LMFDB - Embedded newform 9801.2.a.bv.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9801,2,Mod(1,9801)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9801, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9801.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9801 = 3^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9801.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:	$78.2613790211$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.785052705024.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{6} + 27x^{4} - 13x^{2} + 1 $ x^8 - 10x^6 + 27x^4 - 13*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.35688$ of defining polynomial
Character	$\chi$	$=$	9801.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35688 q^{2} +3.55487 q^{4} +0.424290 q^{5} +0.792469 q^{7} -3.66464 q^{8} +O(q^{10})$	$q-2.35688 q^{2} +3.55487 q^{4} +0.424290 q^{5} +0.792469 q^{7} -3.66464 q^{8} -1.00000 q^{10} -0.527361 q^{13} -1.86775 q^{14} +1.52736 q^{16} +4.51322 q^{17} -3.34734 q^{19} +1.50830 q^{20} -6.02151 q^{23} -4.81998 q^{25} +1.24293 q^{26} +2.81712 q^{28} +9.82185 q^{29} -0.942424 q^{31} +3.72947 q^{32} -10.6371 q^{34} +0.336237 q^{35} -8.40206 q^{37} +7.88927 q^{38} -1.55487 q^{40} -3.47686 q^{41} +3.49729 q^{43} +14.1920 q^{46} +10.7181 q^{47} -6.37199 q^{49} +11.3601 q^{50} -1.87470 q^{52} +1.63124 q^{53} -2.90411 q^{56} -23.1489 q^{58} +3.69953 q^{59} +8.24955 q^{61} +2.22118 q^{62} -11.8446 q^{64} -0.223754 q^{65} -3.73775 q^{67} +16.0439 q^{68} -0.792469 q^{70} +1.31449 q^{71} -13.1948 q^{73} +19.8026 q^{74} -11.8994 q^{76} +13.3966 q^{79} +0.648044 q^{80} +8.19453 q^{82} -3.00987 q^{83} +1.91491 q^{85} -8.24269 q^{86} -8.43277 q^{89} -0.417917 q^{91} -21.4057 q^{92} -25.2612 q^{94} -1.42024 q^{95} +4.31698 q^{97} +15.0180 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4} - 2 q^{7}+O(q^{10}) $ 8 * q + 4 * q^4 - 2 * q^7	$ 8 q + 4 q^{4} - 2 q^{7} - 8 q^{10} + 4 q^{13} + 4 q^{16} + 6 q^{19} - 14 q^{25} - 10 q^{28} - 16 q^{31} - 28 q^{34} - 18 q^{37} + 12 q^{40} + 12 q^{43} + 32 q^{46} + 6 q^{49} + 26 q^{52} - 30 q^{58} - 24 q^{61} - 46 q^{64} - 38 q^{67} + 2 q^{70} - 32 q^{73} - 30 q^{76} + 26 q^{79} + 8 q^{82} + 40 q^{85} - 28 q^{91} - 22 q^{94} + 10 q^{97}+O(q^{100}) $ 8 * q + 4 * q^4 - 2 * q^7 - 8 * q^10 + 4 * q^13 + 4 * q^16 + 6 * q^19 - 14 * q^25 - 10 * q^28 - 16 * q^31 - 28 * q^34 - 18 * q^37 + 12 * q^40 + 12 * q^43 + 32 * q^46 + 6 * q^49 + 26 * q^52 - 30 * q^58 - 24 * q^61 - 46 * q^64 - 38 * q^67 + 2 * q^70 - 32 * q^73 - 30 * q^76 + 26 * q^79 + 8 * q^82 + 40 * q^85 - 28 * q^91 - 22 * q^94 + 10 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.35688	−1.66656	−0.833282	−	0.552848i	$-0.813541\pi$
$2$	−2.35688	−1.66656	−0.833282	+	0.552848i	$0.813541\pi$
$3$	0	0
$3$	0	0
$4$	3.55487	1.77744
$5$	0.424290	0.189748	0.0948742	−	0.995489i	$-0.469755\pi$
$5$	0.424290	0.189748	0.0948742	+	0.995489i	$0.469755\pi$
$6$	0	0
$7$	0.792469	0.299525	0.149763	−	0.988722i	$-0.452149\pi$
$7$	0.792469	0.299525	0.149763	+	0.988722i	$0.452149\pi$
$8$	−3.66464	−1.29565
$9$	0	0
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.527361	−0.146264	−0.0731318	−	0.997322i	$-0.523299\pi$
$13$	−0.527361	−0.146264	−0.0731318	+	0.997322i	$0.523299\pi$
$14$	−1.86775	−0.499178
$15$	0	0
$16$	1.52736	0.381840
$17$	4.51322	1.09462	0.547308	−	0.836931i	$-0.315653\pi$
$17$	4.51322	1.09462	0.547308	+	0.836931i	$0.315653\pi$
$18$	0	0
$19$	−3.34734	−0.767932	−0.383966	−	0.923347i	$-0.625442\pi$
$19$	−3.34734	−0.767932	−0.383966	+	0.923347i	$0.625442\pi$
$20$	1.50830	0.337265
$21$	0	0
$22$	0	0
$23$	−6.02151	−1.25557	−0.627786	−	0.778386i	$-0.716039\pi$
$23$	−6.02151	−1.25557	−0.627786	+	0.778386i	$0.716039\pi$
$24$	0	0
$25$	−4.81998	−0.963996
$26$	1.24293	0.243758
$27$	0	0
$28$	2.81712	0.532386
$29$	9.82185	1.82387	0.911936	−	0.410332i	$-0.134587\pi$
$29$	9.82185	1.82387	0.911936	+	0.410332i	$0.134587\pi$
$30$	0	0
$31$	−0.942424	−0.169264	−0.0846321	−	0.996412i	$-0.526972\pi$
$31$	−0.942424	−0.169264	−0.0846321	+	0.996412i	$0.526972\pi$
$32$	3.72947	0.659284
$33$	0	0
$34$	−10.6371	−1.82425
$35$	0.336237	0.0568344
$36$	0	0
$37$	−8.40206	−1.38129	−0.690645	−	0.723194i	$-0.742673\pi$
$37$	−8.40206	−1.38129	−0.690645	+	0.723194i	$0.742673\pi$
$38$	7.88927	1.27981
$39$	0	0
$40$	−1.55487	−0.245847
$41$	−3.47686	−0.542994	−0.271497	−	0.962439i	$-0.587519\pi$
$41$	−3.47686	−0.542994	−0.271497	+	0.962439i	$0.587519\pi$
$42$	0	0
$43$	3.49729	0.533332	0.266666	−	0.963789i	$-0.414078\pi$
$43$	3.49729	0.533332	0.266666	+	0.963789i	$0.414078\pi$
$44$	0	0
$45$	0	0
$46$	14.1920	2.09249
$47$	10.7181	1.56339	0.781696	−	0.623659i	$-0.214355\pi$
$47$	10.7181	1.56339	0.781696	+	0.623659i	$0.214355\pi$
$48$	0	0
$49$	−6.37199	−0.910285
$50$	11.3601	1.60656
$51$	0	0
$52$	−1.87470	−0.259974
$53$	1.63124	0.224068	0.112034	−	0.993704i	$-0.464263\pi$
$53$	1.63124	0.224068	0.112034	+	0.993704i	$0.464263\pi$
$54$	0	0
$55$	0	0
$56$	−2.90411	−0.388078
