LMFDB - Embedded newform 9702.2.a.cj.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1386)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +0.414214 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +0.414214 q^{5} -1.00000 q^{8} -0.414214 q^{10} -1.00000 q^{11} -1.17157 q^{13} +1.00000 q^{16} +2.17157 q^{17} +0.828427 q^{19} +0.414214 q^{20} +1.00000 q^{22} +3.24264 q^{23} -4.82843 q^{25} +1.17157 q^{26} -2.82843 q^{29} -6.48528 q^{31} -1.00000 q^{32} -2.17157 q^{34} +9.65685 q^{37} -0.828427 q^{38} -0.414214 q^{40} -4.65685 q^{41} -2.82843 q^{43} -1.00000 q^{44} -3.24264 q^{46} +9.24264 q^{47} +4.82843 q^{50} -1.17157 q^{52} +5.17157 q^{53} -0.414214 q^{55} +2.82843 q^{58} +3.65685 q^{59} -1.58579 q^{61} +6.48528 q^{62} +1.00000 q^{64} -0.485281 q^{65} -13.4853 q^{67} +2.17157 q^{68} -13.3137 q^{71} -4.82843 q^{73} -9.65685 q^{74} +0.828427 q^{76} +4.75736 q^{79} +0.414214 q^{80} +4.65685 q^{82} +9.82843 q^{83} +0.899495 q^{85} +2.82843 q^{86} +1.00000 q^{88} -12.4853 q^{89} +3.24264 q^{92} -9.24264 q^{94} +0.343146 q^{95} -10.1716 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 2 q^{11} - 8 q^{13} + 2 q^{16} + 10 q^{17} - 4 q^{19} - 2 q^{20} + 2 q^{22} - 2 q^{23} - 4 q^{25} + 8 q^{26} + 4 q^{31} - 2 q^{32} - 10 q^{34} + 8 q^{37} + 4 q^{38} + 2 q^{40} + 2 q^{41} - 2 q^{44} + 2 q^{46} + 10 q^{47} + 4 q^{50} - 8 q^{52} + 16 q^{53} + 2 q^{55} - 4 q^{59} - 6 q^{61} - 4 q^{62} + 2 q^{64} + 16 q^{65} - 10 q^{67} + 10 q^{68} - 4 q^{71} - 4 q^{73} - 8 q^{74} - 4 q^{76} + 18 q^{79} - 2 q^{80} - 2 q^{82} + 14 q^{83} - 18 q^{85} + 2 q^{88} - 8 q^{89} - 2 q^{92} - 10 q^{94} + 12 q^{95} - 26 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 2 * q^10 - 2 * q^11 - 8 * q^13 + 2 * q^16 + 10 * q^17 - 4 * q^19 - 2 * q^20 + 2 * q^22 - 2 * q^23 - 4 * q^25 + 8 * q^26 + 4 * q^31 - 2 * q^32 - 10 * q^34 + 8 * q^37 + 4 * q^38 + 2 * q^40 + 2 * q^41 - 2 * q^44 + 2 * q^46 + 10 * q^47 + 4 * q^50 - 8 * q^52 + 16 * q^53 + 2 * q^55 - 4 * q^59 - 6 * q^61 - 4 * q^62 + 2 * q^64 + 16 * q^65 - 10 * q^67 + 10 * q^68 - 4 * q^71 - 4 * q^73 - 8 * q^74 - 4 * q^76 + 18 * q^79 - 2 * q^80 - 2 * q^82 + 14 * q^83 - 18 * q^85 + 2 * q^88 - 8 * q^89 - 2 * q^92 - 10 * q^94 + 12 * q^95 - 26 * 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
$5$	0.414214	0.185242	0.0926210	−	0.995701i	$-0.470476\pi$
$5$	0.414214	0.185242	0.0926210	+	0.995701i	$0.470476\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.414214	−0.130986
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.17157	−0.324936	−0.162468	−	0.986714i	$-0.551945\pi$
$13$	−1.17157	−0.324936	−0.162468	+	0.986714i	$0.551945\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	2.17157	0.526684	0.263342	−	0.964703i	$-0.415175\pi$
$17$	2.17157	0.526684	0.263342	+	0.964703i	$0.415175\pi$
$18$	0	0
$19$	0.828427	0.190054	0.0950271	−	0.995475i	$-0.469706\pi$
$19$	0.828427	0.190054	0.0950271	+	0.995475i	$0.469706\pi$
$20$	0.414214	0.0926210
$21$	0	0
$22$	1.00000	0.213201
$23$	3.24264	0.676137	0.338069	−	0.941121i	$-0.390226\pi$
$23$	3.24264	0.676137	0.338069	+	0.941121i	$0.390226\pi$
$24$	0	0
$25$	−4.82843	−0.965685
$26$	1.17157	0.229764
$27$	0	0
$28$	0	0
$29$	−2.82843	−0.525226	−0.262613	−	0.964901i	$-0.584584\pi$
$29$	−2.82843	−0.525226	−0.262613	+	0.964901i	$0.584584\pi$
$30$	0	0
$31$	−6.48528	−1.16479	−0.582395	−	0.812906i	$-0.697884\pi$
$31$	−6.48528	−1.16479	−0.582395	+	0.812906i	$0.697884\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.17157	−0.372422
$35$	0	0
$36$	0	0
$37$	9.65685	1.58758	0.793789	−	0.608194i	$-0.208106\pi$
$37$	9.65685	1.58758	0.793789	+	0.608194i	$0.208106\pi$
$38$	−0.828427	−0.134389
$39$	0	0
$40$	−0.414214	−0.0654929
$41$	−4.65685	−0.727278	−0.363639	−	0.931540i	$-0.618466\pi$
$41$	−4.65685	−0.727278	−0.363639	+	0.931540i	$0.618466\pi$
$42$	0	0
$43$	−2.82843	−0.431331	−0.215666	−	0.976467i	$-0.569192\pi$
$43$	−2.82843	−0.431331	−0.215666	+	0.976467i	$0.569192\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−3.24264	−0.478101
$47$	9.24264	1.34818	0.674089	−	0.738650i	$-0.264536\pi$
$47$	9.24264	1.34818	0.674089	+	0.738650i	$0.264536\pi$
$48$	0	0
$49$	0	0
$50$	4.82843	0.682843
$51$	0	0
$52$	−1.17157	−0.162468
$53$	5.17157	0.710370	0.355185	−	0.934796i	$-0.384418\pi$
$53$	5.17157	0.710370	0.355185	+	0.934796i	$0.384418\pi$
$54$	0	0
$55$	−0.414214	−0.0558525
$56$	0	0
$57$	0	0
$58$	2.82843	0.371391
$59$	3.65685	0.476082	0.238041	−	0.971255i	$-0.423495\pi$
$59$	3.65685	0.476082	0.238041	+	0.971255i	$0.423495\pi$
$60$	0	0
$61$	−1.58579	−0.203039	−0.101520	−	0.994834i	$-0.532370\pi$
$61$	−1.58579	−0.203039	−0.101520	+	0.994834i	$0.532370\pi$
$62$	6.48528	0.823632
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.485281	−0.0601917
$66$	0	0
$67$	−13.4853	−1.64749	−0.823745	−	0.566961i	$-0.808119\pi$
$67$	−13.4853	−1.64749	−0.823745	+	0.566961i	$0.808119\pi$
