LMFDB - Embedded newform 2800.2.k.a.2351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(2351,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2800.2351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.k (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.3581125660$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2351.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2800.2351
Dual form			2800.2.k.a.2351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{3} +(2.00000 - 1.73205i) q^{7} +1.00000 q^{9} +O(q^{10})$	$q-2.00000 q^{3} +(2.00000 - 1.73205i) q^{7} +1.00000 q^{9} -5.19615i q^{11} -3.46410i q^{13} -8.00000 q^{19} +(-4.00000 + 3.46410i) q^{21} -5.19615i q^{23} +4.00000 q^{27} -9.00000 q^{29} +2.00000 q^{31} +10.3923i q^{33} +7.00000 q^{37} +6.92820i q^{39} +10.3923i q^{41} +1.73205i q^{43} +(1.00000 - 6.92820i) q^{49} -6.00000 q^{53} +16.0000 q^{57} +6.00000 q^{59} +3.46410i q^{61} +(2.00000 - 1.73205i) q^{63} -12.1244i q^{67} +10.3923i q^{69} -5.19615i q^{71} +10.3923i q^{73} +(-9.00000 - 10.3923i) q^{77} -5.19615i q^{79} -11.0000 q^{81} +6.00000 q^{83} +18.0000 q^{87} +10.3923i q^{89} +(-6.00000 - 6.92820i) q^{91} -4.00000 q^{93} +3.46410i q^{97} -5.19615i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^3 + 4 * q^7 + 2 * q^9	$ 2 q - 4 q^{3} + 4 q^{7} + 2 q^{9} - 16 q^{19} - 8 q^{21} + 8 q^{27} - 18 q^{29} + 4 q^{31} + 14 q^{37} + 2 q^{49} - 12 q^{53} + 32 q^{57} + 12 q^{59} + 4 q^{63} - 18 q^{77} - 22 q^{81} + 12 q^{83} + 36 q^{87} - 12 q^{91} - 8 q^{93}+O(q^{100}) $ 2 * q - 4 * q^3 + 4 * q^7 + 2 * q^9 - 16 * q^19 - 8 * q^21 + 8 * q^27 - 18 * q^29 + 4 * q^31 + 14 * q^37 + 2 * q^49 - 12 * q^53 + 32 * q^57 + 12 * q^59 + 4 * q^63 - 18 * q^77 - 22 * q^81 + 12 * q^83 + 36 * q^87 - 12 * q^91 - 8 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$.

$n$	$351$	$801$	$2101$	$2577$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−2.00000			−1.15470			−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000			−1.15470			−0.577350	+	0.816497i	$0.695913\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	5.19615i		−	1.56670i	−0.621582	−	0.783349i	$-0.713510\pi$
$11$		−	5.19615i		−	1.56670i	0.621582	−	0.783349i	$-0.286490\pi$
$12$		0			0
$13$		−	3.46410i		−	0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$		−	3.46410i		−	0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	−8.00000			−1.83533			−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000			−1.83533			−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$	−4.00000	+	3.46410i	−0.872872	+	0.755929i
$22$		0			0
$23$		−	5.19615i		−	1.08347i	−0.840548	−	0.541736i	$-0.817767\pi$
$23$		−	5.19615i		−	1.08347i	0.840548	−	0.541736i	$-0.182233\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	4.00000			0.769800
$28$		0			0
$29$	−9.00000			−1.67126			−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000			−1.67126			−0.835629	+	0.549294i	$0.814897\pi$
$30$		0			0
$31$	2.00000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$		0			0
$33$			10.3923i			1.80907i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	7.00000			1.15079			0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000			1.15079			0.575396	+	0.817875i	$0.304848\pi$
$38$		0			0
$39$			6.92820i			1.10940i
$40$		0			0
$41$			10.3923i			1.62301i	0.584349	+	0.811503i	$0.301350\pi$
$41$			10.3923i			1.62301i	−0.584349	+	0.811503i	$0.698650\pi$
$42$		0			0
$43$			1.73205i			0.264135i	0.991241	+	0.132068i	$0.0421616\pi$
$43$			1.73205i			0.264135i	−0.991241	+	0.132068i	$0.957838\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	16.0000			2.11925
$58$		0			0
$59$	6.00000			0.781133			0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000			0.781133			0.390567	+	0.920575i	$0.372279\pi$
$60$		0			0
$61$			3.46410i			0.443533i	0.975100	+	0.221766i	$0.0711822\pi$
$61$			3.46410i			0.443533i	−0.975100	+	0.221766i	$0.928818\pi$
$62$		0			0
$63$	2.00000	−	1.73205i	0.251976	−	0.218218i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	12.1244i		−	1.48123i	−0.671932	−	0.740613i	$-0.734535\pi$
$67$		−	12.1244i		−	1.48123i	0.671932	−	0.740613i	$-0.265465\pi$
$68$		0			0
$69$			10.3923i			1.25109i
$70$		0			0
$71$		−	5.19615i		−	0.616670i	−0.951278	−	0.308335i	$-0.900228\pi$
$71$		−	5.19615i		−	0.616670i	0.951278	−	0.308335i	$-0.0997717\pi$
$72$		0			0
$73$			10.3923i			1.21633i	0.793812	+	0.608164i	$0.208094\pi$
$73$			10.3923i			1.21633i	−0.793812	+	0.608164i	$0.791906\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−9.00000	−	10.3923i	−1.02565	−	1.18431i
$78$		0			0
$79$		−	5.19615i		−	0.584613i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$		−	5.19615i		−	0.584613i	0.956325	−	0.292306i	$-0.0944227\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	18.0000			1.92980
$88$		0			0
$89$			10.3923i			1.10158i	0.834643	+	0.550791i	$0.185674\pi$
