Properties

Label 225.6.a.h
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 15)
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)
 
\(f(q)\) \(=\) \( q + 7 q^{2} + 17 q^{4} - 12 q^{7} - 105 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 7 q^{2} + 17 q^{4} - 12 q^{7} - 105 q^{8} - 112 q^{11} + 974 q^{13} - 84 q^{14} - 1279 q^{16} + 2182 q^{17} + 1420 q^{19} - 784 q^{22} + 3216 q^{23} + 6818 q^{26} - 204 q^{28} + 4150 q^{29} - 5688 q^{31} - 5593 q^{32} + 15274 q^{34} - 6482 q^{37} + 9940 q^{38} - 5402 q^{41} + 21764 q^{43} - 1904 q^{44} + 22512 q^{46} - 368 q^{47} - 16663 q^{49} + 16558 q^{52} + 12586 q^{53} + 1260 q^{56} + 29050 q^{58} + 25520 q^{59} + 11782 q^{61} - 39816 q^{62} + 1777 q^{64} + 13188 q^{67} + 37094 q^{68} + 35968 q^{71} - 73186 q^{73} - 45374 q^{74} + 24140 q^{76} + 1344 q^{77} - 52440 q^{79} - 37814 q^{82} + 69036 q^{83} + 152348 q^{86} + 11760 q^{88} + 33870 q^{89} - 11688 q^{91} + 54672 q^{92} - 2576 q^{94} - 143042 q^{97} - 116641 q^{98}+O(q^{100}) \) 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
7.00000 0 17.0000 0 0 −12.0000 −105.000 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.h 1
3.b odd 2 1 75.6.a.a 1
5.b even 2 1 45.6.a.a 1
5.c odd 4 2 225.6.b.a 2
15.d odd 2 1 15.6.a.b 1
15.e even 4 2 75.6.b.a 2
20.d odd 2 1 720.6.a.q 1
60.h even 2 1 240.6.a.b 1
105.g even 2 1 735.6.a.b 1
120.i odd 2 1 960.6.a.k 1
120.m even 2 1 960.6.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.6.a.b 1 15.d odd 2 1
45.6.a.a 1 5.b even 2 1
75.6.a.a 1 3.b odd 2 1
75.6.b.a 2 15.e even 4 2
225.6.a.h 1 1.a even 1 1 trivial
225.6.b.a 2 5.c odd 4 2
240.6.a.b 1 60.h even 2 1
720.6.a.q 1 20.d odd 2 1
735.6.a.b 1 105.g even 2 1
960.6.a.k 1 120.i odd 2 1
960.6.a.x 1 120.m even 2 1

Hecke kernels

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

\( T_{2} - 7 \) Copy content Toggle raw display
\( T_{7} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 7 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 12 \) Copy content Toggle raw display
$11$ \( T + 112 \) Copy content Toggle raw display
$13$ \( T - 974 \) Copy content Toggle raw display
$17$ \( T - 2182 \) Copy content Toggle raw display
$19$ \( T - 1420 \) Copy content Toggle raw display
$23$ \( T - 3216 \) Copy content Toggle raw display
$29$ \( T - 4150 \) Copy content Toggle raw display
$31$ \( T + 5688 \) Copy content Toggle raw display
$37$ \( T + 6482 \) Copy content Toggle raw display
$41$ \( T + 5402 \) Copy content Toggle raw display
$43$ \( T - 21764 \) Copy content Toggle raw display
$47$ \( T + 368 \) Copy content Toggle raw display
$53$ \( T - 12586 \) Copy content Toggle raw display
$59$ \( T - 25520 \) Copy content Toggle raw display
$61$ \( T - 11782 \) Copy content Toggle raw display
$67$ \( T - 13188 \) Copy content Toggle raw display
$71$ \( T - 35968 \) Copy content Toggle raw display
$73$ \( T + 73186 \) Copy content Toggle raw display
$79$ \( T + 52440 \) Copy content Toggle raw display
$83$ \( T - 69036 \) Copy content Toggle raw display
$89$ \( T - 33870 \) Copy content Toggle raw display
$97$ \( T + 143042 \) Copy content Toggle raw display
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