# Properties

 Label 1620.3.o.f Level $1620$ Weight $3$ Character orbit 1620.o 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.o (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: 8.0.3317760000.8 Defining polynomial: $$x^{8} + 4 x^{6} + 7 x^{4} + 36 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + ( 4 - 4 \beta_{2} - \beta_{7} ) q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + ( 4 - 4 \beta_{2} - \beta_{7} ) q^{7} + ( -\beta_{4} + 6 \beta_{5} ) q^{11} + ( -2 \beta_{2} + \beta_{6} ) q^{13} -4 \beta_{3} q^{17} + ( 8 - 2 \beta_{6} + 2 \beta_{7} ) q^{19} + ( 6 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} ) q^{23} + ( 5 - 5 \beta_{2} ) q^{25} + ( -8 \beta_{4} + 6 \beta_{5} ) q^{29} + ( -2 \beta_{2} + 2 \beta_{6} ) q^{31} + ( 4 \beta_{1} - 5 \beta_{3} + 4 \beta_{5} ) q^{35} + ( -34 + 3 \beta_{6} - 3 \beta_{7} ) q^{37} + ( -24 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} ) q^{41} + ( -20 + 20 \beta_{2} + 2 \beta_{7} ) q^{43} + ( 14 \beta_{4} + 12 \beta_{5} ) q^{47} + ( -57 \beta_{2} - 8 \beta_{6} ) q^{49} + ( 24 \beta_{1} + 10 \beta_{3} + 24 \beta_{5} ) q^{53} + ( -30 - \beta_{6} + \beta_{7} ) q^{55} + ( -18 \beta_{1} - 17 \beta_{3} + 17 \beta_{4} ) q^{59} + ( 10 - 10 \beta_{2} + 6 \beta_{7} ) q^{61} + ( 5 \beta_{4} + 2 \beta_{5} ) q^{65} + 76 \beta_{2} q^{67} + ( -12 \beta_{1} + 12 \beta_{3} - 12 \beta_{5} ) q^{71} + ( 38 + 6 \beta_{6} - 6 \beta_{7} ) q^{73} + ( -42 \beta_{1} + 34 \beta_{3} - 34 \beta_{4} ) q^{77} + ( -50 + 50 \beta_{2} - 6 \beta_{7} ) q^{79} + ( -2 \beta_{4} + 24 \beta_{5} ) q^{83} -4 \beta_{6} q^{85} + ( 12 \beta_{1} + 21 \beta_{3} + 12 \beta_{5} ) q^{89} + ( 82 + 2 \beta_{6} - 2 \beta_{7} ) q^{91} + ( 8 \beta_{1} + 10 \beta_{3} - 10 \beta_{4} ) q^{95} + ( 106 - 106 \beta_{2} - 2 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 16 q^{7} + O(q^{10})$$ $$8 q + 16 q^{7} - 8 q^{13} + 64 q^{19} + 20 q^{25} - 8 q^{31} - 272 q^{37} - 80 q^{43} - 228 q^{49} - 240 q^{55} + 40 q^{61} + 304 q^{67} + 304 q^{73} - 200 q^{79} + 656 q^{91} + 424 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 4 x^{6} + 7 x^{4} + 36 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} + 14 \nu^{4} - 7 \nu^{2} - 36$$$$)/63$$ $$\beta_{2}$$ $$=$$ $$($$$$4 \nu^{6} + 7 \nu^{4} + 28 \nu^{2} + 144$$$$)/63$$ $$\beta_{3}$$ $$=$$ $$($$$$4 \nu^{7} + 7 \nu^{5} - 35 \nu^{3} + 81 \nu$$$$)/63$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{7} + 7 \nu^{5} - 35 \nu^{3} - 180 \nu$$$$)/63$$ $$\beta_{5}$$ $$=$$ $$($$$$-8 \nu^{6} - 14 \nu^{4} + 7 \nu^{2} - 162$$$$)/63$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 13 \nu$$$$)/7$$ $$\beta_{7}$$ $$=$$ $$($$$$19 \nu^{7} + 49 \nu^{5} + 133 \nu^{3} + 684 \nu$$$$)/63$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - \beta_{4} + \beta_{3}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2 \beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} - 7 \beta_{3}$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$\beta_{2} + 4 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{7} + 19 \beta_{4}$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$-7 \beta_{5} - 7 \beta_{1} - 22$$ $$\nu^{7}$$ $$=$$ $$($$$$-29 \beta_{6} - 13 \beta_{4} + 13 \beta_{3}$$$$)/6$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
