LMFDB - Embedded newform 2352.2.k.e.881.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(881,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("2352.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.k (of order $2$, degree $1$, not 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:	$18.7808145554$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2352.881
Dual form			2352.2.k.e.881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +1.73205 q^{5} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} +1.73205 q^{5} +3.00000 q^{9} +3.00000i q^{11} +3.46410i q^{13} -3.00000 q^{15} -3.46410 q^{17} -3.46410i q^{19} +6.00000i q^{23} -2.00000 q^{25} -5.19615 q^{27} +3.00000i q^{29} -1.73205i q^{31} -5.19615i q^{33} -2.00000 q^{37} -6.00000i q^{39} -6.92820 q^{41} +8.00000 q^{43} +5.19615 q^{45} -6.92820 q^{47} +6.00000 q^{51} -9.00000i q^{53} +5.19615i q^{55} +6.00000i q^{57} +1.73205 q^{59} +6.00000i q^{65} -2.00000 q^{67} -10.3923i q^{69} +12.0000i q^{71} -6.92820i q^{73} +3.46410 q^{75} +1.00000 q^{79} +9.00000 q^{81} -8.66025 q^{83} -6.00000 q^{85} -5.19615i q^{87} -10.3923 q^{89} +3.00000i q^{93} -6.00000i q^{95} +5.19615i q^{97} +9.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} - 12 q^{15} - 8 q^{25} - 8 q^{37} + 32 q^{43} + 24 q^{51} - 8 q^{67} + 4 q^{79} + 36 q^{81} - 24 q^{85}+O(q^{100}) $ 4 * q + 12 * q^9 - 12 * q^15 - 8 * q^25 - 8 * q^37 + 32 * q^43 + 24 * q^51 - 8 * q^67 + 4 * q^79 + 36 * q^81 - 24 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	−1.73205	−1.00000
$3$	−1.73205	−1.00000
$4$	0	0
$5$	1.73205	0.774597	0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205	0.774597	0.387298	+	0.921954i	$0.373408\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	3.00000i	0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$	3.00000i	0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$	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
$15$	−3.00000	−0.774597
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	− 3.46410i	− 0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	− 3.46410i	− 0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	0	0
$27$	−5.19615	−1.00000
$28$	0	0
$29$	3.00000i	0.557086i	0.960424	+	0.278543i	$0.0898515\pi$
$29$	3.00000i	0.557086i	−0.960424	+	0.278543i	$0.910149\pi$
$30$	0	0
$31$	− 1.73205i	− 0.311086i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	− 1.73205i	− 0.311086i	0.987829	−	0.155543i	$-0.0497126\pi$
$32$	0	0
$33$	− 5.19615i	− 0.904534i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	− 6.00000i	− 0.960769i
$40$	0	0
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	5.19615	0.774597
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	6.00000	0.840168
$52$	0	0
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	0	0
$55$	5.19615i	0.700649i
$56$	0	0
$57$	6.00000i	0.794719i
$58$	0	0
$59$	1.73205	0.225494	0.112747	−	0.993624i	$-0.464035\pi$
$59$	1.73205	0.225494	0.112747	+	0.993624i	$0.464035\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.00000i	0.744208i
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	− 10.3923i	− 1.25109i
$70$	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
$73$	− 6.92820i	− 0.810885i	−0.914121	−	0.405442i	$-0.867117\pi$
$73$	− 6.92820i	− 0.810885i	0.914121	−	0.405442i	$-0.132883\pi$
$74$	0	0
$75$	3.46410	0.400000
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−8.66025	−0.950586	−0.475293	−	0.879827i	$-0.657658\pi$
$83$	−8.66025	−0.950586	−0.475293	+	0.879827i	$0.657658\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	0	0
$87$	− 5.19615i	− 0.557086i
$88$	0	0
$89$	−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
$93$	3.00000i	0.311086i
$94$	0	0
$95$	− 6.00000i	− 0.615587i
$96$	0	0
$97$	5.19615i	0.527589i	0.964579	+	0.263795i	$0.0849741\pi$
$97$	5.19615i	0.527589i	−0.964579	+	0.263795i	$0.915026\pi$
$98$	0	0
$99$	9.00000i	0.904534i
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	3.46410i	0.341328i	0.985329	+	0.170664i	$0.0545913\pi$
$103$	3.46410i	0.341328i	−0.985329	+	0.170664i	$0.945409\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	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
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	3.46410	0.328798
