Properties

Label 768.3.b.c
Level $768$
Weight $3$
Character orbit 768.b
Analytic conductor $20.926$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 12)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{3} q^{5} + ( -4 + 8 \zeta_{12}^{2} ) q^{7} + 3 q^{9} +O(q^{10})\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{3} q^{5} + ( -4 + 8 \zeta_{12}^{2} ) q^{7} + 3 q^{9} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{11} + 2 \zeta_{12}^{3} q^{13} + ( 2 - 4 \zeta_{12}^{2} ) q^{15} + 10 q^{17} + ( -24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{19} -12 \zeta_{12}^{3} q^{21} + ( -16 + 32 \zeta_{12}^{2} ) q^{23} + 21 q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} -26 \zeta_{12}^{3} q^{29} + ( -4 + 8 \zeta_{12}^{2} ) q^{31} -12 q^{33} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{35} -26 \zeta_{12}^{3} q^{37} + ( 2 - 4 \zeta_{12}^{2} ) q^{39} -58 q^{41} + ( -56 \zeta_{12} + 28 \zeta_{12}^{3} ) q^{43} + 6 \zeta_{12}^{3} q^{45} + ( -40 + 80 \zeta_{12}^{2} ) q^{47} + q^{49} + ( -20 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{51} + 74 \zeta_{12}^{3} q^{53} + ( -8 + 16 \zeta_{12}^{2} ) q^{55} + 36 q^{57} + ( -104 \zeta_{12} + 52 \zeta_{12}^{3} ) q^{59} + 26 \zeta_{12}^{3} q^{61} + ( -12 + 24 \zeta_{12}^{2} ) q^{63} -4 q^{65} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{67} -48 \zeta_{12}^{3} q^{69} + 46 q^{73} + ( -42 \zeta_{12} + 21 \zeta_{12}^{3} ) q^{75} + 48 \zeta_{12}^{3} q^{77} + ( -68 + 136 \zeta_{12}^{2} ) q^{79} + 9 q^{81} + ( -56 \zeta_{12} + 28 \zeta_{12}^{3} ) q^{83} + 20 \zeta_{12}^{3} q^{85} + ( -26 + 52 \zeta_{12}^{2} ) q^{87} -82 q^{89} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{91} -12 \zeta_{12}^{3} q^{93} + ( 24 - 48 \zeta_{12}^{2} ) q^{95} + 2 q^{97} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} + 40q^{17} + 84q^{25} - 48q^{33} - 232q^{41} + 4q^{49} + 144q^{57} - 16q^{65} + 184q^{73} + 36q^{81} - 328q^{89} + 8q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0 −1.73205 0 2.00000i 0 6.92820i 0 3.00000 0
127.2 0 −1.73205 0 2.00000i 0 6.92820i 0 3.00000 0
127.3 0 1.73205 0 2.00000i 0 6.92820i 0 3.00000 0
127.4 0 1.73205 0 2.00000i 0 6.92820i 0 3.00000 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 768.3.b.c 4
3.b odd 2 1 2304.3.b.l 4
4.b odd 2 1 inner 768.3.b.c 4
8.b even 2 1 inner 768.3.b.c 4
8.d odd 2 1 inner 768.3.b.c 4
12.b even 2 1 2304.3.b.l 4
16.e even 4 1 12.3.d.a 2
16.e even 4 1 192.3.g.b 2
16.f odd 4 1 12.3.d.a 2
16.f odd 4 1 192.3.g.b 2
24.f even 2 1 2304.3.b.l 4
24.h odd 2 1 2304.3.b.l 4
48.i odd 4 1 36.3.d.c 2
