# Properties

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

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## 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 180) 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.

 $$f(q)$$ $$=$$ $$q - 5 \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} +O(q^{10})$$ q - 5*z * q^5 + (2*z - 2) * q^7 $$q - 5 \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} + (30 \zeta_{6} - 30) q^{11} + 4 \zeta_{6} q^{13} + 90 q^{17} - 28 q^{19} - 120 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + (210 \zeta_{6} - 210) q^{29} + 4 \zeta_{6} q^{31} + 10 q^{35} + 200 q^{37} - 240 \zeta_{6} q^{41} + ( - 136 \zeta_{6} + 136) q^{43} + ( - 120 \zeta_{6} + 120) q^{47} + 339 \zeta_{6} q^{49} - 30 q^{53} + 150 q^{55} + 450 \zeta_{6} q^{59} + ( - 166 \zeta_{6} + 166) q^{61} + ( - 20 \zeta_{6} + 20) q^{65} - 908 \zeta_{6} q^{67} - 1020 q^{71} - 250 q^{73} - 60 \zeta_{6} q^{77} + ( - 916 \zeta_{6} + 916) q^{79} + ( - 1140 \zeta_{6} + 1140) q^{83} - 450 \zeta_{6} q^{85} - 420 q^{89} - 8 q^{91} + 140 \zeta_{6} q^{95} + (1538 \zeta_{6} - 1538) q^{97} +O(q^{100})$$ q - 5*z * q^5 + (2*z - 2) * q^7 + (30*z - 30) * q^11 + 4*z * q^13 + 90 * q^17 - 28 * q^19 - 120*z * q^23 + (25*z - 25) * q^25 + (210*z - 210) * q^29 + 4*z * q^31 + 10 * q^35 + 200 * q^37 - 240*z * q^41 + (-136*z + 136) * q^43 + (-120*z + 120) * q^47 + 339*z * q^49 - 30 * q^53 + 150 * q^55 + 450*z * q^59 + (-166*z + 166) * q^61 + (-20*z + 20) * q^65 - 908*z * q^67 - 1020 * q^71 - 250 * q^73 - 60*z * q^77 + (-916*z + 916) * q^79 + (-1140*z + 1140) * q^83 - 450*z * q^85 - 420 * q^89 - 8 * q^91 + 140*z * q^95 + (1538*z - 1538) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 5 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q - 5 * q^5 - 2 * q^7 $$2 q - 5 q^{5} - 2 q^{7} - 30 q^{11} + 4 q^{13} + 180 q^{17} - 56 q^{19} - 120 q^{23} - 25 q^{25} - 210 q^{29} + 4 q^{31} + 20 q^{35} + 400 q^{37} - 240 q^{41} + 136 q^{43} + 120 q^{47} + 339 q^{49} - 60 q^{53} + 300 q^{55} + 450 q^{59} + 166 q^{61} + 20 q^{65} - 908 q^{67} - 2040 q^{71} - 500 q^{73} - 60 q^{77} + 916 q^{79} + 1140 q^{83} - 450 q^{85} - 840 q^{89} - 16 q^{91} + 140 q^{95} - 1538 q^{97}+O(q^{100})$$ 2 * q - 5 * q^5 - 2 * q^7 - 30 * q^11 + 4 * q^13 + 180 * q^17 - 56 * q^19 - 120 * q^23 - 25 * q^25 - 210 * q^29 + 4 * q^31 + 20 * q^35 + 400 * q^37 - 240 * q^41 + 136 * q^43 + 120 * q^47 + 339 * q^49 - 60 * q^53 + 300 * q^55 + 450 * q^59 + 166 * q^61 + 20 * q^65 - 908 * q^67 - 2040 * q^71 - 500 * q^73 - 60 * q^77 + 916 * q^79 + 1140 * q^83 - 450 * q^85 - 840 * q^89 - 16 * q^91 + 140 * q^95 - 1538 * 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 −1.00000 + 1.73205i 0 0 0
1081.1 0 0 0 −2.50000 + 4.33013i 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
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.c 2
3.b odd 2 1 1620.4.i.i 2
9.c even 3 1 180.4.a.e yes 1
9.c even 3 1 inner 1620.4.i.c 2
9.d odd 6 1 180.4.a.b 1
9.d odd 6 1 1620.4.i.i 2
36.f odd 6 1 720.4.a.w 1
36.h even 6 1 720.4.a.h 1
45.h odd 6 1 900.4.a.i 1
45.j even 6 1 900.4.a.j 1
45.k odd 12 2 900.4.d.i 2
45.l even 12 2 900.4.d.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.4.a.b 1 9.d odd 6 1
180.4.a.e yes 1 9.c even 3 1
720.4.a.h 1 36.h even 6 1
720.4.a.w 1 36.f odd 6 1
900.4.a.i 1 45.h odd 6 1
900.4.a.j 1 45.j even 6 1
900.4.d.d 2 45.l even 12 2
900.4.d.i 2 45.k odd 12 2
1620.4.i.c 2 1.a even 1 1 trivial
1620.4.i.c 2 9.c even 3 1 inner
1620.4.i.i 2 3.b odd 2 1
1620.4.i.i 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 $$S_{4}^{\mathrm{new}}(1620, [\chi])$$:

 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4 $$T_{11}^{2} + 30T_{11} + 900$$ T11^2 + 30*T11 + 900

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 5T + 25$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} + 30T + 900$$
$13$ $$T^{2} - 4T + 16$$
$17$ $$(T - 90)^{2}$$
$19$ $$(T + 28)^{2}$$
$23$ $$T^{2} + 120T + 14400$$
$29$ $$T^{2} + 210T + 44100$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$(T - 200)^{2}$$
$41$ $$T^{2} + 240T + 57600$$
$43$ $$T^{2} - 136T + 18496$$
$47$ $$T^{2} - 120T + 14400$$
$53$ $$(T + 30)^{2}$$
$59$ $$T^{2} - 450T + 202500$$
$61$ $$T^{2} - 166T + 27556$$
$67$ $$T^{2} + 908T + 824464$$
$71$ $$(T + 1020)^{2}$$
$73$ $$(T + 250)^{2}$$
$79$ $$T^{2} - 916T + 839056$$
$83$ $$T^{2} - 1140 T + 1299600$$
$89$ $$(T + 420)^{2}$$
$97$ $$T^{2} + 1538 T + 2365444$$
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