# Properties

 Label 1620.2.x.b Level $1620$ Weight $2$ Character orbit 1620.x Analytic conductor $12.936$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1620.x (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.9357651274$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: 8.0.12960000.1 Defining polynomial: $$x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} - \beta_{6} ) q^{5} + ( -\beta_{1} - \beta_{4} ) q^{7} +O(q^{10})$$ $$q + ( \beta_{3} - \beta_{6} ) q^{5} + ( -\beta_{1} - \beta_{4} ) q^{7} -2 \beta_{2} q^{11} + ( 3 - 3 \beta_{1} + 3 \beta_{4} - 3 \beta_{5} ) q^{13} + ( -\beta_{3} - \beta_{7} ) q^{17} -2 \beta_{5} q^{19} + ( -\beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} ) q^{23} + 5 \beta_{4} q^{25} -2 \beta_{6} q^{29} + ( -4 - 4 \beta_{4} ) q^{31} + ( \beta_{3} - \beta_{7} ) q^{35} + ( -3 - 3 \beta_{5} ) q^{37} + ( -4 \beta_{2} + 4 \beta_{7} ) q^{41} + ( 3 \beta_{1} - 3 \beta_{4} ) q^{43} + ( -3 \beta_{2} - 3 \beta_{6} ) q^{47} + ( 5 \beta_{1} + 5 \beta_{5} ) q^{49} + ( -\beta_{3} + \beta_{7} ) q^{53} + 10 \beta_{5} q^{55} + ( 4 \beta_{3} - 4 \beta_{6} ) q^{59} -6 \beta_{4} q^{61} + ( -3 \beta_{2} - 3 \beta_{6} ) q^{65} + ( 1 + \beta_{1} + \beta_{4} + \beta_{5} ) q^{67} -2 \beta_{7} q^{71} + ( 1 - \beta_{5} ) q^{73} + ( -2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} ) q^{77} -6 \beta_{1} q^{79} + ( -3 \beta_{2} + 3 \beta_{6} ) q^{83} + ( -5 + 5 \beta_{1} - 5 \beta_{4} + 5 \beta_{5} ) q^{85} -2 \beta_{3} q^{89} + 6 q^{91} + ( -2 \beta_{2} + 2 \beta_{7} ) q^{95} + ( 9 \beta_{1} + 9 \beta_{4} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{7} + O(q^{10})$$ $$8q + 4q^{7} + 12q^{13} - 20q^{25} - 16q^{31} - 24q^{37} + 12q^{43} + 24q^{61} + 4q^{67} + 8q^{73} - 20q^{85} + 48q^{91} - 36q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 13 \nu$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 29 \nu$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 9$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{6} - 8 \nu^{4} + 24 \nu^{2} - 9$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 8 \nu^{5} - 20 \nu^{3} + \nu$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-7 \nu^{6} + 24 \nu^{4} - 56 \nu^{2} + 21$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{7} - 8 \nu^{5} + 22 \nu^{3} - \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + 3 \beta_{4} - \beta_{3} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2 \beta_{5}$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{6} + 7 \beta_{4}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{7} + 11 \beta_{5} - 5 \beta_{2} + 11 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{3} - 9$$ $$\nu^{7}$$ $$=$$ $$($$$$-13 \beta_{2} + 29 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$$.

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$\beta_{5}$$ $$1 + \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 1.40126 − 0.809017i −0.535233 + 0.309017i −1.40126 + 0.809017i 0.535233 − 0.309017i −1.40126 − 0.809017i 0.535233 + 0.309017i 1.40126 + 0.809017i −0.535233 − 0.309017i
0 0 0 −1.11803 + 1.93649i 0 1.36603 + 0.366025i 0 0 0
53.2 0 0 0 1.11803 1.93649i 0 1.36603 + 0.366025i 0 0 0
377.1 0 0 0 −1.11803 + 1.93649i 0 −0.366025 + 1.36603i 0 0 0
377.2 0 0 0 1.11803 1.93649i 0 −0.366025 + 1.36603i 0 0 0
593.1 0 0 0 −1.11803 1.93649i 0 −0.366025 1.36603i 0 0 0
593.2 0 0 0 1.11803 + 1.93649i 0 −0.366025 1.36603i 0 0 0
