Properties

Label 4.12.a.a
Level $4$
Weight $12$
Character orbit 4.a
Self dual yes
Analytic conductor $3.073$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(3.07337272224\)
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

\(f(q)\) \(=\) \( q - 516q^{3} - 10530q^{5} + 49304q^{7} + 89109q^{9} + O(q^{10}) \) \( q - 516q^{3} - 10530q^{5} + 49304q^{7} + 89109q^{9} - 309420q^{11} - 1723594q^{13} + 5433480q^{15} - 2279502q^{17} + 4550444q^{19} - 25440864q^{21} - 7282872q^{23} + 62052775q^{25} + 45427608q^{27} - 69040026q^{29} - 141740704q^{31} + 159660720q^{33} - 519171120q^{35} + 711366974q^{37} + 889374504q^{39} - 1225262214q^{41} - 33606220q^{43} - 938317770q^{45} + 123214608q^{47} + 453557673q^{49} + 1176223032q^{51} + 1106121582q^{53} + 3258192600q^{55} - 2348029104q^{57} - 9062779932q^{59} - 3854150458q^{61} + 4393430136q^{63} + 18149444820q^{65} - 15313764676q^{67} + 3757961952q^{69} + 20619626328q^{71} - 2063718694q^{73} - 32019231900q^{75} - 15255643680q^{77} + 13689871472q^{79} - 39226037751q^{81} + 65570428908q^{83} + 24003156060q^{85} + 35624653416q^{87} - 29715508854q^{89} - 84980078576q^{91} + 73138203264q^{93} - 47916175320q^{95} - 23439626206q^{97} - 27572106780q^{99} + O(q^{100}) \)

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 −516.000 0 −10530.0 0 49304.0 0 89109.0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 4.12.a.a 1
3.b odd 2 1 36.12.a.d 1
4.b odd 2 1 16.12.a.c 1
5.b even 2 1 100.12.a.b 1
5.c odd 4 2 100.12.c.a 2
7.b odd 2 1 196.12.a.a 1
7.c even 3 2 196.12.e.b 2
7.d odd 6 2 196.12.e.a 2
8.b even 2 1 64.12.a.g 1
8.d odd 2 1 64.12.a.a 1
12.b even 2 1 144.12.a.n 1
16.e even 4 2 256.12.b.b 2
16.f odd 4 2 256.12.b.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.12.a.a 1 1.a even 1 1 trivial
16.12.a.c 1 4.b odd 2 1
36.12.a.d 1 3.b odd 2 1
64.12.a.a 1 8.d odd 2 1
64.12.a.g 1 8.b even 2 1
100.12.a.b 1 5.b even 2 1
100.12.c.a 2 5.c odd 4 2
144.12.a.n 1 12.b even 2 1
196.12.a.a 1 7.b odd 2 1
196.12.e.a 2 7.d odd 6 2
196.12.e.b 2 7.c even 3 2
256.12.b.b 2 16.e even 4 2
256.12.b.f 2 16.f odd 4 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 516 + T \)
$5$ \( 10530 + T \)
$7$ \( -49304 + T \)
$11$ \( 309420 + T \)
$13$ \( 1723594 + T \)
$17$ \( 2279502 + T \)
$19$ \( -4550444 + T \)
$23$ \( 7282872 + T \)
$29$ \( 69040026 + T \)
$31$ \( 141740704 + T \)
$37$ \( -711366974 + T \)
$41$ \( 1225262214 + T \)
$43$ \( 33606220 + T \)
$47$ \( -123214608 + T \)
$53$ \( -1106121582 + T \)
$59$ \( 9062779932 + T \)
$61$ \( 3854150458 + T \)
$67$ \( 15313764676 + T \)
$71$ \( -20619626328 + T \)
$73$ \( 2063718694 + T \)
$79$ \( -13689871472 + T \)
$83$ \( -65570428908 + T \)
$89$ \( 29715508854 + T \)
$97$ \( 23439626206 + T \)
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