LMFDB - Embedded newform 3042.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1,3042)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$24.2904922949$
Analytic rank:	$1$
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:	no (minimal twist has level 1014)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -4.04892 q^{5} +0.692021 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -4.04892 q^{5} +0.692021 q^{7} +1.00000 q^{8} -4.04892 q^{10} +4.85086 q^{11} +0.692021 q^{14} +1.00000 q^{16} -7.38404 q^{17} -1.78017 q^{19} -4.04892 q^{20} +4.85086 q^{22} -5.10992 q^{23} +11.3937 q^{25} +0.692021 q^{28} +3.34481 q^{29} +0.972853 q^{31} +1.00000 q^{32} -7.38404 q^{34} -2.80194 q^{35} +1.28621 q^{37} -1.78017 q^{38} -4.04892 q^{40} -1.50604 q^{41} -8.31767 q^{43} +4.85086 q^{44} -5.10992 q^{46} +7.20775 q^{47} -6.52111 q^{49} +11.3937 q^{50} -13.4765 q^{53} -19.6407 q^{55} +0.692021 q^{56} +3.34481 q^{58} -1.30798 q^{59} -0.396125 q^{61} +0.972853 q^{62} +1.00000 q^{64} +6.05429 q^{67} -7.38404 q^{68} -2.80194 q^{70} +1.32975 q^{71} -7.65279 q^{73} +1.28621 q^{74} -1.78017 q^{76} +3.35690 q^{77} -8.33944 q^{79} -4.04892 q^{80} -1.50604 q^{82} -15.3274 q^{83} +29.8974 q^{85} -8.31767 q^{86} +4.85086 q^{88} -3.10992 q^{89} -5.10992 q^{92} +7.20775 q^{94} +7.20775 q^{95} -8.54288 q^{97} -6.52111 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} + 3 q^{8} - 3 q^{10} + q^{11} - 3 q^{14} + 3 q^{16} - 12 q^{17} - 4 q^{19} - 3 q^{20} + q^{22} - 16 q^{23} + 2 q^{25} - 3 q^{28} - 13 q^{29} + 9 q^{31} + 3 q^{32} - 12 q^{34} - 4 q^{35} + 12 q^{37} - 4 q^{38} - 3 q^{40} - 14 q^{41} - 8 q^{43} + q^{44} - 16 q^{46} + 4 q^{47} - 4 q^{49} + 2 q^{50} - 15 q^{53} - 22 q^{55} - 3 q^{56} - 13 q^{58} - 9 q^{59} - 10 q^{61} + 9 q^{62} + 3 q^{64} + 6 q^{67} - 12 q^{68} - 4 q^{70} + 6 q^{71} - 5 q^{73} + 12 q^{74} - 4 q^{76} + 6 q^{77} - 5 q^{79} - 3 q^{80} - 14 q^{82} - 7 q^{83} + 26 q^{85} - 8 q^{86} + q^{88} - 10 q^{89} - 16 q^{92} + 4 q^{94} + 4 q^{95} - 7 q^{97} - 4 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 + 3 * q^8 - 3 * q^10 + q^11 - 3 * q^14 + 3 * q^16 - 12 * q^17 - 4 * q^19 - 3 * q^20 + q^22 - 16 * q^23 + 2 * q^25 - 3 * q^28 - 13 * q^29 + 9 * q^31 + 3 * q^32 - 12 * q^34 - 4 * q^35 + 12 * q^37 - 4 * q^38 - 3 * q^40 - 14 * q^41 - 8 * q^43 + q^44 - 16 * q^46 + 4 * q^47 - 4 * q^49 + 2 * q^50 - 15 * q^53 - 22 * q^55 - 3 * q^56 - 13 * q^58 - 9 * q^59 - 10 * q^61 + 9 * q^62 + 3 * q^64 + 6 * q^67 - 12 * q^68 - 4 * q^70 + 6 * q^71 - 5 * q^73 + 12 * q^74 - 4 * q^76 + 6 * q^77 - 5 * q^79 - 3 * q^80 - 14 * q^82 - 7 * q^83 + 26 * q^85 - 8 * q^86 + q^88 - 10 * q^89 - 16 * q^92 + 4 * q^94 + 4 * q^95 - 7 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−4.04892	−1.81073	−0.905365	−	0.424633i	$-0.860403\pi$
$5$	−4.04892	−1.81073	−0.905365	+	0.424633i	$0.860403\pi$
$6$	0	0
$7$	0.692021	0.261560	0.130780	−	0.991411i	$-0.458252\pi$
$7$	0.692021	0.261560	0.130780	+	0.991411i	$0.458252\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−4.04892	−1.28038
$11$	4.85086	1.46259	0.731294	−	0.682062i	$-0.238917\pi$
$11$	4.85086	1.46259	0.731294	+	0.682062i	$0.238917\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0.692021	0.184951
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.38404	−1.79089	−0.895447	−	0.445169i	$-0.853144\pi$
$17$	−7.38404	−1.79089	−0.895447	+	0.445169i	$0.853144\pi$
$18$	0	0
$19$	−1.78017	−0.408398	−0.204199	−	0.978929i	$-0.565459\pi$
$19$	−1.78017	−0.408398	−0.204199	+	0.978929i	$0.565459\pi$
$20$	−4.04892	−0.905365
$21$	0	0
$22$	4.85086	1.03421
$23$	−5.10992	−1.06549	−0.532746	−	0.846275i	$-0.678840\pi$
$23$	−5.10992	−1.06549	−0.532746	+	0.846275i	$0.678840\pi$
$24$	0	0
$25$	11.3937	2.27875
$26$	0	0
$27$	0	0
$28$	0.692021	0.130780
$29$	3.34481	0.621116	0.310558	−	0.950554i	$-0.399484\pi$
$29$	3.34481	0.621116	0.310558	+	0.950554i	$0.399484\pi$
$30$	0	0
$31$	0.972853	0.174730	0.0873648	−	0.996176i	$-0.472155\pi$
$31$	0.972853	0.174730	0.0873648	+	0.996176i	$0.472155\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−7.38404	−1.26635
$35$	−2.80194	−0.473614
$36$	0	0
$37$	1.28621	0.211451	0.105726	−	0.994395i	$-0.466283\pi$
$37$	1.28621	0.211451	0.105726	+	0.994395i	$0.466283\pi$
$38$	−1.78017	−0.288781
$39$	0	0
$40$	−4.04892	−0.640190
$41$	−1.50604	−0.235204	−0.117602	−	0.993061i	$-0.537521\pi$
$41$	−1.50604	−0.235204	−0.117602	+	0.993061i	$0.537521\pi$
$42$	0	0
$43$	−8.31767	−1.26843	−0.634216	−	0.773156i	$-0.718677\pi$
$43$	−8.31767	−1.26843	−0.634216	+	0.773156i	$0.718677\pi$
$44$	4.85086	0.731294
$45$	0	0
$46$	−5.10992	−0.753416
$47$	7.20775	1.05136	0.525679	−	0.850683i	$-0.323811\pi$
$47$	7.20775	1.05136	0.525679	+	0.850683i	$0.323811\pi$
$48$	0	0
$49$	−6.52111	−0.931587
$50$	11.3937	1.61132
$51$	0	0
$52$	0	0
$53$	−13.4765	−1.85114	−0.925570	−	0.378577i	$-0.876414\pi$
$53$	−13.4765	−1.85114	−0.925570	+	0.378577i	$0.876414\pi$
$54$	0	0
$55$	−19.6407	−2.64835
$56$	0.692021	0.0924753
$57$	0	0
$58$	3.34481	0.439196
$59$	−1.30798	−0.170284	−0.0851422	−	0.996369i	$-0.527134\pi$
$59$	−1.30798	−0.170284	−0.0851422	+	0.996369i	$0.527134\pi$
$60$	0	0
