LMFDB - Embedded newform 3328.2.b.e.1665.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3328,2,Mod(1665,3328)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3328, 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("3328.1665");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.b (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:	$26.5742137927$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1665.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3328.1665
Dual form			3328.2.b.e.1665.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{5} -2.00000 q^{7} +3.00000 q^{9} +O(q^{10})$	$q-2.00000i q^{5} -2.00000 q^{7} +3.00000 q^{9} -2.00000i q^{11} -1.00000i q^{13} +6.00000 q^{17} +6.00000i q^{19} +8.00000 q^{23} +1.00000 q^{25} +2.00000i q^{29} -10.0000 q^{31} +4.00000i q^{35} +6.00000i q^{37} +6.00000 q^{41} +4.00000i q^{43} -6.00000i q^{45} +2.00000 q^{47} -3.00000 q^{49} -6.00000i q^{53} -4.00000 q^{55} -10.0000i q^{59} -2.00000i q^{61} -6.00000 q^{63} -2.00000 q^{65} -10.0000i q^{67} +10.0000 q^{71} -2.00000 q^{73} +4.00000i q^{77} +4.00000 q^{79} +9.00000 q^{81} +6.00000i q^{83} -12.0000i q^{85} +6.00000 q^{89} +2.00000i q^{91} +12.0000 q^{95} +2.00000 q^{97} -6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q - 4 * q^7 + 6 * q^9	$ 2 q - 4 q^{7} + 6 q^{9} + 12 q^{17} + 16 q^{23} + 2 q^{25} - 20 q^{31} + 12 q^{41} + 4 q^{47} - 6 q^{49} - 8 q^{55} - 12 q^{63} - 4 q^{65} + 20 q^{71} - 4 q^{73} + 8 q^{79} + 18 q^{81} + 12 q^{89} + 24 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q - 4 * q^7 + 6 * q^9 + 12 * q^17 + 16 * q^23 + 2 * q^25 - 20 * q^31 + 12 * q^41 + 4 * q^47 - 6 * q^49 - 8 * q^55 - 12 * q^63 - 4 * q^65 + 20 * q^71 - 4 * q^73 + 8 * q^79 + 18 * q^81 + 12 * q^89 + 24 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times$.

$n$	$261$	$769$	$1535$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	− 2.00000i	− 0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	− 2.00000i	− 0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$	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
$9$	3.00000	1.00000
$10$	0	0
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	6.00000i	1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$	6.00000i	1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.00000i	0.676123i
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	− 6.00000i	− 0.894427i
$46$	0	0
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 10.0000i	− 1.30189i	−0.759125	−	0.650945i	$-0.774373\pi$
$59$	− 10.0000i	− 1.30189i	0.759125	−	0.650945i	$-0.225627\pi$
$60$	0	0
$61$	− 2.00000i	− 0.256074i	−0.991769	−	0.128037i	$-0.959132\pi$
$61$	− 2.00000i	− 0.256074i	0.991769	−	0.128037i	$-0.0408676\pi$
$62$	0	0
$63$	−6.00000	−0.755929
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	− 10.0000i	− 1.22169i	−0.791748	−	0.610847i	$-0.790829\pi$
$67$	− 10.0000i	− 1.22169i	0.791748	−	0.610847i	$-0.209171\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.00000i	0.455842i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	− 12.0000i	− 1.30158i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	2.00000i	0.209657i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	12.0000	1.23117
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	− 6.00000i	− 0.603023i
$100$	0	0
$101$	2.00000i	0.199007i	0.995037	+	0.0995037i	$0.0317255\pi$
