Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.06625543762\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{3} ) q^{3} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{7} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{3} ) q^{3} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{7} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} + ( 2 - \beta_{1} + \beta_{3} ) q^{11} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{13} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{15} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{17} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{19} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{21} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{23} + ( 1 + 2 \beta_{1} - 2 \beta_{3} ) q^{25} + ( -3 + \beta_{1} + 2 \beta_{2} ) q^{27} + ( \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{29} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} + ( \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{33} + 4 q^{35} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{37} + ( 6 + 2 \beta_{1} - 2 \beta_{2} ) q^{39} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{41} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{43} + ( -\beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{45} -8 q^{47} + ( -1 - 2 \beta_{1} + 2 \beta_{3} ) q^{49} + ( -4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{51} + ( -\beta_{1} - \beta_{3} ) q^{53} + ( -4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{55} + ( -2 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{57} + ( 2 + \beta_{1} - \beta_{3} ) q^{59} + ( 6 + 2 \beta_{1} - 2 \beta_{3} ) q^{61} + ( 4 + 3 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{63} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{65} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{67} + ( 8 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{69} + ( 4 + 6 \beta_{1} - 6 \beta_{3} ) q^{71} -2 q^{73} + ( -5 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{75} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{77} + ( -3 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} ) q^{79} + ( 1 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{81} + ( 6 + \beta_{1} - \beta_{3} ) q^{83} + 8 q^{85} + ( -2 - 7 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{87} + 2 \beta_{2} q^{89} + ( -2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{91} + ( -2 + 3 \beta_{1} - \beta_{3} ) q^{93} + ( 4 + 2 \beta_{1} - 2 \beta_{3} ) q^{95} + ( -6 - 2 \beta_{1} + 2 \beta_{3} ) q^{97} + ( -2 + 5 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + O(q^{10}) \) \( 4q - 2q^{3} + 12q^{11} + 12q^{15} - 8q^{21} - 8q^{23} - 4q^{25} - 14q^{27} + 4q^{33} + 16q^{35} - 16q^{37} + 20q^{39} + 8q^{45} - 32q^{47} + 4q^{49} - 16q^{51} - 4q^{57} + 4q^{59} + 16q^{61} + 8q^{63} + 24q^{69} - 8q^{71} - 8q^{73} - 18q^{75} + 4q^{81} + 20q^{83} + 32q^{85} + 4q^{87} - 16q^{93} + 8q^{95} - 16q^{97} - 20q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu + 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 4 \nu \)
\(\beta_{3}\)\(=\)\( -\nu^{2} + \nu - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{1} - 2\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{3} + \beta_{2} - 2 \beta_{1}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
383.1
0.618034i
0.618034i
1.61803i
1.61803i
0 −1.61803 0.618034i 0 1.23607i 0 3.23607i 0 2.23607 + 2.00000i 0
383.2 0 −1.61803 + 0.618034i 0 1.23607i 0 3.23607i 0 2.23607 2.00000i 0
383.3 0 0.618034 1.61803i 0 3.23607i 0 1.23607i 0 −2.23607 2.00000i 0
383.4 0 0.618034 + 1.61803i 0 3.23607i 0 1.23607i 0 −2.23607 + 2.00000i 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
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 384.2.c.b yes 4
3.b odd 2 1 384.2.c.c yes 4
4.b odd 2 1 384.2.c.c yes 4
8.b even 2 1 384.2.c.d yes 4
8.d odd 2 1 384.2.c.a 4
12.b even 2 1 inner 384.2.c.b yes 4
16.e even 4 1 768.2.f.b 4
16.e even 4 1 768.2.f.f 4
16.f odd 4 1 768.2.f.c 4
16.f odd 4 1 768.2.f.e 4
24.f even 2 1 384.2.c.d yes 4
24.h odd 2 1 384.2.c.a 4
48.i odd 4 1 768.2.f.c 4
48.i odd 4 1 768.2.f.e 4
48.k even 4 1 768.2.f.b 4
48.k even 4 1 768.2.f.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.c.a 4 8.d odd 2 1
384.2.c.a 4 24.h odd 2 1
384.2.c.b yes 4 1.a even 1 1 trivial
384.2.c.b yes 4 12.b even 2 1 inner
384.2.c.c yes 4 3.b odd 2 1
384.2.c.c yes 4 4.b odd 2 1
384.2.c.d yes 4 8.b even 2 1
384.2.c.d yes 4 24.f even 2 1
768.2.f.b 4 16.e even 4 1
768.2.f.b 4 48.k even 4 1
768.2.f.c 4 16.f odd 4 1
768.2.f.c 4 48.i odd 4 1
768.2.f.e 4 16.f odd 4 1
768.2.f.e 4 48.i odd 4 1
768.2.f.f 4 16.e even 4 1
768.2.f.f 4 48.k even 4 1

Hecke kernels

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

\( T_{11}^{2} - 6 T_{11} + 4 \)
\( T_{23}^{2} + 4 T_{23} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + 6 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( 16 + 12 T^{2} + T^{4} \)
$7$ \( 16 + 12 T^{2} + T^{4} \)
$11$ \( ( 4 - 6 T + T^{2} )^{2} \)
$13$ \( ( -20 + T^{2} )^{2} \)
$17$ \( 256 + 48 T^{2} + T^{4} \)
$19$ \( 16 + 28 T^{2} + T^{4} \)
$23$ \( ( -16 + 4 T + T^{2} )^{2} \)
$29$ \( 1936 + 108 T^{2} + T^{4} \)
$31$ \( 16 + 28 T^{2} + T^{4} \)
$37$ \( ( -4 + 8 T + T^{2} )^{2} \)
$41$ \( 256 + 48 T^{2} + T^{4} \)
$43$ \( 400 + 60 T^{2} + T^{4} \)
$47$ \( ( 8 + T )^{4} \)
$53$ \( 16 + 12 T^{2} + T^{4} \)
$59$ \( ( -4 - 2 T + T^{2} )^{2} \)
$61$ \( ( -4 - 8 T + T^{2} )^{2} \)
$67$ \( 1296 + 108 T^{2} + T^{4} \)
$71$ \( ( -176 + 4 T + T^{2} )^{2} \)
$73$ \( ( 2 + T )^{4} \)
$79$ \( 16 + 188 T^{2} + T^{4} \)
$83$ \( ( 20 - 10 T + T^{2} )^{2} \)
$89$ \( ( 16 + T^{2} )^{2} \)
$97$ \( ( -4 + 8 T + T^{2} )^{2} \)
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