LMFDB - Embedded newform 2646.2.d.a.2645.3

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:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 378)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2645.3
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2646.2645
Dual form			2646.2.d.a.2645.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -1.73205 q^{5} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} -1.73205 q^{5} -1.00000i q^{8} -1.73205i q^{10} +1.00000 q^{16} +3.46410 q^{17} +6.92820i q^{19} +1.73205 q^{20} -6.00000i q^{23} -2.00000 q^{25} -9.00000i q^{29} -3.46410i q^{31} +1.00000i q^{32} +3.46410i q^{34} +4.00000 q^{37} -6.92820 q^{38} +1.73205i q^{40} -3.46410 q^{41} -8.00000 q^{43} +6.00000 q^{46} +3.46410 q^{47} -2.00000i q^{50} +3.00000i q^{53} +9.00000 q^{58} +12.1244 q^{59} +3.46410i q^{61} +3.46410 q^{62} -1.00000 q^{64} +14.0000 q^{67} -3.46410 q^{68} +6.00000i q^{71} +12.1244i q^{73} +4.00000i q^{74} -6.92820i q^{76} +11.0000 q^{79} -1.73205 q^{80} -3.46410i q^{82} +17.3205 q^{83} -6.00000 q^{85} -8.00000i q^{86} +10.3923 q^{89} +6.00000i q^{92} +3.46410i q^{94} -12.0000i q^{95} +6.92820i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 4 q^{16} - 8 q^{25} + 16 q^{37} - 32 q^{43} + 24 q^{46} + 36 q^{58} - 4 q^{64} + 56 q^{67} + 44 q^{79} - 24 q^{85}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^16 - 8 * q^25 + 16 * q^37 - 32 * q^43 + 24 * q^46 + 36 * q^58 - 4 * q^64 + 56 * q^67 + 44 * q^79 - 24 * q^85

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
$5$	−1.73205	−0.774597	−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205	−0.774597	−0.387298	+	0.921954i	$0.626592\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	− 1.73205i	− 0.547723i
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	6.92820i	1.58944i	0.606977	+	0.794719i	$0.292382\pi$
$19$	6.92820i	1.58944i	−0.606977	+	0.794719i	$0.707618\pi$
$20$	1.73205	0.387298
$21$	0	0
$22$	0	0
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 9.00000i	− 1.67126i	−0.549294	−	0.835629i	$-0.685103\pi$
$29$	− 9.00000i	− 1.67126i	0.549294	−	0.835629i	$-0.314897\pi$
$30$	0	0
$31$	− 3.46410i	− 0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$	− 3.46410i	− 0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	3.46410i	0.594089i
$35$	0	0
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	−6.92820	−1.12390
$39$	0	0
$40$	1.73205i	0.273861i
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	0	0
$46$	6.00000	0.884652
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	0	0
$50$	− 2.00000i	− 0.282843i
$51$	0	0
$52$	0	0
$53$	3.00000i	0.412082i	0.978543	+	0.206041i	$0.0660580\pi$
$53$	3.00000i	0.412082i	−0.978543	+	0.206041i	$0.933942\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	9.00000	1.18176
$59$	12.1244	1.57846	0.789228	−	0.614100i	$-0.210481\pi$
$59$	12.1244	1.57846	0.789228	+	0.614100i	$0.210481\pi$
$60$	0	0
$61$	3.46410i	0.443533i	0.975100	+	0.221766i	$0.0711822\pi$
$61$	3.46410i	0.443533i	−0.975100	+	0.221766i	$0.928818\pi$
$62$	3.46410	0.439941
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	14.0000	1.71037	0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	−3.46410	−0.420084
$69$	0	0
$70$	0	0
$71$	6.00000i	0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$	6.00000i	0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$	0	0
$73$	12.1244i	1.41905i	0.704681	+	0.709524i	$0.251090\pi$
$73$	12.1244i	1.41905i	−0.704681	+	0.709524i	$0.748910\pi$
$74$	4.00000i	0.464991i
$75$	0	0
$76$	− 6.92820i	− 0.794719i
$77$	0	0
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	−1.73205	−0.193649
$81$	0	0
$82$	− 3.46410i	− 0.382546i
$83$	17.3205	1.90117	0.950586	−	0.310460i	$-0.100483\pi$
$83$	17.3205	1.90117	0.950586	+	0.310460i	$0.100483\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	− 8.00000i	− 0.862662i
$87$	0	0
$88$	0	0
$89$	10.3923	1.10158	0.550791	−	0.834643i	$-0.314326\pi$
$89$	10.3923	1.10158	0.550791	+	0.834643i	$0.314326\pi$
$90$	0	0
$91$	0	0
$92$	6.00000i	0.625543i
$93$	0	0
$94$	3.46410i	0.357295i
$95$	− 12.0000i	− 1.23117i
$96$	0	0
$97$	6.92820i	0.703452i	0.936103	+	0.351726i	$0.114405\pi$
$97$	6.92820i	0.703452i	−0.936103	+	0.351726i	$0.885595\pi$
$98$	0	0
$99$	0	0
$100$	2.00000	0.200000
$101$	−5.19615	−0.517036	−0.258518	−	0.966006i	$-0.583234\pi$