$112$	0	0
$113$	12.0000i	1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$	12.0000i	1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$	0	0
$115$	10.3923i	0.969087i
$116$	0	0
$117$	10.3923i	0.960769i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2.00000	0.181818
$122$	0	0
$123$	12.0000	1.08200
$124$	0	0
$125$	−12.1244	−1.08444
$126$	0	0
$127$	−11.0000	−0.976092	−0.488046	−	0.872818i	$-0.662290\pi$
$127$	−11.0000	−0.976092	−0.488046	+	0.872818i	$0.662290\pi$
$128$	0	0
$129$	−13.8564	−1.21999
$130$	0	0
$131$	5.19615	0.453990	0.226995	−	0.973896i	$-0.427110\pi$
$131$	5.19615	0.453990	0.226995	+	0.973896i	$0.427110\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−9.00000	−0.774597
$136$	0	0
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	0	0
$139$	− 17.3205i	− 1.46911i	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	− 17.3205i	− 1.46911i	0.678551	−	0.734553i	$-0.262608\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	−10.3923	−0.869048
$144$	0	0
$145$	5.19615i	0.431517i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.0000i	1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$	18.0000i	1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$	0	0
$151$	−7.00000	−0.569652	−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000	−0.569652	−0.284826	+	0.958579i	$0.591936\pi$
$152$	0	0
$153$	−10.3923	−0.840168
$154$	0	0
$155$	− 3.00000i	− 0.240966i
$156$	0	0
$157$	20.7846i	1.65879i	0.558661	+	0.829396i	$0.311315\pi$
$157$	20.7846i	1.65879i	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	15.5885i	1.23625i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	0	0
$165$	− 9.00000i	− 0.700649i
$166$	0	0
$167$	−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
$169$	1.00000	0.0769231
$170$	0	0
$171$	− 10.3923i	− 0.794719i
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.00000	−0.225494
$178$	0	0
$179$	− 12.0000i	− 0.896922i	−0.893802	−	0.448461i	$-0.851972\pi$
$179$	− 12.0000i	− 0.896922i	0.893802	−	0.448461i	$-0.148028\pi$
$180$	0	0
$181$	6.92820i	0.514969i	0.966282	+	0.257485i	$0.0828937\pi$
$181$	6.92820i	0.514969i	−0.966282	+	0.257485i	$0.917106\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.46410	−0.254686
$186$	0	0
$187$	− 10.3923i	− 0.759961i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	0	0
$195$	− 10.3923i	− 0.744208i
$196$	0	0
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	10.3923i	0.736691i	0.929689	+	0.368345i	$0.120076\pi$
$199$	10.3923i	0.736691i	−0.929689	+	0.368345i	$0.879924\pi$
$200$	0	0
$201$	3.46410	0.244339
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	18.0000i	1.25109i
$208$	0	0
$209$	10.3923	0.718851
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	− 20.7846i	− 1.42414i
$214$	0	0
$215$	13.8564	0.944999
$216$	0	0
$217$	0	0
$218$	0	0
$219$	12.0000i	0.810885i
$220$	0	0
$221$	− 12.0000i	− 0.807207i
$222$	0	0
$223$	25.9808i	1.73980i	0.493228	+	0.869900i	$0.335817\pi$
$223$	25.9808i	1.73980i	−0.493228	+	0.869900i	$0.664183\pi$
$224$	0	0
$225$	−6.00000	−0.400000
$226$	0	0
$227$	−5.19615	−0.344881	−0.172440	−	0.985020i	$-0.555165\pi$
$227$	−5.19615	−0.344881	−0.172440	+	0.985020i	$0.555165\pi$
$228$	0	0
$229$	− 13.8564i	− 0.915657i	−0.889041	−	0.457829i	$-0.848627\pi$
$229$	− 13.8564i	− 0.915657i	0.889041	−	0.457829i	$-0.151373\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 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
$235$	−12.0000	−0.782794
$236$	0	0
$237$	−1.73205	−0.112509
$238$	0	0
$239$	− 6.00000i	− 0.388108i	−0.980991	−	0.194054i	$-0.937836\pi$
$239$	− 6.00000i	− 0.388108i	0.980991	−	0.194054i	$-0.0621637\pi$
$240$	0	0
$241$	25.9808i	1.67357i	0.547533	+	0.836784i	$0.315567\pi$
$241$	25.9808i	1.67357i	−0.547533	+	0.836784i	$0.684433\pi$
$242$	0	0
$243$	−15.5885	−1.00000
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.0000	0.763542
$248$	0	0
$249$	15.0000	0.950586
$250$	0	0
$251$	19.0526	1.20259	0.601293	−	0.799028i	$-0.294652\pi$
$251$	19.0526	1.20259	0.601293	+	0.799028i	$0.294652\pi$
$252$	0	0
$253$	−18.0000	−1.13165
$254$	0	0
$255$	10.3923	0.650791
$256$	0	0
$257$	10.3923	0.648254	0.324127	−	0.946014i	$-0.394929\pi$
$257$	10.3923	0.648254	0.324127	+	0.946014i	$0.394929\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	9.00000i	0.557086i
