LMFDB - Embedded newform 6400.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6400,2,Mod(1,6400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6400 = 2^{8} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6400.a (trivial)

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:	yes
Analytic conductor:	$51.1042572936$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 200)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	6400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{3} -4.00000 q^{7} +4.00000 q^{9} +O(q^{10})$	$q+2.64575 q^{3} -4.00000 q^{7} +4.00000 q^{9} +2.64575 q^{11} -3.00000 q^{17} -2.64575 q^{19} -10.5830 q^{21} -4.00000 q^{23} +2.64575 q^{27} +4.00000 q^{31} +7.00000 q^{33} -10.5830 q^{37} +5.00000 q^{41} -5.29150 q^{43} -8.00000 q^{47} +9.00000 q^{49} -7.93725 q^{51} +10.5830 q^{53} -7.00000 q^{57} +5.29150 q^{59} -10.5830 q^{61} -16.0000 q^{63} -7.93725 q^{67} -10.5830 q^{69} -8.00000 q^{71} +7.00000 q^{73} -10.5830 q^{77} +4.00000 q^{79} -5.00000 q^{81} -7.93725 q^{83} +1.00000 q^{89} +10.5830 q^{93} -2.00000 q^{97} +10.5830 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{7} + 8 q^{9}+O(q^{10}) $ 2 * q - 8 * q^7 + 8 * q^9	$ 2 q - 8 q^{7} + 8 q^{9} - 6 q^{17} - 8 q^{23} + 8 q^{31} + 14 q^{33} + 10 q^{41} - 16 q^{47} + 18 q^{49} - 14 q^{57} - 32 q^{63} - 16 q^{71} + 14 q^{73} + 8 q^{79} - 10 q^{81} + 2 q^{89} - 4 q^{97}+O(q^{100}) $ 2 * q - 8 * q^7 + 8 * q^9 - 6 * q^17 - 8 * q^23 + 8 * q^31 + 14 * q^33 + 10 * q^41 - 16 * q^47 + 18 * q^49 - 14 * q^57 - 32 * q^63 - 16 * q^71 + 14 * q^73 + 8 * q^79 - 10 * q^81 + 2 * q^89 - 4 * q^97

Display 100 coefficients 10 coefficients

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
$3$	2.64575	1.52753	0.763763	−	0.645497i	$-0.223350\pi$
$3$	2.64575	1.52753	0.763763	+	0.645497i	$0.223350\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	4.00000	1.33333
$10$	0	0
$11$	2.64575	0.797724	0.398862	−	0.917011i	$-0.369405\pi$
$11$	2.64575	0.797724	0.398862	+	0.917011i	$0.369405\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−2.64575	−0.606977	−0.303488	−	0.952835i	$-0.598151\pi$
$19$	−2.64575	−0.606977	−0.303488	+	0.952835i	$0.598151\pi$
$20$	0	0
$21$	−10.5830	−2.30940
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.64575	0.509175
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	7.00000	1.21854
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.5830	−1.73984	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	−10.5830	−1.73984	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	−5.29150	−0.806947	−0.403473	−	0.914991i	$-0.632197\pi$
$43$	−5.29150	−0.806947	−0.403473	+	0.914991i	$0.632197\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	−7.93725	−1.11144
$52$	0	0
$53$	10.5830	1.45369	0.726844	−	0.686803i	$-0.240986\pi$
$53$	10.5830	1.45369	0.726844	+	0.686803i	$0.240986\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−7.00000	−0.927173
$58$	0	0
$59$	5.29150	0.688895	0.344447	−	0.938806i	$-0.388066\pi$
$59$	5.29150	0.688895	0.344447	+	0.938806i	$0.388066\pi$
$60$	0	0
$61$	−10.5830	−1.35501	−0.677507	−	0.735516i	$-0.736940\pi$
$61$	−10.5830	−1.35501	−0.677507	+	0.735516i	$0.736940\pi$
$62$	0	0
$63$	−16.0000	−2.01581
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.93725	−0.969690	−0.484845	−	0.874600i	$-0.661124\pi$
$67$	−7.93725	−0.969690	−0.484845	+	0.874600i	$0.661124\pi$
$68$	0	0
$69$	−10.5830	−1.27404
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−10.5830	−1.20605
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−7.93725	−0.871227	−0.435613	−	0.900134i	$-0.643469\pi$
