# Properties

 Label 108.3.g.a Level $108$ Weight $3$ Character orbit 108.g Analytic conductor $2.943$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 108.g (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.94278685509$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_{2} - 2) q^{5} + ( - 2 \beta_{3} + \beta_1 - 1) q^{7}+O(q^{10})$$ q + (-b3 + b2 - 2) * q^5 + (-2*b3 + b1 - 1) * q^7 $$q + ( - \beta_{3} + \beta_{2} - 2) q^{5} + ( - 2 \beta_{3} + \beta_1 - 1) q^{7} + (5 \beta_{2} - 2 \beta_1 + 12) q^{11} + ( - \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 2) q^{13} + ( - \beta_{3} - 15 \beta_{2} + \beta_1 - 8) q^{17} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{19} + (\beta_{3} + 17 \beta_{2} - 16) q^{23} + (6 \beta_{3} - 5 \beta_{2} - 3 \beta_1 + 3) q^{25} + (14 \beta_{2} + 7 \beta_1 + 21) q^{29} + (3 \beta_{3} - 5 \beta_{2} - 6 \beta_1 - 2) q^{31} + (5 \beta_{3} - 51 \beta_{2} - 5 \beta_1 - 23) q^{35} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 14) q^{37} + (8 \beta_{3} + \beta_{2} + 7) q^{41} + 23 \beta_{2} q^{43} + (12 \beta_{2} - 3 \beta_1 + 27) q^{47} + (\beta_{3} - 26 \beta_{2} - 2 \beta_1 - 25) q^{49} + ( - 8 \beta_{3} + 24 \beta_{2} + 8 \beta_1 + 8) q^{53} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 21) q^{55} + ( - 8 \beta_{3} + 17 \beta_{2} - 25) q^{59} + ( - 6 \beta_{3} - 22 \beta_{2} + 3 \beta_1 - 3) q^{61} + ( - 32 \beta_{2} - 7 \beta_1 - 57) q^{65} + ( - 6 \beta_{3} + 61 \beta_{2} + 12 \beta_1 + 55) q^{67} + (2 \beta_{3} + 30 \beta_{2} - 2 \beta_1 + 16) q^{71} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 20) q^{73} + ( - 17 \beta_{3} - 55 \beta_{2} + 38) q^{77} + ( - 10 \beta_{3} - 44 \beta_{2} + 5 \beta_1 - 5) q^{79} + (12 \beta_{2} - 3 \beta_1 + 27) q^{83} + ( - 6 \beta_{3} + 12 \beta_{2} + 12 \beta_1 + 6) q^{85} + (8 \beta_{3} + 120 \beta_{2} - 8 \beta_1 + 64) q^{89} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 77) q^{91} + (4 \beta_{3} - 22 \beta_{2} + 26) q^{95} + ( - 4 \beta_{3} + 97 \beta_{2} + 2 \beta_1 - 2) q^{97}+O(q^{100})$$ q + (-b3 + b2 - 2) * q^5 + (-2*b3 + b1 - 1) * q^7 + (5*b2 - 2*b1 + 12) * q^11 + (-b3 + 3*b2 + 2*b1 + 2) * q^13 + (-b3 - 15*b2 + b1 - 8) * q^17 + (-b3 - b2 - b1) * q^19 + (b3 + 17*b2 - 16) * q^23 + (6*b3 - 5*b2 - 3*b1 + 3) * q^25 + (14*b2 + 7*b1 + 21) * q^29 + (3*b3 - 5*b2 - 6*b1 - 2) * q^31 + (5*b3 - 51*b2 - 5*b1 - 23) * q^35 + (4*b3 + 4*b2 + 4*b1 - 14) * q^37 + (8*b3 + b2 + 7) * q^41 + 23*b2 * q^43 + (12*b2 - 3*b1 + 27) * q^47 + (b3 - 26*b2 - 2*b1 - 