LMFDB - Embedded newform 9702.2.a.du.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9702,2,Mod(1,9702)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9702, 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("9702.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9702.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:	$77.4708600410$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	9702.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.74657 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +2.74657 q^{5} -1.00000 q^{8} -2.74657 q^{10} -1.00000 q^{11} -5.49314 q^{13} +1.00000 q^{16} +1.00000 q^{17} +8.03677 q^{19} +2.74657 q^{20} +1.00000 q^{22} +1.29021 q^{23} +2.54364 q^{25} +5.49314 q^{26} -4.54364 q^{29} -5.08727 q^{31} -1.00000 q^{32} -1.00000 q^{34} -4.03677 q^{37} -8.03677 q^{38} -2.74657 q^{40} +5.54364 q^{41} -4.03677 q^{43} -1.00000 q^{44} -1.29021 q^{46} -9.29021 q^{47} -2.54364 q^{50} -5.49314 q^{52} -5.49314 q^{53} -2.74657 q^{55} +4.54364 q^{58} -9.52991 q^{59} +1.65929 q^{61} +5.08727 q^{62} +1.00000 q^{64} -15.0873 q^{65} +3.54364 q^{67} +1.00000 q^{68} -2.54364 q^{71} +8.58041 q^{73} +4.03677 q^{74} +8.03677 q^{76} +12.2397 q^{79} +2.74657 q^{80} -5.54364 q^{82} -14.5299 q^{83} +2.74657 q^{85} +4.03677 q^{86} +1.00000 q^{88} -6.50686 q^{89} +1.29021 q^{92} +9.29021 q^{94} +22.0735 q^{95} +0.0872743 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} - 3 q^{11} + 3 q^{16} + 3 q^{17} + 3 q^{19} + 3 q^{22} - 9 q^{23} + 3 q^{25} - 9 q^{29} - 6 q^{31} - 3 q^{32} - 3 q^{34} + 9 q^{37} - 3 q^{38} + 12 q^{41} + 9 q^{43} - 3 q^{44} + 9 q^{46} - 15 q^{47} - 3 q^{50} + 9 q^{58} + 9 q^{59} + 6 q^{61} + 6 q^{62} + 3 q^{64} - 36 q^{65} + 6 q^{67} + 3 q^{68} - 3 q^{71} - 9 q^{74} + 3 q^{76} + 12 q^{79} - 12 q^{82} - 6 q^{83} - 9 q^{86} + 3 q^{88} - 36 q^{89} - 9 q^{92} + 15 q^{94} + 24 q^{95} - 9 q^{97}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8 - 3 * q^11 + 3 * q^16 + 3 * q^17 + 3 * q^19 + 3 * q^22 - 9 * q^23 + 3 * q^25 - 9 * q^29 - 6 * q^31 - 3 * q^32 - 3 * q^34 + 9 * q^37 - 3 * q^38 + 12 * q^41 + 9 * q^43 - 3 * q^44 + 9 * q^46 - 15 * q^47 - 3 * q^50 + 9 * q^58 + 9 * q^59 + 6 * q^61 + 6 * q^62 + 3 * q^64 - 36 * q^65 + 6 * q^67 + 3 * q^68 - 3 * q^71 - 9 * q^74 + 3 * q^76 + 12 * q^79 - 12 * q^82 - 6 * q^83 - 9 * q^86 + 3 * q^88 - 36 * q^89 - 9 * q^92 + 15 * q^94 + 24 * q^95 - 9 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.74657	1.22830	0.614151	−	0.789188i	$-0.289498\pi$
$5$	2.74657	1.22830	0.614151	+	0.789188i	$0.289498\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.74657	−0.868541
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.49314	−1.52352	−0.761761	−	0.647858i	$-0.775665\pi$
$13$	−5.49314	−1.52352	−0.761761	+	0.647858i	$0.775665\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	8.03677	1.84376	0.921881	−	0.387473i	$-0.126652\pi$
$19$	8.03677	1.84376	0.921881	+	0.387473i	$0.126652\pi$
$20$	2.74657	0.614151
$21$	0	0
$22$	1.00000	0.213201
$23$	1.29021	0.269026	0.134513	−	0.990912i	$-0.457053\pi$
$23$	1.29021	0.269026	0.134513	+	0.990912i	$0.457053\pi$
$24$	0	0
$25$	2.54364	0.508727
$26$	5.49314	1.07729
$27$	0	0
$28$	0	0
$29$	−4.54364	−0.843732	−0.421866	−	0.906658i	$-0.638625\pi$
$29$	−4.54364	−0.843732	−0.421866	+	0.906658i	$0.638625\pi$
$30$	0	0
$31$	−5.08727	−0.913701	−0.456851	−	0.889543i	$-0.651023\pi$
$31$	−5.08727	−0.913701	−0.456851	+	0.889543i	$0.651023\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0	0
$36$	0	0
$37$	−4.03677	−0.663641	−0.331821	−	0.943342i	$-0.607663\pi$
$37$	−4.03677	−0.663641	−0.331821	+	0.943342i	$0.607663\pi$
$38$	−8.03677	−1.30374
$39$	0	0
$40$	−2.74657	−0.434271
$41$	5.54364	0.865771	0.432885	−	0.901449i	$-0.357495\pi$
$41$	5.54364	0.865771	0.432885	+	0.901449i	$0.357495\pi$
$42$	0	0
$43$	−4.03677	−0.615602	−0.307801	−	0.951451i	$-0.599593\pi$
$43$	−4.03677	−0.615602	−0.307801	+	0.951451i	$0.599593\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−1.29021	−0.190230
$47$	−9.29021	−1.35512	−0.677558	−	0.735469i	$-0.736962\pi$
$47$	−9.29021	−1.35512	−0.677558	+	0.735469i	$0.736962\pi$
$48$	0	0
$49$	0	0
$50$	−2.54364	−0.359725
$51$	0	0
$52$	−5.49314	−0.761761
$53$	−5.49314	−0.754540	−0.377270	−	0.926103i	$-0.623137\pi$
$53$	−5.49314	−0.754540	−0.377270	+	0.926103i	$0.623137\pi$
$54$	0	0
$55$	−2.74657	−0.370347
$56$	0	0
$57$	0	0
$58$	4.54364	0.596609
$59$	−9.52991	−1.24069	−0.620344	−	0.784330i	$-0.713007\pi$
$59$	−9.52991	−1.24069	−0.620344	+	0.784330i	$0.713007\pi$
$60$	0	0
$61$	1.65929	0.212451	0.106225	−	0.994342i	$-0.466123\pi$
$61$	1.65929	0.212451	0.106225	+	0.994342i	$0.466123\pi$
$62$	5.08727	0.646084
$63$	0	0
$64$	1.00000	0.125000
$65$	−15.0873	−1.87135
$66$	0	0
$67$	3.54364	0.432924	0.216462	−	0.976291i	$-0.430548\pi$
