# Properties

 Label 950.2.q.f Level $950$ Weight $2$ Character orbit 950.q Analytic conductor $7.586$ Analytic rank $0$ Dimension $32$ 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: $$32$$ Relative dimension: $$8$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 4q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 4q^{6} + 72q^{11} + 16q^{16} + 60q^{21} + 8q^{26} - 28q^{36} - 84q^{41} - 84q^{51} - 52q^{61} - 24q^{71} + 16q^{76} + 64q^{81} - 36q^{86} - 84q^{91} + 8q^{96} + O(q^{100})$$

## 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 −0.824123 + 3.07567i 0.866025 + 0.500000i 0 1.59208 2.75757i −3.29328 3.29328i −0.707107 0.707107i −6.18248 3.56945i 0
107.2 −0.965926 0.258819i −0.313773 + 1.17102i 0.866025 + 0.500000i 0 0.606164 1.04991i 1.88504 + 1.88504i −0.707107 0.707107i 1.32525 + 0.765131i 0
107.3 −0.965926 0.258819i 0.206366 0.770169i 0.866025 + 0.500000i 0 −0.398669 + 0.690515i −0.349095 0.349095i −0.707107 0.707107i 2.04750 + 1.18213i 0
107.4 −0.965926 0.258819i 0.672711 2.51059i 0.866025 + 0.500000i 0 −1.29958 + 2.25093i −0.692149 0.692149i −0.707107 0.707107i −3.25245 1.87780i 0
107.5 0.965926 + 0.258819i −0.672711 + 2.51059i 0.866025 + 0.500000i 0 −1.29958 + 2.25093i 0.692149 + 0.692149i 0.707107 + 0.707107i −3.25245 1.87780i 0
107.6 0.965926 + 0.258819i −0.206366 + 0.770169i 0.866025 + 0.500000i 0 −0.398669 + 0.690515i 0.349095 + 0.349095i 0.707107 + 0.707107i 2.04750 + 1.18213i 0
107.7 0.965926 + 0.258819i 0.313773 1.17102i 0.866025 + 0.500000i 0 0.606164 1.04991i −1.88504 1.88504i 0.707107 + 0.707107i 1.32525 + 0.765131i 0
107.8 0.965926 + 0.258819i 0.824123 3.07567i 0.866025 + 0.500000i 0 1.59208 2.75757i 3.29328 + 3.29328i 0.707107 + 0.707107i −6.18248 3.56945i 0
293.1 −0.965926 + 0.258819i −0.824123 3.07567i 0.866025 0.500000i 0 1.59208 + 2.75757i −3.29328 + 3.29328i −0.707107 + 0.707107i −6.18248 + 3.56945i 0
293.2 −0.965926 + 0.258819i −0.313773 1.17102i 0.866025 0.500000i 0 0.606164 + 1.04991i 1.88504 1.88504i −0.707107 + 0.707107i 1.32525 0.765131i 0
293.3 −0.965926 + 0.258819i 0.206366 + 0.770169i 0.866025 0.500000i 0 −0.398669 0.690515i −0.349095 + 0.349095i −0.707107 + 0.707107i 2.04750 1.18213i 0
293.4 −0.965926 + 0.258819i 0.672711 + 2.51059i 0.866025 0.500000i 0 −1.29958 2.25093i −0.692149 + 0.692149i −0.707107 + 0.707107i −3.25245 + 1.87780i 0
293.5 0.965926 0.258819i −0.672711 2.51059i 0.866025 0.500000i 0 −1.29958 2.25093i 0.692149 0.692149i 0.707107 0.707107i −3.25245 + 1.87780i 0
293.6 0.965926 0.258819i −0.206366 0.770169i 0.866025 0.500000i 0 −0.398669 0.690515i 0.349095 0.349095i 0.707107 0.707107i 2.04750 1.18213i 0
293.7 0.965926 0.258819i 0.313773 + 1.17102i 0.866025 0.500000i 0 0.606164 + 1.04991i −1.88504 + 1.88504i 0.707107 0.707107i 1.32525 0.765131i 0
293.8 0.965926 0.258819i 0.824123 + 3.07567i 0.866025 0.500000i 0 1.59208 + 2.75757i 3.29328 3.29328i 0.707107 0.707107i −6.18248 + 3.56945i 0
407.1 −0.258819 0.965926i −3.07567 + 0.824123i −0.866025 + 0.500000i 0 1.59208 + 2.75757i −3.29328 3.29328i 0.707107 + 0.707107i 6.18248 3.56945i 0
407.2 −0.258819 0.965926i −1.17102 + 0.313773i −0.866025 + 0.500000i 0 0.606164 + 1.04991i 1.88504 + 1.88504i 0.707107 + 0.707107i −1.32525 + 0.765131i 0
407.3 −0.258819 0.965926i 0.770169 0.206366i −0.866025 + 0.500000i 0 −0.398669 0.690515i −0.349095 0.349095i 0.707107 + 0.707107i −2.04750 + 1.18213i 0
407.4 −0.258819 0.965926i 2.51059 0.672711i −0.866025 + 0.500000i 0 −1.29958 2.25093i −0.692149 0.692149i 0.707107 + 0.707107i 3.25245 1.87780i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 943.8 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.f 32
5.b even 2 1 inner 950.2.q.f 32
5.c odd 4 2 inner 950.2.q.f 32
19.d odd 6 1 inner 950.2.q.f 32
95.h odd 6 1 inner 950.2.q.f 32
95.l even 12 2 inner 950.2.q.f 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.q.f 32 1.a even 1 1 trivial
950.2.q.f 32 5.b even 2 1 inner
950.2.q.f 32 5.c odd 4 2 inner
950.2.q.f 32 19.d odd 6 1 inner
950.2.q.f 32 95.h odd 6 1 inner
950.2.q.f 32 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 $$S_{2}^{\mathrm{new}}(950, [\chi])$$:

 $$T_{3}^{32} - \cdots$$ $$T_{7}^{16} + 522 T_{7}^{12} + 24273 T_{7}^{8} + 23256 T_{7}^{4} + 1296$$