LMFDB - Embedded newform 2816.2.c.d.1409.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2816,2,Mod(1409,2816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2816, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2816.1409");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2816 = 2^{8} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2816.c (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:	$22.4858732092$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1409.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2816.1409
Dual form			2816.2.c.d.1409.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -3.00000i q^{5} -2.00000 q^{7} -6.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -3.00000i q^{5} -2.00000 q^{7} -6.00000 q^{9} +1.00000i q^{11} -9.00000 q^{15} -6.00000 q^{17} +4.00000i q^{19} +6.00000i q^{21} +1.00000 q^{23} -4.00000 q^{25} +9.00000i q^{27} +8.00000i q^{29} +7.00000 q^{31} +3.00000 q^{33} +6.00000i q^{35} -1.00000i q^{37} -4.00000 q^{41} -6.00000i q^{43} +18.0000i q^{45} +8.00000 q^{47} -3.00000 q^{49} +18.0000i q^{51} +2.00000i q^{53} +3.00000 q^{55} +12.0000 q^{57} +1.00000i q^{59} -4.00000i q^{61} +12.0000 q^{63} -5.00000i q^{67} -3.00000i q^{69} +3.00000 q^{71} -16.0000 q^{73} +12.0000i q^{75} -2.00000i q^{77} -2.00000 q^{79} +9.00000 q^{81} -2.00000i q^{83} +18.0000i q^{85} +24.0000 q^{87} -15.0000 q^{89} -21.0000i q^{93} +12.0000 q^{95} -7.00000 q^{97} -6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7} - 12 q^{9}+O(q^{10}) $ 2 * q - 4 * q^7 - 12 * q^9	$ 2 q - 4 q^{7} - 12 q^{9} - 18 q^{15} - 12 q^{17} + 2 q^{23} - 8 q^{25} + 14 q^{31} + 6 q^{33} - 8 q^{41} + 16 q^{47} - 6 q^{49} + 6 q^{55} + 24 q^{57} + 24 q^{63} + 6 q^{71} - 32 q^{73} - 4 q^{79} + 18 q^{81} + 48 q^{87} - 30 q^{89} + 24 q^{95} - 14 q^{97}+O(q^{100}) $ 2 * q - 4 * q^7 - 12 * q^9 - 18 * q^15 - 12 * q^17 + 2 * q^23 - 8 * q^25 + 14 * q^31 + 6 * q^33 - 8 * q^41 + 16 * q^47 - 6 * q^49 + 6 * q^55 + 24 * q^57 + 24 * q^63 + 6 * q^71 - 32 * q^73 - 4 * q^79 + 18 * q^81 + 48 * q^87 - 30 * q^89 + 24 * q^95 - 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times$.

$n$	$1025$	$1541$	$2047$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 3.00000i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	− 3.00000i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$4$	0	0
$5$	− 3.00000i	− 1.34164i	−0.741620	−	0.670820i	$-0.765942\pi$
$5$	− 3.00000i	− 1.34164i	0.741620	−	0.670820i	$-0.234058\pi$
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	−6.00000	−2.00000
$10$	0	0
$11$	1.00000i	0.301511i
$11$	1.00000i	0.301511i
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	−9.00000	−2.32379
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	6.00000i	1.30931i
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	9.00000i	1.73205i
$28$	0	0
$29$	8.00000i	1.48556i	0.669534	+	0.742781i	$0.266494\pi$
$29$	8.00000i	1.48556i	−0.669534	+	0.742781i	$0.733506\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	6.00000i	1.01419i
$36$	0	0
$37$	− 1.00000i	− 0.164399i	−0.996616	−	0.0821995i	$-0.973806\pi$
$37$	− 1.00000i	− 0.164399i	0.996616	−	0.0821995i	$-0.0261945\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	− 6.00000i	− 0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$	− 6.00000i	− 0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	0	0
$45$	18.0000i	2.68328i
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	18.0000i	2.52050i
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	12.0000	1.58944
$58$	0	0
$59$	1.00000i	0.130189i	0.997879	+	0.0650945i	$0.0207349\pi$
$59$	1.00000i	0.130189i	−0.997879	+	0.0650945i	$0.979265\pi$
$60$	0	0
$61$	− 4.00000i	− 0.512148i	−0.966657	−	0.256074i	$-0.917571\pi$
$61$	− 4.00000i	− 0.512148i	0.966657	−	0.256074i	$-0.0824290\pi$
$62$	0	0
$63$	12.0000	1.51186
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.00000i	− 0.610847i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	− 5.00000i	− 0.610847i	0.952217	−	0.305424i	$-0.0987981\pi$
$68$	0	0
$69$	− 3.00000i	− 0.361158i
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	12.0000i	1.38564i
$76$	0	0
$77$	− 2.00000i	− 0.227921i
