# Properties

 Label 136.3.t.a Level $136$ Weight $3$ Character orbit 136.t Analytic conductor $3.706$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 136.t (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.70573159530$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$4$$ over $$\Q(\zeta_{16})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 8 q^{3} + 8 q^{7} + 16 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32 q - 8 q^{3} + 8 q^{7} + 16 q^{9} - 24 q^{11} + 48 q^{13} + 96 q^{15} + 40 q^{19} - 80 q^{21} - 48 q^{23} + 112 q^{25} - 80 q^{27} + 56 q^{29} - 24 q^{31} - 96 q^{35} + 48 q^{37} - 72 q^{39} - 160 q^{41} + 112 q^{43} - 504 q^{45} + 48 q^{47} + 208 q^{49} - 400 q^{51} + 304 q^{53} - 368 q^{55} - 264 q^{57} + 192 q^{59} - 288 q^{61} + 56 q^{63} + 8 q^{65} + 32 q^{69} + 352 q^{71} - 184 q^{73} + 24 q^{75} + 688 q^{77} - 424 q^{79} + 312 q^{81} + 600 q^{83} - 512 q^{85} + 1336 q^{87} - 144 q^{89} - 24 q^{91} + 944 q^{93} - 256 q^{95} + 416 q^{97} + 128 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1 0 −2.15859 + 3.23056i 0 −1.60027 8.04508i 0 0.739073 3.71557i 0 −2.33284 5.63197i 0
41.2 0 −1.13351 + 1.69642i 0 0.797063 + 4.00711i 0 0.788753 3.96533i 0 1.85116 + 4.46910i 0
41.3 0 0.265432 0.397247i 0 0.373441 + 1.87742i 0 −2.10274 + 10.5712i 0 3.35680 + 8.10403i 0
41.4 0 2.24344 3.35755i 0 −0.109963 0.552824i 0 1.03372 5.19685i 0 −2.79594 6.74999i 0
57.1 0 −4.93198 + 0.981032i 0 0.275904 0.412920i 0 0.527686 + 0.789737i 0 15.0471 6.23272i 0
57.2 0 −0.721983 + 0.143611i 0 −2.80411 + 4.19665i 0 −0.203529 0.304603i 0 −7.81428 + 3.23678i 0
57.3 0 1.35534 0.269593i 0 4.43942 6.64406i 0 −5.05024 7.55823i 0 −6.55066 + 2.71337i 0
57.4 0 4.38842 0.872911i 0 −0.381945 + 0.571621i 0 4.41953 + 6.61429i 0 10.1813 4.21725i 0
65.1 0 −0.912174 + 4.58581i 0 7.13034 4.76434i 0 7.38570 + 4.93497i 0 −11.8827 4.92196i 0
65.2 0 −0.533765 + 2.68342i 0 −4.10611 + 2.74362i 0 −5.75863 3.84780i 0 1.39909 + 0.579524i 0
65.3 0 0.0843794 0.424204i 0 −1.82111 + 1.21683i 0 5.14990 + 3.44105i 0 8.14209 + 3.37256i 0
65.4 0 0.685983 3.44867i 0 4.33868 2.89902i 0 −4.47041 2.98703i 0 −3.10782 1.28730i 0
73.1 0 −2.15859 3.23056i 0 −1.60027 + 8.04508i 0 0.739073 + 3.71557i 0 −2.33284 + 5.63197i 0
73.2 0 −1.13351 1.69642i 0 0.797063 4.00711i 0 0.788753 + 3.96533i 0 1.85116 4.46910i 0
73.3 0 0.265432 + 0.397247i 0 0.373441 1.87742i 0 −2.10274 10.5712i 0 3.35680 8.10403i 0
73.4 0 2.24344 + 3.35755i 0 −0.109963 + 0.552824i 0 1.03372 + 5.19685i 0 −2.79594 + 6.74999i 0
97.1 0 −3.91211 + 2.61399i 0 −7.19369 1.43091i 0 10.7125 2.13084i 0 5.02753 12.1375i 0
97.2 0 −2.30540 + 1.54042i 0 5.10009 + 1.01447i 0 −1.70555 + 0.339255i 0 −0.502166 + 1.21234i 0
97.3 0 0.705713 0.471542i 0 −6.65787 1.32433i 0 −11.2240 + 2.23259i 0 −3.16847 + 7.64937i 0
97.4 0 2.88082 1.92490i 0 2.22013 + 0.441611i 0 3.75824 0.747561i 0 1.14971 2.77565i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 129.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.3.t.a 32
4.b odd 2 1 272.3.bh.f 32
17.e odd 16 1 inner 136.3.t.a 32
68.i even 16 1 272.3.bh.f 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.t.a 32 1.a even 1 1 trivial
136.3.t.a 32 17.e odd 16 1 inner
272.3.bh.f 32 4.b odd 2 1
272.3.bh.f 32 68.i even 16 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$10\!\cdots\!64$$$$T_{3}^{11} - 303339274544 T_{3}^{10} - 311388469280 T_{3}^{9} +$$$$46\!\cdots\!72$$$$T_{3}^{8} -$$$$28\!\cdots\!08$$$$T_{3}^{7} -$$$$19\!\cdots\!16$$$$T_{3}^{6} +$$$$29\!\cdots\!04$$$$T_{3}^{5} - 681868367200 T_{3}^{4} - 247840975232 T_{3}^{3} + 555995974656 T_{3}^{2} - 194282501888 T_{3} + 68254697536$$">$$T_{3}^{32} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(136, [\chi])$$.