# Properties

 Label 1521.4.a.r Level $1521$ Weight $4$ Character orbit 1521.a Self dual yes Analytic conductor $89.742$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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 13) 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (\beta - 4) q^{4} + (\beta - 2) q^{5} + ( - 11 \beta + 10) q^{7} + ( - 11 \beta + 4) q^{8}+O(q^{10})$$ q + b * q^2 + (b - 4) * q^4 + (b - 2) * q^5 + (-11*b + 10) * q^7 + (-11*b + 4) * q^8 $$q + \beta q^{2} + (\beta - 4) q^{4} + (\beta - 2) q^{5} + ( - 11 \beta + 10) q^{7} + ( - 11 \beta + 4) q^{8} + ( - \beta + 4) q^{10} + (12 \beta + 34) q^{11} + ( - \beta - 44) q^{14} + ( - 15 \beta - 12) q^{16} + (17 \beta - 18) q^{17} + (32 \beta + 26) q^{19} + ( - 5 \beta + 12) q^{20} + (46 \beta + 48) q^{22} + (12 \beta - 104) q^{23} + ( - 3 \beta - 117) q^{25} + (43 \beta - 84) q^{28} + ( - 96 \beta + 70) q^{29} + (34 \beta + 26) q^{31} + (61 \beta - 92) q^{32} + ( - \beta + 68) q^{34} + (21 \beta - 64) q^{35} + ( - 5 \beta - 102) q^{37} + (58 \beta + 128) q^{38} + (15 \beta - 52) q^{40} + (22 \beta - 126) q^{41} + (143 \beta + 72) q^{43} + ( - 2 \beta - 88) q^{44} + ( - 92 \beta + 48) q^{46} + ( - 121 \beta + 278) q^{47} + ( - 99 \beta + 241) q^{49} + ( - 120 \beta - 12) q^{50} + ( - 30 \beta + 74) q^{53} + (22 \beta - 20) q^{55} + ( - 33 \beta + 524) q^{56} + ( - 26 \beta - 384) q^{58} + (124 \beta - 246) q^{59} + ( - 190 \beta - 434) q^{61} + (60 \beta + 136) q^{62} + (89 \beta + 340) q^{64} + (232 \beta - 150) q^{67} + ( - 69 \beta + 140) q^{68} + ( - 43 \beta + 84) q^{70} + ( - 231 \beta + 50) q^{71} + ( - 260 \beta - 98) q^{73} + ( - 107 \beta - 20) q^{74} + ( - 70 \beta + 24) q^{76} + ( - 386 \beta - 188) q^{77} + (40 \beta - 524) q^{79} + (3 \beta - 36) q^{80} + ( - 104 \beta + 88) q^{82} + ( - 182 \beta + 1070) q^{83} + ( - 35 \beta + 104) q^{85} + (215 \beta + 572) q^{86} + ( - 458 \beta - 392) q^{88} + ( - 388 \beta - 166) q^{89} + ( - 140 \beta + 464) q^{92} + (157 \beta - 484) q^{94} + ( - 6 \beta + 76) q^{95} + ( - 508 \beta + 718) q^{97} + (142 \beta - 396) q^{98}+O(q^{100})$$ q + b * q^2 + (b - 4) * q^4 + (b - 2) * q^5 + (-11*b + 10) * q^7 + (-11*b + 4) * q^8 + (-b + 4) * q^10 + (12*b + 34) * q^11 + (-b - 44) * q^14 + (-15*b - 12) * q^16 + (17*b - 18) * q^17 + (32*b + 26) * q^19 + (-5*b + 12) * q^20 + (46*b + 48) * q^22 + (12*b - 104) * q^23 + (-3*b - 117) * q^25 + (43*b - 84) * q^28 + (-96*b + 70) * q^29 + (34*b + 26) * q^31 + (61*b - 92) * q^32 + (-b + 68) * q^34 + (21*b - 64) * q^35 + (-5*b - 102) * q^37 + (58*b + 128) * q^38 + (15*b - 52) * q^40 + (22*b - 126) * q^41 + (143*b + 72) * q^43 + (-2*b - 88) * q^44 + (-92*b + 48) * q^46 + (-121*b + 278) * q^47 + (-99*b + 241) * q^49 + (-120*b - 12) * q^50 + (-30*b + 74) * q^53 + (22*b - 20) * q^55 + (-33*b + 524) * q^56 + (-26*b - 384) * q^58 + (124*b - 246) * q^59 + (-190*b - 434) * q^61 + (60*b + 136) * q^62 + (89*b + 340) * q^64 + (232*b - 150) * q^67 + (-69*b + 140) * q^68 + (-43*b + 84) * q^70 + (-231*b + 50) * q^71 + (-260*b - 98) * q^73 + (-107*b - 20) * q^74 + (-70*b + 24) * q^76 + (-386*b - 188) * q^77 + (40*b - 524) * q^79 + (3*b - 36) * q^80 + (-104*b + 88) * q^82 + (-182*b + 1070) * q^83 + (-35*b + 104) * q^85 + (215*b + 572) * q^86 + (-458*b - 392) * q^88 + (-388*b - 166) * q^89 + (-140*b + 464) * q^92 + (157*b - 484) * q^94 + (-6*b + 76) * q^95 + (-508*b + 718) * q^97 + (142*b - 396) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 7 q^{4} - 3 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10})$$ 2 * q + q^2 - 7 * q^4 - 3 * q^5 + 9 * q^7 - 3 * q^8 $$2 q + q^{2} - 7 q^{4} - 3 q^{5} + 9 q^{7} - 3 q^{8} + 7 q^{10} + 80 q^{11} - 89 q^{14} - 39 q^{16} - 19 q^{17} + 84 q^{19} + 19 q^{20} + 142 q^{22} - 196 q^{23} - 237 q^{25} - 125 q^{28} + 44 q^{29} + 86 q^{31} - 123 q^{32} + 135 q^{34} - 107 q^{35} - 209 q^{37} + 314 q^{38} - 89 q^{40} - 230 q^{41} + 287 q^{43} - 178 q^{44} + 4 q^{46} + 435 q^{47} + 383 q^{49} - 144 q^{50} + 118 q^{53} - 18 q^{55} + 1015 q^{56} - 794 q^{58} - 368 q^{59} - 1058 q^{61} + 332 q^{62} + 769 q^{64} - 68 q^{67} + 211 q^{68} + 125 q^{70} - 131 q^{71} - 456 q^{73} - 147 q^{74} - 22 q^{76} - 762 q^{77} - 1008 q^{79} - 69 q^{80} + 72 q^{82} + 1958 q^{83} + 173 q^{85} + 1359 q^{86} - 1242 q^{88} - 720 q^{89} + 788 q^{92} - 811 q^{94} + 146 q^{95} + 928 q^{97} - 650 q^{98}+O(q^{100})$$ 2 * q + q^2 - 7 * q^4 - 3 * q^5 + 9 * q^7 - 3 * q^8 + 7 * q^10 + 80 * q^11 - 89 * q^14 - 39 * q^16 - 19 * q^17 + 84 * q^19 + 19 * q^20 + 142 * q^22 - 196 * q^23 - 237 * q^25 - 125 * q^28 + 44 * q^29 + 86 * q^31 - 123 * q^32 + 135 * q^34 - 107 * q^35 - 209 * q^37 + 314 * q^38 - 89 * q^40 - 230 * q^41 + 287 * q^43 - 178 * q^44 + 4 * q^46 + 435 * q^47 + 383 * q^49 - 144 * q^50 + 118 * q^53 - 18 * q^55 + 1015 * q^56 - 794 * q^58 - 368 * q^59 - 1058 * q^61 + 332 * q^62 + 769 * q^64 - 68 * q^67 + 211 * q^68 + 125 * q^70 - 131 * q^71 - 456 * q^73 - 147 * q^74 - 22 * q^76 - 762 * q^77 - 1008 * q^79 - 69 * q^80 + 72 * q^82 + 1958 * q^83 + 173 * q^85 + 1359 * q^86 - 1242 * q^88 - 720 * q^89 + 788 * q^92 - 811 * q^94 + 146 * q^95 + 928 * q^97 - 650 * 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.

