LMFDB - Embedded newform 9702.2.a.dk.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:	$2$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 1078)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	9702.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} +1.00000 q^{8} +2.82843 q^{10} -1.00000 q^{11} +5.65685 q^{13} +1.00000 q^{16} -2.82843 q^{17} +8.48528 q^{19} +2.82843 q^{20} -1.00000 q^{22} +8.00000 q^{23} +3.00000 q^{25} +5.65685 q^{26} +6.00000 q^{29} -8.48528 q^{31} +1.00000 q^{32} -2.82843 q^{34} -6.00000 q^{37} +8.48528 q^{38} +2.82843 q^{40} +8.48528 q^{41} -4.00000 q^{43} -1.00000 q^{44} +8.00000 q^{46} +2.82843 q^{47} +3.00000 q^{50} +5.65685 q^{52} -6.00000 q^{53} -2.82843 q^{55} +6.00000 q^{58} +5.65685 q^{59} -5.65685 q^{61} -8.48528 q^{62} +1.00000 q^{64} +16.0000 q^{65} -4.00000 q^{67} -2.82843 q^{68} -8.48528 q^{73} -6.00000 q^{74} +8.48528 q^{76} +2.82843 q^{80} +8.48528 q^{82} +2.82843 q^{83} -8.00000 q^{85} -4.00000 q^{86} -1.00000 q^{88} -11.3137 q^{89} +8.00000 q^{92} +2.82843 q^{94} +24.0000 q^{95} +11.3137 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 16 q^{23} + 6 q^{25} + 12 q^{29} + 2 q^{32} - 12 q^{37} - 8 q^{43} - 2 q^{44} + 16 q^{46} + 6 q^{50} - 12 q^{53} + 12 q^{58} + 2 q^{64} + 32 q^{65} - 8 q^{67} - 12 q^{74} - 16 q^{85} - 8 q^{86} - 2 q^{88} + 16 q^{92} + 48 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 + 6 * q^25 + 12 * q^29 + 2 * q^32 - 12 * q^37 - 8 * q^43 - 2 * q^44 + 16 * q^46 + 6 * q^50 - 12 * q^53 + 12 * q^58 + 2 * q^64 + 32 * q^65 - 8 * q^67 - 12 * q^74 - 16 * q^85 - 8 * q^86 - 2 * q^88 + 16 * q^92 + 48 * 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
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	2.82843	0.894427
$11$	−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
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	8.48528	1.94666	0.973329	−	0.229416i	$-0.0736815\pi$
$19$	8.48528	1.94666	0.973329	+	0.229416i	$0.0736815\pi$
$20$	2.82843	0.632456
$21$	0	0
$22$	−1.00000	−0.213201
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	5.65685	1.10940
$27$	0	0
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.82843	−0.485071
$35$	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	$0.664172\pi$
$38$	8.48528	1.37649
$39$	0	0
$40$	2.82843	0.447214
$41$	8.48528	1.32518	0.662589	−	0.748983i	$-0.269458\pi$
$41$	8.48528	1.32518	0.662589	+	0.748983i	$0.269458\pi$
$42$	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
$46$	8.00000	1.17954
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	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	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−2.82843	−0.381385
$56$	0	0
$57$	0	0
$58$	6.00000	0.787839
$59$	5.65685	0.736460	0.368230	−	0.929735i	$-0.379964\pi$
$59$	5.65685	0.736460	0.368230	+	0.929735i	$0.379964\pi$
$60$	0	0
$61$	−5.65685	−0.724286	−0.362143	−	0.932123i	$-0.617955\pi$
$61$	−5.65685	−0.724286	−0.362143	+	0.932123i	$0.617955\pi$
$62$	−8.48528	−1.07763
$63$	0	0
$64$	1.00000	0.125000
$65$	16.0000	1.98456
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−2.82843	−0.342997
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−8.48528	−0.993127	−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528	−0.993127	−0.496564	+	0.868000i	$0.665405\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	8.48528	0.973329
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	2.82843	0.316228
$81$	0	0
$82$	8.48528	0.937043
$83$	2.82843	0.310460	0.155230	−	0.987878i	$-0.450388\pi$
$83$	2.82843	0.310460	0.155230	+	0.987878i	$0.450388\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−11.3137	−1.19925	−0.599625	−	0.800281i	$-0.704684\pi$
