Properties

 Label 3042.2.b.l Level $3042$ Weight $2$ Character orbit 3042.b Analytic conductor $24.290$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3042 = 2 \cdot 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3042.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$24.2904922949$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 78) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} + (\beta_{2} - 3 \beta_1) q^{7} - \beta_1 q^{8}+O(q^{10})$$ q + b1 * q^2 - q^4 - b2 * q^5 + (b2 - 3*b1) * q^7 - b1 * q^8 $$q + \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} + (\beta_{2} - 3 \beta_1) q^{7} - \beta_1 q^{8} + \beta_{3} q^{10} + ( - \beta_{2} + 3 \beta_1) q^{11} + ( - \beta_{3} + 3) q^{14} + q^{16} + 3 \beta_{3} q^{17} + (\beta_{2} + 3 \beta_1) q^{19} + \beta_{2} q^{20} + (\beta_{3} - 3) q^{22} + ( - 3 \beta_{3} - 3) q^{23} + 2 q^{25} + ( - \beta_{2} + 3 \beta_1) q^{28} + 3 q^{29} + (2 \beta_{2} + 6 \beta_1) q^{31} + \beta_1 q^{32} + 3 \beta_{2} q^{34} + ( - 3 \beta_{3} + 3) q^{35} - 3 \beta_1 q^{37} + ( - \beta_{3} - 3) q^{38} - \beta_{3} q^{40} + ( - 2 \beta_{2} - 3 \beta_1) q^{41} + (3 \beta_{3} - 1) q^{43} + (\beta_{2} - 3 \beta_1) q^{44} + ( - 3 \beta_{2} - 3 \beta_1) q^{46} + ( - \beta_{2} - 3 \beta_1) q^{47} + (6 \beta_{3} - 5) q^{49} + 2 \beta_1 q^{50} - 3 q^{53} + (3 \beta_{3} - 3) q^{55} + (\beta_{3} - 3) q^{56} + 3 \beta_1 q^{58} - 8 \beta_{2} q^{59} + (3 \beta_{3} + 10) q^{61} + ( - 2 \beta_{3} - 6) q^{62} - q^{64} + ( - \beta_{2} + 9 \beta_1) q^{67} - 3 \beta_{3} q^{68} + ( - 3 \beta_{2} + 3 \beta_1) q^{70} + (3 \beta_{2} - 3 \beta_1) q^{71} - 7 \beta_{2} q^{73} + 3 q^{74} + ( - \beta_{2} - 3 \beta_1) q^{76} + ( - 6 \beta_{3} + 12) q^{77} + (6 \beta_{3} - 2) q^{79} - \beta_{2} q^{80} + (2 \beta_{3} + 3) q^{82} + ( - 5 \beta_{2} + 3 \beta_1) q^{83} - 9 \beta_1 q^{85} + (3 \beta_{2} - \beta_1) q^{86} + ( - \beta_{3} + 3) q^{88} + (2 \beta_{2} + 6 \beta_1) q^{89} + (3 \beta_{3} + 3) q^{92} + (\beta_{3} + 3) q^{94} + (3 \beta_{3} + 3) q^{95} + 6 \beta_1 q^{97} + (6 \beta_{2} - 5 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 - q^4 - b2 * q^5 + (b2 - 3*b1) * q^7 - b1 * q^8 + b3 * q^10 + (-b2 + 3*b1) * q^11 + (-b3 + 3) * q^14 + q^16 + 3*b3 * q^17 + (b2 + 3*b1) * q^19 + b2 * q^20 + (b3 - 3) * q^22 + (-3*b3 - 3) * q^23 + 2 * q^25 + (-b2 + 3*b1) * q^28 + 3 * q^29 + (2*b2 + 6*b1) * q^31 + b1 * q^32 + 3*b2 * q^34 + (-3*b3 + 3) * q^35 - 3*b1 * q^37 + (-b3 - 3) * q^38 - b3 * q^40 + (-2*b2 - 3*b1) * q^41 + (3*b3 - 1) * q^43 + (b2 - 3*b1) * q^44 + (-3*b2 - 3*b1) * q^46 + (-b2 - 3*b1) * q^47 + (6*b3 - 5) * q^49 + 2*b1 * q^50 - 3 * q^53 + (3*b3 - 3) * q^55 + (b3 - 3) * q^56 + 3*b1 * q^58 - 8*b2 * q^59 + (3*b3 + 10) * q^61 + (-2*b3 - 6) * q^62 - q^64 + (-b2 + 9*b1) * q^67 - 3*b3 * q^68 + (-3*b2 + 3*b1) * q^70 + (3*b2 - 3*b1) * q^71 - 7*b2 * q^73 + 3 * q^74 + (-b2 - 3*b1) * q^76 + (-6*b3 + 12) * q^77 + (6*b3 - 2) * q^79 - b2 * q^80 + (2*b3 + 3) * q^82 + (-5*b2 + 3*b1) * q^83 - 9*b1 * q^85 + (3*b2 - b1) * q^86 + (-b3 + 3) * q^88 + (2*b2 + 6*b1) * q^89 + (3*b3 + 3) * q^92 + (b3 + 3) * q^94 + (3*b3 + 3) * q^95 + 6*b1 * q^97 + (6*b2 - 5*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} + 12 q^{14} + 4 q^{16} - 12 q^{22} - 12 q^{23} + 8 q^{25} + 12 q^{29} + 12 q^{35} - 12 q^{38} - 4 q^{43} - 20 q^{49} - 12 q^{53} - 12 q^{55} - 12 q^{56} + 40 q^{61} - 24 q^{62} - 4 q^{64} + 12 q^{74} + 48 q^{77} - 8 q^{79} + 12 q^{82} + 12 q^{88} + 12 q^{92} + 12 q^{94} + 12 q^{95}+O(q^{100})$$ 4 * q - 4 * q^4 + 12 * q^14 + 4 * q^16 - 12 * q^22 - 12 * q^23 + 8 * q^25 + 12 * q^29 + 12 * q^35 - 12 * q^38 - 4 * q^43 - 20 * q^49 - 12 * q^53 - 12 * q^55 - 12 * q^56 + 40 * q^61 - 24 * q^62 - 4 * q^64 + 12 * q^74 + 48 * q^77 - 8 * q^79 + 12 * q^82 + 12 * q^88 + 12 * q^92 + 12 * q^94 + 12 * q^95

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$$.

