# Properties

 Label 1521.4.a.bb Level $1521$ Weight $4$ Character orbit 1521.a Self dual yes Analytic conductor $89.742$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 2x^{3} - 25x^{2} + 24x + 78$$ x^4 - 2*x^3 - 25*x^2 + 24*x + 78 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 39) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 5) q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{3} - \beta_1 + 4) q^{7} + (\beta_{3} + 2 \beta_{2} + 5 \beta_1 + 11) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + b1 + 5) * q^4 + (-b2 - b1 - 1) * q^5 + (-b3 - b1 + 4) * q^7 + (b3 + 2*b2 + 5*b1 + 11) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 5) q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{3} - \beta_1 + 4) q^{7} + (\beta_{3} + 2 \beta_{2} + 5 \beta_1 + 11) q^{8} + ( - \beta_{3} - 2 \beta_{2} - 9 \beta_1 - 11) q^{10} + (\beta_{3} - \beta_{2} + 4 \beta_1 + 8) q^{11} + ( - 7 \beta_{2} + 6 \beta_1 - 13) q^{14} + (2 \beta_{3} + 5 \beta_{2} + 19 \beta_1 + 21) q^{16} + (\beta_{3} - 19 \beta_1 - 15) q^{17} + ( - \beta_{3} + 7 \beta_{2} - 6 \beta_1 - 28) q^{19} + ( - 2 \beta_{3} - 9 \beta_{2} - 23 \beta_1 - 105) q^{20} + ( - \beta_{3} + 9 \beta_{2} + 2 \beta_1 + 54) q^{22} + ( - 3 \beta_{3} - \beta_{2} + 4 \beta_1 - 28) q^{23} + (2 \beta_{3} + 5 \beta_{2} + 19 \beta_1 - 24) q^{25} + (\beta_{3} - \beta_{2} - 48 \beta_1 + 60) q^{28} + (6 \beta_{3} - \beta_{2} - 11 \beta_1 - 43) q^{29} + (7 \beta_{2} - 53 \beta_1 + 20) q^{31} + ( - 3 \beta_{3} + 20 \beta_{2} + 29 \beta_1 + 149) q^{32} + ( - 13 \beta_{2} - 37 \beta_1 - 247) q^{34} + ( - 5 \beta_{3} + \beta_{2} + 44 \beta_1 - 44) q^{35} + (\beta_{3} + 2 \beta_{2} - 21 \beta_1 - 15) q^{37} + (7 \beta_{3} - 5 \beta_{2} + 18 \beta_1 - 92) q^{38} + ( - \beta_{3} - 28 \beta_{2} - 113 \beta_1 - 193) q^{40} + (3 \beta_{3} + 43 \beta_1 - 285) q^{41} + ( - \beta_{3} - 14 \beta_{2} + 57 \beta_1 + 84) q^{43} + (\beta_{3} + 13 \beta_{2} + 90 \beta_1 - 56) q^{44} + ( - \beta_{3} - 15 \beta_{2} - 22 \beta_1 + 54) q^{46} + (3 \beta_{3} - 47 \beta_{2} - 8 \beta_1 - 20) q^{47} + ( - 2 \beta_{3} - 29 \beta_{2} - 43 \beta_1 + 289) q^{49} + (5 \beta_{3} + 36 \beta_{2} + 24 \beta_1 + 237) q^{50} + ( - 2 \beta_{3} - 23 \beta_{2} - 37 \beta_1 - 47) q^{53} + (3 \beta_{3} - 17 \beta_{2} - 74 \beta_1 + 88) q^{55} + ( - \beta_{3} + 13 \beta_{2} - 46 \beta_1 - 518) q^{56} + ( - \beta_{3} + 24 \beta_{2} - 79 \beta_1 - 141) q^{58} + (14 \beta_{3} + 24 \beta_{2} + 10 \beta_1 + 72) q^{59} + ( - 3 \beta_{3} - 31 \beta_{2} + 26 \beta_1 - 245) q^{61} + (7 \beta_{3} - 46 \beta_{2} + 16 \beta_1 - 703) q^{62} + (4 \beta_{3} - 9 \beta_{2} + 175 \beta_1 + 169) q^{64} + (\beta_{3} + 14 \beta_{2} - 129 \beta_1 + 348) q^{67} + ( - 21 \beta_{3} - 50 \beta_{2} - 223 \beta_1 - 335) q^{68} + (\beta_{3} + 15 \beta_{2} + 22 \beta_1 + 570) q^{70} + (11 \beta_{3} + 49 \beta_{2} - 76 \beta_1 + 304) q^{71} + (2 \beta_{3} + 24 \beta_{2} + 194 \beta_1 - 335) q^{73} + (2 \beta_{3} - 13 \beta_{2} - 25 \beta_1 - 277) q^{74} + (3 \beta_{3} - \beta_{2} - 82 \beta_1 + 468) q^{76} + ( - 16 \beta_{3} + 2 \beta_{2} + 98 \beta_1 - 676) q^{77} + ( - 2 \beta_{3} - 7 \beta_{2} - 21 \beta_1 - 176) q^{79} + ( - 12 \beta_{3} - 75 \beta_{2} - 315 \beta_1 - 573) q^{80} + (61 \beta_{2} - 251 \beta_1 + 559) q^{82} + (5 \beta_{3} + \beta_{2} - 138 \beta_1 - 32) q^{83} + (25 \beta_{3} + 50 \beta_{2} + 147 \beta_1 + 275) q^{85} + ( - 14 \beta_{3} + 37 \beta_{2} + 46 \beta_1 + 769) q^{86} + (21 \beta_{3} + 37 \beta_{2} + 106 \beta_1 + 712) q^{88} + ( - 28 \beta_{3} - 50 \beta_{2} - 146 \beta_1 + 478) q^{89} + (9 \beta_{3} - 35 \beta_{2} - 102 \beta_1 - 32) q^{92} + ( - 47 \beta_{3} - 37 \beta_{2} - 366 \beta_1 - 10) q^{94} + ( - 7 \beta_{3} + 29 \beta_{2} + 58 \beta_1 - 580) q^{95} + (22 \beta_{3} + 63 \beta_{2} - 71 \beta_1 - 506) q^{97} + ( - 29 \beta_{3} - 84 \beta_{2} + 49 \beta_1 - 501) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + b1 + 5) * q^4 + (-b2 - b1 - 1) * q^5 + (-b3 - b1 + 4) * q^7 + (b3 + 2*b2 + 5*b1 + 11) * q^8 + (-b3 - 2*b2 - 9*b1 - 11) * q^10 + (b3 - b2 + 4*b1 + 8) * q^11 + (-7*b2 + 6*b1 - 13) * q^14 + (2*b3 + 5*b2 + 19*b1 + 21) * q^16 + (b3 - 19*b1 - 15) * q^17 + (-b3 + 7*b2 - 6*b1 - 28) * q^19 + (-2*b3 - 9*b2 - 23*b1 - 105) * q^20 + (-b3 + 9*b2 + 2*b1 + 54) * q^22 + (-3*b3 - b2 + 4*b1 - 28) * q^23 + (2*b3 + 5*b2 + 19*b1 - 24) * q^25 + (b3 - b2 - 48*b1 + 60) * q^28 + (6*b3 - b2 - 11*b1 - 43) * q^29 + (7*b2 - 53*b1 + 20) * q^31 + (-3*b3 + 20*b2 + 29*b1 + 149) * q^32 + (-13*b2 - 37*b1 - 247) * q^34 + (-5*b3 + b2 + 44*b1 - 44) * q^35 + (b3 + 2*b2 - 21*b1 - 15) * q^37 + (7*b3 - 5*b2 + 18*b1 - 92) * q^38 + (-b3 - 28*b2 - 113*b1 - 193) * q^40 + (3*b3 + 43*b1 - 285) * q^41 + (-b3 - 14*b2 + 57*b1 + 84) * q^43 + (b3 + 13*b2 + 90*b1 - 56) * q^44 + (-b3 - 15*b2 - 22*b1 + 54) * q^46 + (3*b3 - 47*b2 - 8*b1 - 20) * q^47 + (-2*b3 - 29*b2 - 43*b1 + 289) * q^49 + (5*b3 + 36*b2 + 24*b1 + 237) * q^50 + (-2*b3 - 23*b2 - 37*b1 - 47) * q^53 + (3*b3 - 17*b2 - 74*b1 + 88) * q^55 + (-b3 + 13*b2 - 46*b1 - 518) * q^56 + (-b3 + 24*b2 - 79*b1 - 141) * q^58 + (14*b3 + 24*b2 + 10*b1 + 72) * q^59 + (-3*b3 - 31*b2 + 26*b1 - 245) * q^61 + (7*b3 - 46*b2 + 16*b1 - 703) * q^62 + (4*b3 - 9*b2 + 175*b1 + 169) * q^64 + (b3 + 14*b2 - 129*b1 + 348) * q^67 + (-21*b3 - 50*b2 - 223*b1 - 335) * q^68 + (b3 + 15*b2 + 22*b1 + 570) * q^70 + (11*b3 + 