LMFDB - Embedded newform 2646.2.d.f.2645.12

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			2645.12
Root			$0.793353 + 0.608761i$ of defining polynomial
Character	$\chi$	$=$	2646.2645
Dual form			2646.2.d.f.2645.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -0.115708 q^{5} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} -0.115708 q^{5} -1.00000i q^{8} -0.115708i q^{10} +3.52607i q^{11} -2.01140i q^{13} +1.00000 q^{16} +2.44949 q^{17} -5.10030i q^{19} +0.115708 q^{20} -3.52607 q^{22} -5.73987i q^{23} -4.98661 q^{25} +2.01140 q^{26} -4.48938i q^{29} -1.03446i q^{31} +1.00000i q^{32} +2.44949i q^{34} +4.69354 q^{37} +5.10030 q^{38} +0.115708i q^{40} -4.06047 q^{41} -6.70319 q^{43} -3.52607i q^{44} +5.73987 q^{46} -2.96561 q^{47} -4.98661i q^{50} +2.01140i q^{52} +9.17738i q^{53} -0.407995i q^{55} +4.48938 q^{58} +10.3422 q^{59} -2.73420i q^{61} +1.03446 q^{62} -1.00000 q^{64} +0.232735i q^{65} -2.54910 q^{67} -2.44949 q^{68} -5.12150i q^{71} -7.60796i q^{73} +4.69354i q^{74} +5.10030i q^{76} -4.88030 q^{79} -0.115708 q^{80} -4.06047i q^{82} +17.9544 q^{83} -0.283426 q^{85} -6.70319i q^{86} +3.52607 q^{88} +10.0274 q^{89} +5.73987i q^{92} -2.96561i q^{94} +0.590146i q^{95} -12.2755i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.115708	−0.0517463	−0.0258732	−	0.999665i	$-0.508237\pi$
$5$	−0.115708	−0.0517463	−0.0258732	+	0.999665i	$0.508237\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	− 0.115708i	− 0.0365902i
$11$	3.52607i	1.06315i	0.847011	+	0.531575i	$0.178399\pi$
$11$	3.52607i	1.06315i	−0.847011	+	0.531575i	$0.821601\pi$
$12$	0	0
$13$	− 2.01140i	− 0.557861i	−0.960311	−	0.278930i	$-0.910020\pi$
$13$	− 2.01140i	− 0.557861i	0.960311	−	0.278930i	$-0.0899799\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	2.44949	0.594089	0.297044	−	0.954864i	$-0.403999\pi$
$17$	2.44949	0.594089	0.297044	+	0.954864i	$0.403999\pi$
$18$	0	0
$19$	− 5.10030i	− 1.17009i	−0.811001	−	0.585044i	$-0.801077\pi$
$19$	− 5.10030i	− 1.17009i	0.811001	−	0.585044i	$-0.198923\pi$
$20$	0.115708	0.0258732
$21$	0	0
$22$	−3.52607	−0.751760
$23$	− 5.73987i	− 1.19685i	−0.801181	−	0.598423i	$-0.795794\pi$
$23$	− 5.73987i	− 1.19685i	0.801181	−	0.598423i	$-0.204206\pi$
$24$	0	0
$25$	−4.98661	−0.997322
$26$	2.01140	0.394467
$27$	0	0
$28$	0	0
$29$	− 4.48938i	− 0.833658i	−0.908985	−	0.416829i	$-0.863141\pi$
$29$	− 4.48938i	− 0.833658i	0.908985	−	0.416829i	$-0.136859\pi$
$30$	0	0
$31$	− 1.03446i	− 0.185795i	−0.995676	−	0.0928976i	$-0.970387\pi$
$31$	− 1.03446i	− 0.185795i	0.995676	−	0.0928976i	$-0.0296129\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	2.44949i	0.420084i
$35$	0	0
$36$	0	0
$37$	4.69354	0.771613	0.385806	−	0.922580i	$-0.373923\pi$
$37$	4.69354	0.771613	0.385806	+	0.922580i	$0.373923\pi$
$38$	5.10030	0.827377
$39$	0	0
$40$	0.115708i	0.0182951i
$41$	−4.06047	−0.634139	−0.317070	−	0.948402i	$-0.602699\pi$
$41$	−4.06047	−0.634139	−0.317070	+	0.948402i	$0.602699\pi$
$42$	0	0
$43$	−6.70319	−1.02223	−0.511113	−	0.859513i	$-0.670767\pi$
$43$	−6.70319	−1.02223	−0.511113	+	0.859513i	$0.670767\pi$
$44$	− 3.52607i	− 0.531575i
$45$	0	0
$46$	5.73987	0.846297
$47$	−2.96561	−0.432579	−0.216289	−	0.976329i	$-0.569395\pi$
$47$	−2.96561	−0.432579	−0.216289	+	0.976329i	$0.569395\pi$
$48$	0	0
$49$	0	0
$50$	− 4.98661i	− 0.705213i
$51$	0	0
$52$	2.01140i	0.278930i
$53$	9.17738i	1.26061i	0.776348	+	0.630305i	$0.217070\pi$
$53$	9.17738i	1.26061i	−0.776348	+	0.630305i	$0.782930\pi$
$54$	0	0
$55$	− 0.407995i	− 0.0550140i
$56$	0	0
$57$	0	0
$58$	4.48938	0.589485
$59$	10.3422	1.34643	0.673217	−	0.739445i	$-0.264912\pi$
$59$	10.3422	1.34643	0.673217	+	0.739445i	$0.264912\pi$
$60$	0	0
$61$	− 2.73420i	− 0.350078i	−0.984562	−	0.175039i	$-0.943995\pi$
$61$	− 2.73420i	− 0.350078i	0.984562	−	0.175039i	$-0.0560051\pi$
$62$	1.03446	0.131377
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0.232735i	0.0288672i
$66$	0	0
$67$	−2.54910	−0.311423	−0.155711	−	0.987803i	$-0.549767\pi$
$67$	−2.54910	−0.311423	−0.155711	+	0.987803i	$0.549767\pi$
$68$	−2.44949	−0.297044
$69$	0	0
$70$	0	0
$71$	− 5.12150i	− 0.607810i	−0.952702	−	0.303905i	$-0.901709\pi$
$71$	− 5.12150i	− 0.607810i	0.952702	−	0.303905i	$-0.0982906\pi$
$72$	0	0
$73$	− 7.60796i	− 0.890445i	−0.895420	−	0.445222i	$-0.853125\pi$
$73$	− 7.60796i	− 0.890445i	0.895420	−	0.445222i	$-0.146875\pi$
$74$	4.69354i	0.545613i
$75$	0	0
$76$	5.10030i	0.585044i
$77$	0	0
$78$	0	0
$79$	−4.88030	−0.549077	−0.274539	−	0.961576i	$-0.588525\pi$
$79$	−4.88030	−0.549077	−0.274539	+	0.961576i	$0.588525\pi$
$80$	−0.115708	−0.0129366
$81$	0	0
$82$	− 4.06047i	− 0.448404i
$83$	17.9544	1.97075	0.985374	−	0.170408i	$-0.0545085\pi$
$83$	17.9544	1.97075	0.985374	+	0.170408i	$0.0545085\pi$
$84$	0	0
$85$	−0.283426	−0.0307419
$86$	− 6.70319i	− 0.722823i
