# Properties

 Label 950.2.u.e Level $950$ Weight $2$ Character orbit 950.u Analytic conductor $7.586$ Analytic rank $0$ Dimension $24$ 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.u (of order $$18$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$4$$ over $$\Q(\zeta_{18})$$ Twist minimal: no (minimal twist has level 190) Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 36q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 36q^{9} + 12q^{11} + 24q^{21} + 12q^{29} + 12q^{31} + 36q^{36} + 72q^{39} - 12q^{41} - 12q^{44} - 24q^{46} + 36q^{49} + 24q^{56} + 48q^{59} - 60q^{61} + 12q^{64} + 48q^{66} - 12q^{69} - 84q^{71} - 12q^{74} + 36q^{76} - 120q^{79} + 36q^{81} + 48q^{84} - 72q^{86} + 24q^{89} + 48q^{91} - 120q^{94} + 24q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1 −0.342020 + 0.939693i −1.31143 + 0.231240i −0.766044 0.642788i 0 0.231240 1.31143i 3.78616 + 2.18594i 0.866025 0.500000i −1.15270 + 0.419550i 0
99.2 −0.342020 + 0.939693i 1.31143 0.231240i −0.766044 0.642788i 0 −0.231240 + 1.31143i −2.05411 1.18594i 0.866025 0.500000i −1.15270 + 0.419550i 0
99.3 0.342020 0.939693i −1.31143 + 0.231240i −0.766044 0.642788i 0 −0.231240 + 1.31143i 2.05411 + 1.18594i −0.866025 + 0.500000i −1.15270 + 0.419550i 0
99.4 0.342020 0.939693i 1.31143 0.231240i −0.766044 0.642788i 0 0.231240 1.31143i −3.78616 2.18594i −0.866025 + 0.500000i −1.15270 + 0.419550i 0
149.1 −0.642788 + 0.766044i −0.931116 2.55822i −0.173648 0.984808i 0 2.55822 + 0.931116i −2.31045 + 1.33394i 0.866025 + 0.500000i −3.37939 + 2.83564i 0
149.2 −0.642788 + 0.766044i 0.931116 + 2.55822i −0.173648 0.984808i 0 −2.55822 0.931116i 4.04250 2.33394i 0.866025 + 0.500000i −3.37939 + 2.83564i 0
149.3 0.642788 0.766044i −0.931116 2.55822i −0.173648 0.984808i 0 −2.55822 0.931116i −4.04250 + 2.33394i −0.866025 0.500000i −3.37939 + 2.83564i 0
149.4 0.642788 0.766044i 0.931116 + 2.55822i −0.173648 0.984808i 0 2.55822 + 0.931116i 2.31045 1.33394i −0.866025 0.500000i −3.37939 + 2.83564i 0
199.1 −0.984808 + 0.173648i −1.07851 1.28531i 0.939693 0.342020i 0 1.28531 + 1.07851i 0.411781 + 0.237742i −0.866025 + 0.500000i 0.0320889 0.181985i 0
199.2 −0.984808 + 0.173648i 1.07851 + 1.28531i 0.939693 0.342020i 0 −1.28531 1.07851i −2.14383 1.23774i −0.866025 + 0.500000i 0.0320889 0.181985i 0
199.3 0.984808 0.173648i −1.07851 1.28531i 0.939693 0.342020i 0 −1.28531 1.07851i 2.14383 + 1.23774i 0.866025 0.500000i 0.0320889 0.181985i 0
199.4 0.984808 0.173648i 1.07851 + 1.28531i 0.939693 0.342020i 0 1.28531 + 1.07851i −0.411781 0.237742i 0.866025 0.500000i 0.0320889 0.181985i 0
499.1 −0.342020 0.939693i −1.31143 0.231240i −0.766044 + 0.642788i 0 0.231240 + 1.31143i 3.78616 2.18594i 0.866025 + 0.500000i −1.15270 0.419550i 0
499.2 −0.342020 0.939693i 1.31143 + 0.231240i −0.766044 + 0.642788i 0 −0.231240 1.31143i −2.05411 + 1.18594i 0.866025 + 0.500000i −1.15270 0.419550i 0
499.3 0.342020 + 0.939693i −1.31143 0.231240i −0.766044 + 0.642788i 0 −0.231240 1.31143i 2.05411 1.18594i −0.866025 0.500000i −1.15270 0.419550i 0
499.4 0.342020 + 0.939693i 1.31143 + 0.231240i −0.766044 + 0.642788i 0 0.231240 + 1.31143i −3.78616 + 2.18594i −0.866025 0.500000i −1.15270 0.419550i 0
549.1 −0.984808 0.173648i −1.07851 + 1.28531i 0.939693 + 0.342020i 0 1.28531 1.07851i 0.411781 0.237742i −0.866025 0.500000i 0.0320889 + 0.181985i 0
549.2 −0.984808 0.173648i 1.07851 1.28531i 0.939693 + 0.342020i 0 −1.28531 + 1.07851i −2.14383 + 1.23774i −0.866025 0.500000i 0.0320889 + 0.181985i 0
549.3 0.984808 + 0.173648i −1.07851 + 1.28531i 0.939693 + 0.342020i 0 −1.28531 + 1.07851i 2.14383 1.23774i 0.866025 + 0.500000i 0.0320889 + 0.181985i 0
549.4 0.984808 + 0.173648i 1.07851 1.28531i 0.939693 + 0.342020i 0 1.28531 1.07851i −0.411781 + 0.237742i 0.866025 + 0.500000i 0.0320889 + 0.181985i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 899.4 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
19.e even 9 1 inner
95.p even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.u.e 24
5.b even 2 1 inner 950.2.u.e 24
5.c odd 4 1 190.2.k.b 12
5.c odd 4 1 950.2.l.h 12
19.e even 9 1 inner 950.2.u.e 24
95.p even 18 1 inner 950.2.u.e 24
95.q odd 36 1 190.2.k.b 12
95.q odd 36 1 950.2.l.h 12
95.q odd 36 1 3610.2.a.bc 6
95.r even 36 1 3610.2.a.be 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.k.b 12 5.c odd 4 1
190.2.k.b 12 95.q odd 36 1
950.2.l.h 12 5.c odd 4 1
950.2.l.h 12 95.q odd 36 1
950.2.u.e 24 1.a even 1 1 trivial
950.2.u.e 24 5.b even 2 1 inner
950.2.u.e 24 19.e even 9 1 inner
950.2.u.e 24 95.p even 18 1 inner
3610.2.a.bc 6 95.q odd 36 1
3610.2.a.be 6 95.r even 36 1

## Hecke kernels

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

 $$T_{3}^{12} + 9 T_{3}^{10} + 36 T_{3}^{8} - 64 T_{3}^{6} + 189 T_{3}^{4} - 999 T_{3}^{2} + 1369$$ $$T_{7}^{24} - \cdots$$