LMFDB - Embedded newform 4368.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.56155 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.56155 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{13} +2.56155 q^{15} -3.12311 q^{17} +6.56155 q^{19} -1.00000 q^{21} +7.68466 q^{23} +1.56155 q^{25} +1.00000 q^{27} +0.561553 q^{29} +2.56155 q^{31} +2.00000 q^{33} -2.56155 q^{35} -7.12311 q^{37} -1.00000 q^{39} -1.12311 q^{41} +5.43845 q^{43} +2.56155 q^{45} -5.68466 q^{47} +1.00000 q^{49} -3.12311 q^{51} +4.56155 q^{53} +5.12311 q^{55} +6.56155 q^{57} +3.12311 q^{59} -6.00000 q^{61} -1.00000 q^{63} -2.56155 q^{65} -11.3693 q^{67} +7.68466 q^{69} +11.1231 q^{71} +14.8078 q^{73} +1.56155 q^{75} -2.00000 q^{77} -8.80776 q^{79} +1.00000 q^{81} +10.8078 q^{83} -8.00000 q^{85} +0.561553 q^{87} +1.43845 q^{89} +1.00000 q^{91} +2.56155 q^{93} +16.8078 q^{95} -3.43845 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{13} + q^{15} + 2 q^{17} + 9 q^{19} - 2 q^{21} + 3 q^{23} - q^{25} + 2 q^{27} - 3 q^{29} + q^{31} + 4 q^{33} - q^{35} - 6 q^{37} - 2 q^{39} + 6 q^{41} + 15 q^{43} + q^{45} + q^{47} + 2 q^{49} + 2 q^{51} + 5 q^{53} + 2 q^{55} + 9 q^{57} - 2 q^{59} - 12 q^{61} - 2 q^{63} - q^{65} + 2 q^{67} + 3 q^{69} + 14 q^{71} + 9 q^{73} - q^{75} - 4 q^{77} + 3 q^{79} + 2 q^{81} + q^{83} - 16 q^{85} - 3 q^{87} + 7 q^{89} + 2 q^{91} + q^{93} + 13 q^{95} - 11 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^13 + q^15 + 2 * q^17 + 9 * q^19 - 2 * q^21 + 3 * q^23 - q^25 + 2 * q^27 - 3 * q^29 + q^31 + 4 * q^33 - q^35 - 6 * q^37 - 2 * q^39 + 6 * q^41 + 15 * q^43 + q^45 + q^47 + 2 * q^49 + 2 * q^51 + 5 * q^53 + 2 * q^55 + 9 * q^57 - 2 * q^59 - 12 * q^61 - 2 * q^63 - q^65 + 2 * q^67 + 3 * q^69 + 14 * q^71 + 9 * q^73 - q^75 - 4 * q^77 + 3 * q^79 + 2 * q^81 + q^83 - 16 * q^85 - 3 * q^87 + 7 * q^89 + 2 * q^91 + q^93 + 13 * q^95 - 11 * q^97 + 4 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.56155	1.14556	0.572781	−	0.819709i	$-0.305865\pi$
$5$	2.56155	1.14556	0.572781	+	0.819709i	$0.305865\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	2.56155	0.661390
$16$	0	0
$17$	−3.12311	−0.757464	−0.378732	−	0.925506i	$-0.623640\pi$
$17$	−3.12311	−0.757464	−0.378732	+	0.925506i	$0.623640\pi$
$18$	0	0
$19$	6.56155	1.50532	0.752662	−	0.658407i	$-0.228770\pi$
$19$	6.56155	1.50532	0.752662	+	0.658407i	$0.228770\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	7.68466	1.60236	0.801181	−	0.598422i	$-0.204205\pi$
$23$	7.68466	1.60236	0.801181	+	0.598422i	$0.204205\pi$
$24$	0	0
$25$	1.56155	0.312311
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.561553	0.104278	0.0521389	−	0.998640i	$-0.483396\pi$
$29$	0.561553	0.104278	0.0521389	+	0.998640i	$0.483396\pi$
$30$	0	0
$31$	2.56155	0.460068	0.230034	−	0.973183i	$-0.426116\pi$
$31$	2.56155	0.460068	0.230034	+	0.973183i	$0.426116\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−2.56155	−0.432981
$36$	0	0
$37$	−7.12311	−1.17103	−0.585516	−	0.810661i	$-0.699108\pi$
$37$	−7.12311	−1.17103	−0.585516	+	0.810661i	$0.699108\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−1.12311	−0.175400	−0.0876998	−	0.996147i	$-0.527952\pi$
$41$	−1.12311	−0.175400	−0.0876998	+	0.996147i	$0.527952\pi$
$42$	0	0
$43$	5.43845	0.829355	0.414678	−	0.909968i	$-0.363894\pi$
$43$	5.43845	0.829355	0.414678	+	0.909968i	$0.363894\pi$
$44$	0	0
$45$	2.56155	0.381854
$46$	0	0
$47$	−5.68466	−0.829193	−0.414596	−	0.910005i	$-0.636077\pi$
$47$	−5.68466	−0.829193	−0.414596	+	0.910005i	$0.636077\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.12311	−0.437322
$52$	0	0
$53$	4.56155	0.626577	0.313289	−	0.949658i	$-0.398569\pi$
$53$	4.56155	0.626577	0.313289	+	0.949658i	$0.398569\pi$
$54$	0	0
$55$	5.12311	0.690799
$56$	0	0
$57$	6.56155	0.869099
$58$	0	0
$59$	3.12311	0.406594	0.203297	−	0.979117i	$-0.434834\pi$
$59$	3.12311	0.406594	0.203297	+	0.979117i	$0.434834\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−2.56155	−0.317722
$66$	0	0
$67$	−11.3693	−1.38898	−0.694492	−	0.719501i	$-0.744371\pi$
$67$	−11.3693	−1.38898	−0.694492	+	0.719501i	$0.744371\pi$
$68$	0	0
$69$	7.68466	0.925124
$70$	0	0
$71$	11.1231	1.32007	0.660035	−	0.751235i	$-0.270541\pi$
$71$	11.1231	1.32007	0.660035	+	0.751235i	$0.270541\pi$
$72$	0	0
$73$	14.8078	1.73312	0.866559	−	0.499075i	$-0.166327\pi$
$73$	14.8078	1.73312	0.866559	+	0.499075i	$0.166327\pi$
$74$	0	0
$75$	1.56155	0.180313
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−8.80776	−0.990951	−0.495475	−	0.868622i	$-0.665006\pi$