$89$	−11.3137	−1.19925	−0.599625	+	0.800281i	$0.704684\pi$
$90$	0	0
$91$	0	0
$92$	8.00000	0.834058
$93$	0	0
$94$	2.82843	0.291730
$95$	24.0000	2.46235
$96$	0	0
$97$	11.3137	1.14873	0.574367	−	0.818598i	$-0.305248\pi$
$97$	11.3137	1.14873	0.574367	+	0.818598i	$0.305248\pi$
$98$	0	0
$99$	0	0
$100$	3.00000	0.300000
$101$	−11.3137	−1.12576	−0.562878	−	0.826540i	$-0.690306\pi$
$101$	−11.3137	−1.12576	−0.562878	+	0.826540i	$0.690306\pi$
$102$	0	0
$103$	−14.1421	−1.39347	−0.696733	−	0.717331i	$-0.745364\pi$
$103$	−14.1421	−1.39347	−0.696733	+	0.717331i	$0.745364\pi$
$104$	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
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	−2.82843	−0.269680
$111$	0	0
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	22.6274	2.11002
$116$	6.00000	0.557086
$117$	0	0
$118$	5.65685	0.520756
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−5.65685	−0.512148
$123$	0	0
$124$	−8.48528	−0.762001
$125$	−5.65685	−0.505964
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	16.0000	1.40329
$131$	−2.82843	−0.247121	−0.123560	−	0.992337i	$-0.539431\pi$
$131$	−2.82843	−0.247121	−0.123560	+	0.992337i	$0.539431\pi$
$132$	0	0
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−2.82843	−0.242536
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−8.48528	−0.719712	−0.359856	−	0.933008i	$-0.617174\pi$
$139$	−8.48528	−0.719712	−0.359856	+	0.933008i	$0.617174\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.65685	−0.473050
$144$	0	0
$145$	16.9706	1.40933
$146$	−8.48528	−0.702247
$147$	0	0
$148$	−6.00000	−0.493197
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	8.48528	0.688247
$153$	0	0
$154$	0	0
$155$	−24.0000	−1.92773
$156$	0	0
$157$	−19.7990	−1.58013	−0.790066	−	0.613022i	$-0.789954\pi$
$157$	−19.7990	−1.58013	−0.790066	+	0.613022i	$0.789954\pi$
$158$	0	0
$159$	0	0
$160$	2.82843	0.223607
$161$	0	0
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	8.48528	0.662589
$165$	0	0
$166$	2.82843	0.219529
$167$	5.65685	0.437741	0.218870	−	0.975754i	$-0.429763\pi$
$167$	5.65685	0.437741	0.218870	+	0.975754i	$0.429763\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	−8.00000	−0.613572
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−5.65685	−0.430083	−0.215041	−	0.976605i	$-0.568989\pi$
$173$	−5.65685	−0.430083	−0.215041	+	0.976605i	$0.568989\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−11.3137	−0.847998
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	8.48528	0.630706	0.315353	−	0.948974i	$-0.397877\pi$
$181$	8.48528	0.630706	0.315353	+	0.948974i	$0.397877\pi$
$182$	0	0
$183$	0	0
$184$	8.00000	0.589768
$185$	−16.9706	−1.24770
$186$	0	0
$187$	2.82843	0.206835
$188$	2.82843	0.206284
$189$	0	0
$190$	24.0000	1.74114
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	11.3137	0.812277
$195$	0	0
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−14.1421	−1.00251	−0.501255	−	0.865300i	$-0.667128\pi$
$199$	−14.1421	−1.00251	−0.501255	+	0.865300i	$0.667128\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	−11.3137	−0.796030
$203$	0	0
$204$	0	0
$205$	24.0000	1.67623
$206$	−14.1421	−0.985329
$207$	0	0
$208$	5.65685	0.392232
$209$	−8.48528	−0.586939
$210$	0	0
$211$	28.0000	1.92760	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000	1.92760	0.963800	+	0.266627i	$0.0859092\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−4.00000	−0.273434
$215$	−11.3137	−0.771589
$216$	0	0
$217$	0	0
$218$	−14.0000	−0.948200
$219$	0	0
$220$	−2.82843	−0.190693
$221$	−16.0000	−1.07628
$222$	0	0
$223$	2.82843	0.189405	0.0947027	−	0.995506i	$-0.469810\pi$
$223$	2.82843	0.189405	0.0947027	+	0.995506i	$0.469810\pi$
$224$	0	0
$225$	0	0
$226$	6.00000	0.399114
