# Properties

 Label 1521.4.a.l 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

## 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ 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

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.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 2) q^{2} + 5 \beta q^{4} + (5 \beta - 10) q^{5} + ( - \beta - 7) q^{7} + ( - 7 \beta - 4) q^{8}+O(q^{10})$$ q + (-b - 2) * q^2 + 5*b * q^4 + (5*b - 10) * q^5 + (-b - 7) * q^7 + (-7*b - 4) * q^8 $$q + ( - \beta - 2) q^{2} + 5 \beta q^{4} + (5 \beta - 10) q^{5} + ( - \beta - 7) q^{7} + ( - 7 \beta - 4) q^{8} - 5 \beta q^{10} + (15 \beta + 1) q^{11} + (10 \beta + 18) q^{14} + ( - 15 \beta + 36) q^{16} + (16 \beta - 43) q^{17} + ( - 5 \beta + 73) q^{19} + ( - 25 \beta + 100) q^{20} + ( - 46 \beta - 62) q^{22} + (33 \beta - 89) q^{23} + ( - 75 \beta + 75) q^{25} + ( - 40 \beta - 20) q^{28} + ( - 60 \beta + 13) q^{29} + ( - 100 \beta + 120) q^{31} + (65 \beta + 20) q^{32} + ( - 5 \beta + 22) q^{34} + ( - 30 \beta + 50) q^{35} + (44 \beta + 73) q^{37} + ( - 58 \beta - 126) q^{38} + (15 \beta - 100) q^{40} + (20 \beta + 259) q^{41} + (97 \beta + 179) q^{43} + (80 \beta + 300) q^{44} + ( - 10 \beta + 46) q^{46} + ( - 140 \beta + 100) q^{47} + (15 \beta - 290) q^{49} + (150 \beta + 150) q^{50} + ( - 165 \beta - 190) q^{53} + ( - 70 \beta + 290) q^{55} + (60 \beta + 56) q^{56} + (167 \beta + 214) q^{58} + ( - 55 \beta - 377) q^{59} + ( - 200 \beta + 351) q^{61} + (180 \beta + 160) q^{62} + ( - 95 \beta - 588) q^{64} + ( - 91 \beta + 283) q^{67} + ( - 135 \beta + 320) q^{68} + (40 \beta + 20) q^{70} + (105 \beta + 11) q^{71} + ( - 85 \beta - 250) q^{73} + ( - 205 \beta - 322) q^{74} + (340 \beta - 100) q^{76} + ( - 121 \beta - 67) q^{77} + ( - 40 \beta + 140) q^{79} + (255 \beta - 660) q^{80} + ( - 319 \beta - 598) q^{82} + ( - 100 \beta + 180) q^{83} + ( - 295 \beta + 750) q^{85} + ( - 470 \beta - 746) q^{86} + ( - 172 \beta - 424) q^{88} + ( - 125 \beta + 523) q^{89} + ( - 280 \beta + 660) q^{92} + (320 \beta + 360) q^{94} + (390 \beta - 830) q^{95} + (469 \beta - 27) q^{97} + (245 \beta + 520) q^{98} +O(q^{100})$$ q + (-b - 2) * q^2 + 5*b * q^4 + (5*b - 10) * q^5 + (-b - 7) * q^7 + (-7*b - 4) * q^8 - 5*b * q^10 + (15*b + 1) * q^11 + (10*b + 18) * q^14 + (-15*b + 36) * q^16 + (16*b - 43) * q^17 + (-5*b + 73) * q^19 + (-25*b + 100) * q^20 + (-46*b - 62) * q^22 + (33*b - 89) * q^23 + (-75*b + 75) * q^25 + (-40*b - 20) * q^28 + (-60*b + 13) * q^29 + (-100*b + 120) * q^31 + (65*b + 20) * q^32 + (-5*b + 22) * q^34 + (-30*b + 50) * q^35 + (44*b + 73) * q^37 + (-58*b - 126) * q^38 + (15*b - 100) * q^40 + (20*b + 259) * q^41 + (97*b + 179) * q^43 + (80*b + 300) * q^44 + (-10*b + 46) * q^46 + (-140*b + 100) * q^47 + (15*b - 290) * q^49 + (150*b + 150) * q^50 + (-165*b - 190) * q^53 + (-70*b + 290) * q^55 + (60*b + 56) * q^56 + (167*b + 214) * q^58 + (-55*b - 377) * q^59 + (-200*b + 351) * q^61 + (180*b + 160) * q^62 + (-95*b - 588) * q^64 + (-91*b + 283) * q^67 + (-135*b + 320) * q^68 + (40*b + 20) * q^70 + (105*b + 11) * q^71 + (-85*b - 250) * q^73 + (-205*b - 322) * q^74 + (340*b - 100) * q^76 + (-121*b - 67) * q^77 + (-40*b + 140) * q^79 + (255*b - 660) * q^80 + (-319*b - 598) * q^82 + (-100*b + 180) * q^83 + (-295*b + 750) * q^85 + (-470*b - 746) * q^86 + (-172*b - 424) * q^88 + (-125*b + 523) * q^89 + (-280*b + 660) * q^92 + (320*b + 360) * q^94 + (390*b - 830) * q^95 + (469*b - 27) * q^97 + (245*b + 520) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 5 q^{2} + 5 q^{4} - 15 q^{5} - 15 q^{7} - 15 q^{8}+O(q^{10})$$ 2 * q - 5 * q^2 + 5 * q^4 - 15 * q^5 - 15 * q^7 - 15 * q^8 $$2 q - 5 q^{2} + 5 q^{4} - 15 q^{5} - 15 q^{7} - 15 q^{8} - 5 q^{10} + 17 q^{11} + 46 q^{14} + 57 q^{16} - 70 q^{17} + 141 q^{19} + 175 q^{20} - 170 q^{22} - 145 q^{23} + 75 q^{25} - 80 q^{28} - 34 q^{29} + 140 q^{31} + 105 q^{32} + 39 q^{34} + 70 q^{35} + 190 q^{37} - 310 q^{38} - 185 q^{40} + 538 q^{41} + 455 q^{43} + 680 q^{44} + 82 q^{46} + 60 q^{47} - 565 q^{49} + 450 q^{50} - 545 q^{53} + 510 q^{55} + 172 q^{56} + 595 q^{58} - 809 q^{59} + 502 q^{61} + 500 q^{62} - 1271 q^{64} + 475 q^{67} + 505 q^{68} + 80 q^{70} + 127 q^{71} - 585 q^{73} - 849 q^{74} + 140 q^{76} - 255 q^{77} + 240 q^{79} - 1065 q^{80} - 1515 q^{82} + 260 q^{83} + 1205 q^{85} - 1962 q^{86} - 1020 q^{88} + 921 q^{89} + 1040 q^{92} + 1040 q^{94} - 1270 q^{95} + 415 q^{97} + 1285 q^{98}+O(q^{100})$$ 2 * q - 5 * q^2 + 5 * q^4 - 15 * q^5 - 15 * q^7 - 15 * q^8 - 5 * q^10 + 17 * q^11 + 46 * q^14 + 57 * q^16 - 70 * q^17 + 141 * q^19 + 175 * q^20 - 170 * q^22 - 145 * q^23 + 75 * q^25 - 80 * q^28 - 34 * q^29 + 140 * q^31 + 105 * q^32 + 39 * q^34 + 70 * q^35 + 190 * q^37 - 310 * q^38 - 185 * q^40 + 538 * q^41 + 455 * q^43 + 680 * q^44 + 82 * q^46 + 60 * q^47 - 565 * q^49 + 450 * q^50 - 545 * q^53 + 510 * q^55 + 172 * q^56 + 595 * q^58 - 809 * q^59 + 502 * q^61 + 500 * q^62 - 1271 * q^64 + 475 * q^67 + 505 * q^68 + 80 * q^70 + 127 * q^71 - 585 * q^73 - 849 * q^74 + 140 * q^76 - 255 * q^77 + 240 * q^79 - 1065 * q^80 - 1515 * q^82 + 260 * q^83 + 1205 * q^85 - 1962 * q^86 - 1020 * q^88 + 921 * q^89 + 1040 * q^92 + 1040 * q^94 - 1270 * q^95 + 415 * q^97 + 1285 * 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
 2.56155 −1.56155
−4.56155 0 12.8078 2.80776 0 −9.56155 −21.9309 0 −12.8078
1.2 −0.438447 0 −7.80776 −17.8078 0 −5.43845 6.93087 0 7.80776
 $$n$$: 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.l 2
3.b odd 2 1 169.4.a.j 2
13.b even 2 1 1521.4.a.t 2
13.e even 6 2 117.4.g.d 4
39.d odd 2 1 169.4.a.f 2
39.f even 4 2 169.4.b.e 4
39.h odd 6 2 13.4.c.b 4
39.i odd 6 2 169.4.c.f 4
39.k even 12 4 169.4.e.g 8
156.r even 6 2 208.4.i.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.b 4 39.h odd 6 2
117.4.g.d 4 13.e even 6 2
169.4.a.f 2 39.d odd 2 1
169.4.a.j 2 3.b odd 2 1
169.4.b.e 4 39.f even 4 2
169.4.c.f 4 39.i odd 6 2
169.4.e.g 8 39.k even 12 4
208.4.i.e 4 156.r even 6 2
1521.4.a.l 2 1.a even 1 1 trivial
1521.4.a.t 2 13.b even 2 1