$67$	3.54364	0.432924	0.216462	+	0.976291i	$0.430548\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	−2.54364	−0.301874	−0.150937	−	0.988543i	$-0.548229\pi$
$71$	−2.54364	−0.301874	−0.150937	+	0.988543i	$0.548229\pi$
$72$	0	0
$73$	8.58041	1.00426	0.502131	−	0.864792i	$-0.332550\pi$
$73$	8.58041	1.00426	0.502131	+	0.864792i	$0.332550\pi$
$74$	4.03677	0.469265
$75$	0	0
$76$	8.03677	0.921881
$77$	0	0
$78$	0	0
$79$	12.2397	1.37707	0.688537	−	0.725201i	$-0.258253\pi$
$79$	12.2397	1.37707	0.688537	+	0.725201i	$0.258253\pi$
$80$	2.74657	0.307076
$81$	0	0
$82$	−5.54364	−0.612192
$83$	−14.5299	−1.59486	−0.797432	−	0.603408i	$-0.793809\pi$
$83$	−14.5299	−1.59486	−0.797432	+	0.603408i	$0.793809\pi$
$84$	0	0
$85$	2.74657	0.297907
$86$	4.03677	0.435296
$87$	0	0
$88$	1.00000	0.106600
$89$	−6.50686	−0.689726	−0.344863	−	0.938653i	$-0.612075\pi$
$89$	−6.50686	−0.689726	−0.344863	+	0.938653i	$0.612075\pi$
$90$	0	0
$91$	0	0
$92$	1.29021	0.134513
$93$	0	0
$94$	9.29021	0.958212
$95$	22.0735	2.26470
$96$	0	0
$97$	0.0872743	0.00886136	0.00443068	−	0.999990i	$-0.498590\pi$
$97$	0.0872743	0.00886136	0.00443068	+	0.999990i	$0.498590\pi$
$98$	0	0
$99$	0	0
$100$	2.54364	0.254364
$101$	13.0230	1.29584	0.647921	−	0.761708i	$-0.275639\pi$
$101$	13.0230	1.29584	0.647921	+	0.761708i	$0.275639\pi$
$102$	0	0
$103$	18.0735	1.78084	0.890420	−	0.455140i	$-0.150411\pi$
$103$	18.0735	1.78084	0.890420	+	0.455140i	$0.150411\pi$
$104$	5.49314	0.538646
$105$	0	0
$106$	5.49314	0.533541
$107$	−17.4426	−1.68624	−0.843122	−	0.537723i	$-0.819285\pi$
$107$	−17.4426	−1.68624	−0.843122	+	0.537723i	$0.819285\pi$
$108$	0	0
$109$	1.65929	0.158932	0.0794658	−	0.996838i	$-0.474679\pi$
$109$	1.65929	0.158932	0.0794658	+	0.996838i	$0.474679\pi$
$110$	2.74657	0.261875
$111$	0	0
$112$	0	0
$113$	7.08727	0.666715	0.333357	−	0.942801i	$-0.391818\pi$
$113$	7.08727	0.666715	0.333357	+	0.942801i	$0.391818\pi$
$114$	0	0
$115$	3.54364	0.330446
$116$	−4.54364	−0.421866
$117$	0	0
$118$	9.52991	0.877299
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.65929	−0.150225
$123$	0	0
$124$	−5.08727	−0.456851
$125$	−6.74657	−0.603431
$126$	0	0
$127$	−6.78334	−0.601924	−0.300962	−	0.953636i	$-0.597308\pi$
$127$	−6.78334	−0.601924	−0.300962	+	0.953636i	$0.597308\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	15.0873	1.32324
$131$	−13.8990	−1.21436	−0.607181	−	0.794564i	$-0.707700\pi$
$131$	−13.8990	−1.21436	−0.607181	+	0.794564i	$0.707700\pi$
$132$	0	0
$133$	0	0
$134$	−3.54364	−0.306124
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−5.49314	−0.469310	−0.234655	−	0.972079i	$-0.575396\pi$
$137$	−5.49314	−0.469310	−0.234655	+	0.972079i	$0.575396\pi$
$138$	0	0
$139$	−13.6309	−1.15616	−0.578079	−	0.815981i	$-0.696198\pi$
$139$	−13.6309	−1.15616	−0.578079	+	0.815981i	$0.696198\pi$
$140$	0	0
$141$	0	0
$142$	2.54364	0.213457
$143$	5.49314	0.459359
$144$	0	0
$145$	−12.4794	−1.03636
$146$	−8.58041	−0.710120
$147$	0	0
$148$	−4.03677	−0.331821
$149$	−16.0368	−1.31378	−0.656892	−	0.753985i	$-0.728129\pi$
$149$	−16.0368	−1.31378	−0.656892	+	0.753985i	$0.728129\pi$
$150$	0	0
$151$	−5.69607	−0.463539	−0.231770	−	0.972771i	$-0.574452\pi$
$151$	−5.69607	−0.463539	−0.231770	+	0.972771i	$0.574452\pi$
$152$	−8.03677	−0.651868
$153$	0	0
$154$	0	0
$155$	−13.9725	−1.12230
$156$	0	0
$157$	−18.1103	−1.44536	−0.722680	−	0.691182i	$-0.757090\pi$
$157$	−18.1103	−1.44536	−0.722680	+	0.691182i	$0.757090\pi$
$158$	−12.2397	−0.973739
$159$	0	0
$160$	−2.74657	−0.217135
$161$	0	0
$162$	0	0
$163$	−13.4426	−1.05291	−0.526454	−	0.850203i	$-0.676479\pi$
$163$	−13.4426	−1.05291	−0.526454	+	0.850203i	$0.676479\pi$
$164$	5.54364	0.432885
$165$	0	0
$166$	14.5299	1.12774
$167$	23.5667	1.82364	0.911822	−	0.410585i	$-0.134675\pi$
$167$	23.5667	1.82364	0.911822	+	0.410585i	$0.134675\pi$
$168$	0	0
$169$	17.1745	1.32112
$170$	−2.74657	−0.210652
$171$	0	0
$172$	−4.03677	−0.307801
$173$	−10.9127	−0.829679	−0.414840	−	0.909895i	$-0.636162\pi$
$173$	−10.9127	−0.829679	−0.414840	+	0.909895i	$0.636162\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	6.50686	0.487710
$179$	12.9495	0.967891	0.483946	−	0.875098i	$-0.339203\pi$
$179$	12.9495	0.967891	0.483946	+	0.875098i	$0.339203\pi$
$180$	0	0
$181$	−7.59414	−0.564468	−0.282234	−	0.959346i	$-0.591075\pi$
$181$	−7.59414	−0.564468	−0.282234	+	0.959346i	$0.591075\pi$
$182$	0	0
$183$	0	0
$184$	−1.29021	−0.0951152
$185$	−11.0873	−0.815153
$186$	0	0
$187$	−1.00000	−0.0731272
$188$	−9.29021	−0.677558
$189$	0	0
$190$	−22.0735	−1.60138
$191$	−18.9863	−1.37380	−0.686899	−	0.726753i	$-0.741029\pi$
$191$	−18.9863	−1.37380	−0.686899	+	0.726753i	$0.741029\pi$
$192$	0	0
$193$	−21.5667	−1.55240	−0.776202	−	0.630484i	$-0.782856\pi$
$193$	−21.5667	−1.55240	−0.776202	+	0.630484i	$0.782856\pi$
$194$	−0.0872743	−0.00626593
$195$	0	0
$196$	0	0
$197$	3.12405	0.222579	0.111290	−	0.993788i	$-0.464502\pi$
$197$	3.12405	0.222579	0.111290	+	0.993788i	$0.464502\pi$
