LMFDB - Embedded newform 9702.2.a.do.1.1

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:	$2$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -4.24264 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -4.24264 q^{5} +1.00000 q^{8} -4.24264 q^{10} +1.00000 q^{11} -5.65685 q^{13} +1.00000 q^{16} +7.07107 q^{17} -1.41421 q^{19} -4.24264 q^{20} +1.00000 q^{22} -8.00000 q^{23} +13.0000 q^{25} -5.65685 q^{26} -8.00000 q^{29} -4.24264 q^{31} +1.00000 q^{32} +7.07107 q^{34} +2.00000 q^{37} -1.41421 q^{38} -4.24264 q^{40} -1.41421 q^{41} +8.00000 q^{43} +1.00000 q^{44} -8.00000 q^{46} -9.89949 q^{47} +13.0000 q^{50} -5.65685 q^{52} -2.00000 q^{53} -4.24264 q^{55} -8.00000 q^{58} +8.48528 q^{59} -4.24264 q^{62} +1.00000 q^{64} +24.0000 q^{65} -2.00000 q^{67} +7.07107 q^{68} +12.0000 q^{71} -7.07107 q^{73} +2.00000 q^{74} -1.41421 q^{76} +14.0000 q^{79} -4.24264 q^{80} -1.41421 q^{82} -12.7279 q^{83} -30.0000 q^{85} +8.00000 q^{86} +1.00000 q^{88} -2.82843 q^{89} -8.00000 q^{92} -9.89949 q^{94} +6.00000 q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 2 q^{11} + 2 q^{16} + 2 q^{22} - 16 q^{23} + 26 q^{25} - 16 q^{29} + 2 q^{32} + 4 q^{37} + 16 q^{43} + 2 q^{44} - 16 q^{46} + 26 q^{50} - 4 q^{53} - 16 q^{58} + 2 q^{64} + 48 q^{65} - 4 q^{67} + 24 q^{71} + 4 q^{74} + 28 q^{79} - 60 q^{85} + 16 q^{86} + 2 q^{88} - 16 q^{92} + 12 q^{95}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 2 * q^11 + 2 * q^16 + 2 * q^22 - 16 * q^23 + 26 * q^25 - 16 * q^29 + 2 * q^32 + 4 * q^37 + 16 * q^43 + 2 * q^44 - 16 * q^46 + 26 * q^50 - 4 * q^53 - 16 * q^58 + 2 * q^64 + 48 * q^65 - 4 * q^67 + 24 * q^71 + 4 * q^74 + 28 * q^79 - 60 * q^85 + 16 * q^86 + 2 * q^88 - 16 * q^92 + 12 * 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
$5$	−4.24264	−1.89737	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	−4.24264	−1.89737	−0.948683	+	0.316228i	$0.897584\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	−4.24264	−1.34164
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.65685	−1.56893	−0.784465	−	0.620174i	$-0.787062\pi$
$13$	−5.65685	−1.56893	−0.784465	+	0.620174i	$0.787062\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	7.07107	1.71499	0.857493	−	0.514496i	$-0.172021\pi$
$17$	7.07107	1.71499	0.857493	+	0.514496i	$0.172021\pi$
$18$	0	0
$19$	−1.41421	−0.324443	−0.162221	−	0.986754i	$-0.551866\pi$
$19$	−1.41421	−0.324443	−0.162221	+	0.986754i	$0.551866\pi$
$20$	−4.24264	−0.948683
$21$	0	0
$22$	1.00000	0.213201
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	13.0000	2.60000
$26$	−5.65685	−1.10940
$27$	0	0
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	−4.24264	−0.762001	−0.381000	−	0.924575i	$-0.624420\pi$
$31$	−4.24264	−0.762001	−0.381000	+	0.924575i	$0.624420\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.07107	1.21268
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−1.41421	−0.229416
$39$	0	0
$40$	−4.24264	−0.670820
$41$	−1.41421	−0.220863	−0.110432	−	0.993884i	$-0.535223\pi$
$41$	−1.41421	−0.220863	−0.110432	+	0.993884i	$0.535223\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−8.00000	−1.17954
$47$	−9.89949	−1.44399	−0.721995	−	0.691898i	$-0.756775\pi$
$47$	−9.89949	−1.44399	−0.721995	+	0.691898i	$0.756775\pi$
$48$	0	0
$49$	0	0
$50$	13.0000	1.83848
$51$	0	0
$52$	−5.65685	−0.784465
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−4.24264	−0.572078
$56$	0	0
$57$	0	0
$58$	−8.00000	−1.05045
$59$	8.48528	1.10469	0.552345	−	0.833616i	$-0.313733\pi$
$59$	8.48528	1.10469	0.552345	+	0.833616i	$0.313733\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	−4.24264	−0.538816
$63$	0	0
$64$	1.00000	0.125000
$65$	24.0000	2.97683
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	7.07107	0.857493
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−7.07107	−0.827606	−0.413803	−	0.910366i	$-0.635800\pi$
