LMFDB - Embedded newform 9702.2.a.dz.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:	$0$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3234)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.74912$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.47363 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -2.47363 q^{5} -1.00000 q^{8} +2.47363 q^{10} -1.00000 q^{11} -3.03127 q^{13} +1.00000 q^{16} -6.64167 q^{17} -0.557647 q^{19} -2.47363 q^{20} +1.00000 q^{22} +2.66981 q^{23} +1.11882 q^{25} +3.03127 q^{26} -3.77568 q^{29} -2.00666 q^{31} -1.00000 q^{32} +6.64167 q^{34} -0.158619 q^{37} +0.557647 q^{38} +2.47363 q^{40} -5.93207 q^{41} +4.32666 q^{43} -1.00000 q^{44} -2.66981 q^{46} -7.38607 q^{47} -1.11882 q^{50} -3.03127 q^{52} -2.83785 q^{53} +2.47363 q^{55} +3.77568 q^{58} -1.48881 q^{59} -3.19932 q^{61} +2.00666 q^{62} +1.00000 q^{64} +7.49824 q^{65} -11.2739 q^{67} -6.64167 q^{68} +11.0098 q^{71} -3.18323 q^{73} +0.158619 q^{74} -0.557647 q^{76} -6.95668 q^{79} -2.47363 q^{80} +5.93207 q^{82} +4.59744 q^{83} +16.4290 q^{85} -4.32666 q^{86} +1.00000 q^{88} -6.62558 q^{89} +2.66981 q^{92} +7.38607 q^{94} +1.37941 q^{95} +1.17656 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 - 4 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{8} + 4 q^{10} - 4 q^{11} + 8 q^{13} + 4 q^{16} - 4 q^{17} + 12 q^{19} - 4 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} - 8 q^{26} + 8 q^{29} + 4 q^{31} - 4 q^{32} + 4 q^{34} + 8 q^{37} - 12 q^{38} + 4 q^{40} - 12 q^{41} - 8 q^{43} - 4 q^{44} - 8 q^{46} - 4 q^{47} - 4 q^{50} + 8 q^{52} + 8 q^{53} + 4 q^{55} - 8 q^{58} + 24 q^{61} - 4 q^{62} + 4 q^{64} + 16 q^{65} - 8 q^{67} - 4 q^{68} - 8 q^{71} + 4 q^{73} - 8 q^{74} + 12 q^{76} - 8 q^{79} - 4 q^{80} + 12 q^{82} - 4 q^{83} - 8 q^{85} + 8 q^{86} + 4 q^{88} - 24 q^{89} + 8 q^{92} + 4 q^{94} - 8 q^{95}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 - 4 * q^8 + 4 * q^10 - 4 * q^11 + 8 * q^13 + 4 * q^16 - 4 * q^17 + 12 * q^19 - 4 * q^20 + 4 * q^22 + 8 * q^23 + 4 * q^25 - 8 * q^26 + 8 * q^29 + 4 * q^31 - 4 * q^32 + 4 * q^34 + 8 * q^37 - 12 * q^38 + 4 * q^40 - 12 * q^41 - 8 * q^43 - 4 * q^44 - 8 * q^46 - 4 * q^47 - 4 * q^50 + 8 * q^52 + 8 * q^53 + 4 * q^55 - 8 * q^58 + 24 * q^61 - 4 * q^62 + 4 * q^64 + 16 * q^65 - 8 * q^67 - 4 * q^68 - 8 * q^71 + 4 * q^73 - 8 * q^74 + 12 * q^76 - 8 * q^79 - 4 * q^80 + 12 * q^82 - 4 * q^83 - 8 * q^85 + 8 * q^86 + 4 * q^88 - 24 * q^89 + 8 * q^92 + 4 * q^94 - 8 * q^95

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
$5$	−2.47363	−1.10624	−0.553120	−	0.833102i	$-0.686563\pi$
$5$	−2.47363	−1.10624	−0.553120	+	0.833102i	$0.686563\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.47363	0.782229
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.03127	−0.840724	−0.420362	−	0.907357i	$-0.638097\pi$
$13$	−3.03127	−0.840724	−0.420362	+	0.907357i	$0.638097\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.64167	−1.61084	−0.805421	−	0.592704i	$-0.798061\pi$
$17$	−6.64167	−1.61084	−0.805421	+	0.592704i	$0.798061\pi$
$18$	0	0
$19$	−0.557647	−0.127933	−0.0639665	−	0.997952i	$-0.520375\pi$
$19$	−0.557647	−0.127933	−0.0639665	+	0.997952i	$0.520375\pi$
$20$	−2.47363	−0.553120
$21$	0	0
$22$	1.00000	0.213201
$23$	2.66981	0.556693	0.278347	−	0.960481i	$-0.410214\pi$
$23$	2.66981	0.556693	0.278347	+	0.960481i	$0.410214\pi$
$24$	0	0
$25$	1.11882	0.223765
$26$	3.03127	0.594482
$27$	0	0
$28$	0	0
$29$	−3.77568	−0.701126	−0.350563	−	0.936539i	$-0.614010\pi$
$29$	−3.77568	−0.701126	−0.350563	+	0.936539i	$0.614010\pi$
$30$	0	0
$31$	−2.00666	−0.360407	−0.180204	−	0.983629i	$-0.557676\pi$
$31$	−2.00666	−0.360407	−0.180204	+	0.983629i	$0.557676\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.64167	1.13904
$35$	0	0
$36$	0	0
$37$	−0.158619	−0.0260768	−0.0130384	−	0.999915i	$-0.504150\pi$
$37$	−0.158619	−0.0260768	−0.0130384	+	0.999915i	$0.504150\pi$
$38$	0.557647	0.0904623
$39$	0	0
$40$	2.47363	0.391115
$41$	−5.93207	−0.926433	−0.463217	−	0.886245i	$-0.653305\pi$
$41$	−5.93207	−0.926433	−0.463217	+	0.886245i	$0.653305\pi$
$42$	0	0
$43$	4.32666	0.659810	0.329905	−	0.944014i	$-0.392983\pi$
$43$	4.32666	0.659810	0.329905	+	0.944014i	$0.392983\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−2.66981	−0.393642
$47$	−7.38607	−1.07737	−0.538685	−	0.842507i	$-0.681079\pi$
$47$	−7.38607	−1.07737	−0.538685	+	0.842507i	$0.681079\pi$
$48$	0	0
$49$	0	0
$50$	−1.11882	−0.158226
$51$	0	0
$52$	−3.03127	−0.420362
$53$	−2.83785	−0.389809	−0.194904	−	0.980822i	$-0.562440\pi$
$53$	−2.83785	−0.389809	−0.194904	+	0.980822i	$0.562440\pi$
$54$	0	0
$55$	2.47363	0.333544
$56$	0	0
$57$	0	0
$58$	3.77568	0.495771
$59$	−1.48881	−0.193827	−0.0969134	−	0.995293i	$-0.530897\pi$
$59$	−1.48881	−0.193827	−0.0969134	+	0.995293i	$0.530897\pi$
$60$	0	0
$61$	−3.19932	−0.409630	−0.204815	−	0.978801i	$-0.565659\pi$
$61$	−3.19932	−0.409630	−0.204815	+	0.978801i	$0.565659\pi$
