Properties

 Label 1521.2.b.h Level $1521$ Weight $2$ Character orbit 1521.b Analytic conductor $12.145$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1521.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$12.1452461474$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} - 3) q^{4} + (2 \beta_{2} + \beta_1) q^{5} + (\beta_{2} - \beta_1) q^{7} + (4 \beta_{2} - \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 - 3) * q^4 + (2*b2 + b1) * q^5 + (b2 - b1) * q^7 + (4*b2 - b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{3} - 3) q^{4} + (2 \beta_{2} + \beta_1) q^{5} + (\beta_{2} - \beta_1) q^{7} + (4 \beta_{2} - \beta_1) q^{8} + ( - \beta_{3} - 3) q^{10} - 2 \beta_{2} q^{11} + ( - 2 \beta_{3} + 6) q^{14} + ( - 3 \beta_{3} + 3) q^{16} + ( - \beta_{3} + 1) q^{17} + (4 \beta_{2} + 2 \beta_1) q^{19} - \beta_1 q^{20} + (2 \beta_{3} - 2) q^{22} + 2 q^{23} - 3 \beta_{3} q^{25} + ( - 6 \beta_{2} + 4 \beta_1) q^{28} + ( - 3 \beta_{3} + 1) q^{29} + (\beta_{2} + \beta_1) q^{31} + ( - 4 \beta_{2} + \beta_1) q^{32} + ( - 4 \beta_{2} + \beta_1) q^{34} + 2 q^{35} + ( - 6 \beta_{2} - \beta_1) q^{37} + ( - 2 \beta_{3} - 6) q^{38} + ( - 3 \beta_{3} - 1) q^{40} + \beta_1 q^{41} + ( - \beta_{3} - 2) q^{43} + (4 \beta_{2} - 2 \beta_1) q^{44} + 2 \beta_1 q^{46} + (2 \beta_{2} + 4 \beta_1) q^{47} + (3 \beta_{3} - 1) q^{49} - 12 \beta_{2} q^{50} + (3 \beta_{3} - 7) q^{53} + (2 \beta_{3} + 2) q^{55} + (6 \beta_{3} - 14) q^{56} + ( - 12 \beta_{2} + \beta_1) q^{58} + ( - 6 \beta_{2} + 2 \beta_1) q^{59} + ( - 2 \beta_{3} + 9) q^{61} - 4 q^{62} + ( - \beta_{3} - 3) q^{64} + (3 \beta_{2} + \beta_1) q^{67} + (3 \beta_{3} - 7) q^{68} + 2 \beta_1 q^{70} - 14 \beta_{2} q^{71} + (7 \beta_{2} + 2 \beta_1) q^{73} + (5 \beta_{3} - 1) q^{74} - 2 \beta_1 q^{76} + ( - 2 \beta_{3} + 4) q^{77} + ( - \beta_{3} + 8) q^{79} + ( - 12 \beta_{2} - 3 \beta_1) q^{80} + (\beta_{3} - 5) q^{82} + (4 \beta_{2} - 2 \beta_1) q^{83} + ( - 4 \beta_{2} - \beta_1) q^{85} + ( - 4 \beta_{2} - 2 \beta_1) q^{86} + ( - 2 \beta_{3} + 10) q^{88} + (8 \beta_{2} - 2 \beta_1) q^{89} + (2 \beta_{3} - 6) q^{92} + (2 \beta_{3} - 18) q^{94} + ( - 6 \beta_{3} - 10) q^{95} + ( - 7 \beta_{2} - \beta_1) q^{97} + (12 \beta_{2} - \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 - 3) * q^4 + (2*b2 + b1) * q^5 + (b2 - b1) * q^7 + (4*b2 - b1) * q^8 + (-b3 - 3) * q^10 - 2*b2 * q^11 + (-2*b3 + 6) * q^14 + (-3*b3 + 3) * q^16 + (-b3 + 1) * q^17 + (4*b2 + 2*b1) * q^19 - b1 * q^20 + (2*b3 - 2) * q^22 + 2 * q^23 - 3*b3 * q^25 + (-6*b2 + 4*b1) * q^28 + (-3*b3 + 1) * q^29 + (b2 + b1) * q^31 + (-4*b2 + b1) * q^32 + (-4*b2 + b1) * q^34 + 2 * q^35 + (-6*b2 - b1) * q^37 + (-2*b3 - 6) * q^38 + (-3*b3 - 1) * q^40 + b1 * q^41 + (-b3 - 2) * q^43 + (4*b2 - 2*b1) * q^44 + 2*b1 * q^46 + (2*b2 + 4*b1) * q^47 + (3*b3 - 1) * q^49 - 12*b2 * q^50 + (3*b3 - 7) * q^53 + (2*b3 + 2) * q^55 + (6*b3 - 14) * q^56 + (-12*b2 + b1) * q^58 + (-6*b2 + 2*b1) * q^59 + (-2*b3 + 9) * q^61 - 4 * q^62 + (-b3 - 3) * q^64 + (3*b2 + b1) * q^67 + (3*b3 - 7) * q^68 + 2*b1 * q^70 - 14*b2 * q^71 + (7*b2 + 2*b1) * q^73 + (5*b3 - 1) * q^74 - 2*b1 * q^76 + (-2*b3 + 4) * q^77 + (-b3 + 8) * q^79 + (-12*b2 - 3*b1) * q^80 + (b3 - 5) * q^82 + (4*b2 - 2*b1) * q^83 + (-4*b2 - b1) * q^85 + (-4*b2 - 2*b1) * q^86 + (-2*b3 + 10) * q^88 + (8*b2 - 2*b1) * q^89 + (2*b3 - 6) * q^92 + (2*b3 - 18) * q^94 + (-6*b3 - 10) * q^95 + (-7*b2 - b1) * q^97 + (12*b2 - b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{4}+O(q^{10})$$ 4 * q - 10 * q^4 $$4 q - 10 q^{4} - 14 q^{10} + 20 q^{14} + 6 q^{16} + 2 q^{17} - 4 q^{22} + 8 q^{23} - 6 q^{25} - 2 q^{29} + 8 q^{35} - 28 q^{38} - 10 q^{40} - 10 q^{43} + 2 q^{49} - 22 q^{53} + 12 q^{55} - 44 q^{56} + 32 q^{61} - 16 q^{62} - 14 q^{64} - 22 q^{68} + 6 q^{74} + 12 q^{77} + 30 q^{79} - 18 q^{82} + 36 q^{88} - 20 q^{92} - 68 q^{94} - 52 q^{95}+O(q^{100})$$ 4 * q - 10 * q^4 - 14 * q^10 + 20 * q^14 + 6 * q^16 + 2 * q^17 - 4 * q^22 + 8 * q^23 - 6 * q^25 - 2 * q^29 + 8 * q^35 - 28 * q^38 - 10 * q^40 - 10 * q^43 + 2 * q^49 - 22 * q^53 + 12 * q^55 - 44 * q^56 + 32 * q^61 - 16 * q^62 - 14 * q^64 - 22 * q^68 + 6 * q^74 + 12 * q^77 + 30 * q^79 - 18 * q^82 + 36 * q^88 - 20 * q^92 - 68 * q^94 - 52 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 5\nu ) / 4$$ (v^3 + 5*v) / 4 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$ v^2 + 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ b3 - 5 $$\nu^{3}$$ $$=$$ $$4\beta_{2} - 5\beta_1$$ 4*b2 - 5*b1

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1351.1
 − 2.56155i − 1.56155i 1.56155i 2.56155i
2.56155i 0 −4.56155 0.561553i 0 3.56155i 6.56155i 0 −1.43845
1351.2 1.56155i 0 −0.438447 3.56155i 0 0.561553i 2.43845i 0 −5.56155
