Properties

 Label 138.6.d.a Level $138$ Weight $6$ Character orbit 138.d Analytic conductor $22.133$ Analytic rank $0$ Dimension $40$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 138.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$22.1329671342$$ Analytic rank: $$0$$ Dimension: $$40$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 4 q^{3} - 640 q^{4} + 80 q^{6} + 504 q^{9}+O(q^{10})$$ 40 * q - 4 * q^3 - 640 * q^4 + 80 * q^6 + 504 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 4 q^{3} - 640 q^{4} + 80 q^{6} + 504 q^{9} + 64 q^{12} - 1048 q^{13} + 10240 q^{16} + 1280 q^{18} - 1280 q^{24} + 30480 q^{25} + 1700 q^{27} - 22576 q^{31} - 8064 q^{36} + 55608 q^{39} + 1088 q^{46} - 1024 q^{48} - 23224 q^{49} + 16768 q^{52} + 25456 q^{54} + 210400 q^{55} - 83168 q^{58} - 163840 q^{64} + 99076 q^{69} + 167520 q^{70} - 20480 q^{72} + 241160 q^{73} - 255604 q^{75} - 233440 q^{78} + 78512 q^{81} - 8832 q^{82} - 460296 q^{85} - 4136 q^{87} + 500704 q^{93} - 138272 q^{94} + 20480 q^{96}+O(q^{100})$$ 40 * q - 4 * q^3 - 640 * q^4 + 80 * q^6 + 504 * q^9 + 64 * q^12 - 1048 * q^13 + 10240 * q^16 + 1280 * q^18 - 1280 * q^24 + 30480 * q^25 + 1700 * q^27 - 22576 * q^31 - 8064 * q^36 + 55608 * q^39 + 1088 * q^46 - 1024 * q^48 - 23224 * q^49 + 16768 * q^52 + 25456 * q^54 + 210400 * q^55 - 83168 * q^58 - 163840 * q^64 + 99076 * q^69 + 167520 * q^70 - 20480 * q^72 + 241160 * q^73 - 255604 * q^75 - 233440 * q^78 + 78512 * q^81 - 8832 * q^82 - 460296 * q^85 - 4136 * q^87 + 500704 * q^93 - 138272 * q^94 + 20480 * q^96

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
137.1 4.00000i −11.7474 + 10.2469i −16.0000 −107.494 40.9875 + 46.9896i 192.413i 64.0000i 33.0032 240.748i 429.977i
137.2 4.00000i −11.7474 + 10.2469i −16.0000 107.494 40.9875 + 46.9896i 192.413i 64.0000i 33.0032 240.748i 429.977i
137.3 4.00000i −11.7474 10.2469i −16.0000 −107.494 40.9875 46.9896i 192.413i 64.0000i 33.0032 + 240.748i 429.977i
137.4 4.00000i −11.7474 10.2469i −16.0000 107.494 40.9875 46.9896i 192.413i 64.0000i 33.0032 + 240.748i 429.977i
137.5 4.00000i −1.79544 + 15.4847i −16.0000 −69.6024 61.9389 + 7.18174i 161.717i 64.0000i −236.553 55.6036i 278.410i
137.6 4.00000i −1.79544 + 15.4847i −16.0000 69.6024 61.9389 + 7.18174i 161.717i 64.0000i −236.553 55.6036i 278.410i
137.7 4.00000i −1.79544 15.4847i −16.0000 −69.6024 61.9389 7.18174i 161.717i 64.0000i −236.553 + 55.6036i 278.410i
137.8 4.00000i −1.79544 15.4847i −16.0000 69.6024 61.9389 7.18174i 161.717i 64.0000i −236.553 + 55.6036i 278.410i
137.9 4.00000i −15.1862 + 3.51856i −16.0000 −0.0328033 14.0742 + 60.7447i 156.501i 64.0000i 218.240 106.867i 0.131213i
137.10 4.00000i −15.1862 + 3.51856i −16.0000 0.0328033 14.0742 + 60.7447i 156.501i 64.0000i 218.240 106.867i 0.131213i
137.11 4.00000i −15.1862 3.51856i −16.0000 −0.0328033 14.0742 60.7447i 156.501i 64.0000i 218.240 + 106.867i 0.131213i
137.12 4.00000i −15.1862 3.51856i −16.0000 0.0328033 14.0742 60.7447i 156.501i 64.0000i 218.240 + 106.867i 0.131213i
137.13 4.00000i 15.5676 + 0.805444i −16.0000 −10.2990 3.22177 62.2705i 139.754i 64.0000i 241.703 + 25.0777i 41.1960i
137.14 4.00000i 15.5676 + 0.805444i −16.0000 10.2990 3.22177 62.2705i 139.754i 64.0000i 241.703 + 25.0777i 41.1960i
137.15 4.00000i 15.5676 0.805444i −16.0000 −10.2990 3.22177 + 62.2705i 139.754i 64.0000i 241.703 25.0777i 41.1960i
137.16 4.00000i 15.5676 0.805444i −16.0000 10.2990 3.22177 + 62.2705i 139.754i 64.0000i 241.703 25.0777i 41.1960i
137.17 4.00000i −14.2654 6.28478i −16.0000 −46.4212 −25.1391 + 57.0616i 104.982i 64.0000i 164.003 + 179.310i 185.685i
137.18 4.00000i −14.2654 6.28478i −16.0000 46.4212 −25.1391 + 57.0616i 104.982i 64.0000i 164.003 + 179.310i 185.685i
137.19 4.00000i −14.2654 + 6.28478i −16.0000 −46.4212 −25.1391 57.0616i 104.982i 64.0000i 164.003 179.310i 185.685i
137.20 4.00000i −14.2654 + 6.28478i −16.0000 46.4212 −25.1391 57.0616i 104.982i 64.0000i 164.003 179.310i 185.685i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 137.40 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.b odd 2 1 inner
69.c even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.6.d.a 40
3.b odd 2 1 inner 138.6.d.a 40
23.b odd 2 1 inner 138.6.d.a 40
69.c even 2 1 inner 138.6.d.a 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.6.d.a 40 1.a even 1 1 trivial
138.6.d.a 40 3.b odd 2 1 inner
138.6.d.a 40 23.b odd 2 1 inner
138.6.d.a 40 69.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(138, [\chi])$$.