LMFDB - Embedded newform 2394.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(1,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2394.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.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:	$19.1161862439$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 266)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.772866$ of defining polynomial
Character	$\chi$	$=$	2394.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.22713 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.22713 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.22713 q^{10} -4.94841 q^{11} -0.454269 q^{13} +1.00000 q^{14} +1.00000 q^{16} -7.25963 q^{17} -1.00000 q^{19} -1.22713 q^{20} +4.94841 q^{22} +1.54573 q^{23} -3.49414 q^{25} +0.454269 q^{26} -1.00000 q^{28} +4.03249 q^{29} +8.35109 q^{31} -1.00000 q^{32} +7.25963 q^{34} +1.22713 q^{35} +8.03249 q^{37} +1.00000 q^{38} +1.22713 q^{40} +2.77287 q^{41} -6.03249 q^{43} -4.94841 q^{44} -1.54573 q^{46} +0.597321 q^{47} +1.00000 q^{49} +3.49414 q^{50} -0.454269 q^{52} +8.20804 q^{53} +6.07236 q^{55} +1.00000 q^{56} -4.03249 q^{58} +2.31860 q^{59} +0.143052 q^{61} -8.35109 q^{62} +1.00000 q^{64} +0.557449 q^{65} -1.25963 q^{67} -7.25963 q^{68} -1.22713 q^{70} +0.948410 q^{71} -7.89682 q^{73} -8.03249 q^{74} -1.00000 q^{76} +4.94841 q^{77} +11.5782 q^{79} -1.22713 q^{80} -2.77287 q^{82} +7.09146 q^{83} +8.90854 q^{85} +6.03249 q^{86} +4.94841 q^{88} -2.94841 q^{89} +0.454269 q^{91} +1.54573 q^{92} -0.597321 q^{94} +1.22713 q^{95} -3.85695 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 5 * q^5 - 3 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} - 3 q^{7} - 3 q^{8} + 5 q^{10} - 3 q^{11} - 4 q^{13} + 3 q^{14} + 3 q^{16} - 6 q^{17} - 3 q^{19} - 5 q^{20} + 3 q^{22} + 2 q^{23} + 4 q^{25} + 4 q^{26} - 3 q^{28} - 5 q^{29} + 4 q^{31} - 3 q^{32} + 6 q^{34} + 5 q^{35} + 7 q^{37} + 3 q^{38} + 5 q^{40} + 7 q^{41} - q^{43} - 3 q^{44} - 2 q^{46} + 11 q^{47} + 3 q^{49} - 4 q^{50} - 4 q^{52} - 3 q^{53} - 16 q^{55} + 3 q^{56} + 5 q^{58} + 3 q^{59} + 7 q^{61} - 4 q^{62} + 3 q^{64} + 28 q^{65} + 12 q^{67} - 6 q^{68} - 5 q^{70} - 9 q^{71} - 7 q^{74} - 3 q^{76} + 3 q^{77} + 15 q^{79} - 5 q^{80} - 7 q^{82} + 16 q^{83} + 32 q^{85} + q^{86} + 3 q^{88} + 3 q^{89} + 4 q^{91} + 2 q^{92} - 11 q^{94} + 5 q^{95} - 5 q^{97} - 3 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 5 * q^5 - 3 * q^7 - 3 * q^8 + 5 * q^10 - 3 * q^11 - 4 * q^13 + 3 * q^14 + 3 * q^16 - 6 * q^17 - 3 * q^19 - 5 * q^20 + 3 * q^22 + 2 * q^23 + 4 * q^25 + 4 * q^26 - 3 * q^28 - 5 * q^29 + 4 * q^31 - 3 * q^32 + 6 * q^34 + 5 * q^35 + 7 * q^37 + 3 * q^38 + 5 * q^40 + 7 * q^41 - q^43 - 3 * q^44 - 2 * q^46 + 11 * q^47 + 3 * q^49 - 4 * q^50 - 4 * q^52 - 3 * q^53 - 16 * q^55 + 3 * q^56 + 5 * q^58 + 3 * q^59 + 7 * q^61 - 4 * q^62 + 3 * q^64 + 28 * q^65 + 12 * q^67 - 6 * q^68 - 5 * q^70 - 9 * q^71 - 7 * q^74 - 3 * q^76 + 3 * q^77 + 15 * q^79 - 5 * q^80 - 7 * q^82 + 16 * q^83 + 32 * q^85 + q^86 + 3 * q^88 + 3 * q^89 + 4 * q^91 + 2 * q^92 - 11 * q^94 + 5 * q^95 - 5 * q^97 - 3 * 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
$5$	−1.22713	−0.548791	−0.274396	−	0.961617i	$-0.588478\pi$
$5$	−1.22713	−0.548791	−0.274396	+	0.961617i	$0.588478\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.22713	0.388054
$11$	−4.94841	−1.49200	−0.746001	−	0.665945i	$-0.768029\pi$
$11$	−4.94841	−1.49200	−0.746001	+	0.665945i	$0.768029\pi$
$12$	0	0
$13$	−0.454269	−0.125992	−0.0629958	−	0.998014i	$-0.520065\pi$
$13$	−0.454269	−0.125992	−0.0629958	+	0.998014i	$0.520065\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.25963	−1.76072	−0.880359	−	0.474308i	$-0.842698\pi$
$17$	−7.25963	−1.76072	−0.880359	+	0.474308i	$0.842698\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.22713	−0.274396
$21$	0	0
$22$	4.94841	1.05500
$23$	1.54573	0.322307	0.161154	−	0.986929i	$-0.448479\pi$
$23$	1.54573	0.322307	0.161154	+	0.986929i	$0.448479\pi$
$24$	0	0
$25$	−3.49414	−0.698828
$26$	0.454269	0.0890895
$27$	0	0
$28$	−1.00000	−0.188982
$29$	4.03249	0.748815	0.374407	−	0.927264i	$-0.377846\pi$
$29$	4.03249	0.748815	0.374407	+	0.927264i	$0.377846\pi$
$30$	0	0
$31$	8.35109	1.49990	0.749950	−	0.661495i	$-0.230078\pi$
$31$	8.35109	1.49990	0.749950	+	0.661495i	$0.230078\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	7.25963	1.24502
$35$	1.22713	0.207424
$36$	0	0
$37$	8.03249	1.32053	0.660267	−	0.751031i	$-0.270443\pi$
$37$	8.03249	1.32053	0.660267	+	0.751031i	$0.270443\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	1.22713	0.194027
$41$	2.77287	0.433049	0.216524	−	0.976277i	$-0.430528\pi$
$41$	2.77287	0.433049	0.216524	+	0.976277i	$0.430528\pi$
$42$	0	0
$43$	−6.03249	−0.919946	−0.459973	−	0.887933i	$-0.652141\pi$
$43$	−6.03249	−0.919946	−0.459973	+	0.887933i	$0.652141\pi$
$44$	−4.94841	−0.746001
$45$	0	0
$46$	−1.54573	−0.227906
$47$	0.597321	0.0871282	0.0435641	−	0.999051i	$-0.486129\pi$
$47$	0.597321	0.0871282	0.0435641	+	0.999051i	$0.486129\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.49414	0.494146
$51$	0	0
$52$	−0.454269	−0.0629958
$53$	8.20804	1.12746	0.563730	−	0.825959i	$-0.309366\pi$
$53$	8.20804	1.12746	0.563730	+	0.825959i	$0.309366\pi$