$61$	−0.396125	−0.0507185	−0.0253593	−	0.999678i	$-0.508073\pi$
$61$	−0.396125	−0.0507185	−0.0253593	+	0.999678i	$0.508073\pi$
$62$	0.972853	0.123552
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	6.05429	0.739650	0.369825	−	0.929101i	$-0.379418\pi$
$67$	6.05429	0.739650	0.369825	+	0.929101i	$0.379418\pi$
$68$	−7.38404	−0.895447
$69$	0	0
$70$	−2.80194	−0.334896
$71$	1.32975	0.157812	0.0789061	−	0.996882i	$-0.474857\pi$
$71$	1.32975	0.157812	0.0789061	+	0.996882i	$0.474857\pi$
$72$	0	0
$73$	−7.65279	−0.895692	−0.447846	−	0.894111i	$-0.647809\pi$
$73$	−7.65279	−0.895692	−0.447846	+	0.894111i	$0.647809\pi$
$74$	1.28621	0.149519
$75$	0	0
$76$	−1.78017	−0.204199
$77$	3.35690	0.382554
$78$	0	0
$79$	−8.33944	−0.938260	−0.469130	−	0.883129i	$-0.655432\pi$
$79$	−8.33944	−0.938260	−0.469130	+	0.883129i	$0.655432\pi$
$80$	−4.04892	−0.452683
$81$	0	0
$82$	−1.50604	−0.166314
$83$	−15.3274	−1.68240	−0.841198	−	0.540727i	$-0.818149\pi$
$83$	−15.3274	−1.68240	−0.841198	+	0.540727i	$0.818149\pi$
$84$	0	0
$85$	29.8974	3.24283
$86$	−8.31767	−0.896917
$87$	0	0
$88$	4.85086	0.517103
$89$	−3.10992	−0.329650	−0.164825	−	0.986323i	$-0.552706\pi$
$89$	−3.10992	−0.329650	−0.164825	+	0.986323i	$0.552706\pi$
$90$	0	0
$91$	0	0
$92$	−5.10992	−0.532746
$93$	0	0
$94$	7.20775	0.743423
$95$	7.20775	0.739500
$96$	0	0
$97$	−8.54288	−0.867398	−0.433699	−	0.901058i	$-0.642792\pi$
$97$	−8.54288	−0.867398	−0.433699	+	0.901058i	$0.642792\pi$
$98$	−6.52111	−0.658731
$99$	0	0
$100$	11.3937	1.13937
$101$	11.9976	1.19381	0.596903	−	0.802313i	$-0.296398\pi$
$101$	11.9976	1.19381	0.596903	+	0.802313i	$0.296398\pi$
$102$	0	0
$103$	−12.3230	−1.21423	−0.607113	−	0.794616i	$-0.707672\pi$
$103$	−12.3230	−1.21423	−0.607113	+	0.794616i	$0.707672\pi$
$104$	0	0
$105$	0	0
$106$	−13.4765	−1.30895
$107$	−5.89977	−0.570353	−0.285176	−	0.958475i	$-0.592052\pi$
$107$	−5.89977	−0.570353	−0.285176	+	0.958475i	$0.592052\pi$
$108$	0	0
$109$	0.792249	0.0758837	0.0379418	−	0.999280i	$-0.487920\pi$
$109$	0.792249	0.0758837	0.0379418	+	0.999280i	$0.487920\pi$
$110$	−19.6407	−1.87267
$111$	0	0
$112$	0.692021	0.0653899
$113$	−6.21983	−0.585113	−0.292556	−	0.956248i	$-0.594506\pi$
$113$	−6.21983	−0.585113	−0.292556	+	0.956248i	$0.594506\pi$
$114$	0	0
$115$	20.6896	1.92932
$116$	3.34481	0.310558
$117$	0	0
$118$	−1.30798	−0.120409
$119$	−5.10992	−0.468425
$120$	0	0
$121$	12.5308	1.13916
$122$	−0.396125	−0.0358634
$123$	0	0
$124$	0.972853	0.0873648
$125$	−25.8877	−2.31547
$126$	0	0
$127$	−6.00538	−0.532891	−0.266446	−	0.963850i	$-0.585849\pi$
$127$	−6.00538	−0.532891	−0.266446	+	0.963850i	$0.585849\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−8.81700	−0.770345	−0.385173	−	0.922845i	$-0.625858\pi$
$131$	−8.81700	−0.770345	−0.385173	+	0.922845i	$0.625858\pi$
$132$	0	0
$133$	−1.23191	−0.106821
$134$	6.05429	0.523011
$135$	0	0
$136$	−7.38404	−0.633176
$137$	15.7560	1.34613	0.673063	−	0.739585i	$-0.264978\pi$
$137$	15.7560	1.34613	0.673063	+	0.739585i	$0.264978\pi$
$138$	0	0
$139$	−6.09783	−0.517212	−0.258606	−	0.965983i	$-0.583263\pi$
$139$	−6.09783	−0.517212	−0.258606	+	0.965983i	$0.583263\pi$
$140$	−2.80194	−0.236807
$141$	0	0
$142$	1.32975	0.111590
$143$	0	0
$144$	0	0
$145$	−13.5429	−1.12467
$146$	−7.65279	−0.633350
$147$	0	0
$148$	1.28621	0.105726
$149$	2.55257	0.209114	0.104557	−	0.994519i	$-0.466657\pi$
$149$	2.55257	0.209114	0.104557	+	0.994519i	$0.466657\pi$
$150$	0	0
$151$	−17.7168	−1.44177	−0.720885	−	0.693054i	$-0.756265\pi$
$151$	−17.7168	−1.44177	−0.720885	+	0.693054i	$0.756265\pi$
$152$	−1.78017	−0.144391
$153$	0	0
$154$	3.35690	0.270506
$155$	−3.93900	−0.316388
$156$	0	0
$157$	−6.31767	−0.504205	−0.252102	−	0.967701i	$-0.581122\pi$
$157$	−6.31767	−0.504205	−0.252102	+	0.967701i	$0.581122\pi$
$158$	−8.33944	−0.663450
$159$	0	0
$160$	−4.04892	−0.320095
$161$	−3.53617	−0.278689
$162$	0	0
$163$	−14.5918	−1.14292	−0.571459	−	0.820631i	$-0.693622\pi$
$163$	−14.5918	−1.14292	−0.571459	+	0.820631i	$0.693622\pi$
$164$	−1.50604	−0.117602
$165$	0	0
$166$	−15.3274	−1.18963
$167$	−19.5013	−1.50905	−0.754526	−	0.656270i	$-0.772133\pi$
$167$	−19.5013	−1.50905	−0.754526	+	0.656270i	$0.772133\pi$
$168$	0	0
$169$	0	0
$170$	29.8974	2.29302
$171$	0	0
$172$	−8.31767	−0.634216
$173$	9.29052	0.706345	0.353173	−	0.935558i	$-0.385103\pi$
$173$	9.29052	0.706345	0.353173	+	0.935558i	$0.385103\pi$
$174$	0	0
$175$	7.88471	0.596028
$176$	4.85086	0.365647
$177$	0	0
$178$	−3.10992	−0.233098
$179$	−22.7928	−1.70362	−0.851808	−	0.523853i	$-0.824494\pi$
$179$	−22.7928	−1.70362	−0.851808	+	0.523853i	$0.824494\pi$
$180$	0	0
$181$	0.537500	0.0399520	0.0199760	−	0.999800i	$-0.493641\pi$
$181$	0.537500	0.0399520	0.0199760	+	0.999800i	$0.493641\pi$
$182$	0	0
$183$	0	0
$184$	−5.10992	−0.376708
$185$	−5.20775	−0.382881
$186$	0	0
$187$	−35.8189	−2.61934
$188$	7.20775	0.525679
$189$	0	0
$190$	7.20775	0.522905
$191$	9.79954	0.709070	0.354535	−	0.935043i	$-0.384639\pi$
$191$	9.79954	0.709070	0.354535	+	0.935043i	$0.384639\pi$
$192$	0	0
$193$	14.1957	1.02183	0.510913	−	0.859632i	$-0.329307\pi$
$193$	14.1957	1.02183	0.510913	+	0.859632i	$0.329307\pi$
$194$	−8.54288	−0.613343
$195$	0	0
$196$	−6.52111	−0.465793