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 2.00000i	− 0.219529i	−0.993958	−	0.109764i	$-0.964990\pi$
$83$	− 2.00000i	− 0.219529i	0.993958	−	0.109764i	$-0.0350096\pi$
$84$	0	0
$85$	18.0000i	1.95237i
$86$	0	0
$87$	24.0000	2.57307
$88$	0	0
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 21.0000i	− 2.17760i
$94$	0	0
$95$	12.0000	1.23117
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	0	0
$99$	− 6.00000i	− 0.603023i
$100$	0	0
$101$	− 10.0000i	− 0.995037i	−0.867453	−	0.497519i	$-0.834245\pi$
$101$	− 10.0000i	− 0.995037i	0.867453	−	0.497519i	$-0.165755\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	18.0000	1.75662
$106$	0	0
$107$	− 2.00000i	− 0.193347i	−0.995316	−	0.0966736i	$-0.969180\pi$
$107$	− 2.00000i	− 0.193347i	0.995316	−	0.0966736i	$-0.0308203\pi$
$108$	0	0
$109$	14.0000i	1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$	14.0000i	1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	0	0
$113$	−7.00000	−0.658505	−0.329252	−	0.944242i	$-0.606797\pi$
$113$	−7.00000	−0.658505	−0.329252	+	0.944242i	$0.606797\pi$
$114$	0	0
$115$	− 3.00000i	− 0.279751i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	12.0000i	1.08200i
$124$	0	0
$125$	− 3.00000i	− 0.268328i
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	−18.0000	−1.58481
$130$	0	0
$131$	− 2.00000i	− 0.174741i	−0.996176	−	0.0873704i	$-0.972154\pi$
$131$	− 2.00000i	− 0.174741i	0.996176	−	0.0873704i	$-0.0278464\pi$
$132$	0	0
$133$	− 8.00000i	− 0.693688i
$134$	0	0
$135$	27.0000	2.32379
$136$	0	0
$137$	15.0000	1.28154	0.640768	−	0.767734i	$-0.278616\pi$
$137$	15.0000	1.28154	0.640768	+	0.767734i	$0.278616\pi$
$138$	0	0
$139$	22.0000i	1.86602i	0.359856	+	0.933008i	$0.382826\pi$
$139$	22.0000i	1.86602i	−0.359856	+	0.933008i	$0.617174\pi$
$140$	0	0
$141$	− 24.0000i	− 2.02116i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	24.0000	1.99309
$146$	0	0
$147$	9.00000i	0.742307i
$148$	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
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	0	0
$153$	36.0000	2.91043
$154$	0	0
$155$	− 21.0000i	− 1.68676i
$156$	0	0
$157$	11.0000i	0.877896i	0.898513	+	0.438948i	$0.144649\pi$
$157$	11.0000i	0.877896i	−0.898513	+	0.438948i	$0.855351\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	− 9.00000i	− 0.700649i
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	− 24.0000i	− 1.83533i
$172$	0	0
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	0	0
$175$	8.00000	0.604743
$176$	0	0
$177$	3.00000	0.225494
$178$	0	0
$179$	− 5.00000i	− 0.373718i	−0.982387	−	0.186859i	$-0.940169\pi$
$179$	− 5.00000i	− 0.373718i	0.982387	−	0.186859i	$-0.0598307\pi$
$180$	0	0
$181$	− 5.00000i	− 0.371647i	−0.982583	−	0.185824i	$-0.940505\pi$
$181$	− 5.00000i	− 0.371647i	0.982583	−	0.185824i	$-0.0594953\pi$
$182$	0	0
$183$	−12.0000	−0.887066
$184$	0	0
$185$	−3.00000	−0.220564
$186$	0	0
$187$	− 6.00000i	− 0.438763i
$188$	0	0
$189$	− 18.0000i	− 1.30931i
$190$	0	0
$191$	9.00000	0.651217	0.325609	−	0.945505i	$-0.394431\pi$
$191$	9.00000	0.651217	0.325609	+	0.945505i	$0.394431\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−15.0000	−1.05802
$202$	0	0
$203$	− 16.0000i	− 1.12298i
$204$	0	0
$205$	12.0000i	0.838116i
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	20.0000i	1.37686i	0.725304	+	0.688428i	$0.241699\pi$
$211$	20.0000i	1.37686i	−0.725304	+	0.688428i	$0.758301\pi$
$212$	0	0
$213$	− 9.00000i	− 0.616670i
$214$	0	0
$215$	−18.0000	−1.22759
$216$	0	0
$217$	−14.0000	−0.950382
$218$	0	0
$219$	48.0000i	3.24354i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−29.0000	−1.94198	−0.970992	−	0.239113i	$-0.923143\pi$
$223$	−29.0000	−1.94198	−0.970992	+	0.239113i	$0.923143\pi$
$224$	0	0
$225$	24.0000	1.60000
$226$	0	0
$227$	18.0000i	1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$	18.0000i	1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$	0	0
$229$	− 21.0000i	− 1.38772i	−0.720110	−	0.693860i	$-0.755909\pi$
$229$	− 21.0000i	− 1.38772i	0.720110	−	0.693860i	$-0.244091\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	16.0000	1.04819	0.524097	−	0.851658i	$-0.324403\pi$
$233$	16.0000	1.04819	0.524097	+	0.851658i	$0.324403\pi$
