# Properties

 Label 1620.4.i.p Level $1620$ Weight $4$ Character orbit 1620.i Analytic conductor $95.583$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{41})$$ Defining polynomial: $$x^{4} - x^{3} + 11x^{2} + 10x + 100$$ x^4 - x^3 + 11*x^2 + 10*x + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3^{2}$$ 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 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.

 $$f(q)$$ $$=$$ $$q + ( - 5 \beta_1 + 5) q^{5} + (\beta_{2} - 7 \beta_1) q^{7}+O(q^{10})$$ q + (-5*b1 + 5) * q^5 + (b2 - 7*b1) * q^7 $$q + ( - 5 \beta_1 + 5) q^{5} + (\beta_{2} - 7 \beta_1) q^{7} + ( - 5 \beta_{2} + 10 \beta_1) q^{11} + ( - 4 \beta_{3} - 4 \beta_{2} + 7 \beta_1 - 3) q^{13} + (5 \beta_{3} - 25) q^{17} + ( - 13 \beta_{3} + 28) q^{19} + ( - 5 \beta_{3} - 5 \beta_{2} - 50 \beta_1 + 55) q^{23} - 25 \beta_1 q^{25} + (5 \beta_{2} + 50 \beta_1) q^{29} + (\beta_{3} + \beta_{2} - 36 \beta_1 + 35) q^{31} + ( - 5 \beta_{3} - 30) q^{35} + (15 \beta_{3} - 82) q^{37} + (30 \beta_{3} + 30 \beta_{2} - 210 \beta_1 + 180) q^{41} + (7 \beta_{2} - 88 \beta_1) q^{43} + (25 \beta_{2} + 160 \beta_1) q^{47} + ( - 13 \beta_{3} - 13 \beta_{2} - 202 \beta_1 + 215) q^{49} + (10 \beta_{3} - 380) q^{53} + (25 \beta_{3} + 25) q^{55} + ( - 20 \beta_{3} - 20 \beta_{2} - 440 \beta_1 + 460) q^{59} + ( - 51 \beta_{2} + 25 \beta_1) q^{61} + ( - 20 \beta_{2} + 35 \beta_1) q^{65} + (63 \beta_{3} + 63 \beta_{2} - 199 \beta_1 + 136) q^{67} + (30 \beta_{3} - 840) q^{71} + ( - 15 \beta_{3} + 86) q^{73} + (40 \beta_{3} + 40 \beta_{2} - 530 \beta_1 + 490) q^{77} + (60 \beta_{2} - 203 \beta_1) q^{79} + ( - 100 \beta_{2} + 350 \beta_1) q^{83} + (25 \beta_{3} + 25 \beta_{2} + 100 \beta_1 - 125) q^{85} + ( - 60 \beta_{3} - 720) q^{89} + (31 \beta_{3} + 386) q^{91} + ( - 65 \beta_{3} - 65 \beta_{2} - 75 \beta_1 + 140) q^{95} + (29 \beta_{2} + 297 \beta_1) q^{97}+O(q^{100})$$ q + (-5*b1 + 5) * q^5 + (b2 - 7*b1) * q^7 + (-5*b2 + 10*b1) * q^11 + (-4*b3 - 4*b2 + 7*b1 - 3) * q^13 + (5*b3 - 25) * q^17 + (-13*b3 + 28) * q^19 + (-5*b3 - 5*b2 - 50*b1 + 55) * q^23 - 25*b1 * q^25 + (5*b2 + 50*b1) * q^29 + (b3 + b2 - 36*b1 + 35) * q^31 + (-5*b3 - 30) * q^35 + (15*b3 - 82) * q^37 + (30*b3 + 30*b2 - 210*b1 + 180) * q^41 + (7*b2 - 88*b1) * q^43 + (25*b2 + 160*b1) * q^47 + (-13*b3 - 13*b2 - 202*b1 + 215) * q^49 + (10*b3 - 380) * q^53 + (25*b3 + 25) * q^55 + (-20*b3 - 20*b2 - 440*b1 + 460) * q^59 + (-51*b2 + 25*b1) * q^61 + (-20*b2 + 35*b1) * q^65 + (63*b3 + 63*b2 - 199*b1 + 136) * q^67 + (30*b3 - 840) * q^71 + (-15*b3 + 86) * q^73 + (40*b3 + 40*b2 - 530*b1 + 490) * q^77 + (60*b2 - 203*b1) * q^79 + (-100*b2 + 350*b1) * q^83 + (25*b3 + 25*b2 + 100*b1 - 125) * q^85 + (-60*b3 - 720) * q^89 + (31*b3 + 386) * q^91 + (-65*b3 - 65*b2 - 75*b1 + 140) * q^95 + (29*b2 + 297*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{5} - 13 q^{7}+O(q^{10})$$ 4 * q + 10 * q^5 - 13 * q^7 $$4 q + 10 q^{5} - 13 q^{7} + 15 q^{11} - 10 q^{13} - 90 q^{17} + 86 q^{19} + 105 q^{23} - 50 q^{25} + 105 q^{29} + 71 q^{31} - 130 q^{35} - 298 q^{37} + 390 q^{41} - 169 q^{43} + 345 q^{47} + 417 q^{49} - 1500 q^{53} + 150 q^{55} + 900 q^{59} - q^{61} + 50 q^{65} + 335 q^{67} - 3300 q^{71} + 314 q^{73} + 1020 q^{77} - 346 q^{79} + 600 q^{83} - 225 q^{85} - 3000 q^{89} + 1606 q^{91} + 215 q^{95} + 623 q^{97}+O(q^{100})$$ 4 * q + 10 * q^5 - 13 * q^7 + 15 * q^11 - 10 * q^13 - 90 * q^17 + 86 * q^19 + 105 * q^23 - 50 * q^25 + 105 * q^29 + 71 * q^31 - 130 * q^35 - 298 * q^37 + 390 * q^41 - 169 * q^43 + 345 * q^47 + 417 * q^49 - 1500 * q^53 + 150 * q^55 + 900 * q^59 - q^61 + 50 * q^65 + 335 * q^67 - 3300 * q^71 + 314 * q^73 + 1020 * q^77 - 346 * q^79 + 600 * q^83 - 225 * q^85 - 3000 * q^89 + 1606 * q^91 + 215 * q^95 + 623 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 11x^{2} + 10x + 100$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 11\nu^{2} - 11\nu + 100 ) / 110$$ (-v^3 + 11*v^2 - 11*v + 100) / 110 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 11\nu^{2} + 341\nu - 100 ) / 110$$ (v^3 - 11*v^2 + 341*v - 100) / 110 $$\beta_{3}$$ $$=$$ $$( 3\nu^{3} + 52 ) / 11$$ (3*v^3 + 52) / 11
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 3$$ (b2 + b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 31\beta _1 - 32 ) / 3$$ (b3 + b2 + 31*b1 - 32) / 3 $$\nu^{3}$$ $$=$$ $$( 11\beta_{3} - 52 ) / 3$$ (11*b3 - 52) / 3

