LMFDB - Embedded newform 1323.2.c.f.1322.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(1322,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1323.1322");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.c (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:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 24x^{14} + 212x^{12} + 872x^{10} + 1815x^{8} + 1928x^{6} + 996x^{4} + 200x^{2} + 1 $ x^16 + 24x^14 + 212x^12 + 872x^10 + 1815x^8 + 1928x^6 + 996x^4 + 200*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1322.11
Root			$0.964652i$ of defining polynomial
Character	$\chi$	$=$	1323.1322
Dual form			1323.2.c.f.1322.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.698626i q^{2} +1.51192 q^{4} -2.26676 q^{5} +2.45352i q^{8} +O(q^{10})$	$q+0.698626i q^{2} +1.51192 q^{4} -2.26676 q^{5} +2.45352i q^{8} -1.58361i q^{10} -0.558767i q^{11} +5.53446i q^{13} +1.30975 q^{16} -5.74871 q^{17} +0.893030i q^{19} -3.42716 q^{20} +0.390369 q^{22} -6.26097i q^{23} +0.138181 q^{25} -3.86651 q^{26} +8.42510i q^{29} +0.233422i q^{31} +5.82206i q^{32} -4.01620i q^{34} +5.37407 q^{37} -0.623893 q^{38} -5.56153i q^{40} +2.37380 q^{41} -11.7212 q^{43} -0.844812i q^{44} +4.37407 q^{46} -12.4185 q^{47} +0.0965369i q^{50} +8.36767i q^{52} +7.77314i q^{53} +1.26659i q^{55} -5.88599 q^{58} +7.15404 q^{59} +12.7942i q^{61} -0.163075 q^{62} -1.44794 q^{64} -12.5453i q^{65} -6.03093 q^{67} -8.69161 q^{68} +11.2259i q^{71} -1.77298i q^{73} +3.75446i q^{74} +1.35019i q^{76} -5.77119 q^{79} -2.96889 q^{80} +1.65840i q^{82} +0.913671 q^{83} +13.0309 q^{85} -8.18871i q^{86} +1.37094 q^{88} +2.80582 q^{89} -9.46610i q^{92} -8.67587i q^{94} -2.02428i q^{95} -12.5775i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 48 q^{16} + 64 q^{22} - 16 q^{25} + 32 q^{37} - 16 q^{43} + 16 q^{46} - 32 q^{64} + 48 q^{67} - 64 q^{79} + 64 q^{85} - 176 q^{88}+O(q^{100}) $ 16 * q - 16 * q^4 + 48 * q^16 + 64 * q^22 - 16 * q^25 + 32 * q^37 - 16 * q^43 + 16 * q^46 - 32 * q^64 + 48 * q^67 - 64 * q^79 + 64 * q^85 - 176 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$-1$	$-1$

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 $

$2$	0.698626i	0.494003i	0.969015	+	0.247001i	$0.0794452\pi$
$2$	0.698626i	0.494003i	−0.969015	+	0.247001i	$0.920555\pi$
$3$	0	0
$3$	0	0
$4$	1.51192	0.755961
$5$	−2.26676	−1.01372	−0.506862	−	0.862027i	$-0.669195\pi$
$5$	−2.26676	−1.01372	−0.506862	+	0.862027i	$0.669195\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.45352i	0.867450i
$9$	0	0
$10$	− 1.58361i	− 0.500783i
$11$	− 0.558767i	− 0.168475i	−0.996446	−	0.0842373i	$-0.973155\pi$
$11$	− 0.558767i	− 0.168475i	0.996446	−	0.0842373i	$-0.0268454\pi$
$12$	0	0
$13$	5.53446i	1.53498i	0.641060	+	0.767491i	$0.278495\pi$
$13$	5.53446i	1.53498i	−0.641060	+	0.767491i	$0.721505\pi$
$14$	0	0
$15$	0	0
$16$	1.30975	0.327438
$17$	−5.74871	−1.39427	−0.697134	−	0.716941i	$-0.745542\pi$
$17$	−5.74871	−1.39427	−0.697134	+	0.716941i	$0.745542\pi$
$18$	0	0
$19$	0.893030i	0.204875i	0.994739	+	0.102438i	$0.0326642\pi$
$19$	0.893030i	0.204875i	−0.994739	+	0.102438i	$0.967336\pi$
$20$	−3.42716	−0.766336
$21$	0	0
$22$	0.390369	0.0832269
$23$	− 6.26097i	− 1.30550i	−0.757573	−	0.652751i	$-0.773615\pi$
$23$	− 6.26097i	− 1.30550i	0.757573	−	0.652751i	$-0.226385\pi$
$24$	0	0
$25$	0.138181	0.0276362
$26$	−3.86651	−0.758286
$27$	0	0
$28$	0	0
$29$	8.42510i	1.56450i	0.622963	+	0.782251i	$0.285929\pi$
$29$	8.42510i	1.56450i	−0.622963	+	0.782251i	$0.714071\pi$
$30$	0	0
$31$	0.233422i	0.0419238i	0.999780	+	0.0209619i	$0.00667287\pi$
$31$	0.233422i	0.0419238i	−0.999780	+	0.0209619i	$0.993327\pi$
$32$	5.82206i	1.02921i
$33$	0	0
$34$	− 4.01620i	− 0.688772i
$35$	0	0
$36$	0	0
$37$	5.37407	0.883492	0.441746	−	0.897140i	$-0.354359\pi$
$37$	5.37407	0.883492	0.441746	+	0.897140i	$0.354359\pi$
$38$	−0.623893	−0.101209
$39$	0	0
$40$	− 5.56153i	− 0.879355i
$41$	2.37380	0.370725	0.185363	−	0.982670i	$-0.440654\pi$
$41$	2.37380	0.370725	0.185363	+	0.982670i	$0.440654\pi$
$42$	0	0
$43$	−11.7212	−1.78746	−0.893731	−	0.448603i	$-0.851922\pi$
$43$	−11.7212	−1.78746	−0.893731	+	0.448603i	$0.851922\pi$
$44$	− 0.844812i	− 0.127360i
$45$	0	0
$46$	4.37407	0.644922
$47$	−12.4185	−1.81142	−0.905711	−	0.423895i	$-0.860663\pi$
$47$	−12.4185	−1.81142	−0.905711	+	0.423895i	$0.860663\pi$
$48$	0	0
$49$	0	0
$50$	0.0965369i	0.0136524i
$51$	0	0
$52$	8.36767i	1.16039i
$53$	7.77314i	1.06772i	0.845572	+	0.533861i	$0.179260\pi$
$53$	7.77314i	1.06772i	−0.845572	+	0.533861i	$0.820740\pi$
$54$	0	0
$55$	1.26659i	0.170787i
$56$	0	0
$57$	0	0
$58$	−5.88599	−0.772869
$59$	7.15404	0.931377	0.465688	−	0.884949i	$-0.345807\pi$
$59$	7.15404	0.931377	0.465688	+	0.884949i	$0.345807\pi$
$60$	0	0
$61$	12.7942i	1.63813i	0.573703	+	0.819063i	$0.305506\pi$
$61$	12.7942i	1.63813i	−0.573703	+	0.819063i	$0.694494\pi$
$62$	−0.163075	−0.0207105
$63$	0	0
$64$	−1.44794	−0.180992
$65$	− 12.5453i	− 1.55605i
$66$	0	0
