LMFDB - Embedded newform 2646.2.d.e.2645.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2646,2,Mod(2645,2646)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2646, 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("2646.2645");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2646 = 2 \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2646.d (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:	$21.1284163748$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2645.14
Root			$0.793353 - 0.608761i$ of defining polynomial
Character	$\chi$	$=$	2646.2645
Dual form			2646.2.d.e.2645.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +1.68412 q^{5} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +1.68412 q^{5} -1.00000i q^{8} +1.68412i q^{10} +3.32565i q^{11} +4.34518i q^{13} +1.00000 q^{16} -3.81123 q^{17} -4.06583i q^{19} -1.68412 q^{20} -3.32565 q^{22} +4.69764i q^{23} -2.16373 q^{25} -4.34518 q^{26} -2.50277i q^{29} +4.09593i q^{31} +1.00000i q^{32} -3.81123i q^{34} +6.27558 q^{37} +4.06583 q^{38} -1.68412i q^{40} -5.46525 q^{41} -10.7920 q^{43} -3.32565i q^{44} -4.69764 q^{46} -0.412881 q^{47} -2.16373i q^{50} -4.34518i q^{52} +9.09436i q^{53} +5.60081i q^{55} +2.50277 q^{58} +12.0887 q^{59} -13.9166i q^{61} -4.09593 q^{62} -1.00000 q^{64} +7.31781i q^{65} -5.94993 q^{67} +3.81123 q^{68} +4.94028i q^{71} +5.78542i q^{73} +6.27558i q^{74} +4.06583i q^{76} -12.9228 q^{79} +1.68412 q^{80} -5.46525i q^{82} +6.04915 q^{83} -6.41857 q^{85} -10.7920i q^{86} +3.32565 q^{88} +6.31400 q^{89} -4.69764i q^{92} -0.412881i q^{94} -6.84736i q^{95} +11.6415i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 16 q^{16} - 32 q^{22} + 48 q^{25} - 32 q^{37} + 16 q^{43} - 48 q^{46} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 32 q^{88}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^16 - 32 * q^22 + 48 * q^25 - 32 * q^37 + 16 * q^43 - 48 * q^46 + 16 * q^58 - 16 * q^64 + 16 * q^67 + 32 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$.

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

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	1.68412	0.753163	0.376581	−	0.926384i	$-0.377100\pi$
$5$	1.68412	0.753163	0.376581	+	0.926384i	$0.377100\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	1.68412i	0.532566i
$11$	3.32565i	1.00272i	0.865238	+	0.501361i	$0.167167\pi$
$11$	3.32565i	1.00272i	−0.865238	+	0.501361i	$0.832833\pi$
$12$	0	0
$13$	4.34518i	1.20514i	0.798068	+	0.602568i	$0.205856\pi$
$13$	4.34518i	1.20514i	−0.798068	+	0.602568i	$0.794144\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.81123	−0.924358	−0.462179	−	0.886787i	$-0.652932\pi$
$17$	−3.81123	−0.924358	−0.462179	+	0.886787i	$0.652932\pi$
$18$	0	0
$19$	− 4.06583i	− 0.932766i	−0.884583	−	0.466383i	$-0.845557\pi$
$19$	− 4.06583i	− 0.932766i	0.884583	−	0.466383i	$-0.154443\pi$
$20$	−1.68412	−0.376581
$21$	0	0
$22$	−3.32565	−0.709032
$23$	4.69764i	0.979526i	0.871856	+	0.489763i	$0.162917\pi$
$23$	4.69764i	0.979526i	−0.871856	+	0.489763i	$0.837083\pi$
$24$	0	0
$25$	−2.16373	−0.432746
$26$	−4.34518	−0.852159
$27$	0	0
$28$	0	0
$29$	− 2.50277i	− 0.464753i	−0.972626	−	0.232377i	$-0.925350\pi$
$29$	− 2.50277i	− 0.464753i	0.972626	−	0.232377i	$-0.0746502\pi$
$30$	0	0
$31$	4.09593i	0.735651i	0.929895	+	0.367826i	$0.119898\pi$
$31$	4.09593i	0.735651i	−0.929895	+	0.367826i	$0.880102\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	− 3.81123i	− 0.653620i
$35$	0	0
$36$	0	0
$37$	6.27558	1.03170	0.515850	−	0.856679i	$-0.327476\pi$
$37$	6.27558	1.03170	0.515850	+	0.856679i	$0.327476\pi$
$38$	4.06583	0.659565
$39$	0	0
$40$	− 1.68412i	− 0.266283i
$41$	−5.46525	−0.853529	−0.426764	−	0.904363i	$-0.640347\pi$
$41$	−5.46525	−0.853529	−0.426764	+	0.904363i	$0.640347\pi$
$42$	0	0
$43$	−10.7920	−1.64576	−0.822882	−	0.568212i	$-0.807635\pi$
$43$	−10.7920	−1.64576	−0.822882	+	0.568212i	$0.807635\pi$
$44$	− 3.32565i	− 0.501361i
$45$	0	0
$46$	−4.69764	−0.692629
$47$	−0.412881	−0.0602249	−0.0301125	−	0.999547i	$-0.509587\pi$
$47$	−0.412881	−0.0602249	−0.0301125	+	0.999547i	$0.509587\pi$
$48$	0	0
$49$	0	0
$50$	− 2.16373i	− 0.305998i
$51$	0	0
$52$	− 4.34518i	− 0.602568i
$53$	9.09436i	1.24921i	0.780942	+	0.624603i	$0.214739\pi$
$53$	9.09436i	1.24921i	−0.780942	+	0.624603i	$0.785261\pi$
$54$	0	0
$55$	5.60081i	0.755213i
$56$	0	0
$57$	0	0
$58$	2.50277	0.328630
$59$	12.0887	1.57381	0.786907	−	0.617071i	$-0.211681\pi$
$59$	12.0887	1.57381	0.786907	+	0.617071i	$0.211681\pi$
$60$	0	0
$61$	− 13.9166i	− 1.78184i	−0.454162	−	0.890919i	$-0.650061\pi$
$61$	− 13.9166i	− 1.78184i	0.454162	−	0.890919i	$-0.349939\pi$
$62$	−4.09593	−0.520184
$63$	0	0
$64$	−1.00000	−0.125000
$65$	7.31781i	0.907663i
$66$	0	0
$67$	−5.94993	−0.726900	−0.363450	−	0.931614i	$-0.618401\pi$
$67$	−5.94993	−0.726900	−0.363450	+	0.931614i	$0.618401\pi$
$68$	3.81123	0.462179
$69$	0	0
$70$	0	0
