LMFDB - Embedded newform 1323.2.c.f.1322.9

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			1322.9
Root			$-2.73296i$ of defining polynomial
Character	$\chi$	$=$	1323.1322
Dual form			1323.2.c.f.1322.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.529484i q^{2} +1.71965 q^{4} -0.753692 q^{5} +1.96949i q^{8} +O(q^{10})$	$q+0.529484i q^{2} +1.71965 q^{4} -0.753692 q^{5} +1.96949i q^{8} -0.399068i q^{10} +5.38922i q^{11} -1.46226i q^{13} +2.39648 q^{16} +5.82053 q^{17} -3.54379i q^{19} -1.29608 q^{20} -2.85351 q^{22} +3.26740i q^{23} -4.43195 q^{25} +0.774245 q^{26} -0.0196202i q^{29} +6.49314i q^{31} +5.20788i q^{32} +3.08188i q^{34} -0.730035 q^{37} +1.87638 q^{38} -1.48439i q^{40} -10.1827 q^{41} +6.78337 q^{43} +9.26756i q^{44} -1.73004 q^{46} +3.42231 q^{47} -2.34664i q^{50} -2.51458i q^{52} +10.3163i q^{53} -4.06182i q^{55} +0.0103886 q^{58} +7.88303 q^{59} +12.3687i q^{61} -3.43801 q^{62} +2.03547 q^{64} +1.10210i q^{65} +11.3869 q^{67} +10.0093 q^{68} -13.6174i q^{71} -6.55599i q^{73} -0.386542i q^{74} -6.09407i q^{76} +10.5281 q^{79} -1.80621 q^{80} -5.39155i q^{82} -16.0159 q^{83} -4.38689 q^{85} +3.59168i q^{86} -10.6140 q^{88} -1.63180 q^{89} +5.61877i q^{92} +1.81206i q^{94} +2.67093i q^{95} +16.8415i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.529484i	0.374402i	0.982322	+	0.187201i	$0.0599415\pi$
$2$	0.529484i	0.374402i	−0.982322	+	0.187201i	$0.940058\pi$
$3$	0	0
$3$	0	0
$4$	1.71965	0.859823
$5$	−0.753692	−0.337061	−0.168531	−	0.985696i	$-0.553902\pi$
$5$	−0.753692	−0.337061	−0.168531	+	0.985696i	$0.553902\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.96949i	0.696321i
$9$	0	0
$10$	− 0.399068i	− 0.126196i
$11$	5.38922i	1.62491i	0.583022	+	0.812456i	$0.301870\pi$
$11$	5.38922i	1.62491i	−0.583022	+	0.812456i	$0.698130\pi$
$12$	0	0
$13$	− 1.46226i	− 0.405559i	−0.979224	−	0.202780i	$-0.935003\pi$
$13$	− 1.46226i	− 0.405559i	0.979224	−	0.202780i	$-0.0649975\pi$
$14$	0	0
$15$	0	0
$16$	2.39648	0.599120
$17$	5.82053	1.41169	0.705843	−	0.708368i	$-0.250568\pi$
$17$	5.82053	1.41169	0.705843	+	0.708368i	$0.250568\pi$
$18$	0	0
$19$	− 3.54379i	− 0.813002i	−0.913650	−	0.406501i	$-0.866749\pi$
$19$	− 3.54379i	− 0.813002i	0.913650	−	0.406501i	$-0.133251\pi$
$20$	−1.29608	−0.289813
$21$	0	0
$22$	−2.85351	−0.608370
$23$	3.26740i	0.681300i	0.940190	+	0.340650i	$0.110647\pi$
$23$	3.26740i	0.681300i	−0.940190	+	0.340650i	$0.889353\pi$
$24$	0	0
$25$	−4.43195	−0.886390
$26$	0.774245	0.151842
$27$	0	0
$28$	0	0
$29$	− 0.0196202i	− 0.00364337i	−0.999998	−	0.00182169i	$-0.999420\pi$
$29$	− 0.0196202i	− 0.00364337i	0.999998	−	0.00182169i	$-0.000579861\pi$
$30$	0	0
$31$	6.49314i	1.16620i	0.812400	+	0.583101i	$0.198161\pi$
$31$	6.49314i	1.16620i	−0.812400	+	0.583101i	$0.801839\pi$
$32$	5.20788i	0.920632i
$33$	0	0
$34$	3.08188i	0.528538i
$35$	0	0
$36$	0	0
$37$	−0.730035	−0.120017	−0.0600085	−	0.998198i	$-0.519113\pi$
$37$	−0.730035	−0.120017	−0.0600085	+	0.998198i	$0.519113\pi$
$38$	1.87638	0.304389
$39$	0	0
$40$	− 1.48439i	− 0.234703i
$41$	−10.1827	−1.59026	−0.795132	−	0.606437i	$-0.792598\pi$
$41$	−10.1827	−1.59026	−0.795132	+	0.606437i	$0.792598\pi$
$42$	0	0
$43$	6.78337	1.03445	0.517227	−	0.855848i	$-0.326964\pi$
$43$	6.78337	1.03445	0.517227	+	0.855848i	$0.326964\pi$
$44$	9.26756i	1.39714i
$45$	0	0
$46$	−1.73004	−0.255080
$47$	3.42231	0.499195	0.249598	−	0.968350i	$-0.419702\pi$
$47$	3.42231	0.499195	0.249598	+	0.968350i	$0.419702\pi$
$48$	0	0
$49$	0	0
$50$	− 2.34664i	− 0.331866i
$51$	0	0
$52$	− 2.51458i	− 0.348709i
$53$	10.3163i	1.41705i	0.705684	+	0.708526i	$0.250640\pi$
$53$	10.3163i	1.41705i	−0.705684	+	0.708526i	$0.749360\pi$
$54$	0	0
$55$	− 4.06182i	− 0.547695i
$56$	0	0
$57$	0	0
$58$	0.0103886	0.00136409
$59$	7.88303	1.02628	0.513142	−	0.858304i	$-0.328482\pi$
$59$	7.88303	1.02628	0.513142	+	0.858304i	$0.328482\pi$
$60$	0	0
$61$	12.3687i	1.58365i	0.610746	+	0.791827i	$0.290870\pi$
$61$	12.3687i	1.58365i	−0.610746	+	0.791827i	$0.709130\pi$
$62$	−3.43801	−0.436628
$63$	0	0
$64$	2.03547	0.254434
$65$	1.10210i	0.136698i
$66$	0	0
$67$	11.3869	1.39113	0.695565	−	0.718463i	$-0.255154\pi$
$67$	11.3869	1.39113	0.695565	+	0.718463i	$0.255154\pi$
$68$	10.0093	1.21380
$69$	0	0
$70$	0	0
$71$	− 13.6174i	− 1.61609i	−0.589119	−	0.808046i	$-0.700525\pi$
$71$	− 13.6174i	− 1.61609i	0.589119	−	0.808046i	$-0.299475\pi$
$72$	0	0
$73$	− 6.55599i	− 0.767320i	−0.923474	−	0.383660i	$-0.874663\pi$
$73$	− 6.55599i	− 0.767320i	0.923474	−	0.383660i	$-0.125337\pi$
$74$	− 0.386542i	− 0.0449346i
$75$	0	0
$76$	− 6.09407i	− 0.699038i
$77$	0	0
$78$	0	0
$79$	10.5281	1.18450	0.592252	−	0.805753i	$-0.298239\pi$
$79$	10.5281	1.18450	0.592252	+	0.805753i	$0.298239\pi$
$80$	−1.80621	−0.201940
$81$	0	0
$82$	− 5.39155i	− 0.595397i
$83$	−16.0159	−1.75797	−0.878987	−	0.476845i	$-0.841780\pi$