$67$	−6.03093	−0.736795	−0.368397	−	0.929668i	$-0.620093\pi$
$67$	−6.03093	−0.736795	−0.368397	+	0.929668i	$0.620093\pi$
$68$	−8.69161	−1.05401
$69$	0	0
$70$	0	0
$71$	11.2259i	1.33227i	0.745830	+	0.666137i	$0.232053\pi$
$71$	11.2259i	1.33227i	−0.745830	+	0.666137i	$0.767947\pi$
$72$	0	0
$73$	− 1.77298i	− 0.207511i	−0.994603	−	0.103756i	$-0.966914\pi$
$73$	− 1.77298i	− 0.207511i	0.994603	−	0.103756i	$-0.0330860\pi$
$74$	3.75446i	0.436447i
$75$	0	0
$76$	1.35019i	0.154878i
$77$	0	0
$78$	0	0
$79$	−5.77119	−0.649310	−0.324655	−	0.945833i	$-0.605248\pi$
$79$	−5.77119	−0.649310	−0.324655	+	0.945833i	$0.605248\pi$
$80$	−2.96889	−0.331932
$81$	0	0
$82$	1.65840i	0.183139i
$83$	0.913671	0.100288	0.0501442	−	0.998742i	$-0.484032\pi$
$83$	0.913671	0.100288	0.0501442	+	0.998742i	$0.484032\pi$
$84$	0	0
$85$	13.0309	1.41340
$86$	− 8.18871i	− 0.883011i
$87$	0	0
$88$	1.37094	0.146143
$89$	2.80582	0.297416	0.148708	−	0.988881i	$-0.452489\pi$
$89$	2.80582	0.297416	0.148708	+	0.988881i	$0.452489\pi$
$90$	0	0
$91$	0	0
$92$	− 9.46610i	− 0.986909i
$93$	0	0
$94$	− 8.67587i	− 0.894848i
$95$	− 2.02428i	− 0.207687i
$96$	0	0
$97$	− 12.5775i	− 1.27705i	−0.769601	−	0.638525i	$-0.779545\pi$
$97$	− 12.5775i	− 1.27705i	0.769601	−	0.638525i	$-0.220455\pi$
$98$	0	0
$99$	0	0
$100$	0.208919	0.0208919
$101$	−0.952418	−0.0947692	−0.0473846	−	0.998877i	$-0.515089\pi$
$101$	−0.952418	−0.0947692	−0.0473846	+	0.998877i	$0.515089\pi$
$102$	0	0
$103$	− 8.44270i	− 0.831884i	−0.909391	−	0.415942i	$-0.863452\pi$
$103$	− 8.44270i	− 0.831884i	0.909391	−	0.415942i	$-0.136548\pi$
$104$	−13.5789	−1.33152
$105$	0	0
$106$	−5.43051	−0.527458
$107$	− 8.71449i	− 0.842461i	−0.906954	−	0.421231i	$-0.861598\pi$
$107$	− 8.71449i	− 0.842461i	0.906954	−	0.421231i	$-0.138402\pi$
$108$	0	0
$109$	−1.81213	−0.173571	−0.0867853	−	0.996227i	$-0.527659\pi$
$109$	−1.81213	−0.173571	−0.0867853	+	0.996227i	$0.527659\pi$
$110$	−0.884871	−0.0843691
$111$	0	0
$112$	0	0
$113$	14.0108i	1.31802i	0.752133	+	0.659011i	$0.229025\pi$
$113$	14.0108i	1.31802i	−0.752133	+	0.659011i	$0.770975\pi$
$114$	0	0
$115$	14.1921i	1.32342i
$116$	12.7381i	1.18270i
$117$	0	0
$118$	4.99800i	0.460103i
$119$	0	0
$120$	0	0
$121$	10.6878	0.971616
$122$	−8.93834	−0.809239
$123$	0	0
$124$	0.352916i	0.0316928i
$125$	11.0206	0.985708
$126$	0	0
$127$	18.8192	1.66993	0.834967	−	0.550299i	$-0.185486\pi$
$127$	18.8192	1.66993	0.834967	+	0.550299i	$0.185486\pi$
$128$	10.6326i	0.939795i
$129$	0	0
$130$	8.76444	0.768692
$131$	−0.836235	−0.0730622	−0.0365311	−	0.999333i	$-0.511631\pi$
$131$	−0.836235	−0.0730622	−0.0365311	+	0.999333i	$0.511631\pi$
$132$	0	0
$133$	0	0
$134$	− 4.21336i	− 0.363979i
$135$	0	0
$136$	− 14.1046i	− 1.20946i
$137$	− 10.2613i	− 0.876678i	−0.898810	−	0.438339i	$-0.855567\pi$
$137$	− 10.2613i	− 0.876678i	0.898810	−	0.438339i	$-0.144433\pi$
$138$	0	0
$139$	12.0668i	1.02350i	0.859136	+	0.511748i	$0.171002\pi$
$139$	12.0668i	1.02350i	−0.859136	+	0.511748i	$0.828998\pi$
$140$	0	0
$141$	0	0
$142$	−7.84272	−0.658147
$143$	3.09247	0.258605
$144$	0	0
$145$	− 19.0977i	− 1.58597i
$146$	1.23865	0.102511
$147$	0	0
$148$	8.12518	0.667886
$149$	20.4060i	1.67172i	0.548940	+	0.835862i	$0.315032\pi$
$149$	20.4060i	1.67172i	−0.548940	+	0.835862i	$0.684968\pi$
$150$	0	0
$151$	−16.9337	−1.37804	−0.689022	−	0.724741i	$-0.741960\pi$
$151$	−16.9337	−1.37804	−0.689022	+	0.724741i	$0.741960\pi$
$152$	−2.19106	−0.177719
$153$	0	0
$154$	0	0
$155$	− 0.529111i	− 0.0424992i
$156$	0	0
$157$	− 18.3192i	− 1.46203i	−0.682361	−	0.731016i	$-0.739047\pi$
$157$	− 18.3192i	− 1.46203i	0.682361	−	0.731016i	$-0.260953\pi$
$158$	− 4.03190i	− 0.320761i
$159$	0	0
$160$	− 13.1972i	− 1.04333i
$161$	0	0
$162$	0	0
$163$	−6.98370	−0.547006	−0.273503	−	0.961871i	$-0.588182\pi$
$163$	−6.98370	−0.547006	−0.273503	+	0.961871i	$0.588182\pi$
$164$	3.58900	0.280254
$165$	0	0
$166$	0.638314i	0.0495427i
$167$	11.5922	0.897032	0.448516	−	0.893775i	$-0.351953\pi$
$167$	11.5922	0.897032	0.448516	+	0.893775i	$0.351953\pi$
$168$	0	0
$169$	−17.6302	−1.35617
$170$	9.10374i	0.698225i
$171$	0	0
$172$	−17.7215	−1.35125
$173$	1.66805	0.126820	0.0634098	−	0.997988i	$-0.479802\pi$
$173$	1.66805	0.126820	0.0634098	+	0.997988i	$0.479802\pi$
$174$	0	0
$175$	0	0
$176$	− 0.731847i	− 0.0551651i
$177$	0	0
$178$	1.96022i	0.146924i
$179$	− 13.2059i	− 0.987059i	−0.869729	−	0.493529i	$-0.835707\pi$
$179$	− 13.2059i	− 0.987059i	0.869729	−	0.493529i	$-0.164293\pi$
$180$	0	0
$181$	− 5.51772i	− 0.410129i	−0.978748	−	0.205064i	$-0.934260\pi$
$181$	− 5.51772i	− 0.410129i	0.978748	−	0.205064i	$-0.0657404\pi$
$182$	0	0
$183$	0	0
$184$	15.3614	1.13246
$185$	−12.1817	−0.895617
$186$	0	0
$187$	3.21219i	0.234899i
$188$	−18.7758	−1.36937
$189$	0	0
$190$	1.41421	0.102598
$191$	− 8.62690i	− 0.624220i	−0.950046	−	0.312110i	$-0.898964\pi$
$191$	− 8.62690i	− 0.624220i	0.950046	−	0.312110i	$-0.101036\pi$
$192$	0	0
$193$	7.95986	0.572963	0.286482	−	0.958086i	$-0.407514\pi$