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	− 15.5885i	− 0.957591i
$266$	0	0
$267$	18.0000	1.10158
$268$	0	0
$269$	29.4449	1.79529	0.897643	−	0.440724i	$-0.145278\pi$
$269$	29.4449	1.79529	0.897643	+	0.440724i	$0.145278\pi$
$270$	0	0
$271$	− 5.19615i	− 0.315644i	−0.987468	−	0.157822i	$-0.949553\pi$
$271$	− 5.19615i	− 0.315644i	0.987468	−	0.157822i	$-0.0504472\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 6.00000i	− 0.361814i
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	0	0
$279$	− 5.19615i	− 0.311086i
$280$	0	0
$281$	30.0000i	1.78965i	0.446417	+	0.894825i	$0.352700\pi$
$281$	30.0000i	1.78965i	−0.446417	+	0.894825i	$0.647300\pi$
$282$	0	0
$283$	27.7128i	1.64736i	0.567058	+	0.823678i	$0.308082\pi$
$283$	27.7128i	1.64736i	−0.567058	+	0.823678i	$0.691918\pi$
$284$	0	0
$285$	10.3923i	0.615587i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	− 9.00000i	− 0.527589i
$292$	0	0
$293$	19.0526	1.11306	0.556531	−	0.830827i	$-0.312132\pi$
$293$	19.0526	1.11306	0.556531	+	0.830827i	$0.312132\pi$
$294$	0	0
$295$	3.00000	0.174667
$296$	0	0
$297$	− 15.5885i	− 0.904534i
$298$	0	0
$299$	−20.7846	−1.20201
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	24.2487i	1.38395i	0.721923	+	0.691974i	$0.243259\pi$
$307$	24.2487i	1.38395i	−0.721923	+	0.691974i	$0.756741\pi$
$308$	0	0
$309$	− 6.00000i	− 0.341328i
$310$	0	0
$311$	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
$313$	1.73205i	0.0979013i	0.998801	+	0.0489506i	$0.0155877\pi$
$313$	1.73205i	0.0979013i	−0.998801	+	0.0489506i	$0.984412\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 15.0000i	− 0.842484i	−0.906948	−	0.421242i	$-0.861594\pi$
$317$	− 15.0000i	− 0.842484i	0.906948	−	0.421242i	$-0.138406\pi$
$318$	0	0
$319$	−9.00000	−0.503903
$320$	0	0
$321$	− 5.19615i	− 0.290021i
$322$	0	0
$323$	12.0000i	0.667698i
$324$	0	0
$325$	− 6.92820i	− 0.384308i
$326$	0	0
$327$	3.46410	0.191565
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	−3.46410	−0.189264
$336$	0	0
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	0	0
$339$	− 20.7846i	− 1.12887i
$340$	0	0
$341$	5.19615	0.281387
$342$	0	0
$343$	0	0
$344$	0	0
$345$	− 18.0000i	− 0.969087i
$346$	0	0
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	10.3923i	0.556287i	0.960539	+	0.278144i	$0.0897191\pi$
$349$	10.3923i	0.556287i	−0.960539	+	0.278144i	$0.910281\pi$
$350$	0	0
$351$	− 18.0000i	− 0.960769i
$352$	0	0
$353$	−34.6410	−1.84376	−0.921878	−	0.387481i	$-0.873345\pi$
$353$	−34.6410	−1.84376	−0.921878	+	0.387481i	$0.873345\pi$
$354$	0	0
$355$	20.7846i	1.10313i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 6.00000i	− 0.316668i	−0.987386	−	0.158334i	$-0.949388\pi$
$359$	− 6.00000i	− 0.316668i	0.987386	−	0.158334i	$-0.0506123\pi$
$360$	0	0
$361$	7.00000	0.368421
$362$	0	0
$363$	−3.46410	−0.181818
$364$	0	0
$365$	− 12.0000i	− 0.628109i
$366$	0	0
$367$	− 22.5167i	− 1.17536i	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	− 22.5167i	− 1.17536i	0.809093	−	0.587680i	$-0.199959\pi$
$368$	0	0
$369$	−20.7846	−1.08200
$370$	0	0
$371$	0	0
$372$	0	0
$373$	32.0000	1.65690	0.828449	−	0.560065i	$-0.189224\pi$
$373$	32.0000	1.65690	0.828449	+	0.560065i	$0.189224\pi$
$374$	0	0
$375$	21.0000	1.08444
$376$	0	0
$377$	−10.3923	−0.535231
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	19.0526	0.976092
$382$	0	0
$383$	−3.46410	−0.177007	−0.0885037	−	0.996076i	$-0.528208\pi$
$383$	−3.46410	−0.177007	−0.0885037	+	0.996076i	$0.528208\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	24.0000	1.21999
$388$	0	0
$389$	− 18.0000i	− 0.912636i	−0.889817	−	0.456318i	$-0.849168\pi$
$389$	− 18.0000i	− 0.912636i	0.889817	−	0.456318i	$-0.150832\pi$
$390$	0	0
$391$	− 20.7846i	− 1.05112i
$392$	0	0
$393$	−9.00000	−0.453990
$394$	0	0
$395$	1.73205	0.0871489
$396$	0	0
$397$	− 27.7128i	− 1.39087i	−0.718591	−	0.695433i	$-0.755213\pi$
$397$	− 27.7128i	− 1.39087i	0.718591	−	0.695433i	$-0.244787\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 12.0000i	− 0.599251i	−0.954057	−	0.299626i	$-0.903138\pi$
$401$	− 12.0000i	− 0.599251i	0.954057	−	0.299626i	$-0.0968618\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	0	0
$405$	15.5885	0.774597
$406$	0	0
$407$	− 6.00000i	− 0.297409i
