# Properties

 Label 1620.4.i.u Level $1620$ Weight $4$ Character orbit 1620.i Analytic conductor $95.583$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - x^{5} + 91x^{4} - 294x^{3} + 8292x^{2} - 17280x + 36864$$ x^6 - x^5 + 91*x^4 - 294*x^3 + 8292*x^2 - 17280*x + 36864 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (5 \beta_{2} + 5) q^{5} + ( - \beta_{4} + 5 \beta_{2}) q^{7}+O(q^{10})$$ q + (5*b2 + 5) * q^5 + (-b4 + 5*b2) * q^7 $$q + (5 \beta_{2} + 5) q^{5} + ( - \beta_{4} + 5 \beta_{2}) q^{7} + (\beta_{5} - 8 \beta_{2}) q^{11} + (\beta_{4} - \beta_{3} - 11 \beta_{2} - 11) q^{13} + ( - 2 \beta_{3} + 2 \beta_1 - 14) q^{17} + ( - 3 \beta_{3} + \beta_1 + 7) q^{19} + ( - 2 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 11 \beta_{2} + 2 \beta_1 - 11) q^{23} + 25 \beta_{2} q^{25} + ( - 2 \beta_{5} + 5 \beta_{4} - 74 \beta_{2}) q^{29} + (3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 44 \beta_{2} - 3 \beta_1 - 44) q^{31} + ( - 5 \beta_{3} - 25) q^{35} + ( - 8 \beta_{3} - 2 \beta_1 + 58) q^{37} + ( - 6 \beta_{4} + 6 \beta_{3} - 33 \beta_{2} - 33) q^{41} + ( - 6 \beta_{5} + 2 \beta_{4} - 40 \beta_{2}) q^{43} + ( - 2 \beta_{5} + 3 \beta_{4} - 179 \beta_{2}) q^{47} + ( - 8 \beta_{5} + 17 \beta_{4} - 17 \beta_{3} - 164 \beta_{2} + 8 \beta_1 - 164) q^{49} + (9 \beta_{3} - 10 \beta_1 - 89) q^{53} + (5 \beta_1 + 40) q^{55} + (9 \beta_{5} + 5 \beta_{4} - 5 \beta_{3} - 75 \beta_{2} - 9 \beta_1 - 75) q^{59} + (12 \beta_{5} + 6 \beta_{4} - 160 \beta_{2}) q^{61} + (5 \beta_{4} - 55 \beta_{2}) q^{65} + (10 \beta_{5} - 20 \beta_{4} + 20 \beta_{3} - 4 \beta_{2} - 10 \beta_1 - 4) q^{67} + ( - 8 \beta_{3} + \beta_1 - 190) q^{71} + (2 \beta_{3} - 10 \beta_1 + 354) q^{73} + (2 \beta_{5} + 16 \beta_{4} - 16 \beta_{3} + 104 \beta_{2} - 2 \beta_1 + 104) q^{77} + (4 \beta_{5} - 26 \beta_{4} - 342 \beta_{2}) q^{79} + (18 \beta_{5} - 32 \beta_{4} - 234 \beta_{2}) q^{83} + ( - 10 \beta_{5} + 10 \beta_{4} - 10 \beta_{3} - 70 \beta_{2} + 10 \beta_1 - 70) q^{85} + (11 \beta_{3} + 8 \beta_1 - 380) q^{89} + (23 \beta_{3} - 8 \beta_1 + 537) q^{91} + ( - 5 \beta_{5} + 15 \beta_{4} - 15 \beta_{3} + 35 \beta_{2} + 5 \beta_1 + 35) q^{95} + ( - 2 \beta_{5} + 26 \beta_{4} - 852 \beta_{2}) q^{97}+O(q^{100})$$ q + (5*b2 + 5) * q^5 + (-b4 + 5*b2) * q^7 + (b5 - 8*b2) * q^11 + (b4 - b3 - 11*b2 - 11) * q^13 + (-2*b3 + 2*b1 - 14) * q^17 + (-3*b3 + b1 + 7) * q^19 + (-2*b5 - 3*b4 + 3*b3 - 11*b2 + 2*b1 - 11) * q^23 + 25*b2 * q^25 + (-2*b5 + 5*b4 - 74*b2) * q^29 + (3*b5 + 2*b4 - 2*b3 - 44*b2 - 3*b1 - 44) * q^31 + (-5*b3 - 25) * q^35 + (-8*b3 - 2*b1 + 58) * q^37 + (-6*b4 + 6*b3 - 33*b2 - 33) * q^41 + (-6*b5 + 2*b4 - 40*b2) * q^43 + (-2*b5 + 3*b4 - 179*b2) * q^47 + (-8*b5 + 17*b4 - 17*b3 - 164*b2 + 8*b1 - 164) * q^49 + (9*b3 - 10*b1 - 89) * q^53 + (5*b1 + 40) * q^55 + (9*b5 + 5*b4 - 5*b3 - 75*b2 - 9*b1 - 75) * q^59 + (12*b5 + 6*b4 - 160*b2) * q^61 + (5*b4 - 55*b2) * q^65 + (10*b5 - 20*b4 + 20*b3 - 4*b2 - 10*b1 - 4) * q^67 + (-8*b3 + b1 - 190) * q^71 + (2*b3 - 10*b1 + 354) * q^73 + (2*b5 + 16*b4 - 16*b3 + 104*b2 - 2*b1 + 104) * q^77 + (4*b5 - 26*b4 - 342*b2) * q^79 + (18*b5 - 32*b4 - 234*b2) * q^83 + (-10*b5 + 10*b4 - 10*b3 - 70*b2 + 10*b1 - 70) * q^85 + (11*b3 + 8*b1 - 380) * q^89 + (23*b3 - 8*b1 + 537) * q^91 + (-5*b5 + 15*b4 - 15*b3 + 35*b2 + 5*b1 + 35) * q^95 + (-2*b5 + 26*b4 - 852*b2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 15 q^{5} - 15 q^{7}+O(q^{10})$$ 6 * q + 15 * q^5 - 15 * q^7 $$6 q + 15 q^{5} - 15 q^{7} + 24 q^{11} - 33 q^{13} - 84 q^{17} + 42 q^{19} - 33 q^{23} - 75 q^{25} + 222 q^{29} - 132 q^{31} - 150 q^{35} + 348 q^{37} - 99 q^{41} + 120 q^{43} + 537 q^{47} - 492 q^{49} - 534 q^{53} + 240 q^{55} - 225 q^{59} + 480 q^{61} + 165 q^{65} - 12 q^{67} - 1140 q^{71} + 2124 q^{73} + 312 q^{77} + 1026 q^{79} + 702 q^{83} - 210 q^{85} - 2280 q^{89} + 3222 q^{91} + 105 q^{95} + 2556 q^{97}+O(q^{100})$$ 6 * q + 15 * q^5 - 15 * q^7 + 24 * q^11 - 33 * q^13 - 84 * q^17 + 42 * q^19 - 33 * q^23 - 75 * q^25 + 222 * q^29 - 132 * q^31 - 150 * q^35 + 348 * q^37 - 99 * q^41 + 120 * q^43 + 537 * q^47 - 492 * q^49 - 534 * q^53 + 240 * q^55 - 225 * q^59 + 480 * q^61 + 165 * q^65 - 12 * q^67 - 1140 * q^71 + 2124 * q^73 + 312 * q^77 + 1026 * q^79 + 702 * q^83 - 210 * q^85 - 2280 * q^89 + 3222 * q^91 + 105 * q^95 + 2556 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 91x^{4} - 294x^{3} + 8292x^{2} - 17280x + 36864$$ :