$79$	−8.80776	−0.990951	−0.495475	+	0.868622i	$0.665006\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.8078	1.18631	0.593153	−	0.805090i	$-0.297883\pi$
$83$	10.8078	1.18631	0.593153	+	0.805090i	$0.297883\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	0	0
$87$	0.561553	0.0602048
$88$	0	0
$89$	1.43845	0.152475	0.0762375	−	0.997090i	$-0.475709\pi$
$89$	1.43845	0.152475	0.0762375	+	0.997090i	$0.475709\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	2.56155	0.265621
$94$	0	0
$95$	16.8078	1.72444
$96$	0	0
$97$	−3.43845	−0.349121	−0.174561	−	0.984646i	$-0.555851\pi$
$97$	−3.43845	−0.349121	−0.174561	+	0.984646i	$0.555851\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	12.2462	1.21854	0.609272	−	0.792961i	$-0.291462\pi$
$101$	12.2462	1.21854	0.609272	+	0.792961i	$0.291462\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	−2.56155	−0.249982
$106$	0	0
$107$	−2.24621	−0.217149	−0.108575	−	0.994088i	$-0.534629\pi$
$107$	−2.24621	−0.217149	−0.108575	+	0.994088i	$0.534629\pi$
$108$	0	0
$109$	−0.246211	−0.0235828	−0.0117914	−	0.999930i	$-0.503753\pi$
$109$	−0.246211	−0.0235828	−0.0117914	+	0.999930i	$0.503753\pi$
$110$	0	0
$111$	−7.12311	−0.676095
$112$	0	0
$113$	1.68466	0.158479	0.0792397	−	0.996856i	$-0.474751\pi$
$113$	1.68466	0.158479	0.0792397	+	0.996856i	$0.474751\pi$
$114$	0	0
$115$	19.6847	1.83560
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	3.12311	0.286295
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−1.12311	−0.101267
$124$	0	0
$125$	−8.80776	−0.787790
$126$	0	0
$127$	18.2462	1.61909	0.809545	−	0.587058i	$-0.199714\pi$
$127$	18.2462	1.61909	0.809545	+	0.587058i	$0.199714\pi$
$128$	0	0
$129$	5.43845	0.478829
$130$	0	0
$131$	−12.4924	−1.09147	−0.545734	−	0.837958i	$-0.683749\pi$
$131$	−12.4924	−1.09147	−0.545734	+	0.837958i	$0.683749\pi$
$132$	0	0
$133$	−6.56155	−0.568959
$134$	0	0
$135$	2.56155	0.220463
$136$	0	0
$137$	14.2462	1.21714	0.608568	−	0.793502i	$-0.291744\pi$
$137$	14.2462	1.21714	0.608568	+	0.793502i	$0.291744\pi$
$138$	0	0
$139$	9.12311	0.773812	0.386906	−	0.922119i	$-0.373544\pi$
$139$	9.12311	0.773812	0.386906	+	0.922119i	$0.373544\pi$
$140$	0	0
$141$	−5.68466	−0.478735
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	1.43845	0.119457
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−13.1231	−1.07509	−0.537543	−	0.843236i	$-0.680648\pi$
$149$	−13.1231	−1.07509	−0.537543	+	0.843236i	$0.680648\pi$
$150$	0	0
$151$	−6.24621	−0.508309	−0.254155	−	0.967164i	$-0.581797\pi$
$151$	−6.24621	−0.508309	−0.254155	+	0.967164i	$0.581797\pi$
$152$	0	0
$153$	−3.12311	−0.252488
$154$	0	0
$155$	6.56155	0.527037
$156$	0	0
$157$	−9.36932	−0.747753	−0.373876	−	0.927479i	$-0.621972\pi$
$157$	−9.36932	−0.747753	−0.373876	+	0.927479i	$0.621972\pi$
$158$	0	0
$159$	4.56155	0.361755
$160$	0	0
$161$	−7.68466	−0.605636
$162$	0	0
$163$	5.75379	0.450672	0.225336	−	0.974281i	$-0.427652\pi$
$163$	5.75379	0.450672	0.225336	+	0.974281i	$0.427652\pi$
$164$	0	0
$165$	5.12311	0.398833
$166$	0	0
$167$	−10.8078	−0.836330	−0.418165	−	0.908371i	$-0.637327\pi$
$167$	−10.8078	−0.836330	−0.418165	+	0.908371i	$0.637327\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.56155	0.501774
$172$	0	0
$173$	17.3693	1.32056	0.660282	−	0.751017i	$-0.270437\pi$
$173$	17.3693	1.32056	0.660282	+	0.751017i	$0.270437\pi$
$174$	0	0
$175$	−1.56155	−0.118042
$176$	0	0
$177$	3.12311	0.234747
$178$	0	0
$179$	−12.8078	−0.957297	−0.478649	−	0.878007i	$-0.658873\pi$
$179$	−12.8078	−0.957297	−0.478649	+	0.878007i	$0.658873\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	−18.2462	−1.34149
$186$	0	0
$187$	−6.24621	−0.456768
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	−2.56155	−0.183437
$196$	0	0
$197$	−17.1231	−1.21997	−0.609985	−	0.792413i	$-0.708825\pi$
$197$	−17.1231	−1.21997	−0.609985	+	0.792413i	$0.708825\pi$
$198$	0	0
$199$	−12.4924	−0.885564	−0.442782	−	0.896629i	$-0.646009\pi$
$199$	−12.4924	−0.885564	−0.442782	+	0.896629i	$0.646009\pi$
$200$	0	0
$201$	−11.3693	−0.801930
$202$	0	0
$203$	−0.561553	−0.0394133
$204$	0	0
$205$	−2.87689	−0.200931
$206$	0	0
$207$	7.68466	0.534121
$208$	0	0
$209$	13.1231	0.907744
$210$	0	0
$211$	28.1771	1.93979	0.969895	−	0.243523i	$-0.0783031\pi$
$211$	28.1771	1.93979	0.969895	+	0.243523i	$0.0783031\pi$
$212$	0	0
$213$	11.1231	0.762143
$214$	0	0
$215$	13.9309	0.950077
$216$	0	0
$217$	−2.56155	−0.173890
$218$	0	0
$219$	14.8078	1.00062
$220$	0	0
