LMFDB - Embedded newform 9702.2.a.dv.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:	3.3.2700.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 15x - 20 $ x^3 - 15*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.80560$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.80560 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +2.80560 q^{5} -1.00000 q^{8} -2.80560 q^{10} +1.00000 q^{11} +1.00000 q^{16} -7.96523 q^{17} -6.48261 q^{19} +2.80560 q^{20} -1.00000 q^{22} +5.28822 q^{23} +2.87141 q^{25} -5.12859 q^{29} -8.96523 q^{31} -1.00000 q^{32} +7.96523 q^{34} +10.4826 q^{37} +6.48261 q^{38} -2.80560 q^{40} +5.09382 q^{41} +11.4478 q^{43} +1.00000 q^{44} -5.28822 q^{46} -0.322990 q^{47} -2.87141 q^{50} -8.00000 q^{53} +2.80560 q^{55} +5.12859 q^{58} +0.871407 q^{59} +2.15962 q^{61} +8.96523 q^{62} +1.00000 q^{64} -7.09382 q^{67} -7.96523 q^{68} +7.44784 q^{71} +12.5764 q^{73} -10.4826 q^{74} -6.48261 q^{76} +6.80560 q^{79} +2.80560 q^{80} -5.09382 q^{82} -15.0938 q^{83} -22.3473 q^{85} -11.4478 q^{86} -1.00000 q^{88} -1.61121 q^{89} +5.28822 q^{92} +0.322990 q^{94} -18.1876 q^{95} +7.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 9 q^{19} - 3 q^{22} - 3 q^{23} + 15 q^{25} - 9 q^{29} - 6 q^{31} - 3 q^{32} + 3 q^{34} + 21 q^{37} + 9 q^{38} - 12 q^{41} + 3 q^{43} + 3 q^{44} + 3 q^{46} - 3 q^{47} - 15 q^{50} - 24 q^{53} + 9 q^{58} + 9 q^{59} - 6 q^{61} + 6 q^{62} + 3 q^{64} + 6 q^{67} - 3 q^{68} - 9 q^{71} - 21 q^{74} - 9 q^{76} + 12 q^{79} + 12 q^{82} - 18 q^{83} - 3 q^{86} - 3 q^{88} + 12 q^{89} - 3 q^{92} + 3 q^{94} + 21 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 - 9 * q^19 - 3 * q^22 - 3 * q^23 + 15 * q^25 - 9 * q^29 - 6 * q^31 - 3 * q^32 + 3 * q^34 + 21 * q^37 + 9 * q^38 - 12 * q^41 + 3 * q^43 + 3 * q^44 + 3 * q^46 - 3 * q^47 - 15 * q^50 - 24 * q^53 + 9 * q^58 + 9 * q^59 - 6 * q^61 + 6 * q^62 + 3 * q^64 + 6 * q^67 - 3 * q^68 - 9 * q^71 - 21 * q^74 - 9 * q^76 + 12 * q^79 + 12 * q^82 - 18 * q^83 - 3 * q^86 - 3 * q^88 + 12 * q^89 - 3 * q^92 + 3 * q^94 + 21 * 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
$5$	2.80560	1.25470	0.627352	−	0.778736i	$-0.284139\pi$
$5$	2.80560	1.25470	0.627352	+	0.778736i	$0.284139\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.80560	−0.887210
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.96523	−1.93185	−0.965926	−	0.258820i	$-0.916666\pi$
$17$	−7.96523	−1.93185	−0.965926	+	0.258820i	$0.916666\pi$
$18$	0	0
$19$	−6.48261	−1.48721	−0.743607	−	0.668617i	$-0.766886\pi$
$19$	−6.48261	−1.48721	−0.743607	+	0.668617i	$0.766886\pi$
$20$	2.80560	0.627352
$21$	0	0
$22$	−1.00000	−0.213201
$23$	5.28822	1.10267	0.551335	−	0.834284i	$-0.314119\pi$
$23$	5.28822	1.10267	0.551335	+	0.834284i	$0.314119\pi$
$24$	0	0
$25$	2.87141	0.574281
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.12859	−0.952356	−0.476178	−	0.879349i	$-0.657978\pi$
$29$	−5.12859	−0.952356	−0.476178	+	0.879349i	$0.657978\pi$
$30$	0	0
$31$	−8.96523	−1.61020	−0.805101	−	0.593138i	$-0.797889\pi$
$31$	−8.96523	−1.61020	−0.805101	+	0.593138i	$0.797889\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	7.96523	1.36602
$35$	0	0
$36$	0	0
$37$	10.4826	1.72333	0.861665	−	0.507477i	$-0.169422\pi$
$37$	10.4826	1.72333	0.861665	+	0.507477i	$0.169422\pi$
$38$	6.48261	1.05162
$39$	0	0
$40$	−2.80560	−0.443605
$41$	5.09382	0.795521	0.397760	−	0.917489i	$-0.369788\pi$
$41$	5.09382	0.795521	0.397760	+	0.917489i	$0.369788\pi$
$42$	0	0
$43$	11.4478	1.74578	0.872890	−	0.487918i	$-0.162243\pi$
$43$	11.4478	1.74578	0.872890	+	0.487918i	$0.162243\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−5.28822	−0.779705
$47$	−0.322990	−0.0471129	−0.0235565	−	0.999723i	$-0.507499\pi$
$47$	−0.322990	−0.0471129	−0.0235565	+	0.999723i	$0.507499\pi$
$48$	0	0
$49$	0	0
$50$	−2.87141	−0.406078
$51$	0	0
$52$	0	0
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	0	0
$55$	2.80560	0.378307
$56$	0	0
$57$	0	0
$58$	5.12859	0.673417
$59$	0.871407	0.113448	0.0567238	−	0.998390i	$-0.481935\pi$
$59$	0.871407	0.113448	0.0567238	+	0.998390i	$0.481935\pi$
$60$	0	0
$61$	2.15962	0.276511	0.138256	−	0.990397i	$-0.455850\pi$
$61$	2.15962	0.276511	0.138256	+	0.990397i	$0.455850\pi$
$62$	8.96523	1.13858
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−7.09382	−0.866648	−0.433324	−	0.901238i	$-0.642659\pi$
$67$	−7.09382	−0.866648	−0.433324	+	0.901238i	$0.642659\pi$
$68$	−7.96523	−0.965926
$69$	0	0
$70$	0	0
$71$	7.44784	0.883896	0.441948	−	0.897041i	$-0.354288\pi$
$71$	7.44784	0.883896	0.441948	+	0.897041i	$0.354288\pi$
$72$	0	0
$73$	12.5764	1.47196	0.735980	−	0.677003i	$-0.236722\pi$
$73$	12.5764	1.47196	0.735980	+	0.677003i	$0.236722\pi$
$74$	−10.4826	−1.21858
$75$	0	0
