LMFDB - Embedded newform 2646.2.d.c.2645.4

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:	$\Z[a_1, \ldots, a_{31}]$
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			2645.4
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2646.2645
Dual form			2646.2.d.c.2645.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{8} -3.00000i q^{11} +3.46410i q^{13} +1.00000 q^{16} +3.00000 q^{22} -5.00000 q^{25} -3.46410 q^{26} +9.00000i q^{29} -1.73205i q^{31} +1.00000i q^{32} -8.00000 q^{37} +10.3923 q^{41} +4.00000 q^{43} +3.00000i q^{44} +10.3923 q^{47} -5.00000i q^{50} -3.46410i q^{52} +6.00000i q^{53} -9.00000 q^{58} +5.19615 q^{59} +13.8564i q^{61} +1.73205 q^{62} -1.00000 q^{64} +2.00000 q^{67} +12.0000i q^{71} +5.19615i q^{73} -8.00000i q^{74} -13.0000 q^{79} +10.3923i q^{82} -5.19615 q^{83} +4.00000i q^{86} -3.00000 q^{88} -10.3923 q^{89} +10.3923i q^{94} -8.66025i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 4 q^{16} + 12 q^{22} - 20 q^{25} - 32 q^{37} + 16 q^{43} - 36 q^{58} - 4 q^{64} + 8 q^{67} - 52 q^{79} - 12 q^{88}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^16 + 12 * q^22 - 20 * q^25 - 32 * q^37 + 16 * q^43 - 36 * q^58 - 4 * q^64 + 8 * q^67 - 52 * q^79 - 12 * 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	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	− 3.00000i	− 0.904534i	−0.891883	−	0.452267i	$-0.850615\pi$
$11$	− 3.00000i	− 0.904534i	0.891883	−	0.452267i	$-0.149385\pi$
$12$	0	0
$13$	3.46410i	0.960769i	0.877058	+	0.480384i	$0.159503\pi$
$13$	3.46410i	0.960769i	−0.877058	+	0.480384i	$0.840497\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	3.00000	0.639602
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	−3.46410	−0.679366
$27$	0	0
$28$	0	0
$29$	9.00000i	1.67126i	0.549294	+	0.835629i	$0.314897\pi$
$29$	9.00000i	1.67126i	−0.549294	+	0.835629i	$0.685103\pi$
$30$	0	0
$31$	− 1.73205i	− 0.311086i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	− 1.73205i	− 0.311086i	0.987829	−	0.155543i	$-0.0497126\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.3923	1.62301	0.811503	−	0.584349i	$-0.198650\pi$
$41$	10.3923	1.62301	0.811503	+	0.584349i	$0.198650\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	3.00000i	0.452267i
$45$	0	0
$46$	0	0
$47$	10.3923	1.51587	0.757937	−	0.652328i	$-0.226208\pi$
$47$	10.3923	1.51587	0.757937	+	0.652328i	$0.226208\pi$
$48$	0	0
$49$	0	0
$50$	− 5.00000i	− 0.707107i
$51$	0	0
$52$	− 3.46410i	− 0.480384i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−9.00000	−1.18176
$59$	5.19615	0.676481	0.338241	−	0.941060i	$-0.390168\pi$
$59$	5.19615	0.676481	0.338241	+	0.941060i	$0.390168\pi$
$60$	0	0
$61$	13.8564i	1.77413i	0.461644	+	0.887066i	$0.347260\pi$
$61$	13.8564i	1.77413i	−0.461644	+	0.887066i	$0.652740\pi$
$62$	1.73205	0.219971
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.0000i	1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$	12.0000i	1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$	0	0
$73$	5.19615i	0.608164i	0.952646	+	0.304082i	$0.0983496\pi$
$73$	5.19615i	0.608164i	−0.952646	+	0.304082i	$0.901650\pi$
$74$	− 8.00000i	− 0.929981i
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−13.0000	−1.46261	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000	−1.46261	−0.731307	+	0.682048i	$0.761089\pi$
$80$	0	0
$81$	0	0
$82$	10.3923i	1.14764i
$83$	−5.19615	−0.570352	−0.285176	−	0.958475i	$-0.592052\pi$
$83$	−5.19615	−0.570352	−0.285176	+	0.958475i	$0.592052\pi$
$84$	0	0
$85$	0	0
$86$	4.00000i	0.431331i
$87$	0	0
$88$	−3.00000	−0.319801
$89$	−10.3923	−1.10158	−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923	−1.10158	−0.550791	+	0.834643i	$0.685674\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	10.3923i	1.07188i
$95$	0	0
$96$	0	0
$97$	− 8.66025i	− 0.879316i	−0.898165	−	0.439658i	$-0.855100\pi$
$97$	− 8.66025i	− 0.879316i	0.898165	−	0.439658i	$-0.144900\pi$
$98$	0	0
$99$	0	0
$100$	5.00000	0.500000
$101$	5.19615	0.517036	0.258518	−	0.966006i	$-0.416766\pi$
