Properties

Label 225.8.a.c
Level $225$
Weight $8$
Character orbit 225.a
Self dual yes
Analytic conductor $70.287$
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] = [225,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(70.2866307339\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q - 13 q^{2} + 41 q^{4} - 1380 q^{7} + 1131 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 13 q^{2} + 41 q^{4} - 1380 q^{7} + 1131 q^{8} + 3304 q^{11} - 8506 q^{13} + 17940 q^{14} - 19951 q^{16} - 9994 q^{17} + 41236 q^{19} - 42952 q^{22} + 84120 q^{23} + 110578 q^{26} - 56580 q^{28} - 132802 q^{29} - 55800 q^{31} + 114595 q^{32} + 129922 q^{34} - 228170 q^{37} - 536068 q^{38} + 139670 q^{41} + 755492 q^{43} + 135464 q^{44} - 1093560 q^{46} + 836984 q^{47} + 1080857 q^{49} - 348746 q^{52} + 1641650 q^{53} - 1560780 q^{56} + 1726426 q^{58} + 989656 q^{59} - 1658162 q^{61} + 725400 q^{62} + 1063993 q^{64} + 4523844 q^{67} - 409754 q^{68} + 389408 q^{71} - 5617330 q^{73} + 2966210 q^{74} + 1690676 q^{76} - 4559520 q^{77} + 3901080 q^{79} - 1815710 q^{82} - 9394116 q^{83} - 9821396 q^{86} + 3736824 q^{88} - 2803746 q^{89} + 11738280 q^{91} + 3448920 q^{92} - 10880792 q^{94} - 5099426 q^{97} - 14051141 q^{98}+O(q^{100}) \) 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
−13.0000 0 41.0000 0 0 −1380.00 1131.00 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 225.8.a.c 1
3.b odd 2 1 75.8.a.b 1
5.b even 2 1 45.8.a.e 1
5.c odd 4 2 225.8.b.c 2
15.d odd 2 1 15.8.a.b 1
15.e even 4 2 75.8.b.b 2
60.h even 2 1 240.8.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.8.a.b 1 15.d odd 2 1
45.8.a.e 1 5.b even 2 1
75.8.a.b 1 3.b odd 2 1
75.8.b.b 2 15.e even 4 2
225.8.a.c 1 1.a even 1 1 trivial
225.8.b.c 2 5.c odd 4 2
240.8.a.h 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} + 13 \) Copy content Toggle raw display
\( T_{7} + 1380 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 13 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 1380 \) Copy content Toggle raw display
$11$ \( T - 3304 \) Copy content Toggle raw display
$13$ \( T + 8506 \) Copy content Toggle raw display
$17$ \( T + 9994 \) Copy content Toggle raw display
$19$ \( T - 41236 \) Copy content Toggle raw display
$23$ \( T - 84120 \) Copy content Toggle raw display
$29$ \( T + 132802 \) Copy content Toggle raw display
$31$ \( T + 55800 \) Copy content Toggle raw display
$37$ \( T + 228170 \) Copy content Toggle raw display
$41$ \( T - 139670 \) Copy content Toggle raw display
$43$ \( T - 755492 \) Copy content Toggle raw display
$47$ \( T - 836984 \) Copy content Toggle raw display
$53$ \( T - 1641650 \) Copy content Toggle raw display
$59$ \( T - 989656 \) Copy content Toggle raw display
$61$ \( T + 1658162 \) Copy content Toggle raw display
$67$ \( T - 4523844 \) Copy content Toggle raw display
$71$ \( T - 389408 \) Copy content Toggle raw display
$73$ \( T + 5617330 \) Copy content Toggle raw display
$79$ \( T - 3901080 \) Copy content Toggle raw display
$83$ \( T + 9394116 \) Copy content Toggle raw display
$89$ \( T + 2803746 \) Copy content Toggle raw display
$97$ \( T + 5099426 \) Copy content Toggle raw display
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