$221$	3.12311	0.210083
$222$	0	0
$223$	15.6847	1.05032	0.525161	−	0.851003i	$-0.324005\pi$
$223$	15.6847	1.05032	0.525161	+	0.851003i	$0.324005\pi$
$224$	0	0
$225$	1.56155	0.104104
$226$	0	0
$227$	5.36932	0.356374	0.178187	−	0.983997i	$-0.442977\pi$
$227$	5.36932	0.356374	0.178187	+	0.983997i	$0.442977\pi$
$228$	0	0
$229$	−20.7386	−1.37045	−0.685224	−	0.728333i	$-0.740296\pi$
$229$	−20.7386	−1.37045	−0.685224	+	0.728333i	$0.740296\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	−1.19224	−0.0781060	−0.0390530	−	0.999237i	$-0.512434\pi$
$233$	−1.19224	−0.0781060	−0.0390530	+	0.999237i	$0.512434\pi$
$234$	0	0
$235$	−14.5616	−0.949891
$236$	0	0
$237$	−8.80776	−0.572126
$238$	0	0
$239$	−4.87689	−0.315460	−0.157730	−	0.987482i	$-0.550418\pi$
$239$	−4.87689	−0.315460	−0.157730	+	0.987482i	$0.550418\pi$
$240$	0	0
$241$	−15.9309	−1.02620	−0.513099	−	0.858330i	$-0.671503\pi$
$241$	−15.9309	−1.02620	−0.513099	+	0.858330i	$0.671503\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.56155	0.163652
$246$	0	0
$247$	−6.56155	−0.417502
$248$	0	0
$249$	10.8078	0.684914
$250$	0	0
$251$	−15.3693	−0.970103	−0.485051	−	0.874486i	$-0.661199\pi$
$251$	−15.3693	−0.970103	−0.485051	+	0.874486i	$0.661199\pi$
$252$	0	0
$253$	15.3693	0.966261
$254$	0	0
$255$	−8.00000	−0.500979
$256$	0	0
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	0	0
$259$	7.12311	0.442608
$260$	0	0
$261$	0.561553	0.0347592
$262$	0	0
$263$	−13.4384	−0.828650	−0.414325	−	0.910129i	$-0.635982\pi$
$263$	−13.4384	−0.828650	−0.414325	+	0.910129i	$0.635982\pi$
$264$	0	0
$265$	11.6847	0.717783
$266$	0	0
$267$	1.43845	0.0880315
$268$	0	0
$269$	−0.876894	−0.0534652	−0.0267326	−	0.999643i	$-0.508510\pi$
$269$	−0.876894	−0.0534652	−0.0267326	+	0.999643i	$0.508510\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	3.12311	0.188330
$276$	0	0
$277$	7.43845	0.446933	0.223466	−	0.974712i	$-0.428263\pi$
$277$	7.43845	0.446933	0.223466	+	0.974712i	$0.428263\pi$
$278$	0	0
$279$	2.56155	0.153356
$280$	0	0
$281$	1.12311	0.0669989	0.0334994	−	0.999439i	$-0.489335\pi$
$281$	1.12311	0.0669989	0.0334994	+	0.999439i	$0.489335\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	0	0
$285$	16.8078	0.995606
$286$	0	0
$287$	1.12311	0.0662948
$288$	0	0
$289$	−7.24621	−0.426248
$290$	0	0
$291$	−3.43845	−0.201565
$292$	0	0
$293$	20.3153	1.18683	0.593417	−	0.804895i	$-0.297778\pi$
$293$	20.3153	1.18683	0.593417	+	0.804895i	$0.297778\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	−7.68466	−0.444415
$300$	0	0
$301$	−5.43845	−0.313467
$302$	0	0
$303$	12.2462	0.703526
$304$	0	0
$305$	−15.3693	−0.880045
$306$	0	0
$307$	−23.0540	−1.31576	−0.657880	−	0.753123i	$-0.728547\pi$
$307$	−23.0540	−1.31576	−0.657880	+	0.753123i	$0.728547\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−33.1231	−1.87824	−0.939120	−	0.343591i	$-0.888357\pi$
$311$	−33.1231	−1.87824	−0.939120	+	0.343591i	$0.888357\pi$
$312$	0	0
$313$	16.2462	0.918290	0.459145	−	0.888361i	$-0.348156\pi$
$313$	16.2462	0.918290	0.459145	+	0.888361i	$0.348156\pi$
$314$	0	0
$315$	−2.56155	−0.144327
$316$	0	0
$317$	−16.4924	−0.926307	−0.463153	−	0.886278i	$-0.653282\pi$
$317$	−16.4924	−0.926307	−0.463153	+	0.886278i	$0.653282\pi$
$318$	0	0
$319$	1.12311	0.0628818
$320$	0	0
$321$	−2.24621	−0.125371
$322$	0	0
$323$	−20.4924	−1.14023
$324$	0	0
$325$	−1.56155	−0.0866194
$326$	0	0
$327$	−0.246211	−0.0136155
$328$	0	0
$329$	5.68466	0.313405
$330$	0	0
$331$	27.3693	1.50435	0.752177	−	0.658961i	$-0.229004\pi$
$331$	27.3693	1.50435	0.752177	+	0.658961i	$0.229004\pi$
$332$	0	0
$333$	−7.12311	−0.390344
$334$	0	0
$335$	−29.1231	−1.59117
$336$	0	0
$337$	1.68466	0.0917692	0.0458846	−	0.998947i	$-0.485389\pi$
$337$	1.68466	0.0917692	0.0458846	+	0.998947i	$0.485389\pi$
$338$	0	0
$339$	1.68466	0.0914981
$340$	0	0
$341$	5.12311	0.277432
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	19.6847	1.05979
$346$	0	0
$347$	18.7386	1.00594	0.502971	−	0.864303i	$-0.332240\pi$
$347$	18.7386	1.00594	0.502971	+	0.864303i	$0.332240\pi$
$348$	0	0
$349$	12.5616	0.672405	0.336202	−	0.941790i	$-0.390857\pi$
$349$	12.5616	0.672405	0.336202	+	0.941790i	$0.390857\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−27.3693	−1.45672	−0.728361	−	0.685194i	$-0.759718\pi$
$353$	−27.3693	−1.45672	−0.728361	+	0.685194i	$0.759718\pi$
$354$	0	0
$355$	28.4924	1.51222
$356$	0	0
$357$	3.12311	0.165292
$358$	0	0
$359$	−25.3693	−1.33894	−0.669471	−	0.742838i	$-0.733479\pi$
