LMFDB - Embedded newform 1323.2.c.e.1322.5

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} - 12x^{14} + 106x^{12} - 384x^{10} + 1005x^{8} - 1200x^{6} + 1030x^{4} - 252x^{2} + 49 $ x^16 - 12x^14 + 106x^12 - 384x^10 + 1005x^8 - 1200x^6 + 1030x^4 - 252*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1322.5
Root			$0.954123 - 0.550863i$ of defining polynomial
Character	$\chi$	$=$	1323.1322
Dual form			1323.2.c.e.1322.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.10173i q^{2} +0.786199 q^{4} -2.47687 q^{5} -3.06963i q^{8} +O(q^{10})$	$q-1.10173i q^{2} +0.786199 q^{4} -2.47687 q^{5} -3.06963i q^{8} +2.72883i q^{10} +3.81257i q^{11} +0.163636i q^{13} -1.80949 q^{16} -5.85222 q^{17} +7.11169i q^{19} -1.94731 q^{20} +4.20041 q^{22} +5.54041i q^{23} +1.13489 q^{25} +0.180282 q^{26} -3.32632i q^{29} -0.133537i q^{31} -4.14569i q^{32} +6.44754i q^{34} -7.98251 q^{37} +7.83514 q^{38} +7.60308i q^{40} -9.65151 q^{41} +12.0944 q^{43} +2.99744i q^{44} +6.10401 q^{46} +5.79697 q^{47} -1.25034i q^{50} +0.128651i q^{52} +8.50795i q^{53} -9.44325i q^{55} -3.66470 q^{58} +3.09830 q^{59} +3.22510i q^{61} -0.147121 q^{62} -8.18640 q^{64} -0.405306i q^{65} +7.59133 q^{67} -4.60101 q^{68} +12.8279i q^{71} +10.5633i q^{73} +8.79454i q^{74} +5.59121i q^{76} -10.3123 q^{79} +4.48188 q^{80} +10.6333i q^{82} +7.08378 q^{83} +14.4952 q^{85} -13.3247i q^{86} +11.7032 q^{88} +4.95619 q^{89} +4.35586i q^{92} -6.38667i q^{94} -17.6147i q^{95} +4.99484i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 48 q^{16} + 16 q^{22} + 32 q^{25} - 16 q^{37} + 32 q^{43} - 80 q^{46} - 96 q^{58} - 176 q^{64} + 96 q^{67} - 64 q^{79} - 32 q^{85} + 112 q^{88}+O(q^{100}) $ 16 * q - 16 * q^4 + 48 * q^16 + 16 * q^22 + 32 * q^25 - 16 * q^37 + 32 * q^43 - 80 * q^46 - 96 * q^58 - 176 * q^64 + 96 * q^67 - 64 * q^79 - 32 * q^85 + 112 * 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 $

$2$	− 1.10173i	− 0.779038i	−0.921018	−	0.389519i	$-0.872641\pi$
$2$	− 1.10173i	− 0.779038i	0.921018	−	0.389519i	$-0.127359\pi$
$3$	0	0
$3$	0	0
$4$	0.786199	0.393100
$5$	−2.47687	−1.10769	−0.553845	−	0.832620i	$-0.686840\pi$
$5$	−2.47687	−1.10769	−0.553845	+	0.832620i	$0.686840\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 3.06963i	− 1.08528i
$9$	0	0
$10$	2.72883i	0.862933i
$11$	3.81257i	1.14953i	0.818317	+	0.574767i	$0.194907\pi$
$11$	3.81257i	1.14953i	−0.818317	+	0.574767i	$0.805093\pi$
$12$	0	0
$13$	0.163636i	0.0453845i	0.999742	+	0.0226923i	$0.00722379\pi$
$13$	0.163636i	0.0453845i	−0.999742	+	0.0226923i	$0.992776\pi$
$14$	0	0
$15$	0	0
$16$	−1.80949	−0.452373
$17$	−5.85222	−1.41937	−0.709686	−	0.704519i	$-0.751163\pi$
$17$	−5.85222	−1.41937	−0.709686	+	0.704519i	$0.751163\pi$
$18$	0	0
$19$	7.11169i	1.63153i	0.578381	+	0.815767i	$0.303685\pi$
$19$	7.11169i	1.63153i	−0.578381	+	0.815767i	$0.696315\pi$
$20$	−1.94731	−0.435433
$21$	0	0
$22$	4.20041	0.895531
$23$	5.54041i	1.15525i	0.816301	+	0.577627i	$0.196021\pi$
$23$	5.54041i	1.15525i	−0.816301	+	0.577627i	$0.803979\pi$
$24$	0	0
$25$	1.13489	0.226978
$26$	0.180282	0.0353563
$27$	0	0
$28$	0	0
$29$	− 3.32632i	− 0.617683i	−0.951114	−	0.308841i	$-0.900059\pi$
$29$	− 3.32632i	− 0.617683i	0.951114	−	0.308841i	$-0.0999412\pi$
$30$	0	0
$31$	− 0.133537i	− 0.0239839i	−0.999928	−	0.0119920i	$-0.996183\pi$
$31$	− 0.133537i	− 0.0239839i	0.999928	−	0.0119920i	$-0.00381725\pi$
$32$	− 4.14569i	− 0.732862i
$33$	0	0
$34$	6.44754i	1.10574i
$35$	0	0
$36$	0	0
$37$	−7.98251	−1.31232	−0.656158	−	0.754623i	$-0.727820\pi$
$37$	−7.98251	−1.31232	−0.656158	+	0.754623i	$0.727820\pi$
$38$	7.83514	1.27103
$39$	0	0
$40$	7.60308i	1.20215i
$41$	−9.65151	−1.50731	−0.753657	−	0.657268i	$-0.771712\pi$
$41$	−9.65151	−1.50731	−0.753657	+	0.657268i	$0.771712\pi$
$42$	0	0
$43$	12.0944	1.84437	0.922186	−	0.386746i	$-0.126401\pi$
$43$	12.0944	1.84437	0.922186	+	0.386746i	$0.126401\pi$
$44$	2.99744i	0.451881i
$45$	0	0
$46$	6.10401	0.899987
$47$	5.79697	0.845575	0.422787	−	0.906229i	$-0.361052\pi$
$47$	5.79697	0.845575	0.422787	+	0.906229i	$0.361052\pi$
$48$	0	0
$49$	0	0
$50$	− 1.25034i	− 0.176825i
$51$	0	0
$52$	0.128651i	0.0178406i
$53$	8.50795i	1.16866i	0.811518	+	0.584328i	$0.198642\pi$
$53$	8.50795i	1.16866i	−0.811518	+	0.584328i	$0.801358\pi$
$54$	0	0
$55$	− 9.44325i	− 1.27333i
$56$	0	0
$57$	0	0
$58$	−3.66470	−0.481198
$59$	3.09830	0.403364	0.201682	−	0.979451i	$-0.435359\pi$
$59$	3.09830	0.403364	0.201682	+	0.979451i	$0.435359\pi$
$60$	0	0
$61$	3.22510i	0.412932i	0.978454	+	0.206466i	$0.0661963\pi$
$61$	3.22510i	0.412932i	−0.978454	+	0.206466i	$0.933804\pi$
$62$	−0.147121	−0.0186844
$63$	0	0
$64$	−8.18640	−1.02330
$65$	− 0.405306i	− 0.0502720i
$66$	0	0
$67$	7.59133	0.927429	0.463714	−	0.885985i	$-0.346516\pi$
$67$	7.59133	0.927429	0.463714	+	0.885985i	$0.346516\pi$
$68$	−4.60101	−0.557954
$69$	0	0
$70$	0	0
