# Properties

 Label 930.2.e.b Level $930$ Weight $2$ Character orbit 930.e Analytic conductor $7.426$ Analytic rank $0$ Dimension $32$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [930,2,Mod(929,930)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("930.929");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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 algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 32 q^{2} + 32 q^{4} - 2 q^{5} + 32 q^{8} + 4 q^{9}+O(q^{10})$$ 32 * q + 32 * q^2 + 32 * q^4 - 2 * q^5 + 32 * q^8 + 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 32 q^{2} + 32 q^{4} - 2 q^{5} + 32 q^{8} + 4 q^{9} - 2 q^{10} + 32 q^{16} + 4 q^{18} + 8 q^{19} - 2 q^{20} + 10 q^{25} - 12 q^{31} + 32 q^{32} + 8 q^{33} - 16 q^{35} + 4 q^{36} + 8 q^{38} - 4 q^{39} - 2 q^{40} - 42 q^{45} + 4 q^{47} - 36 q^{49} + 10 q^{50} - 4 q^{51} - 12 q^{62} + 24 q^{63} + 32 q^{64} + 8 q^{66} - 8 q^{69} - 16 q^{70} + 4 q^{72} + 8 q^{76} - 4 q^{78} - 2 q^{80} + 24 q^{81} + 4 q^{87} - 42 q^{90} - 24 q^{93} + 4 q^{94} + 26 q^{95} - 36 q^{98}+O(q^{100})$$ 32 * q + 32 * q^2 + 32 * q^4 - 2 * q^5 + 32 * q^8 + 4 * q^9 - 2 * q^10 + 32 * q^16 + 4 * q^18 + 8 * q^19 - 2 * q^20 + 10 * q^25 - 12 * q^31 + 32 * q^32 + 8 * q^33 - 16 * q^35 + 4 * q^36 + 8 * q^38 - 4 * q^39 - 2 * q^40 - 42 * q^45 + 4 * q^47 - 36 * q^49 + 10 * q^50 - 4 * q^51 - 12 * q^62 + 24 * q^63 + 32 * q^64 + 8 * q^66 - 8 * q^69 - 16 * q^70 + 4 * q^72 + 8 * q^76 - 4 * q^78 - 2 * q^80 + 24 * q^81 + 4 * q^87 - 42 * q^90 - 24 * q^93 + 4 * q^94 + 26 * q^95 - 36 * q^98

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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.b yes 32
3.b odd 2 1 930.2.e.a 32
5.b even 2 1 930.2.e.a 32
15.d odd 2 1 inner 930.2.e.b yes 32
31.b odd 2 1 inner 930.2.e.b yes 32
93.c even 2 1 930.2.e.a 32
155.c odd 2 1 930.2.e.a 32
465.g even 2 1 inner 930.2.e.b yes 32

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{47}^{8} - T_{47}^{7} - 234T_{47}^{6} + 252T_{47}^{5} + 15056T_{47}^{4} - 12776T_{47}^{3} - 268088T_{47}^{2} + 10752T_{47} + 55296$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.