LMFDB - Embedded newform 8379.2.a.bf.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8379,2,Mod(1,8379)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8379.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8379, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8379 = 3^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8379.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,1,0,3,-6,0,0,6,0,-3,5,0,4,0,0,-3,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$66.9066518536$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 133)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	8379.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30278 q^{2} +3.30278 q^{4} -3.00000 q^{5} +3.00000 q^{8} -6.90833 q^{10} +0.697224 q^{11} +5.60555 q^{13} +0.302776 q^{16} -5.30278 q^{17} -1.00000 q^{19} -9.90833 q^{20} +1.60555 q^{22} +3.00000 q^{23} +4.00000 q^{25} +12.9083 q^{26} -9.90833 q^{29} -1.30278 q^{31} -5.30278 q^{32} -12.2111 q^{34} +3.60555 q^{37} -2.30278 q^{38} -9.00000 q^{40} +0.697224 q^{41} -10.0000 q^{43} +2.30278 q^{44} +6.90833 q^{46} +6.21110 q^{47} +9.21110 q^{50} +18.5139 q^{52} +6.90833 q^{53} -2.09167 q^{55} -22.8167 q^{58} -6.21110 q^{59} +4.21110 q^{61} -3.00000 q^{62} -12.8167 q^{64} -16.8167 q^{65} -1.90833 q^{67} -17.5139 q^{68} -12.2111 q^{71} -1.51388 q^{73} +8.30278 q^{74} -3.30278 q^{76} +11.2111 q^{79} -0.908327 q^{80} +1.60555 q^{82} -12.9083 q^{83} +15.9083 q^{85} -23.0278 q^{86} +2.09167 q^{88} -10.6056 q^{89} +9.90833 q^{92} +14.3028 q^{94} +3.00000 q^{95} -9.60555 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 3 q^{4} - 6 q^{5} + 6 q^{8} - 3 q^{10} + 5 q^{11} + 4 q^{13} - 3 q^{16} - 7 q^{17} - 2 q^{19} - 9 q^{20} - 4 q^{22} + 6 q^{23} + 8 q^{25} + 15 q^{26} - 9 q^{29} + q^{31} - 7 q^{32} - 10 q^{34}+ \cdots - 12 q^{97}+O(q^{100}) $ 2 * q + q^2 + 3 * q^4 - 6 * q^5 + 6 * q^8 - 3 * q^10 + 5 * q^11 + 4 * q^13 - 3 * q^16 - 7 * q^17 - 2 * q^19 - 9 * q^20 - 4 * q^22 + 6 * q^23 + 8 * q^25 + 15 * q^26 - 9 * q^29 + q^31 - 7 * q^32 - 10 * q^34 - q^38 - 18 * q^40 + 5 * q^41 - 20 * q^43 + q^44 + 3 * q^46 - 2 * q^47 + 4 * q^50 + 19 * q^52 + 3 * q^53 - 15 * q^55 - 24 * q^58 + 2 * q^59 - 6 * q^61 - 6 * q^62 - 4 * q^64 - 12 * q^65 + 7 * q^67 - 17 * q^68 - 10 * q^71 + 15 * q^73 + 13 * q^74 - 3 * q^76 + 8 * q^79 + 9 * q^80 - 4 * q^82 - 15 * q^83 + 21 * q^85 - 10 * q^86 + 15 * q^88 - 14 * q^89 + 9 * q^92 + 25 * q^94 + 6 * q^95 - 12 * q^97

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 $

$2$	2.30278	1.62831	0.814154	−	0.580649i	$-0.197201\pi$
$2$	2.30278	1.62831	0.814154	+	0.580649i	$0.197201\pi$
$3$	0	0
$3$	0	0
$4$	3.30278	1.65139
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	3.00000	1.06066
$9$	0	0
$10$	−6.90833	−2.18460
$11$	0.697224	0.210221	0.105111	−	0.994461i	$-0.466480\pi$
$11$	0.697224	0.210221	0.105111	+	0.994461i	$0.466480\pi$
$12$	0	0
$13$	5.60555	1.55470	0.777350	−	0.629068i	$-0.216563\pi$
$13$	5.60555	1.55470	0.777350	+	0.629068i	$0.216563\pi$
$14$	0	0
$15$	0	0
$16$	0.302776	0.0756939
$17$	−5.30278	−1.28611	−0.643056	−	0.765819i	$-0.722334\pi$
$17$	−5.30278	−1.28611	−0.643056	+	0.765819i	$0.722334\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−9.90833	−2.21557
$21$	0	0
$22$	1.60555	0.342305
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	12.9083	2.53153
$27$	0	0
$28$	0	0
$29$	−9.90833	−1.83993	−0.919965	−	0.392000i	$-0.871783\pi$
$29$	−9.90833	−1.83993	−0.919965	+	0.392000i	$0.871783\pi$
$30$	0	0
$31$	−1.30278	−0.233985	−0.116993	−	0.993133i	$-0.537325\pi$
$31$	−1.30278	−0.233985	−0.116993	+	0.993133i	$0.537325\pi$
$32$	−5.30278	−0.937407
$33$	0	0
$34$	−12.2111	−2.09419
$35$	0	0
$36$	0	0
$37$	3.60555	0.592749	0.296374	−	0.955072i	$-0.404222\pi$
$37$	3.60555	0.592749	0.296374	+	0.955072i	$0.404222\pi$
$38$	−2.30278	−0.373560
$39$	0	0
$40$	−9.00000	−1.42302
$41$	0.697224	0.108888	0.0544441	−	0.998517i	$-0.482661\pi$
$41$	0.697224	0.108888	0.0544441	+	0.998517i	$0.482661\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	2.30278	0.347156
$45$	0	0
$46$	6.90833	1.01858
$47$	6.21110	0.905982	0.452991	−	0.891515i	$-0.350357\pi$
$47$	6.21110	0.905982	0.452991	+	0.891515i	$0.350357\pi$
$48$	0	0
$49$	0	0
$50$	9.21110	1.30265
$51$	0	0
$52$	18.5139	2.56741
$53$	6.90833	0.948932	0.474466	−	0.880274i	$-0.342641\pi$
$53$	6.90833	0.948932	0.474466	+	0.880274i	$0.342641\pi$
$54$	0	0
$55$	−2.09167	−0.282041
$56$	0	0
$57$	0	0
$58$	−22.8167	−2.99597
$59$	−6.21110	−0.808617	−0.404308	−	0.914623i	$-0.632488\pi$
$59$	−6.21110	−0.808617	−0.404308	+	0.914623i	$0.632488\pi$
$60$	0	0
$61$	4.21110	0.539176	0.269588	−	0.962976i	$-0.413112\pi$
$61$	4.21110	0.539176	0.269588	+	0.962976i	$0.413112\pi$
$62$	−3.00000	−0.381000
$63$	0	0
$64$	−12.8167	−1.60208
$65$	−16.8167	−2.08585
$66$	0	0
