LMFDB - Embedded newform 2061.2.a.j.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2061,2,Mod(1,2061)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2061, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2061.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2061 = 3^{2} \cdot 229 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2061.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:	$16.4571678566$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	2061.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.754652 q^{2} -1.43050 q^{4} -1.13990 q^{5} -4.86899 q^{7} +2.58883 q^{8} +O(q^{10})$	\(q-0.754652 q^{2} -1.43050 q^{4} -1.13990 q^{5} -4.86899 q^{7} +2.58883 q^{8} +0.860226 q^{10} -4.52452 q^{11} -5.04117 q^{13} +3.67439 q^{14} +0.907335 q^{16} -1.16281 q^{17} +1.20147 q^{19} +1.63062 q^{20} +3.41444 q^{22} -3.74633 q^{23} -3.70063 q^{25} +3.80433 q^{26} +6.96509 q^{28} -9.35204 q^{29} +0.624005 q^{31} -5.86239 q^{32} +0.877514 q^{34} +5.55015 q^{35} -3.07517 q^{37} -0.906694 q^{38} -2.95100 q^{40} -5.08728 q^{41} +3.31430 q^{43} +6.47233 q^{44} +2.82718 q^{46} +0.612741 q^{47} +16.7070 q^{49} +2.79269 q^{50} +7.21140 q^{52} +1.22775 q^{53} +5.15749 q^{55} -12.6050 q^{56} +7.05753 q^{58} -9.45896 q^{59} +4.15041 q^{61} -0.470906 q^{62} +2.60939 q^{64} +5.74642 q^{65} +6.38803 q^{67} +1.66340 q^{68} -4.18843 q^{70} +5.50785 q^{71} +4.25781 q^{73} +2.32068 q^{74} -1.71871 q^{76} +22.0298 q^{77} -13.8023 q^{79} -1.03427 q^{80} +3.83912 q^{82} -7.95930 q^{83} +1.32548 q^{85} -2.50114 q^{86} -11.7132 q^{88} -11.2451 q^{89} +24.5454 q^{91} +5.35913 q^{92} -0.462406 q^{94} -1.36956 q^{95} -2.63245 q^{97} -12.6080 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 36 q^{4} + 12 q^{7}+O(q^{10}) $ 24 * q + 36 * q^4 + 12 * q^7	$ 24 q + 36 q^{4} + 12 q^{7} + 12 q^{10} + 14 q^{13} + 52 q^{16} + 50 q^{19} + 14 q^{22} + 34 q^{25} + 26 q^{28} + 34 q^{31} + 28 q^{34} + 30 q^{37} + 40 q^{40} + 48 q^{43} + 54 q^{46} + 64 q^{49} + 60 q^{52} + 20 q^{55} + 20 q^{58} + 24 q^{61} + 98 q^{64} + 52 q^{67} - 28 q^{70} + 20 q^{73} + 90 q^{76} + 48 q^{79} + 18 q^{82} + 16 q^{85} - 38 q^{88} + 42 q^{91} - 30 q^{94} - 6 q^{97}+O(q^{100}) $ 24 * q + 36 * q^4 + 12 * q^7 + 12 * q^10 + 14 * q^13 + 52 * q^16 + 50 * q^19 + 14 * q^22 + 34 * q^25 + 26 * q^28 + 34 * q^31 + 28 * q^34 + 30 * q^37 + 40 * q^40 + 48 * q^43 + 54 * q^46 + 64 * q^49 + 60 * q^52 + 20 * q^55 + 20 * q^58 + 24 * q^61 + 98 * q^64 + 52 * q^67 - 28 * q^70 + 20 * q^73 + 90 * q^76 + 48 * q^79 + 18 * q^82 + 16 * q^85 - 38 * q^88 + 42 * q^91 - 30 * q^94 - 6 * 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 $

$2$	−0.754652	−0.533619	−0.266810	−	0.963749i	$-0.585969\pi$
$2$	−0.754652	−0.533619	−0.266810	+	0.963749i	$0.585969\pi$
$3$	0	0
$3$	0	0
$4$	−1.43050	−0.715250
$5$	−1.13990	−0.509778	−0.254889	−	0.966970i	$-0.582039\pi$
$5$	−1.13990	−0.509778	−0.254889	+	0.966970i	$0.582039\pi$
$6$	0	0
$7$	−4.86899	−1.84030	−0.920152	−	0.391561i	$-0.871935\pi$
$7$	−4.86899	−1.84030	−0.920152	+	0.391561i	$0.871935\pi$
$8$	2.58883	0.915291
$9$	0	0
$10$	0.860226	0.272027
$11$	−4.52452	−1.36419	−0.682097	−	0.731261i	$-0.738932\pi$
$11$	−4.52452	−1.36419	−0.682097	+	0.731261i	$0.738932\pi$
$12$	0	0
$13$	−5.04117	−1.39817	−0.699085	−	0.715039i	$-0.746409\pi$
$13$	−5.04117	−1.39817	−0.699085	+	0.715039i	$0.746409\pi$
$14$	3.67439	0.982022
$15$	0	0
$16$	0.907335	0.226834
$17$	−1.16281	−0.282022	−0.141011	−	0.990008i	$-0.545035\pi$
$17$	−1.16281	−0.282022	−0.141011	+	0.990008i	$0.545035\pi$
$18$	0	0
$19$	1.20147	0.275637	0.137818	−	0.990458i	$-0.455991\pi$
$19$	1.20147	0.275637	0.137818	+	0.990458i	$0.455991\pi$
$20$	1.63062	0.364619
$21$	0	0
$22$	3.41444	0.727961
$23$	−3.74633	−0.781165	−0.390582	−	0.920568i	$-0.627726\pi$
$23$	−3.74633	−0.781165	−0.390582	+	0.920568i	$0.627726\pi$
$24$	0	0
$25$	−3.70063	−0.740127
$26$	3.80433	0.746090
$27$	0	0
$28$	6.96509	1.31628
$29$	−9.35204	−1.73663	−0.868315	−	0.496014i	$-0.834797\pi$
$29$	−9.35204	−1.73663	−0.868315	+	0.496014i	$0.834797\pi$
$30$	0	0
$31$	0.624005	0.112075	0.0560373	−	0.998429i	$-0.482153\pi$
$31$	0.624005	0.112075	0.0560373	+	0.998429i	$0.482153\pi$
$32$	−5.86239	−1.03633
$33$	0	0
$34$	0.877514	0.150492
$35$	5.55015	0.938146
$36$	0	0
$37$	−3.07517	−0.505554	−0.252777	−	0.967525i	$-0.581344\pi$
$37$	−3.07517	−0.505554	−0.252777	+	0.967525i	$0.581344\pi$
$38$	−0.906694	−0.147085
$39$	0	0
$40$	−2.95100	−0.466595
$41$	−5.08728	−0.794499	−0.397250	−	0.917711i	$-0.630035\pi$
$41$	−5.08728	−0.794499	−0.397250	+	0.917711i	$0.630035\pi$
$42$	0	0
$43$	3.31430	0.505425	0.252713	−	0.967541i	$-0.418677\pi$
$43$	3.31430	0.505425	0.252713	+	0.967541i	$0.418677\pi$
$44$	6.47233	0.975741
$45$	0	0
$46$	2.82718	0.416845
$47$	0.612741	0.0893774	0.0446887	−	0.999001i	$-0.485770\pi$
$47$	0.612741	0.0893774	0.0446887	+	0.999001i	$0.485770\pi$
$48$	0	0
$49$	16.7070	2.38672
$50$	2.79269	0.394946
$51$	0	0
$52$	7.21140	1.00004
$53$	1.22775	0.168644	0.0843220	−	0.996439i	$-0.473128\pi$
$53$	1.22775	0.168644	0.0843220	+	0.996439i	$0.473128\pi$
$54$	0	0
