LMFDB - Embedded newform 1323.2.c.e.1322.2

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			1322.2
Root			$2.38976 - 1.37973i$ of defining polynomial
Character	$\chi$	$=$	1323.1322
Dual form			1323.2.c.e.1322.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75946i q^{2} -5.61463 q^{4} +2.24426 q^{5} +9.97442i q^{8} +O(q^{10})$	$q-2.75946i q^{2} -5.61463 q^{4} +2.24426 q^{5} +9.97442i q^{8} -6.19294i q^{10} -0.797407i q^{11} -5.06262i q^{13} +16.2948 q^{16} -4.46023 q^{17} +1.25139i q^{19} -12.6007 q^{20} -2.20041 q^{22} -3.78594i q^{23} +0.0366824 q^{25} -13.9701 q^{26} -4.04562i q^{29} -5.35979i q^{31} -25.0160i q^{32} +12.3078i q^{34} -5.33120 q^{37} +3.45316 q^{38} +22.3851i q^{40} -4.50330 q^{41} +0.390920 q^{43} +4.47714i q^{44} -10.4472 q^{46} -6.60051 q^{47} -0.101224i q^{50} +28.4247i q^{52} +6.67723i q^{53} -1.78958i q^{55} -11.1637 q^{58} -13.3598 q^{59} -8.12408i q^{61} -14.7901 q^{62} -36.4410 q^{64} -11.3618i q^{65} +12.8939 q^{67} +25.0425 q^{68} -3.19597i q^{71} +9.03259i q^{73} +14.7112i q^{74} -7.02608i q^{76} +5.14069 q^{79} +36.5696 q^{80} +12.4267i q^{82} +9.29454 q^{83} -10.0099 q^{85} -1.07873i q^{86} +7.95367 q^{88} +10.6640 q^{89} +21.2566i q^{92} +18.2139i q^{94} +2.80844i q^{95} +1.93337i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 2.75946i	− 1.95123i	−0.219481	−	0.975617i	$-0.570436\pi$
$2$	− 2.75946i	− 1.95123i	0.219481	−	0.975617i	$-0.429564\pi$
$3$	0	0
$3$	0	0
$4$	−5.61463	−2.80731
$5$	2.24426	1.00366	0.501831	−	0.864966i	$-0.332660\pi$
$5$	2.24426	1.00366	0.501831	+	0.864966i	$0.332660\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	9.97442i	3.52649i
$9$	0	0
$10$	− 6.19294i	− 1.95838i
$11$	− 0.797407i	− 0.240427i	−0.992748	−	0.120214i	$-0.961642\pi$
$11$	− 0.797407i	− 0.240427i	0.992748	−	0.120214i	$-0.0383579\pi$
$12$	0	0
$13$	− 5.06262i	− 1.40412i	−0.712119	−	0.702058i	$-0.752264\pi$
$13$	− 5.06262i	− 1.40412i	0.712119	−	0.702058i	$-0.247736\pi$
$14$	0	0
$15$	0	0
$16$	16.2948	4.07369
$17$	−4.46023	−1.08177	−0.540883	−	0.841098i	$-0.681910\pi$
$17$	−4.46023	−1.08177	−0.540883	+	0.841098i	$0.681910\pi$
$18$	0	0
$19$	1.25139i	0.287088i	0.989644	+	0.143544i	$0.0458499\pi$
$19$	1.25139i	0.287088i	−0.989644	+	0.143544i	$0.954150\pi$
$20$	−12.6007	−2.81759
$21$	0	0
$22$	−2.20041	−0.469130
$23$	− 3.78594i	− 0.789423i	−0.918805	−	0.394712i	$-0.870845\pi$
$23$	− 3.78594i	− 0.789423i	0.918805	−	0.394712i	$-0.129155\pi$
$24$	0	0
$25$	0.0366824	0.00733649
$26$	−13.9701	−2.73976
$27$	0	0
$28$	0	0
$29$	− 4.04562i	− 0.751253i	−0.926771	−	0.375626i	$-0.877428\pi$
$29$	− 4.04562i	− 0.751253i	0.926771	−	0.375626i	$-0.122572\pi$
$30$	0	0
$31$	− 5.35979i	− 0.962646i	−0.876543	−	0.481323i	$-0.840156\pi$
$31$	− 5.35979i	− 0.962646i	0.876543	−	0.481323i	$-0.159844\pi$
$32$	− 25.0160i	− 4.42224i
$33$	0	0
$34$	12.3078i	2.11078i
$35$	0	0
$36$	0	0
$37$	−5.33120	−0.876444	−0.438222	−	0.898867i	$-0.644392\pi$
$37$	−5.33120	−0.876444	−0.438222	+	0.898867i	$0.644392\pi$
$38$	3.45316	0.560177
$39$	0	0
$40$	22.3851i	3.53940i
$41$	−4.50330	−0.703298	−0.351649	−	0.936132i	$-0.614379\pi$
$41$	−4.50330	−0.703298	−0.351649	+	0.936132i	$0.614379\pi$
$42$	0	0
$43$	0.390920	0.0596147	0.0298073	−	0.999556i	$-0.490511\pi$
$43$	0.390920	0.0596147	0.0298073	+	0.999556i	$0.490511\pi$
$44$	4.47714i	0.674954i
$45$	0	0
$46$	−10.4472	−1.54035
$47$	−6.60051	−0.962783	−0.481392	−	0.876506i	$-0.659869\pi$
$47$	−6.60051	−0.962783	−0.481392	+	0.876506i	$0.659869\pi$
$48$	0	0
$49$	0	0
$50$	− 0.101224i	− 0.0143152i
$51$	0	0
$52$	28.4247i	3.94180i
$53$	6.67723i	0.917188i	0.888646	+	0.458594i	$0.151647\pi$
$53$	6.67723i	0.917188i	−0.888646	+	0.458594i	$0.848353\pi$
$54$	0	0
$55$	− 1.78958i	− 0.241307i
$56$	0	0
$57$	0	0
$58$	−11.1637	−1.46587
$59$	−13.3598	−1.73930	−0.869648	−	0.493673i	$-0.835654\pi$
$59$	−13.3598	−1.73930	−0.869648	+	0.493673i	$0.835654\pi$
$60$	0	0
$61$	− 8.12408i	− 1.04018i	−0.854111	−	0.520091i	$-0.825898\pi$
$61$	− 8.12408i	− 1.04018i	0.854111	−	0.520091i	$-0.174102\pi$
$62$	−14.7901	−1.87835
$63$	0	0
$64$	−36.4410	−4.55513
$65$	− 11.3618i	− 1.40926i
$66$	0	0
$67$	12.8939	1.57525	0.787623	−	0.616157i	$-0.211311\pi$
$67$	12.8939	1.57525	0.787623	+	0.616157i	$0.211311\pi$
$68$	25.0425	3.03686
$69$	0	0
$70$	0	0
$71$	− 3.19597i	− 0.379291i	−0.981853	−	0.189646i	$-0.939266\pi$
$71$	− 3.19597i	− 0.379291i	0.981853	−	0.189646i	$-0.0607339\pi$
$72$	0	0
$73$	9.03259i	1.05718i	0.848876	+	0.528592i	$0.177280\pi$
$73$	9.03259i	1.05718i	−0.848876	+	0.528592i	$0.822720\pi$
$74$	14.7112i	1.71015i
$75$	0	0
$76$	− 7.02608i	− 0.805947i
$77$	0	0
$78$	0	0
$79$	5.14069	0.578373	0.289187	−	0.957273i	$-0.406615\pi$
$79$	5.14069	0.578373	0.289187	+	0.957273i	$0.406615\pi$
$80$	36.5696	4.08861
$81$	0	0
$82$	12.4267i	1.37230i
