# Properties

 Label 1520.2.d.h 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: 6.0.16516096.1 Defining polynomial: $$x^{6} + 9 x^{4} + 13 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 95) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{3} + \beta_{3} q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + ( -3 - \beta_{2} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q -\beta_{4} q^{3} + \beta_{3} q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + ( -3 - \beta_{2} - \beta_{5} ) q^{9} + ( \beta_{2} + \beta_{3} ) q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( -2 + \beta_{2} - \beta_{4} - \beta_{5} ) q^{15} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} - q^{19} + ( 4 + \beta_{2} + \beta_{5} ) q^{21} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{23} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{25} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{27} -6 q^{29} + ( \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{31} + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{33} + ( 1 - 2 \beta_{1} + \beta_{5} ) q^{35} + ( \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{37} + ( -2 + 2 \beta_{3} - 2 \beta_{5} ) q^{39} + ( 2 + \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{41} + ( \beta_{1} + \beta_{4} ) q^{43} + ( -2 + 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{45} + ( -3 \beta_{1} + \beta_{4} ) q^{47} + ( 1 + \beta_{2} + \beta_{3} ) q^{49} + ( -2 \beta_{3} + 2 \beta_{5} ) q^{51} + ( -3 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{53} + ( 5 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{55} + \beta_{4} q^{57} + ( 4 + \beta_{2} + \beta_{5} ) q^{59} + ( -2 + 2 \beta_{2} + 3 \beta_{3} - \beta_{5} ) q^{61} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{63} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{65} + ( -2 \beta_{1} - 5 \beta_{4} ) q^{67} + ( 4 + \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{69} + ( -8 + \beta_{2} + \beta_{5} ) q^{71} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{73} + ( 6 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{75} + ( -4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{77} + ( -4 - \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{79} + ( 7 - \beta_{2} - 4 \beta_{3} + 3 \beta_{5} ) q^{81} + ( -\beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{83} + ( 3 - 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{85} + 6 \beta_{4} q^{87} + ( -2 + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} ) q^{89} + ( 4 - \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{91} + ( 2 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} ) q^{93} -\beta_{3} q^{95} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{97} + ( -4 + \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - q^{5} - 14 q^{9} + O(q^{10})$$ $$6 q - q^{5} - 14 q^{9} - 2 q^{11} - 10 q^{15} - 6 q^{19} + 20 q^{21} + 3 q^{25} - 36 q^{29} + 3 q^{35} - 8 q^{39} + 12 q^{41} - 15 q^{45} + 4 q^{49} - 4 q^{51} + 33 q^{55} + 20 q^{59} - 14 q^{61} - 20 q^{65} + 24 q^{69} - 52 q^{71} + 34 q^{75} - 24 q^{79} + 38 q^{81} + 13 q^{85} - 24 q^{89} + 24 q^{91} + q^{95} - 30 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 9 x^{4} + 13 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} - 8 \nu^{3} - 5 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{5} + \nu^{4} + 26 \nu^{3} + 6 \nu^{2} + 33 \nu - 1$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{5} + \nu^{4} - 26 \nu^{3} + 6 \nu^{2} - 33 \nu - 1$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$\nu^{5} + 9 \nu^{3} + 12 \nu$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{5} + 3 \nu^{4} - 26 \nu^{3} + 26 \nu^{2} - 33 \nu + 21$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} - \beta_{3} + \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} - 2 \beta_{3} - \beta_{2} - 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$9 \beta_{4} + 7 \beta_{3} - 7 \beta_{2} - 3 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{5} + 8 \beta_{3} + 5 \beta_{2} + 19$$ $$\nu^{5}$$ $$=$$ $$($$$$-67 \beta_{4} - 51 \beta_{3} + 51 \beta_{2} + 15 \beta_{1}$$$$)/2$$

