Properties

Label 2304.2.l.b
Level $2304$
Weight $2$
Character orbit 2304.l
Analytic conductor $18.398$
Analytic rank $0$
Dimension $8$
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.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{5} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{7} +O(q^{10})\) \( q + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{5} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{7} + ( -2 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{11} + ( -1 + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{13} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{17} + ( 4 \zeta_{24} - 2 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{19} + ( -2 + 4 \zeta_{24}^{4} ) q^{23} + ( 2 - 4 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{25} + ( -5 \zeta_{24} - \zeta_{24}^{3} + 5 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{29} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} - \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{31} + ( 2 - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{35} + ( -3 - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{37} + ( 5 \zeta_{24} + 5 \zeta_{24}^{3} + \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{41} + ( 4 \zeta_{24} + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{43} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{47} -5 q^{49} + ( -5 \zeta_{24} + 3 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{53} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{55} + ( -2 - 8 \zeta_{24}^{2} + 8 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{59} + ( 3 + 3 \zeta_{24}^{6} ) q^{61} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{65} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{67} + 14 \zeta_{24}^{6} q^{71} + ( 4 - 8 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{73} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{77} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 7 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{79} + ( 10 - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + 10 \zeta_{24}^{6} ) q^{83} + ( 2 - 6 \zeta_{24}^{2} + 6 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{85} + ( 5 \zeta_{24} + 5 \zeta_{24}^{3} + 7 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{89} + ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{91} + ( 10 - 12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{95} + ( -8 + 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 8q^{11} + 8q^{35} - 8q^{37} - 40q^{49} + 16q^{59} + 24q^{61} + 72q^{83} + 40q^{85} + 80q^{95} - 64q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{24}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
0.258819 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 0.965926i
0 0 0 −1.93185 + 1.93185i 0 −1.41421 0 0 0
575.2 0 0 0 −0.517638 + 0.517638i 0 1.41421 0 0 0
575.3 0 0 0 0.517638 0.517638i 0 −1.41421 0 0 0
575.4 0 0 0 1.93185 1.93185i 0 1.41421 0 0 0
1727.1 0 0 0 −1.93185 1.93185i 0 −1.41421 0 0 0
1727.2 0 0 0 −0.517638 0.517638i 0 1.41421 0 0 0
1727.3 0 0 0 0.517638 + 0.517638i 0 −1.41421 0 0 0
1727.4 0 0 0 1.93185 + 1.93185i 0 1.41421 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1727.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner
16.e even 4 1 inner
48.k even 4 1 inner

Twists

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

Hecke kernels

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

\( T_{5}^{8} + 56 T_{5}^{4} + 16 \)
\( T_{11}^{4} + 4 T_{11}^{3} + 8 T_{11}^{2} - 16 T_{11} + 16 \)
\( T_{13}^{4} + 36 \)
\( T_{37}^{4} + 4 T_{37}^{3} + 8 T_{37}^{2} - 88 T_{37} + 484 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 4 T^{4} - 474 T^{8} - 2500 T^{12} + 390625 T^{16} \)
$7$ \( ( 1 + 12 T^{2} + 49 T^{4} )^{4} \)
$11$ \( ( 1 + 4 T + 8 T^{2} + 28 T^{3} + 82 T^{4} + 308 T^{5} + 968 T^{6} + 5324 T^{7} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 + 62 T^{4} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 40 T^{2} + 786 T^{4} - 11560 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( 1 + 292 T^{4} - 25242 T^{8} + 38053732 T^{12} + 16983563041 T^{16} \)
$23$ \( ( 1 - 34 T^{2} + 529 T^{4} )^{4} \)
$29$ \( 1 - 964 T^{4} + 566886 T^{8} - 681818884 T^{12} + 500246412961 T^{16} \)
$31$ \( ( 1 - 40 T^{2} + 594 T^{4} - 38440 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 4 T + 8 T^{2} + 60 T^{3} - 34 T^{4} + 2220 T^{5} + 10952 T^{6} + 202612 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 + 40 T^{2} + 2034 T^{4} + 67240 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( 1 - 3548 T^{4} + 6433446 T^{8} - 12129905948 T^{12} + 11688200277601 T^{16} \)
$47$ \( ( 1 - 14 T^{2} + 2209 T^{4} )^{4} \)
$53$ \( 1 - 4868 T^{4} + 16478118 T^{8} - 38410861508 T^{12} + 62259690411361 T^{16} \)
$59$ \( ( 1 - 8 T + 32 T^{2} + 232 T^{3} - 6062 T^{4} + 13688 T^{5} + 111392 T^{6} - 1643032 T^{7} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 6 T + 18 T^{2} - 366 T^{3} + 3721 T^{4} )^{4} \)
$67$ \( 1 + 10276 T^{4} + 56007654 T^{8} + 207072919396 T^{12} + 406067677556641 T^{16} \)
$71$ \( ( 1 + 54 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 188 T^{2} + 18726 T^{4} - 1001852 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 168 T^{2} + 14738 T^{4} - 1048488 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 - 36 T + 648 T^{2} - 8604 T^{3} + 89906 T^{4} - 714132 T^{5} + 4464072 T^{6} - 20584332 T^{7} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 80 T^{2} + 15714 T^{4} - 633680 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 + 16 T + 150 T^{2} + 1552 T^{3} + 9409 T^{4} )^{4} \)
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