Properties

Label 384.5.h.d
Level $384$
Weight $5$
Character orbit 384.h
Self dual yes
Analytic conductor $39.694$
Analytic rank $0$
Dimension $2$
CM discriminant -24
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: yes
Analytic conductor: \(39.6940658242\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 8\sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 q^{3} + \beta q^{5} + 5 \beta q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} + \beta q^{5} + 5 \beta q^{7} + 81 q^{9} + 142 q^{11} + 9 \beta q^{15} + 45 \beta q^{21} - 241 q^{25} + 729 q^{27} - 75 \beta q^{29} - 95 \beta q^{31} + 1278 q^{33} + 1920 q^{35} + 81 \beta q^{45} + 7199 q^{49} - 235 \beta q^{53} + 142 \beta q^{55} - 6862 q^{59} + 405 \beta q^{63} + 8158 q^{73} - 2169 q^{75} + 710 \beta q^{77} - 435 \beta q^{79} + 6561 q^{81} - 4178 q^{83} - 675 \beta q^{87} - 855 \beta q^{93} + 17282 q^{97} + 11502 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 18 q^{3} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 18 q^{3} + 162 q^{9} + 284 q^{11} - 482 q^{25} + 1458 q^{27} + 2556 q^{33} + 3840 q^{35} + 14398 q^{49} - 13724 q^{59} + 16316 q^{73} - 4338 q^{75} + 13122 q^{81} - 8356 q^{83} + 34564 q^{97} + 23004 q^{99}+O(q^{100}) \) 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
−2.44949
2.44949
0 9.00000 0 −19.5959 0 −97.9796 0 81.0000 0
65.2 0 9.00000 0 19.5959 0 97.9796 0 81.0000 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
8.d odd 2 1 inner
12.b even 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.d yes 2
3.b odd 2 1 384.5.h.a 2
4.b odd 2 1 384.5.h.a 2
8.b even 2 1 384.5.h.a 2
8.d odd 2 1 inner 384.5.h.d yes 2
12.b even 2 1 inner 384.5.h.d yes 2
24.f even 2 1 384.5.h.a 2
24.h odd 2 1 CM 384.5.h.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.h.a 2 3.b odd 2 1
384.5.h.a 2 4.b odd 2 1
384.5.h.a 2 8.b even 2 1
384.5.h.a 2 24.f even 2 1
384.5.h.d yes 2 1.a even 1 1 trivial
384.5.h.d yes 2 8.d odd 2 1 inner
384.5.h.d yes 2 12.b even 2 1 inner
384.5.h.d yes 2 24.h odd 2 1 CM

Hecke kernels

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

\( T_{5}^{2} - 384 \) Copy content Toggle raw display
\( T_{11} - 142 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 384 \) Copy content Toggle raw display
$7$ \( T^{2} - 9600 \) Copy content Toggle raw display
$11$ \( (T - 142)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 2160000 \) Copy content Toggle raw display
$31$ \( T^{2} - 3465600 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 21206400 \) Copy content Toggle raw display
$59$ \( (T + 6862)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 8158)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 72662400 \) Copy content Toggle raw display
$83$ \( (T + 4178)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 17282)^{2} \) Copy content Toggle raw display
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