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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 4 q^{5} + 18 q^{6} + 4 q^{7} + 24 q^{8} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 4 q^{5} + 18 q^{6} + 4 q^{7} + 24 q^{8} - 27 q^{9} - 36 q^{10} - 5 q^{11} - 36 q^{12} - 59 q^{13} + 36 q^{14} + 120 q^{15} - 48 q^{16} - 291 q^{17} + 54 q^{18} + 319 q^{19} + 160 q^{20} + 45 q^{21} + 384 q^{22} + 472 q^{23} - 720 q^{24} + 321 q^{25} + 250 q^{26} - 81 q^{27} - 72 q^{28} + 753 q^{29} - 108 q^{30} - 345 q^{31} + 96 q^{32} - 609 q^{33} + 164 q^{34} - 646 q^{35} - 108 q^{36} - 349 q^{37} + 242 q^{38} - 177 q^{39} - 56 q^{40} - 548 q^{41} - 24 q^{42} + 1800 q^{43} - 20 q^{44} - 1026 q^{45} + 46 q^{46} + 2666 q^{47} - 144 q^{48} - 1685 q^{49} + 414 q^{50} + 51 q^{51} - 280 q^{52} + 769 q^{53} + 162 q^{54} - 4188 q^{55} - 32 q^{56} - 1518 q^{57} - 1264 q^{58} + 2649 q^{59} - 48 q^{60} + 876 q^{61} + 8 q^{62} + 36 q^{63} - 192 q^{64} + 906 q^{65} - 300 q^{66} - 451 q^{67} - 1648 q^{68} + 459 q^{69} + 1512 q^{70} - 2161 q^{71} + 216 q^{72} - 1838 q^{73} + 698 q^{74} - 621 q^{75} + 264 q^{76} + 7182 q^{77} - 1098 q^{78} - 4324 q^{79} - 64 q^{80} - 243 q^{81} + 3736 q^{82} + 191 q^{83} - 84 q^{84} - 2734 q^{85} + 1086 q^{86} - 1074 q^{87} + 392 q^{88} + 4073 q^{89} + 72 q^{90} - 1970 q^{91} - 4624 q^{92} + 1506 q^{93} - 954 q^{94} + 2153 q^{95} + 288 q^{96} - 157 q^{97} - 2988 q^{98} - 1827 q^{99}+O(q^{100}) \) 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 −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:
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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.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 \(S_{4}^{\mathrm{new}}(138, [\chi])\). Copy content Toggle raw display