# Properties

 Label 138.3.g.a Level $138$ Weight $3$ Character orbit 138.g Analytic conductor $3.760$ Analytic rank $0$ Dimension $160$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.76022764817$$ Analytic rank: $$0$$ Dimension: $$160$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$160 q - 4 q^{3} + 32 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10})$$ 160 * q - 4 * q^3 + 32 * q^4 + 8 * q^6 + 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$160 q - 4 q^{3} + 32 q^{4} + 8 q^{6} + 4 q^{9} + 8 q^{12} + 8 q^{13} + 126 q^{15} - 64 q^{16} + 160 q^{18} - 40 q^{19} + 62 q^{21} - 16 q^{22} - 16 q^{24} + 192 q^{25} - 250 q^{27} - 328 q^{30} - 136 q^{31} - 158 q^{33} + 16 q^{34} - 8 q^{36} + 488 q^{37} - 156 q^{39} - 128 q^{42} + 16 q^{43} - 4 q^{45} - 16 q^{48} - 752 q^{49} + 4 q^{51} - 16 q^{52} - 132 q^{54} - 916 q^{55} - 566 q^{57} - 440 q^{58} - 120 q^{60} - 664 q^{61} - 754 q^{63} + 128 q^{64} - 32 q^{66} + 260 q^{67} + 110 q^{69} + 352 q^{70} + 208 q^{72} - 188 q^{73} + 1362 q^{75} + 80 q^{76} + 332 q^{78} + 656 q^{79} + 1420 q^{81} + 456 q^{82} + 360 q^{84} + 1212 q^{85} + 532 q^{87} + 32 q^{88} - 32 q^{90} + 72 q^{91} + 108 q^{93} + 32 q^{96} + 2076 q^{97} - 468 q^{99}+O(q^{100})$$ 160 * q - 4 * q^3 + 32 * q^4 + 8 * q^6 + 4 * q^9 + 8 * q^12 + 8 * q^13 + 126 * q^15 - 64 * q^16 + 160 * q^18 - 40 * q^19 + 62 * q^21 - 16 * q^22 - 16 * q^24 + 192 * q^25 - 250 * q^27 - 328 * q^30 - 136 * q^31 - 158 * q^33 + 16 * q^34 - 8 * q^36 + 488 * q^37 - 156 * q^39 - 128 * q^42 + 16 * q^43 - 4 * q^45 - 16 * q^48 - 752 * q^49 + 4 * q^51 - 16 * q^52 - 132 * q^54 - 916 * q^55 - 566 * q^57 - 440 * q^58 - 120 * q^60 - 664 * q^61 - 754 * q^63 + 128 * q^64 - 32 * q^66 + 260 * q^67 + 110 * q^69 + 352 * q^70 + 208 * q^72 - 188 * q^73 + 1362 * q^75 + 80 * q^76 + 332 * q^78 + 656 * q^79 + 1420 * q^81 + 456 * q^82 + 360 * q^84 + 1212 * q^85 + 532 * q^87 + 32 * q^88 - 32 * q^90 + 72 * q^91 + 108 * q^93 + 32 * q^96 + 2076 * q^97 - 468 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −1.28641 0.587486i −2.98078 0.339083i 1.30972 + 1.51150i −3.03728 4.72610i 3.63531 + 2.18736i −0.130023 + 0.904330i −0.796860 2.71386i 8.77005 + 2.02146i 1.13068 + 7.86408i
29.2 −1.28641 0.587486i −2.45333 1.72660i 1.30972 + 1.51150i 2.91988 + 4.54342i 2.14165 + 3.66242i 1.52608 10.6142i −0.796860 2.71386i 3.03768 + 8.47186i −1.08698 7.56011i
29.3 −1.28641 0.587486i −0.901649 2.86130i 1.30972 + 1.51150i 1.85466 + 2.88591i −0.521077 + 4.21052i −1.93124 + 13.4321i −0.796860 2.71386i −7.37406 + 5.15978i −0.690432 4.80206i
29.4 −1.28641 0.587486i −0.671287 + 2.92393i 1.30972 + 1.51150i 3.78542 + 5.89022i 2.58132 3.36701i 0.377062 2.62252i −0.796860 2.71386i −8.09875 3.92560i −1.40919 9.80114i
29.5 −1.28641 0.587486i −0.200212 + 2.99331i 1.30972 + 1.51150i −4.05813 6.31458i 2.01608 3.73302i −0.756114 + 5.25889i −0.796860 2.71386i −8.91983 1.19859i 1.51072 + 10.5073i
29.6 −1.28641 0.587486i 1.85221 2.35994i 1.30972 + 1.51150i −1.93323 3.00816i −3.76914 + 1.94772i 0.370720 2.57842i −0.796860 2.71386i −2.13865 8.74221i 0.719680 + 5.00549i
