# Properties

 Label 1521.4.a.z Level $1521$ Weight $4$ Character orbit 1521.a Self dual yes Analytic conductor $89.742$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - 2x^{3} - 13x^{2} + 14x - 2$$ x^4 - 2*x^3 - 13*x^2 + 14*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + 9 q^{4} + ( - 2 \beta_{3} + 3 \beta_1) q^{5} + ( - 3 \beta_{3} + 11 \beta_1) q^{7} + \beta_{3} q^{8}+O(q^{10})$$ q + b3 * q^2 + 9 * q^4 + (-2*b3 + 3*b1) * q^5 + (-3*b3 + 11*b1) * q^7 + b3 * q^8 $$q + \beta_{3} q^{2} + 9 q^{4} + ( - 2 \beta_{3} + 3 \beta_1) q^{5} + ( - 3 \beta_{3} + 11 \beta_1) q^{7} + \beta_{3} q^{8} + (3 \beta_{2} - 34) q^{10} + ( - \beta_{3} + 21 \beta_1) q^{11} + (11 \beta_{2} - 51) q^{14} - 55 q^{16} + ( - \beta_{2} + 36) q^{17} + (9 \beta_{3} + 37 \beta_1) q^{19} + ( - 18 \beta_{3} + 27 \beta_1) q^{20} + (21 \beta_{2} - 17) q^{22} + (7 \beta_{2} + 69) q^{23} + ( - 12 \beta_{2} - 30) q^{25} + ( - 27 \beta_{3} + 99 \beta_1) q^{28} + ( - 22 \beta_{2} - 3) q^{29} + (42 \beta_{3} - 78 \beta_1) q^{31} - 63 \beta_{3} q^{32} + (36 \beta_{3} - 17 \beta_1) q^{34} + ( - 31 \beta_{2} + 201) q^{35} + (45 \beta_{3} + 82 \beta_1) q^{37} + (37 \beta_{2} + 153) q^{38} + (3 \beta_{2} - 34) q^{40} + (\beta_{3} + 30 \beta_1) q^{41} + ( - 15 \beta_{2} + 235) q^{43} + ( - 9 \beta_{3} + 189 \beta_1) q^{44} + (69 \beta_{3} + 119 \beta_1) q^{46} + ( - 79 \beta_{3} - 111 \beta_1) q^{47} + ( - 66 \beta_{2} + 173) q^{49} + ( - 30 \beta_{3} - 204 \beta_1) q^{50} + (18 \beta_{2} + 567) q^{53} + ( - 45 \beta_{2} + 223) q^{55} + (11 \beta_{2} - 51) q^{56} + ( - 3 \beta_{3} - 374 \beta_1) q^{58} + (8 \beta_{3} + 360 \beta_1) q^{59} + (87 \beta_{2} + 80) q^{61} + ( - 78 \beta_{2} + 714) q^{62} - 631 q^{64} + ( - 21 \beta_{3} + 83 \beta_1) q^{67} + ( - 9 \beta_{2} + 324) q^{68} + (201 \beta_{3} - 527 \beta_1) q^{70} + ( - 17 \beta_{3} + 219 \beta_1) q^{71} + 225 \beta_1 q^{73} + (82 \beta_{2} + 765) q^{74} + (81 \beta_{3} + 333 \beta_1) q^{76} + ( - 74 \beta_{2} + 744) q^{77} + (126 \beta_{2} + 2) q^{79} + (110 \beta_{3} - 165 \beta_1) q^{80} + (30 \beta_{2} + 17) q^{82} + (241 \beta_{3} - 177 \beta_1) q^{83} + ( - 81 \beta_{3} + 142 \beta_1) q^{85} + (235 \beta_{3} - 255 \beta_1) q^{86} + (21 \beta_{2} - 17) q^{88} + (242 \beta_{3} + 42 \beta_1) q^{89} + (63 \beta_{2} + 621) q^{92} + ( - 111 \beta_{2} - 1343) q^{94} + ( - 47 \beta_{2} + 27) q^{95} + ( - 402 \beta_{3} - 56 \beta_1) q^{97} + (173 \beta_{3} - 1122 \beta_1) q^{98}+O(q^{100})$$ q + b3 * q^2 + 9 * q^4 + (-2*b3 + 3*b1) * q^5 + (-3*b3 + 11*b1) * q^7 + b3 * q^8 + (3*b2 - 34) * q^10 + (-b3 + 21*b1) * q^11 + (11*b2 - 51) * q^14 - 55 * q^16 + (-b2 + 36) * q^17 + (9*b3 + 37*b1) * q^19 + (-18*b3 + 27*b1) * q^20 + (21*b2 - 17) * q^22 + (7*b2 + 69) * q^23 + (-12*b2 - 30) * q^25 + (-27*b3 + 99*b1) * q^28 + (-22*b2 - 3) * q^29 + (42*b3 - 78*b1) * q^31 - 63*b3 * q^32 + (36*b3 - 17*b1) * q^34 + (-31*b2 + 201) * q^35 + (45*b3 + 82*b1) * q^37 + (37*b2 + 153) * q^38 + (3*b2 - 34) * q^40 + (b3 + 30*b1) * q^41 + (-15*b2 + 235) * q^43 + (-9*b3 + 189*b1) * q^44 + (69*b3 + 119*b1) * q^46 + (-79*b3 - 111*b1) * q^47 + (-66*b2 + 173) * q^49 + (-30*b3 - 204*b1) * q^50 + (18*b2 + 567) * q^53 + (-45*b2 + 223) * q^55 + (11*b2 - 51) * q^56 + (-3*b3 - 374*b1) * q^58 + (8*b3 + 360*b1) * q^59 + (87*b2 + 80) * q^61 + (-78*b2 + 714) * q^62 - 631 * q^64 + (-21*b3 + 83*b1) * q^67 + (-9*b2 + 324) * q^68 + (201*b3 - 527*b1) * q^70 + (-17*b3 + 219*b1) * q^71 + 225*b1 * q^73 + (82*b2 + 765) * q^74 + (81*b3 + 333*b1) * q^76 + (-74*b2 + 744) * q^77 + (126*b2 + 2) * q^79 + (110*b3 - 165*b1) * q^80 + (30*b2 + 17) * q^82 + (241*b3 - 177*b1) * q^83 + (-81*b3 + 142*b1) * q^85 + (235*b3 - 255*b1) * q^86 + (21*b2 - 17) * q^88 + (242*b3 + 42*b1) * q^89 + (63*b2 + 621) * q^92 + (-111*b2 - 1343) * q^94 + (-47*b2 + 27) * q^95 + (-402*b3 - 56*b1) * q^97 + (173*b3 - 1122*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 36 q^{4}+O(q^{10})$$ 4 * q + 36 * q^4 $$4 q + 36 q^{4} - 136 q^{10} - 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} - 120 q^{25} - 12 q^{29} + 804 q^{35} + 612 q^{38} - 136 q^{40} + 940 q^{43} + 692 q^{49} + 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} - 2524 q^{64} + 1296 q^{68} + 3060 q^{74} + 2976 q^{77} + 8 q^{79} + 68 q^{82} - 68 q^{88} + 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100})$$ 4 * q + 36 * q^4 - 136 * q^10 - 204 * q^14 - 220 * q^16 + 144 * q^17 - 68 * q^22 + 276 * q^23 - 120 * q^25 - 12 * q^29 + 804 * q^35 + 612 * q^38 - 136 * q^40 + 940 * q^43 + 692 * q^49 + 2268 * q^53 + 892 * q^55 - 204 * q^56 + 320 * q^61 + 2856 * q^62 - 2524 * q^64 + 1296 * q^68 + 3060 * q^74 + 2976 * q^77 + 8 * q^79 + 68 * q^82 - 68 * q^88 + 2484 * q^92 - 5372 * q^94 + 108 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 13x^{2} + 14x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{3} - 3\nu^{2} - 25\nu + 13 ) / 5$$ (2*v^3 - 3*v^2 - 25*v + 13) / 5 $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 7$$ v^2 - v - 7 $$\beta_{3}$$ $$=$$ $$( -4\nu^{3} + 6\nu^{2} + 60\nu - 31 ) / 5$$ (-4*v^3 + 6*v^2 + 60*v - 31) / 5
 $$\nu$$ $$=$$ $$( \beta_{3} + 2\beta _1 + 1 ) / 2$$ (b3 + 2*b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 2\beta_{2} + 2\beta _1 + 15 ) / 2$$ (b3 + 2*b2 + 2*b1 + 15) / 2 $$\nu^{3}$$ $$=$$ $$( 14\beta_{3} + 3\beta_{2} + 33\beta _1 + 22 ) / 2$$ (14*b3 + 3*b2 + 33*b1 + 22) / 2