$197$	−3.00969	−0.214431	−0.107216	−	0.994236i	$-0.534194\pi$
$197$	−3.00969	−0.214431	−0.107216	+	0.994236i	$0.534194\pi$
$198$	0	0
$199$	12.8944	0.914059	0.457030	−	0.889451i	$-0.348913\pi$
$199$	12.8944	0.914059	0.457030	+	0.889451i	$0.348913\pi$
$200$	11.3937	0.805658
$201$	0	0
$202$	11.9976	0.844149
$203$	2.31468	0.162459
$204$	0	0
$205$	6.09783	0.425891
$206$	−12.3230	−0.858587
$207$	0	0
$208$	0	0
$209$	−8.63533	−0.597319
$210$	0	0
$211$	7.79954	0.536943	0.268471	−	0.963288i	$-0.413482\pi$
$211$	7.79954	0.536943	0.268471	+	0.963288i	$0.413482\pi$
$212$	−13.4765	−0.925570
$213$	0	0
$214$	−5.89977	−0.403300
$215$	33.6775	2.29679
$216$	0	0
$217$	0.673235	0.0457022
$218$	0.792249	0.0536579
$219$	0	0
$220$	−19.6407	−1.32418
$221$	0	0
$222$	0	0
$223$	12.1957	0.816682	0.408341	−	0.912829i	$-0.366107\pi$
$223$	12.1957	0.816682	0.408341	+	0.912829i	$0.366107\pi$
$224$	0.692021	0.0462376
$225$	0	0
$226$	−6.21983	−0.413737
$227$	6.74333	0.447571	0.223785	−	0.974638i	$-0.428159\pi$
$227$	6.74333	0.447571	0.223785	+	0.974638i	$0.428159\pi$
$228$	0	0
$229$	19.8237	1.30999	0.654994	−	0.755634i	$-0.272671\pi$
$229$	19.8237	1.30999	0.654994	+	0.755634i	$0.272671\pi$
$230$	20.6896	1.36423
$231$	0	0
$232$	3.34481	0.219598
$233$	30.0301	1.96734	0.983670	−	0.179983i	$-0.0576044\pi$
$233$	30.0301	1.96734	0.983670	+	0.179983i	$0.0576044\pi$
$234$	0	0
$235$	−29.1836	−1.90373
$236$	−1.30798	−0.0851422
$237$	0	0
$238$	−5.10992	−0.331227
$239$	22.0978	1.42939	0.714695	−	0.699436i	$-0.246565\pi$
$239$	22.0978	1.42939	0.714695	+	0.699436i	$0.246565\pi$
$240$	0	0
$241$	−10.1274	−0.652362	−0.326181	−	0.945307i	$-0.605762\pi$
$241$	−10.1274	−0.652362	−0.326181	+	0.945307i	$0.605762\pi$
$242$	12.5308	0.805510
$243$	0	0
$244$	−0.396125	−0.0253593
$245$	26.4034	1.68685
$246$	0	0
$247$	0	0
$248$	0.972853	0.0617762
$249$	0	0
$250$	−25.8877	−1.63728
$251$	−5.54719	−0.350135	−0.175068	−	0.984556i	$-0.556014\pi$
$251$	−5.54719	−0.350135	−0.175068	+	0.984556i	$0.556014\pi$
$252$	0	0
$253$	−24.7875	−1.55837
$254$	−6.00538	−0.376811
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.7995	0.860792	0.430396	−	0.902640i	$-0.358374\pi$
$257$	13.7995	0.860792	0.430396	+	0.902640i	$0.358374\pi$
$258$	0	0
$259$	0.890084	0.0553071
$260$	0	0
$261$	0	0
$262$	−8.81700	−0.544716
$263$	22.4698	1.38555	0.692773	−	0.721155i	$-0.256389\pi$
$263$	22.4698	1.38555	0.692773	+	0.721155i	$0.256389\pi$
$264$	0	0
$265$	54.5652	3.35192
$266$	−1.23191	−0.0755335
$267$	0	0
$268$	6.05429	0.369825
$269$	26.0140	1.58610	0.793051	−	0.609156i	$-0.208491\pi$
$269$	26.0140	1.58610	0.793051	+	0.609156i	$0.208491\pi$
$270$	0	0
$271$	−2.88471	−0.175233	−0.0876167	−	0.996154i	$-0.527925\pi$
$271$	−2.88471	−0.175233	−0.0876167	+	0.996154i	$0.527925\pi$
$272$	−7.38404	−0.447723
$273$	0	0
$274$	15.7560	0.951855
$275$	55.2693	3.33287
$276$	0	0
$277$	−1.46250	−0.0878731	−0.0439366	−	0.999034i	$-0.513990\pi$
$277$	−1.46250	−0.0878731	−0.0439366	+	0.999034i	$0.513990\pi$
$278$	−6.09783	−0.365724
$279$	0	0
$280$	−2.80194	−0.167448
$281$	5.68233	0.338980	0.169490	−	0.985532i	$-0.445788\pi$
$281$	5.68233	0.338980	0.169490	+	0.985532i	$0.445788\pi$
$282$	0	0
$283$	−25.2078	−1.49845	−0.749223	−	0.662318i	$-0.769573\pi$
$283$	−25.2078	−1.49845	−0.749223	+	0.662318i	$0.769573\pi$
$284$	1.32975	0.0789061
$285$	0	0
$286$	0	0
$287$	−1.04221	−0.0615199
$288$	0	0
$289$	37.5241	2.20730
$290$	−13.5429	−0.795265
$291$	0	0
$292$	−7.65279	−0.447846
$293$	−7.14914	−0.417658	−0.208829	−	0.977952i	$-0.566965\pi$
$293$	−7.14914	−0.417658	−0.208829	+	0.977952i	$0.566965\pi$
$294$	0	0
$295$	5.29590	0.308339
$296$	1.28621	0.0747593
$297$	0	0
$298$	2.55257	0.147866
$299$	0	0
$300$	0	0
$301$	−5.75600	−0.331771
$302$	−17.7168	−1.01949
$303$	0	0
$304$	−1.78017	−0.102100
$305$	1.60388	0.0918376
$306$	0	0
$307$	17.9952	1.02704	0.513521	−	0.858077i	$-0.328341\pi$
$307$	17.9952	1.02704	0.513521	+	0.858077i	$0.328341\pi$
$308$	3.35690	0.191277
$309$	0	0
$310$	−3.93900	−0.223720
$311$	−3.32975	−0.188813	−0.0944064	−	0.995534i	$-0.530095\pi$
$311$	−3.32975	−0.188813	−0.0944064	+	0.995534i	$0.530095\pi$
$312$	0	0
$313$	−17.8834	−1.01083	−0.505414	−	0.862877i	$-0.668660\pi$
$313$	−17.8834	−1.01083	−0.505414	+	0.862877i	$0.668660\pi$
$314$	−6.31767	−0.356527
$315$	0	0
$316$	−8.33944	−0.469130
$317$	4.39373	0.246777	0.123388	−	0.992358i	$-0.460624\pi$
$317$	4.39373	0.246777	0.123388	+	0.992358i	$0.460624\pi$
$318$	0	0
$319$	16.2252	0.908437
$320$	−4.04892	−0.226341
$321$	0	0
$322$	−3.53617	−0.197063
$323$	13.1448	0.731398
$324$	0	0
$325$	0	0
$326$	−14.5918	−0.808165
$327$	0	0
$328$	−1.50604	−0.0831572
$329$	4.98792	0.274993
$330$	0	0
$331$	25.6775	1.41137	0.705683	−	0.708528i	$-0.250640\pi$
$331$	25.6775	1.41137	0.705683	+	0.708528i	$0.250640\pi$
$332$	−15.3274	−0.841198
$333$	0	0
$334$	−19.5013	−1.06706
$335$	−24.5133	−1.33931
$336$	0	0
$337$	−24.6504	−1.34279	−0.671396	−	0.741098i	$-0.734305\pi$
$337$	−24.6504	−1.34279	−0.671396	+	0.741098i	$0.734305\pi$
$338$	0	0
$339$	0	0
$340$	29.8974	1.62141
$341$	4.71917	0.255557
$342$	0	0
$343$	−9.35690	−0.505225
$344$	−8.31767	−0.448459
$345$	0	0
$346$	9.29052	0.499461