$71$	12.8279i	1.52239i	0.648521	+	0.761197i	$0.275388\pi$
$71$	12.8279i	1.52239i	−0.648521	+	0.761197i	$0.724612\pi$
$72$	0	0
$73$	10.5633i	1.23634i	0.786043	+	0.618172i	$0.212126\pi$
$73$	10.5633i	1.23634i	−0.786043	+	0.618172i	$0.787874\pi$
$74$	8.79454i	1.02234i
$75$	0	0
$76$	5.59121i	0.641355i
$77$	0	0
$78$	0	0
$79$	−10.3123	−1.16022	−0.580110	−	0.814538i	$-0.696991\pi$
$79$	−10.3123	−1.16022	−0.580110	+	0.814538i	$0.696991\pi$
$80$	4.48188	0.501090
$81$	0	0
$82$	10.6333i	1.17425i
$83$	7.08378	0.777546	0.388773	−	0.921333i	$-0.372899\pi$
$83$	7.08378	0.777546	0.388773	+	0.921333i	$0.372899\pi$
$84$	0	0
$85$	14.4952	1.57222
$86$	− 13.3247i	− 1.43684i
$87$	0	0
$88$	11.7032	1.24756
$89$	4.95619	0.525355	0.262677	−	0.964884i	$-0.415394\pi$
$89$	4.95619	0.525355	0.262677	+	0.964884i	$0.415394\pi$
$90$	0	0
$91$	0	0
$92$	4.35586i	0.454130i
$93$	0	0
$94$	− 6.38667i	− 0.658735i
$95$	− 17.6147i	− 1.80723i
$96$	0	0
$97$	4.99484i	0.507149i	0.967316	+	0.253574i	$0.0816062\pi$
$97$	4.99484i	0.507149i	−0.967316	+	0.253574i	$0.918394\pi$
$98$	0	0
$99$	0	0
$100$	0.892250	0.0892250
$101$	−17.3958	−1.73095	−0.865473	−	0.500955i	$-0.832982\pi$
$101$	−17.3958	−1.73095	−0.865473	+	0.500955i	$0.832982\pi$
$102$	0	0
$103$	− 1.12007i	− 0.110364i	−0.998476	−	0.0551820i	$-0.982426\pi$
$103$	− 1.12007i	− 0.110364i	0.998476	−	0.0551820i	$-0.0175739\pi$
$104$	0.502302	0.0492548
$105$	0	0
$106$	9.37343	0.910428
$107$	4.85902i	0.469739i	0.972027	+	0.234870i	$0.0754663\pi$
$107$	4.85902i	0.469739i	−0.972027	+	0.234870i	$0.924534\pi$
$108$	0	0
$109$	−13.2237	−1.26660	−0.633301	−	0.773906i	$-0.718300\pi$
$109$	−13.2237	−1.26660	−0.633301	+	0.773906i	$0.718300\pi$
$110$	−10.4039	−0.991971
$111$	0	0
$112$	0	0
$113$	− 3.69121i	− 0.347240i	−0.984813	−	0.173620i	$-0.944454\pi$
$113$	− 3.69121i	− 0.347240i	0.984813	−	0.173620i	$-0.0555465\pi$
$114$	0	0
$115$	− 13.7229i	− 1.27966i
$116$	− 2.61515i	− 0.242811i
$117$	0	0
$118$	− 3.41348i	− 0.314236i
$119$	0	0
$120$	0	0
$121$	−3.53572	−0.321429
$122$	3.55318	0.321690
$123$	0	0
$124$	− 0.104986i	− 0.00942806i
$125$	9.57338	0.856269
$126$	0	0
$127$	−14.7609	−1.30981	−0.654907	−	0.755709i	$-0.727292\pi$
$127$	−14.7609	−1.30981	−0.654907	+	0.755709i	$0.727292\pi$
$128$	0.727794i	0.0643285i
$129$	0	0
$130$	−0.446536	−0.0391638
$131$	−8.65649	−0.756321	−0.378161	−	0.925740i	$-0.623443\pi$
$131$	−8.65649	−0.756321	−0.378161	+	0.925740i	$0.623443\pi$
$132$	0	0
$133$	0	0
$134$	− 8.36357i	− 0.722502i
$135$	0	0
$136$	17.9641i	1.54041i
$137$	− 21.7650i	− 1.85951i	−0.368184	−	0.929753i	$-0.620020\pi$
$137$	− 21.7650i	− 1.85951i	0.368184	−	0.929753i	$-0.379980\pi$
$138$	0	0
$139$	− 1.55281i	− 0.131707i	−0.997829	−	0.0658537i	$-0.979023\pi$
$139$	− 1.55281i	− 0.131707i	0.997829	−	0.0658537i	$-0.0209771\pi$
$140$	0	0
$141$	0	0
$142$	14.1329	1.18600
$143$	−0.623875	−0.0521710
$144$	0	0
$145$	8.23887i	0.684201i
$146$	11.6379	0.963159
$147$	0	0
$148$	−6.27584	−0.515871
$149$	− 1.99780i	− 0.163666i	−0.996646	−	0.0818332i	$-0.973923\pi$
$149$	− 1.99780i	− 0.163666i	0.996646	−	0.0818332i	$-0.0260775\pi$
$150$	0	0
$151$	−11.5605	−0.940776	−0.470388	−	0.882460i	$-0.655886\pi$
$151$	−11.5605	−0.940776	−0.470388	+	0.882460i	$0.655886\pi$
$152$	21.8303	1.77067
$153$	0	0
$154$	0	0
$155$	0.330753i	0.0265667i
$156$	0	0
$157$	3.06147i	0.244332i	0.992510	+	0.122166i	$0.0389840\pi$
$157$	3.06147i	0.244332i	−0.992510	+	0.122166i	$0.961016\pi$
$158$	11.3613i	0.903856i
$159$	0	0
$160$	10.2683i	0.811784i
$161$	0	0
$162$	0	0
$163$	−11.5048	−0.901128	−0.450564	−	0.892744i	$-0.648777\pi$
$163$	−11.5048	−0.901128	−0.450564	+	0.892744i	$0.648777\pi$
$164$	−7.58801	−0.592524
$165$	0	0
$166$	− 7.80439i	− 0.605738i
$167$	23.0009	1.77986	0.889931	−	0.456095i	$-0.150752\pi$
$167$	23.0009	1.77986	0.889931	+	0.456095i	$0.150752\pi$
$168$	0	0
$169$	12.9732	0.997940
$170$	− 15.9697i	− 1.22482i
$171$	0	0
$172$	9.50858	0.725022
$173$	11.8285	0.899301	0.449650	−	0.893205i	$-0.351549\pi$
$173$	11.8285	0.899301	0.449650	+	0.893205i	$0.351549\pi$
$174$	0	0
$175$	0	0
$176$	− 6.89883i	− 0.520019i
$177$	0	0
$178$	− 5.46036i	− 0.409272i
$179$	1.14825i	0.0858242i	0.999079	+	0.0429121i	$0.0136635\pi$
$179$	1.14825i	0.0858242i	−0.999079	+	0.0429121i	$0.986336\pi$
$180$	0	0
$181$	− 22.8669i	− 1.69969i	−0.527037	−	0.849843i	$-0.676697\pi$
$181$	− 22.8669i	− 1.69969i	0.527037	−	0.849843i	$-0.323303\pi$
$182$	0	0
$183$	0	0
$184$	17.0070	1.25377
$185$	19.7716	1.45364
$186$	0	0
$187$	− 22.3120i	− 1.63162i
$188$	4.55757	0.332395
$189$	0	0
$190$	−19.4066	−1.40790
$191$	4.46247i	0.322893i	0.986881	+	0.161447i	$0.0516159\pi$
$191$	4.46247i	0.322893i	−0.986881	+	0.161447i	$0.948384\pi$
$192$	0	0
$193$	−7.75532	−0.558240	−0.279120	−	0.960256i	$-0.590043\pi$
$193$	−7.75532	−0.558240	−0.279120	+	0.960256i	$0.590043\pi$
$194$	5.50294	0.395088
$195$	0	0
$196$	0	0