$101$	−5.19615	−0.517036	−0.258518	+	0.966006i	$0.583234\pi$
$102$	0	0
$103$	10.3923i	1.02398i	0.858990	+	0.511992i	$0.171092\pi$
$103$	10.3923i	1.02398i	−0.858990	+	0.511992i	$0.828908\pi$
$104$	0	0
$105$	0	0
$106$	−3.00000	−0.291386
$107$	3.00000i	0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$	3.00000i	0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 18.0000i	− 1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$	− 18.0000i	− 1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$	0	0
$115$	10.3923i	0.969087i
$116$	9.00000i	0.835629i
$117$	0	0
$118$	12.1244i	1.11614i
$119$	0	0
$120$	0	0
$121$	11.0000	1.00000
$122$	−3.46410	−0.313625
$123$	0	0
$124$	3.46410i	0.311086i
$125$	12.1244	1.08444
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−10.3923	−0.907980	−0.453990	−	0.891007i	$-0.650000\pi$
$131$	−10.3923	−0.907980	−0.453990	+	0.891007i	$0.650000\pi$
$132$	0	0
$133$	0	0
$134$	14.0000i	1.20942i
$135$	0	0
$136$	− 3.46410i	− 0.297044i
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	10.3923i	0.881464i	0.897639	+	0.440732i	$0.145281\pi$
$139$	10.3923i	0.881464i	−0.897639	+	0.440732i	$0.854719\pi$
$140$	0	0
$141$	0	0
$142$	−6.00000	−0.503509
$143$	0	0
$144$	0	0
$145$	15.5885i	1.29455i
$146$	−12.1244	−1.00342
$147$	0	0
$148$	−4.00000	−0.328798
$149$	15.0000i	1.22885i	0.788976	+	0.614424i	$0.210612\pi$
$149$	15.0000i	1.22885i	−0.788976	+	0.614424i	$0.789388\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	6.92820	0.561951
$153$	0	0
$154$	0	0
$155$	6.00000i	0.481932i
$156$	0	0
$157$	− 10.3923i	− 0.829396i	−0.909959	−	0.414698i	$-0.863887\pi$
$157$	− 10.3923i	− 0.829396i	0.909959	−	0.414698i	$-0.136113\pi$
$158$	11.0000i	0.875113i
$159$	0	0
$160$	− 1.73205i	− 0.136931i
$161$	0	0
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	3.46410	0.270501
$165$	0	0
$166$	17.3205i	1.34433i
$167$	−17.3205	−1.34030	−0.670151	−	0.742225i	$-0.733770\pi$
$167$	−17.3205	−1.34030	−0.670151	+	0.742225i	$0.733770\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	− 6.00000i	− 0.460179i
$171$	0	0
$172$	8.00000	0.609994
$173$	5.19615	0.395056	0.197528	−	0.980297i	$-0.436709\pi$
$173$	5.19615	0.395056	0.197528	+	0.980297i	$0.436709\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	10.3923i	0.778936i
$179$	− 15.0000i	− 1.12115i	−0.828103	−	0.560576i	$-0.810580\pi$
$179$	− 15.0000i	− 1.12115i	0.828103	−	0.560576i	$-0.189420\pi$
$180$	0	0
$181$	− 17.3205i	− 1.28742i	−0.765268	−	0.643712i	$-0.777394\pi$
$181$	− 17.3205i	− 1.28742i	0.765268	−	0.643712i	$-0.222606\pi$
$182$	0	0
$183$	0	0
$184$	−6.00000	−0.442326
$185$	−6.92820	−0.509372
$186$	0	0
$187$	0	0
$188$	−3.46410	−0.252646
$189$	0	0
$190$	12.0000	0.870572
$191$	− 6.00000i	− 0.434145i	−0.976156	−	0.217072i	$-0.930349\pi$
$191$	− 6.00000i	− 0.434145i	0.976156	−	0.217072i	$-0.0696508\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	−6.92820	−0.497416
$195$	0	0
$196$	0	0
$197$	− 3.00000i	− 0.213741i	−0.994273	−	0.106871i	$-0.965917\pi$
$197$	− 3.00000i	− 0.213741i	0.994273	−	0.106871i	$-0.0340831\pi$
$198$	0	0
$199$	− 8.66025i	− 0.613909i	−0.951724	−	0.306955i	$-0.900690\pi$
$199$	− 8.66025i	− 0.613909i	0.951724	−	0.306955i	$-0.0993100\pi$
$200$	2.00000i	0.141421i
$201$	0	0
$202$	− 5.19615i	− 0.365600i
$203$	0	0
$204$	0	0
$205$	6.00000	0.419058
$206$	−10.3923	−0.724066
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	− 3.00000i	− 0.206041i
$213$	0	0
$214$	−3.00000	−0.205076
$215$	13.8564	0.944999
$216$	0	0
$217$	0	0
$218$	16.0000i	1.08366i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	15.5885i	1.04388i	0.852982	+	0.521940i	$0.174792\pi$
$223$	15.5885i	1.04388i	−0.852982	+	0.521940i	$0.825208\pi$
$224$	0	0
$225$	0	0
$226$	18.0000	1.19734
$227$	25.9808	1.72440	0.862202	−	0.506565i	$-0.169085\pi$
$227$	25.9808	1.72440	0.862202	+	0.506565i	$0.169085\pi$
$228$	0	0
$229$	− 3.46410i	− 0.228914i	−0.993428	−	0.114457i	$-0.963487\pi$
$229$	− 3.46410i	− 0.228914i	0.993428	−	0.114457i	$-0.0365129\pi$
$230$	−10.3923	−0.685248
$231$	0	0
$232$	−9.00000	−0.590879
$233$	30.0000i	1.96537i	0.185296	+	0.982683i	$0.440675\pi$
$233$	30.0000i	1.96537i	−0.185296	+	0.982683i	$0.559325\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	−12.1244	−0.789228
$237$	0	0
$238$	0	0
