# Properties

 Label 1521.4.a.m 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: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 8 q^{4} - 3 \beta q^{5} + 6 \beta q^{7} +O(q^{10})$$ q - 8 * q^4 - 3*b * q^5 + 6*b * q^7 $$q - 8 q^{4} - 3 \beta q^{5} + 6 \beta q^{7} + 30 \beta q^{11} + 64 q^{16} - 117 q^{17} + 14 \beta q^{19} + 24 \beta q^{20} + 18 q^{23} - 98 q^{25} - 48 \beta q^{28} + 99 q^{29} + 112 \beta q^{31} - 54 q^{35} - 65 \beta q^{37} + 21 \beta q^{41} + 82 q^{43} - 240 \beta q^{44} - 42 \beta q^{47} - 235 q^{49} + 261 q^{53} - 270 q^{55} + 456 \beta q^{59} - 719 q^{61} - 512 q^{64} - 406 \beta q^{67} + 936 q^{68} - 270 \beta q^{71} - 395 \beta q^{73} - 112 \beta q^{76} + 540 q^{77} - 440 q^{79} - 192 \beta q^{80} - 690 \beta q^{83} + 351 \beta q^{85} + 876 \beta q^{89} - 144 q^{92} - 126 q^{95} + 668 \beta q^{97} +O(q^{100})$$ q - 8 * q^4 - 3*b * q^5 + 6*b * q^7 + 30*b * q^11 + 64 * q^16 - 117 * q^17 + 14*b * q^19 + 24*b * q^20 + 18 * q^23 - 98 * q^25 - 48*b * q^28 + 99 * q^29 + 112*b * q^31 - 54 * q^35 - 65*b * q^37 + 21*b * q^41 + 82 * q^43 - 240*b * q^44 - 42*b * q^47 - 235 * q^49 + 261 * q^53 - 270 * q^55 + 456*b * q^59 - 719 * q^61 - 512 * q^64 - 406*b * q^67 + 936 * q^68 - 270*b * q^71 - 395*b * q^73 - 112*b * q^76 + 540 * q^77 - 440 * q^79 - 192*b * q^80 - 690*b * q^83 + 351*b * q^85 + 876*b * q^89 - 144 * q^92 - 126 * q^95 + 668*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 16 q^{4}+O(q^{10})$$ 2 * q - 16 * q^4 $$2 q - 16 q^{4} + 128 q^{16} - 234 q^{17} + 36 q^{23} - 196 q^{25} + 198 q^{29} - 108 q^{35} + 164 q^{43} - 470 q^{49} + 522 q^{53} - 540 q^{55} - 1438 q^{61} - 1024 q^{64} + 1872 q^{68} + 1080 q^{77} - 880 q^{79} - 288 q^{92} - 252 q^{95}+O(q^{100})$$ 2 * q - 16 * q^4 + 128 * q^16 - 234 * q^17 + 36 * q^23 - 196 * q^25 + 198 * q^29 - 108 * q^35 + 164 * q^43 - 470 * q^49 + 522 * q^53 - 540 * q^55 - 1438 * q^61 - 1024 * q^64 + 1872 * q^68 + 1080 * q^77 - 880 * q^79 - 288 * q^92 - 252 * q^95

## 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
0 0 −8.00000 −5.19615 0 10.3923 0 0 0
1.2 0 0 −8.00000 5.19615 0 −10.3923 0 0 0
 $$n$$: 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

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.m 2
3.b odd 2 1 507.4.a.g 2
13.b even 2 1 inner 1521.4.a.m 2
13.f odd 12 2 117.4.q.b 2
39.d odd 2 1 507.4.a.g 2
39.f even 4 2 507.4.b.a 2
39.k even 12 2 39.4.j.a 2
156.v odd 12 2 624.4.bv.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.a 2 39.k even 12 2
117.4.q.b 2 13.f odd 12 2
507.4.a.g 2 3.b odd 2 1
507.4.a.g 2 39.d odd 2 1
507.4.b.a 2 39.f even 4 2
624.4.bv.a 2 156.v odd 12 2
1521.4.a.m 2 1.a even 1 1 trivial
1521.4.a.m 2 13.b even 2 1 inner

## Hecke kernels

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

 $$T_{2}$$ T2 $$T_{5}^{2} - 27$$ T5^2 - 27 $$T_{7}^{2} - 108$$ T7^2 - 108

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 27$$
$7$ $$T^{2} - 108$$
$11$ $$T^{2} - 2700$$
$13$ $$T^{2}$$
$17$ $$(T + 117)^{2}$$
$19$ $$T^{2} - 588$$
$23$ $$(T - 18)^{2}$$
$29$ $$(T - 99)^{2}$$
$31$ $$T^{2} - 37632$$
$37$ $$T^{2} - 12675$$
$41$ $$T^{2} - 1323$$
$43$ $$(T - 82)^{2}$$
$47$ $$T^{2} - 5292$$
$53$ $$(T - 261)^{2}$$
$59$ $$T^{2} - 623808$$
$61$ $$(T + 719)^{2}$$
$67$ $$T^{2} - 494508$$
$71$ $$T^{2} - 218700$$
$73$ $$T^{2} - 468075$$
$79$ $$(T + 440)^{2}$$
$83$ $$T^{2} - 1428300$$
$89$ $$T^{2} - 2302128$$
$97$ $$T^{2} - 1338672$$