# Properties

 Label 1620.3.t.c 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: 8.0.1485512441856.6 Defining polynomial: $$x^{8} - 24 x^{6} + 455 x^{4} - 2904 x^{2} + 14641$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 180) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{5} + \beta_{6} ) q^{5} + \beta_{3} q^{7} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{5} + \beta_{6} ) q^{5} + \beta_{3} q^{7} -7 \beta_{4} q^{11} + 3 \beta_{7} q^{13} -4 \beta_{2} q^{17} + 12 q^{19} + ( 2 \beta_{2} + 2 \beta_{5} ) q^{23} + ( -21 \beta_{1} + 2 \beta_{3} ) q^{25} + 6 \beta_{4} q^{29} + ( 38 - 38 \beta_{1} ) q^{31} + ( 2 \beta_{2} - 23 \beta_{4} - 23 \beta_{6} ) q^{35} + ( -\beta_{3} - \beta_{7} ) q^{37} -49 \beta_{6} q^{41} -10 \beta_{3} q^{43} + 16 \beta_{5} q^{47} + ( -3 + 3 \beta_{1} ) q^{49} + ( 14 + 7 \beta_{3} + 7 \beta_{7} ) q^{55} + 59 \beta_{6} q^{59} + 70 \beta_{1} q^{61} + ( 69 \beta_{4} + 6 \beta_{5} ) q^{65} -16 \beta_{7} q^{67} + ( 84 \beta_{4} + 84 \beta_{6} ) q^{71} + ( -2 \beta_{3} - 2 \beta_{7} ) q^{73} + ( 14 \beta_{2} + 14 \beta_{5} ) q^{77} -30 \beta_{1} q^{79} + 28 \beta_{5} q^{83} + ( -92 + 92 \beta_{1} + 4 \beta_{7} ) q^{85} + ( 23 \beta_{4} + 23 \beta_{6} ) q^{89} -138 q^{91} + ( 12 \beta_{2} + 12 \beta_{5} + 12 \beta_{6} ) q^{95} + 14 \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q + 96 q^{19} - 84 q^{25} + 152 q^{31} - 12 q^{49} + 112 q^{55} + 280 q^{61} - 120 q^{79} - 368 q^{85} - 1104 q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 24 x^{6} + 455 x^{4} - 2904 x^{2} + 14641$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-24 \nu^{6} + 455 \nu^{4} - 10920 \nu^{2} + 69696$$$$)/55055$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{6} - 2556$$$$)/455$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 13021 \nu$$$$)/5005$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 3011 \nu$$$$)/5005$$ $$\beta_{5}$$ $$=$$ $$($$$$-167 \nu^{6} + 5460 \nu^{4} - 75985 \nu^{2} + 484968$$$$)/55055$$ $$\beta_{6}$$ $$=$$ $$($$$$191 \nu^{7} - 5915 \nu^{5} + 86905 \nu^{3} - 554664 \nu$$$$)/605605$$ $$\beta_{7}$$ $$=$$ $$($$$$719 \nu^{7} - 15925 \nu^{5} + 327145 \nu^{3} - 2087976 \nu$$$$)/605605$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} + \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{2} - 12 \beta_{1} + 12$$ $$\nu^{3}$$ $$=$$ $$($$$$13 \beta_{7} - 35 \beta_{6} - 35 \beta_{4} + 13 \beta_{3}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$24 \beta_{5} - 167 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$($$$$191 \beta_{7} - 719 \beta_{6}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-455 \beta_{2} - 2556$$ $$\nu^{7}$$ $$=$$ $$($$$$13021 \beta_{4} - 3011 \beta_{3}$$$$)/2$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1
 2.32446 + 1.34203i −2.32446 − 1.34203i −3.54921 − 2.04914i 3.54921 + 2.04914i 2.32446 − 1.34203i −2.32446 + 1.34203i −3.54921 + 2.04914i 3.54921 − 2.04914i
0 0 0 −3.62266 3.44621i 0 5.87367 + 3.39116i 0 0 0
269.2 0 0 0 −1.17317 4.86042i 0 −5.87367 3.39116i 0 0 0
269.3 0 0 0 1.17317 + 4.86042i 0 −5.87367 3.39116i 0 0 0
269.4 0 0 0 3.62266 + 3.44621i 0 5.87367 + 3.39116i 0 0 0
1349.1 0 0 0 −3.62266 + 3.44621i 0 5.87367 3.39116i 0 0 0
1349.2 0 0 0 −1.17317 + 4.86042i 0 −5.87367 + 3.39116i 0 0 0
1349.3 0 0 0 1.17317 4.86042i 0 −5.87367 + 3.39116i 0 0 0
1349.4 0 0 0 3.62266 3.44621i 0 5.87367 3.39116i 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.c 8
3.b odd 2 1 inner 1620.3.t.c 8
5.b even 2 1 inner 1620.3.t.c 8
9.c even 3 1 180.3.b.a 4
9.c even 3 1 inner 1620.3.t.c 8
9.d odd 6 1 180.3.b.a 4
9.d odd 6 1 inner 1620.3.t.c 8
15.d odd 2 1 inner 1620.3.t.c 8
36.f odd 6 1 720.3.c.b 4
36.h even 6 1 720.3.c.b 4
45.h odd 6 1 180.3.b.a 4
45.h odd 6 1 inner 1620.3.t.c 8
45.j even 6 1 180.3.b.a 4
45.j even 6 1 inner 1620.3.t.c 8
45.k odd 12 2 900.3.g.c 4
45.l even 12 2 900.3.g.c 4
72.j odd 6 1 2880.3.c.c 4
72.l even 6 1 2880.3.c.f 4
72.n even 6 1 2880.3.c.c 4
72.p odd 6 1 2880.3.c.f 4
180.n even 6 1 720.3.c.b 4
180.p odd 6 1 720.3.c.b 4
180.v odd 12 2 3600.3.l.r 4
180.x even 12 2 3600.3.l.r 4
360.z odd 6 1 2880.3.c.f 4
360.bd even 6 1 2880.3.c.f 4
360.bh odd 6 1 2880.3.c.c 4
360.bk even 6 1 2880.3.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.b.a 4 9.c even 3 1
180.3.b.a 4 9.d odd 6 1
180.3.b.a 4 45.h odd 6 1
180.3.b.a 4 45.j even 6 1
720.3.c.b 4 36.f odd 6 1
720.3.c.b 4 36.h even 6 1
720.3.c.b 4 180.n even 6 1
720.3.c.b 4 180.p odd 6 1
900.3.g.c 4 45.k odd 12 2
900.3.g.c 4 45.l even 12 2
1620.3.t.c 8 1.a even 1 1 trivial
1620.3.t.c 8 3.b odd 2 1 inner
1620.3.t.c 8 5.b even 2 1 inner
1620.3.t.c 8 9.c even 3 1 inner
1620.3.t.c 8 9.d odd 6 1 inner
1620.3.t.c 8 15.d odd 2 1 inner
1620.3.t.c 8 45.h odd 6 1 inner
1620.3.t.c 8 45.j even 6 1 inner
2880.3.c.c 4 72.j odd 6 1
2880.3.c.c 4 72.n even 6 1
2880.3.c.c 4 360.bh odd 6 1
2880.3.c.c 4 360.bk even 6 1
2880.3.c.f 4 72.l even 6 1
2880.3.c.f 4 72.p odd 6 1
2880.3.c.f 4 360.z odd 6 1
2880.3.c.f 4 360.bd even 6 1
3600.3.l.r 4 180.v odd 12 2
3600.3.l.r 4 180.x even 12 2

