Properties

Label 768.4.c.n
Level $768$
Weight $4$
Character orbit 768.c
Analytic conductor $45.313$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(45.3134668844\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-6}, \sqrt{-14})\)
Defining polynomial: \(x^{4} + 10 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( -2 \beta_{1} - \beta_{2} ) q^{5} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{7} + ( 15 + 3 \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} + ( -2 \beta_{1} - \beta_{2} ) q^{5} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{7} + ( 15 + 3 \beta_{2} ) q^{9} + ( -32 + \beta_{1} + 2 \beta_{3} ) q^{11} + ( 28 - 4 \beta_{1} - 8 \beta_{3} ) q^{13} + ( -24 + 9 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} ) q^{15} + ( -16 \beta_{1} + 2 \beta_{2} ) q^{17} + ( -7 \beta_{1} - 16 \beta_{2} ) q^{19} + ( 36 + 18 \beta_{1} - 9 \beta_{2} + 8 \beta_{3} ) q^{21} + ( 80 + 4 \beta_{1} + 8 \beta_{3} ) q^{23} + ( -27 + 16 \beta_{1} + 32 \beta_{3} ) q^{25} + ( -27 \beta_{1} + 3 \beta_{3} ) q^{27} + ( -14 \beta_{1} - 23 \beta_{2} ) q^{29} + ( -7 \beta_{1} - 6 \beta_{2} ) q^{31} + ( 42 + 3 \beta_{2} - 32 \beta_{3} ) q^{33} + ( 32 + 4 \beta_{1} + 8 \beta_{3} ) q^{35} + ( 124 + 28 \beta_{1} + 56 \beta_{3} ) q^{37} + ( -168 - 12 \beta_{2} + 28 \beta_{3} ) q^{39} + ( 16 \beta_{1} - 24 \beta_{2} ) q^{41} + ( 49 \beta_{1} - 16 \beta_{2} ) q^{43} + ( 168 - 54 \beta_{1} - 15 \beta_{2} - 48 \beta_{3} ) q^{45} + ( 32 \beta_{1} + 64 \beta_{3} ) q^{47} + ( -97 - 48 \beta_{1} - 96 \beta_{3} ) q^{49} + ( -192 - 18 \beta_{1} + 48 \beta_{2} - 8 \beta_{3} ) q^{51} + ( 70 \beta_{1} - 45 \beta_{2} ) q^{53} + ( 78 \beta_{1} + 44 \beta_{2} ) q^{55} + ( -84 + 144 \beta_{1} + 21 \beta_{2} + 64 \beta_{3} ) q^{57} + ( -5 \beta_{1} - 10 \beta_{3} ) q^{59} + ( 28 - 68 \beta_{1} - 136 \beta_{3} ) q^{61} + ( 336 + 81 \beta_{1} - 30 \beta_{2} + 72 \beta_{3} ) q^{63} + ( -112 \beta_{1} - 76 \beta_{2} ) q^{65} + ( 91 \beta_{1} + 32 \beta_{2} ) q^{67} + ( 168 + 12 \beta_{2} + 80 \beta_{3} ) q^{69} + ( 80 + 100 \beta_{1} + 200 \beta_{3} ) q^{71} + ( 154 + 48 \beta_{1} + 96 \beta_{3} ) q^{73} + ( 672 + 48 \beta_{2} - 27 \beta_{3} ) q^{75} + ( -68 \beta_{1} + 46 \beta_{2} ) q^{77} + ( 189 \beta_{1} + 2 \beta_{2} ) q^{79} + ( -279 + 90 \beta_{2} ) q^{81} + ( 672 - 37 \beta_{1} - 74 \beta_{3} ) q^{83} + ( -656 + 48 \beta_{1} + 96 \beta_{3} ) q^{85} + ( -168 + 207 \beta_{1} + 42 \beta_{2} + 92 \beta_{3} ) q^{87} + ( -128 \beta_{1} + 26 \beta_{2} ) q^{89} + ( -28 \beta_{1} + 16 \beta_{2} ) q^{91} + ( -84 + 54 \beta_{1} + 21 \beta_{2} + 24 \beta_{3} ) q^{93} + ( -1232 + 156 \beta_{1} + 312 \beta_{3} ) q^{95} + ( 714 - 32 \beta_{1} - 64 \beta_{3} ) q^{97} + ( -480 - 27 \beta_{1} - 96 \beta_{2} + 30 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 60 q^{9} + O(q^{10}) \) \( 4 q + 60 q^{9} - 128 q^{11} + 112 q^{13} - 96 q^{15} + 144 q^{21} + 320 q^{23} - 108 q^{25} + 168 q^{33} + 128 q^{35} + 496 q^{37} - 672 q^{39} + 672 q^{45} - 388 q^{49} - 768 q^{51} - 336 q^{57} + 112 q^{61} + 1344 q^{63} + 672 q^{69} + 320 q^{71} + 616 q^{73} + 2688 q^{75} - 1116 q^{81} + 2688 q^{83} - 2624 q^{85} - 672 q^{87} - 336 q^{93} - 4928 q^{95} + 2856 q^{97} - 1920 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 10 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{3} - 8 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 12 \nu \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} + 8 \nu + 10 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} + \beta_{1} - 10\)\()/2\)
\(\nu^{3}\)\(=\)\(-2 \beta_{2} - 3 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
767.1
3.09557i
3.09557i
0.646084i
0.646084i
0 −4.58258 2.44949i 0 17.2813i 0 0.269691i 0 15.0000 + 22.4499i 0
767.2 0 −4.58258 + 2.44949i 0 17.2813i 0 0.269691i 0 15.0000 22.4499i 0
767.3 0 4.58258 2.44949i 0 2.31464i 0 29.6636i 0 15.0000 22.4499i 0
767.4 0 4.58258 + 2.44949i 0 2.31464i 0 29.6636i 0 15.0000 + 22.4499i 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
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.4.c.n 4
3.b odd 2 1 768.4.c.p 4
4.b odd 2 1 768.4.c.p 4
8.b even 2 1 768.4.c.o 4
8.d odd 2 1 768.4.c.m 4
12.b even 2 1 inner 768.4.c.n 4
16.e even 4 2 384.4.f.g 8
16.f odd 4 2 384.4.f.h yes 8
24.f even 2 1 768.4.c.o 4
24.h odd 2 1 768.4.c.m 4
48.i odd 4 2 384.4.f.h yes 8
48.k even 4 2 384.4.f.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.f.g 8 16.e even 4 2
384.4.f.g 8 48.k even 4 2
384.4.f.h yes 8 16.f odd 4 2
384.4.f.h yes 8 48.i odd 4 2
768.4.c.m 4 8.d odd 2 1
768.4.c.m 4 24.h odd 2 1
768.4.c.n 4 1.a even 1 1 trivial
768.4.c.n 4 12.b even 2 1 inner
768.4.c.o 4 8.b even 2 1
768.4.c.o 4 24.f even 2 1
768.4.c.p 4 3.b odd 2 1
768.4.c.p 4 4.b odd 2 1

