# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.l (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$4$$ over $$\Q(\zeta_{9})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 3q^{7} + 12q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 3q^{7} + 12q^{8} - 6q^{11} + 3q^{12} - 24q^{13} - 15q^{14} + 9q^{17} - 30q^{18} - 15q^{19} - 18q^{21} - 12q^{23} - 9q^{26} + 21q^{27} - 12q^{28} - 12q^{29} + 9q^{31} + 42q^{33} - 9q^{34} - 66q^{37} - 6q^{38} + 66q^{39} + 18q^{41} - 9q^{42} + 3q^{43} - 3q^{46} + 12q^{47} - 27q^{49} - 3q^{51} + 12q^{52} + 45q^{53} + 27q^{54} - 6q^{56} + 27q^{57} + 18q^{58} + 36q^{59} + 12q^{61} + 24q^{62} + 63q^{63} - 12q^{64} + 48q^{66} + 54q^{67} - 3q^{68} + 21q^{69} - 39q^{71} - 48q^{73} + 18q^{74} + 6q^{76} - 48q^{77} + 12q^{78} - 42q^{79} - 36q^{81} - 18q^{82} + 3q^{83} + 9q^{84} - 39q^{86} + 24q^{87} + 6q^{88} - 36q^{89} + 12q^{91} + 15q^{92} + 6q^{93} + 12q^{94} + 6q^{96} + 54q^{97} - 3q^{98} + 51q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1 −0.766044 + 0.642788i −2.56782 0.934611i 0.173648 0.984808i 0 2.56782 0.934611i 1.06844 1.85059i 0.500000 + 0.866025i 3.42208 + 2.87147i 0
101.2 −0.766044 + 0.642788i −0.0801377 0.0291678i 0.173648 0.984808i 0 0.0801377 0.0291678i 0.920368 1.59412i 0.500000 + 0.866025i −2.29256 1.92369i 0
101.3 −0.766044 + 0.642788i 0.799943 + 0.291155i 0.173648 0.984808i 0 −0.799943 + 0.291155i −2.52492 + 4.37330i 0.500000 + 0.866025i −1.74300 1.46255i 0
101.4 −0.766044 + 0.642788i 2.78771 + 1.01464i 0.173648 0.984808i 0 −2.78771 + 1.01464i 0.454865 0.787850i 0.500000 + 0.866025i 4.44370 + 3.72870i 0
251.1 0.939693 + 0.342020i −0.501764 2.84564i 0.766044 + 0.642788i 0 0.501764 2.84564i −2.01830 + 3.49580i 0.500000 + 0.866025i −5.02684 + 1.82962i 0
251.2 0.939693 + 0.342020i −0.227288 1.28901i 0.766044 + 0.642788i 0 0.227288 1.28901i 1.11539 1.93191i 0.500000 + 0.866025i 1.20918 0.440106i 0
251.3 0.939693 + 0.342020i 0.111946 + 0.634880i 0.766044 + 0.642788i 0 −0.111946 + 0.634880i 0.213557 0.369892i 0.500000 + 0.866025i 2.42854 0.883915i 0
251.4 0.939693 + 0.342020i 0.443457 + 2.51497i 0.766044 + 0.642788i 0 −0.443457 + 2.51497i −2.28243 + 3.95328i 0.500000 + 0.866025i −3.30934 + 1.20450i 0
301.1 −0.766044 0.642788i −2.56782 + 0.934611i 0.173648 + 0.984808i 0 2.56782 + 0.934611i 1.06844 + 1.85059i 0.500000 0.866025i 3.42208 2.87147i 0
301.2 −0.766044 0.642788i −0.0801377 + 0.0291678i 0.173648 + 0.984808i 0 0.0801377 + 0.0291678i 0.920368 + 1.59412i 0.500000 0.866025i −2.29256 + 1.92369i 0
301.3 −0.766044 0.642788i 0.799943 0.291155i 0.173648 + 0.984808i 0 −0.799943 0.291155i −2.52492 4.37330i 0.500000 0.866025i −1.74300 + 1.46255i 0
301.4 −0.766044 0.642788i 2.78771 1.01464i 0.173648 + 0.984808i 0 −2.78771 1.01464i 0.454865 + 0.787850i 0.500000 0.866025i 4.44370 3.72870i 0
351.1 −0.173648 0.984808i −2.49196 + 2.09100i −0.939693 + 0.342020i 0 2.49196 + 2.09100i −0.556842 + 0.964479i 0.500000 + 0.866025i 1.31663 7.46696i 0
351.2 −0.173648 0.984808i −0.639791 + 0.536848i −0.939693 + 0.342020i 0 0.639791 + 0.536848i 2.09271 3.62467i 0.500000 + 0.866025i −0.399818 + 2.26748i 0
351.3 −0.173648 0.984808i 0.654394 0.549102i −0.939693 + 0.342020i 0 −0.654394 0.549102i −1.39086 + 2.40903i 0.500000 + 0.866025i −0.394226 + 2.23577i 0
351.4 −0.173648 0.984808i 1.71131 1.43596i −0.939693 + 0.342020i 0 −1.71131 1.43596i 1.40802 2.43877i 0.500000 + 0.866025i 0.345659 1.96033i 0
651.1 0.939693 0.342020i −0.501764 + 2.84564i 0.766044 0.642788i 0 0.501764 + 2.84564i −2.01830 3.49580i 0.500000 0.866025i −5.02684 1.82962i 0
651.2 0.939693 0.342020i −0.227288 + 1.28901i 0.766044 0.642788i 0 0.227288 + 1.28901i 1.11539 + 1.93191i 0.500000 0.866025i 1.20918 + 0.440106i 0
651.3 0.939693 0.342020i 0.111946 0.634880i 0.766044 0.642788i 0 −0.111946 0.634880i 0.213557 + 0.369892i 0.500000 0.866025i 2.42854 + 0.883915i 0
651.4 0.939693 0.342020i 0.443457 2.51497i 0.766044 0.642788i 0 −0.443457 2.51497i −2.28243 3.95328i 0.500000 0.866025i −3.30934 1.20450i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 701.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.l.j 24
5.b even 2 1 950.2.l.k yes 24
5.c odd 4 2 950.2.u.h 48
19.e even 9 1 inner 950.2.l.j 24
95.p even 18 1 950.2.l.k yes 24
95.q odd 36 2 950.2.u.h 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.l.j 24 1.a even 1 1 trivial
950.2.l.j 24 19.e even 9 1 inner
950.2.l.k yes 24 5.b even 2 1
950.2.l.k yes 24 95.p even 18 1
950.2.u.h 48 5.c odd 4 2
950.2.u.h 48 95.q odd 36 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{24} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.