$197$	7.21532i	0.514070i	0.966402	+	0.257035i	$0.0827456\pi$
$197$	7.21532i	0.514070i	−0.966402	+	0.257035i	$0.917254\pi$
$198$	0	0
$199$	4.14608i	0.293908i	0.989143	+	0.146954i	$0.0469469\pi$
$199$	4.14608i	0.293908i	−0.989143	+	0.146954i	$0.953053\pi$
$200$	− 3.48369i	− 0.246334i
$201$	0	0
$202$	19.1654i	1.34847i
$203$	0	0
$204$	0	0
$205$	23.9056	1.66964
$206$	−1.23401	−0.0859778
$207$	0	0
$208$	− 0.296099i	− 0.0205307i
$209$	−27.1138	−1.87550
$210$	0	0
$211$	7.88430	0.542778	0.271389	−	0.962470i	$-0.412517\pi$
$211$	7.88430	0.542778	0.271389	+	0.962470i	$0.412517\pi$
$212$	6.68894i	0.459398i
$213$	0	0
$214$	5.35331	0.365945
$215$	−29.9562	−2.04299
$216$	0	0
$217$	0	0
$218$	14.5689i	0.986731i
$219$	0	0
$220$	− 7.42428i	− 0.500545i
$221$	− 0.957634i	− 0.0644175i
$222$	0	0
$223$	− 7.40398i	− 0.495807i	−0.968785	−	0.247904i	$-0.920258\pi$
$223$	− 7.40398i	− 0.495807i	0.968785	−	0.247904i	$-0.0797416\pi$
$224$	0	0
$225$	0	0
$226$	−4.06671	−0.270513
$227$	4.97338	0.330095	0.165048	−	0.986286i	$-0.447222\pi$
$227$	4.97338	0.330095	0.165048	+	0.986286i	$0.447222\pi$
$228$	0	0
$229$	− 28.2680i	− 1.86800i	−0.357273	−	0.934000i	$-0.616293\pi$
$229$	− 28.2680i	− 1.86800i	0.357273	−	0.934000i	$-0.383707\pi$
$230$	−15.1188	−0.996907
$231$	0	0
$232$	−10.2106	−0.670357
$233$	− 13.3670i	− 0.875699i	−0.899048	−	0.437849i	$-0.855740\pi$
$233$	− 13.3670i	− 0.875699i	0.899048	−	0.437849i	$-0.144260\pi$
$234$	0	0
$235$	−14.3583	−0.936635
$236$	2.43588	0.158562
$237$	0	0
$238$	0	0
$239$	13.1145i	0.848307i	0.905590	+	0.424153i	$0.139428\pi$
$239$	13.1145i	0.848307i	−0.905590	+	0.424153i	$0.860572\pi$
$240$	0	0
$241$	− 13.4107i	− 0.863861i	−0.901907	−	0.431931i	$-0.857833\pi$
$241$	− 13.4107i	− 0.863861i	0.901907	−	0.431931i	$-0.142167\pi$
$242$	3.89539i	0.250405i
$243$	0	0
$244$	2.53557i	0.162323i
$245$	0	0
$246$	0	0
$247$	−1.16373	−0.0740464
$248$	−0.409908	−0.0260292
$249$	0	0
$250$	− 10.5472i	− 0.667066i
$251$	−14.1641	−0.894031	−0.447015	−	0.894526i	$-0.647513\pi$
$251$	−14.1641	−0.894031	−0.447015	+	0.894526i	$0.647513\pi$
$252$	0	0
$253$	−21.1232	−1.32800
$254$	16.2624i	1.02040i
$255$	0	0
$256$	−15.5710	−0.973186
$257$	1.98638	0.123907	0.0619535	−	0.998079i	$-0.480267\pi$
$257$	1.98638	0.123907	0.0619535	+	0.998079i	$0.480267\pi$
$258$	0	0
$259$	0	0
$260$	− 0.318651i	− 0.0197619i
$261$	0	0
$262$	9.53708i	0.589203i
$263$	10.4275i	0.642984i	0.946912	+	0.321492i	$0.104184\pi$
$263$	10.4275i	0.642984i	−0.946912	+	0.321492i	$0.895816\pi$
$264$	0	0
$265$	− 21.0731i	− 1.29451i
$266$	0	0
$267$	0	0
$268$	5.96830	0.364572
$269$	−6.82638	−0.416212	−0.208106	−	0.978106i	$-0.566730\pi$
$269$	−6.82638	−0.416212	−0.208106	+	0.978106i	$0.566730\pi$
$270$	0	0
$271$	− 2.39934i	− 0.145750i	−0.997341	−	0.0728748i	$-0.976783\pi$
$271$	− 2.39934i	− 0.145750i	0.997341	−	0.0728748i	$-0.0232174\pi$
$272$	10.5895	0.642086
$273$	0	0
$274$	−23.9790	−1.44863
$275$	4.32685i	0.260919i
$276$	0	0
$277$	−14.6342	−0.879283	−0.439641	−	0.898173i	$-0.644894\pi$
$277$	−14.6342	−0.879283	−0.439641	+	0.898173i	$0.644894\pi$
$278$	−1.71077	−0.102605
$279$	0	0
$280$	0	0
$281$	11.8537i	0.707130i	0.935410	+	0.353565i	$0.115031\pi$
$281$	11.8537i	0.707130i	−0.935410	+	0.353565i	$0.884969\pi$
$282$	0	0
$283$	− 14.2486i	− 0.846991i	−0.905898	−	0.423496i	$-0.860803\pi$
$283$	− 14.2486i	− 0.846991i	0.905898	−	0.423496i	$-0.139197\pi$
$284$	10.0853i	0.598452i
$285$	0	0
$286$	0.687339i	0.0406432i
$287$	0	0
$288$	0	0
$289$	17.2484	1.01461
$290$	9.07698	0.533019
$291$	0	0
$292$	8.30488i	0.486006i
$293$	−14.7281	−0.860422	−0.430211	−	0.902728i	$-0.641561\pi$
$293$	−14.7281	−0.860422	−0.430211	+	0.902728i	$0.641561\pi$
$294$	0	0
$295$	−7.67409	−0.446803
$296$	24.5033i	1.42423i
$297$	0	0
$298$	−2.20103	−0.127502
$299$	−0.906611	−0.0524307
$300$	0	0
$301$	0	0
$302$	12.7365i	0.732901i
$303$	0	0
$304$	− 12.8686i	− 0.738062i
$305$	− 7.98817i	− 0.457401i
$306$	0	0
$307$	− 5.05349i	− 0.288418i	−0.989547	−	0.144209i	$-0.953936\pi$
$307$	− 5.05349i	− 0.288418i	0.989547	−	0.144209i	$-0.0460637\pi$
$308$	0	0
$309$	0	0
$310$	0.364400	0.0206965
$311$	−24.3933	−1.38322	−0.691609	−	0.722272i	$-0.743098\pi$
$311$	−24.3933	−1.38322	−0.691609	+	0.722272i	$0.743098\pi$
$312$	0	0
$313$	− 1.62059i	− 0.0916010i	−0.998951	−	0.0458005i	$-0.985416\pi$
$313$	− 1.62059i	− 0.0916010i	0.998951	−	0.0458005i	$-0.0145839\pi$
$314$	3.37290	0.190344
$315$	0	0
$316$	−8.10749	−0.456082
$317$	− 11.8623i	− 0.666253i	−0.942882	−	0.333126i	$-0.891896\pi$
$317$	− 11.8623i	− 0.666253i	0.942882	−	0.333126i	$-0.108104\pi$
$318$	0	0
$319$	12.6818	0.710047
$320$	20.2767	1.13350
$321$	0	0
$322$	0	0
$323$	− 41.6192i	− 2.31575i
$324$	0	0
$325$	0.185709i	0.0103013i
$326$	12.6752i	0.702013i
$327$	0	0
$328$	29.6266i	1.63585i
$329$	0	0
$330$	0	0
$331$	−11.4335	−0.628443	−0.314221	−	0.949350i	$-0.601743\pi$