$83$	−7.93725	−0.871227	−0.435613	+	0.900134i	$0.643469\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.00000	0.106000	0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000	0.106000	0.0529999	+	0.998595i	$0.483122\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	10.5830	1.09741
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	10.5830	1.06363
$100$	0	0
$101$	−10.5830	−1.05305	−0.526524	−	0.850160i	$-0.676505\pi$
$101$	−10.5830	−1.05305	−0.526524	+	0.850160i	$0.676505\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.64575	−0.255774	−0.127887	−	0.991789i	$-0.540820\pi$
$107$	−2.64575	−0.255774	−0.127887	+	0.991789i	$0.540820\pi$
$108$	0	0
$109$	−10.5830	−1.01367	−0.506834	−	0.862044i	$-0.669184\pi$
$109$	−10.5830	−1.01367	−0.506834	+	0.862044i	$0.669184\pi$
$110$	0	0
$111$	−28.0000	−2.65764
$112$	0	0
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	−4.00000	−0.363636
$122$	0	0
$123$	13.2288	1.19280
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	−14.0000	−1.23263
$130$	0	0
$131$	15.8745	1.38696	0.693481	−	0.720475i	$-0.256076\pi$
$131$	15.8745	1.38696	0.693481	+	0.720475i	$0.256076\pi$
$132$	0	0
$133$	10.5830	0.917663
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.0000	1.62328	0.811640	−	0.584158i	$-0.198575\pi$
$137$	19.0000	1.62328	0.811640	+	0.584158i	$0.198575\pi$
$138$	0	0
$139$	−18.5203	−1.57087	−0.785434	−	0.618945i	$-0.787560\pi$
$139$	−18.5203	−1.57087	−0.785434	+	0.618945i	$0.787560\pi$
$140$	0	0
$141$	−21.1660	−1.78250
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	23.8118	1.96396
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	−12.0000	−0.970143
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.5830	−0.844616	−0.422308	−	0.906452i	$-0.638780\pi$
$157$	−10.5830	−0.844616	−0.422308	+	0.906452i	$0.638780\pi$
$158$	0	0
$159$	28.0000	2.22054
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	13.2288	1.03616	0.518078	−	0.855333i	$-0.326648\pi$
$163$	13.2288	1.03616	0.518078	+	0.855333i	$0.326648\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	−10.5830	−0.809303
$172$	0	0
$173$	21.1660	1.60922	0.804611	−	0.593802i	$-0.202374\pi$
$173$	21.1660	1.60922	0.804611	+	0.593802i	$0.202374\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.0000	1.05230
$178$	0	0
$179$	−23.8118	−1.77977	−0.889887	−	0.456180i	$-0.849217\pi$
$179$	−23.8118	−1.77977	−0.889887	+	0.456180i	$0.849217\pi$
$180$	0	0
$181$	10.5830	0.786629	0.393314	−	0.919404i	$-0.371328\pi$
$181$	10.5830	0.786629	0.393314	+	0.919404i	$0.371328\pi$
$182$	0	0
$183$	−28.0000	−2.06982
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−7.93725	−0.580429
$188$	0	0
$189$	−10.5830	−0.769800
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.5830	0.754008	0.377004	−	0.926212i	$-0.376954\pi$
$197$	10.5830	0.754008	0.377004	+	0.926212i	$0.376954\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	−21.0000	−1.48123
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−16.0000	−1.11208
$208$	0	0
$209$	−7.00000	−0.484200
$210$	0	0
$211$	7.93725	0.546423	0.273212	−	0.961954i	$-0.411914\pi$
$211$	7.93725	0.546423	0.273212	+	0.961954i	$0.411914\pi$
$212$	0	0
$213$	−21.1660	−1.45027
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	18.5203	1.25148
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.8745	−1.05363	−0.526814	−	0.849981i	$-0.676614\pi$
$227$	−15.8745	−1.05363	−0.526814	+	0.849981i	$0.676614\pi$
$228$	0	0
$229$	21.1660	1.39869	0.699345	−	0.714785i	$-0.253475\pi$
$229$	21.1660	1.39869	0.699345	+	0.714785i	$0.253475\pi$
$230$	0	0
$231$	−28.0000	−1.84226
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.5830	0.687440
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−21.0000	−1.35273	−0.676364	−	0.736567i	$-0.736446\pi$
$241$	−21.0000	−1.35273	−0.676364	+	0.736567i	$0.736446\pi$
$242$	0	0
$243$	−21.1660	−1.35780
