# Properties

 Label 1620.4.i.v Level $1620$ Weight $4$ Character orbit 1620.i Analytic conductor $95.583$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - x^{5} + 91x^{4} + 570x^{3} + 7860x^{2} + 21600x + 57600$$ x^6 - x^5 + 91*x^4 + 570*x^3 + 7860*x^2 + 21600*x + 57600 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.

 $$f(q)$$ $$=$$ $$q + (5 \beta_1 + 5) q^{5} + ( - \beta_{4} - \beta_1) q^{7}+O(q^{10})$$ q + (5*b1 + 5) * q^5 + (-b4 - b1) * q^7 $$q + (5 \beta_1 + 5) q^{5} + ( - \beta_{4} - \beta_1) q^{7} + (\beta_{5} + \beta_{4} + 8 \beta_1) q^{11} + (2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{13} + (2 \beta_{3} + 2 \beta_{2} + 10) q^{17} + ( - \beta_{3} - 4 \beta_{2} - 9) q^{19} + ( - \beta_{4} - \beta_{2} + 33 \beta_1 + 33) q^{23} + 25 \beta_1 q^{25} + (\beta_{4} + 18 \beta_1) q^{29} + (\beta_{5} - 5 \beta_{4} + \beta_{3} - 5 \beta_{2} - 24 \beta_1 - 24) q^{31} + (5 \beta_{2} + 5) q^{35} + (2 \beta_{3} - 8 \beta_{2} - 6) q^{37} + (4 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 137 \beta_1 + 137) q^{41} + (4 \beta_{5} + 10 \beta_{4} + 148 \beta_1) q^{43} + ( - 10 \beta_{5} + 3 \beta_{4} + 25 \beta_1) q^{47} + ( - 10 \beta_{5} - 7 \beta_{4} - 10 \beta_{3} - 7 \beta_{2} - 68 \beta_1 - 68) q^{49} + ( - \beta_{2} - 57) q^{53} + ( - 5 \beta_{3} - 5 \beta_{2} - 40) q^{55} + ( - 9 \beta_{5} - 28 \beta_{4} - 9 \beta_{3} - 28 \beta_{2} + 99 \beta_1 + 99) q^{59} + ( - 16 \beta_{5} - 2 \beta_{4} + 228 \beta_1) q^{61} + (10 \beta_{5} - 5 \beta_{4} - 5 \beta_1) q^{65} + ( - 10 \beta_{5} + 10 \beta_{4} - 10 \beta_{3} + 10 \beta_{2} - 4 \beta_1 - 4) q^{67} + ( - 29 \beta_{3} - \beta_{2} - 214) q^{71} + (18 \beta_{3} + 30 \beta_{2} - 22) q^{73} + (6 \beta_{5} + 32 \beta_{4} + 6 \beta_{3} + 32 \beta_{2} + 348 \beta_1 + 348) q^{77} + (12 \beta_{5} - 6 \beta_{4} + 374 \beta_1) q^{79} + (28 \beta_{4} - 30 \beta_1) q^{83} + (10 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} + 10 \beta_{2} + 50 \beta_1 + 50) q^{85} + (18 \beta_{3} - 33 \beta_{2} - 252) q^{89} + ( - 18 \beta_{3} + 29 \beta_{2} - 551) q^{91} + ( - 5 \beta_{5} - 20 \beta_{4} - 5 \beta_{3} - 20 \beta_{2} - 45 \beta_1 - 45) q^{95} + (6 \beta_{5} - 26 \beta_{4} + 512 \beta_1) q^{97}+O(q^{100})$$ q + (5*b1 + 5) * q^5 + (-b4 - b1) * q^7 + (b5 + b4 + 8*b1) * q^11 + (2*b5 - b4 + 2*b3 - b2 - b1 - 1) * q^13 + (2*b3 + 2*b2 + 10) * q^17 + (-b3 - 4*b2 - 9) * q^19 + (-b4 - b2 + 33*b1 + 33) * q^23 + 25*b1 * q^25 + (b4 + 18*b1) * q^29 + (b5 - 5*b4 + b3 - 5*b2 - 24*b1 - 24) * q^31 + (5*b2 + 5) * q^35 + (2*b3 - 8*b2 - 6) * q^37 + (4*b5 + 2*b4 + 4*b3 + 2*b2 + 137*b1 + 137) * q^41 + (4*b5 + 10*b4 + 148*b1) * q^43 + (-10*b5 + 3*b4 + 25*b1) * q^47 + (-10*b5 - 7*b4 - 10*b3 - 7*b2 - 68*b1 - 68) * q^49 + (-b2 - 57) * q^53 + (-5*b3 - 5*b2 - 40) * q^55 + (-9*b5 - 28*b4 - 9*b3 - 28*b2 + 99*b1 + 99) * q^59 + (-16*b5 - 2*b4 + 228*b1) * q^61 + (10*b5 - 5*b4 - 5*b1) * q^65 + (-10*b5 + 10*b4 - 10*b3 + 10*b2 - 4*b1 - 4) * q^67 + (-29*b3 - b2 - 214) * q^71 + (18*b3 + 30*b2 - 22) * q^73 + (6*b5 + 32*b4 + 6*b3 + 32*b2 + 348*b1 + 348) * q^77 + (12*b5 - 6*b4 + 374*b1) * q^79 + (28*b4 - 30*b1) * q^83 + (10*b5 + 10*b4 + 10*b3 + 10*b2 + 50*b1 + 50) * q^85 + (18*b3 - 33*b2 - 252) * q^89 + (-18*b3 + 29*b2 - 551) * q^91 + (-5*b5 - 20*b4 - 5*b3 - 20*b2 - 45*b1 - 45) * q^95 + (6*b5 - 26*b4 + 512*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 15 q^{5} + 3 q^{7}+O(q^{10})$$ 6 * q + 15 * q^5 + 3 * q^7 $$6 q + 15 q^{5} + 3 q^{7} - 24 q^{11} - 3 q^{13} + 60 q^{17} - 54 q^{19} + 99 q^{23} - 75 q^{25} - 54 q^{29} - 72 q^{31} + 30 q^{35} - 36 q^{37} + 411 q^{41} - 444 q^{43} - 75 q^{47} - 204 q^{49} - 342 q^{53} - 240 q^{55} + 297 q^{59} - 684 q^{61} + 15 q^{65} - 12 q^{67} - 1284 q^{71} - 132 q^{73} + 1044 q^{77} - 1122 q^{79} + 90 q^{83} + 150 q^{85} - 1512 q^{89} - 3306 q^{91} - 135 q^{95} - 1536 q^{97}+O(q^{100})$$ 6 * q + 15 * q^5 + 3 * q^7 - 24 * q^11 - 3 * q^13 + 60 * q^17 - 54 * q^19 + 99 * q^23 - 75 * q^25 - 54 * q^29 - 72 * q^31 + 30 * q^35 - 36 * q^37 + 411 * q^41 - 444 * q^43 - 75 * q^47 - 204 * q^49 - 342 * q^53 - 240 * q^55 + 297 * q^59 - 684 * q^61 + 15 * q^65 - 12 * q^67 - 1284 * q^71 - 132 * q^73 + 1044 * q^77 - 1122 * q^79 + 90 * q^83 + 150 * q^85 - 1512 * q^89 - 3306 * q^91 - 135 * q^95 - 1536 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 91x^{4} + 570x^{3} + 7860x^{2} + 21600x + 57600$$ :