$57$	0	0
$58$	−23.1489	−3.03960
$59$	3.69953	0.481637	0.240819	−	0.970570i	$-0.422584\pi$
$59$	3.69953	0.481637	0.240819	+	0.970570i	$0.422584\pi$
$60$	0	0
$61$	8.24955	1.05625	0.528123	−	0.849168i	$-0.322896\pi$
$61$	8.24955	1.05625	0.528123	+	0.849168i	$0.322896\pi$
$62$	2.22118	0.282090
$63$	0	0
$64$	−11.8446	−1.48058
$65$	−0.223754	−0.0277533
$66$	0	0
$67$	−3.73775	−0.456638	−0.228319	−	0.973586i	$-0.573323\pi$
$67$	−3.73775	−0.456638	−0.228319	+	0.973586i	$0.573323\pi$
$68$	16.0439	1.94561
$69$	0	0
$70$	−0.792469	−0.0947181
$71$	1.31449	0.156001	0.0780005	−	0.996953i	$-0.475146\pi$
$71$	1.31449	0.156001	0.0780005	+	0.996953i	$0.475146\pi$
$72$	0	0
$73$	−13.1948	−1.54434	−0.772169	−	0.635418i	$-0.780828\pi$
$73$	−13.1948	−1.54434	−0.772169	+	0.635418i	$0.780828\pi$
$74$	19.8026	2.30201
$75$	0	0
$76$	−11.8994	−1.36495
$77$	0	0
$78$	0	0
$79$	13.3966	1.50724	0.753620	−	0.657310i	$-0.228306\pi$
$79$	13.3966	1.50724	0.753620	+	0.657310i	$0.228306\pi$
$80$	0.648044	0.0724536
$81$	0	0
$82$	8.19453	0.904934
$83$	−3.00987	−0.330376	−0.165188	−	0.986262i	$-0.552823\pi$
$83$	−3.00987	−0.330376	−0.165188	+	0.986262i	$0.552823\pi$
$84$	0	0
$85$	1.91491	0.207702
$86$	−8.24269	−0.888832
$87$	0	0
$88$	0	0
$89$	−8.43277	−0.893872	−0.446936	−	0.894566i	$-0.647485\pi$
$89$	−8.43277	−0.893872	−0.446936	+	0.894566i	$0.647485\pi$
$90$	0	0
$91$	−0.417917	−0.0438096
$92$	−21.4057	−2.23170
$93$	0	0
$94$	−25.2612	−2.60549
$95$	−1.42024	−0.145714
$96$	0	0
$97$	4.31698	0.438322	0.219161	−	0.975689i	$-0.429668\pi$
$97$	4.31698	0.438322	0.219161	+	0.975689i	$0.429668\pi$
$98$	15.0180	1.51705
$99$	0	0
$100$	−17.1344	−1.71344
$101$	−9.80466	−0.975601	−0.487800	−	0.872955i	$-0.662201\pi$
$101$	−9.80466	−0.975601	−0.487800	+	0.872955i	$0.662201\pi$
$102$	0	0
$103$	14.6187	1.44042	0.720211	−	0.693755i	$-0.244045\pi$
$103$	14.6187	1.44042	0.720211	+	0.693755i	$0.244045\pi$
$104$	1.93259	0.189506
$105$	0	0
$106$	−3.84463	−0.373424
$107$	−13.2383	−1.27980	−0.639898	−	0.768460i	$-0.721023\pi$
$107$	−13.2383	−1.27980	−0.639898	+	0.768460i	$0.721023\pi$
$108$	0	0
$109$	2.84463	0.272466	0.136233	−	0.990677i	$-0.456500\pi$
$109$	2.84463	0.272466	0.136233	+	0.990677i	$0.456500\pi$
$110$	0	0
$111$	0	0
$112$	1.21039	0.114371
$113$	−8.80871	−0.828654	−0.414327	−	0.910128i	$-0.635983\pi$
$113$	−8.80871	−0.828654	−0.414327	+	0.910128i	$0.635983\pi$
$114$	0	0
$115$	−2.55487	−0.238243
$116$	34.9154	3.24181
$117$	0	0
$118$	−8.71933	−0.802680
$119$	3.57658	0.327865
$120$	0	0
$121$	0	0
$122$	−19.4432	−1.76030
$123$	0	0
$124$	−3.35019	−0.300856
$125$	−4.16652	−0.372665
$126$	0	0
$127$	13.5915	1.20605	0.603024	−	0.797723i	$-0.293962\pi$
$127$	13.5915	1.20605	0.603024	+	0.797723i	$0.293962\pi$
$128$	20.4574	1.80820
$129$	0	0
$130$	0.527361	0.0462526
$131$	−5.34904	−0.467348	−0.233674	−	0.972315i	$-0.575075\pi$
$131$	−5.34904	−0.467348	−0.233674	+	0.972315i	$0.575075\pi$
$132$	0	0
$133$	−2.65266	−0.230015
$134$	8.80941	0.761017
$135$	0	0
$136$	−16.5393	−1.41823
$137$	17.0222	1.45430	0.727150	−	0.686478i	$-0.240844\pi$
$137$	17.0222	1.45430	0.727150	+	0.686478i	$0.240844\pi$
$138$	0	0
$139$	−19.8291	−1.68188	−0.840940	−	0.541128i	$-0.817997\pi$
$139$	−19.8291	−1.68188	−0.840940	+	0.541128i	$0.817997\pi$
$140$	1.19528	0.101019
$141$	0	0
$142$	−3.09809	−0.259986
$143$	0	0
$144$	0	0
$145$	4.16732	0.346077
$146$	31.0986	2.57374
$147$	0	0
$148$	−29.8682	−2.45515
$149$	13.3935	1.09724	0.548619	−	0.836073i	$-0.315154\pi$
$149$	13.3935	1.09724	0.548619	+	0.836073i	$0.315154\pi$
$150$	0	0
$151$	−6.89965	−0.561486	−0.280743	−	0.959783i	$-0.590581\pi$
$151$	−6.89965	−0.561486	−0.280743	+	0.959783i	$0.590581\pi$
$152$	12.2668	0.994968
$153$	0	0
$154$	0	0
$155$	−0.399861	−0.0321176
$156$	0	0
$157$	−17.2413	−1.37600	−0.688002	−	0.725709i	$-0.741512\pi$
$157$	−17.2413	−1.37600	−0.688002	+	0.725709i	$0.741512\pi$
$158$	−31.5743	−2.51191
$159$	0	0
$160$	1.58238	0.125098
$161$	−4.77186	−0.376075
$162$	0	0
$163$	0.265108	0.0207648	0.0103824	−	0.999946i	$-0.496695\pi$
$163$	0.265108	0.0207648	0.0103824	+	0.999946i	$0.496695\pi$
$164$	−12.3598	−0.965137
$165$	0	0
$166$	7.09388	0.550592
$167$	11.9134	0.921883	0.460942	−	0.887430i	$-0.347512\pi$
$167$	11.9134	0.921883	0.460942	+	0.887430i	$0.347512\pi$
$168$	0	0
$169$	−12.7219	−0.978607
$170$	−4.51322	−0.346148
$171$	0	0
$172$	12.4324	0.947963
$173$	0.316752	0.0240822	0.0120411	−	0.999928i	$-0.496167\pi$
$173$	0.316752	0.0240822	0.0120411	+	0.999928i	$0.496167\pi$
$174$	0	0
$175$	−3.81968	−0.288741
$176$	0	0
$177$	0	0
$178$	19.8750	1.48969
$179$	3.65861	0.273457	0.136729	−	0.990609i	$-0.456341\pi$
$179$	3.65861	0.273457	0.136729	+	0.990609i	$0.456341\pi$
$180$	0	0
$181$	16.1590	1.20109	0.600546	−	0.799590i	$-0.294950\pi$
$181$	16.1590	1.20109	0.600546	+	0.799590i	$0.294950\pi$
$182$	0.984979	0.0730115
$183$	0	0
$184$	22.0667	1.62678
$185$	−3.56491	−0.262098
$186$	0	0
$187$	0	0
$188$	38.1014	2.77883
$189$	0	0
$190$	3.34734	0.242841