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−21.0000	−1.33082
$250$	0	0
$251$	−7.93725	−0.500995	−0.250498	−	0.968117i	$-0.580594\pi$
$251$	−7.93725	−0.500995	−0.250498	+	0.968117i	$0.580594\pi$
$252$	0	0
$253$	−10.5830	−0.665348
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	42.3320	2.63038
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.64575	0.161917
$268$	0	0
$269$	21.1660	1.29051	0.645257	−	0.763965i	$-0.276750\pi$
$269$	21.1660	1.29051	0.645257	+	0.763965i	$0.276750\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−21.1660	−1.27174	−0.635871	−	0.771795i	$-0.719359\pi$
$277$	−21.1660	−1.27174	−0.635871	+	0.771795i	$0.719359\pi$
$278$	0	0
$279$	16.0000	0.957895
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	−13.2288	−0.786368	−0.393184	−	0.919460i	$-0.628626\pi$
$283$	−13.2288	−0.786368	−0.393184	+	0.919460i	$0.628626\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−20.0000	−1.18056
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−5.29150	−0.310193
$292$	0	0
$293$	−10.5830	−0.618266	−0.309133	−	0.951019i	$-0.600039\pi$
$293$	−10.5830	−0.618266	−0.309133	+	0.951019i	$0.600039\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7.00000	0.406181
$298$	0	0
$299$	0	0
$300$	0	0
$301$	21.1660	1.21999
$302$	0	0
$303$	−28.0000	−1.60856
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.64575	0.151001	0.0755005	−	0.997146i	$-0.475945\pi$
$307$	2.64575	0.151001	0.0755005	+	0.997146i	$0.475945\pi$
$308$	0	0
$309$	−21.1660	−1.20409
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−7.00000	−0.390702
$322$	0	0
$323$	7.93725	0.441641
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−28.0000	−1.54840
$328$	0	0
$329$	32.0000	1.76422
$330$	0	0
$331$	2.64575	0.145424	0.0727118	−	0.997353i	$-0.476835\pi$
$331$	2.64575	0.145424	0.0727118	+	0.997353i	$0.476835\pi$
$332$	0	0
$333$	−42.3320	−2.31978
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−15.0000	−0.817102	−0.408551	−	0.912735i	$-0.633966\pi$
$337$	−15.0000	−0.817102	−0.408551	+	0.912735i	$0.633966\pi$
$338$	0	0
$339$	−39.6863	−2.15546
$340$	0	0
$341$	10.5830	0.573102
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.64575	−0.142031	−0.0710157	−	0.997475i	$-0.522624\pi$
$347$	−2.64575	−0.142031	−0.0710157	+	0.997475i	$0.522624\pi$
$348$	0	0
$349$	10.5830	0.566495	0.283248	−	0.959047i	$-0.408588\pi$
$349$	10.5830	0.566495	0.283248	+	0.959047i	$0.408588\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	31.7490	1.68034
$358$	0	0
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	0	0
$361$	−12.0000	−0.631579
$362$	0	0
$363$	−10.5830	−0.555464
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	20.0000	1.04116
$370$	0	0
$371$	−42.3320	−2.19777
$372$	0	0
$373$	10.5830	0.547967	0.273984	−	0.961734i	$-0.411659\pi$
$373$	10.5830	0.547967	0.273984	+	0.961734i	$0.411659\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−7.93725	−0.407709	−0.203855	−	0.979001i	$-0.565347\pi$
$379$	−7.93725	−0.407709	−0.203855	+	0.979001i	$0.565347\pi$
$380$	0	0
$381$	31.7490	1.62655
$382$	0	0
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−21.1660	−1.07593
$388$	0	0
$389$	10.5830	0.536580	0.268290	−	0.963338i	$-0.413542\pi$
$389$	10.5830	0.536580	0.268290	+	0.963338i	$0.413542\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	42.0000	2.11862
$394$	0	0
$395$	0	0
$396$	0	0
$397$	21.1660	1.06229	0.531146	−	0.847280i	$-0.321762\pi$
$397$	21.1660	1.06229	0.531146	+	0.847280i	$0.321762\pi$
$398$	0	0
$399$	28.0000	1.40175
$400$	0	0
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−28.0000	−1.38791
$408$	0	0
$409$	−3.00000	−0.148340	−0.0741702	−	0.997246i	$-0.523631\pi$
$409$	−3.00000	−0.148340	−0.0741702	+	0.997246i	$0.523631\pi$
