# Properties

 Label 1620.3.t.c Level $1620$ Weight $3$ Character orbit 1620.t Analytic conductor $44.142$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1620.t (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$44.1418028264$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.1485512441856.6 Defining polynomial: $$x^{8} - 24x^{6} + 455x^{4} - 2904x^{2} + 14641$$ x^8 - 24*x^6 + 455*x^4 - 2904*x^2 + 14641 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 180) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} + \beta_{5} + \beta_{2}) q^{5} + \beta_{3} q^{7}+O(q^{10})$$ q + (b6 + b5 + b2) * q^5 + b3 * q^7 $$q + (\beta_{6} + \beta_{5} + \beta_{2}) q^{5} + \beta_{3} q^{7} - 7 \beta_{4} q^{11} + 3 \beta_{7} q^{13} - 4 \beta_{2} q^{17} + 12 q^{19} + (2 \beta_{5} + 2 \beta_{2}) q^{23} + (2 \beta_{3} - 21 \beta_1) q^{25} + 6 \beta_{4} q^{29} + ( - 38 \beta_1 + 38) q^{31} + ( - 23 \beta_{6} - 23 \beta_{4} + 2 \beta_{2}) q^{35} + ( - \beta_{7} - \beta_{3}) q^{37} - 49 \beta_{6} q^{41} - 10 \beta_{3} q^{43} + 16 \beta_{5} q^{47} + (3 \beta_1 - 3) q^{49} + (7 \beta_{7} + 7 \beta_{3} + 14) q^{55} + 59 \beta_{6} q^{59} + 70 \beta_1 q^{61} + (6 \beta_{5} + 69 \beta_{4}) q^{65} - 16 \beta_{7} q^{67} + (84 \beta_{6} + 84 \beta_{4}) q^{71} + ( - 2 \beta_{7} - 2 \beta_{3}) q^{73} + (14 \beta_{5} + 14 \beta_{2}) q^{77} - 30 \beta_1 q^{79} + 28 \beta_{5} q^{83} + (4 \beta_{7} + 92 \beta_1 - 92) q^{85} + (23 \beta_{6} + 23 \beta_{4}) q^{89} - 138 q^{91} + (12 \beta_{6} + 12 \beta_{5} + 12 \beta_{2}) q^{95} + 14 \beta_{3} q^{97}+O(q^{100})$$ q + (b6 + b5 + b2) * q^5 + b3 * q^7 - 7*b4 * q^11 + 3*b7 * q^13 - 4*b2 * q^17 + 12 * q^19 + (2*b5 + 2*b2) * q^23 + (2*b3 - 21*b1) * q^25 + 6*b4 * q^29 + (-38*b1 + 38) * q^31 + (-23*b6 - 23*b4 + 2*b2) * q^35 + (-b7 - b3) * q^37 - 49*b6 * q^41 - 10*b3 * q^43 + 16*b5 * q^47 + (3*b1 - 3) * q^49 + (7*b7 + 7*b3 + 14) * q^55 + 59*b6 * q^59 + 70*b1 * q^61 + (6*b5 + 69*b4) * q^65 - 16*b7 * q^67 + (84*b6 + 84*b4) * q^71 + (-2*b7 - 2*b3) * q^73 + (14*b5 + 14*b2) * q^77 - 30*b1 * q^79 + 28*b5 * q^83 + (4*b7 + 92*b1 - 92) * q^85 + (23*b6 + 23*b4) * q^89 - 138 * q^91 + (12*b6 + 12*b5 + 12*b2) * q^95 + 14*b3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 96 q^{19} - 84 q^{25} + 152 q^{31} - 12 q^{49} + 112 q^{55} + 280 q^{61} - 120 q^{79} - 368 q^{85} - 1104 q^{91}+O(q^{100})$$ 8 * q + 96 * q^19 - 84 * q^25 + 152 * q^31 - 12 * q^49 + 112 * q^55 + 280 * q^61 - 120 * q^79 - 368 * q^85 - 1104 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 24x^{6} + 455x^{4} - 2904x^{2} + 14641$$ :

