LMFDB - Embedded newform 8379.2.a.bq.1.3

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 = [3,-1,0,9,0,0,0,9,0,-10,-4,0,-2,0,0,13,12] 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:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 399)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.210756$ 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)

$f(q)$	$=$	$q+2.74483 q^{2} +5.53407 q^{4} -2.53407 q^{5} +9.70041 q^{8} -6.95558 q^{10} -2.42151 q^{11} +0.421512 q^{13} +15.5578 q^{16} +6.53407 q^{17} +1.00000 q^{19} -14.0237 q^{20} -6.64663 q^{22} +1.57849 q^{23} +1.42151 q^{25} +1.15698 q^{26} -6.02372 q^{29} -3.48965 q^{31} +23.3026 q^{32} +17.9349 q^{34} -2.84302 q^{37} +2.74483 q^{38} -24.5815 q^{40} +8.42151 q^{41} +11.4897 q^{43} -13.4008 q^{44} +4.33268 q^{46} +8.02372 q^{47} +3.90180 q^{50} +2.33268 q^{52} +10.8667 q^{53} +6.13628 q^{55} -16.5341 q^{58} +14.1363 q^{59} -6.84302 q^{61} -9.57849 q^{62} +32.8461 q^{64} -1.06814 q^{65} -9.91116 q^{67} +36.1600 q^{68} +13.3771 q^{71} -7.06814 q^{73} -7.80361 q^{74} +5.53407 q^{76} +14.1363 q^{79} -39.4245 q^{80} +23.1156 q^{82} +2.11256 q^{83} -16.5578 q^{85} +31.5371 q^{86} -23.4897 q^{88} -9.71477 q^{89} +8.73546 q^{92} +22.0237 q^{94} -2.53407 q^{95} +4.97930 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 9 q^{4} + 9 q^{8} - 10 q^{10} - 4 q^{11} - 2 q^{13} + 13 q^{16} + 12 q^{17} + 3 q^{19} - 16 q^{20} - 8 q^{22} + 8 q^{23} + q^{25} + 10 q^{26} + 8 q^{29} + 8 q^{31} + 27 q^{32} + 6 q^{34}+ \cdots - 22 q^{97}+O(q^{100}) $ 3 * q - q^2 + 9 * q^4 + 9 * q^8 - 10 * q^10 - 4 * q^11 - 2 * q^13 + 13 * q^16 + 12 * q^17 + 3 * q^19 - 16 * q^20 - 8 * q^22 + 8 * q^23 + q^25 + 10 * q^26 + 8 * q^29 + 8 * q^31 + 27 * q^32 + 6 * q^34 - 2 * q^37 - q^38 - 14 * q^40 + 22 * q^41 + 16 * q^43 - 12 * q^46 - 2 * q^47 + 9 * q^50 - 18 * q^52 - 12 * q^55 - 42 * q^58 + 12 * q^59 - 14 * q^61 - 32 * q^62 + 29 * q^64 + 12 * q^65 - 8 * q^67 + 52 * q^68 + 26 * q^71 - 6 * q^73 - 18 * q^74 + 9 * q^76 + 12 * q^79 - 52 * q^80 + 2 * q^82 + 2 * q^83 - 16 * q^85 + 24 * q^86 - 52 * q^88 - 2 * q^89 + 36 * q^92 + 40 * q^94 - 22 * 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 $

$2$	2.74483	1.94089	0.970443	−	0.241332i	$-0.0775843\pi$
$2$	2.74483	1.94089	0.970443	+	0.241332i	$0.0775843\pi$
$3$	0	0
$3$	0	0
$4$	5.53407	2.76704
$5$	−2.53407	−1.13327	−0.566635	−	0.823969i	$-0.691755\pi$
$5$	−2.53407	−1.13327	−0.566635	+	0.823969i	$0.691755\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	9.70041	3.42961
$9$	0	0
$10$	−6.95558	−2.19955
$11$	−2.42151	−0.730113	−0.365057	−	0.930985i	$-0.618950\pi$
$11$	−2.42151	−0.730113	−0.365057	+	0.930985i	$0.618950\pi$
$12$	0	0
$13$	0.421512	0.116906	0.0584532	−	0.998290i	$-0.481383\pi$
$13$	0.421512	0.116906	0.0584532	+	0.998290i	$0.481383\pi$
$14$	0	0
$15$	0	0
$16$	15.5578	3.88945
$17$	6.53407	1.58474	0.792372	−	0.610038i	$-0.208846\pi$
$17$	6.53407	1.58474	0.792372	+	0.610038i	$0.208846\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−14.0237	−3.13580
$21$	0	0
$22$	−6.64663	−1.41707
$23$	1.57849	0.329138	0.164569	−	0.986366i	$-0.447377\pi$
$23$	1.57849	0.329138	0.164569	+	0.986366i	$0.447377\pi$
$24$	0	0
$25$	1.42151	0.284302
$26$	1.15698	0.226902
$27$	0	0
$28$	0	0
$29$	−6.02372	−1.11858	−0.559289	−	0.828973i	$-0.688926\pi$
$29$	−6.02372	−1.11858	−0.559289	+	0.828973i	$0.688926\pi$
$30$	0	0
$31$	−3.48965	−0.626760	−0.313380	−	0.949628i	$-0.601461\pi$
$31$	−3.48965	−0.626760	−0.313380	+	0.949628i	$0.601461\pi$
$32$	23.3026	4.11936
$33$	0	0
$34$	17.9349	3.07581
$35$	0	0
$36$	0	0
$37$	−2.84302	−0.467390	−0.233695	−	0.972310i	$-0.575082\pi$
$37$	−2.84302	−0.467390	−0.233695	+	0.972310i	$0.575082\pi$
$38$	2.74483	0.445270
$39$	0	0
$40$	−24.5815	−3.88668
$41$	8.42151	1.31522	0.657610	−	0.753359i	$-0.271568\pi$
$41$	8.42151	1.31522	0.657610	+	0.753359i	$0.271568\pi$
$42$	0	0
$43$	11.4897	1.75216	0.876078	−	0.482170i	$-0.160151\pi$
$43$	11.4897	1.75216	0.876078	+	0.482170i	$0.160151\pi$
$44$	−13.4008	−2.02025
$45$	0	0
$46$	4.33268	0.638818
$47$	8.02372	1.17038	0.585190	−	0.810896i	$-0.301020\pi$
$47$	8.02372	1.17038	0.585190	+	0.810896i	$0.301020\pi$
$48$	0	0
$49$	0	0
$50$	3.90180	0.551798
$51$	0	0
$52$	2.33268	0.323484
$53$	10.8667	1.49266	0.746331	−	0.665575i	$-0.231814\pi$
$53$	10.8667	1.49266	0.746331	+	0.665575i	$0.231814\pi$
$54$	0	0
$55$	6.13628	0.827416
$56$	0	0
$57$	0	0
$58$	−16.5341	−2.17103
$59$	14.1363	1.84039	0.920194	−	0.391464i	$-0.128031\pi$
$59$	14.1363	1.84039	0.920194	+	0.391464i	$0.128031\pi$
$60$	0	0
$61$	−6.84302	−0.876159	−0.438080	−	0.898936i	$-0.644341\pi$
$61$	−6.84302	−0.876159	−0.438080	+	0.898936i	$0.644341\pi$
$62$	−9.57849	−1.21647
$63$	0	0
$64$	32.8461	4.10576
$65$	−1.06814	−0.132487
$66$	0	0
$67$	−9.91116	−1.21084	−0.605421	−	0.795906i	$-0.706995\pi$