$239$	− 24.0000i	− 1.55243i	−0.630468	−	0.776215i	$-0.717137\pi$
$239$	− 24.0000i	− 1.55243i	0.630468	−	0.776215i	$-0.282863\pi$
$240$	0	0
$241$	− 1.73205i	− 0.111571i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	− 1.73205i	− 0.111571i	0.998443	−	0.0557856i	$-0.0177663\pi$
$242$	11.0000i	0.707107i
$243$	0	0
$244$	− 3.46410i	− 0.221766i
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−3.46410	−0.219971
$249$	0	0
$250$	12.1244i	0.766812i
$251$	8.66025	0.546630	0.273315	−	0.961925i	$-0.411880\pi$
$251$	8.66025	0.546630	0.273315	+	0.961925i	$0.411880\pi$
$252$	0	0
$253$	0	0
$254$	− 7.00000i	− 0.439219i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−20.7846	−1.29651	−0.648254	−	0.761424i	$-0.724501\pi$
$257$	−20.7846	−1.29651	−0.648254	+	0.761424i	$0.724501\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	− 10.3923i	− 0.642039i
$263$	− 18.0000i	− 1.10993i	−0.831875	−	0.554964i	$-0.812732\pi$
$263$	− 18.0000i	− 1.10993i	0.831875	−	0.554964i	$-0.187268\pi$
$264$	0	0
$265$	− 5.19615i	− 0.319197i
$266$	0	0
$267$	0	0
$268$	−14.0000	−0.855186
$269$	1.73205	0.105605	0.0528025	−	0.998605i	$-0.483185\pi$
$269$	1.73205	0.105605	0.0528025	+	0.998605i	$0.483185\pi$
$270$	0	0
$271$	12.1244i	0.736502i	0.929726	+	0.368251i	$0.120043\pi$
$271$	12.1244i	0.736502i	−0.929726	+	0.368251i	$0.879957\pi$
$272$	3.46410	0.210042
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−10.3923	−0.623289
$279$	0	0
$280$	0	0
$281$	18.0000i	1.07379i	0.843649	+	0.536895i	$0.180403\pi$
$281$	18.0000i	1.07379i	−0.843649	+	0.536895i	$0.819597\pi$
$282$	0	0
$283$	− 6.92820i	− 0.411839i	−0.978569	−	0.205919i	$-0.933982\pi$
$283$	− 6.92820i	− 0.411839i	0.978569	−	0.205919i	$-0.0660185\pi$
$284$	− 6.00000i	− 0.356034i
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−5.00000	−0.294118
$290$	−15.5885	−0.915386
$291$	0	0
$292$	− 12.1244i	− 0.709524i
$293$	−29.4449	−1.72019	−0.860094	−	0.510136i	$-0.829595\pi$
$293$	−29.4449	−1.72019	−0.860094	+	0.510136i	$0.829595\pi$
$294$	0	0
$295$	−21.0000	−1.22267
$296$	− 4.00000i	− 0.232495i
$297$	0	0
$298$	−15.0000	−0.868927
$299$	0	0
$300$	0	0
$301$	0	0
$302$	− 8.00000i	− 0.460348i
$303$	0	0
$304$	6.92820i	0.397360i
$305$	− 6.00000i	− 0.343559i
$306$	0	0
$307$	6.92820i	0.395413i	0.980261	+	0.197707i	$0.0633494\pi$
$307$	6.92820i	0.395413i	−0.980261	+	0.197707i	$0.936651\pi$
$308$	0	0
$309$	0	0
$310$	−6.00000	−0.340777
$311$	13.8564	0.785725	0.392862	−	0.919597i	$-0.371485\pi$
$311$	13.8564	0.785725	0.392862	+	0.919597i	$0.371485\pi$
$312$	0	0
$313$	− 22.5167i	− 1.27272i	−0.771393	−	0.636358i	$-0.780440\pi$
$313$	− 22.5167i	− 1.27272i	0.771393	−	0.636358i	$-0.219560\pi$
$314$	10.3923	0.586472
$315$	0	0
$316$	−11.0000	−0.618798
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	0	0
$319$	0	0
$320$	1.73205	0.0968246
$321$	0	0
$322$	0	0
$323$	24.0000i	1.33540i
$324$	0	0
$325$	0	0
$326$	2.00000i	0.110770i
$327$	0	0
$328$	3.46410i	0.191273i
$329$	0	0
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	−17.3205	−0.950586
$333$	0	0
$334$	− 17.3205i	− 0.947736i
$335$	−24.2487	−1.32485
$336$	0	0
$337$	−23.0000	−1.25289	−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$338$	13.0000i	0.707107i
$339$	0	0
$340$	6.00000	0.325396
$341$	0	0
$342$	0	0
$343$	0	0
$344$	8.00000i	0.431331i
$345$	0	0
$346$	5.19615i	0.279347i
$347$	− 33.0000i	− 1.77153i	−0.464131	−	0.885766i	$-0.653633\pi$
$347$	− 33.0000i	− 1.77153i	0.464131	−	0.885766i	$-0.346367\pi$
$348$	0	0
$349$	3.46410i	0.185429i	0.995693	+	0.0927146i	$0.0295544\pi$
$349$	3.46410i	0.185429i	−0.995693	+	0.0927146i	$0.970446\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.46410	0.184376	0.0921878	−	0.995742i	$-0.470614\pi$
$353$	3.46410	0.184376	0.0921878	+	0.995742i	$0.470614\pi$
$354$	0	0
$355$	− 10.3923i	− 0.551566i
$356$	−10.3923	−0.550791
$357$	0	0
$358$	15.0000	0.792775
$359$	− 24.0000i	− 1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$	− 24.0000i	− 1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$	0	0
$361$	−29.0000	−1.52632
$362$	17.3205	0.910346
$363$	0	0
$364$	0	0
$365$	− 21.0000i	− 1.09919i
$366$	0	0
$367$	8.66025i	0.452062i	0.974120	+	0.226031i	$0.0725750\pi$
$367$	8.66025i	0.452062i	−0.974120	+	0.226031i	$0.927425\pi$
$368$	− 6.00000i	− 0.312772i
$369$	0	0
$370$	− 6.92820i	− 0.360180i
$371$	0	0