$62$	2.00666	0.254847
$63$	0	0
$64$	1.00000	0.125000
$65$	7.49824	0.930042
$66$	0	0
$67$	−11.2739	−1.37733	−0.688664	−	0.725081i	$-0.741802\pi$
$67$	−11.2739	−1.37733	−0.688664	+	0.725081i	$0.741802\pi$
$68$	−6.64167	−0.805421
$69$	0	0
$70$	0	0
$71$	11.0098	1.30662	0.653311	−	0.757089i	$-0.273379\pi$
$71$	11.0098	1.30662	0.653311	+	0.757089i	$0.273379\pi$
$72$	0	0
$73$	−3.18323	−0.372569	−0.186284	−	0.982496i	$-0.559645\pi$
$73$	−3.18323	−0.372569	−0.186284	+	0.982496i	$0.559645\pi$
$74$	0.158619	0.0184391
$75$	0	0
$76$	−0.557647	−0.0639665
$77$	0	0
$78$	0	0
$79$	−6.95668	−0.782687	−0.391344	−	0.920245i	$-0.627990\pi$
$79$	−6.95668	−0.782687	−0.391344	+	0.920245i	$0.627990\pi$
$80$	−2.47363	−0.276560
$81$	0	0
$82$	5.93207	0.655087
$83$	4.59744	0.504635	0.252317	−	0.967645i	$-0.418807\pi$
$83$	4.59744	0.504635	0.252317	+	0.967645i	$0.418807\pi$
$84$	0	0
$85$	16.4290	1.78198
$86$	−4.32666	−0.466556
$87$	0	0
$88$	1.00000	0.106600
$89$	−6.62558	−0.702310	−0.351155	−	0.936317i	$-0.614211\pi$
$89$	−6.62558	−0.702310	−0.351155	+	0.936317i	$0.614211\pi$
$90$	0	0
$91$	0	0
$92$	2.66981	0.278347
$93$	0	0
$94$	7.38607	0.761815
$95$	1.37941	0.141525
$96$	0	0
$97$	1.17656	0.119462	0.0597310	−	0.998215i	$-0.480976\pi$
$97$	1.17656	0.119462	0.0597310	+	0.998215i	$0.480976\pi$
$98$	0	0
$99$	0	0
$100$	1.11882	0.111882
$101$	−14.5295	−1.44574	−0.722870	−	0.690984i	$-0.757177\pi$
$101$	−14.5295	−1.44574	−0.722870	+	0.690984i	$0.757177\pi$
$102$	0	0
$103$	12.2239	1.20446	0.602230	−	0.798323i	$-0.294279\pi$
$103$	12.2239	1.20446	0.602230	+	0.798323i	$0.294279\pi$
$104$	3.03127	0.297241
$105$	0	0
$106$	2.83785	0.275636
$107$	2.81900	0.272523	0.136262	−	0.990673i	$-0.456491\pi$
$107$	2.81900	0.272523	0.136262	+	0.990673i	$0.456491\pi$
$108$	0	0
$109$	−20.5478	−1.96812	−0.984062	−	0.177823i	$-0.943095\pi$
$109$	−20.5478	−1.96812	−0.984062	+	0.177823i	$0.943095\pi$
$110$	−2.47363	−0.235851
$111$	0	0
$112$	0	0
$113$	−11.7194	−1.10247	−0.551234	−	0.834351i	$-0.685843\pi$
$113$	−11.7194	−1.10247	−0.551234	+	0.834351i	$0.685843\pi$
$114$	0	0
$115$	−6.60411	−0.615836
$116$	−3.77568	−0.350563
$117$	0	0
$118$	1.48881	0.137056
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	3.19932	0.289652
$123$	0	0
$124$	−2.00666	−0.180204
$125$	9.60058	0.858702
$126$	0	0
$127$	0.0397948	0.00353121	0.00176561	−	0.999998i	$-0.499438\pi$
$127$	0.0397948	0.00353121	0.00176561	+	0.999998i	$0.499438\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−7.49824	−0.657639
$131$	−9.60097	−0.838841	−0.419420	−	0.907792i	$-0.637767\pi$
$131$	−9.60097	−0.838841	−0.419420	+	0.907792i	$0.637767\pi$
$132$	0	0
$133$	0	0
$134$	11.2739	0.973918
$135$	0	0
$136$	6.64167	0.569518
$137$	18.6667	1.59480	0.797400	−	0.603451i	$-0.206208\pi$
$137$	18.6667	1.59480	0.797400	+	0.603451i	$0.206208\pi$
$138$	0	0
$139$	8.33333	0.706823	0.353412	−	0.935468i	$-0.385021\pi$
$139$	8.33333	0.706823	0.353412	+	0.935468i	$0.385021\pi$
$140$	0	0
$141$	0	0
$142$	−11.0098	−0.923922
$143$	3.03127	0.253488
$144$	0	0
$145$	9.33962	0.775613
$146$	3.18323	0.263446
$147$	0	0
$148$	−0.158619	−0.0130384
$149$	15.2341	1.24803	0.624014	−	0.781413i	$-0.285501\pi$
$149$	15.2341	1.24803	0.624014	+	0.781413i	$0.285501\pi$
$150$	0	0
$151$	−8.31724	−0.676847	−0.338424	−	0.940994i	$-0.609894\pi$
$151$	−8.31724	−0.676847	−0.338424	+	0.940994i	$0.609894\pi$
$152$	0.557647	0.0452312
$153$	0	0
$154$	0	0
$155$	4.96374	0.398697
$156$	0	0
$157$	−9.13401	−0.728973	−0.364487	−	0.931209i	$-0.618756\pi$
$157$	−9.13401	−0.728973	−0.364487	+	0.931209i	$0.618756\pi$
$158$	6.95668	0.553443
$159$	0	0
$160$	2.47363	0.195557
$161$	0	0
$162$	0	0
$163$	8.33609	0.652933	0.326466	−	0.945209i	$-0.394142\pi$
$163$	8.33609	0.652933	0.326466	+	0.945209i	$0.394142\pi$
$164$	−5.93207	−0.463217
$165$	0	0
$166$	−4.59744	−0.356831
$167$	22.1610	1.71487	0.857434	−	0.514594i	$-0.172057\pi$
$167$	22.1610	1.71487	0.857434	+	0.514594i	$0.172057\pi$
$168$	0	0
$169$	−3.81138	−0.293183
$170$	−16.4290	−1.26005
$171$	0	0
$172$	4.32666	0.329905
$173$	−20.9656	−1.59398	−0.796991	−	0.603991i	$-0.793576\pi$
$173$	−20.9656	−1.59398	−0.796991	+	0.603991i	$0.793576\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	6.62558	0.496608
$179$	12.3759	0.925017	0.462508	−	0.886615i	$-0.346950\pi$
$179$	12.3759	0.925017	0.462508	+	0.886615i	$0.346950\pi$
$180$	0	0
$181$	20.5595	1.52817	0.764087	−	0.645113i	$-0.223190\pi$
$181$	20.5595	1.52817	0.764087	+	0.645113i	$0.223190\pi$
$182$	0	0
$183$	0	0
$184$	−2.66981	−0.196821
$185$	0.392364	0.0288472
$186$	0	0
$187$	6.64167	0.485687
$188$	−7.38607	−0.538685
$189$	0	0
$190$	−1.37941	−0.100073
$191$	20.3267	1.47079	0.735393	−	0.677641i	$-0.236998\pi$
$191$	20.3267	1.47079	0.735393	+	0.677641i	$0.236998\pi$
$192$	0	0
$193$	−13.0761	−0.941235	−0.470618	−	0.882337i	$-0.655969\pi$
$193$	−13.0761	−0.941235	−0.470618	+	0.882337i	$0.655969\pi$
