Properties

Label 384.4.d.f
Level $384$
Weight $4$
Character orbit 384.d
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1534132224.8
Defining polynomial: \(x^{8} + 18 x^{6} + 107 x^{4} + 210 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{6} q^{5} -\beta_{2} q^{7} -9 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{6} q^{5} -\beta_{2} q^{7} -9 q^{9} + ( -4 \beta_{1} - \beta_{3} ) q^{11} + ( -\beta_{6} - \beta_{7} ) q^{13} + ( \beta_{2} + \beta_{4} ) q^{15} + ( 30 - \beta_{5} ) q^{17} + ( 4 \beta_{1} - 3 \beta_{3} ) q^{19} + ( -2 \beta_{6} + \beta_{7} ) q^{21} + ( 2 \beta_{2} - 4 \beta_{4} ) q^{23} + ( -43 - 2 \beta_{5} ) q^{25} -9 \beta_{1} q^{27} + ( 5 \beta_{6} + 2 \beta_{7} ) q^{29} + ( 5 \beta_{2} + 4 \beta_{4} ) q^{31} + ( 36 - \beta_{5} ) q^{33} + ( 40 \beta_{1} + 3 \beta_{3} ) q^{35} + ( 17 \beta_{6} - \beta_{7} ) q^{37} + ( -6 \beta_{2} + 3 \beta_{4} ) q^{39} + ( -102 - 5 \beta_{5} ) q^{41} + ( -44 \beta_{1} - 3 \beta_{3} ) q^{43} + 9 \beta_{6} q^{45} + ( -14 \beta_{2} - 8 \beta_{4} ) q^{47} + ( 209 - 8 \beta_{5} ) q^{49} + ( 30 \beta_{1} + 9 \beta_{3} ) q^{51} + ( -13 \beta_{6} - 2 \beta_{7} ) q^{53} + ( -4 \beta_{2} + 4 \beta_{4} ) q^{55} + ( -36 - 3 \beta_{5} ) q^{57} + ( -156 \beta_{1} + 16 \beta_{3} ) q^{59} + ( -49 \beta_{6} + 5 \beta_{7} ) q^{61} + 9 \beta_{2} q^{63} + ( -144 - 9 \beta_{5} ) q^{65} + ( 196 \beta_{1} - 12 \beta_{3} ) q^{67} + ( -24 \beta_{6} - 6 \beta_{7} ) q^{69} + ( 30 \beta_{2} + 12 \beta_{4} ) q^{71} + ( -242 - 8 \beta_{5} ) q^{73} + ( -43 \beta_{1} + 18 \beta_{3} ) q^{75} + ( 16 \beta_{6} - 12 \beta_{7} ) q^{77} + ( -11 \beta_{2} - 20 \beta_{4} ) q^{79} + 81 q^{81} + ( 284 \beta_{1} + 17 \beta_{3} ) q^{83} + ( 26 \beta_{6} + 8 \beta_{7} ) q^{85} + ( 9 \beta_{2} - 9 \beta_{4} ) q^{87} + ( 918 - 6 \beta_{5} ) q^{89} + ( -432 \beta_{1} - 63 \beta_{3} ) q^{91} + ( 38 \beta_{6} - \beta_{7} ) q^{93} + ( 4 \beta_{2} + 28 \beta_{4} ) q^{95} + ( -370 + 2 \beta_{5} ) q^{97} + ( 36 \beta_{1} + 9 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 72q^{9} + O(q^{10}) \) \( 8q - 72q^{9} + 240q^{17} - 344q^{25} + 288q^{33} - 816q^{41} + 1672q^{49} - 288q^{57} - 1152q^{65} - 1936q^{73} + 648q^{81} + 7344q^{89} - 2960q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 18 x^{6} + 107 x^{4} + 210 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -3 \nu^{5} - 33 \nu^{3} - 87 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\( \nu^{6} + 6 \nu^{4} - 2 \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( 8 \nu^{5} + 104 \nu^{3} + 328 \nu \)
\(\beta_{4}\)\(=\)\( 8 \nu^{6} + 96 \nu^{4} + 272 \nu^{2} - 48 \)
\(\beta_{5}\)\(=\)\( 24 \nu^{6} + 288 \nu^{4} + 864 \nu^{2} + 72 \)
\(\beta_{6}\)\(=\)\( 2 \nu^{7} + 31 \nu^{5} + 155 \nu^{3} + 253 \nu \)
\(\beta_{7}\)\(=\)\( 10 \nu^{7} + 131 \nu^{5} + 487 \nu^{3} + 377 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{7} + 10 \beta_{6} - 3 \beta_{3} + 16 \beta_{1}\)\()/96\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} - 3 \beta_{4} - 216\)\()/48\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{7} - 15 \beta_{6} + 6 \beta_{3} - 16 \beta_{1}\)\()/24\)
\(\nu^{4}\)\(=\)\((\)\(-6 \beta_{5} + 19 \beta_{4} - 8 \beta_{2} + 1320\)\()/48\)