$83$	9.29454	1.02021	0.510104	−	0.860113i	$-0.329607\pi$
$83$	9.29454	1.02021	0.510104	+	0.860113i	$0.329607\pi$
$84$	0	0
$85$	−10.0099	−1.08573
$86$	− 1.07873i	− 0.116322i
$87$	0	0
$88$	7.95367	0.847864
$89$	10.6640	1.13038	0.565190	−	0.824961i	$-0.308803\pi$
$89$	10.6640	1.13038	0.565190	+	0.824961i	$0.308803\pi$
$90$	0	0
$91$	0	0
$92$	21.2566i	2.21616i
$93$	0	0
$94$	18.2139i	1.87862i
$95$	2.80844i	0.288140i
$96$	0	0
$97$	1.93337i	0.196304i	0.995171	+	0.0981519i	$0.0312931\pi$
$97$	1.93337i	0.196304i	−0.995171	+	0.0981519i	$0.968707\pi$
$98$	0	0
$99$	0	0
$100$	−0.205958	−0.0205958
$101$	−10.5034	−1.04513	−0.522566	−	0.852599i	$-0.675025\pi$
$101$	−10.5034	−1.04513	−0.522566	+	0.852599i	$0.675025\pi$
$102$	0	0
$103$	− 7.24301i	− 0.713675i	−0.934167	−	0.356837i	$-0.883855\pi$
$103$	− 7.24301i	− 0.713675i	0.934167	−	0.356837i	$-0.116145\pi$
$104$	50.4967	4.95160
$105$	0	0
$106$	18.4256	1.78965
$107$	− 14.0640i	− 1.35962i	−0.733388	−	0.679811i	$-0.762062\pi$
$107$	− 14.0640i	− 1.35962i	0.733388	−	0.679811i	$-0.237938\pi$
$108$	0	0
$109$	4.88056	0.467473	0.233736	−	0.972300i	$-0.424905\pi$
$109$	4.88056	0.467473	0.233736	+	0.972300i	$0.424905\pi$
$110$	−4.93829	−0.470847
$111$	0	0
$112$	0	0
$113$	− 12.2757i	− 1.15480i	−0.816460	−	0.577402i	$-0.804067\pi$
$113$	− 12.2757i	− 1.15480i	0.816460	−	0.577402i	$-0.195933\pi$
$114$	0	0
$115$	− 8.49662i	− 0.792314i
$116$	22.7146i	2.10900i
$117$	0	0
$118$	36.8658i	3.39377i
$119$	0	0
$120$	0	0
$121$	10.3641	0.942195
$122$	−22.4181	−2.02964
$123$	0	0
$124$	30.0932i	2.70245i
$125$	−11.1390	−0.996298
$126$	0	0
$127$	1.79030	0.158864	0.0794318	−	0.996840i	$-0.474689\pi$
$127$	1.79030	0.158864	0.0794318	+	0.996840i	$0.474689\pi$
$128$	50.5256i	4.46588i
$129$	0	0
$130$	−31.3525	−2.74979
$131$	−3.03602	−0.265258	−0.132629	−	0.991166i	$-0.542342\pi$
$131$	−3.03602	−0.265258	−0.132629	+	0.991166i	$0.542342\pi$
$132$	0	0
$133$	0	0
$134$	− 35.5804i	− 3.07367i
$135$	0	0
$136$	− 44.4883i	− 3.81484i
$137$	5.79063i	0.494727i	0.968923	+	0.247364i	$0.0795642\pi$
$137$	5.79063i	0.494727i	−0.968923	+	0.247364i	$0.920436\pi$
$138$	0	0
$139$	− 3.34617i	− 0.283819i	−0.989880	−	0.141909i	$-0.954676\pi$
$139$	− 3.34617i	− 0.283819i	0.989880	−	0.141909i	$-0.0453241\pi$
$140$	0	0
$141$	0	0
$142$	−8.81914	−0.740086
$143$	−4.03696	−0.337588
$144$	0	0
$145$	− 9.07940i	− 0.754003i
$146$	24.9251	2.06281
$147$	0	0
$148$	29.9327	2.46045
$149$	18.7199i	1.53360i	0.641889	+	0.766798i	$0.278151\pi$
$149$	18.7199i	1.53360i	−0.641889	+	0.766798i	$0.721849\pi$
$150$	0	0
$151$	−1.41011	−0.114753	−0.0573766	−	0.998353i	$-0.518274\pi$
$151$	−1.41011	−0.114753	−0.0573766	+	0.998353i	$0.518274\pi$
$152$	−12.4819	−1.01241
$153$	0	0
$154$	0	0
$155$	− 12.0287i	− 0.966171i
$156$	0	0
$157$	− 3.06147i	− 0.244332i	−0.992510	−	0.122166i	$-0.961016\pi$
$157$	− 3.06147i	− 0.244332i	0.992510	−	0.122166i	$-0.0389840\pi$
$158$	− 14.1855i	− 1.12854i
$159$	0	0
$160$	− 56.1422i	− 4.43843i
$161$	0	0
$162$	0	0
$163$	17.8480	1.39796	0.698981	−	0.715140i	$-0.253637\pi$
$163$	17.8480	1.39796	0.698981	+	0.715140i	$0.253637\pi$
$164$	25.2844	1.97438
$165$	0	0
$166$	− 25.6479i	− 1.99067i
$167$	−13.7738	−1.06585	−0.532926	−	0.846162i	$-0.678908\pi$
$167$	−13.7738	−1.06585	−0.532926	+	0.846162i	$0.678908\pi$
$168$	0	0
$169$	−12.6301	−0.971544
$170$	27.6219i	2.11851i
$171$	0	0
$172$	−2.19487	−0.167357
$173$	−22.3867	−1.70203	−0.851016	−	0.525140i	$-0.824013\pi$
$173$	−22.3867	−1.70203	−0.851016	+	0.525140i	$0.824013\pi$
$174$	0	0
$175$	0	0
$176$	− 12.9936i	− 0.979426i
$177$	0	0
$178$	− 29.4268i	− 2.20563i
$179$	20.5388i	1.53514i	0.640963	+	0.767572i	$0.278535\pi$
$179$	20.5388i	1.53514i	−0.640963	+	0.767572i	$0.721465\pi$
$180$	0	0
$181$	− 19.5428i	− 1.45261i	−0.687374	−	0.726303i	$-0.741237\pi$
$181$	− 19.5428i	− 1.45261i	0.687374	−	0.726303i	$-0.258763\pi$
$182$	0	0
$183$	0	0
$184$	37.7626	2.78389
$185$	−11.9646	−0.879653
$186$	0	0
$187$	3.55662i	0.260086i
$188$	37.0594	2.70283
$189$	0	0
$190$	7.74977	0.562228
$191$	− 27.1791i	− 1.96661i	−0.181963	−	0.983305i	$-0.558245\pi$
$191$	− 27.1791i	− 1.96661i	0.181963	−	0.983305i	$-0.441755\pi$
$192$	0	0
$193$	14.0985	1.01483	0.507415	−	0.861702i	$-0.330601\pi$
$193$	14.0985	1.01483	0.507415	+	0.861702i	$0.330601\pi$
$194$	5.33505	0.383035
$195$	0	0
$196$	0	0
$197$	15.0415i	1.07167i	0.844324	+	0.535833i	$0.180002\pi$
$197$	15.0415i	1.07167i	−0.844324	+	0.535833i	$0.819998\pi$
$198$	0	0
$199$	1.34725i	0.0955037i	0.998859	+	0.0477518i	$0.0152057\pi$
$199$	1.34725i	0.0955037i	−0.998859	+	0.0477518i	$0.984794\pi$
$200$	0.365886i	0.0258721i
$201$	0	0
$202$	28.9839i	2.03930i
$203$	0	0
$204$	0	0
$205$	−10.1066	−0.705873
$206$	−19.9868	−1.39255
$207$	0	0
$208$	− 82.4942i	− 5.71994i
$209$	0.997866	0.0690238
$210$	0	0