$55$	5.15749	0.695436
$56$	−12.6050	−1.68441
$57$	0	0
$58$	7.05753	0.926699
$59$	−9.45896	−1.23145	−0.615726	−	0.787960i	$-0.711137\pi$
$59$	−9.45896	−1.23145	−0.615726	+	0.787960i	$0.711137\pi$
$60$	0	0
$61$	4.15041	0.531406	0.265703	−	0.964055i	$-0.414396\pi$
$61$	4.15041	0.531406	0.265703	+	0.964055i	$0.414396\pi$
$62$	−0.470906	−0.0598052
$63$	0	0
$64$	2.60939	0.326174
$65$	5.74642	0.712756
$66$	0	0
$67$	6.38803	0.780422	0.390211	−	0.920726i	$-0.372402\pi$
$67$	6.38803	0.780422	0.390211	+	0.920726i	$0.372402\pi$
$68$	1.66340	0.201717
$69$	0	0
$70$	−4.18843	−0.500613
$71$	5.50785	0.653662	0.326831	−	0.945083i	$-0.394019\pi$
$71$	5.50785	0.653662	0.326831	+	0.945083i	$0.394019\pi$
$72$	0	0
$73$	4.25781	0.498339	0.249169	−	0.968460i	$-0.419842\pi$
$73$	4.25781	0.498339	0.249169	+	0.968460i	$0.419842\pi$
$74$	2.32068	0.269774
$75$	0	0
$76$	−1.71871	−0.197149
$77$	22.0298	2.51053
$78$	0	0
$79$	−13.8023	−1.55288	−0.776442	−	0.630188i	$-0.782978\pi$
$79$	−13.8023	−1.55288	−0.776442	+	0.630188i	$0.782978\pi$
$80$	−1.03427	−0.115635
$81$	0	0
$82$	3.83912	0.423960
$83$	−7.95930	−0.873647	−0.436823	−	0.899547i	$-0.643897\pi$
$83$	−7.95930	−0.873647	−0.436823	+	0.899547i	$0.643897\pi$
$84$	0	0
$85$	1.32548	0.143769
$86$	−2.50114	−0.269705
$87$	0	0
$88$	−11.7132	−1.24863
$89$	−11.2451	−1.19197	−0.595987	−	0.802994i	$-0.703239\pi$
$89$	−11.2451	−1.19197	−0.595987	+	0.802994i	$0.703239\pi$
$90$	0	0
$91$	24.5454	2.57306
$92$	5.35913	0.558728
$93$	0	0
$94$	−0.462406	−0.0476935
$95$	−1.36956	−0.140514
$96$	0	0
$97$	−2.63245	−0.267285	−0.133642	−	0.991030i	$-0.542667\pi$
$97$	−2.63245	−0.267285	−0.133642	+	0.991030i	$0.542667\pi$
$98$	−12.6080	−1.27360
$99$	0	0
$100$	5.29376	0.529376
$101$	18.2928	1.82020	0.910102	−	0.414385i	$-0.136003\pi$
$101$	18.2928	1.82020	0.910102	+	0.414385i	$0.136003\pi$
$102$	0	0
$103$	4.00663	0.394785	0.197393	−	0.980324i	$-0.436753\pi$
$103$	4.00663	0.394785	0.197393	+	0.980324i	$0.436753\pi$
$104$	−13.0508	−1.27973
$105$	0	0
$106$	−0.926521	−0.0899916
$107$	−14.8156	−1.43228	−0.716140	−	0.697957i	$-0.754093\pi$
$107$	−14.8156	−1.43228	−0.716140	+	0.697957i	$0.754093\pi$
$108$	0	0
$109$	15.4562	1.48044	0.740218	−	0.672367i	$-0.234722\pi$
$109$	15.4562	1.48044	0.740218	+	0.672367i	$0.234722\pi$
$110$	−3.89211	−0.371098
$111$	0	0
$112$	−4.41780	−0.417443
$113$	−16.8900	−1.58888	−0.794438	−	0.607345i	$-0.792234\pi$
$113$	−16.8900	−1.58888	−0.794438	+	0.607345i	$0.792234\pi$
$114$	0	0
$115$	4.27044	0.398220
$116$	13.3781	1.24213
$117$	0	0
$118$	7.13822	0.657126
$119$	5.66169	0.519007
$120$	0	0
$121$	9.47131	0.861028
$122$	−3.13211	−0.283568
$123$	0	0
$124$	−0.892640	−0.0801614
$125$	9.91783	0.887078
$126$	0	0
$127$	5.70894	0.506587	0.253293	−	0.967389i	$-0.418486\pi$
$127$	5.70894	0.506587	0.253293	+	0.967389i	$0.418486\pi$
$128$	9.75560	0.862281
$129$	0	0
$130$	−4.33655	−0.380340
$131$	19.4683	1.70095	0.850475	−	0.526016i	$-0.176315\pi$
$131$	19.4683	1.70095	0.850475	+	0.526016i	$0.176315\pi$
$132$	0	0
$133$	−5.84996	−0.507256
$134$	−4.82074	−0.416448
$135$	0	0
$136$	−3.01031	−0.258132
$137$	10.1404	0.866350	0.433175	−	0.901310i	$-0.357393\pi$
$137$	10.1404	0.866350	0.433175	+	0.901310i	$0.357393\pi$
$138$	0	0
$139$	−0.625859	−0.0530847	−0.0265423	−	0.999648i	$-0.508450\pi$
$139$	−0.625859	−0.0530847	−0.0265423	+	0.999648i	$0.508450\pi$
$140$	−7.93949	−0.671010
$141$	0	0
$142$	−4.15651	−0.348806
$143$	22.8089	1.90738
$144$	0	0
$145$	10.6604	0.885295
$146$	−3.21316	−0.265923
$147$	0	0
$148$	4.39903	0.361598
$149$	−22.0413	−1.80570	−0.902848	−	0.429960i	$-0.858528\pi$
$149$	−22.0413	−1.80570	−0.902848	+	0.429960i	$0.858528\pi$
$150$	0	0
$151$	5.38754	0.438431	0.219216	−	0.975676i	$-0.429650\pi$
$151$	5.38754	0.438431	0.219216	+	0.975676i	$0.429650\pi$
$152$	3.11041	0.252288
$153$	0	0
$154$	−16.6249	−1.33967
$155$	−0.711302	−0.0571331
$156$	0	0
$157$	−20.6565	−1.64857	−0.824285	−	0.566175i	$-0.808423\pi$
$157$	−20.6565	−1.64857	−0.824285	+	0.566175i	$0.808423\pi$
$158$	10.4160	0.828649
$159$	0	0
$160$	6.68252	0.528300
$161$	18.2409	1.43758
$162$	0	0
$163$	−13.0034	−1.01851	−0.509254	−	0.860616i	$-0.670079\pi$
$163$	−13.0034	−1.01851	−0.509254	+	0.860616i	$0.670079\pi$
$164$	7.27735	0.568266
$165$	0	0
$166$	6.00650	0.466195
$167$	2.15688	0.166904	0.0834521	−	0.996512i	$-0.473405\pi$
$167$	2.15688	0.166904	0.0834521	+	0.996512i	$0.473405\pi$
$168$	0	0
$169$	12.4134	0.954879
$170$	−1.00028	−0.0767177
$171$	0	0
$172$	−4.74110	−0.361506
$173$	−18.3762	−1.39711	−0.698557	−	0.715554i	$-0.746174\pi$
$173$	−18.3762	−1.39711	−0.698557	+	0.715554i	$0.746174\pi$
$174$	0	0
$175$	18.0183	1.36206
$176$	−4.10526	−0.309445
$177$	0	0
$178$	8.48610	0.636060
$179$	9.80550	0.732898	0.366449	−	0.930438i	$-0.380573\pi$
$179$	9.80550	0.732898	0.366449	+	0.930438i	$0.380573\pi$
$180$	0	0
$181$	−21.6840	−1.61176	−0.805881	−	0.592078i	$-0.798308\pi$
$181$	−21.6840	−1.61176	−0.805881	+	0.592078i	$0.798308\pi$
$182$	−18.5232	−1.37303
$183$	0	0
$184$	−9.69863	−0.714993
$185$	3.50538	0.257720
$186$	0	0
$187$	5.26115	0.384733
$188$	−0.876526	−0.0639273
$189$	0	0