$198$	0	0
$199$	26.0735	1.84830	0.924152	−	0.382024i	$-0.124773\pi$
$199$	26.0735	1.84830	0.924152	+	0.382024i	$0.124773\pi$
$200$	−2.54364	−0.179862
$201$	0	0
$202$	−13.0230	−0.916298
$203$	0	0
$204$	0	0
$205$	15.2260	1.06343
$206$	−18.0735	−1.25924
$207$	0	0
$208$	−5.49314	−0.380880
$209$	−8.03677	−0.555915
$210$	0	0
$211$	15.6677	1.07861	0.539304	−	0.842111i	$-0.318687\pi$
$211$	15.6677	1.07861	0.539304	+	0.842111i	$0.318687\pi$
$212$	−5.49314	−0.377270
$213$	0	0
$214$	17.4426	1.19235
$215$	−11.0873	−0.756146
$216$	0	0
$217$	0	0
$218$	−1.65929	−0.112382
$219$	0	0
$220$	−2.74657	−0.185174
$221$	−5.49314	−0.369508
$222$	0	0
$223$	−2.40586	−0.161108	−0.0805542	−	0.996750i	$-0.525669\pi$
$223$	−2.40586	−0.161108	−0.0805542	+	0.996750i	$0.525669\pi$
$224$	0	0
$225$	0	0
$226$	−7.08727	−0.471438
$227$	2.45636	0.163035	0.0815173	−	0.996672i	$-0.474023\pi$
$227$	2.45636	0.163035	0.0815173	+	0.996672i	$0.474023\pi$
$228$	0	0
$229$	10.5804	0.699173	0.349587	−	0.936904i	$-0.386322\pi$
$229$	10.5804	0.699173	0.349587	+	0.936904i	$0.386322\pi$
$230$	−3.54364	−0.233661
$231$	0	0
$232$	4.54364	0.298304
$233$	7.07355	0.463403	0.231702	−	0.972787i	$-0.425571\pi$
$233$	7.07355	0.463403	0.231702	+	0.972787i	$0.425571\pi$
$234$	0	0
$235$	−25.5162	−1.66449
$236$	−9.52991	−0.620344
$237$	0	0
$238$	0	0
$239$	−7.66769	−0.495981	−0.247991	−	0.968762i	$-0.579770\pi$
$239$	−7.66769	−0.495981	−0.247991	+	0.968762i	$0.579770\pi$
$240$	0	0
$241$	25.0598	1.61424	0.807122	−	0.590384i	$-0.201024\pi$
$241$	25.0598	1.61424	0.807122	+	0.590384i	$0.201024\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	1.65929	0.106225
$245$	0	0
$246$	0	0
$247$	−44.1471	−2.80901
$248$	5.08727	0.323042
$249$	0	0
$250$	6.74657	0.426690
$251$	−25.0230	−1.57944	−0.789720	−	0.613467i	$-0.789774\pi$
$251$	−25.0230	−1.57944	−0.789720	+	0.613467i	$0.789774\pi$
$252$	0	0
$253$	−1.29021	−0.0811145
$254$	6.78334	0.425625
$255$	0	0
$256$	1.00000	0.0625000
$257$	−10.5069	−0.655400	−0.327700	−	0.944782i	$-0.606274\pi$
$257$	−10.5069	−0.655400	−0.327700	+	0.944782i	$0.606274\pi$
$258$	0	0
$259$	0	0
$260$	−15.0873	−0.935673
$261$	0	0
$262$	13.8990	0.858683
$263$	−23.6402	−1.45772	−0.728860	−	0.684663i	$-0.759949\pi$
$263$	−23.6402	−1.45772	−0.728860	+	0.684663i	$0.759949\pi$
$264$	0	0
$265$	−15.0873	−0.926804
$266$	0	0
$267$	0	0
$268$	3.54364	0.216462
$269$	−15.1524	−0.923860	−0.461930	−	0.886916i	$-0.652843\pi$
$269$	−15.1524	−0.923860	−0.461930	+	0.886916i	$0.652843\pi$
$270$	0	0
$271$	−18.9863	−1.15333	−0.576667	−	0.816979i	$-0.695647\pi$
$271$	−18.9863	−1.15333	−0.576667	+	0.816979i	$0.695647\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	5.49314	0.331853
$275$	−2.54364	−0.153387
$276$	0	0
$277$	24.4794	1.47083	0.735413	−	0.677620i	$-0.236988\pi$
$277$	24.4794	1.47083	0.735413	+	0.677620i	$0.236988\pi$
$278$	13.6309	0.817528
$279$	0	0
$280$	0	0
$281$	12.0873	0.721066	0.360533	−	0.932746i	$-0.382595\pi$
$281$	12.0873	0.721066	0.360533	+	0.932746i	$0.382595\pi$
$282$	0	0
$283$	−4.68141	−0.278281	−0.139141	−	0.990273i	$-0.544434\pi$
$283$	−4.68141	−0.278281	−0.139141	+	0.990273i	$0.544434\pi$
$284$	−2.54364	−0.150937
$285$	0	0
$286$	−5.49314	−0.324816
$287$	0	0
$288$	0	0
$289$	−16.0000	−0.941176
$290$	12.4794	0.732816
$291$	0	0
$292$	8.58041	0.502131
$293$	−11.6309	−0.679485	−0.339743	−	0.940518i	$-0.610340\pi$
$293$	−11.6309	−0.679485	−0.339743	+	0.940518i	$0.610340\pi$
$294$	0	0
$295$	−26.1745	−1.52394
$296$	4.03677	0.234633
$297$	0	0
$298$	16.0368	0.928985
$299$	−7.08727	−0.409868
$300$	0	0
$301$	0	0
$302$	5.69607	0.327772
$303$	0	0
$304$	8.03677	0.460941
$305$	4.55736	0.260954
$306$	0	0
$307$	14.0735	0.803220	0.401610	−	0.915811i	$-0.368451\pi$
$307$	14.0735	0.803220	0.401610	+	0.915811i	$0.368451\pi$
$308$	0	0
$309$	0	0
$310$	13.9725	0.793587
$311$	12.2765	0.696135	0.348068	−	0.937469i	$-0.386838\pi$
$311$	12.2765	0.696135	0.348068	+	0.937469i	$0.386838\pi$
$312$	0	0
$313$	−23.4564	−1.32583	−0.662916	−	0.748694i	$-0.730681\pi$
$313$	−23.4564	−1.32583	−0.662916	+	0.748694i	$0.730681\pi$
$314$	18.1103	1.02202
$315$	0	0
$316$	12.2397	0.688537
$317$	0.572020	0.0321278	0.0160639	−	0.999871i	$-0.494886\pi$
$317$	0.572020	0.0321278	0.0160639	+	0.999871i	$0.494886\pi$
$318$	0	0
$319$	4.54364	0.254395
$320$	2.74657	0.153538
$321$	0	0
$322$	0	0
$323$	8.03677	0.447178
$324$	0	0
$325$	−13.9725	−0.775057
$326$	13.4426	0.744519
$327$	0	0
$328$	−5.54364	−0.306096
$329$	0	0
$330$	0	0
$331$	7.71819	0.424230	0.212115	−	0.977245i	$-0.431965\pi$
$331$	7.71819	0.424230	0.212115	+	0.977245i	$0.431965\pi$
$332$	−14.5299	−0.797432
$333$	0	0
$334$	−23.5667	−1.28951
$335$	9.73284	0.531762
$336$	0	0
$337$	28.5530	1.55538	0.777689	−	0.628649i	$-0.216392\pi$
$337$	28.5530	1.55538	0.777689	+	0.628649i	$0.216392\pi$
$338$	−17.1745	−0.934172
$339$	0	0
$340$	2.74657	0.148954
