LMFDB - Embedded newform 2646.2.d.f.2645.7

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			2645.7
Root			$0.608761 + 0.793353i$ of defining polynomial
Character	$\chi$	$=$	2646.2645
Dual form			2646.2.d.f.2645.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +2.49742 q^{5} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +2.49742 q^{5} +1.00000i q^{8} -2.49742i q^{10} -0.874758i q^{11} +2.76652i q^{13} +1.00000 q^{16} -2.44949 q^{17} -6.86245i q^{19} -2.49742 q^{20} -0.874758 q^{22} +4.78620i q^{23} +1.23710 q^{25} +2.76652 q^{26} -9.26593i q^{29} -8.64243i q^{31} -1.00000i q^{32} +2.44949i q^{34} +4.48384 q^{37} -6.86245 q^{38} +2.49742i q^{40} -0.0376809 q^{41} +5.35449 q^{43} +0.874758i q^{44} +4.78620 q^{46} +4.66756 q^{47} -1.23710i q^{50} -2.76652i q^{52} -10.2756i q^{53} -2.18464i q^{55} -9.26593 q^{58} +8.56928 q^{59} +0.327272i q^{61} -8.64243 q^{62} -1.00000 q^{64} +6.90914i q^{65} +5.72648 q^{67} +2.44949 q^{68} -6.54936i q^{71} +8.24201i q^{73} -4.48384i q^{74} +6.86245i q^{76} +16.5837 q^{79} +2.49742 q^{80} +0.0376809i q^{82} -11.0117 q^{83} -6.11740 q^{85} -5.35449i q^{86} +0.874758 q^{88} +8.82389 q^{89} -4.78620i q^{92} -4.66756i q^{94} -17.1384i q^{95} +8.33787i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 16 q^{16} + 16 q^{22} + 16 q^{37} - 32 q^{43} + 48 q^{46} - 32 q^{58} - 16 q^{64} - 32 q^{67} - 16 q^{88}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^16 + 16 * q^22 + 16 * q^37 - 32 * q^43 + 48 * q^46 - 32 * q^58 - 16 * q^64 - 32 * q^67 - 16 * 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
$5$	2.49742	1.11688	0.558440	−	0.829545i	$-0.311400\pi$
$5$	2.49742	1.11688	0.558440	+	0.829545i	$0.311400\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000i	0.353553i
$9$	0	0
$10$	− 2.49742i	− 0.789753i
$11$	− 0.874758i	− 0.263750i	−0.991266	−	0.131875i	$-0.957900\pi$
$11$	− 0.874758i	− 0.263750i	0.991266	−	0.131875i	$-0.0420997\pi$
$12$	0	0
$13$	2.76652i	0.767293i	0.923480	+	0.383647i	$0.125332\pi$
$13$	2.76652i	0.767293i	−0.923480	+	0.383647i	$0.874668\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.44949	−0.594089	−0.297044	−	0.954864i	$-0.596001\pi$
$17$	−2.44949	−0.594089	−0.297044	+	0.954864i	$0.596001\pi$
$18$	0	0
$19$	− 6.86245i	− 1.57435i	−0.616728	−	0.787177i	$-0.711542\pi$
$19$	− 6.86245i	− 1.57435i	0.616728	−	0.787177i	$-0.288458\pi$
$20$	−2.49742	−0.558440
$21$	0	0
$22$	−0.874758	−0.186499
$23$	4.78620i	0.997991i	0.866604	+	0.498996i	$0.166298\pi$
$23$	4.78620i	0.997991i	−0.866604	+	0.498996i	$0.833702\pi$
$24$	0	0
$25$	1.23710	0.247419
$26$	2.76652	0.542558
$27$	0	0
$28$	0	0
$29$	− 9.26593i	− 1.72064i	−0.509754	−	0.860320i	$-0.670263\pi$
$29$	− 9.26593i	− 1.72064i	0.509754	−	0.860320i	$-0.329737\pi$
$30$	0	0
$31$	− 8.64243i	− 1.55223i	−0.630594	−	0.776113i	$-0.717189\pi$
$31$	− 8.64243i	− 1.55223i	0.630594	−	0.776113i	$-0.282811\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	2.44949i	0.420084i
$35$	0	0
$36$	0	0
$37$	4.48384	0.737139	0.368569	−	0.929600i	$-0.379848\pi$
$37$	4.48384	0.737139	0.368569	+	0.929600i	$0.379848\pi$
$38$	−6.86245	−1.11324
$39$	0	0
$40$	2.49742i	0.394876i
$41$	−0.0376809	−0.00588477	−0.00294238	−	0.999996i	$-0.500937\pi$
$41$	−0.0376809	−0.00588477	−0.00294238	+	0.999996i	$0.500937\pi$
$42$	0	0
$43$	5.35449	0.816553	0.408276	−	0.912858i	$-0.366130\pi$
$43$	5.35449	0.816553	0.408276	+	0.912858i	$0.366130\pi$
$44$	0.874758i	0.131875i
$45$	0	0
$46$	4.78620	0.705687
$47$	4.66756	0.680834	0.340417	−	0.940275i	$-0.389432\pi$
$47$	4.66756	0.680834	0.340417	+	0.940275i	$0.389432\pi$
$48$	0	0
$49$	0	0
$50$	− 1.23710i	− 0.174952i
$51$	0	0
$52$	− 2.76652i	− 0.383647i
$53$	− 10.2756i	− 1.41146i	−0.708481	−	0.705730i	$-0.750619\pi$
$53$	− 10.2756i	− 1.41146i	0.708481	−	0.705730i	$-0.249381\pi$
$54$	0	0
$55$	− 2.18464i	− 0.294576i
$56$	0	0
$57$	0	0
$58$	−9.26593	−1.21668
$59$	8.56928	1.11563	0.557813	−	0.829967i	$-0.311641\pi$
$59$	8.56928	1.11563	0.557813	+	0.829967i	$0.311641\pi$
$60$	0	0
$61$	0.327272i	0.0419029i	0.999780	+	0.0209515i	$0.00666955\pi$
$61$	0.327272i	0.0419029i	−0.999780	+	0.0209515i	$0.993330\pi$
$62$	−8.64243	−1.09759
$63$	0	0
$64$	−1.00000	−0.125000
$65$	6.90914i	0.856974i
$66$	0	0
$67$	5.72648	0.699601	0.349800	−	0.936824i	$-0.386249\pi$
$67$	5.72648	0.699601	0.349800	+	0.936824i	$0.386249\pi$
$68$	2.44949	0.297044
$69$	0	0
$70$	0	0
$71$	− 6.54936i	− 0.777266i	−0.921393	−	0.388633i	$-0.872947\pi$
$71$	− 6.54936i	− 0.777266i	0.921393	−	0.388633i	$-0.127053\pi$
$72$	0	0
$73$	8.24201i	0.964655i	0.875991	+	0.482327i	$0.160208\pi$
$73$	8.24201i	0.964655i	−0.875991	+	0.482327i	$0.839792\pi$
$74$	− 4.48384i	− 0.521236i
$75$	0	0
$76$	6.86245i	0.787177i
$77$	0	0
$78$	0	0
$79$	16.5837	1.86582	0.932909	−	0.360113i	$-0.117262\pi$
$79$	16.5837	1.86582	0.932909	+	0.360113i	$0.117262\pi$