$67$	−9.91116	−1.21084	−0.605421	+	0.795906i	$0.706995\pi$
$68$	36.1600	4.38504
$69$	0	0
$70$	0	0
$71$	13.3771	1.58757	0.793784	−	0.608199i	$-0.208108\pi$
$71$	13.3771	1.58757	0.793784	+	0.608199i	$0.208108\pi$
$72$	0	0
$73$	−7.06814	−0.827263	−0.413632	−	0.910444i	$-0.635740\pi$
$73$	−7.06814	−0.827263	−0.413632	+	0.910444i	$0.635740\pi$
$74$	−7.80361	−0.907151
$75$	0	0
$76$	5.53407	0.634801
$77$	0	0
$78$	0	0
$79$	14.1363	1.59046	0.795228	−	0.606311i	$-0.207351\pi$
$79$	14.1363	1.59046	0.795228	+	0.606311i	$0.207351\pi$
$80$	−39.4245	−4.40780
$81$	0	0
$82$	23.1156	2.55269
$83$	2.11256	0.231883	0.115942	−	0.993256i	$-0.463011\pi$
$83$	2.11256	0.231883	0.115942	+	0.993256i	$0.463011\pi$
$84$	0	0
$85$	−16.5578	−1.79594
$86$	31.5371	3.40073
$87$	0	0
$88$	−23.4897	−2.50401
$89$	−9.71477	−1.02976	−0.514882	−	0.857261i	$-0.672164\pi$
$89$	−9.71477	−1.02976	−0.514882	+	0.857261i	$0.672164\pi$
$90$	0	0
$91$	0	0
$92$	8.73546	0.910735
$93$	0	0
$94$	22.0237	2.27157
$95$	−2.53407	−0.259990
$96$	0	0
$97$	4.97930	0.505572	0.252786	−	0.967522i	$-0.418653\pi$
$97$	4.97930	0.505572	0.252786	+	0.967522i	$0.418653\pi$
$98$	0	0
$99$	0	0
$100$	7.86675	0.786675
$101$	−1.69105	−0.168265	−0.0841327	−	0.996455i	$-0.526812\pi$
$101$	−1.69105	−0.168265	−0.0841327	+	0.996455i	$0.526812\pi$
$102$	0	0
$103$	−3.15698	−0.311066	−0.155533	−	0.987831i	$-0.549710\pi$
$103$	−3.15698	−0.311066	−0.155533	+	0.987831i	$0.549710\pi$
$104$	4.08884	0.400943
$105$	0	0
$106$	29.8273	2.89709
$107$	16.5341	1.59841	0.799204	−	0.601059i	$-0.205254\pi$
$107$	16.5341	1.59841	0.799204	+	0.601059i	$0.205254\pi$
$108$	0	0
$109$	1.77488	0.170003	0.0850015	−	0.996381i	$-0.472910\pi$
$109$	1.77488	0.170003	0.0850015	+	0.996381i	$0.472910\pi$
$110$	16.8430	1.60592
$111$	0	0
$112$	0	0
$113$	−10.0237	−0.942952	−0.471476	−	0.881879i	$-0.656279\pi$
$113$	−10.0237	−0.942952	−0.471476	+	0.881879i	$0.656279\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	−33.3357	−3.09514
$117$	0	0
$118$	38.8016	3.57198
$119$	0	0
$120$	0	0
$121$	−5.13628	−0.466935
$122$	−18.7829	−1.70052
$123$	0	0
$124$	−19.3120	−1.73427
$125$	9.06814	0.811079
$126$	0	0
$127$	2.08884	0.185354	0.0926771	−	0.995696i	$-0.470458\pi$
$127$	2.08884	0.185354	0.0926771	+	0.995696i	$0.470458\pi$
$128$	43.5515	3.84944
$129$	0	0
$130$	−2.93186	−0.257141
$131$	−13.9349	−1.21750	−0.608748	−	0.793363i	$-0.708328\pi$
$131$	−13.9349	−1.21750	−0.608748	+	0.793363i	$0.708328\pi$
$132$	0	0
$133$	0	0
$134$	−27.2044	−2.35010
$135$	0	0
$136$	63.3831	5.43506
$137$	−16.9793	−1.45064	−0.725320	−	0.688412i	$-0.758308\pi$
$137$	−16.9793	−1.45064	−0.725320	+	0.688412i	$0.758308\pi$
$138$	0	0
$139$	1.06814	0.0905985	0.0452992	−	0.998973i	$-0.485576\pi$
$139$	1.06814	0.0905985	0.0452992	+	0.998973i	$0.485576\pi$
$140$	0	0
$141$	0	0
$142$	36.7178	3.08129
$143$	−1.02070	−0.0853549
$144$	0	0
$145$	15.2645	1.26765
$146$	−19.4008	−1.60562
$147$	0	0
$148$	−15.7335	−1.29329
$149$	−14.0474	−1.15081	−0.575406	−	0.817868i	$-0.695156\pi$
$149$	−14.0474	−1.15081	−0.575406	+	0.817868i	$0.695156\pi$
$150$	0	0
$151$	−14.7542	−1.20068	−0.600339	−	0.799745i	$-0.704968\pi$
$151$	−14.7542	−1.20068	−0.600339	+	0.799745i	$0.704968\pi$
$152$	9.70041	0.786807
$153$	0	0
$154$	0	0
$155$	8.84302	0.710289
$156$	0	0
$157$	16.1363	1.28782	0.643908	−	0.765103i	$-0.277312\pi$
$157$	16.1363	1.28782	0.643908	+	0.765103i	$0.277312\pi$
$158$	38.8016	3.08689
$159$	0	0
$160$	−59.0505	−4.66835
$161$	0	0
$162$	0	0
$163$	−6.64663	−0.520604	−0.260302	−	0.965527i	$-0.583822\pi$
$163$	−6.64663	−0.520604	−0.260302	+	0.965527i	$0.583822\pi$
$164$	46.6052	3.63926
$165$	0	0
$166$	5.79861	0.450059
$167$	−14.3614	−1.11132	−0.555659	−	0.831410i	$-0.687534\pi$
$167$	−14.3614	−1.11132	−0.555659	+	0.831410i	$0.687534\pi$
$168$	0	0
$169$	−12.8223	−0.986333
$170$	−45.4483	−3.48572
$171$	0	0
$172$	63.5845	4.84828
$173$	−5.71477	−0.434486	−0.217243	−	0.976118i	$-0.569706\pi$
$173$	−5.71477	−0.434486	−0.217243	+	0.976118i	$0.569706\pi$
$174$	0	0
$175$	0	0
$176$	−37.6734	−2.83974
$177$	0	0
$178$	−26.6654	−1.99865
$179$	9.60221	0.717703	0.358851	−	0.933395i	$-0.383168\pi$
$179$	9.60221	0.717703	0.358851	+	0.933395i	$0.383168\pi$
$180$	0	0
$181$	0.979304	0.0727911	0.0363956	−	0.999337i	$-0.488412\pi$
$181$	0.979304	0.0727911	0.0363956	+	0.999337i	$0.488412\pi$
$182$	0	0
$183$	0	0
$184$	15.3120	1.12881
$185$	7.20442	0.529680
$186$	0	0
$187$	−15.8223	−1.15704
$188$	44.4038	3.23848
$189$	0	0
$190$	−6.95558	−0.504611
$191$	11.7148	0.847651	0.423825	−	0.905744i	$-0.360687\pi$
$191$	11.7148	0.847651	0.423825	+	0.905744i	$0.360687\pi$
$192$	0	0
$193$	17.8223	1.28288	0.641440	−	0.767174i	$-0.278337\pi$
$193$	17.8223	1.28288	0.641440	+	0.767174i	$0.278337\pi$
$194$	13.6673	0.981257
$195$	0	0
$196$	0	0
$197$	19.1156	1.36193	0.680965	−	0.732316i	$-0.261561\pi$
