# Properties

 Label 315.2.cg.e Level 315 Weight 2 Character orbit 315.cg Analytic conductor 2.515 Analytic rank 0 Dimension 160 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.cg (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$160$$ Relative dimension: $$40$$ over $$\Q(\zeta_{12})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$160q - 2q^{2} - 12q^{3} - 24q^{6} + 6q^{7} - 16q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$160q - 2q^{2} - 12q^{3} - 24q^{6} + 6q^{7} - 16q^{8} - 24q^{10} + 32q^{11} - 12q^{12} + 16q^{15} + 76q^{16} - 6q^{17} - 44q^{18} - 60q^{20} - 60q^{21} + 8q^{22} - 16q^{23} - 4q^{25} - 36q^{26} + 36q^{27} + 22q^{28} - 44q^{30} + 48q^{31} - 6q^{32} + 60q^{33} - 36q^{35} - 32q^{36} - 4q^{37} + 12q^{41} + 2q^{42} - 4q^{43} - 24q^{45} - 16q^{46} - 54q^{47} + 18q^{48} - 44q^{50} - 4q^{51} + 8q^{53} - 92q^{56} - 4q^{57} - 56q^{58} - 28q^{60} - 24q^{61} + 54q^{63} + 62q^{65} + 12q^{66} + 12q^{67} + 2q^{70} - 40q^{71} + 28q^{72} + 36q^{73} + 36q^{75} - 96q^{76} - 110q^{77} - 62q^{78} + 36q^{80} - 16q^{81} - 66q^{82} + 138q^{83} - 20q^{85} + 32q^{86} + 48q^{87} - 92q^{88} - 18q^{90} - 48q^{91} - 26q^{92} + 40q^{93} - 94q^{95} + 132q^{96} - 48q^{97} + 102q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
157.1 −2.59469 + 0.695245i 0.350395 1.69624i 4.51700 2.60789i −0.619852 + 2.14844i 0.270133 + 4.64482i 1.50779 + 2.17407i −6.10819 + 6.10819i −2.75445 1.18871i 0.114633 6.00548i
157.2 −2.57821 + 0.690830i 1.61836 + 0.617171i 4.43789 2.56221i −2.21752 0.287408i −4.59885 0.473186i −1.61180 2.09812i −5.89699 + 5.89699i 2.23820 + 1.99762i 5.91579 0.790931i
157.3 −2.57260 + 0.689326i −1.64466 + 0.543229i 4.41104 2.54672i 0.495489 2.18048i 3.85658 2.53122i 2.23839 1.41053i −5.82578 + 5.82578i 2.40980 1.78685i 0.228366 + 5.95105i
157.4 −2.33005 + 0.624336i 1.68393 + 0.405458i 3.30730 1.90947i 2.23418 0.0919089i −4.17678 + 0.106596i 2.03168 + 1.69478i −3.10261 + 3.10261i 2.67121 + 1.36552i −5.14837 + 1.60903i
157.5 −2.27594 + 0.609836i −1.68624 0.395739i 3.07594 1.77589i 0.427382 + 2.19484i 4.07910 0.127650i −2.23213 1.42042i −2.58544 + 2.58544i 2.68678 + 1.33462i −2.31119 4.73470i
157.6 −2.17690 + 0.583298i −0.853583 + 1.50712i 2.66660 1.53956i −2.19471 + 0.428081i 0.979065 3.77873i −1.03156 + 2.43637i −1.71970 + 1.71970i −1.54279 2.57289i 4.52796 2.21206i
157.7 −1.97684 + 0.529692i −1.08684 1.34862i 1.89527 1.09423i 1.45689 1.69631i 2.86285 + 2.09032i −0.793015 + 2.52411i −0.272736 + 0.272736i −0.637576 + 2.93147i −1.98151 + 4.12504i
157.8 −1.89474 + 0.507694i −0.298455 1.70614i 1.60024 0.923896i −1.92207 1.14265i 1.43169 + 3.08117i 0.974477 2.45976i 0.211119 0.211119i −2.82185 + 1.01841i 4.22194 + 1.18920i
157.9 −1.84711 + 0.494931i 1.20304 1.24607i 1.43480 0.828381i 2.21218 + 0.325952i −1.60543 + 2.89705i −1.00496 2.44746i 0.464119 0.464119i −0.105377 2.99815i −4.24746 + 0.492809i
157.10 −1.58439 + 0.424536i 0.483103 + 1.66331i 0.598007 0.345260i −0.829835 2.07638i −1.47156 2.43024i 2.60741 + 0.448788i 1.51881 1.51881i −2.53322 + 1.60710i 2.19628 + 2.93751i
157.11 −1.35792 + 0.363854i −1.38606 + 1.03867i −0.0204859 + 0.0118276i 1.77376 + 1.36153i 1.50424 1.91476i 2.14673 + 1.54646i 2.01165 2.01165i 0.842327 2.87932i −2.90403 1.20346i
