Properties

Label 768.4.f.c
Level $768$
Weight $4$
Character orbit 768.f
Analytic conductor $45.313$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(45.3134668844\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12960000.1
Defining polynomial: \(x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{18} \)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{3} + \beta_{1} q^{5} + \beta_{2} q^{7} + ( 3 - 3 \beta_{7} ) q^{9} +O(q^{10})\) \( q -\beta_{3} q^{3} + \beta_{1} q^{5} + \beta_{2} q^{7} + ( 3 - 3 \beta_{7} ) q^{9} + ( -5 \beta_{3} - 5 \beta_{4} ) q^{11} -5 \beta_{5} q^{13} + ( -4 \beta_{2} - 5 \beta_{6} ) q^{15} -4 \beta_{7} q^{17} + ( -9 \beta_{3} + 9 \beta_{4} ) q^{19} + ( 3 \beta_{1} - 15 \beta_{5} ) q^{21} + 14 \beta_{6} q^{23} -45 q^{25} + ( -6 \beta_{3} - 27 \beta_{4} ) q^{27} + 17 \beta_{1} q^{29} -29 \beta_{2} q^{31} + ( -120 - 15 \beta_{7} ) q^{33} + ( 10 \beta_{3} + 10 \beta_{4} ) q^{35} + 65 \beta_{5} q^{37} + ( 5 \beta_{2} - 5 \beta_{6} ) q^{39} -14 \beta_{7} q^{41} + ( 29 \beta_{3} - 29 \beta_{4} ) q^{43} + ( 3 \beta_{1} + 120 \beta_{5} ) q^{45} -28 \beta_{6} q^{47} + 283 q^{49} + ( -4 \beta_{3} - 36 \beta_{4} ) q^{51} + 61 \beta_{1} q^{53} -40 \beta_{2} q^{55} + ( 270 - 27 \beta_{7} ) q^{57} + ( -25 \beta_{3} - 25 \beta_{4} ) q^{59} -221 \beta_{5} q^{61} + ( 3 \beta_{2} - 30 \beta_{6} ) q^{63} + 10 \beta_{7} q^{65} + ( 95 \beta_{3} - 95 \beta_{4} ) q^{67} + ( -42 \beta_{1} - 168 \beta_{5} ) q^{69} -150 \beta_{6} q^{71} -410 q^{73} + 45 \beta_{3} q^{75} + 30 \beta_{1} q^{77} + 11 \beta_{2} q^{79} + ( -711 - 18 \beta_{7} ) q^{81} + ( -181 \beta_{3} - 181 \beta_{4} ) q^{83} + 160 \beta_{5} q^{85} + ( -68 \beta_{2} - 85 \beta_{6} ) q^{87} + 94 \beta_{7} q^{89} + ( 10 \beta_{3} - 10 \beta_{4} ) q^{91} + ( -87 \beta_{1} + 435 \beta_{5} ) q^{93} -90 \beta_{6} q^{95} + 770 q^{97} + ( 105 \beta_{3} - 135 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{9} + O(q^{10}) \) \( 8q + 24q^{9} - 360q^{25} - 960q^{33} + 2264q^{49} + 2160q^{57} - 3280q^{73} - 5688q^{81} + 6160q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{6} + 9 \)
\(\beta_{2}\)\(=\)\( -4 \nu^{6} + 12 \nu^{4} - 28 \nu^{2} + 6 \)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{7} - 6 \nu^{6} + 16 \nu^{5} + 16 \nu^{4} - 44 \nu^{3} - 48 \nu^{2} + 31 \nu + 10 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{7} - 6 \nu^{6} - 16 \nu^{5} + 16 \nu^{4} + 44 \nu^{3} - 48 \nu^{2} - 31 \nu + 10 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 8 \nu^{5} - 20 \nu^{3} + \nu \)\()/2\)
\(\beta_{6}\)\(=\)\( -2 \nu^{7} + 8 \nu^{5} - 20 \nu^{3} + 14 \nu \)
\(\beta_{7}\)\(=\)\( 6 \nu^{7} - 16 \nu^{5} + 44 \nu^{3} - 2 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3}\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - \beta_{1} + 12\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 4 \beta_{5}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-7 \beta_{4} - 7 \beta_{3} + 6 \beta_{2} + 3 \beta_{1} - 28\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{7} + 11 \beta_{6} + 22 \beta_{5} + 10 \beta_{4} - 10 \beta_{3}\)\()/16\)