$101$	2.00000i	0.199007i	−0.995037	+	0.0995037i	$0.968274\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.0000i	− 1.54678i	−0.633932	−	0.773389i	$-0.718560\pi$
$107$	− 16.0000i	− 1.54678i	0.633932	−	0.773389i	$-0.281440\pi$
$108$	0	0
$109$	− 14.0000i	− 1.34096i	−0.741929	−	0.670478i	$-0.766089\pi$
$109$	− 14.0000i	− 1.34096i	0.741929	−	0.670478i	$-0.233911\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	− 16.0000i	− 1.49201i
$116$	0	0
$117$	− 3.00000i	− 0.277350i
$118$	0	0
$119$	−12.0000	−1.10004
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 12.0000i	− 1.07331i
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.0000i	1.39793i	0.715158	+	0.698963i	$0.246355\pi$
$131$	16.0000i	1.39793i	−0.715158	+	0.698963i	$0.753645\pi$
$132$	0	0
$133$	− 12.0000i	− 1.04053i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	16.0000i	1.35710i	0.734553	+	0.678551i	$0.237392\pi$
$139$	16.0000i	1.35710i	−0.734553	+	0.678551i	$0.762608\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 18.0000i	− 1.47462i	−0.675556	−	0.737309i	$-0.736096\pi$
$149$	− 18.0000i	− 1.47462i	0.675556	−	0.737309i	$-0.263904\pi$
$150$	0	0
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	0	0
$153$	18.0000	1.45521
$154$	0	0
$155$	20.0000i	1.60644i
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	10.0000i	0.783260i	0.920123	+	0.391630i	$0.128089\pi$
$163$	10.0000i	0.783260i	−0.920123	+	0.391630i	$0.871911\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	18.0000i	1.37649i
$172$	0	0
$173$	− 10.0000i	− 0.760286i	−0.924928	−	0.380143i	$-0.875875\pi$
$173$	− 10.0000i	− 0.760286i	0.924928	−	0.380143i	$-0.124125\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	0	0
$178$	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
$181$	6.00000i	0.445976i	0.974821	+	0.222988i	$0.0715812\pi$
$181$	6.00000i	0.445976i	−0.974821	+	0.222988i	$0.928419\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	− 12.0000i	− 0.877527i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	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
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 4.00000i	− 0.280745i
$204$	0	0
$205$	− 12.0000i	− 0.838116i
$206$	0	0
$207$	24.0000	1.66812
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	− 8.00000i	− 0.550743i	−0.961338	−	0.275371i	$-0.911199\pi$
$211$	− 8.00000i	− 0.550743i	0.961338	−	0.275371i	$-0.0888008\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	20.0000	1.35769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 6.00000i	− 0.403604i
$222$	0	0
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	− 18.0000i	− 1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$	− 18.0000i	− 1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$	0	0
$229$	− 18.0000i	− 1.18947i	−0.803921	−	0.594737i	$-0.797256\pi$
$229$	− 18.0000i	− 1.18947i	0.803921	−	0.594737i	$-0.202744\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	− 4.00000i	− 0.260931i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−18.0000	−1.16432	−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000	−1.16432	−0.582162	+	0.813073i	$0.697793\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.00000i	0.383326i
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.0000i	0.757433i	0.925513	+	0.378717i	$0.123635\pi$
$251$	12.0000i	0.757433i	−0.925513	+	0.378717i	$0.876365\pi$
$252$	0	0
