LMFDB - Embedded newform 2800.2.e.b.2799.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(2799,2800)]

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, 1, 1]))

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

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

chi := DirichletCharacter("2800.2799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.e (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:	$22.3581125660$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2799.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2800.2799
Dual form			2800.2.e.b.2799.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +(1.73205 + 2.00000i) q^{7} -1.00000 q^{9} +O(q^{10})$	$q+2.00000i q^{3} +(1.73205 + 2.00000i) q^{7} -1.00000 q^{9} +3.46410i q^{11} -3.46410 q^{13} -2.00000 q^{19} +(-4.00000 + 3.46410i) q^{21} +3.46410 q^{23} +4.00000i q^{27} -6.00000 q^{29} -8.00000 q^{31} -6.92820 q^{33} +2.00000i q^{37} -6.92820i q^{39} -6.92820i q^{41} +10.3923 q^{43} +(-1.00000 + 6.92820i) q^{49} +6.00000i q^{53} -4.00000i q^{57} -6.00000 q^{59} +3.46410i q^{61} +(-1.73205 - 2.00000i) q^{63} +3.46410 q^{67} +6.92820i q^{69} +3.46410i q^{71} -6.92820 q^{73} +(-6.92820 + 6.00000i) q^{77} -3.46410i q^{79} -11.0000 q^{81} -6.00000i q^{83} -12.0000i q^{87} +6.92820i q^{89} +(-6.00000 - 6.92820i) q^{91} -16.0000i q^{93} +13.8564 q^{97} -3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 8 q^{19} - 16 q^{21} - 24 q^{29} - 32 q^{31} - 4 q^{49} - 24 q^{59} - 44 q^{81} - 24 q^{91}+O(q^{100}) $ 4 * q - 4 * q^9 - 8 * q^19 - 16 * q^21 - 24 * q^29 - 32 * q^31 - 4 * q^49 - 24 * q^59 - 44 * q^81 - 24 * q^91

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
$3$			2.00000i			1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$			2.00000i			1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.73205	+	2.00000i	0.654654	+	0.755929i
$7$	1.73205	+	2.00000i	0.654654	+	0.755929i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$		0			0
$13$	−3.46410			−0.960769			−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410			−0.960769			−0.480384	+	0.877058i	$0.659503\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$		0			0
$21$	−4.00000	+	3.46410i	−0.872872	+	0.755929i
$22$		0			0
$23$	3.46410			0.722315			0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410			0.722315			0.361158	+	0.932505i	$0.382382\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			4.00000i			0.769800i
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$		0			0
$33$	−6.92820			−1.20605
$34$		0			0
$35$		0			0
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$		−	6.92820i		−	1.10940i
$40$		0			0
$41$		−	6.92820i		−	1.08200i	−0.841021	−	0.541002i	$-0.818045\pi$
$41$		−	6.92820i		−	1.08200i	0.841021	−	0.541002i	$-0.181955\pi$
$42$		0			0
$43$	10.3923			1.58481			0.792406	−	0.609994i	$-0.208828\pi$
$43$	10.3923			1.58481			0.792406	+	0.609994i	$0.208828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−1.00000	+	6.92820i	−0.142857	+	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	4.00000i		−	0.529813i
$58$		0			0
$59$	−6.00000			−0.781133			−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000			−0.781133			−0.390567	+	0.920575i	$0.627721\pi$
$60$		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
$63$	−1.73205	−	2.00000i	−0.218218	−	0.251976i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	3.46410			0.423207			0.211604	−	0.977356i	$-0.432131\pi$
$67$	3.46410			0.423207			0.211604	+	0.977356i	$0.432131\pi$
$68$		0			0
$69$			6.92820i			0.834058i
$70$		0			0
$71$			3.46410i			0.411113i	0.978645	+	0.205557i	$0.0659005\pi$
$71$			3.46410i			0.411113i	−0.978645	+	0.205557i	$0.934100\pi$
$72$		0			0
$73$	−6.92820			−0.810885			−0.405442	−	0.914121i	$-0.632883\pi$
$73$	−6.92820			−0.810885			−0.405442	+	0.914121i	$0.632883\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.92820	+	6.00000i	−0.789542	+	0.683763i
$78$		0			0
$79$		−	3.46410i		−	0.389742i	−0.980829	−	0.194871i	$-0.937571\pi$
$79$		−	3.46410i		−	0.389742i	0.980829	−	0.194871i	$-0.0624288\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$		−	6.00000i		−	0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$		−	6.00000i		−	0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	12.0000i		−	1.28654i
$88$		0			0
$89$			6.92820i			0.734388i	0.930144	+	0.367194i	$0.119682\pi$
$89$			6.92820i			0.734388i	−0.930144	+	0.367194i	$0.880318\pi$
$90$		0			0
$91$	−6.00000	−	6.92820i	−0.628971	−	0.726273i
$92$		0			0
$93$		−	16.0000i		−	1.65912i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	13.8564			1.40690			0.703452	−	0.710742i	$-0.251641\pi$
$97$	13.8564			1.40690			0.703452	+	0.710742i	$0.251641\pi$
$98$		0			0
$99$		−	3.46410i		−	0.348155i
$100$		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