$211$	7.42941	0.511462	0.255731	−	0.966748i	$-0.417684\pi$
$211$	7.42941	0.511462	0.255731	+	0.966748i	$0.417684\pi$
$212$	− 37.4902i	− 2.57483i
$213$	0	0
$214$	−38.8092	−2.65294
$215$	0.877323	0.0598329
$216$	0	0
$217$	0	0
$218$	− 13.4677i	− 0.912149i
$219$	0	0
$220$	10.0478i	0.677426i
$221$	22.5805i	1.51893i
$222$	0	0
$223$	− 24.6135i	− 1.64824i	−0.566415	−	0.824120i	$-0.691670\pi$
$223$	− 24.6135i	− 1.64824i	0.566415	−	0.824120i	$-0.308330\pi$
$224$	0	0
$225$	0	0
$226$	−33.8744	−2.25329
$227$	20.1800	1.33939	0.669696	−	0.742636i	$-0.266425\pi$
$227$	20.1800	1.33939	0.669696	+	0.742636i	$0.266425\pi$
$228$	0	0
$229$	27.4274i	1.81246i	0.422789	+	0.906228i	$0.361051\pi$
$229$	27.4274i	1.81246i	−0.422789	+	0.906228i	$0.638949\pi$
$230$	−23.4461	−1.54599
$231$	0	0
$232$	40.3527	2.64929
$233$	7.38680i	0.483925i	0.970286	+	0.241963i	$0.0777912\pi$
$233$	7.38680i	0.483925i	−0.970286	+	0.241963i	$0.922209\pi$
$234$	0	0
$235$	−14.8132	−0.966309
$236$	75.0102	4.88275
$237$	0	0
$238$	0	0
$239$	4.83804i	0.312947i	0.987682	+	0.156473i	$0.0500126\pi$
$239$	4.83804i	0.312947i	−0.987682	+	0.156473i	$0.949987\pi$
$240$	0	0
$241$	14.2513i	0.918005i	0.888435	+	0.459002i	$0.151793\pi$
$241$	14.2513i	0.918005i	−0.888435	+	0.459002i	$0.848207\pi$
$242$	− 28.5994i	− 1.83844i
$243$	0	0
$244$	45.6137i	2.92012i
$245$	0	0
$246$	0	0
$247$	6.33530	0.403106
$248$	53.4608	3.39476
$249$	0	0
$250$	30.7375i	1.94401i
$251$	2.83977	0.179245	0.0896224	−	0.995976i	$-0.471434\pi$
$251$	2.83977	0.179245	0.0896224	+	0.995976i	$0.471434\pi$
$252$	0	0
$253$	−3.01893	−0.189799
$254$	− 4.94027i	− 0.309980i
$255$	0	0
$256$	66.5415	4.15885
$257$	20.4218	1.27388	0.636938	−	0.770915i	$-0.280201\pi$
$257$	20.4218	1.27388	0.636938	+	0.770915i	$0.280201\pi$
$258$	0	0
$259$	0	0
$260$	63.7923i	3.95623i
$261$	0	0
$262$	8.37777i	0.517580i
$263$	− 12.2388i	− 0.754676i	−0.926076	−	0.377338i	$-0.876839\pi$
$263$	− 12.2388i	− 0.754676i	0.926076	−	0.377338i	$-0.123161\pi$
$264$	0	0
$265$	14.9854i	0.920547i
$266$	0	0
$267$	0	0
$268$	−72.3947	−4.42221
$269$	−13.8987	−0.847421	−0.423711	−	0.905798i	$-0.639273\pi$
$269$	−13.8987	−0.847421	−0.423711	+	0.905798i	$0.639273\pi$
$270$	0	0
$271$	1.55881i	0.0946909i	0.998879	+	0.0473455i	$0.0150762\pi$
$271$	1.55881i	0.0946909i	−0.998879	+	0.0473455i	$0.984924\pi$
$272$	−72.6785	−4.40678
$273$	0	0
$274$	15.9790	0.965328
$275$	− 0.0292508i	− 0.00176389i
$276$	0	0
$277$	21.1195	1.26895	0.634473	−	0.772945i	$-0.281217\pi$
$277$	21.1195	1.26895	0.634473	+	0.772945i	$0.281217\pi$
$278$	−9.23363	−0.553797
$279$	0	0
$280$	0	0
$281$	− 4.38090i	− 0.261343i	−0.991426	−	0.130671i	$-0.958287\pi$
$281$	− 4.38090i	− 0.261343i	0.991426	−	0.130671i	$-0.0417132\pi$
$282$	0	0
$283$	− 15.1453i	− 0.900293i	−0.892955	−	0.450147i	$-0.851372\pi$
$283$	− 15.1453i	− 0.900293i	0.892955	−	0.450147i	$-0.148628\pi$
$284$	17.9441i	1.06479i
$285$	0	0
$286$	11.1398i	0.658713i
$287$	0	0
$288$	0	0
$289$	2.89369	0.170217
$290$	−25.0543	−1.47124
$291$	0	0
$292$	− 50.7146i	− 2.96785i
$293$	−1.34729	−0.0787094	−0.0393547	−	0.999225i	$-0.512530\pi$
$293$	−1.34729	−0.0787094	−0.0393547	+	0.999225i	$0.512530\pi$
$294$	0	0
$295$	−29.9828	−1.74566
$296$	− 53.1756i	− 3.09077i
$297$	0	0
$298$	51.6569	2.99240
$299$	−19.1668	−1.10844
$300$	0	0
$301$	0	0
$302$	3.89115i	0.223910i
$303$	0	0
$304$	20.3911i	1.16951i
$305$	− 18.2325i	− 1.04399i
$306$	0	0
$307$	− 26.9640i	− 1.53891i	−0.638698	−	0.769457i	$-0.720527\pi$
$307$	− 26.9640i	− 1.53891i	0.638698	−	0.769457i	$-0.279473\pi$
$308$	0	0
$309$	0	0
$310$	−33.1928	−1.88523
$311$	6.06228	0.343760	0.171880	−	0.985118i	$-0.445016\pi$
$311$	6.06228	0.343760	0.171880	+	0.985118i	$0.445016\pi$
$312$	0	0
$313$	− 1.24917i	− 0.0706072i	−0.999377	−	0.0353036i	$-0.988760\pi$
$313$	− 1.24917i	− 0.0706072i	0.999377	−	0.0353036i	$-0.0112398\pi$
$314$	−8.44800	−0.476748
$315$	0	0
$316$	−28.8631	−1.62367
$317$	− 16.3340i	− 0.917411i	−0.888588	−	0.458705i	$-0.848313\pi$
$317$	− 16.3340i	− 0.917411i	0.888588	−	0.458705i	$-0.151687\pi$
$318$	0	0
$319$	−3.22600	−0.180622
$320$	−81.7829	−4.57181
$321$	0	0
$322$	0	0
$323$	− 5.58149i	− 0.310562i
$324$	0	0
$325$	− 0.185709i	− 0.0103013i
$326$	− 49.2508i	− 2.72775i
$327$	0	0
$328$	− 44.9178i	− 2.48017i
$329$	0	0
$330$	0	0
$331$	−1.73806	−0.0955325	−0.0477663	−	0.998859i	$-0.515210\pi$
$331$	−1.73806	−0.0955325	−0.0477663	+	0.998859i	$0.515210\pi$
$332$	−52.1854	−2.86405
$333$	0	0
$334$	38.0084i	2.07973i
$335$	28.9373	1.58101
$336$	0	0
$337$	−17.3682	−0.946109	−0.473054	−	0.881033i	$-0.656849\pi$
$337$	−17.3682	−0.946109	−0.473054	+	0.881033i	$0.656849\pi$
$338$	34.8522i	1.89571i
$339$	0	0
$340$	56.2019	3.04797
$341$	−4.27393	−0.231446
$342$	0	0
$343$	0	0
$344$	3.89920i	0.210231i
$345$	0	0
$346$	61.7753i	3.32106i