$410$	0	0
$411$	50.2693	2.47960
$412$	0	0
$413$	−21.1660	−1.04151
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−49.0000	−2.39954
$418$	0	0
$419$	18.5203	0.904774	0.452387	−	0.891822i	$-0.350573\pi$
$419$	18.5203	0.904774	0.452387	+	0.891822i	$0.350573\pi$
$420$	0	0
$421$	21.1660	1.03157	0.515784	−	0.856719i	$-0.327501\pi$
$421$	21.1660	1.03157	0.515784	+	0.856719i	$0.327501\pi$
$422$	0	0
$423$	−32.0000	−1.55589
$424$	0	0
$425$	0	0
$426$	0	0
$427$	42.3320	2.04859
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	37.0000	1.77811	0.889053	−	0.457804i	$-0.151364\pi$
$433$	37.0000	1.77811	0.889053	+	0.457804i	$0.151364\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.5830	0.506254
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	36.0000	1.71429
$442$	0	0
$443$	29.1033	1.38274	0.691369	−	0.722502i	$-0.257008\pi$
$443$	29.1033	1.38274	0.691369	+	0.722502i	$0.257008\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	27.0000	1.27421	0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000	1.27421	0.637104	+	0.770778i	$0.280132\pi$
$450$	0	0
$451$	13.2288	0.622918
$452$	0	0
$453$	−10.5830	−0.497233
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.0000	1.26301	0.631503	−	0.775373i	$-0.282438\pi$
$457$	27.0000	1.26301	0.631503	+	0.775373i	$0.282438\pi$
$458$	0	0
$459$	−7.93725	−0.370479
$460$	0	0
$461$	42.3320	1.97160	0.985799	−	0.167927i	$-0.0537074\pi$
$461$	42.3320	1.97160	0.985799	+	0.167927i	$0.0537074\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.4575	1.22431	0.612154	−	0.790739i	$-0.290303\pi$
$467$	26.4575	1.22431	0.612154	+	0.790739i	$0.290303\pi$
$468$	0	0
$469$	31.7490	1.46603
$470$	0	0
$471$	−28.0000	−1.29017
$472$	0	0
$473$	−14.0000	−0.643721
$474$	0	0
$475$	0	0
$476$	0	0
$477$	42.3320	1.93825
$478$	0	0
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	42.3320	1.92617
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	35.0000	1.58275
$490$	0	0
$491$	5.29150	0.238802	0.119401	−	0.992846i	$-0.461903\pi$
$491$	5.29150	0.238802	0.119401	+	0.992846i	$0.461903\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	32.0000	1.43540
$498$	0	0
$499$	−26.4575	−1.18440	−0.592200	−	0.805791i	$-0.701741\pi$
$499$	−26.4575	−1.18440	−0.592200	+	0.805791i	$0.701741\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−34.3948	−1.52753
$508$	0	0
$509$	31.7490	1.40725	0.703625	−	0.710571i	$-0.251563\pi$
$509$	31.7490	1.40725	0.703625	+	0.710571i	$0.251563\pi$
$510$	0	0
$511$	−28.0000	−1.23865
$512$	0	0
$513$	−7.00000	−0.309058
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−21.1660	−0.930880
$518$	0	0
$519$	56.0000	2.45813
$520$	0	0
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	0	0
$523$	−2.64575	−0.115691	−0.0578453	−	0.998326i	$-0.518423\pi$
$523$	−2.64575	−0.115691	−0.0578453	+	0.998326i	$0.518423\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	21.1660	0.918527
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−63.0000	−2.71865
$538$	0	0
$539$	23.8118	1.02565
$540$	0	0
$541$	21.1660	0.909998	0.454999	−	0.890492i	$-0.349640\pi$
$541$	21.1660	0.909998	0.454999	+	0.890492i	$0.349640\pi$
$542$	0	0
$543$	28.0000	1.20160
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−18.5203	−0.791869	−0.395935	−	0.918279i	$-0.629579\pi$
$547$	−18.5203	−0.791869	−0.395935	+	0.918279i	$0.629579\pi$
$548$	0	0
$549$	−42.3320	−1.80669
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−16.0000	−0.680389
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−21.0000	−0.886621
$562$	0	0
$563$	−15.8745	−0.669031	−0.334515	−	0.942390i	$-0.608573\pi$
$563$	−15.8745	−0.669031	−0.334515	+	0.942390i	$0.608573\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	20.0000	0.839921