$359$	−25.3693	−1.33894	−0.669471	+	0.742838i	$0.733479\pi$
$360$	0	0
$361$	24.0540	1.26600
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	37.9309	1.98539
$366$	0	0
$367$	−10.8769	−0.567769	−0.283885	−	0.958858i	$-0.591623\pi$
$367$	−10.8769	−0.567769	−0.283885	+	0.958858i	$0.591623\pi$
$368$	0	0
$369$	−1.12311	−0.0584665
$370$	0	0
$371$	−4.56155	−0.236824
$372$	0	0
$373$	−32.2462	−1.66965	−0.834823	−	0.550519i	$-0.814430\pi$
$373$	−32.2462	−1.66965	−0.834823	+	0.550519i	$0.814430\pi$
$374$	0	0
$375$	−8.80776	−0.454831
$376$	0	0
$377$	−0.561553	−0.0289214
$378$	0	0
$379$	−9.61553	−0.493917	−0.246958	−	0.969026i	$-0.579431\pi$
$379$	−9.61553	−0.493917	−0.246958	+	0.969026i	$0.579431\pi$
$380$	0	0
$381$	18.2462	0.934782
$382$	0	0
$383$	−13.3693	−0.683140	−0.341570	−	0.939856i	$-0.610959\pi$
$383$	−13.3693	−0.683140	−0.341570	+	0.939856i	$0.610959\pi$
$384$	0	0
$385$	−5.12311	−0.261098
$386$	0	0
$387$	5.43845	0.276452
$388$	0	0
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	−12.4924	−0.630159
$394$	0	0
$395$	−22.5616	−1.13519
$396$	0	0
$397$	33.0540	1.65893	0.829466	−	0.558558i	$-0.188645\pi$
$397$	33.0540	1.65893	0.829466	+	0.558558i	$0.188645\pi$
$398$	0	0
$399$	−6.56155	−0.328489
$400$	0	0
$401$	−5.75379	−0.287330	−0.143665	−	0.989626i	$-0.545889\pi$
$401$	−5.75379	−0.287330	−0.143665	+	0.989626i	$0.545889\pi$
$402$	0	0
$403$	−2.56155	−0.127600
$404$	0	0
$405$	2.56155	0.127285
$406$	0	0
$407$	−14.2462	−0.706158
$408$	0	0
$409$	6.80776	0.336622	0.168311	−	0.985734i	$-0.446169\pi$
$409$	6.80776	0.336622	0.168311	+	0.985734i	$0.446169\pi$
$410$	0	0
$411$	14.2462	0.702714
$412$	0	0
$413$	−3.12311	−0.153678
$414$	0	0
$415$	27.6847	1.35899
$416$	0	0
$417$	9.12311	0.446760
$418$	0	0
$419$	−21.1231	−1.03193	−0.515966	−	0.856609i	$-0.672567\pi$
$419$	−21.1231	−1.03193	−0.515966	+	0.856609i	$0.672567\pi$
$420$	0	0
$421$	−19.1231	−0.932003	−0.466002	−	0.884784i	$-0.654306\pi$
$421$	−19.1231	−0.932003	−0.466002	+	0.884784i	$0.654306\pi$
$422$	0	0
$423$	−5.68466	−0.276398
$424$	0	0
$425$	−4.87689	−0.236564
$426$	0	0
$427$	6.00000	0.290360
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	−8.24621	−0.397206	−0.198603	−	0.980080i	$-0.563640\pi$
$431$	−8.24621	−0.397206	−0.198603	+	0.980080i	$0.563640\pi$
$432$	0	0
$433$	6.63068	0.318650	0.159325	−	0.987226i	$-0.449068\pi$
$433$	6.63068	0.318650	0.159325	+	0.987226i	$0.449068\pi$
$434$	0	0
$435$	1.43845	0.0689683
$436$	0	0
$437$	50.4233	2.41207
$438$	0	0
$439$	17.6155	0.840743	0.420372	−	0.907352i	$-0.361900\pi$
$439$	17.6155	0.840743	0.420372	+	0.907352i	$0.361900\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	24.1771	1.14869	0.574344	−	0.818614i	$-0.305257\pi$
$443$	24.1771	1.14869	0.574344	+	0.818614i	$0.305257\pi$
$444$	0	0
$445$	3.68466	0.174670
$446$	0	0
$447$	−13.1231	−0.620702
$448$	0	0
$449$	−18.2462	−0.861092	−0.430546	−	0.902569i	$-0.641679\pi$
$449$	−18.2462	−0.861092	−0.430546	+	0.902569i	$0.641679\pi$
$450$	0	0
$451$	−2.24621	−0.105770
$452$	0	0
$453$	−6.24621	−0.293473
$454$	0	0
$455$	2.56155	0.120087
$456$	0	0
$457$	3.12311	0.146093	0.0730464	−	0.997329i	$-0.476728\pi$
$457$	3.12311	0.146093	0.0730464	+	0.997329i	$0.476728\pi$
$458$	0	0
$459$	−3.12311	−0.145774
$460$	0	0
$461$	27.3693	1.27472	0.637358	−	0.770568i	$-0.280027\pi$
$461$	27.3693	1.27472	0.637358	+	0.770568i	$0.280027\pi$
$462$	0	0
$463$	35.8617	1.66664	0.833318	−	0.552794i	$-0.186438\pi$
$463$	35.8617	1.66664	0.833318	+	0.552794i	$0.186438\pi$
$464$	0	0
$465$	6.56155	0.304285
$466$	0	0
$467$	−7.36932	−0.341011	−0.170506	−	0.985357i	$-0.554540\pi$
$467$	−7.36932	−0.341011	−0.170506	+	0.985357i	$0.554540\pi$
$468$	0	0
$469$	11.3693	0.524986
$470$	0	0
$471$	−9.36932	−0.431715
$472$	0	0
$473$	10.8769	0.500120
$474$	0	0
$475$	10.2462	0.470128
$476$	0	0
$477$	4.56155	0.208859
$478$	0	0
$479$	−12.5616	−0.573952	−0.286976	−	0.957938i	$-0.592650\pi$
$479$	−12.5616	−0.573952	−0.286976	+	0.957938i	$0.592650\pi$
$480$	0	0
$481$	7.12311	0.324786
$482$	0	0
$483$	−7.68466	−0.349664
$484$	0	0
$485$	−8.80776	−0.399940
$486$	0	0
$487$	23.8617	1.08128	0.540639	−	0.841255i	$-0.318182\pi$
$487$	23.8617	1.08128	0.540639	+	0.841255i	$0.318182\pi$
$488$	0	0
$489$	5.75379	0.260195
$490$	0	0
$491$	−18.2462	−0.823440	−0.411720	−	0.911310i	$-0.635072\pi$
$491$	−18.2462	−0.823440	−0.411720	+	0.911310i	$0.635072\pi$
$492$	0	0
$493$	−1.75379	−0.0789867
$494$	0	0
$495$	5.12311	0.230266
$496$	0	0
$497$	−11.1231	−0.498939
$498$	0	0