$347$	− 11.0499i	− 0.593188i	−0.955004	−	0.296594i	$-0.904149\pi$
$347$	− 11.0499i	− 0.593188i	0.955004	−	0.296594i	$-0.0958509\pi$
$348$	0	0
$349$	− 2.60574i	− 0.139482i	−0.997565	−	0.0697411i	$-0.977783\pi$
$349$	− 2.60574i	− 0.139482i	0.997565	−	0.0697411i	$-0.0222173\pi$
$350$	0	0
$351$	0	0
$352$	−19.9479	−1.06323
$353$	23.7919	1.26631	0.633157	−	0.774023i	$-0.281759\pi$
$353$	23.7919	1.26631	0.633157	+	0.774023i	$0.281759\pi$
$354$	0	0
$355$	− 7.17256i	− 0.380680i
$356$	−59.8743	−3.17333
$357$	0	0
$358$	56.6761	2.99542
$359$	31.0060i	1.63643i	0.574911	+	0.818216i	$0.305037\pi$
$359$	31.0060i	1.63643i	−0.574911	+	0.818216i	$0.694963\pi$
$360$	0	0
$361$	17.4340	0.917580
$362$	−53.9277	−2.83438
$363$	0	0
$364$	0	0
$365$	20.2714i	1.06106i
$366$	0	0
$367$	− 12.8336i	− 0.669907i	−0.942235	−	0.334953i	$-0.891279\pi$
$367$	− 12.8336i	− 0.669907i	0.942235	−	0.334953i	$-0.108721\pi$
$368$	− 61.6910i	− 3.21587i
$369$	0	0
$370$	33.0158i	1.71641i
$371$	0	0
$372$	0	0
$373$	−26.7014	−1.38255	−0.691273	−	0.722594i	$-0.742950\pi$
$373$	−26.7014	−1.38255	−0.691273	+	0.722594i	$0.742950\pi$
$374$	9.81436	0.507488
$375$	0	0
$376$	− 65.8363i	− 3.39525i
$377$	−20.4814	−1.05485
$378$	0	0
$379$	−18.1788	−0.933783	−0.466892	−	0.884315i	$-0.654626\pi$
$379$	−18.1788	−0.933783	−0.466892	+	0.884315i	$0.654626\pi$
$380$	− 15.7683i	− 0.808898i
$381$	0	0
$382$	−74.9997	−3.83732
$383$	5.47829	0.279927	0.139964	−	0.990157i	$-0.455301\pi$
$383$	5.47829	0.279927	0.139964	+	0.990157i	$0.455301\pi$
$384$	0	0
$385$	0	0
$386$	− 38.9042i	− 1.98017i
$387$	0	0
$388$	− 10.8551i	− 0.551086i
$389$	− 14.7845i	− 0.749603i	−0.927105	−	0.374802i	$-0.877711\pi$
$389$	− 14.7845i	− 0.749603i	0.927105	−	0.374802i	$-0.122289\pi$
$390$	0	0
$391$	16.8862i	0.853971i
$392$	0	0
$393$	0	0
$394$	41.5065	2.09107
$395$	11.5370	0.580491
$396$	0	0
$397$	− 3.32834i	− 0.167045i	−0.996506	−	0.0835224i	$-0.973383\pi$
$397$	− 3.32834i	− 0.167045i	0.996506	−	0.0835224i	$-0.0266170\pi$
$398$	3.71767	0.186350
$399$	0	0
$400$	0.597732	0.0298866
$401$	24.0773i	1.20236i	0.799112	+	0.601182i	$0.205303\pi$
$401$	24.0773i	1.20236i	−0.799112	+	0.601182i	$0.794697\pi$
$402$	0	0
$403$	−27.1345	−1.35167
$404$	58.9729	2.93401
$405$	0	0
$406$	0	0
$407$	4.25113i	0.210721i
$408$	0	0
$409$	7.60844i	0.376213i	0.982149	+	0.188106i	$0.0602350\pi$
$409$	7.60844i	0.376213i	−0.982149	+	0.188106i	$0.939765\pi$
$410$	27.8887i	1.37732i
$411$	0	0
$412$	40.6668i	2.00351i
$413$	0	0
$414$	0	0
$415$	20.8593	1.02394
$416$	−126.646	−6.20934
$417$	0	0
$418$	− 2.75357i	− 0.134682i
$419$	−28.1468	−1.37506	−0.687531	−	0.726155i	$-0.741306\pi$
$419$	−28.1468	−1.37506	−0.687531	+	0.726155i	$0.741306\pi$
$420$	0	0
$421$	6.14273	0.299378	0.149689	−	0.988733i	$-0.452173\pi$
$421$	6.14273	0.299378	0.149689	+	0.988733i	$0.452173\pi$
$422$	− 20.5012i	− 0.997981i
$423$	0	0
$424$	−66.6015	−3.23446
$425$	−0.163612	−0.00793636
$426$	0	0
$427$	0	0
$428$	78.9643i	3.81688i
$429$	0	0
$430$	− 2.42094i	− 0.116748i
$431$	− 3.95970i	− 0.190732i	−0.995442	−	0.0953660i	$-0.969598\pi$
$431$	− 3.95970i	− 0.190732i	0.995442	−	0.0953660i	$-0.0304021\pi$
$432$	0	0
$433$	− 0.409544i	− 0.0196814i	−0.999952	−	0.00984072i	$-0.996868\pi$
$433$	− 0.409544i	− 0.0196814i	0.999952	−	0.00984072i	$-0.00313245\pi$
$434$	0	0
$435$	0	0
$436$	−27.4025	−1.31234
$437$	4.73769	0.226634
$438$	0	0
$439$	− 17.7978i	− 0.849444i	−0.905324	−	0.424722i	$-0.860372\pi$
$439$	− 17.7978i	− 0.849444i	0.905324	−	0.424722i	$-0.139628\pi$
$440$	17.8501	0.850968
$441$	0	0
$442$	62.3099	2.96378
$443$	2.22743i	0.105828i	0.998599	+	0.0529141i	$0.0168510\pi$
$443$	2.22743i	0.105828i	−0.998599	+	0.0529141i	$0.983149\pi$
$444$	0	0
$445$	23.9327	1.13452
$446$	−67.9199	−3.21610
$447$	0	0
$448$	0	0
$449$	− 4.25376i	− 0.200747i	−0.994950	−	0.100374i	$-0.967996\pi$
$449$	− 4.25376i	− 0.200747i	0.994950	−	0.100374i	$-0.0320038\pi$
$450$	0	0
$451$	3.59096i	0.169092i
$452$	68.9237i	3.24190i
$453$	0	0
$454$	− 55.6858i	− 2.61347i
$455$	0	0
$456$	0	0
$457$	37.6032	1.75900	0.879502	−	0.475896i	$-0.157876\pi$
$457$	37.6032	1.75900	0.879502	+	0.475896i	$0.157876\pi$
$458$	75.6849	3.53653
$459$	0	0
$460$	47.7053i	2.22427i
$461$	27.0807	1.26127	0.630637	−	0.776078i	$-0.282794\pi$
$461$	27.0807	1.26127	0.630637	+	0.776078i	$0.282794\pi$
$462$	0	0
$463$	36.6047	1.70117	0.850583	−	0.525840i	$-0.176249\pi$
$463$	36.6047	1.70117	0.850583	+	0.525840i	$0.176249\pi$
$464$	− 65.9225i	− 3.06037i
$465$	0	0
$466$	20.3836	0.944252
$467$	27.9285	1.29238	0.646188	−	0.763178i	$-0.276362\pi$
$467$	27.9285	1.29238	0.646188	+	0.763178i	$0.276362\pi$
$468$	0	0
$469$	0	0
$470$	40.8765i	1.88549i
$471$	0	0
$472$	− 133.256i	− 6.13361i
$473$	− 0.311722i	− 0.0143330i
$474$	0	0
$475$	0.0459040i	0.00210622i
$476$	0	0
$477$	0	0
$478$	13.3504	0.610633