 $$\beta_{1}$$ $$=$$ $$( -24\nu^{6} + 455\nu^{4} - 10920\nu^{2} + 69696 ) / 55055$$ (-24*v^6 + 455*v^4 - 10920*v^2 + 69696) / 55055 $$\beta_{2}$$ $$=$$ $$( -\nu^{6} - 2556 ) / 455$$ (-v^6 - 2556) / 455 $$\beta_{3}$$ $$=$$ $$( \nu^{7} + 13021\nu ) / 5005$$ (v^7 + 13021*v) / 5005 $$\beta_{4}$$ $$=$$ $$( \nu^{7} + 3011\nu ) / 5005$$ (v^7 + 3011*v) / 5005 $$\beta_{5}$$ $$=$$ $$( -167\nu^{6} + 5460\nu^{4} - 75985\nu^{2} + 484968 ) / 55055$$ (-167*v^6 + 5460*v^4 - 75985*v^2 + 484968) / 55055 $$\beta_{6}$$ $$=$$ $$( 191\nu^{7} - 5915\nu^{5} + 86905\nu^{3} - 554664\nu ) / 605605$$ (191*v^7 - 5915*v^5 + 86905*v^3 - 554664*v) / 605605 $$\beta_{7}$$ $$=$$ $$( 719\nu^{7} - 15925\nu^{5} + 327145\nu^{3} - 2087976\nu ) / 605605$$ (719*v^7 - 15925*v^5 + 327145*v^3 - 2087976*v) / 605605
 $$\nu$$ $$=$$ $$( -\beta_{4} + \beta_{3} ) / 2$$ (-b4 + b3) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{2} - 12\beta _1 + 12$$ b5 + b2 - 12*b1 + 12 $$\nu^{3}$$ $$=$$ $$( 13\beta_{7} - 35\beta_{6} - 35\beta_{4} + 13\beta_{3} ) / 2$$ (13*b7 - 35*b6 - 35*b4 + 13*b3) / 2 $$\nu^{4}$$ $$=$$ $$24\beta_{5} - 167\beta_1$$ 24*b5 - 167*b1 $$\nu^{5}$$ $$=$$ $$( 191\beta_{7} - 719\beta_{6} ) / 2$$ (191*b7 - 719*b6) / 2 $$\nu^{6}$$ $$=$$ $$-455\beta_{2} - 2556$$ -455*b2 - 2556 $$\nu^{7}$$ $$=$$ $$( 13021\beta_{4} - 3011\beta_{3} ) / 2$$ (13021*b4 - 3011*b3) / 2

