# Properties

 Label 1512.2.j.c Level 1512 Weight 2 Character orbit 1512.j Analytic conductor 12.073 Analytic rank 0 Dimension 32 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: $$32$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 8q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 8q^{4} - 8q^{10} - 8q^{16} - 64q^{19} + 24q^{22} - 16q^{25} - 8q^{28} + 8q^{34} - 24q^{40} - 48q^{43} - 8q^{46} - 32q^{49} - 24q^{52} - 96q^{58} - 40q^{64} + 16q^{67} - 16q^{70} - 16q^{73} + 16q^{76} + 24q^{82} + 72q^{88} + 16q^{91} - 56q^{94} + 64q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
323.1 −1.40936 0.117111i 0 1.97257 + 0.330101i 1.17792 0 1.00000i −2.74140 0.696239i 0 −1.66010 0.137947i
323.2 −1.40936 + 0.117111i 0 1.97257 0.330101i 1.17792 0 1.00000i −2.74140 + 0.696239i 0 −1.66010 + 0.137947i
323.3 −1.33436 0.468477i 0 1.56106 + 1.25024i −3.47003 0 1.00000i −1.49731 2.39959i 0 4.63028 + 1.62563i
323.4 −1.33436 + 0.468477i 0 1.56106 1.25024i −3.47003 0 1.00000i −1.49731 + 2.39959i 0 4.63028 1.62563i
323.5 −1.30241 0.551119i 0 1.39253 + 1.43556i 3.61017 0 1.00000i −1.02248 2.63714i 0 −4.70192 1.98963i
323.6 −1.30241 + 0.551119i 0 1.39253 1.43556i 3.61017 0 1.00000i −1.02248 + 2.63714i 0 −4.70192 + 1.98963i
323.7 −1.16953 0.795104i 0 0.735620 + 1.85980i 2.09311 0 1.00000i 0.618402 2.76000i 0 −2.44797 1.66424i
323.8 −1.16953 + 0.795104i 0 0.735620 1.85980i 2.09311 0 1.00000i 0.618402 + 2.76000i 0 −2.44797 + 1.66424i
323.9 −0.985454 1.01434i 0 −0.0577616 + 1.99917i 0.206991 0 1.00000i 2.08475 1.91150i 0 −0.203980 0.209959i
323.10 −0.985454 + 1.01434i 0 −0.0577616 1.99917i 0.206991 0 1.00000i 2.08475 + 1.91150i 0 −0.203980 + 0.209959i
323.11 −0.971008 1.02818i 0 −0.114288 + 1.99673i −2.21305 0 1.00000i 2.16396 1.82133i 0 2.14889 + 2.27540i
323.12 −0.971008 + 1.02818i 0 −0.114288 1.99673i −2.21305 0 1.00000i 2.16396 + 1.82133i 0 2.14889 2.27540i
323.13 −0.481042 1.32989i 0 −1.53720 + 1.27946i −0.436221 0 1.00000i 2.44100 + 1.42882i 0 0.209841 + 0.580125i
323.14 −0.481042 + 1.32989i 0 −1.53720 1.27946i −0.436221 0 1.00000i 2.44100 1.42882i 0 0.209841 0.580125i
323.15 −0.154052 1.40580i 0 −1.95254 + 0.433131i −0.162025 0 1.00000i 0.909686 + 2.67815i 0 0.0249602 + 0.227774i
323.16 −0.154052 + 1.40580i 0 −1.95254 0.433131i −0.162025 0 1.00000i 0.909686 2.67815i 0 0.0249602 0.227774i
323.17 0.154052 1.40580i 0 −1.95254 0.433131i 0.162025 0 1.00000i −0.909686 + 2.67815i 0 0.0249602 0.227774i
323.18 0.154052 + 1.40580i 0 −1.95254 + 0.433131i 0.162025 0 1.00000i −0.909686 2.67815i 0 0.0249602 + 0.227774i
323.19 0.481042 1.32989i 0 −1.53720 1.27946i 0.436221 0 1.00000i −2.44100 + 1.42882i 0 0.209841 0.580125i
323.20 0.481042 + 1.32989i 0 −1.53720 + 1.27946i 0.436221 0 1.00000i −2.44100 1.42882i 0 0.209841 + 0.580125i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 323.32 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.c 32
3.b odd 2 1 inner 1512.2.j.c 32
4.b odd 2 1 6048.2.j.c 32
8.b even 2 1 6048.2.j.c 32
8.d odd 2 1 inner 1512.2.j.c 32
12.b even 2 1 6048.2.j.c 32
24.f even 2 1 inner 1512.2.j.c 32
24.h odd 2 1 6048.2.j.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.j.c 32 1.a even 1 1 trivial
1512.2.j.c 32 3.b odd 2 1 inner
1512.2.j.c 32 8.d odd 2 1 inner
1512.2.j.c 32 24.f even 2 1 inner
6048.2.j.c 32 4.b odd 2 1
6048.2.j.c 32 8.b even 2 1
6048.2.j.c 32 12.b even 2 1
6048.2.j.c 32 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1512, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database