 $$\beta_{1}$$ $$=$$ $$( -273\nu^{5} + 281\nu^{4} - 25571\nu^{3} - 89362\nu^{2} - 2208660\nu - 6069600 ) / 5894880$$ (-273*v^5 + 281*v^4 - 25571*v^3 - 89362*v^2 - 2208660*v - 6069600) / 5894880 $$\beta_{2}$$ $$=$$ $$( 91\nu^{5} - 8281\nu^{4} + 16711\nu^{3} - 715260\nu^{2} - 1965600\nu - 42188400 ) / 1473720$$ (91*v^5 - 8281*v^4 + 16711*v^3 - 715260*v^2 - 1965600*v - 42188400) / 1473720 $$\beta_{3}$$ $$=$$ $$( -97\nu^{5} + 8827\nu^{4} - 66397\nu^{3} + 762420\nu^{2} + 2095200\nu + 28921080 ) / 736860$$ (-97*v^5 + 8827*v^4 - 66397*v^3 + 762420*v^2 + 2095200*v + 28921080) / 736860 $$\beta_{4}$$ $$=$$ $$( 4093\nu^{5} - 4033\nu^{4} + 367003\nu^{3} + 2829870\nu^{2} + 31699380\nu + 87112800 ) / 2947440$$ (4093*v^5 - 4033*v^4 + 367003*v^3 + 2829870*v^2 + 31699380*v + 87112800) / 2947440 $$\beta_{5}$$ $$=$$ $$( -7889\nu^{5} + 5601\nu^{4} - 509691\nu^{3} - 5759018\nu^{2} - 44023860\nu - 120981600 ) / 2947440$$ (-7889*v^5 + 5601*v^4 - 509691*v^3 - 5759018*v^2 - 44023860*v - 120981600) / 2947440
 $$\nu$$ $$=$$ $$( \beta_{5} + 2\beta_{4} + \beta_{3} + 2\beta_{2} + 2\beta _1 + 2 ) / 6$$ (b5 + 2*b4 + b3 + 2*b2 + 2*b1 + 2) / 6 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + 14\beta_{4} + 362\beta_1 ) / 6$$ (b5 + 14*b4 + 362*b1) / 6 $$\nu^{3}$$ $$=$$ $$( -91\beta_{3} - 194\beta_{2} - 1982 ) / 6$$ (-91*b3 - 194*b2 - 1982) / 6 $$\nu^{4}$$ $$=$$ $$( -421\beta_{5} - 1934\beta_{4} - 421\beta_{3} - 1934\beta_{2} - 35042\beta _1 - 35042 ) / 6$$ (-421*b5 - 1934*b4 - 421*b3 - 1934*b2 - 35042*b1 - 35042) / 6 $$\nu^{5}$$ $$=$$ $$( -8851\beta_{5} - 22754\beta_{4} - 300302\beta_1 ) / 6$$ (-8851*b5 - 22754*b4 - 300302*b1) / 6