$68$	2.17157	0.263342
$69$	0	0
$70$	0	0
$71$	−13.3137	−1.58005	−0.790023	−	0.613077i	$-0.789932\pi$
$71$	−13.3137	−1.58005	−0.790023	+	0.613077i	$0.789932\pi$
$72$	0	0
$73$	−4.82843	−0.565125	−0.282562	−	0.959249i	$-0.591184\pi$
$73$	−4.82843	−0.565125	−0.282562	+	0.959249i	$0.591184\pi$
$74$	−9.65685	−1.12259
$75$	0	0
$76$	0.828427	0.0950271
$77$	0	0
$78$	0	0
$79$	4.75736	0.535245	0.267622	−	0.963524i	$-0.413762\pi$
$79$	4.75736	0.535245	0.267622	+	0.963524i	$0.413762\pi$
$80$	0.414214	0.0463105
$81$	0	0
$82$	4.65685	0.514264
$83$	9.82843	1.07881	0.539405	−	0.842046i	$-0.318649\pi$
$83$	9.82843	1.07881	0.539405	+	0.842046i	$0.318649\pi$
$84$	0	0
$85$	0.899495	0.0975639
$86$	2.82843	0.304997
$87$	0	0
$88$	1.00000	0.106600
$89$	−12.4853	−1.32344	−0.661719	−	0.749752i	$-0.730173\pi$
$89$	−12.4853	−1.32344	−0.661719	+	0.749752i	$0.730173\pi$
$90$	0	0
$91$	0	0
$92$	3.24264	0.338069
$93$	0	0
$94$	−9.24264	−0.953306
$95$	0.343146	0.0352060
$96$	0	0
$97$	−10.1716	−1.03277	−0.516383	−	0.856358i	$-0.672722\pi$
$97$	−10.1716	−1.03277	−0.516383	+	0.856358i	$0.672722\pi$
$98$	0	0
$99$	0	0
$100$	−4.82843	−0.482843
$101$	14.1421	1.40720	0.703598	−	0.710599i	$-0.251576\pi$
$101$	14.1421	1.40720	0.703598	+	0.710599i	$0.251576\pi$
$102$	0	0
$103$	−9.17157	−0.903702	−0.451851	−	0.892093i	$-0.649236\pi$
$103$	−9.17157	−0.903702	−0.451851	+	0.892093i	$0.649236\pi$
$104$	1.17157	0.114882
$105$	0	0
$106$	−5.17157	−0.502308
$107$	16.6569	1.61028	0.805139	−	0.593086i	$-0.202090\pi$
$107$	16.6569	1.61028	0.805139	+	0.593086i	$0.202090\pi$
$108$	0	0
$109$	3.24264	0.310589	0.155294	−	0.987868i	$-0.450367\pi$
$109$	3.24264	0.310589	0.155294	+	0.987868i	$0.450367\pi$
$110$	0.414214	0.0394937
$111$	0	0
$112$	0	0
$113$	−3.65685	−0.344008	−0.172004	−	0.985096i	$-0.555024\pi$
$113$	−3.65685	−0.344008	−0.172004	+	0.985096i	$0.555024\pi$
$114$	0	0
$115$	1.34315	0.125249
$116$	−2.82843	−0.262613
$117$	0	0
$118$	−3.65685	−0.336641
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.58579	0.143570
$123$	0	0
$124$	−6.48528	−0.582395
$125$	−4.07107	−0.364127
$126$	0	0
$127$	1.24264	0.110267	0.0551333	−	0.998479i	$-0.482442\pi$
$127$	1.24264	0.110267	0.0551333	+	0.998479i	$0.482442\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0.485281	0.0425620
$131$	−15.3137	−1.33796	−0.668982	−	0.743278i	$-0.733270\pi$
$131$	−15.3137	−1.33796	−0.668982	+	0.743278i	$0.733270\pi$
$132$	0	0
$133$	0	0
$134$	13.4853	1.16495
$135$	0	0
$136$	−2.17157	−0.186211
$137$	4.82843	0.412520	0.206260	−	0.978497i	$-0.433871\pi$
$137$	4.82843	0.412520	0.206260	+	0.978497i	$0.433871\pi$
$138$	0	0
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	0	0
$141$	0	0
$142$	13.3137	1.11726
$143$	1.17157	0.0979718
$144$	0	0
$145$	−1.17157	−0.0972938
$146$	4.82843	0.399603
$147$	0	0
$148$	9.65685	0.793789
$149$	−11.3137	−0.926855	−0.463428	−	0.886135i	$-0.653381\pi$
$149$	−11.3137	−0.926855	−0.463428	+	0.886135i	$0.653381\pi$
$150$	0	0
$151$	1.24264	0.101125	0.0505623	−	0.998721i	$-0.483899\pi$
$151$	1.24264	0.101125	0.0505623	+	0.998721i	$0.483899\pi$
$152$	−0.828427	−0.0671943
$153$	0	0
$154$	0	0
$155$	−2.68629	−0.215768
$156$	0	0
$157$	−9.17157	−0.731971	−0.365986	−	0.930621i	$-0.619268\pi$
$157$	−9.17157	−0.731971	−0.365986	+	0.930621i	$0.619268\pi$
$158$	−4.75736	−0.378475
$159$	0	0
$160$	−0.414214	−0.0327465
$161$	0	0
$162$	0	0
$163$	3.00000	0.234978	0.117489	−	0.993074i	$-0.462515\pi$
$163$	3.00000	0.234978	0.117489	+	0.993074i	$0.462515\pi$
$164$	−4.65685	−0.363639
$165$	0	0
$166$	−9.82843	−0.762834
$167$	16.4853	1.27567	0.637835	−	0.770173i	$-0.279830\pi$
$167$	16.4853	1.27567	0.637835	+	0.770173i	$0.279830\pi$
$168$	0	0
$169$	−11.6274	−0.894417
$170$	−0.899495	−0.0689881
$171$	0	0
$172$	−2.82843	−0.215666
$173$	12.1421	0.923149	0.461575	−	0.887101i	$-0.347285\pi$
$173$	12.1421	0.923149	0.461575	+	0.887101i	$0.347285\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	12.4853	0.935811
$179$	3.17157	0.237054	0.118527	−	0.992951i	$-0.462183\pi$
$179$	3.17157	0.237054	0.118527	+	0.992951i	$0.462183\pi$
$180$	0	0
$181$	−2.34315	−0.174165	−0.0870823	−	0.996201i	$-0.527754\pi$
$181$	−2.34315	−0.174165	−0.0870823	+	0.996201i	$0.527754\pi$
$182$	0	0
$183$	0	0
$184$	−3.24264	−0.239051
$185$	4.00000	0.294086
$186$	0	0
$187$	−2.17157	−0.158801
$188$	9.24264	0.674089
$189$	0	0
$190$	−0.343146	−0.0248944
$191$	9.31371	0.673916	0.336958	−	0.941520i	$-0.390602\pi$
$191$	9.31371	0.673916	0.336958	+	0.941520i	$0.390602\pi$
$192$	0	0
$193$	−9.17157	−0.660184	−0.330092	−	0.943949i	$-0.607080\pi$
$193$	−9.17157	−0.660184	−0.330092	+	0.943949i	$0.607080\pi$
$194$	10.1716	0.730276
$195$	0	0
$196$	0	0
$197$	3.51472	0.250413	0.125207	−	0.992131i	$-0.460041\pi$
$197$	3.51472	0.250413	0.125207	+	0.992131i	$0.460041\pi$
$198$	0	0
$199$	−8.34315	−0.591430	−0.295715	−	0.955276i	$-0.595558\pi$
