Properties

Label 384.3.i.b
Level $384$
Weight $3$
Character orbit 384.i
Analytic conductor $10.463$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.i (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.629407744.1
Defining polynomial: \(x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{3} + ( 3 \beta_{4} - \beta_{6} - \beta_{7} ) q^{5} + ( 1 - \beta_{2} - \beta_{3} - \beta_{6} ) q^{7} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{3} + ( 3 \beta_{4} - \beta_{6} - \beta_{7} ) q^{5} + ( 1 - \beta_{2} - \beta_{3} - \beta_{6} ) q^{7} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{9} + 2 \beta_{4} q^{11} + ( 11 + 2 \beta_{2} + 13 \beta_{3} + \beta_{6} + \beta_{7} ) q^{13} + ( -14 + 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{15} + ( \beta_{1} - \beta_{4} - 9 \beta_{5} - 8 \beta_{6} - 8 \beta_{7} ) q^{17} + ( -2 + 8 \beta_{2} + 6 \beta_{3} + 4 \beta_{6} + 4 \beta_{7} ) q^{19} + ( 5 - 3 \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{21} + ( 2 \beta_{1} + 2 \beta_{4} - 10 \beta_{5} ) q^{23} + ( 14 - 14 \beta_{2} + 17 \beta_{3} - 14 \beta_{6} ) q^{25} + ( -5 + 12 \beta_{3} + \beta_{4} + 2 \beta_{6} + 9 \beta_{7} ) q^{27} + ( -7 \beta_{1} - 7 \beta_{5} - 7 \beta_{6} - 7 \beta_{7} ) q^{29} + ( -9 - \beta_{2} - \beta_{3} - \beta_{7} ) q^{31} + ( -8 + 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 6 \beta_{7} ) q^{33} + ( -4 \beta_{1} + 6 \beta_{5} + \beta_{6} + \beta_{7} ) q^{35} + ( -9 + 19 \beta_{3} - 5 \beta_{6} + 5 \beta_{7} ) q^{37} + ( -13 - 10 \beta_{1} + 13 \beta_{2} + 19 \beta_{3} - 10 \beta_{4} - 3 \beta_{5} + 13 \beta_{6} ) q^{39} + ( \beta_{1} + \beta_{4} + 7 \beta_{5} ) q^{41} + ( -34 + 26 \beta_{3} + 4 \beta_{6} - 4 \beta_{7} ) q^{43} + ( 24 + 17 \beta_{1} - 20 \beta_{2} + 4 \beta_{3} - 15 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} ) q^{45} + ( -14 \beta_{1} + 14 \beta_{4} + 24 \beta_{5} + 10 \beta_{6} + 10 \beta_{7} ) q^{47} + ( 41 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} ) q^{49} + ( 6 - 10 \beta_{1} - 20 \beta_{2} - 14 \beta_{3} - 24 \beta_{5} - 27 \beta_{6} - 27 \beta_{7} ) q^{51} + ( 7 \beta_{4} - 15 \beta_{6} - 15 \beta_{7} ) q^{53} + ( 12 - 12 \beta_{2} + 28 \beta_{3} - 12 \beta_{6} ) q^{55} + ( -6 + 6 \beta_{1} + 6 \beta_{2} + 30 \beta_{3} + 6 \beta_{4} - 12 \beta_{5} + 6 \beta_{6} ) q^{57} + ( 10 \beta_{4} + 5 \beta_{6} + 5 \beta_{7} ) q^{59} + ( -31 + 10 \beta_{2} - 21 \beta_{3} + 5 \beta_{6} + 5 \beta_{7} ) q^{61} + ( 13 - 4 \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} - 6 \beta_{5} - 10 \beta_{6} - 9 \beta_{7} ) q^{63} + ( -29 \beta_{1} + 29 \beta_{4} + 5 \beta_{5} - 24 \beta_{6} - 24 \beta_{7} ) q^{65} + ( -44 + 30 \beta_{2} - 14 \beta_{3} + 15 \beta_{6} + 15 \beta_{7} ) q^{67} + ( 12 + 12 \beta_{3} - 12 \beta_{4} + 6 \beta_{6} + 30 \beta_{7} ) q^{69} + ( 16 \beta_{1} + 16 \beta_{4} + 26 \beta_{5} ) q^{71} + ( -14 + 14 \beta_{2} - 22 \beta_{3} + 14 \beta_{6} ) q^{73} + ( 39 - 42 \beta_{3} - 45 \beta_{4} + 3 \beta_{6} ) q^{75} + ( -4 \beta_{1} - 2 \beta_{6} - 2 \beta_{7} ) q^{77} + ( 17 + 13 \beta_{2} + 13 \beta_{3} + 13 \beta_{7} ) q^{79} + ( 23 + 22 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} - 22 \beta_{4} + 6 \beta_{5} + 28 \beta_{6} + 36 \beta_{7} ) q^{81} + ( 38 \beta_{1} - 2 \beta_{5} + 18 \beta_{6} + 18 \beta_{7} ) q^{83} + ( -8 - 36 \beta_{3} + 22 \beta_{6} - 22 \beta_{7} ) q^{85} + ( -42 \beta_{3} - 21 \beta_{5} ) q^{87} + ( -18 \beta_{1} - 18 \beta_{4} + 10 \beta_{5} ) q^{89} + ( 32 - 6 \beta_{3} - 13 \beta_{6} + 13 \beta_{7} ) q^{91} + ( -3 + 7 \beta_{1} - 10 \beta_{2} - 13 \beta_{3} + 3 \beta_{5} ) q^{93} + ( 22 \beta_{1} - 22 \beta_{4} - 26 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{95} + ( -60 - 8 \beta_{2} - 8 \beta_{3} - 8 \beta_{7} ) q^{97} + ( 24 + 14 \beta_{1} - 8 \beta_{2} + 16 \beta_{3} - 6 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{3} + O(q^{10}) \) \( 8q + 4q^{3} + 96q^{13} - 112q^{15} + 16q^{19} + 32q^{21} - 68q^{27} - 72q^{31} - 64q^{33} - 112q^{37} - 240q^{43} + 112q^{45} + 328q^{49} - 32q^{51} - 208q^{61} + 104q^{63} - 232q^{67} + 324q^{75} + 136q^{79} + 184q^{81} + 112q^{85} + 152q^{91} - 64q^{93} - 480q^{97} + 160q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{5} + 10 \nu^{3} + 24 \nu^{2} + 8 \nu \)\()/24\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 2 \nu^{4} - 2 \nu^{2} - 4 \)\()/12\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} - 10 \nu^{3} + 4 \nu \)\()/12\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + \nu^{5} + 2 \nu^{3} + 10 \nu \)\()/6\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 8 \nu^{6} - 2 \nu^{5} - 8 \nu^{4} - 10 \nu^{3} + 8 \nu^{2} - 8 \nu - 32 \)\()/24\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} - 8 \nu^{6} - 2 \nu^{5} + 8 \nu^{4} - 10 \nu^{3} - 8 \nu^{2} - 8 \nu + 32 \)\()/24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + \beta_{6} + 2 \beta_{2}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\(\beta_{7} + 4 \beta_{3} + \beta_{2}\)
\(\nu^{5}\)\(=\)\(-2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} - \beta_{1}\)
\(\nu^{6}\)\(=\)\(-\beta_{7} + \beta_{6} + 4 \beta_{3} + 4\)
\(\nu^{7}\)\(=\)\(-3 \beta_{7} - 3 \beta_{6} - 5 \beta_{5} + \beta_{4} + 3 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-\beta_{3}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.767178 1.18804i
1.38255 + 0.297594i
−0.767178 + 1.18804i
−1.38255 0.297594i
0.767178 + 1.18804i
1.38255 0.297594i
−0.767178 1.18804i
−1.38255 + 0.297594i
0 −2.77809 + 1.13234i 0 6.28651 + 6.28651i 0 1.64575i 0 6.43560 6.29150i 0
161.2 0 0.737922 + 2.90783i 0 1.57472 + 1.57472i 0 3.64575i 0 −7.91094 + 4.29150i 0
161.3 0 1.13234 2.77809i 0 −6.28651 6.28651i 0 1.64575i 0 −6.43560 6.29150i 0
161.4 0 2.90783 + 0.737922i 0 −1.57472 1.57472i 0 3.64575i 0 7.91094 + 4.29150i 0
