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Properties

Label 3528.2.a.bc
Level $3528$
Weight $2$
Character orbit 3528.a
Self dual yes
Analytic conductor $28.171$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3528,2,Mod(1,3528)] 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("3528.1"); S:= CuspForms(chi, 2); 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, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-4,0,0,0,0,0,4,0,0,0,0,0,-12,0,8,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(28.1712218331\)
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 1176)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta - 2) q^{5} + (2 \beta + 2) q^{11} + 3 \beta q^{13} + (\beta - 6) q^{17} + ( - 2 \beta + 4) q^{19} + ( - 2 \beta + 2) q^{23} + ( - 4 \beta + 1) q^{25} + 2 \beta q^{29} + 2 \beta q^{31} + (4 \beta + 4) q^{37}+ \cdots + (3 \beta + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} + 4 q^{11} - 12 q^{17} + 8 q^{19} + 4 q^{23} + 2 q^{25} + 8 q^{37} - 12 q^{41} - 8 q^{47} + 4 q^{53} + 8 q^{61} + 12 q^{65} + 4 q^{71} + 24 q^{73} + 16 q^{79} + 8 q^{83} + 28 q^{85} - 20 q^{89}+ \cdots + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 0 0 −3.41421 0 0 0 0 0
1.2 0 0 0 −0.585786 0 0 0 0 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.2.a.bc 2
3.b odd 2 1 1176.2.a.l ✓ 2
4.b odd 2 1 7056.2.a.ce 2
7.b odd 2 1 3528.2.a.bm 2
7.c even 3 2 3528.2.s.bl 4
7.d odd 6 2 3528.2.s.bc 4
12.b even 2 1 2352.2.a.bg 2
21.c even 2 1 1176.2.a.m yes 2
21.g even 6 2 1176.2.q.m 4
21.h odd 6 2 1176.2.q.n 4
24.f even 2 1 9408.2.a.dh 2
24.h odd 2 1 9408.2.a.dv 2
28.d even 2 1 7056.2.a.cw 2
84.h odd 2 1 2352.2.a.z 2
84.j odd 6 2 2352.2.q.bg 4
84.n even 6 2 2352.2.q.ba 4
168.e odd 2 1 9408.2.a.ed 2
168.i even 2 1 9408.2.a.dr 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.l ✓ 2 3.b odd 2 1
1176.2.a.m yes 2 21.c even 2 1
1176.2.q.m 4 21.g even 6 2
1176.2.q.n 4 21.h odd 6 2
2352.2.a.z 2 84.h odd 2 1
2352.2.a.bg 2 12.b even 2 1
2352.2.q.ba 4 84.n even 6 2
2352.2.q.bg 4 84.j odd 6 2
3528.2.a.bc 2 1.a even 1 1 trivial
3528.2.a.bm 2 7.b odd 2 1
3528.2.s.bc 4 7.d odd 6 2
3528.2.s.bl 4 7.c even 3 2
7056.2.a.ce 2 4.b odd 2 1
7056.2.a.cw 2 28.d even 2 1
9408.2.a.dh 2 24.f even 2 1
9408.2.a.dr 2 168.i even 2 1
9408.2.a.dv 2 24.h odd 2 1
9408.2.a.ed 2 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3528))\):

\( T_{5}^{2} + 4T_{5} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 18 \) Copy content Toggle raw display
\( T_{23}^{2} - 4T_{23} - 4 \) 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
$5$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 18 \) Copy content Toggle raw display
$17$ \( T^{2} + 12T + 34 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$29$ \( T^{2} - 8 \) Copy content Toggle raw display
$31$ \( T^{2} - 8 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 18 \) Copy content Toggle raw display
$43$ \( T^{2} - 128 \) Copy content Toggle raw display
$47$ \( T^{2} + 8T - 56 \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 72 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T - 34 \) Copy content Toggle raw display
$67$ \( T^{2} - 128 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 68 \) Copy content Toggle raw display
$73$ \( T^{2} - 24T + 126 \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 32 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 20T + 82 \) Copy content Toggle raw display
$97$ \( T^{2} - 8T - 2 \) Copy content Toggle raw display
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