# Properties

 Label 1521.4.a.ba 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: 4.4.1520092.1 Defining polynomial: $$x^{4} - x^{3} - 40x^{2} - 9x + 81$$ x^4 - x^3 - 40*x^2 - 9*x + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 117) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 14) q^{4} - \beta_{3} q^{5} + ( - 2 \beta_{2} - 8) q^{7} + (2 \beta_{3} + 9 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 14) * q^4 - b3 * q^5 + (-2*b2 - 8) * q^7 + (2*b3 + 9*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 14) q^{4} - \beta_{3} q^{5} + ( - 2 \beta_{2} - 8) q^{7} + (2 \beta_{3} + 9 \beta_1) q^{8} + ( - 10 \beta_{2} + 4) q^{10} + ( - \beta_{3} + 2 \beta_1) q^{11} + ( - 4 \beta_{3} - 14 \beta_1) q^{14} + (21 \beta_{2} + 78) q^{16} + 8 \beta_1 q^{17} + (14 \beta_{2} - 28) q^{19} + ( - 12 \beta_{3} - 26 \beta_1) q^{20} + ( - 8 \beta_{2} + 48) q^{22} + (2 \beta_{3} + 32 \beta_1) q^{23} + ( - 12 \beta_{2} + 171) q^{25} + ( - 38 \beta_{2} - 228) q^{28} + ( - 10 \beta_{3} + 20 \beta_1) q^{29} + ( - 2 \beta_{2} + 152) q^{31} + (26 \beta_{3} + 69 \beta_1) q^{32} + (8 \beta_{2} + 176) q^{34} + (4 \beta_{3} + 52 \beta_1) q^{35} + ( - 32 \beta_{2} - 30) q^{37} + (28 \beta_{3} + 14 \beta_1) q^{38} + ( - 66 \beta_{2} - 556) q^{40} + (9 \beta_{3} + 4 \beta_1) q^{41} + (8 \beta_{2} + 216) q^{43} + ( - 8 \beta_{3} + 8 \beta_1) q^{44} + (52 \beta_{2} + 696) q^{46} + ( - 5 \beta_{3} + 54 \beta_1) q^{47} + (36 \beta_{2} - 47) q^{49} + ( - 24 \beta_{3} + 135 \beta_1) q^{50} + ( - 8 \beta_{3} + 52 \beta_1) q^{53} + ( - 32 \beta_{2} + 304) q^{55} + ( - 44 \beta_{3} - 230 \beta_1) q^{56} + ( - 80 \beta_{2} + 480) q^{58} + ( - 15 \beta_{3} + 46 \beta_1) q^{59} + (44 \beta_{2} + 142) q^{61} + ( - 4 \beta_{3} + 146 \beta_1) q^{62} + (161 \beta_{2} + 790) q^{64} + (34 \beta_{2} - 780) q^{67} + (16 \beta_{3} + 136 \beta_1) q^{68} + (92 \beta_{2} + 1128) q^{70} + ( - 29 \beta_{3} - 98 \beta_1) q^{71} + (136 \beta_{2} + 10) q^{73} + ( - 64 \beta_{3} - 126 \beta_1) q^{74} + (182 \beta_{2} + 420) q^{76} + ( - 4 \beta_{3} + 24 \beta_1) q^{77} + (112 \beta_{2} - 236) q^{79} + ( - 36 \beta_{3} - 546 \beta_1) q^{80} + (94 \beta_{2} + 52) q^{82} + ( - 5 \beta_{3} + 2 \beta_1) q^{83} + ( - 80 \beta_{2} + 32) q^{85} + (16 \beta_{3} + 240 \beta_1) q^{86} + ( - 8 \beta_{2} - 176) q^{88} + (7 \beta_{3} + 204 \beta_1) q^{89} + (88 \beta_{3} + 596 \beta_1) q^{92} + (4 \beta_{2} + 1208) q^{94} + (56 \beta_{3} - 364 \beta_1) q^{95} + (104 \beta_{2} + 34) q^{97} + (72 \beta_{3} + 61 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + 14) * q^4 - b3 * q^5 + (-2*b2 - 8) * q^7 + (2*b3 + 9*b1) * q^8 + (-10*b2 + 4) * q^10 + (-b3 + 2*b1) * q^11 + (-4*b3 - 14*b1) * q^14 + (21*b2 + 78) * q^16 + 8*b1 * q^17 + (14*b2 - 28) * q^19 + (-12*b3 - 26*b1) * q^20 + (-8*b2 + 48) * q^22 + (2*b3 + 32*b1) * q^23 + (-12*b2 + 171) * q^25 + (-38*b2 - 228) * q^28 + (-10*b3 + 20*b1) * q^29 + (-2*b2 + 152) * q^31 + (26*b3 + 69*b1) * q^32 + (8*b2 + 176) * q^34 + (4*b3 + 52*b1) * q^35 + (-32*b2 - 30) * q^37 + (28*b3 + 14*b1) * q^38 + (-66*b2 - 556) * q^40 + (9*b3 + 4*b1) * q^41 + (8*b2 + 216) * q^43 + (-8*b3 + 8*b1) * q^44 + (52*b2 + 696) * q^46 + (-5*b3 + 54*b1) * q^47 + (36*b2 - 47) * q^49 + (-24*b3 + 135*b1) * q^50 + (-8*b3 + 52*b1) * q^53 + (-32*b2 + 304) * q^55 + (-44*b3 - 230*b1) * q^56 + (-80*b2 + 480) * q^58 + (-15*b3 + 46*b1) * q^59 + (44*b2 + 142) * q^61 + (-4*b3 + 146*b1) * q^62 + (161*b2 + 790) * q^64 + (34*b2 - 780) * q^67 + (16*b3 + 136*b1) * q^68 + (92*b2 + 1128) * q^70 + (-29*b3 - 98*b1) * q^71 + (136*b2 + 10) * q^73 + (-64*b3 - 126*b1) * q^74 + (182*b2 + 420) * q^76 + (-4*b3 + 24*b1) * q^77 + (112*b2 - 236) * q^79 + (-36*b3 - 546*b1) * q^80 + (94*b2 + 52) * q^82 + (-5*b3 + 2*b1) * q^83 + (-80*b2 + 32) * q^85 + (16*b3 + 240*b1) * q^86 + (-8*b2 - 176) * q^88 + (7*b3 + 204*b1) * q^89 + (88*b3 + 596*b1) * q^92 + (4*b2 + 1208) * q^94 + (56*b3 - 364*b1) * q^95 + (104*b2 + 34) * q^97 + (72*b3 + 61*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 58 q^{4} - 36 q^{7}+O(q^{10})$$ 4 * q + 58 * q^4 - 36 * q^7 $$4 q + 58 q^{4} - 36 q^{7} - 4 q^{10} + 354 q^{16} - 84 q^{19} + 176 q^{22} + 660 q^{25} - 988 q^{28} + 604 q^{31} + 720 q^{34} - 184 q^{37} - 2356 q^{40} + 880 q^{43} + 2888 q^{46} - 116 q^{49} + 1152 q^{55} + 1760 q^{58} + 656 q^{61} + 3482 q^{64} - 3052 q^{67} + 4696 q^{70} + 312 q^{73} + 2044 q^{76} - 720 q^{79} + 396 q^{82} - 32 q^{85} - 720 q^{88} + 4840 q^{94} + 344 q^{97}+O(q^{100})$$ 4 * q + 58 * q^4 - 36 * q^7 - 4 * q^10 + 354 * q^16 - 84 * q^19 + 176 * q^22 + 660 * q^25 - 988 * q^28 + 604 * q^31 + 720 * q^34 - 184 * q^37 - 2356 * q^40 + 880 * q^43 + 2888 * q^46 - 116 * q^49 + 1152 * q^55 + 1760 * q^58 + 656 * q^61 + 3482 * q^64 - 3052 * q^67 + 4696 * q^70 + 312 * q^73 + 2044 * q^76 - 720 * q^79 + 396 * q^82 - 32 * q^85 - 720 * q^88 + 4840 * q^94 + 344 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 40x^{2} - 9x + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - \nu^{2} - 31\nu - 9 ) / 9$$ (v^3 - v^2 - 31*v - 9) / 9 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 49\nu + 9 ) / 9$$ (-v^3 + v^2 + 49*v + 9) / 9 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 10\nu^{2} + 22\nu - 171 ) / 9$$ (-v^3 + 10*v^2 + 22*v - 171) / 9
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{3} + \beta_{2} + 3\beta _1 + 40 ) / 2$$ (2*b3 + b2 + 3*b1 + 40) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 16\beta_{2} + 26\beta _1 + 29$$ b3 + 16*b2 + 26*b1 + 29

