Properties

Label 225.8.a.a
Level $225$
Weight $8$
Character orbit 225.a
Self dual yes
Analytic conductor $70.287$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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)
 
\(f(q)\) \(=\) \( q - 22 q^{2} + 356 q^{4} + 420 q^{7} - 5016 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 22 q^{2} + 356 q^{4} + 420 q^{7} - 5016 q^{8} + 2944 q^{11} + 11006 q^{13} - 9240 q^{14} + 64784 q^{16} - 16546 q^{17} - 25364 q^{19} - 64768 q^{22} - 5880 q^{23} - 242132 q^{26} + 149520 q^{28} - 163042 q^{29} - 201600 q^{31} - 783200 q^{32} + 364012 q^{34} - 120530 q^{37} + 558008 q^{38} + 115910 q^{41} + 19148 q^{43} + 1048064 q^{44} + 129360 q^{46} + 841016 q^{47} - 647143 q^{49} + 3918136 q^{52} + 501890 q^{53} - 2106720 q^{56} + 3586924 q^{58} + 1586176 q^{59} - 372962 q^{61} + 4435200 q^{62} + 8938048 q^{64} - 4561044 q^{67} - 5890376 q^{68} - 1512832 q^{71} + 1522910 q^{73} + 2651660 q^{74} - 9029584 q^{76} + 1236480 q^{77} + 4231920 q^{79} - 2550020 q^{82} - 1854204 q^{83} - 421256 q^{86} - 14767104 q^{88} + 6888174 q^{89} + 4622520 q^{91} - 2093280 q^{92} - 18502352 q^{94} - 3700034 q^{97} + 14237146 q^{98}+O(q^{100}) \) 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
−22.0000 0 356.000 0 0 420.000 −5016.00 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.a 1
3.b odd 2 1 75.8.a.c 1
5.b even 2 1 45.8.a.g 1
5.c odd 4 2 225.8.b.a 2
15.d odd 2 1 15.8.a.a 1
15.e even 4 2 75.8.b.a 2
60.h even 2 1 240.8.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.8.a.a 1 15.d odd 2 1
45.8.a.g 1 5.b even 2 1
75.8.a.c 1 3.b odd 2 1
75.8.b.a 2 15.e even 4 2
225.8.a.a 1 1.a even 1 1 trivial
225.8.b.a 2 5.c odd 4 2
240.8.a.c 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 \(S_{8}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} + 22 \) Copy content Toggle raw display
\( T_{7} - 420 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 22 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 420 \) Copy content Toggle raw display
$11$ \( T - 2944 \) Copy content Toggle raw display
$13$ \( T - 11006 \) Copy content Toggle raw display
$17$ \( T + 16546 \) Copy content Toggle raw display
$19$ \( T + 25364 \) Copy content Toggle raw display
$23$ \( T + 5880 \) Copy content Toggle raw display
$29$ \( T + 163042 \) Copy content Toggle raw display
$31$ \( T + 201600 \) Copy content Toggle raw display
$37$ \( T + 120530 \) Copy content Toggle raw display
$41$ \( T - 115910 \) Copy content Toggle raw display
$43$ \( T - 19148 \) Copy content Toggle raw display
$47$ \( T - 841016 \) Copy content Toggle raw display
$53$ \( T - 501890 \) Copy content Toggle raw display
$59$ \( T - 1586176 \) Copy content Toggle raw display
$61$ \( T + 372962 \) Copy content Toggle raw display
$67$ \( T + 4561044 \) Copy content Toggle raw display
$71$ \( T + 1512832 \) Copy content Toggle raw display
$73$ \( T - 1522910 \) Copy content Toggle raw display
$79$ \( T - 4231920 \) Copy content Toggle raw display
$83$ \( T + 1854204 \) Copy content Toggle raw display
$89$ \( T - 6888174 \) Copy content Toggle raw display
$97$ \( T + 3700034 \) Copy content Toggle raw display
show more
show less