Properties

 Label 1512.2.c.g 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 6q^{4} + 24q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 6q^{4} + 24q^{7} - 16q^{10} + 2q^{16} + 16q^{22} - 24q^{25} + 6q^{28} + 8q^{31} + 22q^{34} + 26q^{46} + 24q^{49} - 6q^{52} + 16q^{55} - 58q^{58} + 6q^{64} - 16q^{70} + 60q^{76} + 8q^{79} - 28q^{82} + 12q^{88} + 36q^{94} + O(q^{100})$$

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 −1.40920 0.118957i 0 1.97170 + 0.335269i 3.46300i 0 1.00000 −2.73864 0.707009i 0 −0.411948 + 4.88007i
757.2 −1.40920 + 0.118957i 0 1.97170 0.335269i 3.46300i 0 1.00000 −2.73864 + 0.707009i 0 −0.411948 4.88007i
757.3 −1.35918 0.390688i 0 1.69473 + 1.06203i 1.82986i 0 1.00000 −1.88851 2.10559i 0 0.714906 2.48711i
757.4 −1.35918 + 0.390688i 0 1.69473 1.06203i 1.82986i 0 1.00000 −1.88851 + 2.10559i 0 0.714906 + 2.48711i
757.5 −1.13123 0.848721i 0 0.559347 + 1.92019i 0.940450i 0 1.00000 0.996957 2.64690i 0 0.798179 1.06386i
757.6 −1.13123 + 0.848721i 0 0.559347 1.92019i 0.940450i 0 1.00000 0.996957 + 2.64690i 0 0.798179 + 1.06386i
757.7 −1.10240 0.885841i 0 0.430570 + 1.95310i 2.99001i 0 1.00000 1.25548 2.53452i 0 −2.64867 + 3.29619i
757.8 −1.10240 + 0.885841i 0 0.430570 1.95310i 2.99001i 0 1.00000 1.25548 + 2.53452i 0 −2.64867 3.29619i
757.9 −0.497658 1.32376i 0 −1.50467 + 1.31756i 1.25392i 0 1.00000 2.49294 + 1.33613i 0 1.65989 0.624024i
757.10 −0.497658 + 1.32376i 0 −1.50467 1.31756i 1.25392i 0 1.00000 2.49294 1.33613i 0 1.65989 + 0.624024i
757.11 −0.417332 1.35123i 0 −1.65167 + 1.12783i 3.04340i 0 1.00000 2.21325 + 1.76111i 0 −4.11235 + 1.27011i
757.12 −0.417332 + 1.35123i 0 −1.65167 1.12783i 3.04340i 0 1.00000 2.21325 1.76111i 0 −4.11235 1.27011i
757.13 0.417332 1.35123i 0 −1.65167 1.12783i 3.04340i 0 1.00000 −2.21325 + 1.76111i 0 −4.11235 1.27011i
757.14 0.417332 + 1.35123i 0 −1.65167 + 1.12783i 3.04340i 0 1.00000 −2.21325 1.76111i 0 −4.11235 + 1.27011i
757.15 0.497658 1.32376i 0 −1.50467 1.31756i 1.25392i 0 1.00000 −2.49294 + 1.33613i 0 1.65989 + 0.624024i
757.16 0.497658 + 1.32376i 0 −1.50467 + 1.31756i 1.25392i 0 1.00000 −2.49294 1.33613i 0 1.65989 0.624024i
757.17 1.10240 0.885841i 0 0.430570 1.95310i 2.99001i 0 1.00000 −1.25548 2.53452i 0 −2.64867 3.29619i
757.18 1.10240 + 0.885841i 0 0.430570 + 1.95310i 2.99001i 0 1.00000 −1.25548 + 2.53452i 0 −2.64867 + 3.29619i
757.19 1.13123 0.848721i 0 0.559347 1.92019i 0.940450i 0 1.00000 −0.996957 2.64690i 0 0.798179 + 1.06386i
757.20 1.13123 + 0.848721i 0 0.559347 + 1.92019i 0.940450i 0 1.00000 −0.996957 + 2.64690i 0 0.798179 1.06386i
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.g 24
3.b odd 2 1 inner 1512.2.c.g 24
4.b odd 2 1 6048.2.c.f 24
8.b even 2 1 inner 1512.2.c.g 24
8.d odd 2 1 6048.2.c.f 24
12.b even 2 1 6048.2.c.f 24
24.f even 2 1 6048.2.c.f 24
24.h odd 2 1 inner 1512.2.c.g 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.c.g 24 1.a even 1 1 trivial
1512.2.c.g 24 3.b odd 2 1 inner
1512.2.c.g 24 8.b even 2 1 inner
1512.2.c.g 24 24.h odd 2 1 inner
6048.2.c.f 24 4.b odd 2 1
6048.2.c.f 24 8.d odd 2 1
6048.2.c.f 24 12.b even 2 1
6048.2.c.f 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 $$S_{2}^{\mathrm{new}}(1512, [\chi])$$:

 $$T_{5}^{12} + 36 T_{5}^{10} + 486 T_{5}^{8} + 3036 T_{5}^{6} + 8801 T_{5}^{4} + 10952 T_{5}^{2} + 4624$$ $$T_{17}^{12} - 100 T_{17}^{10} + 3253 T_{17}^{8} - 41296 T_{17}^{6} + 197515 T_{17}^{4} - 252108 T_{17}^{2} + 63$$

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database