Properties

Label 768.2.d.d
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 24)
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.

\(f(q)\) \(=\) \( q + i q^{3} + 2 i q^{5} - q^{9} +O(q^{10})\) \( q + i q^{3} + 2 i q^{5} - q^{9} + 4 i q^{11} -2 i q^{13} -2 q^{15} + 2 q^{17} + 4 i q^{19} -8 q^{23} + q^{25} -i q^{27} + 6 i q^{29} -8 q^{31} -4 q^{33} -6 i q^{37} + 2 q^{39} + 6 q^{41} + 4 i q^{43} -2 i q^{45} -7 q^{49} + 2 i q^{51} + 2 i q^{53} -8 q^{55} -4 q^{57} + 4 i q^{59} -2 i q^{61} + 4 q^{65} + 4 i q^{67} -8 i q^{69} + 8 q^{71} -10 q^{73} + i q^{75} + 8 q^{79} + q^{81} + 4 i q^{83} + 4 i q^{85} -6 q^{87} + 6 q^{89} -8 i q^{93} -8 q^{95} + 2 q^{97} -4 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 4q^{15} + 4q^{17} - 16q^{23} + 2q^{25} - 16q^{31} - 8q^{33} + 4q^{39} + 12q^{41} - 14q^{49} - 16q^{55} - 8q^{57} + 8q^{65} + 16q^{71} - 20q^{73} + 16q^{79} + 2q^{81} - 12q^{87} + 12q^{89} - 16q^{95} + 4q^{97} + O(q^{100}) \)