$197$	19.1156	1.36193	0.680965	+	0.732316i	$0.261561\pi$
$198$	0	0
$199$	−16.0474	−1.13757	−0.568787	−	0.822485i	$-0.692587\pi$
$199$	−16.0474	−1.13757	−0.568787	+	0.822485i	$0.692587\pi$
$200$	13.7892	0.975047
$201$	0	0
$202$	−4.64163	−0.326584
$203$	0	0
$204$	0	0
$205$	−21.3407	−1.49050
$206$	−8.66535	−0.603744
$207$	0	0
$208$	6.55779	0.454701
$209$	−2.42151	−0.167499
$210$	0	0
$211$	−9.29326	−0.639774	−0.319887	−	0.947456i	$-0.603645\pi$
$211$	−9.29326	−0.639774	−0.319887	+	0.947456i	$0.603645\pi$
$212$	60.1373	4.13025
$213$	0	0
$214$	45.3831	3.10233
$215$	−29.1156	−1.98567
$216$	0	0
$217$	0	0
$218$	4.87175	0.329956
$219$	0	0
$220$	33.9586	2.28949
$221$	2.75419	0.185267
$222$	0	0
$223$	8.33268	0.557997	0.278999	−	0.960292i	$-0.409998\pi$
$223$	8.33268	0.557997	0.278999	+	0.960292i	$0.409998\pi$
$224$	0	0
$225$	0	0
$226$	−27.5134	−1.83016
$227$	16.8905	1.12106	0.560530	−	0.828134i	$-0.310597\pi$
$227$	16.8905	1.12106	0.560530	+	0.828134i	$0.310597\pi$
$228$	0	0
$229$	−6.04744	−0.399626	−0.199813	−	0.979834i	$-0.564034\pi$
$229$	−6.04744	−0.399626	−0.199813	+	0.979834i	$0.564034\pi$
$230$	−10.9793	−0.723954
$231$	0	0
$232$	−58.4326	−3.83629
$233$	−21.2044	−1.38915	−0.694574	−	0.719421i	$-0.744407\pi$
$233$	−21.2044	−1.38915	−0.694574	+	0.719421i	$0.744407\pi$
$234$	0	0
$235$	−20.3327	−1.32636
$236$	78.2312	5.09242
$237$	0	0
$238$	0	0
$239$	5.80361	0.375404	0.187702	−	0.982226i	$-0.439896\pi$
$239$	5.80361	0.375404	0.187702	+	0.982226i	$0.439896\pi$
$240$	0	0
$241$	12.9793	0.836070	0.418035	−	0.908431i	$-0.362719\pi$
$241$	12.9793	0.836070	0.418035	+	0.908431i	$0.362719\pi$
$242$	−14.0982	−0.906266
$243$	0	0
$244$	−37.8698	−2.42436
$245$	0	0
$246$	0	0
$247$	0.421512	0.0268202
$248$	−33.8510	−2.14954
$249$	0	0
$250$	24.8905	1.57421
$251$	10.3377	0.652508	0.326254	−	0.945282i	$-0.394213\pi$
$251$	10.3377	0.652508	0.326254	+	0.945282i	$0.394213\pi$
$252$	0	0
$253$	−3.82233	−0.240308
$254$	5.73349	0.359751
$255$	0	0
$256$	53.8491	3.36557
$257$	−15.1757	−0.946634	−0.473317	−	0.880892i	$-0.656944\pi$
$257$	−15.1757	−0.946634	−0.473317	+	0.880892i	$0.656944\pi$
$258$	0	0
$259$	0	0
$260$	−5.91116	−0.366595
$261$	0	0
$262$	−38.2488	−2.36302
$263$	8.51035	0.524771	0.262385	−	0.964963i	$-0.415491\pi$
$263$	8.51035	0.524771	0.262385	+	0.964963i	$0.415491\pi$
$264$	0	0
$265$	−27.5371	−1.69159
$266$	0	0
$267$	0	0
$268$	−54.8491	−3.35044
$269$	15.1757	0.925279	0.462639	−	0.886547i	$-0.346902\pi$
$269$	15.1757	0.925279	0.462639	+	0.886547i	$0.346902\pi$
$270$	0	0
$271$	−0.843024	−0.0512100	−0.0256050	−	0.999672i	$-0.508151\pi$
$271$	−0.843024	−0.0512100	−0.0256050	+	0.999672i	$0.508151\pi$
$272$	101.656	6.16378
$273$	0	0
$274$	−46.6052	−2.81553
$275$	−3.44221	−0.207573
$276$	0	0
$277$	10.5578	0.634356	0.317178	−	0.948366i	$-0.397265\pi$
$277$	10.5578	0.634356	0.317178	+	0.948366i	$0.397265\pi$
$278$	2.93186	0.175841
$279$	0	0
$280$	0	0
$281$	−15.0444	−0.897475	−0.448737	−	0.893664i	$-0.648126\pi$
$281$	−15.0444	−0.897475	−0.448737	+	0.893664i	$0.648126\pi$
$282$	0	0
$283$	−1.24581	−0.0740559	−0.0370279	−	0.999314i	$-0.511789\pi$
$283$	−1.24581	−0.0740559	−0.0370279	+	0.999314i	$0.511789\pi$
$284$	74.0298	4.39286
$285$	0	0
$286$	−2.80163	−0.165664
$287$	0	0
$288$	0	0
$289$	25.6941	1.51142
$290$	41.8985	2.46036
$291$	0	0
$292$	−39.1156	−2.28907
$293$	20.4215	1.19304	0.596519	−	0.802599i	$-0.296550\pi$
$293$	20.4215	1.19304	0.596519	+	0.802599i	$0.296550\pi$
$294$	0	0
$295$	−35.8223	−2.08566
$296$	−27.5785	−1.60297
$297$	0	0
$298$	−38.5578	−2.23359
$299$	0.665351	0.0384783
$300$	0	0
$301$	0	0
$302$	−40.4977	−2.33038
$303$	0	0
$304$	15.5578	0.892301
$305$	17.3407	0.992926
$306$	0	0
$307$	2.64663	0.151051	0.0755255	−	0.997144i	$-0.475937\pi$
$307$	2.64663	0.151051	0.0755255	+	0.997144i	$0.475937\pi$
$308$	0	0
$309$	0	0
$310$	24.2726	1.37859
$311$	−5.09186	−0.288733	−0.144367	−	0.989524i	$-0.546114\pi$
$311$	−5.09186	−0.288733	−0.144367	+	0.989524i	$0.546114\pi$
$312$	0	0
$313$	−19.9586	−1.12813	−0.564064	−	0.825731i	$-0.690763\pi$
$313$	−19.9586	−1.12813	−0.564064	+	0.825731i	$0.690763\pi$
$314$	44.2913	2.49950
$315$	0	0
$316$	78.2312	4.40085
$317$	6.81930	0.383010	0.191505	−	0.981492i	$-0.438663\pi$
$317$	6.81930	0.383010	0.191505	+	0.981492i	$0.438663\pi$
$318$	0	0
$319$	14.5865	0.816688
$320$	−83.2342	−4.65293
$321$	0	0
$322$	0	0
$323$	6.53407	0.363565
$324$	0	0
$325$	0.599184	0.0332367
$326$	−18.2438	−1.01043
$327$	0	0
$328$	81.6921	4.51069
$329$	0	0
$330$	0	0
$331$	−5.91116	−0.324907	−0.162453	−	0.986716i	$-0.551941\pi$
$331$	−5.91116	−0.324907	−0.162453	+	0.986716i	$0.551941\pi$
$332$	11.6910	0.641630
$333$	0	0
$334$	−39.4195	−2.15694
$335$	25.1156	1.37221
$336$	0	0
$337$	4.08884	0.222733	0.111367	−	0.993779i	$-0.464477\pi$
$337$	4.08884	0.222733	0.111367	+	0.993779i	$0.464477\pi$
