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Properties

Label 2772.1.dl.a
Level $2772$
Weight $1$
Character orbit 2772.dl
Analytic conductor $1.383$
Analytic rank $0$
Dimension $4$
Projective image $D_{10}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2772,1,Mod(811,2772)] 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("2772.811"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2772, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 0, 5, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2772 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2772.dl (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-1,0,-1,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.38340821502\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content 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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 308)
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.5797306783837184.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{10}^{2} q^{2} + \zeta_{10}^{4} q^{4} - \zeta_{10}^{3} q^{7} - \zeta_{10} q^{8} + \zeta_{10} q^{11} + q^{14} - \zeta_{10}^{3} q^{16} + \zeta_{10}^{3} q^{22} + ( - \zeta_{10}^{3} - \zeta_{10}^{2}) q^{23} + \cdots - \zeta_{10}^{3} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - q^{4} - q^{7} - q^{8} + q^{11} + 4 q^{14} - q^{16} + q^{22} - q^{25} - q^{28} + 4 q^{32} - 3 q^{37} + 2 q^{43} - 4 q^{44} + 5 q^{46} - q^{49} - q^{50} + 3 q^{53} - q^{56} - q^{64}+ \cdots - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2772\mathbb{Z}\right)^\times\).

\(n\) \(1387\) \(1541\) \(1585\) \(2521\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(\zeta_{10}^{3}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
811.1
0.809017 + 0.587785i
0.809017 − 0.587785i
−0.309017 + 0.951057i
−0.309017 − 0.951057i
0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0 0 0.309017 − 0.951057i −0.809017 − 0.587785i 0 0
1063.1 0.309017 − 0.951057i 0 −0.809017 − 0.587785i 0 0 0.309017 + 0.951057i −0.809017 + 0.587785i 0 0
1315.1 −0.809017 − 0.587785i 0 0.309017 + 0.951057i 0 0 −0.809017 + 0.587785i 0.309017 − 0.951057i 0 0
2323.1 −0.809017 + 0.587785i 0 0.309017 − 0.951057i 0 0 −0.809017 − 0.587785i 0.309017 + 0.951057i 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
44.g even 10 1 inner
308.s odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2772.1.dl.a 4
3.b odd 2 1 308.1.s.b yes 4
4.b odd 2 1 2772.1.dl.b 4
7.b odd 2 1 CM 2772.1.dl.a 4
11.d odd 10 1 2772.1.dl.b 4
12.b even 2 1 308.1.s.a ✓ 4
21.c even 2 1 308.1.s.b yes 4
21.g even 6 2 2156.1.bm.a 8
21.h odd 6 2 2156.1.bm.a 8
28.d even 2 1 2772.1.dl.b 4
33.d even 2 1 3388.1.s.b 4
33.f even 10 1 308.1.s.a ✓ 4
33.f even 10 1 3388.1.g.a 4
33.f even 10 1 3388.1.s.d 4
33.f even 10 1 3388.1.s.h 4
33.h odd 10 1 3388.1.g.b 4
33.h odd 10 1 3388.1.s.a 4
33.h odd 10 1 3388.1.s.e 4
33.h odd 10 1 3388.1.s.g 4
44.g even 10 1 inner 2772.1.dl.a 4
77.l even 10 1 2772.1.dl.b 4
84.h odd 2 1 308.1.s.a ✓ 4
84.j odd 6 2 2156.1.bm.b 8