$80$	2.49742	0.279220
$81$	0	0
$82$	0.0376809i	0.00416116i
$83$	−11.0117	−1.20869	−0.604344	−	0.796724i	$-0.706565\pi$
$83$	−11.0117	−1.20869	−0.604344	+	0.796724i	$0.706565\pi$
$84$	0	0
$85$	−6.11740	−0.663525
$86$	− 5.35449i	− 0.577390i
$87$	0	0
$88$	0.874758	0.0932495
$89$	8.82389	0.935330	0.467665	−	0.883906i	$-0.345095\pi$
$89$	8.82389	0.935330	0.467665	+	0.883906i	$0.345095\pi$
$90$	0	0
$91$	0	0
$92$	− 4.78620i	− 0.498996i
$93$	0	0
$94$	− 4.66756i	− 0.481422i
$95$	− 17.1384i	− 1.75836i
$96$	0	0
$97$	8.33787i	0.846582i	0.905994	+	0.423291i	$0.139125\pi$
$97$	8.33787i	0.846582i	−0.905994	+	0.423291i	$0.860875\pi$
$98$	0	0
$99$	0	0
$100$	−1.23710	−0.123710
$101$	−18.6444	−1.85518	−0.927592	−	0.373595i	$-0.878125\pi$
$101$	−18.6444	−1.85518	−0.927592	+	0.373595i	$0.878125\pi$
$102$	0	0
$103$	9.44011i	0.930162i	0.885268	+	0.465081i	$0.153975\pi$
$103$	9.44011i	0.930162i	−0.885268	+	0.465081i	$0.846025\pi$
$104$	−2.76652	−0.271279
$105$	0	0
$106$	−10.2756	−0.998053
$107$	− 10.5508i	− 1.01998i	−0.860179	−	0.509992i	$-0.829648\pi$
$107$	− 10.5508i	− 1.01998i	0.860179	−	0.509992i	$-0.170352\pi$
$108$	0	0
$109$	12.7591	1.22210	0.611048	−	0.791593i	$-0.290748\pi$
$109$	12.7591	1.22210	0.611048	+	0.791593i	$0.290748\pi$
$110$	−2.18464	−0.208297
$111$	0	0
$112$	0	0
$113$	− 18.3102i	− 1.72248i	−0.508197	−	0.861241i	$-0.669688\pi$
$113$	− 18.3102i	− 1.72248i	0.508197	−	0.861241i	$-0.330312\pi$
$114$	0	0
$115$	11.9531i	1.11464i
$116$	9.26593i	0.860320i
$117$	0	0
$118$	− 8.56928i	− 0.788867i
$119$	0	0
$120$	0	0
$121$	10.2348	0.930436
$122$	0.327272	0.0296298
$123$	0	0
$124$	8.64243i	0.776113i
$125$	−9.39755	−0.840542
$126$	0	0
$127$	6.53017	0.579459	0.289729	−	0.957109i	$-0.406435\pi$
$127$	6.53017	0.579459	0.289729	+	0.957109i	$0.406435\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	6.90914	0.605972
$131$	17.4706	1.52641	0.763205	−	0.646156i	$-0.223625\pi$
$131$	17.4706	1.52641	0.763205	+	0.646156i	$0.223625\pi$
$132$	0	0
$133$	0	0
$134$	− 5.72648i	− 0.494692i
$135$	0	0
$136$	− 2.44949i	− 0.210042i
$137$	19.4678i	1.66325i	0.555341	+	0.831623i	$0.312588\pi$
$137$	19.4678i	1.66325i	−0.555341	+	0.831623i	$0.687412\pi$
$138$	0	0
$139$	− 8.47657i	− 0.718973i	−0.933150	−	0.359487i	$-0.882952\pi$
$139$	− 8.47657i	− 0.718973i	0.933150	−	0.359487i	$-0.117048\pi$
$140$	0	0
$141$	0	0
$142$	−6.54936	−0.549610
$143$	2.42003	0.202373
$144$	0	0
$145$	− 23.1409i	− 1.92175i
$146$	8.24201	0.682114
$147$	0	0
$148$	−4.48384	−0.368569
$149$	− 6.16773i	− 0.505280i	−0.967560	−	0.252640i	$-0.918701\pi$
$149$	− 6.16773i	− 0.505280i	0.967560	−	0.252640i	$-0.0812988\pi$
$150$	0	0
$151$	−3.91144	−0.318309	−0.159154	−	0.987254i	$-0.550877\pi$
$151$	−3.91144	−0.318309	−0.159154	+	0.987254i	$0.550877\pi$
$152$	6.86245	0.556618
$153$	0	0
$154$	0	0
$155$	− 21.5837i	− 1.73365i
$156$	0	0
$157$	12.4746i	0.995583i	0.867297	+	0.497792i	$0.165856\pi$
$157$	12.4746i	0.995583i	−0.867297	+	0.497792i	$0.834144\pi$
$158$	− 16.5837i	− 1.31933i
$159$	0	0
$160$	− 2.49742i	− 0.197438i
$161$	0	0
$162$	0	0
$163$	−7.67989	−0.601536	−0.300768	−	0.953697i	$-0.597243\pi$
$163$	−7.67989	−0.601536	−0.300768	+	0.953697i	$0.597243\pi$
$164$	0.0376809	0.00294238
$165$	0	0
$166$	11.0117i	0.854671i
$167$	−10.6813	−0.826540	−0.413270	−	0.910608i	$-0.635613\pi$
$167$	−10.6813	−0.826540	−0.413270	+	0.910608i	$0.635613\pi$
$168$	0	0
$169$	5.34639	0.411261
$170$	6.11740i	0.469183i
$171$	0	0
$172$	−5.35449	−0.408276
$173$	−18.0772	−1.37438	−0.687191	−	0.726477i	$-0.741157\pi$
$173$	−18.0772	−1.37438	−0.687191	+	0.726477i	$0.741157\pi$
$174$	0	0
$175$	0	0
$176$	− 0.874758i	− 0.0659374i
$177$	0	0
$178$	− 8.82389i	− 0.661379i
$179$	− 21.1177i	− 1.57841i	−0.614131	−	0.789204i	$-0.710493\pi$
$179$	− 21.1177i	− 1.57841i	0.614131	−	0.789204i	$-0.289507\pi$
$180$	0	0
$181$	− 1.11471i	− 0.0828559i	−0.999141	−	0.0414280i	$-0.986809\pi$
$181$	− 1.11471i	− 0.0828559i	0.999141	−	0.0414280i	$-0.0131907\pi$
$182$	0	0
$183$	0	0
$184$	−4.78620	−0.352843
$185$	11.1980	0.823295
$186$	0	0
$187$	2.14271i	0.156691i
$188$	−4.66756	−0.340417
$189$	0	0
$190$	−17.1384	−1.24335
$191$	− 2.30092i	− 0.166489i	−0.996529	−	0.0832443i	$-0.973472\pi$
$191$	− 2.30092i	− 0.166489i	0.996529	−	0.0832443i	$-0.0265282\pi$
$192$	0	0
$193$	−27.1020	−1.95084	−0.975421	−	0.220348i	$-0.929281\pi$
$193$	−27.1020	−1.95084	−0.975421	+	0.220348i	$0.929281\pi$
$194$	8.33787	0.598624
$195$	0	0
$196$	0	0
$197$	3.92253i	0.279469i	0.990189	+	0.139734i	$0.0446249\pi$
$197$	3.92253i	0.279469i	−0.990189	+	0.139734i	$0.955375\pi$
$198$	0	0
$199$	11.6125i	0.823187i	0.911368	+	0.411593i	$0.135028\pi$
$199$	11.6125i	0.823187i	−0.911368	+	0.411593i	$0.864972\pi$
$200$	1.23710i	0.0874758i
$201$	0	0
$202$	18.6444i	1.31181i
$203$	0	0
$204$	0	0
$205$	−0.0941049	−0.00657257
$206$	9.44011	0.657724
$207$	0	0