$499$	−38.2462	−1.71214	−0.856068	−	0.516864i	$-0.827099\pi$
$499$	−38.2462	−1.71214	−0.856068	+	0.516864i	$0.827099\pi$
$500$	0	0
$501$	−10.8078	−0.482855
$502$	0	0
$503$	−27.3693	−1.22034	−0.610169	−	0.792271i	$-0.708898\pi$
$503$	−27.3693	−1.22034	−0.610169	+	0.792271i	$0.708898\pi$
$504$	0	0
$505$	31.3693	1.39592
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−41.9309	−1.85855	−0.929277	−	0.369385i	$-0.879568\pi$
$509$	−41.9309	−1.85855	−0.929277	+	0.369385i	$0.879568\pi$
$510$	0	0
$511$	−14.8078	−0.655057
$512$	0	0
$513$	6.56155	0.289700
$514$	0	0
$515$	20.4924	0.903004
$516$	0	0
$517$	−11.3693	−0.500022
$518$	0	0
$519$	17.3693	0.762428
$520$	0	0
$521$	−15.1231	−0.662555	−0.331278	−	0.943533i	$-0.607480\pi$
$521$	−15.1231	−0.662555	−0.331278	+	0.943533i	$0.607480\pi$
$522$	0	0
$523$	−29.6155	−1.29500	−0.647498	−	0.762067i	$-0.724185\pi$
$523$	−29.6155	−1.29500	−0.647498	+	0.762067i	$0.724185\pi$
$524$	0	0
$525$	−1.56155	−0.0681518
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	36.0540	1.56756
$530$	0	0
$531$	3.12311	0.135531
$532$	0	0
$533$	1.12311	0.0486471
$534$	0	0
$535$	−5.75379	−0.248758
$536$	0	0
$537$	−12.8078	−0.552696
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	27.6155	1.18728	0.593642	−	0.804729i	$-0.297689\pi$
$541$	27.6155	1.18728	0.593642	+	0.804729i	$0.297689\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	0	0
$545$	−0.630683	−0.0270155
$546$	0	0
$547$	15.0540	0.643662	0.321831	−	0.946797i	$-0.395702\pi$
$547$	15.0540	0.643662	0.321831	+	0.946797i	$0.395702\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	3.68466	0.156972
$552$	0	0
$553$	8.80776	0.374544
$554$	0	0
$555$	−18.2462	−0.774509
$556$	0	0
$557$	−12.4924	−0.529321	−0.264660	−	0.964342i	$-0.585260\pi$
$557$	−12.4924	−0.529321	−0.264660	+	0.964342i	$0.585260\pi$
$558$	0	0
$559$	−5.43845	−0.230022
$560$	0	0
$561$	−6.24621	−0.263715
$562$	0	0
$563$	26.7386	1.12690	0.563450	−	0.826150i	$-0.309474\pi$
$563$	26.7386	1.12690	0.563450	+	0.826150i	$0.309474\pi$
$564$	0	0
$565$	4.31534	0.181548
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	19.3002	0.809106	0.404553	−	0.914515i	$-0.367427\pi$
$569$	19.3002	0.809106	0.404553	+	0.914515i	$0.367427\pi$
$570$	0	0
$571$	33.9309	1.41996	0.709981	−	0.704220i	$-0.248703\pi$
$571$	33.9309	1.41996	0.709981	+	0.704220i	$0.248703\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	10.0000	0.415586
$580$	0	0
$581$	−10.8078	−0.448382
$582$	0	0
$583$	9.12311	0.377840
$584$	0	0
$585$	−2.56155	−0.105907
$586$	0	0
$587$	18.8078	0.776280	0.388140	−	0.921601i	$-0.373118\pi$
$587$	18.8078	0.776280	0.388140	+	0.921601i	$0.373118\pi$
$588$	0	0
$589$	16.8078	0.692552
$590$	0	0
$591$	−17.1231	−0.704350
$592$	0	0
$593$	22.5616	0.926492	0.463246	−	0.886230i	$-0.346685\pi$
$593$	22.5616	0.926492	0.463246	+	0.886230i	$0.346685\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	0	0
$597$	−12.4924	−0.511281
$598$	0	0
$599$	−24.3153	−0.993498	−0.496749	−	0.867894i	$-0.665473\pi$
$599$	−24.3153	−0.993498	−0.496749	+	0.867894i	$0.665473\pi$
$600$	0	0
$601$	−20.2462	−0.825860	−0.412930	−	0.910763i	$-0.635495\pi$
$601$	−20.2462	−0.825860	−0.412930	+	0.910763i	$0.635495\pi$
$602$	0	0
$603$	−11.3693	−0.462994
$604$	0	0
$605$	−17.9309	−0.728994
$606$	0	0
$607$	−25.6155	−1.03970	−0.519851	−	0.854257i	$-0.674013\pi$
$607$	−25.6155	−1.03970	−0.519851	+	0.854257i	$0.674013\pi$
$608$	0	0
$609$	−0.561553	−0.0227553
$610$	0	0
$611$	5.68466	0.229977
$612$	0	0
$613$	−9.36932	−0.378423	−0.189212	−	0.981936i	$-0.560593\pi$
$613$	−9.36932	−0.378423	−0.189212	+	0.981936i	$0.560593\pi$
$614$	0	0
$615$	−2.87689	−0.116008
$616$	0	0
$617$	4.63068	0.186424	0.0932121	−	0.995646i	$-0.470287\pi$
$617$	4.63068	0.186424	0.0932121	+	0.995646i	$0.470287\pi$
$618$	0	0
$619$	−28.9848	−1.16500	−0.582500	−	0.812831i	$-0.697925\pi$
$619$	−28.9848	−1.16500	−0.582500	+	0.812831i	$0.697925\pi$
$620$	0	0
$621$	7.68466	0.308375
$622$	0	0
$623$	−1.43845	−0.0576302
$624$	0	0
$625$	−30.3693	−1.21477
$626$	0	0
$627$	13.1231	0.524086
$628$	0	0
$629$	22.2462	0.887015
$630$	0	0
$631$	14.8769	0.592240	0.296120	−	0.955151i	$-0.404307\pi$
$631$	14.8769	0.592240	0.296120	+	0.955151i	$0.404307\pi$
$632$	0	0
$633$	28.1771	1.11994
$634$	0	0
$635$	46.7386	1.85477
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	11.1231	0.440023
$640$	0	0
$641$	−30.8078	−1.21683	−0.608417	−	0.793618i	$-0.708195\pi$
$641$	−30.8078	−1.21683	−0.608417	+	0.793618i	$0.708195\pi$
$642$	0	0