$103$			4.00000i			0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$			4.00000i			0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.3923			−1.00466			−0.502331	−	0.864675i	$-0.667524\pi$
$107$	−10.3923			−1.00466			−0.502331	+	0.864675i	$0.667524\pi$
$108$		0			0
$109$	−14.0000			−1.34096			−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000			−1.34096			−0.670478	+	0.741929i	$0.733911\pi$
$110$		0			0
$111$	−4.00000			−0.379663
$112$		0			0
$113$		−	18.0000i		−	1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$		−	18.0000i		−	1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	3.46410			0.320256
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	13.8564			1.24939
$124$		0			0
$125$		0			0
$126$		0			0
$127$	10.3923			0.922168			0.461084	−	0.887357i	$-0.347461\pi$
$127$	10.3923			0.922168			0.461084	+	0.887357i	$0.347461\pi$
$128$		0			0
$129$			20.7846i			1.82998i
$130$		0			0
$131$	18.0000			1.57267			0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000			1.57267			0.786334	+	0.617802i	$0.211977\pi$
$132$		0			0
$133$	−3.46410	−	4.00000i	−0.300376	−	0.346844i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$	2.00000			0.169638			0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000			0.169638			0.0848189	+	0.996396i	$0.472969\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	12.0000i		−	1.00349i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−13.8564	−	2.00000i	−1.14286	−	0.164957i
$148$		0			0
$149$	−6.00000			−0.491539			−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000			−0.491539			−0.245770	+	0.969328i	$0.579041\pi$
$150$		0			0
$151$		−	3.46410i		−	0.281905i	−0.990016	−	0.140952i	$-0.954984\pi$
$151$		−	3.46410i		−	0.281905i	0.990016	−	0.140952i	$-0.0450164\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−10.3923			−0.829396			−0.414698	−	0.909959i	$-0.636113\pi$
$157$	−10.3923			−0.829396			−0.414698	+	0.909959i	$0.636113\pi$
$158$		0			0
$159$	−12.0000			−0.951662
$160$		0			0
$161$	6.00000	+	6.92820i	0.472866	+	0.546019i
$162$		0			0
$163$	−17.3205			−1.35665			−0.678323	−	0.734763i	$-0.737293\pi$
$163$	−17.3205			−1.35665			−0.678323	+	0.734763i	$0.737293\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	2.00000			0.152944
$172$		0			0
$173$	−3.46410			−0.263371			−0.131685	−	0.991292i	$-0.542039\pi$
$173$	−3.46410			−0.263371			−0.131685	+	0.991292i	$0.542039\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	12.0000i		−	0.901975i
$178$		0			0
$179$			10.3923i			0.776757i	0.921500	+	0.388379i	$0.126965\pi$
$179$			10.3923i			0.776757i	−0.921500	+	0.388379i	$0.873035\pi$
$180$		0			0
$181$		−	17.3205i		−	1.28742i	−0.765268	−	0.643712i	$-0.777394\pi$
$181$		−	17.3205i		−	1.28742i	0.765268	−	0.643712i	$-0.222606\pi$
$182$		0			0
$183$	−6.92820			−0.512148
$184$		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
$191$		−	24.2487i		−	1.75458i	−0.479965	−	0.877288i	$-0.659351\pi$
$191$		−	24.2487i		−	1.75458i	0.479965	−	0.877288i	$-0.340649\pi$
$192$		0			0
$193$		−	2.00000i		−	0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$		−	2.00000i		−	0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$			6.92820i			0.488678i
$202$		0			0
$203$	−10.3923	−	12.0000i	−0.729397	−	0.842235i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−3.46410			−0.240772
$208$		0			0
$209$		−	6.92820i		−	0.479234i
$210$		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
$213$	−6.92820			−0.474713
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−13.8564	−	16.0000i	−0.940634	−	1.08615i
$218$		0			0
$219$		−	13.8564i		−	0.936329i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			6.00000i			0.398234i	0.979976	+	0.199117i	$0.0638074\pi$
$227$			6.00000i			0.398234i	−0.979976	+	0.199117i	$0.936193\pi$
$228$		0			0
$229$		−	10.3923i		−	0.686743i	−0.939200	−	0.343371i	$-0.888431\pi$
$229$		−	10.3923i		−	0.686743i	0.939200	−	0.343371i	$-0.111569\pi$
$230$		0			0
$231$	−12.0000	−	13.8564i	−0.789542	−	0.911685i
$232$		0			0
$233$		−	6.00000i		−	0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$		−	6.00000i		−	0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	6.92820			0.450035
$238$		0			0
$239$		−	10.3923i		−	0.672222i	−0.941822	−	0.336111i	$-0.890888\pi$
$239$		−	10.3923i		−	0.672222i	0.941822	−	0.336111i	$-0.109112\pi$
$240$		0			0
$241$			27.7128i			1.78514i	0.450910	+	0.892570i	$0.351100\pi$
$241$			27.7128i			1.78514i	−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$		−	10.0000i		−	0.641500i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	6.92820			0.440831
$248$		0			0
$249$	12.0000			0.760469
$250$		0			0
$251$	30.0000			1.89358			0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000			1.89358			0.946792	+	0.321847i	$0.104304\pi$
$252$		0			0
$253$			12.0000i			0.754434i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	27.7128			1.72868			0.864339	−	0.502910i	$-0.167737\pi$