$479$	14.5600	0.665264	0.332632	−	0.943057i	$-0.392063\pi$
$479$	14.5600	0.665264	0.332632	+	0.943057i	$0.392063\pi$
$480$	0	0
$481$	26.9898i	1.23063i
$482$	39.3258	1.79124
$483$	0	0
$484$	−58.1908	−2.64504
$485$	4.33897i	0.197023i
$486$	0	0
$487$	−12.7378	−0.577206	−0.288603	−	0.957449i	$-0.593191\pi$
$487$	−12.7378	−0.577206	−0.288603	+	0.957449i	$0.593191\pi$
$488$	81.0330	3.66819
$489$	0	0
$490$	0	0
$491$	15.7391i	0.710294i	0.934810	+	0.355147i	$0.115569\pi$
$491$	15.7391i	0.710294i	−0.934810	+	0.355147i	$0.884431\pi$
$492$	0	0
$493$	18.0444i	0.812679i
$494$	− 17.4820i	− 0.786553i
$495$	0	0
$496$	− 87.3365i	− 3.92153i
$497$	0	0
$498$	0	0
$499$	37.0868	1.66023	0.830116	−	0.557591i	$-0.188274\pi$
$499$	37.0868	1.66023	0.830116	+	0.557591i	$0.188274\pi$
$500$	62.5411	2.79692
$501$	0	0
$502$	− 7.83624i	− 0.349748i
$503$	12.5405	0.559152	0.279576	−	0.960124i	$-0.409806\pi$
$503$	12.5405	0.559152	0.279576	+	0.960124i	$0.409806\pi$
$504$	0	0
$505$	−23.5724	−1.04896
$506$	8.33063i	0.370342i
$507$	0	0
$508$	−10.0519	−0.445980
$509$	5.99402	0.265680	0.132840	−	0.991137i	$-0.457590\pi$
$509$	5.99402	0.265680	0.132840	+	0.991137i	$0.457590\pi$
$510$	0	0
$511$	0	0
$512$	− 82.5675i	− 3.64900i
$513$	0	0
$514$	− 56.3531i	− 2.48563i
$515$	− 16.2552i	− 0.716288i
$516$	0	0
$517$	5.26329i	0.231479i
$518$	0	0
$519$	0	0
$520$	113.327	4.96974
$521$	17.5334	0.768150	0.384075	−	0.923302i	$-0.374520\pi$
$521$	17.5334	0.768150	0.384075	+	0.923302i	$0.374520\pi$
$522$	0	0
$523$	− 40.0001i	− 1.74908i	−0.484950	−	0.874542i	$-0.661162\pi$
$523$	− 40.0001i	− 1.74908i	0.484950	−	0.874542i	$-0.338838\pi$
$524$	17.0461	0.744662
$525$	0	0
$526$	−33.7725	−1.47255
$527$	23.9059i	1.04136i
$528$	0	0
$529$	8.66665	0.376811
$530$	41.3517	1.79620
$531$	0	0
$532$	0	0
$533$	22.7985i	0.987513i
$534$	0	0
$535$	− 31.5633i	− 1.36460i
$536$	128.610i	5.55509i
$537$	0	0
$538$	38.3530i	1.65352i
$539$	0	0
$540$	0	0
$541$	−34.3970	−1.47884	−0.739421	−	0.673244i	$-0.764901\pi$
$541$	−34.3970	−1.47884	−0.739421	+	0.673244i	$0.764901\pi$
$542$	4.30147	0.184764
$543$	0	0
$544$	111.577i	4.78383i
$545$	10.9532	0.469185
$546$	0	0
$547$	35.3009	1.50936	0.754680	−	0.656094i	$-0.227792\pi$
$547$	35.3009	1.50936	0.754680	+	0.656094i	$0.227792\pi$
$548$	− 32.5122i	− 1.38885i
$549$	0	0
$550$	−0.0807165	−0.00344176
$551$	5.06264	0.215676
$552$	0	0
$553$	0	0
$554$	− 58.2783i	− 2.47601i
$555$	0	0
$556$	18.7875i	0.796768i
$557$	− 14.4115i	− 0.610637i	−0.952250	−	0.305318i	$-0.901237\pi$
$557$	− 14.4115i	− 0.610637i	0.952250	−	0.305318i	$-0.0987629\pi$
$558$	0	0
$559$	− 1.97908i	− 0.0837060i
$560$	0	0
$561$	0	0
$562$	−12.0889	−0.509940
$563$	−13.5455	−0.570875	−0.285438	−	0.958397i	$-0.592139\pi$
$563$	−13.5455	−0.570875	−0.285438	+	0.958397i	$0.592139\pi$
$564$	0	0
$565$	− 27.5499i	− 1.15903i
$566$	−41.7928	−1.75668
$567$	0	0
$568$	31.8779	1.33757
$569$	− 10.2750i	− 0.430751i	−0.976531	−	0.215376i	$-0.930902\pi$
$569$	− 10.2750i	− 0.430751i	0.976531	−	0.215376i	$-0.0690976\pi$
$570$	0	0
$571$	15.5872	0.652305	0.326153	−	0.945317i	$-0.394248\pi$
$571$	15.5872	0.652305	0.326153	+	0.945317i	$0.394248\pi$
$572$	22.6660	0.947715
$573$	0	0
$574$	0	0
$575$	− 0.138878i	− 0.00579160i
$576$	0	0
$577$	26.0495i	1.08445i	0.840232	+	0.542227i	$0.182419\pi$
$577$	26.0495i	1.08445i	−0.840232	+	0.542227i	$0.817581\pi$
$578$	− 7.98503i	− 0.332134i
$579$	0	0
$580$	50.9775i	2.11672i
$581$	0	0
$582$	0	0
$583$	5.32447	0.220517
$584$	−90.0949	−3.72815
$585$	0	0
$586$	3.71779i	0.153580i
$587$	32.3864	1.33673	0.668364	−	0.743834i	$-0.266995\pi$
$587$	32.3864	1.33673	0.668364	+	0.743834i	$0.266995\pi$
$588$	0	0
$589$	6.70718	0.276365
$590$	82.7363i	3.40620i
$591$	0	0
$592$	−86.8707	−3.57036
$593$	5.21887	0.214313	0.107157	−	0.994242i	$-0.465825\pi$
$593$	5.21887	0.214313	0.107157	+	0.994242i	$0.465825\pi$
$594$	0	0
$595$	0	0
$596$	− 105.105i	− 4.30528i
$597$	0	0
$598$	52.8899i	2.16283i
$599$	− 32.7107i	− 1.33652i	−0.743926	−	0.668262i	$-0.767039\pi$
$599$	− 32.7107i	− 1.33652i	0.743926	−	0.668262i	$-0.232961\pi$
$600$	0	0
$601$	20.5762i	0.839319i	0.907682	+	0.419659i	$0.137850\pi$
$601$	20.5762i	0.839319i	−0.907682	+	0.419659i	$0.862150\pi$
$602$	0	0
$603$	0	0
$604$	7.91724	0.322148
$605$	23.2598	0.945645
$606$	0	0
$607$	21.3300i	0.865758i	0.901452	+	0.432879i	$0.142502\pi$
$607$	21.3300i	0.865758i	−0.901452	+	0.432879i	$0.857498\pi$
$608$	31.3047	1.26957
$609$	0	0
$610$	−50.3119	−2.03707
$611$	33.4159i	1.35186i
$612$	0	0
$613$	−16.3388	−0.659918	−0.329959	−	0.943995i	$-0.607035\pi$
$613$	−16.3388	−0.659918	−0.329959	+	0.943995i	$0.607035\pi$
$614$	−74.4060	−3.00278
$615$	0	0
$616$	0	0
$617$	− 20.8105i	− 0.837800i	−0.908032	−	0.418900i	$-0.862416\pi$
$617$	− 20.8105i	− 0.837800i	0.908032	−	0.418900i	$-0.137584\pi$