$89$			10.3923i			1.10158i	−0.834643	+	0.550791i	$0.814326\pi$
$90$		0			0
$91$	−6.00000	−	6.92820i	−0.628971	−	0.726273i
$92$		0			0
$93$	−4.00000			−0.414781
$94$		0			0
$95$		0			0
$96$		0			0
$97$			3.46410i			0.351726i	0.984415	+	0.175863i	$0.0562716\pi$
$97$			3.46410i			0.351726i	−0.984415	+	0.175863i	$0.943728\pi$
$98$		0			0
$99$		−	5.19615i		−	0.522233i
$100$		0			0
$101$			10.3923i			1.03407i	0.855963	+	0.517036i	$0.172965\pi$
$101$			10.3923i			1.03407i	−0.855963	+	0.517036i	$0.827035\pi$
$102$		0			0
$103$	−4.00000			−0.394132			−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000			−0.394132			−0.197066	+	0.980390i	$0.563141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			10.3923i			1.00466i	0.864675	+	0.502331i	$0.167524\pi$
$107$			10.3923i			1.00466i	−0.864675	+	0.502331i	$0.832476\pi$
$108$		0			0
$109$	−1.00000			−0.0957826			−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000			−0.0957826			−0.0478913	+	0.998853i	$0.515250\pi$
$110$		0			0
$111$	−14.0000			−1.32882
$112$		0			0
$113$	3.00000			0.282216			0.141108	−	0.989994i	$-0.454933\pi$
$113$	3.00000			0.282216			0.141108	+	0.989994i	$0.454933\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	3.46410i		−	0.320256i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−16.0000			−1.45455
$122$		0			0
$123$		−	20.7846i		−	1.87409i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			15.5885i			1.38325i	0.722256	+	0.691626i	$0.243105\pi$
$127$			15.5885i			1.38325i	−0.722256	+	0.691626i	$0.756895\pi$
$128$		0			0
$129$		−	3.46410i		−	0.304997i
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$	−16.0000	+	13.8564i	−1.38738	+	1.20150i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.00000			0.512615			0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000			0.512615			0.256307	+	0.966595i	$0.417494\pi$
$138$		0			0
$139$	−22.0000			−1.86602			−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−22.0000			−1.86602			−0.933008	+	0.359856i	$0.882826\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−18.0000			−1.50524
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−2.00000	+	13.8564i	−0.164957	+	1.14286i
$148$		0			0
$149$	−9.00000			−0.737309			−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000			−0.737309			−0.368654	+	0.929567i	$0.620181\pi$
$150$		0			0
$151$		−	12.1244i		−	0.986666i	−0.869841	−	0.493333i	$-0.835778\pi$
$151$		−	12.1244i		−	0.986666i	0.869841	−	0.493333i	$-0.164222\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	24.2487i		−	1.93526i	−0.252377	−	0.967629i	$-0.581212\pi$
$157$		−	24.2487i		−	1.93526i	0.252377	−	0.967629i	$-0.418788\pi$
$158$		0			0
$159$	12.0000			0.951662
$160$		0			0
$161$	−9.00000	−	10.3923i	−0.709299	−	0.819028i
$162$		0			0
$163$			17.3205i			1.35665i	0.734763	+	0.678323i	$0.237293\pi$
$163$			17.3205i			1.35665i	−0.734763	+	0.678323i	$0.762707\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−18.0000			−1.39288			−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000			−1.39288			−0.696441	+	0.717614i	$0.745234\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−8.00000			−0.611775
$172$		0			0
$173$		−	20.7846i		−	1.58022i	−0.612962	−	0.790112i	$-0.710022\pi$
$173$		−	20.7846i		−	1.58022i	0.612962	−	0.790112i	$-0.289978\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−12.0000			−0.901975
$178$		0			0
$179$		−	10.3923i		−	0.776757i	−0.921500	−	0.388379i	$-0.873035\pi$
$179$		−	10.3923i		−	0.776757i	0.921500	−	0.388379i	$-0.126965\pi$
$180$		0			0
$181$			17.3205i			1.28742i	0.765268	+	0.643712i	$0.222606\pi$
$181$			17.3205i			1.28742i	−0.765268	+	0.643712i	$0.777394\pi$
$182$		0			0
$183$		−	6.92820i		−	0.512148i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	8.00000	−	6.92820i	0.581914	−	0.503953i
$190$		0			0
$191$			10.3923i			0.751961i	0.926628	+	0.375980i	$0.122694\pi$
$191$			10.3923i			0.751961i	−0.926628	+	0.375980i	$0.877306\pi$
$192$		0			0
$193$	7.00000			0.503871			0.251936	−	0.967744i	$-0.418933\pi$
$193$	7.00000			0.503871			0.251936	+	0.967744i	$0.418933\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3.00000			0.213741			0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000			0.213741			0.106871	+	0.994273i	$0.465917\pi$
$198$		0			0
$199$	10.0000			0.708881			0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000			0.708881			0.354441	+	0.935079i	$0.384671\pi$
$200$		0			0
$201$			24.2487i			1.71037i
$202$		0			0
$203$	−18.0000	+	15.5885i	−1.26335	+	1.09410i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	5.19615i		−	0.361158i
$208$		0			0
$209$			41.5692i			2.87540i
$210$		0			0
$211$			3.46410i			0.238479i	0.992866	+	0.119239i	$0.0380456\pi$
$211$			3.46410i			0.238479i	−0.992866	+	0.119239i	$0.961954\pi$
$212$		0			0
$213$			10.3923i			0.712069i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	4.00000	−	3.46410i	0.271538	−	0.235159i
$218$		0			0
