Properties

Label 384.2.d.a
Level $384$
Weight $2$
Character orbit 384.d
Analytic conductor $3.066$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.06625543762\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{3} -4 q^{7} - q^{9} +O(q^{10})\) \( q + i q^{3} -4 q^{7} - q^{9} + 4 i q^{11} + 4 i q^{13} -2 q^{17} + 4 i q^{19} -4 i q^{21} -8 q^{23} + 5 q^{25} -i q^{27} -8 i q^{29} + 4 q^{31} -4 q^{33} + 4 i q^{37} -4 q^{39} -6 q^{41} + 4 i q^{43} + 8 q^{47} + 9 q^{49} -2 i q^{51} + 8 i q^{53} -4 q^{57} -12 i q^{59} + 12 i q^{61} + 4 q^{63} -12 i q^{67} -8 i q^{69} + 8 q^{71} + 6 q^{73} + 5 i q^{75} -16 i q^{77} + 4 q^{79} + q^{81} + 4 i q^{83} + 8 q^{87} + 6 q^{89} -16 i q^{91} + 4 i q^{93} -2 q^{97} -4 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{7} - 2q^{9} + O(q^{10}) \) \( 2q - 8q^{7} - 2q^{9} - 4q^{17} - 16q^{23} + 10q^{25} + 8q^{31} - 8q^{33} - 8q^{39} - 12q^{41} + 16q^{47} + 18q^{49} - 8q^{57} + 8q^{63} + 16q^{71} + 12q^{73} + 8q^{79} + 2q^{81} + 16q^{87} + 12q^{89} - 4q^{97} + O(q^{100}) \)

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
1.00000i
1.00000i
0 1.00000i 0 0 0 −4.00000 0 −1.00000 0
193.2 0 1.00000i 0 0 0 −4.00000 0 −1.00000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.2.d.a 2
3.b odd 2 1 1152.2.d.a 2
4.b odd 2 1 384.2.d.b yes 2
8.b even 2 1 inner 384.2.d.a 2
8.d odd 2 1 384.2.d.b yes 2
12.b even 2 1 1152.2.d.f 2
16.e even 4 1 768.2.a.c 1
16.e even 4 1 768.2.a.g 1
16.f odd 4 1 768.2.a.b 1
16.f odd 4 1 768.2.a.f 1
24.f even 2 1 1152.2.d.f 2
24.h odd 2 1 1152.2.d.a 2
48.i odd 4 1 2304.2.a.j 1
48.i odd 4 1 2304.2.a.k 1
48.k even 4 1 2304.2.a.f 1
48.k even 4 1 2304.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.d.a 2 1.a even 1 1 trivial
384.2.d.a 2 8.b even 2 1 inner
384.2.d.b yes 2 4.b odd 2 1
384.2.d.b yes 2 8.d odd 2 1
768.2.a.b 1 16.f odd 4 1
768.2.a.c 1 16.e even 4 1
768.2.a.f 1 16.f odd 4 1
768.2.a.g 1 16.e even 4 1
1152.2.d.a 2 3.b odd 2 1
1152.2.d.a 2 24.h odd 2 1
1152.2.d.f 2 12.b even 2 1
1152.2.d.f 2 24.f even 2 1
2304.2.a.f 1 48.k even 4 1
2304.2.a.g 1 48.k even 4 1
2304.2.a.j 1 48.i odd 4 1
2304.2.a.k 1 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(384, [\chi])\):

\( T_{5} \)
\( T_{7} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 4 + T )^{2} \)
$11$ \( 16 + T^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( 16 + T^{2} \)
$23$ \( ( 8 + T )^{2} \)
$29$ \( 64 + T^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 16 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( 64 + T^{2} \)
$59$ \( 144 + T^{2} \)
$61$ \( 144 + T^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( ( -6 + T )^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( ( 2 + T )^{2} \)
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