LMFDB - Embedded newform 4842.2.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4842,2,Mod(1,4842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4842, 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("4842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4842 = 2 \cdot 3^{2} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4842.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:	$38.6635646587$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 $ x^7 - x^6 - 15x^5 + 16x^4 + 49x^3 - 53x^2 - 44*x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 538)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.14366$ of defining polynomial
Character	$\chi$	$=$	4842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -0.725043 q^{5} +1.29978 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -0.725043 q^{5} +1.29978 q^{7} -1.00000 q^{8} +0.725043 q^{10} +3.71687 q^{11} +2.30856 q^{13} -1.29978 q^{14} +1.00000 q^{16} -6.32982 q^{17} +0.0212539 q^{19} -0.725043 q^{20} -3.71687 q^{22} +1.30511 q^{23} -4.47431 q^{25} -2.30856 q^{26} +1.29978 q^{28} +2.06303 q^{29} -5.82316 q^{31} -1.00000 q^{32} +6.32982 q^{34} -0.942394 q^{35} -7.25272 q^{37} -0.0212539 q^{38} +0.725043 q^{40} -8.77454 q^{41} +4.52327 q^{43} +3.71687 q^{44} -1.30511 q^{46} -3.85502 q^{47} -5.31058 q^{49} +4.47431 q^{50} +2.30856 q^{52} +6.86923 q^{53} -2.69489 q^{55} -1.29978 q^{56} -2.06303 q^{58} -7.84518 q^{59} +3.84958 q^{61} +5.82316 q^{62} +1.00000 q^{64} -1.67381 q^{65} -9.31541 q^{67} -6.32982 q^{68} +0.942394 q^{70} +0.643097 q^{71} -2.97920 q^{73} +7.25272 q^{74} +0.0212539 q^{76} +4.83110 q^{77} +0.418245 q^{79} -0.725043 q^{80} +8.77454 q^{82} +15.6322 q^{83} +4.58939 q^{85} -4.52327 q^{86} -3.71687 q^{88} -10.3879 q^{89} +3.00062 q^{91} +1.30511 q^{92} +3.85502 q^{94} -0.0154100 q^{95} +5.07659 q^{97} +5.31058 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} + 7 q^{4} - 7 q^{5} + 6 q^{7} - 7 q^{8}+O(q^{10}) $ 7 * q - 7 * q^2 + 7 * q^4 - 7 * q^5 + 6 * q^7 - 7 * q^8	$ 7 q - 7 q^{2} + 7 q^{4} - 7 q^{5} + 6 q^{7} - 7 q^{8} + 7 q^{10} + 3 q^{11} - 9 q^{13} - 6 q^{14} + 7 q^{16} - 8 q^{17} - 11 q^{19} - 7 q^{20} - 3 q^{22} - 12 q^{23} + 22 q^{25} + 9 q^{26} + 6 q^{28} + 5 q^{29} + 14 q^{31} - 7 q^{32} + 8 q^{34} + 4 q^{35} + 13 q^{37} + 11 q^{38} + 7 q^{40} - 12 q^{41} - 11 q^{43} + 3 q^{44} + 12 q^{46} - 2 q^{47} + 15 q^{49} - 22 q^{50} - 9 q^{52} - 19 q^{53} - 40 q^{55} - 6 q^{56} - 5 q^{58} + 9 q^{59} - 3 q^{61} - 14 q^{62} + 7 q^{64} + 10 q^{65} - 33 q^{67} - 8 q^{68} - 4 q^{70} - 28 q^{71} - 14 q^{73} - 13 q^{74} - 11 q^{76} - 10 q^{77} + 2 q^{79} - 7 q^{80} + 12 q^{82} + 7 q^{83} - 16 q^{85} + 11 q^{86} - 3 q^{88} - 18 q^{89} - 26 q^{91} - 12 q^{92} + 2 q^{94} + 34 q^{95} - 4 q^{97} - 15 q^{98}+O(q^{100}) $ 7 * q - 7 * q^2 + 7 * q^4 - 7 * q^5 + 6 * q^7 - 7 * q^8 + 7 * q^10 + 3 * q^11 - 9 * q^13 - 6 * q^14 + 7 * q^16 - 8 * q^17 - 11 * q^19 - 7 * q^20 - 3 * q^22 - 12 * q^23 + 22 * q^25 + 9 * q^26 + 6 * q^28 + 5 * q^29 + 14 * q^31 - 7 * q^32 + 8 * q^34 + 4 * q^35 + 13 * q^37 + 11 * q^38 + 7 * q^40 - 12 * q^41 - 11 * q^43 + 3 * q^44 + 12 * q^46 - 2 * q^47 + 15 * q^49 - 22 * q^50 - 9 * q^52 - 19 * q^53 - 40 * q^55 - 6 * q^56 - 5 * q^58 + 9 * q^59 - 3 * q^61 - 14 * q^62 + 7 * q^64 + 10 * q^65 - 33 * q^67 - 8 * q^68 - 4 * q^70 - 28 * q^71 - 14 * q^73 - 13 * q^74 - 11 * q^76 - 10 * q^77 + 2 * q^79 - 7 * q^80 + 12 * q^82 + 7 * q^83 - 16 * q^85 + 11 * q^86 - 3 * q^88 - 18 * q^89 - 26 * q^91 - 12 * q^92 + 2 * q^94 + 34 * q^95 - 4 * q^97 - 15 * 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
$5$	−0.725043	−0.324249	−0.162125	−	0.986770i	$-0.551835\pi$
$5$	−0.725043	−0.324249	−0.162125	+	0.986770i	$0.551835\pi$
$6$	0	0
$7$	1.29978	0.491269	0.245635	−	0.969362i	$-0.421004\pi$
$7$	1.29978	0.491269	0.245635	+	0.969362i	$0.421004\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0.725043	0.229279
$11$	3.71687	1.12068	0.560340	−	0.828263i	$-0.310671\pi$
$11$	3.71687	1.12068	0.560340	+	0.828263i	$0.310671\pi$
$12$	0	0
$13$	2.30856	0.640280	0.320140	−	0.947370i	$-0.396270\pi$
$13$	2.30856	0.640280	0.320140	+	0.947370i	$0.396270\pi$
$14$	−1.29978	−0.347380
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.32982	−1.53521	−0.767603	−	0.640925i	$-0.778551\pi$
$17$	−6.32982	−1.53521	−0.767603	+	0.640925i	$0.778551\pi$
$18$	0	0
$19$	0.0212539	0.00487598	0.00243799	−	0.999997i	$-0.499224\pi$
$19$	0.0212539	0.00487598	0.00243799	+	0.999997i	$0.499224\pi$
$20$	−0.725043	−0.162125
$21$	0	0
$22$	−3.71687	−0.792440
$23$	1.30511	0.272134	0.136067	−	0.990700i	$-0.456554\pi$
$23$	1.30511	0.272134	0.136067	+	0.990700i	$0.456554\pi$
$24$	0	0
$25$	−4.47431	−0.894863
$26$	−2.30856	−0.452747
$27$	0	0
$28$	1.29978	0.245635
$29$	2.06303	0.383095	0.191548	−	0.981483i	$-0.438649\pi$
$29$	2.06303	0.383095	0.191548	+	0.981483i	$0.438649\pi$
$30$	0	0
$31$	−5.82316	−1.04587	−0.522935	−	0.852372i	$-0.675163\pi$
$31$	−5.82316	−1.04587	−0.522935	+	0.852372i	$0.675163\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.32982	1.08555
$35$	−0.942394	−0.159294
$36$	0	0
$37$	−7.25272	−1.19234	−0.596170	−	0.802858i	$-0.703312\pi$
$37$	−7.25272	−1.19234	−0.596170	+	0.802858i	$0.703312\pi$
$38$	−0.0212539	−0.00344784
$39$	0	0
$40$	0.725043	0.114639
$41$	−8.77454	−1.37035	−0.685176	−	0.728377i	$-0.740275\pi$