$338$	−35.1951	−1.91436
$339$	0	0
$340$	−91.6320	−4.96944
$341$	8.45023	0.457606
$342$	0	0
$343$	0	0
$344$	111.454	6.00921
$345$	0	0
$346$	−15.6860	−0.843287
$347$	17.3534	0.931578	0.465789	−	0.884896i	$-0.345771\pi$
$347$	17.3534	0.931578	0.465789	+	0.884896i	$0.345771\pi$
$348$	0	0
$349$	4.70674	0.251946	0.125973	−	0.992034i	$-0.459795\pi$
$349$	4.70674	0.251946	0.125973	+	0.992034i	$0.459795\pi$
$350$	0	0
$351$	0	0
$352$	−56.4276	−3.00760
$353$	20.6704	1.10017	0.550086	−	0.835108i	$-0.314595\pi$
$353$	20.6704	1.10017	0.550086	+	0.835108i	$0.314595\pi$
$354$	0	0
$355$	−33.8985	−1.79915
$356$	−53.7622	−2.84939
$357$	0	0
$358$	26.3564	1.39298
$359$	−8.10756	−0.427901	−0.213950	−	0.976845i	$-0.568633\pi$
$359$	−8.10756	−0.427901	−0.213950	+	0.976845i	$0.568633\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	2.68802	0.141279
$363$	0	0
$364$	0	0
$365$	17.9112	0.937513
$366$	0	0
$367$	−28.8905	−1.50807	−0.754035	−	0.656834i	$-0.771895\pi$
$367$	−28.8905	−1.50807	−0.754035	+	0.656834i	$0.771895\pi$
$368$	24.5578	1.28016
$369$	0	0
$370$	19.7749	1.02805
$371$	0	0
$372$	0	0
$373$	11.9586	0.619193	0.309597	−	0.950868i	$-0.399806\pi$
$373$	11.9586	0.619193	0.309597	+	0.950868i	$0.399806\pi$
$374$	−43.4295	−2.24569
$375$	0	0
$376$	77.8334	4.01395
$377$	−2.53907	−0.130769
$378$	0	0
$379$	−15.3821	−0.790125	−0.395063	−	0.918654i	$-0.629277\pi$
$379$	−15.3821	−0.790125	−0.395063	+	0.918654i	$0.629277\pi$
$380$	−14.0237	−0.719402
$381$	0	0
$382$	32.1550	1.64519
$383$	−0.617907	−0.0315736	−0.0157868	−	0.999875i	$-0.505025\pi$
$383$	−0.617907	−0.0315736	−0.0157868	+	0.999875i	$0.505025\pi$
$384$	0	0
$385$	0	0
$386$	48.9192	2.48992
$387$	0	0
$388$	27.5558	1.39893
$389$	11.0681	0.561177	0.280588	−	0.959828i	$-0.409470\pi$
$389$	11.0681	0.561177	0.280588	+	0.959828i	$0.409470\pi$
$390$	0	0
$391$	10.3140	0.521599
$392$	0	0
$393$	0	0
$394$	52.4690	2.64335
$395$	−35.8223	−1.80242
$396$	0	0
$397$	10.4502	0.524482	0.262241	−	0.965002i	$-0.415538\pi$
$397$	10.4502	0.524482	0.262241	+	0.965002i	$0.415538\pi$
$398$	−44.0474	−2.20790
$399$	0	0
$400$	22.1156	1.10578
$401$	−3.26953	−0.163273	−0.0816364	−	0.996662i	$-0.526015\pi$
$401$	−3.26953	−0.163273	−0.0816364	+	0.996662i	$0.526015\pi$
$402$	0	0
$403$	−1.47093	−0.0732722
$404$	−9.35837	−0.465596
$405$	0	0
$406$	0	0
$407$	6.88441	0.341248
$408$	0	0
$409$	−7.40082	−0.365947	−0.182973	−	0.983118i	$-0.558572\pi$
$409$	−7.40082	−0.365947	−0.182973	+	0.983118i	$0.558572\pi$
$410$	−58.5765	−2.89289
$411$	0	0
$412$	−17.4709	−0.860731
$413$	0	0
$414$	0	0
$415$	−5.35337	−0.262787
$416$	9.82233	0.481579
$417$	0	0
$418$	−6.64663	−0.325097
$419$	−31.8461	−1.55578	−0.777891	−	0.628400i	$-0.783710\pi$
$419$	−31.8461	−1.55578	−0.777891	+	0.628400i	$0.783710\pi$
$420$	0	0
$421$	40.0061	1.94978	0.974888	−	0.222696i	$-0.0714858\pi$
$421$	40.0061	1.94978	0.974888	+	0.222696i	$0.0714858\pi$
$422$	−25.5084	−1.24173
$423$	0	0
$424$	105.412	5.11925
$425$	9.28826	0.450547
$426$	0	0
$427$	0	0
$428$	91.5007	4.42285
$429$	0	0
$430$	−79.9172	−3.85395
$431$	−3.91616	−0.188635	−0.0943175	−	0.995542i	$-0.530067\pi$
$431$	−3.91616	−0.188635	−0.0943175	+	0.995542i	$0.530067\pi$
$432$	0	0
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	0	0
$435$	0	0
$436$	9.82233	0.470404
$437$	1.57849	0.0755093
$438$	0	0
$439$	−17.4483	−0.832760	−0.416380	−	0.909191i	$-0.636701\pi$
$439$	−17.4483	−0.832760	−0.416380	+	0.909191i	$0.636701\pi$
$440$	59.5244	2.83772
$441$	0	0
$442$	7.55977	0.359581
$443$	3.03942	0.144407	0.0722036	−	0.997390i	$-0.476997\pi$
$443$	3.03942	0.144407	0.0722036	+	0.997390i	$0.476997\pi$
$444$	0	0
$445$	24.6179	1.16700
$446$	22.8717	1.08301
$447$	0	0
$448$	0	0
$449$	−35.9349	−1.69587	−0.847936	−	0.530099i	$-0.822155\pi$
$449$	−35.9349	−1.69587	−0.847936	+	0.530099i	$0.822155\pi$
$450$	0	0
$451$	−20.3928	−0.960259
$452$	−55.4720	−2.60918
$453$	0	0
$454$	46.3614	2.17585
$455$	0	0
$456$	0	0
$457$	−24.4215	−1.14239	−0.571195	−	0.820814i	$-0.693520\pi$
$457$	−24.4215	−1.14239	−0.571195	+	0.820814i	$0.693520\pi$
$458$	−16.5992	−0.775629
$459$	0	0
$460$	−22.1363	−1.03211
$461$	10.1313	0.471861	0.235930	−	0.971770i	$-0.424186\pi$
$461$	10.1313	0.471861	0.235930	+	0.971770i	$0.424186\pi$
$462$	0	0
$463$	−5.86372	−0.272510	−0.136255	−	0.990674i	$-0.543507\pi$
$463$	−5.86372	−0.272510	−0.136255	+	0.990674i	$0.543507\pi$
$464$	−93.7158	−4.35065
$465$	0	0
$466$	−58.2024	−2.69618
$467$	−14.9556	−0.692062	−0.346031	−	0.938223i	$-0.612471\pi$
$467$	−14.9556	−0.692062	−0.346031	+	0.938223i	$0.612471\pi$
$468$	0	0
$469$	0	0
$470$	−55.8097	−2.57431
$471$	0	0
$472$	137.128	6.31181
$473$	−27.8223	−1.27927
$474$	0	0
$475$	1.42151	0.0652234
$476$	0	0
$477$	0	0
$478$	15.9299	0.728616
$479$	−33.7098	−1.54024	−0.770119	−	0.637900i	$-0.779803\pi$
