Properties

Label 63.6.a.a.1.1
Level $63$
Weight $6$
Character 63.1
Self dual yes
Analytic conductor $10.104$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [63,6,Mod(1,63)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("63.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.1041806482\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 63.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-10.0000 q^{2} +68.0000 q^{4} +106.000 q^{5} -49.0000 q^{7} -360.000 q^{8} -1060.00 q^{10} -92.0000 q^{11} +670.000 q^{13} +490.000 q^{14} +1424.00 q^{16} +222.000 q^{17} -908.000 q^{19} +7208.00 q^{20} +920.000 q^{22} +1176.00 q^{23} +8111.00 q^{25} -6700.00 q^{26} -3332.00 q^{28} -1118.00 q^{29} +3696.00 q^{31} -2720.00 q^{32} -2220.00 q^{34} -5194.00 q^{35} +4182.00 q^{37} +9080.00 q^{38} -38160.0 q^{40} +6662.00 q^{41} -3700.00 q^{43} -6256.00 q^{44} -11760.0 q^{46} +7056.00 q^{47} +2401.00 q^{49} -81110.0 q^{50} +45560.0 q^{52} +37578.0 q^{53} -9752.00 q^{55} +17640.0 q^{56} +11180.0 q^{58} -32700.0 q^{59} -10802.0 q^{61} -36960.0 q^{62} -18368.0 q^{64} +71020.0 q^{65} +64996.0 q^{67} +15096.0 q^{68} +51940.0 q^{70} +61320.0 q^{71} +38922.0 q^{73} -41820.0 q^{74} -61744.0 q^{76} +4508.00 q^{77} -88096.0 q^{79} +150944. q^{80} -66620.0 q^{82} -71892.0 q^{83} +23532.0 q^{85} +37000.0 q^{86} +33120.0 q^{88} -111818. q^{89} -32830.0 q^{91} +79968.0 q^{92} -70560.0 q^{94} -96248.0 q^{95} -150846. q^{97} -24010.0 q^{98} +O(q^{100})\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.0000 −1.76777 −0.883883 − 0.467707i \(-0.845080\pi\)
−0.883883 + 0.467707i \(0.845080\pi\)
\(3\) 0 0
\(4\) 68.0000 2.12500
\(5\) 106.000 1.89619 0.948093 − 0.317994i \(-0.103009\pi\)
0.948093 + 0.317994i \(0.103009\pi\)
\(6\) 0 0
\(7\) −49.0000 −0.377964
\(8\) −360.000 −1.98874
\(9\) 0 0
\(10\) −1060.00 −3.35201
\(11\) −92.0000 −0.229248 −0.114624 − 0.993409i \(-0.536566\pi\)
−0.114624 + 0.993409i \(0.536566\pi\)
\(12\) 0 0
\(13\) 670.000 1.09955 0.549777 − 0.835312i \(-0.314713\pi\)
0.549777 + 0.835312i \(0.314713\pi\)
\(14\) 490.000 0.668153
\(15\) 0 0
\(16\) 1424.00 1.39062
\(17\) 222.000 0.186308 0.0931538 − 0.995652i \(-0.470305\pi\)
0.0931538 + 0.995652i \(0.470305\pi\)
\(18\) 0 0
\(19\) −908.000 −0.577035 −0.288517 − 0.957475i \(-0.593162\pi\)
−0.288517 + 0.957475i \(0.593162\pi\)
\(20\) 7208.00 4.02939
\(21\) 0 0
\(22\) 920.000 0.405258
\(23\) 1176.00 0.463541 0.231770 − 0.972771i \(-0.425548\pi\)
0.231770 + 0.972771i \(0.425548\pi\)
\(24\) 0 0
\(25\) 8111.00 2.59552
\(26\) −6700.00 −1.94375
\(27\) 0 0
\(28\) −3332.00 −0.803175
\(29\) −1118.00 −0.246858 −0.123429 − 0.992353i \(-0.539389\pi\)
−0.123429 + 0.992353i \(0.539389\pi\)
\(30\) 0 0
\(31\) 3696.00 0.690761 0.345380 − 0.938463i \(-0.387750\pi\)
0.345380 + 0.938463i \(0.387750\pi\)
\(32\) −2720.00 −0.469563
\(33\) 0 0
\(34\) −2220.00 −0.329348
\(35\) −5194.00 −0.716691
\(36\) 0 0
\(37\) 4182.00 0.502203 0.251102 − 0.967961i \(-0.419207\pi\)
0.251102 + 0.967961i \(0.419207\pi\)
\(38\) 9080.00 1.02006
\(39\) 0 0