$253$	− 16.0000i	− 1.00591i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	− 12.0000i	− 0.745644i
$260$	0	0
$261$	6.00000i	0.371391i
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.0000i	1.09748i	0.835993	+	0.548740i	$0.184892\pi$
$269$	18.0000i	1.09748i	−0.835993	+	0.548740i	$0.815108\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2.00000i	− 0.120605i
$276$	0	0
$277$	− 30.0000i	− 1.80253i	−0.433273	−	0.901263i	$-0.642641\pi$
$277$	− 30.0000i	− 1.80253i	0.433273	−	0.901263i	$-0.357359\pi$
$278$	0	0
$279$	−30.0000	−1.79605
$280$	0	0
$281$	14.0000	0.835170	0.417585	−	0.908638i	$-0.362877\pi$
$281$	14.0000	0.835170	0.417585	+	0.908638i	$0.362877\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	0	0
$295$	−20.0000	−1.16445
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 8.00000i	− 0.462652i
$300$	0	0
$301$	− 8.00000i	− 0.461112i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.00000	−0.229039
$306$	0	0
$307$	34.0000i	1.94048i	0.242140	+	0.970241i	$0.422151\pi$
$307$	34.0000i	1.94048i	−0.242140	+	0.970241i	$0.577849\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	12.0000i	0.676123i
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	0	0
$322$	0	0
$323$	36.0000i	2.00309i
$324$	0	0
$325$	− 1.00000i	− 0.0554700i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	− 22.0000i	− 1.20923i	−0.796518	−	0.604615i	$-0.793327\pi$
$331$	− 22.0000i	− 1.20923i	0.796518	−	0.604615i	$-0.206673\pi$
$332$	0	0
$333$	18.0000i	0.986394i
$334$	0	0
$335$	−20.0000	−1.09272
$336$	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
$339$	0	0
$340$	0	0
$341$	20.0000i	1.08306i
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 24.0000i	− 1.28839i	−0.764862	−	0.644194i	$-0.777193\pi$
$347$	− 24.0000i	− 1.28839i	0.764862	−	0.644194i	$-0.222807\pi$
$348$	0	0
$349$	10.0000i	0.535288i	0.963518	+	0.267644i	$0.0862451\pi$
$349$	10.0000i	0.535288i	−0.963518	+	0.267644i	$0.913755\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	− 20.0000i	− 1.06149i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.00000i	0.209370i
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	0	0
$369$	18.0000	0.937043
$370$	0	0
$371$	12.0000i	0.623009i
$372$	0	0
$373$	22.0000i	1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$	22.0000i	1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	10.0000i	0.513665i	0.966456	+	0.256833i	$0.0826790\pi$
$379$	10.0000i	0.513665i	−0.966456	+	0.256833i	$0.917321\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	0	0
$385$	8.00000	0.407718
$386$	0	0
$387$	12.0000i	0.609994i
$388$	0	0
$389$	6.00000i	0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$	6.00000i	0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$	0	0
$391$	48.0000	2.42746
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 8.00000i	− 0.402524i
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	0	0
$399$	0	0
$400$	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
$403$	10.0000i	0.498135i
$404$	0	0
$405$	− 18.0000i	− 0.894427i
$406$	0	0
$407$	12.0000	0.594818
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	20.0000i	0.984136i
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 24.0000i	− 1.17248i	−0.810139	−	0.586238i	$-0.800608\pi$
$419$	− 24.0000i	− 1.17248i	0.810139	−	0.586238i	$-0.199392\pi$
$420$	0	0