## Hecke kernels

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

 $$T_{2}^{2} + 5T_{2} + 2$$ T2^2 + 5*T2 + 2 $$T_{5}^{2} + 15T_{5} - 50$$ T5^2 + 15*T5 - 50 $$T_{7}^{2} + 15T_{7} + 52$$ T7^2 + 15*T7 + 52

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 5T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 15T - 50$$
$7$ $$T^{2} + 15T + 52$$
$11$ $$T^{2} - 17T - 884$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 70T + 137$$
$19$ $$T^{2} - 141T + 4864$$
$23$ $$T^{2} + 145T + 628$$
$29$ $$T^{2} + 34T - 15011$$
$31$ $$T^{2} - 140T - 37600$$
$37$ $$T^{2} - 190T + 797$$
$41$ $$T^{2} - 538T + 70661$$
$43$ $$T^{2} - 455T + 11768$$
$47$ $$T^{2} - 60T - 82400$$
$53$ $$T^{2} + 545T - 41450$$
$59$ $$T^{2} + 809T + 150764$$
$61$ $$T^{2} - 502T - 106999$$
$67$ $$T^{2} - 475T + 21212$$
$71$ $$T^{2} - 127T - 42824$$
$73$ $$T^{2} + 585T + 54850$$
$79$ $$T^{2} - 240T + 7600$$
$83$ $$T^{2} - 260T - 25600$$
$89$ $$T^{2} - 921T + 145654$$
$97$ $$T^{2} - 415T - 891778$$