# Properties

 Label 108.2.b.a 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: $$\Q(\sqrt{3}, \sqrt{-5})$$ Defining polynomial: $$x^{4} + x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + ( -1 - 2 \beta_{2} ) q^{7} + ( -\beta_{1} + 2 \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + ( -1 - 2 \beta_{2} ) q^{7} + ( -\beta_{1} + 2 \beta_{3} ) q^{8} + ( -2 + \beta_{2} ) q^{10} + ( \beta_{1} - \beta_{3} ) q^{11} + 2 q^{13} + ( \beta_{1} - 4 \beta_{3} ) q^{14} + ( -4 - \beta_{2} ) q^{16} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{17} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{20} + ( 2 + \beta_{2} ) q^{22} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{23} + 2 \beta_{1} q^{26} + ( 8 + \beta_{2} ) q^{28} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{29} + ( 1 + 2 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{32} + ( 4 - 2 \beta_{2} ) q^{34} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{35} -4 q^{37} + ( -4 - 3 \beta_{2} ) q^{40} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{41} + ( 2 + 4 \beta_{2} ) q^{43} + ( \beta_{1} + 2 \beta_{3} ) q^{44} + ( -8 - 4 \beta_{2} ) q^{46} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{47} -8 q^{49} + 2 \beta_{2} q^{52} + ( \beta_{1} + \beta_{3} ) q^{53} + ( 1 + 2 \beta_{2} ) q^{55} + ( 7 \beta_{1} + 2 \beta_{3} ) q^{56} + ( 4 - 2 \beta_{2} ) q^{58} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{59} -4 q^{61} + ( -\beta_{1} + 4 \beta_{3} ) q^{62} + ( 4 - 3 \beta_{2} ) q^{64} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{65} + ( -2 - 4 \beta_{2} ) q^{67} + ( 6 \beta_{1} - 4 \beta_{3} ) q^{68} + ( 10 + 5 \beta_{2} ) q^{70} + ( -6 \beta_{1} + 6 \beta_{3} ) q^{71} + 5 q^{73} -4 \beta_{1} q^{74} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{77} + ( 2 + 4 \beta_{2} ) q^{79} + ( -\beta_{1} - 6 \beta_{3} ) q^{80} + ( -8 + 4 \beta_{2} ) q^{82} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{83} + 10 q^{85} + ( -2 \beta_{1} + 8 \beta_{3} ) q^{86} + ( -4 + \beta_{2} ) q^{88} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{89} + ( -2 - 4 \beta_{2} ) q^{91} + ( -4 \beta_{1} - 8 \beta_{3} ) q^{92} + ( -4 - 2 \beta_{2} ) q^{94} + 11 q^{97} -8 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{4} + O(q^{10})$$ $$4 q - 2 q^{4} - 10 q^{10} + 8 q^{13} - 14 q^{16} + 6 q^{22} + 30 q^{28} + 20 q^{34} - 16 q^{37} - 10 q^{40} - 24 q^{46} - 32 q^{49} - 4 q^{52} + 20 q^{58} - 16 q^{61} + 22 q^{64} + 30 q^{70} + 20 q^{73} - 40 q^{82} + 40 q^{85} - 18 q^{88} - 12 q^{94} + 44 q^{97} + O(q^{100})$$

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

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

## 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
 −0.866025 − 1.11803i −0.866025 + 1.11803i 0.866025 − 1.11803i 0.866025 + 1.11803i
−0.866025 1.11803i 0 −0.500000 + 1.93649i 2.23607i 0 3.87298i 2.59808 1.11803i 0 −2.50000 + 1.93649i
107.2 −0.866025 + 1.11803i 0 −0.500000 1.93649i 2.23607i 0 3.87298i 2.59808 + 1.11803i 0 −2.50000 1.93649i
107.3 0.866025 1.11803i 0 −0.500000 1.93649i 2.23607i 0 3.87298i −2.59808 1.11803i 0 −2.50000 1.93649i
107.4 0.866025 + 1.11803i 0 −0.500000 + 1.93649i 2.23607i 0 3.87298i −2.59808 + 1.11803i 0 −2.50000 + 1.93649i
 $$n$$: 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.a 4
3.b odd 2 1 inner 108.2.b.a 4
4.b odd 2 1 inner 108.2.b.a 4
8.b even 2 1 1728.2.c.c 4
8.d odd 2 1 1728.2.c.c 4
9.c even 3 2 324.2.h.d 8
9.d odd 6 2 324.2.h.d 8
12.b even 2 1 inner 108.2.b.a 4
24.f even 2 1 1728.2.c.c 4
24.h odd 2 1 1728.2.c.c 4
36.f odd 6 2 324.2.h.d 8
36.h even 6 2 324.2.h.d 8

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$( 15 + T^{2} )^{2}$$
$11$ $$( -3 + T^{2} )^{2}$$
$13$ $$( -2 + T )^{4}$$
$17$ $$( 20 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$( -48 + T^{2} )^{2}$$
$29$ $$( 20 + T^{2} )^{2}$$
$31$ $$( 15 + T^{2} )^{2}$$
$37$ $$( 4 + T )^{4}$$
$41$ $$( 80 + T^{2} )^{2}$$
$43$ $$( 60 + T^{2} )^{2}$$
$47$ $$( -12 + T^{2} )^{2}$$
$53$ $$( 5 + T^{2} )^{2}$$
$59$ $$( -12 + T^{2} )^{2}$$
$61$ $$( 4 + T )^{4}$$
$67$ $$( 60 + T^{2} )^{2}$$
$71$ $$( -108 + T^{2} )^{2}$$
$73$ $$( -5 + T )^{4}$$
$79$ $$( 60 + T^{2} )^{2}$$
$83$ $$( -147 + T^{2} )^{2}$$
$89$ $$( 20 + T^{2} )^{2}$$
$97$ $$( -11 + T )^{4}$$