$193$	7.95986	0.572963	0.286482	+	0.958086i	$0.407514\pi$
$194$	8.78696	0.630867
$195$	0	0
$196$	0	0
$197$	− 9.95188i	− 0.709042i	−0.935048	−	0.354521i	$-0.884644\pi$
$197$	− 9.95188i	− 0.709042i	0.935048	−	0.354521i	$-0.115356\pi$
$198$	0	0
$199$	11.3677i	0.805834i	0.915236	+	0.402917i	$0.132004\pi$
$199$	11.3677i	0.805834i	−0.915236	+	0.402917i	$0.867996\pi$
$200$	0.339030i	0.0239730i
$201$	0	0
$202$	− 0.665384i	− 0.0468162i
$203$	0	0
$204$	0	0
$205$	−5.38082	−0.375813
$206$	5.89829	0.410953
$207$	0	0
$208$	7.24878i	0.502612i
$209$	0.498995	0.0345162
$210$	0	0
$211$	13.8860	0.955951	0.477976	−	0.878373i	$-0.341371\pi$
$211$	13.8860	0.955951	0.477976	+	0.878373i	$0.341371\pi$
$212$	11.7524i	0.807157i
$213$	0	0
$214$	6.08816	0.416178
$215$	26.5690	1.81199
$216$	0	0
$217$	0	0
$218$	− 1.26600i	− 0.0857443i
$219$	0	0
$220$	1.91498i	0.129108i
$221$	− 31.8160i	− 2.14018i
$222$	0	0
$223$	− 10.2139i	− 0.683974i	−0.939705	−	0.341987i	$-0.888900\pi$
$223$	− 10.2139i	− 0.683974i	0.939705	−	0.341987i	$-0.111100\pi$
$224$	0	0
$225$	0	0
$226$	−9.78828	−0.651107
$227$	6.73372	0.446932	0.223466	−	0.974712i	$-0.428263\pi$
$227$	6.73372	0.446932	0.223466	+	0.974712i	$0.428263\pi$
$228$	0	0
$229$	11.1127i	0.734349i	0.930152	+	0.367175i	$0.119675\pi$
$229$	11.1127i	0.734349i	−0.930152	+	0.367175i	$0.880325\pi$
$230$	−9.91495	−0.653772
$231$	0	0
$232$	−20.6712	−1.35713
$233$	− 1.15453i	− 0.0756356i	−0.999285	−	0.0378178i	$-0.987959\pi$
$233$	− 1.15453i	− 0.0756356i	0.999285	−	0.0378178i	$-0.0120407\pi$
$234$	0	0
$235$	28.1497	1.83628
$236$	10.8164	0.704085
$237$	0	0
$238$	0	0
$239$	− 25.1538i	− 1.62707i	−0.581518	−	0.813533i	$-0.697541\pi$
$239$	− 25.1538i	− 1.62707i	0.581518	−	0.813533i	$-0.302459\pi$
$240$	0	0
$241$	30.3705i	1.95634i	0.207812	+	0.978169i	$0.433366\pi$
$241$	30.3705i	1.95634i	−0.207812	+	0.978169i	$0.566634\pi$
$242$	7.46676i	0.479981i
$243$	0	0
$244$	19.3438i	1.23836i
$245$	0	0
$246$	0	0
$247$	−4.94243	−0.314480
$248$	−0.572705	−0.0363668
$249$	0	0
$250$	7.69924i	0.486943i
$251$	22.0663	1.39281	0.696406	−	0.717648i	$-0.254781\pi$
$251$	22.0663	1.39281	0.696406	+	0.717648i	$0.254781\pi$
$252$	0	0
$253$	−3.49842	−0.219944
$254$	13.1476i	0.824953i
$255$	0	0
$256$	−10.3241	−0.645253
$257$	12.9488	0.807726	0.403863	−	0.914819i	$-0.367667\pi$
$257$	12.9488	0.807726	0.403863	+	0.914819i	$0.367667\pi$
$258$	0	0
$259$	0	0
$260$	− 18.9675i	− 1.17631i
$261$	0	0
$262$	− 0.584215i	− 0.0360929i
$263$	9.40750i	0.580091i	0.957013	+	0.290046i	$0.0936705\pi$
$263$	9.40750i	0.580091i	−0.957013	+	0.290046i	$0.906330\pi$
$264$	0	0
$265$	− 17.6198i	− 1.08238i
$266$	0	0
$267$	0	0
$268$	−9.11829	−0.556988
$269$	5.96896	0.363934	0.181967	−	0.983305i	$-0.441754\pi$
$269$	5.96896	0.363934	0.181967	+	0.983305i	$0.441754\pi$
$270$	0	0
$271$	− 6.63814i	− 0.403238i	−0.979464	−	0.201619i	$-0.935380\pi$
$271$	− 6.63814i	− 0.403238i	0.979464	−	0.201619i	$-0.0646204\pi$
$272$	−7.52940	−0.456537
$273$	0	0
$274$	7.16878	0.433082
$275$	− 0.0772110i	− 0.00465600i
$276$	0	0
$277$	−8.37374	−0.503129	−0.251565	−	0.967840i	$-0.580945\pi$
$277$	−8.37374	−0.503129	−0.251565	+	0.967840i	$0.580945\pi$
$278$	−8.43020	−0.505610
$279$	0	0
$280$	0	0
$281$	5.56729i	0.332117i	0.986116	+	0.166058i	$0.0531040\pi$
$281$	5.56729i	0.332117i	−0.986116	+	0.166058i	$0.946896\pi$
$282$	0	0
$283$	− 15.2040i	− 0.903781i	−0.892073	−	0.451891i	$-0.850750\pi$
$283$	− 15.2040i	− 0.903781i	0.892073	−	0.451891i	$-0.149250\pi$
$284$	16.9727i	1.00715i
$285$	0	0
$286$	2.16048i	0.127752i
$287$	0	0
$288$	0	0
$289$	16.0477	0.943982
$290$	13.3421	0.783476
$291$	0	0
$292$	− 2.68060i	− 0.156870i
$293$	16.8511	0.984453	0.492226	−	0.870467i	$-0.336183\pi$
$293$	16.8511	0.984453	0.492226	+	0.870467i	$0.336183\pi$
$294$	0	0
$295$	−16.2165	−0.944159
$296$	13.1854i	0.766385i
$297$	0	0
$298$	−14.2561	−0.825836
$299$	34.6510	2.00392
$300$	0	0
$301$	0	0
$302$	− 11.8303i	− 0.680757i
$303$	0	0
$304$	1.16965i	0.0670840i
$305$	− 29.0013i	− 1.66061i
$306$	0	0
$307$	13.9504i	0.796194i	0.917343	+	0.398097i	$0.130329\pi$
$307$	13.9504i	0.796194i	−0.917343	+	0.398097i	$0.869671\pi$
$308$	0	0
$309$	0	0
$310$	0.369650	0.0209947
$311$	27.8051	1.57668	0.788342	−	0.615237i	$-0.210940\pi$
$311$	27.8051	1.57668	0.788342	+	0.615237i	$0.210940\pi$
$312$	0	0
$313$	24.2403i	1.37014i	0.728476	+	0.685071i	$0.240229\pi$
$313$	24.2403i	1.37014i	−0.728476	+	0.685071i	$0.759771\pi$
$314$	12.7983	0.722248
$315$	0	0
$316$	−8.72559	−0.490853
$317$	21.7139i	1.21958i	0.792565	+	0.609788i	$0.208745\pi$
$317$	21.7139i	1.21958i	−0.792565	+	0.609788i	$0.791255\pi$
$318$	0	0
$319$	4.70767	0.263579
$320$	3.28212	0.183476
$321$	0	0
$322$	0	0
$323$	− 5.13377i	− 0.285651i
$324$	0	0
$325$	0.764757i	0.0424211i
$326$	− 4.87899i	− 0.270222i
$327$	0	0
$328$	5.82416i	0.321585i
$329$	0	0
$330$	0	0
$331$	12.4384	0.683676	0.341838	−	0.939759i	$-0.388951\pi$
$331$	12.4384	0.683676	0.341838	+	0.939759i	$0.388951\pi$
