Properties

Label 784.3.c.e
Level $784$
Weight $3$
Character orbit 784.c
Analytic conductor $21.362$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 784.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.3624527258\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 14)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{2} ) q^{3} + ( 2 \beta_{1} + \beta_{2} ) q^{5} + 2 \beta_{3} q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{2} ) q^{3} + ( 2 \beta_{1} + \beta_{2} ) q^{5} + 2 \beta_{3} q^{9} + ( 9 - \beta_{3} ) q^{11} + ( -2 \beta_{1} - 6 \beta_{2} ) q^{13} + ( 9 - \beta_{3} ) q^{15} + ( 2 \beta_{1} - 5 \beta_{2} ) q^{17} + ( \beta_{1} + \beta_{2} ) q^{19} + ( 15 + 3 \beta_{3} ) q^{23} + ( -2 - 4 \beta_{3} ) q^{25} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{27} + ( 12 - 2 \beta_{3} ) q^{29} + ( 15 \beta_{1} + 7 \beta_{2} ) q^{31} + ( -12 \beta_{1} + 15 \beta_{2} ) q^{33} + ( 31 - 8 \beta_{3} ) q^{37} + ( 6 - 4 \beta_{3} ) q^{39} + ( 10 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 2 + 2 \beta_{3} ) q^{43} + ( 6 \beta_{1} + 24 \beta_{2} ) q^{45} + ( \beta_{1} + 29 \beta_{2} ) q^{47} + ( 27 - 7 \beta_{3} ) q^{51} + ( 39 + 4 \beta_{3} ) q^{53} + ( 15 \beta_{1} - 3 \beta_{2} ) q^{55} + 3 q^{57} + ( -25 \beta_{1} + 13 \beta_{2} ) q^{59} + ( 32 \beta_{1} + 7 \beta_{2} ) q^{61} + ( 42 + 14 \beta_{3} ) q^{65} + ( -29 + 15 \beta_{3} ) q^{67} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{69} + ( 6 - 10 \beta_{3} ) q^{71} + ( -16 \beta_{1} + 53 \beta_{2} ) q^{73} + ( -10 \beta_{1} + 22 \beta_{2} ) q^{75} + ( 55 - 5 \beta_{3} ) q^{79} + ( -9 + 18 \beta_{3} ) q^{81} + ( -4 \beta_{1} - 68 \beta_{2} ) q^{83} + ( -9 + 8 \beta_{3} ) q^{85} + ( -18 \beta_{1} + 24 \beta_{2} ) q^{87} + ( -24 \beta_{1} + 63 \beta_{2} ) q^{89} + ( 69 - 8 \beta_{3} ) q^{93} + ( -15 - 3 \beta_{3} ) q^{95} + ( -26 \beta_{1} - 22 \beta_{2} ) q^{97} + ( -36 + 18 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q + 36 q^{11} + 36 q^{15} + 60 q^{23} - 8 q^{25} + 48 q^{29} + 124 q^{37} + 24 q^{39} + 8 q^{43} + 108 q^{51} + 156 q^{53} + 12 q^{57} + 168 q^{65} - 116 q^{67} + 24 q^{71} + 220 q^{79} - 36 q^{81} - 36 q^{85} + 276 q^{93} - 60 q^{95} - 144 q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{3} \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{3}\)\()/3\)