$568$	0	0
$569$	−11.0000	−0.461144	−0.230572	−	0.973055i	$-0.574060\pi$
$569$	−11.0000	−0.461144	−0.230572	+	0.973055i	$0.574060\pi$
$570$	0	0
$571$	−37.0405	−1.55010	−0.775049	−	0.631901i	$-0.782275\pi$
$571$	−37.0405	−1.55010	−0.775049	+	0.631901i	$0.782275\pi$
$572$	0	0
$573$	10.5830	0.442111
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−7.00000	−0.291414	−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000	−0.291414	−0.145707	+	0.989328i	$0.546546\pi$
$578$	0	0
$579$	13.2288	0.549768
$580$	0	0
$581$	31.7490	1.31717
$582$	0	0
$583$	28.0000	1.15964
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.93725	0.327606	0.163803	−	0.986493i	$-0.447624\pi$
$587$	7.93725	0.327606	0.163803	+	0.986493i	$0.447624\pi$
$588$	0	0
$589$	−10.5830	−0.436065
$590$	0	0
$591$	28.0000	1.15177
$592$	0	0
$593$	41.0000	1.68367	0.841834	−	0.539736i	$-0.181476\pi$
$593$	41.0000	1.68367	0.841834	+	0.539736i	$0.181476\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	63.4980	2.59880
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−7.00000	−0.285536	−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000	−0.285536	−0.142768	+	0.989756i	$0.545600\pi$
$602$	0	0
$603$	−31.7490	−1.29292
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−10.5830	−0.427444	−0.213722	−	0.976895i	$-0.568559\pi$
$613$	−10.5830	−0.427444	−0.213722	+	0.976895i	$0.568559\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	5.29150	0.212683	0.106342	−	0.994330i	$-0.466086\pi$
$619$	5.29150	0.212683	0.106342	+	0.994330i	$0.466086\pi$
$620$	0	0
$621$	−10.5830	−0.424681
$622$	0	0
$623$	−4.00000	−0.160257
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−18.5203	−0.739628
$628$	0	0
$629$	31.7490	1.26592
$630$	0	0
$631$	−36.0000	−1.43314	−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000	−1.43314	−0.716569	+	0.697517i	$0.754288\pi$
$632$	0	0
$633$	21.0000	0.834675
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−32.0000	−1.26590
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	−15.8745	−0.626029	−0.313015	−	0.949748i	$-0.601339\pi$
$643$	−15.8745	−0.626029	−0.313015	+	0.949748i	$0.601339\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	14.0000	0.549548
$650$	0	0
$651$	−42.3320	−1.65912
$652$	0	0
$653$	31.7490	1.24243	0.621217	−	0.783638i	$-0.286638\pi$
$653$	31.7490	1.24243	0.621217	+	0.783638i	$0.286638\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	28.0000	1.09238
$658$	0	0
$659$	7.93725	0.309192	0.154596	−	0.987978i	$-0.450592\pi$
$659$	7.93725	0.309192	0.154596	+	0.987978i	$0.450592\pi$
$660$	0	0
$661$	21.1660	0.823262	0.411631	−	0.911351i	$-0.364959\pi$
$661$	21.1660	0.823262	0.411631	+	0.911351i	$0.364959\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−42.3320	−1.63665
$670$	0	0
$671$	−28.0000	−1.08093
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−42.3320	−1.62695	−0.813476	−	0.581599i	$-0.802427\pi$
$677$	−42.3320	−1.62695	−0.813476	+	0.581599i	$0.802427\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	−42.0000	−1.60944
$682$	0	0
$683$	−23.8118	−0.911132	−0.455566	−	0.890202i	$-0.650563\pi$
$683$	−23.8118	−0.911132	−0.455566	+	0.890202i	$0.650563\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	56.0000	2.13653
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−23.8118	−0.905842	−0.452921	−	0.891551i	$-0.649618\pi$
$691$	−23.8118	−0.905842	−0.452921	+	0.891551i	$0.649618\pi$
$692$	0	0
$693$	−42.3320	−1.60806
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−15.0000	−0.568166
$698$	0	0
$699$	−15.8745	−0.600429
$700$	0	0
$701$	−21.1660	−0.799429	−0.399715	−	0.916640i	$-0.630891\pi$
$701$	−21.1660	−0.799429	−0.399715	+	0.916640i	$0.630891\pi$
$702$	0	0
$703$	28.0000	1.05604
$704$	0	0