$219$		−	20.7846i		−	1.40449i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−14.0000			−0.937509			−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000			−0.937509			−0.468755	+	0.883328i	$0.655297\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−24.0000			−1.59294			−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000			−1.59294			−0.796468	+	0.604681i	$0.793301\pi$
$228$		0			0
$229$		−	6.92820i		−	0.457829i	−0.973447	−	0.228914i	$-0.926482\pi$
$229$		−	6.92820i		−	0.457829i	0.973447	−	0.228914i	$-0.0735176\pi$
$230$		0			0
$231$	18.0000	+	20.7846i	1.18431	+	1.36753i
$232$		0			0
$233$	−9.00000			−0.589610			−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000			−0.589610			−0.294805	+	0.955557i	$0.595255\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			10.3923i			0.675053i
$238$		0			0
$239$			10.3923i			0.672222i	0.941822	+	0.336111i	$0.109112\pi$
$239$			10.3923i			0.672222i	−0.941822	+	0.336111i	$0.890888\pi$
$240$		0			0
$241$		−	6.92820i		−	0.446285i	−0.974786	−	0.223142i	$-0.928369\pi$
$241$		−	6.92820i		−	0.446285i	0.974786	−	0.223142i	$-0.0716315\pi$
$242$		0			0
$243$	10.0000			0.641500
$244$		0			0
$245$		0			0
$246$		0			0
$247$			27.7128i			1.76332i
$248$		0			0
$249$	−12.0000			−0.760469
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	−27.0000			−1.69748
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	10.3923i		−	0.648254i	−0.946014	−	0.324127i	$-0.894929\pi$
$257$		−	10.3923i		−	0.648254i	0.946014	−	0.324127i	$-0.105071\pi$
$258$		0			0
$259$	14.0000	−	12.1244i	0.869918	−	0.753371i
$260$		0			0
$261$	−9.00000			−0.557086
$262$		0			0
$263$			15.5885i			0.961225i	0.876933	+	0.480613i	$0.159586\pi$
$263$			15.5885i			0.961225i	−0.876933	+	0.480613i	$0.840414\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	20.7846i		−	1.27200i
$268$		0			0
$269$		−	31.1769i		−	1.90089i	−0.310893	−	0.950445i	$-0.600628\pi$
$269$		−	31.1769i		−	1.90089i	0.310893	−	0.950445i	$-0.399372\pi$
$270$		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
$273$	12.0000	+	13.8564i	0.726273	+	0.838628i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	22.0000			1.32185			0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000			1.32185			0.660926	+	0.750451i	$0.270164\pi$
$278$		0			0
$279$	2.00000			0.119737
$280$		0			0
$281$	−21.0000			−1.25275			−0.626377	−	0.779520i	$-0.715463\pi$
$281$	−21.0000			−1.25275			−0.626377	+	0.779520i	$0.715463\pi$
$282$		0			0
$283$	16.0000			0.951101			0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000			0.951101			0.475551	+	0.879688i	$0.342249\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	18.0000	+	20.7846i	1.06251	+	1.22688i
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$		−	6.92820i		−	0.406138i
$292$		0			0
$293$		−	10.3923i		−	0.607125i	−0.952812	−	0.303562i	$-0.901824\pi$
$293$		−	10.3923i		−	0.607125i	0.952812	−	0.303562i	$-0.0981761\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	20.7846i		−	1.20605i
$298$		0			0
$299$	−18.0000			−1.04097
$300$		0			0
$301$	3.00000	+	3.46410i	0.172917	+	0.199667i
$302$		0			0
$303$		−	20.7846i		−	1.19404i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−8.00000			−0.456584			−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000			−0.456584			−0.228292	+	0.973593i	$0.573314\pi$
$308$		0			0
$309$	8.00000			0.455104
$310$		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
$313$		−	6.92820i		−	0.391605i	−0.980643	−	0.195803i	$-0.937269\pi$
$313$		−	6.92820i		−	0.391605i	0.980643	−	0.195803i	$-0.0627312\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−21.0000			−1.17948			−0.589739	−	0.807594i	$-0.700769\pi$
$317$	−21.0000			−1.17948			−0.589739	+	0.807594i	$0.700769\pi$
$318$		0			0
$319$			46.7654i			2.61836i
$320$		0			0
$321$		−	20.7846i		−	1.16008i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	2.00000			0.110600
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	5.19615i		−	0.285606i	−0.989751	−	0.142803i	$-0.954388\pi$
$331$		−	5.19615i		−	0.285606i	0.989751	−	0.142803i	$-0.0456116\pi$
$332$		0			0
$333$	7.00000			0.383598
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−34.0000			−1.85210			−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000			−1.85210			−0.926049	+	0.377403i	$0.876817\pi$
$338$		0			0
$339$	−6.00000			−0.325875
$340$		0			0
$341$		−	10.3923i		−	0.562775i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	25.9808i		−	1.39472i	−0.716721	−	0.697360i	$-0.754358\pi$
$347$		−	25.9808i		−	1.39472i	0.716721	−	0.697360i	$-0.245642\pi$
$348$		0			0
$349$			20.7846i			1.11257i	0.830990	+	0.556287i	$0.187775\pi$
$349$			20.7846i			1.11257i	−0.830990	+	0.556287i	$0.812225\pi$
$350$		0			0
$351$		−	13.8564i		−	0.739600i
$352$		0			0
$353$			10.3923i			0.553127i	0.960996	+	0.276563i	$0.0891955\pi$