$227$	−2.82843	−0.187729	−0.0938647	−	0.995585i	$-0.529922\pi$
$227$	−2.82843	−0.187729	−0.0938647	+	0.995585i	$0.529922\pi$
$228$	0	0
$229$	−14.1421	−0.934539	−0.467269	−	0.884115i	$-0.654762\pi$
$229$	−14.1421	−0.934539	−0.467269	+	0.884115i	$0.654762\pi$
$230$	22.6274	1.49201
$231$	0	0
$232$	6.00000	0.393919
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	5.65685	0.368230
$237$	0	0
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	2.82843	0.182195	0.0910975	−	0.995842i	$-0.470963\pi$
$241$	2.82843	0.182195	0.0910975	+	0.995842i	$0.470963\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−5.65685	−0.362143
$245$	0	0
$246$	0	0
$247$	48.0000	3.05417
$248$	−8.48528	−0.538816
$249$	0	0
$250$	−5.65685	−0.357771
$251$	5.65685	0.357057	0.178529	−	0.983935i	$-0.442866\pi$
$251$	5.65685	0.357057	0.178529	+	0.983935i	$0.442866\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−11.3137	−0.705730	−0.352865	−	0.935674i	$-0.614792\pi$
$257$	−11.3137	−0.705730	−0.352865	+	0.935674i	$0.614792\pi$
$258$	0	0
$259$	0	0
$260$	16.0000	0.992278
$261$	0	0
$262$	−2.82843	−0.174741
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	−16.9706	−1.04249
$266$	0	0
$267$	0	0
$268$	−4.00000	−0.244339
$269$	25.4558	1.55207	0.776035	−	0.630690i	$-0.217228\pi$
$269$	25.4558	1.55207	0.776035	+	0.630690i	$0.217228\pi$
$270$	0	0
$271$	28.2843	1.71815	0.859074	−	0.511852i	$-0.171040\pi$
$271$	28.2843	1.71815	0.859074	+	0.511852i	$0.171040\pi$
$272$	−2.82843	−0.171499
$273$	0	0
$274$	18.0000	1.08742
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−8.48528	−0.508913
$279$	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
$283$	14.1421	0.840663	0.420331	−	0.907371i	$-0.361914\pi$
$283$	14.1421	0.840663	0.420331	+	0.907371i	$0.361914\pi$
$284$	0	0
$285$	0	0
$286$	−5.65685	−0.334497
$287$	0	0
$288$	0	0
$289$	−9.00000	−0.529412
$290$	16.9706	0.996546
$291$	0	0
$292$	−8.48528	−0.496564
$293$	−5.65685	−0.330477	−0.165238	−	0.986254i	$-0.552839\pi$
$293$	−5.65685	−0.330477	−0.165238	+	0.986254i	$0.552839\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	−6.00000	−0.348743
$297$	0	0
$298$	10.0000	0.579284
$299$	45.2548	2.61715
$300$	0	0
$301$	0	0
$302$	8.00000	0.460348
$303$	0	0
$304$	8.48528	0.486664
$305$	−16.0000	−0.916157
$306$	0	0
$307$	14.1421	0.807134	0.403567	−	0.914950i	$-0.367770\pi$
$307$	14.1421	0.807134	0.403567	+	0.914950i	$0.367770\pi$
$308$	0	0
$309$	0	0
$310$	−24.0000	−1.36311
$311$	−14.1421	−0.801927	−0.400963	−	0.916094i	$-0.631325\pi$
$311$	−14.1421	−0.801927	−0.400963	+	0.916094i	$0.631325\pi$
$312$	0	0
$313$	−16.9706	−0.959233	−0.479616	−	0.877478i	$-0.659224\pi$
$313$	−16.9706	−0.959233	−0.479616	+	0.877478i	$0.659224\pi$
$314$	−19.7990	−1.11732
$315$	0	0
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	2.82843	0.158114
$321$	0	0
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	16.9706	0.941357
$326$	12.0000	0.664619
$327$	0	0
$328$	8.48528	0.468521
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	2.82843	0.155230
$333$	0	0
$334$	5.65685	0.309529
$335$	−11.3137	−0.618134
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	19.0000	1.03346
$339$	0	0
$340$	−8.00000	−0.433861
$341$	8.48528	0.459504
$342$	0	0
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−5.65685	−0.304114
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	−11.3137	−0.599625
$357$	0	0
$358$	12.0000	0.634220
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	53.0000	2.78947
$362$	8.48528	0.445976
$363$	0	0
$364$	0	0
$365$	−24.0000	−1.25622
$366$	0	0
$367$	25.4558	1.32878	0.664392	−	0.747384i	$-0.268691\pi$
$367$	25.4558	1.32878	0.664392	+	0.747384i	$0.268691\pi$
$368$	8.00000	0.417029
