LMFDB - Embedded newform 1458.2.a.g.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1458,2,Mod(1,1458)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1458, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1458.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1458 = 2 \cdot 3^{6} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1458.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:	$11.6421886147$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.6357609.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} - x^{3} + 18x^{2} - 8 $ x^6 - 9x^4 - x^3 + 18x^2 - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 54)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.758254$ of defining polynomial
Character	$\chi$	$=$	1458.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.55368 q^{5} -0.319501 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +3.55368 q^{5} -0.319501 q^{7} +1.00000 q^{8} +3.55368 q^{10} +4.45206 q^{11} -4.67874 q^{13} -0.319501 q^{14} +1.00000 q^{16} +1.13471 q^{17} +1.85779 q^{19} +3.55368 q^{20} +4.45206 q^{22} +0.144212 q^{23} +7.62865 q^{25} -4.67874 q^{26} -0.319501 q^{28} -4.35924 q^{29} +0.671692 q^{31} +1.00000 q^{32} +1.13471 q^{34} -1.13540 q^{35} +7.58877 q^{37} +1.85779 q^{38} +3.55368 q^{40} +2.17729 q^{41} -7.09771 q^{43} +4.45206 q^{44} +0.144212 q^{46} -10.3265 q^{47} -6.89792 q^{49} +7.62865 q^{50} -4.67874 q^{52} +0.805554 q^{53} +15.8212 q^{55} -0.319501 q^{56} -4.35924 q^{58} +2.98285 q^{59} -3.49464 q^{61} +0.671692 q^{62} +1.00000 q^{64} -16.6267 q^{65} +7.44325 q^{67} +1.13471 q^{68} -1.13540 q^{70} -8.09856 q^{71} +14.6013 q^{73} +7.58877 q^{74} +1.85779 q^{76} -1.42244 q^{77} -12.5870 q^{79} +3.55368 q^{80} +2.17729 q^{82} -5.41363 q^{83} +4.03239 q^{85} -7.09771 q^{86} +4.45206 q^{88} +5.05248 q^{89} +1.49486 q^{91} +0.144212 q^{92} -10.3265 q^{94} +6.60200 q^{95} +18.6741 q^{97} -6.89792 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 3 q^{5} + 6 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + 6 * q^2 + 6 * q^4 + 3 * q^5 + 6 * q^7 + 6 * q^8	$ 6 q + 6 q^{2} + 6 q^{4} + 3 q^{5} + 6 q^{7} + 6 q^{8} + 3 q^{10} + 3 q^{11} + 9 q^{13} + 6 q^{14} + 6 q^{16} + 6 q^{17} + 9 q^{19} + 3 q^{20} + 3 q^{22} - 3 q^{23} + 15 q^{25} + 9 q^{26} + 6 q^{28} + 3 q^{29} + 12 q^{31} + 6 q^{32} + 6 q^{34} - 3 q^{35} + 15 q^{37} + 9 q^{38} + 3 q^{40} + 3 q^{41} + 12 q^{43} + 3 q^{44} - 3 q^{46} - 9 q^{47} + 12 q^{49} + 15 q^{50} + 9 q^{52} - 6 q^{53} + 9 q^{55} + 6 q^{56} + 3 q^{58} - 3 q^{59} + 12 q^{61} + 12 q^{62} + 6 q^{64} - 3 q^{65} + 21 q^{67} + 6 q^{68} - 3 q^{70} - 12 q^{71} + 21 q^{73} + 15 q^{74} + 9 q^{76} - 3 q^{77} + 3 q^{80} + 3 q^{82} - 27 q^{83} + 12 q^{86} + 3 q^{88} - 12 q^{89} + 6 q^{91} - 3 q^{92} - 9 q^{94} - 30 q^{95} + 18 q^{97} + 12 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 3 * q^5 + 6 * q^7 + 6 * q^8 + 3 * q^10 + 3 * q^11 + 9 * q^13 + 6 * q^14 + 6 * q^16 + 6 * q^17 + 9 * q^19 + 3 * q^20 + 3 * q^22 - 3 * q^23 + 15 * q^25 + 9 * q^26 + 6 * q^28 + 3 * q^29 + 12 * q^31 + 6 * q^32 + 6 * q^34 - 3 * q^35 + 15 * q^37 + 9 * q^38 + 3 * q^40 + 3 * q^41 + 12 * q^43 + 3 * q^44 - 3 * q^46 - 9 * q^47 + 12 * q^49 + 15 * q^50 + 9 * q^52 - 6 * q^53 + 9 * q^55 + 6 * q^56 + 3 * q^58 - 3 * q^59 + 12 * q^61 + 12 * q^62 + 6 * q^64 - 3 * q^65 + 21 * q^67 + 6 * q^68 - 3 * q^70 - 12 * q^71 + 21 * q^73 + 15 * q^74 + 9 * q^76 - 3 * q^77 + 3 * q^80 + 3 * q^82 - 27 * q^83 + 12 * q^86 + 3 * q^88 - 12 * q^89 + 6 * q^91 - 3 * q^92 - 9 * q^94 - 30 * q^95 + 18 * q^97 + 12 * 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
$5$	3.55368	1.58925	0.794627	−	0.607098i	$-0.207666\pi$
$5$	3.55368	1.58925	0.794627	+	0.607098i	$0.207666\pi$
$6$	0	0
$7$	−0.319501	−0.120760	−0.0603800	−	0.998175i	$-0.519231\pi$
$7$	−0.319501	−0.120760	−0.0603800	+	0.998175i	$0.519231\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.55368	1.12377
$11$	4.45206	1.34235	0.671173	−	0.741301i	$-0.265791\pi$
$11$	4.45206	1.34235	0.671173	+	0.741301i	$0.265791\pi$
$12$	0	0
$13$	−4.67874	−1.29765	−0.648824	−	0.760938i	$-0.724739\pi$
$13$	−4.67874	−1.29765	−0.648824	+	0.760938i	$0.724739\pi$
$14$	−0.319501	−0.0853902
$15$	0	0
$16$	1.00000	0.250000
$17$	1.13471	0.275207	0.137604	−	0.990487i	$-0.456060\pi$
$17$	1.13471	0.275207	0.137604	+	0.990487i	$0.456060\pi$
$18$	0	0
$19$	1.85779	0.426207	0.213103	−	0.977030i	$-0.431643\pi$
$19$	1.85779	0.426207	0.213103	+	0.977030i	$0.431643\pi$
$20$	3.55368	0.794627
$21$	0	0
$22$	4.45206	0.949181
$23$	0.144212	0.0300702	0.0150351	−	0.999887i	$-0.495214\pi$
$23$	0.144212	0.0300702	0.0150351	+	0.999887i	$0.495214\pi$
$24$	0	0
$25$	7.62865	1.52573
$26$	−4.67874	−0.917576
$27$	0	0
$28$	−0.319501	−0.0603800
$29$	−4.35924	−0.809490	−0.404745	−	0.914430i	$-0.632640\pi$
$29$	−4.35924	−0.809490	−0.404745	+	0.914430i	$0.632640\pi$
$30$	0	0
$31$	0.671692	0.120639	0.0603197	−	0.998179i	$-0.480788\pi$
$31$	0.671692	0.120639	0.0603197	+	0.998179i	$0.480788\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.13471	0.194601
$35$	−1.13540	−0.191918
$36$	0	0
$37$	7.58877	1.24759	0.623793	−	0.781590i	$-0.285591\pi$
$37$	7.58877	1.24759	0.623793	+	0.781590i	$0.285591\pi$
$38$	1.85779	0.301374
$39$	0	0
$40$	3.55368	0.561886
$41$	2.17729	0.340036	0.170018	−	0.985441i	$-0.445617\pi$
$41$	2.17729	0.340036	0.170018	+	0.985441i	$0.445617\pi$
$42$	0	0
$43$	−7.09771	−1.08239	−0.541195	−	0.840897i	$-0.682028\pi$
