Properties

Label 225.10.a.b
Level $225$
Weight $10$
Character orbit 225.a
Self dual yes
Analytic conductor $115.883$
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] = [225,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
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: \(115.883063137\)
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)
 
\(f(q)\) \(=\) \( q - 8 q^{2} - 448 q^{4} - 4242 q^{7} + 7680 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} - 448 q^{4} - 4242 q^{7} + 7680 q^{8} + 46208 q^{11} + 115934 q^{13} + 33936 q^{14} + 167936 q^{16} + 494842 q^{17} - 1008740 q^{19} - 369664 q^{22} - 532554 q^{23} - 927472 q^{26} + 1900416 q^{28} - 4196390 q^{29} - 3365028 q^{31} - 5275648 q^{32} - 3958736 q^{34} + 14931358 q^{37} + 8069920 q^{38} - 11056262 q^{41} + 6396794 q^{43} - 20701184 q^{44} + 4260432 q^{46} - 35559158 q^{47} - 22359043 q^{49} - 51938432 q^{52} + 39738586 q^{53} - 32578560 q^{56} + 33571120 q^{58} + 85185620 q^{59} + 45748642 q^{61} + 26920224 q^{62} - 43778048 q^{64} + 45286158 q^{67} - 221689216 q^{68} + 189967468 q^{71} - 412170946 q^{73} - 119450864 q^{74} + 451915520 q^{76} - 196014336 q^{77} + 95040840 q^{79} + 88450096 q^{82} + 261706326 q^{83} - 51174352 q^{86} + 354877440 q^{88} + 19938630 q^{89} - 491792028 q^{91} + 238584192 q^{92} + 284473264 q^{94} + 19503358 q^{97} + 178872344 q^{98}+O(q^{100}) \) 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
−8.00000 0 −448.000 0 0 −4242.00 7680.00 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 225.10.a.b 1
3.b odd 2 1 25.10.a.a 1
5.b even 2 1 45.10.a.c 1
5.c odd 4 2 225.10.b.d 2
12.b even 2 1 400.10.a.c 1
15.d odd 2 1 5.10.a.a 1
15.e even 4 2 25.10.b.a 2
60.h even 2 1 80.10.a.d 1
60.l odd 4 2 400.10.c.e 2
105.g even 2 1 245.10.a.a 1
120.i odd 2 1 320.10.a.h 1
120.m even 2 1 320.10.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.a.a 1 15.d odd 2 1
25.10.a.a 1 3.b odd 2 1
25.10.b.a 2 15.e even 4 2
45.10.a.c 1 5.b even 2 1
80.10.a.d 1 60.h even 2 1
225.10.a.b 1 1.a even 1 1 trivial
225.10.b.d 2 5.c odd 4 2
245.10.a.a 1 105.g even 2 1
320.10.a.c 1 120.m even 2 1
320.10.a.h 1 120.i odd 2 1
400.10.a.c 1 12.b even 2 1
400.10.c.e 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} + 8 \) Copy content Toggle raw display
\( T_{7} + 4242 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 8 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 4242 \) Copy content Toggle raw display
$11$ \( T - 46208 \) Copy content Toggle raw display
$13$ \( T - 115934 \) Copy content Toggle raw display
$17$ \( T - 494842 \) Copy content Toggle raw display
$19$ \( T + 1008740 \) Copy content Toggle raw display
$23$ \( T + 532554 \) Copy content Toggle raw display
$29$ \( T + 4196390 \) Copy content Toggle raw display
$31$ \( T + 3365028 \) Copy content Toggle raw display
$37$ \( T - 14931358 \) Copy content Toggle raw display
$41$ \( T + 11056262 \) Copy content Toggle raw display
$43$ \( T - 6396794 \) Copy content Toggle raw display
$47$ \( T + 35559158 \) Copy content Toggle raw display
$53$ \( T - 39738586 \) Copy content Toggle raw display
$59$ \( T - 85185620 \) Copy content Toggle raw display
$61$ \( T - 45748642 \) Copy content Toggle raw display
$67$ \( T - 45286158 \) Copy content Toggle raw display
$71$ \( T - 189967468 \) Copy content Toggle raw display
$73$ \( T + 412170946 \) Copy content Toggle raw display
$79$ \( T - 95040840 \) Copy content Toggle raw display
$83$ \( T - 261706326 \) Copy content Toggle raw display
$89$ \( T - 19938630 \) Copy content Toggle raw display
$97$ \( T - 19503358 \) Copy content Toggle raw display
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