$101$	5.19615	0.517036	0.258518	+	0.966006i	$0.416766\pi$
$102$	0	0
$103$	17.3205i	1.70664i	0.521387	+	0.853320i	$0.325415\pi$
$103$	17.3205i	1.70664i	−0.521387	+	0.853320i	$0.674585\pi$
$104$	3.46410	0.339683
$105$	0	0
$106$	−6.00000	−0.582772
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	0	0
$115$	0	0
$116$	− 9.00000i	− 0.835629i
$117$	0	0
$118$	5.19615i	0.478345i
$119$	0	0
$120$	0	0
$121$	2.00000	0.181818
$122$	−13.8564	−1.25450
$123$	0	0
$124$	1.73205i	0.155543i
$125$	0	0
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	15.5885	1.36197	0.680985	−	0.732297i	$-0.261552\pi$
$131$	15.5885	1.36197	0.680985	+	0.732297i	$0.261552\pi$
$132$	0	0
$133$	0	0
$134$	2.00000i	0.172774i
$135$	0	0
$136$	0	0
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	0	0
$139$	6.92820i	0.587643i	0.955860	+	0.293821i	$0.0949270\pi$
$139$	6.92820i	0.587643i	−0.955860	+	0.293821i	$0.905073\pi$
$140$	0	0
$141$	0	0
$142$	−12.0000	−1.00702
$143$	10.3923	0.869048
$144$	0	0
$145$	0	0
$146$	−5.19615	−0.430037
$147$	0	0
$148$	8.00000	0.657596
$149$	− 15.0000i	− 1.22885i	−0.788976	−	0.614424i	$-0.789388\pi$
$149$	− 15.0000i	− 1.22885i	0.788976	−	0.614424i	$-0.210612\pi$
$150$	0	0
$151$	−23.0000	−1.87171	−0.935857	−	0.352381i	$-0.885372\pi$
$151$	−23.0000	−1.87171	−0.935857	+	0.352381i	$0.885372\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.3923i	0.829396i	0.909959	+	0.414698i	$0.136113\pi$
$157$	10.3923i	0.829396i	−0.909959	+	0.414698i	$0.863887\pi$
$158$	− 13.0000i	− 1.03422i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	−10.3923	−0.811503
$165$	0	0
$166$	− 5.19615i	− 0.403300i
$167$	10.3923	0.804181	0.402090	−	0.915600i	$-0.368284\pi$
$167$	10.3923	0.804181	0.402090	+	0.915600i	$0.368284\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−25.9808	−1.97528	−0.987640	−	0.156737i	$-0.949903\pi$
$173$	−25.9808	−1.97528	−0.987640	+	0.156737i	$0.949903\pi$
$174$	0	0
$175$	0	0
$176$	− 3.00000i	− 0.226134i
$177$	0	0
$178$	− 10.3923i	− 0.778936i
$179$	9.00000i	0.672692i	0.941739	+	0.336346i	$0.109191\pi$
$179$	9.00000i	0.672692i	−0.941739	+	0.336346i	$0.890809\pi$
$180$	0	0
$181$	20.7846i	1.54491i	0.635071	+	0.772454i	$0.280971\pi$
$181$	20.7846i	1.54491i	−0.635071	+	0.772454i	$0.719029\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−10.3923	−0.757937
$189$	0	0
$190$	0	0
$191$	− 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
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	8.66025	0.621770
$195$	0	0
$196$	0	0
$197$	15.0000i	1.06871i	0.845262	+	0.534353i	$0.179445\pi$
$197$	15.0000i	1.06871i	−0.845262	+	0.534353i	$0.820555\pi$
$198$	0	0
$199$	1.73205i	0.122782i	0.998114	+	0.0613909i	$0.0195536\pi$
$199$	1.73205i	0.122782i	−0.998114	+	0.0613909i	$0.980446\pi$
$200$	5.00000i	0.353553i
$201$	0	0
$202$	5.19615i	0.365600i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−17.3205	−1.20678
$207$	0	0
$208$	3.46410i	0.240192i
$209$	0	0
$210$	0	0
$211$	−2.00000	−0.137686	−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000	−0.137686	−0.0688428	+	0.997628i	$0.521931\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 8.00000i	− 0.541828i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	5.19615i	0.347960i	0.984749	+	0.173980i	$0.0556628\pi$
$223$	5.19615i	0.347960i	−0.984749	+	0.173980i	$0.944337\pi$
$224$	0	0
$225$	0	0
$226$	12.0000	0.798228
$227$	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
$229$	− 6.92820i	− 0.457829i	−0.973447	−	0.228914i	$-0.926482\pi$
$229$	− 6.92820i	− 0.457829i	0.973447	−	0.228914i	$-0.0735176\pi$
$230$	0	0
$231$	0	0
$232$	9.00000	0.590879
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	0	0
$235$	0	0
$236$	−5.19615	−0.338241
$237$	0	0
$238$	0	0
$239$	− 12.0000i	− 0.776215i	−0.921614	−	0.388108i	$-0.873129\pi$
$239$	− 12.0000i	− 0.776215i	0.921614	−	0.388108i	$-0.126871\pi$
$240$	0	0
$241$	− 12.1244i	− 0.780998i	−0.920603	−	0.390499i	$-0.872302\pi$
$241$	− 12.1244i	− 0.780998i	0.920603	−	0.390499i	$-0.127698\pi$
$242$	2.00000i	0.128565i
$243$	0	0
$244$	− 13.8564i	− 0.887066i