\(\nu^{5}\)\(=\)\((\)\(-74 \beta_{7} + 370 \beta_{6} - 177 \beta_{3} + 176 \beta_{1}\)\()/96\)
\(\nu^{6}\)\(=\)\((\)\(19 \beta_{5} - 60 \beta_{4} + 48 \beta_{2} - 4104\)\()/24\)
\(\nu^{7}\)\(=\)\((\)\(470 \beta_{7} - 2302 \beta_{6} + 1263 \beta_{3} + 208 \beta_{1}\)\()/96\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.0690906i
2.63019i
2.21597i
2.48330i
2.48330i
2.21597i
2.63019i
0.0690906i
0 3.00000i 0 17.4288i 0 2.99032 0 −9.00000 0
193.2 0 3.00000i 0 5.67763i 0 33.0917 0 −9.00000 0
193.3 0 3.00000i 0 5.67763i 0 −33.0917 0 −9.00000 0
193.4 0 3.00000i 0 17.4288i 0 −2.99032 0 −9.00000 0
193.5 0 3.00000i 0 17.4288i 0 −2.99032 0 −9.00000 0
193.6 0 3.00000i 0 5.67763i 0 −33.0917 0 −9.00000 0
193.7 0 3.00000i 0 5.67763i 0 33.0917 0 −9.00000 0
193.8 0 3.00000i 0 17.4288i 0 2.99032 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.4.d.f 8
3.b odd 2 1 1152.4.d.p 8
4.b odd 2 1 inner 384.4.d.f 8
8.b even 2 1 inner 384.4.d.f 8
8.d odd 2 1 inner 384.4.d.f 8
12.b even 2 1 1152.4.d.p 8
16.e even 4 1 768.4.a.u 4
16.e even 4 1 768.4.a.v 4
16.f odd 4 1 768.4.a.u 4
16.f odd 4 1 768.4.a.v 4
24.f even 2 1 1152.4.d.p 8
24.h odd 2 1 1152.4.d.p 8
48.i odd 4 1 2304.4.a.by 4
48.i odd 4 1 2304.4.a.cb 4
48.k even 4 1 2304.4.a.by 4
48.k even 4 1 2304.4.a.cb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.f 8 1.a even 1 1 trivial
384.4.d.f 8 4.b odd 2 1 inner
384.4.d.f 8 8.b even 2 1 inner
384.4.d.f 8 8.d odd 2 1 inner
768.4.a.u 4 16.e even 4 1
768.4.a.u 4 16.f odd 4 1
768.4.a.v 4 16.e even 4 1
768.4.a.v 4 16.f odd 4 1
1152.4.d.p 8 3.b odd 2 1
1152.4.d.p 8 12.b even 2 1
1152.4.d.p 8 24.f even 2 1
1152.4.d.p 8 24.h odd 2 1
2304.4.a.by 4 48.i odd 4 1
2304.4.a.by 4 48.k even 4 1
2304.4.a.cb 4 48.i odd 4 1
2304.4.a.cb 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 \(S_{4}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{4} + 336 T_{5}^{2} + 9792 \)
\( T_{7}^{4} - 1104 T_{7}^{2} + 9792 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 9 + T^{2} )^{4} \)
$5$ \( ( 9792 + 336 T^{2} + T^{4} )^{2} \)
$7$ \( ( 9792 - 1104 T^{2} + T^{4} )^{2} \)
$11$ \( ( 135424 + 1312 T^{2} + T^{4} )^{2} \)
$13$ \( ( 12690432 + 8640 T^{2} + T^{4} )^{2} \)
$17$ \( ( -3708 - 60 T + T^{2} )^{4} \)
$19$ \( ( 19927296 + 9504 T^{2} + T^{4} )^{2} \)
$23$ \( ( 621831168 - 53568 T^{2} + T^{4} )^{2} \)
$29$ \( ( 419577408 + 41040 T^{2} + T^{4} )^{2} \)
$31$ \( ( 461095488 - 55248 T^{2} + T^{4} )^{2} \)
$37$ \( ( 2487324672 + 107136 T^{2} + T^{4} )^{2} \)
$41$ \( ( -104796 + 204 T + T^{2} )^{4} \)
$43$ \( ( 164249856 + 44064 T^{2} + T^{4} )^{2} \)
$47$ \( ( 21332616192 - 302400 T^{2} + T^{4} )^{2} \)
$53$ \( ( 806557248 + 87888 T^{2} + T^{4} )^{2} \)
$59$ \( ( 7735554304 + 700192 T^{2} + T^{4} )^{2} \)
$61$ \( ( 270509248512 + 1040256 T^{2} + T^{4} )^{2} \)
$67$ \( ( 73992704256 + 838944 T^{2} + T^{4} )^{2} \)
$71$ \( ( 297070322688 - 1104192 T^{2} + T^{4} )^{2} \)
$73$ \( ( -236348 + 484 T + T^{2} )^{4} \)
$79$ \( ( 1904357952 - 1039824 T^{2} + T^{4} )^{2} \)
$83$ \( ( 334010020096 + 1747744 T^{2} + T^{4} )^{2} \)
$89$ \( ( 676836 - 1836 T + T^{2} )^{4} \)
$97$ \( ( 118468 + 740 T + T^{2} )^{4} \)
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