$191$	−21.2932	−1.54072	−0.770362	−	0.637607i	$-0.779924\pi$
$191$	−21.2932	−1.54072	−0.770362	+	0.637607i	$0.779924\pi$
$192$	0	0
$193$	−16.2986	−1.17320	−0.586598	−	0.809878i	$-0.699533\pi$
$193$	−16.2986	−1.17320	−0.586598	+	0.809878i	$0.699533\pi$
$194$	−10.1746	−0.730492
$195$	0	0
$196$	−22.6516	−1.61797
$197$	12.0119	0.855811	0.427905	−	0.903824i	$-0.359252\pi$
$197$	12.0119	0.855811	0.427905	+	0.903824i	$0.359252\pi$
$198$	0	0
$199$	−25.8892	−1.83524	−0.917619	−	0.397462i	$-0.869891\pi$
$199$	−25.8892	−1.83524	−0.917619	+	0.397462i	$0.869891\pi$
$200$	17.6635	1.24900
$201$	0	0
$202$	23.1084	1.62590
$203$	7.78351	0.546295
$204$	0	0
$205$	−1.47520	−0.103032
$206$	−34.4544	−2.40056
$207$	0	0
$208$	−0.805471	−0.0558493
$209$	0	0
$210$	0	0
$211$	−12.6187	−0.868706	−0.434353	−	0.900743i	$-0.643023\pi$
$211$	−12.6187	−0.868706	−0.434353	+	0.900743i	$0.643023\pi$
$212$	5.79885	0.398266
$213$	0	0
$214$	31.2011	2.13286
$215$	1.48387	0.101199
$216$	0	0
$217$	−0.746841	−0.0506989
$218$	−6.70445	−0.454083
$219$	0	0
$220$	0	0
$221$	−2.38010	−0.160103
$222$	0	0
$223$	6.80983	0.456020	0.228010	−	0.973659i	$-0.426778\pi$
$223$	6.80983	0.456020	0.228010	+	0.973659i	$0.426778\pi$
$224$	2.95549	0.197472
$225$	0	0
$226$	20.7611	1.38100
$227$	−12.4257	−0.824723	−0.412361	−	0.911020i	$-0.635296\pi$
$227$	−12.4257	−0.824723	−0.412361	+	0.911020i	$0.635296\pi$
$228$	0	0
$229$	−16.4296	−1.08570	−0.542848	−	0.839831i	$-0.682654\pi$
$229$	−16.4296	−1.08570	−0.542848	+	0.839831i	$0.682654\pi$
$230$	6.02151	0.397047
$231$	0	0
$232$	−35.9935	−2.36309
$233$	−2.65593	−0.173996	−0.0869978	−	0.996209i	$-0.527727\pi$
$233$	−2.65593	−0.173996	−0.0869978	+	0.996209i	$0.527727\pi$
$234$	0	0
$235$	4.54758	0.296651
$236$	13.1513	0.856079
$237$	0	0
$238$	−8.42957	−0.546408
$239$	−26.1483	−1.69139	−0.845697	−	0.533663i	$-0.820815\pi$
$239$	−26.1483	−1.69139	−0.845697	+	0.533663i	$0.820815\pi$
$240$	0	0
$241$	−17.9294	−1.15494	−0.577468	−	0.816413i	$-0.695959\pi$
$241$	−17.9294	−1.15494	−0.577468	+	0.816413i	$0.695959\pi$
$242$	0	0
$243$	0	0
$244$	29.3261	1.87741
$245$	−2.70357	−0.172725
$246$	0	0
$247$	1.76526	0.112321
$248$	3.45364	0.219306
$249$	0	0
$250$	9.81998	0.621070
$251$	−21.5470	−1.36004	−0.680019	−	0.733195i	$-0.738028\pi$
$251$	−21.5470	−1.36004	−0.680019	+	0.733195i	$0.738028\pi$
$252$	0	0
$253$	0	0
$254$	−32.0334	−2.00996
$255$	0	0
$256$	−24.5263	−1.53289
$257$	5.88017	0.366795	0.183398	−	0.983039i	$-0.441290\pi$
$257$	5.88017	0.366795	0.183398	+	0.983039i	$0.441290\pi$
$258$	0	0
$259$	−6.65837	−0.413731
$260$	−0.795417	−0.0493297
$261$	0	0
$262$	12.6070	0.778865
$263$	15.4674	0.953762	0.476881	−	0.878968i	$-0.341767\pi$
$263$	15.4674	0.953762	0.476881	+	0.878968i	$0.341767\pi$
$264$	0	0
$265$	0.692119	0.0425165
$266$	6.25200	0.383335
$267$	0	0
$268$	−13.2872	−0.811645
$269$	20.7655	1.26609	0.633047	−	0.774114i	$-0.281804\pi$
$269$	20.7655	1.26609	0.633047	+	0.774114i	$0.281804\pi$
$270$	0	0
$271$	28.2311	1.71492	0.857460	−	0.514551i	$-0.172041\pi$
$271$	28.2311	1.71492	0.857460	+	0.514551i	$0.172041\pi$
$272$	6.89331	0.417969
$273$	0	0
$274$	−40.1191	−2.42369
$275$	0	0
$276$	0	0
$277$	8.59659	0.516519	0.258260	−	0.966076i	$-0.416851\pi$
$277$	8.59659	0.516519	0.258260	+	0.966076i	$0.416851\pi$
$278$	46.7347	2.80296
$279$	0	0
$280$	−1.23219	−0.0736372
$281$	4.15606	0.247930	0.123965	−	0.992287i	$-0.460439\pi$
$281$	4.15606	0.247930	0.123965	+	0.992287i	$0.460439\pi$
$282$	0	0
$283$	−6.78450	−0.403297	−0.201648	−	0.979458i	$-0.564630\pi$
$283$	−6.78450	−0.403297	−0.201648	+	0.979458i	$0.564630\pi$
$284$	4.67283	0.277282
$285$	0	0
$286$	0	0
$287$	−2.75530	−0.162640
$288$	0	0
$289$	3.36914	0.198185
$290$	−9.82185	−0.576759
$291$	0	0
$292$	−46.9059	−2.74496
$293$	4.58903	0.268094	0.134047	−	0.990975i	$-0.457203\pi$
$293$	4.58903	0.268094	0.134047	+	0.990975i	$0.457203\pi$
$294$	0	0
$295$	1.56967	0.0913899
$296$	30.7905	1.78966
$297$	0	0
$298$	−31.5668	−1.82862
$299$	3.17551	0.183645
$300$	0	0
$301$	2.77150	0.159746
$302$	16.2616	0.935752
$303$	0	0
$304$	−5.11259	−0.293227
$305$	3.50020	0.200421
$306$	0	0
$307$	7.80442	0.445422	0.222711	−	0.974885i	$-0.428509\pi$
$307$	7.80442	0.445422	0.222711	+	0.974885i	$0.428509\pi$
$308$	0	0
$309$	0	0
$310$	0.942424	0.0535261
$311$	28.5472	1.61876	0.809382	−	0.587283i	$-0.199802\pi$
$311$	28.5472	1.61876	0.809382	+	0.587283i	$0.199802\pi$
$312$	0	0
$313$	−33.1333	−1.87281	−0.936404	−	0.350925i	$-0.885867\pi$
$313$	−33.1333	−1.87281	−0.936404	+	0.350925i	$0.885867\pi$
$314$	40.6356	2.29320
$315$	0	0
$316$	47.6233	2.67902
$317$	−29.2462	−1.64263	−0.821316	−	0.570473i	$-0.806760\pi$
$317$	−29.2462	−1.64263	−0.821316	+	0.570473i	$0.806760\pi$
$318$	0	0
$319$	0	0
$320$	−5.02556	−0.280937
$321$	0	0
$322$	11.2467	0.626754
$323$	−15.1073	−0.840591
$324$	0	0
$325$	2.54187	0.140997
$326$	−0.624826	−0.0346059
$327$	0	0
$328$	12.7414	0.703528
$329$	8.49374	0.468275
$330$	0	0