$618$	0	0
$619$	− 2.84787i	− 0.114465i	−0.998361	−	0.0572327i	$-0.981772\pi$
$619$	− 2.84787i	− 0.114465i	0.998361	−	0.0572327i	$-0.0182277\pi$
$620$	67.5368i	2.71235i
$621$	0	0
$622$	− 16.7286i	− 0.670757i
$623$	0	0
$624$	0	0
$625$	−25.1821	−1.00728
$626$	−3.44703	−0.137771
$627$	0	0
$628$	17.1890i	0.685916i
$629$	23.7784	0.948107
$630$	0	0
$631$	24.5713	0.978167	0.489084	−	0.872237i	$-0.337331\pi$
$631$	24.5713	0.978167	0.489084	+	0.872237i	$0.337331\pi$
$632$	51.2754i	2.03963i
$633$	0	0
$634$	−45.0731	−1.79008
$635$	4.01790	0.159445
$636$	0	0
$637$	0	0
$638$	8.90203i	0.352435i
$639$	0	0
$640$	113.392i	4.48223i
$641$	− 37.6347i	− 1.48648i	−0.669025	−	0.743240i	$-0.733288\pi$
$641$	− 37.6347i	− 1.48648i	0.669025	−	0.743240i	$-0.266712\pi$
$642$	0	0
$643$	− 13.4783i	− 0.531534i	−0.964037	−	0.265767i	$-0.914375\pi$
$643$	− 13.4783i	− 0.531534i	0.964037	−	0.265767i	$-0.0856252\pi$
$644$	0	0
$645$	0	0
$646$	−15.4019	−0.605980
$647$	−4.47576	−0.175960	−0.0879802	−	0.996122i	$-0.528041\pi$
$647$	−4.47576	−0.175960	−0.0879802	+	0.996122i	$0.528041\pi$
$648$	0	0
$649$	10.6532i	0.418174i
$650$	−0.512457	−0.0201002
$651$	0	0
$652$	−100.210	−3.92452
$653$	28.3537i	1.10957i	0.831995	+	0.554783i	$0.187199\pi$
$653$	28.3537i	1.10957i	−0.831995	+	0.554783i	$0.812801\pi$
$654$	0	0
$655$	−6.81360	−0.266229
$656$	−73.3803	−2.86502
$657$	0	0
$658$	0	0
$659$	9.66270i	0.376405i	0.982130	+	0.188203i	$0.0602662\pi$
$659$	9.66270i	0.376405i	−0.982130	+	0.188203i	$0.939734\pi$
$660$	0	0
$661$	− 7.68551i	− 0.298932i	−0.988767	−	0.149466i	$-0.952245\pi$
$661$	− 7.68551i	− 0.298932i	0.988767	−	0.149466i	$-0.0477554\pi$
$662$	4.79611i	0.186406i
$663$	0	0
$664$	92.7077i	3.59776i
$665$	0	0
$666$	0	0
$667$	−15.3165	−0.593056
$668$	77.3350	2.99218
$669$	0	0
$670$	− 79.8514i	− 3.08493i
$671$	−6.47820	−0.250088
$672$	0	0
$673$	−14.4935	−0.558683	−0.279341	−	0.960192i	$-0.590116\pi$
$673$	−14.4935	−0.558683	−0.279341	+	0.960192i	$0.590116\pi$
$674$	47.9270i	1.84608i
$675$	0	0
$676$	70.9132	2.72743
$677$	20.4462	0.785813	0.392907	−	0.919578i	$-0.371470\pi$
$677$	20.4462	0.785813	0.392907	+	0.919578i	$0.371470\pi$
$678$	0	0
$679$	0	0
$680$	− 99.8430i	− 3.82880i
$681$	0	0
$682$	11.7937i	0.451606i
$683$	− 22.0854i	− 0.845073i	−0.906346	−	0.422536i	$-0.861140\pi$
$683$	− 22.0854i	− 0.845073i	0.906346	−	0.422536i	$-0.138860\pi$
$684$	0	0
$685$	12.9957i	0.496539i
$686$	0	0
$687$	0	0
$688$	6.36995	0.242852
$689$	33.8043	1.28784
$690$	0	0
$691$	31.4512i	1.19646i	0.801325	+	0.598230i	$0.204129\pi$
$691$	31.4512i	1.19646i	−0.801325	+	0.598230i	$0.795871\pi$
$692$	125.693	4.77814
$693$	0	0
$694$	−30.4917	−1.15745
$695$	− 7.50967i	− 0.284858i
$696$	0	0
$697$	20.0858	0.760804
$698$	−7.19044	−0.272162
$699$	0	0
$700$	0	0
$701$	4.78252i	0.180633i	0.995913	+	0.0903167i	$0.0287879\pi$
$701$	4.78252i	0.180633i	−0.995913	+	0.0903167i	$0.971212\pi$
$702$	0	0
$703$	− 6.67141i	− 0.251617i
$704$	29.0583i	1.09518i
$705$	0	0
$706$	− 65.6528i	− 2.47088i
$707$	0	0
$708$	0	0
$709$	−37.4050	−1.40477	−0.702386	−	0.711796i	$-0.747882\pi$
$709$	−37.4050	−1.40477	−0.702386	+	0.711796i	$0.747882\pi$
$710$	−19.7924	−0.742796
$711$	0	0
$712$	106.367i	3.98627i
$713$	−20.2918	−0.759935
$714$	0	0
$715$	−9.05998	−0.338824
$716$	− 115.318i	− 4.30963i
$717$	0	0
$718$	85.5598	3.19306
$719$	−35.4881	−1.32348	−0.661742	−	0.749732i	$-0.730183\pi$
$719$	−35.4881	−1.32348	−0.661742	+	0.749732i	$0.730183\pi$
$720$	0	0
$721$	0	0
$722$	− 48.1085i	− 1.79041i
$723$	0	0
$724$	109.726i	4.07792i
$725$	− 0.148403i	− 0.00551156i
$726$	0	0
$727$	− 34.7768i	− 1.28980i	−0.764267	−	0.644900i	$-0.776899\pi$
$727$	− 34.7768i	− 1.28980i	0.764267	−	0.644900i	$-0.223101\pi$
$728$	0	0
$729$	0	0
$730$	55.9383	2.07037
$731$	−1.74359	−0.0644891
$732$	0	0
$733$	− 16.4602i	− 0.607971i	−0.952677	−	0.303986i	$-0.901683\pi$
$733$	− 16.4602i	− 0.607971i	0.952677	−	0.303986i	$-0.0983175\pi$
$734$	−35.4137	−1.30715
$735$	0	0
$736$	−94.7089	−3.49102
$737$	− 10.2817i	− 0.378732i
$738$	0	0
$739$	−7.37549	−0.271312	−0.135656	−	0.990756i	$-0.543314\pi$
$739$	−7.37549	−0.271312	−0.135656	+	0.990756i	$0.543314\pi$
$740$	67.1766	2.46946
$741$	0	0
$742$	0	0
$743$	25.9252i	0.951103i	0.879688	+	0.475552i	$0.157752\pi$
$743$	25.9252i	0.951103i	−0.879688	+	0.475552i	$0.842248\pi$
$744$	0	0
$745$	42.0123i	1.53921i
$746$	73.6815i	2.69767i
$747$	0	0
$748$	− 19.9691i	− 0.730142i
$749$	0	0
$750$	0	0
$751$	22.4705	0.819958	0.409979	−	0.912095i	$-0.365536\pi$
$751$	22.4705	0.819958	0.409979	+	0.912095i	$0.365536\pi$
$752$	−107.554	−3.92208
$753$	0	0
$754$	56.5177i	2.05825i
$755$	−3.16465	−0.115173
$756$	0	0
$757$	−5.00011	−0.181732	−0.0908660	−	0.995863i	$-0.528963\pi$
$757$	−5.00011	−0.181732	−0.0908660	+	0.995863i	$0.528963\pi$
$758$	50.1637i	1.82203i