$347$	−14.2959	−0.767444	−0.383722	−	0.923449i	$-0.625358\pi$
$347$	−14.2959	−0.767444	−0.383722	+	0.923449i	$0.625358\pi$
$348$	0	0
$349$	−11.0616	−0.592113	−0.296057	−	0.955170i	$-0.595672\pi$
$349$	−11.0616	−0.592113	−0.296057	+	0.955170i	$0.595672\pi$
$350$	7.88471	0.421455
$351$	0	0
$352$	4.85086	0.258551
$353$	−10.5047	−0.559109	−0.279555	−	0.960130i	$-0.590187\pi$
$353$	−10.5047	−0.559109	−0.279555	+	0.960130i	$0.590187\pi$
$354$	0	0
$355$	−5.38404	−0.285755
$356$	−3.10992	−0.164825
$357$	0	0
$358$	−22.7928	−1.20464
$359$	5.10992	0.269691	0.134846	−	0.990867i	$-0.456946\pi$
$359$	5.10992	0.269691	0.134846	+	0.990867i	$0.456946\pi$
$360$	0	0
$361$	−15.8310	−0.833211
$362$	0.537500	0.0282504
$363$	0	0
$364$	0	0
$365$	30.9855	1.62186
$366$	0	0
$367$	−8.44803	−0.440983	−0.220492	−	0.975389i	$-0.570766\pi$
$367$	−8.44803	−0.440983	−0.220492	+	0.975389i	$0.570766\pi$
$368$	−5.10992	−0.266373
$369$	0	0
$370$	−5.20775	−0.270738
$371$	−9.32603	−0.484183
$372$	0	0
$373$	7.69096	0.398223	0.199111	−	0.979977i	$-0.436194\pi$
$373$	7.69096	0.398223	0.199111	+	0.979977i	$0.436194\pi$
$374$	−35.8189	−1.85215
$375$	0	0
$376$	7.20775	0.371711
$377$	0	0
$378$	0	0
$379$	11.4034	0.585754	0.292877	−	0.956150i	$-0.405387\pi$
$379$	11.4034	0.585754	0.292877	+	0.956150i	$0.405387\pi$
$380$	7.20775	0.369750
$381$	0	0
$382$	9.79954	0.501388
$383$	20.6703	1.05620	0.528100	−	0.849182i	$-0.322905\pi$
$383$	20.6703	1.05620	0.528100	+	0.849182i	$0.322905\pi$
$384$	0	0
$385$	−13.5918	−0.692702
$386$	14.1957	0.722541
$387$	0	0
$388$	−8.54288	−0.433699
$389$	−17.4776	−0.886148	−0.443074	−	0.896485i	$-0.646112\pi$
$389$	−17.4776	−0.886148	−0.443074	+	0.896485i	$0.646112\pi$
$390$	0	0
$391$	37.7318	1.90818
$392$	−6.52111	−0.329366
$393$	0	0
$394$	−3.00969	−0.151626
$395$	33.7657	1.69894
$396$	0	0
$397$	19.3599	0.971645	0.485822	−	0.874058i	$-0.338520\pi$
$397$	19.3599	0.971645	0.485822	+	0.874058i	$0.338520\pi$
$398$	12.8944	0.646338
$399$	0	0
$400$	11.3937	0.569687
$401$	−14.4832	−0.723257	−0.361628	−	0.932322i	$-0.617779\pi$
$401$	−14.4832	−0.723257	−0.361628	+	0.932322i	$0.617779\pi$
$402$	0	0
$403$	0	0
$404$	11.9976	0.596903
$405$	0	0
$406$	2.31468	0.114876
$407$	6.23921	0.309266
$408$	0	0
$409$	18.8984	0.934468	0.467234	−	0.884134i	$-0.345251\pi$
$409$	18.8984	0.934468	0.467234	+	0.884134i	$0.345251\pi$
$410$	6.09783	0.301151
$411$	0	0
$412$	−12.3230	−0.607113
$413$	−0.905149	−0.0445395
$414$	0	0
$415$	62.0592	3.04637
$416$	0	0
$417$	0	0
$418$	−8.63533	−0.422368
$419$	21.7603	1.06306	0.531531	−	0.847039i	$-0.321617\pi$
$419$	21.7603	1.06306	0.531531	+	0.847039i	$0.321617\pi$
$420$	0	0
$421$	−20.5918	−1.00358	−0.501791	−	0.864989i	$-0.667325\pi$
$421$	−20.5918	−1.00358	−0.501791	+	0.864989i	$0.667325\pi$
$422$	7.79954	0.379676
$423$	0	0
$424$	−13.4765	−0.654477
$425$	−84.1318	−4.08099
$426$	0	0
$427$	−0.274127	−0.0132659
$428$	−5.89977	−0.285176
$429$	0	0
$430$	33.6775	1.62408
$431$	34.5133	1.66245	0.831224	−	0.555937i	$-0.187640\pi$
$431$	34.5133	1.66245	0.831224	+	0.555937i	$0.187640\pi$
$432$	0	0
$433$	−2.12631	−0.102184	−0.0510920	−	0.998694i	$-0.516270\pi$
$433$	−2.12631	−0.102184	−0.0510920	+	0.998694i	$0.516270\pi$
$434$	0.673235	0.0323163
$435$	0	0
$436$	0.792249	0.0379418
$437$	9.09651	0.435145
$438$	0	0
$439$	21.8321	1.04199	0.520994	−	0.853560i	$-0.325561\pi$
$439$	21.8321	1.04199	0.520994	+	0.853560i	$0.325561\pi$
$440$	−19.6407	−0.936334
$441$	0	0
$442$	0	0
$443$	7.54048	0.358259	0.179130	−	0.983825i	$-0.442672\pi$
$443$	7.54048	0.358259	0.179130	+	0.983825i	$0.442672\pi$
$444$	0	0
$445$	12.5918	0.596908
$446$	12.1957	0.577482
$447$	0	0
$448$	0.692021	0.0326949
$449$	19.7560	0.932343	0.466172	−	0.884694i	$-0.345633\pi$
$449$	19.7560	0.932343	0.466172	+	0.884694i	$0.345633\pi$
$450$	0	0
$451$	−7.30559	−0.344007
$452$	−6.21983	−0.292556
$453$	0	0
$454$	6.74333	0.316480
$455$	0	0
$456$	0	0
$457$	−23.8582	−1.11604	−0.558019	−	0.829828i	$-0.688438\pi$
$457$	−23.8582	−1.11604	−0.558019	+	0.829828i	$0.688438\pi$
$458$	19.8237	0.926301
$459$	0	0
$460$	20.6896	0.964659
$461$	−17.5773	−0.818657	−0.409329	−	0.912387i	$-0.634237\pi$
$461$	−17.5773	−0.818657	−0.409329	+	0.912387i	$0.634237\pi$
$462$	0	0
$463$	−23.8431	−1.10808	−0.554041	−	0.832489i	$-0.686915\pi$
$463$	−23.8431	−1.10808	−0.554041	+	0.832489i	$0.686915\pi$
$464$	3.34481	0.155279
$465$	0	0
$466$	30.0301	1.39112
$467$	−8.61058	−0.398450	−0.199225	−	0.979954i	$-0.563842\pi$
$467$	−8.61058	−0.398450	−0.199225	+	0.979954i	$0.563842\pi$
$468$	0	0
$469$	4.18970	0.193462
$470$	−29.1836	−1.34614
$471$	0	0
$472$	−1.30798	−0.0602046
$473$	−40.3478	−1.85519
$474$	0	0
$475$	−20.2828	−0.930636
$476$	−5.10992	−0.234213
$477$	0	0
$478$	22.0978	1.01073
$479$	6.58104	0.300695	0.150348	−	0.988633i	$-0.451961\pi$
$479$	6.58104	0.300695	0.150348	+	0.988633i	$0.451961\pi$
$480$	0	0
$481$	0	0
$482$	−10.1274	−0.461289
$483$	0	0
$484$	12.5308	0.569582
$485$	34.5894	1.57062
$486$	0	0
$487$	23.2760	1.05474	0.527369	−	0.849636i	$-0.323178\pi$
$487$	23.2760	1.05474	0.527369	+	0.849636i	$0.323178\pi$
$488$	−0.396125	−0.0179317
$489$	0	0
$490$	26.4034	1.19278
