# Properties

 Label 1620.4.i.h 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 540) 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.

 $$f(q)$$ $$=$$ $$q + 5 \zeta_{6} q^{5} + (17 \zeta_{6} - 17) q^{7}+O(q^{10})$$ q + 5*z * q^5 + (17*z - 17) * q^7 $$q + 5 \zeta_{6} q^{5} + (17 \zeta_{6} - 17) q^{7} + (30 \zeta_{6} - 30) q^{11} + 61 \zeta_{6} q^{13} + 120 q^{17} - 43 q^{19} + 90 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + (90 \zeta_{6} - 90) q^{29} - 8 \zeta_{6} q^{31} - 85 q^{35} + 317 q^{37} - 30 \zeta_{6} q^{41} + ( - 220 \zeta_{6} + 220) q^{43} + (180 \zeta_{6} - 180) q^{47} + 54 \zeta_{6} q^{49} + 630 q^{53} - 150 q^{55} - 840 \zeta_{6} q^{59} + (599 \zeta_{6} - 599) q^{61} + (305 \zeta_{6} - 305) q^{65} - 107 \zeta_{6} q^{67} + 210 q^{71} - 421 q^{73} - 510 \zeta_{6} q^{77} + (353 \zeta_{6} - 353) q^{79} + (1350 \zeta_{6} - 1350) q^{83} + 600 \zeta_{6} q^{85} - 1020 q^{89} - 1037 q^{91} - 215 \zeta_{6} q^{95} + ( - 997 \zeta_{6} + 997) q^{97} +O(q^{100})$$ q + 5*z * q^5 + (17*z - 17) * q^7 + (30*z - 30) * q^11 + 61*z * q^13 + 120 * q^17 - 43 * q^19 + 90*z * q^23 + (25*z - 25) * q^25 + (90*z - 90) * q^29 - 8*z * q^31 - 85 * q^35 + 317 * q^37 - 30*z * q^41 + (-220*z + 220) * q^43 + (180*z - 180) * q^47 + 54*z * q^49 + 630 * q^53 - 150 * q^55 - 840*z * q^59 + (599*z - 599) * q^61 + (305*z - 305) * q^65 - 107*z * q^67 + 210 * q^71 - 421 * q^73 - 510*z * q^77 + (353*z - 353) * q^79 + (1350*z - 1350) * q^83 + 600*z * q^85 - 1020 * q^89 - 1037 * q^91 - 215*z * q^95 + (-997*z + 997) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{5} - 17 q^{7}+O(q^{10})$$ 2 * q + 5 * q^5 - 17 * q^7 $$2 q + 5 q^{5} - 17 q^{7} - 30 q^{11} + 61 q^{13} + 240 q^{17} - 86 q^{19} + 90 q^{23} - 25 q^{25} - 90 q^{29} - 8 q^{31} - 170 q^{35} + 634 q^{37} - 30 q^{41} + 220 q^{43} - 180 q^{47} + 54 q^{49} + 1260 q^{53} - 300 q^{55} - 840 q^{59} - 599 q^{61} - 305 q^{65} - 107 q^{67} + 420 q^{71} - 842 q^{73} - 510 q^{77} - 353 q^{79} - 1350 q^{83} + 600 q^{85} - 2040 q^{89} - 2074 q^{91} - 215 q^{95} + 997 q^{97}+O(q^{100})$$ 2 * q + 5 * q^5 - 17 * q^7 - 30 * q^11 + 61 * q^13 + 240 * q^17 - 86 * q^19 + 90 * q^23 - 25 * q^25 - 90 * q^29 - 8 * q^31 - 170 * q^35 + 634 * q^37 - 30 * q^41 + 220 * q^43 - 180 * q^47 + 54 * q^49 + 1260 * q^53 - 300 * q^55 - 840 * q^59 - 599 * q^61 - 305 * q^65 - 107 * q^67 + 420 * q^71 - 842 * q^73 - 510 * q^77 - 353 * q^79 - 1350 * q^83 + 600 * q^85 - 2040 * q^89 - 2074 * q^91 - 215 * q^95 + 997 * 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 −8.50000 + 14.7224i 0 0 0
1081.1 0 0 0 2.50000 4.33013i 0 −8.50000 14.7224i 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.h 2
3.b odd 2 1 1620.4.i.b 2
9.c even 3 1 540.4.a.b 1
9.c even 3 1 inner 1620.4.i.h 2
9.d odd 6 1 540.4.a.d yes 1
9.d odd 6 1 1620.4.i.b 2
36.f odd 6 1 2160.4.a.a 1
36.h even 6 1 2160.4.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.b 1 9.c even 3 1
540.4.a.d yes 1 9.d odd 6 1
1620.4.i.b 2 3.b odd 2 1
1620.4.i.b 2 9.d odd 6 1
1620.4.i.h 2 1.a even 1 1 trivial
1620.4.i.h 2 9.c even 3 1 inner
2160.4.a.a 1 36.f odd 6 1
2160.4.a.k 1 36.h even 6 1

## Hecke kernels

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

 $$T_{7}^{2} + 17T_{7} + 289$$ T7^2 + 17*T7 + 289 $$T_{11}^{2} + 30T_{11} + 900$$ T11^2 + 30*T11 + 900

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 5T + 25$$
$7$ $$T^{2} + 17T + 289$$
$11$ $$T^{2} + 30T + 900$$
$13$ $$T^{2} - 61T + 3721$$
$17$ $$(T - 120)^{2}$$
$19$ $$(T + 43)^{2}$$
$23$ $$T^{2} - 90T + 8100$$
$29$ $$T^{2} + 90T + 8100$$
$31$ $$T^{2} + 8T + 64$$
$37$ $$(T - 317)^{2}$$
$41$ $$T^{2} + 30T + 900$$
$43$ $$T^{2} - 220T + 48400$$
$47$ $$T^{2} + 180T + 32400$$
$53$ $$(T - 630)^{2}$$
$59$ $$T^{2} + 840T + 705600$$
$61$ $$T^{2} + 599T + 358801$$
$67$ $$T^{2} + 107T + 11449$$
$71$ $$(T - 210)^{2}$$
$73$ $$(T + 421)^{2}$$
$79$ $$T^{2} + 353T + 124609$$
$83$ $$T^{2} + 1350 T + 1822500$$
$89$ $$(T + 1020)^{2}$$
$97$ $$T^{2} - 997T + 994009$$
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