$41$	−8.77454	−1.37035	−0.685176	+	0.728377i	$0.740275\pi$
$42$	0	0
$43$	4.52327	0.689792	0.344896	−	0.938641i	$-0.387914\pi$
$43$	4.52327	0.689792	0.344896	+	0.938641i	$0.387914\pi$
$44$	3.71687	0.560340
$45$	0	0
$46$	−1.30511	−0.192428
$47$	−3.85502	−0.562313	−0.281156	−	0.959662i	$-0.590718\pi$
$47$	−3.85502	−0.562313	−0.281156	+	0.959662i	$0.590718\pi$
$48$	0	0
$49$	−5.31058	−0.758654
$50$	4.47431	0.632763
$51$	0	0
$52$	2.30856	0.320140
$53$	6.86923	0.943562	0.471781	−	0.881716i	$-0.343611\pi$
$53$	6.86923	0.943562	0.471781	+	0.881716i	$0.343611\pi$
$54$	0	0
$55$	−2.69489	−0.363379
$56$	−1.29978	−0.173690
$57$	0	0
$58$	−2.06303	−0.270889
$59$	−7.84518	−1.02136	−0.510678	−	0.859772i	$-0.670605\pi$
$59$	−7.84518	−1.02136	−0.510678	+	0.859772i	$0.670605\pi$
$60$	0	0
$61$	3.84958	0.492888	0.246444	−	0.969157i	$-0.420738\pi$
$61$	3.84958	0.492888	0.246444	+	0.969157i	$0.420738\pi$
$62$	5.82316	0.739542
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.67381	−0.207610
$66$	0	0
$67$	−9.31541	−1.13806	−0.569029	−	0.822317i	$-0.692681\pi$
$67$	−9.31541	−1.13806	−0.569029	+	0.822317i	$0.692681\pi$
$68$	−6.32982	−0.767603
$69$	0	0
$70$	0.942394	0.112638
$71$	0.643097	0.0763215	0.0381608	−	0.999272i	$-0.487850\pi$
$71$	0.643097	0.0763215	0.0381608	+	0.999272i	$0.487850\pi$
$72$	0	0
$73$	−2.97920	−0.348689	−0.174345	−	0.984685i	$-0.555781\pi$
$73$	−2.97920	−0.348689	−0.174345	+	0.984685i	$0.555781\pi$
$74$	7.25272	0.843112
$75$	0	0
$76$	0.0212539	0.00243799
$77$	4.83110	0.550555
$78$	0	0
$79$	0.418245	0.0470563	0.0235281	−	0.999723i	$-0.492510\pi$
$79$	0.418245	0.0470563	0.0235281	+	0.999723i	$0.492510\pi$
$80$	−0.725043	−0.0810623
$81$	0	0
$82$	8.77454	0.968985
$83$	15.6322	1.71585	0.857927	−	0.513771i	$-0.171752\pi$
$83$	15.6322	1.71585	0.857927	+	0.513771i	$0.171752\pi$
$84$	0	0
$85$	4.58939	0.497789
$86$	−4.52327	−0.487757
$87$	0	0
$88$	−3.71687	−0.396220
$89$	−10.3879	−1.10111	−0.550556	−	0.834798i	$-0.685584\pi$
$89$	−10.3879	−1.10111	−0.550556	+	0.834798i	$0.685584\pi$
$90$	0	0
$91$	3.00062	0.314550
$92$	1.30511	0.136067
$93$	0	0
$94$	3.85502	0.397615
$95$	−0.0154100	−0.00158103
$96$	0	0
$97$	5.07659	0.515450	0.257725	−	0.966218i	$-0.417027\pi$
$97$	5.07659	0.515450	0.257725	+	0.966218i	$0.417027\pi$
$98$	5.31058	0.536450
$99$	0	0
$100$	−4.47431	−0.447431
$101$	−12.5321	−1.24699	−0.623495	−	0.781827i	$-0.714288\pi$
$101$	−12.5321	−1.24699	−0.623495	+	0.781827i	$0.714288\pi$
$102$	0	0
$103$	5.43374	0.535403	0.267701	−	0.963502i	$-0.413736\pi$
$103$	5.43374	0.535403	0.267701	+	0.963502i	$0.413736\pi$
$104$	−2.30856	−0.226373
$105$	0	0
$106$	−6.86923	−0.667199
$107$	3.45500	0.334007	0.167004	−	0.985956i	$-0.446591\pi$
$107$	3.45500	0.334007	0.167004	+	0.985956i	$0.446591\pi$
$108$	0	0
$109$	−6.80801	−0.652089	−0.326044	−	0.945354i	$-0.605716\pi$
$109$	−6.80801	−0.652089	−0.326044	+	0.945354i	$0.605716\pi$
$110$	2.69489	0.256948
$111$	0	0
$112$	1.29978	0.122817
$113$	−0.407582	−0.0383421	−0.0191710	−	0.999816i	$-0.506103\pi$
$113$	−0.407582	−0.0383421	−0.0191710	+	0.999816i	$0.506103\pi$
$114$	0	0
$115$	−0.946259	−0.0882391
$116$	2.06303	0.191548
$117$	0	0
$118$	7.84518	0.722207
$119$	−8.22735	−0.754200
$120$	0	0
$121$	2.81514	0.255922
$122$	−3.84958	−0.348524
$123$	0	0
$124$	−5.82316	−0.522935
$125$	6.86928	0.614407
$126$	0	0
$127$	−9.52589	−0.845286	−0.422643	−	0.906296i	$-0.638898\pi$
$127$	−9.52589	−0.845286	−0.422643	+	0.906296i	$0.638898\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	1.67381	0.146803
$131$	21.9325	1.91625	0.958127	−	0.286342i	$-0.0924394\pi$
$131$	21.9325	1.91625	0.958127	+	0.286342i	$0.0924394\pi$
$132$	0	0
$133$	0.0276253	0.00239542
$134$	9.31541	0.804729
$135$	0	0
$136$	6.32982	0.542777
$137$	2.48849	0.212606	0.106303	−	0.994334i	$-0.466099\pi$
$137$	2.48849	0.212606	0.106303	+	0.994334i	$0.466099\pi$
$138$	0	0
$139$	21.6244	1.83416	0.917078	−	0.398708i	$-0.130541\pi$
$139$	21.6244	1.83416	0.917078	+	0.398708i	$0.130541\pi$
$140$	−0.942394	−0.0796468
$141$	0	0
$142$	−0.643097	−0.0539675
$143$	8.58064	0.717549
$144$	0	0
$145$	−1.49579	−0.124218
$146$	2.97920	0.246561
$147$	0	0
$148$	−7.25272	−0.596170
$149$	−20.9106	−1.71306	−0.856532	−	0.516093i	$-0.827386\pi$
$149$	−20.9106	−1.71306	−0.856532	+	0.516093i	$0.827386\pi$
$150$	0	0
$151$	−2.07209	−0.168624	−0.0843120	−	0.996439i	$-0.526869\pi$
$151$	−2.07209	−0.168624	−0.0843120	+	0.996439i	$0.526869\pi$
$152$	−0.0212539	−0.00172392
$153$	0	0
$154$	−4.83110	−0.389301
$155$	4.22204	0.339123
$156$	0	0
$157$	−2.60975	−0.208281	−0.104140	−	0.994563i	$-0.533209\pi$
$157$	−2.60975	−0.208281	−0.104140	+	0.994563i	$0.533209\pi$
$158$	−0.418245	−0.0332738
$159$	0	0
$160$	0.725043	0.0573197
$161$	1.69635	0.133691
$162$	0	0
$163$	−6.63147	−0.519417	−0.259708	−	0.965687i	$-0.583626\pi$
$163$	−6.63147	−0.519417	−0.259708	+	0.965687i	$0.583626\pi$
$164$	−8.77454	−0.685176
$165$	0	0
$166$	−15.6322	−1.21329
$167$	−22.6369	−1.75170	−0.875850	−	0.482584i	$-0.839698\pi$
$167$	−22.6369	−1.75170	−0.875850	+	0.482584i	$0.839698\pi$
$168$	0	0
$169$	−7.67053	−0.590041
$170$	−4.58939	−0.351990
$171$	0	0
$172$	4.52327	0.344896
$173$	−12.2456	−0.931018	−0.465509	−	0.885043i	$-0.654129\pi$
$173$	−12.2456	−0.931018	−0.465509	+	0.885043i	$0.654129\pi$
$174$	0	0
$175$	−5.81561	−0.439619
$176$	3.71687	0.280170
$177$	0	0
