# Properties

 Label 950.2.bb.c Level $950$ Weight $2$ Character orbit 950.bb Analytic conductor $7.586$ Analytic rank $0$ Dimension $72$ 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: $$72$$ Relative dimension: $$6$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$72q + 24q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q + 24q^{6} + 24q^{21} + 12q^{26} + 36q^{31} - 24q^{36} + 48q^{41} - 36q^{46} + 156q^{51} + 168q^{61} - 36q^{66} - 84q^{71} - 48q^{76} - 60q^{81} - 60q^{86} - 264q^{91} + 24q^{96} + O(q^{100})$$

## 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 −0.181419 + 2.07363i −0.342020 0.939693i 0 −1.59456 1.33799i 1.21655 + 4.54023i 0.965926 + 0.258819i −1.31260 0.231446i 0
143.2 −0.573576 + 0.819152i 0.00886944 0.101378i −0.342020 0.939693i 0 0.0779568 + 0.0654135i −0.0402147 0.150083i 0.965926 + 0.258819i 2.94422 + 0.519146i 0
143.3 −0.573576 + 0.819152i 0.218924 2.50231i −0.342020 0.939693i 0 1.92421 + 1.61460i 0.801272 + 2.99039i 0.965926 + 0.258819i −3.25922 0.574689i 0
143.4 0.573576 0.819152i −0.218924 + 2.50231i −0.342020 0.939693i 0 1.92421 + 1.61460i −0.801272 2.99039i −0.965926 0.258819i −3.25922 0.574689i 0
143.5 0.573576 0.819152i −0.00886944 + 0.101378i −0.342020 0.939693i 0 0.0779568 + 0.0654135i 0.0402147 + 0.150083i −0.965926 0.258819i 2.94422 + 0.519146i 0
143.6 0.573576 0.819152i 0.181419 2.07363i −0.342020 0.939693i 0 −1.59456 1.33799i −1.21655 4.54023i −0.965926 0.258819i −1.31260 0.231446i 0
193.1 −0.0871557 0.996195i −1.14235 + 2.44977i −0.984808 + 0.173648i 0 2.54001 + 0.924487i −1.86007 0.498404i 0.258819 + 0.965926i −2.76804 3.29882i 0
193.2 −0.0871557 0.996195i −0.475045 + 1.01874i −0.984808 + 0.173648i 0 1.05626 + 0.384448i 3.66894 + 0.983089i 0.258819 + 0.965926i 1.11621 + 1.33024i 0
193.3 −0.0871557 0.996195i 0.400509 0.858895i −0.984808 + 0.173648i 0 −0.890533 0.324128i 0.243409 + 0.0652212i 0.258819 + 0.965926i 1.35107 + 1.61014i 0
193.4 0.0871557 + 0.996195i −0.400509 + 0.858895i −0.984808 + 0.173648i 0 −0.890533 0.324128i −0.243409 0.0652212i −0.258819 0.965926i 1.35107 + 1.61014i 0
193.5 0.0871557 + 0.996195i 0.475045 1.01874i −0.984808 + 0.173648i 0 1.05626 + 0.384448i −3.66894 0.983089i −0.258819 0.965926i 1.11621 + 1.33024i 0
193.6 0.0871557 + 0.996195i 1.14235 2.44977i −0.984808 + 0.173648i 0 2.54001 + 0.924487i 1.86007 + 0.498404i −0.258819 0.965926i −2.76804 3.29882i 0
243.1 −0.422618 0.906308i −2.29045 1.60379i −0.642788 + 0.766044i 0 −0.485541 + 2.75364i 0.367125 + 1.37013i 0.965926 + 0.258819i 1.64795 + 4.52771i 0
243.2 −0.422618 0.906308i −0.0850875 0.0595789i −0.642788 + 0.766044i 0 −0.0180373 + 0.102295i 0.497281 + 1.85588i 0.965926 + 0.258819i −1.02237 2.80894i 0
243.3 −0.422618 0.906308i 1.84087 + 1.28899i −0.642788 + 0.766044i 0 0.390238 2.21315i −0.947246 3.53517i 0.965926 + 0.258819i 0.701247 + 1.92666i 0
243.4 0.422618 + 0.906308i −1.84087 1.28899i −0.642788 + 0.766044i 0 0.390238 2.21315i 0.947246 + 3.53517i −0.965926 0.258819i 0.701247 + 1.92666i 0
243.5 0.422618 + 0.906308i 0.0850875 + 0.0595789i −0.642788 + 0.766044i 0 −0.0180373 + 0.102295i −0.497281 1.85588i −0.965926 0.258819i −1.02237 2.80894i 0
243.6 0.422618 + 0.906308i 2.29045 + 1.60379i −0.642788 + 0.766044i 0 −0.485541 + 2.75364i −0.367125 1.37013i −0.965926 0.258819i 1.64795 + 4.52771i 0
257.1 −0.819152 0.573576i −2.50231 0.218924i 0.342020 + 0.939693i 0 1.92421 + 1.61460i −2.99039 + 0.801272i 0.258819 0.965926i 3.25922 + 0.574689i 0
257.2 −0.819152 0.573576i −0.101378 0.00886944i 0.342020 + 0.939693i 0 0.0779568 + 0.0654135i 0.150083 0.0402147i 0.258819 0.965926i −2.94422 0.519146i 0
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 907.6 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.c 72
5.b even 2 1 inner 950.2.bb.c 72
5.c odd 4 2 inner 950.2.bb.c 72
19.f odd 18 1 inner 950.2.bb.c 72
95.o odd 18 1 inner 950.2.bb.c 72
95.r even 36 2 inner 950.2.bb.c 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.bb.c 72 1.a even 1 1 trivial
950.2.bb.c 72 5.b even 2 1 inner
950.2.bb.c 72 5.c odd 4 2 inner
950.2.bb.c 72 19.f odd 18 1 inner
950.2.bb.c 72 95.o odd 18 1 inner
950.2.bb.c 72 95.r even 36 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$40\!\cdots\!80$$$$T_{3}^{40} -$$$$21\!\cdots\!40$$$$T_{3}^{36} +$$$$36\!\cdots\!82$$$$T_{3}^{32} +$$$$25\!\cdots\!04$$$$T_{3}^{28} +$$$$12\!\cdots\!86$$$$T_{3}^{24} +$$$$38\!\cdots\!04$$$$T_{3}^{20} +$$$$64\!\cdots\!37$$$$T_{3}^{16} - 148923767649 T_{3}^{12} - 69896577 T_{3}^{8} - 4362 T_{3}^{4} + 1$$">$$T_{3}^{72} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.