# Properties

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

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## 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 4 q^{5} + 18 q^{6} + 4 q^{7} + 24 q^{8} - 27 q^{9}+O(q^{10})$$ 30 * q + 6 * q^2 - 9 * q^3 - 12 * q^4 - 4 * q^5 + 18 * q^6 + 4 * q^7 + 24 * q^8 - 27 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$30 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 4 q^{5} + 18 q^{6} + 4 q^{7} + 24 q^{8} - 27 q^{9} - 36 q^{10} - 5 q^{11} - 36 q^{12} - 59 q^{13} + 36 q^{14} + 120 q^{15} - 48 q^{16} - 291 q^{17} + 54 q^{18} + 319 q^{19} + 160 q^{20} + 45 q^{21} + 384 q^{22} + 472 q^{23} - 720 q^{24} + 321 q^{25} + 250 q^{26} - 81 q^{27} - 72 q^{28} + 753 q^{29} - 108 q^{30} - 345 q^{31} + 96 q^{32} - 609 q^{33} + 164 q^{34} - 646 q^{35} - 108 q^{36} - 349 q^{37} + 242 q^{38} - 177 q^{39} - 56 q^{40} - 548 q^{41} - 24 q^{42} + 1800 q^{43} - 20 q^{44} - 1026 q^{45} + 46 q^{46} + 2666 q^{47} - 144 q^{48} - 1685 q^{49} + 414 q^{50} + 51 q^{51} - 280 q^{52} + 769 q^{53} + 162 q^{54} - 4188 q^{55} - 32 q^{56} - 1518 q^{57} - 1264 q^{58} + 2649 q^{59} - 48 q^{60} + 876 q^{61} + 8 q^{62} + 36 q^{63} - 192 q^{64} + 906 q^{65} - 300 q^{66} - 451 q^{67} - 1648 q^{68} + 459 q^{69} + 1512 q^{70} - 2161 q^{71} + 216 q^{72} - 1838 q^{73} + 698 q^{74} - 621 q^{75} + 264 q^{76} + 7182 q^{77} - 1098 q^{78} - 4324 q^{79} - 64 q^{80} - 243 q^{81} + 3736 q^{82} + 191 q^{83} - 84 q^{84} - 2734 q^{85} + 1086 q^{86} - 1074 q^{87} + 392 q^{88} + 4073 q^{89} + 72 q^{90} - 1970 q^{91} - 4624 q^{92} + 1506 q^{93} - 954 q^{94} + 2153 q^{95} + 288 q^{96} - 157 q^{97} - 2988 q^{98} - 1827 q^{99}+O(q^{100})$$ 30 * q + 6 * q^2 - 9 * q^3 - 12 * q^4 - 4 * q^5 + 18 * q^6 + 4 * q^7 + 24 * q^8 - 27 * q^9 - 36 * q^10 - 5 * q^11 - 36 * q^12 - 59 * q^13 + 36 * q^14 + 120 * q^15 - 48 * q^16 - 291 * q^17 + 54 * q^18 + 319 * q^19 + 160 * q^20 + 45 * q^21 + 384 * q^22 + 472 * q^23 - 720 * q^24 + 321 * q^25 + 250 * q^26 - 81 * q^27 - 72 * q^28 + 753 * q^29 - 108 * q^30 - 345 * q^31 + 96 * q^32 - 609 * q^33 + 164 * q^34 - 646 * q^35 - 108 * q^36 - 349 * q^37 + 242 * q^38 - 177 * q^39 - 56 * q^40 - 548 * q^41 - 24 * q^42 + 1800 * q^43 - 20 * q^44 - 1026 * q^45 + 46 * q^46 + 2666 * q^47 - 144 * q^48 - 1685 * q^49 + 414 * q^50 + 51 * q^51 - 280 * q^52 + 769 * q^53 + 162 * q^54 - 4188 * q^55 - 32 * q^56 - 1518 * q^57 - 1264 * q^58 + 2649 * q^59 - 48 * q^60 + 876 * q^61 + 8 * q^62 + 36 * q^63 - 192 * q^64 + 906 * q^65 - 300 * q^66 - 451 * q^67 - 1648 * q^68 + 459 * q^69 + 1512 * q^70 - 2161 * q^71 + 216 * q^72 - 1838 * q^73 + 698 * q^74 - 621 * q^75 + 264 * q^76 + 7182 * q^77 - 1098 * q^78 - 4324 * q^79 - 64 * q^80 - 243 * q^81 + 3736 * q^82 + 191 * q^83 - 84 * q^84 - 2734 * q^85 + 1086 * q^86 - 1074 * q^87 + 392 * q^88 + 4073 * q^89 + 72 * q^90 - 1970 * q^91 - 4624 * q^92 + 1506 * q^93 - 954 * q^94 + 2153 * q^95 + 288 * q^96 - 157 * q^97 - 2988 * q^98 - 1827 * 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 −4.97931 + 10.9032i 0.853889 + 5.93893i 10.5666 + 3.10264i −6.73003 + 4.32513i 3.73874 + 8.18669i −23.0016 + 6.75388i
13.2 1.30972 + 1.51150i 2.52376 + 1.62192i −0.569259 + 3.95929i −3.70472 + 8.11221i 0.853889 + 5.93893i −24.4006 7.16466i −6.73003 + 4.32513i 3.73874 + 8.18669i −17.1138 + 5.02505i
13.3 1.30972 + 1.51150i 2.52376 + 1.62192i −0.569259 + 3.95929i 5.61304 12.2908i 0.853889 + 5.93893i 18.2993 + 5.37317i −6.73003 + 4.32513i 3.73874 + 8.18669i 25.9291 7.61348i