$43$	−7.09771	−1.08239	−0.541195	+	0.840897i	$0.682028\pi$
$44$	4.45206	0.671173
$45$	0	0
$46$	0.144212	0.0212628
$47$	−10.3265	−1.50628	−0.753141	−	0.657859i	$-0.771462\pi$
$47$	−10.3265	−1.50628	−0.753141	+	0.657859i	$0.771462\pi$
$48$	0	0
$49$	−6.89792	−0.985417
$50$	7.62865	1.07885
$51$	0	0
$52$	−4.67874	−0.648824
$53$	0.805554	0.110651	0.0553257	−	0.998468i	$-0.482380\pi$
$53$	0.805554	0.110651	0.0553257	+	0.998468i	$0.482380\pi$
$54$	0	0
$55$	15.8212	2.13333
$56$	−0.319501	−0.0426951
$57$	0	0
$58$	−4.35924	−0.572396
$59$	2.98285	0.388334	0.194167	−	0.980969i	$-0.437800\pi$
$59$	2.98285	0.388334	0.194167	+	0.980969i	$0.437800\pi$
$60$	0	0
$61$	−3.49464	−0.447443	−0.223721	−	0.974653i	$-0.571821\pi$
$61$	−3.49464	−0.447443	−0.223721	+	0.974653i	$0.571821\pi$
$62$	0.671692	0.0853050
$63$	0	0
$64$	1.00000	0.125000
$65$	−16.6267	−2.06229
$66$	0	0
$67$	7.44325	0.909337	0.454669	−	0.890661i	$-0.349758\pi$
$67$	7.44325	0.909337	0.454669	+	0.890661i	$0.349758\pi$
$68$	1.13471	0.137604
$69$	0	0
$70$	−1.13540	−0.135707
$71$	−8.09856	−0.961122	−0.480561	−	0.876961i	$-0.659567\pi$
$71$	−8.09856	−0.961122	−0.480561	+	0.876961i	$0.659567\pi$
$72$	0	0
$73$	14.6013	1.70895	0.854477	−	0.519489i	$-0.173878\pi$
$73$	14.6013	1.70895	0.854477	+	0.519489i	$0.173878\pi$
$74$	7.58877	0.882176
$75$	0	0
$76$	1.85779	0.213103
$77$	−1.42244	−0.162102
$78$	0	0
$79$	−12.5870	−1.41615	−0.708074	−	0.706138i	$-0.750436\pi$
$79$	−12.5870	−1.41615	−0.708074	+	0.706138i	$0.750436\pi$
$80$	3.55368	0.397314
$81$	0	0
$82$	2.17729	0.240442
$83$	−5.41363	−0.594223	−0.297111	−	0.954843i	$-0.596023\pi$
$83$	−5.41363	−0.594223	−0.297111	+	0.954843i	$0.596023\pi$
$84$	0	0
$85$	4.03239	0.437374
$86$	−7.09771	−0.765366
$87$	0	0
$88$	4.45206	0.474591
$89$	5.05248	0.535562	0.267781	−	0.963480i	$-0.413710\pi$
$89$	5.05248	0.535562	0.267781	+	0.963480i	$0.413710\pi$
$90$	0	0
$91$	1.49486	0.156704
$92$	0.144212	0.0150351
$93$	0	0
$94$	−10.3265	−1.06510
$95$	6.60200	0.677351
$96$	0	0
$97$	18.6741	1.89607	0.948035	−	0.318166i	$-0.103067\pi$
$97$	18.6741	1.89607	0.948035	+	0.318166i	$0.103067\pi$
$98$	−6.89792	−0.696795
$99$	0	0
$100$	7.62865	0.762865
$101$	−2.41967	−0.240766	−0.120383	−	0.992728i	$-0.538412\pi$
$101$	−2.41967	−0.240766	−0.120383	+	0.992728i	$0.538412\pi$
$102$	0	0
$103$	1.99535	0.196608	0.0983039	−	0.995156i	$-0.468658\pi$
$103$	1.99535	0.196608	0.0983039	+	0.995156i	$0.468658\pi$
$104$	−4.67874	−0.458788
$105$	0	0
$106$	0.805554	0.0782423
$107$	8.87072	0.857565	0.428783	−	0.903408i	$-0.358943\pi$
$107$	8.87072	0.857565	0.428783	+	0.903408i	$0.358943\pi$
$108$	0	0
$109$	−1.97204	−0.188887	−0.0944434	−	0.995530i	$-0.530107\pi$
$109$	−1.97204	−0.188887	−0.0944434	+	0.995530i	$0.530107\pi$
$110$	15.8212	1.50849
$111$	0	0
$112$	−0.319501	−0.0301900
$113$	−9.62865	−0.905787	−0.452894	−	0.891565i	$-0.649608\pi$
$113$	−9.62865	−0.905787	−0.452894	+	0.891565i	$0.649608\pi$
$114$	0	0
$115$	0.512482	0.0477892
$116$	−4.35924	−0.404745
$117$	0	0
$118$	2.98285	0.274593
$119$	−0.362540	−0.0332340
$120$	0	0
$121$	8.82079	0.801890
$122$	−3.49464	−0.316390
$123$	0	0
$124$	0.671692	0.0603197
$125$	9.34139	0.835520
$126$	0	0
$127$	0.159210	0.0141276	0.00706379	−	0.999975i	$-0.497752\pi$
$127$	0.159210	0.0141276	0.00706379	+	0.999975i	$0.497752\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−16.6267	−1.45826
$131$	−9.48583	−0.828781	−0.414391	−	0.910099i	$-0.636005\pi$
$131$	−9.48583	−0.828781	−0.414391	+	0.910099i	$0.636005\pi$
$132$	0	0
$133$	−0.593566	−0.0514687
$134$	7.44325	0.642999
$135$	0	0
$136$	1.13471	0.0973004
$137$	11.7094	1.00040	0.500199	−	0.865910i	$-0.333260\pi$
$137$	11.7094	1.00040	0.500199	+	0.865910i	$0.333260\pi$
$138$	0	0
$139$	−5.25027	−0.445322	−0.222661	−	0.974896i	$-0.571474\pi$
$139$	−5.25027	−0.445322	−0.222661	+	0.974896i	$0.571474\pi$
$140$	−1.13540	−0.0959592
$141$	0	0
$142$	−8.09856	−0.679616
$143$	−20.8300	−1.74189
$144$	0	0
$145$	−15.4913	−1.28649
$146$	14.6013	1.20841
$147$	0	0
$148$	7.58877	0.623793
$149$	−16.2922	−1.33471	−0.667356	−	0.744739i	$-0.732574\pi$
$149$	−16.2922	−1.33471	−0.667356	+	0.744739i	$0.732574\pi$
$150$	0	0
$151$	−15.6179	−1.27097	−0.635485	−	0.772113i	$-0.719200\pi$
$151$	−15.6179	−1.27097	−0.635485	+	0.772113i	$0.719200\pi$
$152$	1.85779	0.150687
$153$	0	0
$154$	−1.42244	−0.114623
$155$	2.38698	0.191727
$156$	0	0
$157$	17.6315	1.40715	0.703575	−	0.710621i	$-0.251586\pi$
$157$	17.6315	1.40715	0.703575	+	0.710621i	$0.251586\pi$
$158$	−12.5870	−1.00137
$159$	0	0
$160$	3.55368	0.280943
$161$	−0.0460757	−0.00363128
$162$	0	0
$163$	−15.8801	−1.24382	−0.621912	−	0.783087i	$-0.713644\pi$
$163$	−15.8801	−1.24382	−0.621912	+	0.783087i	$0.713644\pi$
$164$	2.17729	0.170018
$165$	0	0
$166$	−5.41363	−0.420179
$167$	−13.5784	−1.05073	−0.525365	−	0.850877i	$-0.676071\pi$
$167$	−13.5784	−1.05073	−0.525365	+	0.850877i	$0.676071\pi$
$168$	0	0
$169$	8.89058	0.683890
$170$	4.03239	0.309270
$171$	0	0
$172$	−7.09771	−0.541195
$173$	6.44441	0.489959	0.244980	−	0.969528i	$-0.421219\pi$
$173$	6.44441	0.489959	0.244980	+	0.969528i	$0.421219\pi$
$174$	0	0
$175$	−2.43736	−0.184247
$176$	4.45206	0.335586
$177$	0	0
$178$	5.05248	0.378699
$179$	−12.5123	−0.935210	−0.467605	−	0.883938i	$-0.654883\pi$
$179$	−12.5123	−0.935210	−0.467605	+	0.883938i	$0.654883\pi$
$180$	0	0
$181$	0.304503	0.0226335	0.0113168	−	0.999936i	$-0.496398\pi$
$181$	0.304503	0.0226335	0.0113168	+	0.999936i	$0.496398\pi$
$182$	1.49486	0.110806