$199$	−8.34315	−0.591430	−0.295715	+	0.955276i	$0.595558\pi$
$200$	4.82843	0.341421
$201$	0	0
$202$	−14.1421	−0.995037
$203$	0	0
$204$	0	0
$205$	−1.92893	−0.134722
$206$	9.17157	0.639014
$207$	0	0
$208$	−1.17157	−0.0812340
$209$	−0.828427	−0.0573035
$210$	0	0
$211$	14.8284	1.02083	0.510416	−	0.859928i	$-0.329492\pi$
$211$	14.8284	1.02083	0.510416	+	0.859928i	$0.329492\pi$
$212$	5.17157	0.355185
$213$	0	0
$214$	−16.6569	−1.13864
$215$	−1.17157	−0.0799006
$216$	0	0
$217$	0	0
$218$	−3.24264	−0.219619
$219$	0	0
$220$	−0.414214	−0.0279263
$221$	−2.54416	−0.171138
$222$	0	0
$223$	−13.3137	−0.891552	−0.445776	−	0.895145i	$-0.647072\pi$
$223$	−13.3137	−0.891552	−0.445776	+	0.895145i	$0.647072\pi$
$224$	0	0
$225$	0	0
$226$	3.65685	0.243250
$227$	−12.3137	−0.817290	−0.408645	−	0.912694i	$-0.633998\pi$
$227$	−12.3137	−0.817290	−0.408645	+	0.912694i	$0.633998\pi$
$228$	0	0
$229$	−11.3137	−0.747631	−0.373815	−	0.927503i	$-0.621951\pi$
$229$	−11.3137	−0.747631	−0.373815	+	0.927503i	$0.621951\pi$
$230$	−1.34315	−0.0885644
$231$	0	0
$232$	2.82843	0.185695
$233$	−8.31371	−0.544649	−0.272325	−	0.962205i	$-0.587792\pi$
$233$	−8.31371	−0.544649	−0.272325	+	0.962205i	$0.587792\pi$
$234$	0	0
$235$	3.82843	0.249739
$236$	3.65685	0.238041
$237$	0	0
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	24.9706	1.60850	0.804248	−	0.594294i	$-0.202569\pi$
$241$	24.9706	1.60850	0.804248	+	0.594294i	$0.202569\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−1.58579	−0.101520
$245$	0	0
$246$	0	0
$247$	−0.970563	−0.0617554
$248$	6.48528	0.411816
$249$	0	0
$250$	4.07107	0.257477
$251$	−26.1421	−1.65008	−0.825038	−	0.565077i	$-0.808847\pi$
$251$	−26.1421	−1.65008	−0.825038	+	0.565077i	$0.808847\pi$
$252$	0	0
$253$	−3.24264	−0.203863
$254$	−1.24264	−0.0779702
$255$	0	0
$256$	1.00000	0.0625000
$257$	−25.1716	−1.57016	−0.785080	−	0.619395i	$-0.787378\pi$
$257$	−25.1716	−1.57016	−0.785080	+	0.619395i	$0.787378\pi$
$258$	0	0
$259$	0	0
$260$	−0.485281	−0.0300959
$261$	0	0
$262$	15.3137	0.946084
$263$	−9.31371	−0.574308	−0.287154	−	0.957884i	$-0.592709\pi$
$263$	−9.31371	−0.574308	−0.287154	+	0.957884i	$0.592709\pi$
$264$	0	0
$265$	2.14214	0.131590
$266$	0	0
$267$	0	0
$268$	−13.4853	−0.823745
$269$	8.07107	0.492102	0.246051	−	0.969257i	$-0.420867\pi$
$269$	8.07107	0.492102	0.246051	+	0.969257i	$0.420867\pi$
$270$	0	0
$271$	−13.3137	−0.808750	−0.404375	−	0.914593i	$-0.632511\pi$
$271$	−13.3137	−0.808750	−0.404375	+	0.914593i	$0.632511\pi$
$272$	2.17157	0.131671
$273$	0	0
$274$	−4.82843	−0.291696
$275$	4.82843	0.291165
$276$	0	0
$277$	−6.82843	−0.410280	−0.205140	−	0.978733i	$-0.565765\pi$
$277$	−6.82843	−0.410280	−0.205140	+	0.978733i	$0.565765\pi$
$278$	6.00000	0.359856
$279$	0	0
$280$	0	0
$281$	−2.51472	−0.150016	−0.0750078	−	0.997183i	$-0.523898\pi$
$281$	−2.51472	−0.150016	−0.0750078	+	0.997183i	$0.523898\pi$
$282$	0	0
$283$	−1.85786	−0.110439	−0.0552193	−	0.998474i	$-0.517586\pi$
$283$	−1.85786	−0.110439	−0.0552193	+	0.998474i	$0.517586\pi$
$284$	−13.3137	−0.790023
$285$	0	0
$286$	−1.17157	−0.0692766
$287$	0	0
$288$	0	0
$289$	−12.2843	−0.722604
$290$	1.17157	0.0687971
$291$	0	0
$292$	−4.82843	−0.282562
$293$	−12.8284	−0.749445	−0.374722	−	0.927137i	$-0.622262\pi$
$293$	−12.8284	−0.749445	−0.374722	+	0.927137i	$0.622262\pi$
$294$	0	0
$295$	1.51472	0.0881903
$296$	−9.65685	−0.561293
$297$	0	0
$298$	11.3137	0.655386
$299$	−3.79899	−0.219701
$300$	0	0
$301$	0	0
$302$	−1.24264	−0.0715059
$303$	0	0
$304$	0.828427	0.0475136
$305$	−0.656854	−0.0376114
$306$	0	0
$307$	−34.6274	−1.97629	−0.988146	−	0.153520i	$-0.950939\pi$
$307$	−34.6274	−1.97629	−0.988146	+	0.153520i	$0.950939\pi$
$308$	0	0
$309$	0	0
$310$	2.68629	0.152571
$311$	−10.2132	−0.579138	−0.289569	−	0.957157i	$-0.593512\pi$
$311$	−10.2132	−0.579138	−0.289569	+	0.957157i	$0.593512\pi$
$312$	0	0
$313$	−29.3137	−1.65691	−0.828454	−	0.560057i	$-0.810779\pi$
$313$	−29.3137	−1.65691	−0.828454	+	0.560057i	$0.810779\pi$
$314$	9.17157	0.517582
$315$	0	0
$316$	4.75736	0.267622
$317$	33.8701	1.90233	0.951166	−	0.308680i	$-0.0998872\pi$
$317$	33.8701	1.90233	0.951166	+	0.308680i	$0.0998872\pi$
$318$	0	0
$319$	2.82843	0.158362
$320$	0.414214	0.0231552
$321$	0	0
$322$	0	0
$323$	1.79899	0.100098
$324$	0	0
$325$	5.65685	0.313786
$326$	−3.00000	−0.166155
$327$	0	0
$328$	4.65685	0.257132
$329$	0	0
$330$	0	0
$331$	6.31371	0.347033	0.173516	−	0.984831i	$-0.444487\pi$
$331$	6.31371	0.347033	0.173516	+	0.984831i	$0.444487\pi$
$332$	9.82843	0.539405
$333$	0	0
$334$	−16.4853	−0.902034
$335$	−5.58579	−0.305184
$336$	0	0
$337$	9.51472	0.518300	0.259150	−	0.965837i	$-0.416558\pi$
$337$	9.51472	0.518300	0.259150	+	0.965837i	$0.416558\pi$
$338$	11.6274	0.632448
$339$	0	0
$340$	0.899495	0.0487820
$341$	6.48528	0.351198
$342$	0	0
$343$	0	0
$344$	2.82843	0.152499
$345$	0	0
$346$	−12.1421	−0.652765
$347$	−4.85786	−0.260784	−0.130392	−	0.991463i	$-0.541624\pi$
