# Properties

 Label 1449.2.a.j Level $1449$ Weight $2$ Character orbit 1449.a Self dual yes Analytic conductor $11.570$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1449 = 3^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1449.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.5703232529$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 483) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( 1 + \beta ) q^{4} + ( 3 - \beta ) q^{5} - q^{7} + 3 q^{8} +O(q^{10})$$ $$q + \beta q^{2} + ( 1 + \beta ) q^{4} + ( 3 - \beta ) q^{5} - q^{7} + 3 q^{8} + ( -3 + 2 \beta ) q^{10} + 5 q^{11} + \beta q^{13} -\beta q^{14} + ( -2 + \beta ) q^{16} + ( 1 + 2 \beta ) q^{17} + ( 3 - 2 \beta ) q^{19} + \beta q^{20} + 5 \beta q^{22} - q^{23} + ( 7 - 5 \beta ) q^{25} + ( 3 + \beta ) q^{26} + ( -1 - \beta ) q^{28} + ( 3 - 4 \beta ) q^{29} + 3 q^{31} + ( -3 - \beta ) q^{32} + ( 6 + 3 \beta ) q^{34} + ( -3 + \beta ) q^{35} -9 q^{37} + ( -6 + \beta ) q^{38} + ( 9 - 3 \beta ) q^{40} + ( 3 + 4 \beta ) q^{41} + ( -6 + 5 \beta ) q^{43} + ( 5 + 5 \beta ) q^{44} -\beta q^{46} + ( 4 + 2 \beta ) q^{47} + q^{49} + ( -15 + 2 \beta ) q^{50} + ( 3 + 2 \beta ) q^{52} + ( 1 + 5 \beta ) q^{53} + ( 15 - 5 \beta ) q^{55} -3 q^{56} + ( -12 - \beta ) q^{58} + ( 3 - 3 \beta ) q^{59} + ( -8 + 3 \beta ) q^{61} + 3 \beta q^{62} + ( 1 - 6 \beta ) q^{64} + ( -3 + 2 \beta ) q^{65} + ( -5 - 3 \beta ) q^{67} + ( 7 + 5 \beta ) q^{68} + ( 3 - 2 \beta ) q^{70} + ( 6 - 3 \beta ) q^{71} + ( 7 - 4 \beta ) q^{73} -9 \beta q^{74} + ( -3 - \beta ) q^{76} -5 q^{77} - q^{79} + ( -9 + 4 \beta ) q^{80} + ( 12 + 7 \beta ) q^{82} + ( -1 - 2 \beta ) q^{83} + ( -3 + 3 \beta ) q^{85} + ( 15 - \beta ) q^{86} + 15 q^{88} + ( -4 - 3 \beta ) q^{89} -\beta q^{91} + ( -1 - \beta ) q^{92} + ( 6 + 6 \beta ) q^{94} + ( 15 - 7 \beta ) q^{95} + ( -13 - 2 \beta ) q^{97} + \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 3 q^{4} + 5 q^{5} - 2 q^{7} + 6 q^{8} + O(q^{10})$$ $$2 q + q^{2} + 3 q^{4} + 5 q^{5} - 2 q^{7} + 6 q^{8} - 4 q^{10} + 10 q^{11} + q^{13} - q^{14} - 3 q^{16} + 4 q^{17} + 4 q^{19} + q^{20} + 5 q^{22} - 2 q^{23} + 9 q^{25} + 7 q^{26} - 3 q^{28} + 2 q^{29} + 6 q^{31} - 7 q^{32} + 15 q^{34} - 5 q^{35} - 18 q^{37} - 11 q^{38} + 15 q^{40} + 10 q^{41} - 7 q^{43} + 15 q^{44} - q^{46} + 10 q^{47} + 2 q^{49} - 28 q^{50} + 8 q^{52} + 7 q^{53} + 25 q^{55} - 6 q^{56} - 25 q^{58} + 3 q^{59} - 13 q^{61} + 3 q^{62} - 4 q^{64} - 4 q^{65} - 13 q^{67} + 19 q^{68} + 4 q^{70} + 9 q^{71} + 10 q^{73} - 9 q^{74} - 7 q^{76} - 10 q^{77} - 2 q^{79} - 14 q^{80} + 31 q^{82} - 4 q^{83} - 3 q^{85} + 29 q^{86} + 30 q^{88} - 11 q^{89} - q^{91} - 3 q^{92} + 18 q^{94} + 23 q^{95} - 28 q^{97} + q^{98} + O(q^{100})$$

## 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
 −1.30278 2.30278
−1.30278 0 −0.302776 4.30278 0 −1.00000 3.00000 0 −5.60555
1.2 2.30278 0 3.30278 0.697224 0 −1.00000 3.00000 0 1.60555
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$
$$23$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1449.2.a.j 2
3.b odd 2 1 483.2.a.f 2
12.b even 2 1 7728.2.a.x 2
21.c even 2 1 3381.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.f 2 3.b odd 2 1
1449.2.a.j 2 1.a even 1 1 trivial
3381.2.a.p 2 21.c even 2 1
7728.2.a.x 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1449))$$:

 $$T_{2}^{2} - T_{2} - 3$$ $$T_{5}^{2} - 5 T_{5} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$3 - 5 T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( -5 + T )^{2}$$
$13$ $$-3 - T + T^{2}$$
$17$ $$-9 - 4 T + T^{2}$$
$19$ $$-9 - 4 T + T^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$-51 - 2 T + T^{2}$$
$31$ $$( -3 + T )^{2}$$
$37$ $$( 9 + T )^{2}$$
$41$ $$-27 - 10 T + T^{2}$$
$43$ $$-69 + 7 T + T^{2}$$
$47$ $$12 - 10 T + T^{2}$$
$53$ $$-69 - 7 T + T^{2}$$
$59$ $$-27 - 3 T + T^{2}$$
$61$ $$13 + 13 T + T^{2}$$
$67$ $$13 + 13 T + T^{2}$$
$71$ $$-9 - 9 T + T^{2}$$
$73$ $$-27 - 10 T + T^{2}$$
$79$ $$( 1 + T )^{2}$$
$83$ $$-9 + 4 T + T^{2}$$
$89$ $$1 + 11 T + T^{2}$$
$97$ $$183 + 28 T + T^{2}$$