$353$			10.3923i			0.553127i	−0.960996	+	0.276563i	$0.910804\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	5.19615i		−	0.274242i	−0.990554	−	0.137121i	$-0.956215\pi$
$359$		−	5.19615i		−	0.274242i	0.990554	−	0.137121i	$-0.0437850\pi$
$360$		0			0
$361$	45.0000			2.36842
$362$		0			0
$363$	32.0000			1.67956
$364$		0			0
$365$		0			0
$366$		0			0
$367$	22.0000			1.14839			0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000			1.14839			0.574195	+	0.818718i	$0.305315\pi$
$368$		0			0
$369$			10.3923i			0.541002i
$370$		0			0
$371$	−12.0000	+	10.3923i	−0.623009	+	0.539542i
$372$		0			0
$373$	11.0000			0.569558			0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000			0.569558			0.284779	+	0.958593i	$0.408080\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			31.1769i			1.60569i
$378$		0			0
$379$			1.73205i			0.0889695i	0.999010	+	0.0444847i	$0.0141646\pi$
$379$			1.73205i			0.0889695i	−0.999010	+	0.0444847i	$0.985835\pi$
$380$		0			0
$381$		−	31.1769i		−	1.59724i
$382$		0			0
$383$	6.00000			0.306586			0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000			0.306586			0.153293	+	0.988181i	$0.451012\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			1.73205i			0.0880451i
$388$		0			0
$389$	−33.0000			−1.67317			−0.836583	−	0.547840i	$-0.815450\pi$
$389$	−33.0000			−1.67317			−0.836583	+	0.547840i	$0.815450\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	24.0000			1.21064
$394$		0			0
$395$		0			0
$396$		0			0
$397$			27.7128i			1.39087i	0.718591	+	0.695433i	$0.244787\pi$
$397$			27.7128i			1.39087i	−0.718591	+	0.695433i	$0.755213\pi$
$398$		0			0
$399$	32.0000	−	27.7128i	1.60200	−	1.38738i
$400$		0			0
$401$	15.0000			0.749064			0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000			0.749064			0.374532	+	0.927214i	$0.377803\pi$
$402$		0			0
$403$		−	6.92820i		−	0.345118i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	36.3731i		−	1.80295i
$408$		0			0
$409$		−	27.7128i		−	1.37031i	−0.728397	−	0.685155i	$-0.759734\pi$
$409$		−	27.7128i		−	1.37031i	0.728397	−	0.685155i	$-0.240266\pi$
$410$		0			0
$411$	−12.0000			−0.591916
$412$		0			0
$413$	12.0000	−	10.3923i	0.590481	−	0.511372i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	44.0000			2.15469
$418$		0			0
$419$	−30.0000			−1.46560			−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000			−1.46560			−0.732798	+	0.680446i	$0.761786\pi$
$420$		0			0
$421$	19.0000			0.926003			0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000			0.926003			0.463002	+	0.886357i	$0.346772\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	6.00000	+	6.92820i	0.290360	+	0.335279i
$428$		0			0
$429$	36.0000			1.73810
$430$		0			0
$431$		−	31.1769i		−	1.50174i	−0.660451	−	0.750870i	$-0.729635\pi$
$431$		−	31.1769i		−	1.50174i	0.660451	−	0.750870i	$-0.270365\pi$
$432$		0			0
$433$		−	34.6410i		−	1.66474i	−0.554220	−	0.832370i	$-0.686983\pi$
$433$		−	34.6410i		−	1.66474i	0.554220	−	0.832370i	$-0.313017\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			41.5692i			1.98853i
$438$		0			0
$439$	22.0000			1.05000			0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000			1.05000			0.525001	+	0.851101i	$0.324065\pi$
$440$		0			0
$441$	1.00000	−	6.92820i	0.0476190	−	0.329914i
$442$		0			0
$443$			31.1769i			1.48126i	0.671913	+	0.740630i	$0.265473\pi$
$443$			31.1769i			1.48126i	−0.671913	+	0.740630i	$0.734527\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	18.0000			0.851371
$448$		0			0
$449$	15.0000			0.707894			0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000			0.707894			0.353947	+	0.935266i	$0.384839\pi$
$450$		0			0
$451$	54.0000			2.54276
$452$		0			0
$453$			24.2487i			1.13930i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.00000			−0.0467780			−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000			−0.0467780			−0.0233890	+	0.999726i	$0.507446\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			31.1769i			1.45205i	0.687666	+	0.726027i	$0.258635\pi$
$461$			31.1769i			1.45205i	−0.687666	+	0.726027i	$0.741365\pi$
$462$		0			0
$463$		−	10.3923i		−	0.482971i	−0.970404	−	0.241486i	$-0.922365\pi$
$463$		−	10.3923i		−	0.482971i	0.970404	−	0.241486i	$-0.0776347\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	−21.0000	−	24.2487i	−0.969690	−	1.11970i
$470$		0			0
$471$			48.4974i			2.23464i
$472$		0			0
$473$	9.00000			0.413820
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$	−6.00000			−0.274147			−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000			−0.274147			−0.137073	+	0.990561i	$0.543770\pi$
$480$		0			0
$481$		−	24.2487i		−	1.10565i
$482$		0			0
$483$	18.0000	+	20.7846i	0.819028	+	0.945732i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	32.9090i		−	1.49125i	−0.666367	−	0.745624i	$-0.732152\pi$
$487$		−	32.9090i		−	1.49125i	0.666367	−	0.745624i	$-0.267848\pi$
$488$		0			0
$489$		−	34.6410i		−	1.56652i
$490$		0			0
$491$		−	25.9808i		−	1.17250i	−0.810132	−	0.586248i	$-0.800605\pi$