$331$	16.9635	0.932396	0.466198	−	0.884680i	$-0.345623\pi$
$331$	16.9635	0.932396	0.466198	+	0.884680i	$0.345623\pi$
$332$	−10.6997	−0.587221
$333$	0	0
$334$	−28.0783	−1.53638
$335$	−1.58589	−0.0866464
$336$	0	0
$337$	8.05246	0.438645	0.219323	−	0.975652i	$-0.429615\pi$
$337$	8.05246	0.438645	0.219323	+	0.975652i	$0.429615\pi$
$338$	29.9839	1.63091
$339$	0	0
$340$	6.80727	0.369176
$341$	0	0
$342$	0	0
$343$	−10.5969	−0.572178
$344$	−12.8163	−0.691009
$345$	0	0
$346$	−0.746545	−0.0401345
$347$	−17.0319	−0.914321	−0.457161	−	0.889384i	$-0.651134\pi$
$347$	−17.0319	−0.914321	−0.457161	+	0.889384i	$0.651134\pi$
$348$	0	0
$349$	25.7497	1.37835	0.689175	−	0.724595i	$-0.257973\pi$
$349$	25.7497	1.37835	0.689175	+	0.724595i	$0.257973\pi$
$350$	9.00252	0.481205
$351$	0	0
$352$	0	0
$353$	−14.3167	−0.762000	−0.381000	−	0.924575i	$-0.624420\pi$
$353$	−14.3167	−0.762000	−0.381000	+	0.924575i	$0.624420\pi$
$354$	0	0
$355$	0.557725	0.0296009
$356$	−29.9774	−1.58880
$357$	0	0
$358$	−8.62289	−0.455734
$359$	−18.5941	−0.981358	−0.490679	−	0.871340i	$-0.663251\pi$
$359$	−18.5941	−0.981358	−0.490679	+	0.871340i	$0.663251\pi$
$360$	0	0
$361$	−7.79532	−0.410280
$362$	−38.0849	−2.00170
$363$	0	0
$364$	−1.48564	−0.0778687
$365$	−5.59844	−0.293036
$366$	0	0
$367$	−30.1290	−1.57272	−0.786360	−	0.617768i	$-0.788037\pi$
$367$	−30.1290	−1.57272	−0.786360	+	0.617768i	$0.788037\pi$
$368$	−9.19703	−0.479428
$369$	0	0
$370$	8.40206	0.436802
$371$	1.29271	0.0671140
$372$	0	0
$373$	11.1097	0.575241	0.287620	−	0.957745i	$-0.407136\pi$
$373$	11.1097	0.575241	0.287620	+	0.957745i	$0.407136\pi$
$374$	0	0
$375$	0	0
$376$	−39.2779	−2.02560
$377$	−5.17966	−0.266766
$378$	0	0
$379$	−3.74060	−0.192142	−0.0960709	−	0.995374i	$-0.530628\pi$
$379$	−3.74060	−0.192142	−0.0960709	+	0.995374i	$0.530628\pi$
$380$	−5.04878	−0.258997
$381$	0	0
$382$	50.1855	2.56771
$383$	−25.8241	−1.31955	−0.659776	−	0.751462i	$-0.729349\pi$
$383$	−25.8241	−1.31955	−0.659776	+	0.751462i	$0.729349\pi$
$384$	0	0
$385$	0	0
$386$	38.4137	1.95521
$387$	0	0
$388$	15.3463	0.779090
$389$	9.09357	0.461062	0.230531	−	0.973065i	$-0.425954\pi$
$389$	9.09357	0.461062	0.230531	+	0.973065i	$0.425954\pi$
$390$	0	0
$391$	−27.1764	−1.37437
$392$	23.3510	1.17941
$393$	0	0
$394$	−28.3105	−1.42626
$395$	5.68407	0.285996
$396$	0	0
$397$	14.4115	0.723290	0.361645	−	0.932316i	$-0.382215\pi$
$397$	14.4115	0.723290	0.361645	+	0.932316i	$0.382215\pi$
$398$	61.0177	3.05854
$399$	0	0
$400$	−7.36185	−0.368092
$401$	30.8846	1.54230	0.771151	−	0.636652i	$-0.219681\pi$
$401$	30.8846	1.54230	0.771151	+	0.636652i	$0.219681\pi$
$402$	0	0
$403$	0.496997	0.0247572
$404$	−34.8543	−1.73407
$405$	0	0
$406$	−18.3448	−0.910436
$407$	0	0
$408$	0	0
$409$	−30.8986	−1.52784	−0.763919	−	0.645313i	$-0.776727\pi$
$409$	−30.8986	−1.52784	−0.763919	+	0.645313i	$0.776727\pi$
$410$	3.47686	0.171710
$411$	0	0
$412$	51.9675	2.56026
$413$	2.93176	0.144262
$414$	0	0
$415$	−1.27706	−0.0626882
$416$	−1.96678	−0.0964293
$417$	0	0
$418$	0	0
$419$	−14.1108	−0.689358	−0.344679	−	0.938721i	$-0.612012\pi$
$419$	−14.1108	−0.689358	−0.344679	+	0.938721i	$0.612012\pi$
$420$	0	0
$421$	15.4444	0.752713	0.376356	−	0.926475i	$-0.377177\pi$
$421$	15.4444	0.752713	0.376356	+	0.926475i	$0.377177\pi$
$422$	29.7407	1.44775
$423$	0	0
$424$	−5.97790	−0.290313
$425$	−21.7536	−1.05521
$426$	0	0
$427$	6.53751	0.316372
$428$	−47.0605	−2.27475
$429$	0	0
$430$	−3.49729	−0.168654
$431$	−23.9835	−1.15524	−0.577621	−	0.816305i	$-0.696019\pi$
$431$	−23.9835	−1.15524	−0.577621	+	0.816305i	$0.696019\pi$
$432$	0	0
$433$	−9.53022	−0.457993	−0.228996	−	0.973427i	$-0.573544\pi$
$433$	−9.53022	−0.457993	−0.228996	+	0.973427i	$0.573544\pi$
$434$	1.76021	0.0844929
$435$	0	0
$436$	10.1123	0.484291
$437$	20.1561	0.964195
$438$	0	0
$439$	14.5447	0.694182	0.347091	−	0.937832i	$-0.387170\pi$
$439$	14.5447	0.694182	0.347091	+	0.937832i	$0.387170\pi$
$440$	0	0
$441$	0	0
$442$	5.60959	0.266821
$443$	−23.0479	−1.09504	−0.547520	−	0.836793i	$-0.684428\pi$
$443$	−23.0479	−1.09504	−0.547520	+	0.836793i	$0.684428\pi$
$444$	0	0
$445$	−3.57794	−0.169611
$446$	−16.0499	−0.759986
$447$	0	0
$448$	−9.38650	−0.443470
$449$	20.0956	0.948372	0.474186	−	0.880425i	$-0.342742\pi$
$449$	20.0956	0.948372	0.474186	+	0.880425i	$0.342742\pi$
$450$	0	0
$451$	0	0
$452$	−31.3138	−1.47288
$453$	0	0
$454$	29.2859	1.37445
$455$	−0.177318	−0.00831280
$456$	0	0
$457$	28.4159	1.32924	0.664620	−	0.747182i	$-0.268594\pi$
$457$	28.4159	1.32924	0.664620	+	0.747182i	$0.268594\pi$
$458$	38.7225	1.80938
$459$	0	0
$460$	−9.08223	−0.423461
$461$	18.5199	0.862556	0.431278	−	0.902219i	$-0.358063\pi$
$461$	18.5199	0.862556	0.431278	+	0.902219i	$0.358063\pi$
$462$	0	0
$463$	−1.05984	−0.0492549	−0.0246274	−	0.999697i	$-0.507840\pi$
$463$	−1.05984	−0.0492549	−0.0246274	+	0.999697i	$0.507840\pi$
$464$	15.0015	0.696428
$465$	0	0
$466$	6.25969	0.289975
$467$	−22.9649	−1.06269	−0.531344	−	0.847156i	$-0.678313\pi$
$467$	−22.9649	−1.06269	−0.531344	+	0.847156i	$0.678313\pi$