 $$\beta_{1}$$ $$=$$ $$( 79\nu^{5} - 7189\nu^{4} - 83093\nu^{3} - 655068\nu^{2} + 1365120\nu - 29852544 ) / 1474584$$ (79*v^5 - 7189*v^4 - 83093*v^3 - 655068*v^2 + 1365120*v - 29852544) / 1474584 $$\beta_{2}$$ $$=$$ $$( -1365\nu^{5} + 1333\nu^{4} - 121303\nu^{3} + 136318\nu^{2} - 11053236\nu - 559104 ) / 23593344$$ (-1365*v^5 + 1333*v^4 - 121303*v^3 + 136318*v^2 - 11053236*v - 559104) / 23593344 $$\beta_{3}$$ $$=$$ $$( -91\nu^{5} + 8281\nu^{4} - 16279\nu^{3} + 754572\nu^{2} - 1572480\nu + 45773136 ) / 1474584$$ (-91*v^5 + 8281*v^4 - 16279*v^3 + 754572*v^2 - 1572480*v + 45773136) / 1474584 $$\beta_{4}$$ $$=$$ $$( 6585\nu^{5} + 15175\nu^{4} + 585187\nu^{3} - 657622\nu^{2} + 53085092\nu + 2697216 ) / 3932224$$ (6585*v^5 + 15175*v^4 + 585187*v^3 - 657622*v^2 + 53085092*v + 2697216) / 3932224 $$\beta_{5}$$ $$=$$ $$( -660\nu^{5} - 1381\nu^{4} - 58652\nu^{3} + 65912\nu^{2} - 3110265\nu - 270336 ) / 368646$$ (-660*v^5 - 1381*v^4 - 58652*v^3 + 65912*v^2 - 3110265*v - 270336) / 368646
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} - 2\beta_{2} ) / 6$$ (b5 + b4 - 2*b2) / 6 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - 11\beta_{4} + 11\beta_{3} - 362\beta_{2} - \beta _1 - 362 ) / 6$$ (b5 - 11*b4 + 11*b3 - 362*b2 - b1 - 362) / 6 $$\nu^{3}$$ $$=$$ $$( -79\beta_{3} - 91\beta _1 + 610 ) / 6$$ (-79*b3 - 91*b1 + 610) / 6 $$\nu^{4}$$ $$=$$ $$( 11\beta_{5} + 1103\beta_{4} + 31586\beta_{2} ) / 6$$ (11*b5 + 1103*b4 + 31586*b2) / 6 $$\nu^{5}$$ $$=$$ $$( -7987\beta_{5} - 8119\beta_{4} + 8119\beta_{3} - 92818\beta_{2} + 7987\beta _1 - 92818 ) / 6$$ (-7987*b5 - 8119*b4 + 8119*b3 - 92818*b2 + 7987*b1 - 92818) / 6