$332$	1.38140	0.0758141
$333$	0	0
$334$	8.09861i	0.443136i
$335$	13.6706	0.746907
$336$	0	0
$337$	−3.16911	−0.172632	−0.0863161	−	0.996268i	$-0.527510\pi$
$337$	−3.16911	−0.172632	−0.0863161	+	0.996268i	$0.527510\pi$
$338$	− 12.3169i	− 0.669952i
$339$	0	0
$340$	19.7017	1.06848
$341$	0.130428	0.00706310
$342$	0	0
$343$	0	0
$344$	− 28.7581i	− 1.55053i
$345$	0	0
$346$	1.16534i	0.0626493i
$347$	23.5645i	1.26501i	0.774558	+	0.632503i	$0.217972\pi$
$347$	23.5645i	1.26501i	−0.774558	+	0.632503i	$0.782028\pi$
$348$	0	0
$349$	16.0140i	0.857211i	0.903492	+	0.428606i	$0.140995\pi$
$349$	16.0140i	0.857211i	−0.903492	+	0.428606i	$0.859005\pi$
$350$	0	0
$351$	0	0
$352$	3.25318	0.173395
$353$	−3.36316	−0.179003	−0.0895015	−	0.995987i	$-0.528527\pi$
$353$	−3.36316	−0.179003	−0.0895015	+	0.995987i	$0.528527\pi$
$354$	0	0
$355$	− 25.4465i	− 1.35056i
$356$	4.24218	0.224835
$357$	0	0
$358$	9.22601	0.487610
$359$	− 3.38359i	− 0.178579i	−0.996006	−	0.0892896i	$-0.971540\pi$
$359$	− 3.38359i	− 0.178579i	0.996006	−	0.0892896i	$-0.0284597\pi$
$360$	0	0
$361$	18.2025	0.958026
$362$	3.85482	0.202605
$363$	0	0
$364$	0	0
$365$	4.01890i	0.210359i
$366$	0	0
$367$	− 15.4675i	− 0.807396i	−0.914892	−	0.403698i	$-0.867725\pi$
$367$	− 15.4675i	− 0.807396i	0.914892	−	0.403698i	$-0.132275\pi$
$368$	− 8.20033i	− 0.427472i
$369$	0	0
$370$	− 8.51045i	− 0.442437i
$371$	0	0
$372$	0	0
$373$	14.8113	0.766902	0.383451	−	0.923561i	$-0.374736\pi$
$373$	14.8113	0.766902	0.383451	+	0.923561i	$0.374736\pi$
$374$	−2.24412	−0.116041
$375$	0	0
$376$	− 30.4690i	− 1.57132i
$377$	−46.6284	−2.40148
$378$	0	0
$379$	−0.481657	−0.0247410	−0.0123705	−	0.999923i	$-0.503938\pi$
$379$	−0.481657	−0.0247410	−0.0123705	+	0.999923i	$0.503938\pi$
$380$	− 3.06055i	− 0.157003i
$381$	0	0
$382$	6.02697	0.308367
$383$	17.8417	0.911669	0.455835	−	0.890065i	$-0.349341\pi$
$383$	17.8417	0.911669	0.455835	+	0.890065i	$0.349341\pi$
$384$	0	0
$385$	0	0
$386$	5.56096i	0.283045i
$387$	0	0
$388$	− 19.0162i	− 0.965401i
$389$	− 26.0373i	− 1.32014i	−0.751204	−	0.660071i	$-0.770526\pi$
$389$	− 26.0373i	− 1.32014i	0.751204	−	0.660071i	$-0.229474\pi$
$390$	0	0
$391$	35.9925i	1.82022i
$392$	0	0
$393$	0	0
$394$	6.95264	0.350269
$395$	13.0819	0.658221
$396$	0	0
$397$	− 17.6508i	− 0.885867i	−0.896554	−	0.442934i	$-0.853938\pi$
$397$	− 17.6508i	− 0.885867i	0.896554	−	0.442934i	$-0.146062\pi$
$398$	−7.94176	−0.398084
$399$	0	0
$400$	0.180983	0.00904916
$401$	− 9.71282i	− 0.485035i	−0.970147	−	0.242518i	$-0.922027\pi$
$401$	− 9.71282i	− 0.485035i	0.970147	−	0.242518i	$-0.0779732\pi$
$402$	0	0
$403$	−1.29186	−0.0643523
$404$	−1.43998	−0.0716418
$405$	0	0
$406$	0	0
$407$	− 3.00285i	− 0.148846i
$408$	0	0
$409$	1.29391i	0.0639798i	0.999488	+	0.0319899i	$0.0101844\pi$
$409$	1.29391i	0.0639798i	−0.999488	+	0.0319899i	$0.989816\pi$
$410$	− 3.75918i	− 0.185653i
$411$	0	0
$412$	− 12.7647i	− 0.628872i
$413$	0	0
$414$	0	0
$415$	−2.07107	−0.101665
$416$	−32.2220	−1.57981
$417$	0	0
$418$	0.348611i	0.0170511i
$419$	8.20815	0.400994	0.200497	−	0.979694i	$-0.435744\pi$
$419$	8.20815	0.400994	0.200497	+	0.979694i	$0.435744\pi$
$420$	0	0
$421$	16.0477	0.782117	0.391058	−	0.920366i	$-0.372109\pi$
$421$	16.0477	0.782117	0.391058	+	0.920366i	$0.372109\pi$
$422$	9.70111i	0.472243i
$423$	0	0
$424$	−19.0715	−0.926196
$425$	−0.794363	−0.0385323
$426$	0	0
$427$	0	0
$428$	− 13.1756i	− 0.636868i
$429$	0	0
$430$	18.5618i	0.895130i
$431$	− 16.2409i	− 0.782296i	−0.920328	−	0.391148i	$-0.872078\pi$
$431$	− 16.2409i	− 0.782296i	0.920328	−	0.391148i	$-0.127922\pi$
$432$	0	0
$433$	− 7.47525i	− 0.359238i	−0.983736	−	0.179619i	$-0.942514\pi$
$433$	− 7.47525i	− 0.359238i	0.983736	−	0.179619i	$-0.0574864\pi$
$434$	0	0
$435$	0	0
$436$	−2.73980	−0.131213
$437$	5.59123	0.267465
$438$	0	0
$439$	12.7742i	0.609677i	0.952404	+	0.304839i	$0.0986026\pi$
$439$	12.7742i	0.609677i	−0.952404	+	0.304839i	$0.901397\pi$
$440$	−3.10760	−0.148149
$441$	0	0
$442$	22.2275	1.05725
$443$	− 6.43543i	− 0.305756i	−0.988245	−	0.152878i	$-0.951146\pi$
$443$	− 6.43543i	− 0.305756i	0.988245	−	0.152878i	$-0.0488542\pi$
$444$	0	0
$445$	−6.36010	−0.301498
$446$	7.13570	0.337885
$447$	0	0
$448$	0	0
$449$	6.29690i	0.297169i	0.988900	+	0.148585i	$0.0474717\pi$
$449$	6.29690i	0.297169i	−0.988900	+	0.148585i	$0.952528\pi$
$450$	0	0
$451$	− 1.32640i	− 0.0624577i
$452$	21.1832i	0.996374i
$453$	0	0
$454$	4.70435i	0.220786i
$455$	0	0
$456$	0	0
$457$	−24.0714	−1.12601	−0.563006	−	0.826453i	$-0.690355\pi$
$457$	−24.0714	−1.12601	−0.563006	+	0.826453i	$0.690355\pi$
$458$	−7.76363	−0.362771
$459$	0	0
$460$	21.4573i	1.00045i
$461$	29.8904	1.39213	0.696067	−	0.717977i	$-0.254932\pi$
$461$	29.8904	1.39213	0.696067	+	0.717977i	$0.254932\pi$
$462$	0	0
$463$	−25.1319	−1.16798	−0.583989	−	0.811761i	$-0.698509\pi$
$463$	−25.1319	−1.16798	−0.583989	+	0.811761i	$0.698509\pi$
$464$	11.0348i	0.512278i
$465$	0	0
$466$	0.806582	0.0373642
$467$	−22.1660	−1.02572	−0.512860	−	0.858472i	$-0.671414\pi$