$369$	0	0
$370$	−16.9706	−0.882258
$371$	0	0
$372$	0	0
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	2.82843	0.146254
$375$	0	0
$376$	2.82843	0.145865
$377$	33.9411	1.74806
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	24.0000	1.23117
$381$	0	0
$382$	8.00000	0.409316
$383$	−14.1421	−0.722629	−0.361315	−	0.932444i	$-0.617672\pi$
$383$	−14.1421	−0.722629	−0.361315	+	0.932444i	$0.617672\pi$
$384$	0	0
$385$	0	0
$386$	−14.0000	−0.712581
$387$	0	0
$388$	11.3137	0.574367
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	−22.6274	−1.14432
$392$	0	0
$393$	0	0
$394$	18.0000	0.906827
$395$	0	0
$396$	0	0
$397$	−8.48528	−0.425864	−0.212932	−	0.977067i	$-0.568301\pi$
$397$	−8.48528	−0.425864	−0.212932	+	0.977067i	$0.568301\pi$
$398$	−14.1421	−0.708881
$399$	0	0
$400$	3.00000	0.150000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	−48.0000	−2.39105
$404$	−11.3137	−0.562878
$405$	0	0
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	8.48528	0.419570	0.209785	−	0.977748i	$-0.432724\pi$
$409$	8.48528	0.419570	0.209785	+	0.977748i	$0.432724\pi$
$410$	24.0000	1.18528
$411$	0	0
$412$	−14.1421	−0.696733
$413$	0	0
$414$	0	0
$415$	8.00000	0.392705
$416$	5.65685	0.277350
$417$	0	0
$418$	−8.48528	−0.415029
$419$	−22.6274	−1.10542	−0.552711	−	0.833373i	$-0.686407\pi$
$419$	−22.6274	−1.10542	−0.552711	+	0.833373i	$0.686407\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	28.0000	1.36302
$423$	0	0
$424$	−6.00000	−0.291386
$425$	−8.48528	−0.411597
$426$	0	0
$427$	0	0
$428$	−4.00000	−0.193347
$429$	0	0
$430$	−11.3137	−0.545595
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	−33.9411	−1.63111	−0.815553	−	0.578682i	$-0.803567\pi$
$433$	−33.9411	−1.63111	−0.815553	+	0.578682i	$0.803567\pi$
$434$	0	0
$435$	0	0
$436$	−14.0000	−0.670478
$437$	67.8823	3.24725
$438$	0	0
$439$	−28.2843	−1.34993	−0.674967	−	0.737848i	$-0.735842\pi$
$439$	−28.2843	−1.34993	−0.674967	+	0.737848i	$0.735842\pi$
$440$	−2.82843	−0.134840
$441$	0	0
$442$	−16.0000	−0.761042
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	0	0
$445$	−32.0000	−1.51695
$446$	2.82843	0.133930
$447$	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
$451$	−8.48528	−0.399556
$452$	6.00000	0.282216
$453$	0	0
$454$	−2.82843	−0.132745
$455$	0	0
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−14.1421	−0.660819
$459$	0	0
$460$	22.6274	1.05501
$461$	−11.3137	−0.526932	−0.263466	−	0.964669i	$-0.584866\pi$
$461$	−11.3137	−0.526932	−0.263466	+	0.964669i	$0.584866\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	26.0000	1.20443
$467$	−33.9411	−1.57061	−0.785304	−	0.619110i	$-0.787493\pi$
$467$	−33.9411	−1.57061	−0.785304	+	0.619110i	$0.787493\pi$
$468$	0	0
$469$	0	0
$470$	8.00000	0.369012
$471$	0	0
$472$	5.65685	0.260378
$473$	4.00000	0.183920
$474$	0	0
$475$	25.4558	1.16799
$476$	0	0
$477$	0	0
$478$	−24.0000	−1.09773
$479$	−11.3137	−0.516937	−0.258468	−	0.966020i	$-0.583218\pi$
$479$	−11.3137	−0.516937	−0.258468	+	0.966020i	$0.583218\pi$
$480$	0	0
$481$	−33.9411	−1.54758
$482$	2.82843	0.128831
$483$	0	0
$484$	1.00000	0.0454545
$485$	32.0000	1.45305
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−5.65685	−0.256074
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−16.9706	−0.764316
$494$	48.0000	2.15962
$495$	0	0
$496$	−8.48528	−0.381000
$497$	0	0
$498$	0	0
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	−5.65685	−0.252982
$501$	0	0
$502$	5.65685	0.252478
$503$	5.65685	0.252227	0.126113	−	0.992016i	$-0.459750\pi$
$503$	5.65685	0.252227	0.126113	+	0.992016i	$0.459750\pi$
$504$	0	0
$505$	−32.0000	−1.42398
$506$	−8.00000	−0.355643
$507$	0	0
$508$	8.00000	0.354943
$509$	−31.1127	−1.37905	−0.689523	−	0.724264i	$-0.742180\pi$
