LMFDB - Embedded newform 9702.2.a.dg.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{8} -1.41421 q^{10} -1.00000 q^{11} +5.65685 q^{13} +1.00000 q^{16} +1.41421 q^{17} -4.24264 q^{19} -1.41421 q^{20} -1.00000 q^{22} -4.00000 q^{23} -3.00000 q^{25} +5.65685 q^{26} -4.24264 q^{31} +1.00000 q^{32} +1.41421 q^{34} -6.00000 q^{37} -4.24264 q^{38} -1.41421 q^{40} +4.24264 q^{41} -4.00000 q^{43} -1.00000 q^{44} -4.00000 q^{46} -1.41421 q^{47} -3.00000 q^{50} +5.65685 q^{52} +6.00000 q^{53} +1.41421 q^{55} -2.82843 q^{59} -5.65685 q^{61} -4.24264 q^{62} +1.00000 q^{64} -8.00000 q^{65} -10.0000 q^{67} +1.41421 q^{68} -4.24264 q^{73} -6.00000 q^{74} -4.24264 q^{76} -6.00000 q^{79} -1.41421 q^{80} +4.24264 q^{82} +15.5563 q^{83} -2.00000 q^{85} -4.00000 q^{86} -1.00000 q^{88} +14.1421 q^{89} -4.00000 q^{92} -1.41421 q^{94} +6.00000 q^{95} -5.65685 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 8 q^{23} - 6 q^{25} + 2 q^{32} - 12 q^{37} - 8 q^{43} - 2 q^{44} - 8 q^{46} - 6 q^{50} + 12 q^{53} + 2 q^{64} - 16 q^{65} - 20 q^{67} - 12 q^{74} - 12 q^{79} - 4 q^{85} - 8 q^{86} - 2 q^{88} - 8 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 - 8 * q^23 - 6 * q^25 + 2 * q^32 - 12 * q^37 - 8 * q^43 - 2 * q^44 - 8 * q^46 - 6 * q^50 + 12 * q^53 + 2 * q^64 - 16 * q^65 - 20 * q^67 - 12 * q^74 - 12 * q^79 - 4 * q^85 - 8 * q^86 - 2 * q^88 - 8 * 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
$5$	−1.41421	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.41421	−0.447214
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.65685	1.56893	0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685	1.56893	0.784465	+	0.620174i	$0.212938\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	1.41421	0.342997	0.171499	−	0.985184i	$-0.445139\pi$
$17$	1.41421	0.342997	0.171499	+	0.985184i	$0.445139\pi$
$18$	0	0
$19$	−4.24264	−0.973329	−0.486664	−	0.873589i	$-0.661786\pi$
$19$	−4.24264	−0.973329	−0.486664	+	0.873589i	$0.661786\pi$
$20$	−1.41421	−0.316228
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	5.65685	1.10940
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	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
$34$	1.41421	0.242536
$35$	0	0
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−4.24264	−0.688247
$39$	0	0
$40$	−1.41421	−0.223607
$41$	4.24264	0.662589	0.331295	−	0.943527i	$-0.392515\pi$
$41$	4.24264	0.662589	0.331295	+	0.943527i	$0.392515\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−4.00000	−0.589768
$47$	−1.41421	−0.206284	−0.103142	−	0.994667i	$-0.532890\pi$
$47$	−1.41421	−0.206284	−0.103142	+	0.994667i	$0.532890\pi$
$48$	0	0
$49$	0	0
$50$	−3.00000	−0.424264
$51$	0	0
$52$	5.65685	0.784465
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	1.41421	0.190693
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.82843	−0.368230	−0.184115	−	0.982905i	$-0.558942\pi$
$59$	−2.82843	−0.368230	−0.184115	+	0.982905i	$0.558942\pi$
$60$	0	0
$61$	−5.65685	−0.724286	−0.362143	−	0.932123i	$-0.617955\pi$
$61$	−5.65685	−0.724286	−0.362143	+	0.932123i	$0.617955\pi$
$62$	−4.24264	−0.538816
$63$	0	0
$64$	1.00000	0.125000
$65$	−8.00000	−0.992278
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	1.41421	0.171499
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−4.24264	−0.496564	−0.248282	−	0.968688i	$-0.579866\pi$
$73$	−4.24264	−0.496564	−0.248282	+	0.968688i	$0.579866\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−4.24264	−0.486664
$77$	0	0
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	−1.41421	−0.158114
$81$	0	0
$82$	4.24264	0.468521
$83$	15.5563	1.70753	0.853766	−	0.520658i	$-0.174313\pi$
$83$	15.5563	1.70753	0.853766	+	0.520658i	$0.174313\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−1.00000	−0.106600