$71$	4.94028i	0.586303i	0.956066	+	0.293152i	$0.0947041\pi$
$71$	4.94028i	0.586303i	−0.956066	+	0.293152i	$0.905296\pi$
$72$	0	0
$73$	5.78542i	0.677132i	0.940943	+	0.338566i	$0.109942\pi$
$73$	5.78542i	0.677132i	−0.940943	+	0.338566i	$0.890058\pi$
$74$	6.27558i	0.729522i
$75$	0	0
$76$	4.06583i	0.466383i
$77$	0	0
$78$	0	0
$79$	−12.9228	−1.45393	−0.726964	−	0.686676i	$-0.759069\pi$
$79$	−12.9228	−1.45393	−0.726964	+	0.686676i	$0.759069\pi$
$80$	1.68412	0.188291
$81$	0	0
$82$	− 5.46525i	− 0.603536i
$83$	6.04915	0.663981	0.331990	−	0.943283i	$-0.392280\pi$
$83$	6.04915	0.663981	0.331990	+	0.943283i	$0.392280\pi$
$84$	0	0
$85$	−6.41857	−0.696192
$86$	− 10.7920i	− 1.16373i
$87$	0	0
$88$	3.32565	0.354516
$89$	6.31400	0.669283	0.334642	−	0.942345i	$-0.391385\pi$
$89$	6.31400	0.669283	0.334642	+	0.942345i	$0.391385\pi$
$90$	0	0
$91$	0	0
$92$	− 4.69764i	− 0.489763i
$93$	0	0
$94$	− 0.412881i	− 0.0425854i
$95$	− 6.84736i	− 0.702524i
$96$	0	0
$97$	11.6415i	1.18201i	0.806667	+	0.591006i	$0.201269\pi$
$97$	11.6415i	1.18201i	−0.806667	+	0.591006i	$0.798731\pi$
$98$	0	0
$99$	0	0
$100$	2.16373	0.216373
$101$	−6.89588	−0.686166	−0.343083	−	0.939305i	$-0.611471\pi$
$101$	−6.89588	−0.686166	−0.343083	+	0.939305i	$0.611471\pi$
$102$	0	0
$103$	4.65462i	0.458633i	0.973352	+	0.229317i	$0.0736491\pi$
$103$	4.65462i	0.458633i	−0.973352	+	0.229317i	$0.926351\pi$
$104$	4.34518	0.426080
$105$	0	0
$106$	−9.09436	−0.883323
$107$	13.1098i	1.26737i	0.773590	+	0.633687i	$0.218459\pi$
$107$	13.1098i	1.26737i	−0.773590	+	0.633687i	$0.781541\pi$
$108$	0	0
$109$	−17.7553	−1.70065	−0.850326	−	0.526257i	$-0.823595\pi$
$109$	−17.7553	−1.70065	−0.850326	+	0.526257i	$0.823595\pi$
$110$	−5.60081	−0.534016
$111$	0	0
$112$	0	0
$113$	18.3234i	1.72372i	0.507149	+	0.861858i	$0.330699\pi$
$113$	18.3234i	1.72372i	−0.507149	+	0.861858i	$0.669301\pi$
$114$	0	0
$115$	7.91140i	0.737742i
$116$	2.50277i	0.232377i
$117$	0	0
$118$	12.0887i	1.11285i
$119$	0	0
$120$	0	0
$121$	−0.0599765	−0.00545241
$122$	13.9166	1.25995
$123$	0	0
$124$	− 4.09593i	− 0.367826i
$125$	−12.0646	−1.07909
$126$	0	0
$127$	−18.6723	−1.65690	−0.828450	−	0.560064i	$-0.810777\pi$
$127$	−18.6723	−1.65690	−0.828450	+	0.560064i	$0.810777\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	−7.31781	−0.641815
$131$	15.5954	1.36257	0.681287	−	0.732017i	$-0.261421\pi$
$131$	15.5954	1.36257	0.681287	+	0.732017i	$0.261421\pi$
$132$	0	0
$133$	0	0
$134$	− 5.94993i	− 0.513996i
$135$	0	0
$136$	3.81123i	0.326810i
$137$	− 12.7650i	− 1.09058i	−0.838246	−	0.545292i	$-0.816419\pi$
$137$	− 12.7650i	− 1.09058i	0.838246	−	0.545292i	$-0.183581\pi$
$138$	0	0
$139$	13.0451i	1.10647i	0.833024	+	0.553237i	$0.186608\pi$
$139$	13.0451i	1.10647i	−0.833024	+	0.553237i	$0.813392\pi$
$140$	0	0
$141$	0	0
$142$	−4.94028	−0.414579
$143$	−14.4506	−1.20842
$144$	0	0
$145$	− 4.21498i	− 0.350035i
$146$	−5.78542	−0.478804
$147$	0	0
$148$	−6.27558	−0.515850
$149$	5.81360i	0.476268i	0.971232	+	0.238134i	$0.0765358\pi$
$149$	5.81360i	0.476268i	−0.971232	+	0.238134i	$0.923464\pi$
$150$	0	0
$151$	4.17302	0.339595	0.169798	−	0.985479i	$-0.445689\pi$
$151$	4.17302	0.339595	0.169798	+	0.985479i	$0.445689\pi$
$152$	−4.06583	−0.329783
$153$	0	0
$154$	0	0
$155$	6.89805i	0.554065i
$156$	0	0
$157$	− 11.2201i	− 0.895461i	−0.894169	−	0.447730i	$-0.852232\pi$
$157$	− 11.2201i	− 0.895461i	0.894169	−	0.447730i	$-0.147768\pi$
$158$	− 12.9228i	− 1.02808i
$159$	0	0
$160$	1.68412i	0.133142i
$161$	0	0
$162$	0	0
$163$	12.0941	0.947283	0.473642	−	0.880718i	$-0.342939\pi$
$163$	12.0941	0.947283	0.473642	+	0.880718i	$0.342939\pi$
$164$	5.46525	0.426764
$165$	0	0
$166$	6.04915i	0.469505i
$167$	7.76449	0.600834	0.300417	−	0.953808i	$-0.402874\pi$
$167$	7.76449	0.600834	0.300417	+	0.953808i	$0.402874\pi$
$168$	0	0
$169$	−5.88056	−0.452351
$170$	− 6.41857i	− 0.492282i
$171$	0	0
$172$	10.7920	0.822882
$173$	7.28069	0.553541	0.276770	−	0.960936i	$-0.410736\pi$
$173$	7.28069	0.553541	0.276770	+	0.960936i	$0.410736\pi$
$174$	0	0
$175$	0	0
$176$	3.32565i	0.250681i
$177$	0	0
$178$	6.31400i	0.473255i
$179$	14.8165i	1.10744i	0.832704	+	0.553718i	$0.186791\pi$
$179$	14.8165i	1.10744i	−0.832704	+	0.553718i	$0.813209\pi$
$180$	0	0
$181$	− 11.9676i	− 0.889547i	−0.895643	−	0.444773i	$-0.853284\pi$
$181$	− 11.9676i	− 0.889547i	0.895643	−	0.444773i	$-0.146716\pi$
$182$	0	0
$183$	0	0
$184$	4.69764	0.346315
$185$	10.5689	0.777038
$186$	0	0
$187$	− 12.6748i	− 0.926875i
$188$	0.412881	0.0301125
$189$	0	0
$190$	6.84736	0.496760
$191$	− 3.07927i	− 0.222808i	−0.993775	−	0.111404i	$-0.964465\pi$
$191$	− 3.07927i	− 0.222808i	0.993775	−	0.111404i	$-0.0355348\pi$
$192$	0	0
$193$	−18.6569	−1.34295	−0.671475	−	0.741027i	$-0.734339\pi$
$193$	−18.6569	−1.34295	−0.671475	+	0.741027i	$0.734339\pi$
$194$	−11.6415	−0.835809
$195$	0	0
$196$	0	0
$197$	− 23.6108i	− 1.68220i	−0.540881	−	0.841099i	$-0.681909\pi$