$347$	−4.85786	−0.260784	−0.130392	+	0.991463i	$0.541624\pi$
$348$	0	0
$349$	−15.7279	−0.841896	−0.420948	−	0.907085i	$-0.638303\pi$
$349$	−15.7279	−0.841896	−0.420948	+	0.907085i	$0.638303\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	16.8284	0.895687	0.447843	−	0.894112i	$-0.352192\pi$
$353$	16.8284	0.895687	0.447843	+	0.894112i	$0.352192\pi$
$354$	0	0
$355$	−5.51472	−0.292691
$356$	−12.4853	−0.661719
$357$	0	0
$358$	−3.17157	−0.167623
$359$	8.48528	0.447836	0.223918	−	0.974608i	$-0.428115\pi$
$359$	8.48528	0.447836	0.223918	+	0.974608i	$0.428115\pi$
$360$	0	0
$361$	−18.3137	−0.963879
$362$	2.34315	0.123153
$363$	0	0
$364$	0	0
$365$	−2.00000	−0.104685
$366$	0	0
$367$	5.51472	0.287866	0.143933	−	0.989587i	$-0.454025\pi$
$367$	5.51472	0.287866	0.143933	+	0.989587i	$0.454025\pi$
$368$	3.24264	0.169034
$369$	0	0
$370$	−4.00000	−0.207950
$371$	0	0
$372$	0	0
$373$	−15.7279	−0.814361	−0.407180	−	0.913348i	$-0.633488\pi$
$373$	−15.7279	−0.814361	−0.407180	+	0.913348i	$0.633488\pi$
$374$	2.17157	0.112289
$375$	0	0
$376$	−9.24264	−0.476653
$377$	3.31371	0.170665
$378$	0	0
$379$	19.3431	0.993591	0.496795	−	0.867868i	$-0.334510\pi$
$379$	19.3431	0.993591	0.496795	+	0.867868i	$0.334510\pi$
$380$	0.343146	0.0176030
$381$	0	0
$382$	−9.31371	−0.476531
$383$	34.2843	1.75184	0.875922	−	0.482452i	$-0.160254\pi$
$383$	34.2843	1.75184	0.875922	+	0.482452i	$0.160254\pi$
$384$	0	0
$385$	0	0
$386$	9.17157	0.466821
$387$	0	0
$388$	−10.1716	−0.516383
$389$	1.72792	0.0876091	0.0438046	−	0.999040i	$-0.486052\pi$
$389$	1.72792	0.0876091	0.0438046	+	0.999040i	$0.486052\pi$
$390$	0	0
$391$	7.04163	0.356111
$392$	0	0
$393$	0	0
$394$	−3.51472	−0.177069
$395$	1.97056	0.0991498
$396$	0	0
$397$	12.4853	0.626618	0.313309	−	0.949651i	$-0.398562\pi$
$397$	12.4853	0.626618	0.313309	+	0.949651i	$0.398562\pi$
$398$	8.34315	0.418204
$399$	0	0
$400$	−4.82843	−0.241421
$401$	15.7990	0.788964	0.394482	−	0.918904i	$-0.370924\pi$
$401$	15.7990	0.788964	0.394482	+	0.918904i	$0.370924\pi$
$402$	0	0
$403$	7.59798	0.378482
$404$	14.1421	0.703598
$405$	0	0
$406$	0	0
$407$	−9.65685	−0.478672
$408$	0	0
$409$	−11.4558	−0.566455	−0.283228	−	0.959053i	$-0.591405\pi$
$409$	−11.4558	−0.566455	−0.283228	+	0.959053i	$0.591405\pi$
$410$	1.92893	0.0952632
$411$	0	0
$412$	−9.17157	−0.451851
$413$	0	0
$414$	0	0
$415$	4.07107	0.199841
$416$	1.17157	0.0574411
$417$	0	0
$418$	0.828427	0.0405197
$419$	25.7990	1.26036	0.630182	−	0.776448i	$-0.282980\pi$
$419$	25.7990	1.26036	0.630182	+	0.776448i	$0.282980\pi$
$420$	0	0
$421$	14.8284	0.722693	0.361347	−	0.932432i	$-0.382317\pi$
$421$	14.8284	0.722693	0.361347	+	0.932432i	$0.382317\pi$
$422$	−14.8284	−0.721837
$423$	0	0
$424$	−5.17157	−0.251154
$425$	−10.4853	−0.508611
$426$	0	0
$427$	0	0
$428$	16.6569	0.805139
$429$	0	0
$430$	1.17157	0.0564983
$431$	19.1716	0.923462	0.461731	−	0.887020i	$-0.347229\pi$
$431$	19.1716	0.923462	0.461731	+	0.887020i	$0.347229\pi$
$432$	0	0
$433$	−26.6569	−1.28105	−0.640523	−	0.767939i	$-0.721283\pi$
$433$	−26.6569	−1.28105	−0.640523	+	0.767939i	$0.721283\pi$
$434$	0	0
$435$	0	0
$436$	3.24264	0.155294
$437$	2.68629	0.128503
$438$	0	0
$439$	3.38478	0.161547	0.0807733	−	0.996733i	$-0.474261\pi$
$439$	3.38478	0.161547	0.0807733	+	0.996733i	$0.474261\pi$
$440$	0.414214	0.0197469
$441$	0	0
$442$	2.54416	0.121013
$443$	−14.3431	−0.681463	−0.340732	−	0.940161i	$-0.610675\pi$
$443$	−14.3431	−0.681463	−0.340732	+	0.940161i	$0.610675\pi$
$444$	0	0
$445$	−5.17157	−0.245156
$446$	13.3137	0.630422
$447$	0	0
$448$	0	0
$449$	−23.1127	−1.09076	−0.545378	−	0.838190i	$-0.683614\pi$
$449$	−23.1127	−1.09076	−0.545378	+	0.838190i	$0.683614\pi$
$450$	0	0
$451$	4.65685	0.219283
$452$	−3.65685	−0.172004
$453$	0	0
$454$	12.3137	0.577911
$455$	0	0
$456$	0	0
$457$	34.6274	1.61980	0.809901	−	0.586566i	$-0.199521\pi$
$457$	34.6274	1.61980	0.809901	+	0.586566i	$0.199521\pi$
$458$	11.3137	0.528655
$459$	0	0
$460$	1.34315	0.0626245
$461$	23.6569	1.10181	0.550905	−	0.834568i	$-0.314283\pi$
$461$	23.6569	1.10181	0.550905	+	0.834568i	$0.314283\pi$
$462$	0	0
$463$	23.1127	1.07414	0.537069	−	0.843538i	$-0.319531\pi$
$463$	23.1127	1.07414	0.537069	+	0.843538i	$0.319531\pi$
$464$	−2.82843	−0.131306
$465$	0	0
$466$	8.31371	0.385125
$467$	2.97056	0.137461	0.0687306	−	0.997635i	$-0.478105\pi$
$467$	2.97056	0.137461	0.0687306	+	0.997635i	$0.478105\pi$
$468$	0	0
$469$	0	0
$470$	−3.82843	−0.176592
$471$	0	0
$472$	−3.65685	−0.168320
$473$	2.82843	0.130051
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	0	0
$478$	20.0000	0.914779
$479$	−5.17157	−0.236295	−0.118148	−	0.992996i	$-0.537696\pi$
$479$	−5.17157	−0.236295	−0.118148	+	0.992996i	$0.537696\pi$
$480$	0	0
$481$	−11.3137	−0.515861
$482$	−24.9706	−1.13738
$483$	0	0
$484$	1.00000	0.0454545
$485$	−4.21320	−0.191312
$486$	0	0
$487$	−21.3137	−0.965816	−0.482908	−	0.875671i	$-0.660420\pi$