$87$	0	0
$88$	3.52607	0.375880
$89$	10.0274	1.06290	0.531449	−	0.847091i	$-0.321648\pi$
$89$	10.0274	1.06290	0.531449	+	0.847091i	$0.321648\pi$
$90$	0	0
$91$	0	0
$92$	5.73987i	0.598423i
$93$	0	0
$94$	− 2.96561i	− 0.305879i
$95$	0.590146i	0.0605477i
$96$	0	0
$97$	− 12.2755i	− 1.24639i	−0.782066	−	0.623195i	$-0.785834\pi$
$97$	− 12.2755i	− 1.24639i	0.782066	−	0.623195i	$-0.214166\pi$
$98$	0	0
$99$	0	0
$100$	4.98661	0.498661
$101$	18.5530	1.84609	0.923044	−	0.384695i	$-0.125693\pi$
$101$	18.5530	1.84609	0.923044	+	0.384695i	$0.125693\pi$
$102$	0	0
$103$	− 9.25440i	− 0.911863i	−0.890015	−	0.455932i	$-0.849306\pi$
$103$	− 9.25440i	− 0.911863i	0.890015	−	0.455932i	$-0.150694\pi$
$104$	−2.01140	−0.197234
$105$	0	0
$106$	−9.17738	−0.891386
$107$	− 18.3003i	− 1.76916i	−0.466390	−	0.884579i	$-0.654446\pi$
$107$	− 18.3003i	− 1.76916i	0.466390	−	0.884579i	$-0.345554\pi$
$108$	0	0
$109$	−14.7842	−1.41607	−0.708033	−	0.706180i	$-0.750417\pi$
$109$	−14.7842	−1.41607	−0.708033	+	0.706180i	$0.750417\pi$
$110$	0.407995	0.0389008
$111$	0	0
$112$	0	0
$113$	− 11.4294i	− 1.07519i	−0.843204	−	0.537594i	$-0.819333\pi$
$113$	− 11.4294i	− 1.07519i	0.843204	−	0.537594i	$-0.180667\pi$
$114$	0	0
$115$	0.664150i	0.0619323i
$116$	4.48938i	0.416829i
$117$	0	0
$118$	10.3422i	0.952073i
$119$	0	0
$120$	0	0
$121$	−1.43315	−0.130286
$122$	2.73420	0.247542
$123$	0	0
$124$	1.03446i	0.0928976i
$125$	1.15553	0.103354
$126$	0	0
$127$	15.0480	1.33530	0.667648	−	0.744477i	$-0.267301\pi$
$127$	15.0480	1.33530	0.667648	+	0.744477i	$0.267301\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	−0.232735	−0.0204122
$131$	13.1615	1.14992	0.574962	−	0.818180i	$-0.305017\pi$
$131$	13.1615	1.14992	0.574962	+	0.818180i	$0.305017\pi$
$132$	0	0
$133$	0	0
$134$	− 2.54910i	− 0.220209i
$135$	0	0
$136$	− 2.44949i	− 0.210042i
$137$	13.3425i	1.13993i	0.821669	+	0.569965i	$0.193043\pi$
$137$	13.3425i	1.13993i	−0.821669	+	0.569965i	$0.806957\pi$
$138$	0	0
$139$	− 21.8991i	− 1.85746i	−0.370757	−	0.928730i	$-0.620902\pi$
$139$	− 21.8991i	− 1.85746i	0.370757	−	0.928730i	$-0.379098\pi$
$140$	0	0
$141$	0	0
$142$	5.12150	0.429787
$143$	7.09231	0.593089
$144$	0	0
$145$	0.519459i	0.0431387i
$146$	7.60796	0.629639
$147$	0	0
$148$	−4.69354	−0.385806
$149$	− 21.9423i	− 1.79759i	−0.438372	−	0.898793i	$-0.644445\pi$
$149$	− 21.9423i	− 1.79759i	0.438372	−	0.898793i	$-0.355555\pi$
$150$	0	0
$151$	−2.21380	−0.180157	−0.0900783	−	0.995935i	$-0.528712\pi$
$151$	−2.21380	−0.180157	−0.0900783	+	0.995935i	$0.528712\pi$
$152$	−5.10030	−0.413689
$153$	0	0
$154$	0	0
$155$	0.119696i	0.00961422i
$156$	0	0
$157$	18.3494i	1.46444i	0.681067	+	0.732221i	$0.261516\pi$
$157$	18.3494i	1.46444i	−0.681067	+	0.732221i	$0.738484\pi$
$158$	− 4.88030i	− 0.388256i
$159$	0	0
$160$	− 0.115708i	− 0.00914754i
$161$	0	0
$162$	0	0
$163$	−9.94438	−0.778904	−0.389452	−	0.921047i	$-0.627336\pi$
$163$	−9.94438	−0.778904	−0.389452	+	0.921047i	$0.627336\pi$
$164$	4.06047	0.317070
$165$	0	0
$166$	17.9544i	1.39353i
$167$	−6.88516	−0.532790	−0.266395	−	0.963864i	$-0.585833\pi$
$167$	−6.88516	−0.532790	−0.266395	+	0.963864i	$0.585833\pi$
$168$	0	0
$169$	8.95429	0.688791
$170$	− 0.283426i	− 0.0217378i
$171$	0	0
$172$	6.70319	0.511113
$173$	−16.2961	−1.23897	−0.619485	−	0.785009i	$-0.712658\pi$
$173$	−16.2961	−1.23897	−0.619485	+	0.785009i	$0.712658\pi$
$174$	0	0
$175$	0	0
$176$	3.52607i	0.265787i
$177$	0	0
$178$	10.0274i	0.751582i
$179$	− 3.56456i	− 0.266427i	−0.991087	−	0.133214i	$-0.957470\pi$
$179$	− 3.56456i	− 0.266427i	0.991087	−	0.133214i	$-0.0425297\pi$
$180$	0	0
$181$	14.7498i	1.09634i	0.836367	+	0.548170i	$0.184675\pi$
$181$	14.7498i	1.09634i	−0.836367	+	0.548170i	$0.815325\pi$
$182$	0	0
$183$	0	0
$184$	−5.73987	−0.423149
$185$	−0.543081	−0.0399281
$186$	0	0
$187$	8.63706i	0.631605i
$188$	2.96561	0.216289
$189$	0	0
$190$	−0.590146	−0.0428137
$191$	20.2251i	1.46344i	0.681605	+	0.731720i	$0.261282\pi$
$191$	20.2251i	1.46344i	−0.681605	+	0.731720i	$0.738718\pi$
$192$	0	0
$193$	3.94557	0.284008	0.142004	−	0.989866i	$-0.454645\pi$
$193$	3.94557	0.284008	0.142004	+	0.989866i	$0.454645\pi$
$194$	12.2755	0.881331
$195$	0	0
$196$	0	0
$197$	2.29826i	0.163744i	0.996643	+	0.0818720i	$0.0260899\pi$
$197$	2.29826i	0.163744i	−0.996643	+	0.0818720i	$0.973910\pi$
$198$	0	0
$199$	− 25.9021i	− 1.83615i	−0.396408	−	0.918074i	$-0.629744\pi$
$199$	− 25.9021i	− 1.83615i	0.396408	−	0.918074i	$-0.370256\pi$
$200$	4.98661i	0.352607i
$201$	0	0
$202$	18.5530i	1.30538i
$203$	0	0
$204$	0	0
$205$	0.469830	0.0328144
$206$	9.25440	0.644785
$207$	0	0
$208$	− 2.01140i	− 0.139465i
$209$	17.9840	1.24398
$210$	0	0
$211$	2.20067	0.151500	0.0757502	−	0.997127i	$-0.475865\pi$
$211$	2.20067	0.151500	0.0757502	+	0.997127i	$0.475865\pi$
$212$	− 9.17738i	− 0.630305i
$213$	0	0