$197$	− 23.6108i	− 1.68220i	0.540881	−	0.841099i	$-0.318091\pi$
$198$	0	0
$199$	− 15.4496i	− 1.09519i	−0.836743	−	0.547595i	$-0.815543\pi$
$199$	− 15.4496i	− 1.09519i	0.836743	−	0.547595i	$-0.184457\pi$
$200$	2.16373i	0.152999i
$201$	0	0
$202$	− 6.89588i	− 0.485193i
$203$	0	0
$204$	0	0
$205$	−9.20415	−0.642846
$206$	−4.65462	−0.324303
$207$	0	0
$208$	4.34518i	0.301284i
$209$	13.5216	0.935305
$210$	0	0
$211$	−13.7399	−0.945892	−0.472946	−	0.881092i	$-0.656809\pi$
$211$	−13.7399	−0.945892	−0.472946	+	0.881092i	$0.656809\pi$
$212$	− 9.09436i	− 0.624603i
$213$	0	0
$214$	−13.1098	−0.896168
$215$	−18.1751	−1.23953
$216$	0	0
$217$	0	0
$218$	− 17.7553i	− 1.20254i
$219$	0	0
$220$	− 5.60081i	− 0.377607i
$221$	− 16.5605i	− 1.11398i
$222$	0	0
$223$	− 4.65684i	− 0.311845i	−0.987769	−	0.155923i	$-0.950165\pi$
$223$	− 4.65684i	− 0.311845i	0.987769	−	0.155923i	$-0.0498350\pi$
$224$	0	0
$225$	0	0
$226$	−18.3234	−1.21885
$227$	−27.6540	−1.83546	−0.917729	−	0.397207i	$-0.869980\pi$
$227$	−27.6540	−1.83546	−0.917729	+	0.397207i	$0.869980\pi$
$228$	0	0
$229$	− 12.6394i	− 0.835232i	−0.908624	−	0.417616i	$-0.862866\pi$
$229$	− 12.6394i	− 0.835232i	0.908624	−	0.417616i	$-0.137134\pi$
$230$	−7.91140	−0.521662
$231$	0	0
$232$	−2.50277	−0.164315
$233$	− 4.67061i	− 0.305982i	−0.988228	−	0.152991i	$-0.951110\pi$
$233$	− 4.67061i	− 0.305982i	0.988228	−	0.152991i	$-0.0488905\pi$
$234$	0	0
$235$	−0.695343	−0.0453592
$236$	−12.0887	−0.786907
$237$	0	0
$238$	0	0
$239$	23.7883i	1.53873i	0.638807	+	0.769367i	$0.279428\pi$
$239$	23.7883i	1.53873i	−0.638807	+	0.769367i	$0.720572\pi$
$240$	0	0
$241$	2.77410i	0.178695i	0.996001	+	0.0893476i	$0.0284782\pi$
$241$	2.77410i	0.178695i	−0.996001	+	0.0893476i	$0.971522\pi$
$242$	− 0.0599765i	− 0.00385544i
$243$	0	0
$244$	13.9166i	0.890919i
$245$	0	0
$246$	0	0
$247$	17.6668	1.12411
$248$	4.09593	0.260092
$249$	0	0
$250$	− 12.0646i	− 0.763032i
$251$	23.2943	1.47032	0.735162	−	0.677892i	$-0.237106\pi$
$251$	23.2943	1.47032	0.735162	+	0.677892i	$0.237106\pi$
$252$	0	0
$253$	−15.6227	−0.982192
$254$	− 18.6723i	− 1.17160i
$255$	0	0
$256$	1.00000	0.0625000
$257$	31.1100	1.94059	0.970296	−	0.241923i	$-0.0777780\pi$
$257$	31.1100	1.94059	0.970296	+	0.241923i	$0.0777780\pi$
$258$	0	0
$259$	0	0
$260$	− 7.31781i	− 0.453831i
$261$	0	0
$262$	15.5954i	0.963485i
$263$	11.1363i	0.686696i	0.939208	+	0.343348i	$0.111561\pi$
$263$	11.1363i	0.686696i	−0.939208	+	0.343348i	$0.888439\pi$
$264$	0	0
$265$	15.3160i	0.940856i
$266$	0	0
$267$	0	0
$268$	5.94993	0.363450
$269$	−30.5496	−1.86264	−0.931320	−	0.364201i	$-0.881342\pi$
$269$	−30.5496	−1.86264	−0.931320	+	0.364201i	$0.881342\pi$
$270$	0	0
$271$	15.8834i	0.964848i	0.875938	+	0.482424i	$0.160244\pi$
$271$	15.8834i	0.964848i	−0.875938	+	0.482424i	$0.839756\pi$
$272$	−3.81123	−0.231090
$273$	0	0
$274$	12.7650	0.771160
$275$	− 7.19582i	− 0.433924i
$276$	0	0
$277$	27.6470	1.66115	0.830573	−	0.556910i	$-0.188013\pi$
$277$	27.6470	1.66115	0.830573	+	0.556910i	$0.188013\pi$
$278$	−13.0451	−0.782395
$279$	0	0
$280$	0	0
$281$	− 13.2618i	− 0.791135i	−0.918437	−	0.395567i	$-0.870548\pi$
$281$	− 13.2618i	− 0.791135i	0.918437	−	0.395567i	$-0.129452\pi$
$282$	0	0
$283$	25.4310i	1.51172i	0.654735	+	0.755858i	$0.272780\pi$
$283$	25.4310i	1.51172i	−0.654735	+	0.755858i	$0.727220\pi$
$284$	− 4.94028i	− 0.293152i
$285$	0	0
$286$	− 14.4506i	− 0.854479i
$287$	0	0
$288$	0	0
$289$	−2.47455	−0.145562
$290$	4.21498	0.247512
$291$	0	0
$292$	− 5.78542i	− 0.338566i
$293$	4.40649	0.257430	0.128715	−	0.991682i	$-0.458915\pi$
$293$	4.40649	0.257430	0.128715	+	0.991682i	$0.458915\pi$
$294$	0	0
$295$	20.3589	1.18534
$296$	− 6.27558i	− 0.364761i
$297$	0	0
$298$	−5.81360	−0.336773
$299$	−20.4121	−1.18046
$300$	0	0
$301$	0	0
$302$	4.17302i	0.240130i
$303$	0	0
$304$	− 4.06583i	− 0.233191i
$305$	− 23.4373i	− 1.34201i
$306$	0	0
$307$	− 1.72649i	− 0.0985362i	−0.998786	−	0.0492681i	$-0.984311\pi$
$307$	− 1.72649i	− 0.0985362i	0.998786	−	0.0492681i	$-0.0156889\pi$
$308$	0	0
$309$	0	0
$310$	−6.89805	−0.391783
$311$	13.0155	0.738040	0.369020	−	0.929421i	$-0.379693\pi$
$311$	13.0155	0.738040	0.369020	+	0.929421i	$0.379693\pi$
$312$	0	0
$313$	− 2.91566i	− 0.164803i	−0.996599	−	0.0824014i	$-0.973741\pi$
$313$	− 2.91566i	− 0.164803i	0.996599	−	0.0824014i	$-0.0262590\pi$
$314$	11.2201	0.633186
$315$	0	0
$316$	12.9228	0.726964
$317$	28.3391i	1.59168i	0.605507	+	0.795840i	$0.292971\pi$
$317$	28.3391i	1.59168i	−0.605507	+	0.795840i	$0.707029\pi$
$318$	0	0
$319$	8.32336	0.466019
$320$	−1.68412	−0.0941453
$321$	0	0
$322$	0	0
$323$	15.4958i	0.862210i
$324$	0	0
$325$	− 9.40179i	− 0.521517i
$326$	12.0941i	0.669831i
$327$	0	0
$328$	5.46525i	0.301768i
$329$	0	0
$330$	0	0
$331$	7.60934	0.418247	0.209124	−	0.977889i	$-0.432939\pi$
$331$	7.60934	0.418247	0.209124	+	0.977889i	$0.432939\pi$
$332$	−6.04915	−0.331990
$333$	0	0
$334$	7.76449i	0.424854i
$335$	−10.0204	−0.547474
