Properties

Label 768.2.d.c
Level $768$
Weight $2$
Character orbit 768.d
Analytic conductor $6.133$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.13251087523\)
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: no (minimal twist has level 384)
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.

\(f(q)\) \(=\) \( q -i q^{3} + 4 i q^{5} -2 q^{7} - q^{9} +O(q^{10})\) \( q -i q^{3} + 4 i q^{5} -2 q^{7} - q^{9} -4 i q^{11} + 2 i q^{13} + 4 q^{15} -2 q^{17} + 8 i q^{19} + 2 i q^{21} -4 q^{23} -11 q^{25} + i q^{27} -6 q^{31} -4 q^{33} -8 i q^{35} + 2 i q^{37} + 2 q^{39} -6 q^{41} -4 i q^{45} + 4 q^{47} -3 q^{49} + 2 i q^{51} + 16 q^{55} + 8 q^{57} + 4 i q^{59} + 14 i q^{61} + 2 q^{63} -8 q^{65} + 4 i q^{67} + 4 i q^{69} -12 q^{71} + 10 q^{73} + 11 i q^{75} + 8 i q^{77} + 10 q^{79} + q^{81} -12 i q^{83} -8 i q^{85} + 14 q^{89} -4 i q^{91} + 6 i q^{93} -32 q^{95} + 10 q^{97} + 4 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{7} - 2q^{9} + O(q^{10}) \) \( 2q - 4q^{7} - 2q^{9} + 8q^{15} - 4q^{17} - 8q^{23} - 22q^{25} - 12q^{31} - 8q^{33} + 4q^{39} - 12q^{41} + 8q^{47} - 6q^{49} + 32q^{55} + 16q^{57} + 4q^{63} - 16q^{65} - 24q^{71} + 20q^{73} + 20q^{79} + 2q^{81} + 28q^{89} - 64q^{95} + 20q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
385.1
1.00000i
1.00000i
0 1.00000i 0 4.00000i 0 −2.00000 0 −1.00000 0
385.2 0 1.00000i 0 4.00000i 0 −2.00000 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 768.2.d.c 2
3.b odd 2 1 2304.2.d.f 2
4.b odd 2 1 768.2.d.f 2
8.b even 2 1 inner 768.2.d.c 2
8.d odd 2 1 768.2.d.f 2
12.b even 2 1 2304.2.d.o 2
16.e even 4 1 384.2.a.a 1
16.e even 4 1 384.2.a.h yes 1
16.f odd 4 1 384.2.a.d yes 1
16.f odd 4 1 384.2.a.e yes 1
24.f even 2 1 2304.2.d.o 2
24.h odd 2 1 2304.2.d.f 2
48.i odd 4 1 1152.2.a.b 1
48.i odd 4 1 1152.2.a.t 1
48.k even 4 1 1152.2.a.a 1
48.k even 4 1 1152.2.a.s 1
80.k odd 4 1 9600.2.a.t 1
80.k odd 4 1 9600.2.a.bz 1
80.q even 4 1 9600.2.a.e 1
80.q even 4 1 9600.2.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.a.a 1 16.e even 4 1
384.2.a.d yes 1 16.f odd 4 1
384.2.a.e yes 1 16.f odd 4 1
384.2.a.h yes 1 16.e even 4 1
768.2.d.c 2 1.a even 1 1 trivial
768.2.d.c 2 8.b even 2 1 inner
768.2.d.f 2 4.b odd 2 1
768.2.d.f 2 8.d odd 2 1
1152.2.a.a 1 48.k even 4 1
1152.2.a.b 1 48.i odd 4 1
1152.2.a.s 1 48.k even 4 1
1152.2.a.t 1 48.i odd 4 1
2304.2.d.f 2 3.b odd 2 1
2304.2.d.f 2 24.h odd 2 1
2304.2.d.o 2 12.b even 2 1
2304.2.d.o 2 24.f even 2 1
9600.2.a.e 1 80.q even 4 1
9600.2.a.t 1 80.k odd 4 1
9600.2.a.bk 1 80.q even 4 1
9600.2.a.bz 1 80.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} + 16 \)
\( T_{7} + 2 \)
\( T_{23} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 16 + T^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( 16 + T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( 64 + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 6 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( ( -4 + T )^{2} \)
$53$ \( T^{2} \)
$59$ \( 16 + T^{2} \)
$61$ \( 196 + T^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( ( -10 + T )^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( -14 + T )^{2} \)
$97$ \( ( -10 + T )^{2} \)
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