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: \( x^{2} - 2 \) Copy content Toggle raw display
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.

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

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(784))\):

\( T_{3}^{2} - 50 \) Copy content Toggle raw display
\( T_{5}^{2} - 392 \) Copy content Toggle raw display

Hecke characteristic polynomials

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