$194$	−1.17656	−0.0844724
$195$	0	0
$196$	0	0
$197$	3.80159	0.270852	0.135426	−	0.990787i	$-0.456760\pi$
$197$	3.80159	0.270852	0.135426	+	0.990787i	$0.456760\pi$
$198$	0	0
$199$	−6.87098	−0.487071	−0.243536	−	0.969892i	$-0.578307\pi$
$199$	−6.87098	−0.487071	−0.243536	+	0.969892i	$0.578307\pi$
$200$	−1.11882	−0.0791128
$201$	0	0
$202$	14.5295	1.02229
$203$	0	0
$204$	0	0
$205$	14.6737	1.02486
$206$	−12.2239	−0.851681
$207$	0	0
$208$	−3.03127	−0.210181
$209$	0.557647	0.0385733
$210$	0	0
$211$	11.7984	0.812237	0.406119	−	0.913820i	$-0.366882\pi$
$211$	11.7984	0.812237	0.406119	+	0.913820i	$0.366882\pi$
$212$	−2.83785	−0.194904
$213$	0	0
$214$	−2.81900	−0.192703
$215$	−10.7025	−0.729907
$216$	0	0
$217$	0	0
$218$	20.5478	1.39167
$219$	0	0
$220$	2.47363	0.166772
$221$	20.1327	1.35427
$222$	0	0
$223$	−21.0031	−1.40647	−0.703237	−	0.710956i	$-0.748263\pi$
$223$	−21.0031	−1.40647	−0.703237	+	0.710956i	$0.748263\pi$
$224$	0	0
$225$	0	0
$226$	11.7194	0.779563
$227$	29.4884	1.95721	0.978607	−	0.205736i	$-0.0659590\pi$
$227$	29.4884	1.95721	0.978607	+	0.205736i	$0.0659590\pi$
$228$	0	0
$229$	−17.3254	−1.14489	−0.572446	−	0.819942i	$-0.694005\pi$
$229$	−17.3254	−1.14489	−0.572446	+	0.819942i	$0.694005\pi$
$230$	6.60411	0.435462
$231$	0	0
$232$	3.77568	0.247885
$233$	−29.4388	−1.92860	−0.964300	−	0.264812i	$-0.914690\pi$
$233$	−29.4388	−1.92860	−0.964300	+	0.264812i	$0.914690\pi$
$234$	0	0
$235$	18.2704	1.19183
$236$	−1.48881	−0.0969134
$237$	0	0
$238$	0	0
$239$	−24.2047	−1.56567	−0.782835	−	0.622229i	$-0.786227\pi$
$239$	−24.2047	−1.56567	−0.782835	+	0.622229i	$0.786227\pi$
$240$	0	0
$241$	−15.3083	−0.986096	−0.493048	−	0.870002i	$-0.664117\pi$
$241$	−15.3083	−0.986096	−0.493048	+	0.870002i	$0.664117\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−3.19932	−0.204815
$245$	0	0
$246$	0	0
$247$	1.69038	0.107556
$248$	2.00666	0.127423
$249$	0	0
$250$	−9.60058	−0.607194
$251$	−7.79453	−0.491986	−0.245993	−	0.969272i	$-0.579114\pi$
$251$	−7.79453	−0.491986	−0.245993	+	0.969272i	$0.579114\pi$
$252$	0	0
$253$	−2.66981	−0.167849
$254$	−0.0397948	−0.00249695
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.0973	−0.879368	−0.439684	−	0.898152i	$-0.644910\pi$
$257$	−14.0973	−0.879368	−0.439684	+	0.898152i	$0.644910\pi$
$258$	0	0
$259$	0	0
$260$	7.49824	0.465021
$261$	0	0
$262$	9.60097	0.593150
$263$	−4.61353	−0.284482	−0.142241	−	0.989832i	$-0.545431\pi$
$263$	−4.61353	−0.284482	−0.142241	+	0.989832i	$0.545431\pi$
$264$	0	0
$265$	7.01978	0.431222
$266$	0	0
$267$	0	0
$268$	−11.2739	−0.688664
$269$	3.18323	0.194085	0.0970424	−	0.995280i	$-0.469062\pi$
$269$	3.18323	0.194085	0.0970424	+	0.995280i	$0.469062\pi$
$270$	0	0
$271$	0.722930	0.0439149	0.0219574	−	0.999759i	$-0.493010\pi$
$271$	0.722930	0.0439149	0.0219574	+	0.999759i	$0.493010\pi$
$272$	−6.64167	−0.402710
$273$	0	0
$274$	−18.6667	−1.12769
$275$	−1.11882	−0.0674676
$276$	0	0
$277$	1.18490	0.0711938	0.0355969	−	0.999366i	$-0.488667\pi$
$277$	1.18490	0.0711938	0.0355969	+	0.999366i	$0.488667\pi$
$278$	−8.33333	−0.499800
$279$	0	0
$280$	0	0
$281$	−2.89097	−0.172461	−0.0862305	−	0.996275i	$-0.527482\pi$
$281$	−2.89097	−0.172461	−0.0862305	+	0.996275i	$0.527482\pi$
$282$	0	0
$283$	−5.32353	−0.316451	−0.158225	−	0.987403i	$-0.550577\pi$
$283$	−5.32353	−0.316451	−0.158225	+	0.987403i	$0.550577\pi$
$284$	11.0098	0.653311
$285$	0	0
$286$	−3.03127	−0.179243
$287$	0	0
$288$	0	0
$289$	27.1118	1.59481
$290$	−9.33962	−0.548441
$291$	0	0
$292$	−3.18323	−0.186284
$293$	−10.6413	−0.621670	−0.310835	−	0.950464i	$-0.600609\pi$
$293$	−10.6413	−0.621670	−0.310835	+	0.950464i	$0.600609\pi$
$294$	0	0
$295$	3.68276	0.214419
$296$	0.158619	0.00921955
$297$	0	0
$298$	−15.2341	−0.882489
$299$	−8.09292	−0.468025
$300$	0	0
$301$	0	0
$302$	8.31724	0.478603
$303$	0	0
$304$	−0.557647	−0.0319833
$305$	7.91391	0.453149
$306$	0	0
$307$	28.9733	1.65359	0.826797	−	0.562500i	$-0.190160\pi$
$307$	28.9733	1.65359	0.826797	+	0.562500i	$0.190160\pi$
$308$	0	0
$309$	0	0
$310$	−4.96374	−0.281921
$311$	−2.62019	−0.148578	−0.0742888	−	0.997237i	$-0.523669\pi$
$311$	−2.62019	−0.148578	−0.0742888	+	0.997237i	$0.523669\pi$
$312$	0	0
$313$	−31.7566	−1.79499	−0.897494	−	0.441027i	$-0.854614\pi$
$313$	−31.7566	−1.79499	−0.897494	+	0.441027i	$0.854614\pi$
$314$	9.13401	0.515462
$315$	0	0
$316$	−6.95668	−0.391344
$317$	24.8308	1.39464	0.697318	−	0.716762i	$-0.254377\pi$
$317$	24.8308	1.39464	0.697318	+	0.716762i	$0.254377\pi$
$318$	0	0
$319$	3.77568	0.211397
$320$	−2.47363	−0.138280
$321$	0	0
$322$	0	0
$323$	3.70371	0.206080
$324$	0	0
$325$	−3.39146	−0.188124
$326$	−8.33609	−0.461693
$327$	0	0
$328$	5.93207	0.327544
$329$	0	0
$330$	0	0
$331$	21.2669	1.16893	0.584466	−	0.811418i	$-0.301304\pi$
$331$	21.2669	1.16893	0.584466	+	0.811418i	$0.301304\pi$
$332$	4.59744	0.252317
$333$	0	0
$334$	−22.1610	−1.21260
$335$	27.8874	1.52365
$336$	0	0