29.7 −1.28641 0.587486i 2.39931 + 1.80092i 1.30972 + 1.51150i −1.08843 1.69363i −2.02849 3.72629i 1.71626 11.9369i −0.796860 2.71386i 2.51336 + 8.64193i 0.405187 + 2.81814i
29.8 −1.28641 0.587486i 2.89781 + 0.776326i 1.30972 + 1.51150i 1.55712 + 2.42292i −3.27171 2.70110i −1.17275 + 8.15664i −0.796860 2.71386i 7.79464 + 4.49930i −0.579665 4.03166i
29.9 1.28641 + 0.587486i −2.99735 + 0.126139i 1.30972 + 1.51150i −2.91988 4.54342i −3.92993 1.59863i 1.52608 10.6142i 0.796860 + 2.71386i 8.96818 0.756167i −1.08698 7.56011i
29.10 1.28641 + 0.587486i −2.69091 1.32627i 1.30972 + 1.51150i 3.03728 + 4.72610i −2.68246 3.28701i −0.130023 + 0.904330i 0.796860 + 2.71386i 5.48199 + 7.13777i 1.13068 + 7.86408i
29.11 1.28641 + 0.587486i −2.30545 + 1.91961i 1.30972 + 1.51150i −1.85466 2.88591i −4.09351 + 1.11499i −1.93124 + 13.4321i 0.796860 + 2.71386i 1.63020 8.85113i −0.690432 4.80206i
29.12 1.28641 + 0.587486i 0.282296 + 2.98669i 1.30972 + 1.51150i 1.93323 + 3.00816i −1.39149 + 4.00796i 0.370720 2.57842i 0.796860 + 2.71386i −8.84062 + 1.68626i 0.719680 + 5.00549i
29.13 1.28641 + 0.587486i 1.01607 2.82269i 1.30972 + 1.51150i −3.78542 5.89022i 2.96538 3.03422i 0.377062 2.62252i 0.796860 + 2.71386i −6.93519 5.73613i −1.40919 9.80114i
29.14 1.28641 + 0.587486i 1.44988 2.62638i 1.30972 + 1.51150i 4.05813 + 6.31458i 3.40810 2.52683i −0.756114 + 5.25889i 0.796860 + 2.71386i −4.79571 7.61585i 1.51072 + 10.5073i
29.15 1.28641 + 0.587486i 2.85751 + 0.913589i 1.30972 + 1.51150i −1.55712 2.42292i 3.13922 + 2.85400i −1.17275 + 8.15664i 0.796860 + 2.71386i 7.33071 + 5.22117i −0.579665 4.03166i
29.16 1.28641 + 0.587486i 2.99208 0.217869i 1.30972 + 1.51150i 1.08843 + 1.69363i 3.97705 + 1.47753i 1.71626 11.9369i 0.796860 + 2.71386i 8.90507 1.30376i 0.405187 + 2.81814i
35.1 −0.764582 1.18971i −2.91647 + 0.703009i −0.830830 + 1.81926i −0.00374340 0.0127488i 3.06625 + 2.93225i 1.68160 + 1.94067i 2.79964 0.402527i 8.01156 4.10060i −0.0123053 + 0.0142011i
35.2 −0.764582 1.18971i −1.64224 2.51059i −0.830830 + 1.81926i −2.24222 7.63629i −1.73126 + 3.87334i 0.444889 + 0.513429i 2.79964 0.402527i −3.60612 + 8.24596i −7.37063 + 8.50616i
35.3 −0.764582 1.18971i −0.894326 + 2.86360i −0.830830 + 1.81926i −0.674627 2.29757i 4.09064 1.12546i 1.86579 + 2.15324i 2.79964 0.402527i −7.40036 5.12197i −2.21764 + 2.55929i
35.4 −0.764582 1.18971i −0.618173 2.93562i −0.830830 + 1.81926i 2.12160 + 7.22551i −3.01990 + 2.97997i 7.03927 + 8.12375i 2.79964 0.402527i −8.23573 + 3.62944i 6.97414 8.04859i
See next 80 embeddings (of 160 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 131.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.c even 11 1 inner
69.h 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.g.a 160
3.b odd 2 1 inner 138.3.g.a 160
23.c even 11 1 inner 138.3.g.a 160
69.h odd 22 1 inner 138.3.g.a 160

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.3.g.a 160 1.a even 1 1 trivial
138.3.g.a 160 3.b odd 2 1 inner
138.3.g.a 160 23.c even 11 1 inner
138.3.g.a 160 69.h odd 22 1 inner

## Hecke kernels

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