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	0	0
$376$	− 3.46410i	− 0.178647i
$377$	0	0
$378$	0	0
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	12.0000i	0.615587i
$381$	0	0
$382$	6.00000	0.306987
$383$	17.3205	0.885037	0.442518	−	0.896759i	$-0.354085\pi$
$383$	17.3205	0.885037	0.442518	+	0.896759i	$0.354085\pi$
$384$	0	0
$385$	0	0
$386$	10.0000i	0.508987i
$387$	0	0
$388$	− 6.92820i	− 0.351726i
$389$	21.0000i	1.06474i	0.846511	+	0.532371i	$0.178699\pi$
$389$	21.0000i	1.06474i	−0.846511	+	0.532371i	$0.821301\pi$
$390$	0	0
$391$	− 20.7846i	− 1.05112i
$392$	0	0
$393$	0	0
$394$	3.00000	0.151138
$395$	−19.0526	−0.958638
$396$	0	0
$397$	− 27.7128i	− 1.39087i	−0.718591	−	0.695433i	$-0.755213\pi$
$397$	− 27.7128i	− 1.39087i	0.718591	−	0.695433i	$-0.244787\pi$
$398$	8.66025	0.434099
$399$	0	0
$400$	−2.00000	−0.100000
$401$	− 6.00000i	− 0.299626i	−0.988714	−	0.149813i	$-0.952133\pi$
$401$	− 6.00000i	− 0.299626i	0.988714	−	0.149813i	$-0.0478671\pi$
$402$	0	0
$403$	0	0
$404$	5.19615	0.258518
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	15.5885i	0.770800i	0.922750	+	0.385400i	$0.125936\pi$
$409$	15.5885i	0.770800i	−0.922750	+	0.385400i	$0.874064\pi$
$410$	6.00000i	0.296319i
$411$	0	0
$412$	− 10.3923i	− 0.511992i
$413$	0	0
$414$	0	0
$415$	−30.0000	−1.47264
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.46410	0.169232	0.0846162	−	0.996414i	$-0.473034\pi$
$419$	3.46410	0.169232	0.0846162	+	0.996414i	$0.473034\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	16.0000i	0.778868i
$423$	0	0
$424$	3.00000	0.145693
$425$	−6.92820	−0.336067
$426$	0	0
$427$	0	0
$428$	− 3.00000i	− 0.145010i
$429$	0	0
$430$	13.8564i	0.668215i
$431$	36.0000i	1.73406i	0.498257	+	0.867029i	$0.333974\pi$
$431$	36.0000i	1.73406i	−0.498257	+	0.867029i	$0.666026\pi$
$432$	0	0
$433$	− 1.73205i	− 0.0832370i	−0.999134	−	0.0416185i	$-0.986749\pi$
$433$	− 1.73205i	− 0.0832370i	0.999134	−	0.0416185i	$-0.0132514\pi$
$434$	0	0
$435$	0	0
$436$	−16.0000	−0.766261
$437$	41.5692	1.98853
$438$	0	0
$439$	− 31.1769i	− 1.48799i	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	− 31.1769i	− 1.48799i	0.668184	−	0.743996i	$-0.267072\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9.00000i	0.427603i	0.976877	+	0.213801i	$0.0685846\pi$
$443$	9.00000i	0.427603i	−0.976877	+	0.213801i	$0.931415\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	−15.5885	−0.738135
$447$	0	0
$448$	0	0
$449$	6.00000i	0.283158i	0.989927	+	0.141579i	$0.0452178\pi$
$449$	6.00000i	0.283158i	−0.989927	+	0.141579i	$0.954782\pi$
$450$	0	0
$451$	0	0
$452$	18.0000i	0.846649i
$453$	0	0
$454$	25.9808i	1.21934i
$455$	0	0
$456$	0	0
$457$	−41.0000	−1.91790	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	−41.0000	−1.91790	−0.958950	+	0.283577i	$0.908479\pi$
$458$	3.46410	0.161867
$459$	0	0
$460$	− 10.3923i	− 0.484544i
$461$	22.5167	1.04871	0.524353	−	0.851501i	$-0.324307\pi$
$461$	22.5167	1.04871	0.524353	+	0.851501i	$0.324307\pi$
$462$	0	0
$463$	23.0000	1.06890	0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000	1.06890	0.534450	+	0.845200i	$0.320519\pi$
$464$	− 9.00000i	− 0.417815i
$465$	0	0
$466$	−30.0000	−1.38972
$467$	5.19615	0.240449	0.120225	−	0.992747i	$-0.461639\pi$
$467$	5.19615	0.240449	0.120225	+	0.992747i	$0.461639\pi$
$468$	0	0
$469$	0	0
$470$	− 6.00000i	− 0.276759i
$471$	0	0
$472$	− 12.1244i	− 0.558069i
$473$	0	0
$474$	0	0
$475$	− 13.8564i	− 0.635776i
$476$	0	0
$477$	0	0
$478$	24.0000	1.09773
$479$	−13.8564	−0.633115	−0.316558	−	0.948573i	$-0.602527\pi$
$479$	−13.8564	−0.633115	−0.316558	+	0.948573i	$0.602527\pi$
$480$	0	0
$481$	0	0
$482$	1.73205	0.0788928
$483$	0	0
$484$	−11.0000	−0.500000
$485$	− 12.0000i	− 0.544892i
$486$	0	0
$487$	1.00000	0.0453143	0.0226572	−	0.999743i	$-0.492787\pi$
$487$	1.00000	0.0453143	0.0226572	+	0.999743i	$0.492787\pi$
$488$	3.46410	0.156813
$489$	0	0
$490$	0	0
$491$	3.00000i	0.135388i	0.997706	+	0.0676941i	$0.0215642\pi$
$491$	3.00000i	0.135388i	−0.997706	+	0.0676941i	$0.978436\pi$
$492$	0	0
$493$	− 31.1769i	− 1.40414i
$494$	0	0
$495$	0	0
$496$	− 3.46410i	− 0.155543i
$497$	0	0
$498$	0	0
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	−12.1244	−0.542218
$501$	0	0
$502$	8.66025i	0.386526i
$503$	27.7128	1.23565	0.617827	−	0.786314i	$-0.288013\pi$
$503$	27.7128	1.23565	0.617827	+	0.786314i	$0.288013\pi$
