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Properties

Label 328.1.k.a
Level $328$
Weight $1$
Character orbit 328.k
Analytic conductor $0.164$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [328,1,Mod(91,328)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(328, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3])) 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("328.91"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 328 = 2^{3} \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 328.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.163693324144\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.551368.1
Artin image: $C_4\wr C_2$
Artin field: Galois closure of 8.0.282300416.3

$q$-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.

\(f(q)\) \(=\) \( q - i q^{2} + ( - i + 1) q^{3} - q^{4} + ( - i - 1) q^{6} + i q^{8} - i q^{9} + (i - 1) q^{11} + (i - 1) q^{12} + q^{16} + (i + 1) q^{17} - q^{18} + ( - i - 1) q^{19} + (i + 1) q^{22} + (i + 1) q^{24} + \cdots + (i + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{4} - 2 q^{6} - 2 q^{11} - 2 q^{12} + 2 q^{16} + 2 q^{17} - 2 q^{18} - 2 q^{19} + 2 q^{22} + 2 q^{24} - 2 q^{25} + 2 q^{34} - 2 q^{38} + 2 q^{44} + 2 q^{48} + 4 q^{51} - 4 q^{57} - 2 q^{64}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(129\) \(165\) \(247\)
\(\chi(n)\) \(i\) \(-1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
− 1.00000i
1.00000i
1.00000i 1.00000 + 1.00000i −1.00000 0 −1.00000 + 1.00000i 0 − 1.00000i 1.00000i 0
155.1 − 1.00000i 1.00000 − 1.00000i −1.00000 0 −1.00000 − 1.00000i 0 1.00000i − 1.00000i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
41.c even 4 1 inner
328.k odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 328.1.k.a ✓ 2
3.b odd 2 1 2952.1.v.a 2
4.b odd 2 1 1312.1.q.a 2
8.b even 2 1 1312.1.q.a 2
8.d odd 2 1 CM 328.1.k.a ✓ 2
24.f even 2 1 2952.1.v.a 2
41.c even 4 1 inner 328.1.k.a ✓ 2
123.f odd 4 1 2952.1.v.a 2
164.e odd 4 1 1312.1.q.a 2
328.k odd 4 1 inner 328.1.k.a ✓ 2
328.l even 4 1 1312.1.q.a 2
984.s even 4 1 2952.1.v.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
328.1.k.a ✓ 2 1.a even 1 1 trivial
328.1.k.a ✓ 2 8.d odd 2 1 CM
328.1.k.a ✓ 2 41.c even 4 1 inner
328.1.k.a ✓ 2 328.k odd 4 1 inner
1312.1.q.a 2 4.b odd 2 1
1312.1.q.a 2 8.b even 2 1
1312.1.q.a 2 164.e odd 4 1
1312.1.q.a 2 328.l even 4 1
2952.1.v.a 2 3.b odd 2 1
2952.1.v.a 2 24.f even 2 1
2952.1.v.a 2 123.f odd 4 1
2952.1.v.a 2 984.s even 4 1

Hecke kernels

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

Hecke characteristic polynomials

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