$214$	18.3003	1.25098
$215$	0.775614	0.0528964
$216$	0	0
$217$	0	0
$218$	− 14.7842i	− 1.00131i
$219$	0	0
$220$	0.407995i	0.0275070i
$221$	− 4.92689i	− 0.331419i
$222$	0	0
$223$	− 5.12818i	− 0.343408i	−0.985149	−	0.171704i	$-0.945073\pi$
$223$	− 5.12818i	− 0.343408i	0.985149	−	0.171704i	$-0.0549273\pi$
$224$	0	0
$225$	0	0
$226$	11.4294	0.760273
$227$	18.1149	1.20233	0.601163	−	0.799127i	$-0.294704\pi$
$227$	18.1149	1.20233	0.601163	+	0.799127i	$0.294704\pi$
$228$	0	0
$229$	− 6.45493i	− 0.426554i	−0.976992	−	0.213277i	$-0.931586\pi$
$229$	− 6.45493i	− 0.426554i	0.976992	−	0.213277i	$-0.0684136\pi$
$230$	−0.664150	−0.0437928
$231$	0	0
$232$	−4.48938	−0.294743
$233$	− 12.0288i	− 0.788035i	−0.919103	−	0.394018i	$-0.871085\pi$
$233$	− 12.0288i	− 0.788035i	0.919103	−	0.394018i	$-0.128915\pi$
$234$	0	0
$235$	0.343146	0.0223844
$236$	−10.3422	−0.673217
$237$	0	0
$238$	0	0
$239$	− 1.19342i	− 0.0771962i	−0.999255	−	0.0385981i	$-0.987711\pi$
$239$	− 1.19342i	− 0.0771962i	0.999255	−	0.0385981i	$-0.0122892\pi$
$240$	0	0
$241$	11.9940i	0.772599i	0.922373	+	0.386299i	$0.126247\pi$
$241$	11.9940i	0.772599i	−0.922373	+	0.386299i	$0.873753\pi$
$242$	− 1.43315i	− 0.0921262i
$243$	0	0
$244$	2.73420i	0.175039i
$245$	0	0
$246$	0	0
$247$	−10.2587	−0.652746
$248$	−1.03446	−0.0656885
$249$	0	0
$250$	1.15553i	0.0730824i
$251$	−7.66816	−0.484010	−0.242005	−	0.970275i	$-0.577805\pi$
$251$	−7.66816	−0.484010	−0.242005	+	0.970275i	$0.577805\pi$
$252$	0	0
$253$	20.2392	1.27242
$254$	15.0480i	0.944197i
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.49305	0.0931341	0.0465670	−	0.998915i	$-0.485172\pi$
$257$	1.49305	0.0931341	0.0465670	+	0.998915i	$0.485172\pi$
$258$	0	0
$259$	0	0
$260$	− 0.232735i	− 0.0144336i
$261$	0	0
$262$	13.1615i	0.813119i
$263$	14.9939i	0.924561i	0.886734	+	0.462280i	$0.152969\pi$
$263$	14.9939i	0.924561i	−0.886734	+	0.462280i	$0.847031\pi$
$264$	0	0
$265$	− 1.06190i	− 0.0652319i
$266$	0	0
$267$	0	0
$268$	2.54910	0.155711
$269$	−12.2100	−0.744455	−0.372227	−	0.928142i	$-0.621406\pi$
$269$	−12.2100	−0.744455	−0.372227	+	0.928142i	$0.621406\pi$
$270$	0	0
$271$	11.3211i	0.687709i	0.939023	+	0.343854i	$0.111733\pi$
$271$	11.3211i	0.687709i	−0.939023	+	0.343854i	$0.888267\pi$
$272$	2.44949	0.148522
$273$	0	0
$274$	−13.3425	−0.806053
$275$	− 17.5831i	− 1.06030i
$276$	0	0
$277$	−19.6283	−1.17935	−0.589674	−	0.807641i	$-0.700744\pi$
$277$	−19.6283	−1.17935	−0.589674	+	0.807641i	$0.700744\pi$
$278$	21.8991	1.31342
$279$	0	0
$280$	0	0
$281$	9.26424i	0.552658i	0.961063	+	0.276329i	$0.0891179\pi$
$281$	9.26424i	0.552658i	−0.961063	+	0.276329i	$0.910882\pi$
$282$	0	0
$283$	− 6.48744i	− 0.385638i	−0.981234	−	0.192819i	$-0.938237\pi$
$283$	− 6.48744i	− 0.385638i	0.981234	−	0.192819i	$-0.0617631\pi$
$284$	5.12150i	0.303905i
$285$	0	0
$286$	7.09231i	0.419377i
$287$	0	0
$288$	0	0
$289$	−11.0000	−0.647059
$290$	−0.519459	−0.0305037
$291$	0	0
$292$	7.60796i	0.445222i
$293$	−12.8799	−0.752453	−0.376226	−	0.926528i	$-0.622779\pi$
$293$	−12.8799	−0.752453	−0.376226	+	0.926528i	$0.622779\pi$
$294$	0	0
$295$	−1.19667	−0.0696730
$296$	− 4.69354i	− 0.272806i
$297$	0	0
$298$	21.9423	1.27109
$299$	−11.5451	−0.667673
$300$	0	0
$301$	0	0
$302$	− 2.21380i	− 0.127390i
$303$	0	0
$304$	− 5.10030i	− 0.292522i
$305$	0.316369i	0.0181152i
$306$	0	0
$307$	− 12.1063i	− 0.690945i	−0.938429	−	0.345472i	$-0.887719\pi$
$307$	− 12.1063i	− 0.690945i	0.938429	−	0.345472i	$-0.112281\pi$
$308$	0	0
$309$	0	0
$310$	−0.119696	−0.00679828
$311$	−28.1723	−1.59751	−0.798753	−	0.601660i	$-0.794506\pi$
$311$	−28.1723	−1.59751	−0.798753	+	0.601660i	$0.794506\pi$
$312$	0	0
$313$	24.9784i	1.41186i	0.708280	+	0.705932i	$0.249472\pi$
$313$	24.9784i	1.41186i	−0.708280	+	0.705932i	$0.750528\pi$
$314$	−18.3494	−1.03552
$315$	0	0
$316$	4.88030	0.274539
$317$	28.0688i	1.57650i	0.615356	+	0.788250i	$0.289012\pi$
$317$	28.0688i	1.57650i	−0.615356	+	0.788250i	$0.710988\pi$
$318$	0	0
$319$	15.8299	0.886303
$320$	0.115708	0.00646829
$321$	0	0
$322$	0	0
$323$	− 12.4931i	− 0.695136i
$324$	0	0
$325$	10.0300i	0.556367i
$326$	− 9.94438i	− 0.550768i
$327$	0	0
$328$	4.06047i	0.224202i
$329$	0	0
$330$	0	0
$331$	−30.4124	−1.67162	−0.835809	−	0.549020i	$-0.815001\pi$
$331$	−30.4124	−1.67162	−0.835809	+	0.549020i	$0.815001\pi$
$332$	−17.9544	−0.985374
$333$	0	0
$334$	− 6.88516i	− 0.376739i
$335$	0.294952	0.0161150
$336$	0	0
$337$	32.5762	1.77454	0.887268	−	0.461254i	$-0.152600\pi$
$337$	32.5762	1.77454	0.887268	+	0.461254i	$0.152600\pi$
$338$	8.95429i	0.487049i
$339$	0	0
$340$	0.283426	0.0153709
$341$	3.64759	0.197528
$342$	0	0
$343$	0	0
$344$	6.70319i	0.361412i
$345$	0	0
$346$	− 16.2961i	− 0.876083i
$347$	12.5588i	0.674189i	0.941471	+	0.337095i	$0.109444\pi$
$347$	12.5588i	0.674189i	−0.941471	+	0.337095i	$0.890556\pi$
$348$	0	0