$67$	−1.90833	−0.233139	−0.116570	−	0.993183i	$-0.537190\pi$
$67$	−1.90833	−0.233139	−0.116570	+	0.993183i	$0.537190\pi$
$68$	−17.5139	−2.12387
$69$	0	0
$70$	0	0
$71$	−12.2111	−1.44919	−0.724596	−	0.689174i	$-0.757973\pi$
$71$	−12.2111	−1.44919	−0.724596	+	0.689174i	$0.757973\pi$
$72$	0	0
$73$	−1.51388	−0.177186	−0.0885930	−	0.996068i	$-0.528237\pi$
$73$	−1.51388	−0.177186	−0.0885930	+	0.996068i	$0.528237\pi$
$74$	8.30278	0.965178
$75$	0	0
$76$	−3.30278	−0.378854
$77$	0	0
$78$	0	0
$79$	11.2111	1.26135	0.630674	−	0.776048i	$-0.282779\pi$
$79$	11.2111	1.26135	0.630674	+	0.776048i	$0.282779\pi$
$80$	−0.908327	−0.101554
$81$	0	0
$82$	1.60555	0.177303
$83$	−12.9083	−1.41687	−0.708436	−	0.705775i	$-0.750599\pi$
$83$	−12.9083	−1.41687	−0.708436	+	0.705775i	$0.750599\pi$
$84$	0	0
$85$	15.9083	1.72550
$86$	−23.0278	−2.48315
$87$	0	0
$88$	2.09167	0.222973
$89$	−10.6056	−1.12419	−0.562093	−	0.827074i	$-0.690004\pi$
$89$	−10.6056	−1.12419	−0.562093	+	0.827074i	$0.690004\pi$
$90$	0	0
$91$	0	0
$92$	9.90833	1.03301
$93$	0	0
$94$	14.3028	1.47522
$95$	3.00000	0.307794
$96$	0	0
$97$	−9.60555	−0.975296	−0.487648	−	0.873040i	$-0.662145\pi$
$97$	−9.60555	−0.975296	−0.487648	+	0.873040i	$0.662145\pi$
$98$	0	0
$99$	0	0
$100$	13.2111	1.32111
$101$	−4.39445	−0.437264	−0.218632	−	0.975807i	$-0.570159\pi$
$101$	−4.39445	−0.437264	−0.218632	+	0.975807i	$0.570159\pi$
$102$	0	0
$103$	16.2111	1.59733	0.798664	−	0.601778i	$-0.205541\pi$
$103$	16.2111	1.59733	0.798664	+	0.601778i	$0.205541\pi$
$104$	16.8167	1.64901
$105$	0	0
$106$	15.9083	1.54515
$107$	5.78890	0.559634	0.279817	−	0.960053i	$-0.409726\pi$
$107$	5.78890	0.559634	0.279817	+	0.960053i	$0.409726\pi$
$108$	0	0
$109$	−10.2111	−0.978046	−0.489023	−	0.872271i	$-0.662647\pi$
$109$	−10.2111	−0.978046	−0.489023	+	0.872271i	$0.662647\pi$
$110$	−4.81665	−0.459250
$111$	0	0
$112$	0	0
$113$	−9.69722	−0.912238	−0.456119	−	0.889919i	$-0.650761\pi$
$113$	−9.69722	−0.912238	−0.456119	+	0.889919i	$0.650761\pi$
$114$	0	0
$115$	−9.00000	−0.839254
$116$	−32.7250	−3.03844
$117$	0	0
$118$	−14.3028	−1.31668
$119$	0	0
$120$	0	0
$121$	−10.5139	−0.955807
$122$	9.69722	0.877945
$123$	0	0
$124$	−4.30278	−0.386401
$125$	3.00000	0.268328
$126$	0	0
$127$	−11.8167	−1.04856	−0.524279	−	0.851546i	$-0.675665\pi$
$127$	−11.8167	−1.04856	−0.524279	+	0.851546i	$0.675665\pi$
$128$	−18.9083	−1.67128
$129$	0	0
$130$	−38.7250	−3.39641
$131$	−20.5139	−1.79231	−0.896153	−	0.443745i	$-0.853649\pi$
$131$	−20.5139	−1.79231	−0.896153	+	0.443745i	$0.853649\pi$
$132$	0	0
$133$	0	0
$134$	−4.39445	−0.379623
$135$	0	0
$136$	−15.9083	−1.36413
$137$	−21.6333	−1.84826	−0.924129	−	0.382080i	$-0.875208\pi$
$137$	−21.6333	−1.84826	−0.924129	+	0.382080i	$0.875208\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	0	0
$142$	−28.1194	−2.35973
$143$	3.90833	0.326831
$144$	0	0
$145$	29.7250	2.46853
$146$	−3.48612	−0.288513
$147$	0	0
$148$	11.9083	0.978858
$149$	6.21110	0.508833	0.254417	−	0.967095i	$-0.418117\pi$
$149$	6.21110	0.508833	0.254417	+	0.967095i	$0.418117\pi$
$150$	0	0
$151$	5.90833	0.480813	0.240406	−	0.970672i	$-0.422719\pi$
$151$	5.90833	0.480813	0.240406	+	0.970672i	$0.422719\pi$
$152$	−3.00000	−0.243332
$153$	0	0
$154$	0	0
$155$	3.90833	0.313924
$156$	0	0
$157$	−4.51388	−0.360247	−0.180123	−	0.983644i	$-0.557650\pi$
$157$	−4.51388	−0.360247	−0.180123	+	0.983644i	$0.557650\pi$
$158$	25.8167	2.05386
$159$	0	0
$160$	15.9083	1.25766
$161$	0	0
$162$	0	0
$163$	5.69722	0.446241	0.223121	−	0.974791i	$-0.428376\pi$
$163$	5.69722	0.446241	0.223121	+	0.974791i	$0.428376\pi$
$164$	2.30278	0.179817
$165$	0	0
$166$	−29.7250	−2.30711
$167$	4.39445	0.340053	0.170026	−	0.985440i	$-0.445615\pi$
$167$	4.39445	0.340053	0.170026	+	0.985440i	$0.445615\pi$
$168$	0	0
$169$	18.4222	1.41709
$170$	36.6333	2.80965
$171$	0	0
$172$	−33.0278	−2.51834
$173$	7.81665	0.594289	0.297145	−	0.954832i	$-0.403966\pi$
$173$	7.81665	0.594289	0.297145	+	0.954832i	$0.403966\pi$
$174$	0	0
$175$	0	0
$176$	0.211103	0.0159125
$177$	0	0
$178$	−24.4222	−1.83052
$179$	11.7250	0.876366	0.438183	−	0.898886i	$-0.355622\pi$
$179$	11.7250	0.876366	0.438183	+	0.898886i	$0.355622\pi$
$180$	0	0
$181$	10.6972	0.795118	0.397559	−	0.917577i	$-0.369857\pi$
$181$	10.6972	0.795118	0.397559	+	0.917577i	$0.369857\pi$
$182$	0	0
$183$	0	0
$184$	9.00000	0.663489
$185$	−10.8167	−0.795256
$186$	0	0
$187$	−3.69722	−0.270368
$188$	20.5139	1.49613
$189$	0	0
$190$	6.90833	0.501183
$191$	−25.3305	−1.83285	−0.916426	−	0.400203i	$-0.868940\pi$
$191$	−25.3305	−1.83285	−0.916426	+	0.400203i	$0.868940\pi$
$192$	0	0
$193$	−23.1194	−1.66417	−0.832086	−	0.554646i	$-0.812854\pi$
$193$	−23.1194	−1.66417	−0.832086	+	0.554646i	$0.812854\pi$
$194$	−22.1194	−1.58808
$195$	0	0
$196$	0	0
$197$	6.90833	0.492198	0.246099	−	0.969245i	$-0.420851\pi$
$197$	6.90833	0.492198	0.246099	+	0.969245i	$0.420851\pi$