$73$	−7.07107	−0.827606	−0.413803	+	0.910366i	$0.635800\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−1.41421	−0.162221
$77$	0	0
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	−4.24264	−0.474342
$81$	0	0
$82$	−1.41421	−0.156174
$83$	−12.7279	−1.39707	−0.698535	−	0.715575i	$-0.746165\pi$
$83$	−12.7279	−1.39707	−0.698535	+	0.715575i	$0.746165\pi$
$84$	0	0
$85$	−30.0000	−3.25396
$86$	8.00000	0.862662
$87$	0	0
$88$	1.00000	0.106600
$89$	−2.82843	−0.299813	−0.149906	−	0.988700i	$-0.547897\pi$
$89$	−2.82843	−0.299813	−0.149906	+	0.988700i	$0.547897\pi$
$90$	0	0
$91$	0	0
$92$	−8.00000	−0.834058
$93$	0	0
$94$	−9.89949	−1.02105
$95$	6.00000	0.615587
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	0	0
$100$	13.0000	1.30000
$101$	5.65685	0.562878	0.281439	−	0.959579i	$-0.409188\pi$
$101$	5.65685	0.562878	0.281439	+	0.959579i	$0.409188\pi$
$102$	0	0
$103$	7.07107	0.696733	0.348367	−	0.937358i	$-0.386736\pi$
$103$	7.07107	0.696733	0.348367	+	0.937358i	$0.386736\pi$
$104$	−5.65685	−0.554700
$105$	0	0
$106$	−2.00000	−0.194257
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	−4.24264	−0.404520
$111$	0	0
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	33.9411	3.16503
$116$	−8.00000	−0.742781
$117$	0	0
$118$	8.48528	0.781133
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	−4.24264	−0.381000
$125$	−33.9411	−3.03579
$126$	0	0
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	24.0000	2.10494
$131$	−9.89949	−0.864923	−0.432461	−	0.901652i	$-0.642355\pi$
$131$	−9.89949	−0.864923	−0.432461	+	0.901652i	$0.642355\pi$
$132$	0	0
$133$	0	0
$134$	−2.00000	−0.172774
$135$	0	0
$136$	7.07107	0.606339
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	0	0
$139$	−12.7279	−1.07957	−0.539784	−	0.841803i	$-0.681494\pi$
$139$	−12.7279	−1.07957	−0.539784	+	0.841803i	$0.681494\pi$
$140$	0	0
$141$	0	0
$142$	12.0000	1.00702
$143$	−5.65685	−0.473050
$144$	0	0
$145$	33.9411	2.81866
$146$	−7.07107	−0.585206
$147$	0	0
$148$	2.00000	0.164399
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−1.41421	−0.114708
$153$	0	0
$154$	0	0
$155$	18.0000	1.44579
$156$	0	0
$157$	7.07107	0.564333	0.282166	−	0.959366i	$-0.408947\pi$
$157$	7.07107	0.564333	0.282166	+	0.959366i	$0.408947\pi$
$158$	14.0000	1.11378
$159$	0	0
$160$	−4.24264	−0.335410
$161$	0	0
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−1.41421	−0.110432
$165$	0	0
$166$	−12.7279	−0.987878
$167$	14.1421	1.09435	0.547176	−	0.837018i	$-0.315703\pi$
$167$	14.1421	1.09435	0.547176	+	0.837018i	$0.315703\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	−30.0000	−2.30089
$171$	0	0
$172$	8.00000	0.609994
$173$	19.7990	1.50529	0.752645	−	0.658427i	$-0.228778\pi$
$173$	19.7990	1.50529	0.752645	+	0.658427i	$0.228778\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	−2.82843	−0.212000
$179$	−14.0000	−1.04641	−0.523205	−	0.852207i	$-0.675264\pi$
$179$	−14.0000	−1.04641	−0.523205	+	0.852207i	$0.675264\pi$
$180$	0	0
$181$	−1.41421	−0.105118	−0.0525588	−	0.998618i	$-0.516738\pi$
$181$	−1.41421	−0.105118	−0.0525588	+	0.998618i	$0.516738\pi$
$182$	0	0
$183$	0	0
$184$	−8.00000	−0.589768
$185$	−8.48528	−0.623850
$186$	0	0
$187$	7.07107	0.517088
$188$	−9.89949	−0.721995
$189$	0	0
$190$	6.00000	0.435286
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	0	0
$199$	15.5563	1.10276	0.551380	−	0.834254i	$-0.314101\pi$
$199$	15.5563	1.10276	0.551380	+	0.834254i	$0.314101\pi$
$200$	13.0000	0.919239
$201$	0	0
$202$	5.65685	0.398015
$203$	0	0
$204$	0	0
$205$	6.00000	0.419058
$206$	7.07107	0.492665
$207$	0	0
$208$	−5.65685	−0.392232
$209$	−1.41421	−0.0978232
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	12.0000	0.820303
$215$	−33.9411	−2.31477
$216$	0	0
$217$	0	0
$218$	8.00000	0.541828
$219$	0	0
$220$	−4.24264	−0.286039