$349$	− 17.1062i	− 0.915676i	−0.889036	−	0.457838i	$-0.848624\pi$
$349$	− 17.1062i	− 0.915676i	0.889036	−	0.457838i	$-0.151376\pi$
$350$	0	0
$351$	0	0
$352$	−3.52607	−0.187940
$353$	9.61557	0.511785	0.255893	−	0.966705i	$-0.417631\pi$
$353$	9.61557	0.511785	0.255893	+	0.966705i	$0.417631\pi$
$354$	0	0
$355$	0.592600i	0.0314519i
$356$	−10.0274	−0.531449
$357$	0	0
$358$	3.56456	0.188393
$359$	15.0521i	0.794421i	0.917727	+	0.397211i	$0.130022\pi$
$359$	15.0521i	0.794421i	−0.917727	+	0.397211i	$0.869978\pi$
$360$	0	0
$361$	−7.01303	−0.369107
$362$	−14.7498	−0.775230
$363$	0	0
$364$	0	0
$365$	0.880304i	0.0460772i
$366$	0	0
$367$	− 11.2943i	− 0.589560i	−0.955565	−	0.294780i	$-0.904754\pi$
$367$	− 11.2943i	− 0.589560i	0.955565	−	0.294780i	$-0.0952464\pi$
$368$	− 5.73987i	− 0.299211i
$369$	0	0
$370$	− 0.543081i	− 0.0282334i
$371$	0	0
$372$	0	0
$373$	−5.56600	−0.288196	−0.144098	−	0.989563i	$-0.546028\pi$
$373$	−5.56600	−0.288196	−0.144098	+	0.989563i	$0.546028\pi$
$374$	−8.63706	−0.446612
$375$	0	0
$376$	2.96561i	0.152940i
$377$	−9.02993	−0.465065
$378$	0	0
$379$	−5.80720	−0.298296	−0.149148	−	0.988815i	$-0.547653\pi$
$379$	−5.80720	−0.298296	−0.149148	+	0.988815i	$0.547653\pi$
$380$	− 0.590146i	− 0.0302739i
$381$	0	0
$382$	−20.2251	−1.03481
$383$	7.28335	0.372162	0.186081	−	0.982534i	$-0.440421\pi$
$383$	7.28335	0.372162	0.186081	+	0.982534i	$0.440421\pi$
$384$	0	0
$385$	0	0
$386$	3.94557i	0.200824i
$387$	0	0
$388$	12.2755i	0.623195i
$389$	− 23.7553i	− 1.20444i	−0.798329	−	0.602221i	$-0.794283\pi$
$389$	− 23.7553i	− 1.20444i	0.798329	−	0.602221i	$-0.205717\pi$
$390$	0	0
$391$	− 14.0597i	− 0.711032i
$392$	0	0
$393$	0	0
$394$	−2.29826	−0.115785
$395$	0.564691	0.0284127
$396$	0	0
$397$	− 15.8429i	− 0.795131i	−0.917574	−	0.397565i	$-0.869855\pi$
$397$	− 15.8429i	− 0.795131i	0.917574	−	0.397565i	$-0.130145\pi$
$398$	25.9021	1.29835
$399$	0	0
$400$	−4.98661	−0.249331
$401$	8.33412i	0.416186i	0.978109	+	0.208093i	$0.0667257\pi$
$401$	8.33412i	0.416186i	−0.978109	+	0.208093i	$0.933274\pi$
$402$	0	0
$403$	−2.08072	−0.103648
$404$	−18.5530	−0.923044
$405$	0	0
$406$	0	0
$407$	16.5497i	0.820339i
$408$	0	0
$409$	28.1151i	1.39020i	0.718913	+	0.695100i	$0.244640\pi$
$409$	28.1151i	1.39020i	−0.718913	+	0.695100i	$0.755360\pi$
$410$	0.469830i	0.0232033i
$411$	0	0
$412$	9.25440i	0.455932i
$413$	0	0
$414$	0	0
$415$	−2.07747	−0.101979
$416$	2.01140	0.0986168
$417$	0	0
$418$	17.9840i	0.879626i
$419$	−15.3664	−0.750700	−0.375350	−	0.926883i	$-0.622478\pi$
$419$	−15.3664	−0.750700	−0.375350	+	0.926883i	$0.622478\pi$
$420$	0	0
$421$	−2.33930	−0.114010	−0.0570052	−	0.998374i	$-0.518155\pi$
$421$	−2.33930	−0.114010	−0.0570052	+	0.998374i	$0.518155\pi$
$422$	2.20067i	0.107127i
$423$	0	0
$424$	9.17738	0.445693
$425$	−12.2147	−0.592498
$426$	0	0
$427$	0	0
$428$	18.3003i	0.884579i
$429$	0	0
$430$	0.775614i	0.0374034i
$431$	− 24.4954i	− 1.17990i	−0.807438	−	0.589952i	$-0.799147\pi$
$431$	− 24.4954i	− 1.17990i	0.807438	−	0.589952i	$-0.200853\pi$
$432$	0	0
$433$	− 4.66332i	− 0.224105i	−0.993702	−	0.112052i	$-0.964258\pi$
$433$	− 4.66332i	− 0.224105i	0.993702	−	0.112052i	$-0.0357424\pi$
$434$	0	0
$435$	0	0
$436$	14.7842	0.708033
$437$	−29.2750	−1.40041
$438$	0	0
$439$	− 16.9027i	− 0.806722i	−0.915041	−	0.403361i	$-0.867842\pi$
$439$	− 16.9027i	− 0.806722i	0.915041	−	0.403361i	$-0.132158\pi$
$440$	−0.407995	−0.0194504
$441$	0	0
$442$	4.92689	0.234348
$443$	34.2190i	1.62579i	0.582407	+	0.812897i	$0.302111\pi$
$443$	34.2190i	1.62579i	−0.582407	+	0.812897i	$0.697889\pi$
$444$	0	0
$445$	−1.16025	−0.0550010
$446$	5.12818	0.242826
$447$	0	0
$448$	0	0
$449$	1.38574i	0.0653970i	0.999465	+	0.0326985i	$0.0104101\pi$
$449$	1.38574i	0.0653970i	−0.999465	+	0.0326985i	$0.989590\pi$
$450$	0	0
$451$	− 14.3175i	− 0.674184i
$452$	11.4294i	0.537594i
$453$	0	0
$454$	18.1149i	0.850172i
$455$	0	0
$456$	0	0
$457$	30.3990	1.42201	0.711004	−	0.703188i	$-0.248241\pi$
$457$	30.3990	1.42201	0.711004	+	0.703188i	$0.248241\pi$
$458$	6.45493	0.301619
$459$	0	0
$460$	− 0.664150i	− 0.0309662i
$461$	16.3888	0.763303	0.381651	−	0.924306i	$-0.375355\pi$
$461$	16.3888	0.763303	0.381651	+	0.924306i	$0.375355\pi$
$462$	0	0
$463$	−7.18677	−0.333997	−0.166999	−	0.985957i	$-0.553408\pi$
$463$	−7.18677	−0.333997	−0.166999	+	0.985957i	$0.553408\pi$
$464$	− 4.48938i	− 0.208414i
$465$	0	0
$466$	12.0288	0.557225
$467$	−13.7962	−0.638412	−0.319206	−	0.947685i	$-0.603416\pi$
$467$	−13.7962	−0.638412	−0.319206	+	0.947685i	$0.603416\pi$
$468$	0	0
$469$	0	0
$470$	0.343146i	0.0158281i
$471$	0	0
$472$	− 10.3422i	− 0.476036i
$473$	− 23.6359i	− 1.08678i
$474$	0	0
$475$	25.4332i	1.16696i
$476$	0	0
$477$	0	0
$478$	1.19342	0.0545860
$479$	−14.1443	−0.646267	−0.323134	−	0.946353i	$-0.604736\pi$
$479$	−14.1443	−0.646267	−0.323134	+	0.946353i	$0.604736\pi$
