3932	−	4.14677i	−0.585228	−	0.213005i	0.0324014	−	0.999475i	$-0.489684\pi$
$379$	−11.3932	−	4.14677i	−0.585228	−	0.213005i	−0.617629	+	0.786469i	$0.711907\pi$
$380$		0			0
$381$	12.9613	−	7.48319i	0.664026	−	0.383375i
$382$		0			0
$383$		0			0		−0.906308	−	0.422618i	$-0.861111\pi$
$383$		0			0		0.906308	+	0.422618i	$0.138889\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.751122	−	1.61079i	−0.0381817	−	0.0818808i
$388$	−22.2729	−	31.8090i	−1.13074	−	1.61486i
$389$		0			0		0.819152	−	0.573576i	$-0.194444\pi$
$389$		0			0		−0.819152	+	0.573576i	$0.805556\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	34.1263	+	19.7028i	1.71275	+	0.988855i	0.930814	+	0.365493i	$0.119100\pi$
$397$	34.1263	+	19.7028i	1.71275	+	0.988855i	0.781933	+	0.623362i	$0.214234\pi$
$398$		0			0
$399$	−52.8357	−	14.1573i	−2.64509	−	0.708750i
$400$	12.8558	−	15.3209i	0.642788	−	0.766044i
$401$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$401$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$402$		0			0
$403$	−51.0826	+	18.5925i	−2.54460	+	0.926160i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	31.9986	+	22.4057i	1.58223	+	1.10789i	0.939895	+	0.341463i	$0.110922\pi$
$409$	31.9986	+	22.4057i	1.58223	+	1.10789i	0.642333	+	0.766426i	$0.277967\pi$
$410$		0			0
$411$		0			0
$412$	−0.319045	−	3.64670i	−0.0157182	−	0.179660i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	20.2564	−	35.0852i	0.991962	−	1.71813i
$418$		0			0
$419$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$419$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$420$		0			0
$421$	−10.1340	−	37.8205i	−0.493900	−	1.84326i	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−10.1340	−	37.8205i	−0.493900	−	1.84326i	0.0422075	−	0.999109i	$-0.486561\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−20.9527	+	9.77040i	−1.01397	+	0.472823i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.422618	−	0.906308i	$-0.361111\pi$
$431$		0			0		−0.422618	+	0.906308i	$0.638889\pi$
$432$	20.4688	−	3.60921i	0.984808	−	0.173648i
$433$	9.52628	+	16.5000i	0.457804	+	0.792939i	0.998845	−	0.0480569i	$-0.0153029\pi$
$433$	9.52628	+	16.5000i	0.457804	+	0.792939i	−0.541041	+	0.840996i	$0.681970\pi$
$434$		0			0
$435$		0			0
$436$	20.4710	+	5.48519i	0.980383	+	0.262693i
$437$		0			0
$438$		0			0
$439$	−39.9366	+	3.49400i	−1.90607	+	0.166760i	−0.978412	−	0.206666i	$-0.933739\pi$
$439$	−39.9366	+	3.49400i	−1.90607	+	0.166760i	−0.927660	+	0.373425i	$0.878183\pi$
$440$		0			0
$441$	−7.48839	+	42.4688i	−0.356590	+	2.02232i
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	−20.9817	+	1.94087i	−0.995749	+	0.0921098i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−12.6500	−	34.7556i	−0.597657	−	1.64205i
$449$		0			0		−0.0871557	−	0.996195i	$-0.527778\pi$
$449$		0			0		0.0871557	+	0.996195i	$0.472222\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−13.2649	−	4.82802i	−0.623238	−	0.226840i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	20.2313	+	9.43403i	0.946382	+	0.441305i	0.833633	−	0.552318i	$-0.186257\pi$
$457$	20.2313	+	9.43403i	0.946382	+	0.441305i	0.112749	+	0.993624i	$0.464034\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.422618	−	0.906308i	$-0.638889\pi$
$461$		0			0		0.422618	+	0.906308i	$0.361111\pi$
$462$		0			0
$463$	31.3238	−	21.9331i	1.45574	−	1.01932i	0.464739	−	0.885448i	$-0.346148\pi$
