# Properties

 Label 930.2.e.a Level $930$ Weight $2$ Character orbit 930.e Analytic conductor $7.426$ Analytic rank $0$ Dimension $32$ 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.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$32$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 32q^{2} + 32q^{4} + 2q^{5} - 32q^{8} + 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 32q^{2} + 32q^{4} + 2q^{5} - 32q^{8} + 4q^{9} - 2q^{10} + 32q^{16} - 4q^{18} + 8q^{19} + 2q^{20} + 10q^{25} - 12q^{31} - 32q^{32} - 8q^{33} + 16q^{35} + 4q^{36} - 8q^{38} - 4q^{39} - 2q^{40} + 10q^{45} - 4q^{47} - 36q^{49} - 10q^{50} - 4q^{51} + 12q^{62} - 24q^{63} + 32q^{64} + 8q^{66} - 8q^{69} - 16q^{70} - 4q^{72} + 8q^{76} + 4q^{78} + 2q^{80} + 24q^{81} - 4q^{87} - 10q^{90} + 24q^{93} + 4q^{94} - 26q^{95} + 36q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
929.1 −1.00000 −1.70787 0.288390i 1.00000 2.10277 0.760496i 1.70787 + 0.288390i 4.63015i −1.00000 2.83366 + 0.985067i −2.10277 + 0.760496i
929.2 −1.00000 −1.70787 + 0.288390i 1.00000 2.10277 + 0.760496i 1.70787 0.288390i 4.63015i −1.00000 2.83366 0.985067i −2.10277 0.760496i
929.3 −1.00000 −1.69795 0.341995i 1.00000 −1.23971 1.86095i 1.69795 + 0.341995i 0.843688i −1.00000 2.76608 + 1.16138i 1.23971 + 1.86095i
929.4 −1.00000 −1.69795 + 0.341995i 1.00000 −1.23971 + 1.86095i 1.69795 0.341995i 0.843688i −1.00000 2.76608 1.16138i 1.23971 1.86095i
929.5 −1.00000 −1.63502 0.571593i 1.00000 1.46321 1.69086i 1.63502 + 0.571593i 2.83992i −1.00000 2.34656 + 1.86913i −1.46321 + 1.69086i
929.6 −1.00000 −1.63502 + 0.571593i 1.00000 1.46321 + 1.69086i 1.63502 0.571593i 2.83992i −1.00000 2.34656 1.86913i −1.46321 1.69086i
929.7 −1.00000 −1.36636 1.06445i 1.00000 −2.16820 + 0.546716i 1.36636 + 1.06445i 1.97405i −1.00000 0.733873 + 2.90885i 2.16820 0.546716i
929.8 −1.00000 −1.36636 + 1.06445i 1.00000 −2.16820 0.546716i 1.36636 1.06445i 1.97405i −1.00000 0.733873 2.90885i 2.16820 + 0.546716i
929.9 −1.00000 −1.02858 1.39357i 1.00000 0.467590 + 2.18663i 1.02858 + 1.39357i 1.43029i −1.00000 −0.884063 + 2.86678i −0.467590 2.18663i
929.10 −1.00000 −1.02858 + 1.39357i 1.00000 0.467590 2.18663i 1.02858 1.39357i 1.43029i −1.00000 −0.884063 2.86678i −0.467590 + 2.18663i
929.11 −1.00000 −0.880669 1.49145i 1.00000 2.09732 + 0.775395i 0.880669 + 1.49145i 0.657746i −1.00000 −1.44884 + 2.62695i −2.09732 0.775395i
929.12 −1.00000 −0.880669 + 1.49145i 1.00000 2.09732 0.775395i 0.880669 1.49145i 0.657746i −1.00000 −1.44884 2.62695i −2.09732 + 0.775395i
929.13 −1.00000 −0.522710 1.65129i 1.00000 −1.93647 + 1.11808i 0.522710 + 1.65129i 3.41776i −1.00000 −2.45355 + 1.72630i 1.93647 1.11808i
929.14 −1.00000 −0.522710 + 1.65129i 1.00000 −1.93647 1.11808i 0.522710 1.65129i 3.41776i −1.00000 −2.45355 1.72630i 1.93647 + 1.11808i
929.15 −1.00000 −0.230519 1.71664i 1.00000 −0.286520 2.21764i 0.230519 + 1.71664i 4.09004i −1.00000 −2.89372 + 0.791438i 0.286520 + 2.21764i
929.16 −1.00000 −0.230519 + 1.71664i 1.00000 −0.286520 + 2.21764i 0.230519 1.71664i 4.09004i −1.00000 −2.89372 0.791438i 0.286520 2.21764i
929.17 −1.00000 0.230519 1.71664i 1.00000 −0.286520 + 2.21764i −0.230519 + 1.71664i 4.09004i −1.00000 −2.89372 0.791438i 0.286520 2.21764i
929.18 −1.00000 0.230519 + 1.71664i 1.00000 −0.286520 2.21764i −0.230519 1.71664i 4.09004i −1.00000 −2.89372 + 0.791438i 0.286520 + 2.21764i
929.19 −1.00000 0.522710 1.65129i 1.00000 −1.93647 1.11808i −0.522710 + 1.65129i 3.41776i −1.00000 −2.45355 1.72630i 1.93647 + 1.11808i
929.20 −1.00000 0.522710 + 1.65129i 1.00000 −1.93647 + 1.11808i −0.522710 1.65129i 3.41776i −1.00000 −2.45355 + 1.72630i 1.93647 1.11808i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 929.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
31.b odd 2 1 inner
465.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.e.a 32
3.b odd 2 1 930.2.e.b yes 32
5.b even 2 1 930.2.e.b yes 32
15.d odd 2 1 inner 930.2.e.a 32
31.b odd 2 1 inner 930.2.e.a 32
93.c even 2 1 930.2.e.b yes 32
155.c odd 2 1 930.2.e.b yes 32
465.g even 2 1 inner 930.2.e.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.e.a 32 1.a even 1 1 trivial
930.2.e.a 32 15.d odd 2 1 inner
930.2.e.a 32 31.b odd 2 1 inner
930.2.e.a 32 465.g even 2 1 inner
930.2.e.b yes 32 3.b odd 2 1
930.2.e.b yes 32 5.b even 2 1
930.2.e.b yes 32 93.c even 2 1
930.2.e.b yes 32 155.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{47}^{8} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.