$234$	0	0
$235$	− 24.0000i	− 1.56559i
$236$	0	0
$237$	6.00000i	0.389742i
$238$	0	0
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	9.00000i	0.574989i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	13.0000i	0.820553i	0.911961	+	0.410276i	$0.134568\pi$
$251$	13.0000i	0.820553i	−0.911961	+	0.410276i	$0.865432\pi$
$252$	0	0
$253$	1.00000i	0.0628695i
$254$	0	0
$255$	54.0000	3.38161
$256$	0	0
$257$	−10.0000	−0.623783	−0.311891	−	0.950118i	$-0.600963\pi$
$257$	−10.0000	−0.623783	−0.311891	+	0.950118i	$0.600963\pi$
$258$	0	0
$259$	2.00000i	0.124274i
$260$	0	0
$261$	− 48.0000i	− 2.97113i
$262$	0	0
$263$	14.0000	0.863277	0.431638	−	0.902047i	$-0.357936\pi$
$263$	14.0000	0.863277	0.431638	+	0.902047i	$0.357936\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	45.0000i	2.75396i
$268$	0	0
$269$	− 26.0000i	− 1.58525i	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	− 26.0000i	− 1.58525i	0.609711	−	0.792624i	$-0.291286\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 4.00000i	− 0.241209i
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	0	0
$279$	−42.0000	−2.51447
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	− 36.0000i	− 2.13246i
$286$	0	0
$287$	8.00000	0.472225
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	21.0000i	1.23104i
$292$	0	0
$293$	12.0000i	0.701047i	0.936554	+	0.350524i	$0.113996\pi$
$293$	12.0000i	0.701047i	−0.936554	+	0.350524i	$0.886004\pi$
$294$	0	0
$295$	3.00000	0.174667
$296$	0	0
$297$	−9.00000	−0.522233
$298$	0	0
$299$	0	0
$300$	0	0
$301$	12.0000i	0.691669i
$302$	0	0
$303$	−30.0000	−1.72345
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	48.0000i	2.73062i
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	9.00000	0.508710	0.254355	−	0.967111i	$-0.418137\pi$
$313$	9.00000	0.508710	0.254355	+	0.967111i	$0.418137\pi$
$314$	0	0
$315$	− 36.0000i	− 2.02837i
$316$	0	0
$317$	15.0000i	0.842484i	0.906948	+	0.421242i	$0.138406\pi$
$317$	15.0000i	0.842484i	−0.906948	+	0.421242i	$0.861594\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	42.0000	2.32261
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	35.0000i	1.92377i	0.273447	+	0.961887i	$0.411836\pi$
$331$	35.0000i	1.92377i	−0.273447	+	0.961887i	$0.588164\pi$
$332$	0	0
$333$	6.00000i	0.328798i
$334$	0	0
$335$	−15.0000	−0.819538
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	0	0
$339$	21.0000i	1.14056i
$340$	0	0
$341$	7.00000i	0.379071i
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	−9.00000	−0.484544
$346$	0	0
$347$	32.0000i	1.71785i	0.512101	+	0.858925i	$0.328867\pi$
$347$	32.0000i	1.71785i	−0.512101	+	0.858925i	$0.671133\pi$
$348$	0	0
$349$	14.0000i	0.749403i	0.927146	+	0.374701i	$0.122255\pi$
$349$	14.0000i	0.749403i	−0.927146	+	0.374701i	$0.877745\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	0	0
$355$	− 9.00000i	− 0.477670i
$356$	0	0
$357$	− 36.0000i	− 1.90532i
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	3.00000i	0.157459i
$364$	0	0
$365$	48.0000i	2.51243i
$366$	0	0
$367$	−33.0000	−1.72259	−0.861293	−	0.508109i	$-0.830345\pi$
$367$	−33.0000	−1.72259	−0.861293	+	0.508109i	$0.830345\pi$
$368$	0	0
$369$	24.0000	1.24939
$370$	0	0
$371$	− 4.00000i	− 0.207670i
$372$	0	0
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 25.0000i	− 1.28416i	−0.766636	−	0.642082i	$-0.778071\pi$
$379$	− 25.0000i	− 1.28416i	0.766636	−	0.642082i	$-0.221929\pi$
$380$	0	0
$381$	12.0000i	0.614779i
$382$	0	0
$383$	−1.00000	−0.0510976	−0.0255488	−	0.999674i	$-0.508133\pi$
$383$	−1.00000	−0.0510976	−0.0255488	+	0.999674i	$0.508133\pi$
$384$	0	0
$385$	−6.00000	−0.305788
$386$	0	0
$387$	36.0000i	1.82998i
$388$	0	0
$389$	13.0000i	0.659126i	0.944134	+	0.329563i	$0.106901\pi$
$389$	13.0000i	0.659126i	−0.944134	+	0.329563i	$0.893099\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	6.00000i	0.301893i
$396$	0	0
$397$	26.0000i	1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$	26.0000i	1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$	0	0
$399$	−24.0000	−1.20150
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 27.0000i	− 1.34164i
$406$	0	0
$407$	1.00000	0.0495682
$408$	0	0