Character values

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

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−0.707107 + 1.22474i
0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0 4.18154i 0 3.16693i 0 0 0 −8.48528 0
97.2 0 0.717439i 0 6.63103i 0 0 0 8.48528 0
97.3 0 0.717439i 0 6.63103i 0 0 0 8.48528 0
97.4 0 4.18154i 0 3.16693i 0 0 0 −8.48528 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
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.3.c.e 4
4.b odd 2 1 98.3.b.b 4
7.b odd 2 1 inner 784.3.c.e 4
7.c even 3 1 112.3.s.b 4
7.c even 3 1 784.3.s.c 4
7.d odd 6 1 112.3.s.b 4
7.d odd 6 1 784.3.s.c 4
12.b even 2 1 882.3.c.f 4
21.g even 6 1 1008.3.cg.l 4
21.h odd 6 1 1008.3.cg.l 4
28.d even 2 1 98.3.b.b 4
28.f even 6 1 14.3.d.a 4
28.f even 6 1 98.3.d.a 4
28.g odd 6 1 14.3.d.a 4
28.g odd 6 1 98.3.d.a 4
56.j odd 6 1 448.3.s.c 4
56.k odd 6 1 448.3.s.d 4
56.m even 6 1 448.3.s.d 4
56.p even 6 1 448.3.s.c 4
84.h odd 2 1 882.3.c.f 4
84.j odd 6 1 126.3.n.c 4
84.j odd 6 1 882.3.n.b 4
84.n even 6 1 126.3.n.c 4
84.n even 6 1 882.3.n.b 4
140.p odd 6 1 350.3.k.a 4
140.s even 6 1 350.3.k.a 4
140.w even 12 2 350.3.i.a 8
140.x odd 12 2 350.3.i.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.3.d.a 4 28.f even 6 1
14.3.d.a 4 28.g odd 6 1
98.3.b.b 4 4.b odd 2 1
98.3.b.b 4 28.d even 2 1
98.3.d.a 4 28.f even 6 1
98.3.d.a 4 28.g odd 6 1
112.3.s.b 4 7.c even 3 1
112.3.s.b 4 7.d odd 6 1
126.3.n.c 4 84.j odd 6 1
126.3.n.c 4 84.n even 6 1
350.3.i.a 8 140.w even 12 2
350.3.i.a 8 140.x odd 12 2
350.3.k.a 4 140.p odd 6 1
350.3.k.a 4 140.s even 6 1
448.3.s.c 4 56.j odd 6 1
448.3.s.c 4 56.p even 6 1
448.3.s.d 4 56.k odd 6 1
448.3.s.d 4 56.m even 6 1
784.3.c.e 4 1.a even 1 1 trivial
784.3.c.e 4 7.b odd 2 1 inner
784.3.s.c 4 7.c even 3 1
784.3.s.c 4 7.d odd 6 1
882.3.c.f 4 12.b even 2 1
882.3.c.f 4 84.h odd 2 1
882.3.n.b 4 84.j odd 6 1
882.3.n.b 4 84.n even 6 1
1008.3.cg.l 4 21.g even 6 1
1008.3.cg.l 4 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 18 T_{3}^{2} + 9 \) acting on \(S_{3}^{\mathrm{new}}(784, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + 18 T^{2} + T^{4} \)
$5$ \( 441 + 54 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 63 - 18 T + T^{2} )^{2} \)
$13$ \( 7056 + 264 T^{2} + T^{4} \)
$17$ \( 2601 + 198 T^{2} + T^{4} \)
$19$ \( 9 + 18 T^{2} + T^{4} \)
$23$ \( ( 63 - 30 T + T^{2} )^{2} \)
$29$ \( ( 72 - 24 T + T^{2} )^{2} \)
$31$ \( 1447209 + 2994 T^{2} + T^{4} \)
$37$ \( ( -191 - 62 T + T^{2} )^{2} \)
$41$ \( 345744 + 1224 T^{2} + T^{4} \)
$43$ \( ( -68 - 4 T + T^{2} )^{2} \)
$47$ \( 6335289 + 5058 T^{2} + T^{4} \)
$53$ \( ( 1233 - 78 T + T^{2} )^{2} \)
$59$ \( 10517049 + 8514 T^{2} + T^{4} \)
$61$ \( 35964009 + 12582 T^{2} + T^{4} \)
$67$ \( ( -3209 + 58 T + T^{2} )^{2} \)
$71$ \( ( -1764 - 12 T + T^{2} )^{2} \)
$73$ \( 47485881 + 19926 T^{2} + T^{4} \)
$79$ \( ( 2575 - 110 T + T^{2} )^{2} \)
$83$ \( 189778176 + 27936 T^{2} + T^{4} \)
$89$ \( 71419401 + 30726 T^{2} + T^{4} \)
$97$ \( 6780816 + 11016 T^{2} + T^{4} \)
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