Properties

Label 784.4.a.z
Level $784$
Weight $4$
Character orbit 784.a
Self dual yes
Analytic conductor $46.257$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{22}) \)
Defining polynomial: \( x^{2} - 22 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2\sqrt{22}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + \beta q^{5} + 61 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + \beta q^{5} + 61 q^{9} - 20 q^{11} + 7 \beta q^{13} + 88 q^{15} + 6 \beta q^{17} - \beta q^{19} - 48 q^{23} - 37 q^{25} + 34 \beta q^{27} - 166 q^{29} + 22 \beta q^{31} - 20 \beta q^{33} - 78 q^{37} + 616 q^{39} + 42 \beta q^{41} - 436 q^{43} + 61 \beta q^{45} - 22 \beta q^{47} + 528 q^{51} + 62 q^{53} - 20 \beta q^{55} - 88 q^{57} + 71 \beta q^{59} + 29 \beta q^{61} + 616 q^{65} - 580 q^{67} - 48 \beta q^{69} + 544 q^{71} - 64 \beta q^{73} - 37 \beta q^{75} + 680 q^{79} + 1345 q^{81} - 21 \beta q^{83} + 528 q^{85} - 166 \beta q^{87} - 160 \beta q^{89} + 1936 q^{93} - 88 q^{95} - 70 \beta q^{97} - 1220 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 122 q^{9} - 40 q^{11} + 176 q^{15} - 96 q^{23} - 74 q^{25} - 332 q^{29} - 156 q^{37} + 1232 q^{39} - 872 q^{43} + 1056 q^{51} + 124 q^{53} - 176 q^{57} + 1232 q^{65} - 1160 q^{67} + 1088 q^{71} + 1360 q^{79} + 2690 q^{81} + 1056 q^{85} + 3872 q^{93} - 176 q^{95} - 2440 q^{99}+O(q^{100}) \) 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
−4.69042
4.69042
0 −9.38083 0 −9.38083 0 0 0 61.0000 0
1.2 0 9.38083 0 9.38083 0 0 0 61.0000 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.z 2
4.b odd 2 1 98.4.a.h 2
7.b odd 2 1 inner 784.4.a.z 2
12.b even 2 1 882.4.a.w 2
20.d odd 2 1 2450.4.a.bs 2
28.d even 2 1 98.4.a.h 2
28.f even 6 2 98.4.c.g 4
28.g odd 6 2 98.4.c.g 4
84.h odd 2 1 882.4.a.w 2
84.j odd 6 2 882.4.g.bi 4
84.n even 6 2 882.4.g.bi 4
140.c even 2 1 2450.4.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.h 2 4.b odd 2 1
98.4.a.h 2 28.d even 2 1
98.4.c.g 4 28.f even 6 2
98.4.c.g 4 28.g odd 6 2
784.4.a.z 2 1.a even 1 1 trivial
784.4.a.z 2 7.b odd 2 1 inner
882.4.a.w 2 12.b even 2 1
882.4.a.w 2 84.h odd 2 1
882.4.g.bi 4 84.j odd 6 2
882.4.g.bi 4 84.n even 6 2
2450.4.a.bs 2 20.d odd 2 1
2450.4.a.bs 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(784))\):

\( T_{3}^{2} - 88 \) Copy content Toggle raw display
\( T_{5}^{2} - 88 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 88 \) Copy content Toggle raw display
$5$ \( T^{2} - 88 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 20)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 4312 \) Copy content Toggle raw display
$17$ \( T^{2} - 3168 \) Copy content Toggle raw display
$19$ \( T^{2} - 88 \) Copy content Toggle raw display
$23$ \( (T + 48)^{2} \) Copy content Toggle raw display
$29$ \( (T + 166)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 42592 \) Copy content Toggle raw display
$37$ \( (T + 78)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 155232 \) Copy content Toggle raw display
$43$ \( (T + 436)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 42592 \) Copy content Toggle raw display
$53$ \( (T - 62)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 443608 \) Copy content Toggle raw display
$61$ \( T^{2} - 74008 \) Copy content Toggle raw display
$67$ \( (T + 580)^{2} \) Copy content Toggle raw display
$71$ \( (T - 544)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 360448 \) Copy content Toggle raw display
$79$ \( (T - 680)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 38808 \) Copy content Toggle raw display
$89$ \( T^{2} - 2252800 \) Copy content Toggle raw display
$97$ \( T^{2} - 431200 \) Copy content Toggle raw display
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