# Properties

 Label 138.5.h.a Level $138$ Weight $5$ Character orbit 138.h Analytic conductor $14.265$ Analytic rank $0$ Dimension $160$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 138.h (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.2650549056$$ Analytic rank: $$0$$ Dimension: $$160$$ Relative dimension: $$16$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$160 q - 128 q^{4} - 432 q^{9}+O(q^{10})$$ 160 * q - 128 * q^4 - 432 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$160 q - 128 q^{4} - 432 q^{9} + 208 q^{13} - 1024 q^{16} - 1980 q^{17} - 3828 q^{19} - 3168 q^{20} + 3120 q^{23} + 9768 q^{25} + 4416 q^{26} + 5280 q^{28} - 828 q^{29} - 7748 q^{31} + 16464 q^{35} - 3456 q^{36} - 3520 q^{37} - 9072 q^{39} - 5472 q^{41} - 7216 q^{43} - 1280 q^{46} + 6960 q^{47} + 13888 q^{49} - 7296 q^{50} + 17424 q^{51} + 1664 q^{52} + 14784 q^{53} + 37316 q^{55} - 15840 q^{57} - 9984 q^{58} - 29808 q^{59} - 44968 q^{61} - 8192 q^{64} - 5852 q^{67} - 10584 q^{69} + 4352 q^{70} + 30048 q^{71} + 17084 q^{73} + 4176 q^{75} - 14160 q^{77} - 6912 q^{78} + 28424 q^{79} - 11664 q^{81} - 128 q^{82} - 35904 q^{83} - 91892 q^{85} + 8064 q^{87} - 15312 q^{89} - 6720 q^{92} + 11952 q^{93} + 21248 q^{94} + 224268 q^{95} + 41712 q^{97} + 62976 q^{98} + 17820 q^{99}+O(q^{100})$$ 160 * q - 128 * q^4 - 432 * q^9 + 208 * q^13 - 1024 * q^16 - 1980 * q^17 - 3828 * q^19 - 3168 * q^20 + 3120 * q^23 + 9768 * q^25 + 4416 * q^26 + 5280 * q^28 - 828 * q^29 - 7748 * q^31 + 16464 * q^35 - 3456 * q^36 - 3520 * q^37 - 9072 * q^39 - 5472 * q^41 - 7216 * q^43 - 1280 * q^46 + 6960 * q^47 + 13888 * q^49 - 7296 * q^50 + 17424 * q^51 + 1664 * q^52 + 14784 * q^53 + 37316 * q^55 - 15840 * q^57 - 9984 * q^58 - 29808 * q^59 - 44968 * q^61 - 8192 * q^64 - 5852 * q^67 - 10584 * q^69 + 4352 * q^70 + 30048 * q^71 + 17084 * q^73 + 4176 * q^75 - 14160 * q^77 - 6912 * q^78 + 28424 * q^79 - 11664 * q^81 - 128 * q^82 - 35904 * q^83 - 91892 * q^85 + 8064 * q^87 - 15312 * q^89 - 6720 * q^92 + 11952 * q^93 + 21248 * q^94 + 224268 * q^95 + 41712 * q^97 + 62976 * q^98 + 17820 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −1.85223 + 2.13758i −4.37128 + 2.80925i −1.13852 7.91857i −31.8139 + 14.5289i 2.09159 14.5473i −15.1574 51.6215i 19.0354 + 12.2333i 11.2162 24.5601i 27.8698 94.9158i
7.2 −1.85223 + 2.13758i −4.37128 + 2.80925i −1.13852 7.91857i −16.9332 + 7.73311i 2.09159 14.5473i 26.3243 + 89.6523i 19.0354 + 12.2333i 11.2162 24.5601i 14.8339 50.5195i
7.3 −1.85223 + 2.13758i −4.37128 + 2.80925i −1.13852 7.91857i 0.0789724 0.0360655i 2.09159 14.5473i −7.37685 25.1232i 19.0354 + 12.2333i 11.2162 24.5601i −0.0691817 + 0.235611i
7.4 −1.85223 + 2.13758i −4.37128 + 2.80925i −1.13852 7.91857i 33.6623 15.3731i 2.09159 14.5473i −9.29808 31.6664i 19.0354 + 12.2333i 11.2162 24.5601i −29.4890 + 100.430i
7.5 −1.85223 + 2.13758i 4.37128 2.80925i −1.13852 7.91857i −19.2776 + 8.80377i −2.09159 + 14.5473i 14.6430 + 49.8696i 19.0354 + 12.2333i 11.2162 24.5601i 16.8876 57.5140i