$178$	10.3879	0.778603
$179$	18.0181	1.34673	0.673367	−	0.739308i	$-0.264847\pi$
$179$	18.0181	1.34673	0.673367	+	0.739308i	$0.264847\pi$
$180$	0	0
$181$	7.77690	0.578052	0.289026	−	0.957321i	$-0.406669\pi$
$181$	7.77690	0.578052	0.289026	+	0.957321i	$0.406669\pi$
$182$	−3.00062	−0.222421
$183$	0	0
$184$	−1.30511	−0.0962138
$185$	5.25854	0.386615
$186$	0	0
$187$	−23.5271	−1.72047
$188$	−3.85502	−0.281156
$189$	0	0
$190$	0.0154100	0.00111796
$191$	−0.312243	−0.0225931	−0.0112965	−	0.999936i	$-0.503596\pi$
$191$	−0.312243	−0.0225931	−0.0112965	+	0.999936i	$0.503596\pi$
$192$	0	0
$193$	−0.336949	−0.0242541	−0.0121271	−	0.999926i	$-0.503860\pi$
$193$	−0.336949	−0.0242541	−0.0121271	+	0.999926i	$0.503860\pi$
$194$	−5.07659	−0.364478
$195$	0	0
$196$	−5.31058	−0.379327
$197$	−26.8275	−1.91138	−0.955691	−	0.294370i	$-0.904890\pi$
$197$	−26.8275	−1.91138	−0.955691	+	0.294370i	$0.904890\pi$
$198$	0	0
$199$	4.03560	0.286076	0.143038	−	0.989717i	$-0.454313\pi$
$199$	4.03560	0.286076	0.143038	+	0.989717i	$0.454313\pi$
$200$	4.47431	0.316382
$201$	0	0
$202$	12.5321	0.881755
$203$	2.68148	0.188203
$204$	0	0
$205$	6.36192	0.444335
$206$	−5.43374	−0.378587
$207$	0	0
$208$	2.30856	0.160070
$209$	0.0789980	0.00546440
$210$	0	0
$211$	9.22688	0.635205	0.317602	−	0.948224i	$-0.397122\pi$
$211$	9.22688	0.635205	0.317602	+	0.948224i	$0.397122\pi$
$212$	6.86923	0.471781
$213$	0	0
$214$	−3.45500	−0.236179
$215$	−3.27956	−0.223664
$216$	0	0
$217$	−7.56881	−0.513804
$218$	6.80801	0.461096
$219$	0	0
$220$	−2.69489	−0.181690
$221$	−14.6128	−0.982963
$222$	0	0
$223$	−9.44842	−0.632713	−0.316357	−	0.948640i	$-0.602460\pi$
$223$	−9.44842	−0.632713	−0.316357	+	0.948640i	$0.602460\pi$
$224$	−1.29978	−0.0868450
$225$	0	0
$226$	0.407582	0.0271119
$227$	−15.2061	−1.00926	−0.504632	−	0.863335i	$-0.668372\pi$
$227$	−15.2061	−1.00926	−0.504632	+	0.863335i	$0.668372\pi$
$228$	0	0
$229$	28.6411	1.89266	0.946329	−	0.323205i	$-0.104760\pi$
$229$	28.6411	1.89266	0.946329	+	0.323205i	$0.104760\pi$
$230$	0.946259	0.0623945
$231$	0	0
$232$	−2.06303	−0.135445
$233$	−3.14875	−0.206281	−0.103141	−	0.994667i	$-0.532889\pi$
$233$	−3.14875	−0.206281	−0.103141	+	0.994667i	$0.532889\pi$
$234$	0	0
$235$	2.79506	0.182329
$236$	−7.84518	−0.510678
$237$	0	0
$238$	8.22735	0.533300
$239$	−12.5059	−0.808940	−0.404470	−	0.914551i	$-0.632544\pi$
$239$	−12.5059	−0.808940	−0.404470	+	0.914551i	$0.632544\pi$
$240$	0	0
$241$	3.73027	0.240287	0.120144	−	0.992757i	$-0.461664\pi$
$241$	3.73027	0.240287	0.120144	+	0.992757i	$0.461664\pi$
$242$	−2.81514	−0.180964
$243$	0	0
$244$	3.84958	0.246444
$245$	3.85040	0.245993
$246$	0	0
$247$	0.0490660	0.00312199
$248$	5.82316	0.369771
$249$	0	0
$250$	−6.86928	−0.434452
$251$	26.2795	1.65875	0.829374	−	0.558694i	$-0.188697\pi$
$251$	26.2795	1.65875	0.829374	+	0.558694i	$0.188697\pi$
$252$	0	0
$253$	4.85092	0.304975
$254$	9.52589	0.597708
$255$	0	0
$256$	1.00000	0.0625000
$257$	−28.2237	−1.76055	−0.880273	−	0.474468i	$-0.842641\pi$
$257$	−28.2237	−1.76055	−0.880273	+	0.474468i	$0.842641\pi$
$258$	0	0
$259$	−9.42692	−0.585760
$260$	−1.67381	−0.103805
$261$	0	0
$262$	−21.9325	−1.35500
$263$	−25.3865	−1.56540	−0.782701	−	0.622398i	$-0.786158\pi$
$263$	−25.3865	−1.56540	−0.782701	+	0.622398i	$0.786158\pi$
$264$	0	0
$265$	−4.98049	−0.305949
$266$	−0.0276253	−0.00169382
$267$	0	0
$268$	−9.31541	−0.569029
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	0	0
$271$	−17.9078	−1.08782	−0.543911	−	0.839143i	$-0.683057\pi$
$271$	−17.9078	−1.08782	−0.543911	+	0.839143i	$0.683057\pi$
$272$	−6.32982	−0.383802
$273$	0	0
$274$	−2.48849	−0.150335
$275$	−16.6304	−1.00285
$276$	0	0
$277$	−6.51988	−0.391741	−0.195871	−	0.980630i	$-0.562753\pi$
$277$	−6.51988	−0.391741	−0.195871	+	0.980630i	$0.562753\pi$
$278$	−21.6244	−1.29694
$279$	0	0
$280$	0.942394	0.0563188
$281$	2.61394	0.155934	0.0779672	−	0.996956i	$-0.475157\pi$
$281$	2.61394	0.155934	0.0779672	+	0.996956i	$0.475157\pi$
$282$	0	0
$283$	−9.65338	−0.573834	−0.286917	−	0.957955i	$-0.592630\pi$
$283$	−9.65338	−0.573834	−0.286917	+	0.957955i	$0.592630\pi$
$284$	0.643097	0.0381608
$285$	0	0
$286$	−8.58064	−0.507384
$287$	−11.4049	−0.673212
$288$	0	0
$289$	23.0666	1.35686
$290$	1.49579	0.0878356
$291$	0	0
$292$	−2.97920	−0.174345
$293$	−2.04724	−0.119601	−0.0598005	−	0.998210i	$-0.519046\pi$
$293$	−2.04724	−0.119601	−0.0598005	+	0.998210i	$0.519046\pi$
$294$	0	0
$295$	5.68809	0.331173
$296$	7.25272	0.421556
$297$	0	0
$298$	20.9106	1.21132
$299$	3.01293	0.174242
$300$	0	0
$301$	5.87924	0.338874
$302$	2.07209	0.119235
$303$	0	0
$304$	0.0212539	0.00121899
$305$	−2.79111	−0.159818
$306$	0	0
$307$	−18.6159	−1.06246	−0.531231	−	0.847227i	$-0.678271\pi$
$307$	−18.6159	−1.06246	−0.531231	+	0.847227i	$0.678271\pi$
$308$	4.83110	0.275278
$309$	0	0
$310$	−4.22204	−0.239796
$311$	9.88503	0.560529	0.280264	−	0.959923i	$-0.409578\pi$
$311$	9.88503	0.560529	0.280264	+	0.959923i	$0.409578\pi$
$312$	0	0
$313$	17.4438	0.985983	0.492991	−	0.870034i	$-0.335903\pi$
$313$	17.4438	0.985983	0.492991	+	0.870034i	$0.335903\pi$
$314$	2.60975	0.147277
$315$	0	0
$316$	0.418245	0.0235281
$317$	−2.92828	−0.164468	−0.0822342	−	0.996613i	$-0.526206\pi$
$317$	−2.92828	−0.164468	−0.0822342	+	0.996613i	$0.526206\pi$
$318$	0	0
$319$	7.66802	0.429327
$320$	−0.725043	−0.0405311
$321$	0	0