$487$	−21.3137	−0.965816	−0.482908	+	0.875671i	$0.660420\pi$
$488$	1.58579	0.0717852
$489$	0	0
$490$	0	0
$491$	15.6274	0.705255	0.352628	−	0.935764i	$-0.385288\pi$
$491$	15.6274	0.705255	0.352628	+	0.935764i	$0.385288\pi$
$492$	0	0
$493$	−6.14214	−0.276628
$494$	0.970563	0.0436677
$495$	0	0
$496$	−6.48528	−0.291198
$497$	0	0
$498$	0	0
$499$	14.3431	0.642087	0.321044	−	0.947064i	$-0.395966\pi$
$499$	14.3431	0.642087	0.321044	+	0.947064i	$0.395966\pi$
$500$	−4.07107	−0.182064
$501$	0	0
$502$	26.1421	1.16678
$503$	−8.97056	−0.399978	−0.199989	−	0.979798i	$-0.564091\pi$
$503$	−8.97056	−0.399978	−0.199989	+	0.979798i	$0.564091\pi$
$504$	0	0
$505$	5.85786	0.260672
$506$	3.24264	0.144153
$507$	0	0
$508$	1.24264	0.0551333
$509$	−25.1716	−1.11571	−0.557855	−	0.829938i	$-0.688376\pi$
$509$	−25.1716	−1.11571	−0.557855	+	0.829938i	$0.688376\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	25.1716	1.11027
$515$	−3.79899	−0.167403
$516$	0	0
$517$	−9.24264	−0.406491
$518$	0	0
$519$	0	0
$520$	0.485281	0.0212810
$521$	−29.3137	−1.28426	−0.642128	−	0.766597i	$-0.721948\pi$
$521$	−29.3137	−1.28426	−0.642128	+	0.766597i	$0.721948\pi$
$522$	0	0
$523$	−31.4558	−1.37547	−0.687734	−	0.725963i	$-0.741394\pi$
$523$	−31.4558	−1.37547	−0.687734	+	0.725963i	$0.741394\pi$
$524$	−15.3137	−0.668982
$525$	0	0
$526$	9.31371	0.406097
$527$	−14.0833	−0.613476
$528$	0	0
$529$	−12.4853	−0.542838
$530$	−2.14214	−0.0930484
$531$	0	0
$532$	0	0
$533$	5.45584	0.236319
$534$	0	0
$535$	6.89949	0.298291
$536$	13.4853	0.582475
$537$	0	0
$538$	−8.07107	−0.347968
$539$	0	0
$540$	0	0
$541$	−5.44365	−0.234041	−0.117020	−	0.993130i	$-0.537334\pi$
$541$	−5.44365	−0.234041	−0.117020	+	0.993130i	$0.537334\pi$
$542$	13.3137	0.571873
$543$	0	0
$544$	−2.17157	−0.0931054
$545$	1.34315	0.0575340
$546$	0	0
$547$	−36.1421	−1.54533	−0.772663	−	0.634816i	$-0.781076\pi$
$547$	−36.1421	−1.54533	−0.772663	+	0.634816i	$0.781076\pi$
$548$	4.82843	0.206260
$549$	0	0
$550$	−4.82843	−0.205885
$551$	−2.34315	−0.0998214
$552$	0	0
$553$	0	0
$554$	6.82843	0.290112
$555$	0	0
$556$	−6.00000	−0.254457
$557$	4.48528	0.190047	0.0950237	−	0.995475i	$-0.469707\pi$
$557$	4.48528	0.190047	0.0950237	+	0.995475i	$0.469707\pi$
$558$	0	0
$559$	3.31371	0.140155
$560$	0	0
$561$	0	0
$562$	2.51472	0.106077
$563$	−25.6569	−1.08131	−0.540654	−	0.841245i	$-0.681823\pi$
$563$	−25.6569	−1.08131	−0.540654	+	0.841245i	$0.681823\pi$
$564$	0	0
$565$	−1.51472	−0.0637247
$566$	1.85786	0.0780919
$567$	0	0
$568$	13.3137	0.558631
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	−21.6569	−0.906311	−0.453156	−	0.891431i	$-0.649702\pi$
$571$	−21.6569	−0.906311	−0.453156	+	0.891431i	$0.649702\pi$
$572$	1.17157	0.0489859
$573$	0	0
$574$	0	0
$575$	−15.6569	−0.652936
$576$	0	0
$577$	5.14214	0.214070	0.107035	−	0.994255i	$-0.465864\pi$
$577$	5.14214	0.214070	0.107035	+	0.994255i	$0.465864\pi$
$578$	12.2843	0.510958
$579$	0	0
$580$	−1.17157	−0.0486469
$581$	0	0
$582$	0	0
$583$	−5.17157	−0.214185
$584$	4.82843	0.199802
$585$	0	0
$586$	12.8284	0.529937
$587$	28.1421	1.16155	0.580775	−	0.814064i	$-0.302750\pi$
$587$	28.1421	1.16155	0.580775	+	0.814064i	$0.302750\pi$
$588$	0	0
$589$	−5.37258	−0.221373
$590$	−1.51472	−0.0623600
$591$	0	0
$592$	9.65685	0.396894
$593$	−39.9411	−1.64019	−0.820093	−	0.572231i	$-0.806078\pi$
$593$	−39.9411	−1.64019	−0.820093	+	0.572231i	$0.806078\pi$
$594$	0	0
$595$	0	0
$596$	−11.3137	−0.463428
$597$	0	0
$598$	3.79899	0.155352
$599$	−29.0416	−1.18661	−0.593304	−	0.804978i	$-0.702177\pi$
$599$	−29.0416	−1.18661	−0.593304	+	0.804978i	$0.702177\pi$
$600$	0	0
$601$	−20.0000	−0.815817	−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000	−0.815817	−0.407909	+	0.913023i	$0.633742\pi$
$602$	0	0
$603$	0	0
$604$	1.24264	0.0505623
$605$	0.414214	0.0168402
$606$	0	0
$607$	46.2132	1.87574	0.937868	−	0.346992i	$-0.112797\pi$
$607$	46.2132	1.87574	0.937868	+	0.346992i	$0.112797\pi$
$608$	−0.828427	−0.0335972
$609$	0	0
$610$	0.656854	0.0265953
$611$	−10.8284	−0.438071
$612$	0	0
$613$	−30.8995	−1.24802	−0.624009	−	0.781417i	$-0.714497\pi$
$613$	−30.8995	−1.24802	−0.624009	+	0.781417i	$0.714497\pi$
$614$	34.6274	1.39745
$615$	0	0
$616$	0	0
$617$	24.4853	0.985740	0.492870	−	0.870103i	$-0.335948\pi$
$617$	24.4853	0.985740	0.492870	+	0.870103i	$0.335948\pi$
$618$	0	0
$619$	0.514719	0.0206883	0.0103441	−	0.999946i	$-0.496707\pi$
$619$	0.514719	0.0206883	0.0103441	+	0.999946i	$0.496707\pi$
$620$	−2.68629	−0.107884
$621$	0	0
$622$	10.2132	0.409512
$623$	0	0
$624$	0	0
$625$	22.4558	0.898234
$626$	29.3137	1.17161
$627$	0	0
$628$	−9.17157	−0.365986
$629$	20.9706	0.836151
$630$	0	0
$631$	−1.02944	−0.0409812	−0.0204906	−	0.999790i	$-0.506523\pi$
$631$	−1.02944	−0.0409812	−0.0204906	+	0.999790i	$0.506523\pi$
$632$	−4.75736	−0.189238
$633$	0	0
$634$	−33.8701	−1.34515
$635$	0.514719	0.0204260
$636$	0	0
$637$	0	0
$638$	−2.82843	−0.111979
$639$	0	0