$337$	23.3927	1.27428	0.637142	−	0.770747i	$-0.280117\pi$
$337$	23.3927	1.27428	0.637142	+	0.770747i	$0.280117\pi$
$338$	3.81138	0.207312
$339$	0	0
$340$	16.4290	0.890988
$341$	2.00666	0.108667
$342$	0	0
$343$	0	0
$344$	−4.32666	−0.233278
$345$	0	0
$346$	20.9656	1.12712
$347$	−6.60020	−0.354317	−0.177159	−	0.984182i	$-0.556691\pi$
$347$	−6.60020	−0.354317	−0.177159	+	0.984182i	$0.556691\pi$
$348$	0	0
$349$	5.06164	0.270944	0.135472	−	0.990781i	$-0.456745\pi$
$349$	5.06164	0.270944	0.135472	+	0.990781i	$0.456745\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	11.5166	0.612964	0.306482	−	0.951876i	$-0.400848\pi$
$353$	11.5166	0.612964	0.306482	+	0.951876i	$0.400848\pi$
$354$	0	0
$355$	−27.2341	−1.44544
$356$	−6.62558	−0.351155
$357$	0	0
$358$	−12.3759	−0.654086
$359$	10.5080	0.554593	0.277296	−	0.960784i	$-0.410562\pi$
$359$	10.5080	0.554593	0.277296	+	0.960784i	$0.410562\pi$
$360$	0	0
$361$	−18.6890	−0.983633
$362$	−20.5595	−1.08058
$363$	0	0
$364$	0	0
$365$	7.87412	0.412150
$366$	0	0
$367$	10.8976	0.568852	0.284426	−	0.958698i	$-0.408197\pi$
$367$	10.8976	0.568852	0.284426	+	0.958698i	$0.408197\pi$
$368$	2.66981	0.139173
$369$	0	0
$370$	−0.392364	−0.0203981
$371$	0	0
$372$	0	0
$373$	21.3459	1.10525	0.552624	−	0.833431i	$-0.313627\pi$
$373$	21.3459	1.10525	0.552624	+	0.833431i	$0.313627\pi$
$374$	−6.64167	−0.343433
$375$	0	0
$376$	7.38607	0.380908
$377$	11.4451	0.589453
$378$	0	0
$379$	−21.3066	−1.09445	−0.547225	−	0.836986i	$-0.684316\pi$
$379$	−21.3066	−1.09445	−0.547225	+	0.836986i	$0.684316\pi$
$380$	1.37941	0.0707623
$381$	0	0
$382$	−20.3267	−1.04000
$383$	34.6202	1.76901	0.884505	−	0.466531i	$-0.154497\pi$
$383$	34.6202	1.76901	0.884505	+	0.466531i	$0.154497\pi$
$384$	0	0
$385$	0	0
$386$	13.0761	0.665554
$387$	0	0
$388$	1.17656	0.0597310
$389$	9.68276	0.490936	0.245468	−	0.969405i	$-0.421058\pi$
$389$	9.68276	0.490936	0.245468	+	0.969405i	$0.421058\pi$
$390$	0	0
$391$	−17.7320	−0.896745
$392$	0	0
$393$	0	0
$394$	−3.80159	−0.191521
$395$	17.2082	0.865839
$396$	0	0
$397$	−11.8928	−0.596884	−0.298442	−	0.954428i	$-0.596467\pi$
$397$	−11.8928	−0.596884	−0.298442	+	0.954428i	$0.596467\pi$
$398$	6.87098	0.344411
$399$	0	0
$400$	1.11882	0.0559412
$401$	−33.9804	−1.69690	−0.848449	−	0.529277i	$-0.822463\pi$
$401$	−33.9804	−1.69690	−0.848449	+	0.529277i	$0.822463\pi$
$402$	0	0
$403$	6.08275	0.303003
$404$	−14.5295	−0.722870
$405$	0	0
$406$	0	0
$407$	0.158619	0.00786246
$408$	0	0
$409$	9.51305	0.470390	0.235195	−	0.971948i	$-0.424427\pi$
$409$	9.51305	0.470390	0.235195	+	0.971948i	$0.424427\pi$
$410$	−14.6737	−0.724683
$411$	0	0
$412$	12.2239	0.602230
$413$	0	0
$414$	0	0
$415$	−11.3724	−0.558247
$416$	3.03127	0.148620
$417$	0	0
$418$	−0.557647	−0.0272754
$419$	15.9249	0.777981	0.388990	−	0.921242i	$-0.372824\pi$
$419$	15.9249	0.777981	0.388990	+	0.921242i	$0.372824\pi$
$420$	0	0
$421$	3.55842	0.173427	0.0867133	−	0.996233i	$-0.472364\pi$
$421$	3.55842	0.173427	0.0867133	+	0.996233i	$0.472364\pi$
$422$	−11.7984	−0.574339
$423$	0	0
$424$	2.83785	0.137818
$425$	−7.43086	−0.360450
$426$	0	0
$427$	0	0
$428$	2.81900	0.136262
$429$	0	0
$430$	10.7025	0.516122
$431$	−20.2774	−0.976730	−0.488365	−	0.872639i	$-0.662407\pi$
$431$	−20.2774	−0.976730	−0.488365	+	0.872639i	$0.662407\pi$
$432$	0	0
$433$	27.7144	1.33187	0.665935	−	0.746010i	$-0.268033\pi$
$433$	27.7144	1.33187	0.665935	+	0.746010i	$0.268033\pi$
$434$	0	0
$435$	0	0
$436$	−20.5478	−0.984062
$437$	−1.48881	−0.0712195
$438$	0	0
$439$	4.85433	0.231685	0.115842	−	0.993268i	$-0.463043\pi$
$439$	4.85433	0.231685	0.115842	+	0.993268i	$0.463043\pi$
$440$	−2.47363	−0.117925
$441$	0	0
$442$	−20.1327	−0.957615
$443$	30.5269	1.45038	0.725188	−	0.688551i	$-0.241753\pi$
$443$	30.5269	1.45038	0.725188	+	0.688551i	$0.241753\pi$
$444$	0	0
$445$	16.3892	0.776923
$446$	21.0031	0.994527
$447$	0	0
$448$	0	0
$449$	30.3986	1.43460	0.717300	−	0.696764i	$-0.245378\pi$
$449$	30.3986	1.43460	0.717300	+	0.696764i	$0.245378\pi$
$450$	0	0
$451$	5.93207	0.279330
$452$	−11.7194	−0.551234
$453$	0	0
$454$	−29.4884	−1.38396
$455$	0	0
$456$	0	0
$457$	11.1020	0.519328	0.259664	−	0.965699i	$-0.416388\pi$
$457$	11.1020	0.519328	0.259664	+	0.965699i	$0.416388\pi$
$458$	17.3254	0.809561
$459$	0	0
$460$	−6.60411	−0.307918
$461$	−38.1693	−1.77772	−0.888861	−	0.458177i	$-0.848503\pi$
$461$	−38.1693	−1.77772	−0.888861	+	0.458177i	$0.848503\pi$
$462$	0	0
$463$	−31.4388	−1.46108	−0.730542	−	0.682867i	$-0.760733\pi$
$463$	−31.4388	−1.46108	−0.730542	+	0.682867i	$0.760733\pi$
$464$	−3.77568	−0.175281
$465$	0	0
$466$	29.4388	1.36373
$467$	11.9204	0.551611	0.275805	−	0.961213i	$-0.411055\pi$
$467$	11.9204	0.551611	0.275805	+	0.961213i	$0.411055\pi$
$468$	0	0
$469$	0	0
$470$	−18.2704	−0.842750
$471$	0	0
$472$	1.48881	0.0685281
$473$	−4.32666	−0.198940
$474$	0	0
$475$	−0.623909	−0.0286269
$476$	0	0
$477$	0	0
$478$	24.2047	1.10710
$479$	39.4155	1.80094	0.900470	−	0.434918i	$-0.143223\pi$