$208$	2.76652i	0.191823i
$209$	−6.00298	−0.415235
$210$	0	0
$211$	8.17918	0.563078	0.281539	−	0.959550i	$-0.409155\pi$
$211$	8.17918	0.563078	0.281539	+	0.959550i	$0.409155\pi$
$212$	10.2756i	0.705730i
$213$	0	0
$214$	−10.5508	−0.721238
$215$	13.3724	0.911991
$216$	0	0
$217$	0	0
$218$	− 12.7591i	− 0.864153i
$219$	0	0
$220$	2.18464i	0.147288i
$221$	− 6.77655i	− 0.455840i
$222$	0	0
$223$	− 5.68754i	− 0.380866i	−0.981700	−	0.190433i	$-0.939011\pi$
$223$	− 5.68754i	− 0.380866i	0.981700	−	0.190433i	$-0.0609891\pi$
$224$	0	0
$225$	0	0
$226$	−18.3102	−1.21798
$227$	−13.4284	−0.891272	−0.445636	−	0.895214i	$-0.647022\pi$
$227$	−13.4284	−0.891272	−0.445636	+	0.895214i	$0.647022\pi$
$228$	0	0
$229$	− 14.0172i	− 0.926281i	−0.886285	−	0.463141i	$-0.846722\pi$
$229$	− 14.0172i	− 0.926281i	0.886285	−	0.463141i	$-0.153278\pi$
$230$	11.9531	0.788167
$231$	0	0
$232$	9.26593	0.608338
$233$	1.84592i	0.120930i	0.998170	+	0.0604651i	$0.0192584\pi$
$233$	1.84592i	0.120930i	−0.998170	+	0.0604651i	$0.980742\pi$
$234$	0	0
$235$	11.6569	0.760409
$236$	−8.56928	−0.557813
$237$	0	0
$238$	0	0
$239$	30.0091i	1.94112i	0.240851	+	0.970562i	$0.422573\pi$
$239$	30.0091i	1.94112i	−0.240851	+	0.970562i	$0.577427\pi$
$240$	0	0
$241$	− 11.5097i	− 0.741404i	−0.928752	−	0.370702i	$-0.879117\pi$
$241$	− 11.5097i	− 0.741404i	0.928752	−	0.370702i	$-0.120883\pi$
$242$	− 10.2348i	− 0.657918i
$243$	0	0
$244$	− 0.327272i	− 0.0209515i
$245$	0	0
$246$	0	0
$247$	18.9851	1.20799
$248$	8.64243	0.548795
$249$	0	0
$250$	9.39755i	0.594353i
$251$	−2.17928	−0.137555	−0.0687773	−	0.997632i	$-0.521910\pi$
$251$	−2.17928	−0.137555	−0.0687773	+	0.997632i	$0.521910\pi$
$252$	0	0
$253$	4.18677	0.263220
$254$	− 6.53017i	− 0.409739i
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.19087	0.573311	0.286655	−	0.958034i	$-0.407457\pi$
$257$	9.19087	0.573311	0.286655	+	0.958034i	$0.407457\pi$
$258$	0	0
$259$	0	0
$260$	− 6.90914i	− 0.428487i
$261$	0	0
$262$	− 17.4706i	− 1.07933i
$263$	14.0670i	0.867406i	0.901056	+	0.433703i	$0.142793\pi$
$263$	14.0670i	0.867406i	−0.901056	+	0.433703i	$0.857207\pi$
$264$	0	0
$265$	− 25.6624i	− 1.57643i
$266$	0	0
$267$	0	0
$268$	−5.72648	−0.349800
$269$	−12.6583	−0.771791	−0.385895	−	0.922543i	$-0.626107\pi$
$269$	−12.6583	−0.771791	−0.385895	+	0.922543i	$0.626107\pi$
$270$	0	0
$271$	3.63846i	0.221021i	0.993875	+	0.110510i	$0.0352485\pi$
$271$	3.63846i	0.221021i	−0.993875	+	0.110510i	$0.964751\pi$
$272$	−2.44949	−0.148522
$273$	0	0
$274$	19.4678	1.17609
$275$	− 1.08216i	− 0.0652567i
$276$	0	0
$277$	−29.0021	−1.74257	−0.871283	−	0.490782i	$-0.836711\pi$
$277$	−29.0021	−1.74257	−0.871283	+	0.490782i	$0.836711\pi$
$278$	−8.47657	−0.508391
$279$	0	0
$280$	0	0
$281$	− 13.4331i	− 0.801354i	−0.916219	−	0.400677i	$-0.868775\pi$
$281$	− 13.4331i	− 0.801354i	0.916219	−	0.400677i	$-0.131225\pi$
$282$	0	0
$283$	− 1.84442i	− 0.109640i	−0.998496	−	0.0548198i	$-0.982542\pi$
$283$	− 1.84442i	− 0.109640i	0.998496	−	0.0548198i	$-0.0174584\pi$
$284$	6.54936i	0.388633i
$285$	0	0
$286$	− 2.42003i	− 0.143099i
$287$	0	0
$288$	0	0
$289$	−11.0000	−0.647059
$290$	−23.1409	−1.35888
$291$	0	0
$292$	− 8.24201i	− 0.482327i
$293$	−20.6424	−1.20594	−0.602970	−	0.797764i	$-0.706016\pi$
$293$	−20.6424	−1.20594	−0.602970	+	0.797764i	$0.706016\pi$
$294$	0	0
$295$	21.4011	1.24602
$296$	4.48384i	0.260618i
$297$	0	0
$298$	−6.16773	−0.357287
$299$	−13.2411	−0.765752
$300$	0	0
$301$	0	0
$302$	3.91144i	0.225078i
$303$	0	0
$304$	− 6.86245i	− 0.393588i
$305$	0.817336i	0.0468005i
$306$	0	0
$307$	− 16.3467i	− 0.932956i	−0.884533	−	0.466478i	$-0.845523\pi$
$307$	− 16.3467i	− 0.932956i	0.884533	−	0.466478i	$-0.154477\pi$
$308$	0	0
$309$	0	0
$310$	−21.5837	−1.22587
$311$	7.63584	0.432989	0.216494	−	0.976284i	$-0.430538\pi$
$311$	7.63584	0.432989	0.216494	+	0.976284i	$0.430538\pi$
$312$	0	0
$313$	− 7.97514i	− 0.450781i	−0.974268	−	0.225391i	$-0.927634\pi$
$313$	− 7.97514i	− 0.450781i	0.974268	−	0.225391i	$-0.0723658\pi$
$314$	12.4746	0.703984
$315$	0	0
$316$	−16.5837	−0.932909
$317$	22.4235i	1.25943i	0.776826	+	0.629715i	$0.216828\pi$
$317$	22.4235i	1.25943i	−0.776826	+	0.629715i	$0.783172\pi$
$318$	0	0
$319$	−8.10545	−0.453818
$320$	−2.49742	−0.139610
$321$	0	0
$322$	0	0
$323$	16.8095i	0.935305i
$324$	0	0
$325$	3.42244i	0.189843i
$326$	7.67989i	0.425350i
$327$	0	0
$328$	− 0.0376809i	− 0.00208058i
$329$	0	0
$330$	0	0
$331$	−12.2430	−0.672936	−0.336468	−	0.941695i	$-0.609232\pi$
$331$	−12.2430	−0.672936	−0.336468	+	0.941695i	$0.609232\pi$
$332$	11.0117	0.604344
$333$	0	0
$334$	10.6813i	0.584452i
$335$	14.3014	0.781370
$336$	0	0
$337$	−1.22334	−0.0666398	−0.0333199	−	0.999445i	$-0.510608\pi$
$337$	−1.22334	−0.0666398	−0.0333199	+	0.999445i	$0.510608\pi$
$338$	− 5.34639i	− 0.290806i
$339$	0	0
$340$	6.11740	0.331763
$341$	−7.56003	−0.409399
$342$	0	0
$343$	0	0