$341$	5.08727	0.275491
$342$	0	0
$343$	0	0
$344$	4.03677	0.217648
$345$	0	0
$346$	10.9127	0.586672
$347$	23.4426	1.25847	0.629233	−	0.777216i	$-0.283369\pi$
$347$	23.4426	1.25847	0.629233	+	0.777216i	$0.283369\pi$
$348$	0	0
$349$	−25.7328	−1.37745	−0.688724	−	0.725024i	$-0.741829\pi$
$349$	−25.7328	−1.37745	−0.688724	+	0.725024i	$0.741829\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	−18.4059	−0.979645	−0.489823	−	0.871822i	$-0.662938\pi$
$353$	−18.4059	−0.979645	−0.489823	+	0.871822i	$0.662938\pi$
$354$	0	0
$355$	−6.98627	−0.370793
$356$	−6.50686	−0.344863
$357$	0	0
$358$	−12.9495	−0.684402
$359$	31.9725	1.68745	0.843723	−	0.536778i	$-0.180359\pi$
$359$	31.9725	1.68745	0.843723	+	0.536778i	$0.180359\pi$
$360$	0	0
$361$	45.5897	2.39946
$362$	7.59414	0.399139
$363$	0	0
$364$	0	0
$365$	23.5667	1.23354
$366$	0	0
$367$	−36.4794	−1.90421	−0.952105	−	0.305772i	$-0.901086\pi$
$367$	−36.4794	−1.90421	−0.952105	+	0.305772i	$0.901086\pi$
$368$	1.29021	0.0672566
$369$	0	0
$370$	11.0873	0.576400
$371$	0	0
$372$	0	0
$373$	−20.3133	−1.05178	−0.525890	−	0.850552i	$-0.676268\pi$
$373$	−20.3133	−1.05178	−0.525890	+	0.850552i	$0.676268\pi$
$374$	1.00000	0.0517088
$375$	0	0
$376$	9.29021	0.479106
$377$	24.9588	1.28544
$378$	0	0
$379$	9.44264	0.485036	0.242518	−	0.970147i	$-0.422027\pi$
$379$	9.44264	0.485036	0.242518	+	0.970147i	$0.422027\pi$
$380$	22.0735	1.13235
$381$	0	0
$382$	18.9863	0.971422
$383$	23.7045	1.21124	0.605621	−	0.795754i	$-0.292925\pi$
$383$	23.7045	1.21124	0.605621	+	0.795754i	$0.292925\pi$
$384$	0	0
$385$	0	0
$386$	21.5667	1.09772
$387$	0	0
$388$	0.0872743	0.00443068
$389$	1.25343	0.0635515	0.0317758	−	0.999495i	$-0.489884\pi$
$389$	1.25343	0.0635515	0.0317758	+	0.999495i	$0.489884\pi$
$390$	0	0
$391$	1.29021	0.0652485
$392$	0	0
$393$	0	0
$394$	−3.12405	−0.157387
$395$	33.6172	1.69146
$396$	0	0
$397$	−28.6172	−1.43626	−0.718128	−	0.695911i	$-0.755001\pi$
$397$	−28.6172	−1.43626	−0.718128	+	0.695911i	$0.755001\pi$
$398$	−26.0735	−1.30695
$399$	0	0
$400$	2.54364	0.127182
$401$	−0.405862	−0.0202678	−0.0101339	−	0.999949i	$-0.503226\pi$
$401$	−0.405862	−0.0202678	−0.0101339	+	0.999949i	$0.503226\pi$
$402$	0	0
$403$	27.9451	1.39204
$404$	13.0230	0.647921
$405$	0	0
$406$	0	0
$407$	4.03677	0.200095
$408$	0	0
$409$	32.7550	1.61963	0.809814	−	0.586686i	$-0.199568\pi$
$409$	32.7550	1.61963	0.809814	+	0.586686i	$0.199568\pi$
$410$	−15.2260	−0.751957
$411$	0	0
$412$	18.0735	0.890420
$413$	0	0
$414$	0	0
$415$	−39.9074	−1.95898
$416$	5.49314	0.269323
$417$	0	0
$418$	8.03677	0.393091
$419$	−15.9358	−0.778513	−0.389257	−	0.921129i	$-0.627268\pi$
$419$	−15.9358	−0.778513	−0.389257	+	0.921129i	$0.627268\pi$
$420$	0	0
$421$	−27.1240	−1.32195	−0.660973	−	0.750410i	$-0.729856\pi$
$421$	−27.1240	−1.32195	−0.660973	+	0.750410i	$0.729856\pi$
$422$	−15.6677	−0.762691
$423$	0	0
$424$	5.49314	0.266770
$425$	2.54364	0.123385
$426$	0	0
$427$	0	0
$428$	−17.4426	−0.843122
$429$	0	0
$430$	11.0873	0.534676
$431$	40.6265	1.95691	0.978455	−	0.206461i	$-0.0661945\pi$
$431$	40.6265	1.95691	0.978455	+	0.206461i	$0.0661945\pi$
$432$	0	0
$433$	−1.10100	−0.0529107	−0.0264554	−	0.999650i	$-0.508422\pi$
$433$	−1.10100	−0.0529107	−0.0264554	+	0.999650i	$0.508422\pi$
$434$	0	0
$435$	0	0
$436$	1.65929	0.0794658
$437$	10.3691	0.496021
$438$	0	0
$439$	−22.8569	−1.09090	−0.545450	−	0.838143i	$-0.683641\pi$
$439$	−22.8569	−1.09090	−0.545450	+	0.838143i	$0.683641\pi$
$440$	2.74657	0.130938
$441$	0	0
$442$	5.49314	0.261282
$443$	9.55736	0.454084	0.227042	−	0.973885i	$-0.427095\pi$
$443$	9.55736	0.454084	0.227042	+	0.973885i	$0.427095\pi$
$444$	0	0
$445$	−17.8715	−0.847192
$446$	2.40586	0.113921
$447$	0	0
$448$	0	0
$449$	−0.681412	−0.0321578	−0.0160789	−	0.999871i	$-0.505118\pi$
$449$	−0.681412	−0.0321578	−0.0160789	+	0.999871i	$0.505118\pi$
$450$	0	0
$451$	−5.54364	−0.261040
$452$	7.08727	0.333357
$453$	0	0
$454$	−2.45636	−0.115283
$455$	0	0
$456$	0	0
$457$	18.1471	0.848885	0.424443	−	0.905455i	$-0.360470\pi$
$457$	18.1471	0.848885	0.424443	+	0.905455i	$0.360470\pi$
$458$	−10.5804	−0.494390
$459$	0	0
$460$	3.54364	0.165223
$461$	−41.7045	−1.94237	−0.971185	−	0.238326i	$-0.923401\pi$
$461$	−41.7045	−1.94237	−0.971185	+	0.238326i	$0.923401\pi$
$462$	0	0
$463$	−28.5804	−1.32824	−0.664122	−	0.747624i	$-0.731195\pi$
$463$	−28.5804	−1.32824	−0.664122	+	0.747624i	$0.731195\pi$
$464$	−4.54364	−0.210933
$465$	0	0
$466$	−7.07355	−0.327676
$467$	5.38282	0.249087	0.124544	−	0.992214i	$-0.460253\pi$
$467$	5.38282	0.249087	0.124544	+	0.992214i	$0.460253\pi$
$468$	0	0
$469$	0	0
$470$	25.5162	1.17697
$471$	0	0
$472$	9.52991	0.438650
$473$	4.03677	0.185611
$474$	0	0
$475$	20.4426	0.937972
$476$	0	0
$477$	0	0
$478$	7.66769	0.350712
$479$	14.4794	0.661581	0.330791	−	0.943704i	$-0.392685\pi$
$479$	14.4794	0.661581	0.330791	+	0.943704i	$0.392685\pi$
$480$	0	0
$481$	22.1745	1.01107