$705$	0	0
$706$	0	0
$707$	42.3320	1.59206
$708$	0	0
$709$	21.1660	0.794906	0.397453	−	0.917622i	$-0.369894\pi$
$709$	21.1660	0.794906	0.397453	+	0.917622i	$0.369894\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	−16.0000	−0.599205
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−21.1660	−0.790459
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	0	0
$723$	−55.5608	−2.06633
$724$	0	0
$725$	0	0
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	−41.0000	−1.51852
$730$	0	0
$731$	15.8745	0.587140
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−21.0000	−0.773545
$738$	0	0
$739$	15.8745	0.583953	0.291977	−	0.956425i	$-0.405687\pi$
$739$	15.8745	0.583953	0.291977	+	0.956425i	$0.405687\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−31.7490	−1.16164
$748$	0	0
$749$	10.5830	0.386695
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	−21.0000	−0.765283
$754$	0	0
$755$	0	0
$756$	0	0
$757$	21.1660	0.769292	0.384646	−	0.923064i	$-0.374324\pi$
$757$	21.1660	0.769292	0.384646	+	0.923064i	$0.374324\pi$
$758$	0	0
$759$	−28.0000	−1.01634
$760$	0	0
$761$	29.0000	1.05125	0.525625	−	0.850717i	$-0.323832\pi$
$761$	29.0000	1.05125	0.525625	+	0.850717i	$0.323832\pi$
$762$	0	0
$763$	42.3320	1.53252
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−21.0000	−0.757279	−0.378640	−	0.925544i	$-0.623608\pi$
$769$	−21.0000	−0.757279	−0.378640	+	0.925544i	$0.623608\pi$
$770$	0	0
$771$	37.0405	1.33398
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	112.000	4.01798
$778$	0	0
$779$	−13.2288	−0.473969
$780$	0	0
$781$	−21.1660	−0.757379
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26.4575	0.943108	0.471554	−	0.881837i	$-0.343693\pi$
$787$	26.4575	0.943108	0.471554	+	0.881837i	$0.343693\pi$
$788$	0	0
$789$	−31.7490	−1.13029
$790$	0	0
$791$	60.0000	2.13335
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31.7490	−1.12461	−0.562304	−	0.826931i	$-0.690085\pi$
$797$	−31.7490	−1.12461	−0.562304	+	0.826931i	$0.690085\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	4.00000	0.141333
$802$	0	0
$803$	18.5203	0.653566
$804$	0	0
$805$	0	0
$806$	0	0
$807$	56.0000	1.97129
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	5.29150	0.185810	0.0929049	−	0.995675i	$-0.470385\pi$
$811$	5.29150	0.185810	0.0929049	+	0.995675i	$0.470385\pi$
$812$	0	0
$813$	−52.9150	−1.85581
$814$	0	0
$815$	0	0
$816$	0	0
$817$	14.0000	0.489798
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.5830	−0.369349	−0.184675	−	0.982800i	$-0.559123\pi$
$821$	−10.5830	−0.369349	−0.184675	+	0.982800i	$0.559123\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13.2288	−0.460009	−0.230004	−	0.973190i	$-0.573874\pi$
$827$	−13.2288	−0.460009	−0.230004	+	0.973190i	$0.573874\pi$
$828$	0	0
$829$	21.1660	0.735126	0.367563	−	0.929999i	$-0.380192\pi$
$829$	21.1660	0.735126	0.367563	+	0.929999i	$0.380192\pi$
$830$	0	0
$831$	−56.0000	−1.94262
$832$	0	0
$833$	−27.0000	−0.935495
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10.5830	0.365802
$838$	0	0
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	58.2065	2.00474
$844$	0	0
$845$	0	0
$846$	0	0
$847$	16.0000	0.549767
$848$	0	0
$849$	−35.0000	−1.20120
$850$	0	0
$851$	42.3320	1.45112
$852$	0	0
$853$	52.9150	1.81178	0.905888	−	0.423517i	$-0.139205\pi$
$853$	52.9150	1.81178	0.905888	+	0.423517i	$0.139205\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.0000	−0.717346	−0.358673	−	0.933463i	$-0.616771\pi$
$857$	−21.0000	−0.717346	−0.358673	+	0.933463i	$0.616771\pi$
$858$	0	0
$859$	2.64575	0.0902719	0.0451359	−	0.998981i	$-0.485628\pi$
$859$	2.64575	0.0902719	0.0451359	+	0.998981i	$0.485628\pi$
$860$	0	0
$861$	−52.9150	−1.80334
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−21.1660	−0.718835