$221$	−40.0000	−2.69069
$222$	0	0
$223$	24.0416	1.60995	0.804973	−	0.593311i	$-0.202180\pi$
$223$	24.0416	1.60995	0.804973	+	0.593311i	$0.202180\pi$
$224$	0	0
$225$	0	0
$226$	10.0000	0.665190
$227$	−1.41421	−0.0938647	−0.0469323	−	0.998898i	$-0.514945\pi$
$227$	−1.41421	−0.0938647	−0.0469323	+	0.998898i	$0.514945\pi$
$228$	0	0
$229$	18.3848	1.21490	0.607450	−	0.794358i	$-0.292192\pi$
$229$	18.3848	1.21490	0.607450	+	0.794358i	$0.292192\pi$
$230$	33.9411	2.23801
$231$	0	0
$232$	−8.00000	−0.525226
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	42.0000	2.73978
$236$	8.48528	0.552345
$237$	0	0
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	24.0416	1.54866	0.774329	−	0.632783i	$-0.218088\pi$
$241$	24.0416	1.54866	0.774329	+	0.632783i	$0.218088\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.00000	0.509028
$248$	−4.24264	−0.269408
$249$	0	0
$250$	−33.9411	−2.14663
$251$	19.7990	1.24970	0.624851	−	0.780744i	$-0.285160\pi$
$251$	19.7990	1.24970	0.624851	+	0.780744i	$0.285160\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	6.00000	0.376473
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.82843	−0.176432	−0.0882162	−	0.996101i	$-0.528117\pi$
$257$	−2.82843	−0.176432	−0.0882162	+	0.996101i	$0.528117\pi$
$258$	0	0
$259$	0	0
$260$	24.0000	1.48842
$261$	0	0
$262$	−9.89949	−0.611593
$263$	22.0000	1.35658	0.678289	−	0.734795i	$-0.262722\pi$
$263$	22.0000	1.35658	0.678289	+	0.734795i	$0.262722\pi$
$264$	0	0
$265$	8.48528	0.521247
$266$	0	0
$267$	0	0
$268$	−2.00000	−0.122169
$269$	15.5563	0.948487	0.474244	−	0.880394i	$-0.342722\pi$
$269$	15.5563	0.948487	0.474244	+	0.880394i	$0.342722\pi$
$270$	0	0
$271$	−25.4558	−1.54633	−0.773166	−	0.634203i	$-0.781328\pi$
$271$	−25.4558	−1.54633	−0.773166	+	0.634203i	$0.781328\pi$
$272$	7.07107	0.428746
$273$	0	0
$274$	−22.0000	−1.32907
$275$	13.0000	0.783929
$276$	0	0
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	−12.7279	−0.763370
$279$	0	0
$280$	0	0
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	0	0
$283$	9.89949	0.588464	0.294232	−	0.955734i	$-0.404936\pi$
$283$	9.89949	0.588464	0.294232	+	0.955734i	$0.404936\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	−5.65685	−0.334497
$287$	0	0
$288$	0	0
$289$	33.0000	1.94118
$290$	33.9411	1.99309
$291$	0	0
$292$	−7.07107	−0.413803
$293$	14.1421	0.826192	0.413096	−	0.910687i	$-0.364447\pi$
$293$	14.1421	0.826192	0.413096	+	0.910687i	$0.364447\pi$
$294$	0	0
$295$	−36.0000	−2.09600
$296$	2.00000	0.116248
$297$	0	0
$298$	−18.0000	−1.04271
$299$	45.2548	2.61715
$300$	0	0
$301$	0	0
$302$	−8.00000	−0.460348
$303$	0	0
$304$	−1.41421	−0.0811107
$305$	0	0
$306$	0	0
$307$	1.41421	0.0807134	0.0403567	−	0.999185i	$-0.487151\pi$
$307$	1.41421	0.0807134	0.0403567	+	0.999185i	$0.487151\pi$
$308$	0	0
$309$	0	0
$310$	18.0000	1.02233
$311$	−26.8701	−1.52366	−0.761831	−	0.647776i	$-0.775699\pi$
$311$	−26.8701	−1.52366	−0.761831	+	0.647776i	$0.775699\pi$
$312$	0	0
$313$	19.7990	1.11911	0.559553	−	0.828795i	$-0.310973\pi$
$313$	19.7990	1.11911	0.559553	+	0.828795i	$0.310973\pi$
$314$	7.07107	0.399043
$315$	0	0
$316$	14.0000	0.787562
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	−4.24264	−0.237171
$321$	0	0
$322$	0	0
$323$	−10.0000	−0.556415
$324$	0	0
$325$	−73.5391	−4.07922
$326$	−4.00000	−0.221540
$327$	0	0
$328$	−1.41421	−0.0780869
$329$	0	0
$330$	0	0
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	−12.7279	−0.698535
$333$	0	0
$334$	14.1421	0.773823
$335$	8.48528	0.463600
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	19.0000	1.03346
$339$	0	0
$340$	−30.0000	−1.62698
$341$	−4.24264	−0.229752
$342$	0	0
$343$	0	0
$344$	8.00000	0.431331
$345$	0	0
$346$	19.7990	1.06440
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	25.4558	1.36262	0.681310	−	0.731995i	$-0.261411\pi$
