# Properties

 Label 1521.4.a.bc Level $1521$ Weight $4$ Character orbit 1521.a Self dual yes Analytic conductor $89.742$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 52x^{6} + 805x^{4} - 4210x^{2} + 4992$$ x^8 - 52*x^6 + 805*x^4 - 4210*x^2 + 4992 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + \beta_{3} q^{5} + (\beta_{5} - 3) q^{7} + (\beta_{4} + \beta_{3} + 5 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 5) * q^4 + b3 * q^5 + (b5 - 3) * q^7 + (b4 + b3 + 5*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + \beta_{3} q^{5} + (\beta_{5} - 3) q^{7} + (\beta_{4} + \beta_{3} + 5 \beta_1) q^{8} + (\beta_{6} + \beta_{5} + 3 \beta_{2} + 4) q^{10} + (\beta_{7} + \beta_{3} + \beta_1) q^{11} + (\beta_{7} - \beta_{4} + 2 \beta_{3} - 7 \beta_1) q^{14} + (2 \beta_{6} + 9 \beta_{2} + 25) q^{16} + (\beta_{7} - 2 \beta_{4} - 13 \beta_1) q^{17} + ( - \beta_{6} - \beta_{5} + 6 \beta_{2} + 31) q^{19} + (\beta_{7} + 3 \beta_{4} + 6 \beta_{3} + 18 \beta_1) q^{20} + (\beta_{6} + 7 \beta_{5} + 8 \beta_{2} + 15) q^{22} + ( - \beta_{7} - 2 \beta_{4} + 5 \beta_{3} - 11 \beta_1) q^{23} + ( - \beta_{6} - 6 \beta_{5} + 46) q^{25} + (\beta_{6} + \beta_{5} - 6 \beta_{2} - 57) q^{28} + (6 \beta_{4} - \beta_{3} + 10 \beta_1) q^{29} + (3 \beta_{6} + 6 \beta_{5} + 8 \beta_{2} + 28) q^{31} + (3 \beta_{4} + 19 \beta_{3} + 45 \beta_1) q^{32} + ( - 2 \beta_{6} + 8 \beta_{5} - 27 \beta_{2} - 163) q^{34} + ( - \beta_{7} - 17 \beta_{3} + 27 \beta_1) q^{35} + ( - 2 \beta_{6} + 7 \beta_{5} + 12 \beta_{2} + 126) q^{37} + ( - \beta_{7} + 6 \beta_{4} - 5 \beta_{3} + 89 \beta_1) q^{38} + (\beta_{6} + \beta_{5} + 43 \beta_{2} + 212) q^{40} + (\beta_{7} - 4 \beta_{4} + 6 \beta_{3} + 49 \beta_1) q^{41} + ( - 7 \beta_{5} - 26 \beta_{2} + 11) q^{43} + ( - \beta_{7} + 2 \beta_{4} + 23 \beta_{3} + 37 \beta_1) q^{44} + (3 \beta_{6} + \beta_{5} - 18 \beta_{2} - 113) q^{46} + (\beta_{7} + 2 \beta_{4} + 11 \beta_{3} - 33 \beta_1) q^{47} + ( - 7 \beta_{6} + 8 \beta_{5} - 10 \beta_{2} + 37) q^{49} + ( - 6 \beta_{7} + 5 \beta_{4} - 21 \beta_{3} + 76 \beta_1) q^{50} + ( - 6 \beta_{7} + 2 \beta_{4} + 19 \beta_{3} - 60 \beta_1) q^{53} + ( - 11 \beta_{6} - 7 \beta_{5} + 24 \beta_{2} + 167) q^{55} + ( - 7 \beta_{7} + 2 \beta_{4} - 11 \beta_{3} - 59 \beta_1) q^{56} + (5 \beta_{6} - 7 \beta_{5} + 61 \beta_{2} + 102) q^{58} + ( - 6 \beta_{7} - 4 \beta_{4} + 6 \beta_{3} + 2 \beta_1) q^{59} + ( - 3 \beta_{6} - 15 \beta_{5} - 10 \beta_{2} + 148) q^{61} + (6 \beta_{7} + 5 \beta_{4} + 47 \beta_{3} + 50 \beta_1) q^{62} + (6 \beta_{6} + 16 \beta_{5} + 57 \beta_{2} + 449) q^{64} + ( - 8 \beta_{6} + \beta_{5} - 30 \beta_{2} - 273) q^{67} + ( - 21 \beta_{4} - 29 \beta_{3} - 295 \beta_1) q^{68} + ( - 17 \beta_{6} - 23 \beta_{5} - 28 \beta_{2} + 285) q^{70} + (\beta_{7} - 