$54$	0	0
$55$	6.07236	0.818797
$56$	1.00000	0.133631
$57$	0	0
$58$	−4.03249	−0.529492
$59$	2.31860	0.301856	0.150928	−	0.988545i	$-0.451774\pi$
$59$	2.31860	0.301856	0.150928	+	0.988545i	$0.451774\pi$
$60$	0	0
$61$	0.143052	0.0183160	0.00915798	−	0.999958i	$-0.497085\pi$
$61$	0.143052	0.0183160	0.00915798	+	0.999958i	$0.497085\pi$
$62$	−8.35109	−1.06059
$63$	0	0
$64$	1.00000	0.125000
$65$	0.557449	0.0691430
$66$	0	0
$67$	−1.25963	−0.153888	−0.0769439	−	0.997035i	$-0.524516\pi$
$67$	−1.25963	−0.153888	−0.0769439	+	0.997035i	$0.524516\pi$
$68$	−7.25963	−0.880359
$69$	0	0
$70$	−1.22713	−0.146671
$71$	0.948410	0.112556	0.0562778	−	0.998415i	$-0.482077\pi$
$71$	0.948410	0.112556	0.0562778	+	0.998415i	$0.482077\pi$
$72$	0	0
$73$	−7.89682	−0.924253	−0.462126	−	0.886814i	$-0.652913\pi$
$73$	−7.89682	−0.924253	−0.462126	+	0.886814i	$0.652913\pi$
$74$	−8.03249	−0.933758
$75$	0	0
$76$	−1.00000	−0.114708
$77$	4.94841	0.563924
$78$	0	0
$79$	11.5782	1.30265	0.651326	−	0.758798i	$-0.274213\pi$
$79$	11.5782	1.30265	0.651326	+	0.758798i	$0.274213\pi$
$80$	−1.22713	−0.137198
$81$	0	0
$82$	−2.77287	−0.306212
$83$	7.09146	0.778389	0.389195	−	0.921156i	$-0.372753\pi$
$83$	7.09146	0.778389	0.389195	+	0.921156i	$0.372753\pi$
$84$	0	0
$85$	8.90854	0.966267
$86$	6.03249	0.650500
$87$	0	0
$88$	4.94841	0.527502
$89$	−2.94841	−0.312531	−0.156265	−	0.987715i	$-0.549946\pi$
$89$	−2.94841	−0.312531	−0.156265	+	0.987715i	$0.549946\pi$
$90$	0	0
$91$	0.454269	0.0476203
$92$	1.54573	0.161154
$93$	0	0
$94$	−0.597321	−0.0616090
$95$	1.22713	0.125901
$96$	0	0
$97$	−3.85695	−0.391614	−0.195807	−	0.980642i	$-0.562733\pi$
$97$	−3.85695	−0.391614	−0.195807	+	0.980642i	$0.562733\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−3.49414	−0.349414
$101$	−4.18292	−0.416217	−0.208108	−	0.978106i	$-0.566731\pi$
$101$	−4.18292	−0.416217	−0.208108	+	0.978106i	$0.566731\pi$
$102$	0	0
$103$	15.1564	1.49341	0.746705	−	0.665156i	$-0.231635\pi$
$103$	15.1564	1.49341	0.746705	+	0.665156i	$0.231635\pi$
$104$	0.454269	0.0445447
$105$	0	0
$106$	−8.20804	−0.797235
$107$	−3.15645	−0.305145	−0.152573	−	0.988292i	$-0.548756\pi$
$107$	−3.15645	−0.305145	−0.152573	+	0.988292i	$0.548756\pi$
$108$	0	0
$109$	−0.208036	−0.0199263	−0.00996314	−	0.999950i	$-0.503171\pi$
$109$	−0.208036	−0.0199263	−0.00996314	+	0.999950i	$0.503171\pi$
$110$	−6.07236	−0.578977
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	0.168164	0.0158196	0.00790978	−	0.999969i	$-0.497482\pi$
$113$	0.168164	0.0158196	0.00790978	+	0.999969i	$0.497482\pi$
$114$	0	0
$115$	−1.89682	−0.176879
$116$	4.03249	0.374407
$117$	0	0
$118$	−2.31860	−0.213444
$119$	7.25963	0.665489
$120$	0	0
$121$	13.4868	1.22607
$122$	−0.143052	−0.0129513
$123$	0	0
$124$	8.35109	0.749950
$125$	10.4235	0.932302
$126$	0	0
$127$	−5.85695	−0.519720	−0.259860	−	0.965646i	$-0.583676\pi$
$127$	−5.85695	−0.519720	−0.259860	+	0.965646i	$0.583676\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−0.557449	−0.0488915
$131$	5.19464	0.453858	0.226929	−	0.973911i	$-0.427131\pi$
$131$	5.19464	0.453858	0.226929	+	0.973911i	$0.427131\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	1.25963	0.108815
$135$	0	0
$136$	7.25963	0.622508
$137$	15.1889	1.29768	0.648839	−	0.760925i	$-0.275255\pi$
$137$	15.1889	1.29768	0.648839	+	0.760925i	$0.275255\pi$
$138$	0	0
$139$	−9.61072	−0.815170	−0.407585	−	0.913167i	$-0.633629\pi$
$139$	−9.61072	−0.815170	−0.407585	+	0.913167i	$0.633629\pi$
$140$	1.22713	0.103712
$141$	0	0
$142$	−0.948410	−0.0795888
$143$	2.24791	0.187980
$144$	0	0
$145$	−4.94841	−0.410943
$146$	7.89682	0.653545
$147$	0	0
$148$	8.03249	0.660267
$149$	15.2596	1.25012	0.625059	−	0.780578i	$-0.285075\pi$
$149$	15.2596	1.25012	0.625059	+	0.780578i	$0.285075\pi$
$150$	0	0
$151$	−8.70218	−0.708173	−0.354087	−	0.935213i	$-0.615208\pi$
$151$	−8.70218	−0.708173	−0.354087	+	0.935213i	$0.615208\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−4.94841	−0.398754
$155$	−10.2479	−0.823132
$156$	0	0
$157$	−13.4677	−1.07484	−0.537418	−	0.843316i	$-0.680600\pi$
$157$	−13.4677	−1.07484	−0.537418	+	0.843316i	$0.680600\pi$
$158$	−11.5782	−0.921114
$159$	0	0
$160$	1.22713	0.0970135
$161$	−1.54573	−0.121821
$162$	0	0
$163$	−0.662305	−0.0518758	−0.0259379	−	0.999664i	$-0.508257\pi$
$163$	−0.662305	−0.0518758	−0.0259379	+	0.999664i	$0.508257\pi$
$164$	2.77287	0.216524
$165$	0	0
$166$	−7.09146	−0.550404
$167$	−13.9618	−1.08040	−0.540198	−	0.841538i	$-0.681651\pi$
$167$	−13.9618	−1.08040	−0.540198	+	0.841538i	$0.681651\pi$
$168$	0	0
$169$	−12.7936	−0.984126
$170$	−8.90854	−0.683254
$171$	0	0
$172$	−6.03249	−0.459973
$173$	11.8968	0.904498	0.452249	−	0.891892i	$-0.350622\pi$
$173$	11.8968	0.904498	0.452249	+	0.891892i	$0.350622\pi$
$174$	0	0
$175$	3.49414	0.264132
$176$	−4.94841	−0.373000
$177$	0	0
$178$	2.94841	0.220993
$179$	22.2479	1.66289	0.831443	−	0.555609i	$-0.187515\pi$
$179$	22.2479	1.66289	0.831443	+	0.555609i	$0.187515\pi$
$180$	0	0
$181$	−14.4161	−1.07154	−0.535769	−	0.844365i	$-0.679978\pi$
$181$	−14.4161	−1.07154	−0.535769	+	0.844365i	$0.679978\pi$
$182$	−0.454269	−0.0336727
$183$	0	0
$184$	−1.54573	−0.113953
$185$	−9.85695	−0.724697
$186$	0	0
$187$	35.9236	2.62699
$188$	0.597321	0.0435641
$189$	0	0
$190$	−1.22713	−0.0890257
