# Properties

 Label 522.4.a.k Level $522$ Weight $4$ Character orbit 522.a Self dual yes Analytic conductor $30.799$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$522 = 2 \cdot 3^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 522.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$30.7989970230$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.19816.1 Defining polynomial: $$x^{3} - x^{2} - 42x - 54$$ x^3 - x^2 - 42*x - 54 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 58) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 4 q^{4} + ( - \beta_{2} - 2 \beta_1 - 6) q^{5} + ( - 4 \beta_{2} + 8) q^{7} - 8 q^{8}+O(q^{10})$$ q - 2 * q^2 + 4 * q^4 + (-b2 - 2*b1 - 6) * q^5 + (-4*b2 + 8) * q^7 - 8 * q^8 $$q - 2 q^{2} + 4 q^{4} + ( - \beta_{2} - 2 \beta_1 - 6) q^{5} + ( - 4 \beta_{2} + 8) q^{7} - 8 q^{8} + (2 \beta_{2} + 4 \beta_1 + 12) q^{10} + (2 \beta_{2} - \beta_1 - 3) q^{11} + (11 \beta_{2} + 8 \beta_1 - 4) q^{13} + (8 \beta_{2} - 16) q^{14} + 16 q^{16} + (16 \beta_{2} + 12 \beta_1 + 18) q^{17} + (10 \beta_{2} - 8 \beta_1 - 52) q^{19} + ( - 4 \beta_{2} - 8 \beta_1 - 24) q^{20} + ( - 4 \beta_{2} + 2 \beta_1 + 6) q^{22} + ( - 12 \beta_{2} + 6 \beta_1 + 66) q^{23} + (11 \beta_{2} + 40 \beta_1 + 13) q^{25} + ( - 22 \beta_{2} - 16 \beta_1 + 8) q^{26} + ( - 16 \beta_{2} + 32) q^{28} - 29 q^{29} + ( - 36 \beta_{2} - 11 \beta_1 - 25) q^{31} - 32 q^{32} + ( - 32 \beta_{2} - 24 \beta_1 - 36) q^{34} + ( - 12 \beta_{2} - 24 \beta_1) q^{35} + (38 \beta_{2} - 12 \beta_1 - 10) q^{37} + ( - 20 \beta_{2} + 16 \beta_1 + 104) q^{38} + (8 \beta_{2} + 16 \beta_1 + 48) q^{40} + (6 \beta_{2} - 4 \beta_1 - 186) q^{41} + (2 \beta_{2} - 15 \beta_1 + 11) q^{43} + (8 \beta_{2} - 4 \beta_1 - 12) q^{44} + (24 \beta_{2} - 12 \beta_1 - 132) q^{46} + ( - 2 \beta_{2} - 33 \beta_1 - 207) q^{47} + ( - 80 \beta_{2} - 64 \beta_1 + 201) q^{49} + ( - 22 \beta_{2} - 80 \beta_1 - 26) q^{50} + (44 \beta_{2} + 32 \beta_1 - 16) q^{52} + (27 \beta_{2} + 116 \beta_1 - 276) q^{53} + (8 \beta_{2} + 25 \beta_1 + 39) q^{55} + (32 \beta_{2} - 64) q^{56} + 58 q^{58} + ( - 4 \beta_{2} + 86 \beta_1 - 90) q^{59} + (40 \beta_{2} - 80 \beta_1 + 134) q^{61} + (72 \beta_{2} + 22 \beta_1 + 50) q^{62} + 64 q^{64} + ( - 9 \beta_{2} - 90 \beta_1 - 468) q^{65} + ( - 72 \beta_{2} + 36 \beta_1 - 88) q^{67} + (64 \beta_{2} + 48 \beta_1 + 72) q^{68} + (24 \beta_{2} + 48 \beta_1) q^{70} + (4 \beta_{2} - 2 \beta_1 + 18) q^{71} + (30 \beta_{2} + 40 \beta_1 - 178) q^{73} + ( - 76 \beta_{2} + 24 \beta_1 + 20) q^{74} + (40 \beta_{2} - 32 \beta_1 - 208) q^{76} + (24 \beta_{2} + 28 \beta_1 - 300) q^{77} + ( - 38 \beta_{2} - 37 \beta_1 - 691) q^{79} + ( - 16 \beta_{2} - 32 \beta_1 - 96) q^{80} + ( - 12 \beta_{2} + 8 \beta_1 + 372) q^{82} + ( - 12 \beta_{2} - 162 \beta_1 + 150) q^{83} + ( - 38 \beta_{2} - 184 \beta_1 - 840) q^{85} + ( - 4 \beta_{2} + 30 \beta_1 - 22) q^{86} + ( - 16 \beta_{2} + 8 \beta_1 + 24) q^{88} + ( - 154 \beta_{2} - 68 \beta_1 - 282) q^{89} + (244 \beta_{2} + 208 \beta_1 - 1064) q^{91} + ( - 48 \beta_{2} + 24 \beta_1 + 264) q^{92} + (4 \beta_{2} + 66 \beta_1 + 414) q^{94} + (86 \beta_{2} + 244 \beta_1 + 552) q^{95} + ( - 34 \beta_{2} + 176 \beta_1 + 14) q^{97} + (160 \beta_{2} + 128 \beta_1 - 402) q^{98}+O(q^{100})$$ q - 2 * q^2 + 4 * q^4 + (-b2 - 2*b1 - 6) * q^5 + (-4*b2 + 8) * q^7 - 8 * q^8 + (2*b2 + 4*b1 + 12) * q^10 + (2*b2 - b1 - 3) * q^11 + (11*b2 + 8*b1 - 4) * q^13 + (8*b2 - 16) * q^14 + 16 * q^16 + (16*b2 + 12*b1 + 18) * q^17 + (10*b2 - 8*b1 - 52) * q^19 + (-4*b2 - 8*b1 - 24) * q^20 + (-4*b2 + 2*b1 + 6) * q^22 + (-12*b2 + 6*b1 + 66) * q^23 + (11*b2 + 40*b1 + 13) * q^25 + (-22*b2 - 16*b1 + 8) * q^26 + (-16*b2 + 32) * q^28 - 29 * q^29 + (-36*b2 - 11*b1 - 25) * q^31 - 32 * q^32 + (-32*b2 - 24*b1 - 36) * q^34 + (-12*b2 - 24*b1) * q^35 + (38*b2 - 12*b1 - 10) * q^37 + (-20*b2 + 16*b1 + 104) * q^38 + (8*b2 + 16*b1 + 48) * q^40 + (6*b2 - 4*b1 - 186) * q^41 + (2*b2 - 15*b1 + 11) * q^43 + (8*b2 - 4*b1 - 12) * q^44 + (24*b2 - 12*b1 - 132) * q^46 + (-2*b2 - 33*b1 - 207) * q^47 + (-80*b2 - 64*b1 + 201) * q^49 + (-22*b2 - 80*b1 - 26) * q^50 + (44*b2 + 32*b1 - 16) * q^52 + (27*b2 + 116*b1 - 276) * q^53 + (8*b2 + 25*b1 + 39) * q^55 + (32*b2 - 64) * q^56 + 58 * q^58 + (-4*b2 + 86*b1 - 90) * q^59 + (40*b2 - 80*b1 + 134) * q^61 + (72*b2 + 22*b1 + 50) * q^62 + 64 * q^64 + (-9*b2 - 90*b1 - 468) * q^65 + (-72*b2 + 36*b1 - 88) * q^67 + (64*b2 + 48*b1 + 72) * q^68 + (24*b2 + 48*b1) * q^70 + (4*b2 - 2*b1 + 18) * q^71 + (30*b2 + 40*b1 - 178) * q^73 + (-76*b2 + 24*b1 + 20) * q^74 + (40*b2 - 32*b1 - 208) * q^76 + (24*b2 + 28*b1 - 300) * q^77 + (-38*b2 - 37*b1 - 691) * q^79 + (-16*b2 - 32*b1 - 96) * q^80 + (-12*b2 + 8*b1 + 372) * q^82 + (-12*b2 - 162*b1 + 150) * q^83 + (-38*b2 - 184*b1 - 840) * q^85 + (-4*b2 + 30*b1 - 22) * q^86 + (-16*b2 + 8*b1 + 24) * q^88 + (-154*b2 - 68*b1 - 282) * q^89 + (244*b2 + 208*b1 - 1064) * q^91 + (-48*b2 + 24*b1 + 264) * q^92 + (4*b2 + 66*b1 + 414) * q^94 + (86*b2 + 244*b1 + 552) * q^95 + (-34*b2 + 176*b1 + 14) * q^97 + (160*b2 + 128*b1 - 402) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 6 q^{2} + 12 q^{4} - 20 q^{5} + 24 q^{7} - 24 q^{8}+O(q^{10})$$ 3 * q - 6 * q^2 + 12 * q^4 - 20 * q^5 + 24 * q^7 - 24 * q^8 $$3 q - 6 q^{2} + 12 q^{4} - 20 q^{5} + 24 q^{7} - 24 q^{8} + 40 q^{10} - 10 q^{11} - 4 q^{13} - 48 q^{14} + 48 q^{16} + 66 q^{17} - 164 q^{19} - 80 q^{20} + 20 q^{22} + 204 q^{23} + 79 q^{25} + 8 q^{26} + 96 q^{28} - 87 q^{29} - 86 q^{31} - 96 q^{32} - 132 q^{34} - 24 q^{35} - 42 q^{37} + 328 q^{38} + 160 q^{40} - 562 q^{41} + 18 q^{43} - 40 q^{44} - 408 q^{46} - 654 q^{47} + 539 q^{49} - 158 q^{50} - 16 q^{52} - 712 q^{53} + 142 q^{55} - 192 q^{56} + 174 q^{58} - 184 q^{59} + 322 q^{61} + 172 q^{62} + 192 q^{64} - 1494 q^{65} - 228 q^{67} + 264 q^{68} + 48 q^{70} + 52 q^{71} - 494 q^{73} + 84 q^{74} - 656 q^{76} - 872 q^{77} - 2110 q^{79} - 320 q^{80} + 1124 q^{82} + 288 q^{83} - 2704 q^{85} - 36 q^{86} + 80 q^{88} - 914 q^{89} - 2984 q^{91} + 816 q^{92} + 1308 q^{94} + 1900 q^{95} + 218 q^{97} - 1078 q^{98}+O(q^{100})$$ 3 * q - 6 * q^2 + 12 * q^4 - 20 * q^5 + 24 * q^7 - 