Properties

 Label 1512.2.c.a Level 1512 Weight 2 Character orbit 1512.c Analytic conductor 12.073 Analytic rank 0 Dimension 2 CM no Inner twists 2

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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - i ) q^{2} + 2 i q^{4} + 2 i q^{5} - q^{7} + ( 2 - 2 i ) q^{8} +O(q^{10})$$ $$q + ( -1 - i ) q^{2} + 2 i q^{4} + 2 i q^{5} - q^{7} + ( 2 - 2 i ) q^{8} + ( 2 - 2 i ) q^{10} + 5 i q^{13} + ( 1 + i ) q^{14} -4 q^{16} + q^{17} + 4 i q^{19} -4 q^{20} -5 q^{23} + q^{25} + ( 5 - 5 i ) q^{26} -2 i q^{28} -9 i q^{29} -7 q^{31} + ( 4 + 4 i ) q^{32} + ( -1 - i ) q^{34} -2 i q^{35} + 2 i q^{37} + ( 4 - 4 i ) q^{38} + ( 4 + 4 i ) q^{40} + 2 q^{41} + 5 i q^{43} + ( 5 + 5 i ) q^{46} + q^{49} + ( -1 - i ) q^{50} -10 q^{52} -9 i q^{53} + ( -2 + 2 i ) q^{56} + ( -9 + 9 i ) q^{58} + i q^{59} + 6 i q^{61} + ( 7 + 7 i ) q^{62} -8 i q^{64} -10 q^{65} -9 i q^{67} + 2 i q^{68} + ( -2 + 2 i ) q^{70} -15 q^{71} + ( 2 - 2 i ) q^{74} -8 q^{76} -14 q^{79} -8 i q^{80} + ( -2 - 2 i ) q^{82} + 4 i q^{83} + 2 i q^{85} + ( 5 - 5 i ) q^{86} -13 q^{89} -5 i q^{91} -10 i q^{92} -8 q^{95} -14 q^{97} + ( -1 - i ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{7} + 4q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{7} + 4q^{8} + 4q^{10} + 2q^{14} - 8q^{16} + 2q^{17} - 8q^{20} - 10q^{23} + 2q^{25} + 10q^{26} - 14q^{31} + 8q^{32} - 2q^{34} + 8q^{38} + 8q^{40} + 4q^{41} + 10q^{46} + 2q^{49} - 2q^{50} - 20q^{52} - 4q^{56} - 18q^{58} + 14q^{62} - 20q^{65} - 4q^{70} - 30q^{71} + 4q^{74} - 16q^{76} - 28q^{79} - 4q^{82} + 10q^{86} - 26q^{89} - 16q^{95} - 28q^{97} - 2q^{98} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$$.

 $$n$$ $$757$$ $$785$$ $$1081$$ $$1135$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
757.1
 1.00000i − 1.00000i
−1.00000 1.00000i 0 2.00000i 2.00000i 0 −1.00000 2.00000 2.00000i 0 2.00000 2.00000i
757.2 −1.00000 + 1.00000i 0 2.00000i 2.00000i 0 −1.00000 2.00000 + 2.00000i 0 2.00000 + 2.00000i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b 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.c.a 2
3.b odd 2 1 1512.2.c.b yes 2
4.b odd 2 1 6048.2.c.b 2
8.b even 2 1 inner 1512.2.c.a 2
8.d odd 2 1 6048.2.c.b 2
12.b even 2 1 6048.2.c.a 2
24.f even 2 1 6048.2.c.a 2
24.h odd 2 1 1512.2.c.b yes 2

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

Hecke kernels

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

 $$T_{5}^{2} + 4$$ $$T_{17} - 1$$

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2}$$
$3$ 
$5$ $$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( 1 - 11 T^{2} )^{2}$$
$13$ $$1 - T^{2} + 169 T^{4}$$
$17$ $$( 1 - T + 17 T^{2} )^{2}$$
$19$ $$1 - 22 T^{2} + 361 T^{4}$$
$23$ $$( 1 + 5 T + 23 T^{2} )^{2}$$
$29$ $$1 + 23 T^{2} + 841 T^{4}$$
$31$ $$( 1 + 7 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 - 2 T + 41 T^{2} )^{2}$$
$43$ $$1 - 61 T^{2} + 1849 T^{4}$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$1 - 25 T^{2} + 2809 T^{4}$$
$59$ $$1 - 117 T^{2} + 3481 T^{4}$$
$61$ $$1 - 86 T^{2} + 3721 T^{4}$$
$67$ $$1 - 53 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 15 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 73 T^{2} )^{2}$$
$79$ $$( 1 + 14 T + 79 T^{2} )^{2}$$
$83$ $$1 - 150 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 13 T + 89 T^{2} )^{2}$$
$97$ $$( 1 + 14 T + 97 T^{2} )^{2}$$