$509$	−31.1127	−1.37905	−0.689523	+	0.724264i	$0.742180\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−11.3137	−0.499026
$515$	−40.0000	−1.76261
$516$	0	0
$517$	−2.82843	−0.124394
$518$	0	0
$519$	0	0
$520$	16.0000	0.701646
$521$	−11.3137	−0.495663	−0.247831	−	0.968803i	$-0.579718\pi$
$521$	−11.3137	−0.495663	−0.247831	+	0.968803i	$0.579718\pi$
$522$	0	0
$523$	−31.1127	−1.36046	−0.680232	−	0.732997i	$-0.738121\pi$
$523$	−31.1127	−1.36046	−0.680232	+	0.732997i	$0.738121\pi$
$524$	−2.82843	−0.123560
$525$	0	0
$526$	16.0000	0.697633
$527$	24.0000	1.04546
$528$	0	0
$529$	41.0000	1.78261
$530$	−16.9706	−0.737154
$531$	0	0
$532$	0	0
$533$	48.0000	2.07911
$534$	0	0
$535$	−11.3137	−0.489134
$536$	−4.00000	−0.172774
$537$	0	0
$538$	25.4558	1.09748
$539$	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	$0.568964\pi$
$542$	28.2843	1.21491
$543$	0	0
$544$	−2.82843	−0.121268
$545$	−39.5980	−1.69619
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	−3.00000	−0.127920
$551$	50.9117	2.16891
$552$	0	0
$553$	0	0
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−8.48528	−0.359856
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	0	0
$559$	−22.6274	−0.957038
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	19.7990	0.834428	0.417214	−	0.908808i	$-0.363007\pi$
$563$	19.7990	0.834428	0.417214	+	0.908808i	$0.363007\pi$
$564$	0	0
$565$	16.9706	0.713957
$566$	14.1421	0.594438
$567$	0	0
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	−5.65685	−0.236525
$573$	0	0
$574$	0	0
$575$	24.0000	1.00087
$576$	0	0
$577$	16.9706	0.706494	0.353247	−	0.935530i	$-0.385078\pi$
$577$	16.9706	0.706494	0.353247	+	0.935530i	$0.385078\pi$
$578$	−9.00000	−0.374351
$579$	0	0
$580$	16.9706	0.704664
$581$	0	0
$582$	0	0
$583$	6.00000	0.248495
$584$	−8.48528	−0.351123
$585$	0	0
$586$	−5.65685	−0.233682
$587$	39.5980	1.63438	0.817192	−	0.576366i	$-0.195530\pi$
$587$	39.5980	1.63438	0.817192	+	0.576366i	$0.195530\pi$
$588$	0	0
$589$	−72.0000	−2.96671
$590$	16.0000	0.658710
$591$	0	0
$592$	−6.00000	−0.246598
$593$	−36.7696	−1.50994	−0.754972	−	0.655757i	$-0.772350\pi$
$593$	−36.7696	−1.50994	−0.754972	+	0.655757i	$0.772350\pi$
$594$	0	0
$595$	0	0
$596$	10.0000	0.409616
$597$	0	0
$598$	45.2548	1.85061
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	19.7990	0.807618	0.403809	−	0.914843i	$-0.367686\pi$
$601$	19.7990	0.807618	0.403809	+	0.914843i	$0.367686\pi$
$602$	0	0
$603$	0	0
$604$	8.00000	0.325515
$605$	2.82843	0.114992
$606$	0	0
$607$	33.9411	1.37763	0.688814	−	0.724938i	$-0.258132\pi$
$607$	33.9411	1.37763	0.688814	+	0.724938i	$0.258132\pi$
$608$	8.48528	0.344124
$609$	0	0
$610$	−16.0000	−0.647821
$611$	16.0000	0.647291
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	14.1421	0.570730
$615$	0	0
$616$	0	0
$617$	−38.0000	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$618$	0	0
$619$	11.3137	0.454736	0.227368	−	0.973809i	$-0.426988\pi$
$619$	11.3137	0.454736	0.227368	+	0.973809i	$0.426988\pi$
$620$	−24.0000	−0.963863
$621$	0	0
$622$	−14.1421	−0.567048
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	−16.9706	−0.678280
$627$	0	0
$628$	−19.7990	−0.790066
$629$	16.9706	0.676661
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	0	0
$633$	0	0
$634$	2.00000	0.0794301
$635$	22.6274	0.897942
$636$	0	0
$637$	0	0
$638$	−6.00000	−0.237542
$639$	0	0
$640$	2.82843	0.111803
$641$	−46.0000	−1.81689	−0.908445	−	0.418004i	$-0.862730\pi$
$641$	−46.0000	−1.81689	−0.908445	+	0.418004i	$0.862730\pi$
$642$	0	0
$643$	−5.65685	−0.223085	−0.111542	−	0.993760i	$-0.535579\pi$
$643$	−5.65685	−0.223085	−0.111542	+	0.993760i	$0.535579\pi$
$644$	0	0
$645$	0	0
$646$	−24.0000	−0.944267
