LMFDB - Embedded newform 5700.2.f.p.3649.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5700,2,Mod(3649,5700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5700, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5700.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5700.f (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:	$45.5147291521$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.37161216.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 15x^{4} + 51x^{2} + 1 $ x^6 + 15x^4 + 51x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.3
Root			$-3.13264i$ of defining polynomial
Character	$\chi$	$=$	5700.3649
Dual form			5700.2.f.p.3649.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +4.81342i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +4.81342i q^{7} -1.00000 q^{9} -1.45186 q^{11} -2.81342i q^{13} -6.26527i q^{17} -1.00000 q^{19} +4.81342 q^{21} +6.26527i q^{23} +1.00000i q^{27} -0.548143 q^{29} +8.26527 q^{31} +1.45186i q^{33} -6.81342i q^{37} -2.81342 q^{39} +4.54814 q^{41} -7.71713i q^{43} +10.2653i q^{47} -16.1690 q^{49} -6.26527 q^{51} -0.265275i q^{53} +1.00000i q^{57} -2.90371 q^{59} +11.3616 q^{61} -4.81342i q^{63} +6.26527 q^{69} +6.90371 q^{71} -6.53055i q^{73} -6.98840i q^{77} -10.7231 q^{79} +1.00000 q^{81} +9.16899i q^{83} +0.548143i q^{87} +18.7055 q^{89} +13.5422 q^{91} -8.26527i q^{93} +6.81342i q^{97} +1.45186 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^9	$ 6 q - 6 q^{9} - 4 q^{11} - 6 q^{19} - 8 q^{29} + 16 q^{31} + 12 q^{39} + 32 q^{41} - 54 q^{49} - 4 q^{51} - 8 q^{59} + 44 q^{61} + 4 q^{69} + 32 q^{71} - 16 q^{79} + 6 q^{81} - 8 q^{89} + 96 q^{91} + 4 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 - 4 * q^11 - 6 * q^19 - 8 * q^29 + 16 * q^31 + 12 * q^39 + 32 * q^41 - 54 * q^49 - 4 * q^51 - 8 * q^59 + 44 * q^61 + 4 * q^69 + 32 * q^71 - 16 * q^79 + 6 * q^81 - 8 * q^89 + 96 * q^91 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5700\mathbb{Z}\right)^\times$.

$n$	$1901$	$2851$	$3877$	$4201$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.81342i	1.81930i	0.415375	+	0.909650i	$0.363650\pi$
$7$	4.81342i	1.81930i	−0.415375	+	0.909650i	$0.636350\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.45186	−0.437751	−0.218876	−	0.975753i	$-0.570239\pi$
$11$	−1.45186	−0.437751	−0.218876	+	0.975753i	$0.570239\pi$
$12$	0	0
$13$	− 2.81342i	− 0.780302i	−0.920751	−	0.390151i	$-0.872423\pi$
$13$	− 2.81342i	− 0.780302i	0.920751	−	0.390151i	$-0.127577\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.26527i	− 1.51955i	−0.650185	−	0.759776i	$-0.725308\pi$
$17$	− 6.26527i	− 1.51955i	0.650185	−	0.759776i	$-0.274692\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	4.81342	1.05037
$22$	0	0
$23$	6.26527i	1.30640i	0.757185	+	0.653200i	$0.226574\pi$
$23$	6.26527i	1.30640i	−0.757185	+	0.653200i	$0.773426\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−0.548143	−0.101788	−0.0508938	−	0.998704i	$-0.516207\pi$
$29$	−0.548143	−0.101788	−0.0508938	+	0.998704i	$0.516207\pi$
$30$	0	0
$31$	8.26527	1.48449	0.742244	−	0.670130i	$-0.233762\pi$
$31$	8.26527	1.48449	0.742244	+	0.670130i	$0.233762\pi$
$32$	0	0
$33$	1.45186i	0.252736i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.81342i	− 1.12012i	−0.828452	−	0.560059i	$-0.810778\pi$
$37$	− 6.81342i	− 1.12012i	0.828452	−	0.560059i	$-0.189222\pi$
$38$	0	0
$39$	−2.81342	−0.450507
$40$	0	0
$41$	4.54814	0.710301	0.355150	−	0.934809i	$-0.384430\pi$
$41$	4.54814	0.710301	0.355150	+	0.934809i	$0.384430\pi$
$42$	0	0
$43$	− 7.71713i	− 1.17685i	−0.808551	−	0.588426i	$-0.799748\pi$
$43$	− 7.71713i	− 1.17685i	0.808551	−	0.588426i	$-0.200252\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.2653i	1.49734i	0.662940	+	0.748672i	$0.269308\pi$
$47$	10.2653i	1.49734i	−0.662940	+	0.748672i	$0.730692\pi$
$48$	0	0
$49$	−16.1690	−2.30986
$50$	0	0
$51$	−6.26527	−0.877314
$52$	0	0
$53$	− 0.265275i	− 0.0364383i	−0.999834	−	0.0182192i	$-0.994200\pi$
$53$	− 0.265275i	− 0.0364383i	0.999834	−	0.0182192i	$-0.00579966\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.00000i	0.132453i
$58$	0	0
$59$	−2.90371	−0.378031	−0.189016	−	0.981974i	$-0.560530\pi$
$59$	−2.90371	−0.378031	−0.189016	+	0.981974i	$0.560530\pi$
$60$	0	0
$61$	11.3616	1.45470	0.727349	−	0.686267i	$-0.240752\pi$
$61$	11.3616	1.45470	0.727349	+	0.686267i	$0.240752\pi$
$62$	0	0
$63$	− 4.81342i	− 0.606434i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	6.26527	0.754250
$70$	0	0
$71$	6.90371	0.819320	0.409660	−	0.912238i	$-0.365647\pi$
$71$	6.90371	0.819320	0.409660	+	0.912238i	$0.365647\pi$
$72$	0	0