$331$	−11.4335	−0.628443	−0.314221	+	0.949350i	$0.601743\pi$
$332$	5.56926	0.305653
$333$	0	0
$334$	− 25.3407i	− 1.38658i
$335$	−18.8028	−1.02730
$336$	0	0
$337$	6.68195	0.363989	0.181995	−	0.983300i	$-0.441745\pi$
$337$	6.68195	0.363989	0.181995	+	0.983300i	$0.441745\pi$
$338$	− 14.2929i	− 0.777434i
$339$	0	0
$340$	11.3961	0.618040
$341$	0.509119	0.0275703
$342$	0	0
$343$	0	0
$344$	− 37.1252i	− 2.00166i
$345$	0	0
$346$	− 13.0317i	− 0.700590i
$347$	28.4282i	1.52611i	0.646335	+	0.763054i	$0.276301\pi$
$347$	28.4282i	1.52611i	−0.646335	+	0.763054i	$0.723699\pi$
$348$	0	0
$349$	6.31603i	0.338089i	0.985608	+	0.169045i	$0.0540682\pi$
$349$	6.31603i	0.338089i	−0.985608	+	0.169045i	$0.945932\pi$
$350$	0	0
$351$	0	0
$352$	15.8058	0.842449
$353$	21.3042	1.13391	0.566953	−	0.823750i	$-0.308122\pi$
$353$	21.3042	1.13391	0.566953	+	0.823750i	$0.308122\pi$
$354$	0	0
$355$	− 31.7731i	− 1.68634i
$356$	3.89655	0.206517
$357$	0	0
$358$	1.26506	0.0668604
$359$	21.1423i	1.11585i	0.829892	+	0.557924i	$0.188402\pi$
$359$	21.1423i	1.11585i	−0.829892	+	0.557924i	$0.811598\pi$
$360$	0	0
$361$	−31.5762	−1.66190
$362$	−25.1931	−1.32412
$363$	0	0
$364$	0	0
$365$	− 26.1640i	− 1.36949i
$366$	0	0
$367$	− 13.0962i	− 0.683616i	−0.939770	−	0.341808i	$-0.888961\pi$
$367$	− 13.0962i	− 0.683616i	0.939770	−	0.341808i	$-0.111039\pi$
$368$	− 10.0253i	− 0.522606i
$369$	0	0
$370$	− 21.7829i	− 1.13244i
$371$	0	0
$372$	0	0
$373$	18.5593	0.960962	0.480481	−	0.877005i	$-0.340462\pi$
$373$	18.5593	0.960962	0.480481	+	0.877005i	$0.340462\pi$
$374$	−24.5817	−1.27109
$375$	0	0
$376$	− 17.7945i	− 0.917683i
$377$	0.544307	0.0280332
$378$	0	0
$379$	−19.2770	−0.990194	−0.495097	−	0.868838i	$-0.664868\pi$
$379$	−19.2770	−0.990194	−0.495097	+	0.868838i	$0.664868\pi$
$380$	− 13.8487i	− 0.710423i
$381$	0	0
$382$	4.91642	0.251546
$383$	8.06498	0.412101	0.206051	−	0.978541i	$-0.433939\pi$
$383$	8.06498	0.412101	0.206051	+	0.978541i	$0.433939\pi$
$384$	0	0
$385$	0	0
$386$	8.54424i	0.434890i
$387$	0	0
$388$	3.92693i	0.199360i
$389$	0.648024i	0.0328561i	0.999865	+	0.0164281i	$0.00522945\pi$
$389$	0.648024i	0.0328561i	−0.999865	+	0.0164281i	$0.994771\pi$
$390$	0	0
$391$	− 32.4237i	− 1.63973i
$392$	0	0
$393$	0	0
$394$	7.94931	0.400480
$395$	25.5422	1.28517
$396$	0	0
$397$	7.38679i	0.370732i	0.982670	+	0.185366i	$0.0593471\pi$
$397$	7.38679i	0.370732i	−0.982670	+	0.185366i	$0.940653\pi$
$398$	4.56785	0.228965
$399$	0	0
$400$	−2.05358	−0.102679
$401$	28.8502i	1.44071i	0.693606	+	0.720354i	$0.256021\pi$
$401$	28.8502i	1.44071i	−0.693606	+	0.720354i	$0.743979\pi$
$402$	0	0
$403$	0.0218514	0.00108850
$404$	−13.6766	−0.680434
$405$	0	0
$406$	0	0
$407$	− 30.4339i	− 1.50855i
$408$	0	0
$409$	35.6418i	1.76238i	0.472766	+	0.881188i	$0.343256\pi$
$409$	35.6418i	1.76238i	−0.472766	+	0.881188i	$0.656744\pi$
$410$	− 26.3374i	− 1.30071i
$411$	0	0
$412$	− 0.880600i	− 0.0433841i
$413$	0	0
$414$	0	0
$415$	−17.5456	−0.861281
$416$	0.678385	0.0332606
$417$	0	0
$418$	29.8720i	1.46109i
$419$	22.0324	1.07635	0.538177	−	0.842832i	$-0.319113\pi$
$419$	22.0324	1.07635	0.538177	+	0.842832i	$0.319113\pi$
$420$	0	0
$421$	12.5436	0.611336	0.305668	−	0.952138i	$-0.401120\pi$
$421$	12.5436	0.611336	0.305668	+	0.952138i	$0.401120\pi$
$422$	− 8.68634i	− 0.422845i
$423$	0	0
$424$	26.1162	1.26832
$425$	−6.64162	−0.322166
$426$	0	0
$427$	0	0
$428$	3.82016i	0.184654i
$429$	0	0
$430$	33.0035i	1.59157i
$431$	− 7.92413i	− 0.381692i	−0.981620	−	0.190846i	$-0.938877\pi$
$431$	− 7.92413i	− 0.381692i	0.981620	−	0.190846i	$-0.0611231\pi$
$432$	0	0
$433$	26.0931i	1.25396i	0.779037	+	0.626978i	$0.215708\pi$
$433$	26.0931i	1.25396i	−0.779037	+	0.626978i	$0.784292\pi$
$434$	0	0
$435$	0	0
$436$	−10.3965	−0.497900
$437$	−39.4017	−1.88484
$438$	0	0
$439$	10.8696i	0.518779i	0.965773	+	0.259390i	$0.0835214\pi$
$439$	10.8696i	0.518779i	−0.965773	+	0.259390i	$0.916479\pi$
$440$	−28.9873	−1.38191
$441$	0	0
$442$	−1.05505	−0.0501837
$443$	− 21.4765i	− 1.02038i	−0.860062	−	0.510189i	$-0.829575\pi$
$443$	− 21.4765i	− 1.02038i	0.860062	−	0.510189i	$-0.170425\pi$
$444$	0	0
$445$	−12.2758	−0.581931
$446$	−8.15716	−0.386253
$447$	0	0
$448$	0	0
$449$	29.4305i	1.38891i	0.719535	+	0.694456i	$0.244355\pi$
$449$	29.4305i	1.38891i	−0.719535	+	0.694456i	$0.755645\pi$
$450$	0	0
$451$	− 36.7971i	− 1.73271i
$452$	− 2.90203i	− 0.136500i
$453$	0	0
$454$	− 5.47931i	− 0.257157i
$455$	0	0
$456$	0	0
$457$	−5.46106	−0.255458	−0.127729	−	0.991809i	$-0.540769\pi$
$457$	−5.46106	−0.255458	−0.127729	+	0.991809i	$0.540769\pi$
$458$	−31.1436	−1.45524
$459$	0	0
$460$	− 10.7889i	− 0.503035i
$461$	31.1693	1.45170	0.725850	−	0.687853i	$-0.241447\pi$
$461$	31.1693	1.45170	0.725850	+	0.687853i	$0.241447\pi$
$462$	0	0
$463$	0.851101	0.0395540	0.0197770	−	0.999804i	$-0.493704\pi$
$463$	0.851101	0.0395540	0.0197770	+	0.999804i	$0.493704\pi$
$464$	6.01896i	0.279423i
