LMFDB - Embedded newform 8379.2.a.bu.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 = [4,0,0,2,-8,0,0,-6,0,10,6,0,2,0,0,2,-8] 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:	$4$
Coefficient field:	4.4.5744.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} - 2x + 1 $ x^4 - 5x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 931)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.291367$ 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)

$f(q)$	$=$	$q-0.291367 q^{2} -1.91511 q^{4} -2.06574 q^{5} +1.14073 q^{8} +0.601888 q^{10} +4.94909 q^{11} -1.62374 q^{13} +3.49784 q^{16} -6.99010 q^{17} -1.00000 q^{19} +3.95611 q^{20} -1.44200 q^{22} +7.41294 q^{23} -0.732718 q^{25} +0.473104 q^{26} -3.88335 q^{29} -0.808361 q^{31} -3.30062 q^{32} +2.03668 q^{34} +4.10674 q^{37} +0.291367 q^{38} -2.35646 q^{40} -8.50709 q^{41} +7.31545 q^{43} -9.47803 q^{44} -2.15989 q^{46} -5.41950 q^{47} +0.213490 q^{50} +3.10963 q^{52} +1.99010 q^{53} -10.2235 q^{55} +1.13148 q^{58} -5.75522 q^{59} +11.3812 q^{61} +0.235529 q^{62} -6.03399 q^{64} +3.35422 q^{65} +13.9950 q^{67} +13.3868 q^{68} -6.51969 q^{71} +2.08982 q^{73} -1.19657 q^{74} +1.91511 q^{76} -2.31977 q^{79} -7.22563 q^{80} +2.47868 q^{82} +5.01753 q^{83} +14.4397 q^{85} -2.13148 q^{86} +5.64559 q^{88} +6.69441 q^{89} -14.1966 q^{92} +1.57906 q^{94} +2.06574 q^{95} -14.0663 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 8 q^{5} - 6 q^{8} + 10 q^{10} + 6 q^{11} + 2 q^{13} + 2 q^{16} - 8 q^{17} - 4 q^{19} - 14 q^{22} + 8 q^{23} + 20 q^{25} - 16 q^{26} - 2 q^{29} - 2 q^{32} - 22 q^{34} + 10 q^{37} + 22 q^{40}+ \cdots + 2 q^{97}+O(q^{100}) $ 4 * q + 2 * q^4 - 8 * q^5 - 6 * q^8 + 10 * q^10 + 6 * q^11 + 2 * q^13 + 2 * q^16 - 8 * q^17 - 4 * q^19 - 14 * q^22 + 8 * q^23 + 20 * q^25 - 16 * q^26 - 2 * q^29 - 2 * q^32 - 22 * q^34 + 10 * q^37 + 22 * q^40 - 12 * q^41 + 4 * q^43 + 14 * q^44 - 8 * q^46 - 16 * q^47 - 12 * q^50 + 28 * q^52 - 12 * q^53 - 8 * q^55 + 4 * q^58 - 14 * q^59 + 20 * q^61 + 20 * q^62 - 20 * q^64 - 10 * q^65 + 2 * q^67 + 24 * q^68 + 2 * q^71 - 16 * q^73 + 26 * q^74 - 2 * q^76 - 8 * q^79 - 28 * q^80 - 12 * q^82 - 20 * q^83 - 14 * q^85 - 8 * q^86 - 2 * q^88 - 16 * q^89 - 26 * q^92 + 10 * q^94 + 8 * q^95 + 2 * 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 $

$2$	−0.291367	−0.206028	−0.103014	−	0.994680i	$-0.532849\pi$
$2$	−0.291367	−0.206028	−0.103014	+	0.994680i	$0.532849\pi$
$3$	0	0
$3$	0	0
$4$	−1.91511	−0.957553
$5$	−2.06574	−0.923827	−0.461914	−	0.886925i	$-0.652837\pi$
$5$	−2.06574	−0.923827	−0.461914	+	0.886925i	$0.652837\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.14073	0.403310
$9$	0	0
$10$	0.601888	0.190334
$11$	4.94909	1.49221	0.746104	−	0.665830i	$-0.231922\pi$
$11$	4.94909	1.49221	0.746104	+	0.665830i	$0.231922\pi$
$12$	0	0
$13$	−1.62374	−0.450344	−0.225172	−	0.974319i	$-0.572294\pi$
$13$	−1.62374	−0.450344	−0.225172	+	0.974319i	$0.572294\pi$
$14$	0	0
$15$	0	0
$16$	3.49784	0.874460
$17$	−6.99010	−1.69535	−0.847674	−	0.530518i	$-0.821998\pi$
$17$	−6.99010	−1.69535	−0.847674	+	0.530518i	$0.821998\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	3.95611	0.884613
$21$	0	0
$22$	−1.44200	−0.307436
$23$	7.41294	1.54571	0.772853	−	0.634585i	$-0.218829\pi$
$23$	7.41294	1.54571	0.772853	+	0.634585i	$0.218829\pi$
$24$	0	0
$25$	−0.732718	−0.146544
$26$	0.473104	0.0927833
$27$	0	0
$28$	0	0
$29$	−3.88335	−0.721120	−0.360560	−	0.932736i	$-0.617414\pi$
$29$	−3.88335	−0.721120	−0.360560	+	0.932736i	$0.617414\pi$
$30$	0	0
$31$	−0.808361	−0.145186	−0.0725929	−	0.997362i	$-0.523127\pi$
$31$	−0.808361	−0.145186	−0.0725929	+	0.997362i	$0.523127\pi$
$32$	−3.30062	−0.583472
$33$	0	0
$34$	2.03668	0.349288
$35$	0	0
$36$	0	0
$37$	4.10674	0.675145	0.337572	−	0.941300i	$-0.390394\pi$
$37$	4.10674	0.675145	0.337572	+	0.941300i	$0.390394\pi$
$38$	0.291367	0.0472660
$39$	0	0
$40$	−2.35646	−0.372588
$41$	−8.50709	−1.32858	−0.664292	−	0.747473i	$-0.731267\pi$
$41$	−8.50709	−1.32858	−0.664292	+	0.747473i	$0.731267\pi$
$42$	0	0
$43$	7.31545	1.11560	0.557798	−	0.829977i	$-0.311646\pi$
$43$	7.31545	1.11560	0.557798	+	0.829977i	$0.311646\pi$
$44$	−9.47803	−1.42887
$45$	0	0
$46$	−2.15989	−0.318458
$47$	−5.41950	−0.790515	−0.395258	−	0.918570i	$-0.629345\pi$
$47$	−5.41950	−0.790515	−0.395258	+	0.918570i	$0.629345\pi$
$48$	0	0
$49$	0	0
$50$	0.213490	0.0301920
$51$	0	0
$52$	3.10963	0.431228
$53$	1.99010	0.273361	0.136680	−	0.990615i	$-0.456357\pi$
$53$	1.99010	0.273361	0.136680	+	0.990615i	$0.456357\pi$
$54$	0	0
$55$	−10.2235	−1.37854
$56$	0	0
$57$	0	0
$58$	1.13148	0.148571
$59$	−5.75522	−0.749266	−0.374633	−	0.927173i	$-0.622231\pi$
$59$	−5.75522	−0.749266	−0.374633	+	0.927173i	$0.622231\pi$
$60$	0	0
$61$	11.3812	1.45721	0.728606	−	0.684933i	$-0.240169\pi$
$61$	11.3812	1.45721	0.728606	+	0.684933i	$0.240169\pi$
$62$	0.235529	0.0299123
$63$	0	0
$64$	−6.03399	−0.754248
