# Properties

 Label 138.4.e.b 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30 q - 6 q^{2} + 9 q^{3} - 12 q^{4} - 6 q^{5} + 18 q^{6} + 22 q^{7} - 24 q^{8} - 27 q^{9}+O(q^{10})$$ 30 * q - 6 * q^2 + 9 * q^3 - 12 * q^4 - 6 * q^5 + 18 * q^6 + 22 * q^7 - 24 * q^8 - 27 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$30 q - 6 q^{2} + 9 q^{3} - 12 q^{4} - 6 q^{5} + 18 q^{6} + 22 q^{7} - 24 q^{8} - 27 q^{9} - 56 q^{10} - 105 q^{11} + 36 q^{12} - 21 q^{13} - 114 q^{15} - 48 q^{16} + 41 q^{17} - 54 q^{18} - 149 q^{19} + 152 q^{20} - 33 q^{21} - 584 q^{22} + 472 q^{23} - 720 q^{24} + 281 q^{25} + 90 q^{26} + 81 q^{27} - 1505 q^{29} + 168 q^{30} - 991 q^{31} - 96 q^{32} + 315 q^{33} - 1392 q^{34} + 646 q^{35} - 108 q^{36} + 103 q^{37} - 606 q^{38} + 63 q^{39} + 40 q^{40} + 966 q^{41} - 132 q^{42} + 1532 q^{43} - 420 q^{44} - 54 q^{45} - 46 q^{46} + 1718 q^{47} + 144 q^{48} + 843 q^{49} + 122 q^{50} + 273 q^{51} - 40 q^{52} + 911 q^{53} + 162 q^{54} + 2112 q^{55} + 176 q^{56} - 972 q^{57} + 1060 q^{58} + 415 q^{59} + 72 q^{60} - 1424 q^{61} - 464 q^{62} + 198 q^{63} - 192 q^{64} + 5246 q^{65} + 300 q^{66} - 5 q^{67} - 144 q^{68} - 1449 q^{69} + 2744 q^{70} + 4415 q^{71} - 216 q^{72} + 2890 q^{73} + 206 q^{74} - 183 q^{75} - 464 q^{76} - 5116 q^{77} + 1050 q^{78} - 3436 q^{79} - 96 q^{80} - 243 q^{81} - 4668 q^{82} + 5757 q^{83} - 132 q^{84} + 568 q^{85} + 710 q^{86} - 138 q^{87} + 1624 q^{88} + 375 q^{89} - 108 q^{90} - 8002 q^{91} - 48 q^{92} - 690 q^{93} + 1082 q^{94} - 5577 q^{95} + 288 q^{96} + 3179 q^{97} - 4100 q^{98} - 945 q^{99}+O(q^{100})$$ 30 * q - 6 * q^2 + 9 * q^3 - 12 * q^4 - 6 * q^5 + 18 * q^6 + 22 * q^7 - 24 * q^8 - 27 * q^9 - 56 * q^10 - 105 * q^11 + 36 * q^12 - 21 * q^13 - 114 * q^15 - 48 * q^16 + 41 * q^17 - 54 * q^18 - 149 * q^19 + 152 * q^20 - 33 * q^21 - 584 * q^22 + 472 * q^23 - 720 * q^24 + 281 * q^25 + 90 * q^26 + 81 * q^27 - 1505 * q^29 + 168 * q^30 - 991 * q^31 - 96 * q^32 + 315 * q^33 - 1392 * q^34 + 646 * q^35 - 108 * q^36 + 103 * q^37 - 606 * q^38 + 63 * q^39 + 40 * q^40 + 966 * q^41 - 132 * q^42 + 1532 * q^43 - 420 * q^44 - 54 * q^45 - 46 * q^46 + 1718 * q^47 + 144 * q^48 + 843 * q^49 + 122 * q^50 + 273 * q^51 - 40 * q^52 + 911 * q^53 + 162 * q^54 + 2112 * q^55 + 176 * q^56 - 972 * q^57 + 1060 * q^58 + 415 * q^59 + 72 * q^60 - 1424 * q^61 - 464 * q^62 + 198 * q^63 - 192 * q^64 + 5246 * q^65 + 300 * q^66 - 5 * q^67 - 144 * q^68 - 1449 * q^69 + 2744 * q^70 + 4415 * q^71 - 216 * q^72 + 2890 * q^73 + 206 * q^74 - 183 * q^75 - 464 * q^76 - 5116 * q^77 + 1050 * q^78 - 3436 * q^79 - 96 * q^80 - 243 * q^81 - 4668 * q^82 + 5757 * q^83 - 132 * q^84 + 568 * q^85 + 710 * q^86 - 138 * q^87 + 1624 * q^88 + 375 * q^89 - 108 * q^90 - 8002 * q^91 - 48 * q^92 - 690 * q^93 + 1082 * q^94 - 5577 * q^95 + 288 * q^96 + 3179 * q^97 - 4100 * q^98 - 945 * 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 −3.80011 + 8.32109i 0.853889 + 5.93893i 2.38181 + 0.699363i 6.73003 4.32513i 3.73874 + 8.18669i 17.5544 5.15444i
13.2 −1.30972 1.51150i −2.52376 1.62192i −0.569259 + 3.95929i 2.02822 4.44117i 0.853889 + 5.93893i −16.2703 4.77739i 6.73003 4.32513i 3.73874 + 8.18669i −9.36923 + 2.75105i
13.3 −1.30972 1.51150i −2.52376 1.62192i −0.569259 + 3.95929i 6.04376 13.2340i 0.853889 + 5.93893i 29.2734 + 8.59546i 6.73003 4.32513i 3.73874 + 8.18669i −27.9188 + 8.19771i