$491$		−	25.9808i		−	1.17250i	0.810132	−	0.586248i	$-0.199395\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.00000	−	10.3923i	−0.403705	−	0.466159i
$498$		0			0
$499$			24.2487i			1.08552i	0.839887	+	0.542761i	$0.182621\pi$
$499$			24.2487i			1.08552i	−0.839887	+	0.542761i	$0.817379\pi$
$500$		0			0
$501$	36.0000			1.60836
$502$		0			0
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−2.00000			−0.0888231
$508$		0			0
$509$			31.1769i			1.38189i	0.722906	+	0.690946i	$0.242806\pi$
$509$			31.1769i			1.38189i	−0.722906	+	0.690946i	$0.757194\pi$
$510$		0			0
$511$	18.0000	+	20.7846i	0.796273	+	0.919457i
$512$		0			0
$513$	−32.0000			−1.41283
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			41.5692i			1.82469i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−28.0000			−1.22435			−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000			−1.22435			−0.612177	+	0.790721i	$0.709706\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−4.00000			−0.173913
$530$		0			0
$531$	6.00000			0.260378
$532$		0			0
$533$	36.0000			1.55933
$534$		0			0
$535$		0			0
$536$		0			0
$537$			20.7846i			0.896922i
$538$		0			0
$539$	−36.0000	−	5.19615i	−1.55063	−	0.223814i
$540$		0			0
$541$	−17.0000			−0.730887			−0.365444	−	0.930834i	$-0.619083\pi$
$541$	−17.0000			−0.730887			−0.365444	+	0.930834i	$0.619083\pi$
$542$		0			0
$543$		−	34.6410i		−	1.48659i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	25.9808i		−	1.11086i	−0.831564	−	0.555429i	$-0.812554\pi$
$547$		−	25.9808i		−	1.11086i	0.831564	−	0.555429i	$-0.187446\pi$
$548$		0			0
$549$			3.46410i			0.147844i
$550$		0			0
$551$	72.0000			3.06730
$552$		0			0
$553$	−9.00000	−	10.3923i	−0.382719	−	0.441926i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.00000			0.127114			0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000			0.127114			0.0635570	+	0.997978i	$0.479756\pi$
$558$		0			0
$559$	6.00000			0.253773
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12.0000			0.505740			0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000			0.505740			0.252870	+	0.967500i	$0.418626\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−22.0000	+	19.0526i	−0.923913	+	0.800132i
$568$		0			0
$569$	−33.0000			−1.38343			−0.691716	−	0.722170i	$-0.743145\pi$
$569$	−33.0000			−1.38343			−0.691716	+	0.722170i	$0.743145\pi$
$570$		0			0
$571$			1.73205i			0.0724841i	0.999343	+	0.0362420i	$0.0115387\pi$
$571$			1.73205i			0.0724841i	−0.999343	+	0.0362420i	$0.988461\pi$
$572$		0			0
$573$		−	20.7846i		−	0.868290i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	6.92820i		−	0.288425i	−0.989547	−	0.144212i	$-0.953935\pi$
$577$		−	6.92820i		−	0.288425i	0.989547	−	0.144212i	$-0.0460649\pi$
$578$		0			0
$579$	−14.0000			−0.581820
$580$		0			0
$581$	12.0000	−	10.3923i	0.497844	−	0.431145i
$582$		0			0
$583$			31.1769i			1.29122i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−42.0000			−1.73353			−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000			−1.73353			−0.866763	+	0.498721i	$0.833803\pi$
$588$		0			0
$589$	−16.0000			−0.659269
$590$		0			0
$591$	−6.00000			−0.246807
$592$		0			0
$593$		−	20.7846i		−	0.853522i	−0.904365	−	0.426761i	$-0.859655\pi$
$593$		−	20.7846i		−	0.853522i	0.904365	−	0.426761i	$-0.140345\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−20.0000			−0.818546
$598$		0			0
$599$			36.3731i			1.48616i	0.669201	+	0.743082i	$0.266637\pi$
$599$			36.3731i			1.48616i	−0.669201	+	0.743082i	$0.733363\pi$
$600$		0			0
$601$		−	13.8564i		−	0.565215i	−0.959236	−	0.282607i	$-0.908801\pi$
$601$		−	13.8564i		−	0.565215i	0.959236	−	0.282607i	$-0.0911993\pi$
$602$		0			0
$603$		−	12.1244i		−	0.493742i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−14.0000			−0.568242			−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000			−0.568242			−0.284121	+	0.958788i	$0.591702\pi$
$608$		0			0
$609$	36.0000	−	31.1769i	1.45879	−	1.26335i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	47.0000			1.89831			0.949156	−	0.314806i	$-0.101939\pi$
$613$	47.0000			1.89831			0.949156	+	0.314806i	$0.101939\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.00000			0.120775			0.0603877	−	0.998175i	$-0.480766\pi$
$617$	3.00000			0.120775			0.0603877	+	0.998175i	$0.480766\pi$
$618$		0			0
$619$	14.0000			0.562708			0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000			0.562708			0.281354	+	0.959604i	$0.409217\pi$
$620$		0			0
$621$		−	20.7846i		−	0.834058i
$622$		0			0
$623$	18.0000	+	20.7846i	0.721155	+	0.832718i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		−	83.1384i		−	3.32023i
$628$		0			0
$629$		0			0
$630$		0			0
$631$			29.4449i			1.17218i	0.810245	+	0.586091i	$0.199334\pi$
$631$			29.4449i			1.17218i	−0.810245	+	0.586091i	$0.800666\pi$
$632$		0			0
$633$		−	6.92820i		−	0.275371i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−24.0000	−	3.46410i	−0.950915	−	0.137253i