$479$	−33.7098	−1.54024	−0.770119	+	0.637900i	$0.779803\pi$
$480$	0	0
$481$	−1.19837	−0.0546409
$482$	35.6259	1.62272
$483$	0	0
$484$	−28.4245	−1.29202
$485$	−12.6179	−0.572950
$486$	0	0
$487$	21.5084	0.974637	0.487319	−	0.873224i	$-0.337975\pi$
$487$	21.5084	0.974637	0.487319	+	0.873224i	$0.337975\pi$
$488$	−66.3801	−3.00489
$489$	0	0
$490$	0	0
$491$	17.3534	0.783147	0.391573	−	0.920147i	$-0.371931\pi$
$491$	17.3534	0.783147	0.391573	+	0.920147i	$0.371931\pi$
$492$	0	0
$493$	−39.3594	−1.77266
$494$	1.15698	0.0520548
$495$	0	0
$496$	−54.2913	−2.43775
$497$	0	0
$498$	0	0
$499$	−32.9379	−1.47450	−0.737252	−	0.675618i	$-0.763877\pi$
$499$	−32.9379	−1.47450	−0.737252	+	0.675618i	$0.763877\pi$
$500$	50.1837	2.24428
$501$	0	0
$502$	28.3751	1.26644
$503$	−38.2074	−1.70359	−0.851793	−	0.523879i	$-0.824485\pi$
$503$	−38.2074	−1.70359	−0.851793	+	0.523879i	$0.824485\pi$
$504$	0	0
$505$	4.28523	0.190690
$506$	−10.4916	−0.466410
$507$	0	0
$508$	11.5598	0.512882
$509$	−8.82430	−0.391130	−0.195565	−	0.980691i	$-0.562654\pi$
$509$	−8.82430	−0.391130	−0.195565	+	0.980691i	$0.562654\pi$
$510$	0	0
$511$	0	0
$512$	60.7034	2.68274
$513$	0	0
$514$	−41.6547	−1.83731
$515$	8.00000	0.352522
$516$	0	0
$517$	−19.4295	−0.854510
$518$	0	0
$519$	0	0
$520$	−10.3614	−0.454377
$521$	−24.2913	−1.06422	−0.532110	−	0.846675i	$-0.678601\pi$
$521$	−24.2913	−1.06422	−0.532110	+	0.846675i	$0.678601\pi$
$522$	0	0
$523$	−19.0969	−0.835047	−0.417524	−	0.908666i	$-0.637102\pi$
$523$	−19.0969	−0.835047	−0.417524	+	0.908666i	$0.637102\pi$
$524$	−77.1166	−3.36886
$525$	0	0
$526$	23.3594	1.01852
$527$	−22.8016	−0.993255
$528$	0	0
$529$	−20.5084	−0.891668
$530$	−75.5845	−3.28318
$531$	0	0
$532$	0	0
$533$	3.54977	0.153757
$534$	0	0
$535$	−41.8985	−1.81143
$536$	−96.1423	−4.15272
$537$	0	0
$538$	41.6547	1.79586
$539$	0	0
$540$	0	0
$541$	−36.2312	−1.55770	−0.778850	−	0.627210i	$-0.784197\pi$
$541$	−36.2312	−1.55770	−0.778850	+	0.627210i	$0.784197\pi$
$542$	−2.31395	−0.0993928
$543$	0	0
$544$	152.261	6.52813
$545$	−4.49768	−0.192659
$546$	0	0
$547$	−29.9586	−1.28094	−0.640469	−	0.767984i	$-0.721260\pi$
$547$	−29.9586	−1.28094	−0.640469	+	0.767984i	$0.721260\pi$
$548$	−93.9647	−4.01397
$549$	0	0
$550$	−9.44826	−0.402875
$551$	−6.02372	−0.256619
$552$	0	0
$553$	0	0
$554$	28.9793	1.23121
$555$	0	0
$556$	5.91116	0.250689
$557$	14.4402	0.611852	0.305926	−	0.952055i	$-0.401034\pi$
$557$	14.4402	0.611852	0.305926	+	0.952055i	$0.401034\pi$
$558$	0	0
$559$	4.84302	0.204838
$560$	0	0
$561$	0	0
$562$	−41.2943	−1.74190
$563$	−17.9112	−0.754866	−0.377433	−	0.926037i	$-0.623193\pi$
$563$	−17.9112	−0.754866	−0.377433	+	0.926037i	$0.623193\pi$
$564$	0	0
$565$	25.4008	1.06862
$566$	−3.41954	−0.143734
$567$	0	0
$568$	129.763	5.44475
$569$	8.50535	0.356563	0.178281	−	0.983980i	$-0.442946\pi$
$569$	8.50535	0.356563	0.178281	+	0.983980i	$0.442946\pi$
$570$	0	0
$571$	−6.80163	−0.284639	−0.142320	−	0.989821i	$-0.545456\pi$
$571$	−6.80163	−0.284639	−0.142320	+	0.989821i	$0.545456\pi$
$572$	−5.64860	−0.236180
$573$	0	0
$574$	0	0
$575$	2.24384	0.0935746
$576$	0	0
$577$	14.2726	0.594175	0.297087	−	0.954850i	$-0.403985\pi$
$577$	14.2726	0.594175	0.297087	+	0.954850i	$0.403985\pi$
$578$	70.5258	2.93348
$579$	0	0
$580$	84.4750	3.50763
$581$	0	0
$582$	0	0
$583$	−26.3140	−1.08981
$584$	−68.5638	−2.83719
$585$	0	0
$586$	56.0535	2.31555
$587$	40.2963	1.66321	0.831603	−	0.555371i	$-0.187424\pi$
$587$	40.2963	1.66321	0.831603	+	0.555371i	$0.187424\pi$
$588$	0	0
$589$	−3.48965	−0.143789
$590$	−98.3261	−4.04802
$591$	0	0
$592$	−44.2312	−1.81789
$593$	26.1313	1.07308	0.536542	−	0.843874i	$-0.319730\pi$
$593$	26.1313	1.07308	0.536542	+	0.843874i	$0.319730\pi$
$594$	0	0
$595$	0	0
$596$	−77.7395	−3.18434
$597$	0	0
$598$	1.82627	0.0746819
$599$	−3.06314	−0.125157	−0.0625783	−	0.998040i	$-0.519932\pi$
$599$	−3.06314	−0.125157	−0.0625783	+	0.998040i	$0.519932\pi$
$600$	0	0
$601$	−28.1363	−1.14770	−0.573851	−	0.818959i	$-0.694551\pi$
$601$	−28.1363	−1.14770	−0.573851	+	0.818959i	$0.694551\pi$
$602$	0	0
$603$	0	0
$604$	−81.6507	−3.32232
$605$	13.0157	0.529163
$606$	0	0
$607$	−28.8430	−1.17070	−0.585351	−	0.810780i	$-0.699043\pi$
$607$	−28.8430	−1.17070	−0.585351	+	0.810780i	$0.699043\pi$
$608$	23.3026	0.945046
$609$	0	0
$610$	47.5972	1.92715
$611$	3.38209	0.136825
$612$	0	0
$613$	22.2852	0.900092	0.450046	−	0.893005i	$-0.351408\pi$
$613$	22.2852	0.900092	0.450046	+	0.893005i	$0.351408\pi$
$614$	7.26454	0.293173
$615$	0	0
$616$	0	0
$617$	−4.13628	−0.166520	−0.0832602	−	0.996528i	$-0.526533\pi$
$617$	−4.13628	−0.166520	−0.0832602	+	0.996528i	$0.526533\pi$
$618$	0	0
$619$	9.06814	0.364479	0.182240	−	0.983254i	$-0.441665\pi$
$619$	9.06814	0.364479	0.182240	+	0.983254i	$0.441665\pi$
$620$	48.9379	1.96539
$621$	0	0
$622$	−13.9763	−0.560398
$623$	0	0
$624$	0	0