$191$	26.2479	1.89923	0.949616	−	0.313416i	$-0.101473\pi$
$191$	26.2479	1.89923	0.949616	+	0.313416i	$0.101473\pi$
$192$	0	0
$193$	−9.71390	−0.699221	−0.349611	−	0.936895i	$-0.613686\pi$
$193$	−9.71390	−0.699221	−0.349611	+	0.936895i	$0.613686\pi$
$194$	3.85695	0.276913
$195$	0	0
$196$	1.00000	0.0714286
$197$	−16.5193	−1.17695	−0.588474	−	0.808516i	$-0.700271\pi$
$197$	−16.5193	−1.17695	−0.588474	+	0.808516i	$0.700271\pi$
$198$	0	0
$199$	21.7464	1.54156	0.770780	−	0.637101i	$-0.219867\pi$
$199$	21.7464	1.54156	0.770780	+	0.637101i	$0.219867\pi$
$200$	3.49414	0.247073
$201$	0	0
$202$	4.18292	0.294310
$203$	−4.03249	−0.283025
$204$	0	0
$205$	−3.40268	−0.237653
$206$	−15.1564	−1.05600
$207$	0	0
$208$	−0.454269	−0.0314979
$209$	4.94841	0.342289
$210$	0	0
$211$	4.70218	0.323711	0.161856	−	0.986814i	$-0.448252\pi$
$211$	4.70218	0.323711	0.161856	+	0.986814i	$0.448252\pi$
$212$	8.20804	0.563730
$213$	0	0
$214$	3.15645	0.215770
$215$	7.40268	0.504859
$216$	0	0
$217$	−8.35109	−0.566909
$218$	0.208036	0.0140900
$219$	0	0
$220$	6.07236	0.409399
$221$	3.29782	0.221836
$222$	0	0
$223$	28.4161	1.90288	0.951440	−	0.307833i	$-0.0996037\pi$
$223$	28.4161	1.90288	0.951440	+	0.307833i	$0.0996037\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−0.168164	−0.0111861
$227$	14.5193	0.963677	0.481838	−	0.876260i	$-0.339969\pi$
$227$	14.5193	0.963677	0.481838	+	0.876260i	$0.339969\pi$
$228$	0	0
$229$	17.2271	1.13840	0.569201	−	0.822199i	$-0.307253\pi$
$229$	17.2271	1.13840	0.569201	+	0.822199i	$0.307253\pi$
$230$	1.89682	0.125073
$231$	0	0
$232$	−4.03249	−0.264746
$233$	−6.31122	−0.413462	−0.206731	−	0.978398i	$-0.566282\pi$
$233$	−6.31122	−0.413462	−0.206731	+	0.978398i	$0.566282\pi$
$234$	0	0
$235$	−0.732993	−0.0478152
$236$	2.31860	0.150928
$237$	0	0
$238$	−7.25963	−0.470572
$239$	1.89682	0.122695	0.0613475	−	0.998116i	$-0.480460\pi$
$239$	1.89682	0.122695	0.0613475	+	0.998116i	$0.480460\pi$
$240$	0	0
$241$	−3.61642	−0.232954	−0.116477	−	0.993193i	$-0.537160\pi$
$241$	−3.61642	−0.232954	−0.116477	+	0.993193i	$0.537160\pi$
$242$	−13.4868	−0.866962
$243$	0	0
$244$	0.143052	0.00915798
$245$	−1.22713	−0.0783987
$246$	0	0
$247$	0.454269	0.0289044
$248$	−8.35109	−0.530295
$249$	0	0
$250$	−10.4235	−0.659237
$251$	−8.98828	−0.567335	−0.283668	−	0.958923i	$-0.591551\pi$
$251$	−8.98828	−0.567335	−0.283668	+	0.958923i	$0.591551\pi$
$252$	0	0
$253$	−7.64891	−0.480883
$254$	5.85695	0.367498
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.68140	0.479153	0.239576	−	0.970878i	$-0.422991\pi$
$257$	7.68140	0.479153	0.239576	+	0.970878i	$0.422991\pi$
$258$	0	0
$259$	−8.03249	−0.499115
$260$	0.557449	0.0345715
$261$	0	0
$262$	−5.19464	−0.320926
$263$	−14.2331	−0.877654	−0.438827	−	0.898572i	$-0.644606\pi$
$263$	−14.2331	−0.877654	−0.438827	+	0.898572i	$0.644606\pi$
$264$	0	0
$265$	−10.0724	−0.618740
$266$	−1.00000	−0.0613139
$267$	0	0
$268$	−1.25963	−0.0769439
$269$	−16.5193	−1.00720	−0.503598	−	0.863938i	$-0.667991\pi$
$269$	−16.5193	−1.00720	−0.503598	+	0.863938i	$0.667991\pi$
$270$	0	0
$271$	−19.9293	−1.21062	−0.605310	−	0.795990i	$-0.706951\pi$
$271$	−19.9293	−1.21062	−0.605310	+	0.795990i	$0.706951\pi$
$272$	−7.25963	−0.440180
$273$	0	0
$274$	−15.1889	−0.917597
$275$	17.2904	1.04265
$276$	0	0
$277$	28.9353	1.73856	0.869278	−	0.494324i	$-0.164584\pi$
$277$	28.9353	1.73856	0.869278	+	0.494324i	$0.164584\pi$
$278$	9.61072	0.576412
$279$	0	0
$280$	−1.22713	−0.0733353
$281$	13.6489	0.814226	0.407113	−	0.913378i	$-0.366536\pi$
$281$	13.6489	0.814226	0.407113	+	0.913378i	$0.366536\pi$
$282$	0	0
$283$	4.70218	0.279515	0.139758	−	0.990186i	$-0.455368\pi$
$283$	4.70218	0.279515	0.139758	+	0.990186i	$0.455368\pi$
$284$	0.948410	0.0562778
$285$	0	0
$286$	−2.24791	−0.132922
$287$	−2.77287	−0.163677
$288$	0	0
$289$	35.7022	2.10013
$290$	4.94841	0.290581
$291$	0	0
$292$	−7.89682	−0.462126
$293$	22.0650	1.28905	0.644525	−	0.764583i	$-0.277055\pi$
$293$	22.0650	1.28905	0.644525	+	0.764583i	$0.277055\pi$
$294$	0	0
$295$	−2.84523	−0.165656
$296$	−8.03249	−0.466879
$297$	0	0
$298$	−15.2596	−0.883966
$299$	−0.702178	−0.0406080
$300$	0	0
$301$	6.03249	0.347707
$302$	8.70218	0.500754
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	−0.175544	−0.0100516
$306$	0	0
$307$	−10.0473	−0.573427	−0.286713	−	0.958016i	$-0.592563\pi$
$307$	−10.0473	−0.573427	−0.286713	+	0.958016i	$0.592563\pi$
$308$	4.94841	0.281962
$309$	0	0
$310$	10.2479	0.582042
$311$	−10.4941	−0.595068	−0.297534	−	0.954711i	$-0.596164\pi$
$311$	−10.4941	−0.595068	−0.297534	+	0.954711i	$0.596164\pi$
$312$	0	0
$313$	−5.77888	−0.326642	−0.163321	−	0.986573i	$-0.552221\pi$
$313$	−5.77888	−0.326642	−0.163321	+	0.986573i	$0.552221\pi$
$314$	13.4677	0.760024
$315$	0	0
$316$	11.5782	0.651326
$317$	−12.9102	−0.725110	−0.362555	−	0.931962i	$-0.618095\pi$
$317$	−12.9102	−0.725110	−0.362555	+	0.931962i	$0.618095\pi$
$318$	0	0
$319$	−19.9544	−1.11723
$320$	−1.22713	−0.0685989
$321$	0	0
$322$	1.54573	0.0861402
$323$	7.25963	0.403936
$324$	0	0
$325$	1.58728	0.0880464
$326$	0.662305	0.0366817
$327$	0	0
$328$	−2.77287	−0.153106
$329$	−0.597321	−0.0329314
$330$	0	0
$331$	30.8703	1.69679	0.848394	−	0.529366i	$-0.177570\pi$
$331$	30.8703	1.69679	0.848394	+	0.529366i	$0.177570\pi$