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

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
 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
1.00000i 0 −1.00000 1.73205i 0 4.73205i 1.00000i 0 −1.73205
1351.2 1.00000i 0 −1.00000 1.73205i 0 1.26795i 1.00000i 0 1.73205
1351.3 1.00000i 0 −1.00000 1.73205i 0 1.26795i 1.00000i 0 1.73205
1351.4 1.00000i 0 −1.00000 1.73205i 0 4.73205i 1.00000i 0 −1.73205
 $$n$$: 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 3042.2.b.l 4
3.b odd 2 1 1014.2.b.d 4
13.b even 2 1 inner 3042.2.b.l 4
13.c even 3 1 234.2.l.a 4
13.d odd 4 1 3042.2.a.s 2
13.d odd 4 1 3042.2.a.v 2
13.e even 6 1 234.2.l.a 4
39.d odd 2 1 1014.2.b.d 4
39.f even 4 1 1014.2.a.h 2
39.f even 4 1 1014.2.a.j 2
39.h odd 6 1 78.2.i.b 4
39.h odd 6 1 1014.2.i.f 4
39.i odd 6 1 78.2.i.b 4
39.i odd 6 1 1014.2.i.f 4
39.k even 12 2 1014.2.e.h 4
39.k even 12 2 1014.2.e.j 4
52.i odd 6 1 1872.2.by.k 4
52.j odd 6 1 1872.2.by.k 4
156.l odd 4 1 8112.2.a.bq 2
156.l odd 4 1 8112.2.a.bx 2
156.p even 6 1 624.2.bv.d 4
156.r even 6 1 624.2.bv.d 4
195.x odd 6 1 1950.2.bc.c 4
195.y odd 6 1 1950.2.bc.c 4
195.bf even 12 1 1950.2.y.a 4
195.bf even 12 1 1950.2.y.h 4
195.bl even 12 1 1950.2.y.a 4
195.bl even 12 1 1950.2.y.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.b 4 39.h odd 6 1
78.2.i.b 4 39.i odd 6 1
234.2.l.a 4 13.c even 3 1
234.2.l.a 4 13.e even 6 1
624.2.bv.d 4 156.p even 6 1
624.2.bv.d 4 156.r even 6 1
1014.2.a.h 2 39.f even 4 1
1014.2.a.j 2 39.f even 4 1
1014.2.b.d 4 3.b odd 2 1
1014.2.b.d 4 39.d odd 2 1
1014.2.e.h 4 39.k even 12 2
1014.2.e.j 4 39.k even 12 2
1014.2.i.f 4 39.h odd 6 1
1014.2.i.f 4 39.i odd 6 1
1872.2.by.k 4 52.i odd 6 1
1872.2.by.k 4 52.j odd 6 1
1950.2.y.a 4 195.bf even 12 1
1950.2.y.a 4 195.bl even 12 1
1950.2.y.h 4 195.bf even 12 1
1950.2.y.h 4 195.bl even 12 1
1950.2.bc.c 4 195.x odd 6 1
1950.2.bc.c 4 195.y odd 6 1
3042.2.a.s 2 13.d odd 4 1
3042.2.a.v 2 13.d odd 4 1
3042.2.b.l 4 1.a even 1 1 trivial
3042.2.b.l 4 13.b even 2 1 inner
8112.2.a.bq 2 156.l odd 4 1
8112.2.a.bx 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 $$S_{2}^{\mathrm{new}}(3042, [\chi])$$:

 $$T_{5}^{2} + 3$$ T5^2 + 3 $$T_{7}^{4} + 24T_{7}^{2} + 36$$ T7^4 + 24*T7^2 + 36 $$T_{17}^{2} - 27$$ T17^2 - 27 $$T_{23}^{2} + 6T_{23} - 18$$ T23^2 + 6*T23 - 18

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 3)^{2}$$
$7$ $$T^{4} + 24T^{2} + 36$$
$11$ $$T^{4} + 24T^{2} + 36$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 27)^{2}$$
$19$ $$T^{4} + 24T^{2} + 36$$
$23$ $$(T^{2} + 6 T - 18)^{2}$$
$29$ $$(T - 3)^{4}$$
$31$ $$T^{4} + 96T^{2} + 576$$
$37$ $$(T^{2} + 9)^{2}$$
$41$ $$T^{4} + 42T^{2} + 9$$
$43$ $$(T^{2} + 2 T - 26)^{2}$$
$47$ $$T^{4} + 24T^{2} + 36$$
$53$ $$(T + 3)^{4}$$
$59$ $$(T^{2} + 192)^{2}$$
$61$ $$(T^{2} - 20 T + 73)^{2}$$
$67$ $$T^{4} + 168T^{2} + 6084$$
$71$ $$T^{4} + 72T^{2} + 324$$
$73$ $$(T^{2} + 147)^{2}$$
$79$ $$(T^{2} + 4 T - 104)^{2}$$
$83$ $$T^{4} + 168T^{2} + 4356$$
$89$ $$T^{4} + 96T^{2} + 576$$
$97$ $$(T^{2} + 36)^{2}$$