# Properties

 Label 1512.2.j.d Level 1512 Weight 2 Character orbit 1512.j Analytic conductor 12.073 Analytic rank 0 Dimension 48 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1512 = 2^{3} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1512.j (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.0733807856$$ Analytic rank: $$0$$ Dimension: $$48$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 4q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 4q^{4} - 12q^{10} - 12q^{16} + 16q^{19} - 24q^{22} + 48q^{25} + 8q^{28} + 12q^{34} + 32q^{40} + 64q^{43} + 60q^{46} - 48q^{49} + 16q^{52} + 36q^{58} - 4q^{64} + 32q^{67} + 12q^{70} - 32q^{76} + 36q^{82} + 24q^{88} - 60q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
323.1 −1.41047 0.102782i 0 1.97887 + 0.289943i −2.90802 0 1.00000i −2.76135 0.612349i 0 4.10168 + 0.298892i
323.2 −1.41047 + 0.102782i 0 1.97887 0.289943i −2.90802 0 1.00000i −2.76135 + 0.612349i 0 4.10168 0.298892i
323.3 −1.37397 0.334962i 0 1.77560 + 0.920458i 4.18263 0 1.00000i −2.13131 1.85944i 0 −5.74682 1.40102i
323.4 −1.37397 + 0.334962i 0 1.77560 0.920458i 4.18263 0 1.00000i −2.13131 + 1.85944i 0 −5.74682 + 1.40102i
323.5 −1.29821 0.560940i 0 1.37069 + 1.45643i 0.530075 0 1.00000i −0.962474 2.65963i 0 −0.688148 0.297340i
323.6 −1.29821 + 0.560940i 0 1.37069 1.45643i 0.530075 0 1.00000i −0.962474 + 2.65963i 0 −0.688148 + 0.297340i
323.7 −1.25654 0.648937i 0 1.15776 + 1.63082i −1.27600 0 1.00000i −0.396469 2.80050i 0 1.60334 + 0.828043i
323.8 −1.25654 + 0.648937i 0 1.15776 1.63082i −1.27600 0 1.00000i −0.396469 + 2.80050i 0 1.60334 0.828043i
323.9 −1.06881 0.926094i 0 0.284699 + 1.97963i −2.85878 0 1.00000i 1.52904 2.37950i 0 3.05549 + 2.64750i
323.10 −1.06881 + 0.926094i 0 0.284699 1.97963i −2.85878 0 1.00000i 1.52904 + 2.37950i 0 3.05549 2.64750i
323.11 −1.04493 0.952952i 0 0.183763 + 1.99154i 2.35025 0 1.00000i 1.70582 2.25614i 0 −2.45584 2.23967i
323.12 −1.04493 + 0.952952i 0 0.183763 1.99154i 2.35025 0 1.00000i 1.70582 + 2.25614i 0 −2.45584 + 2.23967i
323.13 −0.908924 1.08345i 0 −0.347713 + 1.96954i 0.863507 0 1.00000i 2.44994 1.41344i 0 −0.784863 0.935564i
323.14 −0.908924 + 1.08345i 0 −0.347713 1.96954i 0.863507 0 1.00000i 2.44994 + 1.41344i 0 −0.784863 + 0.935564i
323.15 −0.743762 1.20284i 0 −0.893635 + 1.78925i 3.69622 0 1.00000i 2.81683 0.255879i 0 −2.74911 4.44595i
323.16 −0.743762 + 1.20284i 0 −0.893635 1.78925i 3.69622 0 1.00000i 2.81683 + 0.255879i 0 −2.74911 + 4.44595i
323.17 −0.682616 1.23856i 0 −1.06807 + 1.69092i 0.381416 0 1.00000i 2.82340 + 0.168622i 0 −0.260361 0.472408i
323.18 −0.682616 + 1.23856i 0 −1.06807 1.69092i 0.381416 0 1.00000i 2.82340 0.168622i 0 −0.260361 + 0.472408i
323.19 −0.508876 1.31949i 0 −1.48209 + 1.34291i −2.29653 0 1.00000i 2.52615 + 1.27222i 0 1.16865 + 3.03024i
323.20 −0.508876 + 1.31949i 0 −1.48209 1.34291i −2.29653 0 1.00000i 2.52615 1.27222i 0 1.16865 3.03024i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 323.48 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
8.d odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.j.d 48
3.b odd 2 1 inner 1512.2.j.d 48
4.b odd 2 1 6048.2.j.d 48
8.b even 2 1 6048.2.j.d 48
8.d odd 2 1 inner 1512.2.j.d 48
12.b even 2 1 6048.2.j.d 48
24.f even 2 1 inner 1512.2.j.d 48
24.h odd 2 1 6048.2.j.d 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.j.d 48 1.a even 1 1 trivial
1512.2.j.d 48 3.b odd 2 1 inner
1512.2.j.d 48 8.d odd 2 1 inner
1512.2.j.d 48 24.f even 2 1 inner
6048.2.j.d 48 4.b odd 2 1
6048.2.j.d 48 8.b even 2 1
6048.2.j.d 48 12.b even 2 1
6048.2.j.d 48 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{24} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1512, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database