7.6 −1.85223 + 2.13758i 4.37128 2.80925i −1.13852 7.91857i 2.65171 1.21100i −2.09159 + 14.5473i −17.2943 58.8990i 19.0354 + 12.2333i 11.2162 24.5601i −2.32296 + 7.91129i
7.7 −1.85223 + 2.13758i 4.37128 2.80925i −1.13852 7.91857i 5.66414 2.58673i −2.09159 + 14.5473i 9.37267 + 31.9204i 19.0354 + 12.2333i 11.2162 24.5601i −4.96193 + 16.8988i
7.8 −1.85223 + 2.13758i 4.37128 2.80925i −1.13852 7.91857i 43.6718 19.9442i −2.09159 + 14.5473i −0.290384 0.988957i 19.0354 + 12.2333i 11.2162 24.5601i −38.2575 + 130.293i
7.9 1.85223 2.13758i −4.37128 + 2.80925i −1.13852 7.91857i −23.8564 + 10.8948i −2.09159 + 14.5473i 2.81212 + 9.57718i −19.0354 12.2333i 11.2162 24.5601i −20.8988 + 71.1747i
7.10 1.85223 2.13758i −4.37128 + 2.80925i −1.13852 7.91857i 0.678234 0.309739i −2.09159 + 14.5473i −6.16270 20.9882i −19.0354 12.2333i 11.2162 24.5601i 0.594150 2.02349i
7.11 1.85223 2.13758i −4.37128 + 2.80925i −1.13852 7.91857i 1.14894 0.524701i −2.09159 + 14.5473i 1.77751 + 6.05364i −19.0354 12.2333i 11.2162 24.5601i 1.00650 3.42781i
7.12 1.85223 2.13758i −4.37128 + 2.80925i −1.13852 7.91857i 42.3107 19.3226i −2.09159 + 14.5473i 22.4575 + 76.4833i −19.0354 12.2333i 11.2162 24.5601i 37.0652 126.232i
7.13 1.85223 2.13758i 4.37128 2.80925i −1.13852 7.91857i −26.7504 + 12.2165i 2.09159 14.5473i 18.8860 + 64.3200i −19.0354 12.2333i 11.2162 24.5601i −23.4340 + 79.8088i
7.14 1.85223 2.13758i 4.37128 2.80925i −1.13852 7.91857i −22.2415 + 10.1573i 2.09159 14.5473i −7.02266 23.9170i −19.0354 12.2333i 11.2162 24.5601i −19.4841 + 66.3567i
7.15 1.85223 2.13758i 4.37128 2.80925i −1.13852 7.91857i 21.6210 9.87396i 2.09159 14.5473i 14.7757 + 50.3214i −19.0354 12.2333i 11.2162 24.5601i 18.9405 64.5054i
7.16 1.85223 2.13758i 4.37128 2.80925i −1.13852 7.91857i 24.7936 11.3229i 2.09159 14.5473i −17.6937 60.2593i −19.0354 12.2333i 11.2162 24.5601i 21.7198 73.9710i
19.1 −1.17497 2.57283i −4.98567 + 1.46393i −5.23889 + 6.04600i −20.3659 + 31.6899i 9.62445 + 11.1072i −81.9918 + 11.7886i 21.7108 + 6.37488i 22.7138 14.5973i 105.462 + 15.1631i
19.2 −1.17497 2.57283i −4.98567 + 1.46393i −5.23889 + 6.04600i −14.7644 + 22.9738i 9.62445 + 11.1072i 56.7330 8.15697i 21.7108 + 6.37488i 22.7138 14.5973i 76.4554 + 10.9926i
19.3 −1.17497 2.57283i −4.98567 + 1.46393i −5.23889 + 6.04600i 7.54204 11.7356i 9.62445 + 11.1072i −12.2007 + 1.75419i 21.7108 + 6.37488i 22.7138 14.5973i −39.0554 5.61533i
19.4 −1.17497 2.57283i −4.98567 + 1.46393i −5.23889 + 6.04600i 19.8768 30.9289i 9.62445 + 11.1072i −20.0835 + 2.88758i 21.7108 + 6.37488i 22.7138 14.5973i −102.929 14.7990i
See next 80 embeddings (of 160 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.d odd 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.5.h.a 160
23.d odd 22 1 inner 138.5.h.a 160

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.5.h.a 160 1.a even 1 1 trivial
138.5.h.a 160 23.d odd 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(138, [\chi])$$.