$465$	0	0
$466$	−14.7267	−0.682203
$467$	4.11541	0.190438	0.0952192	−	0.995456i	$-0.469645\pi$
$467$	4.11541	0.190438	0.0952192	+	0.995456i	$0.469645\pi$
$468$	0	0
$469$	0	0
$470$	15.8190i	0.729674i
$471$	0	0
$472$	− 9.51063i	− 0.437762i
$473$	46.1106i	2.12017i
$474$	0	0
$475$	8.07099i	0.370322i
$476$	0	0
$477$	0	0
$478$	14.4486	0.660864
$479$	−1.34185	−0.0613107	−0.0306554	−	0.999530i	$-0.509759\pi$
$479$	−1.34185	−0.0613107	−0.0306554	+	0.999530i	$0.509759\pi$
$480$	0	0
$481$	− 1.30623i	− 0.0595588i
$482$	−14.7750	−0.672981
$483$	0	0
$484$	−2.77978	−0.126353
$485$	− 12.3716i	− 0.561764i
$486$	0	0
$487$	11.7673	0.533226	0.266613	−	0.963804i	$-0.414096\pi$
$487$	11.7673	0.533226	0.266613	+	0.963804i	$0.414096\pi$
$488$	9.89987	0.448146
$489$	0	0
$490$	0	0
$491$	18.1755i	0.820251i	0.912029	+	0.410125i	$0.134515\pi$
$491$	18.1755i	0.820251i	−0.912029	+	0.410125i	$0.865485\pi$
$492$	0	0
$493$	19.4664i	0.876721i
$494$	1.28211i	0.0576850i
$495$	0	0
$496$	0.241634i	0.0108497i
$497$	0	0
$498$	0	0
$499$	−1.77307	−0.0793734	−0.0396867	−	0.999212i	$-0.512636\pi$
$499$	−1.77307	−0.0793734	−0.0396867	+	0.999212i	$0.512636\pi$
$500$	7.52658	0.336599
$501$	0	0
$502$	15.6050i	0.696484i
$503$	−12.0974	−0.539395	−0.269697	−	0.962945i	$-0.586924\pi$
$503$	−12.0974	−0.539395	−0.269697	+	0.962945i	$0.586924\pi$
$504$	0	0
$505$	43.0871	1.91735
$506$	23.2720i	1.03457i
$507$	0	0
$508$	−11.6050	−0.514888
$509$	1.69459	0.0751112	0.0375556	−	0.999295i	$-0.488043\pi$
$509$	1.69459	0.0751112	0.0375556	+	0.999295i	$0.488043\pi$
$510$	0	0
$511$	0	0
$512$	18.6105i	0.822478i
$513$	0	0
$514$	− 2.18845i	− 0.0965283i
$515$	2.77428i	0.122249i
$516$	0	0
$517$	22.1014i	0.972017i
$518$	0	0
$519$	0	0
$520$	−1.24414	−0.0545591
$521$	22.8769	1.00225	0.501127	−	0.865374i	$-0.332919\pi$
$521$	22.8769	1.00225	0.501127	+	0.865374i	$0.332919\pi$
$522$	0	0
$523$	16.5920i	0.725515i	0.931883	+	0.362758i	$0.118165\pi$
$523$	16.5920i	0.725515i	−0.931883	+	0.362758i	$0.881835\pi$
$524$	−6.80572	−0.297309
$525$	0	0
$526$	11.4882	0.500909
$527$	0.781486i	0.0340421i
$528$	0	0
$529$	−7.69609	−0.334613
$530$	−23.2168	−1.00847
$531$	0	0
$532$	0	0
$533$	− 1.57934i	− 0.0684087i
$534$	0	0
$535$	− 12.0352i	− 0.520325i
$536$	− 23.3026i	− 1.00652i
$537$	0	0
$538$	7.52080i	0.324245i
$539$	0	0
$540$	0	0
$541$	−12.0883	−0.519717	−0.259858	−	0.965647i	$-0.583676\pi$
$541$	−12.0883	−0.519717	−0.259858	+	0.965647i	$0.583676\pi$
$542$	−2.64342	−0.113545
$543$	0	0
$544$	24.2615i	1.04020i
$545$	32.7534	1.40300
$546$	0	0
$547$	−24.9578	−1.06712	−0.533559	−	0.845763i	$-0.679146\pi$
$547$	−24.9578	−1.06712	−0.533559	+	0.845763i	$0.679146\pi$
$548$	− 17.1116i	− 0.730971i
$549$	0	0
$550$	4.76701	0.203266
$551$	23.6558	1.00777
$552$	0	0
$553$	0	0
$554$	16.1229i	0.684995i
$555$	0	0
$556$	− 1.22082i	− 0.0517741i
$557$	− 22.0700i	− 0.935138i	−0.883957	−	0.467569i	$-0.845130\pi$
$557$	− 22.0700i	− 0.935138i	0.883957	−	0.467569i	$-0.154870\pi$
$558$	0	0
$559$	1.97908i	0.0837060i
$560$	0	0
$561$	0	0
$562$	13.0595	0.550881
$563$	45.4251	1.91444	0.957220	−	0.289362i	$-0.0934431\pi$
$563$	45.4251	1.91444	0.957220	+	0.289362i	$0.0934431\pi$
$564$	0	0
$565$	9.14266i	0.384635i
$566$	−15.6981	−0.659838
$567$	0	0
$568$	39.3769	1.65222
$569$	− 36.9423i	− 1.54870i	−0.632757	−	0.774351i	$-0.718077\pi$
$569$	− 36.9423i	− 1.54870i	0.632757	−	0.774351i	$-0.281923\pi$
$570$	0	0
$571$	31.0402	1.29899	0.649496	−	0.760365i	$-0.274980\pi$
$571$	31.0402	1.29899	0.649496	+	0.760365i	$0.274980\pi$
$572$	−0.490490	−0.0205084
$573$	0	0
$574$	0	0
$575$	6.28775i	0.262217i
$576$	0	0
$577$	26.1582i	1.08898i	0.838767	+	0.544491i	$0.183277\pi$
$577$	26.1582i	1.08898i	−0.838767	+	0.544491i	$0.816723\pi$
$578$	− 19.0031i	− 0.790423i
$579$	0	0
$580$	6.47739i	0.268959i
$581$	0	0
$582$	0	0
$583$	−32.4372	−1.34341
$584$	32.4255	1.34178
$585$	0	0
$586$	16.2263i	0.670302i
$587$	−17.7853	−0.734076	−0.367038	−	0.930206i	$-0.619628\pi$
$587$	−17.7853	−0.734076	−0.367038	+	0.930206i	$0.619628\pi$
$588$	0	0
$589$	0.949672	0.0391306
$590$	8.45475i	0.348076i
$591$	0	0
$592$	14.4443	0.593657
$593$	29.6455	1.21740	0.608698	−	0.793402i	$-0.291692\pi$
$593$	29.6455	1.21740	0.608698	+	0.793402i	$0.291692\pi$
$594$	0	0
$595$	0	0
$596$	− 1.57067i	− 0.0643372i
$597$	0	0
$598$	0.998837i	0.0408455i
$599$	− 2.58574i	− 0.105651i	−0.998604	−	0.0528253i	$-0.983177\pi$
$599$	− 2.58574i	− 0.105651i	0.998604	−	0.0528253i	$-0.0168226\pi$
$600$	0	0
$601$	− 35.2731i	− 1.43882i	−0.694586	−	0.719410i	$-0.744412\pi$
$601$	− 35.2731i	− 1.43882i	0.694586	−	0.719410i	$-0.255588\pi$
$602$	0	0
$603$	0	0
$604$	−9.08882	−0.369819
$605$	8.75751	0.356043
$606$	0	0
$607$	11.7742i	0.477899i	0.971032	+	0.238949i	$0.0768030\pi$
$607$	11.7742i	0.477899i	−0.971032	+	0.238949i	$0.923197\pi$
$608$	29.4829	1.19569
$609$	0	0
$610$	−8.80077	−0.356333
