# Properties

 Label 315.2.bs.e Level 315 Weight 2 Character orbit 315.bs 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.bs (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.

 $$\operatorname{Tr}(f)(q) =$$ $$160q + 4q^{2} - 18q^{3} - 6q^{5} + 24q^{6} - 16q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$160q + 4q^{2} - 18q^{3} - 6q^{5} + 24q^{6} - 16q^{8} - 24q^{10} - 16q^{11} - 30q^{12} + 16q^{15} - 152q^{16} - 6q^{17} + 58q^{18} + 60q^{20} - 36q^{21} + 8q^{22} + 8q^{23} + 2q^{25} - 36q^{26} - 36q^{27} + 22q^{28} - 26q^{30} + 12q^{32} - 6q^{33} - 36q^{35} - 32q^{36} - 4q^{37} - 18q^{38} - 6q^{40} - 12q^{41} - 28q^{42} - 4q^{43} - 54q^{45} - 16q^{46} - 18q^{48} - 44q^{50} + 80q^{51} + 54q^{52} + 8q^{53} + 148q^{56} - 4q^{57} + 28q^{58} + 104q^{60} - 60q^{63} - 124q^{65} + 36q^{66} - 24q^{67} + 42q^{68} - 34q^{70} - 40q^{71} + 70q^{72} + 36q^{73} - 60q^{75} + 96q^{76} + 58q^{77} - 62q^{78} + 36q^{80} + 8q^{81} - 66q^{82} - 138q^{83} - 20q^{85} - 16q^{86} + 102q^{87} + 46q^{88} + 18q^{90} - 48q^{91} - 26q^{92} + 82q^{93} + 188q^{95} - 48q^{96} + 48q^{97} + 102q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
52.1 −1.89043 1.89043i 0.147691 1.72574i 5.14743i 1.96858 1.06052i −3.54159 + 2.98319i 1.93775 + 1.80142i 5.94998 5.94998i −2.95637 0.509752i −5.72629 1.71661i
52.2 −1.88273 1.88273i −1.73169 0.0355078i 5.08933i −2.21562 0.301714i 3.19344 + 3.32715i 2.48041 0.920630i 5.81636 5.81636i 2.99748 + 0.122977i 3.60336 + 4.73945i
52.3 −1.82176 1.82176i −0.791575 + 1.54059i 4.63759i 1.54912 1.61252i 4.24863 1.36452i −2.63707 0.214125i 4.80505 4.80505i −1.74682 2.43898i −5.75974 + 0.115499i
52.4 −1.69178 1.69178i 1.41922 + 0.992886i 3.72426i −1.82828 1.28740i −0.721260 4.08075i 0.203370 + 2.63792i 2.91707 2.91707i 1.02836 + 2.81824i 0.915043 + 5.27105i
52.5 −1.55445 1.55445i −0.951641 1.44720i 2.83266i 1.24343 + 1.85846i −0.770319 + 3.72889i −1.65840 2.06148i 1.29433 1.29433i −1.18876 + 2.75442i 0.956046 4.82175i
52.6 −1.53214 1.53214i −1.70780 + 0.288849i 2.69492i 0.485164 + 2.18280i 3.05914 + 2.17403i −1.33658 + 2.28332i 1.06472 1.06472i 2.83313 0.986590i 2.60102 4.08770i
52.7 −1.48808 1.48808i 0.377509 + 1.69041i 2.42876i −1.63729 + 1.52292i 1.95370 3.07723i −1.04527 2.43052i 0.638021 0.638021i −2.71497 + 1.27629i 4.70264 + 0.170180i
52.8 −1.45962 1.45962i 1.30818 + 1.13520i 2.26101i 2.22626 + 0.209158i −0.252491 3.56641i 2.27033 1.35853i 0.380972 0.380972i 0.422664 + 2.97008i −2.94422 3.55480i
52.9 −1.21497 1.21497i −1.20986 1.23945i 0.952318i −0.123740 2.23264i −0.0359577 + 2.97585i 0.124742 2.64281i −1.27291 + 1.27291i −0.0724885 + 2.99912i −2.56226 + 2.86294i
52.10 −1.07573 1.07573i 1.65548 0.509290i 0.314407i 2.21353 0.316708i −2.32872 1.23300i −2.59482 0.516641i −1.81325 + 1.81325i 2.48125 1.68624i −2.72186 2.04047i