48.i odd 4 1 576.3.g.e 2
48.k even 4 1 36.3.d.c 2
48.k even 4 1 576.3.g.e 2
80.i odd 4 1 300.3.f.a 4
80.j even 4 1 300.3.f.a 4
80.k odd 4 1 300.3.c.b 2
80.q even 4 1 300.3.c.b 2
80.s even 4 1 300.3.f.a 4
80.t odd 4 1 300.3.f.a 4
112.j even 4 1 588.3.g.b 2
112.l odd 4 1 588.3.g.b 2
144.u even 12 1 324.3.f.a 2
144.u even 12 1 324.3.f.g 2
144.v odd 12 1 324.3.f.d 2
144.v odd 12 1 324.3.f.j 2
144.w odd 12 1 324.3.f.a 2
144.w odd 12 1 324.3.f.g 2
144.x even 12 1 324.3.f.d 2
144.x even 12 1 324.3.f.j 2
240.t even 4 1 900.3.c.e 2
240.z odd 4 1 900.3.f.c 4
240.bb even 4 1 900.3.f.c 4
240.bd odd 4 1 900.3.f.c 4
240.bf even 4 1 900.3.f.c 4
240.bm odd 4 1 900.3.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.d.a 2 16.e even 4 1
12.3.d.a 2 16.f odd 4 1
36.3.d.c 2 48.i odd 4 1
36.3.d.c 2 48.k even 4 1
192.3.g.b 2 16.e even 4 1
192.3.g.b 2 16.f odd 4 1
300.3.c.b 2 80.k odd 4 1
300.3.c.b 2 80.q even 4 1
300.3.f.a 4 80.i odd 4 1
300.3.f.a 4 80.j even 4 1
300.3.f.a 4 80.s even 4 1
300.3.f.a 4 80.t odd 4 1
324.3.f.a 2 144.u even 12 1
324.3.f.a 2 144.w odd 12 1
324.3.f.d 2 144.v odd 12 1
324.3.f.d 2 144.x even 12 1
324.3.f.g 2 144.u even 12 1
324.3.f.g 2 144.w odd 12 1
324.3.f.j 2 144.v odd 12 1
324.3.f.j 2 144.x even 12 1
576.3.g.e 2 48.i odd 4 1
576.3.g.e 2 48.k even 4 1
588.3.g.b 2 112.j even 4 1
588.3.g.b 2 112.l odd 4 1
768.3.b.c 4 1.a even 1 1 trivial
768.3.b.c 4 4.b odd 2 1 inner
768.3.b.c 4 8.b even 2 1 inner
768.3.b.c 4 8.d odd 2 1 inner
900.3.c.e 2 240.t even 4 1
900.3.c.e 2 240.bm odd 4 1
900.3.f.c 4 240.z odd 4 1
900.3.f.c 4 240.bb even 4 1
900.3.f.c 4 240.bd odd 4 1
900.3.f.c 4 240.bf even 4 1
2304.3.b.l 4 3.b odd 2 1
2304.3.b.l 4 12.b even 2 1
2304.3.b.l 4 24.f even 2 1
2304.3.b.l 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} + 4 \)
\( T_{11}^{2} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( ( 4 + T^{2} )^{2} \)
$7$ \( ( 48 + T^{2} )^{2} \)
$11$ \( ( -48 + T^{2} )^{2} \)
$13$ \( ( 4 + T^{2} )^{2} \)
$17$ \( ( -10 + T )^{4} \)
$19$ \( ( -432 + T^{2} )^{2} \)
$23$ \( ( 768 + T^{2} )^{2} \)
$29$ \( ( 676 + T^{2} )^{2} \)
$31$ \( ( 48 + T^{2} )^{2} \)
$37$ \( ( 676 + T^{2} )^{2} \)
$41$ \( ( 58 + T )^{4} \)
$43$ \( ( -2352 + T^{2} )^{2} \)
$47$ \( ( 4800 + T^{2} )^{2} \)
$53$ \( ( 5476 + T^{2} )^{2} \)
$59$ \( ( -8112 + T^{2} )^{2} \)
$61$ \( ( 676 + T^{2} )^{2} \)
$67$ \( ( -48 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( ( -46 + T )^{4} \)
$79$ \( ( 13872 + T^{2} )^{2} \)
$83$ \( ( -2352 + T^{2} )^{2} \)
$89$ \( ( 82 + T )^{4} \)
$97$ \( ( -2 + T )^{4} \)
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