# Properties

 Label 1521.4.a.c Level $1521$ Weight $4$ Character orbit 1521.a Self dual yes Analytic conductor $89.742$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - 3 q^{2} + q^{4} + 9 q^{5} + 2 q^{7} + 21 q^{8}+O(q^{10})$$ q - 3 * q^2 + q^4 + 9 * q^5 + 2 * q^7 + 21 * q^8 $$q - 3 q^{2} + q^{4} + 9 q^{5} + 2 q^{7} + 21 q^{8} - 27 q^{10} - 30 q^{11} - 6 q^{14} - 71 q^{16} + 111 q^{17} - 46 q^{19} + 9 q^{20} + 90 q^{22} + 6 q^{23} - 44 q^{25} + 2 q^{28} + 105 q^{29} - 100 q^{31} + 45 q^{32} - 333 q^{34} + 18 q^{35} + 17 q^{37} + 138 q^{38} + 189 q^{40} + 231 q^{41} - 514 q^{43} - 30 q^{44} - 18 q^{46} + 162 q^{47} - 339 q^{49} + 132 q^{50} - 639 q^{53} - 270 q^{55} + 42 q^{56} - 315 q^{58} - 600 q^{59} + 233 q^{61} + 300 q^{62} + 433 q^{64} + 926 q^{67} + 111 q^{68} - 54 q^{70} + 930 q^{71} - 253 q^{73} - 51 q^{74} - 46 q^{76} - 60 q^{77} - 1324 q^{79} - 639 q^{80} - 693 q^{82} - 810 q^{83} + 999 q^{85} + 1542 q^{86} - 630 q^{88} - 498 q^{89} + 6 q^{92} - 486 q^{94} - 414 q^{95} + 1358 q^{97} + 1017 q^{98}+O(q^{100})$$ q - 3 * q^2 + q^4 + 9 * q^5 + 2 * q^7 + 21 * q^8 - 27 * q^10 - 30 * q^11 - 6 * q^14 - 71 * q^16 + 111 * q^17 - 46 * q^19 + 9 * q^20 + 90 * q^22 + 6 * q^23 - 44 * q^25 + 2 * q^28 + 105 * q^29 - 100 * q^31 + 45 * q^32 - 333 * q^34 + 18 * q^35 + 17 * q^37 + 138 * q^38 + 189 * q^40 + 231 * q^41 - 514 * q^43 - 30 * q^44 - 18 * q^46 + 162 * q^47 - 339 * q^49 + 132 * q^50 - 639 * q^53 - 270 * q^55 + 42 * q^56 - 315 * q^58 - 600 * q^59 + 233 * q^61 + 300 * q^62 + 433 * q^64 + 926 * q^67 + 111 * q^68 - 54 * q^70 + 930 * q^71 - 253 * q^73 - 51 * q^74 - 46 * q^76 - 60 * q^77 - 1324 * q^79 - 639 * q^80 - 693 * q^82 - 810 * q^83 + 999 * q^85 + 1542 * q^86 - 630 * q^88 - 498 * q^89 + 6 * q^92 - 486 * q^94 - 414 * q^95 + 1358 * q^97 + 1017 * 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
−3.00000 0 1.00000 9.00000 0 2.00000 21.0000 0 −27.0000
 $$n$$: 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 1521.4.a.c 1
3.b odd 2 1 507.4.a.e 1
13.b even 2 1 1521.4.a.j 1
13.c even 3 2 117.4.g.b 2
39.d odd 2 1 507.4.a.a 1
39.f even 4 2 507.4.b.c 2
39.i odd 6 2 39.4.e.a 2
156.p even 6 2 624.4.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.a 2 39.i odd 6 2
117.4.g.b 2 13.c even 3 2
507.4.a.a 1 39.d odd 2 1
507.4.a.e 1 3.b odd 2 1
507.4.b.c 2 39.f even 4 2
624.4.q.b 2 156.p even 6 2
1521.4.a.c 1 1.a even 1 1 trivial
1521.4.a.j 1 13.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1521))$$:

 $$T_{2} + 3$$ T2 + 3 $$T_{5} - 9$$ T5 - 9 $$T_{7} - 2$$ T7 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 3$$
$3$ $$T$$
$5$ $$T - 9$$
$7$ $$T - 2$$
$11$ $$T + 30$$
$13$ $$T$$
$17$ $$T - 111$$
$19$ $$T + 46$$
$23$ $$T - 6$$
$29$ $$T - 105$$
$31$ $$T + 100$$
$37$ $$T - 17$$
$41$ $$T - 231$$
$43$ $$T + 514$$
$47$ $$T - 162$$
$53$ $$T + 639$$
$59$ $$T + 600$$
$61$ $$T - 233$$
$67$ $$T - 926$$
$71$ $$T - 930$$
$73$ $$T + 253$$
$79$ $$T + 1324$$
$83$ $$T + 810$$
$89$ $$T + 498$$
$97$ $$T - 1358$$