$643$	22.2462	0.877305	0.438652	−	0.898657i	$-0.355456\pi$
$643$	22.2462	0.877305	0.438652	+	0.898657i	$0.355456\pi$
$644$	0	0
$645$	13.9309	0.548527
$646$	0	0
$647$	−8.63068	−0.339307	−0.169654	−	0.985504i	$-0.554265\pi$
$647$	−8.63068	−0.339307	−0.169654	+	0.985504i	$0.554265\pi$
$648$	0	0
$649$	6.24621	0.245185
$650$	0	0
$651$	−2.56155	−0.100395
$652$	0	0
$653$	−10.4924	−0.410600	−0.205300	−	0.978699i	$-0.565817\pi$
$653$	−10.4924	−0.410600	−0.205300	+	0.978699i	$0.565817\pi$
$654$	0	0
$655$	−32.0000	−1.25034
$656$	0	0
$657$	14.8078	0.577706
$658$	0	0
$659$	36.8078	1.43383	0.716913	−	0.697162i	$-0.245554\pi$
$659$	36.8078	1.43383	0.716913	+	0.697162i	$0.245554\pi$
$660$	0	0
$661$	−45.6847	−1.77693	−0.888464	−	0.458947i	$-0.848227\pi$
$661$	−45.6847	−1.77693	−0.888464	+	0.458947i	$0.848227\pi$
$662$	0	0
$663$	3.12311	0.121291
$664$	0	0
$665$	−16.8078	−0.651777
$666$	0	0
$667$	4.31534	0.167091
$668$	0	0
$669$	15.6847	0.606404
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	−4.56155	−0.175835	−0.0879175	−	0.996128i	$-0.528021\pi$
$673$	−4.56155	−0.175835	−0.0879175	+	0.996128i	$0.528021\pi$
$674$	0	0
$675$	1.56155	0.0601042
$676$	0	0
$677$	−22.4924	−0.864454	−0.432227	−	0.901765i	$-0.642272\pi$
$677$	−22.4924	−0.864454	−0.432227	+	0.901765i	$0.642272\pi$
$678$	0	0
$679$	3.43845	0.131955
$680$	0	0
$681$	5.36932	0.205753
$682$	0	0
$683$	−15.6155	−0.597512	−0.298756	−	0.954330i	$-0.596572\pi$
$683$	−15.6155	−0.597512	−0.298756	+	0.954330i	$0.596572\pi$
$684$	0	0
$685$	36.4924	1.39430
$686$	0	0
$687$	−20.7386	−0.791228
$688$	0	0
$689$	−4.56155	−0.173781
$690$	0	0
$691$	39.6847	1.50968	0.754838	−	0.655912i	$-0.227716\pi$
$691$	39.6847	1.50968	0.754838	+	0.655912i	$0.227716\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	23.3693	0.886449
$696$	0	0
$697$	3.50758	0.132859
$698$	0	0
$699$	−1.19224	−0.0450945
$700$	0	0
$701$	−32.5616	−1.22983	−0.614916	−	0.788592i	$-0.710810\pi$
$701$	−32.5616	−1.22983	−0.614916	+	0.788592i	$0.710810\pi$
$702$	0	0
$703$	−46.7386	−1.76278
$704$	0	0
$705$	−14.5616	−0.548420
$706$	0	0
$707$	−12.2462	−0.460566
$708$	0	0
$709$	−28.7386	−1.07930	−0.539651	−	0.841889i	$-0.681444\pi$
$709$	−28.7386	−1.07930	−0.539651	+	0.841889i	$0.681444\pi$
$710$	0	0
$711$	−8.80776	−0.330317
$712$	0	0
$713$	19.6847	0.737196
$714$	0	0
$715$	−5.12311	−0.191593
$716$	0	0
$717$	−4.87689	−0.182131
$718$	0	0
$719$	2.38447	0.0889258	0.0444629	−	0.999011i	$-0.485842\pi$
$719$	2.38447	0.0889258	0.0444629	+	0.999011i	$0.485842\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	−15.9309	−0.592475
$724$	0	0
$725$	0.876894	0.0325670
$726$	0	0
$727$	−5.12311	−0.190005	−0.0950027	−	0.995477i	$-0.530286\pi$
$727$	−5.12311	−0.190005	−0.0950027	+	0.995477i	$0.530286\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.9848	−0.628207
$732$	0	0
$733$	−6.17708	−0.228156	−0.114078	−	0.993472i	$-0.536391\pi$
$733$	−6.17708	−0.228156	−0.114078	+	0.993472i	$0.536391\pi$
$734$	0	0
$735$	2.56155	0.0944843
$736$	0	0
$737$	−22.7386	−0.837588
$738$	0	0
$739$	52.3542	1.92588	0.962939	−	0.269718i	$-0.0869303\pi$
$739$	52.3542	1.92588	0.962939	+	0.269718i	$0.0869303\pi$
$740$	0	0
$741$	−6.56155	−0.241045
$742$	0	0
$743$	−35.6155	−1.30661	−0.653304	−	0.757096i	$-0.726617\pi$
$743$	−35.6155	−1.30661	−0.653304	+	0.757096i	$0.726617\pi$
$744$	0	0
$745$	−33.6155	−1.23158
$746$	0	0
$747$	10.8078	0.395435
$748$	0	0
$749$	2.24621	0.0820748
$750$	0	0
$751$	−27.0540	−0.987214	−0.493607	−	0.869685i	$-0.664322\pi$
$751$	−27.0540	−0.987214	−0.493607	+	0.869685i	$0.664322\pi$
$752$	0	0
$753$	−15.3693	−0.560089
$754$	0	0
$755$	−16.0000	−0.582300
$756$	0	0
$757$	−18.3153	−0.665682	−0.332841	−	0.942983i	$-0.608007\pi$
$757$	−18.3153	−0.665682	−0.332841	+	0.942983i	$0.608007\pi$
$758$	0	0
$759$	15.3693	0.557871
$760$	0	0
$761$	−17.9309	−0.649994	−0.324997	−	0.945715i	$-0.605363\pi$
$761$	−17.9309	−0.649994	−0.324997	+	0.945715i	$0.605363\pi$
$762$	0	0
$763$	0.246211	0.00891345
$764$	0	0
$765$	−8.00000	−0.289241
$766$	0	0
$767$	−3.12311	−0.112769
$768$	0	0
$769$	−2.31534	−0.0834934	−0.0417467	−	0.999128i	$-0.513292\pi$
$769$	−2.31534	−0.0834934	−0.0417467	+	0.999128i	$0.513292\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	0	0
$773$	47.3693	1.70376	0.851878	−	0.523740i	$-0.175464\pi$
$773$	47.3693	1.70376	0.851878	+	0.523740i	$0.175464\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	7.12311	0.255540
$778$	0	0
$779$	−7.36932	−0.264033
$780$	0	0
$781$	22.2462	0.796032