$647$	−36.7696	−1.44556	−0.722780	−	0.691078i	$-0.757136\pi$
$647$	−36.7696	−1.44556	−0.722780	+	0.691078i	$0.757136\pi$
$648$	0	0
$649$	−5.65685	−0.222051
$650$	16.9706	0.665640
$651$	0	0
$652$	12.0000	0.469956
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	8.48528	0.331295
$657$	0	0
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	8.48528	0.330039	0.165020	−	0.986290i	$-0.447231\pi$
$661$	8.48528	0.330039	0.165020	+	0.986290i	$0.447231\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	2.82843	0.109764
$665$	0	0
$666$	0	0
$667$	48.0000	1.85857
$668$	5.65685	0.218870
$669$	0	0
$670$	−11.3137	−0.437087
$671$	5.65685	0.218380
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	19.0000	0.730769
$677$	28.2843	1.08705	0.543526	−	0.839392i	$-0.317089\pi$
$677$	28.2843	1.08705	0.543526	+	0.839392i	$0.317089\pi$
$678$	0	0
$679$	0	0
$680$	−8.00000	−0.306786
$681$	0	0
$682$	8.48528	0.324918
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	0	0
$685$	50.9117	1.94524
$686$	0	0
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−33.9411	−1.29305
$690$	0	0
$691$	50.9117	1.93677	0.968386	−	0.249457i	$-0.0802520\pi$
$691$	50.9117	1.93677	0.968386	+	0.249457i	$0.0802520\pi$
$692$	−5.65685	−0.215041
$693$	0	0
$694$	4.00000	0.151838
$695$	−24.0000	−0.910372
$696$	0	0
$697$	−24.0000	−0.909065
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	−50.9117	−1.92017
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	0	0
$712$	−11.3137	−0.423999
$713$	−67.8823	−2.54221
$714$	0	0
$715$	−16.0000	−0.598366
$716$	12.0000	0.448461
$717$	0	0
$718$	−24.0000	−0.895672
$719$	−25.4558	−0.949343	−0.474671	−	0.880163i	$-0.657433\pi$
$719$	−25.4558	−0.949343	−0.474671	+	0.880163i	$0.657433\pi$
$720$	0	0
$721$	0	0
$722$	53.0000	1.97246
$723$	0	0
$724$	8.48528	0.315353
$725$	18.0000	0.668503
$726$	0	0
$727$	14.1421	0.524503	0.262251	−	0.965000i	$-0.415535\pi$
$727$	14.1421	0.524503	0.262251	+	0.965000i	$0.415535\pi$
$728$	0	0
$729$	0	0
$730$	−24.0000	−0.888280
$731$	11.3137	0.418453
$732$	0	0
$733$	16.9706	0.626822	0.313411	−	0.949618i	$-0.398528\pi$
$733$	16.9706	0.626822	0.313411	+	0.949618i	$0.398528\pi$
$734$	25.4558	0.939592
$735$	0	0
$736$	8.00000	0.294884
$737$	4.00000	0.147342
$738$	0	0
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	−16.9706	−0.623850
$741$	0	0
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	28.2843	1.03626
$746$	−2.00000	−0.0732252
$747$	0	0
$748$	2.82843	0.103418
$749$	0	0
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	2.82843	0.103142
$753$	0	0
$754$	33.9411	1.23606
$755$	22.6274	0.823496
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	−28.0000	−1.01701
$759$	0	0
$760$	24.0000	0.870572
$761$	2.82843	0.102530	0.0512652	−	0.998685i	$-0.483675\pi$
$761$	2.82843	0.102530	0.0512652	+	0.998685i	$0.483675\pi$
$762$	0	0
$763$	0	0
$764$	8.00000	0.289430
$765$	0	0
$766$	−14.1421	−0.510976
$767$	32.0000	1.15545
$768$	0	0
$769$	−36.7696	−1.32594	−0.662972	−	0.748644i	$-0.730705\pi$
$769$	−36.7696	−1.32594	−0.662972	+	0.748644i	$0.730705\pi$
$770$	0	0
$771$	0	0
$772$	−14.0000	−0.503871
$773$	−2.82843	−0.101731	−0.0508657	−	0.998706i	$-0.516198\pi$
$773$	−2.82843	−0.101731	−0.0508657	+	0.998706i	$0.516198\pi$
$774$	0	0
$775$	−25.4558	−0.914401
$776$	11.3137	0.406138
$777$	0	0
$778$	−10.0000	−0.358517
$779$	72.0000	2.57967
$780$	0	0
$781$	0	0
$782$	−22.6274	−0.809155
$783$	0	0
$784$	0	0
$785$	−56.0000	−1.99873
$786$	0	0
$787$	−25.4558	−0.907403	−0.453701	−	0.891154i	$-0.649897\pi$
$787$	−25.4558	−0.907403	−0.453701	+	0.891154i	$0.649897\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−32.0000	−1.13635
$794$	−8.48528	−0.301131
$795$	0	0
$796$	−14.1421	−0.501255
