# Properties

 Label 1521.4.a.o Level 1521 Weight 4 Character orbit 1521.a Self dual yes Analytic conductor 89.742 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$89.7419051187$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} -5 q^{4} + \beta q^{5} + 8 \beta q^{7} -13 \beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} -5 q^{4} + \beta q^{5} + 8 \beta q^{7} -13 \beta q^{8} + 3 q^{10} -8 \beta q^{11} + 24 q^{14} + q^{16} + 117 q^{17} -66 \beta q^{19} -5 \beta q^{20} -24 q^{22} -78 q^{23} -122 q^{25} -40 \beta q^{28} + 141 q^{29} + 90 \beta q^{31} + 105 \beta q^{32} + 117 \beta q^{34} + 24 q^{35} -83 \beta q^{37} -198 q^{38} -39 q^{40} + 157 \beta q^{41} -104 q^{43} + 40 \beta q^{44} -78 \beta q^{46} + 174 \beta q^{47} -151 q^{49} -122 \beta q^{50} -93 q^{53} -24 q^{55} -312 q^{56} + 141 \beta q^{58} -164 \beta q^{59} + 145 q^{61} + 270 q^{62} + 307 q^{64} + 454 \beta q^{67} -585 q^{68} + 24 \beta q^{70} -610 \beta q^{71} -265 \beta q^{73} -249 q^{74} + 330 \beta q^{76} -192 q^{77} + 1276 q^{79} + \beta q^{80} + 471 q^{82} + 456 \beta q^{83} + 117 \beta q^{85} -104 \beta q^{86} + 312 q^{88} + 564 \beta q^{89} + 390 q^{92} + 522 q^{94} -198 q^{95} + 116 \beta q^{97} -151 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 10q^{4} + O(q^{10})$$ $$2q - 10q^{4} + 6q^{10} + 48q^{14} + 2q^{16} + 234q^{17} - 48q^{22} - 156q^{23} - 244q^{25} + 282q^{29} + 48q^{35} - 396q^{38} - 78q^{40} - 208q^{43} - 302q^{49} - 186q^{53} - 48q^{55} - 624q^{56} + 290q^{61} + 540q^{62} + 614q^{64} - 1170q^{68} - 498q^{74} - 384q^{77} + 2552q^{79} + 942q^{82} + 624q^{88} + 780q^{92} + 1044q^{94} - 396q^{95} + O(q^{100})$$

## 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
 −1.73205 1.73205
−1.73205 0 −5.00000 −1.73205 0 −13.8564 22.5167 0 3.00000
1.2 1.73205 0 −5.00000 1.73205 0 13.8564 −22.5167 0 3.00000
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.4.a.o 2
3.b odd 2 1 169.4.a.i 2
13.b even 2 1 inner 1521.4.a.o 2
13.f odd 12 2 117.4.q.a 2
39.d odd 2 1 169.4.a.i 2
39.f even 4 2 169.4.b.d 2
39.h odd 6 2 169.4.c.h 4
39.i odd 6 2 169.4.c.h 4
39.k even 12 2 13.4.e.b 2
39.k even 12 2 169.4.e.a 2
156.v odd 12 2 208.4.w.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.b 2 39.k even 12 2
117.4.q.a 2 13.f odd 12 2
169.4.a.i 2 3.b odd 2 1
169.4.a.i 2 39.d odd 2 1
169.4.b.d 2 39.f even 4 2
169.4.c.h 4 39.h odd 6 2
169.4.c.h 4 39.i odd 6 2
169.4.e.a 2 39.k even 12 2
208.4.w.b 2 156.v odd 12 2
1521.4.a.o 2 1.a even 1 1 trivial
1521.4.a.o 2 13.b even 2 1 inner

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1521))$$:

 $$T_{2}^{2} - 3$$ $$T_{5}^{2} - 3$$ $$T_{7}^{2} - 192$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 13 T^{2} + 64 T^{4}$$
$3$ 1
$5$ $$1 + 247 T^{2} + 15625 T^{4}$$
$7$ $$1 + 494 T^{2} + 117649 T^{4}$$
$11$ $$1 + 2470 T^{2} + 1771561 T^{4}$$
$13$ 1
$17$ $$( 1 - 117 T + 4913 T^{2} )^{2}$$
$19$ $$1 + 650 T^{2} + 47045881 T^{4}$$
$23$ $$( 1 + 78 T + 12167 T^{2} )^{2}$$
$29$ $$( 1 - 141 T + 24389 T^{2} )^{2}$$
$31$ $$1 + 35282 T^{2} + 887503681 T^{4}$$
$37$ $$1 + 80639 T^{2} + 2565726409 T^{4}$$
$41$ $$1 + 63895 T^{2} + 4750104241 T^{4}$$
$43$ $$( 1 + 104 T + 79507 T^{2} )^{2}$$
$47$ $$1 + 116818 T^{2} + 10779215329 T^{4}$$
$53$ $$( 1 + 93 T + 148877 T^{2} )^{2}$$
$59$ $$1 + 330070 T^{2} + 42180533641 T^{4}$$
$61$ $$( 1 - 145 T + 226981 T^{2} )^{2}$$
$67$ $$1 - 16822 T^{2} + 90458382169 T^{4}$$
$71$ $$1 - 400478 T^{2} + 128100283921 T^{4}$$
$73$ $$1 + 567359 T^{2} + 151334226289 T^{4}$$
$79$ $$( 1 - 1276 T + 493039 T^{2} )^{2}$$
$83$ $$1 + 519766 T^{2} + 326940373369 T^{4}$$
$89$ $$1 + 455650 T^{2} + 496981290961 T^{4}$$
$97$ $$1 + 1784978 T^{2} + 832972004929 T^{4}$$