$491$	15.5405	0.701332	0.350666	−	0.936501i	$-0.385955\pi$
$491$	15.5405	0.701332	0.350666	+	0.936501i	$0.385955\pi$
$492$	0	0
$493$	−24.6983	−1.11235
$494$	0	0
$495$	0	0
$496$	0.972853	0.0436824
$497$	0.920215	0.0412773
$498$	0	0
$499$	−9.53617	−0.426898	−0.213449	−	0.976954i	$-0.568470\pi$
$499$	−9.53617	−0.426898	−0.213449	+	0.976954i	$0.568470\pi$
$500$	−25.8877	−1.15773
$501$	0	0
$502$	−5.54719	−0.247583
$503$	13.8345	0.616848	0.308424	−	0.951249i	$-0.400198\pi$
$503$	13.8345	0.616848	0.308424	+	0.951249i	$0.400198\pi$
$504$	0	0
$505$	−48.5773	−2.16166
$506$	−24.7875	−1.10194
$507$	0	0
$508$	−6.00538	−0.266446
$509$	−40.9638	−1.81569	−0.907843	−	0.419310i	$-0.862272\pi$
$509$	−40.9638	−1.81569	−0.907843	+	0.419310i	$0.862272\pi$
$510$	0	0
$511$	−5.29590	−0.234277
$512$	1.00000	0.0441942
$513$	0	0
$514$	13.7995	0.608672
$515$	49.8950	2.19864
$516$	0	0
$517$	34.9638	1.53770
$518$	0.890084	0.0391080
$519$	0	0
$520$	0	0
$521$	−36.3672	−1.59327	−0.796637	−	0.604457i	$-0.793390\pi$
$521$	−36.3672	−1.59327	−0.796637	+	0.604457i	$0.793390\pi$
$522$	0	0
$523$	−6.03013	−0.263679	−0.131840	−	0.991271i	$-0.542088\pi$
$523$	−6.03013	−0.263679	−0.131840	+	0.991271i	$0.542088\pi$
$524$	−8.81700	−0.385173
$525$	0	0
$526$	22.4698	0.979730
$527$	−7.18359	−0.312922
$528$	0	0
$529$	3.11124	0.135271
$530$	54.5652	2.37016
$531$	0	0
$532$	−1.23191	−0.0534103
$533$	0	0
$534$	0	0
$535$	23.8877	1.03275
$536$	6.05429	0.261506
$537$	0	0
$538$	26.0140	1.12154
$539$	−31.6329	−1.36253
$540$	0	0
$541$	7.92154	0.340574	0.170287	−	0.985395i	$-0.445531\pi$
$541$	7.92154	0.340574	0.170287	+	0.985395i	$0.445531\pi$
$542$	−2.88471	−0.123909
$543$	0	0
$544$	−7.38404	−0.316588
$545$	−3.20775	−0.137405
$546$	0	0
$547$	18.4155	0.787390	0.393695	−	0.919241i	$-0.371197\pi$
$547$	18.4155	0.787390	0.393695	+	0.919241i	$0.371197\pi$
$548$	15.7560	0.673063
$549$	0	0
$550$	55.2693	2.35669
$551$	−5.95433	−0.253663
$552$	0	0
$553$	−5.77107	−0.245411
$554$	−1.46250	−0.0621357
$555$	0	0
$556$	−6.09783	−0.258606
$557$	−23.9758	−1.01589	−0.507944	−	0.861390i	$-0.669594\pi$
$557$	−23.9758	−1.01589	−0.507944	+	0.861390i	$0.669594\pi$
$558$	0	0
$559$	0	0
$560$	−2.80194	−0.118403
$561$	0	0
$562$	5.68233	0.239695
$563$	2.29291	0.0966348	0.0483174	−	0.998832i	$-0.484614\pi$
$563$	2.29291	0.0966348	0.0483174	+	0.998832i	$0.484614\pi$
$564$	0	0
$565$	25.1836	1.05948
$566$	−25.2078	−1.05956
$567$	0	0
$568$	1.32975	0.0557950
$569$	44.3430	1.85896	0.929478	−	0.368878i	$-0.120258\pi$
$569$	44.3430	1.85896	0.929478	+	0.368878i	$0.120258\pi$
$570$	0	0
$571$	−15.2707	−0.639058	−0.319529	−	0.947577i	$-0.603525\pi$
$571$	−15.2707	−0.639058	−0.319529	+	0.947577i	$0.603525\pi$
$572$	0	0
$573$	0	0
$574$	−1.04221	−0.0435011
$575$	−58.2210	−2.42798
$576$	0	0
$577$	−8.77048	−0.365120	−0.182560	−	0.983195i	$-0.558438\pi$
$577$	−8.77048	−0.365120	−0.182560	+	0.983195i	$0.558438\pi$
$578$	37.5241	1.56080
$579$	0	0
$580$	−13.5429	−0.562337
$581$	−10.6069	−0.440047
$582$	0	0
$583$	−65.3726	−2.70745
$584$	−7.65279	−0.316675
$585$	0	0
$586$	−7.14914	−0.295328
$587$	−38.1430	−1.57433	−0.787166	−	0.616742i	$-0.788452\pi$
$587$	−38.1430	−1.57433	−0.787166	+	0.616742i	$0.788452\pi$
$588$	0	0
$589$	−1.73184	−0.0713593
$590$	5.29590	0.218029
$591$	0	0
$592$	1.28621	0.0528628
$593$	−37.9517	−1.55849	−0.779244	−	0.626720i	$-0.784397\pi$
$593$	−37.9517	−1.55849	−0.779244	+	0.626720i	$0.784397\pi$
$594$	0	0
$595$	20.6896	0.848192
$596$	2.55257	0.104557
$597$	0	0
$598$	0	0
$599$	3.57971	0.146263	0.0731315	−	0.997322i	$-0.476701\pi$
$599$	3.57971	0.146263	0.0731315	+	0.997322i	$0.476701\pi$
$600$	0	0
$601$	−5.71678	−0.233192	−0.116596	−	0.993179i	$-0.537198\pi$
$601$	−5.71678	−0.233192	−0.116596	+	0.993179i	$0.537198\pi$
$602$	−5.75600	−0.234597
$603$	0	0
$604$	−17.7168	−0.720885
$605$	−50.7362	−2.06272
$606$	0	0
$607$	22.4286	0.910351	0.455175	−	0.890402i	$-0.349577\pi$
$607$	22.4286	0.910351	0.455175	+	0.890402i	$0.349577\pi$
$608$	−1.78017	−0.0721953
$609$	0	0
$610$	1.60388	0.0649390
$611$	0	0
$612$	0	0
$613$	39.9603	1.61398	0.806991	−	0.590564i	$-0.201095\pi$
$613$	39.9603	1.61398	0.806991	+	0.590564i	$0.201095\pi$
$614$	17.9952	0.726228
$615$	0	0
$616$	3.35690	0.135253
$617$	31.4470	1.26601	0.633003	−	0.774149i	$-0.281822\pi$
$617$	31.4470	1.26601	0.633003	+	0.774149i	$0.281822\pi$
$618$	0	0
$619$	29.3685	1.18042	0.590210	−	0.807250i	$-0.299045\pi$
$619$	29.3685	1.18042	0.590210	+	0.807250i	$0.299045\pi$
$620$	−3.93900	−0.158194
$621$	0	0
$622$	−3.32975	−0.133511
$623$	−2.15213	−0.0862232
$624$	0	0
$625$	47.8485	1.91394
$626$	−17.8834	−0.714764
$627$	0	0
$628$	−6.31767	−0.252102
$629$	−9.49742	−0.378687
$630$	0	0
$631$	−21.6799	−0.863065	−0.431532	−	0.902097i	$-0.642027\pi$
$631$	−21.6799	−0.863065	−0.431532	+	0.902097i	$0.642027\pi$
$632$	−8.33944	−0.331725
$633$	0	0
$634$	4.39373	0.174497
$635$	24.3153	0.964922
$636$	0	0
$637$	0	0
$638$	16.2252	0.642362
$639$	0	0
$640$	−4.04892	−0.160048
$641$	−14.0108	−0.553391	−0.276696	−	0.960958i	$-0.589239\pi$
$641$	−14.0108	−0.553391	−0.276696	+	0.960958i	$0.589239\pi$
$642$	0	0
$643$	−30.7875	−1.21414	−0.607070	−	0.794649i	$-0.707655\pi$