## 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
 0.285442i − 2.68667i − 1.30397i 1.30397i 2.68667i − 0.285442i
0 3.21789i 0 −0.370556 2.20515i 0 2.59637i 0 −7.35482 0
609.2 0 2.31446i 0 1.94827 + 1.09737i 0 1.45033i 0 −2.35673 0
609.3 0 0.537080i 0 −2.07772 0.826491i 0 3.18676i 0 2.71155 0
609.4 0 0.537080i 0 −2.07772 + 0.826491i 0 3.18676i 0 2.71155 0
609.5 0 2.31446i 0 1.94827 1.09737i 0 1.45033i 0 −2.35673 0
609.6 0 3.21789i 0 −0.370556 + 2.20515i 0 2.59637i 0 −7.35482 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.h 6
4.b odd 2 1 95.2.b.b 6
5.b even 2 1 inner 1520.2.d.h 6
5.c odd 4 2 7600.2.a.ck 6
12.b even 2 1 855.2.c.d 6
20.d odd 2 1 95.2.b.b 6
20.e even 4 2 475.2.a.j 6
60.h even 2 1 855.2.c.d 6
60.l odd 4 2 4275.2.a.br 6
76.d even 2 1 1805.2.b.e 6
380.d even 2 1 1805.2.b.e 6
380.j odd 4 2 9025.2.a.bx 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.b 6 4.b odd 2 1
95.2.b.b 6 20.d odd 2 1
475.2.a.j 6 20.e even 4 2
855.2.c.d 6 12.b even 2 1
855.2.c.d 6 60.h even 2 1
1520.2.d.h 6 1.a even 1 1 trivial
1520.2.d.h 6 5.b even 2 1 inner
1805.2.b.e 6 76.d even 2 1
1805.2.b.e 6 380.d even 2 1
4275.2.a.br 6 60.l odd 4 2
7600.2.a.ck 6 5.c odd 4 2
9025.2.a.bx 6 380.j odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1520, [\chi])$$:

 $$T_{3}^{6} + 16 T_{3}^{4} + 60 T_{3}^{2} + 16$$ $$T_{7}^{6} + 19 T_{7}^{4} + 104 T_{7}^{2} + 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$16 + 60 T^{2} + 16 T^{4} + T^{6}$$
$5$ $$125 + 25 T - 5 T^{2} - 2 T^{3} - T^{4} + T^{5} + T^{6}$$
$7$ $$144 + 104 T^{2} + 19 T^{4} + T^{6}$$
$11$ $$( -12 - 16 T + T^{2} + T^{3} )^{2}$$
$13$ $$576 + 236 T^{2} + 28 T^{4} + T^{6}$$
$17$ $$5184 + 1008 T^{2} + 59 T^{4} + T^{6}$$
$19$ $$( 1 + T )^{6}$$
$23$ $$64 + 208 T^{2} + 36 T^{4} + T^{6}$$
$29$ $$( 6 + T )^{6}$$
$31$ $$( -128 - 56 T + T^{3} )^{2}$$
$37$ $$1296 + 764 T^{2} + 56 T^{4} + T^{6}$$
$41$ $$( -24 - 44 T - 6 T^{2} + T^{3} )^{2}$$
$43$ $$144 + 104 T^{2} + 19 T^{4} + T^{6}$$
$47$ $$85264 + 7464 T^{2} + 187 T^{4} + T^{6}$$
$53$ $$64 + 2476 T^{2} + 156 T^{4} + T^{6}$$
$59$ $$( 48 + 8 T - 10 T^{2} + T^{3} )^{2}$$
$61$ $$( -776 - 104 T + 7 T^{2} + T^{3} )^{2}$$
$67$ $$484416 + 28556 T^{2} + 340 T^{4} + T^{6}$$
$71$ $$( 432 + 200 T + 26 T^{2} + T^{3} )^{2}$$
$73$ $$5184 + 1616 T^{2} + 131 T^{4} + T^{6}$$
$79$ $$( -32 - 8 T + 12 T^{2} + T^{3} )^{2}$$
$83$ $$141376 + 11728 T^{2} + 228 T^{4} + T^{6}$$
$89$ $$( -3456 - 284 T + 12 T^{2} + T^{3} )^{2}$$
$97$ $$576 + 236 T^{2} + 28 T^{4} + T^{6}$$