25) * q^49 + (-8*b3 + 24*b2 + 8*b1 + 8) * q^53 + (-3*b3 - 3*b2 - 3*b1 + 21) * q^55 + (-8*b3 + 17*b2 - 25) * q^59 + (-6*b3 - 22*b2 + 3*b1 - 3) * q^61 + (-32*b2 - 7*b1 - 57) * q^65 + (-6*b3 + 61*b2 + 12*b1 + 55) * q^67 + (2*b3 + 30*b2 - 2*b1 + 16) * q^71 + (-3*b3 - 3*b2 - 3*b1 + 20) * q^73 + (-17*b3 - 55*b2 + 38) * q^77 + (-10*b3 - 44*b2 + 5*b1 - 5) * q^79 + (12*b2 - 3*b1 + 27) * q^83 + (-6*b3 + 12*b2 + 12*b1 + 6) * q^85 + (8*b3 + 120*b2 - 8*b1 + 64) * q^89 + (-3*b3 - 3*b2 - 3*b1 - 77) * q^91 + (4*b3 - 22*b2 + 26) * q^95 + (-4*b3 + 97*b2 + 2*b1 - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 9 q^{5} - q^{7}+O(q^{10})$$ 4 * q - 9 * q^5 - q^7 $$4 q - 9 q^{5} - q^{7} + 36 q^{11} + 5 q^{13} + 2 q^{19} - 99 q^{23} + 13 q^{25} + 63 q^{29} - 7 q^{31} - 64 q^{37} + 18 q^{41} - 46 q^{43} + 81 q^{47} - 51 q^{49} + 90 q^{55} - 126 q^{59} + 41 q^{61} - 171 q^{65} + 116 q^{67} + 86 q^{73} + 279 q^{77} + 83 q^{79} + 81 q^{83} + 18 q^{85} - 302 q^{91} + 144 q^{95} - 196 q^{97}+O(q^{100})$$ 4 * q - 9 * q^5 - q^7 + 36 * q^11 + 5 * q^13 + 2 * q^19 - 99 * q^23 + 13 * q^25 + 63 * q^29 - 7 * q^31 - 64 * q^37 + 18 * q^41 - 46 * q^43 + 81 * q^47 - 51 * q^49 + 90 * q^55 - 126 * q^59 + 41 * q^61 - 171 * q^65 + 116 * q^67 + 86 * q^73 + 279 * q^77 + 83 * q^79 + 81 * q^83 + 18 * q^85 - 302 * q^91 + 144 * q^95 - 196 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} + 16\nu - 9 ) / 6$$ (v^3 + 2*v^2 + 16*v - 9) / 6 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 9 ) / 6$$ (v^3 + 2*v^2 - 2*v - 9) / 6 $$\beta_{3}$$ $$=$$ $$( -4\nu^{3} + \nu^{2} + 8\nu + 12 ) / 3$$ (-4*v^3 + v^2 + 8*v + 12) / 3
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta_1 ) / 3$$ (-b2 + b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 8\beta_{2} + 8 ) / 3$$ (b3 + 8*b2 + 8) / 3 $$\nu^{3}$$ $$=$$ $$( -2\beta_{3} + 2\beta _1 + 11 ) / 3$$ (-2*b3 + 2*b1 + 11) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$55$$ $$\chi(n)$$ $$1 + \beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 1.68614 − 0.396143i −1.18614 + 1.26217i 1.68614 + 0.396143i −1.18614 − 1.26217i
0 0 0 −6.55842 3.78651i 0 −4.55842 7.89542i 0 0 0
17.2 0 0 0 2.05842 + 1.18843i 0 4.05842 + 7.02939i 0 0 0
89.1 0 0 0 −6.55842 + 3.78651i 0 −4.55842 + 7.89542i 0 0 0
89.2 0 0 0 2.05842 1.18843i 0 4.05842 7.02939i 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.3.g.a 4
3.b odd 2 1 36.3.g.a 4
4.b odd 2 1 432.3.q.b 4
5.b even 2 1 2700.3.p.b 4
5.c odd 4 2 2700.3.u.b 8
8.b even 2 1 1728.3.q.g 4
8.d odd 2 1 1728.3.q.h 4
9.c even 3 1 36.3.g.a 4
