# Properties

 Label 1512.2.c.f Level 1512 Weight 2 Character orbit 1512.c Analytic conductor 12.073 Analytic rank 0 Dimension 24 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.0733807856$$ Analytic rank: $$0$$ Dimension: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 24q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 24q^{7} + 20q^{10} - 4q^{16} + 4q^{22} - 24q^{25} - 16q^{31} + 4q^{34} + 12q^{40} - 52q^{46} + 24q^{49} + 12q^{52} - 8q^{55} - 28q^{58} + 24q^{64} + 20q^{70} - 24q^{76} + 32q^{79} + 44q^{82} - 60q^{88} + 12q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
757.1 −1.41290 0.0608900i 0 1.99258 + 0.172063i 3.11390i 0 1.00000 −2.80485 0.364437i 0 −0.189605 + 4.39964i
757.2 −1.41290 + 0.0608900i 0 1.99258 0.172063i 3.11390i 0 1.00000 −2.80485 + 0.364437i 0 −0.189605 4.39964i
757.3 −1.29659 0.564669i 0 1.36230 + 1.46429i 1.53368i 0 1.00000 −0.939505 2.66783i 0 0.866019 1.98855i
757.4 −1.29659 + 0.564669i 0 1.36230 1.46429i 1.53368i 0 1.00000 −0.939505 + 2.66783i 0 0.866019 + 1.98855i
757.5 −1.09864 0.890504i 0 0.414007 + 1.95668i 1.58470i 0 1.00000 1.28759 2.51836i 0 −1.41118 + 1.74101i
757.6 −1.09864 + 0.890504i 0 0.414007 1.95668i 1.58470i 0 1.00000 1.28759 + 2.51836i 0 −1.41118 1.74101i
757.7 −0.796644 1.16849i 0 −0.730716 + 1.86173i 2.52623i 0 1.00000 2.75753 0.629309i 0 2.95186 2.01251i
757.8 −0.796644 + 1.16849i 0 −0.730716 1.86173i 2.52623i 0 1.00000 2.75753 + 0.629309i 0 2.95186 + 2.01251i
757.9 −0.671239 1.24476i 0 −1.09888 + 1.67107i 3.66698i 0 1.00000 2.81770 + 0.246156i 0 4.56453 2.46142i
757.10 −0.671239 + 1.24476i 0 −1.09888 1.67107i 3.66698i 0 1.00000 2.81770 0.246156i 0 4.56453 + 2.46142i
757.11 −0.174217 1.40344i 0 −1.93930 + 0.489007i 1.26947i 0 1.00000 1.02415 + 2.63650i 0 −1.78162 + 0.221163i
757.12 −0.174217 + 1.40344i 0 −1.93930 0.489007i 1.26947i 0 1.00000 1.02415 2.63650i 0 −1.78162 0.221163i
757.13 0.174217 1.40344i 0 −1.93930 0.489007i 1.26947i 0 1.00000 −1.02415 + 2.63650i 0 −1.78162 0.221163i
757.14 0.174217 + 1.40344i 0 −1.93930 + 0.489007i 1.26947i 0 1.00000 −1.02415 2.63650i 0 −1.78162 + 0.221163i
757.15 0.671239 1.24476i 0 −1.09888 1.67107i 3.66698i 0 1.00000 −2.81770 + 0.246156i 0 4.56453 + 2.46142i
757.16 0.671239 + 1.24476i 0 −1.09888 + 1.67107i 3.66698i 0 1.00000 −2.81770 0.246156i 0 4.56453 2.46142i
757.17 0.796644 1.16849i 0 −0.730716 1.86173i 2.52623i 0 1.00000 −2.75753 0.629309i 0 2.95186 + 2.01251i
757.18 0.796644 + 1.16849i 0 −0.730716 + 1.86173i 2.52623i 0 1.00000 −2.75753 + 0.629309i 0 2.95186 2.01251i
757.19 1.09864 0.890504i 0 0.414007 1.95668i 1.58470i 0 1.00000 −1.28759 2.51836i 0 −1.41118 1.74101i
757.20 1.09864 + 0.890504i 0 0.414007 + 1.95668i 1.58470i 0 1.00000 −1.28759 + 2.51836i 0 −1.41118 + 1.74101i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 757.24 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.b even 2 1 inner
24.h odd 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1512, [\chi])$$:

 $$T_{5}^{12} + 36 T_{5}^{10} + 483 T_{5}^{8} + 3048 T_{5}^{6} + 9491 T_{5}^{4} + 14084 T_{5}^{2} + 7921$$ $$T_{17}^{12} - 124 T_{17}^{10} + 5788 T_{17}^{8} - 128800 T_{17}^{6} + 1407472 T_{17}^{4} - 6700992 T_{17}^{2} + 8156736$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database