# Properties

 Label 324.2.b.b Level $324$ Weight $2$ Character orbit 324.b Analytic conductor $2.587$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 324.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.58715302549$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.170772624.1 Defining polynomial: $$x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 6 x^{4} - 12 x^{3} + 20 x^{2} - 24 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 6 x^{4} - 12 x^{3} + 20 x^{2} - 24 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} - \nu^{5} + 2 \nu^{4} + 8 \nu^{2} - 4 \nu + 4$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{5} - \nu^{4} + 2 \nu^{3} - 6 \nu^{2} + 4 \nu - 12$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{7} + 3 \nu^{6} - 5 \nu^{5} + 5 \nu^{4} - 6 \nu^{3} + 14 \nu^{2} - 16 \nu + 20$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} + 3 \nu^{5} - 2 \nu^{4} + 3 \nu^{3} - 8 \nu^{2} + 12 \nu - 12$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} - 4 \nu^{5} + 3 \nu^{4} - 4 \nu^{3} + 10 \nu^{2} - 12 \nu + 16$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{7} + 4 \nu^{6} - 8 \nu^{5} + 5 \nu^{4} - 8 \nu^{3} + 22 \nu^{2} - 20 \nu + 32$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{7} - 5 \nu^{6} + 7 \nu^{5} - 6 \nu^{4} + 10 \nu^{3} - 24 \nu^{2} + 28 \nu - 28$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + \beta_{4} + \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{4} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} + 4 \beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-3 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{1} + 8$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{6} - \beta_{5} + 4 \beta_{2} + 4$$ $$\nu^{7}$$ $$=$$ $$($$$$-3 \beta_{7} - 2 \beta_{6} + 9 \beta_{5} + 5 \beta_{4} - 8 \beta_{3} + 4 \beta_{2} + \beta_{1} + 4$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$$.

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
323.1
 −1.02187 − 0.977642i −1.02187 + 0.977642i 0.774115 + 1.18353i 0.774115 − 1.18353i 1.41203 + 0.0786378i 1.41203 − 0.0786378i 0.335728 + 1.37379i 0.335728 − 1.37379i
−1.35760 0.396143i 0 1.68614 + 1.07561i 2.52434i 0 1.27582i −1.86301 2.12819i 0 −1.00000 + 3.42703i
323.2 −1.35760 + 0.396143i 0 1.68614 1.07561i 2.52434i 0 1.27582i −1.86301 + 2.12819i 0 −1.00000 3.42703i
323.3 −0.637910 1.26217i 0 −1.18614 + 1.61030i 0.792287i 0 2.71519i 2.78912 + 0.469882i 0 −1.00000 + 0.505408i
323.4 −0.637910 + 1.26217i 0 −1.18614 1.61030i 0.792287i 0 2.71519i 2.78912 0.469882i 0 −1.00000 0.505408i
323.5 0.637910 1.26217i 0 −1.18614 1.61030i 0.792287i 0 2.71519i −2.78912 + 0.469882i 0 −1.00000 0.505408i
323.6 0.637910 + 1.26217i 0 −1.18614 + 1.61030i 0.792287i 0 2.71519i −2.78912 0.469882i 0 −1.00000 + 0.505408i
323.7 1.35760 0.396143i 0 1.68614 1.07561i 2.52434i 0 1.27582i 1.86301 2.12819i 0 −1.00000 3.42703i
