# Properties

 Label 1872.2.t.r Level $1872$ Weight $2$ Character orbit 1872.t Analytic conductor $14.948$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1872 = 2^{4} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1872.t (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.9479952584$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ x^4 - x^3 + 5*x^2 + 4*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + 1) q^{5} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{7}+O(q^{10})$$ q + (-b3 + 1) * q^5 + (-b3 + 2*b2 - b1 - 2) * q^7 $$q + ( - \beta_{3} + 1) q^{5} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{7} + 2 \beta_{2} q^{11} + ( - \beta_{3} + \beta_{2} - 2 \beta_1) q^{13} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{17} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{19} - 2 \beta_{2} q^{23} - 3 \beta_{3} q^{25} + ( - \beta_{2} + 3 \beta_1) q^{29} + \beta_{3} q^{31} + ( - 2 \beta_{2} + 2) q^{35} + ( - 5 \beta_{2} - \beta_1) q^{37} + (\beta_{2} - \beta_1) q^{41} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{43} + ( - 4 \beta_{3} - 2) q^{47} + ( - \beta_{2} + 3 \beta_1) q^{49} + ( - 3 \beta_{3} - 7) q^{53} + (2 \beta_{2} + 2 \beta_1) q^{55} + (2 \beta_{3} - 8 \beta_{2} + 2 \beta_1 + 8) q^{59} + ( - 2 \beta_{3} + 9 \beta_{2} - 2 \beta_1 - 9) q^{61} + ( - 2 \beta_{3} - 7 \beta_{2} - 3 \beta_1 + 4) q^{65} + (2 \beta_{2} + \beta_1) q^{67} + (14 \beta_{2} - 14) q^{71} + (2 \beta_{3} - 5) q^{73} + ( - 2 \beta_{3} - 4) q^{77} + ( - \beta_{3} - 8) q^{79} + ( - 2 \beta_{3} - 6) q^{83} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 3) q^{85} + (10 \beta_{2} - 2 \beta_1) q^{89} + ( - 2 \beta_{3} - 4 \beta_{2} + \beta_1 - 6) q^{91} + ( - 6 \beta_{3} - 10 \beta_{2} - 6 \beta_1 + 10) q^{95} + ( - \beta_{3} - 6 \beta_{2} - \beta_1 + 6) q^{97}+O(q^{100})$$ q + (-b3 + 1) * q^5 + (-b3 + 2*b2 - b1 - 2) * q^7 + 2*b2 * q^11 + (-b3 + b2 - 2*b1) * q^13 + (b3 - b2 + b1 + 1) * q^17 + (-2*b3 - 2*b2 - 2*b1 + 2) * q^19 - 2*b2 * q^23 - 3*b3 * q^25 + (-b2 + 3*b1) * q^29 + b3 * q^31 + (-2*b2 + 2) * q^35 + (-5*b2 - b1) * q^37 + (b2 - b1) * q^41 + (-b3 - 2*b2 - b1 + 2) * q^43 + (-4*b3 - 2) * q^47 + (-b2 + 3*b1) * q^49 + (-3*b3 - 7) * q^53 + (2*b2 + 2*b1) * q^55 + (2*b3 - 8*b2 + 2*b1 + 8) * q^59 + (-2*b3 + 9*b2 - 2*b1 - 9) * q^61 + (-2*b3 - 7*b2 - 3*b1 + 4) * q^65 + (2*b2 + b1) * q^67 + (14*b2 - 14) * q^71 + (2*b3 - 5) * q^73 + (-2*b3 - 4) * q^77 + (-b3 - 8) * q^79 + (-2*b3 - 6) * q^83 + (b3 + 3*b2 + b1 - 3) * q^85 + (10*b2 - 2*b1) * q^89 + (-2*b3 - 4*b2 + b1 - 6) * q^91 + (-6*b3 - 10*b2 - 6*b1 + 10) * q^95 + (-b3 - 6*b2 - b1 + 6) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{5} - 3 q^{7}+O(q^{10})$$ 4 * q + 6 * q^5 - 3 * q^7 $$4 q + 6 q^{5} - 3 q^{7} + 4 q^{11} + 2 q^{13} + q^{17} + 6 q^{19} - 4 q^{23} + 6 q^{25} + q^{29} - 2 q^{31} + 4 q^{35} - 11 q^{37} + q^{41} + 5 q^{43} + q^{49} - 22 q^{53} + 6 q^{55} + 14 q^{59} - 16 q^{61} + 3 q^{65} + 5 q^{67} - 28 q^{71} - 24 q^{73} - 12 q^{77} - 30 q^{79} - 20 q^{83} - 7 q^{85} + 18 q^{89} - 27 q^{91} + 26 q^{95} + 13 q^{97}+O(q^{100})$$ 4 * q + 6 * q^5 - 3 * q^7 + 4 * q^11 + 2 * q^13 + q^17 + 6 * q^19 - 4 * q^23 + 6 * q^25 + q^29 - 2 * q^31 + 4 * q^35 - 11 * q^37 + q^41 + 5 * q^43 + q^49 - 22 * q^53 + 6 * q^55 + 14 * q^59 - 16 * q^61 + 3 * q^65 + 5 * q^67 - 28 * q^71 - 24 * q^73 - 12 * q^77 - 30 * q^79 - 20 * q^83 - 7 * q^85 + 18 * q^89 - 27 * q^91 + 26 * q^95 + 13 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20$$ (-v^3 + 5*v^2 - 5*v + 16) / 20 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 4 ) / 5$$ (v^3 + 4) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4\beta_{2} + \beta _1 - 4$$ b3 + 4*b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$5\beta_{3} - 4$$ 5*b3 - 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times$$.