$463$	31.3238	−	21.9331i	1.45574	−	1.01932i	0.990999	−	0.133871i	$-0.0427409\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$467$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$468$	13.7787	−	29.5484i	0.636919	−	1.36588i
$469$	−2.88254	+	0.508270i	−0.133103	+	0.0234697i
$470$		0			0
$471$	−26.0948	−	15.0658i	−1.20238	−	0.694197i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−24.1506	+	24.1506i	−1.10811	+	1.10811i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.573576	−	0.819152i	$-0.305556\pi$
$479$		0			0		−0.573576	+	0.819152i	$0.694444\pi$
$480$		0			0
$481$	−16.3838	+	28.7064i	−0.747038	+	1.30890i
$482$		0			0
$483$		0			0
$484$	21.6658	+	3.82026i	0.984808	+	0.173648i
$485$		0			0
$486$		0			0
$487$	20.2679	+	20.2679i	0.918428	+	0.918428i	0.996915	−	0.0784867i	$-0.0250088\pi$
$487$	20.2679	+	20.2679i	0.918428	+	0.918428i	−0.0784867	+	0.996915i	$0.525009\pi$
$488$		0			0
$489$	11.4282	−	42.6506i	0.516801	−	1.92873i
$490$		0			0
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−3.48767	+	39.8643i	−0.156601	+	1.78996i
$497$		0			0
$498$		0			0
$499$	12.5225	+	17.8840i	0.560586	+	0.800599i	0.994975	−	0.100126i	$-0.0319246\pi$
$499$	12.5225	+	17.8840i	0.560586	+	0.800599i	−0.434389	+	0.900725i	$0.643036\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.996195	−	0.0871557i	$-0.972222\pi$
$503$		0			0		0.996195	+	0.0871557i	$0.0277778\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−14.3125	−	24.7901i	−0.635642	−	1.10097i
$508$	−14.9664	−	8.64084i	−0.664026	−	0.383375i
$509$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$509$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$510$		0			0
$511$	45.7131	−	54.4787i	2.02223	−	2.41000i
$512$		0			0
$513$	−35.3590	+	3.09351i	−1.56114	+	0.136582i
$514$		0			0
$515$		0			0
$516$	−1.17713	+	1.68111i	−0.0518201	+	0.0740067i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$521$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$522$		0			0
$523$	−1.02682	−	11.7366i	−0.0448997	−	0.513206i	−0.985029	−	0.172388i	$-0.944852\pi$
$523$	−1.02682	−	11.7366i	−0.0448997	−	0.513206i	0.940129	−	0.340818i	$-0.110704\pi$
$524$		0			0
$525$	30.6714	+	25.7364i	1.33861	+	1.12323i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	19.9186	−	11.5000i	0.866025	−	0.500000i
$530$		0			0
$531$		0			0
$532$	16.3474	+	61.0094i	0.708750	+	2.64509i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−20.0036	+	5.35995i	−0.860023	+	0.230442i	−0.661768	−	0.749708i	$-0.730194\pi$
$541$	−20.0036	+	5.35995i	−0.860023	+	0.230442i	−0.198254	+	0.980151i	$0.563527\pi$
$542$		0			0
$543$	37.5600	−	6.62284i	1.61185	−	0.284213i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	26.7008	+	7.15445i	1.14164	+	0.305902i	0.779611	−	0.626264i	$-0.215417\pi$
$547$	26.7008	+	7.15445i	1.14164	+	0.305902i	0.362031	+	0.932166i	$0.382083\pi$
$548$		0			0
$549$	−10.6077	+	10.6077i	−0.452725	+	0.452725i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−31.5400	+	45.0438i	−1.34122	+	1.91546i
$554$		0			0
$555$		0			0
$556$	−46.7802			−1.98392
$557$		0			0		−0.819152	−	0.573576i	$-0.805556\pi$
$557$		0			0		0.819152	+	0.573576i	$0.194444\pi$
$558$		0			0
$559$	1.10103	+	3.02506i	0.0465687	+	0.127946i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$563$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	7.22540	+	40.9773i	0.303438	+	1.72088i
