# Properties

 Label 1620.3.t.b Level $1620$ Weight $3$ Character orbit 1620.t Analytic conductor $44.142$ 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: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1620.t (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$44.1418028264$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.12960000.1 Defining polynomial: $$x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{5} -2 \beta_{4} q^{7} +O(q^{10})$$ $$q -\beta_{5} q^{5} -2 \beta_{4} q^{7} + ( \beta_{2} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{11} + 3 \beta_{1} q^{13} + 7 \beta_{2} q^{17} -6 q^{19} -\beta_{7} q^{23} + ( 15 \beta_{3} + 5 \beta_{4} ) q^{25} + ( 3 \beta_{2} + 6 \beta_{5} + 6 \beta_{6} - 3 \beta_{7} ) q^{29} + ( -34 + 34 \beta_{3} ) q^{31} + ( -6 \beta_{2} + 4 \beta_{6} ) q^{35} + ( 11 \beta_{1} + 11 \beta_{4} ) q^{37} + ( 4 \beta_{5} - 2 \beta_{7} ) q^{41} -7 \beta_{4} q^{43} + ( \beta_{2} + \beta_{7} ) q^{47} + ( 15 - 15 \beta_{3} ) q^{49} -9 \beta_{2} q^{53} + ( -40 - 5 \beta_{1} - 5 \beta_{4} ) q^{55} + ( 22 \beta_{5} - 11 \beta_{7} ) q^{59} -74 \beta_{3} q^{61} + ( -9 \beta_{2} + 6 \beta_{5} + 6 \beta_{6} - 15 \beta_{7} ) q^{65} + 23 \beta_{1} q^{67} + ( 6 \beta_{2} + 12 \beta_{6} ) q^{71} + ( -14 \beta_{1} - 14 \beta_{4} ) q^{73} -16 \beta_{7} q^{77} + 78 \beta_{3} q^{79} + ( -23 \beta_{2} - 23 \beta_{7} ) q^{83} + ( 70 - 35 \beta_{1} - 70 \beta_{3} ) q^{85} + ( -2 \beta_{2} - 4 \beta_{6} ) q^{89} + 96 q^{91} + 6 \beta_{5} q^{95} + 8 \beta_{4} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 48q^{19} + 60q^{25} - 136q^{31} + 60q^{49} - 320q^{55} - 296q^{61} + 312q^{79} + 280q^{85} + 768q^{91} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} - 13 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{6} - 9$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{6} - 8 \nu^{4} + 24 \nu^{2} - 1$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{7} + 16 \nu^{5} - 40 \nu^{3} + 15 \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{7} - 7 \nu^{6} + 24 \nu^{4} - 56 \nu^{2} + 58 \nu + 21$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$-12 \nu^{7} + \nu^{6} + 32 \nu^{5} - 88 \nu^{3} + 4 \nu + 9$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$-7 \nu^{6} + 24 \nu^{4} - 56 \nu^{2} + 21$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + 2 \beta_{5} + \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + 6 \beta_{3} + \beta_{2}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{6} + 2 \beta_{4} - \beta_{2} + 2 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{7} + 14 \beta_{3} - 14$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} + 11 \beta_{4} - 5 \beta_{2}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{2} - 9$$ $$\nu^{7}$$ $$=$$ $$($$$$13 \beta_{7} - 26 \beta_{5} - 29 \beta_{1}$$$$)/8$$

