Properties

Label 384.5.h.e
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: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 3) q^{3} + \beta_{2} q^{5} + \beta_{2} q^{7} + (6 \beta_1 - 63) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 3) q^{3} + \beta_{2} q^{5} + \beta_{2} q^{7} + (6 \beta_1 - 63) q^{9} + 134 q^{11} - \beta_{3} q^{13} + ( - \beta_{3} - 3 \beta_{2}) q^{15} - 28 \beta_1 q^{17} + 58 \beta_1 q^{19} + ( - \beta_{3} - 3 \beta_{2}) q^{21} + 2 \beta_{3} q^{23} + 815 q^{25} + (45 \beta_1 + 621) q^{27} + 21 \beta_{2} q^{29} + 29 \beta_{2} q^{31} + ( - 134 \beta_1 - 402) q^{33} + 1440 q^{35} + 5 \beta_{3} q^{37} + (3 \beta_{3} - 72 \beta_{2}) q^{39} - 152 \beta_1 q^{41} - 50 \beta_1 q^{43} + (6 \beta_{3} - 63 \beta_{2}) q^{45} - 8 \beta_{3} q^{47} - 961 q^{49} + (84 \beta_1 - 2016) q^{51} - 43 \beta_{2} q^{53} + 134 \beta_{2} q^{55} + ( - 174 \beta_1 + 4176) q^{57} - 1382 q^{59} - \beta_{3} q^{61} + (6 \beta_{3} - 63 \beta_{2}) q^{63} - 1440 \beta_1 q^{65} + 22 \beta_1 q^{67} + ( - 6 \beta_{3} + 144 \beta_{2}) q^{69} - 6 \beta_{3} q^{71} + 4894 q^{73} + ( - 815 \beta_1 - 2445) q^{75} + 134 \beta_{2} q^{77} - 39 \beta_{2} q^{79} + ( - 756 \beta_1 + 1377) q^{81} - 5914 q^{83} - 28 \beta_{3} q^{85} + ( - 21 \beta_{3} - 63 \beta_{2}) q^{87} - 1068 \beta_1 q^{89} - 1440 \beta_1 q^{91} + ( - 29 \beta_{3} - 87 \beta_{2}) q^{93} + 58 \beta_{3} q^{95} + 5858 q^{97} + (804 \beta_1 - 8442) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} - 252 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} - 252 q^{9} + 536 q^{11} + 3260 q^{25} + 2484 q^{27} - 1608 q^{33} + 5760 q^{35} - 3844 q^{49} - 8064 q^{51} + 16704 q^{57} - 5528 q^{59} + 19576 q^{73} - 9780 q^{75} + 5508 q^{81} - 23656 q^{83} + 23432 q^{97} - 33768 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu^{3} - 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -4\nu^{3} + 28\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 144\nu^{2} - 288 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 288 ) / 144 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} + 14\beta_1 ) / 24 \) Copy content Toggle raw display

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.e 4
3.b odd 2 1 384.5.h.f yes 4
4.b odd 2 1 384.5.h.f yes 4
8.b even 2 1 384.5.h.f yes 4
8.d odd 2 1 inner 384.5.h.e 4
12.b even 2 1 inner 384.5.h.e 4
24.f even 2 1 384.5.h.f yes 4
24.h odd 2 1 inner 384.5.h.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.h.e 4 1.a even 1 1 trivial
384.5.h.e 4 8.d odd 2 1 inner
384.5.h.e 4 12.b even 2 1 inner
384.5.h.e 4 24.h odd 2 1 inner
384.5.h.f yes 4 3.b odd 2 1
384.5.h.f yes 4 4.b odd 2 1
384.5.h.f yes 4 8.b even 2 1
384.5.h.f yes 4 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{2} - 1440 \) Copy content Toggle raw display
\( T_{11} - 134 \) Copy content Toggle raw display

Hecke characteristic polynomials

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