 $$n$$ $$145$$ $$209$$ $$469$$ $$703$$ $$\chi(n)$$ $$-1 + \beta_{2}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 −0.780776 − 1.35234i 1.28078 + 2.21837i −0.780776 + 1.35234i 1.28078 − 2.21837i
0 0 0 −0.561553 0 −1.78078 + 3.08440i 0 0 0
289.2 0 0 0 3.56155 0 0.280776 0.486319i 0 0 0
1153.1 0 0 0 −0.561553 0 −1.78078 3.08440i 0 0 0
1153.2 0 0 0 3.56155 0 0.280776 + 0.486319i 0 0 0
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.2.t.r 4
3.b odd 2 1 624.2.q.h 4
4.b odd 2 1 117.2.g.c 4
12.b even 2 1 39.2.e.b 4
13.c even 3 1 inner 1872.2.t.r 4
39.h odd 6 1 8112.2.a.bo 2
39.i odd 6 1 624.2.q.h 4
39.i odd 6 1 8112.2.a.bk 2
52.i odd 6 1 1521.2.a.m 2
52.j odd 6 1 117.2.g.c 4
52.j odd 6 1 1521.2.a.g 2
52.l even 12 2 1521.2.b.h 4
60.h even 2 1 975.2.i.k 4
60.l odd 4 2 975.2.bb.i 8
156.h even 2 1 507.2.e.g 4
156.l odd 4 2 507.2.j.g 8
156.p even 6 1 39.2.e.b 4
156.p even 6 1 507.2.a.g 2
156.r even 6 1 507.2.a.d 2
156.r even 6 1 507.2.e.g 4
156.v odd 12 2 507.2.b.d 4
156.v odd 12 2 507.2.j.g 8
780.br even 6 1 975.2.i.k 4
780.cj 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 12.b even 2 1
39.2.e.b 4 156.p even 6 1
117.2.g.c 4 4.b odd 2 1
117.2.g.c 4 52.j odd 6 1
507.2.a.d 2 156.r even 6 1
507.2.a.g 2 156.p even 6 1
507.2.b.d 4 156.v odd 12 2
507.2.e.g 4 156.h even 2 1
507.2.e.g 4 156.r even 6 1
507.2.j.g 8 156.l odd 4 2
507.2.j.g 8 156.v odd 12 2
624.2.q.h 4 3.b odd 2 1
624.2.q.h 4 39.i odd 6 1
975.2.i.k 4 60.h even 2 1
975.2.i.k 4 780.br even 6 1
975.2.bb.i 8 60.l odd 4 2
975.2.bb.i 8 780.cj odd 12 2
1521.2.a.g 2 52.j odd 6 1
1521.2.a.m 2 52.i odd 6 1
1521.2.b.h 4 52.l even 12 2
1872.2.t.r 4 1.a even 1 1 trivial
1872.2.t.r 4 13.c even 3 1 inner
8112.2.a.bk 2 39.i odd 6 1
8112.2.a.bo 2 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1872, [\chi])$$:

 $$T_{5}^{2} - 3T_{5} - 2$$ T5^2 - 3*T5 - 2 $$T_{7}^{4} + 3T_{7}^{3} + 11T_{7}^{2} - 6T_{7} + 4$$ T7^4 + 3*T7^3 + 11*T7^2 - 6*T7 + 4 $$T_{11}^{2} - 2T_{11} + 4$$ T11^2 - 2*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 3 T - 2)^{2}$$
$7$ $$T^{4} + 3 T^{3} + 11 T^{2} - 6 T + 4$$
$11$ $$(T^{2} - 2 T + 4)^{2}$$
$13$ $$(T^{2} - T + 13)^{2}$$
$17$ $$T^{4} - T^{3} + 5 T^{2} + 4 T + 16$$
$19$ $$T^{4} - 6 T^{3} + 44 T^{2} + 48 T + 64$$
$23$ $$(T^{2} + 2 T + 4)^{2}$$
$29$ $$T^{4} - T^{3} + 39 T^{2} + 38 T + 1444$$
$31$ $$(T^{2} + T - 4)^{2}$$
$37$ $$T^{4} + 11 T^{3} + 95 T^{2} + \cdots + 676$$
$41$ $$T^{4} - T^{3} + 5 T^{2} + 4 T + 16$$
$43$ $$T^{4} - 5 T^{3} + 23 T^{2} - 10 T + 4$$
$47$ $$(T^{2} - 68)^{2}$$
$53$ $$(T^{2} + 11 T - 8)^{2}$$
$59$ $$T^{4} - 14 T^{3} + 164 T^{2} + \cdots + 1024$$
$61$ $$T^{4} + 16 T^{3} + 209 T^{2} + \cdots + 2209$$
$67$ $$T^{4} - 5 T^{3} + 23 T^{2} - 10 T + 4$$
$71$ $$(T^{2} + 14 T + 196)^{2}$$
$73$ $$(T^{2} + 12 T + 19)^{2}$$
$79$ $$(T^{2} + 15 T + 52)^{2}$$
$83$ $$(T^{2} + 10 T + 8)^{2}$$
$89$ $$T^{4} - 18 T^{3} + 260 T^{2} + \cdots + 4096$$
$97$ $$T^{4} - 13 T^{3} + 131 T^{2} + \cdots + 1444$$