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−1.73205	−0.109985
$249$	0	0
$250$	0	0
$251$	−25.9808	−1.63989	−0.819946	−	0.572441i	$-0.805996\pi$
$251$	−25.9808	−1.63989	−0.819946	+	0.572441i	$0.805996\pi$
$252$	0	0
$253$	0	0
$254$	8.00000i	0.501965i
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.7846	1.29651	0.648254	−	0.761424i	$-0.275499\pi$
$257$	20.7846	1.29651	0.648254	+	0.761424i	$0.275499\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	15.5885i	0.963058i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−2.00000	−0.122169
$269$	15.5885	0.950445	0.475223	−	0.879866i	$-0.342368\pi$
$269$	15.5885	0.950445	0.475223	+	0.879866i	$0.342368\pi$
$270$	0	0
$271$	− 3.46410i	− 0.210429i	−0.994450	−	0.105215i	$-0.966447\pi$
$271$	− 3.46410i	− 0.210429i	0.994450	−	0.105215i	$-0.0335529\pi$
$272$	0	0
$273$	0	0
$274$	−12.0000	−0.724947
$275$	15.0000i	0.904534i
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	−6.92820	−0.415526
$279$	0	0
$280$	0	0
$281$	− 6.00000i	− 0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$	− 6.00000i	− 0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$	0	0
$283$	6.92820i	0.411839i	0.978569	+	0.205919i	$0.0660185\pi$
$283$	6.92820i	0.411839i	−0.978569	+	0.205919i	$0.933982\pi$
$284$	− 12.0000i	− 0.712069i
$285$	0	0
$286$	10.3923i	0.614510i
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	− 5.19615i	− 0.304082i
$293$	5.19615	0.303562	0.151781	−	0.988414i	$-0.451499\pi$
$293$	5.19615	0.303562	0.151781	+	0.988414i	$0.451499\pi$
$294$	0	0
$295$	0	0
$296$	8.00000i	0.464991i
$297$	0	0
$298$	15.0000	0.868927
$299$	0	0
$300$	0	0
$301$	0	0
$302$	− 23.0000i	− 1.32350i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 17.3205i	− 0.988534i	−0.869310	−	0.494267i	$-0.835437\pi$
$307$	− 17.3205i	− 0.988534i	0.869310	−	0.494267i	$-0.164563\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−31.1769	−1.76788	−0.883940	−	0.467600i	$-0.845119\pi$
$311$	−31.1769	−1.76788	−0.883940	+	0.467600i	$0.845119\pi$
$312$	0	0
$313$	20.7846i	1.17482i	0.809291	+	0.587408i	$0.199852\pi$
$313$	20.7846i	1.17482i	−0.809291	+	0.587408i	$0.800148\pi$
$314$	−10.3923	−0.586472
$315$	0	0
$316$	13.0000	0.731307
$317$	− 9.00000i	− 0.505490i	−0.967533	−	0.252745i	$-0.918667\pi$
$317$	− 9.00000i	− 0.505490i	0.967533	−	0.252745i	$-0.0813334\pi$
$318$	0	0
$319$	27.0000	1.51171
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 17.3205i	− 0.960769i
$326$	− 10.0000i	− 0.553849i
$327$	0	0
$328$	− 10.3923i	− 0.573819i
$329$	0	0
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	5.19615	0.285176
$333$	0	0
$334$	10.3923i	0.568642i
$335$	0	0
$336$	0	0
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	1.00000i	0.0543928i
$339$	0	0
$340$	0	0
$341$	−5.19615	−0.281387
$342$	0	0
$343$	0	0
$344$	− 4.00000i	− 0.215666i
$345$	0	0
$346$	− 25.9808i	− 1.39673i
$347$	3.00000i	0.161048i	0.996753	+	0.0805242i	$0.0256594\pi$
$347$	3.00000i	0.161048i	−0.996753	+	0.0805242i	$0.974341\pi$
$348$	0	0
$349$	− 27.7128i	− 1.48343i	−0.670714	−	0.741716i	$-0.734012\pi$
$349$	− 27.7128i	− 1.48343i	0.670714	−	0.741716i	$-0.265988\pi$
$350$	0	0
$351$	0	0
$352$	3.00000	0.159901
$353$	−10.3923	−0.553127	−0.276563	−	0.960996i	$-0.589196\pi$
$353$	−10.3923	−0.553127	−0.276563	+	0.960996i	$0.589196\pi$
$354$	0	0
$355$	0	0
$356$	10.3923	0.550791
$357$	0	0
$358$	−9.00000	−0.475665
$359$	− 12.0000i	− 0.633336i	−0.948536	−	0.316668i	$-0.897436\pi$
$359$	− 12.0000i	− 0.633336i	0.948536	−	0.316668i	$-0.102564\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	−20.7846	−1.09241
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.3923i	0.542474i	0.962513	+	0.271237i	$0.0874327\pi$
$367$	10.3923i	0.542474i	−0.962513	+	0.271237i	$0.912567\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	0	0
$376$	− 10.3923i	− 0.535942i
$377$	−31.1769	−1.60569
$378$	0	0
$379$	−26.0000	−1.33553	−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000	−1.33553	−0.667765	+	0.744372i	$0.732749\pi$
$380$	0	0
$381$	0	0
$382$	6.00000	0.306987