$480$	0	0
$481$	− 9.44056i	− 0.430452i
$482$	−11.9940	−0.546310
$483$	0	0
$484$	1.43315	0.0651431
$485$	1.42038i	0.0644961i
$486$	0	0
$487$	−9.92664	−0.449819	−0.224909	−	0.974380i	$-0.572209\pi$
$487$	−9.92664	−0.449819	−0.224909	+	0.974380i	$0.572209\pi$
$488$	−2.73420	−0.123771
$489$	0	0
$490$	0	0
$491$	− 1.46802i	− 0.0662510i	−0.999451	−	0.0331255i	$-0.989454\pi$
$491$	− 1.46802i	− 0.0662510i	0.999451	−	0.0331255i	$-0.0105461\pi$
$492$	0	0
$493$	− 10.9967i	− 0.495267i
$494$	− 10.2587i	− 0.461561i
$495$	0	0
$496$	− 1.03446i	− 0.0464488i
$497$	0	0
$498$	0	0
$499$	8.29707	0.371428	0.185714	−	0.982604i	$-0.440540\pi$
$499$	8.29707	0.371428	0.185714	+	0.982604i	$0.440540\pi$
$500$	−1.15553	−0.0516770
$501$	0	0
$502$	− 7.66816i	− 0.342247i
$503$	−28.1114	−1.25343	−0.626714	−	0.779250i	$-0.715600\pi$
$503$	−28.1114	−1.25343	−0.626714	+	0.779250i	$0.715600\pi$
$504$	0	0
$505$	−2.14673	−0.0955282
$506$	20.2392i	0.899740i
$507$	0	0
$508$	−15.0480	−0.667648
$509$	43.9940	1.95000	0.975000	−	0.222204i	$-0.0713252\pi$
$509$	43.9940	1.95000	0.975000	+	0.222204i	$0.0713252\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	1.49305i	0.0658557i
$515$	1.07081i	0.0471856i
$516$	0	0
$517$	− 10.4569i	− 0.459896i
$518$	0	0
$519$	0	0
$520$	0.232735	0.0102061
$521$	−31.8306	−1.39452	−0.697262	−	0.716816i	$-0.745599\pi$
$521$	−31.8306	−1.39452	−0.697262	+	0.716816i	$0.745599\pi$
$522$	0	0
$523$	18.8678i	0.825029i	0.910951	+	0.412515i	$0.135349\pi$
$523$	18.8678i	0.825029i	−0.910951	+	0.412515i	$0.864651\pi$
$524$	−13.1615	−0.574962
$525$	0	0
$526$	−14.9939	−0.653763
$527$	− 2.53391i	− 0.110379i
$528$	0	0
$529$	−9.94608	−0.432438
$530$	1.06190	0.0461259
$531$	0	0
$532$	0	0
$533$	8.16721i	0.353761i
$534$	0	0
$535$	2.11750i	0.0915474i
$536$	2.54910i	0.110105i
$537$	0	0
$538$	− 12.2100i	− 0.526409i
$539$	0	0
$540$	0	0
$541$	28.4031	1.22115	0.610573	−	0.791960i	$-0.290939\pi$
$541$	28.4031	1.22115	0.610573	+	0.791960i	$0.290939\pi$
$542$	−11.3211	−0.486283
$543$	0	0
$544$	2.44949i	0.105021i
$545$	1.71065	0.0732762
$546$	0	0
$547$	−5.64525	−0.241373	−0.120687	−	0.992691i	$-0.538510\pi$
$547$	−5.64525	−0.241373	−0.120687	+	0.992691i	$0.538510\pi$
$548$	− 13.3425i	− 0.569965i
$549$	0	0
$550$	17.5831	0.749747
$551$	−22.8972	−0.975453
$552$	0	0
$553$	0	0
$554$	− 19.6283i	− 0.833925i
$555$	0	0
$556$	21.8991i	0.928730i
$557$	− 25.8164i	− 1.09387i	−0.837173	−	0.546937i	$-0.815794\pi$
$557$	− 25.8164i	− 1.09387i	0.837173	−	0.546937i	$-0.184206\pi$
$558$	0	0
$559$	13.4828i	0.570260i
$560$	0	0
$561$	0	0
$562$	−9.26424	−0.390788
$563$	−28.1008	−1.18431	−0.592153	−	0.805826i	$-0.701722\pi$
$563$	−28.1008	−1.18431	−0.592153	+	0.805826i	$0.701722\pi$
$564$	0	0
$565$	1.32248i	0.0556370i
$566$	6.48744	0.272688
$567$	0	0
$568$	−5.12150	−0.214893
$569$	31.0489i	1.30164i	0.759233	+	0.650819i	$0.225574\pi$
$569$	31.0489i	1.30164i	−0.759233	+	0.650819i	$0.774426\pi$
$570$	0	0
$571$	45.0250	1.88424	0.942119	−	0.335278i	$-0.108830\pi$
$571$	45.0250	1.88424	0.942119	+	0.335278i	$0.108830\pi$
$572$	−7.09231	−0.296545
$573$	0	0
$574$	0	0
$575$	28.6225i	1.19364i
$576$	0	0
$577$	12.5714i	0.523354i	0.965156	+	0.261677i	$0.0842754\pi$
$577$	12.5714i	0.523354i	−0.965156	+	0.261677i	$0.915725\pi$
$578$	− 11.0000i	− 0.457540i
$579$	0	0
$580$	− 0.519459i	− 0.0215694i
$581$	0	0
$582$	0	0
$583$	−32.3600	−1.34022
$584$	−7.60796	−0.314820
$585$	0	0
$586$	− 12.8799i	− 0.532065i
$587$	11.1722	0.461125	0.230562	−	0.973058i	$-0.425943\pi$
$587$	11.1722	0.461125	0.230562	+	0.973058i	$0.425943\pi$
$588$	0	0
$589$	−5.27607	−0.217397
$590$	− 1.19667i	− 0.0492662i
$591$	0	0
$592$	4.69354	0.192903
$593$	10.1941	0.418623	0.209311	−	0.977849i	$-0.432878\pi$
$593$	10.1941	0.418623	0.209311	+	0.977849i	$0.432878\pi$
$594$	0	0
$595$	0	0
$596$	21.9423i	0.898793i
$597$	0	0
$598$	− 11.5451i	− 0.472116i
$599$	− 31.6315i	− 1.29243i	−0.763156	−	0.646214i	$-0.776351\pi$
$599$	− 31.6315i	− 1.29243i	0.763156	−	0.646214i	$-0.223649\pi$
$600$	0	0
$601$	26.4742i	1.07990i	0.841696	+	0.539952i	$0.181558\pi$
$601$	26.4742i	1.07990i	−0.841696	+	0.539952i	$0.818442\pi$
$602$	0	0
$603$	0	0
$604$	2.21380	0.0900783
$605$	0.165827	0.00674183
$606$	0	0
$607$	− 24.2960i	− 0.986144i	−0.869988	−	0.493072i	$-0.835874\pi$
$607$	− 24.2960i	− 0.986144i	0.869988	−	0.493072i	$-0.164126\pi$
$608$	5.10030	0.206844
$609$	0	0
$610$	−0.316369	−0.0128094
$611$	5.96502i	0.241319i
$612$	0	0
$613$	−42.5646	−1.71917	−0.859584	−	0.510995i	$-0.829277\pi$
$613$	−42.5646	−1.71917	−0.859584	+	0.510995i	$0.829277\pi$
$614$	12.1063	0.488572
$615$	0	0
$616$	0	0
$617$	45.9424i	1.84957i	0.380490	+	0.924785i	$0.375755\pi$
$617$	45.9424i	1.84957i	−0.380490	+	0.924785i	$0.624245\pi$
$618$	0	0
$619$	− 21.6196i	− 0.868964i	−0.900680	−	0.434482i	$-0.856931\pi$