$89$	14.1421	1.49906	0.749532	−	0.661968i	$-0.230279\pi$
$89$	14.1421	1.49906	0.749532	+	0.661968i	$0.230279\pi$
$90$	0	0
$91$	0	0
$92$	−4.00000	−0.417029
$93$	0	0
$94$	−1.41421	−0.145865
$95$	6.00000	0.615587
$96$	0	0
$97$	−5.65685	−0.574367	−0.287183	−	0.957876i	$-0.592719\pi$
$97$	−5.65685	−0.574367	−0.287183	+	0.957876i	$0.592719\pi$
$98$	0	0
$99$	0	0
$100$	−3.00000	−0.300000
$101$	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
$106$	6.00000	0.582772
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	1.41421	0.134840
$111$	0	0
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	5.65685	0.527504
$116$	0	0
$117$	0	0
$118$	−2.82843	−0.260378
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−5.65685	−0.512148
$123$	0	0
$124$	−4.24264	−0.381000
$125$	11.3137	1.01193
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−8.00000	−0.701646
$131$	−15.5563	−1.35916	−0.679582	−	0.733599i	$-0.737839\pi$
$131$	−15.5563	−1.35916	−0.679582	+	0.733599i	$0.737839\pi$
$132$	0	0
$133$	0	0
$134$	−10.0000	−0.863868
$135$	0	0
$136$	1.41421	0.121268
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−4.24264	−0.359856	−0.179928	−	0.983680i	$-0.557586\pi$
$139$	−4.24264	−0.359856	−0.179928	+	0.983680i	$0.557586\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.65685	−0.473050
$144$	0	0
$145$	0	0
$146$	−4.24264	−0.351123
$147$	0	0
$148$	−6.00000	−0.493197
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−4.24264	−0.344124
$153$	0	0
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	18.3848	1.46726	0.733632	−	0.679546i	$-0.237823\pi$
$157$	18.3848	1.46726	0.733632	+	0.679546i	$0.237823\pi$
$158$	−6.00000	−0.477334
$159$	0	0
$160$	−1.41421	−0.111803
$161$	0	0
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	4.24264	0.331295
$165$	0	0
$166$	15.5563	1.20741
$167$	−19.7990	−1.53209	−0.766046	−	0.642786i	$-0.777779\pi$
$167$	−19.7990	−1.53209	−0.766046	+	0.642786i	$0.777779\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	−2.00000	−0.153393
$171$	0	0
$172$	−4.00000	−0.304997
$173$	2.82843	0.215041	0.107521	−	0.994203i	$-0.465709\pi$
$173$	2.82843	0.215041	0.107521	+	0.994203i	$0.465709\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	14.1421	1.06000
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−12.7279	−0.946059	−0.473029	−	0.881047i	$-0.656840\pi$
$181$	−12.7279	−0.946059	−0.473029	+	0.881047i	$0.656840\pi$
$182$	0	0
$183$	0	0
$184$	−4.00000	−0.294884
$185$	8.48528	0.623850
$186$	0	0
$187$	−1.41421	−0.103418
$188$	−1.41421	−0.103142
$189$	0	0
$190$	6.00000	0.435286
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−5.65685	−0.406138
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−18.3848	−1.30326	−0.651631	−	0.758536i	$-0.725915\pi$
$199$	−18.3848	−1.30326	−0.651631	+	0.758536i	$0.725915\pi$
$200$	−3.00000	−0.212132
$201$	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
$209$	4.24264	0.293470
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−4.00000	−0.273434
$215$	5.65685	0.385794
$216$	0	0
$217$	0	0
$218$	−8.00000	−0.541828
$219$	0	0
$220$	1.41421	0.0953463
$221$	8.00000	0.538138
$222$	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
$226$	−18.0000	−1.19734
$227$	−7.07107	−0.469323	−0.234662	−	0.972077i	$-0.575398\pi$
$227$	−7.07107	−0.469323	−0.234662	+	0.972077i	$0.575398\pi$
$228$	0	0
$229$	7.07107	0.467269	0.233635	−	0.972324i	$-0.424938\pi$
$229$	7.07107	0.467269	0.233635	+	0.972324i	$0.424938\pi$
$230$	5.65685	0.373002
$231$	0	0
$232$	0	0
$233$	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
$235$	2.00000	0.130466
$236$	−2.82843	−0.184115
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−1.41421	−0.0910975	−0.0455488	−	0.998962i	$-0.514504\pi$