24 * q^8 + 40 * q^10 - 10 * q^11 - 4 * q^13 - 48 * q^14 + 48 * q^16 + 66 * q^17 - 164 * q^19 - 80 * q^20 + 20 * q^22 + 204 * q^23 + 79 * q^25 + 8 * q^26 + 96 * q^28 - 87 * q^29 - 86 * q^31 - 96 * q^32 - 132 * q^34 - 24 * q^35 - 42 * q^37 + 328 * q^38 + 160 * q^40 - 562 * q^41 + 18 * q^43 - 40 * q^44 - 408 * q^46 - 654 * q^47 + 539 * q^49 - 158 * q^50 - 16 * q^52 - 712 * q^53 + 142 * q^55 - 192 * q^56 + 174 * q^58 - 184 * q^59 + 322 * q^61 + 172 * q^62 + 192 * q^64 - 1494 * q^65 - 228 * q^67 + 264 * q^68 + 48 * q^70 + 52 * q^71 - 494 * q^73 + 84 * q^74 - 656 * q^76 - 872 * q^77 - 2110 * q^79 - 320 * q^80 + 1124 * q^82 + 288 * q^83 - 2704 * q^85 - 36 * q^86 + 80 * q^88 - 914 * q^89 - 2984 * q^91 + 816 * q^92 + 1308 * q^94 + 1900 * q^95 + 218 * q^97 - 1078 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 42x - 54$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 4\nu - 27 ) / 3$$ (v^2 - 4*v - 27) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2} + 4\beta _1 + 27$$ 3*b2 + 4*b1 + 27

## 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
 7.53003 −5.13291 −1.39712
−2.00000 0 4.00000 −20.9205 0 8.55839 −8.00000 0 41.8409
1.2 −2.00000 0 4.00000 −2.36031 0 −18.5045 −8.00000 0 4.72062
1.3 −2.00000 0 4.00000 3.28077 0 33.9461 −8.00000 0 −6.56153
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$29$$ $$1$$

## 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 522.4.a.k 3
3.b odd 2 1 58.4.a.d 3
12.b even 2 1 464.4.a.i 3
15.d odd 2 1 1450.4.a.h 3
24.f even 2 1 1856.4.a.s 3
24.h odd 2 1 1856.4.a.r 3
87.d odd 2 1 1682.4.a.d 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.d 3 3.b odd 2 1
464.4.a.i 3 12.b even 2 1
522.4.a.k 3 1.a even 1 1 trivial
1450.4.a.h 3 15.d odd 2 1
1682.4.a.d 3 87.d odd 2 1
1856.4.a.r 3 24.h odd 2 1
1856.4.a.s 3 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} + 20T_{5}^{2} - 27T_{5} - 162$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(522))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 2)^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3} + 20 T^{2} - 27 T - 162$$
$7$ $$T^{3} - 24 T^{2} - 496 T + 5376$$
$11$ $$T^{3} + 10 T^{2} - 233 T - 2424$$
$13$ $$T^{3} + 4 T^{2} - 5619 T + 131706$$
$17$ $$T^{3} - 66 T^{2} - 10660 T + 679368$$
$19$ $$T^{3} + 164 T^{2} - 124 T - 664448$$
$23$ $$T^{3} - 204 T^{2} + 4284 T + 677376$$
$29$ $$(T + 29)^{3}$$
$31$ $$T^{3} + 86 T^{2} - 48089 T - 4766172$$
$37$ $$T^{3} + 42 T^{2} - 79456 T - 7684896$$
$41$ $$T^{3} + 562 T^{2} + 102432 T + 5982048$$
$43$ $$T^{3} - 18 T^{2} - 10369 T + 196488$$
$47$ $$T^{3} + 654 T^{2} + 98015 T + 3425124$$
$53$ $$T^{3} + 712 T^{2} + \cdots - 252120546$$
$59$ $$T^{3} + 184 T^{2} + \cdots - 57362928$$
$61$ $$T^{3} - 322 T^{2} - 388372 T - 5254424$$
$67$ $$T^{3} + 228 T^{2} + \cdots + 47608192$$
$71$ $$T^{3} - 52 T^{2} - 164 T + 672$$
$73$ $$T^{3} + 494 T^{2} + 6112 T - 9410208$$
$79$ $$T^{3} + 2110 T^{2} + \cdots + 285187172$$
$83$ $$T^{3} - 288 T^{2} + \cdots + 437606064$$
$89$ $$T^{3} + 914 T^{2} + \cdots - 598011552$$
$97$ $$T^{3} - 218 T^{2} + \cdots - 17006112$$