Properties

 Label 1521.2.b.e Level $1521$ Weight $2$ Character orbit 1521.b Analytic conductor $12.145$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1521.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$12.1452461474$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 117) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + 2 q^{4} + i q^{7}+O(q^{10})$$ q + 2 * q^4 + i * q^7 $$q + 2 q^{4} + i q^{7} + 4 q^{16} + 8 i q^{19} + 5 q^{25} + 2 i q^{28} - 7 i q^{31} + 10 i q^{37} + 13 q^{43} + 6 q^{49} - 13 q^{61} + 8 q^{64} + 11 i q^{67} - 17 i q^{73} + 16 i q^{76} - 13 q^{79} + 5 i q^{97} +O(q^{100})$$ q + 2 * q^4 + i * q^7 + 4 * q^16 + 8*i * q^19 + 5 * q^25 + 2*i * q^28 - 7*i * q^31 + 10*i * q^37 + 13 * q^43 + 6 * q^49 - 13 * q^61 + 8 * q^64 + 11*i * q^67 - 17*i * q^73 + 16*i * q^76 - 13 * q^79 + 5*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4}+O(q^{10})$$ 2 * q + 4 * q^4 $$2 q + 4 q^{4} + 8 q^{16} + 10 q^{25} + 26 q^{43} + 12 q^{49} - 26 q^{61} + 16 q^{64} - 26 q^{79}+O(q^{100})$$ 2 * q + 4 * q^4 + 8 * q^16 + 10 * q^25 + 26 * q^43 + 12 * q^49 - 26 * q^61 + 16 * q^64 - 26 * q^79

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1351.1
 − 1.00000i 1.00000i
0 0 2.00000 0 0 1.00000i 0 0 0
1351.2 0 0 2.00000 0 0 1.00000i 0 0 0
 $$n$$: 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
13.b even 2 1 inner
39.d odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.2.b.e 2
3.b odd 2 1 CM 1521.2.b.e 2
13.b even 2 1 inner 1521.2.b.e 2
13.d odd 4 1 1521.2.a.b 1
13.d odd 4 1 1521.2.a.c 1
13.f odd 12 2 117.2.g.a 2
39.d odd 2 1 inner 1521.2.b.e 2
39.f even 4 1 1521.2.a.b 1
39.f even 4 1 1521.2.a.c 1
39.k even 12 2 117.2.g.a 2
52.l even 12 2 1872.2.t.g 2
156.v odd 12 2 1872.2.t.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.g.a 2 13.f odd 12 2
117.2.g.a 2 39.k even 12 2
1521.2.a.b 1 13.d odd 4 1
1521.2.a.b 1 39.f even 4 1
1521.2.a.c 1 13.d odd 4 1
1521.2.a.c 1 39.f even 4 1
1521.2.b.e 2 1.a even 1 1 trivial
1521.2.b.e 2 3.b odd 2 1 CM
1521.2.b.e 2 13.b even 2 1 inner
1521.2.b.e 2 39.d odd 2 1 inner
1872.2.t.g 2 52.l even 12 2
1872.2.t.g 2 156.v odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1521, [\chi])$$:

 $$T_{2}$$ T2 $$T_{5}$$ T5 $$T_{7}^{2} + 1$$ T7^2 + 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 64$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 49$$
$37$ $$T^{2} + 100$$
$41$ $$T^{2}$$
$43$ $$(T - 13)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 13)^{2}$$
$67$ $$T^{2} + 121$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 289$$
$79$ $$(T + 13)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 25$$