# Properties

 Label 324.4.e.i Level $324$ Weight $4$ Character orbit 324.e Analytic conductor $19.117$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 324.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.1166188419$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16$$ x^8 + 3*x^6 + 5*x^4 + 12*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{8}\cdot 3^{10}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{5} + ( - \beta_{7} + \beta_{4} - 4 \beta_1) q^{7}+O(q^{10})$$ q - b3 * q^5 + (-b7 + b4 - 4*b1) * q^7 $$q - \beta_{3} q^{5} + ( - \beta_{7} + \beta_{4} - 4 \beta_1) q^{7} + (\beta_{6} - \beta_{5} - \beta_{2}) q^{11} + ( - \beta_{7} - 29 \beta_1 - 29) q^{13} + (5 \beta_{6} + 2 \beta_{5} + 2 \beta_{3}) q^{17} + (\beta_{4} + 56) q^{19} + (7 \beta_{3} + 5 \beta_{2}) q^{23} + (6 \beta_{7} - 6 \beta_{4} + 154 \beta_1) q^{25} + (12 \beta_{6} + 5 \beta_{5} - 12 \beta_{2}) q^{29} + (6 \beta_{7} - 92 \beta_1 - 92) q^{31} + (25 \beta_{6} - 13 \beta_{5} - 13 \beta_{3}) q^{35} + ( - 7 \beta_{4} + 167) q^{37} + ( - 16 \beta_{3} + 4 \beta_{2}) q^{41} + ( - 9 \beta_{7} + 9 \beta_{4} + 164 \beta_1) q^{43} + (20 \beta_{6} - 20 \beta_{2}) q^{47} + ( - 8 \beta_{7} - 429 \beta_1 - 429) q^{49} + (30 \beta_{6} + 22 \beta_{5} + 22 \beta_{3}) q^{53} + (15 \beta_{4} + 360) q^{55} + (4 \beta_{3} + 28 \beta_{2}) q^{59} + ( - 9 \beta_{7} + 9 \beta_{4} + 251 \beta_1) q^{61} + (25 \beta_{6} - 38 \beta_{5} - 25 \beta_{2}) q^{65} + ( - 15 \beta_{7} - 80 \beta_1 - 80) q^{67} + (41 \beta_{6} + 23 \beta_{5} + 23 \beta_{3}) q^{71} + 305 q^{73} + (40 \beta_{3} - 20 \beta_{2}) q^{77} + (25 \beta_{7} - 25 \beta_{4} - 16 \beta_1) q^{79} + ( - 10 \beta_{6} + 54 \beta_{5} + 10 \beta_{2}) q^{83} + (33 \beta_{7} - 153 \beta_1 - 153) q^{85} + ( - 43 \beta_{6} - 88 \beta_{5} - 88 \beta_{3}) q^{89} + ( - 33 \beta_{4} - 872) q^{91} + ( - 65 \beta_{3} + 25 \beta_{2}) q^{95} + (4 \beta_{7} - 4 \beta_{4} + 506 \beta_1) q^{97}+O(q^{100})$$ q - b3 * q^5 + (-b7 + b4 - 4*b1) * q^7 + (b6 - b5 - b2) * q^11 + (-b7 - 29*b1 - 29) * q^13 + (5*b6 + 2*b5 + 2*b3) * q^17 + (b4 + 56) * q^19 + (7*b3 + 5*b2) * q^23 + (6*b7 - 6*b4 + 154*b1) * q^25 + (12*b6 + 5*b5 - 12*b2) * q^29 + (6*b7 - 92*b1 - 92) * q^31 + (25*b6 - 13*b5 - 13*b3) * q^35 + (-7*b4 + 167) * q^37 + (-16*b3 + 4*b2) * q^41 + (-9*b7 + 9*b4 + 164*b1) * q^43 + (20*b6 - 20*b2) * q^47 + (-8*b7 - 429*b1 - 429) * q^49 + (30*b6 + 22*b5 + 22*b3) * q^53 + (15*b4 + 360) * q^55 + (4*b3 + 28*b2) * q^59 + (-9*b7 + 9*b4 + 251*b1) * q^61 + (25*b6 - 38*b5 - 25*b2) * q^65 + (-15*b7 - 80*b1 - 80) * q^67 + (41*b6 + 23*b5 + 23*b3) * q^71 + 305 * q^73 + (40*b3 - 20*b2) * q^77 + (25*b7 - 25*b4 - 16*b1) * q^79 + (-10*b6 + 54*b5 + 10*b2) * q^83 + (33*b7 - 153*b1 - 153) * q^85 + (-43*b6 - 88*b5 - 88*b3) * q^89 + (-33*b4 - 872) * q^91 + (-65*b3 + 25*b2) * q^95 + (4*b7 - 4*b4 + 506*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 16 q^{7}+O(q^{10})$$ 8 * q + 16 * q^7 $$8 q + 16 q^{7} - 116 q^{13} + 448 q^{19} - 616 q^{25} - 368 q^{31} + 1336 q^{37} - 656 q^{43} - 1716 q^{49} + 2880 q^{55} - 1004 q^{61} - 320 q^{67} + 2440 q^{73} + 64 q^{79} - 612 q^{85} - 6976 q^{91} - 2024 q^{97}+O(q^{100})$$ 8 * q + 16 * q^7 - 116 * q^13 + 448 * q^19 - 616 * q^25 - 368 * q^31 + 1336 * q^37 - 656 * q^43 - 1716 * q^49 + 2880 * q^55 - 1004 * q^61 - 320 * q^67 + 2440 * q^73 + 64 * q^79 - 612 * q^85 - 6976 * q^91 - 2024 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( 3\nu^{6} + 5\nu^{4} + 15\nu^{2} + 16 ) / 20$$ (3*v^6 + 5*v^4 + 15*v^2 + 16) / 20 $$\beta_{2}$$ $$=$$ $$( -9\nu^{7} - 45\nu^{5} + 45\nu^{3} - 18\nu ) / 20$$ (-9*v^7 - 45*v^5 + 45*v^3 - 18*v) / 20 $$\beta_{3}$$ $$=$$ $$( 3\nu^{7} + 3\nu^{5} - 3\nu^{3} - 30\nu ) / 4$$ (3*v^7 + 3*v^5 - 3*v^3 - 30*v) / 4 $$\beta_{4}$$ $$=$$ $$-3\nu^{6} - 9\nu^{4} - 3\nu^{2} - 18$$ -3*v^6 - 9*v^4 - 3*v^2 - 18 $$\beta_{5}$$ $$=$$ $$( 9\nu^{7} + 27\nu^{5} + 69\nu^{3} + 132\nu ) / 8$$ (9*v^7 + 27*v^5 + 69*v^3 + 132*v) / 8 $$\beta_{6}$$ $$=$$ $$( 9\nu^{7} - 9\nu^{5} + 9\nu^{3} + 72\nu ) / 8$$ (9*v^7 - 9*v^5 + 9*v^3 + 72*v) / 8 $$\beta_{7}$$ $$=$$ $$( -51\nu^{6} - 45\nu^{4} - 15\nu^{2} - 252 ) / 10$$ (-51*v^6 - 45*v^4 - 15*v^2 - 252) / 10
 $$\nu$$ $$=$$ $$( 4\beta_{6} - 9\beta_{3} - 5\beta_{2} ) / 108$$ (4*b6 - 9*b3 - 5*b2) / 108 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{4} + 54\beta_1 ) / 36$$ (b7 + b4 + 54*b1) / 36 $$\nu^{3}$$ $$=$$ $$( -7\beta_{6} + 9\beta_{5} + 9\beta_{3} + 20\beta_{2} ) / 108$$ (-7*b6 + 9*b5 + 9*b3 + 20*b2) / 108 $$\nu^{4}$$ $$=$$ $$( \beta_{7} - 2\beta_{4} - 6\beta _1 - 6 ) / 12$$ (b7 - 2*b4 - 6*b1 - 6) / 12 $$\nu^{5}$$ $$=$$ $$( -19\beta_{6} + 9\beta_{5} - 25\beta_{2} ) / 108$$ (-19*b6 + 9*b5 - 25*b2) / 108 $$\nu^{6}$$ $$=$$ $$( -10\beta_{7} + 5\beta_{4} - 162 ) / 36$$ (-10*b7 + 5*b4 - 162) / 36 $$\nu^{7}$$ $$=$$ $$( 52\beta_{6} + 63\beta_{3} - 5\beta_{2} ) / 108$$ (52*b6 + 63*b3 - 5*b2) / 108

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$$.