$332$	7.09146	0.389195
$333$	0	0
$334$	13.9618	0.763956
$335$	1.54573	0.0844523
$336$	0	0
$337$	19.6757	1.07180	0.535902	−	0.844280i	$-0.319972\pi$
$337$	19.6757	1.07180	0.535902	+	0.844280i	$0.319972\pi$
$338$	12.7936	0.695882
$339$	0	0
$340$	8.90854	0.483133
$341$	−41.3246	−2.23785
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	6.03249	0.325250
$345$	0	0
$346$	−11.8968	−0.639577
$347$	0.908538	0.0487729	0.0243864	−	0.999703i	$-0.492237\pi$
$347$	0.908538	0.0487729	0.0243864	+	0.999703i	$0.492237\pi$
$348$	0	0
$349$	12.1829	0.652137	0.326068	−	0.945346i	$-0.394276\pi$
$349$	12.1829	0.652137	0.326068	+	0.945346i	$0.394276\pi$
$350$	−3.49414	−0.186770
$351$	0	0
$352$	4.94841	0.263751
$353$	−23.9618	−1.27536	−0.637679	−	0.770302i	$-0.720105\pi$
$353$	−23.9618	−1.27536	−0.637679	+	0.770302i	$0.720105\pi$
$354$	0	0
$355$	−1.16383	−0.0617695
$356$	−2.94841	−0.156265
$357$	0	0
$358$	−22.2479	−1.17584
$359$	12.7672	0.673825	0.336913	−	0.941536i	$-0.390617\pi$
$359$	12.7672	0.673825	0.336913	+	0.941536i	$0.390617\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	14.4161	0.757692
$363$	0	0
$364$	0.454269	0.0238102
$365$	9.69046	0.507222
$366$	0	0
$367$	−3.68878	−0.192553	−0.0962765	−	0.995355i	$-0.530693\pi$
$367$	−3.68878	−0.192553	−0.0962765	+	0.995355i	$0.530693\pi$
$368$	1.54573	0.0805768
$369$	0	0
$370$	9.85695	0.512438
$371$	−8.20804	−0.426140
$372$	0	0
$373$	25.0060	1.29476	0.647381	−	0.762166i	$-0.275864\pi$
$373$	25.0060	1.29476	0.647381	+	0.762166i	$0.275864\pi$
$374$	−35.9236	−1.85757
$375$	0	0
$376$	−0.597321	−0.0308045
$377$	−1.83184	−0.0943443
$378$	0	0
$379$	0.286105	0.0146962	0.00734810	−	0.999973i	$-0.497661\pi$
$379$	0.286105	0.0146962	0.00734810	+	0.999973i	$0.497661\pi$
$380$	1.22713	0.0629507
$381$	0	0
$382$	−26.2479	−1.34296
$383$	0.973522	0.0497446	0.0248723	−	0.999691i	$-0.492082\pi$
$383$	0.973522	0.0497446	0.0248723	+	0.999691i	$0.492082\pi$
$384$	0	0
$385$	−6.07236	−0.309476
$386$	9.71390	0.494424
$387$	0	0
$388$	−3.85695	−0.195807
$389$	−0.519253	−0.0263272	−0.0131636	−	0.999913i	$-0.504190\pi$
$389$	−0.519253	−0.0263272	−0.0131636	+	0.999913i	$0.504190\pi$
$390$	0	0
$391$	−11.2214	−0.567492
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	16.5193	0.832228
$395$	−14.2080	−0.714884
$396$	0	0
$397$	−23.2847	−1.16863	−0.584314	−	0.811528i	$-0.698636\pi$
$397$	−23.2847	−1.16863	−0.584314	+	0.811528i	$0.698636\pi$
$398$	−21.7464	−1.09005
$399$	0	0
$400$	−3.49414	−0.174707
$401$	−14.6874	−0.733455	−0.366727	−	0.930328i	$-0.619522\pi$
$401$	−14.6874	−0.733455	−0.366727	+	0.930328i	$0.619522\pi$
$402$	0	0
$403$	−3.79364	−0.188975
$404$	−4.18292	−0.208108
$405$	0	0
$406$	4.03249	0.200129
$407$	−39.7481	−1.97024
$408$	0	0
$409$	−9.46766	−0.468146	−0.234073	−	0.972219i	$-0.575205\pi$
$409$	−9.46766	−0.468146	−0.234073	+	0.972219i	$0.575205\pi$
$410$	3.40268	0.168046
$411$	0	0
$412$	15.1564	0.746705
$413$	−2.31860	−0.114091
$414$	0	0
$415$	−8.70218	−0.427173
$416$	0.454269	0.0222724
$417$	0	0
$418$	−4.94841	−0.242035
$419$	33.3896	1.63119	0.815594	−	0.578624i	$-0.196410\pi$
$419$	33.3896	1.63119	0.815594	+	0.578624i	$0.196410\pi$
$420$	0	0
$421$	−22.9085	−1.11649	−0.558247	−	0.829675i	$-0.688526\pi$
$421$	−22.9085	−1.11649	−0.558247	+	0.829675i	$0.688526\pi$
$422$	−4.70218	−0.228898
$423$	0	0
$424$	−8.20804	−0.398617
$425$	25.3662	1.23044
$426$	0	0
$427$	−0.143052	−0.00692279
$428$	−3.15645	−0.152573
$429$	0	0
$430$	−7.40268	−0.356989
$431$	32.5859	1.56961	0.784804	−	0.619744i	$-0.212764\pi$
$431$	32.5859	1.56961	0.784804	+	0.619744i	$0.212764\pi$
$432$	0	0
$433$	−3.61642	−0.173794	−0.0868970	−	0.996217i	$-0.527695\pi$
$433$	−3.61642	−0.173794	−0.0868970	+	0.996217i	$0.527695\pi$
$434$	8.35109	0.400865
$435$	0	0
$436$	−0.208036	−0.00996314
$437$	−1.54573	−0.0739423
$438$	0	0
$439$	−19.4426	−0.927942	−0.463971	−	0.885850i	$-0.653576\pi$
$439$	−19.4426	−0.927942	−0.463971	+	0.885850i	$0.653576\pi$
$440$	−6.07236	−0.289489
$441$	0	0
$442$	−3.29782	−0.156861
$443$	5.76115	0.273720	0.136860	−	0.990590i	$-0.456299\pi$
$443$	5.76115	0.273720	0.136860	+	0.990590i	$0.456299\pi$
$444$	0	0
$445$	3.61810	0.171514
$446$	−28.4161	−1.34554
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−1.72866	−0.0815803	−0.0407902	−	0.999168i	$-0.512988\pi$
$449$	−1.72866	−0.0815803	−0.0407902	+	0.999168i	$0.512988\pi$
$450$	0	0
$451$	−13.7213	−0.646110
$452$	0.168164	0.00790978
$453$	0	0
$454$	−14.5193	−0.681422
$455$	−0.557449	−0.0261336
$456$	0	0
$457$	−10.9336	−0.511455	−0.255727	−	0.966749i	$-0.582315\pi$
$457$	−10.9336	−0.511455	−0.255727	+	0.966749i	$0.582315\pi$
$458$	−17.2271	−0.804971
$459$	0	0
$460$	−1.89682	−0.0884397
$461$	27.0784	1.26117	0.630583	−	0.776122i	$-0.282816\pi$
$461$	27.0784	1.26117	0.630583	+	0.776122i	$0.282816\pi$
$462$	0	0
$463$	−9.32461	−0.433351	−0.216676	−	0.976244i	$-0.569521\pi$
$463$	−9.32461	−0.433351	−0.216676	+	0.976244i	$0.569521\pi$
$464$	4.03249	0.187204
$465$	0	0
$466$	6.31122	0.292361
$467$	−26.5842	−1.23017	−0.615086	−	0.788460i	$-0.710879\pi$
$467$	−26.5842	−1.23017	−0.615086	+	0.788460i	$0.710879\pi$
$468$	0	0
$469$	1.25963	0.0581641
$470$	0.732993	0.0338105
$471$	0	0
$472$	−2.31860	−0.106722
$473$	29.8512	1.37256
$474$	0	0