25.1 −1.68251 1.08128i −0.426945 2.96946i 1.66166 + 3.63853i −11.8265 3.47259i −2.49249 + 5.45779i 13.0481 15.0583i 1.13852 7.91857i −8.63544 + 2.53559i 16.1434 + 18.6305i
25.2 −1.68251 1.08128i −0.426945 2.96946i 1.66166 + 3.63853i 4.71256 + 1.38373i −2.49249 + 5.45779i 3.70690 4.27800i 1.13852 7.91857i −8.63544 + 2.53559i −6.43272 7.42375i
25.3 −1.68251 1.08128i −0.426945 2.96946i 1.66166 + 3.63853i 19.5175 + 5.73085i −2.49249 + 5.45779i −20.5565 + 23.7235i 1.13852 7.91857i −8.63544 + 2.53559i −26.6416 30.7461i
31.1 0.284630 1.97964i 1.24625 + 2.72890i −3.83797 1.12693i −7.03109 8.11432i 5.75696 1.69040i −5.50559 3.53823i −3.32332 + 7.27706i −5.89375 + 6.80175i −18.0647 + 11.6095i
31.2 0.284630 1.97964i 1.24625 + 2.72890i −3.83797 1.12693i −2.69854 3.11428i 5.75696 1.69040i 22.2299 + 14.2863i −3.32332 + 7.27706i −5.89375 + 6.80175i −6.93325 + 4.45573i
31.3 0.284630 1.97964i 1.24625 + 2.72890i −3.83797 1.12693i 13.1550 + 15.1817i 5.75696 1.69040i −15.4591 9.93493i −3.32332 + 7.27706i −5.89375 + 6.80175i 33.7986 21.7210i
49.1 0.284630 + 1.97964i 1.24625 2.72890i −3.83797 + 1.12693i −7.03109 + 8.11432i 5.75696 + 1.69040i −5.50559 + 3.53823i −3.32332 7.27706i −5.89375 6.80175i −18.0647 11.6095i
49.2 0.284630 + 1.97964i 1.24625 2.72890i −3.83797 + 1.12693i −2.69854 + 3.11428i 5.75696 + 1.69040i 22.2299 14.2863i −3.32332 7.27706i −5.89375 6.80175i −6.93325 4.45573i
49.3 0.284630 + 1.97964i 1.24625 2.72890i −3.83797 + 1.12693i 13.1550 15.1817i 5.75696 + 1.69040i −15.4591 + 9.93493i −3.32332 7.27706i −5.89375 6.80175i 33.7986 + 21.7210i
55.1 1.91899 0.563465i −1.96458 2.26725i 3.36501 2.16256i −1.52360 10.5969i −5.04752 3.24384i 0.567261 1.24213i 5.23889 6.04600i −1.28083 + 8.90839i −8.89475 19.4768i
55.2 1.91899 0.563465i −1.96458 2.26725i 3.36501 2.16256i 1.45978 + 10.1530i −5.04752 3.24384i −13.4424 + 29.4347i 5.23889 6.04600i −1.28083 + 8.90839i 8.52215 + 18.6609i
55.3 1.91899 0.563465i −1.96458 2.26725i 3.36501 2.16256i 1.75662 + 12.2176i −5.04752 3.24384i 12.8585 28.1562i 5.23889 6.04600i −1.28083 + 8.90839i 10.2551 + 22.4555i
73.1 −0.830830 1.81926i −2.87848 + 0.845198i −2.61944 + 3.02300i −16.6025 10.6698i 3.92916 + 4.53450i 3.87830 + 26.9742i 7.67594 + 2.25386i 7.57128 4.86577i −5.61729 + 39.0691i
73.2 −0.830830 1.81926i −2.87848 + 0.845198i −2.61944 + 3.02300i −3.38842 2.17761i 3.92916 + 4.53450i −3.30636 22.9962i 7.67594 + 2.25386i 7.57128 4.86577i −1.14644 + 7.97366i
73.3 −0.830830 1.81926i −2.87848 + 0.845198i −2.61944 + 3.02300i 3.54028 + 2.27520i 3.92916 + 4.53450i −0.484390 3.36901i 7.67594 + 2.25386i 7.57128 4.86577i 1.19782 8.33101i
85.1 1.30972 1.51150i 2.52376 1.62192i −0.569259 3.95929i −4.97931 10.9032i 0.853889 5.93893i 10.5666 3.10264i −6.73003 4.32513i 3.73874 8.18669i −23.0016 6.75388i
85.2 1.30972 1.51150i 2.52376 1.62192i −0.569259 3.95929i −3.70472 8.11221i 0.853889 5.93893i −24.4006 + 7.16466i −6.73003 4.32513i 3.73874 8.18669i −17.1138 5.02505i
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.c 30
23.c even 11 1 inner 138.4.e.c 30

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{30} + 4 T_{5}^{29} + 35 T_{5}^{28} + 797 T_{5}^{27} + 81670 T_{5}^{26} - 589815 T_{5}^{25} + 11136436 T_{5}^{24} - 319966486 T_{5}^{23} + 1800165068 T_{5}^{22} + 70972833116 T_{5}^{21} + \cdots + 51\!\cdots\!21$$ acting on $$S_{4}^{\mathrm{new}}(138, [\chi])$$.