Properties

Label 2304.3.b.k
Level $2304$
Weight $3$
Character orbit 2304.b
Analytic conductor $62.779$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4 - 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} + ( -4 + 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{7} +O(q^{10})\) \( q + ( 4 - 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} + ( -4 + 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{7} + ( -8 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{11} + ( 8 - 16 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{13} + ( 6 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{17} + ( 24 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{19} + 8 \zeta_{12}^{3} q^{23} + ( -27 + 32 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{25} + ( 4 - 8 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{29} + ( -20 + 40 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{31} + ( 56 - 48 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{35} + ( 16 - 32 \zeta_{12}^{2} + 18 \zeta_{12}^{3} ) q^{37} + ( -22 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{41} + ( -40 + 40 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{43} + ( 8 - 16 \zeta_{12}^{2} - 56 \zeta_{12}^{3} ) q^{47} + ( -15 + 64 \zeta_{12} - 32 \zeta_{12}^{3} ) q^{49} + ( -4 + 8 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{53} + ( -24 + 48 \zeta_{12}^{2} + 32 \zeta_{12}^{3} ) q^{55} + ( -32 - 88 \zeta_{12} + 44 \zeta_{12}^{3} ) q^{59} + 14 \zeta_{12}^{3} q^{61} + ( -76 - 48 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{65} + ( -32 + 56 \zeta_{12} - 28 \zeta_{12}^{3} ) q^{67} + ( 48 - 96 \zeta_{12}^{2} + 40 \zeta_{12}^{3} ) q^{71} + ( 30 - 64 \zeta_{12} + 32 \zeta_{12}^{3} ) q^{73} + ( 16 - 32 \zeta_{12}^{2} - 16 \zeta_{12}^{3} ) q^{77} + ( 12 - 24 \zeta_{12}^{2} - 76 \zeta_{12}^{3} ) q^{79} + ( -56 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{83} + ( 8 - 16 \zeta_{12}^{2} - 84 \zeta_{12}^{3} ) q^{85} + ( -78 + 32 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{89} + ( 56 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{91} + ( 88 - 176 \zeta_{12}^{2} ) q^{95} + ( -62 - 96 \zeta_{12} + 48 \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 32q^{11} + 24q^{17} + 96q^{19} - 108q^{25} + 224q^{35} - 88q^{41} - 160q^{43} - 60q^{49} - 128q^{59} - 304q^{65} - 128q^{67} + 120q^{73} - 224q^{83} - 312q^{89} + 224q^{91} - 248q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0 0 8.92820i 0 10.9282i 0 0 0
127.2 0 0 0 4.92820i 0 2.92820i 0 0 0
127.3 0 0 0 4.92820i 0 2.92820i 0 0 0
127.4 0 0 0 8.92820i 0 10.9282i 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.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.b.k 4
3.b odd 2 1 768.3.b.d 4
4.b odd 2 1 2304.3.b.o 4
8.b even 2 1 2304.3.b.o 4
8.d odd 2 1 inner 2304.3.b.k 4
12.b even 2 1 768.3.b.a 4
16.e even 4 1 288.3.g.d 4
16.e even 4 1 576.3.g.j 4
16.f odd 4 1 288.3.g.d 4
16.f odd 4 1 576.3.g.j 4
24.f even 2 1 768.3.b.d 4
24.h odd 2 1 768.3.b.a 4
48.i odd 4 1 96.3.g.a 4
48.i odd 4 1 192.3.g.c 4
48.k even 4 1 96.3.g.a 4
48.k even 4 1 192.3.g.c 4
240.t even 4 1 2400.3.e.a 4
240.z odd 4 1 2400.3.j.a 4
240.bb even 4 1 2400.3.j.b 4
240.bd odd 4 1 2400.3.j.b 4
240.bf even 4 1 2400.3.j.a 4
240.bm odd 4 1 2400.3.e.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.3.g.a 4 48.i odd 4 1
96.3.g.a 4 48.k even 4 1
192.3.g.c 4 48.i odd 4 1
192.3.g.c 4 48.k even 4 1
288.3.g.d 4 16.e even 4 1
288.3.g.d 4 16.f odd 4 1
576.3.g.j 4 16.e even 4 1
576.3.g.j 4 16.f odd 4 1
768.3.b.a 4 12.b even 2 1
768.3.b.a 4 24.h odd 2 1
768.3.b.d 4 3.b odd 2 1
768.3.b.d 4 24.f even 2 1
2304.3.b.k 4 1.a even 1 1 trivial
2304.3.b.k 4 8.d odd 2 1 inner
2304.3.b.o 4 4.b odd 2 1
2304.3.b.o 4 8.b even 2 1
2400.3.e.a 4 240.t even 4 1
2400.3.e.a 4 240.bm odd 4 1
2400.3.j.a 4 240.z odd 4 1
2400.3.j.a 4 240.bf even 4 1
2400.3.j.b 4 240.bb even 4 1
2400.3.j.b 4 240.bd odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{4} + 104 T_{5}^{2} + 1936 \)
\( T_{7}^{4} + 128 T_{7}^{2} + 1024 \)
\( T_{11}^{2} + 16 T_{11} + 16 \)
\( T_{17}^{2} - 12 T_{17} - 156 \)
\( T_{19}^{2} - 48 T_{19} + 528 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 1936 + 104 T^{2} + T^{4} \)
$7$ \( 1024 + 128 T^{2} + T^{4} \)
$11$ \( ( 16 + 16 T + T^{2} )^{2} \)
$13$ \( 8464 + 584 T^{2} + T^{4} \)
$17$ \( ( -156 - 12 T + T^{2} )^{2} \)
$19$ \( ( 528 - 48 T + T^{2} )^{2} \)
$23$ \( ( 64 + T^{2} )^{2} \)
$29$ \( 2704 + 296 T^{2} + T^{4} \)
$31$ \( 1401856 + 2432 T^{2} + T^{4} \)
$37$ \( 197136 + 2184 T^{2} + T^{4} \)
$41$ \( ( 292 + 44 T + T^{2} )^{2} \)
$43$ \( ( 400 + 80 T + T^{2} )^{2} \)
$47$ \( 8667136 + 6656 T^{2} + T^{4} \)
$53$ \( 144 + 168 T^{2} + T^{4} \)
$59$ \( ( -4784 + 64 T + T^{2} )^{2} \)
$61$ \( ( 196 + T^{2} )^{2} \)
$67$ \( ( -1328 + 64 T + T^{2} )^{2} \)
$71$ \( 28217344 + 17024 T^{2} + T^{4} \)
$73$ \( ( -2172 - 60 T + T^{2} )^{2} \)
$79$ \( 28558336 + 12416 T^{2} + T^{4} \)
$83$ \( ( 3088 + 112 T + T^{2} )^{2} \)
$89$ \( ( 5316 + 156 T + T^{2} )^{2} \)
$97$ \( ( -3068 + 124 T + T^{2} )^{2} \)
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