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: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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:

Inner twists

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

Atkin-Lehner signs

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

Hecke kernels

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