$241$	−1.41421	−0.0910975	−0.0455488	+	0.998962i	$0.514504\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−5.65685	−0.362143
$245$	0	0
$246$	0	0
$247$	−24.0000	−1.52708
$248$	−4.24264	−0.269408
$249$	0	0
$250$	11.3137	0.715542
$251$	−19.7990	−1.24970	−0.624851	−	0.780744i	$-0.714840\pi$
$251$	−19.7990	−1.24970	−0.624851	+	0.780744i	$0.714840\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	31.1127	1.94076	0.970378	−	0.241590i	$-0.0776689\pi$
$257$	31.1127	1.94076	0.970378	+	0.241590i	$0.0776689\pi$
$258$	0	0
$259$	0	0
$260$	−8.00000	−0.496139
$261$	0	0
$262$	−15.5563	−0.961074
$263$	−2.00000	−0.123325	−0.0616626	−	0.998097i	$-0.519640\pi$
$263$	−2.00000	−0.123325	−0.0616626	+	0.998097i	$0.519640\pi$
$264$	0	0
$265$	−8.48528	−0.521247
$266$	0	0
$267$	0	0
$268$	−10.0000	−0.610847
$269$	12.7279	0.776035	0.388018	−	0.921652i	$-0.373160\pi$
$269$	12.7279	0.776035	0.388018	+	0.921652i	$0.373160\pi$
$270$	0	0
$271$	2.82843	0.171815	0.0859074	−	0.996303i	$-0.472621\pi$
$271$	2.82843	0.171815	0.0859074	+	0.996303i	$0.472621\pi$
$272$	1.41421	0.0857493
$273$	0	0
$274$	6.00000	0.362473
$275$	3.00000	0.180907
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	−4.24264	−0.254457
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	−15.5563	−0.924729	−0.462364	−	0.886690i	$-0.652999\pi$
$283$	−15.5563	−0.924729	−0.462364	+	0.886690i	$0.652999\pi$
$284$	0	0
$285$	0	0
$286$	−5.65685	−0.334497
$287$	0	0
$288$	0	0
$289$	−15.0000	−0.882353
$290$	0	0
$291$	0	0
$292$	−4.24264	−0.248282
$293$	−31.1127	−1.81762	−0.908812	−	0.417207i	$-0.863009\pi$
$293$	−31.1127	−1.81762	−0.908812	+	0.417207i	$0.863009\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	−6.00000	−0.348743
$297$	0	0
$298$	−2.00000	−0.115857
$299$	−22.6274	−1.30858
$300$	0	0
$301$	0	0
$302$	−16.0000	−0.920697
$303$	0	0
$304$	−4.24264	−0.243332
$305$	8.00000	0.458079
$306$	0	0
$307$	−7.07107	−0.403567	−0.201784	−	0.979430i	$-0.564674\pi$
$307$	−7.07107	−0.403567	−0.201784	+	0.979430i	$0.564674\pi$
$308$	0	0
$309$	0	0
$310$	6.00000	0.340777
$311$	−18.3848	−1.04251	−0.521253	−	0.853402i	$-0.674535\pi$
$311$	−18.3848	−1.04251	−0.521253	+	0.853402i	$0.674535\pi$
$312$	0	0
$313$	−8.48528	−0.479616	−0.239808	−	0.970820i	$-0.577085\pi$
$313$	−8.48528	−0.479616	−0.239808	+	0.970820i	$0.577085\pi$
$314$	18.3848	1.03751
$315$	0	0
$316$	−6.00000	−0.337526
$317$	−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
$319$	0	0
$320$	−1.41421	−0.0790569
$321$	0	0
$322$	0	0
$323$	−6.00000	−0.333849
$324$	0	0
$325$	−16.9706	−0.941357
$326$	−12.0000	−0.664619
$327$	0	0
$328$	4.24264	0.234261
$329$	0	0
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	15.5563	0.853766
$333$	0	0
$334$	−19.7990	−1.08335
$335$	14.1421	0.772667
$336$	0	0
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	19.0000	1.03346
$339$	0	0
$340$	−2.00000	−0.108465
$341$	4.24264	0.229752
$342$	0	0
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	2.82843	0.152057
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	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
$353$	−25.4558	−1.35488	−0.677439	−	0.735579i	$-0.736910\pi$
$353$	−25.4558	−1.35488	−0.677439	+	0.735579i	$0.736910\pi$
$354$	0	0
$355$	0	0
$356$	14.1421	0.749532
$357$	0	0
$358$	6.00000	0.317110
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	−12.7279	−0.668965
$363$	0	0
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	21.2132	1.10732	0.553660	−	0.832743i	$-0.313231\pi$
$367$	21.2132	1.10732	0.553660	+	0.832743i	$0.313231\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	8.48528	0.441129
$371$	0	0
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	−1.41421	−0.0731272
$375$	0	0
$376$	−1.41421	−0.0729325
