# Properties

 Label 950.2.bb.b Level $950$ Weight $2$ Character orbit 950.bb Analytic conductor $7.586$ Analytic rank $0$ Dimension $48$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.bb (of order $$36$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$4$$ over $$\Q(\zeta_{36})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 48q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 48q^{6} - 12q^{11} + 36q^{21} - 72q^{31} + 48q^{36} + 96q^{41} + 72q^{46} - 48q^{51} - 108q^{61} + 24q^{66} - 60q^{71} - 48q^{76} - 168q^{81} - 48q^{86} + 252q^{91} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1 −0.573576 + 0.819152i −0.133530 + 1.52626i −0.342020 0.939693i 0 −1.17365 0.984808i −1.06982 3.99264i 0.965926 + 0.258819i 0.642788 + 0.113341i 0
143.2 −0.573576 + 0.819152i −0.133530 + 1.52626i −0.342020 0.939693i 0 −1.17365 0.984808i 0.227319 + 0.848367i 0.965926 + 0.258819i 0.642788 + 0.113341i 0
143.3 0.573576 0.819152i 0.133530 1.52626i −0.342020 0.939693i 0 −1.17365 0.984808i −0.227319 0.848367i −0.965926 0.258819i 0.642788 + 0.113341i 0
143.4 0.573576 0.819152i 0.133530 1.52626i −0.342020 0.939693i 0 −1.17365 0.984808i 1.06982 + 3.99264i −0.965926 0.258819i 0.642788 + 0.113341i 0
193.1 −0.0871557 0.996195i 0.794263 1.70330i −0.984808 + 0.173648i 0 −1.76604 0.642788i −3.76160 1.00792i 0.258819 + 0.965926i −0.342020 0.407604i 0
193.2 −0.0871557 0.996195i 0.794263 1.70330i −0.984808 + 0.173648i 0 −1.76604 0.642788i 3.18056 + 0.852228i 0.258819 + 0.965926i −0.342020 0.407604i 0
193.3 0.0871557 + 0.996195i −0.794263 + 1.70330i −0.984808 + 0.173648i 0 −1.76604 0.642788i −3.18056 0.852228i −0.258819 0.965926i −0.342020 0.407604i 0
193.4 0.0871557 + 0.996195i −0.794263 + 1.70330i −0.984808 + 0.173648i 0 −1.76604 0.642788i 3.76160 + 1.00792i −0.258819 0.965926i −0.342020 0.407604i 0
243.1 −0.422618 0.906308i −0.284489 0.199201i −0.642788 + 0.766044i 0 −0.0603074 + 0.342020i −0.127266 0.474963i 0.965926 + 0.258819i −0.984808 2.70574i 0
243.2 −0.422618 0.906308i −0.284489 0.199201i −0.642788 + 0.766044i 0 −0.0603074 + 0.342020i 0.814083 + 3.03820i 0.965926 + 0.258819i −0.984808 2.70574i 0
243.3 0.422618 + 0.906308i 0.284489 + 0.199201i −0.642788 + 0.766044i 0 −0.0603074 + 0.342020i −0.814083 3.03820i −0.965926 0.258819i −0.984808 2.70574i 0
243.4 0.422618 + 0.906308i 0.284489 + 0.199201i −0.642788 + 0.766044i 0 −0.0603074 + 0.342020i 0.127266 + 0.474963i −0.965926 0.258819i −0.984808 2.70574i 0
257.1 −0.819152 0.573576i 1.52626 + 0.133530i 0.342020 + 0.939693i 0 −1.17365 0.984808i −0.848367 + 0.227319i 0.258819 0.965926i −0.642788 0.113341i 0
257.2 −0.819152 0.573576i 1.52626 + 0.133530i 0.342020 + 0.939693i 0 −1.17365 0.984808i 3.99264 1.06982i 0.258819 0.965926i −0.642788 0.113341i 0
257.3 0.819152 + 0.573576i −1.52626 0.133530i 0.342020 + 0.939693i 0 −1.17365 0.984808i −3.99264 + 1.06982i −0.258819 + 0.965926i −0.642788 0.113341i 0
257.4 0.819152 + 0.573576i −1.52626 0.133530i 0.342020 + 0.939693i 0 −1.17365 0.984808i 0.848367 0.227319i −0.258819 + 0.965926i −0.642788 0.113341i 0
307.1 −0.996195 + 0.0871557i 1.70330 + 0.794263i 0.984808 0.173648i 0 −1.76604 0.642788i −1.00792 + 3.76160i −0.965926 + 0.258819i 0.342020 + 0.407604i 0
307.2 −0.996195 + 0.0871557i 1.70330 + 0.794263i 0.984808 0.173648i 0 −1.76604 0.642788i 0.852228 3.18056i −0.965926 + 0.258819i 0.342020 + 0.407604i 0
307.3 0.996195 0.0871557i −1.70330 0.794263i 0.984808 0.173648i 0 −1.76604 0.642788i −0.852228 + 3.18056i 0.965926 0.258819i 0.342020 + 0.407604i 0
307.4 0.996195 0.0871557i −1.70330 0.794263i 0.984808 0.173648i 0 −1.76604 0.642788i 1.00792 3.76160i 0.965926 0.258819i 0.342020 + 0.407604i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 907.4 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
5.c odd 4 2 inner
19.f odd 18 1 inner
95.o odd 18 1 inner
95.r even 36 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.bb.b 48
5.b even 2 1 inner 950.2.bb.b 48
5.c odd 4 2 inner 950.2.bb.b 48
19.f odd 18 1 inner 950.2.bb.b 48
95.o odd 18 1 inner 950.2.bb.b 48
95.r even 36 2 inner 950.2.bb.b 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.bb.b 48 1.a even 1 1 trivial
950.2.bb.b 48 5.b even 2 1 inner
950.2.bb.b 48 5.c odd 4 2 inner
950.2.bb.b 48 19.f odd 18 1 inner
950.2.bb.b 48 95.o odd 18 1 inner
950.2.bb.b 48 95.r even 36 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{24} - 6 T_{3}^{20} + 141 T_{3}^{16} - 1477 T_{3}^{12} + 4692 T_{3}^{8} + 105 T_{3}^{4} + 1$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.