$475$	3.49414	0.160322
$476$	7.25963	0.332744
$477$	0	0
$478$	−1.89682	−0.0867585
$479$	−36.9028	−1.68613	−0.843067	−	0.537809i	$-0.819252\pi$
$479$	−36.9028	−1.68613	−0.843067	+	0.537809i	$0.819252\pi$
$480$	0	0
$481$	−3.64891	−0.166376
$482$	3.61642	0.164723
$483$	0	0
$484$	13.4868	0.613035
$485$	4.73299	0.214914
$486$	0	0
$487$	34.4338	1.56034	0.780172	−	0.625565i	$-0.215131\pi$
$487$	34.4338	1.56034	0.780172	+	0.625565i	$0.215131\pi$
$488$	−0.143052	−0.00647567
$489$	0	0
$490$	1.22713	0.0554363
$491$	−33.0385	−1.49101	−0.745503	−	0.666502i	$-0.767791\pi$
$491$	−33.0385	−1.49101	−0.745503	+	0.666502i	$0.767791\pi$
$492$	0	0
$493$	−29.2744	−1.31845
$494$	−0.454269	−0.0204385
$495$	0	0
$496$	8.35109	0.374975
$497$	−0.948410	−0.0425420
$498$	0	0
$499$	38.7347	1.73400	0.867001	−	0.498306i	$-0.166045\pi$
$499$	38.7347	1.73400	0.867001	+	0.498306i	$0.166045\pi$
$500$	10.4235	0.466151
$501$	0	0
$502$	8.98828	0.401167
$503$	10.5250	0.469285	0.234642	−	0.972082i	$-0.424608\pi$
$503$	10.5250	0.469285	0.234642	+	0.972082i	$0.424608\pi$
$504$	0	0
$505$	5.13301	0.228416
$506$	7.64891	0.340036
$507$	0	0
$508$	−5.85695	−0.259860
$509$	14.9883	0.664344	0.332172	−	0.943219i	$-0.392219\pi$
$509$	14.9883	0.664344	0.332172	+	0.943219i	$0.392219\pi$
$510$	0	0
$511$	7.89682	0.349335
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−7.68140	−0.338812
$515$	−18.5990	−0.819570
$516$	0	0
$517$	−2.95579	−0.129995
$518$	8.03249	0.352927
$519$	0	0
$520$	−0.557449	−0.0244458
$521$	−34.1300	−1.49526	−0.747631	−	0.664115i	$-0.768809\pi$
$521$	−34.1300	−1.49526	−0.747631	+	0.664115i	$0.768809\pi$
$522$	0	0
$523$	−17.4807	−0.764380	−0.382190	−	0.924084i	$-0.624830\pi$
$523$	−17.4807	−0.764380	−0.382190	+	0.924084i	$0.624830\pi$
$524$	5.19464	0.226929
$525$	0	0
$526$	14.2331	0.620595
$527$	−60.6258	−2.64090
$528$	0	0
$529$	−20.6107	−0.896118
$530$	10.0724	0.437516
$531$	0	0
$532$	1.00000	0.0433555
$533$	−1.25963	−0.0545605
$534$	0	0
$535$	3.87338	0.167461
$536$	1.25963	0.0544076
$537$	0	0
$538$	16.5193	0.712196
$539$	−4.94841	−0.213143
$540$	0	0
$541$	28.8054	1.23844	0.619220	−	0.785218i	$-0.287449\pi$
$541$	28.8054	1.23844	0.619220	+	0.785218i	$0.287449\pi$
$542$	19.9293	0.856037
$543$	0	0
$544$	7.25963	0.311254
$545$	0.255289	0.0109354
$546$	0	0
$547$	25.4693	1.08899	0.544495	−	0.838764i	$-0.316721\pi$
$547$	25.4693	1.08899	0.544495	+	0.838764i	$0.316721\pi$
$548$	15.1889	0.648839
$549$	0	0
$550$	−17.2904	−0.737267
$551$	−4.03249	−0.171790
$552$	0	0
$553$	−11.5782	−0.492356
$554$	−28.9353	−1.22934
$555$	0	0
$556$	−9.61072	−0.407585
$557$	−30.4308	−1.28940	−0.644698	−	0.764437i	$-0.723017\pi$
$557$	−30.4308	−1.28940	−0.644698	+	0.764437i	$0.723017\pi$
$558$	0	0
$559$	2.74037	0.115905
$560$	1.22713	0.0518559
$561$	0	0
$562$	−13.6489	−0.575745
$563$	5.84219	0.246219	0.123109	−	0.992393i	$-0.460713\pi$
$563$	5.84219	0.246219	0.123109	+	0.992393i	$0.460713\pi$
$564$	0	0
$565$	−0.206360	−0.00868164
$566$	−4.70218	−0.197647
$567$	0	0
$568$	−0.948410	−0.0397944
$569$	10.9883	0.460653	0.230326	−	0.973113i	$-0.426021\pi$
$569$	10.9883	0.460653	0.230326	+	0.973113i	$0.426021\pi$
$570$	0	0
$571$	0.662305	0.0277166	0.0138583	−	0.999904i	$-0.495589\pi$
$571$	0.662305	0.0277166	0.0138583	+	0.999904i	$0.495589\pi$
$572$	2.24791	0.0939898
$573$	0	0
$574$	2.77287	0.115737
$575$	−5.40100	−0.225237
$576$	0	0
$577$	−39.9766	−1.66425	−0.832123	−	0.554591i	$-0.812875\pi$
$577$	−39.9766	−1.66425	−0.832123	+	0.554591i	$0.812875\pi$
$578$	−35.7022	−1.48501
$579$	0	0
$580$	−4.94841	−0.205472
$581$	−7.09146	−0.294203
$582$	0	0
$583$	−40.6167	−1.68217
$584$	7.89682	0.326773
$585$	0	0
$586$	−22.0650	−0.911496
$587$	24.6372	1.01689	0.508443	−	0.861096i	$-0.330221\pi$
$587$	24.6372	1.01689	0.508443	+	0.861096i	$0.330221\pi$
$588$	0	0
$589$	−8.35109	−0.344101
$590$	2.84523	0.117136
$591$	0	0
$592$	8.03249	0.330133
$593$	−16.8054	−0.690113	−0.345057	−	0.938582i	$-0.612140\pi$
$593$	−16.8054	−0.690113	−0.345057	+	0.938582i	$0.612140\pi$
$594$	0	0
$595$	−8.90854	−0.365214
$596$	15.2596	0.625059
$597$	0	0
$598$	0.702178	0.0287142
$599$	4.03987	0.165065	0.0825324	−	0.996588i	$-0.473699\pi$
$599$	4.03987	0.165065	0.0825324	+	0.996588i	$0.473699\pi$
$600$	0	0
$601$	−16.5193	−0.673834	−0.336917	−	0.941534i	$-0.609384\pi$
$601$	−16.5193	−0.673834	−0.336917	+	0.941534i	$0.609384\pi$
$602$	−6.03249	−0.245866
$603$	0	0
$604$	−8.70218	−0.354087
$605$	−16.5501	−0.672856
$606$	0	0
$607$	−24.6224	−0.999394	−0.499697	−	0.866200i	$-0.666555\pi$
$607$	−24.6224	−0.999394	−0.499697	+	0.866200i	$0.666555\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0.175544	0.00710758
$611$	−0.271344	−0.0109774
$612$	0	0
$613$	−7.12966	−0.287964	−0.143982	−	0.989580i	$-0.545991\pi$
$613$	−7.12966	−0.287964	−0.143982	+	0.989580i	$0.545991\pi$
$614$	10.0473	0.405474
$615$	0	0
$616$	−4.94841	−0.199377
$617$	32.1625	1.29481	0.647406	−	0.762145i	$-0.275854\pi$
$617$	32.1625	1.29481	0.647406	+	0.762145i	$0.275854\pi$
$618$	0	0
$619$	−22.9501	−0.922442	−0.461221	−	0.887285i	$-0.652588\pi$
$619$	−22.9501	−0.922442	−0.461221	+	0.887285i	$0.652588\pi$
$620$	−10.2479	−0.411566
$621$	0	0
$622$	10.4941	0.420777
$623$	2.94841	0.118126
$624$	0	0
$625$	4.67973	0.187189
