Properties

Label 225.12.a.a
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 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)
 
\(f(q)\) \(=\) \( q - 56 q^{2} + 1088 q^{4} - 27984 q^{7} + 53760 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 56 q^{2} + 1088 q^{4} - 27984 q^{7} + 53760 q^{8} + 112028 q^{11} + 1096922 q^{13} + 1567104 q^{14} - 5238784 q^{16} - 249566 q^{17} - 13712420 q^{19} - 6273568 q^{22} + 41395728 q^{23} - 61427632 q^{26} - 30446592 q^{28} + 4533850 q^{29} - 265339008 q^{31} + 183271424 q^{32} + 13975696 q^{34} + 212136946 q^{37} + 767895520 q^{38} + 1266969958 q^{41} - 14129548 q^{43} + 121886464 q^{44} - 2318160768 q^{46} - 2657273336 q^{47} - 1194222487 q^{49} + 1193451136 q^{52} + 2402699278 q^{53} - 1504419840 q^{56} - 253895600 q^{58} - 7498737220 q^{59} - 4064828858 q^{61} + 14858984448 q^{62} + 465829888 q^{64} - 6871514244 q^{67} - 271527808 q^{68} + 13283734648 q^{71} + 28875844262 q^{73} - 11879668976 q^{74} - 14919112960 q^{76} - 3134991552 q^{77} + 27100302240 q^{79} - 70950317648 q^{82} - 34365255132 q^{83} + 791254688 q^{86} + 6022625280 q^{88} + 63500412630 q^{89} - 30696265248 q^{91} + 45038552064 q^{92} + 148807306816 q^{94} - 19634495234 q^{97} + 66876459272 q^{98}+O(q^{100}) \) 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
−56.0000 0 1088.00 0 0 −27984.0 53760.0 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.a 1
3.b odd 2 1 75.12.a.b 1
5.b even 2 1 45.12.a.b 1
5.c odd 4 2 225.12.b.b 2
15.d odd 2 1 15.12.a.a 1
15.e even 4 2 75.12.b.b 2
60.h even 2 1 240.12.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.12.a.a 1 15.d odd 2 1
45.12.a.b 1 5.b even 2 1
75.12.a.b 1 3.b odd 2 1
75.12.b.b 2 15.e even 4 2
225.12.a.a 1 1.a even 1 1 trivial
225.12.b.b 2 5.c odd 4 2
240.12.a.e 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 \(S_{12}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} + 56 \) Copy content Toggle raw display
\( T_{7} + 27984 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 56 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 27984 \) Copy content Toggle raw display
$11$ \( T - 112028 \) Copy content Toggle raw display
$13$ \( T - 1096922 \) Copy content Toggle raw display
$17$ \( T + 249566 \) Copy content Toggle raw display
$19$ \( T + 13712420 \) Copy content Toggle raw display
$23$ \( T - 41395728 \) Copy content Toggle raw display
$29$ \( T - 4533850 \) Copy content Toggle raw display
$31$ \( T + 265339008 \) Copy content Toggle raw display
$37$ \( T - 212136946 \) Copy content Toggle raw display
$41$ \( T - 1266969958 \) Copy content Toggle raw display
$43$ \( T + 14129548 \) Copy content Toggle raw display
$47$ \( T + 2657273336 \) Copy content Toggle raw display
$53$ \( T - 2402699278 \) Copy content Toggle raw display
$59$ \( T + 7498737220 \) Copy content Toggle raw display
$61$ \( T + 4064828858 \) Copy content Toggle raw display
$67$ \( T + 6871514244 \) Copy content Toggle raw display
$71$ \( T - 13283734648 \) Copy content Toggle raw display
$73$ \( T - 28875844262 \) Copy content Toggle raw display
$79$ \( T - 27100302240 \) Copy content Toggle raw display
$83$ \( T + 34365255132 \) Copy content Toggle raw display
$89$ \( T - 63500412630 \) Copy content Toggle raw display
$97$ \( T + 19634495234 \) Copy content Toggle raw display
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