$65$	3.35422	0.416040
$66$	0	0
$67$	13.9950	1.70976	0.854882	−	0.518822i	$-0.173629\pi$
$67$	13.9950	1.70976	0.854882	+	0.518822i	$0.173629\pi$
$68$	13.3868	1.62338
$69$	0	0
$70$	0	0
$71$	−6.51969	−0.773745	−0.386872	−	0.922133i	$-0.626445\pi$
$71$	−6.51969	−0.773745	−0.386872	+	0.922133i	$0.626445\pi$
$72$	0	0
$73$	2.08982	0.244595	0.122298	−	0.992493i	$-0.460974\pi$
$73$	2.08982	0.244595	0.122298	+	0.992493i	$0.460974\pi$
$74$	−1.19657	−0.139098
$75$	0	0
$76$	1.91511	0.219678
$77$	0	0
$78$	0	0
$79$	−2.31977	−0.260995	−0.130497	−	0.991449i	$-0.541657\pi$
$79$	−2.31977	−0.260995	−0.130497	+	0.991449i	$0.541657\pi$
$80$	−7.22563	−0.807850
$81$	0	0
$82$	2.47868	0.273725
$83$	5.01753	0.550745	0.275373	−	0.961338i	$-0.411199\pi$
$83$	5.01753	0.550745	0.275373	+	0.961338i	$0.411199\pi$
$84$	0	0
$85$	14.4397	1.56621
$86$	−2.13148	−0.229843
$87$	0	0
$88$	5.64559	0.601822
$89$	6.69441	0.709606	0.354803	−	0.934941i	$-0.384548\pi$
$89$	6.69441	0.709606	0.354803	+	0.934941i	$0.384548\pi$
$90$	0	0
$91$	0	0
$92$	−14.1966	−1.48009
$93$	0	0
$94$	1.57906	0.162868
$95$	2.06574	0.211940
$96$	0	0
$97$	−14.0663	−1.42822	−0.714111	−	0.700033i	$-0.753169\pi$
$97$	−14.0663	−1.42822	−0.714111	+	0.700033i	$0.753169\pi$
$98$	0	0
$99$	0	0
$100$	1.40323	0.140323
$101$	1.79414	0.178523	0.0892616	−	0.996008i	$-0.471549\pi$
$101$	1.79414	0.178523	0.0892616	+	0.996008i	$0.471549\pi$
$102$	0	0
$103$	16.9009	1.66529	0.832647	−	0.553805i	$-0.186825\pi$
$103$	16.9009	1.66529	0.832647	+	0.553805i	$0.186825\pi$
$104$	−1.85225	−0.181628
$105$	0	0
$106$	−0.579848	−0.0563199
$107$	−9.00925	−0.870957	−0.435479	−	0.900199i	$-0.643421\pi$
$107$	−9.00925	−0.870957	−0.435479	+	0.900199i	$0.643421\pi$
$108$	0	0
$109$	7.00925	0.671365	0.335682	−	0.941975i	$-0.391033\pi$
$109$	7.00925	0.671365	0.335682	+	0.941975i	$0.391033\pi$
$110$	2.97880	0.284018
$111$	0	0
$112$	0	0
$113$	0.261701	0.0246188	0.0123094	−	0.999924i	$-0.496082\pi$
$113$	0.261701	0.0246188	0.0123094	+	0.999924i	$0.496082\pi$
$114$	0	0
$115$	−15.3132	−1.42796
$116$	7.43703	0.690511
$117$	0	0
$118$	1.67688	0.154369
$119$	0	0
$120$	0	0
$121$	13.4935	1.22668
$122$	−3.31610	−0.300226
$123$	0	0
$124$	1.54810	0.139023
$125$	11.8423	1.05921
$126$	0	0
$127$	−8.07337	−0.716395	−0.358198	−	0.933646i	$-0.616609\pi$
$127$	−8.07337	−0.716395	−0.358198	+	0.933646i	$0.616609\pi$
$128$	8.35934	0.738868
$129$	0	0
$130$	−0.977309	−0.0857157
$131$	19.4327	1.69784	0.848922	−	0.528519i	$-0.177252\pi$
$131$	19.4327	1.69784	0.848922	+	0.528519i	$0.177252\pi$
$132$	0	0
$133$	0	0
$134$	−4.07769	−0.352259
$135$	0	0
$136$	−7.97383	−0.683750
$137$	6.28146	0.536662	0.268331	−	0.963327i	$-0.413528\pi$
$137$	6.28146	0.536662	0.268331	+	0.963327i	$0.413528\pi$
$138$	0	0
$139$	18.1959	1.54336	0.771679	−	0.636012i	$-0.219417\pi$
$139$	18.1959	1.54336	0.771679	+	0.636012i	$0.219417\pi$
$140$	0	0
$141$	0	0
$142$	1.89962	0.159413
$143$	−8.03603	−0.672007
$144$	0	0
$145$	8.02200	0.666191
$146$	−0.608906	−0.0503934
$147$	0	0
$148$	−7.86485	−0.646487
$149$	−8.75806	−0.717488	−0.358744	−	0.933436i	$-0.616795\pi$
$149$	−8.75806	−0.717488	−0.358744	+	0.933436i	$0.616795\pi$
$150$	0	0
$151$	−12.0445	−0.980167	−0.490084	−	0.871675i	$-0.663034\pi$
$151$	−12.0445	−0.980167	−0.490084	+	0.871675i	$0.663034\pi$
$152$	−1.14073	−0.0925256
$153$	0	0
$154$	0	0
$155$	1.66986	0.134127
$156$	0	0
$157$	−4.27588	−0.341253	−0.170626	−	0.985336i	$-0.554579\pi$
$157$	−4.27588	−0.341253	−0.170626	+	0.985336i	$0.554579\pi$
$158$	0.675905	0.0537721
$159$	0	0
$160$	6.81822	0.539028
$161$	0	0
$162$	0	0
$163$	−9.02678	−0.707032	−0.353516	−	0.935428i	$-0.615014\pi$
$163$	−9.02678	−0.707032	−0.353516	+	0.935428i	$0.615014\pi$
$164$	16.2920	1.27219
$165$	0	0
$166$	−1.46194	−0.113469
$167$	−21.2318	−1.64297	−0.821484	−	0.570232i	$-0.806853\pi$
$167$	−21.2318	−1.64297	−0.821484	+	0.570232i	$0.806853\pi$
$168$	0	0
$169$	−10.3635	−0.797190
$170$	−4.20726	−0.322682
$171$	0	0
$172$	−14.0099	−1.06824
$173$	6.80069	0.517047	0.258524	−	0.966005i	$-0.416764\pi$
$173$	6.80069	0.517047	0.258524	+	0.966005i	$0.416764\pi$
$174$	0	0
$175$	0	0
$176$	17.3111	1.30488
$177$	0	0
$178$	−1.95053	−0.146198
$179$	−19.2722	−1.44047	−0.720235	−	0.693730i	$-0.755966\pi$
$179$	−19.2722	−1.44047	−0.720235	+	0.693730i	$0.755966\pi$
$180$	0	0
$181$	20.5508	1.52753	0.763764	−	0.645495i	$-0.223349\pi$
$181$	20.5508	1.52753	0.763764	+	0.645495i	$0.223349\pi$
$182$	0	0
$183$	0	0
$184$	8.45618	0.623398
$185$	−8.48347	−0.623717
$186$	0	0
$187$	−34.5946	−2.52981
$188$	10.3789	0.756960
$189$	0	0
$190$	−0.601888	−0.0436656
$191$	4.48366	0.324426	0.162213	−	0.986756i	$-0.448137\pi$
$191$	4.48366	0.324426	0.162213	+	0.986756i	$0.448137\pi$
$192$	0	0
$193$	−25.0290	−1.80163	−0.900814	−	0.434205i	$-0.857029\pi$
$193$	−25.0290	−1.80163	−0.900814	+	0.434205i	$0.857029\pi$
$194$	4.09847	0.294253
$195$	0	0
