# Properties

 Label 1620.3.t.a Level $1620$ Weight $3$ Character orbit 1620.t Analytic conductor $44.142$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: 8.0.154550410641.1 Defining polynomial: $$x^{8} - 15 x^{6} + 221 x^{4} - 60 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 540) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} - \beta_{7} ) q^{5} + \beta_{1} q^{7} +O(q^{10})$$ $$q + ( \beta_{2} - \beta_{7} ) q^{5} + \beta_{1} q^{7} + ( -3 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{11} + ( -3 \beta_{3} + 3 \beta_{4} + 3 \beta_{6} + 3 \beta_{7} ) q^{13} + ( 1 - \beta_{3} + \beta_{5} - \beta_{7} ) q^{17} + ( 3 + 3 \beta_{3} - 3 \beta_{5} + 3 \beta_{7} ) q^{19} + ( -7 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} ) q^{23} + ( -3 + 5 \beta_{1} - 3 \beta_{2} + 3 \beta_{5} ) q^{25} + ( 9 \beta_{1} + 3 \beta_{4} + 3 \beta_{5} ) q^{29} + ( 7 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 6 \beta_{7} ) q^{31} + ( 10 - 5 \beta_{3} ) q^{35} + ( 2 \beta_{1} + 6 \beta_{3} + 6 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} ) q^{37} + ( \beta_{3} - \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{41} + ( -\beta_{1} - 9 \beta_{4} - 9 \beta_{5} ) q^{43} + ( -14 - 14 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} ) q^{47} + ( 21 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{7} ) q^{49} + ( -51 - 6 \beta_{3} + 6 \beta_{5} - 6 \beta_{7} ) q^{53} + ( 14 + 10 \beta_{1} + 15 \beta_{3} + 6 \beta_{5} - 10 \beta_{6} - 6 \beta_{7} ) q^{55} + ( -11 \beta_{3} + 11 \beta_{4} + 9 \beta_{6} + 11 \beta_{7} ) q^{59} + ( -11 - 11 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} ) q^{61} + ( -36 - 15 \beta_{1} - 36 \beta_{2} - 15 \beta_{4} - 9 \beta_{5} ) q^{65} + ( -6 \beta_{3} + 6 \beta_{4} - 8 \beta_{6} + 6 \beta_{7} ) q^{67} + ( -3 \beta_{1} - 15 \beta_{3} - 15 \beta_{5} + 3 \beta_{6} + 15 \beta_{7} ) q^{71} + ( -17 \beta_{1} - 12 \beta_{3} - 12 \beta_{5} + 17 \beta_{6} + 12 \beta_{7} ) q^{73} + ( 44 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} ) q^{77} + ( -3 - 3 \beta_{2} - 9 \beta_{4} + 9 \beta_{5} ) q^{79} + ( 67 + 67 \beta_{2} - 10 \beta_{4} + 10 \beta_{5} ) q^{83} + ( -25 \beta_{2} + 5 \beta_{6} ) q^{85} + ( 9 \beta_{1} - 11 \beta_{3} - 11 \beta_{5} - 9 \beta_{6} + 11 \beta_{7} ) q^{89} + ( 24 + 6 \beta_{3} - 6 \beta_{5} + 6 \beta_{7} ) q^{91} + ( 81 \beta_{2} - 15 \beta_{6} - 6 \beta_{7} ) q^{95} + ( -28 \beta_{1} - 3 \beta_{4} - 3 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 3 q^{5} + O(q^{10})$$ $$8 q - 3 q^{5} + 12 q^{17} + 12 q^{19} + 30 q^{23} - 9 q^{25} - 16 q^{31} + 90 q^{35} - 48 q^{47} - 78 q^{49} - 384 q^{53} + 94 q^{55} - 38 q^{61} - 138 q^{65} - 174 q^{77} + 6 q^{79} + 288 q^{83} + 100 q^{85} + 168 q^{91} - 318 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 15 x^{6} + 221 x^{4} - 60 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} - 3034 \nu$$$$)/221$$ $$\beta_{2}$$ $$=$$ $$($$$$-15 \nu^{6} + 221 \nu^{4} - 3315 \nu^{2} + 16$$$$)/884$$ $$\beta_{3}$$ $$=$$ $$($$$$60 \nu^{7} - \nu^{6} - 884 \nu^{5} + 13039 \nu^{3} - 64 \nu - 1708$$$$)/442$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 29 \nu^{6} - 442 \nu^{4} + 6409 \nu^{2} + 3476 \nu - 1740$$$$)/442$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 29 \nu^{6} + 442 \nu^{4} - 6409 \nu^{2} + 3476 \nu + 1740$$$$)/442$$ $$\beta_{6}$$ $$=$$ $$($$$$103 \nu^{7} - 1547 \nu^{5} + 22763 \nu^{3} - 6180 \nu$$$$)/442$$ $$\beta_{7}$$ $$=$$ $$($$$$-59 \nu^{7} - 30 \nu^{6} + 884 \nu^{5} + 442 \nu^{4} - 13039 \nu^{3} - 6409 \nu^{2} + 3540 \nu + 32$$$$)/442$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{4} + \beta_{3} - 8 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-7 \beta_{7} - 8 \beta_{6} + 7 \beta_{5} + 7 \beta_{3} + 8 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$15 \beta_{5} - 15 \beta_{4} - 116 \beta_{2} - 116$$ $$\nu^{5}$$ $$=$$ $$-103 \beta_{7} - 118 \beta_{6} - 103 \beta_{4} + 103 \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$-221 \beta_{7} + 221 \beta_{5} - 221 \beta_{3} - 1708$$ $$\nu^{7}$$ $$=$$ $$-1517 \beta_{5} - 1517 \beta_{4} - 1738 \beta_{1}$$