\(\nu^{6}\)\(=\)\(\beta_{1} - 9\)
\(\nu^{7}\)\(=\)\((\)\(-13 \beta_{7} + 29 \beta_{6} - 58 \beta_{5} + 26 \beta_{4} - 26 \beta_{3}\)\()/16\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
383.1
1.40126 0.809017i
0.535233 0.309017i
1.40126 + 0.809017i
0.535233 + 0.309017i
−1.40126 + 0.809017i
−0.535233 + 0.309017i
−1.40126 0.809017i
−0.535233 0.309017i
0 −3.87298 3.46410i 0 −8.94427 0 7.74597i 0 3.00000 + 26.8328i 0
383.2 0 −3.87298 3.46410i 0 8.94427 0 7.74597i 0 3.00000 + 26.8328i 0
383.3 0 −3.87298 + 3.46410i 0 −8.94427 0 7.74597i 0 3.00000 26.8328i 0
383.4 0 −3.87298 + 3.46410i 0 8.94427 0 7.74597i 0 3.00000 26.8328i 0
383.5 0 3.87298 3.46410i 0 −8.94427 0 7.74597i 0 3.00000 26.8328i 0
383.6 0 3.87298 3.46410i 0 8.94427 0 7.74597i 0 3.00000 26.8328i 0
383.7 0 3.87298 + 3.46410i 0 −8.94427 0 7.74597i 0 3.00000 + 26.8328i 0
383.8 0 3.87298 + 3.46410i 0 8.94427 0 7.74597i 0 3.00000 + 26.8328i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 383.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.4.f.c 8
3.b odd 2 1 inner 768.4.f.c 8
4.b odd 2 1 inner 768.4.f.c 8
8.b even 2 1 inner 768.4.f.c 8
8.d odd 2 1 inner 768.4.f.c 8
12.b even 2 1 inner 768.4.f.c 8
16.e even 4 1 12.4.b.a 4
16.e even 4 1 192.4.c.b 4
16.f odd 4 1 12.4.b.a 4
16.f odd 4 1 192.4.c.b 4
24.f even 2 1 inner 768.4.f.c 8
24.h odd 2 1 inner 768.4.f.c 8
48.i odd 4 1 12.4.b.a 4
48.i odd 4 1 192.4.c.b 4
48.k even 4 1 12.4.b.a 4
48.k even 4 1 192.4.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.b.a 4 16.e even 4 1
12.4.b.a 4 16.f odd 4 1
12.4.b.a 4 48.i odd 4 1
12.4.b.a 4 48.k even 4 1
192.4.c.b 4 16.e even 4 1
192.4.c.b 4 16.f odd 4 1
192.4.c.b 4 48.i odd 4 1
192.4.c.b 4 48.k even 4 1
768.4.f.c 8 1.a even 1 1 trivial
768.4.f.c 8 3.b odd 2 1 inner
768.4.f.c 8 4.b odd 2 1 inner
768.4.f.c 8 8.b even 2 1 inner
768.4.f.c 8 8.d odd 2 1 inner
768.4.f.c 8 12.b even 2 1 inner
768.4.f.c 8 24.f even 2 1 inner
768.4.f.c 8 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} - 80 \)
\( T_{19}^{2} - 4860 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 729 - 6 T^{2} + T^{4} )^{2} \)
$5$ \( ( -80 + T^{2} )^{4} \)
$7$ \( ( 60 + T^{2} )^{4} \)
$11$ \( ( 1200 + T^{2} )^{4} \)
$13$ \( ( 100 + T^{2} )^{4} \)
$17$ \( ( 1280 + T^{2} )^{4} \)
$19$ \( ( -4860 + T^{2} )^{4} \)
$23$ \( ( -9408 + T^{2} )^{4} \)
$29$ \( ( -23120 + T^{2} )^{4} \)
$31$ \( ( 50460 + T^{2} )^{4} \)
$37$ \( ( 16900 + T^{2} )^{4} \)
$41$ \( ( 15680 + T^{2} )^{4} \)
$43$ \( ( -50460 + T^{2} )^{4} \)
$47$ \( ( -37632 + T^{2} )^{4} \)
$53$ \( ( -297680 + T^{2} )^{4} \)
$59$ \( ( 30000 + T^{2} )^{4} \)
$61$ \( ( 195364 + T^{2} )^{4} \)
$67$ \( ( -541500 + T^{2} )^{4} \)
$71$ \( ( -1080000 + T^{2} )^{4} \)
$73$ \( ( 410 + T )^{8} \)
$79$ \( ( 7260 + T^{2} )^{4} \)
$83$ \( ( 1572528 + T^{2} )^{4} \)
$89$ \( ( 706880 + T^{2} )^{4} \)
$97$ \( ( -770 + T )^{8} \)
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