# Properties

 Label 1323.2.a.w Level $1323$ Weight $2$ Character orbit 1323.a Self dual yes Analytic conductor $10.564$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.5642081874$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 189) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 5 q^{4} + \beta q^{5} + 3 \beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} + 5 q^{4} + \beta q^{5} + 3 \beta q^{8} + 7 q^{10} -\beta q^{11} + 2 q^{13} + 11 q^{16} -7 q^{19} + 5 \beta q^{20} -7 q^{22} -3 \beta q^{23} + 2 q^{25} + 2 \beta q^{26} + 2 \beta q^{29} -3 q^{31} + 5 \beta q^{32} -3 q^{37} -7 \beta q^{38} + 21 q^{40} -\beta q^{41} + 8 q^{43} -5 \beta q^{44} -21 q^{46} + 2 \beta q^{50} + 10 q^{52} -7 q^{55} + 14 q^{58} + 8 q^{61} -3 \beta q^{62} + 13 q^{64} + 2 \beta q^{65} -2 q^{67} + 3 \beta q^{71} -3 \beta q^{74} -35 q^{76} -4 q^{79} + 11 \beta q^{80} -7 q^{82} -6 \beta q^{83} + 8 \beta q^{86} -21 q^{88} + 7 \beta q^{89} -15 \beta q^{92} -7 \beta q^{95} + 12 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 10q^{4} + O(q^{10})$$ $$2q + 10q^{4} + 14q^{10} + 4q^{13} + 22q^{16} - 14q^{19} - 14q^{22} + 4q^{25} - 6q^{31} - 6q^{37} + 42q^{40} + 16q^{43} - 42q^{46} + 20q^{52} - 14q^{55} + 28q^{58} + 16q^{61} + 26q^{64} - 4q^{67} - 70q^{76} - 8q^{79} - 14q^{82} - 42q^{88} + 24q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.64575 2.64575
−2.64575 0 5.00000 −2.64575 0 0 −7.93725 0 7.00000
1.2 2.64575 0 5.00000 2.64575 0 0 7.93725 0 7.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.a.w 2
3.b odd 2 1 inner 1323.2.a.w 2
7.b odd 2 1 189.2.a.f 2
21.c even 2 1 189.2.a.f 2
28.d even 2 1 3024.2.a.bi 2
35.c odd 2 1 4725.2.a.bb 2
63.l odd 6 2 567.2.f.i 4
63.o even 6 2 567.2.f.i 4
84.h odd 2 1 3024.2.a.bi 2
105.g even 2 1 4725.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.f 2 7.b odd 2 1
189.2.a.f 2 21.c even 2 1
567.2.f.i 4 63.l odd 6 2
567.2.f.i 4 63.o even 6 2
1323.2.a.w 2 1.a even 1 1 trivial
1323.2.a.w 2 3.b odd 2 1 inner
3024.2.a.bi 2 28.d even 2 1
3024.2.a.bi 2 84.h odd 2 1
4725.2.a.bb 2 35.c odd 2 1
4725.2.a.bb 2 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1323))$$:

 $$T_{2}^{2} - 7$$ $$T_{5}^{2} - 7$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-7 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$-7 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-7 + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$( 7 + T )^{2}$$
$23$ $$-63 + T^{2}$$
$29$ $$-28 + T^{2}$$
$31$ $$( 3 + T )^{2}$$
$37$ $$( 3 + T )^{2}$$
$41$ $$-7 + T^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -8 + T )^{2}$$
$67$ $$( 2 + T )^{2}$$
$71$ $$-63 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$-252 + T^{2}$$
$89$ $$-343 + T^{2}$$
$97$ $$( -12 + T )^{2}$$