Properties

Label 225.8.a.b
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 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)
 
\(f(q)\) \(=\) \( q - 14 q^{2} + 68 q^{4} + 1644 q^{7} + 840 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 14 q^{2} + 68 q^{4} + 1644 q^{7} + 840 q^{8} - 172 q^{11} - 3862 q^{13} - 23016 q^{14} - 20464 q^{16} - 12254 q^{17} - 25940 q^{19} + 2408 q^{22} + 12972 q^{23} + 54068 q^{26} + 111792 q^{28} + 81610 q^{29} - 156888 q^{31} + 178976 q^{32} + 171556 q^{34} - 110126 q^{37} + 363160 q^{38} - 467882 q^{41} + 499208 q^{43} - 11696 q^{44} - 181608 q^{46} - 396884 q^{47} + 1879193 q^{49} - 262616 q^{52} - 1280498 q^{53} + 1380960 q^{56} - 1142540 q^{58} + 1337420 q^{59} - 923978 q^{61} + 2196432 q^{62} + 113728 q^{64} + 797304 q^{67} - 833272 q^{68} - 5103392 q^{71} + 4267478 q^{73} + 1541764 q^{74} - 1763920 q^{76} - 282768 q^{77} - 960 q^{79} + 6550348 q^{82} + 6140832 q^{83} - 6988912 q^{86} - 144480 q^{88} - 2010570 q^{89} - 6349128 q^{91} + 882096 q^{92} + 5556376 q^{94} + 4881934 q^{97} - 26308702 q^{98}+O(q^{100}) \) 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
−14.0000 0 68.0000 0 0 1644.00 840.000 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 225.8.a.b 1
3.b odd 2 1 25.8.a.a 1
5.b even 2 1 45.8.a.f 1
5.c odd 4 2 225.8.b.b 2
12.b even 2 1 400.8.a.e 1
15.d odd 2 1 5.8.a.a 1
15.e even 4 2 25.8.b.a 2
60.h even 2 1 80.8.a.d 1
60.l odd 4 2 400.8.c.e 2
105.g even 2 1 245.8.a.a 1
120.i odd 2 1 320.8.a.h 1
120.m even 2 1 320.8.a.a 1
165.d even 2 1 605.8.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.8.a.a 1 15.d odd 2 1
25.8.a.a 1 3.b odd 2 1
25.8.b.a 2 15.e even 4 2
45.8.a.f 1 5.b even 2 1
80.8.a.d 1 60.h even 2 1
225.8.a.b 1 1.a even 1 1 trivial
225.8.b.b 2 5.c odd 4 2
245.8.a.a 1 105.g even 2 1
320.8.a.a 1 120.m even 2 1
320.8.a.h 1 120.i odd 2 1
400.8.a.e 1 12.b even 2 1
400.8.c.e 2 60.l odd 4 2
605.8.a.c 1 165.d even 2 1

Hecke kernels

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

\( T_{2} + 14 \) Copy content Toggle raw display
\( T_{7} - 1644 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 14 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 1644 \) Copy content Toggle raw display
$11$ \( T + 172 \) Copy content Toggle raw display
$13$ \( T + 3862 \) Copy content Toggle raw display
$17$ \( T + 12254 \) Copy content Toggle raw display
$19$ \( T + 25940 \) Copy content Toggle raw display
$23$ \( T - 12972 \) Copy content Toggle raw display
$29$ \( T - 81610 \) Copy content Toggle raw display
$31$ \( T + 156888 \) Copy content Toggle raw display
$37$ \( T + 110126 \) Copy content Toggle raw display
$41$ \( T + 467882 \) Copy content Toggle raw display
$43$ \( T - 499208 \) Copy content Toggle raw display
$47$ \( T + 396884 \) Copy content Toggle raw display
$53$ \( T + 1280498 \) Copy content Toggle raw display
$59$ \( T - 1337420 \) Copy content Toggle raw display
$61$ \( T + 923978 \) Copy content Toggle raw display
$67$ \( T - 797304 \) Copy content Toggle raw display
$71$ \( T + 5103392 \) Copy content Toggle raw display
$73$ \( T - 4267478 \) Copy content Toggle raw display
$79$ \( T + 960 \) Copy content Toggle raw display
$83$ \( T - 6140832 \) Copy content Toggle raw display
$89$ \( T + 2010570 \) Copy content Toggle raw display
$97$ \( T - 4881934 \) Copy content Toggle raw display
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