# Properties

 Label 138.4.e.d Level $138$ Weight $4$ Character orbit 138.e Analytic conductor $8.142$ Analytic rank $0$ Dimension $30$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 138.e (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.14226358079$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$3$$ over $$\Q(\zeta_{11})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$30 q + 6 q^{2} + 9 q^{3} - 12 q^{4} - 2 q^{5} - 18 q^{6} + 24 q^{8} - 27 q^{9}+O(q^{10})$$ 30 * q + 6 * q^2 + 9 * q^3 - 12 * q^4 - 2 * q^5 - 18 * q^6 + 24 * q^8 - 27 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$30 q + 6 q^{2} + 9 q^{3} - 12 q^{4} - 2 q^{5} - 18 q^{6} + 24 q^{8} - 27 q^{9} + 48 q^{10} + 51 q^{11} + 36 q^{12} - 61 q^{13} + 44 q^{14} - 126 q^{15} - 48 q^{16} + 45 q^{17} + 54 q^{18} + 305 q^{19} + 168 q^{20} - 33 q^{21} + 8 q^{22} + 282 q^{23} + 720 q^{24} + 709 q^{25} + 210 q^{26} + 81 q^{27} - 88 q^{28} - 471 q^{29} - 144 q^{30} - 463 q^{31} + 96 q^{32} + 771 q^{33} + 724 q^{34} - 1424 q^{35} - 108 q^{36} - 483 q^{37} + 270 q^{38} + 183 q^{39} + 104 q^{40} + 886 q^{41} - 974 q^{43} + 204 q^{44} - 18 q^{45} + 382 q^{46} - 122 q^{47} + 144 q^{48} + 791 q^{49} - 450 q^{50} - 729 q^{51} - 200 q^{52} - 1117 q^{53} - 162 q^{54} - 2104 q^{55} - 354 q^{57} + 788 q^{58} - 4103 q^{59} + 24 q^{60} - 870 q^{61} - 592 q^{62} - 192 q^{64} - 2058 q^{65} - 24 q^{66} + 1365 q^{67} - 304 q^{68} + 2091 q^{69} - 584 q^{70} - 119 q^{71} + 216 q^{72} - 3314 q^{73} + 966 q^{74} - 675 q^{75} + 208 q^{76} + 606 q^{77} + 1218 q^{78} + 4040 q^{79} - 32 q^{80} - 243 q^{81} - 2300 q^{82} - 2365 q^{83} - 132 q^{84} + 4242 q^{85} - 1946 q^{86} - 402 q^{87} - 1992 q^{88} - 4963 q^{89} + 36 q^{90} + 8054 q^{91} + 3768 q^{92} - 2406 q^{93} - 1450 q^{94} + 1623 q^{95} - 288 q^{96} + 2287 q^{97} - 2748 q^{98} - 2313 q^{99}+O(q^{100})$$ 30 * q + 6 * q^2 + 9 * q^3 - 12 * q^4 - 2 * q^5 - 18 * q^6 + 24 * q^8 - 27 * q^9 + 48 * q^10 + 51 * q^11 + 36 * q^12 - 61 * q^13 + 44 * q^14 - 126 * q^15 - 48 * q^16 + 45 * q^17 + 54 * q^18 + 305 * q^19 + 168 * q^20 - 33 * q^21 + 8 * q^22 + 282 * q^23 + 720 * q^24 + 709 * q^25 + 210 * q^26 + 81 * q^27 - 88 * q^28 - 471 * q^29 - 144 * q^30 - 463 * q^31 + 96 * q^32 + 771 * q^33 + 724 * q^34 - 1424 * q^35 - 108 * q^36 - 483 * q^37 + 270 * q^38 + 183 * q^39 + 104 * q^40 + 886 * q^41 - 974 * q^43 + 204 * q^44 - 18 * q^45 + 382 * q^46 - 122 * q^47 + 144 * q^48 + 791 * q^49 - 450 * q^50 - 729 * q^51 - 200 * q^52 - 1117 * q^53 - 162 * q^54 - 2104 * q^55 - 354 * q^57 + 788 * q^58 - 4103 * q^59 + 24 * q^60 - 870 * q^61 - 592 * q^62 - 192 * q^64 - 2058 * q^65 - 24 * q^66 + 1365 * q^67 - 304 * q^68 + 2091 * q^69 - 584 * q^70 - 119 * q^71 + 216 * q^72 - 3314 * q^73 + 966 * q^74 - 675 * q^75 + 208 * q^76 + 606 * q^77 + 1218 * q^78 + 4040 * q^79 - 32 * q^80 - 243 * q^81 - 2300 * q^82 - 2365 * q^83 - 132 * q^84 + 4242 * q^85 - 1946 * q^86 - 402 * q^87 - 1992 * q^88 - 4963 * q^89 + 36 * q^90 + 8054 * q^91 + 3768 * q^92 - 2406 * q^93 - 1450 * q^94 + 1623 * q^95 - 288 * q^96 + 2287 * q^97 - 2748 * q^98 - 2313 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 1.30972 + 1.51150i −2.52376 1.62192i −0.569259 + 3.95929i −1.00284 + 2.19592i −0.853889 5.93893i 19.8790 + 5.83701i −6.73003 + 4.32513i 3.73874 + 8.18669i −4.63257 + 1.36025i
13.2 1.30972 + 1.51150i −2.52376 1.62192i −0.569259 + 3.95929i 2.20263 4.82309i −0.853889 5.93893i −18.0866 5.31070i −6.73003 + 4.32513i 3.73874 + 8.18669i 10.1749 2.98763i
13.3 1.30972 + 1.51150i −2.52376 1.62192i −0.569259 + 3.95929i 3.30855 7.24472i −0.853889 5.93893i −0.555799 0.163197i −6.73003 + 4.32513i 3.73874 + 8.18669i 15.2837 4.48769i