$377$	0	0
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	−16.0000	−0.818631
$383$	−18.3848	−0.939418	−0.469709	−	0.882821i	$-0.655641\pi$
$383$	−18.3848	−0.939418	−0.469709	+	0.882821i	$0.655641\pi$
$384$	0	0
$385$	0	0
$386$	−2.00000	−0.101797
$387$	0	0
$388$	−5.65685	−0.287183
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	−5.65685	−0.286079
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.48528	0.426941
$396$	0	0
$397$	−29.6985	−1.49052	−0.745262	−	0.666772i	$-0.767676\pi$
$397$	−29.6985	−1.49052	−0.745262	+	0.666772i	$0.767676\pi$
$398$	−18.3848	−0.921546
$399$	0	0
$400$	−3.00000	−0.150000
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	−24.0000	−1.19553
$404$	5.65685	0.281439
$405$	0	0
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	−4.24264	−0.209785	−0.104893	−	0.994484i	$-0.533450\pi$
$409$	−4.24264	−0.209785	−0.104893	+	0.994484i	$0.533450\pi$
$410$	−6.00000	−0.296319
$411$	0	0
$412$	7.07107	0.348367
$413$	0	0
$414$	0	0
$415$	−22.0000	−1.07994
$416$	5.65685	0.277350
$417$	0	0
$418$	4.24264	0.207514
$419$	11.3137	0.552711	0.276355	−	0.961056i	$-0.410873\pi$
$419$	11.3137	0.552711	0.276355	+	0.961056i	$0.410873\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−8.00000	−0.389434
$423$	0	0
$424$	6.00000	0.291386
$425$	−4.24264	−0.205798
$426$	0	0
$427$	0	0
$428$	−4.00000	−0.193347
$429$	0	0
$430$	5.65685	0.272798
$431$	26.0000	1.25238	0.626188	−	0.779672i	$-0.284614\pi$
$431$	26.0000	1.25238	0.626188	+	0.779672i	$0.284614\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−8.00000	−0.383131
$437$	16.9706	0.811812
$438$	0	0
$439$	−19.7990	−0.944954	−0.472477	−	0.881343i	$-0.656640\pi$
$439$	−19.7990	−0.944954	−0.472477	+	0.881343i	$0.656640\pi$
$440$	1.41421	0.0674200
$441$	0	0
$442$	8.00000	0.380521
$443$	−10.0000	−0.475114	−0.237557	−	0.971374i	$-0.576347\pi$
$443$	−10.0000	−0.475114	−0.237557	+	0.971374i	$0.576347\pi$
$444$	0	0
$445$	−20.0000	−0.948091
$446$	24.0416	1.13840
$447$	0	0
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−4.24264	−0.199778
$452$	−18.0000	−0.846649
$453$	0	0
$454$	−7.07107	−0.331862
$455$	0	0
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	7.07107	0.330409
$459$	0	0
$460$	5.65685	0.263752
$461$	−28.2843	−1.31733	−0.658665	−	0.752436i	$-0.728879\pi$
$461$	−28.2843	−1.31733	−0.658665	+	0.752436i	$0.728879\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	0	0
$465$	0	0
$466$	14.0000	0.648537
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	2.00000	0.0922531
$471$	0	0
$472$	−2.82843	−0.130189
$473$	4.00000	0.183920
$474$	0	0
$475$	12.7279	0.583997
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.65685	0.258468	0.129234	−	0.991614i	$-0.458748\pi$
$479$	5.65685	0.258468	0.129234	+	0.991614i	$0.458748\pi$
$480$	0	0
$481$	−33.9411	−1.54758
$482$	−1.41421	−0.0644157
$483$	0	0
$484$	1.00000	0.0454545
$485$	8.00000	0.363261
$486$	0	0
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	−5.65685	−0.256074
$489$	0	0
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	0	0
$494$	−24.0000	−1.07981
$495$	0	0
$496$	−4.24264	−0.190500
$497$	0	0
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	11.3137	0.505964
$501$	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
$505$	−8.00000	−0.355995
$506$	4.00000	0.177822
$507$	0	0
$508$	2.00000	0.0887357
$509$	−26.8701	−1.19099	−0.595497	−	0.803357i	$-0.703045\pi$
$509$	−26.8701	−1.19099	−0.595497	+	0.803357i	$0.703045\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	31.1127	1.37232
$515$	−10.0000	−0.440653
$516$	0	0
$517$	1.41421	0.0621970
$518$	0	0
$519$	0	0
$520$	−8.00000	−0.350823
$521$	−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
$523$	41.0122	1.79334	0.896669	−	0.442702i	$-0.145980\pi$