$638$		0			0
$639$		−	5.19615i		−	0.205557i
$640$		0			0
$641$	15.0000			0.592464			0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000			0.592464			0.296232	+	0.955116i	$0.404270\pi$
$642$		0			0
$643$	4.00000			0.157745			0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000			0.157745			0.0788723	+	0.996885i	$0.474868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	12.0000			0.471769			0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000			0.471769			0.235884	+	0.971781i	$0.424201\pi$
$648$		0			0
$649$		−	31.1769i		−	1.22380i
$650$		0			0
$651$	−8.00000	+	6.92820i	−0.313545	+	0.271538i
$652$		0			0
$653$	18.0000			0.704394			0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000			0.704394			0.352197	+	0.935926i	$0.385435\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			10.3923i			0.405442i
$658$		0			0
$659$		−	31.1769i		−	1.21448i	−0.794518	−	0.607240i	$-0.792277\pi$
$659$		−	31.1769i		−	1.21448i	0.794518	−	0.607240i	$-0.207723\pi$
$660$		0			0
$661$		−	20.7846i		−	0.808428i	−0.914665	−	0.404214i	$-0.867545\pi$
$661$		−	20.7846i		−	0.808428i	0.914665	−	0.404214i	$-0.132455\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			46.7654i			1.81076i
$668$		0			0
$669$	28.0000			1.08254
$670$		0			0
$671$	18.0000			0.694882
$672$		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
$677$			20.7846i			0.798817i	0.916773	+	0.399409i	$0.130785\pi$
$677$			20.7846i			0.798817i	−0.916773	+	0.399409i	$0.869215\pi$
$678$		0			0
$679$	6.00000	+	6.92820i	0.230259	+	0.265880i
$680$		0			0
$681$	48.0000			1.83936
$682$		0			0
$683$		−	5.19615i		−	0.198825i	−0.995046	−	0.0994126i	$-0.968304\pi$
$683$		−	5.19615i		−	0.198825i	0.995046	−	0.0994126i	$-0.0316964\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			13.8564i			0.528655i
$688$		0			0
$689$			20.7846i			0.791831i
$690$		0			0
$691$	−10.0000			−0.380418			−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000			−0.380418			−0.190209	+	0.981744i	$0.560917\pi$
$692$		0			0
$693$	−9.00000	−	10.3923i	−0.341882	−	0.394771i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	18.0000			0.680823
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$	−56.0000			−2.11208
$704$		0			0
$705$		0			0
$706$		0			0
$707$	18.0000	+	20.7846i	0.676960	+	0.781686i
$708$		0			0
$709$	50.0000			1.87779			0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000			1.87779			0.938895	+	0.344204i	$0.111851\pi$
$710$		0			0
$711$		−	5.19615i		−	0.194871i
$712$		0			0
$713$		−	10.3923i		−	0.389195i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	20.7846i		−	0.776215i
$718$		0			0
$719$	−18.0000			−0.671287			−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000			−0.671287			−0.335643	+	0.941989i	$0.608954\pi$
$720$		0			0
$721$	−8.00000	+	6.92820i	−0.297936	+	0.258020i
$722$		0			0
$723$			13.8564i			0.515325i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	32.0000			1.18681			0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000			1.18681			0.593407	+	0.804902i	$0.297782\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−63.0000			−2.32063
$738$		0			0
$739$			15.5885i			0.573431i	0.958016	+	0.286715i	$0.0925634\pi$
$739$			15.5885i			0.573431i	−0.958016	+	0.286715i	$0.907437\pi$
$740$		0			0
$741$		−	55.4256i		−	2.03611i
$742$		0			0
$743$			10.3923i			0.381257i	0.981662	+	0.190628i	$0.0610525\pi$
$743$			10.3923i			0.381257i	−0.981662	+	0.190628i	$0.938947\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	6.00000			0.219529
$748$		0			0
$749$	18.0000	+	20.7846i	0.657706	+	0.759453i
$750$		0			0
$751$		−	10.3923i		−	0.379221i	−0.981859	−	0.189610i	$-0.939278\pi$
$751$		−	10.3923i		−	0.379221i	0.981859	−	0.189610i	$-0.0607225\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	35.0000			1.27210			0.636048	−	0.771649i	$-0.280568\pi$
$757$	35.0000			1.27210			0.636048	+	0.771649i	$0.280568\pi$
$758$		0			0
$759$	54.0000			1.96008
$760$		0			0
$761$		−	20.7846i		−	0.753442i	−0.926327	−	0.376721i	$-0.877052\pi$
$761$		−	20.7846i		−	0.753442i	0.926327	−	0.376721i	$-0.122948\pi$
$762$		0			0
$763$	−2.00000	+	1.73205i	−0.0724049	+	0.0627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	20.7846i		−	0.750489i
$768$		0			0
$769$		−	3.46410i		−	0.124919i	−0.998048	−	0.0624593i	$-0.980106\pi$
$769$		−	3.46410i		−	0.124919i	0.998048	−	0.0624593i	$-0.0198944\pi$
$770$		0			0
$771$			20.7846i			0.748539i
$772$		0			0
$773$		−	31.1769i		−	1.12136i	−0.828034	−	0.560678i	$-0.810541\pi$
$773$		−	31.1769i		−	1.12136i	0.828034	−	0.560678i	$-0.189459\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−28.0000	+	24.2487i	−1.00449	+	0.869918i
$778$		0			0
$779$		−	83.1384i		−	2.97874i
$780$		0			0
$781$	−27.0000			−0.966136
$782$		0			0
$783$	−36.0000			−1.28654
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−34.0000			−1.21197			−0.605985	−	0.795476i	$-0.707221\pi$