$619$	− 21.6196i	− 0.868964i	0.900680	−	0.434482i	$-0.143069\pi$
$620$	− 0.119696i	− 0.00480711i
$621$	0	0
$622$	− 28.1723i	− 1.12961i
$623$	0	0
$624$	0	0
$625$	24.7994	0.991974
$626$	−24.9784	−0.998338
$627$	0	0
$628$	− 18.3494i	− 0.732221i
$629$	11.4968	0.458406
$630$	0	0
$631$	−46.4078	−1.84747	−0.923733	−	0.383036i	$-0.874879\pi$
$631$	−46.4078	−1.84747	−0.923733	+	0.383036i	$0.874879\pi$
$632$	4.88030i	0.194128i
$633$	0	0
$634$	−28.0688	−1.11475
$635$	−1.74118	−0.0690967
$636$	0	0
$637$	0	0
$638$	15.8299i	0.626711i
$639$	0	0
$640$	0.115708i	0.00457377i
$641$	0.107493i	0.00424573i	0.999998	+	0.00212287i	$0.000675730\pi$
$641$	0.107493i	0.00424573i	−0.999998	+	0.00212287i	$0.999324\pi$
$642$	0	0
$643$	− 16.3117i	− 0.643270i	−0.946864	−	0.321635i	$-0.895768\pi$
$643$	− 16.3117i	− 0.643270i	0.946864	−	0.321635i	$-0.104232\pi$
$644$	0	0
$645$	0	0
$646$	12.4931	0.491535
$647$	−27.2436	−1.07106	−0.535528	−	0.844518i	$-0.679887\pi$
$647$	−27.2436	−1.07106	−0.535528	+	0.844518i	$0.679887\pi$
$648$	0	0
$649$	36.4671i	1.43146i
$650$	−10.0300	−0.393411
$651$	0	0
$652$	9.94438	0.389452
$653$	3.85546i	0.150876i	0.997151	+	0.0754379i	$0.0240355\pi$
$653$	3.85546i	0.150876i	−0.997151	+	0.0754379i	$0.975965\pi$
$654$	0	0
$655$	−1.52289	−0.0595043
$656$	−4.06047	−0.158535
$657$	0	0
$658$	0	0
$659$	− 3.47215i	− 0.135256i	−0.997711	−	0.0676279i	$-0.978457\pi$
$659$	− 3.47215i	− 0.135256i	0.997711	−	0.0676279i	$-0.0215431\pi$
$660$	0	0
$661$	6.78526i	0.263916i	0.991255	+	0.131958i	$0.0421264\pi$
$661$	6.78526i	0.263916i	−0.991255	+	0.131958i	$0.957874\pi$
$662$	− 30.4124i	− 1.18201i
$663$	0	0
$664$	− 17.9544i	− 0.696764i
$665$	0	0
$666$	0	0
$667$	−25.7685	−0.997759
$668$	6.88516	0.266395
$669$	0	0
$670$	0.294952i	0.0113950i
$671$	9.64095	0.372185
$672$	0	0
$673$	22.2776	0.858739	0.429370	−	0.903129i	$-0.358736\pi$
$673$	22.2776	0.858739	0.429370	+	0.903129i	$0.358736\pi$
$674$	32.5762i	1.25479i
$675$	0	0
$676$	−8.95429	−0.344396
$677$	−38.0215	−1.46129	−0.730643	−	0.682760i	$-0.760780\pi$
$677$	−38.0215	−1.46129	−0.730643	+	0.682760i	$0.760780\pi$
$678$	0	0
$679$	0	0
$680$	0.283426i	0.0108689i
$681$	0	0
$682$	3.64759i	0.139673i
$683$	12.9592i	0.495871i	0.968777	+	0.247935i	$0.0797520\pi$
$683$	12.9592i	0.495871i	−0.968777	+	0.247935i	$0.920248\pi$
$684$	0	0
$685$	− 1.54384i	− 0.0589872i
$686$	0	0
$687$	0	0
$688$	−6.70319	−0.255557
$689$	18.4593	0.703244
$690$	0	0
$691$	− 49.4231i	− 1.88014i	−0.340978	−	0.940071i	$-0.610758\pi$
$691$	− 49.4231i	− 1.88014i	0.340978	−	0.940071i	$-0.389242\pi$
$692$	16.2961	0.619485
$693$	0	0
$694$	−12.5588	−0.476724
$695$	2.53391i	0.0961167i
$696$	0	0
$697$	−9.94608	−0.376735
$698$	17.1062	0.647481
$699$	0	0
$700$	0	0
$701$	4.29078i	0.162060i	0.996712	+	0.0810302i	$0.0258210\pi$
$701$	4.29078i	0.162060i	−0.996712	+	0.0810302i	$0.974179\pi$
$702$	0	0
$703$	− 23.9384i	− 0.902855i
$704$	− 3.52607i	− 0.132894i
$705$	0	0
$706$	9.61557i	0.361887i
$707$	0	0
$708$	0	0
$709$	47.6242	1.78856	0.894282	−	0.447504i	$-0.147687\pi$
$709$	47.6242	1.78856	0.894282	+	0.447504i	$0.147687\pi$
$710$	−0.592600	−0.0222399
$711$	0	0
$712$	− 10.0274i	− 0.375791i
$713$	−5.93769	−0.222368
$714$	0	0
$715$	−0.820639	−0.0306902
$716$	3.56456i	0.133214i
$717$	0	0
$718$	−15.0521	−0.561741
$719$	11.5753	0.431685	0.215842	−	0.976428i	$-0.430750\pi$
$719$	11.5753	0.431685	0.215842	+	0.976428i	$0.430750\pi$
$720$	0	0
$721$	0	0
$722$	− 7.01303i	− 0.260998i
$723$	0	0
$724$	− 14.7498i	− 0.548170i
$725$	22.3868i	0.831425i
$726$	0	0
$727$	− 25.4165i	− 0.942646i	−0.881961	−	0.471323i	$-0.843777\pi$
$727$	− 25.4165i	− 0.942646i	0.881961	−	0.471323i	$-0.156223\pi$
$728$	0	0
$729$	0	0
$730$	−0.880304	−0.0325815
$731$	−16.4194	−0.607293
$732$	0	0
$733$	− 52.2922i	− 1.93146i	−0.259557	−	0.965728i	$-0.583577\pi$
$733$	− 52.2922i	− 1.93146i	0.259557	−	0.965728i	$-0.416423\pi$
$734$	11.2943	0.416882
$735$	0	0
$736$	5.73987	0.211574
$737$	− 8.98831i	− 0.331089i
$738$	0	0
$739$	−13.1369	−0.483250	−0.241625	−	0.970370i	$-0.577680\pi$
$739$	−13.1369	−0.483250	−0.241625	+	0.970370i	$0.577680\pi$
$740$	0.543081	0.0199641
$741$	0	0
$742$	0	0
$743$	− 5.94789i	− 0.218207i	−0.994030	−	0.109103i	$-0.965202\pi$
$743$	− 5.94789i	− 0.218207i	0.994030	−	0.109103i	$-0.0347980\pi$
$744$	0	0
$745$	2.53891i	0.0930185i
$746$	− 5.56600i	− 0.203786i
$747$	0	0
$748$	− 8.63706i	− 0.315802i
$749$	0	0
$750$	0	0
$751$	4.68990	0.171137	0.0855685	−	0.996332i	$-0.472729\pi$
$751$	4.68990	0.171137	0.0855685	+	0.996332i	$0.472729\pi$
$752$	−2.96561	−0.108145
$753$	0	0
$754$	− 9.02993i	− 0.328851i
$755$	0.256155	0.00932244
$756$	0	0
$757$	−32.9356	−1.19706	−0.598532	−	0.801099i	$-0.704249\pi$
$757$	−32.9356	−1.19706	−0.598532	+	0.801099i	$0.704249\pi$
$758$	− 5.80720i	− 0.210927i
$759$	0	0