$83$	−16.0159	−1.75797	−0.878987	+	0.476845i	$0.841780\pi$
$84$	0	0
$85$	−4.38689	−0.475825
$86$	3.59168i	0.387301i
$87$	0	0
$88$	−10.6140	−1.13146
$89$	−1.63180	−0.172971	−0.0864854	−	0.996253i	$-0.527564\pi$
$89$	−1.63180	−0.172971	−0.0864854	+	0.996253i	$0.527564\pi$
$90$	0	0
$91$	0	0
$92$	5.61877i	0.585798i
$93$	0	0
$94$	1.81206i	0.186900i
$95$	2.67093i	0.274032i
$96$	0	0
$97$	16.8415i	1.70999i	0.518634	+	0.854996i	$0.326441\pi$
$97$	16.8415i	1.70999i	−0.518634	+	0.854996i	$0.673559\pi$
$98$	0	0
$99$	0	0
$100$	−7.62139	−0.762139
$101$	10.4508	1.03989	0.519945	−	0.854200i	$-0.325952\pi$
$101$	10.4508	1.03989	0.519945	+	0.854200i	$0.325952\pi$
$102$	0	0
$103$	− 11.8014i	− 1.16283i	−0.813607	−	0.581415i	$-0.802499\pi$
$103$	− 11.8014i	− 1.16283i	0.813607	−	0.581415i	$-0.197501\pi$
$104$	2.87992	0.282399
$105$	0	0
$106$	−5.46231	−0.530547
$107$	1.29791i	0.125473i	0.998030	+	0.0627367i	$0.0199828\pi$
$107$	1.29791i	0.125473i	−0.998030	+	0.0627367i	$0.980017\pi$
$108$	0	0
$109$	6.70496	0.642218	0.321109	−	0.947042i	$-0.395944\pi$
$109$	6.70496	0.642218	0.321109	+	0.947042i	$0.395944\pi$
$110$	2.15067	0.205058
$111$	0	0
$112$	0	0
$113$	1.61620i	0.152039i	0.997106	+	0.0760196i	$0.0242212\pi$
$113$	1.61620i	0.152039i	−0.997106	+	0.0760196i	$0.975779\pi$
$114$	0	0
$115$	− 2.46261i	− 0.229640i
$116$	− 0.0337398i	− 0.00313266i
$117$	0	0
$118$	4.17394i	0.384242i
$119$	0	0
$120$	0	0
$121$	−18.0437	−1.64034
$122$	−6.54904	−0.592923
$123$	0	0
$124$	11.1659i	1.00273i
$125$	7.10879	0.635829
$126$	0	0
$127$	−7.53114	−0.668280	−0.334140	−	0.942523i	$-0.608446\pi$
$127$	−7.53114	−0.668280	−0.334140	+	0.942523i	$0.608446\pi$
$128$	11.4935i	1.01589i
$129$	0	0
$130$	−0.583543	−0.0511801
$131$	−7.70964	−0.673594	−0.336797	−	0.941577i	$-0.609344\pi$
$131$	−7.70964	−0.673594	−0.336797	+	0.941577i	$0.609344\pi$
$132$	0	0
$133$	0	0
$134$	6.02917i	0.520841i
$135$	0	0
$136$	11.4635i	0.982987i
$137$	7.43594i	0.635295i	0.948209	+	0.317647i	$0.102893\pi$
$137$	7.43594i	0.635295i	−0.948209	+	0.317647i	$0.897107\pi$
$138$	0	0
$139$	− 10.0573i	− 0.853053i	−0.904475	−	0.426526i	$-0.859737\pi$
$139$	− 10.0573i	− 0.853053i	0.904475	−	0.426526i	$-0.140263\pi$
$140$	0	0
$141$	0	0
$142$	7.21021	0.605068
$143$	7.88047	0.658998
$144$	0	0
$145$	0.0147876i	0.00122804i
$146$	3.47129	0.287286
$147$	0	0
$148$	−1.25540	−0.103194
$149$	− 15.3579i	− 1.25817i	−0.777338	−	0.629083i	$-0.783431\pi$
$149$	− 15.3579i	− 1.25817i	0.777338	−	0.629083i	$-0.216569\pi$
$150$	0	0
$151$	−11.8682	−0.965820	−0.482910	−	0.875670i	$-0.660420\pi$
$151$	−11.8682	−0.965820	−0.482910	+	0.875670i	$0.660420\pi$
$152$	6.97947	0.566110
$153$	0	0
$154$	0	0
$155$	− 4.89383i	− 0.393082i
$156$	0	0
$157$	− 15.4765i	− 1.23516i	−0.786507	−	0.617581i	$-0.788113\pi$
$157$	− 15.4765i	− 1.23516i	0.786507	−	0.617581i	$-0.211887\pi$
$158$	5.57446i	0.443480i
$159$	0	0
$160$	− 3.92514i	− 0.310310i
$161$	0	0
$162$	0	0
$163$	−4.12347	−0.322975	−0.161488	−	0.986875i	$-0.551629\pi$
$163$	−4.12347	−0.322975	−0.161488	+	0.986875i	$0.551629\pi$
$164$	−17.5106	−1.36735
$165$	0	0
$166$	− 8.48017i	− 0.658188i
$167$	8.29844	0.642153	0.321076	−	0.947053i	$-0.395955\pi$
$167$	8.29844	0.642153	0.321076	+	0.947053i	$0.395955\pi$
$168$	0	0
$169$	10.8618	0.835522
$170$	− 2.32279i	− 0.178150i
$171$	0	0
$172$	11.6650	0.889448
$173$	−18.0294	−1.37075	−0.685376	−	0.728189i	$-0.740362\pi$
$173$	−18.0294	−1.37075	−0.685376	+	0.728189i	$0.740362\pi$
$174$	0	0
$175$	0	0
$176$	12.9152i	0.973517i
$177$	0	0
$178$	− 0.864014i	− 0.0647606i
$179$	− 16.5491i	− 1.23694i	−0.785810	−	0.618468i	$-0.787754\pi$
$179$	− 16.5491i	− 1.23694i	0.785810	−	0.618468i	$-0.212246\pi$
$180$	0	0
$181$	− 21.2548i	− 1.57986i	−0.613199	−	0.789928i	$-0.710118\pi$
$181$	− 21.2548i	− 1.57986i	0.613199	−	0.789928i	$-0.289882\pi$
$182$	0	0
$183$	0	0
$184$	−6.43512	−0.474403
$185$	0.550222	0.0404531
$186$	0	0
$187$	31.3682i	2.29387i
$188$	5.88517	0.429220
$189$	0	0
$190$	−1.41421	−0.102598
$191$	− 23.2742i	− 1.68406i	−0.539431	−	0.842030i	$-0.681360\pi$
$191$	− 23.2742i	− 1.68406i	0.539431	−	0.842030i	$-0.318640\pi$
$192$	0	0
$193$	4.68418	0.337175	0.168587	−	0.985687i	$-0.446080\pi$
$193$	4.68418	0.337175	0.168587	+	0.985687i	$0.446080\pi$
$194$	−8.91729	−0.640224
$195$	0	0
$196$	0	0
$197$	8.99030i	0.640532i	0.947328	+	0.320266i	$0.103772\pi$
$197$	8.99030i	0.640532i	−0.947328	+	0.320266i	$0.896228\pi$
$198$	0	0
$199$	2.34807i	0.166450i	0.996531	+	0.0832250i	$0.0265220\pi$
$199$	2.34807i	0.166450i	−0.996531	+	0.0832250i	$0.973478\pi$
$200$	− 8.72869i	− 0.617212i
$201$	0	0
$202$	5.53351i	0.389336i
$203$	0	0
$204$	0	0
$205$	7.67459	0.536017
$206$	6.24867	0.435365
$207$	0	0
$208$	− 3.50429i	− 0.242979i
$209$	19.0983	1.32106
$210$	0	0
$211$	7.98961	0.550028	0.275014	−	0.961440i	$-0.411318\pi$