$322$	−1.69635	−0.0945338
$323$	−0.134533	−0.00748563
$324$	0	0
$325$	−10.3292	−0.572963
$326$	6.63147	0.367283
$327$	0	0
$328$	8.77454	0.484493
$329$	−5.01067	−0.276247
$330$	0	0
$331$	5.44140	0.299087	0.149543	−	0.988755i	$-0.452220\pi$
$331$	5.44140	0.299087	0.149543	+	0.988755i	$0.452220\pi$
$332$	15.6322	0.857927
$333$	0	0
$334$	22.6369	1.23864
$335$	6.75407	0.369014
$336$	0	0
$337$	−3.73840	−0.203644	−0.101822	−	0.994803i	$-0.532467\pi$
$337$	−3.73840	−0.203644	−0.101822	+	0.994803i	$0.532467\pi$
$338$	7.67053	0.417222
$339$	0	0
$340$	4.58939	0.248895
$341$	−21.6439	−1.17209
$342$	0	0
$343$	−16.0010	−0.863973
$344$	−4.52327	−0.243878
$345$	0	0
$346$	12.2456	0.658329
$347$	−10.8089	−0.580252	−0.290126	−	0.956988i	$-0.593697\pi$
$347$	−10.8089	−0.580252	−0.290126	+	0.956988i	$0.593697\pi$
$348$	0	0
$349$	19.8978	1.06510	0.532552	−	0.846397i	$-0.321233\pi$
$349$	19.8978	1.06510	0.532552	+	0.846397i	$0.321233\pi$
$350$	5.81561	0.310857
$351$	0	0
$352$	−3.71687	−0.198110
$353$	11.8505	0.630736	0.315368	−	0.948969i	$-0.397872\pi$
$353$	11.8505	0.630736	0.315368	+	0.948969i	$0.397872\pi$
$354$	0	0
$355$	−0.466273	−0.0247472
$356$	−10.3879	−0.550556
$357$	0	0
$358$	−18.0181	−0.952285
$359$	−28.6632	−1.51279	−0.756393	−	0.654118i	$-0.773040\pi$
$359$	−28.6632	−1.51279	−0.756393	+	0.654118i	$0.773040\pi$
$360$	0	0
$361$	−18.9995	−0.999976
$362$	−7.77690	−0.408745
$363$	0	0
$364$	3.00062	0.157275
$365$	2.16005	0.113062
$366$	0	0
$367$	13.1083	0.684246	0.342123	−	0.939655i	$-0.388854\pi$
$367$	13.1083	0.684246	0.342123	+	0.939655i	$0.388854\pi$
$368$	1.30511	0.0680335
$369$	0	0
$370$	−5.25854	−0.273378
$371$	8.92847	0.463543
$372$	0	0
$373$	−17.2738	−0.894406	−0.447203	−	0.894432i	$-0.647580\pi$
$373$	−17.2738	−0.894406	−0.447203	+	0.894432i	$0.647580\pi$
$374$	23.5271	1.21656
$375$	0	0
$376$	3.85502	0.198808
$377$	4.76264	0.245288
$378$	0	0
$379$	6.30803	0.324022	0.162011	−	0.986789i	$-0.448202\pi$
$379$	6.30803	0.324022	0.162011	+	0.986789i	$0.448202\pi$
$380$	−0.0154100	−0.000790515 0
$381$	0	0
$382$	0.312243	0.0159757
$383$	28.2196	1.44196	0.720978	−	0.692958i	$-0.243693\pi$
$383$	28.2196	1.44196	0.720978	+	0.692958i	$0.243693\pi$
$384$	0	0
$385$	−3.50276	−0.178517
$386$	0.336949	0.0171502
$387$	0	0
$388$	5.07659	0.257725
$389$	3.15029	0.159726	0.0798629	−	0.996806i	$-0.474552\pi$
$389$	3.15029	0.159726	0.0798629	+	0.996806i	$0.474552\pi$
$390$	0	0
$391$	−8.26110	−0.417782
$392$	5.31058	0.268225
$393$	0	0
$394$	26.8275	1.35155
$395$	−0.303246	−0.0152579
$396$	0	0
$397$	31.2926	1.57053	0.785265	−	0.619159i	$-0.212527\pi$
$397$	31.2926	1.57053	0.785265	+	0.619159i	$0.212527\pi$
$398$	−4.03560	−0.202286
$399$	0	0
$400$	−4.47431	−0.223716
$401$	−0.744072	−0.0371572	−0.0185786	−	0.999827i	$-0.505914\pi$
$401$	−0.744072	−0.0371572	−0.0185786	+	0.999827i	$0.505914\pi$
$402$	0	0
$403$	−13.4431	−0.669651
$404$	−12.5321	−0.623495
$405$	0	0
$406$	−2.68148	−0.133080
$407$	−26.9574	−1.33623
$408$	0	0
$409$	3.60242	0.178128	0.0890641	−	0.996026i	$-0.471612\pi$
$409$	3.60242	0.178128	0.0890641	+	0.996026i	$0.471612\pi$
$410$	−6.36192	−0.314193
$411$	0	0
$412$	5.43374	0.267701
$413$	−10.1970	−0.501760
$414$	0	0
$415$	−11.3340	−0.556364
$416$	−2.30856	−0.113187
$417$	0	0
$418$	−0.0789980	−0.00386392
$419$	17.1323	0.836966	0.418483	−	0.908225i	$-0.362562\pi$
$419$	17.1323	0.836966	0.418483	+	0.908225i	$0.362562\pi$
$420$	0	0
$421$	24.1651	1.17773	0.588866	−	0.808230i	$-0.299574\pi$
$421$	24.1651	1.17773	0.588866	+	0.808230i	$0.299574\pi$
$422$	−9.22688	−0.449158
$423$	0	0
$424$	−6.86923	−0.333600
$425$	28.3216	1.37380
$426$	0	0
$427$	5.00359	0.242141
$428$	3.45500	0.167004
$429$	0	0
$430$	3.27956	0.158155
$431$	−16.8823	−0.813193	−0.406596	−	0.913608i	$-0.633284\pi$
$431$	−16.8823	−0.813193	−0.406596	+	0.913608i	$0.633284\pi$
$432$	0	0
$433$	−13.0644	−0.627834	−0.313917	−	0.949450i	$-0.601641\pi$
$433$	−13.0644	−0.627834	−0.313917	+	0.949450i	$0.601641\pi$
$434$	7.56881	0.363314
$435$	0	0
$436$	−6.80801	−0.326044
$437$	0.0277386	0.00132692
$438$	0	0
$439$	3.91599	0.186900	0.0934499	−	0.995624i	$-0.470211\pi$
$439$	3.91599	0.186900	0.0934499	+	0.995624i	$0.470211\pi$
$440$	2.69489	0.128474
$441$	0	0
$442$	14.6128	0.695060
$443$	−25.2673	−1.20049	−0.600243	−	0.799818i	$-0.704929\pi$
$443$	−25.2673	−1.20049	−0.600243	+	0.799818i	$0.704929\pi$
$444$	0	0
$445$	7.53165	0.357034
$446$	9.44842	0.447396
$447$	0	0
$448$	1.29978	0.0614087
$449$	−27.4706	−1.29642	−0.648210	−	0.761462i	$-0.724482\pi$
$449$	−27.4706	−1.29642	−0.648210	+	0.761462i	$0.724482\pi$
$450$	0	0
$451$	−32.6138	−1.53573
$452$	−0.407582	−0.0191710
$453$	0	0
$454$	15.2061	0.713657
$455$	−2.17558	−0.101993
$456$	0	0
$457$	−14.5345	−0.679896	−0.339948	−	0.940444i	$-0.610409\pi$
$457$	−14.5345	−0.679896	−0.339948	+	0.940444i	$0.610409\pi$
$458$	−28.6411	−1.33831
$459$	0	0
$460$	−0.946259	−0.0441196
$461$	10.4630	0.487312	0.243656	−	0.969862i	$-0.421653\pi$
$461$	10.4630	0.487312	0.243656	+	0.969862i	$0.421653\pi$
$462$	0	0
$463$	−27.7519	−1.28974	−0.644871	−	0.764292i	$-0.723089\pi$
$463$	−27.7519	−1.28974	−0.644871	+	0.764292i	$0.723089\pi$
$464$	2.06303	0.0957738
$465$	0	0
$466$	3.14875	0.145863
$467$	12.3322	0.570665	0.285333	−	0.958429i	$-0.407896\pi$
$467$	12.3322	0.570665	0.285333	+	0.958429i	$0.407896\pi$
$468$	0	0
$469$	−12.1080	−0.559093