$643$	−30.7875	−1.21414	−0.607070	+	0.794649i	$0.707655\pi$
$644$	−3.53617	−0.139345
$645$	0	0
$646$	13.1448	0.517177
$647$	16.6025	0.652713	0.326357	−	0.945247i	$-0.394179\pi$
$647$	16.6025	0.652713	0.326357	+	0.945247i	$0.394179\pi$
$648$	0	0
$649$	−6.34481	−0.249056
$650$	0	0
$651$	0	0
$652$	−14.5918	−0.571459
$653$	−28.5459	−1.11709	−0.558543	−	0.829476i	$-0.688639\pi$
$653$	−28.5459	−1.11709	−0.558543	+	0.829476i	$0.688639\pi$
$654$	0	0
$655$	35.6993	1.39489
$656$	−1.50604	−0.0588010
$657$	0	0
$658$	4.98792	0.194449
$659$	−27.7187	−1.07977	−0.539884	−	0.841740i	$-0.681532\pi$
$659$	−27.7187	−1.07977	−0.539884	+	0.841740i	$0.681532\pi$
$660$	0	0
$661$	−10.8009	−0.420105	−0.210053	−	0.977690i	$-0.567364\pi$
$661$	−10.8009	−0.420105	−0.210053	+	0.977690i	$0.567364\pi$
$662$	25.6775	0.997986
$663$	0	0
$664$	−15.3274	−0.594817
$665$	4.98792	0.193423
$666$	0	0
$667$	−17.0917	−0.661794
$668$	−19.5013	−0.754526
$669$	0	0
$670$	−24.5133	−0.947033
$671$	−1.92154	−0.0741803
$672$	0	0
$673$	16.3260	0.629322	0.314661	−	0.949204i	$-0.398109\pi$
$673$	16.3260	0.629322	0.314661	+	0.949204i	$0.398109\pi$
$674$	−24.6504	−0.949498
$675$	0	0
$676$	0	0
$677$	−41.4252	−1.59210	−0.796050	−	0.605231i	$-0.793081\pi$
$677$	−41.4252	−1.59210	−0.796050	+	0.605231i	$0.793081\pi$
$678$	0	0
$679$	−5.91185	−0.226876
$680$	29.8974	1.14651
$681$	0	0
$682$	4.71917	0.180706
$683$	31.2325	1.19508	0.597539	−	0.801840i	$-0.296145\pi$
$683$	31.2325	1.19508	0.597539	+	0.801840i	$0.296145\pi$
$684$	0	0
$685$	−63.7948	−2.43747
$686$	−9.35690	−0.357248
$687$	0	0
$688$	−8.31767	−0.317108
$689$	0	0
$690$	0	0
$691$	−24.9638	−0.949666	−0.474833	−	0.880076i	$-0.657492\pi$
$691$	−24.9638	−0.949666	−0.474833	+	0.880076i	$0.657492\pi$
$692$	9.29052	0.353173
$693$	0	0
$694$	−14.2959	−0.542665
$695$	24.6896	0.936531
$696$	0	0
$697$	11.1207	0.421225
$698$	−11.0616	−0.418687
$699$	0	0
$700$	7.88471	0.298014
$701$	−8.17151	−0.308634	−0.154317	−	0.988021i	$-0.549318\pi$
$701$	−8.17151	−0.308634	−0.154317	+	0.988021i	$0.549318\pi$
$702$	0	0
$703$	−2.28967	−0.0863564
$704$	4.85086	0.182823
$705$	0	0
$706$	−10.5047	−0.395350
$707$	8.30260	0.312251
$708$	0	0
$709$	36.7982	1.38199	0.690993	−	0.722861i	$-0.257174\pi$
$709$	36.7982	1.38199	0.690993	+	0.722861i	$0.257174\pi$
$710$	−5.38404	−0.202060
$711$	0	0
$712$	−3.10992	−0.116549
$713$	−4.97120	−0.186173
$714$	0	0
$715$	0	0
$716$	−22.7928	−0.851808
$717$	0	0
$718$	5.10992	0.190700
$719$	−35.2223	−1.31357	−0.656786	−	0.754077i	$-0.728084\pi$
$719$	−35.2223	−1.31357	−0.656786	+	0.754077i	$0.728084\pi$
$720$	0	0
$721$	−8.52781	−0.317592
$722$	−15.8310	−0.589169
$723$	0	0
$724$	0.537500	0.0199760
$725$	38.1099	1.41537
$726$	0	0
$727$	40.6872	1.50901	0.754503	−	0.656297i	$-0.227878\pi$
$727$	40.6872	1.50901	0.754503	+	0.656297i	$0.227878\pi$
$728$	0	0
$729$	0	0
$730$	30.9855	1.14683
$731$	61.4180	2.27163
$732$	0	0
$733$	−27.1400	−1.00244	−0.501220	−	0.865320i	$-0.667115\pi$
$733$	−27.1400	−1.00244	−0.501220	+	0.865320i	$0.667115\pi$
$734$	−8.44803	−0.311822
$735$	0	0
$736$	−5.10992	−0.188354
$737$	29.3685	1.08180
$738$	0	0
$739$	−3.72587	−0.137058	−0.0685292	−	0.997649i	$-0.521831\pi$
$739$	−3.72587	−0.137058	−0.0685292	+	0.997649i	$0.521831\pi$
$740$	−5.20775	−0.191441
$741$	0	0
$742$	−9.32603	−0.342369
$743$	21.8586	0.801915	0.400958	−	0.916097i	$-0.368677\pi$
$743$	21.8586	0.801915	0.400958	+	0.916097i	$0.368677\pi$
$744$	0	0
$745$	−10.3351	−0.378650
$746$	7.69096	0.281586
$747$	0	0
$748$	−35.8189	−1.30967
$749$	−4.08277	−0.149181
$750$	0	0
$751$	53.3642	1.94729	0.973644	−	0.228075i	$-0.0732432\pi$
$751$	53.3642	1.94729	0.973644	+	0.228075i	$0.0732432\pi$
$752$	7.20775	0.262840
$753$	0	0
$754$	0	0
$755$	71.7338	2.61066
$756$	0	0
$757$	−4.63533	−0.168474	−0.0842370	−	0.996446i	$-0.526845\pi$
$757$	−4.63533	−0.168474	−0.0842370	+	0.996446i	$0.526845\pi$
$758$	11.4034	0.414191
$759$	0	0
$760$	7.20775	0.261453
$761$	11.5603	0.419062	0.209531	−	0.977802i	$-0.432806\pi$
$761$	11.5603	0.419062	0.209531	+	0.977802i	$0.432806\pi$
$762$	0	0
$763$	0.548253	0.0198481
$764$	9.79954	0.354535
$765$	0	0
$766$	20.6703	0.746847
$767$	0	0
$768$	0	0
$769$	−14.7439	−0.531679	−0.265840	−	0.964017i	$-0.585649\pi$
$769$	−14.7439	−0.531679	−0.265840	+	0.964017i	$0.585649\pi$
$770$	−13.5918	−0.489814
$771$	0	0
$772$	14.1957	0.510913
$773$	−6.42268	−0.231008	−0.115504	−	0.993307i	$-0.536848\pi$
$773$	−6.42268	−0.231008	−0.115504	+	0.993307i	$0.536848\pi$
$774$	0	0
$775$	11.0844	0.398164
$776$	−8.54288	−0.306671
$777$	0	0
$778$	−17.4776	−0.626601
$779$	2.68100	0.0960570
$780$	0	0
$781$	6.45042	0.230814
$782$	37.7318	1.34929
$783$	0	0
$784$	−6.52111	−0.232897
$785$	25.5797	0.912979
$786$	0	0
$787$	−43.4336	−1.54824	−0.774119	−	0.633040i	$-0.781807\pi$
$787$	−43.4336	−1.54824	−0.774119	+	0.633040i	$0.781807\pi$
$788$	−3.00969	−0.107216
$789$	0	0
$790$	33.7657	1.20133
$791$	−4.30426	−0.153042
$792$	0	0
$793$	0	0
$794$	19.3599	0.687056
$795$	0	0
$796$	12.8944	0.457030
$797$	−21.0164	−0.744439	−0.372219	−	0.928145i	$-0.621403\pi$
$797$	−21.0164	−0.744439	−0.372219	+	0.928145i	$0.621403\pi$
$798$	0	0
$799$	−53.2223	−1.88287
$800$	11.3937	0.402829