## Character values

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

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1 - \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1
 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 −1.40126 − 0.809017i
0 0 0 −4.99102 + 0.299576i 0 −6.92820 4.00000i 0 0 0
269.2 0 0 0 −2.75495 + 4.17256i 0 6.92820 + 4.00000i 0 0 0
269.3 0 0 0 2.75495 4.17256i 0 6.92820 + 4.00000i 0 0 0
269.4 0 0 0 4.99102 0.299576i 0 −6.92820 4.00000i 0 0 0
1349.1 0 0 0 −4.99102 0.299576i 0 −6.92820 + 4.00000i 0 0 0
1349.2 0 0 0 −2.75495 4.17256i 0 6.92820 4.00000i 0 0 0
1349.3 0 0 0 2.75495 + 4.17256i 0 6.92820 4.00000i 0 0 0
1349.4 0 0 0 4.99102 + 0.299576i 0 −6.92820 + 4.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1349.4 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.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.d odd 2 1 inner
45.h odd 6 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.3.t.b 8
3.b odd 2 1 inner 1620.3.t.b 8
5.b even 2 1 inner 1620.3.t.b 8
9.c even 3 1 60.3.b.a 4
9.c even 3 1 inner 1620.3.t.b 8
9.d odd 6 1 60.3.b.a 4
9.d odd 6 1 inner 1620.3.t.b 8
15.d odd 2 1 inner 1620.3.t.b 8
36.f odd 6 1 240.3.c.d 4
36.h even 6 1 240.3.c.d 4
45.h odd 6 1 60.3.b.a 4
45.h odd 6 1 inner 1620.3.t.b 8
45.j even 6 1 60.3.b.a 4
45.j even 6 1 inner 1620.3.t.b 8
45.k odd 12 1 300.3.g.e 2
45.k odd 12 1 300.3.g.h 2
45.l even 12 1 300.3.g.e 2
45.l even 12 1 300.3.g.h 2
72.j odd 6 1 960.3.c.h 4
72.l even 6 1 960.3.c.g 4
72.n even 6 1 960.3.c.h 4
72.p odd 6 1 960.3.c.g 4
180.n even 6 1 240.3.c.d 4
180.p odd 6 1 240.3.c.d 4
180.v odd 12 1 1200.3.l.h 2
180.v odd 12 1 1200.3.l.q 2
180.x even 12 1 1200.3.l.h 2
180.x even 12 1 1200.3.l.q 2
360.z odd 6 1 960.3.c.g 4
360.bd even 6 1 960.3.c.g 4
360.bh odd 6 1 960.3.c.h 4
360.bk even 6 1 960.3.c.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.b.a 4 9.c even 3 1
60.3.b.a 4 9.d odd 6 1
60.3.b.a 4 45.h odd 6 1
60.3.b.a 4 45.j even 6 1
240.3.c.d 4 36.f odd 6 1
240.3.c.d 4 36.h even 6 1
240.3.c.d 4 180.n even 6 1
240.3.c.d 4 180.p odd 6 1
300.3.g.e 2 45.k odd 12 1
300.3.g.e 2 45.l even 12 1
300.3.g.h 2 45.k odd 12 1
300.3.g.h 2 45.l even 12 1
960.3.c.g 4 72.l even 6 1
960.3.c.g 4 72.p odd 6 1
960.3.c.g 4 360.z odd 6 1
960.3.c.g 4 360.bd even 6 1
960.3.c.h 4 72.j odd 6 1
960.3.c.h 4 72.n even 6 1
960.3.c.h 4 360.bh odd 6 1
960.3.c.h 4 360.bk even 6 1
1200.3.l.h 2 180.v odd 12 1
1200.3.l.h 2 180.x even 12 1
1200.3.l.q 2 180.v odd 12 1
1200.3.l.q 2 180.x even 12 1
1620.3.t.b 8 1.a even 1 1 trivial
1620.3.t.b 8 3.b odd 2 1 inner
1620.3.t.b 8 5.b even 2 1 inner
1620.3.t.b 8 9.c even 3 1 inner
1620.3.t.b 8 9.d odd 6 1 inner
1620.3.t.b 8 15.d odd 2 1 inner
1620.3.t.b 8 45.h odd 6 1 inner
1620.3.t.b 8 45.j even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1620, [\chi])$$:

 $$T_{7}^{4} - 64 T_{7}^{2} + 4096$$ $$T_{17}^{2} - 980$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$390625 - 18750 T^{2} + 275 T^{4} - 30 T^{6} + T^{8}$$
$7$ $$( 4096 - 64 T^{2} + T^{4} )^{2}$$
$11$ $$( 6400 - 80 T^{2} + T^{4} )^{2}$$
$13$ $$( 20736 - 144 T^{2} + T^{4} )^{2}$$
$17$ $$( -980 + T^{2} )^{4}$$
$19$ $$( 6 + T )^{8}$$
$23$ $$( 400 + 20 T^{2} + T^{4} )^{2}$$
$29$ $$( 518400 - 720 T^{2} + T^{4} )^{2}$$
$31$ $$( 1156 + 34 T + T^{2} )^{4}$$
$37$ $$( 1936 + T^{2} )^{4}$$
$41$ $$( 102400 - 320 T^{2} + T^{4} )^{2}$$
$43$ $$( 614656 - 784 T^{2} + T^{4} )^{2}$$
$47$ $$( 400 + 20 T^{2} + T^{4} )^{2}$$
$53$ $$( -1620 + T^{2} )^{4}$$
$59$ $$( 93702400 - 9680 T^{2} + T^{4} )^{2}$$
$61$ $$( 5476 + 74 T + T^{2} )^{4}$$
$67$ $$( 71639296 - 8464 T^{2} + T^{4} )^{2}$$
$71$ $$( 2880 + T^{2} )^{4}$$
$73$ $$( 3136 + T^{2} )^{4}$$
$79$ $$( 6084 - 78 T + T^{2} )^{4}$$
$83$ $$( 111936400 + 10580 T^{2} + T^{4} )^{2}$$
$89$ $$( 320 + T^{2} )^{4}$$
$97$ $$( 1048576 - 1024 T^{2} + T^{4} )^{2}$$