$797$	19.7990	0.701316	0.350658	−	0.936504i	$-0.385958\pi$
$797$	19.7990	0.701316	0.350658	+	0.936504i	$0.385958\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	3.00000	0.106066
$801$	0	0
$802$	10.0000	0.353112
$803$	8.48528	0.299439
$804$	0	0
$805$	0	0
$806$	−48.0000	−1.69073
$807$	0	0
$808$	−11.3137	−0.398015
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−8.48528	−0.297959	−0.148979	−	0.988840i	$-0.547599\pi$
$811$	−8.48528	−0.297959	−0.148979	+	0.988840i	$0.547599\pi$
$812$	0	0
$813$	0	0
$814$	6.00000	0.210300
$815$	33.9411	1.18891
$816$	0	0
$817$	−33.9411	−1.18745
$818$	8.48528	0.296681
$819$	0	0
$820$	24.0000	0.838116
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	−14.1421	−0.492665
$825$	0	0
$826$	0	0
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	0	0
$829$	−25.4558	−0.884118	−0.442059	−	0.896986i	$-0.645752\pi$
$829$	−25.4558	−0.884118	−0.442059	+	0.896986i	$0.645752\pi$
$830$	8.00000	0.277684
$831$	0	0
$832$	5.65685	0.196116
$833$	0	0
$834$	0	0
$835$	16.0000	0.553703
$836$	−8.48528	−0.293470
$837$	0	0
$838$	−22.6274	−0.781651
$839$	53.7401	1.85531	0.927657	−	0.373432i	$-0.121819\pi$
$839$	53.7401	1.85531	0.927657	+	0.373432i	$0.121819\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−26.0000	−0.896019
$843$	0	0
$844$	28.0000	0.963800
$845$	53.7401	1.84872
$846$	0	0
$847$	0	0
$848$	−6.00000	−0.206041
$849$	0	0
$850$	−8.48528	−0.291043
$851$	−48.0000	−1.64542
$852$	0	0
$853$	−50.9117	−1.74318	−0.871592	−	0.490233i	$-0.836912\pi$
$853$	−50.9117	−1.74318	−0.871592	+	0.490233i	$0.836912\pi$
$854$	0	0
$855$	0	0
$856$	−4.00000	−0.136717
$857$	2.82843	0.0966172	0.0483086	−	0.998832i	$-0.484617\pi$
$857$	2.82843	0.0966172	0.0483086	+	0.998832i	$0.484617\pi$
$858$	0	0
$859$	39.5980	1.35107	0.675533	−	0.737330i	$-0.263914\pi$
$859$	39.5980	1.35107	0.675533	+	0.737330i	$0.263914\pi$
$860$	−11.3137	−0.385794
$861$	0	0
$862$	−16.0000	−0.544962
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	−16.0000	−0.544016
$866$	−33.9411	−1.15337
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−22.6274	−0.766701
$872$	−14.0000	−0.474100
$873$	0	0
$874$	67.8823	2.29615
$875$	0	0
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	−28.2843	−0.954548
$879$	0	0
$880$	−2.82843	−0.0953463
$881$	−5.65685	−0.190584	−0.0952921	−	0.995449i	$-0.530379\pi$
$881$	−5.65685	−0.190584	−0.0952921	+	0.995449i	$0.530379\pi$
$882$	0	0
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	−16.0000	−0.538138
$885$	0	0
$886$	20.0000	0.671913
$887$	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
$890$	−32.0000	−1.07264
$891$	0	0
$892$	2.82843	0.0947027
$893$	24.0000	0.803129
$894$	0	0
$895$	33.9411	1.13453
$896$	0	0
$897$	0	0
$898$	6.00000	0.200223
$899$	−50.9117	−1.69800
$900$	0	0
$901$	16.9706	0.565371
$902$	−8.48528	−0.282529
$903$	0	0
$904$	6.00000	0.199557
$905$	24.0000	0.797787
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	−2.82843	−0.0938647
$909$	0	0
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	0	0
$913$	−2.82843	−0.0936073
$914$	10.0000	0.330771
$915$	0	0
$916$	−14.1421	−0.467269
$917$	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	$0.585007\pi$
$920$	22.6274	0.746004
$921$	0	0
$922$	−11.3137	−0.372597
$923$	0	0
$924$	0	0
$925$	−18.0000	−0.591836
$926$	−24.0000	−0.788689
$927$	0	0
$928$	6.00000	0.196960
$929$	11.3137	0.371191	0.185595	−	0.982626i	$-0.440579\pi$
$929$	11.3137	0.371191	0.185595	+	0.982626i	$0.440579\pi$
$930$	0	0
$931$	0	0
$932$	26.0000	0.851658
$933$	0	0
$934$	−33.9411	−1.11059
$935$	8.00000	0.261628
$936$	0	0
$937$	−19.7990	−0.646805	−0.323402	−	0.946262i	$-0.604827\pi$
$937$	−19.7990	−0.646805	−0.323402	+	0.946262i	$0.604827\pi$