$468$	0	0
$469$	−2.96205	−0.136775
$470$	−10.7181	−0.494388
$471$	0	0
$472$	−13.5574	−0.624031
$473$	0	0
$474$	0	0
$475$	16.1341	0.740283
$476$	12.7143	0.582759
$477$	0	0
$478$	61.6284	2.81882
$479$	12.9918	0.593613	0.296806	−	0.954938i	$-0.404078\pi$
$479$	12.9918	0.593613	0.296806	+	0.954938i	$0.404078\pi$
$480$	0	0
$481$	4.43092	0.202033
$482$	42.2574	1.92477
$483$	0	0
$484$	0	0
$485$	1.83165	0.0831710
$486$	0	0
$487$	27.0888	1.22751	0.613755	−	0.789497i	$-0.289658\pi$
$487$	27.0888	1.22751	0.613755	+	0.789497i	$0.289658\pi$
$488$	−30.2316	−1.36852
$489$	0	0
$490$	6.37199	0.287857
$491$	21.8222	0.984821	0.492411	−	0.870363i	$-0.336116\pi$
$491$	21.8222	0.984821	0.492411	+	0.870363i	$0.336116\pi$
$492$	0	0
$493$	44.3282	1.99644
$494$	−4.16049	−0.187189
$495$	0	0
$496$	−1.43942	−0.0646319
$497$	1.04169	0.0467262
$498$	0	0
$499$	−10.5954	−0.474314	−0.237157	−	0.971471i	$-0.576216\pi$
$499$	−10.5954	−0.474314	−0.237157	+	0.971471i	$0.576216\pi$
$500$	−14.8114	−0.662388
$501$	0	0
$502$	50.7837	2.26659
$503$	9.22927	0.411513	0.205756	−	0.978603i	$-0.434035\pi$
$503$	9.22927	0.411513	0.205756	+	0.978603i	$0.434035\pi$
$504$	0	0
$505$	−4.16002	−0.185119
$506$	0	0
$507$	0	0
$508$	48.3159	2.14367
$509$	20.5467	0.910715	0.455357	−	0.890309i	$-0.349511\pi$
$509$	20.5467	0.910715	0.455357	+	0.890309i	$0.349511\pi$
$510$	0	0
$511$	−10.4565	−0.462568
$512$	16.8907	0.746471
$513$	0	0
$514$	−13.8588	−0.611287
$515$	6.20257	0.273318
$516$	0	0
$517$	0	0
$518$	15.6930	0.689509
$519$	0	0
$520$	0.819978	0.0359584
$521$	16.8438	0.737938	0.368969	−	0.929442i	$-0.379711\pi$
$521$	16.8438	0.737938	0.368969	+	0.929442i	$0.379711\pi$
$522$	0	0
$523$	−40.9033	−1.78858	−0.894289	−	0.447490i	$-0.852318\pi$
$523$	−40.9033	−1.78858	−0.894289	+	0.447490i	$0.852318\pi$
$524$	−19.0151	−0.830681
$525$	0	0
$526$	−36.4548	−1.58951
$527$	−4.25336	−0.185279
$528$	0	0
$529$	13.2586	0.576463
$530$	−1.63124	−0.0708565
$531$	0	0
$532$	−9.42987	−0.408837
$533$	1.83356	0.0794203
$534$	0	0
$535$	−5.61688	−0.242839
$536$	13.6975	0.591641
$537$	0	0
$538$	−48.9417	−2.11003
$539$	0	0
$540$	0	0
$541$	−28.4032	−1.22115	−0.610574	−	0.791959i	$-0.709061\pi$
$541$	−28.4032	−1.22115	−0.610574	+	0.791959i	$0.709061\pi$
$542$	−66.5373	−2.85802
$543$	0	0
$544$	16.8319	0.721663
$545$	1.20695	0.0517000
$546$	0	0
$547$	39.1982	1.67599	0.837997	−	0.545674i	$-0.183726\pi$
$547$	39.1982	1.67599	0.837997	+	0.545674i	$0.183726\pi$
$548$	60.5116	2.58493
$549$	0	0
$550$	0	0
$551$	−32.8771	−1.40061
$552$	0	0
$553$	10.6164	0.451456
$554$	−20.2611	−0.860812
$555$	0	0
$556$	−70.4898	−2.98943
$557$	9.73341	0.412418	0.206209	−	0.978508i	$-0.433887\pi$
$557$	9.73341	0.412418	0.206209	+	0.978508i	$0.433887\pi$
$558$	0	0
$559$	−1.84434	−0.0780071
$560$	0.513555	0.0217017
$561$	0	0
$562$	−9.79532	−0.413191
$563$	−43.0907	−1.81606	−0.908029	−	0.418907i	$-0.862413\pi$
$563$	−43.0907	−1.81606	−0.908029	+	0.418907i	$0.862413\pi$
$564$	0	0
$565$	−3.73745	−0.157236
$566$	15.9902	0.672119
$567$	0	0
$568$	−4.81712	−0.202122
$569$	−28.3767	−1.18961	−0.594807	−	0.803868i	$-0.702772\pi$
$569$	−28.3767	−1.18961	−0.594807	+	0.803868i	$0.702772\pi$
$570$	0	0
$571$	3.84148	0.160761	0.0803805	−	0.996764i	$-0.474386\pi$
$571$	3.84148	0.160761	0.0803805	+	0.996764i	$0.474386\pi$
$572$	0	0
$573$	0	0
$574$	6.49391	0.271050
$575$	29.0236	1.21037
$576$	0	0
$577$	−1.67505	−0.0697334	−0.0348667	−	0.999392i	$-0.511101\pi$
$577$	−1.67505	−0.0697334	−0.0348667	+	0.999392i	$0.511101\pi$
$578$	−7.94065	−0.330287
$579$	0	0
$580$	14.8143	0.615129
$581$	−2.38522	−0.0989558
$582$	0	0
$583$	0	0
$584$	48.3543	2.00091
$585$	0	0
$586$	−10.8158	−0.446796
$587$	42.1933	1.74150	0.870752	−	0.491722i	$-0.163632\pi$
$587$	42.1933	1.74150	0.870752	+	0.491722i	$0.163632\pi$
$588$	0	0
$589$	3.15461	0.129983
$590$	−3.69953	−0.152307
$591$	0	0
$592$	−12.8330	−0.527432
$593$	31.5266	1.29464	0.647321	−	0.762217i	$-0.275889\pi$
$593$	31.5266	1.29464	0.647321	+	0.762217i	$0.275889\pi$
$594$	0	0
$595$	1.51751	0.0622118
$596$	47.6121	1.95027
$597$	0	0
$598$	−7.48429	−0.306055
$599$	12.8811	0.526310	0.263155	−	0.964754i	$-0.415237\pi$
$599$	12.8811	0.526310	0.263155	+	0.964754i	$0.415237\pi$
$600$	0	0
$601$	9.33463	0.380768	0.190384	−	0.981710i	$-0.439027\pi$
$601$	9.33463	0.380768	0.190384	+	0.981710i	$0.439027\pi$
$602$	−6.53208	−0.266228
$603$	0	0
$604$	−24.5274	−0.998004
$605$	0	0
$606$	0	0
$607$	−45.9834	−1.86641	−0.933204	−	0.359347i	$-0.882999\pi$
$607$	−45.9834	−1.86641	−0.933204	+	0.359347i	$0.882999\pi$
$608$	−12.4838	−0.506285
$609$	0	0
$610$	−8.24955	−0.334014
$611$	−5.65230	−0.228667
$612$	0	0
$613$	22.3422	0.902394	0.451197	−	0.892424i	$-0.350997\pi$
$613$	22.3422	0.902394	0.451197	+	0.892424i	$0.350997\pi$
$614$	−18.3941	−0.742324
$615$	0	0
$616$	0	0
$617$	0.777017	0.0312815	0.0156408	−	0.999878i	$-0.495021\pi$
$617$	0.777017	0.0312815	0.0156408	+	0.999878i	$0.495021\pi$
$618$	0	0
