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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4,12,Mod(1,4)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 4.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 516 q^{3} - 10530 q^{5} + 49304 q^{7} + 89109 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 516 q^{3} - 10530 q^{5} + 49304 q^{7} + 89109 q^{9} - 309420 q^{11} - 1723594 q^{13} + 5433480 q^{15} - 2279502 q^{17} + 4550444 q^{19} - 25440864 q^{21} - 7282872 q^{23} + 62052775 q^{25} + 45427608 q^{27} - 69040026 q^{29} - 141740704 q^{31} + 159660720 q^{33} - 519171120 q^{35} + 711366974 q^{37} + 889374504 q^{39} - 1225262214 q^{41} - 33606220 q^{43} - 938317770 q^{45} + 123214608 q^{47} + 453557673 q^{49} + 1176223032 q^{51} + 1106121582 q^{53} + 3258192600 q^{55} - 2348029104 q^{57} - 9062779932 q^{59} - 3854150458 q^{61} + 4393430136 q^{63} + 18149444820 q^{65} - 15313764676 q^{67} + 3757961952 q^{69} + 20619626328 q^{71} - 2063718694 q^{73} - 32019231900 q^{75} - 15255643680 q^{77} + 13689871472 q^{79} - 39226037751 q^{81} + 65570428908 q^{83} + 24003156060 q^{85} + 35624653416 q^{87} - 29715508854 q^{89} - 84980078576 q^{91} + 73138203264 q^{93} - 47916175320 q^{95} - 23439626206 q^{97} - 27572106780 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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

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