$383$	20.7846	1.06204	0.531022	−	0.847358i	$-0.321808\pi$
$383$	20.7846	1.06204	0.531022	+	0.847358i	$0.321808\pi$
$384$	0	0
$385$	0	0
$386$	− 11.0000i	− 0.559885i
$387$	0	0
$388$	8.66025i	0.439658i
$389$	− 9.00000i	− 0.456318i	−0.973624	−	0.228159i	$-0.926729\pi$
$389$	− 9.00000i	− 0.456318i	0.973624	−	0.228159i	$-0.0732706\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	−15.0000	−0.755689
$395$	0	0
$396$	0	0
$397$	− 24.2487i	− 1.21701i	−0.793551	−	0.608504i	$-0.791770\pi$
$397$	− 24.2487i	− 1.21701i	0.793551	−	0.608504i	$-0.208230\pi$
$398$	−1.73205	−0.0868199
$399$	0	0
$400$	−5.00000	−0.250000
$401$	− 30.0000i	− 1.49813i	−0.662497	−	0.749064i	$-0.730503\pi$
$401$	− 30.0000i	− 1.49813i	0.662497	−	0.749064i	$-0.269497\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	−5.19615	−0.258518
$405$	0	0
$406$	0	0
$407$	24.0000i	1.18964i
$408$	0	0
$409$	6.92820i	0.342578i	0.985221	+	0.171289i	$0.0547931\pi$
$409$	6.92820i	0.342578i	−0.985221	+	0.171289i	$0.945207\pi$
$410$	0	0
$411$	0	0
$412$	− 17.3205i	− 0.853320i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−3.46410	−0.169842
$417$	0	0
$418$	0	0
$419$	31.1769	1.52309	0.761546	−	0.648111i	$-0.224441\pi$
$419$	31.1769	1.52309	0.761546	+	0.648111i	$0.224441\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	− 2.00000i	− 0.0973585i
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	18.0000i	0.867029i	0.901146	+	0.433515i	$0.142727\pi$
$431$	18.0000i	0.867029i	−0.901146	+	0.433515i	$0.857273\pi$
$432$	0	0
$433$	12.1244i	0.582659i	0.956623	+	0.291330i	$0.0940977\pi$
$433$	12.1244i	0.582659i	−0.956623	+	0.291330i	$0.905902\pi$
$434$	0	0
$435$	0	0
$436$	8.00000	0.383131
$437$	0	0
$438$	0	0
$439$	− 15.5885i	− 0.743996i	−0.928233	−	0.371998i	$-0.878673\pi$
$439$	− 15.5885i	− 0.743996i	0.928233	−	0.371998i	$-0.121327\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	27.0000i	1.28281i	0.767203	+	0.641404i	$0.221648\pi$
$443$	27.0000i	1.28281i	−0.767203	+	0.641404i	$0.778352\pi$
$444$	0	0
$445$	0	0
$446$	−5.19615	−0.246045
$447$	0	0
$448$	0	0
$449$	30.0000i	1.41579i	0.706319	+	0.707894i	$0.250354\pi$
$449$	30.0000i	1.41579i	−0.706319	+	0.707894i	$0.749646\pi$
$450$	0	0
$451$	− 31.1769i	− 1.46806i
$452$	12.0000i	0.564433i
$453$	0	0
$454$	25.9808i	1.21934i
$455$	0	0
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	6.92820	0.323734
$459$	0	0
$460$	0	0
$461$	−15.5885	−0.726027	−0.363013	−	0.931784i	$-0.618252\pi$
$461$	−15.5885	−0.726027	−0.363013	+	0.931784i	$0.618252\pi$
$462$	0	0
$463$	5.00000	0.232370	0.116185	−	0.993228i	$-0.462933\pi$
$463$	5.00000	0.232370	0.116185	+	0.993228i	$0.462933\pi$
$464$	9.00000i	0.417815i
$465$	0	0
$466$	18.0000	0.833834
$467$	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
$470$	0	0
$471$	0	0
$472$	− 5.19615i	− 0.239172i
$473$	− 12.0000i	− 0.551761i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	12.0000	0.548867
$479$	−10.3923	−0.474837	−0.237418	−	0.971408i	$-0.576301\pi$
$479$	−10.3923	−0.474837	−0.237418	+	0.971408i	$0.576301\pi$
$480$	0	0
$481$	− 27.7128i	− 1.26360i
$482$	12.1244	0.552249
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	0	0
$487$	−11.0000	−0.498458	−0.249229	−	0.968445i	$-0.580177\pi$
$487$	−11.0000	−0.498458	−0.249229	+	0.968445i	$0.580177\pi$
$488$	13.8564	0.627250
$489$	0	0
$490$	0	0
$491$	− 36.0000i	− 1.62466i	−0.583200	−	0.812329i	$-0.698200\pi$
$491$	− 36.0000i	− 1.62466i	0.583200	−	0.812329i	$-0.301800\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	− 1.73205i	− 0.0777714i
$497$	0	0
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	0	0
$502$	− 25.9808i	− 1.15958i
$503$	31.1769	1.39011	0.695055	−	0.718957i	$-0.255380\pi$
$503$	31.1769	1.39011	0.695055	+	0.718957i	$0.255380\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−8.00000	−0.354943
$509$	−5.19615	−0.230315	−0.115158	−	0.993347i	$-0.536737\pi$
$509$	−5.19615	−0.230315	−0.115158	+	0.993347i	$0.536737\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	20.7846i	0.916770i
$515$	0	0
$516$	0	0