$568$		0			0
$569$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$569$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$570$		0			0
$571$	−18.5482	+	15.5638i	−0.776217	+	0.651323i	−0.942293	−	0.334790i	$-0.891335\pi$
$571$	−18.5482	+	15.5638i	−0.776217	+	0.651323i	0.166076	+	0.986113i	$0.446890\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−15.4269	−	18.3851i	−0.642788	−	0.766044i
$577$	42.1638	+	3.68886i	1.75530	+	0.153569i	0.918885	−	0.394524i	$-0.129091\pi$
$577$	42.1638	+	3.68886i	1.75530	+	0.153569i	0.836417	+	0.548094i	$0.184646\pi$
$578$		0			0
$579$	3.22643	−	6.91911i	0.134086	−	0.287548i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.996195	−	0.0871557i	$-0.0277778\pi$
$587$		0			0		−0.996195	+	0.0871557i	$0.972222\pi$
$588$	46.7922	−	17.0310i	1.92968	−	0.702345i
$589$	11.8665	−	67.2984i	0.488951	−	2.77298i
$590$		0			0
$591$		0			0
$592$	14.0547	+	19.8612i	0.577644	+	0.816289i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	4.25632	+	48.6500i	0.174200	+	1.99111i
$598$		0			0
$599$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$599$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$600$		0			0
$601$	−28.3503	−	10.3187i	−1.15643	−	0.420907i	−0.308611	−	0.951188i	$-0.599864\pi$
$601$	−28.3503	−	10.3187i	−1.15643	−	0.420907i	−0.847822	+	0.530281i	$0.822086\pi$
$602$		0			0
$603$	−1.64485	+	0.949655i	−0.0669835	+	0.0386729i
$604$	2.83046	+	16.0523i	0.115170	+	0.653160i
$605$		0			0
$606$		0			0
$607$	4.00790	−	45.8105i	0.162676	−	1.85939i	−0.276531	−	0.961005i	$-0.589185\pi$
$607$	4.00790	−	45.8105i	0.162676	−	1.85939i	0.439206	−	0.898386i	$-0.355260\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	21.1535	+	25.2097i	0.854381	+	1.01821i	0.999585	+	0.0288097i	$0.00917168\pi$
$613$	21.1535	+	25.2097i	0.854381	+	1.01821i	−0.145204	+	0.989402i	$0.546384\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$617$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$618$		0			0
$619$	43.0929	+	24.8797i	1.73205	+	1.00000i	0.866226	+	0.499653i	$0.166539\pi$
$619$	43.0929	+	24.8797i	1.73205	+	1.00000i	0.865825	+	0.500347i	$0.166794\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−37.5035	+	3.28114i	−1.50134	+	0.131351i
$625$	23.4923	−	8.55050i	0.939693	−	0.342020i
$626$		0			0
$627$		0			0
$628$			34.7930i			1.38839i
$629$		0			0
$630$		0			0
$631$	−2.77004	−	1.93960i	−0.110273	−	0.0772143i	0.517139	−	0.855901i	$-0.326997\pi$
$631$	−2.77004	−	1.93960i	−0.110273	−	0.0772143i	−0.627412	+	0.778687i	$0.715886\pi$
$632$		0			0
$633$	−3.99366	−	10.9725i	−0.158734	−	0.436117i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	20.2163	−	75.4481i	0.800997	−	2.98936i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$641$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$642$		0			0
$643$	−12.7930	−	47.7440i	−0.504506	−	1.88284i	−0.468449	−	0.883491i	$-0.655187\pi$
$643$	−12.7930	−	47.7440i	−0.504506	−	1.88284i	−0.0360565	−	0.999350i	$-0.511480\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.573576	−	0.819152i	$-0.694444\pi$
$647$		0			0		0.573576	+	0.819152i	$0.305556\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−79.8057	−	6.98209i	−3.12783	−	0.273650i
$652$	−49.2487	+	13.1962i	−1.92873	+	0.516801i
$653$		0			0		0.422618	−	0.906308i	$-0.361111\pi$