## Hecke kernels

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

 $$T_{7}^{4} - 46 T_{7}^{2} + 2116$$ $$T_{17}^{2} - 368$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$390625 + 26250 T^{2} + 1139 T^{4} + 42 T^{6} + T^{8}$$
$7$ $$( 2116 - 46 T^{2} + T^{4} )^{2}$$
$11$ $$( 9604 - 98 T^{2} + T^{4} )^{2}$$
$13$ $$( 171396 - 414 T^{2} + T^{4} )^{2}$$
$17$ $$( -368 + T^{2} )^{4}$$
$19$ $$( -12 + T )^{8}$$
$23$ $$( 8464 + 92 T^{2} + T^{4} )^{2}$$
$29$ $$( 5184 - 72 T^{2} + T^{4} )^{2}$$
$31$ $$( 1444 - 38 T + T^{2} )^{4}$$
$37$ $$( 46 + T^{2} )^{4}$$
$41$ $$( 23059204 - 4802 T^{2} + T^{4} )^{2}$$
$43$ $$( 21160000 - 4600 T^{2} + T^{4} )^{2}$$
$47$ $$( 34668544 + 5888 T^{2} + T^{4} )^{2}$$
$53$ $$T^{8}$$
$59$ $$( 48469444 - 6962 T^{2} + T^{4} )^{2}$$
$61$ $$( 4900 - 70 T + T^{2} )^{4}$$
$67$ $$( 138674176 - 11776 T^{2} + T^{4} )^{2}$$
$71$ $$( 14112 + T^{2} )^{4}$$
$73$ $$( 184 + T^{2} )^{4}$$
$79$ $$( 900 + 30 T + T^{2} )^{4}$$
$83$ $$( 325153024 + 18032 T^{2} + T^{4} )^{2}$$
$89$ $$( 1058 + T^{2} )^{4}$$
$97$ $$( 81288256 - 9016 T^{2} + T^{4} )^{2}$$