$349$	25.4558	1.36262	0.681310	+	0.731995i	$0.261411\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	−2.82843	−0.150542	−0.0752710	−	0.997163i	$-0.523982\pi$
$353$	−2.82843	−0.150542	−0.0752710	+	0.997163i	$0.523982\pi$
$354$	0	0
$355$	−50.9117	−2.70211
$356$	−2.82843	−0.149906
$357$	0	0
$358$	−14.0000	−0.739923
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	−1.41421	−0.0743294
$363$	0	0
$364$	0	0
$365$	30.0000	1.57027
$366$	0	0
$367$	21.2132	1.10732	0.553660	−	0.832743i	$-0.313231\pi$
$367$	21.2132	1.10732	0.553660	+	0.832743i	$0.313231\pi$
$368$	−8.00000	−0.417029
$369$	0	0
$370$	−8.48528	−0.441129
$371$	0	0
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	7.07107	0.365636
$375$	0	0
$376$	−9.89949	−0.510527
$377$	45.2548	2.33074
$378$	0	0
$379$	−36.0000	−1.84920	−0.924598	−	0.380945i	$-0.875599\pi$
$379$	−36.0000	−1.84920	−0.924598	+	0.380945i	$0.875599\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	20.0000	1.02329
$383$	18.3848	0.939418	0.469709	−	0.882821i	$-0.344359\pi$
$383$	18.3848	0.939418	0.469709	+	0.882821i	$0.344359\pi$
$384$	0	0
$385$	0	0
$386$	22.0000	1.11977
$387$	0	0
$388$	0	0
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	−56.5685	−2.86079
$392$	0	0
$393$	0	0
$394$	−8.00000	−0.403034
$395$	−59.3970	−2.98859
$396$	0	0
$397$	26.8701	1.34857	0.674285	−	0.738471i	$-0.264452\pi$
$397$	26.8701	1.34857	0.674285	+	0.738471i	$0.264452\pi$
$398$	15.5563	0.779769
$399$	0	0
$400$	13.0000	0.650000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	24.0000	1.19553
$404$	5.65685	0.281439
$405$	0	0
$406$	0	0
$407$	2.00000	0.0991363
$408$	0	0
$409$	4.24264	0.209785	0.104893	−	0.994484i	$-0.466550\pi$
$409$	4.24264	0.209785	0.104893	+	0.994484i	$0.466550\pi$
$410$	6.00000	0.296319
$411$	0	0
$412$	7.07107	0.348367
$413$	0	0
$414$	0	0
$415$	54.0000	2.65076
$416$	−5.65685	−0.277350
$417$	0	0
$418$	−1.41421	−0.0691714
$419$	16.9706	0.829066	0.414533	−	0.910034i	$-0.363945\pi$
$419$	16.9706	0.829066	0.414533	+	0.910034i	$0.363945\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	91.9239	4.45896
$426$	0	0
$427$	0	0
$428$	12.0000	0.580042
$429$	0	0
$430$	−33.9411	−1.63679
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	0	0
$433$	33.9411	1.63111	0.815553	−	0.578682i	$-0.196433\pi$
$433$	33.9411	1.63111	0.815553	+	0.578682i	$0.196433\pi$
$434$	0	0
$435$	0	0
$436$	8.00000	0.383131
$437$	11.3137	0.541208
$438$	0	0
$439$	−8.48528	−0.404980	−0.202490	−	0.979284i	$-0.564903\pi$
$439$	−8.48528	−0.404980	−0.202490	+	0.979284i	$0.564903\pi$
$440$	−4.24264	−0.202260
$441$	0	0
$442$	−40.0000	−1.90261
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	24.0416	1.13840
$447$	0	0
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	−1.41421	−0.0665927
$452$	10.0000	0.470360
$453$	0	0
$454$	−1.41421	−0.0663723
$455$	0	0
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	18.3848	0.859064
$459$	0	0
$460$	33.9411	1.58251
$461$	5.65685	0.263466	0.131733	−	0.991285i	$-0.457946\pi$
$461$	5.65685	0.263466	0.131733	+	0.991285i	$0.457946\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	14.0000	0.648537
$467$	−16.9706	−0.785304	−0.392652	−	0.919687i	$-0.628442\pi$
$467$	−16.9706	−0.785304	−0.392652	+	0.919687i	$0.628442\pi$
$468$	0	0
$469$	0	0
$470$	42.0000	1.93732
$471$	0	0
$472$	8.48528	0.390567
$473$	8.00000	0.367840
$474$	0	0
$475$	−18.3848	−0.843551
$476$	0	0
$477$	0	0
$478$	−8.00000	−0.365911
$479$	33.9411	1.55081	0.775405	−	0.631464i	$-0.217546\pi$
$479$	33.9411	1.55081	0.775405	+	0.631464i	$0.217546\pi$
$480$	0	0
$481$	−11.3137	−0.515861
$482$	24.0416	1.09507
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	−56.5685	−2.54772
$494$	8.00000	0.359937
$495$	0	0
$496$	−4.24264	−0.190500
$497$	0	0
$498$	0	0
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	−33.9411	−1.51789