$183$	0	0
$184$	0.144212	0.0106314
$185$	26.9681	1.98273
$186$	0	0
$187$	5.05178	0.369423
$188$	−10.3265	−0.753141
$189$	0	0
$190$	6.60200	0.478960
$191$	−14.3927	−1.04142	−0.520710	−	0.853734i	$-0.674333\pi$
$191$	−14.3927	−1.04142	−0.520710	+	0.853734i	$0.674333\pi$
$192$	0	0
$193$	3.05063	0.219589	0.109794	−	0.993954i	$-0.464981\pi$
$193$	3.05063	0.219589	0.109794	+	0.993954i	$0.464981\pi$
$194$	18.6741	1.34072
$195$	0	0
$196$	−6.89792	−0.492709
$197$	22.2642	1.58626	0.793129	−	0.609053i	$-0.208450\pi$
$197$	22.2642	1.58626	0.793129	+	0.609053i	$0.208450\pi$
$198$	0	0
$199$	24.5634	1.74125	0.870626	−	0.491945i	$-0.163714\pi$
$199$	24.5634	1.74125	0.870626	+	0.491945i	$0.163714\pi$
$200$	7.62865	0.539427
$201$	0	0
$202$	−2.41967	−0.170247
$203$	1.39278	0.0977540
$204$	0	0
$205$	7.73741	0.540404
$206$	1.99535	0.139023
$207$	0	0
$208$	−4.67874	−0.324412
$209$	8.27099	0.572117
$210$	0	0
$211$	−3.01931	−0.207858	−0.103929	−	0.994585i	$-0.533141\pi$
$211$	−3.01931	−0.207858	−0.103929	+	0.994585i	$0.533141\pi$
$212$	0.805554	0.0553257
$213$	0	0
$214$	8.87072	0.606390
$215$	−25.2230	−1.72019
$216$	0	0
$217$	−0.214606	−0.0145684
$218$	−1.97204	−0.133563
$219$	0	0
$220$	15.8212	1.06666
$221$	−5.30900	−0.357122
$222$	0	0
$223$	9.98040	0.668337	0.334168	−	0.942513i	$-0.391545\pi$
$223$	9.98040	0.668337	0.334168	+	0.942513i	$0.391545\pi$
$224$	−0.319501	−0.0213476
$225$	0	0
$226$	−9.62865	−0.640488
$227$	−21.1224	−1.40194	−0.700970	−	0.713190i	$-0.747250\pi$
$227$	−21.1224	−1.40194	−0.700970	+	0.713190i	$0.747250\pi$
$228$	0	0
$229$	−23.2807	−1.53843	−0.769217	−	0.638987i	$-0.779354\pi$
$229$	−23.2807	−1.53843	−0.769217	+	0.638987i	$0.779354\pi$
$230$	0.512482	0.0337921
$231$	0	0
$232$	−4.35924	−0.286198
$233$	26.7243	1.75077	0.875383	−	0.483429i	$-0.160609\pi$
$233$	26.7243	1.75077	0.875383	+	0.483429i	$0.160609\pi$
$234$	0	0
$235$	−36.6973	−2.39386
$236$	2.98285	0.194167
$237$	0	0
$238$	−0.362540	−0.0235000
$239$	−24.0599	−1.55630	−0.778152	−	0.628076i	$-0.783843\pi$
$239$	−24.0599	−1.55630	−0.778152	+	0.628076i	$0.783843\pi$
$240$	0	0
$241$	−1.49152	−0.0960772	−0.0480386	−	0.998845i	$-0.515297\pi$
$241$	−1.49152	−0.0960772	−0.0480386	+	0.998845i	$0.515297\pi$
$242$	8.82079	0.567022
$243$	0	0
$244$	−3.49464	−0.223721
$245$	−24.5130	−1.56608
$246$	0	0
$247$	−8.69212	−0.553066
$248$	0.671692	0.0426525
$249$	0	0
$250$	9.34139	0.590802
$251$	14.5750	0.919963	0.459982	−	0.887928i	$-0.347856\pi$
$251$	14.5750	0.919963	0.459982	+	0.887928i	$0.347856\pi$
$252$	0	0
$253$	0.642038	0.0403646
$254$	0.159210	0.00998971
$255$	0	0
$256$	1.00000	0.0625000
$257$	−11.0246	−0.687697	−0.343849	−	0.939025i	$-0.611731\pi$
$257$	−11.0246	−0.687697	−0.343849	+	0.939025i	$0.611731\pi$
$258$	0	0
$259$	−2.42462	−0.150658
$260$	−16.6267	−1.03115
$261$	0	0
$262$	−9.48583	−0.586037
$263$	−13.1401	−0.810250	−0.405125	−	0.914261i	$-0.632772\pi$
$263$	−13.1401	−0.810250	−0.405125	+	0.914261i	$0.632772\pi$
$264$	0	0
$265$	2.86268	0.175853
$266$	−0.593566	−0.0363939
$267$	0	0
$268$	7.44325	0.454669
$269$	−10.3086	−0.628528	−0.314264	−	0.949336i	$-0.601758\pi$
$269$	−10.3086	−0.628528	−0.314264	+	0.949336i	$0.601758\pi$
$270$	0	0
$271$	1.01454	0.0616288	0.0308144	−	0.999525i	$-0.490190\pi$
$271$	1.01454	0.0616288	0.0308144	+	0.999525i	$0.490190\pi$
$272$	1.13471	0.0688018
$273$	0	0
$274$	11.7094	0.707388
$275$	33.9632	2.04806
$276$	0	0
$277$	−10.8317	−0.650813	−0.325406	−	0.945574i	$-0.605501\pi$
$277$	−10.8317	−0.650813	−0.325406	+	0.945574i	$0.605501\pi$
$278$	−5.25027	−0.314890
$279$	0	0
$280$	−1.13540	−0.0678534
$281$	−8.47673	−0.505679	−0.252840	−	0.967508i	$-0.581364\pi$
$281$	−8.47673	−0.505679	−0.252840	+	0.967508i	$0.581364\pi$
$282$	0	0
$283$	6.30428	0.374750	0.187375	−	0.982288i	$-0.440002\pi$
$283$	6.30428	0.374750	0.187375	+	0.982288i	$0.440002\pi$
$284$	−8.09856	−0.480561
$285$	0	0
$286$	−20.8300	−1.23170
$287$	−0.695647	−0.0410628
$288$	0	0
$289$	−15.7124	−0.924261
$290$	−15.4913	−0.909683
$291$	0	0
$292$	14.6013	0.854477
$293$	−2.43305	−0.142141	−0.0710703	−	0.997471i	$-0.522641\pi$
$293$	−2.43305	−0.142141	−0.0710703	+	0.997471i	$0.522641\pi$
$294$	0	0
$295$	10.6001	0.617161
$296$	7.58877	0.441088
$297$	0	0
$298$	−16.2922	−0.943784
$299$	−0.674728	−0.0390205
$300$	0	0
$301$	2.26772	0.130709
$302$	−15.6179	−0.898711
$303$	0	0
$304$	1.85779	0.106552
$305$	−12.4188	−0.711101
$306$	0	0
$307$	−23.7258	−1.35410	−0.677050	−	0.735937i	$-0.736742\pi$
$307$	−23.7258	−1.35410	−0.677050	+	0.735937i	$0.736742\pi$
$308$	−1.42244	−0.0810508
$309$	0	0
$310$	2.38698	0.135571
$311$	−10.8006	−0.612446	−0.306223	−	0.951960i	$-0.599065\pi$
$311$	−10.8006	−0.612446	−0.306223	+	0.951960i	$0.599065\pi$
$312$	0	0
$313$	−15.8912	−0.898223	−0.449112	−	0.893476i	$-0.648259\pi$
$313$	−15.8912	−0.898223	−0.449112	+	0.893476i	$0.648259\pi$
$314$	17.6315	0.995005
$315$	0	0
$316$	−12.5870	−0.708074
$317$	−3.23504	−0.181698	−0.0908490	−	0.995865i	$-0.528958\pi$
$317$	−3.23504	−0.181698	−0.0908490	+	0.995865i	$0.528958\pi$
$318$	0	0
$319$	−19.4076	−1.08661
$320$	3.55368	0.198657
$321$	0	0
$322$	−0.0460757	−0.00256770
$323$	2.10805	0.117295
$324$	0	0
$325$	−35.6925	−1.97986
$326$	−15.8801	−0.879516
$327$	0	0
$328$	2.17729	0.120221
$329$	3.29934	0.181899
$330$	0	0
$331$	−17.6637	−0.970883	−0.485441	−	0.874269i	$-0.661341\pi$
$331$	−17.6637	−0.970883	−0.485441	+	0.874269i	$0.661341\pi$
$332$	−5.41363	−0.297111
$333$	0	0