$467$	−22.1660	−1.02572	−0.512860	+	0.858472i	$0.671414\pi$
$468$	0	0
$469$	0	0
$470$	19.6661i	0.907129i
$471$	0	0
$472$	17.5526i	0.807923i
$473$	6.54940i	0.301142i
$474$	0	0
$475$	0.123400i	0.00566197i
$476$	0	0
$477$	0	0
$478$	17.5731	0.803775
$479$	−15.8987	−0.726430	−0.363215	−	0.931705i	$-0.618321\pi$
$479$	−15.8987	−0.726430	−0.363215	+	0.931705i	$0.618321\pi$
$480$	0	0
$481$	29.7426i	1.35614i
$482$	−21.2176	−0.966436
$483$	0	0
$484$	16.1591	0.734504
$485$	28.5101i	1.29458i
$486$	0	0
$487$	3.39399	0.153797	0.0768983	−	0.997039i	$-0.475498\pi$
$487$	3.39399	0.153797	0.0768983	+	0.997039i	$0.475498\pi$
$488$	−31.3908	−1.42099
$489$	0	0
$490$	0	0
$491$	30.2097i	1.36335i	0.731657	+	0.681673i	$0.238747\pi$
$491$	30.2097i	1.36335i	−0.731657	+	0.681673i	$0.761253\pi$
$492$	0	0
$493$	− 48.4335i	− 2.18134i
$494$	− 3.45291i	− 0.155354i
$495$	0	0
$496$	0.305725i	0.0137275i
$497$	0	0
$498$	0	0
$499$	−36.0289	−1.61288	−0.806438	−	0.591319i	$-0.798608\pi$
$499$	−36.0289	−1.61288	−0.806438	+	0.591319i	$0.798608\pi$
$500$	16.6622	0.745157
$501$	0	0
$502$	15.4161i	0.688053i
$503$	9.73396	0.434016	0.217008	−	0.976170i	$-0.430370\pi$
$503$	9.73396	0.434016	0.217008	+	0.976170i	$0.430370\pi$
$504$	0	0
$505$	2.15890	0.0960698
$506$	− 2.44409i	− 0.108653i
$507$	0	0
$508$	28.4532	1.26241
$509$	−17.0295	−0.754819	−0.377410	−	0.926046i	$-0.623185\pi$
$509$	−17.0295	−0.754819	−0.377410	+	0.926046i	$0.623185\pi$
$510$	0	0
$511$	0	0
$512$	14.0525i	0.621038i
$513$	0	0
$514$	9.04639i	0.399019i
$515$	19.1375i	0.843301i
$516$	0	0
$517$	6.93904i	0.305179i
$518$	0	0
$519$	0	0
$520$	30.7800	1.34979
$521$	−27.3357	−1.19760	−0.598799	−	0.800899i	$-0.704355\pi$
$521$	−27.3357	−1.19760	−0.598799	+	0.800899i	$0.704355\pi$
$522$	0	0
$523$	5.82076i	0.254524i	0.991869	+	0.127262i	$0.0406189\pi$
$523$	5.82076i	0.254524i	−0.991869	+	0.127262i	$0.959381\pi$
$524$	−1.26432	−0.0552322
$525$	0	0
$526$	−6.57232	−0.286567
$527$	− 1.34188i	− 0.0584530i
$528$	0	0
$529$	−16.1997	−0.704335
$530$	12.3096	0.534697
$531$	0	0
$532$	0	0
$533$	13.1377i	0.569056i
$534$	0	0
$535$	19.7536i	0.854023i
$536$	− 14.7970i	− 0.639133i
$537$	0	0
$538$	4.17007i	0.179784i
$539$	0	0
$540$	0	0
$541$	−15.5802	−0.669845	−0.334922	−	0.942246i	$-0.608710\pi$
$541$	−15.5802	−0.669845	−0.334922	+	0.942246i	$0.608710\pi$
$542$	4.63758	0.199201
$543$	0	0
$544$	− 33.4694i	− 1.43499i
$545$	4.10765	0.175953
$546$	0	0
$547$	−2.86923	−0.122679	−0.0613397	−	0.998117i	$-0.519537\pi$
$547$	−2.86923	−0.122679	−0.0613397	+	0.998117i	$0.519537\pi$
$548$	− 15.5142i	− 0.662735i
$549$	0	0
$550$	0.0539416	0.00230008
$551$	−7.52387	−0.320528
$552$	0	0
$553$	0	0
$554$	− 5.85011i	− 0.248547i
$555$	0	0
$556$	18.2441i	0.773723i
$557$	5.56301i	0.235712i	0.993031	+	0.117856i	$0.0376022\pi$
$557$	5.56301i	0.235712i	−0.993031	+	0.117856i	$0.962398\pi$
$558$	0	0
$559$	− 64.8703i	− 2.74372i
$560$	0	0
$561$	0	0
$562$	−3.88945	−0.164067
$563$	−13.4984	−0.568891	−0.284445	−	0.958692i	$-0.591809\pi$
$563$	−13.4984	−0.568891	−0.284445	+	0.958692i	$0.591809\pi$
$564$	0	0
$565$	− 31.7590i	− 1.33611i
$566$	10.6219	0.446470
$567$	0	0
$568$	−27.5430	−1.15568
$569$	− 1.11892i	− 0.0469074i	−0.999725	−	0.0234537i	$-0.992534\pi$
$569$	− 1.11892i	− 0.0469074i	0.999725	−	0.0234537i	$-0.00746622\pi$
$570$	0	0
$571$	−45.2921	−1.89542	−0.947709	−	0.319137i	$-0.896607\pi$
$571$	−45.2921	−1.89542	−0.947709	+	0.319137i	$0.896607\pi$
$572$	4.67558	0.195496
$573$	0	0
$574$	0	0
$575$	− 0.865147i	− 0.0360791i
$576$	0	0
$577$	28.8678i	1.20178i	0.799331	+	0.600891i	$0.205187\pi$
$577$	28.8678i	1.20178i	−0.799331	+	0.600891i	$0.794813\pi$
$578$	11.2113i	0.466330i
$579$	0	0
$580$	− 28.8742i	− 1.19893i
$581$	0	0
$582$	0	0
$583$	4.34337	0.179884
$584$	4.35003	0.180006
$585$	0	0
$586$	11.7726i	0.486323i
$587$	−29.1477	−1.20305	−0.601527	−	0.798852i	$-0.705441\pi$
$587$	−29.1477	−1.20305	−0.601527	+	0.798852i	$0.705441\pi$
$588$	0	0
$589$	−0.208453	−0.00858914
$590$	− 11.3292i	− 0.466417i
$591$	0	0
$592$	7.03871	0.289289
$593$	−36.9631	−1.51789	−0.758946	−	0.651154i	$-0.774285\pi$
$593$	−36.9631	−1.51789	−0.758946	+	0.651154i	$0.774285\pi$
$594$	0	0
$595$	0	0
$596$	30.8523i	1.26376i
$597$	0	0
$598$	24.2081i	0.989943i
$599$	26.3316i	1.07588i	0.842983	+	0.537940i	$0.180797\pi$
$599$	26.3316i	1.07588i	−0.842983	+	0.537940i	$0.819203\pi$
$600$	0	0
$601$	7.86645i	0.320879i	0.987046	+	0.160440i	$0.0512912\pi$
$601$	7.86645i	0.320879i	−0.987046	+	0.160440i	$0.948709\pi$
$602$	0	0
$603$	0	0
$604$	−25.6024	−1.04175
$605$	−24.2266	−0.984951
$606$	0	0
$607$	22.7603i	0.923812i	0.886929	+	0.461906i	$0.152834\pi$
$607$	22.7603i	0.923812i	−0.886929	+	0.461906i	$0.847166\pi$
$608$	−5.19928	−0.210858
$609$	0	0
$610$	20.2610	0.820345
$611$	− 68.7296i	− 2.78050i
$612$	0	0
$613$	14.4090	0.581972	0.290986	−	0.956727i	$-0.406017\pi$
$613$	14.4090	0.581972	0.290986	+	0.956727i	$0.406017\pi$
$614$	−9.74614	−0.393322
