# Properties

 Label 1521.4.a.i Level $1521$ Weight $4$ Character orbit 1521.a Self dual yes Analytic conductor $89.742$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3 q^{2} + q^{4} - 9 q^{5} - 15 q^{7} - 21 q^{8} + O(q^{10})$$ $$q + 3 q^{2} + q^{4} - 9 q^{5} - 15 q^{7} - 21 q^{8} - 27 q^{10} - 48 q^{11} - 45 q^{14} - 71 q^{16} - 45 q^{17} - 6 q^{19} - 9 q^{20} - 144 q^{22} + 162 q^{23} - 44 q^{25} - 15 q^{28} + 144 q^{29} - 264 q^{31} - 45 q^{32} - 135 q^{34} + 135 q^{35} - 303 q^{37} - 18 q^{38} + 189 q^{40} - 192 q^{41} + 97 q^{43} - 48 q^{44} + 486 q^{46} + 111 q^{47} - 118 q^{49} - 132 q^{50} + 414 q^{53} + 432 q^{55} + 315 q^{56} + 432 q^{58} + 522 q^{59} + 376 q^{61} - 792 q^{62} + 433 q^{64} + 36 q^{67} - 45 q^{68} + 405 q^{70} + 357 q^{71} + 1098 q^{73} - 909 q^{74} - 6 q^{76} + 720 q^{77} - 830 q^{79} + 639 q^{80} - 576 q^{82} - 438 q^{83} + 405 q^{85} + 291 q^{86} + 1008 q^{88} - 438 q^{89} + 162 q^{92} + 333 q^{94} + 54 q^{95} + 852 q^{97} - 354 q^{98} + O(q^{100})$$

## 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
 0
3.00000 0 1.00000 −9.00000 0 −15.0000 −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.i 1
3.b odd 2 1 169.4.a.b 1
13.b even 2 1 1521.4.a.d 1
13.d odd 4 2 117.4.b.a 2
39.d odd 2 1 169.4.a.c 1
39.f even 4 2 13.4.b.a 2
39.h odd 6 2 169.4.c.b 2
39.i odd 6 2 169.4.c.c 2
39.k even 12 4 169.4.e.d 4
156.l odd 4 2 208.4.f.b 2
195.j odd 4 2 325.4.d.a 2
195.n even 4 2 325.4.c.b 2
195.u odd 4 2 325.4.d.b 2
312.w odd 4 2 832.4.f.c 2
312.y even 4 2 832.4.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 39.f even 4 2
117.4.b.a 2 13.d odd 4 2
169.4.a.b 1 3.b odd 2 1
169.4.a.c 1 39.d odd 2 1
169.4.c.b 2 39.h odd 6 2
169.4.c.c 2 39.i odd 6 2
169.4.e.d 4 39.k even 12 4
208.4.f.b 2 156.l odd 4 2
325.4.c.b 2 195.n even 4 2
325.4.d.a 2 195.j odd 4 2
325.4.d.b 2 195.u odd 4 2
832.4.f.c 2 312.w odd 4 2
832.4.f.e 2 312.y even 4 2
1521.4.a.d 1 13.b even 2 1
1521.4.a.i 1 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{2} - 3$$ $$T_{5} + 9$$ $$T_{7} + 15$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T$$
$3$ $$T$$
$5$ $$9 + T$$
$7$ $$15 + T$$
$11$ $$48 + T$$
$13$ $$T$$
$17$ $$45 + T$$
$19$ $$6 + T$$
$23$ $$-162 + T$$
$29$ $$-144 + T$$
$31$ $$264 + T$$
$37$ $$303 + T$$
$41$ $$192 + T$$
$43$ $$-97 + T$$
$47$ $$-111 + T$$
$53$ $$-414 + T$$
$59$ $$-522 + T$$
$61$ $$-376 + T$$
$67$ $$-36 + T$$
$71$ $$-357 + T$$
$73$ $$-1098 + T$$
$79$ $$830 + T$$
$83$ $$438 + T$$
$89$ $$438 + T$$
$97$ $$-852 + T$$