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: \(x^{4} + 1\)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \zeta_{8}^{2} q^{3} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} + ( -10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{7} -9 q^{9} +O(q^{10})\) \( q + 3 \zeta_{8}^{2} q^{3} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} + ( -10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{7} -9 q^{9} + 20 \zeta_{8}^{2} q^{11} + ( -28 \zeta_{8} - 28 \zeta_{8}^{3} ) q^{13} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{15} -34 q^{17} -52 \zeta_{8}^{2} q^{19} + ( -30 \zeta_{8} - 30 \zeta_{8}^{3} ) q^{21} + ( -44 \zeta_{8} + 44 \zeta_{8}^{3} ) q^{23} + 117 q^{25} -27 \zeta_{8}^{2} q^{27} + ( -142 \zeta_{8} - 142 \zeta_{8}^{3} ) q^{29} + ( -78 \zeta_{8} + 78 \zeta_{8}^{3} ) q^{31} -60 q^{33} -40 \zeta_{8}^{2} q^{35} + ( -192 \zeta_{8} - 192 \zeta_{8}^{3} ) q^{37} + ( 84 \zeta_{8} - 84 \zeta_{8}^{3} ) q^{39} + 26 q^{41} + 252 \zeta_{8}^{2} q^{43} + ( -18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{45} + ( 244 \zeta_{8} - 244 \zeta_{8}^{3} ) q^{47} -143 q^{49} -102 \zeta_{8}^{2} q^{51} + ( -482 \zeta_{8} - 482 \zeta_{8}^{3} ) q^{53} + ( -40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{55} + 156 q^{57} + 364 \zeta_{8}^{2} q^{59} + ( -520 \zeta_{8} - 520 \zeta_{8}^{3} ) q^{61} + ( 90 \zeta_{8} - 90 \zeta_{8}^{3} ) q^{63} + 112 q^{65} -628 \zeta_{8}^{2} q^{67} + ( -132 \zeta_{8} - 132 \zeta_{8}^{3} ) q^{69} + ( 236 \zeta_{8} - 236 \zeta_{8}^{3} ) q^{71} -338 q^{73} + 351 \zeta_{8}^{2} q^{75} + ( -200 \zeta_{8} - 200 \zeta_{8}^{3} ) q^{77} + ( -558 \zeta_{8} + 558 \zeta_{8}^{3} ) q^{79} + 81 q^{81} -1036 \zeta_{8}^{2} q^{83} + ( -68 \zeta_{8} - 68 \zeta_{8}^{3} ) q^{85} + ( 426 \zeta_{8} - 426 \zeta_{8}^{3} ) q^{87} -234 q^{89} + 560 \zeta_{8}^{2} q^{91} + ( -234 \zeta_{8} - 234 \zeta_{8}^{3} ) q^{93} + ( 104 \zeta_{8} - 104 \zeta_{8}^{3} ) q^{95} -178 q^{97} -180 \zeta_{8}^{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 36q^{9} + O(q^{10}) \) \( 4q - 36q^{9} - 136q^{17} + 468q^{25} - 240q^{33} + 104q^{41} - 572q^{49} + 624q^{57} + 448q^{65} - 1352q^{73} + 324q^{81} - 936q^{89} - 712q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 \(S_{4}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{2} + 8 \)
\( T_{7}^{2} - 200 \)

Hecke characteristic polynomials

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