$198$	0	0
$199$	13.4222	0.951475	0.475737	−	0.879587i	$-0.342181\pi$
$199$	13.4222	0.951475	0.475737	+	0.879587i	$0.342181\pi$
$200$	12.0000	0.848528
$201$	0	0
$202$	−10.1194	−0.712001
$203$	0	0
$204$	0	0
$205$	−2.09167	−0.146089
$206$	37.3305	2.60094
$207$	0	0
$208$	1.69722	0.117681
$209$	−0.697224	−0.0482280
$210$	0	0
$211$	−7.69722	−0.529899	−0.264949	−	0.964262i	$-0.585355\pi$
$211$	−7.69722	−0.529899	−0.264949	+	0.964262i	$0.585355\pi$
$212$	22.8167	1.56705
$213$	0	0
$214$	13.3305	0.911256
$215$	30.0000	2.04598
$216$	0	0
$217$	0	0
$218$	−23.5139	−1.59256
$219$	0	0
$220$	−6.90833	−0.465759
$221$	−29.7250	−1.99952
$222$	0	0
$223$	−12.8167	−0.858267	−0.429133	−	0.903241i	$-0.641181\pi$
$223$	−12.8167	−0.858267	−0.429133	+	0.903241i	$0.641181\pi$
$224$	0	0
$225$	0	0
$226$	−22.3305	−1.48540
$227$	−25.1194	−1.66724	−0.833618	−	0.552342i	$-0.813734\pi$
$227$	−25.1194	−1.66724	−0.833618	+	0.552342i	$0.813734\pi$
$228$	0	0
$229$	0.788897	0.0521318	0.0260659	−	0.999660i	$-0.491702\pi$
$229$	0.788897	0.0521318	0.0260659	+	0.999660i	$0.491702\pi$
$230$	−20.7250	−1.36656
$231$	0	0
$232$	−29.7250	−1.95154
$233$	2.09167	0.137030	0.0685150	−	0.997650i	$-0.478174\pi$
$233$	2.09167	0.137030	0.0685150	+	0.997650i	$0.478174\pi$
$234$	0	0
$235$	−18.6333	−1.21550
$236$	−20.5139	−1.33534
$237$	0	0
$238$	0	0
$239$	27.4222	1.77379	0.886897	−	0.461966i	$-0.152856\pi$
$239$	27.4222	1.77379	0.886897	+	0.461966i	$0.152856\pi$
$240$	0	0
$241$	20.6056	1.32732	0.663660	−	0.748034i	$-0.269002\pi$
$241$	20.6056	1.32732	0.663660	+	0.748034i	$0.269002\pi$
$242$	−24.2111	−1.55635
$243$	0	0
$244$	13.9083	0.890389
$245$	0	0
$246$	0	0
$247$	−5.60555	−0.356673
$248$	−3.90833	−0.248179
$249$	0	0
$250$	6.90833	0.436921
$251$	−15.9083	−1.00412	−0.502062	−	0.864831i	$-0.667425\pi$
$251$	−15.9083	−1.00412	−0.502062	+	0.864831i	$0.667425\pi$
$252$	0	0
$253$	2.09167	0.131502
$254$	−27.2111	−1.70738
$255$	0	0
$256$	−17.9083	−1.11927
$257$	−0.908327	−0.0566599	−0.0283299	−	0.999599i	$-0.509019\pi$
$257$	−0.908327	−0.0566599	−0.0283299	+	0.999599i	$0.509019\pi$
$258$	0	0
$259$	0	0
$260$	−55.5416	−3.44455
$261$	0	0
$262$	−47.2389	−2.91843
$263$	9.69722	0.597956	0.298978	−	0.954260i	$-0.403354\pi$
$263$	9.69722	0.597956	0.298978	+	0.954260i	$0.403354\pi$
$264$	0	0
$265$	−20.7250	−1.27313
$266$	0	0
$267$	0	0
$268$	−6.30278	−0.385003
$269$	20.7250	1.26362	0.631812	−	0.775122i	$-0.282311\pi$
$269$	20.7250	1.26362	0.631812	+	0.775122i	$0.282311\pi$
$270$	0	0
$271$	−17.9083	−1.08785	−0.543927	−	0.839133i	$-0.683063\pi$
$271$	−17.9083	−1.08785	−0.543927	+	0.839133i	$0.683063\pi$
$272$	−1.60555	−0.0973508
$273$	0	0
$274$	−49.8167	−3.00953
$275$	2.78890	0.168177
$276$	0	0
$277$	0.605551	0.0363840	0.0181920	−	0.999835i	$-0.494209\pi$
$277$	0.605551	0.0363840	0.0181920	+	0.999835i	$0.494209\pi$
$278$	−11.5139	−0.690557
$279$	0	0
$280$	0	0
$281$	−3.00000	−0.178965	−0.0894825	−	0.995988i	$-0.528521\pi$
$281$	−3.00000	−0.178965	−0.0894825	+	0.995988i	$0.528521\pi$
$282$	0	0
$283$	−27.3305	−1.62463	−0.812316	−	0.583218i	$-0.801793\pi$
$283$	−27.3305	−1.62463	−0.812316	+	0.583218i	$0.801793\pi$
$284$	−40.3305	−2.39318
$285$	0	0
$286$	9.00000	0.532181
$287$	0	0
$288$	0	0
$289$	11.1194	0.654084
$290$	68.4500	4.01952
$291$	0	0
$292$	−5.00000	−0.292603
$293$	−12.6333	−0.738046	−0.369023	−	0.929420i	$-0.620308\pi$
$293$	−12.6333	−0.738046	−0.369023	+	0.929420i	$0.620308\pi$
$294$	0	0
$295$	18.6333	1.08487
$296$	10.8167	0.628705
$297$	0	0
$298$	14.3028	0.828538
$299$	16.8167	0.972532
$300$	0	0
$301$	0	0
$302$	13.6056	0.782911
$303$	0	0
$304$	−0.302776	−0.0173654
$305$	−12.6333	−0.723381
$306$	0	0
$307$	−14.6972	−0.838815	−0.419407	−	0.907798i	$-0.637762\pi$
$307$	−14.6972	−0.838815	−0.419407	+	0.907798i	$0.637762\pi$
$308$	0	0
$309$	0	0
$310$	9.00000	0.511166
$311$	−2.51388	−0.142549	−0.0712745	−	0.997457i	$-0.522707\pi$
$311$	−2.51388	−0.142549	−0.0712745	+	0.997457i	$0.522707\pi$
$312$	0	0
$313$	27.0278	1.52770	0.763850	−	0.645394i	$-0.223307\pi$
$313$	27.0278	1.52770	0.763850	+	0.645394i	$0.223307\pi$
$314$	−10.3944	−0.586593
$315$	0	0
$316$	37.0278	2.08297
$317$	−8.78890	−0.493634	−0.246817	−	0.969062i	$-0.579385\pi$
$317$	−8.78890	−0.493634	−0.246817	+	0.969062i	$0.579385\pi$
$318$	0	0
$319$	−6.90833	−0.386792
$320$	38.4500	2.14942
$321$	0	0
$322$	0	0
$323$	5.30278	0.295054
$324$	0	0
$325$	22.4222	1.24376
$326$	13.1194	0.726618
$327$	0	0
$328$	2.09167	0.115493
$329$	0	0
$330$	0	0
$331$	11.6972	0.642938	0.321469	−	0.946920i	$-0.395823\pi$
$331$	11.6972	0.642938	0.321469	+	0.946920i	$0.395823\pi$
$332$	−42.6333	−2.33981
$333$	0	0
$334$	10.1194	0.553711
$335$	5.72498	0.312789
$336$	0	0
$337$	7.72498	0.420807	0.210403	−	0.977615i	$-0.432522\pi$
$337$	7.72498	0.420807	0.210403	+	0.977615i	$0.432522\pi$
