# Properties

 Label 1521.2.a.m 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: $$x^{2} - x - 4$$ 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.

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

## 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
 −1.56155 2.56155
−1.56155 0 0.438447 −3.56155 0 −0.561553 2.43845 0 5.56155
1.2 2.56155 0 4.56155 0.561553 0 3.56155 6.56155 0 1.43845
 $$n$$: 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.m 2
3.b odd 2 1 507.2.a.d 2
12.b even 2 1 8112.2.a.bo 2
13.b even 2 1 1521.2.a.g 2
13.d odd 4 2 1521.2.b.h 4
13.e even 6 2 117.2.g.c 4
39.d odd 2 1 507.2.a.g 2
39.f even 4 2 507.2.b.d 4
39.h odd 6 2 39.2.e.b 4
39.i odd 6 2 507.2.e.g 4
39.k even 12 4 507.2.j.g 8
52.i odd 6 2 1872.2.t.r 4
156.h even 2 1 8112.2.a.bk 2
156.r even 6 2 624.2.q.h 4
195.y odd 6 2 975.2.i.k 4
195.bf 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.h odd 6 2
117.2.g.c 4 13.e even 6 2
507.2.a.d 2 3.b odd 2 1
507.2.a.g 2 39.d odd 2 1
507.2.b.d 4 39.f even 4 2
507.2.e.g 4 39.i odd 6 2
507.2.j.g 8 39.k even 12 4
624.2.q.h 4 156.r even 6 2
975.2.i.k 4 195.y odd 6 2
975.2.bb.i 8 195.bf even 12 4
1521.2.a.g 2 13.b even 2 1
1521.2.a.m 2 1.a even 1 1 trivial
1521.2.b.h 4 13.d odd 4 2
1872.2.t.r 4 52.i odd 6 2
8112.2.a.bk 2 156.h even 2 1
8112.2.a.bo 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1521))$$:

 $$T_{2}^{2} - T_{2} - 4$$ T2^2 - T2 - 4 $$T_{5}^{2} + 3T_{5} - 2$$ T5^2 + 3*T5 - 2 $$T_{7}^{2} - 3T_{7} - 2$$ T7^2 - 3*T7 - 2

## Hecke characteristic polynomials

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