Properties

Label 784.4.a.y
Level $784$
Weight $4$
Character orbit 784.a
Self dual yes
Analytic conductor $46.257$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.2574974445\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 98)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 \beta q^{3} -14 \beta q^{5} + 23 q^{9} +O(q^{10})\) \( q + 5 \beta q^{3} -14 \beta q^{5} + 23 q^{9} + 14 q^{11} + 36 \beta q^{13} -140 q^{15} + \beta q^{17} + \beta q^{19} -140 q^{23} + 267 q^{25} -20 \beta q^{27} -286 q^{29} + 66 \beta q^{31} + 70 \beta q^{33} -38 q^{37} + 360 q^{39} -89 \beta q^{41} + 34 q^{43} -322 \beta q^{45} -370 \beta q^{47} + 10 q^{51} -74 q^{53} -196 \beta q^{55} + 10 q^{57} -307 \beta q^{59} + 10 \beta q^{61} -1008 q^{65} -684 q^{67} -700 \beta q^{69} -588 q^{71} -191 \beta q^{73} + 1335 \beta q^{75} -1220 q^{79} -821 q^{81} -299 \beta q^{83} -28 q^{85} -1430 \beta q^{87} + 437 \beta q^{89} + 660 q^{93} -28 q^{95} + 1049 \beta q^{97} + 322 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 46q^{9} + O(q^{10}) \) \( 2q + 46q^{9} + 28q^{11} - 280q^{15} - 280q^{23} + 534q^{25} - 572q^{29} - 76q^{37} + 720q^{39} + 68q^{43} + 20q^{51} - 148q^{53} + 20q^{57} - 2016q^{65} - 1368q^{67} - 1176q^{71} - 2440q^{79} - 1642q^{81} - 56q^{85} + 1320q^{93} - 56q^{95} + 644q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 −7.07107 0 19.7990 0 0 0 23.0000 0
1.2 0 7.07107 0 −19.7990 0 0 0 23.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

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 784.4.a.y 2
4.b odd 2 1 98.4.a.g 2
7.b odd 2 1 inner 784.4.a.y 2
12.b even 2 1 882.4.a.bg 2
20.d odd 2 1 2450.4.a.bx 2
28.d even 2 1 98.4.a.g 2
28.f even 6 2 98.4.c.h 4
28.g odd 6 2 98.4.c.h 4
84.h odd 2 1 882.4.a.bg 2
84.j odd 6 2 882.4.g.ba 4
84.n even 6 2 882.4.g.ba 4
140.c even 2 1 2450.4.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.g 2 4.b odd 2 1
98.4.a.g 2 28.d even 2 1
98.4.c.h 4 28.f even 6 2
98.4.c.h 4 28.g odd 6 2
784.4.a.y 2 1.a even 1 1 trivial
784.4.a.y 2 7.b odd 2 1 inner
882.4.a.bg 2 12.b even 2 1
882.4.a.bg 2 84.h odd 2 1
882.4.g.ba 4 84.j odd 6 2
882.4.g.ba 4 84.n even 6 2
2450.4.a.bx 2 20.d odd 2 1
2450.4.a.bx 2 140.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(784))\):

\( T_{3}^{2} - 50 \)
\( T_{5}^{2} - 392 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -50 + T^{2} \)
$5$ \( -392 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -14 + T )^{2} \)
$13$ \( -2592 + T^{2} \)
$17$ \( -2 + T^{2} \)
$19$ \( -2 + T^{2} \)
$23$ \( ( 140 + T )^{2} \)
$29$ \( ( 286 + T )^{2} \)
$31$ \( -8712 + T^{2} \)
$37$ \( ( 38 + T )^{2} \)
$41$ \( -15842 + T^{2} \)
$43$ \( ( -34 + T )^{2} \)
$47$ \( -273800 + T^{2} \)
$53$ \( ( 74 + T )^{2} \)
$59$ \( -188498 + T^{2} \)
$61$ \( -200 + T^{2} \)
$67$ \( ( 684 + T )^{2} \)
$71$ \( ( 588 + T )^{2} \)
$73$ \( -72962 + T^{2} \)
$79$ \( ( 1220 + T )^{2} \)
$83$ \( -178802 + T^{2} \)
$89$ \( -381938 + T^{2} \)
$97$ \( -2200802 + T^{2} \)
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