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Properties

Label 760.1.bm.a
Level $760$
Weight $1$
Character orbit 760.bm
Analytic conductor $0.379$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -40
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [760,1,Mod(539,760)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(760, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 3, 2])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("760.539"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 760.bm (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.379289409601\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.14440.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.23104000.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - \zeta_{6} q^{5} - q^{7} + q^{8} + \zeta_{6}^{2} q^{9} + \zeta_{6}^{2} q^{10} - q^{11} + 2 \zeta_{6}^{2} q^{13} + \zeta_{6} q^{14} - \zeta_{6} q^{16} + q^{18} + \cdots - \zeta_{6}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - q^{5} - 2 q^{7} + 2 q^{8} - q^{9} - q^{10} - 2 q^{11} - 2 q^{13} + q^{14} - q^{16} + 2 q^{18} - q^{19} + 2 q^{20} + q^{22} + q^{23} - q^{25} + 4 q^{26} + q^{28} - q^{32}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/760\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(381\) \(401\) \(457\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{6}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
539.1
0.500000 + 0.866025i
0.500000 − 0.866025i
−0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.500000 − 0.866025i 0 −1.00000 1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
619.1 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.500000 + 0.866025i 0 −1.00000 1.00000 −0.500000 − 0.866025i −0.500000 − 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
19.c even 3 1 inner
760.bm odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.1.bm.a ✓ 2
4.b odd 2 1 3040.1.cc.a 2
5.b even 2 1 760.1.bm.b yes 2
5.c odd 4 2 3800.1.bd.f 4
8.b even 2 1 3040.1.cc.b 2
8.d odd 2 1 760.1.bm.b yes 2
19.c even 3 1 inner 760.1.bm.a ✓ 2
20.d odd 2 1 3040.1.cc.b 2
40.e odd 2 1 CM 760.1.bm.a ✓ 2
40.f even 2 1 3040.1.cc.a 2
40.k even 4 2 3800.1.bd.f 4
76.g odd 6 1 3040.1.cc.a 2
95.i even 6 1 760.1.bm.b yes 2
95.m odd 12 2 3800.1.bd.f 4
152.k odd 6 1 760.1.bm.b yes 2
152.p even 6 1 3040.1.cc.b 2
380.p odd 6 1 3040.1.cc.b 2
760.z even 6 1 3040.1.cc.a 2
760.bm odd 6 1 inner 760.1.bm.a ✓ 2
760.bw even 12 2 3800.1.bd.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.1.bm.a ✓ 2 1.a even 1 1 trivial
760.1.bm.a ✓ 2 19.c even 3 1 inner
760.1.bm.a ✓ 2 40.e odd 2 1 CM
760.1.bm.a ✓ 2 760.bm odd 6 1 inner
760.1.bm.b yes 2 5.b even 2 1
760.1.bm.b yes 2 8.d odd 2 1
760.1.bm.b yes 2 95.i even 6 1
760.1.bm.b yes 2 152.k odd 6 1
3040.1.cc.a 2 4.b odd 2 1
3040.1.cc.a 2 40.f even 2 1
3040.1.cc.a 2 76.g odd 6 1
3040.1.cc.a 2 760.z even 6 1
3040.1.cc.b 2 8.b even 2 1
3040.1.cc.b 2 20.d odd 2 1
3040.1.cc.b 2 152.p even 6 1
3040.1.cc.b 2 380.p odd 6 1
3800.1.bd.f 4 5.c odd 4 2
3800.1.bd.f 4 40.k even 4 2
3800.1.bd.f 4 95.m odd 12 2
3800.1.bd.f 4 760.bw even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 1 \) acting on \(S_{1}^{\mathrm{new}}(760, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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