9.c even 3 1 324.3.c.b 4
9.d odd 6 1 inner 108.3.g.a 4
9.d odd 6 1 324.3.c.b 4
12.b even 2 1 144.3.q.b 4
15.d odd 2 1 900.3.p.a 4
15.e even 4 2 900.3.u.a 8
24.f even 2 1 576.3.q.g 4
24.h odd 2 1 576.3.q.d 4
36.f odd 6 1 144.3.q.b 4
36.f odd 6 1 1296.3.e.e 4
36.h even 6 1 432.3.q.b 4
36.h even 6 1 1296.3.e.e 4
45.h odd 6 1 2700.3.p.b 4
45.j even 6 1 900.3.p.a 4
45.k odd 12 2 900.3.u.a 8
45.l even 12 2 2700.3.u.b 8
72.j odd 6 1 1728.3.q.g 4
72.l even 6 1 1728.3.q.h 4
72.n even 6 1 576.3.q.d 4
72.p odd 6 1 576.3.q.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.g.a 4 3.b odd 2 1
36.3.g.a 4 9.c even 3 1
108.3.g.a 4 1.a even 1 1 trivial
108.3.g.a 4 9.d odd 6 1 inner
144.3.q.b 4 12.b even 2 1
144.3.q.b 4 36.f odd 6 1
324.3.c.b 4 9.c even 3 1
324.3.c.b 4 9.d odd 6 1
432.3.q.b 4 4.b odd 2 1
432.3.q.b 4 36.h even 6 1
576.3.q.d 4 24.h odd 2 1
576.3.q.d 4 72.n even 6 1
576.3.q.g 4 24.f even 2 1
576.3.q.g 4 72.p odd 6 1
900.3.p.a 4 15.d odd 2 1
900.3.p.a 4 45.j even 6 1
900.3.u.a 8 15.e even 4 2
900.3.u.a 8 45.k odd 12 2
1296.3.e.e 4 36.f odd 6 1
1296.3.e.e 4 36.h even 6 1
1728.3.q.g 4 8.b even 2 1
1728.3.q.g 4 72.j odd 6 1
1728.3.q.h 4 8.d odd 2 1
1728.3.q.h 4 72.l even 6 1
2700.3.p.b 4 5.b even 2 1
2700.3.p.b 4 45.h odd 6 1
2700.3.u.b 8 5.c odd 4 2
2700.3.u.b 8 45.l even 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(108, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 9 T^{3} + 9 T^{2} - 162 T + 324$$
$7$ $$T^{4} + T^{3} + 75 T^{2} - 74 T + 5476$$
$11$ $$T^{4} - 36 T^{3} + 441 T^{2} + \cdots + 81$$
$13$ $$T^{4} - 5 T^{3} + 93 T^{2} + \cdots + 4624$$
$17$ $$T^{4} + 387 T^{2} + 20736$$
$19$ $$(T^{2} - T - 74)^{2}$$
$23$ $$T^{4} + 99 T^{3} + 4059 T^{2} + \cdots + 627264$$
$29$ $$T^{4} - 63 T^{3} + 441 T^{2} + \cdots + 777924$$
$31$ $$T^{4} + 7 T^{3} + 705 T^{2} + \cdots + 430336$$
$37$ $$(T^{2} + 32 T - 932)^{2}$$
$41$ $$T^{4} - 18 T^{3} - 1449 T^{2} + \cdots + 2424249$$
$43$ $$(T^{2} + 23 T + 529)^{2}$$
$47$ $$T^{4} - 81 T^{3} + 2511 T^{2} + \cdots + 104976$$
$53$ $$T^{4} + 4032 T^{2} + \cdots + 1327104$$
$59$ $$T^{4} + 126 T^{3} + 5031 T^{2} + \cdots + 68121$$
$61$ $$T^{4} - 41 T^{3} + 1929 T^{2} + \cdots + 61504$$
$67$ $$T^{4} - 116 T^{3} + 12765 T^{2} + \cdots + 477481$$
$71$ $$T^{4} + 1548 T^{2} + 331776$$
$73$ $$(T^{2} - 43 T - 206)^{2}$$
$79$ $$T^{4} - 83 T^{3} + 7023 T^{2} + \cdots + 17956$$
$83$ $$T^{4} - 81 T^{3} + 2511 T^{2} + \cdots + 104976$$
$89$ $$T^{4} + 24768 T^{2} + \cdots + 84934656$$
$97$ $$T^{4} + 196 T^{3} + \cdots + 86620249$$