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 $2$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 6 q^{2} + 9 q^{3} - 12 q^{4} - 2 q^{5} - 18 q^{6} + 24 q^{8} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 6 q^{2} + 9 q^{3} - 12 q^{4} - 2 q^{5} - 18 q^{6} + 24 q^{8} - 27 q^{9} + 48 q^{10} + 51 q^{11} + 36 q^{12} - 61 q^{13} + 44 q^{14} - 126 q^{15} - 48 q^{16} + 45 q^{17} + 54 q^{18} + 305 q^{19} + 168 q^{20} - 33 q^{21} + 8 q^{22} + 282 q^{23} + 720 q^{24} + 709 q^{25} + 210 q^{26} + 81 q^{27} - 88 q^{28} - 471 q^{29} - 144 q^{30} - 463 q^{31} + 96 q^{32} + 771 q^{33} + 724 q^{34} - 1424 q^{35} - 108 q^{36} - 483 q^{37} + 270 q^{38} + 183 q^{39} + 104 q^{40} + 886 q^{41} - 974 q^{43} + 204 q^{44} - 18 q^{45} + 382 q^{46} - 122 q^{47} + 144 q^{48} + 791 q^{49} - 450 q^{50} - 729 q^{51} - 200 q^{52} - 1117 q^{53} - 162 q^{54} - 2104 q^{55} - 354 q^{57} + 788 q^{58} - 4103 q^{59} + 24 q^{60} - 870 q^{61} - 592 q^{62} - 192 q^{64} - 2058 q^{65} - 24 q^{66} + 1365 q^{67} - 304 q^{68} + 2091 q^{69} - 584 q^{70} - 119 q^{71} + 216 q^{72} - 3314 q^{73} + 966 q^{74} - 675 q^{75} + 208 q^{76} + 606 q^{77} + 1218 q^{78} + 4040 q^{79} - 32 q^{80} - 243 q^{81} - 2300 q^{82} - 2365 q^{83} - 132 q^{84} + 4242 q^{85} - 1946 q^{86} - 402 q^{87} - 1992 q^{88} - 4963 q^{89} + 36 q^{90} + 8054 q^{91} + 3768 q^{92} - 2406 q^{93} - 1450 q^{94} + 1623 q^{95} - 288 q^{96} + 2287 q^{97} - 2748 q^{98} - 2313 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
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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 \(S_{4}^{\mathrm{new}}(138, [\chi])\). Copy content Toggle raw display