$421$	− 10.0000i	− 0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$	− 10.0000i	− 0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	4.00000i	0.193574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	48.0000i	2.29615i
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−9.00000	−0.428571
$442$	0	0
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	0	0
$445$	− 12.0000i	− 0.568855i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−38.0000	−1.79333	−0.896665	−	0.442709i	$-0.854018\pi$
$449$	−38.0000	−1.79333	−0.896665	+	0.442709i	$0.854018\pi$
$450$	0	0
$451$	− 12.0000i	− 0.565058i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 6.00000i	− 0.279448i	−0.990190	−	0.139724i	$-0.955378\pi$
$461$	− 6.00000i	− 0.279448i	0.990190	−	0.139724i	$-0.0446215\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 20.0000i	− 0.925490i	−0.886492	−	0.462745i	$-0.846865\pi$
$467$	− 20.0000i	− 0.925490i	0.886492	−	0.462745i	$-0.153135\pi$
$468$	0	0
$469$	20.0000i	0.923514i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	6.00000i	0.275299i
$476$	0	0
$477$	− 18.0000i	− 0.824163i
$478$	0	0
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 4.00000i	− 0.181631i
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 20.0000i	− 0.902587i	−0.892375	−	0.451294i	$-0.850963\pi$
$491$	− 20.0000i	− 0.902587i	0.892375	−	0.451294i	$-0.149037\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	−12.0000	−0.539360
$496$	0	0
$497$	−20.0000	−0.897123
$498$	0	0
$499$	14.0000i	0.626726i	0.949633	+	0.313363i	$0.101456\pi$
$499$	14.0000i	0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 14.0000i	− 0.620539i	−0.950649	−	0.310270i	$-0.899581\pi$
$509$	− 14.0000i	− 0.620539i	0.950649	−	0.310270i	$-0.100419\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16.0000i	0.705044i
$516$	0	0
$517$	− 4.00000i	− 0.175920i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	24.0000i	1.04945i	0.851273	+	0.524723i	$0.175831\pi$
$523$	24.0000i	1.04945i	−0.851273	+	0.524723i	$0.824169\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−60.0000	−2.61364
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	− 30.0000i	− 1.30189i
$532$	0	0
$533$	− 6.00000i	− 0.259889i
$534$	0	0
$535$	−32.0000	−1.38348
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.00000i	0.258438i
$540$	0	0
$541$	26.0000i	1.11783i	0.829226	+	0.558914i	$0.188782\pi$
$541$	26.0000i	1.11783i	−0.829226	+	0.558914i	$0.811218\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−28.0000	−1.19939
$546$	0	0
$547$	32.0000i	1.36822i	0.729378	+	0.684111i	$0.239809\pi$
$547$	32.0000i	1.36822i	−0.729378	+	0.684111i	$0.760191\pi$
$548$	0	0
$549$	− 6.00000i	− 0.256074i
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.0000i	1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$	42.0000i	1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	− 28.0000i	− 1.17797i
$566$	0	0
$567$	−18.0000	−0.755929
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	− 12.0000i	− 0.502184i	−0.967963	−	0.251092i	$-0.919210\pi$
$571$	− 12.0000i	− 0.502184i	0.967963	−	0.251092i	$-0.0807897\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.00000	0.333623
$576$	0	0
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 12.0000i	− 0.497844i
$582$	0	0
$583$	−12.0000	−0.496989
$584$	0	0
$585$	−6.00000	−0.248069
$586$	0	0
$587$	26.0000i	1.07313i	0.843857	+	0.536567i	$0.180279\pi$
$587$	26.0000i	1.07313i	−0.843857	+	0.536567i	$0.819721\pi$
$588$	0	0