$479$	39.4155	1.80094	0.900470	+	0.434918i	$0.143223\pi$
$480$	0	0
$481$	0.480818	0.0219234
$482$	15.3083	0.697275
$483$	0	0
$484$	1.00000	0.0454545
$485$	−2.91038	−0.132154
$486$	0	0
$487$	23.1551	1.04926	0.524629	−	0.851331i	$-0.324204\pi$
$487$	23.1551	1.04926	0.524629	+	0.851331i	$0.324204\pi$
$488$	3.19932	0.144826
$489$	0	0
$490$	0	0
$491$	4.47530	0.201967	0.100984	−	0.994888i	$-0.467801\pi$
$491$	4.47530	0.201967	0.100984	+	0.994888i	$0.467801\pi$
$492$	0	0
$493$	25.0768	1.12940
$494$	−1.69038	−0.0760538
$495$	0	0
$496$	−2.00666	−0.0901019
$497$	0	0
$498$	0	0
$499$	−16.7727	−0.750850	−0.375425	−	0.926853i	$-0.622503\pi$
$499$	−16.7727	−0.750850	−0.375425	+	0.926853i	$0.622503\pi$
$500$	9.60058	0.429351
$501$	0	0
$502$	7.79453	0.347887
$503$	41.1653	1.83547	0.917734	−	0.397195i	$-0.130016\pi$
$503$	41.1653	1.83547	0.917734	+	0.397195i	$0.130016\pi$
$504$	0	0
$505$	35.9406	1.59933
$506$	2.66981	0.118687
$507$	0	0
$508$	0.0397948	0.00176561
$509$	−14.5987	−0.647077	−0.323538	−	0.946215i	$-0.604872\pi$
$509$	−14.5987	−0.647077	−0.323538	+	0.946215i	$0.604872\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	14.0973	0.621807
$515$	−30.2374	−1.33242
$516$	0	0
$517$	7.38607	0.324839
$518$	0	0
$519$	0	0
$520$	−7.49824	−0.328819
$521$	−17.0054	−0.745019	−0.372509	−	0.928028i	$-0.621503\pi$
$521$	−17.0054	−0.745019	−0.372509	+	0.928028i	$0.621503\pi$
$522$	0	0
$523$	32.8874	1.43807	0.719033	−	0.694976i	$-0.244585\pi$
$523$	32.8874	1.43807	0.719033	+	0.694976i	$0.244585\pi$
$524$	−9.60097	−0.419420
$525$	0	0
$526$	4.61353	0.201159
$527$	13.3276	0.580559
$528$	0	0
$529$	−15.8721	−0.690092
$530$	−7.01978	−0.304920
$531$	0	0
$532$	0	0
$533$	17.9817	0.778874
$534$	0	0
$535$	−6.97316	−0.301476
$536$	11.2739	0.486959
$537$	0	0
$538$	−3.18323	−0.137239
$539$	0	0
$540$	0	0
$541$	−3.11529	−0.133937	−0.0669685	−	0.997755i	$-0.521333\pi$
$541$	−3.11529	−0.133937	−0.0669685	+	0.997755i	$0.521333\pi$
$542$	−0.722930	−0.0310525
$543$	0	0
$544$	6.64167	0.284759
$545$	50.8276	2.17722
$546$	0	0
$547$	−4.58706	−0.196129	−0.0980643	−	0.995180i	$-0.531265\pi$
$547$	−4.58706	−0.196129	−0.0980643	+	0.995180i	$0.531265\pi$
$548$	18.6667	0.797400
$549$	0	0
$550$	1.11882	0.0477068
$551$	2.10550	0.0896972
$552$	0	0
$553$	0	0
$554$	−1.18490	−0.0503416
$555$	0	0
$556$	8.33333	0.353412
$557$	−18.4290	−0.780862	−0.390431	−	0.920632i	$-0.627674\pi$
$557$	−18.4290	−0.780862	−0.390431	+	0.920632i	$0.627674\pi$
$558$	0	0
$559$	−13.1153	−0.554718
$560$	0	0
$561$	0	0
$562$	2.89097	0.121948
$563$	−7.03903	−0.296660	−0.148330	−	0.988938i	$-0.547390\pi$
$563$	−7.03903	−0.296660	−0.148330	+	0.988938i	$0.547390\pi$
$564$	0	0
$565$	28.9894	1.21959
$566$	5.32353	0.223765
$567$	0	0
$568$	−11.0098	−0.461961
$569$	20.2913	0.850657	0.425328	−	0.905039i	$-0.360159\pi$
$569$	20.2913	0.850657	0.425328	+	0.905039i	$0.360159\pi$
$570$	0	0
$571$	10.8517	0.454131	0.227066	−	0.973879i	$-0.427087\pi$
$571$	10.8517	0.454131	0.227066	+	0.973879i	$0.427087\pi$
$572$	3.03127	0.126744
$573$	0	0
$574$	0	0
$575$	2.98705	0.124568
$576$	0	0
$577$	−15.0944	−0.628387	−0.314194	−	0.949359i	$-0.601734\pi$
$577$	−15.0944	−0.628387	−0.314194	+	0.949359i	$0.601734\pi$
$578$	−27.1118	−1.12770
$579$	0	0
$580$	9.33962	0.387806
$581$	0	0
$582$	0	0
$583$	2.83785	0.117532
$584$	3.18323	0.131723
$585$	0	0
$586$	10.6413	0.439587
$587$	−8.46197	−0.349263	−0.174631	−	0.984634i	$-0.555873\pi$
$587$	−8.46197	−0.349263	−0.174631	+	0.984634i	$0.555873\pi$
$588$	0	0
$589$	1.11901	0.0461080
$590$	−3.68276	−0.151617
$591$	0	0
$592$	−0.158619	−0.00651921
$593$	−28.2234	−1.15900	−0.579498	−	0.814974i	$-0.696751\pi$
$593$	−28.2234	−1.15900	−0.579498	+	0.814974i	$0.696751\pi$
$594$	0	0
$595$	0	0
$596$	15.2341	0.624014
$597$	0	0
$598$	8.09292	0.330944
$599$	−15.4396	−0.630845	−0.315422	−	0.948951i	$-0.602146\pi$
$599$	−15.4396	−0.630845	−0.315422	+	0.948951i	$0.602146\pi$
$600$	0	0
$601$	−10.8464	−0.442432	−0.221216	−	0.975225i	$-0.571003\pi$
$601$	−10.8464	−0.442432	−0.221216	+	0.975225i	$0.571003\pi$
$602$	0	0
$603$	0	0
$604$	−8.31724	−0.338424
$605$	−2.47363	−0.100567
$606$	0	0
$607$	−13.9134	−0.564725	−0.282363	−	0.959308i	$-0.591118\pi$
$607$	−13.9134	−0.564725	−0.282363	+	0.959308i	$0.591118\pi$
$608$	0.557647	0.0226156
$609$	0	0
$610$	−7.91391	−0.320425
$611$	22.3892	0.905770
$612$	0	0
$613$	6.82843	0.275798	0.137899	−	0.990446i	$-0.455965\pi$
$613$	6.82843	0.275798	0.137899	+	0.990446i	$0.455965\pi$
$614$	−28.9733	−1.16927
$615$	0	0
$616$	0	0
$617$	32.0688	1.29104	0.645521	−	0.763743i	$-0.276640\pi$
$617$	32.0688	1.29104	0.645521	+	0.763743i	$0.276640\pi$
$618$	0	0
$619$	−18.0492	−0.725459	−0.362730	−	0.931894i	$-0.618155\pi$
$619$	−18.0492	−0.725459	−0.362730	+	0.931894i	$0.618155\pi$
$620$	4.96374	0.199348
$621$	0	0
$622$	2.62019	0.105060
$623$	0	0
$624$	0	0
$625$	−29.3424	−1.17369
$626$	31.7566	1.26925