$615$	0	0
$616$	0	0
$617$	14.8252i	0.596841i	0.954435	+	0.298420i	$0.0964597\pi$
$617$	14.8252i	0.596841i	−0.954435	+	0.298420i	$0.903540\pi$
$618$	0	0
$619$	− 34.0864i	− 1.37005i	−0.728520	−	0.685025i	$-0.759791\pi$
$619$	− 34.0864i	− 1.37005i	0.728520	−	0.685025i	$-0.240209\pi$
$620$	− 0.799974i	− 0.0321277i
$621$	0	0
$622$	19.4254i	0.778886i
$623$	0	0
$624$	0	0
$625$	−25.6718	−1.02687
$626$	−16.9349	−0.676855
$627$	0	0
$628$	− 27.6972i	− 1.10524i
$629$	−30.8940	−1.23182
$630$	0	0
$631$	−7.10726	−0.282935	−0.141468	−	0.989943i	$-0.545182\pi$
$631$	−7.10726	−0.282935	−0.141468	+	0.989943i	$0.545182\pi$
$632$	− 14.1597i	− 0.563244i
$633$	0	0
$634$	−15.1699	−0.602474
$635$	−42.6586	−1.69285
$636$	0	0
$637$	0	0
$638$	3.28890i	0.130209i
$639$	0	0
$640$	− 24.1014i	− 0.952693i
$641$	− 8.59556i	− 0.339504i	−0.985487	−	0.169752i	$-0.945703\pi$
$641$	− 8.59556i	− 0.339504i	0.985487	−	0.169752i	$-0.0542967\pi$
$642$	0	0
$643$	− 47.3240i	− 1.86628i	−0.359516	−	0.933139i	$-0.617058\pi$
$643$	− 47.3240i	− 1.86628i	0.359516	−	0.933139i	$-0.382942\pi$
$644$	0	0
$645$	0	0
$646$	3.58658	0.141112
$647$	37.5030	1.47439	0.737197	−	0.675678i	$-0.236149\pi$
$647$	37.5030	1.47439	0.737197	+	0.675678i	$0.236149\pi$
$648$	0	0
$649$	− 3.99744i	− 0.156913i
$650$	−0.534279	−0.0209562
$651$	0	0
$652$	−10.5588	−0.413515
$653$	− 2.66764i	− 0.104393i	−0.998637	−	0.0521964i	$-0.983378\pi$
$653$	− 2.66764i	− 0.104393i	0.998637	−	0.0521964i	$-0.0166222\pi$
$654$	0	0
$655$	1.89554	0.0740649
$656$	3.10909	0.121390
$657$	0	0
$658$	0	0
$659$	− 39.7218i	− 1.54734i	−0.633589	−	0.773670i	$-0.718419\pi$
$659$	− 39.7218i	− 1.54734i	0.633589	−	0.773670i	$-0.281581\pi$
$660$	0	0
$661$	− 7.68591i	− 0.298947i	−0.988766	−	0.149474i	$-0.952242\pi$
$661$	− 7.68591i	− 0.298947i	0.988766	−	0.149474i	$-0.0477579\pi$
$662$	8.68978i	0.337738i
$663$	0	0
$664$	2.24171i	0.0869951i
$665$	0	0
$666$	0	0
$667$	52.7493	2.04246
$668$	17.5265	0.678121
$669$	0	0
$670$	9.55065i	0.368974i
$671$	7.14896	0.275983
$672$	0	0
$673$	7.64404	0.294656	0.147328	−	0.989088i	$-0.452933\pi$
$673$	7.64404	0.294656	0.147328	+	0.989088i	$0.452933\pi$
$674$	− 2.21402i	− 0.0852808i
$675$	0	0
$676$	−26.6555	−1.02521
$677$	14.8718	0.571571	0.285785	−	0.958294i	$-0.407746\pi$
$677$	14.8718	0.571571	0.285785	+	0.958294i	$0.407746\pi$
$678$	0	0
$679$	0	0
$680$	31.9716i	1.22606i
$681$	0	0
$682$	0.0911207i	0.00348919i
$683$	38.1933i	1.46142i	0.682685	+	0.730712i	$0.260812\pi$
$683$	38.1933i	1.46142i	−0.682685	+	0.730712i	$0.739188\pi$
$684$	0	0
$685$	23.2598i	0.888710i
$686$	0	0
$687$	0	0
$688$	−15.3519	−0.585284
$689$	−43.0201	−1.63893
$690$	0	0
$691$	10.5130i	0.399935i	0.979803	+	0.199968i	$0.0640837\pi$
$691$	10.5130i	0.399935i	−0.979803	+	0.199968i	$0.935916\pi$
$692$	2.52197	0.0958708
$693$	0	0
$694$	−16.4627	−0.624917
$695$	− 27.3526i	− 1.03754i
$696$	0	0
$697$	−13.6463	−0.516890
$698$	−11.1878	−0.423465
$699$	0	0
$700$	0	0
$701$	26.6270i	1.00569i	0.864378	+	0.502843i	$0.167713\pi$
$701$	26.6270i	1.00569i	−0.864378	+	0.502843i	$0.832287\pi$
$702$	0	0
$703$	4.79920i	0.181005i
$704$	0.809058i	0.0304925i
$705$	0	0
$706$	− 2.34959i	− 0.0884280i
$707$	0	0
$708$	0	0
$709$	13.5562	0.509114	0.254557	−	0.967058i	$-0.418070\pi$
$709$	13.5562	0.509114	0.254557	+	0.967058i	$0.418070\pi$
$710$	17.7775	0.667179
$711$	0	0
$712$	6.88413i	0.257994i
$713$	1.46145	0.0547316
$714$	0	0
$715$	−7.00988	−0.262155
$716$	− 19.9664i	− 0.746178i
$717$	0	0
$718$	2.36386	0.0882186
$719$	−25.4247	−0.948180	−0.474090	−	0.880476i	$-0.657223\pi$
$719$	−25.4247	−0.948180	−0.474090	+	0.880476i	$0.657223\pi$
$720$	0	0
$721$	0	0
$722$	12.7167i	0.473268i
$723$	0	0
$724$	− 8.34236i	− 0.310042i
$725$	1.16419i	0.0432369i
$726$	0	0
$727$	42.5225i	1.57707i	0.614988	+	0.788536i	$0.289161\pi$
$727$	42.5225i	1.57707i	−0.614988	+	0.788536i	$0.710839\pi$
$728$	0	0
$729$	0	0
$730$	−2.80771	−0.103918
$731$	67.3816	2.49220
$732$	0	0
$733$	− 18.4165i	− 0.680229i	−0.940384	−	0.340114i	$-0.889534\pi$
$733$	− 18.4165i	− 0.680229i	0.940384	−	0.340114i	$-0.110466\pi$
$734$	10.8060	0.398856
$735$	0	0
$736$	36.4518	1.34363
$737$	3.36988i	0.124131i
$738$	0	0
$739$	8.71395	0.320548	0.160274	−	0.987073i	$-0.448762\pi$
$739$	8.71395	0.320548	0.160274	+	0.987073i	$0.448762\pi$
$740$	−18.4178	−0.677052
$741$	0	0
$742$	0	0
$743$	1.30005i	0.0476941i	0.999716	+	0.0238470i	$0.00759147\pi$
$743$	1.30005i	0.0476941i	−0.999716	+	0.0238470i	$0.992409\pi$
$744$	0	0
$745$	− 46.2554i	− 1.69467i
$746$	10.3476i	0.378852i
$747$	0	0
$748$	4.85658i	0.177574i
$749$	0	0
$750$	0	0
$751$	21.4698	0.783443	0.391722	−	0.920084i	$-0.371880\pi$
$751$	21.4698	0.783443	0.391722	+	0.920084i	$0.371880\pi$
$752$	−16.2652	−0.593130
$753$	0	0
$754$	− 32.5758i	− 1.18634i
$755$	38.3845	1.39696
$756$	0	0
$757$	16.7524	0.608877	0.304439	−	0.952532i	$-0.401531\pi$
$757$	16.7524	0.608877	0.304439	+	0.952532i	$0.401531\pi$
$758$	− 0.336498i	− 0.0122221i
$759$	0	0