$640$	−0.414214	−0.0163732
$641$	27.3137	1.07883	0.539413	−	0.842041i	$-0.318646\pi$
$641$	27.3137	1.07883	0.539413	+	0.842041i	$0.318646\pi$
$642$	0	0
$643$	17.6569	0.696318	0.348159	−	0.937435i	$-0.386807\pi$
$643$	17.6569	0.696318	0.348159	+	0.937435i	$0.386807\pi$
$644$	0	0
$645$	0	0
$646$	−1.79899	−0.0707803
$647$	−42.6985	−1.67865	−0.839325	−	0.543629i	$-0.817050\pi$
$647$	−42.6985	−1.67865	−0.839325	+	0.543629i	$0.817050\pi$
$648$	0	0
$649$	−3.65685	−0.143544
$650$	−5.65685	−0.221880
$651$	0	0
$652$	3.00000	0.117489
$653$	−44.3553	−1.73576	−0.867879	−	0.496775i	$-0.834517\pi$
$653$	−44.3553	−1.73576	−0.867879	+	0.496775i	$0.834517\pi$
$654$	0	0
$655$	−6.34315	−0.247847
$656$	−4.65685	−0.181820
$657$	0	0
$658$	0	0
$659$	36.9411	1.43902	0.719511	−	0.694481i	$-0.244366\pi$
$659$	36.9411	1.43902	0.719511	+	0.694481i	$0.244366\pi$
$660$	0	0
$661$	−38.9706	−1.51578	−0.757890	−	0.652383i	$-0.773769\pi$
$661$	−38.9706	−1.51578	−0.757890	+	0.652383i	$0.773769\pi$
$662$	−6.31371	−0.245389
$663$	0	0
$664$	−9.82843	−0.381417
$665$	0	0
$666$	0	0
$667$	−9.17157	−0.355125
$668$	16.4853	0.637835
$669$	0	0
$670$	5.58579	0.215798
$671$	1.58579	0.0612186
$672$	0	0
$673$	24.6274	0.949317	0.474659	−	0.880170i	$-0.342572\pi$
$673$	24.6274	0.949317	0.474659	+	0.880170i	$0.342572\pi$
$674$	−9.51472	−0.366493
$675$	0	0
$676$	−11.6274	−0.447208
$677$	−42.7696	−1.64377	−0.821884	−	0.569655i	$-0.807077\pi$
$677$	−42.7696	−1.64377	−0.821884	+	0.569655i	$0.807077\pi$
$678$	0	0
$679$	0	0
$680$	−0.899495	−0.0344941
$681$	0	0
$682$	−6.48528	−0.248334
$683$	−17.6569	−0.675621	−0.337810	−	0.941214i	$-0.609686\pi$
$683$	−17.6569	−0.675621	−0.337810	+	0.941214i	$0.609686\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	0	0
$688$	−2.82843	−0.107833
$689$	−6.05887	−0.230825
$690$	0	0
$691$	29.1421	1.10862	0.554310	−	0.832311i	$-0.312982\pi$
$691$	29.1421	1.10862	0.554310	+	0.832311i	$0.312982\pi$
$692$	12.1421	0.461575
$693$	0	0
$694$	4.85786	0.184402
$695$	−2.48528	−0.0942721
$696$	0	0
$697$	−10.1127	−0.383046
$698$	15.7279	0.595311
$699$	0	0
$700$	0	0
$701$	−28.1421	−1.06291	−0.531457	−	0.847085i	$-0.678355\pi$
$701$	−28.1421	−1.06291	−0.531457	+	0.847085i	$0.678355\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−16.8284	−0.633346
$707$	0	0
$708$	0	0
$709$	3.85786	0.144885	0.0724426	−	0.997373i	$-0.476921\pi$
$709$	3.85786	0.144885	0.0724426	+	0.997373i	$0.476921\pi$
$710$	5.51472	0.206964
$711$	0	0
$712$	12.4853	0.467906
$713$	−21.0294	−0.787559
$714$	0	0
$715$	0.485281	0.0181485
$716$	3.17157	0.118527
$717$	0	0
$718$	−8.48528	−0.316668
$719$	16.2132	0.604650	0.302325	−	0.953205i	$-0.402237\pi$
$719$	16.2132	0.604650	0.302325	+	0.953205i	$0.402237\pi$
$720$	0	0
$721$	0	0
$722$	18.3137	0.681566
$723$	0	0
$724$	−2.34315	−0.0870823
$725$	13.6569	0.507203
$726$	0	0
$727$	−32.4853	−1.20481	−0.602406	−	0.798190i	$-0.705791\pi$
$727$	−32.4853	−1.20481	−0.602406	+	0.798190i	$0.705791\pi$
$728$	0	0
$729$	0	0
$730$	2.00000	0.0740233
$731$	−6.14214	−0.227175
$732$	0	0
$733$	−14.7574	−0.545076	−0.272538	−	0.962145i	$-0.587863\pi$
$733$	−14.7574	−0.545076	−0.272538	+	0.962145i	$0.587863\pi$
$734$	−5.51472	−0.203552
$735$	0	0
$736$	−3.24264	−0.119525
$737$	13.4853	0.496737
$738$	0	0
$739$	−34.0000	−1.25071	−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000	−1.25071	−0.625355	+	0.780340i	$0.715046\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	0	0
$743$	13.6569	0.501021	0.250511	−	0.968114i	$-0.419401\pi$
$743$	13.6569	0.501021	0.250511	+	0.968114i	$0.419401\pi$
$744$	0	0
$745$	−4.68629	−0.171692
$746$	15.7279	0.575840
$747$	0	0
$748$	−2.17157	−0.0794006
$749$	0	0
$750$	0	0
$751$	29.7990	1.08738	0.543690	−	0.839286i	$-0.317027\pi$
$751$	29.7990	1.08738	0.543690	+	0.839286i	$0.317027\pi$
$752$	9.24264	0.337044
$753$	0	0
$754$	−3.31371	−0.120678
$755$	0.514719	0.0187325
$756$	0	0
$757$	19.6569	0.714441	0.357220	−	0.934020i	$-0.383725\pi$
$757$	19.6569	0.714441	0.357220	+	0.934020i	$0.383725\pi$
$758$	−19.3431	−0.702575
$759$	0	0
$760$	−0.343146	−0.0124472
$761$	−24.4558	−0.886524	−0.443262	−	0.896392i	$-0.646179\pi$
$761$	−24.4558	−0.886524	−0.443262	+	0.896392i	$0.646179\pi$
$762$	0	0
$763$	0	0
$764$	9.31371	0.336958
$765$	0	0
$766$	−34.2843	−1.23874
$767$	−4.28427	−0.154696
$768$	0	0
$769$	26.4853	0.955084	0.477542	−	0.878609i	$-0.341528\pi$
$769$	26.4853	0.955084	0.477542	+	0.878609i	$0.341528\pi$
$770$	0	0
$771$	0	0
$772$	−9.17157	−0.330092
$773$	−31.2426	−1.12372	−0.561860	−	0.827232i	$-0.689914\pi$
$773$	−31.2426	−1.12372	−0.561860	+	0.827232i	$0.689914\pi$
$774$	0	0
$775$	31.3137	1.12482
$776$	10.1716	0.365138
$777$	0	0
$778$	−1.72792	−0.0619490
$779$	−3.85786	−0.138222
$780$	0	0
$781$	13.3137	0.476402
$782$	−7.04163	−0.251808
$783$	0	0
$784$	0	0
$785$	−3.79899	−0.135592
$786$	0	0
$787$	−38.7696	−1.38199	−0.690993	−	0.722862i	$-0.742826\pi$