$257$	27.7128			1.72868			0.864339	+	0.502910i	$0.167737\pi$
$258$		0			0
$259$	−4.00000	+	3.46410i	−0.248548	+	0.215249i
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$	24.2487			1.49524			0.747620	−	0.664127i	$-0.231197\pi$
$263$	24.2487			1.49524			0.747620	+	0.664127i	$0.231197\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−13.8564			−0.847998
$268$		0			0
$269$		−	3.46410i		−	0.211210i	−0.994408	−	0.105605i	$-0.966322\pi$
$269$		−	3.46410i		−	0.211210i	0.994408	−	0.105605i	$-0.0336779\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$		0			0
$273$	13.8564	−	12.0000i	0.838628	−	0.726273i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			2.00000i			0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$			2.00000i			0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$		0			0
$279$	8.00000			0.478947
$280$		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
$283$			14.0000i			0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$			14.0000i			0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	13.8564	−	12.0000i	0.817918	−	0.708338i
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$			27.7128i			1.62455i
$292$		0			0
$293$	−10.3923			−0.607125			−0.303562	−	0.952812i	$-0.598176\pi$
$293$	−10.3923			−0.607125			−0.303562	+	0.952812i	$0.598176\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−13.8564			−0.804030
$298$		0			0
$299$	−12.0000			−0.693978
$300$		0			0
$301$	18.0000	+	20.7846i	1.03750	+	1.19800i
$302$		0			0
$303$	−20.7846			−1.19404
$304$		0			0
$305$		0			0
$306$		0			0
$307$			22.0000i			1.25561i	0.778372	+	0.627803i	$0.216046\pi$
$307$			22.0000i			1.25561i	−0.778372	+	0.627803i	$0.783954\pi$
$308$		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
$313$	−6.92820			−0.391605			−0.195803	−	0.980643i	$-0.562731\pi$
$313$	−6.92820			−0.391605			−0.195803	+	0.980643i	$0.562731\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	6.00000i		−	0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$		−	6.00000i		−	0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$		0			0
$319$		−	20.7846i		−	1.16371i
$320$		0			0
$321$		−	20.7846i		−	1.16008i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	28.0000i		−	1.54840i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			3.46410i			0.190404i	0.995458	+	0.0952021i	$0.0303497\pi$
$331$			3.46410i			0.190404i	−0.995458	+	0.0952021i	$0.969650\pi$
$332$		0			0
$333$		−	2.00000i		−	0.109599i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	14.0000i		−	0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$		−	14.0000i		−	0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$		0			0
$339$	36.0000			1.95525
$340$		0			0
$341$		−	27.7128i		−	1.50073i
$342$		0			0
$343$	−15.5885	+	10.0000i	−0.841698	+	0.539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.3205			0.929814			0.464907	−	0.885360i	$-0.346088\pi$
$347$	17.3205			0.929814			0.464907	+	0.885360i	$0.346088\pi$
$348$		0			0
$349$		−	3.46410i		−	0.185429i	−0.995693	−	0.0927146i	$-0.970446\pi$
$349$		−	3.46410i		−	0.185429i	0.995693	−	0.0927146i	$-0.0295544\pi$
$350$		0			0
$351$		−	13.8564i		−	0.739600i
$352$		0			0
$353$	27.7128			1.47500			0.737502	−	0.675345i	$-0.236005\pi$
$353$	27.7128			1.47500			0.737502	+	0.675345i	$0.236005\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			31.1769i			1.64545i	0.568436	+	0.822727i	$0.307549\pi$
$359$			31.1769i			1.64545i	−0.568436	+	0.822727i	$0.692451\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		−	2.00000i		−	0.104973i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	8.00000i		−	0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$		−	8.00000i		−	0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$		0			0
$369$			6.92820i			0.360668i
$370$		0			0
$371$	−12.0000	+	10.3923i	−0.623009	+	0.539542i
$372$		0			0
$373$			14.0000i			0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$			14.0000i			0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	20.7846			1.07046
$378$		0			0
$379$			24.2487i			1.24557i	0.782392	+	0.622786i	$0.213999\pi$
$379$			24.2487i			1.24557i	−0.782392	+	0.622786i	$0.786001\pi$
$380$		0			0
$381$			20.7846i			1.06483i
$382$		0			0
$383$			24.0000i			1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$			24.0000i			1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−10.3923			−0.528271
$388$		0			0
$389$	18.0000			0.912636			0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000			0.912636			0.456318	+	0.889817i	$0.349168\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			36.0000i			1.81596i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−10.3923			−0.521575			−0.260787	−	0.965396i	$-0.583982\pi$
$397$	−10.3923			−0.521575			−0.260787	+	0.965396i	$0.583982\pi$
$398$		0			0
$399$	8.00000	−	6.92820i	0.400501	−	0.346844i
$400$		0			0
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$		0			0