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
541.1
 1.09880 − 1.90317i 4.38373 − 7.59284i −4.98253 + 8.62999i 1.09880 + 1.90317i 4.38373 + 7.59284i −4.98253 − 8.62999i
0 0 0 2.50000 + 4.33013i 0 −16.8420 + 29.1713i 0 0 0
541.2 0 0 0 2.50000 + 4.33013i 0 −0.474792 + 0.822364i 0 0 0
541.3 0 0 0 2.50000 + 4.33013i 0 9.81683 17.0033i 0 0 0
1081.1 0 0 0 2.50000 4.33013i 0 −16.8420 29.1713i 0 0 0
1081.2 0 0 0 2.50000 4.33013i 0 −0.474792 0.822364i 0 0 0
1081.3 0 0 0 2.50000 4.33013i 0 9.81683 + 17.0033i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1081.3 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.u 6
3.b odd 2 1 1620.4.i.s 6
9.c even 3 1 1620.4.a.d 3
9.c even 3 1 inner 1620.4.i.u 6
9.d odd 6 1 1620.4.a.f yes 3
9.d odd 6 1 1620.4.i.s 6

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

## Hecke kernels

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

 $$T_{7}^{6} + 15T_{7}^{5} + 873T_{7}^{4} - 8464T_{7}^{3} + 429324T_{7}^{2} + 406944T_{7} + 394384$$ T7^6 + 15*T7^5 + 873*T7^4 - 8464*T7^3 + 429324*T7^2 + 406944*T7 + 394384 $$T_{11}^{6} - 24T_{11}^{5} + 2721T_{11}^{4} + 86184T_{11}^{3} + 4184577T_{11}^{2} + 37220040T_{11} + 301091904$$ T11^6 - 24*T11^5 + 2721*T11^4 + 86184*T11^3 + 4184577*T11^2 + 37220040*T11 + 301091904

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$(T^{2} - 5 T + 25)^{3}$$
$7$ $$T^{6} + 15 T^{5} + 873 T^{4} + \cdots + 394384$$
$11$ $$T^{6} - 24 T^{5} + \cdots + 301091904$$
$13$ $$T^{6} + 33 T^{5} + 1449 T^{4} + \cdots + 14137600$$
$17$ $$(T^{3} + 42 T^{2} - 10884 T - 429480)^{2}$$
$19$ $$(T^{3} - 21 T^{2} - 8121 T + 311725)^{2}$$
$23$ $$T^{6} + 33 T^{5} + \cdots + 301915083024$$
$29$ $$T^{6} - 222 T^{5} + \cdots + 124366254336$$
$31$ $$T^{6} + 132 T^{5} + \cdots + 1300238478400$$
$37$ $$(T^{3} - 174 T^{2} - 48600 T - 849920)^{2}$$
$41$ $$T^{6} + 99 T^{5} + \cdots + 2076996910041$$
$43$ $$T^{6} + \cdots + 163840000000000$$
$47$ $$T^{6} - 537 T^{5} + \cdots + 9594134553600$$
$53$ $$(T^{3} + 267 T^{2} - 251220 T + 14075604)^{2}$$
$59$ $$T^{6} + \cdots + 162393249576321$$
$61$ $$T^{6} - 480 T^{5} + \cdots + 17\!\cdots\!00$$
$67$ $$T^{6} + 12 T^{5} + \cdots + 34\!\cdots\!96$$
$71$ $$(T^{3} + 570 T^{2} + 61227 T + 1353402)^{2}$$
$73$ $$(T^{3} - 1062 T^{2} + 143196 T + 79568728)^{2}$$
$79$ $$T^{6} - 1026 T^{5} + \cdots + 69\!\cdots\!00$$
$83$ $$T^{6} - 702 T^{5} + \cdots + 22\!\cdots\!00$$
$89$ $$(T^{3} + 1140 T^{2} + 179253 T - 2459646)^{2}$$
$97$ $$T^{6} - 2556 T^{5} + \cdots + 88\!\cdots\!00$$
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