$196$	0	0
$197$	0.100378	0.00715167	0.00357583	−	0.999994i	$-0.498862\pi$
$197$	0.100378	0.00715167	0.00357583	+	0.999994i	$0.498862\pi$
$198$	0	0
$199$	−26.9299	−1.90901	−0.954506	−	0.298191i	$-0.903617\pi$
$199$	−26.9299	−1.90901	−0.954506	+	0.298191i	$0.903617\pi$
$200$	−0.835834	−0.0591024
$201$	0	0
$202$	−0.522752	−0.0367807
$203$	0	0
$204$	0	0
$205$	17.5734	1.22738
$206$	−4.92436	−0.343096
$207$	0	0
$208$	−5.67958	−0.393808
$209$	−4.94909	−0.342336
$210$	0	0
$211$	14.5345	1.00060	0.500299	−	0.865853i	$-0.333223\pi$
$211$	14.5345	1.00060	0.500299	+	0.865853i	$0.333223\pi$
$212$	−3.81125	−0.261757
$213$	0	0
$214$	2.62500	0.179441
$215$	−15.1118	−1.03062
$216$	0	0
$217$	0	0
$218$	−2.04226	−0.138320
$219$	0	0
$220$	19.5792	1.32003
$221$	11.3501	0.763490
$222$	0	0
$223$	−2.30829	−0.154574	−0.0772872	−	0.997009i	$-0.524626\pi$
$223$	−2.30829	−0.154574	−0.0772872	+	0.997009i	$0.524626\pi$
$224$	0	0
$225$	0	0
$226$	−0.0762511	−0.00507215
$227$	−19.8522	−1.31764	−0.658819	−	0.752302i	$-0.728944\pi$
$227$	−19.8522	−1.31764	−0.658819	+	0.752302i	$0.728944\pi$
$228$	0	0
$229$	17.4271	1.15162	0.575808	−	0.817585i	$-0.304687\pi$
$229$	17.4271	1.15162	0.575808	+	0.817585i	$0.304687\pi$
$230$	4.46176	0.294200
$231$	0	0
$232$	−4.42986	−0.290835
$233$	−8.89228	−0.582553	−0.291276	−	0.956639i	$-0.594080\pi$
$233$	−8.89228	−0.582553	−0.291276	+	0.956639i	$0.594080\pi$
$234$	0	0
$235$	11.1953	0.730300
$236$	11.0219	0.717461
$237$	0	0
$238$	0	0
$239$	10.9234	0.706575	0.353287	−	0.935515i	$-0.385064\pi$
$239$	10.9234	0.706575	0.353287	+	0.935515i	$0.385064\pi$
$240$	0	0
$241$	5.18397	0.333929	0.166964	−	0.985963i	$-0.446603\pi$
$241$	5.18397	0.333929	0.166964	+	0.985963i	$0.446603\pi$
$242$	−3.93156	−0.252731
$243$	0	0
$244$	−21.7962	−1.39536
$245$	0	0
$246$	0	0
$247$	1.62374	0.103316
$248$	−0.922123	−0.0585549
$249$	0	0
$250$	−3.45046	−0.218226
$251$	−12.2672	−0.774301	−0.387151	−	0.922016i	$-0.626541\pi$
$251$	−12.2672	−0.774301	−0.387151	+	0.922016i	$0.626541\pi$
$252$	0	0
$253$	36.6873	2.30651
$254$	2.35231	0.147597
$255$	0	0
$256$	9.63234	0.602021
$257$	22.3740	1.39565	0.697825	−	0.716268i	$-0.254151\pi$
$257$	22.3740	1.39565	0.697825	+	0.716268i	$0.254151\pi$
$258$	0	0
$259$	0	0
$260$	−6.42369	−0.398380
$261$	0	0
$262$	−5.66205	−0.349802
$263$	−9.07625	−0.559666	−0.279833	−	0.960049i	$-0.590279\pi$
$263$	−9.07625	−0.559666	−0.279833	+	0.960049i	$0.590279\pi$
$264$	0	0
$265$	−4.11102	−0.252538
$266$	0	0
$267$	0	0
$268$	−26.8020	−1.63719
$269$	−18.8198	−1.14747	−0.573733	−	0.819042i	$-0.694505\pi$
$269$	−18.8198	−1.14747	−0.573733	+	0.819042i	$0.694505\pi$
$270$	0	0
$271$	24.9815	1.51752	0.758758	−	0.651373i	$-0.225807\pi$
$271$	24.9815	1.51752	0.758758	+	0.651373i	$0.225807\pi$
$272$	−24.4502	−1.48251
$273$	0	0
$274$	−1.83021	−0.110567
$275$	−3.62629	−0.218673
$276$	0	0
$277$	20.8741	1.25420	0.627100	−	0.778939i	$-0.284242\pi$
$277$	20.8741	1.25420	0.627100	+	0.778939i	$0.284242\pi$
$278$	−5.30169	−0.317974
$279$	0	0
$280$	0	0
$281$	−29.3117	−1.74859	−0.874296	−	0.485393i	$-0.838676\pi$
$281$	−29.3117	−1.74859	−0.874296	+	0.485393i	$0.838676\pi$
$282$	0	0
$283$	0.566275	0.0336616	0.0168308	−	0.999858i	$-0.494642\pi$
$283$	0.566275	0.0336616	0.0168308	+	0.999858i	$0.494642\pi$
$284$	12.4859	0.740901
$285$	0	0
$286$	2.34143	0.138452
$287$	0	0
$288$	0	0
$289$	31.8615	1.87420
$290$	−2.33734	−0.137254
$291$	0	0
$292$	−4.00223	−0.234213
$293$	−25.2701	−1.47630	−0.738148	−	0.674639i	$-0.764300\pi$
$293$	−25.2701	−1.47630	−0.738148	+	0.674639i	$0.764300\pi$
$294$	0	0
$295$	11.8888	0.692192
$296$	4.68470	0.272292
$297$	0	0
$298$	2.55181	0.147822
$299$	−12.0367	−0.696099
$300$	0	0
$301$	0	0
$302$	3.50937	0.201941
$303$	0	0
$304$	−3.49784	−0.200615
$305$	−23.5106	−1.34621
$306$	0	0
$307$	−6.37114	−0.363620	−0.181810	−	0.983334i	$-0.558196\pi$
$307$	−6.37114	−0.363620	−0.181810	+	0.983334i	$0.558196\pi$
$308$	0	0
$309$	0	0
$310$	−0.486543	−0.0276338
$311$	−0.295689	−0.0167670	−0.00838348	−	0.999965i	$-0.502669\pi$
$311$	−0.295689	−0.0167670	−0.00838348	+	0.999965i	$0.502669\pi$
$312$	0	0
$313$	−20.3679	−1.15126	−0.575632	−	0.817709i	$-0.695244\pi$
$313$	−20.3679	−1.15126	−0.575632	+	0.817709i	$0.695244\pi$
$314$	1.24585	0.0703074
$315$	0	0
$316$	4.44261	0.249916
$317$	30.5160	1.71395	0.856974	−	0.515360i	$-0.172342\pi$
$317$	30.5160	1.71395	0.856974	+	0.515360i	$0.172342\pi$
$318$	0	0
$319$	−19.2191	−1.07606
$320$	12.4646	0.696795
$321$	0	0
$322$	0	0
$323$	6.99010	0.388939
$324$	0	0
$325$	1.18974	0.0659950
$326$	2.63010	0.145668
$327$	0	0
$328$	−9.70431	−0.535831
$329$	0	0
$330$	0	0
$331$	−28.2814	−1.55449	−0.777244	−	0.629200i	$-0.783383\pi$
$331$	−28.2814	−1.55449	−0.777244	+	0.629200i	$0.783383\pi$
$332$	−9.60910	−0.527368
$333$	0	0
$334$	6.18625	0.338496
$335$	−28.9101	−1.57953
$336$	0	0
$337$	0.0981441	0.00534625	0.00267312	−	0.999996i	$-0.499149\pi$
