Properties

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

Newform invariants

Self dual: no
Analytic conductor: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + 3 \beta_1 q^{3} + \beta_{2} q^{5} - 5 \beta_{3} q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_1 q^{3} + \beta_{2} q^{5} - 5 \beta_{3} q^{7} - 9 q^{9} + 20 \beta_1 q^{11} - 14 \beta_{2} q^{13} - 3 \beta_{3} q^{15} - 34 q^{17} - 52 \beta_1 q^{19} - 15 \beta_{2} q^{21} - 22 \beta_{3} q^{23} + 117 q^{25} - 27 \beta_1 q^{27} - 71 \beta_{2} q^{29} - 39 \beta_{3} q^{31} - 60 q^{33} - 40 \beta_1 q^{35} - 96 \beta_{2} q^{37} + 42 \beta_{3} q^{39} + 26 q^{41} + 252 \beta_1 q^{43} - 9 \beta_{2} q^{45} + 122 \beta_{3} q^{47} - 143 q^{49} - 102 \beta_1 q^{51} - 241 \beta_{2} q^{53} - 20 \beta_{3} q^{55} + 156 q^{57} + 364 \beta_1 q^{59} - 260 \beta_{2} q^{61} + 45 \beta_{3} q^{63} + 112 q^{65} - 628 \beta_1 q^{67} - 66 \beta_{2} q^{69} + 118 \beta_{3} q^{71} - 338 q^{73} + 351 \beta_1 q^{75} - 100 \beta_{2} q^{77} - 279 \beta_{3} q^{79} + 81 q^{81} - 1036 \beta_1 q^{83} - 34 \beta_{2} q^{85} + 213 \beta_{3} q^{87} - 234 q^{89} + 560 \beta_1 q^{91} - 117 \beta_{2} q^{93} + 52 \beta_{3} q^{95} - 178 q^{97} - 180 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 36 q^{9} - 136 q^{17} + 468 q^{25} - 240 q^{33} + 104 q^{41} - 572 q^{49} + 624 q^{57} + 448 q^{65} - 1352 q^{73} + 324 q^{81} - 936 q^{89} - 712 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display

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
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0 3.00000i 0 2.82843i 0 −14.1421 0 −9.00000 0
193.2 0 3.00000i 0 2.82843i 0 14.1421 0 −9.00000 0
193.3 0 3.00000i 0 2.82843i 0 14.1421 0 −9.00000 0
193.4 0 3.00000i 0 2.82843i 0 −14.1421 0 −9.00000 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
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.d 4
3.b odd 2 1 1152.4.d.n 4
4.b odd 2 1 inner 384.4.d.d 4
8.b even 2 1 inner 384.4.d.d 4
8.d odd 2 1 inner 384.4.d.d 4
12.b even 2 1 1152.4.d.n 4
16.e even 4 1 768.4.a.h 2
16.e even 4 1 768.4.a.m 2
16.f odd 4 1 768.4.a.h 2
16.f odd 4 1 768.4.a.m 2
24.f even 2 1 1152.4.d.n 4
24.h odd 2 1 1152.4.d.n 4
48.i odd 4 1 2304.4.a.bb 2
48.i odd 4 1 2304.4.a.bh 2
48.k even 4 1 2304.4.a.bb 2
48.k even 4 1 2304.4.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.d 4 1.a even 1 1 trivial
384.4.d.d 4 4.b odd 2 1 inner
384.4.d.d 4 8.b even 2 1 inner
384.4.d.d 4 8.d odd 2 1 inner
768.4.a.h 2 16.e even 4 1
768.4.a.h 2 16.f odd 4 1
768.4.a.m 2 16.e even 4 1
768.4.a.m 2 16.f odd 4 1
1152.4.d.n 4 3.b odd 2 1
1152.4.d.n 4 12.b even 2 1
1152.4.d.n 4 24.f even 2 1
1152.4.d.n 4 24.h odd 2 1
2304.4.a.bb 2 48.i odd 4 1
2304.4.a.bb 2 48.k even 4 1
2304.4.a.bh 2 48.i odd 4 1
2304.4.a.bh 2 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 \(S_{4}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{7}^{2} - 200 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 200)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1568)^{2} \) Copy content Toggle raw display
$17$ \( (T + 34)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2704)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 3872)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 40328)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12168)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 73728)^{2} \) Copy content Toggle raw display
$41$ \( (T - 26)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 63504)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 119072)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 464648)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 132496)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 540800)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 394384)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 111392)^{2} \) Copy content Toggle raw display
$73$ \( (T + 338)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 622728)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 1073296)^{2} \) Copy content Toggle raw display
$89$ \( (T + 234)^{4} \) Copy content Toggle raw display
$97$ \( (T + 178)^{4} \) Copy content Toggle raw display
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