$627$	0	0
$628$	−9.13401	−0.364487
$629$	1.05350	0.0420056
$630$	0	0
$631$	9.28724	0.369719	0.184860	−	0.982765i	$-0.440817\pi$
$631$	9.28724	0.369719	0.184860	+	0.982765i	$0.440817\pi$
$632$	6.95668	0.276722
$633$	0	0
$634$	−24.8308	−0.986157
$635$	−0.0984373	−0.00390637
$636$	0	0
$637$	0	0
$638$	−3.77568	−0.149481
$639$	0	0
$640$	2.47363	0.0977786
$641$	15.3700	0.607078	0.303539	−	0.952819i	$-0.401832\pi$
$641$	15.3700	0.607078	0.303539	+	0.952819i	$0.401832\pi$
$642$	0	0
$643$	−11.3833	−0.448914	−0.224457	−	0.974484i	$-0.572061\pi$
$643$	−11.3833	−0.448914	−0.224457	+	0.974484i	$0.572061\pi$
$644$	0	0
$645$	0	0
$646$	−3.70371	−0.145720
$647$	20.1464	0.792038	0.396019	−	0.918242i	$-0.370391\pi$
$647$	20.1464	0.792038	0.396019	+	0.918242i	$0.370391\pi$
$648$	0	0
$649$	1.48881	0.0584410
$650$	3.39146	0.133024
$651$	0	0
$652$	8.33609	0.326466
$653$	38.1062	1.49121	0.745606	−	0.666387i	$-0.232160\pi$
$653$	38.1062	1.49121	0.745606	+	0.666387i	$0.232160\pi$
$654$	0	0
$655$	23.7492	0.927958
$656$	−5.93207	−0.231608
$657$	0	0
$658$	0	0
$659$	−21.4653	−0.836168	−0.418084	−	0.908408i	$-0.637298\pi$
$659$	−21.4653	−0.836168	−0.418084	+	0.908408i	$0.637298\pi$
$660$	0	0
$661$	6.01166	0.233826	0.116913	−	0.993142i	$-0.462700\pi$
$661$	6.01166	0.233826	0.116913	+	0.993142i	$0.462700\pi$
$662$	−21.2669	−0.826560
$663$	0	0
$664$	−4.59744	−0.178415
$665$	0	0
$666$	0	0
$667$	−10.0803	−0.390312
$668$	22.1610	0.857434
$669$	0	0
$670$	−27.8874	−1.07739
$671$	3.19932	0.123508
$672$	0	0
$673$	0.415116	0.0160015	0.00800077	−	0.999968i	$-0.497453\pi$
$673$	0.415116	0.0160015	0.00800077	+	0.999968i	$0.497453\pi$
$674$	−23.3927	−0.901055
$675$	0	0
$676$	−3.81138	−0.146592
$677$	18.2067	0.699742	0.349871	−	0.936798i	$-0.386225\pi$
$677$	18.2067	0.699742	0.349871	+	0.936798i	$0.386225\pi$
$678$	0	0
$679$	0	0
$680$	−16.4290	−0.630024
$681$	0	0
$682$	−2.00666	−0.0768391
$683$	44.9763	1.72097	0.860485	−	0.509476i	$-0.170161\pi$
$683$	44.9763	1.72097	0.860485	+	0.509476i	$0.170161\pi$
$684$	0	0
$685$	−46.1743	−1.76423
$686$	0	0
$687$	0	0
$688$	4.32666	0.164952
$689$	8.60230	0.327722
$690$	0	0
$691$	−30.7910	−1.17134	−0.585672	−	0.810548i	$-0.699169\pi$
$691$	−30.7910	−1.17134	−0.585672	+	0.810548i	$0.699169\pi$
$692$	−20.9656	−0.796991
$693$	0	0
$694$	6.60020	0.250540
$695$	−20.6135	−0.781916
$696$	0	0
$697$	39.3988	1.49234
$698$	−5.06164	−0.191586
$699$	0	0
$700$	0	0
$701$	7.83376	0.295877	0.147939	−	0.988997i	$-0.452736\pi$
$701$	7.83376	0.295877	0.147939	+	0.988997i	$0.452736\pi$
$702$	0	0
$703$	0.0884535	0.00333609
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−11.5166	−0.433431
$707$	0	0
$708$	0	0
$709$	44.2507	1.66187	0.830936	−	0.556368i	$-0.187806\pi$
$709$	44.2507	1.66187	0.830936	+	0.556368i	$0.187806\pi$
$710$	27.2341	1.02208
$711$	0	0
$712$	6.62558	0.248304
$713$	−5.35741	−0.200636
$714$	0	0
$715$	−7.49824	−0.280418
$716$	12.3759	0.462508
$717$	0	0
$718$	−10.5080	−0.392156
$719$	6.50843	0.242723	0.121362	−	0.992608i	$-0.461274\pi$
$719$	6.50843	0.242723	0.121362	+	0.992608i	$0.461274\pi$
$720$	0	0
$721$	0	0
$722$	18.6890	0.695534
$723$	0	0
$724$	20.5595	0.764087
$725$	−4.22432	−0.156887
$726$	0	0
$727$	18.9110	0.701369	0.350684	−	0.936494i	$-0.385949\pi$
$727$	18.9110	0.701369	0.350684	+	0.936494i	$0.385949\pi$
$728$	0	0
$729$	0	0
$730$	−7.87412	−0.291434
$731$	−28.7363	−1.06285
$732$	0	0
$733$	−10.2146	−0.377286	−0.188643	−	0.982046i	$-0.560409\pi$
$733$	−10.2146	−0.377286	−0.188643	+	0.982046i	$0.560409\pi$
$734$	−10.8976	−0.402239
$735$	0	0
$736$	−2.66981	−0.0984104
$737$	11.2739	0.415280
$738$	0	0
$739$	7.08939	0.260787	0.130394	−	0.991462i	$-0.458376\pi$
$739$	7.08939	0.260787	0.130394	+	0.991462i	$0.458376\pi$
$740$	0.392364	0.0144236
$741$	0	0
$742$	0	0
$743$	32.5408	1.19380	0.596902	−	0.802314i	$-0.296398\pi$
$743$	32.5408	1.19380	0.596902	+	0.802314i	$0.296398\pi$
$744$	0	0
$745$	−37.6835	−1.38062
$746$	−21.3459	−0.781528
$747$	0	0
$748$	6.64167	0.242843
$749$	0	0
$750$	0	0
$751$	−29.0496	−1.06003	−0.530017	−	0.847987i	$-0.677815\pi$
$751$	−29.0496	−1.06003	−0.530017	+	0.847987i	$0.677815\pi$
$752$	−7.38607	−0.269342
$753$	0	0
$754$	−11.4451	−0.416806
$755$	20.5737	0.748755
$756$	0	0
$757$	28.4151	1.03276	0.516382	−	0.856358i	$-0.327278\pi$
$757$	28.4151	1.03276	0.516382	+	0.856358i	$0.327278\pi$
$758$	21.3066	0.773892
$759$	0	0
$760$	−1.37941	−0.0500365
$761$	25.7999	0.935245	0.467622	−	0.883928i	$-0.345111\pi$
$761$	25.7999	0.935245	0.467622	+	0.883928i	$0.345111\pi$
$762$	0	0
$763$	0	0
$764$	20.3267	0.735393
$765$	0	0
$766$	−34.6202	−1.25088
$767$	4.51299	0.162955
$768$	0	0
$769$	27.1932	0.980612	0.490306	−	0.871550i	$-0.336885\pi$
$769$	27.1932	0.980612	0.490306	+	0.871550i	$0.336885\pi$
$770$	0	0
$771$	0	0
$772$	−13.0761	−0.470618
$773$	24.5032	0.881319	0.440660	−	0.897674i	$-0.354745\pi$
$773$	24.5032	0.881319	0.440660	+	0.897674i	$0.354745\pi$