Hecke kernels

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

\( T_{5}^{4} + 304 T_{5}^{2} + 1600 \)
\( T_{7}^{4} + 880 T_{7}^{2} + 64 \)
\( T_{11}^{2} + 64 T_{11} + 940 \)
\( T_{13}^{2} - 56 T_{13} - 560 \)
\( T_{23}^{2} - 160 T_{23} + 5056 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 729 - 30 T^{2} + T^{4} \)
$5$ \( 1600 + 304 T^{2} + T^{4} \)
$7$ \( 64 + 880 T^{2} + T^{4} \)
$11$ \( ( 940 + 64 T + T^{2} )^{2} \)
$13$ \( ( -560 - 56 T + T^{2} )^{2} \)
$17$ \( 35046400 + 12736 T^{2} + T^{4} \)
$19$ \( 173185600 + 31024 T^{2} + T^{4} \)
$23$ \( ( 5056 - 160 T + T^{2} )^{2} \)
$29$ \( 621006400 + 68656 T^{2} + T^{4} \)
$31$ \( 705600 + 6384 T^{2} + T^{4} \)
$37$ \( ( -50480 - 248 T + T^{2} )^{2} \)
$41$ \( 681836544 + 76800 T^{2} + T^{4} \)
$43$ \( 1873850944 + 143920 T^{2} + T^{4} \)
$47$ \( ( -86016 + T^{2} )^{2} \)
$53$ \( 17640000 + 462000 T^{2} + T^{4} \)
$59$ \( ( -2100 + T^{2} )^{2} \)
$61$ \( ( -387632 - 56 T + T^{2} )^{2} \)
$67$ \( 19993960000 + 512176 T^{2} + T^{4} \)
$71$ \( ( -833600 - 160 T + T^{2} )^{2} \)
$73$ \( ( -169820 - 308 T + T^{2} )^{2} \)
$79$ \( 734586126400 + 1715056 T^{2} + T^{4} \)
$83$ \( ( 336588 - 1344 T + T^{2} )^{2} \)
$89$ \( 126280729600 + 862144 T^{2} + T^{4} \)
$97$ \( ( 423780 - 1428 T + T^{2} )^{2} \)
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