# Properties

 Label 950.2.q.e Level $950$ Weight $2$ Character orbit 950.q Analytic conductor $7.586$ Analytic rank $0$ Dimension $24$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.q (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 12q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 12q^{6} - 48q^{11} + 12q^{16} - 84q^{21} + 24q^{26} - 24q^{36} + 48q^{41} + 12q^{51} + 12q^{61} + 24q^{71} + 36q^{76} + 12q^{81} - 36q^{86} - 228q^{91} - 24q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1 −0.965926 0.258819i −0.431989 + 1.61221i 0.866025 + 0.500000i 0 0.834540 1.44546i 2.81008 + 2.81008i −0.707107 0.707107i 0.185481 + 0.107087i 0
107.2 −0.965926 0.258819i 0.394434 1.47205i 0.866025 + 0.500000i 0 −0.761988 + 1.31980i 1.69567 + 1.69567i −0.707107 0.707107i 0.586729 + 0.338748i 0
107.3 −0.965926 0.258819i 0.814012 3.03794i 0.866025 + 0.500000i 0 −1.57255 + 2.72374i −2.05625 2.05625i −0.707107 0.707107i −5.96836 3.44584i 0
107.4 0.965926 + 0.258819i −0.814012 + 3.03794i 0.866025 + 0.500000i 0 −1.57255 + 2.72374i 2.05625 + 2.05625i 0.707107 + 0.707107i −5.96836 3.44584i 0
107.5 0.965926 + 0.258819i −0.394434 + 1.47205i 0.866025 + 0.500000i 0 −0.761988 + 1.31980i −1.69567 1.69567i 0.707107 + 0.707107i 0.586729 + 0.338748i 0
107.6 0.965926 + 0.258819i 0.431989 1.61221i 0.866025 + 0.500000i 0 0.834540 1.44546i −2.81008 2.81008i 0.707107 + 0.707107i 0.185481 + 0.107087i 0
293.1 −0.965926 + 0.258819i −0.431989 1.61221i 0.866025 0.500000i 0 0.834540 + 1.44546i 2.81008 2.81008i −0.707107 + 0.707107i 0.185481 0.107087i 0
293.2 −0.965926 + 0.258819i 0.394434 + 1.47205i 0.866025 0.500000i 0 −0.761988 1.31980i 1.69567 1.69567i −0.707107 + 0.707107i 0.586729 0.338748i 0
293.3 −0.965926 + 0.258819i 0.814012 + 3.03794i 0.866025 0.500000i 0 −1.57255 2.72374i −2.05625 + 2.05625i −0.707107 + 0.707107i −5.96836 + 3.44584i 0
293.4 0.965926 0.258819i −0.814012 3.03794i 0.866025 0.500000i 0 −1.57255 2.72374i 2.05625 2.05625i 0.707107 0.707107i −5.96836 + 3.44584i 0
293.5 0.965926 0.258819i −0.394434 1.47205i 0.866025 0.500000i 0 −0.761988 1.31980i −1.69567 + 1.69567i 0.707107 0.707107i 0.586729 0.338748i 0
293.6 0.965926 0.258819i 0.431989 + 1.61221i 0.866025 0.500000i 0 0.834540 + 1.44546i −2.81008 + 2.81008i 0.707107 0.707107i 0.185481 0.107087i 0
407.1 −0.258819 0.965926i −1.61221 + 0.431989i −0.866025 + 0.500000i 0 0.834540 + 1.44546i 2.81008 + 2.81008i 0.707107 + 0.707107i −0.185481 + 0.107087i 0
407.2 −0.258819 0.965926i 1.47205 0.394434i −0.866025 + 0.500000i 0 −0.761988 1.31980i 1.69567 + 1.69567i 0.707107 + 0.707107i −0.586729 + 0.338748i 0
407.3 −0.258819 0.965926i 3.03794 0.814012i −0.866025 + 0.500000i 0 −1.57255 2.72374i −2.05625 2.05625i 0.707107 + 0.707107i 5.96836 3.44584i 0
407.4 0.258819 + 0.965926i −3.03794 + 0.814012i −0.866025 + 0.500000i 0 −1.57255 2.72374i 2.05625 + 2.05625i −0.707107 0.707107i 5.96836 3.44584i 0
407.5 0.258819 + 0.965926i −1.47205 + 0.394434i −0.866025 + 0.500000i 0 −0.761988 1.31980i −1.69567 1.69567i −0.707107 0.707107i −0.586729 + 0.338748i 0
407.6 0.258819 + 0.965926i 1.61221 0.431989i −0.866025 + 0.500000i 0 0.834540 + 1.44546i −2.81008 2.81008i −0.707107 0.707107i −0.185481 + 0.107087i 0
943.1 −0.258819 + 0.965926i −1.61221 0.431989i −0.866025 0.500000i 0 0.834540 1.44546i 2.81008 2.81008i 0.707107 0.707107i −0.185481 0.107087i 0
943.2 −0.258819 + 0.965926i 1.47205 + 0.394434i −0.866025 0.500000i 0 −0.761988 + 1.31980i 1.69567 1.69567i 0.707107 0.707107i −0.586729 0.338748i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 943.6 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
5.c odd 4 2 inner
19.d odd 6 1 inner
95.h odd 6 1 inner
95.l even 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.q.e 24
5.b even 2 1 inner 950.2.q.e 24
5.c odd 4 2 inner 950.2.q.e 24
19.d odd 6 1 inner 950.2.q.e 24
95.h odd 6 1 inner 950.2.q.e 24
95.l even 12 2 inner 950.2.q.e 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.q.e 24 1.a even 1 1 trivial
950.2.q.e 24 5.b even 2 1 inner
950.2.q.e 24 5.c odd 4 2 inner
950.2.q.e 24 19.d odd 6 1 inner
950.2.q.e 24 95.h odd 6 1 inner
950.2.q.e 24 95.l even 12 2 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$:

 $$T_{3}^{24} - 111 T_{3}^{20} + 10992 T_{3}^{16} - 139327 T_{3}^{12} + 1311585 T_{3}^{8} - 5443584 T_{3}^{4} + 16777216$$ $$T_{7}^{12} + 354 T_{7}^{8} + 28449 T_{7}^{4} + 589824$$