$403$	27.7128			1.38047
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−6.92820			−0.343418
$408$		0			0
$409$		−	6.92820i		−	0.342578i	−0.985221	−	0.171289i	$-0.945207\pi$
$409$		−	6.92820i		−	0.342578i	0.985221	−	0.171289i	$-0.0547931\pi$
$410$		0			0
$411$	−12.0000			−0.591916
$412$		0			0
$413$	−10.3923	−	12.0000i	−0.511372	−	0.590481i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			4.00000i			0.195881i
$418$		0			0
$419$	30.0000			1.46560			0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000			1.46560			0.732798	+	0.680446i	$0.238214\pi$
$420$		0			0
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−6.92820	+	6.00000i	−0.335279	+	0.290360i
$428$		0			0
$429$	24.0000			1.15873
$430$		0			0
$431$			38.1051i			1.83546i	0.397206	+	0.917729i	$0.369980\pi$
$431$			38.1051i			1.83546i	−0.397206	+	0.917729i	$0.630020\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.92820			−0.331421
$438$		0			0
$439$	28.0000			1.33637			0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000			1.33637			0.668184	+	0.743996i	$0.267072\pi$
$440$		0			0
$441$	1.00000	−	6.92820i	0.0476190	−	0.329914i
$442$		0			0
$443$	−3.46410			−0.164584			−0.0822922	−	0.996608i	$-0.526224\pi$
$443$	−3.46410			−0.164584			−0.0822922	+	0.996608i	$0.526224\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	12.0000i		−	0.567581i
$448$		0			0
$449$	30.0000			1.41579			0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000			1.41579			0.707894	+	0.706319i	$0.249646\pi$
$450$		0			0
$451$	24.0000			1.13012
$452$		0			0
$453$	6.92820			0.325515
$454$		0			0
$455$		0			0
$456$		0			0
$457$			34.0000i			1.59045i	0.606313	+	0.795226i	$0.292648\pi$
$457$			34.0000i			1.59045i	−0.606313	+	0.795226i	$0.707352\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	38.1051i		−	1.77473i	−0.461065	−	0.887366i	$-0.652533\pi$
$461$		−	38.1051i		−	1.77473i	0.461065	−	0.887366i	$-0.347467\pi$
$462$		0			0
$463$	24.2487			1.12693			0.563467	−	0.826139i	$-0.309467\pi$
$463$	24.2487			1.12693			0.563467	+	0.826139i	$0.309467\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			30.0000i			1.38823i	0.719862	+	0.694117i	$0.244205\pi$
$467$			30.0000i			1.38823i	−0.719862	+	0.694117i	$0.755795\pi$
$468$		0			0
$469$	6.00000	+	6.92820i	0.277054	+	0.319915i
$470$		0			0
$471$		−	20.7846i		−	0.957704i
$472$		0			0
$473$			36.0000i			1.65528i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	6.00000i		−	0.274721i
$478$		0			0
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$		0			0
$481$		−	6.92820i		−	0.315899i
$482$		0			0
$483$	−13.8564	+	12.0000i	−0.630488	+	0.546019i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	24.2487			1.09881			0.549407	−	0.835555i	$-0.314854\pi$
$487$	24.2487			1.09881			0.549407	+	0.835555i	$0.314854\pi$
$488$		0			0
$489$		−	34.6410i		−	1.56652i
$490$		0			0
$491$			17.3205i			0.781664i	0.920462	+	0.390832i	$0.127813\pi$
$491$			17.3205i			0.781664i	−0.920462	+	0.390832i	$0.872187\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.92820	+	6.00000i	−0.310772	+	0.269137i
$498$		0			0
$499$			10.3923i			0.465223i	0.972570	+	0.232612i	$0.0747271\pi$
$499$			10.3923i			0.465223i	−0.972570	+	0.232612i	$0.925273\pi$
$500$		0			0
$501$	−24.0000			−1.07224
$502$		0			0
$503$			36.0000i			1.60516i	0.596544	+	0.802580i	$0.296540\pi$
$503$			36.0000i			1.60516i	−0.596544	+	0.802580i	$0.703460\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	2.00000i		−	0.0888231i
$508$		0			0
$509$		−	31.1769i		−	1.38189i	−0.722906	−	0.690946i	$-0.757194\pi$
$509$		−	31.1769i		−	1.38189i	0.722906	−	0.690946i	$-0.242806\pi$
$510$		0			0
$511$	−12.0000	−	13.8564i	−0.530849	−	0.612971i
$512$		0			0
$513$		−	8.00000i		−	0.353209i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	6.92820i		−	0.304114i
$520$		0			0
$521$		−	34.6410i		−	1.51765i	−0.651294	−	0.758825i	$-0.725774\pi$
$521$		−	34.6410i		−	1.51765i	0.651294	−	0.758825i	$-0.274226\pi$
$522$		0			0
$523$		−	2.00000i		−	0.0874539i	−0.999044	−	0.0437269i	$-0.986077\pi$
$523$		−	2.00000i		−	0.0874539i	0.999044	−	0.0437269i	$-0.0139232\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−11.0000			−0.478261
$530$		0			0
$531$	6.00000			0.260378
$532$		0			0
$533$			24.0000i			1.03956i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−20.7846			−0.896922
$538$		0			0
$539$	−24.0000	−	3.46410i	−1.03375	−	0.149209i
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$	34.6410			1.48659
$544$		0			0
$545$		0			0
$546$		0			0
$547$	17.3205			0.740571			0.370286	−	0.928918i	$-0.379260\pi$
$547$	17.3205			0.740571			0.370286	+	0.928918i	$0.379260\pi$
$548$		0			0
$549$		−	3.46410i		−	0.147844i
$550$		0			0
$551$	12.0000			0.511217
$552$		0			0
$553$	6.92820	−	6.00000i	0.294617	−	0.255146i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			18.0000i			0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$			18.0000i			0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$		0			0