157.12 −1.33121 + 0.356696i 1.23400 + 1.21542i −0.0871697 + 0.0503274i 0.497684 + 2.17998i −2.07625 1.17781i −2.13219 + 1.56645i 2.04711 2.04711i 0.0455283 + 2.99965i −1.44011 2.72448i
157.13 −1.30553 + 0.349816i 1.70101 0.326450i −0.150007 + 0.0866067i −1.14755 + 1.91915i −2.10652 + 1.02123i 2.22736 1.42789i 2.07698 2.07698i 2.78686 1.11059i 0.826810 2.90694i
157.14 −1.21428 + 0.325365i −0.985125 + 1.42462i −0.363441 + 0.209833i 2.19872 0.406980i 0.732697 2.05040i −1.07100 2.41929i 2.15087 2.15087i −1.05906 2.80685i −2.53744 + 1.20957i
157.15 −1.02124 + 0.273642i −1.58003 0.709585i −0.763991 + 0.441090i −1.46795 + 1.68675i 1.80777 + 0.292298i 2.16930 + 1.51465i 2.15473 2.15473i 1.99298 + 2.24233i 1.03757 2.12428i
157.16 −1.01365 + 0.271606i −1.72630 0.141011i −0.778338 + 0.449374i −1.53386 1.62705i 1.78816 0.325938i −2.60738 0.448985i 2.15099 2.15099i 2.96023 + 0.486856i 1.99671 + 1.23265i
157.17 −0.649326 + 0.173986i −0.195991 1.72093i −1.34070 + 0.774052i 1.84508 + 1.26321i 0.426680 + 1.08334i −2.09066 + 1.62146i 1.68656 1.68656i −2.92317 + 0.674573i −1.41784 0.499216i
157.18 −0.409914 + 0.109836i 1.73004 + 0.0834602i −1.57608 + 0.909953i 0.861800 2.06332i −0.718335 + 0.155809i −2.64547 0.0385118i 1.14627 1.14627i 2.98607 + 0.288779i −0.126636 + 0.940443i
157.19 −0.0182967 + 0.00490258i −0.782329 1.54530i −1.73174 + 0.999821i 0.587591 + 2.15748i 0.0218900 + 0.0244385i 0.705053 2.55008i 0.0535716 0.0535716i −1.77592 + 2.41787i −0.0213282 0.0365941i
157.20 0.112795 0.0302234i −0.803249 + 1.53453i −1.72024 + 0.993182i 0.634953 2.14402i −0.0442239 + 0.197365i −0.960898 + 2.46509i −0.329161 + 0.329161i −1.70958 2.46523i 0.00682007 0.261026i
See next 80 embeddings (of 160 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 313.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
63.k odd 6 1 inner
315.cg even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.cg.e yes 160
3.b odd 2 1 945.2.cj.e 160
5.c odd 4 1 inner 315.2.cg.e yes 160
7.d odd 6 1 315.2.bs.e 160
9.c even 3 1 315.2.bs.e 160
9.d odd 6 1 945.2.bv.e 160
15.e even 4 1 945.2.cj.e 160
21.g even 6 1 945.2.bv.e 160
35.k even 12 1 315.2.bs.e 160
45.k odd 12 1 315.2.bs.e 160
45.l even 12 1 945.2.bv.e 160
63.k odd 6 1 inner 315.2.cg.e yes 160
63.s even 6 1 945.2.cj.e 160
105.w odd 12 1 945.2.bv.e 160
315.bw odd 12 1 945.2.cj.e 160
315.cg even 12 1 inner 315.2.cg.e yes 160

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bs.e 160 7.d odd 6 1
315.2.bs.e 160 9.c even 3 1
315.2.bs.e 160 35.k even 12 1
315.2.bs.e 160 45.k odd 12 1
315.2.cg.e yes 160 1.a even 1 1 trivial
315.2.cg.e yes 160 5.c odd 4 1 inner
315.2.cg.e yes 160 63.k odd 6 1 inner
315.2.cg.e yes 160 315.cg even 12 1 inner
945.2.bv.e 160 9.d odd 6 1
945.2.bv.e 160 21.g even 6 1
945.2.bv.e 160 45.l even 12 1
945.2.bv.e 160 105.w odd 12 1
945.2.cj.e 160 3.b odd 2 1
945.2.cj.e 160 15.e even 4 1
945.2.cj.e 160 63.s even 6 1
945.2.cj.e 160 315.bw odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$:

 $$T_{2}^{160} + \cdots$$ $$T_{11}^{40} - \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database