$76$	−6.48261	−0.743607
$77$	0	0
$78$	0	0
$79$	6.80560	0.765690	0.382845	−	0.923813i	$-0.374944\pi$
$79$	6.80560	0.765690	0.382845	+	0.923813i	$0.374944\pi$
$80$	2.80560	0.313676
$81$	0	0
$82$	−5.09382	−0.562518
$83$	−15.0938	−1.65676	−0.828381	−	0.560165i	$-0.810738\pi$
$83$	−15.0938	−1.65676	−0.828381	+	0.560165i	$0.810738\pi$
$84$	0	0
$85$	−22.3473	−2.42390
$86$	−11.4478	−1.23445
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−1.61121	−0.170787	−0.0853937	−	0.996347i	$-0.527215\pi$
$89$	−1.61121	−0.170787	−0.0853937	+	0.996347i	$0.527215\pi$
$90$	0	0
$91$	0	0
$92$	5.28822	0.551335
$93$	0	0
$94$	0.322990	0.0333139
$95$	−18.1876	−1.86601
$96$	0	0
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	0	0
$99$	0	0
$100$	2.87141	0.287141
$101$	−10.7398	−1.06865	−0.534325	−	0.845279i	$-0.679434\pi$
$101$	−10.7398	−1.06865	−0.534325	+	0.845279i	$0.679434\pi$
$102$	0	0
$103$	2.96523	0.292172	0.146086	−	0.989272i	$-0.453332\pi$
$103$	2.96523	0.292172	0.146086	+	0.989272i	$0.453332\pi$
$104$	0	0
$105$	0	0
$106$	8.00000	0.777029
$107$	−4.83663	−0.467575	−0.233787	−	0.972288i	$-0.575112\pi$
$107$	−4.83663	−0.467575	−0.233787	+	0.972288i	$0.575112\pi$
$108$	0	0
$109$	−10.1596	−0.973115	−0.486558	−	0.873648i	$-0.661748\pi$
$109$	−10.1596	−0.973115	−0.486558	+	0.873648i	$0.661748\pi$
$110$	−2.80560	−0.267504
$111$	0	0
$112$	0	0
$113$	−10.9652	−1.03152	−0.515761	−	0.856733i	$-0.672491\pi$
$113$	−10.9652	−1.03152	−0.515761	+	0.856733i	$0.672491\pi$
$114$	0	0
$115$	14.8366	1.38352
$116$	−5.12859	−0.476178
$117$	0	0
$118$	−0.871407	−0.0802195
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.15962	−0.195523
$123$	0	0
$124$	−8.96523	−0.805101
$125$	−5.97199	−0.534151
$126$	0	0
$127$	−9.28822	−0.824196	−0.412098	−	0.911140i	$-0.635204\pi$
$127$	−9.28822	−0.824196	−0.412098	+	0.911140i	$0.635204\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−12.9652	−1.13278	−0.566389	−	0.824138i	$-0.691660\pi$
$131$	−12.9652	−1.13278	−0.566389	+	0.824138i	$0.691660\pi$
$132$	0	0
$133$	0	0
$134$	7.09382	0.612813
$135$	0	0
$136$	7.96523	0.683012
$137$	−5.61121	−0.479398	−0.239699	−	0.970847i	$-0.577049\pi$
$137$	−5.61121	−0.479398	−0.239699	+	0.970847i	$0.577049\pi$
$138$	0	0
$139$	13.1286	1.11355	0.556776	−	0.830662i	$-0.312038\pi$
$139$	13.1286	1.11355	0.556776	+	0.830662i	$0.312038\pi$
$140$	0	0
$141$	0	0
$142$	−7.44784	−0.625009
$143$	0	0
$144$	0	0
$145$	−14.3888	−1.19492
$146$	−12.5764	−1.04083
$147$	0	0
$148$	10.4826	0.861665
$149$	−16.7398	−1.37138	−0.685689	−	0.727895i	$-0.740499\pi$
$149$	−16.7398	−1.37138	−0.685689	+	0.727895i	$0.740499\pi$
$150$	0	0
$151$	10.8994	0.886982	0.443491	−	0.896279i	$-0.353740\pi$
$151$	10.8994	0.886982	0.443491	+	0.896279i	$0.353740\pi$
$152$	6.48261	0.525809
$153$	0	0
$154$	0	0
$155$	−25.1529	−2.02033
$156$	0	0
$157$	1.77457	0.141626	0.0708132	−	0.997490i	$-0.477441\pi$
$157$	1.77457	0.141626	0.0708132	+	0.997490i	$0.477441\pi$
$158$	−6.80560	−0.541425
$159$	0	0
$160$	−2.80560	−0.221802
$161$	0	0
$162$	0	0
$163$	−4.05904	−0.317929	−0.158964	−	0.987284i	$-0.550816\pi$
$163$	−4.05904	−0.317929	−0.158964	+	0.987284i	$0.550816\pi$
$164$	5.09382	0.397760
$165$	0	0
$166$	15.0938	1.17151
$167$	−13.8684	−1.07317	−0.536584	−	0.843847i	$-0.680286\pi$
$167$	−13.8684	−1.07317	−0.536584	+	0.843847i	$0.680286\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	22.3473	1.71396
$171$	0	0
$172$	11.4478	0.872890
$173$	−16.9652	−1.28984	−0.644921	−	0.764249i	$-0.723110\pi$
$173$	−16.9652	−1.28984	−0.644921	+	0.764249i	$0.723110\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	1.61121	0.120765
$179$	12.6703	0.947019	0.473509	−	0.880789i	$-0.342987\pi$
$179$	12.6703	0.947019	0.473509	+	0.880789i	$0.342987\pi$
$180$	0	0
$181$	−2.57643	−0.191505	−0.0957523	−	0.995405i	$-0.530526\pi$
$181$	−2.57643	−0.191505	−0.0957523	+	0.995405i	$0.530526\pi$
$182$	0	0
$183$	0	0
$184$	−5.28822	−0.389852
$185$	29.4100	2.16227
$186$	0	0
$187$	−7.96523	−0.582475
$188$	−0.322990	−0.0235565
$189$	0	0
$190$	18.1876	1.31947
$191$	2.38879	0.172847	0.0864235	−	0.996258i	$-0.472456\pi$
$191$	2.38879	0.172847	0.0864235	+	0.996258i	$0.472456\pi$
$192$	0	0
$193$	−14.5764	−1.04923	−0.524617	−	0.851338i	$-0.675792\pi$
$193$	−14.5764	−1.04923	−0.524617	+	0.851338i	$0.675792\pi$
$194$	−7.00000	−0.502571
$195$	0	0
$196$	0	0
$197$	12.4826	0.889349	0.444675	−	0.895692i	$-0.353319\pi$
$197$	12.4826	0.889349	0.444675	+	0.895692i	$0.353319\pi$
$198$	0	0
$199$	−14.1876	−1.00573	−0.502867	−	0.864364i	$-0.667722\pi$
$199$	−14.1876	−1.00573	−0.502867	+	0.864364i	$0.667722\pi$
$200$	−2.87141	−0.203039
$201$	0	0
$202$	10.7398	0.755650
$203$	0	0
$204$	0	0
$205$	14.2912	0.998143
$206$	−2.96523	−0.206597
$207$	0	0
