# Properties

 Label 950.2.u.f 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 6q^{6} - 18q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 6q^{6} - 18q^{9} - 12q^{11} + 12q^{14} - 12q^{19} - 72q^{21} - 6q^{24} - 6q^{26} - 72q^{29} - 48q^{31} + 12q^{34} + 18q^{36} + 24q^{39} - 24q^{41} + 18q^{44} - 36q^{46} - 54q^{51} - 18q^{54} + 24q^{56} + 54q^{59} + 108q^{61} + 12q^{64} - 78q^{66} + 48q^{69} + 48q^{71} - 30q^{74} + 18q^{76} + 72q^{79} - 18q^{81} - 24q^{84} + 48q^{86} - 36q^{89} + 24q^{91} - 36q^{94} - 36q^{99} + O(q^{100})$$

## 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 −2.25357 + 0.397366i −0.766044 0.642788i 0 0.397366 2.25357i 2.39766 + 1.38429i 0.866025 0.500000i 2.10161 0.764925i 0
99.2 −0.342020 + 0.939693i 0.402740 0.0710139i −0.766044 0.642788i 0 −0.0710139 + 0.402740i −1.99244 1.15033i 0.866025 0.500000i −2.66192 + 0.968860i 0
99.3 0.342020 0.939693i −0.402740 + 0.0710139i −0.766044 0.642788i 0 −0.0710139 + 0.402740i 1.99244 + 1.15033i −0.866025 + 0.500000i −2.66192 + 0.968860i 0
99.4 0.342020 0.939693i 2.25357 0.397366i −0.766044 0.642788i 0 0.397366 2.25357i −2.39766 1.38429i −0.866025 + 0.500000i 2.10161 0.764925i 0
149.1 −0.642788 + 0.766044i −0.936765 2.57374i −0.173648 0.984808i 0 2.57374 + 0.936765i 3.33331 1.92448i 0.866025 + 0.500000i −3.44848 + 2.89362i 0
149.2 −0.642788 + 0.766044i 0.412760 + 1.13405i −0.173648 0.984808i 0 −1.13405 0.412760i −1.90202 + 1.09813i 0.866025 + 0.500000i 1.18244 0.992183i 0
149.3 0.642788 0.766044i −0.412760 1.13405i −0.173648 0.984808i 0 −1.13405 0.412760i 1.90202 1.09813i −0.866025 0.500000i 1.18244 0.992183i 0
149.4 0.642788 0.766044i 0.936765 + 2.57374i −0.173648 0.984808i 0 2.57374 + 0.936765i −3.33331 + 1.92448i −0.866025 0.500000i −3.44848 + 2.89362i 0
199.1 −0.984808 + 0.173648i −1.68231 2.00490i 0.939693 0.342020i 0 2.00490 + 1.68231i −0.840422 0.485218i −0.866025 + 0.500000i −0.668514 + 3.79133i 0
199.2 −0.984808 + 0.173648i 1.90555 + 2.27095i 0.939693 0.342020i 0 −2.27095 1.90555i −2.51922 1.45447i −0.866025 + 0.500000i −1.00513 + 5.70040i 0
199.3 0.984808 0.173648i −1.90555 2.27095i 0.939693 0.342020i 0 −2.27095 1.90555i 2.51922 + 1.45447i 0.866025 0.500000i −1.00513 + 5.70040i 0
199.4 0.984808 0.173648i 1.68231 + 2.00490i 0.939693 0.342020i 0 2.00490 + 1.68231i 0.840422 + 0.485218i 0.866025 0.500000i −0.668514 + 3.79133i 0
499.1 −0.342020 0.939693i −2.25357 0.397366i −0.766044 + 0.642788i 0 0.397366 + 2.25357i 2.39766 1.38429i 0.866025 + 0.500000i 2.10161 + 0.764925i 0
499.2 −0.342020 0.939693i 0.402740 + 0.0710139i −0.766044 + 0.642788i 0 −0.0710139 0.402740i −1.99244 + 1.15033i 0.866025 + 0.500000i −2.66192 0.968860i 0
499.3 0.342020 + 0.939693i −0.402740 0.0710139i −0.766044 + 0.642788i 0 −0.0710139 0.402740i 1.99244 1.15033i −0.866025 0.500000i −2.66192 0.968860i 0
499.4 0.342020 + 0.939693i 2.25357 + 0.397366i −0.766044 + 0.642788i 0 0.397366 + 2.25357i −2.39766 + 1.38429i −0.866025 0.500000i 2.10161 + 0.764925i 0
549.1 −0.984808 0.173648i −1.68231 + 2.00490i 0.939693 + 0.342020i 0 2.00490 1.68231i −0.840422 + 0.485218i −0.866025 0.500000i −0.668514 3.79133i 0
549.2 −0.984808 0.173648i 1.90555 2.27095i 0.939693 + 0.342020i 0 −2.27095 + 1.90555i −2.51922 + 1.45447i −0.866025 0.500000i −1.00513 5.70040i 0
549.3 0.984808 + 0.173648i −1.90555 + 2.27095i 0.939693 + 0.342020i 0 −2.27095 + 1.90555i 2.51922 1.45447i 0.866025 + 0.500000i −1.00513 5.70040i 0
549.4 0.984808 + 0.173648i 1.68231 2.00490i 0.939693 + 0.342020i 0 2.00490 1.68231i 0.840422 0.485218i 0.866025 + 0.500000i −0.668514 3.79133i 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.f 24
5.b even 2 1 inner 950.2.u.f 24
5.c odd 4 1 190.2.k.c 12
5.c odd 4 1 950.2.l.g 12
19.e even 9 1 inner 950.2.u.f 24
95.p even 18 1 inner 950.2.u.f 24
95.q odd 36 1 190.2.k.c 12
95.q odd 36 1 950.2.l.g 12
95.q odd 36 1 3610.2.a.bf 6
95.r even 36 1 3610.2.a.bd 6

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

## Hecke kernels

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

 $$T_{3}^{24} + \cdots$$ $$T_{7}^{24} - \cdots$$