25.1 1.68251 + 1.08128i 0.426945 + 2.96946i 1.66166 + 3.63853i −20.3614 5.97866i −2.49249 + 5.45779i 5.13322 5.92406i −1.13852 + 7.91857i −8.63544 + 2.53559i −27.7936 32.0756i
25.2 1.68251 + 1.08128i 0.426945 + 2.96946i 1.66166 + 3.63853i 2.07213 + 0.608431i −2.49249 + 5.45779i −14.7194 + 16.9870i −1.13852 + 7.91857i −8.63544 + 2.53559i 2.82848 + 3.26424i
25.3 1.68251 + 1.08128i 0.426945 + 2.96946i 1.66166 + 3.63853i 16.0712 + 4.71893i −2.49249 + 5.45779i 17.1563 19.7994i −1.13852 + 7.91857i −8.63544 + 2.53559i 21.9374 + 25.3171i
31.1 −0.284630 + 1.97964i −1.24625 2.72890i −3.83797 1.12693i −11.4557 13.2206i 5.75696 1.69040i 16.3172 + 10.4864i 3.32332 7.27706i −5.89375 + 6.80175i 29.4328 18.9153i
31.2 −0.284630 + 1.97964i −1.24625 2.72890i −3.83797 1.12693i 1.48009 + 1.70811i 5.75696 1.69040i 3.78550 + 2.43279i 3.32332 7.27706i −5.89375 + 6.80175i −3.80273 + 2.44386i
31.3 −0.284630 + 1.97964i −1.24625 2.72890i −3.83797 1.12693i 11.2596 + 12.9943i 5.75696 1.69040i −30.1361 19.3673i 3.32332 7.27706i −5.89375 + 6.80175i −28.9289 + 18.5915i
49.1 −0.284630 1.97964i −1.24625 + 2.72890i −3.83797 + 1.12693i −11.4557 + 13.2206i 5.75696 + 1.69040i 16.3172 10.4864i 3.32332 + 7.27706i −5.89375 6.80175i 29.4328 + 18.9153i
49.2 −0.284630 1.97964i −1.24625 + 2.72890i −3.83797 + 1.12693i 1.48009 1.70811i 5.75696 + 1.69040i 3.78550 2.43279i 3.32332 + 7.27706i −5.89375 6.80175i −3.80273 2.44386i
49.3 −0.284630 1.97964i −1.24625 + 2.72890i −3.83797 + 1.12693i 11.2596 12.9943i 5.75696 + 1.69040i −30.1361 + 19.3673i 3.32332 + 7.27706i −5.89375 6.80175i −28.9289 18.5915i
55.1 −1.91899 + 0.563465i 1.96458 + 2.26725i 3.36501 2.16256i −1.40880 9.79842i −5.04752 3.24384i −9.01039 + 19.7300i −5.23889 + 6.04600i −1.28083 + 8.90839i 8.22454 + 18.0092i
55.2 −1.91899 + 0.563465i 1.96458 + 2.26725i 3.36501 2.16256i −0.966187 6.71998i −5.04752 3.24384i 5.61851 12.3028i −5.23889 + 6.04600i −1.28083 + 8.90839i 5.64057 + 12.3511i
55.3 −1.91899 + 0.563465i 1.96458 + 2.26725i 3.36501 2.16256i 3.11759 + 21.6833i −5.04752 3.24384i −0.846192 + 1.85290i −5.23889 + 6.04600i −1.28083 + 8.90839i −18.2004 39.8533i
73.1 0.830830 + 1.81926i 2.87848 0.845198i −2.61944 + 3.02300i −11.8567 7.61986i 3.92916 + 4.53450i −2.71532 18.8855i −7.67594 2.25386i 7.57128 4.86577i 4.01160 27.9013i
73.2 0.830830 + 1.81926i 2.87848 0.845198i −2.61944 + 3.02300i −8.49581 5.45993i 3.92916 + 4.53450i 4.18194 + 29.0860i −7.67594 2.25386i 7.57128 4.86577i 2.87447 19.9924i
73.3 0.830830 + 1.81926i 2.87848 0.845198i −2.61944 + 3.02300i 13.2722 + 8.52953i 3.92916 + 4.53450i 0.849820 + 5.91063i −7.67594 2.25386i 7.57128 4.86577i −4.49052 + 31.2322i
85.1 −1.30972 + 1.51150i −2.52376 + 1.62192i −0.569259 3.95929i −3.80011 8.32109i 0.853889 5.93893i 2.38181 0.699363i 6.73003 + 4.32513i 3.73874 8.18669i 17.5544 + 5.15444i
85.2 −1.30972 + 1.51150i −2.52376 + 1.62192i −0.569259 3.95929i 2.02822 + 4.44117i 0.853889 5.93893i −16.2703 + 4.77739i 6.73003 + 4.32513i 3.73874 8.18669i −9.36923 2.75105i
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.b 30
23.c even 11 1 inner 138.4.e.b 30

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{30} + 6 T_{5}^{29} + 65 T_{5}^{28} + 6409 T_{5}^{27} + 5482 T_{5}^{26} - 1016815 T_{5}^{25} + 38176278 T_{5}^{24} + 432306806 T_{5}^{23} - 3451756324 T_{5}^{22} - 163702456662 T_{5}^{21} + \cdots + 12\!\cdots\!69$$ acting on $$S_{4}^{\mathrm{new}}(138, [\chi])$$.