Properties

Label 768.3.e.m
Level $768$
Weight $3$
Character orbit 768.e
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.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + ( 2 + \beta_{1} ) q^{3} -\beta_{2} q^{5} + \beta_{3} q^{7} + ( -1 + 4 \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( 2 + \beta_{1} ) q^{3} -\beta_{2} q^{5} + \beta_{3} q^{7} + ( -1 + 4 \beta_{1} ) q^{9} + 2 \beta_{1} q^{11} -2 \beta_{3} q^{13} + ( -2 \beta_{2} - \beta_{3} ) q^{15} + 8 \beta_{1} q^{17} -20 q^{19} + ( -5 \beta_{2} + 2 \beta_{3} ) q^{21} -4 \beta_{2} q^{23} + 9 q^{25} + ( -22 + 7 \beta_{1} ) q^{27} + 13 \beta_{2} q^{29} + 3 \beta_{3} q^{31} + ( -10 + 4 \beta_{1} ) q^{33} + 16 \beta_{1} q^{35} + 6 \beta_{3} q^{37} + ( 10 \beta_{2} - 4 \beta_{3} ) q^{39} + 16 \beta_{1} q^{41} -36 q^{43} + ( \beta_{2} - 4 \beta_{3} ) q^{45} + 16 \beta_{2} q^{47} + 31 q^{49} + ( -40 + 16 \beta_{1} ) q^{51} -5 \beta_{2} q^{53} -2 \beta_{3} q^{55} + ( -40 - 20 \beta_{1} ) q^{57} + 46 \beta_{1} q^{59} -2 \beta_{3} q^{61} + ( -20 \beta_{2} - \beta_{3} ) q^{63} -32 \beta_{1} q^{65} -44 q^{67} + ( -8 \beta_{2} - 4 \beta_{3} ) q^{69} -20 \beta_{2} q^{71} + 50 q^{73} + ( 18 + 9 \beta_{1} ) q^{75} -10 \beta_{2} q^{77} -9 \beta_{3} q^{79} + ( -79 - 8 \beta_{1} ) q^{81} + 46 \beta_{1} q^{83} -8 \beta_{3} q^{85} + ( 26 \beta_{2} + 13 \beta_{3} ) q^{87} -72 \beta_{1} q^{89} -160 q^{91} + ( -15 \beta_{2} + 6 \beta_{3} ) q^{93} + 20 \beta_{2} q^{95} + 50 q^{97} + ( -40 - 2 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{3} - 4q^{9} + O(q^{10}) \) \( 4q + 8q^{3} - 4q^{9} - 80q^{19} + 36q^{25} - 88q^{27} - 40q^{33} - 144q^{43} + 124q^{49} - 160q^{51} - 160q^{57} - 176q^{67} + 200q^{73} + 72q^{75} - 316q^{81} - 640q^{91} + 200q^{97} - 160q^{99} + O(q^{100}) \)

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.618034i
1.61803i
1.61803i
0.618034i
0 2.00000 2.23607i 0 4.00000i 0 8.94427 0 −1.00000 8.94427i 0
257.2 0 2.00000 2.23607i 0 4.00000i 0 −8.94427 0 −1.00000 8.94427i 0
257.3 0 2.00000 + 2.23607i 0 4.00000i 0 −8.94427 0 −1.00000 + 8.94427i 0
257.4 0 2.00000 + 2.23607i 0 4.00000i 0 8.94427 0 −1.00000 + 8.94427i 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
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.3.e.m 4
3.b odd 2 1 inner 768.3.e.m 4
4.b odd 2 1 768.3.e.h 4
8.b even 2 1 768.3.e.h 4
8.d odd 2 1 inner 768.3.e.m 4
12.b even 2 1 768.3.e.h 4
16.e even 4 1 384.3.h.e 4
16.e even 4 1 384.3.h.f yes 4
16.f odd 4 1 384.3.h.e 4
16.f odd 4 1 384.3.h.f yes 4
24.f even 2 1 inner 768.3.e.m 4
24.h odd 2 1 768.3.e.h 4
48.i odd 4 1 384.3.h.e 4
48.i odd 4 1 384.3.h.f yes 4
48.k even 4 1 384.3.h.e 4
48.k even 4 1 384.3.h.f yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.h.e 4 16.e even 4 1
384.3.h.e 4 16.f odd 4 1
384.3.h.e 4 48.i odd 4 1
384.3.h.e 4 48.k even 4 1
384.3.h.f yes 4 16.e even 4 1
384.3.h.f yes 4 16.f odd 4 1
384.3.h.f yes 4 48.i odd 4 1
384.3.h.f yes 4 48.k even 4 1
768.3.e.h 4 4.b odd 2 1
768.3.e.h 4 8.b even 2 1
768.3.e.h 4 12.b even 2 1
768.3.e.h 4 24.h odd 2 1
768.3.e.m 4 1.a even 1 1 trivial
768.3.e.m 4 3.b odd 2 1 inner
768.3.e.m 4 8.d odd 2 1 inner
768.3.e.m 4 24.f even 2 1 inner

Hecke kernels

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

\( T_{5}^{2} + 16 \)
\( T_{7}^{2} - 80 \)
\( T_{11}^{2} + 20 \)
\( T_{19} + 20 \)
\( T_{37}^{2} - 2880 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 9 - 4 T + T^{2} )^{2} \)
$5$ \( ( 16 + T^{2} )^{2} \)
$7$ \( ( -80 + T^{2} )^{2} \)
$11$ \( ( 20 + T^{2} )^{2} \)
$13$ \( ( -320 + T^{2} )^{2} \)
$17$ \( ( 320 + T^{2} )^{2} \)
$19$ \( ( 20 + T )^{4} \)
$23$ \( ( 256 + T^{2} )^{2} \)
$29$ \( ( 2704 + T^{2} )^{2} \)
$31$ \( ( -720 + T^{2} )^{2} \)
$37$ \( ( -2880 + T^{2} )^{2} \)
$41$ \( ( 1280 + T^{2} )^{2} \)
$43$ \( ( 36 + T )^{4} \)
$47$ \( ( 4096 + T^{2} )^{2} \)
$53$ \( ( 400 + T^{2} )^{2} \)
$59$ \( ( 10580 + T^{2} )^{2} \)
$61$ \( ( -320 + T^{2} )^{2} \)
$67$ \( ( 44 + T )^{4} \)
$71$ \( ( 6400 + T^{2} )^{2} \)
$73$ \( ( -50 + T )^{4} \)
$79$ \( ( -6480 + T^{2} )^{2} \)
$83$ \( ( 10580 + T^{2} )^{2} \)
$89$ \( ( 25920 + T^{2} )^{2} \)
$97$ \( ( -50 + T )^{4} \)
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