Properties

Label 784.3.s.g
Level $784$
Weight $3$
Character orbit 784.s
Analytic conductor $21.362$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 784.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.3624527258\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
Defining polynomial: \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 7^{4} \)
Twist minimal: no (minimal twist has level 196)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{3} + ( \beta_{1} - \beta_{2} + 2 \beta_{6} ) q^{5} + ( 1 - \beta_{4} - \beta_{5} - \beta_{7} ) q^{9} +O(q^{10})\) \( q -\beta_{3} q^{3} + ( \beta_{1} - \beta_{2} + 2 \beta_{6} ) q^{5} + ( 1 - \beta_{4} - \beta_{5} - \beta_{7} ) q^{9} + ( -6 \beta_{4} + \beta_{7} ) q^{11} + ( -3 \beta_{1} + 4 \beta_{3} + 4 \beta_{6} ) q^{13} + ( 18 - 2 \beta_{5} ) q^{15} + 3 \beta_{3} q^{17} + ( -5 \beta_{1} + 5 \beta_{2} + \beta_{6} ) q^{19} + ( 26 - 26 \beta_{4} ) q^{23} + ( 27 \beta_{4} + 2 \beta_{7} ) q^{25} + ( -\beta_{1} - 2 \beta_{3} - 2 \beta_{6} ) q^{27} + ( 8 + 2 \beta_{5} ) q^{29} + ( 2 \beta_{2} - 8 \beta_{3} ) q^{31} + ( \beta_{1} - \beta_{2} + 4 \beta_{6} ) q^{33} + ( 32 - 32 \beta_{4} ) q^{37} + ( 46 \beta_{4} + 4 \beta_{7} ) q^{39} + ( -3 \beta_{1} - 5 \beta_{3} - 5 \beta_{6} ) q^{41} + ( -10 + 7 \beta_{5} ) q^{43} + ( 7 \beta_{2} - 20 \beta_{3} ) q^{45} + ( 12 \beta_{1} - 12 \beta_{2} - 2 \beta_{6} ) q^{47} + ( -30 + 30 \beta_{4} + 3 \beta_{5} + 3 \beta_{7} ) q^{51} + ( -6 \beta_{4} + 2 \beta_{7} ) q^{53} + ( -14 \beta_{1} + 6 \beta_{3} + 6 \beta_{6} ) q^{55} + ( 20 - \beta_{5} ) q^{57} + ( 11 \beta_{2} + 3 \beta_{3} ) q^{59} + ( -9 \beta_{1} + 9 \beta_{2} - 8 \beta_{6} ) q^{61} + ( -24 + 24 \beta_{4} + 14 \beta_{5} + 14 \beta_{7} ) q^{65} + ( 28 \beta_{4} - 6 \beta_{7} ) q^{67} + ( -26 \beta_{3} - 26 \beta_{6} ) q^{69} + ( -56 + 4 \beta_{5} ) q^{71} + ( 11 \beta_{2} - 3 \beta_{3} ) q^{73} + ( 2 \beta_{1} - 2 \beta_{2} + 47 \beta_{6} ) q^{75} + ( 60 - 60 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{79} + ( -9 \beta_{4} + 7 \beta_{7} ) q^{81} + ( -13 \beta_{1} - 33 \beta_{3} - 33 \beta_{6} ) q^{83} + ( -54 + 6 \beta_{5} ) q^{85} + ( 2 \beta_{2} + 12 \beta_{3} ) q^{87} + ( -11 \beta_{1} + 11 \beta_{2} + 25 \beta_{6} ) q^{89} + ( 84 - 84 \beta_{4} - 8 \beta_{5} - 8 \beta_{7} ) q^{93} + ( -62 \beta_{4} + 12 \beta_{7} ) q^{95} + ( -22 \beta_{1} - 33 \beta_{3} - 33 \beta_{6} ) q^{97} + ( 92 + 5 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{9} + O(q^{10}) \) \( 8q + 4q^{9} - 24q^{11} + 144q^{15} + 104q^{23} + 108q^{25} + 64q^{29} + 128q^{37} + 184q^{39} - 80q^{43} - 120q^{51} - 24q^{53} + 160q^{57} - 96q^{65} + 112q^{67} - 448q^{71} + 240q^{79} - 36q^{81} - 432q^{85} + 336q^{93} - 248q^{95} + 736q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 8 \nu \)\()/14\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - 41 \nu \)\()/7\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{6} + 7 \nu^{4} - 28 \nu^{2} + 16 \)\()/14\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} - 20 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -12 \nu^{7} + 49 \nu^{5} - 168 \nu^{3} + 96 \nu \)\()/14\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{6} + 14 \nu^{4} - 42 \nu^{2} + 24 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2}\)\()/7\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + \beta_{5} - 14 \beta_{4} + 14\)\()/7\)
\(\nu^{3}\)\(=\)\(\beta_{1}\)
\(\nu^{4}\)\(=\)\((\)\(4 \beta_{7} - 42 \beta_{4}\)\()/7\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{6} - 24 \beta_{2} + 24 \beta_{1}\)\()/7\)
\(\nu^{6}\)\(=\)\(-2 \beta_{5} - 20\)