$338$	42.4222	2.30746
$339$	0	0
$340$	52.5416	2.84947
$341$	−0.908327	−0.0491887
$342$	0	0
$343$	0	0
$344$	−30.0000	−1.61749
$345$	0	0
$346$	18.0000	0.967686
$347$	28.5416	1.53220	0.766098	−	0.642724i	$-0.222196\pi$
$347$	28.5416	1.53220	0.766098	+	0.642724i	$0.222196\pi$
$348$	0	0
$349$	−7.51388	−0.402209	−0.201104	−	0.979570i	$-0.564453\pi$
$349$	−7.51388	−0.402209	−0.201104	+	0.979570i	$0.564453\pi$
$350$	0	0
$351$	0	0
$352$	−3.69722	−0.197063
$353$	9.90833	0.527367	0.263684	−	0.964609i	$-0.415063\pi$
$353$	9.90833	0.527367	0.263684	+	0.964609i	$0.415063\pi$
$354$	0	0
$355$	36.6333	1.94429
$356$	−35.0278	−1.85647
$357$	0	0
$358$	27.0000	1.42699
$359$	25.5416	1.34804	0.674018	−	0.738715i	$-0.264567\pi$
$359$	25.5416	1.34804	0.674018	+	0.738715i	$0.264567\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	24.6333	1.29470
$363$	0	0
$364$	0	0
$365$	4.54163	0.237720
$366$	0	0
$367$	0.0277564	0.00144887	0.000724436	−	1.00000i	$-0.499769\pi$
$367$	0.0277564	0.00144887	0.000724436		1.00000i	$0.499769\pi$
$368$	0.908327	0.0473498
$369$	0	0
$370$	−24.9083	−1.29492
$371$	0	0
$372$	0	0
$373$	24.1194	1.24886	0.624428	−	0.781082i	$-0.285332\pi$
$373$	24.1194	1.24886	0.624428	+	0.781082i	$0.285332\pi$
$374$	−8.51388	−0.440242
$375$	0	0
$376$	18.6333	0.960939
$377$	−55.5416	−2.86054
$378$	0	0
$379$	6.81665	0.350148	0.175074	−	0.984555i	$-0.443984\pi$
$379$	6.81665	0.350148	0.175074	+	0.984555i	$0.443984\pi$
$380$	9.90833	0.508286
$381$	0	0
$382$	−58.3305	−2.98445
$383$	6.63331	0.338946	0.169473	−	0.985535i	$-0.445793\pi$
$383$	6.63331	0.338946	0.169473	+	0.985535i	$0.445793\pi$
$384$	0	0
$385$	0	0
$386$	−53.2389	−2.70979
$387$	0	0
$388$	−31.7250	−1.61059
$389$	24.1472	1.22431	0.612155	−	0.790737i	$-0.290303\pi$
$389$	24.1472	1.22431	0.612155	+	0.790737i	$0.290303\pi$
$390$	0	0
$391$	−15.9083	−0.804519
$392$	0	0
$393$	0	0
$394$	15.9083	0.801450
$395$	−33.6333	−1.69228
$396$	0	0
$397$	−37.0278	−1.85837	−0.929185	−	0.369615i	$-0.879490\pi$
$397$	−37.0278	−1.85837	−0.929185	+	0.369615i	$0.879490\pi$
$398$	30.9083	1.54929
$399$	0	0
$400$	1.21110	0.0605551
$401$	3.48612	0.174089	0.0870443	−	0.996204i	$-0.472258\pi$
$401$	3.48612	0.174089	0.0870443	+	0.996204i	$0.472258\pi$
$402$	0	0
$403$	−7.30278	−0.363777
$404$	−14.5139	−0.722092
$405$	0	0
$406$	0	0
$407$	2.51388	0.124608
$408$	0	0
$409$	−20.9083	−1.03385	−0.516925	−	0.856031i	$-0.672923\pi$
$409$	−20.9083	−1.03385	−0.516925	+	0.856031i	$0.672923\pi$
$410$	−4.81665	−0.237878
$411$	0	0
$412$	53.5416	2.63781
$413$	0	0
$414$	0	0
$415$	38.7250	1.90093
$416$	−29.7250	−1.45739
$417$	0	0
$418$	−1.60555	−0.0785301
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	0	0
$421$	−29.6056	−1.44289	−0.721443	−	0.692474i	$-0.756521\pi$
$421$	−29.6056	−1.44289	−0.721443	+	0.692474i	$0.756521\pi$
$422$	−17.7250	−0.862839
$423$	0	0
$424$	20.7250	1.00649
$425$	−21.2111	−1.02889
$426$	0	0
$427$	0	0
$428$	19.1194	0.924173
$429$	0	0
$430$	69.0833	3.33149
$431$	3.21110	0.154673	0.0773367	−	0.997005i	$-0.475358\pi$
$431$	3.21110	0.154673	0.0773367	+	0.997005i	$0.475358\pi$
$432$	0	0
$433$	0.577795	0.0277671	0.0138835	−	0.999904i	$-0.495581\pi$
$433$	0.577795	0.0277671	0.0138835	+	0.999904i	$0.495581\pi$
$434$	0	0
$435$	0	0
$436$	−33.7250	−1.61513
$437$	−3.00000	−0.143509
$438$	0	0
$439$	−22.7889	−1.08765	−0.543827	−	0.839197i	$-0.683025\pi$
$439$	−22.7889	−1.08765	−0.543827	+	0.839197i	$0.683025\pi$
$440$	−6.27502	−0.299150
$441$	0	0
$442$	−68.4500	−3.25583
$443$	16.1194	0.765857	0.382929	−	0.923778i	$-0.374916\pi$
$443$	16.1194	0.765857	0.382929	+	0.923778i	$0.374916\pi$
$444$	0	0
$445$	31.8167	1.50825
$446$	−29.5139	−1.39752
$447$	0	0
$448$	0	0
$449$	0.486122	0.0229415	0.0114708	−	0.999934i	$-0.496349\pi$
$449$	0.486122	0.0229415	0.0114708	+	0.999934i	$0.496349\pi$
$450$	0	0
$451$	0.486122	0.0228906
$452$	−32.0278	−1.50646
$453$	0	0
$454$	−57.8444	−2.71477
$455$	0	0
$456$	0	0
$457$	22.3028	1.04328	0.521640	−	0.853166i	$-0.325320\pi$
$457$	22.3028	1.04328	0.521640	+	0.853166i	$0.325320\pi$
$458$	1.81665	0.0848867
$459$	0	0
$460$	−29.7250	−1.38593
$461$	0.908327	0.0423050	0.0211525	−	0.999776i	$-0.493266\pi$
$461$	0.908327	0.0423050	0.0211525	+	0.999776i	$0.493266\pi$
$462$	0	0
$463$	−36.4500	−1.69397	−0.846987	−	0.531614i	$-0.821586\pi$
$463$	−36.4500	−1.69397	−0.846987	+	0.531614i	$0.821586\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	4.81665	0.223127
$467$	16.5416	0.765456	0.382728	−	0.923861i	$-0.374985\pi$
$467$	16.5416	0.765456	0.382728	+	0.923861i	$0.374985\pi$
$468$	0	0
$469$	0	0
$470$	−42.9083	−1.97921
$471$	0	0
$472$	−18.6333	−0.857668
$473$	−6.97224	−0.320584
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	0	0
$478$	63.1472	2.88829
$479$	−0.697224	−0.0318570	−0.0159285	−	0.999873i	$-0.505070\pi$
