Properties

 Label 387.2.y.c Level $387$ Weight $2$ Character orbit 387.y Analytic conductor $3.090$ Analytic rank $0$ Dimension $36$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$387 = 3^{2} \cdot 43$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 387.y (of order $$21$$, degree $$12$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.09021055822$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$3$$ over $$\Q(\zeta_{21})$$ Twist minimal: no (minimal twist has level 43) Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$36q + 10q^{2} - 18q^{4} + 17q^{5} + 6q^{7} - 18q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q + 10q^{2} - 18q^{4} + 17q^{5} + 6q^{7} - 18q^{8} - 7q^{10} + 4q^{11} - 18q^{14} - 10q^{16} + 10q^{17} + 10q^{19} + 3q^{20} - 3q^{22} - 4q^{23} - 2q^{25} + 15q^{26} + 20q^{28} - 9q^{29} + 40q^{31} - 48q^{32} - 42q^{34} - 11q^{35} - 19q^{37} + 21q^{38} - 97q^{40} + 28q^{41} - 8q^{43} - 14q^{44} - 61q^{46} + 30q^{47} + 6q^{49} + 3q^{50} - 8q^{52} + 24q^{53} + 14q^{55} - 39q^{56} + 64q^{58} + q^{59} - 14q^{61} - 33q^{62} + 48q^{64} - 38q^{65} + 66q^{67} - 66q^{68} + 47q^{70} + 33q^{71} + 29q^{73} + 40q^{74} - 39q^{76} + 27q^{77} - 17q^{79} - 8q^{80} - 54q^{82} + 23q^{83} - 56q^{85} + 45q^{86} - 17q^{88} + 19q^{89} - 13q^{91} + 18q^{92} + 44q^{94} - q^{95} - 31q^{97} + 5q^{98} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1 −0.581275 2.54673i 0 −4.34603 + 2.09294i −2.95419 + 0.445272i 0 −0.339884 0.588696i 4.59900 + 5.76696i 0 2.85118 + 7.26470i
10.2 −0.188565 0.826155i 0 1.15496 0.556200i 3.39819 0.512194i 0 −0.134521 0.232998i −1.73398 2.17435i 0 −1.06393 2.71085i
10.3 0.483758 + 2.11948i 0 −2.45624 + 1.18286i −0.0260188 + 0.00392170i 0 1.56464 + 2.71003i −0.984367 1.23436i 0 −0.0208988 0.0532492i
100.1 −0.993512 + 0.478450i 0 −0.488828 + 0.612971i 3.17479 0.979294i 0 1.23273 2.13515i 0.683135 2.99301i 0 −2.68565 + 2.49192i
100.2 0.982954 0.473366i 0 −0.504856 + 0.633069i −1.29085 + 0.398175i 0 −0.108163 + 0.187343i −0.682116 + 2.98855i 0 −1.08036 + 1.00243i
100.3 2.05399 0.989151i 0 1.99349 2.49975i 0.131558 0.0405802i 0 0.934721 1.61898i 0.607386 2.66113i 0 0.230079 0.213482i
109.1 0.0594739 + 0.0745779i 0 0.443017 1.94098i 0.284956 3.80248i 0 1.30981 + 2.26866i 0.342987 0.165174i 0 0.300528 0.204897i
109.2 0.651405 + 0.816837i 0 0.202149 0.885672i 0.0373507 0.498411i 0 −1.65334 2.86367i 2.73775 1.31843i 0 0.431451 0.294158i
109.3 1.72039 + 2.15730i 0 −1.24917 + 5.47296i −0.0684907 + 0.913945i 0 −0.971539 1.68276i −8.98382 + 4.32638i 0 −2.08949 + 1.42459i
154.1 −1.94880 0.938491i 0 1.67006 + 2.09419i 2.00127 1.85691i 0 1.01083 1.75082i −0.326607 1.43096i 0 −5.64276 + 1.74056i
154.2 −0.118393 0.0570152i 0 −1.23621 1.55016i 1.30537 1.21121i 0 −0.00749281 + 0.0129779i 0.116458 + 0.510236i 0 −0.223604 + 0.0689728i
154.3 1.06783 + 0.514239i 0 −0.371164 0.465425i −2.37710 + 2.20563i 0 −1.38418 + 2.39748i −0.684463 2.99883i 0 −3.67256 + 1.13283i
181.1 −1.06016 1.32939i 0 −0.198316 + 0.868879i 1.45700 + 0.993363i 0 −0.297283 + 0.514909i −1.69861 + 0.818009i 0 −0.224073 2.99004i
181.2 1.11243 + 1.39495i 0 −0.263330 + 1.15372i 0.492700 + 0.335917i 0 2.18154 3.77854i 1.31271 0.632167i 0 0.0795092 + 1.06098i
181.3 1.44188 + 1.80806i 0 −0.745019 + 3.26414i 2.09843 + 1.43068i 0 −1.09365 + 1.89425i −2.80884 + 1.35267i 0 0.438918 + 5.85695i
253.1 −0.377390 + 1.65345i 0 −0.789549 0.380227i −0.140805 0.358765i 0 1.74586 + 3.02391i −1.18819 + 1.48995i 0 0.646339 0.0974200i
253.2 0.178150 0.780524i 0 1.22446 + 0.589667i −0.511972 1.30448i 0 −2.37928 4.12103i 1.67671 2.10253i 0 −1.10939 + 0.167213i
253.3 0.515822 2.25996i 0 −3.03942 1.46371i 1.48781 + 3.79089i 0 1.38920 + 2.40617i −1.98513 + 2.48927i 0 9.33472 1.40698i
271.1 −0.581275 + 2.54673i 0 −4.34603 2.09294i −2.95419 0.445272i 0 −0.339884 + 0.588696i 4.59900 5.76696i 0 2.85118 7.26470i
271.2 −0.188565 + 0.826155i 0 1.15496 + 0.556200i 3.39819 + 0.512194i 0 −0.134521 + 0.232998i −1.73398 + 2.17435i 0 −1.06393 + 2.71085i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.g even 21 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.2.y.c 36
3.b odd 2 1 43.2.g.a 36
12.b even 2 1 688.2.bg.c 36
43.g even 21 1 inner 387.2.y.c 36
129.n even 42 1 1849.2.a.o 18
129.o odd 42 1 43.2.g.a 36
129.o odd 42 1 1849.2.a.n 18
516.bb even 42 1 688.2.bg.c 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.g.a 36 3.b odd 2 1
43.2.g.a 36 129.o odd 42 1
387.2.y.c 36 1.a even 1 1 trivial
387.2.y.c 36 43.g even 21 1 inner
688.2.bg.c 36 12.b even 2 1
688.2.bg.c 36 516.bb even 42 1
1849.2.a.n 18 129.o odd 42 1
1849.2.a.o 18 129.n even 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{36} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(387, [\chi])$$.