Properties

Label 2304.2.k.b
Level $2304$
Weight $2$
Character orbit 2304.k
Analytic conductor $18.398$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 768)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{7} + (\beta_1 - 1) q^{13} - 6 q^{17} + (3 \beta_{3} - 3 \beta_{2}) q^{19} + 6 \beta_{2} q^{23} - 5 \beta_1 q^{25} + ( - 6 \beta_1 + 6) q^{29} + \beta_{3} q^{31} + ( - 5 \beta_1 - 5) q^{37} - 6 \beta_1 q^{41} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{43} - 6 \beta_{3} q^{47} + 5 q^{49} + ( - 6 \beta_1 - 6) q^{53} + (5 \beta_1 - 5) q^{61} + (2 \beta_{3} - 2 \beta_{2}) q^{67} + 6 \beta_{2} q^{71} - \beta_{3} q^{79} + (6 \beta_{3} - 6 \beta_{2}) q^{83} - 6 \beta_1 q^{89} + (\beta_{3} + \beta_{2}) q^{91} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{13} - 24 q^{17} + 24 q^{29} - 20 q^{37} + 20 q^{49} - 24 q^{53} - 20 q^{61} - 48 q^{97}+O(q^{100}) \) 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
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) 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

Character values

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

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{1}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
0 0 0 0 0 1.41421i 0 0 0
577.2 0 0 0 0 0 1.41421i 0 0 0
1729.1 0 0 0 0 0 1.41421i 0 0 0
1729.2 0 0 0 0 0 1.41421i 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
4.b odd 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.k.b 4
3.b odd 2 1 768.2.j.b 4
4.b odd 2 1 inner 2304.2.k.b 4
8.b even 2 1 2304.2.k.c 4
8.d odd 2 1 2304.2.k.c 4
12.b even 2 1 768.2.j.b 4
16.e even 4 1 inner 2304.2.k.b 4
16.e even 4 1 2304.2.k.c 4
16.f odd 4 1 inner 2304.2.k.b 4
16.f odd 4 1 2304.2.k.c 4
24.f even 2 1 768.2.j.c yes 4
24.h odd 2 1 768.2.j.c yes 4
32.g even 8 1 9216.2.a.n 2
32.g even 8 1 9216.2.a.o 2
32.h odd 8 1 9216.2.a.n 2
32.h odd 8 1 9216.2.a.o 2
48.i odd 4 1 768.2.j.b 4
48.i odd 4 1 768.2.j.c yes 4
48.k even 4 1 768.2.j.b 4
48.k even 4 1 768.2.j.c yes 4
96.o even 8 1 3072.2.a.a 2
96.o even 8 1 3072.2.a.g 2
96.o even 8 2 3072.2.d.a 4
96.p odd 8 1 3072.2.a.a 2
96.p odd 8 1 3072.2.a.g 2
96.p odd 8 2 3072.2.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.b 4 3.b odd 2 1
768.2.j.b 4 12.b even 2 1
768.2.j.b 4 48.i odd 4 1
768.2.j.b 4 48.k even 4 1
768.2.j.c yes 4 24.f even 2 1
768.2.j.c yes 4 24.h odd 2 1
768.2.j.c yes 4 48.i odd 4 1
768.2.j.c yes 4 48.k even 4 1
2304.2.k.b 4 1.a even 1 1 trivial
2304.2.k.b 4 4.b odd 2 1 inner
2304.2.k.b 4 16.e even 4 1 inner
2304.2.k.b 4 16.f odd 4 1 inner
2304.2.k.c 4 8.b even 2 1
2304.2.k.c 4 8.d odd 2 1
2304.2.k.c 4 16.e even 4 1
2304.2.k.c 4 16.f odd 4 1
3072.2.a.a 2 96.o even 8 1
3072.2.a.a 2 96.p odd 8 1
3072.2.a.g 2 96.o even 8 1
3072.2.a.g 2 96.p odd 8 1
3072.2.d.a 4 96.o even 8 2
3072.2.d.a 4 96.p odd 8 2
9216.2.a.n 2 32.g even 8 1
9216.2.a.n 2 32.h odd 8 1
9216.2.a.o 2 32.g even 8 1
9216.2.a.o 2 32.h odd 8 1

Hecke kernels

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

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} + 2 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} + 2 \) 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
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$17$ \( (T + 6)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 1296 \) Copy content Toggle raw display
$23$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 12 T + 72)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 1296 \) Copy content Toggle raw display
$47$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 12 T + 72)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 256 \) Copy content Toggle raw display
$71$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 20736 \) Copy content Toggle raw display
$89$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$97$ \( (T + 12)^{4} \) Copy content Toggle raw display
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