1351.3 1.56155i 0 −0.438447 3.56155i 0 0.561553i 2.43845i 0 −5.56155
1351.4 2.56155i 0 −4.56155 0.561553i 0 3.56155i 6.56155i 0 −1.43845
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.2.b.h 4
3.b odd 2 1 507.2.b.d 4
13.b even 2 1 inner 1521.2.b.h 4
13.d odd 4 1 1521.2.a.g 2
13.d odd 4 1 1521.2.a.m 2
13.f odd 12 2 117.2.g.c 4
39.d odd 2 1 507.2.b.d 4
39.f even 4 1 507.2.a.d 2
39.f even 4 1 507.2.a.g 2
39.h odd 6 2 507.2.j.g 8
39.i odd 6 2 507.2.j.g 8
39.k even 12 2 39.2.e.b 4
39.k even 12 2 507.2.e.g 4
52.l even 12 2 1872.2.t.r 4
156.l odd 4 1 8112.2.a.bk 2
156.l odd 4 1 8112.2.a.bo 2
156.v odd 12 2 624.2.q.h 4
195.bc odd 12 2 975.2.bb.i 8
195.bh even 12 2 975.2.i.k 4
195.bn odd 12 2 975.2.bb.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 39.k even 12 2
117.2.g.c 4 13.f odd 12 2
507.2.a.d 2 39.f even 4 1
507.2.a.g 2 39.f even 4 1
507.2.b.d 4 3.b odd 2 1
507.2.b.d 4 39.d odd 2 1
507.2.e.g 4 39.k even 12 2
507.2.j.g 8 39.h odd 6 2
507.2.j.g 8 39.i odd 6 2
624.2.q.h 4 156.v odd 12 2
975.2.i.k 4 195.bh even 12 2
975.2.bb.i 8 195.bc odd 12 2
975.2.bb.i 8 195.bn odd 12 2
1521.2.a.g 2 13.d odd 4 1
1521.2.a.m 2 13.d odd 4 1
1521.2.b.h 4 1.a even 1 1 trivial
1521.2.b.h 4 13.b even 2 1 inner
1872.2.t.r 4 52.l even 12 2
8112.2.a.bk 2 156.l odd 4 1
8112.2.a.bo 2 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1521, [\chi])$$:

 $$T_{2}^{4} + 9T_{2}^{2} + 16$$ T2^4 + 9*T2^2 + 16 $$T_{5}^{4} + 13T_{5}^{2} + 4$$ T5^4 + 13*T5^2 + 4 $$T_{7}^{4} + 13T_{7}^{2} + 4$$ T7^4 + 13*T7^2 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 9T^{2} + 16$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 13T^{2} + 4$$
$7$ $$T^{4} + 13T^{2} + 4$$
$11$ $$(T^{2} + 4)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - T - 4)^{2}$$
$19$ $$T^{4} + 52T^{2} + 64$$
$23$ $$(T - 2)^{4}$$
$29$ $$(T^{2} + T - 38)^{2}$$
$31$ $$T^{4} + 9T^{2} + 16$$
$37$ $$T^{4} + 69T^{2} + 676$$
$41$ $$T^{4} + 9T^{2} + 16$$
$43$ $$(T^{2} + 5 T + 2)^{2}$$
$47$ $$(T^{2} + 68)^{2}$$
$53$ $$(T^{2} + 11 T - 8)^{2}$$
$59$ $$T^{4} + 132T^{2} + 1024$$
$61$ $$(T^{2} - 16 T + 47)^{2}$$
$67$ $$T^{4} + 21T^{2} + 4$$
$71$ $$(T^{2} + 196)^{2}$$
$73$ $$T^{4} + 106T^{2} + 361$$
$79$ $$(T^{2} - 15 T + 52)^{2}$$
$83$ $$T^{4} + 84T^{2} + 64$$
$89$ $$T^{4} + 196T^{2} + 4096$$
$97$ $$T^{4} + 93T^{2} + 1444$$