$801$	0	0
$802$	−14.4832	−0.511420
$803$	−37.1226	−1.31003
$804$	0	0
$805$	14.3177	0.504631
$806$	0	0
$807$	0	0
$808$	11.9976	0.422074
$809$	8.32245	0.292602	0.146301	−	0.989240i	$-0.453263\pi$
$809$	8.32245	0.292602	0.146301	+	0.989240i	$0.453263\pi$
$810$	0	0
$811$	−14.4638	−0.507894	−0.253947	−	0.967218i	$-0.581729\pi$
$811$	−14.4638	−0.507894	−0.253947	+	0.967218i	$0.581729\pi$
$812$	2.31468	0.0812295
$813$	0	0
$814$	6.23921	0.218684
$815$	59.0810	2.06952
$816$	0	0
$817$	14.8068	0.518026
$818$	18.8984	0.660769
$819$	0	0
$820$	6.09783	0.212946
$821$	38.4161	1.34073	0.670365	−	0.742031i	$-0.266137\pi$
$821$	38.4161	1.34073	0.670365	+	0.742031i	$0.266137\pi$
$822$	0	0
$823$	−40.9748	−1.42829	−0.714145	−	0.699997i	$-0.753184\pi$
$823$	−40.9748	−1.42829	−0.714145	+	0.699997i	$0.753184\pi$
$824$	−12.3230	−0.429294
$825$	0	0
$826$	−0.905149	−0.0314942
$827$	−3.51035	−0.122067	−0.0610335	−	0.998136i	$-0.519440\pi$
$827$	−3.51035	−0.122067	−0.0610335	+	0.998136i	$0.519440\pi$
$828$	0	0
$829$	−13.2185	−0.459098	−0.229549	−	0.973297i	$-0.573725\pi$
$829$	−13.2185	−0.459098	−0.229549	+	0.973297i	$0.573725\pi$
$830$	62.0592	2.15411
$831$	0	0
$832$	0	0
$833$	48.1521	1.66837
$834$	0	0
$835$	78.9590	2.73249
$836$	−8.63533	−0.298659
$837$	0	0
$838$	21.7603	0.751698
$839$	−55.8491	−1.92812	−0.964062	−	0.265678i	$-0.914404\pi$
$839$	−55.8491	−1.92812	−0.964062	+	0.265678i	$0.914404\pi$
$840$	0	0
$841$	−17.8122	−0.614214
$842$	−20.5918	−0.709640
$843$	0	0
$844$	7.79954	0.268471
$845$	0	0
$846$	0	0
$847$	8.67158	0.297959
$848$	−13.4765	−0.462785
$849$	0	0
$850$	−84.1318	−2.88570
$851$	−6.57242	−0.225300
$852$	0	0
$853$	21.8103	0.746770	0.373385	−	0.927676i	$-0.378197\pi$
$853$	21.8103	0.746770	0.373385	+	0.927676i	$0.378197\pi$
$854$	−0.274127	−0.00938042
$855$	0	0
$856$	−5.89977	−0.201650
$857$	−28.8961	−0.987070	−0.493535	−	0.869726i	$-0.664296\pi$
$857$	−28.8961	−0.987070	−0.493535	+	0.869726i	$0.664296\pi$
$858$	0	0
$859$	−17.2755	−0.589431	−0.294715	−	0.955585i	$-0.595225\pi$
$859$	−17.2755	−0.589431	−0.294715	+	0.955585i	$0.595225\pi$
$860$	33.6775	1.14839
$861$	0	0
$862$	34.5133	1.17553
$863$	−44.7741	−1.52413	−0.762063	−	0.647503i	$-0.775813\pi$
$863$	−44.7741	−1.52413	−0.762063	+	0.647503i	$0.775813\pi$
$864$	0	0
$865$	−37.6165	−1.27900
$866$	−2.12631	−0.0722549
$867$	0	0
$868$	0.673235	0.0228511
$869$	−40.4534	−1.37229
$870$	0	0
$871$	0	0
$872$	0.792249	0.0268289
$873$	0	0
$874$	9.09651	0.307694
$875$	−17.9148	−0.605632
$876$	0	0
$877$	40.2741	1.35996	0.679980	−	0.733230i	$-0.261988\pi$
$877$	40.2741	1.35996	0.679980	+	0.733230i	$0.261988\pi$
$878$	21.8321	0.736797
$879$	0	0
$880$	−19.6407	−0.662088
$881$	−9.40821	−0.316971	−0.158485	−	0.987361i	$-0.550661\pi$
$881$	−9.40821	−0.316971	−0.158485	+	0.987361i	$0.550661\pi$
$882$	0	0
$883$	−51.2271	−1.72393	−0.861965	−	0.506968i	$-0.830766\pi$
$883$	−51.2271	−1.72393	−0.861965	+	0.506968i	$0.830766\pi$
$884$	0	0
$885$	0	0
$886$	7.54048	0.253328
$887$	39.9215	1.34043	0.670217	−	0.742165i	$-0.266201\pi$
$887$	39.9215	1.34043	0.670217	+	0.742165i	$0.266201\pi$
$888$	0	0
$889$	−4.15585	−0.139383
$890$	12.5918	0.422078
$891$	0	0
$892$	12.1957	0.408341
$893$	−12.8310	−0.429373
$894$	0	0
$895$	92.2863	3.08479
$896$	0.692021	0.0231188
$897$	0	0
$898$	19.7560	0.659266
$899$	3.25401	0.108527
$900$	0	0
$901$	99.5111	3.31519
$902$	−7.30559	−0.243249
$903$	0	0
$904$	−6.21983	−0.206869
$905$	−2.17629	−0.0723424
$906$	0	0
$907$	−34.8310	−1.15654	−0.578272	−	0.815844i	$-0.696273\pi$
$907$	−34.8310	−1.15654	−0.578272	+	0.815844i	$0.696273\pi$
$908$	6.74333	0.223785
$909$	0	0
$910$	0	0
$911$	−13.2125	−0.437751	−0.218875	−	0.975753i	$-0.570239\pi$
$911$	−13.2125	−0.437751	−0.218875	+	0.975753i	$0.570239\pi$
$912$	0	0
$913$	−74.3508	−2.46065
$914$	−23.8582	−0.789157
$915$	0	0
$916$	19.8237	0.654994
$917$	−6.10156	−0.201491
$918$	0	0
$919$	21.0175	0.693302	0.346651	−	0.937994i	$-0.387319\pi$
$919$	21.0175	0.693302	0.346651	+	0.937994i	$0.387319\pi$
$920$	20.6896	0.682117
$921$	0	0
$922$	−17.5773	−0.578878
$923$	0	0
$924$	0	0
$925$	14.6547	0.481844
$926$	−23.8431	−0.783532
$927$	0	0
$928$	3.34481	0.109799
$929$	−26.9965	−0.885728	−0.442864	−	0.896589i	$-0.646038\pi$
$929$	−26.9965	−0.885728	−0.442864	+	0.896589i	$0.646038\pi$
$930$	0	0
$931$	11.6087	0.380459
$932$	30.0301	0.983670
$933$	0	0
$934$	−8.61058	−0.281747
$935$	145.028	4.74292
$936$	0	0
$937$	23.1745	0.757078	0.378539	−	0.925585i	$-0.376427\pi$
$937$	23.1745	0.757078	0.378539	+	0.925585i	$0.376427\pi$
$938$	4.18970	0.136799
$939$	0	0
$940$	−29.1836	−0.951864
$941$	7.54048	0.245813	0.122906	−	0.992418i	$-0.460779\pi$
$941$	7.54048	0.245813	0.122906	+	0.992418i	$0.460779\pi$
$942$	0	0
$943$	7.69574	0.250608
$944$	−1.30798	−0.0425711
$945$	0	0
$946$	−40.3478	−1.31182
$947$	20.6601	0.671363	0.335681	−	0.941976i	$-0.391033\pi$
$947$	20.6601	0.671363	0.335681	+	0.941976i	$0.391033\pi$
$948$	0	0
$949$	0	0
$950$	−20.2828	−0.658059
$951$	0	0
$952$	−5.10992	−0.165613
$953$	30.2935	0.981303	0.490651	−	0.871356i	$-0.336759\pi$
$953$	30.2935	0.981303	0.490651	+	0.871356i	$0.336759\pi$
$954$	0	0
$955$	−39.6775	−1.28394