$501$	0	0
$502$	19.7990	0.883672
$503$	−19.7990	−0.882793	−0.441397	−	0.897312i	$-0.645517\pi$
$503$	−19.7990	−0.882793	−0.441397	+	0.897312i	$0.645517\pi$
$504$	0	0
$505$	−24.0000	−1.06799
$506$	−8.00000	−0.355643
$507$	0	0
$508$	6.00000	0.266207
$509$	−12.7279	−0.564155	−0.282078	−	0.959392i	$-0.591024\pi$
$509$	−12.7279	−0.564155	−0.282078	+	0.959392i	$0.591024\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−2.82843	−0.124757
$515$	−30.0000	−1.32196
$516$	0	0
$517$	−9.89949	−0.435379
$518$	0	0
$519$	0	0
$520$	24.0000	1.05247
$521$	−19.7990	−0.867409	−0.433705	−	0.901055i	$-0.642794\pi$
$521$	−19.7990	−0.867409	−0.433705	+	0.901055i	$0.642794\pi$
$522$	0	0
$523$	21.2132	0.927589	0.463794	−	0.885943i	$-0.346488\pi$
$523$	21.2132	0.927589	0.463794	+	0.885943i	$0.346488\pi$
$524$	−9.89949	−0.432461
$525$	0	0
$526$	22.0000	0.959246
$527$	−30.0000	−1.30682
$528$	0	0
$529$	41.0000	1.78261
$530$	8.48528	0.368577
$531$	0	0
$532$	0	0
$533$	8.00000	0.346518
$534$	0	0
$535$	−50.9117	−2.20110
$536$	−2.00000	−0.0863868
$537$	0	0
$538$	15.5563	0.670682
$539$	0	0
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	−25.4558	−1.09342
$543$	0	0
$544$	7.07107	0.303170
$545$	−33.9411	−1.45388
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	−22.0000	−0.939793
$549$	0	0
$550$	13.0000	0.554322
$551$	11.3137	0.481980
$552$	0	0
$553$	0	0
$554$	4.00000	0.169944
$555$	0	0
$556$	−12.7279	−0.539784
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−45.2548	−1.91408
$560$	0	0
$561$	0	0
$562$	−14.0000	−0.590554
$563$	−1.41421	−0.0596020	−0.0298010	−	0.999556i	$-0.509487\pi$
$563$	−1.41421	−0.0596020	−0.0298010	+	0.999556i	$0.509487\pi$
$564$	0	0
$565$	−42.4264	−1.78489
$566$	9.89949	0.416107
$567$	0	0
$568$	12.0000	0.503509
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	−5.65685	−0.236525
$573$	0	0
$574$	0	0
$575$	−104.000	−4.33710
$576$	0	0
$577$	19.7990	0.824243	0.412121	−	0.911129i	$-0.364788\pi$
$577$	19.7990	0.824243	0.412121	+	0.911129i	$0.364788\pi$
$578$	33.0000	1.37262
$579$	0	0
$580$	33.9411	1.40933
$581$	0	0
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	−7.07107	−0.292603
$585$	0	0
$586$	14.1421	0.584206
$587$	−14.1421	−0.583708	−0.291854	−	0.956463i	$-0.594272\pi$
$587$	−14.1421	−0.583708	−0.291854	+	0.956463i	$0.594272\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	−36.0000	−1.48210
$591$	0	0
$592$	2.00000	0.0821995
$593$	−12.7279	−0.522673	−0.261337	−	0.965248i	$-0.584163\pi$
$593$	−12.7279	−0.522673	−0.261337	+	0.965248i	$0.584163\pi$
$594$	0	0
$595$	0	0
$596$	−18.0000	−0.737309
$597$	0	0
$598$	45.2548	1.85061
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−12.7279	−0.519183	−0.259591	−	0.965719i	$-0.583588\pi$
$601$	−12.7279	−0.519183	−0.259591	+	0.965719i	$0.583588\pi$
$602$	0	0
$603$	0	0
$604$	−8.00000	−0.325515
$605$	−4.24264	−0.172488
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	−1.41421	−0.0573539
$609$	0	0
$610$	0	0
$611$	56.0000	2.26552
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	1.41421	0.0570730
$615$	0	0
$616$	0	0
$617$	−20.0000	−0.805170	−0.402585	−	0.915383i	$-0.631888\pi$
$617$	−20.0000	−0.805170	−0.402585	+	0.915383i	$0.631888\pi$
$618$	0	0
$619$	45.2548	1.81895	0.909473	−	0.415764i	$-0.136486\pi$
$619$	45.2548	1.81895	0.909473	+	0.415764i	$0.136486\pi$
$620$	18.0000	0.722897
$621$	0	0
$622$	−26.8701	−1.07739
$623$	0	0
$624$	0	0
$625$	79.0000	3.16000
$626$	19.7990	0.791327
$627$	0	0
$628$	7.07107	0.282166
$629$	14.1421	0.563884
$630$	0	0
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	14.0000	0.556890
$633$	0	0
$634$	−22.0000	−0.873732
$635$	−25.4558	−1.01018
$636$	0	0
$637$	0	0
$638$	−8.00000	−0.316723
$639$	0	0
$640$	−4.24264	−0.167705
$641$	44.0000	1.73790	0.868948	−	0.494904i	$-0.164797\pi$
$641$	44.0000	1.73790	0.868948	+	0.494904i	$0.164797\pi$