$337$	0.0981441	0.00534625	0.00267312	+	0.999996i	$0.499149\pi$
$338$	3.01957	0.164243
$339$	0	0
$340$	−27.6536	−1.49973
$341$	−4.00065	−0.216647
$342$	0	0
$343$	0	0
$344$	8.34497	0.449931
$345$	0	0
$346$	−1.98150	−0.106526
$347$	1.07373	0.0576410	0.0288205	−	0.999585i	$-0.490825\pi$
$347$	1.07373	0.0576410	0.0288205	+	0.999585i	$0.490825\pi$
$348$	0	0
$349$	−7.94431	−0.425249	−0.212625	−	0.977134i	$-0.568201\pi$
$349$	−7.94431	−0.425249	−0.212625	+	0.977134i	$0.568201\pi$
$350$	0	0
$351$	0	0
$352$	−16.3351	−0.870662
$353$	−8.75810	−0.466147	−0.233073	−	0.972459i	$-0.574878\pi$
$353$	−8.75810	−0.466147	−0.233073	+	0.972459i	$0.574878\pi$
$354$	0	0
$355$	13.4680	0.714806
$356$	−12.8205	−0.679485
$357$	0	0
$358$	5.61527	0.296776
$359$	−8.88559	−0.468963	−0.234482	−	0.972121i	$-0.575339\pi$
$359$	−8.88559	−0.468963	−0.234482	+	0.972121i	$0.575339\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−5.98782	−0.314713
$363$	0	0
$364$	0	0
$365$	−4.31703	−0.225964
$366$	0	0
$367$	−14.0644	−0.734158	−0.367079	−	0.930190i	$-0.619642\pi$
$367$	−14.0644	−0.734158	−0.367079	+	0.930190i	$0.619642\pi$
$368$	25.9293	1.35166
$369$	0	0
$370$	2.47180	0.128503
$371$	0	0
$372$	0	0
$373$	−36.8568	−1.90837	−0.954187	−	0.299212i	$-0.903276\pi$
$373$	−36.8568	−1.90837	−0.954187	+	0.299212i	$0.903276\pi$
$374$	10.0797	0.521211
$375$	0	0
$376$	−6.18220	−0.318823
$377$	6.30555	0.324752
$378$	0	0
$379$	−17.2907	−0.888164	−0.444082	−	0.895986i	$-0.646470\pi$
$379$	−17.2907	−0.888164	−0.444082	+	0.895986i	$0.646470\pi$
$380$	−3.95611	−0.202944
$381$	0	0
$382$	−1.30639	−0.0668407
$383$	−23.6224	−1.20705	−0.603525	−	0.797344i	$-0.706238\pi$
$383$	−23.6224	−1.20705	−0.603525	+	0.797344i	$0.706238\pi$
$384$	0	0
$385$	0	0
$386$	7.29263	0.371185
$387$	0	0
$388$	26.9385	1.36760
$389$	2.81761	0.142859	0.0714293	−	0.997446i	$-0.477244\pi$
$389$	2.81761	0.142859	0.0714293	+	0.997446i	$0.477244\pi$
$390$	0	0
$391$	−51.8172	−2.62051
$392$	0	0
$393$	0	0
$394$	−0.0292469	−0.00147344
$395$	4.79205	0.241114
$396$	0	0
$397$	−24.0012	−1.20459	−0.602293	−	0.798275i	$-0.705746\pi$
$397$	−24.0012	−1.20459	−0.602293	+	0.798275i	$0.705746\pi$
$398$	7.84649	0.393309
$399$	0	0
$400$	−2.56293	−0.128146
$401$	−33.3740	−1.66662	−0.833309	−	0.552808i	$-0.813556\pi$
$401$	−33.3740	−1.66662	−0.833309	+	0.552808i	$0.813556\pi$
$402$	0	0
$403$	1.31257	0.0653836
$404$	−3.43596	−0.170945
$405$	0	0
$406$	0	0
$407$	20.3247	1.00746
$408$	0	0
$409$	28.4148	1.40502	0.702511	−	0.711673i	$-0.252062\pi$
$409$	28.4148	1.40502	0.702511	+	0.711673i	$0.252062\pi$
$410$	−5.12032	−0.252874
$411$	0	0
$412$	−32.3670	−1.59461
$413$	0	0
$414$	0	0
$415$	−10.3649	−0.508793
$416$	5.35934	0.262763
$417$	0	0
$418$	1.44200	0.0705306
$419$	5.95276	0.290812	0.145406	−	0.989372i	$-0.453551\pi$
$419$	5.95276	0.290812	0.145406	+	0.989372i	$0.453551\pi$
$420$	0	0
$421$	−2.95244	−0.143893	−0.0719465	−	0.997408i	$-0.522921\pi$
$421$	−2.95244	−0.143893	−0.0719465	+	0.997408i	$0.522921\pi$
$422$	−4.23488	−0.206151
$423$	0	0
$424$	2.27017	0.110249
$425$	5.12177	0.248442
$426$	0	0
$427$	0	0
$428$	17.2537	0.833987
$429$	0	0
$430$	4.40308	0.212336
$431$	−9.97033	−0.480254	−0.240127	−	0.970741i	$-0.577189\pi$
$431$	−9.97033	−0.480254	−0.240127	+	0.970741i	$0.577189\pi$
$432$	0	0
$433$	19.1090	0.918319	0.459159	−	0.888354i	$-0.348151\pi$
$433$	19.1090	0.918319	0.459159	+	0.888354i	$0.348151\pi$
$434$	0	0
$435$	0	0
$436$	−13.4235	−0.642867
$437$	−7.41294	−0.354609
$438$	0	0
$439$	−7.30681	−0.348735	−0.174367	−	0.984681i	$-0.555788\pi$
$439$	−7.30681	−0.348735	−0.174367	+	0.984681i	$0.555788\pi$
$440$	−11.6623	−0.555979
$441$	0	0
$442$	−3.30704	−0.157300
$443$	16.4639	0.782221	0.391111	−	0.920344i	$-0.372091\pi$
$443$	16.4639	0.782221	0.391111	+	0.920344i	$0.372091\pi$
$444$	0	0
$445$	−13.8289	−0.655553
$446$	0.672558	0.0318466
$447$	0	0
$448$	0	0
$449$	9.71849	0.458644	0.229322	−	0.973351i	$-0.426349\pi$
$449$	9.71849	0.458644	0.229322	+	0.973351i	$0.426349\pi$
$450$	0	0
$451$	−42.1024	−1.98252
$452$	−0.501186	−0.0235738
$453$	0	0
$454$	5.78428	0.271470
$455$	0	0
$456$	0	0
$457$	−11.0016	−0.514632	−0.257316	−	0.966327i	$-0.582838\pi$
$457$	−11.0016	−0.514632	−0.257316	+	0.966327i	$0.582838\pi$
$458$	−5.07769	−0.237265
$459$	0	0
$460$	29.3264	1.36735
$461$	−28.6779	−1.33567	−0.667833	−	0.744311i	$-0.732778\pi$
$461$	−28.6779	−1.33567	−0.667833	+	0.744311i	$0.732778\pi$
$462$	0	0
$463$	−33.7722	−1.56953	−0.784765	−	0.619794i	$-0.787216\pi$
$463$	−33.7722	−1.56953	−0.784765	+	0.619794i	$0.787216\pi$
$464$	−13.5833	−0.630591
$465$	0	0
$466$	2.59092	0.120022
$467$	−16.8477	−0.779620	−0.389810	−	0.920895i	$-0.627459\pi$
$467$	−16.8477	−0.779620	−0.389810	+	0.920895i	$0.627459\pi$
$468$	0	0
$469$	0	0
$470$	−3.26193	−0.150462
$471$	0	0
$472$	−6.56516	−0.302186
$473$	36.2048	1.66470
$474$	0	0
$475$	0.732718	0.0336194
$476$	0	0