$482$	−25.0598	−1.14144
$483$	0	0
$484$	1.00000	0.0454545
$485$	0.239705	0.0108844
$486$	0	0
$487$	−8.43332	−0.382150	−0.191075	−	0.981575i	$-0.561197\pi$
$487$	−8.43332	−0.382150	−0.191075	+	0.981575i	$0.561197\pi$
$488$	−1.65929	−0.0751127
$489$	0	0
$490$	0	0
$491$	38.6770	1.74547	0.872734	−	0.488195i	$-0.162345\pi$
$491$	38.6770	1.74547	0.872734	+	0.488195i	$0.162345\pi$
$492$	0	0
$493$	−4.54364	−0.204635
$494$	44.1471	1.98627
$495$	0	0
$496$	−5.08727	−0.228425
$497$	0	0
$498$	0	0
$499$	38.0461	1.70318	0.851589	−	0.524211i	$-0.175640\pi$
$499$	38.0461	1.70318	0.851589	+	0.524211i	$0.175640\pi$
$500$	−6.74657	−0.301716
$501$	0	0
$502$	25.0230	1.11683
$503$	21.5667	0.961611	0.480805	−	0.876827i	$-0.340344\pi$
$503$	21.5667	0.961611	0.480805	+	0.876827i	$0.340344\pi$
$504$	0	0
$505$	35.7687	1.59169
$506$	1.29021	0.0573566
$507$	0	0
$508$	−6.78334	−0.300962
$509$	−26.6540	−1.18142	−0.590708	−	0.806885i	$-0.701151\pi$
$509$	−26.6540	−1.18142	−0.590708	+	0.806885i	$0.701151\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	10.5069	0.463438
$515$	49.6402	2.18741
$516$	0	0
$517$	9.29021	0.408583
$518$	0	0
$519$	0	0
$520$	15.0873	0.661621
$521$	−23.9725	−1.05026	−0.525128	−	0.851023i	$-0.675983\pi$
$521$	−23.9725	−1.05026	−0.525128	+	0.851023i	$0.675983\pi$
$522$	0	0
$523$	−20.6814	−0.904335	−0.452168	−	0.891933i	$-0.649349\pi$
$523$	−20.6814	−0.904335	−0.452168	+	0.891933i	$0.649349\pi$
$524$	−13.8990	−0.607181
$525$	0	0
$526$	23.6402	1.03076
$527$	−5.08727	−0.221605
$528$	0	0
$529$	−21.3354	−0.927625
$530$	15.0873	0.655349
$531$	0	0
$532$	0	0
$533$	−30.4520	−1.31902
$534$	0	0
$535$	−47.9074	−2.07122
$536$	−3.54364	−0.153062
$537$	0	0
$538$	15.1524	0.653268
$539$	0	0
$540$	0	0
$541$	35.9074	1.54378	0.771890	−	0.635757i	$-0.219312\pi$
$541$	35.9074	1.54378	0.771890	+	0.635757i	$0.219312\pi$
$542$	18.9863	0.815530
$543$	0	0
$544$	−1.00000	−0.0428746
$545$	4.55736	0.195216
$546$	0	0
$547$	−5.12405	−0.219088	−0.109544	−	0.993982i	$-0.534939\pi$
$547$	−5.12405	−0.219088	−0.109544	+	0.993982i	$0.534939\pi$
$548$	−5.49314	−0.234655
$549$	0	0
$550$	2.54364	0.108461
$551$	−36.5162	−1.55564
$552$	0	0
$553$	0	0
$554$	−24.4794	−1.04003
$555$	0	0
$556$	−13.6309	−0.578079
$557$	12.8760	0.545572	0.272786	−	0.962075i	$-0.412055\pi$
$557$	12.8760	0.545572	0.272786	+	0.962075i	$0.412055\pi$
$558$	0	0
$559$	22.1745	0.937883
$560$	0	0
$561$	0	0
$562$	−12.0873	−0.509871
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	19.4657	0.818927
$566$	4.68141	0.196774
$567$	0	0
$568$	2.54364	0.106729
$569$	−21.3554	−0.895263	−0.447632	−	0.894218i	$-0.647732\pi$
$569$	−21.3554	−0.895263	−0.447632	+	0.894218i	$0.647732\pi$
$570$	0	0
$571$	7.45636	0.312039	0.156020	−	0.987754i	$-0.450134\pi$
$571$	7.45636	0.312039	0.156020	+	0.987754i	$0.450134\pi$
$572$	5.49314	0.229680
$573$	0	0
$574$	0	0
$575$	3.28181	0.136861
$576$	0	0
$577$	−5.64464	−0.234989	−0.117495	−	0.993074i	$-0.537486\pi$
$577$	−5.64464	−0.234989	−0.117495	+	0.993074i	$0.537486\pi$
$578$	16.0000	0.665512
$579$	0	0
$580$	−12.4794	−0.518179
$581$	0	0
$582$	0	0
$583$	5.49314	0.227502
$584$	−8.58041	−0.355060
$585$	0	0
$586$	11.6309	0.480469
$587$	−34.4059	−1.42008	−0.710041	−	0.704160i	$-0.751324\pi$
$587$	−34.4059	−1.42008	−0.710041	+	0.704160i	$0.751324\pi$
$588$	0	0
$589$	−40.8853	−1.68465
$590$	26.1745	1.07759
$591$	0	0
$592$	−4.03677	−0.165910
$593$	−18.7182	−0.768664	−0.384332	−	0.923195i	$-0.625568\pi$
$593$	−18.7182	−0.768664	−0.384332	+	0.923195i	$0.625568\pi$
$594$	0	0
$595$	0	0
$596$	−16.0368	−0.656892
$597$	0	0
$598$	7.08727	0.289820
$599$	−10.4143	−0.425515	−0.212757	−	0.977105i	$-0.568244\pi$
$599$	−10.4143	−0.425515	−0.212757	+	0.977105i	$0.568244\pi$
$600$	0	0
$601$	−25.0873	−1.02333	−0.511666	−	0.859185i	$-0.670971\pi$
$601$	−25.0873	−1.02333	−0.511666	+	0.859185i	$0.670971\pi$
$602$	0	0
$603$	0	0
$604$	−5.69607	−0.231770
$605$	2.74657	0.111664
$606$	0	0
$607$	−31.9074	−1.29508	−0.647541	−	0.762031i	$-0.724202\pi$
$607$	−31.9074	−1.29508	−0.647541	+	0.762031i	$0.724202\pi$
$608$	−8.03677	−0.325934
$609$	0	0
$610$	−4.55736	−0.184522
$611$	51.0324	2.06455
$612$	0	0
$613$	3.42798	0.138455	0.0692274	−	0.997601i	$-0.477947\pi$
$613$	3.42798	0.138455	0.0692274	+	0.997601i	$0.477947\pi$
$614$	−14.0735	−0.567962
$615$	0	0
$616$	0	0
$617$	−12.7550	−0.513495	−0.256748	−	0.966478i	$-0.582651\pi$
$617$	−12.7550	−0.513495	−0.256748	+	0.966478i	$0.582651\pi$
$618$	0	0
$619$	21.5897	0.867765	0.433882	−	0.900970i	$-0.357143\pi$
$619$	21.5897	0.867765	0.433882	+	0.900970i	$0.357143\pi$
$620$	−13.9725	−0.561151
$621$	0	0
$622$	−12.2765	−0.492242
$623$	0	0
$624$	0	0
$625$	−31.2481	−1.24992
$626$	23.4564	0.937505
$627$	0	0
$628$	−18.1103	−0.722680
$629$	−4.03677	−0.160957
$630$	0	0
$631$	4.23131	0.168446	0.0842230	−	0.996447i	$-0.473159\pi$
$631$	4.23131	0.168446	0.0842230	+	0.996447i	$0.473159\pi$