$589$	− 60.0000i	− 2.47226i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	0	0
$595$	24.0000i	0.983904i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	− 30.0000i	− 1.22169i
$604$	0	0
$605$	− 14.0000i	− 0.569181i
$606$	0	0
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 2.00000i	− 0.0809113i
$612$	0	0
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−34.0000	−1.36879	−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000	−1.36879	−0.684394	+	0.729112i	$0.739933\pi$
$618$	0	0
$619$	− 2.00000i	− 0.0803868i	−0.999192	−	0.0401934i	$-0.987203\pi$
$619$	− 2.00000i	− 0.0803868i	0.999192	−	0.0401934i	$-0.0127974\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	36.0000i	1.43541i
$630$	0	0
$631$	22.0000	0.875806	0.437903	−	0.899022i	$-0.355721\pi$
$631$	22.0000	0.875806	0.437903	+	0.899022i	$0.355721\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 16.0000i	− 0.634941i
$636$	0	0
$637$	3.00000i	0.118864i
$638$	0	0
$639$	30.0000	1.18678
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	− 2.00000i	− 0.0788723i	−0.999222	−	0.0394362i	$-0.987444\pi$
$643$	− 2.00000i	− 0.0788723i	0.999222	−	0.0394362i	$-0.0125562\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	0	0
$655$	32.0000	1.25034
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	− 2.00000i	− 0.0777910i	−0.999243	−	0.0388955i	$-0.987616\pi$
$661$	− 2.00000i	− 0.0777910i	0.999243	−	0.0388955i	$-0.0123839\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−24.0000	−0.930680
$666$	0	0
$667$	16.0000i	0.619522i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 2.00000i	− 0.0765279i	−0.999268	−	0.0382639i	$-0.987817\pi$
$683$	− 2.00000i	− 0.0765279i	0.999268	−	0.0382639i	$-0.0121828\pi$
$684$	0	0
$685$	36.0000i	1.37549i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	− 34.0000i	− 1.29342i	−0.762736	−	0.646710i	$-0.776144\pi$
$691$	− 34.0000i	− 1.29342i	0.762736	−	0.646710i	$-0.223856\pi$
$692$	0	0
$693$	12.0000i	0.455842i
$694$	0	0
$695$	32.0000	1.21383
$696$	0	0
$697$	36.0000	1.36360
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.0000i	0.679851i	0.940452	+	0.339925i	$0.110402\pi$
$701$	18.0000i	0.679851i	−0.940452	+	0.339925i	$0.889598\pi$
$702$	0	0
$703$	−36.0000	−1.35777
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.00000i	− 0.150435i
$708$	0	0
$709$	22.0000i	0.826227i	0.910679	+	0.413114i	$0.135559\pi$
$709$	22.0000i	0.826227i	−0.910679	+	0.413114i	$0.864441\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	0	0
$713$	−80.0000	−2.99602
$714$	0	0
$715$	4.00000i	0.149592i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.00000i	0.0742781i
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	24.0000i	0.887672i
$732$	0	0
$733$	26.0000i	0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$	26.0000i	0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20.0000	−0.736709
$738$	0	0
$739$	− 6.00000i	− 0.220714i	−0.993892	−	0.110357i	$-0.964801\pi$
$739$	− 6.00000i	− 0.220714i	0.993892	−	0.110357i	$-0.0351994\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.0000	−0.513610	−0.256805	−	0.966463i	$-0.582670\pi$
$743$	−14.0000	−0.513610	−0.256805	+	0.966463i	$0.582670\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	0	0
$747$	18.0000i	0.658586i
$748$	0	0
$749$	32.0000i	1.16925i
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 12.0000i	− 0.436725i
$756$	0	0
$757$	42.0000i	1.52652i	0.646094	+	0.763258i	$0.276401\pi$