$344$	5.35449i	0.288695i
$345$	0	0
$346$	18.0772i	0.971835i
$347$	7.56481i	0.406100i	0.979168	+	0.203050i	$0.0650855\pi$
$347$	7.56481i	0.406100i	−0.979168	+	0.203050i	$0.934915\pi$
$348$	0	0
$349$	20.0057i	1.07088i	0.844574	+	0.535439i	$0.179854\pi$
$349$	20.0057i	1.07088i	−0.844574	+	0.535439i	$0.820146\pi$
$350$	0	0
$351$	0	0
$352$	−0.874758	−0.0466248
$353$	−32.5454	−1.73221	−0.866107	−	0.499858i	$-0.833386\pi$
$353$	−32.5454	−1.73221	−0.866107	+	0.499858i	$0.833386\pi$
$354$	0	0
$355$	− 16.3565i	− 0.868112i
$356$	−8.82389	−0.467665
$357$	0	0
$358$	−21.1177	−1.11610
$359$	− 9.74952i	− 0.514560i	−0.966337	−	0.257280i	$-0.917174\pi$
$359$	− 9.74952i	− 0.514560i	0.966337	−	0.257280i	$-0.0828262\pi$
$360$	0	0
$361$	−28.0932	−1.47859
$362$	−1.11471	−0.0585880
$363$	0	0
$364$	0	0
$365$	20.5837i	1.07740i
$366$	0	0
$367$	24.7519i	1.29204i	0.763321	+	0.646019i	$0.223567\pi$
$367$	24.7519i	1.29204i	−0.763321	+	0.646019i	$0.776433\pi$
$368$	4.78620i	0.249498i
$369$	0	0
$370$	− 11.1980i	− 0.582157i
$371$	0	0
$372$	0	0
$373$	37.3947	1.93622	0.968112	−	0.250519i	$-0.0806013\pi$
$373$	37.3947	1.93622	0.968112	+	0.250519i	$0.0806013\pi$
$374$	2.14271	0.110797
$375$	0	0
$376$	4.66756i	0.240711i
$377$	25.6343	1.32024
$378$	0	0
$379$	27.3603	1.40540	0.702702	−	0.711484i	$-0.251977\pi$
$379$	27.3603	1.40540	0.702702	+	0.711484i	$0.251977\pi$
$380$	17.1384i	0.879181i
$381$	0	0
$382$	−2.30092	−0.117725
$383$	15.1669	0.774991	0.387496	−	0.921872i	$-0.373340\pi$
$383$	15.1669	0.774991	0.387496	+	0.921872i	$0.373340\pi$
$384$	0	0
$385$	0	0
$386$	27.1020i	1.37945i
$387$	0	0
$388$	− 8.33787i	− 0.423291i
$389$	6.39502i	0.324241i	0.986771	+	0.162120i	$0.0518333\pi$
$389$	6.39502i	0.324241i	−0.986771	+	0.162120i	$0.948167\pi$
$390$	0	0
$391$	− 11.7237i	− 0.592895i
$392$	0	0
$393$	0	0
$394$	3.92253	0.197614
$395$	41.4165	2.08389
$396$	0	0
$397$	1.98618i	0.0996836i	0.998757	+	0.0498418i	$0.0158717\pi$
$397$	1.98618i	0.0996836i	−0.998757	+	0.0498418i	$0.984128\pi$
$398$	11.6125	0.582081
$399$	0	0
$400$	1.23710	0.0618548
$401$	− 27.1439i	− 1.35550i	−0.735291	−	0.677752i	$-0.762954\pi$
$401$	− 27.1439i	− 1.35550i	0.735291	−	0.677752i	$-0.237046\pi$
$402$	0	0
$403$	23.9094	1.19101
$404$	18.6444	0.927592
$405$	0	0
$406$	0	0
$407$	− 3.92228i	− 0.194420i
$408$	0	0
$409$	15.0349i	0.743430i	0.928347	+	0.371715i	$0.121230\pi$
$409$	15.0349i	0.743430i	−0.928347	+	0.371715i	$0.878770\pi$
$410$	0.0941049i	0.00464751i
$411$	0	0
$412$	− 9.44011i	− 0.465081i
$413$	0	0
$414$	0	0
$415$	−27.5007	−1.34996
$416$	2.76652	0.135640
$417$	0	0
$418$	6.00298i	0.293615i
$419$	24.9078	1.21682	0.608412	−	0.793621i	$-0.291807\pi$
$419$	24.9078	1.21682	0.608412	+	0.793621i	$0.291807\pi$
$420$	0	0
$421$	−20.9423	−1.02067	−0.510334	−	0.859976i	$-0.670478\pi$
$421$	−20.9423	−1.02067	−0.510334	+	0.859976i	$0.670478\pi$
$422$	− 8.17918i	− 0.398156i
$423$	0	0
$424$	10.2756	0.499026
$425$	−3.03025	−0.146989
$426$	0	0
$427$	0	0
$428$	10.5508i	0.509992i
$429$	0	0
$430$	− 13.3724i	− 0.644875i
$431$	16.6861i	0.803739i	0.915697	+	0.401870i	$0.131639\pi$
$431$	16.6861i	0.803739i	−0.915697	+	0.401870i	$0.868361\pi$
$432$	0	0
$433$	27.9188i	1.34169i	0.741596	+	0.670846i	$0.234069\pi$
$433$	27.9188i	1.34169i	−0.741596	+	0.670846i	$0.765931\pi$
$434$	0	0
$435$	0	0
$436$	−12.7591	−0.611048
$437$	32.8450	1.57119
$438$	0	0
$439$	26.1754i	1.24928i	0.780911	+	0.624642i	$0.214755\pi$
$439$	26.1754i	1.24928i	−0.780911	+	0.624642i	$0.785245\pi$
$440$	2.18464	0.104148
$441$	0	0
$442$	−6.77655	−0.322328
$443$	12.7660i	0.606533i	0.952906	+	0.303267i	$0.0980773\pi$
$443$	12.7660i	0.606533i	−0.952906	+	0.303267i	$0.901923\pi$
$444$	0	0
$445$	22.0369	1.04465
$446$	−5.68754	−0.269313
$447$	0	0
$448$	0	0
$449$	6.11621i	0.288642i	0.989531	+	0.144321i	$0.0460998\pi$
$449$	6.11621i	0.288642i	−0.989531	+	0.144321i	$0.953900\pi$
$450$	0	0
$451$	0.0329617i	0.00155210i
$452$	18.3102i	0.861241i
$453$	0	0
$454$	13.4284i	0.630224i
$455$	0	0
$456$	0	0
$457$	6.00591	0.280945	0.140472	−	0.990085i	$-0.455138\pi$
$457$	6.00591	0.280945	0.140472	+	0.990085i	$0.455138\pi$
$458$	−14.0172	−0.654980
$459$	0	0
$460$	− 11.9531i	− 0.557318i
$461$	24.1872	1.12651	0.563255	−	0.826283i	$-0.309549\pi$
$461$	24.1872	1.12651	0.563255	+	0.826283i	$0.309549\pi$
$462$	0	0
$463$	14.0676	0.653776	0.326888	−	0.945063i	$-0.394000\pi$
$463$	14.0676	0.653776	0.326888	+	0.945063i	$0.394000\pi$
$464$	− 9.26593i	− 0.430160i
$465$	0	0
$466$	1.84592	0.0855105
$467$	−19.9191	−0.921748	−0.460874	−	0.887466i	$-0.652464\pi$
$467$	−19.9191	−0.921748	−0.460874	+	0.887466i	$0.652464\pi$
$468$	0	0
$469$	0	0
$470$	− 11.6569i	− 0.537691i
$471$	0	0
$472$	8.56928i	0.394433i
$473$	− 4.68389i	− 0.215365i
$474$	0	0
$475$	− 8.48950i	− 0.389525i
$476$	0	0
$477$	0	0
$478$	30.0091	1.37258
$479$	−22.5817	−1.03179	−0.515893	−	0.856653i	$-0.672540\pi$