$479$	−0.697224	−0.0318570	−0.0159285	+	0.999873i	$0.505070\pi$
$480$	0	0
$481$	20.2111	0.921547
$482$	47.4500	2.16129
$483$	0	0
$484$	−34.7250	−1.57841
$485$	28.8167	1.30850
$486$	0	0
$487$	4.57779	0.207440	0.103720	−	0.994607i	$-0.466925\pi$
$487$	4.57779	0.207440	0.103720	+	0.994607i	$0.466925\pi$
$488$	12.6333	0.571883
$489$	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
$493$	52.5416	2.36636
$494$	−12.9083	−0.580773
$495$	0	0
$496$	−0.394449	−0.0177113
$497$	0	0
$498$	0	0
$499$	19.9361	0.892462	0.446231	−	0.894918i	$-0.352766\pi$
$499$	19.9361	0.892462	0.446231	+	0.894918i	$0.352766\pi$
$500$	9.90833	0.443114
$501$	0	0
$502$	−36.6333	−1.63502
$503$	6.42221	0.286352	0.143176	−	0.989697i	$-0.454269\pi$
$503$	6.42221	0.286352	0.143176	+	0.989697i	$0.454269\pi$
$504$	0	0
$505$	13.1833	0.586651
$506$	4.81665	0.214126
$507$	0	0
$508$	−39.0278	−1.73158
$509$	33.6333	1.49077	0.745385	−	0.666634i	$-0.232266\pi$
$509$	33.6333	1.49077	0.745385	+	0.666634i	$0.232266\pi$
$510$	0	0
$511$	0	0
$512$	−3.42221	−0.151242
$513$	0	0
$514$	−2.09167	−0.0922597
$515$	−48.6333	−2.14304
$516$	0	0
$517$	4.33053	0.190457
$518$	0	0
$519$	0	0
$520$	−50.4500	−2.21238
$521$	36.6333	1.60493	0.802467	−	0.596696i	$-0.203520\pi$
$521$	36.6333	1.60493	0.802467	+	0.596696i	$0.203520\pi$
$522$	0	0
$523$	−37.6611	−1.64680	−0.823402	−	0.567459i	$-0.807927\pi$
$523$	−37.6611	−1.64680	−0.823402	+	0.567459i	$0.807927\pi$
$524$	−67.7527	−2.95979
$525$	0	0
$526$	22.3305	0.973657
$527$	6.90833	0.300931
$528$	0	0
$529$	−14.0000	−0.608696
$530$	−47.7250	−2.07304
$531$	0	0
$532$	0	0
$533$	3.90833	0.169288
$534$	0	0
$535$	−17.3667	−0.750828
$536$	−5.72498	−0.247282
$537$	0	0
$538$	47.7250	2.05757
$539$	0	0
$540$	0	0
$541$	16.7889	0.721811	0.360906	−	0.932602i	$-0.382468\pi$
$541$	16.7889	0.721811	0.360906	+	0.932602i	$0.382468\pi$
$542$	−41.2389	−1.77136
$543$	0	0
$544$	28.1194	1.20561
$545$	30.6333	1.31219
$546$	0	0
$547$	−10.4861	−0.448354	−0.224177	−	0.974548i	$-0.571969\pi$
$547$	−10.4861	−0.448354	−0.224177	+	0.974548i	$0.571969\pi$
$548$	−71.4500	−3.05219
$549$	0	0
$550$	6.42221	0.273844
$551$	9.90833	0.422109
$552$	0	0
$553$	0	0
$554$	1.39445	0.0592444
$555$	0	0
$556$	−16.5139	−0.700344
$557$	1.11943	0.0474317	0.0237159	−	0.999719i	$-0.492450\pi$
$557$	1.11943	0.0474317	0.0237159	+	0.999719i	$0.492450\pi$
$558$	0	0
$559$	−56.0555	−2.37090
$560$	0	0
$561$	0	0
$562$	−6.90833	−0.291410
$563$	−24.2111	−1.02038	−0.510188	−	0.860063i	$-0.670424\pi$
$563$	−24.2111	−1.02038	−0.510188	+	0.860063i	$0.670424\pi$
$564$	0	0
$565$	29.0917	1.22390
$566$	−62.9361	−2.64540
$567$	0	0
$568$	−36.6333	−1.53710
$569$	−7.18335	−0.301142	−0.150571	−	0.988599i	$-0.548111\pi$
$569$	−7.18335	−0.301142	−0.150571	+	0.988599i	$0.548111\pi$
$570$	0	0
$571$	−2.39445	−0.100205	−0.0501023	−	0.998744i	$-0.515955\pi$
$571$	−2.39445	−0.100205	−0.0501023	+	0.998744i	$0.515955\pi$
$572$	12.9083	0.539724
$573$	0	0
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	−19.5139	−0.812373	−0.406187	−	0.913790i	$-0.633142\pi$
$577$	−19.5139	−0.812373	−0.406187	+	0.913790i	$0.633142\pi$
$578$	25.6056	1.06505
$579$	0	0
$580$	98.1749	4.07649
$581$	0	0
$582$	0	0
$583$	4.81665	0.199485
$584$	−4.54163	−0.187934
$585$	0	0
$586$	−29.0917	−1.20177
$587$	0.422205	0.0174263	0.00871313	−	0.999962i	$-0.497226\pi$
$587$	0.422205	0.0174263	0.00871313	+	0.999962i	$0.497226\pi$
$588$	0	0
$589$	1.30278	0.0536799
$590$	42.9083	1.76651
$591$	0	0
$592$	1.09167	0.0448675
$593$	−4.81665	−0.197796	−0.0988981	−	0.995098i	$-0.531532\pi$
$593$	−4.81665	−0.197796	−0.0988981	+	0.995098i	$0.531532\pi$
$594$	0	0
$595$	0	0
$596$	20.5139	0.840281
$597$	0	0
$598$	38.7250	1.58358
$599$	0.908327	0.0371132	0.0185566	−	0.999828i	$-0.494093\pi$
$599$	0.908327	0.0371132	0.0185566	+	0.999828i	$0.494093\pi$
$600$	0	0
$601$	27.5139	1.12231	0.561157	−	0.827709i	$-0.310356\pi$
$601$	27.5139	1.12231	0.561157	+	0.827709i	$0.310356\pi$
$602$	0	0
$603$	0	0
$604$	19.5139	0.794008
$605$	31.5416	1.28235
$606$	0	0
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	5.30278	0.215056
$609$	0	0
$610$	−29.0917	−1.17789
$611$	34.8167	1.40853
$612$	0	0
$613$	−0.724981	−0.0292817	−0.0146408	−	0.999893i	$-0.504660\pi$
$613$	−0.724981	−0.0292817	−0.0146408	+	0.999893i	$0.504660\pi$
$614$	−33.8444	−1.36585
$615$	0	0
$616$	0	0
$617$	10.8806	0.438035	0.219018	−	0.975721i	$-0.429715\pi$
$617$	10.8806	0.438035	0.219018	+	0.975721i	$0.429715\pi$
$618$	0	0
$619$	41.3305	1.66121	0.830607	−	0.556859i	$-0.187994\pi$
$619$	41.3305	1.66121	0.830607	+	0.556859i	$0.187994\pi$
$620$	12.9083	0.518411
$621$	0	0
$622$	−5.78890	−0.232114
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	62.2389	2.48757
$627$	0	0
$628$	−14.9083	−0.594907
$629$	−19.1194	−0.762342
$630$	0	0
$631$	13.2389	0.527031	0.263515	−	0.964655i	$-0.415118\pi$
