Properties

Label 6.12.a.c
Level $6$
Weight $12$
Character orbit 6.a
Self dual yes
Analytic conductor $4.610$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.61005908336\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 32q^{2} + 243q^{3} + 1024q^{4} + 3630q^{5} + 7776q^{6} + 32936q^{7} + 32768q^{8} + 59049q^{9} + O(q^{10}) \) \( q + 32q^{2} + 243q^{3} + 1024q^{4} + 3630q^{5} + 7776q^{6} + 32936q^{7} + 32768q^{8} + 59049q^{9} + 116160q^{10} - 758748q^{11} + 248832q^{12} - 2482858q^{13} + 1053952q^{14} + 882090q^{15} + 1048576q^{16} + 8290386q^{17} + 1889568q^{18} - 10867300q^{19} + 3717120q^{20} + 8003448q^{21} - 24279936q^{22} + 20539272q^{23} + 7962624q^{24} - 35651225q^{25} - 79451456q^{26} + 14348907q^{27} + 33726464q^{28} + 28814550q^{29} + 28226880q^{30} + 150501392q^{31} + 33554432q^{32} - 184375764q^{33} + 265292352q^{34} + 119557680q^{35} + 60466176q^{36} - 319891714q^{37} - 347753600q^{38} - 603334494q^{39} + 118947840q^{40} - 368008998q^{41} + 256110336q^{42} + 620469572q^{43} - 776957952q^{44} + 214347870q^{45} + 657256704q^{46} + 2763110256q^{47} + 254803968q^{48} - 892546647q^{49} - 1140839200q^{50} + 2014563798q^{51} - 2542446592q^{52} - 268284258q^{53} + 459165024q^{54} - 2754255240q^{55} + 1079246848q^{56} - 2640753900q^{57} + 922065600q^{58} + 1672894740q^{59} + 903260160q^{60} - 7787197498q^{61} + 4816044544q^{62} + 1944837864q^{63} + 1073741824q^{64} - 9012774540q^{65} - 5900024448q^{66} + 18706694156q^{67} + 8489355264q^{68} + 4991043096q^{69} + 3825845760q^{70} - 8346990888q^{71} + 1934917632q^{72} + 19641746522q^{73} - 10236534848q^{74} - 8663247675q^{75} - 11128115200q^{76} - 24990124128q^{77} - 19306703808q^{78} - 5873807200q^{79} + 3806330880q^{80} + 3486784401q^{81} - 11776287936q^{82} + 8492558172q^{83} + 8195530752q^{84} + 30094101180q^{85} + 19855026304q^{86} + 7001935650q^{87} - 24862654464q^{88} + 75527864010q^{89} + 6859131840q^{90} - 81775411088q^{91} + 21032214528q^{92} + 36571838256q^{93} + 88419528192q^{94} - 39448299000q^{95} + 8153726976q^{96} - 82356782494q^{97} - 28561492704q^{98} - 44803310652q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
32.0000 243.000 1024.00 3630.00 7776.00 32936.0 32768.0 59049.0 116160.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6.12.a.c 1
3.b odd 2 1 18.12.a.a 1
4.b odd 2 1 48.12.a.d 1
5.b even 2 1 150.12.a.a 1
5.c odd 4 2 150.12.c.e 2
8.b even 2 1 192.12.a.c 1
8.d odd 2 1 192.12.a.m 1
9.c even 3 2 162.12.c.b 2
9.d odd 6 2 162.12.c.i 2
12.b even 2 1 144.12.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.12.a.c 1 1.a even 1 1 trivial
18.12.a.a 1 3.b odd 2 1
48.12.a.d 1 4.b odd 2 1
144.12.a.e 1 12.b even 2 1
150.12.a.a 1 5.b even 2 1
150.12.c.e 2 5.c odd 4 2
162.12.c.b 2 9.c even 3 2
162.12.c.i 2 9.d odd 6 2
192.12.a.c 1 8.b even 2 1
192.12.a.m 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 3630 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -32 + T \)
$3$ \( -243 + T \)
$5$ \( -3630 + T \)
$7$ \( -32936 + T \)
$11$ \( 758748 + T \)
$13$ \( 2482858 + T \)
$17$ \( -8290386 + T \)
$19$ \( 10867300 + T \)
$23$ \( -20539272 + T \)
$29$ \( -28814550 + T \)
$31$ \( -150501392 + T \)
$37$ \( 319891714 + T \)
$41$ \( 368008998 + T \)
$43$ \( -620469572 + T \)
$47$ \( -2763110256 + T \)
$53$ \( 268284258 + T \)
$59$ \( -1672894740 + T \)
$61$ \( 7787197498 + T \)
$67$ \( -18706694156 + T \)
$71$ \( 8346990888 + T \)
$73$ \( -19641746522 + T \)
$79$ \( 5873807200 + T \)
$83$ \( -8492558172 + T \)
$89$ \( -75527864010 + T \)
$97$ \( 82356782494 + T \)
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