$190$	1.03354	0.0749807
$191$	−8.22399	−0.595067	−0.297534	−	0.954711i	$-0.596164\pi$
$191$	−8.22399	−0.595067	−0.297534	+	0.954711i	$0.596164\pi$
$192$	0	0
$193$	−1.23581	−0.0889555	−0.0444778	−	0.999010i	$-0.514162\pi$
$193$	−1.23581	−0.0889555	−0.0444778	+	0.999010i	$0.514162\pi$
$194$	1.98658	0.142628
$195$	0	0
$196$	−23.8994	−1.70710
$197$	−2.48865	−0.177309	−0.0886546	−	0.996062i	$-0.528257\pi$
$197$	−2.48865	−0.177309	−0.0886546	+	0.996062i	$0.528257\pi$
$198$	0	0
$199$	18.9496	1.34330	0.671650	−	0.740869i	$-0.265586\pi$
$199$	18.9496	1.34330	0.671650	+	0.740869i	$0.265586\pi$
$200$	−9.58032	−0.677431
$201$	0	0
$202$	−13.8047	−0.971296
$203$	45.5349	3.19593
$204$	0	0
$205$	5.79898	0.405018
$206$	−3.02361	−0.210665
$207$	0	0
$208$	−4.57403	−0.317152
$209$	−5.43609	−0.376022
$210$	0	0
$211$	23.8174	1.63966	0.819829	−	0.572609i	$-0.194068\pi$
$211$	23.8174	1.63966	0.819829	+	0.572609i	$0.194068\pi$
$212$	−1.75629	−0.120623
$213$	0	0
$214$	11.1806	0.764292
$215$	−3.77796	−0.257655
$216$	0	0
$217$	−3.03827	−0.206251
$218$	−11.6640	−0.789989
$219$	0	0
$220$	−7.37780	−0.497411
$221$	5.86191	0.394315
$222$	0	0
$223$	−13.7157	−0.918472	−0.459236	−	0.888314i	$-0.651877\pi$
$223$	−13.7157	−0.918472	−0.459236	+	0.888314i	$0.651877\pi$
$224$	28.5439	1.90717
$225$	0	0
$226$	12.7461	0.847855
$227$	13.2991	0.882695	0.441347	−	0.897336i	$-0.354501\pi$
$227$	13.2991	0.882695	0.441347	+	0.897336i	$0.354501\pi$
$228$	0	0
$229$	1.00000	0.0660819
$229$	1.00000	0.0660819
$230$	−3.22269	−0.212498
$231$	0	0
$232$	−24.2109	−1.58952
$233$	−11.9943	−0.785776	−0.392888	−	0.919586i	$-0.628524\pi$
$233$	−11.9943	−0.785776	−0.392888	+	0.919586i	$0.628524\pi$
$234$	0	0
$235$	−0.698462	−0.0455626
$236$	13.5311	0.880797
$237$	0	0
$238$	−4.27261	−0.276952
$239$	−3.37225	−0.218133	−0.109067	−	0.994034i	$-0.534786\pi$
$239$	−3.37225	−0.218133	−0.109067	+	0.994034i	$0.534786\pi$
$240$	0	0
$241$	−2.57731	−0.166019	−0.0830096	−	0.996549i	$-0.526453\pi$
$241$	−2.57731	−0.166019	−0.0830096	+	0.996549i	$0.526453\pi$
$242$	−7.14754	−0.459461
$243$	0	0
$244$	−5.93717	−0.380088
$245$	−19.0443	−1.21670
$246$	0	0
$247$	−6.05683	−0.385387
$248$	1.61544	0.102581
$249$	0	0
$250$	−7.48451	−0.473362
$251$	−24.5119	−1.54718	−0.773588	−	0.633689i	$-0.781540\pi$
$251$	−24.5119	−1.54718	−0.773588	+	0.633689i	$0.781540\pi$
$252$	0	0
$253$	16.9504	1.06566
$254$	−4.30826	−0.270325
$255$	0	0
$256$	−12.5809	−0.786303
$257$	−13.1742	−0.821784	−0.410892	−	0.911684i	$-0.634783\pi$
$257$	−13.1742	−0.821784	−0.410892	+	0.911684i	$0.634783\pi$
$258$	0	0
$259$	14.9730	0.930374
$260$	−8.22026	−0.509799
$261$	0	0
$262$	−14.6918	−0.907660
$263$	−21.7119	−1.33881	−0.669406	−	0.742896i	$-0.733451\pi$
$263$	−21.7119	−1.33881	−0.669406	+	0.742896i	$0.733451\pi$
$264$	0	0
$265$	−1.39951	−0.0859709
$266$	4.41468	0.270681
$267$	0	0
$268$	−9.13808	−0.558197
$269$	−6.29822	−0.384009	−0.192004	−	0.981394i	$-0.561499\pi$
$269$	−6.29822	−0.384009	−0.192004	+	0.981394i	$0.561499\pi$
$270$	0	0
$271$	4.09179	0.248559	0.124279	−	0.992247i	$-0.460338\pi$
$271$	4.09179	0.248559	0.124279	+	0.992247i	$0.460338\pi$
$272$	−1.05506	−0.0639722
$273$	0	0
$274$	−7.65244	−0.462301
$275$	16.7436	1.00968
$276$	0	0
$277$	−31.6953	−1.90438	−0.952192	−	0.305500i	$-0.901176\pi$
$277$	−31.6953	−1.90438	−0.952192	+	0.305500i	$0.901176\pi$
$278$	0.472305	0.0283270
$279$	0	0
$280$	14.3684	0.858676
$281$	22.9375	1.36833	0.684167	−	0.729325i	$-0.260166\pi$
$281$	22.9375	1.36833	0.684167	+	0.729325i	$0.260166\pi$
$282$	0	0
$283$	33.6286	1.99901	0.999507	−	0.0314043i	$-0.00999794\pi$
$283$	33.6286	1.99901	0.999507	+	0.0314043i	$0.00999794\pi$
$284$	−7.87899	−0.467532
$285$	0	0
$286$	−17.2128	−1.01781
$287$	24.7699	1.46212
$288$	0	0
$289$	−15.6479	−0.920463
$290$	−8.04486	−0.472411
$291$	0	0
$292$	−6.09080	−0.356437
$293$	26.1430	1.52729	0.763644	−	0.645638i	$-0.223408\pi$
$293$	26.1430	1.52729	0.763644	+	0.645638i	$0.223408\pi$
$294$	0	0
$295$	10.7822	0.627767
$296$	−7.96110	−0.462729
$297$	0	0
$298$	16.6335	0.963554
$299$	18.8859	1.09220
$300$	0	0
$301$	−16.1373	−0.930137
$302$	−4.06571	−0.233955
$303$	0	0
$304$	1.09014	0.0625237
$305$	−4.73104	−0.270899
$306$	0	0
$307$	11.0433	0.630272	0.315136	−	0.949047i	$-0.397950\pi$
$307$	11.0433	0.630272	0.315136	+	0.949047i	$0.397950\pi$
$308$	−31.5137	−1.79566
$309$	0	0
$310$	0.536785	0.0304873
$311$	−24.4387	−1.38579	−0.692897	−	0.721037i	$-0.743666\pi$
$311$	−24.4387	−1.38579	−0.692897	+	0.721037i	$0.743666\pi$
$312$	0	0
$313$	17.4749	0.987738	0.493869	−	0.869536i	$-0.335582\pi$
$313$	17.4749	0.987738	0.493869	+	0.869536i	$0.335582\pi$
$314$	15.5885	0.879709
$315$	0	0
$316$	19.7443	1.11070
$317$	−6.95951	−0.390885	−0.195443	−	0.980715i	$-0.562614\pi$
$317$	−6.95951	−0.390885	−0.195443	+	0.980715i	$0.562614\pi$
$318$	0	0
$319$	42.3135	2.36910
$320$	−2.97444	−0.166276
$321$	0	0
$322$	−13.7655	−0.767121
$323$	−1.39708	−0.0777357
$324$	0	0
$325$	18.6555	1.03482
$326$	9.81306	0.543495
$327$	0	0
$328$	−13.1701	−0.727198
$329$	−2.98343	−0.164482
$330$	0	0
$331$	14.9873	0.823776	0.411888	−	0.911234i	$-0.364869\pi$