## Character values

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

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\beta_{2}$$

## 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
 3.32360 + 1.91888i 0.451318 + 0.260569i −3.32360 − 1.91888i −0.451318 − 0.260569i 3.32360 − 1.91888i 0.451318 − 0.260569i −3.32360 + 1.91888i −0.451318 + 0.260569i
0 0 0 −4.29522 2.55951i 0 2.42096 + 1.39774i 0 0 0
269.2 0 0 0 −2.11715 + 4.52964i 0 −6.19588 3.57719i 0 0 0
269.3 0 0 0 −0.0689865 4.99952i 0 −2.42096 1.39774i 0 0 0
269.4 0 0 0 4.98136 + 0.431312i 0 6.19588 + 3.57719i 0 0 0
1349.1 0 0 0 −4.29522 + 2.55951i 0 2.42096 1.39774i 0 0 0
1349.2 0 0 0 −2.11715 4.52964i 0 −6.19588 + 3.57719i 0 0 0
1349.3 0 0 0 −0.0689865 + 4.99952i 0 −2.42096 + 1.39774i 0 0 0
1349.4 0 0 0 4.98136 0.431312i 0 6.19588 3.57719i 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
9.c even 3 1 inner
15.d odd 2 1 inner
45.h odd 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.a 8
3.b odd 2 1 1620.3.t.d 8
5.b even 2 1 1620.3.t.d 8
9.c even 3 1 540.3.b.b yes 4
9.c even 3 1 inner 1620.3.t.a 8
9.d odd 6 1 540.3.b.a 4
9.d odd 6 1 1620.3.t.d 8
15.d odd 2 1 inner 1620.3.t.a 8
36.f odd 6 1 2160.3.c.l 4
36.h even 6 1 2160.3.c.h 4
45.h odd 6 1 540.3.b.b yes 4
45.h odd 6 1 inner 1620.3.t.a 8
45.j even 6 1 540.3.b.a 4
45.j even 6 1 1620.3.t.d 8
45.k odd 12 2 2700.3.g.s 8
45.l even 12 2 2700.3.g.s 8
180.n even 6 1 2160.3.c.l 4
180.p odd 6 1 2160.3.c.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.b.a 4 9.d odd 6 1
540.3.b.a 4 45.j even 6 1
540.3.b.b yes 4 9.c even 3 1
540.3.b.b yes 4 45.h odd 6 1
1620.3.t.a 8 1.a even 1 1 trivial
1620.3.t.a 8 9.c even 3 1 inner
1620.3.t.a 8 15.d odd 2 1 inner
1620.3.t.a 8 45.h odd 6 1 inner
1620.3.t.d 8 3.b odd 2 1
1620.3.t.d 8 5.b even 2 1
1620.3.t.d 8 9.d odd 6 1
1620.3.t.d 8 45.j even 6 1
2160.3.c.h 4 36.h even 6 1
2160.3.c.h 4 180.p odd 6 1
2160.3.c.l 4 36.f odd 6 1
2160.3.c.l 4 180.n even 6 1
2700.3.g.s 8 45.k odd 12 2
2700.3.g.s 8 45.l even 12 2

## Hecke kernels

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

 $$T_{7}^{8} - 59 T_{7}^{6} + 3081 T_{7}^{4} - 23600 T_{7}^{2} + 160000$$ $$T_{17}^{2} - 3 T_{17} - 50$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$390625 + 46875 T + 5625 T^{2} - 3750 T^{3} - 850 T^{4} - 150 T^{5} + 9 T^{6} + 3 T^{7} + T^{8}$$
$7$ $$160000 - 23600 T^{2} + 3081 T^{4} - 59 T^{6} + T^{8}$$
$11$ $$71639296 - 3004720 T^{2} + 117561 T^{4} - 355 T^{6} + T^{8}$$
$13$ $$26873856 - 2799360 T^{2} + 286416 T^{4} - 540 T^{6} + T^{8}$$
$17$ $$( -50 - 3 T + T^{2} )^{4}$$
$19$ $$( -468 - 3 T + T^{2} )^{4}$$
$23$ $$( 16 - 60 T + 221 T^{2} - 15 T^{3} + T^{4} )^{2}$$
$29$ $$( 2509056 - 1584 T^{2} + T^{4} )^{2}$$
$31$ $$( 3478225 - 14920 T + 1929 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$37$ $$( 1216 + T^{2} )^{4}$$
$41$ $$( 30976 - 176 T^{2} + T^{4} )^{2}$$
$43$ $$93790392487936 - 64111681280 T^{2} + 34139856 T^{4} - 6620 T^{6} + T^{8}$$
$47$ $$( 478864 - 16608 T + 1268 T^{2} + 24 T^{3} + T^{4} )^{2}$$
$53$ $$( 423 + 96 T + T^{2} )^{4}$$
$59$ $$17048839192576 - 28407685120 T^{2} + 43205376 T^{4} - 6880 T^{6} + T^{8}$$
$61$ $$( 144400 - 7220 T + 741 T^{2} + 19 T^{3} + T^{4} )^{2}$$
$67$ $$293434556416 - 6160166912 T^{2} + 128780688 T^{4} - 11372 T^{6} + T^{8}$$
$71$ $$( 68558400 + 16956 T^{2} + T^{4} )^{2}$$
$73$ $$( 6091024 + 11795 T^{2} + T^{4} )^{2}$$
$79$ $$( 17892900 + 12690 T + 4239 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$83$ $$( 1681 + 5904 T + 20777 T^{2} - 144 T^{3} + T^{4} )^{2}$$
$89$ $$( 19430464 + 24700 T^{2} + T^{4} )^{2}$$
$97$ $$71077471001645056 - 10534840621760 T^{2} + 1294831641 T^{4} - 39515 T^{6} + T^{8}$$