## 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
 −3.29360 0.170498 0.829502 4.29360
−4.12311 0 9.00000 3.05006 0 −6.68324 −4.12311 0 −12.5757
1.2 −4.12311 0 9.00000 13.4424 0 31.4219 −4.12311 0 −55.4243
1.3 4.12311 0 9.00000 −13.4424 0 −31.4219 4.12311 0 −55.4243
1.4 4.12311 0 9.00000 −3.05006 0 6.68324 4.12311 0 −12.5757
 $$n$$: 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
13.b even 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.z 4
3.b odd 2 1 507.4.a.k 4
13.b even 2 1 inner 1521.4.a.z 4
13.f odd 12 2 117.4.q.d 4
39.d odd 2 1 507.4.a.k 4
39.f even 4 2 507.4.b.e 4
39.k even 12 2 39.4.j.b 4
156.v odd 12 2 624.4.bv.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.b 4 39.k even 12 2
117.4.q.d 4 13.f odd 12 2
507.4.a.k 4 3.b odd 2 1
507.4.a.k 4 39.d odd 2 1
507.4.b.e 4 39.f even 4 2
624.4.bv.c 4 156.v odd 12 2
1521.4.a.z 4 1.a even 1 1 trivial
1521.4.a.z 4 13.b even 2 1 inner

## Hecke kernels

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

 $$T_{2}^{2} - 17$$ T2^2 - 17 $$T_{5}^{4} - 190T_{5}^{2} + 1681$$ T5^4 - 190*T5^2 + 1681 $$T_{7}^{4} - 1032T_{7}^{2} + 44100$$ T7^4 - 1032*T7^2 + 44100

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 17)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 190T^{2} + 1681$$
$7$ $$T^{4} - 1032 T^{2} + 44100$$
$11$ $$T^{4} - 2680 T^{2} + \cdots + 1705636$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 72 T + 1245)^{2}$$
$19$ $$T^{4} - 10968 T^{2} + \cdots + 7452900$$
$23$ $$(T^{2} - 138 T + 2262)^{2}$$
$29$ $$(T^{2} + 6 T - 24675)^{2}$$
$31$ $$T^{4} - 96480 T^{2} + \cdots + 137733696$$
$37$ $$T^{4} - 109194 T^{2} + \cdots + 203148009$$
$41$ $$T^{4} - 5434 T^{2} + \cdots + 7198489$$
$43$ $$(T^{2} - 470 T + 43750)^{2}$$
$47$ $$T^{4} - 286120 T^{2} + \cdots + 4779509956$$
$53$ $$(T^{2} - 1134 T + 304965)^{2}$$
$59$ $$T^{4} - 779776 T^{2} + \cdots + 150320594944$$
$61$ $$(T^{2} - 160 T - 379619)^{2}$$
$67$ $$T^{4} - 56328 T^{2} + \cdots + 173448900$$
$71$ $$T^{4} - 297592 T^{2} + \cdots + 19312660900$$
$73$ $$(T^{2} - 151875)^{2}$$
$79$ $$(T^{2} - 4 T - 809672)^{2}$$
$83$ $$T^{4} - 2162728 T^{2} + \cdots + 798145692100$$
$89$ $$T^{4} - 2001760 T^{2} + \cdots + 980686167616$$
$97$ $$T^{4} - 5513352 T^{2} + \cdots + 7495877379600$$