$938$	0	0
$939$	0	0
$940$	8.00000	0.260931
$941$	−50.9117	−1.65967	−0.829837	−	0.558006i	$-0.811567\pi$
$941$	−50.9117	−1.65967	−0.829837	+	0.558006i	$0.811567\pi$
$942$	0	0
$943$	67.8823	2.21055
$944$	5.65685	0.184115
$945$	0	0
$946$	4.00000	0.130051
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	0	0
$949$	−48.0000	−1.55815
$950$	25.4558	0.825897
$951$	0	0
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	0	0
$955$	22.6274	0.732206
$956$	−24.0000	−0.776215
$957$	0	0
$958$	−11.3137	−0.365529
$959$	0	0
$960$	0	0
$961$	41.0000	1.32258
$962$	−33.9411	−1.09431
$963$	0	0
$964$	2.82843	0.0910975
$965$	−39.5980	−1.27470
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	32.0000	1.02746
$971$	−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
$974$	8.00000	0.256337
$975$	0	0
$976$	−5.65685	−0.181071
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	11.3137	0.361588
$980$	0	0
$981$	0	0
$982$	12.0000	0.382935
$983$	48.0833	1.53362	0.766809	−	0.641875i	$-0.221843\pi$
$983$	48.0833	1.53362	0.766809	+	0.641875i	$0.221843\pi$
$984$	0	0
$985$	50.9117	1.62218
$986$	−16.9706	−0.540453
$987$	0	0
$988$	48.0000	1.52708
$989$	−32.0000	−1.01754
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	−8.48528	−0.269408
$993$	0	0
$994$	0	0
$995$	−40.0000	−1.26809
$996$	0	0
$997$	11.3137	0.358309	0.179154	−	0.983821i	$-0.442664\pi$
$997$	11.3137	0.358309	0.179154	+	0.983821i	$0.442664\pi$
$998$	44.0000	1.39280
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.dk.1.2		2
3.2	odd	2		1078.2.a.o.1.1	✓	2
7.6	odd	2	inner	9702.2.a.dk.1.1		2
12.11	even	2		8624.2.a.bo.1.1		2
21.2	odd	6		1078.2.e.u.67.2		4
21.5	even	6		1078.2.e.u.67.1		4
21.11	odd	6		1078.2.e.u.177.2		4
21.17	even	6		1078.2.e.u.177.1		4
21.20	even	2		1078.2.a.o.1.2	yes	2
84.83	odd	2		8624.2.a.bo.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1078.2.a.o.1.1	✓	2	3.2	odd	2
1078.2.a.o.1.2	yes	2	21.20	even	2
1078.2.e.u.67.1		4	21.5	even	6
1078.2.e.u.67.2		4	21.2	odd	6
1078.2.e.u.177.1		4	21.17	even	6
1078.2.e.u.177.2		4	21.11	odd	6
8624.2.a.bo.1.1		2	12.11	even	2
8624.2.a.bo.1.2		2	84.83	odd	2
9702.2.a.dk.1.1		2	7.6	odd	2	inner
9702.2.a.dk.1.2		2	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} +1.00000 q^{8} +2.82843 q^{10} -1.00000 q^{11} +5.65685 q^{13} +1.00000 q^{16} -2.82843 q^{17} +8.48528 q^{19} +2.82843 q^{20} -1.00000 q^{22} +8.00000 q^{23} +3.00000 q^{25} +5.65685 q^{26} +6.00000 q^{29} -8.48528 q^{31} +1.00000 q^{32} -2.82843 q^{34} -6.00000 q^{37} +8.48528 q^{38} +2.82843 q^{40} +8.48528 q^{41} -4.00000 q^{43} -1.00000 q^{44} +8.00000 q^{46} +2.82843 q^{47} +3.00000 q^{50} +5.65685 q^{52} -6.00000 q^{53} -2.82843 q^{55} +6.00000 q^{58} +5.65685 q^{59} -5.65685 q^{61} -8.48528 q^{62} +1.00000 q^{64} +16.0000 q^{65} -4.00000 q^{67} -2.82843 q^{68} -8.48528 q^{73} -6.00000 q^{74} +8.48528 q^{76} +2.82843 q^{80} +8.48528 q^{82} +2.82843 q^{83} -8.00000 q^{85} -4.00000 q^{86} -1.00000 q^{88} -11.3137 q^{89} +8.00000 q^{92} +2.82843 q^{94} +24.0000 q^{95} +11.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} + 2 q^{16} - 2 q^{22} + 16 q^{23} + 6 q^{25} + 12 q^{29} + 2 q^{32} - 12 q^{37} - 8 q^{43} - 2 q^{44} + 16 q^{46} + 6 q^{50} - 12 q^{53} + 12 q^{58} + 2 q^{64} + 32 q^{65} - 8 q^{67} - 12 q^{74} - 16 q^{85} - 8 q^{86} - 2 q^{88} + 16 q^{92} + 48 q^{95}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^11 + 2 * q^16 - 2 * q^22 + 16 * q^23 + 6 * q^25 + 12 * q^29 + 2 * q^32 - 12 * q^37 - 8 * q^43 - 2 * q^44 + 16 * q^46 + 6 * q^50 - 12 * q^53 + 12 * q^58 + 2 * q^64 + 32 * q^65 - 8 * q^67 - 12 * q^74 - 16 * q^85 - 8 * q^86 - 2 * q^88 + 16 * q^92 + 48 * q^95

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