Properties

Label 384.5.b.b
Level $384$
Weight $5$
Character orbit 384.b
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.6940658242\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{3} + 12 \zeta_{12}^{3} q^{5} + ( 36 - 72 \zeta_{12}^{2} ) q^{7} + 27 q^{9} +O(q^{10})\) \( q + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{3} + 12 \zeta_{12}^{3} q^{5} + ( 36 - 72 \zeta_{12}^{2} ) q^{7} + 27 q^{9} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{11} + 264 \zeta_{12}^{3} q^{13} + ( -36 + 72 \zeta_{12}^{2} ) q^{15} + 110 q^{17} + ( 120 \zeta_{12} - 60 \zeta_{12}^{3} ) q^{19} -324 \zeta_{12}^{3} q^{21} + ( -72 + 144 \zeta_{12}^{2} ) q^{23} + 481 q^{25} + ( 162 \zeta_{12} - 81 \zeta_{12}^{3} ) q^{27} -228 \zeta_{12}^{3} q^{29} + ( 828 - 1656 \zeta_{12}^{2} ) q^{31} + 108 q^{33} + ( 864 \zeta_{12} - 432 \zeta_{12}^{3} ) q^{35} -1392 \zeta_{12}^{3} q^{37} + ( -792 + 1584 \zeta_{12}^{2} ) q^{39} + 1282 q^{41} + ( 2904 \zeta_{12} - 1452 \zeta_{12}^{3} ) q^{43} + 324 \zeta_{12}^{3} q^{45} + ( -1512 + 3024 \zeta_{12}^{2} ) q^{47} -1487 q^{49} + ( 660 \zeta_{12} - 330 \zeta_{12}^{3} ) q^{51} + 4500 \zeta_{12}^{3} q^{53} + ( -144 + 288 \zeta_{12}^{2} ) q^{55} + 540 q^{57} + ( 7320 \zeta_{12} - 3660 \zeta_{12}^{3} ) q^{59} -960 \zeta_{12}^{3} q^{61} + ( 972 - 1944 \zeta_{12}^{2} ) q^{63} -3168 q^{65} + ( 3768 \zeta_{12} - 1884 \zeta_{12}^{3} ) q^{67} + 648 \zeta_{12}^{3} q^{69} + ( 3528 - 7056 \zeta_{12}^{2} ) q^{71} -3170 q^{73} + ( 2886 \zeta_{12} - 1443 \zeta_{12}^{3} ) q^{75} -1296 \zeta_{12}^{3} q^{77} + ( -900 + 1800 \zeta_{12}^{2} ) q^{79} + 729 q^{81} + ( 5112 \zeta_{12} - 2556 \zeta_{12}^{3} ) q^{83} + 1320 \zeta_{12}^{3} q^{85} + ( 684 - 1368 \zeta_{12}^{2} ) q^{87} + 1550 q^{89} + ( 19008 \zeta_{12} - 9504 \zeta_{12}^{3} ) q^{91} -7452 \zeta_{12}^{3} q^{93} + ( -720 + 1440 \zeta_{12}^{2} ) q^{95} -8018 q^{97} + ( 648 \zeta_{12} - 324 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 108q^{9} + O(q^{10}) \) \( 4q + 108q^{9} + 440q^{17} + 1924q^{25} + 432q^{33} + 5128q^{41} - 5948q^{49} + 2160q^{57} - 12672q^{65} - 12680q^{73} + 2916q^{81} + 6200q^{89} - 32072q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
0 −5.19615 0 12.0000i 0 62.3538i 0 27.0000 0
319.2 0 −5.19615 0 12.0000i 0 62.3538i 0 27.0000 0
319.3 0 5.19615 0 12.0000i 0 62.3538i 0 27.0000 0
319.4 0 5.19615 0 12.0000i 0 62.3538i 0 27.0000 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
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 384.5.b.b 4
3.b odd 2 1 1152.5.b.e 4
4.b odd 2 1 inner 384.5.b.b 4
8.b even 2 1 inner 384.5.b.b 4
8.d odd 2 1 inner 384.5.b.b 4
12.b even 2 1 1152.5.b.e 4
16.e even 4 1 768.5.g.a 2
16.e even 4 1 768.5.g.b 2
16.f odd 4 1 768.5.g.a 2
16.f odd 4 1 768.5.g.b 2
24.f even 2 1 1152.5.b.e 4
24.h odd 2 1 1152.5.b.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.b.b 4 1.a even 1 1 trivial
384.5.b.b 4 4.b odd 2 1 inner
384.5.b.b 4 8.b even 2 1 inner
384.5.b.b 4 8.d odd 2 1 inner
768.5.g.a 2 16.e even 4 1
768.5.g.a 2 16.f odd 4 1
768.5.g.b 2 16.e even 4 1
768.5.g.b 2 16.f odd 4 1
1152.5.b.e 4 3.b odd 2 1
1152.5.b.e 4 12.b even 2 1
1152.5.b.e 4 24.f even 2 1
1152.5.b.e 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 144 \) acting on \(S_{5}^{\mathrm{new}}(384, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -27 + T^{2} )^{2} \)
$5$ \( ( 144 + T^{2} )^{2} \)
$7$ \( ( 3888 + T^{2} )^{2} \)
$11$ \( ( -432 + T^{2} )^{2} \)
$13$ \( ( 69696 + T^{2} )^{2} \)
$17$ \( ( -110 + T )^{4} \)
$19$ \( ( -10800 + T^{2} )^{2} \)
$23$ \( ( 15552 + T^{2} )^{2} \)
$29$ \( ( 51984 + T^{2} )^{2} \)
$31$ \( ( 2056752 + T^{2} )^{2} \)
$37$ \( ( 1937664 + T^{2} )^{2} \)
$41$ \( ( -1282 + T )^{4} \)
$43$ \( ( -6324912 + T^{2} )^{2} \)
$47$ \( ( 6858432 + T^{2} )^{2} \)
$53$ \( ( 20250000 + T^{2} )^{2} \)
$59$ \( ( -40186800 + T^{2} )^{2} \)
$61$ \( ( 921600 + T^{2} )^{2} \)
$67$ \( ( -10648368 + T^{2} )^{2} \)
$71$ \( ( 37340352 + T^{2} )^{2} \)
$73$ \( ( 3170 + T )^{4} \)
$79$ \( ( 2430000 + T^{2} )^{2} \)
$83$ \( ( -19599408 + T^{2} )^{2} \)
$89$ \( ( -1550 + T )^{4} \)
$97$ \( ( 8018 + T )^{4} \)
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