LMFDB - Embedded newform 3042.2.a.bc.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3042,2,Mod(1,3042)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3042, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3042.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$3042 = 2 \cdot 3^{2} \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3042.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,0,3,1,0,1,-3,0,-1,9,0,0,-1,0,3,14] 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:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 2x + 1$ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	3042.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.80194 q^{5} +4.85086 q^{7} -1.00000 q^{8} -1.80194 q^{10} +6.04892 q^{11} -4.85086 q^{14} +1.00000 q^{16} +4.89008 q^{17} -4.49396 q^{19} +1.80194 q^{20} -6.04892 q^{22} +6.71379 q^{23} -1.75302 q^{25} +4.85086 q^{28} +3.55496 q^{29} +3.82908 q^{31} -1.00000 q^{32} -4.89008 q^{34} +8.74094 q^{35} -3.78017 q^{37} +4.49396 q^{38} -1.80194 q^{40} +5.70171 q^{41} -3.78017 q^{43} +6.04892 q^{44} -6.71379 q^{46} -3.82371 q^{47} +16.5308 q^{49} +1.75302 q^{50} -2.78986 q^{53} +10.8998 q^{55} -4.85086 q^{56} -3.55496 q^{58} -8.41119 q^{59} +0.219833 q^{61} -3.82908 q^{62} +1.00000 q^{64} -11.4819 q^{67} +4.89008 q^{68} -8.74094 q^{70} -13.5254 q^{71} -0.417895 q^{73} +3.78017 q^{74} -4.49396 q^{76} +29.3424 q^{77} -10.5526 q^{79} +1.80194 q^{80} -5.70171 q^{82} +3.42327 q^{83} +8.81163 q^{85} +3.78017 q^{86} -6.04892 q^{88} -17.3599 q^{89} +6.71379 q^{92} +3.82371 q^{94} -8.09783 q^{95} -1.00969 q^{97} -16.5308 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 3 q^{2} + 3 q^{4} + q^{5} + q^{7} - 3 q^{8} - q^{10} + 9 q^{11} - q^{14} + 3 q^{16} + 14 q^{17} - 4 q^{19} + q^{20} - 9 q^{22} + 12 q^{23} - 10 q^{25} + q^{28} + 11 q^{29} + q^{31} - 3 q^{32}+ \cdots - 12 q^{98}+O(q^{100})$ 3 * q - 3 * q^2 + 3 * q^4 + q^5 + q^7 - 3 * q^8 - q^10 + 9 * q^11 - q^14 + 3 * q^16 + 14 * q^17 - 4 * q^19 + q^20 - 9 * q^22 + 12 * q^23 - 10 * q^25 + q^28 + 11 * q^29 + q^31 - 3 * q^32 - 14 * q^34 + 12 * q^35 - 10 * q^37 + 4 * q^38 - q^40 - 10 * q^41 - 10 * q^43 + 9 * q^44 - 12 * q^46 - 4 * q^47 + 12 * q^49 + 10 * q^50 + 15 * q^53 + 10 * q^55 - q^56 - 11 * q^58 - 9 * q^59 + 2 * q^61 - q^62 + 3 * q^64 - 6 * q^67 + 14 * q^68 - 12 * q^70 - 6 * q^71 - 7 * q^73 + 10 * q^74 - 4 * q^76 + 24 * q^77 + 9 * q^79 + q^80 + 10 * q^82 + 13 * q^83 + 10 * q^86 - 9 * q^88 - 4 * q^89 + 12 * q^92 + 4 * q^94 - 6 * q^95 + 19 * q^97 - 12 * q^98

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.80194	0.805851	0.402926	−	0.915233i	$-0.367993\pi$
$5$	1.80194	0.805851	0.402926	+	0.915233i	$0.367993\pi$
$6$	0	0
$7$	4.85086	1.83345	0.916725	−	0.399518i	$-0.130822\pi$
$7$	4.85086	1.83345	0.916725	+	0.399518i	$0.130822\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.80194	−0.569823
$11$	6.04892	1.82382	0.911909	−	0.410393i	$-0.134609\pi$
$11$	6.04892	1.82382	0.911909	+	0.410393i	$0.134609\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−4.85086	−1.29645
$15$	0	0
$16$	1.00000	0.250000
$17$	4.89008	1.18602	0.593010	−	0.805195i	$-0.297940\pi$
$17$	4.89008	1.18602	0.593010	+	0.805195i	$0.297940\pi$
$18$	0	0
$19$	−4.49396	−1.03098	−0.515492	−	0.856894i	$-0.672391\pi$
$19$	−4.49396	−1.03098	−0.515492	+	0.856894i	$0.672391\pi$
$20$	1.80194	0.402926
$21$	0	0
$22$	−6.04892	−1.28963
$23$	6.71379	1.39992	0.699961	−	0.714181i	$-0.253201\pi$
$23$	6.71379	1.39992	0.699961	+	0.714181i	$0.253201\pi$
$24$	0	0
$25$	−1.75302	−0.350604
$26$	0	0
$27$	0	0
$28$	4.85086	0.916725
$29$	3.55496	0.660139	0.330070	−	0.943957i	$-0.392928\pi$
$29$	3.55496	0.660139	0.330070	+	0.943957i	$0.392928\pi$
$30$	0	0
$31$	3.82908	0.687724	0.343862	−	0.939020i	$-0.388265\pi$
$31$	3.82908	0.687724	0.343862	+	0.939020i	$0.388265\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.89008	−0.838642
$35$	8.74094	1.47749
$36$	0	0
$37$	−3.78017	−0.621456	−0.310728	−	0.950499i	$-0.600573\pi$
$37$	−3.78017	−0.621456	−0.310728	+	0.950499i	$0.600573\pi$
$38$	4.49396	0.729016
$39$	0	0
$40$	−1.80194	−0.284911
$41$	5.70171	0.890458	0.445229	−	0.895417i	$-0.353122\pi$
$41$	5.70171	0.890458	0.445229	+	0.895417i	$0.353122\pi$
$42$	0	0
$43$	−3.78017	−0.576470	−0.288235	−	0.957560i	$-0.593068\pi$
$43$	−3.78017	−0.576470	−0.288235	+	0.957560i	$0.593068\pi$
$44$	6.04892	0.911909
$45$	0	0
$46$	−6.71379	−0.989895
$47$	−3.82371	−0.557745	−0.278873	−	0.960328i	$-0.589961\pi$
$47$	−3.82371	−0.557745	−0.278873	+	0.960328i	$0.589961\pi$
$48$	0	0
$49$	16.5308	2.36154
$50$	1.75302	0.247915
$51$	0	0
$52$	0	0
$53$	−2.78986	−0.383216	−0.191608	−	0.981472i	$-0.561370\pi$
$53$	−2.78986	−0.383216	−0.191608	+	0.981472i	$0.561370\pi$
$54$	0	0
$55$	10.8998	1.46973
$56$	−4.85086	−0.648223
