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Properties

Label 2475.1.cr.b
Level $2475$
Weight $1$
Character orbit 2475.cr
Analytic conductor $1.235$
Analytic rank $0$
Dimension $4$
Projective image $D_{12}$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2475,1,Mod(32,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 3, 6]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2475.32");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2475.cr (of order \(12\), degree \(4\), 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: \(4\)
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: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{3} - \zeta_{12} q^{4} + \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{3} - \zeta_{12} q^{4} + \zeta_{12}^{2} q^{9} - \zeta_{12}^{5} q^{11} + \zeta_{12}^{2} q^{12} + \zeta_{12}^{2} q^{16} + (\zeta_{12}^{4} + \zeta_{12}) q^{23} - \zeta_{12}^{3} q^{27} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{31} - q^{33} - \zeta_{12}^{3} q^{36} + ( - \zeta_{12}^{5} - \zeta_{12}^{4}) q^{37} - q^{44} + ( - \zeta_{12} + 1) q^{47} - \zeta_{12}^{3} q^{48} + \zeta_{12} q^{49} + ( - \zeta_{12}^{5} - \zeta_{12}^{4}) q^{53} - \zeta_{12}^{4} q^{59} - \zeta_{12}^{3} q^{64} + (\zeta_{12}^{5} + 1) q^{67} + ( - \zeta_{12}^{5} - \zeta_{12}^{2}) q^{69} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{71} + \zeta_{12}^{4} q^{81} + ( - \zeta_{12}^{5} - \zeta_{12}^{2}) q^{92} + ( - \zeta_{12}^{4} + 1) q^{93} + (\zeta_{12}^{4} - \zeta_{12}^{3}) q^{97} + \zeta_{12} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{9} + 2 q^{12} + 2 q^{16} - 2 q^{23} - 4 q^{33} + 2 q^{37} - 4 q^{44} + 4 q^{47} + 2 q^{53} + 2 q^{59} + 4 q^{67} - 2 q^{69} - 2 q^{81} - 2 q^{92} + 6 q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(-\zeta_{12}^{4}\) \(-1\) \(-\zeta_{12}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
−0.866025 − 0.500000i
0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 0.866025 + 0.500000i 0.866025 + 0.500000i 0 0 0 0 0.500000 + 0.866025i 0
857.1 0 −0.866025 + 0.500000i −0.866025 + 0.500000i 0 0 0 0 0.500000 − 0.866025i 0
1418.1 0 −0.866025 − 0.500000i −0.866025 − 0.500000i 0 0 0 0 0.500000 + 0.866025i 0
2243.1 0 0.866025 − 0.500000i 0.866025 − 0.500000i 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 \(\Q(\sqrt{-11}) \)
45.l even 12 1 inner
495.bd odd 12 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{4} + 2T_{23}^{3} + 2T_{23}^{2} + 4T_{23} + 4 \) 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
$11$ \( T^{4} - T^{2} + 1 \) 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
$23$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$37$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( (T^{2} + 3)^{2} \) 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
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
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