$211$	7.98961	0.550028	0.275014	+	0.961440i	$0.411318\pi$
$212$	17.7404i	1.21842i
$213$	0	0
$214$	−0.687221	−0.0469775
$215$	−5.11257	−0.348675
$216$	0	0
$217$	0	0
$218$	3.55017i	0.240448i
$219$	0	0
$220$	− 6.98489i	− 0.470921i
$221$	− 8.51116i	− 0.572522i
$222$	0	0
$223$	23.6546i	1.58403i	0.610504	+	0.792014i	$0.290967\pi$
$223$	23.6546i	1.58403i	−0.610504	+	0.792014i	$0.709033\pi$
$224$	0	0
$225$	0	0
$226$	−0.855751	−0.0569237
$227$	−12.6961	−0.842672	−0.421336	−	0.906905i	$-0.638439\pi$
$227$	−12.6961	−0.842672	−0.421336	+	0.906905i	$0.638439\pi$
$228$	0	0
$229$	− 22.6949i	− 1.49972i	−0.661594	−	0.749862i	$-0.730120\pi$
$229$	− 22.6949i	− 1.49972i	0.661594	−	0.749862i	$-0.269880\pi$
$230$	1.30391	0.0859776
$231$	0	0
$232$	0.0386418	0.00253696
$233$	− 13.2027i	− 0.864935i	−0.901650	−	0.432468i	$-0.857643\pi$
$233$	− 13.2027i	− 0.864935i	0.901650	−	0.432468i	$-0.142357\pi$
$234$	0	0
$235$	−2.57937	−0.168260
$236$	13.5560	0.882422
$237$	0	0
$238$	0	0
$239$	− 8.89381i	− 0.575293i	−0.957737	−	0.287646i	$-0.907127\pi$
$239$	− 8.89381i	− 0.575293i	0.957737	−	0.287646i	$-0.0928728\pi$
$240$	0	0
$241$	11.4794i	0.739452i	0.929141	+	0.369726i	$0.120549\pi$
$241$	11.4794i	0.739452i	−0.929141	+	0.369726i	$0.879451\pi$
$242$	− 9.55387i	− 0.614146i
$243$	0	0
$244$	21.2699i	1.36166i
$245$	0	0
$246$	0	0
$247$	−5.18196	−0.329720
$248$	−12.7882	−0.812051
$249$	0	0
$250$	3.76399i	0.238055i
$251$	−6.44193	−0.406611	−0.203306	−	0.979115i	$-0.565169\pi$
$251$	−6.44193	−0.406611	−0.203306	+	0.979115i	$0.565169\pi$
$252$	0	0
$253$	−17.6088	−1.10705
$254$	− 3.98762i	− 0.250205i
$255$	0	0
$256$	−2.01469	−0.125918
$257$	−2.99353	−0.186731	−0.0933657	−	0.995632i	$-0.529763\pi$
$257$	−2.99353	−0.186731	−0.0933657	+	0.995632i	$0.529763\pi$
$258$	0	0
$259$	0	0
$260$	1.89522i	0.117536i
$261$	0	0
$262$	− 4.08213i	− 0.252195i
$263$	− 20.3938i	− 1.25754i	−0.777593	−	0.628768i	$-0.783560\pi$
$263$	− 20.3938i	− 1.25754i	0.777593	−	0.628768i	$-0.216440\pi$
$264$	0	0
$265$	− 7.77532i	− 0.477634i
$266$	0	0
$267$	0	0
$268$	19.5814	1.19613
$269$	−25.7032	−1.56715	−0.783577	−	0.621295i	$-0.786607\pi$
$269$	−25.7032	−1.56715	−0.783577	+	0.621295i	$0.786607\pi$
$270$	0	0
$271$	0.405904i	0.0246569i	0.999924	+	0.0123285i	$0.00392437\pi$
$271$	0.405904i	0.0246569i	−0.999924	+	0.0123285i	$0.996076\pi$
$272$	13.9488	0.845769
$273$	0	0
$274$	−3.93721	−0.237855
$275$	− 23.8848i	− 1.44031i
$276$	0	0
$277$	−13.1516	−0.790203	−0.395101	−	0.918638i	$-0.629291\pi$
$277$	−13.1516	−0.790203	−0.395101	+	0.918638i	$0.629291\pi$
$278$	5.32520	0.319384
$279$	0	0
$280$	0	0
$281$	− 13.2356i	− 0.789567i	−0.918774	−	0.394784i	$-0.870820\pi$
$281$	− 13.2356i	− 0.789567i	0.918774	−	0.394784i	$-0.129180\pi$
$282$	0	0
$283$	11.4608i	0.681274i	0.940195	+	0.340637i	$0.110643\pi$
$283$	11.4608i	0.681274i	−0.940195	+	0.340637i	$0.889357\pi$
$284$	− 23.4172i	− 1.38955i
$285$	0	0
$286$	4.17258i	0.246730i
$287$	0	0
$288$	0	0
$289$	16.8786	0.992858
$290$	−0.00782978	−0.000459780 0
$291$	0	0
$292$	− 11.2740i	− 0.659760i
$293$	11.0114	0.643294	0.321647	−	0.946860i	$-0.395764\pi$
$293$	11.0114	0.643294	0.321647	+	0.946860i	$0.395764\pi$
$294$	0	0
$295$	−5.94138	−0.345921
$296$	− 1.43780i	− 0.0835704i
$297$	0	0
$298$	8.13175	0.471059
$299$	4.77780	0.276307
$300$	0	0
$301$	0	0
$302$	− 6.28402i	− 0.361605i
$303$	0	0
$304$	− 8.49262i	− 0.487085i
$305$	− 9.32222i	− 0.533789i
$306$	0	0
$307$	− 21.0426i	− 1.20096i	−0.799639	−	0.600481i	$-0.794976\pi$
$307$	− 21.0426i	− 1.20096i	0.799639	−	0.600481i	$-0.205024\pi$
$308$	0	0
$309$	0	0
$310$	2.59120	0.147170
$311$	3.16299	0.179356	0.0896782	−	0.995971i	$-0.471416\pi$
$311$	3.16299	0.179356	0.0896782	+	0.995971i	$0.471416\pi$
$312$	0	0
$313$	− 0.147318i	− 0.00832691i	−0.999991	−	0.00416345i	$-0.998675\pi$
$313$	− 0.147318i	− 0.00832691i	0.999991	−	0.00416345i	$-0.00132527\pi$
$314$	8.19458	0.462447
$315$	0	0
$316$	18.1046	1.01846
$317$	− 4.36561i	− 0.245197i	−0.992456	−	0.122598i	$-0.960877\pi$
$317$	− 4.36561i	− 0.245197i	0.992456	−	0.122598i	$-0.0391227\pi$
$318$	0	0
$319$	0.105738	0.00592016
$320$	−1.53412	−0.0857598
$321$	0	0
$322$	0	0
$323$	− 20.6268i	− 1.14770i
$324$	0	0
$325$	6.48068i	0.359483i
$326$	− 2.18331i	− 0.120922i
$327$	0	0
$328$	− 20.0547i	− 1.10733i
$329$	0	0
$330$	0	0
$331$	−0.856550	−0.0470802	−0.0235401	−	0.999723i	$-0.507494\pi$
$331$	−0.856550	−0.0470802	−0.0235401	+	0.999723i	$0.507494\pi$
$332$	−27.5417	−1.51155
$333$	0	0
$334$	4.39389i	0.240423i
$335$	−8.58221	−0.468896
$336$	0	0
$337$	18.8188	1.02513	0.512564	−	0.858649i	$-0.328696\pi$
$337$	18.8188	1.02513	0.512564	+	0.858649i	$0.328696\pi$
$338$	5.75114i	0.312821i
$339$	0	0
$340$	−7.54390	−0.409125
$341$	−34.9930	−1.89498
$342$	0	0
$343$	0	0
$344$	13.3598i	0.720312i
$345$	0	0