$208$	0	0
$209$	−6.48261	−0.448412
$210$	0	0
$211$	−20.8336	−1.43425	−0.717123	−	0.696947i	$-0.754541\pi$
$211$	−20.8336	−1.43425	−0.717123	+	0.696947i	$0.754541\pi$
$212$	−8.00000	−0.549442
$213$	0	0
$214$	4.83663	0.330625
$215$	32.1181	2.19044
$216$	0	0
$217$	0	0
$218$	10.1596	0.688096
$219$	0	0
$220$	2.80560	0.189154
$221$	0	0
$222$	0	0
$223$	12.5764	0.842180	0.421090	−	0.907019i	$-0.361648\pi$
$223$	12.5764	0.842180	0.421090	+	0.907019i	$0.361648\pi$
$224$	0	0
$225$	0	0
$226$	10.9652	0.729396
$227$	−20.0590	−1.33137	−0.665683	−	0.746235i	$-0.731860\pi$
$227$	−20.0590	−1.33137	−0.665683	+	0.746235i	$0.731860\pi$
$228$	0	0
$229$	−22.5764	−1.49189	−0.745946	−	0.666006i	$-0.768002\pi$
$229$	−22.5764	−1.49189	−0.745946	+	0.666006i	$0.768002\pi$
$230$	−14.8366	−0.978299
$231$	0	0
$232$	5.12859	0.336709
$233$	0.742815	0.0486634	0.0243317	−	0.999704i	$-0.492254\pi$
$233$	0.742815	0.0486634	0.0243317	+	0.999704i	$0.492254\pi$
$234$	0	0
$235$	−0.906181	−0.0591128
$236$	0.871407	0.0567238
$237$	0	0
$238$	0	0
$239$	−4.31925	−0.279389	−0.139694	−	0.990195i	$-0.544612\pi$
$239$	−4.31925	−0.279389	−0.139694	+	0.990195i	$0.544612\pi$
$240$	0	0
$241$	−18.1876	−1.17157	−0.585784	−	0.810467i	$-0.699213\pi$
$241$	−18.1876	−1.17157	−0.585784	+	0.810467i	$0.699213\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	2.15962	0.138256
$245$	0	0
$246$	0	0
$247$	0	0
$248$	8.96523	0.569292
$249$	0	0
$250$	5.97199	0.377702
$251$	−28.6703	−1.80965	−0.904825	−	0.425784i	$-0.859999\pi$
$251$	−28.6703	−1.80965	−0.904825	+	0.425784i	$0.859999\pi$
$252$	0	0
$253$	5.28822	0.332467
$254$	9.28822	0.582794
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.6112	0.849044	0.424522	−	0.905418i	$-0.360442\pi$
$257$	13.6112	0.849044	0.424522	+	0.905418i	$0.360442\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	12.9652	0.800994
$263$	3.61121	0.222676	0.111338	−	0.993783i	$-0.464486\pi$
$263$	3.61121	0.222676	0.111338	+	0.993783i	$0.464486\pi$
$264$	0	0
$265$	−22.4448	−1.37877
$266$	0	0
$267$	0	0
$268$	−7.09382	−0.433324
$269$	9.38203	0.572033	0.286016	−	0.958225i	$-0.407669\pi$
$269$	9.38203	0.572033	0.286016	+	0.958225i	$0.407669\pi$
$270$	0	0
$271$	5.61121	0.340856	0.170428	−	0.985370i	$-0.445485\pi$
$271$	5.61121	0.340856	0.170428	+	0.985370i	$0.445485\pi$
$272$	−7.96523	−0.482963
$273$	0	0
$274$	5.61121	0.338985
$275$	2.87141	0.173152
$276$	0	0
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	−13.1286	−0.787401
$279$	0	0
$280$	0	0
$281$	−24.1529	−1.44084	−0.720420	−	0.693539i	$-0.756051\pi$
$281$	−24.1529	−1.44084	−0.720420	+	0.693539i	$0.756051\pi$
$282$	0	0
$283$	7.54166	0.448305	0.224152	−	0.974554i	$-0.428039\pi$
$283$	7.54166	0.448305	0.224152	+	0.974554i	$0.428039\pi$
$284$	7.44784	0.441948
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	46.4448	2.73205
$290$	14.3888	0.844939
$291$	0	0
$292$	12.5764	0.735980
$293$	−7.90618	−0.461884	−0.230942	−	0.972968i	$-0.574181\pi$
$293$	−7.90618	−0.461884	−0.230942	+	0.972968i	$0.574181\pi$
$294$	0	0
$295$	2.44482	0.142343
$296$	−10.4826	−0.609289
$297$	0	0
$298$	16.7398	0.969710
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−10.8994	−0.627191
$303$	0	0
$304$	−6.48261	−0.371803
$305$	6.05904	0.346940
$306$	0	0
$307$	12.8957	0.735995	0.367998	−	0.929827i	$-0.380043\pi$
$307$	12.8957	0.735995	0.367998	+	0.929827i	$0.380043\pi$
$308$	0	0
$309$	0	0
$310$	25.1529	1.42859
$311$	−10.8994	−0.618049	−0.309025	−	0.951054i	$-0.600003\pi$
$311$	−10.8994	−0.618049	−0.309025	+	0.951054i	$0.600003\pi$
$312$	0	0
$313$	−9.12859	−0.515979	−0.257989	−	0.966148i	$-0.583060\pi$
$313$	−9.12859	−0.515979	−0.257989	+	0.966148i	$0.583060\pi$
$314$	−1.77457	−0.100145
$315$	0	0
$316$	6.80560	0.382845
$317$	9.51364	0.534339	0.267170	−	0.963649i	$-0.413912\pi$
$317$	9.51364	0.534339	0.267170	+	0.963649i	$0.413912\pi$
$318$	0	0
$319$	−5.12859	−0.287146
$320$	2.80560	0.156838
$321$	0	0
$322$	0	0
$323$	51.6355	2.87307
$324$	0	0
$325$	0	0
$326$	4.05904	0.224810
$327$	0	0
$328$	−5.09382	−0.281259
$329$	0	0
$330$	0	0
$331$	1.09382	0.0601217	0.0300609	−	0.999548i	$-0.490430\pi$
$331$	1.09382	0.0601217	0.0300609	+	0.999548i	$0.490430\pi$
$332$	−15.0938	−0.828381
$333$	0	0
$334$	13.8684	0.758845
$335$	−19.9024	−1.08739
$336$	0	0
$337$	−25.7988	−1.40535	−0.702676	−	0.711510i	$-0.748012\pi$
$337$	−25.7988	−1.40535	−0.702676	+	0.711510i	$0.748012\pi$
$338$	13.0000	0.707107
$339$	0	0
$340$	−22.3473	−1.21195
$341$	−8.96523	−0.485494
$342$	0	0
$343$	0	0
$344$	−11.4478	−0.617226
$345$	0	0
$346$	16.9652	0.912056
$347$	−20.7671	−1.11484	−0.557418	−	0.830232i	$-0.688208\pi$
$347$	−20.7671	−1.11484	−0.557418	+	0.830232i	$0.688208\pi$
$348$	0	0
$349$	−9.51364	−0.509254	−0.254627	−	0.967039i	$-0.581953\pi$