$868$	0	0
$869$	10.5830	0.359004
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−8.00000	−0.270759
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−42.3320	−1.42945	−0.714725	−	0.699405i	$-0.753448\pi$
$877$	−42.3320	−1.42945	−0.714725	+	0.699405i	$0.753448\pi$
$878$	0	0
$879$	−28.0000	−0.944417
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	44.9778	1.51362	0.756811	−	0.653633i	$-0.226756\pi$
$883$	44.9778	1.51362	0.756811	+	0.653633i	$0.226756\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−56.0000	−1.88030	−0.940148	−	0.340766i	$-0.889313\pi$
$887$	−56.0000	−1.88030	−0.940148	+	0.340766i	$0.889313\pi$
$888$	0	0
$889$	−48.0000	−1.60987
$890$	0	0
$891$	−13.2288	−0.443180
$892$	0	0
$893$	21.1660	0.708294
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−31.7490	−1.05771
$902$	0	0
$903$	56.0000	1.86356
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−5.29150	−0.175701	−0.0878507	−	0.996134i	$-0.528000\pi$
$907$	−5.29150	−0.175701	−0.0878507	+	0.996134i	$0.528000\pi$
$908$	0	0
$909$	−42.3320	−1.40406
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	−21.0000	−0.694999
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−63.4980	−2.09689
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	7.00000	0.230658
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−32.0000	−1.05102
$928$	0	0
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	−23.8118	−0.780399
$932$	0	0
$933$	10.5830	0.346472
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.00000	0.228680	0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000	0.228680	0.114340	+	0.993442i	$0.463525\pi$
$938$	0	0
$939$	−15.8745	−0.518045
$940$	0	0
$941$	−42.3320	−1.37998	−0.689992	−	0.723817i	$-0.742386\pi$
$941$	−42.3320	−1.37998	−0.689992	+	0.723817i	$0.742386\pi$
$942$	0	0
$943$	−20.0000	−0.651290
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−15.8745	−0.515852	−0.257926	−	0.966165i	$-0.583039\pi$
$947$	−15.8745	−0.515852	−0.257926	+	0.966165i	$0.583039\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−5.00000	−0.161966	−0.0809829	−	0.996715i	$-0.525806\pi$
$953$	−5.00000	−0.161966	−0.0809829	+	0.996715i	$0.525806\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−76.0000	−2.45417
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−10.5830	−0.341033
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	21.0000	0.674617
$970$	0	0
$971$	23.8118	0.764156	0.382078	−	0.924130i	$-0.375209\pi$
$971$	23.8118	0.764156	0.382078	+	0.924130i	$0.375209\pi$
$972$	0	0
$973$	74.0810	2.37493
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37.0000	1.18373	0.591867	−	0.806035i	$-0.298391\pi$
$977$	37.0000	1.18373	0.591867	+	0.806035i	$0.298391\pi$
$978$	0	0
$979$	2.64575	0.0845586
$980$	0	0
$981$	−42.3320	−1.35156
$982$	0	0
$983$	28.0000	0.893061	0.446531	−	0.894768i	$-0.352659\pi$
$983$	28.0000	0.893061	0.446531	+	0.894768i	$0.352659\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	84.6640	2.69489
$988$	0	0
$989$	21.1660	0.673040
$990$	0	0
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	0	0
$993$	7.00000	0.222138
$994$	0	0
$995$	0	0
$996$	0	0
$997$	42.3320	1.34067	0.670334	−	0.742059i	$-0.266151\pi$
$997$	42.3320	1.34067	0.670334	+	0.742059i	$0.266151\pi$
$998$	0	0
$999$	−28.0000	−0.885881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6400.2.a.bg.1.2		2
4.3	odd	2		6400.2.a.cb.1.1		2
5.4	even	2		6400.2.a.cc.1.1		2
8.3	odd	2		6400.2.a.cb.1.2		2
8.5	even	2	inner	6400.2.a.bg.1.1		2
16.3	odd	4		800.2.d.a.401.1		2
16.5	even	4		200.2.d.c.101.1	yes	2
16.11	odd	4		800.2.d.a.401.2		2
16.13	even	4		200.2.d.c.101.2	yes	2
20.19	odd	2		6400.2.a.bh.1.2		2
40.19	odd	2		6400.2.a.bh.1.1		2
40.29	even	2		6400.2.a.cc.1.2		2
