# Properties

 Label 930.2.s.c Level $930$ Weight $2$ Character orbit 930.s Analytic conductor $7.426$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.s (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 24q^{4} + 4q^{5} - 12q^{6} + 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 24q^{4} + 4q^{5} - 12q^{6} + 12q^{9} - 4q^{11} + 12q^{14} + 24q^{16} - 4q^{19} - 4q^{20} - 12q^{21} + 12q^{24} - 4q^{25} + 16q^{26} + 32q^{29} - 8q^{30} + 24q^{31} - 20q^{34} + 48q^{35} - 12q^{36} + 32q^{39} - 12q^{41} + 4q^{44} - 4q^{45} + 56q^{46} + 24q^{49} - 16q^{50} + 20q^{51} - 24q^{54} + 36q^{55} - 12q^{56} - 4q^{59} - 40q^{61} - 24q^{64} - 4q^{65} + 8q^{66} + 28q^{69} - 32q^{70} - 56q^{71} + 12q^{74} + 16q^{75} + 4q^{76} - 72q^{79} + 4q^{80} - 12q^{81} + 12q^{84} + 40q^{85} - 24q^{86} + 128q^{89} - 160q^{91} + 56q^{94} - 56q^{95} - 12q^{96} + 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
439.1 1.00000i −0.866025 0.500000i −1.00000 −1.89131 + 1.19288i −0.500000 + 0.866025i 2.10840 + 1.21729i 1.00000i 0.500000 + 0.866025i 1.19288 + 1.89131i
439.2 1.00000i −0.866025 0.500000i −1.00000 −0.968769 + 2.01531i −0.500000 + 0.866025i −3.61895 2.08940i 1.00000i 0.500000 + 0.866025i 2.01531 + 0.968769i
439.3 1.00000i −0.866025 0.500000i −1.00000 −0.291362 2.21700i −0.500000 + 0.866025i 3.11266 + 1.79709i 1.00000i 0.500000 + 0.866025i −2.21700 + 0.291362i
439.4 1.00000i −0.866025 0.500000i −1.00000 0.875319 2.05762i −0.500000 + 0.866025i 0.765960 + 0.442227i 1.00000i 0.500000 + 0.866025i −2.05762 0.875319i
439.5 1.00000i −0.866025 0.500000i −1.00000 1.24897 + 1.85475i −0.500000 + 0.866025i −0.669647 0.386621i 1.00000i 0.500000 + 0.866025i 1.85475 1.24897i
439.6 1.00000i −0.866025 0.500000i −1.00000 2.02715 + 0.943742i −0.500000 + 0.866025i 3.49773 + 2.01941i 1.00000i 0.500000 + 0.866025i 0.943742 2.02715i
439.7 1.00000i 0.866025 + 0.500000i −1.00000 −2.21961 0.270763i −0.500000 + 0.866025i −0.765960 0.442227i 1.00000i 0.500000 + 0.866025i 0.270763 2.21961i
439.8 1.00000i 0.866025 + 0.500000i −1.00000 −1.77430 1.36083i −0.500000 + 0.866025i −3.11266 1.79709i 1.00000i 0.500000 + 0.866025i 1.36083 1.77430i
439.9 1.00000i 0.866025 + 0.500000i −1.00000 −0.196273 + 2.22744i −0.500000 + 0.866025i −3.49773 2.01941i 1.00000i 0.500000 + 0.866025i −2.22744 0.196273i
439.10 1.00000i 0.866025 + 0.500000i −1.00000 0.981775 + 2.00901i −0.500000 + 0.866025i 0.669647 + 0.386621i 1.00000i 0.500000 + 0.866025i −2.00901 + 0.981775i
439.11 1.00000i 0.866025 + 0.500000i −1.00000 1.97872 1.04148i −0.500000 + 0.866025i −2.10840 1.21729i 1.00000i 0.500000 + 0.866025i 1.04148 + 1.97872i
439.12 1.00000i 0.866025 + 0.500000i −1.00000 2.22970 + 0.168678i −0.500000 + 0.866025i 3.61895 + 2.08940i 1.00000i 0.500000 + 0.866025i −0.168678 + 2.22970i
769.1 1.00000i 0.866025 0.500000i −1.00000 −2.21961 + 0.270763i −0.500000 0.866025i −0.765960 + 0.442227i 1.00000i 0.500000 0.866025i 0.270763 + 2.21961i
769.2 1.00000i 0.866025 0.500000i −1.00000 −1.77430 + 1.36083i −0.500000 0.866025i −3.11266 + 1.79709i 1.00000i 0.500000 0.866025i 1.36083 + 1.77430i
769.3 1.00000i 0.866025 0.500000i −1.00000 −0.196273 2.22744i −0.500000 0.866025i −3.49773 + 2.01941i 1.00000i 0.500000 0.866025i −2.22744 + 0.196273i
769.4 1.00000i 0.866025 0.500000i −1.00000 0.981775 2.00901i −0.500000 0.866025i 0.669647 0.386621i 1.00000i 0.500000 0.866025i −2.00901 0.981775i
769.5 1.00000i 0.866025 0.500000i −1.00000 1.97872 + 1.04148i −0.500000 0.866025i −2.10840 + 1.21729i 1.00000i 0.500000 0.866025i 1.04148 1.97872i
769.6 1.00000i 0.866025 0.500000i −1.00000 2.22970 0.168678i −0.500000 0.866025i 3.61895 2.08940i 1.00000i 0.500000 0.866025i −0.168678 2.22970i
769.7 1.00000i −0.866025 + 0.500000i −1.00000 −1.89131 1.19288i −0.500000 0.866025i 2.10840 1.21729i 1.00000i 0.500000 0.866025i 1.19288 1.89131i
769.8 1.00000i −0.866025 + 0.500000i −1.00000 −0.968769 2.01531i −0.500000 0.866025i −3.61895 + 2.08940i 1.00000i 0.500000 0.866025i 2.01531 0.968769i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 769.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.c even 3 1 inner
155.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.s.c 24
5.b even 2 1 inner 930.2.s.c 24
31.c even 3 1 inner 930.2.s.c 24
155.j even 6 1 inner 930.2.s.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.s.c 24 1.a even 1 1 trivial
930.2.s.c 24 5.b even 2 1 inner
930.2.s.c 24 31.c even 3 1 inner
930.2.s.c 24 155.j even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{24} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.