$349$	−9.51364	−0.509254	−0.254627	+	0.967039i	$0.581953\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	12.5764	0.669376	0.334688	−	0.942329i	$-0.391369\pi$
$353$	12.5764	0.669376	0.334688	+	0.942329i	$0.391369\pi$
$354$	0	0
$355$	20.8957	1.10903
$356$	−1.61121	−0.0853937
$357$	0	0
$358$	−12.6703	−0.669644
$359$	0.0695483	0.00367062	0.00183531	−	0.999998i	$-0.499416\pi$
$359$	0.0695483	0.00367062	0.00183531	+	0.999998i	$0.499416\pi$
$360$	0	0
$361$	23.0243	1.21180
$362$	2.57643	0.135414
$363$	0	0
$364$	0	0
$365$	35.2845	1.84687
$366$	0	0
$367$	−31.5417	−1.64646	−0.823231	−	0.567707i	$-0.807831\pi$
$367$	−31.5417	−1.64646	−0.823231	+	0.567707i	$0.807831\pi$
$368$	5.28822	0.275667
$369$	0	0
$370$	−29.4100	−1.52896
$371$	0	0
$372$	0	0
$373$	15.3125	0.792850	0.396425	−	0.918067i	$-0.370251\pi$
$373$	15.3125	0.792850	0.396425	+	0.918067i	$0.370251\pi$
$374$	7.96523	0.411872
$375$	0	0
$376$	0.322990	0.0166569
$377$	0	0
$378$	0	0
$379$	16.0590	0.824898	0.412449	−	0.910981i	$-0.364674\pi$
$379$	16.0590	0.824898	0.412449	+	0.910981i	$0.364674\pi$
$380$	−18.1876	−0.933006
$381$	0	0
$382$	−2.38879	−0.122221
$383$	0.739798	0.0378019	0.0189010	−	0.999821i	$-0.493983\pi$
$383$	0.739798	0.0378019	0.0189010	+	0.999821i	$0.493983\pi$
$384$	0	0
$385$	0	0
$386$	14.5764	0.741921
$387$	0	0
$388$	7.00000	0.355371
$389$	−3.58319	−0.181675	−0.0908375	−	0.995866i	$-0.528954\pi$
$389$	−3.58319	−0.181675	−0.0908375	+	0.995866i	$0.528954\pi$
$390$	0	0
$391$	−42.1218	−2.13019
$392$	0	0
$393$	0	0
$394$	−12.4826	−0.628865
$395$	19.0938	0.960714
$396$	0	0
$397$	−25.1286	−1.26117	−0.630584	−	0.776121i	$-0.717185\pi$
$397$	−25.1286	−1.26117	−0.630584	+	0.776121i	$0.717185\pi$
$398$	14.1876	0.711162
$399$	0	0
$400$	2.87141	0.143570
$401$	3.86839	0.193178	0.0965891	−	0.995324i	$-0.469207\pi$
$401$	3.86839	0.193178	0.0965891	+	0.995324i	$0.469207\pi$
$402$	0	0
$403$	0	0
$404$	−10.7398	−0.534325
$405$	0	0
$406$	0	0
$407$	10.4826	0.519604
$408$	0	0
$409$	4.64598	0.229729	0.114864	−	0.993381i	$-0.463357\pi$
$409$	4.64598	0.229729	0.114864	+	0.993381i	$0.463357\pi$
$410$	−14.2912	−0.705794
$411$	0	0
$412$	2.96523	0.146086
$413$	0	0
$414$	0	0
$415$	−42.3473	−2.07875
$416$	0	0
$417$	0	0
$418$	6.48261	0.317075
$419$	19.5174	0.953487	0.476743	−	0.879043i	$-0.341817\pi$
$419$	19.5174	0.953487	0.476743	+	0.879043i	$0.341817\pi$
$420$	0	0
$421$	15.5174	0.756271	0.378136	−	0.925750i	$-0.376565\pi$
$421$	15.5174	0.756271	0.378136	+	0.925750i	$0.376565\pi$
$422$	20.8336	1.01416
$423$	0	0
$424$	8.00000	0.388514
$425$	−22.8714	−1.10943
$426$	0	0
$427$	0	0
$428$	−4.83663	−0.233787
$429$	0	0
$430$	−32.1181	−1.54887
$431$	14.3888	0.693084	0.346542	−	0.938034i	$-0.387356\pi$
$431$	14.3888	0.693084	0.346542	+	0.938034i	$0.387356\pi$
$432$	0	0
$433$	2.03477	0.0977850	0.0488925	−	0.998804i	$-0.484431\pi$
$433$	2.03477	0.0977850	0.0488925	+	0.998804i	$0.484431\pi$
$434$	0	0
$435$	0	0
$436$	−10.1596	−0.486558
$437$	−34.2815	−1.63990
$438$	0	0
$439$	3.22616	0.153976	0.0769880	−	0.997032i	$-0.475470\pi$
$439$	3.22616	0.153976	0.0769880	+	0.997032i	$0.475470\pi$
$440$	−2.80560	−0.133752
$441$	0	0
$442$	0	0
$443$	22.0938	1.04971	0.524854	−	0.851192i	$-0.324120\pi$
$443$	22.0938	1.04971	0.524854	+	0.851192i	$0.324120\pi$
$444$	0	0
$445$	−4.52040	−0.214288
$446$	−12.5764	−0.595511
$447$	0	0
$448$	0	0
$449$	−22.2497	−1.05003	−0.525014	−	0.851094i	$-0.675940\pi$
$449$	−22.2497	−1.05003	−0.525014	+	0.851094i	$0.675940\pi$
$450$	0	0
$451$	5.09382	0.239859
$452$	−10.9652	−0.515761
$453$	0	0
$454$	20.0590	0.941418
$455$	0	0
$456$	0	0
$457$	−18.6385	−0.871872	−0.435936	−	0.899978i	$-0.643583\pi$
$457$	−18.6385	−0.871872	−0.435936	+	0.899978i	$0.643583\pi$
$458$	22.5764	1.05493
$459$	0	0
$460$	14.8366	0.691762
$461$	20.9895	0.977578	0.488789	−	0.872402i	$-0.337439\pi$
$461$	20.9895	0.977578	0.488789	+	0.872402i	$0.337439\pi$
$462$	0	0
$463$	−5.35402	−0.248822	−0.124411	−	0.992231i	$-0.539704\pi$
$463$	−5.35402	−0.248822	−0.124411	+	0.992231i	$0.539704\pi$
$464$	−5.12859	−0.238089
$465$	0	0
$466$	−0.742815	−0.0344102
$467$	−26.0243	−1.20426	−0.602130	−	0.798398i	$-0.705681\pi$
$467$	−26.0243	−1.20426	−0.602130	+	0.798398i	$0.705681\pi$
$468$	0	0
$469$	0	0
$470$	0.906181	0.0417990
$471$	0	0
$472$	−0.871407	−0.0401098
$473$	11.4478	0.526372
$474$	0	0
$475$	−18.6142	−0.854079
$476$	0	0
$477$	0	0
$478$	4.31925	0.197558
$479$	3.61121	0.165000	0.0825001	−	0.996591i	$-0.473710\pi$
$479$	3.61121	0.165000	0.0825001	+	0.996591i	$0.473710\pi$
$480$	0	0
$481$	0	0
$482$	18.1876	0.828424
$483$	0	0
$484$	1.00000	0.0454545
$485$	19.6392	0.891771
$486$	0	0
$487$	−8.06206	−0.365327	−0.182663	−	0.983176i	$-0.558472\pi$
