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Properties

Label 3528.1.d.b
Level $3528$
Weight $1$
Character orbit 3528.d
Analytic conductor $1.761$
Analytic rank $0$
Dimension $2$
Projective image $S_{4}$
CM/RM no
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3528,1,Mod(1961,3528)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3528.1961"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3528, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3528.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.76070136457\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.21168.2
Artin image: $\GL(2,3)$
Artin field: Galois closure of 8.2.263473523712.2

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} - \beta q^{11} + q^{13} + q^{19} - q^{25} + q^{31} + q^{37} + \beta q^{41} - q^{43} - \beta q^{47} + 2 q^{55} + \beta q^{65} - q^{67} + \beta q^{71} - q^{73} + q^{79} + \beta q^{83} + \cdots + \beta q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{13} + 2 q^{19} - 2 q^{25} + 2 q^{31} + 2 q^{37} - 2 q^{43} + 4 q^{55} - 2 q^{67} - 2 q^{73} + 2 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1961.1
− 1.41421i
1.41421i
0 0 0 − 1.41421i 0 0 0 0 0
1961.2 0 0 0 1.41421i 0 0 0 0 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.d.b 2
3.b odd 2 1 inner 3528.1.d.b 2
7.b odd 2 1 3528.1.d.a 2
7.c even 3 2 3528.1.cu.a 4
7.d odd 6 2 504.1.cu.a ✓ 4
21.c even 2 1 3528.1.d.a 2
21.g even 6 2 504.1.cu.a ✓ 4
21.h odd 6 2 3528.1.cu.a 4
28.f even 6 2 1008.1.dc.a 4
84.j odd 6 2 1008.1.dc.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.cu.a ✓ 4 7.d odd 6 2
504.1.cu.a ✓ 4 21.g even 6 2
1008.1.dc.a 4 28.f even 6 2
1008.1.dc.a 4 84.j odd 6 2
3528.1.d.a 2 7.b odd 2 1
3528.1.d.a 2 21.c even 2 1
3528.1.d.b 2 1.a even 1 1 trivial
3528.1.d.b 2 3.b odd 2 1 inner
3528.1.cu.a 4 7.c even 3 2
3528.1.cu.a 4 21.h odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13} - 1 \) acting on \(S_{1}^{\mathrm{new}}(3528, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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