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	− 45.0000i	− 2.21969i
$412$	0	0
$413$	− 2.00000i	− 0.0984136i
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	66.0000	3.23203
$418$	0	0
$419$	− 28.0000i	− 1.36789i	−0.729534	−	0.683945i	$-0.760263\pi$
$419$	− 28.0000i	− 1.36789i	0.729534	−	0.683945i	$-0.239737\pi$
$420$	0	0
$421$	30.0000i	1.46211i	0.682318	+	0.731055i	$0.260972\pi$
$421$	30.0000i	1.46211i	−0.682318	+	0.731055i	$0.739028\pi$
$422$	0	0
$423$	−48.0000	−2.33384
$424$	0	0
$425$	24.0000	1.16417
$426$	0	0
$427$	8.00000i	0.387147i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	26.0000	1.25238	0.626188	−	0.779672i	$-0.284614\pi$
$431$	26.0000	1.25238	0.626188	+	0.779672i	$0.284614\pi$
$432$	0	0
$433$	13.0000	0.624740	0.312370	−	0.949960i	$-0.398877\pi$
$433$	13.0000	0.624740	0.312370	+	0.949960i	$0.398877\pi$
$434$	0	0
$435$	− 72.0000i	− 3.45214i
$436$	0	0
$437$	4.00000i	0.191346i
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	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
$445$	45.0000i	2.13320i
$446$	0	0
$447$	54.0000	2.55411
$448$	0	0
$449$	−21.0000	−0.991051	−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000	−0.991051	−0.495526	+	0.868593i	$0.665025\pi$
$450$	0	0
$451$	− 4.00000i	− 0.188353i
$452$	0	0
$453$	54.0000i	2.53714i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12.0000	−0.561336	−0.280668	−	0.959805i	$-0.590556\pi$
$457$	−12.0000	−0.561336	−0.280668	+	0.959805i	$0.590556\pi$
$458$	0	0
$459$	− 54.0000i	− 2.52050i
$460$	0	0
$461$	28.0000i	1.30409i	0.758180	+	0.652045i	$0.226089\pi$
$461$	28.0000i	1.30409i	−0.758180	+	0.652045i	$0.773911\pi$
$462$	0	0
$463$	−27.0000	−1.25480	−0.627398	−	0.778699i	$-0.715880\pi$
$463$	−27.0000	−1.25480	−0.627398	+	0.778699i	$0.715880\pi$
$464$	0	0
$465$	−63.0000	−2.92156
$466$	0	0
$467$	− 33.0000i	− 1.52706i	−0.645774	−	0.763529i	$-0.723465\pi$
$467$	− 33.0000i	− 1.52706i	0.645774	−	0.763529i	$-0.276535\pi$
$468$	0	0
$469$	10.0000i	0.461757i
$470$	0	0
$471$	33.0000	1.52056
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	− 16.0000i	− 0.734130i
$476$	0	0
$477$	− 12.0000i	− 0.549442i
$478$	0	0
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	6.00000i	0.273009i
$484$	0	0
$485$	21.0000i	0.953561i
$486$	0	0
$487$	9.00000	0.407829	0.203914	−	0.978989i	$-0.434634\pi$
$487$	9.00000	0.407829	0.203914	+	0.978989i	$0.434634\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	0	0
$491$	− 8.00000i	− 0.361035i	−0.983572	−	0.180517i	$-0.942223\pi$
$491$	− 8.00000i	− 0.361035i	0.983572	−	0.180517i	$-0.0577772\pi$
$492$	0	0
$493$	− 48.0000i	− 2.16181i
$494$	0	0
$495$	−18.0000	−0.809040
$496$	0	0
$497$	−6.00000	−0.269137
$498$	0	0
$499$	− 20.0000i	− 0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$	− 20.0000i	− 0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$	0	0
$501$	− 48.0000i	− 2.14448i
$502$	0	0
$503$	−30.0000	−1.33763	−0.668817	−	0.743427i	$-0.733199\pi$
$503$	−30.0000	−1.33763	−0.668817	+	0.743427i	$0.733199\pi$
$504$	0	0
$505$	−30.0000	−1.33498
$506$	0	0
$507$	− 39.0000i	− 1.73205i
$508$	0	0
$509$	13.0000i	0.576215i	0.957598	+	0.288107i	$0.0930260\pi$
$509$	13.0000i	0.576215i	−0.957598	+	0.288107i	$0.906974\pi$
$510$	0	0
$511$	32.0000	1.41560
$512$	0	0
$513$	−36.0000	−1.58944
$514$	0	0
$515$	48.0000i	2.11513i
$516$	0	0
$517$	8.00000i	0.351840i
$518$	0	0
$519$	−54.0000	−2.37034
$520$	0	0
$521$	−37.0000	−1.62100	−0.810500	−	0.585739i	$-0.800804\pi$
$521$	−37.0000	−1.62100	−0.810500	+	0.585739i	$0.800804\pi$
$522$	0	0
$523$	− 44.0000i	− 1.92399i	−0.273075	−	0.961993i	$-0.588041\pi$
$523$	− 44.0000i	− 1.92399i	0.273075	−	0.961993i	$-0.411959\pi$
$524$	0	0
$525$	− 24.0000i	− 1.04745i
$526$	0	0
$527$	−42.0000	−1.82955
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	− 6.00000i	− 0.260378i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−6.00000	−0.259403
$536$	0	0
$537$	−15.0000	−0.647298
$538$	0	0
$539$	− 3.00000i	− 0.129219i
$540$	0	0
$541$	− 12.0000i	− 0.515920i	−0.966156	−	0.257960i	$-0.916950\pi$
$541$	− 12.0000i	− 0.515920i	0.966156	−	0.257960i	$-0.0830503\pi$
$542$	0	0
$543$	−15.0000	−0.643712
$544$	0	0
$545$	42.0000	1.79908
