# Properties

 Label 1520.2.d.j Level $1520$ Weight $2$ Character orbit 1520.d Analytic conductor $12.137$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.1372611072$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.5161984.1 Defining polynomial: $$x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 190) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{3} + ( \beta_{2} + \beta_{3} ) q^{5} + ( -\beta_{4} + \beta_{5} ) q^{7} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{5} q^{3} + ( \beta_{2} + \beta_{3} ) q^{5} + ( -\beta_{4} + \beta_{5} ) q^{7} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + ( \beta_{1} - \beta_{2} ) q^{11} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{13} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{15} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{17} + q^{19} + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{21} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{4} - 3 \beta_{5} ) q^{23} + ( 2 + 2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{25} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{5} ) q^{27} + ( 3 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{29} + ( -2 - 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{31} + ( -\beta_{1} - \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{33} + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{35} + ( 3 \beta_{4} - 2 \beta_{5} ) q^{37} + ( 5 + 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} ) q^{39} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{41} + ( -\beta_{1} - \beta_{2} + 3 \beta_{4} ) q^{43} + ( 4 + \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{45} + ( 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} ) q^{47} + ( 2 - \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{49} + ( 3 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{51} + ( -\beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{53} + ( -1 - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{55} + \beta_{5} q^{57} + ( 1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{59} + ( -10 - \beta_{1} + \beta_{2} ) q^{61} + ( \beta_{4} - 2 \beta_{5} ) q^{63} + ( -2 - 2 \beta_{1} - \beta_{2} + 4 \beta_{5} ) q^{65} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{5} ) q^{67} + ( 9 + 5 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{69} + ( 4 - \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{71} + ( 3 \beta_{1} + 3 \beta_{2} - 5 \beta_{5} ) q^{73} + ( -7 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 4 \beta_{5} ) q^{75} + ( \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{77} + ( -2 + 6 \beta_{3} ) q^{79} + ( -5 - 2 \beta_{3} ) q^{81} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{83} + ( -2 - \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{85} + ( -4 \beta_{1} - 4 \beta_{2} - 4 \beta_{4} + 9 \beta_{5} ) q^{87} + ( 4 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{89} + ( -1 - 3 \beta_{3} ) q^{91} + ( 5 \beta_{1} + 5 \beta_{2} + 2 \beta_{4} - 8 \beta_{5} ) q^{93} + ( \beta_{2} + \beta_{3} ) q^{95} + ( 4 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} - 4 \beta_{5} ) q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 2q^{5} - 10q^{9} + O(q^{10})$$ $$6q + 2q^{5} - 10q^{9} - 8q^{15} + 6q^{19} - 20q^{21} + 10q^{25} + 16q^{29} - 8q^{31} - 8q^{35} + 20q^{39} + 4q^{41} + 18q^{45} + 6q^{49} + 12q^{51} - 4q^{55} + 4q^{59} - 60q^{61} - 12q^{65} + 44q^{69} + 16q^{71} - 