$625$	−30.0869	−1.20347
$626$	−54.7829	−2.18957
$627$	0	0
$628$	89.2993	3.56343
$629$	−18.5765	−0.740694
$630$	0	0
$631$	−41.6259	−1.65710	−0.828551	−	0.559913i	$-0.810834\pi$
$631$	−41.6259	−1.65710	−0.828551	+	0.559913i	$0.810834\pi$
$632$	137.128	5.45465
$633$	0	0
$634$	18.7178	0.743379
$635$	−5.29326	−0.210057
$636$	0	0
$637$	0	0
$638$	40.0374	1.58510
$639$	0	0
$640$	−110.362	−4.36246
$641$	−3.09186	−0.122121	−0.0610606	−	0.998134i	$-0.519448\pi$
$641$	−3.09186	−0.122121	−0.0610606	+	0.998134i	$0.519448\pi$
$642$	0	0
$643$	−2.93186	−0.115621	−0.0578106	−	0.998328i	$-0.518412\pi$
$643$	−2.93186	−0.115621	−0.0578106	+	0.998328i	$0.518412\pi$
$644$	0	0
$645$	0	0
$646$	17.9349	0.705639
$647$	−5.93489	−0.233324	−0.116662	−	0.993172i	$-0.537219\pi$
$647$	−5.93489	−0.233324	−0.116662	+	0.993172i	$0.537219\pi$
$648$	0	0
$649$	−34.2312	−1.34369
$650$	1.64466	0.0645087
$651$	0	0
$652$	−36.7829	−1.44053
$653$	−1.82233	−0.0713132	−0.0356566	−	0.999364i	$-0.511352\pi$
$653$	−1.82233	−0.0713132	−0.0356566	+	0.999364i	$0.511352\pi$
$654$	0	0
$655$	35.3120	1.37975
$656$	131.020	5.11548
$657$	0	0
$658$	0	0
$659$	−10.3978	−0.405040	−0.202520	−	0.979278i	$-0.564913\pi$
$659$	−10.3978	−0.405040	−0.202520	+	0.979278i	$0.564913\pi$
$660$	0	0
$661$	−23.2231	−0.903276	−0.451638	−	0.892201i	$-0.649160\pi$
$661$	−23.2231	−0.903276	−0.451638	+	0.892201i	$0.649160\pi$
$662$	−16.2251	−0.630607
$663$	0	0
$664$	20.4927	0.795270
$665$	0	0
$666$	0	0
$667$	−9.50837	−0.368166
$668$	−79.4770	−3.07506
$669$	0	0
$670$	68.9379	2.66330
$671$	16.5705	0.639696
$672$	0	0
$673$	13.5498	0.522305	0.261153	−	0.965298i	$-0.415897\pi$
$673$	13.5498	0.522305	0.261153	+	0.965298i	$0.415897\pi$
$674$	11.2231	0.432299
$675$	0	0
$676$	−70.9597	−2.72922
$677$	−12.6466	−0.486049	−0.243025	−	0.970020i	$-0.578140\pi$
$677$	−12.6466	−0.486049	−0.243025	+	0.970020i	$0.578140\pi$
$678$	0	0
$679$	0	0
$680$	−160.617	−6.15939
$681$	0	0
$682$	23.1944	0.888160
$683$	13.7799	0.527273	0.263636	−	0.964622i	$-0.415078\pi$
$683$	13.7799	0.527273	0.263636	+	0.964622i	$0.415078\pi$
$684$	0	0
$685$	43.0267	1.64397
$686$	0	0
$687$	0	0
$688$	178.754	6.81492
$689$	4.58046	0.174502
$690$	0	0
$691$	−1.11559	−0.0424389	−0.0212194	−	0.999775i	$-0.506755\pi$
$691$	−1.11559	−0.0424389	−0.0212194	+	0.999775i	$0.506755\pi$
$692$	−31.6259	−1.20224
$693$	0	0
$694$	47.6320	1.80809
$695$	−2.70674	−0.102673
$696$	0	0
$697$	55.0267	2.08429
$698$	12.9192	0.488999
$699$	0	0
$700$	0	0
$701$	−16.7067	−0.631005	−0.315502	−	0.948925i	$-0.602173\pi$
$701$	−16.7067	−0.631005	−0.315502	+	0.948925i	$0.602173\pi$
$702$	0	0
$703$	−2.84302	−0.107227
$704$	−79.5371	−2.99767
$705$	0	0
$706$	56.7365	2.13531
$707$	0	0
$708$	0	0
$709$	−25.1443	−0.944314	−0.472157	−	0.881514i	$-0.656525\pi$
$709$	−25.1443	−0.944314	−0.472157	+	0.881514i	$0.656525\pi$
$710$	−93.0455	−3.49193
$711$	0	0
$712$	−94.2372	−3.53169
$713$	−5.50837	−0.206290
$714$	0	0
$715$	2.58651	0.0967302
$716$	53.1393	1.98591
$717$	0	0
$718$	−22.2538	−0.830506
$719$	32.2488	1.20268	0.601339	−	0.798994i	$-0.294634\pi$
$719$	32.2488	1.20268	0.601339	+	0.798994i	$0.294634\pi$
$720$	0	0
$721$	0	0
$722$	2.74483	0.102152
$723$	0	0
$724$	5.41954	0.201416
$725$	−8.56279	−0.318014
$726$	0	0
$727$	30.2312	1.12121	0.560606	−	0.828083i	$-0.310568\pi$
$727$	30.2312	1.12121	0.560606	+	0.828083i	$0.310568\pi$
$728$	0	0
$729$	0	0
$730$	49.1630	1.81961
$731$	75.0742	2.77672
$732$	0	0
$733$	−25.0267	−0.924384	−0.462192	−	0.886780i	$-0.652937\pi$
$733$	−25.0267	−0.924384	−0.462192	+	0.886780i	$0.652937\pi$
$734$	−79.2993	−2.92699
$735$	0	0
$736$	36.7829	1.35584
$737$	24.0000	0.884051
$738$	0	0
$739$	5.53104	0.203463	0.101731	−	0.994812i	$-0.467562\pi$
$739$	5.53104	0.203463	0.101731	+	0.994812i	$0.467562\pi$
$740$	39.8698	1.46564
$741$	0	0
$742$	0	0
$743$	−21.6971	−0.795989	−0.397995	−	0.917388i	$-0.630294\pi$
$743$	−21.6971	−0.795989	−0.397995	+	0.917388i	$0.630294\pi$
$744$	0	0
$745$	35.5972	1.30418
$746$	32.8243	1.20178
$747$	0	0
$748$	−87.5619	−3.20158
$749$	0	0
$750$	0	0
$751$	−48.9854	−1.78750	−0.893751	−	0.448564i	$-0.851935\pi$
$751$	−48.9854	−1.78750	−0.893751	+	0.448564i	$0.851935\pi$
$752$	124.831	4.55213
$753$	0	0
$754$	−6.96931	−0.253807
$755$	37.3881	1.36069
$756$	0	0
$757$	−4.04139	−0.146887	−0.0734434	−	0.997299i	$-0.523399\pi$
$757$	−4.04139	−0.146887	−0.0734434	+	0.997299i	$0.523399\pi$
$758$	−42.2212	−1.53354
$759$	0	0
$760$	−24.5815	−0.891665
$761$	−36.9429	−1.33918	−0.669590	−	0.742731i	$-0.733530\pi$
$761$	−36.9429	−1.33918	−0.669590	+	0.742731i	$0.733530\pi$
$762$	0	0
$763$	0	0
$764$	64.8304	2.34548
$765$	0	0
$766$	−1.69605	−0.0612806
$767$	5.95861	0.215153
$768$	0	0
$769$	−24.1363	−0.870377	−0.435188	−	0.900339i	$-0.643318\pi$
$769$	−24.1363	−0.870377	−0.435188	+	0.900339i	$0.643318\pi$
$770$	0	0
$771$	0	0
