Properties

Label 12.12.a.a
Level 12
Weight 12
Character orbit 12.a
Self dual yes
Analytic conductor 9.220
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(9.22011816672\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 243q^{3} + 9990q^{5} - 86128q^{7} + 59049q^{9} + O(q^{10}) \) \( q - 243q^{3} + 9990q^{5} - 86128q^{7} + 59049q^{9} - 806004q^{11} - 960250q^{13} - 2427570q^{15} - 4306878q^{17} + 401300q^{19} + 20929104q^{21} + 17751528q^{23} + 50971975q^{25} - 14348907q^{27} - 84704994q^{29} + 140930504q^{31} + 195858972q^{33} - 860418720q^{35} - 413506594q^{37} + 233340750q^{39} + 150094890q^{41} + 706702028q^{43} + 589899510q^{45} - 2475725472q^{47} + 5440705641q^{49} + 1046571354q^{51} + 1600124166q^{53} - 8051979960q^{55} - 97515900q^{57} + 3945492396q^{59} - 885973498q^{61} - 5085772272q^{63} - 9592897500q^{65} - 4881597772q^{67} - 4313621304q^{69} + 12631469400q^{71} + 1423335194q^{73} - 12386189925q^{75} + 69419512512q^{77} + 667407512q^{79} + 3486784401q^{81} + 5716071828q^{83} - 43025711220q^{85} + 20583313542q^{87} - 85738736790q^{89} + 82704412000q^{91} - 34246112472q^{93} + 4008987000q^{95} - 52302647806q^{97} - 47593730196q^{99} + O(q^{100}) \)

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
0 −243.000 0 9990.00 0 −86128.0 0 59049.0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 12.12.a.a 1
3.b odd 2 1 36.12.a.a 1
4.b odd 2 1 48.12.a.i 1
8.b even 2 1 192.12.a.k 1
8.d odd 2 1 192.12.a.a 1
12.b even 2 1 144.12.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.12.a.a 1 1.a even 1 1 trivial
36.12.a.a 1 3.b odd 2 1
48.12.a.i 1 4.b odd 2 1
144.12.a.a 1 12.b even 2 1
192.12.a.a 1 8.d odd 2 1
192.12.a.k 1 8.b even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 9990 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(12))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 243 T \)
$5$ \( 1 - 9990 T + 48828125 T^{2} \)
$7$ \( 1 + 86128 T + 1977326743 T^{2} \)
$11$ \( 1 + 806004 T + 285311670611 T^{2} \)
$13$ \( 1 + 960250 T + 1792160394037 T^{2} \)
$17$ \( 1 + 4306878 T + 34271896307633 T^{2} \)
$19$ \( 1 - 401300 T + 116490258898219 T^{2} \)
$23$ \( 1 - 17751528 T + 952809757913927 T^{2} \)
$29$ \( 1 + 84704994 T + 12200509765705829 T^{2} \)
$31$ \( 1 - 140930504 T + 25408476896404831 T^{2} \)
$37$ \( 1 + 413506594 T + 177917621779460413 T^{2} \)
$41$ \( 1 - 150094890 T + 550329031716248441 T^{2} \)
$43$ \( 1 - 706702028 T + 929293739471222707 T^{2} \)
$47$ \( 1 + 2475725472 T + 2472159215084012303 T^{2} \)
$53$ \( 1 - 1600124166 T + 9269035929372191597 T^{2} \)
$59$ \( 1 - 3945492396 T + 30155888444737842659 T^{2} \)
$61$ \( 1 + 885973498 T + 43513917611435838661 T^{2} \)
$67$ \( 1 + 4881597772 T + \)\(12\!\cdots\!83\)\( T^{2} \)
$71$ \( 1 - 12631469400 T + \)\(23\!\cdots\!71\)\( T^{2} \)
$73$ \( 1 - 1423335194 T + \)\(31\!\cdots\!77\)\( T^{2} \)
$79$ \( 1 - 667407512 T + \)\(74\!\cdots\!79\)\( T^{2} \)
$83$ \( 1 - 5716071828 T + \)\(12\!\cdots\!67\)\( T^{2} \)
$89$ \( 1 + 85738736790 T + \)\(27\!\cdots\!89\)\( T^{2} \)
$97$ \( 1 + 52302647806 T + \)\(71\!\cdots\!53\)\( T^{2} \)
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