$331$	14.9873	0.823776	0.411888	+	0.911234i	$0.364869\pi$
$332$	11.3858	0.624876
$333$	0	0
$334$	−1.62769	−0.0890633
$335$	−7.28170	−0.397842
$336$	0	0
$337$	−17.9914	−0.980053	−0.490027	−	0.871708i	$-0.663013\pi$
$337$	−17.9914	−0.980053	−0.490027	+	0.871708i	$0.663013\pi$
$338$	−9.36781	−0.509542
$339$	0	0
$340$	−1.89610	−0.102831
$341$	−2.82332	−0.152892
$342$	0	0
$343$	−47.2635	−2.55199
$344$	8.58016	0.462611
$345$	0	0
$346$	13.8676	0.745527
$347$	−1.98067	−0.106328	−0.0531639	−	0.998586i	$-0.516931\pi$
$347$	−1.98067	−0.106328	−0.0531639	+	0.998586i	$0.516931\pi$
$348$	0	0
$349$	−4.24688	−0.227330	−0.113665	−	0.993519i	$-0.536259\pi$
$349$	−4.24688	−0.227330	−0.113665	+	0.993519i	$0.536259\pi$
$350$	−13.5976	−0.726820
$351$	0	0
$352$	26.5245	1.41376
$353$	24.5516	1.30675	0.653375	−	0.757034i	$-0.273352\pi$
$353$	24.5516	1.30675	0.653375	+	0.757034i	$0.273352\pi$
$354$	0	0
$355$	−6.27839	−0.333222
$356$	16.0861	0.852560
$357$	0	0
$358$	−7.39974	−0.391088
$359$	−29.9907	−1.58285	−0.791425	−	0.611266i	$-0.790660\pi$
$359$	−29.9907	−1.58285	−0.791425	+	0.611266i	$0.790660\pi$
$360$	0	0
$361$	−17.5565	−0.924024
$362$	16.3639	0.860067
$363$	0	0
$364$	−35.1122	−1.84038
$365$	−4.85346	−0.254042
$366$	0	0
$367$	−27.0427	−1.41162	−0.705810	−	0.708401i	$-0.749417\pi$
$367$	−27.0427	−1.41162	−0.705810	+	0.708401i	$0.749417\pi$
$368$	−3.39918	−0.177195
$369$	0	0
$370$	−2.64534	−0.137525
$371$	−5.97788	−0.310356
$372$	0	0
$373$	1.02594	0.0531214	0.0265607	−	0.999647i	$-0.491544\pi$
$373$	1.02594	0.0531214	0.0265607	+	0.999647i	$0.491544\pi$
$374$	−3.97033	−0.205301
$375$	0	0
$376$	1.58628	0.0818063
$377$	47.1452	2.42810
$378$	0	0
$379$	34.5732	1.77591	0.887954	−	0.459933i	$-0.152127\pi$
$379$	34.5732	1.77591	0.887954	+	0.459933i	$0.152127\pi$
$380$	1.95915	0.100502
$381$	0	0
$382$	6.20625	0.317539
$383$	−6.53729	−0.334040	−0.167020	−	0.985954i	$-0.553414\pi$
$383$	−6.53729	−0.334040	−0.167020	+	0.985954i	$0.553414\pi$
$384$	0	0
$385$	−25.1118	−1.27981
$386$	0.932606	0.0474684
$387$	0	0
$388$	3.76572	0.191176
$389$	−22.9266	−1.16242	−0.581211	−	0.813753i	$-0.697421\pi$
$389$	−22.9266	−1.16242	−0.581211	+	0.813753i	$0.697421\pi$
$390$	0	0
$391$	4.35626	0.220306
$392$	43.2517	2.18454
$393$	0	0
$394$	1.87807	0.0946156
$395$	15.7333	0.791626
$396$	0	0
$397$	−18.7800	−0.942542	−0.471271	−	0.881988i	$-0.656205\pi$
$397$	−18.7800	−0.942542	−0.471271	+	0.881988i	$0.656205\pi$
$398$	−14.3003	−0.716810
$399$	0	0
$400$	−3.35771	−0.167886
$401$	7.56715	0.377885	0.188943	−	0.981988i	$-0.439494\pi$
$401$	7.56715	0.377885	0.188943	+	0.981988i	$0.439494\pi$
$402$	0	0
$403$	−3.14572	−0.156699
$404$	−26.1679	−1.30190
$405$	0	0
$406$	−34.3630	−1.70541
$407$	13.9137	0.689675
$408$	0	0
$409$	−16.1989	−0.800986	−0.400493	−	0.916300i	$-0.631161\pi$
$409$	−16.1989	−0.800986	−0.400493	+	0.916300i	$0.631161\pi$
$410$	−4.37621	−0.216125
$411$	0	0
$412$	−5.73149	−0.282370
$413$	46.0556	2.26625
$414$	0	0
$415$	9.07279	0.445366
$416$	29.5533	1.44897
$417$	0	0
$418$	4.10236	0.200653
$419$	10.1414	0.495440	0.247720	−	0.968832i	$-0.420319\pi$
$419$	10.1414	0.495440	0.247720	+	0.968832i	$0.420319\pi$
$420$	0	0
$421$	−29.3299	−1.42945	−0.714725	−	0.699405i	$-0.753448\pi$
$421$	−29.3299	−1.42945	−0.714725	+	0.699405i	$0.753448\pi$
$422$	−17.9738	−0.874953
$423$	0	0
$424$	3.17843	0.154358
$425$	4.30312	0.208732
$426$	0	0
$427$	−20.2083	−0.977948
$428$	21.1937	1.02444
$429$	0	0
$430$	2.85104	0.137489
$431$	−26.2939	−1.26653	−0.633267	−	0.773933i	$-0.718287\pi$
$431$	−26.2939	−1.26653	−0.633267	+	0.773933i	$0.718287\pi$
$432$	0	0
$433$	−26.4654	−1.27185	−0.635923	−	0.771752i	$-0.719381\pi$
$433$	−26.4654	−1.27185	−0.635923	+	0.771752i	$0.719381\pi$
$434$	2.29284	0.110060
$435$	0	0
$436$	−22.1101	−1.05888
$437$	−4.50112	−0.215318
$438$	0	0
$439$	2.42715	0.115841	0.0579207	−	0.998321i	$-0.481553\pi$
$439$	2.42715	0.115841	0.0579207	+	0.998321i	$0.481553\pi$
$440$	13.3519	0.636526
$441$	0	0
$442$	−4.42370	−0.210414
$443$	−11.7983	−0.560556	−0.280278	−	0.959919i	$-0.590427\pi$
$443$	−11.7983	−0.560556	−0.280278	+	0.959919i	$0.590427\pi$
$444$	0	0
$445$	12.8182	0.607642
$446$	10.3506	0.490114
$447$	0	0
$448$	−12.7051	−0.600259
$449$	−1.83911	−0.0867930	−0.0433965	−	0.999058i	$-0.513818\pi$
$449$	−1.83911	−0.0867930	−0.0433965	+	0.999058i	$0.513818\pi$
$450$	0	0
$451$	23.0175	1.08385
$452$	24.1611	1.13644
$453$	0	0
$454$	−10.0362	−0.471023
$455$	−27.9793	−1.31169
$456$	0	0
$457$	−13.3839	−0.626070	−0.313035	−	0.949742i	$-0.601346\pi$
$457$	−13.3839	−0.626070	−0.313035	+	0.949742i	$0.601346\pi$
$458$	−0.754652	−0.0352626
$459$	0	0
$460$	−6.10887	−0.284827
$461$	29.5240	1.37507	0.687535	−	0.726151i	$-0.258693\pi$
$461$	29.5240	1.37507	0.687535	+	0.726151i	$0.258693\pi$
$462$	0	0
$463$	9.16455	0.425913	0.212956	−	0.977062i	$-0.431691\pi$
$463$	9.16455	0.425913	0.212956	+	0.977062i	$0.431691\pi$
$464$	−8.48543	−0.393926
$465$	0	0
$466$	9.05155	0.419305
$467$	16.4910	0.763111	0.381556	−	0.924346i	$-0.375388\pi$
$467$	16.4910	0.763111	0.381556	+	0.924346i	$0.375388\pi$
$468$	0	0