## 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
 1.32145 −5.49403 −1.63814 6.81072
−5.48928 0 22.1322 14.0859 0 −24.2643 −77.5754 0 −77.3217
1.2 −3.85588 0 6.86783 −19.5342 0 6.26434 4.36551 0 75.3217
1.3 3.85588 0 6.86783 19.5342 0 6.26434 −4.36551 0 75.3217
1.4 5.48928 0 22.1322 −14.0859 0 −24.2643 77.5754 0 −77.3217
 $$n$$: 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.ba 4
3.b odd 2 1 inner 1521.4.a.ba 4
13.b even 2 1 117.4.a.g 4
39.d odd 2 1 117.4.a.g 4
52.b odd 2 1 1872.4.a.bo 4
156.h even 2 1 1872.4.a.bo 4

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

 $$T_{2}^{4} - 45T_{2}^{2} + 448$$ T2^4 - 45*T2^2 + 448 $$T_{5}^{4} - 580T_{5}^{2} + 75712$$ T5^4 - 580*T5^2 + 75712 $$T_{7}^{2} + 18T_{7} - 152$$ T7^2 + 18*T7 - 152

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 45T^{2} + 448$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 580 T^{2} + 75712$$
$7$ $$(T^{2} + 18 T - 152)^{2}$$
$11$ $$T^{4} - 752T^{2} + 7168$$
$13$ $$T^{4}$$
$17$ $$T^{4} - 2880 T^{2} + \cdots + 1835008$$
$19$ $$(T^{2} + 42 T - 10976)^{2}$$
$23$ $$T^{4} - 48656 T^{2} + \cdots + 295386112$$
$29$ $$T^{4} - 75200 T^{2} + \cdots + 71680000$$
$31$ $$(T^{2} - 302 T + 22568)^{2}$$
$37$ $$(T^{2} + 92 T - 57532)^{2}$$
$41$ $$T^{4} - 47844 T^{2} + \cdots + 569017792$$
$43$ $$(T^{2} - 440 T + 44672)^{2}$$
$47$ $$T^{4} - 144640 T^{2} + \cdots + 4778706688$$
$53$ $$T^{4} - 157136 T^{2} + \cdots + 3798925312$$
$59$ $$T^{4} - 222960 T^{2} + \cdots + 375897088$$
$61$ $$(T^{2} - 328 T - 85876)^{2}$$
$67$ $$(T^{2} + 1526 T + 514832)^{2}$$
$71$ $$T^{4} - 931328 T^{2} + \cdots + 31867295488$$
$73$ $$(T^{2} - 156 T - 1071308)^{2}$$
$79$ $$(T^{2} + 360 T - 698288)^{2}$$
$83$ $$T^{4} - 14640 T^{2} + \cdots + 39251968$$
$89$ $$T^{4} - 1906852 T^{2} + \cdots + 626946109888$$
$97$ $$(T^{2} - 172 T - 622636)^{2}$$