# Properties

 Label 950.2.q.g Level $950$ Weight $2$ Character orbit 950.q Analytic conductor $7.586$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# 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: no (minimal twist has level 190) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 12q^{3} - 24q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 12q^{3} - 24q^{7} - 16q^{11} - 24q^{13} + 16q^{16} + 8q^{17} + 12q^{22} - 4q^{23} - 16q^{26} + 12q^{28} + 24q^{33} - 8q^{36} - 16q^{38} + 24q^{41} - 20q^{42} + 24q^{43} + 36q^{47} + 12q^{48} + 24q^{51} - 24q^{52} + 72q^{53} + 24q^{57} - 24q^{58} - 48q^{61} + 4q^{62} - 16q^{63} + 32q^{66} - 36q^{67} - 16q^{68} + 24q^{71} - 8q^{73} + 24q^{77} + 24q^{78} + 56q^{81} - 8q^{82} - 24q^{83} - 104q^{87} - 24q^{91} + 4q^{92} - 52q^{93} + 24q^{97} - 72q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1 −0.965926 0.258819i −0.570807 + 2.13028i 0.866025 + 0.500000i 0 1.10271 1.90996i −0.526345 0.526345i −0.707107 0.707107i −1.61420 0.931958i 0
107.2 −0.965926 0.258819i 0.186170 0.694795i 0.866025 + 0.500000i 0 −0.359652 + 0.622936i −1.01416 1.01416i −0.707107 0.707107i 2.15000 + 1.24130i 0
107.3 −0.965926 0.258819i 0.338196 1.26216i 0.866025 + 0.500000i 0 −0.653344 + 1.13162i −1.48691 1.48691i −0.707107 0.707107i 1.11940 + 0.646284i 0
107.4 −0.965926 0.258819i 0.680416 2.53935i 0.866025 + 0.500000i 0 −1.31446 + 2.27672i 2.47691 + 2.47691i −0.707107 0.707107i −3.38724 1.95563i 0
107.5 0.965926 + 0.258819i −0.407975 + 1.52258i 0.866025 + 0.500000i 0 −0.788148 + 1.36511i −2.78714 2.78714i 0.707107 + 0.707107i 0.446257 + 0.257647i 0
107.6 0.965926 + 0.258819i 0.0641036 0.239238i 0.866025 + 0.500000i 0 0.123839 0.214495i 2.17543 + 2.17543i 0.707107 + 0.707107i 2.54495 + 1.46933i 0
107.7 0.965926 + 0.258819i 0.108675 0.405580i 0.866025 + 0.500000i 0 0.209944 0.363633i −2.14890 2.14890i 0.707107 + 0.707107i 2.44539 + 1.41185i 0
107.8 0.965926 + 0.258819i 0.869171 3.24379i 0.866025 + 0.500000i 0 1.67911 2.90830i −2.68888 2.68888i 0.707107 + 0.707107i −7.16865 4.13882i 0
293.1 −0.965926 + 0.258819i −0.570807 2.13028i 0.866025 0.500000i 0 1.10271 + 1.90996i −0.526345 + 0.526345i −0.707107 + 0.707107i −1.61420 + 0.931958i 0
293.2 −0.965926 + 0.258819i 0.186170 + 0.694795i 0.866025 0.500000i 0 −0.359652 0.622936i −1.01416 + 1.01416i −0.707107 + 0.707107i 2.15000 1.24130i 0
293.3 −0.965926 + 0.258819i 0.338196 + 1.26216i 0.866025 0.500000i 0 −0.653344 1.13162i −1.48691 + 1.48691i −0.707107 + 0.707107i 1.11940 0.646284i 0
293.4 −0.965926 + 0.258819i 0.680416 + 2.53935i 0.866025 0.500000i 0 −1.31446 2.27672i 2.47691 2.47691i −0.707107 + 0.707107i −3.38724 + 1.95563i 0
293.5 0.965926 0.258819i −0.407975 1.52258i 0.866025 0.500000i 0 −0.788148 1.36511i −2.78714 + 2.78714i 0.707107 0.707107i 0.446257 0.257647i 0
293.6 0.965926 0.258819i 0.0641036 + 0.239238i 0.866025 0.500000i 0 0.123839 + 0.214495i 2.17543 2.17543i 0.707107 0.707107i 2.54495 1.46933i 0
293.7 0.965926 0.258819i 0.108675 + 0.405580i 0.866025 0.500000i 0 0.209944 + 0.363633i −2.14890 + 2.14890i 0.707107 0.707107i 2.44539 1.41185i 0
293.8 0.965926 0.258819i 0.869171 + 3.24379i 0.866025 0.500000i 0 1.67911 + 2.90830i −2.68888 + 2.68888i 0.707107 0.707107i −7.16865 + 4.13882i 0
407.1 −0.258819 0.965926i −2.13028 + 0.570807i −0.866025 + 0.500000i 0 1.10271 + 1.90996i −0.526345 0.526345i 0.707107 + 0.707107i 1.61420 0.931958i 0
407.2 −0.258819 0.965926i 0.694795 0.186170i −0.866025 + 0.500000i 0 −0.359652 0.622936i −1.01416 1.01416i 0.707107 + 0.707107i −2.15000 + 1.24130i 0
407.3 −0.258819 0.965926i 1.26216 0.338196i −0.866025 + 0.500000i 0 −0.653344 1.13162i −1.48691 1.48691i 0.707107 + 0.707107i −1.11940 + 0.646284i 0
407.4 −0.258819 0.965926i 2.53935 0.680416i −0.866025 + 0.500000i 0 −1.31446 2.27672i 2.47691 + 2.47691i 0.707107 + 0.707107i 3.38724 1.95563i 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.c odd 4 1 inner
19.d odd 6 1 inner
95.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.q.g 32
5.b even 2 1 190.2.m.b 32
5.c odd 4 1 190.2.m.b 32
5.c odd 4 1 inner 950.2.q.g 32
19.d odd 6 1 inner 950.2.q.g 32
95.h odd 6 1 190.2.m.b 32
95.l even 12 1 190.2.m.b 32
95.l even 12 1 inner 950.2.q.g 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.m.b 32 5.b even 2 1
190.2.m.b 32 5.c odd 4 1
190.2.m.b 32 95.h odd 6 1
190.2.m.b 32 95.l even 12 1
950.2.q.g 32 1.a even 1 1 trivial
950.2.q.g 32 5.c odd 4 1 inner
950.2.q.g 32 19.d odd 6 1 inner
950.2.q.g 32 95.l even 12 1 inner

## Hecke kernels

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

 $$T_{3}^{32} - \cdots$$ $$T_{7}^{16} + \cdots$$