$479$	−22.5817	−1.03179	−0.515893	+	0.856653i	$0.672540\pi$
$480$	0	0
$481$	12.4046i	0.565601i
$482$	−11.5097	−0.524252
$483$	0	0
$484$	−10.2348	−0.465218
$485$	20.8231i	0.945530i
$486$	0	0
$487$	12.2814	0.556523	0.278261	−	0.960505i	$-0.410242\pi$
$487$	12.2814	0.556523	0.278261	+	0.960505i	$0.410242\pi$
$488$	−0.327272	−0.0148149
$489$	0	0
$490$	0	0
$491$	34.0390i	1.53616i	0.640355	+	0.768079i	$0.278787\pi$
$491$	34.0390i	1.53616i	−0.640355	+	0.768079i	$0.721213\pi$
$492$	0	0
$493$	22.6968i	1.02221i
$494$	− 18.9851i	− 0.854178i
$495$	0	0
$496$	− 8.64243i	− 0.388056i
$497$	0	0
$498$	0	0
$499$	30.8593	1.38145	0.690727	−	0.723116i	$-0.257291\pi$
$499$	30.8593	1.38145	0.690727	+	0.723116i	$0.257291\pi$
$500$	9.39755	0.420271
$501$	0	0
$502$	2.17928i	0.0972658i
$503$	12.9435	0.577120	0.288560	−	0.957462i	$-0.406823\pi$
$503$	12.9435	0.577120	0.288560	+	0.957462i	$0.406823\pi$
$504$	0	0
$505$	−46.5628	−2.07202
$506$	− 4.18677i	− 0.186125i
$507$	0	0
$508$	−6.53017	−0.289729
$509$	−40.0693	−1.77604	−0.888020	−	0.459804i	$-0.847920\pi$
$509$	−40.0693	−1.77604	−0.888020	+	0.459804i	$0.847920\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	− 9.19087i	− 0.405392i
$515$	23.5759i	1.03888i
$516$	0	0
$517$	− 4.08299i	− 0.179570i
$518$	0	0
$519$	0	0
$520$	−6.90914	−0.302986
$521$	11.7113	0.513080	0.256540	−	0.966534i	$-0.417417\pi$
$521$	11.7113	0.513080	0.256540	+	0.966534i	$0.417417\pi$
$522$	0	0
$523$	18.8990i	0.826394i	0.910642	+	0.413197i	$0.135588\pi$
$523$	18.8990i	0.826394i	−0.910642	+	0.413197i	$0.864412\pi$
$524$	−17.4706	−0.763205
$525$	0	0
$526$	14.0670	0.613349
$527$	21.1695i	0.922159i
$528$	0	0
$529$	0.0922990	0.00401300
$530$	−25.6624	−1.11470
$531$	0	0
$532$	0	0
$533$	− 0.104245i	− 0.00451534i
$534$	0	0
$535$	− 26.3498i	− 1.13920i
$536$	5.72648i	0.247346i
$537$	0	0
$538$	12.6583i	0.545738i
$539$	0	0
$540$	0	0
$541$	7.22526	0.310638	0.155319	−	0.987864i	$-0.450359\pi$
$541$	7.22526	0.310638	0.155319	+	0.987864i	$0.450359\pi$
$542$	3.63846	0.156285
$543$	0	0
$544$	2.44949i	0.105021i
$545$	31.8647	1.36493
$546$	0	0
$547$	−3.44883	−0.147461	−0.0737307	−	0.997278i	$-0.523491\pi$
$547$	−3.44883	−0.147461	−0.0737307	+	0.997278i	$0.523491\pi$
$548$	− 19.4678i	− 0.831623i
$549$	0	0
$550$	−1.08216	−0.0461434
$551$	−63.5870	−2.70890
$552$	0	0
$553$	0	0
$554$	29.0021i	1.23218i
$555$	0	0
$556$	8.47657i	0.359487i
$557$	21.0184i	0.890577i	0.895387	+	0.445288i	$0.146899\pi$
$557$	21.0184i	0.890577i	−0.895387	+	0.445288i	$0.853101\pi$
$558$	0	0
$559$	14.8133i	0.626535i
$560$	0	0
$561$	0	0
$562$	−13.4331	−0.566643
$563$	−20.6738	−0.871296	−0.435648	−	0.900117i	$-0.643481\pi$
$563$	−20.6738	−0.871296	−0.435648	+	0.900117i	$0.643481\pi$
$564$	0	0
$565$	− 45.7283i	− 1.92380i
$566$	−1.84442	−0.0775268
$567$	0	0
$568$	6.54936	0.274805
$569$	− 28.6838i	− 1.20249i	−0.799066	−	0.601243i	$-0.794672\pi$
$569$	− 28.6838i	− 1.20249i	0.799066	−	0.601243i	$-0.205328\pi$
$570$	0	0
$571$	−6.34448	−0.265508	−0.132754	−	0.991149i	$-0.542382\pi$
$571$	−6.34448	−0.265508	−0.132754	+	0.991149i	$0.542382\pi$
$572$	−2.42003	−0.101187
$573$	0	0
$574$	0	0
$575$	5.92098i	0.246922i
$576$	0	0
$577$	30.5427i	1.27151i	0.771892	+	0.635754i	$0.219311\pi$
$577$	30.5427i	1.27151i	−0.771892	+	0.635754i	$0.780689\pi$
$578$	11.0000i	0.457540i
$579$	0	0
$580$	23.1409i	0.960874i
$581$	0	0
$582$	0	0
$583$	−8.98865	−0.372272
$584$	−8.24201	−0.341057
$585$	0	0
$586$	20.6424i	0.852729i
$587$	−4.05825	−0.167502	−0.0837510	−	0.996487i	$-0.526690\pi$
$587$	−4.05825	−0.167502	−0.0837510	+	0.996487i	$0.526690\pi$
$588$	0	0
$589$	−59.3082	−2.44375
$590$	− 21.4011i	− 0.881069i
$591$	0	0
$592$	4.48384	0.184285
$593$	18.3043	0.751670	0.375835	−	0.926687i	$-0.377356\pi$
$593$	18.3043	0.751670	0.375835	+	0.926687i	$0.377356\pi$
$594$	0	0
$595$	0	0
$596$	6.16773i	0.252640i
$597$	0	0
$598$	13.2411i	0.541469i
$599$	− 10.4081i	− 0.425262i	−0.977133	−	0.212631i	$-0.931797\pi$
$599$	− 10.4081i	− 0.425262i	0.977133	−	0.212631i	$-0.0682033\pi$
$600$	0	0
$601$	− 4.91967i	− 0.200678i	−0.994953	−	0.100339i	$-0.968007\pi$
$601$	− 4.91967i	− 0.200678i	0.994953	−	0.100339i	$-0.0319927\pi$
$602$	0	0
$603$	0	0
$604$	3.91144	0.159154
$605$	25.5606	1.03918
$606$	0	0
$607$	− 7.32730i	− 0.297406i	−0.988882	−	0.148703i	$-0.952490\pi$
$607$	− 7.32730i	− 0.297406i	0.988882	−	0.148703i	$-0.0475099\pi$
$608$	−6.86245	−0.278309
$609$	0	0
$610$	0.817336	0.0330930
$611$	12.9129i	0.522399i
$612$	0	0
$613$	30.5935	1.23566	0.617831	−	0.786311i	$-0.288012\pi$
$613$	30.5935	1.23566	0.617831	+	0.786311i	$0.288012\pi$
$614$	−16.3467	−0.659700
$615$	0	0
$616$	0	0
$617$	30.6436i	1.23367i	0.787094	+	0.616833i	$0.211585\pi$
$617$	30.6436i	1.23367i	−0.787094	+	0.616833i	$0.788415\pi$
$618$	0	0
$619$	− 28.9214i	− 1.16245i	−0.813743	−	0.581225i	$-0.802573\pi$
