# Properties

 Label 1521.2.b.j 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: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{3} - 1) q^{4} + (\beta_{2} - \beta_1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (-b3 - 1) * q^4 + (b2 - b1) * q^5 + (-b2 + b1) * q^7 + (-b2 - 2*b1) * q^8 $$q + \beta_1 q^{2} + ( - \beta_{3} - 1) q^{4} + (\beta_{2} - \beta_1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} + (\beta_{3} + 4) q^{10} + ( - \beta_{2} - \beta_1) q^{11} + ( - \beta_{3} - 4) q^{14} + 3 q^{16} + ( - 2 \beta_{3} + 2) q^{17} + ( - \beta_{2} + \beta_1) q^{19} + (3 \beta_{2} + 5 \beta_1) q^{20} + (\beta_{3} + 2) q^{22} - 4 q^{23} - 3 q^{25} + ( - 3 \beta_{2} - 5 \beta_1) q^{28} - 2 q^{29} + ( - \beta_{2} - 3 \beta_1) q^{31} + ( - 2 \beta_{2} - \beta_1) q^{32} + ( - 2 \beta_{2} - 4 \beta_1) q^{34} + 8 q^{35} + (3 \beta_{2} - \beta_1) q^{37} + ( - \beta_{3} - 4) q^{38} + ( - 3 \beta_{3} - 4) q^{40} + ( - 3 \beta_{2} - 5 \beta_1) q^{41} + ( - 2 \beta_{3} - 4) q^{43} + ( - \beta_{2} + 3 \beta_1) q^{44} - 4 \beta_1 q^{46} + ( - 5 \beta_{2} - \beta_1) q^{47} - q^{49} - 3 \beta_1 q^{50} + 2 q^{53} - 2 \beta_{3} q^{55} + (3 \beta_{3} + 4) q^{56} - 2 \beta_1 q^{58} + (3 \beta_{2} - \beta_1) q^{59} + ( - 4 \beta_{3} + 2) q^{61} + (3 \beta_{3} + 8) q^{62} + (\beta_{3} + 7) q^{64} + (3 \beta_{2} + \beta_1) q^{67} + 14 q^{68} + 8 \beta_1 q^{70} + ( - \beta_{2} - \beta_1) q^{71} + ( - \beta_{2} - 5 \beta_1) q^{73} + (\beta_{3} + 6) q^{74} + ( - 3 \beta_{2} - 5 \beta_1) q^{76} + 2 \beta_{3} q^{77} + 4 \beta_{3} q^{79} + (3 \beta_{2} - 3 \beta_1) q^{80} + (5 \beta_{3} + 12) q^{82} + ( - \beta_{2} + 3 \beta_1) q^{83} + (10 \beta_{2} + 6 \beta_1) q^{85} + ( - 2 \beta_{2} - 10 \beta_1) q^{86} + ( - \beta_{3} - 6) q^{88} + (7 \beta_{2} + 5 \beta_1) q^{89} + (4 \beta_{3} + 4) q^{92} + (\beta_{3} - 2) q^{94} + 8 q^{95} + (\beta_{2} - 3 \beta_1) q^{97} - \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + (-b3 - 1) * q^4 + (b2 - b1) * q^5 + (-b2 + b1) * q^7 + (-b2 - 2*b1) * q^8 + (b3 + 4) * q^10 + (-b2 - b1) * q^11 + (-b3 - 4) * q^14 + 3 * q^16 + (-2*b3 + 2) * q^17 + (-b2 + b1) * q^19 + (3*b2 + 5*b1) * q^20 + (b3 + 2) * q^22 - 4 * q^23 - 3 * q^25 + (-3*b2 - 5*b1) * q^28 - 2 * q^29 + (-b2 - 3*b1) * q^31 + (-2*b2 - b1) * q^32 + (-2*b2 - 4*b1) * q^34 + 8 * q^35 + (3*b2 - b1) * q^37 + (-b3 - 4) * q^38 + (-3*b3 - 4) * q^40 + (-3*b2 - 5*b1) * q^41 + (-2*b3 - 4) * q^43 + (-b2 + 3*b1) * q^44 - 4*b1 * q^46 + (-5*b2 - b1) * q^47 - q^49 - 3*b1 * q^50 + 2 * q^53 - 2*b3 * q^55 + (3*b3 + 4) * q^56 - 2*b1 * q^58 + (3*b2 - b1) * q^59 + (-4*b3 + 2) * q^61 + (3*b3 + 8) * q^62 + (b3 + 7) * q^64 + (3*b2 + b1) * q^67 + 14 * q^68 + 8*b1 * q^70 + (-b2 - b1) * q^71 + (-b2 - 5*b1) * q^73 + (b3 + 6) * q^74 + (-3*b2 - 5*b1) * q^76 + 2*b3 * q^77 + 4*b3 * q^79 + (3*b2 - 3*b1) * q^80 + (5*b3 + 12) * q^82 + (-b2 + 3*b1) * q^83 + (10*b2 + 6*b1) * q^85 + (-2*b2 - 10*b1) * q^86 + (-b3 - 6) * q^88 + (7*b2 + 5*b1) * q^89 + (4*b3 + 4) * q^92 + (b3 - 2) * q^94 + 8 * q^95 + (b2 - 3*b1) * q^97 - b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} + 16 q^{10} - 16 q^{14} + 12 q^{16} + 8 q^{17} + 8 q^{22} - 16 q^{23} - 12 q^{25} - 8 q^{29} + 32 q^{35} - 16 q^{38} - 16 q^{40} - 16 q^{43} - 4 q^{49} + 8 q^{53} + 16 q^{56} + 8 q^{61} + 32 q^{62} + 28 q^{64} + 56 q^{68} + 24 q^{74} + 48 q^{82} - 24 q^{88} + 16 q^{92} - 8 q^{94} + 32 q^{95}+O(q^{100})$$ 4 * q - 4 * q^4 + 16 * q^10 - 16 * q^14 + 12 * q^16 + 8 * q^17 + 8 * q^22 - 16 * q^23 - 12 * q^25 - 8 * q^29 + 32 * q^35 - 16 * q^38 - 16 * q^40 - 16 * q^43 - 4 * q^49 + 8 * q^53 + 16 * q^56 + 8 * q^61 + 32 * q^62 + 28 * q^64 + 56 * q^68 + 24 * q^74 + 48 * q^82 - 24 * q^88 + 16 * q^92 - 8 * q^94 + 32 * q^95

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}$$ v^3 + v^2 + v $$\beta_{2}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8}$$ -v^3 + v^2 - v $$\beta_{3}$$ $$=$$ $$-2\zeta_{8}^{3} + 2\zeta_{8}$$ -2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (b3 - b2 + b1) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (-b3 - b2 + b1) / 4

## 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
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
2.41421i 0 −3.82843 2.82843i 0 2.82843i 4.41421i 0 6.82843
1351.2 0.414214i 0 1.82843 2.82843i 0 2.82843i 1.58579i 0 1.17157
1351.3 0.414214i 0 1.82843 2.82843i 0 2.82843i 1.58579i 0 1.17157
1351.4 2.41421i 0 −3.82843 2.82843i 0 2.82843i 4.41421i 0 6.82843
 $$n$$: 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.j 4
3.b odd 2 1 507.2.b.e 4
13.b even 2 1 inner 1521.2.b.j 4
13.d odd 4 1 117.2.a.c 2
13.d odd 4 1 1521.2.a.f 2
39.d odd 2 1 507.2.b.e 4
39.f even 4 1 39.2.a.b 2
39.f even 4 1 507.2.a.h 2