$772$	98.6300	3.54977
$773$	−25.5371	−0.918506	−0.459253	−	0.888306i	$-0.651883\pi$
$773$	−25.5371	−0.918506	−0.459253	+	0.888306i	$0.651883\pi$
$774$	0	0
$775$	−4.96058	−0.178189
$776$	48.3013	1.73392
$777$	0	0
$778$	30.3801	1.08918
$779$	8.42151	0.301732
$780$	0	0
$781$	−32.3928	−1.15911
$782$	28.3100	1.01236
$783$	0	0
$784$	0	0
$785$	−40.8905	−1.45944
$786$	0	0
$787$	19.0969	0.680730	0.340365	−	0.940293i	$-0.389449\pi$
$787$	19.0969	0.680730	0.340365	+	0.940293i	$0.389449\pi$
$788$	105.787	3.76851
$789$	0	0
$790$	−98.3261	−3.49828
$791$	0	0
$792$	0	0
$793$	−2.88441	−0.102429
$794$	28.6841	1.01796
$795$	0	0
$796$	−88.8077	−3.14770
$797$	8.02872	0.284392	0.142196	−	0.989839i	$-0.454584\pi$
$797$	8.02872	0.284392	0.142196	+	0.989839i	$0.454584\pi$
$798$	0	0
$799$	52.4276	1.85475
$800$	33.1249	1.17114
$801$	0	0
$802$	−8.97430	−0.316894
$803$	17.1156	0.603996
$804$	0	0
$805$	0	0
$806$	−4.03745	−0.142213
$807$	0	0
$808$	−16.4038	−0.577085
$809$	29.0742	1.02219	0.511097	−	0.859523i	$-0.329239\pi$
$809$	29.0742	1.02219	0.511097	+	0.859523i	$0.329239\pi$
$810$	0	0
$811$	−4.66535	−0.163823	−0.0819113	−	0.996640i	$-0.526102\pi$
$811$	−4.66535	−0.163823	−0.0819113	+	0.996640i	$0.526102\pi$
$812$	0	0
$813$	0	0
$814$	18.8965	0.662323
$815$	16.8430	0.589985
$816$	0	0
$817$	11.4897	0.401972
$818$	−20.3140	−0.710261
$819$	0	0
$820$	−118.101	−4.12426
$821$	17.5972	0.614147	0.307073	−	0.951686i	$-0.400650\pi$
$821$	17.5972	0.614147	0.307073	+	0.951686i	$0.400650\pi$
$822$	0	0
$823$	−7.93989	−0.276767	−0.138384	−	0.990379i	$-0.544191\pi$
$823$	−7.93989	−0.276767	−0.138384	+	0.990379i	$0.544191\pi$
$824$	−30.6240	−1.06684
$825$	0	0
$826$	0	0
$827$	31.7859	1.10531	0.552653	−	0.833412i	$-0.313616\pi$
$827$	31.7859	1.10531	0.552653	+	0.833412i	$0.313616\pi$
$828$	0	0
$829$	48.9793	1.70112	0.850561	−	0.525877i	$-0.176263\pi$
$829$	48.9793	1.70112	0.850561	+	0.525877i	$0.176263\pi$
$830$	−14.6941	−0.510039
$831$	0	0
$832$	13.8450	0.479989
$833$	0	0
$834$	0	0
$835$	36.3928	1.25942
$836$	−13.4008	−0.463477
$837$	0	0
$838$	−87.4119	−3.01959
$839$	36.2726	1.25227	0.626134	−	0.779716i	$-0.284636\pi$
$839$	36.2726	1.25227	0.626134	+	0.779716i	$0.284636\pi$
$840$	0	0
$841$	7.28523	0.251215
$842$	109.810	3.78429
$843$	0	0
$844$	−51.4295	−1.77028
$845$	32.4927	1.11778
$846$	0	0
$847$	0	0
$848$	169.063	5.80563
$849$	0	0
$850$	25.4947	0.874459
$851$	−4.48768	−0.153836
$852$	0	0
$853$	34.6654	1.18692	0.593460	−	0.804864i	$-0.297762\pi$
$853$	34.6654	1.18692	0.593460	+	0.804864i	$0.297762\pi$
$854$	0	0
$855$	0	0
$856$	160.387	5.48192
$857$	−4.87175	−0.166416	−0.0832078	−	0.996532i	$-0.526517\pi$
$857$	−4.87175	−0.166416	−0.0832078	+	0.996532i	$0.526517\pi$
$858$	0	0
$859$	37.1156	1.26637	0.633184	−	0.774002i	$-0.281748\pi$
$859$	37.1156	1.26637	0.633184	+	0.774002i	$0.281748\pi$
$860$	−161.128	−5.49441
$861$	0	0
$862$	−10.7492	−0.366119
$863$	29.1520	0.992345	0.496172	−	0.868224i	$-0.334738\pi$
$863$	29.1520	0.992345	0.496172	+	0.868224i	$0.334738\pi$
$864$	0	0
$865$	14.4816	0.492390
$866$	−49.4069	−1.67891
$867$	0	0
$868$	0	0
$869$	−34.2312	−1.16121
$870$	0	0
$871$	−4.17767	−0.141555
$872$	17.2171	0.583044
$873$	0	0
$874$	4.33268	0.146555
$875$	0	0
$876$	0	0
$877$	14.4502	0.487950	0.243975	−	0.969782i	$-0.421549\pi$
$877$	14.4502	0.487950	0.243975	+	0.969782i	$0.421549\pi$
$878$	−47.8924	−1.61629
$879$	0	0
$880$	95.4670	3.21819
$881$	10.5341	0.354902	0.177451	−	0.984130i	$-0.443215\pi$
$881$	10.5341	0.354902	0.177451	+	0.984130i	$0.443215\pi$
$882$	0	0
$883$	−5.11559	−0.172153	−0.0860766	−	0.996289i	$-0.527433\pi$
$883$	−5.11559	−0.172153	−0.0860766	+	0.996289i	$0.527433\pi$
$884$	15.2419	0.512639
$885$	0	0
$886$	8.34267	0.280278
$887$	−0.617907	−0.0207473	−0.0103736	−	0.999946i	$-0.503302\pi$
$887$	−0.617907	−0.0207473	−0.0103736	+	0.999946i	$0.503302\pi$
$888$	0	0
$889$	0	0
$890$	67.5719	2.26501
$891$	0	0
$892$	46.1136	1.54400
$893$	8.02372	0.268504
$894$	0	0
$895$	−24.3327	−0.813352
$896$	0	0
$897$	0	0
$898$	−98.6350	−3.29149
$899$	21.0207	0.701079
$900$	0	0
$901$	71.0041	2.36549
$902$	−55.9747	−1.86375
$903$	0	0
$904$	−97.2342	−3.23396
$905$	−2.48163	−0.0824920
$906$	0	0
$907$	−8.27256	−0.274686	−0.137343	−	0.990524i	$-0.543856\pi$
$907$	−8.27256	−0.274686	−0.137343	+	0.990524i	$0.543856\pi$
$908$	93.4730	3.10201
$909$	0	0
$910$	0	0
$911$	5.77988	0.191496	0.0957480	−	0.995406i	$-0.469476\pi$
$911$	5.77988	0.191496	0.0957480	+	0.995406i	$0.469476\pi$
$912$	0	0
$913$	−5.11559	−0.169301
$914$	−67.0328	−2.21725
$915$	0	0
$916$	−33.4670	−1.10578
$917$	0	0
$918$	0	0
$919$	4.62791	0.152661	0.0763303	−	0.997083i	$-0.475680\pi$
$919$	4.62791	0.152661	0.0763303	+	0.997083i	$0.475680\pi$
$920$	−38.8016	−1.27925
$921$	0	0
$922$	27.8086	0.915828
$923$	5.63860	0.185597
$924$	0	0
$925$	−4.04139	−0.132880