$523$	41.0122	1.79334	0.896669	+	0.442702i	$0.145980\pi$
$524$	−15.5563	−0.679582
$525$	0	0
$526$	−2.00000	−0.0872041
$527$	−6.00000	−0.261364
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−8.48528	−0.368577
$531$	0	0
$532$	0	0
$533$	24.0000	1.03956
$534$	0	0
$535$	5.65685	0.244567
$536$	−10.0000	−0.431934
$537$	0	0
$538$	12.7279	0.548740
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	2.82843	0.121491
$543$	0	0
$544$	1.41421	0.0606339
$545$	11.3137	0.484626
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	3.00000	0.127920
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−28.0000	−1.18961
$555$	0	0
$556$	−4.24264	−0.179928
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	0	0
$559$	−22.6274	−0.957038
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	15.5563	0.655622	0.327811	−	0.944743i	$-0.393689\pi$
$563$	15.5563	0.655622	0.327811	+	0.944743i	$0.393689\pi$
$564$	0	0
$565$	25.4558	1.07094
$566$	−15.5563	−0.653882
$567$	0	0
$568$	0	0
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	0	0
$571$	−24.0000	−1.00437	−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000	−1.00437	−0.502184	+	0.864761i	$0.667470\pi$
$572$	−5.65685	−0.236525
$573$	0	0
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	8.48528	0.353247	0.176623	−	0.984278i	$-0.443483\pi$
$577$	8.48528	0.353247	0.176623	+	0.984278i	$0.443483\pi$
$578$	−15.0000	−0.623918
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6.00000	−0.248495
$584$	−4.24264	−0.175562
$585$	0	0
$586$	−31.1127	−1.28525
$587$	31.1127	1.28416	0.642079	−	0.766638i	$-0.278072\pi$
$587$	31.1127	1.28416	0.642079	+	0.766638i	$0.278072\pi$
$588$	0	0
$589$	18.0000	0.741677
$590$	4.00000	0.164677
$591$	0	0
$592$	−6.00000	−0.246598
$593$	−7.07107	−0.290374	−0.145187	−	0.989404i	$-0.546378\pi$
$593$	−7.07107	−0.290374	−0.145187	+	0.989404i	$0.546378\pi$
$594$	0	0
$595$	0	0
$596$	−2.00000	−0.0819232
$597$	0	0
$598$	−22.6274	−0.925304
$599$	−20.0000	−0.817178	−0.408589	−	0.912719i	$-0.633979\pi$
$599$	−20.0000	−0.817178	−0.408589	+	0.912719i	$0.633979\pi$
$600$	0	0
$601$	−43.8406	−1.78830	−0.894148	−	0.447771i	$-0.852218\pi$
$601$	−43.8406	−1.78830	−0.894148	+	0.447771i	$0.852218\pi$
$602$	0	0
$603$	0	0
$604$	−16.0000	−0.651031
$605$	−1.41421	−0.0574960
$606$	0	0
$607$	16.9706	0.688814	0.344407	−	0.938820i	$-0.388080\pi$
$607$	16.9706	0.688814	0.344407	+	0.938820i	$0.388080\pi$
$608$	−4.24264	−0.172062
$609$	0	0
$610$	8.00000	0.323911
$611$	−8.00000	−0.323645
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	−7.07107	−0.285365
$615$	0	0
$616$	0	0
$617$	28.0000	1.12724	0.563619	−	0.826035i	$-0.309409\pi$
$617$	28.0000	1.12724	0.563619	+	0.826035i	$0.309409\pi$
$618$	0	0
$619$	11.3137	0.454736	0.227368	−	0.973809i	$-0.426988\pi$
$619$	11.3137	0.454736	0.227368	+	0.973809i	$0.426988\pi$
$620$	6.00000	0.240966
$621$	0	0
$622$	−18.3848	−0.737162
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	−8.48528	−0.339140
$627$	0	0
$628$	18.3848	0.733632
$629$	−8.48528	−0.338330
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	−6.00000	−0.238667
$633$	0	0
$634$	−22.0000	−0.873732
$635$	−2.82843	−0.112243
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	−1.41421	−0.0559017
$641$	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
$643$	36.7696	1.45005	0.725025	−	0.688723i	$-0.241828\pi$
$643$	36.7696	1.45005	0.725025	+	0.688723i	$0.241828\pi$
$644$	0	0
$645$	0	0
$646$	−6.00000	−0.236067
$647$	−7.07107	−0.277992	−0.138996	−	0.990293i	$-0.544388\pi$
$647$	−7.07107	−0.277992	−0.138996	+	0.990293i	$0.544388\pi$
$648$	0	0
$649$	2.82843	0.111025
$650$	−16.9706	−0.665640
$651$	0	0
$652$	−12.0000	−0.469956
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	22.0000	0.859611
