# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 108.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.94278685509$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} + (2 \beta - 2) q^{4} + 7 q^{5} + 5 \beta q^{7} + 8 q^{8}+O(q^{10})$$ q + (-b - 1) * q^2 + (2*b - 2) * q^4 + 7 * q^5 + 5*b * q^7 + 8 * q^8 $$q + ( - \beta - 1) q^{2} + (2 \beta - 2) q^{4} + 7 q^{5} + 5 \beta q^{7} + 8 q^{8} + ( - 7 \beta - 7) q^{10} - 5 \beta q^{11} + 20 q^{13} + ( - 5 \beta + 15) q^{14} + ( - 8 \beta - 8) q^{16} - 8 q^{17} - 6 \beta q^{19} + (14 \beta - 14) q^{20} + (5 \beta - 15) q^{22} + 2 \beta q^{23} + 24 q^{25} + ( - 20 \beta - 20) q^{26} + ( - 10 \beta - 30) q^{28} + 10 q^{29} + 31 \beta q^{31} + (16 \beta - 16) q^{32} + (8 \beta + 8) q^{34} + 35 \beta q^{35} - 10 q^{37} + (6 \beta - 18) q^{38} + 56 q^{40} - 50 q^{41} - 10 \beta q^{43} + (10 \beta + 30) q^{44} + ( - 2 \beta + 6) q^{46} - 50 \beta q^{47} - 26 q^{49} + ( - 24 \beta - 24) q^{50} + (40 \beta - 40) q^{52} - 47 q^{53} - 35 \beta q^{55} + 40 \beta q^{56} + ( - 10 \beta - 10) q^{58} + 20 \beta q^{59} - 64 q^{61} + ( - 31 \beta + 93) q^{62} + 64 q^{64} + 140 q^{65} - 50 \beta q^{67} + ( - 16 \beta + 16) q^{68} + ( - 35 \beta + 105) q^{70} - 55 q^{73} + (10 \beta + 10) q^{74} + (12 \beta + 36) q^{76} + 75 q^{77} - 4 \beta q^{79} + ( - 56 \beta - 56) q^{80} + (50 \beta + 50) q^{82} - 17 \beta q^{83} - 56 q^{85} + (10 \beta - 30) q^{86} - 40 \beta q^{88} + 10 q^{89} + 100 \beta q^{91} + ( - 4 \beta - 12) q^{92} + (50 \beta - 150) q^{94} - 42 \beta q^{95} - 25 q^{97} + (26 \beta + 26) q^{98} +O(q^{100})$$ q + (-b - 1) * q^2 + (2*b - 2) * q^4 + 7 * q^5 + 5*b * q^7 + 8 * q^8 + (-7*b - 7) * q^10 - 5*b * q^11 + 20 * q^13 + (-5*b + 15) * q^14 + (-8*b - 8) * q^16 - 8 * q^17 - 6*b * q^19 + (14*b - 14) * q^20 + (5*b - 15) * q^22 + 2*b * q^23 + 24 * q^25 + (-20*b - 20) * q^26 + (-10*b - 30) * q^28 + 10 * q^29 + 31*b * q^31 + (16*b - 16) * q^32 + (8*b + 8) * q^34 + 35*b * q^35 - 10 * q^37 + (6*b - 18) * q^38 + 56 * q^40 - 50 * q^41 - 10*b * q^43 + (10*b + 30) * q^44 + (-2*b + 6) * q^46 - 50*b * q^47 - 26 * q^49 + (-24*b - 24) * q^50 + (40*b - 40) * q^52 - 47 * q^53 - 35*b * q^55 + 40*b * q^56 + (-10*b - 10) * q^58 + 20*b * q^59 - 64 * q^61 + (-31*b + 93) * q^62 + 64 * q^64 + 140 * q^65 - 50*b * q^67 + (-16*b + 16) * q^68 + (-35*b + 105) * q^70 - 55 * q^73 + (10*b + 10) * q^74 + (12*b + 36) * q^76 + 75 * q^77 - 4*b * q^79 + (-56*b - 56) * q^80 + (50*b + 50) * q^82 - 17*b * q^83 - 56 * q^85 + (10*b - 30) * q^86 - 40*b * q^88 + 10 * q^89 + 100*b * q^91 + (-4*b - 12) * q^92 + (50*b - 150) * q^94 - 42*b * q^95 - 25 * q^97 + (26*b + 26) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} + 14 q^{5} + 16 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^4 + 14 * q^5 + 16 * q^8 $$2 q - 2 q^{2} - 4 q^{4} + 14 q^{5} + 16 q^{8} - 14 q^{10} + 40 q^{13} + 30 q^{14} - 16 q^{16} - 16 q^{17} - 28 q^{20} - 30 q^{22} + 48 q^{25} - 40 q^{26} - 60 q^{28} + 20 q^{29} - 32 q^{32} + 16 q^{34} - 20 q^{37} - 36 q^{38} + 112 q^{40} - 100 q^{41} + 60 q^{44} + 12 q^{46} - 52 q^{49} - 48 q^{50} - 80 q^{52} - 94 q^{53} - 20 q^{58} - 128 q^{61} + 186 q^{62} + 128 q^{64} + 280 q^{65} + 32 q^{68} + 210 q^{70} - 110 q^{73} + 20 q^{74} + 72 q^{76} + 150 q^{77} - 112 q^{80} + 100 q^{82} - 112 q^{85} - 60 q^{86} + 20 q^{89} - 24 q^{92} - 300 q^{94} - 50 q^{97} + 52 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^4 + 14 * q^5 + 16 * q^8 - 14 * q^10 + 40 * q^13 + 30 * q^14 - 16 * q^16 - 16 * q^17 - 28 * q^20 - 30 * q^22 + 48 * q^25 - 40 * q^26 - 60 * q^28 + 20 * q^29 - 32 * q^32 + 16 * q^34 - 20 * q^37 - 36 * q^38 + 112 * q^40 - 100 * q^41 + 60 * q^44 + 12 * q^46 - 52 * q^49 - 48 * q^50 - 80 * q^52 - 94 * q^53 - 20 * q^58 - 128 * q^61 + 186 * q^62 + 128 * q^64 + 280 * q^65 + 32 * q^68 + 210 * q^70 - 110 * q^73 + 20 * q^74 + 72 * q^76 + 150 * q^77 - 112 * q^80 + 100 * q^82 - 112 * q^85 - 60 * q^86 + 20 * q^89 - 24 * q^92 - 300 * q^94 - 50 * q^97 + 52 * q^98

