Properties

Label 1152.5.b.i
Level 1152
Weight 5
Character orbit 1152.b
Analytic conductor 119.082
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1152.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(119.082197473\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + \beta_{3} q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + \beta_{3} q^{7} -11 \beta_{2} q^{11} -27 \beta_{1} q^{13} + 162 q^{17} -45 \beta_{2} q^{19} + 9 \beta_{3} q^{23} + 561 q^{25} + 163 \beta_{1} q^{29} -8 \beta_{3} q^{31} -64 \beta_{2} q^{35} -189 \beta_{1} q^{37} -1890 q^{41} -297 \beta_{2} q^{43} -18 \beta_{3} q^{47} -3743 q^{49} -247 \beta_{1} q^{53} -11 \beta_{3} q^{55} -231 \beta_{2} q^{59} + 297 \beta_{1} q^{61} + 1728 q^{65} + 171 \beta_{2} q^{67} + 99 \beta_{3} q^{71} + 2750 q^{73} -1056 \beta_{1} q^{77} -102 \beta_{3} q^{79} + 953 \beta_{2} q^{83} + 162 \beta_{1} q^{85} -2430 q^{89} + 1728 \beta_{2} q^{91} -45 \beta_{3} q^{95} + 7454 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 648q^{17} + 2244q^{25} - 7560q^{41} - 14972q^{49} + 6912q^{65} + 11000q^{73} - 9720q^{89} + 29816q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 8 \nu^{2} \)\(/3\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{3} + 12 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 32 \nu^{3} + 96 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 8 \beta_{2}\)\()/64\)
\(\nu^{2}\)\(=\)\(3 \beta_{1}\)\(/8\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{3} - 24 \beta_{2}\)\()/64\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
1.22474 1.22474i
−1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
0 0 0 8.00000i 0 78.3837i 0 0 0
703.2 0 0 0 8.00000i 0 78.3837i 0 0 0
703.3 0 0 0 8.00000i 0 78.3837i 0 0 0
703.4 0 0 0 8.00000i 0 78.3837i 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
8.b even 2 1 inner
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 1152.5.b.i 4
3.b odd 2 1 128.5.d.c 4
4.b odd 2 1 inner 1152.5.b.i 4
8.b even 2 1 inner 1152.5.b.i 4
8.d odd 2 1 inner 1152.5.b.i 4
12.b even 2 1 128.5.d.c 4
24.f even 2 1 128.5.d.c 4
24.h odd 2 1 128.5.d.c 4
48.i odd 4 1 256.5.c.c 2
48.i odd 4 1 256.5.c.f 2
48.k even 4 1 256.5.c.c 2
48.k even 4 1 256.5.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.5.d.c 4 3.b odd 2 1
128.5.d.c 4 12.b even 2 1
128.5.d.c 4 24.f even 2 1
128.5.d.c 4 24.h odd 2 1
256.5.c.c 2 48.i odd 4 1
256.5.c.c 2 48.k even 4 1
256.5.c.f 2 48.i odd 4 1
256.5.c.f 2 48.k even 4 1
1152.5.b.i 4 1.a even 1 1 trivial
1152.5.b.i 4 4.b odd 2 1 inner
1152.5.b.i 4 8.b even 2 1 inner
1152.5.b.i 4 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{2} + 64 \)
\( T_{17} - 162 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 1186 T^{2} + 390625 T^{4} )^{2} \)
$7$ \( ( 1 + 1342 T^{2} + 5764801 T^{4} )^{2} \)
$11$ \( ( 1 + 17666 T^{2} + 214358881 T^{4} )^{2} \)
$13$ \( ( 1 - 10466 T^{2} + 815730721 T^{4} )^{2} \)
$17$ \( ( 1 - 162 T + 83521 T^{2} )^{4} \)
$19$ \( ( 1 + 66242 T^{2} + 16983563041 T^{4} )^{2} \)
$23$ \( ( 1 - 62018 T^{2} + 78310985281 T^{4} )^{2} \)
$29$ \( ( 1 + 285854 T^{2} + 500246412961 T^{4} )^{2} \)
$31$ \( ( 1 - 1453826 T^{2} + 852891037441 T^{4} )^{2} \)
$37$ \( ( 1 - 1462178 T^{2} + 3512479453921 T^{4} )^{2} \)
$41$ \( ( 1 + 1890 T + 2825761 T^{2} )^{4} \)
$43$ \( ( 1 - 1630462 T^{2} + 11688200277601 T^{4} )^{2} \)
$47$ \( ( 1 - 7768706 T^{2} + 23811286661761 T^{4} )^{2} \)
$53$ \( ( 1 - 11876386 T^{2} + 62259690411361 T^{4} )^{2} \)
$59$ \( ( 1 + 19112066 T^{2} + 146830437604321 T^{4} )^{2} \)
$61$ \( ( 1 - 22046306 T^{2} + 191707312997281 T^{4} )^{2} \)
$67$ \( ( 1 + 37495106 T^{2} + 406067677556641 T^{4} )^{2} \)
$71$ \( ( 1 + 9393982 T^{2} + 645753531245761 T^{4} )^{2} \)
$73$ \( ( 1 - 2750 T + 28398241 T^{2} )^{4} \)
$79$ \( ( 1 - 13977986 T^{2} + 1517108809906561 T^{4} )^{2} \)
$83$ \( ( 1 + 7728578 T^{2} + 2252292232139041 T^{4} )^{2} \)
$89$ \( ( 1 + 2430 T + 62742241 T^{2} )^{4} \)
$97$ \( ( 1 - 7454 T + 88529281 T^{2} )^{4} \)
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