$787$	−34.0000			−1.21197			−0.605985	+	0.795476i	$0.707221\pi$
$788$		0			0
$789$		−	31.1769i		−	1.10993i
$790$		0			0
$791$	6.00000	−	5.19615i	0.213335	−	0.184754i
$792$		0			0
$793$	12.0000			0.426132
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$			10.3923i			0.367194i
$802$		0			0
$803$	54.0000			1.90562
$804$		0			0
$805$		0			0
$806$		0			0
$807$			62.3538i			2.19496i
$808$		0			0
$809$	−45.0000			−1.58212			−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000			−1.58212			−0.791058	+	0.611741i	$0.790469\pi$
$810$		0			0
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$		0			0
$813$	−4.00000			−0.140286
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	13.8564i		−	0.484774i
$818$		0			0
$819$	−6.00000	−	6.92820i	−0.209657	−	0.242091i
$820$		0			0
$821$	−30.0000			−1.04701			−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000			−1.04701			−0.523504	+	0.852023i	$0.675375\pi$
$822$		0			0
$823$		−	12.1244i		−	0.422628i	−0.977418	−	0.211314i	$-0.932226\pi$
$823$		−	12.1244i		−	0.422628i	0.977418	−	0.211314i	$-0.0677743\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			15.5885i			0.542064i	0.962570	+	0.271032i	$0.0873649\pi$
$827$			15.5885i			0.542064i	−0.962570	+	0.271032i	$0.912635\pi$
$828$		0			0
$829$			20.7846i			0.721879i	0.932589	+	0.360940i	$0.117544\pi$
$829$			20.7846i			0.721879i	−0.932589	+	0.360940i	$0.882456\pi$
$830$		0			0
$831$	−44.0000			−1.52634
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	8.00000			0.276520
$838$		0			0
$839$	18.0000			0.621429			0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000			0.621429			0.310715	+	0.950503i	$0.399432\pi$
$840$		0			0
$841$	52.0000			1.79310
$842$		0			0
$843$	42.0000			1.44656
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−32.0000	+	27.7128i	−1.09953	+	0.952224i
$848$		0			0
$849$	−32.0000			−1.09824
$850$		0			0
$851$		−	36.3731i		−	1.24685i
$852$		0			0
$853$			55.4256i			1.89774i	0.315671	+	0.948869i	$0.397770\pi$
$853$			55.4256i			1.89774i	−0.315671	+	0.948869i	$0.602230\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			51.9615i			1.77497i	0.460835	+	0.887486i	$0.347550\pi$
$857$			51.9615i			1.77497i	−0.460835	+	0.887486i	$0.652450\pi$
$858$		0			0
$859$	4.00000			0.136478			0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000			0.136478			0.0682391	+	0.997669i	$0.478262\pi$
$860$		0			0
$861$	−36.0000	−	41.5692i	−1.22688	−	1.41668i
$862$		0			0
$863$		−	5.19615i		−	0.176879i	−0.996082	−	0.0884395i	$-0.971812\pi$
$863$		−	5.19615i		−	0.176879i	0.996082	−	0.0884395i	$-0.0281880\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−34.0000			−1.15470
$868$		0			0
$869$	−27.0000			−0.915912
$870$		0			0
$871$	−42.0000			−1.42312
$872$		0			0
$873$			3.46410i			0.117242i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	2.00000			0.0675352			0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000			0.0675352			0.0337676	+	0.999430i	$0.489249\pi$
$878$		0			0
$879$			20.7846i			0.701047i
$880$		0			0
$881$			31.1769i			1.05038i	0.850986	+	0.525188i	$0.176005\pi$
$881$			31.1769i			1.05038i	−0.850986	+	0.525188i	$0.823995\pi$
$882$		0			0
$883$			22.5167i			0.757746i	0.925449	+	0.378873i	$0.123688\pi$
$883$			22.5167i			0.757746i	−0.925449	+	0.378873i	$0.876312\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−42.0000			−1.41022			−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000			−1.41022			−0.705111	+	0.709097i	$0.749103\pi$
$888$		0			0
$889$	27.0000	+	31.1769i	0.905551	+	1.04564i
$890$		0			0
$891$			57.1577i			1.91485i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	36.0000			1.20201
$898$		0			0
$899$	−18.0000			−0.600334
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−6.00000	−	6.92820i	−0.199667	−	0.230556i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	10.3923i		−	0.345071i	−0.985003	−	0.172535i	$-0.944804\pi$
$907$		−	10.3923i		−	0.345071i	0.985003	−	0.172535i	$-0.0551959\pi$
$908$		0			0
$909$			10.3923i			0.344691i
$910$		0			0
$911$			36.3731i			1.20509i	0.798084	+	0.602547i	$0.205847\pi$
$911$			36.3731i			1.20509i	−0.798084	+	0.602547i	$0.794153\pi$
$912$		0			0
$913$		−	31.1769i		−	1.03181i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−24.0000	+	20.7846i	−0.792550	+	0.686368i
$918$		0			0
$919$		−	12.1244i		−	0.399946i	−0.979801	−	0.199973i	$-0.935915\pi$
$919$		−	12.1244i		−	0.399946i	0.979801	−	0.199973i	$-0.0640854\pi$
$920$		0			0
$921$	16.0000			0.527218
$922$		0			0
$923$	−18.0000			−0.592477
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−4.00000			−0.131377
$928$		0			0
$929$		−	10.3923i		−	0.340960i	−0.985361	−	0.170480i	$-0.945468\pi$
$929$		−	10.3923i		−	0.340960i	0.985361	−	0.170480i	$-0.0545319\pi$
$930$		0			0
$931$	−8.00000	+	55.4256i	−0.262189	+	1.81650i
$932$		0			0
$933$	24.0000			0.785725