## Character values

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

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

## 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
 −1.55365 − 2.69100i −3.48579 − 6.03757i 5.53944 + 9.59460i −1.55365 + 2.69100i −3.48579 + 6.03757i 5.53944 − 9.59460i
0 0 0 2.50000 + 4.33013i 0 −11.3093 + 19.5883i 0 0 0
541.2 0 0 0 2.50000 + 4.33013i 0 −0.606369 + 1.05026i 0 0 0
541.3 0 0 0 2.50000 + 4.33013i 0 13.4157 23.2367i 0 0 0
1081.1 0 0 0 2.50000 4.33013i 0 −11.3093 19.5883i 0 0 0
1081.2 0 0 0 2.50000 4.33013i 0 −0.606369 1.05026i 0 0 0
1081.3 0 0 0 2.50000 4.33013i 0 13.4157 + 23.2367i 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.v 6
3.b odd 2 1 1620.4.i.t 6
9.c even 3 1 1620.4.a.c 3
9.c even 3 1 inner 1620.4.i.v 6
9.d odd 6 1 1620.4.a.e yes 3
9.d odd 6 1 1620.4.i.t 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.4.a.c 3 9.c even 3 1
1620.4.a.e yes 3 9.d odd 6 1
1620.4.i.t 6 3.b odd 2 1
1620.4.i.t 6 9.d odd 6 1
1620.4.i.v 6 1.a even 1 1 trivial
1620.4.i.v 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 $$S_{4}^{\mathrm{new}}(1620, [\chi])$$:

 $$T_{7}^{6} - 3T_{7}^{5} + 621T_{7}^{4} + 3308T_{7}^{3} + 372336T_{7}^{2} + 450432T_{7} + 541696$$ T7^6 - 3*T7^5 + 621*T7^4 + 3308*T7^3 + 372336*T7^2 + 450432*T7 + 541696 $$T_{11}^{6} + 24T_{11}^{5} + 2001T_{11}^{4} + 216T_{11}^{3} + 2443617T_{11}^{2} + 24521400T_{11} + 296115264$$ T11^6 + 24*T11^5 + 2001*T11^4 + 216*T11^3 + 2443617*T11^2 + 24521400*T11 + 296115264

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$(T^{2} - 5 T + 25)^{3}$$
$7$ $$T^{6} - 3 T^{5} + 621 T^{4} + \cdots + 541696$$
$11$ $$T^{6} + 24 T^{5} + \cdots + 296115264$$
$13$ $$T^{6} + 3 T^{5} + 5889 T^{4} + \cdots + 50694400$$
$17$ $$(T^{3} - 30 T^{2} - 6168 T + 101952)^{2}$$
$19$ $$(T^{3} + 27 T^{2} - 9969 T + 289237)^{2}$$
$23$ $$T^{6} - 99 T^{5} + \cdots + 204261264$$
$29$ $$T^{6} + 54 T^{5} + 2559 T^{4} + \cdots + 15116544$$
$31$ $$T^{6} + 72 T^{5} + \cdots + 785953171600$$
$37$ $$(T^{3} + 18 T^{2} - 47460 T - 2963720)^{2}$$
$41$ $$T^{6} - 411 T^{5} + \cdots + 1388407386249$$
$43$ $$T^{6} + 444 T^{5} + \cdots + 69780505600$$
$47$ $$T^{6} + 75 T^{5} + \cdots + 10844507610000$$
$53$ $$(T^{3} + 171 T^{2} + 9132 T + 151488)^{2}$$
$59$ $$T^{6} - 297 T^{5} + \cdots + 39\!\cdots\!89$$
$61$ $$T^{6} + 684 T^{5} + \cdots + 13780131865600$$
$67$ $$T^{6} + \cdots + 552309409597696$$
$71$ $$(T^{3} + 642 T^{2} - 876429 T - 547252470)^{2}$$
$73$ $$(T^{3} + 66 T^{2} - 831336 T - 197601920)^{2}$$
$79$ $$T^{6} + \cdots + 709033342182400$$
$83$ $$T^{6} - 90 T^{5} + \cdots + 19\!\cdots\!00$$
$89$ $$(T^{3} + 756 T^{2} - 996651 T - 780931530)^{2}$$
$97$ $$T^{6} + 1536 T^{5} + \cdots + 12\!\cdots\!00$$