$619$	− 28.9214i	− 1.16245i	0.813743	−	0.581225i	$-0.197427\pi$
$620$	21.5837i	0.866824i
$621$	0	0
$622$	− 7.63584i	− 0.306169i
$623$	0	0
$624$	0	0
$625$	−29.6551	−1.18620
$626$	−7.97514	−0.318751
$627$	0	0
$628$	− 12.4746i	− 0.497792i
$629$	−10.9831	−0.437926
$630$	0	0
$631$	44.4619	1.77000	0.885000	−	0.465591i	$-0.154158\pi$
$631$	44.4619	1.77000	0.885000	+	0.465591i	$0.154158\pi$
$632$	16.5837i	0.659666i
$633$	0	0
$634$	22.4235	0.890552
$635$	16.3086	0.647186
$636$	0	0
$637$	0	0
$638$	8.10545i	0.320898i
$639$	0	0
$640$	2.49742i	0.0987191i
$641$	13.4352i	0.530659i	0.964158	+	0.265329i	$0.0854807\pi$
$641$	13.4352i	0.530659i	−0.964158	+	0.265329i	$0.914519\pi$
$642$	0	0
$643$	− 31.9927i	− 1.26167i	−0.775918	−	0.630834i	$-0.782713\pi$
$643$	− 31.9927i	− 1.26167i	0.775918	−	0.630834i	$-0.217287\pi$
$644$	0	0
$645$	0	0
$646$	16.8095	0.661361
$647$	40.4659	1.59088	0.795440	−	0.606032i	$-0.207240\pi$
$647$	40.4659	1.59088	0.795440	+	0.606032i	$0.207240\pi$
$648$	0	0
$649$	− 7.49605i	− 0.294246i
$650$	3.42244	0.134239
$651$	0	0
$652$	7.67989	0.300768
$653$	− 21.1506i	− 0.827687i	−0.910348	−	0.413843i	$-0.864186\pi$
$653$	− 21.1506i	− 0.827687i	0.910348	−	0.413843i	$-0.135814\pi$
$654$	0	0
$655$	43.6313	1.70482
$656$	−0.0376809	−0.00147119
$657$	0	0
$658$	0	0
$659$	− 9.21754i	− 0.359064i	−0.983752	−	0.179532i	$-0.942542\pi$
$659$	− 9.21754i	− 0.359064i	0.983752	−	0.179532i	$-0.0574584\pi$
$660$	0	0
$661$	− 45.8560i	− 1.78359i	−0.452437	−	0.891796i	$-0.649445\pi$
$661$	− 45.8560i	− 1.78359i	0.452437	−	0.891796i	$-0.350555\pi$
$662$	12.2430i	0.475838i
$663$	0	0
$664$	− 11.0117i	− 0.427336i
$665$	0	0
$666$	0	0
$667$	44.3486	1.71718
$668$	10.6813	0.413270
$669$	0	0
$670$	− 14.3014i	− 0.552512i
$671$	0.286284	0.0110519
$672$	0	0
$673$	32.6702	1.25935	0.629673	−	0.776861i	$-0.283189\pi$
$673$	32.6702	1.25935	0.629673	+	0.776861i	$0.283189\pi$
$674$	1.22334i	0.0471214i
$675$	0	0
$676$	−5.34639	−0.205631
$677$	−13.9545	−0.536314	−0.268157	−	0.963375i	$-0.586415\pi$
$677$	−13.9545	−0.536314	−0.268157	+	0.963375i	$0.586415\pi$
$678$	0	0
$679$	0	0
$680$	− 6.11740i	− 0.234592i
$681$	0	0
$682$	7.56003i	0.289489i
$683$	1.36004i	0.0520405i	0.999661	+	0.0260202i	$0.00828343\pi$
$683$	1.36004i	0.0520405i	−0.999661	+	0.0260202i	$0.991717\pi$
$684$	0	0
$685$	48.6192i	1.85764i
$686$	0	0
$687$	0	0
$688$	5.35449	0.204138
$689$	28.4276	1.08300
$690$	0	0
$691$	28.5181i	1.08488i	0.840095	+	0.542439i	$0.182499\pi$
$691$	28.5181i	1.08488i	−0.840095	+	0.542439i	$0.817501\pi$
$692$	18.0772	0.687191
$693$	0	0
$694$	7.56481	0.287156
$695$	− 21.1695i	− 0.803006i
$696$	0	0
$697$	0.0922990	0.00349607
$698$	20.0057	0.757225
$699$	0	0
$700$	0	0
$701$	38.0734i	1.43801i	0.695004	+	0.719006i	$0.255403\pi$
$701$	38.0734i	1.43801i	−0.695004	+	0.719006i	$0.744597\pi$
$702$	0	0
$703$	− 30.7701i	− 1.16052i
$704$	0.874758i	0.0329687i
$705$	0	0
$706$	32.5454i	1.22486i
$707$	0	0
$708$	0	0
$709$	5.30683	0.199302	0.0996510	−	0.995022i	$-0.468227\pi$
$709$	5.30683	0.199302	0.0996510	+	0.995022i	$0.468227\pi$
$710$	−16.3565	−0.613848
$711$	0	0
$712$	8.82389i	0.330689i
$713$	41.3644	1.54911
$714$	0	0
$715$	6.04383	0.226026
$716$	21.1177i	0.789204i
$717$	0	0
$718$	−9.74952	−0.363849
$719$	−16.0252	−0.597641	−0.298820	−	0.954309i	$-0.596593\pi$
$719$	−16.0252	−0.597641	−0.298820	+	0.954309i	$0.596593\pi$
$720$	0	0
$721$	0	0
$722$	28.0932i	1.04552i
$723$	0	0
$724$	1.11471i	0.0414280i
$725$	− 11.4628i	− 0.425719i
$726$	0	0
$727$	2.75913i	0.102330i	0.998690	+	0.0511652i	$0.0162935\pi$
$727$	2.75913i	0.102330i	−0.998690	+	0.0511652i	$0.983706\pi$
$728$	0	0
$729$	0	0
$730$	20.5837	0.761839
$731$	−13.1158	−0.485105
$732$	0	0
$733$	0.421567i	0.0155709i	0.999970	+	0.00778546i	$0.00247821\pi$
$733$	0.421567i	0.0155709i	−0.999970	+	0.00778546i	$0.997522\pi$
$734$	24.7519	0.913609
$735$	0	0
$736$	4.78620	0.176422
$737$	− 5.00929i	− 0.184519i
$738$	0	0
$739$	−33.5354	−1.23362	−0.616809	−	0.787113i	$-0.711575\pi$
$739$	−33.5354	−1.23362	−0.616809	+	0.787113i	$0.711575\pi$
$740$	−11.1980	−0.411647
$741$	0	0
$742$	0	0
$743$	− 37.2254i	− 1.36567i	−0.730574	−	0.682834i	$-0.760747\pi$
$743$	− 37.2254i	− 1.36567i	0.730574	−	0.682834i	$-0.239253\pi$
$744$	0	0
$745$	− 15.4034i	− 0.564337i
$746$	− 37.3947i	− 1.36912i
$747$	0	0
$748$	− 2.14271i	− 0.0783453i
$749$	0	0
$750$	0	0
$751$	−38.9525	−1.42140	−0.710699	−	0.703497i	$-0.751621\pi$
$751$	−38.9525	−1.42140	−0.710699	+	0.703497i	$0.751621\pi$
$752$	4.66756	0.170209
$753$	0	0
$754$	− 25.6343i	− 0.933548i
$755$	−9.76850	−0.355512
$756$	0	0
$757$	19.8834	0.722676	0.361338	−	0.932435i	$-0.382320\pi$
$757$	19.8834	0.722676	0.361338	+	0.932435i	$0.382320\pi$
$758$	− 27.3603i	− 0.993771i
$759$	0	0
$760$	17.1384	0.621675
$761$	−4.42009	−0.160228	−0.0801141	−	0.996786i	$-0.525528\pi$
$761$	−4.42009	−0.160228	−0.0801141	+	0.996786i	$0.525528\pi$
