Properties

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

Related objects

Downloads

Learn more about

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: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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 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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} -2 \beta_{1} q^{5} -4 q^{7} + ( 5 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} -2 \beta_{1} q^{5} -4 q^{7} + ( 5 + \beta_{2} ) q^{9} + 3 \beta_{1} q^{11} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{13} + ( -8 + 2 \beta_{2} ) q^{15} + 2 \beta_{2} q^{17} + ( -\beta_{1} - 2 \beta_{3} ) q^{19} -4 \beta_{3} q^{21} + 4 \beta_{2} q^{23} -7 q^{25} + ( -9 \beta_{1} + \beta_{3} ) q^{27} -6 \beta_{1} q^{29} -4 q^{31} + ( 12 - 3 \beta_{2} ) q^{33} + 8 \beta_{1} q^{35} + ( -10 \beta_{1} - 20 \beta_{3} ) q^{37} + ( -28 - 2 \beta_{2} ) q^{39} -4 \beta_{2} q^{41} + ( -\beta_{1} - 2 \beta_{3} ) q^{43} + ( -18 \beta_{1} - 16 \beta_{3} ) q^{45} -33 q^{49} + ( -18 \beta_{1} - 8 \beta_{3} ) q^{51} -18 \beta_{1} q^{53} + 48 q^{55} + ( -14 - \beta_{2} ) q^{57} + 17 \beta_{1} q^{59} + ( -18 \beta_{1} - 36 \beta_{3} ) q^{61} + ( -20 - 4 \beta_{2} ) q^{63} -8 \beta_{2} q^{65} + ( 9 \beta_{1} + 18 \beta_{3} ) q^{67} + ( -36 \beta_{1} - 16 \beta_{3} ) q^{69} -12 \beta_{2} q^{71} + 6 q^{73} -7 \beta_{3} q^{75} -12 \beta_{1} q^{77} + 124 q^{79} + ( -31 + 10 \beta_{2} ) q^{81} + \beta_{1} q^{83} + ( -16 \beta_{1} - 32 \beta_{3} ) q^{85} + ( -24 + 6 \beta_{2} ) q^{87} + 14 \beta_{2} q^{89} + ( 8 \beta_{1} + 16 \beta_{3} ) q^{91} -4 \beta_{3} q^{93} -4 \beta_{2} q^{95} + 118 q^{97} + ( 27 \beta_{1} + 24 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{7} + 20q^{9} + O(q^{10}) \) \( 4q - 16q^{7} + 20q^{9} - 32q^{15} - 28q^{25} - 16q^{31} + 48q^{33} - 112q^{39} - 132q^{49} + 192q^{55} - 56q^{57} - 80q^{63} + 24q^{73} + 496q^{79} - 124q^{81} - 96q^{87} + 472q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 8 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{3} - 10 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{3} + 22 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 3 \nu^{2} + 5 \nu + 12 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} + \beta_{1} - 8\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{2} - 11 \beta_{1}\)\()/4\)

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
2.57794i
2.57794i
1.16372i
1.16372i
0 −2.64575 1.41421i 0 5.65685i 0 −4.00000 0 5.00000 + 7.48331i 0
257.2 0 −2.64575 + 1.41421i 0 5.65685i 0 −4.00000 0 5.00000 7.48331i 0
257.3 0 2.64575 1.41421i 0 5.65685i 0 −4.00000 0 5.00000 7.48331i 0
257.4 0 2.64575 + 1.41421i 0 5.65685i 0 −4.00000 0 5.00000 + 7.48331i 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.b 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 768.3.e.i 4
3.b odd 2 1 inner 768.3.e.i 4
4.b odd 2 1 768.3.e.l 4
8.b even 2 1 inner 768.3.e.i 4
8.d odd 2 1 768.3.e.l 4
12.b even 2 1 768.3.e.l 4
16.e even 4 2 24.3.h.c 4
16.f odd 4 2 96.3.h.c 4
24.f even 2 1 768.3.e.l 4
24.h odd 2 1 inner 768.3.e.i 4
48.i odd 4 2 24.3.h.c 4
48.k even 4 2 96.3.h.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.h.c 4 16.e even 4 2
24.3.h.c 4 48.i odd 4 2
96.3.h.c 4 16.f odd 4 2
96.3.h.c 4 48.k even 4 2
768.3.e.i 4 1.a even 1 1 trivial
768.3.e.i 4 3.b odd 2 1 inner
768.3.e.i 4 8.b even 2 1 inner
768.3.e.i 4 24.h odd 2 1 inner
768.3.e.l 4 4.b odd 2 1
768.3.e.l 4 8.d odd 2 1
768.3.e.l 4 12.b even 2 1
768.3.e.l 4 24.f even 2 1

Hecke kernels

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

\( T_{5}^{2} + 32 \)
\( T_{7} + 4 \)
\( T_{11}^{2} + 72 \)
\( T_{19}^{2} - 28 \)
\( T_{37}^{2} - 2800 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 10 T^{2} + 81 T^{4} \)
$5$ \( ( 1 - 18 T^{2} + 625 T^{4} )^{2} \)
$7$ \( ( 1 + 4 T + 49 T^{2} )^{4} \)
$11$ \( ( 1 - 170 T^{2} + 14641 T^{4} )^{2} \)
$13$ \( ( 1 + 226 T^{2} + 28561 T^{4} )^{2} \)
$17$ \( ( 1 - 354 T^{2} + 83521 T^{4} )^{2} \)
$19$ \( ( 1 + 694 T^{2} + 130321 T^{4} )^{2} \)
$23$ \( ( 1 - 162 T^{2} + 279841 T^{4} )^{2} \)
$29$ \( ( 1 - 1394 T^{2} + 707281 T^{4} )^{2} \)
$31$ \( ( 1 + 4 T + 961 T^{2} )^{4} \)
$37$ \( ( 1 - 62 T^{2} + 1874161 T^{4} )^{2} \)
$41$ \( ( 1 - 2466 T^{2} + 2825761 T^{4} )^{2} \)
$43$ \( ( 1 + 3670 T^{2} + 3418801 T^{4} )^{2} \)
$47$ \( ( 1 - 47 T )^{4}( 1 + 47 T )^{4} \)
$53$ \( ( 1 - 3026 T^{2} + 7890481 T^{4} )^{2} \)
$59$ \( ( 1 - 4650 T^{2} + 12117361 T^{4} )^{2} \)
$61$ \( ( 1 - 1630 T^{2} + 13845841 T^{4} )^{2} \)
$67$ \( ( 1 + 6710 T^{2} + 20151121 T^{4} )^{2} \)
$71$ \( ( 1 - 110 T + 5041 T^{2} )^{2}( 1 + 110 T + 5041 T^{2} )^{2} \)
$73$ \( ( 1 - 6 T + 5329 T^{2} )^{4} \)
$79$ \( ( 1 - 124 T + 6241 T^{2} )^{4} \)
$83$ \( ( 1 - 13770 T^{2} + 47458321 T^{4} )^{2} \)
$89$ \( ( 1 - 4866 T^{2} + 62742241 T^{4} )^{2} \)
$97$ \( ( 1 - 118 T + 9409 T^{2} )^{4} \)
show more
show less