$346$	− 9.54629i	− 0.513212i
$347$	− 28.5747i	− 1.53397i	−0.641666	−	0.766984i	$-0.721757\pi$
$347$	− 28.5747i	− 1.53397i	0.641666	−	0.766984i	$-0.278243\pi$
$348$	0	0
$349$	− 2.84518i	− 0.152299i	−0.997096	−	0.0761494i	$-0.975737\pi$
$349$	− 2.84518i	− 0.152299i	0.997096	−	0.0761494i	$-0.0242626\pi$
$350$	0	0
$351$	0	0
$352$	−28.0665	−1.49595
$353$	18.4654	0.982814	0.491407	−	0.870930i	$-0.336483\pi$
$353$	18.4654	0.982814	0.491407	+	0.870930i	$0.336483\pi$
$354$	0	0
$355$	10.2634i	0.544723i
$356$	−2.80613	−0.148724
$357$	0	0
$358$	8.76246	0.463110
$359$	− 15.5097i	− 0.818572i	−0.912406	−	0.409286i	$-0.865778\pi$
$359$	− 15.5097i	− 0.818572i	0.912406	−	0.409286i	$-0.134222\pi$
$360$	0	0
$361$	6.44154	0.339028
$362$	11.2541	0.591501
$363$	0	0
$364$	0	0
$365$	4.94120i	0.258634i
$366$	0	0
$367$	2.76607i	0.144387i	0.997391	+	0.0721937i	$0.0230000\pi$
$367$	2.76607i	0.144387i	−0.997391	+	0.0721937i	$0.977000\pi$
$368$	7.83026i	0.408180i
$369$	0	0
$370$	0.291334i	0.0151457i
$371$	0	0
$372$	0	0
$373$	1.78773	0.0925649	0.0462825	−	0.998928i	$-0.485263\pi$
$373$	1.78773	0.0925649	0.0462825	+	0.998928i	$0.485263\pi$
$374$	−16.6089	−0.858827
$375$	0	0
$376$	6.74022i	0.347600i
$377$	−0.0286899	−0.00147760
$378$	0	0
$379$	3.65672	0.187833	0.0939166	−	0.995580i	$-0.470061\pi$
$379$	3.65672	0.187833	0.0939166	+	0.995580i	$0.470061\pi$
$380$	4.59305i	0.235619i
$381$	0	0
$382$	12.3233	0.630515
$383$	24.9733	1.27608	0.638039	−	0.770004i	$-0.279746\pi$
$383$	24.9733	1.27608	0.638039	+	0.770004i	$0.279746\pi$
$384$	0	0
$385$	0	0
$386$	2.48020i	0.126239i
$387$	0	0
$388$	28.9614i	1.47029i
$389$	− 11.5686i	− 0.586553i	−0.956028	−	0.293276i	$-0.905254\pi$
$389$	− 11.5686i	− 0.586553i	0.956028	−	0.293276i	$-0.0947456\pi$
$390$	0	0
$391$	19.0180i	0.961782i
$392$	0	0
$393$	0	0
$394$	−4.76022	−0.239816
$395$	−7.93494	−0.399250
$396$	0	0
$397$	− 21.7551i	− 1.09186i	−0.837832	−	0.545929i	$-0.816177\pi$
$397$	− 21.7551i	− 1.09186i	0.837832	−	0.545929i	$-0.183823\pi$
$398$	−1.24326	−0.0623191
$399$	0	0
$400$	−10.6211	−0.531054
$401$	− 13.9016i	− 0.694215i	−0.937825	−	0.347108i	$-0.887164\pi$
$401$	− 13.9016i	− 0.694215i	0.937825	−	0.347108i	$-0.112836\pi$
$402$	0	0
$403$	9.49469	0.472964
$404$	17.9716	0.894122
$405$	0	0
$406$	0	0
$407$	− 3.93433i	− 0.195017i
$408$	0	0
$409$	18.9915i	0.939068i	0.882914	+	0.469534i	$0.155578\pi$
$409$	18.9915i	0.939068i	−0.882914	+	0.469534i	$0.844422\pi$
$410$	4.06357i	0.200685i
$411$	0	0
$412$	− 20.2943i	− 0.999828i
$413$	0	0
$414$	0	0
$415$	12.0711	0.592545
$416$	7.61530	0.373371
$417$	0	0
$418$	10.1122i	0.494606i
$419$	11.1604	0.545220	0.272610	−	0.962125i	$-0.412113\pi$
$419$	11.1604	0.545220	0.272610	+	0.962125i	$0.412113\pi$
$420$	0	0
$421$	16.8786	0.822612	0.411306	−	0.911497i	$-0.365073\pi$
$421$	16.8786	0.822612	0.411306	+	0.911497i	$0.365073\pi$
$422$	4.23037i	0.205931i
$423$	0	0
$424$	−20.3179	−0.986724
$425$	−25.7963	−1.25130
$426$	0	0
$427$	0	0
$428$	2.23194i	0.107885i
$429$	0	0
$430$	− 2.70702i	− 0.130544i
$431$	10.5020i	0.505865i	0.967484	+	0.252932i	$0.0813950\pi$
$431$	10.5020i	0.505865i	−0.967484	+	0.252932i	$0.918605\pi$
$432$	0	0
$433$	7.29923i	0.350778i	0.984499	+	0.175389i	$0.0561184\pi$
$433$	7.29923i	0.350778i	−0.984499	+	0.175389i	$0.943882\pi$
$434$	0	0
$435$	0	0
$436$	11.5302	0.552194
$437$	11.5790	0.553898
$438$	0	0
$439$	− 28.8315i	− 1.37605i	−0.725685	−	0.688027i	$-0.758477\pi$
$439$	− 28.8315i	− 1.37605i	0.725685	−	0.688027i	$-0.241523\pi$
$440$	7.99972	0.381372
$441$	0	0
$442$	4.50652	0.214353
$443$	17.0704i	0.811040i	0.914086	+	0.405520i	$0.132910\pi$
$443$	17.0704i	0.811040i	−0.914086	+	0.405520i	$0.867090\pi$
$444$	0	0
$445$	1.22988	0.0583018
$446$	−12.5247	−0.593062
$447$	0	0
$448$	0	0
$449$	35.1390i	1.65831i	0.559018	+	0.829156i	$0.311178\pi$
$449$	35.1390i	1.65831i	−0.559018	+	0.829156i	$0.688822\pi$
$450$	0	0
$451$	− 54.8766i	− 2.58404i
$452$	2.77929i	0.130727i
$453$	0	0
$454$	− 6.72240i	− 0.315498i
$455$	0	0
$456$	0	0
$457$	0.952698	0.0445653	0.0222827	−	0.999752i	$-0.492907\pi$
$457$	0.952698	0.0445653	0.0222827	+	0.999752i	$0.492907\pi$
$458$	12.0166	0.561499
$459$	0	0
$460$	− 4.23483i	− 0.197450i
$461$	29.3645	1.36764	0.683821	−	0.729650i	$-0.260317\pi$
$461$	29.3645	1.36764	0.683821	+	0.729650i	$0.260317\pi$
$462$	0	0
$463$	−42.9016	−1.99380	−0.996902	−	0.0786517i	$-0.974939\pi$
$463$	−42.9016	−1.99380	−0.996902	+	0.0786517i	$0.974939\pi$
$464$	− 0.0470193i	− 0.00218282i
$465$	0	0
$466$	6.99059	0.323833
$467$	6.75710	0.312681	0.156341	−	0.987703i	$-0.450030\pi$
$467$	6.75710	0.312681	0.156341	+	0.987703i	$0.450030\pi$
$468$	0	0
$469$	0	0
$470$	− 1.36573i	− 0.0629966i
$471$	0	0
$472$	15.5256i	0.714622i
$473$	36.5571i	1.68090i
$474$	0	0
$475$	15.7059i	0.720636i
$476$	0	0
$477$	0	0
$478$	4.70913	0.215391