$762$	0	0
$763$	0	0
$764$	2.30092i	0.0832443i
$765$	0	0
$766$	− 15.1669i	− 0.548002i
$767$	23.7071i	0.856012i
$768$	0	0
$769$	11.6504i	0.420126i	0.977688	+	0.210063i	$0.0673668\pi$
$769$	11.6504i	0.420126i	−0.977688	+	0.210063i	$0.932633\pi$
$770$	0	0
$771$	0	0
$772$	27.1020	0.975421
$773$	14.6412	0.526606	0.263303	−	0.964713i	$-0.415188\pi$
$773$	14.6412	0.526606	0.263303	+	0.964713i	$0.415188\pi$
$774$	0	0
$775$	− 10.6915i	− 0.384050i
$776$	−8.33787	−0.299312
$777$	0	0
$778$	6.39502	0.229273
$779$	0.258583i	0.00926470i
$780$	0	0
$781$	−5.72911	−0.205004
$782$	−11.7237	−0.419240
$783$	0	0
$784$	0	0
$785$	31.1543i	1.11195i
$786$	0	0
$787$	− 41.3648i	− 1.47450i	−0.675622	−	0.737248i	$-0.736125\pi$
$787$	− 41.3648i	− 1.47450i	0.675622	−	0.737248i	$-0.263875\pi$
$788$	− 3.92253i	− 0.139734i
$789$	0	0
$790$	− 41.4165i	− 1.47353i
$791$	0	0
$792$	0	0
$793$	−0.905404	−0.0321518
$794$	1.98618	0.0704870
$795$	0	0
$796$	− 11.6125i	− 0.411593i
$797$	−23.0667	−0.817064	−0.408532	−	0.912744i	$-0.633959\pi$
$797$	−23.0667	−0.817064	−0.408532	+	0.912744i	$0.633959\pi$
$798$	0	0
$799$	−11.4331	−0.404476
$800$	− 1.23710i	− 0.0437379i
$801$	0	0
$802$	−27.1439	−0.958486
$803$	7.20977	0.254427
$804$	0	0
$805$	0	0
$806$	− 23.9094i	− 0.842173i
$807$	0	0
$808$	− 18.6444i	− 0.655907i
$809$	22.3423i	0.785513i	0.919642	+	0.392757i	$0.128479\pi$
$809$	22.3423i	0.785513i	−0.919642	+	0.392757i	$0.871521\pi$
$810$	0	0
$811$	34.1127i	1.19786i	0.800802	+	0.598929i	$0.204407\pi$
$811$	34.1127i	1.19786i	−0.800802	+	0.598929i	$0.795593\pi$
$812$	0	0
$813$	0	0
$814$	−3.92228	−0.137476
$815$	−19.1799	−0.671842
$816$	0	0
$817$	− 36.7449i	− 1.28554i
$818$	15.0349	0.525684
$819$	0	0
$820$	0.0941049	0.00328629
$821$	37.0847i	1.29427i	0.762377	+	0.647133i	$0.224032\pi$
$821$	37.0847i	1.29427i	−0.762377	+	0.647133i	$0.775968\pi$
$822$	0	0
$823$	13.9447	0.486083	0.243042	−	0.970016i	$-0.421855\pi$
$823$	13.9447	0.486083	0.243042	+	0.970016i	$0.421855\pi$
$824$	−9.44011	−0.328862
$825$	0	0
$826$	0	0
$827$	− 22.7009i	− 0.789387i	−0.918813	−	0.394694i	$-0.870851\pi$
$827$	− 22.7009i	− 0.789387i	0.918813	−	0.394694i	$-0.129149\pi$
$828$	0	0
$829$	9.38016i	0.325786i	0.986644	+	0.162893i	$0.0520826\pi$
$829$	9.38016i	0.325786i	−0.986644	+	0.162893i	$0.947917\pi$
$830$	27.5007i	0.954564i
$831$	0	0
$832$	− 2.76652i	− 0.0959117i
$833$	0	0
$834$	0	0
$835$	−26.6756	−0.923146
$836$	6.00298	0.207617
$837$	0	0
$838$	− 24.9078i	− 0.860425i
$839$	40.8155	1.40911	0.704553	−	0.709651i	$-0.251147\pi$
$839$	40.8155	1.40911	0.704553	+	0.709651i	$0.251147\pi$
$840$	0	0
$841$	−56.8575	−1.96061
$842$	20.9423i	0.721721i
$843$	0	0
$844$	−8.17918	−0.281539
$845$	13.3522	0.459329
$846$	0	0
$847$	0	0
$848$	− 10.2756i	− 0.352865i
$849$	0	0
$850$	3.03025i	0.103937i
$851$	21.4605i	0.735658i
$852$	0	0
$853$	31.7961i	1.08868i	0.838866	+	0.544339i	$0.183219\pi$
$853$	31.7961i	1.08868i	−0.838866	+	0.544339i	$0.816781\pi$
$854$	0	0
$855$	0	0
$856$	10.5508	0.360619
$857$	−18.1392	−0.619622	−0.309811	−	0.950798i	$-0.600266\pi$
$857$	−18.1392	−0.619622	−0.309811	+	0.950798i	$0.600266\pi$
$858$	0	0
$859$	43.4012i	1.48083i	0.672151	+	0.740414i	$0.265371\pi$
$859$	43.4012i	1.48083i	−0.672151	+	0.740414i	$0.734629\pi$
$860$	−13.3724	−0.455995
$861$	0	0
$862$	16.6861	0.568329
$863$	− 20.0136i	− 0.681272i	−0.940195	−	0.340636i	$-0.889358\pi$
$863$	− 20.0136i	− 0.681272i	0.940195	−	0.340636i	$-0.110642\pi$
$864$	0	0
$865$	−45.1463	−1.53502
$866$	27.9188	0.948720
$867$	0	0
$868$	0	0
$869$	− 14.5068i	− 0.492108i
$870$	0	0
$871$	15.8424i	0.536799i
$872$	12.7591i	0.432076i
$873$	0	0
$874$	− 32.8450i	− 1.11100i
$875$	0	0
$876$	0	0
$877$	−1.74590	−0.0589550	−0.0294775	−	0.999565i	$-0.509384\pi$
$877$	−1.74590	−0.0589550	−0.0294775	+	0.999565i	$0.509384\pi$
$878$	26.1754	0.883377
$879$	0	0
$880$	− 2.18464i	− 0.0736441i
$881$	6.71644	0.226283	0.113141	−	0.993579i	$-0.463909\pi$
$881$	6.71644	0.226283	0.113141	+	0.993579i	$0.463909\pi$
$882$	0	0
$883$	24.7725	0.833660	0.416830	−	0.908984i	$-0.363141\pi$
$883$	24.7725	0.833660	0.416830	+	0.908984i	$0.363141\pi$
$884$	6.77655i	0.227920i
$885$	0	0
$886$	12.7660	0.428884
$887$	−15.2832	−0.513161	−0.256580	−	0.966523i	$-0.582596\pi$
$887$	−15.2832	−0.513161	−0.256580	+	0.966523i	$0.582596\pi$
$888$	0	0
$889$	0	0
$890$	− 22.0369i	− 0.738680i
$891$	0	0
$892$	5.68754i	0.190433i
$893$	− 32.0309i	− 1.07187i
$894$	0	0
$895$	− 52.7396i	− 1.76289i
$896$	0	0
$897$	0	0
$898$	6.11621	0.204101
$899$	−80.0802	−2.67082
$900$	0	0
$901$	25.1699i	0.838532i
$902$	0.0329617	0.00109750
$903$	0	0
$904$	18.3102	0.608989
$905$	− 2.78390i	− 0.0925400i
$906$	0	0
$907$	38.3370	1.27296	0.636480	−	0.771294i	$-0.280390\pi$
$907$	38.3370	1.27296	0.636480	+	0.771294i	$0.280390\pi$
$908$	13.4284	0.445636
$909$	0	0
$910$	0	0