$57$	0	0
$58$	−3.55496	−0.466789
$59$	−8.41119	−1.09504	−0.547522	−	0.836791i	$-0.684429\pi$
$59$	−8.41119	−1.09504	−0.547522	+	0.836791i	$0.684429\pi$
$60$	0	0
$61$	0.219833	0.0281467	0.0140733	−	0.999901i	$-0.495520\pi$
$61$	0.219833	0.0281467	0.0140733	+	0.999901i	$0.495520\pi$
$62$	−3.82908	−0.486294
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−11.4819	−1.40273	−0.701367	−	0.712800i	$-0.747427\pi$
$67$	−11.4819	−1.40273	−0.701367	+	0.712800i	$0.747427\pi$
$68$	4.89008	0.593010
$69$	0	0
$70$	−8.74094	−1.04474
$71$	−13.5254	−1.60517	−0.802586	−	0.596537i	$-0.796543\pi$
$71$	−13.5254	−1.60517	−0.802586	+	0.596537i	$0.796543\pi$
$72$	0	0
$73$	−0.417895	−0.0489109	−0.0244554	−	0.999701i	$-0.507785\pi$
$73$	−0.417895	−0.0489109	−0.0244554	+	0.999701i	$0.507785\pi$
$74$	3.78017	0.439436
$75$	0	0
$76$	−4.49396	−0.515492
$77$	29.3424	3.34388
$78$	0	0
$79$	−10.5526	−1.18726	−0.593628	−	0.804739i	$-0.702305\pi$
$79$	−10.5526	−1.18726	−0.593628	+	0.804739i	$0.702305\pi$
$80$	1.80194	0.201463
$81$	0	0
$82$	−5.70171	−0.629649
$83$	3.42327	0.375753	0.187876	−	0.982193i	$-0.439840\pi$
$83$	3.42327	0.375753	0.187876	+	0.982193i	$0.439840\pi$
$84$	0	0
$85$	8.81163	0.955755
$86$	3.78017	0.407626
$87$	0	0
$88$	−6.04892	−0.644817
$89$	−17.3599	−1.84014	−0.920072	−	0.391750i	$-0.871870\pi$
$89$	−17.3599	−1.84014	−0.920072	+	0.391750i	$0.871870\pi$
$90$	0	0
$91$	0	0
$92$	6.71379	0.699961
$93$	0	0
$94$	3.82371	0.394385
$95$	−8.09783	−0.830820
$96$	0	0
$97$	−1.00969	−0.102518	−0.0512592	−	0.998685i	$-0.516323\pi$
$97$	−1.00969	−0.102518	−0.0512592	+	0.998685i	$0.516323\pi$
$98$	−16.5308	−1.66986
$99$	0	0
$100$	−1.75302	−0.175302
$101$	5.68664	0.565842	0.282921	−	0.959143i	$-0.408697\pi$
$101$	5.68664	0.565842	0.282921	+	0.959143i	$0.408697\pi$
$102$	0	0
$103$	−6.73556	−0.663675	−0.331837	−	0.943337i	$-0.607669\pi$
$103$	−6.73556	−0.663675	−0.331837	+	0.943337i	$0.607669\pi$
$104$	0	0
$105$	0	0
$106$	2.78986	0.270975
$107$	−0.0586060	−0.00566566	−0.00283283	−	0.999996i	$-0.500902\pi$
$107$	−0.0586060	−0.00566566	−0.00283283	+	0.999996i	$0.500902\pi$
$108$	0	0
$109$	−15.5254	−1.48707	−0.743533	−	0.668700i	$-0.766851\pi$
$109$	−15.5254	−1.48707	−0.743533	+	0.668700i	$0.766851\pi$
$110$	−10.8998	−1.03925
$111$	0	0
$112$	4.85086	0.458363
$113$	1.95646	0.184048	0.0920241	−	0.995757i	$-0.470666\pi$
$113$	1.95646	0.184048	0.0920241	+	0.995757i	$0.470666\pi$
$114$	0	0
$115$	12.0978	1.12813
$116$	3.55496	0.330070
$117$	0	0
$118$	8.41119	0.774313
$119$	23.7211	2.17451
$120$	0	0
$121$	25.5894	2.32631
$122$	−0.219833	−0.0199027
$123$	0	0
$124$	3.82908	0.343862
$125$	−12.1685	−1.08839
$126$	0	0
$127$	−8.11960	−0.720498	−0.360249	−	0.932856i	$-0.617308\pi$
$127$	−8.11960	−0.720498	−0.360249	+	0.932856i	$0.617308\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−12.6679	−1.10680	−0.553398	−	0.832917i	$-0.686669\pi$
$131$	−12.6679	−1.10680	−0.553398	+	0.832917i	$0.686669\pi$
$132$	0	0
$133$	−21.7995	−1.89026
$134$	11.4819	0.991883
$135$	0	0
$136$	−4.89008	−0.419321
$137$	−15.3599	−1.31228	−0.656142	−	0.754638i	$-0.727813\pi$
$137$	−15.3599	−1.31228	−0.656142	+	0.754638i	$0.727813\pi$
$138$	0	0
$139$	−11.7995	−1.00082	−0.500412	−	0.865787i	$-0.666818\pi$
$139$	−11.7995	−1.00082	−0.500412	+	0.865787i	$0.666818\pi$
$140$	8.74094	0.738744
$141$	0	0
$142$	13.5254	1.13503
$143$	0	0
$144$	0	0
$145$	6.40581	0.531974
$146$	0.417895	0.0345852
$147$	0	0
$148$	−3.78017	−0.310728
$149$	3.21313	0.263230	0.131615	−	0.991301i	$-0.457984\pi$
$149$	3.21313	0.263230	0.131615	+	0.991301i	$0.457984\pi$
$150$	0	0
$151$	9.15883	0.745335	0.372668	−	0.927965i	$-0.378443\pi$
$151$	9.15883	0.745335	0.372668	+	0.927965i	$0.378443\pi$
$152$	4.49396	0.364508
$153$	0	0
$154$	−29.3424	−2.36448
$155$	6.89977	0.554203
$156$	0	0
$157$	−17.9323	−1.43115	−0.715577	−	0.698534i	$-0.753836\pi$
$157$	−17.9323	−1.43115	−0.715577	+	0.698534i	$0.753836\pi$
$158$	10.5526	0.839517
$159$	0	0
$160$	−1.80194	−0.142456
$161$	32.5676	2.56669
$162$	0	0
$163$	−0.591794	−0.0463529	−0.0231764	−	0.999731i	$-0.507378\pi$
$163$	−0.591794	−0.0463529	−0.0231764	+	0.999731i	$0.507378\pi$
$164$	5.70171	0.445229
$165$	0	0
$166$	−3.42327	−0.265697
$167$	14.0000	1.08335	0.541676	−	0.840587i	$-0.317790\pi$
$167$	14.0000	1.08335	0.541676	+	0.840587i	$0.317790\pi$
$168$	0	0
$169$	0	0
$170$	−8.81163	−0.675821
$171$	0	0
$172$	−3.78017	−0.288235
$173$	−0.982542	−0.0747013	−0.0373506	−	0.999302i	$-0.511892\pi$
$173$	−0.982542	−0.0747013	−0.0373506	+	0.999302i	$0.511892\pi$
$174$	0	0
$175$	−8.50365	−0.642815
$176$	6.04892	0.455954
$177$	0	0
$178$	17.3599	1.30118
$179$	9.94331	0.743198	0.371599	−	0.928393i	$-0.378810\pi$
$179$	9.94331	0.743198	0.371599	+	0.928393i	$0.378810\pi$
$180$	0	0
$181$	−5.87800	−0.436908	−0.218454	−	0.975847i	$-0.570101\pi$