$653$		0			0		−0.422618	+	0.906308i	$0.638889\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	15.7832	−	43.3641i	0.615763	−	1.69180i
$658$		0			0
$659$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$659$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$660$		0			0
$661$	−31.0786	+	2.71902i	−1.20882	+	0.105758i	−0.673690	−	0.739014i	$-0.735292\pi$
$661$	−31.0786	+	2.71902i	−1.20882	+	0.105758i	−0.535126	+	0.844772i	$0.679736\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−38.8175	−	6.84457i	−1.50077	−	0.264626i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−27.8634	−	23.3802i	−1.07405	−	0.901239i	−0.0786409	−	0.996903i	$-0.525058\pi$
$673$	−27.8634	−	23.3802i	−1.07405	−	0.901239i	−0.995414	+	0.0956642i	$0.969503\pi$
$674$		0			0
$675$	24.4139	+	8.88594i	0.939693	+	0.342020i
$676$	−16.5267	+	28.6251i	−0.635642	+	1.10097i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	−81.3544	−	37.9362i	−3.12210	−	1.45586i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.422618	−	0.906308i	$-0.638889\pi$
$683$		0			0		0.422618	+	0.906308i	$0.361111\pi$
$684$	23.5080	+	33.5729i	0.898852	+	1.28369i
$685$		0			0
$686$		0			0
$687$	18.5355	+	22.0898i	0.707175	+	0.842779i
$688$	2.36072	+	0.206537i	0.0900017	+	0.00787413i
$689$		0			0
$690$		0			0
$691$	24.9052	−	4.39146i	0.947439	−	0.167059i	0.321481	−	0.946916i	$-0.395819\pi$
$691$	24.9052	−	4.39146i	0.947439	−	0.167059i	0.625958	+	0.779857i	$0.284708\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	8.02823	−	45.5303i	0.303438	−	1.72088i
$701$		0			0		0.573576	−	0.819152i	$-0.305556\pi$
$701$		0			0		−0.573576	+	0.819152i	$0.694444\pi$
$702$		0			0
$703$	−20.9539	−	35.8799i	−0.790291	−	1.35323i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	18.4860	+	18.4860i	0.694255	+	0.694255i	0.963165	−	0.268910i	$-0.0866634\pi$
$709$	18.4860	+	18.4860i	0.694255	+	0.694255i	−0.268910	+	0.963165i	$0.586663\pi$
$710$		0			0
$711$	−9.23504	+	34.4657i	−0.346341	+	1.29256i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$719$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$720$		0			0
$721$	−4.85363	−	6.93170i	−0.180759	−	0.258150i
$722$		0			0
$723$	45.2526	−	21.1017i	1.68296	−	0.784779i
$724$	−28.3081	−	33.7363i	−1.05206	−	1.25380i
$725$		0			0
$726$		0			0
$727$	−17.3761	+	37.2631i	−0.644443	+	1.38201i	0.264211	+	0.964465i	$0.414888\pi$
$727$	−17.3761	+	37.2631i	−0.644443	+	1.38201i	−0.908655	+	0.417548i	$0.862889\pi$
$728$		0			0
$729$	13.5000	+	23.3827i	0.500000	+	0.866025i
$730$		0			0
$731$		0			0
$732$	16.7321	+	4.48334i	0.618434	+	0.165709i
$733$	−1.56206	+	1.86159i	−0.0576961	+	0.0687595i	−0.794121	−	0.607760i	$-0.792068\pi$
$733$	−1.56206	+	1.86159i	−0.0576961	+	0.0687595i	0.736425	+	0.676520i	$0.236513\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$			39.8372i			1.46543i	0.680534	+	0.732717i	$0.261748\pi$
$739$			39.8372i			1.46543i	−0.680534	+	0.732717i	$0.738252\pi$
$740$		0			0
$741$	64.2897			2.36174
$742$		0			0
$743$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$743$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	47.4478	−	27.3940i	1.73139	−	0.999621i	0.851930	−	0.523655i	$-0.175432\pi$
$751$	47.4478	−	27.3940i	1.73139	−	0.999621i	0.879464	−	0.475965i	$-0.157901\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	36.8057	−	30.8837i	1.33861	−	1.12323i
$757$	10.4868	+	22.4889i	0.381148	+	0.817374i	0.999505	+	0.0314762i	$0.0100208\pi$
