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Properties

Label 1225.1.i.b
Level $1225$
Weight $1$
Character orbit 1225.i
Analytic conductor $0.611$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1225,1,Mod(276,1225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1225.276");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1225.i (of order \(6\), degree \(2\), not minimal)

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: no
Analytic conductor: \(0.611354640475\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.175.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.10504375.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{2} + q^{8} + \zeta_{6}^{2} q^{9} + \zeta_{6} q^{11} - \zeta_{6}^{2} q^{16} + \zeta_{6} q^{18} + q^{22} - \zeta_{6}^{2} q^{23} - q^{29} - \zeta_{6}^{2} q^{37} - q^{43} - \zeta_{6} q^{46} + \cdots - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{8} - q^{9} + q^{11} + q^{16} + q^{18} + 2 q^{22} + q^{23} - 2 q^{29} + q^{37} - 2 q^{43} - q^{46} - 2 q^{53} - q^{58} + 2 q^{64} + q^{67} - 2 q^{71} - q^{72} - q^{74} + q^{79}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(-\zeta_{6}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
276.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0 0 0 0 0 1.00000 −0.500000 − 0.866025i 0
901.1 0.500000 − 0.866025i 0 0 0 0 0 1.00000 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.1.i.b 2
5.b even 2 1 1225.1.i.a 2
5.c odd 4 2 1225.1.j.a 4
7.b odd 2 1 CM 1225.1.i.b 2
7.c even 3 1 175.1.d.a ✓ 1
7.c even 3 1 inner 1225.1.i.b 2
7.d odd 6 1 175.1.d.a ✓ 1
7.d odd 6 1 inner 1225.1.i.b 2
21.g even 6 1 1575.1.h.c 1
21.h odd 6 1 1575.1.h.c 1
28.f even 6 1 2800.1.f.a 1
28.g odd 6 1 2800.1.f.a 1
35.c odd 2 1 1225.1.i.a 2
35.f even 4 2 1225.1.j.a 4
35.i odd 6 1 175.1.d.b yes 1
35.i odd 6 1 1225.1.i.a 2
35.j even 6 1 175.1.d.b yes 1
35.j even 6 1 1225.1.i.a 2
35.k even 12 2 175.1.c.a 2
35.k even 12 2 1225.1.j.a 4
35.l odd 12 2 175.1.c.a 2
35.l odd 12 2 1225.1.j.a 4
105.o odd 6 1 1575.1.h.a 1
105.p even 6 1 1575.1.h.a 1
105.w odd 12 2 1575.1.e.a 2
105.x even 12 2 1575.1.e.a 2
140.p odd 6 1 2800.1.f.b 1
140.s even 6 1 2800.1.f.b 1
140.w even 12 2 2800.1.p.a 2
140.x odd 12 2 2800.1.p.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.1.c.a 2 35.k even 12 2
175.1.c.a 2 35.l odd 12 2
175.1.d.a ✓ 1 7.c even 3 1
175.1.d.a ✓ 1 7.d odd 6 1
175.1.d.b yes 1 35.i odd 6 1
175.1.d.b yes 1 35.j even 6 1
1225.1.i.a 2 5.b even 2 1
1225.1.i.a 2 35.c odd 2 1
1225.1.i.a 2 35.i odd 6 1
1225.1.i.a 2 35.j even 6 1
1225.1.i.b 2 1.a even 1 1 trivial
1225.1.i.b 2 7.b odd 2 1 CM
1225.1.i.b 2 7.c even 3 1 inner
1225.1.i.b 2 7.d odd 6 1 inner
1225.1.j.a 4 5.c odd 4 2
1225.1.j.a 4 35.f even 4 2
1225.1.j.a 4 35.k even 12 2
1225.1.j.a 4 35.l odd 12 2
1575.1.e.a 2 105.w odd 12 2
1575.1.e.a 2 105.x even 12 2
1575.1.h.a 1 105.o odd 6 1
1575.1.h.a 1 105.p even 6 1
1575.1.h.c 1 21.g even 6 1
1575.1.h.c 1 21.h odd 6 1
2800.1.f.a 1 28.f even 6 1
2800.1.f.a 1 28.g odd 6 1
2800.1.f.b 1 140.p odd 6 1
2800.1.f.b 1 140.s even 6 1
2800.1.p.a 2 140.w even 12 2
2800.1.p.a 2 140.x odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$71$ \( (T + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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