$559$	−36.0000			−1.52264
$560$		0			0
$561$		0			0
$562$		0			0
$563$			18.0000i			0.758610i	0.925272	+	0.379305i	$0.123837\pi$
$563$			18.0000i			0.758610i	−0.925272	+	0.379305i	$0.876163\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−19.0526	−	22.0000i	−0.800132	−	0.923913i
$568$		0			0
$569$	18.0000			0.754599			0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000			0.754599			0.377300	+	0.926091i	$0.376853\pi$
$570$		0			0
$571$			45.0333i			1.88459i	0.334790	+	0.942293i	$0.391335\pi$
$571$			45.0333i			1.88459i	−0.334790	+	0.942293i	$0.608665\pi$
$572$		0			0
$573$	48.4974			2.02601
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−27.7128			−1.15370			−0.576850	−	0.816850i	$-0.695718\pi$
$577$	−27.7128			−1.15370			−0.576850	+	0.816850i	$0.695718\pi$
$578$		0			0
$579$	4.00000			0.166234
$580$		0			0
$581$	12.0000	−	10.3923i	0.497844	−	0.431145i
$582$		0			0
$583$	−20.7846			−0.860811
$584$		0			0
$585$		0			0
$586$		0			0
$587$			18.0000i			0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$			18.0000i			0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$		0			0
$589$	16.0000			0.659269
$590$		0			0
$591$	−36.0000			−1.48084
$592$		0			0
$593$	13.8564			0.569014			0.284507	−	0.958674i	$-0.408170\pi$
$593$	13.8564			0.569014			0.284507	+	0.958674i	$0.408170\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	40.0000i		−	1.63709i
$598$		0			0
$599$			24.2487i			0.990775i	0.868672	+	0.495388i	$0.164974\pi$
$599$			24.2487i			0.990775i	−0.868672	+	0.495388i	$0.835026\pi$
$600$		0			0
$601$			20.7846i			0.847822i	0.905704	+	0.423911i	$0.139343\pi$
$601$			20.7846i			0.847822i	−0.905704	+	0.423911i	$0.860657\pi$
$602$		0			0
$603$	−3.46410			−0.141069
$604$		0			0
$605$		0			0
$606$		0			0
$607$			16.0000i			0.649420i	0.945814	+	0.324710i	$0.105267\pi$
$607$			16.0000i			0.649420i	−0.945814	+	0.324710i	$0.894733\pi$
$608$		0			0
$609$	24.0000	−	20.7846i	0.972529	−	0.842235i
$610$		0			0
$611$		0			0
$612$		0			0
$613$			38.0000i			1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$			38.0000i			1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			18.0000i			0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$			18.0000i			0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$		0			0
$619$	26.0000			1.04503			0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000			1.04503			0.522514	+	0.852631i	$0.324994\pi$
$620$		0			0
$621$			13.8564i			0.556038i
$622$		0			0
$623$	−13.8564	+	12.0000i	−0.555145	+	0.480770i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	13.8564			0.553372
$628$		0			0
$629$		0			0
$630$		0			0
$631$			3.46410i			0.137904i	0.997620	+	0.0689519i	$0.0219655\pi$
$631$			3.46410i			0.137904i	−0.997620	+	0.0689519i	$0.978035\pi$
$632$		0			0
$633$	−6.92820			−0.275371
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.46410	−	24.0000i	0.137253	−	0.950915i
$638$		0			0
$639$		−	3.46410i		−	0.137038i
$640$		0			0
$641$	30.0000			1.18493			0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000			1.18493			0.592464	+	0.805597i	$0.298155\pi$
$642$		0			0
$643$			26.0000i			1.02534i	0.858586	+	0.512670i	$0.171344\pi$
$643$			26.0000i			1.02534i	−0.858586	+	0.512670i	$0.828656\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			12.0000i			0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$			12.0000i			0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$		0			0
$649$		−	20.7846i		−	0.815867i
$650$		0			0
$651$	32.0000	−	27.7128i	1.25418	−	1.08615i
$652$		0			0
$653$		−	18.0000i		−	0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$		−	18.0000i		−	0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	6.92820			0.270295
$658$		0			0
$659$		−	3.46410i		−	0.134942i	−0.997721	−	0.0674711i	$-0.978507\pi$
$659$		−	3.46410i		−	0.134942i	0.997721	−	0.0674711i	$-0.0214931\pi$
$660$		0			0
$661$		−	3.46410i		−	0.134738i	−0.997728	−	0.0673690i	$-0.978540\pi$
$661$		−	3.46410i		−	0.134738i	0.997728	−	0.0673690i	$-0.0214605\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−20.7846			−0.804783
$668$		0			0
$669$	32.0000			1.23719
$670$		0			0
$671$	−12.0000			−0.463255
$672$		0			0
$673$		−	14.0000i		−	0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$		−	14.0000i		−	0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.46410			−0.133136			−0.0665681	−	0.997782i	$-0.521205\pi$
$677$	−3.46410			−0.133136			−0.0665681	+	0.997782i	$0.521205\pi$
$678$		0			0
$679$	24.0000	+	27.7128i	0.921035	+	1.06352i
$680$		0			0
$681$	−12.0000			−0.459841
$682$		0			0
$683$	38.1051			1.45805			0.729026	−	0.684486i	$-0.239973\pi$
$683$	38.1051			1.45805			0.729026	+	0.684486i	$0.239973\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	20.7846			0.792982
$688$		0			0
$689$		−	20.7846i		−	0.791831i