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	38.1051i		−	1.24484i	−0.782683	−	0.622420i	$-0.786150\pi$
$937$		−	38.1051i		−	1.24484i	0.782683	−	0.622420i	$-0.213850\pi$
$938$		0			0
$939$			13.8564i			0.452187i
$940$		0			0
$941$			10.3923i			0.338779i	0.985549	+	0.169390i	$0.0541797\pi$
$941$			10.3923i			0.338779i	−0.985549	+	0.169390i	$0.945820\pi$
$942$		0			0
$943$	54.0000			1.75848
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	10.3923i		−	0.337705i	−0.985641	−	0.168852i	$-0.945994\pi$
$947$		−	10.3923i		−	0.337705i	0.985641	−	0.168852i	$-0.0540061\pi$
$948$		0			0
$949$	36.0000			1.16861
$950$		0			0
$951$	42.0000			1.36194
$952$		0			0
$953$	51.0000			1.65205			0.826026	−	0.563632i	$-0.190596\pi$
$953$	51.0000			1.65205			0.826026	+	0.563632i	$0.190596\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	93.5307i		−	3.02342i
$958$		0			0
$959$	12.0000	−	10.3923i	0.387500	−	0.335585i
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$			10.3923i			0.334887i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			10.3923i			0.334194i	0.985940	+	0.167097i	$0.0534393\pi$
$967$			10.3923i			0.334194i	−0.985940	+	0.167097i	$0.946561\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−48.0000			−1.54039			−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000			−1.54039			−0.770197	+	0.637806i	$0.779842\pi$
$972$		0			0
$973$	−44.0000	+	38.1051i	−1.41058	+	1.22159i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−45.0000			−1.43968			−0.719839	−	0.694141i	$-0.755784\pi$
$977$	−45.0000			−1.43968			−0.719839	+	0.694141i	$0.755784\pi$
$978$		0			0
$979$	54.0000			1.72585
$980$		0			0
$981$	−1.00000			−0.0319275
$982$		0			0
$983$	6.00000			0.191370			0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000			0.191370			0.0956851	+	0.995412i	$0.469496\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	9.00000			0.286183
$990$		0			0
$991$		−	39.8372i		−	1.26547i	−0.774369	−	0.632735i	$-0.781932\pi$
$991$		−	39.8372i		−	1.26547i	0.774369	−	0.632735i	$-0.218068\pi$
$992$		0			0
$993$			10.3923i			0.329790i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$	28.0000			0.885881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.k.a.2351.1	✓	2
4.3	odd	2		2800.2.k.f.2351.2	yes	2
5.2	odd	4		2800.2.e.d.2799.4		4
5.3	odd	4		2800.2.e.d.2799.1		4
5.4	even	2		2800.2.k.d.2351.2	yes	2
7.6	odd	2		2800.2.k.f.2351.1	yes	2
20.3	even	4		2800.2.e.a.2799.4		4
20.7	even	4		2800.2.e.a.2799.1		4
20.19	odd	2		2800.2.k.c.2351.1	yes	2
28.27	even	2	inner	2800.2.k.a.2351.2	yes	2
35.13	even	4		2800.2.e.a.2799.3		4
35.27	even	4		2800.2.e.a.2799.2		4
35.34	odd	2		2800.2.k.c.2351.2	yes	2
140.27	odd	4		2800.2.e.d.2799.3		4
140.83	odd	4		2800.2.e.d.2799.2		4
140.139	even	2		2800.2.k.d.2351.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2800.2.e.a.2799.1		4	20.7	even	4
2800.2.e.a.2799.2		4	35.27	even	4
2800.2.e.a.2799.3		4	35.13	even	4
2800.2.e.a.2799.4		4	20.3	even	4
2800.2.e.d.2799.1		4	5.3	odd	4
2800.2.e.d.2799.2		4	140.83	odd	4
2800.2.e.d.2799.3		4	140.27	odd	4
2800.2.e.d.2799.4		4	5.2	odd	4
2800.2.k.a.2351.1	✓	2	1.1	even	1	trivial
2800.2.k.a.2351.2	yes	2	28.27	even	2	inner
2800.2.k.c.2351.1	yes	2	20.19	odd	2
2800.2.k.c.2351.2	yes	2	35.34	odd	2
2800.2.k.d.2351.1	yes	2	140.139	even	2
2800.2.k.d.2351.2	yes	2	5.4	even	2
2800.2.k.f.2351.1	yes	2	7.6	odd	2
2800.2.k.f.2351.2	yes	2	4.3	odd	2

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:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{3} +(2.00000 - 1.73205i) q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{3} +(2.00000 - 1.73205i) q^{7} +1.00000 q^{9} -5.19615i q^{11} -3.46410i q^{13} -8.00000 q^{19} +(-4.00000 + 3.46410i) q^{21} -5.19615i q^{23} +4.00000 q^{27} -9.00000 q^{29} +2.00000 q^{31} +10.3923i q^{33} +7.00000 q^{37} +6.92820i q^{39} +10.3923i q^{41} +1.73205i q^{43} +(1.00000 - 6.92820i) q^{49} -6.00000 q^{53} +16.0000 q^{57} +6.00000 q^{59} +3.46410i q^{61} +(2.00000 - 1.73205i) q^{63} -12.1244i q^{67} +10.3923i q^{69} -5.19615i q^{71} +10.3923i q^{73} +(-9.00000 - 10.3923i) q^{77} -5.19615i q^{79} -11.0000 q^{81} +6.00000 q^{83} +18.0000 q^{87} +10.3923i q^{89} +(-6.00000 - 6.92820i) q^{91} -4.00000 q^{93} +3.46410i q^{97} -5.19615i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^3 + 4 * q^7 + 2 * q^9	\( 2 q - 4 q^{3} + 4 q^{7} + 2 q^{9} - 16 q^{19} - 8 q^{21} + 8 q^{27} - 18 q^{29} + 4 q^{31} + 14 q^{37} + 2 q^{49} - 12 q^{53} + 32 q^{57} + 12 q^{59} + 4 q^{63} - 18 q^{77} - 22 q^{81} + 12 q^{83} + 36 q^{87} - 12 q^{91} - 8 q^{93}+O(q^{100}) \) 2 * q - 4 * q^3 + 4 * q^7 + 2 * q^9 - 16 * q^19 - 8 * q^21 + 8 * q^27 - 18 * q^29 + 4 * q^31 + 14 * q^37 + 2 * q^49 - 12 * q^53 + 32 * q^57 + 12 * q^59 + 4 * q^63 - 18 * q^77 - 22 * q^81 + 12 * q^83 + 36 * q^87 - 12 * q^91 - 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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