44q^{75} - 34q^{81} - 12q^{85} + 28q^{89} - 12q^{91} + 2q^{95} - 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$5 \nu^{5} + 2 \nu^{4} + 25 \nu^{3} - 10 \nu^{2} + 121 \nu - 100$$$$)/121$$ $$\beta_{3}$$ $$=$$ $$($$$$-7 \nu^{5} - 27 \nu^{4} - 35 \nu^{3} + 14 \nu^{2} - 223$$$$)/121$$ $$\beta_{4}$$ $$=$$ $$($$$$-25 \nu^{5} - 10 \nu^{4} - 4 \nu^{3} + 50 \nu^{2} - 605 \nu + 258$$$$)/121$$ $$\beta_{5}$$ $$=$$ $$($$$$-45 \nu^{5} - 18 \nu^{4} + 17 \nu^{3} + 211 \nu^{2} - 968 \nu + 416$$$$)/121$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + 5 \beta_{2} + 2$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{3} - 7 \beta_{2} + 7 \beta_{1} - 15$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} - 29 \beta_{1} + 16$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$401$$ $$1141$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
609.1
 0.432320 + 0.432320i 1.32001 − 1.32001i −1.75233 − 1.75233i −1.75233 + 1.75233i 1.32001 + 1.32001i 0.432320 − 0.432320i
0 2.76156i 0 −2.19388 + 0.432320i 0 0.761557i 0 −4.62620 0
609.2 0 2.12489i 0 1.80487 1.32001i 0 4.12489i 0 −1.51514 0
609.3 0 1.36333i 0 1.38900 1.75233i 0 0.636672i 0 1.14134 0
609.4 0 1.36333i 0 1.38900 + 1.75233i 0 0.636672i 0 1.14134 0
609.5 0 2.12489i 0 1.80487 + 1.32001i 0 4.12489i 0 −1.51514 0
609.6 0 2.76156i 0 −2.19388 0.432320i 0 0.761557i 0 −4.62620 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 609.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.d.j 6
4.b odd 2 1 190.2.b.b 6
5.b even 2 1 inner 1520.2.d.j 6
5.c odd 4 1 7600.2.a.bi 3
5.c odd 4 1 7600.2.a.cd 3
12.b even 2 1 1710.2.d.d 6
20.d odd 2 1 190.2.b.b 6
20.e even 4 1 950.2.a.i 3
20.e even 4 1 950.2.a.n 3
60.h even 2 1 1710.2.d.d 6
60.l odd 4 1 8550.2.a.ck 3
60.l odd 4 1 8550.2.a.cl 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.b 6 4.b odd 2 1
190.2.b.b 6 20.d odd 2 1
950.2.a.i 3 20.e even 4 1
950.2.a.n 3 20.e even 4 1
1520.2.d.j 6 1.a even 1 1 trivial
1520.2.d.j 6 5.b even 2 1 inner
1710.2.d.d 6 12.b even 2 1
1710.2.d.d 6 60.h even 2 1
7600.2.a.bi 3 5.c odd 4 1
7600.2.a.cd 3 5.c odd 4 1
8550.2.a.ck 3 60.l odd 4 1
8550.2.a.cl 3 60.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 $$S_{2}^{\mathrm{new}}(1520, [\chi])$$:

 $$T_{3}^{6} + 14 T_{3}^{4} + 57 T_{3}^{2} + 64$$ $$T_{7}^{6} + 18 T_{7}^{4} + 17 T_{7}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$64 + 57 T^{2} + 14 T^{4} + T^{6}$$
$5$ $$125 - 50 T - 15 T^{2} + 24 T^{3} - 3 T^{4} - 2 T^{5} + T^{6}$$
$7$ $$4 + 17 T^{2} + 18 T^{4} + T^{6}$$
$11$ $$( 8 - 10 T + T^{3} )^{2}$$
$13$ $$4 + 201 T^{2} + 38 T^{4} + T^{6}$$
$17$ $$16 + 65 T^{2} + 18 T^{4} + T^{6}$$
$19$ $$( -1 + T )^{6}$$
$23$ $$14884 + 2401 T^{2} + 98 T^{4} + T^{6}$$
$29$ $$( 410 - 51 T - 8 T^{2} + T^{3} )^{2}$$
$31$ $$( -232 - 62 T + 4 T^{2} + T^{3} )^{2}$$
$37$ $$256 + 1152 T^{2} + 116 T^{4} + T^{6}$$
$41$ $$( -100 - 50 T - 2 T^{2} + T^{3} )^{2}$$
$43$ $$21904 + 4276 T^{2} + 128 T^{4} + T^{6}$$
$47$ $$4096 + 2816 T^{2} + 132 T^{4} + T^{6}$$
$53$ $$11236 + 2537 T^{2} + 102 T^{4} + T^{6}$$
$59$ $$( 80 - 29 T - 2 T^{2} + T^{3} )^{2}$$
$61$ $$( 892 + 290 T + 30 T^{2} + T^{3} )^{2}$$
$67$ $$4096 + 3977 T^{2} + 126 T^{4} + T^{6}$$
$71$ $$( 1016 - 122 T - 8 T^{2} + T^{3} )^{2}$$
$73$ $$26896 + 5745 T^{2} + 290 T^{4} + T^{6}$$
$79$ $$( -880 - 228 T + T^{3} )^{2}$$
$83$ $$64 + 880 T^{2} + 92 T^{4} + T^{6}$$
$89$ $$( 20 + 46 T - 14 T^{2} + T^{3} )^{2}$$
$97$ $$238144 + 19760 T^{2} + 300 T^{4} + T^{6}$$