Properties

Label 384.4.f.e
Level $384$
Weight $4$
Character orbit 384.f
Analytic conductor $22.657$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{26})\)
Defining polynomial: \(x^{4} + 169\)
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 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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{2} ) q^{3} + ( \beta_{1} + 2 \beta_{2} ) q^{5} + \beta_{3} q^{7} + ( 25 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{2} ) q^{3} + ( \beta_{1} + 2 \beta_{2} ) q^{5} + \beta_{3} q^{7} + ( 25 + \beta_{3} ) q^{9} -23 \beta_{1} q^{11} + 22 \beta_{1} q^{13} + ( -52 - \beta_{3} ) q^{15} + 2 \beta_{3} q^{17} + ( \beta_{1} + 2 \beta_{2} ) q^{19} + ( -25 \beta_{1} + 2 \beta_{2} ) q^{21} -88 q^{23} -21 q^{25} + ( -50 \beta_{1} - 23 \beta_{2} ) q^{27} + ( 25 \beta_{1} + 50 \beta_{2} ) q^{29} + 21 \beta_{3} q^{31} + ( -46 + 23 \beta_{3} ) q^{33} + 52 \beta_{1} q^{35} -166 \beta_{1} q^{37} + ( 44 - 22 \beta_{3} ) q^{39} + 48 \beta_{3} q^{41} + ( 23 \beta_{1} + 46 \beta_{2} ) q^{43} + ( 77 \beta_{1} + 50 \beta_{2} ) q^{45} + 384 q^{47} + 239 q^{49} + ( -50 \beta_{1} + 4 \beta_{2} ) q^{51} + ( 45 \beta_{1} + 90 \beta_{2} ) q^{53} -46 \beta_{3} q^{55} + ( -52 - \beta_{3} ) q^{57} + 315 \beta_{1} q^{59} + 118 \beta_{1} q^{61} + ( -104 + 25 \beta_{3} ) q^{63} + 44 \beta_{3} q^{65} + ( -5 \beta_{1} - 10 \beta_{2} ) q^{67} + ( 88 \beta_{1} + 88 \beta_{2} ) q^{69} + 680 q^{71} + 422 q^{73} + ( 21 \beta_{1} + 21 \beta_{2} ) q^{75} + ( 46 \beta_{1} + 92 \beta_{2} ) q^{77} + 73 \beta_{3} q^{79} + ( 521 + 50 \beta_{3} ) q^{81} + 93 \beta_{1} q^{83} + 104 \beta_{1} q^{85} + ( -1300 - 25 \beta_{3} ) q^{87} + 94 \beta_{3} q^{89} + ( -44 \beta_{1} - 88 \beta_{2} ) q^{91} + ( -525 \beta_{1} + 42 \beta_{2} ) q^{93} + 104 q^{95} -1062 q^{97} + ( -529 \beta_{1} + 92 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 100q^{9} + O(q^{10}) \) \( 4q + 100q^{9} - 208q^{15} - 352q^{23} - 84q^{25} - 184q^{33} + 176q^{39} + 1536q^{47} + 956q^{49} - 208q^{57} - 416q^{63} + 2720q^{71} + 1688q^{73} + 2084q^{81} - 5200q^{87} + 416q^{95} - 4248q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 169\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{2} \)\(/13\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} - \nu^{2} + 13 \nu \)\()/13\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} + 26 \nu \)\()/13\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\(13 \beta_{1}\)\(/2\)
\(\nu^{3}\)\(=\)\((\)\(13 \beta_{3} - 26 \beta_{2} - 13 \beta_{1}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
2.54951 + 2.54951i
2.54951 2.54951i
−2.54951 2.54951i
−2.54951 + 2.54951i
0 −5.09902 1.00000i 0 10.1980 0 10.1980i 0 25.0000 + 10.1980i 0
191.2 0 −5.09902 + 1.00000i 0 10.1980 0 10.1980i 0 25.0000 10.1980i 0
191.3 0 5.09902 1.00000i 0 −10.1980 0 10.1980i 0 25.0000 10.1980i 0
191.4 0 5.09902 + 1.00000i 0 −10.1980 0 10.1980i 0 25.0000 + 10.1980i 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 384.4.f.e 4
3.b odd 2 1 384.4.f.f yes 4
4.b odd 2 1 384.4.f.f yes 4
8.b even 2 1 inner 384.4.f.e 4
8.d odd 2 1 384.4.f.f yes 4
12.b even 2 1 inner 384.4.f.e 4
16.e even 4 1 768.4.c.e 2
16.e even 4 1 768.4.c.f 2
16.f odd 4 1 768.4.c.d 2
16.f odd 4 1 768.4.c.g 2
24.f even 2 1 inner 384.4.f.e 4
24.h odd 2 1 384.4.f.f yes 4
48.i odd 4 1 768.4.c.d 2
48.i odd 4 1 768.4.c.g 2
48.k even 4 1 768.4.c.e 2
48.k even 4 1 768.4.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.f.e 4 1.a even 1 1 trivial
384.4.f.e 4 8.b even 2 1 inner
384.4.f.e 4 12.b even 2 1 inner
384.4.f.e 4 24.f even 2 1 inner
384.4.f.f yes 4 3.b odd 2 1
384.4.f.f yes 4 4.b odd 2 1
384.4.f.f yes 4 8.d odd 2 1
384.4.f.f yes 4 24.h odd 2 1
768.4.c.d 2 16.f odd 4 1
768.4.c.d 2 48.i odd 4 1
768.4.c.e 2 16.e even 4 1
768.4.c.e 2 48.k even 4 1
768.4.c.f 2 16.e even 4 1
768.4.c.f 2 48.k even 4 1
768.4.c.g 2 16.f odd 4 1
768.4.c.g 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 \(S_{4}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{2} - 104 \)
\( T_{23} + 88 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 729 - 50 T^{2} + T^{4} \)
$5$ \( ( -104 + T^{2} )^{2} \)
$7$ \( ( 104 + T^{2} )^{2} \)
$11$ \( ( 2116 + T^{2} )^{2} \)
$13$ \( ( 1936 + T^{2} )^{2} \)
$17$ \( ( 416 + T^{2} )^{2} \)
$19$ \( ( -104 + T^{2} )^{2} \)
$23$ \( ( 88 + T )^{4} \)
$29$ \( ( -65000 + T^{2} )^{2} \)
$31$ \( ( 45864 + T^{2} )^{2} \)
$37$ \( ( 110224 + T^{2} )^{2} \)
$41$ \( ( 239616 + T^{2} )^{2} \)
$43$ \( ( -55016 + T^{2} )^{2} \)
$47$ \( ( -384 + T )^{4} \)
$53$ \( ( -210600 + T^{2} )^{2} \)
$59$ \( ( 396900 + T^{2} )^{2} \)
$61$ \( ( 55696 + T^{2} )^{2} \)
$67$ \( ( -2600 + T^{2} )^{2} \)
$71$ \( ( -680 + T )^{4} \)
$73$ \( ( -422 + T )^{4} \)
$79$ \( ( 554216 + T^{2} )^{2} \)
$83$ \( ( 34596 + T^{2} )^{2} \)
$89$ \( ( 918944 + T^{2} )^{2} \)
$97$ \( ( 1062 + T )^{4} \)
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