# Properties

 Label 1521.2.b.m Level $1521$ Weight $2$ Character orbit 1521.b Analytic conductor $12.145$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.153664.1 Defining polynomial: $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 507) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( -\beta_{3} + \beta_{5} ) q^{5} + ( -\beta_{3} + 3 \beta_{5} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( -\beta_{3} + \beta_{5} ) q^{5} + ( -\beta_{3} + 3 \beta_{5} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + ( 1 - 2 \beta_{4} ) q^{10} + ( 3 \beta_{1} - 4 \beta_{3} - 2 \beta_{5} ) q^{11} + ( 1 - 4 \beta_{4} ) q^{14} + ( -3 + 3 \beta_{2} + \beta_{4} ) q^{16} + ( -2 - \beta_{2} ) q^{17} + ( -3 \beta_{1} - \beta_{3} - 3 \beta_{5} ) q^{19} -\beta_{1} q^{20} + ( -2 + 3 \beta_{2} - 2 \beta_{4} ) q^{22} + ( -2 + 5 \beta_{2} + 3 \beta_{4} ) q^{23} + ( 2 \beta_{2} + 3 \beta_{4} ) q^{25} + ( -3 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{28} + ( 1 + 2 \beta_{2} + 3 \beta_{4} ) q^{29} + ( 5 \beta_{1} - 2 \beta_{3} - 5 \beta_{5} ) q^{31} + ( -3 \beta_{1} + 4 \beta_{3} + \beta_{5} ) q^{32} + ( -\beta_{1} - \beta_{3} ) q^{34} + ( -9 + 4 \beta_{2} + 5 \beta_{4} ) q^{35} + ( \beta_{1} - \beta_{3} + 4 \beta_{5} ) q^{37} + ( 7 - 3 \beta_{2} + 2 \beta_{4} ) q^{38} + ( 4 - \beta_{2} - 4 \beta_{4} ) q^{40} + \beta_{1} q^{41} + ( 1 - 2 \beta_{2} + 2 \beta_{4} ) q^{43} + ( -\beta_{1} - 3 \beta_{3} - 6 \beta_{5} ) q^{44} + ( -4 \beta_{1} + 2 \beta_{3} + 3 \beta_{5} ) q^{46} + ( -9 \beta_{1} + 3 \beta_{3} + 7 \beta_{5} ) q^{47} + ( -10 + 6 \beta_{2} + 7 \beta_{4} ) q^{49} + ( \beta_{1} - \beta_{3} + 3 \beta_{5} ) q^{50} + ( 6 - 2 \beta_{2} - 3 \beta_{4} ) q^{53} + ( -5 + 2 \beta_{2} ) q^{55} + ( 6 - 3 \beta_{2} - 8 \beta_{4} ) q^{56} + ( 2 \beta_{1} - \beta_{3} + 3 \beta_{5} ) q^{58} + ( -4 \beta_{1} + 6 \beta_{3} + 8 \beta_{5} ) q^{59} + ( -4 - 3 \beta_{2} + 2 \beta_{4} ) q^{61} + ( -8 + 5 \beta_{2} + 3 \beta_{4} ) q^{62} + ( -4 + 3 \beta_{2} + 5 \beta_{4} ) q^{64} + ( 4 \beta_{1} - 3 \beta_{3} - 4 \beta_{5} ) q^{67} + ( -1 - 3 \beta_{2} - \beta_{4} ) q^{68} + ( -8 \beta_{1} - \beta_{3} + 5 \beta_{5} ) q^{70} + ( -5 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{71} + ( -2 \beta_{1} + \beta_{3} + 7 \beta_{5} ) q^{73} + ( -1 + \beta_{2} - 5 \beta_{4} ) q^{74} + ( 6 \beta_{1} - 7 \beta_{3} - 4 \beta_{5} ) q^{76} + ( -9 + 10 \beta_{2} + 2 \beta_{4} ) q^{77} + ( -5 + 5 \beta_{2} + \beta_{4} ) q^{79} + ( -\beta_{1} + 3 \beta_{3} - 4 \beta_{5} ) q^{80} + ( -2 + \beta_{2} ) q^{82} + ( -\beta_{1} - 3 \beta_{3} - 6 \beta_{5} ) q^{83} + ( \beta_{1} + 2 \beta_{3} - 2 \beta_{5} ) q^{85} + ( 5 \beta_{1} - 4 \beta_{3} + 2 \beta_{5} ) q^{86} + ( 1 + 5 \beta_{2} - \beta_{4} ) q^{88} + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{5} ) q^{89} + ( 2 + 6 \beta_{2} + 5 \beta_{4} ) q^{92} + ( 15 - 9 \beta_{2} - 4 \beta_{4} ) q^{94} + ( -2 \beta_{2} + 5 \beta_{4} ) q^{95} + ( -4 \beta_{1} + 10 \beta_{3} + 3 \beta_{5} ) q^{97} + ( -9 \beta_{1} - \beta_{3} + 7 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 2q^{4} + O(q^{10})$$ $$6q + 2q^{4} + 2q^{10} - 2q^{14} - 10q^{16} - 14q^{17} - 10q^{22} + 4q^{23} + 10q^{25} + 16q^{29} - 36q^{35} + 40q^{38} + 14q^{40} + 6q^{43} - 34q^{49} + 26q^{53} - 26q^{55} + 14q^{56} - 26q^{61} - 32q^{62} - 8q^{64} - 14q^{68} - 14q^{74} - 30q^{77} - 18q^{79} - 10q^{82} + 14q^{88} + 34q^{92} + 64q^{94} + 6q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} + 3 \nu^{2} + 1$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} + 4 \nu^{3} + 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} - 3 \beta_{2} + 5$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{3} + 9 \beta_{1}$$