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	24.0000i	1.02430i
$550$	0	0
$551$	−32.0000	−1.36325
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	9.00000i	0.382029i
$556$	0	0
$557$	− 42.0000i	− 1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$	− 42.0000i	− 1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−18.0000	−0.759961
$562$	0	0
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	21.0000i	0.883477i
$566$	0	0
$567$	−18.0000	−0.755929
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	4.00000i	0.167395i	0.996491	+	0.0836974i	$0.0266729\pi$
$571$	4.00000i	0.167395i	−0.996491	+	0.0836974i	$0.973327\pi$
$572$	0	0
$573$	− 27.0000i	− 1.12794i
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−23.0000	−0.957503	−0.478751	−	0.877951i	$-0.658910\pi$
$577$	−23.0000	−0.957503	−0.478751	+	0.877951i	$0.658910\pi$
$578$	0	0
$579$	− 12.0000i	− 0.498703i
$580$	0	0
$581$	4.00000i	0.165948i
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	0	0
$589$	28.0000i	1.15372i
$590$	0	0
$591$	18.0000	0.740421
$592$	0	0
$593$	−4.00000	−0.164260	−0.0821302	−	0.996622i	$-0.526172\pi$
$593$	−4.00000	−0.164260	−0.0821302	+	0.996622i	$0.526172\pi$
$594$	0	0
$595$	− 36.0000i	− 1.47586i
$596$	0	0
$597$	− 24.0000i	− 0.982255i
$598$	0	0
$599$	48.0000	1.96123	0.980613	−	0.195952i	$-0.0627798\pi$
$599$	48.0000	1.96123	0.980613	+	0.195952i	$0.0627798\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	0	0
$603$	30.0000i	1.22169i
$604$	0	0
$605$	3.00000i	0.121967i
$606$	0	0
$607$	18.0000	0.730597	0.365299	−	0.930890i	$-0.380967\pi$
$607$	18.0000	0.730597	0.365299	+	0.930890i	$0.380967\pi$
$608$	0	0
$609$	−48.0000	−1.94506
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	0	0
$615$	36.0000	1.45166
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	3.00000i	0.120580i	0.998181	+	0.0602901i	$0.0192026\pi$
$619$	3.00000i	0.120580i	−0.998181	+	0.0602901i	$0.980797\pi$
$620$	0	0
$621$	9.00000i	0.361158i
$622$	0	0
$623$	30.0000	1.20192
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	12.0000i	0.479234i
$628$	0	0
$629$	6.00000i	0.239236i
$630$	0	0
$631$	9.00000	0.358284	0.179142	−	0.983823i	$-0.442668\pi$
$631$	9.00000	0.358284	0.179142	+	0.983823i	$0.442668\pi$
$632$	0	0
$633$	60.0000	2.38479
$634$	0	0
$635$	12.0000i	0.476205i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−18.0000	−0.712069
$640$	0	0
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	0	0
$643$	7.00000i	0.276053i	0.990429	+	0.138027i	$0.0440759\pi$
$643$	7.00000i	0.276053i	−0.990429	+	0.138027i	$0.955924\pi$
$644$	0	0
$645$	54.0000i	2.12625i
$646$	0	0
$647$	15.0000	0.589711	0.294855	−	0.955542i	$-0.404729\pi$
$647$	15.0000	0.589711	0.294855	+	0.955542i	$0.404729\pi$
$648$	0	0
$649$	−1.00000	−0.0392534
$650$	0	0
$651$	42.0000i	1.64611i
$652$	0	0
$653$	− 11.0000i	− 0.430463i	−0.976563	−	0.215232i	$-0.930949\pi$
$653$	− 11.0000i	− 0.430463i	0.976563	−	0.215232i	$-0.0690506\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	0	0
$657$	96.0000	3.74532
$658$	0	0
$659$	22.0000i	0.856998i	0.903542	+	0.428499i	$0.140958\pi$
$659$	22.0000i	0.856998i	−0.903542	+	0.428499i	$0.859042\pi$
$660$	0	0
$661$	− 7.00000i	− 0.272268i	−0.990690	−	0.136134i	$-0.956532\pi$
$661$	− 7.00000i	− 0.272268i	0.990690	−	0.136134i	$-0.0434678\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−24.0000	−0.930680
$666$	0	0
$667$	8.00000i	0.309761i
$668$	0	0
$669$	87.0000i	3.36361i
$670$	0	0
$671$	4.00000	0.154418
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	0	0
$675$	− 36.0000i	− 1.38564i
$676$	0	0
$677$	− 30.0000i	− 1.15299i	−0.817099	−	0.576497i	$-0.804419\pi$
$677$	− 30.0000i	− 1.15299i	0.817099	−	0.576497i	$-0.195581\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	0	0
$681$	54.0000	2.06928
$682$	0	0
$683$	− 8.00000i	− 0.306111i	−0.988218	−	0.153056i	$-0.951089\pi$
$683$	− 8.00000i	− 0.306111i	0.988218	−	0.153056i	$-0.0489114\pi$
$684$	0	0
$685$	− 45.0000i	− 1.71936i
$686$	0	0
$687$	−63.0000	−2.40360
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 29.0000i	− 1.10321i	−0.834105	−	0.551606i	$-0.814015\pi$
$691$	− 29.0000i	− 1.10321i	0.834105	−	0.551606i	$-0.185985\pi$
$692$	0	0