$774$	0	0
$775$	−2.24510	−0.0806465
$776$	−1.17656	−0.0422362
$777$	0	0
$778$	−9.68276	−0.347144
$779$	3.30800	0.118521
$780$	0	0
$781$	−11.0098	−0.393962
$782$	17.7320	0.634094
$783$	0	0
$784$	0	0
$785$	22.5941	0.806419
$786$	0	0
$787$	−52.8741	−1.88476	−0.942379	−	0.334547i	$-0.891417\pi$
$787$	−52.8741	−1.88476	−0.942379	+	0.334547i	$0.891417\pi$
$788$	3.80159	0.135426
$789$	0	0
$790$	−17.2082	−0.612241
$791$	0	0
$792$	0	0
$793$	9.69800	0.344386
$794$	11.8928	0.422061
$795$	0	0
$796$	−6.87098	−0.243536
$797$	−27.1028	−0.960032	−0.480016	−	0.877260i	$-0.659369\pi$
$797$	−27.1028	−0.960032	−0.480016	+	0.877260i	$0.659369\pi$
$798$	0	0
$799$	49.0559	1.73547
$800$	−1.11882	−0.0395564
$801$	0	0
$802$	33.9804	1.19989
$803$	3.18323	0.112334
$804$	0	0
$805$	0	0
$806$	−6.08275	−0.214256
$807$	0	0
$808$	14.5295	0.511146
$809$	3.58488	0.126038	0.0630189	−	0.998012i	$-0.479927\pi$
$809$	3.58488	0.126038	0.0630189	+	0.998012i	$0.479927\pi$
$810$	0	0
$811$	50.2933	1.76604	0.883018	−	0.469339i	$-0.155508\pi$
$811$	50.2933	1.76604	0.883018	+	0.469339i	$0.155508\pi$
$812$	0	0
$813$	0	0
$814$	−0.158619	−0.00555960
$815$	−20.6204	−0.722300
$816$	0	0
$817$	−2.41275	−0.0844115
$818$	−9.51305	−0.332616
$819$	0	0
$820$	14.6737	0.512428
$821$	11.1556	0.389335	0.194667	−	0.980869i	$-0.437637\pi$
$821$	11.1556	0.389335	0.194667	+	0.980869i	$0.437637\pi$
$822$	0	0
$823$	22.7860	0.794271	0.397136	−	0.917760i	$-0.370004\pi$
$823$	22.7860	0.794271	0.397136	+	0.917760i	$0.370004\pi$
$824$	−12.2239	−0.425841
$825$	0	0
$826$	0	0
$827$	−17.0490	−0.592853	−0.296426	−	0.955056i	$-0.595795\pi$
$827$	−17.0490	−0.592853	−0.296426	+	0.955056i	$0.595795\pi$
$828$	0	0
$829$	−11.6952	−0.406190	−0.203095	−	0.979159i	$-0.565100\pi$
$829$	−11.6952	−0.406190	−0.203095	+	0.979159i	$0.565100\pi$
$830$	11.3724	0.394740
$831$	0	0
$832$	−3.03127	−0.105090
$833$	0	0
$834$	0	0
$835$	−54.8180	−1.89705
$836$	0.557647	0.0192866
$837$	0	0
$838$	−15.9249	−0.550116
$839$	−30.2833	−1.04550	−0.522748	−	0.852487i	$-0.675093\pi$
$839$	−30.2833	−1.04550	−0.522748	+	0.852487i	$0.675093\pi$
$840$	0	0
$841$	−14.7443	−0.508422
$842$	−3.55842	−0.122631
$843$	0	0
$844$	11.7984	0.406119
$845$	9.42794	0.324331
$846$	0	0
$847$	0	0
$848$	−2.83785	−0.0974522
$849$	0	0
$850$	7.43086	0.254876
$851$	−0.423483	−0.0145168
$852$	0	0
$853$	−34.7318	−1.18920	−0.594598	−	0.804023i	$-0.702689\pi$
$853$	−34.7318	−1.18920	−0.594598	+	0.804023i	$0.702689\pi$
$854$	0	0
$855$	0	0
$856$	−2.81900	−0.0963515
$857$	−21.7344	−0.742433	−0.371216	−	0.928546i	$-0.621059\pi$
$857$	−21.7344	−0.742433	−0.371216	+	0.928546i	$0.621059\pi$
$858$	0	0
$859$	−39.9241	−1.36219	−0.681096	−	0.732194i	$-0.738496\pi$
$859$	−39.9241	−1.36219	−0.681096	+	0.732194i	$0.738496\pi$
$860$	−10.7025	−0.364954
$861$	0	0
$862$	20.2774	0.690652
$863$	−57.8380	−1.96883	−0.984414	−	0.175866i	$-0.943727\pi$
$863$	−57.8380	−1.96883	−0.984414	+	0.175866i	$0.943727\pi$
$864$	0	0
$865$	51.8610	1.76333
$866$	−27.7144	−0.941774
$867$	0	0
$868$	0	0
$869$	6.95668	0.235989
$870$	0	0
$871$	34.1743	1.15795
$872$	20.5478	0.695837
$873$	0	0
$874$	1.48881	0.0503598
$875$	0	0
$876$	0	0
$877$	−38.1717	−1.28897	−0.644484	−	0.764618i	$-0.722928\pi$
$877$	−38.1717	−1.28897	−0.644484	+	0.764618i	$0.722928\pi$
$878$	−4.85433	−0.163826
$879$	0	0
$880$	2.47363	0.0833859
$881$	50.5818	1.70415	0.852073	−	0.523423i	$-0.175345\pi$
$881$	50.5818	1.70415	0.852073	+	0.523423i	$0.175345\pi$
$882$	0	0
$883$	−45.3333	−1.52559	−0.762794	−	0.646642i	$-0.776173\pi$
$883$	−45.3333	−1.52559	−0.762794	+	0.646642i	$0.776173\pi$
$884$	20.1327	0.677136
$885$	0	0
$886$	−30.5269	−1.02557
$887$	14.6337	0.491353	0.245676	−	0.969352i	$-0.420990\pi$
$887$	14.6337	0.491353	0.245676	+	0.969352i	$0.420990\pi$
$888$	0	0
$889$	0	0
$890$	−16.3892	−0.549368
$891$	0	0
$892$	−21.0031	−0.703237
$893$	4.11882	0.137831
$894$	0	0
$895$	−30.6133	−1.02329
$896$	0	0
$897$	0	0
$898$	−30.3986	−1.01442
$899$	7.57652	0.252691
$900$	0	0
$901$	18.8481	0.627920
$902$	−5.93207	−0.197516
$903$	0	0
$904$	11.7194	0.389781
$905$	−50.8565	−1.69053
$906$	0	0
$907$	51.4597	1.70869	0.854346	−	0.519704i	$-0.173958\pi$
$907$	51.4597	1.70869	0.854346	+	0.519704i	$0.173958\pi$
$908$	29.4884	0.978607
$909$	0	0
$910$	0	0
$911$	−4.27373	−0.141595	−0.0707975	−	0.997491i	$-0.522554\pi$
$911$	−4.27373	−0.141595	−0.0707975	+	0.997491i	$0.522554\pi$
$912$	0	0
$913$	−4.59744	−0.152153
$914$	−11.1020	−0.367220
$915$	0	0
$916$	−17.3254	−0.572446
$917$	0	0
$918$	0	0
$919$	46.0992	1.52067	0.760336	−	0.649530i	$-0.225034\pi$
$919$	46.0992	1.52067	0.760336	+	0.649530i	$0.225034\pi$
$920$	6.60411	0.217731
$921$	0	0
$922$	38.1693	1.25704
$923$	−33.3737	−1.09851
$924$	0	0
$925$	−0.177467	−0.00583508
$926$	31.4388	1.03314
$927$	0	0
$928$	3.77568	0.123943
$929$	−46.3287	−1.51999	−0.759997	−	0.649926i	$-0.774800\pi$
$929$	−46.3287	−1.51999	−0.759997	+	0.649926i	$0.774800\pi$
