Properties

Label 784.2.bb.b
Level $784$
Weight $2$
Character orbit 784.bb
Analytic conductor $6.260$
Analytic rank $0$
Dimension $120$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.bb (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 24 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 24 q^{9} - 14 q^{17} + 16 q^{21} + 40 q^{25} + 32 q^{29} - 62 q^{37} - 28 q^{41} - 60 q^{49} + 14 q^{53} - 34 q^{57} - 112 q^{61} - 32 q^{65} + 112 q^{69} + 42 q^{73} + 66 q^{77} - 44 q^{81} - 12 q^{85} + 28 q^{89} - 58 q^{93} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1 0 −2.07723 2.60477i 0 −0.165391 + 0.131895i 0 2.53455 0.758996i 0 −1.80236 + 7.89664i 0
111.2 0 −1.94065 2.43350i 0 −2.10945 + 1.68223i 0 −1.58011 2.12208i 0 −1.48823 + 6.52037i 0
111.3 0 −1.44547 1.81256i 0 2.77297 2.21137i 0 −1.94347 1.79525i 0 −0.528427 + 2.31519i 0
111.4 0 −1.36747 1.71475i 0 1.64860 1.31472i 0 −1.20605 + 2.35488i 0 −0.402843 + 1.76497i 0
111.5 0 −1.31050 1.64332i 0 −0.883194 + 0.704324i 0 2.18004 + 1.49914i 0 −0.315513 + 1.38235i 0
111.6 0 −0.977383 1.22560i 0 −0.470082 + 0.374878i 0 1.48039 + 2.19281i 0 0.120747 0.529025i 0
111.7 0 −0.912783 1.14459i 0 −0.583523 + 0.465344i 0 −2.34056 1.23360i 0 0.190641 0.835251i 0
111.8 0 −0.881327 1.10515i 0 2.63324 2.09994i 0 0.910280 2.48423i 0 0.222946 0.976789i 0
111.9 0 −0.0556550 0.0697892i 0 −3.27604 + 2.61256i 0 1.51276 2.17061i 0 0.665790 2.91702i 0
111.10 0 −0.0314318 0.0394142i 0 −0.245577 + 0.195841i 0 −1.27396 + 2.31884i 0 0.666997 2.92231i 0
111.11 0 0.0314318 + 0.0394142i 0 −0.245577 + 0.195841i 0 1.27396 2.31884i 0 0.666997 2.92231i 0
111.12 0 0.0556550 + 0.0697892i 0 −3.27604 + 2.61256i 0 −1.51276 + 2.17061i 0 0.665790 2.91702i 0
111.13 0 0.881327 + 1.10515i 0 2.63324 2.09994i 0 −0.910280 + 2.48423i 0 0.222946 0.976789i 0
111.14 0 0.912783 + 1.14459i 0 −0.583523 + 0.465344i 0 2.34056 + 1.23360i 0 0.190641 0.835251i 0
111.15 0 0.977383 + 1.22560i 0 −0.470082 + 0.374878i 0 −1.48039 2.19281i 0 0.120747 0.529025i 0
111.16 0 1.31050 + 1.64332i 0 −0.883194 + 0.704324i 0 −2.18004 1.49914i 0 −0.315513 + 1.38235i 0
111.17 0 1.36747 + 1.71475i 0 1.64860 1.31472i 0 1.20605 2.35488i 0 −0.402843 + 1.76497i 0
111.18 0 1.44547 + 1.81256i 0 2.77297 2.21137i 0 1.94347 + 1.79525i 0 −0.528427 + 2.31519i 0
111.19 0 1.94065 + 2.43350i 0 −2.10945 + 1.68223i 0 1.58011 + 2.12208i 0 −1.48823 + 6.52037i 0
111.20 0 2.07723 + 2.60477i 0 −0.165391 + 0.131895i 0 −2.53455 + 0.758996i 0 −1.80236 + 7.89664i 0