$334$	−13.5784	−0.742978
$335$	26.4509	1.44517
$336$	0	0
$337$	−10.5429	−0.574307	−0.287154	−	0.957885i	$-0.592709\pi$
$337$	−10.5429	−0.574307	−0.287154	+	0.957885i	$0.592709\pi$
$338$	8.89058	0.483584
$339$	0	0
$340$	4.03239	0.218687
$341$	2.99041	0.161940
$342$	0	0
$343$	4.44040	0.239759
$344$	−7.09771	−0.382683
$345$	0	0
$346$	6.44441	0.346453
$347$	32.5047	1.74494	0.872471	−	0.488666i	$-0.162516\pi$
$347$	32.5047	1.74494	0.872471	+	0.488666i	$0.162516\pi$
$348$	0	0
$349$	−20.0703	−1.07434	−0.537169	−	0.843475i	$-0.680506\pi$
$349$	−20.0703	−1.07434	−0.537169	+	0.843475i	$0.680506\pi$
$350$	−2.43736	−0.130282
$351$	0	0
$352$	4.45206	0.237295
$353$	11.0361	0.587390	0.293695	−	0.955899i	$-0.405115\pi$
$353$	11.0361	0.587390	0.293695	+	0.955899i	$0.405115\pi$
$354$	0	0
$355$	−28.7797	−1.52747
$356$	5.05248	0.267781
$357$	0	0
$358$	−12.5123	−0.661293
$359$	−3.92087	−0.206936	−0.103468	−	0.994633i	$-0.532994\pi$
$359$	−3.92087	−0.206936	−0.103468	+	0.994633i	$0.532994\pi$
$360$	0	0
$361$	−15.5486	−0.818348
$362$	0.304503	0.0160043
$363$	0	0
$364$	1.49486	0.0783520
$365$	51.8884	2.71596
$366$	0	0
$367$	29.1512	1.52168	0.760841	−	0.648938i	$-0.224787\pi$
$367$	29.1512	1.52168	0.760841	+	0.648938i	$0.224787\pi$
$368$	0.144212	0.00751755
$369$	0	0
$370$	26.9681	1.40200
$371$	−0.257375	−0.0133623
$372$	0	0
$373$	6.27181	0.324742	0.162371	−	0.986730i	$-0.448086\pi$
$373$	6.27181	0.324742	0.162371	+	0.986730i	$0.448086\pi$
$374$	5.05178	0.261222
$375$	0	0
$376$	−10.3265	−0.532551
$377$	20.3957	1.05043
$378$	0	0
$379$	−26.9562	−1.38465	−0.692324	−	0.721587i	$-0.743413\pi$
$379$	−26.9562	−1.38465	−0.692324	+	0.721587i	$0.743413\pi$
$380$	6.60200	0.338676
$381$	0	0
$382$	−14.3927	−0.736395
$383$	−3.25627	−0.166388	−0.0831938	−	0.996533i	$-0.526512\pi$
$383$	−3.25627	−0.166388	−0.0831938	+	0.996533i	$0.526512\pi$
$384$	0	0
$385$	−5.05488	−0.257621
$386$	3.05063	0.155273
$387$	0	0
$388$	18.6741	0.948035
$389$	1.38968	0.0704596	0.0352298	−	0.999379i	$-0.488784\pi$
$389$	1.38968	0.0704596	0.0352298	+	0.999379i	$0.488784\pi$
$390$	0	0
$391$	0.163638	0.00827554
$392$	−6.89792	−0.348398
$393$	0	0
$394$	22.2642	1.12165
$395$	−44.7302	−2.25062
$396$	0	0
$397$	2.14020	0.107414	0.0537069	−	0.998557i	$-0.482896\pi$
$397$	2.14020	0.107414	0.0537069	+	0.998557i	$0.482896\pi$
$398$	24.5634	1.23125
$399$	0	0
$400$	7.62865	0.381433
$401$	−8.55331	−0.427132	−0.213566	−	0.976929i	$-0.568508\pi$
$401$	−8.55331	−0.427132	−0.213566	+	0.976929i	$0.568508\pi$
$402$	0	0
$403$	−3.14267	−0.156548
$404$	−2.41967	−0.120383
$405$	0	0
$406$	1.39278	0.0691225
$407$	33.7856	1.67469
$408$	0	0
$409$	27.2719	1.34851	0.674254	−	0.738500i	$-0.264465\pi$
$409$	27.2719	1.34851	0.674254	+	0.738500i	$0.264465\pi$
$410$	7.73741	0.382123
$411$	0	0
$412$	1.99535	0.0983039
$413$	−0.953022	−0.0468952
$414$	0	0
$415$	−19.2383	−0.944372
$416$	−4.67874	−0.229394
$417$	0	0
$418$	8.27099	0.404548
$419$	16.0153	0.782400	0.391200	−	0.920306i	$-0.372060\pi$
$419$	16.0153	0.782400	0.391200	+	0.920306i	$0.372060\pi$
$420$	0	0
$421$	−18.1470	−0.884432	−0.442216	−	0.896909i	$-0.645808\pi$
$421$	−18.1470	−0.884432	−0.442216	+	0.896909i	$0.645808\pi$
$422$	−3.01931	−0.146978
$423$	0	0
$424$	0.805554	0.0391212
$425$	8.65630	0.419892
$426$	0	0
$427$	1.11654	0.0540332
$428$	8.87072	0.428783
$429$	0	0
$430$	−25.2230	−1.21636
$431$	2.13698	0.102935	0.0514673	−	0.998675i	$-0.483610\pi$
$431$	2.13698	0.102935	0.0514673	+	0.998675i	$0.483610\pi$
$432$	0	0
$433$	25.1733	1.20975	0.604876	−	0.796320i	$-0.293223\pi$
$433$	25.1733	1.20975	0.604876	+	0.796320i	$0.293223\pi$
$434$	−0.214606	−0.0103014
$435$	0	0
$436$	−1.97204	−0.0944434
$437$	0.267915	0.0128161
$438$	0	0
$439$	25.8519	1.23384	0.616922	−	0.787025i	$-0.288380\pi$
$439$	25.8519	1.23384	0.616922	+	0.787025i	$0.288380\pi$
$440$	15.8212	0.754245
$441$	0	0
$442$	−5.30900	−0.252523
$443$	13.9386	0.662242	0.331121	−	0.943588i	$-0.392573\pi$
$443$	13.9386	0.662242	0.331121	+	0.943588i	$0.392573\pi$
$444$	0	0
$445$	17.9549	0.851144
$446$	9.98040	0.472585
$447$	0	0
$448$	−0.319501	−0.0150950
$449$	29.1538	1.37585	0.687927	−	0.725780i	$-0.258521\pi$
$449$	29.1538	1.37585	0.687927	+	0.725780i	$0.258521\pi$
$450$	0	0
$451$	9.69343	0.456446
$452$	−9.62865	−0.452894
$453$	0	0
$454$	−21.1224	−0.991322
$455$	5.31226	0.249043
$456$	0	0
$457$	23.7982	1.11323	0.556615	−	0.830770i	$-0.312100\pi$
$457$	23.7982	1.11323	0.556615	+	0.830770i	$0.312100\pi$
$458$	−23.2807	−1.08784
$459$	0	0
$460$	0.512482	0.0238946
$461$	30.8133	1.43512	0.717559	−	0.696497i	$-0.245259\pi$
$461$	30.8133	1.43512	0.717559	+	0.696497i	$0.245259\pi$
$462$	0	0
$463$	36.1401	1.67957	0.839787	−	0.542916i	$-0.182680\pi$
$463$	36.1401	1.67957	0.839787	+	0.542916i	$0.182680\pi$
$464$	−4.35924	−0.202372
$465$	0	0
$466$	26.7243	1.23798
$467$	−36.9754	−1.71102	−0.855509	−	0.517788i	$-0.826756\pi$
$467$	−36.9754	−1.71102	−0.855509	+	0.517788i	$0.826756\pi$
$468$	0	0
$469$	−2.37812	−0.109812
$470$	−36.6973	−1.69272
$471$	0	0
$472$	2.98285	0.137297
$473$	−31.5994	−1.45294
$474$	0	0
$475$	14.1725	0.650277
$476$	−0.362540	−0.0166170
$477$	0	0
$478$	−24.0599	−1.10047
$479$	38.3809	1.75367	0.876835	−	0.480792i	$-0.159651\pi$
$479$	38.3809	1.75367	0.876835	+	0.480792i	$0.159651\pi$
$480$	0	0
$481$	−35.5058	−1.61893
$482$	−1.49152	−0.0679369
$483$	0	0
$484$	8.82079	0.400945
$485$	66.3619	3.01334
$486$	0	0
$487$	34.3088	1.55468	0.777339	−	0.629082i	$-0.216569\pi$
