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Properties

Label 760.1.b.a
Level $760$
Weight $1$
Character orbit 760.b
Analytic conductor $0.379$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -95
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [760,1,Mod(189,760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(760, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("760.189");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 760.b (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.66724352000.2

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{16} q^{2} + (\zeta_{16}^{5} + \zeta_{16}^{3}) q^{3} + \zeta_{16}^{2} q^{4} - \zeta_{16}^{4} q^{5} + ( - \zeta_{16}^{6} - \zeta_{16}^{4}) q^{6} - \zeta_{16}^{3} q^{8} + (\zeta_{16}^{6} - \zeta_{16}^{2} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{16} q^{2} + (\zeta_{16}^{5} + \zeta_{16}^{3}) q^{3} + \zeta_{16}^{2} q^{4} - \zeta_{16}^{4} q^{5} + ( - \zeta_{16}^{6} - \zeta_{16}^{4}) q^{6} - \zeta_{16}^{3} q^{8} + (\zeta_{16}^{6} - \zeta_{16}^{2} - 1) q^{9} + \zeta_{16}^{5} q^{10} + (\zeta_{16}^{6} + \zeta_{16}^{2}) q^{11} + (\zeta_{16}^{7} + \zeta_{16}^{5}) q^{12} + (\zeta_{16}^{7} + \zeta_{16}) q^{13} + ( - \zeta_{16}^{7} + \zeta_{16}) q^{15} + \zeta_{16}^{4} q^{16} + ( - \zeta_{16}^{7} + \zeta_{16}^{3} + \zeta_{16}) q^{18} + \zeta_{16}^{4} q^{19} - \zeta_{16}^{6} q^{20} + ( - \zeta_{16}^{7} - \zeta_{16}^{3}) q^{22} + ( - \zeta_{16}^{6} + 1) q^{24} - q^{25} + ( - \zeta_{16}^{2} + 1) q^{26} + ( - \zeta_{16}^{7} - \zeta_{16}^{5} + \cdots - \zeta_{16}) q^{27} + \cdots + ( - \zeta_{16}^{6} + \cdots - \zeta_{16}^{2}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{9} + 8 q^{24} - 8 q^{25} + 8 q^{26} - 8 q^{30} - 8 q^{36} - 8 q^{44} + 8 q^{49} - 8 q^{54} + 8 q^{66} + 8 q^{80} + 8 q^{81} + 8 q^{95} + 8 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(191\) \(381\) \(401\) \(457\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
189.1
0.923880 + 0.382683i
0.923880 − 0.382683i
0.382683 + 0.923880i
0.382683 − 0.923880i
−0.382683 + 0.923880i
−0.382683 − 0.923880i
−0.923880 + 0.382683i
−0.923880 − 0.382683i
−0.923880 − 0.382683i 1.84776i 0.707107 + 0.707107i − 1.00000i 0.707107 − 1.70711i 0 −0.382683 − 0.923880i −2.41421 −0.382683 + 0.923880i
189.2 −0.923880 + 0.382683i − 1.84776i 0.707107 − 0.707107i 1.00000i 0.707107 + 1.70711i 0 −0.382683 + 0.923880i −2.41421 −0.382683 − 0.923880i
189.3 −0.382683 − 0.923880i − 0.765367i −0.707107 + 0.707107i 1.00000i −0.707107 + 0.292893i 0 0.923880 + 0.382683i 0.414214 0.923880 − 0.382683i
189.4 −0.382683 + 0.923880i 0.765367i −0.707107 − 0.707107i − 1.00000i −0.707107 − 0.292893i 0 0.923880 − 0.382683i 0.414214 0.923880 + 0.382683i
189.5 0.382683 − 0.923880i − 0.765367i −0.707107 − 0.707107i − 1.00000i −0.707107 − 0.292893i 0 −0.923880 + 0.382683i 0.414214 −0.923880 − 0.382683i
189.6 0.382683 + 0.923880i 0.765367i −0.707107 + 0.707107i 1.00000i −0.707107 + 0.292893i 0 −0.923880 − 0.382683i 0.414214 −0.923880 + 0.382683i
189.7 0.923880 − 0.382683i 1.84776i 0.707107 − 0.707107i 1.00000i 0.707107 + 1.70711i 0 0.382683 − 0.923880i −2.41421 0.382683 + 0.923880i
189.8 0.923880 + 0.382683i − 1.84776i 0.707107 + 0.707107i − 1.00000i 0.707107 − 1.70711i 0 0.382683 + 0.923880i −2.41421 0.382683 − 0.923880i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 189.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
5.b even 2 1 inner
8.b even 2 1 inner
19.b odd 2 1 inner
40.f even 2 1 inner
152.g odd 2 1 inner
760.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.1.b.a ✓ 8
4.b odd 2 1 3040.1.b.a 8
5.b even 2 1 inner 760.1.b.a ✓ 8
5.c odd 4 2 3800.1.o.g 8
8.b even 2 1 inner 760.1.b.a ✓ 8
8.d odd 2 1 3040.1.b.a 8
19.b odd 2 1 inner 760.1.b.a ✓ 8
20.d odd 2 1 3040.1.b.a 8
40.e odd 2 1 3040.1.b.a 8
40.f even 2 1 inner 760.1.b.a ✓ 8
40.i odd 4 2 3800.1.o.g 8
76.d even 2 1 3040.1.b.a 8
95.d odd 2 1 CM 760.1.b.a ✓ 8
95.g even 4 2 3800.1.o.g 8
152.b even 2 1 3040.1.b.a 8
152.g odd 2 1 inner 760.1.b.a ✓ 8
380.d even 2 1 3040.1.b.a 8
760.b odd 2 1 inner 760.1.b.a ✓ 8
760.p even 2 1 3040.1.b.a 8
760.t even 4 2 3800.1.o.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.1.b.a ✓ 8 1.a even 1 1 trivial
760.1.b.a ✓ 8 5.b even 2 1 inner
760.1.b.a ✓ 8 8.b even 2 1 inner
760.1.b.a ✓ 8 19.b odd 2 1 inner
760.1.b.a ✓ 8 40.f even 2 1 inner
760.1.b.a ✓ 8 95.d odd 2 1 CM
760.1.b.a ✓ 8 152.g odd 2 1 inner
760.1.b.a ✓ 8 760.b odd 2 1 inner
3040.1.b.a 8 4.b odd 2 1
3040.1.b.a 8 8.d odd 2 1
3040.1.b.a 8 20.d odd 2 1
3040.1.b.a 8 40.e odd 2 1
3040.1.b.a 8 76.d even 2 1
3040.1.b.a 8 152.b even 2 1
3040.1.b.a 8 380.d even 2 1
3040.1.b.a 8 760.p even 2 1
3800.1.o.g 8 5.c odd 4 2
3800.1.o.g 8 40.i odd 4 2
3800.1.o.g 8 95.g even 4 2
3800.1.o.g 8 760.t even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(760, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
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