# Properties

 Label 76.4.i.a Level $76$ Weight $4$ Character orbit 76.i Analytic conductor $4.484$ Analytic rank $0$ Dimension $30$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 76.i (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.48414516044$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$5$$ over $$\Q(\zeta_{9})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$30q - 3q^{3} + 6q^{7} + 15q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q - 3q^{3} + 6q^{7} + 15q^{9} + 42q^{11} - 42q^{13} + 78q^{15} + 30q^{17} + 282q^{19} + 198q^{21} - 300q^{23} - 276q^{25} + 219q^{27} + 216q^{29} + 30q^{31} - 597q^{33} - 636q^{35} + 60q^{37} - 2172q^{39} - 63q^{41} - 246q^{43} - 882q^{45} + 762q^{47} - 525q^{49} + 2613q^{51} + 882q^{53} + 1350q^{55} + 924q^{57} + 2085q^{59} + 1530q^{61} + 2424q^{63} + 1530q^{65} - 3609q^{67} + 756q^{69} - 4962q^{71} - 2394q^{73} - 3516q^{77} - 630q^{79} - 3723q^{81} - 2382q^{83} + 3228q^{85} - 1110q^{87} + 2196q^{89} + 6036q^{91} + 5010q^{93} + 6204q^{95} + 6459q^{97} + 6189q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 0 −1.53481 + 8.70432i 0 10.3935 + 8.72119i 0 −7.14447 12.3746i 0 −48.0379 17.4843i 0
5.2 0 −0.580234 + 3.29067i 0 −11.9065 9.99077i 0 −13.7659 23.8433i 0 14.8799 + 5.41582i 0
5.3 0 −0.267608 + 1.51768i 0 −1.63306 1.37030i 0 13.0616 + 22.6234i 0 23.1400 + 8.42226i 0
5.4 0 0.629037 3.56745i 0 12.9592 + 10.8741i 0 −4.22583 7.31935i 0 13.0407 + 4.74643i 0
5.5 0 1.42726 8.09439i 0 −6.74894 5.66304i 0 0.266189 + 0.461053i 0 −38.1104 13.8710i 0
9.1 0 −7.47864 + 6.27533i 0 −3.19346 1.16233i 0 4.69206 8.12689i 0 11.8619 67.2720i 0
9.2 0 −1.65540 + 1.38904i 0 8.99353 + 3.27338i 0 −0.0466734 + 0.0808407i 0 −3.87760 + 21.9910i 0
9.3 0 −0.487209 + 0.408817i 0 −11.6536 4.24157i 0 −11.7723 + 20.3903i 0 −4.61826 + 26.1915i 0
9.4 0 3.64834 3.06132i 0 −10.2365 3.72578i 0 16.4303 28.4581i 0 −0.749805 + 4.25235i 0
9.5 0 6.23895 5.23510i 0 12.3313 + 4.48821i 0 −4.34015 + 7.51736i 0 6.82973 38.7333i 0
17.1 0 −7.47864 6.27533i 0 −3.19346 + 1.16233i 0 4.69206 + 8.12689i 0 11.8619 + 67.2720i 0
17.2 0 −1.65540 1.38904i 0 8.99353 3.27338i 0 −0.0466734 0.0808407i 0 −3.87760 21.9910i 0
17.3 0 −0.487209 0.408817i 0 −11.6536 + 4.24157i 0 −11.7723 20.3903i 0 −4.61826 26.1915i 0
17.4 0 3.64834 + 3.06132i 0 −10.2365 + 3.72578i 0 16.4303 + 28.4581i 0 −0.749805 4.25235i 0
17.5 0 6.23895 + 5.23510i 0 12.3313 4.48821i 0 −4.34015 7.51736i 0 6.82973 + 38.7333i 0
25.1 0 −8.30252 3.02187i 0 2.68735 + 15.2407i 0 13.2377 22.9284i 0 39.1169 + 32.8230i 0
25.2 0 −3.29897 1.20073i 0 −0.0631039 0.357880i 0 −13.5658 + 23.4967i 0 −11.2417 9.43292i 0
25.3 0 −1.97311 0.718152i 0 −1.01111 5.73427i 0 2.37147 4.10751i 0 −17.3058 14.5213i 0
25.4 0 5.87446 + 2.13813i 0 −3.07163 17.4201i 0 12.3773 21.4380i 0 9.25452 + 7.76546i 0
25.5 0 6.26044 + 2.27861i 0 2.15308 + 12.2107i 0 −4.57544 + 7.92489i 0 13.3179 + 11.1750i 0
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.4.i.a 30
19.e even 9 1 inner 76.4.i.a 30
19.e even 9 1 1444.4.a.j 15
19.f odd 18 1 1444.4.a.k 15

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.i.a 30 1.a even 1 1 trivial
76.4.i.a 30 19.e even 9 1 inner
1444.4.a.j 15 19.e even 9 1
1444.4.a.k 15 19.f odd 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(76, [\chi])$$.