# Properties

 Label 315.2.ce.a Level 315 Weight 2 Character orbit 315.ce Analytic conductor 2.515 Analytic rank 0 Dimension 64 CM no Inner twists 8

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q + 8q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q + 8q^{7} + 8q^{10} + 32q^{16} - 48q^{22} - 16q^{25} + 88q^{28} + 32q^{31} - 16q^{37} - 40q^{40} - 16q^{43} - 80q^{52} - 32q^{55} - 88q^{58} + 48q^{61} - 32q^{67} - 112q^{70} - 88q^{73} - 320q^{76} - 56q^{82} + 16q^{85} + 120q^{88} - 128q^{91} + 208q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1 −0.695956 + 2.59734i 0 −4.52979 2.61528i 1.27254 1.83865i 0 −1.50727 2.17442i 6.14253 6.14253i 0 3.88997 + 4.58485i
53.2 −0.604483 + 2.25596i 0 −2.99191 1.72738i −0.254820 + 2.22150i 0 −2.19240 + 1.48100i 2.40251 2.40251i 0 −4.85759 1.91772i
53.3 −0.543545 + 2.02854i 0 −2.08747 1.20520i −1.61611 1.54538i 0 1.07923 2.41563i 0.609443 0.609443i 0 4.01329 2.43835i
53.4 −0.494677 + 1.84616i 0 −1.43155 0.826508i 0.541988 + 2.16939i 0 2.64533 + 0.0471386i −0.468944 + 0.468944i 0 −4.27315 0.0725500i
53.5 −0.364480 + 1.36026i 0 0.0145936 + 0.00842564i −2.23601 0.0163814i 0 −2.58285 0.573503i −2.00834 + 2.00834i 0 0.837263 3.03558i
53.6 −0.271660 + 1.01385i 0 0.777962 + 0.449157i 0.310035 2.21447i 0 −1.26800 + 2.32211i −2.15109 + 2.15109i 0 2.16091 + 0.915911i
53.7 −0.0991680 + 0.370100i 0 1.60491 + 0.926596i 1.78935 + 1.34098i 0 2.39134 + 1.13202i −1.04395 + 1.04395i 0 −0.673742 + 0.529258i
53.8 −0.0362581 + 0.135317i 0 1.71505 + 0.990187i −1.87468 + 1.21884i 0 0.702570 2.55076i −0.394292 + 0.394292i 0 −0.0969571 0.297870i
53.9 0.0362581 0.135317i 0 1.71505 + 0.990187i 1.87468 1.21884i 0 0.702570 2.55076i 0.394292 0.394292i 0 −0.0969571 0.297870i
53.10 0.0991680 0.370100i 0 1.60491 + 0.926596i −1.78935 1.34098i 0 2.39134 + 1.13202i 1.04395 1.04395i 0 −0.673742 + 0.529258i
53.11 0.271660 1.01385i 0 0.777962 + 0.449157i −0.310035 + 2.21447i 0 −1.26800 + 2.32211i 2.15109 2.15109i 0 2.16091 + 0.915911i
53.12 0.364480 1.36026i 0 0.0145936 + 0.00842564i 2.23601 + 0.0163814i 0 −2.58285 0.573503i 2.00834 2.00834i 0 0.837263 3.03558i
53.13 0.494677 1.84616i 0 −1.43155 0.826508i −0.541988 2.16939i 0 2.64533 + 0.0471386i 0.468944 0.468944i 0 −4.27315 0.0725500i
53.14 0.543545 2.02854i 0 −2.08747 1.20520i 1.61611 + 1.54538i 0 1.07923 2.41563i −0.609443 + 0.609443i 0 4.01329 2.43835i
53.15 0.604483 2.25596i 0 −2.99191 1.72738i 0.254820 2.22150i 0 −2.19240 + 1.48100i −2.40251 + 2.40251i 0 −4.85759 1.91772i
53.16 0.695956 2.59734i 0 −4.52979 2.61528i −1.27254 + 1.83865i 0 −1.50727 2.17442i −6.14253 + 6.14253i 0 3.88997 + 4.58485i
107.1 −0.695956 2.59734i 0 −4.52979 + 2.61528i 1.27254 + 1.83865i 0 −1.50727 + 2.17442i 6.14253 + 6.14253i 0 3.88997 4.58485i
107.2 −0.604483 2.25596i 0 −2.99191 + 1.72738i −0.254820 2.22150i 0 −2.19240 1.48100i 2.40251 + 2.40251i 0 −4.85759 + 1.91772i
107.3 −0.543545 2.02854i 0 −2.08747 + 1.20520i −1.61611 + 1.54538i 0 1.07923 + 2.41563i 0.609443 + 0.609443i 0 4.01329 + 2.43835i
107.4 −0.494677 1.84616i 0 −1.43155 + 0.826508i 0.541988 2.16939i 0 2.64533 0.0471386i −0.468944 0.468944i 0 −4.27315 + 0.0725500i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 242.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.c even 3 1 inner
15.e even 4 1 inner
21.h odd 6 1 inner
35.l odd 12 1 inner
105.x 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.ce.a 64
3.b odd 2 1 inner 315.2.ce.a 64
5.c odd 4 1 inner 315.2.ce.a 64
7.c even 3 1 inner 315.2.ce.a 64
15.e even 4 1 inner 315.2.ce.a 64
21.h odd 6 1 inner 315.2.ce.a 64
35.l odd 12 1 inner 315.2.ce.a 64
105.x even 12 1 inner 315.2.ce.a 64

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database