$757$	10.4868	+	22.4889i	0.381148	+	0.817374i	−0.618357	+	0.785897i	$0.712201\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$761$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$762$		0			0
$763$	47.3215	−	12.6798i	1.71315	−	0.459038i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−9.47834	+	26.0415i	−0.342020	+	0.939693i
$769$	−44.9630	−	12.0478i	−1.62141	−	0.434455i	−0.669994	−	0.742367i	$-0.733703\pi$
$769$	−44.9630	−	12.0478i	−1.62141	−	0.434455i	−0.951415	+	0.307912i	$0.900370\pi$
$770$		0			0
$771$		0			0
$772$	−8.78189	+	0.768316i	−0.316067	+	0.0276523i
$773$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$773$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$774$		0			0
$775$	−28.6907	+	40.9745i	−1.03060	+	1.47185i
$776$		0			0
$777$	−39.7608	+	28.1366i	−1.42641	+	1.00939i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−44.0464	−	36.9594i	−1.57309	−	1.31998i
$785$		0			0
$786$		0			0
$787$	−20.3359	+	35.2228i	−0.724897	+	1.25556i	0.234120	+	0.972208i	$0.424779\pi$
$787$	−20.3359	+	35.2228i	−0.724897	+	1.25556i	−0.959017	+	0.283350i	$0.908554\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	20.8150	−	17.4659i	0.739162	−	0.620231i
$794$		0			0
$795$		0			0
$796$	46.1926	−	32.3444i	1.63725	−	1.14642i
$797$		0			0		0.906308	−	0.422618i	$-0.138889\pi$
$797$		0			0		−0.906308	+	0.422618i	$0.861111\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	1.89931	+	1.09657i	0.0669835	+	0.0386729i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.996195	−	0.0871557i	$-0.0277778\pi$
$809$		0			0		−0.996195	+	0.0871557i	$0.972222\pi$
$810$		0			0
$811$	−7.51919	+	42.6434i	−0.264034	+	1.49741i	0.507736	+	0.861512i	$0.330482\pi$
$811$	−7.51919	+	42.6434i	−0.264034	+	1.49741i	−0.771771	+	0.635901i	$0.780629\pi$
$812$		0			0
$813$			54.1549i			1.89929i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−3.98534	−	0.702724i	−0.139430	−	0.0245852i
$818$		0			0
$819$	−6.56862	−	75.0797i	−0.229526	−	2.62350i
$820$		0			0
$821$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$821$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$822$		0			0
$823$	52.0788	+	18.9551i	1.81535	+	0.660734i	0.996194	+	0.0871670i	$0.0277814\pi$
$823$	52.0788	+	18.9551i	1.81535	+	0.660734i	0.819159	+	0.573567i	$0.194441\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.906308	−	0.422618i	$-0.861111\pi$
$827$		0			0		0.906308	+	0.422618i	$0.138889\pi$
$828$		0			0
$829$	2.35758	−	26.9473i	0.0818822	−	0.935918i	−0.838501	−	0.544900i	$-0.816567\pi$
$829$	2.35758	−	26.9473i	0.0818822	−	0.935918i	0.920383	−	0.391018i	$-0.127877\pi$
$830$		0			0
$831$	22.3211	+	47.8679i	0.774312	+	1.66052i
$832$	24.9338	+	35.6092i	0.864425	+	1.23453i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−50.2117	+	13.4542i	−1.73557	+	0.465045i
$838$		0			0
$839$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$839$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$840$		0			0
$841$	25.1147	+	14.5000i	0.866025	+	0.500000i
$842$		0			0
$843$		0			0
$844$	−8.66675	+	10.3286i	−0.298322	+	0.355526i
$845$		0			0
$846$		0			0
$847$	47.7890	−	17.3938i	1.64205	−	0.597657i
$848$		0			0
$849$	9.77724	−	13.9633i	0.335554	−	0.479221i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−47.3608	−	33.1624i	−1.62160	−	1.13546i	−0.892493	−	0.451061i	$-0.851046\pi$