$487$	−8.06206	−0.365327	−0.182663	+	0.983176i	$0.558472\pi$
$488$	−2.15962	−0.0977615
$489$	0	0
$490$	0	0
$491$	−20.9062	−0.943483	−0.471741	−	0.881737i	$-0.656374\pi$
$491$	−20.9062	−0.943483	−0.471741	+	0.881737i	$0.656374\pi$
$492$	0	0
$493$	40.8504	1.83981
$494$	0	0
$495$	0	0
$496$	−8.96523	−0.402551
$497$	0	0
$498$	0	0
$499$	4.96523	0.222274	0.111137	−	0.993805i	$-0.464551\pi$
$499$	4.96523	0.222274	0.111137	+	0.993805i	$0.464551\pi$
$500$	−5.97199	−0.267075
$501$	0	0
$502$	28.6703	1.27962
$503$	17.7988	0.793611	0.396806	−	0.917903i	$-0.370119\pi$
$503$	17.7988	0.793611	0.396806	+	0.917903i	$0.370119\pi$
$504$	0	0
$505$	−30.1316	−1.34084
$506$	−5.28822	−0.235090
$507$	0	0
$508$	−9.28822	−0.412098
$509$	−4.51437	−0.200096	−0.100048	−	0.994983i	$-0.531900\pi$
$509$	−4.51437	−0.200096	−0.100048	+	0.994983i	$0.531900\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−13.6112	−0.600365
$515$	8.31925	0.366590
$516$	0	0
$517$	−0.322990	−0.0142051
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	6.90317	0.301854	0.150927	−	0.988545i	$-0.451774\pi$
$523$	6.90317	0.301854	0.150927	+	0.988545i	$0.451774\pi$
$524$	−12.9652	−0.566389
$525$	0	0
$526$	−3.61121	−0.157456
$527$	71.4100	3.11067
$528$	0	0
$529$	4.96523	0.215879
$530$	22.4448	0.974941
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−13.5697	−0.586668
$536$	7.09382	0.306406
$537$	0	0
$538$	−9.38203	−0.404488
$539$	0	0
$540$	0	0
$541$	2.80560	0.120622	0.0603111	−	0.998180i	$-0.480791\pi$
$541$	2.80560	0.120622	0.0603111	+	0.998180i	$0.480791\pi$
$542$	−5.61121	−0.241022
$543$	0	0
$544$	7.96523	0.341506
$545$	−28.5039	−1.22097
$546$	0	0
$547$	32.9274	1.40788	0.703938	−	0.710262i	$-0.251423\pi$
$547$	32.9274	1.40788	0.703938	+	0.710262i	$0.251423\pi$
$548$	−5.61121	−0.239699
$549$	0	0
$550$	−2.87141	−0.122437
$551$	33.2467	1.41636
$552$	0	0
$553$	0	0
$554$	16.0000	0.679775
$555$	0	0
$556$	13.1286	0.556776
$557$	−23.7050	−1.00441	−0.502207	−	0.864747i	$-0.667478\pi$
$557$	−23.7050	−1.00441	−0.502207	+	0.864747i	$0.667478\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	24.1529	1.01883
$563$	−30.4448	−1.28310	−0.641548	−	0.767083i	$-0.721708\pi$
$563$	−30.4448	−1.28310	−0.641548	+	0.767083i	$0.721708\pi$
$564$	0	0
$565$	−30.7641	−1.29425
$566$	−7.54166	−0.317000
$567$	0	0
$568$	−7.44784	−0.312504
$569$	−22.2815	−0.934087	−0.467044	−	0.884234i	$-0.654681\pi$
$569$	−22.2815	−0.934087	−0.467044	+	0.884234i	$0.654681\pi$
$570$	0	0
$571$	−2.87141	−0.120165	−0.0600823	−	0.998193i	$-0.519136\pi$
$571$	−2.87141	−0.120165	−0.0600823	+	0.998193i	$0.519136\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	15.1846	0.633242
$576$	0	0
$577$	−15.0243	−0.625469	−0.312734	−	0.949841i	$-0.601245\pi$
$577$	−15.0243	−0.625469	−0.312734	+	0.949841i	$0.601245\pi$
$578$	−46.4448	−1.93185
$579$	0	0
$580$	−14.3888	−0.597462
$581$	0	0
$582$	0	0
$583$	−8.00000	−0.331326
$584$	−12.5764	−0.520416
$585$	0	0
$586$	7.90618	0.326601
$587$	42.5069	1.75445	0.877223	−	0.480082i	$-0.159393\pi$
$587$	42.5069	1.75445	0.877223	+	0.480082i	$0.159393\pi$
$588$	0	0
$589$	58.1181	2.39471
$590$	−2.44482	−0.100652
$591$	0	0
$592$	10.4826	0.430833
$593$	−15.3858	−0.631818	−0.315909	−	0.948789i	$-0.602309\pi$
$593$	−15.3858	−0.631818	−0.315909	+	0.948789i	$0.602309\pi$
$594$	0	0
$595$	0	0
$596$	−16.7398	−0.685689
$597$	0	0
$598$	0	0
$599$	39.6392	1.61961	0.809807	−	0.586696i	$-0.199572\pi$
$599$	39.6392	1.61961	0.809807	+	0.586696i	$0.199572\pi$
$600$	0	0
$601$	33.3405	1.35999	0.679994	−	0.733218i	$-0.261983\pi$
$601$	33.3405	1.35999	0.679994	+	0.733218i	$0.261983\pi$
$602$	0	0
$603$	0	0
$604$	10.8994	0.443491
$605$	2.80560	0.114064
$606$	0	0
$607$	15.9585	0.647734	0.323867	−	0.946103i	$-0.395017\pi$
$607$	15.9585	0.647734	0.323867	+	0.946103i	$0.395017\pi$
$608$	6.48261	0.262905
$609$	0	0
$610$	−6.05904	−0.245324
$611$	0	0
$612$	0	0
$613$	46.6665	1.88484	0.942421	−	0.334428i	$-0.108543\pi$
$613$	46.6665	1.88484	0.942421	+	0.334428i	$0.108543\pi$
$614$	−12.8957	−0.520427
$615$	0	0
$616$	0	0
$617$	9.28447	0.373779	0.186889	−	0.982381i	$-0.440159\pi$
$617$	9.28447	0.373779	0.186889	+	0.982381i	$0.440159\pi$
$618$	0	0
$619$	33.9895	1.36615	0.683077	−	0.730347i	$-0.260642\pi$
$619$	33.9895	1.36615	0.683077	+	0.730347i	$0.260642\pi$
$620$	−25.1529	−1.01016
$621$	0	0
$622$	10.8994	0.437027
$623$	0	0
$624$	0	0
$625$	−31.1121	−1.24448
$626$	9.12859	0.364852
$627$	0	0
$628$	1.77457	0.0708132
$629$	−83.4964	−3.32922
$630$	0	0
$631$	−14.6460	−0.583047	−0.291524	−	0.956564i	$-0.594162\pi$
$631$	−14.6460	−0.583047	−0.291524	+	0.956564i	$0.594162\pi$
$632$	−6.80560	−0.270712
$633$	0	0
$634$	−9.51364	−0.377835
$635$	−26.0590	−1.03412
$636$	0	0
$637$	0	0
$638$	5.12859	0.203043