$408$	0	0
$409$	− 8.66025i	− 0.428222i	−0.976809	−	0.214111i	$-0.931315\pi$
$409$	− 8.66025i	− 0.428222i	0.976809	−	0.214111i	$-0.0686854\pi$
$410$	0	0
$411$	− 31.1769i	− 1.53784i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−15.0000	−0.736321
$416$	0	0
$417$	30.0000i	1.46911i
$418$	0	0
$419$	24.2487	1.18463	0.592314	−	0.805708i	$-0.298215\pi$
$419$	24.2487	1.18463	0.592314	+	0.805708i	$0.298215\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	0	0
$423$	−20.7846	−1.01058
$424$	0	0
$425$	6.92820	0.336067
$426$	0	0
$427$	0	0
$428$	0	0
$429$	18.0000	0.869048
$430$	0	0
$431$	24.0000i	1.15604i	0.816023	+	0.578020i	$0.196174\pi$
$431$	24.0000i	1.15604i	−0.816023	+	0.578020i	$0.803826\pi$
$432$	0	0
$433$	34.6410i	1.66474i	0.554220	+	0.832370i	$0.313017\pi$
$433$	34.6410i	1.66474i	−0.554220	+	0.832370i	$0.686983\pi$
$434$	0	0
$435$	− 9.00000i	− 0.431517i
$436$	0	0
$437$	20.7846	0.994263
$438$	0	0
$439$	− 36.3731i	− 1.73599i	−0.496571	−	0.867996i	$-0.665408\pi$
$439$	− 36.3731i	− 1.73599i	0.496571	−	0.867996i	$-0.334592\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 9.00000i	− 0.427603i	−0.976877	−	0.213801i	$-0.931415\pi$
$443$	− 9.00000i	− 0.427603i	0.976877	−	0.213801i	$-0.0685846\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	0	0
$447$	− 31.1769i	− 1.47462i
$448$	0	0
$449$	− 30.0000i	− 1.41579i	−0.706319	−	0.707894i	$-0.749646\pi$
$449$	− 30.0000i	− 1.41579i	0.706319	−	0.707894i	$-0.250354\pi$
$450$	0	0
$451$	− 20.7846i	− 0.978709i
$452$	0	0
$453$	12.1244	0.569652
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.00000	−0.233890	−0.116945	−	0.993138i	$-0.537310\pi$
$457$	−5.00000	−0.233890	−0.116945	+	0.993138i	$0.537310\pi$
$458$	0	0
$459$	18.0000	0.840168
$460$	0	0
$461$	13.8564	0.645357	0.322679	−	0.946509i	$-0.395417\pi$
$461$	13.8564	0.645357	0.322679	+	0.946509i	$0.395417\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	0	0
$465$	5.19615i	0.240966i
$466$	0	0
$467$	31.1769	1.44270	0.721348	−	0.692573i	$-0.243523\pi$
$467$	31.1769	1.44270	0.721348	+	0.692573i	$0.243523\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 36.0000i	− 1.65879i
$472$	0	0
$473$	24.0000i	1.10352i
$474$	0	0
$475$	6.92820i	0.317888i
$476$	0	0
$477$	− 27.0000i	− 1.23625i
$478$	0	0
$479$	6.92820	0.316558	0.158279	−	0.987394i	$-0.449406\pi$
$479$	6.92820	0.316558	0.158279	+	0.987394i	$0.449406\pi$
$480$	0	0
$481$	− 6.92820i	− 0.315899i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9.00000i	0.408669i
$486$	0	0
$487$	−1.00000	−0.0453143	−0.0226572	−	0.999743i	$-0.507213\pi$
$487$	−1.00000	−0.0453143	−0.0226572	+	0.999743i	$0.507213\pi$
$488$	0	0
$489$	24.2487	1.09656
$490$	0	0
$491$	− 33.0000i	− 1.48927i	−0.667472	−	0.744635i	$-0.732624\pi$
$491$	− 33.0000i	− 1.48927i	0.667472	−	0.744635i	$-0.267376\pi$
$492$	0	0
$493$	− 10.3923i	− 0.468046i
$494$	0	0
$495$	15.5885i	0.700649i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	0	0
$501$	30.0000	1.34030
$502$	0	0
$503$	38.1051	1.69902	0.849512	−	0.527570i	$-0.176897\pi$
$503$	38.1051	1.69902	0.849512	+	0.527570i	$0.176897\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.73205	−0.0769231
$508$	0	0
$509$	19.0526	0.844490	0.422245	−	0.906482i	$-0.361242\pi$
$509$	19.0526	0.844490	0.422245	+	0.906482i	$0.361242\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	18.0000i	0.794719i
$514$	0	0
$515$	6.00000i	0.264392i
$516$	0	0
$517$	− 20.7846i	− 0.914106i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−27.7128	−1.21412	−0.607060	−	0.794656i	$-0.707651\pi$
$521$	−27.7128	−1.21412	−0.607060	+	0.794656i	$0.707651\pi$
$522$	0	0
$523$	38.1051i	1.66622i	0.553107	+	0.833110i	$0.313442\pi$
$523$	38.1051i	1.66622i	−0.553107	+	0.833110i	$0.686558\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.00000i	0.261364i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	5.19615	0.225494
$532$	0	0
$533$	− 24.0000i	− 1.03956i
$534$	0	0
$535$	5.19615i	0.224649i
$536$	0	0
$537$	20.7846i	0.896922i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−16.0000	−0.687894	−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000	−0.687894	−0.343947	+	0.938989i	$0.611764\pi$
$542$	0	0
$543$	− 12.0000i	− 0.514969i
$544$	0	0
$545$	−3.46410	−0.148386