$479$	−21.4622	−0.980632	−0.490316	−	0.871545i	$-0.663119\pi$
$479$	−21.4622	−0.980632	−0.490316	+	0.871545i	$0.663119\pi$
$480$	0	0
$481$	1.06750i	0.0486740i
$482$	−6.07815	−0.276852
$483$	0	0
$484$	−31.0289	−1.41040
$485$	− 12.6933i	− 0.576373i
$486$	0	0
$487$	−12.6821	−0.574679	−0.287340	−	0.957829i	$-0.592771\pi$
$487$	−12.6821	−0.574679	−0.287340	+	0.957829i	$0.592771\pi$
$488$	−24.3601	−1.10273
$489$	0	0
$490$	0	0
$491$	− 23.4283i	− 1.05730i	−0.848839	−	0.528651i	$-0.822698\pi$
$491$	− 23.4283i	− 1.05730i	0.848839	−	0.528651i	$-0.177302\pi$
$492$	0	0
$493$	− 0.114200i	− 0.00514330i
$494$	− 2.74376i	− 0.123448i
$495$	0	0
$496$	15.5607i	0.698695i
$497$	0	0
$498$	0	0
$499$	−6.35479	−0.284480	−0.142240	−	0.989832i	$-0.545430\pi$
$499$	−6.35479	−0.284480	−0.142240	+	0.989832i	$0.545430\pi$
$500$	12.2246	0.546701
$501$	0	0
$502$	− 3.41090i	− 0.152236i
$503$	−29.5594	−1.31799	−0.658995	−	0.752147i	$-0.729018\pi$
$503$	−29.5594	−1.31799	−0.658995	+	0.752147i	$0.729018\pi$
$504$	0	0
$505$	−7.87666	−0.350507
$506$	− 9.32355i	− 0.414482i
$507$	0	0
$508$	−12.9509	−0.574603
$509$	−9.21537	−0.408464	−0.204232	−	0.978923i	$-0.565470\pi$
$509$	−9.21537	−0.408464	−0.204232	+	0.978923i	$0.565470\pi$
$510$	0	0
$511$	0	0
$512$	21.9203i	0.968749i
$513$	0	0
$514$	− 1.58503i	− 0.0699125i
$515$	8.89465i	0.391945i
$516$	0	0
$517$	18.4436i	0.811149i
$518$	0	0
$519$	0	0
$520$	−2.17057	−0.0951859
$521$	38.0020	1.66490	0.832448	−	0.554103i	$-0.186939\pi$
$521$	38.0020	1.66490	0.832448	+	0.554103i	$0.186939\pi$
$522$	0	0
$523$	9.12546i	0.399029i	0.979895	+	0.199514i	$0.0639364\pi$
$523$	9.12546i	0.399029i	−0.979895	+	0.199514i	$0.936064\pi$
$524$	−13.2579	−0.579172
$525$	0	0
$526$	10.7982	0.470823
$527$	37.7935i	1.64631i
$528$	0	0
$529$	12.3241	0.535830
$530$	4.11690	0.178827
$531$	0	0
$532$	0	0
$533$	14.8897i	0.644946i
$534$	0	0
$535$	− 0.978223i	− 0.0422923i
$536$	22.4264i	0.968673i
$537$	0	0
$538$	− 13.6094i	− 0.586745i
$539$	0	0
$540$	0	0
$541$	15.1171	0.649933	0.324966	−	0.945726i	$-0.394647\pi$
$541$	15.1171	0.649933	0.324966	+	0.945726i	$0.394647\pi$
$542$	−0.214920	−0.00923160
$543$	0	0
$544$	30.3126i	1.29964i
$545$	−5.05347	−0.216467
$546$	0	0
$547$	21.2759	0.909690	0.454845	−	0.890571i	$-0.349695\pi$
$547$	21.2759	0.909690	0.454845	+	0.890571i	$0.349695\pi$
$548$	12.7872i	0.546241i
$549$	0	0
$550$	12.6466	0.539253
$551$	−0.0695298	−0.00296207
$552$	0	0
$553$	0	0
$554$	− 6.96356i	− 0.295853i
$555$	0	0
$556$	− 17.2951i	− 0.733475i
$557$	25.2672i	1.07060i	0.844661	+	0.535302i	$0.179802\pi$
$557$	25.2672i	1.07060i	−0.844661	+	0.535302i	$0.820198\pi$
$558$	0	0
$559$	− 9.91908i	− 0.419532i
$560$	0	0
$561$	0	0
$562$	7.00801	0.295615
$563$	4.79617	0.202135	0.101067	−	0.994880i	$-0.467774\pi$
$563$	4.79617	0.202135	0.101067	+	0.994880i	$0.467774\pi$
$564$	0	0
$565$	− 1.21812i	− 0.0512465i
$566$	−6.06831	−0.255070
$567$	0	0
$568$	26.8195	1.12532
$569$	17.0555i	0.715005i	0.933912	+	0.357503i	$0.116372\pi$
$569$	17.0555i	0.715005i	−0.933912	+	0.357503i	$0.883628\pi$
$570$	0	0
$571$	24.6031	1.02961	0.514805	−	0.857308i	$-0.327864\pi$
$571$	24.6031	1.02961	0.514805	+	0.857308i	$0.327864\pi$
$572$	13.5516	0.566622
$573$	0	0
$574$	0	0
$575$	− 14.4809i	− 0.603897i
$576$	0	0
$577$	3.79959i	0.158179i	0.996868	+	0.0790895i	$0.0252013\pi$
$577$	3.79959i	0.158179i	−0.996868	+	0.0790895i	$0.974799\pi$
$578$	8.93694i	0.371728i
$579$	0	0
$580$	0.0254294i	0.00105590i
$581$	0	0
$582$	0	0
$583$	−55.5969	−2.30259
$584$	12.9120	0.534301
$585$	0	0
$586$	5.83037i	0.240850i
$587$	−19.3497	−0.798649	−0.399324	−	0.916810i	$-0.630755\pi$
$587$	−19.3497	−0.798649	−0.399324	+	0.916810i	$0.630755\pi$
$588$	0	0
$589$	23.0103	0.948124
$590$	− 3.14586i	− 0.129513i
$591$	0	0
$592$	−1.74951	−0.0719046
$593$	26.1790	1.07504	0.537521	−	0.843250i	$-0.319361\pi$
$593$	26.1790	1.07504	0.537521	+	0.843250i	$0.319361\pi$
$594$	0	0
$595$	0	0
$596$	− 26.4101i	− 1.08180i
$597$	0	0
$598$	2.52977i	0.103450i
$599$	0.851665i	0.0347981i	0.999849	+	0.0173990i	$0.00553857\pi$
$599$	0.851665i	0.0347981i	−0.999849	+	0.0173990i	$0.994461\pi$
$600$	0	0
$601$	− 0.300521i	− 0.0122585i	−0.999981	−	0.00612924i	$-0.998049\pi$
$601$	− 0.300521i	− 0.0122585i	0.999981	−	0.00612924i	$-0.00195101\pi$
$602$	0	0
$603$	0	0
$604$	−20.4091	−0.830435
$605$	13.5994	0.552895
$606$	0	0
$607$	38.6216i	1.56760i	0.621011	+	0.783802i	$0.286722\pi$
$607$	38.6216i	1.56760i	−0.621011	+	0.783802i	$0.713278\pi$
$608$	18.4557	0.748476
$609$	0	0
$610$	4.93596	0.199851
$611$	− 5.00433i	− 0.202453i
$612$	0	0
$613$	−32.8271	−1.32587	−0.662937	−	0.748675i	$-0.730690\pi$
$613$	−32.8271	−1.32587	−0.662937	+	0.748675i	$0.730690\pi$
$614$	11.1417	0.449642
$615$	0	0
$616$	0	0
$617$	37.9728i	1.52873i	0.644785	+	0.764364i	$0.276947\pi$
