Properties

Label 138.3.h.a
Level $138$
Weight $3$
Character orbit 138.h
Analytic conductor $3.760$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 138.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.76022764817\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(8\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 16 q^{4} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 16 q^{4} - 24 q^{9} - 16 q^{13} - 32 q^{16} + 220 q^{17} + 132 q^{19} + 88 q^{20} - 104 q^{23} - 336 q^{25} - 208 q^{26} - 264 q^{28} - 164 q^{29} - 268 q^{31} + 552 q^{35} - 48 q^{36} + 352 q^{37} + 216 q^{39} + 192 q^{41} + 88 q^{43} + 80 q^{46} - 64 q^{47} - 40 q^{49} + 160 q^{50} - 264 q^{51} - 32 q^{52} - 352 q^{53} + 196 q^{55} - 528 q^{57} + 312 q^{58} - 696 q^{59} + 616 q^{61} + 96 q^{62} - 64 q^{64} + 44 q^{67} + 72 q^{69} - 32 q^{70} - 32 q^{71} - 284 q^{73} - 48 q^{75} - 224 q^{77} + 144 q^{78} - 440 q^{79} - 72 q^{81} - 616 q^{82} + 352 q^{83} - 532 q^{85} - 96 q^{87} + 88 q^{89} - 32 q^{92} - 192 q^{93} + 16 q^{94} + 372 q^{95} - 264 q^{97} + 1184 q^{98} + 660 q^{99}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.926113 + 1.06879i −1.45709 + 0.936417i −0.284630 1.97964i 1.12201 0.512407i 0.348599 2.42456i 2.20763 + 7.51851i 2.37942 + 1.52916i 1.24625 2.72890i −0.491456 + 1.67375i
7.2 −0.926113 + 1.06879i −1.45709 + 0.936417i −0.284630 1.97964i 3.19308 1.45823i 0.348599 2.42456i −1.79462 6.11193i 2.37942 + 1.52916i 1.24625 2.72890i −1.39861 + 4.76323i
7.3 −0.926113 + 1.06879i 1.45709 0.936417i −0.284630 1.97964i −6.36427 + 2.90646i −0.348599 + 2.42456i −3.11450 10.6070i 2.37942 + 1.52916i 1.24625 2.72890i 2.78763 9.49379i
7.4 −0.926113 + 1.06879i 1.45709 0.936417i −0.284630 1.97964i 0.0820350 0.0374641i −0.348599 + 2.42456i 2.03512 + 6.93100i 2.37942 + 1.52916i 1.24625 2.72890i −0.0359323 + 0.122374i
7.5 0.926113 1.06879i −1.45709 + 0.936417i −0.284630 1.97964i −7.01760 + 3.20483i −0.348599 + 2.42456i −1.16439 3.96555i −2.37942 1.52916i 1.24625 2.72890i −3.07380 + 10.4684i
7.6 0.926113 1.06879i −1.45709 + 0.936417i −0.284630 1.97964i 2.51088 1.14668i −0.348599 + 2.42456i −2.32390 7.91446i −2.37942 1.52916i 1.24625 2.72890i 1.09980 3.74556i
7.7 0.926113 1.06879i 1.45709 0.936417i −0.284630 1.97964i −2.15873 + 0.985859i 0.348599 2.42456i −3.47983 11.8512i −2.37942 1.52916i 1.24625 2.72890i −0.945551 + 3.22025i
7.8 0.926113 1.06879i 1.45709 0.936417i −0.284630 1.97964i 4.69832 2.14565i 0.348599 2.42456i 1.48393 + 5.05379i −2.37942 1.52916i 1.24625 2.72890i 2.05792 7.00863i
19.1 −0.587486 1.28641i −1.66189 + 0.487975i −1.30972 + 1.51150i −0.634924 + 0.987962i 1.60407 + 1.85120i 1.01096 0.145354i 2.71386 + 0.796860i 2.52376 1.62192i 1.64394 + 0.236362i
19.2 −0.587486 1.28641i −1.66189 + 0.487975i −1.30972 + 1.51150i 0.791659 1.23185i 1.60407 + 1.85120i 8.65930 1.24502i 2.71386 + 0.796860i 2.52376 1.62192i −2.04975 0.294710i
19.3 −0.587486 1.28641i 1.66189 0.487975i −1.30972 + 1.51150i −3.49672 + 5.44100i −1.60407 1.85120i 9.55110 1.37324i 2.71386 + 0.796860i 2.52376 1.62192i 9.05366 + 1.30172i
19.4 −0.587486 1.28641i 1.66189 0.487975i −1.30972 + 1.51150i 3.94925 6.14515i −1.60407 1.85120i −2.61314 + 0.375713i 2.71386 + 0.796860i 2.52376 1.62192i −10.2253 1.47018i
19.5 0.587486 + 1.28641i −1.66189 + 0.487975i −1.30972 + 1.51150i −0.858668 + 1.33611i −1.60407 1.85120i −8.36490 + 1.20269i −2.71386 0.796860i 2.52376 1.62192i −2.22325 0.319655i
19.6 0.587486 + 1.28641i −1.66189 + 0.487975i −1.30972 + 1.51150i 5.01874 7.80931i −1.60407 1.85120i 5.11630 0.735612i −2.71386 0.796860i 2.52376 1.62192i 12.9944 + 1.86832i
19.7 0.587486 + 1.28641i 1.66189 0.487975i −1.30972 + 1.51150i −4.44262 + 6.91285i 1.60407 + 1.85120i −10.3360 + 1.48610i −2.71386 0.796860i 2.52376 1.62192i −11.5028 1.65385i
19.8 0.587486 + 1.28641i 1.66189 0.487975i −1.30972 + 1.51150i 0.891807 1.38768i 1.60407 + 1.85120i 9.81972 1.41186i −2.71386 0.796860i 2.52376 1.62192i 2.30905 + 0.331992i
37.1 −1.35693 + 0.398430i −1.13425 1.30900i 1.68251 1.08128i −3.69869 + 0.531791i 2.06064 + 1.32429i 1.55981 + 0.712339i −1.85223 + 2.13758i −0.426945 + 2.96946i 4.80697 2.19527i
37.2 −1.35693 + 0.398430i −1.13425 1.30900i 1.68251 1.08128i 4.69745 0.675391i 2.06064 + 1.32429i −5.04392 2.30348i −1.85223 + 2.13758i −0.426945 + 2.96946i −6.10500 + 2.78806i
37.3 −1.35693 + 0.398430i 1.13425 + 1.30900i 1.68251 1.08128i −3.60161 + 0.517833i −2.06064 1.32429i −10.7332 4.90168i −1.85223 + 2.13758i −0.426945 + 2.96946i 4.68080 2.13765i
37.4 −1.35693 + 0.398430i 1.13425 + 1.30900i 1.68251 1.08128i 5.59508 0.804450i −2.06064 1.32429i 0.513410 + 0.234466i −1.85223 + 2.13758i −0.426945 + 2.96946i −7.27160 + 3.32083i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.d odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.3.h.a 80
3.b odd 2 1 414.3.l.b 80
23.d odd 22 1 inner 138.3.h.a 80
69.g even 22 1 414.3.l.b 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.3.h.a 80 1.a even 1 1 trivial
138.3.h.a 80 23.d odd 22 1 inner
414.3.l.b 80 3.b odd 2 1
414.3.l.b 80 69.g even 22 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(138, [\chi])\).