$759$	0	0
$760$	−28.0125	−1.01612
$761$	4.87680	0.176784	0.0883920	−	0.996086i	$-0.471827\pi$
$761$	4.87680	0.176784	0.0883920	+	0.996086i	$0.471827\pi$
$762$	0	0
$763$	0	0
$764$	152.601i	5.52089i
$765$	0	0
$766$	− 15.1171i	− 0.546204i
$767$	67.6354i	2.44217i
$768$	0	0
$769$	− 21.9517i	− 0.791600i	−0.918337	−	0.395800i	$-0.870467\pi$
$769$	− 21.9517i	− 0.791600i	0.918337	−	0.395800i	$-0.129533\pi$
$770$	0	0
$771$	0	0
$772$	−79.1576	−2.84894
$773$	13.6225	0.489968	0.244984	−	0.969527i	$-0.421217\pi$
$773$	13.6225	0.489968	0.244984	+	0.969527i	$0.421217\pi$
$774$	0	0
$775$	− 0.196610i	− 0.00706245i
$776$	−19.2842	−0.692263
$777$	0	0
$778$	−40.7972	−1.46265
$779$	− 5.63539i	− 0.201909i
$780$	0	0
$781$	−2.54848	−0.0911919
$782$	46.5968	1.66630
$783$	0	0
$784$	0	0
$785$	− 6.87071i	− 0.245226i
$786$	0	0
$787$	− 23.1804i	− 0.826293i	−0.910665	−	0.413146i	$-0.864430\pi$
$787$	− 23.1804i	− 0.826293i	0.910665	−	0.413146i	$-0.135570\pi$
$788$	− 84.4526i	− 3.00850i
$789$	0	0
$790$	− 31.8360i	− 1.13267i
$791$	0	0
$792$	0	0
$793$	−41.1291	−1.46054
$794$	−9.18444	−0.325943
$795$	0	0
$796$	− 7.56428i	− 0.268109i
$797$	29.1588	1.03286	0.516428	−	0.856331i	$-0.327261\pi$
$797$	29.1588	1.03286	0.516428	+	0.856331i	$0.327261\pi$
$798$	0	0
$799$	29.4398	1.04151
$800$	− 0.917647i	− 0.0324437i
$801$	0	0
$802$	66.4404	2.34609
$803$	7.20265	0.254176
$804$	0	0
$805$	0	0
$806$	74.8767i	2.63742i
$807$	0	0
$808$	− 104.766i	− 3.68565i
$809$	− 5.69368i	− 0.200179i	−0.994978	−	0.100090i	$-0.968087\pi$
$809$	− 5.69368i	− 0.200179i	0.994978	−	0.100090i	$-0.0319130\pi$
$810$	0	0
$811$	45.3695i	1.59314i	0.604547	+	0.796570i	$0.293354\pi$
$811$	45.3695i	1.59314i	−0.604547	+	0.796570i	$0.706646\pi$
$812$	0	0
$813$	0	0
$814$	11.7308	0.411166
$815$	40.0554	1.40308
$816$	0	0
$817$	0.489192i	0.0171147i
$818$	20.9952	0.734079
$819$	0	0
$820$	56.7446	1.98161
$821$	− 2.41885i	− 0.0844183i	−0.999109	−	0.0422091i	$-0.986560\pi$
$821$	− 2.41885i	− 0.0844183i	0.999109	−	0.0422091i	$-0.0134396\pi$
$822$	0	0
$823$	7.82407	0.272730	0.136365	−	0.990659i	$-0.456458\pi$
$823$	7.82407	0.272730	0.136365	+	0.990659i	$0.456458\pi$
$824$	72.2448	2.51677
$825$	0	0
$826$	0	0
$827$	23.3143i	0.810717i	0.914158	+	0.405359i	$0.132853\pi$
$827$	23.3143i	0.810717i	−0.914158	+	0.405359i	$0.867147\pi$
$828$	0	0
$829$	28.2751i	0.982034i	0.871150	+	0.491017i	$0.163375\pi$
$829$	28.2751i	0.982034i	−0.871150	+	0.491017i	$0.836625\pi$
$830$	− 57.5605i	− 1.99795i
$831$	0	0
$832$	184.487i	6.39593i
$833$	0	0
$834$	0	0
$835$	−30.9120	−1.06975
$836$	−5.60264	−0.193772
$837$	0	0
$838$	77.6701i	2.68307i
$839$	−32.3512	−1.11689	−0.558444	−	0.829542i	$-0.688601\pi$
$839$	−32.3512	−1.11689	−0.558444	+	0.829542i	$0.688601\pi$
$840$	0	0
$841$	12.6330	0.435619
$842$	− 16.9506i	− 0.584157i
$843$	0	0
$844$	−41.7133	−1.43583
$845$	−28.3451	−0.975102
$846$	0	0
$847$	0	0
$848$	108.804i	3.73634i
$849$	0	0
$850$	0.451482i	0.0154857i
$851$	20.1836i	0.691885i
$852$	0	0
$853$	14.1269i	0.483694i	0.970314	+	0.241847i	$0.0777533\pi$
$853$	14.1269i	0.483694i	−0.970314	+	0.241847i	$0.922247\pi$
$854$	0	0
$855$	0	0
$856$	140.281	4.79469
$857$	29.5897	1.01076	0.505382	−	0.862896i	$-0.331352\pi$
$857$	29.5897	1.01076	0.505382	+	0.862896i	$0.331352\pi$
$858$	0	0
$859$	11.3240i	0.386370i	0.981162	+	0.193185i	$0.0618818\pi$
$859$	11.3240i	0.386370i	−0.981162	+	0.193185i	$0.938118\pi$
$860$	−4.92584	−0.167970
$861$	0	0
$862$	−10.9266	−0.372163
$863$	30.6289i	1.04262i	0.853368	+	0.521309i	$0.174556\pi$
$863$	30.6289i	1.04262i	−0.853368	+	0.521309i	$0.825444\pi$
$864$	0	0
$865$	−50.2416	−1.70826
$866$	−1.13012	−0.0384031
$867$	0	0
$868$	0	0
$869$	− 4.09922i	− 0.139057i
$870$	0	0
$871$	− 65.2771i	− 2.21183i
$872$	48.6808i	1.64854i
$873$	0	0
$874$	− 13.0735i	− 0.442216i
$875$	0	0
$876$	0	0
$877$	−16.5491	−0.558824	−0.279412	−	0.960171i	$-0.590140\pi$
$877$	−16.5491	−0.558824	−0.279412	+	0.960171i	$0.590140\pi$
$878$	−49.1124	−1.65746
$879$	0	0
$880$	− 29.1609i	− 0.983013i
$881$	−33.8234	−1.13954	−0.569770	−	0.821804i	$-0.692968\pi$
$881$	−33.8234	−1.13954	−0.569770	+	0.821804i	$0.692968\pi$
$882$	0	0
$883$	10.8964	0.366691	0.183346	−	0.983048i	$-0.441307\pi$
$883$	10.8964	0.366691	0.183346	+	0.983048i	$0.441307\pi$
$884$	− 126.781i	− 4.26410i
$885$	0	0
$886$	6.14650	0.206496
$887$	55.7874	1.87316	0.936578	−	0.350459i	$-0.113974\pi$
$887$	55.7874	1.87316	0.936578	+	0.350459i	$0.113974\pi$
$888$	0	0
$889$	0	0
$890$	− 66.0413i	− 2.21371i
$891$	0	0
$892$	138.195i	4.62712i
$893$	− 8.25981i	− 0.276404i
$894$	0	0
$895$	46.0944i	1.54076i
$896$	0	0
$897$	0	0
$898$	−11.7381	−0.391705
$899$	−21.6837	−0.723191
$900$	0	0
$901$	− 29.7820i	− 0.992183i
$902$	9.90913	0.329938
$903$	0	0
$904$	122.443	4.07241
$905$	− 43.8591i	− 1.45793i
$906$	0	0
