# Properties

 Label 930.2.s.d Level $930$ Weight $2$ Character orbit 930.s Analytic conductor $7.426$ Analytic rank $0$ Dimension $28$ 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: $$28$$ Relative dimension: $$14$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q - 28q^{4} + 2q^{5} + 14q^{6} + 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - 28q^{4} + 2q^{5} + 14q^{6} + 14q^{9} + 4q^{10} + 2q^{11} - 8q^{14} - 8q^{15} + 28q^{16} + 4q^{19} - 2q^{20} - 8q^{21} - 14q^{24} - 14q^{25} - 12q^{26} + 24q^{29} + 4q^{30} + 10q^{34} + 8q^{35} - 14q^{36} + 24q^{39} - 4q^{40} + 20q^{41} - 2q^{44} - 2q^{45} + 12q^{46} + 2q^{49} - 8q^{50} + 10q^{51} + 28q^{54} + 6q^{55} + 8q^{56} - 20q^{59} + 8q^{60} + 32q^{61} - 28q^{64} - 12q^{65} + 4q^{66} - 6q^{69} - 16q^{70} - 16q^{71} - 6q^{74} - 8q^{75} - 4q^{76} + 10q^{79} + 2q^{80} - 14q^{81} + 8q^{84} + 72q^{85} + 18q^{86} - 80q^{89} - 4q^{90} + 32q^{91} - 12q^{94} - 32q^{95} + 14q^{96} - 2q^{99} + O(q^{100})$$

## 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 −2.16404 0.562979i 0.500000 0.866025i 0.759514 + 0.438506i 1.00000i 0.500000 + 0.866025i −0.562979 + 2.16404i
439.2 1.00000i 0.866025 + 0.500000i −1.00000 −1.82738 + 1.28868i 0.500000 0.866025i −2.60552 1.50429i 1.00000i 0.500000 + 0.866025i 1.28868 + 1.82738i
439.3 1.00000i 0.866025 + 0.500000i −1.00000 −0.670801 2.13308i 0.500000 0.866025i −1.77376 1.02408i 1.00000i 0.500000 + 0.866025i −2.13308 + 0.670801i
439.4 1.00000i 0.866025 + 0.500000i −1.00000 −0.347530 + 2.20890i 0.500000 0.866025i 3.10811 + 1.79447i 1.00000i 0.500000 + 0.866025i 2.20890 + 0.347530i
439.5 1.00000i 0.866025 + 0.500000i −1.00000 0.836743 2.07361i 0.500000 0.866025i 1.24405 + 0.718253i 1.00000i 0.500000 + 0.866025i −2.07361 0.836743i
439.6 1.00000i 0.866025 + 0.500000i −1.00000 1.02320 + 1.98823i 0.500000 0.866025i −3.96572 2.28961i 1.00000i 0.500000 + 0.866025i 1.98823 1.02320i
439.7 1.00000i 0.866025 + 0.500000i −1.00000 1.91775 + 1.14989i 0.500000 0.866025i −0.230788 0.133246i 1.00000i 0.500000 + 0.866025i 1.14989 1.91775i
439.8 1.00000i −0.866025 0.500000i −1.00000 −2.21417 0.312164i 0.500000 0.866025i −1.24405 0.718253i 1.00000i 0.500000 + 0.866025i 0.312164 2.21417i
439.9 1.00000i −0.866025 0.500000i −1.00000 −1.51190 1.64747i 0.500000 0.866025i 1.77376 + 1.02408i 1.00000i 0.500000 + 0.866025i 1.64747 1.51190i
439.10 1.00000i −0.866025 0.500000i −1.00000 0.0369558 + 2.23576i 0.500000 0.866025i 0.230788 + 0.133246i 1.00000i 0.500000 + 0.866025i −2.23576 + 0.0369558i
439.11 1.00000i −0.866025 0.500000i −1.00000 0.594464 2.15560i 0.500000 0.866025i −0.759514 0.438506i 1.00000i 0.500000 + 0.866025i 2.15560 + 0.594464i
439.12 1.00000i −0.866025 0.500000i −1.00000 1.21026 + 1.88023i 0.500000 0.866025i 3.96572 + 2.28961i 1.00000i 0.500000 + 0.866025i −1.88023 + 1.21026i
439.13 1.00000i −0.866025 0.500000i −1.00000 2.02972 0.938214i 0.500000 0.866025i 2.60552 + 1.50429i 1.00000i 0.500000 + 0.866025i 0.938214 + 2.02972i
439.14 1.00000i −0.866025 0.500000i −1.00000 2.08673 + 0.803479i 0.500000 0.866025i −3.10811 1.79447i 1.00000i 0.500000 + 0.866025i −0.803479 + 2.08673i
769.1 1.00000i −0.866025 + 0.500000i −1.00000 −2.21417 + 0.312164i 0.500000 + 0.866025i −1.24405 + 0.718253i 1.00000i 0.500000 0.866025i 0.312164 + 2.21417i
769.2 1.00000i −0.866025 + 0.500000i −1.00000 −1.51190 + 1.64747i 0.500000 + 0.866025i 1.77376 1.02408i 1.00000i 0.500000 0.866025i 1.64747 + 1.51190i
769.3 1.00000i −0.866025 + 0.500000i −1.00000 0.0369558 2.23576i 0.500000 + 0.866025i 0.230788 0.133246i 1.00000i 0.500000 0.866025i −2.23576 0.0369558i
769.4 1.00000i −0.866025 + 0.500000i −1.00000 0.594464 + 2.15560i 0.500000 + 0.866025i −0.759514 + 0.438506i 1.00000i 0.500000 0.866025i 2.15560 0.594464i
769.5 1.00000i −0.866025 + 0.500000i −1.00000 1.21026 1.88023i 0.500000 + 0.866025i 3.96572 2.28961i 1.00000i 0.500000 0.866025i −1.88023 1.21026i
769.6 1.00000i −0.866025 + 0.500000i −1.00000 2.02972 + 0.938214i 0.500000 + 0.866025i 2.60552 1.50429i 1.00000i 0.500000 0.866025i 0.938214 2.02972i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 769.14 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.d 28
5.b even 2 1 inner 930.2.s.d 28
31.c even 3 1 inner 930.2.s.d 28
155.j even 6 1 inner 930.2.s.d 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.s.d 28 1.a even 1 1 trivial
930.2.s.d 28 5.b even 2 1 inner
930.2.s.d 28 31.c even 3 1 inner
930.2.s.d 28 155.j even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{28} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.