353.1 0 −2.77809 1.13234i 0 6.28651 6.28651i 0 1.64575i 0 6.43560 + 6.29150i 0
353.2 0 0.737922 2.90783i 0 1.57472 1.57472i 0 3.64575i 0 −7.91094 4.29150i 0
353.3 0 1.13234 + 2.77809i 0 −6.28651 + 6.28651i 0 1.64575i 0 −6.43560 + 6.29150i 0
353.4 0 2.90783 0.737922i 0 −1.57472 + 1.57472i 0 3.64575i 0 7.91094 4.29150i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 353.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.3.i.b 8
3.b odd 2 1 inner 384.3.i.b 8
4.b odd 2 1 384.3.i.a 8
8.b even 2 1 192.3.i.a 8
8.d odd 2 1 48.3.i.a 8
12.b even 2 1 384.3.i.a 8
16.e even 4 1 192.3.i.a 8
16.e even 4 1 inner 384.3.i.b 8
16.f odd 4 1 48.3.i.a 8
16.f odd 4 1 384.3.i.a 8
24.f even 2 1 48.3.i.a 8
24.h odd 2 1 192.3.i.a 8
48.i odd 4 1 192.3.i.a 8
48.i odd 4 1 inner 384.3.i.b 8
48.k even 4 1 48.3.i.a 8
48.k even 4 1 384.3.i.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.i.a 8 8.d odd 2 1
48.3.i.a 8 16.f odd 4 1
48.3.i.a 8 24.f even 2 1
48.3.i.a 8 48.k even 4 1
192.3.i.a 8 8.b even 2 1
192.3.i.a 8 16.e even 4 1
192.3.i.a 8 24.h odd 2 1
192.3.i.a 8 48.i odd 4 1
384.3.i.a 8 4.b odd 2 1
384.3.i.a 8 12.b even 2 1
384.3.i.a 8 16.f odd 4 1
384.3.i.a 8 48.k even 4 1
384.3.i.b 8 1.a even 1 1 trivial
384.3.i.b 8 3.b odd 2 1 inner
384.3.i.b 8 16.e even 4 1 inner
384.3.i.b 8 48.i odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{8} + 6272 T_{5}^{4} + 153664 \)
\( T_{19}^{4} - 8 T_{19}^{3} + 32 T_{19}^{2} + 1728 T_{19} + 46656 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 6561 - 2916 T + 648 T^{2} + 108 T^{3} - 126 T^{4} + 12 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} \)
$5$ \( 153664 + 6272 T^{4} + T^{8} \)
$7$ \( ( 36 + 16 T^{2} + T^{4} )^{2} \)
$11$ \( 16384 + 2048 T^{4} + T^{8} \)
$13$ \( ( 75076 - 13152 T + 1152 T^{2} - 48 T^{3} + T^{4} )^{2} \)
$17$ \( ( 103968 + 920 T^{2} + T^{4} )^{2} \)
$19$ \( ( 46656 + 1728 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$23$ \( ( 225792 - 1120 T^{2} + T^{4} )^{2} \)
$29$ \( 29884728384 + 614656 T^{4} + T^{8} \)
$31$ \( ( 74 + 18 T + T^{2} )^{4} \)
$37$ \( ( 1764 + 2352 T + 1568 T^{2} + 56 T^{3} + T^{4} )^{2} \)
$41$ \( ( 2592 - 664 T^{2} + T^{4} )^{2} \)
$43$ \( ( 2483776 + 189120 T + 7200 T^{2} + 120 T^{3} + T^{4} )^{2} \)
$47$ \( ( 14193792 + 8144 T^{2} + T^{4} )^{2} \)
$53$ \( 18847994944 + 18284288 T^{4} + T^{8} \)
$59$ \( 2624400000000 + 3520000 T^{4} + T^{8} \)
$61$ \( ( 1004004 + 104208 T + 5408 T^{2} + 104 T^{3} + T^{4} )^{2} \)
$67$ \( ( 2155024 - 170288 T + 6728 T^{2} + 116 T^{3} + T^{4} )^{2} \)
$71$ \( ( 24668288 - 16560 T^{2} + T^{4} )^{2} \)
$73$ \( ( 788544 + 3712 T^{2} + T^{4} )^{2} \)
$79$ \( ( -894 - 34 T + T^{2} )^{4} \)
$83$ \( 1048798070063104 + 135735296 T^{4} + T^{8} \)
$89$ \( ( 184832 - 6240 T^{2} + T^{4} )^{2} \)
$97$ \( ( 3152 + 120 T + T^{2} )^{4} \)
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