# Properties

 Label 950.2.bb.a Level $950$ Weight $2$ Character orbit 950.bb Analytic conductor $7.586$ Analytic rank $0$ Dimension $24$ 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: $$24$$ Relative dimension: $$2$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 36q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 36q^{6} + 12q^{11} - 12q^{21} + 12q^{26} + 108q^{31} - 36q^{36} - 84q^{41} - 36q^{46} - 12q^{51} - 12q^{61} - 60q^{66} - 24q^{71} + 72q^{76} - 216q^{81} + 12q^{86} - 12q^{91} - 24q^{96} + O(q^{100})$$

## 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 0.190417 2.17648i −0.342020 0.939693i 0 1.67365 + 1.40436i 0.115556 + 0.431262i 0.965926 + 0.258819i −1.74638 0.307934i 0
143.2 0.573576 0.819152i −0.190417 + 2.17648i −0.342020 0.939693i 0 1.67365 + 1.40436i −0.115556 0.431262i −0.965926 0.258819i −1.74638 0.307934i 0
193.1 −0.0871557 0.996195i −1.01913 + 2.18554i −0.984808 + 0.173648i 0 2.26604 + 0.824773i −1.01230 0.271245i 0.258819 + 0.965926i −1.80958 2.15657i 0
193.2 0.0871557 + 0.996195i 1.01913 2.18554i −0.984808 + 0.173648i 0 2.26604 + 0.824773i 1.01230 + 0.271245i −0.258819 0.965926i −1.80958 2.15657i 0
243.1 −0.422618 0.906308i 2.64314 + 1.85075i −0.642788 + 0.766044i 0 0.560307 3.17766i 0.958062 + 3.57554i 0.965926 + 0.258819i 2.53487 + 6.96451i 0
243.2 0.422618 + 0.906308i −2.64314 1.85075i −0.642788 + 0.766044i 0 0.560307 3.17766i −0.958062 3.57554i −0.965926 0.258819i 2.53487 + 6.96451i 0
257.1 −0.819152 0.573576i −2.17648 0.190417i 0.342020 + 0.939693i 0 1.67365 + 1.40436i −0.431262 + 0.115556i 0.258819 0.965926i 1.74638 + 0.307934i 0
257.2 0.819152 + 0.573576i 2.17648 + 0.190417i 0.342020 + 0.939693i 0 1.67365 + 1.40436i 0.431262 0.115556i −0.258819 + 0.965926i 1.74638 + 0.307934i 0
307.1 −0.996195 + 0.0871557i −2.18554 1.01913i 0.984808 0.173648i 0 2.26604 + 0.824773i −0.271245 + 1.01230i −0.965926 + 0.258819i 1.80958 + 2.15657i 0
307.2 0.996195 0.0871557i 2.18554 + 1.01913i 0.984808 0.173648i 0 2.26604 + 0.824773i 0.271245 1.01230i 0.965926 0.258819i 1.80958 + 2.15657i 0
357.1 −0.906308 + 0.422618i −1.85075 + 2.64314i 0.642788 0.766044i 0 0.560307 3.17766i 3.57554 0.958062i −0.258819 + 0.965926i −2.53487 6.96451i 0
357.2 0.906308 0.422618i 1.85075 2.64314i 0.642788 0.766044i 0 0.560307 3.17766i −3.57554 + 0.958062i 0.258819 0.965926i −2.53487 6.96451i 0
393.1 −0.996195 0.0871557i −2.18554 + 1.01913i 0.984808 + 0.173648i 0 2.26604 0.824773i −0.271245 1.01230i −0.965926 0.258819i 1.80958 2.15657i 0
393.2 0.996195 + 0.0871557i 2.18554 1.01913i 0.984808 + 0.173648i 0 2.26604 0.824773i 0.271245 + 1.01230i 0.965926 + 0.258819i 1.80958 2.15657i 0
507.1 −0.0871557 + 0.996195i −1.01913 2.18554i −0.984808 0.173648i 0 2.26604 0.824773i −1.01230 + 0.271245i 0.258819 0.965926i −1.80958 + 2.15657i 0
507.2 0.0871557 0.996195i 1.01913 + 2.18554i −0.984808 0.173648i 0 2.26604 0.824773i 1.01230 0.271245i −0.258819 + 0.965926i −1.80958 + 2.15657i 0
743.1 −0.819152 + 0.573576i −2.17648 + 0.190417i 0.342020 0.939693i 0 1.67365 1.40436i −0.431262 0.115556i 0.258819 + 0.965926i 1.74638 0.307934i 0
743.2 0.819152 0.573576i 2.17648 0.190417i 0.342020 0.939693i 0 1.67365 1.40436i 0.431262 + 0.115556i −0.258819 0.965926i 1.74638 0.307934i 0
793.1 −0.906308 0.422618i −1.85075 2.64314i 0.642788 + 0.766044i 0 0.560307 + 3.17766i 3.57554 + 0.958062i −0.258819 0.965926i −2.53487 + 6.96451i 0
793.2 0.906308 + 0.422618i 1.85075 + 2.64314i 0.642788 + 0.766044i 0 0.560307 + 3.17766i −3.57554 0.958062i 0.258819 + 0.965926i −2.53487 + 6.96451i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 907.2 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.a 24
5.b even 2 1 inner 950.2.bb.a 24
5.c odd 4 2 inner 950.2.bb.a 24
19.f odd 18 1 inner 950.2.bb.a 24
95.o odd 18 1 inner 950.2.bb.a 24
95.r even 36 2 inner 950.2.bb.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.bb.a 24 1.a even 1 1 trivial
950.2.bb.a 24 5.b even 2 1 inner
950.2.bb.a 24 5.c odd 4 2 inner
950.2.bb.a 24 19.f odd 18 1 inner
950.2.bb.a 24 95.o odd 18 1 inner
950.2.bb.a 24 95.r even 36 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{24} + 135 T_{3}^{20} + 7749 T_{3}^{16} - 215416 T_{3}^{12} + 7101630 T_{3}^{8} - 405160371 T_{3}^{4} + 6975757441$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.