$181$	−5.87800	−0.436908	−0.218454	+	0.975847i	$0.570101\pi$
$182$	0	0
$183$	0	0
$184$	−6.71379	−0.494947
$185$	−6.81163	−0.500801
$186$	0	0
$187$	29.5797	2.16308
$188$	−3.82371	−0.278873
$189$	0	0
$190$	8.09783	0.587479
$191$	11.0422	0.798986	0.399493	−	0.916736i	$-0.369186\pi$
$191$	11.0422	0.798986	0.399493	+	0.916736i	$0.369186\pi$
$192$	0	0
$193$	20.9638	1.50900	0.754502	−	0.656298i	$-0.227878\pi$
$193$	20.9638	1.50900	0.754502	+	0.656298i	$0.227878\pi$
$194$	1.00969	0.0724914
$195$	0	0
$196$	16.5308	1.18077
$197$	−15.7409	−1.12150	−0.560748	−	0.827987i	$-0.689486\pi$
$197$	−15.7409	−1.12150	−0.560748	+	0.827987i	$0.689486\pi$
$198$	0	0
$199$	−1.51142	−0.107142	−0.0535708	−	0.998564i	$-0.517060\pi$
$199$	−1.51142	−0.107142	−0.0535708	+	0.998564i	$0.517060\pi$
$200$	1.75302	0.123957
$201$	0	0
$202$	−5.68664	−0.400111
$203$	17.2446	1.21033
$204$	0	0
$205$	10.2741	0.717576
$206$	6.73556	0.469289
$207$	0	0
$208$	0	0
$209$	−27.1836	−1.88033
$210$	0	0
$211$	5.65817	0.389524	0.194762	−	0.980850i	$-0.437606\pi$
$211$	5.65817	0.389524	0.194762	+	0.980850i	$0.437606\pi$
$212$	−2.78986	−0.191608
$213$	0	0
$214$	0.0586060	0.00400622
$215$	−6.81163	−0.464549
$216$	0	0
$217$	18.5743	1.26091
$218$	15.5254	1.05151
$219$	0	0
$220$	10.8998	0.734863
$221$	0	0
$222$	0	0
$223$	−1.97584	−0.132312	−0.0661559	−	0.997809i	$-0.521073\pi$
$223$	−1.97584	−0.132312	−0.0661559	+	0.997809i	$0.521073\pi$
$224$	−4.85086	−0.324111
$225$	0	0
$226$	−1.95646	−0.130142
$227$	−25.0640	−1.66355	−0.831777	−	0.555109i	$-0.812676\pi$
$227$	−25.0640	−1.66355	−0.831777	+	0.555109i	$0.812676\pi$
$228$	0	0
$229$	−24.5133	−1.61989	−0.809943	−	0.586508i	$-0.800502\pi$
$229$	−24.5133	−1.61989	−0.809943	+	0.586508i	$0.800502\pi$
$230$	−12.0978	−0.797708
$231$	0	0
$232$	−3.55496	−0.233394
$233$	−8.37196	−0.548465	−0.274233	−	0.961663i	$-0.588424\pi$
$233$	−8.37196	−0.548465	−0.274233	+	0.961663i	$0.588424\pi$
$234$	0	0
$235$	−6.89008	−0.449460
$236$	−8.41119	−0.547522
$237$	0	0
$238$	−23.7211	−1.53761
$239$	14.0194	0.906838	0.453419	−	0.891297i	$-0.350204\pi$
$239$	14.0194	0.906838	0.453419	+	0.891297i	$0.350204\pi$
$240$	0	0
$241$	−21.1183	−1.36035	−0.680174	−	0.733051i	$-0.738096\pi$
$241$	−21.1183	−1.36035	−0.680174	+	0.733051i	$0.738096\pi$
$242$	−25.5894	−1.64495
$243$	0	0
$244$	0.219833	0.0140733
$245$	29.7875	1.90305
$246$	0	0
$247$	0	0
$248$	−3.82908	−0.243147
$249$	0	0
$250$	12.1685	0.769605
$251$	26.1129	1.64823	0.824116	−	0.566421i	$-0.191672\pi$
$251$	26.1129	1.64823	0.824116	+	0.566421i	$0.191672\pi$
$252$	0	0
$253$	40.6112	2.55320
$254$	8.11960	0.509469
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.5918	1.03497	0.517484	−	0.855693i	$-0.326869\pi$
$257$	16.5918	1.03497	0.517484	+	0.855693i	$0.326869\pi$
$258$	0	0
$259$	−18.3370	−1.13941
$260$	0	0
$261$	0	0
$262$	12.6679	0.782623
$263$	9.64742	0.594885	0.297443	−	0.954740i	$-0.403866\pi$
$263$	9.64742	0.594885	0.297443	+	0.954740i	$0.403866\pi$
$264$	0	0
$265$	−5.02715	−0.308815
$266$	21.7995	1.33662
$267$	0	0
$268$	−11.4819	−0.701367
$269$	10.1981	0.621787	0.310893	−	0.950445i	$-0.399372\pi$
$269$	10.1981	0.621787	0.310893	+	0.950445i	$0.399372\pi$
$270$	0	0
$271$	27.6799	1.68144	0.840718	−	0.541473i	$-0.182133\pi$
$271$	27.6799	1.68144	0.840718	+	0.541473i	$0.182133\pi$
$272$	4.89008	0.296505
$273$	0	0
$274$	15.3599	0.927924
$275$	−10.6039	−0.639438
$276$	0	0
$277$	6.32842	0.380238	0.190119	−	0.981761i	$-0.439113\pi$
$277$	6.32842	0.380238	0.190119	+	0.981761i	$0.439113\pi$
$278$	11.7995	0.707690
$279$	0	0
$280$	−8.74094	−0.522371
$281$	9.26205	0.552527	0.276264	−	0.961082i	$-0.410904\pi$
$281$	9.26205	0.552527	0.276264	+	0.961082i	$0.410904\pi$
$282$	0	0
$283$	0.561663	0.0333874	0.0166937	−	0.999861i	$-0.494686\pi$
$283$	0.561663	0.0333874	0.0166937	+	0.999861i	$0.494686\pi$
$284$	−13.5254	−0.802586
$285$	0	0
$286$	0	0
$287$	27.6582	1.63261
$288$	0	0
$289$	6.91292	0.406642
$290$	−6.40581	−0.376162
$291$	0	0
$292$	−0.417895	−0.0244554
$293$	11.7506	0.686479	0.343239	−	0.939248i	$-0.388476\pi$
$293$	11.7506	0.686479	0.343239	+	0.939248i	$0.388476\pi$
$294$	0	0
$295$	−15.1564	−0.882442
$296$	3.78017	0.219718
$297$	0	0
$298$	−3.21313	−0.186131
$299$	0	0
$300$	0	0
$301$	−18.3370	−1.05693
$302$	−9.15883	−0.527032
$303$	0	0
$304$	−4.49396	−0.257746
$305$	0.396125	0.0226820
$306$	0	0
$307$	1.09054	0.0622403	0.0311202	−	0.999516i	$-0.490093\pi$
$307$	1.09054	0.0622403	0.0311202	+	0.999516i	$0.490093\pi$
$308$	29.3424	1.67194
$309$	0	0
$310$	−6.89977	−0.391881
$311$	8.09783	0.459186	0.229593	−	0.973287i	$-0.426260\pi$
$311$	8.09783	0.459186	0.229593	+	0.973287i	$0.426260\pi$
$312$	0	0
$313$	19.4252	1.09798	0.548988	−	0.835830i	$-0.315013\pi$
$313$	19.4252	1.09798	0.548988	+	0.835830i	$0.315013\pi$