$517$	− 31.1769i	− 1.37116i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.7846	0.910590	0.455295	−	0.890341i	$-0.349534\pi$
$521$	20.7846	0.910590	0.455295	+	0.890341i	$0.349534\pi$
$522$	0	0
$523$	− 38.1051i	− 1.66622i	−0.553107	−	0.833110i	$-0.686558\pi$
$523$	− 38.1051i	− 1.66622i	0.553107	−	0.833110i	$-0.313442\pi$
$524$	−15.5885	−0.680985
$525$	0	0
$526$	−24.0000	−1.04645
$527$	0	0
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	36.0000i	1.55933i
$534$	0	0
$535$	0	0
$536$	− 2.00000i	− 0.0863868i
$537$	0	0
$538$	15.5885i	0.672066i
$539$	0	0
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	3.46410	0.148796
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.0000	0.598597	0.299298	−	0.954160i	$-0.403247\pi$
$547$	14.0000	0.598597	0.299298	+	0.954160i	$0.403247\pi$
$548$	− 12.0000i	− 0.512615i
$549$	0	0
$550$	−15.0000	−0.639602
$551$	0	0
$552$	0	0
$553$	0	0
$554$	14.0000i	0.594803i
$555$	0	0
$556$	− 6.92820i	− 0.293821i
$557$	− 21.0000i	− 0.889799i	−0.895581	−	0.444899i	$-0.853239\pi$
$557$	− 21.0000i	− 0.889799i	0.895581	−	0.444899i	$-0.146761\pi$
$558$	0	0
$559$	13.8564i	0.586064i
$560$	0	0
$561$	0	0
$562$	6.00000	0.253095
$563$	31.1769	1.31395	0.656975	−	0.753912i	$-0.271836\pi$
$563$	31.1769	1.31395	0.656975	+	0.753912i	$0.271836\pi$
$564$	0	0
$565$	0	0
$566$	−6.92820	−0.291214
$567$	0	0
$568$	12.0000	0.503509
$569$	− 12.0000i	− 0.503066i	−0.967849	−	0.251533i	$-0.919065\pi$
$569$	− 12.0000i	− 0.503066i	0.967849	−	0.251533i	$-0.0809347\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−10.3923	−0.434524
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 8.66025i	− 0.360531i	−0.983618	−	0.180266i	$-0.942304\pi$
$577$	− 8.66025i	− 0.360531i	0.983618	−	0.180266i	$-0.0576957\pi$
$578$	− 17.0000i	− 0.707107i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	18.0000	0.745484
$584$	5.19615	0.215018
$585$	0	0
$586$	5.19615i	0.214651i
$587$	−10.3923	−0.428936	−0.214468	−	0.976731i	$-0.568802\pi$
$587$	−10.3923	−0.428936	−0.214468	+	0.976731i	$0.568802\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−8.00000	−0.328798
$593$	−20.7846	−0.853522	−0.426761	−	0.904365i	$-0.640345\pi$
$593$	−20.7846	−0.853522	−0.426761	+	0.904365i	$0.640345\pi$
$594$	0	0
$595$	0	0
$596$	15.0000i	0.614424i
$597$	0	0
$598$	0	0
$599$	30.0000i	1.22577i	0.790173	+	0.612883i	$0.209990\pi$
$599$	30.0000i	1.22577i	−0.790173	+	0.612883i	$0.790010\pi$
$600$	0	0
$601$	34.6410i	1.41304i	0.707695	+	0.706518i	$0.249735\pi$
$601$	34.6410i	1.41304i	−0.707695	+	0.706518i	$0.750265\pi$
$602$	0	0
$603$	0	0
$604$	23.0000	0.935857
$605$	0	0
$606$	0	0
$607$	5.19615i	0.210905i	0.994424	+	0.105453i	$0.0336291\pi$
$607$	5.19615i	0.210905i	−0.994424	+	0.105453i	$0.966371\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	36.0000i	1.45640i
$612$	0	0
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	17.3205	0.698999
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	34.6410i	1.39234i	0.717877	+	0.696170i	$0.245114\pi$
$619$	34.6410i	1.39234i	−0.717877	+	0.696170i	$0.754886\pi$
$620$	0	0
$621$	0	0
$622$	− 31.1769i	− 1.25008i
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	−20.7846	−0.830720
$627$	0	0
$628$	− 10.3923i	− 0.414698i
$629$	0	0
$630$	0	0
$631$	−37.0000	−1.47295	−0.736473	−	0.676467i	$-0.763510\pi$
$631$	−37.0000	−1.47295	−0.736473	+	0.676467i	$0.763510\pi$
$632$	13.0000i	0.517112i
$633$	0	0
$634$	9.00000	0.357436
$635$	0	0
$636$	0	0
$637$	0	0
$638$	27.0000i	1.06894i
$639$	0	0
$640$	0	0
$641$	− 42.0000i	− 1.65890i	−0.558581	−	0.829450i	$-0.688654\pi$
$641$	− 42.0000i	− 1.65890i	0.558581	−	0.829450i	$-0.311346\pi$
$642$	0	0
$643$	− 3.46410i	− 0.136611i	−0.997664	−	0.0683054i	$-0.978241\pi$
$643$	− 3.46410i	− 0.136611i	0.997664	−	0.0683054i	$-0.0217592\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	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
$649$	− 15.5885i	− 0.611900i
$650$	17.3205	0.679366
$651$	0	0
$652$	10.0000	0.391630
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	0	0
$655$	0	0
