Properties

Label 12.12.a.b
Level 12
Weight 12
Character orbit 12.a
Self dual yes
Analytic conductor 9.220
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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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: \(0\)
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

\(f(q)\) \(=\) \( q + 243q^{3} + 2862q^{5} + 9128q^{7} + 59049q^{9} + O(q^{10}) \) \( q + 243q^{3} + 2862q^{5} + 9128q^{7} + 59049q^{9} + 668196q^{11} + 2052950q^{13} + 695466q^{15} + 1604178q^{17} - 230500q^{19} + 2218104q^{21} - 43012728q^{23} - 40637081q^{25} + 14348907q^{27} - 141745194q^{29} + 233221904q^{31} + 162371628q^{33} + 26124336q^{35} + 278269694q^{37} + 498866850q^{39} - 1181577510q^{41} + 856975172q^{43} + 168998238q^{45} - 1664054928q^{47} - 1894006359q^{49} + 389815254q^{51} - 3851181666q^{53} + 1912376952q^{55} - 56011500q^{57} + 10339000596q^{59} + 185948102q^{61} + 538999272q^{63} + 5875542900q^{65} + 2915010572q^{67} - 10452092904q^{69} + 12662314200q^{71} - 15201270694q^{73} - 9874810683q^{75} + 6099293088q^{77} - 36644027488q^{79} + 3486784401q^{81} - 9217637028q^{83} + 4591157436q^{85} - 34444082142q^{87} + 30573828810q^{89} + 18739327600q^{91} + 56672922672q^{93} - 659691000q^{95} + 145701815906q^{97} + 39456305604q^{99} + O(q^{100}) \)

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 2862.00 0 9128.00 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.b 1
3.b odd 2 1 36.12.a.b 1
4.b odd 2 1 48.12.a.c 1
8.b even 2 1 192.12.a.d 1
8.d odd 2 1 192.12.a.n 1
12.b even 2 1 144.12.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.12.a.b 1 1.a even 1 1 trivial
36.12.a.b 1 3.b odd 2 1
48.12.a.c 1 4.b odd 2 1
144.12.a.f 1 12.b even 2 1
192.12.a.d 1 8.b even 2 1
192.12.a.n 1 8.d odd 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 \( T_{5} - 2862 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(12))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - 243 T \)
$5$ \( 1 - 2862 T + 48828125 T^{2} \)
$7$ \( 1 - 9128 T + 1977326743 T^{2} \)
$11$ \( 1 - 668196 T + 285311670611 T^{2} \)
$13$ \( 1 - 2052950 T + 1792160394037 T^{2} \)
$17$ \( 1 - 1604178 T + 34271896307633 T^{2} \)
$19$ \( 1 + 230500 T + 116490258898219 T^{2} \)
$23$ \( 1 + 43012728 T + 952809757913927 T^{2} \)
$29$ \( 1 + 141745194 T + 12200509765705829 T^{2} \)
$31$ \( 1 - 233221904 T + 25408476896404831 T^{2} \)
$37$ \( 1 - 278269694 T + 177917621779460413 T^{2} \)
$41$ \( 1 + 1181577510 T + 550329031716248441 T^{2} \)
$43$ \( 1 - 856975172 T + 929293739471222707 T^{2} \)
$47$ \( 1 + 1664054928 T + 2472159215084012303 T^{2} \)
$53$ \( 1 + 3851181666 T + 9269035929372191597 T^{2} \)
$59$ \( 1 - 10339000596 T + 30155888444737842659 T^{2} \)
$61$ \( 1 - 185948102 T + 43513917611435838661 T^{2} \)
$67$ \( 1 - 2915010572 T + \)\(12\!\cdots\!83\)\( T^{2} \)
$71$ \( 1 - 12662314200 T + \)\(23\!\cdots\!71\)\( T^{2} \)
$73$ \( 1 + 15201270694 T + \)\(31\!\cdots\!77\)\( T^{2} \)
$79$ \( 1 + 36644027488 T + \)\(74\!\cdots\!79\)\( T^{2} \)
$83$ \( 1 + 9217637028 T + \)\(12\!\cdots\!67\)\( T^{2} \)
$89$ \( 1 - 30573828810 T + \)\(27\!\cdots\!89\)\( T^{2} \)
$97$ \( 1 - 145701815906 T + \)\(71\!\cdots\!53\)\( T^{2} \)
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