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10.3923	0.442727
$552$	0	0
$553$	0	0
$554$	0	0
$555$	6.00000	0.254686
$556$	0	0
$557$	− 3.00000i	− 0.127114i	−0.997978	−	0.0635570i	$-0.979756\pi$
$557$	− 3.00000i	− 0.127114i	0.997978	−	0.0635570i	$-0.0202445\pi$
$558$	0	0
$559$	27.7128i	1.17213i
$560$	0	0
$561$	18.0000i	0.759961i
$562$	0	0
$563$	25.9808	1.09496	0.547479	−	0.836819i	$-0.315587\pi$
$563$	25.9808	1.09496	0.547479	+	0.836819i	$0.315587\pi$
$564$	0	0
$565$	20.7846i	0.874415i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 6.00000i	− 0.251533i	−0.992060	−	0.125767i	$-0.959861\pi$
$569$	− 6.00000i	− 0.251533i	0.992060	−	0.125767i	$-0.0401390\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 12.0000i	− 0.500435i
$576$	0	0
$577$	1.73205i	0.0721062i	0.999350	+	0.0360531i	$0.0114785\pi$
$577$	1.73205i	0.0721062i	−0.999350	+	0.0360531i	$0.988521\pi$
$578$	0	0
$579$	39.8372	1.65558
$580$	0	0
$581$	0	0
$582$	0	0
$583$	27.0000	1.11823
$584$	0	0
$585$	18.0000i	0.744208i
$586$	0	0
$587$	−15.5885	−0.643404	−0.321702	−	0.946841i	$-0.604255\pi$
$587$	−15.5885	−0.643404	−0.321702	+	0.946841i	$0.604255\pi$
$588$	0	0
$589$	−6.00000	−0.247226
$590$	0	0
$591$	31.1769i	1.28245i
$592$	0	0
$593$	−38.1051	−1.56479	−0.782395	−	0.622783i	$-0.786002\pi$
$593$	−38.1051	−1.56479	−0.782395	+	0.622783i	$0.786002\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 18.0000i	− 0.736691i
$598$	0	0
$599$	− 30.0000i	− 1.22577i	−0.790173	−	0.612883i	$-0.790010\pi$
$599$	− 30.0000i	− 1.22577i	0.790173	−	0.612883i	$-0.209990\pi$
$600$	0	0
$601$	− 29.4449i	− 1.20108i	−0.799594	−	0.600541i	$-0.794952\pi$
$601$	− 29.4449i	− 1.20108i	0.799594	−	0.600541i	$-0.205048\pi$
$602$	0	0
$603$	−6.00000	−0.244339
$604$	0	0
$605$	3.46410	0.140836
$606$	0	0
$607$	− 22.5167i	− 0.913923i	−0.889486	−	0.456962i	$-0.848938\pi$
$607$	− 22.5167i	− 0.913923i	0.889486	−	0.456962i	$-0.151062\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 24.0000i	− 0.970936i
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	20.7846	0.838116
$616$	0	0
$617$	− 12.0000i	− 0.483102i	−0.970388	−	0.241551i	$-0.922344\pi$
$617$	− 12.0000i	− 0.483102i	0.970388	−	0.241551i	$-0.0776561\pi$
$618$	0	0
$619$	6.92820i	0.278468i	0.990260	+	0.139234i	$0.0444640\pi$
$619$	6.92820i	0.278468i	−0.990260	+	0.139234i	$0.955536\pi$
$620$	0	0
$621$	− 31.1769i	− 1.25109i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	−18.0000	−0.718851
$628$	0	0
$629$	6.92820	0.276246
$630$	0	0
$631$	31.0000	1.23409	0.617045	−	0.786928i	$-0.288330\pi$
$631$	31.0000	1.23409	0.617045	+	0.786928i	$0.288330\pi$
$632$	0	0
$633$	6.92820	0.275371
$634$	0	0
$635$	−19.0526	−0.756078
$636$	0	0
$637$	0	0
$638$	0	0
$639$	36.0000i	1.42414i
$640$	0	0
$641$	24.0000i	0.947943i	0.880540	+	0.473972i	$0.157180\pi$
$641$	24.0000i	0.947943i	−0.880540	+	0.473972i	$0.842820\pi$
$642$	0	0
$643$	17.3205i	0.683054i	0.939872	+	0.341527i	$0.110944\pi$
$643$	17.3205i	0.683054i	−0.939872	+	0.341527i	$0.889056\pi$
$644$	0	0
$645$	−24.0000	−0.944999
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	5.19615i	0.203967i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 3.00000i	− 0.117399i	−0.998276	−	0.0586995i	$-0.981305\pi$
$653$	− 3.00000i	− 0.117399i	0.998276	−	0.0586995i	$-0.0186954\pi$
$654$	0	0
$655$	9.00000	0.351659
$656$	0	0
$657$	− 20.7846i	− 0.810885i
$658$	0	0
$659$	− 12.0000i	− 0.467454i	−0.972302	−	0.233727i	$-0.924908\pi$
$659$	− 12.0000i	− 0.467454i	0.972302	−	0.233727i	$-0.0750921\pi$
$660$	0	0
$661$	6.92820i	0.269476i	0.990881	+	0.134738i	$0.0430193\pi$
$661$	6.92820i	0.269476i	−0.990881	+	0.134738i	$0.956981\pi$
$662$	0	0
$663$	20.7846i	0.807207i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−18.0000	−0.696963
$668$	0	0
$669$	− 45.0000i	− 1.73980i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−41.0000	−1.58043	−0.790217	−	0.612827i	$-0.790032\pi$
$673$	−41.0000	−1.58043	−0.790217	+	0.612827i	$0.790032\pi$
$674$	0	0
$675$	10.3923	0.400000
$676$	0	0
$677$	−5.19615	−0.199704	−0.0998522	−	0.995002i	$-0.531837\pi$
$677$	−5.19615	−0.199704	−0.0998522	+	0.995002i	$0.531837\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	9.00000	0.344881
$682$	0	0
$683$	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
$685$	31.1769i	1.19121i