$487$	34.3088	1.55468	0.777339	+	0.629082i	$0.216569\pi$
$488$	−3.49464	−0.158195
$489$	0	0
$490$	−24.5130	−1.10738
$491$	−17.4076	−0.785596	−0.392798	−	0.919625i	$-0.628493\pi$
$491$	−17.4076	−0.785596	−0.392798	+	0.919625i	$0.628493\pi$
$492$	0	0
$493$	−4.94646	−0.222777
$494$	−8.69212	−0.391077
$495$	0	0
$496$	0.671692	0.0301599
$497$	2.58750	0.116065
$498$	0	0
$499$	26.2832	1.17660	0.588299	−	0.808643i	$-0.299798\pi$
$499$	26.2832	1.17660	0.588299	+	0.808643i	$0.299798\pi$
$500$	9.34139	0.417760
$501$	0	0
$502$	14.5750	0.650512
$503$	−18.1786	−0.810544	−0.405272	−	0.914196i	$-0.632823\pi$
$503$	−18.1786	−0.810544	−0.405272	+	0.914196i	$0.632823\pi$
$504$	0	0
$505$	−8.59873	−0.382639
$506$	0.642038	0.0285421
$507$	0	0
$508$	0.159210	0.00706379
$509$	−24.6390	−1.09210	−0.546052	−	0.837751i	$-0.683870\pi$
$509$	−24.6390	−1.09210	−0.546052	+	0.837751i	$0.683870\pi$
$510$	0	0
$511$	−4.66513	−0.206373
$512$	1.00000	0.0441942
$513$	0	0
$514$	−11.0246	−0.486275
$515$	7.09084	0.312460
$516$	0	0
$517$	−45.9743	−2.02195
$518$	−2.42462	−0.106532
$519$	0	0
$520$	−16.6267	−0.729131
$521$	30.0131	1.31490	0.657449	−	0.753499i	$-0.271636\pi$
$521$	30.0131	1.31490	0.657449	+	0.753499i	$0.271636\pi$
$522$	0	0
$523$	−27.3437	−1.19566	−0.597829	−	0.801623i	$-0.703970\pi$
$523$	−27.3437	−1.19566	−0.597829	+	0.801623i	$0.703970\pi$
$524$	−9.48583	−0.414391
$525$	0	0
$526$	−13.1401	−0.572934
$527$	0.762174	0.0332008
$528$	0	0
$529$	−22.9792	−0.999096
$530$	2.86268	0.124347
$531$	0	0
$532$	−0.593566	−0.0257344
$533$	−10.1870	−0.441247
$534$	0	0
$535$	31.5237	1.36289
$536$	7.44325	0.321499
$537$	0	0
$538$	−10.3086	−0.444437
$539$	−30.7099	−1.32277
$540$	0	0
$541$	8.65632	0.372164	0.186082	−	0.982534i	$-0.440421\pi$
$541$	8.65632	0.372164	0.186082	+	0.982534i	$0.440421\pi$
$542$	1.01454	0.0435781
$543$	0	0
$544$	1.13471	0.0486502
$545$	−7.00799	−0.300189
$546$	0	0
$547$	27.5448	1.17773	0.588866	−	0.808231i	$-0.299575\pi$
$547$	27.5448	1.17773	0.588866	+	0.808231i	$0.299575\pi$
$548$	11.7094	0.500199
$549$	0	0
$550$	33.9632	1.44819
$551$	−8.09856	−0.345010
$552$	0	0
$553$	4.02156	0.171014
$554$	−10.8317	−0.460194
$555$	0	0
$556$	−5.25027	−0.222661
$557$	−26.1805	−1.10931	−0.554653	−	0.832082i	$-0.687149\pi$
$557$	−26.1805	−1.10931	−0.554653	+	0.832082i	$0.687149\pi$
$558$	0	0
$559$	33.2083	1.40456
$560$	−1.13540	−0.0479796
$561$	0	0
$562$	−8.47673	−0.357569
$563$	−3.97226	−0.167411	−0.0837055	−	0.996491i	$-0.526675\pi$
$563$	−3.97226	−0.167411	−0.0837055	+	0.996491i	$0.526675\pi$
$564$	0	0
$565$	−34.2172	−1.43953
$566$	6.30428	0.264988
$567$	0	0
$568$	−8.09856	−0.339808
$569$	−40.4911	−1.69748	−0.848738	−	0.528813i	$-0.822637\pi$
$569$	−40.4911	−1.69748	−0.848738	+	0.528813i	$0.822637\pi$
$570$	0	0
$571$	38.3015	1.60287	0.801434	−	0.598083i	$-0.204071\pi$
$571$	38.3015	1.60287	0.801434	+	0.598083i	$0.204071\pi$
$572$	−20.8300	−0.870946
$573$	0	0
$574$	−0.695647	−0.0290358
$575$	1.10014	0.0458790
$576$	0	0
$577$	8.27856	0.344641	0.172320	−	0.985041i	$-0.444874\pi$
$577$	8.27856	0.344641	0.172320	+	0.985041i	$0.444874\pi$
$578$	−15.7124	−0.653551
$579$	0	0
$580$	−15.4913	−0.643243
$581$	1.72966	0.0717584
$582$	0	0
$583$	3.58637	0.148532
$584$	14.6013	0.604206
$585$	0	0
$586$	−2.43305	−0.100509
$587$	−13.9428	−0.575482	−0.287741	−	0.957708i	$-0.592904\pi$
$587$	−13.9428	−0.575482	−0.287741	+	0.957708i	$0.592904\pi$
$588$	0	0
$589$	1.24786	0.0514174
$590$	10.6001	0.436399
$591$	0	0
$592$	7.58877	0.311896
$593$	−41.2342	−1.69329	−0.846644	−	0.532160i	$-0.821380\pi$
$593$	−41.2342	−1.69329	−0.846644	+	0.532160i	$0.821380\pi$
$594$	0	0
$595$	−1.28835	−0.0528173
$596$	−16.2922	−0.667356
$597$	0	0
$598$	−0.674728	−0.0275917
$599$	−3.13624	−0.128143	−0.0640716	−	0.997945i	$-0.520409\pi$
$599$	−3.13624	−0.128143	−0.0640716	+	0.997945i	$0.520409\pi$
$600$	0	0
$601$	−34.8355	−1.42097	−0.710485	−	0.703712i	$-0.751524\pi$
$601$	−34.8355	−1.42097	−0.710485	+	0.703712i	$0.751524\pi$
$602$	2.26772	0.0924256
$603$	0	0
$604$	−15.6179	−0.635485
$605$	31.3463	1.27441
$606$	0	0
$607$	−0.773204	−0.0313834	−0.0156917	−	0.999877i	$-0.504995\pi$
$607$	−0.773204	−0.0313834	−0.0156917	+	0.999877i	$0.504995\pi$
$608$	1.85779	0.0753434
$609$	0	0
$610$	−12.4188	−0.502824
$611$	48.3152	1.95462
$612$	0	0
$613$	13.9984	0.565389	0.282695	−	0.959210i	$-0.408772\pi$
$613$	13.9984	0.565389	0.282695	+	0.959210i	$0.408772\pi$
$614$	−23.7258	−0.957494
$615$	0	0
$616$	−1.42244	−0.0573116
$617$	11.7134	0.471566	0.235783	−	0.971806i	$-0.424235\pi$
$617$	11.7134	0.471566	0.235783	+	0.971806i	$0.424235\pi$
$618$	0	0
$619$	48.0396	1.93088	0.965438	−	0.260633i	$-0.0839312\pi$
$619$	48.0396	1.93088	0.965438	+	0.260633i	$0.0839312\pi$
$620$	2.38698	0.0958634
$621$	0	0
$622$	−10.8006	−0.433065
$623$	−1.61427	−0.0646744
$624$	0	0
$625$	−4.94692	−0.197877
$626$	−15.8912	−0.635140
$627$	0	0
$628$	17.6315	0.703575
$629$	8.61104	0.343345
$630$	0	0
$631$	39.5449	1.57426	0.787130	−	0.616788i	$-0.211566\pi$
$631$	39.5449	1.57426	0.787130	+	0.616788i	$0.211566\pi$
$632$	−12.5870	−0.500684
$633$	0	0
$634$	−3.23504	−0.128480
$635$	0.565781	0.0224523
$636$	0	0
$637$	32.2735	1.27872
$638$	−19.4076	−0.768353
$639$	0	0
$640$	3.55368	0.140472
$641$	0.661314	0.0261203	0.0130602	−	0.999915i	$-0.495843\pi$
$641$	0.661314	0.0261203	0.0130602	+	0.999915i	$0.495843\pi$
$642$	0	0
$643$	1.81344	0.0715152	0.0357576	−	0.999360i	$-0.488616\pi$
$643$	1.81344	0.0715152	0.0357576	+	0.999360i	$0.488616\pi$