$757$	42.0000i	1.52652i	−0.646094	+	0.763258i	$0.723599\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	28.0000i	1.01367i
$764$	0	0
$765$	− 36.0000i	− 1.30158i
$766$	0	0
$767$	−10.0000	−0.361079
$768$	0	0
$769$	−54.0000	−1.94729	−0.973645	−	0.228069i	$-0.926759\pi$
$769$	−54.0000	−1.94729	−0.973645	+	0.228069i	$0.926759\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	46.0000i	1.65451i	0.561830	+	0.827253i	$0.310097\pi$
$773$	46.0000i	1.65451i	−0.561830	+	0.827253i	$0.689903\pi$
$774$	0	0
$775$	−10.0000	−0.359211
$776$	0	0
$777$	0	0
$778$	0	0
$779$	36.0000i	1.28983i
$780$	0	0
$781$	− 20.0000i	− 0.715656i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.00000	0.142766
$786$	0	0
$787$	− 38.0000i	− 1.35455i	−0.735728	−	0.677277i	$-0.763160\pi$
$787$	− 38.0000i	− 1.35455i	0.735728	−	0.677277i	$-0.236840\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−28.0000	−0.995565
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	4.00000i	0.141157i
$804$	0	0
$805$	32.0000i	1.12785i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	− 22.0000i	− 0.772524i	−0.922389	−	0.386262i	$-0.873766\pi$
$811$	− 22.0000i	− 0.772524i	0.922389	−	0.386262i	$-0.126234\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	20.0000	0.700569
$816$	0	0
$817$	−24.0000	−0.839654
$818$	0	0
$819$	6.00000i	0.209657i
$820$	0	0
$821$	− 2.00000i	− 0.0698005i	−0.999391	−	0.0349002i	$-0.988889\pi$
$821$	− 2.00000i	− 0.0698005i	0.999391	−	0.0349002i	$-0.0111113\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.00000i	0.208640i	0.994544	+	0.104320i	$0.0332667\pi$
$827$	6.00000i	0.208640i	−0.994544	+	0.104320i	$0.966733\pi$
$828$	0	0
$829$	22.0000i	0.764092i	0.924143	+	0.382046i	$0.124780\pi$
$829$	22.0000i	0.764092i	−0.924143	+	0.382046i	$0.875220\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	− 12.0000i	− 0.415277i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−10.0000	−0.345238	−0.172619	−	0.984989i	$-0.555223\pi$
$839$	−10.0000	−0.345238	−0.172619	+	0.984989i	$0.555223\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.00000i	0.0688021i
$846$	0	0
$847$	−14.0000	−0.481046
$848$	0	0
$849$	0	0
$850$	0	0
$851$	48.0000i	1.64542i
$852$	0	0
$853$	38.0000i	1.30110i	0.759465	+	0.650548i	$0.225461\pi$
$853$	38.0000i	1.30110i	−0.759465	+	0.650548i	$0.774539\pi$
$854$	0	0
$855$	36.0000	1.23117
$856$	0	0
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	8.00000i	0.272956i	0.990643	+	0.136478i	$0.0435784\pi$
$859$	8.00000i	0.272956i	−0.990643	+	0.136478i	$0.956422\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−34.0000	−1.15737	−0.578687	−	0.815550i	$-0.696435\pi$
$863$	−34.0000	−1.15737	−0.578687	+	0.815550i	$0.696435\pi$
$864$	0	0
$865$	−20.0000	−0.680020
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 8.00000i	− 0.271381i
$870$	0	0
$871$	−10.0000	−0.338837
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	24.0000i	0.811348i
$876$	0	0
$877$	18.0000i	0.607817i	0.952701	+	0.303908i	$0.0982917\pi$
$877$	18.0000i	0.607817i	−0.952701	+	0.303908i	$0.901708\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	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
$885$	0	0
$886$	0	0
$887$	4.00000	0.134307	0.0671534	−	0.997743i	$-0.478608\pi$
$887$	4.00000	0.134307	0.0671534	+	0.997743i	$0.478608\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	− 18.0000i	− 0.603023i
$892$	0	0
$893$	12.0000i	0.401565i
