Properties

Label 384.4.f.g
Level $384$
Weight $4$
Character orbit 384.f
Analytic conductor $22.657$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.12745506816.1
Defining polynomial: \(x^{8} + 23 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{3} + \beta_{4} ) q^{3} + ( -2 \beta_{3} + \beta_{6} ) q^{5} + ( -3 \beta_{2} + 2 \beta_{5} ) q^{7} + ( -15 - 3 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( \beta_{3} + \beta_{4} ) q^{3} + ( -2 \beta_{3} + \beta_{6} ) q^{5} + ( -3 \beta_{2} + 2 \beta_{5} ) q^{7} + ( -15 - 3 \beta_{5} ) q^{9} + ( -8 \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{11} + ( -7 \beta_{1} - 4 \beta_{3} - 8 \beta_{4} ) q^{13} + ( -24 + 7 \beta_{2} + 6 \beta_{5} + \beta_{7} ) q^{15} + ( -16 \beta_{2} + 2 \beta_{5} ) q^{17} + ( 7 \beta_{3} - 16 \beta_{6} ) q^{19} + ( 9 \beta_{1} + 10 \beta_{3} - 8 \beta_{4} + 9 \beta_{6} ) q^{21} + ( -80 - 2 \beta_{7} ) q^{23} + ( 27 - 8 \beta_{7} ) q^{25} + ( -30 \beta_{3} - 3 \beta_{4} ) q^{27} + ( 14 \beta_{3} - 23 \beta_{6} ) q^{29} + ( -7 \beta_{2} - 6 \beta_{5} ) q^{31} + ( 42 + 16 \beta_{2} + 3 \beta_{5} - 8 \beta_{7} ) q^{33} + ( -8 \beta_{1} + 4 \beta_{3} + 8 \beta_{4} ) q^{35} + ( 31 \beta_{1} - 28 \beta_{3} - 56 \beta_{4} ) q^{37} + ( 168 + 14 \beta_{2} + 12 \beta_{5} - 7 \beta_{7} ) q^{39} + ( -16 \beta_{2} + 24 \beta_{5} ) q^{41} + ( 49 \beta_{3} + 16 \beta_{6} ) q^{43} + ( -42 \beta_{1} + 6 \beta_{3} - 48 \beta_{4} - 15 \beta_{6} ) q^{45} + 16 \beta_{7} q^{47} + ( -97 - 24 \beta_{7} ) q^{49} + ( 48 \beta_{1} + 10 \beta_{3} - 8 \beta_{4} + 48 \beta_{6} ) q^{51} + ( 70 \beta_{3} + 45 \beta_{6} ) q^{53} + ( -78 \beta_{2} - 44 \beta_{5} ) q^{55} + ( 84 - 112 \beta_{2} - 21 \beta_{5} - 16 \beta_{7} ) q^{57} + ( 5 \beta_{3} + 10 \beta_{4} ) q^{59} + ( -7 \beta_{1} - 68 \beta_{3} - 136 \beta_{4} ) q^{61} + ( 336 + 45 \beta_{2} - 30 \beta_{5} + 18 \beta_{7} ) q^{63} + ( -112 \beta_{2} - 76 \beta_{5} ) q^{65} + ( -91 \beta_{3} + 32 \beta_{6} ) q^{67} + ( 42 \beta_{1} - 80 \beta_{3} - 80 \beta_{4} - 12 \beta_{6} ) q^{69} + ( -80 - 50 \beta_{7} ) q^{71} + ( -154 - 24 \beta_{7} ) q^{73} + ( 168 \beta_{1} + 27 \beta_{3} + 27 \beta_{4} - 48 \beta_{6} ) q^{75} + ( 68 \beta_{3} + 46 \beta_{6} ) q^{77} + ( 189 \beta_{2} + 2 \beta_{5} ) q^{79} + ( -279 + 90 \beta_{5} ) q^{81} + ( -168 \beta_{1} - 37 \beta_{3} - 74 \beta_{4} ) q^{83} + ( -164 \beta_{1} - 48 \beta_{3} - 96 \beta_{4} ) q^{85} + ( 168 - 161 \beta_{2} - 42 \beta_{5} - 23 \beta_{7} ) q^{87} + ( 128 \beta_{2} - 26 \beta_{5} ) q^{89} + ( -28 \beta_{3} - 16 \beta_{6} ) q^{91} + ( 21 \beta_{1} - 30 \beta_{3} + 24 \beta_{4} + 21 \beta_{6} ) q^{93} + ( -1232 + 78 \beta_{7} ) q^{95} + ( 714 - 16 \beta_{7} ) q^{97} + ( 120 \beta_{1} + 57 \beta_{3} + 30 \beta_{4} - 96 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 120q^{9} + O(q^{10}) \) \( 8q - 120q^{9} - 192q^{15} - 640q^{23} + 216q^{25} + 336q^{33} + 1344q^{39} - 776q^{49} + 672q^{57} + 2688q^{63} - 640q^{71} - 1232q^{73} - 2232q^{81} + 1344q^{87} - 9856q^{95} + 5712q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 23 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 4 \nu^{6} + 96 \nu^{2} \)\()/5\)
\(\beta_{2}\)\(=\)\((\)\( -8 \nu^{7} - 2 \nu^{5} - 182 \nu^{3} - 38 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( -8 \nu^{7} + 2 \nu^{5} - 182 \nu^{3} + 38 \nu \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( 4 \nu^{7} - 5 \nu^{6} - \nu^{5} + 91 \nu^{3} - 110 \nu^{2} - 19 \nu \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( 12 \nu^{7} + 2 \nu^{5} + 278 \nu^{3} + 58 \nu \)\()/5\)
\(\beta_{6}\)\(=\)\((\)\( -12 \nu^{7} + 2 \nu^{5} - 278 \nu^{3} + 58 \nu \)\()/5\)
\(\beta_{7}\)\(=\)\((\)\( 8 \nu^{4} + 92 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} + \beta_{5} - \beta_{3} + \beta_{2}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(4 \beta_{4} + 2 \beta_{3} + 5 \beta_{1}\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{6} + 2 \beta_{5} + 3 \beta_{3} + 3 \beta_{2}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(5 \beta_{7} - 92\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(-19 \beta_{6} - 19 \beta_{5} + 29 \beta_{3} - 29 \beta_{2}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-48 \beta_{4} - 24 \beta_{3} - 55 \beta_{1}\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(91 \beta_{6} - 91 \beta_{5} - 139 \beta_{3} - 139 \beta_{2}\)\()/8\)

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
−0.323042 0.323042i
1.54779 1.54779i
−0.323042 + 0.323042i
1.54779 + 1.54779i
−1.54779 + 1.54779i
0.323042 + 0.323042i
−1.54779 1.54779i
0.323042 0.323042i
0 −2.44949 4.58258i 0 2.31464 0 29.6636i 0 −15.0000 + 22.4499i 0
191.2 0 −2.44949 4.58258i 0 17.2813 0 0.269691i 0 −15.0000 + 22.4499i 0
191.3 0 −2.44949 + 4.58258i 0 2.31464 0 29.6636i 0 −15.0000 22.4499i 0
191.4 0 −2.44949 + 4.58258i 0 17.2813 0 0.269691i 0 −15.0000 22.4499i 0
191.5 0 2.44949 4.58258i 0 −17.2813 0 0.269691i 0 −15.0000 22.4499i 0
191.6 0 2.44949 4.58258i 0 −2.31464 0 29.6636i 0 −15.0000 22.4499i 0
191.7 0 2.44949 + 4.58258i 0 −17.2813 0 0.269691i 0 −15.0000 + 22.4499i 0
