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Properties

Label 3276.2.z.d
Level $3276$
Weight $2$
Character orbit 3276.z
Analytic conductor $26.159$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3276,2,Mod(757,3276)] 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("3276.757"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3276, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3276.z (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-6,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.1589917022\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 364)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 2) q^{5} + (\beta_{2} + 1) q^{7} + (\beta_{3} - 5 \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{2} + 2 \beta_1 - 1) q^{13} + (\beta_{2} + 2 \beta_1 + 1) q^{17} + ( - 5 \beta_{2} - \beta_1 - 5) q^{19}+ \cdots + (10 \beta_{2} + 3 \beta_1 + 10) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{5} + 2 q^{7} + 9 q^{11} + 4 q^{17} - 11 q^{19} + 4 q^{23} + 2 q^{25} + 3 q^{29} - 16 q^{31} - 3 q^{35} - 4 q^{37} - 10 q^{41} + q^{43} - 32 q^{47} - 2 q^{49} - 7 q^{55} - 4 q^{59} - 13 q^{65}+ \cdots + 23 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 7 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} - 7 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3276\mathbb{Z}\right)^\times\).

\(n\) \(1639\) \(2017\) \(2341\) \(2549\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
1.15139 − 1.99426i
−0.651388 + 1.12824i
1.15139 + 1.99426i
−0.651388 − 1.12824i
0 0 0 −3.30278 0 0.500000 − 0.866025i 0 0 0
757.2 0 0 0 0.302776 0 0.500000 − 0.866025i 0 0 0
3025.1 0 0 0 −3.30278 0 0.500000 + 0.866025i 0 0 0
3025.2 0 0 0 0.302776 0 0.500000 + 0.866025i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3276.2.z.d 4
3.b odd 2 1 364.2.k.c ✓ 4
12.b even 2 1 1456.2.s.m 4
13.c even 3 1 inner 3276.2.z.d 4
21.c even 2 1 2548.2.k.f 4
21.g even 6 1 2548.2.i.l 4
21.g even 6 1 2548.2.l.i 4
21.h odd 6 1 2548.2.i.j 4
21.h odd 6 1 2548.2.l.k 4
39.h odd 6 1 4732.2.a.j 2
39.i odd 6 1 364.2.k.c ✓ 4
39.i odd 6 1 4732.2.a.k 2
39.k even 12 2 4732.2.g.g 4
156.p even 6 1 1456.2.s.m 4
273.r even 6 1 2548.2.l.i 4
273.s odd 6 1 2548.2.l.k 4
273.bf even 6 1 2548.2.i.l 4
273.bm odd 6 1 2548.2.i.j 4
273.bn even 6 1 2548.2.k.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.k.c ✓ 4 3.b odd 2 1
364.2.k.c ✓ 4 39.i odd 6 1
1456.2.s.m 4 12.b even 2 1
1456.2.s.m 4 156.p even 6 1
2548.2.i.j 4 21.h odd 6 1
2548.2.i.j 4 273.bm odd 6 1
2548.2.i.l 4 21.g even 6 1
2548.2.i.l 4 273.bf even 6 1
2548.2.k.f 4 21.c even 2 1
2548.2.k.f 4 273.bn even 6 1
2548.2.l.i 4 21.g even 6 1
2548.2.l.i 4 273.r even 6 1
2548.2.l.k 4 21.h odd 6 1
2548.2.l.k 4 273.s odd 6 1
3276.2.z.d 4 1.a even 1 1 trivial
3276.2.z.d 4 13.c even 3 1 inner
4732.2.a.j 2 39.h odd 6 1
4732.2.a.k 2 39.i odd 6 1
4732.2.g.g 4 39.k even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3276, [\chi])\):

\( T_{5}^{2} + 3T_{5} - 1 \) Copy content Toggle raw display
\( T_{11}^{4} - 9T_{11}^{3} + 64T_{11}^{2} - 153T_{11} + 289 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 9 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$13$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{4} + 11 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$41$ \( T^{4} + 10 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$43$ \( T^{4} - T^{3} + \cdots + 841 \) Copy content Toggle raw display
$47$ \( (T^{2} + 16 T + 51)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 117)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$61$ \( T^{4} + 52T^{2} + 2704 \) Copy content Toggle raw display
$67$ \( (T^{2} + 13 T + 169)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 2 T^{3} + \cdots + 13456 \) Copy content Toggle raw display
$73$ \( (T^{2} - 8 T - 36)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 16 T + 51)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 21 T^{3} + \cdots + 6561 \) Copy content Toggle raw display
$97$ \( T^{4} - 23 T^{3} + \cdots + 10609 \) Copy content Toggle raw display
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