$632$	−12.2397	−0.486869
$633$	0	0
$634$	−0.572020	−0.0227178
$635$	−18.6309	−0.739345
$636$	0	0
$637$	0	0
$638$	−4.54364	−0.179884
$639$	0	0
$640$	−2.74657	−0.108568
$641$	−4.07355	−0.160895	−0.0804477	−	0.996759i	$-0.525635\pi$
$641$	−4.07355	−0.160895	−0.0804477	+	0.996759i	$0.525635\pi$
$642$	0	0
$643$	19.2618	0.759612	0.379806	−	0.925066i	$-0.375991\pi$
$643$	19.2618	0.759612	0.379806	+	0.925066i	$0.375991\pi$
$644$	0	0
$645$	0	0
$646$	−8.03677	−0.316203
$647$	−18.8937	−0.742787	−0.371393	−	0.928476i	$-0.621120\pi$
$647$	−18.8937	−0.742787	−0.371393	+	0.928476i	$0.621120\pi$
$648$	0	0
$649$	9.52991	0.374082
$650$	13.9725	0.548048
$651$	0	0
$652$	−13.4426	−0.526454
$653$	8.92112	0.349110	0.174555	−	0.984647i	$-0.444151\pi$
$653$	8.92112	0.349110	0.174555	+	0.984647i	$0.444151\pi$
$654$	0	0
$655$	−38.1745	−1.49160
$656$	5.54364	0.216443
$657$	0	0
$658$	0	0
$659$	−45.4426	−1.77019	−0.885097	−	0.465407i	$-0.845908\pi$
$659$	−45.4426	−1.77019	−0.885097	+	0.465407i	$0.845908\pi$
$660$	0	0
$661$	29.0230	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$661$	29.0230	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$662$	−7.71819	−0.299976
$663$	0	0
$664$	14.5299	0.563870
$665$	0	0
$666$	0	0
$667$	−5.86223	−0.226986
$668$	23.5667	0.911822
$669$	0	0
$670$	−9.73284	−0.376012
$671$	−1.65929	−0.0640563
$672$	0	0
$673$	−16.2481	−0.626318	−0.313159	−	0.949701i	$-0.601387\pi$
$673$	−16.2481	−0.626318	−0.313159	+	0.949701i	$0.601387\pi$
$674$	−28.5530	−1.09982
$675$	0	0
$676$	17.1745	0.660560
$677$	−3.88968	−0.149493	−0.0747463	−	0.997203i	$-0.523815\pi$
$677$	−3.88968	−0.149493	−0.0747463	+	0.997203i	$0.523815\pi$
$678$	0	0
$679$	0	0
$680$	−2.74657	−0.105326
$681$	0	0
$682$	−5.08727	−0.194802
$683$	−5.80546	−0.222140	−0.111070	−	0.993813i	$-0.535428\pi$
$683$	−5.80546	−0.222140	−0.111070	+	0.993813i	$0.535428\pi$
$684$	0	0
$685$	−15.0873	−0.576455
$686$	0	0
$687$	0	0
$688$	−4.03677	−0.153901
$689$	30.1745	1.14956
$690$	0	0
$691$	6.35536	0.241769	0.120885	−	0.992667i	$-0.461427\pi$
$691$	6.35536	0.241769	0.120885	+	0.992667i	$0.461427\pi$
$692$	−10.9127	−0.414840
$693$	0	0
$694$	−23.4426	−0.889870
$695$	−37.4382	−1.42011
$696$	0	0
$697$	5.54364	0.209980
$698$	25.7328	0.974002
$699$	0	0
$700$	0	0
$701$	−5.86223	−0.221413	−0.110707	−	0.993853i	$-0.535311\pi$
$701$	−5.86223	−0.221413	−0.110707	+	0.993853i	$0.535311\pi$
$702$	0	0
$703$	−32.4426	−1.22360
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	18.4059	0.692714
$707$	0	0
$708$	0	0
$709$	−13.4564	−0.505364	−0.252682	−	0.967549i	$-0.581313\pi$
$709$	−13.4564	−0.505364	−0.252682	+	0.967549i	$0.581313\pi$
$710$	6.98627	0.262190
$711$	0	0
$712$	6.50686	0.243855
$713$	−6.56363	−0.245810
$714$	0	0
$715$	15.0873	0.564232
$716$	12.9495	0.483946
$717$	0	0
$718$	−31.9725	−1.19320
$719$	−24.2765	−0.905360	−0.452680	−	0.891673i	$-0.649532\pi$
$719$	−24.2765	−0.905360	−0.452680	+	0.891673i	$0.649532\pi$
$720$	0	0
$721$	0	0
$722$	−45.5897	−1.69667
$723$	0	0
$724$	−7.59414	−0.282234
$725$	−11.5574	−0.429230
$726$	0	0
$727$	22.1471	0.821390	0.410695	−	0.911773i	$-0.365286\pi$
$727$	22.1471	0.821390	0.410695	+	0.911773i	$0.365286\pi$
$728$	0	0
$729$	0	0
$730$	−23.5667	−0.872242
$731$	−4.03677	−0.149305
$732$	0	0
$733$	5.32698	0.196756	0.0983782	−	0.995149i	$-0.468635\pi$
$733$	5.32698	0.196756	0.0983782	+	0.995149i	$0.468635\pi$
$734$	36.4794	1.34648
$735$	0	0
$736$	−1.29021	−0.0475576
$737$	−3.54364	−0.130532
$738$	0	0
$739$	−25.2618	−0.929271	−0.464636	−	0.885502i	$-0.653815\pi$
$739$	−25.2618	−0.929271	−0.464636	+	0.885502i	$0.653815\pi$
$740$	−11.0873	−0.407576
$741$	0	0
$742$	0	0
$743$	25.4196	0.932554	0.466277	−	0.884639i	$-0.345595\pi$
$743$	25.4196	0.932554	0.466277	+	0.884639i	$0.345595\pi$
$744$	0	0
$745$	−44.0461	−1.61372
$746$	20.3133	0.743721
$747$	0	0
$748$	−1.00000	−0.0365636
$749$	0	0
$750$	0	0
$751$	−40.5530	−1.47980	−0.739899	−	0.672718i	$-0.765127\pi$
$751$	−40.5530	−1.47980	−0.739899	+	0.672718i	$0.765127\pi$
$752$	−9.29021	−0.338779
$753$	0	0
$754$	−24.9588	−0.908947
$755$	−15.6446	−0.569367
$756$	0	0
$757$	13.3554	0.485409	0.242704	−	0.970100i	$-0.421965\pi$
$757$	13.3554	0.485409	0.242704	+	0.970100i	$0.421965\pi$
$758$	−9.44264	−0.342972
$759$	0	0
$760$	−22.0735	−0.800692
$761$	18.4564	0.669043	0.334521	−	0.942388i	$-0.391425\pi$
$761$	18.4564	0.669043	0.334521	+	0.942388i	$0.391425\pi$
$762$	0	0
$763$	0	0
$764$	−18.9863	−0.686899
$765$	0	0
$766$	−23.7045	−0.856477
$767$	52.3491	1.89022
$768$	0	0
$769$	−49.9158	−1.80001	−0.900005	−	0.435881i	$-0.856437\pi$
$769$	−49.9158	−1.80001	−0.900005	+	0.435881i	$0.856437\pi$
$770$	0	0
$771$	0	0
$772$	−21.5667	−0.776202
$773$	−42.1387	−1.51562	−0.757812	−	0.652473i	$-0.773732\pi$
$773$	−42.1387	−1.51562	−0.757812	+	0.652473i	$0.773732\pi$
$774$	0	0
$775$	−12.9402	−0.464825
$776$	−0.0872743	−0.00313296