25.1 −1.68251 1.08128i 0.426945 + 2.96946i 1.66166 + 3.63853i −17.7871 5.22276i 2.49249 5.45779i 3.76192 4.34149i 1.13852 7.91857i −8.63544 + 2.53559i 24.2796 + 28.0202i
25.2 −1.68251 1.08128i 0.426945 + 2.96946i 1.66166 + 3.63853i 1.85372 + 0.544301i 2.49249 5.45779i −14.6603 + 16.9188i 1.13852 7.91857i −8.63544 + 2.53559i −2.53035 2.92018i
25.3 −1.68251 1.08128i 0.426945 + 2.96946i 1.66166 + 3.63853i 15.3096 + 4.49530i 2.49249 5.45779i 7.46958 8.62035i 1.13852 7.91857i −8.63544 + 2.53559i −20.8978 24.1174i
31.1 0.284630 1.97964i −1.24625 2.72890i −3.83797 1.12693i −10.3477 11.9419i −5.75696 + 1.69040i −5.39451 3.46684i −3.32332 + 7.27706i −5.89375 + 6.80175i −26.5860 + 17.0858i
31.2 0.284630 1.97964i −1.24625 2.72890i −3.83797 1.12693i 0.957858 + 1.10543i −5.75696 + 1.69040i −12.8395 8.25145i −3.32332 + 7.27706i −5.89375 + 6.80175i 2.46098 1.58158i
31.3 0.284630 1.97964i −1.24625 2.72890i −3.83797 1.12693i 8.16051 + 9.41773i −5.75696 + 1.69040i 20.8971 + 13.4298i −3.32332 + 7.27706i −5.89375 + 6.80175i 20.9665 13.4743i
49.1 0.284630 + 1.97964i −1.24625 + 2.72890i −3.83797 + 1.12693i −10.3477 + 11.9419i −5.75696 1.69040i −5.39451 + 3.46684i −3.32332 7.27706i −5.89375 6.80175i −26.5860 17.0858i
49.2 0.284630 + 1.97964i −1.24625 + 2.72890i −3.83797 + 1.12693i 0.957858 1.10543i −5.75696 1.69040i −12.8395 + 8.25145i −3.32332 7.27706i −5.89375 6.80175i 2.46098 + 1.58158i
49.3 0.284630 + 1.97964i −1.24625 + 2.72890i −3.83797 + 1.12693i 8.16051 9.41773i −5.75696 1.69040i 20.8971 13.4298i −3.32332 7.27706i −5.89375 6.80175i 20.9665 + 13.4743i
55.1 1.91899 0.563465i 1.96458 + 2.26725i 3.36501 2.16256i −1.77045 12.3137i 5.04752 + 3.24384i 8.86624 19.4144i 5.23889 6.04600i −1.28083 + 8.90839i −10.3358 22.6323i
55.2 1.91899 0.563465i 1.96458 + 2.26725i 3.36501 2.16256i 1.07844 + 7.50075i 5.04752 + 3.24384i 1.48666 3.25533i 5.23889 6.04600i −1.28083 + 8.90839i 6.29593 + 13.7862i
55.3 1.91899 0.563465i 1.96458 + 2.26725i 3.36501 2.16256i 1.91350 + 13.3087i 5.04752 + 3.24384i −11.4577 + 25.0888i 5.23889 6.04600i −1.28083 + 8.90839i 11.1710 + 24.4610i
73.1 −0.830830 1.81926i 2.87848 0.845198i −2.61944 + 3.02300i −14.1264 9.07847i −3.92916 4.53450i −1.02856 7.15376i 7.67594 + 2.25386i 7.57128 4.86577i −4.77952 + 33.2423i
73.2 −0.830830 1.81926i 2.87848 0.845198i −2.61944 + 3.02300i 0.809240 + 0.520067i −3.92916 4.53450i −0.360364 2.50638i 7.67594 + 2.25386i 7.57128 4.86577i 0.273798 1.90431i
73.3 −0.830830 1.81926i 2.87848 0.845198i −2.61944 + 3.02300i 8.44042 + 5.42433i −3.92916 4.53450i 2.02267 + 14.0680i 7.67594 + 2.25386i 7.57128 4.86577i 2.85573 19.8621i
85.1 1.30972 1.51150i −2.52376 + 1.62192i −0.569259 3.95929i −1.00284 2.19592i −0.853889 + 5.93893i 19.8790 5.83701i −6.73003 4.32513i 3.73874 8.18669i −4.63257 1.36025i
85.2 1.30972 1.51150i −2.52376 + 1.62192i −0.569259 3.95929i 2.20263 + 4.82309i −0.853889 + 5.93893i −18.0866 + 5.31070i −6.73003 4.32513i 3.73874 8.18669i 10.1749 + 2.98763i
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 133.3 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.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.4.e.d 30
23.c even 11 1 inner 138.4.e.d 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.4.e.d 30 1.a even 1 1 trivial
138.4.e.d 30 23.c even 11 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{30} + 2 T_{5}^{29} - 165 T_{5}^{28} - 137 T_{5}^{27} + 21974 T_{5}^{26} - 565839 T_{5}^{25} + 6329128 T_{5}^{24} + 51001204 T_{5}^{23} + 638237736 T_{5}^{22} - 25323100482 T_{5}^{21} + \cdots + 11\!\cdots\!49$$ acting on $$S_{4}^{\mathrm{new}}(138, [\chi])$$.