$611$	0.948594i	0.0383760i
$612$	0	0
$613$	7.71139	0.311460	0.155730	−	0.987800i	$-0.450227\pi$
$613$	7.71139	0.311460	0.155730	+	0.987800i	$0.450227\pi$
$614$	−5.56756	−0.224688
$615$	0	0
$616$	0	0
$617$	22.8202i	0.918705i	0.888254	+	0.459352i	$0.151919\pi$
$617$	22.8202i	0.918705i	−0.888254	+	0.459352i	$0.848081\pi$
$618$	0	0
$619$	16.1099i	0.647513i	0.946140	+	0.323756i	$0.104946\pi$
$619$	16.1099i	0.647513i	−0.946140	+	0.323756i	$0.895054\pi$
$620$	0.260038i	0.0104434i
$621$	0	0
$622$	26.8747i	1.07758i
$623$	0	0
$624$	0	0
$625$	−29.3865	−1.17546
$626$	−1.78544	−0.0713607
$627$	0	0
$628$	2.40692i	0.0960467i
$629$	46.7154	1.86266
$630$	0	0
$631$	44.6835	1.77882	0.889412	−	0.457107i	$-0.151114\pi$
$631$	44.6835	1.77882	0.889412	+	0.457107i	$0.151114\pi$
$632$	31.6548i	1.25916i
$633$	0	0
$634$	−13.0690	−0.519036
$635$	36.5608	1.45087
$636$	0	0
$637$	0	0
$638$	− 13.9719i	− 0.553154i
$639$	0	0
$640$	− 1.80265i	− 0.0712561i
$641$	− 11.1024i	− 0.438520i	−0.975666	−	0.219260i	$-0.929636\pi$
$641$	− 11.1024i	− 0.438520i	0.975666	−	0.219260i	$-0.0703643\pi$
$642$	0	0
$643$	− 38.9756i	− 1.53705i	−0.639822	−	0.768523i	$-0.720992\pi$
$643$	− 38.9756i	− 1.53705i	0.639822	−	0.768523i	$-0.279008\pi$
$644$	0	0
$645$	0	0
$646$	−45.8529	−1.80406
$647$	−11.7858	−0.463348	−0.231674	−	0.972793i	$-0.574420\pi$
$647$	−11.7858	−0.463348	−0.231674	+	0.972793i	$0.574420\pi$
$648$	0	0
$649$	11.8125i	0.463681i
$650$	0.204601	0.00802510
$651$	0	0
$652$	−9.04509	−0.354233
$653$	− 22.9371i	− 0.897599i	−0.893633	−	0.448799i	$-0.851852\pi$
$653$	− 22.9371i	− 0.897599i	0.893633	−	0.448799i	$-0.148148\pi$
$654$	0	0
$655$	21.4410	0.837770
$656$	17.4643	0.681868
$657$	0	0
$658$	0	0
$659$	43.0410i	1.67664i	0.545179	+	0.838320i	$0.316462\pi$
$659$	43.0410i	1.67664i	−0.545179	+	0.838320i	$0.683538\pi$
$660$	0	0
$661$	48.9066i	1.90225i	0.308813	+	0.951123i	$0.400068\pi$
$661$	48.9066i	1.90225i	−0.308813	+	0.951123i	$0.599932\pi$
$662$	12.5966i	0.489581i
$663$	0	0
$664$	− 21.7446i	− 0.843854i
$665$	0	0
$666$	0	0
$667$	18.4292	0.713580
$668$	18.0833	0.699663
$669$	0	0
$670$	20.7155i	0.800309i
$671$	−12.2959	−0.474680
$672$	0	0
$673$	5.80720	0.223851	0.111925	−	0.993717i	$-0.464298\pi$
$673$	5.80720	0.223851	0.111925	+	0.993717i	$0.464298\pi$
$674$	− 7.36169i	− 0.283562i
$675$	0	0
$676$	10.1995	0.392290
$677$	−19.2291	−0.739036	−0.369518	−	0.929224i	$-0.620477\pi$
$677$	−19.2291	−0.739036	−0.369518	+	0.929224i	$0.620477\pi$
$678$	0	0
$679$	0	0
$680$	− 44.4948i	− 1.70630i
$681$	0	0
$682$	− 0.560909i	− 0.0214783i
$683$	9.41718i	0.360339i	0.983636	+	0.180169i	$0.0576646\pi$
$683$	9.41718i	0.360339i	−0.983636	+	0.180169i	$0.942335\pi$
$684$	0	0
$685$	53.9090i	2.05976i
$686$	0	0
$687$	0	0
$688$	−21.8847	−0.834345
$689$	−1.39221	−0.0530389
$690$	0	0
$691$	30.5545i	1.16235i	0.813779	+	0.581174i	$0.197406\pi$
$691$	30.5545i	1.16235i	−0.813779	+	0.581174i	$0.802594\pi$
$692$	9.29952	0.353515
$693$	0	0
$694$	31.3201	1.18890
$695$	3.84610i	0.145891i
$696$	0	0
$697$	56.4828	2.13944
$698$	6.95854	0.263384
$699$	0	0
$700$	0	0
$701$	− 45.5356i	− 1.71986i	−0.510416	−	0.859928i	$-0.670509\pi$
$701$	− 45.5356i	− 1.71986i	0.510416	−	0.859928i	$-0.329491\pi$
$702$	0	0
$703$	− 56.7691i	− 2.14109i
$704$	− 31.2113i	− 1.17632i
$705$	0	0
$706$	− 23.4714i	− 0.883356i
$707$	0	0
$708$	0	0
$709$	39.4050	1.47988	0.739942	−	0.672671i	$-0.234853\pi$
$709$	39.4050	1.47988	0.739942	+	0.672671i	$0.234853\pi$
$710$	−35.0052	−1.31372
$711$	0	0
$712$	− 15.2137i	− 0.570156i
$713$	0.739848	0.0277075
$714$	0	0
$715$	1.54526	0.0577894
$716$	0.902753i	0.0337375i
$717$	0	0
$718$	23.2930	0.869289
$719$	5.60977	0.209209	0.104604	−	0.994514i	$-0.466642\pi$
$719$	5.60977	0.209209	0.104604	+	0.994514i	$0.466642\pi$
$720$	0	0
$721$	0	0
$722$	34.7883i	1.29469i
$723$	0	0
$724$	− 17.9780i	− 0.668145i
$725$	− 3.77501i	− 0.140200i
$726$	0	0
$727$	47.7927i	1.77253i	0.463176	+	0.886266i	$0.346710\pi$
$727$	47.7927i	1.77253i	−0.463176	+	0.886266i	$0.653290\pi$
$728$	0	0
$729$	0	0
$730$	−28.8256	−1.06688
$731$	−70.7788	−2.61785
$732$	0	0
$733$	5.82170i	0.215029i	0.994203	+	0.107515i	$0.0342893\pi$
$733$	5.82170i	0.215029i	−0.994203	+	0.107515i	$0.965711\pi$
$734$	−14.4284	−0.532563
$735$	0	0
$736$	22.9688	0.846642
$737$	28.9425i	1.06611i
$738$	0	0
$739$	15.5765	0.572991	0.286495	−	0.958082i	$-0.407510\pi$
$739$	15.5765	0.572991	0.286495	+	0.958082i	$0.407510\pi$
$740$	15.5444	0.571425
$741$	0	0
$742$	0	0
$743$	12.0618i	0.442506i	0.975216	+	0.221253i	$0.0710146\pi$
$743$	12.0618i	0.442506i	−0.975216	+	0.221253i	$0.928985\pi$
$744$	0	0
$745$	4.94830i	0.181292i
$746$	− 20.4472i	− 0.748626i
$747$	0	0
$748$	− 17.5417i	− 0.641387i
$749$	0	0
$750$	0	0
$751$	−5.78416	−0.211067	−0.105533	−	0.994416i	$-0.533655\pi$
$751$	−5.78416	−0.211067	−0.105533	+	0.994416i	$0.533655\pi$
$752$	−10.4896	−0.382515