$782$	0	0
$783$	0.561553	0.0200683
$784$	0	0
$785$	−24.0000	−0.856597
$786$	0	0
$787$	−15.6847	−0.559098	−0.279549	−	0.960131i	$-0.590185\pi$
$787$	−15.6847	−0.559098	−0.279549	+	0.960131i	$0.590185\pi$
$788$	0	0
$789$	−13.4384	−0.478421
$790$	0	0
$791$	−1.68466	−0.0598996
$792$	0	0
$793$	6.00000	0.213066
$794$	0	0
$795$	11.6847	0.414412
$796$	0	0
$797$	46.9848	1.66429	0.832144	−	0.554559i	$-0.187113\pi$
$797$	46.9848	1.66429	0.832144	+	0.554559i	$0.187113\pi$
$798$	0	0
$799$	17.7538	0.628084
$800$	0	0
$801$	1.43845	0.0508250
$802$	0	0
$803$	29.6155	1.04511
$804$	0	0
$805$	−19.6847	−0.693793
$806$	0	0
$807$	−0.876894	−0.0308681
$808$	0	0
$809$	−48.5616	−1.70733	−0.853667	−	0.520820i	$-0.825626\pi$
$809$	−48.5616	−1.70733	−0.853667	+	0.520820i	$0.825626\pi$
$810$	0	0
$811$	−5.75379	−0.202043	−0.101021	−	0.994884i	$-0.532211\pi$
$811$	−5.75379	−0.202043	−0.101021	+	0.994884i	$0.532211\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	0	0
$815$	14.7386	0.516272
$816$	0	0
$817$	35.6847	1.24845
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	−42.8769	−1.49641	−0.748207	−	0.663465i	$-0.769085\pi$
$821$	−42.8769	−1.49641	−0.748207	+	0.663465i	$0.769085\pi$
$822$	0	0
$823$	−52.4924	−1.82977	−0.914885	−	0.403714i	$-0.867719\pi$
$823$	−52.4924	−1.82977	−0.914885	+	0.403714i	$0.867719\pi$
$824$	0	0
$825$	3.12311	0.108733
$826$	0	0
$827$	32.1080	1.11650	0.558251	−	0.829672i	$-0.311472\pi$
$827$	32.1080	1.11650	0.558251	+	0.829672i	$0.311472\pi$
$828$	0	0
$829$	−18.4924	−0.642268	−0.321134	−	0.947034i	$-0.604064\pi$
$829$	−18.4924	−0.642268	−0.321134	+	0.947034i	$0.604064\pi$
$830$	0	0
$831$	7.43845	0.258037
$832$	0	0
$833$	−3.12311	−0.108209
$834$	0	0
$835$	−27.6847	−0.958067
$836$	0	0
$837$	2.56155	0.0885402
$838$	0	0
$839$	−19.1231	−0.660203	−0.330101	−	0.943945i	$-0.607083\pi$
$839$	−19.1231	−0.660203	−0.330101	+	0.943945i	$0.607083\pi$
$840$	0	0
$841$	−28.6847	−0.989126
$842$	0	0
$843$	1.12311	0.0386818
$844$	0	0
$845$	2.56155	0.0881201
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	12.0000	0.411839
$850$	0	0
$851$	−54.7386	−1.87642
$852$	0	0
$853$	21.1922	0.725608	0.362804	−	0.931865i	$-0.381819\pi$
$853$	21.1922	0.725608	0.362804	+	0.931865i	$0.381819\pi$
$854$	0	0
$855$	16.8078	0.574813
$856$	0	0
$857$	−31.7538	−1.08469	−0.542344	−	0.840156i	$-0.682463\pi$
$857$	−31.7538	−1.08469	−0.542344	+	0.840156i	$0.682463\pi$
$858$	0	0
$859$	6.87689	0.234637	0.117318	−	0.993094i	$-0.462570\pi$
$859$	6.87689	0.234637	0.117318	+	0.993094i	$0.462570\pi$
$860$	0	0
$861$	1.12311	0.0382753
$862$	0	0
$863$	−12.3845	−0.421572	−0.210786	−	0.977532i	$-0.567602\pi$
$863$	−12.3845	−0.421572	−0.210786	+	0.977532i	$0.567602\pi$
$864$	0	0
$865$	44.4924	1.51279
$866$	0	0
$867$	−7.24621	−0.246094
$868$	0	0
$869$	−17.6155	−0.597566
$870$	0	0
$871$	11.3693	0.385235
$872$	0	0
$873$	−3.43845	−0.116374
$874$	0	0
$875$	8.80776	0.297757
$876$	0	0
$877$	7.61553	0.257158	0.128579	−	0.991699i	$-0.458958\pi$
$877$	7.61553	0.257158	0.128579	+	0.991699i	$0.458958\pi$
$878$	0	0
$879$	20.3153	0.685219
$880$	0	0
$881$	−48.7386	−1.64205	−0.821023	−	0.570895i	$-0.806596\pi$
$881$	−48.7386	−1.64205	−0.821023	+	0.570895i	$0.806596\pi$
$882$	0	0
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	0	0
$885$	8.00000	0.268917
$886$	0	0
$887$	−24.9848	−0.838909	−0.419454	−	0.907776i	$-0.637779\pi$
$887$	−24.9848	−0.838909	−0.419454	+	0.907776i	$0.637779\pi$
$888$	0	0
$889$	−18.2462	−0.611958
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	−37.3002	−1.24820
$894$	0	0
$895$	−32.8078	−1.09664
$896$	0	0
$897$	−7.68466	−0.256583
$898$	0	0
$899$	1.43845	0.0479749
$900$	0	0
$901$	−14.2462	−0.474610
$902$	0	0
$903$	−5.43845	−0.180980
$904$	0	0
$905$	−25.6155	−0.851489
$906$	0	0
$907$	−23.0540	−0.765495	−0.382747	−	0.923853i	$-0.625022\pi$
$907$	−23.0540	−0.765495	−0.382747	+	0.923853i	$0.625022\pi$
$908$	0	0
$909$	12.2462	0.406181
$910$	0	0
$911$	−43.6847	−1.44734	−0.723669	−	0.690148i	$-0.757546\pi$
$911$	−43.6847	−1.44734	−0.723669	+	0.690148i	$0.757546\pi$
$912$	0	0
$913$	21.6155	0.715370
$914$	0	0
$915$	−15.3693	−0.508094
$916$	0	0
$917$	12.4924	0.412536
$918$	0	0
$919$	−52.4924	−1.73157	−0.865783	−	0.500420i	$-0.833179\pi$
$919$	−52.4924	−1.73157	−0.865783	+	0.500420i	$0.833179\pi$
$920$	0	0
$921$	−23.0540	−0.759654
$922$	0	0
$923$	−11.1231	−0.366121
$924$	0	0
$925$	−11.1231	−0.365725
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	37.7926	1.23994	0.619968	−	0.784627i	$-0.287146\pi$