$690$		0			0
$691$	10.0000			0.380418			0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000			0.380418			0.190209	+	0.981744i	$0.439083\pi$
$692$		0			0
$693$	6.92820	−	6.00000i	0.263181	−	0.227921i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	12.0000			0.453882
$700$		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
$703$		−	4.00000i		−	0.150863i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−20.7846	+	18.0000i	−0.781686	+	0.676960i
$708$		0			0
$709$	10.0000			0.375558			0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000			0.375558			0.187779	+	0.982211i	$0.439871\pi$
$710$		0			0
$711$			3.46410i			0.129914i
$712$		0			0
$713$	−27.7128			−1.03785
$714$		0			0
$715$		0			0
$716$		0			0
$717$	20.7846			0.776215
$718$		0			0
$719$	48.0000			1.79010			0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000			1.79010			0.895049	+	0.445968i	$0.147140\pi$
$720$		0			0
$721$	−8.00000	+	6.92820i	−0.297936	+	0.258020i
$722$		0			0
$723$	−55.4256			−2.06130
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	28.0000i		−	1.03846i	−0.854634	−	0.519231i	$-0.826218\pi$
$727$		−	28.0000i		−	1.03846i	0.854634	−	0.519231i	$-0.173782\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$		0			0
$732$		0			0
$733$	51.9615			1.91924			0.959621	−	0.281295i	$-0.0907639\pi$
$733$	51.9615			1.91924			0.959621	+	0.281295i	$0.0907639\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			12.0000i			0.442026i
$738$		0			0
$739$			10.3923i			0.382287i	0.981562	+	0.191144i	$0.0612196\pi$
$739$			10.3923i			0.382287i	−0.981562	+	0.191144i	$0.938780\pi$
$740$		0			0
$741$			13.8564i			0.509028i
$742$		0			0
$743$	−24.2487			−0.889599			−0.444799	−	0.895630i	$-0.646725\pi$
$743$	−24.2487			−0.889599			−0.444799	+	0.895630i	$0.646725\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			6.00000i			0.219529i
$748$		0			0
$749$	−18.0000	−	20.7846i	−0.657706	−	0.759453i
$750$		0			0
$751$		−	45.0333i		−	1.64329i	−0.570000	−	0.821645i	$-0.693057\pi$
$751$		−	45.0333i		−	1.64329i	0.570000	−	0.821645i	$-0.306943\pi$
$752$		0			0
$753$			60.0000i			2.18652i
$754$		0			0
$755$		0			0
$756$		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
$759$	−24.0000			−0.871145
$760$		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
$763$	−24.2487	−	28.0000i	−0.877862	−	1.01367i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	20.7846			0.750489
$768$		0			0
$769$		−	13.8564i		−	0.499675i	−0.968288	−	0.249837i	$-0.919623\pi$
$769$		−	13.8564i		−	0.499675i	0.968288	−	0.249837i	$-0.0803772\pi$
$770$		0			0
$771$			55.4256i			1.99611i
$772$		0			0
$773$	3.46410			0.124595			0.0622975	−	0.998058i	$-0.480157\pi$
$773$	3.46410			0.124595			0.0622975	+	0.998058i	$0.480157\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−6.92820	−	8.00000i	−0.248548	−	0.286998i
$778$		0			0
$779$			13.8564i			0.496457i
$780$		0			0
$781$	−12.0000			−0.429394
$782$		0			0
$783$		−	24.0000i		−	0.857690i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	34.0000i		−	1.21197i	−0.795476	−	0.605985i	$-0.792779\pi$
$787$		−	34.0000i		−	1.21197i	0.795476	−	0.605985i	$-0.207221\pi$
$788$		0			0
$789$			48.4974i			1.72655i
$790$		0			0
$791$	36.0000	−	31.1769i	1.28001	−	1.10852i
$792$		0			0
$793$		−	12.0000i		−	0.426132i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.3205			0.613524			0.306762	−	0.951786i	$-0.400754\pi$
$797$	17.3205			0.613524			0.306762	+	0.951786i	$0.400754\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		−	6.92820i		−	0.244796i
$802$		0			0
$803$		−	24.0000i		−	0.846942i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	6.92820			0.243884
$808$		0			0
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$	22.0000			0.772524			0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000			0.772524			0.386262	+	0.922389i	$0.373766\pi$
$812$		0			0
$813$		−	16.0000i		−	0.561144i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−20.7846			−0.727161
$818$		0			0
$819$	6.00000	+	6.92820i	0.209657	+	0.242091i
$820$		0			0
$821$	30.0000			1.04701			0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000			1.04701			0.523504	+	0.852023i	$0.324625\pi$
$822$		0			0
$823$	−3.46410			−0.120751			−0.0603755	−	0.998176i	$-0.519230\pi$
$823$	−3.46410			−0.120751			−0.0603755	+	0.998176i	$0.519230\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24.2487			−0.843210			−0.421605	−	0.906780i	$-0.638533\pi$
$827$	−24.2487			−0.843210			−0.421605	+	0.906780i	$0.638533\pi$
$828$		0			0
$829$		−	3.46410i		−	0.120313i	−0.998189	−	0.0601566i	$-0.980840\pi$
$829$		−	3.46410i		−	0.120313i	0.998189	−	0.0601566i	$-0.0191600\pi$
$830$		0			0
$831$	−4.00000			−0.138758
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	32.0000i		−	1.10608i
$838$		0			0