$642$	0	0
$643$	−8.48528	−0.334627	−0.167313	−	0.985904i	$-0.553509\pi$
$643$	−8.48528	−0.334627	−0.167313	+	0.985904i	$0.553509\pi$
$644$	0	0
$645$	0	0
$646$	−10.0000	−0.393445
$647$	−26.8701	−1.05637	−0.528185	−	0.849129i	$-0.677127\pi$
$647$	−26.8701	−1.05637	−0.528185	+	0.849129i	$0.677127\pi$
$648$	0	0
$649$	8.48528	0.333076
$650$	−73.5391	−2.88444
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	0	0
$655$	42.0000	1.64108
$656$	−1.41421	−0.0552158
$657$	0	0
$658$	0	0
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	0	0
$661$	−4.24264	−0.165020	−0.0825098	−	0.996590i	$-0.526294\pi$
$661$	−4.24264	−0.165020	−0.0825098	+	0.996590i	$0.526294\pi$
$662$	26.0000	1.01052
$663$	0	0
$664$	−12.7279	−0.493939
$665$	0	0
$666$	0	0
$667$	64.0000	2.47809
$668$	14.1421	0.547176
$669$	0	0
$670$	8.48528	0.327815
$671$	0	0
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	19.0000	0.730769
$677$	−19.7990	−0.760937	−0.380468	−	0.924794i	$-0.624237\pi$
$677$	−19.7990	−0.760937	−0.380468	+	0.924794i	$0.624237\pi$
$678$	0	0
$679$	0	0
$680$	−30.0000	−1.15045
$681$	0	0
$682$	−4.24264	−0.162459
$683$	38.0000	1.45403	0.727015	−	0.686622i	$-0.240907\pi$
$683$	38.0000	1.45403	0.727015	+	0.686622i	$0.240907\pi$
$684$	0	0
$685$	93.3381	3.56627
$686$	0	0
$687$	0	0
$688$	8.00000	0.304997
$689$	11.3137	0.431018
$690$	0	0
$691$	−8.48528	−0.322795	−0.161398	−	0.986889i	$-0.551600\pi$
$691$	−8.48528	−0.322795	−0.161398	+	0.986889i	$0.551600\pi$
$692$	19.7990	0.752645
$693$	0	0
$694$	−4.00000	−0.151838
$695$	54.0000	2.04834
$696$	0	0
$697$	−10.0000	−0.378777
$698$	25.4558	0.963518
$699$	0	0
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	−2.82843	−0.106676
$704$	1.00000	0.0376889
$705$	0	0
$706$	−2.82843	−0.106449
$707$	0	0
$708$	0	0
$709$	2.00000	0.0751116	0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000	0.0751116	0.0375558	+	0.999295i	$0.488043\pi$
$710$	−50.9117	−1.91068
$711$	0	0
$712$	−2.82843	−0.106000
$713$	33.9411	1.27111
$714$	0	0
$715$	24.0000	0.897549
$716$	−14.0000	−0.523205
$717$	0	0
$718$	10.0000	0.373197
$719$	24.0416	0.896602	0.448301	−	0.893883i	$-0.352029\pi$
$719$	24.0416	0.896602	0.448301	+	0.893883i	$0.352029\pi$
$720$	0	0
$721$	0	0
$722$	−17.0000	−0.632674
$723$	0	0
$724$	−1.41421	−0.0525588
$725$	−104.000	−3.86246
$726$	0	0
$727$	−49.4975	−1.83576	−0.917880	−	0.396858i	$-0.870100\pi$
$727$	−49.4975	−1.83576	−0.917880	+	0.396858i	$0.870100\pi$
$728$	0	0
$729$	0	0
$730$	30.0000	1.11035
$731$	56.5685	2.09226
$732$	0	0
$733$	28.2843	1.04470	0.522352	−	0.852730i	$-0.325055\pi$
$733$	28.2843	1.04470	0.522352	+	0.852730i	$0.325055\pi$
$734$	21.2132	0.782994
$735$	0	0
$736$	−8.00000	−0.294884
$737$	−2.00000	−0.0736709
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−8.48528	−0.311925
$741$	0	0
$742$	0	0
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	76.3675	2.79789
$746$	26.0000	0.951928
$747$	0	0
$748$	7.07107	0.258544
$749$	0	0
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−9.89949	−0.360997
$753$	0	0
$754$	45.2548	1.64808
$755$	33.9411	1.23524
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−36.0000	−1.30758
$759$	0	0
$760$	6.00000	0.217643
$761$	−21.2132	−0.768978	−0.384489	−	0.923130i	$-0.625622\pi$
$761$	−21.2132	−0.768978	−0.384489	+	0.923130i	$0.625622\pi$
$762$	0	0
$763$	0	0
$764$	20.0000	0.723575
$765$	0	0
$766$	18.3848	0.664269
$767$	−48.0000	−1.73318
$768$	0	0
$769$	7.07107	0.254989	0.127495	−	0.991839i	$-0.459306\pi$
$769$	7.07107	0.254989	0.127495	+	0.991839i	$0.459306\pi$
$770$	0	0
$771$	0	0
$772$	22.0000	0.791797
$773$	7.07107	0.254329	0.127164	−	0.991882i	$-0.459412\pi$
$773$	7.07107	0.254329	0.127164	+	0.991882i	$0.459412\pi$
$774$	0	0
$775$	−55.1543	−1.98120
$776$	0	0
$777$	0	0
$778$	−2.00000	−0.0717035
$779$	2.00000	0.0716574
