Properties

Label 2304.2.l.h
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: 8.0.157351936.1
Defining polynomial: \(x^{8} + x^{4} + 16\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{4} ) q^{5} + ( \beta_{1} - \beta_{7} ) q^{7} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{4} ) q^{5} + ( \beta_{1} - \beta_{7} ) q^{7} + ( 1 - \beta_{2} + \beta_{6} ) q^{11} + ( 2 + 2 \beta_{2} + \beta_{5} ) q^{13} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} ) q^{17} + ( \beta_{3} - \beta_{4} ) q^{19} + 6 \beta_{2} q^{23} + ( 3 \beta_{2} + \beta_{5} - \beta_{6} ) q^{25} + ( 2 \beta_{3} - 2 \beta_{4} + \beta_{7} ) q^{29} + ( \beta_{1} - \beta_{3} + 3 \beta_{4} + \beta_{7} ) q^{31} + ( -7 - 7 \beta_{2} - \beta_{5} ) q^{35} + ( 1 - \beta_{2} ) q^{37} + ( -\beta_{1} + 2 \beta_{3} + \beta_{7} ) q^{41} + ( -2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{43} + 6 q^{47} + 7 q^{49} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{53} + ( -2 \beta_{1} - 8 \beta_{3} + 2 \beta_{7} ) q^{55} + ( 4 - 4 \beta_{2} - 2 \beta_{6} ) q^{59} + ( 5 + 5 \beta_{2} - 2 \beta_{5} ) q^{61} + ( -3 \beta_{1} + 3 \beta_{3} - 12 \beta_{4} - 3 \beta_{7} ) q^{65} + ( 4 \beta_{3} - 4 \beta_{4} + 2 \beta_{7} ) q^{67} + ( 8 \beta_{2} + \beta_{5} - \beta_{6} ) q^{71} + ( 2 \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{73} + ( 8 \beta_{3} - 8 \beta_{4} - 2 \beta_{7} ) q^{77} + ( -3 \beta_{1} + 3 \beta_{3} - \beta_{4} - 3 \beta_{7} ) q^{79} + ( -3 - 3 \beta_{2} + 3 \beta_{5} ) q^{83} + ( 5 - 5 \beta_{2} - \beta_{6} ) q^{85} + ( 2 \beta_{1} + 5 \beta_{3} - 2 \beta_{7} ) q^{89} + ( 4 \beta_{1} + 5 \beta_{3} + 9 \beta_{4} ) q^{91} + ( -2 - \beta_{5} - \beta_{6} ) q^{95} + ( 8 + \beta_{5} + \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{11} + 16q^{13} - 56q^{35} + 8q^{37} + 48q^{47} + 56q^{49} + 32q^{59} + 40q^{61} - 24q^{83} + 40q^{85} - 16q^{95} + 64q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 5 \nu^{2} \)\()/12\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - 4 \nu^{5} + 7 \nu^{3} + 4 \nu \)\()/24\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} + 7 \nu^{3} - 4 \nu \)\()/24\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{6} + 8 \nu^{4} + 9 \nu^{2} + 4 \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{6} - 8 \nu^{4} + 9 \nu^{2} - 4 \)\()/12\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{7} - 4 \nu^{5} + 13 \nu^{3} + 4 \nu \)\()/24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + \beta_{5} + 6 \beta_{2}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 3 \beta_{4} + 2 \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-3 \beta_{6} + 3 \beta_{5} - 2\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(6 \beta_{4} - 6 \beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{6} - 5 \beta_{5} + 18 \beta_{2}\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(7 \beta_{7} - 3 \beta_{4} - 10 \beta_{3}\)\()/2\)

Character values

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

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(-1\) \(-\beta_{2}\)

