# Properties

 Label 1620.3.o.c 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: $$x^{4} - 5x^{2} + 25$$ 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 540) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{5} + 4 \beta_{2} q^{7}+O(q^{10})$$ q - b1 * q^5 + 4*b2 * q^7 $$q - \beta_1 q^{5} + 4 \beta_{2} q^{7} + (3 \beta_{3} - 3 \beta_1) q^{11} + ( - 7 \beta_{2} + 7) q^{13} - 9 \beta_{3} q^{17} + 8 q^{19} + 3 \beta_1 q^{23} + 5 \beta_{2} q^{25} + (21 \beta_{3} - 21 \beta_1) q^{29} + (29 \beta_{2} - 29) q^{31} - 4 \beta_{3} q^{35} + 2 q^{37} + 6 \beta_1 q^{41} + 7 \beta_{2} q^{43} + (15 \beta_{3} - 15 \beta_1) q^{47} + ( - 33 \beta_{2} + 33) q^{49} - 42 \beta_{3} q^{53} + 15 q^{55} - 18 \beta_1 q^{59} - 62 \beta_{2} q^{61} + (7 \beta_{3} - 7 \beta_1) q^{65} + ( - 58 \beta_{2} + 58) q^{67} - 24 \beta_{3} q^{71} - 52 q^{73} - 12 \beta_1 q^{77} + 49 \beta_{2} q^{79} + ( - 6 \beta_{3} + 6 \beta_1) q^{83} + (45 \beta_{2} - 45) q^{85} - 48 \beta_{3} q^{89} + 28 q^{91} - 8 \beta_1 q^{95} + 34 \beta_{2} q^{97}+O(q^{100})$$ q - b1 * q^5 + 4*b2 * q^7 + (3*b3 - 3*b1) * q^11 + (-7*b2 + 7) * q^13 - 9*b3 * q^17 + 8 * q^19 + 3*b1 * q^23 + 5*b2 * q^25 + (21*b3 - 21*b1) * q^29 + (29*b2 - 29) * q^31 - 4*b3 * q^35 + 2 * q^37 + 6*b1 * q^41 + 7*b2 * q^43 + (15*b3 - 15*b1) * q^47 + (-33*b2 + 33) * q^49 - 42*b3 * q^53 + 15 * q^55 - 18*b1 * q^59 - 62*b2 * q^61 + (7*b3 - 7*b1) * q^65 + (-58*b2 + 58) * q^67 - 24*b3 * q^71 - 52 * q^73 - 12*b1 * q^77 + 49*b2 * q^79 + (-6*b3 + 6*b1) * q^83 + (45*b2 - 45) * q^85 - 48*b3 * q^89 + 28 * q^91 - 8*b1 * q^95 + 34*b2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{7}+O(q^{10})$$ 4 * q + 8 * q^7 $$4 q + 8 q^{7} + 14 q^{13} + 32 q^{19} + 10 q^{25} - 58 q^{31} + 8 q^{37} + 14 q^{43} + 66 q^{49} + 60 q^{55} - 124 q^{61} + 116 q^{67} - 208 q^{73} + 98 q^{79} - 90 q^{85} + 112 q^{91} + 68 q^{97}+O(q^{100})$$ 4 * q + 8 * q^7 + 14 * q^13 + 32 * q^19 + 10 * q^25 - 58 * q^31 + 8 * q^37 + 14 * q^43 + 66 * q^49 + 60 * q^55 - 124 * q^61 + 116 * q^67 - 208 * q^73 + 98 * q^79 - 90 * q^85 + 112 * q^91 + 68 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 5$$ (v^2) / 5 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 5$$ (v^3) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$5\beta_{2}$$ 5*b2 $$\nu^{3}$$ $$=$$ $$5\beta_{3}$$ 5*b3

## Character values

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

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

## 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 2.00000 + 3.46410i 0 0 0
701.2 0 0 0 1.93649 + 1.11803i 0 2.00000 + 3.46410i 0 0 0
1241.1 0 0 0 −1.93649 + 1.11803i 0 2.00000 3.46410i 0 0 0
1241.2 0 0 0 1.93649 1.11803i 0 2.00000 3.46410i 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.c 4
3.b odd 2 1 inner 1620.3.o.c 4
9.c even 3 1 540.3.g.b 2
9.c even 3 1 inner 1620.3.o.c 4
9.d odd 6 1 540.3.g.b 2
9.d odd 6 1 inner 1620.3.o.c 4
36.f odd 6 1 2160.3.l.c 2
36.h even 6 1 2160.3.l.c 2
45.h odd 6 1 2700.3.g.k 2
45.j even 6 1 2700.3.g.k 2
45.k odd 12 2 2700.3.b.i 4
45.l even 12 2 2700.3.b.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.g.b 2 9.c even 3 1
540.3.g.b 2 9.d odd 6 1
1620.3.o.c 4 1.a even 1 1 trivial
1620.3.o.c 4 3.b odd 2 1 inner
1620.3.o.c 4 9.c even 3 1 inner
1620.3.o.c 4 9.d odd 6 1 inner
2160.3.l.c 2 36.f odd 6 1
2160.3.l.c 2 36.h even 6 1
2700.3.b.i 4 45.k odd 12 2
2700.3.b.i 4 45.l even 12 2
2700.3.g.k 2 45.h odd 6 1
2700.3.g.k 2 45.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 4T_{7} + 16$$ acting on $$S_{3}^{\mathrm{new}}(1620, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 5T^{2} + 25$$
$7$ $$(T^{2} - 4 T + 16)^{2}$$
$11$ $$T^{4} - 45T^{2} + 2025$$
$13$ $$(T^{2} - 7 T + 49)^{2}$$
$17$ $$(T^{2} + 405)^{2}$$
$19$ $$(T - 8)^{4}$$
$23$ $$T^{4} - 45T^{2} + 2025$$
$29$ $$T^{4} - 2205 T^{2} + \cdots + 4862025$$
$31$ $$(T^{2} + 29 T + 841)^{2}$$
$37$ $$(T - 2)^{4}$$
$41$ $$T^{4} - 180 T^{2} + 32400$$
$43$ $$(T^{2} - 7 T + 49)^{2}$$
$47$ $$T^{4} - 1125 T^{2} + \cdots + 1265625$$
$53$ $$(T^{2} + 8820)^{2}$$
$59$ $$T^{4} - 1620 T^{2} + \cdots + 2624400$$
$61$ $$(T^{2} + 62 T + 3844)^{2}$$
$67$ $$(T^{2} - 58 T + 3364)^{2}$$
$71$ $$(T^{2} + 2880)^{2}$$
$73$ $$(T + 52)^{4}$$
$79$ $$(T^{2} - 49 T + 2401)^{2}$$
$83$ $$T^{4} - 180 T^{2} + 32400$$
$89$ $$(T^{2} + 11520)^{2}$$
$97$ $$(T^{2} - 34 T + 1156)^{2}$$