$477$	0	0
$478$	−3.18271	−0.145574
$479$	−23.0450	−1.05295	−0.526476	−	0.850190i	$-0.676487\pi$
$479$	−23.0450	−1.05295	−0.526476	+	0.850190i	$0.676487\pi$
$480$	0	0
$481$	−6.66828	−0.304047
$482$	−1.51044	−0.0687985
$483$	0	0
$484$	−25.8415	−1.17461
$485$	29.0574	1.31943
$486$	0	0
$487$	−21.9468	−0.994505	−0.497253	−	0.867606i	$-0.665658\pi$
$487$	−21.9468	−0.994505	−0.497253	+	0.867606i	$0.665658\pi$
$488$	12.9829	0.587708
$489$	0	0
$490$	0	0
$491$	−10.1883	−0.459791	−0.229896	−	0.973215i	$-0.573838\pi$
$491$	−10.1883	−0.459791	−0.229896	+	0.973215i	$0.573838\pi$
$492$	0	0
$493$	27.1450	1.22255
$494$	−0.473104	−0.0212859
$495$	0	0
$496$	−2.82752	−0.126959
$497$	0	0
$498$	0	0
$499$	16.4300	0.735508	0.367754	−	0.929923i	$-0.380127\pi$
$499$	16.4300	0.735508	0.367754	+	0.929923i	$0.380127\pi$
$500$	−22.6793	−1.01425
$501$	0	0
$502$	3.57427	0.159527
$503$	2.13595	0.0952373	0.0476186	−	0.998866i	$-0.484837\pi$
$503$	2.13595	0.0952373	0.0476186	+	0.998866i	$0.484837\pi$
$504$	0	0
$505$	−3.70622	−0.164925
$506$	−10.6895	−0.475205
$507$	0	0
$508$	15.4613	0.685986
$509$	−24.2247	−1.07374	−0.536869	−	0.843665i	$-0.680393\pi$
$509$	−24.2247	−1.07374	−0.536869	+	0.843665i	$0.680393\pi$
$510$	0	0
$511$	0	0
$512$	−19.5252	−0.862901
$513$	0	0
$514$	−6.51904	−0.287542
$515$	−34.9128	−1.53844
$516$	0	0
$517$	−26.8216	−1.17961
$518$	0	0
$519$	0	0
$520$	3.82627	0.167793
$521$	−7.56441	−0.331403	−0.165701	−	0.986176i	$-0.552989\pi$
$521$	−7.56441	−0.331403	−0.165701	+	0.986176i	$0.552989\pi$
$522$	0	0
$523$	11.7441	0.513532	0.256766	−	0.966474i	$-0.417343\pi$
$523$	11.7441	0.513532	0.256766	+	0.966474i	$0.417343\pi$
$524$	−37.2157	−1.62577
$525$	0	0
$526$	2.64452	0.115307
$527$	5.65052	0.246140
$528$	0	0
$529$	31.9517	1.38921
$530$	1.19782	0.0520298
$531$	0	0
$532$	0	0
$533$	13.8133	0.598320
$534$	0	0
$535$	18.6108	0.804614
$536$	15.9646	0.689565
$537$	0	0
$538$	5.48348	0.236410
$539$	0	0
$540$	0	0
$541$	−4.44391	−0.191059	−0.0955293	−	0.995427i	$-0.530454\pi$
$541$	−4.44391	−0.191059	−0.0955293	+	0.995427i	$0.530454\pi$
$542$	−7.27877	−0.312650
$543$	0	0
$544$	23.0716	0.989189
$545$	−14.4793	−0.620225
$546$	0	0
$547$	−8.15542	−0.348700	−0.174350	−	0.984684i	$-0.555782\pi$
$547$	−8.15542	−0.348700	−0.174350	+	0.984684i	$0.555782\pi$
$548$	−12.0297	−0.513882
$549$	0	0
$550$	1.05658	0.0450527
$551$	3.88335	0.165436
$552$	0	0
$553$	0	0
$554$	−6.08201	−0.258400
$555$	0	0
$556$	−34.8471	−1.47785
$557$	−5.23473	−0.221803	−0.110901	−	0.993831i	$-0.535374\pi$
$557$	−5.23473	−0.221803	−0.110901	+	0.993831i	$0.535374\pi$
$558$	0	0
$559$	−11.8784	−0.502402
$560$	0	0
$561$	0	0
$562$	8.54047	0.360258
$563$	25.3904	1.07008	0.535040	−	0.844827i	$-0.320297\pi$
$563$	25.3904	1.07008	0.535040	+	0.844827i	$0.320297\pi$
$564$	0	0
$565$	−0.540607	−0.0227435
$566$	−0.164994	−0.00693521
$567$	0	0
$568$	−7.43722	−0.312059
$569$	7.18156	0.301067	0.150533	−	0.988605i	$-0.451901\pi$
$569$	7.18156	0.301067	0.150533	+	0.988605i	$0.451901\pi$
$570$	0	0
$571$	8.94189	0.374206	0.187103	−	0.982340i	$-0.440090\pi$
$571$	8.94189	0.374206	0.187103	+	0.982340i	$0.440090\pi$
$572$	15.3898	0.643482
$573$	0	0
$574$	0	0
$575$	−5.43159	−0.226513
$576$	0	0
$577$	−12.7644	−0.531390	−0.265695	−	0.964057i	$-0.585601\pi$
$577$	−12.7644	−0.531390	−0.265695	+	0.964057i	$0.585601\pi$
$578$	−9.28337	−0.386137
$579$	0	0
$580$	−15.3630	−0.637913
$581$	0	0
$582$	0	0
$583$	9.84918	0.407911
$584$	2.38393	0.0986477
$585$	0	0
$586$	7.36288	0.304158
$587$	24.2518	1.00098	0.500489	−	0.865743i	$-0.333153\pi$
$587$	24.2518	1.00098	0.500489	+	0.865743i	$0.333153\pi$
$588$	0	0
$589$	0.808361	0.0333079
$590$	−3.46400	−0.142611
$591$	0	0
$592$	14.3647	0.590387
$593$	−3.85095	−0.158140	−0.0790698	−	0.996869i	$-0.525195\pi$
$593$	−3.85095	−0.158140	−0.0790698	+	0.996869i	$0.525195\pi$
$594$	0	0
$595$	0	0
$596$	16.7726	0.687033
$597$	0	0
$598$	3.50709	0.143416
$599$	−37.2717	−1.52288	−0.761440	−	0.648235i	$-0.775507\pi$
$599$	−37.2717	−1.52288	−0.761440	+	0.648235i	$0.775507\pi$
$600$	0	0
$601$	2.99633	0.122223	0.0611114	−	0.998131i	$-0.480536\pi$
$601$	2.99633	0.122223	0.0611114	+	0.998131i	$0.480536\pi$
$602$	0	0
$603$	0	0
$604$	23.0665	0.938562
$605$	−27.8741	−1.13324
$606$	0	0
$607$	−25.3423	−1.02861	−0.514306	−	0.857607i	$-0.671950\pi$
$607$	−25.3423	−1.02861	−0.514306	+	0.857607i	$0.671950\pi$
$608$	3.30062	0.133858
$609$	0	0
$610$	6.85021	0.277357
$611$	8.79985	0.356004
$612$	0	0
$613$	11.2393	0.453953	0.226976	−	0.973900i	$-0.427116\pi$
$613$	11.2393	0.453953	0.226976	+	0.973900i	$0.427116\pi$
$614$	1.85634	0.0749158
$615$	0	0
$616$	0	0
$617$	16.1879	0.651701	0.325851	−	0.945421i	$-0.394349\pi$
$617$	16.1879	0.651701	0.325851	+	0.945421i	$0.394349\pi$
$618$	0	0
$619$	−14.6528	−0.588944	−0.294472	−	0.955660i	$-0.595144\pi$
$619$	−14.6528	−0.588944	−0.294472	+	0.955660i	$0.595144\pi$
$620$	−3.19796	−0.128433