$336$	0	0
$337$	31.4203	1.71157	0.855786	−	0.517331i	$-0.173074\pi$
$337$	31.4203	1.71157	0.855786	+	0.517331i	$0.173074\pi$
$338$	− 5.88056i	− 0.319860i
$339$	0	0
$340$	6.41857	0.348096
$341$	−13.6217	−0.737654
$342$	0	0
$343$	0	0
$344$	10.7920i	0.581866i
$345$	0	0
$346$	7.28069i	0.391412i
$347$	− 13.8398i	− 0.742958i	−0.928441	−	0.371479i	$-0.878851\pi$
$347$	− 13.8398i	− 0.742958i	0.928441	−	0.371479i	$-0.121149\pi$
$348$	0	0
$349$	− 20.6085i	− 1.10315i	−0.834126	−	0.551574i	$-0.814027\pi$
$349$	− 20.6085i	− 1.10315i	0.834126	−	0.551574i	$-0.185973\pi$
$350$	0	0
$351$	0	0
$352$	−3.32565	−0.177258
$353$	−21.5019	−1.14443	−0.572214	−	0.820104i	$-0.693915\pi$
$353$	−21.5019	−1.14443	−0.572214	+	0.820104i	$0.693915\pi$
$354$	0	0
$355$	8.32004i	0.441582i
$356$	−6.31400	−0.334642
$357$	0	0
$358$	−14.8165	−0.783075
$359$	25.9574i	1.36998i	0.728552	+	0.684990i	$0.240194\pi$
$359$	25.9574i	1.36998i	−0.728552	+	0.684990i	$0.759806\pi$
$360$	0	0
$361$	2.46901	0.129948
$362$	11.9676	0.629005
$363$	0	0
$364$	0	0
$365$	9.74335i	0.509990i
$366$	0	0
$367$	7.42133i	0.387390i	0.981062	+	0.193695i	$0.0620473\pi$
$367$	7.42133i	0.387390i	−0.981062	+	0.193695i	$0.937953\pi$
$368$	4.69764i	0.244881i
$369$	0	0
$370$	10.5689i	0.549449i
$371$	0	0
$372$	0	0
$373$	−6.97671	−0.361240	−0.180620	−	0.983553i	$-0.557810\pi$
$373$	−6.97671	−0.361240	−0.180620	+	0.983553i	$0.557810\pi$
$374$	12.6748	0.655399
$375$	0	0
$376$	0.412881i	0.0212927i
$377$	10.8750	0.560090
$378$	0	0
$379$	7.68980	0.394998	0.197499	−	0.980303i	$-0.436718\pi$
$379$	7.68980	0.394998	0.197499	+	0.980303i	$0.436718\pi$
$380$	6.84736i	0.351262i
$381$	0	0
$382$	3.07927	0.157549
$383$	7.15314	0.365509	0.182754	−	0.983159i	$-0.441499\pi$
$383$	7.15314	0.365509	0.182754	+	0.983159i	$0.441499\pi$
$384$	0	0
$385$	0	0
$386$	− 18.6569i	− 0.949609i
$387$	0	0
$388$	− 11.6415i	− 0.591006i
$389$	6.21550i	0.315138i	0.987508	+	0.157569i	$0.0503657\pi$
$389$	6.21550i	0.315138i	−0.987508	+	0.157569i	$0.949634\pi$
$390$	0	0
$391$	− 17.9038i	− 0.905432i
$392$	0	0
$393$	0	0
$394$	23.6108	1.18949
$395$	−21.7636	−1.09504
$396$	0	0
$397$	15.8118i	0.793574i	0.917911	+	0.396787i	$0.129875\pi$
$397$	15.8118i	0.793574i	−0.917911	+	0.396787i	$0.870125\pi$
$398$	15.4496	0.774416
$399$	0	0
$400$	−2.16373	−0.108187
$401$	− 13.7471i	− 0.686498i	−0.939244	−	0.343249i	$-0.888473\pi$
$401$	− 13.7471i	− 0.686498i	0.939244	−	0.343249i	$-0.111527\pi$
$402$	0	0
$403$	−17.7975	−0.886559
$404$	6.89588	0.343083
$405$	0	0
$406$	0	0
$407$	20.8704i	1.03451i
$408$	0	0
$409$	− 15.3972i	− 0.761342i	−0.924711	−	0.380671i	$-0.875693\pi$
$409$	− 15.3972i	− 0.761342i	0.924711	−	0.380671i	$-0.124307\pi$
$410$	− 9.20415i	− 0.454561i
$411$	0	0
$412$	− 4.65462i	− 0.229317i
$413$	0	0
$414$	0	0
$415$	10.1875	0.500085
$416$	−4.34518	−0.213040
$417$	0	0
$418$	13.5216i	0.661361i
$419$	−3.80384	−0.185830	−0.0929149	−	0.995674i	$-0.529618\pi$
$419$	−3.80384	−0.185830	−0.0929149	+	0.995674i	$0.529618\pi$
$420$	0	0
$421$	20.2583	0.987332	0.493666	−	0.869652i	$-0.335657\pi$
$421$	20.2583	0.987332	0.493666	+	0.869652i	$0.335657\pi$
$422$	− 13.7399i	− 0.668846i
$423$	0	0
$424$	9.09436	0.441661
$425$	8.24647	0.400012
$426$	0	0
$427$	0	0
$428$	− 13.1098i	− 0.633687i
$429$	0	0
$430$	− 18.1751i	− 0.876479i
$431$	11.3141i	0.544980i	0.962159	+	0.272490i	$0.0878471\pi$
$431$	11.3141i	0.544980i	−0.962159	+	0.272490i	$0.912153\pi$
$432$	0	0
$433$	− 18.3925i	− 0.883885i	−0.897043	−	0.441943i	$-0.854290\pi$
$433$	− 18.3925i	− 0.883885i	0.897043	−	0.441943i	$-0.145710\pi$
$434$	0	0
$435$	0	0
$436$	17.7553	0.850326
$437$	19.0998	0.913668
$438$	0	0
$439$	3.97200i	0.189573i	0.995498	+	0.0947866i	$0.0302169\pi$
$439$	3.97200i	0.189573i	−0.995498	+	0.0947866i	$0.969783\pi$
$440$	5.60081	0.267008
$441$	0	0
$442$	16.5605	0.787700
$443$	− 24.5779i	− 1.16773i	−0.811850	−	0.583867i	$-0.801539\pi$
$443$	− 24.5779i	− 1.16773i	0.811850	−	0.583867i	$-0.198461\pi$
$444$	0	0
$445$	10.6336	0.504079
$446$	4.65684	0.220508
$447$	0	0
$448$	0	0
$449$	− 19.7867i	− 0.933793i	−0.884312	−	0.466896i	$-0.845372\pi$
$449$	− 19.7867i	− 0.933793i	0.884312	−	0.466896i	$-0.154628\pi$
$450$	0	0
$451$	− 18.1755i	− 0.855852i
$452$	− 18.3234i	− 0.861858i
$453$	0	0
$454$	− 27.6540i	− 1.29786i
$455$	0	0
$456$	0	0
$457$	3.24264	0.151684	0.0758422	−	0.997120i	$-0.475835\pi$
$457$	3.24264	0.151684	0.0758422	+	0.997120i	$0.475835\pi$
$458$	12.6394	0.590599
$459$	0	0
$460$	− 7.91140i	− 0.368871i
$461$	−12.9170	−0.601603	−0.300801	−	0.953687i	$-0.597254\pi$
$461$	−12.9170	−0.601603	−0.300801	+	0.953687i	$0.597254\pi$
$462$	0	0
$463$	25.9016	1.20375	0.601874	−	0.798591i	$-0.294421\pi$
$463$	25.9016	1.20375	0.601874	+	0.798591i	$0.294421\pi$
$464$	− 2.50277i	− 0.116188i
$465$	0	0
$466$	4.67061	0.216362
$467$	7.30432	0.338004	0.169002	−	0.985616i	$-0.445946\pi$
$467$	7.30432	0.338004	0.169002	+	0.985616i	$0.445946\pi$