$469$	−31.1032	−1.43621
$470$	0.527095	0.0243131
$471$	0	0
$472$	−24.4877	−1.12714
$473$	−14.9956	−0.689499
$474$	0	0
$475$	−4.44621	−0.204006
$476$	−8.09906	−0.371220
$477$	0	0
$478$	2.54488	0.116400
$479$	−15.9823	−0.730250	−0.365125	−	0.930958i	$-0.618974\pi$
$479$	−15.9823	−0.730250	−0.365125	+	0.930958i	$0.618974\pi$
$480$	0	0
$481$	15.5025	0.706851
$482$	1.94497	0.0885910
$483$	0	0
$484$	−13.5487	−0.615851
$485$	3.00073	0.136256
$486$	0	0
$487$	−9.92673	−0.449823	−0.224911	−	0.974379i	$-0.572209\pi$
$487$	−9.92673	−0.449823	−0.224911	+	0.974379i	$0.572209\pi$
$488$	10.7447	0.486391
$489$	0	0
$490$	14.3718	0.649253
$491$	1.89826	0.0856673	0.0428337	−	0.999082i	$-0.486361\pi$
$491$	1.89826	0.0856673	0.0428337	+	0.999082i	$0.486361\pi$
$492$	0	0
$493$	10.8746	0.489768
$494$	4.57080	0.205650
$495$	0	0
$496$	0.566182	0.0254223
$497$	−26.8177	−1.20294
$498$	0	0
$499$	−21.6048	−0.967165	−0.483583	−	0.875299i	$-0.660665\pi$
$499$	−21.6048	−0.967165	−0.483583	+	0.875299i	$0.660665\pi$
$500$	−14.1875	−0.634483
$501$	0	0
$502$	18.4979	0.825603
$503$	−38.5257	−1.71778	−0.858889	−	0.512162i	$-0.828845\pi$
$503$	−38.5257	−1.71778	−0.858889	+	0.512162i	$0.828845\pi$
$504$	0	0
$505$	−20.8519	−0.927899
$506$	−12.7916	−0.568657
$507$	0	0
$508$	−8.16665	−0.362337
$509$	−41.4496	−1.83722	−0.918610	−	0.395165i	$-0.870687\pi$
$509$	−41.4496	−1.83722	−0.918610	+	0.395165i	$0.870687\pi$
$510$	0	0
$511$	−20.7312	−0.917095
$512$	−10.0170	−0.442694
$513$	0	0
$514$	9.94193	0.438520
$515$	−4.56715	−0.201253
$516$	0	0
$517$	−2.77236	−0.121928
$518$	−11.2994	−0.496466
$519$	0	0
$520$	14.8765	0.652379
$521$	−18.5249	−0.811590	−0.405795	−	0.913964i	$-0.633005\pi$
$521$	−18.5249	−0.811590	−0.405795	+	0.913964i	$0.633005\pi$
$522$	0	0
$523$	−23.0716	−1.00885	−0.504426	−	0.863455i	$-0.668296\pi$
$523$	−23.0716	−1.00885	−0.504426	+	0.863455i	$0.668296\pi$
$524$	−27.8494	−1.21661
$525$	0	0
$526$	16.3849	0.714416
$527$	−0.725598	−0.0316075
$528$	0	0
$529$	−8.96498	−0.389782
$530$	1.05614	0.0458757
$531$	0	0
$532$	8.36837	0.362815
$533$	25.6458	1.11084
$534$	0	0
$535$	16.8883	0.730144
$536$	16.5375	0.714313
$537$	0	0
$538$	4.75296	0.204915
$539$	−75.5914	−3.25595
$540$	0	0
$541$	4.03456	0.173459	0.0867297	−	0.996232i	$-0.472358\pi$
$541$	4.03456	0.173459	0.0867297	+	0.996232i	$0.472358\pi$
$542$	−3.08788	−0.132636
$543$	0	0
$544$	6.81683	0.292269
$545$	−17.6185	−0.754693
$546$	0	0
$547$	−4.40498	−0.188343	−0.0941717	−	0.995556i	$-0.530020\pi$
$547$	−4.40498	−0.188343	−0.0941717	+	0.995556i	$0.530020\pi$
$548$	−14.5058	−0.619657
$549$	0	0
$550$	−12.6356	−0.538783
$551$	−11.2362	−0.478679
$552$	0	0
$553$	67.2034	2.85778
$554$	23.9189	1.01622
$555$	0	0
$556$	0.895292	0.0379688
$557$	3.83095	0.162323	0.0811613	−	0.996701i	$-0.474137\pi$
$557$	3.83095	0.162323	0.0811613	+	0.996701i	$0.474137\pi$
$558$	0	0
$559$	−16.7079	−0.706671
$560$	5.03584	0.212803
$561$	0	0
$562$	−17.3098	−0.730170
$563$	9.83510	0.414500	0.207250	−	0.978288i	$-0.433549\pi$
$563$	9.83510	0.414500	0.207250	+	0.978288i	$0.433549\pi$
$564$	0	0
$565$	19.2528	0.809973
$566$	−25.3779	−1.06671
$567$	0	0
$568$	14.2589	0.598290
$569$	24.2138	1.01510	0.507549	−	0.861623i	$-0.330552\pi$
$569$	24.2138	1.01510	0.507549	+	0.861623i	$0.330552\pi$
$570$	0	0
$571$	27.2297	1.13953	0.569765	−	0.821808i	$-0.307034\pi$
$571$	27.2297	1.13953	0.569765	+	0.821808i	$0.307034\pi$
$572$	−32.6282	−1.36425
$573$	0	0
$574$	−18.6926	−0.780215
$575$	13.8638	0.578161
$576$	0	0
$577$	5.63786	0.234707	0.117353	−	0.993090i	$-0.462559\pi$
$577$	5.63786	0.234707	0.117353	+	0.993090i	$0.462559\pi$
$578$	11.8087	0.491177
$579$	0	0
$580$	−15.2497	−0.633208
$581$	38.7537	1.60778
$582$	0	0
$583$	−5.55497	−0.230063
$584$	11.0228	0.456125
$585$	0	0
$586$	−19.7288	−0.814990
$587$	−27.0049	−1.11461	−0.557306	−	0.830307i	$-0.688165\pi$
$587$	−27.0049	−1.11461	−0.557306	+	0.830307i	$0.688165\pi$
$588$	0	0
$589$	0.749725	0.0308919
$590$	−8.13684	−0.334988
$591$	0	0
$592$	−2.79021	−0.114677
$593$	−11.9012	−0.488722	−0.244361	−	0.969684i	$-0.578578\pi$
$593$	−11.9012	−0.488722	−0.244361	+	0.969684i	$0.578578\pi$
$594$	0	0
$595$	−6.45375	−0.264578
$596$	31.5301	1.29153
$597$	0	0
$598$	−14.2523	−0.582819
$599$	2.46529	0.100729	0.0503645	−	0.998731i	$-0.483962\pi$
$599$	2.46529	0.100729	0.0503645	+	0.998731i	$0.483962\pi$
$600$	0	0
$601$	35.8707	1.46320	0.731599	−	0.681736i	$-0.238775\pi$
$601$	35.8707	1.46320	0.731599	+	0.681736i	$0.238775\pi$
$602$	12.1780	0.496339
$603$	0	0
$604$	−7.70687	−0.313588
$605$	−10.7963	−0.438933
$606$	0	0
$607$	−2.10441	−0.0854153	−0.0427076	−	0.999088i	$-0.513598\pi$
$607$	−2.10441	−0.0854153	−0.0427076	+	0.999088i	$0.513598\pi$
$608$	−7.04350	−0.285652
$609$	0	0
$610$	3.57029	0.144557
$611$	−3.08893	−0.124965
$612$	0	0
$613$	−11.9729	−0.483582	−0.241791	−	0.970328i	$-0.577735\pi$
$613$	−11.9729	−0.483582	−0.241791	+	0.970328i	$0.577735\pi$
$614$	−8.33381	−0.336325
$615$	0	0
$616$	57.0316	2.29787
$617$	−31.2275	−1.25717	−0.628586	−	0.777740i	$-0.716366\pi$
$617$	−31.2275	−1.25717	−0.628586	+	0.777740i	$0.716366\pi$
$618$	0	0