$760$	4.96661	0.180158
$761$	−34.3655	−1.24575	−0.622874	−	0.782322i	$-0.714035\pi$
$761$	−34.3655	−1.24575	−0.622874	+	0.782322i	$0.714035\pi$
$762$	0	0
$763$	0	0
$764$	− 13.0432i	− 0.471886i
$765$	0	0
$766$	12.4647i	0.450367i
$767$	39.5937i	1.42965i
$768$	0	0
$769$	− 14.6402i	− 0.527938i	−0.964531	−	0.263969i	$-0.914968\pi$
$769$	− 14.6402i	− 0.527938i	0.964531	−	0.263969i	$-0.0850317\pi$
$770$	0	0
$771$	0	0
$772$	12.0347	0.433138
$773$	−43.2233	−1.55464	−0.777318	−	0.629108i	$-0.783420\pi$
$773$	−43.2233	−1.55464	−0.777318	+	0.629108i	$0.783420\pi$
$774$	0	0
$775$	0.0322545i	0.00115862i
$776$	30.8591	1.10778
$777$	0	0
$778$	18.1903	0.652154
$779$	2.11987i	0.0759523i
$780$	0	0
$781$	6.27268	0.224454
$782$	−25.1453	−0.899193
$783$	0	0
$784$	0	0
$785$	41.5252i	1.48210i
$786$	0	0
$787$	28.5985i	1.01942i	0.860345	+	0.509712i	$0.170248\pi$
$787$	28.5985i	1.01942i	−0.860345	+	0.509712i	$0.829752\pi$
$788$	− 15.0465i	− 0.536009i
$789$	0	0
$790$	9.13933i	0.325163i
$791$	0	0
$792$	0	0
$793$	−70.8088	−2.51449
$794$	12.3313	0.437621
$795$	0	0
$796$	17.1871i	0.609179i
$797$	−41.0119	−1.45272	−0.726358	−	0.687317i	$-0.758789\pi$
$797$	−41.0119	−1.45272	−0.726358	+	0.687317i	$0.758789\pi$
$798$	0	0
$799$	71.3903	2.52561
$800$	0.804499i	0.0284434i
$801$	0	0
$802$	6.78562	0.239609
$803$	−0.990680	−0.0349603
$804$	0	0
$805$	0	0
$806$	− 0.902529i	− 0.0317902i
$807$	0	0
$808$	− 2.33678i	− 0.0822075i
$809$	50.4495i	1.77371i	0.462048	+	0.886855i	$0.347115\pi$
$809$	50.4495i	1.77371i	−0.462048	+	0.886855i	$0.652885\pi$
$810$	0	0
$811$	− 24.9315i	− 0.875465i	−0.899105	−	0.437732i	$-0.855782\pi$
$811$	− 24.9315i	− 0.875465i	0.899105	−	0.437732i	$-0.144218\pi$
$812$	0	0
$813$	0	0
$814$	2.09787	0.0735303
$815$	15.8303	0.554513
$816$	0	0
$817$	− 10.4674i	− 0.366206i
$818$	−0.903959	−0.0316062
$819$	0	0
$820$	−8.13538	−0.284100
$821$	7.86728i	0.274570i	0.990532	+	0.137285i	$0.0438376\pi$
$821$	7.86728i	0.274570i	−0.990532	+	0.137285i	$0.956162\pi$
$822$	0	0
$823$	36.7198	1.27997	0.639986	−	0.768386i	$-0.278940\pi$
$823$	36.7198	1.27997	0.639986	+	0.768386i	$0.278940\pi$
$824$	20.7143	0.721618
$825$	0	0
$826$	0	0
$827$	− 19.7296i	− 0.686067i	−0.939323	−	0.343033i	$-0.888546\pi$
$827$	− 19.7296i	− 0.686067i	0.939323	−	0.343033i	$-0.111454\pi$
$828$	0	0
$829$	26.1288i	0.907492i	0.891131	+	0.453746i	$0.149913\pi$
$829$	26.1288i	0.907492i	−0.891131	+	0.453746i	$0.850087\pi$
$830$	− 1.44690i	− 0.0502227i
$831$	0	0
$832$	− 8.01353i	− 0.277819i
$833$	0	0
$834$	0	0
$835$	−26.2767	−0.909342
$836$	0.754442	0.0260929
$837$	0	0
$838$	5.73442i	0.198092i
$839$	−53.3357	−1.84135	−0.920676	−	0.390328i	$-0.872362\pi$
$839$	−53.3357	−1.84135	−0.920676	+	0.390328i	$0.872362\pi$
$840$	0	0
$841$	−41.9824	−1.44767
$842$	11.2113i	0.386368i
$843$	0	0
$844$	20.9945	0.722662
$845$	39.9634	1.37478
$846$	0	0
$847$	0	0
$848$	10.1809i	0.349613i
$849$	0	0
$850$	− 0.554963i	− 0.0190351i
$851$	− 33.6469i	− 1.15340i
$852$	0	0
$853$	44.6277i	1.52802i	0.645202	+	0.764012i	$0.276773\pi$
$853$	44.6277i	1.52802i	−0.645202	+	0.764012i	$0.723227\pi$
$854$	0	0
$855$	0	0
$856$	21.3812	0.730793
$857$	−39.5137	−1.34976	−0.674882	−	0.737926i	$-0.735805\pi$
$857$	−39.5137	−1.34976	−0.674882	+	0.737926i	$0.735805\pi$
$858$	0	0
$859$	41.5978i	1.41930i	0.704555	+	0.709650i	$0.251147\pi$
$859$	41.5978i	1.41930i	−0.704555	+	0.709650i	$0.748853\pi$
$860$	40.1703	1.36980
$861$	0	0
$862$	11.3463	0.386457
$863$	− 21.8415i	− 0.743493i	−0.928334	−	0.371747i	$-0.878759\pi$
$863$	− 21.8415i	− 0.743493i	0.928334	−	0.371747i	$-0.121241\pi$
$864$	0	0
$865$	−3.78107	−0.128560
$866$	5.22240	0.177464
$867$	0	0
$868$	0	0
$869$	3.22475i	0.109392i
$870$	0	0
$871$	− 33.3779i	− 1.13097i
$872$	− 4.44609i	− 0.150564i
$873$	0	0
$874$	3.90617i	0.132128i
$875$	0	0
$876$	0	0
$877$	39.2454	1.32522	0.662611	−	0.748963i	$-0.269448\pi$
$877$	39.2454	1.32522	0.662611	+	0.748963i	$0.269448\pi$
$878$	−8.92436	−0.301182
$879$	0	0
$880$	1.65892i	0.0559221i
$881$	2.29892	0.0774524	0.0387262	−	0.999250i	$-0.487670\pi$
$881$	2.29892	0.0774524	0.0387262	+	0.999250i	$0.487670\pi$
$882$	0	0
$883$	−5.14326	−0.173085	−0.0865423	−	0.996248i	$-0.527582\pi$
$883$	−5.14326	−0.173085	−0.0865423	+	0.996248i	$0.527582\pi$
$884$	− 48.1033i	− 1.61789i
$885$	0	0
$886$	4.49595	0.151045
$887$	−13.2017	−0.443269	−0.221634	−	0.975130i	$-0.571139\pi$
$887$	−13.2017	−0.443269	−0.221634	+	0.975130i	$0.571139\pi$
$888$	0	0
$889$	0	0
$890$	− 4.44333i	− 0.148941i
$891$	0	0
$892$	− 15.4426i	− 0.517058i
$893$	− 11.0901i	− 0.371115i
$894$	0	0
$895$	29.9347i	1.00061i
$896$	0	0
$897$	0	0
$898$	−4.39918	−0.146802
$899$	−1.96660	−0.0655899
$900$	0	0
$901$	− 44.6855i	− 1.48869i
$902$	0.926657	0.0308543
$903$	0	0
$904$	−34.3757	−1.14332
$905$	12.5073i	0.415758i
$906$	0	0
$907$	12.1283	0.402714	0.201357	−	0.979518i	$-0.435465\pi$
$907$	12.1283	0.402714	0.201357	+	0.979518i	$0.435465\pi$
$908$	10.1809	0.337864
$909$	0	0