$504$	0	0
$505$	9.00000	0.400495
$506$	0	0
$507$	0	0
$508$	7.00000	0.310575
$509$	−34.6410	−1.53544	−0.767718	−	0.640788i	$-0.778608\pi$
$509$	−34.6410	−1.53544	−0.767718	+	0.640788i	$0.778608\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	− 20.7846i	− 0.916770i
$515$	− 18.0000i	− 0.793175i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3.46410	−0.151765	−0.0758825	−	0.997117i	$-0.524177\pi$
$521$	−3.46410	−0.151765	−0.0758825	+	0.997117i	$0.524177\pi$
$522$	0	0
$523$	20.7846i	0.908848i	0.890786	+	0.454424i	$0.150155\pi$
$523$	20.7846i	0.908848i	−0.890786	+	0.454424i	$0.849845\pi$
$524$	10.3923	0.453990
$525$	0	0
$526$	18.0000	0.784837
$527$	− 12.0000i	− 0.522728i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	5.19615	0.225706
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 5.19615i	− 0.224649i
$536$	− 14.0000i	− 0.604708i
$537$	0	0
$538$	1.73205i	0.0746740i
$539$	0	0
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	−12.1244	−0.520786
$543$	0	0
$544$	3.46410i	0.148522i
$545$	−27.7128	−1.18709
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	62.3538	2.65636
$552$	0	0
$553$	0	0
$554$	8.00000i	0.339887i
$555$	0	0
$556$	− 10.3923i	− 0.440732i
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−18.0000	−0.759284
$563$	−5.19615	−0.218992	−0.109496	−	0.993987i	$-0.534924\pi$
$563$	−5.19615	−0.218992	−0.109496	+	0.993987i	$0.534924\pi$
$564$	0	0
$565$	31.1769i	1.31162i
$566$	6.92820	0.291214
$567$	0	0
$568$	6.00000	0.251754
$569$	42.0000i	1.76073i	0.474295	+	0.880366i	$0.342703\pi$
$569$	42.0000i	1.76073i	−0.474295	+	0.880366i	$0.657297\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	12.0000i	0.500435i
$576$	0	0
$577$	15.5885i	0.648956i	0.945893	+	0.324478i	$0.105189\pi$
$577$	15.5885i	0.648956i	−0.945893	+	0.324478i	$0.894811\pi$
$578$	− 5.00000i	− 0.207973i
$579$	0	0
$580$	− 15.5885i	− 0.647275i
$581$	0	0
$582$	0	0
$583$	0	0
$584$	12.1244	0.501709
$585$	0	0
$586$	− 29.4449i	− 1.21636i
$587$	36.3731	1.50128	0.750639	−	0.660713i	$-0.229746\pi$
$587$	36.3731	1.50128	0.750639	+	0.660713i	$0.229746\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	− 21.0000i	− 0.864556i
$591$	0	0
$592$	4.00000	0.164399
$593$	−24.2487	−0.995775	−0.497888	−	0.867242i	$-0.665891\pi$
$593$	−24.2487	−0.995775	−0.497888	+	0.867242i	$0.665891\pi$
$594$	0	0
$595$	0	0
$596$	− 15.0000i	− 0.614424i
$597$	0	0
$598$	0	0
$599$	24.0000i	0.980613i	0.871550	+	0.490307i	$0.163115\pi$
$599$	24.0000i	0.980613i	−0.871550	+	0.490307i	$0.836885\pi$
$600$	0	0
$601$	1.73205i	0.0706518i	0.999376	+	0.0353259i	$0.0112469\pi$
$601$	1.73205i	0.0706518i	−0.999376	+	0.0353259i	$0.988753\pi$
$602$	0	0
$603$	0	0
$604$	8.00000	0.325515
$605$	−19.0526	−0.774597
$606$	0	0
$607$	− 32.9090i	− 1.33573i	−0.744281	−	0.667867i	$-0.767208\pi$
$607$	− 32.9090i	− 1.33573i	0.744281	−	0.667867i	$-0.232792\pi$
$608$	−6.92820	−0.280976
$609$	0	0
$610$	6.00000	0.242933
$611$	0	0
$612$	0	0
$613$	8.00000	0.323117	0.161558	−	0.986863i	$-0.448348\pi$
$613$	8.00000	0.323117	0.161558	+	0.986863i	$0.448348\pi$
$614$	−6.92820	−0.279600
$615$	0	0
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	0	0
$619$	− 20.7846i	− 0.835404i	−0.908584	−	0.417702i	$-0.862836\pi$
$619$	− 20.7846i	− 0.835404i	0.908584	−	0.417702i	$-0.137164\pi$
$620$	− 6.00000i	− 0.240966i
$621$	0	0
$622$	13.8564i	0.555591i
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	22.5167	0.899947
$627$	0	0
$628$	10.3923i	0.414698i
$629$	13.8564	0.552491
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	− 11.0000i	− 0.437557i
$633$	0	0
$634$	−6.00000	−0.238290
$635$	12.1244	0.481140
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	1.73205i	0.0684653i
$641$	12.0000i	0.473972i	0.971513	+	0.236986i	$0.0761595\pi$
$641$	12.0000i	0.473972i	−0.971513	+	0.236986i	$0.923841\pi$
$642$	0	0
$643$	− 27.7128i	− 1.09289i	−0.837496	−	0.546443i	$-0.815981\pi$
$643$	− 27.7128i	− 1.09289i	0.837496	−	0.546443i	$-0.184019\pi$
$644$	0	0
$645$	0	0
$646$	−24.0000	−0.944267
$647$	10.3923	0.408564	0.204282	−	0.978912i	$-0.434514\pi$
$647$	10.3923	0.408564	0.204282	+	0.978912i	$0.434514\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−2.00000	−0.0783260
$653$	− 15.0000i	− 0.586995i	−0.955960	−	0.293498i	$-0.905181\pi$