$930$	0	0
$931$	0	0
$932$	−29.4388	−0.964300
$933$	0	0
$934$	−11.9204	−0.390048
$935$	−16.4290	−0.537286
$936$	0	0
$937$	50.7016	1.65635	0.828175	−	0.560470i	$-0.189379\pi$
$937$	50.7016	1.65635	0.828175	+	0.560470i	$0.189379\pi$
$938$	0	0
$939$	0	0
$940$	18.2704	0.595914
$941$	−23.8736	−0.778257	−0.389128	−	0.921184i	$-0.627224\pi$
$941$	−23.8736	−0.778257	−0.389128	+	0.921184i	$0.627224\pi$
$942$	0	0
$943$	−15.8375	−0.515739
$944$	−1.48881	−0.0484567
$945$	0	0
$946$	4.32666	0.140672
$947$	21.8678	0.710610	0.355305	−	0.934751i	$-0.384377\pi$
$947$	21.8678	0.710610	0.355305	+	0.934751i	$0.384377\pi$
$948$	0	0
$949$	9.64923	0.313227
$950$	0.623909	0.0202423
$951$	0	0
$952$	0	0
$953$	−4.42292	−0.143273	−0.0716363	−	0.997431i	$-0.522822\pi$
$953$	−4.42292	−0.143273	−0.0716363	+	0.997431i	$0.522822\pi$
$954$	0	0
$955$	−50.2806	−1.62704
$956$	−24.2047	−0.782835
$957$	0	0
$958$	−39.4155	−1.27346
$959$	0	0
$960$	0	0
$961$	−26.9733	−0.870106
$962$	−0.480818	−0.0155022
$963$	0	0
$964$	−15.3083	−0.493048
$965$	32.3453	1.04123
$966$	0	0
$967$	17.1894	0.552773	0.276386	−	0.961047i	$-0.410863\pi$
$967$	17.1894	0.552773	0.276386	+	0.961047i	$0.410863\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	2.91038	0.0934467
$971$	25.4614	0.817094	0.408547	−	0.912737i	$-0.366036\pi$
$971$	25.4614	0.817094	0.408547	+	0.912737i	$0.366036\pi$
$972$	0	0
$973$	0	0
$974$	−23.1551	−0.741937
$975$	0	0
$976$	−3.19932	−0.102408
$977$	−16.9143	−0.541136	−0.270568	−	0.962701i	$-0.587211\pi$
$977$	−16.9143	−0.541136	−0.270568	+	0.962701i	$0.587211\pi$
$978$	0	0
$979$	6.62558	0.211754
$980$	0	0
$981$	0	0
$982$	−4.47530	−0.142812
$983$	−20.3763	−0.649902	−0.324951	−	0.945731i	$-0.605348\pi$
$983$	−20.3763	−0.649902	−0.324951	+	0.945731i	$0.605348\pi$
$984$	0	0
$985$	−9.40370	−0.299627
$986$	−25.0768	−0.798608
$987$	0	0
$988$	1.69038	0.0537782
$989$	11.5514	0.367312
$990$	0	0
$991$	35.1482	1.11652	0.558260	−	0.829666i	$-0.311469\pi$
$991$	35.1482	1.11652	0.558260	+	0.829666i	$0.311469\pi$
$992$	2.00666	0.0637116
$993$	0	0
$994$	0	0
$995$	16.9962	0.538817
$996$	0	0
$997$	43.8660	1.38925	0.694625	−	0.719372i	$-0.255571\pi$
$997$	43.8660	1.38925	0.694625	+	0.719372i	$0.255571\pi$
$998$	16.7727	0.530931
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.dz.1.2		4
3.2	odd	2		3234.2.a.bm.1.3	yes	4
7.6	odd	2		9702.2.a.ea.1.3		4
21.20	even	2		3234.2.a.bl.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.a.bl.1.2	✓	4	21.20	even	2
3234.2.a.bm.1.3	yes	4	3.2	odd	2
9702.2.a.dz.1.2		4	1.1	even	1	trivial
9702.2.a.ea.1.3		4	7.6	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.47363 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.47363 q^{5} -1.00000 q^{8} +2.47363 q^{10} -1.00000 q^{11} -3.03127 q^{13} +1.00000 q^{16} -6.64167 q^{17} -0.557647 q^{19} -2.47363 q^{20} +1.00000 q^{22} +2.66981 q^{23} +1.11882 q^{25} +3.03127 q^{26} -3.77568 q^{29} -2.00666 q^{31} -1.00000 q^{32} +6.64167 q^{34} -0.158619 q^{37} +0.557647 q^{38} +2.47363 q^{40} -5.93207 q^{41} +4.32666 q^{43} -1.00000 q^{44} -2.66981 q^{46} -7.38607 q^{47} -1.11882 q^{50} -3.03127 q^{52} -2.83785 q^{53} +2.47363 q^{55} +3.77568 q^{58} -1.48881 q^{59} -3.19932 q^{61} +2.00666 q^{62} +1.00000 q^{64} +7.49824 q^{65} -11.2739 q^{67} -6.64167 q^{68} +11.0098 q^{71} -3.18323 q^{73} +0.158619 q^{74} -0.557647 q^{76} -6.95668 q^{79} -2.47363 q^{80} +5.93207 q^{82} +4.59744 q^{83} +16.4290 q^{85} -4.32666 q^{86} +1.00000 q^{88} -6.62558 q^{89} +2.66981 q^{92} +7.38607 q^{94} +1.37941 q^{95} +1.17656 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 - 4 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{8} + 4 q^{10} - 4 q^{11} + 8 q^{13} + 4 q^{16} - 4 q^{17} + 12 q^{19} - 4 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} - 8 q^{26} + 8 q^{29} + 4 q^{31} - 4 q^{32} + 4 q^{34} + 8 q^{37} - 12 q^{38} + 4 q^{40} - 12 q^{41} - 8 q^{43} - 4 q^{44} - 8 q^{46} - 4 q^{47} - 4 q^{50} + 8 q^{52} + 8 q^{53} + 4 q^{55} - 8 q^{58} + 24 q^{61} - 4 q^{62} + 4 q^{64} + 16 q^{65} - 8 q^{67} - 4 q^{68} - 8 q^{71} + 4 q^{73} - 8 q^{74} + 12 q^{76} - 8 q^{79} - 4 q^{80} + 12 q^{82} - 4 q^{83} - 8 q^{85} + 8 q^{86} + 4 q^{88} - 24 q^{89} + 8 q^{92} + 4 q^{94} - 8 q^{95}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 - 4 * q^8 + 4 * q^10 - 4 * q^11 + 8 * q^13 + 4 * q^16 - 4 * q^17 + 12 * q^19 - 4 * q^20 + 4 * q^22 + 8 * q^23 + 4 * q^25 - 8 * q^26 + 8 * q^29 + 4 * q^31 - 4 * q^32 + 4 * q^34 + 8 * q^37 - 12 * q^38 + 4 * q^40 - 12 * q^41 - 8 * q^43 - 4 * q^44 - 8 * q^46 - 4 * q^47 - 4 * q^50 + 8 * q^52 + 8 * q^53 + 4 * q^55 - 8 * q^58 + 24 * q^61 - 4 * q^62 + 4 * q^64 + 16 * q^65 - 8 * q^67 - 4 * q^68 - 8 * q^71 + 4 * q^73 - 8 * q^74 + 12 * q^76 - 8 * q^79 - 4 * q^80 + 12 * q^82 - 4 * q^83 - 8 * q^85 + 8 * q^86 + 4 * q^88 - 24 * q^89 + 8 * q^92 + 4 * q^94 - 8 * q^95

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