# Properties

 Label 1521.2.a.g Level $1521$ Weight $2$ Character orbit 1521.a Self dual yes Analytic conductor $12.145$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1521.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.1452461474$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + ( 2 + \beta ) q^{4} + ( 2 - \beta ) q^{5} + ( -1 - \beta ) q^{7} + ( -4 - \beta ) q^{8} +O(q^{10})$$ $$q -\beta q^{2} + ( 2 + \beta ) q^{4} + ( 2 - \beta ) q^{5} + ( -1 - \beta ) q^{7} + ( -4 - \beta ) q^{8} + ( 4 - \beta ) q^{10} + 2 q^{11} + ( 4 + 2 \beta ) q^{14} + 3 \beta q^{16} -\beta q^{17} + ( 4 - 2 \beta ) q^{19} -\beta q^{20} -2 \beta q^{22} -2 q^{23} + ( 3 - 3 \beta ) q^{25} + ( -6 - 4 \beta ) q^{28} + ( -2 + 3 \beta ) q^{29} + ( 1 - \beta ) q^{31} + ( -4 - \beta ) q^{32} + ( 4 + \beta ) q^{34} + 2 q^{35} + ( 6 - \beta ) q^{37} + ( 8 - 2 \beta ) q^{38} + ( -4 + 3 \beta ) q^{40} -\beta q^{41} + ( 3 - \beta ) q^{43} + ( 4 + 2 \beta ) q^{44} + 2 \beta q^{46} + ( -2 + 4 \beta ) q^{47} + ( -2 + 3 \beta ) q^{49} + 12 q^{50} + ( -4 - 3 \beta ) q^{53} + ( 4 - 2 \beta ) q^{55} + ( 8 + 6 \beta ) q^{56} + ( -12 - \beta ) q^{58} + ( 6 + 2 \beta ) q^{59} + ( 7 + 2 \beta ) q^{61} + 4 q^{62} + ( 4 - \beta ) q^{64} + ( 3 - \beta ) q^{67} + ( -4 - 3 \beta ) q^{68} -2 \beta q^{70} -14 q^{71} + ( -7 + 2 \beta ) q^{73} + ( 4 - 5 \beta ) q^{74} -2 \beta q^{76} + ( -2 - 2 \beta ) q^{77} + ( 7 + \beta ) q^{79} + ( -12 + 3 \beta ) q^{80} + ( 4 + \beta ) q^{82} + ( 4 + 2 \beta ) q^{83} + ( 4 - \beta ) q^{85} + ( 4 - 2 \beta ) q^{86} + ( -8 - 2 \beta ) q^{88} + ( -8 - 2 \beta ) q^{89} + ( -4 - 2 \beta ) q^{92} + ( -16 - 2 \beta ) q^{94} + ( 16 - 6 \beta ) q^{95} + ( -7 + \beta ) q^{97} + ( -12 - \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + 5q^{4} + 3q^{5} - 3q^{7} - 9q^{8} + O(q^{10})$$ $$2q - q^{2} + 5q^{4} + 3q^{5} - 3q^{7} - 9q^{8} + 7q^{10} + 4q^{11} + 10q^{14} + 3q^{16} - q^{17} + 6q^{19} - q^{20} - 2q^{22} - 4q^{23} + 3q^{25} - 16q^{28} - q^{29} + q^{31} - 9q^{32} + 9q^{34} + 4q^{35} + 11q^{37} + 14q^{38} - 5q^{40} - q^{41} + 5q^{43} + 10q^{44} + 2q^{46} - q^{49} + 24q^{50} - 11q^{53} + 6q^{55} + 22q^{56} - 25q^{58} + 14q^{59} + 16q^{61} + 8q^{62} + 7q^{64} + 5q^{67} - 11q^{68} - 2q^{70} - 28q^{71} - 12q^{73} + 3q^{74} - 2q^{76} - 6q^{77} + 15q^{79} - 21q^{80} + 9q^{82} + 10q^{83} + 7q^{85} + 6q^{86} - 18q^{88} - 18q^{89} - 10q^{92} - 34q^{94} + 26q^{95} - 13q^{97} - 25q^{98} + O(q^{100})$$

## 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
 2.56155 −1.56155
−2.56155 0 4.56155 −0.561553 0 −3.56155 −6.56155 0 1.43845
1.2 1.56155 0 0.438447 3.56155 0 0.561553 −2.43845 0 5.56155
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$13$$ $$1$$

## 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 1521.2.a.g 2
3.b odd 2 1 507.2.a.g 2
12.b even 2 1 8112.2.a.bk 2
13.b even 2 1 1521.2.a.m 2
13.c even 3 2 117.2.g.c 4
13.d odd 4 2 1521.2.b.h 4
39.d odd 2 1 507.2.a.d 2
39.f even 4 2 507.2.b.d 4
39.h odd 6 2 507.2.e.g 4
39.i odd 6 2 39.2.e.b 4
39.k even 12 4 507.2.j.g 8
52.j odd 6 2 1872.2.t.r 4
156.h even 2 1 8112.2.a.bo 2
156.p even 6 2 624.2.q.h 4
195.x odd 6 2 975.2.i.k 4
195.bl even 12 4 975.2.bb.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 39.i odd 6 2
117.2.g.c 4 13.c even 3 2
507.2.a.d 2 39.d odd 2 1
507.2.a.g 2 3.b odd 2 1
507.2.b.d 4 39.f even 4 2
507.2.e.g 4 39.h odd 6 2
507.2.j.g 8 39.k even 12 4
624.2.q.h 4 156.p even 6 2
975.2.i.k 4 195.x odd 6 2
975.2.bb.i 8 195.bl even 12 4
1521.2.a.g 2 1.a even 1 1 trivial
1521.2.a.m 2 13.b even 2 1
1521.2.b.h 4 13.d odd 4 2
1872.2.t.r 4 52.j odd 6 2
8112.2.a.bk 2 12.b even 2 1
8112.2.a.bo 2 156.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1521))$$:

 $$T_{2}^{2} + T_{2} - 4$$ $$T_{5}^{2} - 3 T_{5} - 2$$ $$T_{7}^{2} + 3 T_{7} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$-2 - 3 T + T^{2}$$
$7$ $$-2 + 3 T + T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$-4 + T + T^{2}$$
$19$ $$-8 - 6 T + T^{2}$$
$23$ $$( 2 + T )^{2}$$
$29$ $$-38 + T + T^{2}$$
$31$ $$-4 - T + T^{2}$$
$37$ $$26 - 11 T + T^{2}$$
$41$ $$-4 + T + T^{2}$$
$43$ $$2 - 5 T + T^{2}$$
$47$ $$-68 + T^{2}$$
$53$ $$-8 + 11 T + T^{2}$$
$59$ $$32 - 14 T + T^{2}$$
$61$ $$47 - 16 T + T^{2}$$
$67$ $$2 - 5 T + T^{2}$$
$71$ $$( 14 + T )^{2}$$
$73$ $$19 + 12 T + T^{2}$$
$79$ $$52 - 15 T + T^{2}$$
$83$ $$8 - 10 T + T^{2}$$
$89$ $$64 + 18 T + T^{2}$$
$97$ $$38 + 13 T + T^{2}$$