$693$	12.0000i	0.455842i
$694$	0	0
$695$	66.0000	2.50352
$696$	0	0
$697$	24.0000	0.909065
$698$	0	0
$699$	− 48.0000i	− 1.81553i
$700$	0	0
$701$	10.0000i	0.377695i	0.982006	+	0.188847i	$0.0604752\pi$
$701$	10.0000i	0.377695i	−0.982006	+	0.188847i	$0.939525\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	0	0
$705$	−72.0000	−2.71168
$706$	0	0
$707$	20.0000i	0.752177i
$708$	0	0
$709$	19.0000i	0.713560i	0.934188	+	0.356780i	$0.116125\pi$
$709$	19.0000i	0.713560i	−0.934188	+	0.356780i	$0.883875\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	0	0
$713$	7.00000	0.262152
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6.00000i	− 0.224074i
$718$	0	0
$719$	23.0000	0.857755	0.428878	−	0.903363i	$-0.358909\pi$
$719$	23.0000	0.857755	0.428878	+	0.903363i	$0.358909\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 32.0000i	− 1.18845i
$726$	0	0
$727$	−35.0000	−1.29808	−0.649039	−	0.760755i	$-0.724829\pi$
$727$	−35.0000	−1.29808	−0.649039	+	0.760755i	$0.724829\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	36.0000i	1.33151i
$732$	0	0
$733$	− 4.00000i	− 0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	− 4.00000i	− 0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	0	0
$735$	27.0000	0.995910
$736$	0	0
$737$	5.00000	0.184177
$738$	0	0
$739$	− 26.0000i	− 0.956425i	−0.878244	−	0.478213i	$-0.841285\pi$
$739$	− 26.0000i	− 0.956425i	0.878244	−	0.478213i	$-0.158715\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	54.0000	1.97841
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	4.00000i	0.146157i
$750$	0	0
$751$	−23.0000	−0.839282	−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000	−0.839282	−0.419641	+	0.907690i	$0.637844\pi$
$752$	0	0
$753$	39.0000	1.42124
$754$	0	0
$755$	54.0000i	1.96526i
$756$	0	0
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	0	0
$759$	3.00000	0.108893
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	− 28.0000i	− 1.01367i
$764$	0	0
$765$	− 108.000i	− 3.90475i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	0	0
$771$	30.0000i	1.08042i
$772$	0	0
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	0	0
$775$	−28.0000	−1.00579
$776$	0	0
$777$	6.00000	0.215249
$778$	0	0
$779$	− 16.0000i	− 0.573259i
$780$	0	0
$781$	3.00000i	0.107348i
$782$	0	0
$783$	−72.0000	−2.57307
$784$	0	0
$785$	33.0000	1.17782
$786$	0	0
$787$	24.0000i	0.855508i	0.903895	+	0.427754i	$0.140695\pi$
$787$	24.0000i	0.855508i	−0.903895	+	0.427754i	$0.859305\pi$
$788$	0	0
$789$	− 42.0000i	− 1.49524i
$790$	0	0
$791$	14.0000	0.497783
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 18.0000i	− 0.638394i
$796$	0	0
$797$	− 1.00000i	− 0.0354218i	−0.999843	−	0.0177109i	$-0.994362\pi$
$797$	− 1.00000i	− 0.0354218i	0.999843	−	0.0177109i	$-0.00563785\pi$
$798$	0	0
$799$	−48.0000	−1.69812
$800$	0	0
$801$	90.0000	3.17999
$802$	0	0
$803$	− 16.0000i	− 0.564628i
$804$	0	0
$805$	6.00000i	0.211472i
$806$	0	0
$807$	−78.0000	−2.74573
$808$	0	0
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	26.0000i	0.912983i	0.889728	+	0.456492i	$0.150894\pi$
$811$	26.0000i	0.912983i	−0.889728	+	0.456492i	$0.849106\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12.0000	−0.420342
$816$	0	0
$817$	24.0000	0.839654
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 34.0000i	− 1.18661i	−0.804978	−	0.593304i	$-0.797823\pi$
$821$	− 34.0000i	− 1.18661i	0.804978	−	0.593304i	$-0.202177\pi$
$822$	0	0
$823$	−31.0000	−1.08059	−0.540296	−	0.841475i	$-0.681688\pi$
$823$	−31.0000	−1.08059	−0.540296	+	0.841475i	$0.681688\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	16.0000i	0.556375i	0.960527	+	0.278187i	$0.0897336\pi$
$827$	16.0000i	0.556375i	−0.960527	+	0.278187i	$0.910266\pi$
$828$	0	0
$829$	35.0000i	1.21560i	0.794090	+	0.607800i	$0.207948\pi$
$829$	35.0000i	1.21560i	−0.794090	+	0.607800i	$0.792052\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	− 48.0000i	− 1.66111i
$836$	0	0
$837$	63.0000i	2.17760i
$838$	0	0
$839$	21.0000	0.725001	0.362500	−	0.931984i	$-0.381923\pi$
$839$	21.0000	0.725001	0.362500	+	0.931984i	$0.381923\pi$
$840$	0	0
$841$	−35.0000	−1.20690
$842$	0	0
$843$	18.0000i	0.619953i
$844$	0	0
$845$	− 39.0000i	− 1.34164i