$621$	0	0
$622$	0.0861539	0.00345446
$623$	0	0
$624$	0	0
$625$	−20.7995	−0.831981
$626$	5.93454	0.237192
$627$	0	0
$628$	8.18877	0.326767
$629$	−28.7065	−1.14460
$630$	0	0
$631$	−23.3309	−0.928790	−0.464395	−	0.885628i	$-0.653728\pi$
$631$	−23.3309	−0.928790	−0.464395	+	0.885628i	$0.653728\pi$
$632$	−2.64624	−0.105262
$633$	0	0
$634$	−8.89134	−0.353120
$635$	16.6775	0.661825
$636$	0	0
$637$	0	0
$638$	5.59980	0.221698
$639$	0	0
$640$	−17.2682	−0.682587
$641$	−15.1832	−0.599699	−0.299850	−	0.953986i	$-0.596937\pi$
$641$	−15.1832	−0.599699	−0.299850	+	0.953986i	$0.596937\pi$
$642$	0	0
$643$	31.7842	1.25345	0.626723	−	0.779242i	$-0.284396\pi$
$643$	31.7842	1.25345	0.626723	+	0.779242i	$0.284396\pi$
$644$	0	0
$645$	0	0
$646$	−2.03668	−0.0801322
$647$	−13.3767	−0.525893	−0.262947	−	0.964810i	$-0.584694\pi$
$647$	−13.3767	−0.525893	−0.262947	+	0.964810i	$0.584694\pi$
$648$	0	0
$649$	−28.4831	−1.11806
$650$	−0.346651	−0.0135968
$651$	0	0
$652$	17.2872	0.677020
$653$	27.3705	1.07109	0.535544	−	0.844507i	$-0.320106\pi$
$653$	27.3705	1.07109	0.535544	+	0.844507i	$0.320106\pi$
$654$	0	0
$655$	−40.1429	−1.56851
$656$	−29.7564	−1.16179
$657$	0	0
$658$	0	0
$659$	−47.6905	−1.85776	−0.928879	−	0.370383i	$-0.879226\pi$
$659$	−47.6905	−1.85776	−0.928879	+	0.370383i	$0.879226\pi$
$660$	0	0
$661$	−16.8217	−0.654289	−0.327144	−	0.944974i	$-0.606086\pi$
$661$	−16.8217	−0.654289	−0.327144	+	0.944974i	$0.606086\pi$
$662$	8.24027	0.320267
$663$	0	0
$664$	5.72366	0.222121
$665$	0	0
$666$	0	0
$667$	−28.7871	−1.11464
$668$	40.6612	1.57323
$669$	0	0
$670$	8.42344	0.325426
$671$	56.3266	2.17446
$672$	0	0
$673$	14.1611	0.545870	0.272935	−	0.962032i	$-0.412006\pi$
$673$	14.1611	0.545870	0.272935	+	0.962032i	$0.412006\pi$
$674$	−0.0285959	−0.00110147
$675$	0	0
$676$	19.8471	0.763352
$677$	5.05314	0.194208	0.0971040	−	0.995274i	$-0.469042\pi$
$677$	5.05314	0.194208	0.0971040	+	0.995274i	$0.469042\pi$
$678$	0	0
$679$	0	0
$680$	16.4719	0.631667
$681$	0	0
$682$	1.16566	0.0446353
$683$	−20.3359	−0.778130	−0.389065	−	0.921210i	$-0.627202\pi$
$683$	−20.3359	−0.778130	−0.389065	+	0.921210i	$0.627202\pi$
$684$	0	0
$685$	−12.9759	−0.495783
$686$	0	0
$687$	0	0
$688$	25.5883	0.975544
$689$	−3.23140	−0.123106
$690$	0	0
$691$	−23.1140	−0.879300	−0.439650	−	0.898169i	$-0.644897\pi$
$691$	−23.1140	−0.879300	−0.439650	+	0.898169i	$0.644897\pi$
$692$	−13.0240	−0.495100
$693$	0	0
$694$	−0.312850	−0.0118756
$695$	−37.5880	−1.42580
$696$	0	0
$697$	59.4654	2.25241
$698$	2.31471	0.0876130
$699$	0	0
$700$	0	0
$701$	50.1400	1.89376	0.946882	−	0.321583i	$-0.104215\pi$
$701$	50.1400	1.89376	0.946882	+	0.321583i	$0.104215\pi$
$702$	0	0
$703$	−4.10674	−0.154889
$704$	−29.8628	−1.12550
$705$	0	0
$706$	2.55182	0.0960391
$707$	0	0
$708$	0	0
$709$	−3.80739	−0.142989	−0.0714947	−	0.997441i	$-0.522777\pi$
$709$	−3.80739	−0.142989	−0.0714947	+	0.997441i	$0.522777\pi$
$710$	−3.92412	−0.147270
$711$	0	0
$712$	7.63653	0.286191
$713$	−5.99233	−0.224415
$714$	0	0
$715$	16.6004	0.620818
$716$	36.9082	1.37933
$717$	0	0
$718$	2.58897	0.0966193
$719$	10.7018	0.399108	0.199554	−	0.979887i	$-0.436051\pi$
$719$	10.7018	0.399108	0.199554	+	0.979887i	$0.436051\pi$
$720$	0	0
$721$	0	0
$722$	−0.291367	−0.0108436
$723$	0	0
$724$	−39.3569	−1.46269
$725$	2.84540	0.105676
$726$	0	0
$727$	−22.1589	−0.821829	−0.410914	−	0.911674i	$-0.634790\pi$
$727$	−22.1589	−0.821829	−0.410914	+	0.911674i	$0.634790\pi$
$728$	0	0
$729$	0	0
$730$	1.25784	0.0465548
$731$	−51.1357	−1.89132
$732$	0	0
$733$	37.3419	1.37925	0.689627	−	0.724165i	$-0.257775\pi$
$733$	37.3419	1.37925	0.689627	+	0.724165i	$0.257775\pi$
$734$	4.09791	0.151257
$735$	0	0
$736$	−24.4673	−0.901877
$737$	69.2627	2.55132
$738$	0	0
$739$	−16.0538	−0.590548	−0.295274	−	0.955413i	$-0.595411\pi$
$739$	−16.0538	−0.590548	−0.295274	+	0.955413i	$0.595411\pi$
$740$	16.2467	0.597242
$741$	0	0
$742$	0	0
$743$	−24.6993	−0.906131	−0.453065	−	0.891477i	$-0.649670\pi$
$743$	−24.6993	−0.906131	−0.453065	+	0.891477i	$0.649670\pi$
$744$	0	0
$745$	18.0919	0.662835
$746$	10.7389	0.393177
$747$	0	0
$748$	66.2524	2.42243
$749$	0	0
$750$	0	0
$751$	26.1921	0.955762	0.477881	−	0.878425i	$-0.341405\pi$
$751$	26.1921	0.955762	0.477881	+	0.878425i	$0.341405\pi$
$752$	−18.9565	−0.691274
$753$	0	0
$754$	−1.83723	−0.0669079
$755$	24.8808	0.905505
$756$	0	0
$757$	23.2718	0.845826	0.422913	−	0.906170i	$-0.361008\pi$
$757$	23.2718	0.845826	0.422913	+	0.906170i	$0.361008\pi$
$758$	5.03794	0.182986
$759$	0	0
$760$	2.35646	0.0854776
$761$	−34.5434	−1.25220	−0.626099	−	0.779744i	$-0.715349\pi$
$761$	−34.5434	−1.25220	−0.626099	+	0.779744i	$0.715349\pi$
$762$	0	0
$763$	0	0
$764$	−8.58668	−0.310655
$765$	0	0
$766$	6.88280	0.248686
$767$	9.34497	0.337427
$768$	0	0
$769$	11.1106	0.400659	0.200329	−	0.979729i	$-0.435799\pi$
$769$	11.1106	0.400659	0.200329	+	0.979729i	$0.435799\pi$
$770$	0	0
$771$	0	0