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 −1.09445 + 0.895644i −0.228425 + 1.39564i 0.228425 − 1.39564i 1.09445 − 0.895644i −1.09445 − 0.895644i −0.228425 − 1.39564i 0.228425 + 1.39564i 1.09445 + 0.895644i
0 0 0 −10.5353 + 18.2477i 0 15.7477 + 27.2759i 0 0 0
109.2 0 0 0 −5.33918 + 9.24773i 0 −11.7477 20.3477i 0 0 0
109.3 0 0 0 5.33918 9.24773i 0 −11.7477 20.3477i 0 0 0
109.4 0 0 0 10.5353 18.2477i 0 15.7477 + 27.2759i 0 0 0
217.1 0 0 0 −10.5353 18.2477i 0 15.7477 27.2759i 0 0 0
217.2 0 0 0 −5.33918 9.24773i 0 −11.7477 + 20.3477i 0 0 0
217.3 0 0 0 5.33918 + 9.24773i 0 −11.7477 + 20.3477i 0 0 0
217.4 0 0 0 10.5353 + 18.2477i 0 15.7477 27.2759i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 217.4 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
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.4.e.i 8
3.b odd 2 1 inner 324.4.e.i 8
9.c even 3 1 324.4.a.e 4
9.c even 3 1 inner 324.4.e.i 8
9.d odd 6 1 324.4.a.e 4
9.d odd 6 1 inner 324.4.e.i 8
36.f odd 6 1 1296.4.a.z 4
36.h even 6 1 1296.4.a.z 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.4.a.e 4 9.c even 3 1
324.4.a.e 4 9.d odd 6 1
324.4.e.i 8 1.a even 1 1 trivial
324.4.e.i 8 3.b odd 2 1 inner
324.4.e.i 8 9.c even 3 1 inner
324.4.e.i 8 9.d odd 6 1 inner
1296.4.a.z 4 36.f odd 6 1
1296.4.a.z 4 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 $$S_{4}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}^{8} + 558T_{5}^{6} + 260739T_{5}^{4} + 28248750T_{5}^{2} + 2562890625$$ T5^8 + 558*T5^6 + 260739*T5^4 + 28248750*T5^2 + 2562890625 $$T_{7}^{4} - 8T_{7}^{3} + 804T_{7}^{2} + 5920T_{7} + 547600$$ T7^4 - 8*T7^3 + 804*T7^2 + 5920*T7 + 547600

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 558 T^{6} + \cdots + 2562890625$$
$7$ $$(T^{4} - 8 T^{3} + 804 T^{2} + \cdots + 547600)^{2}$$
$11$ $$T^{8} + 1368 T^{6} + \cdots + 1049760000$$
$13$ $$(T^{4} + 58 T^{3} + 3279 T^{2} + \cdots + 7225)^{2}$$
$17$ $$(T^{4} - 11142 T^{2} + 12638025)^{2}$$
$19$ $$(T^{2} - 112 T + 2380)^{4}$$
$23$ $$T^{8} + 28152 T^{6} + \cdots + 12\!\cdots\!00$$
$29$ $$T^{8} + 64494 T^{6} + \cdots + 14\!\cdots\!81$$
$31$ $$(T^{4} + 184 T^{3} + 52608 T^{2} + \cdots + 351637504)^{2}$$
$37$ $$(T^{2} - 334 T - 9155)^{4}$$
$41$ $$T^{8} + 171360 T^{6} + \cdots + 35\!\cdots\!96$$
$43$ $$(T^{4} + 328 T^{3} + 141924 T^{2} + \cdots + 1179235600)^{2}$$
$47$ $$(T^{4} + 97200 T^{2} + \cdots + 9447840000)^{2}$$
$53$ $$(T^{4} - 493632 T^{2} + 8294400)^{2}$$
$59$ $$T^{8} + 353664 T^{6} + \cdots + 81\!\cdots\!76$$
$61$ $$(T^{4} + 502 T^{3} + 250239 T^{2} + \cdots + 3115225)^{2}$$
$67$ $$(T^{4} + 160 T^{3} + \cdots + 26797690000)^{2}$$
$71$ $$(T^{4} - 806616 T^{2} + \cdots + 18684702864)^{2}$$
$73$ $$(T - 305)^{8}$$
$79$ $$(T^{4} - 32 T^{3} + \cdots + 223014395536)^{2}$$
$83$ $$T^{8} + 1850688 T^{6} + \cdots + 87\!\cdots\!00$$
$89$ $$(T^{4} - 3993750 T^{2} + \cdots + 3633221022201)^{2}$$
$97$ $$(T^{4} + 1012 T^{3} + \cdots + 59506723600)^{2}$$