84.n even 6 2 2156.1.bm.b 8
132.d odd 2 1 3388.1.s.g 4
132.n odd 10 1 308.1.s.b yes 4
132.n odd 10 1 3388.1.g.b 4
132.n odd 10 1 3388.1.s.a 4
132.n odd 10 1 3388.1.s.e 4
132.o even 10 1 3388.1.g.a 4
132.o even 10 1 3388.1.s.b 4
132.o even 10 1 3388.1.s.d 4
132.o even 10 1 3388.1.s.h 4
231.h odd 2 1 3388.1.s.b 4
231.r odd 10 1 308.1.s.a ✓ 4
231.r odd 10 1 3388.1.g.a 4
231.r odd 10 1 3388.1.s.d 4
231.r odd 10 1 3388.1.s.h 4
231.u even 10 1 3388.1.g.b 4
231.u even 10 1 3388.1.s.a 4
231.u even 10 1 3388.1.s.e 4
231.u even 10 1 3388.1.s.g 4
231.be even 30 2 2156.1.bm.b 8
231.bf odd 30 2 2156.1.bm.b 8
308.s odd 10 1 inner 2772.1.dl.a 4
924.n even 2 1 3388.1.s.g 4
924.bj even 10 1 308.1.s.b yes 4
924.bj even 10 1 3388.1.g.b 4
924.bj even 10 1 3388.1.s.a 4
924.bj even 10 1 3388.1.s.e 4
924.bk odd 10 1 3388.1.g.a 4
924.bk odd 10 1 3388.1.s.b 4
924.bk odd 10 1 3388.1.s.d 4
924.bk odd 10 1 3388.1.s.h 4
924.ce odd 30 2 2156.1.bm.a 8
924.cf even 30 2 2156.1.bm.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.1.s.a ✓ 4 12.b even 2 1
308.1.s.a ✓ 4 33.f even 10 1
308.1.s.a ✓ 4 84.h odd 2 1
308.1.s.a ✓ 4 231.r odd 10 1
308.1.s.b yes 4 3.b odd 2 1
308.1.s.b yes 4 21.c even 2 1
308.1.s.b yes 4 132.n odd 10 1
308.1.s.b yes 4 924.bj even 10 1
2156.1.bm.a 8 21.g even 6 2
2156.1.bm.a 8 21.h odd 6 2
2156.1.bm.a 8 924.ce odd 30 2
2156.1.bm.a 8 924.cf even 30 2
2156.1.bm.b 8 84.j odd 6 2
2156.1.bm.b 8 84.n even 6 2
2156.1.bm.b 8 231.be even 30 2
2156.1.bm.b 8 231.bf odd 30 2
2772.1.dl.a 4 1.a even 1 1 trivial
2772.1.dl.a 4 7.b odd 2 1 CM
2772.1.dl.a 4 44.g even 10 1 inner
2772.1.dl.a 4 308.s odd 10 1 inner
2772.1.dl.b 4 4.b odd 2 1
2772.1.dl.b 4 11.d odd 10 1
2772.1.dl.b 4 28.d even 2 1
2772.1.dl.b 4 77.l even 10 1
3388.1.g.a 4 33.f even 10 1
3388.1.g.a 4 132.o even 10 1
3388.1.g.a 4 231.r odd 10 1
3388.1.g.a 4 924.bk odd 10 1
3388.1.g.b 4 33.h odd 10 1
3388.1.g.b 4 132.n odd 10 1
3388.1.g.b 4 231.u even 10 1
3388.1.g.b 4 924.bj even 10 1
3388.1.s.a 4 33.h odd 10 1
3388.1.s.a 4 132.n odd 10 1
3388.1.s.a 4 231.u even 10 1
3388.1.s.a 4 924.bj even 10 1
3388.1.s.b 4 33.d even 2 1
3388.1.s.b 4 132.o even 10 1
3388.1.s.b 4 231.h odd 2 1
3388.1.s.b 4 924.bk odd 10 1
3388.1.s.d 4 33.f even 10 1
3388.1.s.d 4 132.o even 10 1
3388.1.s.d 4 231.r odd 10 1
3388.1.s.d 4 924.bk odd 10 1
3388.1.s.e 4 33.h odd 10 1
3388.1.s.e 4 132.n odd 10 1
3388.1.s.e 4 231.u even 10 1
3388.1.s.e 4 924.bj even 10 1
3388.1.s.g 4 33.h odd 10 1
3388.1.s.g 4 132.d odd 2 1
3388.1.s.g 4 231.u even 10 1
3388.1.s.g 4 924.n even 2 1
3388.1.s.h 4 33.f even 10 1
3388.1.s.h 4 132.o even 10 1
3388.1.s.h 4 231.r odd 10 1
3388.1.s.h 4 924.bk odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2772, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{23}^{4} + 5T_{23}^{2} + 5 \) Copy content Toggle raw display
\( T_{43}^{2} - T_{43} - 1 \) Copy content Toggle raw display
\( T_{107}^{4} + 3T_{107}^{3} + 4T_{107}^{2} + 2T_{107} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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