$853$	−47.3608	−	33.1624i	−1.62160	−	1.13546i	−0.729110	−	0.684397i	$-0.760066\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$	−3.20259	+	11.9522i	−0.109271	+	0.407805i	−0.998795	−	0.0490840i	$-0.984370\pi$
$859$	−3.20259	+	11.9522i	−0.109271	+	0.407805i	0.889524	+	0.456889i	$0.151036\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$863$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	22.5561	−	18.9268i	0.766044	−	0.642788i
$868$	39.0937	+	83.8368i	1.32693	+	2.84561i
$869$		0			0
$870$		0			0
$871$	3.11787	−	1.45389i	0.105645	−	0.0492630i
$872$		0			0
$873$	−58.0259	−	5.07661i	−1.96388	−	0.171817i
$874$		0			0
$875$		0			0
$876$	−52.4766	+	9.25304i	−1.77302	+	0.312631i
$877$	−29.5030	−	51.1007i	−0.996245	−	1.72555i	−0.573103	−	0.819483i	$-0.694261\pi$
$877$	−29.5030	−	51.1007i	−0.996245	−	1.72555i	−0.423141	−	0.906064i	$-0.639073\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$881$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$882$		0			0
$883$	35.8475	−	3.13625i	1.20637	−	0.105543i	0.533820	−	0.845598i	$-0.320756\pi$
$883$	35.8475	−	3.13625i	1.20637	−	0.105543i	0.672546	+	0.740055i	$0.265201\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	−39.9490			−1.33985
$890$		0			0
$891$		0			0
$892$	15.5667	+	42.7692i	0.521212	+	1.43202i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−5.20945	−	29.5442i	−0.173648	−	0.984808i
$901$		0			0
$902$		0			0
$903$	−0.413473	+	4.72601i	−0.0137595	+	0.157272i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	11.4240	−	7.99914i	0.379326	−	0.265607i	−0.368327	−	0.929696i	$-0.620069\pi$
$907$	11.4240	−	7.99914i	0.379326	−	0.265607i	0.747653	+	0.664089i	$0.231180\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$911$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$912$	20.0006	−	42.8914i	0.662285	−	1.42028i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	11.3883	−	31.2891i	0.376280	−	1.03382i
$917$		0			0
$918$		0			0
$919$	39.1487	−	39.1487i	1.29140	−	1.29140i	0.357472	−	0.933924i	$-0.383639\pi$
$919$	39.1487	−	39.1487i	1.29140	−	1.29140i	0.933924	−	0.357472i	$-0.116361\pi$
$920$		0			0
$921$	16.8344	−	6.12722i	0.554712	−	0.201899i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	2.80141	+	30.2845i	0.0921098	+	0.995749i
$926$		0			0
$927$	−4.49792	−	3.14948i	−0.147731	−	0.103442i
$928$		0			0
$929$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$929$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$930$		0			0
$931$	69.4313	+	69.4313i	2.27552	+	2.27552i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.950781	−	5.39215i	−0.0310607	−	0.176154i	0.965331	−	0.261029i	$-0.0840619\pi$
$937$	−0.950781	−	5.39215i	−0.0310607	−	0.176154i	−0.996392	+	0.0848755i	$0.972951\pi$
$938$		0			0
$939$	3.68799	+	13.7638i	0.120353	+	0.449163i
$940$		0			0
$941$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$941$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.996195	−	0.0871557i	$-0.972222\pi$
$947$		0			0		0.996195	+	0.0871557i	$0.0277778\pi$
$948$	39.7975	−	10.6637i	1.29256	−	0.346341i
$949$	−35.3248	+	75.7542i	−1.14669	+	2.45909i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$953$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	69.0827i		−	2.22848i
$962$		0			0
$963$		0			0
$964$	−47.2283	−	33.0696i	−1.52112	−	1.06510i
$965$		0			0
$966$		0			0
$967$	−5.38837	−	61.5893i	−0.173278	−	1.98058i	−0.194946	−	0.980814i	$-0.562453\pi$
