Properties

Label 336.6.a.a.1.1
Level $336$
Weight $6$
Character 336.1
Self dual yes
Analytic conductor $53.889$
Analytic rank $1$
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] = [336,6,Mod(1,336)] 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("336.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 336.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-9,0,-106,0,49] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(53.8889634572\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 336.1

$q$-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)
 
\(f(q)\) \(=\) \(q-9.00000 q^{3} -106.000 q^{5} +49.0000 q^{7} +81.0000 q^{9} -92.0000 q^{11} +670.000 q^{13} +954.000 q^{15} -222.000 q^{17} +908.000 q^{19} -441.000 q^{21} +1176.00 q^{23} +8111.00 q^{25} -729.000 q^{27} +1118.00 q^{29} -3696.00 q^{31} +828.000 q^{33} -5194.00 q^{35} +4182.00 q^{37} -6030.00 q^{39} -6662.00 q^{41} +3700.00 q^{43} -8586.00 q^{45} +7056.00 q^{47} +2401.00 q^{49} +1998.00 q^{51} -37578.0 q^{53} +9752.00 q^{55} -8172.00 q^{57} -32700.0 q^{59} -10802.0 q^{61} +3969.00 q^{63} -71020.0 q^{65} -64996.0 q^{67} -10584.0 q^{69} +61320.0 q^{71} +38922.0 q^{73} -72999.0 q^{75} -4508.00 q^{77} +88096.0 q^{79} +6561.00 q^{81} -71892.0 q^{83} +23532.0 q^{85} -10062.0 q^{87} +111818. q^{89} +32830.0 q^{91} +33264.0 q^{93} -96248.0 q^{95} -150846. q^{97} -7452.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −9.00000 −0.577350
\(4\) 0 0
\(5\) −106.000 −1.89619 −0.948093 − 0.317994i \(-0.896991\pi\)
−0.948093 + 0.317994i \(0.896991\pi\)
\(6\) 0 0
\(7\) 49.0000 0.377964
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −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 \(0.314713\pi\)
\(14\) 0 0
\(15\) 954.000 1.09476
\(16\) 0 0
\(17\) −222.000 −0.186308 −0.0931538 − 0.995652i \(-0.529695\pi\)
−0.0931538 + 0.995652i \(0.529695\pi\)
\(18\) 0 0
\(19\) 908.000 0.577035 0.288517 − 0.957475i \(-0.406838\pi\)
0.288517 + 0.957475i \(0.406838\pi\)
\(20\) 0 0
\(21\) −441.000 −0.218218
\(22\) 0 0
\(23\) 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
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) 1118.00 0.246858 0.123429 − 0.992353i \(-0.460611\pi\)
0.123429 + 0.992353i \(0.460611\pi\)
\(30\) 0 0
\(31\) −3696.00 −0.690761 −0.345380 − 0.938463i \(-0.612250\pi\)
−0.345380 + 0.938463i \(0.612250\pi\)
\(32\) 0 0
\(33\) 828.000 0.132357
\(34\) 0 0
\(35\) −5194.00 −0.716691
\(36\) 0 0
\(37\) 4182.00 0.502203 0.251102 − 0.967961i \(-0.419207\pi\)
0.251102 + 0.967961i \(0.419207\pi\)
\(38\) 0 0
\(39\) −6030.00 −0.634828
\(40\) 0 0
\(41\) −6662.00 −0.618935 −0.309467 − 0.950910i \(-0.600151\pi\)
−0.309467 + 0.950910i \(0.600151\pi\)
\(42\) 0 0
\(43\) 3700.00 0.305162 0.152581 − 0.988291i \(-0.451241\pi\)
0.152581 + 0.988291i \(0.451241\pi\)
\(44\) 0 0
\(45\) −8586.00 −0.632062
\(46\) 0 0
\(47\) 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
\(50\) 0 0
\(51\) 1998.00 0.107565
\(52\) 0 0
\(53\) −37578.0 −1.83757 −0.918785 − 0.394758i \(-0.870828\pi\)
−0.918785 + 0.394758i \(0.870828\pi\)
\(54\) 0 0
\(55\) 9752.00 0.434697
\(56\) 0 0
\(57\) −8172.00 −0.333151
\(58\) 0 0
\(59\) −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 \(0.559502\pi\)
\(62\) 0 0
\(63\) 3969.00 0.125988
\(64\) 0 0
\(65\) −71020.0 −2.08496
\(66\) 0 0
\(67\) −64996.0 −1.76889 −0.884443 − 0.466649i \(-0.845461\pi\)
−0.884443 + 0.466649i \(0.845461\pi\)
\(68\) 0 0
\(69\) −10584.0 −0.267625
\(70\) 0 0
\(71\) 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 \(0.359421\pi\)
\(74\) 0 0
\(75\) −72999.0 −1.49852
\(76\) 0 0
\(77\) −4508.00 −0.0866477
\(78\) 0 0
\(79\) 88096.0 1.58814 0.794069 − 0.607827i \(-0.207959\pi\)
0.794069 + 0.607827i \(0.207959\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) −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
\(86\) 0 0
\(87\) −10062.0 −0.142523
\(88\) 0 0
\(89\) 111818. 1.49636 0.748181 − 0.663495i \(-0.230927\pi\)
0.748181 + 0.663495i \(0.230927\pi\)
\(90\) 0 0
\(91\) 32830.0 0.415592
\(92\) 0 0
\(93\) 33264.0 0.398811
\(94\) 0 0
\(95\) −96248.0 −1.09416
\(96\) 0 0
\(97\) −150846. −1.62781 −0.813906 − 0.580996i \(-0.802663\pi\)
−0.813906 + 0.580996i \(0.802663\pi\)
\(98\) 0 0
\(99\) −7452.00 −0.0764161
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 336.6.a.a.1.1 1
3.2 odd 2 1008.6.a.bc.1.1 1
4.3 odd 2 21.6.a.d.1.1 ✓ 1
12.11 even 2 63.6.a.a.1.1 1
20.3 even 4 525.6.d.a.274.1 2
20.7 even 4 525.6.d.a.274.2 2
20.19 odd 2 525.6.a.a.1.1 1
28.3 even 6 147.6.e.b.79.1 2
28.11 odd 6 147.6.e.a.79.1 2
28.19 even 6 147.6.e.b.67.1 2
28.23 odd 6 147.6.e.a.67.1 2
28.27 even 2 147.6.a.g.1.1 1
84.83 odd 2 441.6.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.d.1.1 ✓ 1 4.3 odd 2
63.6.a.a.1.1 1 12.11 even 2
147.6.a.g.1.1 1 28.27 even 2
147.6.e.a.67.1 2 28.23 odd 6
147.6.e.a.79.1 2 28.11 odd 6
147.6.e.b.67.1 2 28.19 even 6
147.6.e.b.79.1 2 28.3 even 6
336.6.a.a.1.1 1 1.1 even 1 trivial
441.6.a.b.1.1 1 84.83 odd 2
525.6.a.a.1.1 1 20.19 odd 2
525.6.d.a.274.1 2 20.3 even 4
525.6.d.a.274.2 2 20.7 even 4
1008.6.a.bc.1.1 1 3.2 odd 2