$468$	0	0
$469$	0	0
$470$	− 0.695343i	− 0.0320738i
$471$	0	0
$472$	− 12.0887i	− 0.556427i
$473$	− 35.8905i	− 1.65025i
$474$	0	0
$475$	8.79736i	0.403651i
$476$	0	0
$477$	0	0
$478$	−23.7883	−1.08805
$479$	15.0285	0.686668	0.343334	−	0.939213i	$-0.388444\pi$
$479$	15.0285	0.686668	0.343334	+	0.939213i	$0.388444\pi$
$480$	0	0
$481$	27.2685i	1.24334i
$482$	−2.77410	−0.126357
$483$	0	0
$484$	0.0599765	0.00272620
$485$	19.6057i	0.890248i
$486$	0	0
$487$	8.14335	0.369011	0.184505	−	0.982832i	$-0.440932\pi$
$487$	8.14335	0.369011	0.184505	+	0.982832i	$0.440932\pi$
$488$	−13.9166	−0.629975
$489$	0	0
$490$	0	0
$491$	26.6719i	1.20369i	0.798614	+	0.601844i	$0.205567\pi$
$491$	26.6719i	1.20369i	−0.798614	+	0.601844i	$0.794433\pi$
$492$	0	0
$493$	9.53863i	0.429598i
$494$	17.6668i	0.794865i
$495$	0	0
$496$	4.09593i	0.183913i
$497$	0	0
$498$	0	0
$499$	−31.5456	−1.41218	−0.706088	−	0.708124i	$-0.749542\pi$
$499$	−31.5456	−1.41218	−0.706088	+	0.708124i	$0.749542\pi$
$500$	12.0646	0.539545
$501$	0	0
$502$	23.2943i	1.03968i
$503$	9.08274	0.404979	0.202490	−	0.979284i	$-0.435097\pi$
$503$	9.08274	0.404979	0.202490	+	0.979284i	$0.435097\pi$
$504$	0	0
$505$	−11.6135	−0.516795
$506$	− 15.6227i	− 0.694515i
$507$	0	0
$508$	18.6723	0.828450
$509$	41.2308	1.82752	0.913762	−	0.406249i	$-0.133164\pi$
$509$	41.2308	1.82752	0.913762	+	0.406249i	$0.133164\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	31.1100i	1.37221i
$515$	7.83895i	0.345426i
$516$	0	0
$517$	− 1.37310i	− 0.0603889i
$518$	0	0
$519$	0	0
$520$	7.31781	0.320907
$521$	24.4010	1.06903	0.534513	−	0.845160i	$-0.320495\pi$
$521$	24.4010	1.06903	0.534513	+	0.845160i	$0.320495\pi$
$522$	0	0
$523$	− 15.4857i	− 0.677141i	−0.940941	−	0.338570i	$-0.890057\pi$
$523$	− 15.4857i	− 0.677141i	0.940941	−	0.338570i	$-0.109943\pi$
$524$	−15.5954	−0.681287
$525$	0	0
$526$	−11.1363	−0.485567
$527$	− 15.6105i	− 0.680005i
$528$	0	0
$529$	0.932181	0.0405296
$530$	−15.3160	−0.665286
$531$	0	0
$532$	0	0
$533$	− 23.7475i	− 1.02862i
$534$	0	0
$535$	22.0785i	0.954538i
$536$	5.94993i	0.256998i
$537$	0	0
$538$	− 30.5496i	− 1.31709i
$539$	0	0
$540$	0	0
$541$	−8.12150	−0.349171	−0.174585	−	0.984642i	$-0.555858\pi$
$541$	−8.12150	−0.349171	−0.174585	+	0.984642i	$0.555858\pi$
$542$	−15.8834	−0.682251
$543$	0	0
$544$	− 3.81123i	− 0.163405i
$545$	−29.9021	−1.28087
$546$	0	0
$547$	29.1593	1.24676	0.623381	−	0.781919i	$-0.285759\pi$
$547$	29.1593	1.24676	0.623381	+	0.781919i	$0.285759\pi$
$548$	12.7650i	0.545292i
$549$	0	0
$550$	7.19582	0.306831
$551$	−10.1759	−0.433506
$552$	0	0
$553$	0	0
$554$	27.6470i	1.17461i
$555$	0	0
$556$	− 13.0451i	− 0.553237i
$557$	33.5006i	1.41947i	0.704470	+	0.709734i	$0.251185\pi$
$557$	33.5006i	1.41947i	−0.704470	+	0.709734i	$0.748815\pi$
$558$	0	0
$559$	− 46.8932i	− 1.98337i
$560$	0	0
$561$	0	0
$562$	13.2618	0.559417
$563$	36.7289	1.54794	0.773970	−	0.633222i	$-0.218268\pi$
$563$	36.7289	1.54794	0.773970	+	0.633222i	$0.218268\pi$
$564$	0	0
$565$	30.8588i	1.29824i
$566$	−25.4310	−1.06894
$567$	0	0
$568$	4.94028	0.207290
$569$	5.10811i	0.214143i	0.994251	+	0.107072i	$0.0341474\pi$
$569$	5.10811i	0.214143i	−0.994251	+	0.107072i	$0.965853\pi$
$570$	0	0
$571$	40.0982	1.67806	0.839029	−	0.544086i	$-0.183124\pi$
$571$	40.0982	1.67806	0.839029	+	0.544086i	$0.183124\pi$
$572$	14.4506	0.604208
$573$	0	0
$574$	0	0
$575$	− 10.1644i	− 0.423886i
$576$	0	0
$577$	− 23.0600i	− 0.960001i	−0.877268	−	0.480001i	$-0.840636\pi$
$577$	− 23.0600i	− 0.960001i	0.877268	−	0.480001i	$-0.159364\pi$
$578$	− 2.47455i	− 0.102928i
$579$	0	0
$580$	4.21498i	0.175017i
$581$	0	0
$582$	0	0
$583$	−30.2447	−1.25261
$584$	5.78542	0.239402
$585$	0	0
$586$	4.40649i	0.182031i
$587$	5.62333	0.232100	0.116050	−	0.993243i	$-0.462977\pi$
$587$	5.62333	0.232100	0.116050	+	0.993243i	$0.462977\pi$
$588$	0	0
$589$	16.6534	0.686190
$590$	20.3589i	0.838161i
$591$	0	0
$592$	6.27558	0.257925
$593$	−11.9230	−0.489621	−0.244810	−	0.969571i	$-0.578726\pi$
$593$	−11.9230	−0.489621	−0.244810	+	0.969571i	$0.578726\pi$
$594$	0	0
$595$	0	0
$596$	− 5.81360i	− 0.238134i
$597$	0	0
$598$	− 20.4121i	− 0.834712i
$599$	5.73868i	0.234476i	0.993104	+	0.117238i	$0.0374041\pi$
$599$	5.73868i	0.234476i	−0.993104	+	0.117238i	$0.962596\pi$
$600$	0	0
$601$	42.3941i	1.72929i	0.502382	+	0.864646i	$0.332457\pi$
$601$	42.3941i	1.72929i	−0.502382	+	0.864646i	$0.667543\pi$
$602$	0	0
$603$	0	0
$604$	−4.17302	−0.169798
$605$	−0.101008	−0.00410655
$606$	0	0
$607$	7.60327i	0.308607i	0.988023	+	0.154304i	$0.0493134\pi$
$607$	7.60327i	0.308607i	−0.988023	+	0.154304i	$0.950687\pi$
$608$	4.06583	0.164891
$609$	0	0
$610$	23.4373	0.948947
$611$	− 1.79404i	− 0.0725792i
$612$	0	0
$613$	29.9823	1.21097	0.605486	−	0.795856i	$-0.292979\pi$
$613$	29.9823	1.21097	0.605486	+	0.795856i	$0.292979\pi$
$614$	1.72649	0.0696756
$615$	0	0
$616$	0	0
$617$	− 12.6950i	− 0.511081i	−0.966798	−	0.255540i	$-0.917747\pi$