49*b2 - 76*b1 + 304) * q^71 + (2*b3 + 24*b2 + 194*b1 - 335) * q^73 + (2*b3 - 13*b2 - 25*b1 - 277) * q^74 + (3*b3 - b2 - 82*b1 + 468) * q^76 + (-16*b3 + 2*b2 + 98*b1 - 676) * q^77 + (-2*b3 - 7*b2 - 21*b1 - 176) * q^79 + (-12*b3 - 75*b2 - 315*b1 - 573) * q^80 + (61*b2 - 251*b1 + 559) * q^82 + (5*b3 + b2 - 138*b1 - 32) * q^83 + (25*b3 + 50*b2 + 147*b1 + 275) * q^85 + (-14*b3 + 37*b2 + 46*b1 + 769) * q^86 + (21*b3 + 37*b2 + 106*b1 + 712) * q^88 + (-28*b3 - 50*b2 - 146*b1 + 478) * q^89 + (9*b3 - 35*b2 - 102*b1 - 32) * q^92 + (-47*b3 - 37*b2 - 366*b1 - 10) * q^94 + (-7*b3 + 29*b2 + 58*b1 - 580) * q^95 + (22*b3 + 63*b2 - 71*b1 - 506) * q^97 + (-29*b3 - 84*b2 + 49*b1 - 501) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 22 q^{4} - 6 q^{5} + 14 q^{7} + 54 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 + 22 * q^4 - 6 * q^5 + 14 * q^7 + 54 * q^8 $$4 q + 2 q^{2} + 22 q^{4} - 6 q^{5} + 14 q^{7} + 54 q^{8} - 62 q^{10} + 40 q^{11} - 40 q^{14} + 122 q^{16} - 98 q^{17} - 124 q^{19} - 466 q^{20} + 220 q^{22} - 104 q^{23} - 58 q^{25} + 144 q^{28} - 194 q^{29} - 26 q^{31} + 654 q^{32} - 1062 q^{34} - 88 q^{35} - 102 q^{37} - 332 q^{38} - 998 q^{40} - 1054 q^{41} + 450 q^{43} - 44 q^{44} + 172 q^{46} - 96 q^{47} + 1070 q^{49} + 996 q^{50} - 262 q^{53} + 204 q^{55} - 2164 q^{56} - 722 q^{58} + 308 q^{59} - 928 q^{61} - 2780 q^{62} + 1026 q^{64} + 1134 q^{67} - 1786 q^{68} + 2324 q^{70} + 1064 q^{71} - 952 q^{73} - 1158 q^{74} + 1708 q^{76} - 2508 q^{77} - 746 q^{79} - 2922 q^{80} + 1734 q^{82} - 404 q^{83} + 1394 q^{85} + 3168 q^{86} + 3060 q^{88} + 1620 q^{89} - 332 q^{92} - 772 q^{94} - 2204 q^{95} - 2166 q^{97} - 1906 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 + 22 * q^4 - 6 * q^5 + 14 * q^7 + 54 * q^8 - 62 * q^10 + 40 * q^11 - 40 * q^14 + 122 * q^16 - 98 * q^17 - 124 * q^19 - 466 * q^20 + 220 * q^22 - 104 * q^23 - 58 * q^25 + 144 * q^28 - 194 * q^29 - 26 * q^31 + 654 * q^32 - 1062 * q^34 - 88 * q^35 - 102 * q^37 - 332 * q^38 - 998 * q^40 - 1054 * q^41 + 450 * q^43 - 44 * q^44 + 172 * q^46 - 96 * q^47 + 1070 * q^49 + 996 * q^50 - 262 * q^53 + 204 * q^55 - 2164 * q^56 - 722 * q^58 + 308 * q^59 - 928 * q^61 - 2780 * q^62 + 1026 * q^64 + 1134 * q^67 - 1786 * q^68 + 2324 * q^70 + 1064 * q^71 - 952 * q^73 - 1158 * q^74 + 1708 * q^76 - 2508 * q^77 - 746 * q^79 - 2922 * q^80 + 1734 * q^82 - 404 * q^83 + 1394 * q^85 + 3168 * q^86 + 3060 * q^88 + 1620 * q^89 - 332 * q^92 - 772 * q^94 - 2204 * q^95 - 2166 * q^97 - 1906 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 25x^{2} + 24x + 78$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 13$$ v^2 - v - 13 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 19\nu + 15$$ v^3 - 2*v^2 - 19*v + 15