$626$	5.77888	0.230970
$627$	0	0
$628$	−13.4677	−0.537418
$629$	−58.3129	−2.32509
$630$	0	0
$631$	3.66367	0.145848	0.0729242	−	0.997337i	$-0.476767\pi$
$631$	3.66367	0.145848	0.0729242	+	0.997337i	$0.476767\pi$
$632$	−11.5782	−0.460557
$633$	0	0
$634$	12.9102	0.512730
$635$	7.18726	0.285218
$636$	0	0
$637$	−0.454269	−0.0179988
$638$	19.9544	0.790003
$639$	0	0
$640$	1.22713	0.0485067
$641$	34.5460	1.36449	0.682243	−	0.731125i	$-0.261004\pi$
$641$	34.5460	1.36449	0.682243	+	0.731125i	$0.261004\pi$
$642$	0	0
$643$	28.7022	1.13190	0.565952	−	0.824438i	$-0.308509\pi$
$643$	28.7022	1.13190	0.565952	+	0.824438i	$0.308509\pi$
$644$	−1.54573	−0.0609103
$645$	0	0
$646$	−7.25963	−0.285626
$647$	12.2007	0.479657	0.239829	−	0.970815i	$-0.422909\pi$
$647$	12.2007	0.479657	0.239829	+	0.970815i	$0.422909\pi$
$648$	0	0
$649$	−11.4734	−0.450369
$650$	−1.58728	−0.0622582
$651$	0	0
$652$	−0.662305	−0.0259379
$653$	−33.5223	−1.31183	−0.655914	−	0.754835i	$-0.727717\pi$
$653$	−33.5223	−1.31183	−0.655914	+	0.754835i	$0.727717\pi$
$654$	0	0
$655$	−6.37452	−0.249073
$656$	2.77287	0.108262
$657$	0	0
$658$	0.597321	0.0232860
$659$	33.7407	1.31435	0.657175	−	0.753738i	$-0.271751\pi$
$659$	33.7407	1.31435	0.657175	+	0.753738i	$0.271751\pi$
$660$	0	0
$661$	15.2449	0.592957	0.296478	−	0.955040i	$-0.404188\pi$
$661$	15.2449	0.592957	0.296478	+	0.955040i	$0.404188\pi$
$662$	−30.8703	−1.19981
$663$	0	0
$664$	−7.09146	−0.275202
$665$	−1.22713	−0.0475862
$666$	0	0
$667$	6.23315	0.241348
$668$	−13.9618	−0.540198
$669$	0	0
$670$	−1.54573	−0.0597168
$671$	−0.707881	−0.0273275
$672$	0	0
$673$	−46.4308	−1.78978	−0.894889	−	0.446290i	$-0.852745\pi$
$673$	−46.4308	−1.78978	−0.894889	+	0.446290i	$0.852745\pi$
$674$	−19.6757	−0.757880
$675$	0	0
$676$	−12.7936	−0.492063
$677$	−44.3129	−1.70308	−0.851541	−	0.524287i	$-0.824332\pi$
$677$	−44.3129	−1.70308	−0.851541	+	0.524287i	$0.824332\pi$
$678$	0	0
$679$	3.85695	0.148016
$680$	−8.90854	−0.341627
$681$	0	0
$682$	41.3246	1.58240
$683$	−27.7789	−1.06293	−0.531465	−	0.847080i	$-0.678358\pi$
$683$	−27.7789	−1.06293	−0.531465	+	0.847080i	$0.678358\pi$
$684$	0	0
$685$	−18.6389	−0.712155
$686$	1.00000	0.0381802
$687$	0	0
$688$	−6.03249	−0.229987
$689$	−3.72866	−0.142050
$690$	0	0
$691$	−19.2864	−0.733690	−0.366845	−	0.930282i	$-0.619562\pi$
$691$	−19.2864	−0.733690	−0.366845	+	0.930282i	$0.619562\pi$
$692$	11.8968	0.452249
$693$	0	0
$694$	−0.908538	−0.0344876
$695$	11.7936	0.447358
$696$	0	0
$697$	−20.1300	−0.762477
$698$	−12.1829	−0.461130
$699$	0	0
$700$	3.49414	0.132066
$701$	−40.0268	−1.51179	−0.755895	−	0.654692i	$-0.772798\pi$
$701$	−40.0268	−1.51179	−0.755895	+	0.654692i	$0.772798\pi$
$702$	0	0
$703$	−8.03249	−0.302951
$704$	−4.94841	−0.186500
$705$	0	0
$706$	23.9618	0.901814
$707$	4.18292	0.157315
$708$	0	0
$709$	−17.8586	−0.670695	−0.335347	−	0.942095i	$-0.608854\pi$
$709$	−17.8586	−0.670695	−0.335347	+	0.942095i	$0.608854\pi$
$710$	1.16383	0.0436776
$711$	0	0
$712$	2.94841	0.110496
$713$	12.9085	0.483429
$714$	0	0
$715$	−2.75849	−0.103162
$716$	22.2479	0.831443
$717$	0	0
$718$	−12.7672	−0.476466
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	0	0
$721$	−15.1564	−0.564456
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−14.4161	−0.535769
$725$	−14.0901	−0.523293
$726$	0	0
$727$	−17.8717	−0.662825	−0.331412	−	0.943486i	$-0.607525\pi$
$727$	−17.8717	−0.662825	−0.331412	+	0.943486i	$0.607525\pi$
$728$	−0.454269	−0.0168363
$729$	0	0
$730$	−9.69046	−0.358660
$731$	43.7936	1.61977
$732$	0	0
$733$	−2.43654	−0.0899955	−0.0449978	−	0.998987i	$-0.514328\pi$
$733$	−2.43654	−0.0899955	−0.0449978	+	0.998987i	$0.514328\pi$
$734$	3.68878	0.136155
$735$	0	0
$736$	−1.54573	−0.0569764
$737$	6.23315	0.229601
$738$	0	0
$739$	−17.8261	−0.655745	−0.327872	−	0.944722i	$-0.606332\pi$
$739$	−17.8261	−0.655745	−0.327872	+	0.944722i	$0.606332\pi$
$740$	−9.85695	−0.362349
$741$	0	0
$742$	8.20804	0.301326
$743$	−28.4218	−1.04269	−0.521347	−	0.853345i	$-0.674570\pi$
$743$	−28.4218	−1.04269	−0.521347	+	0.853345i	$0.674570\pi$
$744$	0	0
$745$	−18.7256	−0.686053
$746$	−25.0060	−0.915535
$747$	0	0
$748$	35.9236	1.31350
$749$	3.15645	0.115334
$750$	0	0
$751$	21.1240	0.770824	0.385412	−	0.922745i	$-0.374059\pi$
$751$	21.1240	0.770824	0.385412	+	0.922745i	$0.374059\pi$
$752$	0.597321	0.0217821
$753$	0	0
$754$	1.83184	0.0667115
$755$	10.6787	0.388639
$756$	0	0
$757$	6.92330	0.251632	0.125816	−	0.992054i	$-0.459845\pi$
$757$	6.92330	0.251632	0.125816	+	0.992054i	$0.459845\pi$
$758$	−0.286105	−0.0103918
$759$	0	0
$760$	−1.22713	−0.0445128
$761$	13.4278	0.486757	0.243379	−	0.969931i	$-0.421744\pi$
$761$	13.4278	0.486757	0.243379	+	0.969931i	$0.421744\pi$
$762$	0	0
$763$	0.208036	0.00753143
$764$	26.2479	0.949616
$765$	0	0
$766$	−0.973522	−0.0351748
$767$	−1.05327	−0.0380312
$768$	0	0
$769$	38.2861	1.38063	0.690316	−	0.723508i	$-0.257471\pi$
$769$	38.2861	1.38063	0.690316	+	0.723508i	$0.257471\pi$
$770$	6.07236	0.218833
$771$	0	0
$772$	−9.71390	−0.349611
$773$	24.9353	0.896861	0.448431	−	0.893818i	$-0.351983\pi$
$773$	24.9353	0.896861	0.448431	+	0.893818i	$0.351983\pi$
$774$	0	0
$775$	−29.1799	−1.04817
$776$	3.85695	0.138456
$777$	0	0
$778$	0.519253	0.0186161
$779$	−2.77287	−0.0993482
$780$	0	0
$781$	−4.69312	−0.167933
