Properties

Label 304.4.a.b
Level $304$
Weight $4$
Character orbit 304.a
Self dual yes
Analytic conductor $17.937$
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] = [304,4,Mod(1,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 304.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: \(17.9365806417\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
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)
 
\(f(q)\) \(=\) \( q + 5 q^{3} - 12 q^{5} - 11 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{3} - 12 q^{5} - 11 q^{7} - 2 q^{9} + 54 q^{11} + 11 q^{13} - 60 q^{15} - 93 q^{17} - 19 q^{19} - 55 q^{21} - 183 q^{23} + 19 q^{25} - 145 q^{27} - 249 q^{29} - 56 q^{31} + 270 q^{33} + 132 q^{35} - 250 q^{37} + 55 q^{39} + 240 q^{41} + 196 q^{43} + 24 q^{45} + 168 q^{47} - 222 q^{49} - 465 q^{51} + 435 q^{53} - 648 q^{55} - 95 q^{57} - 195 q^{59} - 358 q^{61} + 22 q^{63} - 132 q^{65} + 961 q^{67} - 915 q^{69} + 246 q^{71} + 353 q^{73} + 95 q^{75} - 594 q^{77} + 34 q^{79} - 671 q^{81} - 234 q^{83} + 1116 q^{85} - 1245 q^{87} - 168 q^{89} - 121 q^{91} - 280 q^{93} + 228 q^{95} + 758 q^{97} - 108 q^{99}+O(q^{100}) \) 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 5.00000 0 −12.0000 0 −11.0000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(19\) \( +1 \)

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 304.4.a.b 1
4.b odd 2 1 19.4.a.a 1
8.b even 2 1 1216.4.a.a 1
8.d odd 2 1 1216.4.a.f 1
12.b even 2 1 171.4.a.d 1
20.d odd 2 1 475.4.a.e 1
20.e even 4 2 475.4.b.c 2
28.d even 2 1 931.4.a.a 1
44.c even 2 1 2299.4.a.b 1
76.d even 2 1 361.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.a 1 4.b odd 2 1
171.4.a.d 1 12.b even 2 1
304.4.a.b 1 1.a even 1 1 trivial
361.4.a.b 1 76.d even 2 1
475.4.a.e 1 20.d odd 2 1
475.4.b.c 2 20.e even 4 2
931.4.a.a 1 28.d even 2 1
1216.4.a.a 1 8.b even 2 1
1216.4.a.f 1 8.d odd 2 1
2299.4.a.b 1 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(304))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 5 \) Copy content Toggle raw display
$5$ \( T + 12 \) Copy content Toggle raw display
$7$ \( T + 11 \) Copy content Toggle raw display
$11$ \( T - 54 \) Copy content Toggle raw display
$13$ \( T - 11 \) Copy content Toggle raw display
$17$ \( T + 93 \) Copy content Toggle raw display
$19$ \( T + 19 \) Copy content Toggle raw display
$23$ \( T + 183 \) Copy content Toggle raw display
$29$ \( T + 249 \) Copy content Toggle raw display
$31$ \( T + 56 \) Copy content Toggle raw display
$37$ \( T + 250 \) Copy content Toggle raw display
$41$ \( T - 240 \) Copy content Toggle raw display
$43$ \( T - 196 \) Copy content Toggle raw display
$47$ \( T - 168 \) Copy content Toggle raw display
$53$ \( T - 435 \) Copy content Toggle raw display
$59$ \( T + 195 \) Copy content Toggle raw display
$61$ \( T + 358 \) Copy content Toggle raw display
$67$ \( T - 961 \) Copy content Toggle raw display
$71$ \( T - 246 \) Copy content Toggle raw display
$73$ \( T - 353 \) Copy content Toggle raw display
$79$ \( T - 34 \) Copy content Toggle raw display
$83$ \( T + 234 \) Copy content Toggle raw display
$89$ \( T + 168 \) Copy content Toggle raw display
$97$ \( T - 758 \) Copy content Toggle raw display
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