$787$	−38.7696	−1.38199	−0.690993	+	0.722862i	$0.742826\pi$
$788$	3.51472	0.125207
$789$	0	0
$790$	−1.97056	−0.0701095
$791$	0	0
$792$	0	0
$793$	1.85786	0.0659747
$794$	−12.4853	−0.443086
$795$	0	0
$796$	−8.34315	−0.295715
$797$	−52.8995	−1.87380	−0.936898	−	0.349602i	$-0.886317\pi$
$797$	−52.8995	−1.87380	−0.936898	+	0.349602i	$0.886317\pi$
$798$	0	0
$799$	20.0711	0.710063
$800$	4.82843	0.170711
$801$	0	0
$802$	−15.7990	−0.557882
$803$	4.82843	0.170391
$804$	0	0
$805$	0	0
$806$	−7.59798	−0.267627
$807$	0	0
$808$	−14.1421	−0.497519
$809$	9.34315	0.328488	0.164244	−	0.986420i	$-0.447482\pi$
$809$	9.34315	0.328488	0.164244	+	0.986420i	$0.447482\pi$
$810$	0	0
$811$	−7.65685	−0.268869	−0.134434	−	0.990923i	$-0.542922\pi$
$811$	−7.65685	−0.268869	−0.134434	+	0.990923i	$0.542922\pi$
$812$	0	0
$813$	0	0
$814$	9.65685	0.338473
$815$	1.24264	0.0435278
$816$	0	0
$817$	−2.34315	−0.0819763
$818$	11.4558	0.400544
$819$	0	0
$820$	−1.92893	−0.0673612
$821$	−21.5147	−0.750869	−0.375434	−	0.926849i	$-0.622506\pi$
$821$	−21.5147	−0.750869	−0.375434	+	0.926849i	$0.622506\pi$
$822$	0	0
$823$	−21.5147	−0.749956	−0.374978	−	0.927034i	$-0.622350\pi$
$823$	−21.5147	−0.749956	−0.374978	+	0.927034i	$0.622350\pi$
$824$	9.17157	0.319507
$825$	0	0
$826$	0	0
$827$	−2.17157	−0.0755130	−0.0377565	−	0.999287i	$-0.512021\pi$
$827$	−2.17157	−0.0755130	−0.0377565	+	0.999287i	$0.512021\pi$
$828$	0	0
$829$	−28.0000	−0.972480	−0.486240	−	0.873825i	$-0.661632\pi$
$829$	−28.0000	−0.972480	−0.486240	+	0.873825i	$0.661632\pi$
$830$	−4.07107	−0.141309
$831$	0	0
$832$	−1.17157	−0.0406170
$833$	0	0
$834$	0	0
$835$	6.82843	0.236307
$836$	−0.828427	−0.0286518
$837$	0	0
$838$	−25.7990	−0.891211
$839$	−8.21320	−0.283551	−0.141776	−	0.989899i	$-0.545281\pi$
$839$	−8.21320	−0.283551	−0.141776	+	0.989899i	$0.545281\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	−14.8284	−0.511021
$843$	0	0
$844$	14.8284	0.510416
$845$	−4.81623	−0.165683
$846$	0	0
$847$	0	0
$848$	5.17157	0.177593
$849$	0	0
$850$	10.4853	0.359642
$851$	31.3137	1.07342
$852$	0	0
$853$	−23.2426	−0.795813	−0.397906	−	0.917426i	$-0.630263\pi$
$853$	−23.2426	−0.795813	−0.397906	+	0.917426i	$0.630263\pi$
$854$	0	0
$855$	0	0
$856$	−16.6569	−0.569320
$857$	−21.6863	−0.740790	−0.370395	−	0.928874i	$-0.620778\pi$
$857$	−21.6863	−0.740790	−0.370395	+	0.928874i	$0.620778\pi$
$858$	0	0
$859$	−50.9411	−1.73809	−0.869044	−	0.494734i	$-0.835265\pi$
$859$	−50.9411	−1.73809	−0.869044	+	0.494734i	$0.835265\pi$
$860$	−1.17157	−0.0399503
$861$	0	0
$862$	−19.1716	−0.652986
$863$	39.1838	1.33383	0.666915	−	0.745133i	$-0.267614\pi$
$863$	39.1838	1.33383	0.666915	+	0.745133i	$0.267614\pi$
$864$	0	0
$865$	5.02944	0.171006
$866$	26.6569	0.905837
$867$	0	0
$868$	0	0
$869$	−4.75736	−0.161382
$870$	0	0
$871$	15.7990	0.535328
$872$	−3.24264	−0.109810
$873$	0	0
$874$	−2.68629	−0.0908652
$875$	0	0
$876$	0	0
$877$	24.5563	0.829209	0.414604	−	0.910002i	$-0.363920\pi$
$877$	24.5563	0.829209	0.414604	+	0.910002i	$0.363920\pi$
$878$	−3.38478	−0.114231
$879$	0	0
$880$	−0.414214	−0.0139631
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	0	0
$883$	0.514719	0.0173217	0.00866083	−	0.999962i	$-0.497243\pi$
$883$	0.514719	0.0173217	0.00866083	+	0.999962i	$0.497243\pi$
$884$	−2.54416	−0.0855692
$885$	0	0
$886$	14.3431	0.481867
$887$	−8.34315	−0.280135	−0.140068	−	0.990142i	$-0.544732\pi$
$887$	−8.34315	−0.280135	−0.140068	+	0.990142i	$0.544732\pi$
$888$	0	0
$889$	0	0
$890$	5.17157	0.173352
$891$	0	0
$892$	−13.3137	−0.445776
$893$	7.65685	0.256227
$894$	0	0
$895$	1.31371	0.0439124
$896$	0	0
$897$	0	0
$898$	23.1127	0.771281
$899$	18.3431	0.611778
$900$	0	0
$901$	11.2304	0.374140
$902$	−4.65685	−0.155056
$903$	0	0
$904$	3.65685	0.121625
$905$	−0.970563	−0.0322626
$906$	0	0
$907$	10.8579	0.360529	0.180265	−	0.983618i	$-0.442305\pi$
$907$	10.8579	0.360529	0.180265	+	0.983618i	$0.442305\pi$
$908$	−12.3137	−0.408645
$909$	0	0
$910$	0	0
$911$	−50.2132	−1.66364	−0.831819	−	0.555047i	$-0.812700\pi$
$911$	−50.2132	−1.66364	−0.831819	+	0.555047i	$0.812700\pi$
$912$	0	0
$913$	−9.82843	−0.325273
$914$	−34.6274	−1.14537
$915$	0	0
$916$	−11.3137	−0.373815
$917$	0	0
$918$	0	0
$919$	−22.8995	−0.755385	−0.377692	−	0.925931i	$-0.623282\pi$
$919$	−22.8995	−0.755385	−0.377692	+	0.925931i	$0.623282\pi$
$920$	−1.34315	−0.0442822
$921$	0	0
$922$	−23.6569	−0.779097
$923$	15.5980	0.513414
$924$	0	0
$925$	−46.6274	−1.53310
$926$	−23.1127	−0.759530
$927$	0	0
$928$	2.82843	0.0928477
$929$	−7.45584	−0.244618	−0.122309	−	0.992492i	$-0.539030\pi$
$929$	−7.45584	−0.244618	−0.122309	+	0.992492i	$0.539030\pi$
$930$	0	0
$931$	0	0
$932$	−8.31371	−0.272325
$933$	0	0
$934$	−2.97056	−0.0971998
$935$	−0.899495	−0.0294166
$936$	0	0
$937$	11.6569	0.380813	0.190406	−	0.981705i	$-0.439019\pi$
$937$	11.6569	0.380813	0.190406	+	0.981705i	$0.439019\pi$
$938$	0	0
$939$	0	0
$940$	3.82843	0.124870