$782$	11.2214	0.401278
$783$	0	0
$784$	1.00000	0.0357143
$785$	16.5266	0.589861
$786$	0	0
$787$	31.9293	1.13816	0.569079	−	0.822283i	$-0.307300\pi$
$787$	31.9293	1.13816	0.569079	+	0.822283i	$0.307300\pi$
$788$	−16.5193	−0.588474
$789$	0	0
$790$	14.2080	0.505499
$791$	−0.168164	−0.00597923
$792$	0	0
$793$	−0.0649842	−0.00230766
$794$	23.2847	0.826344
$795$	0	0
$796$	21.7464	0.770780
$797$	−41.3012	−1.46296	−0.731481	−	0.681861i	$-0.761171\pi$
$797$	−41.3012	−1.46296	−0.731481	+	0.681861i	$0.761171\pi$
$798$	0	0
$799$	−4.33633	−0.153408
$800$	3.49414	0.123537
$801$	0	0
$802$	14.6874	0.518631
$803$	39.0767	1.37899
$804$	0	0
$805$	1.89682	0.0668541
$806$	3.79364	0.133625
$807$	0	0
$808$	4.18292	0.147155
$809$	−26.3454	−0.926254	−0.463127	−	0.886292i	$-0.653273\pi$
$809$	−26.3454	−0.926254	−0.463127	+	0.886292i	$0.653273\pi$
$810$	0	0
$811$	15.0664	0.529051	0.264526	−	0.964379i	$-0.414785\pi$
$811$	15.0664	0.529051	0.264526	+	0.964379i	$0.414785\pi$
$812$	−4.03249	−0.141513
$813$	0	0
$814$	39.7481	1.39317
$815$	0.812738	0.0284690
$816$	0	0
$817$	6.03249	0.211050
$818$	9.46766	0.331029
$819$	0	0
$820$	−3.40268	−0.118827
$821$	50.1300	1.74955	0.874774	−	0.484531i	$-0.161010\pi$
$821$	50.1300	1.74955	0.874774	+	0.484531i	$0.161010\pi$
$822$	0	0
$823$	−22.5340	−0.785486	−0.392743	−	0.919648i	$-0.628474\pi$
$823$	−22.5340	−0.785486	−0.392743	+	0.919648i	$0.628474\pi$
$824$	−15.1564	−0.528000
$825$	0	0
$826$	2.31860	0.0806743
$827$	33.3394	1.15932	0.579662	−	0.814857i	$-0.303185\pi$
$827$	33.3394	1.15932	0.579662	+	0.814857i	$0.303185\pi$
$828$	0	0
$829$	42.2747	1.46826	0.734130	−	0.679008i	$-0.237590\pi$
$829$	42.2747	1.46826	0.734130	+	0.679008i	$0.237590\pi$
$830$	8.70218	0.302057
$831$	0	0
$832$	−0.454269	−0.0157489
$833$	−7.25963	−0.251531
$834$	0	0
$835$	17.1330	0.592912
$836$	4.94841	0.171144
$837$	0	0
$838$	−33.3896	−1.15342
$839$	19.4426	0.671231	0.335616	−	0.941999i	$-0.391056\pi$
$839$	19.4426	0.671231	0.335616	+	0.941999i	$0.391056\pi$
$840$	0	0
$841$	−12.7390	−0.439276
$842$	22.9085	0.789480
$843$	0	0
$844$	4.70218	0.161856
$845$	15.6995	0.540080
$846$	0	0
$847$	−13.4868	−0.463411
$848$	8.20804	0.281865
$849$	0	0
$850$	−25.3662	−0.870052
$851$	12.4161	0.425618
$852$	0	0
$853$	9.08576	0.311090	0.155545	−	0.987829i	$-0.450287\pi$
$853$	9.08576	0.311090	0.155545	+	0.987829i	$0.450287\pi$
$854$	0.143052	0.00489515
$855$	0	0
$856$	3.15645	0.107885
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	24.7672	0.845045	0.422522	−	0.906353i	$-0.361145\pi$
$859$	24.7672	0.845045	0.422522	+	0.906353i	$0.361145\pi$
$860$	7.40268	0.252429
$861$	0	0
$862$	−32.5859	−1.10988
$863$	19.0060	0.646972	0.323486	−	0.946233i	$-0.395145\pi$
$863$	19.0060	0.646972	0.323486	+	0.946233i	$0.395145\pi$
$864$	0	0
$865$	−14.5990	−0.496381
$866$	3.61642	0.122891
$867$	0	0
$868$	−8.35109	−0.283454
$869$	−57.2938	−1.94356
$870$	0	0
$871$	0.572209	0.0193886
$872$	0.208036	0.00704500
$873$	0	0
$874$	1.54573	0.0522851
$875$	−10.4235	−0.352377
$876$	0	0
$877$	16.4941	0.556968	0.278484	−	0.960441i	$-0.410168\pi$
$877$	16.4941	0.556968	0.278484	+	0.960441i	$0.410168\pi$
$878$	19.4426	0.656154
$879$	0	0
$880$	6.07236	0.204699
$881$	38.5608	1.29915	0.649573	−	0.760299i	$-0.274948\pi$
$881$	38.5608	1.29915	0.649573	+	0.760299i	$0.274948\pi$
$882$	0	0
$883$	56.1698	1.89027	0.945133	−	0.326686i	$-0.105932\pi$
$883$	56.1698	1.89027	0.945133	+	0.326686i	$0.105932\pi$
$884$	3.29782	0.110918
$885$	0	0
$886$	−5.76115	−0.193550
$887$	17.7554	0.596169	0.298085	−	0.954539i	$-0.403652\pi$
$887$	17.7554	0.596169	0.298085	+	0.954539i	$0.403652\pi$
$888$	0	0
$889$	5.85695	0.196436
$890$	−3.61810	−0.121279
$891$	0	0
$892$	28.4161	0.951440
$893$	−0.597321	−0.0199886
$894$	0	0
$895$	−27.3012	−0.912578
$896$	1.00000	0.0334077
$897$	0	0
$898$	1.72866	0.0576860
$899$	33.6757	1.12315
$900$	0	0
$901$	−59.5873	−1.98514
$902$	13.7213	0.456869
$903$	0	0
$904$	−0.168164	−0.00559306
$905$	17.6905	0.588051
$906$	0	0
$907$	23.7139	0.787407	0.393703	−	0.919237i	$-0.371194\pi$
$907$	23.7139	0.787407	0.393703	+	0.919237i	$0.371194\pi$
$908$	14.5193	0.481838
$909$	0	0
$910$	0.557449	0.0184793
$911$	39.5018	1.30875	0.654377	−	0.756168i	$-0.272931\pi$
$911$	39.5018	1.30875	0.654377	+	0.756168i	$0.272931\pi$
$912$	0	0
$913$	−35.0915	−1.16136
$914$	10.9336	0.361653
$915$	0	0
$916$	17.2271	0.569201
$917$	−5.19464	−0.171542
$918$	0	0
$919$	−33.7052	−1.11183	−0.555916	−	0.831238i	$-0.687633\pi$
$919$	−33.7052	−1.11183	−0.555916	+	0.831238i	$0.687633\pi$
$920$	1.89682	0.0625363
$921$	0	0
$922$	−27.0784	−0.891779
$923$	−0.430833	−0.0141810
$924$	0	0
$925$	−28.0667	−0.922826
$926$	9.32461	0.306426
$927$	0	0
$928$	−4.03249	−0.132373
$929$	−2.43083	−0.0797530	−0.0398765	−	0.999205i	$-0.512696\pi$
$929$	−2.43083	−0.0797530	−0.0398765	+	0.999205i	$0.512696\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	−6.31122	−0.206731
$933$	0	0
$934$	26.5842	0.869863
$935$	−44.0831	−1.44167
$936$	0	0
$937$	−49.7025	−1.62371	−0.811855	−	0.583859i	$-0.801542\pi$
$937$	−49.7025	−1.62371	−0.811855	+	0.583859i	$0.801542\pi$
$938$	−1.25963	−0.0411283
$939$	0	0
$940$	−0.732993	−0.0239076
$941$	11.6757	0.380617	0.190308	−	0.981724i	$-0.439051\pi$