191.8 0 2.44949 + 4.58258i 0 −2.31464 0 29.6636i 0 −15.0000 + 22.4499i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.8
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.g 8
3.b odd 2 1 384.4.f.h yes 8
4.b odd 2 1 384.4.f.h yes 8
8.b even 2 1 inner 384.4.f.g 8
8.d odd 2 1 384.4.f.h yes 8
12.b even 2 1 inner 384.4.f.g 8
16.e even 4 1 768.4.c.n 4
16.e even 4 1 768.4.c.o 4
16.f odd 4 1 768.4.c.m 4
16.f odd 4 1 768.4.c.p 4
24.f even 2 1 inner 384.4.f.g 8
24.h odd 2 1 384.4.f.h yes 8
48.i odd 4 1 768.4.c.m 4
48.i odd 4 1 768.4.c.p 4
48.k even 4 1 768.4.c.n 4
48.k even 4 1 768.4.c.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.f.g 8 1.a even 1 1 trivial
384.4.f.g 8 8.b even 2 1 inner
384.4.f.g 8 12.b even 2 1 inner
384.4.f.g 8 24.f even 2 1 inner
384.4.f.h yes 8 3.b odd 2 1
384.4.f.h yes 8 4.b odd 2 1
384.4.f.h yes 8 8.d odd 2 1
384.4.f.h yes 8 24.h odd 2 1
768.4.c.m 4 16.f odd 4 1
768.4.c.m 4 48.i odd 4 1
768.4.c.n 4 16.e even 4 1
768.4.c.n 4 48.k even 4 1
768.4.c.o 4 16.e even 4 1
768.4.c.o 4 48.k even 4 1
768.4.c.p 4 16.f odd 4 1
768.4.c.p 4 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 \(S_{4}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{4} - 304 T_{5}^{2} + 1600 \)
\( T_{23}^{2} + 160 T_{23} + 5056 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 729 + 30 T^{2} + T^{4} )^{2} \)
$5$ \( ( 1600 - 304 T^{2} + T^{4} )^{2} \)
$7$ \( ( 64 + 880 T^{2} + T^{4} )^{2} \)
$11$ \( ( 883600 + 2216 T^{2} + T^{4} )^{2} \)
$13$ \( ( 313600 + 4256 T^{2} + T^{4} )^{2} \)
$17$ \( ( 35046400 + 12736 T^{2} + T^{4} )^{2} \)
$19$ \( ( 173185600 - 31024 T^{2} + T^{4} )^{2} \)
$23$ \( ( 5056 + 160 T + T^{2} )^{4} \)
$29$ \( ( 621006400 - 68656 T^{2} + T^{4} )^{2} \)
$31$ \( ( 705600 + 6384 T^{2} + T^{4} )^{2} \)
$37$ \( ( 2548230400 + 162464 T^{2} + T^{4} )^{2} \)
$41$ \( ( 681836544 + 76800 T^{2} + T^{4} )^{2} \)
$43$ \( ( 1873850944 - 143920 T^{2} + T^{4} )^{2} \)
$47$ \( ( -86016 + T^{2} )^{4} \)
$53$ \( ( 17640000 - 462000 T^{2} + T^{4} )^{2} \)
$59$ \( ( 2100 + T^{2} )^{4} \)
$61$ \( ( 150258567424 + 778400 T^{2} + T^{4} )^{2} \)
$67$ \( ( 19993960000 - 512176 T^{2} + T^{4} )^{2} \)
$71$ \( ( -833600 + 160 T + T^{2} )^{4} \)
$73$ \( ( -169820 + 308 T + T^{2} )^{4} \)
$79$ \( ( 734586126400 + 1715056 T^{2} + T^{4} )^{2} \)
$83$ \( ( 113291481744 + 1133160 T^{2} + T^{4} )^{2} \)
$89$ \( ( 126280729600 + 862144 T^{2} + T^{4} )^{2} \)
$97$ \( ( 423780 - 1428 T + T^{2} )^{4} \)
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