$753$	0	0
$754$	− 0.599677i	− 0.0218389i
$755$	28.6338	1.04209
$756$	0	0
$757$	3.14224	0.114207	0.0571034	−	0.998368i	$-0.481814\pi$
$757$	3.14224	0.114207	0.0571034	+	0.998368i	$0.481814\pi$
$758$	21.2380i	0.771399i
$759$	0	0
$760$	−54.0707	−1.96135
$761$	−43.6573	−1.58257	−0.791287	−	0.611444i	$-0.790589\pi$
$761$	−43.6573	−1.58257	−0.791287	+	0.611444i	$0.790589\pi$
$762$	0	0
$763$	0	0
$764$	3.50839i	0.126929i
$765$	0	0
$766$	− 8.88540i	− 0.321043i
$767$	0.506994i	0.0183065i
$768$	0	0
$769$	39.8666i	1.43763i	0.695203	+	0.718813i	$0.255314\pi$
$769$	39.8666i	1.43763i	−0.695203	+	0.718813i	$0.744686\pi$
$770$	0	0
$771$	0	0
$772$	−6.09722	−0.219444
$773$	14.5696	0.524034	0.262017	−	0.965063i	$-0.415612\pi$
$773$	14.5696	0.524034	0.262017	+	0.965063i	$0.415612\pi$
$774$	0	0
$775$	− 0.151550i	− 0.00544382i
$776$	15.3323	0.550397
$777$	0	0
$778$	0.713946	0.0255962
$779$	− 68.6386i	− 2.45923i
$780$	0	0
$781$	−48.9074	−1.75004
$782$	−35.7220	−1.27742
$783$	0	0
$784$	0	0
$785$	− 7.58286i	− 0.270644i
$786$	0	0
$787$	18.6296i	0.664073i	0.943266	+	0.332037i	$0.107736\pi$
$787$	18.6296i	0.664073i	−0.943266	+	0.332037i	$0.892264\pi$
$788$	5.67268i	0.202081i
$789$	0	0
$790$	− 28.1405i	− 1.00119i
$791$	0	0
$792$	0	0
$793$	−0.527744	−0.0187407
$794$	8.13822	0.288815
$795$	0	0
$796$	3.25964i	0.115535i
$797$	−25.8693	−0.916339	−0.458170	−	0.888865i	$-0.651495\pi$
$797$	−25.8693	−0.916339	−0.458170	+	0.888865i	$0.651495\pi$
$798$	0	0
$799$	−33.9251	−1.20018
$800$	− 4.70491i	− 0.166344i
$801$	0	0
$802$	31.7850	1.12237
$803$	−40.2735	−1.42122
$804$	0	0
$805$	0	0
$806$	− 0.0240743i	0 0.000847981i
$807$	0	0
$808$	53.3986i	1.87856i
$809$	14.9593i	0.525941i	0.964804	+	0.262970i	$0.0847022\pi$
$809$	14.9593i	0.525941i	−0.964804	+	0.262970i	$0.915298\pi$
$810$	0	0
$811$	− 7.26441i	− 0.255088i	−0.991833	−	0.127544i	$-0.959291\pi$
$811$	− 7.26441i	− 0.255088i	0.991833	−	0.127544i	$-0.0407094\pi$
$812$	0	0
$813$	0	0
$814$	−33.5298	−1.17522
$815$	28.4960	0.998171
$816$	0	0
$817$	86.0114i	3.00916i
$818$	39.2676	1.37296
$819$	0	0
$820$	18.7945	0.656333
$821$	16.7680i	0.585206i	0.956234	+	0.292603i	$0.0945214\pi$
$821$	16.7680i	0.585206i	−0.956234	+	0.292603i	$0.905479\pi$
$822$	0	0
$823$	37.6318	1.31176	0.655881	−	0.754865i	$-0.272297\pi$
$823$	37.6318	1.31176	0.655881	+	0.754865i	$0.272297\pi$
$824$	−3.43821	−0.119776
$825$	0	0
$826$	0	0
$827$	− 13.5586i	− 0.471480i	−0.971816	−	0.235740i	$-0.924249\pi$
$827$	− 13.5586i	− 0.471480i	0.971816	−	0.235740i	$-0.0757514\pi$
$828$	0	0
$829$	− 45.0012i	− 1.56296i	−0.623932	−	0.781479i	$-0.714466\pi$
$829$	− 45.0012i	− 1.56296i	0.623932	−	0.781479i	$-0.285534\pi$
$830$	19.3305i	0.670971i
$831$	0	0
$832$	− 1.33959i	− 0.0464420i
$833$	0	0
$834$	0	0
$835$	−56.9702	−1.97154
$836$	−21.3169	−0.737260
$837$	0	0
$838$	− 24.2737i	− 0.838521i
$839$	4.35734	0.150432	0.0752160	−	0.997167i	$-0.476035\pi$
$839$	4.35734	0.150432	0.0752160	+	0.997167i	$0.476035\pi$
$840$	0	0
$841$	17.9356	0.618468
$842$	− 13.8196i	− 0.476254i
$843$	0	0
$844$	6.19863	0.213366
$845$	−32.1330	−1.10541
$846$	0	0
$847$	0	0
$848$	− 15.3951i	− 0.528669i
$849$	0	0
$850$	7.31725i	0.250980i
$851$	− 44.2263i	− 1.51606i
$852$	0	0
$853$	− 45.8981i	− 1.57152i	−0.618530	−	0.785761i	$-0.712272\pi$
$853$	− 45.8981i	− 1.57152i	0.618530	−	0.785761i	$-0.287728\pi$
$854$	0	0
$855$	0	0
$856$	14.9154	0.509797
$857$	0.178142	0.00608522	0.00304261	−	0.999995i	$-0.499032\pi$
$857$	0.178142	0.00608522	0.00304261	+	0.999995i	$0.499032\pi$
$858$	0	0
$859$	46.3770i	1.58236i	0.611581	+	0.791182i	$0.290534\pi$
$859$	46.3770i	1.58236i	−0.611581	+	0.791182i	$0.709466\pi$
$860$	−23.5515	−0.803100
$861$	0	0
$862$	−8.73022	−0.297352
$863$	− 48.1103i	− 1.63769i	−0.574012	−	0.818847i	$-0.694614\pi$
$863$	− 48.1103i	− 1.63769i	0.574012	−	0.818847i	$-0.305386\pi$
$864$	0	0
$865$	−29.2976	−0.996147
$866$	28.7475	0.976880
$867$	0	0
$868$	0	0
$869$	− 39.3163i	− 1.33371i
$870$	0	0
$871$	1.24222i	0.0420909i
$872$	40.5919i	1.37461i
$873$	0	0
$874$	43.4098i	1.46836i
$875$	0	0
$876$	0	0
$877$	38.4070	1.29691	0.648456	−	0.761252i	$-0.275415\pi$
$877$	38.4070	1.29691	0.648456	+	0.761252i	$0.275415\pi$
$878$	11.9754	0.404149
$879$	0	0
$880$	17.0875i	0.576020i
$881$	−40.1980	−1.35430	−0.677152	−	0.735843i	$-0.736786\pi$
$881$	−40.1980	−1.35430	−0.677152	+	0.735843i	$0.736786\pi$
$882$	0	0
$883$	−0.352197	−0.0118524	−0.00592618	−	0.999982i	$-0.501886\pi$
$883$	−0.352197	−0.0118524	−0.00592618	+	0.999982i	$0.501886\pi$
$884$	− 0.752891i	− 0.0253225i
$885$	0	0
$886$	−23.6612	−0.794914
$887$	−5.69776	−0.191312	−0.0956560	−	0.995414i	$-0.530495\pi$
$887$	−5.69776	−0.191312	−0.0956560	+	0.995414i	$0.530495\pi$
$888$	0	0
$889$	0	0
$890$	13.5246i	0.453346i
$891$	0	0
$892$	− 5.82100i	− 0.194902i
$893$	41.2262i	1.37958i
$894$	0	0
$895$	− 2.84407i	− 0.0950667i
$896$	0	0