$686$	0	0
$687$	24.0000i	0.915657i
$688$	0	0
$689$	31.1769	1.18775
$690$	0	0
$691$	6.92820i	0.263561i	0.991279	+	0.131781i	$0.0420694\pi$
$691$	6.92820i	0.263561i	−0.991279	+	0.131781i	$0.957931\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 30.0000i	− 1.13796i
$696$	0	0
$697$	24.0000	0.909065
$698$	0	0
$699$	31.1769i	1.17922i
$700$	0	0
$701$	− 3.00000i	− 0.113308i	−0.998394	−	0.0566542i	$-0.981957\pi$
$701$	− 3.00000i	− 0.113308i	0.998394	−	0.0566542i	$-0.0180433\pi$
$702$	0	0
$703$	6.92820i	0.261302i
$704$	0	0
$705$	20.7846	0.782794
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	3.00000	0.112509
$712$	0	0
$713$	10.3923	0.389195
$714$	0	0
$715$	−18.0000	−0.673162
$716$	0	0
$717$	10.3923i	0.388108i
$718$	0	0
$719$	−45.0333	−1.67946	−0.839730	−	0.543005i	$-0.817287\pi$
$719$	−45.0333	−1.67946	−0.839730	+	0.543005i	$0.817287\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 45.0000i	− 1.67357i
$724$	0	0
$725$	− 6.00000i	− 0.222834i
$726$	0	0
$727$	− 29.4449i	− 1.09205i	−0.837769	−	0.546025i	$-0.816140\pi$
$727$	− 29.4449i	− 1.09205i	0.837769	−	0.546025i	$-0.183860\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	−27.7128	−1.02500
$732$	0	0
$733$	34.6410i	1.27950i	0.768585	+	0.639748i	$0.220961\pi$
$733$	34.6410i	1.27950i	−0.768585	+	0.639748i	$0.779039\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 6.00000i	− 0.221013i
$738$	0	0
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	0	0
$741$	−20.7846	−0.763542
$742$	0	0
$743$	18.0000i	0.660356i	0.943919	+	0.330178i	$0.107109\pi$
$743$	18.0000i	0.660356i	−0.943919	+	0.330178i	$0.892891\pi$
$744$	0	0
$745$	31.1769i	1.14223i
$746$	0	0
$747$	−25.9808	−0.950586
$748$	0	0
$749$	0	0
$750$	0	0
$751$	11.0000	0.401396	0.200698	−	0.979653i	$-0.435679\pi$
$751$	11.0000	0.401396	0.200698	+	0.979653i	$0.435679\pi$
$752$	0	0
$753$	−33.0000	−1.20259
$754$	0	0
$755$	−12.1244	−0.441250
$756$	0	0
$757$	4.00000	0.145382	0.0726912	−	0.997354i	$-0.476841\pi$
$757$	4.00000	0.145382	0.0726912	+	0.997354i	$0.476841\pi$
$758$	0	0
$759$	31.1769	1.13165
$760$	0	0
$761$	−17.3205	−0.627868	−0.313934	−	0.949445i	$-0.601647\pi$
$761$	−17.3205	−0.627868	−0.313934	+	0.949445i	$0.601647\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−18.0000	−0.650791
$766$	0	0
$767$	6.00000i	0.216647i
$768$	0	0
$769$	19.0526i	0.687053i	0.939143	+	0.343526i	$0.111621\pi$
$769$	19.0526i	0.687053i	−0.939143	+	0.343526i	$0.888379\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	0	0
$773$	27.7128	0.996761	0.498380	−	0.866959i	$-0.333928\pi$
$773$	27.7128	0.996761	0.498380	+	0.866959i	$0.333928\pi$
$774$	0	0
$775$	3.46410i	0.124434i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	24.0000i	0.859889i
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	− 15.5885i	− 0.557086i
$784$	0	0
$785$	36.0000i	1.28490i
$786$	0	0
$787$	41.5692i	1.48178i	0.671625	+	0.740891i	$0.265597\pi$
$787$	41.5692i	1.48178i	−0.671625	+	0.740891i	$0.734403\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	27.0000i	0.957591i
$796$	0	0
$797$	25.9808	0.920286	0.460143	−	0.887845i	$-0.347798\pi$
$797$	25.9808	0.920286	0.460143	+	0.887845i	$0.347798\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	−31.1769	−1.10158
$802$	0	0
$803$	20.7846	0.733473
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−51.0000	−1.79529
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	− 31.1769i	− 1.09477i	−0.836881	−	0.547385i	$-0.815623\pi$
$811$	− 31.1769i	− 1.09477i	0.836881	−	0.547385i	$-0.184377\pi$
$812$	0	0
$813$	9.00000i	0.315644i
$814$	0	0
$815$	−24.2487	−0.849395
$816$	0	0
$817$	− 27.7128i	− 0.969549i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.00000i	0.104701i	0.998629	+	0.0523504i	$0.0166713\pi$
$821$	3.00000i	0.104701i	−0.998629	+	0.0523504i	$0.983329\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	0	0
$825$	10.3923i	0.361814i
$826$	0	0
$827$	9.00000i	0.312961i	0.987681	+	0.156480i	$0.0500148\pi$
$827$	9.00000i	0.312961i	−0.987681	+	0.156480i	$0.949985\pi$
$828$	0	0
$829$	17.3205i	0.601566i	0.953693	+	0.300783i	$0.0972480\pi$
$829$	17.3205i	0.601566i	−0.953693	+	0.300783i	$0.902752\pi$
$830$	0	0
$831$	−13.8564	−0.480673
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−30.0000	−1.03819