$639$	0	0
$640$	−2.80560	−0.110901
$641$	−20.9652	−0.828077	−0.414038	−	0.910259i	$-0.635882\pi$
$641$	−20.9652	−0.828077	−0.414038	+	0.910259i	$0.635882\pi$
$642$	0	0
$643$	−23.6733	−0.933582	−0.466791	−	0.884368i	$-0.654590\pi$
$643$	−23.6733	−0.933582	−0.466791	+	0.884368i	$0.654590\pi$
$644$	0	0
$645$	0	0
$646$	−51.6355	−2.03157
$647$	12.4168	0.488155	0.244078	−	0.969756i	$-0.421515\pi$
$647$	12.4168	0.488155	0.244078	+	0.969756i	$0.421515\pi$
$648$	0	0
$649$	0.871407	0.0342057
$650$	0	0
$651$	0	0
$652$	−4.05904	−0.158964
$653$	0.0975625	0.00381792	0.00190896	−	0.999998i	$-0.499392\pi$
$653$	0.0975625	0.00381792	0.00190896	+	0.999998i	$0.499392\pi$
$654$	0	0
$655$	−36.3753	−1.42130
$656$	5.09382	0.198880
$657$	0	0
$658$	0	0
$659$	35.0243	1.36435	0.682176	−	0.731188i	$-0.261034\pi$
$659$	35.0243	1.36435	0.682176	+	0.731188i	$0.261034\pi$
$660$	0	0
$661$	29.6355	1.15269	0.576343	−	0.817208i	$-0.304479\pi$
$661$	29.6355	1.15269	0.576343	+	0.817208i	$0.304479\pi$
$662$	−1.09382	−0.0425125
$663$	0	0
$664$	15.0938	0.585754
$665$	0	0
$666$	0	0
$667$	−27.1211	−1.05013
$668$	−13.8684	−0.536584
$669$	0	0
$670$	19.9024	0.768898
$671$	2.15962	0.0833713
$672$	0	0
$673$	16.8957	0.651281	0.325640	−	0.945494i	$-0.394420\pi$
$673$	16.8957	0.651281	0.325640	+	0.945494i	$0.394420\pi$
$674$	25.7988	0.993734
$675$	0	0
$676$	−13.0000	−0.500000
$677$	−13.1907	−0.506958	−0.253479	−	0.967341i	$-0.581575\pi$
$677$	−13.1907	−0.506958	−0.253479	+	0.967341i	$0.581575\pi$
$678$	0	0
$679$	0	0
$680$	22.3473	0.856978
$681$	0	0
$682$	8.96523	0.343296
$683$	−9.83663	−0.376388	−0.188194	−	0.982132i	$-0.560263\pi$
$683$	−9.83663	−0.376388	−0.188194	+	0.982132i	$0.560263\pi$
$684$	0	0
$685$	−15.7428	−0.601502
$686$	0	0
$687$	0	0
$688$	11.4478	0.436445
$689$	0	0
$690$	0	0
$691$	−49.9895	−1.90169	−0.950845	−	0.309667i	$-0.899782\pi$
$691$	−49.9895	−1.90169	−0.950845	+	0.309667i	$0.899782\pi$
$692$	−16.9652	−0.644921
$693$	0	0
$694$	20.7671	0.788308
$695$	36.8336	1.39718
$696$	0	0
$697$	−40.5734	−1.53683
$698$	9.51364	0.360097
$699$	0	0
$700$	0	0
$701$	17.7050	0.668710	0.334355	−	0.942447i	$-0.391482\pi$
$701$	17.7050	0.668710	0.334355	+	0.942447i	$0.391482\pi$
$702$	0	0
$703$	−67.9547	−2.56296
$704$	1.00000	0.0376889
$705$	0	0
$706$	−12.5764	−0.473320
$707$	0	0
$708$	0	0
$709$	−31.9547	−1.20008	−0.600042	−	0.799968i	$-0.704850\pi$
$709$	−31.9547	−1.20008	−0.600042	+	0.799968i	$0.704850\pi$
$710$	−20.8957	−0.784201
$711$	0	0
$712$	1.61121	0.0603825
$713$	−47.4100	−1.77552
$714$	0	0
$715$	0	0
$716$	12.6703	0.473509
$717$	0	0
$718$	−0.0695483	−0.00259552
$719$	−21.7951	−0.812820	−0.406410	−	0.913691i	$-0.633220\pi$
$719$	−21.7951	−0.812820	−0.406410	+	0.913691i	$0.633220\pi$
$720$	0	0
$721$	0	0
$722$	−23.0243	−0.856875
$723$	0	0
$724$	−2.57643	−0.0957523
$725$	−14.7263	−0.546920
$726$	0	0
$727$	−4.84714	−0.179770	−0.0898852	−	0.995952i	$-0.528650\pi$
$727$	−4.84714	−0.179770	−0.0898852	+	0.995952i	$0.528650\pi$
$728$	0	0
$729$	0	0
$730$	−35.2845	−1.30594
$731$	−91.1846	−3.37259
$732$	0	0
$733$	12.2292	0.451695	0.225847	−	0.974163i	$-0.427485\pi$
$733$	12.2292	0.451695	0.225847	+	0.974163i	$0.427485\pi$
$734$	31.5417	1.16422
$735$	0	0
$736$	−5.28822	−0.194926
$737$	−7.09382	−0.261304
$738$	0	0
$739$	−1.03477	−0.0380648	−0.0190324	−	0.999819i	$-0.506059\pi$
$739$	−1.03477	−0.0380648	−0.0190324	+	0.999819i	$0.506059\pi$
$740$	29.4100	1.08113
$741$	0	0
$742$	0	0
$743$	23.9925	0.880200	0.440100	−	0.897949i	$-0.354943\pi$
$743$	23.9925	0.880200	0.440100	+	0.897949i	$0.354943\pi$
$744$	0	0
$745$	−46.9652	−1.72067
$746$	−15.3125	−0.560630
$747$	0	0
$748$	−7.96523	−0.291238
$749$	0	0
$750$	0	0
$751$	20.5069	0.748307	0.374153	−	0.927367i	$-0.377933\pi$
$751$	20.5069	0.748307	0.374153	+	0.927367i	$0.377933\pi$
$752$	−0.322990	−0.0117782
$753$	0	0
$754$	0	0
$755$	30.5794	1.11290
$756$	0	0
$757$	−16.8019	−0.610674	−0.305337	−	0.952244i	$-0.598769\pi$
$757$	−16.8019	−0.610674	−0.305337	+	0.952244i	$0.598769\pi$
$758$	−16.0590	−0.583291
$759$	0	0
$760$	18.1876	0.659735
$761$	14.5099	0.525983	0.262992	−	0.964798i	$-0.415291\pi$
$761$	14.5099	0.525983	0.262992	+	0.964798i	$0.415291\pi$
$762$	0	0
$763$	0	0
$764$	2.38879	0.0864235
$765$	0	0
$766$	−0.739798	−0.0267300
$767$	0	0
$768$	0	0
$769$	3.35402	0.120949	0.0604745	−	0.998170i	$-0.480739\pi$
$769$	3.35402	0.120949	0.0604745	+	0.998170i	$0.480739\pi$
$770$	0	0
$771$	0	0
$772$	−14.5764	−0.524617
$773$	−1.84038	−0.0661938	−0.0330969	−	0.999452i	$-0.510537\pi$
$773$	−1.84038	−0.0661938	−0.0330969	+	0.999452i	$0.510537\pi$
$774$	0	0
$775$	−25.7428	−0.924709
$776$	−7.00000	−0.251285
$777$	0	0
$778$	3.58319	0.128464
$779$	−33.0213	−1.18311
$780$	0	0
$781$	7.44784	0.266505