$314$	17.9323	1.01198
$315$	0	0
$316$	−10.5526	−0.593628
$317$	−24.5719	−1.38010	−0.690049	−	0.723763i	$-0.742411\pi$
$317$	−24.5719	−1.38010	−0.690049	+	0.723763i	$0.742411\pi$
$318$	0	0
$319$	21.5036	1.20397
$320$	1.80194	0.100731
$321$	0	0
$322$	−32.5676	−1.81492
$323$	−21.9758	−1.22277
$324$	0	0
$325$	0	0
$326$	0.591794	0.0327764
$327$	0	0
$328$	−5.70171	−0.314824
$329$	−18.5483	−1.02260
$330$	0	0
$331$	25.9758	1.42776	0.713881	−	0.700267i	$-0.246936\pi$
$331$	25.9758	1.42776	0.713881	+	0.700267i	$0.246936\pi$
$332$	3.42327	0.187876
$333$	0	0
$334$	−14.0000	−0.766046
$335$	−20.6896	−1.13040
$336$	0	0
$337$	1.87263	0.102008	0.0510042	−	0.998698i	$-0.483758\pi$
$337$	1.87263	0.102008	0.0510042	+	0.998698i	$0.483758\pi$
$338$	0	0
$339$	0	0
$340$	8.81163	0.477878
$341$	23.1618	1.25428
$342$	0	0
$343$	46.2325	2.49632
$344$	3.78017	0.203813
$345$	0	0
$346$	0.982542	0.0528218
$347$	−16.6853	−0.895715	−0.447857	−	0.894105i	$-0.647813\pi$
$347$	−16.6853	−0.895715	−0.447857	+	0.894105i	$0.647813\pi$
$348$	0	0
$349$	4.01938	0.215152	0.107576	−	0.994197i	$-0.465691\pi$
$349$	4.01938	0.215152	0.107576	+	0.994197i	$0.465691\pi$
$350$	8.50365	0.454539
$351$	0	0
$352$	−6.04892	−0.322408
$353$	21.6039	1.14986	0.574929	−	0.818203i	$-0.305030\pi$
$353$	21.6039	1.14986	0.574929	+	0.818203i	$0.305030\pi$
$354$	0	0
$355$	−24.3720	−1.29353
$356$	−17.3599	−0.920072
$357$	0	0
$358$	−9.94331	−0.525520
$359$	0.835790	0.0441113	0.0220556	−	0.999757i	$-0.492979\pi$
$359$	0.835790	0.0441113	0.0220556	+	0.999757i	$0.492979\pi$
$360$	0	0
$361$	1.19567	0.0629300
$362$	5.87800	0.308941
$363$	0	0
$364$	0	0
$365$	−0.753020	−0.0394149
$366$	0	0
$367$	−21.1250	−1.10272	−0.551358	−	0.834269i	$-0.685890\pi$
$367$	−21.1250	−1.10272	−0.551358	+	0.834269i	$0.685890\pi$
$368$	6.71379	0.349981
$369$	0	0
$370$	6.81163	0.354120
$371$	−13.5332	−0.702608
$372$	0	0
$373$	21.3840	1.10722	0.553612	−	0.832775i	$-0.313249\pi$
$373$	21.3840	1.10722	0.553612	+	0.832775i	$0.313249\pi$
$374$	−29.5797	−1.52953
$375$	0	0
$376$	3.82371	0.197193
$377$	0	0
$378$	0	0
$379$	13.5496	0.695995	0.347998	−	0.937495i	$-0.386862\pi$
$379$	13.5496	0.695995	0.347998	+	0.937495i	$0.386862\pi$
$380$	−8.09783	−0.415410
$381$	0	0
$382$	−11.0422	−0.564969
$383$	9.10992	0.465495	0.232747	−	0.972537i	$-0.425228\pi$
$383$	9.10992	0.465495	0.232747	+	0.972537i	$0.425228\pi$
$384$	0	0
$385$	52.8732	2.69467
$386$	−20.9638	−1.06703
$387$	0	0
$388$	−1.00969	−0.0512592
$389$	−14.8498	−0.752914	−0.376457	−	0.926434i	$-0.622858\pi$
$389$	−14.8498	−0.752914	−0.376457	+	0.926434i	$0.622858\pi$
$390$	0	0
$391$	32.8310	1.66034
$392$	−16.5308	−0.834931
$393$	0	0
$394$	15.7409	0.793017
$395$	−19.0151	−0.956752
$396$	0	0
$397$	33.7453	1.69363	0.846813	−	0.531891i	$-0.178518\pi$
$397$	33.7453	1.69363	0.846813	+	0.531891i	$0.178518\pi$
$398$	1.51142	0.0757605
$399$	0	0
$400$	−1.75302	−0.0876510
$401$	−19.7017	−0.983856	−0.491928	−	0.870636i	$-0.663708\pi$
$401$	−19.7017	−0.983856	−0.491928	+	0.870636i	$0.663708\pi$
$402$	0	0
$403$	0	0
$404$	5.68664	0.282921
$405$	0	0
$406$	−17.2446	−0.855834
$407$	−22.8659	−1.13342
$408$	0	0
$409$	−35.9366	−1.77695	−0.888475	−	0.458924i	$-0.848235\pi$
$409$	−35.9366	−1.77695	−0.888475	+	0.458924i	$0.848235\pi$
$410$	−10.2741	−0.507403
$411$	0	0
$412$	−6.73556	−0.331837
$413$	−40.8015	−2.00771
$414$	0	0
$415$	6.16852	0.302801
$416$	0	0
$417$	0	0
$418$	27.1836	1.32959
$419$	18.7928	0.918090	0.459045	−	0.888413i	$-0.348192\pi$
$419$	18.7928	0.918090	0.459045	+	0.888413i	$0.348192\pi$
$420$	0	0
$421$	3.90217	0.190180	0.0950900	−	0.995469i	$-0.469686\pi$
$421$	3.90217	0.190180	0.0950900	+	0.995469i	$0.469686\pi$
$422$	−5.65817	−0.275435
$423$	0	0
$424$	2.78986	0.135487
$425$	−8.57242	−0.415823
$426$	0	0
$427$	1.06638	0.0516055
$428$	−0.0586060	−0.00283283
$429$	0	0
$430$	6.81163	0.328486
$431$	−18.8310	−0.907058	−0.453529	−	0.891242i	$-0.649835\pi$
$431$	−18.8310	−0.907058	−0.453529	+	0.891242i	$0.649835\pi$
$432$	0	0
$433$	24.8364	1.19356	0.596780	−	0.802405i	$-0.296446\pi$
$433$	24.8364	1.19356	0.596780	+	0.802405i	$0.296446\pi$
$434$	−18.5743	−0.891597
$435$	0	0
$436$	−15.5254	−0.743533
$437$	−30.1715	−1.44330
$438$	0	0
$439$	22.2784	1.06329	0.531646	−	0.846967i	$-0.321574\pi$
$439$	22.2784	1.06329	0.531646	+	0.846967i	$0.321574\pi$
$440$	−10.8998	−0.519626
$441$	0	0
$442$	0	0
$443$	26.6359	1.26551	0.632756	−	0.774352i	$-0.281924\pi$
$443$	26.6359	1.26551	0.632756	+	0.774352i	$0.281924\pi$
$444$	0	0
$445$	−31.2814	−1.48288
$446$	1.97584	0.0935586
$447$	0	0
$448$	4.85086	0.229181
$449$	21.8538	1.03135	0.515673	−	0.856785i	$-0.327542\pi$
$449$	21.8538	1.03135	0.515673	+	0.856785i	$0.327542\pi$
$450$	0	0
$451$	34.4892	1.62403
$452$	1.95646	0.0920241
$453$	0	0
$454$	25.0640	1.17631