14 \beta_{4} + 11 \beta_{3} + 131 \beta_1) q^{71} + (12 \beta_{6} - 8 \beta_{5} + 6 \beta_{2} - 273) q^{73} + (7 \beta_{7} + 3 \beta_{4} + 8 \beta_{3} + 206 \beta_1) q^{74} + (9 \beta_{6} - 9 \beta_{5} + 76 \beta_{2} + 867) q^{76} + (6 \beta_{7} - 26 \beta_{4} - 28 \beta_{3} + 176 \beta_1) q^{77} + ( - 5 \beta_{6} + 16 \beta_{2} + 234) q^{79} + ( - 7 \beta_{7} + 19 \beta_{4} + 6 \beta_{3} + 402 \beta_1) q^{80} + (2 \beta_{6} + 16 \beta_{5} + 35 \beta_{2} + 675) q^{82} + (7 \beta_{7} - 8 \beta_{4} - 13 \beta_{3} + 51 \beta_1) q^{83} + ( - 18 \beta_{6} - 19 \beta_{5} - 74 \beta_{2} - 102) q^{85} + ( - 7 \beta_{7} - 19 \beta_{4} - 40 \beta_{3} - 169 \beta_1) q^{86} + (17 \beta_{6} - 41 \beta_{5} + 56 \beta_{2} + 447) q^{88} + (8 \beta_{7} - 24 \beta_{4} + 32 \beta_{3} - 44 \beta_1) q^{89} + (9 \beta_{7} - 29 \beta_{3} - 191 \beta_1) q^{92} + (13 \beta_{6} + 15 \beta_{5} + 22 \beta_{2} - 395) q^{94} + (15 \beta_{7} + 22 \beta_{4} + 65 \beta_{3} - 39 \beta_1) q^{95} + ( - \beta_{6} + 48 \beta_{5} + 58 \beta_{2} - 558) q^{97} + (8 \beta_{7} - 25 \beta_{4} - 57 \beta_{3} - 33 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + 5) * q^4 + b3 * q^5 + (b5 - 3) * q^7 + (b4 + b3 + 5*b1) * q^8 + (b6 + b5 + 3*b2 + 4) * q^10 + (b7 + b3 + b1) * q^11 + (b7 - b4 + 2*b3 - 7*b1) * q^14 + (2*b6 + 9*b2 + 25) * q^16 + (b7 - 2*b4 - 13*b1) * q^17 + (-b6 - b5 + 6*b2 + 31) * q^19 + (b7 + 3*b4 + 6*b3 + 18*b1) * q^20 + (b6 + 7*b5 + 8*b2 + 15) * q^22 + (-b7 - 2*b4 + 5*b3 - 11*b1) * q^23 + (-b6 - 6*b5 + 46) * q^25 + (b6 + b5 - 6*b2 - 57) * q^28 + (6*b4 - b3 + 10*b1) * q^29 + (3*b6 + 6*b5 + 8*b2 + 28) * q^31 + (3*b4 + 19*b3 + 45*b1) * q^32 + (-2*b6 + 8*b5 - 27*b2 - 163) * q^34 + (-b7 - 17*b3 + 27*b1) * q^35 + (-2*b6 + 7*b5 + 12*b2 + 126) * q^37 + (-b7 + 6*b4 - 5*b3 + 89*b1) * q^38 + (b6 + b5 + 43*b2 + 212) * q^40 + (b7 - 4*b4 + 6*b3 + 49*b1) * q^41 + (-7*b5 - 26*b2 + 11) * q^43 + (-b7 + 2*b4 + 23*b3 + 37*b1) * q^44 + (3*b6 + b5 - 18*b2 - 113) * q^46 + (b7 + 2*b4 + 11*b3 - 33*b1) * q^47 + (-7*b6 + 8*b5 - 10*b2 + 37) * q^49 + (-6*b7 + 5*b4 - 21*b3 + 76*b1) * q^50 + (-6*b7 + 2*b4 + 19*b3 - 60*b1) * q^53 + (-11*b6 - 7*b5 + 24*b2 + 167) * q^55 + (-7*b7 + 2*b4 - 11*b3 - 59*b1) * q^56 + (5*b6 - 7*b5 + 61*b2 + 102) * q^58 + (-6*b7 - 4*b4 + 6*b3 + 2*b1) * q^59 + (-3*b6 - 15*b5 - 10*b2 + 148) * q^61 + (6*b7 + 5*b4 + 47*b3 + 50*b1) * q^62 + (6*b6 + 16*b5 + 57*b2 + 449) * q^64 + (-8*b6 + b5 - 30*b2 - 273) * q^67 + (-21*b4 - 29*b3 - 295*b1) * q^68 + (-17*b6 - 23*b5 - 28*b2 + 285) * q^70 + (b7 - 14*b4 + 11*b3 + 131*b1) * q^71 + (12*b6 - 8*b5 + 6*b2 - 273) * q^73 + (7*b7 + 3*b4 + 8*b3 + 206*b1) * q^74 + (9*b6 - 9*b5 + 76*b2 + 867) * q^76 + (6*b7 - 26*b4 - 28*b3 + 176*b1) * q^77 + (-5*b6 + 16*b2 + 234) * q^79 + (-7*b7 + 19*b4 + 6*b3 + 