Properties

Label 1225.4.a.r
Level $1225$
Weight $4$
Character orbit 1225.a
Self dual yes
Analytic conductor $72.277$
Analytic rank $0$
Dimension $2$
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] = [1225,4,Mod(1,1225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1225.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.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: \(72.2773397570\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + ( - \beta - 2) q^{3} + (\beta + 2) q^{4} + (3 \beta + 10) q^{6} + (5 \beta - 10) q^{8} + (5 \beta - 13) q^{9} + ( - 10 \beta - 23) q^{11} + ( - 5 \beta - 14) q^{12} + ( - 8 \beta + 30) q^{13} + \cdots + ( - 35 \beta - 201) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 5 q^{3} + 5 q^{4} + 23 q^{6} - 15 q^{8} - 21 q^{9} - 56 q^{11} - 33 q^{12} + 52 q^{13} - 135 q^{16} - 103 q^{17} - 92 q^{18} + 57 q^{19} + 233 q^{22} + 31 q^{23} - 65 q^{24} + 138 q^{26}+ \cdots - 437 q^{99}+O(q^{100}) \) 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
3.70156
−2.70156
−3.70156 −5.70156 5.70156 0 21.1047 0 8.50781 5.50781 0
1.2 2.70156 0.701562 −0.701562 0 1.89531 0 −23.5078 −26.5078 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(7\) \( -1 \)

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 1225.4.a.r 2
5.b even 2 1 1225.4.a.t 2
7.b odd 2 1 175.4.a.d 2
21.c even 2 1 1575.4.a.v 2
35.c odd 2 1 175.4.a.e yes 2
35.f even 4 2 175.4.b.d 4
105.g even 2 1 1575.4.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.4.a.d 2 7.b odd 2 1
175.4.a.e yes 2 35.c odd 2 1
175.4.b.d 4 35.f even 4 2
1225.4.a.r 2 1.a even 1 1 trivial
1225.4.a.t 2 5.b even 2 1
1575.4.a.s 2 105.g even 2 1
1575.4.a.v 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2}^{2} + T_{2} - 10 \) Copy content Toggle raw display
\( T_{3}^{2} + 5T_{3} - 4 \) Copy content Toggle raw display
\( T_{19}^{2} - 57T_{19} + 310 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 10 \) Copy content Toggle raw display
$3$ \( T^{2} + 5T - 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 56T - 241 \) Copy content Toggle raw display
$13$ \( T^{2} - 52T + 20 \) Copy content Toggle raw display
$17$ \( T^{2} + 103T + 2396 \) Copy content Toggle raw display
$19$ \( T^{2} - 57T + 310 \) Copy content Toggle raw display
$23$ \( T^{2} - 31T - 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 413T + 35170 \) Copy content Toggle raw display
$31$ \( T^{2} - 162T - 38088 \) Copy content Toggle raw display
$37$ \( T^{2} + 75T - 31896 \) Copy content Toggle raw display
$41$ \( T^{2} - 505T + 25616 \) Copy content Toggle raw display
$43$ \( T^{2} - 73T - 59440 \) Copy content Toggle raw display
$47$ \( T^{2} - 224T - 107012 \) Copy content Toggle raw display
$53$ \( T^{2} - 262T - 58648 \) Copy content Toggle raw display
$59$ \( T^{2} + 190T - 443000 \) Copy content Toggle raw display
$61$ \( T^{2} + 990T + 215136 \) Copy content Toggle raw display
$67$ \( T^{2} + 908T - 76333 \) Copy content Toggle raw display
$71$ \( T^{2} - 127T - 937010 \) Copy content Toggle raw display
$73$ \( T^{2} + 337T - 713308 \) Copy content Toggle raw display
$79$ \( T^{2} + 1119 T + 295810 \) Copy content Toggle raw display
$83$ \( T^{2} + 1517 T + 526522 \) Copy content Toggle raw display
$89$ \( T^{2} - 1713 T + 730630 \) Copy content Toggle raw display
$97$ \( T^{2} - 1764T + 19588 \) Copy content Toggle raw display
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