$772$	47.9332	1.72515
$773$	−8.30414	−0.298679	−0.149340	−	0.988786i	$-0.547715\pi$
$773$	−8.30414	−0.298679	−0.149340	+	0.988786i	$0.547715\pi$
$774$	0	0
$775$	0.592300	0.0212760
$776$	−16.0459	−0.576015
$777$	0	0
$778$	−0.820959	−0.0294328
$779$	8.50709	0.304798
$780$	0	0
$781$	−32.2665	−1.15459
$782$	15.0978	0.539897
$783$	0	0
$784$	0	0
$785$	8.83286	0.315258
$786$	0	0
$787$	−16.6191	−0.592407	−0.296203	−	0.955125i	$-0.595721\pi$
$787$	−16.6191	−0.592407	−0.296203	+	0.955125i	$0.595721\pi$
$788$	−0.192235	−0.00684810
$789$	0	0
$790$	−1.39624	−0.0496761
$791$	0	0
$792$	0	0
$793$	−18.4801	−0.656247
$794$	6.99316	0.248178
$795$	0	0
$796$	51.5737	1.82798
$797$	−24.5479	−0.869532	−0.434766	−	0.900543i	$-0.643169\pi$
$797$	−24.5479	−0.869532	−0.434766	+	0.900543i	$0.643169\pi$
$798$	0	0
$799$	37.8828	1.34020
$800$	2.41842	0.0855041
$801$	0	0
$802$	9.72407	0.343369
$803$	10.3427	0.364987
$804$	0	0
$805$	0	0
$806$	−0.382438	−0.0134708
$807$	0	0
$808$	2.04663	0.0720001
$809$	3.99029	0.140291	0.0701455	−	0.997537i	$-0.477654\pi$
$809$	3.99029	0.140291	0.0701455	+	0.997537i	$0.477654\pi$
$810$	0	0
$811$	41.1357	1.44447	0.722235	−	0.691648i	$-0.243115\pi$
$811$	41.1357	1.44447	0.722235	+	0.691648i	$0.243115\pi$
$812$	0	0
$813$	0	0
$814$	−5.92193	−0.207564
$815$	18.6470	0.653175
$816$	0	0
$817$	−7.31545	−0.255935
$818$	−8.27914	−0.289473
$819$	0	0
$820$	−33.6550	−1.17528
$821$	2.43814	0.0850917	0.0425459	−	0.999095i	$-0.486453\pi$
$821$	2.43814	0.0850917	0.0425459	+	0.999095i	$0.486453\pi$
$822$	0	0
$823$	35.8018	1.24797	0.623985	−	0.781436i	$-0.285512\pi$
$823$	35.8018	1.24797	0.623985	+	0.781436i	$0.285512\pi$
$824$	19.2794	0.671629
$825$	0	0
$826$	0	0
$827$	−3.66665	−0.127502	−0.0637510	−	0.997966i	$-0.520306\pi$
$827$	−3.66665	−0.127502	−0.0637510	+	0.997966i	$0.520306\pi$
$828$	0	0
$829$	48.5906	1.68762	0.843811	−	0.536640i	$-0.180307\pi$
$829$	48.5906	1.68762	0.843811	+	0.536640i	$0.180307\pi$
$830$	3.01999	0.104825
$831$	0	0
$832$	9.79762	0.339671
$833$	0	0
$834$	0	0
$835$	43.8594	1.51782
$836$	9.47803	0.327805
$837$	0	0
$838$	−1.73444	−0.0599152
$839$	9.11600	0.314719	0.157360	−	0.987541i	$-0.449702\pi$
$839$	9.11600	0.314719	0.157360	+	0.987541i	$0.449702\pi$
$840$	0	0
$841$	−13.9196	−0.479985
$842$	0.860243	0.0296459
$843$	0	0
$844$	−27.8351	−0.958125
$845$	21.4082	0.736466
$846$	0	0
$847$	0	0
$848$	6.96104	0.239043
$849$	0	0
$850$	−1.49231	−0.0511859
$851$	30.4431	1.04358
$852$	0	0
$853$	−1.01631	−0.0347979	−0.0173989	−	0.999849i	$-0.505539\pi$
$853$	−1.01631	−0.0347979	−0.0173989	+	0.999849i	$0.505539\pi$
$854$	0	0
$855$	0	0
$856$	−10.2771	−0.351265
$857$	15.8851	0.542626	0.271313	−	0.962491i	$-0.412542\pi$
$857$	15.8851	0.542626	0.271313	+	0.962491i	$0.412542\pi$
$858$	0	0
$859$	−5.72412	−0.195304	−0.0976522	−	0.995221i	$-0.531133\pi$
$859$	−5.72412	−0.195304	−0.0976522	+	0.995221i	$0.531133\pi$
$860$	28.9407	0.986871
$861$	0	0
$862$	2.90503	0.0989456
$863$	−13.8950	−0.472990	−0.236495	−	0.971633i	$-0.575999\pi$
$863$	−13.8950	−0.472990	−0.236495	+	0.971633i	$0.575999\pi$
$864$	0	0
$865$	−14.0485	−0.477662
$866$	−5.56772	−0.189199
$867$	0	0
$868$	0	0
$869$	−11.4808	−0.389459
$870$	0	0
$871$	−22.7243	−0.769982
$872$	7.99568	0.270768
$873$	0	0
$874$	2.15989	0.0730593
$875$	0	0
$876$	0	0
$877$	−40.0210	−1.35141	−0.675707	−	0.737170i	$-0.736162\pi$
$877$	−40.0210	−1.35141	−0.675707	+	0.737170i	$0.736162\pi$
$878$	2.12896	0.0718490
$879$	0	0
$880$	−35.7603	−1.20548
$881$	−8.38440	−0.282478	−0.141239	−	0.989976i	$-0.545109\pi$
$881$	−8.38440	−0.282478	−0.141239	+	0.989976i	$0.545109\pi$
$882$	0	0
$883$	−20.0109	−0.673421	−0.336711	−	0.941608i	$-0.609314\pi$
$883$	−20.0109	−0.673421	−0.336711	+	0.941608i	$0.609314\pi$
$884$	−21.7366	−0.731082
$885$	0	0
$886$	−4.79702	−0.161159
$887$	29.1581	0.979032	0.489516	−	0.871994i	$-0.337173\pi$
$887$	29.1581	0.979032	0.489516	+	0.871994i	$0.337173\pi$
$888$	0	0
$889$	0	0
$890$	4.02929	0.135062
$891$	0	0
$892$	4.42061	0.148013
$893$	5.41950	0.181357
$894$	0	0
$895$	39.8113	1.33074
$896$	0	0
$897$	0	0
$898$	−2.83165	−0.0944933
$899$	3.13915	0.104696
$900$	0	0
$901$	−13.9110	−0.463442
$902$	12.2672	0.408454
$903$	0	0
$904$	0.298531	0.00992900
$905$	−42.4526	−1.41117
$906$	0	0
$907$	50.6610	1.68217	0.841086	−	0.540902i	$-0.181917\pi$
$907$	50.6610	1.68217	0.841086	+	0.540902i	$0.181917\pi$
$908$	38.0191	1.26171
$909$	0	0
$910$	0	0
$911$	28.9746	0.959970	0.479985	−	0.877277i	$-0.340642\pi$
$911$	28.9746	0.959970	0.479985	+	0.877277i	$0.340642\pi$
$912$	0	0
$913$	24.8322	0.821826
$914$	3.20550	0.106028
$915$	0	0
$916$	−33.3748	−1.10273
$917$	0	0
$918$	0	0
$919$	50.1894	1.65559	0.827797	−	0.561028i	$-0.189594\pi$
$919$	50.1894	1.65559	0.827797	+	0.561028i	$0.189594\pi$
$920$	−17.4683	−0.575912
$921$	0	0
$922$	8.35581	0.275184
$923$	10.5863	0.348451
$924$	0	0
$925$	−3.00908	−0.0989381