$656$	10.3923	0.405751
$657$	0	0
$658$	0	0
$659$	− 15.0000i	− 0.584317i	−0.956370	−	0.292159i	$-0.905627\pi$
$659$	− 15.0000i	− 0.584317i	0.956370	−	0.292159i	$-0.0943735\pi$
$660$	0	0
$661$	10.3923i	0.404214i	0.979363	+	0.202107i	$0.0647788\pi$
$661$	10.3923i	0.404214i	−0.979363	+	0.202107i	$0.935221\pi$
$662$	28.0000i	1.08825i
$663$	0	0
$664$	5.19615i	0.201650i
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−10.3923	−0.402090
$669$	0	0
$670$	0	0
$671$	41.5692	1.60476
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	13.0000i	0.500741i
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	15.5885	0.599113	0.299557	−	0.954079i	$-0.403161\pi$
$677$	15.5885	0.599113	0.299557	+	0.954079i	$0.403161\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	− 5.19615i	− 0.198971i
$683$	21.0000i	0.803543i	0.915740	+	0.401771i	$0.131605\pi$
$683$	21.0000i	0.803543i	−0.915740	+	0.401771i	$0.868395\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	4.00000	0.152499
$689$	−20.7846	−0.791831
$690$	0	0
$691$	− 13.8564i	− 0.527123i	−0.964643	−	0.263561i	$-0.915103\pi$
$691$	− 13.8564i	− 0.527123i	0.964643	−	0.263561i	$-0.0848971\pi$
$692$	25.9808	0.987640
$693$	0	0
$694$	−3.00000	−0.113878
$695$	0	0
$696$	0	0
$697$	0	0
$698$	27.7128	1.04895
$699$	0	0
$700$	0	0
$701$	− 6.00000i	− 0.226617i	−0.993560	−	0.113308i	$-0.963855\pi$
$701$	− 6.00000i	− 0.226617i	0.993560	−	0.113308i	$-0.0361448\pi$
$702$	0	0
$703$	0	0
$704$	3.00000i	0.113067i
$705$	0	0
$706$	− 10.3923i	− 0.391120i
$707$	0	0
$708$	0	0
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	0	0
$711$	0	0
$712$	10.3923i	0.389468i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	− 9.00000i	− 0.336346i
$717$	0	0
$718$	12.0000	0.447836
$719$	−10.3923	−0.387568	−0.193784	−	0.981044i	$-0.562076\pi$
$719$	−10.3923	−0.387568	−0.193784	+	0.981044i	$0.562076\pi$
$720$	0	0
$721$	0	0
$722$	19.0000i	0.707107i
$723$	0	0
$724$	− 20.7846i	− 0.772454i
$725$	− 45.0000i	− 1.67126i
$726$	0	0
$727$	− 24.2487i	− 0.899335i	−0.893196	−	0.449667i	$-0.851542\pi$
$727$	− 24.2487i	− 0.899335i	0.893196	−	0.449667i	$-0.148458\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 13.8564i	− 0.511798i	−0.966704	−	0.255899i	$-0.917629\pi$
$733$	− 13.8564i	− 0.511798i	0.966704	−	0.255899i	$-0.0823715\pi$
$734$	−10.3923	−0.383587
$735$	0	0
$736$	0	0
$737$	− 6.00000i	− 0.221013i
$738$	0	0
$739$	26.0000	0.956425	0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000	0.956425	0.478213	+	0.878244i	$0.341285\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 6.00000i	− 0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$	− 6.00000i	− 0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$	0	0
$745$	0	0
$746$	− 4.00000i	− 0.146450i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	10.3923	0.378968
$753$	0	0
$754$	− 31.1769i	− 1.13540i
$755$	0	0
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	− 26.0000i	− 0.944363i
$759$	0	0
$760$	0	0
$761$	−41.5692	−1.50688	−0.753442	−	0.657515i	$-0.771608\pi$
$761$	−41.5692	−1.50688	−0.753442	+	0.657515i	$0.771608\pi$
$762$	0	0
$763$	0	0
$764$	6.00000i	0.217072i
$765$	0	0
$766$	20.7846i	0.750978i
$767$	18.0000i	0.649942i
$768$	0	0
$769$	− 15.5885i	− 0.562134i	−0.959688	−	0.281067i	$-0.909312\pi$
$769$	− 15.5885i	− 0.562134i	0.959688	−	0.281067i	$-0.0906883\pi$
$770$	0	0
$771$	0	0
$772$	11.0000	0.395899
$773$	−41.5692	−1.49514	−0.747570	−	0.664183i	$-0.768780\pi$
$773$	−41.5692	−1.49514	−0.747570	+	0.664183i	$0.768780\pi$
$774$	0	0
$775$	8.66025i	0.311086i
$776$	−8.66025	−0.310885
$777$	0	0
$778$	9.00000	0.322666
$779$	0	0
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 24.2487i	− 0.864373i	−0.901784	−	0.432187i	$-0.857742\pi$
$787$	− 24.2487i	− 0.864373i	0.901784	−	0.432187i	$-0.142258\pi$
$788$	− 15.0000i	− 0.534353i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−48.0000	−1.70453
$794$	24.2487	0.860555
$795$	0	0
$796$	− 1.73205i	− 0.0613909i
$797$	15.5885	0.552171	0.276086	−	0.961133i	$-0.410963\pi$