$941$	−11.3726	−0.370736	−0.185368	−	0.982669i	$-0.559348\pi$
$941$	−11.3726	−0.370736	−0.185368	+	0.982669i	$0.559348\pi$
$942$	0	0
$943$	−15.1005	−0.491740
$944$	3.65685	0.119020
$945$	0	0
$946$	−2.82843	−0.0919601
$947$	−12.6274	−0.410336	−0.205168	−	0.978727i	$-0.565774\pi$
$947$	−12.6274	−0.410336	−0.205168	+	0.978727i	$0.565774\pi$
$948$	0	0
$949$	5.65685	0.183629
$950$	4.00000	0.129777
$951$	0	0
$952$	0	0
$953$	6.31371	0.204521	0.102261	−	0.994758i	$-0.467392\pi$
$953$	6.31371	0.204521	0.102261	+	0.994758i	$0.467392\pi$
$954$	0	0
$955$	3.85786	0.124838
$956$	−20.0000	−0.646846
$957$	0	0
$958$	5.17157	0.167086
$959$	0	0
$960$	0	0
$961$	11.0589	0.356738
$962$	11.3137	0.364769
$963$	0	0
$964$	24.9706	0.804248
$965$	−3.79899	−0.122294
$966$	0	0
$967$	3.10051	0.0997055	0.0498528	−	0.998757i	$-0.484125\pi$
$967$	3.10051	0.0997055	0.0498528	+	0.998757i	$0.484125\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	4.21320	0.135278
$971$	10.0000	0.320915	0.160458	−	0.987043i	$-0.448703\pi$
$971$	10.0000	0.320915	0.160458	+	0.987043i	$0.448703\pi$
$972$	0	0
$973$	0	0
$974$	21.3137	0.682935
$975$	0	0
$976$	−1.58579	−0.0507598
$977$	−31.9411	−1.02189	−0.510944	−	0.859614i	$-0.670704\pi$
$977$	−31.9411	−1.02189	−0.510944	+	0.859614i	$0.670704\pi$
$978$	0	0
$979$	12.4853	0.399031
$980$	0	0
$981$	0	0
$982$	−15.6274	−0.498691
$983$	−36.7574	−1.17238	−0.586189	−	0.810174i	$-0.699372\pi$
$983$	−36.7574	−1.17238	−0.586189	+	0.810174i	$0.699372\pi$
$984$	0	0
$985$	1.45584	0.0463871
$986$	6.14214	0.195605
$987$	0	0
$988$	−0.970563	−0.0308777
$989$	−9.17157	−0.291639
$990$	0	0
$991$	22.3431	0.709753	0.354877	−	0.934913i	$-0.384523\pi$
$991$	22.3431	0.709753	0.354877	+	0.934913i	$0.384523\pi$
$992$	6.48528	0.205908
$993$	0	0
$994$	0	0
$995$	−3.45584	−0.109558
$996$	0	0
$997$	47.7990	1.51381	0.756905	−	0.653525i	$-0.226711\pi$
$997$	47.7990	1.51381	0.756905	+	0.653525i	$0.226711\pi$
$998$	−14.3431	−0.454024
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.cj.1.2		2
3.2	odd	2		9702.2.a.ds.1.1		2
7.2	even	3		1386.2.k.u.991.1	yes	4
7.4	even	3		1386.2.k.u.793.1	yes	4
7.6	odd	2		9702.2.a.cv.1.1		2
21.2	odd	6		1386.2.k.q.991.2	yes	4
21.11	odd	6		1386.2.k.q.793.2	✓	4
21.20	even	2		9702.2.a.da.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.k.q.793.2	✓	4	21.11	odd	6
1386.2.k.q.991.2	yes	4	21.2	odd	6
1386.2.k.u.793.1	yes	4	7.4	even	3
1386.2.k.u.991.1	yes	4	7.2	even	3
9702.2.a.cj.1.2		2	1.1	even	1	trivial
9702.2.a.cv.1.1		2	7.6	odd	2
9702.2.a.da.1.2		2	21.20	even	2
9702.2.a.ds.1.1		2	3.2	odd	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:	\( N \)	\(=\)	\( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9702.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.414214 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.414214 q^{5} -1.00000 q^{8} -0.414214 q^{10} -1.00000 q^{11} -1.17157 q^{13} +1.00000 q^{16} +2.17157 q^{17} +0.828427 q^{19} +0.414214 q^{20} +1.00000 q^{22} +3.24264 q^{23} -4.82843 q^{25} +1.17157 q^{26} -2.82843 q^{29} -6.48528 q^{31} -1.00000 q^{32} -2.17157 q^{34} +9.65685 q^{37} -0.828427 q^{38} -0.414214 q^{40} -4.65685 q^{41} -2.82843 q^{43} -1.00000 q^{44} -3.24264 q^{46} +9.24264 q^{47} +4.82843 q^{50} -1.17157 q^{52} +5.17157 q^{53} -0.414214 q^{55} +2.82843 q^{58} +3.65685 q^{59} -1.58579 q^{61} +6.48528 q^{62} +1.00000 q^{64} -0.485281 q^{65} -13.4853 q^{67} +2.17157 q^{68} -13.3137 q^{71} -4.82843 q^{73} -9.65685 q^{74} +0.828427 q^{76} +4.75736 q^{79} +0.414214 q^{80} +4.65685 q^{82} +9.82843 q^{83} +0.899495 q^{85} +2.82843 q^{86} +1.00000 q^{88} -12.4853 q^{89} +3.24264 q^{92} -9.24264 q^{94} +0.343146 q^{95} -10.1716 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 2 q^{11} - 8 q^{13} + 2 q^{16} + 10 q^{17} - 4 q^{19} - 2 q^{20} + 2 q^{22} - 2 q^{23} - 4 q^{25} + 8 q^{26} + 4 q^{31} - 2 q^{32} - 10 q^{34} + 8 q^{37} + 4 q^{38} + 2 q^{40} + 2 q^{41} - 2 q^{44} + 2 q^{46} + 10 q^{47} + 4 q^{50} - 8 q^{52} + 16 q^{53} + 2 q^{55} - 4 q^{59} - 6 q^{61} - 4 q^{62} + 2 q^{64} + 16 q^{65} - 10 q^{67} + 10 q^{68} - 4 q^{71} - 4 q^{73} - 8 q^{74} - 4 q^{76} + 18 q^{79} - 2 q^{80} - 2 q^{82} + 14 q^{83} - 18 q^{85} + 2 q^{88} - 8 q^{89} - 2 q^{92} - 10 q^{94} + 12 q^{95} - 26 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 2 * q^10 - 2 * q^11 - 8 * q^13 + 2 * q^16 + 10 * q^17 - 4 * q^19 - 2 * q^20 + 2 * q^22 - 2 * q^23 - 4 * q^25 + 8 * q^26 + 4 * q^31 - 2 * q^32 - 10 * q^34 + 8 * q^37 + 4 * q^38 + 2 * q^40 + 2 * q^41 - 2 * q^44 + 2 * q^46 + 10 * q^47 + 4 * q^50 - 8 * q^52 + 16 * q^53 + 2 * q^55 - 4 * q^59 - 6 * q^61 - 4 * q^62 + 2 * q^64 + 16 * q^65 - 10 * q^67 + 10 * q^68 - 4 * q^71 - 4 * q^73 - 8 * q^74 - 4 * q^76 + 18 * q^79 - 2 * q^80 - 2 * q^82 + 14 * q^83 - 18 * q^85 + 2 * q^88 - 8 * q^89 - 2 * q^92 - 10 * q^94 + 12 * q^95 - 26 * q^97

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