# Properties

 Label 950.2.l.l Level $950$ Weight $2$ Character orbit 950.l Analytic conductor $7.586$ Analytic rank $0$ Dimension $30$ 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: $$30$$ Relative dimension: $$5$$ over $$\Q(\zeta_{9})$$ Twist minimal: no (minimal twist has level 190) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30q - 12q^{7} + 15q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q - 12q^{7} + 15q^{8} + 6q^{11} + 6q^{14} - 30q^{18} + 24q^{19} + 24q^{21} + 3q^{22} + 3q^{23} + 3q^{26} - 18q^{27} + 3q^{28} + 12q^{29} - 30q^{33} + 24q^{37} - 12q^{38} - 24q^{39} - 3q^{41} + 12q^{42} + 6q^{43} - 3q^{44} + 48q^{47} + 15q^{49} - 90q^{51} - 18q^{53} + 18q^{54} - 24q^{56} - 42q^{57} + 36q^{58} - 18q^{59} - 60q^{61} - 24q^{62} - 21q^{63} - 15q^{64} - 78q^{66} - 30q^{67} - 12q^{68} + 24q^{69} + 30q^{73} - 9q^{74} - 3q^{76} + 78q^{77} - 6q^{79} + 60q^{81} + 3q^{82} - 42q^{83} - 6q^{84} + 12q^{86} - 54q^{87} - 6q^{88} - 30q^{89} - 6q^{91} - 6q^{92} + 72q^{93} - 78q^{94} - 42q^{97} + 6q^{98} - 99q^{99} + O(q^{100})$$

## 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 −3.03100 1.10319i 0.173648 0.984808i 0 3.03100 1.10319i −1.36166 + 2.35847i 0.500000 + 0.866025i 5.67178 + 4.75918i 0
101.2 −0.766044 + 0.642788i −0.476729 0.173515i 0.173648 0.984808i 0 0.476729 0.173515i 0.753583 1.30524i 0.500000 + 0.866025i −2.10097 1.76292i 0
101.3 −0.766044 + 0.642788i −0.0808150 0.0294142i 0.173648 0.984808i 0 0.0808150 0.0294142i −1.91879 + 3.32344i 0.500000 + 0.866025i −2.29247 1.92361i 0
101.4 −0.766044 + 0.642788i 0.845354 + 0.307684i 0.173648 0.984808i 0 −0.845354 + 0.307684i 1.06731 1.84863i 0.500000 + 0.866025i −1.67818 1.40816i 0
101.5 −0.766044 + 0.642788i 2.74319 + 0.998438i 0.173648 0.984808i 0 −2.74319 + 0.998438i −0.366794 + 0.635305i 0.500000 + 0.866025i 4.23006 + 3.54944i 0
251.1 0.939693 + 0.342020i −0.419293 2.37793i 0.766044 + 0.642788i 0 0.419293 2.37793i −1.31398 + 2.27588i 0.500000 + 0.866025i −2.65966 + 0.968036i 0
251.2 0.939693 + 0.342020i −0.260779 1.47895i 0.766044 + 0.642788i 0 0.260779 1.47895i 1.51251 2.61975i 0.500000 + 0.866025i 0.699782 0.254700i 0
251.3 0.939693 + 0.342020i −0.0989733 0.561305i 0.766044 + 0.642788i 0 0.0989733 0.561305i −2.10163 + 3.64013i 0.500000 + 0.866025i 2.51381 0.914952i 0
251.4 0.939693 + 0.342020i 0.237849 + 1.34891i 0.766044 + 0.642788i 0 −0.237849 + 1.34891i 0.816213 1.41372i 0.500000 + 0.866025i 1.05609 0.384387i 0
251.5 0.939693 + 0.342020i 0.541196 + 3.06928i 0.766044 + 0.642788i 0 −0.541196 + 3.06928i −0.147073 + 0.254737i 0.500000 + 0.866025i −6.30849 + 2.29610i 0
301.1 −0.766044 0.642788i −3.03100 + 1.10319i 0.173648 + 0.984808i 0 3.03100 + 1.10319i −1.36166 2.35847i 0.500000 0.866025i 5.67178 4.75918i 0
301.2 −0.766044 0.642788i −0.476729 + 0.173515i 0.173648 + 0.984808i 0 0.476729 + 0.173515i 0.753583 + 1.30524i 0.500000 0.866025i −2.10097 + 1.76292i 0
301.3 −0.766044 0.642788i −0.0808150 + 0.0294142i 0.173648 + 0.984808i 0 0.0808150 + 0.0294142i −1.91879 3.32344i 0.500000 0.866025i −2.29247 + 1.92361i 0
301.4 −0.766044 0.642788i 0.845354 0.307684i 0.173648 + 0.984808i 0 −0.845354 0.307684i 1.06731 + 1.84863i 0.500000 0.866025i −1.67818 + 1.40816i 0
301.5 −0.766044 0.642788i 2.74319 0.998438i 0.173648 + 0.984808i 0 −2.74319 0.998438i −0.366794 0.635305i 0.500000 0.866025i 4.23006 3.54944i 0
351.1 −0.173648 0.984808i −2.12694 + 1.78472i −0.939693 + 0.342020i 0 2.12694 + 1.78472i −0.303737 + 0.526088i 0.500000 + 0.866025i 0.817727 4.63756i 0
351.2 −0.173648 0.984808i −1.28288 + 1.07646i −0.939693 + 0.342020i 0 1.28288 + 1.07646i −1.16183 + 2.01234i 0.500000 + 0.866025i −0.0339381 + 0.192472i 0
351.3 −0.173648 0.984808i 0.0864812 0.0725664i −0.939693 + 0.342020i 0 −0.0864812 0.0725664i 0.772138 1.33738i 0.500000 + 0.866025i −0.518731 + 2.94187i 0
351.4 −0.173648 0.984808i 1.52559 1.28013i −0.939693 + 0.342020i 0 −1.52559 1.28013i −1.97239 + 3.41629i 0.500000 + 0.866025i 0.167771 0.951479i 0
351.5 −0.173648 0.984808i 1.79775 1.50849i −0.939693 + 0.342020i 0 −1.79775 1.50849i −0.273873 + 0.474362i 0.500000 + 0.866025i 0.435412 2.46935i 0
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 701.5 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.l 30
5.b even 2 1 950.2.l.m 30
5.c odd 4 2 190.2.p.a 60
19.e even 9 1 inner 950.2.l.l 30
95.p even 18 1 950.2.l.m 30
95.q odd 36 2 190.2.p.a 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.p.a 60 5.c odd 4 2
190.2.p.a 60 95.q odd 36 2
950.2.l.l 30 1.a even 1 1 trivial
950.2.l.l 30 19.e even 9 1 inner
950.2.l.m 30 5.b even 2 1
950.2.l.m 30 95.p even 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{30} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.