## Character values

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

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

## 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
 −1.35078 + 2.33962i 1.85078 − 3.20565i −1.35078 − 2.33962i 1.85078 + 3.20565i
0 0 0 2.50000 + 4.33013i 0 −8.05234 + 13.9471i 0 0 0
541.2 0 0 0 2.50000 + 4.33013i 0 1.55234 2.68874i 0 0 0
1081.1 0 0 0 2.50000 4.33013i 0 −8.05234 13.9471i 0 0 0
1081.2 0 0 0 2.50000 4.33013i 0 1.55234 + 2.68874i 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.p 4
3.b odd 2 1 1620.4.i.m 4
9.c even 3 1 540.4.a.g 2
9.c even 3 1 inner 1620.4.i.p 4
9.d odd 6 1 540.4.a.j yes 2
9.d odd 6 1 1620.4.i.m 4
36.f odd 6 1 2160.4.a.u 2
36.h even 6 1 2160.4.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.g 2 9.c even 3 1
540.4.a.j yes 2 9.d odd 6 1
1620.4.i.m 4 3.b odd 2 1
1620.4.i.m 4 9.d odd 6 1
1620.4.i.p 4 1.a even 1 1 trivial
1620.4.i.p 4 9.c even 3 1 inner
2160.4.a.u 2 36.f odd 6 1
2160.4.a.z 2 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 $$S_{4}^{\mathrm{new}}(1620, [\chi])$$:

 $$T_{7}^{4} + 13T_{7}^{3} + 219T_{7}^{2} - 650T_{7} + 2500$$ T7^4 + 13*T7^3 + 219*T7^2 - 650*T7 + 2500 $$T_{11}^{4} - 15T_{11}^{3} + 2475T_{11}^{2} + 33750T_{11} + 5062500$$ T11^4 - 15*T11^3 + 2475*T11^2 + 33750*T11 + 5062500

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 5 T + 25)^{2}$$
$7$ $$T^{4} + 13 T^{3} + 219 T^{2} + \cdots + 2500$$
$11$ $$T^{4} - 15 T^{3} + 2475 T^{2} + \cdots + 5062500$$
$13$ $$T^{4} + 10 T^{3} + 1551 T^{2} + \cdots + 2105401$$
$17$ $$(T^{2} + 45 T - 1800)^{2}$$
$19$ $$(T^{2} - 43 T - 15128)^{2}$$
$23$ $$T^{4} - 105 T^{3} + 10575 T^{2} + \cdots + 202500$$
$29$ $$T^{4} - 105 T^{3} + 10575 T^{2} + \cdots + 202500$$
$31$ $$T^{4} - 71 T^{3} + 3873 T^{2} + \cdots + 1364224$$
$37$ $$(T^{2} + 149 T - 15206)^{2}$$
$41$ $$T^{4} - 390 T^{3} + \cdots + 2025000000$$
$43$ $$T^{4} + 169 T^{3} + 25941 T^{2} + \cdots + 6864400$$
$47$ $$T^{4} - 345 T^{3} + \cdots + 778410000$$
$53$ $$(T^{2} + 750 T + 131400)^{2}$$
$59$ $$T^{4} - 900 T^{3} + \cdots + 27423360000$$
$61$ $$T^{4} + T^{3} + 239943 T^{2} + \cdots + 57572163364$$
$67$ $$T^{4} - 335 T^{3} + \cdots + 114300791056$$
$71$ $$(T^{2} + 1650 T + 597600)^{2}$$
$73$ $$(T^{2} - 157 T - 14594)^{2}$$
$79$ $$T^{4} + 346 T^{3} + \cdots + 91307313241$$
$83$ $$T^{4} - 600 T^{3} + \cdots + 693056250000$$
$89$ $$(T^{2} + 1500 T + 230400)^{2}$$
$97$ $$T^{4} - 623 T^{3} + \cdots + 378302500$$