$617$	− 12.6950i	− 0.511081i	0.966798	−	0.255540i	$-0.0822534\pi$
$618$	0	0
$619$	− 2.36054i	− 0.0948783i	−0.998874	−	0.0474391i	$-0.984894\pi$
$619$	− 2.36054i	− 0.0948783i	0.998874	−	0.0474391i	$-0.0151060\pi$
$620$	− 6.89805i	− 0.277032i
$621$	0	0
$622$	13.0155i	0.521873i
$623$	0	0
$624$	0	0
$625$	−9.49962	−0.379985
$626$	2.91566	0.116533
$627$	0	0
$628$	11.2201i	0.447730i
$629$	−23.9177	−0.953660
$630$	0	0
$631$	42.1986	1.67990	0.839950	−	0.542663i	$-0.182584\pi$
$631$	42.1986	1.67990	0.839950	+	0.542663i	$0.182584\pi$
$632$	12.9228i	0.514041i
$633$	0	0
$634$	−28.3391	−1.12549
$635$	−31.4465	−1.24791
$636$	0	0
$637$	0	0
$638$	8.32336i	0.329525i
$639$	0	0
$640$	− 1.68412i	− 0.0665708i
$641$	4.40611i	0.174031i	0.996207	+	0.0870155i	$0.0277330\pi$
$641$	4.40611i	0.174031i	−0.996207	+	0.0870155i	$0.972267\pi$
$642$	0	0
$643$	− 10.9171i	− 0.430529i	−0.976556	−	0.215264i	$-0.930939\pi$
$643$	− 10.9171i	− 0.430529i	0.976556	−	0.215264i	$-0.0690613\pi$
$644$	0	0
$645$	0	0
$646$	−15.4958	−0.609674
$647$	−4.73821	−0.186278	−0.0931390	−	0.995653i	$-0.529690\pi$
$647$	−4.73821	−0.186278	−0.0931390	+	0.995653i	$0.529690\pi$
$648$	0	0
$649$	40.2028i	1.57810i
$650$	9.40179	0.368769
$651$	0	0
$652$	−12.0941	−0.473642
$653$	14.2119i	0.556153i	0.960559	+	0.278077i	$0.0896969\pi$
$653$	14.2119i	0.556153i	−0.960559	+	0.278077i	$0.910303\pi$
$654$	0	0
$655$	26.2645	1.02624
$656$	−5.46525	−0.213382
$657$	0	0
$658$	0	0
$659$	13.5374i	0.527343i	0.964613	+	0.263671i	$0.0849335\pi$
$659$	13.5374i	0.527343i	−0.964613	+	0.263671i	$0.915067\pi$
$660$	0	0
$661$	0.345298i	0.0134305i	0.999977	+	0.00671526i	$0.00213755\pi$
$661$	0.345298i	0.0134305i	−0.999977	+	0.00671526i	$0.997862\pi$
$662$	7.60934i	0.295745i
$663$	0	0
$664$	− 6.04915i	− 0.234753i
$665$	0	0
$666$	0	0
$667$	11.7571	0.455238
$668$	−7.76449	−0.300417
$669$	0	0
$670$	− 10.0204i	− 0.387122i
$671$	46.2818	1.78669
$672$	0	0
$673$	16.4060	0.632405	0.316203	−	0.948692i	$-0.397592\pi$
$673$	16.4060	0.632405	0.316203	+	0.948692i	$0.397592\pi$
$674$	31.4203i	1.21026i
$675$	0	0
$676$	5.88056	0.226175
$677$	25.0364	0.962228	0.481114	−	0.876658i	$-0.340232\pi$
$677$	25.0364	0.962228	0.481114	+	0.876658i	$0.340232\pi$
$678$	0	0
$679$	0	0
$680$	6.41857i	0.246141i
$681$	0	0
$682$	− 13.6217i	− 0.521600i
$683$	29.5011i	1.12883i	0.825492	+	0.564414i	$0.190898\pi$
$683$	29.5011i	1.12883i	−0.825492	+	0.564414i	$0.809102\pi$
$684$	0	0
$685$	− 21.4978i	− 0.821388i
$686$	0	0
$687$	0	0
$688$	−10.7920	−0.411441
$689$	−39.5166	−1.50546
$690$	0	0
$691$	22.7916i	0.867032i	0.901146	+	0.433516i	$0.142727\pi$
$691$	22.7916i	0.867032i	−0.901146	+	0.433516i	$0.857273\pi$
$692$	−7.28069	−0.276770
$693$	0	0
$694$	13.8398	0.525351
$695$	21.9696i	0.833355i
$696$	0	0
$697$	20.8293	0.788966
$698$	20.6085	0.780043
$699$	0	0
$700$	0	0
$701$	− 31.6849i	− 1.19672i	−0.801227	−	0.598361i	$-0.795819\pi$
$701$	− 31.6849i	− 1.19672i	0.801227	−	0.598361i	$-0.204181\pi$
$702$	0	0
$703$	− 25.5155i	− 0.962334i
$704$	− 3.32565i	− 0.125340i
$705$	0	0
$706$	− 21.5019i	− 0.809233i
$707$	0	0
$708$	0	0
$709$	−12.9692	−0.487070	−0.243535	−	0.969892i	$-0.578307\pi$
$709$	−12.9692	−0.487070	−0.243535	+	0.969892i	$0.578307\pi$
$710$	−8.32004	−0.312246
$711$	0	0
$712$	− 6.31400i	− 0.236627i
$713$	−19.2412	−0.720589
$714$	0	0
$715$	−24.3365	−0.910134
$716$	− 14.8165i	− 0.553718i
$717$	0	0
$718$	−25.9574	−0.968723
$719$	18.5782	0.692849	0.346424	−	0.938078i	$-0.387396\pi$
$719$	18.5782	0.692849	0.346424	+	0.938078i	$0.387396\pi$
$720$	0	0
$721$	0	0
$722$	2.46901i	0.0918870i
$723$	0	0
$724$	11.9676i	0.444773i
$725$	5.41532i	0.201120i
$726$	0	0
$727$	35.5818i	1.31966i	0.751416	+	0.659828i	$0.229371\pi$
$727$	35.5818i	1.31966i	−0.751416	+	0.659828i	$0.770629\pi$
$728$	0	0
$729$	0	0
$730$	−9.74335	−0.360618
$731$	41.1308	1.52128
$732$	0	0
$733$	− 42.4329i	− 1.56729i	−0.621206	−	0.783647i	$-0.713357\pi$
$733$	− 42.4329i	− 1.56729i	0.621206	−	0.783647i	$-0.286643\pi$
$734$	−7.42133	−0.273926
$735$	0	0
$736$	−4.69764	−0.173157
$737$	− 19.7874i	− 0.728879i
$738$	0	0
$739$	−25.3081	−0.930972	−0.465486	−	0.885055i	$-0.654120\pi$
$739$	−25.3081	−0.930972	−0.465486	+	0.885055i	$0.654120\pi$
$740$	−10.5689	−0.388519
$741$	0	0
$742$	0	0
$743$	33.1867i	1.21750i	0.793362	+	0.608751i	$0.208329\pi$
$743$	33.1867i	1.21750i	−0.793362	+	0.608751i	$0.791671\pi$
$744$	0	0
$745$	9.79081i	0.358708i
$746$	− 6.97671i	− 0.255435i
$747$	0	0
$748$	12.6748i	0.463437i
$749$	0	0
$750$	0	0
$751$	12.1858	0.444666	0.222333	−	0.974971i	$-0.428633\pi$
$751$	12.1858	0.444666	0.222333	+	0.974971i	$0.428633\pi$
$752$	−0.412881	−0.0150562
$753$	0	0
$754$	10.8750i	0.396044i
$755$	7.02787	0.255770
$756$	0	0
$757$	−9.64439	−0.350531	−0.175266	−	0.984521i	$-0.556078\pi$
$757$	−9.64439	−0.350531	−0.175266	+	0.984521i	$0.556078\pi$
$758$	7.68980i	0.279306i
$759$	0	0
$760$	−6.84736	−0.248380