$836$	0	0
$837$	9.00000i	0.311086i
$838$	0	0
$839$	3.46410	0.119594	0.0597970	−	0.998211i	$-0.480955\pi$
$839$	3.46410	0.119594	0.0597970	+	0.998211i	$0.480955\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	0	0
$843$	− 51.9615i	− 1.78965i
$844$	0	0
$845$	1.73205	0.0595844
$846$	0	0
$847$	0	0
$848$	0	0
$849$	− 48.0000i	− 1.64736i
$850$	0	0
$851$	− 12.0000i	− 0.411355i
$852$	0	0
$853$	− 24.2487i	− 0.830260i	−0.909762	−	0.415130i	$-0.863736\pi$
$853$	− 24.2487i	− 0.830260i	0.909762	−	0.415130i	$-0.136264\pi$
$854$	0	0
$855$	− 18.0000i	− 0.615587i
$856$	0	0
$857$	−13.8564	−0.473326	−0.236663	−	0.971592i	$-0.576054\pi$
$857$	−13.8564	−0.473326	−0.236663	+	0.971592i	$0.576054\pi$
$858$	0	0
$859$	20.7846i	0.709162i	0.935025	+	0.354581i	$0.115376\pi$
$859$	20.7846i	0.709162i	−0.935025	+	0.354581i	$0.884624\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	54.0000i	1.83818i	0.394046	+	0.919091i	$0.371075\pi$
$863$	54.0000i	1.83818i	−0.394046	+	0.919091i	$0.628925\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.66025	0.294118
$868$	0	0
$869$	3.00000i	0.101768i
$870$	0	0
$871$	− 6.92820i	− 0.234753i
$872$	0	0
$873$	15.5885i	0.527589i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	44.0000	1.48577	0.742887	−	0.669417i	$-0.233456\pi$
$877$	44.0000	1.48577	0.742887	+	0.669417i	$0.233456\pi$
$878$	0	0
$879$	−33.0000	−1.11306
$880$	0	0
$881$	−10.3923	−0.350126	−0.175063	−	0.984557i	$-0.556013\pi$
$881$	−10.3923	−0.350126	−0.175063	+	0.984557i	$0.556013\pi$
$882$	0	0
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	0	0
$885$	−5.19615	−0.174667
$886$	0	0
$887$	24.2487	0.814192	0.407096	−	0.913385i	$-0.366541\pi$
$887$	24.2487	0.814192	0.407096	+	0.913385i	$0.366541\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	27.0000i	0.904534i
$892$	0	0
$893$	24.0000i	0.803129i
$894$	0	0
$895$	− 20.7846i	− 0.694753i
$896$	0	0
$897$	36.0000	1.20201
$898$	0	0
$899$	5.19615	0.173301
$900$	0	0
$901$	31.1769i	1.03865i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.0000i	0.398893i
$906$	0	0
$907$	26.0000	0.863316	0.431658	−	0.902037i	$-0.357929\pi$
$907$	26.0000	0.863316	0.431658	+	0.902037i	$0.357929\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	− 25.9808i	− 0.859838i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	− 42.0000i	− 1.38395i
$922$	0	0
$923$	−41.5692	−1.36827
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	10.3923i	0.341328i
$928$	0	0
$929$	48.4974	1.59115	0.795574	−	0.605856i	$-0.207169\pi$
$929$	48.4974	1.59115	0.795574	+	0.605856i	$0.207169\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	− 18.0000i	− 0.588663i
$936$	0	0
$937$	22.5167i	0.735587i	0.929907	+	0.367794i	$0.119887\pi$
$937$	22.5167i	0.735587i	−0.929907	+	0.367794i	$0.880113\pi$
$938$	0	0
$939$	− 3.00000i	− 0.0979013i
$940$	0	0
$941$	−32.9090	−1.07280	−0.536401	−	0.843963i	$-0.680216\pi$
$941$	−32.9090	−1.07280	−0.536401	+	0.843963i	$0.680216\pi$
$942$	0	0
$943$	− 41.5692i	− 1.35368i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	25.9808i	0.842484i
$952$	0	0
$953$	24.0000i	0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$953$	24.0000i	0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	15.5885	0.503903
$958$	0	0
$959$	0	0
$960$	0	0
$961$	28.0000	0.903226
$962$	0	0
$963$	9.00000i	0.290021i
$964$	0	0
$965$	−39.8372	−1.28240
$966$	0	0
$967$	7.00000	0.225105	0.112552	−	0.993646i	$-0.464097\pi$
$967$	7.00000	0.225105	0.112552	+	0.993646i	$0.464097\pi$
$968$	0	0
$969$	− 20.7846i	− 0.667698i
$970$	0	0
$971$	−8.66025	−0.277921	−0.138960	−	0.990298i	$-0.544376\pi$
$971$	−8.66025	−0.277921	−0.138960	+	0.990298i	$0.544376\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	12.0000i	0.384308i
$976$	0	0
$977$	− 24.0000i	− 0.767828i	−0.923369	−	0.383914i	$-0.874576\pi$
$977$	− 24.0000i	− 0.767828i	0.923369	−	0.383914i	$-0.125424\pi$
$978$	0	0
$979$	− 31.1769i	− 0.996419i
$980$	0	0
$981$	−6.00000	−0.191565
$982$	0	0
$983$	−13.8564	−0.441951	−0.220975	−	0.975279i	$-0.570924\pi$
$983$	−13.8564	−0.441951	−0.220975	+	0.975279i	$0.570924\pi$
$984$	0	0
$985$	− 31.1769i	− 0.993379i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	48.0000i	1.52631i
$990$	0	0
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	0	0