$470$	−2.79506	−0.128926
$471$	0	0
$472$	7.84518	0.361104
$473$	16.8124	0.773036
$474$	0	0
$475$	−0.0950965	−0.00436333
$476$	−8.22735	−0.377100
$477$	0	0
$478$	12.5059	0.572007
$479$	2.92366	0.133586	0.0667928	−	0.997767i	$-0.478723\pi$
$479$	2.92366	0.133586	0.0667928	+	0.997767i	$0.478723\pi$
$480$	0	0
$481$	−16.7434	−0.763432
$482$	−3.73027	−0.169909
$483$	0	0
$484$	2.81514	0.127961
$485$	−3.68075	−0.167134
$486$	0	0
$487$	10.1739	0.461025	0.230513	−	0.973069i	$-0.425960\pi$
$487$	10.1739	0.461025	0.230513	+	0.973069i	$0.425960\pi$
$488$	−3.84958	−0.174262
$489$	0	0
$490$	−3.85040	−0.173943
$491$	16.8181	0.758989	0.379494	−	0.925194i	$-0.376098\pi$
$491$	16.8181	0.758989	0.379494	+	0.925194i	$0.376098\pi$
$492$	0	0
$493$	−13.0586	−0.588130
$494$	−0.0490660	−0.00220758
$495$	0	0
$496$	−5.82316	−0.261468
$497$	0.835882	0.0374944
$498$	0	0
$499$	−30.0514	−1.34529	−0.672643	−	0.739967i	$-0.734841\pi$
$499$	−30.0514	−1.34529	−0.672643	+	0.739967i	$0.734841\pi$
$500$	6.86928	0.307204
$501$	0	0
$502$	−26.2795	−1.17291
$503$	6.83648	0.304823	0.152412	−	0.988317i	$-0.451296\pi$
$503$	6.83648	0.304823	0.152412	+	0.988317i	$0.451296\pi$
$504$	0	0
$505$	9.08630	0.404335
$506$	−4.85092	−0.215650
$507$	0	0
$508$	−9.52589	−0.422643
$509$	16.9547	0.751502	0.375751	−	0.926721i	$-0.377385\pi$
$509$	16.9547	0.751502	0.375751	+	0.926721i	$0.377385\pi$
$510$	0	0
$511$	−3.87230	−0.171300
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	28.2237	1.24489
$515$	−3.93970	−0.173604
$516$	0	0
$517$	−14.3286	−0.630172
$518$	9.42692	0.414195
$519$	0	0
$520$	1.67381	0.0734013
$521$	−37.1136	−1.62597	−0.812987	−	0.582282i	$-0.802160\pi$
$521$	−37.1136	−1.62597	−0.812987	+	0.582282i	$0.802160\pi$
$522$	0	0
$523$	−9.80481	−0.428734	−0.214367	−	0.976753i	$-0.568769\pi$
$523$	−9.80481	−0.428734	−0.214367	+	0.976753i	$0.568769\pi$
$524$	21.9325	0.958127
$525$	0	0
$526$	25.3865	1.10691
$527$	36.8596	1.60563
$528$	0	0
$529$	−21.2967	−0.925943
$530$	4.98049	0.216339
$531$	0	0
$532$	0.0276253	0.00119771
$533$	−20.2566	−0.877410
$534$	0	0
$535$	−2.50502	−0.108302
$536$	9.31541	0.402365
$537$	0	0
$538$	−1.00000	−0.0431131
$539$	−19.7388	−0.850208
$540$	0	0
$541$	−15.4647	−0.664878	−0.332439	−	0.943125i	$-0.607871\pi$
$541$	−15.4647	−0.664878	−0.332439	+	0.943125i	$0.607871\pi$
$542$	17.9078	0.769206
$543$	0	0
$544$	6.32982	0.271389
$545$	4.93610	0.211439
$546$	0	0
$547$	−31.5080	−1.34719	−0.673593	−	0.739103i	$-0.735250\pi$
$547$	−31.5080	−1.34719	−0.673593	+	0.739103i	$0.735250\pi$
$548$	2.48849	0.106303
$549$	0	0
$550$	16.6304	0.709125
$551$	0.0438474	0.00186796
$552$	0	0
$553$	0.543625	0.0231173
$554$	6.51988	0.277003
$555$	0	0
$556$	21.6244	0.917078
$557$	5.45649	0.231199	0.115599	−	0.993296i	$-0.463121\pi$
$557$	5.45649	0.231199	0.115599	+	0.993296i	$0.463121\pi$
$558$	0	0
$559$	10.4423	0.441660
$560$	−0.942394	−0.0398234
$561$	0	0
$562$	−2.61394	−0.110262
$563$	5.91106	0.249121	0.124561	−	0.992212i	$-0.460248\pi$
$563$	5.91106	0.249121	0.124561	+	0.992212i	$0.460248\pi$
$564$	0	0
$565$	0.295514	0.0124324
$566$	9.65338	0.405762
$567$	0	0
$568$	−0.643097	−0.0269837
$569$	−31.9474	−1.33930	−0.669652	−	0.742675i	$-0.733557\pi$
$569$	−31.9474	−1.33930	−0.669652	+	0.742675i	$0.733557\pi$
$570$	0	0
$571$	−13.7554	−0.575647	−0.287824	−	0.957683i	$-0.592932\pi$
$571$	−13.7554	−0.575647	−0.287824	+	0.957683i	$0.592932\pi$
$572$	8.58064	0.358774
$573$	0	0
$574$	11.4049	0.476033
$575$	−5.83946	−0.243522
$576$	0	0
$577$	−2.93964	−0.122379	−0.0611893	−	0.998126i	$-0.519489\pi$
$577$	−2.93964	−0.122379	−0.0611893	+	0.998126i	$0.519489\pi$
$578$	−23.0666	−0.959444
$579$	0	0
$580$	−1.49579	−0.0621091
$581$	20.3183	0.842947
$582$	0	0
$583$	25.5321	1.05743
$584$	2.97920	0.123280
$585$	0	0
$586$	2.04724	0.0845707
$587$	−15.2876	−0.630989	−0.315494	−	0.948927i	$-0.602170\pi$
$587$	−15.2876	−0.630989	−0.315494	+	0.948927i	$0.602170\pi$
$588$	0	0
$589$	−0.123765	−0.00509964
$590$	−5.68809	−0.234175
$591$	0	0
$592$	−7.25272	−0.298085
$593$	33.5005	1.37570	0.687850	−	0.725853i	$-0.258555\pi$
$593$	33.5005	1.37570	0.687850	+	0.725853i	$0.258555\pi$
$594$	0	0
$595$	5.96518	0.244549
$596$	−20.9106	−0.856532
$597$	0	0
$598$	−3.01293	−0.123208
$599$	−0.150988	−0.00616920	−0.00308460	−	0.999995i	$-0.500982\pi$
$599$	−0.150988	−0.00616920	−0.00308460	+	0.999995i	$0.500982\pi$
$600$	0	0
$601$	−34.9546	−1.42583	−0.712915	−	0.701251i	$-0.752625\pi$
$601$	−34.9546	−1.42583	−0.712915	+	0.701251i	$0.752625\pi$
$602$	−5.87924	−0.239620
$603$	0	0
$604$	−2.07209	−0.0843120
$605$	−2.04110	−0.0829824
$606$	0	0
$607$	10.5301	0.427404	0.213702	−	0.976899i	$-0.431448\pi$
$607$	10.5301	0.427404	0.213702	+	0.976899i	$0.431448\pi$
$608$	−0.0212539	−0.000861959 0
$609$	0	0
$610$	2.79111	0.113009
$611$	−8.89957	−0.360038
$612$	0	0
$613$	−20.2271	−0.816965	−0.408483	−	0.912766i	$-0.633942\pi$
$613$	−20.2271	−0.816965	−0.408483	+	0.912766i	$0.633942\pi$
$614$	18.6159	0.751275
$615$	0	0
$616$	−4.83110	−0.194651
$617$	−7.82969	−0.315211	−0.157606	−	0.987502i	$-0.550378\pi$
$617$	−7.82969	−0.315211	−0.157606	+	0.987502i	$0.550378\pi$
$618$	0	0
$619$	−23.6992	−0.952549	−0.476275	−	0.879297i	$-0.658013\pi$
$619$	−23.6992	−0.952549	−0.476275	+	0.879297i	$0.658013\pi$
$620$	4.22204	0.169561
$621$	0	0
$622$	−9.88503	−0.396354
$623$	−13.5019	−0.540942