$631$	13.2389	0.527031	0.263515	+	0.964655i	$0.415118\pi$
$632$	33.6333	1.33786
$633$	0	0
$634$	−20.2389	−0.803788
$635$	35.4500	1.40679
$636$	0	0
$637$	0	0
$638$	−15.9083	−0.629817
$639$	0	0
$640$	56.7250	2.24225
$641$	−9.90833	−0.391355	−0.195678	−	0.980668i	$-0.562691\pi$
$641$	−9.90833	−0.391355	−0.195678	+	0.980668i	$0.562691\pi$
$642$	0	0
$643$	−30.3944	−1.19864	−0.599320	−	0.800510i	$-0.704562\pi$
$643$	−30.3944	−1.19864	−0.599320	+	0.800510i	$0.704562\pi$
$644$	0	0
$645$	0	0
$646$	12.2111	0.480439
$647$	15.4222	0.606309	0.303155	−	0.952941i	$-0.401960\pi$
$647$	15.4222	0.606309	0.303155	+	0.952941i	$0.401960\pi$
$648$	0	0
$649$	−4.33053	−0.169988
$650$	51.6333	2.02522
$651$	0	0
$652$	18.8167	0.736917
$653$	16.8167	0.658087	0.329043	−	0.944315i	$-0.393274\pi$
$653$	16.8167	0.658087	0.329043	+	0.944315i	$0.393274\pi$
$654$	0	0
$655$	61.5416	2.40463
$656$	0.211103	0.00824217
$657$	0	0
$658$	0	0
$659$	−16.3305	−0.636147	−0.318074	−	0.948066i	$-0.603036\pi$
$659$	−16.3305	−0.636147	−0.318074	+	0.948066i	$0.603036\pi$
$660$	0	0
$661$	0.788897	0.0306846	0.0153423	−	0.999882i	$-0.495116\pi$
$661$	0.788897	0.0306846	0.0153423	+	0.999882i	$0.495116\pi$
$662$	26.9361	1.04690
$663$	0	0
$664$	−38.7250	−1.50282
$665$	0	0
$666$	0	0
$667$	−29.7250	−1.15096
$668$	14.5139	0.561559
$669$	0	0
$670$	13.1833	0.509317
$671$	2.93608	0.113346
$672$	0	0
$673$	14.9083	0.574674	0.287337	−	0.957830i	$-0.407230\pi$
$673$	14.9083	0.574674	0.287337	+	0.957830i	$0.407230\pi$
$674$	17.7889	0.685203
$675$	0	0
$676$	60.8444	2.34017
$677$	17.9361	0.689340	0.344670	−	0.938724i	$-0.387991\pi$
$677$	17.9361	0.689340	0.344670	+	0.938724i	$0.387991\pi$
$678$	0	0
$679$	0	0
$680$	47.7250	1.83017
$681$	0	0
$682$	−2.09167	−0.0800943
$683$	−12.2111	−0.467245	−0.233622	−	0.972327i	$-0.575058\pi$
$683$	−12.2111	−0.467245	−0.233622	+	0.972327i	$0.575058\pi$
$684$	0	0
$685$	64.8999	2.47970
$686$	0	0
$687$	0	0
$688$	−3.02776	−0.115432
$689$	38.7250	1.47530
$690$	0	0
$691$	41.8167	1.59078	0.795390	−	0.606098i	$-0.207266\pi$
$691$	41.8167	1.59078	0.795390	+	0.606098i	$0.207266\pi$
$692$	25.8167	0.981402
$693$	0	0
$694$	65.7250	2.49489
$695$	15.0000	0.568982
$696$	0	0
$697$	−3.69722	−0.140042
$698$	−17.3028	−0.654920
$699$	0	0
$700$	0	0
$701$	−21.2111	−0.801132	−0.400566	−	0.916268i	$-0.631187\pi$
$701$	−21.2111	−0.801132	−0.400566	+	0.916268i	$0.631187\pi$
$702$	0	0
$703$	−3.60555	−0.135986
$704$	−8.93608	−0.336791
$705$	0	0
$706$	22.8167	0.858716
$707$	0	0
$708$	0	0
$709$	−11.6056	−0.435856	−0.217928	−	0.975965i	$-0.569930\pi$
$709$	−11.6056	−0.435856	−0.217928	+	0.975965i	$0.569930\pi$
$710$	84.3583	3.16591
$711$	0	0
$712$	−31.8167	−1.19238
$713$	−3.90833	−0.146368
$714$	0	0
$715$	−11.7250	−0.438489
$716$	38.7250	1.44722
$717$	0	0
$718$	58.8167	2.19502
$719$	−27.6333	−1.03055	−0.515274	−	0.857025i	$-0.672310\pi$
$719$	−27.6333	−1.03055	−0.515274	+	0.857025i	$0.672310\pi$
$720$	0	0
$721$	0	0
$722$	2.30278	0.0857004
$723$	0	0
$724$	35.3305	1.31305
$725$	−39.6333	−1.47194
$726$	0	0
$727$	27.6611	1.02589	0.512946	−	0.858421i	$-0.328554\pi$
$727$	27.6611	1.02589	0.512946	+	0.858421i	$0.328554\pi$
$728$	0	0
$729$	0	0
$730$	10.4584	0.387081
$731$	53.0278	1.96130
$732$	0	0
$733$	−32.0000	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$733$	−32.0000	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$734$	0.0639167	0.00235921
$735$	0	0
$736$	−15.9083	−0.586389
$737$	−1.33053	−0.0490108
$738$	0	0
$739$	−18.5778	−0.683395	−0.341698	−	0.939810i	$-0.611002\pi$
$739$	−18.5778	−0.683395	−0.341698	+	0.939810i	$0.611002\pi$
$740$	−35.7250	−1.31328
$741$	0	0
$742$	0	0
$743$	12.6333	0.463471	0.231736	−	0.972779i	$-0.425560\pi$
$743$	12.6333	0.463471	0.231736	+	0.972779i	$0.425560\pi$
$744$	0	0
$745$	−18.6333	−0.682672
$746$	55.5416	2.03352
$747$	0	0
$748$	−12.2111	−0.446482
$749$	0	0
$750$	0	0
$751$	26.3583	0.961828	0.480914	−	0.876768i	$-0.340305\pi$
$751$	26.3583	0.961828	0.480914	+	0.876768i	$0.340305\pi$
$752$	1.88057	0.0685774
$753$	0	0
$754$	−127.900	−4.65784
$755$	−17.7250	−0.645078
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	15.6972	0.570149
$759$	0	0
$760$	9.00000	0.326464
$761$	−24.2111	−0.877652	−0.438826	−	0.898572i	$-0.644606\pi$
$761$	−24.2111	−0.877652	−0.438826	+	0.898572i	$0.644606\pi$
$762$	0	0
$763$	0	0
$764$	−83.6611	−3.02675
$765$	0	0
$766$	15.2750	0.551909
$767$	−34.8167	−1.25716
$768$	0	0
$769$	25.2111	0.909136	0.454568	−	0.890712i	$-0.349794\pi$
$769$	25.2111	0.909136	0.454568	+	0.890712i	$0.349794\pi$
$770$	0	0
$771$	0	0
$772$	−76.3583	−2.74819
$773$	−22.1833	−0.797880	−0.398940	−	0.916977i	$-0.630622\pi$
$773$	−22.1833	−0.797880	−0.398940	+	0.916977i	$0.630622\pi$
$774$	0	0
$775$	−5.21110	−0.187188
$776$	−28.8167	−1.03446
$777$	0	0
$778$	55.6056	1.99356
