# Properties

 Label 1620.4.i.a Level $1620$ Weight $4$ Character orbit 1620.i Analytic conductor $95.583$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1620.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$95.5830942093$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 5 \zeta_{6} q^{5} + (32 \zeta_{6} - 32) q^{7} +O(q^{10})$$ q - 5*z * q^5 + (32*z - 32) * q^7 $$q - 5 \zeta_{6} q^{5} + (32 \zeta_{6} - 32) q^{7} + (36 \zeta_{6} - 36) q^{11} + 10 \zeta_{6} q^{13} - 78 q^{17} + 140 q^{19} + 192 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + (6 \zeta_{6} - 6) q^{29} + 16 \zeta_{6} q^{31} + 160 q^{35} - 34 q^{37} + 390 \zeta_{6} q^{41} + ( - 52 \zeta_{6} + 52) q^{43} + (408 \zeta_{6} - 408) q^{47} - 681 \zeta_{6} q^{49} - 114 q^{53} + 180 q^{55} - 516 \zeta_{6} q^{59} + ( - 58 \zeta_{6} + 58) q^{61} + ( - 50 \zeta_{6} + 50) q^{65} + 892 \zeta_{6} q^{67} - 120 q^{71} - 646 q^{73} - 1152 \zeta_{6} q^{77} + ( - 1168 \zeta_{6} + 1168) q^{79} + ( - 732 \zeta_{6} + 732) q^{83} + 390 \zeta_{6} q^{85} - 1590 q^{89} - 320 q^{91} - 700 \zeta_{6} q^{95} + (194 \zeta_{6} - 194) q^{97} +O(q^{100})$$ q - 5*z * q^5 + (32*z - 32) * q^7 + (36*z - 36) * q^11 + 10*z * q^13 - 78 * q^17 + 140 * q^19 + 192*z * q^23 + (25*z - 25) * q^25 + (6*z - 6) * q^29 + 16*z * q^31 + 160 * q^35 - 34 * q^37 + 390*z * q^41 + (-52*z + 52) * q^43 + (408*z - 408) * q^47 - 681*z * q^49 - 114 * q^53 + 180 * q^55 - 516*z * q^59 + (-58*z + 58) * q^61 + (-50*z + 50) * q^65 + 892*z * q^67 - 120 * q^71 - 646 * q^73 - 1152*z * q^77 + (-1168*z + 1168) * q^79 + (-732*z + 732) * q^83 + 390*z * q^85 - 1590 * q^89 - 320 * q^91 - 700*z * q^95 + (194*z - 194) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 5 q^{5} - 32 q^{7}+O(q^{10})$$ 2 * q - 5 * q^5 - 32 * q^7 $$2 q - 5 q^{5} - 32 q^{7} - 36 q^{11} + 10 q^{13} - 156 q^{17} + 280 q^{19} + 192 q^{23} - 25 q^{25} - 6 q^{29} + 16 q^{31} + 320 q^{35} - 68 q^{37} + 390 q^{41} + 52 q^{43} - 408 q^{47} - 681 q^{49} - 228 q^{53} + 360 q^{55} - 516 q^{59} + 58 q^{61} + 50 q^{65} + 892 q^{67} - 240 q^{71} - 1292 q^{73} - 1152 q^{77} + 1168 q^{79} + 732 q^{83} + 390 q^{85} - 3180 q^{89} - 640 q^{91} - 700 q^{95} - 194 q^{97}+O(q^{100})$$ 2 * q - 5 * q^5 - 32 * q^7 - 36 * q^11 + 10 * q^13 - 156 * q^17 + 280 * q^19 + 192 * q^23 - 25 * q^25 - 6 * q^29 + 16 * q^31 + 320 * q^35 - 68 * q^37 + 390 * q^41 + 52 * q^43 - 408 * q^47 - 681 * q^49 - 228 * q^53 + 360 * q^55 - 516 * q^59 + 58 * q^61 + 50 * q^65 + 892 * q^67 - 240 * q^71 - 1292 * q^73 - 1152 * q^77 + 1168 * q^79 + 732 * q^83 + 390 * q^85 - 3180 * q^89 - 640 * q^91 - 700 * q^95 - 194 * q^97

## Character values

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

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
541.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −2.50000 4.33013i 0 −16.0000 + 27.7128i 0 0 0
1081.1 0 0 0 −2.50000 + 4.33013i 0 −16.0000 27.7128i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.4.i.a 2
3.b odd 2 1 1620.4.i.g 2
9.c even 3 1 60.4.a.b 1
9.c even 3 1 inner 1620.4.i.a 2
9.d odd 6 1 180.4.a.c 1
9.d odd 6 1 1620.4.i.g 2
36.f odd 6 1 240.4.a.j 1
36.h even 6 1 720.4.a.c 1
45.h odd 6 1 900.4.a.b 1
45.j even 6 1 300.4.a.e 1
45.k odd 12 2 300.4.d.d 2
45.l even 12 2 900.4.d.b 2
72.n even 6 1 960.4.a.bb 1
72.p odd 6 1 960.4.a.a 1
180.p odd 6 1 1200.4.a.s 1
180.x even 12 2 1200.4.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.a.b 1 9.c even 3 1
180.4.a.c 1 9.d odd 6 1
240.4.a.j 1 36.f odd 6 1
300.4.a.e 1 45.j even 6 1
300.4.d.d 2 45.k odd 12 2
720.4.a.c 1 36.h even 6 1
900.4.a.b 1 45.h odd 6 1
900.4.d.b 2 45.l even 12 2
960.4.a.a 1 72.p odd 6 1
960.4.a.bb 1 72.n even 6 1
1200.4.a.s 1 180.p odd 6 1
1200.4.f.e 2 180.x even 12 2
1620.4.i.a 2 1.a even 1 1 trivial
1620.4.i.a 2 9.c even 3 1 inner
1620.4.i.g 2 3.b odd 2 1
1620.4.i.g 2 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1620, [\chi])$$:

 $$T_{7}^{2} + 32T_{7} + 1024$$ T7^2 + 32*T7 + 1024 $$T_{11}^{2} + 36T_{11} + 1296$$ T11^2 + 36*T11 + 1296

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 5T + 25$$
$7$ $$T^{2} + 32T + 1024$$
$11$ $$T^{2} + 36T + 1296$$
$13$ $$T^{2} - 10T + 100$$
$17$ $$(T + 78)^{2}$$
$19$ $$(T - 140)^{2}$$
$23$ $$T^{2} - 192T + 36864$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2} - 16T + 256$$
$37$ $$(T + 34)^{2}$$
$41$ $$T^{2} - 390T + 152100$$
$43$ $$T^{2} - 52T + 2704$$
$47$ $$T^{2} + 408T + 166464$$
$53$ $$(T + 114)^{2}$$
$59$ $$T^{2} + 516T + 266256$$
$61$ $$T^{2} - 58T + 3364$$
$67$ $$T^{2} - 892T + 795664$$
$71$ $$(T + 120)^{2}$$
$73$ $$(T + 646)^{2}$$
$79$ $$T^{2} - 1168 T + 1364224$$
$83$ $$T^{2} - 732T + 535824$$
$89$ $$(T + 1590)^{2}$$
$97$ $$T^{2} + 194T + 37636$$