Properties

Label 225.6.a.f
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 5)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 28 q^{4} - 192 q^{7} - 120 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 28 q^{4} - 192 q^{7} - 120 q^{8} + 148 q^{11} - 286 q^{13} - 384 q^{14} + 656 q^{16} - 1678 q^{17} + 1060 q^{19} + 296 q^{22} + 2976 q^{23} - 572 q^{26} + 5376 q^{28} + 3410 q^{29} - 2448 q^{31} + 5152 q^{32} - 3356 q^{34} - 182 q^{37} + 2120 q^{38} + 9398 q^{41} + 1244 q^{43} - 4144 q^{44} + 5952 q^{46} - 12088 q^{47} + 20057 q^{49} + 8008 q^{52} + 23846 q^{53} + 23040 q^{56} + 6820 q^{58} + 20020 q^{59} + 32302 q^{61} - 4896 q^{62} - 10688 q^{64} - 60972 q^{67} + 46984 q^{68} + 32648 q^{71} + 38774 q^{73} - 364 q^{74} - 29680 q^{76} - 28416 q^{77} - 33360 q^{79} + 18796 q^{82} + 16716 q^{83} + 2488 q^{86} - 17760 q^{88} - 101370 q^{89} + 54912 q^{91} - 83328 q^{92} - 24176 q^{94} + 119038 q^{97} + 40114 q^{98}+O(q^{100}) \) 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
2.00000 0 −28.0000 0 0 −192.000 −120.000 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.f 1
3.b odd 2 1 25.6.a.a 1
5.b even 2 1 45.6.a.b 1
5.c odd 4 2 225.6.b.e 2
12.b even 2 1 400.6.a.g 1
15.d odd 2 1 5.6.a.a 1
15.e even 4 2 25.6.b.a 2
20.d odd 2 1 720.6.a.a 1
60.h even 2 1 80.6.a.e 1
60.l odd 4 2 400.6.c.j 2
105.g even 2 1 245.6.a.b 1
120.i odd 2 1 320.6.a.j 1
120.m even 2 1 320.6.a.g 1
165.d even 2 1 605.6.a.a 1
195.e odd 2 1 845.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.a.a 1 15.d odd 2 1
25.6.a.a 1 3.b odd 2 1
25.6.b.a 2 15.e even 4 2
45.6.a.b 1 5.b even 2 1
80.6.a.e 1 60.h even 2 1
225.6.a.f 1 1.a even 1 1 trivial
225.6.b.e 2 5.c odd 4 2
245.6.a.b 1 105.g even 2 1
320.6.a.g 1 120.m even 2 1
320.6.a.j 1 120.i odd 2 1
400.6.a.g 1 12.b even 2 1
400.6.c.j 2 60.l odd 4 2
605.6.a.a 1 165.d even 2 1
720.6.a.a 1 20.d odd 2 1
845.6.a.b 1 195.e odd 2 1

Hecke kernels

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

\( T_{2} - 2 \) Copy content Toggle raw display
\( T_{7} + 192 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 192 \) Copy content Toggle raw display
$11$ \( T - 148 \) Copy content Toggle raw display
$13$ \( T + 286 \) Copy content Toggle raw display
$17$ \( T + 1678 \) Copy content Toggle raw display
$19$ \( T - 1060 \) Copy content Toggle raw display
$23$ \( T - 2976 \) Copy content Toggle raw display
$29$ \( T - 3410 \) Copy content Toggle raw display
$31$ \( T + 2448 \) Copy content Toggle raw display
$37$ \( T + 182 \) Copy content Toggle raw display
$41$ \( T - 9398 \) Copy content Toggle raw display
$43$ \( T - 1244 \) Copy content Toggle raw display
$47$ \( T + 12088 \) Copy content Toggle raw display
$53$ \( T - 23846 \) Copy content Toggle raw display
$59$ \( T - 20020 \) Copy content Toggle raw display
$61$ \( T - 32302 \) Copy content Toggle raw display
$67$ \( T + 60972 \) Copy content Toggle raw display
$71$ \( T - 32648 \) Copy content Toggle raw display
$73$ \( T - 38774 \) Copy content Toggle raw display
$79$ \( T + 33360 \) Copy content Toggle raw display
$83$ \( T - 16716 \) Copy content Toggle raw display
$89$ \( T + 101370 \) Copy content Toggle raw display
$97$ \( T - 119038 \) Copy content Toggle raw display
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