$455$	0	0
$456$	0	0
$457$	−20.7278	−0.969605	−0.484803	−	0.874624i	$-0.661109\pi$
$457$	−20.7278	−0.969605	−0.484803	+	0.874624i	$0.661109\pi$
$458$	24.5133	1.14543
$459$	0	0
$460$	12.0978	0.564064
$461$	−10.7832	−0.502221	−0.251111	−	0.967958i	$-0.580796\pi$
$461$	−10.7832	−0.502221	−0.251111	+	0.967958i	$0.580796\pi$
$462$	0	0
$463$	−23.1594	−1.07631	−0.538155	−	0.842846i	$-0.680878\pi$
$463$	−23.1594	−1.07631	−0.538155	+	0.842846i	$0.680878\pi$
$464$	3.55496	0.165035
$465$	0	0
$466$	8.37196	0.387824
$467$	13.3515	0.617835	0.308917	−	0.951089i	$-0.400033\pi$
$467$	13.3515	0.617835	0.308917	+	0.951089i	$0.400033\pi$
$468$	0	0
$469$	−55.6969	−2.57185
$470$	6.89008	0.317816
$471$	0	0
$472$	8.41119	0.387156
$473$	−22.8659	−1.05138
$474$	0	0
$475$	7.87800	0.361468
$476$	23.7211	1.08725
$477$	0	0
$478$	−14.0194	−0.641231
$479$	−4.38537	−0.200373	−0.100186	−	0.994969i	$-0.531944\pi$
$479$	−4.38537	−0.200373	−0.100186	+	0.994969i	$0.531944\pi$
$480$	0	0
$481$	0	0
$482$	21.1183	0.961911
$483$	0	0
$484$	25.5894	1.16315
$485$	−1.81940	−0.0826145
$486$	0	0
$487$	7.33214	0.332251	0.166126	−	0.986105i	$-0.446874\pi$
$487$	7.33214	0.332251	0.166126	+	0.986105i	$0.446874\pi$
$488$	−0.219833	−0.00995135
$489$	0	0
$490$	−29.7875	−1.34566
$491$	−16.8062	−0.758455	−0.379228	−	0.925303i	$-0.623810\pi$
$491$	−16.8062	−0.758455	−0.379228	+	0.925303i	$0.623810\pi$
$492$	0	0
$493$	17.3840	0.782938
$494$	0	0
$495$	0	0
$496$	3.82908	0.171931
$497$	−65.6098	−2.94300
$498$	0	0
$499$	−6.59658	−0.295303	−0.147652	−	0.989039i	$-0.547171\pi$
$499$	−6.59658	−0.295303	−0.147652	+	0.989039i	$0.547171\pi$
$500$	−12.1685	−0.544193
$501$	0	0
$502$	−26.1129	−1.16548
$503$	−27.8297	−1.24086	−0.620432	−	0.784260i	$-0.713043\pi$
$503$	−27.8297	−1.24086	−0.620432	+	0.784260i	$0.713043\pi$
$504$	0	0
$505$	10.2470	0.455985
$506$	−40.6112	−1.80539
$507$	0	0
$508$	−8.11960	−0.360249
$509$	4.13275	0.183181	0.0915905	−	0.995797i	$-0.470805\pi$
$509$	4.13275	0.183181	0.0915905	+	0.995797i	$0.470805\pi$
$510$	0	0
$511$	−2.02715	−0.0896757
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−16.5918	−0.731833
$515$	−12.1371	−0.534823
$516$	0	0
$517$	−23.1293	−1.01723
$518$	18.3370	0.805683
$519$	0	0
$520$	0	0
$521$	11.7995	0.516947	0.258474	−	0.966018i	$-0.416780\pi$
$521$	11.7995	0.516947	0.258474	+	0.966018i	$0.416780\pi$
$522$	0	0
$523$	38.2887	1.67425	0.837124	−	0.547013i	$-0.184235\pi$
$523$	38.2887	1.67425	0.837124	+	0.547013i	$0.184235\pi$
$524$	−12.6679	−0.553398
$525$	0	0
$526$	−9.64742	−0.420647
$527$	18.7245	0.815654
$528$	0	0
$529$	22.0750	0.959783
$530$	5.02715	0.218365
$531$	0	0
$532$	−21.7995	−0.945130
$533$	0	0
$534$	0	0
$535$	−0.105604	−0.00456568
$536$	11.4819	0.495942
$537$	0	0
$538$	−10.1981	−0.439670
$539$	99.9934	4.30702
$540$	0	0
$541$	17.4142	0.748694	0.374347	−	0.927289i	$-0.377867\pi$
$541$	17.4142	0.748694	0.374347	+	0.927289i	$0.377867\pi$
$542$	−27.6799	−1.18896
$543$	0	0
$544$	−4.89008	−0.209661
$545$	−27.9758	−1.19835
$546$	0	0
$547$	16.9444	0.724489	0.362245	−	0.932083i	$-0.382010\pi$
$547$	16.9444	0.724489	0.362245	+	0.932083i	$0.382010\pi$
$548$	−15.3599	−0.656142
$549$	0	0
$550$	10.6039	0.452151
$551$	−15.9758	−0.680594
$552$	0	0
$553$	−51.1890	−2.17678
$554$	−6.32842	−0.268869
$555$	0	0
$556$	−11.7995	−0.500412
$557$	11.9758	0.507432	0.253716	−	0.967279i	$-0.418347\pi$
$557$	11.9758	0.507432	0.253716	+	0.967279i	$0.418347\pi$
$558$	0	0
$559$	0	0
$560$	8.74094	0.369372
$561$	0	0
$562$	−9.26205	−0.390696
$563$	42.2911	1.78236	0.891179	−	0.453652i	$-0.149879\pi$
$563$	42.2911	1.78236	0.891179	+	0.453652i	$0.149879\pi$
$564$	0	0
$565$	3.52542	0.148315
$566$	−0.561663	−0.0236085
$567$	0	0
$568$	13.5254	0.567514
$569$	8.26934	0.346669	0.173334	−	0.984863i	$-0.444546\pi$
$569$	8.26934	0.346669	0.173334	+	0.984863i	$0.444546\pi$
$570$	0	0
$571$	−16.8552	−0.705367	−0.352683	−	0.935743i	$-0.614731\pi$
$571$	−16.8552	−0.705367	−0.352683	+	0.935743i	$0.614731\pi$
$572$	0	0
$573$	0	0
$574$	−27.6582	−1.15443
$575$	−11.7694	−0.490818
$576$	0	0
$577$	−37.9995	−1.58194	−0.790970	−	0.611854i	$-0.790424\pi$
$577$	−37.9995	−1.58194	−0.790970	+	0.611854i	$0.790424\pi$
$578$	−6.91292	−0.287540
$579$	0	0
$580$	6.40581	0.265987
$581$	16.6058	0.688924
$582$	0	0
$583$	−16.8756	−0.698916
$584$	0.417895	0.0172926
$585$	0	0
$586$	−11.7506	−0.485414
$587$	−31.4282	−1.29718	−0.648590	−	0.761138i	$-0.724641\pi$
$587$	−31.4282	−1.29718	−0.648590	+	0.761138i	$0.724641\pi$
$588$	0	0
$589$	−17.2078	−0.709033
$590$	15.1564	0.623981
$591$	0	0
$592$	−3.78017	−0.155364
$593$	41.8866	1.72008	0.860039	−	0.510229i	$-0.170439\pi$
$593$	41.8866	1.72008	0.860039	+	0.510229i	$0.170439\pi$
$594$	0	0
$595$	42.7439	1.75233
$596$	3.21313	0.131615
$597$	0	0
$598$	0	0