$993$	−13.8564	−0.439720
$994$	0	0
$995$	18.0000i	0.570638i
$996$	0	0
$997$	− 17.3205i	− 0.548546i	−0.961652	−	0.274273i	$-0.911563\pi$
$997$	− 17.3205i	− 0.548546i	0.961652	−	0.274273i	$-0.0884372\pi$
$998$	0	0
$999$	10.3923	0.328798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.k.e.881.2		4
3.2	odd	2	inner	2352.2.k.e.881.3		4
4.3	odd	2		294.2.d.a.293.4		4
7.2	even	3		336.2.bc.e.17.2		4
7.3	odd	6		336.2.bc.e.257.1		4
7.6	odd	2	inner	2352.2.k.e.881.4		4
12.11	even	2		294.2.d.a.293.1		4
21.2	odd	6		336.2.bc.e.17.1		4
21.17	even	6		336.2.bc.e.257.2		4
21.20	even	2	inner	2352.2.k.e.881.1		4
28.3	even	6		42.2.f.a.5.1	✓	4
28.11	odd	6		294.2.f.a.215.1		4
28.19	even	6		294.2.f.a.227.2		4
28.23	odd	6		42.2.f.a.17.2	yes	4
28.27	even	2		294.2.d.a.293.3		4
84.11	even	6		294.2.f.a.215.2		4
84.23	even	6		42.2.f.a.17.1	yes	4
84.47	odd	6		294.2.f.a.227.1		4
84.59	odd	6		42.2.f.a.5.2	yes	4
84.83	odd	2		294.2.d.a.293.2		4
140.3	odd	12		1050.2.u.a.299.1		4
140.23	even	12		1050.2.u.a.899.2		4
140.59	even	6		1050.2.s.b.551.2		4
140.79	odd	6		1050.2.s.b.101.1		4
140.87	odd	12		1050.2.u.d.299.2		4
140.107	even	12		1050.2.u.d.899.1		4
252.23	even	6		1134.2.l.c.269.1		4
252.31	even	6		1134.2.t.d.593.2		4
252.59	odd	6		1134.2.t.d.593.1		4
252.79	odd	6		1134.2.t.d.1025.1		4
252.115	even	6		1134.2.l.c.215.2		4
252.191	even	6		1134.2.t.d.1025.2		4
252.227	odd	6		1134.2.l.c.215.1		4
252.247	odd	6		1134.2.l.c.269.2		4
420.23	odd	12		1050.2.u.d.899.2		4
420.59	odd	6		1050.2.s.b.551.1		4
420.107	odd	12		1050.2.u.a.899.1		4
420.143	even	12		1050.2.u.d.299.1		4
420.227	even	12		1050.2.u.a.299.2		4
420.359	even	6		1050.2.s.b.101.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.f.a.5.1	✓	4	28.3	even	6
42.2.f.a.5.2	yes	4	84.59	odd	6
42.2.f.a.17.1	yes	4	84.23	even	6
42.2.f.a.17.2	yes	4	28.23	odd	6
294.2.d.a.293.1		4	12.11	even	2
294.2.d.a.293.2		4	84.83	odd	2
294.2.d.a.293.3		4	28.27	even	2
294.2.d.a.293.4		4	4.3	odd	2
294.2.f.a.215.1		4	28.11	odd	6
294.2.f.a.215.2		4	84.11	even	6
294.2.f.a.227.1		4	84.47	odd	6
294.2.f.a.227.2		4	28.19	even	6
336.2.bc.e.17.1		4	21.2	odd	6
336.2.bc.e.17.2		4	7.2	even	3
336.2.bc.e.257.1		4	7.3	odd	6
336.2.bc.e.257.2		4	21.17	even	6
1050.2.s.b.101.1		4	140.79	odd	6
1050.2.s.b.101.2		4	420.359	even	6
1050.2.s.b.551.1		4	420.59	odd	6
1050.2.s.b.551.2		4	140.59	even	6
1050.2.u.a.299.1		4	140.3	odd	12
1050.2.u.a.299.2		4	420.227	even	12
1050.2.u.a.899.1		4	420.107	odd	12
1050.2.u.a.899.2		4	140.23	even	12
1050.2.u.d.299.1		4	420.143	even	12
1050.2.u.d.299.2		4	140.87	odd	12
1050.2.u.d.899.1		4	140.107	even	12
1050.2.u.d.899.2		4	420.23	odd	12
1134.2.l.c.215.1		4	252.227	odd	6
1134.2.l.c.215.2		4	252.115	even	6
1134.2.l.c.269.1		4	252.23	even	6
1134.2.l.c.269.2		4	252.247	odd	6
1134.2.t.d.593.1		4	252.59	odd	6
1134.2.t.d.593.2		4	252.31	even	6
1134.2.t.d.1025.1		4	252.79	odd	6
1134.2.t.d.1025.2		4	252.191	even	6
2352.2.k.e.881.1		4	21.20	even	2	inner
2352.2.k.e.881.2		4	1.1	even	1	trivial
2352.2.k.e.881.3		4	3.2	odd	2	inner
2352.2.k.e.881.4		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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +1.73205 q^{5} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} +1.73205 q^{5} +3.00000 q^{9} +3.00000i q^{11} +3.46410i q^{13} -3.00000 q^{15} -3.46410 q^{17} -3.46410i q^{19} +6.00000i q^{23} -2.00000 q^{25} -5.19615 q^{27} +3.00000i q^{29} -1.73205i q^{31} -5.19615i q^{33} -2.00000 q^{37} -6.00000i q^{39} -6.92820 q^{41} +8.00000 q^{43} +5.19615 q^{45} -6.92820 q^{47} +6.00000 q^{51} -9.00000i q^{53} +5.19615i q^{55} +6.00000i q^{57} +1.73205 q^{59} +6.00000i q^{65} -2.00000 q^{67} -10.3923i q^{69} +12.0000i q^{71} -6.92820i q^{73} +3.46410 q^{75} +1.00000 q^{79} +9.00000 q^{81} -8.66025 q^{83} -6.00000 q^{85} -5.19615i q^{87} -10.3923 q^{89} +3.00000i q^{93} -6.00000i q^{95} +5.19615i q^{97} +9.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} - 12 q^{15} - 8 q^{25} - 8 q^{37} + 32 q^{43} + 24 q^{51} - 8 q^{67} + 4 q^{79} + 36 q^{81} - 24 q^{85}+O(q^{100}) \) 4 * q + 12 * q^9 - 12 * q^15 - 8 * q^25 - 8 * q^37 + 32 * q^43 + 24 * q^51 - 8 * q^67 + 4 * q^79 + 36 * q^81 - 24 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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