# Properties

 Label 324.5.g.b Level 324 Weight 5 Character orbit 324.g Analytic conductor 33.492 Analytic rank 0 Dimension 2 CM discriminant -3 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 324.g (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.4918680392$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 12) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 94 - 94 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 94 - 94 \zeta_{6} ) q^{7} -146 \zeta_{6} q^{13} -46 q^{19} + ( -625 + 625 \zeta_{6} ) q^{25} -194 \zeta_{6} q^{31} -2062 q^{37} + ( 3214 - 3214 \zeta_{6} ) q^{43} -6435 \zeta_{6} q^{49} + ( 1966 - 1966 \zeta_{6} ) q^{61} -5906 \zeta_{6} q^{67} -8542 q^{73} + ( -7682 + 7682 \zeta_{6} ) q^{79} -13724 q^{91} + ( 18814 - 18814 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 94q^{7} + O(q^{10})$$ $$2q + 94q^{7} - 146q^{13} - 92q^{19} - 625q^{25} - 194q^{31} - 4124q^{37} + 3214q^{43} - 6435q^{49} + 1966q^{61} - 5906q^{67} - 17084q^{73} - 7682q^{79} - 27448q^{91} + 18814q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 47.0000 + 81.4064i 0 0 0
269.1 0 0 0 0 0 47.0000 81.4064i 0 0 0
 $$n$$: 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 CM by $$\Q(\sqrt{-3})$$
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.5.g.b 2
3.b odd 2 1 CM 324.5.g.b 2
9.c even 3 1 12.5.c.a 1
9.c even 3 1 inner 324.5.g.b 2
9.d odd 6 1 12.5.c.a 1
9.d odd 6 1 inner 324.5.g.b 2
36.f odd 6 1 48.5.e.a 1
36.h even 6 1 48.5.e.a 1
45.h odd 6 1 300.5.g.b 1
45.j even 6 1 300.5.g.b 1
45.k odd 12 2 300.5.b.a 2
45.l even 12 2 300.5.b.a 2
63.l odd 6 1 588.5.c.a 1
63.o even 6 1 588.5.c.a 1
72.j odd 6 1 192.5.e.a 1
72.l even 6 1 192.5.e.b 1
72.n even 6 1 192.5.e.a 1
72.p odd 6 1 192.5.e.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.5.c.a 1 9.c even 3 1
12.5.c.a 1 9.d odd 6 1
48.5.e.a 1 36.f odd 6 1
48.5.e.a 1 36.h even 6 1
192.5.e.a 1 72.j odd 6 1
192.5.e.a 1 72.n even 6 1
192.5.e.b 1 72.l even 6 1
192.5.e.b 1 72.p odd 6 1
300.5.b.a 2 45.k odd 12 2
300.5.b.a 2 45.l even 12 2
300.5.g.b 1 45.h odd 6 1
300.5.g.b 1 45.j even 6 1
324.5.g.b 2 1.a even 1 1 trivial
324.5.g.b 2 3.b odd 2 1 CM
324.5.g.b 2 9.c even 3 1 inner
324.5.g.b 2 9.d odd 6 1 inner
588.5.c.a 1 63.l odd 6 1
588.5.c.a 1 63.o even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{5}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}$$ $$T_{7}^{2} - 94 T_{7} + 8836$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 25 T + 625 T^{2} )( 1 + 25 T + 625 T^{2} )$$
$7$ $$( 1 - 71 T + 2401 T^{2} )( 1 - 23 T + 2401 T^{2} )$$
$11$ $$( 1 - 121 T + 14641 T^{2} )( 1 + 121 T + 14641 T^{2} )$$
$13$ $$( 1 - 191 T + 28561 T^{2} )( 1 + 337 T + 28561 T^{2} )$$
$17$ $$( 1 - 289 T )^{2}( 1 + 289 T )^{2}$$
$19$ $$( 1 + 46 T + 130321 T^{2} )^{2}$$
$23$ $$( 1 - 529 T + 279841 T^{2} )( 1 + 529 T + 279841 T^{2} )$$
$29$ $$( 1 - 841 T + 707281 T^{2} )( 1 + 841 T + 707281 T^{2} )$$
$31$ $$( 1 - 1559 T + 923521 T^{2} )( 1 + 1753 T + 923521 T^{2} )$$
$37$ $$( 1 + 2062 T + 1874161 T^{2} )^{2}$$
$41$ $$( 1 - 1681 T + 2825761 T^{2} )( 1 + 1681 T + 2825761 T^{2} )$$
$43$ $$( 1 - 3191 T + 3418801 T^{2} )( 1 - 23 T + 3418801 T^{2} )$$
$47$ $$( 1 - 2209 T + 4879681 T^{2} )( 1 + 2209 T + 4879681 T^{2} )$$
$53$ $$( 1 - 2809 T )^{2}( 1 + 2809 T )^{2}$$
$59$ $$( 1 - 3481 T + 12117361 T^{2} )( 1 + 3481 T + 12117361 T^{2} )$$
$61$ $$( 1 - 7199 T + 13845841 T^{2} )( 1 + 5233 T + 13845841 T^{2} )$$
$67$ $$( 1 - 2903 T + 20151121 T^{2} )( 1 + 8809 T + 20151121 T^{2} )$$
$71$ $$( 1 - 5041 T )^{2}( 1 + 5041 T )^{2}$$
$73$ $$( 1 + 8542 T + 28398241 T^{2} )^{2}$$
$79$ $$( 1 - 4679 T + 38950081 T^{2} )( 1 + 12361 T + 38950081 T^{2} )$$
$83$ $$( 1 - 6889 T + 47458321 T^{2} )( 1 + 6889 T + 47458321 T^{2} )$$
$89$ $$( 1 - 7921 T )^{2}( 1 + 7921 T )^{2}$$
$97$ $$( 1 - 9743 T + 88529281 T^{2} )( 1 - 9071 T + 88529281 T^{2} )$$