## 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
 − 1.80194i − 1.24698i − 0.445042i 0.445042i 1.24698i 1.80194i
1.80194i 0 −1.24698 1.44504i 0 3.44504i 1.35690i 0 −2.60388
1351.2 1.24698i 0 0.445042 2.80194i 0 4.80194i 3.04892i 0 3.49396
1351.3 0.445042i 0 1.80194 0.246980i 0 1.75302i 1.69202i 0 0.109916
1351.4 0.445042i 0 1.80194 0.246980i 0 1.75302i 1.69202i 0 0.109916
1351.5 1.24698i 0 0.445042 2.80194i 0 4.80194i 3.04892i 0 3.49396
1351.6 1.80194i 0 −1.24698 1.44504i 0 3.44504i 1.35690i 0 −2.60388
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1351.6 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.m 6
3.b odd 2 1 507.2.b.g 6
13.b even 2 1 inner 1521.2.b.m 6
13.d odd 4 1 1521.2.a.p 3
13.d odd 4 1 1521.2.a.q 3
39.d odd 2 1 507.2.b.g 6
39.f even 4 1 507.2.a.j 3
39.f even 4 1 507.2.a.k yes 3
39.h odd 6 2 507.2.j.h 12
39.i odd 6 2 507.2.j.h 12
39.k even 12 2 507.2.e.j 6
39.k even 12 2 507.2.e.k 6
156.l odd 4 1 8112.2.a.by 3
156.l odd 4 1 8112.2.a.cf 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.j 3 39.f even 4 1
507.2.a.k yes 3 39.f even 4 1
507.2.b.g 6 3.b odd 2 1
507.2.b.g 6 39.d odd 2 1
507.2.e.j 6 39.k even 12 2
507.2.e.k 6 39.k even 12 2
507.2.j.h 12 39.h odd 6 2
507.2.j.h 12 39.i odd 6 2
1521.2.a.p 3 13.d odd 4 1
1521.2.a.q 3 13.d odd 4 1
1521.2.b.m 6 1.a even 1 1 trivial
1521.2.b.m 6 13.b even 2 1 inner
8112.2.a.by 3 156.l odd 4 1
8112.2.a.cf 3 156.l odd 4 1

## Hecke kernels

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

 $$T_{2}^{6} + 5 T_{2}^{4} + 6 T_{2}^{2} + 1$$ $$T_{5}^{6} + 10 T_{5}^{4} + 17 T_{5}^{2} + 1$$ $$T_{7}^{6} + 38 T_{7}^{4} + 381 T_{7}^{2} + 841$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$1 + 17 T^{2} + 10 T^{4} + T^{6}$$
$7$ $$841 + 381 T^{2} + 38 T^{4} + T^{6}$$
$11$ $$1849 + 986 T^{2} + 61 T^{4} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$( 7 + 14 T + 7 T^{2} + T^{3} )^{2}$$
$19$ $$12769 + 2586 T^{2} + 101 T^{4} + T^{6}$$
$23$ $$( -83 - 43 T - 2 T^{2} + T^{3} )^{2}$$
$29$ $$( 43 + 5 T - 8 T^{2} + T^{3} )^{2}$$
$31$ $$38809 + 3681 T^{2} + 110 T^{4} + T^{6}$$
$37$ $$8281 + 1421 T^{2} + 70 T^{4} + T^{6}$$
$41$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$
$43$ $$( -29 - 25 T - 3 T^{2} + T^{3} )^{2}$$
$47$ $$829921 + 30798 T^{2} + 321 T^{4} + T^{6}$$
$53$ $$( -29 + 40 T - 13 T^{2} + T^{3} )^{2}$$
$59$ $$3136 + 1568 T^{2} + 196 T^{4} + T^{6}$$
$61$ $$( -223 + 12 T + 13 T^{2} + T^{3} )^{2}$$
$67$ $$9409 + 1454 T^{2} + 69 T^{4} + T^{6}$$
$71$ $$212521 + 11773 T^{2} + 194 T^{4} + T^{6}$$
$73$ $$27889 + 4189 T^{2} + 122 T^{4} + T^{6}$$
$79$ $$( -169 - 22 T + 9 T^{2} + T^{3} )^{2}$$
$83$ $$1849 + 4401 T^{2} + 146 T^{4} + T^{6}$$
$89$ $$1 + 54 T^{2} + 41 T^{4} + T^{6}$$
$97$ $$1413721 + 40451 T^{2} + 363 T^{4} + T^{6}$$