$73$	− 6.53055i	− 0.764343i	−0.924091	−	0.382172i	$-0.875176\pi$
$73$	− 6.53055i	− 0.764343i	0.924091	−	0.382172i	$-0.124824\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 6.98840i	− 0.796402i
$78$	0	0
$79$	−10.7231	−1.20645	−0.603223	−	0.797573i	$-0.706117\pi$
$79$	−10.7231	−1.20645	−0.603223	+	0.797573i	$0.706117\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.16899i	1.00643i	0.864162	+	0.503214i	$0.167849\pi$
$83$	9.16899i	1.00643i	−0.864162	+	0.503214i	$0.832151\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.548143i	0.0587671i
$88$	0	0
$89$	18.7055	1.98278	0.991391	−	0.130935i	$-0.0417978\pi$
$89$	18.7055	1.98278	0.991391	+	0.130935i	$0.0417978\pi$
$90$	0	0
$91$	13.5422	1.41960
$92$	0	0
$93$	− 8.26527i	− 0.857069i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.81342i	0.691798i	0.938272	+	0.345899i	$0.112426\pi$
$97$	6.81342i	0.691798i	−0.938272	+	0.345899i	$0.887574\pi$
$98$	0	0
$99$	1.45186	0.145917
$100$	0	0
$101$	17.4343	1.73477	0.867387	−	0.497634i	$-0.165798\pi$
$101$	17.4343	1.73477	0.867387	+	0.497634i	$0.165798\pi$
$102$	0	0
$103$	16.5305i	1.62880i	0.580301	+	0.814402i	$0.302935\pi$
$103$	16.5305i	1.62880i	−0.580301	+	0.814402i	$0.697065\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0000i	1.54678i	0.633932	+	0.773389i	$0.281440\pi$
$107$	16.0000i	1.54678i	−0.633932	+	0.773389i	$0.718560\pi$
$108$	0	0
$109$	−3.09629	−0.296570	−0.148285	−	0.988945i	$-0.547375\pi$
$109$	−3.09629	−0.296570	−0.148285	+	0.988945i	$0.547375\pi$
$110$	0	0
$111$	−6.81342	−0.646701
$112$	0	0
$113$	13.3616i	1.25695i	0.777830	+	0.628475i	$0.216321\pi$
$113$	13.3616i	1.25695i	−0.777830	+	0.628475i	$0.783679\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.81342i	0.260101i
$118$	0	0
$119$	30.1574	2.76452
$120$	0	0
$121$	−8.89211	−0.808374
$122$	0	0
$123$	− 4.54814i	− 0.410092i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 4.00000i	− 0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	0	0
$129$	−7.71713	−0.679456
$130$	0	0
$131$	−4.35557	−0.380548	−0.190274	−	0.981731i	$-0.560938\pi$
$131$	−4.35557	−0.380548	−0.190274	+	0.981731i	$0.560938\pi$
$132$	0	0
$133$	− 4.81342i	− 0.417376i
$134$	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
$139$	2.90371	0.246290	0.123145	−	0.992389i	$-0.460702\pi$
$139$	2.90371	0.246290	0.123145	+	0.992389i	$0.460702\pi$
$140$	0	0
$141$	10.2653	0.864492
$142$	0	0
$143$	4.08468i	0.341578i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	16.1690i	1.33360i
$148$	0	0
$149$	−6.53055	−0.535003	−0.267502	−	0.963557i	$-0.586198\pi$
$149$	−6.53055	−0.535003	−0.267502	+	0.963557i	$0.586198\pi$
$150$	0	0
$151$	18.9884	1.54525	0.772627	−	0.634860i	$-0.218942\pi$
$151$	18.9884	1.54525	0.772627	+	0.634860i	$0.218942\pi$
$152$	0	0
$153$	6.26527i	0.506517i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 16.1574i	− 1.28950i	−0.764394	−	0.644750i	$-0.776962\pi$
$157$	− 16.1574i	− 1.28950i	0.764394	−	0.644750i	$-0.223038\pi$
$158$	0	0
$159$	−0.265275	−0.0210377
$160$	0	0
$161$	−30.1574	−2.37673
$162$	0	0
$163$	10.4403i	0.817744i	0.912592	+	0.408872i	$0.134078\pi$
$163$	10.4403i	0.817744i	−0.912592	+	0.408872i	$0.865922\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.8921i	1.07500i	0.843263	+	0.537502i	$0.180632\pi$
$167$	13.8921i	1.07500i	−0.843263	+	0.537502i	$0.819368\pi$
$168$	0	0
$169$	5.08468	0.391129
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	− 2.63844i	− 0.200597i	−0.994957	−	0.100298i	$-0.968020\pi$
$173$	− 2.63844i	− 0.200597i	0.994957	−	0.100298i	$-0.0319798\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.90371i	0.218257i
$178$	0	0
$179$	2.37316	0.177379	0.0886893	−	0.996059i	$-0.471732\pi$
$179$	2.37316	0.177379	0.0886893	+	0.996059i	$0.471732\pi$
$180$	0	0
$181$	0.373165	0.0277371	0.0138686	−	0.999904i	$-0.495585\pi$
$181$	0.373165	0.0277371	0.0138686	+	0.999904i	$0.495585\pi$
$182$	0	0
$183$	− 11.3616i	− 0.839871i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.09629i	0.665186i
$188$	0	0
$189$	−4.81342	−0.350125
$190$	0	0
$191$	4.17498	0.302091	0.151045	−	0.988527i	$-0.451736\pi$
$191$	4.17498	0.302091	0.151045	+	0.988527i	$0.451736\pi$
$192$	0	0
$193$	15.3440i	1.10448i	0.833684	+	0.552241i	$0.186227\pi$
$193$	15.3440i	1.10448i	−0.833684	+	0.552241i	$0.813773\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.2653i	1.58634i	0.609003	+	0.793168i	$0.291570\pi$
$197$	22.2653i	1.58634i	−0.609003	+	0.793168i	$0.708430\pi$
$198$	0	0
$199$	−6.15739	−0.436485	−0.218243	−	0.975895i	$-0.570032\pi$
$199$	−6.15739	−0.436485	−0.218243	+	0.975895i	$0.570032\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2.63844i	− 0.185182i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 6.26527i	− 0.435467i