$599$	−36.0194	−1.47171	−0.735856	−	0.677138i	$-0.763220\pi$
$599$	−36.0194	−1.47171	−0.735856	+	0.677138i	$0.763220\pi$
$600$	0	0
$601$	5.13946	0.209643	0.104821	−	0.994491i	$-0.466573\pi$
$601$	5.13946	0.209643	0.104821	+	0.994491i	$0.466573\pi$
$602$	18.3370	0.747362
$603$	0	0
$604$	9.15883	0.372668
$605$	46.1105	1.87466
$606$	0	0
$607$	−17.6517	−0.716462	−0.358231	−	0.933633i	$-0.616620\pi$
$607$	−17.6517	−0.716462	−0.358231	+	0.933633i	$0.616620\pi$
$608$	4.49396	0.182254
$609$	0	0
$610$	−0.396125	−0.0160386
$611$	0	0
$612$	0	0
$613$	−45.8974	−1.85378	−0.926889	−	0.375336i	$-0.877527\pi$
$613$	−45.8974	−1.85378	−0.926889	+	0.375336i	$0.877527\pi$
$614$	−1.09054	−0.0440106
$615$	0	0
$616$	−29.3424	−1.18224
$617$	5.12929	0.206498	0.103249	−	0.994656i	$-0.467076\pi$
$617$	5.12929	0.206498	0.103249	+	0.994656i	$0.467076\pi$
$618$	0	0
$619$	−43.5448	−1.75021	−0.875107	−	0.483930i	$-0.839209\pi$
$619$	−43.5448	−1.75021	−0.875107	+	0.483930i	$0.839209\pi$
$620$	6.89977	0.277102
$621$	0	0
$622$	−8.09783	−0.324694
$623$	−84.2103	−3.37381
$624$	0	0
$625$	−13.1618	−0.526473
$626$	−19.4252	−0.776387
$627$	0	0
$628$	−17.9323	−0.715577
$629$	−18.4853	−0.737059
$630$	0	0
$631$	30.4655	1.21281	0.606406	−	0.795155i	$-0.292611\pi$
$631$	30.4655	1.21281	0.606406	+	0.795155i	$0.292611\pi$
$632$	10.5526	0.419759
$633$	0	0
$634$	24.5719	0.975877
$635$	−14.6310	−0.580614
$636$	0	0
$637$	0	0
$638$	−21.5036	−0.851338
$639$	0	0
$640$	−1.80194	−0.0712278
$641$	−32.3370	−1.27724	−0.638618	−	0.769524i	$-0.720494\pi$
$641$	−32.3370	−1.27724	−0.638618	+	0.769524i	$0.720494\pi$
$642$	0	0
$643$	1.79092	0.0706270	0.0353135	−	0.999376i	$-0.488757\pi$
$643$	1.79092	0.0706270	0.0353135	+	0.999376i	$0.488757\pi$
$644$	32.5676	1.28334
$645$	0	0
$646$	21.9758	0.864628
$647$	41.4685	1.63029	0.815147	−	0.579254i	$-0.196656\pi$
$647$	41.4685	1.63029	0.815147	+	0.579254i	$0.196656\pi$
$648$	0	0
$649$	−50.8786	−1.99716
$650$	0	0
$651$	0	0
$652$	−0.591794	−0.0231764
$653$	−2.39181	−0.0935989	−0.0467994	−	0.998904i	$-0.514902\pi$
$653$	−2.39181	−0.0935989	−0.0467994	+	0.998904i	$0.514902\pi$
$654$	0	0
$655$	−22.8267	−0.891913
$656$	5.70171	0.222614
$657$	0	0
$658$	18.5483	0.723086
$659$	1.76032	0.0685722	0.0342861	−	0.999412i	$-0.489084\pi$
$659$	1.76032	0.0685722	0.0342861	+	0.999412i	$0.489084\pi$
$660$	0	0
$661$	−44.7138	−1.73916	−0.869582	−	0.493788i	$-0.835612\pi$
$661$	−44.7138	−1.73916	−0.869582	+	0.493788i	$0.835612\pi$
$662$	−25.9758	−1.00958
$663$	0	0
$664$	−3.42327	−0.132849
$665$	−39.2814	−1.52327
$666$	0	0
$667$	23.8672	0.924144
$668$	14.0000	0.541676
$669$	0	0
$670$	20.6896	0.799310
$671$	1.32975	0.0513344
$672$	0	0
$673$	−5.84356	−0.225253	−0.112626	−	0.993637i	$-0.535926\pi$
$673$	−5.84356	−0.225253	−0.112626	+	0.993637i	$0.535926\pi$
$674$	−1.87263	−0.0721308
$675$	0	0
$676$	0	0
$677$	27.8883	1.07183	0.535917	−	0.844271i	$-0.319966\pi$
$677$	27.8883	1.07183	0.535917	+	0.844271i	$0.319966\pi$
$678$	0	0
$679$	−4.89785	−0.187962
$680$	−8.81163	−0.337910
$681$	0	0
$682$	−23.1618	−0.886912
$683$	28.3803	1.08594	0.542971	−	0.839751i	$-0.317299\pi$
$683$	28.3803	1.08594	0.542971	+	0.839751i	$0.317299\pi$
$684$	0	0
$685$	−27.6775	−1.05750
$686$	−46.2325	−1.76517
$687$	0	0
$688$	−3.78017	−0.144118
$689$	0	0
$690$	0	0
$691$	31.3142	1.19125	0.595624	−	0.803263i	$-0.296905\pi$
$691$	31.3142	1.19125	0.595624	+	0.803263i	$0.296905\pi$
$692$	−0.982542	−0.0373506
$693$	0	0
$694$	16.6853	0.633366
$695$	−21.2620	−0.806515
$696$	0	0
$697$	27.8818	1.05610
$698$	−4.01938	−0.152136
$699$	0	0
$700$	−8.50365	−0.321408
$701$	−22.6568	−0.855737	−0.427869	−	0.903841i	$-0.640735\pi$
$701$	−22.6568	−0.855737	−0.427869	+	0.903841i	$0.640735\pi$
$702$	0	0
$703$	16.9879	0.640711
$704$	6.04892	0.227977
$705$	0	0
$706$	−21.6039	−0.813073
$707$	27.5851	1.03744
$708$	0	0
$709$	−24.4155	−0.916943	−0.458472	−	0.888709i	$-0.651603\pi$
$709$	−24.4155	−0.916943	−0.458472	+	0.888709i	$0.651603\pi$
$710$	24.3720	0.914663
$711$	0	0
$712$	17.3599	0.650589
$713$	25.7077	0.962760
$714$	0	0
$715$	0	0
$716$	9.94331	0.371599
$717$	0	0
$718$	−0.835790	−0.0311914
$719$	13.2707	0.494912	0.247456	−	0.968899i	$-0.420405\pi$
$719$	13.2707	0.494912	0.247456	+	0.968899i	$0.420405\pi$
$720$	0	0
$721$	−32.6732	−1.21681
$722$	−1.19567	−0.0444982
$723$	0	0
$724$	−5.87800	−0.218454
$725$	−6.23191	−0.231447
$726$	0	0
$727$	25.7904	0.956515	0.478257	−	0.878220i	$-0.341269\pi$
$727$	25.7904	0.956515	0.478257	+	0.878220i	$0.341269\pi$
$728$	0	0
$729$	0	0
$730$	0.753020	0.0278705
$731$	−18.4853	−0.683705
$732$	0	0
$733$	13.1400	0.485339	0.242669	−	0.970109i	$-0.421977\pi$
$733$	13.1400	0.485339	0.242669	+	0.970109i	$0.421977\pi$
$734$	21.1250	0.779737
$735$	0	0
$736$	−6.71379	−0.247474
$737$	−69.4529	−2.55833
$738$	0	0
