# Properties

 Label 1521.2.b.f Level $1521$ Weight $2$ Character orbit 1521.b Analytic conductor $12.145$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 q^{4} + 2 \beta q^{5} + \beta q^{7}+O(q^{10})$$ q + 2 * q^4 + 2*b * q^5 + b * q^7 $$q + 2 q^{4} + 2 \beta q^{5} + \beta q^{7} + 2 \beta q^{11} + 4 q^{16} - 2 \beta q^{19} + 4 \beta q^{20} + 6 q^{23} - 7 q^{25} + 2 \beta q^{28} - 6 q^{29} - \beta q^{31} - 6 q^{35} + 4 \beta q^{41} - q^{43} + 4 \beta q^{44} + 2 \beta q^{47} + 4 q^{49} - 12 q^{53} - 12 q^{55} - 2 \beta q^{59} + q^{61} + 8 q^{64} + 5 \beta q^{67} - 6 \beta q^{71} - \beta q^{73} - 4 \beta q^{76} - 6 q^{77} - 11 q^{79} + 8 \beta q^{80} - 8 \beta q^{83} - 4 \beta q^{89} + 12 q^{92} + 12 q^{95} + 3 \beta q^{97} +O(q^{100})$$ q + 2 * q^4 + 2*b * q^5 + b * q^7 + 2*b * q^11 + 4 * q^16 - 2*b * q^19 + 4*b * q^20 + 6 * q^23 - 7 * q^25 + 2*b * q^28 - 6 * q^29 - b * q^31 - 6 * q^35 + 4*b * q^41 - q^43 + 4*b * q^44 + 2*b * q^47 + 4 * q^49 - 12 * q^53 - 12 * q^55 - 2*b * q^59 + q^61 + 8 * q^64 + 5*b * q^67 - 6*b * q^71 - b * q^73 - 4*b * q^76 - 6 * q^77 - 11 * q^79 + 8*b * q^80 - 8*b * q^83 - 4*b * q^89 + 12 * q^92 + 12 * q^95 + 3*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4}+O(q^{10})$$ 2 * q + 4 * q^4 $$2 q + 4 q^{4} + 8 q^{16} + 12 q^{23} - 14 q^{25} - 12 q^{29} - 12 q^{35} - 2 q^{43} + 8 q^{49} - 24 q^{53} - 24 q^{55} + 2 q^{61} + 16 q^{64} - 12 q^{77} - 22 q^{79} + 24 q^{92} + 24 q^{95}+O(q^{100})$$ 2 * q + 4 * q^4 + 8 * q^16 + 12 * q^23 - 14 * q^25 - 12 * q^29 - 12 * q^35 - 2 * q^43 + 8 * q^49 - 24 * q^53 - 24 * q^55 + 2 * q^61 + 16 * q^64 - 12 * q^77 - 22 * q^79 + 24 * q^92 + 24 * q^95

## 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 3.46410i 0 1.73205i 0 0 0
1351.2 0 0 2.00000 3.46410i 0 1.73205i 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
13.b even 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.f 2
3.b odd 2 1 507.2.b.c 2
13.b even 2 1 inner 1521.2.b.f 2
13.c even 3 1 117.2.q.a 2
13.d odd 4 2 1521.2.a.h 2
13.e even 6 1 117.2.q.a 2
39.d odd 2 1 507.2.b.c 2
39.f even 4 2 507.2.a.e 2
39.h odd 6 1 39.2.j.a 2
39.h odd 6 1 507.2.j.b 2
39.i odd 6 1 39.2.j.a 2
39.i odd 6 1 507.2.j.b 2
39.k even 12 4 507.2.e.f 4
52.i odd 6 1 1872.2.by.f 2
52.j odd 6 1 1872.2.by.f 2
156.l odd 4 2 8112.2.a.bu 2
156.p even 6 1 624.2.bv.b 2
156.r even 6 1 624.2.bv.b 2
195.x odd 6 1 975.2.bc.c 2
195.y odd 6 1 975.2.bc.c 2
195.bf even 12 2 975.2.w.d 4
195.bl even 12 2 975.2.w.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.j.a 2 39.h odd 6 1
39.2.j.a 2 39.i odd 6 1
117.2.q.a 2 13.c even 3 1
117.2.q.a 2 13.e even 6 1
507.2.a.e 2 39.f even 4 2
507.2.b.c 2 3.b odd 2 1
507.2.b.c 2 39.d odd 2 1
507.2.e.f 4 39.k even 12 4
507.2.j.b 2 39.h odd 6 1
507.2.j.b 2 39.i odd 6 1
624.2.bv.b 2 156.p even 6 1
624.2.bv.b 2 156.r even 6 1
975.2.w.d 4 195.bf even 12 2
975.2.w.d 4 195.bl even 12 2
975.2.bc.c 2 195.x odd 6 1
975.2.bc.c 2 195.y odd 6 1
1521.2.a.h 2 13.d odd 4 2
1521.2.b.f 2 1.a even 1 1 trivial
1521.2.b.f 2 13.b even 2 1 inner
1872.2.by.f 2 52.i odd 6 1
1872.2.by.f 2 52.j odd 6 1
8112.2.a.bu 2 156.l odd 4 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 $$T_{5}^{2} + 12$$ T5^2 + 12 $$T_{7}^{2} + 3$$ T7^2 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 12$$
$7$ $$T^{2} + 3$$
$11$ $$T^{2} + 12$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 12$$
$23$ $$(T - 6)^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 3$$
$37$ $$T^{2}$$
$41$ $$T^{2} + 48$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2} + 12$$
$53$ $$(T + 12)^{2}$$
$59$ $$T^{2} + 12$$
$61$ $$(T - 1)^{2}$$
$67$ $$T^{2} + 75$$
$71$ $$T^{2} + 108$$
$73$ $$T^{2} + 3$$
$79$ $$(T + 11)^{2}$$
$83$ $$T^{2} + 192$$
$89$ $$T^{2} + 48$$
$97$ $$T^{2} + 27$$