Properties

Label 784.4.a.ba
Level $784$
Weight $4$
Character orbit 784.a
Self dual yes
Analytic conductor $46.257$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{37}) \)
Defining polynomial: \(x^{2} - x - 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{37}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} + ( 7 - 2 \beta ) q^{5} + 10 q^{9} +O(q^{10})\) \( q -\beta q^{3} + ( 7 - 2 \beta ) q^{5} + 10 q^{9} + ( -16 - 7 \beta ) q^{11} + ( 14 + 4 \beta ) q^{13} + ( 74 - 7 \beta ) q^{15} + ( 77 - 4 \beta ) q^{17} + ( -112 - 3 \beta ) q^{19} + ( -34 - 7 \beta ) q^{23} + ( 72 - 28 \beta ) q^{25} + 17 \beta q^{27} + ( -118 - 28 \beta ) q^{29} + ( -98 - 7 \beta ) q^{31} + ( 259 + 16 \beta ) q^{33} + ( 173 - 42 \beta ) q^{37} + ( -148 - 14 \beta ) q^{39} + ( 210 + 12 \beta ) q^{41} + 172 q^{43} + ( 70 - 20 \beta ) q^{45} + ( 42 + 15 \beta ) q^{47} + ( 148 - 77 \beta ) q^{51} + ( -219 + 42 \beta ) q^{53} + ( 406 - 17 \beta ) q^{55} + ( 111 + 112 \beta ) q^{57} + ( -28 - 37 \beta ) q^{59} + ( -49 - 74 \beta ) q^{61} -198 q^{65} + ( -168 - 7 \beta ) q^{67} + ( 259 + 34 \beta ) q^{69} + ( -448 + 56 \beta ) q^{71} + ( -483 + 20 \beta ) q^{73} + ( 1036 - 72 \beta ) q^{75} + ( 26 + 133 \beta ) q^{79} -899 q^{81} + ( 196 + 88 \beta ) q^{83} + ( 835 - 182 \beta ) q^{85} + ( 1036 + 118 \beta ) q^{87} + ( -147 + 60 \beta ) q^{89} + ( 259 + 98 \beta ) q^{93} + ( -562 + 203 \beta ) q^{95} + ( 210 - 76 \beta ) q^{97} + ( -160 - 70 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 14q^{5} + 20q^{9} + O(q^{10}) \) \( 2q + 14q^{5} + 20q^{9} - 32q^{11} + 28q^{13} + 148q^{15} + 154q^{17} - 224q^{19} - 68q^{23} + 144q^{25} - 236q^{29} - 196q^{31} + 518q^{33} + 346q^{37} - 296q^{39} + 420q^{41} + 344q^{43} + 140q^{45} + 84q^{47} + 296q^{51} - 438q^{53} + 812q^{55} + 222q^{57} - 56q^{59} - 98q^{61} - 396q^{65} - 336q^{67} + 518q^{69} - 896q^{71} - 966q^{73} + 2072q^{75} + 52q^{79} - 1798q^{81} + 392q^{83} + 1670q^{85} + 2072q^{87} - 294q^{89} + 518q^{93} - 1124q^{95} + 420q^{97} - 320q^{99} + O(q^{100}) \)

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
3.54138
−2.54138
0 −6.08276 0 −5.16553 0 0 0 10.0000 0
1.2 0 6.08276 0 19.1655 0 0 0 10.0000 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.ba 2
4.b odd 2 1 196.4.a.g 2
7.b odd 2 1 784.4.a.u 2
7.d odd 6 2 112.4.i.d 4
12.b even 2 1 1764.4.a.n 2
28.d even 2 1 196.4.a.e 2
28.f even 6 2 28.4.e.a 4
28.g odd 6 2 196.4.e.g 4
56.j odd 6 2 448.4.i.g 4
56.m even 6 2 448.4.i.h 4
84.h odd 2 1 1764.4.a.z 2
84.j odd 6 2 252.4.k.c 4
84.n even 6 2 1764.4.k.ba 4
140.s even 6 2 700.4.i.g 4
140.x odd 12 4 700.4.r.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.e.a 4 28.f even 6 2
112.4.i.d 4 7.d odd 6 2
196.4.a.e 2 28.d even 2 1
196.4.a.g 2 4.b odd 2 1
196.4.e.g 4 28.g odd 6 2
252.4.k.c 4 84.j odd 6 2
448.4.i.g 4 56.j odd 6 2
448.4.i.h 4 56.m even 6 2
700.4.i.g 4 140.s even 6 2
700.4.r.d 8 140.x odd 12 4
784.4.a.u 2 7.b odd 2 1
784.4.a.ba 2 1.a even 1 1 trivial
1764.4.a.n 2 12.b even 2 1
1764.4.a.z 2 84.h odd 2 1
1764.4.k.ba 4 84.n even 6 2

Hecke kernels

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

\( T_{3}^{2} - 37 \)
\( T_{5}^{2} - 14 T_{5} - 99 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -37 + T^{2} \)
$5$ \( -99 - 14 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -1557 + 32 T + T^{2} \)
$13$ \( -396 - 28 T + T^{2} \)
$17$ \( 5337 - 154 T + T^{2} \)
$19$ \( 12211 + 224 T + T^{2} \)
$23$ \( -657 + 68 T + T^{2} \)
$29$ \( -15084 + 236 T + T^{2} \)
$31$ \( 7791 + 196 T + T^{2} \)
$37$ \( -35339 - 346 T + T^{2} \)
$41$ \( 38772 - 420 T + T^{2} \)
$43$ \( ( -172 + T )^{2} \)
$47$ \( -6561 - 84 T + T^{2} \)
$53$ \( -17307 + 438 T + T^{2} \)
$59$ \( -49869 + 56 T + T^{2} \)
$61$ \( -200211 + 98 T + T^{2} \)
$67$ \( 26411 + 336 T + T^{2} \)
$71$ \( 84672 + 896 T + T^{2} \)
$73$ \( 218489 + 966 T + T^{2} \)
$79$ \( -653817 - 52 T + T^{2} \)
$83$ \( -248112 - 392 T + T^{2} \)
$89$ \( -111591 + 294 T + T^{2} \)
$97$ \( -169612 - 420 T + T^{2} \)
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