$926$	9.84011	0.323366
$927$	0	0
$928$	12.8175	0.420754
$929$	−33.9101	−1.11255	−0.556277	−	0.830997i	$-0.687771\pi$
$929$	−33.9101	−1.11255	−0.556277	+	0.830997i	$0.687771\pi$
$930$	0	0
$931$	0	0
$932$	17.0297	0.557825
$933$	0	0
$934$	4.90887	0.160623
$935$	71.4635	2.33711
$936$	0	0
$937$	−17.4372	−0.569648	−0.284824	−	0.958580i	$-0.591935\pi$
$937$	−17.4372	−0.569648	−0.284824	+	0.958580i	$0.591935\pi$
$938$	0	0
$939$	0	0
$940$	−21.4401	−0.699300
$941$	−0.598364	−0.0195061	−0.00975306	−	0.999952i	$-0.503105\pi$
$941$	−0.598364	−0.0195061	−0.00975306	+	0.999952i	$0.503105\pi$
$942$	0	0
$943$	−63.0626	−2.05360
$944$	−20.1308	−0.655203
$945$	0	0
$946$	−10.5489	−0.342974
$947$	−33.3136	−1.08255	−0.541274	−	0.840846i	$-0.682058\pi$
$947$	−33.3136	−1.08255	−0.541274	+	0.840846i	$0.682058\pi$
$948$	0	0
$949$	−3.39333	−0.110152
$950$	−0.213490	−0.00692652
$951$	0	0
$952$	0	0
$953$	−8.41792	−0.272683	−0.136342	−	0.990662i	$-0.543534\pi$
$953$	−8.41792	−0.272683	−0.136342	+	0.990662i	$0.543534\pi$
$954$	0	0
$955$	−9.26207	−0.299714
$956$	−20.9194	−0.676583
$957$	0	0
$958$	6.71454	0.216937
$959$	0	0
$960$	0	0
$961$	−30.3466	−0.978921
$962$	1.94292	0.0626421
$963$	0	0
$964$	−9.92785	−0.319755
$965$	51.7034	1.66439
$966$	0	0
$967$	−17.5459	−0.564237	−0.282118	−	0.959380i	$-0.591037\pi$
$967$	−17.5459	−0.564237	−0.282118	+	0.959380i	$0.591037\pi$
$968$	15.3925	0.494733
$969$	0	0
$970$	−8.46637	−0.271839
$971$	9.87266	0.316829	0.158414	−	0.987373i	$-0.449362\pi$
$971$	9.87266	0.316829	0.158414	+	0.987373i	$0.449362\pi$
$972$	0	0
$973$	0	0
$974$	6.39458	0.204895
$975$	0	0
$976$	39.8096	1.27427
$977$	14.9641	0.478743	0.239371	−	0.970928i	$-0.423059\pi$
$977$	14.9641	0.478743	0.239371	+	0.970928i	$0.423059\pi$
$978$	0	0
$979$	33.1312	1.05888
$980$	0	0
$981$	0	0
$982$	2.96853	0.0947296
$983$	6.58111	0.209905	0.104952	−	0.994477i	$-0.466531\pi$
$983$	6.58111	0.209905	0.104952	+	0.994477i	$0.466531\pi$
$984$	0	0
$985$	−0.207356	−0.00660690
$986$	−7.90916	−0.251879
$987$	0	0
$988$	−3.10963	−0.0989305
$989$	54.2290	1.72438
$990$	0	0
$991$	15.1300	0.480621	0.240311	−	0.970696i	$-0.422751\pi$
$991$	15.1300	0.480621	0.240311	+	0.970696i	$0.422751\pi$
$992$	2.66809	0.0847119
$993$	0	0
$994$	0	0
$995$	55.6303	1.76360
$996$	0	0
$997$	−28.4545	−0.901164	−0.450582	−	0.892735i	$-0.648783\pi$
$997$	−28.4545	−0.901164	−0.450582	+	0.892735i	$0.648783\pi$
$998$	−4.78716	−0.151535
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.bu.1.2		4
3.2	odd	2		931.2.a.m.1.3	yes	4
7.6	odd	2		8379.2.a.bv.1.2		4
21.2	odd	6		931.2.f.n.704.2		8
21.5	even	6		931.2.f.o.704.2		8
21.11	odd	6		931.2.f.n.324.2		8
21.17	even	6		931.2.f.o.324.2		8
21.20	even	2		931.2.a.l.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
931.2.a.l.1.3	✓	4	21.20	even	2
931.2.a.m.1.3	yes	4	3.2	odd	2
931.2.f.n.324.2		8	21.11	odd	6
931.2.f.n.704.2		8	21.2	odd	6
931.2.f.o.324.2		8	21.17	even	6
931.2.f.o.704.2		8	21.5	even	6
8379.2.a.bu.1.2		4	1.1	even	1	trivial
8379.2.a.bv.1.2		4	7.6	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-0.291367 q^{2} -1.91511 q^{4} -2.06574 q^{5} +1.14073 q^{8} +0.601888 q^{10} +4.94909 q^{11} -1.62374 q^{13} +3.49784 q^{16} -6.99010 q^{17} -1.00000 q^{19} +3.95611 q^{20} -1.44200 q^{22} +7.41294 q^{23} -0.732718 q^{25} +0.473104 q^{26} -3.88335 q^{29} -0.808361 q^{31} -3.30062 q^{32} +2.03668 q^{34} +4.10674 q^{37} +0.291367 q^{38} -2.35646 q^{40} -8.50709 q^{41} +7.31545 q^{43} -9.47803 q^{44} -2.15989 q^{46} -5.41950 q^{47} +0.213490 q^{50} +3.10963 q^{52} +1.99010 q^{53} -10.2235 q^{55} +1.13148 q^{58} -5.75522 q^{59} +11.3812 q^{61} +0.235529 q^{62} -6.03399 q^{64} +3.35422 q^{65} +13.9950 q^{67} +13.3868 q^{68} -6.51969 q^{71} +2.08982 q^{73} -1.19657 q^{74} +1.91511 q^{76} -2.31977 q^{79} -7.22563 q^{80} +2.47868 q^{82} +5.01753 q^{83} +14.4397 q^{85} -2.13148 q^{86} +5.64559 q^{88} +6.69441 q^{89} -14.1966 q^{92} +1.57906 q^{94} +2.06574 q^{95} -14.0663 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 8 q^{5} - 6 q^{8} + 10 q^{10} + 6 q^{11} + 2 q^{13} + 2 q^{16} - 8 q^{17} - 4 q^{19} - 14 q^{22} + 8 q^{23} + 20 q^{25} - 16 q^{26} - 2 q^{29} - 2 q^{32} - 22 q^{34} + 10 q^{37} + 22 q^{40}+ \cdots + 2 q^{97}+O(q^{100}) \) 4 * q + 2 * q^4 - 8 * q^5 - 6 * q^8 + 10 * q^10 + 6 * q^11 + 2 * q^13 + 2 * q^16 - 8 * q^17 - 4 * q^19 - 14 * q^22 + 8 * q^23 + 20 * q^25 - 16 * q^26 - 2 * q^29 - 2 * q^32 - 22 * q^34 + 10 * q^37 + 22 * q^40 - 12 * q^41 + 4 * q^43 + 14 * q^44 - 8 * q^46 - 16 * q^47 - 12 * q^50 + 28 * q^52 - 12 * q^53 - 8 * q^55 + 4 * q^58 - 14 * q^59 + 20 * q^61 + 20 * q^62 - 20 * q^64 - 10 * q^65 + 2 * q^67 + 24 * q^68 + 2 * q^71 - 16 * q^73 + 26 * q^74 - 2 * q^76 - 8 * q^79 - 28 * q^80 - 12 * q^82 - 20 * q^83 - 14 * q^85 - 8 * q^86 - 2 * q^88 - 16 * q^89 - 26 * q^92 + 10 * q^94 + 8 * q^95 + 2 * q^97

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