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	− 1.00000i	− 0.0342796i
$852$	0	0
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	0	0
$855$	−72.0000	−2.46235
$856$	0	0
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	0	0
$859$	− 11.0000i	− 0.375315i	−0.982235	−	0.187658i	$-0.939910\pi$
$859$	− 11.0000i	− 0.375315i	0.982235	−	0.187658i	$-0.0600895\pi$
$860$	0	0
$861$	− 24.0000i	− 0.817918i
$862$	0	0
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	0	0
$865$	−54.0000	−1.83606
$866$	0	0
$867$	− 57.0000i	− 1.93582i
$868$	0	0
$869$	− 2.00000i	− 0.0678454i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	42.0000	1.42148
$874$	0	0
$875$	6.00000i	0.202837i
$876$	0	0
$877$	− 56.0000i	− 1.89099i	−0.325643	−	0.945493i	$-0.605581\pi$
$877$	− 56.0000i	− 1.89099i	0.325643	−	0.945493i	$-0.394419\pi$
$878$	0	0
$879$	36.0000	1.21425
$880$	0	0
$881$	−19.0000	−0.640126	−0.320063	−	0.947396i	$-0.603704\pi$
$881$	−19.0000	−0.640126	−0.320063	+	0.947396i	$0.603704\pi$
$882$	0	0
$883$	− 28.0000i	− 0.942275i	−0.882060	−	0.471138i	$-0.843844\pi$
$883$	− 28.0000i	− 0.942275i	0.882060	−	0.471138i	$-0.156156\pi$
$884$	0	0
$885$	− 9.00000i	− 0.302532i
$886$	0	0
$887$	−34.0000	−1.14161	−0.570804	−	0.821086i	$-0.693368\pi$
$887$	−34.0000	−1.14161	−0.570804	+	0.821086i	$0.693368\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	9.00000i	0.301511i
$892$	0	0
$893$	32.0000i	1.07084i
$894$	0	0
$895$	−15.0000	−0.501395
$896$	0	0
$897$	0	0
$898$	0	0
$899$	56.0000i	1.86770i
$900$	0	0
$901$	− 12.0000i	− 0.399778i
$902$	0	0
$903$	36.0000	1.19800
$904$	0	0
$905$	−15.0000	−0.498617
$906$	0	0
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	0	0
$909$	60.0000i	1.99007i
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	2.00000	0.0661903
$914$	0	0
$915$	36.0000i	1.19012i
$916$	0	0
$917$	4.00000i	0.132092i
$918$	0	0
$919$	−50.0000	−1.64935	−0.824674	−	0.565608i	$-0.808641\pi$
$919$	−50.0000	−1.64935	−0.824674	+	0.565608i	$0.808641\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	0	0
$924$	0	0
$925$	4.00000i	0.131519i
$926$	0	0
$927$	96.0000	3.15305
$928$	0	0
$929$	−22.0000	−0.721797	−0.360898	−	0.932605i	$-0.617530\pi$
$929$	−22.0000	−0.721797	−0.360898	+	0.932605i	$0.617530\pi$
$930$	0	0
$931$	− 12.0000i	− 0.393284i
$932$	0	0
$933$	36.0000i	1.17859i
$934$	0	0
$935$	−18.0000	−0.588663
$936$	0	0
$937$	−4.00000	−0.130674	−0.0653372	−	0.997863i	$-0.520812\pi$
$937$	−4.00000	−0.130674	−0.0653372	+	0.997863i	$0.520812\pi$
$938$	0	0
$939$	− 27.0000i	− 0.881112i
$940$	0	0
$941$	− 10.0000i	− 0.325991i	−0.986627	−	0.162995i	$-0.947884\pi$
$941$	− 10.0000i	− 0.325991i	0.986627	−	0.162995i	$-0.0521156\pi$
$942$	0	0
$943$	−4.00000	−0.130258
$944$	0	0
$945$	−54.0000	−1.75662
$946$	0	0
$947$	47.0000i	1.52729i	0.645634	+	0.763647i	$0.276593\pi$
$947$	47.0000i	1.52729i	−0.645634	+	0.763647i	$0.723407\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	45.0000	1.45922
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	0	0
$955$	− 27.0000i	− 0.873699i
$956$	0	0
$957$	24.0000i	0.775810i
$958$	0	0
$959$	−30.0000	−0.968751
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	12.0000i	0.386695i
$964$	0	0
$965$	− 12.0000i	− 0.386294i
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	−72.0000	−2.31297
$970$	0	0
$971$	− 53.0000i	− 1.70085i	−0.526096	−	0.850425i	$-0.676345\pi$
$971$	− 53.0000i	− 1.70085i	0.526096	−	0.850425i	$-0.323655\pi$
$972$	0	0
$973$	− 44.0000i	− 1.41058i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−11.0000	−0.351921	−0.175961	−	0.984397i	$-0.556303\pi$
$977$	−11.0000	−0.351921	−0.175961	+	0.984397i	$0.556303\pi$
$978$	0	0
$979$	− 15.0000i	− 0.479402i
$980$	0	0
$981$	− 84.0000i	− 2.68191i
$982$	0	0
$983$	25.0000	0.797376	0.398688	−	0.917087i	$-0.369466\pi$
$983$	25.0000	0.797376	0.398688	+	0.917087i	$0.369466\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	48.0000i	1.52786i
$988$	0	0
$989$	− 6.00000i	− 0.190789i
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	105.000	3.33207
$994$	0	0
$995$	− 24.0000i	− 0.760851i
$996$	0	0
$997$	2.00000i	0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$	2.00000i	0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$	0	0
