# Properties

 Label 1521.2.a.a Level $1521$ Weight $2$ Character orbit 1521.a Self dual yes Analytic conductor $12.145$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + q^{5} + 2 q^{7} + 3 q^{8}+O(q^{10})$$ q - q^2 - q^4 + q^5 + 2 * q^7 + 3 * q^8 $$q - q^{2} - q^{4} + q^{5} + 2 q^{7} + 3 q^{8} - q^{10} + 2 q^{11} - 2 q^{14} - q^{16} + 7 q^{17} - 6 q^{19} - q^{20} - 2 q^{22} + 6 q^{23} - 4 q^{25} - 2 q^{28} + q^{29} + 4 q^{31} - 5 q^{32} - 7 q^{34} + 2 q^{35} + q^{37} + 6 q^{38} + 3 q^{40} - 9 q^{41} + 6 q^{43} - 2 q^{44} - 6 q^{46} - 6 q^{47} - 3 q^{49} + 4 q^{50} + 9 q^{53} + 2 q^{55} + 6 q^{56} - q^{58} + q^{61} - 4 q^{62} + 7 q^{64} - 2 q^{67} - 7 q^{68} - 2 q^{70} - 6 q^{71} + 11 q^{73} - q^{74} + 6 q^{76} + 4 q^{77} - 4 q^{79} - q^{80} + 9 q^{82} + 14 q^{83} + 7 q^{85} - 6 q^{86} + 6 q^{88} + 14 q^{89} - 6 q^{92} + 6 q^{94} - 6 q^{95} - 2 q^{97} + 3 q^{98}+O(q^{100})$$ q - q^2 - q^4 + q^5 + 2 * q^7 + 3 * q^8 - q^10 + 2 * q^11 - 2 * q^14 - q^16 + 7 * q^17 - 6 * q^19 - q^20 - 2 * q^22 + 6 * q^23 - 4 * q^25 - 2 * q^28 + q^29 + 4 * q^31 - 5 * q^32 - 7 * q^34 + 2 * q^35 + q^37 + 6 * q^38 + 3 * q^40 - 9 * q^41 + 6 * q^43 - 2 * q^44 - 6 * q^46 - 6 * q^47 - 3 * q^49 + 4 * q^50 + 9 * q^53 + 2 * q^55 + 6 * q^56 - q^58 + q^61 - 4 * q^62 + 7 * q^64 - 2 * q^67 - 7 * q^68 - 2 * q^70 - 6 * q^71 + 11 * q^73 - q^74 + 6 * q^76 + 4 * q^77 - 4 * q^79 - q^80 + 9 * q^82 + 14 * q^83 + 7 * q^85 - 6 * q^86 + 6 * q^88 + 14 * q^89 - 6 * q^92 + 6 * q^94 - 6 * q^95 - 2 * q^97 + 3 * 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
 0
−1.00000 0 −1.00000 1.00000 0 2.00000 3.00000 0 −1.00000
 $$n$$: 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.a 1
3.b odd 2 1 507.2.a.c 1
12.b even 2 1 8112.2.a.w 1
13.b even 2 1 1521.2.a.d 1
13.c even 3 2 117.2.g.b 2
13.d odd 4 2 1521.2.b.c 2
39.d odd 2 1 507.2.a.b 1
39.f even 4 2 507.2.b.b 2
39.h odd 6 2 507.2.e.c 2
39.i odd 6 2 39.2.e.a 2
39.k even 12 4 507.2.j.d 4
52.j odd 6 2 1872.2.t.j 2
156.h even 2 1 8112.2.a.bc 1
156.p even 6 2 624.2.q.c 2
195.x odd 6 2 975.2.i.f 2
195.bl even 12 4 975.2.bb.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 39.i odd 6 2
117.2.g.b 2 13.c even 3 2
507.2.a.b 1 39.d odd 2 1
507.2.a.c 1 3.b odd 2 1
507.2.b.b 2 39.f even 4 2
507.2.e.c 2 39.h odd 6 2
507.2.j.d 4 39.k even 12 4
624.2.q.c 2 156.p even 6 2
975.2.i.f 2 195.x odd 6 2
975.2.bb.d 4 195.bl even 12 4
1521.2.a.a 1 1.a even 1 1 trivial
1521.2.a.d 1 13.b even 2 1
1521.2.b.c 2 13.d odd 4 2
1872.2.t.j 2 52.j odd 6 2
8112.2.a.w 1 12.b even 2 1
8112.2.a.bc 1 156.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1521))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{5} - 1$$ T5 - 1 $$T_{7} - 2$$ T7 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T - 2$$
$11$ $$T - 2$$
$13$ $$T$$
$17$ $$T - 7$$
$19$ $$T + 6$$
$23$ $$T - 6$$
$29$ $$T - 1$$
$31$ $$T - 4$$
$37$ $$T - 1$$
$41$ $$T + 9$$
$43$ $$T - 6$$
$47$ $$T + 6$$
$53$ $$T - 9$$
$59$ $$T$$
$61$ $$T - 1$$
$67$ $$T + 2$$
$71$ $$T + 6$$
$73$ $$T - 11$$
$79$ $$T + 4$$
$83$ $$T - 14$$
$89$ $$T - 14$$
$97$ $$T + 2$$