# Properties

 Label 315.3.ca.b Level 315 Weight 3 Character orbit 315.ca Analytic conductor 8.583 Analytic rank 0 Dimension 64 CM no Inner twists 4

# Learn more about

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$8.58312832735$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$16$$ over $$\Q(\zeta_{12})$$ Twist minimal: no (minimal twist has level 105) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q - 4q^{5} - 4q^{7} - 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q - 4q^{5} - 4q^{7} - 24q^{8} - 16q^{10} - 16q^{11} + 80q^{16} - 56q^{17} - 96q^{22} - 72q^{23} - 4q^{25} + 288q^{26} - 380q^{28} - 136q^{31} + 48q^{32} - 76q^{35} - 28q^{37} + 68q^{38} + 164q^{40} - 128q^{41} + 344q^{43} + 240q^{46} - 412q^{47} + 72q^{50} + 388q^{52} + 40q^{53} - 8q^{55} + 864q^{56} + 56q^{58} - 216q^{61} + 912q^{62} - 20q^{65} - 368q^{67} + 492q^{68} + 416q^{70} - 784q^{71} - 316q^{73} - 32q^{76} - 844q^{77} - 908q^{80} + 556q^{82} - 1408q^{83} - 536q^{85} - 1024q^{86} + 372q^{88} - 1064q^{91} + 1704q^{92} - 260q^{95} + 352q^{97} - 272q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −3.52852 + 0.945463i 0 8.09242 4.67216i −3.83301 3.21062i 0 −6.80203 + 1.65300i −13.8047 + 13.8047i 0 16.5603 + 7.70474i
37.2 −3.41166 + 0.914152i 0 7.33966 4.23756i 4.98672 0.364163i 0 −6.35750 2.92953i −11.1766 + 11.1766i 0 −16.6801 + 5.80102i
37.3 −2.88660 + 0.773463i 0 4.27013 2.46536i −0.160937 + 4.99741i 0 5.19830 4.68804i −1.96674 + 1.96674i 0 −3.40075 14.5500i
37.4 −2.56059 + 0.686107i 0 2.62176 1.51367i −1.32982 + 4.81992i 0 0.158915 + 6.99820i 1.82323 1.82323i 0 0.0981374 13.2542i
37.5 −2.13226 + 0.571337i 0 0.756006 0.436480i −2.78563 4.15214i 0 2.73668 6.44287i 4.88107 4.88107i 0 8.31195 + 7.26192i
37.6 −0.984292 + 0.263740i 0 −2.56483 + 1.48081i 4.04958 + 2.93273i 0 −5.81621 + 3.89508i 5.01620 5.01620i 0 −4.75945 1.81862i
37.7 −0.901161 + 0.241465i 0 −2.71032 + 1.56480i −2.44472 4.36158i 0 5.41162 + 4.44009i 4.70337 4.70337i 0 3.25625 + 3.34017i
37.8 −0.232416 + 0.0622758i 0 −3.41396 + 1.97105i 2.65242 4.23847i 0 −4.06422 + 5.69931i 1.35127 1.35127i 0 −0.352512 + 1.15027i
37.9 0.435396 0.116664i 0 −3.28814 + 1.89841i −3.62365 + 3.44517i 0 6.87993 + 1.29096i −2.48509 + 2.48509i 0 −1.17579 + 1.92276i
37.10 0.445275 0.119311i 0 −3.28007 + 1.89375i −4.59550 + 1.97013i 0 −0.171680 6.99789i −2.53844 + 2.53844i 0 −1.81120 + 1.42554i
37.11 1.84280 0.493776i 0 −0.312015 + 0.180142i 1.82066 + 4.65674i 0 −6.63712 2.22456i −5.88211 + 5.88211i 0 5.65448 + 7.68243i
37.12 1.91023 0.511845i 0 −0.0771041 + 0.0445161i 4.99333 + 0.258093i 0 6.99587 + 0.240410i −5.71805 + 5.71805i 0 9.67053 2.06280i
37.13 2.38023 0.637781i 0 1.79464 1.03614i −2.91424 + 4.06291i 0 −4.25379 + 5.55925i −3.35897 + 3.35897i 0 −4.34532 + 11.5293i
37.14 2.87375 0.770020i 0 4.20143 2.42570i −3.78688 3.26489i 0 −1.65502 6.80154i 1.79111 1.79111i 0 −13.3966 6.46653i
37.15 3.18498 0.853412i 0 5.95167 3.43620i 4.91224 0.932701i 0 5.00478 + 4.89410i 6.69718 6.69718i 0 14.8494 7.16279i
37.16 3.56483 0.955194i 0 8.33154 4.81021i 1.05941 4.88648i 0 −6.28878 + 3.07429i 14.6673 14.6673i 0 −0.890898 18.4314i
163.1 −0.955194 3.56483i 0 −8.33154 + 4.81021i 3.70210 3.36072i 0 3.07429 + 6.28878i 14.6673 + 14.6673i 0 −15.5166 9.98725i
163.2 −0.853412 3.18498i 0 −5.95167 + 3.43620i −1.64838 4.72047i 0 4.89410 5.00478i 6.69718 + 6.69718i 0 −13.6279 + 9.27855i
163.3 −0.770020 2.87375i 0 −4.20143 + 2.42570i 4.72092 + 1.64709i 0 −6.80154 + 1.65502i 1.79111 + 1.79111i 0 1.09812 14.8351i
163.4 −0.637781 2.38023i 0 −1.79464 + 1.03614i −2.06146 + 4.55526i 0 5.55925 + 4.25379i −3.35897 3.35897i 0 12.1573 + 2.00150i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 298.16 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
7.c even 3 1 inner
35.l odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.3.ca.b 64
3.b odd 2 1 105.3.v.a 64
5.c odd 4 1 inner 315.3.ca.b 64
7.c even 3 1 inner 315.3.ca.b 64
15.e even 4 1 105.3.v.a 64
21.h odd 6 1 105.3.v.a 64
35.l odd 12 1 inner 315.3.ca.b 64
105.x even 12 1 105.3.v.a 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.v.a 64 3.b odd 2 1
105.3.v.a 64 15.e even 4 1
105.3.v.a 64 21.h odd 6 1
105.3.v.a 64 105.x even 12 1
315.3.ca.b 64 1.a even 1 1 trivial
315.3.ca.b 64 5.c odd 4 1 inner
315.3.ca.b 64 7.c even 3 1 inner
315.3.ca.b 64 35.l odd 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{64} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(315, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database