$929$	37.7926	1.23994	0.619968	+	0.784627i	$0.287146\pi$
$930$	0	0
$931$	6.56155	0.215046
$932$	0	0
$933$	−33.1231	−1.08440
$934$	0	0
$935$	−16.0000	−0.523256
$936$	0	0
$937$	−26.6307	−0.869986	−0.434993	−	0.900434i	$-0.643249\pi$
$937$	−26.6307	−0.869986	−0.434993	+	0.900434i	$0.643249\pi$
$938$	0	0
$939$	16.2462	0.530175
$940$	0	0
$941$	21.9309	0.714926	0.357463	−	0.933927i	$-0.383642\pi$
$941$	21.9309	0.714926	0.357463	+	0.933927i	$0.383642\pi$
$942$	0	0
$943$	−8.63068	−0.281054
$944$	0	0
$945$	−2.56155	−0.0833273
$946$	0	0
$947$	1.50758	0.0489897	0.0244948	−	0.999700i	$-0.492202\pi$
$947$	1.50758	0.0489897	0.0244948	+	0.999700i	$0.492202\pi$
$948$	0	0
$949$	−14.8078	−0.480680
$950$	0	0
$951$	−16.4924	−0.534803
$952$	0	0
$953$	−16.5616	−0.536481	−0.268241	−	0.963352i	$-0.586442\pi$
$953$	−16.5616	−0.536481	−0.268241	+	0.963352i	$0.586442\pi$
$954$	0	0
$955$	20.4924	0.663119
$956$	0	0
$957$	1.12311	0.0363048
$958$	0	0
$959$	−14.2462	−0.460034
$960$	0	0
$961$	−24.4384	−0.788337
$962$	0	0
$963$	−2.24621	−0.0723831
$964$	0	0
$965$	25.6155	0.824593
$966$	0	0
$967$	−54.6004	−1.75583	−0.877915	−	0.478817i	$-0.841066\pi$
$967$	−54.6004	−1.75583	−0.877915	+	0.478817i	$0.841066\pi$
$968$	0	0
$969$	−20.4924	−0.658311
$970$	0	0
$971$	35.3693	1.13506	0.567528	−	0.823354i	$-0.307900\pi$
$971$	35.3693	1.13506	0.567528	+	0.823354i	$0.307900\pi$
$972$	0	0
$973$	−9.12311	−0.292473
$974$	0	0
$975$	−1.56155	−0.0500097
$976$	0	0
$977$	−24.6307	−0.788005	−0.394003	−	0.919109i	$-0.628910\pi$
$977$	−24.6307	−0.788005	−0.394003	+	0.919109i	$0.628910\pi$
$978$	0	0
$979$	2.87689	0.0919459
$980$	0	0
$981$	−0.246211	−0.00786092
$982$	0	0
$983$	−38.8078	−1.23778	−0.618888	−	0.785479i	$-0.712416\pi$
$983$	−38.8078	−1.23778	−0.618888	+	0.785479i	$0.712416\pi$
$984$	0	0
$985$	−43.8617	−1.39755
$986$	0	0
$987$	5.68466	0.180945
$988$	0	0
$989$	41.7926	1.32893
$990$	0	0
$991$	−21.7538	−0.691032	−0.345516	−	0.938413i	$-0.612296\pi$
$991$	−21.7538	−0.691032	−0.345516	+	0.938413i	$0.612296\pi$
$992$	0	0
$993$	27.3693	0.868539
$994$	0	0
$995$	−32.0000	−1.01447
$996$	0	0
$997$	7.61553	0.241186	0.120593	−	0.992702i	$-0.461520\pi$
$997$	7.61553	0.241186	0.120593	+	0.992702i	$0.461520\pi$
$998$	0	0
$999$	−7.12311	−0.225365

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bk.1.2		2
4.3	odd	2		2184.2.a.p.1.2	✓	2
12.11	even	2		6552.2.a.be.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.a.p.1.2	✓	2	4.3	odd	2
4368.2.a.bk.1.2		2	1.1	even	1	trivial
6552.2.a.be.1.1		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.56155 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.56155 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{13} +2.56155 q^{15} -3.12311 q^{17} +6.56155 q^{19} -1.00000 q^{21} +7.68466 q^{23} +1.56155 q^{25} +1.00000 q^{27} +0.561553 q^{29} +2.56155 q^{31} +2.00000 q^{33} -2.56155 q^{35} -7.12311 q^{37} -1.00000 q^{39} -1.12311 q^{41} +5.43845 q^{43} +2.56155 q^{45} -5.68466 q^{47} +1.00000 q^{49} -3.12311 q^{51} +4.56155 q^{53} +5.12311 q^{55} +6.56155 q^{57} +3.12311 q^{59} -6.00000 q^{61} -1.00000 q^{63} -2.56155 q^{65} -11.3693 q^{67} +7.68466 q^{69} +11.1231 q^{71} +14.8078 q^{73} +1.56155 q^{75} -2.00000 q^{77} -8.80776 q^{79} +1.00000 q^{81} +10.8078 q^{83} -8.00000 q^{85} +0.561553 q^{87} +1.43845 q^{89} +1.00000 q^{91} +2.56155 q^{93} +16.8078 q^{95} -3.43845 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{13} + q^{15} + 2 q^{17} + 9 q^{19} - 2 q^{21} + 3 q^{23} - q^{25} + 2 q^{27} - 3 q^{29} + q^{31} + 4 q^{33} - q^{35} - 6 q^{37} - 2 q^{39} + 6 q^{41} + 15 q^{43} + q^{45} + q^{47} + 2 q^{49} + 2 q^{51} + 5 q^{53} + 2 q^{55} + 9 q^{57} - 2 q^{59} - 12 q^{61} - 2 q^{63} - q^{65} + 2 q^{67} + 3 q^{69} + 14 q^{71} + 9 q^{73} - q^{75} - 4 q^{77} + 3 q^{79} + 2 q^{81} + q^{83} - 16 q^{85} - 3 q^{87} + 7 q^{89} + 2 q^{91} + q^{93} + 13 q^{95} - 11 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^13 + q^15 + 2 * q^17 + 9 * q^19 - 2 * q^21 + 3 * q^23 - q^25 + 2 * q^27 - 3 * q^29 + q^31 + 4 * q^33 - q^35 - 6 * q^37 - 2 * q^39 + 6 * q^41 + 15 * q^43 + q^45 + q^47 + 2 * q^49 + 2 * q^51 + 5 * q^53 + 2 * q^55 + 9 * q^57 - 2 * q^59 - 12 * q^61 - 2 * q^63 - q^65 + 2 * q^67 + 3 * q^69 + 14 * q^71 + 9 * q^73 - q^75 - 4 * q^77 + 3 * q^79 + 2 * q^81 + q^83 - 16 * q^85 - 3 * q^87 + 7 * q^89 + 2 * q^91 + q^93 + 13 * q^95 - 11 * q^97 + 4 * q^99

Introduction

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