$911$	37.9493i	1.25732i	0.777681	+	0.628659i	$0.216396\pi$
$911$	37.9493i	1.25732i	−0.777681	+	0.628659i	$0.783604\pi$
$912$	0	0
$913$	9.63255i	0.318791i
$914$	− 6.00591i	− 0.198658i
$915$	0	0
$916$	14.0172i	0.463141i
$917$	0	0
$918$	0	0
$919$	6.16603	0.203398	0.101699	−	0.994815i	$-0.467572\pi$
$919$	6.16603	0.203398	0.101699	+	0.994815i	$0.467572\pi$
$920$	−11.9531	−0.394083
$921$	0	0
$922$	− 24.1872i	− 0.796563i
$923$	18.1189	0.596391
$924$	0	0
$925$	5.54693	0.182382
$926$	− 14.0676i	− 0.462290i
$927$	0	0
$928$	−9.26593	−0.304169
$929$	45.5784	1.49538	0.747690	−	0.664048i	$-0.231163\pi$
$929$	45.5784	1.49538	0.747690	+	0.664048i	$0.231163\pi$
$930$	0	0
$931$	0	0
$932$	− 1.84592i	− 0.0604651i
$933$	0	0
$934$	19.9191i	0.651774i
$935$	5.35125i	0.175004i
$936$	0	0
$937$	13.0215i	0.425393i	0.977118	+	0.212697i	$0.0682246\pi$
$937$	13.0215i	0.425393i	−0.977118	+	0.212697i	$0.931775\pi$
$938$	0	0
$939$	0	0
$940$	−11.6569	−0.380205
$941$	4.74829	0.154790	0.0773948	−	0.997001i	$-0.475340\pi$
$941$	4.74829	0.154790	0.0773948	+	0.997001i	$0.475340\pi$
$942$	0	0
$943$	− 0.180348i	− 0.00587295i
$944$	8.56928	0.278906
$945$	0	0
$946$	−4.68389	−0.152286
$947$	− 0.388750i	− 0.0126327i	−0.999980	−	0.00631634i	$-0.997989\pi$
$947$	− 0.388750i	− 0.0126327i	0.999980	−	0.00631634i	$-0.00201057\pi$
$948$	0	0
$949$	−22.8017	−0.740173
$950$	−8.48950	−0.275436
$951$	0	0
$952$	0	0
$953$	21.1130i	0.683918i	0.939715	+	0.341959i	$0.111090\pi$
$953$	21.1130i	0.683918i	−0.939715	+	0.341959i	$0.888910\pi$
$954$	0	0
$955$	− 5.74635i	− 0.185948i
$956$	− 30.0091i	− 0.970562i
$957$	0	0
$958$	22.5817i	0.729583i
$959$	0	0
$960$	0	0
$961$	−43.6915	−1.40940
$962$	12.4046	0.399941
$963$	0	0
$964$	11.5097i	0.370702i
$965$	−67.6849	−2.17886
$966$	0	0
$967$	−1.73961	−0.0559421	−0.0279711	−	0.999609i	$-0.508905\pi$
$967$	−1.73961	−0.0559421	−0.0279711	+	0.999609i	$0.508905\pi$
$968$	10.2348i	0.328959i
$969$	0	0
$970$	20.8231	0.668591
$971$	−24.7113	−0.793025	−0.396512	−	0.918029i	$-0.629780\pi$
$971$	−24.7113	−0.793025	−0.396512	+	0.918029i	$0.629780\pi$
$972$	0	0
$973$	0	0
$974$	− 12.2814i	− 0.393521i
$975$	0	0
$976$	0.327272i	0.0104757i
$977$	22.3495i	0.715023i	0.933909	+	0.357511i	$0.116375\pi$
$977$	22.3495i	0.715023i	−0.933909	+	0.357511i	$0.883625\pi$
$978$	0	0
$979$	− 7.71877i	− 0.246693i
$980$	0	0
$981$	0	0
$982$	34.0390	1.08623
$983$	10.9844	0.350349	0.175174	−	0.984537i	$-0.443951\pi$
$983$	10.9844	0.350349	0.175174	+	0.984537i	$0.443951\pi$
$984$	0	0
$985$	9.79620i	0.312133i
$986$	22.6968	0.722814
$987$	0	0
$988$	−18.9851	−0.603995
$989$	25.6277i	0.814913i
$990$	0	0
$991$	56.0823	1.78151	0.890756	−	0.454481i	$-0.150175\pi$
$991$	56.0823	1.78151	0.890756	+	0.454481i	$0.150175\pi$
$992$	−8.64243	−0.274397
$993$	0	0
$994$	0	0
$995$	29.0012i	0.919400i
$996$	0	0
$997$	− 48.2219i	− 1.52720i	−0.645688	−	0.763601i	$-0.723429\pi$
$997$	− 48.2219i	− 1.52720i	0.645688	−	0.763601i	$-0.276571\pi$
$998$	− 30.8593i	− 0.976835i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.d.f.2645.7	yes	16
3.2	odd	2	inner	2646.2.d.f.2645.10	yes	16
7.6	odd	2	inner	2646.2.d.f.2645.2	✓	16
21.20	even	2	inner	2646.2.d.f.2645.15	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2646.2.d.f.2645.2	✓	16	7.6	odd	2	inner
2646.2.d.f.2645.7	yes	16	1.1	even	1	trivial
2646.2.d.f.2645.10	yes	16	3.2	odd	2	inner
2646.2.d.f.2645.15	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:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.49742 q^{5} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.49742 q^{5} +1.00000i q^{8} -2.49742i q^{10} -0.874758i q^{11} +2.76652i q^{13} +1.00000 q^{16} -2.44949 q^{17} -6.86245i q^{19} -2.49742 q^{20} -0.874758 q^{22} +4.78620i q^{23} +1.23710 q^{25} +2.76652 q^{26} -9.26593i q^{29} -8.64243i q^{31} -1.00000i q^{32} +2.44949i q^{34} +4.48384 q^{37} -6.86245 q^{38} +2.49742i q^{40} -0.0376809 q^{41} +5.35449 q^{43} +0.874758i q^{44} +4.78620 q^{46} +4.66756 q^{47} -1.23710i q^{50} -2.76652i q^{52} -10.2756i q^{53} -2.18464i q^{55} -9.26593 q^{58} +8.56928 q^{59} +0.327272i q^{61} -8.64243 q^{62} -1.00000 q^{64} +6.90914i q^{65} +5.72648 q^{67} +2.44949 q^{68} -6.54936i q^{71} +8.24201i q^{73} -4.48384i q^{74} +6.86245i q^{76} +16.5837 q^{79} +2.49742 q^{80} +0.0376809i q^{82} -11.0117 q^{83} -6.11740 q^{85} -5.35449i q^{86} +0.874758 q^{88} +8.82389 q^{89} -4.78620i q^{92} -4.66756i q^{94} -17.1384i q^{95} +8.33787i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 16 q^{16} + 16 q^{22} + 16 q^{37} - 32 q^{43} + 48 q^{46} - 32 q^{58} - 16 q^{64} - 32 q^{67} - 16 q^{88}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^16 + 16 * q^22 + 16 * q^37 - 32 * q^43 + 48 * q^46 - 32 * q^58 - 16 * q^64 - 32 * q^67 - 16 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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