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 671.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
49.f odd 14 1 inner
196.j even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.bb.b 120
4.b odd 2 1 inner 784.2.bb.b 120
49.f odd 14 1 inner 784.2.bb.b 120
196.j even 14 1 inner 784.2.bb.b 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.2.bb.b 120 1.a even 1 1 trivial
784.2.bb.b 120 4.b odd 2 1 inner
784.2.bb.b 120 49.f odd 14 1 inner
784.2.bb.b 120 196.j even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(11\!\cdots\!62\)\( T_{3}^{100} + \)\(11\!\cdots\!83\)\( T_{3}^{98} + \)\(10\!\cdots\!66\)\( T_{3}^{96} + \)\(89\!\cdots\!57\)\( T_{3}^{94} + \)\(73\!\cdots\!18\)\( T_{3}^{92} + \)\(56\!\cdots\!03\)\( T_{3}^{90} + \)\(42\!\cdots\!55\)\( T_{3}^{88} + \)\(29\!\cdots\!66\)\( T_{3}^{86} + \)\(19\!\cdots\!98\)\( T_{3}^{84} + \)\(12\!\cdots\!70\)\( T_{3}^{82} + \)\(75\!\cdots\!20\)\( T_{3}^{80} + \)\(43\!\cdots\!91\)\( T_{3}^{78} + \)\(23\!\cdots\!42\)\( T_{3}^{76} + \)\(12\!\cdots\!38\)\( T_{3}^{74} + \)\(58\!\cdots\!55\)\( T_{3}^{72} + \)\(26\!\cdots\!73\)\( T_{3}^{70} + \)\(11\!\cdots\!29\)\( T_{3}^{68} + \)\(46\!\cdots\!59\)\( T_{3}^{66} + \)\(17\!\cdots\!29\)\( T_{3}^{64} + \)\(60\!\cdots\!46\)\( T_{3}^{62} + \)\(19\!\cdots\!73\)\( T_{3}^{60} + \)\(53\!\cdots\!89\)\( T_{3}^{58} + \)\(13\!\cdots\!63\)\( T_{3}^{56} + \)\(31\!\cdots\!35\)\( T_{3}^{54} + \)\(65\!\cdots\!86\)\( T_{3}^{52} + \)\(12\!\cdots\!71\)\( T_{3}^{50} + \)\(21\!\cdots\!11\)\( T_{3}^{48} + \)\(35\!\cdots\!51\)\( T_{3}^{46} + \)\(53\!\cdots\!63\)\( T_{3}^{44} + \)\(73\!\cdots\!99\)\( T_{3}^{42} + \)\(92\!\cdots\!08\)\( T_{3}^{40} + \)\(10\!\cdots\!22\)\( T_{3}^{38} + \)\(11\!\cdots\!20\)\( T_{3}^{36} + \)\(11\!\cdots\!55\)\( T_{3}^{34} + \)\(10\!\cdots\!48\)\( T_{3}^{32} + \)\(87\!\cdots\!65\)\( T_{3}^{30} + \)\(59\!\cdots\!88\)\( T_{3}^{28} + \)\(39\!\cdots\!46\)\( T_{3}^{26} + \)\(29\!\cdots\!23\)\( T_{3}^{24} + \)\(16\!\cdots\!86\)\( T_{3}^{22} + \)\(40\!\cdots\!05\)\( T_{3}^{20} - \)\(14\!\cdots\!36\)\( T_{3}^{18} + \)\(22\!\cdots\!38\)\( T_{3}^{16} + \)\(34\!\cdots\!90\)\( T_{3}^{14} + \)\(12\!\cdots\!65\)\( T_{3}^{12} + \)\(73\!\cdots\!49\)\( T_{3}^{10} + \)\(81\!\cdots\!08\)\( T_{3}^{8} + \)\(10\!\cdots\!01\)\( T_{3}^{6} + \)\(42\!\cdots\!49\)\( T_{3}^{4} - 47281603654 T_{3}^{2} + 3418801 \)">\(T_{3}^{120} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\).