# Properties

 Label 522.4.a.j Level $522$ Weight $4$ Character orbit 522.a Self dual yes Analytic conductor $30.799$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ x^2 - 6 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 $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 4 q^{4} + (6 \beta + 5) q^{5} + (8 \beta - 8) q^{7} + 8 q^{8}+O(q^{10})$$ q + 2 * q^2 + 4 * q^4 + (6*b + 5) * q^5 + (8*b - 8) * q^7 + 8 * q^8 $$q + 2 q^{2} + 4 q^{4} + (6 \beta + 5) q^{5} + (8 \beta - 8) q^{7} + 8 q^{8} + (12 \beta + 10) q^{10} + ( - 3 \beta + 45) q^{11} + ( - 8 \beta - 25) q^{13} + (16 \beta - 16) q^{14} + 16 q^{16} + (16 \beta + 22) q^{17} + (4 \beta + 54) q^{19} + (24 \beta + 20) q^{20} + ( - 6 \beta + 90) q^{22} + ( - 78 \beta + 14) q^{23} + (60 \beta + 116) q^{25} + ( - 16 \beta - 50) q^{26} + (32 \beta - 32) q^{28} - 29 q^{29} + ( - 109 \beta + 33) q^{31} + 32 q^{32} + (32 \beta + 44) q^{34} + ( - 8 \beta + 248) q^{35} + ( - 4 \beta + 20) q^{37} + (8 \beta + 108) q^{38} + (48 \beta + 40) q^{40} + ( - 80 \beta - 152) q^{41} + ( - 53 \beta - 65) q^{43} + ( - 12 \beta + 180) q^{44} + ( - 156 \beta + 28) q^{46} + ( - 99 \beta + 257) q^{47} + ( - 128 \beta + 105) q^{49} + (120 \beta + 232) q^{50} + ( - 32 \beta - 100) q^{52} + (52 \beta + 479) q^{53} + (255 \beta + 117) q^{55} + (64 \beta - 64) q^{56} - 58 q^{58} + (250 \beta + 90) q^{59} + (12 \beta + 514) q^{61} + ( - 218 \beta + 66) q^{62} + 64 q^{64} + ( - 190 \beta - 413) q^{65} + (20 \beta - 456) q^{67} + (64 \beta + 88) q^{68} + ( - 16 \beta + 496) q^{70} + (34 \beta - 398) q^{71} + ( - 176 \beta - 428) q^{73} + ( - 8 \beta + 40) q^{74} + (16 \beta + 216) q^{76} + (384 \beta - 504) q^{77} + (361 \beta - 159) q^{79} + (96 \beta + 80) q^{80} + ( - 160 \beta - 304) q^{82} + ( - 38 \beta + 914) q^{83} + (212 \beta + 686) q^{85} + ( - 106 \beta - 130) q^{86} + ( - 24 \beta + 360) q^{88} + (72 \beta + 472) q^{89} + ( - 136 \beta - 184) q^{91} + ( - 312 \beta + 56) q^{92} + ( - 198 \beta + 514) q^{94} + (344 \beta + 414) q^{95} + ( - 100 \beta + 184) q^{97} + ( - 256 \beta + 210) q^{98}+O(q^{100})$$ q + 2 * q^2 + 4 * q^4 + (6*b + 5) * q^5 + (8*b - 8) * q^7 + 8 * q^8 + (12*b + 10) * q^10 + (-3*b + 45) * q^11 + (-8*b - 25) * q^13 + (16*b - 16) * q^14 + 16 * q^16 + (16*b + 22) * q^17 + (4*b + 54) * q^19 + (24*b + 20) * q^20 + (-6*b + 90) * q^22 + (-78*b + 14) * q^23 + (60*b + 116) * q^25 + (-16*b - 50) * q^26 + (32*b - 32) * q^28 - 29 * q^29 + (-109*b + 33) * q^31 + 32 * q^32 + (32*b + 44) * q^34 + (-8*b + 248) * q^35 + (-4*b + 20) * q^37 + (8*b + 108) * q^38 + (48*b + 40) * q^40 + (-80*b - 152) * q^41 + (-53*b - 65) * q^43 + (-12*b + 180) * q^44 + (-156*b + 28) * q^46 + (-99*b + 257) * q^47 + (-128*b + 105) * q^49 + (120*b + 232) * q^50 + (-32*b - 100) * q^52 + (52*b + 479) * q^53 + (255*b + 117) * q^55 + (64*b - 64) * q^56 - 58 * q^58 + (250*b + 90) * q^59 + (12*b + 514) * q^61 + (-218*b + 66) * q^62 + 64 * q^64 + (-190*b - 413) * q^65 + (20*b - 456) * q^67 + (64*b + 88) * q^68 + (-16*b + 496) * q^70 + (34*b - 398) * q^71 + (-176*b - 428) * q^73 + (-8*b + 40) * q^74 + (16*b + 216) * q^76 + (384*b - 504) * q^77 + (361*b - 159) * q^79 + (96*b + 80) * q^80 + (-160*b - 304) * q^82 + (-38*b + 914) * q^83 + (212*b + 686) * q^85 + (-106*b - 130) * q^86 + (-24*b + 360) * q^88 + (72*b + 472) * q^89 + (-136*b - 184) * q^91 + (-312*b + 56) * q^92 + (-198*b + 514) * q^94 + (344*b + 414) * q^95 + (-100*b + 184) * q^97 + (-256*b + 210) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 8 q^{4} + 10 q^{5} - 16 q^{7} + 16 q^{8}+O(q^{10})$$ 2 * q + 4 * q^2 + 8 * q^4 + 10 * q^5 - 16 * q^7 + 16 * q^8 $$2 q + 4 q^{2} + 8 q^{4} + 10 q^{5} - 16 q^{7} + 16 q^{8} + 20 q^{10} + 90 q^{11} - 50 q^{13} - 32 q^{14} + 32 q^{16} + 44 q^{17} + 108 q^{19} + 40 q^{20} + 180 q^{22} + 28 q^{23} + 232 q^{25} - 100 q^{26} - 64 q^{28} - 58 q^{29} + 66 q^{31} + 64 q^{32} + 88 q^{34} + 496 q^{35} + 40 q^{37} + 216 q^{38} + 80 q^{40} - 304 q^{41} - 130 q^{43} + 360 q^{44} + 56 q^{46} + 514 q^{47} + 210 q^{49} + 464 q^{50} - 200 q^{52} + 958 q^{53} + 234 q^{55} - 128 q^{56} - 116 q^{58} + 180 q^{59} + 1028 q^{61} + 132 q^{62} + 128 q^{64} - 826 q^{65} - 912 q^{67} + 176 q^{68} + 992 q^{70} - 796 q^{71} - 856 q^{73} + 80 q^{74} + 432 q^{76} - 1008 q^{77} - 318 q^{79} + 160 q^{80} - 608 q^{82} + 1828 q^{83} + 1372 q^{85} - 260 q^{86} + 720 q^{88} + 944 q^{89} - 368 q^{91} + 112 q^{92} + 1028 q^{94} + 828 q^{95} + 368 q^{97} + 420 q^{98}+O(q^{100})$$ 2 * q + 4 * q^2 + 8 * q^4 + 10 * q^5 - 16 * q^7 + 16 * q^8 + 20 * q^10 + 90 * q^11 - 50 * q^13 - 32 * q^14 + 32 * q^16 + 44 * q^17 + 108 * q^19 + 40 * q^20 + 180 * q^22 + 28 * q^23 + 232 * q^25 - 100 * q^26 - 64 * q^28 - 58 * q^29 + 66 * q^31 + 64 * q^32 + 88 * q^34 + 496 * q^35 + 40 * q^37 + 216 * q^38 + 80 * q^40 - 304 * q^41 - 130 * q^43 + 360 * q^44 + 56 * q^46 + 514 * q^47 + 210 * q^49 + 464 * q^50 - 200 * q^52 + 958 * q^53 + 234 * q^55 - 128 * q^56 - 116 * q^58 + 180 * q^59 + 1028 * q^61 + 132 * q^62 + 128 * q^64 - 826 * q^65 - 912 * q^67 + 176 * q^68 + 992 * q^70 - 796 * q^71 - 856 * q^73 + 80 * q^74 + 432 * q^76 - 1008 * q^77 - 318 * q^79 + 160 * q^80 - 608 * q^82 + 1828 * q^83 + 1372 * q^85 - 260 * q^86 + 720 * q^88 + 944 * q^89 - 368 * q^91 + 112 * q^92 + 1028 * q^94 + 828 * q^95 + 368 * q^97 + 420 * 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.44949 2.44949
2.00000 0 4.00000 −9.69694 0 −27.5959 8.00000 0 −19.3939
1.2 2.00000 0 4.00000 19.6969 0 11.5959 8.00000 0 39.3939
 $$n$$: 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.j 2
3.b odd 2 1 58.4.a.c 2
12.b even 2 1 464.4.a.e 2
15.d odd 2 1 1450.4.a.g 2
24.f even 2 1 1856.4.a.i 2
24.h odd 2 1 1856.4.a.l 2
87.d odd 2 1 1682.4.a.c 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 10T_{5} - 191$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(522))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 2)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 10T - 191$$
$7$ $$T^{2} + 16T - 320$$
$11$ $$T^{2} - 90T + 1971$$
$13$ $$T^{2} + 50T + 241$$
$17$ $$T^{2} - 44T - 1052$$
$19$ $$T^{2} - 108T + 2820$$
$23$ $$T^{2} - 28T - 36308$$
$29$ $$(T + 29)^{2}$$
$31$ $$T^{2} - 66T - 70197$$
$37$ $$T^{2} - 40T + 304$$
$41$ $$T^{2} + 304T - 15296$$
$43$ $$T^{2} + 130T - 12629$$
$47$ $$T^{2} - 514T + 7243$$
$53$ $$T^{2} - 958T + 213217$$
$59$ $$T^{2} - 180T - 366900$$
$61$ $$T^{2} - 1028 T + 263332$$
$67$ $$T^{2} + 912T + 205536$$
$71$ $$T^{2} + 796T + 151468$$
$73$ $$T^{2} + 856T - 2672$$
$79$ $$T^{2} + 318T - 756645$$
$83$ $$T^{2} - 1828 T + 826732$$
$89$ $$T^{2} - 944T + 191680$$
$97$ $$T^{2} - 368T - 26144$$