Properties

Label 760.1.bz.b
Level $760$
Weight $1$
Character orbit 760.bz
Analytic conductor $0.379$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -40
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [760,1,Mod(99,760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(760, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 9, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("760.99");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 760.bz (of order \(18\), degree \(6\), 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: \(0.379289409601\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.43477921384960000.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{18}^{2} q^{2} + \zeta_{18}^{4} q^{4} + \zeta_{18}^{5} q^{5} + ( - \zeta_{18}^{8} + \zeta_{18}^{7}) q^{7} - \zeta_{18}^{6} q^{8} - \zeta_{18}^{7} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{18}^{2} q^{2} + \zeta_{18}^{4} q^{4} + \zeta_{18}^{5} q^{5} + ( - \zeta_{18}^{8} + \zeta_{18}^{7}) q^{7} - \zeta_{18}^{6} q^{8} - \zeta_{18}^{7} q^{9} - \zeta_{18}^{7} q^{10} + (\zeta_{18}^{8} + \zeta_{18}^{4}) q^{11} - \zeta_{18} q^{13} + ( - \zeta_{18} + 1) q^{14} + \zeta_{18}^{8} q^{16} - q^{18} + \zeta_{18}^{2} q^{19} - q^{20} + ( - \zeta_{18}^{6} + \zeta_{18}) q^{22} + (\zeta_{18}^{5} + \zeta_{18}^{3}) q^{23} - \zeta_{18} q^{25} + \zeta_{18}^{3} q^{26} + (\zeta_{18}^{3} - \zeta_{18}^{2}) q^{28} + \zeta_{18} q^{32} + (\zeta_{18}^{4} - \zeta_{18}^{3}) q^{35} + \zeta_{18}^{2} q^{36} + ( - \zeta_{18}^{8} + \zeta_{18}) q^{37} - \zeta_{18}^{4} q^{38} + \zeta_{18}^{2} q^{40} + (\zeta_{18}^{6} - \zeta_{18}) q^{41} + (\zeta_{18}^{8} - \zeta_{18}^{3}) q^{44} + \zeta_{18}^{3} q^{45} + ( - \zeta_{18}^{7} - \zeta_{18}^{5}) q^{46} - \zeta_{18}^{7} q^{47} + ( - \zeta_{18}^{7} + \cdots - \zeta_{18}^{5}) q^{49} + \cdots + (\zeta_{18}^{6} + \zeta_{18}^{2}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{8} + 6 q^{14} - 6 q^{18} - 6 q^{20} + 3 q^{22} + 3 q^{23} + 3 q^{26} + 3 q^{28} - 3 q^{35} - 3 q^{41} - 3 q^{44} + 3 q^{45} - 3 q^{49} + 3 q^{50} - 6 q^{53} - 6 q^{55} + 3 q^{63} - 3 q^{64} + 3 q^{65} + 3 q^{70} - 3 q^{74} - 3 q^{76} + 6 q^{77} + 3 q^{82} + 6 q^{89} - 6 q^{91} - 6 q^{92} - 6 q^{94} - 6 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(381\) \(401\) \(457\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{18}^{4}\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
−0.173648 + 0.984808i
0.939693 0.342020i
0.939693 + 0.342020i
−0.173648 0.984808i
−0.766044 0.642788i
−0.766044 + 0.642788i
0.939693 + 0.342020i 0 0.766044 + 0.642788i −0.766044 + 0.642788i 0 0.766044 1.32683i 0.500000 + 0.866025i −0.939693 + 0.342020i −0.939693 + 0.342020i
139.1 −0.766044 + 0.642788i 0 0.173648 0.984808i −0.173648 0.984808i 0 0.173648 0.300767i 0.500000 + 0.866025i 0.766044 + 0.642788i 0.766044 + 0.642788i
339.1 −0.766044 0.642788i 0 0.173648 + 0.984808i −0.173648 + 0.984808i 0 0.173648 + 0.300767i 0.500000 0.866025i 0.766044 0.642788i 0.766044 0.642788i
499.1 0.939693 0.342020i 0 0.766044 0.642788i −0.766044 0.642788i 0 0.766044 + 1.32683i 0.500000 0.866025i −0.939693 0.342020i −0.939693 0.342020i
579.1 −0.173648 0.984808i 0 −0.939693 + 0.342020i 0.939693 + 0.342020i 0 −0.939693 + 1.62760i 0.500000 + 0.866025i 0.173648 0.984808i 0.173648 0.984808i
739.1 −0.173648 + 0.984808i 0 −0.939693 0.342020i 0.939693 0.342020i 0 −0.939693 1.62760i 0.500000 0.866025i 0.173648 + 0.984808i 0.173648 + 0.984808i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.1
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.e even 9 1 inner
760.bz odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.1.bz.b yes 6
4.b odd 2 1 3040.1.dv.a 6
5.b even 2 1 760.1.bz.a 6
5.c odd 4 2 3800.1.cv.f 12
8.b even 2 1 3040.1.dv.b 6
8.d odd 2 1 760.1.bz.a 6
19.e even 9 1 inner 760.1.bz.b yes 6
20.d odd 2 1 3040.1.dv.b 6
40.e odd 2 1 CM 760.1.bz.b yes 6
40.f even 2 1 3040.1.dv.a 6
40.k even 4 2 3800.1.cv.f 12
76.l odd 18 1 3040.1.dv.a 6
95.p even 18 1 760.1.bz.a 6
95.q odd 36 2 3800.1.cv.f 12
152.t even 18 1 3040.1.dv.b 6
152.u odd 18 1 760.1.bz.a 6
380.ba odd 18 1 3040.1.dv.b 6
760.bz odd 18 1 inner 760.1.bz.b yes 6
760.cj even 18 1 3040.1.dv.a 6
760.cp even 36 2 3800.1.cv.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.1.bz.a 6 5.b even 2 1
760.1.bz.a 6 8.d odd 2 1
760.1.bz.a 6 95.p even 18 1
760.1.bz.a 6 152.u odd 18 1
760.1.bz.b yes 6 1.a even 1 1 trivial
760.1.bz.b yes 6 19.e even 9 1 inner
760.1.bz.b yes 6 40.e odd 2 1 CM
760.1.bz.b yes 6 760.bz odd 18 1 inner
3040.1.dv.a 6 4.b odd 2 1
3040.1.dv.a 6 40.f even 2 1
3040.1.dv.a 6 76.l odd 18 1
3040.1.dv.a 6 760.cj even 18 1
3040.1.dv.b 6 8.b even 2 1
3040.1.dv.b 6 20.d odd 2 1
3040.1.dv.b 6 152.t even 18 1
3040.1.dv.b 6 380.ba odd 18 1
3800.1.cv.f 12 5.c odd 4 2
3800.1.cv.f 12 40.k even 4 2
3800.1.cv.f 12 95.q odd 36 2
3800.1.cv.f 12 760.cp even 36 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 3T_{7}^{4} - 2T_{7}^{3} + 9T_{7}^{2} - 3T_{7} + 1 \) acting on \(S_{1}^{\mathrm{new}}(760, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$23$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( (T^{3} - 3 T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{6} \) Copy content Toggle raw display
$47$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$53$ \( T^{6} + 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( T^{6} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} - 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{6} \) Copy content Toggle raw display
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