$926$	−16.0949	−0.528911
$927$	0	0
$928$	−140.369	−4.60782
$929$	−29.9536	−0.982746	−0.491373	−	0.870949i	$-0.663505\pi$
$929$	−29.9536	−0.982746	−0.491373	+	0.870949i	$0.663505\pi$
$930$	0	0
$931$	0	0
$932$	−117.347	−3.84382
$933$	0	0
$934$	−41.0505	−1.34321
$935$	40.0949	1.31124
$936$	0	0
$937$	−4.58651	−0.149835	−0.0749174	−	0.997190i	$-0.523869\pi$
$937$	−4.58651	−0.149835	−0.0749174	+	0.997190i	$0.523869\pi$
$938$	0	0
$939$	0	0
$940$	−112.522	−3.67008
$941$	7.41082	0.241586	0.120793	−	0.992678i	$-0.461456\pi$
$941$	7.41082	0.241586	0.120793	+	0.992678i	$0.461456\pi$
$942$	0	0
$943$	13.2933	0.432888
$944$	219.929	7.15809
$945$	0	0
$946$	−76.3675	−2.48292
$947$	−59.6320	−1.93778	−0.968890	−	0.247493i	$-0.920393\pi$
$947$	−59.6320	−1.93778	−0.968890	+	0.247493i	$0.920393\pi$
$948$	0	0
$949$	−2.97930	−0.0967123
$950$	3.90180	0.126591
$951$	0	0
$952$	0	0
$953$	−25.7986	−0.835699	−0.417849	−	0.908516i	$-0.637216\pi$
$953$	−25.7986	−0.835699	−0.417849	+	0.908516i	$0.637216\pi$
$954$	0	0
$955$	−29.6860	−0.960618
$956$	32.1176	1.03876
$957$	0	0
$958$	−92.5275	−2.98943
$959$	0	0
$960$	0	0
$961$	−18.8223	−0.607172
$962$	−3.28931	−0.106052
$963$	0	0
$964$	71.8284	2.31344
$965$	−45.1630	−1.45385
$966$	0	0
$967$	55.5845	1.78748	0.893739	−	0.448587i	$-0.148073\pi$
$967$	55.5845	1.78748	0.893739	+	0.448587i	$0.148073\pi$
$968$	−49.8240	−1.60140
$969$	0	0
$970$	−34.6340	−1.11203
$971$	−48.4877	−1.55604	−0.778022	−	0.628237i	$-0.783777\pi$
$971$	−48.4877	−1.55604	−0.778022	+	0.628237i	$0.783777\pi$
$972$	0	0
$973$	0	0
$974$	59.0367	1.89166
$975$	0	0
$976$	−106.462	−3.40778
$977$	−53.0505	−1.69723	−0.848617	−	0.529007i	$-0.822565\pi$
$977$	−53.0505	−1.69723	−0.848617	+	0.529007i	$0.822565\pi$
$978$	0	0
$979$	23.5244	0.751844
$980$	0	0
$981$	0	0
$982$	47.6320	1.52000
$983$	12.1777	0.388407	0.194204	−	0.980961i	$-0.437788\pi$
$983$	12.1777	0.388407	0.194204	+	0.980961i	$0.437788\pi$
$984$	0	0
$985$	−48.4402	−1.54343
$986$	−108.035	−3.44053
$987$	0	0
$988$	2.33268	0.0742123
$989$	18.1363	0.576700
$990$	0	0
$991$	−2.04139	−0.0648469	−0.0324235	−	0.999474i	$-0.510323\pi$
$991$	−2.04139	−0.0648469	−0.0324235	+	0.999474i	$0.510323\pi$
$992$	−81.3180	−2.58185
$993$	0	0
$994$	0	0
$995$	40.6654	1.28918
$996$	0	0
$997$	45.6447	1.44558	0.722790	−	0.691067i	$-0.242859\pi$
$997$	45.6447	1.44558	0.722790	+	0.691067i	$0.242859\pi$
$998$	−90.4088	−2.86184
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.bq.1.3		3
3.2	odd	2		2793.2.a.w.1.1		3
7.6	odd	2		1197.2.a.m.1.3		3
21.20	even	2		399.2.a.e.1.1	✓	3
84.83	odd	2		6384.2.a.bu.1.1		3
105.104	even	2		9975.2.a.x.1.3		3
399.398	odd	2		7581.2.a.l.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.e.1.1	✓	3	21.20	even	2
1197.2.a.m.1.3		3	7.6	odd	2
2793.2.a.w.1.1		3	3.2	odd	2
6384.2.a.bu.1.1		3	84.83	odd	2
7581.2.a.l.1.3		3	399.398	odd	2
8379.2.a.bq.1.3		3	1.1	even	1	trivial
9975.2.a.x.1.3		3	105.104	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)

\(f(q)\)	\(=\)	\(q+2.74483 q^{2} +5.53407 q^{4} -2.53407 q^{5} +9.70041 q^{8} -6.95558 q^{10} -2.42151 q^{11} +0.421512 q^{13} +15.5578 q^{16} +6.53407 q^{17} +1.00000 q^{19} -14.0237 q^{20} -6.64663 q^{22} +1.57849 q^{23} +1.42151 q^{25} +1.15698 q^{26} -6.02372 q^{29} -3.48965 q^{31} +23.3026 q^{32} +17.9349 q^{34} -2.84302 q^{37} +2.74483 q^{38} -24.5815 q^{40} +8.42151 q^{41} +11.4897 q^{43} -13.4008 q^{44} +4.33268 q^{46} +8.02372 q^{47} +3.90180 q^{50} +2.33268 q^{52} +10.8667 q^{53} +6.13628 q^{55} -16.5341 q^{58} +14.1363 q^{59} -6.84302 q^{61} -9.57849 q^{62} +32.8461 q^{64} -1.06814 q^{65} -9.91116 q^{67} +36.1600 q^{68} +13.3771 q^{71} -7.06814 q^{73} -7.80361 q^{74} +5.53407 q^{76} +14.1363 q^{79} -39.4245 q^{80} +23.1156 q^{82} +2.11256 q^{83} -16.5578 q^{85} +31.5371 q^{86} -23.4897 q^{88} -9.71477 q^{89} +8.73546 q^{92} +22.0237 q^{94} -2.53407 q^{95} +4.97930 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 9 q^{4} + 9 q^{8} - 10 q^{10} - 4 q^{11} - 2 q^{13} + 13 q^{16} + 12 q^{17} + 3 q^{19} - 16 q^{20} - 8 q^{22} + 8 q^{23} + q^{25} + 10 q^{26} + 8 q^{29} + 8 q^{31} + 27 q^{32} + 6 q^{34}+ \cdots - 22 q^{97}+O(q^{100}) \) 3 * q - q^2 + 9 * q^4 + 9 * q^8 - 10 * q^10 - 4 * q^11 - 2 * q^13 + 13 * q^16 + 12 * q^17 + 3 * q^19 - 16 * q^20 - 8 * q^22 + 8 * q^23 + q^25 + 10 * q^26 + 8 * q^29 + 8 * q^31 + 27 * q^32 + 6 * q^34 - 2 * q^37 - q^38 - 14 * q^40 + 22 * q^41 + 16 * q^43 - 12 * q^46 - 2 * q^47 + 9 * q^50 - 18 * q^52 - 12 * q^55 - 42 * q^58 + 12 * q^59 - 14 * q^61 - 32 * q^62 + 29 * q^64 + 12 * q^65 - 8 * q^67 + 52 * q^68 + 26 * q^71 - 6 * q^73 - 18 * q^74 + 9 * q^76 + 12 * q^79 - 52 * q^80 + 2 * q^82 + 2 * q^83 - 16 * q^85 + 24 * q^86 - 52 * q^88 - 2 * q^89 + 36 * q^92 + 40 * q^94 - 22 * q^97

Introduction

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