$761$	−15.8333	−0.573955	−0.286978	−	0.957937i	$-0.592651\pi$
$761$	−15.8333	−0.573955	−0.286978	+	0.957937i	$0.592651\pi$
$762$	0	0
$763$	0	0
$764$	3.07927i	0.111404i
$765$	0	0
$766$	7.15314i	0.258454i
$767$	52.5275i	1.89666i
$768$	0	0
$769$	− 32.1428i	− 1.15910i	−0.814938	−	0.579549i	$-0.803229\pi$
$769$	− 32.1428i	− 1.15910i	0.814938	−	0.579549i	$-0.196771\pi$
$770$	0	0
$771$	0	0
$772$	18.6569	0.671475
$773$	−11.4606	−0.412211	−0.206105	−	0.978530i	$-0.566079\pi$
$773$	−11.4606	−0.412211	−0.206105	+	0.978530i	$0.566079\pi$
$774$	0	0
$775$	− 8.86249i	− 0.318350i
$776$	11.6415	0.417905
$777$	0	0
$778$	−6.21550	−0.222836
$779$	22.2208i	0.796142i
$780$	0	0
$781$	−16.4297	−0.587900
$782$	17.9038	0.640237
$783$	0	0
$784$	0	0
$785$	− 18.8960i	− 0.674428i
$786$	0	0
$787$	− 54.3428i	− 1.93711i	−0.248794	−	0.968557i	$-0.580034\pi$
$787$	− 54.3428i	− 1.93711i	0.248794	−	0.968557i	$-0.419966\pi$
$788$	23.6108i	0.841099i
$789$	0	0
$790$	− 21.7636i	− 0.774313i
$791$	0	0
$792$	0	0
$793$	60.4701	2.14736
$794$	−15.8118	−0.561141
$795$	0	0
$796$	15.4496i	0.547595i
$797$	−38.3049	−1.35683	−0.678415	−	0.734679i	$-0.737333\pi$
$797$	−38.3049	−1.35683	−0.678415	+	0.734679i	$0.737333\pi$
$798$	0	0
$799$	1.57358	0.0556694
$800$	− 2.16373i	− 0.0764994i
$801$	0	0
$802$	13.7471	0.485427
$803$	−19.2403	−0.678975
$804$	0	0
$805$	0	0
$806$	− 17.7975i	− 0.626892i
$807$	0	0
$808$	6.89588i	0.242596i
$809$	− 31.0615i	− 1.09207i	−0.837764	−	0.546033i	$-0.816137\pi$
$809$	− 31.0615i	− 1.09207i	0.837764	−	0.546033i	$-0.183863\pi$
$810$	0	0
$811$	− 40.8494i	− 1.43442i	−0.696860	−	0.717208i	$-0.745420\pi$
$811$	− 40.8494i	− 1.43442i	0.696860	−	0.717208i	$-0.254580\pi$
$812$	0	0
$813$	0	0
$814$	−20.8704	−0.731508
$815$	20.3680	0.713458
$816$	0	0
$817$	43.8785i	1.53511i
$818$	15.3972	0.538350
$819$	0	0
$820$	9.20415	0.321423
$821$	− 20.2714i	− 0.707476i	−0.935345	−	0.353738i	$-0.884910\pi$
$821$	− 20.2714i	− 0.707476i	0.935345	−	0.353738i	$-0.115090\pi$
$822$	0	0
$823$	−2.80986	−0.0979454	−0.0489727	−	0.998800i	$-0.515595\pi$
$823$	−2.80986	−0.0979454	−0.0489727	+	0.998800i	$0.515595\pi$
$824$	4.65462	0.162151
$825$	0	0
$826$	0	0
$827$	24.7733i	0.861453i	0.902483	+	0.430726i	$0.141743\pi$
$827$	24.7733i	0.861453i	−0.902483	+	0.430726i	$0.858257\pi$
$828$	0	0
$829$	52.8045i	1.83398i	0.398914	+	0.916988i	$0.369387\pi$
$829$	52.8045i	1.83398i	−0.398914	+	0.916988i	$0.630613\pi$
$830$	10.1875i	0.353614i
$831$	0	0
$832$	− 4.34518i	− 0.150642i
$833$	0	0
$834$	0	0
$835$	13.0764	0.452526
$836$	−13.5216	−0.467653
$837$	0	0
$838$	− 3.80384i	− 0.131402i
$839$	1.13410	0.0391536	0.0195768	−	0.999808i	$-0.493768\pi$
$839$	1.13410	0.0391536	0.0195768	+	0.999808i	$0.493768\pi$
$840$	0	0
$841$	22.7361	0.784004
$842$	20.2583i	0.698149i
$843$	0	0
$844$	13.7399	0.472946
$845$	−9.90359	−0.340694
$846$	0	0
$847$	0	0
$848$	9.09436i	0.312302i
$849$	0	0
$850$	8.24647i	0.282851i
$851$	29.4804i	1.01058i
$852$	0	0
$853$	4.16705i	0.142677i	0.997452	+	0.0713385i	$0.0227271\pi$
$853$	4.16705i	0.142677i	−0.997452	+	0.0713385i	$0.977273\pi$
$854$	0	0
$855$	0	0
$856$	13.1098	0.448084
$857$	−19.5114	−0.666498	−0.333249	−	0.942839i	$-0.608145\pi$
$857$	−19.5114	−0.666498	−0.333249	+	0.942839i	$0.608145\pi$
$858$	0	0
$859$	31.6734i	1.08068i	0.841445	+	0.540342i	$0.181705\pi$
$859$	31.6734i	1.08068i	−0.841445	+	0.540342i	$0.818295\pi$
$860$	18.1751	0.619764
$861$	0	0
$862$	−11.3141	−0.385359
$863$	− 26.6082i	− 0.905755i	−0.891573	−	0.452877i	$-0.850398\pi$
$863$	− 26.6082i	− 0.905755i	0.891573	−	0.452877i	$-0.149602\pi$
$864$	0	0
$865$	12.2616	0.416906
$866$	18.3925	0.625001
$867$	0	0
$868$	0	0
$869$	− 42.9767i	− 1.45789i
$870$	0	0
$871$	− 25.8535i	− 0.876012i
$872$	17.7553i	0.601271i
$873$	0	0
$874$	19.0998i	0.646061i
$875$	0	0
$876$	0	0
$877$	−16.5348	−0.558339	−0.279169	−	0.960242i	$-0.590059\pi$
$877$	−16.5348	−0.558339	−0.279169	+	0.960242i	$0.590059\pi$
$878$	−3.97200	−0.134049
$879$	0	0
$880$	5.60081i	0.188803i
$881$	−37.3162	−1.25721	−0.628607	−	0.777723i	$-0.716375\pi$
$881$	−37.3162	−1.25721	−0.628607	+	0.777723i	$0.716375\pi$
$882$	0	0
$883$	−15.2822	−0.514287	−0.257144	−	0.966373i	$-0.582781\pi$
$883$	−15.2822	−0.514287	−0.257144	+	0.966373i	$0.582781\pi$
$884$	16.5605i	0.556988i
$885$	0	0
$886$	24.5779	0.825712
$887$	−52.7708	−1.77187	−0.885936	−	0.463808i	$-0.846483\pi$
$887$	−52.7708	−1.77187	−0.885936	+	0.463808i	$0.846483\pi$
$888$	0	0
$889$	0	0
$890$	10.6336i	0.356438i
$891$	0	0
$892$	4.65684i	0.155923i
$893$	1.67871i	0.0561757i
$894$	0	0
$895$	24.9528i	0.834079i
$896$	0	0
$897$	0	0
$898$	19.7867	0.660291
$899$	10.2512	0.341896
$900$	0	0
$901$	− 34.6607i	− 1.15471i
$902$	18.1755	0.605179
$903$	0	0
$904$	18.3234	0.609426
$905$	− 20.1550i	− 0.669973i
$906$	0	0
$907$	5.12346	0.170122	0.0850608	−	0.996376i	$-0.472892\pi$
$907$	5.12346	0.170122	0.0850608	+	0.996376i	$0.472892\pi$
