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Properties

Label 3200.2.c.y
Level $3200$
Weight $2$
Character orbit 3200.c
Analytic conductor $25.552$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3200,2,Mod(2049,3200)] 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("3200.2049"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,12,0,-8,0,0,0,0,0,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{3} + (2 \beta_{2} - 2 \beta_1) q^{7} - 2 \beta_{3} q^{9} + ( - 3 \beta_{3} - 1) q^{11} + 4 \beta_{2} q^{13} + (2 \beta_{2} - 3 \beta_1) q^{17} + ( - \beta_{3} + 3) q^{19} - 2 q^{21}+ \cdots + (2 \beta_{3} + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{11} + 12 q^{19} - 8 q^{21} - 32 q^{29} + 8 q^{31} - 32 q^{39} + 20 q^{41} - 20 q^{49} - 4 q^{51} + 8 q^{59} + 24 q^{61} - 24 q^{69} - 16 q^{71} - 40 q^{79} - 4 q^{81} - 36 q^{89} - 64 q^{91}+ \cdots + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display

Display \(\zeta_{8}^j\) in terms of \(\beta_i\)

\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{8}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2049.1
0.707107 − 0.707107i
−0.707107 − 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0 − 2.41421i 0 0 0 − 0.828427i 0 −2.82843 0
2049.2 0 − 0.414214i 0 0 0 − 4.82843i 0 2.82843 0
2049.3 0 0.414214i 0 0 0 4.82843i 0 2.82843 0
2049.4 0 2.41421i 0 0 0 0.828427i 0 −2.82843 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.2.c.y 4
4.b odd 2 1 3200.2.c.ba 4
5.b even 2 1 inner 3200.2.c.y 4
5.c odd 4 1 3200.2.a.bc ✓ 2
5.c odd 4 1 3200.2.a.bm yes 2
8.b even 2 1 3200.2.c.bb 4
8.d odd 2 1 3200.2.c.z 4
20.d odd 2 1 3200.2.c.ba 4
20.e even 4 1 3200.2.a.bd yes 2
20.e even 4 1 3200.2.a.bn yes 2
40.e odd 2 1 3200.2.c.z 4
40.f even 2 1 3200.2.c.bb 4
40.i odd 4 1 3200.2.a.bh yes 2
40.i odd 4 1 3200.2.a.bj yes 2
40.k even 4 1 3200.2.a.bg yes 2
40.k even 4 1 3200.2.a.bi yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.a.bc ✓ 2 5.c odd 4 1
3200.2.a.bd yes 2 20.e even 4 1
3200.2.a.bg yes 2 40.k even 4 1
3200.2.a.bh yes 2 40.i odd 4 1
3200.2.a.bi yes 2 40.k even 4 1
3200.2.a.bj yes 2 40.i odd 4 1
3200.2.a.bm yes 2 5.c odd 4 1
3200.2.a.bn yes 2 20.e even 4 1
3200.2.c.y 4 1.a even 1 1 trivial
3200.2.c.y 4 5.b even 2 1 inner
3200.2.c.z 4 8.d odd 2 1
3200.2.c.z 4 40.e odd 2 1
3200.2.c.ba 4 4.b odd 2 1
3200.2.c.ba 4 20.d odd 2 1
3200.2.c.bb 4 8.b even 2 1
3200.2.c.bb 4 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3200, [\chi])\):

\( T_{3}^{4} + 6T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 24T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 17 \) Copy content Toggle raw display
\( T_{29} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T - 17)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 34T^{2} + 1 \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T + 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T + 8)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$41$ \( (T^{2} - 10 T - 7)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 96T^{2} + 256 \) Copy content Toggle raw display
$53$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$59$ \( (T^{2} - 4 T - 28)^{2} \) Copy content Toggle raw display
$61$ \( (T - 6)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 134T^{2} + 961 \) Copy content Toggle raw display
$71$ \( (T^{2} + 8 T - 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 34T^{2} + 1 \) Copy content Toggle raw display
$79$ \( (T^{2} + 20 T + 92)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 54T^{2} + 81 \) Copy content Toggle raw display
$89$ \( (T^{2} + 18 T + 73)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 328T^{2} + 8464 \) Copy content Toggle raw display
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