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 20.0000i	− 0.667037i
$900$	0	0
$901$	− 36.0000i	− 1.19933i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.0000	0.398893
$906$	0	0
$907$	36.0000i	1.19536i	0.801735	+	0.597680i	$0.203911\pi$
$907$	36.0000i	1.19536i	−0.801735	+	0.597680i	$0.796089\pi$
$908$	0	0
$909$	6.00000i	0.199007i
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 32.0000i	− 1.05673i
$918$	0	0
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 10.0000i	− 0.329154i
$924$	0	0
$925$	6.00000i	0.197279i
$926$	0	0
$927$	−24.0000	−0.788263
$928$	0	0
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	0	0
$931$	− 18.0000i	− 0.589926i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−24.0000	−0.784884
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.0000i	0.325991i	0.986627	+	0.162995i	$0.0521156\pi$
$941$	10.0000i	0.325991i	−0.986627	+	0.162995i	$0.947884\pi$
$942$	0	0
$943$	48.0000	1.56310
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.0000i	1.36482i	0.730971	+	0.682408i	$0.239067\pi$
$947$	42.0000i	1.36482i	−0.730971	+	0.682408i	$0.760933\pi$
$948$	0	0
$949$	2.00000i	0.0649227i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	8.00000i	0.258874i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.0000	1.16250
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	− 48.0000i	− 1.54678i
$964$	0	0
$965$	− 4.00000i	− 0.128765i
$966$	0	0
$967$	50.0000	1.60789	0.803946	−	0.594703i	$-0.202730\pi$
$967$	50.0000	1.60789	0.803946	+	0.594703i	$0.202730\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 48.0000i	− 1.54039i	−0.637806	−	0.770197i	$-0.720158\pi$
$971$	− 48.0000i	− 1.54039i	0.637806	−	0.770197i	$-0.279842\pi$
$972$	0	0
$973$	− 32.0000i	− 1.02587i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	− 12.0000i	− 0.383522i
$980$	0	0
$981$	− 42.0000i	− 1.34096i
$982$	0	0
$983$	14.0000	0.446531	0.223265	−	0.974758i	$-0.428328\pi$
$983$	14.0000	0.446531	0.223265	+	0.974758i	$0.428328\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	0	0
$988$	0	0
$989$	32.0000i	1.01754i
$990$	0	0
$991$	−12.0000	−0.381193	−0.190596	−	0.981669i	$-0.561042\pi$
$991$	−12.0000	−0.381193	−0.190596	+	0.981669i	$0.561042\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	32.0000i	1.01447i
$996$	0	0
$997$	58.0000i	1.83688i	0.395562	+	0.918439i	$0.370550\pi$
$997$	58.0000i	1.83688i	−0.395562	+	0.918439i	$0.629450\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.b.e.1665.1		2
4.3	odd	2		3328.2.b.q.1665.1		2
8.3	odd	2		3328.2.b.q.1665.2		2
8.5	even	2	inner	3328.2.b.e.1665.2		2
16.3	odd	4		832.2.a.e.1.1		1
16.5	even	4		208.2.a.c.1.1		1
16.11	odd	4		52.2.a.a.1.1	✓	1
16.13	even	4		832.2.a.f.1.1		1
48.5	odd	4		1872.2.a.f.1.1		1
48.11	even	4		468.2.a.b.1.1		1
48.29	odd	4		7488.2.a.bw.1.1		1
48.35	even	4		7488.2.a.bn.1.1		1
80.27	even	4		1300.2.c.c.1249.1		2
80.43	even	4		1300.2.c.c.1249.2		2
80.59	odd	4		1300.2.a.d.1.1		1
80.69	even	4		5200.2.a.q.1.1		1
112.11	odd	12		2548.2.j.e.1353.1		2
112.27	even	4		2548.2.a.e.1.1		1
112.59	even	12		2548.2.j.f.1353.1		2
112.75	even	12		2548.2.j.f.1145.1		2
112.107	odd	12		2548.2.j.e.1145.1		2
144.11	even	12		4212.2.i.i.1405.1		2
144.43	odd	12		4212.2.i.d.1405.1		2
144.59	even	12		4212.2.i.i.2809.1		2
144.139	odd	12		4212.2.i.d.2809.1		2
176.43	even	4		6292.2.a.g.1.1		1
208.5	odd	4		2704.2.f.f.337.1		2
208.11	even	12		676.2.h.c.485.2		4
