Properties

 Label 136.4.c.b Level $136$ Weight $4$ Character orbit 136.c Analytic conductor $8.024$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 136.c (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$8.02425976078$$ Analytic rank: $$0$$ Dimension: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + q^{2} + 5 q^{4} + 40 q^{6} + 84 q^{7} + 13 q^{8} - 216 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24 q + q^{2} + 5 q^{4} + 40 q^{6} + 84 q^{7} + 13 q^{8} - 216 q^{9} - 16 q^{10} - 18 q^{12} + 26 q^{14} - 180 q^{15} - 103 q^{16} + 408 q^{17} - 5 q^{18} - 24 q^{20} + 172 q^{22} + 624 q^{23} - 310 q^{24} - 600 q^{25} - 180 q^{26} - 132 q^{28} - 80 q^{30} - 744 q^{31} - 39 q^{32} - 280 q^{33} + 17 q^{34} + 791 q^{36} - 162 q^{38} + 1236 q^{39} - 1396 q^{40} + 1132 q^{42} + 886 q^{44} - 1246 q^{46} - 956 q^{47} - 2226 q^{48} + 1688 q^{49} + 2505 q^{50} + 2544 q^{52} - 3264 q^{54} + 1684 q^{55} - 1100 q^{56} - 168 q^{57} + 2800 q^{58} + 2520 q^{60} - 1946 q^{62} - 2520 q^{63} - 2143 q^{64} - 72 q^{65} + 3660 q^{66} + 85 q^{68} - 5084 q^{70} + 1532 q^{71} - 2277 q^{72} + 216 q^{73} + 2188 q^{74} + 2094 q^{76} - 4748 q^{78} - 3176 q^{79} + 116 q^{80} + 2040 q^{81} + 3198 q^{82} + 5916 q^{84} - 2066 q^{86} - 236 q^{87} - 3598 q^{88} - 424 q^{89} + 2180 q^{90} + 1940 q^{92} - 2156 q^{94} - 1248 q^{95} - 4674 q^{96} - 1304 q^{97} + 3705 q^{98} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
69.1 −2.79453 0.436572i 3.77193i 7.61881 + 2.44003i 1.90087i −1.64672 + 10.5408i 11.0142 −20.2258 10.1449i 12.7726 0.829868 5.31205i
69.2 −2.79453 + 0.436572i 3.77193i 7.61881 2.44003i 1.90087i −1.64672 10.5408i 11.0142 −20.2258 + 10.1449i 12.7726 0.829868 + 5.31205i
69.3 −2.56626 1.18926i 7.88789i 5.17134 + 6.10387i 14.0362i 9.38073 20.2424i −13.6466 −6.01189 21.8142i −35.2189 −16.6927 + 36.0206i
69.4 −2.56626 + 1.18926i 7.88789i 5.17134 6.10387i 14.0362i 9.38073 + 20.2424i −13.6466 −6.01189 + 21.8142i −35.2189 −16.6927 36.0206i
69.5 −2.13166 1.85904i 3.55140i 1.08798 + 7.92567i 10.8958i −6.60218 + 7.57039i −8.16814 12.4149 18.9175i 14.3876 20.2557 23.2263i
69.6 −2.13166 + 1.85904i 3.55140i 1.08798 7.92567i 10.8958i −6.60218 7.57039i −8.16814 12.4149 + 18.9175i 14.3876 20.2557 + 23.2263i
69.7 −1.85990 2.13091i 3.39972i −1.08156 + 7.92655i 21.1389i −7.24451 + 6.32314i 36.0029 18.9024 12.4379i 15.4419 −45.0451 + 39.3162i
69.8 −1.85990 + 2.13091i 3.39972i −1.08156 7.92655i 21.1389i −7.24451 6.32314i 36.0029 18.9024 + 12.4379i 15.4419 −45.0451 39.3162i
69.9 −1.20282 2.55993i 8.56686i −5.10645 + 6.15826i 20.6960i 21.9305 10.3044i −8.05854 21.9068 + 5.66486i −46.3911 52.9801 24.8935i
69.10 −1.20282 + 2.55993i 8.56686i −5.10645 6.15826i 20.6960i 21.9305 + 10.3044i −8.05854 21.9068 5.66486i −46.3911 52.9801 + 24.8935i
69.11 −0.246793 2.81764i 7.31011i −7.87819 + 1.39075i 6.82683i −20.5973 + 1.80408i −13.0377 5.86290 + 21.8547i −26.4377 −19.2356 + 1.68481i
69.12 −0.246793 + 2.81764i 7.31011i −7.87819 1.39075i 6.82683i −20.5973 1.80408i −13.0377 5.86290 21.8547i −26.4377 −19.2356 1.68481i
69.13 0.705834 2.73894i 9.36589i −7.00360 3.86648i 7.78969i 25.6526 + 6.61076i 32.6671 −15.5334 + 16.4533i −60.7198 −21.3355 5.49823i
69.14 0.705834 + 2.73894i 9.36589i −7.00360 + 3.86648i 7.78969i 25.6526 6.61076i 32.6671 −15.5334 16.4533i −60.7198 −21.3355 + 5.49823i
69.15 0.804094 2.71172i 0.717271i −6.70687 4.36096i 13.2408i 1.94504 + 0.576753i −11.9399 −17.2187 + 14.6805i 26.4855 35.9053 + 10.6468i
69.16 0.804094 + 2.71172i 0.717271i −6.70687 + 4.36096i 13.2408i 1.94504 0.576753i −11.9399 −17.2187 14.6805i 26.4855 35.9053 10.6468i
69.17 2.15286 1.83444i 0.0638194i 1.26962 7.89861i 9.14582i −0.117073 0.137394i 22.9868 −11.7562 19.3337i 26.9959 16.7775 + 19.6897i
69.18 2.15286 + 1.83444i 0.0638194i 1.26962 + 7.89861i 9.14582i −0.117073 + 0.137394i 22.9868 −11.7562 + 19.3337i 26.9959 16.7775 19.6897i
69.19 2.27335 1.68282i 4.93150i 2.33624 7.65127i 13.0027i 8.29881 + 11.2110i 6.24020 −7.56460 21.3255i 2.68034 −21.8811 29.5596i
69.20 2.27335 + 1.68282i 4.93150i 2.33624 + 7.65127i 13.0027i 8.29881 11.2110i 6.24020 −7.56460 + 21.3255i 2.68034 −21.8811 + 29.5596i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 69.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.4.c.b 24
4.b odd 2 1 544.4.c.a 24
8.b even 2 1 inner 136.4.c.b 24
8.d odd 2 1 544.4.c.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.c.b 24 1.a even 1 1 trivial
136.4.c.b 24 8.b even 2 1 inner
544.4.c.a 24 4.b odd 2 1
544.4.c.a 24 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$89\!\cdots\!56$$$$T_{3}^{10} +$$$$91\!\cdots\!28$$$$T_{3}^{8} +$$$$52\!\cdots\!76$$$$T_{3}^{6} +$$$$13\!\cdots\!64$$$$T_{3}^{4} +$$$$58\!\cdots\!92$$$$T_{3}^{2} +$$$$23\!\cdots\!64$$">$$T_{3}^{24} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(136, [\chi])$$.