$619$	−38.5289	−1.54861	−0.774303	−	0.632815i	$-0.781899\pi$
$619$	−38.5289	−1.54861	−0.774303	+	0.632815i	$0.781899\pi$
$620$	−1.42145	−0.0570870
$621$	0	0
$622$	−67.2822	−2.69777
$623$	−6.68270	−0.267737
$624$	0	0
$625$	22.3321	0.893283
$626$	78.0912	3.12115
$627$	0	0
$628$	−61.2905	−2.44576
$629$	−37.9203	−1.51198
$630$	0	0
$631$	−12.3440	−0.491405	−0.245703	−	0.969345i	$-0.579019\pi$
$631$	−12.3440	−0.491405	−0.245703	+	0.969345i	$0.579019\pi$
$632$	−49.0939	−1.95285
$633$	0	0
$634$	68.9298	2.73755
$635$	5.76673	0.228846
$636$	0	0
$637$	3.36034	0.133142
$638$	0	0
$639$	0	0
$640$	8.67987	0.343102
$641$	−19.2322	−0.759625	−0.379813	−	0.925063i	$-0.624012\pi$
$641$	−19.2322	−0.759625	−0.379813	+	0.925063i	$0.624012\pi$
$642$	0	0
$643$	33.7081	1.32932	0.664659	−	0.747146i	$-0.268577\pi$
$643$	33.7081	1.32932	0.664659	+	0.747146i	$0.268577\pi$
$644$	−16.9633	−0.668450
$645$	0	0
$646$	35.6060	1.40090
$647$	−6.50799	−0.255856	−0.127928	−	0.991783i	$-0.540833\pi$
$647$	−6.50799	−0.255856	−0.127928	+	0.991783i	$0.540833\pi$
$648$	0	0
$649$	0	0
$650$	−5.99087	−0.234981
$651$	0	0
$652$	0.942424	0.0369082
$653$	31.0603	1.21548	0.607742	−	0.794135i	$-0.292075\pi$
$653$	31.0603	1.21548	0.607742	+	0.794135i	$0.292075\pi$
$654$	0	0
$655$	−2.26955	−0.0886785
$656$	−5.31042	−0.207337
$657$	0	0
$658$	−20.0187	−0.780411
$659$	−30.9512	−1.20569	−0.602844	−	0.797859i	$-0.705966\pi$
$659$	−30.9512	−1.20569	−0.602844	+	0.797859i	$0.705966\pi$
$660$	0	0
$661$	−26.4836	−1.03009	−0.515047	−	0.857162i	$-0.672225\pi$
$661$	−26.4836	−1.03009	−0.515047	+	0.857162i	$0.672225\pi$
$662$	−39.9808	−1.55390
$663$	0	0
$664$	11.0301	0.428050
$665$	−1.12550	−0.0436450
$666$	0	0
$667$	−59.1424	−2.29000
$668$	42.3504	1.63859
$669$	0	0
$670$	3.73775	0.144402
$671$	0	0
$672$	0	0
$673$	8.47632	0.326738	0.163369	−	0.986565i	$-0.447764\pi$
$673$	8.47632	0.326738	0.163369	+	0.986565i	$0.447764\pi$
$674$	−18.9787	−0.731031
$675$	0	0
$676$	−45.2247	−1.73941
$677$	−3.35726	−0.129030	−0.0645150	−	0.997917i	$-0.520550\pi$
$677$	−3.35726	−0.129030	−0.0645150	+	0.997917i	$0.520550\pi$
$678$	0	0
$679$	3.42107	0.131289
$680$	−7.01747	−0.269108
$681$	0	0
$682$	0	0
$683$	−31.8662	−1.21933	−0.609664	−	0.792660i	$-0.708695\pi$
$683$	−31.8662	−1.21933	−0.609664	+	0.792660i	$0.708695\pi$
$684$	0	0
$685$	7.22233	0.275951
$686$	24.9756	0.953571
$687$	0	0
$688$	5.34163	0.203648
$689$	−0.860252	−0.0327730
$690$	0	0
$691$	12.6723	0.482076	0.241038	−	0.970516i	$-0.422512\pi$
$691$	12.6723	0.482076	0.241038	+	0.970516i	$0.422512\pi$
$692$	1.12601	0.0428045
$693$	0	0
$694$	40.1421	1.52377
$695$	−8.41328	−0.319134
$696$	0	0
$697$	−15.6918	−0.594370
$698$	−60.6889	−2.29711
$699$	0	0
$700$	−13.5785	−0.513218
$701$	1.23849	0.0467773	0.0233886	−	0.999726i	$-0.492554\pi$
$701$	1.23849	0.0467773	0.0233886	+	0.999726i	$0.492554\pi$
$702$	0	0
$703$	28.1245	1.06074
$704$	0	0
$705$	0	0
$706$	33.7426	1.26992
$707$	−7.76989	−0.292217
$708$	0	0
$709$	−2.65605	−0.0997499	−0.0498750	−	0.998755i	$-0.515882\pi$
$709$	−2.65605	−0.0997499	−0.0498750	+	0.998755i	$0.515882\pi$
$710$	−1.31449	−0.0493319
$711$	0	0
$712$	30.9030	1.15814
$713$	5.67482	0.212524
$714$	0	0
$715$	0	0
$716$	13.0059	0.486052
$717$	0	0
$718$	43.8240	1.63550
$719$	4.00869	0.149499	0.0747494	−	0.997202i	$-0.476184\pi$
$719$	4.00869	0.149499	0.0747494	+	0.997202i	$0.476184\pi$
$720$	0	0
$721$	11.5849	0.431442
$722$	18.3726	0.683758
$723$	0	0
$724$	57.4433	2.13486
$725$	−47.3411	−1.75820
$726$	0	0
$727$	18.5889	0.689425	0.344712	−	0.938708i	$-0.387976\pi$
$727$	18.5889	0.689425	0.344712	+	0.938708i	$0.387976\pi$
$728$	1.53151	0.0567617
$729$	0	0
$730$	13.1948	0.488362
$731$	15.7840	0.583794
$732$	0	0
$733$	−23.3049	−0.860785	−0.430392	−	0.902642i	$-0.641625\pi$
$733$	−23.3049	−0.860785	−0.430392	+	0.902642i	$0.641625\pi$
$734$	71.0103	2.62104
$735$	0	0
$736$	−22.4571	−0.827779
$737$	0	0
$738$	0	0
$739$	−29.1214	−1.07125	−0.535624	−	0.844457i	$-0.679924\pi$
$739$	−29.1214	−1.07125	−0.535624	+	0.844457i	$0.679924\pi$
$740$	−12.6728	−0.465861
$741$	0	0
$742$	−3.04675	−0.111850
$743$	2.37952	0.0872962	0.0436481	−	0.999047i	$-0.486102\pi$
$743$	2.37952	0.0872962	0.0436481	+	0.999047i	$0.486102\pi$
$744$	0	0
$745$	5.68273	0.208199
$746$	−26.1843	−0.958675
$747$	0	0
$748$	0	0
$749$	−10.4909	−0.383331
$750$	0	0
$751$	8.54631	0.311859	0.155930	−	0.987768i	$-0.450163\pi$
$751$	8.54631	0.311859	0.155930	+	0.987768i	$0.450163\pi$
$752$	16.3704	0.596966
$753$	0	0
$754$	12.2078	0.444583
$755$	−2.92745	−0.106541
$756$	0	0
$757$	−14.9964	−0.545053	−0.272527	−	0.962148i	$-0.587859\pi$
$757$	−14.9964	−0.545053	−0.272527	+	0.962148i	$0.587859\pi$
$758$	8.81614	0.320217
$759$	0	0
$760$	5.20468	0.188793
$761$	24.4366	0.885827	0.442913	−	0.896564i	$-0.353945\pi$
$761$	24.4366	0.885827	0.442913	+	0.896564i	$0.353945\pi$
$762$	0	0
$763$	2.25428	0.0816105
$764$	−75.6946	−2.73854
$765$	0	0
$766$	60.8643	2.19912
$767$	−1.95099	−0.0704460
$768$	0	0
$769$	15.3814	0.554667	0.277333	−	0.960774i	$-0.410549\pi$
