Properties

Label 12.8.a.b
Level 12
Weight 8
Character orbit 12.a
Self dual yes
Analytic conductor 3.749
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 12.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.74862030581\)
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

\(f(q)\) \(=\) \( q + 27q^{3} + 270q^{5} + 1112q^{7} + 729q^{9} + O(q^{10}) \) \( q + 27q^{3} + 270q^{5} + 1112q^{7} + 729q^{9} - 5724q^{11} - 4570q^{13} + 7290q^{15} - 36558q^{17} + 51740q^{19} + 30024q^{21} + 22248q^{23} - 5225q^{25} + 19683q^{27} - 157194q^{29} - 103936q^{31} - 154548q^{33} + 300240q^{35} - 94834q^{37} - 123390q^{39} + 659610q^{41} - 75772q^{43} + 196830q^{45} + 405648q^{47} + 413001q^{49} - 987066q^{51} - 1346274q^{53} - 1545480q^{55} + 1396980q^{57} - 1303884q^{59} + 1833782q^{61} + 810648q^{63} - 1233900q^{65} + 1369388q^{67} + 600696q^{69} + 2714040q^{71} + 2868794q^{73} - 141075q^{75} - 6365088q^{77} - 1129648q^{79} + 531441q^{81} + 5912028q^{83} - 9870660q^{85} - 4244238q^{87} - 897750q^{89} - 5081840q^{91} - 2806272q^{93} + 13969800q^{95} + 13719074q^{97} - 4172796q^{99} + O(q^{100}) \)

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 27.0000 0 270.000 0 1112.00 0 729.000 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 12.8.a.b 1
3.b odd 2 1 36.8.a.a 1
4.b odd 2 1 48.8.a.d 1
5.b even 2 1 300.8.a.a 1
5.c odd 4 2 300.8.d.a 2
7.b odd 2 1 588.8.a.a 1
7.c even 3 2 588.8.i.b 2
7.d odd 6 2 588.8.i.g 2
8.b even 2 1 192.8.a.b 1
8.d odd 2 1 192.8.a.j 1
9.c even 3 2 324.8.e.b 2
9.d odd 6 2 324.8.e.e 2
12.b even 2 1 144.8.a.c 1
24.f even 2 1 576.8.a.u 1
24.h odd 2 1 576.8.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.8.a.b 1 1.a even 1 1 trivial
36.8.a.a 1 3.b odd 2 1
48.8.a.d 1 4.b odd 2 1
144.8.a.c 1 12.b even 2 1
192.8.a.b 1 8.b even 2 1
192.8.a.j 1 8.d odd 2 1
300.8.a.a 1 5.b even 2 1
300.8.d.a 2 5.c odd 4 2
324.8.e.b 2 9.c even 3 2
324.8.e.e 2 9.d odd 6 2
576.8.a.u 1 24.f even 2 1
576.8.a.v 1 24.h odd 2 1
588.8.a.a 1 7.b odd 2 1
588.8.i.b 2 7.c even 3 2
588.8.i.g 2 7.d odd 6 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 270 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(12))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - 27 T \)
$5$ \( 1 - 270 T + 78125 T^{2} \)
$7$ \( 1 - 1112 T + 823543 T^{2} \)
$11$ \( 1 + 5724 T + 19487171 T^{2} \)
$13$ \( 1 + 4570 T + 62748517 T^{2} \)
$17$ \( 1 + 36558 T + 410338673 T^{2} \)
$19$ \( 1 - 51740 T + 893871739 T^{2} \)
$23$ \( 1 - 22248 T + 3404825447 T^{2} \)
$29$ \( 1 + 157194 T + 17249876309 T^{2} \)
$31$ \( 1 + 103936 T + 27512614111 T^{2} \)
$37$ \( 1 + 94834 T + 94931877133 T^{2} \)
$41$ \( 1 - 659610 T + 194754273881 T^{2} \)
$43$ \( 1 + 75772 T + 271818611107 T^{2} \)
$47$ \( 1 - 405648 T + 506623120463 T^{2} \)
$53$ \( 1 + 1346274 T + 1174711139837 T^{2} \)
$59$ \( 1 + 1303884 T + 2488651484819 T^{2} \)
$61$ \( 1 - 1833782 T + 3142742836021 T^{2} \)
$67$ \( 1 - 1369388 T + 6060711605323 T^{2} \)
$71$ \( 1 - 2714040 T + 9095120158391 T^{2} \)
$73$ \( 1 - 2868794 T + 11047398519097 T^{2} \)
$79$ \( 1 + 1129648 T + 19203908986159 T^{2} \)
$83$ \( 1 - 5912028 T + 27136050989627 T^{2} \)
$89$ \( 1 + 897750 T + 44231334895529 T^{2} \)
$97$ \( 1 - 13719074 T + 80798284478113 T^{2} \)
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