$644$	−0.0460757	−0.00181564
$645$	0	0
$646$	2.10805	0.0829402
$647$	4.13765	0.162668	0.0813339	−	0.996687i	$-0.474082\pi$
$647$	4.13765	0.162668	0.0813339	+	0.996687i	$0.474082\pi$
$648$	0	0
$649$	13.2798	0.521278
$650$	−35.6925	−1.39997
$651$	0	0
$652$	−15.8801	−0.621912
$653$	26.8740	1.05166	0.525831	−	0.850589i	$-0.323755\pi$
$653$	26.8740	1.05166	0.525831	+	0.850589i	$0.323755\pi$
$654$	0	0
$655$	−33.7096	−1.31714
$656$	2.17729	0.0850090
$657$	0	0
$658$	3.29934	0.128622
$659$	14.4576	0.563190	0.281595	−	0.959533i	$-0.409137\pi$
$659$	14.4576	0.563190	0.281595	+	0.959533i	$0.409137\pi$
$660$	0	0
$661$	22.2816	0.866652	0.433326	−	0.901237i	$-0.357340\pi$
$661$	22.2816	0.866652	0.433326	+	0.901237i	$0.357340\pi$
$662$	−17.6637	−0.686518
$663$	0	0
$664$	−5.41363	−0.210090
$665$	−2.10935	−0.0817969
$666$	0	0
$667$	−0.628652	−0.0243415
$668$	−13.5784	−0.525365
$669$	0	0
$670$	26.4509	1.02189
$671$	−15.5583	−0.600623
$672$	0	0
$673$	24.9020	0.959902	0.479951	−	0.877295i	$-0.340655\pi$
$673$	24.9020	0.959902	0.479951	+	0.877295i	$0.340655\pi$
$674$	−10.5429	−0.406096
$675$	0	0
$676$	8.89058	0.341945
$677$	7.98388	0.306845	0.153423	−	0.988161i	$-0.450970\pi$
$677$	7.98388	0.306845	0.153423	+	0.988161i	$0.450970\pi$
$678$	0	0
$679$	−5.96640	−0.228969
$680$	4.03239	0.154635
$681$	0	0
$682$	2.99041	0.114509
$683$	−35.0612	−1.34158	−0.670789	−	0.741648i	$-0.734045\pi$
$683$	−35.0612	−1.34158	−0.670789	+	0.741648i	$0.734045\pi$
$684$	0	0
$685$	41.6113	1.58989
$686$	4.44040	0.169535
$687$	0	0
$688$	−7.09771	−0.270598
$689$	−3.76898	−0.143587
$690$	0	0
$691$	−1.98204	−0.0754003	−0.0377001	−	0.999289i	$-0.512003\pi$
$691$	−1.98204	−0.0754003	−0.0377001	+	0.999289i	$0.512003\pi$
$692$	6.44441	0.244980
$693$	0	0
$694$	32.5047	1.23386
$695$	−18.6578	−0.707730
$696$	0	0
$697$	2.47059	0.0935804
$698$	−20.0703	−0.759672
$699$	0	0
$700$	−2.43736	−0.0921236
$701$	42.8694	1.61916	0.809578	−	0.587013i	$-0.199696\pi$
$701$	42.8694	1.61916	0.809578	+	0.587013i	$0.199696\pi$
$702$	0	0
$703$	14.0984	0.531730
$704$	4.45206	0.167793
$705$	0	0
$706$	11.0361	0.415347
$707$	0.773086	0.0290749
$708$	0	0
$709$	35.1176	1.31887	0.659434	−	0.751763i	$-0.270796\pi$
$709$	35.1176	1.31887	0.659434	+	0.751763i	$0.270796\pi$
$710$	−28.7797	−1.08008
$711$	0	0
$712$	5.05248	0.189350
$713$	0.0968658	0.00362765
$714$	0	0
$715$	−74.0232	−2.76831
$716$	−12.5123	−0.467605
$717$	0	0
$718$	−3.92087	−0.146326
$719$	15.0484	0.561212	0.280606	−	0.959823i	$-0.409465\pi$
$719$	15.0484	0.561212	0.280606	+	0.959823i	$0.409465\pi$
$720$	0	0
$721$	−0.637516	−0.0237424
$722$	−15.5486	−0.578659
$723$	0	0
$724$	0.304503	0.0113168
$725$	−33.2551	−1.23506
$726$	0	0
$727$	14.6256	0.542434	0.271217	−	0.962518i	$-0.412574\pi$
$727$	14.6256	0.542434	0.271217	+	0.962518i	$0.412574\pi$
$728$	1.49486	0.0554032
$729$	0	0
$730$	51.8884	1.92048
$731$	−8.05383	−0.297882
$732$	0	0
$733$	15.0750	0.556807	0.278403	−	0.960464i	$-0.410195\pi$
$733$	15.0750	0.556807	0.278403	+	0.960464i	$0.410195\pi$
$734$	29.1512	1.07599
$735$	0	0
$736$	0.144212	0.00531571
$737$	33.1377	1.22064
$738$	0	0
$739$	47.8062	1.75858	0.879290	−	0.476288i	$-0.158018\pi$
$739$	47.8062	1.75858	0.879290	+	0.476288i	$0.158018\pi$
$740$	26.9681	0.991366
$741$	0	0
$742$	−0.257375	−0.00944854
$743$	−46.5179	−1.70658	−0.853288	−	0.521440i	$-0.825395\pi$
$743$	−46.5179	−1.70658	−0.853288	+	0.521440i	$0.825395\pi$
$744$	0	0
$745$	−57.8974	−2.12120
$746$	6.27181	0.229627
$747$	0	0
$748$	5.05178	0.184712
$749$	−2.83420	−0.103560
$750$	0	0
$751$	1.76579	0.0644346	0.0322173	−	0.999481i	$-0.489743\pi$
$751$	1.76579	0.0644346	0.0322173	+	0.999481i	$0.489743\pi$
$752$	−10.3265	−0.376570
$753$	0	0
$754$	20.3957	0.742768
$755$	−55.5012	−2.01989
$756$	0	0
$757$	3.34143	0.121446	0.0607232	−	0.998155i	$-0.480659\pi$
$757$	3.34143	0.121446	0.0607232	+	0.998155i	$0.480659\pi$
$758$	−26.9562	−0.979094
$759$	0	0
$760$	6.60200	0.239480
$761$	47.7521	1.73101	0.865507	−	0.500897i	$-0.166997\pi$
$761$	47.7521	1.73101	0.865507	+	0.500897i	$0.166997\pi$
$762$	0	0
$763$	0.630067	0.0228100
$764$	−14.3927	−0.520710
$765$	0	0
$766$	−3.25627	−0.117654
$767$	−13.9560	−0.503920
$768$	0	0
$769$	−6.78515	−0.244679	−0.122339	−	0.992488i	$-0.539040\pi$
$769$	−6.78515	−0.244679	−0.122339	+	0.992488i	$0.539040\pi$
$770$	−5.05488	−0.182165
$771$	0	0
$772$	3.05063	0.109794
$773$	−30.1367	−1.08394	−0.541970	−	0.840398i	$-0.682322\pi$
$773$	−30.1367	−1.08394	−0.541970	+	0.840398i	$0.682322\pi$
$774$	0	0
$775$	5.12410	0.184063
$776$	18.6741	0.670362
$777$	0	0
$778$	1.38968	0.0498224
$779$	4.04496	0.144926
$780$	0	0
$781$	−36.0552	−1.29016
$782$	0.163638	0.00585169
$783$	0	0
$784$	−6.89792	−0.246354
$785$	62.6569	2.23632
$786$	0	0
$787$	16.1546	0.575850	0.287925	−	0.957653i	$-0.407035\pi$
$787$	16.1546	0.575850	0.287925	+	0.957653i	$0.407035\pi$
$788$	22.2642	0.793129
$789$	0	0
$790$	−44.7302	−1.59143
$791$	3.07636	0.109383
$792$	0	0
$793$	16.3505	0.580623
$794$	2.14020	0.0759530
$795$	0	0
$796$	24.5634	0.870626
$797$	−0.551279	−0.0195273	−0.00976365	−	0.999952i	$-0.503108\pi$
$797$	−0.551279	−0.0195273	−0.00976365	+	0.999952i	$0.503108\pi$
$798$	0	0
$799$	−11.7176	−0.414540
$800$	7.62865	0.269714
$801$	0	0
$802$	−8.55331	−0.302028
$803$	65.0058	2.29401
$804$	0	0
$805$	−0.163739	−0.00577102
$806$	−3.14267	−0.110696
$807$	0	0
$808$	−2.41967	−0.0851237
$809$	−3.04561	−0.107078	−0.0535389	−	0.998566i	$-0.517050\pi$