$656$	4.24264	0.165647
$657$	0	0
$658$	0	0
$659$	32.0000	1.24654	0.623272	−	0.782006i	$-0.285803\pi$
$659$	32.0000	1.24654	0.623272	+	0.782006i	$0.285803\pi$
$660$	0	0
$661$	29.6985	1.15514	0.577569	−	0.816342i	$-0.304002\pi$
$661$	29.6985	1.15514	0.577569	+	0.816342i	$0.304002\pi$
$662$	2.00000	0.0777322
$663$	0	0
$664$	15.5563	0.603703
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−19.7990	−0.766046
$669$	0	0
$670$	14.1421	0.546358
$671$	5.65685	0.218380
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	6.00000	0.231111
$675$	0	0
$676$	19.0000	0.730769
$677$	2.82843	0.108705	0.0543526	−	0.998522i	$-0.482690\pi$
$677$	2.82843	0.108705	0.0543526	+	0.998522i	$0.482690\pi$
$678$	0	0
$679$	0	0
$680$	−2.00000	−0.0766965
$681$	0	0
$682$	4.24264	0.162459
$683$	−46.0000	−1.76014	−0.880071	−	0.474843i	$-0.842505\pi$
$683$	−46.0000	−1.76014	−0.880071	+	0.474843i	$0.842505\pi$
$684$	0	0
$685$	−8.48528	−0.324206
$686$	0	0
$687$	0	0
$688$	−4.00000	−0.152499
$689$	33.9411	1.29305
$690$	0	0
$691$	25.4558	0.968386	0.484193	−	0.874961i	$-0.339113\pi$
$691$	25.4558	0.968386	0.484193	+	0.874961i	$0.339113\pi$
$692$	2.82843	0.107521
$693$	0	0
$694$	28.0000	1.06287
$695$	6.00000	0.227593
$696$	0	0
$697$	6.00000	0.227266
$698$	25.4558	0.963518
$699$	0	0
$700$	0	0
$701$	46.0000	1.73740	0.868698	−	0.495342i	$-0.164957\pi$
$701$	46.0000	1.73740	0.868698	+	0.495342i	$0.164957\pi$
$702$	0	0
$703$	25.4558	0.960085
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−25.4558	−0.958043
$707$	0	0
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	0	0
$712$	14.1421	0.529999
$713$	16.9706	0.635553
$714$	0	0
$715$	8.00000	0.299183
$716$	6.00000	0.224231
$717$	0	0
$718$	−30.0000	−1.11959
$719$	21.2132	0.791119	0.395559	−	0.918440i	$-0.370551\pi$
$719$	21.2132	0.791119	0.395559	+	0.918440i	$0.370551\pi$
$720$	0	0
$721$	0	0
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−12.7279	−0.473029
$725$	0	0
$726$	0	0
$727$	18.3848	0.681854	0.340927	−	0.940090i	$-0.389259\pi$
$727$	18.3848	0.681854	0.340927	+	0.940090i	$0.389259\pi$
$728$	0	0
$729$	0	0
$730$	6.00000	0.222070
$731$	−5.65685	−0.209226
$732$	0	0
$733$	33.9411	1.25364	0.626822	−	0.779162i	$-0.284355\pi$
$733$	33.9411	1.25364	0.626822	+	0.779162i	$0.284355\pi$
$734$	21.2132	0.782994
$735$	0	0
$736$	−4.00000	−0.147442
$737$	10.0000	0.368355
$738$	0	0
$739$	−24.0000	−0.882854	−0.441427	−	0.897297i	$-0.645528\pi$
$739$	−24.0000	−0.882854	−0.441427	+	0.897297i	$0.645528\pi$
$740$	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
$745$	2.82843	0.103626
$746$	−26.0000	−0.951928
$747$	0	0
$748$	−1.41421	−0.0517088
$749$	0	0
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	−1.41421	−0.0515711
$753$	0	0
$754$	0	0
$755$	22.6274	0.823496
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	6.00000	0.217643
$761$	7.07107	0.256326	0.128163	−	0.991753i	$-0.459092\pi$
$761$	7.07107	0.256326	0.128163	+	0.991753i	$0.459092\pi$
$762$	0	0
$763$	0	0
$764$	−16.0000	−0.578860
$765$	0	0
$766$	−18.3848	−0.664269
$767$	−16.0000	−0.577727
$768$	0	0
$769$	26.8701	0.968959	0.484480	−	0.874803i	$-0.339009\pi$
$769$	26.8701	0.968959	0.484480	+	0.874803i	$0.339009\pi$
$770$	0	0
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	9.89949	0.356060	0.178030	−	0.984025i	$-0.443028\pi$
$773$	9.89949	0.356060	0.178030	+	0.984025i	$0.443028\pi$
$774$	0	0
$775$	12.7279	0.457200
$776$	−5.65685	−0.203069
$777$	0	0
$778$	14.0000	0.501924
$779$	−18.0000	−0.644917
$780$	0	0
$781$	0	0
$782$	−5.65685	−0.202289
$783$	0	0
$784$	0	0
$785$	−26.0000	−0.927980
$786$	0	0
$787$	−21.2132	−0.756169	−0.378085	−	0.925771i	$-0.623417\pi$
$787$	−21.2132	−0.756169	−0.378085	+	0.925771i	$0.623417\pi$