\(40\) −38160.0 −3.77102
\(41\) 6662.00 0.618935 0.309467 − 0.950910i \(-0.399849\pi\)
0.309467 + 0.950910i \(0.399849\pi\)
\(42\) 0 0
\(43\) −3700.00 −0.305162 −0.152581 − 0.988291i \(-0.548759\pi\)
−0.152581 + 0.988291i \(0.548759\pi\)
\(44\) −6256.00 −0.487153
\(45\) 0 0
\(46\) −11760.0 −0.819432
\(47\) 7056.00 0.465923 0.232961 − 0.972486i \(-0.425158\pi\)
0.232961 + 0.972486i \(0.425158\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) −81110.0 −4.58827
\(51\) 0 0
\(52\) 45560.0 2.33655
\(53\) 37578.0 1.83757 0.918785 − 0.394758i \(-0.129172\pi\)
0.918785 + 0.394758i \(0.129172\pi\)
\(54\) 0 0
\(55\) −9752.00 −0.434697
\(56\) 17640.0 0.751672
\(57\) 0 0
\(58\) 11180.0 0.436387
\(59\) −32700.0 −1.22298 −0.611488 − 0.791254i \(-0.709429\pi\)
−0.611488 + 0.791254i \(0.709429\pi\)
\(60\) 0 0
\(61\) −10802.0 −0.371689 −0.185844 − 0.982579i \(-0.559502\pi\)
−0.185844 + 0.982579i \(0.559502\pi\)
\(62\) −36960.0 −1.22110
\(63\) 0 0
\(64\) −18368.0 −0.560547
\(65\) 71020.0 2.08496
\(66\) 0 0
\(67\) 64996.0 1.76889 0.884443 − 0.466649i \(-0.154539\pi\)
0.884443 + 0.466649i \(0.154539\pi\)
\(68\) 15096.0 0.395904
\(69\) 0 0
\(70\) 51940.0 1.26694
\(71\) 61320.0 1.44363 0.721816 − 0.692085i \(-0.243308\pi\)
0.721816 + 0.692085i \(0.243308\pi\)
\(72\) 0 0
\(73\) 38922.0 0.854846 0.427423 − 0.904052i \(-0.359421\pi\)
0.427423 + 0.904052i \(0.359421\pi\)
\(74\) −41820.0 −0.887779
\(75\) 0 0
\(76\) −61744.0 −1.22620
\(77\) 4508.00 0.0866477
\(78\) 0 0
\(79\) −88096.0 −1.58814 −0.794069 − 0.607827i \(-0.792041\pi\)
−0.794069 + 0.607827i \(0.792041\pi\)
\(80\) 150944. 2.63688
\(81\) 0 0
\(82\) −66620.0 −1.09413
\(83\) −71892.0 −1.14547 −0.572737 − 0.819739i \(-0.694118\pi\)
−0.572737 + 0.819739i \(0.694118\pi\)
\(84\) 0 0
\(85\) 23532.0 0.353274
\(86\) 37000.0 0.539455
\(87\) 0 0
\(88\) 33120.0 0.455915
\(89\) −111818. −1.49636 −0.748181 − 0.663495i \(-0.769073\pi\)
−0.748181 + 0.663495i \(0.769073\pi\)
\(90\) 0 0
\(91\) −32830.0 −0.415592
\(92\) 79968.0 0.985024
\(93\) 0 0
\(94\) −70560.0 −0.823643
\(95\) −96248.0 −1.09416
\(96\) 0 0
\(97\) −150846. −1.62781 −0.813906 − 0.580996i \(-0.802663\pi\)
−0.813906 + 0.580996i \(0.802663\pi\)
\(98\) −24010.0 −0.252538
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 63.6.a.a.1.1 1
3.2 odd 2 21.6.a.d.1.1 ✓ 1
4.3 odd 2 1008.6.a.bc.1.1 1
7.6 odd 2 441.6.a.b.1.1 1
12.11 even 2 336.6.a.a.1.1 1
15.2 even 4 525.6.d.a.274.2 2
15.8 even 4 525.6.d.a.274.1 2
15.14 odd 2 525.6.a.a.1.1 1
21.2 odd 6 147.6.e.a.67.1 2
21.5 even 6 147.6.e.b.67.1 2
21.11 odd 6 147.6.e.a.79.1 2
21.17 even 6 147.6.e.b.79.1 2
21.20 even 2 147.6.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.d.1.1 ✓ 1 3.2 odd 2
63.6.a.a.1.1 1 1.1 even 1 trivial
147.6.a.g.1.1 1 21.20 even 2
147.6.e.a.67.1 2 21.2 odd 6
147.6.e.a.79.1 2 21.11 odd 6
147.6.e.b.67.1 2 21.5 even 6
147.6.e.b.79.1 2 21.17 even 6
336.6.a.a.1.1 1 12.11 even 2
441.6.a.b.1.1 1 7.6 odd 2
525.6.a.a.1.1 1 15.14 odd 2
525.6.d.a.274.1 2 15.8 even 4
525.6.d.a.274.2 2 15.2 even 4
1008.6.a.bc.1.1 1 4.3 odd 2