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 13$$ b2 + b1 + 13 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 21\beta _1 + 11$$ b3 + 2*b2 + 21*b1 + 11

## 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
 −4.22605 −1.46610 2.36176 5.33039
−4.22605 0 9.85953 −5.85953 0 24.1254 −7.85849 0 24.7627
1.2 −1.46610 0 −5.85055 9.85055 0 −29.9396 20.3063 0 −14.4419
1.3 2.36176 0 −2.42208 6.42208 0 29.4938 −24.6145 0 15.1674
1.4 5.33039 0 20.4131 −16.4131 0 −9.67968 66.1667 0 −87.4882
 $$n$$: 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.bb 4
3.b odd 2 1 507.4.a.i 4
13.b even 2 1 1521.4.a.v 4
13.e even 6 2 117.4.g.e 8
39.d odd 2 1 507.4.a.m 4
39.f even 4 2 507.4.b.h 8
39.h odd 6 2 39.4.e.c 8
156.r even 6 2 624.4.q.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.c 8 39.h odd 6 2
117.4.g.e 8 13.e even 6 2
507.4.a.i 4 3.b odd 2 1
507.4.a.m 4 39.d odd 2 1
507.4.b.h 8 39.f even 4 2
624.4.q.i 8 156.r even 6 2
1521.4.a.v 4 13.b even 2 1
1521.4.a.bb 4 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{2}^{4} - 2T_{2}^{3} - 25T_{2}^{2} + 24T_{2} + 78$$ T2^4 - 2*T2^3 - 25*T2^2 + 24*T2 + 78 $$T_{5}^{4} + 6T_{5}^{3} - 203T_{5}^{2} - 156T_{5} + 6084$$ T5^4 + 6*T5^3 - 203*T5^2 - 156*T5 + 6084 $$T_{7}^{4} - 14T_{7}^{3} - 1123T_{7}^{2} + 12652T_{7} + 206212$$ T7^4 - 14*T7^3 - 1123*T7^2 + 12652*T7 + 206212

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} - 25 T^{2} + 24 T + 78$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 6 T^{3} - 203 T^{2} + \cdots + 6084$$
$7$ $$T^{4} - 14 T^{3} - 1123 T^{2} + \cdots + 206212$$
$11$ $$T^{4} - 40 T^{3} - 1256 T^{2} + \cdots + 27408$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 98 T^{3} - 7255 T^{2} + \cdots + 22571952$$
$19$ $$T^{4} + 124 T^{3} - 5932 T^{2} + \cdots + 1720384$$
$23$ $$T^{4} + 104 T^{3} - 6824 T^{2} + \cdots - 2571504$$
$29$ $$T^{4} + 194 T^{3} + \cdots - 274591068$$
$31$ $$T^{4} + 26 T^{3} + \cdots + 328187792$$
$37$ $$T^{4} + 102 T^{3} + \cdots + 27176708$$
$41$ $$T^{4} + 1054 T^{3} + \cdots + 1021233552$$
$43$ $$T^{4} - 450 T^{3} + \cdots - 2362804828$$
$47$ $$T^{4} + 96 T^{3} + \cdots + 42871452048$$
$53$ $$T^{4} + 262 T^{3} + \cdots + 744728256$$
$59$ $$T^{4} - 308 T^{3} + \cdots - 2116598016$$
$61$ $$T^{4} + 928 T^{3} + \cdots - 5230543711$$
$67$ $$T^{4} - 1134 T^{3} + \cdots + 10235224388$$
$71$ $$T^{4} - 1064 T^{3} + \cdots - 99058755696$$
$73$ $$T^{4} + 952 T^{3} + \cdots - 120133390247$$
$79$ $$T^{4} + 746 T^{3} + \cdots + 680937616$$
$83$ $$T^{4} + 404 T^{3} + \cdots + 58964273856$$
$89$ $$T^{4} - 1620 T^{3} + \cdots - 268616132736$$
$97$ $$T^{4} + 2166 T^{3} + \cdots - 445091164$$