# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(784, [\chi])$$.