Properties

Label 768.2.d.b
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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + i q^{3} - 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} - 2 q^{7} - q^{9} - 4 i q^{11} - 6 i q^{13} + 6 q^{17} - 2 i q^{21} - 4 q^{23} + 5 q^{25} - i q^{27} - 4 i q^{29} + 10 q^{31} + 4 q^{33} + 2 i q^{37} + 6 q^{39} + 2 q^{41} + 8 i q^{43} - 12 q^{47} - 3 q^{49} + 6 i q^{51} - 12 i q^{53} - 4 i q^{59} - 2 i q^{61} + 2 q^{63} - 4 i q^{67} - 4 i q^{69} + 4 q^{71} + 10 q^{73} + 5 i q^{75} + 8 i q^{77} - 6 q^{79} + q^{81} - 12 i q^{83} + 4 q^{87} - 2 q^{89} + 12 i q^{91} + 10 i q^{93} - 6 q^{97} + 4 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{7} - 2 q^{9} + 12 q^{17} - 8 q^{23} + 10 q^{25} + 20 q^{31} + 8 q^{33} + 12 q^{39} + 4 q^{41} - 24 q^{47} - 6 q^{49} + 4 q^{63} + 8 q^{71} + 20 q^{73} - 12 q^{79} + 2 q^{81} + 8 q^{87} - 4 q^{89} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 0 0 −2.00000 0 −1.00000 0
385.2 0 1.00000i 0 0 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.b 2
3.b odd 2 1 2304.2.d.d 2
4.b odd 2 1 768.2.d.g 2
8.b even 2 1 inner 768.2.d.b 2
8.d odd 2 1 768.2.d.g 2
12.b even 2 1 2304.2.d.m 2
16.e even 4 1 384.2.a.c yes 1
16.e even 4 1 384.2.a.g yes 1
16.f odd 4 1 384.2.a.b 1
16.f odd 4 1 384.2.a.f yes 1
24.f even 2 1 2304.2.d.m 2
24.h odd 2 1 2304.2.d.d 2
48.i odd 4 1 1152.2.a.k 1
48.i odd 4 1 1152.2.a.l 1
48.k even 4 1 1152.2.a.i 1
48.k even 4 1 1152.2.a.j 1
80.k odd 4 1 9600.2.a.w 1
80.k odd 4 1 9600.2.a.bw 1
80.q even 4 1 9600.2.a.h 1
80.q even 4 1 9600.2.a.bh 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.a.b 1 16.f odd 4 1
384.2.a.c yes 1 16.e even 4 1
384.2.a.f yes 1 16.f odd 4 1
384.2.a.g yes 1 16.e even 4 1
768.2.d.b 2 1.a even 1 1 trivial
768.2.d.b 2 8.b even 2 1 inner
768.2.d.g 2 4.b odd 2 1
768.2.d.g 2 8.d odd 2 1
1152.2.a.i 1 48.k even 4 1
1152.2.a.j 1 48.k even 4 1
1152.2.a.k 1 48.i odd 4 1
1152.2.a.l 1 48.i odd 4 1
2304.2.d.d 2 3.b odd 2 1
2304.2.d.d 2 24.h odd 2 1
2304.2.d.m 2 12.b even 2 1
2304.2.d.m 2 24.f even 2 1
9600.2.a.h 1 80.q even 4 1
9600.2.a.w 1 80.k odd 4 1
9600.2.a.bh 1 80.q even 4 1
9600.2.a.bw 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 \(S_{2}^{\mathrm{new}}(768, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{23} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 16 \) Copy content Toggle raw display
$31$ \( (T - 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 64 \) Copy content Toggle raw display
$47$ \( (T + 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 144 \) Copy content Toggle raw display
$59$ \( T^{2} + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T - 4)^{2} \) Copy content Toggle raw display
$73$ \( (T - 10)^{2} \) Copy content Toggle raw display
$79$ \( (T + 6)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T + 2)^{2} \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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