$809$	−3.04561	−0.107078	−0.0535389	+	0.998566i	$0.517050\pi$
$810$	0	0
$811$	−7.71732	−0.270992	−0.135496	−	0.990778i	$-0.543263\pi$
$811$	−7.71732	−0.270992	−0.135496	+	0.990778i	$0.543263\pi$
$812$	1.39278	0.0488770
$813$	0	0
$814$	33.7856	1.18419
$815$	−56.4327	−1.97675
$816$	0	0
$817$	−13.1861	−0.461322
$818$	27.2719	0.953539
$819$	0	0
$820$	7.73741	0.270202
$821$	23.4544	0.818565	0.409282	−	0.912408i	$-0.365779\pi$
$821$	23.4544	0.818565	0.409282	+	0.912408i	$0.365779\pi$
$822$	0	0
$823$	−29.8783	−1.04149	−0.520746	−	0.853711i	$-0.674346\pi$
$823$	−29.8783	−1.04149	−0.520746	+	0.853711i	$0.674346\pi$
$824$	1.99535	0.0695114
$825$	0	0
$826$	−0.953022	−0.0331599
$827$	−19.4419	−0.676060	−0.338030	−	0.941135i	$-0.609760\pi$
$827$	−19.4419	−0.676060	−0.338030	+	0.941135i	$0.609760\pi$
$828$	0	0
$829$	−10.2719	−0.356757	−0.178379	−	0.983962i	$-0.557085\pi$
$829$	−10.2719	−0.356757	−0.178379	+	0.983962i	$0.557085\pi$
$830$	−19.2383	−0.667772
$831$	0	0
$832$	−4.67874	−0.162206
$833$	−7.82713	−0.271194
$834$	0	0
$835$	−48.2534	−1.66988
$836$	8.27099	0.286058
$837$	0	0
$838$	16.0153	0.553241
$839$	−13.4756	−0.465231	−0.232615	−	0.972569i	$-0.574728\pi$
$839$	−13.4756	−0.465231	−0.232615	+	0.972569i	$0.574728\pi$
$840$	0	0
$841$	−9.99706	−0.344726
$842$	−18.1470	−0.625388
$843$	0	0
$844$	−3.01931	−0.103929
$845$	31.5943	1.08688
$846$	0	0
$847$	−2.81825	−0.0968363
$848$	0.805554	0.0276628
$849$	0	0
$850$	8.65630	0.296908
$851$	1.09439	0.0375152
$852$	0	0
$853$	47.5125	1.62680	0.813398	−	0.581708i	$-0.197615\pi$
$853$	47.5125	1.62680	0.813398	+	0.581708i	$0.197615\pi$
$854$	1.11654	0.0382072
$855$	0	0
$856$	8.87072	0.303195
$857$	23.2163	0.793055	0.396528	−	0.918023i	$-0.370215\pi$
$857$	23.2163	0.793055	0.396528	+	0.918023i	$0.370215\pi$
$858$	0	0
$859$	18.4705	0.630204	0.315102	−	0.949058i	$-0.397961\pi$
$859$	18.4705	0.630204	0.315102	+	0.949058i	$0.397961\pi$
$860$	−25.2230	−0.860097
$861$	0	0
$862$	2.13698	0.0727857
$863$	−21.1288	−0.719233	−0.359616	−	0.933100i	$-0.617092\pi$
$863$	−21.1288	−0.719233	−0.359616	+	0.933100i	$0.617092\pi$
$864$	0	0
$865$	22.9014	0.778670
$866$	25.1733	0.855424
$867$	0	0
$868$	−0.214606	−0.00728421
$869$	−56.0380	−1.90096
$870$	0	0
$871$	−34.8250	−1.18000
$872$	−1.97204	−0.0667816
$873$	0	0
$874$	0.267915	0.00906237
$875$	−2.98458	−0.100897
$876$	0	0
$877$	16.2134	0.547488	0.273744	−	0.961803i	$-0.411738\pi$
$877$	16.2134	0.547488	0.273744	+	0.961803i	$0.411738\pi$
$878$	25.8519	0.872459
$879$	0	0
$880$	15.8212	0.533332
$881$	44.5020	1.49931	0.749655	−	0.661829i	$-0.230220\pi$
$881$	44.5020	1.49931	0.749655	+	0.661829i	$0.230220\pi$
$882$	0	0
$883$	16.2665	0.547412	0.273706	−	0.961813i	$-0.411751\pi$
$883$	16.2665	0.547412	0.273706	+	0.961813i	$0.411751\pi$
$884$	−5.30900	−0.178561
$885$	0	0
$886$	13.9386	0.468276
$887$	20.5687	0.690629	0.345314	−	0.938487i	$-0.387772\pi$
$887$	20.5687	0.690629	0.345314	+	0.938487i	$0.387772\pi$
$888$	0	0
$889$	−0.0508676	−0.00170605
$890$	17.9549	0.601850
$891$	0	0
$892$	9.98040	0.334168
$893$	−19.1846	−0.641987
$894$	0	0
$895$	−44.4646	−1.48629
$896$	−0.319501	−0.0106738
$897$	0	0
$898$	29.1538	0.972875
$899$	−2.92806	−0.0976564
$900$	0	0
$901$	0.914069	0.0304521
$902$	9.69343	0.322756
$903$	0	0
$904$	−9.62865	−0.320244
$905$	1.08211	0.0359704
$906$	0	0
$907$	16.5475	0.549452	0.274726	−	0.961523i	$-0.411413\pi$
$907$	16.5475	0.549452	0.274726	+	0.961523i	$0.411413\pi$
$908$	−21.1224	−0.700970
$909$	0	0
$910$	5.31226	0.176100
$911$	26.8332	0.889025	0.444513	−	0.895773i	$-0.353377\pi$
$911$	26.8332	0.889025	0.444513	+	0.895773i	$0.353377\pi$
$912$	0	0
$913$	−24.1018	−0.797652
$914$	23.7982	0.787173
$915$	0	0
$916$	−23.2807	−0.769217
$917$	3.03073	0.100084
$918$	0	0
$919$	−26.3725	−0.869947	−0.434974	−	0.900443i	$-0.643242\pi$
$919$	−26.3725	−0.869947	−0.434974	+	0.900443i	$0.643242\pi$
$920$	0.512482	0.0168960
$921$	0	0
$922$	30.8133	1.01478
$923$	37.8910	1.24720
$924$	0	0
$925$	57.8921	1.90348
$926$	36.1401	1.18764
$927$	0	0
$928$	−4.35924	−0.143099
$929$	42.8113	1.40459	0.702297	−	0.711884i	$-0.252158\pi$
$929$	42.8113	1.40459	0.702297	+	0.711884i	$0.252158\pi$
$930$	0	0
$931$	−12.8149	−0.419991
$932$	26.7243	0.875383
$933$	0	0
$934$	−36.9754	−1.20987
$935$	17.9524	0.587107
$936$	0	0
$937$	−14.1482	−0.462201	−0.231100	−	0.972930i	$-0.574233\pi$
$937$	−14.1482	−0.462201	−0.231100	+	0.972930i	$0.574233\pi$
$938$	−2.37812	−0.0776485
$939$	0	0
$940$	−36.6973	−1.19693
$941$	25.9186	0.844924	0.422462	−	0.906381i	$-0.361166\pi$
$941$	25.9186	0.844924	0.422462	+	0.906381i	$0.361166\pi$
$942$	0	0
$943$	0.313991	0.0102250
$944$	2.98285	0.0970834
$945$	0	0
$946$	−31.5994	−1.02738
$947$	16.4240	0.533710	0.266855	−	0.963737i	$-0.414016\pi$
$947$	16.4240	0.533710	0.266855	+	0.963737i	$0.414016\pi$
$948$	0	0
$949$	−68.3157	−2.21762
$950$	14.1725	0.459815
$951$	0	0
$952$	−0.362540	−0.0117500
$953$	−37.2534	−1.20676	−0.603379	−	0.797455i	$-0.706179\pi$
$953$	−37.2534	−1.20676	−0.603379	+	0.797455i	$0.706179\pi$
$954$	0	0
$955$	−51.1471	−1.65508
$956$	−24.0599	−0.778152
$957$	0	0
$958$	38.3809	1.24003
$959$	−3.74115	−0.120808
$960$	0	0
$961$	−30.5488	−0.985446
$962$	−35.5058	−1.14475
$963$	0	0
$964$	−1.49152	−0.0480386
$965$	10.8410	0.348983
$966$	0	0
$967$	−9.95715	−0.320200	−0.160100	−	0.987101i	$-0.551182\pi$
$967$	−9.95715	−0.320200	−0.160100	+	0.987101i	$0.551182\pi$
$968$	8.82079	0.283511
$969$	0	0
