# Properties

 Label 315.3.o.b Level 315 Weight 3 Character orbit 315.o Analytic conductor 8.583 Analytic rank 0 Dimension 24 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$8.58312832735$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 8q^{2} - 16q^{5} + 48q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 8q^{2} - 16q^{5} + 48q^{8} - 40q^{10} + 64q^{13} - 184q^{16} - 24q^{17} - 72q^{20} + 8q^{22} - 8q^{23} - 136q^{25} + 80q^{26} + 96q^{31} - 56q^{32} + 8q^{37} - 56q^{38} + 232q^{40} - 320q^{41} - 112q^{43} + 320q^{46} - 64q^{47} + 256q^{50} + 96q^{52} + 72q^{53} - 80q^{55} + 336q^{56} - 512q^{58} - 496q^{61} + 776q^{62} - 312q^{65} - 192q^{67} - 568q^{68} + 112q^{70} + 144q^{71} + 224q^{73} + 416q^{76} - 112q^{77} + 528q^{80} + 352q^{82} + 32q^{83} + 24q^{85} - 240q^{86} + 216q^{88} - 1304q^{92} - 376q^{95} - 816q^{97} + 56q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1 −2.74240 2.74240i 0 11.0415i 0.683416 + 4.95307i 0 1.87083 + 1.87083i 19.3105 19.3105i 0 11.7091 15.4575i
127.2 −2.41688 2.41688i 0 7.68258i −4.18124 + 2.74175i 0 −1.87083 1.87083i 8.90034 8.90034i 0 16.7320 + 3.47908i
127.3 −2.24469 2.24469i 0 6.07726i 3.05058 3.96156i 0 1.87083 + 1.87083i 4.66280 4.66280i 0 −15.7401 + 2.04487i
127.4 −2.08980 2.08980i 0 4.73454i −0.137153 4.99812i 0 −1.87083 1.87083i 1.53505 1.53505i 0 −10.1585 + 10.7317i
127.5 −0.992944 0.992944i 0 2.02813i 2.01954 + 4.57400i 0 1.87083 + 1.87083i −5.98559 + 5.98559i 0 2.53644 6.54701i
127.6 −0.675544 0.675544i 0 3.08728i −3.39488 3.67080i 0 −1.87083 1.87083i −4.78777 + 4.78777i 0 −0.186396 + 4.77318i
127.7 −0.408558 0.408558i 0 3.66616i −0.563288 + 4.96817i 0 −1.87083 1.87083i −3.13207 + 3.13207i 0 2.25992 1.79965i
127.8 0.867675 + 0.867675i 0 2.49428i 4.93004 + 0.833478i 0 −1.87083 1.87083i 5.63493 5.63493i 0 3.55449 + 5.00086i
127.9 1.01289 + 1.01289i 0 1.94811i −3.97454 + 3.03365i 0 1.87083 + 1.87083i 6.02478 6.02478i 0 −7.09851 0.953024i
127.10 1.36784 + 1.36784i 0 0.258033i −3.39663 3.66919i 0 1.87083 + 1.87083i 5.82430 5.82430i 0 0.372817 9.66489i
127.11 1.59930 + 1.59930i 0 1.11554i 1.35929 + 4.81169i 0 1.87083 + 1.87083i 4.61313 4.61313i 0 −5.52142 + 9.86926i
127.12 2.72310 + 2.72310i 0 10.8306i −4.39513 + 2.38387i 0 −1.87083 1.87083i −18.6004 + 18.6004i 0 −18.4599 5.47689i
253.1 −2.74240 + 2.74240i 0 11.0415i 0.683416 4.95307i 0 1.87083 1.87083i 19.3105 + 19.3105i 0 11.7091 + 15.4575i
253.2 −2.41688 + 2.41688i 0 7.68258i −4.18124 2.74175i 0 −1.87083 + 1.87083i 8.90034 + 8.90034i 0 16.7320 3.47908i
253.3 −2.24469 + 2.24469i 0 6.07726i 3.05058 + 3.96156i 0 1.87083 1.87083i 4.66280 + 4.66280i 0 −15.7401 2.04487i
253.4 −2.08980 + 2.08980i 0 4.73454i −0.137153 + 4.99812i 0 −1.87083 + 1.87083i 1.53505 + 1.53505i 0 −10.1585 10.7317i
253.5 −0.992944 + 0.992944i 0 2.02813i 2.01954 4.57400i 0 1.87083 1.87083i −5.98559 5.98559i 0 2.53644 + 6.54701i
253.6 −0.675544 + 0.675544i 0 3.08728i −3.39488 + 3.67080i 0 −1.87083 + 1.87083i −4.78777 4.78777i 0 −0.186396 4.77318i
253.7 −0.408558 + 0.408558i 0 3.66616i −0.563288 4.96817i 0 −1.87083 + 1.87083i −3.13207 3.13207i 0 2.25992 + 1.79965i
253.8 0.867675 0.867675i 0 2.49428i 4.93004 0.833478i 0 −1.87083 + 1.87083i 5.63493 + 5.63493i 0 3.55449 5.00086i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 253.12 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.3.o.b 24
3.b odd 2 1 105.3.l.a 24
5.c odd 4 1 inner 315.3.o.b 24
15.d odd 2 1 525.3.l.e 24
15.e even 4 1 105.3.l.a 24
15.e even 4 1 525.3.l.e 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.l.a 24 3.b odd 2 1
105.3.l.a 24 15.e even 4 1
315.3.o.b 24 1.a even 1 1 trivial
315.3.o.b 24 5.c odd 4 1 inner
525.3.l.e 24 15.d odd 2 1
525.3.l.e 24 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(315, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database