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
 −1.56155 2.56155
−1.56155 0 −5.56155 −3.56155 0 27.1771 21.1771 0 5.56155
1.2 2.56155 0 −1.43845 0.561553 0 −18.1771 −24.1771 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 1521.4.a.r 2
3.b odd 2 1 169.4.a.g 2
13.b even 2 1 117.4.a.d 2
39.d odd 2 1 13.4.a.b 2
39.f even 4 2 169.4.b.f 4
39.h odd 6 2 169.4.c.g 4
39.i odd 6 2 169.4.c.j 4
39.k even 12 4 169.4.e.f 8
52.b odd 2 1 1872.4.a.bb 2
156.h even 2 1 208.4.a.h 2
195.e odd 2 1 325.4.a.f 2
195.s even 4 2 325.4.b.e 4
273.g even 2 1 637.4.a.b 2
312.b odd 2 1 832.4.a.s 2
312.h even 2 1 832.4.a.z 2
429.e even 2 1 1573.4.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 39.d odd 2 1
117.4.a.d 2 13.b even 2 1
169.4.a.g 2 3.b odd 2 1
169.4.b.f 4 39.f even 4 2
169.4.c.g 4 39.h odd 6 2
169.4.c.j 4 39.i odd 6 2
169.4.e.f 8 39.k even 12 4
208.4.a.h 2 156.h even 2 1
325.4.a.f 2 195.e odd 2 1
325.4.b.e 4 195.s even 4 2
637.4.a.b 2 273.g even 2 1
832.4.a.s 2 312.b odd 2 1
832.4.a.z 2 312.h even 2 1
1521.4.a.r 2 1.a even 1 1 trivial
1573.4.a.b 2 429.e even 2 1
1872.4.a.bb 2 52.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} - 4$$ T2^2 - T2 - 4 $$T_{5}^{2} + 3T_{5} - 2$$ T5^2 + 3*T5 - 2 $$T_{7}^{2} - 9T_{7} - 494$$ T7^2 - 9*T7 - 494

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 3T - 2$$
$7$ $$T^{2} - 9T - 494$$
$11$ $$T^{2} - 80T + 988$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 19T - 1138$$
$19$ $$T^{2} - 84T - 2588$$
$23$ $$T^{2} + 196T + 8992$$
$29$ $$T^{2} - 44T - 38684$$
$31$ $$T^{2} - 86T - 3064$$
$37$ $$T^{2} + 209T + 10814$$
$41$ $$T^{2} + 230T + 11168$$
$43$ $$T^{2} - 287T - 66316$$
$47$ $$T^{2} - 435T - 14918$$
$53$ $$T^{2} - 118T - 344$$
$59$ $$T^{2} + 368T - 31492$$
$61$ $$T^{2} + 1058 T + 126416$$
$67$ $$T^{2} + 68T - 227596$$
$71$ $$T^{2} + 131T - 222494$$
$73$ $$T^{2} + 456T - 235316$$
$79$ $$T^{2} + 1008 T + 247216$$
$83$ $$T^{2} - 1958 T + 817664$$
$89$ $$T^{2} + 720T - 510212$$
$97$ $$T^{2} - 928T - 881476$$