# Properties

 Label 108.2.b.b Level 108 Weight 2 Character orbit 108.b Analytic conductor 0.862 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.862384341830$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + \beta_{2} q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + \beta_{2} q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + ( 2 - 2 \beta_{2} ) q^{10} + ( -3 \beta_{1} + \beta_{3} ) q^{11} - q^{13} + \beta_{3} q^{14} + ( -2 + 2 \beta_{2} ) q^{16} + ( -\beta_{1} - \beta_{3} ) q^{17} -3 \beta_{2} q^{19} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{20} + ( -6 - 2 \beta_{2} ) q^{22} + ( 3 \beta_{1} - \beta_{3} ) q^{23} -3 q^{25} -\beta_{1} q^{26} + ( -3 + \beta_{2} ) q^{28} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{29} + 2 \beta_{2} q^{31} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{32} + ( 2 - 2 \beta_{2} ) q^{34} + ( 3 \beta_{1} - \beta_{3} ) q^{35} - q^{37} -3 \beta_{3} q^{38} + 8 q^{40} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{41} -2 \beta_{2} q^{43} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{44} + ( 6 + 2 \beta_{2} ) q^{46} + ( -3 \beta_{1} + \beta_{3} ) q^{47} + 4 q^{49} -3 \beta_{1} q^{50} + ( -1 - \beta_{2} ) q^{52} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{53} + 8 \beta_{2} q^{55} + ( -3 \beta_{1} + \beta_{3} ) q^{56} + ( -4 + 4 \beta_{2} ) q^{58} + ( 3 \beta_{1} - \beta_{3} ) q^{59} + 11 q^{61} + 2 \beta_{3} q^{62} -8 q^{64} + ( \beta_{1} + \beta_{3} ) q^{65} -7 \beta_{2} q^{67} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{68} + ( 6 + 2 \beta_{2} ) q^{70} - q^{73} -\beta_{1} q^{74} + ( 9 - 3 \beta_{2} ) q^{76} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{77} + \beta_{2} q^{79} + 8 \beta_{1} q^{80} + ( -4 + 4 \beta_{2} ) q^{82} + ( 6 \beta_{1} - 2 \beta_{3} ) q^{83} -8 q^{85} -2 \beta_{3} q^{86} -8 \beta_{2} q^{88} + ( -\beta_{1} - \beta_{3} ) q^{89} -\beta_{2} q^{91} + ( 6 \beta_{1} + 2 \beta_{3} ) q^{92} + ( -6 - 2 \beta_{2} ) q^{94} + ( -9 \beta_{1} + 3 \beta_{3} ) q^{95} -13 q^{97} + 4 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} + O(q^{10})$$ $$4q + 4q^{4} + 8q^{10} - 4q^{13} - 8q^{16} - 24q^{22} - 12q^{25} - 12q^{28} + 8q^{34} - 4q^{37} + 32q^{40} + 24q^{46} + 16q^{49} - 4q^{52} - 16q^{58} + 44q^{61} - 32q^{64} + 24q^{70} - 4q^{73} + 36q^{76} - 16q^{82} - 32q^{85} - 24q^{94} - 52q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 −1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i 1.22474 + 0.707107i
−1.22474 0.707107i 0 1.00000 + 1.73205i 2.82843i 0 1.73205i 2.82843i 0 2.00000 3.46410i
107.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 2.82843i 0 1.73205i 2.82843i 0 2.00000 + 3.46410i
107.3 1.22474 0.707107i 0 1.00000 1.73205i 2.82843i 0 1.73205i 2.82843i 0 2.00000 + 3.46410i
107.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 2.82843i 0 1.73205i 2.82843i 0 2.00000 3.46410i
 $$n$$: 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 108.2.b.b 4
3.b odd 2 1 inner 108.2.b.b 4
4.b odd 2 1 inner 108.2.b.b 4
8.b even 2 1 1728.2.c.d 4
8.d odd 2 1 1728.2.c.d 4
9.c even 3 1 324.2.h.a 4
9.c even 3 1 324.2.h.b 4
9.d odd 6 1 324.2.h.a 4
9.d odd 6 1 324.2.h.b 4
12.b even 2 1 inner 108.2.b.b 4
24.f even 2 1 1728.2.c.d 4
24.h odd 2 1 1728.2.c.d 4
36.f odd 6 1 324.2.h.a 4
36.f odd 6 1 324.2.h.b 4
36.h even 6 1 324.2.h.a 4
36.h even 6 1 324.2.h.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.b.b 4 1.a even 1 1 trivial
108.2.b.b 4 3.b odd 2 1 inner
108.2.b.b 4 4.b odd 2 1 inner
108.2.b.b 4 12.b even 2 1 inner
324.2.h.a 4 9.c even 3 1
324.2.h.a 4 9.d odd 6 1
324.2.h.a 4 36.f odd 6 1
324.2.h.a 4 36.h even 6 1
324.2.h.b 4 9.c even 3 1
324.2.h.b 4 9.d odd 6 1
324.2.h.b 4 36.f odd 6 1
324.2.h.b 4 36.h even 6 1
1728.2.c.d 4 8.b even 2 1
1728.2.c.d 4 8.d odd 2 1
1728.2.c.d 4 24.f even 2 1
1728.2.c.d 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 8$$ acting on $$S_{2}^{\mathrm{new}}(108, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 4 T^{4}$$
$3$ 1
$5$ $$( 1 - 2 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 - 5 T + 7 T^{2} )^{2}( 1 + 5 T + 7 T^{2} )^{2}$$
$11$ $$( 1 - 2 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 + T + 13 T^{2} )^{4}$$
$17$ $$( 1 - 26 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2}$$
$23$ $$( 1 + 22 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 26 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 50 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 + T + 37 T^{2} )^{4}$$
$41$ $$( 1 - 50 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 74 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 70 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 74 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 94 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 11 T + 61 T^{2} )^{4}$$
$67$ $$( 1 - 11 T + 67 T^{2} )^{2}( 1 + 11 T + 67 T^{2} )^{2}$$
$71$ $$( 1 + 71 T^{2} )^{4}$$
$73$ $$( 1 + T + 73 T^{2} )^{4}$$
$79$ $$( 1 - 155 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 70 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 170 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 13 T + 97 T^{2} )^{4}$$