$619$	12.2824	0.493673	0.246836	−	0.969057i	$-0.420609\pi$
$619$	12.2824	0.493673	0.246836	+	0.969057i	$0.420609\pi$
$620$	1.01752	0.0408645
$621$	0	0
$622$	18.4427	0.739486
$623$	54.7521	2.19359
$624$	0	0
$625$	7.19785	0.287914
$626$	−13.1874	−0.527076
$627$	0	0
$628$	29.5492	1.17914
$629$	3.57583	0.142578
$630$	0	0
$631$	44.4614	1.76998	0.884990	−	0.465611i	$-0.154165\pi$
$631$	44.4614	1.76998	0.884990	+	0.465611i	$0.154165\pi$
$632$	−35.7320	−1.42134
$633$	0	0
$634$	5.25201	0.208584
$635$	−6.50761	−0.258247
$636$	0	0
$637$	−84.2231	−3.33704
$638$	−31.9320	−1.26420
$639$	0	0
$640$	−11.1204	−0.439572
$641$	−4.67244	−0.184550	−0.0922752	−	0.995734i	$-0.529414\pi$
$641$	−4.67244	−0.184550	−0.0922752	+	0.995734i	$0.529414\pi$
$642$	0	0
$643$	−1.99886	−0.0788273	−0.0394137	−	0.999223i	$-0.512549\pi$
$643$	−1.99886	−0.0788273	−0.0394137	+	0.999223i	$0.512549\pi$
$644$	−26.0936	−1.02823
$645$	0	0
$646$	1.05431	0.0414813
$647$	15.2362	0.598996	0.299498	−	0.954097i	$-0.403181\pi$
$647$	15.2362	0.598996	0.299498	+	0.954097i	$0.403181\pi$
$648$	0	0
$649$	42.7973	1.67994
$650$	−14.0784	−0.552201
$651$	0	0
$652$	18.6014	0.728488
$653$	−12.5022	−0.489250	−0.244625	−	0.969618i	$-0.578665\pi$
$653$	−12.5022	−0.489250	−0.244625	+	0.969618i	$0.578665\pi$
$654$	0	0
$655$	−22.1918	−0.867106
$656$	−4.61586	−0.180219
$657$	0	0
$658$	2.25145	0.0877706
$659$	42.4556	1.65383	0.826917	−	0.562323i	$-0.190092\pi$
$659$	42.4556	1.65383	0.826917	+	0.562323i	$0.190092\pi$
$660$	0	0
$661$	−21.1689	−0.823373	−0.411686	−	0.911326i	$-0.635060\pi$
$661$	−21.1689	−0.823373	−0.411686	+	0.911326i	$0.635060\pi$
$662$	−11.3102	−0.439583
$663$	0	0
$664$	−20.6053	−0.799641
$665$	6.66835	0.258588
$666$	0	0
$667$	35.0359	1.35659
$668$	−3.08542	−0.119378
$669$	0	0
$670$	5.49515	0.212296
$671$	−18.7786	−0.724941
$672$	0	0
$673$	−4.78286	−0.184366	−0.0921829	−	0.995742i	$-0.529384\pi$
$673$	−4.78286	−0.184366	−0.0921829	+	0.995742i	$0.529384\pi$
$674$	13.5772	0.522975
$675$	0	0
$676$	−17.7574	−0.682977
$677$	11.5409	0.443551	0.221776	−	0.975098i	$-0.428815\pi$
$677$	11.5409	0.443551	0.221776	+	0.975098i	$0.428815\pi$
$678$	0	0
$679$	12.8174	0.491886
$680$	3.43145	0.131590
$681$	0	0
$682$	2.13063	0.0815859
$683$	8.26146	0.316116	0.158058	−	0.987430i	$-0.449477\pi$
$683$	8.26146	0.316116	0.158058	+	0.987430i	$0.449477\pi$
$684$	0	0
$685$	−11.5590	−0.441646
$686$	35.6674	1.36179
$687$	0	0
$688$	3.00718	0.114648
$689$	−6.18928	−0.235793
$690$	0	0
$691$	28.0360	1.06654	0.533269	−	0.845946i	$-0.320963\pi$
$691$	28.0360	1.06654	0.533269	+	0.845946i	$0.320963\pi$
$692$	26.2871	0.999287
$693$	0	0
$694$	1.49471	0.0567386
$695$	0.713415	0.0270614
$696$	0	0
$697$	5.91552	0.224066
$698$	3.20492	0.121308
$699$	0	0
$700$	−25.7752	−0.974213
$701$	32.2430	1.21780	0.608902	−	0.793246i	$-0.291610\pi$
$701$	32.2430	1.21780	0.608902	+	0.793246i	$0.291610\pi$
$702$	0	0
$703$	−3.69473	−0.139349
$704$	−11.8062	−0.444965
$705$	0	0
$706$	−18.5279	−0.697307
$707$	−89.0675	−3.34973
$708$	0	0
$709$	−16.6726	−0.626151	−0.313076	−	0.949728i	$-0.601359\pi$
$709$	−16.6726	−0.626151	−0.313076	+	0.949728i	$0.601359\pi$
$710$	4.73799	0.177814
$711$	0	0
$712$	−29.1116	−1.09100
$713$	−2.33773	−0.0875487
$714$	0	0
$715$	−25.9998	−0.972338
$716$	−14.0268	−0.524205
$717$	0	0
$718$	22.6325	0.844639
$719$	−12.4630	−0.464791	−0.232396	−	0.972621i	$-0.574656\pi$
$719$	−12.4630	−0.464791	−0.232396	+	0.972621i	$0.574656\pi$
$720$	0	0
$721$	−19.5083	−0.726525
$722$	13.2490	0.493077
$723$	0	0
$724$	31.0190	1.15281
$725$	34.6085	1.28533
$726$	0	0
$727$	13.7480	0.509883	0.254942	−	0.966956i	$-0.417944\pi$
$727$	13.7480	0.509883	0.254942	+	0.966956i	$0.417944\pi$
$728$	63.5440	2.35510
$729$	0	0
$730$	3.66267	0.135562
$731$	−3.85389	−0.142541
$732$	0	0
$733$	41.3655	1.52787	0.763935	−	0.645294i	$-0.223265\pi$
$733$	41.3655	1.52787	0.763935	+	0.645294i	$0.223265\pi$
$734$	20.4079	0.753268
$735$	0	0
$736$	21.9625	0.809547
$737$	−28.9028	−1.06465
$738$	0	0
$739$	42.4619	1.56199	0.780994	−	0.624539i	$-0.214713\pi$
$739$	42.4619	1.56199	0.780994	+	0.624539i	$0.214713\pi$
$740$	−5.01444	−0.184335
$741$	0	0
$742$	4.51122	0.165612
$743$	18.1337	0.665262	0.332631	−	0.943057i	$-0.392064\pi$
$743$	18.1337	0.665262	0.332631	+	0.943057i	$0.392064\pi$
$744$	0	0
$745$	25.1249	0.920504
$746$	−0.774230	−0.0283466
$747$	0	0
$748$	−7.52608	−0.275181
$749$	72.1370	2.63583
$750$	0	0
$751$	33.5914	1.22577	0.612884	−	0.790173i	$-0.290009\pi$
$751$	33.5914	1.22577	0.612884	+	0.790173i	$0.290009\pi$
$752$	0.555961	0.0202738
$753$	0	0
$754$	−35.5782	−1.29568
$755$	−6.14124	−0.223503
$756$	0	0
$757$	−41.9347	−1.52414	−0.762071	−	0.647494i	$-0.775817\pi$
$757$	−41.9347	−1.52414	−0.762071	+	0.647494i	$0.775817\pi$
$758$	−26.0907	−0.947658
$759$	0	0
$760$	−3.54555	−0.128611
$761$	−1.28524	−0.0465899	−0.0232949	−	0.999729i	$-0.507416\pi$
$761$	−1.28524	−0.0465899	−0.0232949	+	0.999729i	$0.507416\pi$
$762$	0	0
$763$	−75.2561	−2.72445
$764$	11.7644	0.425622
$765$	0	0
$766$	4.93338	0.178250
$767$	47.6843	1.72178
$768$	0	0
$769$	7.94880	0.286641	0.143321	−	0.989676i	$-0.454222\pi$