917.1 0 0 0 −1.11803 1.93649i 0 1.36603 0.366025i 0 0 0
917.2 0 0 0 1.11803 + 1.93649i 0 1.36603 0.366025i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 917.2 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 inner
5.c odd 4 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.e even 4 1 inner
45.k odd 12 1 inner
45.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.x.b 8
3.b odd 2 1 inner 1620.2.x.b 8
5.c odd 4 1 inner 1620.2.x.b 8
9.c even 3 1 60.2.i.a 4
9.c even 3 1 inner 1620.2.x.b 8
9.d odd 6 1 60.2.i.a 4
9.d odd 6 1 inner 1620.2.x.b 8
15.e even 4 1 inner 1620.2.x.b 8
36.f odd 6 1 240.2.v.b 4
36.h even 6 1 240.2.v.b 4
45.h odd 6 1 300.2.i.a 4
45.j even 6 1 300.2.i.a 4
45.k odd 12 1 60.2.i.a 4
45.k odd 12 1 300.2.i.a 4
45.k odd 12 1 inner 1620.2.x.b 8
45.l even 12 1 60.2.i.a 4
45.l even 12 1 300.2.i.a 4
45.l even 12 1 inner 1620.2.x.b 8
72.j odd 6 1 960.2.v.e 4
72.l even 6 1 960.2.v.h 4
72.n even 6 1 960.2.v.e 4
72.p odd 6 1 960.2.v.h 4
180.n even 6 1 1200.2.v.i 4
180.p odd 6 1 1200.2.v.i 4
180.v odd 12 1 240.2.v.b 4
180.v odd 12 1 1200.2.v.i 4
180.x even 12 1 240.2.v.b 4
180.x even 12 1 1200.2.v.i 4
360.bo even 12 1 960.2.v.h 4
360.br even 12 1 960.2.v.e 4
360.bt odd 12 1 960.2.v.h 4
360.bu odd 12 1 960.2.v.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.i.a 4 9.c even 3 1
60.2.i.a 4 9.d odd 6 1
60.2.i.a 4 45.k odd 12 1
60.2.i.a 4 45.l even 12 1
240.2.v.b 4 36.f odd 6 1
240.2.v.b 4 36.h even 6 1
240.2.v.b 4 180.v odd 12 1
240.2.v.b 4 180.x even 12 1
300.2.i.a 4 45.h odd 6 1
300.2.i.a 4 45.j even 6 1
300.2.i.a 4 45.k odd 12 1
300.2.i.a 4 45.l even 12 1
960.2.v.e 4 72.j odd 6 1
960.2.v.e 4 72.n even 6 1
960.2.v.e 4 360.br even 12 1
960.2.v.e 4 360.bu odd 12 1
960.2.v.h 4 72.l even 6 1
960.2.v.h 4 72.p odd 6 1
960.2.v.h 4 360.bo even 12 1
960.2.v.h 4 360.bt odd 12 1
1200.2.v.i 4 180.n even 6 1
1200.2.v.i 4 180.p odd 6 1
1200.2.v.i 4 180.v odd 12 1
1200.2.v.i 4 180.x even 12 1
1620.2.x.b 8 1.a even 1 1 trivial
1620.2.x.b 8 3.b odd 2 1 inner
1620.2.x.b 8 5.c odd 4 1 inner
1620.2.x.b 8 9.c even 3 1 inner
1620.2.x.b 8 9.d odd 6 1 inner
1620.2.x.b 8 15.e even 4 1 inner
1620.2.x.b 8 45.k odd 12 1 inner
1620.2.x.b 8 45.l even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 2 T_{7}^{3} + 2 T_{7}^{2} - 4 T_{7} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1620, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 25 + 5 T^{2} + T^{4} )^{2}$$
$7$ $$( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$11$ $$( 400 - 20 T^{2} + T^{4} )^{2}$$
$13$ $$( 324 - 108 T + 18 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$17$ $$( 100 + T^{4} )^{2}$$
$19$ $$( 4 + T^{2} )^{4}$$
$23$ $$10000 - 100 T^{4} + T^{8}$$
$29$ $$( 400 + 20 T^{2} + T^{4} )^{2}$$
$31$ $$( 16 + 4 T + T^{2} )^{4}$$
$37$ $$( 18 + 6 T + T^{2} )^{4}$$
$41$ $$( 6400 - 80 T^{2} + T^{4} )^{2}$$
$43$ $$( 324 - 108 T + 18 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$47$ $$65610000 - 8100 T^{4} + T^{8}$$
$53$ $$( 100 + T^{4} )^{2}$$
$59$ $$( 6400 + 80 T^{2} + T^{4} )^{2}$$
$61$ $$( 36 - 6 T + T^{2} )^{4}$$
$67$ $$( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$71$ $$( 20 + T^{2} )^{4}$$
$73$ $$( 2 - 2 T + T^{2} )^{4}$$
$79$ $$( 1296 - 36 T^{2} + T^{4} )^{2}$$
$83$ $$65610000 - 8100 T^{4} + T^{8}$$
$89$ $$( -20 + T^{2} )^{4}$$
$97$ $$( 26244 + 2916 T + 162 T^{2} + 18 T^{3} + T^{4} )^{2}$$