$653$	− 15.0000i	− 0.586995i	0.955960	−	0.293498i	$-0.0948193\pi$
$654$	0	0
$655$	18.0000	0.703318
$656$	−3.46410	−0.135250
$657$	0	0
$658$	0	0
$659$	− 27.0000i	− 1.05177i	−0.850555	−	0.525885i	$-0.823734\pi$
$659$	− 27.0000i	− 1.05177i	0.850555	−	0.525885i	$-0.176266\pi$
$660$	0	0
$661$	13.8564i	0.538952i	0.963007	+	0.269476i	$0.0868504\pi$
$661$	13.8564i	0.538952i	−0.963007	+	0.269476i	$0.913150\pi$
$662$	10.0000i	0.388661i
$663$	0	0
$664$	− 17.3205i	− 0.672166i
$665$	0	0
$666$	0	0
$667$	−54.0000	−2.09089
$668$	17.3205	0.670151
$669$	0	0
$670$	− 24.2487i	− 0.936809i
$671$	0	0
$672$	0	0
$673$	−5.00000	−0.192736	−0.0963679	−	0.995346i	$-0.530723\pi$
$673$	−5.00000	−0.192736	−0.0963679	+	0.995346i	$0.530723\pi$
$674$	− 23.0000i	− 0.885927i
$675$	0	0
$676$	−13.0000	−0.500000
$677$	−25.9808	−0.998522	−0.499261	−	0.866452i	$-0.666395\pi$
$677$	−25.9808	−0.998522	−0.499261	+	0.866452i	$0.666395\pi$
$678$	0	0
$679$	0	0
$680$	6.00000i	0.230089i
$681$	0	0
$682$	0	0
$683$	− 9.00000i	− 0.344375i	−0.985064	−	0.172188i	$-0.944916\pi$
$683$	− 9.00000i	− 0.344375i	0.985064	−	0.172188i	$-0.0550836\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−8.00000	−0.304997
$689$	0	0
$690$	0	0
$691$	− 27.7128i	− 1.05425i	−0.849789	−	0.527123i	$-0.823271\pi$
$691$	− 27.7128i	− 1.05425i	0.849789	−	0.527123i	$-0.176729\pi$
$692$	−5.19615	−0.197528
$693$	0	0
$694$	33.0000	1.25266
$695$	− 18.0000i	− 0.682779i
$696$	0	0
$697$	−12.0000	−0.454532
$698$	−3.46410	−0.131118
$699$	0	0
$700$	0	0
$701$	− 27.0000i	− 1.01978i	−0.860241	−	0.509888i	$-0.829687\pi$
$701$	− 27.0000i	− 1.01978i	0.860241	−	0.509888i	$-0.170313\pi$
$702$	0	0
$703$	27.7128i	1.04521i
$704$	0	0
$705$	0	0
$706$	3.46410i	0.130373i
$707$	0	0
$708$	0	0
$709$	28.0000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$710$	10.3923	0.390016
$711$	0	0
$712$	− 10.3923i	− 0.389468i
$713$	−20.7846	−0.778390
$714$	0	0
$715$	0	0
$716$	15.0000i	0.560576i
$717$	0	0
$718$	24.0000	0.895672
$719$	48.4974	1.80865	0.904324	−	0.426846i	$-0.140375\pi$
$719$	48.4974	1.80865	0.904324	+	0.426846i	$0.140375\pi$
$720$	0	0
$721$	0	0
$722$	− 29.0000i	− 1.07927i
$723$	0	0
$724$	17.3205i	0.643712i
$725$	18.0000i	0.668503i
$726$	0	0
$727$	− 12.1244i	− 0.449667i	−0.974397	−	0.224834i	$-0.927816\pi$
$727$	− 12.1244i	− 0.449667i	0.974397	−	0.224834i	$-0.0721839\pi$
$728$	0	0
$729$	0	0
$730$	21.0000	0.777245
$731$	−27.7128	−1.02500
$732$	0	0
$733$	− 6.92820i	− 0.255899i	−0.991781	−	0.127950i	$-0.959160\pi$
$733$	− 6.92820i	− 0.255899i	0.991781	−	0.127950i	$-0.0408395\pi$
$734$	−8.66025	−0.319656
$735$	0	0
$736$	6.00000	0.221163
$737$	0	0
$738$	0	0
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	6.92820	0.254686
$741$	0	0
$742$	0	0
$743$	− 36.0000i	− 1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$743$	− 36.0000i	− 1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$744$	0	0
$745$	− 25.9808i	− 0.951861i
$746$	14.0000i	0.512576i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	7.00000	0.255434	0.127717	−	0.991811i	$-0.459235\pi$
$751$	7.00000	0.255434	0.127717	+	0.991811i	$0.459235\pi$
$752$	3.46410	0.126323
$753$	0	0
$754$	0	0
$755$	13.8564	0.504286
$756$	0	0
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	− 14.0000i	− 0.508503i
$759$	0	0
$760$	−12.0000	−0.435286
$761$	−34.6410	−1.25574	−0.627868	−	0.778320i	$-0.716072\pi$
$761$	−34.6410	−1.25574	−0.627868	+	0.778320i	$0.716072\pi$
$762$	0	0
$763$	0	0
$764$	6.00000i	0.217072i
$765$	0	0
$766$	17.3205i	0.625815i
$767$	0	0
$768$	0	0
$769$	48.4974i	1.74886i	0.485150	+	0.874431i	$0.338765\pi$
$769$	48.4974i	1.74886i	−0.485150	+	0.874431i	$0.661235\pi$
$770$	0	0
$771$	0	0
$772$	−10.0000	−0.359908
$773$	13.8564	0.498380	0.249190	−	0.968455i	$-0.419836\pi$
$773$	13.8564	0.498380	0.249190	+	0.968455i	$0.419836\pi$
$774$	0	0
$775$	6.92820i	0.248868i
$776$	6.92820	0.248708
$777$	0	0
$778$	−21.0000	−0.752886
$779$	− 24.0000i	− 0.859889i
$780$	0	0
$781$	0	0
$782$	20.7846	0.743256
$783$	0	0
$784$	0	0
$785$	18.0000i	0.642448i
$786$	0	0
$787$	− 3.46410i	− 0.123482i	−0.998092	−	0.0617409i	$-0.980335\pi$
$787$	− 3.46410i	− 0.123482i	0.998092	−	0.0617409i	$-0.0196653\pi$
$788$	3.00000i	0.106871i
$789$	0	0
$790$	− 19.0526i	− 0.677860i
$791$	0	0
$792$	0	0
$793$	0	0
$794$	27.7128	0.983491
$795$	0	0
$796$	8.66025i	0.306955i