$779$	−0.697224	−0.0249807
$780$	0	0
$781$	−8.51388	−0.304651
$782$	−36.6333	−1.31000
$783$	0	0
$784$	0	0
$785$	13.5416	0.483322
$786$	0	0
$787$	31.6333	1.12761	0.563803	−	0.825909i	$-0.309338\pi$
$787$	31.6333	1.12761	0.563803	+	0.825909i	$0.309338\pi$
$788$	22.8167	0.812810
$789$	0	0
$790$	−77.4500	−2.75555
$791$	0	0
$792$	0	0
$793$	23.6056	0.838258
$794$	−85.2666	−3.02600
$795$	0	0
$796$	44.3305	1.57125
$797$	26.0917	0.924214	0.462107	−	0.886824i	$-0.347094\pi$
$797$	26.0917	0.924214	0.462107	+	0.886824i	$0.347094\pi$
$798$	0	0
$799$	−32.9361	−1.16519
$800$	−21.2111	−0.749926
$801$	0	0
$802$	8.02776	0.283470
$803$	−1.05551	−0.0372482
$804$	0	0
$805$	0	0
$806$	−16.8167	−0.592341
$807$	0	0
$808$	−13.1833	−0.463788
$809$	26.0278	0.915087	0.457544	−	0.889187i	$-0.348729\pi$
$809$	26.0278	0.915087	0.457544	+	0.889187i	$0.348729\pi$
$810$	0	0
$811$	32.0555	1.12562	0.562811	−	0.826586i	$-0.309720\pi$
$811$	32.0555	1.12562	0.562811	+	0.826586i	$0.309720\pi$
$812$	0	0
$813$	0	0
$814$	5.78890	0.202901
$815$	−17.0917	−0.598695
$816$	0	0
$817$	10.0000	0.349856
$818$	−48.1472	−1.68343
$819$	0	0
$820$	−6.90833	−0.241249
$821$	51.6333	1.80201	0.901007	−	0.433804i	$-0.142829\pi$
$821$	51.6333	1.80201	0.901007	+	0.433804i	$0.142829\pi$
$822$	0	0
$823$	16.2389	0.566051	0.283026	−	0.959112i	$-0.408662\pi$
$823$	16.2389	0.566051	0.283026	+	0.959112i	$0.408662\pi$
$824$	48.6333	1.69422
$825$	0	0
$826$	0	0
$827$	45.0000	1.56480	0.782402	−	0.622774i	$-0.213994\pi$
$827$	45.0000	1.56480	0.782402	+	0.622774i	$0.213994\pi$
$828$	0	0
$829$	−54.0555	−1.87743	−0.938713	−	0.344700i	$-0.887981\pi$
$829$	−54.0555	−1.87743	−0.938713	+	0.344700i	$0.887981\pi$
$830$	89.1749	3.09531
$831$	0	0
$832$	−71.8444	−2.49076
$833$	0	0
$834$	0	0
$835$	−13.1833	−0.456229
$836$	−2.30278	−0.0796432
$837$	0	0
$838$	34.5416	1.19322
$839$	−2.78890	−0.0962834	−0.0481417	−	0.998841i	$-0.515330\pi$
$839$	−2.78890	−0.0962834	−0.0481417	+	0.998841i	$0.515330\pi$
$840$	0	0
$841$	69.1749	2.38534
$842$	−68.1749	−2.34946
$843$	0	0
$844$	−25.4222	−0.875068
$845$	−55.2666	−1.90123
$846$	0	0
$847$	0	0
$848$	2.09167	0.0718283
$849$	0	0
$850$	−48.8444	−1.67535
$851$	10.8167	0.370790
$852$	0	0
$853$	−3.33053	−0.114035	−0.0570176	−	0.998373i	$-0.518159\pi$
$853$	−3.33053	−0.114035	−0.0570176	+	0.998373i	$0.518159\pi$
$854$	0	0
$855$	0	0
$856$	17.3667	0.593581
$857$	32.0917	1.09623	0.548115	−	0.836403i	$-0.315345\pi$
$857$	32.0917	1.09623	0.548115	+	0.836403i	$0.315345\pi$
$858$	0	0
$859$	27.3028	0.931559	0.465779	−	0.884901i	$-0.345774\pi$
$859$	27.3028	0.931559	0.465779	+	0.884901i	$0.345774\pi$
$860$	99.0833	3.37871
$861$	0	0
$862$	7.39445	0.251856
$863$	16.5416	0.563084	0.281542	−	0.959549i	$-0.409154\pi$
$863$	16.5416	0.563084	0.281542	+	0.959549i	$0.409154\pi$
$864$	0	0
$865$	−23.4500	−0.797323
$866$	1.33053	0.0452133
$867$	0	0
$868$	0	0
$869$	7.81665	0.265162
$870$	0	0
$871$	−10.6972	−0.362462
$872$	−30.6333	−1.03737
$873$	0	0
$874$	−6.90833	−0.233678
$875$	0	0
$876$	0	0
$877$	−8.18335	−0.276332	−0.138166	−	0.990409i	$-0.544121\pi$
$877$	−8.18335	−0.276332	−0.138166	+	0.990409i	$0.544121\pi$
$878$	−52.4777	−1.77104
$879$	0	0
$880$	−0.633308	−0.0213488
$881$	−19.7527	−0.665487	−0.332743	−	0.943017i	$-0.607974\pi$
$881$	−19.7527	−0.665487	−0.332743	+	0.943017i	$0.607974\pi$
$882$	0	0
$883$	−46.2111	−1.55513	−0.777564	−	0.628804i	$-0.783545\pi$
$883$	−46.2111	−1.55513	−0.777564	+	0.628804i	$0.783545\pi$
$884$	−98.1749	−3.30198
$885$	0	0
$886$	37.1194	1.24705
$887$	19.1833	0.644114	0.322057	−	0.946720i	$-0.395626\pi$
$887$	19.1833	0.644114	0.322057	+	0.946720i	$0.395626\pi$
$888$	0	0
$889$	0	0
$890$	73.2666	2.45590
$891$	0	0
$892$	−42.3305	−1.41733
$893$	−6.21110	−0.207847
$894$	0	0
$895$	−35.1749	−1.17577
$896$	0	0
$897$	0	0
$898$	1.11943	0.0373558
$899$	12.9083	0.430517
$900$	0	0
$901$	−36.6333	−1.22043
$902$	1.11943	0.0372729
$903$	0	0
$904$	−29.0917	−0.967575
$905$	−32.0917	−1.06676
$906$	0	0
$907$	−5.18335	−0.172110	−0.0860551	−	0.996290i	$-0.527426\pi$
$907$	−5.18335	−0.172110	−0.0860551	+	0.996290i	$0.527426\pi$
$908$	−82.9638	−2.75325
$909$	0	0
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	0	0
$913$	−9.00000	−0.297857
$914$	51.3583	1.69878
$915$	0	0
$916$	2.60555	0.0860898
$917$	0	0
$918$	0	0
$919$	−7.97224	−0.262980	−0.131490	−	0.991317i	$-0.541976\pi$
$919$	−7.97224	−0.262980	−0.131490	+	0.991317i	$0.541976\pi$
$920$	−27.0000	−0.890164
$921$	0	0
$922$	2.09167	0.0688856
$923$	−68.4500	−2.25306
$924$	0	0
$925$	14.4222	0.474199
$926$	−83.9361	−2.75831
$927$	0	0
$928$	52.5416	1.72476
$929$	11.5139	0.377758	0.188879	−	0.982000i	$-0.439515\pi$
$929$	11.5139	0.377758	0.188879	+	0.982000i	$0.439515\pi$
$930$	0	0
$931$	0	0
$932$	6.90833	0.226290
$933$	0	0
$934$	38.0917	1.24640
$935$	11.0917	0.362736
$936$	0	0