$999$	9.00000	0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2816.2.c.d.1409.1		2
4.3	odd	2		2816.2.c.i.1409.2		2
8.3	odd	2		2816.2.c.i.1409.1		2
8.5	even	2	inner	2816.2.c.d.1409.2		2
16.3	odd	4		88.2.a.a.1.1	✓	1
16.5	even	4		704.2.a.b.1.1		1
16.11	odd	4		704.2.a.l.1.1		1
16.13	even	4		176.2.a.c.1.1		1
48.5	odd	4		6336.2.a.k.1.1		1
48.11	even	4		6336.2.a.h.1.1		1
48.29	odd	4		1584.2.a.q.1.1		1
48.35	even	4		792.2.a.g.1.1		1
80.3	even	4		2200.2.b.a.1849.1		2
80.13	odd	4		4400.2.b.b.4049.2		2
80.19	odd	4		2200.2.a.k.1.1		1
80.29	even	4		4400.2.a.a.1.1		1
80.67	even	4		2200.2.b.a.1849.2		2
80.77	odd	4		4400.2.b.b.4049.1		2
112.13	odd	4		8624.2.a.c.1.1		1
112.83	even	4		4312.2.a.l.1.1		1
176.3	odd	20		968.2.i.j.9.1		4
176.19	even	20		968.2.i.i.9.1		4
176.21	odd	4		7744.2.a.b.1.1		1
176.35	even	20		968.2.i.i.81.1		4
176.43	even	4		7744.2.a.bk.1.1		1
176.51	even	20		968.2.i.i.753.1		4
176.83	even	20		968.2.i.i.729.1		4
176.109	odd	4		1936.2.a.l.1.1		1
176.115	odd	20		968.2.i.j.729.1		4
176.131	even	4		968.2.a.a.1.1		1
176.147	odd	20		968.2.i.j.753.1		4
176.163	odd	20		968.2.i.j.81.1		4
528.131	odd	4		8712.2.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.a.a.1.1	✓	1	16.3	odd	4
176.2.a.c.1.1		1	16.13	even	4
704.2.a.b.1.1		1	16.5	even	4
704.2.a.l.1.1		1	16.11	odd	4
792.2.a.g.1.1		1	48.35	even	4
968.2.a.a.1.1		1	176.131	even	4
968.2.i.i.9.1		4	176.19	even	20
968.2.i.i.81.1		4	176.35	even	20
968.2.i.i.729.1		4	176.83	even	20
968.2.i.i.753.1		4	176.51	even	20
968.2.i.j.9.1		4	176.3	odd	20
968.2.i.j.81.1		4	176.163	odd	20
968.2.i.j.729.1		4	176.115	odd	20
968.2.i.j.753.1		4	176.147	odd	20
1584.2.a.q.1.1		1	48.29	odd	4
1936.2.a.l.1.1		1	176.109	odd	4
2200.2.a.k.1.1		1	80.19	odd	4
2200.2.b.a.1849.1		2	80.3	even	4
2200.2.b.a.1849.2		2	80.67	even	4
2816.2.c.d.1409.1		2	1.1	even	1	trivial
2816.2.c.d.1409.2		2	8.5	even	2	inner
2816.2.c.i.1409.1		2	8.3	odd	2
2816.2.c.i.1409.2		2	4.3	odd	2
4312.2.a.l.1.1		1	112.83	even	4
4400.2.a.a.1.1		1	80.29	even	4
4400.2.b.b.4049.1		2	80.77	odd	4
4400.2.b.b.4049.2		2	80.13	odd	4
6336.2.a.h.1.1		1	48.11	even	4
6336.2.a.k.1.1		1	48.5	odd	4
7744.2.a.b.1.1		1	176.21	odd	4
7744.2.a.bk.1.1		1	176.43	even	4
8624.2.a.c.1.1		1	112.13	odd	4
8712.2.a.x.1.1		1	528.131	odd	4

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:	\( N \)	\(=\)	\( 2816 = 2^{8} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2816.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -3.00000i q^{5} -2.00000 q^{7} -6.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -3.00000i q^{5} -2.00000 q^{7} -6.00000 q^{9} +1.00000i q^{11} -9.00000 q^{15} -6.00000 q^{17} +4.00000i q^{19} +6.00000i q^{21} +1.00000 q^{23} -4.00000 q^{25} +9.00000i q^{27} +8.00000i q^{29} +7.00000 q^{31} +3.00000 q^{33} +6.00000i q^{35} -1.00000i q^{37} -4.00000 q^{41} -6.00000i q^{43} +18.0000i q^{45} +8.00000 q^{47} -3.00000 q^{49} +18.0000i q^{51} +2.00000i q^{53} +3.00000 q^{55} +12.0000 q^{57} +1.00000i q^{59} -4.00000i q^{61} +12.0000 q^{63} -5.00000i q^{67} -3.00000i q^{69} +3.00000 q^{71} -16.0000 q^{73} +12.0000i q^{75} -2.00000i q^{77} -2.00000 q^{79} +9.00000 q^{81} -2.00000i q^{83} +18.0000i q^{85} +24.0000 q^{87} -15.0000 q^{89} -21.0000i q^{93} +12.0000 q^{95} -7.00000 q^{97} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7} - 12 q^{9}+O(q^{10}) \) 2 * q - 4 * q^7 - 12 * q^9	\( 2 q - 4 q^{7} - 12 q^{9} - 18 q^{15} - 12 q^{17} + 2 q^{23} - 8 q^{25} + 14 q^{31} + 6 q^{33} - 8 q^{41} + 16 q^{47} - 6 q^{49} + 6 q^{55} + 24 q^{57} + 24 q^{63} + 6 q^{71} - 32 q^{73} - 4 q^{79} + 18 q^{81} + 48 q^{87} - 30 q^{89} + 24 q^{95} - 14 q^{97}+O(q^{100}) \) 2 * q - 4 * q^7 - 12 * q^9 - 18 * q^15 - 12 * q^17 + 2 * q^23 - 8 * q^25 + 14 * q^31 + 6 * q^33 - 8 * q^41 + 16 * q^47 - 6 * q^49 + 6 * q^55 + 24 * q^57 + 24 * q^63 + 6 * q^71 - 32 * q^73 - 4 * q^79 + 18 * q^81 + 48 * q^87 - 30 * q^89 + 24 * q^95 - 14 * q^97

Introduction

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Modular forms

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Database

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