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Properties

Label 2304.2.d.b
Level $2304$
Weight $2$
Character orbit 2304.d
Analytic conductor $18.398$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2304,2,Mod(1153,2304)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2304.1153"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,-8] 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: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 128)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} - 4 q^{7} + \beta q^{11} + \beta q^{13} + 2 q^{17} - \beta q^{19} - 4 q^{23} + q^{25} + 3 \beta q^{29} - 4 \beta q^{35} - 5 \beta q^{37} - 6 q^{41} + 3 \beta q^{43} - 8 q^{47} + 9 q^{49} + \cdots - 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{7} + 4 q^{17} - 8 q^{23} + 2 q^{25} - 12 q^{41} - 16 q^{47} + 18 q^{49} - 8 q^{55} - 8 q^{65} - 24 q^{71} - 28 q^{73} + 16 q^{79} - 4 q^{89} + 8 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1153.1
− 1.00000i
1.00000i
0 0 0 − 2.00000i 0 −4.00000 0 0 0
1153.2 0 0 0 2.00000i 0 −4.00000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.d.b 2
3.b odd 2 1 256.2.b.a 2
4.b odd 2 1 2304.2.d.r 2
8.b even 2 1 inner 2304.2.d.b 2
8.d odd 2 1 2304.2.d.r 2
12.b even 2 1 256.2.b.c 2
16.e even 4 1 1152.2.a.h 1
16.e even 4 1 1152.2.a.r 1
16.f odd 4 1 1152.2.a.c 1
16.f odd 4 1 1152.2.a.m 1
24.f even 2 1 256.2.b.c 2
24.h odd 2 1 256.2.b.a 2
48.i odd 4 1 128.2.a.b yes 1
48.i odd 4 1 128.2.a.c yes 1
48.k even 4 1 128.2.a.a ✓ 1
48.k even 4 1 128.2.a.d yes 1
96.o even 8 4 1024.2.e.i 4
96.p odd 8 4 1024.2.e.m 4
240.t even 4 1 3200.2.a.h 1
240.t even 4 1 3200.2.a.x 1
240.z odd 4 1 3200.2.c.e 2
240.z odd 4 1 3200.2.c.l 2
240.bb even 4 1 3200.2.c.f 2
240.bb even 4 1 3200.2.c.k 2
240.bd odd 4 1 3200.2.c.e 2
240.bd odd 4 1 3200.2.c.l 2
240.bf even 4 1 3200.2.c.f 2
240.bf even 4 1 3200.2.c.k 2
240.bm odd 4 1 3200.2.a.e 1
240.bm odd 4 1 3200.2.a.u 1
336.v odd 4 1 6272.2.a.a 1
336.v odd 4 1 6272.2.a.h 1
336.y even 4 1 6272.2.a.b 1
336.y even 4 1 6272.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a ✓ 1 48.k even 4 1
128.2.a.b yes 1 48.i odd 4 1
128.2.a.c yes 1 48.i odd 4 1
128.2.a.d yes 1 48.k even 4 1
256.2.b.a 2 3.b odd 2 1
256.2.b.a 2 24.h odd 2 1
256.2.b.c 2 12.b even 2 1
256.2.b.c 2 24.f even 2 1
1024.2.e.i 4 96.o even 8 4
1024.2.e.m 4 96.p odd 8 4
1152.2.a.c 1 16.f odd 4 1
1152.2.a.h 1 16.e even 4 1
1152.2.a.m 1 16.f odd 4 1
1152.2.a.r 1 16.e even 4 1
2304.2.d.b 2 1.a even 1 1 trivial
2304.2.d.b 2 8.b even 2 1 inner
2304.2.d.r 2 4.b odd 2 1
2304.2.d.r 2 8.d odd 2 1
3200.2.a.e 1 240.bm odd 4 1
3200.2.a.h 1 240.t even 4 1
3200.2.a.u 1 240.bm odd 4 1
3200.2.a.x 1 240.t even 4 1
3200.2.c.e 2 240.z odd 4 1
3200.2.c.e 2 240.bd odd 4 1
3200.2.c.f 2 240.bb even 4 1
3200.2.c.f 2 240.bf even 4 1
3200.2.c.k 2 240.bb even 4 1
3200.2.c.k 2 240.bf even 4 1
3200.2.c.l 2 240.z odd 4 1
3200.2.c.l 2 240.bd odd 4 1
6272.2.a.a 1 336.v odd 4 1
6272.2.a.b 1 336.y even 4 1
6272.2.a.g 1 336.y even 4 1
6272.2.a.h 1 336.v odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 4 \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display
\( T_{23} + 4 \) 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
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 196 \) Copy content Toggle raw display
$61$ \( T^{2} + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 100 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 14)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T + 2)^{2} \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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