$970$	66.3619	2.13075
$971$	2.62332	0.0841864	0.0420932	−	0.999114i	$-0.486597\pi$
$971$	2.62332	0.0841864	0.0420932	+	0.999114i	$0.486597\pi$
$972$	0	0
$973$	1.67746	0.0537771
$974$	34.3088	1.09932
$975$	0	0
$976$	−3.49464	−0.111861
$977$	−6.63804	−0.212370	−0.106185	−	0.994346i	$-0.533864\pi$
$977$	−6.63804	−0.212370	−0.106185	+	0.994346i	$0.533864\pi$
$978$	0	0
$979$	22.4939	0.718909
$980$	−24.5130	−0.783039
$981$	0	0
$982$	−17.4076	−0.555500
$983$	−47.3962	−1.51170	−0.755851	−	0.654743i	$-0.772777\pi$
$983$	−47.3962	−1.51170	−0.755851	+	0.654743i	$0.772777\pi$
$984$	0	0
$985$	79.1199	2.52097
$986$	−4.94646	−0.157527
$987$	0	0
$988$	−8.69212	−0.276533
$989$	−1.02357	−0.0325477
$990$	0	0
$991$	44.1149	1.40136	0.700678	−	0.713478i	$-0.252881\pi$
$991$	44.1149	1.40136	0.700678	+	0.713478i	$0.252881\pi$
$992$	0.671692	0.0213262
$993$	0	0
$994$	2.58750	0.0820704
$995$	87.2905	2.76729
$996$	0	0
$997$	49.8147	1.57765	0.788823	−	0.614620i	$-0.210691\pi$
$997$	49.8147	1.57765	0.788823	+	0.614620i	$0.210691\pi$
$998$	26.2832	0.831981
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.2.a.g.1.6		6
3.2	odd	2		1458.2.a.f.1.1		6
9.2	odd	6		1458.2.c.g.973.6		12
9.4	even	3		1458.2.c.f.487.1		12
9.5	odd	6		1458.2.c.g.487.6		12
9.7	even	3		1458.2.c.f.973.1		12
27.2	odd	18		162.2.e.b.145.1		12
27.4	even	9		486.2.e.f.379.1		12
27.5	odd	18		486.2.e.e.217.1		12
27.7	even	9		486.2.e.f.109.1		12
27.11	odd	18		486.2.e.e.271.1		12
27.13	even	9		54.2.e.b.7.1	✓	12
27.14	odd	18		162.2.e.b.19.1		12
27.16	even	9		486.2.e.h.271.2		12
27.20	odd	18		486.2.e.g.109.2		12
27.22	even	9		486.2.e.h.217.2		12
27.23	odd	18		486.2.e.g.379.2		12
27.25	even	9		54.2.e.b.31.1	yes	12
108.67	odd	18		432.2.u.b.385.2		12
108.79	odd	18		432.2.u.b.193.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.2.e.b.7.1	✓	12	27.13	even	9
54.2.e.b.31.1	yes	12	27.25	even	9
162.2.e.b.19.1		12	27.14	odd	18
162.2.e.b.145.1		12	27.2	odd	18
432.2.u.b.193.2		12	108.79	odd	18
432.2.u.b.385.2		12	108.67	odd	18
486.2.e.e.217.1		12	27.5	odd	18
486.2.e.e.271.1		12	27.11	odd	18
486.2.e.f.109.1		12	27.7	even	9
486.2.e.f.379.1		12	27.4	even	9
486.2.e.g.109.2		12	27.20	odd	18
486.2.e.g.379.2		12	27.23	odd	18
486.2.e.h.217.2		12	27.22	even	9
486.2.e.h.271.2		12	27.16	even	9
1458.2.a.f.1.1		6	3.2	odd	2
1458.2.a.g.1.6		6	1.1	even	1	trivial
1458.2.c.f.487.1		12	9.4	even	3
1458.2.c.f.973.1		12	9.7	even	3
1458.2.c.g.487.6		12	9.5	odd	6
1458.2.c.g.973.6		12	9.2	odd	6

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:	\( N \)	\(=\)	\( 1458 = 2 \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1458.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.55368 q^{5} -0.319501 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.55368 q^{5} -0.319501 q^{7} +1.00000 q^{8} +3.55368 q^{10} +4.45206 q^{11} -4.67874 q^{13} -0.319501 q^{14} +1.00000 q^{16} +1.13471 q^{17} +1.85779 q^{19} +3.55368 q^{20} +4.45206 q^{22} +0.144212 q^{23} +7.62865 q^{25} -4.67874 q^{26} -0.319501 q^{28} -4.35924 q^{29} +0.671692 q^{31} +1.00000 q^{32} +1.13471 q^{34} -1.13540 q^{35} +7.58877 q^{37} +1.85779 q^{38} +3.55368 q^{40} +2.17729 q^{41} -7.09771 q^{43} +4.45206 q^{44} +0.144212 q^{46} -10.3265 q^{47} -6.89792 q^{49} +7.62865 q^{50} -4.67874 q^{52} +0.805554 q^{53} +15.8212 q^{55} -0.319501 q^{56} -4.35924 q^{58} +2.98285 q^{59} -3.49464 q^{61} +0.671692 q^{62} +1.00000 q^{64} -16.6267 q^{65} +7.44325 q^{67} +1.13471 q^{68} -1.13540 q^{70} -8.09856 q^{71} +14.6013 q^{73} +7.58877 q^{74} +1.85779 q^{76} -1.42244 q^{77} -12.5870 q^{79} +3.55368 q^{80} +2.17729 q^{82} -5.41363 q^{83} +4.03239 q^{85} -7.09771 q^{86} +4.45206 q^{88} +5.05248 q^{89} +1.49486 q^{91} +0.144212 q^{92} -10.3265 q^{94} +6.60200 q^{95} +18.6741 q^{97} -6.89792 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 3 q^{5} + 6 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + 6 * q^2 + 6 * q^4 + 3 * q^5 + 6 * q^7 + 6 * q^8	\( 6 q + 6 q^{2} + 6 q^{4} + 3 q^{5} + 6 q^{7} + 6 q^{8} + 3 q^{10} + 3 q^{11} + 9 q^{13} + 6 q^{14} + 6 q^{16} + 6 q^{17} + 9 q^{19} + 3 q^{20} + 3 q^{22} - 3 q^{23} + 15 q^{25} + 9 q^{26} + 6 q^{28} + 3 q^{29} + 12 q^{31} + 6 q^{32} + 6 q^{34} - 3 q^{35} + 15 q^{37} + 9 q^{38} + 3 q^{40} + 3 q^{41} + 12 q^{43} + 3 q^{44} - 3 q^{46} - 9 q^{47} + 12 q^{49} + 15 q^{50} + 9 q^{52} - 6 q^{53} + 9 q^{55} + 6 q^{56} + 3 q^{58} - 3 q^{59} + 12 q^{61} + 12 q^{62} + 6 q^{64} - 3 q^{65} + 21 q^{67} + 6 q^{68} - 3 q^{70} - 12 q^{71} + 21 q^{73} + 15 q^{74} + 9 q^{76} - 3 q^{77} + 3 q^{80} + 3 q^{82} - 27 q^{83} + 12 q^{86} + 3 q^{88} - 12 q^{89} + 6 q^{91} - 3 q^{92} - 9 q^{94} - 30 q^{95} + 18 q^{97} + 12 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 3 * q^5 + 6 * q^7 + 6 * q^8 + 3 * q^10 + 3 * q^11 + 9 * q^13 + 6 * q^14 + 6 * q^16 + 6 * q^17 + 9 * q^19 + 3 * q^20 + 3 * q^22 - 3 * q^23 + 15 * q^25 + 9 * q^26 + 6 * q^28 + 3 * q^29 + 12 * q^31 + 6 * q^32 + 6 * q^34 - 3 * q^35 + 15 * q^37 + 9 * q^38 + 3 * q^40 + 3 * q^41 + 12 * q^43 + 3 * q^44 - 3 * q^46 - 9 * q^47 + 12 * q^49 + 15 * q^50 + 9 * q^52 - 6 * q^53 + 9 * q^55 + 6 * q^56 + 3 * q^58 - 3 * q^59 + 12 * q^61 + 12 * q^62 + 6 * q^64 - 3 * q^65 + 21 * q^67 + 6 * q^68 - 3 * q^70 - 12 * q^71 + 21 * q^73 + 15 * q^74 + 9 * q^76 - 3 * q^77 + 3 * q^80 + 3 * q^82 - 27 * q^83 + 12 * q^86 + 3 * q^88 - 12 * q^89 + 6 * q^91 - 3 * q^92 - 9 * q^94 - 30 * q^95 + 18 * q^97 + 12 * q^98

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