Properties

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

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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: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{57}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta ) q^{3} + ( -11 - \beta ) q^{5} + ( 31 + 2 \beta ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta ) q^{3} + ( -11 - \beta ) q^{5} + ( 31 + 2 \beta ) q^{9} + ( -18 - 6 \beta ) q^{11} + ( -21 + 5 \beta ) q^{13} + ( 68 + 12 \beta ) q^{15} + ( -4 + 6 \beta ) q^{17} + ( -59 + 13 \beta ) q^{19} + ( 52 - 4 \beta ) q^{23} + ( 53 + 22 \beta ) q^{25} + ( -118 - 6 \beta ) q^{27} + ( -28 + 22 \beta ) q^{29} + ( 10 - 14 \beta ) q^{31} + ( 360 + 24 \beta ) q^{33} + ( 252 - 10 \beta ) q^{37} + ( -264 + 16 \beta ) q^{39} + ( 272 + 18 \beta ) q^{41} + ( -206 + 14 \beta ) q^{43} + ( -455 - 53 \beta ) q^{45} + ( -250 + 14 \beta ) q^{47} + ( -338 - 2 \beta ) q^{51} + ( 134 - 72 \beta ) q^{53} + ( 540 + 84 \beta ) q^{55} + ( -682 + 46 \beta ) q^{57} + ( -99 + 13 \beta ) q^{59} + ( 173 + 31 \beta ) q^{61} + ( -54 - 34 \beta ) q^{65} + ( -504 + 68 \beta ) q^{67} + ( 176 - 48 \beta ) q^{69} + ( 612 + 44 \beta ) q^{71} + ( -358 + 36 \beta ) q^{73} + ( -1307 - 75 \beta ) q^{75} + ( -292 - 36 \beta ) q^{79} + ( -377 + 70 \beta ) q^{81} + ( -615 - 47 \beta ) q^{83} + ( -298 - 62 \beta ) q^{85} + ( -1226 + 6 \beta ) q^{87} + ( -298 - 160 \beta ) q^{89} + ( 788 + 4 \beta ) q^{93} + ( -92 - 84 \beta ) q^{95} + ( 428 + 70 \beta ) q^{97} + ( -1242 - 222 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 22q^{5} + 62q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 22q^{5} + 62q^{9} - 36q^{11} - 42q^{13} + 136q^{15} - 8q^{17} - 118q^{19} + 104q^{23} + 106q^{25} - 236q^{27} - 56q^{29} + 20q^{31} + 720q^{33} + 504q^{37} - 528q^{39} + 544q^{41} - 412q^{43} - 910q^{45} - 500q^{47} - 676q^{51} + 268q^{53} + 1080q^{55} - 1364q^{57} - 198q^{59} + 346q^{61} - 108q^{65} - 1008q^{67} + 352q^{69} + 1224q^{71} - 716q^{73} - 2614q^{75} - 584q^{79} - 754q^{81} - 1230q^{83} - 596q^{85} - 2452q^{87} - 596q^{89} + 1576q^{93} - 184q^{95} + 856q^{97} - 2484q^{99} + O(q^{100}) \)

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
4.27492
−3.27492
0 −8.54983 0 −18.5498 0 0 0 46.0997 0
1.2 0 6.54983 0 −3.45017 0 0 0 15.9003 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.4.a.t 2
4.b odd 2 1 392.4.a.h 2
7.b odd 2 1 112.4.a.h 2
21.c even 2 1 1008.4.a.x 2
28.d even 2 1 56.4.a.c 2
28.f even 6 2 392.4.i.l 4
28.g odd 6 2 392.4.i.i 4
56.e even 2 1 448.4.a.s 2
56.h odd 2 1 448.4.a.r 2
84.h odd 2 1 504.4.a.i 2
140.c even 2 1 1400.4.a.i 2
140.j odd 4 2 1400.4.g.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.a.c 2 28.d even 2 1
112.4.a.h 2 7.b odd 2 1
392.4.a.h 2 4.b odd 2 1
392.4.i.i 4 28.g odd 6 2
392.4.i.l 4 28.f even 6 2
448.4.a.r 2 56.h odd 2 1
448.4.a.s 2 56.e even 2 1
504.4.a.i 2 84.h odd 2 1
784.4.a.t 2 1.a even 1 1 trivial
1008.4.a.x 2 21.c even 2 1
1400.4.a.i 2 140.c even 2 1
1400.4.g.h 4 140.j odd 4 2

Hecke kernels

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

\( T_{3}^{2} + 2 T_{3} - 56 \)
\( T_{5}^{2} + 22 T_{5} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -56 + 2 T + T^{2} \)
$5$ \( 64 + 22 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -1728 + 36 T + T^{2} \)
$13$ \( -984 + 42 T + T^{2} \)
$17$ \( -2036 + 8 T + T^{2} \)
$19$ \( -6152 + 118 T + T^{2} \)
$23$ \( 1792 - 104 T + T^{2} \)
$29$ \( -26804 + 56 T + T^{2} \)
$31$ \( -11072 - 20 T + T^{2} \)
$37$ \( 57804 - 504 T + T^{2} \)
$41$ \( 55516 - 544 T + T^{2} \)
$43$ \( 31264 + 412 T + T^{2} \)
$47$ \( 51328 + 500 T + T^{2} \)
$53$ \( -277532 - 268 T + T^{2} \)
$59$ \( 168 + 198 T + T^{2} \)
$61$ \( -24848 - 346 T + T^{2} \)
$67$ \( -9552 + 1008 T + T^{2} \)
$71$ \( 264192 - 1224 T + T^{2} \)
$73$ \( 54292 + 716 T + T^{2} \)
$79$ \( 11392 + 584 T + T^{2} \)
$83$ \( 252312 + 1230 T + T^{2} \)
$89$ \( -1370396 + 596 T + T^{2} \)
$97$ \( -96116 - 856 T + T^{2} \)
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