402*b1) * q^80 + (2*b6 + 16*b5 + 35*b2 + 675) * q^82 + (7*b7 - 8*b4 - 13*b3 + 51*b1) * q^83 + (-18*b6 - 19*b5 - 74*b2 - 102) * q^85 + (-7*b7 - 19*b4 - 40*b3 - 169*b1) * q^86 + (17*b6 - 41*b5 + 56*b2 + 447) * q^88 + (8*b7 - 24*b4 + 32*b3 - 44*b1) * q^89 + (9*b7 - 29*b3 - 191*b1) * q^92 + (13*b6 + 15*b5 + 22*b2 - 395) * q^94 + (15*b7 + 22*b4 + 65*b3 - 39*b1) * q^95 + (-b6 + 48*b5 + 58*b2 - 558) * q^97 + (8*b7 - 25*b4 - 57*b3 - 33*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 40 q^{4} - 22 q^{7}+O(q^{10})$$ 8 * q + 40 * q^4 - 22 * q^7 $$8 q + 40 q^{4} - 22 q^{7} + 36 q^{10} + 204 q^{16} + 244 q^{19} + 136 q^{22} + 354 q^{25} - 452 q^{28} + 242 q^{31} - 1292 q^{34} + 1018 q^{37} + 1700 q^{40} + 74 q^{43} - 896 q^{46} + 298 q^{49} + 1300 q^{55} + 812 q^{58} + 1148 q^{61} + 3636 q^{64} - 2198 q^{67} + 2200 q^{70} - 2176 q^{73} + 6936 q^{76} + 1862 q^{79} + 5436 q^{82} - 890 q^{85} + 3528 q^{88} - 3104 q^{94} - 4370 q^{97}+O(q^{100})$$ 8 * q + 40 * q^4 - 22 * q^7 + 36 * q^10 + 204 * q^16 + 244 * q^19 + 136 * q^22 + 354 * q^25 - 452 * q^28 + 242 * q^31 - 1292 * q^34 + 1018 * q^37 + 1700 * q^40 + 74 * q^43 - 896 * q^46 + 298 * q^49 + 1300 * q^55 + 812 * q^58 + 1148 * q^61 + 3636 * q^64 - 2198 * q^67 + 2200 * q^70 - 2176 * q^73 + 6936 * q^76 + 1862 * q^79 + 5436 * q^82 - 890 * q^85 + 3528 * q^88 - 3104 * q^94 - 4370 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 52x^{6} + 805x^{4} - 4210x^{2} + 4992$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 13$$ v^2 - 13 $$\beta_{3}$$ $$=$$ $$( \nu^{5} - 35\nu^{3} + 210\nu ) / 16$$ (v^5 - 35*v^3 + 210*v) / 16 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} + 51\nu^{3} - 546\nu ) / 16$$ (-v^5 + 51*v^3 - 546*v) / 16 $$\beta_{5}$$ $$=$$ $$( \nu^{6} - 43\nu^{4} + 426\nu^{2} - 688 ) / 16$$ (v^6 - 43*v^4 + 426*v^2 - 688) / 16 $$\beta_{6}$$ $$=$$ $$( \nu^{4} - 33\nu^{2} + 156 ) / 2$$ (v^4 - 33*v^2 + 156) / 2 $$\beta_{7}$$ $$=$$ $$( \nu^{7} - 46\nu^{5} + 547\nu^{3} - 1590\nu ) / 16$$ (v^7 - 46*v^5 + 547*v^3 - 1590*v) / 16
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 13$$ b2 + 13 $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{3} + 21\beta_1$$ b4 + b3 + 21*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{6} + 33\beta_{2} + 273$$ 2*b6 + 33*b2 + 273 $$\nu^{5}$$ $$=$$ $$35\beta_{4} + 51\beta_{3} + 525\beta_1$$ 35*b4 + 51*b3 + 525*b1 $$\nu^{6}$$ $$=$$ $$86\beta_{6} + 16\beta_{5} + 993\beta_{2} + 6889$$ 86*b6 + 16*b5 + 993*b2 + 6889 $$\nu^{7}$$ $$=$$ $$16\beta_{7} + 1063\beta_{4} + 1799\beta_{3} + 14253\beta_1$$ 16*b7 + 1063*b4 + 1799*b3 + 14253*b1

## 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
 −5.39589 −3.69212 −2.75628 −1.28670 1.28670 2.75628 3.69212 5.39589
−5.39589 0 21.1156 −13.0421 0 −6.42494 −70.7705 0 70.3740