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1
 2.32446 + 1.34203i −2.32446 − 1.34203i −3.54921 − 2.04914i 3.54921 + 2.04914i 2.32446 − 1.34203i −2.32446 + 1.34203i −3.54921 + 2.04914i 3.54921 − 2.04914i
0 0 0 −3.62266 3.44621i 0 5.87367 + 3.39116i 0 0 0
269.2 0 0 0 −1.17317 4.86042i 0 −5.87367 3.39116i 0 0 0
269.3 0 0 0 1.17317 + 4.86042i 0 −5.87367 3.39116i 0 0 0
269.4 0 0 0 3.62266 + 3.44621i 0 5.87367 + 3.39116i 0 0 0
1349.1 0 0 0 −3.62266 + 3.44621i 0 5.87367 3.39116i 0 0 0
1349.2 0 0 0 −1.17317 + 4.86042i 0 −5.87367 + 3.39116i 0 0 0
1349.3 0 0 0 1.17317 4.86042i 0 −5.87367 + 3.39116i 0 0 0
1349.4 0 0 0 3.62266 3.44621i 0 5.87367 3.39116i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1349.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.d odd 2 1 inner
45.h odd 6 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.3.t.c 8
3.b odd 2 1 inner 1620.3.t.c 8
5.b even 2 1 inner 1620.3.t.c 8
9.c even 3 1 180.3.b.a 4
9.c even 3 1 inner 1620.3.t.c 8
9.d odd 6 1 180.3.b.a 4
9.d odd 6 1 inner 1620.3.t.c 8
15.d odd 2 1 inner 1620.3.t.c 8
36.f odd 6 1 720.3.c.b 4
36.h even 6 1 720.3.c.b 4
45.h odd 6 1 180.3.b.a 4
45.h odd 6 1 inner 1620.3.t.c 8
45.j even 6 1 180.3.b.a 4
45.j even 6 1 inner 1620.3.t.c 8
45.k odd 12 2 900.3.g.c 4
45.l even 12 2 900.3.g.c 4
72.j odd 6 1 2880.3.c.c 4
72.l even 6 1 2880.3.c.f 4
72.n even 6 1 2880.3.c.c 4
72.p odd 6 1 2880.3.c.f 4
180.n even 6 1 720.3.c.b 4
180.p odd 6 1 720.3.c.b 4
180.v odd 12 2 3600.3.l.r 4
180.x even 12 2 3600.3.l.r 4
360.z odd 6 1 2880.3.c.f 4
360.bd even 6 1 2880.3.c.f 4
360.bh odd 6 1 2880.3.c.c 4
360.bk even 6 1 2880.3.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.b.a 4 9.c even 3 1
180.3.b.a 4 9.d odd 6 1
180.3.b.a 4 45.h odd 6 1
180.3.b.a 4 45.j even 6 1
720.3.c.b 4 36.f odd 6 1
720.3.c.b 4 36.h even 6 1
720.3.c.b 4 180.n even 6 1
720.3.c.b 4 180.p odd 6 1
900.3.g.c 4 45.k odd 12 2
900.3.g.c 4 45.l even 12 2
1620.3.t.c 8 1.a even 1 1 trivial
1620.3.t.c 8 3.b odd 2 1 inner
1620.3.t.c 8 5.b even 2 1 inner
1620.3.t.c 8 9.c even 3 1 inner
1620.3.t.c 8 9.d odd 6 1 inner
1620.3.t.c 8 15.d odd 2 1 inner
1620.3.t.c 8 45.h odd 6 1 inner
1620.3.t.c 8 45.j even 6 1 inner
2880.3.c.c 4 72.j odd 6 1
2880.3.c.c 4 72.n even 6 1
2880.3.c.c 4 360.bh odd 6 1
2880.3.c.c 4 360.bk even 6 1
2880.3.c.f 4 72.l even 6 1
2880.3.c.f 4 72.p odd 6 1
2880.3.c.f 4 360.z odd 6 1
2880.3.c.f 4 360.bd even 6 1
3600.3.l.r 4 180.v odd 12 2
3600.3.l.r 4 180.x even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1620, [\chi])$$:

 $$T_{7}^{4} - 46T_{7}^{2} + 2116$$ T7^4 - 46*T7^2 + 2116 $$T_{17}^{2} - 368$$ T17^2 - 368

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 42 T^{6} + 1139 T^{4} + \cdots + 390625$$
$7$ $$(T^{4} - 46 T^{2} + 2116)^{2}$$
$11$ $$(T^{4} - 98 T^{2} + 9604)^{2}$$
$13$ $$(T^{4} - 414 T^{2} + 171396)^{2}$$
$17$ $$(T^{2} - 368)^{4}$$
$19$ $$(T - 12)^{8}$$
$23$ $$(T^{4} + 92 T^{2} + 8464)^{2}$$
$29$ $$(T^{4} - 72 T^{2} + 5184)^{2}$$
$31$ $$(T^{2} - 38 T + 1444)^{4}$$
$37$ $$(T^{2} + 46)^{4}$$
$41$ $$(T^{4} - 4802 T^{2} + 23059204)^{2}$$
$43$ $$(T^{4} - 4600 T^{2} + 21160000)^{2}$$
$47$ $$(T^{4} + 5888 T^{2} + 34668544)^{2}$$
$53$ $$T^{8}$$
$59$ $$(T^{4} - 6962 T^{2} + 48469444)^{2}$$
$61$ $$(T^{2} - 70 T + 4900)^{4}$$
$67$ $$(T^{4} - 11776 T^{2} + \cdots + 138674176)^{2}$$
$71$ $$(T^{2} + 14112)^{4}$$
$73$ $$(T^{2} + 184)^{4}$$
$79$ $$(T^{2} + 30 T + 900)^{4}$$
$83$ $$(T^{4} + 18032 T^{2} + \cdots + 325153024)^{2}$$
$89$ $$(T^{2} + 1058)^{4}$$
$97$ $$(T^{4} - 9016 T^{2} + 81288256)^{2}$$