Properties

Label 12.9.c.a
Level 12
Weight 9
Character orbit 12.c
Self dual yes
Analytic conductor 4.889
Analytic rank 0
Dimension 1
CM discriminant -3
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 12.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(4.88854332073\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 81q^{3} + 4034q^{7} + 6561q^{9} + O(q^{10}) \) \( q + 81q^{3} + 4034q^{7} + 6561q^{9} - 35806q^{13} - 258526q^{19} + 326754q^{21} + 390625q^{25} + 531441q^{27} - 1809406q^{31} + 503522q^{37} - 2900286q^{39} + 3492194q^{43} + 10508355q^{49} - 20940606q^{57} - 23826526q^{61} + 26467074q^{63} - 5421406q^{67} + 16169282q^{73} + 31640625q^{75} - 18887038q^{79} + 43046721q^{81} - 144441404q^{91} - 146561886q^{93} + 176908034q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0
0 81.0000 0 0 0 4034.00 0 6561.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.9.c.a 1
3.b odd 2 1 CM 12.9.c.a 1
4.b odd 2 1 48.9.e.a 1
5.b even 2 1 300.9.g.a 1
5.c odd 4 2 300.9.b.b 2
8.b even 2 1 192.9.e.a 1
8.d odd 2 1 192.9.e.b 1
9.c even 3 2 324.9.g.a 2
9.d odd 6 2 324.9.g.a 2
12.b even 2 1 48.9.e.a 1
15.d odd 2 1 300.9.g.a 1
15.e even 4 2 300.9.b.b 2
24.f even 2 1 192.9.e.b 1
24.h odd 2 1 192.9.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.9.c.a 1 1.a even 1 1 trivial
12.9.c.a 1 3.b odd 2 1 CM
48.9.e.a 1 4.b odd 2 1
48.9.e.a 1 12.b even 2 1
192.9.e.a 1 8.b even 2 1
192.9.e.a 1 24.h odd 2 1
192.9.e.b 1 8.d odd 2 1
192.9.e.b 1 24.f even 2 1
300.9.b.b 2 5.c odd 4 2
300.9.b.b 2 15.e even 4 2
300.9.g.a 1 5.b even 2 1
300.9.g.a 1 15.d odd 2 1
324.9.g.a 2 9.c even 3 2
324.9.g.a 2 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{9}^{\mathrm{new}}(12, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - 81 T \)
$5$ \( ( 1 - 625 T )( 1 + 625 T ) \)
$7$ \( 1 - 4034 T + 5764801 T^{2} \)
$11$ \( ( 1 - 14641 T )( 1 + 14641 T ) \)
$13$ \( 1 + 35806 T + 815730721 T^{2} \)
$17$ \( ( 1 - 83521 T )( 1 + 83521 T ) \)
$19$ \( 1 + 258526 T + 16983563041 T^{2} \)
$23$ \( ( 1 - 279841 T )( 1 + 279841 T ) \)
$29$ \( ( 1 - 707281 T )( 1 + 707281 T ) \)
$31$ \( 1 + 1809406 T + 852891037441 T^{2} \)
$37$ \( 1 - 503522 T + 3512479453921 T^{2} \)
$41$ \( ( 1 - 2825761 T )( 1 + 2825761 T ) \)
$43$ \( 1 - 3492194 T + 11688200277601 T^{2} \)
$47$ \( ( 1 - 4879681 T )( 1 + 4879681 T ) \)
$53$ \( ( 1 - 7890481 T )( 1 + 7890481 T ) \)
$59$ \( ( 1 - 12117361 T )( 1 + 12117361 T ) \)
$61$ \( 1 + 23826526 T + 191707312997281 T^{2} \)
$67$ \( 1 + 5421406 T + 406067677556641 T^{2} \)
$71$ \( ( 1 - 25411681 T )( 1 + 25411681 T ) \)
$73$ \( 1 - 16169282 T + 806460091894081 T^{2} \)
$79$ \( 1 + 18887038 T + 1517108809906561 T^{2} \)
$83$ \( ( 1 - 47458321 T )( 1 + 47458321 T ) \)
$89$ \( ( 1 - 62742241 T )( 1 + 62742241 T ) \)
$97$ \( 1 - 176908034 T + 7837433594376961 T^{2} \)
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