$208$	0	0
$209$	1.45186	0.100427
$210$	0	0
$211$	−23.7842	−1.63737	−0.818687	−	0.574241i	$-0.805297\pi$
$211$	−23.7842	−1.63737	−0.818687	+	0.574241i	$0.805297\pi$
$212$	0	0
$213$	− 6.90371i	− 0.473035i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	39.7842i	2.70073i
$218$	0	0
$219$	−6.53055	−0.441294
$220$	0	0
$221$	−17.6268	−1.18571
$222$	0	0
$223$	16.5305i	1.10697i	0.832860	+	0.553484i	$0.186702\pi$
$223$	16.5305i	1.10697i	−0.832860	+	0.553484i	$0.813298\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.0847i	1.33307i	0.745475	+	0.666534i	$0.232223\pi$
$227$	20.0847i	1.33307i	−0.745475	+	0.666534i	$0.767777\pi$
$228$	0	0
$229$	27.8921	1.84316	0.921581	−	0.388185i	$-0.126898\pi$
$229$	27.8921	1.84316	0.921581	+	0.388185i	$0.126898\pi$
$230$	0	0
$231$	−6.98840	−0.459803
$232$	0	0
$233$	20.3380i	1.33239i	0.745780	+	0.666193i	$0.232077\pi$
$233$	20.3380i	1.33239i	−0.745780	+	0.666193i	$0.767923\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.7231i	0.696542i
$238$	0	0
$239$	0.921307	0.0595944	0.0297972	−	0.999556i	$-0.490514\pi$
$239$	0.921307	0.0595944	0.0297972	+	0.999556i	$0.490514\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.81342i	0.179013i
$248$	0	0
$249$	9.16899	0.581061
$250$	0	0
$251$	15.6092	0.985247	0.492623	−	0.870243i	$-0.336038\pi$
$251$	15.6092	0.985247	0.492623	+	0.870243i	$0.336038\pi$
$252$	0	0
$253$	− 9.09629i	− 0.571879i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 7.16899i	− 0.447189i	−0.974682	−	0.223595i	$-0.928221\pi$
$257$	− 7.16899i	− 0.447189i	0.974682	−	0.223595i	$-0.0717792\pi$
$258$	0	0
$259$	32.7958	2.03783
$260$	0	0
$261$	0.548143	0.0339292
$262$	0	0
$263$	− 24.4227i	− 1.50597i	−0.658040	−	0.752983i	$-0.728614\pi$
$263$	− 24.4227i	− 1.50597i	0.658040	−	0.752983i	$-0.271386\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 18.7055i	− 1.14476i
$268$	0	0
$269$	10.1750	0.620379	0.310190	−	0.950675i	$-0.399607\pi$
$269$	10.1750	0.620379	0.310190	+	0.950675i	$0.399607\pi$
$270$	0	0
$271$	2.37316	0.144159	0.0720797	−	0.997399i	$-0.477036\pi$
$271$	2.37316	0.144159	0.0720797	+	0.997399i	$0.477036\pi$
$272$	0	0
$273$	− 13.5422i	− 0.819608i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	16.3380i	0.981654i	0.871257	+	0.490827i	$0.163305\pi$
$277$	16.3380i	0.981654i	−0.871257	+	0.490827i	$0.836695\pi$
$278$	0	0
$279$	−8.26527	−0.494829
$280$	0	0
$281$	5.64443	0.336718	0.168359	−	0.985726i	$-0.446153\pi$
$281$	5.64443	0.336718	0.168359	+	0.985726i	$0.446153\pi$
$282$	0	0
$283$	− 21.3440i	− 1.26877i	−0.773018	−	0.634384i	$-0.781254\pi$
$283$	− 21.3440i	− 1.26877i	0.773018	−	0.634384i	$-0.218746\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	21.8921i	1.29225i
$288$	0	0
$289$	−22.2537	−1.30904
$290$	0	0
$291$	6.81342	0.399410
$292$	0	0
$293$	10.6384i	0.621504i	0.950491	+	0.310752i	$0.100581\pi$
$293$	10.6384i	0.621504i	−0.950491	+	0.310752i	$0.899419\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.45186i	− 0.0842453i
$298$	0	0
$299$	17.6268	1.01939
$300$	0	0
$301$	37.1458	2.14105
$302$	0	0
$303$	− 17.4343i	− 1.00157i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 22.1574i	− 1.26459i	−0.774728	−	0.632294i	$-0.782113\pi$
$307$	− 22.1574i	− 1.26459i	0.774728	−	0.632294i	$-0.217887\pi$
$308$	0	0
$309$	16.5305	0.940390
$310$	0	0
$311$	4.17498	0.236741	0.118371	−	0.992969i	$-0.462233\pi$
$311$	4.17498	0.236741	0.118371	+	0.992969i	$0.462233\pi$
$312$	0	0
$313$	− 12.3732i	− 0.699373i	−0.936867	−	0.349686i	$-0.886288\pi$
$313$	− 12.3732i	− 0.699373i	0.936867	−	0.349686i	$-0.113712\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.73473i	0.209763i	0.994485	+	0.104882i	$0.0334463\pi$
$317$	3.73473i	0.209763i	−0.994485	+	0.104882i	$0.966554\pi$
$318$	0	0
$319$	0.795825	0.0445576
$320$	0	0
$321$	16.0000	0.893033
$322$	0	0
$323$	6.26527i	0.348609i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.09629i	0.171225i
$328$	0	0
$329$	−49.4111	−2.72412
$330$	0	0
$331$	14.9884	0.823837	0.411918	−	0.911221i	$-0.364859\pi$
$331$	14.9884	0.823837	0.411918	+	0.911221i	$0.364859\pi$
$332$	0	0
$333$	6.81342i	0.373373i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 32.9708i	− 1.79603i	−0.439961	−	0.898017i	$-0.645008\pi$
$337$	− 32.9708i	− 1.79603i	0.439961	−	0.898017i	$-0.354992\pi$
$338$	0	0
$339$	13.3616	0.725700
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	− 44.1342i	− 2.38302i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 35.1458i	− 1.88672i	−0.331765	−	0.943362i	$-0.607644\pi$
$347$	− 35.1458i	− 1.88672i	0.331765	−	0.943362i	$-0.392356\pi$
