Properties

Label 384.6.f.b
Level $384$
Weight $6$
Character orbit 384.f
Analytic conductor $61.587$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 384.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(61.5873868082\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -11 \zeta_{8} - \zeta_{8}^{2} + 11 \zeta_{8}^{3} ) q^{3} + ( 241 + 22 \zeta_{8} + 22 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -11 \zeta_{8} - \zeta_{8}^{2} + 11 \zeta_{8}^{3} ) q^{3} + ( 241 + 22 \zeta_{8} + 22 \zeta_{8}^{3} ) q^{9} + 474 \zeta_{8}^{2} q^{11} + ( -1004 \zeta_{8} - 1004 \zeta_{8}^{3} ) q^{17} + ( 894 \zeta_{8} - 894 \zeta_{8}^{3} ) q^{19} -3125 q^{25} + ( -2629 \zeta_{8} - 725 \zeta_{8}^{2} + 2629 \zeta_{8}^{3} ) q^{27} + ( 474 - 5214 \zeta_{8} - 5214 \zeta_{8}^{3} ) q^{33} + ( -11608 \zeta_{8} - 11608 \zeta_{8}^{3} ) q^{41} + ( 6306 \zeta_{8} - 6306 \zeta_{8}^{3} ) q^{43} + 16807 q^{49} + ( -1004 \zeta_{8} + 22088 \zeta_{8}^{2} + 1004 \zeta_{8}^{3} ) q^{51} + ( -19668 - 894 \zeta_{8} - 894 \zeta_{8}^{3} ) q^{57} + 48486 \zeta_{8}^{2} q^{59} + ( -21054 \zeta_{8} + 21054 \zeta_{8}^{3} ) q^{67} -50402 q^{73} + ( 34375 \zeta_{8} + 3125 \zeta_{8}^{2} - 34375 \zeta_{8}^{3} ) q^{75} + ( 57113 + 10604 \zeta_{8} + 10604 \zeta_{8}^{3} ) q^{81} + 89298 \zeta_{8}^{2} q^{83} + ( 105556 \zeta_{8} + 105556 \zeta_{8}^{3} ) q^{89} + 85450 q^{97} + ( -10428 \zeta_{8} + 114234 \zeta_{8}^{2} + 10428 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 964q^{9} + O(q^{10}) \) \( 4q + 964q^{9} - 12500q^{25} + 1896q^{33} + 67228q^{49} - 78672q^{57} - 201608q^{73} + 228452q^{81} + 341800q^{97} + O(q^{100}) \)

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
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0 −15.5563 1.00000i 0 0 0 0 0 241.000 + 31.1127i 0
191.2 0 −15.5563 + 1.00000i 0 0 0 0 0 241.000 31.1127i 0
191.3 0 15.5563 1.00000i 0 0 0 0 0 241.000 31.1127i 0
191.4 0 15.5563 + 1.00000i 0 0 0 0 0 241.000 + 31.1127i 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.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.6.f.b 4
3.b odd 2 1 inner 384.6.f.b 4
4.b odd 2 1 inner 384.6.f.b 4
8.b even 2 1 inner 384.6.f.b 4
8.d odd 2 1 CM 384.6.f.b 4
12.b even 2 1 inner 384.6.f.b 4
24.f even 2 1 inner 384.6.f.b 4
24.h odd 2 1 inner 384.6.f.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.6.f.b 4 1.a even 1 1 trivial
384.6.f.b 4 3.b odd 2 1 inner
384.6.f.b 4 4.b odd 2 1 inner
384.6.f.b 4 8.b even 2 1 inner
384.6.f.b 4 8.d odd 2 1 CM
384.6.f.b 4 12.b even 2 1 inner
384.6.f.b 4 24.f even 2 1 inner
384.6.f.b 4 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(384, [\chi])\):

\( T_{5} \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 59049 - 482 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 224676 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( 2016032 + T^{2} )^{2} \)
$19$ \( ( -1598472 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 269491328 + T^{2} )^{2} \)
$43$ \( ( -79531272 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( ( 2350892196 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( -886541832 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( ( 50402 + T )^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 7974132804 + T^{2} )^{2} \)
$89$ \( ( 22284138272 + T^{2} )^{2} \)
$97$ \( ( -85450 + T )^{4} \)
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