# Properties

 Label 108.3.f.a Level 108 Weight 3 Character orbit 108.f 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.f (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.94278685509$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 4 q^{4} + ( 4 - 4 \zeta_{6} ) q^{5} + ( -4 + 2 \zeta_{6} ) q^{7} -8 q^{8} +O(q^{10})$$ $$q -2 q^{2} + 4 q^{4} + ( 4 - 4 \zeta_{6} ) q^{5} + ( -4 + 2 \zeta_{6} ) q^{7} -8 q^{8} + ( -8 + 8 \zeta_{6} ) q^{10} + ( 14 - 7 \zeta_{6} ) q^{11} + ( 22 - 22 \zeta_{6} ) q^{13} + ( 8 - 4 \zeta_{6} ) q^{14} + 16 q^{16} + 11 q^{17} + ( 9 - 18 \zeta_{6} ) q^{19} + ( 16 - 16 \zeta_{6} ) q^{20} + ( -28 + 14 \zeta_{6} ) q^{22} + ( -14 - 14 \zeta_{6} ) q^{23} + 9 \zeta_{6} q^{25} + ( -44 + 44 \zeta_{6} ) q^{26} + ( -16 + 8 \zeta_{6} ) q^{28} + 34 \zeta_{6} q^{29} + ( 4 + 4 \zeta_{6} ) q^{31} -32 q^{32} -22 q^{34} + ( -8 + 16 \zeta_{6} ) q^{35} -16 q^{37} + ( -18 + 36 \zeta_{6} ) q^{38} + ( -32 + 32 \zeta_{6} ) q^{40} + ( 13 - 13 \zeta_{6} ) q^{41} + ( -58 + 29 \zeta_{6} ) q^{43} + ( 56 - 28 \zeta_{6} ) q^{44} + ( 28 + 28 \zeta_{6} ) q^{46} + ( -4 + 2 \zeta_{6} ) q^{47} + ( -37 + 37 \zeta_{6} ) q^{49} -18 \zeta_{6} q^{50} + ( 88 - 88 \zeta_{6} ) q^{52} -52 q^{53} + ( 28 - 56 \zeta_{6} ) q^{55} + ( 32 - 16 \zeta_{6} ) q^{56} -68 \zeta_{6} q^{58} + ( 31 + 31 \zeta_{6} ) q^{59} + 16 \zeta_{6} q^{61} + ( -8 - 8 \zeta_{6} ) q^{62} + 64 q^{64} -88 \zeta_{6} q^{65} + ( 67 + 67 \zeta_{6} ) q^{67} + 44 q^{68} + ( 16 - 32 \zeta_{6} ) q^{70} -25 q^{73} + 32 q^{74} + ( 36 - 72 \zeta_{6} ) q^{76} + ( -42 + 42 \zeta_{6} ) q^{77} + ( 32 - 16 \zeta_{6} ) q^{79} + ( 64 - 64 \zeta_{6} ) q^{80} + ( -26 + 26 \zeta_{6} ) q^{82} + ( -40 + 20 \zeta_{6} ) q^{83} + ( 44 - 44 \zeta_{6} ) q^{85} + ( 116 - 58 \zeta_{6} ) q^{86} + ( -112 + 56 \zeta_{6} ) q^{88} + 2 q^{89} + ( -44 + 88 \zeta_{6} ) q^{91} + ( -56 - 56 \zeta_{6} ) q^{92} + ( 8 - 4 \zeta_{6} ) q^{94} + ( -36 - 36 \zeta_{6} ) q^{95} + 43 \zeta_{6} q^{97} + ( 74 - 74 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 8q^{4} + 4q^{5} - 6q^{7} - 16q^{8} + O(q^{10})$$ $$2q - 4q^{2} + 8q^{4} + 4q^{5} - 6q^{7} - 16q^{8} - 8q^{10} + 21q^{11} + 22q^{13} + 12q^{14} + 32q^{16} + 22q^{17} + 16q^{20} - 42q^{22} - 42q^{23} + 9q^{25} - 44q^{26} - 24q^{28} + 34q^{29} + 12q^{31} - 64q^{32} - 44q^{34} - 32q^{37} - 32q^{40} + 13q^{41} - 87q^{43} + 84q^{44} + 84q^{46} - 6q^{47} - 37q^{49} - 18q^{50} + 88q^{52} - 104q^{53} + 48q^{56} - 68q^{58} + 93q^{59} + 16q^{61} - 24q^{62} + 128q^{64} - 88q^{65} + 201q^{67} + 88q^{68} - 50q^{73} + 64q^{74} - 42q^{77} + 48q^{79} + 64q^{80} - 26q^{82} - 60q^{83} + 44q^{85} + 174q^{86} - 168q^{88} + 4q^{89} - 168q^{92} + 12q^{94} - 108q^{95} + 43q^{97} + 74q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$29$$ $$55$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.5 − 0.866025i 0.5 + 0.866025i
