Properties

Label 384.3.h.e
Level $384$
Weight $3$
Character orbit 384.h
Analytic conductor $10.463$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} -4 q^{5} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{7} + ( 1 - \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} -4 q^{5} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{7} + ( 1 - \beta_{2} ) q^{9} + ( -\beta_{1} - \beta_{3} ) q^{11} + 2 \beta_{2} q^{13} -4 \beta_{3} q^{15} + 2 \beta_{2} q^{17} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{19} + ( -20 + 2 \beta_{2} ) q^{21} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{23} -9 q^{25} + ( -9 \beta_{1} + 2 \beta_{3} ) q^{27} -52 q^{29} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{31} + ( -10 + \beta_{2} ) q^{33} + ( 8 \beta_{1} + 8 \beta_{3} ) q^{35} + 6 \beta_{2} q^{37} + ( 18 \beta_{1} - 2 \beta_{3} ) q^{39} -4 \beta_{2} q^{41} + ( -9 \beta_{1} + 9 \beta_{3} ) q^{43} + ( -4 + 4 \beta_{2} ) q^{45} + ( -16 \beta_{1} + 16 \beta_{3} ) q^{47} + 31 q^{49} + ( 18 \beta_{1} - 2 \beta_{3} ) q^{51} -20 q^{53} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{55} + ( 40 + 5 \beta_{2} ) q^{57} + ( -23 \beta_{1} - 23 \beta_{3} ) q^{59} + 2 \beta_{2} q^{61} + ( 18 \beta_{1} - 22 \beta_{3} ) q^{63} -8 \beta_{2} q^{65} + ( 11 \beta_{1} - 11 \beta_{3} ) q^{67} + ( -32 - 4 \beta_{2} ) q^{69} + ( -20 \beta_{1} + 20 \beta_{3} ) q^{71} -50 q^{73} -9 \beta_{3} q^{75} + 40 q^{77} + ( -18 \beta_{1} - 18 \beta_{3} ) q^{79} + ( -79 - 2 \beta_{2} ) q^{81} + ( 23 \beta_{1} + 23 \beta_{3} ) q^{83} -8 \beta_{2} q^{85} -52 \beta_{3} q^{87} + 18 \beta_{2} q^{89} + ( -40 \beta_{1} + 40 \beta_{3} ) q^{91} + ( 60 - 6 \beta_{2} ) q^{93} + ( -20 \beta_{1} + 20 \beta_{3} ) q^{95} + 50 q^{97} + ( 9 \beta_{1} - 11 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{5} + 4q^{9} + O(q^{10}) \) \( 4q - 16q^{5} + 4q^{9} - 80q^{21} - 36q^{25} - 208q^{29} - 40q^{33} - 16q^{45} + 124q^{49} - 80q^{53} + 160q^{57} - 128q^{69} - 200q^{73} + 160q^{77} - 316q^{81} + 240q^{93} + 200q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{3} + 2 \nu^{2} + 4 \nu + 3 \)
\(\beta_{2}\)\(=\)\( 4 \nu^{3} + 16 \nu \)
\(\beta_{3}\)\(=\)\( -2 \nu^{3} + 2 \nu^{2} - 4 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{1} - 6\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{3} - \beta_{2} + 2 \beta_{1}\)\()/4\)

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
1.61803i
1.61803i
0.618034i
0.618034i
0 −2.23607 2.00000i 0 −4.00000 0 8.94427 0 1.00000 + 8.94427i 0
65.2 0 −2.23607 + 2.00000i 0 −4.00000 0 8.94427 0 1.00000 8.94427i 0
65.3 0 2.23607 2.00000i 0 −4.00000 0 −8.94427 0 1.00000 8.94427i 0
65.4 0 2.23607 + 2.00000i 0 −4.00000 0 −8.94427 0 1.00000 + 8.94427i 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
24.f 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 384.3.h.e 4
3.b odd 2 1 384.3.h.f yes 4
4.b odd 2 1 inner 384.3.h.e 4
8.b even 2 1 384.3.h.f yes 4
8.d odd 2 1 384.3.h.f yes 4
12.b even 2 1 384.3.h.f yes 4
16.e even 4 1 768.3.e.h 4
16.e even 4 1 768.3.e.m 4
16.f odd 4 1 768.3.e.h 4
16.f odd 4 1 768.3.e.m 4
24.f even 2 1 inner 384.3.h.e 4
24.h odd 2 1 inner 384.3.h.e 4
48.i odd 4 1 768.3.e.h 4
48.i odd 4 1 768.3.e.m 4
48.k even 4 1 768.3.e.h 4
48.k even 4 1 768.3.e.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.h.e 4 1.a even 1 1 trivial
384.3.h.e 4 4.b odd 2 1 inner
384.3.h.e 4 24.f even 2 1 inner
384.3.h.e 4 24.h odd 2 1 inner
384.3.h.f yes 4 3.b odd 2 1
384.3.h.f yes 4 8.b even 2 1
384.3.h.f yes 4 8.d odd 2 1
384.3.h.f yes 4 12.b even 2 1
768.3.e.h 4 16.e even 4 1
768.3.e.h 4 16.f odd 4 1
768.3.e.h 4 48.i odd 4 1
768.3.e.h 4 48.k even 4 1
768.3.e.m 4 16.e even 4 1
768.3.e.m 4 16.f odd 4 1
768.3.e.m 4 48.i odd 4 1
768.3.e.m 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 \(S_{3}^{\mathrm{new}}(384, [\chi])\):

\( T_{5} + 4 \)
\( T_{11}^{2} - 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 81 - 2 T^{2} + T^{4} \)
$5$ \( ( 4 + T )^{4} \)
$7$ \( ( -80 + T^{2} )^{2} \)
$11$ \( ( -20 + T^{2} )^{2} \)
$13$ \( ( 320 + T^{2} )^{2} \)
$17$ \( ( 320 + T^{2} )^{2} \)
$19$ \( ( 400 + T^{2} )^{2} \)
$23$ \( ( 256 + T^{2} )^{2} \)
$29$ \( ( 52 + T )^{4} \)
$31$ \( ( -720 + T^{2} )^{2} \)
$37$ \( ( 2880 + T^{2} )^{2} \)
$41$ \( ( 1280 + T^{2} )^{2} \)
$43$ \( ( 1296 + T^{2} )^{2} \)
$47$ \( ( 4096 + T^{2} )^{2} \)
$53$ \( ( 20 + T )^{4} \)
$59$ \( ( -10580 + T^{2} )^{2} \)
$61$ \( ( 320 + T^{2} )^{2} \)
$67$ \( ( 1936 + T^{2} )^{2} \)
$71$ \( ( 6400 + T^{2} )^{2} \)
$73$ \( ( 50 + T )^{4} \)
$79$ \( ( -6480 + T^{2} )^{2} \)
$83$ \( ( -10580 + T^{2} )^{2} \)
$89$ \( ( 25920 + T^{2} )^{2} \)
$97$ \( ( -50 + T )^{4} \)
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