$777$	0	0
$778$	−1.25343	−0.0449377
$779$	44.5530	1.59628
$780$	0	0
$781$	2.54364	0.0910185
$782$	−1.29021	−0.0461377
$783$	0	0
$784$	0	0
$785$	−49.7412	−1.77534
$786$	0	0
$787$	34.3584	1.22475	0.612373	−	0.790569i	$-0.290215\pi$
$787$	34.3584	1.22475	0.612373	+	0.790569i	$0.290215\pi$
$788$	3.12405	0.111290
$789$	0	0
$790$	−33.6172	−1.19605
$791$	0	0
$792$	0	0
$793$	−9.11473	−0.323673
$794$	28.6172	1.01559
$795$	0	0
$796$	26.0735	0.924152
$797$	−18.1387	−0.642506	−0.321253	−	0.946993i	$-0.604104\pi$
$797$	−18.1387	−0.642506	−0.321253	+	0.946993i	$0.604104\pi$
$798$	0	0
$799$	−9.29021	−0.328664
$800$	−2.54364	−0.0899312
$801$	0	0
$802$	0.405862	0.0143315
$803$	−8.58041	−0.302796
$804$	0	0
$805$	0	0
$806$	−27.9451	−0.984324
$807$	0	0
$808$	−13.0230	−0.458149
$809$	17.5436	0.616801	0.308401	−	0.951257i	$-0.400206\pi$
$809$	17.5436	0.616801	0.308401	+	0.951257i	$0.400206\pi$
$810$	0	0
$811$	43.0873	1.51300	0.756499	−	0.653994i	$-0.226908\pi$
$811$	43.0873	1.51300	0.756499	+	0.653994i	$0.226908\pi$
$812$	0	0
$813$	0	0
$814$	−4.03677	−0.141489
$815$	−36.9211	−1.29329
$816$	0	0
$817$	−32.4426	−1.13502
$818$	−32.7550	−1.14525
$819$	0	0
$820$	15.2260	0.531714
$821$	−2.10100	−0.0733255	−0.0366627	−	0.999328i	$-0.511673\pi$
$821$	−2.10100	−0.0733255	−0.0366627	+	0.999328i	$0.511673\pi$
$822$	0	0
$823$	19.7412	0.688136	0.344068	−	0.938945i	$-0.388195\pi$
$823$	19.7412	0.688136	0.344068	+	0.938945i	$0.388195\pi$
$824$	−18.0735	−0.629622
$825$	0	0
$826$	0	0
$827$	−23.7182	−0.824762	−0.412381	−	0.911011i	$-0.635303\pi$
$827$	−23.7182	−0.824762	−0.412381	+	0.911011i	$0.635303\pi$
$828$	0	0
$829$	−39.9632	−1.38798	−0.693990	−	0.719985i	$-0.744149\pi$
$829$	−39.9632	−1.38798	−0.693990	+	0.719985i	$0.744149\pi$
$830$	39.9074	1.38521
$831$	0	0
$832$	−5.49314	−0.190440
$833$	0	0
$834$	0	0
$835$	64.7275	2.23999
$836$	−8.03677	−0.277958
$837$	0	0
$838$	15.9358	0.550492
$839$	29.0682	1.00355	0.501773	−	0.864999i	$-0.332681\pi$
$839$	29.0682	1.00355	0.501773	+	0.864999i	$0.332681\pi$
$840$	0	0
$841$	−8.35536	−0.288116
$842$	27.1240	0.934756
$843$	0	0
$844$	15.6677	0.539304
$845$	47.1711	1.62273
$846$	0	0
$847$	0	0
$848$	−5.49314	−0.188635
$849$	0	0
$850$	−2.54364	−0.0872460
$851$	−5.20827	−0.178537
$852$	0	0
$853$	−20.2397	−0.692994	−0.346497	−	0.938051i	$-0.612629\pi$
$853$	−20.2397	−0.692994	−0.346497	+	0.938051i	$0.612629\pi$
$854$	0	0
$855$	0	0
$856$	17.4426	0.596177
$857$	−12.1608	−0.415406	−0.207703	−	0.978192i	$-0.566599\pi$
$857$	−12.1608	−0.415406	−0.207703	+	0.978192i	$0.566599\pi$
$858$	0	0
$859$	14.5299	0.495754	0.247877	−	0.968791i	$-0.420267\pi$
$859$	14.5299	0.495754	0.247877	+	0.968791i	$0.420267\pi$
$860$	−11.0873	−0.378073
$861$	0	0
$862$	−40.6265	−1.38374
$863$	13.5857	0.462464	0.231232	−	0.972899i	$-0.425724\pi$
$863$	13.5857	0.462464	0.231232	+	0.972899i	$0.425724\pi$
$864$	0	0
$865$	−29.9725	−1.01910
$866$	1.10100	0.0374135
$867$	0	0
$868$	0	0
$869$	−12.2397	−0.415204
$870$	0	0
$871$	−19.4657	−0.659569
$872$	−1.65929	−0.0561908
$873$	0	0
$874$	−10.3691	−0.350740
$875$	0	0
$876$	0	0
$877$	−2.06516	−0.0697354	−0.0348677	−	0.999392i	$-0.511101\pi$
$877$	−2.06516	−0.0697354	−0.0348677	+	0.999392i	$0.511101\pi$
$878$	22.8569	0.771383
$879$	0	0
$880$	−2.74657	−0.0925868
$881$	−3.08727	−0.104013	−0.0520065	−	0.998647i	$-0.516562\pi$
$881$	−3.08727	−0.104013	−0.0520065	+	0.998647i	$0.516562\pi$
$882$	0	0
$883$	21.8927	0.736749	0.368375	−	0.929677i	$-0.379914\pi$
$883$	21.8927	0.736749	0.368375	+	0.929677i	$0.379914\pi$
$884$	−5.49314	−0.184754
$885$	0	0
$886$	−9.55736	−0.321086
$887$	37.7412	1.26723	0.633613	−	0.773650i	$-0.281571\pi$
$887$	37.7412	1.26723	0.633613	+	0.773650i	$0.281571\pi$
$888$	0	0
$889$	0	0
$890$	17.8715	0.599056
$891$	0	0
$892$	−2.40586	−0.0805542
$893$	−74.6633	−2.49851
$894$	0	0
$895$	35.5667	1.18886
$896$	0	0
$897$	0	0
$898$	0.681412	0.0227390
$899$	23.1147	0.770919
$900$	0	0
$901$	−5.49314	−0.183003
$902$	5.54364	0.184583
$903$	0	0
$904$	−7.08727	−0.235719
$905$	−20.8578	−0.693337
$906$	0	0
$907$	31.5897	1.04892	0.524460	−	0.851435i	$-0.324267\pi$
$907$	31.5897	1.04892	0.524460	+	0.851435i	$0.324267\pi$
$908$	2.45636	0.0815173
$909$	0	0
$910$	0	0
$911$	27.7422	0.919139	0.459569	−	0.888142i	$-0.348004\pi$
$911$	27.7422	0.919139	0.459569	+	0.888142i	$0.348004\pi$
$912$	0	0
$913$	14.5299	0.480870
$914$	−18.1471	−0.600253
$915$	0	0
$916$	10.5804	0.349587
$917$	0	0
$918$	0	0
$919$	18.9304	0.624457	0.312229	−	0.950007i	$-0.398924\pi$
$919$	18.9304	0.624457	0.312229	+	0.950007i	$0.398924\pi$
$920$	−3.54364	−0.116830
$921$	0	0
$922$	41.7045	1.37346
$923$	13.9725	0.459912
$924$	0	0
$925$	−10.2681	−0.337613
$926$	28.5804	0.939211
$927$	0	0
$928$	4.54364	0.149152
$929$	13.7687	0.451736	0.225868	−	0.974158i	$-0.427478\pi$
$929$	13.7687	0.451736	0.225868	+	0.974158i	$0.427478\pi$
$930$	0	0
$931$	0	0
