Properties

Label 384.5.h.f
Level $384$
Weight $5$
Character orbit 384.h
Analytic conductor $39.694$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 384.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.6940658242\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( 3 - \beta_{1} ) q^{3} -\beta_{2} q^{5} + \beta_{2} q^{7} + ( -63 - 6 \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( 3 - \beta_{1} ) q^{3} -\beta_{2} q^{5} + \beta_{2} q^{7} + ( -63 - 6 \beta_{1} ) q^{9} -134 q^{11} -\beta_{3} q^{13} + ( -3 \beta_{2} + \beta_{3} ) q^{15} + 28 \beta_{1} q^{17} + 58 \beta_{1} q^{19} + ( 3 \beta_{2} - \beta_{3} ) q^{21} -2 \beta_{3} q^{23} + 815 q^{25} + ( -621 + 45 \beta_{1} ) q^{27} -21 \beta_{2} q^{29} + 29 \beta_{2} q^{31} + ( -402 + 134 \beta_{1} ) q^{33} -1440 q^{35} + 5 \beta_{3} q^{37} + ( -72 \beta_{2} - 3 \beta_{3} ) q^{39} + 152 \beta_{1} q^{41} -50 \beta_{1} q^{43} + ( 63 \beta_{2} + 6 \beta_{3} ) q^{45} + 8 \beta_{3} q^{47} -961 q^{49} + ( 2016 + 84 \beta_{1} ) q^{51} + 43 \beta_{2} q^{53} + 134 \beta_{2} q^{55} + ( 4176 + 174 \beta_{1} ) q^{57} + 1382 q^{59} -\beta_{3} q^{61} + ( -63 \beta_{2} - 6 \beta_{3} ) q^{63} + 1440 \beta_{1} q^{65} + 22 \beta_{1} q^{67} + ( -144 \beta_{2} - 6 \beta_{3} ) q^{69} + 6 \beta_{3} q^{71} + 4894 q^{73} + ( 2445 - 815 \beta_{1} ) q^{75} -134 \beta_{2} q^{77} -39 \beta_{2} q^{79} + ( 1377 + 756 \beta_{1} ) q^{81} + 5914 q^{83} -28 \beta_{3} q^{85} + ( -63 \beta_{2} + 21 \beta_{3} ) q^{87} + 1068 \beta_{1} q^{89} -1440 \beta_{1} q^{91} + ( 87 \beta_{2} - 29 \beta_{3} ) q^{93} -58 \beta_{3} q^{95} + 5858 q^{97} + ( 8442 + 804 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{3} - 252q^{9} + O(q^{10}) \) \( 4q + 12q^{3} - 252q^{9} - 536q^{11} + 3260q^{25} - 2484q^{27} - 1608q^{33} - 5760q^{35} - 3844q^{49} + 8064q^{51} + 16704q^{57} + 5528q^{59} + 19576q^{73} + 9780q^{75} + 5508q^{81} + 23656q^{83} + 23432q^{97} + 33768q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 4 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{3} - 2 \nu \)
\(\beta_{2}\)\(=\)\( -4 \nu^{3} + 28 \nu \)
\(\beta_{3}\)\(=\)\( 144 \nu^{2} - 288 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1}\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 288\)\()/144\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{2} + 14 \beta_{1}\)\()/24\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.58114 + 0.707107i
−1.58114 + 0.707107i
1.58114 0.707107i
−1.58114 0.707107i
0 3.00000 8.48528i 0 −37.9473 0 37.9473 0 −63.0000 50.9117i 0
65.2 0 3.00000 8.48528i 0 37.9473 0 −37.9473 0 −63.0000 50.9117i 0
65.3 0 3.00000 + 8.48528i 0 −37.9473 0 37.9473 0 −63.0000 + 50.9117i 0
65.4 0 3.00000 + 8.48528i 0 37.9473 0 −37.9473 0 −63.0000 + 50.9117i 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
12.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.5.h.f yes 4
3.b odd 2 1 384.5.h.e 4
4.b odd 2 1 384.5.h.e 4
8.b even 2 1 384.5.h.e 4
8.d odd 2 1 inner 384.5.h.f yes 4
12.b even 2 1 inner 384.5.h.f yes 4
24.f even 2 1 384.5.h.e 4
24.h odd 2 1 inner 384.5.h.f yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.h.e 4 3.b odd 2 1
384.5.h.e 4 4.b odd 2 1
384.5.h.e 4 8.b even 2 1
384.5.h.e 4 24.f even 2 1
384.5.h.f yes 4 1.a even 1 1 trivial
384.5.h.f yes 4 8.d odd 2 1 inner
384.5.h.f yes 4 12.b even 2 1 inner
384.5.h.f yes 4 24.h 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 \(S_{5}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{2} - 1440 \)
\( T_{11} + 134 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 81 - 6 T + T^{2} )^{2} \)
$5$ \( ( -1440 + T^{2} )^{2} \)
$7$ \( ( -1440 + T^{2} )^{2} \)
$11$ \( ( 134 + T )^{4} \)
$13$ \( ( 103680 + T^{2} )^{2} \)
$17$ \( ( 56448 + T^{2} )^{2} \)
$19$ \( ( 242208 + T^{2} )^{2} \)
$23$ \( ( 414720 + T^{2} )^{2} \)
$29$ \( ( -635040 + T^{2} )^{2} \)
$31$ \( ( -1211040 + T^{2} )^{2} \)
$37$ \( ( 2592000 + T^{2} )^{2} \)
$41$ \( ( 1663488 + T^{2} )^{2} \)
$43$ \( ( 180000 + T^{2} )^{2} \)
$47$ \( ( 6635520 + T^{2} )^{2} \)
$53$ \( ( -2662560 + T^{2} )^{2} \)
$59$ \( ( -1382 + T )^{4} \)
$61$ \( ( 103680 + T^{2} )^{2} \)
$67$ \( ( 34848 + T^{2} )^{2} \)
$71$ \( ( 3732480 + T^{2} )^{2} \)
$73$ \( ( -4894 + T )^{4} \)
$79$ \( ( -2190240 + T^{2} )^{2} \)
$83$ \( ( -5914 + T )^{4} \)
$89$ \( ( 82124928 + T^{2} )^{2} \)
$97$ \( ( -5858 + T )^{4} \)
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