Properties

Label 384.2.f.d
Level $384$
Weight $2$
Character orbit 384.f
Analytic conductor $3.066$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.06625543762\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{7} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{7} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} -2 \zeta_{8}^{2} q^{11} -4 \zeta_{8}^{2} q^{13} + ( 4 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{15} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{17} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{19} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{21} -8 q^{23} + 3 q^{25} + ( \zeta_{8} + 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{27} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{31} + ( 2 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{33} -8 \zeta_{8}^{2} q^{35} + 4 \zeta_{8}^{2} q^{37} + ( 4 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{39} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{43} + ( -2 \zeta_{8} + 8 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{45} - q^{49} + ( 4 \zeta_{8} + 8 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{51} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{53} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{55} + ( -4 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{57} -6 \zeta_{8}^{2} q^{59} -4 \zeta_{8}^{2} q^{61} + ( 8 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{63} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{65} + ( -10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{67} + ( 8 \zeta_{8} - 8 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{69} -8 q^{71} -10 q^{73} + ( -3 \zeta_{8} + 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{75} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{77} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{79} + ( -7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} + 6 \zeta_{8}^{2} q^{83} + 16 \zeta_{8}^{2} q^{85} + ( 4 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{87} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{89} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{91} + ( 6 \zeta_{8} + 12 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{93} -8 q^{95} -6 q^{97} + ( -4 \zeta_{8} - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{9} + 16q^{15} - 32q^{23} + 12q^{25} + 8q^{33} + 16q^{39} - 4q^{49} - 16q^{57} + 32q^{63} - 32q^{71} - 40q^{73} - 28q^{81} + 16q^{87} - 32q^{95} - 24q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0 −1.41421 1.00000i 0 −2.82843 0 2.82843i 0 1.00000 + 2.82843i 0
191.2 0 −1.41421 + 1.00000i 0 −2.82843 0 2.82843i 0 1.00000 2.82843i 0
191.3 0 1.41421 1.00000i 0 2.82843 0 2.82843i 0 1.00000 2.82843i 0
191.4 0 1.41421 + 1.00000i 0 2.82843 0 2.82843i 0 1.00000 + 2.82843i 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.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.2.f.d yes 4
3.b odd 2 1 384.2.f.b 4
4.b odd 2 1 384.2.f.b 4
8.b even 2 1 inner 384.2.f.d yes 4
8.d odd 2 1 384.2.f.b 4
12.b even 2 1 inner 384.2.f.d yes 4
16.e even 4 1 768.2.c.b 2
16.e even 4 1 768.2.c.e 2
16.f odd 4 1 768.2.c.a 2
16.f odd 4 1 768.2.c.f 2
24.f even 2 1 inner 384.2.f.d yes 4
24.h odd 2 1 384.2.f.b 4
48.i odd 4 1 768.2.c.a 2
48.i odd 4 1 768.2.c.f 2
48.k even 4 1 768.2.c.b 2
48.k even 4 1 768.2.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.f.b 4 3.b odd 2 1
384.2.f.b 4 4.b odd 2 1
384.2.f.b 4 8.d odd 2 1
384.2.f.b 4 24.h odd 2 1
384.2.f.d yes 4 1.a even 1 1 trivial
384.2.f.d yes 4 8.b even 2 1 inner
384.2.f.d yes 4 12.b even 2 1 inner
384.2.f.d yes 4 24.f even 2 1 inner
768.2.c.a 2 16.f odd 4 1
768.2.c.a 2 48.i odd 4 1
768.2.c.b 2 16.e even 4 1
768.2.c.b 2 48.k even 4 1
768.2.c.e 2 16.e even 4 1
768.2.c.e 2 48.k even 4 1
768.2.c.f 2 16.f odd 4 1
768.2.c.f 2 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{2} - 8 \)
\( T_{7}^{2} + 8 \)
\( T_{23} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 - 2 T^{2} + T^{4} \)
$5$ \( ( -8 + T^{2} )^{2} \)
$7$ \( ( 8 + T^{2} )^{2} \)
$11$ \( ( 4 + T^{2} )^{2} \)
$13$ \( ( 16 + T^{2} )^{2} \)
$17$ \( ( 32 + T^{2} )^{2} \)
$19$ \( ( -8 + T^{2} )^{2} \)
$23$ \( ( 8 + T )^{4} \)
$29$ \( ( -8 + T^{2} )^{2} \)
$31$ \( ( 72 + T^{2} )^{2} \)
$37$ \( ( 16 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( -8 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( ( -72 + T^{2} )^{2} \)
$59$ \( ( 36 + T^{2} )^{2} \)
$61$ \( ( 16 + T^{2} )^{2} \)
$67$ \( ( -200 + T^{2} )^{2} \)
$71$ \( ( 8 + T )^{4} \)
$73$ \( ( 10 + T )^{4} \)
$79$ \( ( 8 + T^{2} )^{2} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( ( 32 + T^{2} )^{2} \)
$97$ \( ( 6 + T )^{4} \)
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