$617$	37.9728i	1.52873i	−0.644785	+	0.764364i	$0.723053\pi$
$618$	0	0
$619$	18.7804i	0.754850i	0.926040	+	0.377425i	$0.123190\pi$
$619$	18.7804i	0.754850i	−0.926040	+	0.377425i	$0.876810\pi$
$620$	− 8.41566i	− 0.337981i
$621$	0	0
$622$	1.67475i	0.0671513i
$623$	0	0
$624$	0	0
$625$	16.8019	0.672076
$626$	0.0780025	0.00311761
$627$	0	0
$628$	− 26.6142i	− 1.06202i
$629$	−4.24919	−0.169426
$630$	0	0
$631$	−19.9549	−0.794394	−0.397197	−	0.917733i	$-0.630017\pi$
$631$	−19.9549	−0.794394	−0.397197	+	0.917733i	$0.630017\pi$
$632$	20.7350i	0.824794i
$633$	0	0
$634$	2.31152	0.0918021
$635$	5.67616	0.225252
$636$	0	0
$637$	0	0
$638$	0.0559863i	0.00221652i
$639$	0	0
$640$	− 8.66257i	− 0.342418i
$641$	45.4868i	1.79662i	0.439364	+	0.898309i	$0.355204\pi$
$641$	45.4868i	1.79662i	−0.439364	+	0.898309i	$0.644796\pi$
$642$	0	0
$643$	5.44558i	0.214753i	0.994218	+	0.107376i	$0.0342450\pi$
$643$	5.44558i	0.214753i	−0.994218	+	0.107376i	$0.965755\pi$
$644$	0	0
$645$	0	0
$646$	10.9215	0.429702
$647$	−1.91003	−0.0750912	−0.0375456	−	0.999295i	$-0.511954\pi$
$647$	−1.91003	−0.0750912	−0.0375456	+	0.999295i	$0.511954\pi$
$648$	0	0
$649$	42.4834i	1.66762i
$650$	−3.43142	−0.134591
$651$	0	0
$652$	−7.09092	−0.277702
$653$	14.3164i	0.560245i	0.959964	+	0.280123i	$0.0903751\pi$
$653$	14.3164i	0.560245i	−0.959964	+	0.280123i	$0.909625\pi$
$654$	0	0
$655$	5.81069	0.227043
$656$	−24.4025	−0.952758
$657$	0	0
$658$	0	0
$659$	− 17.1595i	− 0.668440i	−0.942495	−	0.334220i	$-0.891527\pi$
$659$	− 17.1595i	− 0.668440i	0.942495	−	0.334220i	$-0.108473\pi$
$660$	0	0
$661$	2.69907i	0.104982i	0.998621	+	0.0524909i	$0.0167161\pi$
$661$	2.69907i	0.104982i	−0.998621	+	0.0524909i	$0.983284\pi$
$662$	− 0.453529i	− 0.0176269i
$663$	0	0
$664$	− 31.5432i	− 1.22411i
$665$	0	0
$666$	0	0
$667$	0.0641069	0.00248223
$668$	14.2704	0.552138
$669$	0	0
$670$	− 4.54414i	− 0.175556i
$671$	−66.6579	−2.57330
$672$	0	0
$673$	−45.6325	−1.75900	−0.879502	−	0.475895i	$-0.842124\pi$
$673$	−45.6325	−1.75900	−0.879502	+	0.475895i	$0.842124\pi$
$674$	9.96427i	0.383809i
$675$	0	0
$676$	18.6784	0.718401
$677$	46.4244	1.78424	0.892118	−	0.451802i	$-0.149219\pi$
$677$	46.4244	1.78424	0.892118	+	0.451802i	$0.149219\pi$
$678$	0	0
$679$	0	0
$680$	− 8.63995i	− 0.331327i
$681$	0	0
$682$	− 18.5282i	− 0.709482i
$683$	11.4914i	0.439707i	0.975533	+	0.219853i	$0.0705579\pi$
$683$	11.4914i	0.439707i	−0.975533	+	0.219853i	$0.929442\pi$
$684$	0	0
$685$	− 5.60441i	− 0.214133i
$686$	0	0
$687$	0	0
$688$	16.2562	0.619762
$689$	15.0852	0.574699
$690$	0	0
$691$	− 46.8170i	− 1.78100i	−0.454979	−	0.890502i	$-0.650353\pi$
$691$	− 46.8170i	− 1.78100i	0.454979	−	0.890502i	$-0.349647\pi$
$692$	−31.0043	−1.17861
$693$	0	0
$694$	15.1298	0.574320
$695$	7.58014i	0.287531i
$696$	0	0
$697$	−59.2685	−2.24495
$698$	1.50648	0.0570209
$699$	0	0
$700$	0	0
$701$	− 24.2906i	− 0.917442i	−0.888580	−	0.458721i	$-0.848308\pi$
$701$	− 24.2906i	− 0.917442i	0.888580	−	0.458721i	$-0.151692\pi$
$702$	0	0
$703$	2.58709i	0.0975741i
$704$	10.9696i	0.413432i
$705$	0	0
$706$	9.77713i	0.367967i
$707$	0	0
$708$	0	0
$709$	−43.8269	−1.64595	−0.822977	−	0.568074i	$-0.807689\pi$
$709$	−43.8269	−1.64595	−0.822977	+	0.568074i	$0.807689\pi$
$710$	−5.43428	−0.203945
$711$	0	0
$712$	− 3.21383i	− 0.120443i
$713$	−21.2157	−0.794534
$714$	0	0
$715$	−5.93945	−0.222123
$716$	− 28.4585i	− 1.06355i
$717$	0	0
$718$	8.21214	0.306474
$719$	19.0904	0.711953	0.355976	−	0.934495i	$-0.384148\pi$
$719$	19.0904	0.711953	0.355976	+	0.934495i	$0.384148\pi$
$720$	0	0
$721$	0	0
$722$	3.41069i	0.126933i
$723$	0	0
$724$	− 36.5508i	− 1.35840i
$725$	0.0869556i	0.00322945i
$726$	0	0
$727$	− 23.2606i	− 0.862689i	−0.902187	−	0.431344i	$-0.858039\pi$
$727$	− 23.2606i	− 0.862689i	0.902187	−	0.431344i	$-0.141961\pi$
$728$	0	0
$729$	0	0
$730$	−2.61628	−0.0968330
$731$	39.4828	1.46032
$732$	0	0
$733$	8.52772i	0.314979i	0.987521	+	0.157489i	$0.0503400\pi$
$733$	8.52772i	0.314979i	−0.987521	+	0.157489i	$0.949660\pi$
$734$	−1.46459	−0.0540589
$735$	0	0
$736$	−17.0162	−0.627227
$737$	61.3665i	2.26046i
$738$	0	0
$739$	−18.2278	−0.670519	−0.335260	−	0.942126i	$-0.608824\pi$
$739$	−18.2278	−0.670519	−0.335260	+	0.942126i	$0.608824\pi$
$740$	0.946188	0.0347826
$741$	0	0
$742$	0	0
$743$	− 25.6584i	− 0.941317i	−0.882315	−	0.470659i	$-0.844016\pi$
$743$	− 25.6584i	− 0.941317i	0.882315	−	0.470659i	$-0.155984\pi$
$744$	0	0
$745$	11.5751i	0.424079i
$746$	0.946572i	0.0346565i
$747$	0	0
$748$	53.9421i	1.97232i
$749$	0	0
$750$	0	0
$751$	6.14551	0.224253	0.112126	−	0.993694i	$-0.464234\pi$
$751$	6.14551	0.224253	0.112126	+	0.993694i	$0.464234\pi$
$752$	8.20150	0.299078
$753$	0	0
$754$	− 0.0151908i	0 0.000553217i
$755$	8.94497	0.325541
$756$	0	0
$757$	−30.0519	−1.09225	−0.546127	−	0.837702i	$-0.683898\pi$
$757$	−30.0519	−1.09225	−0.546127	+	0.837702i	$0.683898\pi$