$739$	11.8130	0.434547	0.217273	−	0.976111i	$-0.430284\pi$
$739$	11.8130	0.434547	0.217273	+	0.976111i	$0.430284\pi$
$740$	−6.81163	−0.250400
$741$	0	0
$742$	13.5332	0.496819
$743$	−3.50125	−0.128449	−0.0642243	−	0.997935i	$-0.520457\pi$
$743$	−3.50125	−0.128449	−0.0642243	+	0.997935i	$0.520457\pi$
$744$	0	0
$745$	5.78986	0.212124
$746$	−21.3840	−0.782925
$747$	0	0
$748$	29.5797	1.08154
$749$	−0.284289	−0.0103877
$750$	0	0
$751$	−11.6722	−0.425924	−0.212962	−	0.977061i	$-0.568311\pi$
$751$	−11.6722	−0.425924	−0.212962	+	0.977061i	$0.568311\pi$
$752$	−3.82371	−0.139436
$753$	0	0
$754$	0	0
$755$	16.5036	0.600629
$756$	0	0
$757$	0.444451	0.0161538	0.00807692	−	0.999967i	$-0.497429\pi$
$757$	0.444451	0.0161538	0.00807692	+	0.999967i	$0.497429\pi$
$758$	−13.5496	−0.492143
$759$	0	0
$760$	8.09783	0.293739
$761$	−29.4905	−1.06903	−0.534515	−	0.845159i	$-0.679506\pi$
$761$	−29.4905	−1.06903	−0.534515	+	0.845159i	$0.679506\pi$
$762$	0	0
$763$	−75.3116	−2.72646
$764$	11.0422	0.399493
$765$	0	0
$766$	−9.10992	−0.329155
$767$	0	0
$768$	0	0
$769$	−13.4517	−0.485082	−0.242541	−	0.970141i	$-0.577981\pi$
$769$	−13.4517	−0.485082	−0.242541	+	0.970141i	$0.577981\pi$
$770$	−52.8732	−1.90542
$771$	0	0
$772$	20.9638	0.754502
$773$	24.1021	0.866894	0.433447	−	0.901179i	$-0.357297\pi$
$773$	24.1021	0.866894	0.433447	+	0.901179i	$0.357297\pi$
$774$	0	0
$775$	−6.71246	−0.241119
$776$	1.00969	0.0362457
$777$	0	0
$778$	14.8498	0.532391
$779$	−25.6233	−0.918048
$780$	0	0
$781$	−81.8141	−2.92754
$782$	−32.8310	−1.17403
$783$	0	0
$784$	16.5308	0.590386
$785$	−32.3129	−1.15330
$786$	0	0
$787$	16.2258	0.578387	0.289194	−	0.957271i	$-0.406613\pi$
$787$	16.2258	0.578387	0.289194	+	0.957271i	$0.406613\pi$
$788$	−15.7409	−0.560748
$789$	0	0
$790$	19.0151	0.676526
$791$	9.49050	0.337443
$792$	0	0
$793$	0	0
$794$	−33.7453	−1.19757
$795$	0	0
$796$	−1.51142	−0.0535708
$797$	37.2760	1.32039	0.660193	−	0.751096i	$-0.270475\pi$
$797$	37.2760	1.32039	0.660193	+	0.751096i	$0.270475\pi$
$798$	0	0
$799$	−18.6983	−0.661497
$800$	1.75302	0.0619786
$801$	0	0
$802$	19.7017	0.695692
$803$	−2.52781	−0.0892045
$804$	0	0
$805$	58.6848	2.06837
$806$	0	0
$807$	0	0
$808$	−5.68664	−0.200055
$809$	24.0844	0.846763	0.423382	−	0.905951i	$-0.360843\pi$
$809$	24.0844	0.846763	0.423382	+	0.905951i	$0.360843\pi$
$810$	0	0
$811$	9.87800	0.346864	0.173432	−	0.984846i	$-0.444514\pi$
$811$	9.87800	0.346864	0.173432	+	0.984846i	$0.444514\pi$
$812$	17.2446	0.605166
$813$	0	0
$814$	22.8659	0.801450
$815$	−1.06638	−0.0373535
$816$	0	0
$817$	16.9879	0.594332
$818$	35.9366	1.25649
$819$	0	0
$820$	10.2741	0.358788
$821$	31.2731	1.09144	0.545719	−	0.837968i	$-0.316257\pi$
$821$	31.2731	1.09144	0.545719	+	0.837968i	$0.316257\pi$
$822$	0	0
$823$	33.8388	1.17955	0.589773	−	0.807569i	$-0.299217\pi$
$823$	33.8388	1.17955	0.589773	+	0.807569i	$0.299217\pi$
$824$	6.73556	0.234644
$825$	0	0
$826$	40.8015	1.41966
$827$	17.4028	0.605156	0.302578	−	0.953125i	$-0.402153\pi$
$827$	17.4028	0.605156	0.302578	+	0.953125i	$0.402153\pi$
$828$	0	0
$829$	−50.9724	−1.77034	−0.885172	−	0.465264i	$-0.845959\pi$
$829$	−50.9724	−1.77034	−0.885172	+	0.465264i	$0.845959\pi$
$830$	−6.16852	−0.214113
$831$	0	0
$832$	0	0
$833$	80.8370	2.80084
$834$	0	0
$835$	25.2271	0.873021
$836$	−27.1836	−0.940164
$837$	0	0
$838$	−18.7928	−0.649188
$839$	34.6983	1.19792	0.598958	−	0.800780i	$-0.295581\pi$
$839$	34.6983	1.19792	0.598958	+	0.800780i	$0.295581\pi$
$840$	0	0
$841$	−16.3623	−0.564216
$842$	−3.90217	−0.134477
$843$	0	0
$844$	5.65817	0.194762
$845$	0	0
$846$	0	0
$847$	124.130	4.26517
$848$	−2.78986	−0.0958041
$849$	0	0
$850$	8.57242	0.294031
$851$	−25.3793	−0.869990
$852$	0	0
$853$	−8.37675	−0.286814	−0.143407	−	0.989664i	$-0.545806\pi$
$853$	−8.37675	−0.286814	−0.143407	+	0.989664i	$0.545806\pi$
$854$	−1.06638	−0.0364906
$855$	0	0
$856$	0.0586060	0.00200311
$857$	−17.7888	−0.607654	−0.303827	−	0.952727i	$-0.598264\pi$
$857$	−17.7888	−0.607654	−0.303827	+	0.952727i	$0.598264\pi$
$858$	0	0
$859$	50.1473	1.71101	0.855503	−	0.517798i	$-0.173248\pi$
$859$	50.1473	1.71101	0.855503	+	0.517798i	$0.173248\pi$
$860$	−6.81163	−0.232275
$861$	0	0
$862$	18.8310	0.641387
$863$	−50.1172	−1.70601	−0.853005	−	0.521903i	$-0.825222\pi$
$863$	−50.1172	−1.70601	−0.853005	+	0.521903i	$0.825222\pi$
$864$	0	0
$865$	−1.77048	−0.0601981
$866$	−24.8364	−0.843975
$867$	0	0
$868$	18.5743	0.630454
$869$	−63.8316	−2.16534
$870$	0	0
$871$	0	0
$872$	15.5254	0.525757
$873$	0	0
$874$	30.1715	1.02057
$875$	−59.0277	−1.99550
$876$	0	0
$877$	−38.3827	−1.29609	−0.648046	−	0.761601i	$-0.724414\pi$
$877$	−38.3827	−1.29609	−0.648046	+	0.761601i	$0.724414\pi$
$878$	−22.2784	−0.751861
$879$	0	0
$880$	10.8998	0.367431
$881$	−31.8383	−1.07266	−0.536330	−	0.844009i	$-0.680190\pi$
