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Properties

Label 328.1.z.a
Level $328$
Weight $1$
Character orbit 328.z
Analytic conductor $0.164$
Analytic rank $0$
Dimension $8$
Projective image $D_{20}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [328,1,Mod(43,328)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(328, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 10, 13]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("328.43");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 328 = 2^{3} \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 328.z (of order \(20\), degree \(8\), 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: \(0.163693324144\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{20}^{7} q^{2} + (\zeta_{20}^{3} - \zeta_{20}^{2}) q^{3} - \zeta_{20}^{4} q^{4} + ( - \zeta_{20}^{9} - 1) q^{6} + \zeta_{20} q^{8} + (\zeta_{20}^{6} + \cdots + \zeta_{20}^{4}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{20}^{7} q^{2} + (\zeta_{20}^{3} - \zeta_{20}^{2}) q^{3} - \zeta_{20}^{4} q^{4} + ( - \zeta_{20}^{9} - 1) q^{6} + \zeta_{20} q^{8} + (\zeta_{20}^{6} + \cdots + \zeta_{20}^{4}) q^{9} + \cdots + (\zeta_{20}^{9} + \zeta_{20}^{8} + \cdots + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 2 q^{4} - 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} + 2 q^{4} - 8 q^{6} + 2 q^{11} + 2 q^{12} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 8 q^{19} - 2 q^{22} - 2 q^{24} + 2 q^{25} - 2 q^{34} + 10 q^{36} + 2 q^{38} - 2 q^{44} + 8 q^{48} - 4 q^{51} + 4 q^{57} + 2 q^{64} - 4 q^{66} + 8 q^{67} - 8 q^{68} - 2 q^{72} - 8 q^{75} - 2 q^{76} - 12 q^{81} - 2 q^{82} + 4 q^{83} + 4 q^{86} + 2 q^{88} - 2 q^{89} + 2 q^{96} + 2 q^{97} + 2 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/328\mathbb{Z}\right)^\times\).

\(n\) \(129\) \(165\) \(247\)
\(\chi(n)\) \(\zeta_{20}^{9}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.587785 − 0.809017i
0.951057 + 0.309017i
−0.951057 − 0.309017i
−0.587785 + 0.809017i
0.951057 − 0.309017i
0.587785 + 0.809017i
−0.587785 − 0.809017i
−0.951057 + 0.309017i
0.951057 − 0.309017i −0.642040 + 0.642040i 0.809017 − 0.587785i 0 −0.412215 + 0.809017i 0 0.587785 − 0.809017i 0.175571i 0
115.1 −0.587785 + 0.809017i −0.221232 + 0.221232i −0.309017 − 0.951057i 0 −0.0489435 − 0.309017i 0 0.951057 + 0.309017i 0.902113i 0
131.1 0.587785 − 0.809017i −1.39680 − 1.39680i −0.309017 − 0.951057i 0 −1.95106 + 0.309017i 0 −0.951057 − 0.309017i 2.90211i 0
203.1 −0.951057 + 0.309017i 1.26007 + 1.26007i 0.809017 − 0.587785i 0 −1.58779 − 0.809017i 0 −0.587785 + 0.809017i 2.17557i 0
251.1 −0.587785 − 0.809017i −0.221232 − 0.221232i −0.309017 + 0.951057i 0 −0.0489435 + 0.309017i 0 0.951057 − 0.309017i − 0.902113i 0
267.1 0.951057 + 0.309017i −0.642040 − 0.642040i 0.809017 + 0.587785i 0 −0.412215 − 0.809017i 0 0.587785 + 0.809017i − 0.175571i 0
307.1 −0.951057 − 0.309017i 1.26007 − 1.26007i 0.809017 + 0.587785i 0 −1.58779 + 0.809017i 0 −0.587785 − 0.809017i − 2.17557i 0
323.1 0.587785 + 0.809017i −1.39680 + 1.39680i −0.309017 + 0.951057i 0 −1.95106 − 0.309017i 0 −0.951057 + 0.309017i − 2.90211i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
41.g even 20 1 inner
328.z odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 328.1.z.a ✓ 8
3.b odd 2 1 2952.1.cy.a 8
4.b odd 2 1 1312.1.cd.a 8
8.b even 2 1 1312.1.cd.a 8
8.d odd 2 1 CM 328.1.z.a ✓ 8
24.f even 2 1 2952.1.cy.a 8
41.g even 20 1 inner 328.1.z.a ✓ 8
123.m odd 20 1 2952.1.cy.a 8
164.n odd 20 1 1312.1.cd.a 8
328.y even 20 1 1312.1.cd.a 8
328.z odd 20 1 inner 328.1.z.a ✓ 8
984.cb even 20 1 2952.1.cy.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
328.1.z.a ✓ 8 1.a even 1 1 trivial
328.1.z.a ✓ 8 8.d odd 2 1 CM
328.1.z.a ✓ 8 41.g even 20 1 inner
328.1.z.a ✓ 8 328.z odd 20 1 inner
1312.1.cd.a 8 4.b odd 2 1
1312.1.cd.a 8 8.b even 2 1
1312.1.cd.a 8 164.n odd 20 1
1312.1.cd.a 8 328.y even 20 1
2952.1.cy.a 8 3.b odd 2 1
2952.1.cy.a 8 24.f even 2 1
2952.1.cy.a 8 123.m odd 20 1
2952.1.cy.a 8 984.cb even 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(328, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{6} + T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{8} + 8 T^{7} + \cdots + 1 \) 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
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} - T^{6} + T^{4} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{8} + T^{6} + 6 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} - 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{2} - T - 1)^{4} \) Copy content Toggle raw display
$89$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
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