# Properties

 Label 950.2.u.h Level $950$ Weight $2$ Character orbit 950.u Analytic conductor $7.586$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.u (of order $$18$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$8$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 12q^{11} + 30q^{14} + 30q^{19} - 36q^{21} - 18q^{26} + 24q^{29} + 18q^{31} + 18q^{34} - 132q^{39} + 36q^{41} - 6q^{46} + 54q^{49} - 6q^{51} - 54q^{54} - 12q^{56} - 72q^{59} + 24q^{61} + 24q^{64} + 96q^{66} - 42q^{69} - 78q^{71} - 36q^{74} + 12q^{76} + 84q^{79} - 72q^{81} - 18q^{84} - 78q^{86} + 72q^{89} + 24q^{91} - 24q^{94} + 12q^{96} - 102q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1 −0.342020 + 0.939693i −2.84564 + 0.501764i −0.766044 0.642788i 0 0.501764 2.84564i −3.49580 2.01830i 0.866025 0.500000i 5.02684 1.82962i 0
99.2 −0.342020 + 0.939693i −1.28901 + 0.227288i −0.766044 0.642788i 0 0.227288 1.28901i 1.93191 + 1.11539i 0.866025 0.500000i −1.20918 + 0.440106i 0
99.3 −0.342020 + 0.939693i 0.634880 0.111946i −0.766044 0.642788i 0 −0.111946 + 0.634880i 0.369892 + 0.213557i 0.866025 0.500000i −2.42854 + 0.883915i 0
99.4 −0.342020 + 0.939693i 2.51497 0.443457i −0.766044 0.642788i 0 −0.443457 + 2.51497i −3.95328 2.28243i 0.866025 0.500000i 3.30934 1.20450i 0
99.5 0.342020 0.939693i −2.51497 + 0.443457i −0.766044 0.642788i 0 −0.443457 + 2.51497i 3.95328 + 2.28243i −0.866025 + 0.500000i 3.30934 1.20450i 0
99.6 0.342020 0.939693i −0.634880 + 0.111946i −0.766044 0.642788i 0 −0.111946 + 0.634880i −0.369892 0.213557i −0.866025 + 0.500000i −2.42854 + 0.883915i 0
99.7 0.342020 0.939693i 1.28901 0.227288i −0.766044 0.642788i 0 0.227288 1.28901i −1.93191 1.11539i −0.866025 + 0.500000i −1.20918 + 0.440106i 0
99.8 0.342020 0.939693i 2.84564 0.501764i −0.766044 0.642788i 0 0.501764 2.84564i 3.49580 + 2.01830i −0.866025 + 0.500000i 5.02684 1.82962i 0
149.1 −0.642788 + 0.766044i −0.934611 2.56782i −0.173648 0.984808i 0 2.56782 + 0.934611i 1.85059 1.06844i 0.866025 + 0.500000i −3.42208 + 2.87147i 0
149.2 −0.642788 + 0.766044i −0.0291678 0.0801377i −0.173648 0.984808i 0 0.0801377 + 0.0291678i 1.59412 0.920368i 0.866025 + 0.500000i 2.29256 1.92369i 0
149.3 −0.642788 + 0.766044i 0.291155 + 0.799943i −0.173648 0.984808i 0 −0.799943 0.291155i −4.37330 + 2.52492i 0.866025 + 0.500000i 1.74300 1.46255i 0
149.4 −0.642788 + 0.766044i 1.01464 + 2.78771i −0.173648 0.984808i 0 −2.78771 1.01464i 0.787850 0.454865i 0.866025 + 0.500000i −4.44370 + 3.72870i 0
149.5 0.642788 0.766044i −1.01464 2.78771i −0.173648 0.984808i 0 −2.78771 1.01464i −0.787850 + 0.454865i −0.866025 0.500000i −4.44370 + 3.72870i 0
149.6 0.642788 0.766044i −0.291155 0.799943i −0.173648 0.984808i 0 −0.799943 0.291155i 4.37330 2.52492i −0.866025 0.500000i 1.74300 1.46255i 0
149.7 0.642788 0.766044i 0.0291678 + 0.0801377i −0.173648 0.984808i 0 0.0801377 + 0.0291678i −1.59412 + 0.920368i −0.866025 0.500000i 2.29256 1.92369i 0
149.8 0.642788 0.766044i 0.934611 + 2.56782i −0.173648 0.984808i 0 2.56782 + 0.934611i −1.85059 + 1.06844i −0.866025 0.500000i −3.42208 + 2.87147i 0
199.1 −0.984808 + 0.173648i −2.09100 2.49196i 0.939693 0.342020i 0 2.49196 + 2.09100i 0.964479 + 0.556842i −0.866025 + 0.500000i −1.31663 + 7.46696i 0
199.2 −0.984808 + 0.173648i −0.536848 0.639791i 0.939693 0.342020i 0 0.639791 + 0.536848i −3.62467 2.09271i −0.866025 + 0.500000i 0.399818 2.26748i 0
199.3 −0.984808 + 0.173648i 0.549102 + 0.654394i 0.939693 0.342020i 0 −0.654394 0.549102i 2.40903 + 1.39086i −0.866025 + 0.500000i 0.394226 2.23577i 0
199.4 −0.984808 + 0.173648i 1.43596 + 1.71131i 0.939693 0.342020i 0 −1.71131 1.43596i −2.43877 1.40802i −0.866025 + 0.500000i −0.345659 + 1.96033i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 899.8 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
19.e even 9 1 inner
95.p even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.u.h 48
5.b even 2 1 inner 950.2.u.h 48
5.c odd 4 1 950.2.l.j 24
5.c odd 4 1 950.2.l.k yes 24
19.e even 9 1 inner 950.2.u.h 48
95.p even 18 1 inner 950.2.u.h 48
95.q odd 36 1 950.2.l.j 24
95.q odd 36 1 950.2.l.k yes 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.l.j 24 5.c odd 4 1
950.2.l.j 24 95.q odd 36 1
950.2.l.k yes 24 5.c odd 4 1
950.2.l.k yes 24 95.q odd 36 1
950.2.u.h 48 1.a even 1 1 trivial
950.2.u.h 48 5.b even 2 1 inner
950.2.u.h 48 19.e even 9 1 inner
950.2.u.h 48 95.p even 18 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$:

 $$T_{3}^{48} + \cdots$$ $$11\!\cdots\!31$$$$T_{7}^{32} -$$$$12\!\cdots\!37$$$$T_{7}^{30} +$$$$11\!\cdots\!92$$$$T_{7}^{28} -$$$$85\!\cdots\!63$$$$T_{7}^{26} +$$$$53\!\cdots\!97$$$$T_{7}^{24} -$$$$27\!\cdots\!96$$$$T_{7}^{22} +$$$$11\!\cdots\!80$$$$T_{7}^{20} -$$$$40\!\cdots\!16$$$$T_{7}^{18} +$$$$11\!\cdots\!72$$$$T_{7}^{16} -$$$$24\!\cdots\!72$$$$T_{7}^{14} +$$$$40\!\cdots\!36$$$$T_{7}^{12} -$$$$48\!\cdots\!52$$$$T_{7}^{10} +$$$$42\!\cdots\!00$$$$T_{7}^{8} -$$$$24\!\cdots\!52$$$$T_{7}^{6} +$$$$96\!\cdots\!64$$$$T_{7}^{4} -$$$$15\!\cdots\!44$$$$T_{7}^{2} +$$$$18\!\cdots\!36$$">$$T_{7}^{48} - \cdots$$