$782$	42.1218	1.50627
$783$	0	0
$784$	0	0
$785$	4.97875	0.177699
$786$	0	0
$787$	11.5869	0.413030	0.206515	−	0.978443i	$-0.433788\pi$
$787$	11.5869	0.413030	0.206515	+	0.978443i	$0.433788\pi$
$788$	12.4826	0.444675
$789$	0	0
$790$	−19.0938	−0.679328
$791$	0	0
$792$	0	0
$793$	0	0
$794$	25.1286	0.891780
$795$	0	0
$796$	−14.1876	−0.502867
$797$	48.4653	1.71673	0.858365	−	0.513039i	$-0.171480\pi$
$797$	48.4653	1.71673	0.858365	+	0.513039i	$0.171480\pi$
$798$	0	0
$799$	2.57269	0.0910151
$800$	−2.87141	−0.101520
$801$	0	0
$802$	−3.86839	−0.136598
$803$	12.5764	0.443813
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	10.7398	0.377825
$809$	2.76708	0.0972855	0.0486428	−	0.998816i	$-0.484510\pi$
$809$	2.76708	0.0972855	0.0486428	+	0.998816i	$0.484510\pi$
$810$	0	0
$811$	4.89568	0.171910	0.0859552	−	0.996299i	$-0.472606\pi$
$811$	4.89568	0.171910	0.0859552	+	0.996299i	$0.472606\pi$
$812$	0	0
$813$	0	0
$814$	−10.4826	−0.367415
$815$	−11.3881	−0.398907
$816$	0	0
$817$	−74.2119	−2.59635
$818$	−4.64598	−0.162443
$819$	0	0
$820$	14.2912	0.499071
$821$	−3.54915	−0.123866	−0.0619330	−	0.998080i	$-0.519727\pi$
$821$	−3.54915	−0.123866	−0.0619330	+	0.998080i	$0.519727\pi$
$822$	0	0
$823$	32.5069	1.13312	0.566559	−	0.824021i	$-0.308274\pi$
$823$	32.5069	1.13312	0.566559	+	0.824021i	$0.308274\pi$
$824$	−2.96523	−0.103299
$825$	0	0
$826$	0	0
$827$	−19.5447	−0.679635	−0.339817	−	0.940491i	$-0.610365\pi$
$827$	−19.5447	−0.679635	−0.339817	+	0.940491i	$0.610365\pi$
$828$	0	0
$829$	25.1907	0.874908	0.437454	−	0.899241i	$-0.355880\pi$
$829$	25.1907	0.874908	0.437454	+	0.899241i	$0.355880\pi$
$830$	42.3473	1.46989
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−38.9092	−1.34651
$836$	−6.48261	−0.224206
$837$	0	0
$838$	−19.5174	−0.674217
$839$	50.4033	1.74011	0.870057	−	0.492950i	$-0.164082\pi$
$839$	50.4033	1.74011	0.870057	+	0.492950i	$0.164082\pi$
$840$	0	0
$841$	−2.69754	−0.0930185
$842$	−15.5174	−0.534764
$843$	0	0
$844$	−20.8336	−0.717123
$845$	−36.4728	−1.25470
$846$	0	0
$847$	0	0
$848$	−8.00000	−0.274721
$849$	0	0
$850$	22.8714	0.784483
$851$	55.4343	1.90026
$852$	0	0
$853$	−16.8752	−0.577794	−0.288897	−	0.957360i	$-0.593289\pi$
$853$	−16.8752	−0.577794	−0.288897	+	0.957360i	$0.593289\pi$
$854$	0	0
$855$	0	0
$856$	4.83663	0.165313
$857$	−43.3057	−1.47930	−0.739648	−	0.672994i	$-0.765008\pi$
$857$	−43.3057	−1.47930	−0.739648	+	0.672994i	$0.765008\pi$
$858$	0	0
$859$	3.61422	0.123316	0.0616578	−	0.998097i	$-0.480361\pi$
$859$	3.61422	0.123316	0.0616578	+	0.998097i	$0.480361\pi$
$860$	32.1181	1.09522
$861$	0	0
$862$	−14.3888	−0.490084
$863$	−4.41681	−0.150350	−0.0751750	−	0.997170i	$-0.523952\pi$
$863$	−4.41681	−0.150350	−0.0751750	+	0.997170i	$0.523952\pi$
$864$	0	0
$865$	−47.5977	−1.61837
$866$	−2.03477	−0.0691444
$867$	0	0
$868$	0	0
$869$	6.80560	0.230864
$870$	0	0
$871$	0	0
$872$	10.1596	0.344048
$873$	0	0
$874$	34.2815	1.15959
$875$	0	0
$876$	0	0
$877$	44.2777	1.49515	0.747576	−	0.664176i	$-0.231218\pi$
$877$	44.2777	1.49515	0.747576	+	0.664176i	$0.231218\pi$
$878$	−3.22616	−0.108877
$879$	0	0
$880$	2.80560	0.0945769
$881$	18.1876	0.612757	0.306379	−	0.951910i	$-0.400883\pi$
$881$	18.1876	0.612757	0.306379	+	0.951910i	$0.400883\pi$
$882$	0	0
$883$	−4.57945	−0.154111	−0.0770553	−	0.997027i	$-0.524552\pi$
$883$	−4.57945	−0.154111	−0.0770553	+	0.997027i	$0.524552\pi$
$884$	0	0
$885$	0	0
$886$	−22.0938	−0.742256
$887$	0.576432	0.0193547	0.00967734	−	0.999953i	$-0.496920\pi$
$887$	0.576432	0.0193547	0.00967734	+	0.999953i	$0.496920\pi$
$888$	0	0
$889$	0	0
$890$	4.52040	0.151524
$891$	0	0
$892$	12.5764	0.421090
$893$	2.09382	0.0700670
$894$	0	0
$895$	35.5477	1.18823
$896$	0	0
$897$	0	0
$898$	22.2497	0.742482
$899$	45.9790	1.53349
$900$	0	0
$901$	63.7218	2.12288
$902$	−5.09382	−0.169606
$903$	0	0
$904$	10.9652	0.364698
$905$	−7.22844	−0.240282
$906$	0	0
$907$	−22.0590	−0.732459	−0.366229	−	0.930525i	$-0.619351\pi$
$907$	−22.0590	−0.732459	−0.366229	+	0.930525i	$0.619351\pi$
$908$	−20.0590	−0.665683
$909$	0	0
$910$	0	0
$911$	−9.93420	−0.329135	−0.164567	−	0.986366i	$-0.552623\pi$
$911$	−9.93420	−0.329135	−0.164567	+	0.986366i	$0.552623\pi$
$912$	0	0
$913$	−15.0938	−0.499532
$914$	18.6385	0.616507
$915$	0	0
$916$	−22.5764	−0.745946
$917$	0	0
$918$	0	0
$919$	44.6347	1.47236	0.736182	−	0.676783i	$-0.236627\pi$
$919$	44.6347	1.47236	0.736182	+	0.676783i	$0.236627\pi$
$920$	−14.8366	−0.489149
$921$	0	0
$922$	−20.9895	−0.691252
$923$	0	0
$924$	0	0
$925$	30.0999	0.989677
$926$	5.35402	0.175944
$927$	0	0
$928$	5.12859	0.168354
$929$	42.5764	1.39689	0.698444	−	0.715665i	$-0.253876\pi$
$929$	42.5764	1.39689	0.698444	+	0.715665i	$0.253876\pi$
$930$	0	0
$931$	0	0
$932$	0.742815	0.0243317
