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Properties

Label 2775.1.bb.a
Level $2775$
Weight $1$
Character orbit 2775.bb
Analytic conductor $1.385$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -3
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2775,1,Mod(899,2775)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2775.899"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2775, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 5])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2775 = 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2775.bb (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.38490541006\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content 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 111)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.624095613.1

$q$-expansion

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

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{3} - \zeta_{12}^{2} q^{4} - \zeta_{12}^{5} q^{7} - \zeta_{12}^{4} q^{9} + \zeta_{12} q^{12} + (\zeta_{12}^{3} + \zeta_{12}) q^{13} + \zeta_{12}^{4} q^{16} + \zeta_{12}^{4} q^{21} + \cdots + (\zeta_{12}^{5} - \zeta_{12}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} + 2 q^{9} - 2 q^{16} - 2 q^{21} - 4 q^{36} - 6 q^{39} + 4 q^{64} + 6 q^{79} - 2 q^{81} + 4 q^{84} + 6 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2775\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(926\) \(1777\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(-1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
899.1
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 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0
899.2 0 0.866025 + 0.500000i −0.500000 + 0.866025i 0 0 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0
2099.1 0 −0.866025 + 0.500000i −0.500000 − 0.866025i 0 0 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0
2099.2 0 0.866025 − 0.500000i −0.500000 − 0.866025i 0 0 −0.866025 + 0.500000i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
15.d odd 2 1 inner
37.e even 6 1 inner
111.h odd 6 1 inner
185.l even 6 1 inner
555.ba odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2775.1.bb.a 4
3.b odd 2 1 CM 2775.1.bb.a 4
5.b even 2 1 inner 2775.1.bb.a 4
5.c odd 4 1 111.1.h.a ✓ 2
5.c odd 4 1 2775.1.w.a 2
15.d odd 2 1 inner 2775.1.bb.a 4
15.e even 4 1 111.1.h.a ✓ 2
15.e even 4 1 2775.1.w.a 2
20.e even 4 1 1776.1.bq.a 2
37.e even 6 1 inner 2775.1.bb.a 4
45.k odd 12 1 2997.1.o.a 2
45.k odd 12 1 2997.1.v.a 2
45.l even 12 1 2997.1.o.a 2
45.l even 12 1 2997.1.v.a 2
60.l odd 4 1 1776.1.bq.a 2
111.h odd 6 1 inner 2775.1.bb.a 4
185.l even 6 1 inner 2775.1.bb.a 4
185.r odd 12 1 111.1.h.a ✓ 2
185.r odd 12 1 2775.1.w.a 2
555.ba odd 6 1 inner 2775.1.bb.a 4
555.bh even 12 1 111.1.h.a ✓ 2
555.bh even 12 1 2775.1.w.a 2
740.bh even 12 1 1776.1.bq.a 2
1665.cn odd 12 1 2997.1.o.a 2
1665.cq even 12 1 2997.1.o.a 2
1665.db odd 12 1 2997.1.v.a 2
1665.dh even 12 1 2997.1.v.a 2
2220.cw odd 12 1 1776.1.bq.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.h.a ✓ 2 5.c odd 4 1
111.1.h.a ✓ 2 15.e even 4 1
111.1.h.a ✓ 2 185.r odd 12 1
111.1.h.a ✓ 2 555.bh even 12 1
1776.1.bq.a 2 20.e even 4 1
1776.1.bq.a 2 60.l odd 4 1
1776.1.bq.a 2 740.bh even 12 1
1776.1.bq.a 2 2220.cw odd 12 1
2775.1.w.a 2 5.c odd 4 1
2775.1.w.a 2 15.e even 4 1
2775.1.w.a 2 185.r odd 12 1
2775.1.w.a 2 555.bh even 12 1
2775.1.bb.a 4 1.a even 1 1 trivial
2775.1.bb.a 4 3.b odd 2 1 CM
2775.1.bb.a 4 5.b even 2 1 inner
2775.1.bb.a 4 15.d odd 2 1 inner
2775.1.bb.a 4 37.e even 6 1 inner
2775.1.bb.a 4 111.h odd 6 1 inner
2775.1.bb.a 4 185.l even 6 1 inner
2775.1.bb.a 4 555.ba odd 6 1 inner
2997.1.o.a 2 45.k odd 12 1
2997.1.o.a 2 45.l even 12 1
2997.1.o.a 2 1665.cn odd 12 1
2997.1.o.a 2 1665.cq even 12 1
2997.1.v.a 2 45.k odd 12 1
2997.1.v.a 2 45.l even 12 1
2997.1.v.a 2 1665.db odd 12 1
2997.1.v.a 2 1665.dh even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2775, [\chi])\).

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