$758$	1.93618i	0.0703251i
$759$	0	0
$760$	−5.26038	−0.190814
$761$	−31.9480	−1.15811	−0.579056	−	0.815288i	$-0.696579\pi$
$761$	−31.9480	−1.15811	−0.579056	+	0.815288i	$0.696579\pi$
$762$	0	0
$763$	0	0
$764$	− 40.0234i	− 1.44799i
$765$	0	0
$766$	13.2230i	0.477766i
$767$	− 11.5271i	− 0.416219i
$768$	0	0
$769$	28.8779i	1.04136i	0.853751	+	0.520682i	$0.174322\pi$
$769$	28.8779i	1.04136i	−0.853751	+	0.520682i	$0.825678\pi$
$770$	0	0
$771$	0	0
$772$	8.05513	0.289911
$773$	−36.5807	−1.31572	−0.657859	−	0.753141i	$-0.728538\pi$
$773$	−36.5807	−1.31572	−0.657859	+	0.753141i	$0.728538\pi$
$774$	0	0
$775$	− 28.7773i	− 1.03371i
$776$	−33.1692	−1.19070
$777$	0	0
$778$	6.12540	0.219606
$779$	36.0852i	1.29289i
$780$	0	0
$781$	73.3875	2.62601
$782$	−10.0697	−0.360093
$783$	0	0
$784$	0	0
$785$	11.6645i	0.416326i
$786$	0	0
$787$	35.1572i	1.25322i	0.779333	+	0.626609i	$0.215558\pi$
$787$	35.1572i	1.25322i	−0.779333	+	0.626609i	$0.784442\pi$
$788$	15.4601i	0.550745i
$789$	0	0
$790$	− 4.20142i	− 0.149480i
$791$	0	0
$792$	0	0
$793$	18.0864	0.642265
$794$	11.5190	0.408793
$795$	0	0
$796$	4.03784i	0.143118i
$797$	25.7466	0.911993	0.455996	−	0.889982i	$-0.349283\pi$
$797$	25.7466	0.911993	0.455996	+	0.889982i	$0.349283\pi$
$798$	0	0
$799$	19.9197	0.704707
$800$	− 23.0811i	− 0.816039i
$801$	0	0
$802$	7.36070	0.259915
$803$	35.3317	1.24683
$804$	0	0
$805$	0	0
$806$	5.02728i	0.177078i
$807$	0	0
$808$	20.5827i	0.724097i
$809$	32.5835i	1.14557i	0.819704	+	0.572787i	$0.194138\pi$
$809$	32.5835i	1.14557i	−0.819704	+	0.572787i	$0.805862\pi$
$810$	0	0
$811$	− 13.9227i	− 0.488892i	−0.969663	−	0.244446i	$-0.921394\pi$
$811$	− 13.9227i	− 0.488892i	0.969663	−	0.244446i	$-0.0786062\pi$
$812$	0	0
$813$	0	0
$814$	2.08316	0.0730148
$815$	3.10783	0.108862
$816$	0	0
$817$	− 24.0388i	− 0.841013i
$818$	−10.0557	−0.351588
$819$	0	0
$820$	13.1976	0.460880
$821$	− 35.7332i	− 1.24710i	−0.781785	−	0.623548i	$-0.785690\pi$
$821$	− 35.7332i	− 1.24710i	0.781785	−	0.623548i	$-0.214310\pi$
$822$	0	0
$823$	−15.8049	−0.550926	−0.275463	−	0.961312i	$-0.588831\pi$
$823$	−15.8049	−0.550926	−0.275463	+	0.961312i	$0.588831\pi$
$824$	23.2428	0.809703
$825$	0	0
$826$	0	0
$827$	− 3.97062i	− 0.138072i	−0.997614	−	0.0690360i	$-0.978008\pi$
$827$	− 3.97062i	− 0.138072i	0.997614	−	0.0690360i	$-0.0219923\pi$
$828$	0	0
$829$	− 18.4073i	− 0.639312i	−0.947534	−	0.319656i	$-0.896433\pi$
$829$	− 18.4073i	− 0.639312i	0.947534	−	0.319656i	$-0.103567\pi$
$830$	6.39144i	0.221850i
$831$	0	0
$832$	− 2.97639i	− 0.103188i
$833$	0	0
$834$	0	0
$835$	−6.25447	−0.216445
$836$	32.8423	1.13588
$837$	0	0
$838$	5.90924i	0.204131i
$839$	14.2526	0.492055	0.246027	−	0.969263i	$-0.420875\pi$
$839$	14.2526	0.492055	0.246027	+	0.969263i	$0.420875\pi$
$840$	0	0
$841$	28.9996	0.999987
$842$	8.93694i	0.307987i
$843$	0	0
$844$	13.7393	0.472927
$845$	−8.18644	−0.281622
$846$	0	0
$847$	0	0
$848$	24.7228i	0.848984i
$849$	0	0
$850$	− 13.6587i	− 0.468490i
$851$	− 2.38532i	− 0.0817676i
$852$	0	0
$853$	− 12.3104i	− 0.421500i	−0.977540	−	0.210750i	$-0.932409\pi$
$853$	− 12.3104i	− 0.421500i	0.977540	−	0.210750i	$-0.0675906\pi$
$854$	0	0
$855$	0	0
$856$	−2.55622	−0.0873698
$857$	−38.8030	−1.32549	−0.662743	−	0.748847i	$-0.730608\pi$
$857$	−38.8030	−1.32549	−0.662743	+	0.748847i	$0.730608\pi$
$858$	0	0
$859$	0.731986i	0.0249751i	0.999922	+	0.0124875i	$0.00397501\pi$
$859$	0.731986i	0.0249751i	−0.999922	+	0.0124875i	$0.996025\pi$
$860$	−8.79182	−0.299799
$861$	0	0
$862$	−5.56065	−0.189397
$863$	23.2724i	0.792201i	0.918207	+	0.396101i	$0.129637\pi$
$863$	23.2724i	0.792201i	−0.918207	+	0.396101i	$0.870363\pi$
$864$	0	0
$865$	13.5886	0.462028
$866$	−3.86482	−0.131332
$867$	0	0
$868$	0	0
$869$	56.7383i	1.92471i
$870$	0	0
$871$	− 16.6506i	− 0.564186i
$872$	13.2054i	0.447190i
$873$	0	0
$874$	6.13089i	0.207380i
$875$	0	0
$876$	0	0
$877$	−0.703825	−0.0237665	−0.0118832	−	0.999929i	$-0.503783\pi$
$877$	−0.703825	−0.0237665	−0.0118832	+	0.999929i	$0.503783\pi$
$878$	15.2658	0.515197
$879$	0	0
$880$	− 9.73406i	− 0.328135i
$881$	8.73862	0.294412	0.147206	−	0.989106i	$-0.452972\pi$
$881$	8.73862	0.294412	0.147206	+	0.989106i	$0.452972\pi$
$882$	0	0
$883$	29.5164	0.993308	0.496654	−	0.867949i	$-0.334562\pi$
$883$	29.5164	0.993308	0.496654	+	0.867949i	$0.334562\pi$
$884$	− 14.6362i	− 0.492268i
$885$	0	0
$886$	−9.03851	−0.303655
$887$	19.3011	0.648067	0.324033	−	0.946046i	$-0.394961\pi$
$887$	19.3011	0.648067	0.324033	+	0.946046i	$0.394961\pi$
$888$	0	0
$889$	0	0
$890$	0.651201i	0.0218283i
$891$	0	0
$892$	40.6775i	1.36198i
$893$	− 12.1280i	− 0.405847i
$894$	0	0
$895$	12.4729i	0.416923i
$896$	0	0
$897$	0	0
$898$	−18.6055	−0.620874
$899$	0.127396	0.00424891
$900$	0	0
$901$	60.0464i	2.00043i
$902$	29.0563	0.967468
$903$	0	0
$904$	−3.18309	−0.105868