$348$	0	0
$349$	4.19257	0.224423	0.112212	−	0.993684i	$-0.464207\pi$
$349$	4.19257	0.224423	0.112212	+	0.993684i	$0.464207\pi$
$350$	0	0
$351$	2.81342	0.150169
$352$	0	0
$353$	− 3.80743i	− 0.202649i	−0.994853	−	0.101325i	$-0.967692\pi$
$353$	− 3.80743i	− 0.202649i	0.994853	−	0.101325i	$-0.0323080\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 30.1574i	− 1.59610i
$358$	0	0
$359$	1.27126	0.0670947	0.0335474	−	0.999437i	$-0.489320\pi$
$359$	1.27126	0.0670947	0.0335474	+	0.999437i	$0.489320\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	8.89211i	0.466715i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 10.0903i	− 0.526709i	−0.964699	−	0.263355i	$-0.915171\pi$
$367$	− 10.0903i	− 0.526709i	0.964699	−	0.263355i	$-0.0848289\pi$
$368$	0	0
$369$	−4.54814	−0.236767
$370$	0	0
$371$	1.27688	0.0662923
$372$	0	0
$373$	− 14.2829i	− 0.739539i	−0.929124	−	0.369769i	$-0.879437\pi$
$373$	− 14.2829i	− 0.739539i	0.929124	−	0.369769i	$-0.120563\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.54215i	0.0794250i
$378$	0	0
$379$	−20.7958	−1.06821	−0.534105	−	0.845418i	$-0.679351\pi$
$379$	−20.7958	−1.06821	−0.534105	+	0.845418i	$0.679351\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	3.46945i	0.177281i	0.996064	+	0.0886403i	$0.0282522\pi$
$383$	3.46945i	0.177281i	−0.996064	+	0.0886403i	$0.971748\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.71713i	0.392284i
$388$	0	0
$389$	−8.72312	−0.442280	−0.221140	−	0.975242i	$-0.570978\pi$
$389$	−8.72312	−0.442280	−0.221140	+	0.975242i	$0.570978\pi$
$390$	0	0
$391$	39.2537	1.98514
$392$	0	0
$393$	4.35557i	0.219710i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 19.4462i	− 0.975979i	−0.872850	−	0.487989i	$-0.837730\pi$
$397$	− 19.4462i	− 0.975979i	0.872850	−	0.487989i	$-0.162270\pi$
$398$	0	0
$399$	−4.81342	−0.240972
$400$	0	0
$401$	27.9824	1.39737	0.698687	−	0.715427i	$-0.253768\pi$
$401$	27.9824	1.39737	0.698687	+	0.715427i	$0.253768\pi$
$402$	0	0
$403$	− 23.2537i	− 1.15835i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.89211i	0.490334i
$408$	0	0
$409$	25.9648	1.28388	0.641939	−	0.766756i	$-0.278130\pi$
$409$	25.9648	1.28388	0.641939	+	0.766756i	$0.278130\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	− 13.9768i	− 0.687753i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 2.90371i	− 0.142196i
$418$	0	0
$419$	28.7055	1.40236	0.701178	−	0.712986i	$-0.252658\pi$
$419$	28.7055	1.40236	0.701178	+	0.712986i	$0.252658\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	− 10.2653i	− 0.499115i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	54.6879i	2.64653i
$428$	0	0
$429$	4.08468	0.197210
$430$	0	0
$431$	22.6879	1.09284	0.546420	−	0.837512i	$-0.315990\pi$
$431$	22.6879	1.09284	0.546420	+	0.837512i	$0.315990\pi$
$432$	0	0
$433$	31.3440i	1.50629i	0.657852	+	0.753147i	$0.271465\pi$
$433$	31.3440i	1.50629i	−0.657852	+	0.753147i	$0.728535\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6.26527i	− 0.299709i
$438$	0	0
$439$	10.1926	0.486465	0.243232	−	0.969968i	$-0.421792\pi$
$439$	10.1926	0.486465	0.243232	+	0.969968i	$0.421792\pi$
$440$	0	0
$441$	16.1690	0.769952
$442$	0	0
$443$	− 13.5189i	− 0.642304i	−0.947028	−	0.321152i	$-0.895930\pi$
$443$	− 13.5189i	− 0.642304i	0.947028	−	0.321152i	$-0.104070\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.53055i	0.308884i
$448$	0	0
$449$	15.9824	0.754256	0.377128	−	0.926161i	$-0.376912\pi$
$449$	15.9824	0.754256	0.377128	+	0.926161i	$0.376912\pi$
$450$	0	0
$451$	−6.60325	−0.310935
$452$	0	0
$453$	− 18.9884i	− 0.892153i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.0963i	1.08040i	0.841537	+	0.540199i	$0.181651\pi$
$457$	23.0963i	1.08040i	−0.841537	+	0.540199i	$0.818349\pi$
$458$	0	0
$459$	6.26527	0.292438
$460$	0	0
$461$	−29.2537	−1.36248	−0.681240	−	0.732060i	$-0.738559\pi$
$461$	−29.2537	−1.36248	−0.681240	+	0.732060i	$0.738559\pi$
$462$	0	0
$463$	15.5365i	0.722044i	0.932557	+	0.361022i	$0.117572\pi$
$463$	15.5365i	0.722044i	−0.932557	+	0.361022i	$0.882428\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.5422i	0.534107i	0.963682	+	0.267054i	$0.0860501\pi$
$467$	11.5422i	0.534107i	−0.963682	+	0.267054i	$0.913950\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−16.1574	−0.744493
$472$	0	0
$473$	11.2042i	0.515169i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.265275i	0.0121461i
$478$	0	0
$479$	16.3556	0.747305	0.373653	−	0.927569i	$-0.378105\pi$
$479$	16.3556	0.747305	0.373653	+	0.927569i	$0.378105\pi$
$480$	0	0
$481$	−19.1690	−0.874031
$482$	0	0
$483$	30.1574i	1.37221i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 41.4111i	− 1.87651i	−0.345939	−	0.938257i	$-0.612440\pi$
