# Properties

 Label 108.3.d.b 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: $$\Q(\sqrt{-3})$$ 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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} -7 q^{5} + ( -5 + 10 \zeta_{6} ) q^{7} -8 q^{8} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} -7 q^{5} + ( -5 + 10 \zeta_{6} ) q^{7} -8 q^{8} -14 \zeta_{6} q^{10} + ( -5 + 10 \zeta_{6} ) q^{11} + 20 q^{13} + ( -20 + 10 \zeta_{6} ) q^{14} -16 \zeta_{6} q^{16} + 8 q^{17} + ( 6 - 12 \zeta_{6} ) q^{19} + ( 28 - 28 \zeta_{6} ) q^{20} + ( -20 + 10 \zeta_{6} ) q^{22} + ( 2 - 4 \zeta_{6} ) q^{23} + 24 q^{25} + 40 \zeta_{6} q^{26} + ( -20 - 20 \zeta_{6} ) q^{28} -10 q^{29} + ( -31 + 62 \zeta_{6} ) q^{31} + ( 32 - 32 \zeta_{6} ) q^{32} + 16 \zeta_{6} q^{34} + ( 35 - 70 \zeta_{6} ) q^{35} -10 q^{37} + ( 24 - 12 \zeta_{6} ) q^{38} + 56 q^{40} + 50 q^{41} + ( 10 - 20 \zeta_{6} ) q^{43} + ( -20 - 20 \zeta_{6} ) q^{44} + ( 8 - 4 \zeta_{6} ) q^{46} + ( -50 + 100 \zeta_{6} ) q^{47} -26 q^{49} + 48 \zeta_{6} q^{50} + ( -80 + 80 \zeta_{6} ) q^{52} + 47 q^{53} + ( 35 - 70 \zeta_{6} ) q^{55} + ( 40 - 80 \zeta_{6} ) q^{56} -20 \zeta_{6} q^{58} + ( 20 - 40 \zeta_{6} ) q^{59} -64 q^{61} + ( -124 + 62 \zeta_{6} ) q^{62} + 64 q^{64} -140 q^{65} + ( 50 - 100 \zeta_{6} ) q^{67} + ( -32 + 32 \zeta_{6} ) q^{68} + ( 140 - 70 \zeta_{6} ) q^{70} -55 q^{73} -20 \zeta_{6} q^{74} + ( 24 + 24 \zeta_{6} ) q^{76} -75 q^{77} + ( 4 - 8 \zeta_{6} ) q^{79} + 112 \zeta_{6} q^{80} + 100 \zeta_{6} q^{82} + ( -17 + 34 \zeta_{6} ) q^{83} -56 q^{85} + ( 40 - 20 \zeta_{6} ) q^{86} + ( 40 - 80 \zeta_{6} ) q^{88} -10 q^{89} + ( -100 + 200 \zeta_{6} ) q^{91} + ( 8 + 8 \zeta_{6} ) q^{92} + ( -200 + 100 \zeta_{6} ) q^{94} + ( -42 + 84 \zeta_{6} ) q^{95} -25 q^{97} -52 \zeta_{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 4q^{4} - 14q^{5} - 16q^{8} + O(q^{10})$$ $$2q + 2q^{2} - 4q^{4} - 14q^{5} - 16q^{8} - 14q^{10} + 40q^{13} - 30q^{14} - 16q^{16} + 16q^{17} + 28q^{20} - 30q^{22} + 48q^{25} + 40q^{26} - 60q^{28} - 20q^{29} + 32q^{32} + 16q^{34} - 20q^{37} + 36q^{38} + 112q^{40} + 100q^{41} - 60q^{44} + 12q^{46} - 52q^{49} + 48q^{50} - 80q^{52} + 94q^{53} - 20q^{58} - 128q^{61} - 186q^{62} + 128q^{64} - 280q^{65} - 32q^{68} + 210q^{70} - 110q^{73} - 20q^{74} + 72q^{76} - 150q^{77} + 112q^{80} + 100q^{82} - 112q^{85} + 60q^{86} - 20q^{89} + 24q^{92} - 300q^{94} - 50q^{97} - 52q^{98} + O(q^{100})$$

## Character values

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

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

## 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.b yes 2
3.b odd 2 1 108.3.d.a 2
4.b odd 2 1 inner 108.3.d.b yes 2
8.b even 2 1 1728.3.g.f 2
8.d odd 2 1 1728.3.g.f 2
9.c even 3 1 324.3.f.b 2
9.c even 3 1 324.3.f.h 2
9.d odd 6 1 324.3.f.c 2
9.d odd 6 1 324.3.f.i 2
12.b even 2 1 108.3.d.a 2
24.f even 2 1 1728.3.g.a 2
24.h odd 2 1 1728.3.g.a 2
36.f odd 6 1 324.3.f.b 2
36.f odd 6 1 324.3.f.h 2
36.h even 6 1 324.3.f.c 2
36.h even 6 1 324.3.f.i 2

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

$p$ $F_p(T)$
$2$ $$1 - 2 T + 4 T^{2}$$
$3$ 1
$5$ $$( 1 + 7 T + 25 T^{2} )^{2}$$
$7$ $$( 1 - 11 T + 49 T^{2} )( 1 + 11 T + 49 T^{2} )$$
$11$ $$1 - 167 T^{2} + 14641 T^{4}$$
$13$ $$( 1 - 20 T + 169 T^{2} )^{2}$$
$17$ $$( 1 - 8 T + 289 T^{2} )^{2}$$
$19$ $$1 - 614 T^{2} + 130321 T^{4}$$
$23$ $$1 - 1046 T^{2} + 279841 T^{4}$$
$29$ $$( 1 + 10 T + 841 T^{2} )^{2}$$
$31$ $$( 1 - 31 T + 961 T^{2} )( 1 + 31 T + 961 T^{2} )$$
$37$ $$( 1 + 10 T + 1369 T^{2} )^{2}$$
$41$ $$( 1 - 50 T + 1681 T^{2} )^{2}$$
$43$ $$1 - 3398 T^{2} + 3418801 T^{4}$$
$47$ $$1 + 3082 T^{2} + 4879681 T^{4}$$
$53$ $$( 1 - 47 T + 2809 T^{2} )^{2}$$
$59$ $$1 - 5762 T^{2} + 12117361 T^{4}$$
$61$ $$( 1 + 64 T + 3721 T^{2} )^{2}$$
$67$ $$1 - 1478 T^{2} + 20151121 T^{4}$$
$71$ $$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$
$73$ $$( 1 + 55 T + 5329 T^{2} )^{2}$$
$79$ $$1 - 12434 T^{2} + 38950081 T^{4}$$
$83$ $$1 - 12911 T^{2} + 47458321 T^{4}$$
$89$ $$( 1 + 10 T + 7921 T^{2} )^{2}$$
$97$ $$( 1 + 25 T + 9409 T^{2} )^{2}$$