$769$	7.94880	0.286641	0.143321	+	0.989676i	$0.454222\pi$
$770$	18.9506	0.682934
$771$	0	0
$772$	1.76783	0.0636255
$773$	−12.4997	−0.449581	−0.224791	−	0.974407i	$-0.572170\pi$
$773$	−12.4997	−0.449581	−0.224791	+	0.974407i	$0.572170\pi$
$774$	0	0
$775$	−2.30921	−0.0829494
$776$	−6.81498	−0.244643
$777$	0	0
$778$	17.3016	0.620291
$779$	−6.11223	−0.218993
$780$	0	0
$781$	−24.9204	−0.891722
$782$	−3.28746	−0.117559
$783$	0	0
$784$	15.1589	0.541389
$785$	23.5463	0.840404
$786$	0	0
$787$	−30.5370	−1.08853	−0.544264	−	0.838914i	$-0.683191\pi$
$787$	−30.5370	−1.08853	−0.544264	+	0.838914i	$0.683191\pi$
$788$	3.56002	0.126820
$789$	0	0
$790$	−11.8731	−0.422427
$791$	82.2371	2.92401
$792$	0	0
$793$	−20.9229	−0.742995
$794$	14.1724	0.502959
$795$	0	0
$796$	−27.1074	−0.960795
$797$	−14.4504	−0.511860	−0.255930	−	0.966695i	$-0.582382\pi$
$797$	−14.4504	−0.511860	−0.255930	+	0.966695i	$0.582382\pi$
$798$	0	0
$799$	−0.712500	−0.0252064
$800$	21.6945	0.767018
$801$	0	0
$802$	−5.71056	−0.201647
$803$	−19.2645	−0.679831
$804$	0	0
$805$	−20.7927	−0.732847
$806$	2.37392	0.0836178
$807$	0	0
$808$	47.3571	1.66602
$809$	−2.88973	−0.101597	−0.0507987	−	0.998709i	$-0.516177\pi$
$809$	−2.88973	−0.101597	−0.0507987	+	0.998709i	$0.516177\pi$
$810$	0	0
$811$	−12.9231	−0.453791	−0.226895	−	0.973919i	$-0.572858\pi$
$811$	−12.9231	−0.453791	−0.226895	+	0.973919i	$0.572858\pi$
$812$	−65.1378	−2.28589
$813$	0	0
$814$	−10.5000	−0.368024
$815$	14.8226	0.519212
$816$	0	0
$817$	3.98204	0.139314
$818$	12.2246	0.427421
$819$	0	0
$820$	−8.29544	−0.289689
$821$	−24.6672	−0.860890	−0.430445	−	0.902617i	$-0.641643\pi$
$821$	−24.6672	−0.860890	−0.430445	+	0.902617i	$0.641643\pi$
$822$	0	0
$823$	24.7453	0.862567	0.431284	−	0.902216i	$-0.358061\pi$
$823$	24.7453	0.862567	0.431284	+	0.902216i	$0.358061\pi$
$824$	10.3725	0.361343
$825$	0	0
$826$	−34.7559	−1.20931
$827$	12.9135	0.449047	0.224523	−	0.974469i	$-0.427917\pi$
$827$	12.9135	0.449047	0.224523	+	0.974469i	$0.427917\pi$
$828$	0	0
$829$	−13.8639	−0.481514	−0.240757	−	0.970585i	$-0.577396\pi$
$829$	−13.8639	−0.481514	−0.240757	+	0.970585i	$0.577396\pi$
$830$	−6.84679	−0.237656
$831$	0	0
$832$	−13.1544	−0.456046
$833$	−19.4271	−0.673108
$834$	0	0
$835$	−2.45862	−0.0850841
$836$	7.77634	0.268950
$837$	0	0
$838$	−7.65323	−0.264377
$839$	22.7769	0.786347	0.393173	−	0.919464i	$-0.371377\pi$
$839$	22.7769	0.786347	0.393173	+	0.919464i	$0.371377\pi$
$840$	0	0
$841$	58.4606	2.01588
$842$	22.1338	0.762782
$843$	0	0
$844$	−34.0708	−1.17277
$845$	−14.1500	−0.486776
$846$	0	0
$847$	−46.1157	−1.58455
$848$	1.11398	0.0382541
$849$	0	0
$850$	−3.24736	−0.111383
$851$	11.5206	0.394921
$852$	0	0
$853$	12.6184	0.432046	0.216023	−	0.976388i	$-0.430691\pi$
$853$	12.6184	0.432046	0.216023	+	0.976388i	$0.430691\pi$
$854$	15.2502	0.521852
$855$	0	0
$856$	−38.3551	−1.31095
$857$	−26.1052	−0.891735	−0.445868	−	0.895099i	$-0.647105\pi$
$857$	−26.1052	−0.891735	−0.445868	+	0.895099i	$0.647105\pi$
$858$	0	0
$859$	−13.5819	−0.463408	−0.231704	−	0.972786i	$-0.574430\pi$
$859$	−13.5819	−0.463408	−0.231704	+	0.972786i	$0.574430\pi$
$860$	5.40437	0.184288
$861$	0	0
$862$	19.8428	0.675847
$863$	43.2285	1.47151	0.735757	−	0.677245i	$-0.236826\pi$
$863$	43.2285	1.47151	0.735757	+	0.677245i	$0.236826\pi$
$864$	0	0
$865$	20.9470	0.712218
$866$	19.9722	0.678682
$867$	0	0
$868$	4.34625	0.147521
$869$	62.4490	2.11844
$870$	0	0
$871$	−32.2032	−1.09116
$872$	40.0135	1.35503
$873$	0	0
$874$	3.39678	0.114898
$875$	−48.2898	−1.63249
$876$	0	0
$877$	−5.28930	−0.178607	−0.0893036	−	0.996004i	$-0.528464\pi$
$877$	−5.28930	−0.178607	−0.0893036	+	0.996004i	$0.528464\pi$
$878$	−1.83165	−0.0618152
$879$	0	0
$880$	4.67957	0.157748
$881$	2.38446	0.0803346	0.0401673	−	0.999193i	$-0.487211\pi$
$881$	2.38446	0.0803346	0.0401673	+	0.999193i	$0.487211\pi$
$882$	0	0
$883$	40.4988	1.36289	0.681447	−	0.731867i	$-0.261351\pi$
$883$	40.4988	1.36289	0.681447	+	0.731867i	$0.261351\pi$
$884$	−8.38547	−0.282034
$885$	0	0
$886$	8.90364	0.299124
$887$	−32.7922	−1.10105	−0.550526	−	0.834818i	$-0.685573\pi$
$887$	−32.7922	−1.10105	−0.550526	+	0.834818i	$0.685573\pi$
$888$	0	0
$889$	−27.7968	−0.932274
$890$	−9.67329	−0.324249
$891$	0	0
$892$	19.6203	0.656937
$893$	0.736192	0.0246357
$894$	0	0
$895$	−11.1773	−0.373615
$896$	−47.4999	−1.58686
$897$	0	0
$898$	1.38789	0.0463144
$899$	−5.83572	−0.194632
$900$	0	0
$901$	−1.42763	−0.0475613
$902$	−17.3702	−0.578364
$903$	0	0
$904$	−43.7253	−1.45428
$905$	24.7176	0.821640
$906$	0	0
$907$	−31.2456	−1.03749	−0.518746	−	0.854928i	$-0.673601\pi$
$907$	−31.2456	−1.03749	−0.518746	+	0.854928i	$0.673601\pi$
$908$	−19.0244	−0.631348
$909$	0	0
$910$	21.1146	0.699942
$911$	−30.3720	−1.00627	−0.503134	−	0.864208i	$-0.667820\pi$
$911$	−30.3720	−1.00627	−0.503134	+	0.864208i	$0.667820\pi$
$912$	0	0
$913$	36.0120	1.19182
$914$	10.1001	0.334083
$915$	0	0
$916$	−1.43050	−0.0472651
$917$	−94.7907	−3.13027
$918$	0	0
$919$	36.7009	1.21065	0.605325	−	0.795979i	$-0.293043\pi$
$919$	36.7009	1.21065	0.605325	+	0.795979i	$0.293043\pi$
$920$	11.0555	0.364487
$921$	0	0