\(\nu^{7}\)\(=\)\((\)\(-8 \beta_{3} - 82 \beta_{2}\)\()/7\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
−0.662827 0.382683i
−1.60021 0.923880i
1.60021 + 0.923880i
0.662827 + 0.382683i
−0.662827 + 0.382683i
−1.60021 + 0.923880i
1.60021 0.923880i
0.662827 0.382683i
0 −3.86324 2.23044i 0 −7.33820 + 4.23671i 0 0 0 5.44975 + 9.43924i 0
129.2 0 −0.274552 0.158513i 0 4.91434 2.83730i 0 0 0 −4.44975 7.70719i 0
129.3 0 0.274552 + 0.158513i 0 −4.91434 + 2.83730i 0 0 0 −4.44975 7.70719i 0
129.4 0 3.86324 + 2.23044i 0 7.33820 4.23671i 0 0 0 5.44975 + 9.43924i 0
705.1 0 −3.86324 + 2.23044i 0 −7.33820 4.23671i 0 0 0 5.44975 9.43924i 0
705.2 0 −0.274552 + 0.158513i 0 4.91434 + 2.83730i 0 0 0 −4.44975 + 7.70719i 0
705.3 0 0.274552 0.158513i 0 −4.91434 2.83730i 0 0 0 −4.44975 + 7.70719i 0
705.4 0 3.86324 2.23044i 0 7.33820 + 4.23671i 0 0 0 5.44975 9.43924i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 705.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.3.s.g 8
4.b odd 2 1 196.3.h.c 8
7.b odd 2 1 inner 784.3.s.g 8
7.c even 3 1 784.3.c.d 4
7.c even 3 1 inner 784.3.s.g 8
7.d odd 6 1 784.3.c.d 4
7.d odd 6 1 inner 784.3.s.g 8
12.b even 2 1 1764.3.z.k 8
28.d even 2 1 196.3.h.c 8
28.f even 6 1 196.3.b.b 4
28.f even 6 1 196.3.h.c 8
28.g odd 6 1 196.3.b.b 4
28.g odd 6 1 196.3.h.c 8
84.h odd 2 1 1764.3.z.k 8
84.j odd 6 1 1764.3.d.e 4
84.j odd 6 1 1764.3.z.k 8
84.n even 6 1 1764.3.d.e 4
84.n even 6 1 1764.3.z.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.3.b.b 4 28.f even 6 1
196.3.b.b 4 28.g odd 6 1
196.3.h.c 8 4.b odd 2 1
196.3.h.c 8 28.d even 2 1
196.3.h.c 8 28.f even 6 1
196.3.h.c 8 28.g odd 6 1
784.3.c.d 4 7.c even 3 1
784.3.c.d 4 7.d odd 6 1
784.3.s.g 8 1.a even 1 1 trivial
784.3.s.g 8 7.b odd 2 1 inner
784.3.s.g 8 7.c even 3 1 inner
784.3.s.g 8 7.d odd 6 1 inner
1764.3.d.e 4 84.j odd 6 1
1764.3.d.e 4 84.n even 6 1
1764.3.z.k 8 12.b even 2 1
1764.3.z.k 8 84.h odd 2 1
1764.3.z.k 8 84.j odd 6 1
1764.3.z.k 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 20 T_{3}^{6} + 398 T_{3}^{4} - 40 T_{3}^{2} + 4 \) acting on \(S_{3}^{\mathrm{new}}(784, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 4 - 40 T^{2} + 398 T^{4} - 20 T^{6} + T^{8} \)
$5$ \( 5345344 - 240448 T^{2} + 8504 T^{4} - 104 T^{6} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( ( 3844 - 744 T + 206 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$13$ \( ( 150152 + 776 T^{2} + T^{4} )^{2} \)
$17$ \( 26244 - 29160 T^{2} + 32238 T^{4} - 180 T^{6} + T^{8} \)
$19$ \( 2079542404 - 48338120 T^{2} + 1077998 T^{4} - 1060 T^{6} + T^{8} \)
$23$ \( ( 676 - 26 T + T^{2} )^{4} \)
$29$ \( ( -328 - 16 T + T^{2} )^{4} \)
$31$ \( 94450499584 - 481890304 T^{2} + 2151296 T^{4} - 1568 T^{6} + T^{8} \)
$37$ \( ( 1024 - 32 T + T^{2} )^{4} \)
$41$ \( ( 132098 + 740 T^{2} + T^{4} )^{2} \)
$43$ \( ( -4702 + 20 T + T^{2} )^{4} \)
$47$ \( 1420787713024 - 7189950976 T^{2} + 35193056 T^{4} - 6032 T^{6} + T^{8} \)
$53$ \( ( 126736 - 4272 T + 500 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$59$ \( 111931731844 - 1591176872 T^{2} + 22284974 T^{4} - 4756 T^{6} + T^{8} \)
$61$ \( 8688298598464 - 11625302848 T^{2} + 12607544 T^{4} - 3944 T^{6} + T^{8} \)
$67$ \( ( 7529536 + 153664 T + 5880 T^{2} - 56 T^{3} + T^{4} )^{2} \)
$71$ \( ( 1568 + 112 T + T^{2} )^{4} \)
$73$ \( 2755075464964 - 8770605128 T^{2} + 26260814 T^{4} - 5284 T^{6} + T^{8} \)
$79$ \( ( 10291264 - 384960 T + 11192 T^{2} - 120 T^{3} + T^{4} )^{2} \)
$83$ \( ( 102330818 + 25108 T^{2} + T^{4} )^{2} \)
$89$ \( 6573541223354884 - 1584251966120 T^{2} + 300734222 T^{4} - 19540 T^{6} + T^{8} \)
$97$ \( ( 310652738 + 35332 T^{2} + T^{4} )^{2} \)
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