$487$	− 41.4111i	− 1.87651i	0.345939	−	0.938257i	$-0.387560\pi$
$488$	0	0
$489$	10.4403	0.472125
$490$	0	0
$491$	−11.0787	−0.499974	−0.249987	−	0.968249i	$-0.580426\pi$
$491$	−11.0787	−0.499974	−0.249987	+	0.968249i	$0.580426\pi$
$492$	0	0
$493$	3.43426i	0.154671i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	33.2305i	1.49059i
$498$	0	0
$499$	16.3500	0.731925	0.365962	−	0.930630i	$-0.380740\pi$
$499$	16.3500	0.731925	0.365962	+	0.930630i	$0.380740\pi$
$500$	0	0
$501$	13.8921	0.620654
$502$	0	0
$503$	11.8921i	0.530243i	0.964215	+	0.265121i	$0.0854121\pi$
$503$	11.8921i	0.530243i	−0.964215	+	0.265121i	$0.914588\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 5.08468i	− 0.225819i
$508$	0	0
$509$	−32.8981	−1.45818	−0.729091	−	0.684416i	$-0.760057\pi$
$509$	−32.8981	−1.45818	−0.729091	+	0.684416i	$0.760057\pi$
$510$	0	0
$511$	31.4343	1.39057
$512$	0	0
$513$	− 1.00000i	− 0.0441511i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 14.9037i	− 0.655465i
$518$	0	0
$519$	−2.63844	−0.115815
$520$	0	0
$521$	−25.0787	−1.09872	−0.549359	−	0.835587i	$-0.685128\pi$
$521$	−25.0787	−1.09872	−0.549359	+	0.835587i	$0.685128\pi$
$522$	0	0
$523$	23.9648i	1.04791i	0.851747	+	0.523954i	$0.175544\pi$
$523$	23.9648i	1.04791i	−0.851747	+	0.523954i	$0.824456\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 51.7842i	− 2.25576i
$528$	0	0
$529$	−16.2537	−0.706681
$530$	0	0
$531$	2.90371	0.126010
$532$	0	0
$533$	− 12.7958i	− 0.554249i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 2.37316i	− 0.102410i
$538$	0	0
$539$	23.4751	1.01114
$540$	0	0
$541$	11.0611	0.475554	0.237777	−	0.971320i	$-0.423581\pi$
$541$	11.0611	0.475554	0.237777	+	0.971320i	$0.423581\pi$
$542$	0	0
$543$	− 0.373165i	− 0.0160140i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 21.0963i	− 0.902012i	−0.892521	−	0.451006i	$-0.851065\pi$
$547$	− 21.0963i	− 0.902012i	0.892521	−	0.451006i	$-0.148935\pi$
$548$	0	0
$549$	−11.3616	−0.484900
$550$	0	0
$551$	0.548143	0.0233517
$552$	0	0
$553$	− 51.6149i	− 2.19489i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	39.5916i	1.67755i	0.544477	+	0.838776i	$0.316728\pi$
$557$	39.5916i	1.67755i	−0.544477	+	0.838776i	$0.683272\pi$
$558$	0	0
$559$	−21.7115	−0.918299
$560$	0	0
$561$	9.09629	0.384045
$562$	0	0
$563$	− 28.0847i	− 1.18363i	−0.806074	−	0.591814i	$-0.798412\pi$
$563$	− 28.0847i	− 1.18363i	0.806074	−	0.591814i	$-0.201588\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.81342i	0.202145i
$568$	0	0
$569$	−41.0787	−1.72211	−0.861054	−	0.508513i	$-0.830195\pi$
$569$	−41.0787	−1.72211	−0.861054	+	0.508513i	$0.830195\pi$
$570$	0	0
$571$	−11.8194	−0.494627	−0.247313	−	0.968936i	$-0.579548\pi$
$571$	−11.8194	−0.494627	−0.247313	+	0.968936i	$0.579548\pi$
$572$	0	0
$573$	− 4.17498i	− 0.174412i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	29.4343i	1.22536i	0.790329	+	0.612682i	$0.209909\pi$
$577$	29.4343i	1.22536i	−0.790329	+	0.612682i	$0.790091\pi$
$578$	0	0
$579$	15.3440	0.637674
$580$	0	0
$581$	−44.1342	−1.83099
$582$	0	0
$583$	0.385141i	0.0159509i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 31.7115i	− 1.30887i	−0.756116	−	0.654437i	$-0.772906\pi$
$587$	− 31.7115i	− 1.30887i	0.756116	−	0.654437i	$-0.227094\pi$
$588$	0	0
$589$	−8.26527	−0.340565
$590$	0	0
$591$	22.2653	0.915871
$592$	0	0
$593$	14.5305i	0.596698i	0.954457	+	0.298349i	$0.0964360\pi$
$593$	14.5305i	0.596698i	−0.954457	+	0.298349i	$0.903564\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.15739i	0.252005i
$598$	0	0
$599$	−17.8074	−0.727592	−0.363796	−	0.931479i	$-0.618519\pi$
$599$	−17.8074	−0.727592	−0.363796	+	0.931479i	$0.618519\pi$
$600$	0	0
$601$	20.6879	0.843878	0.421939	−	0.906624i	$-0.361350\pi$
$601$	20.6879	0.843878	0.421939	+	0.906624i	$0.361350\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.2417i	0.699819i	0.936784	+	0.349909i	$0.113788\pi$
$607$	17.2417i	0.699819i	−0.936784	+	0.349909i	$0.886212\pi$
$608$	0	0
$609$	−2.63844	−0.106915
$610$	0	0
$611$	28.8805	1.16838
$612$	0	0
$613$	− 3.62684i	− 0.146486i	−0.997314	−	0.0732432i	$-0.976665\pi$
$613$	− 3.62684i	− 0.146486i	0.997314	−	0.0732432i	$-0.0233349\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 23.3264i	− 0.939084i	−0.882910	−	0.469542i	$-0.844419\pi$
$617$	− 23.3264i	− 0.939084i	0.882910	−	0.469542i	$-0.155581\pi$
$618$	0	0
$619$	24.8805	1.00003	0.500016	−	0.866016i	$-0.333327\pi$
$619$	24.8805	1.00003	0.500016	+	0.866016i	$0.333327\pi$
$620$	0	0
$621$	−6.26527	−0.251417
$622$	0	0
$623$	90.0375i	3.60728i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.45186i	− 0.0579816i