$769$	15.3814	0.554667	0.277333	+	0.960774i	$0.410549\pi$
$770$	0	0
$771$	0	0
$772$	−57.9393	−2.08528
$773$	−22.5473	−0.810970	−0.405485	−	0.914102i	$-0.632897\pi$
$773$	−22.5473	−0.810970	−0.405485	+	0.914102i	$0.632897\pi$
$774$	0	0
$775$	4.54246	0.163170
$776$	−15.8202	−0.567910
$777$	0	0
$778$	−21.4324	−0.768390
$779$	11.6382	0.416983
$780$	0	0
$781$	0	0
$782$	64.0515	2.29048
$783$	0	0
$784$	−9.73233	−0.347583
$785$	−7.31531	−0.261095
$786$	0	0
$787$	51.7191	1.84359	0.921794	−	0.387681i	$-0.126724\pi$
$787$	51.7191	1.84359	0.921794	+	0.387681i	$0.126724\pi$
$788$	42.7006	1.52115
$789$	0	0
$790$	−13.3966	−0.476631
$791$	−6.98063	−0.248203
$792$	0	0
$793$	−4.35049	−0.154490
$794$	−33.9660	−1.20541
$795$	0	0
$796$	−92.0328	−3.26201
$797$	22.0508	0.781078	0.390539	−	0.920586i	$-0.372289\pi$
$797$	22.0508	0.781078	0.390539	+	0.920586i	$0.372289\pi$
$798$	0	0
$799$	48.3730	1.71132
$800$	−17.9760	−0.635547
$801$	0	0
$802$	−72.7912	−2.57035
$803$	0	0
$804$	0	0
$805$	−2.02465	−0.0713597
$806$	−1.17136	−0.0412595
$807$	0	0
$808$	35.9305	1.26403
$809$	−8.77573	−0.308538	−0.154269	−	0.988029i	$-0.549302\pi$
$809$	−8.77573	−0.308538	−0.154269	+	0.988029i	$0.549302\pi$
$810$	0	0
$811$	3.89876	0.136904	0.0684520	−	0.997654i	$-0.478194\pi$
$811$	3.89876	0.136904	0.0684520	+	0.997654i	$0.478194\pi$
$812$	27.6694	0.971005
$813$	0	0
$814$	0	0
$815$	0.112483	0.00394009
$816$	0	0
$817$	−11.7066	−0.409563
$818$	72.8242	2.54624
$819$	0	0
$820$	−5.24413	−0.183133
$821$	−11.6970	−0.408229	−0.204115	−	0.978947i	$-0.565432\pi$
$821$	−11.6970	−0.408229	−0.204115	+	0.978947i	$0.565432\pi$
$822$	0	0
$823$	−26.8636	−0.936405	−0.468203	−	0.883621i	$-0.655098\pi$
$823$	−26.8636	−0.936405	−0.468203	+	0.883621i	$0.655098\pi$
$824$	−53.5722	−1.86628
$825$	0	0
$826$	−6.90980	−0.240423
$827$	−44.5261	−1.54833	−0.774163	−	0.632986i	$-0.781829\pi$
$827$	−44.5261	−1.54833	−0.774163	+	0.632986i	$0.781829\pi$
$828$	0	0
$829$	11.2166	0.389570	0.194785	−	0.980846i	$-0.437599\pi$
$829$	11.2166	0.389570	0.194785	+	0.980846i	$0.437599\pi$
$830$	3.00987	0.104474
$831$	0	0
$832$	6.24640	0.216555
$833$	−28.7582	−0.996412
$834$	0	0
$835$	5.05472	0.174926
$836$	0	0
$837$	0	0
$838$	33.2574	1.14886
$839$	−0.785529	−0.0271195	−0.0135597	−	0.999908i	$-0.504316\pi$
$839$	−0.785529	−0.0271195	−0.0135597	+	0.999908i	$0.504316\pi$
$840$	0	0
$841$	67.4688	2.32651
$842$	−36.4005	−1.25444
$843$	0	0
$844$	−44.8578	−1.54407
$845$	−5.39777	−0.185689
$846$	0	0
$847$	0	0
$848$	2.49149	0.0855582
$849$	0	0
$850$	51.2706	1.75857
$851$	50.5931	1.73431
$852$	0	0
$853$	−29.1188	−0.997010	−0.498505	−	0.866887i	$-0.666117\pi$
$853$	−29.1188	−0.997010	−0.498505	+	0.866887i	$0.666117\pi$
$854$	−15.4081	−0.527254
$855$	0	0
$856$	48.5136	1.65816
$857$	30.0488	1.02645	0.513223	−	0.858255i	$-0.328451\pi$
$857$	30.0488	1.02645	0.513223	+	0.858255i	$0.328451\pi$
$858$	0	0
$859$	−56.7657	−1.93682	−0.968410	−	0.249361i	$-0.919779\pi$
$859$	−56.7657	−1.93682	−0.968410	+	0.249361i	$0.919779\pi$
$860$	5.27496	0.179875
$861$	0	0
$862$	56.5261	1.92529
$863$	45.5345	1.55001	0.775006	−	0.631954i	$-0.217747\pi$
$863$	45.5345	1.55001	0.775006	+	0.631954i	$0.217747\pi$
$864$	0	0
$865$	0.134395	0.00456956
$866$	22.4615	0.763274
$867$	0	0
$868$	−2.65492	−0.0901140
$869$	0	0
$870$	0	0
$871$	1.97114	0.0667896
$872$	−10.4245	−0.353020
$873$	0	0
$874$	−47.5053	−1.60689
$875$	−3.30184	−0.111622
$876$	0	0
$877$	−24.8830	−0.840240	−0.420120	−	0.907468i	$-0.638012\pi$
$877$	−24.8830	−0.840240	−0.420120	+	0.907468i	$0.638012\pi$
$878$	−34.2801	−1.15690
$879$	0	0
$880$	0	0
$881$	4.67925	0.157648	0.0788240	−	0.996889i	$-0.474883\pi$
$881$	4.67925	0.157648	0.0788240	+	0.996889i	$0.474883\pi$
$882$	0	0
$883$	−36.7730	−1.23751	−0.618755	−	0.785584i	$-0.712363\pi$
$883$	−36.7730	−1.23751	−0.618755	+	0.785584i	$0.712363\pi$
$884$	−8.46093	−0.284572
$885$	0	0
$886$	54.3211	1.82495
$887$	−13.9498	−0.468387	−0.234193	−	0.972190i	$-0.575245\pi$
$887$	−13.9498	−0.468387	−0.234193	+	0.972190i	$0.575245\pi$
$888$	0	0
$889$	10.7708	0.361242
$890$	8.43277	0.282667
$891$	0	0
$892$	24.2081	0.810546
$893$	−35.8770	−1.20058
$894$	0	0
$895$	1.55231	0.0518881
$896$	16.2118	0.541600
$897$	0	0
$898$	−47.3630	−1.58052
$899$	−9.25635	−0.308716
$900$	0	0
$901$	7.36214	0.245269
$902$	0	0
$903$	0	0
$904$	32.2807	1.07364
$905$	6.85613	0.227905
$906$	0	0
$907$	−55.6979	−1.84942	−0.924709	−	0.380674i	$-0.875692\pi$
$907$	−55.6979	−1.84942	−0.924709	+	0.380674i	$0.875692\pi$
$908$	−44.1718	−1.46589
$909$	0	0
$910$	0.417917	0.0138538
$911$	−36.5274	−1.21021	−0.605103	−	0.796147i	$-0.706868\pi$
$911$	−36.5274	−1.21021	−0.605103	+	0.796147i	$0.706868\pi$
$912$	0	0
$913$	0	0
$914$	−66.9728	−2.21526
$915$	0	0
$916$	−58.4050	−1.92976
$917$	−4.23895	−0.139982
$918$	0	0
$919$	−11.9762	−0.395057	−0.197529	−	0.980297i	$-0.563292\pi$
$919$	−11.9762	−0.395057	−0.197529	+	0.980297i	$0.563292\pi$
$920$	9.36267	0.308678
$921$	0	0