$780$	0	0
$781$	12.0000	0.429394
$782$	−56.5685	−2.02289
$783$	0	0
$784$	0	0
$785$	−30.0000	−1.07075
$786$	0	0
$787$	−41.0122	−1.46193	−0.730963	−	0.682417i	$-0.760929\pi$
$787$	−41.0122	−1.46193	−0.730963	+	0.682417i	$0.760929\pi$
$788$	−8.00000	−0.284988
$789$	0	0
$790$	−59.3970	−2.11325
$791$	0	0
$792$	0	0
$793$	0	0
$794$	26.8701	0.953583
$795$	0	0
$796$	15.5563	0.551380
$797$	−32.5269	−1.15216	−0.576081	−	0.817392i	$-0.695419\pi$
$797$	−32.5269	−1.15216	−0.576081	+	0.817392i	$0.695419\pi$
$798$	0	0
$799$	−70.0000	−2.47642
$800$	13.0000	0.459619
$801$	0	0
$802$	18.0000	0.635602
$803$	−7.07107	−0.249533
$804$	0	0
$805$	0	0
$806$	24.0000	0.845364
$807$	0	0
$808$	5.65685	0.199007
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	9.89949	0.347618	0.173809	−	0.984779i	$-0.444392\pi$
$811$	9.89949	0.347618	0.173809	+	0.984779i	$0.444392\pi$
$812$	0	0
$813$	0	0
$814$	2.00000	0.0701000
$815$	16.9706	0.594453
$816$	0	0
$817$	−11.3137	−0.395817
$818$	4.24264	0.148340
$819$	0	0
$820$	6.00000	0.209529
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	7.07107	0.246332
$825$	0	0
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	9.89949	0.343824	0.171912	−	0.985112i	$-0.445006\pi$
$829$	9.89949	0.343824	0.171912	+	0.985112i	$0.445006\pi$
$830$	54.0000	1.87437
$831$	0	0
$832$	−5.65685	−0.196116
$833$	0	0
$834$	0	0
$835$	−60.0000	−2.07639
$836$	−1.41421	−0.0489116
$837$	0	0
$838$	16.9706	0.586238
$839$	18.3848	0.634713	0.317356	−	0.948306i	$-0.397205\pi$
$839$	18.3848	0.634713	0.317356	+	0.948306i	$0.397205\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−10.0000	−0.344623
$843$	0	0
$844$	4.00000	0.137686
$845$	−80.6102	−2.77307
$846$	0	0
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	91.9239	3.15296
$851$	−16.0000	−0.548473
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	4.24264	0.144926	0.0724629	−	0.997371i	$-0.476914\pi$
$857$	4.24264	0.144926	0.0724629	+	0.997371i	$0.476914\pi$
$858$	0	0
$859$	−25.4558	−0.868542	−0.434271	−	0.900782i	$-0.642994\pi$
$859$	−25.4558	−0.868542	−0.434271	+	0.900782i	$0.642994\pi$
$860$	−33.9411	−1.15738
$861$	0	0
$862$	−30.0000	−1.02180
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	0	0
$865$	−84.0000	−2.85609
$866$	33.9411	1.15337
$867$	0	0
$868$	0	0
$869$	14.0000	0.474917
$870$	0	0
$871$	11.3137	0.383350
$872$	8.00000	0.270914
$873$	0	0
$874$	11.3137	0.382692
$875$	0	0
$876$	0	0
$877$	−28.0000	−0.945493	−0.472746	−	0.881199i	$-0.656737\pi$
$877$	−28.0000	−0.945493	−0.472746	+	0.881199i	$0.656737\pi$
$878$	−8.48528	−0.286364
$879$	0	0
$880$	−4.24264	−0.143019
$881$	45.2548	1.52467	0.762337	−	0.647180i	$-0.224052\pi$
$881$	45.2548	1.52467	0.762337	+	0.647180i	$0.224052\pi$
$882$	0	0
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	−40.0000	−1.34535
$885$	0	0
$886$	−6.00000	−0.201574
$887$	−5.65685	−0.189939	−0.0949693	−	0.995480i	$-0.530275\pi$
$887$	−5.65685	−0.189939	−0.0949693	+	0.995480i	$0.530275\pi$
$888$	0	0
$889$	0	0
$890$	12.0000	0.402241
$891$	0	0
$892$	24.0416	0.804973
$893$	14.0000	0.468492
$894$	0	0
$895$	59.3970	1.98542
$896$	0	0
$897$	0	0
$898$	2.00000	0.0667409
$899$	33.9411	1.13200
$900$	0	0
$901$	−14.1421	−0.471143
$902$	−1.41421	−0.0470882
$903$	0	0
$904$	10.0000	0.332595
$905$	6.00000	0.199447
$906$	0	0
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	−1.41421	−0.0469323
$909$	0	0
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	−12.7279	−0.421233
$914$	−10.0000	−0.330771
$915$	0	0
$916$	18.3848	0.607450
$917$	0	0
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	33.9411	1.11901
$921$	0	0
$922$	5.65685	0.186299
$923$	−67.8823	−2.23437
$924$	0	0
$925$	26.0000	0.854875
$926$	16.0000	0.525793
$927$	0	0
$928$	−8.00000	−0.262613
$929$	−2.82843	−0.0927977	−0.0463988	−	0.998923i	$-0.514775\pi$