$624$	0	0
$625$	17.3910	0.695642
$626$	−17.4438	−0.697195
$627$	0	0
$628$	−2.60975	−0.104140
$629$	45.9084	1.83049
$630$	0	0
$631$	−31.8687	−1.26867	−0.634336	−	0.773058i	$-0.718726\pi$
$631$	−31.8687	−1.26867	−0.634336	+	0.773058i	$0.718726\pi$
$632$	−0.418245	−0.0166369
$633$	0	0
$634$	2.92828	0.116297
$635$	6.90668	0.274083
$636$	0	0
$637$	−12.2598	−0.485752
$638$	−7.66802	−0.303580
$639$	0	0
$640$	0.725043	0.0286598
$641$	−10.3578	−0.409108	−0.204554	−	0.978855i	$-0.565574\pi$
$641$	−10.3578	−0.409108	−0.204554	+	0.978855i	$0.565574\pi$
$642$	0	0
$643$	−21.9761	−0.866653	−0.433326	−	0.901237i	$-0.642660\pi$
$643$	−21.9761	−0.866653	−0.433326	+	0.901237i	$0.642660\pi$
$644$	1.69635	0.0668455
$645$	0	0
$646$	0.134533	0.00529314
$647$	−9.10376	−0.357906	−0.178953	−	0.983858i	$-0.557271\pi$
$647$	−9.10376	−0.357906	−0.178953	+	0.983858i	$0.557271\pi$
$648$	0	0
$649$	−29.1595	−1.14461
$650$	10.3292	0.405146
$651$	0	0
$652$	−6.63147	−0.259708
$653$	−6.84892	−0.268019	−0.134010	−	0.990980i	$-0.542785\pi$
$653$	−6.84892	−0.268019	−0.134010	+	0.990980i	$0.542785\pi$
$654$	0	0
$655$	−15.9020	−0.621344
$656$	−8.77454	−0.342588
$657$	0	0
$658$	5.01067	0.195336
$659$	10.9102	0.424999	0.212500	−	0.977161i	$-0.431840\pi$
$659$	10.9102	0.424999	0.212500	+	0.977161i	$0.431840\pi$
$660$	0	0
$661$	−9.26687	−0.360439	−0.180220	−	0.983626i	$-0.557681\pi$
$661$	−9.26687	−0.360439	−0.180220	+	0.983626i	$0.557681\pi$
$662$	−5.44140	−0.211486
$663$	0	0
$664$	−15.6322	−0.606646
$665$	−0.0200295	−0.000776712 0
$666$	0	0
$667$	2.69248	0.104253
$668$	−22.6369	−0.875850
$669$	0	0
$670$	−6.75407	−0.260933
$671$	14.3084	0.552369
$672$	0	0
$673$	−15.0272	−0.579255	−0.289627	−	0.957139i	$-0.593531\pi$
$673$	−15.0272	−0.579255	−0.289627	+	0.957139i	$0.593531\pi$
$674$	3.73840	0.143998
$675$	0	0
$676$	−7.67053	−0.295020
$677$	46.3947	1.78309	0.891547	−	0.452929i	$-0.149621\pi$
$677$	46.3947	1.78309	0.891547	+	0.452929i	$0.149621\pi$
$678$	0	0
$679$	6.59843	0.253225
$680$	−4.58939	−0.175995
$681$	0	0
$682$	21.6439	0.828790
$683$	34.2506	1.31056	0.655281	−	0.755385i	$-0.272550\pi$
$683$	34.2506	1.31056	0.655281	+	0.755385i	$0.272550\pi$
$684$	0	0
$685$	−1.80426	−0.0689373
$686$	16.0010	0.610921
$687$	0	0
$688$	4.52327	0.172448
$689$	15.8581	0.604144
$690$	0	0
$691$	−1.19208	−0.0453489	−0.0226744	−	0.999743i	$-0.507218\pi$
$691$	−1.19208	−0.0453489	−0.0226744	+	0.999743i	$0.507218\pi$
$692$	−12.2456	−0.465509
$693$	0	0
$694$	10.8089	0.410300
$695$	−15.6786	−0.594723
$696$	0	0
$697$	55.5412	2.10377
$698$	−19.8978	−0.753142
$699$	0	0
$700$	−5.81561	−0.219809
$701$	24.6698	0.931767	0.465883	−	0.884846i	$-0.345737\pi$
$701$	24.6698	0.931767	0.465883	+	0.884846i	$0.345737\pi$
$702$	0	0
$703$	−0.154149	−0.00581382
$704$	3.71687	0.140085
$705$	0	0
$706$	−11.8505	−0.445998
$707$	−16.2889	−0.612608
$708$	0	0
$709$	23.3891	0.878398	0.439199	−	0.898390i	$-0.355262\pi$
$709$	23.3891	0.878398	0.439199	+	0.898390i	$0.355262\pi$
$710$	0.466273	0.0174989
$711$	0	0
$712$	10.3879	0.389302
$713$	−7.59986	−0.284617
$714$	0	0
$715$	−6.22133	−0.232665
$716$	18.0181	0.673367
$717$	0	0
$718$	28.6632	1.06970
$719$	1.52136	0.0567371	0.0283686	−	0.999598i	$-0.490969\pi$
$719$	1.52136	0.0567371	0.0283686	+	0.999598i	$0.490969\pi$
$720$	0	0
$721$	7.06265	0.263027
$722$	18.9995	0.707090
$723$	0	0
$724$	7.77690	0.289026
$725$	−9.23065	−0.342818
$726$	0	0
$727$	36.3855	1.34946	0.674732	−	0.738063i	$-0.264259\pi$
$727$	36.3855	1.34946	0.674732	+	0.738063i	$0.264259\pi$
$728$	−3.00062	−0.111210
$729$	0	0
$730$	−2.16005	−0.0799470
$731$	−28.6315	−1.05897
$732$	0	0
$733$	−18.7869	−0.693908	−0.346954	−	0.937882i	$-0.612784\pi$
$733$	−18.7869	−0.693908	−0.346954	+	0.937882i	$0.612784\pi$
$734$	−13.1083	−0.483835
$735$	0	0
$736$	−1.30511	−0.0481069
$737$	−34.6242	−1.27540
$738$	0	0
$739$	−38.7776	−1.42646	−0.713228	−	0.700932i	$-0.752767\pi$
$739$	−38.7776	−1.42646	−0.713228	+	0.700932i	$0.752767\pi$
$740$	5.25854	0.193308
$741$	0	0
$742$	−8.92847	−0.327774
$743$	6.80190	0.249537	0.124769	−	0.992186i	$-0.460181\pi$
$743$	6.80190	0.249537	0.124769	+	0.992186i	$0.460181\pi$
$744$	0	0
$745$	15.1611	0.555460
$746$	17.2738	0.632441
$747$	0	0
$748$	−23.5271	−0.860237
$749$	4.49073	0.164088
$750$	0	0
$751$	10.5672	0.385601	0.192801	−	0.981238i	$-0.438243\pi$
$751$	10.5672	0.385601	0.192801	+	0.981238i	$0.438243\pi$
$752$	−3.85502	−0.140578
$753$	0	0
$754$	−4.76264	−0.173445
$755$	1.50235	0.0546762
$756$	0	0
$757$	−22.6389	−0.822825	−0.411412	−	0.911449i	$-0.634964\pi$
$757$	−22.6389	−0.822825	−0.411412	+	0.911449i	$0.634964\pi$
$758$	−6.30803	−0.229118
$759$	0	0
$760$	0.0154100	0.000558979 0
$761$	−1.83584	−0.0665490	−0.0332745	−	0.999446i	$-0.510594\pi$
$761$	−1.83584	−0.0665490	−0.0332745	+	0.999446i	$0.510594\pi$
$762$	0	0
$763$	−8.84889	−0.320351
$764$	−0.312243	−0.0112965
$765$	0	0
$766$	−28.2196	−1.01962
$767$	−18.1111	−0.653954
$768$	0	0
$769$	50.0137	1.80354	0.901770	−	0.432215i	$-0.142268\pi$
$769$	50.0137	1.80354	0.901770	+	0.432215i	$0.142268\pi$
$770$	3.50276	0.126231
$771$	0	0
$772$	−0.336949	−0.0121271
$773$	−40.2988	−1.44945	−0.724724	−	0.689039i	$-0.758033\pi$
$773$	−40.2988	−1.44945	−0.724724	+	0.689039i	$0.758033\pi$
$774$	0	0
$775$	26.0546	0.935911
$776$	−5.07659	−0.182239
$777$	0	0
$778$	−3.15029	−0.112943