$922$	−43.6490	−1.43750
$923$	−0.693210	−0.0228173
$924$	0	0
$925$	40.4977	1.33156
$926$	2.49791	0.0820864
$927$	0	0
$928$	36.6303	1.20245
$929$	−41.5313	−1.36260	−0.681299	−	0.732005i	$-0.738585\pi$
$929$	−41.5313	−1.36260	−0.681299	+	0.732005i	$0.738585\pi$
$930$	0	0
$931$	21.3292	0.699037
$932$	−9.44148	−0.309266
$933$	0	0
$934$	54.1254	1.77104
$935$	0	0
$936$	0	0
$937$	−17.4713	−0.570762	−0.285381	−	0.958414i	$-0.592120\pi$
$937$	−17.4713	−0.570762	−0.285381	+	0.958414i	$0.592120\pi$
$938$	6.98118	0.227944
$939$	0	0
$940$	16.1660	0.527278
$941$	−44.3021	−1.44421	−0.722104	−	0.691784i	$-0.756825\pi$
$941$	−44.3021	−1.44421	−0.722104	+	0.691784i	$0.756825\pi$
$942$	0	0
$943$	20.9360	0.681769
$944$	5.65051	0.183909
$945$	0	0
$946$	0	0
$947$	1.93419	0.0628526	0.0314263	−	0.999506i	$-0.489995\pi$
$947$	1.93419	0.0628526	0.0314263	+	0.999506i	$0.489995\pi$
$948$	0	0
$949$	6.95844	0.225880
$950$	−38.0261	−1.23373
$951$	0	0
$952$	−13.1069	−0.424797
$953$	−49.8941	−1.61623	−0.808114	−	0.589026i	$-0.799512\pi$
$953$	−49.8941	−1.61623	−0.808114	+	0.589026i	$0.799512\pi$
$954$	0	0
$955$	−9.03451	−0.292350
$956$	−92.9539	−3.00634
$957$	0	0
$958$	−30.6202	−0.989293
$959$	13.4895	0.435600
$960$	0	0
$961$	−30.1118	−0.971350
$962$	−10.4431	−0.336700
$963$	0	0
$964$	−63.7368	−2.05282
$965$	−6.91532	−0.222612
$966$	0	0
$967$	24.6839	0.793782	0.396891	−	0.917866i	$-0.370089\pi$
$967$	24.6839	0.793782	0.396891	+	0.917866i	$0.370089\pi$
$968$	0	0
$969$	0	0
$970$	−4.31698	−0.138610
$971$	−55.2266	−1.77231	−0.886154	−	0.463391i	$-0.846632\pi$
$971$	−55.2266	−1.77231	−0.886154	+	0.463391i	$0.846632\pi$
$972$	0	0
$973$	−15.7139	−0.503765
$974$	−63.8449	−2.04572
$975$	0	0
$976$	12.6000	0.403317
$977$	−20.5511	−0.657488	−0.328744	−	0.944419i	$-0.606625\pi$
$977$	−20.5511	−0.657488	−0.328744	+	0.944419i	$0.606625\pi$
$978$	0	0
$979$	0	0
$980$	−9.61086	−0.307008
$981$	0	0
$982$	−51.4322	−1.64127
$983$	5.88157	0.187593	0.0937964	−	0.995591i	$-0.470100\pi$
$983$	5.88157	0.187593	0.0937964	+	0.995591i	$0.470100\pi$
$984$	0	0
$985$	5.09652	0.162389
$986$	−104.476	−3.32720
$987$	0	0
$988$	6.27526	0.199642
$989$	−21.0590	−0.669637
$990$	0	0
$991$	−34.1182	−1.08380	−0.541901	−	0.840442i	$-0.682295\pi$
$991$	−34.1182	−1.08380	−0.541901	+	0.840442i	$0.682295\pi$
$992$	−3.51474	−0.111593
$993$	0	0
$994$	−2.45514	−0.0778722
$995$	−10.9845	−0.348233
$996$	0	0
$997$	−45.0818	−1.42775	−0.713877	−	0.700271i	$-0.753062\pi$
$997$	−45.0818	−1.42775	−0.713877	+	0.700271i	$0.753062\pi$
$998$	24.9720	0.790475
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9801.2.a.bv.1.1	✓	8
3.2	odd	2	inner	9801.2.a.bv.1.8	yes	8
11.10	odd	2		9801.2.a.bw.1.8	yes	8
33.32	even	2		9801.2.a.bw.1.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9801.2.a.bv.1.1	✓	8	1.1	even	1	trivial
9801.2.a.bv.1.8	yes	8	3.2	odd	2	inner
9801.2.a.bw.1.1	yes	8	33.32	even	2
9801.2.a.bw.1.8	yes	8	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9801 = 3^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9801.a (trivial)

\(f(q)\)	\(=\)	\(q-2.35688 q^{2} +3.55487 q^{4} +0.424290 q^{5} +0.792469 q^{7} -3.66464 q^{8} +O(q^{10})\)	\(q-2.35688 q^{2} +3.55487 q^{4} +0.424290 q^{5} +0.792469 q^{7} -3.66464 q^{8} -1.00000 q^{10} -0.527361 q^{13} -1.86775 q^{14} +1.52736 q^{16} +4.51322 q^{17} -3.34734 q^{19} +1.50830 q^{20} -6.02151 q^{23} -4.81998 q^{25} +1.24293 q^{26} +2.81712 q^{28} +9.82185 q^{29} -0.942424 q^{31} +3.72947 q^{32} -10.6371 q^{34} +0.336237 q^{35} -8.40206 q^{37} +7.88927 q^{38} -1.55487 q^{40} -3.47686 q^{41} +3.49729 q^{43} +14.1920 q^{46} +10.7181 q^{47} -6.37199 q^{49} +11.3601 q^{50} -1.87470 q^{52} +1.63124 q^{53} -2.90411 q^{56} -23.1489 q^{58} +3.69953 q^{59} +8.24955 q^{61} +2.22118 q^{62} -11.8446 q^{64} -0.223754 q^{65} -3.73775 q^{67} +16.0439 q^{68} -0.792469 q^{70} +1.31449 q^{71} -13.1948 q^{73} +19.8026 q^{74} -11.8994 q^{76} +13.3966 q^{79} +0.648044 q^{80} +8.19453 q^{82} -3.00987 q^{83} +1.91491 q^{85} -8.24269 q^{86} -8.43277 q^{89} -0.417917 q^{91} -21.4057 q^{92} -25.2612 q^{94} -1.42024 q^{95} +4.31698 q^{97} +15.0180 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} - 2 q^{7}+O(q^{10}) \) 8 * q + 4 * q^4 - 2 * q^7	\( 8 q + 4 q^{4} - 2 q^{7} - 8 q^{10} + 4 q^{13} + 4 q^{16} + 6 q^{19} - 14 q^{25} - 10 q^{28} - 16 q^{31} - 28 q^{34} - 18 q^{37} + 12 q^{40} + 12 q^{43} + 32 q^{46} + 6 q^{49} + 26 q^{52} - 30 q^{58} - 24 q^{61} - 46 q^{64} - 38 q^{67} + 2 q^{70} - 32 q^{73} - 30 q^{76} + 26 q^{79} + 8 q^{82} + 40 q^{85} - 28 q^{91} - 22 q^{94} + 10 q^{97}+O(q^{100}) \) 8 * q + 4 * q^4 - 2 * q^7 - 8 * q^10 + 4 * q^13 + 4 * q^16 + 6 * q^19 - 14 * q^25 - 10 * q^28 - 16 * q^31 - 28 * q^34 - 18 * q^37 + 12 * q^40 + 12 * q^43 + 32 * q^46 + 6 * q^49 + 26 * q^52 - 30 * q^58 - 24 * q^61 - 46 * q^64 - 38 * q^67 + 2 * q^70 - 32 * q^73 - 30 * q^76 + 26 * q^79 + 8 * q^82 + 40 * q^85 - 28 * q^91 - 22 * q^94 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more