1.2 −3.69212 0 5.63172 18.7574 0 −24.1383 8.74396 0 −69.2546
1.3 −2.75628 0 −0.402937 −0.313209 0 28.5660 23.1608 0 0.863291
1.4 −1.28670 0 −6.34441 −12.4484 0 −9.00273 18.4569 0 16.0173
1.5 1.28670 0 −6.34441 12.4484 0 −9.00273 −18.4569 0 16.0173
1.6 2.75628 0 −0.402937 0.313209 0 28.5660 −23.1608 0 0.863291
1.7 3.69212 0 5.63172 −18.7574 0 −24.1383 −8.74396 0 −69.2546
1.8 5.39589 0 21.1156 13.0421 0 −6.42494 70.7705 0 70.3740
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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.bc 8
3.b odd 2 1 inner 1521.4.a.bc 8
13.b even 2 1 1521.4.a.bd 8
13.c even 3 2 117.4.g.f 16
39.d odd 2 1 1521.4.a.bd 8
39.i odd 6 2 117.4.g.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.g.f 16 13.c even 3 2
117.4.g.f 16 39.i odd 6 2
1521.4.a.bc 8 1.a even 1 1 trivial
1521.4.a.bc 8 3.b odd 2 1 inner
1521.4.a.bd 8 13.b even 2 1
1521.4.a.bd 8 39.d odd 2 1

## Hecke kernels

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

 $$T_{2}^{8} - 52T_{2}^{6} + 805T_{2}^{4} - 4210T_{2}^{2} + 4992$$ T2^8 - 52*T2^6 + 805*T2^4 - 4210*T2^2 + 4992 $$T_{5}^{8} - 677T_{5}^{6} + 140795T_{5}^{4} - 9287935T_{5}^{2} + 909792$$ T5^8 - 677*T5^6 + 140795*T5^4 - 9287935*T5^2 + 909792 $$T_{7}^{4} + 11T_{7}^{3} - 700T_{7}^{2} - 10894T_{7} - 39884$$ T7^4 + 11*T7^3 - 700*T7^2 - 10894*T7 - 39884

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 52 T^{6} + 805 T^{4} + \cdots + 4992$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 677 T^{6} + 140795 T^{4} + \cdots + 909792$$
$7$ $$(T^{4} + 11 T^{3} - 700 T^{2} + \cdots - 39884)^{2}$$
$11$ $$T^{8} - 9720 T^{6} + \cdots + 17384221390848$$
$13$ $$T^{8}$$
$17$ $$T^{8} - 29425 T^{6} + \cdots + 30103640387808$$
$19$ $$(T^{4} - 122 T^{3} - 8174 T^{2} + \cdots + 14520480)^{2}$$
$23$ $$T^{8} + \cdots + 203262208261632$$
$29$ $$T^{8} - 139365 T^{6} + \cdots + 24\!\cdots\!48$$
$31$ $$(T^{4} - 121 T^{3} - 54236 T^{2} + \cdots - 349762400)^{2}$$
$37$ $$(T^{4} - 509 T^{3} + 30151 T^{2} + \cdots - 1178984222)^{2}$$
$41$ $$T^{8} - 225537 T^{6} + \cdots + 14\!\cdots\!92$$
$43$ $$(T^{4} - 37 T^{3} - 146172 T^{2} + \cdots + 1323590892)^{2}$$
$47$ $$T^{8} - 157456 T^{6} + \cdots + 36\!\cdots\!52$$
$53$ $$T^{8} - 733517 T^{6} + \cdots + 44\!\cdots\!28$$
$59$ $$T^{8} - 445056 T^{6} + \cdots + 18\!\cdots\!88$$
$61$ $$(T^{4} - 574 T^{3} - 70694 T^{2} + \cdots + 684529261)^{2}$$
$67$ $$(T^{4} + 1099 T^{3} + \cdots + 2700497644)^{2}$$
$71$ $$T^{8} - 1791632 T^{6} + \cdots + 12\!\cdots\!00$$
$73$ $$(T^{4} + 1088 T^{3} + \cdots + 20997802657)^{2}$$
$79$ $$(T^{4} - 931 T^{3} + 176852 T^{2} + \cdots - 9861650240)^{2}$$
$83$ $$T^{8} - 870232 T^{6} + \cdots + 23\!\cdots\!32$$
$89$ $$T^{8} - 3056192 T^{6} + \cdots + 14\!\cdots\!68$$
$97$ $$(T^{4} + 2185 T^{3} + \cdots + 291511171744)^{2}$$