701.1
 −1.40294 + 1.01575i 1.40294 − 1.01575i −0.178197 + 1.72286i 0.178197 − 1.72286i −1.40294 − 1.01575i 1.40294 + 1.01575i −0.178197 − 1.72286i 0.178197 + 1.72286i
0 0 0 −1.93649 1.11803i 0 −2.74342 4.75174i 0 0 0
701.2 0 0 0 −1.93649 1.11803i 0 6.74342 + 11.6799i 0 0 0
701.3 0 0 0 1.93649 + 1.11803i 0 −2.74342 4.75174i 0 0 0
701.4 0 0 0 1.93649 + 1.11803i 0 6.74342 + 11.6799i 0 0 0
1241.1 0 0 0 −1.93649 + 1.11803i 0 −2.74342 + 4.75174i 0 0 0
1241.2 0 0 0 −1.93649 + 1.11803i 0 6.74342 11.6799i 0 0 0
1241.3 0 0 0 1.93649 1.11803i 0 −2.74342 + 4.75174i 0 0 0
1241.4 0 0 0 1.93649 1.11803i 0 6.74342 11.6799i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1241.4 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
9.c even 3 1 inner
9.d 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.o.f 8
3.b odd 2 1 inner 1620.3.o.f 8
9.c even 3 1 180.3.g.a 4
9.c even 3 1 inner 1620.3.o.f 8
9.d odd 6 1 180.3.g.a 4
9.d odd 6 1 inner 1620.3.o.f 8
36.f odd 6 1 720.3.l.c 4
36.h even 6 1 720.3.l.c 4
45.h odd 6 1 900.3.g.d 4
45.j even 6 1 900.3.g.d 4
45.k odd 12 2 900.3.b.b 8
45.l even 12 2 900.3.b.b 8
72.j odd 6 1 2880.3.l.b 4
72.l even 6 1 2880.3.l.f 4
72.n even 6 1 2880.3.l.b 4
72.p odd 6 1 2880.3.l.f 4
180.n even 6 1 3600.3.l.n 4
180.p odd 6 1 3600.3.l.n 4
180.v odd 12 2 3600.3.c.k 8
180.x even 12 2 3600.3.c.k 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 8 T_{7}^{3} + 138 T_{7}^{2} + 592 T_{7} + 5476$$ acting on $$S_{3}^{\mathrm{new}}(1620, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 25 - 5 T^{2} + T^{4} )^{2}$$
$7$ $$( 5476 + 592 T + 138 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$11$ $$688747536 - 10392624 T^{2} + 130572 T^{4} - 396 T^{6} + T^{8}$$
$13$ $$( 7396 - 344 T + 102 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$17$ $$( 288 + T^{2} )^{4}$$
$19$ $$( -296 - 16 T + T^{2} )^{4}$$
$23$ $$136048896 - 5878656 T^{2} + 242352 T^{4} - 504 T^{6} + T^{8}$$
$29$ $$892616806656 - 2516904576 T^{2} + 6152112 T^{4} - 2664 T^{6} + T^{8}$$
$31$ $$( 126736 - 1424 T + 372 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$37$ $$( 346 + 68 T + T^{2} )^{4}$$
$41$ $$54575510850576 - 44945696016 T^{2} + 29627532 T^{4} - 6084 T^{6} + T^{8}$$
$43$ $$( 1600 + 1600 T + 1560 T^{2} + 40 T^{3} + T^{4} )^{2}$$
$47$ $$62171080298496 - 66989804544 T^{2} + 64297152 T^{4} - 8496 T^{6} + T^{8}$$
$53$ $$( 1166400 + 9360 T^{2} + T^{4} )^{2}$$
$59$ $$164627478364176 - 175062398256 T^{2} + 173328012 T^{4} - 13644 T^{6} + T^{8}$$
$61$ $$( 9859600 + 62800 T + 3540 T^{2} - 20 T^{3} + T^{4} )^{2}$$
$67$ $$( 5776 - 76 T + T^{2} )^{4}$$
$71$ $$( 3504384 + 6624 T^{2} + T^{4} )^{2}$$
$73$ $$( -1796 - 76 T + T^{2} )^{4}$$
$79$ $$( 547600 - 74000 T + 10740 T^{2} + 100 T^{3} + T^{4} )^{2}$$
$83$ $$62171080298496 - 46552237056 T^{2} + 26972352 T^{4} - 5904 T^{6} + T^{8}$$
$89$ $$( 52099524 + 17316 T^{2} + T^{4} )^{2}$$
$97$ $$( 118287376 - 2305712 T + 34068 T^{2} - 212 T^{3} + T^{4} )^{2}$$