$905$	16.0196i	0.532509i
$906$	0	0
$907$	8.62860	0.286508	0.143254	−	0.989686i	$-0.454243\pi$
$907$	8.62860	0.286508	0.143254	+	0.989686i	$0.454243\pi$
$908$	−21.8329	−0.724549
$909$	0	0
$910$	0	0
$911$	− 16.9309i	− 0.560947i	−0.959862	−	0.280474i	$-0.909508\pi$
$911$	− 16.9309i	− 0.560947i	0.959862	−	0.280474i	$-0.0904915\pi$
$912$	0	0
$913$	− 86.3133i	− 2.85655i
$914$	0.504438i	0.0166853i
$915$	0	0
$916$	− 39.0273i	− 1.28950i
$917$	0	0
$918$	0	0
$919$	56.6555	1.86889	0.934447	−	0.356102i	$-0.115895\pi$
$919$	56.6555	1.86889	0.934447	+	0.356102i	$0.115895\pi$
$920$	4.85010	0.159903
$921$	0	0
$922$	15.5480i	0.512047i
$923$	−19.9123	−0.655421
$924$	0	0
$925$	3.23548	0.106382
$926$	− 22.7157i	− 0.746484i
$927$	0	0
$928$	0.102180	0.00335421
$929$	−41.3922	−1.35803	−0.679017	−	0.734122i	$-0.737594\pi$
$929$	−41.3922	−1.35803	−0.679017	+	0.734122i	$0.737594\pi$
$930$	0	0
$931$	0	0
$932$	− 22.7039i	− 0.743691i
$933$	0	0
$934$	3.57778i	0.117068i
$935$	− 23.6419i	− 0.773174i
$936$	0	0
$937$	− 42.6326i	− 1.39275i	−0.717680	−	0.696373i	$-0.754796\pi$
$937$	− 42.6326i	− 1.39275i	0.717680	−	0.696373i	$-0.245204\pi$
$938$	0	0
$939$	0	0
$940$	−4.43561	−0.144674
$941$	15.8612	0.517060	0.258530	−	0.966003i	$-0.416762\pi$
$941$	15.8612	0.517060	0.258530	+	0.966003i	$0.416762\pi$
$942$	0	0
$943$	− 33.2708i	− 1.08345i
$944$	18.8915	0.614867
$945$	0	0
$946$	−19.3564	−0.629331
$947$	23.4734i	0.762782i	0.924414	+	0.381391i	$0.124555\pi$
$947$	23.4734i	0.762782i	−0.924414	+	0.381391i	$0.875445\pi$
$948$	0	0
$949$	−9.58659	−0.311194
$950$	−8.31602	−0.269807
$951$	0	0
$952$	0	0
$953$	38.4020i	1.24396i	0.783032	+	0.621982i	$0.213672\pi$
$953$	38.4020i	1.24396i	−0.783032	+	0.621982i	$0.786328\pi$
$954$	0	0
$955$	17.5416i	0.567632i
$956$	− 15.2942i	− 0.494650i
$957$	0	0
$958$	− 11.3639i	− 0.367150i
$959$	0	0
$960$	0	0
$961$	−11.1609	−0.360028
$962$	−0.565227	−0.0182236
$963$	0	0
$964$	19.7405i	0.635799i
$965$	−3.53043	−0.113649
$966$	0	0
$967$	53.0002	1.70437	0.852186	−	0.523239i	$-0.175276\pi$
$967$	53.0002	1.70437	0.852186	+	0.523239i	$0.175276\pi$
$968$	− 35.5370i	− 1.14220i
$969$	0	0
$970$	6.72089	0.215795
$971$	27.1779	0.872179	0.436090	−	0.899903i	$-0.356363\pi$
$971$	27.1779	0.872179	0.436090	+	0.899903i	$0.356363\pi$
$972$	0	0
$973$	0	0
$974$	− 6.71495i	− 0.215161i
$975$	0	0
$976$	29.6414i	0.948798i
$977$	33.1408i	1.06027i	0.847913	+	0.530135i	$0.177859\pi$
$977$	33.1408i	1.06027i	−0.847913	+	0.530135i	$0.822141\pi$
$978$	0	0
$979$	− 8.79416i	− 0.281063i
$980$	0	0
$981$	0	0
$982$	12.4049	0.395856
$983$	−30.8040	−0.982495	−0.491247	−	0.871020i	$-0.663459\pi$
$983$	−30.8040	−0.982495	−0.491247	+	0.871020i	$0.663459\pi$
$984$	0	0
$985$	− 6.77592i	− 0.215899i
$986$	0.0604670	0.00192566
$987$	0	0
$988$	−8.91114	−0.283501
$989$	22.1640i	0.704774i
$990$	0	0
$991$	−31.1680	−0.990083	−0.495041	−	0.868869i	$-0.664847\pi$
$991$	−31.1680	−0.990083	−0.495041	+	0.868869i	$0.664847\pi$
$992$	−33.8155	−1.07364
$993$	0	0
$994$	0	0
$995$	− 1.76972i	− 0.0561038i
$996$	0	0
$997$	60.2047i	1.90670i	0.301863	+	0.953351i	$0.402391\pi$
$997$	60.2047i	1.90670i	−0.301863	+	0.953351i	$0.597609\pi$
$998$	− 3.36476i	− 0.106510i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.c.f.1322.9	yes	16
3.2	odd	2	inner	1323.2.c.f.1322.8	yes	16
7.6	odd	2	inner	1323.2.c.f.1322.10	yes	16
21.20	even	2	inner	1323.2.c.f.1322.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.2.c.f.1322.7	✓	16	21.20	even	2	inner
1323.2.c.f.1322.8	yes	16	3.2	odd	2	inner
1323.2.c.f.1322.9	yes	16	1.1	even	1	trivial
1323.2.c.f.1322.10	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)

\(f(q)\)	\(=\)	\(q+0.529484i q^{2} +1.71965 q^{4} -0.753692 q^{5} +1.96949i q^{8} +O(q^{10})\)	\(q+0.529484i q^{2} +1.71965 q^{4} -0.753692 q^{5} +1.96949i q^{8} -0.399068i q^{10} +5.38922i q^{11} -1.46226i q^{13} +2.39648 q^{16} +5.82053 q^{17} -3.54379i q^{19} -1.29608 q^{20} -2.85351 q^{22} +3.26740i q^{23} -4.43195 q^{25} +0.774245 q^{26} -0.0196202i q^{29} +6.49314i q^{31} +5.20788i q^{32} +3.08188i q^{34} -0.730035 q^{37} +1.87638 q^{38} -1.48439i q^{40} -10.1827 q^{41} +6.78337 q^{43} +9.26756i q^{44} -1.73004 q^{46} +3.42231 q^{47} -2.34664i q^{50} -2.51458i q^{52} +10.3163i q^{53} -4.06182i q^{55} +0.0103886 q^{58} +7.88303 q^{59} +12.3687i q^{61} -3.43801 q^{62} +2.03547 q^{64} +1.10210i q^{65} +11.3869 q^{67} +10.0093 q^{68} -13.6174i q^{71} -6.55599i q^{73} -0.386542i q^{74} -6.09407i q^{76} +10.5281 q^{79} -1.80621 q^{80} -5.39155i q^{82} -16.0159 q^{83} -4.38689 q^{85} +3.59168i q^{86} -10.6140 q^{88} -1.63180 q^{89} +5.61877i q^{92} +1.81206i q^{94} +2.67093i q^{95} +16.8415i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 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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