# Properties

 Label 950.2.bb.d Level $950$ Weight $2$ Character orbit 950.bb Analytic conductor $7.586$ Analytic rank $0$ Dimension $96$ 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: $$96$$ Relative dimension: $$8$$ 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.

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

## 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 −0.282278 + 3.22645i −0.342020 0.939693i 0 −2.48104 2.08184i −0.0261014 0.0974119i 0.965926 + 0.258819i −7.37587 1.30056i 0
143.2 −0.573576 + 0.819152i −0.0476128 + 0.544217i −0.342020 0.939693i 0 −0.418487 0.351152i −0.123323 0.460247i 0.965926 + 0.258819i 2.66052 + 0.469121i 0
143.3 −0.573576 + 0.819152i 0.134247 1.53445i −0.342020 0.939693i 0 1.17995 + 0.990096i −1.26063 4.70472i 0.965926 + 0.258819i 0.617895 + 0.108952i 0
143.4 −0.573576 + 0.819152i 0.225912 2.58219i −0.342020 0.939693i 0 1.98563 + 1.66614i 0.690517 + 2.57704i 0.965926 + 0.258819i −3.66223 0.645749i 0
143.5 0.573576 0.819152i −0.225912 + 2.58219i −0.342020 0.939693i 0 1.98563 + 1.66614i −0.690517 2.57704i −0.965926 0.258819i −3.66223 0.645749i 0
143.6 0.573576 0.819152i −0.134247 + 1.53445i −0.342020 0.939693i 0 1.17995 + 0.990096i 1.26063 + 4.70472i −0.965926 0.258819i 0.617895 + 0.108952i 0
143.7 0.573576 0.819152i 0.0476128 0.544217i −0.342020 0.939693i 0 −0.418487 0.351152i 0.123323 + 0.460247i −0.965926 0.258819i 2.66052 + 0.469121i 0
143.8 0.573576 0.819152i 0.282278 3.22645i −0.342020 0.939693i 0 −2.48104 2.08184i 0.0261014 + 0.0974119i −0.965926 0.258819i −7.37587 1.30056i 0
193.1 −0.0871557 0.996195i −1.07287 + 2.30077i −0.984808 + 0.173648i 0 2.38552 + 0.868257i 3.67829 + 0.985594i 0.258819 + 0.965926i −2.21412 2.63869i 0
193.2 −0.0871557 0.996195i −0.0142429 + 0.0305440i −0.984808 + 0.173648i 0 0.0316692 + 0.0115266i −2.42021 0.648494i 0.258819 + 0.965926i 1.92763 + 2.29726i 0
193.3 −0.0871557 0.996195i 0.368687 0.790651i −0.984808 + 0.173648i 0 −0.819776 0.298374i 1.54311 + 0.413475i 0.258819 + 0.965926i 1.43916 + 1.71513i 0
193.4 −0.0871557 0.996195i 1.36591 2.92920i −0.984808 + 0.173648i 0 −3.03710 1.10542i 2.44739 + 0.655775i 0.258819 + 0.965926i −4.78616 5.70392i 0
193.5 0.0871557 + 0.996195i −1.36591 + 2.92920i −0.984808 + 0.173648i 0 −3.03710 1.10542i −2.44739 0.655775i −0.258819 0.965926i −4.78616 5.70392i 0
193.6 0.0871557 + 0.996195i −0.368687 + 0.790651i −0.984808 + 0.173648i 0 −0.819776 0.298374i −1.54311 0.413475i −0.258819 0.965926i 1.43916 + 1.71513i 0
193.7 0.0871557 + 0.996195i 0.0142429 0.0305440i −0.984808 + 0.173648i 0 0.0316692 + 0.0115266i 2.42021 + 0.648494i −0.258819 0.965926i 1.92763 + 2.29726i 0
193.8 0.0871557 + 0.996195i 1.07287 2.30077i −0.984808 + 0.173648i 0 2.38552 + 0.868257i −3.67829 0.985594i −0.258819 0.965926i −2.21412 2.63869i 0
243.1 −0.422618 0.906308i −2.62162 1.83568i −0.642788 + 0.766044i 0 −0.555745 + 3.15179i −0.894036 3.33659i 0.965926 + 0.258819i 2.47712 + 6.80584i 0
243.2 −0.422618 0.906308i −1.05824 0.740990i −0.642788 + 0.766044i 0 −0.224332 + 1.27225i 1.24089 + 4.63106i 0.965926 + 0.258819i −0.455247 1.25078i 0
243.3 −0.422618 0.906308i 0.451001 + 0.315794i −0.642788 + 0.766044i 0 0.0956055 0.542206i −0.410083 1.53045i 0.965926 + 0.258819i −0.922385 2.53423i 0
243.4 −0.422618 0.906308i 1.68936 + 1.18290i −0.642788 + 0.766044i 0 0.358120 2.03100i −0.500613 1.86831i 0.965926 + 0.258819i 0.428624 + 1.17763i 0
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 907.8 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.d 96
5.b even 2 1 inner 950.2.bb.d 96
5.c odd 4 2 inner 950.2.bb.d 96
19.f odd 18 1 inner 950.2.bb.d 96
95.o odd 18 1 inner 950.2.bb.d 96
95.r even 36 2 inner 950.2.bb.d 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.bb.d 96 1.a even 1 1 trivial
950.2.bb.d 96 5.b even 2 1 inner
950.2.bb.d 96 5.c odd 4 2 inner
950.2.bb.d 96 19.f odd 18 1 inner
950.2.bb.d 96 95.o odd 18 1 inner
950.2.bb.d 96 95.r even 36 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$16\!\cdots\!60$$$$T_{3}^{72} -$$$$78\!\cdots\!33$$$$T_{3}^{68} +$$$$17\!\cdots\!38$$$$T_{3}^{64} -$$$$11\!\cdots\!83$$$$T_{3}^{60} +$$$$24\!\cdots\!81$$$$T_{3}^{56} +$$$$84\!\cdots\!87$$$$T_{3}^{52} +$$$$12\!\cdots\!20$$$$T_{3}^{48} -$$$$11\!\cdots\!20$$$$T_{3}^{44} -$$$$66\!\cdots\!12$$$$T_{3}^{40} +$$$$11\!\cdots\!01$$$$T_{3}^{36} +$$$$46\!\cdots\!74$$$$T_{3}^{32} +$$$$11\!\cdots\!49$$$$T_{3}^{28} +$$$$14\!\cdots\!21$$$$T_{3}^{24} -$$$$50\!\cdots\!08$$$$T_{3}^{20} -$$$$10\!\cdots\!24$$$$T_{3}^{16} -$$$$35\!\cdots\!80$$$$T_{3}^{12} +$$$$10\!\cdots\!96$$$$T_{3}^{8} +$$$$45\!\cdots\!92$$$$T_{3}^{4} + 16777216$$">$$T_{3}^{96} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.