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 2.00000i 0 0 0 −1.00000 0
385.2 0 1.00000i 0 2.00000i 0 0 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.d 2
3.b odd 2 1 2304.2.d.k 2
4.b odd 2 1 768.2.d.e 2
8.b even 2 1 inner 768.2.d.d 2
8.d odd 2 1 768.2.d.e 2
12.b even 2 1 2304.2.d.i 2
16.e even 4 1 48.2.a.a 1
16.e even 4 1 192.2.a.b 1
16.f odd 4 1 24.2.a.a 1
16.f odd 4 1 192.2.a.d 1
24.f even 2 1 2304.2.d.i 2
24.h odd 2 1 2304.2.d.k 2
48.i odd 4 1 144.2.a.b 1
48.i odd 4 1 576.2.a.b 1
48.k even 4 1 72.2.a.a 1
48.k even 4 1 576.2.a.d 1
80.i odd 4 1 1200.2.f.b 2
80.i odd 4 1 4800.2.f.bg 2
80.j even 4 1 600.2.f.e 2
80.j even 4 1 4800.2.f.d 2
80.k odd 4 1 600.2.a.h 1
80.k odd 4 1 4800.2.a.q 1
80.q even 4 1 1200.2.a.d 1
80.q even 4 1 4800.2.a.cc 1
80.s even 4 1 600.2.f.e 2
80.s even 4 1 4800.2.f.d 2
80.t odd 4 1 1200.2.f.b 2
80.t odd 4 1 4800.2.f.bg 2
112.j even 4 1 1176.2.a.i 1
112.j even 4 1 9408.2.a.h 1
112.l odd 4 1 2352.2.a.i 1
112.l odd 4 1 9408.2.a.cc 1
112.u odd 12 2 1176.2.q.i 2
112.v even 12 2 1176.2.q.a 2
112.w even 12 2 2352.2.q.l 2
112.x odd 12 2 2352.2.q.r 2
144.u even 12 2 648.2.i.b 2
144.v odd 12 2 648.2.i.g 2
144.w odd 12 2 1296.2.i.e 2
144.x even 12 2 1296.2.i.m 2
176.i even 4 1 2904.2.a.c 1
176.l odd 4 1 5808.2.a.s 1
208.l even 4 1 4056.2.c.e 2
208.o odd 4 1 4056.2.a.i 1
208.p even 4 1 8112.2.a.be 1
208.s even 4 1 4056.2.c.e 2
240.t even 4 1 1800.2.a.m 1
240.z odd 4 1 1800.2.f.c 2
240.bb even 4 1 3600.2.f.r 2
240.bd odd 4 1 1800.2.f.c 2
240.bf even 4 1 3600.2.f.r 2
240.bm odd 4 1 3600.2.a.v 1
272.k odd 4 1 6936.2.a.p 1
304.m even 4 1 8664.2.a.j 1
336.v odd 4 1 3528.2.a.d 1
336.y even 4 1 7056.2.a.q 1
336.br odd 12 2 3528.2.s.y 2
336.bu even 12 2 3528.2.s.j 2
528.s odd 4 1 8712.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 16.f odd 4 1
48.2.a.a 1 16.e even 4 1
72.2.a.a 1 48.k even 4 1
144.2.a.b 1 48.i odd 4 1
192.2.a.b 1 16.e even 4 1
192.2.a.d 1 16.f odd 4 1
576.2.a.b 1 48.i odd 4 1
576.2.a.d 1 48.k even 4 1
600.2.a.h 1 80.k odd 4 1
600.2.f.e 2 80.j even 4 1
600.2.f.e 2 80.s even 4 1
648.2.i.b 2 144.u even 12 2
648.2.i.g 2 144.v odd 12 2
768.2.d.d 2 1.a even 1 1 trivial
768.2.d.d 2 8.b even 2 1 inner
768.2.d.e 2 4.b odd 2 1
768.2.d.e 2 8.d odd 2 1
1176.2.a.i 1 112.j even 4 1
1176.2.q.a 2 112.v even 12 2
1176.2.q.i 2 112.u odd 12 2
1200.2.a.d 1 80.q even 4 1
1200.2.f.b 2 80.i odd 4 1
1200.2.f.b 2 80.t odd 4 1
1296.2.i.e 2 144.w odd 12 2
1296.2.i.m 2 144.x even 12 2
1800.2.a.m 1 240.t even 4 1
1800.2.f.c 2 240.z odd 4 1
1800.2.f.c 2 240.bd odd 4 1
2304.2.d.i 2 12.b even 2 1
2304.2.d.i 2 24.f even 2 1
2304.2.d.k 2 3.b odd 2 1
2304.2.d.k 2 24.h odd 2 1
2352.2.a.i 1 112.l odd 4 1
2352.2.q.l 2 112.w even 12 2
2352.2.q.r 2 112.x odd 12 2
2904.2.a.c 1 176.i even 4 1
3528.2.a.d 1 336.v odd 4 1
3528.2.s.j 2 336.bu even 12 2
3528.2.s.y 2 336.br odd 12 2
3600.2.a.v 1 240.bm odd 4 1
3600.2.f.r 2 240.bb even 4 1
3600.2.f.r 2 240.bf even 4 1
4056.2.a.i 1 208.o odd 4 1
4056.2.c.e 2 208.l even 4 1
4056.2.c.e 2 208.s even 4 1
4800.2.a.q 1 80.k odd 4 1
4800.2.a.cc 1 80.q even 4 1
4800.2.f.d 2 80.j even 4 1
4800.2.f.d 2 80.s even 4 1
4800.2.f.bg 2 80.i odd 4 1
4800.2.f.bg 2 80.t odd 4 1
5808.2.a.s 1 176.l odd 4 1
6936.2.a.p 1 272.k odd 4 1
7056.2.a.q 1 336.y even 4 1
8112.2.a.be 1 208.p even 4 1
8664.2.a.j 1 304.m even 4 1
8712.2.a.u 1 528.s odd 4 1
9408.2.a.h 1 112.j even 4 1
9408.2.a.cc 1 112.l odd 4 1

Hecke kernels

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

\( T_{5}^{2} + 4 \)
\( T_{7} \)
\( T_{23} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T^{2} \)
$5$ \( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \)
$7$ \( ( 1 + 7 T^{2} )^{2} \)
$11$ \( 1 - 6 T^{2} + 121 T^{4} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( ( 1 - 2 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 22 T^{2} + 361 T^{4} \)
$23$ \( ( 1 + 8 T + 23 T^{2} )^{2} \)
$29$ \( 1 - 22 T^{2} + 841 T^{4} \)
$31$ \( ( 1 + 8 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 38 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( 1 - 102 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 102 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 118 T^{2} + 3721 T^{4} \)
$67$ \( 1 - 118 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 8 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 10 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 150 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 2 T + 97 T^{2} )^{2} \)
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