48.5	odd	4		1800.2.k.d.901.2		2
48.11	even	4		7200.2.k.b.3601.2		2
48.29	odd	4		1800.2.k.d.901.1		2
48.35	even	4		7200.2.k.b.3601.1		2
80.3	even	4		800.2.f.d.49.4		4
80.13	odd	4		200.2.f.d.149.3		4
80.19	odd	4		800.2.d.d.401.2		2
80.27	even	4		800.2.f.d.49.3		4
80.29	even	4		200.2.d.b.101.1	✓	2
80.37	odd	4		200.2.f.d.149.4		4
80.43	even	4		800.2.f.d.49.2		4
80.53	odd	4		200.2.f.d.149.1		4
80.59	odd	4		800.2.d.d.401.1		2
80.67	even	4		800.2.f.d.49.1		4
80.69	even	4		200.2.d.b.101.2	yes	2
80.77	odd	4		200.2.f.d.149.2		4
240.29	odd	4		1800.2.k.f.901.2		2
240.53	even	4		1800.2.d.m.1549.4		4
240.59	even	4		7200.2.k.i.3601.2		2
240.77	even	4		1800.2.d.m.1549.3		4
240.83	odd	4		7200.2.d.m.2449.3		4
240.107	odd	4		7200.2.d.m.2449.2		4
240.149	odd	4		1800.2.k.f.901.1		2
240.173	even	4		1800.2.d.m.1549.2		4
240.179	even	4		7200.2.k.i.3601.1		2
240.197	even	4		1800.2.d.m.1549.1		4
240.203	odd	4		7200.2.d.m.2449.4		4
240.227	odd	4		7200.2.d.m.2449.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.d.b.101.1	✓	2	80.29	even	4
200.2.d.b.101.2	yes	2	80.69	even	4
200.2.d.c.101.1	yes	2	16.5	even	4
200.2.d.c.101.2	yes	2	16.13	even	4
200.2.f.d.149.1		4	80.53	odd	4
200.2.f.d.149.2		4	80.77	odd	4
200.2.f.d.149.3		4	80.13	odd	4
200.2.f.d.149.4		4	80.37	odd	4
800.2.d.a.401.1		2	16.3	odd	4
800.2.d.a.401.2		2	16.11	odd	4
800.2.d.d.401.1		2	80.59	odd	4
800.2.d.d.401.2		2	80.19	odd	4
800.2.f.d.49.1		4	80.67	even	4
800.2.f.d.49.2		4	80.43	even	4
800.2.f.d.49.3		4	80.27	even	4
800.2.f.d.49.4		4	80.3	even	4
1800.2.d.m.1549.1		4	240.197	even	4
1800.2.d.m.1549.2		4	240.173	even	4
1800.2.d.m.1549.3		4	240.77	even	4
1800.2.d.m.1549.4		4	240.53	even	4
1800.2.k.d.901.1		2	48.29	odd	4
1800.2.k.d.901.2		2	48.5	odd	4
1800.2.k.f.901.1		2	240.149	odd	4
1800.2.k.f.901.2		2	240.29	odd	4
6400.2.a.bg.1.1		2	8.5	even	2	inner
6400.2.a.bg.1.2		2	1.1	even	1	trivial
6400.2.a.bh.1.1		2	40.19	odd	2
6400.2.a.bh.1.2		2	20.19	odd	2
6400.2.a.cb.1.1		2	4.3	odd	2
6400.2.a.cb.1.2		2	8.3	odd	2
6400.2.a.cc.1.1		2	5.4	even	2
6400.2.a.cc.1.2		2	40.29	even	2
7200.2.d.m.2449.1		4	240.227	odd	4
7200.2.d.m.2449.2		4	240.107	odd	4
7200.2.d.m.2449.3		4	240.83	odd	4
7200.2.d.m.2449.4		4	240.203	odd	4
7200.2.k.b.3601.1		2	48.35	even	4
7200.2.k.b.3601.2		2	48.11	even	4
7200.2.k.i.3601.1		2	240.179	even	4
7200.2.k.i.3601.2		2	240.59	even	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:	\( N \)	\(=\)	\( 6400 = 2^{8} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6400.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64575 q^{3} -4.00000 q^{7} +4.00000 q^{9} +O(q^{10})\)	\(q+2.64575 q^{3} -4.00000 q^{7} +4.00000 q^{9} +2.64575 q^{11} -3.00000 q^{17} -2.64575 q^{19} -10.5830 q^{21} -4.00000 q^{23} +2.64575 q^{27} +4.00000 q^{31} +7.00000 q^{33} -10.5830 q^{37} +5.00000 q^{41} -5.29150 q^{43} -8.00000 q^{47} +9.00000 q^{49} -7.93725 q^{51} +10.5830 q^{53} -7.00000 q^{57} +5.29150 q^{59} -10.5830 q^{61} -16.0000 q^{63} -7.93725 q^{67} -10.5830 q^{69} -8.00000 q^{71} +7.00000 q^{73} -10.5830 q^{77} +4.00000 q^{79} -5.00000 q^{81} -7.93725 q^{83} +1.00000 q^{89} +10.5830 q^{93} -2.00000 q^{97} +10.5830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{7} + 8 q^{9}+O(q^{10}) \) 2 * q - 8 * q^7 + 8 * q^9	\( 2 q - 8 q^{7} + 8 q^{9} - 6 q^{17} - 8 q^{23} + 8 q^{31} + 14 q^{33} + 10 q^{41} - 16 q^{47} + 18 q^{49} - 14 q^{57} - 32 q^{63} - 16 q^{71} + 14 q^{73} + 8 q^{79} - 10 q^{81} + 2 q^{89} - 4 q^{97}+O(q^{100}) \) 2 * q - 8 * q^7 + 8 * q^9 - 6 * q^17 - 8 * q^23 + 8 * q^31 + 14 * q^33 + 10 * q^41 - 16 * q^47 + 18 * q^49 - 14 * q^57 - 32 * q^63 - 16 * q^71 + 14 * q^73 + 8 * q^79 - 10 * q^81 + 2 * q^89 - 4 * q^97

Introduction

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