$932$	7.07355	0.231702
$933$	0	0
$934$	−5.38282	−0.176131
$935$	−2.74657	−0.0898224
$936$	0	0
$937$	17.2618	0.563919	0.281960	−	0.959426i	$-0.409016\pi$
$937$	17.2618	0.563919	0.281960	+	0.959426i	$0.409016\pi$
$938$	0	0
$939$	0	0
$940$	−25.5162	−0.832246
$941$	−37.6309	−1.22673	−0.613366	−	0.789799i	$-0.710185\pi$
$941$	−37.6309	−1.22673	−0.613366	+	0.789799i	$0.710185\pi$
$942$	0	0
$943$	7.15243	0.232915
$944$	−9.52991	−0.310172
$945$	0	0
$946$	−4.03677	−0.131247
$947$	7.73191	0.251253	0.125627	−	0.992078i	$-0.459906\pi$
$947$	7.73191	0.251253	0.125627	+	0.992078i	$0.459906\pi$
$948$	0	0
$949$	−47.1334	−1.53001
$950$	−20.4426	−0.663247
$951$	0	0
$952$	0	0
$953$	20.6770	0.669794	0.334897	−	0.942255i	$-0.391298\pi$
$953$	20.6770	0.669794	0.334897	+	0.942255i	$0.391298\pi$
$954$	0	0
$955$	−52.1471	−1.68744
$956$	−7.66769	−0.247991
$957$	0	0
$958$	−14.4794	−0.467808
$959$	0	0
$960$	0	0
$961$	−5.11964	−0.165150
$962$	−22.1745	−0.714936
$963$	0	0
$964$	25.0598	0.807122
$965$	−59.2344	−1.90682
$966$	0	0
$967$	15.7235	0.505634	0.252817	−	0.967514i	$-0.418643\pi$
$967$	15.7235	0.505634	0.252817	+	0.967514i	$0.418643\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−0.239705	−0.00769646
$971$	8.24810	0.264694	0.132347	−	0.991203i	$-0.457749\pi$
$971$	8.24810	0.264694	0.132347	+	0.991203i	$0.457749\pi$
$972$	0	0
$973$	0	0
$974$	8.43332	0.270221
$975$	0	0
$976$	1.65929	0.0531127
$977$	35.3354	1.13048	0.565239	−	0.824927i	$-0.308784\pi$
$977$	35.3354	1.13048	0.565239	+	0.824927i	$0.308784\pi$
$978$	0	0
$979$	6.50686	0.207960
$980$	0	0
$981$	0	0
$982$	−38.6770	−1.23423
$983$	−9.14311	−0.291620	−0.145810	−	0.989313i	$-0.546579\pi$
$983$	−9.14311	−0.291620	−0.145810	+	0.989313i	$0.546579\pi$
$984$	0	0
$985$	8.58041	0.273395
$986$	4.54364	0.144699
$987$	0	0
$988$	−44.1471	−1.40451
$989$	−5.20827	−0.165613
$990$	0	0
$991$	35.4657	1.12660	0.563302	−	0.826251i	$-0.309531\pi$
$991$	35.4657	1.12660	0.563302	+	0.826251i	$0.309531\pi$
$992$	5.08727	0.161521
$993$	0	0
$994$	0	0
$995$	71.6128	2.27028
$996$	0	0
$997$	−45.4931	−1.44078	−0.720391	−	0.693568i	$-0.756038\pi$
$997$	−45.4931	−1.44078	−0.720391	+	0.693568i	$0.756038\pi$
$998$	−38.0461	−1.20433
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.du.1.3		3
3.2	odd	2		3234.2.a.bg.1.1		3
7.3	odd	6		1386.2.k.w.793.3		6
7.5	odd	6		1386.2.k.w.991.3		6
7.6	odd	2		9702.2.a.dt.1.1		3
21.5	even	6		462.2.i.f.67.1	✓	6
21.17	even	6		462.2.i.f.331.1	yes	6
21.20	even	2		3234.2.a.bi.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.i.f.67.1	✓	6	21.5	even	6
462.2.i.f.331.1	yes	6	21.17	even	6
1386.2.k.w.793.3		6	7.3	odd	6
1386.2.k.w.991.3		6	7.5	odd	6
3234.2.a.bg.1.1		3	3.2	odd	2
3234.2.a.bi.1.3		3	21.20	even	2
9702.2.a.dt.1.1		3	7.6	odd	2
9702.2.a.du.1.3		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9702.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.74657 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.74657 q^{5} -1.00000 q^{8} -2.74657 q^{10} -1.00000 q^{11} -5.49314 q^{13} +1.00000 q^{16} +1.00000 q^{17} +8.03677 q^{19} +2.74657 q^{20} +1.00000 q^{22} +1.29021 q^{23} +2.54364 q^{25} +5.49314 q^{26} -4.54364 q^{29} -5.08727 q^{31} -1.00000 q^{32} -1.00000 q^{34} -4.03677 q^{37} -8.03677 q^{38} -2.74657 q^{40} +5.54364 q^{41} -4.03677 q^{43} -1.00000 q^{44} -1.29021 q^{46} -9.29021 q^{47} -2.54364 q^{50} -5.49314 q^{52} -5.49314 q^{53} -2.74657 q^{55} +4.54364 q^{58} -9.52991 q^{59} +1.65929 q^{61} +5.08727 q^{62} +1.00000 q^{64} -15.0873 q^{65} +3.54364 q^{67} +1.00000 q^{68} -2.54364 q^{71} +8.58041 q^{73} +4.03677 q^{74} +8.03677 q^{76} +12.2397 q^{79} +2.74657 q^{80} -5.54364 q^{82} -14.5299 q^{83} +2.74657 q^{85} +4.03677 q^{86} +1.00000 q^{88} -6.50686 q^{89} +1.29021 q^{92} +9.29021 q^{94} +22.0735 q^{95} +0.0872743 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} - 3 q^{11} + 3 q^{16} + 3 q^{17} + 3 q^{19} + 3 q^{22} - 9 q^{23} + 3 q^{25} - 9 q^{29} - 6 q^{31} - 3 q^{32} - 3 q^{34} + 9 q^{37} - 3 q^{38} + 12 q^{41} + 9 q^{43} - 3 q^{44} + 9 q^{46} - 15 q^{47} - 3 q^{50} + 9 q^{58} + 9 q^{59} + 6 q^{61} + 6 q^{62} + 3 q^{64} - 36 q^{65} + 6 q^{67} + 3 q^{68} - 3 q^{71} - 9 q^{74} + 3 q^{76} + 12 q^{79} - 12 q^{82} - 6 q^{83} - 9 q^{86} + 3 q^{88} - 36 q^{89} - 9 q^{92} + 15 q^{94} + 24 q^{95} - 9 q^{97}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8 - 3 * q^11 + 3 * q^16 + 3 * q^17 + 3 * q^19 + 3 * q^22 - 9 * q^23 + 3 * q^25 - 9 * q^29 - 6 * q^31 - 3 * q^32 - 3 * q^34 + 9 * q^37 - 3 * q^38 + 12 * q^41 + 9 * q^43 - 3 * q^44 + 9 * q^46 - 15 * q^47 - 3 * q^50 + 9 * q^58 + 9 * q^59 + 6 * q^61 + 6 * q^62 + 3 * q^64 - 36 * q^65 + 6 * q^67 + 3 * q^68 - 3 * q^71 - 9 * q^74 + 3 * q^76 + 12 * q^79 - 12 * q^82 - 6 * q^83 - 9 * q^86 + 3 * q^88 - 36 * q^89 - 9 * q^92 + 15 * q^94 + 24 * q^95 - 9 * q^97

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