$933$	0	0
$934$	26.0243	0.851540
$935$	−22.3473	−0.730834
$936$	0	0
$937$	−36.2572	−1.18447	−0.592235	−	0.805765i	$-0.701754\pi$
$937$	−36.2572	−1.18447	−0.592235	+	0.805765i	$0.701754\pi$
$938$	0	0
$939$	0	0
$940$	−0.906181	−0.0295564
$941$	−2.68377	−0.0874884	−0.0437442	−	0.999043i	$-0.513929\pi$
$941$	−2.68377	−0.0874884	−0.0437442	+	0.999043i	$0.513929\pi$
$942$	0	0
$943$	26.9372	0.877196
$944$	0.871407	0.0283619
$945$	0	0
$946$	−11.4478	−0.372201
$947$	−49.5039	−1.60866	−0.804330	−	0.594183i	$-0.797475\pi$
$947$	−49.5039	−1.60866	−0.804330	+	0.594183i	$0.797475\pi$
$948$	0	0
$949$	0	0
$950$	18.6142	0.603925
$951$	0	0
$952$	0	0
$953$	−15.8019	−0.511872	−0.255936	−	0.966694i	$-0.582384\pi$
$953$	−15.8019	−0.511872	−0.255936	+	0.966694i	$0.582384\pi$
$954$	0	0
$955$	6.70201	0.216872
$956$	−4.31925	−0.139694
$957$	0	0
$958$	−3.61121	−0.116673
$959$	0	0
$960$	0	0
$961$	49.3753	1.59275
$962$	0	0
$963$	0	0
$964$	−18.1876	−0.585784
$965$	−40.8957	−1.31648
$966$	0	0
$967$	−12.6497	−0.406788	−0.203394	−	0.979097i	$-0.565197\pi$
$967$	−12.6497	−0.406788	−0.203394	+	0.979097i	$0.565197\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−19.6392	−0.630577
$971$	−49.5342	−1.58963	−0.794814	−	0.606854i	$-0.792431\pi$
$971$	−49.5342	−1.58963	−0.794814	+	0.606854i	$0.792431\pi$
$972$	0	0
$973$	0	0
$974$	8.06206	0.258325
$975$	0	0
$976$	2.15962	0.0691278
$977$	−34.4448	−1.10199	−0.550994	−	0.834509i	$-0.685751\pi$
$977$	−34.4448	−1.10199	−0.550994	+	0.834509i	$0.685751\pi$
$978$	0	0
$979$	−1.61121	−0.0514944
$980$	0	0
$981$	0	0
$982$	20.9062	0.667143
$983$	3.35776	0.107096	0.0535480	−	0.998565i	$-0.482947\pi$
$983$	3.35776	0.107096	0.0535480	+	0.998565i	$0.482947\pi$
$984$	0	0
$985$	35.0213	1.11587
$986$	−40.8504	−1.30094
$987$	0	0
$988$	0	0
$989$	60.5386	1.92502
$990$	0	0
$991$	−45.8474	−1.45639	−0.728195	−	0.685370i	$-0.759641\pi$
$991$	−45.8474	−1.45639	−0.728195	+	0.685370i	$0.759641\pi$
$992$	8.96523	0.284646
$993$	0	0
$994$	0	0
$995$	−39.8049	−1.26190
$996$	0	0
$997$	−43.8609	−1.38909	−0.694544	−	0.719450i	$-0.744394\pi$
$997$	−43.8609	−1.38909	−0.694544	+	0.719450i	$0.744394\pi$
$998$	−4.96523	−0.157171
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.dv.1.3		3
3.2	odd	2		3234.2.a.bf.1.1		3
7.2	even	3		1386.2.k.v.991.1		6
7.4	even	3		1386.2.k.v.793.1		6
7.6	odd	2		9702.2.a.dw.1.1		3
21.2	odd	6		462.2.i.g.67.3	✓	6
21.11	odd	6		462.2.i.g.331.3	yes	6
21.20	even	2		3234.2.a.bh.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.i.g.67.3	✓	6	21.2	odd	6
462.2.i.g.331.3	yes	6	21.11	odd	6
1386.2.k.v.793.1		6	7.4	even	3
1386.2.k.v.991.1		6	7.2	even	3
3234.2.a.bf.1.1		3	3.2	odd	2
3234.2.a.bh.1.3		3	21.20	even	2
9702.2.a.dv.1.3		3	1.1	even	1	trivial
9702.2.a.dw.1.1		3	7.6	odd	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.80560 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.80560 q^{5} -1.00000 q^{8} -2.80560 q^{10} +1.00000 q^{11} +1.00000 q^{16} -7.96523 q^{17} -6.48261 q^{19} +2.80560 q^{20} -1.00000 q^{22} +5.28822 q^{23} +2.87141 q^{25} -5.12859 q^{29} -8.96523 q^{31} -1.00000 q^{32} +7.96523 q^{34} +10.4826 q^{37} +6.48261 q^{38} -2.80560 q^{40} +5.09382 q^{41} +11.4478 q^{43} +1.00000 q^{44} -5.28822 q^{46} -0.322990 q^{47} -2.87141 q^{50} -8.00000 q^{53} +2.80560 q^{55} +5.12859 q^{58} +0.871407 q^{59} +2.15962 q^{61} +8.96523 q^{62} +1.00000 q^{64} -7.09382 q^{67} -7.96523 q^{68} +7.44784 q^{71} +12.5764 q^{73} -10.4826 q^{74} -6.48261 q^{76} +6.80560 q^{79} +2.80560 q^{80} -5.09382 q^{82} -15.0938 q^{83} -22.3473 q^{85} -11.4478 q^{86} -1.00000 q^{88} -1.61121 q^{89} +5.28822 q^{92} +0.322990 q^{94} -18.1876 q^{95} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} + 3 q^{11} + 3 q^{16} - 3 q^{17} - 9 q^{19} - 3 q^{22} - 3 q^{23} + 15 q^{25} - 9 q^{29} - 6 q^{31} - 3 q^{32} + 3 q^{34} + 21 q^{37} + 9 q^{38} - 12 q^{41} + 3 q^{43} + 3 q^{44} + 3 q^{46} - 3 q^{47} - 15 q^{50} - 24 q^{53} + 9 q^{58} + 9 q^{59} - 6 q^{61} + 6 q^{62} + 3 q^{64} + 6 q^{67} - 3 q^{68} - 9 q^{71} - 21 q^{74} - 9 q^{76} + 12 q^{79} + 12 q^{82} - 18 q^{83} - 3 q^{86} - 3 q^{88} + 12 q^{89} - 3 q^{92} + 3 q^{94} + 21 q^{97}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8 + 3 * q^11 + 3 * q^16 - 3 * q^17 - 9 * q^19 - 3 * q^22 - 3 * q^23 + 15 * q^25 - 9 * q^29 - 6 * q^31 - 3 * q^32 + 3 * q^34 + 21 * q^37 + 9 * q^38 - 12 * q^41 + 3 * q^43 + 3 * q^44 + 3 * q^46 - 3 * q^47 - 15 * q^50 - 24 * q^53 + 9 * q^58 + 9 * q^59 - 6 * q^61 + 6 * q^62 + 3 * q^64 + 6 * q^67 - 3 * q^68 - 9 * q^71 - 21 * q^74 - 9 * q^76 + 12 * q^79 + 12 * q^82 - 18 * q^83 - 3 * q^86 - 3 * q^88 + 12 * q^89 - 3 * q^92 + 3 * q^94 + 21 * q^97

Introduction

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