$967$	−5.38837	−	61.5893i	−0.173278	−	1.98058i	0.0216683	−	0.999765i	$-0.493102\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$971$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$972$	15.5885	−	27.0000i	0.500000	−	0.866025i
$973$	−93.6510	+	54.0694i	−3.00231	+	1.73339i
$974$		0			0
$975$	−42.6495	−	19.8878i	−1.36588	−	0.636919i
$976$	−5.17691	−	19.3205i	−0.165709	−	0.618434i
$977$		0			0		0.0871557	−	0.996195i	$-0.472222\pi$
$977$		0			0		−0.0871557	+	0.996195i	$0.527778\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	26.0406	−	18.2338i	0.831413	−	0.582161i
$982$		0			0
$983$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$983$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−37.1177	−	64.2897i	−1.18087	−	2.04533i
$989$		0			0
$990$		0			0
$991$	−52.3109	−	14.0167i	−1.66171	−	0.445254i	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−52.3109	−	14.0167i	−1.66171	−	0.445254i	−0.962857	+	0.270011i	$0.912973\pi$
$992$		0			0
$993$	−35.7889	+	35.7889i	−1.13573	+	1.13573i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−36.0306	+	51.4570i	−1.14110	+	1.62966i	−0.486507	+	0.873677i	$0.661729\pi$
$997$	−36.0306	+	51.4570i	−1.14110	+	1.62966i	−0.654593	+	0.755982i	$0.727160\pi$
$998$		0			0
$999$	−18.0000	+	25.9808i	−0.569495	+	0.821995i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	111.2.q.a.17.1	✓	12
3.2	odd	2	CM	111.2.q.a.17.1	✓	12
37.24	odd	36	inner	111.2.q.a.98.1	yes	12
111.98	even	36	inner	111.2.q.a.98.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.2.q.a.17.1	✓	12	1.1	even	1	trivial
111.2.q.a.17.1	✓	12	3.2	odd	2	CM
111.2.q.a.98.1	yes	12	37.24	odd	36	inner
111.2.q.a.98.1	yes	12	111.98	even	36	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:	\( N \)	\(=\)	\( 111 = 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	111.q (of order \(36\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.70574 - 0.300767i) q^{3} +(0.684040 + 1.87939i) q^{4} +(3.54163 + 2.97178i) q^{7} +(2.81908 + 1.02606i) q^{9} +O(q^{10})\)	\(q+(-1.70574 - 0.300767i) q^{3} +(0.684040 + 1.87939i) q^{4} +(3.54163 + 2.97178i) q^{7} +(2.81908 + 1.02606i) q^{9} +(-0.601535 - 3.41147i) q^{12} +(-4.92474 - 2.29644i) q^{13} +(-3.06418 + 2.57115i) q^{16} +(5.59549 - 3.91800i) q^{19} +(-5.14728 - 6.13429i) q^{21} +(-4.92404 + 0.868241i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(-3.16250 + 8.68891i) q^{28} +(7.07399 - 7.07399i) q^{31} +6.00000i q^{36} +(0.500000 - 6.06218i) q^{37} +(7.70961 + 5.39833i) q^{39} +(-0.418915 - 0.418915i) q^{43} +(6.00000 - 3.46410i) q^{48} +(2.49613 + 14.1563i) q^{49} +(0.947182 - 10.8263i) q^{52} +(-10.7228 + 5.00014i) q^{57} +(-2.11331 + 4.53201i) q^{61} +(6.93491 + 12.0116i) q^{63} +(-6.92820 - 4.00000i) q^{64} +(-0.406951 + 0.484985i) q^{67} -15.3824i q^{73} +8.66025 q^{75} +(11.1910 + 7.83601i) q^{76} +(1.03662 + 11.8486i) q^{79} +(6.89440 + 5.78509i) q^{81} +(8.00774 - 13.8698i) q^{84} +(-10.6171 - 22.7684i) q^{91} +(-14.1940 + 9.93874i) q^{93} +(-18.7543 + 5.02519i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 54 q^{27} - 48 q^{28} + 6 q^{37} + 18 q^{39} + 72 q^{48} + 78 q^{49} - 114 q^{91}+O(q^{100}) \) 12 * q - 54 * q^27 - 48 * q^28 + 6 * q^37 + 18 * q^39 + 72 * q^48 + 78 * q^49 - 114 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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