$922$	−22.2803	−0.733764
$923$	−27.7660	−0.913930
$924$	0	0
$925$	11.3801	0.374174
$926$	−6.91604	−0.227275
$927$	0	0
$928$	54.8253	1.79973
$929$	−22.2117	−0.728741	−0.364370	−	0.931254i	$-0.618716\pi$
$929$	−22.2117	−0.728741	−0.364370	+	0.931254i	$0.618716\pi$
$930$	0	0
$931$	20.0731	0.657868
$932$	17.1579	0.562026
$933$	0	0
$934$	−12.4449	−0.407211
$935$	−5.99717	−0.196128
$936$	0	0
$937$	43.6257	1.42519	0.712594	−	0.701576i	$-0.247520\pi$
$937$	43.6257	1.42519	0.712594	+	0.701576i	$0.247520\pi$
$938$	23.4721	0.766391
$939$	0	0
$940$	0.999151	0.0325887
$941$	−46.9114	−1.52927	−0.764634	−	0.644465i	$-0.777080\pi$
$941$	−46.9114	−1.52927	−0.764634	+	0.644465i	$0.777080\pi$
$942$	0	0
$943$	19.0586	0.620635
$944$	−8.58245	−0.279335
$945$	0	0
$946$	11.3165	0.367930
$947$	9.27001	0.301235	0.150617	−	0.988592i	$-0.451874\pi$
$947$	9.27001	0.301235	0.150617	+	0.988592i	$0.451874\pi$
$948$	0	0
$949$	−21.4643	−0.696762
$950$	3.35534	0.108862
$951$	0	0
$952$	14.6572	0.475042
$953$	4.16084	0.134783	0.0673915	−	0.997727i	$-0.478532\pi$
$953$	4.16084	0.134783	0.0673915	+	0.997727i	$0.478532\pi$
$954$	0	0
$955$	9.37451	0.303352
$956$	4.82401	0.156020
$957$	0	0
$958$	12.0611	0.389676
$959$	−49.3733	−1.59435
$960$	0	0
$961$	−30.6106	−0.987439
$962$	−11.6989	−0.377189
$963$	0	0
$964$	3.68684	0.118745
$965$	1.40870	0.0453476
$966$	0	0
$967$	18.8108	0.604914	0.302457	−	0.953163i	$-0.402193\pi$
$967$	18.8108	0.604914	0.302457	+	0.953163i	$0.402193\pi$
$968$	24.5196	0.788091
$969$	0	0
$970$	−2.26450	−0.0727088
$971$	−8.41135	−0.269933	−0.134966	−	0.990850i	$-0.543093\pi$
$971$	−8.41135	−0.269933	−0.134966	+	0.990850i	$0.543093\pi$
$972$	0	0
$973$	3.04730	0.0976919
$974$	7.49122	0.240034
$975$	0	0
$976$	3.76581	0.120541
$977$	−51.1418	−1.63617	−0.818086	−	0.575096i	$-0.804965\pi$
$977$	−51.1418	−1.63617	−0.818086	+	0.575096i	$0.804965\pi$
$978$	0	0
$979$	50.8785	1.62608
$980$	27.2429	0.870243
$981$	0	0
$982$	−1.43253	−0.0457137
$983$	25.1843	0.803253	0.401627	−	0.915804i	$-0.368445\pi$
$983$	25.1843	0.803253	0.401627	+	0.915804i	$0.368445\pi$
$984$	0	0
$985$	2.83681	0.0903883
$986$	−8.20655	−0.261350
$987$	0	0
$988$	8.66431	0.275648
$989$	−12.4165	−0.394821
$990$	0	0
$991$	7.66110	0.243363	0.121682	−	0.992569i	$-0.461171\pi$
$991$	7.66110	0.243363	0.121682	+	0.992569i	$0.461171\pi$
$992$	−3.65816	−0.116147
$993$	0	0
$994$	20.2380	0.641910
$995$	−21.6006	−0.684784
$996$	0	0
$997$	−2.93908	−0.0930818	−0.0465409	−	0.998916i	$-0.514820\pi$
$997$	−2.93908	−0.0930818	−0.0465409	+	0.998916i	$0.514820\pi$
$998$	16.3041	0.516098
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2061.2.a.j.1.10	✓	24
3.2	odd	2	inner	2061.2.a.j.1.15	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2061.2.a.j.1.10	✓	24	1.1	even	1	trivial
2061.2.a.j.1.15	yes	24	3.2	odd	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:	\( N \)	\(=\)	\( 2061 = 3^{2} \cdot 229 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2061.a (trivial)

\(f(q)\)	\(=\)	\(q-0.754652 q^{2} -1.43050 q^{4} -1.13990 q^{5} -4.86899 q^{7} +2.58883 q^{8} +O(q^{10})\)	\(q-0.754652 q^{2} -1.43050 q^{4} -1.13990 q^{5} -4.86899 q^{7} +2.58883 q^{8} +0.860226 q^{10} -4.52452 q^{11} -5.04117 q^{13} +3.67439 q^{14} +0.907335 q^{16} -1.16281 q^{17} +1.20147 q^{19} +1.63062 q^{20} +3.41444 q^{22} -3.74633 q^{23} -3.70063 q^{25} +3.80433 q^{26} +6.96509 q^{28} -9.35204 q^{29} +0.624005 q^{31} -5.86239 q^{32} +0.877514 q^{34} +5.55015 q^{35} -3.07517 q^{37} -0.906694 q^{38} -2.95100 q^{40} -5.08728 q^{41} +3.31430 q^{43} +6.47233 q^{44} +2.82718 q^{46} +0.612741 q^{47} +16.7070 q^{49} +2.79269 q^{50} +7.21140 q^{52} +1.22775 q^{53} +5.15749 q^{55} -12.6050 q^{56} +7.05753 q^{58} -9.45896 q^{59} +4.15041 q^{61} -0.470906 q^{62} +2.60939 q^{64} +5.74642 q^{65} +6.38803 q^{67} +1.66340 q^{68} -4.18843 q^{70} +5.50785 q^{71} +4.25781 q^{73} +2.32068 q^{74} -1.71871 q^{76} +22.0298 q^{77} -13.8023 q^{79} -1.03427 q^{80} +3.83912 q^{82} -7.95930 q^{83} +1.32548 q^{85} -2.50114 q^{86} -11.7132 q^{88} -11.2451 q^{89} +24.5454 q^{91} +5.35913 q^{92} -0.462406 q^{94} -1.36956 q^{95} -2.63245 q^{97} -12.6080 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 36 q^{4} + 12 q^{7}+O(q^{10}) \) 24 * q + 36 * q^4 + 12 * q^7	\( 24 q + 36 q^{4} + 12 q^{7} + 12 q^{10} + 14 q^{13} + 52 q^{16} + 50 q^{19} + 14 q^{22} + 34 q^{25} + 26 q^{28} + 34 q^{31} + 28 q^{34} + 30 q^{37} + 40 q^{40} + 48 q^{43} + 54 q^{46} + 64 q^{49} + 60 q^{52} + 20 q^{55} + 20 q^{58} + 24 q^{61} + 98 q^{64} + 52 q^{67} - 28 q^{70} + 20 q^{73} + 90 q^{76} + 48 q^{79} + 18 q^{82} + 16 q^{85} - 38 q^{88} + 42 q^{91} - 30 q^{94} - 6 q^{97}+O(q^{100}) \) 24 * q + 36 * q^4 + 12 * q^7 + 12 * q^10 + 14 * q^13 + 52 * q^16 + 50 * q^19 + 14 * q^22 + 34 * q^25 + 26 * q^28 + 34 * q^31 + 28 * q^34 + 30 * q^37 + 40 * q^40 + 48 * q^43 + 54 * q^46 + 64 * q^49 + 60 * q^52 + 20 * q^55 + 20 * q^58 + 24 * q^61 + 98 * q^64 + 52 * q^67 - 28 * q^70 + 20 * q^73 + 90 * q^76 + 48 * q^79 + 18 * q^82 + 16 * q^85 - 38 * q^88 + 42 * q^91 - 30 * q^94 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more