$839$	12.0000			0.414286			0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000			0.414286			0.207143	+	0.978311i	$0.433583\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		−	12.0000i		−	0.413302i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−1.73205	−	2.00000i	−0.0595140	−	0.0687208i
$848$		0			0
$849$	−28.0000			−0.960958
$850$		0			0
$851$			6.92820i			0.237496i
$852$		0			0
$853$	3.46410			0.118609			0.0593043	−	0.998240i	$-0.481112\pi$
$853$	3.46410			0.118609			0.0593043	+	0.998240i	$0.481112\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	34.6410			1.18331			0.591657	−	0.806190i	$-0.298474\pi$
$857$	34.6410			1.18331			0.591657	+	0.806190i	$0.298474\pi$
$858$		0			0
$859$	−14.0000			−0.477674			−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000			−0.477674			−0.238837	+	0.971060i	$0.576766\pi$
$860$		0			0
$861$	24.0000	+	27.7128i	0.817918	+	0.944450i
$862$		0			0
$863$	−31.1769			−1.06127			−0.530637	−	0.847599i	$-0.678047\pi$
$863$	−31.1769			−1.06127			−0.530637	+	0.847599i	$0.678047\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	34.0000i		−	1.15470i
$868$		0			0
$869$	12.0000			0.407072
$870$		0			0
$871$	−12.0000			−0.406604
$872$		0			0
$873$	−13.8564			−0.468968
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	38.0000i		−	1.28317i	−0.767052	−	0.641584i	$-0.778277\pi$
$877$		−	38.0000i		−	1.28317i	0.767052	−	0.641584i	$-0.221723\pi$
$878$		0			0
$879$		−	20.7846i		−	0.701047i
$880$		0			0
$881$			13.8564i			0.466834i	0.972377	+	0.233417i	$0.0749907\pi$
$881$			13.8564i			0.466834i	−0.972377	+	0.233417i	$0.925009\pi$
$882$		0			0
$883$	−3.46410			−0.116576			−0.0582882	−	0.998300i	$-0.518564\pi$
$883$	−3.46410			−0.116576			−0.0582882	+	0.998300i	$0.518564\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	12.0000i		−	0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$		−	12.0000i		−	0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$		0			0
$889$	18.0000	+	20.7846i	0.603701	+	0.697093i
$890$		0			0
$891$		−	38.1051i		−	1.27657i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	24.0000i		−	0.801337i
$898$		0			0
$899$	48.0000			1.60089
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−41.5692	+	36.0000i	−1.38334	+	1.19800i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−24.2487			−0.805165			−0.402583	−	0.915384i	$-0.631887\pi$
$907$	−24.2487			−0.805165			−0.402583	+	0.915384i	$0.631887\pi$
$908$		0			0
$909$		−	10.3923i		−	0.344691i
$910$		0			0
$911$			10.3923i			0.344312i	0.985070	+	0.172156i	$0.0550734\pi$
$911$			10.3923i			0.344312i	−0.985070	+	0.172156i	$0.944927\pi$
$912$		0			0
$913$	20.7846			0.687870
$914$		0			0
$915$		0			0
$916$		0			0
$917$	31.1769	+	36.0000i	1.02955	+	1.18882i
$918$		0			0
$919$		−	31.1769i		−	1.02843i	−0.857661	−	0.514216i	$-0.828083\pi$
$919$		−	31.1769i		−	1.02843i	0.857661	−	0.514216i	$-0.171917\pi$
$920$		0			0
$921$	−44.0000			−1.44985
$922$		0			0
$923$		−	12.0000i		−	0.394985i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	4.00000i		−	0.131377i
$928$		0			0
$929$		−	41.5692i		−	1.36384i	−0.731426	−	0.681921i	$-0.761145\pi$
$929$		−	41.5692i		−	1.36384i	0.731426	−	0.681921i	$-0.238855\pi$
$930$		0			0
$931$	2.00000	−	13.8564i	0.0655474	−	0.454125i
$932$		0			0
$933$		−	24.0000i		−	0.785725i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−48.4974			−1.58434			−0.792171	−	0.610299i	$-0.791049\pi$
$937$	−48.4974			−1.58434			−0.792171	+	0.610299i	$0.791049\pi$
$938$		0			0
$939$		−	13.8564i		−	0.452187i
$940$		0			0
$941$		−	24.2487i		−	0.790485i	−0.918577	−	0.395243i	$-0.870660\pi$
$941$		−	24.2487i		−	0.790485i	0.918577	−	0.395243i	$-0.129340\pi$
$942$		0			0
$943$		−	24.0000i		−	0.781548i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−24.2487			−0.787977			−0.393989	−	0.919115i	$-0.628905\pi$
$947$	−24.2487			−0.787977			−0.393989	+	0.919115i	$0.628905\pi$
$948$		0			0
$949$	24.0000			0.779073
$950$		0			0
$951$	12.0000			0.389127
$952$		0			0
$953$		−	6.00000i		−	0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$		−	6.00000i		−	0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	41.5692			1.34374
$958$		0			0
$959$	−12.0000	+	10.3923i	−0.387500	+	0.335585i
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$	10.3923			0.334887
$964$		0			0
$965$		0			0
$966$		0			0
$967$	24.2487			0.779786			0.389893	−	0.920860i	$-0.372512\pi$
$967$	24.2487			0.779786			0.389893	+	0.920860i	$0.372512\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−18.0000			−0.577647			−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000			−0.577647			−0.288824	+	0.957382i	$0.593264\pi$
$972$		0			0
$973$	3.46410	+	4.00000i	0.111054	+	0.128234i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			30.0000i			0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$			30.0000i			0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$		0			0
$979$	−24.0000			−0.767043
$980$		0			0
$981$	14.0000			0.446986
