# Properties

 Label 108.3.d.c Level 108 Weight 3 Character orbit 108.d Analytic conductor 2.943 Analytic rank 0 Dimension 4 CM no Inner twists 4

# 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: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{13})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{2} + ( 3 + \beta_{3} ) q^{4} + ( -2 \beta_{1} - \beta_{2} ) q^{5} + ( -1 - 2 \beta_{3} ) q^{7} + ( -\beta_{1} - 5 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{2} + ( 3 + \beta_{3} ) q^{4} + ( -2 \beta_{1} - \beta_{2} ) q^{5} + ( -1 - 2 \beta_{3} ) q^{7} + ( -\beta_{1} - 5 \beta_{2} ) q^{8} + ( 7 + \beta_{3} ) q^{10} -7 \beta_{2} q^{11} -16 q^{13} + ( -3 \beta_{1} + 5 \beta_{2} ) q^{14} + ( -1 + 5 \beta_{3} ) q^{16} + ( -8 \beta_{1} - 4 \beta_{2} ) q^{17} + ( -6 - 12 \beta_{3} ) q^{19} + ( -5 \beta_{1} - 9 \beta_{2} ) q^{20} + ( -7 + 7 \beta_{3} ) q^{22} + 10 \beta_{2} q^{23} -12 q^{25} + ( 16 \beta_{1} + 16 \beta_{2} ) q^{26} + ( 17 - 5 \beta_{3} ) q^{28} + ( 28 \beta_{1} + 14 \beta_{2} ) q^{29} + ( 1 + 2 \beta_{3} ) q^{31} + ( 11 \beta_{1} - 9 \beta_{2} ) q^{32} + ( 28 + 4 \beta_{3} ) q^{34} + 13 \beta_{2} q^{35} + 26 q^{37} + ( -18 \beta_{1} + 30 \beta_{2} ) q^{38} + ( 11 + 9 \beta_{3} ) q^{40} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 2 + 4 \beta_{3} ) q^{43} + ( 21 \beta_{1} - 7 \beta_{2} ) q^{44} + ( 10 - 10 \beta_{3} ) q^{46} + 2 \beta_{2} q^{47} + 10 q^{49} + ( 12 \beta_{1} + 12 \beta_{2} ) q^{50} + ( -48 - 16 \beta_{3} ) q^{52} + ( -38 \beta_{1} - 19 \beta_{2} ) q^{53} + ( 7 + 14 \beta_{3} ) q^{55} + ( -27 \beta_{1} - 7 \beta_{2} ) q^{56} + ( -98 - 14 \beta_{3} ) q^{58} -44 \beta_{2} q^{59} + 8 q^{61} + ( 3 \beta_{1} - 5 \beta_{2} ) q^{62} + ( -53 + 9 \beta_{3} ) q^{64} + ( 32 \beta_{1} + 16 \beta_{2} ) q^{65} + ( 10 + 20 \beta_{3} ) q^{67} + ( -20 \beta_{1} - 36 \beta_{2} ) q^{68} + ( 13 - 13 \beta_{3} ) q^{70} + 36 \beta_{2} q^{71} -19 q^{73} + ( -26 \beta_{1} - 26 \beta_{2} ) q^{74} + ( 102 - 30 \beta_{3} ) q^{76} + ( -42 \beta_{1} - 21 \beta_{2} ) q^{77} + ( 8 + 16 \beta_{3} ) q^{79} + ( 7 \beta_{1} - 29 \beta_{2} ) q^{80} + ( -14 - 2 \beta_{3} ) q^{82} -67 \beta_{2} q^{83} + 52 q^{85} + ( 6 \beta_{1} - 10 \beta_{2} ) q^{86} + ( -91 + 7 \beta_{3} ) q^{88} + ( -44 \beta_{1} - 22 \beta_{2} ) q^{89} + ( 16 + 32 \beta_{3} ) q^{91} + ( -30 \beta_{1} + 10 \beta_{2} ) q^{92} + ( 2 - 2 \beta_{3} ) q^{94} + 78 \beta_{2} q^{95} + 119 q^{97} + ( -10 \beta_{1} - 10 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 10q^{4} + O(q^{10})$$ $$4q + 10q^{4} + 26q^{10} - 64q^{13} - 14q^{16} - 42q^{22} - 48q^{25} + 78q^{28} + 104q^{34} + 104q^{37} + 26q^{40} + 60q^{46} + 40q^{49} - 160q^{52} - 364q^{58} + 32q^{61} - 230q^{64} + 78q^{70} - 76q^{73} + 468q^{76} - 52q^{82} + 208q^{85} - 378q^{88} + 12q^{94} + 476q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} - 2 \nu^{2} + 2 \nu - 9$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu^{2} - 4 \nu + 3$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} + 7 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 3 \beta_{2} - \beta_{1} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{2} - 4 \beta_{1} - 5$$