−2.00000 0 4.00000 2.00000 + 3.46410i 0 −3.00000 1.73205i −8.00000 0 −4.00000 6.92820i
91.1 −2.00000 0 4.00000 2.00000 3.46410i 0 −3.00000 + 1.73205i −8.00000 0 −4.00000 + 6.92820i
 $$n$$: 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
36.f 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.f.a 2
3.b odd 2 1 36.3.f.b yes 2
4.b odd 2 1 108.3.f.b 2
8.b even 2 1 1728.3.o.a 2
8.d odd 2 1 1728.3.o.b 2
9.c even 3 1 108.3.f.b 2
9.c even 3 1 324.3.d.c 2
9.d odd 6 1 36.3.f.a 2
9.d odd 6 1 324.3.d.b 2
12.b even 2 1 36.3.f.a 2
24.f even 2 1 576.3.o.a 2
24.h odd 2 1 576.3.o.b 2
36.f odd 6 1 inner 108.3.f.a 2
36.f odd 6 1 324.3.d.c 2
36.h even 6 1 36.3.f.b yes 2
36.h even 6 1 324.3.d.b 2
72.j odd 6 1 576.3.o.a 2
72.l even 6 1 576.3.o.b 2
72.n even 6 1 1728.3.o.b 2
72.p odd 6 1 1728.3.o.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.a 2 9.d odd 6 1
36.3.f.a 2 12.b even 2 1
36.3.f.b yes 2 3.b odd 2 1
36.3.f.b yes 2 36.h even 6 1
108.3.f.a 2 1.a even 1 1 trivial
108.3.f.a 2 36.f odd 6 1 inner
108.3.f.b 2 4.b odd 2 1
108.3.f.b 2 9.c even 3 1
324.3.d.b 2 9.d odd 6 1
324.3.d.b 2 36.h even 6 1
324.3.d.c 2 9.c even 3 1
324.3.d.c 2 36.f odd 6 1
576.3.o.a 2 24.f even 2 1
576.3.o.a 2 72.j odd 6 1
576.3.o.b 2 24.h odd 2 1
576.3.o.b 2 72.l even 6 1
1728.3.o.a 2 8.b even 2 1
1728.3.o.a 2 72.p odd 6 1
1728.3.o.b 2 8.d odd 2 1
1728.3.o.b 2 72.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(108, [\chi])$$:

 $$T_{5}^{2} - 4 T_{5} + 16$$ $$T_{7}^{2} + 6 T_{7} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T )^{2}$$
$3$ 1
$5$ $$1 - 4 T - 9 T^{2} - 100 T^{3} + 625 T^{4}$$
$7$ $$1 + 6 T + 61 T^{2} + 294 T^{3} + 2401 T^{4}$$
$11$ $$1 - 21 T + 268 T^{2} - 2541 T^{3} + 14641 T^{4}$$
$13$ $$( 1 - 23 T + 169 T^{2} )( 1 + T + 169 T^{2} )$$
$17$ $$( 1 - 11 T + 289 T^{2} )^{2}$$
$19$ $$1 - 479 T^{2} + 130321 T^{4}$$
$23$ $$1 + 42 T + 1117 T^{2} + 22218 T^{3} + 279841 T^{4}$$
$29$ $$1 - 34 T + 315 T^{2} - 28594 T^{3} + 707281 T^{4}$$
$31$ $$1 - 12 T + 1009 T^{2} - 11532 T^{3} + 923521 T^{4}$$
$37$ $$( 1 + 16 T + 1369 T^{2} )^{2}$$
$41$ $$1 - 13 T - 1512 T^{2} - 21853 T^{3} + 2825761 T^{4}$$
$43$ $$1 + 87 T + 4372 T^{2} + 160863 T^{3} + 3418801 T^{4}$$
$47$ $$1 + 6 T + 2221 T^{2} + 13254 T^{3} + 4879681 T^{4}$$
$53$ $$( 1 + 52 T + 2809 T^{2} )^{2}$$
$59$ $$1 - 93 T + 6364 T^{2} - 323733 T^{3} + 12117361 T^{4}$$
$61$ $$1 - 16 T - 3465 T^{2} - 59536 T^{3} + 13845841 T^{4}$$
$67$ $$( 1 - 67 T )^{2}( 1 - 67 T + 4489 T^{2} )$$
$71$ $$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$
$73$ $$( 1 + 25 T + 5329 T^{2} )^{2}$$
$79$ $$1 - 48 T + 7009 T^{2} - 299568 T^{3} + 38950081 T^{4}$$
$83$ $$1 + 60 T + 8089 T^{2} + 413340 T^{3} + 47458321 T^{4}$$
$89$ $$( 1 - 2 T + 7921 T^{2} )^{2}$$
$97$ $$1 - 43 T - 7560 T^{2} - 404587 T^{3} + 88529281 T^{4}$$