$797$	41.5692	1.47246	0.736229	−	0.676733i	$-0.236605\pi$
$797$	41.5692	1.47246	0.736229	+	0.676733i	$0.236605\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	− 2.00000i	− 0.0707107i
$801$	0	0
$802$	6.00000	0.211867
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	5.19615i	0.182800i
$809$	24.0000i	0.843795i	0.906644	+	0.421898i	$0.138636\pi$
$809$	24.0000i	0.843795i	−0.906644	+	0.421898i	$0.861364\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.46410	−0.121342
$816$	0	0
$817$	− 55.4256i	− 1.93910i
$818$	−15.5885	−0.545038
$819$	0	0
$820$	−6.00000	−0.209529
$821$	− 9.00000i	− 0.314102i	−0.987590	−	0.157051i	$-0.949801\pi$
$821$	− 9.00000i	− 0.314102i	0.987590	−	0.157051i	$-0.0501987\pi$
$822$	0	0
$823$	5.00000	0.174289	0.0871445	−	0.996196i	$-0.472226\pi$
$823$	5.00000	0.174289	0.0871445	+	0.996196i	$0.472226\pi$
$824$	10.3923	0.362033
$825$	0	0
$826$	0	0
$827$	3.00000i	0.104320i	0.998639	+	0.0521601i	$0.0166106\pi$
$827$	3.00000i	0.104320i	−0.998639	+	0.0521601i	$0.983389\pi$
$828$	0	0
$829$	34.6410i	1.20313i	0.798823	+	0.601566i	$0.205456\pi$
$829$	34.6410i	1.20313i	−0.798823	+	0.601566i	$0.794544\pi$
$830$	− 30.0000i	− 1.04132i
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	30.0000	1.03819
$836$	0	0
$837$	0	0
$838$	3.46410i	0.119665i
$839$	−6.92820	−0.239188	−0.119594	−	0.992823i	$-0.538159\pi$
$839$	−6.92820	−0.239188	−0.119594	+	0.992823i	$0.538159\pi$
$840$	0	0
$841$	−52.0000	−1.79310
$842$	32.0000i	1.10279i
$843$	0	0
$844$	−16.0000	−0.550743
$845$	−22.5167	−0.774597
$846$	0	0
$847$	0	0
$848$	3.00000i	0.103020i
$849$	0	0
$850$	− 6.92820i	− 0.237635i
$851$	− 24.0000i	− 0.822709i
$852$	0	0
$853$	− 6.92820i	− 0.237217i	−0.992941	−	0.118609i	$-0.962157\pi$
$853$	− 6.92820i	− 0.237217i	0.992941	−	0.118609i	$-0.0378434\pi$
$854$	0	0
$855$	0	0
$856$	3.00000	0.102538
$857$	−6.92820	−0.236663	−0.118331	−	0.992974i	$-0.537755\pi$
$857$	−6.92820	−0.236663	−0.118331	+	0.992974i	$0.537755\pi$
$858$	0	0
$859$	− 24.2487i	− 0.827355i	−0.910423	−	0.413678i	$-0.864244\pi$
$859$	− 24.2487i	− 0.827355i	0.910423	−	0.413678i	$-0.135756\pi$
$860$	−13.8564	−0.472500
$861$	0	0
$862$	−36.0000	−1.22616
$863$	− 18.0000i	− 0.612727i	−0.951915	−	0.306364i	$-0.900888\pi$
$863$	− 18.0000i	− 0.612727i	0.951915	−	0.306364i	$-0.0991123\pi$
$864$	0	0
$865$	−9.00000	−0.306009
$866$	1.73205	0.0588575
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 16.0000i	− 0.541828i
$873$	0	0
$874$	41.5692i	1.40610i
$875$	0	0
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	31.1769	1.05217
$879$	0	0
$880$	0	0
$881$	−31.1769	−1.05038	−0.525188	−	0.850986i	$-0.676005\pi$
$881$	−31.1769	−1.05038	−0.525188	+	0.850986i	$0.676005\pi$
$882$	0	0
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	0	0
$885$	0	0
$886$	−9.00000	−0.302361
$887$	−17.3205	−0.581566	−0.290783	−	0.956789i	$-0.593916\pi$
$887$	−17.3205	−0.581566	−0.290783	+	0.956789i	$0.593916\pi$
$888$	0	0
$889$	0	0
$890$	− 18.0000i	− 0.603361i
$891$	0	0
$892$	− 15.5885i	− 0.521940i
$893$	24.0000i	0.803129i
$894$	0	0
$895$	25.9808i	0.868441i
$896$	0	0
$897$	0	0
$898$	−6.00000	−0.200223
$899$	−31.1769	−1.03981
$900$	0	0
$901$	10.3923i	0.346218i
$902$	0	0
$903$	0	0
$904$	−18.0000	−0.598671
$905$	30.0000i	0.997234i
$906$	0	0
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	−25.9808	−0.862202
$909$	0	0
$910$	0	0
$911$	30.0000i	0.993944i	0.867766	+	0.496972i	$0.165555\pi$
$911$	30.0000i	0.993944i	−0.867766	+	0.496972i	$0.834445\pi$
$912$	0	0
$913$	0	0
$914$	− 41.0000i	− 1.35616i
$915$	0	0
$916$	3.46410i	0.114457i
$917$	0	0
$918$	0	0
$919$	7.00000	0.230909	0.115454	−	0.993313i	$-0.463168\pi$
$919$	7.00000	0.230909	0.115454	+	0.993313i	$0.463168\pi$
$920$	10.3923	0.342624
$921$	0	0
$922$	22.5167i	0.741547i
$923$	0	0
$924$	0	0
$925$	−8.00000	−0.263038
$926$	23.0000i	0.755827i
$927$	0	0
$928$	9.00000	0.295439
$929$	−38.1051	−1.25019	−0.625094	−	0.780549i	$-0.714939\pi$
$929$	−38.1051	−1.25019	−0.625094	+	0.780549i	$0.714939\pi$
$930$	0	0
$931$	0	0
$932$	− 30.0000i	− 0.982683i
$933$	0	0
$934$	5.19615i	0.170023i
$935$	0	0
$936$	0	0
$937$	− 57.1577i	− 1.86726i	−0.358239	−	0.933630i	$-0.616623\pi$
$937$	− 57.1577i	− 1.86726i	0.358239	−	0.933630i	$-0.383377\pi$
$938$	0	0
$939$	0	0
$940$	6.00000	0.195698
$941$	−55.4256	−1.80682	−0.903412	−	0.428774i	$-0.858946\pi$