$881$	−31.8383	−1.07266	−0.536330	+	0.844009i	$0.680190\pi$
$882$	0	0
$883$	4.74871	0.159807	0.0799034	−	0.996803i	$-0.474539\pi$
$883$	4.74871	0.159807	0.0799034	+	0.996803i	$0.474539\pi$
$884$	0	0
$885$	0	0
$886$	−26.6359	−0.894851
$887$	−22.9288	−0.769875	−0.384938	−	0.922943i	$-0.625777\pi$
$887$	−22.9288	−0.769875	−0.384938	+	0.922943i	$0.625777\pi$
$888$	0	0
$889$	−39.3870	−1.32100
$890$	31.2814	1.04856
$891$	0	0
$892$	−1.97584	−0.0661559
$893$	17.1836	0.575027
$894$	0	0
$895$	17.9172	0.598907
$896$	−4.85086	−0.162056
$897$	0	0
$898$	−21.8538	−0.729272
$899$	13.6122	0.453993
$900$	0	0
$901$	−13.6426	−0.454502
$902$	−34.4892	−1.14836
$903$	0	0
$904$	−1.95646	−0.0650709
$905$	−10.5918	−0.352083
$906$	0	0
$907$	43.0267	1.42868	0.714339	−	0.699800i	$-0.246728\pi$
$907$	43.0267	1.42868	0.714339	+	0.699800i	$0.246728\pi$
$908$	−25.0640	−0.831777
$909$	0	0
$910$	0	0
$911$	−6.65087	−0.220353	−0.110177	−	0.993912i	$-0.535142\pi$
$911$	−6.65087	−0.220353	−0.110177	+	0.993912i	$0.535142\pi$
$912$	0	0
$913$	20.7071	0.685305
$914$	20.7278	0.685614
$915$	0	0
$916$	−24.5133	−0.809943
$917$	−61.4499	−2.02926
$918$	0	0
$919$	−9.11231	−0.300587	−0.150294	−	0.988641i	$-0.548022\pi$
$919$	−9.11231	−0.300587	−0.150294	+	0.988641i	$0.548022\pi$
$920$	−12.0978	−0.398854
$921$	0	0
$922$	10.7832	0.355124
$923$	0	0
$924$	0	0
$925$	6.62671	0.217885
$926$	23.1594	0.761066
$927$	0	0
$928$	−3.55496	−0.116697
$929$	−11.8672	−0.389352	−0.194676	−	0.980868i	$-0.562366\pi$
$929$	−11.8672	−0.389352	−0.194676	+	0.980868i	$0.562366\pi$
$930$	0	0
$931$	−74.2887	−2.43471
$932$	−8.37196	−0.274233
$933$	0	0
$934$	−13.3515	−0.436875
$935$	53.3008	1.74312
$936$	0	0
$937$	19.1153	0.624469	0.312235	−	0.950005i	$-0.398922\pi$
$937$	19.1153	0.624469	0.312235	+	0.950005i	$0.398922\pi$
$938$	55.6969	1.81857
$939$	0	0
$940$	−6.89008	−0.224730
$941$	0.445042	0.0145080	0.00725398	−	0.999974i	$-0.497691\pi$
$941$	0.445042	0.0145080	0.00725398	+	0.999974i	$0.497691\pi$
$942$	0	0
$943$	38.2801	1.24657
$944$	−8.41119	−0.273761
$945$	0	0
$946$	22.8659	0.743435
$947$	−8.99894	−0.292426	−0.146213	−	0.989253i	$-0.546709\pi$
$947$	−8.99894	−0.292426	−0.146213	+	0.989253i	$0.546709\pi$
$948$	0	0
$949$	0	0
$950$	−7.87800	−0.255596
$951$	0	0
$952$	−23.7211	−0.768805
$953$	37.2573	1.20688	0.603441	−	0.797408i	$-0.293796\pi$
$953$	37.2573	1.20688	0.603441	+	0.797408i	$0.293796\pi$
$954$	0	0
$955$	19.8974	0.643864
$956$	14.0194	0.453419
$957$	0	0
$958$	4.38537	0.141685
$959$	−74.5086	−2.40601
$960$	0	0
$961$	−16.3381	−0.527036
$962$	0	0
$963$	0	0
$964$	−21.1183	−0.680174
$965$	37.7754	1.21603
$966$	0	0
$967$	37.2567	1.19809	0.599047	−	0.800714i	$-0.295546\pi$
$967$	37.2567	1.19809	0.599047	+	0.800714i	$0.295546\pi$
$968$	−25.5894	−0.822474
$969$	0	0
$970$	1.81940	0.0584173
$971$	−54.0393	−1.73421	−0.867103	−	0.498130i	$-0.834020\pi$
$971$	−54.0393	−1.73421	−0.867103	+	0.498130i	$0.834020\pi$
$972$	0	0
$973$	−57.2379	−1.83496
$974$	−7.33214	−0.234937
$975$	0	0
$976$	0.219833	0.00703667
$977$	−11.4470	−0.366221	−0.183110	−	0.983092i	$-0.558617\pi$
$977$	−11.4470	−0.366221	−0.183110	+	0.983092i	$0.558617\pi$
$978$	0	0
$979$	−105.008	−3.35609
$980$	29.7875	0.951526
$981$	0	0
$982$	16.8062	0.536309
$983$	−0.111244	−0.00354814	−0.00177407	−	0.999998i	$-0.500565\pi$
$983$	−0.111244	−0.00354814	−0.00177407	+	0.999998i	$0.500565\pi$
$984$	0	0
$985$	−28.3642	−0.903758
$986$	−17.3840	−0.553621
$987$	0	0
$988$	0	0
$989$	−25.3793	−0.807013
$990$	0	0
$991$	11.3709	0.361208	0.180604	−	0.983556i	$-0.442195\pi$
$991$	11.3709	0.361208	0.180604	+	0.983556i	$0.442195\pi$
$992$	−3.82908	−0.121574
$993$	0	0
$994$	65.6098	2.08102
$995$	−2.72348	−0.0863401
$996$	0	0
$997$	6.45042	0.204287	0.102143	−	0.994770i	$-0.467430\pi$
$997$	6.45042	0.204287	0.102143	+	0.994770i	$0.467430\pi$
$998$	6.59658	0.208811
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.a.bc.1.3	yes	3
3.2	odd	2		3042.2.a.bg.1.1	yes	3
13.5	odd	4		3042.2.b.p.1351.4		6
13.8	odd	4		3042.2.b.p.1351.3		6
13.12	even	2		3042.2.a.bf.1.1	yes	3
39.5	even	4		3042.2.b.q.1351.3		6
39.8	even	4		3042.2.b.q.1351.4		6
39.38	odd	2		3042.2.a.bb.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3042.2.a.bb.1.3	✓	3	39.38	odd	2
3042.2.a.bc.1.3	yes	3	1.1	even	1	trivial
3042.2.a.bf.1.1	yes	3	13.12	even	2
3042.2.a.bg.1.1	yes	3	3.2	odd	2
3042.2.b.p.1351.3		6	13.8	odd	4
3042.2.b.p.1351.4		6	13.5	odd	4
3042.2.b.q.1351.3		6	39.5	even	4
3042.2.b.q.1351.4		6	39.8	even	4

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

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Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3042.2.a.bc.1.3

Level	$3042$
Weight	$2$
Character	3042.1
Self dual	yes
Analytic conductor	$24.290$
Analytic rank	$0$
Dimension	$3$
CM	no
Inner twists	$1$