Properties

Label 12.4.a.a
Level $12$
Weight $4$
Character orbit 12.a
Self dual yes
Analytic conductor $0.708$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 12.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.708022920069\)
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

\(f(q)\) \(=\) \( q + 3 q^{3} - 18 q^{5} + 8 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - 18 q^{5} + 8 q^{7} + 9 q^{9} + 36 q^{11} - 10 q^{13} - 54 q^{15} + 18 q^{17} - 100 q^{19} + 24 q^{21} + 72 q^{23} + 199 q^{25} + 27 q^{27} - 234 q^{29} - 16 q^{31} + 108 q^{33} - 144 q^{35} - 226 q^{37} - 30 q^{39} + 90 q^{41} + 452 q^{43} - 162 q^{45} + 432 q^{47} - 279 q^{49} + 54 q^{51} + 414 q^{53} - 648 q^{55} - 300 q^{57} - 684 q^{59} + 422 q^{61} + 72 q^{63} + 180 q^{65} + 332 q^{67} + 216 q^{69} - 360 q^{71} + 26 q^{73} + 597 q^{75} + 288 q^{77} + 512 q^{79} + 81 q^{81} - 1188 q^{83} - 324 q^{85} - 702 q^{87} - 630 q^{89} - 80 q^{91} - 48 q^{93} + 1800 q^{95} - 1054 q^{97} + 324 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 3.00000 0 −18.0000 0 8.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 12.4.a.a 1
3.b odd 2 1 36.4.a.a 1
4.b odd 2 1 48.4.a.a 1
5.b even 2 1 300.4.a.b 1
5.c odd 4 2 300.4.d.e 2
7.b odd 2 1 588.4.a.c 1
7.c even 3 2 588.4.i.d 2
7.d odd 6 2 588.4.i.e 2
8.b even 2 1 192.4.a.f 1
8.d odd 2 1 192.4.a.l 1
9.c even 3 2 324.4.e.h 2
9.d odd 6 2 324.4.e.a 2
11.b odd 2 1 1452.4.a.d 1
12.b even 2 1 144.4.a.g 1
13.b even 2 1 2028.4.a.c 1
13.d odd 4 2 2028.4.b.c 2
15.d odd 2 1 900.4.a.g 1
15.e even 4 2 900.4.d.c 2
16.e even 4 2 768.4.d.g 2
16.f odd 4 2 768.4.d.j 2
20.d odd 2 1 1200.4.a.be 1
20.e even 4 2 1200.4.f.d 2
21.c even 2 1 1764.4.a.b 1
21.g even 6 2 1764.4.k.o 2
21.h odd 6 2 1764.4.k.b 2
24.f even 2 1 576.4.a.a 1
24.h odd 2 1 576.4.a.b 1
28.d even 2 1 2352.4.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 1.a even 1 1 trivial
36.4.a.a 1 3.b odd 2 1
48.4.a.a 1 4.b odd 2 1
144.4.a.g 1 12.b even 2 1
192.4.a.f 1 8.b even 2 1
192.4.a.l 1 8.d odd 2 1
300.4.a.b 1 5.b even 2 1
300.4.d.e 2 5.c odd 4 2
324.4.e.a 2 9.d odd 6 2
324.4.e.h 2 9.c even 3 2
576.4.a.a 1 24.f even 2 1
576.4.a.b 1 24.h odd 2 1
588.4.a.c 1 7.b odd 2 1
588.4.i.d 2 7.c even 3 2
588.4.i.e 2 7.d odd 6 2
768.4.d.g 2 16.e even 4 2
768.4.d.j 2 16.f odd 4 2
900.4.a.g 1 15.d odd 2 1
900.4.d.c 2 15.e even 4 2
1200.4.a.be 1 20.d odd 2 1
1200.4.f.d 2 20.e even 4 2
1452.4.a.d 1 11.b odd 2 1
1764.4.a.b 1 21.c even 2 1
1764.4.k.b 2 21.h odd 6 2
1764.4.k.o 2 21.g even 6 2
2028.4.a.c 1 13.b even 2 1
2028.4.b.c 2 13.d odd 4 2
2352.4.a.bk 1 28.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(\Gamma_0(12))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T + 18 \) Copy content Toggle raw display
$7$ \( T - 8 \) Copy content Toggle raw display
$11$ \( T - 36 \) Copy content Toggle raw display
$13$ \( T + 10 \) Copy content Toggle raw display
$17$ \( T - 18 \) Copy content Toggle raw display
$19$ \( T + 100 \) Copy content Toggle raw display
$23$ \( T - 72 \) Copy content Toggle raw display
$29$ \( T + 234 \) Copy content Toggle raw display
$31$ \( T + 16 \) Copy content Toggle raw display
$37$ \( T + 226 \) Copy content Toggle raw display
$41$ \( T - 90 \) Copy content Toggle raw display
$43$ \( T - 452 \) Copy content Toggle raw display
$47$ \( T - 432 \) Copy content Toggle raw display
$53$ \( T - 414 \) Copy content Toggle raw display
$59$ \( T + 684 \) Copy content Toggle raw display
$61$ \( T - 422 \) Copy content Toggle raw display
$67$ \( T - 332 \) Copy content Toggle raw display
$71$ \( T + 360 \) Copy content Toggle raw display
$73$ \( T - 26 \) Copy content Toggle raw display
$79$ \( T - 512 \) Copy content Toggle raw display
$83$ \( T + 1188 \) Copy content Toggle raw display
$89$ \( T + 630 \) Copy content Toggle raw display
$97$ \( T + 1054 \) Copy content Toggle raw display
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