$779$	−0.186493	−0.00668180
$780$	0	0
$781$	2.39031	0.0855319
$782$	8.26110	0.295416
$783$	0	0
$784$	−5.31058	−0.189664
$785$	1.89218	0.0675349
$786$	0	0
$787$	38.2500	1.36347	0.681733	−	0.731601i	$-0.261227\pi$
$787$	38.2500	1.36347	0.681733	+	0.731601i	$0.261227\pi$
$788$	−26.8275	−0.955691
$789$	0	0
$790$	0.303246	0.0107890
$791$	−0.529765	−0.0188363
$792$	0	0
$793$	8.88700	0.315586
$794$	−31.2926	−1.11053
$795$	0	0
$796$	4.03560	0.143038
$797$	33.4247	1.18397	0.591983	−	0.805951i	$-0.298345\pi$
$797$	33.4247	1.18397	0.591983	+	0.805951i	$0.298345\pi$
$798$	0	0
$799$	24.4016	0.863266
$800$	4.47431	0.158191
$801$	0	0
$802$	0.744072	0.0262741
$803$	−11.0733	−0.390769
$804$	0	0
$805$	−1.22993	−0.0433492
$806$	13.4431	0.473514
$807$	0	0
$808$	12.5321	0.440877
$809$	39.8841	1.40225	0.701125	−	0.713038i	$-0.252681\pi$
$809$	39.8841	1.40225	0.701125	+	0.713038i	$0.252681\pi$
$810$	0	0
$811$	34.7237	1.21931	0.609657	−	0.792665i	$-0.291307\pi$
$811$	34.7237	1.21931	0.609657	+	0.792665i	$0.291307\pi$
$812$	2.68148	0.0941015
$813$	0	0
$814$	26.9574	0.944858
$815$	4.80810	0.168420
$816$	0	0
$817$	0.0961370	0.00336341
$818$	−3.60242	−0.125956
$819$	0	0
$820$	6.36192	0.222168
$821$	48.5169	1.69325	0.846626	−	0.532189i	$-0.178630\pi$
$821$	48.5169	1.69325	0.846626	+	0.532189i	$0.178630\pi$
$822$	0	0
$823$	19.5287	0.680728	0.340364	−	0.940294i	$-0.389450\pi$
$823$	19.5287	0.680728	0.340364	+	0.940294i	$0.389450\pi$
$824$	−5.43374	−0.189293
$825$	0	0
$826$	10.1970	0.354798
$827$	−18.5400	−0.644698	−0.322349	−	0.946621i	$-0.604472\pi$
$827$	−18.5400	−0.644698	−0.322349	+	0.946621i	$0.604472\pi$
$828$	0	0
$829$	−20.2344	−0.702769	−0.351385	−	0.936231i	$-0.614289\pi$
$829$	−20.2344	−0.702769	−0.351385	+	0.936231i	$0.614289\pi$
$830$	11.3340	0.393409
$831$	0	0
$832$	2.30856	0.0800351
$833$	33.6150	1.16469
$834$	0	0
$835$	16.4128	0.567987
$836$	0.0789980	0.00273220
$837$	0	0
$838$	−17.1323	−0.591824
$839$	−46.3931	−1.60167	−0.800833	−	0.598887i	$-0.795610\pi$
$839$	−46.3931	−1.60167	−0.800833	+	0.598887i	$0.795610\pi$
$840$	0	0
$841$	−24.7439	−0.853238
$842$	−24.1651	−0.832783
$843$	0	0
$844$	9.22688	0.317602
$845$	5.56146	0.191320
$846$	0	0
$847$	3.65905	0.125727
$848$	6.86923	0.235890
$849$	0	0
$850$	−28.3216	−0.971422
$851$	−9.46559	−0.324476
$852$	0	0
$853$	−0.133230	−0.00456172	−0.00228086	−	0.999997i	$-0.500726\pi$
$853$	−0.133230	−0.00456172	−0.00228086	+	0.999997i	$0.500726\pi$
$854$	−5.00359	−0.171219
$855$	0	0
$856$	−3.45500	−0.118089
$857$	10.2942	0.351642	0.175821	−	0.984422i	$-0.443742\pi$
$857$	10.2942	0.351642	0.175821	+	0.984422i	$0.443742\pi$
$858$	0	0
$859$	35.5814	1.21402	0.607011	−	0.794694i	$-0.292368\pi$
$859$	35.5814	1.21402	0.607011	+	0.794694i	$0.292368\pi$
$860$	−3.27956	−0.111832
$861$	0	0
$862$	16.8823	0.575014
$863$	−17.9896	−0.612372	−0.306186	−	0.951972i	$-0.599053\pi$
$863$	−17.9896	−0.612372	−0.306186	+	0.951972i	$0.599053\pi$
$864$	0	0
$865$	8.87861	0.301882
$866$	13.0644	0.443946
$867$	0	0
$868$	−7.56881	−0.256902
$869$	1.55456	0.0527350
$870$	0	0
$871$	−21.5052	−0.728677
$872$	6.80801	0.230548
$873$	0	0
$874$	−0.0277386	−0.000938273 0
$875$	8.92853	0.301839
$876$	0	0
$877$	47.6131	1.60778	0.803890	−	0.594778i	$-0.202760\pi$
$877$	47.6131	1.60778	0.803890	+	0.594778i	$0.202760\pi$
$878$	−3.91599	−0.132158
$879$	0	0
$880$	−2.69489	−0.0908448
$881$	15.7287	0.529914	0.264957	−	0.964260i	$-0.414642\pi$
$881$	15.7287	0.529914	0.264957	+	0.964260i	$0.414642\pi$
$882$	0	0
$883$	−20.9111	−0.703713	−0.351856	−	0.936054i	$-0.614449\pi$
$883$	−20.9111	−0.703713	−0.351856	+	0.936054i	$0.614449\pi$
$884$	−14.6128	−0.491481
$885$	0	0
$886$	25.2673	0.848871
$887$	−3.47543	−0.116694	−0.0583468	−	0.998296i	$-0.518583\pi$
$887$	−3.47543	−0.116694	−0.0583468	+	0.998296i	$0.518583\pi$
$888$	0	0
$889$	−12.3815	−0.415263
$890$	−7.53165	−0.252461
$891$	0	0
$892$	−9.44842	−0.316357
$893$	−0.0819342	−0.00274182
$894$	0	0
$895$	−13.0639	−0.436677
$896$	−1.29978	−0.0434225
$897$	0	0
$898$	27.4706	0.916707
$899$	−12.0134	−0.400668
$900$	0	0
$901$	−43.4810	−1.44856
$902$	32.6138	1.08592
$903$	0	0
$904$	0.407582	0.0135560
$905$	−5.63859	−0.187433
$906$	0	0
$907$	5.99370	0.199018	0.0995088	−	0.995037i	$-0.468273\pi$
$907$	5.99370	0.199018	0.0995088	+	0.995037i	$0.468273\pi$
$908$	−15.2061	−0.504632
$909$	0	0
$910$	2.17558	0.0721196
$911$	−3.66404	−0.121395	−0.0606976	−	0.998156i	$-0.519333\pi$
$911$	−3.66404	−0.121395	−0.0606976	+	0.998156i	$0.519333\pi$
$912$	0	0
$913$	58.1028	1.92292
$914$	14.5345	0.480759
$915$	0	0
$916$	28.6411	0.946329
$917$	28.5074	0.941397
$918$	0	0
$919$	44.8323	1.47888	0.739441	−	0.673222i	$-0.235090\pi$
$919$	44.8323	1.47888	0.739441	+	0.673222i	$0.235090\pi$
$920$	0.946259	0.0311972
$921$	0	0
$922$	−10.4630	−0.344581
$923$	1.48463	0.0488672
$924$	0	0
$925$	32.4510	1.06698
$926$	27.7519	0.911985
$927$	0	0
$928$	−2.06303	−0.0677223
$929$	35.1823	1.15429	0.577146	−	0.816641i	$-0.304166\pi$
$929$	35.1823	1.15429	0.577146	+	0.816641i	$0.304166\pi$
$930$	0	0
$931$	−0.112870	−0.00369918
$932$	−3.14875	−0.103141
$933$	0	0
$934$	−12.3322	−0.403521
$935$	17.0582	0.557862
$936$	0	0
$937$	37.7329	1.23268	0.616339	−	0.787481i	$-0.288615\pi$
$937$	37.7329	1.23268	0.616339	+	0.787481i	$0.288615\pi$
$938$	12.1080	0.395339
$939$	0	0
$940$	2.79506	0.0911647