$908$	27.6540	0.917729
$909$	0	0
$910$	0	0
$911$	− 56.0918i	− 1.85841i	−0.369571	−	0.929203i	$-0.620495\pi$
$911$	− 56.0918i	− 1.85841i	0.369571	−	0.929203i	$-0.379505\pi$
$912$	0	0
$913$	20.1174i	0.665788i
$914$	3.24264i	0.107257i
$915$	0	0
$916$	12.6394i	0.417616i
$917$	0	0
$918$	0	0
$919$	−18.3448	−0.605141	−0.302570	−	0.953127i	$-0.597845\pi$
$919$	−18.3448	−0.605141	−0.302570	+	0.953127i	$0.597845\pi$
$920$	7.91140	0.260831
$921$	0	0
$922$	− 12.9170i	− 0.425397i
$923$	−21.4664	−0.706575
$924$	0	0
$925$	−13.5787	−0.446464
$926$	25.9016i	0.851178i
$927$	0	0
$928$	2.50277	0.0821575
$929$	7.51183	0.246455	0.123228	−	0.992378i	$-0.460676\pi$
$929$	7.51183	0.246455	0.123228	+	0.992378i	$0.460676\pi$
$930$	0	0
$931$	0	0
$932$	4.67061i	0.152991i
$933$	0	0
$934$	7.30432i	0.239005i
$935$	− 21.3460i	− 0.698087i
$936$	0	0
$937$	− 17.8522i	− 0.583207i	−0.956539	−	0.291603i	$-0.905811\pi$
$937$	− 17.8522i	− 0.583207i	0.956539	−	0.291603i	$-0.0941887\pi$
$938$	0	0
$939$	0	0
$940$	0.695343	0.0226796
$941$	47.7965	1.55812	0.779061	−	0.626949i	$-0.215696\pi$
$941$	47.7965	1.55812	0.779061	+	0.626949i	$0.215696\pi$
$942$	0	0
$943$	− 25.6738i	− 0.836053i
$944$	12.0887	0.393454
$945$	0	0
$946$	35.8905	1.16690
$947$	29.3751i	0.954563i	0.878750	+	0.477282i	$0.158378\pi$
$947$	29.3751i	0.954563i	−0.878750	+	0.477282i	$0.841622\pi$
$948$	0	0
$949$	−25.1387	−0.816035
$950$	−8.79736	−0.285424
$951$	0	0
$952$	0	0
$953$	− 7.66147i	− 0.248179i	−0.992271	−	0.124090i	$-0.960399\pi$
$953$	− 7.66147i	− 0.248179i	0.992271	−	0.124090i	$-0.0396011\pi$
$954$	0	0
$955$	− 5.18588i	− 0.167811i
$956$	− 23.7883i	− 0.769367i
$957$	0	0
$958$	15.0285i	0.485547i
$959$	0	0
$960$	0	0
$961$	14.2233	0.458818
$962$	−27.2685	−0.879172
$963$	0	0
$964$	− 2.77410i	− 0.0893476i
$965$	−31.4204	−1.01146
$966$	0	0
$967$	−9.64987	−0.310319	−0.155159	−	0.987889i	$-0.549589\pi$
$967$	−9.64987	−0.310319	−0.155159	+	0.987889i	$0.549589\pi$
$968$	0.0599765i	0.00192772i
$969$	0	0
$970$	−19.6057	−0.629500
$971$	−53.0563	−1.70266	−0.851328	−	0.524633i	$-0.824202\pi$
$971$	−53.0563	−1.70266	−0.851328	+	0.524633i	$0.824202\pi$
$972$	0	0
$973$	0	0
$974$	8.14335i	0.260930i
$975$	0	0
$976$	− 13.9166i	− 0.445460i
$977$	− 38.1550i	− 1.22069i	−0.792138	−	0.610343i	$-0.791032\pi$
$977$	− 38.1550i	− 1.22069i	0.792138	−	0.610343i	$-0.208968\pi$
$978$	0	0
$979$	20.9982i	0.671105i
$980$	0	0
$981$	0	0
$982$	−26.6719	−0.851136
$983$	6.95322	0.221773	0.110887	−	0.993833i	$-0.464631\pi$
$983$	6.95322	0.221773	0.110887	+	0.993833i	$0.464631\pi$
$984$	0	0
$985$	− 39.7635i	− 1.26697i
$986$	−9.53863	−0.303772
$987$	0	0
$988$	−17.6668	−0.562055
$989$	− 50.6969i	− 1.61207i
$990$	0	0
$991$	−38.9327	−1.23674	−0.618369	−	0.785888i	$-0.712206\pi$
$991$	−38.9327	−1.23674	−0.618369	+	0.785888i	$0.712206\pi$
$992$	−4.09593	−0.130046
$993$	0	0
$994$	0	0
$995$	− 26.0190i	− 0.824856i
$996$	0	0
$997$	8.66626i	0.274463i	0.990539	+	0.137232i	$0.0438204\pi$
$997$	8.66626i	0.274463i	−0.990539	+	0.137232i	$0.956180\pi$
$998$	− 31.5456i	− 0.998559i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.d.e.2645.14	yes	16
3.2	odd	2	inner	2646.2.d.e.2645.3	✓	16
7.6	odd	2	inner	2646.2.d.e.2645.11	yes	16
21.20	even	2	inner	2646.2.d.e.2645.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2646.2.d.e.2645.3	✓	16	3.2	odd	2	inner
2646.2.d.e.2645.6	yes	16	21.20	even	2	inner
2646.2.d.e.2645.11	yes	16	7.6	odd	2	inner
2646.2.d.e.2645.14	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.68412 q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.68412 q^{5} -1.00000i q^{8} +1.68412i q^{10} +3.32565i q^{11} +4.34518i q^{13} +1.00000 q^{16} -3.81123 q^{17} -4.06583i q^{19} -1.68412 q^{20} -3.32565 q^{22} +4.69764i q^{23} -2.16373 q^{25} -4.34518 q^{26} -2.50277i q^{29} +4.09593i q^{31} +1.00000i q^{32} -3.81123i q^{34} +6.27558 q^{37} +4.06583 q^{38} -1.68412i q^{40} -5.46525 q^{41} -10.7920 q^{43} -3.32565i q^{44} -4.69764 q^{46} -0.412881 q^{47} -2.16373i q^{50} -4.34518i q^{52} +9.09436i q^{53} +5.60081i q^{55} +2.50277 q^{58} +12.0887 q^{59} -13.9166i q^{61} -4.09593 q^{62} -1.00000 q^{64} +7.31781i q^{65} -5.94993 q^{67} +3.81123 q^{68} +4.94028i q^{71} +5.78542i q^{73} +6.27558i q^{74} +4.06583i q^{76} -12.9228 q^{79} +1.68412 q^{80} -5.46525i q^{82} +6.04915 q^{83} -6.41857 q^{85} -10.7920i q^{86} +3.32565 q^{88} +6.31400 q^{89} -4.69764i q^{92} -0.412881i q^{94} -6.84736i q^{95} +11.6415i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 16 q^{16} - 32 q^{22} + 48 q^{25} - 32 q^{37} + 16 q^{43} - 48 q^{46} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 32 q^{88}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^16 - 32 * q^22 + 48 * q^25 - 32 * q^37 + 16 * q^43 - 48 * q^46 + 16 * q^58 - 16 * q^64 + 16 * q^67 + 32 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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