Properties

Label 2304.2.f.g
Level $2304$
Weight $2$
Character orbit 2304.f
Analytic conductor $18.398$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1152)
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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{7} +O(q^{10})\) \( q + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{7} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{11} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{13} + ( -\zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{17} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{19} + ( -4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{23} + ( 1 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{25} + ( -2 + \zeta_{8} - \zeta_{8}^{3} ) q^{29} + ( -2 \zeta_{8} + 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{31} + ( 6 \zeta_{8} + 8 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{35} + ( -4 \zeta_{8} + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{37} + ( -\zeta_{8} + 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{41} + ( -4 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{43} + ( 4 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{47} + ( -5 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{49} + ( 2 - 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{53} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{55} + ( 4 \zeta_{8} - 8 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{59} + ( 4 \zeta_{8} + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{61} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{65} -12 q^{67} + ( -4 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{71} -4 q^{73} + ( -8 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{77} + ( -6 \zeta_{8} + 2 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{79} + ( 2 \zeta_{8} - 8 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{83} + ( -6 \zeta_{8} - 10 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{85} + ( 3 \zeta_{8} + 8 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{89} + ( -8 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{91} + ( 8 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{95} + ( 8 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{5} + O(q^{10}) \) \( 4q + 8q^{5} - 16q^{23} + 4q^{25} - 8q^{29} - 16q^{43} + 16q^{47} - 20q^{49} + 8q^{53} - 48q^{67} - 16q^{71} - 16q^{73} - 32q^{77} - 32q^{91} + 32q^{95} + 32q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
0 0 0 0.585786 0 0.828427i 0 0 0
1151.2 0 0 0 0.585786 0 0.828427i 0 0 0
1151.3 0 0 0 3.41421 0 4.82843i 0 0 0
1151.4 0 0 0 3.41421 0 4.82843i 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
24.f 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.f.g 4
3.b odd 2 1 2304.2.f.b 4
4.b odd 2 1 2304.2.f.h 4
8.b even 2 1 2304.2.f.a 4
8.d odd 2 1 2304.2.f.b 4
12.b even 2 1 2304.2.f.a 4
16.e even 4 1 1152.2.c.c yes 4
16.e even 4 1 1152.2.c.d yes 4
16.f odd 4 1 1152.2.c.a 4
16.f odd 4 1 1152.2.c.b yes 4
24.f even 2 1 inner 2304.2.f.g 4
24.h odd 2 1 2304.2.f.h 4
48.i odd 4 1 1152.2.c.a 4
48.i odd 4 1 1152.2.c.b yes 4
48.k even 4 1 1152.2.c.c yes 4
48.k even 4 1 1152.2.c.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.c.a 4 16.f odd 4 1
1152.2.c.a 4 48.i odd 4 1
1152.2.c.b yes 4 16.f odd 4 1
1152.2.c.b yes 4 48.i odd 4 1
1152.2.c.c yes 4 16.e even 4 1
1152.2.c.c yes 4 48.k even 4 1
1152.2.c.d yes 4 16.e even 4 1
1152.2.c.d yes 4 48.k even 4 1
2304.2.f.a 4 8.b even 2 1
2304.2.f.a 4 12.b even 2 1
2304.2.f.b 4 3.b odd 2 1
2304.2.f.b 4 8.d odd 2 1
2304.2.f.g 4 1.a even 1 1 trivial
2304.2.f.g 4 24.f even 2 1 inner
2304.2.f.h 4 4.b odd 2 1
2304.2.f.h 4 24.h odd 2 1

Hecke kernels

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

\( T_{5}^{2} - 4 T_{5} + 2 \)
\( T_{7}^{4} + 24 T_{7}^{2} + 16 \)
\( T_{23}^{2} + 8 T_{23} + 8 \)
\( T_{43}^{2} + 8 T_{43} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 4 T + 12 T^{2} - 20 T^{3} + 25 T^{4} )^{2} \)
$7$ \( 1 - 4 T^{2} - 26 T^{4} - 196 T^{6} + 2401 T^{8} \)
$11$ \( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 18 T^{2} + 169 T^{4} )^{2} \)
$17$ \( 1 - 32 T^{2} + 706 T^{4} - 9248 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 + 6 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 8 T + 54 T^{2} + 184 T^{3} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 4 T + 60 T^{2} + 116 T^{3} + 841 T^{4} )^{2} \)
$31$ \( 1 - 36 T^{2} + 1094 T^{4} - 34596 T^{6} + 923521 T^{8} \)
$37$ \( 1 - 76 T^{2} + 3670 T^{4} - 104044 T^{6} + 1874161 T^{8} \)
$41$ \( 1 - 128 T^{2} + 7330 T^{4} - 215168 T^{6} + 2825761 T^{8} \)
$43$ \( ( 1 + 8 T + 70 T^{2} + 344 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 8 T + 38 T^{2} - 376 T^{3} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 4 T + 60 T^{2} - 212 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( 1 - 44 T^{2} - 746 T^{4} - 153164 T^{6} + 12117361 T^{8} \)
$61$ \( 1 - 172 T^{2} + 14326 T^{4} - 640012 T^{6} + 13845841 T^{8} \)
$67$ \( ( 1 + 12 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 4 T + 73 T^{2} )^{4} \)
$79$ \( 1 - 164 T^{2} + 18054 T^{4} - 1023524 T^{6} + 38950081 T^{8} \)
$83$ \( 1 - 188 T^{2} + 20566 T^{4} - 1295132 T^{6} + 47458321 T^{8} \)
$89$ \( 1 - 192 T^{2} + 20450 T^{4} - 1520832 T^{6} + 62742241 T^{8} \)
$97$ \( ( 1 - 16 T + 226 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \)
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