$907$	−52.8615	−1.75524	−0.877619	−	0.479358i	$-0.840869\pi$
$907$	−52.8615	−1.75524	−0.877619	+	0.479358i	$0.840869\pi$
$908$	−113.303	−3.76009
$909$	0	0
$910$	0	0
$911$	− 10.0952i	− 0.334470i	−0.985917	−	0.167235i	$-0.946516\pi$
$911$	− 10.0952i	− 0.334470i	0.985917	−	0.167235i	$-0.0534838\pi$
$912$	0	0
$913$	− 7.41153i	− 0.245286i
$914$	− 103.765i	− 3.43223i
$915$	0	0
$916$	− 153.995i	− 5.08813i
$917$	0	0
$918$	0	0
$919$	7.74711	0.255554	0.127777	−	0.991803i	$-0.459216\pi$
$919$	7.74711	0.255554	0.127777	+	0.991803i	$0.459216\pi$
$920$	84.7488	2.79409
$921$	0	0
$922$	− 74.7281i	− 2.46104i
$923$	−16.1799	−0.532569
$924$	0	0
$925$	−0.195561	−0.00643002
$926$	− 101.009i	− 3.31937i
$927$	0	0
$928$	−101.205	−3.32222
$929$	−46.5611	−1.52762	−0.763810	−	0.645441i	$-0.776673\pi$
$929$	−46.5611	−1.52762	−0.763810	+	0.645441i	$0.776673\pi$
$930$	0	0
$931$	0	0
$932$	− 41.4741i	− 1.35853i
$933$	0	0
$934$	− 77.0675i	− 2.52173i
$935$	7.98196i	0.261038i
$936$	0	0
$937$	1.77445i	0.0579688i	0.999580	+	0.0289844i	$0.00922731\pi$
$937$	1.77445i	0.0579688i	−0.999580	+	0.0289844i	$0.990773\pi$
$938$	0	0
$939$	0	0
$940$	83.1708	2.71273
$941$	−56.0365	−1.82674	−0.913370	−	0.407132i	$-0.866529\pi$
$941$	−56.0365	−1.82674	−0.913370	+	0.407132i	$0.866529\pi$
$942$	0	0
$943$	17.0492i	0.555200i
$944$	−217.695	−7.08536
$945$	0	0
$946$	−0.860184	−0.0279670
$947$	22.2139i	0.721853i	0.932594	+	0.360927i	$0.117540\pi$
$947$	22.2139i	0.721853i	−0.932594	+	0.360927i	$0.882460\pi$
$948$	0	0
$949$	45.7285	1.48441
$950$	0.126670	0.00410973
$951$	0	0
$952$	0	0
$953$	− 29.4738i	− 0.954750i	−0.878700	−	0.477375i	$-0.841588\pi$
$953$	− 29.4738i	− 0.954750i	0.878700	−	0.477375i	$-0.158412\pi$
$954$	0	0
$955$	− 60.9969i	− 1.97381i
$956$	− 27.1638i	− 0.878540i
$957$	0	0
$958$	− 40.1778i	− 1.29809i
$959$	0	0
$960$	0	0
$961$	2.27267	0.0733118
$962$	74.4774	2.40125
$963$	0	0
$964$	− 80.0155i	− 2.57713i
$965$	31.6406	1.01855
$966$	0	0
$967$	−15.5115	−0.498815	−0.249407	−	0.968399i	$-0.580236\pi$
$967$	−15.5115	−0.498815	−0.249407	+	0.968399i	$0.580236\pi$
$968$	103.376i	3.32264i
$969$	0	0
$970$	11.9732	0.384437
$971$	12.0421	0.386450	0.193225	−	0.981154i	$-0.438105\pi$
$971$	12.0421	0.386450	0.193225	+	0.981154i	$0.438105\pi$
$972$	0	0
$973$	0	0
$974$	35.1495i	1.12626i
$975$	0	0
$976$	− 132.380i	− 4.23738i
$977$	33.0576i	1.05761i	0.848745	+	0.528803i	$0.177359\pi$
$977$	33.0576i	1.05761i	−0.848745	+	0.528803i	$0.822641\pi$
$978$	0	0
$979$	− 8.50353i	− 0.271774i
$980$	0	0
$981$	0	0
$982$	43.4314	1.38595
$983$	−54.5392	−1.73953	−0.869765	−	0.493467i	$-0.835729\pi$
$983$	−54.5392	−1.73953	−0.869765	+	0.493467i	$0.835729\pi$
$984$	0	0
$985$	33.7570i	1.07559i
$986$	49.7929	1.58573
$987$	0	0
$988$	−35.5704	−1.13164
$989$	− 1.48000i	− 0.0470612i
$990$	0	0
$991$	−44.9904	−1.42917	−0.714583	−	0.699550i	$-0.753384\pi$
$991$	−44.9904	−1.42917	−0.714583	+	0.699550i	$0.753384\pi$
$992$	−134.080	−4.25705
$993$	0	0
$994$	0	0
$995$	3.02356i	0.0958534i
$996$	0	0
$997$	44.6129i	1.41290i	0.707761	+	0.706452i	$0.249705\pi$
$997$	44.6129i	1.41290i	−0.707761	+	0.706452i	$0.750295\pi$
$998$	− 102.340i	− 3.23950i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.c.e.1322.2	yes	16
3.2	odd	2	inner	1323.2.c.e.1322.15	yes	16
7.6	odd	2	inner	1323.2.c.e.1322.1	✓	16
21.20	even	2	inner	1323.2.c.e.1322.16	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.2.c.e.1322.1	✓	16	7.6	odd	2	inner
1323.2.c.e.1322.2	yes	16	1.1	even	1	trivial
1323.2.c.e.1322.15	yes	16	3.2	odd	2	inner
1323.2.c.e.1322.16	yes	16	21.20	even	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)

\(f(q)\)	\(=\)	\(q-2.75946i q^{2} -5.61463 q^{4} +2.24426 q^{5} +9.97442i q^{8} +O(q^{10})\)	\(q-2.75946i q^{2} -5.61463 q^{4} +2.24426 q^{5} +9.97442i q^{8} -6.19294i q^{10} -0.797407i q^{11} -5.06262i q^{13} +16.2948 q^{16} -4.46023 q^{17} +1.25139i q^{19} -12.6007 q^{20} -2.20041 q^{22} -3.78594i q^{23} +0.0366824 q^{25} -13.9701 q^{26} -4.04562i q^{29} -5.35979i q^{31} -25.0160i q^{32} +12.3078i q^{34} -5.33120 q^{37} +3.45316 q^{38} +22.3851i q^{40} -4.50330 q^{41} +0.390920 q^{43} +4.47714i q^{44} -10.4472 q^{46} -6.60051 q^{47} -0.101224i q^{50} +28.4247i q^{52} +6.67723i q^{53} -1.78958i q^{55} -11.1637 q^{58} -13.3598 q^{59} -8.12408i q^{61} -14.7901 q^{62} -36.4410 q^{64} -11.3618i q^{65} +12.8939 q^{67} +25.0425 q^{68} -3.19597i q^{71} +9.03259i q^{73} +14.7112i q^{74} -7.02608i q^{76} +5.14069 q^{79} +36.5696 q^{80} +12.4267i q^{82} +9.29454 q^{83} -10.0099 q^{85} -1.07873i q^{86} +7.95367 q^{88} +10.6640 q^{89} +21.2566i q^{92} +18.2139i q^{94} +2.80844i q^{95} +1.93337i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 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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