# Properties

 Label 1620.3.o.b Level $1620$ Weight $3$ Character orbit 1620.o Analytic conductor $44.142$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{3} ) q^{5} + ( -2 + 2 \beta_{2} ) q^{7} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{3} ) q^{5} + ( -2 + 2 \beta_{2} ) q^{7} -6 \beta_{1} q^{11} -8 \beta_{2} q^{13} + 6 \beta_{3} q^{17} -34 q^{19} + ( 18 \beta_{1} - 18 \beta_{3} ) q^{23} + ( 5 - 5 \beta_{2} ) q^{25} + 18 \beta_{1} q^{29} -14 \beta_{2} q^{31} -2 \beta_{3} q^{35} + 56 q^{37} + ( -12 \beta_{1} + 12 \beta_{3} ) q^{41} + ( -8 + 8 \beta_{2} ) q^{43} + 18 \beta_{1} q^{47} + 45 \beta_{2} q^{49} -18 \beta_{3} q^{53} + 30 q^{55} + ( 6 \beta_{1} - 6 \beta_{3} ) q^{59} + ( 46 - 46 \beta_{2} ) q^{61} + 8 \beta_{1} q^{65} -32 \beta_{2} q^{67} + 24 \beta_{3} q^{71} -106 q^{73} + ( 12 \beta_{1} - 12 \beta_{3} ) q^{77} + ( 22 - 22 \beta_{2} ) q^{79} + 54 \beta_{1} q^{83} -30 \beta_{2} q^{85} + 48 \beta_{3} q^{89} + 16 q^{91} + ( 34 \beta_{1} - 34 \beta_{3} ) q^{95} + ( -122 + 122 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{7} + O(q^{10})$$ $$4q - 4q^{7} - 16q^{13} - 136q^{19} + 10q^{25} - 28q^{31} + 224q^{37} - 16q^{43} + 90q^{49} + 120q^{55} + 92q^{61} - 64q^{67} - 424q^{73} + 44q^{79} - 60q^{85} + 64q^{91} - 244q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3}$$

## 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
 1.93649 − 1.11803i −1.93649 + 1.11803i 1.93649 + 1.11803i −1.93649 − 1.11803i
0 0 0 −1.93649 1.11803i 0 −1.00000 1.73205i 0 0 0
701.2 0 0 0 1.93649 + 1.11803i 0 −1.00000 1.73205i 0 0 0
1241.1 0 0 0 −1.93649 + 1.11803i 0 −1.00000 + 1.73205i 0 0 0
1241.2 0 0 0 1.93649 1.11803i 0 −1.00000 + 1.73205i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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.b 4
3.b odd 2 1 inner 1620.3.o.b 4
9.c even 3 1 60.3.g.a 2
9.c even 3 1 inner 1620.3.o.b 4
9.d odd 6 1 60.3.g.a 2
9.d odd 6 1 inner 1620.3.o.b 4
36.f odd 6 1 240.3.l.a 2
36.h even 6 1 240.3.l.a 2
45.h odd 6 1 300.3.g.d 2
45.j even 6 1 300.3.g.d 2
45.k odd 12 2 300.3.b.c 4
45.l even 12 2 300.3.b.c 4
72.j odd 6 1 960.3.l.a 2
72.l even 6 1 960.3.l.d 2
72.n even 6 1 960.3.l.a 2
72.p odd 6 1 960.3.l.d 2
180.n even 6 1 1200.3.l.r 2
180.p odd 6 1 1200.3.l.r 2
180.v odd 12 2 1200.3.c.e 4
180.x even 12 2 1200.3.c.e 4

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$25 - 5 T^{2} + T^{4}$$
$7$ $$( 4 + 2 T + T^{2} )^{2}$$
$11$ $$32400 - 180 T^{2} + T^{4}$$
$13$ $$( 64 + 8 T + T^{2} )^{2}$$
$17$ $$( 180 + T^{2} )^{2}$$
$19$ $$( 34 + T )^{4}$$
$23$ $$2624400 - 1620 T^{2} + T^{4}$$
$29$ $$2624400 - 1620 T^{2} + T^{4}$$
$31$ $$( 196 + 14 T + T^{2} )^{2}$$
$37$ $$( -56 + T )^{4}$$
$41$ $$518400 - 720 T^{2} + T^{4}$$
$43$ $$( 64 + 8 T + T^{2} )^{2}$$
$47$ $$2624400 - 1620 T^{2} + T^{4}$$
$53$ $$( 1620 + T^{2} )^{2}$$
$59$ $$32400 - 180 T^{2} + T^{4}$$
$61$ $$( 2116 - 46 T + T^{2} )^{2}$$
$67$ $$( 1024 + 32 T + T^{2} )^{2}$$
$71$ $$( 2880 + T^{2} )^{2}$$
$73$ $$( 106 + T )^{4}$$
$79$ $$( 484 - 22 T + T^{2} )^{2}$$
$83$ $$212576400 - 14580 T^{2} + T^{4}$$
$89$ $$( 11520 + T^{2} )^{2}$$
$97$ $$( 14884 + 122 T + T^{2} )^{2}$$