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Properties

Label 2475.1.bs.a
Level $2475$
Weight $1$
Character orbit 2475.bs
Analytic conductor $1.235$
Analytic rank $0$
Dimension $4$
Projective image $D_{10}$
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(109,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 7, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2475.109");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2475.bs (of order \(10\), 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: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.11170196533203125.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{10}^{4} q^{4} + q^{5} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{10}^{4} q^{4} + q^{5} + \zeta_{10}^{2} q^{11} - \zeta_{10}^{3} q^{16} - \zeta_{10}^{4} q^{20} + (\zeta_{10}^{3} - \zeta_{10}) q^{23} + q^{25} + ( - \zeta_{10}^{4} + \zeta_{10}^{3}) q^{31} + \zeta_{10} q^{44} - q^{49} + ( - \zeta_{10}^{2} - \zeta_{10}) q^{53} + \zeta_{10}^{2} q^{55} + (\zeta_{10} - 1) q^{59} - \zeta_{10}^{2} q^{64} + (\zeta_{10}^{2} - 1) q^{67} + ( - \zeta_{10}^{2} + \zeta_{10}) q^{71} - \zeta_{10}^{3} q^{80} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{89} + (\zeta_{10}^{2} - 1) q^{92} + ( - \zeta_{10}^{2} - \zeta_{10}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{4} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{4} + 4 q^{5} - q^{11} - q^{16} + q^{20} + 4 q^{25} + 2 q^{31} + q^{44} - 4 q^{49} - q^{55} - 3 q^{59} + q^{64} - 5 q^{67} + 2 q^{71} - q^{80} - 2 q^{89} - 5 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{10}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−0.309017 + 0.951057i
0.809017 − 0.587785i
0.809017 + 0.587785i
−0.309017 − 0.951057i
0 0 −0.309017 − 0.951057i 1.00000 0 0 0 0 0
604.1 0 0 0.809017 + 0.587785i 1.00000 0 0 0 0 0
1594.1 0 0 0.809017 − 0.587785i 1.00000 0 0 0 0 0
2089.1 0 0 −0.309017 + 0.951057i 1.00000 0 0 0 0 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 \(\Q(\sqrt{-11}) \)
25.e even 10 1 inner
275.s odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2475.1.bs.a 4
3.b odd 2 1 275.1.s.a ✓ 4
11.b odd 2 1 CM 2475.1.bs.a 4
15.d odd 2 1 1375.1.s.a 4
15.e even 4 2 1375.1.v.b 8
25.e even 10 1 inner 2475.1.bs.a 4
33.d even 2 1 275.1.s.a ✓ 4
33.f even 10 1 3025.1.o.a 4
33.f even 10 1 3025.1.p.a 4
33.f even 10 1 3025.1.r.a 4
33.f even 10 1 3025.1.bd.a 4
33.h odd 10 1 3025.1.o.a 4
33.h odd 10 1 3025.1.p.a 4
33.h odd 10 1 3025.1.r.a 4
33.h odd 10 1 3025.1.bd.a 4
75.h odd 10 1 275.1.s.a ✓ 4
75.j odd 10 1 1375.1.s.a 4
75.l even 20 2 1375.1.v.b 8
165.d even 2 1 1375.1.s.a 4
165.l odd 4 2 1375.1.v.b 8
275.s odd 10 1 inner 2475.1.bs.a 4
825.s even 10 1 3025.1.p.a 4
825.x even 10 1 1375.1.s.a 4
825.bc odd 10 1 3025.1.r.a 4
825.be odd 10 1 3025.1.o.a 4
825.bg odd 10 1 3025.1.bd.a 4
825.bo even 10 1 3025.1.o.a 4
825.br even 10 1 3025.1.bd.a 4
825.bu even 10 1 3025.1.r.a 4
825.ce even 10 1 275.1.s.a ✓ 4
825.ch odd 10 1 3025.1.p.a 4
825.co odd 20 2 1375.1.v.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.1.s.a ✓ 4 3.b odd 2 1
275.1.s.a ✓ 4 33.d even 2 1
275.1.s.a ✓ 4 75.h odd 10 1
275.1.s.a ✓ 4 825.ce even 10 1
1375.1.s.a 4 15.d odd 2 1
1375.1.s.a 4 75.j odd 10 1
1375.1.s.a 4 165.d even 2 1
1375.1.s.a 4 825.x even 10 1
1375.1.v.b 8 15.e even 4 2
1375.1.v.b 8 75.l even 20 2
1375.1.v.b 8 165.l odd 4 2
1375.1.v.b 8 825.co odd 20 2
2475.1.bs.a 4 1.a even 1 1 trivial
2475.1.bs.a 4 11.b odd 2 1 CM
2475.1.bs.a 4 25.e even 10 1 inner
2475.1.bs.a 4 275.s odd 10 1 inner
3025.1.o.a 4 33.f even 10 1
3025.1.o.a 4 33.h odd 10 1
3025.1.o.a 4 825.be odd 10 1
3025.1.o.a 4 825.bo even 10 1
3025.1.p.a 4 33.f even 10 1
3025.1.p.a 4 33.h odd 10 1
3025.1.p.a 4 825.s even 10 1
3025.1.p.a 4 825.ch odd 10 1
3025.1.r.a 4 33.f even 10 1
3025.1.r.a 4 33.h odd 10 1
3025.1.r.a 4 825.bc odd 10 1
3025.1.r.a 4 825.bu even 10 1
3025.1.bd.a 4 33.f even 10 1
3025.1.bd.a 4 33.h odd 10 1
3025.1.bd.a 4 825.bg odd 10 1
3025.1.bd.a 4 825.br even 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2475, [\chi])\).

Hecke characteristic polynomials

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