$956$	22.0978	0.714695
$957$	0	0
$958$	6.58104	0.212624
$959$	10.9035	0.352092
$960$	0	0
$961$	−30.0536	−0.969470
$962$	0	0
$963$	0	0
$964$	−10.1274	−0.326181
$965$	−57.4771	−1.85025
$966$	0	0
$967$	20.7289	0.666595	0.333298	−	0.942822i	$-0.391839\pi$
$967$	20.7289	0.666595	0.333298	+	0.942822i	$0.391839\pi$
$968$	12.5308	0.402755
$969$	0	0
$970$	34.5894	1.11060
$971$	39.1094	1.25508	0.627541	−	0.778584i	$-0.284062\pi$
$971$	39.1094	1.25508	0.627541	+	0.778584i	$0.284062\pi$
$972$	0	0
$973$	−4.21983	−0.135282
$974$	23.2760	0.745813
$975$	0	0
$976$	−0.396125	−0.0126796
$977$	7.27545	0.232762	0.116381	−	0.993205i	$-0.462871\pi$
$977$	7.27545	0.232762	0.116381	+	0.993205i	$0.462871\pi$
$978$	0	0
$979$	−15.0858	−0.482143
$980$	26.4034	0.843426
$981$	0	0
$982$	15.5405	0.495917
$983$	53.4857	1.70593	0.852965	−	0.521969i	$-0.174802\pi$
$983$	53.4857	1.70593	0.852965	+	0.521969i	$0.174802\pi$
$984$	0	0
$985$	12.1860	0.388278
$986$	−24.6983	−0.786553
$987$	0	0
$988$	0	0
$989$	42.5026	1.35150
$990$	0	0
$991$	16.0575	0.510085	0.255042	−	0.966930i	$-0.417911\pi$
$991$	16.0575	0.510085	0.255042	+	0.966930i	$0.417911\pi$
$992$	0.972853	0.0308881
$993$	0	0
$994$	0.920215	0.0291874
$995$	−52.2083	−1.65512
$996$	0	0
$997$	22.4263	0.710247	0.355123	−	0.934819i	$-0.384439\pi$
$997$	22.4263	0.710247	0.355123	+	0.934819i	$0.384439\pi$
$998$	−9.53617	−0.301862
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.a.be.1.1		3
3.2	odd	2		1014.2.a.m.1.3	✓	3
12.11	even	2		8112.2.a.ce.1.3		3
13.5	odd	4		3042.2.b.r.1351.3		6
13.8	odd	4		3042.2.b.r.1351.4		6
13.12	even	2		3042.2.a.bd.1.3		3
39.2	even	12		1014.2.i.g.823.1		12
39.5	even	4		1014.2.b.g.337.4		6
39.8	even	4		1014.2.b.g.337.3		6
39.11	even	12		1014.2.i.g.823.6		12
39.17	odd	6		1014.2.e.k.991.1		6
39.20	even	12		1014.2.i.g.361.3		12
39.23	odd	6		1014.2.e.k.529.1		6
39.29	odd	6		1014.2.e.m.529.3		6
39.32	even	12		1014.2.i.g.361.4		12
39.35	odd	6		1014.2.e.m.991.3		6
39.38	odd	2		1014.2.a.o.1.1	yes	3
156.155	even	2		8112.2.a.bz.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1014.2.a.m.1.3	✓	3	3.2	odd	2
1014.2.a.o.1.1	yes	3	39.38	odd	2
1014.2.b.g.337.3		6	39.8	even	4
1014.2.b.g.337.4		6	39.5	even	4
1014.2.e.k.529.1		6	39.23	odd	6
1014.2.e.k.991.1		6	39.17	odd	6
1014.2.e.m.529.3		6	39.29	odd	6
1014.2.e.m.991.3		6	39.35	odd	6
1014.2.i.g.361.3		12	39.20	even	12
1014.2.i.g.361.4		12	39.32	even	12
1014.2.i.g.823.1		12	39.2	even	12
1014.2.i.g.823.6		12	39.11	even	12
3042.2.a.bd.1.3		3	13.12	even	2
3042.2.a.be.1.1		3	1.1	even	1	trivial
3042.2.b.r.1351.3		6	13.5	odd	4
3042.2.b.r.1351.4		6	13.8	odd	4
8112.2.a.bz.1.1		3	156.155	even	2
8112.2.a.ce.1.3		3	12.11	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -4.04892 q^{5} +0.692021 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -4.04892 q^{5} +0.692021 q^{7} +1.00000 q^{8} -4.04892 q^{10} +4.85086 q^{11} +0.692021 q^{14} +1.00000 q^{16} -7.38404 q^{17} -1.78017 q^{19} -4.04892 q^{20} +4.85086 q^{22} -5.10992 q^{23} +11.3937 q^{25} +0.692021 q^{28} +3.34481 q^{29} +0.972853 q^{31} +1.00000 q^{32} -7.38404 q^{34} -2.80194 q^{35} +1.28621 q^{37} -1.78017 q^{38} -4.04892 q^{40} -1.50604 q^{41} -8.31767 q^{43} +4.85086 q^{44} -5.10992 q^{46} +7.20775 q^{47} -6.52111 q^{49} +11.3937 q^{50} -13.4765 q^{53} -19.6407 q^{55} +0.692021 q^{56} +3.34481 q^{58} -1.30798 q^{59} -0.396125 q^{61} +0.972853 q^{62} +1.00000 q^{64} +6.05429 q^{67} -7.38404 q^{68} -2.80194 q^{70} +1.32975 q^{71} -7.65279 q^{73} +1.28621 q^{74} -1.78017 q^{76} +3.35690 q^{77} -8.33944 q^{79} -4.04892 q^{80} -1.50604 q^{82} -15.3274 q^{83} +29.8974 q^{85} -8.31767 q^{86} +4.85086 q^{88} -3.10992 q^{89} -5.10992 q^{92} +7.20775 q^{94} +7.20775 q^{95} -8.54288 q^{97} -6.52111 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} + 3 q^{8} - 3 q^{10} + q^{11} - 3 q^{14} + 3 q^{16} - 12 q^{17} - 4 q^{19} - 3 q^{20} + q^{22} - 16 q^{23} + 2 q^{25} - 3 q^{28} - 13 q^{29} + 9 q^{31} + 3 q^{32} - 12 q^{34} - 4 q^{35} + 12 q^{37} - 4 q^{38} - 3 q^{40} - 14 q^{41} - 8 q^{43} + q^{44} - 16 q^{46} + 4 q^{47} - 4 q^{49} + 2 q^{50} - 15 q^{53} - 22 q^{55} - 3 q^{56} - 13 q^{58} - 9 q^{59} - 10 q^{61} + 9 q^{62} + 3 q^{64} + 6 q^{67} - 12 q^{68} - 4 q^{70} + 6 q^{71} - 5 q^{73} + 12 q^{74} - 4 q^{76} + 6 q^{77} - 5 q^{79} - 3 q^{80} - 14 q^{82} - 7 q^{83} + 26 q^{85} - 8 q^{86} + q^{88} - 10 q^{89} - 16 q^{92} + 4 q^{94} + 4 q^{95} - 7 q^{97} - 4 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 + 3 * q^8 - 3 * q^10 + q^11 - 3 * q^14 + 3 * q^16 - 12 * q^17 - 4 * q^19 - 3 * q^20 + q^22 - 16 * q^23 + 2 * q^25 - 3 * q^28 - 13 * q^29 + 9 * q^31 + 3 * q^32 - 12 * q^34 - 4 * q^35 + 12 * q^37 - 4 * q^38 - 3 * q^40 - 14 * q^41 - 8 * q^43 + q^44 - 16 * q^46 + 4 * q^47 - 4 * q^49 + 2 * q^50 - 15 * q^53 - 22 * q^55 - 3 * q^56 - 13 * q^58 - 9 * q^59 - 10 * q^61 + 9 * q^62 + 3 * q^64 + 6 * q^67 - 12 * q^68 - 4 * q^70 + 6 * q^71 - 5 * q^73 + 12 * q^74 - 4 * q^76 + 6 * q^77 - 5 * q^79 - 3 * q^80 - 14 * q^82 - 7 * q^83 + 26 * q^85 - 8 * q^86 + q^88 - 10 * q^89 - 16 * q^92 + 4 * q^94 + 4 * q^95 - 7 * q^97 - 4 * q^98

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