$941$	11.6757	0.380617	0.190308	+	0.981724i	$0.439051\pi$
$942$	0	0
$943$	4.28610	0.139575
$944$	2.31860	0.0754639
$945$	0	0
$946$	−29.8512	−0.970548
$947$	36.9484	1.20066	0.600331	−	0.799752i	$-0.295036\pi$
$947$	36.9484	1.20066	0.600331	+	0.799752i	$0.295036\pi$
$948$	0	0
$949$	3.58728	0.116448
$950$	−3.49414	−0.113365
$951$	0	0
$952$	−7.25963	−0.235286
$953$	−49.4278	−1.60112	−0.800562	−	0.599250i	$-0.795465\pi$
$953$	−49.4278	−1.60112	−0.800562	+	0.599250i	$0.795465\pi$
$954$	0	0
$955$	−32.2097	−1.04228
$956$	1.89682	0.0613475
$957$	0	0
$958$	36.9028	1.19228
$959$	−15.1889	−0.490476
$960$	0	0
$961$	38.7407	1.24970
$962$	3.64891	0.117646
$963$	0	0
$964$	−3.61642	−0.116477
$965$	11.9203	0.383727
$966$	0	0
$967$	58.1065	1.86858	0.934290	−	0.356514i	$-0.116035\pi$
$967$	58.1065	1.86858	0.934290	+	0.356514i	$0.116035\pi$
$968$	−13.4868	−0.433481
$969$	0	0
$970$	−4.73299	−0.151967
$971$	−17.6653	−0.566908	−0.283454	−	0.958986i	$-0.591480\pi$
$971$	−17.6653	−0.566908	−0.283454	+	0.958986i	$0.591480\pi$
$972$	0	0
$973$	9.61072	0.308105
$974$	−34.4338	−1.10333
$975$	0	0
$976$	0.143052	0.00457899
$977$	−10.6224	−0.339842	−0.169921	−	0.985458i	$-0.554351\pi$
$977$	−10.6224	−0.339842	−0.169921	+	0.985458i	$0.554351\pi$
$978$	0	0
$979$	14.5899	0.466297
$980$	−1.22713	−0.0391994
$981$	0	0
$982$	33.0385	1.05430
$983$	4.90854	0.156558	0.0782790	−	0.996931i	$-0.475057\pi$
$983$	4.90854	0.156558	0.0782790	+	0.996931i	$0.475057\pi$
$984$	0	0
$985$	20.2713	0.645899
$986$	29.2744	0.932286
$987$	0	0
$988$	0.454269	0.0144522
$989$	−9.32461	−0.296505
$990$	0	0
$991$	−4.61208	−0.146508	−0.0732538	−	0.997313i	$-0.523338\pi$
$991$	−4.61208	−0.146508	−0.0732538	+	0.997313i	$0.523338\pi$
$992$	−8.35109	−0.265147
$993$	0	0
$994$	0.948410	0.0300817
$995$	−26.6857	−0.845995
$996$	0	0
$997$	5.73436	0.181609	0.0908045	−	0.995869i	$-0.471056\pi$
$997$	5.73436	0.181609	0.0908045	+	0.995869i	$0.471056\pi$
$998$	−38.7347	−1.22612
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.a.ba.1.2		3
3.2	odd	2		266.2.a.d.1.1	✓	3
12.11	even	2		2128.2.a.s.1.3		3
15.14	odd	2		6650.2.a.cd.1.3		3
21.20	even	2		1862.2.a.r.1.3		3
24.5	odd	2		8512.2.a.bm.1.3		3
24.11	even	2		8512.2.a.bj.1.1		3
57.56	even	2		5054.2.a.r.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
266.2.a.d.1.1	✓	3	3.2	odd	2
1862.2.a.r.1.3		3	21.20	even	2
2128.2.a.s.1.3		3	12.11	even	2
2394.2.a.ba.1.2		3	1.1	even	1	trivial
5054.2.a.r.1.3		3	57.56	even	2
6650.2.a.cd.1.3		3	15.14	odd	2
8512.2.a.bj.1.1		3	24.11	even	2
8512.2.a.bm.1.3		3	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.22713 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.22713 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.22713 q^{10} -4.94841 q^{11} -0.454269 q^{13} +1.00000 q^{14} +1.00000 q^{16} -7.25963 q^{17} -1.00000 q^{19} -1.22713 q^{20} +4.94841 q^{22} +1.54573 q^{23} -3.49414 q^{25} +0.454269 q^{26} -1.00000 q^{28} +4.03249 q^{29} +8.35109 q^{31} -1.00000 q^{32} +7.25963 q^{34} +1.22713 q^{35} +8.03249 q^{37} +1.00000 q^{38} +1.22713 q^{40} +2.77287 q^{41} -6.03249 q^{43} -4.94841 q^{44} -1.54573 q^{46} +0.597321 q^{47} +1.00000 q^{49} +3.49414 q^{50} -0.454269 q^{52} +8.20804 q^{53} +6.07236 q^{55} +1.00000 q^{56} -4.03249 q^{58} +2.31860 q^{59} +0.143052 q^{61} -8.35109 q^{62} +1.00000 q^{64} +0.557449 q^{65} -1.25963 q^{67} -7.25963 q^{68} -1.22713 q^{70} +0.948410 q^{71} -7.89682 q^{73} -8.03249 q^{74} -1.00000 q^{76} +4.94841 q^{77} +11.5782 q^{79} -1.22713 q^{80} -2.77287 q^{82} +7.09146 q^{83} +8.90854 q^{85} +6.03249 q^{86} +4.94841 q^{88} -2.94841 q^{89} +0.454269 q^{91} +1.54573 q^{92} -0.597321 q^{94} +1.22713 q^{95} -3.85695 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 5 * q^5 - 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} - 3 q^{7} - 3 q^{8} + 5 q^{10} - 3 q^{11} - 4 q^{13} + 3 q^{14} + 3 q^{16} - 6 q^{17} - 3 q^{19} - 5 q^{20} + 3 q^{22} + 2 q^{23} + 4 q^{25} + 4 q^{26} - 3 q^{28} - 5 q^{29} + 4 q^{31} - 3 q^{32} + 6 q^{34} + 5 q^{35} + 7 q^{37} + 3 q^{38} + 5 q^{40} + 7 q^{41} - q^{43} - 3 q^{44} - 2 q^{46} + 11 q^{47} + 3 q^{49} - 4 q^{50} - 4 q^{52} - 3 q^{53} - 16 q^{55} + 3 q^{56} + 5 q^{58} + 3 q^{59} + 7 q^{61} - 4 q^{62} + 3 q^{64} + 28 q^{65} + 12 q^{67} - 6 q^{68} - 5 q^{70} - 9 q^{71} - 7 q^{74} - 3 q^{76} + 3 q^{77} + 15 q^{79} - 5 q^{80} - 7 q^{82} + 16 q^{83} + 32 q^{85} + q^{86} + 3 q^{88} + 3 q^{89} + 4 q^{91} + 2 q^{92} - 11 q^{94} + 5 q^{95} - 5 q^{97} - 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 5 * q^5 - 3 * q^7 - 3 * q^8 + 5 * q^10 - 3 * q^11 - 4 * q^13 + 3 * q^14 + 3 * q^16 - 6 * q^17 - 3 * q^19 - 5 * q^20 + 3 * q^22 + 2 * q^23 + 4 * q^25 + 4 * q^26 - 3 * q^28 - 5 * q^29 + 4 * q^31 - 3 * q^32 + 6 * q^34 + 5 * q^35 + 7 * q^37 + 3 * q^38 + 5 * q^40 + 7 * q^41 - q^43 - 3 * q^44 - 2 * q^46 + 11 * q^47 + 3 * q^49 - 4 * q^50 - 4 * q^52 - 3 * q^53 - 16 * q^55 + 3 * q^56 + 5 * q^58 + 3 * q^59 + 7 * q^61 - 4 * q^62 + 3 * q^64 + 28 * q^65 + 12 * q^67 - 6 * q^68 - 5 * q^70 - 9 * q^71 - 7 * q^74 - 3 * q^76 + 3 * q^77 + 15 * q^79 - 5 * q^80 - 7 * q^82 + 16 * q^83 + 32 * q^85 + q^86 + 3 * q^88 + 3 * q^89 + 4 * q^91 + 2 * q^92 - 11 * q^94 + 5 * q^95 - 5 * q^97 - 3 * q^98

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