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Properties

Label 2475.1.y.b
Level $2475$
Weight $1$
Character orbit 2475.y
Analytic conductor $1.235$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -11
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2475,1,Mod(76,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2475.76");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2475.y (of order \(6\), degree \(2\), 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: \(1.23518590627\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 495)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.99235125.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{3} + \zeta_{12}^{4} q^{4} + \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{3} + \zeta_{12}^{4} q^{4} + \zeta_{12}^{2} q^{9} + \zeta_{12}^{2} q^{11} - \zeta_{12}^{5} q^{12} - \zeta_{12}^{2} q^{16} - \zeta_{12}^{3} q^{27} + \zeta_{12}^{4} q^{31} - \zeta_{12}^{3} q^{33} - q^{36} + (\zeta_{12}^{5} - \zeta_{12}) q^{37} - q^{44} + (\zeta_{12}^{3} + \zeta_{12}) q^{47} + \zeta_{12}^{3} q^{48} + \zeta_{12}^{4} q^{49} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{53} + \zeta_{12}^{4} q^{59} + q^{64} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{67} - q^{71} + \zeta_{12}^{4} q^{81} + q^{89} - \zeta_{12}^{5} q^{93} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{97} + \zeta_{12}^{4} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{4} + 2 q^{9} + 2 q^{11} - 2 q^{16} - 2 q^{31} - 4 q^{36} - 4 q^{44} - 2 q^{49} - 2 q^{59} + 4 q^{64} - 4 q^{71} - 2 q^{81} + 8 q^{89} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(\zeta_{12}^{4}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0 −0.866025 − 0.500000i −0.500000 + 0.866025i 0 0 0 0 0.500000 + 0.866025i 0
76.2 0 0.866025 + 0.500000i −0.500000 + 0.866025i 0 0 0 0 0.500000 + 0.866025i 0
1726.1 0 −0.866025 + 0.500000i −0.500000 − 0.866025i 0 0 0 0 0.500000 − 0.866025i 0
1726.2 0 0.866025 − 0.500000i −0.500000 − 0.866025i 0 0 0 0 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner
55.d odd 2 1 inner
99.h odd 6 1 inner
495.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2475.1.y.b 4
5.b even 2 1 inner 2475.1.y.b 4
5.c odd 4 1 495.1.o.a ✓ 2
5.c odd 4 1 495.1.o.b yes 2
9.c even 3 1 inner 2475.1.y.b 4
11.b odd 2 1 CM 2475.1.y.b 4
15.e even 4 1 1485.1.o.a 2
15.e even 4 1 1485.1.o.b 2
45.j even 6 1 inner 2475.1.y.b 4
45.k odd 12 1 495.1.o.a ✓ 2
45.k odd 12 1 495.1.o.b yes 2
45.l even 12 1 1485.1.o.a 2
45.l even 12 1 1485.1.o.b 2
55.d odd 2 1 inner 2475.1.y.b 4
55.e even 4 1 495.1.o.a ✓ 2
55.e even 4 1 495.1.o.b yes 2
99.h odd 6 1 inner 2475.1.y.b 4
165.l odd 4 1 1485.1.o.a 2
165.l odd 4 1 1485.1.o.b 2
495.o odd 6 1 inner 2475.1.y.b 4
495.bd odd 12 1 1485.1.o.a 2
495.bd odd 12 1 1485.1.o.b 2
495.bf even 12 1 495.1.o.a ✓ 2
495.bf even 12 1 495.1.o.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.1.o.a ✓ 2 5.c odd 4 1
495.1.o.a ✓ 2 45.k odd 12 1
495.1.o.a ✓ 2 55.e even 4 1
495.1.o.a ✓ 2 495.bf even 12 1
495.1.o.b yes 2 5.c odd 4 1
495.1.o.b yes 2 45.k odd 12 1
495.1.o.b yes 2 55.e even 4 1
495.1.o.b yes 2 495.bf even 12 1
1485.1.o.a 2 15.e even 4 1
1485.1.o.a 2 45.l even 12 1
1485.1.o.a 2 165.l odd 4 1
1485.1.o.a 2 495.bd odd 12 1
1485.1.o.b 2 15.e even 4 1
1485.1.o.b 2 45.l even 12 1
1485.1.o.b 2 165.l odd 4 1
1485.1.o.b 2 495.bd odd 12 1
2475.1.y.b 4 1.a even 1 1 trivial
2475.1.y.b 4 5.b even 2 1 inner
2475.1.y.b 4 9.c even 3 1 inner
2475.1.y.b 4 11.b odd 2 1 CM
2475.1.y.b 4 45.j even 6 1 inner
2475.1.y.b 4 55.d odd 2 1 inner
2475.1.y.b 4 99.h odd 6 1 inner
2475.1.y.b 4 495.o odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23} \) acting on \(S_{1}^{\mathrm{new}}(2475, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$53$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$71$ \( (T + 1)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T - 2)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
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