# Properties

 Label 1521.4.a.p 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$89.7419051187$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 117) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - q^{4} - 4 \beta q^{5} + 22 q^{7} - 9 \beta q^{8} +O(q^{10})$$ q + b * q^2 - q^4 - 4*b * q^5 + 22 * q^7 - 9*b * q^8 $$q + \beta q^{2} - q^{4} - 4 \beta q^{5} + 22 q^{7} - 9 \beta q^{8} - 28 q^{10} - 2 \beta q^{11} + 22 \beta q^{14} - 55 q^{16} - 44 \beta q^{17} + 126 q^{19} + 4 \beta q^{20} - 14 q^{22} - 12 \beta q^{23} - 13 q^{25} - 22 q^{28} - 20 \beta q^{29} + 182 q^{31} + 17 \beta q^{32} - 308 q^{34} - 88 \beta q^{35} + 86 q^{37} + 126 \beta q^{38} + 252 q^{40} - 168 \beta q^{41} + 96 q^{43} + 2 \beta q^{44} - 84 q^{46} + 138 \beta q^{47} + 141 q^{49} - 13 \beta q^{50} + 72 \beta q^{53} + 56 q^{55} - 198 \beta q^{56} - 140 q^{58} - 222 \beta q^{59} + 574 q^{61} + 182 \beta q^{62} + 559 q^{64} + 530 q^{67} + 44 \beta q^{68} - 616 q^{70} + 306 \beta q^{71} + 154 q^{73} + 86 \beta q^{74} - 126 q^{76} - 44 \beta q^{77} - 460 q^{79} + 220 \beta q^{80} - 1176 q^{82} - 122 \beta q^{83} + 1232 q^{85} + 96 \beta q^{86} + 126 q^{88} + 544 \beta q^{89} + 12 \beta q^{92} + 966 q^{94} - 504 \beta q^{95} - 70 q^{97} + 141 \beta q^{98} +O(q^{100})$$ q + b * q^2 - q^4 - 4*b * q^5 + 22 * q^7 - 9*b * q^8 - 28 * q^10 - 2*b * q^11 + 22*b * q^14 - 55 * q^16 - 44*b * q^17 + 126 * q^19 + 4*b * q^20 - 14 * q^22 - 12*b * q^23 - 13 * q^25 - 22 * q^28 - 20*b * q^29 + 182 * q^31 + 17*b * q^32 - 308 * q^34 - 88*b * q^35 + 86 * q^37 + 126*b * q^38 + 252 * q^40 - 168*b * q^41 + 96 * q^43 + 2*b * q^44 - 84 * q^46 + 138*b * q^47 + 141 * q^49 - 13*b * q^50 + 72*b * q^53 + 56 * q^55 - 198*b * q^56 - 140 * q^58 - 222*b * q^59 + 574 * q^61 + 182*b * q^62 + 559 * q^64 + 530 * q^67 + 44*b * q^68 - 616 * q^70 + 306*b * q^71 + 154 * q^73 + 86*b * q^74 - 126 * q^76 - 44*b * q^77 - 460 * q^79 + 220*b * q^80 - 1176 * q^82 - 122*b * q^83 + 1232 * q^85 + 96*b * q^86 + 126 * q^88 + 544*b * q^89 + 12*b * q^92 + 966 * q^94 - 504*b * q^95 - 70 * q^97 + 141*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 44 q^{7}+O(q^{10})$$ 2 * q - 2 * q^4 + 44 * q^7 $$2 q - 2 q^{4} + 44 q^{7} - 56 q^{10} - 110 q^{16} + 252 q^{19} - 28 q^{22} - 26 q^{25} - 44 q^{28} + 364 q^{31} - 616 q^{34} + 172 q^{37} + 504 q^{40} + 192 q^{43} - 168 q^{46} + 282 q^{49} + 112 q^{55} - 280 q^{58} + 1148 q^{61} + 1118 q^{64} + 1060 q^{67} - 1232 q^{70} + 308 q^{73} - 252 q^{76} - 920 q^{79} - 2352 q^{82} + 2464 q^{85} + 252 q^{88} + 1932 q^{94} - 140 q^{97}+O(q^{100})$$ 2 * q - 2 * q^4 + 44 * q^7 - 56 * q^10 - 110 * q^16 + 252 * q^19 - 28 * q^22 - 26 * q^25 - 44 * q^28 + 364 * q^31 - 616 * q^34 + 172 * q^37 + 504 * q^40 + 192 * q^43 - 168 * q^46 + 282 * q^49 + 112 * q^55 - 280 * q^58 + 1148 * q^61 + 1118 * q^64 + 1060 * q^67 - 1232 * q^70 + 308 * q^73 - 252 * q^76 - 920 * q^79 - 2352 * q^82 + 2464 * q^85 + 252 * q^88 + 1932 * q^94 - 140 * q^97

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.64575 2.64575
−2.64575 0 −1.00000 10.5830 0 22.0000 23.8118 0 −28.0000
1.2 2.64575 0 −1.00000 −10.5830 0 22.0000 −23.8118 0 −28.0000
 $$n$$: 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 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.p 2
3.b odd 2 1 inner 1521.4.a.p 2
13.b even 2 1 117.4.a.e 2
39.d odd 2 1 117.4.a.e 2
52.b odd 2 1 1872.4.a.ba 2
156.h even 2 1 1872.4.a.ba 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.a.e 2 13.b even 2 1
117.4.a.e 2 39.d odd 2 1
1521.4.a.p 2 1.a even 1 1 trivial
1521.4.a.p 2 3.b odd 2 1 inner
1872.4.a.ba 2 52.b odd 2 1
1872.4.a.ba 2 156.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1521))$$:

 $$T_{2}^{2} - 7$$ T2^2 - 7 $$T_{5}^{2} - 112$$ T5^2 - 112 $$T_{7} - 22$$ T7 - 22

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 7$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 112$$
$7$ $$(T - 22)^{2}$$
$11$ $$T^{2} - 28$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 13552$$
$19$ $$(T - 126)^{2}$$
$23$ $$T^{2} - 1008$$
$29$ $$T^{2} - 2800$$
$31$ $$(T - 182)^{2}$$
$37$ $$(T - 86)^{2}$$
$41$ $$T^{2} - 197568$$
$43$ $$(T - 96)^{2}$$
$47$ $$T^{2} - 133308$$
$53$ $$T^{2} - 36288$$
$59$ $$T^{2} - 344988$$
$61$ $$(T - 574)^{2}$$
$67$ $$(T - 530)^{2}$$
$71$ $$T^{2} - 655452$$
$73$ $$(T - 154)^{2}$$
$79$ $$(T + 460)^{2}$$
$83$ $$T^{2} - 104188$$
$89$ $$T^{2} - 2071552$$
$97$ $$(T + 70)^{2}$$