$788$	0	0
$789$	0	0
$790$	8.48528	0.301893
$791$	0	0
$792$	0	0
$793$	−32.0000	−1.13635
$794$	−29.6985	−1.05396
$795$	0	0
$796$	−18.3848	−0.651631
$797$	−18.3848	−0.651222	−0.325611	−	0.945504i	$-0.605570\pi$
$797$	−18.3848	−0.651222	−0.325611	+	0.945504i	$0.605570\pi$
$798$	0	0
$799$	−2.00000	−0.0707549
$800$	−3.00000	−0.106066
$801$	0	0
$802$	−2.00000	−0.0706225
$803$	4.24264	0.149720
$804$	0	0
$805$	0	0
$806$	−24.0000	−0.845364
$807$	0	0
$808$	5.65685	0.199007
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	−4.24264	−0.148979	−0.0744896	−	0.997222i	$-0.523733\pi$
$811$	−4.24264	−0.148979	−0.0744896	+	0.997222i	$0.523733\pi$
$812$	0	0
$813$	0	0
$814$	6.00000	0.210300
$815$	16.9706	0.594453
$816$	0	0
$817$	16.9706	0.593725
$818$	−4.24264	−0.148340
$819$	0	0
$820$	−6.00000	−0.209529
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	0	0
$823$	−48.0000	−1.67317	−0.836587	−	0.547833i	$-0.815453\pi$
$823$	−48.0000	−1.67317	−0.836587	+	0.547833i	$0.815453\pi$
$824$	7.07107	0.246332
$825$	0	0
$826$	0	0
$827$	52.0000	1.80822	0.904109	−	0.427303i	$-0.140536\pi$
$827$	52.0000	1.80822	0.904109	+	0.427303i	$0.140536\pi$
$828$	0	0
$829$	21.2132	0.736765	0.368383	−	0.929674i	$-0.379912\pi$
$829$	21.2132	0.736765	0.368383	+	0.929674i	$0.379912\pi$
$830$	−22.0000	−0.763631
$831$	0	0
$832$	5.65685	0.196116
$833$	0	0
$834$	0	0
$835$	28.0000	0.968980
$836$	4.24264	0.146735
$837$	0	0
$838$	11.3137	0.390826
$839$	15.5563	0.537065	0.268532	−	0.963271i	$-0.413461\pi$
$839$	15.5563	0.537065	0.268532	+	0.963271i	$0.413461\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	22.0000	0.758170
$843$	0	0
$844$	−8.00000	−0.275371
$845$	−26.8701	−0.924358
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	−4.24264	−0.145521
$851$	24.0000	0.822709
$852$	0	0
$853$	−33.9411	−1.16212	−0.581061	−	0.813860i	$-0.697362\pi$
$853$	−33.9411	−1.16212	−0.581061	+	0.813860i	$0.697362\pi$
$854$	0	0
$855$	0	0
$856$	−4.00000	−0.136717
$857$	−1.41421	−0.0483086	−0.0241543	−	0.999708i	$-0.507689\pi$
$857$	−1.41421	−0.0483086	−0.0241543	+	0.999708i	$0.507689\pi$
$858$	0	0
$859$	−19.7990	−0.675533	−0.337766	−	0.941230i	$-0.609671\pi$
$859$	−19.7990	−0.675533	−0.337766	+	0.941230i	$0.609671\pi$
$860$	5.65685	0.192897
$861$	0	0
$862$	26.0000	0.885564
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	0	0
$865$	−4.00000	−0.136004
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.00000	0.203536
$870$	0	0
$871$	−56.5685	−1.91675
$872$	−8.00000	−0.270914
$873$	0	0
$874$	16.9706	0.574038
$875$	0	0
$876$	0	0
$877$	4.00000	0.135070	0.0675352	−	0.997717i	$-0.478487\pi$
$877$	4.00000	0.135070	0.0675352	+	0.997717i	$0.478487\pi$
$878$	−19.7990	−0.668184
$879$	0	0
$880$	1.41421	0.0476731
$881$	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
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	−10.0000	−0.335957
$887$	45.2548	1.51951	0.759754	−	0.650210i	$-0.225319\pi$
$887$	45.2548	1.51951	0.759754	+	0.650210i	$0.225319\pi$
$888$	0	0
$889$	0	0
$890$	−20.0000	−0.670402
$891$	0	0
$892$	24.0416	0.804973
$893$	6.00000	0.200782
$894$	0	0
$895$	−8.48528	−0.283632
$896$	0	0
$897$	0	0
$898$	6.00000	0.200223
$899$	0	0
$900$	0	0
$901$	8.48528	0.282686
$902$	−4.24264	−0.141264
$903$	0	0
$904$	−18.0000	−0.598671
$905$	18.0000	0.598340
$906$	0	0
$907$	6.00000	0.199227	0.0996134	−	0.995026i	$-0.468239\pi$
$907$	6.00000	0.199227	0.0996134	+	0.995026i	$0.468239\pi$
$908$	−7.07107	−0.234662
$909$	0	0
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	0	0
$913$	−15.5563	−0.514840
$914$	−26.0000	−0.860004
$915$	0	0
$916$	7.07107	0.233635
$917$	0	0
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	5.65685	0.186501
$921$	0	0
$922$	−28.2843	−0.931493
$923$	0	0
$924$	0	0