$628$	0	0
$629$	−42.6879	−1.70208
$630$	0	0
$631$	−36.3148	−1.44567	−0.722834	−	0.691022i	$-0.757161\pi$
$631$	−36.3148	−1.44567	−0.722834	+	0.691022i	$0.757161\pi$
$632$	0	0
$633$	23.7842i	0.945338i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	45.4901i	1.80238i
$638$	0	0
$639$	−6.90371	−0.273107
$640$	0	0
$641$	14.3556	0.567011	0.283506	−	0.958971i	$-0.408503\pi$
$641$	14.3556	0.567011	0.283506	+	0.958971i	$0.408503\pi$
$642$	0	0
$643$	37.1634i	1.46558i	0.680455	+	0.732790i	$0.261782\pi$
$643$	37.1634i	1.46558i	−0.680455	+	0.732790i	$0.738218\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 40.9532i	− 1.61004i	−0.593250	−	0.805018i	$-0.702155\pi$
$647$	− 40.9532i	− 1.61004i	0.593250	−	0.805018i	$-0.297845\pi$
$648$	0	0
$649$	4.21578	0.165484
$650$	0	0
$651$	39.7842	1.55927
$652$	0	0
$653$	− 18.7958i	− 0.735537i	−0.929917	−	0.367769i	$-0.880122\pi$
$653$	− 18.7958i	− 0.735537i	0.929917	−	0.367769i	$-0.119878\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.53055i	0.254781i
$658$	0	0
$659$	−8.53055	−0.332303	−0.166152	−	0.986100i	$-0.553134\pi$
$659$	−8.53055	−0.332303	−0.166152	+	0.986100i	$0.553134\pi$
$660$	0	0
$661$	13.4343	0.522532	0.261266	−	0.965267i	$-0.415860\pi$
$661$	13.4343	0.522532	0.261266	+	0.965267i	$0.415860\pi$
$662$	0	0
$663$	17.6268i	0.684570i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.43426i	− 0.132975i
$668$	0	0
$669$	16.5305	0.639108
$670$	0	0
$671$	−16.4954	−0.636796
$672$	0	0
$673$	− 6.81342i	− 0.262638i	−0.991340	−	0.131319i	$-0.958079\pi$
$673$	− 6.81342i	− 0.262638i	0.991340	−	0.131319i	$-0.0419212\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 47.3032i	− 1.81801i	−0.416787	−	0.909004i	$-0.636844\pi$
$677$	− 47.3032i	− 1.81801i	0.416787	−	0.909004i	$-0.363156\pi$
$678$	0	0
$679$	−32.7958	−1.25859
$680$	0	0
$681$	20.0847	0.769647
$682$	0	0
$683$	− 11.2537i	− 0.430610i	−0.976547	−	0.215305i	$-0.930925\pi$
$683$	− 11.2537i	− 0.430610i	0.976547	−	0.215305i	$-0.0690745\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 27.8921i	− 1.06415i
$688$	0	0
$689$	−0.746329	−0.0284329
$690$	0	0
$691$	9.62684	0.366222	0.183111	−	0.983092i	$-0.441383\pi$
$691$	9.62684	0.366222	0.183111	+	0.983092i	$0.441383\pi$
$692$	0	0
$693$	6.98840i	0.265467i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 28.4954i	− 1.07934i
$698$	0	0
$699$	20.3380	0.769253
$700$	0	0
$701$	16.7231	0.631624	0.315812	−	0.948822i	$-0.397723\pi$
$701$	16.7231	0.631624	0.315812	+	0.948822i	$0.397723\pi$
$702$	0	0
$703$	6.81342i	0.256973i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	83.9184i	3.15608i
$708$	0	0
$709$	32.9532	1.23758	0.618792	−	0.785555i	$-0.287622\pi$
$709$	32.9532	1.23758	0.618792	+	0.785555i	$0.287622\pi$
$710$	0	0
$711$	10.7231	0.402148
$712$	0	0
$713$	51.7842i	1.93933i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 0.921307i	− 0.0344069i
$718$	0	0
$719$	15.6444	0.583439	0.291719	−	0.956504i	$-0.405773\pi$
$719$	15.6444	0.583439	0.291719	+	0.956504i	$0.405773\pi$
$720$	0	0
$721$	−79.5684	−2.96328
$722$	0	0
$723$	2.00000i	0.0743808i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 24.2829i	− 0.900602i	−0.892877	−	0.450301i	$-0.851317\pi$
$727$	− 24.2829i	− 0.900602i	0.892877	−	0.450301i	$-0.148683\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−48.3500	−1.78829
$732$	0	0
$733$	− 18.5305i	− 0.684441i	−0.939620	−	0.342221i	$-0.888821\pi$
$733$	− 18.5305i	− 0.684441i	0.939620	−	0.342221i	$-0.111179\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	9.44624	0.347486	0.173743	−	0.984791i	$-0.444414\pi$
$739$	9.44624	0.347486	0.173743	+	0.984791i	$0.444414\pi$
$740$	0	0
$741$	2.81342	0.103353
$742$	0	0
$743$	41.9768i	1.53998i	0.638056	+	0.769990i	$0.279739\pi$
$743$	41.9768i	1.53998i	−0.638056	+	0.769990i	$0.720261\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 9.16899i	− 0.335476i
$748$	0	0
$749$	−77.0147	−2.81406
$750$	0	0
$751$	26.4578	0.965461	0.482730	−	0.875769i	$-0.339645\pi$
$751$	26.4578	0.965461	0.482730	+	0.875769i	$0.339645\pi$
$752$	0	0
$753$	− 15.6092i	− 0.568832i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 12.9037i	− 0.468993i	−0.972117	−	0.234497i	$-0.924656\pi$
$757$	− 12.9037i	− 0.468993i	0.972117	−	0.234497i	$-0.0753442\pi$
$758$	0	0
$759$	−9.09629	−0.330174
$760$	0	0
$761$	−40.7231	−1.47621	−0.738106	−	0.674685i	$-0.764280\pi$
$761$	−40.7231	−1.47621	−0.738106	+	0.674685i	$0.764280\pi$
$762$	0	0
$763$	− 14.9037i	− 0.539551i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.16936i	0.294979i
$768$	0	0
$769$	5.25367	0.189452	0.0947261	−	0.995503i	$-0.469802\pi$