$760$	0.590146	0.0214069
$761$	−41.4163	−1.50134	−0.750669	−	0.660678i	$-0.770269\pi$
$761$	−41.4163	−1.50134	−0.750669	+	0.660678i	$0.770269\pi$
$762$	0	0
$763$	0	0
$764$	− 20.2251i	− 0.731720i
$765$	0	0
$766$	7.28335i	0.263158i
$767$	− 20.8022i	− 0.751123i
$768$	0	0
$769$	19.2896i	0.695601i	0.937569	+	0.347801i	$0.113071\pi$
$769$	19.2896i	0.695601i	−0.937569	+	0.347801i	$0.886929\pi$
$770$	0	0
$771$	0	0
$772$	−3.94557	−0.142004
$773$	20.8000	0.748125	0.374062	−	0.927404i	$-0.377965\pi$
$773$	20.8000	0.748125	0.374062	+	0.927404i	$0.377965\pi$
$774$	0	0
$775$	5.15847i	0.185298i
$776$	−12.2755	−0.440666
$777$	0	0
$778$	23.7553	0.851669
$779$	20.7096i	0.741999i
$780$	0	0
$781$	18.0588	0.646193
$782$	14.0597	0.502776
$783$	0	0
$784$	0	0
$785$	− 2.12318i	− 0.0757795i
$786$	0	0
$787$	− 38.5833i	− 1.37535i	−0.726020	−	0.687674i	$-0.758632\pi$
$787$	− 38.5833i	− 1.37535i	0.726020	−	0.687674i	$-0.241368\pi$
$788$	− 2.29826i	− 0.0818720i
$789$	0	0
$790$	0.564691i	0.0200908i
$791$	0	0
$792$	0	0
$793$	−5.49955	−0.195295
$794$	15.8429	0.562242
$795$	0	0
$796$	25.9021i	0.918074i
$797$	−42.3609	−1.50050	−0.750250	−	0.661155i	$-0.770067\pi$
$797$	−42.3609	−1.50050	−0.750250	+	0.661155i	$0.770067\pi$
$798$	0	0
$799$	−7.26424	−0.256990
$800$	− 4.98661i	− 0.176303i
$801$	0	0
$802$	−8.33412	−0.294288
$803$	26.8262	0.946675
$804$	0	0
$805$	0	0
$806$	− 2.08072i	− 0.0732901i
$807$	0	0
$808$	− 18.5530i	− 0.652691i
$809$	− 11.0315i	− 0.387847i	−0.981017	−	0.193923i	$-0.937879\pi$
$809$	− 11.0315i	− 0.387847i	0.981017	−	0.193923i	$-0.0621213\pi$
$810$	0	0
$811$	− 2.99413i	− 0.105138i	−0.998617	−	0.0525691i	$-0.983259\pi$
$811$	− 2.99413i	− 0.105138i	0.998617	−	0.0525691i	$-0.0167410\pi$
$812$	0	0
$813$	0	0
$814$	−16.5497	−0.580068
$815$	1.15065	0.0403054
$816$	0	0
$817$	34.1882i	1.19609i
$818$	−28.1151	−0.983020
$819$	0	0
$820$	−0.469830	−0.0164072
$821$	− 5.01543i	− 0.175040i	−0.996163	−	0.0875198i	$-0.972106\pi$
$821$	− 5.01543i	− 0.175040i	0.996163	−	0.0875198i	$-0.0278941\pi$
$822$	0	0
$823$	46.2889	1.61353	0.806765	−	0.590873i	$-0.201217\pi$
$823$	46.2889	1.61353	0.806765	+	0.590873i	$0.201217\pi$
$824$	−9.25440	−0.322392
$825$	0	0
$826$	0	0
$827$	14.2511i	0.495559i	0.968816	+	0.247780i	$0.0797009\pi$
$827$	14.2511i	0.495559i	−0.968816	+	0.247780i	$0.920299\pi$
$828$	0	0
$829$	43.4959i	1.51068i	0.655336	+	0.755338i	$0.272527\pi$
$829$	43.4959i	1.51068i	−0.655336	+	0.755338i	$0.727473\pi$
$830$	− 2.07747i	− 0.0721100i
$831$	0	0
$832$	2.01140i	0.0697326i
$833$	0	0
$834$	0	0
$835$	0.796670	0.0275699
$836$	−17.9840	−0.621989
$837$	0	0
$838$	− 15.3664i	− 0.530825i
$839$	24.1805	0.834802	0.417401	−	0.908722i	$-0.362941\pi$
$839$	24.1805	0.834802	0.417401	+	0.908722i	$0.362941\pi$
$840$	0	0
$841$	8.84543	0.305015
$842$	− 2.33930i	− 0.0806176i
$843$	0	0
$844$	−2.20067	−0.0757502
$845$	−1.03609	−0.0356424
$846$	0	0
$847$	0	0
$848$	9.17738i	0.315152i
$849$	0	0
$850$	− 12.2147i	− 0.418959i
$851$	− 26.9403i	− 0.923501i
$852$	0	0
$853$	31.9963i	1.09553i	0.836632	+	0.547766i	$0.184522\pi$
$853$	31.9963i	1.09553i	−0.836632	+	0.547766i	$0.815478\pi$
$854$	0	0
$855$	0	0
$856$	−18.3003	−0.625492
$857$	−53.7174	−1.83495	−0.917476	−	0.397790i	$-0.869777\pi$
$857$	−53.7174	−1.83495	−0.917476	+	0.397790i	$0.869777\pi$
$858$	0	0
$859$	− 17.3109i	− 0.590640i	−0.955398	−	0.295320i	$-0.904574\pi$
$859$	− 17.3109i	− 0.590640i	0.955398	−	0.295320i	$-0.0954263\pi$
$860$	−0.775614	−0.0264482
$861$	0	0
$862$	24.4954	0.834318
$863$	2.08650i	0.0710252i	0.999369	+	0.0355126i	$0.0113064\pi$
$863$	2.08650i	0.0710252i	−0.999369	+	0.0355126i	$0.988694\pi$
$864$	0	0
$865$	1.88559	0.0641121
$866$	4.66332	0.158466
$867$	0	0
$868$	0	0
$869$	− 17.2083i	− 0.583751i
$870$	0	0
$871$	5.12726i	0.173730i
$872$	14.7842i	0.500655i
$873$	0	0
$874$	− 29.2750i	− 0.990243i
$875$	0	0
$876$	0	0
$877$	11.9004	0.401847	0.200924	−	0.979607i	$-0.435606\pi$
$877$	11.9004	0.401847	0.200924	+	0.979607i	$0.435606\pi$
$878$	16.9027	0.570439
$879$	0	0
$880$	− 0.407995i	− 0.0137535i
$881$	−31.5992	−1.06460	−0.532302	−	0.846554i	$-0.678673\pi$
$881$	−31.5992	−1.06460	−0.532302	+	0.846554i	$0.678673\pi$
$882$	0	0
$883$	−35.5159	−1.19521	−0.597603	−	0.801792i	$-0.703880\pi$
$883$	−35.5159	−1.19521	−0.597603	+	0.801792i	$0.703880\pi$
$884$	4.92689i	0.165709i
$885$	0	0
$886$	−34.2190	−1.14961
$887$	−8.49754	−0.285319	−0.142660	−	0.989772i	$-0.545565\pi$
$887$	−8.49754	−0.285319	−0.142660	+	0.989772i	$0.545565\pi$
$888$	0	0
$889$	0	0
$890$	− 1.16025i	− 0.0388916i
$891$	0	0
$892$	5.12818i	0.171704i
$893$	15.1255i	0.506156i
$894$	0	0
$895$	0.412448i	0.0137866i
$896$	0	0
$897$	0	0
$898$	−1.38574	−0.0462426
$899$	−4.64411	−0.154890
$900$	0	0
$901$	22.4799i	0.748914i
$902$	14.3175	0.476720
$903$	0	0
$904$	−11.4294	−0.380136
$905$	− 1.70667i	− 0.0567316i