$941$	−55.4256	−1.80682	−0.903412	+	0.428774i	$0.858946\pi$
$942$	0	0
$943$	20.7846i	0.676840i
$944$	12.1244	0.394614
$945$	0	0
$946$	0	0
$947$	− 27.0000i	− 0.877382i	−0.898638	−	0.438691i	$-0.855442\pi$
$947$	− 27.0000i	− 0.877382i	0.898638	−	0.438691i	$-0.144558\pi$
$948$	0	0
$949$	0	0
$950$	13.8564	0.449561
$951$	0	0
$952$	0	0
$953$	12.0000i	0.388718i	0.980930	+	0.194359i	$0.0622627\pi$
$953$	12.0000i	0.388718i	−0.980930	+	0.194359i	$0.937737\pi$
$954$	0	0
$955$	10.3923i	0.336287i
$956$	24.0000i	0.776215i
$957$	0	0
$958$	− 13.8564i	− 0.447680i
$959$	0	0
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	0	0
$964$	1.73205i	0.0557856i
$965$	−17.3205	−0.557567
$966$	0	0
$967$	23.0000	0.739630	0.369815	−	0.929105i	$-0.379421\pi$
$967$	23.0000	0.739630	0.369815	+	0.929105i	$0.379421\pi$
$968$	− 11.0000i	− 0.353553i
$969$	0	0
$970$	12.0000	0.385297
$971$	1.73205	0.0555842	0.0277921	−	0.999614i	$-0.491152\pi$
$971$	1.73205	0.0555842	0.0277921	+	0.999614i	$0.491152\pi$
$972$	0	0
$973$	0	0
$974$	1.00000i	0.0320421i
$975$	0	0
$976$	3.46410i	0.110883i
$977$	12.0000i	0.383914i	0.981403	+	0.191957i	$0.0614834\pi$
$977$	12.0000i	0.383914i	−0.981403	+	0.191957i	$0.938517\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−3.00000	−0.0957338
$983$	27.7128	0.883901	0.441951	−	0.897039i	$-0.354287\pi$
$983$	27.7128	0.883901	0.441951	+	0.897039i	$0.354287\pi$
$984$	0	0
$985$	5.19615i	0.165563i
$986$	31.1769	0.992875
$987$	0	0
$988$	0	0
$989$	48.0000i	1.52631i
$990$	0	0
$991$	−5.00000	−0.158830	−0.0794151	−	0.996842i	$-0.525305\pi$
$991$	−5.00000	−0.158830	−0.0794151	+	0.996842i	$0.525305\pi$
$992$	3.46410	0.109985
$993$	0	0
$994$	0	0
$995$	15.0000i	0.475532i
$996$	0	0
$997$	− 48.4974i	− 1.53593i	−0.640493	−	0.767964i	$-0.721270\pi$
$997$	− 48.4974i	− 1.53593i	0.640493	−	0.767964i	$-0.278730\pi$
$998$	− 14.0000i	− 0.443162i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.d.a.2645.3		4
3.2	odd	2	inner	2646.2.d.a.2645.2		4
7.2	even	3		378.2.k.c.269.2	yes	4
7.3	odd	6		378.2.k.c.215.1	✓	4
7.6	odd	2	inner	2646.2.d.a.2645.4		4
21.2	odd	6		378.2.k.c.269.1	yes	4
21.17	even	6		378.2.k.c.215.2	yes	4
21.20	even	2	inner	2646.2.d.a.2645.1		4
63.2	odd	6		1134.2.t.c.1025.2		4
63.16	even	3		1134.2.t.c.1025.1		4
63.23	odd	6		1134.2.l.b.269.1		4
63.31	odd	6		1134.2.t.c.593.2		4
63.38	even	6		1134.2.l.b.215.1		4
63.52	odd	6		1134.2.l.b.215.2		4
63.58	even	3		1134.2.l.b.269.2		4
63.59	even	6		1134.2.t.c.593.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.k.c.215.1	✓	4	7.3	odd	6
378.2.k.c.215.2	yes	4	21.17	even	6
378.2.k.c.269.1	yes	4	21.2	odd	6
378.2.k.c.269.2	yes	4	7.2	even	3
1134.2.l.b.215.1		4	63.38	even	6
1134.2.l.b.215.2		4	63.52	odd	6
1134.2.l.b.269.1		4	63.23	odd	6
1134.2.l.b.269.2		4	63.58	even	3
1134.2.t.c.593.1		4	63.59	even	6
1134.2.t.c.593.2		4	63.31	odd	6
1134.2.t.c.1025.1		4	63.16	even	3
1134.2.t.c.1025.2		4	63.2	odd	6
2646.2.d.a.2645.1		4	21.20	even	2	inner
2646.2.d.a.2645.2		4	3.2	odd	2	inner
2646.2.d.a.2645.3		4	1.1	even	1	trivial
2646.2.d.a.2645.4		4	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.73205 q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.73205 q^{5} -1.00000i q^{8} -1.73205i q^{10} +1.00000 q^{16} +3.46410 q^{17} +6.92820i q^{19} +1.73205 q^{20} -6.00000i q^{23} -2.00000 q^{25} -9.00000i q^{29} -3.46410i q^{31} +1.00000i q^{32} +3.46410i q^{34} +4.00000 q^{37} -6.92820 q^{38} +1.73205i q^{40} -3.46410 q^{41} -8.00000 q^{43} +6.00000 q^{46} +3.46410 q^{47} -2.00000i q^{50} +3.00000i q^{53} +9.00000 q^{58} +12.1244 q^{59} +3.46410i q^{61} +3.46410 q^{62} -1.00000 q^{64} +14.0000 q^{67} -3.46410 q^{68} +6.00000i q^{71} +12.1244i q^{73} +4.00000i q^{74} -6.92820i q^{76} +11.0000 q^{79} -1.73205 q^{80} -3.46410i q^{82} +17.3205 q^{83} -6.00000 q^{85} -8.00000i q^{86} +10.3923 q^{89} +6.00000i q^{92} +3.46410i q^{94} -12.0000i q^{95} +6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 4 q^{16} - 8 q^{25} + 16 q^{37} - 32 q^{43} + 24 q^{46} + 36 q^{58} - 4 q^{64} + 56 q^{67} + 44 q^{79} - 24 q^{85}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^16 - 8 * q^25 + 16 * q^37 - 32 * q^43 + 24 * q^46 + 36 * q^58 - 4 * q^64 + 56 * q^67 + 44 * q^79 - 24 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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