$925$	18.0000	0.591836
$926$	24.0000	0.788689
$927$	0	0
$928$	0	0
$929$	2.82843	0.0927977	0.0463988	−	0.998923i	$-0.485225\pi$
$929$	2.82843	0.0927977	0.0463988	+	0.998923i	$0.485225\pi$
$930$	0	0
$931$	0	0
$932$	14.0000	0.458585
$933$	0	0
$934$	0	0
$935$	2.00000	0.0654070
$936$	0	0
$937$	−15.5563	−0.508204	−0.254102	−	0.967177i	$-0.581780\pi$
$937$	−15.5563	−0.508204	−0.254102	+	0.967177i	$0.581780\pi$
$938$	0	0
$939$	0	0
$940$	2.00000	0.0652328
$941$	−25.4558	−0.829837	−0.414918	−	0.909859i	$-0.636190\pi$
$941$	−25.4558	−0.829837	−0.414918	+	0.909859i	$0.636190\pi$
$942$	0	0
$943$	−16.9706	−0.552638
$944$	−2.82843	−0.0920575
$945$	0	0
$946$	4.00000	0.130051
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	0	0
$949$	−24.0000	−0.779073
$950$	12.7279	0.412948
$951$	0	0
$952$	0	0
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	0	0
$955$	22.6274	0.732206
$956$	0	0
$957$	0	0
$958$	5.65685	0.182765
$959$	0	0
$960$	0	0
$961$	−13.0000	−0.419355
$962$	−33.9411	−1.09431
$963$	0	0
$964$	−1.41421	−0.0455488
$965$	2.82843	0.0910503
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	8.00000	0.256865
$971$	22.6274	0.726148	0.363074	−	0.931760i	$-0.381727\pi$
$971$	22.6274	0.726148	0.363074	+	0.931760i	$0.381727\pi$
$972$	0	0
$973$	0	0
$974$	−4.00000	−0.128168
$975$	0	0
$976$	−5.65685	−0.181071
$977$	48.0000	1.53566	0.767828	−	0.640656i	$-0.221338\pi$
$977$	48.0000	1.53566	0.767828	+	0.640656i	$0.221338\pi$
$978$	0	0
$979$	−14.1421	−0.451985
$980$	0	0
$981$	0	0
$982$	−12.0000	−0.382935
$983$	9.89949	0.315745	0.157872	−	0.987460i	$-0.449537\pi$
$983$	9.89949	0.315745	0.157872	+	0.987460i	$0.449537\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−24.0000	−0.763542
$989$	16.0000	0.508770
$990$	0	0
$991$	36.0000	1.14358	0.571789	−	0.820401i	$-0.306250\pi$
$991$	36.0000	1.14358	0.571789	+	0.820401i	$0.306250\pi$
$992$	−4.24264	−0.134704
$993$	0	0
$994$	0	0
$995$	26.0000	0.824255
$996$	0	0
$997$	2.82843	0.0895772	0.0447886	−	0.998996i	$-0.485739\pi$
$997$	2.82843	0.0895772	0.0447886	+	0.998996i	$0.485739\pi$
$998$	−22.0000	−0.696398
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9702.2.a.ct.1.1	✓	2	21.20	even	2
9702.2.a.ct.1.2	yes	2	3.2	odd	2
9702.2.a.dg.1.1	yes	2	1.1	even	1	trivial
9702.2.a.dg.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{8} -1.41421 q^{10} -1.00000 q^{11} +5.65685 q^{13} +1.00000 q^{16} +1.41421 q^{17} -4.24264 q^{19} -1.41421 q^{20} -1.00000 q^{22} -4.00000 q^{23} -3.00000 q^{25} +5.65685 q^{26} -4.24264 q^{31} +1.00000 q^{32} +1.41421 q^{34} -6.00000 q^{37} -4.24264 q^{38} -1.41421 q^{40} +4.24264 q^{41} -4.00000 q^{43} -1.00000 q^{44} -4.00000 q^{46} -1.41421 q^{47} -3.00000 q^{50} +5.65685 q^{52} +6.00000 q^{53} +1.41421 q^{55} -2.82843 q^{59} -5.65685 q^{61} -4.24264 q^{62} +1.00000 q^{64} -8.00000 q^{65} -10.0000 q^{67} +1.41421 q^{68} -4.24264 q^{73} -6.00000 q^{74} -4.24264 q^{76} -6.00000 q^{79} -1.41421 q^{80} +4.24264 q^{82} +15.5563 q^{83} -2.00000 q^{85} -4.00000 q^{86} -1.00000 q^{88} +14.1421 q^{89} -4.00000 q^{92} -1.41421 q^{94} +6.00000 q^{95} -5.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} + 2 q^{16} - 2 q^{22} - 8 q^{23} - 6 q^{25} + 2 q^{32} - 12 q^{37} - 8 q^{43} - 2 q^{44} - 8 q^{46} - 6 q^{50} + 12 q^{53} + 2 q^{64} - 16 q^{65} - 20 q^{67} - 12 q^{74} - 12 q^{79} - 4 q^{85} - 8 q^{86} - 2 q^{88} - 8 q^{92} + 12 q^{95}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^11 + 2 * q^16 - 2 * q^22 - 8 * q^23 - 6 * q^25 + 2 * q^32 - 12 * q^37 - 8 * q^43 - 2 * q^44 - 8 * q^46 - 6 * q^50 + 12 * q^53 + 2 * q^64 - 16 * q^65 - 20 * q^67 - 12 * q^74 - 12 * q^79 - 4 * q^85 - 8 * q^86 - 2 * q^88 - 8 * q^92 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more