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: \(x^{4} + 1\)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{8}\)

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 \(S_{2}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5} \)
\( T_{7}^{2} + 2 \)
\( T_{13}^{2} + 2 T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 12 T^{2} + 49 T^{4} )^{2} \)
$11$ \( ( 1 + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 + 6 T + 17 T^{2} )^{4} \)
$19$ \( 1 - 718 T^{4} + 130321 T^{8} \)
$23$ \( ( 1 + 26 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 12 T + 72 T^{2} - 348 T^{3} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 60 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 - 46 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( 1 - 1198 T^{4} + 3418801 T^{8} \)
$47$ \( ( 1 + 22 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 12 T + 72 T^{2} + 636 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 10 T + 50 T^{2} + 610 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 + 4946 T^{4} + 20151121 T^{8} \)
$71$ \( ( 1 - 70 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{4} \)
$79$ \( ( 1 + 156 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( 1 - 13294 T^{4} + 47458321 T^{8} \)
$89$ \( ( 1 - 142 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 12 T + 97 T^{2} )^{4} \)
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