$929$	−2.82843	−0.0927977	−0.0463988	+	0.998923i	$0.514775\pi$
$930$	0	0
$931$	0	0
$932$	14.0000	0.458585
$933$	0	0
$934$	−16.9706	−0.555294
$935$	−30.0000	−0.981105
$936$	0	0
$937$	38.1838	1.24741	0.623705	−	0.781660i	$-0.285627\pi$
$937$	38.1838	1.24741	0.623705	+	0.781660i	$0.285627\pi$
$938$	0	0
$939$	0	0
$940$	42.0000	1.36989
$941$	19.7990	0.645429	0.322714	−	0.946496i	$-0.395405\pi$
$941$	19.7990	0.645429	0.322714	+	0.946496i	$0.395405\pi$
$942$	0	0
$943$	11.3137	0.368425
$944$	8.48528	0.276172
$945$	0	0
$946$	8.00000	0.260102
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	0	0
$949$	40.0000	1.29845
$950$	−18.3848	−0.596481
$951$	0	0
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	−84.8528	−2.74577
$956$	−8.00000	−0.258738
$957$	0	0
$958$	33.9411	1.09659
$959$	0	0
$960$	0	0
$961$	−13.0000	−0.419355
$962$	−11.3137	−0.364769
$963$	0	0
$964$	24.0416	0.774329
$965$	−93.3381	−3.00466
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	0	0
$971$	−11.3137	−0.363074	−0.181537	−	0.983384i	$-0.558107\pi$
$971$	−11.3137	−0.363074	−0.181537	+	0.983384i	$0.558107\pi$
$972$	0	0
$973$	0	0
$974$	−20.0000	−0.640841
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	−2.82843	−0.0903969
$980$	0	0
$981$	0	0
$982$	−20.0000	−0.638226
$983$	57.9828	1.84936	0.924681	−	0.380742i	$-0.124331\pi$
$983$	57.9828	1.84936	0.924681	+	0.380742i	$0.124331\pi$
$984$	0	0
$985$	33.9411	1.08145
$986$	−56.5685	−1.80151
$987$	0	0
$988$	8.00000	0.254514
$989$	−64.0000	−2.03508
$990$	0	0
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	−4.24264	−0.134704
$993$	0	0
$994$	0	0
$995$	−66.0000	−2.09234
$996$	0	0
$997$	−59.3970	−1.88112	−0.940560	−	0.339626i	$-0.889699\pi$
$997$	−59.3970	−1.88112	−0.940560	+	0.339626i	$0.889699\pi$
$998$	−6.00000	−0.189927
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.do.1.1	yes	2
3.2	odd	2		9702.2.a.cm.1.2	yes	2
7.6	odd	2	inner	9702.2.a.do.1.2	yes	2
21.20	even	2		9702.2.a.cm.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9702.2.a.cm.1.1	✓	2	21.20	even	2
9702.2.a.cm.1.2	yes	2	3.2	odd	2
9702.2.a.do.1.1	yes	2	1.1	even	1	trivial
9702.2.a.do.1.2	yes	2	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -4.24264 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -4.24264 q^{5} +1.00000 q^{8} -4.24264 q^{10} +1.00000 q^{11} -5.65685 q^{13} +1.00000 q^{16} +7.07107 q^{17} -1.41421 q^{19} -4.24264 q^{20} +1.00000 q^{22} -8.00000 q^{23} +13.0000 q^{25} -5.65685 q^{26} -8.00000 q^{29} -4.24264 q^{31} +1.00000 q^{32} +7.07107 q^{34} +2.00000 q^{37} -1.41421 q^{38} -4.24264 q^{40} -1.41421 q^{41} +8.00000 q^{43} +1.00000 q^{44} -8.00000 q^{46} -9.89949 q^{47} +13.0000 q^{50} -5.65685 q^{52} -2.00000 q^{53} -4.24264 q^{55} -8.00000 q^{58} +8.48528 q^{59} -4.24264 q^{62} +1.00000 q^{64} +24.0000 q^{65} -2.00000 q^{67} +7.07107 q^{68} +12.0000 q^{71} -7.07107 q^{73} +2.00000 q^{74} -1.41421 q^{76} +14.0000 q^{79} -4.24264 q^{80} -1.41421 q^{82} -12.7279 q^{83} -30.0000 q^{85} +8.00000 q^{86} +1.00000 q^{88} -2.82843 q^{89} -8.00000 q^{92} -9.89949 q^{94} +6.00000 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 2 q^{11} + 2 q^{16} + 2 q^{22} - 16 q^{23} + 26 q^{25} - 16 q^{29} + 2 q^{32} + 4 q^{37} + 16 q^{43} + 2 q^{44} - 16 q^{46} + 26 q^{50} - 4 q^{53} - 16 q^{58} + 2 q^{64} + 48 q^{65} - 4 q^{67} + 24 q^{71} + 4 q^{74} + 28 q^{79} - 60 q^{85} + 16 q^{86} + 2 q^{88} - 16 q^{92} + 12 q^{95}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 2 * q^11 + 2 * q^16 + 2 * q^22 - 16 * q^23 + 26 * q^25 - 16 * q^29 + 2 * q^32 + 4 * q^37 + 16 * q^43 + 2 * q^44 - 16 * q^46 + 26 * q^50 - 4 * q^53 - 16 * q^58 + 2 * q^64 + 48 * q^65 - 4 * q^67 + 24 * q^71 + 4 * q^74 + 28 * q^79 - 60 * q^85 + 16 * q^86 + 2 * q^88 - 16 * q^92 + 12 * q^95

Introduction

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