$937$	−21.1194	−0.689942	−0.344971	−	0.938613i	$-0.612111\pi$
$937$	−21.1194	−0.689942	−0.344971	+	0.938613i	$0.612111\pi$
$938$	0	0
$939$	0	0
$940$	−61.5416	−2.00727
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	2.09167	0.0681142
$944$	−1.88057	−0.0612074
$945$	0	0
$946$	−16.0555	−0.522010
$947$	−36.9083	−1.19936	−0.599680	−	0.800240i	$-0.704705\pi$
$947$	−36.9083	−1.19936	−0.599680	+	0.800240i	$0.704705\pi$
$948$	0	0
$949$	−8.48612	−0.275471
$950$	−9.21110	−0.298848
$951$	0	0
$952$	0	0
$953$	−4.33053	−0.140280	−0.0701398	−	0.997537i	$-0.522345\pi$
$953$	−4.33053	−0.140280	−0.0701398	+	0.997537i	$0.522345\pi$
$954$	0	0
$955$	75.9916	2.45903
$956$	90.5694	2.92922
$957$	0	0
$958$	−1.60555	−0.0518730
$959$	0	0
$960$	0	0
$961$	−29.3028	−0.945251
$962$	46.5416	1.50056
$963$	0	0
$964$	68.0555	2.19192
$965$	69.3583	2.23272
$966$	0	0
$967$	26.6972	0.858525	0.429262	−	0.903180i	$-0.358774\pi$
$967$	26.6972	0.858525	0.429262	+	0.903180i	$0.358774\pi$
$968$	−31.5416	−1.01379
$969$	0	0
$970$	66.3583	2.13064
$971$	−18.6333	−0.597971	−0.298986	−	0.954258i	$-0.596648\pi$
$971$	−18.6333	−0.597971	−0.298986	+	0.954258i	$0.596648\pi$
$972$	0	0
$973$	0	0
$974$	10.5416	0.337776
$975$	0	0
$976$	1.27502	0.0408124
$977$	17.4500	0.558274	0.279137	−	0.960251i	$-0.409952\pi$
$977$	17.4500	0.558274	0.279137	+	0.960251i	$0.409952\pi$
$978$	0	0
$979$	−7.39445	−0.236328
$980$	0	0
$981$	0	0
$982$	27.6333	0.881814
$983$	−49.0555	−1.56463	−0.782314	−	0.622884i	$-0.785961\pi$
$983$	−49.0555	−1.56463	−0.782314	+	0.622884i	$0.785961\pi$
$984$	0	0
$985$	−20.7250	−0.660353
$986$	120.992	3.85316
$987$	0	0
$988$	−18.5139	−0.589005
$989$	−30.0000	−0.953945
$990$	0	0
$991$	−16.9722	−0.539141	−0.269571	−	0.962981i	$-0.586882\pi$
$991$	−16.9722	−0.539141	−0.269571	+	0.962981i	$0.586882\pi$
$992$	6.90833	0.219340
$993$	0	0
$994$	0	0
$995$	−40.2666	−1.27654
$996$	0	0
$997$	−5.27502	−0.167062	−0.0835308	−	0.996505i	$-0.526620\pi$
$997$	−5.27502	−0.167062	−0.0835308	+	0.996505i	$0.526620\pi$
$998$	45.9083	1.45320
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.bf.1.2		2
3.2	odd	2		931.2.a.g.1.1		2
7.6	odd	2		1197.2.a.h.1.2		2
21.2	odd	6		931.2.f.g.704.2		4
21.5	even	6		931.2.f.h.704.2		4
21.11	odd	6		931.2.f.g.324.2		4
21.17	even	6		931.2.f.h.324.2		4
21.20	even	2		133.2.a.b.1.1	✓	2
84.83	odd	2		2128.2.a.l.1.1		2
105.104	even	2		3325.2.a.n.1.2		2
168.83	odd	2		8512.2.a.l.1.2		2
168.125	even	2		8512.2.a.bh.1.1		2
399.398	odd	2		2527.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
133.2.a.b.1.1	✓	2	21.20	even	2
931.2.a.g.1.1		2	3.2	odd	2
931.2.f.g.324.2		4	21.11	odd	6
931.2.f.g.704.2		4	21.2	odd	6
931.2.f.h.324.2		4	21.17	even	6
931.2.f.h.704.2		4	21.5	even	6
1197.2.a.h.1.2		2	7.6	odd	2
2128.2.a.l.1.1		2	84.83	odd	2
2527.2.a.d.1.2		2	399.398	odd	2
3325.2.a.n.1.2		2	105.104	even	2
8379.2.a.bf.1.2		2	1.1	even	1	trivial
8512.2.a.l.1.2		2	168.83	odd	2
8512.2.a.bh.1.1		2	168.125	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30278 q^{2} +3.30278 q^{4} -3.00000 q^{5} +3.00000 q^{8} -6.90833 q^{10} +0.697224 q^{11} +5.60555 q^{13} +0.302776 q^{16} -5.30278 q^{17} -1.00000 q^{19} -9.90833 q^{20} +1.60555 q^{22} +3.00000 q^{23} +4.00000 q^{25} +12.9083 q^{26} -9.90833 q^{29} -1.30278 q^{31} -5.30278 q^{32} -12.2111 q^{34} +3.60555 q^{37} -2.30278 q^{38} -9.00000 q^{40} +0.697224 q^{41} -10.0000 q^{43} +2.30278 q^{44} +6.90833 q^{46} +6.21110 q^{47} +9.21110 q^{50} +18.5139 q^{52} +6.90833 q^{53} -2.09167 q^{55} -22.8167 q^{58} -6.21110 q^{59} +4.21110 q^{61} -3.00000 q^{62} -12.8167 q^{64} -16.8167 q^{65} -1.90833 q^{67} -17.5139 q^{68} -12.2111 q^{71} -1.51388 q^{73} +8.30278 q^{74} -3.30278 q^{76} +11.2111 q^{79} -0.908327 q^{80} +1.60555 q^{82} -12.9083 q^{83} +15.9083 q^{85} -23.0278 q^{86} +2.09167 q^{88} -10.6056 q^{89} +9.90833 q^{92} +14.3028 q^{94} +3.00000 q^{95} -9.60555 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 3 q^{4} - 6 q^{5} + 6 q^{8} - 3 q^{10} + 5 q^{11} + 4 q^{13} - 3 q^{16} - 7 q^{17} - 2 q^{19} - 9 q^{20} - 4 q^{22} + 6 q^{23} + 8 q^{25} + 15 q^{26} - 9 q^{29} + q^{31} - 7 q^{32} - 10 q^{34}+ \cdots - 12 q^{97}+O(q^{100}) \) 2 * q + q^2 + 3 * q^4 - 6 * q^5 + 6 * q^8 - 3 * q^10 + 5 * q^11 + 4 * q^13 - 3 * q^16 - 7 * q^17 - 2 * q^19 - 9 * q^20 - 4 * q^22 + 6 * q^23 + 8 * q^25 + 15 * q^26 - 9 * q^29 + q^31 - 7 * q^32 - 10 * q^34 - q^38 - 18 * q^40 + 5 * q^41 - 20 * q^43 + q^44 + 3 * q^46 - 2 * q^47 + 4 * q^50 + 19 * q^52 + 3 * q^53 - 15 * q^55 - 24 * q^58 + 2 * q^59 - 6 * q^61 - 6 * q^62 - 4 * q^64 - 12 * q^65 + 7 * q^67 - 17 * q^68 - 10 * q^71 + 15 * q^73 + 13 * q^74 - 3 * q^76 + 8 * q^79 + 9 * q^80 - 4 * q^82 - 15 * q^83 + 21 * q^85 - 10 * q^86 + 15 * q^88 - 14 * q^89 + 9 * q^92 + 25 * q^94 + 6 * q^95 - 12 * q^97

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