$897$	0	0
$898$	32.4244	1.08202
$899$	−0.444186	−0.0148144
$900$	0	0
$901$	− 49.7903i	− 1.65876i
$902$	−40.5403	−1.34985
$903$	0	0
$904$	−11.3307	−0.376852
$905$	56.6384i	1.88273i
$906$	0	0
$907$	29.0626	0.965006	0.482503	−	0.875894i	$-0.339728\pi$
$907$	29.0626	0.965006	0.482503	+	0.875894i	$0.339728\pi$
$908$	3.91007	0.129760
$909$	0	0
$910$	0	0
$911$	− 12.0849i	− 0.400389i	−0.979756	−	0.200194i	$-0.935843\pi$
$911$	− 12.0849i	− 0.400389i	0.979756	−	0.200194i	$-0.0641574\pi$
$912$	0	0
$913$	27.0074i	0.893816i
$914$	6.01659i	0.199011i
$915$	0	0
$916$	− 22.2242i	− 0.734310i
$917$	0	0
$918$	0	0
$919$	6.19401	0.204322	0.102161	−	0.994768i	$-0.467424\pi$
$919$	6.19401	0.204322	0.102161	+	0.994768i	$0.467424\pi$
$920$	−42.1241	−1.38879
$921$	0	0
$922$	− 34.3401i	− 1.13093i
$923$	−2.09911	−0.0690931
$924$	0	0
$925$	−9.05927	−0.297867
$926$	− 0.937680i	− 0.0308141i
$927$	0	0
$928$	−13.7899	−0.452676
$929$	38.8694	1.27526	0.637632	−	0.770341i	$-0.279914\pi$
$929$	38.8694	1.27526	0.637632	+	0.770341i	$0.279914\pi$
$930$	0	0
$931$	0	0
$932$	− 10.5091i	− 0.344237i
$933$	0	0
$934$	− 4.53406i	− 0.148359i
$935$	55.2640i	1.80732i
$936$	0	0
$937$	42.3164i	1.38242i	0.722656	+	0.691208i	$0.242921\pi$
$937$	42.3164i	1.38242i	−0.722656	+	0.691208i	$0.757079\pi$
$938$	0	0
$939$	0	0
$940$	−11.2885	−0.368191
$941$	40.1457	1.30871	0.654356	−	0.756187i	$-0.272940\pi$
$941$	40.1457	1.30871	0.654356	+	0.756187i	$0.272940\pi$
$942$	0	0
$943$	− 53.4733i	− 1.74133i
$944$	−5.60635	−0.182471
$945$	0	0
$946$	50.8013	1.65169
$947$	24.1828i	0.785836i	0.919574	+	0.392918i	$0.128534\pi$
$947$	24.1828i	0.785836i	−0.919574	+	0.392918i	$0.871466\pi$
$948$	0	0
$949$	−1.72854	−0.0561109
$950$	8.89202	0.288495
$951$	0	0
$952$	0	0
$953$	15.7315i	0.509594i	0.966995	+	0.254797i	$0.0820087\pi$
$953$	15.7315i	0.509594i	−0.966995	+	0.254797i	$0.917991\pi$
$954$	0	0
$955$	− 11.0530i	− 0.357665i
$956$	10.3106i	0.333469i
$957$	0	0
$958$	1.47835i	0.0477634i
$959$	0	0
$960$	0	0
$961$	30.9822	0.999425
$962$	−1.43911	−0.0463986
$963$	0	0
$964$	− 10.5435i	− 0.339583i
$965$	19.2089	0.618357
$966$	0	0
$967$	−34.7139	−1.11632	−0.558162	−	0.829732i	$-0.688493\pi$
$967$	−34.7139	−1.11632	−0.558162	+	0.829732i	$0.688493\pi$
$968$	10.8533i	0.348839i
$969$	0	0
$970$	−13.6301	−0.437635
$971$	56.7985	1.82275	0.911375	−	0.411578i	$-0.135022\pi$
$971$	56.7985	1.82275	0.911375	+	0.411578i	$0.135022\pi$
$972$	0	0
$973$	0	0
$974$	− 12.9643i	− 0.415403i
$975$	0	0
$976$	− 5.83580i	− 0.186799i
$977$	− 8.16509i	− 0.261224i	−0.991434	−	0.130612i	$-0.958306\pi$
$977$	− 8.16509i	− 0.261224i	0.991434	−	0.130612i	$-0.0416943\pi$
$978$	0	0
$979$	18.8958i	0.603913i
$980$	0	0
$981$	0	0
$982$	20.0245	0.639007
$983$	41.5123	1.32404	0.662018	−	0.749488i	$-0.269700\pi$
$983$	41.5123	1.32404	0.662018	+	0.749488i	$0.269700\pi$
$984$	0	0
$985$	− 17.8714i	− 0.569431i
$986$	21.4466	0.682999
$987$	0	0
$988$	−0.914923	−0.0291076
$989$	67.0077i	2.13072i
$990$	0	0
$991$	4.01981	0.127694	0.0638468	−	0.997960i	$-0.479663\pi$
$991$	4.01981	0.127694	0.0638468	+	0.997960i	$0.479663\pi$
$992$	−0.553602	−0.0175769
$993$	0	0
$994$	0	0
$995$	− 10.2693i	− 0.325559i
$996$	0	0
$997$	29.5682i	0.936433i	0.883614	+	0.468216i	$0.155103\pi$
$997$	29.5682i	0.936433i	−0.883614	+	0.468216i	$0.844897\pi$
$998$	1.95344i	0.0618349i
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.10173i q^{2} +0.786199 q^{4} -2.47687 q^{5} -3.06963i q^{8} +O(q^{10})\)	\(q-1.10173i q^{2} +0.786199 q^{4} -2.47687 q^{5} -3.06963i q^{8} +2.72883i q^{10} +3.81257i q^{11} +0.163636i q^{13} -1.80949 q^{16} -5.85222 q^{17} +7.11169i q^{19} -1.94731 q^{20} +4.20041 q^{22} +5.54041i q^{23} +1.13489 q^{25} +0.180282 q^{26} -3.32632i q^{29} -0.133537i q^{31} -4.14569i q^{32} +6.44754i q^{34} -7.98251 q^{37} +7.83514 q^{38} +7.60308i q^{40} -9.65151 q^{41} +12.0944 q^{43} +2.99744i q^{44} +6.10401 q^{46} +5.79697 q^{47} -1.25034i q^{50} +0.128651i q^{52} +8.50795i q^{53} -9.44325i q^{55} -3.66470 q^{58} +3.09830 q^{59} +3.22510i q^{61} -0.147121 q^{62} -8.18640 q^{64} -0.405306i q^{65} +7.59133 q^{67} -4.60101 q^{68} +12.8279i q^{71} +10.5633i q^{73} +8.79454i q^{74} +5.59121i q^{76} -10.3123 q^{79} +4.48188 q^{80} +10.6333i q^{82} +7.08378 q^{83} +14.4952 q^{85} -13.3247i q^{86} +11.7032 q^{88} +4.95619 q^{89} +4.35586i q^{92} -6.38667i q^{94} -17.6147i q^{95} +4.99484i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 48 q^{16} + 16 q^{22} + 32 q^{25} - 16 q^{37} + 32 q^{43} - 80 q^{46} - 96 q^{58} - 176 q^{64} + 96 q^{67} - 64 q^{79} - 32 q^{85} + 112 q^{88}+O(q^{100}) \) 16 * q - 16 * q^4 + 48 * q^16 + 16 * q^22 + 32 * q^25 - 16 * q^37 + 32 * q^43 - 80 * q^46 - 96 * q^58 - 176 * q^64 + 96 * q^67 - 64 * q^79 - 32 * q^85 + 112 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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