Properties

Label 225.6.a.g
Level $225$
Weight $6$
Character orbit 225.a
Self dual yes
Analytic conductor $36.086$
Analytic rank $0$
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] = [225,6,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.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: \(36.0863594579\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} - 16 q^{4} + 225 q^{7} - 192 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} - 16 q^{4} + 225 q^{7} - 192 q^{8} + 434 q^{11} - 613 q^{13} + 900 q^{14} - 256 q^{16} - 878 q^{17} - 731 q^{19} + 1736 q^{22} + 2850 q^{23} - 2452 q^{26} - 3600 q^{28} + 7582 q^{29} + 2175 q^{31} + 5120 q^{32} - 3512 q^{34} + 9310 q^{37} - 2924 q^{38} + 12040 q^{41} + 1121 q^{43} - 6944 q^{44} + 11400 q^{46} + 29878 q^{47} + 33818 q^{49} + 9808 q^{52} + 5740 q^{53} - 43200 q^{56} + 30328 q^{58} + 5174 q^{59} - 38717 q^{61} + 8700 q^{62} + 28672 q^{64} - 31707 q^{67} + 14048 q^{68} - 64472 q^{71} + 19790 q^{73} + 37240 q^{74} + 11696 q^{76} + 97650 q^{77} - 105000 q^{79} + 48160 q^{82} - 3318 q^{83} + 4484 q^{86} - 83328 q^{88} + 65376 q^{89} - 137925 q^{91} - 45600 q^{92} + 119512 q^{94} + 89143 q^{97} + 135272 q^{98}+O(q^{100}) \) 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
4.00000 0 −16.0000 0 0 225.000 −192.000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)

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 225.6.a.g 1
3.b odd 2 1 75.6.a.b 1
5.b even 2 1 225.6.a.b 1
5.c odd 4 2 225.6.b.c 2
15.d odd 2 1 75.6.a.d yes 1
15.e even 4 2 75.6.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.6.a.b 1 3.b odd 2 1
75.6.a.d yes 1 15.d odd 2 1
75.6.b.c 2 15.e even 4 2
225.6.a.b 1 5.b even 2 1
225.6.a.g 1 1.a even 1 1 trivial
225.6.b.c 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} - 4 \) Copy content Toggle raw display
\( T_{7} - 225 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 225 \) Copy content Toggle raw display
$11$ \( T - 434 \) Copy content Toggle raw display
$13$ \( T + 613 \) Copy content Toggle raw display
$17$ \( T + 878 \) Copy content Toggle raw display
$19$ \( T + 731 \) Copy content Toggle raw display
$23$ \( T - 2850 \) Copy content Toggle raw display
$29$ \( T - 7582 \) Copy content Toggle raw display
$31$ \( T - 2175 \) Copy content Toggle raw display
$37$ \( T - 9310 \) Copy content Toggle raw display
$41$ \( T - 12040 \) Copy content Toggle raw display
$43$ \( T - 1121 \) Copy content Toggle raw display
$47$ \( T - 29878 \) Copy content Toggle raw display
$53$ \( T - 5740 \) Copy content Toggle raw display
$59$ \( T - 5174 \) Copy content Toggle raw display
$61$ \( T + 38717 \) Copy content Toggle raw display
$67$ \( T + 31707 \) Copy content Toggle raw display
$71$ \( T + 64472 \) Copy content Toggle raw display
$73$ \( T - 19790 \) Copy content Toggle raw display
$79$ \( T + 105000 \) Copy content Toggle raw display
$83$ \( T + 3318 \) Copy content Toggle raw display
$89$ \( T - 65376 \) Copy content Toggle raw display
$97$ \( T - 89143 \) Copy content Toggle raw display
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