323.8 1.35760 + 0.396143i 0 1.68614 + 1.07561i 2.52434i 0 1.27582i 1.86301 + 2.12819i 0 −1.00000 + 3.42703i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 323.8 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 324.2.b.b 8
3.b odd 2 1 inner 324.2.b.b 8
4.b odd 2 1 inner 324.2.b.b 8
8.b even 2 1 5184.2.c.j 8
8.d odd 2 1 5184.2.c.j 8
9.c even 3 1 36.2.h.a 8
9.c even 3 1 108.2.h.a 8
9.d odd 6 1 36.2.h.a 8
9.d odd 6 1 108.2.h.a 8
12.b even 2 1 inner 324.2.b.b 8
24.f even 2 1 5184.2.c.j 8
24.h odd 2 1 5184.2.c.j 8
36.f odd 6 1 36.2.h.a 8
36.f odd 6 1 108.2.h.a 8
36.h even 6 1 36.2.h.a 8
36.h even 6 1 108.2.h.a 8
45.h odd 6 1 900.2.r.c 8
45.j even 6 1 900.2.r.c 8
45.k odd 12 2 900.2.o.a 16
45.l even 12 2 900.2.o.a 16
72.j odd 6 1 576.2.s.f 8
72.j odd 6 1 1728.2.s.f 8
72.l even 6 1 576.2.s.f 8
72.l even 6 1 1728.2.s.f 8
72.n even 6 1 576.2.s.f 8
72.n even 6 1 1728.2.s.f 8
72.p odd 6 1 576.2.s.f 8
72.p odd 6 1 1728.2.s.f 8
180.n even 6 1 900.2.r.c 8
180.p odd 6 1 900.2.r.c 8
180.v odd 12 2 900.2.o.a 16
180.x even 12 2 900.2.o.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.h.a 8 9.c even 3 1
36.2.h.a 8 9.d odd 6 1
36.2.h.a 8 36.f odd 6 1
36.2.h.a 8 36.h even 6 1
108.2.h.a 8 9.c even 3 1
108.2.h.a 8 9.d odd 6 1
108.2.h.a 8 36.f odd 6 1
108.2.h.a 8 36.h even 6 1
324.2.b.b 8 1.a even 1 1 trivial
324.2.b.b 8 3.b odd 2 1 inner
324.2.b.b 8 4.b odd 2 1 inner
324.2.b.b 8 12.b even 2 1 inner
576.2.s.f 8 72.j odd 6 1
576.2.s.f 8 72.l even 6 1
576.2.s.f 8 72.n even 6 1
576.2.s.f 8 72.p odd 6 1
900.2.o.a 16 45.k odd 12 2
900.2.o.a 16 45.l even 12 2
900.2.o.a 16 180.v odd 12 2
900.2.o.a 16 180.x even 12 2
900.2.r.c 8 45.h odd 6 1
900.2.r.c 8 45.j even 6 1
900.2.r.c 8 180.n even 6 1
900.2.r.c 8 180.p odd 6 1
1728.2.s.f 8 72.j odd 6 1
1728.2.s.f 8 72.l even 6 1
1728.2.s.f 8 72.n even 6 1
1728.2.s.f 8 72.p odd 6 1
5184.2.c.j 8 8.b even 2 1
5184.2.c.j 8 8.d odd 2 1
5184.2.c.j 8 24.f even 2 1
5184.2.c.j 8 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 7 T_{5}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(324, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 4 T^{2} - T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 4 + 7 T^{2} + T^{4} )^{2}$$
$7$ $$( 12 + 9 T^{2} + T^{4} )^{2}$$
$11$ $$( 3 - 12 T^{2} + T^{4} )^{2}$$
$13$ $$( -8 - T + T^{2} )^{4}$$
$17$ $$( 4 + 7 T^{2} + T^{4} )^{2}$$
$19$ $$( 108 + 27 T^{2} + T^{4} )^{2}$$
$23$ $$( 48 - 15 T^{2} + T^{4} )^{2}$$
$29$ $$( 4 + 7 T^{2} + T^{4} )^{2}$$
$31$ $$( 192 + 69 T^{2} + T^{4} )^{2}$$
$37$ $$( -32 + 2 T + T^{2} )^{4}$$
$41$ $$( 1 + 46 T^{2} + T^{4} )^{2}$$
$43$ $$( 2883 + 108 T^{2} + T^{4} )^{2}$$
$47$ $$( 192 - 135 T^{2} + T^{4} )^{2}$$
$53$ $$( 256 + 76 T^{2} + T^{4} )^{2}$$
$59$ $$( 4107 - 180 T^{2} + T^{4} )^{2}$$
$61$ $$( -8 - T + T^{2} )^{4}$$
$67$ $$( 2883 + 108 T^{2} + T^{4} )^{2}$$
$71$ $$( 432 - 144 T^{2} + T^{4} )^{2}$$
$73$ $$( -8 - T + T^{2} )^{4}$$
$79$ $$( 10092 + 201 T^{2} + T^{4} )^{2}$$
$83$ $$( 3072 - 111 T^{2} + T^{4} )^{2}$$
$89$ $$( 4096 + 172 T^{2} + T^{4} )^{2}$$
$97$ $$( -131 + 2 T + T^{2} )^{4}$$