$769$	5.25367	0.189452	0.0947261	+	0.995503i	$0.469802\pi$
$770$	0	0
$771$	−7.16899	−0.258185
$772$	0	0
$773$	− 38.9884i	− 1.40232i	−0.713006	−	0.701158i	$-0.752667\pi$
$773$	− 38.9884i	− 1.40232i	0.713006	−	0.701158i	$-0.247333\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 32.7958i	− 1.17654i
$778$	0	0
$779$	−4.54814	−0.162954
$780$	0	0
$781$	−10.0232	−0.358659
$782$	0	0
$783$	− 0.548143i	− 0.0195890i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	11.2889i	0.402404i	0.979550	+	0.201202i	$0.0644848\pi$
$787$	11.2889i	0.402404i	−0.979550	+	0.201202i	$0.935515\pi$
$788$	0	0
$789$	−24.4227	−0.869470
$790$	0	0
$791$	−64.3148	−2.28677
$792$	0	0
$793$	− 31.9648i	− 1.13510i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.6995i	0.981168i	0.871394	+	0.490584i	$0.163217\pi$
$797$	27.6995i	0.981168i	−0.871394	+	0.490584i	$0.836783\pi$
$798$	0	0
$799$	64.3148	2.27529
$800$	0	0
$801$	−18.7055	−0.660927
$802$	0	0
$803$	9.48143i	0.334592i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 10.1750i	− 0.358176i
$808$	0	0
$809$	0.723121	0.0254236	0.0127118	−	0.999919i	$-0.495954\pi$
$809$	0.723121	0.0254236	0.0127118	+	0.999919i	$0.495954\pi$
$810$	0	0
$811$	−42.5073	−1.49263	−0.746317	−	0.665590i	$-0.768180\pi$
$811$	−42.5073	−1.49263	−0.746317	+	0.665590i	$0.768180\pi$
$812$	0	0
$813$	− 2.37316i	− 0.0832305i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.71713i	0.269988i
$818$	0	0
$819$	−13.5422	−0.473201
$820$	0	0
$821$	−4.90371	−0.171141	−0.0855704	−	0.996332i	$-0.527271\pi$
$821$	−4.90371	−0.171141	−0.0855704	+	0.996332i	$0.527271\pi$
$822$	0	0
$823$	− 1.37915i	− 0.0480743i	−0.999711	−	0.0240371i	$-0.992348\pi$
$823$	− 1.37915i	− 0.0480743i	0.999711	−	0.0240371i	$-0.00765199\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.9764i	0.590328i	0.955447	+	0.295164i	$0.0953743\pi$
$827$	16.9764i	0.590328i	−0.955447	+	0.295164i	$0.904626\pi$
$828$	0	0
$829$	13.4343	0.466591	0.233296	−	0.972406i	$-0.425049\pi$
$829$	13.4343	0.466591	0.233296	+	0.972406i	$0.425049\pi$
$830$	0	0
$831$	16.3380	0.566758
$832$	0	0
$833$	101.303i	3.50995i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.26527i	0.285690i
$838$	0	0
$839$	43.2185	1.49207	0.746034	−	0.665908i	$-0.231956\pi$
$839$	43.2185	1.49207	0.746034	+	0.665908i	$0.231956\pi$
$840$	0	0
$841$	−28.6995	−0.989639
$842$	0	0
$843$	− 5.64443i	− 0.194404i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 42.8014i	− 1.47067i
$848$	0	0
$849$	−21.3440	−0.732523
$850$	0	0
$851$	42.6879	1.46332
$852$	0	0
$853$	− 19.0963i	− 0.653844i	−0.945051	−	0.326922i	$-0.893988\pi$
$853$	− 19.0963i	− 0.653844i	0.945051	−	0.326922i	$-0.106012\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.5422i	0.872503i	0.899825	+	0.436252i	$0.143694\pi$
$857$	25.5422i	0.872503i	−0.899825	+	0.436252i	$0.856306\pi$
$858$	0	0
$859$	−51.3991	−1.75371	−0.876857	−	0.480751i	$-0.840364\pi$
$859$	−51.3991	−1.75371	−0.876857	+	0.480751i	$0.840364\pi$
$860$	0	0
$861$	21.8921	0.746081
$862$	0	0
$863$	44.0000i	1.49778i	0.662696	+	0.748889i	$0.269412\pi$
$863$	44.0000i	1.49778i	−0.662696	+	0.748889i	$0.730588\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	22.2537i	0.755774i
$868$	0	0
$869$	15.5684	0.528123
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 6.81342i	− 0.230599i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	49.6819i	1.67764i	0.544409	+	0.838820i	$0.316754\pi$
$877$	49.6819i	1.67764i	−0.544409	+	0.838820i	$0.683246\pi$
$878$	0	0
$879$	10.6384	0.358826
$880$	0	0
$881$	−21.2889	−0.717240	−0.358620	−	0.933484i	$-0.616753\pi$
$881$	−21.2889	−0.717240	−0.358620	+	0.933484i	$0.616753\pi$
$882$	0	0
$883$	− 30.6208i	− 1.03047i	−0.857048	−	0.515237i	$-0.827704\pi$
$883$	− 30.6208i	− 1.03047i	0.857048	−	0.515237i	$-0.172296\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	33.0611i	1.11008i	0.831823	+	0.555042i	$0.187298\pi$
$887$	33.0611i	1.11008i	−0.831823	+	0.555042i	$0.812702\pi$
$888$	0	0
$889$	19.2537	0.645747
$890$	0	0
$891$	−1.45186	−0.0486391
$892$	0	0
$893$	− 10.2653i	− 0.343514i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 17.6268i	− 0.588543i
$898$	0	0
$899$	−4.53055	−0.151102
$900$	0	0
$901$	−1.66202	−0.0553699
$902$	0	0
$903$	− 37.1458i	− 1.23613i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 51.7490i	− 1.71830i	−0.511725	−	0.859149i	$-0.670993\pi$
$907$	− 51.7490i	− 1.71830i	0.511725	−	0.859149i	$-0.329007\pi$
$908$	0	0
$909$	−17.4343	−0.578258
$910$	0	0
$911$	3.78422	0.125377	0.0626884	−	0.998033i	$-0.480033\pi$
$911$	3.78422	0.125377	0.0626884	+	0.998033i	$0.480033\pi$
$912$	0	0
$913$	− 13.3121i	− 0.440565i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 20.9652i	− 0.692331i
