# Properties

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

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## 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: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{4} + 3 \beta q^{7}+O(q^{10})$$ q + 2 * q^4 + 3*b * q^7 $$q + 2 q^{4} + 3 \beta q^{7} + 4 q^{16} - 2 \beta q^{19} + 5 q^{25} + 6 \beta q^{28} + 5 \beta q^{31} + 4 \beta q^{37} - 13 q^{43} - 20 q^{49} + 13 q^{61} + 8 q^{64} + 7 \beta q^{67} + \beta q^{73} - 4 \beta q^{76} + 13 q^{79} - 11 \beta q^{97} +O(q^{100})$$ q + 2 * q^4 + 3*b * q^7 + 4 * q^16 - 2*b * q^19 + 5 * q^25 + 6*b * q^28 + 5*b * q^31 + 4*b * q^37 - 13 * q^43 - 20 * q^49 + 13 * q^61 + 8 * q^64 + 7*b * q^67 + b * q^73 - 4*b * q^76 + 13 * q^79 - 11*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4}+O(q^{10})$$ 2 * q + 4 * q^4 $$2 q + 4 q^{4} + 8 q^{16} + 10 q^{25} - 26 q^{43} - 40 q^{49} + 26 q^{61} + 16 q^{64} + 26 q^{79}+O(q^{100})$$ 2 * q + 4 * q^4 + 8 * q^16 + 10 * q^25 - 26 * q^43 - 40 * 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
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 2.00000 0 0 5.19615i 0 0 0
1351.2 0 0 2.00000 0 0 5.19615i 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.d 2
3.b odd 2 1 CM 1521.2.b.d 2
13.b even 2 1 inner 1521.2.b.d 2
13.c even 3 1 117.2.q.b 2
13.d odd 4 2 1521.2.a.i 2
13.e even 6 1 117.2.q.b 2
39.d odd 2 1 inner 1521.2.b.d 2
39.f even 4 2 1521.2.a.i 2
39.h odd 6 1 117.2.q.b 2
39.i odd 6 1 117.2.q.b 2
52.i odd 6 1 1872.2.by.a 2
52.j odd 6 1 1872.2.by.a 2
156.p even 6 1 1872.2.by.a 2
156.r even 6 1 1872.2.by.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.q.b 2 13.c even 3 1
117.2.q.b 2 13.e even 6 1
117.2.q.b 2 39.h odd 6 1
117.2.q.b 2 39.i odd 6 1
1521.2.a.i 2 13.d odd 4 2
1521.2.a.i 2 39.f even 4 2
1521.2.b.d 2 1.a even 1 1 trivial
1521.2.b.d 2 3.b odd 2 1 CM
1521.2.b.d 2 13.b even 2 1 inner
1521.2.b.d 2 39.d odd 2 1 inner
1872.2.by.a 2 52.i odd 6 1
1872.2.by.a 2 52.j odd 6 1
1872.2.by.a 2 156.p even 6 1
1872.2.by.a 2 156.r even 6 1

## Hecke kernels

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

 $$T_{2}$$ T2 $$T_{5}$$ T5 $$T_{7}^{2} + 27$$ T7^2 + 27

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 27$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 12$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 75$$
$37$ $$T^{2} + 48$$
$41$ $$T^{2}$$
$43$ $$(T + 13)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 13)^{2}$$
$67$ $$T^{2} + 147$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 3$$
$79$ $$(T - 13)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 363$$
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