Properties

Label 225.12.a.e
Level $225$
Weight $12$
Character orbit 225.a
Self dual yes
Analytic conductor $172.877$
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] = [225,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
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: \(172.877215626\)
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)
 
\(f(q)\) \(=\) \( q + 34 q^{2} - 892 q^{4} + 17556 q^{7} - 99960 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 34 q^{2} - 892 q^{4} + 17556 q^{7} - 99960 q^{8} + 468788 q^{11} + 374042 q^{13} + 596904 q^{14} - 1571824 q^{16} - 3724286 q^{17} - 379460 q^{19} + 15938792 q^{22} - 32458092 q^{23} + 12717428 q^{26} - 15659952 q^{28} - 69696710 q^{29} + 171448632 q^{31} + 151276064 q^{32} - 126625724 q^{34} + 291340546 q^{37} - 12901640 q^{38} - 191343242 q^{41} + 1759857392 q^{43} - 418158896 q^{44} - 1103575128 q^{46} + 1623469924 q^{47} - 1669113607 q^{49} - 333645464 q^{52} - 644888642 q^{53} - 1754897760 q^{56} - 2369688140 q^{58} - 925569220 q^{59} - 10898589338 q^{61} + 5829253488 q^{62} + 8362481728 q^{64} - 3795674064 q^{67} + 3322063112 q^{68} + 22966943728 q^{71} - 9880820458 q^{73} + 9905578564 q^{74} + 338478320 q^{76} + 8230042128 q^{77} - 20768886240 q^{79} - 6505670228 q^{82} + 3204862008 q^{83} + 59835151328 q^{86} - 46860048480 q^{88} - 63176321130 q^{89} + 6566681352 q^{91} + 28952618064 q^{92} + 55197977416 q^{94} - 126494473874 q^{97} - 56749862638 q^{98}+O(q^{100}) \) 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
34.0000 0 −892.000 0 0 17556.0 −99960.0 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 225.12.a.e 1
3.b odd 2 1 25.12.a.a 1
5.b even 2 1 45.12.a.a 1
5.c odd 4 2 225.12.b.c 2
15.d odd 2 1 5.12.a.a 1
15.e even 4 2 25.12.b.a 2
60.h even 2 1 80.12.a.f 1
105.g even 2 1 245.12.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.12.a.a 1 15.d odd 2 1
25.12.a.a 1 3.b odd 2 1
25.12.b.a 2 15.e even 4 2
45.12.a.a 1 5.b even 2 1
80.12.a.f 1 60.h even 2 1
225.12.a.e 1 1.a even 1 1 trivial
225.12.b.c 2 5.c odd 4 2
245.12.a.a 1 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} - 34 \) Copy content Toggle raw display
\( T_{7} - 17556 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 34 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 17556 \) Copy content Toggle raw display
$11$ \( T - 468788 \) Copy content Toggle raw display
$13$ \( T - 374042 \) Copy content Toggle raw display
$17$ \( T + 3724286 \) Copy content Toggle raw display
$19$ \( T + 379460 \) Copy content Toggle raw display
$23$ \( T + 32458092 \) Copy content Toggle raw display
$29$ \( T + 69696710 \) Copy content Toggle raw display
$31$ \( T - 171448632 \) Copy content Toggle raw display
$37$ \( T - 291340546 \) Copy content Toggle raw display
$41$ \( T + 191343242 \) Copy content Toggle raw display
$43$ \( T - 1759857392 \) Copy content Toggle raw display
$47$ \( T - 1623469924 \) Copy content Toggle raw display
$53$ \( T + 644888642 \) Copy content Toggle raw display
$59$ \( T + 925569220 \) Copy content Toggle raw display
$61$ \( T + 10898589338 \) Copy content Toggle raw display
$67$ \( T + 3795674064 \) Copy content Toggle raw display
$71$ \( T - 22966943728 \) Copy content Toggle raw display
$73$ \( T + 9880820458 \) Copy content Toggle raw display
$79$ \( T + 20768886240 \) Copy content Toggle raw display
$83$ \( T - 3204862008 \) Copy content Toggle raw display
$89$ \( T + 63176321130 \) Copy content Toggle raw display
$97$ \( T + 126494473874 \) Copy content Toggle raw display
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