$906$	0	0
$907$	10.4395	0.346639	0.173320	−	0.984866i	$-0.444551\pi$
$907$	10.4395	0.346639	0.173320	+	0.984866i	$0.444551\pi$
$908$	−18.1149	−0.601163
$909$	0	0
$910$	0	0
$911$	0.0955475i	0.00316563i	0.999999	+	0.00158282i	$0.000503826\pi$
$911$	0.0955475i	0.00316563i	−0.999999	+	0.00158282i	$0.999496\pi$
$912$	0	0
$913$	63.3083i	2.09520i
$914$	30.3990i	1.00551i
$915$	0	0
$916$	6.45493i	0.213277i
$917$	0	0
$918$	0	0
$919$	14.0845	0.464603	0.232302	−	0.972644i	$-0.425374\pi$
$919$	14.0845	0.464603	0.232302	+	0.972644i	$0.425374\pi$
$920$	0.664150	0.0218964
$921$	0	0
$922$	16.3888i	0.539737i
$923$	−10.3014	−0.339074
$924$	0	0
$925$	−23.4048	−0.769547
$926$	− 7.18677i	− 0.236172i
$927$	0	0
$928$	4.48938	0.147371
$929$	40.3080	1.32246	0.661232	−	0.750182i	$-0.270034\pi$
$929$	40.3080	1.32246	0.661232	+	0.750182i	$0.270034\pi$
$930$	0	0
$931$	0	0
$932$	12.0288i	0.394018i
$933$	0	0
$934$	− 13.7962i	− 0.451426i
$935$	− 0.999380i	− 0.0326832i
$936$	0	0
$937$	33.1386i	1.08259i	0.840833	+	0.541295i	$0.182066\pi$
$937$	33.1386i	1.08259i	−0.840833	+	0.541295i	$0.817934\pi$
$938$	0	0
$939$	0	0
$940$	−0.343146	−0.0111922
$941$	−2.92082	−0.0952161	−0.0476081	−	0.998866i	$-0.515160\pi$
$941$	−2.92082	−0.0952161	−0.0476081	+	0.998866i	$0.515160\pi$
$942$	0	0
$943$	23.3066i	0.758966i
$944$	10.3422	0.336609
$945$	0	0
$946$	23.6359	0.768469
$947$	− 33.2988i	− 1.08207i	−0.841002	−	0.541033i	$-0.818034\pi$
$947$	− 33.2988i	− 1.08207i	0.841002	−	0.541033i	$-0.181966\pi$
$948$	0	0
$949$	−15.3026	−0.496744
$950$	−25.4332	−0.825162
$951$	0	0
$952$	0	0
$953$	− 19.2086i	− 0.622228i	−0.950373	−	0.311114i	$-0.899298\pi$
$953$	− 19.2086i	− 0.622228i	0.950373	−	0.311114i	$-0.100702\pi$
$954$	0	0
$955$	− 2.34022i	− 0.0757276i
$956$	1.19342i	0.0385981i
$957$	0	0
$958$	− 14.1443i	− 0.456980i
$959$	0	0
$960$	0	0
$961$	29.9299	0.965480
$962$	9.44056	0.304376
$963$	0	0
$964$	− 11.9940i	− 0.386299i
$965$	−0.456535	−0.0146964
$966$	0	0
$967$	−13.2334	−0.425556	−0.212778	−	0.977101i	$-0.568251\pi$
$967$	−13.2334	−0.425556	−0.212778	+	0.977101i	$0.568251\pi$
$968$	1.43315i	0.0460631i
$969$	0	0
$970$	−1.42038	−0.0456056
$971$	14.4446	0.463548	0.231774	−	0.972770i	$-0.425547\pi$
$971$	14.4446	0.463548	0.231774	+	0.972770i	$0.425547\pi$
$972$	0	0
$973$	0	0
$974$	− 9.92664i	− 0.318070i
$975$	0	0
$976$	− 2.73420i	− 0.0875195i
$977$	25.1431i	0.804400i	0.915552	+	0.402200i	$0.131754\pi$
$977$	25.1431i	0.804400i	−0.915552	+	0.402200i	$0.868246\pi$
$978$	0	0
$979$	35.3571i	1.13002i
$980$	0	0
$981$	0	0
$982$	1.46802	0.0468465
$983$	55.0721	1.75653	0.878263	−	0.478178i	$-0.158702\pi$
$983$	55.0721	1.75653	0.878263	+	0.478178i	$0.158702\pi$
$984$	0	0
$985$	− 0.265927i	− 0.00847315i
$986$	10.9967	0.350206
$987$	0	0
$988$	10.2587	0.326373
$989$	38.4754i	1.22345i
$990$	0	0
$991$	49.2537	1.56459	0.782297	−	0.622905i	$-0.214048\pi$
$991$	49.2537	1.56459	0.782297	+	0.622905i	$0.214048\pi$
$992$	1.03446	0.0328443
$993$	0	0
$994$	0	0
$995$	2.99708i	0.0950139i
$996$	0	0
$997$	− 35.6075i	− 1.12770i	−0.825877	−	0.563850i	$-0.809319\pi$
$997$	− 35.6075i	− 1.12770i	0.825877	−	0.563850i	$-0.190681\pi$
$998$	8.29707i	0.262639i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.d.f.2645.12	yes	16
3.2	odd	2	inner	2646.2.d.f.2645.5	yes	16
7.6	odd	2	inner	2646.2.d.f.2645.13	yes	16
21.20	even	2	inner	2646.2.d.f.2645.4	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2646.2.d.f.2645.4	✓	16	21.20	even	2	inner
2646.2.d.f.2645.5	yes	16	3.2	odd	2	inner
2646.2.d.f.2645.12	yes	16	1.1	even	1	trivial
2646.2.d.f.2645.13	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:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -0.115708 q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -0.115708 q^{5} -1.00000i q^{8} -0.115708i q^{10} +3.52607i q^{11} -2.01140i q^{13} +1.00000 q^{16} +2.44949 q^{17} -5.10030i q^{19} +0.115708 q^{20} -3.52607 q^{22} -5.73987i q^{23} -4.98661 q^{25} +2.01140 q^{26} -4.48938i q^{29} -1.03446i q^{31} +1.00000i q^{32} +2.44949i q^{34} +4.69354 q^{37} +5.10030 q^{38} +0.115708i q^{40} -4.06047 q^{41} -6.70319 q^{43} -3.52607i q^{44} +5.73987 q^{46} -2.96561 q^{47} -4.98661i q^{50} +2.01140i q^{52} +9.17738i q^{53} -0.407995i q^{55} +4.48938 q^{58} +10.3422 q^{59} -2.73420i q^{61} +1.03446 q^{62} -1.00000 q^{64} +0.232735i q^{65} -2.54910 q^{67} -2.44949 q^{68} -5.12150i q^{71} -7.60796i q^{73} +4.69354i q^{74} +5.10030i q^{76} -4.88030 q^{79} -0.115708 q^{80} -4.06047i q^{82} +17.9544 q^{83} -0.283426 q^{85} -6.70319i q^{86} +3.52607 q^{88} +10.0274 q^{89} +5.73987i q^{92} -2.96561i q^{94} +0.590146i q^{95} -12.2755i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 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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