$918$	0	0
$919$	2.72312	0.0898275	0.0449137	−	0.998991i	$-0.485699\pi$
$919$	2.72312	0.0898275	0.0449137	+	0.998991i	$0.485699\pi$
$920$	0	0
$921$	−22.1574	−0.730111
$922$	0	0
$923$	− 19.4230i	− 0.639317i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 16.5305i	− 0.542934i
$928$	0	0
$929$	−19.8426	−0.651015	−0.325508	−	0.945539i	$-0.605535\pi$
$929$	−19.8426	−0.651015	−0.325508	+	0.945539i	$0.605535\pi$
$930$	0	0
$931$	16.1690	0.529917
$932$	0	0
$933$	− 4.17498i	− 0.136683i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.28886i	0.0421051i	0.999778	+	0.0210525i	$0.00670173\pi$
$937$	1.28886i	0.0421051i	−0.999778	+	0.0210525i	$0.993298\pi$
$938$	0	0
$939$	−12.3732	−0.403783
$940$	0	0
$941$	−21.7898	−0.710328	−0.355164	−	0.934804i	$-0.615575\pi$
$941$	−21.7898	−0.710328	−0.355164	+	0.934804i	$0.615575\pi$
$942$	0	0
$943$	28.4954i	0.927937i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 12.6384i	− 0.410694i	−0.978689	−	0.205347i	$-0.934168\pi$
$947$	− 12.6384i	− 0.410694i	0.978689	−	0.205347i	$-0.0658323\pi$
$948$	0	0
$949$	−18.3732	−0.596418
$950$	0	0
$951$	3.73473	0.121107
$952$	0	0
$953$	29.6763i	0.961311i	0.876910	+	0.480655i	$0.159601\pi$
$953$	29.6763i	0.961311i	−0.876910	+	0.480655i	$0.840399\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 0.795825i	− 0.0257254i
$958$	0	0
$959$	−28.8805	−0.932600
$960$	0	0
$961$	37.3148	1.20370
$962$	0	0
$963$	− 16.0000i	− 0.515593i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 10.2597i	− 0.329928i	−0.986300	−	0.164964i	$-0.947249\pi$
$967$	− 10.2597i	− 0.329928i	0.986300	−	0.164964i	$-0.0527509\pi$
$968$	0	0
$969$	6.26527	0.201270
$970$	0	0
$971$	−49.5916	−1.59147	−0.795736	−	0.605644i	$-0.792916\pi$
$971$	−49.5916	−1.59147	−0.795736	+	0.605644i	$0.792916\pi$
$972$	0	0
$973$	13.9768i	0.448075i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9.71152i	0.310699i	0.987860	+	0.155349i	$0.0496504\pi$
$977$	9.71152i	0.310699i	−0.987860	+	0.155349i	$0.950350\pi$
$978$	0	0
$979$	−27.1578	−0.867966
$980$	0	0
$981$	3.09629	0.0988568
$982$	0	0
$983$	4.83101i	0.154085i	0.997028	+	0.0770427i	$0.0245478\pi$
$983$	4.83101i	0.154085i	−0.997028	+	0.0770427i	$0.975452\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	49.4111i	1.57277i
$988$	0	0
$989$	48.3500	1.53744
$990$	0	0
$991$	−10.1926	−0.323778	−0.161889	−	0.986809i	$-0.551759\pi$
$991$	−10.1926	−0.323778	−0.161889	+	0.986809i	$0.551759\pi$
$992$	0	0
$993$	− 14.9884i	− 0.475642i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	17.9648i	0.568951i	0.958683	+	0.284476i	$0.0918195\pi$
$997$	17.9648i	0.568951i	−0.958683	+	0.284476i	$0.908180\pi$
$998$	0	0
$999$	6.81342	0.215567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5700.2.f.p.3649.3		6
5.2	odd	4		1140.2.a.f.1.1	✓	3
5.3	odd	4		5700.2.a.z.1.3		3
5.4	even	2	inner	5700.2.f.p.3649.4		6
15.2	even	4		3420.2.a.k.1.1		3
20.7	even	4		4560.2.a.bu.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.a.f.1.1	✓	3	5.2	odd	4
3420.2.a.k.1.1		3	15.2	even	4
4560.2.a.bu.1.3		3	20.7	even	4
5700.2.a.z.1.3		3	5.3	odd	4
5700.2.f.p.3649.3		6	1.1	even	1	trivial
5700.2.f.p.3649.4		6	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5700.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +4.81342i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +4.81342i q^{7} -1.00000 q^{9} -1.45186 q^{11} -2.81342i q^{13} -6.26527i q^{17} -1.00000 q^{19} +4.81342 q^{21} +6.26527i q^{23} +1.00000i q^{27} -0.548143 q^{29} +8.26527 q^{31} +1.45186i q^{33} -6.81342i q^{37} -2.81342 q^{39} +4.54814 q^{41} -7.71713i q^{43} +10.2653i q^{47} -16.1690 q^{49} -6.26527 q^{51} -0.265275i q^{53} +1.00000i q^{57} -2.90371 q^{59} +11.3616 q^{61} -4.81342i q^{63} +6.26527 q^{69} +6.90371 q^{71} -6.53055i q^{73} -6.98840i q^{77} -10.7231 q^{79} +1.00000 q^{81} +9.16899i q^{83} +0.548143i q^{87} +18.7055 q^{89} +13.5422 q^{91} -8.26527i q^{93} +6.81342i q^{97} +1.45186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^9	\( 6 q - 6 q^{9} - 4 q^{11} - 6 q^{19} - 8 q^{29} + 16 q^{31} + 12 q^{39} + 32 q^{41} - 54 q^{49} - 4 q^{51} - 8 q^{59} + 44 q^{61} + 4 q^{69} + 32 q^{71} - 16 q^{79} + 6 q^{81} - 8 q^{89} + 96 q^{91} + 4 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 - 4 * q^11 - 6 * q^19 - 8 * q^29 + 16 * q^31 + 12 * q^39 + 32 * q^41 - 54 * q^49 - 4 * q^51 - 8 * q^59 + 44 * q^61 + 4 * q^69 + 32 * q^71 - 16 * q^79 + 6 * q^81 - 8 * q^89 + 96 * q^91 + 4 * q^99

Introduction

L-functions

Modular forms

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Database

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