$941$	31.0321	1.01162	0.505809	−	0.862646i	$-0.331194\pi$
$941$	31.0321	1.01162	0.505809	+	0.862646i	$0.331194\pi$
$942$	0	0
$943$	−11.4517	−0.372919
$944$	−7.84518	−0.255339
$945$	0	0
$946$	−16.8124	−0.546619
$947$	−19.5388	−0.634924	−0.317462	−	0.948271i	$-0.602831\pi$
$947$	−19.5388	−0.634924	−0.317462	+	0.948271i	$0.602831\pi$
$948$	0	0
$949$	−6.87768	−0.223259
$950$	0.0950965	0.00308534
$951$	0	0
$952$	8.22735	0.266650
$953$	−15.9894	−0.517948	−0.258974	−	0.965884i	$-0.583384\pi$
$953$	−15.9894	−0.517948	−0.258974	+	0.965884i	$0.583384\pi$
$954$	0	0
$955$	0.226389	0.00732579
$956$	−12.5059	−0.404470
$957$	0	0
$958$	−2.92366	−0.0944592
$959$	3.23448	0.104447
$960$	0	0
$961$	2.90921	0.0938456
$962$	16.7434	0.539828
$963$	0	0
$964$	3.73027	0.120144
$965$	0.244302	0.00786437
$966$	0	0
$967$	38.1787	1.22774	0.613872	−	0.789406i	$-0.289611\pi$
$967$	38.1787	1.22774	0.613872	+	0.789406i	$0.289611\pi$
$968$	−2.81514	−0.0904820
$969$	0	0
$970$	3.68075	0.118182
$971$	−2.16411	−0.0694497	−0.0347248	−	0.999397i	$-0.511055\pi$
$971$	−2.16411	−0.0694497	−0.0347248	+	0.999397i	$0.511055\pi$
$972$	0	0
$973$	28.1069	0.901064
$974$	−10.1739	−0.325994
$975$	0	0
$976$	3.84958	0.123222
$977$	14.7297	0.471243	0.235622	−	0.971845i	$-0.424287\pi$
$977$	14.7297	0.471243	0.235622	+	0.971845i	$0.424287\pi$
$978$	0	0
$979$	−38.6104	−1.23399
$980$	3.85040	0.122996
$981$	0	0
$982$	−16.8181	−0.536686
$983$	−36.4217	−1.16167	−0.580836	−	0.814020i	$-0.697274\pi$
$983$	−36.4217	−1.16167	−0.580836	+	0.814020i	$0.697274\pi$
$984$	0	0
$985$	19.4511	0.619764
$986$	13.0586	0.415871
$987$	0	0
$988$	0.0490660	0.00156100
$989$	5.90336	0.187716
$990$	0	0
$991$	−24.5429	−0.779632	−0.389816	−	0.920893i	$-0.627461\pi$
$991$	−24.5429	−0.779632	−0.389816	+	0.920893i	$0.627461\pi$
$992$	5.82316	0.184886
$993$	0	0
$994$	−0.835882	−0.0265126
$995$	−2.92598	−0.0927598
$996$	0	0
$997$	−52.7413	−1.67033	−0.835167	−	0.549996i	$-0.814629\pi$
$997$	−52.7413	−1.67033	−0.835167	+	0.549996i	$0.814629\pi$
$998$	30.0514	0.951260
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4842.2.a.n.1.4		7
3.2	odd	2		538.2.a.e.1.5	✓	7
12.11	even	2		4304.2.a.h.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.e.1.5	✓	7	3.2	odd	2
4304.2.a.h.1.3		7	12.11	even	2
4842.2.a.n.1.4		7	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4842 = 2 \cdot 3^{2} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4842.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.725043 q^{5} +1.29978 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.725043 q^{5} +1.29978 q^{7} -1.00000 q^{8} +0.725043 q^{10} +3.71687 q^{11} +2.30856 q^{13} -1.29978 q^{14} +1.00000 q^{16} -6.32982 q^{17} +0.0212539 q^{19} -0.725043 q^{20} -3.71687 q^{22} +1.30511 q^{23} -4.47431 q^{25} -2.30856 q^{26} +1.29978 q^{28} +2.06303 q^{29} -5.82316 q^{31} -1.00000 q^{32} +6.32982 q^{34} -0.942394 q^{35} -7.25272 q^{37} -0.0212539 q^{38} +0.725043 q^{40} -8.77454 q^{41} +4.52327 q^{43} +3.71687 q^{44} -1.30511 q^{46} -3.85502 q^{47} -5.31058 q^{49} +4.47431 q^{50} +2.30856 q^{52} +6.86923 q^{53} -2.69489 q^{55} -1.29978 q^{56} -2.06303 q^{58} -7.84518 q^{59} +3.84958 q^{61} +5.82316 q^{62} +1.00000 q^{64} -1.67381 q^{65} -9.31541 q^{67} -6.32982 q^{68} +0.942394 q^{70} +0.643097 q^{71} -2.97920 q^{73} +7.25272 q^{74} +0.0212539 q^{76} +4.83110 q^{77} +0.418245 q^{79} -0.725043 q^{80} +8.77454 q^{82} +15.6322 q^{83} +4.58939 q^{85} -4.52327 q^{86} -3.71687 q^{88} -10.3879 q^{89} +3.00062 q^{91} +1.30511 q^{92} +3.85502 q^{94} -0.0154100 q^{95} +5.07659 q^{97} +5.31058 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} + 7 q^{4} - 7 q^{5} + 6 q^{7} - 7 q^{8}+O(q^{10}) \) 7 * q - 7 * q^2 + 7 * q^4 - 7 * q^5 + 6 * q^7 - 7 * q^8	\( 7 q - 7 q^{2} + 7 q^{4} - 7 q^{5} + 6 q^{7} - 7 q^{8} + 7 q^{10} + 3 q^{11} - 9 q^{13} - 6 q^{14} + 7 q^{16} - 8 q^{17} - 11 q^{19} - 7 q^{20} - 3 q^{22} - 12 q^{23} + 22 q^{25} + 9 q^{26} + 6 q^{28} + 5 q^{29} + 14 q^{31} - 7 q^{32} + 8 q^{34} + 4 q^{35} + 13 q^{37} + 11 q^{38} + 7 q^{40} - 12 q^{41} - 11 q^{43} + 3 q^{44} + 12 q^{46} - 2 q^{47} + 15 q^{49} - 22 q^{50} - 9 q^{52} - 19 q^{53} - 40 q^{55} - 6 q^{56} - 5 q^{58} + 9 q^{59} - 3 q^{61} - 14 q^{62} + 7 q^{64} + 10 q^{65} - 33 q^{67} - 8 q^{68} - 4 q^{70} - 28 q^{71} - 14 q^{73} - 13 q^{74} - 11 q^{76} - 10 q^{77} + 2 q^{79} - 7 q^{80} + 12 q^{82} + 7 q^{83} - 16 q^{85} + 11 q^{86} - 3 q^{88} - 18 q^{89} - 26 q^{91} - 12 q^{92} + 2 q^{94} + 34 q^{95} - 4 q^{97} - 15 q^{98}+O(q^{100}) \) 7 * q - 7 * q^2 + 7 * q^4 - 7 * q^5 + 6 * q^7 - 7 * q^8 + 7 * q^10 + 3 * q^11 - 9 * q^13 - 6 * q^14 + 7 * q^16 - 8 * q^17 - 11 * q^19 - 7 * q^20 - 3 * q^22 - 12 * q^23 + 22 * q^25 + 9 * q^26 + 6 * q^28 + 5 * q^29 + 14 * q^31 - 7 * q^32 + 8 * q^34 + 4 * q^35 + 13 * q^37 + 11 * q^38 + 7 * q^40 - 12 * q^41 - 11 * q^43 + 3 * q^44 + 12 * q^46 - 2 * q^47 + 15 * q^49 - 22 * q^50 - 9 * q^52 - 19 * q^53 - 40 * q^55 - 6 * q^56 - 5 * q^58 + 9 * q^59 - 3 * q^61 - 14 * q^62 + 7 * q^64 + 10 * q^65 - 33 * q^67 - 8 * q^68 - 4 * q^70 - 28 * q^71 - 14 * q^73 - 13 * q^74 - 11 * q^76 - 10 * q^77 + 2 * q^79 - 7 * q^80 + 12 * q^82 + 7 * q^83 - 16 * q^85 + 11 * q^86 - 3 * q^88 - 18 * q^89 - 26 * q^91 - 12 * q^92 + 2 * q^94 + 34 * q^95 - 4 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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