208.21	odd	4		2704.2.f.f.337.2		2
208.43	odd	12		676.2.e.b.653.1		2
208.59	even	12		676.2.h.c.361.2		4
208.75	odd	12		676.2.e.b.529.1		2
208.107	odd	12		676.2.e.c.529.1		2
208.123	even	12		676.2.h.c.361.1		4
208.139	odd	12		676.2.e.c.653.1		2
208.155	odd	4		676.2.a.c.1.1		1
208.171	even	12		676.2.h.c.485.1		4
208.181	even	4		2704.2.a.g.1.1		1
208.187	even	4		676.2.d.c.337.1		2
208.203	even	4		676.2.d.c.337.2		2
624.155	even	4		6084.2.a.m.1.1		1
624.203	odd	4		6084.2.b.m.4393.1		2
624.395	odd	4		6084.2.b.m.4393.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.2.a.a.1.1	✓	1	16.11	odd	4
208.2.a.c.1.1		1	16.5	even	4
468.2.a.b.1.1		1	48.11	even	4
676.2.a.c.1.1		1	208.155	odd	4
676.2.d.c.337.1		2	208.187	even	4
676.2.d.c.337.2		2	208.203	even	4
676.2.e.b.529.1		2	208.75	odd	12
676.2.e.b.653.1		2	208.43	odd	12
676.2.e.c.529.1		2	208.107	odd	12
676.2.e.c.653.1		2	208.139	odd	12
676.2.h.c.361.1		4	208.123	even	12
676.2.h.c.361.2		4	208.59	even	12
676.2.h.c.485.1		4	208.171	even	12
676.2.h.c.485.2		4	208.11	even	12
832.2.a.e.1.1		1	16.3	odd	4
832.2.a.f.1.1		1	16.13	even	4
1300.2.a.d.1.1		1	80.59	odd	4
1300.2.c.c.1249.1		2	80.27	even	4
1300.2.c.c.1249.2		2	80.43	even	4
1872.2.a.f.1.1		1	48.5	odd	4
2548.2.a.e.1.1		1	112.27	even	4
2548.2.j.e.1145.1		2	112.107	odd	12
2548.2.j.e.1353.1		2	112.11	odd	12
2548.2.j.f.1145.1		2	112.75	even	12
2548.2.j.f.1353.1		2	112.59	even	12
2704.2.a.g.1.1		1	208.181	even	4
2704.2.f.f.337.1		2	208.5	odd	4
2704.2.f.f.337.2		2	208.21	odd	4
3328.2.b.e.1665.1		2	1.1	even	1	trivial
3328.2.b.e.1665.2		2	8.5	even	2	inner
3328.2.b.q.1665.1		2	4.3	odd	2
3328.2.b.q.1665.2		2	8.3	odd	2
4212.2.i.d.1405.1		2	144.43	odd	12
4212.2.i.d.2809.1		2	144.139	odd	12
4212.2.i.i.1405.1		2	144.11	even	12
4212.2.i.i.2809.1		2	144.59	even	12
5200.2.a.q.1.1		1	80.69	even	4
6084.2.a.m.1.1		1	624.155	even	4
6084.2.b.m.4393.1		2	624.203	odd	4
6084.2.b.m.4393.2		2	624.395	odd	4
6292.2.a.g.1.1		1	176.43	even	4
7488.2.a.bn.1.1		1	48.35	even	4
7488.2.a.bw.1.1		1	48.29	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:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{5} -2.00000 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{5} -2.00000 q^{7} +3.00000 q^{9} -2.00000i q^{11} -1.00000i q^{13} +6.00000 q^{17} +6.00000i q^{19} +8.00000 q^{23} +1.00000 q^{25} +2.00000i q^{29} -10.0000 q^{31} +4.00000i q^{35} +6.00000i q^{37} +6.00000 q^{41} +4.00000i q^{43} -6.00000i q^{45} +2.00000 q^{47} -3.00000 q^{49} -6.00000i q^{53} -4.00000 q^{55} -10.0000i q^{59} -2.00000i q^{61} -6.00000 q^{63} -2.00000 q^{65} -10.0000i q^{67} +10.0000 q^{71} -2.00000 q^{73} +4.00000i q^{77} +4.00000 q^{79} +9.00000 q^{81} +6.00000i q^{83} -12.0000i q^{85} +6.00000 q^{89} +2.00000i q^{91} +12.0000 q^{95} +2.00000 q^{97} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q - 4 * q^7 + 6 * q^9	\( 2 q - 4 q^{7} + 6 q^{9} + 12 q^{17} + 16 q^{23} + 2 q^{25} - 20 q^{31} + 12 q^{41} + 4 q^{47} - 6 q^{49} - 8 q^{55} - 12 q^{63} - 4 q^{65} + 20 q^{71} - 4 q^{73} + 8 q^{79} + 18 q^{81} + 12 q^{89} + 24 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q - 4 * q^7 + 6 * q^9 + 12 * q^17 + 16 * q^23 + 2 * q^25 - 20 * q^31 + 12 * q^41 + 4 * q^47 - 6 * q^49 - 8 * q^55 - 12 * q^63 - 4 * q^65 + 20 * q^71 - 4 * q^73 + 8 * q^79 + 18 * q^81 + 12 * q^89 + 24 * q^95 + 4 * q^97

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