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
1.28897 0.581861i
0.581861 1.28897i
−0.581861 + 1.28897i
−1.28897 + 0.581861i
1.28897 + 0.581861i
0.581861 + 1.28897i
−0.581861 1.28897i
−1.28897 0.581861i
0 0 0 −2.57794 + 2.57794i 0 3.74166 0 0 0
575.2 0 0 0 −1.16372 + 1.16372i 0 3.74166 0 0 0
575.3 0 0 0 1.16372 1.16372i 0 −3.74166 0 0 0
575.4 0 0 0 2.57794 2.57794i 0 −3.74166 0 0 0
1727.1 0 0 0 −2.57794 2.57794i 0 3.74166 0 0 0
1727.2 0 0 0 −1.16372 1.16372i 0 3.74166 0 0 0
1727.3 0 0 0 1.16372 + 1.16372i 0 −3.74166 0 0 0
1727.4 0 0 0 2.57794 + 2.57794i 0 −3.74166 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.h yes 8
3.b odd 2 1 2304.2.l.d yes 8
4.b odd 2 1 2304.2.l.d yes 8
8.b even 2 1 2304.2.l.a 8
8.d odd 2 1 2304.2.l.e yes 8
12.b even 2 1 inner 2304.2.l.h yes 8
16.e even 4 1 2304.2.l.a 8
16.e even 4 1 inner 2304.2.l.h yes 8
16.f odd 4 1 2304.2.l.d yes 8
16.f odd 4 1 2304.2.l.e yes 8
24.f even 2 1 2304.2.l.a 8
24.h odd 2 1 2304.2.l.e yes 8
48.i odd 4 1 2304.2.l.d yes 8
48.i odd 4 1 2304.2.l.e yes 8
48.k even 4 1 2304.2.l.a 8
48.k even 4 1 inner 2304.2.l.h yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2304.2.l.a 8 8.b even 2 1
2304.2.l.a 8 16.e even 4 1
2304.2.l.a 8 24.f even 2 1
2304.2.l.a 8 48.k even 4 1
2304.2.l.d yes 8 3.b odd 2 1
2304.2.l.d yes 8 4.b odd 2 1
2304.2.l.d yes 8 16.f odd 4 1
2304.2.l.d yes 8 48.i odd 4 1
2304.2.l.e yes 8 8.d odd 2 1
2304.2.l.e yes 8 16.f odd 4 1
2304.2.l.e yes 8 24.h odd 2 1
2304.2.l.e yes 8 48.i odd 4 1
2304.2.l.h yes 8 1.a even 1 1 trivial
2304.2.l.h yes 8 12.b even 2 1 inner
2304.2.l.h yes 8 16.e even 4 1 inner
2304.2.l.h yes 8 48.k even 4 1 inner

Hecke kernels

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

\( T_{5}^{8} + 184 T_{5}^{4} + 1296 \)
\( T_{11}^{4} - 4 T_{11}^{3} + 8 T_{11}^{2} + 48 T_{11} + 144 \)
\( T_{13}^{4} - 8 T_{13}^{3} + 32 T_{13}^{2} + 48 T_{13} + 36 \)
\( T_{37}^{2} - 2 T_{37} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 36 T^{4} + 1126 T^{8} - 22500 T^{12} + 390625 T^{16} \)
$7$ \( ( 1 + 49 T^{4} )^{4} \)
$11$ \( ( 1 - 4 T + 8 T^{2} + 4 T^{3} - 142 T^{4} + 44 T^{5} + 968 T^{6} - 5324 T^{7} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 8 T + 32 T^{2} - 56 T^{3} + 62 T^{4} - 728 T^{5} + 5408 T^{6} - 17576 T^{7} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 24 T^{2} + 274 T^{4} - 6936 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 12 T + 72 T^{2} - 228 T^{3} + 361 T^{4} )^{2}( 1 + 12 T + 72 T^{2} + 228 T^{3} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 10 T^{2} + 529 T^{4} )^{4} \)
$29$ \( 1 - 612 T^{4} - 384602 T^{8} - 432855972 T^{12} + 500246412961 T^{16} \)
$31$ \( ( 1 - 80 T^{2} + 3074 T^{4} - 76880 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 - 2 T + 2 T^{2} - 74 T^{3} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 + 120 T^{2} + 6514 T^{4} + 201720 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 334 T^{4} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 - 6 T + 47 T^{2} )^{8} \)
$53$ \( 1 - 4388 T^{4} + 10107366 T^{8} - 34623430628 T^{12} + 62259690411361 T^{16} \)
$59$ \( ( 1 - 16 T + 128 T^{2} - 560 T^{3} + 1874 T^{4} - 33040 T^{5} + 445568 T^{6} - 3286064 T^{7} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 20 T + 200 T^{2} - 1100 T^{3} + 6014 T^{4} - 67100 T^{5} + 744200 T^{6} - 4539620 T^{7} + 13845841 T^{8} )^{2} \)
$67$ \( 1 + 4516 T^{4} + 43788006 T^{8} + 91002462436 T^{12} + 406067677556641 T^{16} \)
$71$ \( ( 1 - 100 T^{2} + 5414 T^{4} - 504100 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 60 T^{2} + 9766 T^{4} - 319740 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 48 T^{2} + 9026 T^{4} - 299568 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 + 12 T + 72 T^{2} - 300 T^{3} - 10414 T^{4} - 24900 T^{5} + 496008 T^{6} + 6861444 T^{7} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 144 T^{2} + 9826 T^{4} + 1140624 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 16 T + 230 T^{2} - 1552 T^{3} + 9409 T^{4} )^{4} \)
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