$797$	15.5885	0.552171	0.276086	+	0.961133i	$0.410963\pi$
$798$	0	0
$799$	0	0
$800$	− 5.00000i	− 0.176777i
$801$	0	0
$802$	30.0000	1.05934
$803$	15.5885	0.550105
$804$	0	0
$805$	0	0
$806$	6.00000i	0.211341i
$807$	0	0
$808$	− 5.19615i	− 0.182800i
$809$	− 6.00000i	− 0.210949i	−0.994422	−	0.105474i	$-0.966364\pi$
$809$	− 6.00000i	− 0.210949i	0.994422	−	0.105474i	$-0.0336361\pi$
$810$	0	0
$811$	− 10.3923i	− 0.364923i	−0.983213	−	0.182462i	$-0.941593\pi$
$811$	− 10.3923i	− 0.364923i	0.983213	−	0.182462i	$-0.0584065\pi$
$812$	0	0
$813$	0	0
$814$	−24.0000	−0.841200
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−6.92820	−0.242239
$819$	0	0
$820$	0	0
$821$	33.0000i	1.15171i	0.817553	+	0.575854i	$0.195330\pi$
$821$	33.0000i	1.15171i	−0.817553	+	0.575854i	$0.804670\pi$
$822$	0	0
$823$	11.0000	0.383436	0.191718	−	0.981450i	$-0.438594\pi$
$823$	11.0000	0.383436	0.191718	+	0.981450i	$0.438594\pi$
$824$	17.3205	0.603388
$825$	0	0
$826$	0	0
$827$	− 3.00000i	− 0.104320i	−0.998639	−	0.0521601i	$-0.983389\pi$
$827$	− 3.00000i	− 0.104320i	0.998639	−	0.0521601i	$-0.0166106\pi$
$828$	0	0
$829$	− 31.1769i	− 1.08282i	−0.840759	−	0.541409i	$-0.817891\pi$
$829$	− 31.1769i	− 1.08282i	0.840759	−	0.541409i	$-0.182109\pi$
$830$	0	0
$831$	0	0
$832$	− 3.46410i	− 0.120096i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	31.1769i	1.07699i
$839$	20.7846	0.717564	0.358782	−	0.933421i	$-0.383192\pi$
$839$	20.7846	0.717564	0.358782	+	0.933421i	$0.383192\pi$
$840$	0	0
$841$	−52.0000	−1.79310
$842$	− 10.0000i	− 0.344623i
$843$	0	0
$844$	2.00000	0.0688428
$845$	0	0
$846$	0	0
$847$	0	0
$848$	6.00000i	0.206041i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	10.3923i	0.355826i	0.984046	+	0.177913i	$0.0569345\pi$
$853$	10.3923i	0.355826i	−0.984046	+	0.177913i	$0.943065\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	10.3923	0.354994	0.177497	−	0.984121i	$-0.443200\pi$
$857$	10.3923	0.354994	0.177497	+	0.984121i	$0.443200\pi$
$858$	0	0
$859$	48.4974i	1.65471i	0.561679	+	0.827355i	$0.310156\pi$
$859$	48.4974i	1.65471i	−0.561679	+	0.827355i	$0.689844\pi$
$860$	0	0
$861$	0	0
$862$	−18.0000	−0.613082
$863$	6.00000i	0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$	6.00000i	0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$	0	0
$865$	0	0
$866$	−12.1244	−0.412002
$867$	0	0
$868$	0	0
$869$	39.0000i	1.32298i
$870$	0	0
$871$	6.92820i	0.234753i
$872$	8.00000i	0.270914i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	15.5885	0.526085
$879$	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
$883$	−2.00000	−0.0673054	−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000	−0.0673054	−0.0336527	+	0.999434i	$0.510714\pi$
$884$	0	0
$885$	0	0
$886$	−27.0000	−0.907083
$887$	31.1769	1.04682	0.523409	−	0.852081i	$-0.324660\pi$
$887$	31.1769	1.04682	0.523409	+	0.852081i	$0.324660\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	− 5.19615i	− 0.173980i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−30.0000	−1.00111
$899$	15.5885	0.519904
$900$	0	0
$901$	0	0
$902$	31.1769	1.03808
$903$	0	0
$904$	−12.0000	−0.399114
$905$	0	0
$906$	0	0
$907$	22.0000	0.730498	0.365249	−	0.930910i	$-0.380984\pi$
$907$	22.0000	0.730498	0.365249	+	0.930910i	$0.380984\pi$
$908$	−25.9808	−0.862202
$909$	0	0
$910$	0	0
$911$	− 36.0000i	− 1.19273i	−0.802712	−	0.596367i	$-0.796610\pi$
$911$	− 36.0000i	− 1.19273i	0.802712	−	0.596367i	$-0.203390\pi$
$912$	0	0
$913$	15.5885i	0.515903i
$914$	22.0000i	0.727695i
$915$	0	0
$916$	6.92820i	0.228914i
$917$	0	0
$918$	0	0
$919$	−35.0000	−1.15454	−0.577272	−	0.816552i	$-0.695883\pi$
$919$	−35.0000	−1.15454	−0.577272	+	0.816552i	$0.695883\pi$
$920$	0	0
$921$	0	0
$922$	− 15.5885i	− 0.513378i
$923$	−41.5692	−1.36827
$924$	0	0
$925$	40.0000	1.31519
$926$	5.00000i	0.164310i
$927$	0	0
$928$	−9.00000	−0.295439
$929$	−41.5692	−1.36384	−0.681921	−	0.731426i	$-0.738855\pi$
$929$	−41.5692	−1.36384	−0.681921	+	0.731426i	$0.738855\pi$
$930$	0	0
$931$	0	0
$932$	18.0000i	0.589610i
$933$	0	0
$934$	5.19615i	0.170023i
$935$	0	0
$936$	0	0