$982$		0			0
$983$		−	36.0000i		−	1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$		−	36.0000i		−	1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	36.0000			1.14473
$990$		0			0
$991$			3.46410i			0.110041i	0.998485	+	0.0550204i	$0.0175224\pi$
$991$			3.46410i			0.110041i	−0.998485	+	0.0550204i	$0.982478\pi$
$992$		0			0
$993$	−6.92820			−0.219860
$994$		0			0
$995$		0			0
$996$		0			0
$997$	51.9615			1.64564			0.822819	−	0.568304i	$-0.192400\pi$
$997$	51.9615			1.64564			0.822819	+	0.568304i	$0.192400\pi$
$998$		0			0
$999$	−8.00000			−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.e.b.2799.4		4
4.3	odd	2		2800.2.e.c.2799.1		4
5.2	odd	4		112.2.f.b.111.1	yes	2
5.3	odd	4		2800.2.k.b.2351.1		2
5.4	even	2	inner	2800.2.e.b.2799.1		4
7.6	odd	2		2800.2.e.c.2799.2		4
15.2	even	4		1008.2.b.b.559.2		2
20.3	even	4		2800.2.k.e.2351.2		2
20.7	even	4		112.2.f.a.111.1	✓	2
20.19	odd	2		2800.2.e.c.2799.4		4
28.27	even	2	inner	2800.2.e.b.2799.3		4
35.2	odd	12		784.2.p.a.31.1		2
35.12	even	12		784.2.p.f.31.1		2
35.13	even	4		2800.2.k.e.2351.1		2
35.17	even	12		784.2.p.e.607.1		2
35.27	even	4		112.2.f.a.111.2	yes	2
35.32	odd	12		784.2.p.b.607.1		2
35.34	odd	2		2800.2.e.c.2799.3		4
40.27	even	4		448.2.f.c.447.2		2
40.37	odd	4		448.2.f.a.447.2		2
60.47	odd	4		1008.2.b.g.559.2		2
80.27	even	4		1792.2.e.a.895.4		4
80.37	odd	4		1792.2.e.c.895.2		4
80.67	even	4		1792.2.e.a.895.1		4
80.77	odd	4		1792.2.e.c.895.3		4
105.62	odd	4		1008.2.b.g.559.1		2
120.77	even	4		4032.2.b.b.3583.1		2
120.107	odd	4		4032.2.b.h.3583.1		2
140.27	odd	4		112.2.f.b.111.2	yes	2
140.47	odd	12		784.2.p.b.31.1		2
140.67	even	12		784.2.p.f.607.1		2
140.83	odd	4		2800.2.k.b.2351.2		2
140.87	odd	12		784.2.p.a.607.1		2
140.107	even	12		784.2.p.e.31.1		2
140.139	even	2	inner	2800.2.e.b.2799.2		4
280.27	odd	4		448.2.f.a.447.1		2
280.237	even	4		448.2.f.c.447.1		2
420.167	even	4		1008.2.b.b.559.1		2
560.27	odd	4		1792.2.e.c.895.1		4
560.237	even	4		1792.2.e.a.895.2		4
560.307	odd	4		1792.2.e.c.895.4		4
560.517	even	4		1792.2.e.a.895.3		4
840.587	even	4		4032.2.b.b.3583.2		2
840.797	odd	4		4032.2.b.h.3583.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.f.a.111.1	✓	2	20.7	even	4
112.2.f.a.111.2	yes	2	35.27	even	4
112.2.f.b.111.1	yes	2	5.2	odd	4
112.2.f.b.111.2	yes	2	140.27	odd	4
448.2.f.a.447.1		2	280.27	odd	4
448.2.f.a.447.2		2	40.37	odd	4
448.2.f.c.447.1		2	280.237	even	4
448.2.f.c.447.2		2	40.27	even	4
784.2.p.a.31.1		2	35.2	odd	12
784.2.p.a.607.1		2	140.87	odd	12
784.2.p.b.31.1		2	140.47	odd	12
784.2.p.b.607.1		2	35.32	odd	12
784.2.p.e.31.1		2	140.107	even	12
784.2.p.e.607.1		2	35.17	even	12
784.2.p.f.31.1		2	35.12	even	12
784.2.p.f.607.1		2	140.67	even	12
1008.2.b.b.559.1		2	420.167	even	4
1008.2.b.b.559.2		2	15.2	even	4
1008.2.b.g.559.1		2	105.62	odd	4
1008.2.b.g.559.2		2	60.47	odd	4
1792.2.e.a.895.1		4	80.67	even	4
1792.2.e.a.895.2		4	560.237	even	4
1792.2.e.a.895.3		4	560.517	even	4
1792.2.e.a.895.4		4	80.27	even	4
1792.2.e.c.895.1		4	560.27	odd	4
1792.2.e.c.895.2		4	80.37	odd	4
1792.2.e.c.895.3		4	80.77	odd	4
1792.2.e.c.895.4		4	560.307	odd	4
2800.2.e.b.2799.1		4	5.4	even	2	inner
2800.2.e.b.2799.2		4	140.139	even	2	inner
2800.2.e.b.2799.3		4	28.27	even	2	inner
2800.2.e.b.2799.4		4	1.1	even	1	trivial
2800.2.e.c.2799.1		4	4.3	odd	2
2800.2.e.c.2799.2		4	7.6	odd	2
2800.2.e.c.2799.3		4	35.34	odd	2
2800.2.e.c.2799.4		4	20.19	odd	2
2800.2.k.b.2351.1		2	5.3	odd	4
2800.2.k.b.2351.2		2	140.83	odd	4
2800.2.k.e.2351.1		2	35.13	even	4
2800.2.k.e.2351.2		2	20.3	even	4
4032.2.b.b.3583.1		2	120.77	even	4
4032.2.b.b.3583.2		2	840.587	even	4
4032.2.b.h.3583.1		2	120.107	odd	4
4032.2.b.h.3583.2		2	840.797	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:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +(1.73205 + 2.00000i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{3} +(1.73205 + 2.00000i) q^{7} -1.00000 q^{9} +3.46410i q^{11} -3.46410 q^{13} -2.00000 q^{19} +(-4.00000 + 3.46410i) q^{21} +3.46410 q^{23} +4.00000i q^{27} -6.00000 q^{29} -8.00000 q^{31} -6.92820 q^{33} +2.00000i q^{37} -6.92820i q^{39} -6.92820i q^{41} +10.3923 q^{43} +(-1.00000 + 6.92820i) q^{49} +6.00000i q^{53} -4.00000i q^{57} -6.00000 q^{59} +3.46410i q^{61} +(-1.73205 - 2.00000i) q^{63} +3.46410 q^{67} +6.92820i q^{69} +3.46410i q^{71} -6.92820 q^{73} +(-6.92820 + 6.00000i) q^{77} -3.46410i q^{79} -11.0000 q^{81} -6.00000i q^{83} -12.0000i q^{87} +6.92820i q^{89} +(-6.00000 - 6.92820i) q^{91} -16.0000i q^{93} +13.8564 q^{97} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 8 q^{19} - 16 q^{21} - 24 q^{29} - 32 q^{31} - 4 q^{49} - 24 q^{59} - 44 q^{81} - 24 q^{91}+O(q^{100}) \) 4 * q - 4 * q^9 - 8 * q^19 - 16 * q^21 - 24 * q^29 - 32 * q^31 - 4 * q^49 - 24 * q^59 - 44 * q^81 - 24 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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