## Character values

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

 $$n$$ $$29$$ $$55$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 1.73205i 0 −2.00000 + 3.46410i 7.00000 0 8.66025i 8.00000 0 −7.00000 12.1244i
55.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 7.00000 0 8.66025i 8.00000 0 −7.00000 + 12.1244i
 $$n$$: 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.3.d.a 2
3.b odd 2 1 108.3.d.b yes 2
4.b odd 2 1 inner 108.3.d.a 2
8.b even 2 1 1728.3.g.a 2
8.d odd 2 1 1728.3.g.a 2
9.c even 3 1 324.3.f.c 2
9.c even 3 1 324.3.f.i 2
9.d odd 6 1 324.3.f.b 2
9.d odd 6 1 324.3.f.h 2
12.b even 2 1 108.3.d.b yes 2
24.f even 2 1 1728.3.g.f 2
24.h odd 2 1 1728.3.g.f 2
36.f odd 6 1 324.3.f.c 2
36.f odd 6 1 324.3.f.i 2
36.h even 6 1 324.3.f.b 2
36.h even 6 1 324.3.f.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.d.a 2 1.a even 1 1 trivial
108.3.d.a 2 4.b odd 2 1 inner
108.3.d.b yes 2 3.b odd 2 1
108.3.d.b yes 2 12.b even 2 1
324.3.f.b 2 9.d odd 6 1
324.3.f.b 2 36.h even 6 1
324.3.f.c 2 9.c even 3 1
324.3.f.c 2 36.f odd 6 1
324.3.f.h 2 9.d odd 6 1
324.3.f.h 2 36.h even 6 1
324.3.f.i 2 9.c even 3 1
324.3.f.i 2 36.f odd 6 1
1728.3.g.a 2 8.b even 2 1
1728.3.g.a 2 8.d odd 2 1
1728.3.g.f 2 24.f even 2 1
1728.3.g.f 2 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 7$$ acting on $$S_{3}^{\mathrm{new}}(108, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$(T - 7)^{2}$$
$7$ $$T^{2} + 75$$
$11$ $$T^{2} + 75$$
$13$ $$(T - 20)^{2}$$
$17$ $$(T + 8)^{2}$$
$19$ $$T^{2} + 108$$
$23$ $$T^{2} + 12$$
$29$ $$(T - 10)^{2}$$
$31$ $$T^{2} + 2883$$
$37$ $$(T + 10)^{2}$$
$41$ $$(T + 50)^{2}$$
$43$ $$T^{2} + 300$$
$47$ $$T^{2} + 7500$$
$53$ $$(T + 47)^{2}$$
$59$ $$T^{2} + 1200$$
$61$ $$(T + 64)^{2}$$
$67$ $$T^{2} + 7500$$
$71$ $$T^{2}$$
$73$ $$(T + 55)^{2}$$
$79$ $$T^{2} + 48$$
$83$ $$T^{2} + 867$$
$89$ $$(T - 10)^{2}$$
$97$ $$(T + 25)^{2}$$