39.h odd 6 2 507.2.j.f 8
39.i odd 6 2 507.2.j.f 8
39.k even 12 2 507.2.e.d 4
39.k even 12 2 507.2.e.h 4
52.f even 4 1 1872.2.a.w 2
65.f even 4 1 2925.2.c.u 4
65.g odd 4 1 2925.2.a.v 2
65.k even 4 1 2925.2.c.u 4
91.i even 4 1 5733.2.a.u 2
104.j odd 4 1 7488.2.a.cl 2
104.m even 4 1 7488.2.a.co 2
117.y odd 12 2 1053.2.e.e 4
117.z even 12 2 1053.2.e.m 4
156.l odd 4 1 624.2.a.k 2
156.l odd 4 1 8112.2.a.bm 2
195.j odd 4 1 975.2.c.h 4
195.n even 4 1 975.2.a.l 2
195.u odd 4 1 975.2.c.h 4
273.o odd 4 1 1911.2.a.h 2
312.w odd 4 1 2496.2.a.bi 2
312.y even 4 1 2496.2.a.bf 2
429.l odd 4 1 4719.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 39.f even 4 1
117.2.a.c 2 13.d odd 4 1
507.2.a.h 2 39.f even 4 1
507.2.b.e 4 3.b odd 2 1
507.2.b.e 4 39.d odd 2 1
507.2.e.d 4 39.k even 12 2
507.2.e.h 4 39.k even 12 2
507.2.j.f 8 39.h odd 6 2
507.2.j.f 8 39.i odd 6 2
624.2.a.k 2 156.l odd 4 1
975.2.a.l 2 195.n even 4 1
975.2.c.h 4 195.j odd 4 1
975.2.c.h 4 195.u odd 4 1
1053.2.e.e 4 117.y odd 12 2
1053.2.e.m 4 117.z even 12 2
1521.2.a.f 2 13.d odd 4 1
1521.2.b.j 4 1.a even 1 1 trivial
1521.2.b.j 4 13.b even 2 1 inner
1872.2.a.w 2 52.f even 4 1
1911.2.a.h 2 273.o odd 4 1
2496.2.a.bf 2 312.y even 4 1
2496.2.a.bi 2 312.w odd 4 1
2925.2.a.v 2 65.g odd 4 1
2925.2.c.u 4 65.f even 4 1
2925.2.c.u 4 65.k even 4 1
4719.2.a.p 2 429.l odd 4 1
5733.2.a.u 2 91.i even 4 1
7488.2.a.cl 2 104.j odd 4 1
7488.2.a.co 2 104.m even 4 1
8112.2.a.bm 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 $$S_{2}^{\mathrm{new}}(1521, [\chi])$$:

 $$T_{2}^{4} + 6T_{2}^{2} + 1$$ T2^4 + 6*T2^2 + 1 $$T_{5}^{2} + 8$$ T5^2 + 8 $$T_{7}^{2} + 8$$ T7^2 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 8)^{2}$$
$7$ $$(T^{2} + 8)^{2}$$
$11$ $$(T^{2} + 4)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 4 T - 28)^{2}$$
$19$ $$(T^{2} + 8)^{2}$$
$23$ $$(T + 4)^{4}$$
$29$ $$(T + 2)^{4}$$
$31$ $$T^{4} + 48T^{2} + 64$$
$37$ $$T^{4} + 72T^{2} + 784$$
$41$ $$T^{4} + 144T^{2} + 3136$$
$43$ $$(T^{2} + 8 T - 16)^{2}$$
$47$ $$T^{4} + 136T^{2} + 16$$
$53$ $$(T - 2)^{4}$$
$59$ $$T^{4} + 72T^{2} + 784$$
$61$ $$(T^{2} - 4 T - 124)^{2}$$
$67$ $$T^{4} + 48T^{2} + 64$$
$71$ $$(T^{2} + 4)^{2}$$
$73$ $$T^{4} + 136T^{2} + 16$$
$79$ $$(T^{2} - 128)^{2}$$
$83$ $$T^{4} + 72T^{2} + 784$$
$89$ $$T^{4} + 304 T^{2} + 18496$$
$97$ $$T^{4} + 72T^{2} + 784$$