$910$	0	0
$911$	− 7.54730i	− 0.250053i	−0.992153	−	0.125027i	$-0.960098\pi$
$911$	− 7.54730i	− 0.250053i	0.992153	−	0.125027i	$-0.0399016\pi$
$912$	0	0
$913$	− 0.510529i	− 0.0168960i
$914$	− 16.8169i	− 0.556253i
$915$	0	0
$916$	16.8016i	0.555140i
$917$	0	0
$918$	0	0
$919$	3.76262	0.124117	0.0620587	−	0.998072i	$-0.480233\pi$
$919$	3.76262	0.124117	0.0620587	+	0.998072i	$0.480233\pi$
$920$	−34.8205	−1.14800
$921$	0	0
$922$	20.8822i	0.687718i
$923$	−62.1295	−2.04502
$924$	0	0
$925$	0.742595	0.0244164
$926$	− 17.5578i	− 0.576985i
$927$	0	0
$928$	−49.0515	−1.61019
$929$	−40.9953	−1.34501	−0.672507	−	0.740091i	$-0.734782\pi$
$929$	−40.9953	−1.34501	−0.672507	+	0.740091i	$0.734782\pi$
$930$	0	0
$931$	0	0
$932$	− 1.74556i	− 0.0571776i
$933$	0	0
$934$	− 15.4857i	− 0.506709i
$935$	− 7.28125i	− 0.238122i
$936$	0	0
$937$	− 13.0152i	− 0.425187i	−0.977141	−	0.212593i	$-0.931809\pi$
$937$	− 13.0152i	− 0.425187i	0.977141	−	0.212593i	$-0.0681910\pi$
$938$	0	0
$939$	0	0
$940$	42.5601	1.38816
$941$	10.7228	0.349554	0.174777	−	0.984608i	$-0.444080\pi$
$941$	10.7228	0.349554	0.174777	+	0.984608i	$0.444080\pi$
$942$	0	0
$943$	− 14.8623i	− 0.483982i
$944$	9.37004	0.304969
$945$	0	0
$946$	−4.57558	−0.148765
$947$	35.9063i	1.16680i	0.812186	+	0.583398i	$0.198277\pi$
$947$	35.9063i	1.16680i	−0.812186	+	0.583398i	$0.801723\pi$
$948$	0	0
$949$	9.81246	0.318526
$950$	−0.0862103	−0.00279703
$951$	0	0
$952$	0	0
$953$	17.6691i	0.572360i	0.958176	+	0.286180i	$0.0923855\pi$
$953$	17.6691i	0.572360i	−0.958176	+	0.286180i	$0.907615\pi$
$954$	0	0
$955$	19.5551i	0.632787i
$956$	− 38.0307i	− 1.23000i
$957$	0	0
$958$	− 11.1072i	− 0.358859i
$959$	0	0
$960$	0	0
$961$	30.9455	0.998242
$962$	−20.7789	−0.669939
$963$	0	0
$964$	45.9179i	1.47892i
$965$	−18.0431	−0.580826
$966$	0	0
$967$	−50.3563	−1.61935	−0.809675	−	0.586878i	$-0.800357\pi$
$967$	−50.3563	−1.61935	−0.809675	+	0.586878i	$0.800357\pi$
$968$	26.2227i	0.842828i
$969$	0	0
$970$	−19.9179	−0.639525
$971$	51.1649	1.64196	0.820980	−	0.570956i	$-0.193427\pi$
$971$	51.1649	1.64196	0.820980	+	0.570956i	$0.193427\pi$
$972$	0	0
$973$	0	0
$974$	2.37113i	0.0759759i
$975$	0	0
$976$	16.7572i	0.536386i
$977$	27.9975i	0.895720i	0.894104	+	0.447860i	$0.147814\pi$
$977$	27.9975i	0.895720i	−0.894104	+	0.447860i	$0.852186\pi$
$978$	0	0
$979$	− 1.56780i	− 0.0501070i
$980$	0	0
$981$	0	0
$982$	−21.1053	−0.673497
$983$	24.0255	0.766295	0.383147	−	0.923687i	$-0.374840\pi$
$983$	24.0255	0.766295	0.383147	+	0.923687i	$0.374840\pi$
$984$	0	0
$985$	22.5585i	0.718773i
$986$	33.8369	1.07759
$987$	0	0
$988$	−7.47258	−0.237734
$989$	73.3859i	2.33353i
$990$	0	0
$991$	34.2532	1.08809	0.544045	−	0.839056i	$-0.316892\pi$
$991$	34.2532	1.08809	0.544045	+	0.839056i	$0.316892\pi$
$992$	−1.35900	−0.0431482
$993$	0	0
$994$	0	0
$995$	− 25.7678i	− 0.816893i
$996$	0	0
$997$	39.1980i	1.24141i	0.784043	+	0.620706i	$0.213154\pi$
$997$	39.1980i	1.24141i	−0.784043	+	0.620706i	$0.786846\pi$
$998$	− 25.1707i	− 0.796765i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.c.f.1322.11	yes	16
3.2	odd	2	inner	1323.2.c.f.1322.6	yes	16
7.6	odd	2	inner	1323.2.c.f.1322.12	yes	16
21.20	even	2	inner	1323.2.c.f.1322.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.2.c.f.1322.5	✓	16	21.20	even	2	inner
1323.2.c.f.1322.6	yes	16	3.2	odd	2	inner
1323.2.c.f.1322.11	yes	16	1.1	even	1	trivial
1323.2.c.f.1322.12	yes	16	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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.698626i q^{2} +1.51192 q^{4} -2.26676 q^{5} +2.45352i q^{8} +O(q^{10})\)	\(q+0.698626i q^{2} +1.51192 q^{4} -2.26676 q^{5} +2.45352i q^{8} -1.58361i q^{10} -0.558767i q^{11} +5.53446i q^{13} +1.30975 q^{16} -5.74871 q^{17} +0.893030i q^{19} -3.42716 q^{20} +0.390369 q^{22} -6.26097i q^{23} +0.138181 q^{25} -3.86651 q^{26} +8.42510i q^{29} +0.233422i q^{31} +5.82206i q^{32} -4.01620i q^{34} +5.37407 q^{37} -0.623893 q^{38} -5.56153i q^{40} +2.37380 q^{41} -11.7212 q^{43} -0.844812i q^{44} +4.37407 q^{46} -12.4185 q^{47} +0.0965369i q^{50} +8.36767i q^{52} +7.77314i q^{53} +1.26659i q^{55} -5.88599 q^{58} +7.15404 q^{59} +12.7942i q^{61} -0.163075 q^{62} -1.44794 q^{64} -12.5453i q^{65} -6.03093 q^{67} -8.69161 q^{68} +11.2259i q^{71} -1.77298i q^{73} +3.75446i q^{74} +1.35019i q^{76} -5.77119 q^{79} -2.96889 q^{80} +1.65840i q^{82} +0.913671 q^{83} +13.0309 q^{85} -8.18871i q^{86} +1.37094 q^{88} +2.80582 q^{89} -9.46610i q^{92} -8.67587i q^{94} -2.02428i q^{95} -12.5775i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 48 q^{16} + 64 q^{22} - 16 q^{25} + 32 q^{37} - 16 q^{43} + 16 q^{46} - 32 q^{64} + 48 q^{67} - 64 q^{79} + 64 q^{85} - 176 q^{88}+O(q^{100}) \) 16 * q - 16 * q^4 + 48 * q^16 + 64 * q^22 - 16 * q^25 + 32 * q^37 - 16 * q^43 + 16 * q^46 - 32 * q^64 + 48 * q^67 - 64 * q^79 + 64 * q^85 - 176 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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