$937$	− 34.6410i	− 1.13167i	−0.824518	−	0.565836i	$-0.808553\pi$
$937$	− 34.6410i	− 1.13167i	0.824518	−	0.565836i	$-0.191447\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.19615	0.169390	0.0846949	−	0.996407i	$-0.473008\pi$
$941$	5.19615	0.169390	0.0846949	+	0.996407i	$0.473008\pi$
$942$	0	0
$943$	0	0
$944$	5.19615	0.169120
$945$	0	0
$946$	12.0000	0.390154
$947$	9.00000i	0.292461i	0.989251	+	0.146230i	$0.0467141\pi$
$947$	9.00000i	0.292461i	−0.989251	+	0.146230i	$0.953286\pi$
$948$	0	0
$949$	−18.0000	−0.584305
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 18.0000i	− 0.583077i	−0.956559	−	0.291539i	$-0.905833\pi$
$953$	− 18.0000i	− 0.583077i	0.956559	−	0.291539i	$-0.0941672\pi$
$954$	0	0
$955$	0	0
$956$	12.0000i	0.388108i
$957$	0	0
$958$	− 10.3923i	− 0.335760i
$959$	0	0
$960$	0	0
$961$	28.0000	0.903226
$962$	27.7128	0.893497
$963$	0	0
$964$	12.1244i	0.390499i
$965$	0	0
$966$	0	0
$967$	56.0000	1.80084	0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000	1.80084	0.900419	+	0.435023i	$0.143260\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	0	0
$970$	0	0
$971$	31.1769	1.00051	0.500257	−	0.865877i	$-0.333239\pi$
$971$	31.1769	1.00051	0.500257	+	0.865877i	$0.333239\pi$
$972$	0	0
$973$	0	0
$974$	− 11.0000i	− 0.352463i
$975$	0	0
$976$	13.8564i	0.443533i
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	0	0
$979$	31.1769i	0.996419i
$980$	0	0
$981$	0	0
$982$	36.0000	1.14881
$983$	−20.7846	−0.662926	−0.331463	−	0.943468i	$-0.607542\pi$
$983$	−20.7846	−0.662926	−0.331463	+	0.943468i	$0.607542\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	1.73205	0.0549927
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	38.1051i	1.20680i	0.797438	+	0.603401i	$0.206188\pi$
$997$	38.1051i	1.20680i	−0.797438	+	0.603401i	$0.793812\pi$
$998$	16.0000i	0.506471i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.d.c.2645.4		4
3.2	odd	2	inner	2646.2.d.c.2645.2		4
7.4	even	3		378.2.k.a.215.1	✓	4
7.5	odd	6		378.2.k.a.269.2	yes	4
7.6	odd	2	inner	2646.2.d.c.2645.3		4
21.5	even	6		378.2.k.a.269.1	yes	4
21.11	odd	6		378.2.k.a.215.2	yes	4
21.20	even	2	inner	2646.2.d.c.2645.1		4
63.4	even	3		1134.2.t.a.593.2		4
63.5	even	6		1134.2.l.d.269.1		4
63.11	odd	6		1134.2.l.d.215.1		4
63.25	even	3		1134.2.l.d.215.2		4
63.32	odd	6		1134.2.t.a.593.1		4
63.40	odd	6		1134.2.l.d.269.2		4
63.47	even	6		1134.2.t.a.1025.2		4
63.61	odd	6		1134.2.t.a.1025.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.k.a.215.1	✓	4	7.4	even	3
378.2.k.a.215.2	yes	4	21.11	odd	6
378.2.k.a.269.1	yes	4	21.5	even	6
378.2.k.a.269.2	yes	4	7.5	odd	6
1134.2.l.d.215.1		4	63.11	odd	6
1134.2.l.d.215.2		4	63.25	even	3
1134.2.l.d.269.1		4	63.5	even	6
1134.2.l.d.269.2		4	63.40	odd	6
1134.2.t.a.593.1		4	63.32	odd	6
1134.2.t.a.593.2		4	63.4	even	3
1134.2.t.a.1025.1		4	63.61	odd	6
1134.2.t.a.1025.2		4	63.47	even	6
2646.2.d.c.2645.1		4	21.20	even	2	inner
2646.2.d.c.2645.2		4	3.2	odd	2	inner
2646.2.d.c.2645.3		4	7.6	odd	2	inner
2646.2.d.c.2645.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{8} -3.00000i q^{11} +3.46410i q^{13} +1.00000 q^{16} +3.00000 q^{22} -5.00000 q^{25} -3.46410 q^{26} +9.00000i q^{29} -1.73205i q^{31} +1.00000i q^{32} -8.00000 q^{37} +10.3923 q^{41} +4.00000 q^{43} +3.00000i q^{44} +10.3923 q^{47} -5.00000i q^{50} -3.46410i q^{52} +6.00000i q^{53} -9.00000 q^{58} +5.19615 q^{59} +13.8564i q^{61} +1.73205 q^{62} -1.00000 q^{64} +2.00000 q^{67} +12.0000i q^{71} +5.19615i q^{73} -8.00000i q^{74} -13.0000 q^{79} +10.3923i q^{82} -5.19615 q^{83} +4.00000i q^{86} -3.00000 q^{88} -10.3923 q^{89} +10.3923i q^{94} -8.66025i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 4 q^{16} + 12 q^{22} - 20 q^{25} - 32 q^{37} + 16 q^{43} - 36 q^{58} - 4 q^{64} + 8 q^{67} - 52 q^{79} - 12 q^{88}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^16 + 12 * q^22 - 20 * q^25 - 32 * q^37 + 16 * q^43 - 36 * q^58 - 4 * q^64 + 8 * q^67 - 52 * q^79 - 12 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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