52.11 −1.03040 1.03040i −0.767189 + 1.55288i 0.123432i −0.154472 2.23073i 2.39058 0.809568i 2.47377 + 0.938328i −1.93361 + 1.93361i −1.82284 2.38270i −2.13936 + 2.45770i
52.12 −0.899204 0.899204i −0.810966 1.53047i 0.382866i −2.02048 + 0.957954i −0.646980 + 2.10543i 1.52253 + 2.16377i −2.14268 + 2.14268i −1.68467 + 2.48232i 2.67822 + 0.955424i
52.13 −0.690838 0.690838i 0.0911190 + 1.72965i 1.04549i 1.77477 + 1.36022i 1.13196 1.25786i −0.704044 + 2.55036i −2.10394 + 2.10394i −2.98339 + 0.315208i −0.286383 2.16577i
52.14 −0.664590 0.664590i −1.71542 + 0.239440i 1.11664i −2.09469 0.782473i 1.29918 + 0.980921i −2.43118 + 1.04372i −2.07129 + 2.07129i 2.88534 0.821482i 0.872087 + 1.91213i
52.15 −0.634954 0.634954i −1.27037 + 1.17735i 1.19367i −0.623828 + 2.14729i 1.55419 + 0.0590616i 1.63520 2.07993i −2.02783 + 2.02783i 0.227680 2.99135i 1.75953 0.967326i
52.16 −0.565391 0.565391i 1.73112 0.0568332i 1.36066i −2.17953 0.499648i −1.01089 0.946626i 1.69248 2.03359i −1.90009 + 1.90009i 2.99354 0.196770i 0.949791 + 1.51478i
52.17 −0.333535 0.333535i 0.667654 1.59820i 1.77751i 1.47915 1.67694i −0.755740 + 0.310369i 2.25720 1.38023i −1.25993 + 1.25993i −2.10848 2.13409i −1.05266 + 0.0659706i
52.18 −0.309887 0.309887i −1.71259 + 0.258881i 1.80794i 2.07755 0.826926i 0.610935 + 0.450487i −2.09180 1.61999i −1.18003 + 1.18003i 2.86596 0.886717i −0.900059 0.387551i
52.19 −0.141728 0.141728i 0.505641 1.65660i 1.95983i −1.15314 + 1.91579i −0.306450 + 0.163123i −2.32441 1.26377i −0.561218 + 0.561218i −2.48865 1.67529i 0.434954 0.108090i
52.20 −0.108614 0.108614i 1.61761 + 0.619140i 1.97641i 0.847505 2.06924i −0.108448 0.242943i 0.0705295 + 2.64481i −0.431895 + 0.431895i 2.23333 + 2.00306i −0.316800 + 0.132697i
See next 80 embeddings (of 160 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 292.40 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.t odd 6 1 inner
315.bs 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.bs.e 160
3.b odd 2 1 945.2.bv.e 160
5.c odd 4 1 inner 315.2.bs.e 160
7.d odd 6 1 315.2.cg.e yes 160
9.c even 3 1 315.2.cg.e yes 160
9.d odd 6 1 945.2.cj.e 160
15.e even 4 1 945.2.bv.e 160
21.g even 6 1 945.2.cj.e 160
35.k even 12 1 315.2.cg.e yes 160
45.k odd 12 1 315.2.cg.e yes 160
45.l even 12 1 945.2.cj.e 160
63.i even 6 1 945.2.bv.e 160
63.t odd 6 1 inner 315.2.bs.e 160
105.w odd 12 1 945.2.cj.e 160
315.bs even 12 1 inner 315.2.bs.e 160
315.bu odd 12 1 945.2.bv.e 160

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

## Hecke kernels

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

 $$T_{2}^{80} - \cdots$$ $$T_{11}^{80} + \cdots$$ $$T_{13}^{160} - \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database