## 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
 1.15139 + 1.99426i 1.15139 − 1.99426i −0.651388 − 1.12824i −0.651388 + 1.12824i
−1.80278 0.866025i 0 2.50000 + 3.12250i −3.60555 0 6.24500i −1.80278 7.79423i 0 6.50000 + 3.12250i
55.2 −1.80278 + 0.866025i 0 2.50000 3.12250i −3.60555 0 6.24500i −1.80278 + 7.79423i 0 6.50000 3.12250i
55.3 1.80278 0.866025i 0 2.50000 3.12250i 3.60555 0 6.24500i 1.80278 7.79423i 0 6.50000 3.12250i
55.4 1.80278 + 0.866025i 0 2.50000 + 3.12250i 3.60555 0 6.24500i 1.80278 + 7.79423i 0 6.50000 + 3.12250i
 $$n$$: 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
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 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.c 4
3.b odd 2 1 inner 108.3.d.c 4
4.b odd 2 1 inner 108.3.d.c 4
8.b even 2 1 1728.3.g.i 4
8.d odd 2 1 1728.3.g.i 4
9.c even 3 1 324.3.f.l 4
9.c even 3 1 324.3.f.m 4
9.d odd 6 1 324.3.f.l 4
9.d odd 6 1 324.3.f.m 4
12.b even 2 1 inner 108.3.d.c 4
24.f even 2 1 1728.3.g.i 4
24.h odd 2 1 1728.3.g.i 4
36.f odd 6 1 324.3.f.l 4
36.f odd 6 1 324.3.f.m 4
36.h even 6 1 324.3.f.l 4
36.h even 6 1 324.3.f.m 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 13$$ acting on $$S_{3}^{\mathrm{new}}(108, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 5 T^{2} + 16 T^{4}$$
$3$ 1
$5$ $$( 1 + 37 T^{2} + 625 T^{4} )^{2}$$
$7$ $$( 1 - 59 T^{2} + 2401 T^{4} )^{2}$$
$11$ $$( 1 - 95 T^{2} + 14641 T^{4} )^{2}$$
$13$ $$( 1 + 16 T + 169 T^{2} )^{4}$$
$17$ $$( 1 + 370 T^{2} + 83521 T^{4} )^{2}$$
$19$ $$( 1 + 682 T^{2} + 130321 T^{4} )^{2}$$
$23$ $$( 1 - 758 T^{2} + 279841 T^{4} )^{2}$$
$29$ $$( 1 - 866 T^{2} + 707281 T^{4} )^{2}$$
$31$ $$( 1 - 1883 T^{2} + 923521 T^{4} )^{2}$$
$37$ $$( 1 - 26 T + 1369 T^{2} )^{4}$$
$41$ $$( 1 + 3310 T^{2} + 2825761 T^{4} )^{2}$$
$43$ $$( 1 - 3542 T^{2} + 3418801 T^{4} )^{2}$$
$47$ $$( 1 - 4406 T^{2} + 4879681 T^{4} )^{2}$$
$53$ $$( 1 + 925 T^{2} + 7890481 T^{4} )^{2}$$
$59$ $$( 1 - 1154 T^{2} + 12117361 T^{4} )^{2}$$
$61$ $$( 1 - 8 T + 3721 T^{2} )^{4}$$
$67$ $$( 1 - 5078 T^{2} + 20151121 T^{4} )^{2}$$
$71$ $$( 1 - 6194 T^{2} + 25411681 T^{4} )^{2}$$
$73$ $$( 1 + 19 T + 5329 T^{2} )^{4}$$
$79$ $$( 1 - 9986 T^{2} + 38950081 T^{4} )^{2}$$
$83$ $$( 1 - 311 T^{2} + 47458321 T^{4} )^{2}$$
$89$ $$( 1 + 9550 T^{2} + 62742241 T^{4} )^{2}$$
$97$ $$( 1 - 119 T + 9409 T^{2} )^{4}$$