# Properties

 Label 3042.2.a.bd Level $3042$ Weight $2$ Character orbit 3042.a Self dual yes Analytic conductor $24.290$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3042 = 2 \cdot 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3042.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.2904922949$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1014) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (\beta_{2} + \beta_1 + 1) q^{5} + ( - 2 \beta_{2} + \beta_1) q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (b2 + b1 + 1) * q^5 + (-2*b2 + b1) * q^7 - q^8 $$q - q^{2} + q^{4} + (\beta_{2} + \beta_1 + 1) q^{5} + ( - 2 \beta_{2} + \beta_1) q^{7} - q^{8} + ( - \beta_{2} - \beta_1 - 1) q^{10} + ( - \beta_{2} - 2 \beta_1) q^{11} + (2 \beta_{2} - \beta_1) q^{14} + q^{16} + ( - 4 \beta_{2} + 2 \beta_1 - 6) q^{17} + (4 \beta_{2} - 4 \beta_1 + 4) q^{19} + (\beta_{2} + \beta_1 + 1) q^{20} + (\beta_{2} + 2 \beta_1) q^{22} + (2 \beta_{2} - 2 \beta_1 - 4) q^{23} + (4 \beta_{2} + 3 \beta_1 + 1) q^{25} + ( - 2 \beta_{2} + \beta_1) q^{28} + (3 \beta_{2} + 2 \beta_1 - 4) q^{29} + (5 \beta_{2} - 4 \beta_1) q^{31} - q^{32} + (4 \beta_{2} - 2 \beta_1 + 6) q^{34} + ( - \beta_1 - 1) q^{35} + ( - 2 \beta_{2} + 4 \beta_1 - 6) q^{37} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{38} + ( - \beta_{2} - \beta_1 - 1) q^{40} + ( - 2 \beta_{2} + 4) q^{41} + (2 \beta_{2} - 6 \beta_1) q^{43} + ( - \beta_{2} - 2 \beta_1) q^{44} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{46} - 4 \beta_1 q^{47} + ( - 7 \beta_{2} + 4 \beta_1 - 5) q^{49} + ( - 4 \beta_{2} - 3 \beta_1 - 1) q^{50} + (3 \beta_{2} - 9 \beta_1 - 1) q^{53} + ( - 5 \beta_{2} - 3 \beta_1 - 8) q^{55} + (2 \beta_{2} - \beta_1) q^{56} + ( - 3 \beta_{2} - 2 \beta_1 + 4) q^{58} + ( - 2 \beta_{2} + \beta_1 + 2) q^{59} + (2 \beta_1 - 4) q^{61} + ( - 5 \beta_{2} + 4 \beta_1) q^{62} + q^{64} + ( - 10 \beta_{2} + 8 \beta_1 - 8) q^{67} + ( - 4 \beta_{2} + 2 \beta_1 - 6) q^{68} + (\beta_1 + 1) q^{70} + (6 \beta_{2} - 6 \beta_1 + 2) q^{71} + (\beta_{2} + 3 \beta_1 + 1) q^{73} + (2 \beta_{2} - 4 \beta_1 + 6) q^{74} + (4 \beta_{2} - 4 \beta_1 + 4) q^{76} + ( - \beta_{2} + 2 \beta_1 + 1) q^{77} + (6 \beta_{2} - 11 \beta_1 + 4) q^{79} + (\beta_{2} + \beta_1 + 1) q^{80} + (2 \beta_{2} - 4) q^{82} + ( - 2 \beta_{2} + 11 \beta_1 - 2) q^{83} + ( - 6 \beta_{2} - 8 \beta_1 - 8) q^{85} + ( - 2 \beta_{2} + 6 \beta_1) q^{86} + (\beta_{2} + 2 \beta_1) q^{88} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{89} + (2 \beta_{2} - 2 \beta_1 - 4) q^{92} + 4 \beta_1 q^{94} + 4 \beta_1 q^{95} + (3 \beta_{2} + \beta_1 + 3) q^{97} + (7 \beta_{2} - 4 \beta_1 + 5) q^{98}+O(q^{100})$$ q - q^2 + q^4 + (b2 + b1 + 1) * q^5 + (-2*b2 + b1) * q^7 - q^8 + (-b2 - b1 - 1) * q^10 + (-b2 - 2*b1) * q^11 + (2*b2 - b1) * q^14 + q^16 + (-4*b2 + 2*b1 - 6) * q^17 + (4*b2 - 4*b1 + 4) * q^19 + (b2 + b1 + 1) * q^20 + (b2 + 2*b1) * q^22 + (2*b2 - 2*b1 - 4) * q^23 + (4*b2 + 3*b1 + 1) * q^25 + (-2*b2 + b1) * q^28 + (3*b2 + 2*b1 - 4) * q^29 + (5*b2 - 4*b1) * q^31 - q^32 + (4*b2 - 2*b1 + 6) * q^34 + (-b1 - 1) * q^35 + (-2*b2 + 4*b1 - 6) * q^37 + (-4*b2 + 4*b1 - 4) * q^38 + (-b2 - b1 - 1) * q^40 + (-2*b2 + 4) * q^41 + (2*b2 - 6*b1) * q^43 + (-b2 - 2*b1) * q^44 + (-2*b2 + 2*b1 + 4) * q^46 - 4*b1 * q^47 + (-7*b2 + 4*b1 - 5) * q^49 + (-4*b2 - 3*b1 - 1) * q^50 + (3*b2 - 9*b1 - 1) * q^53 + (-5*b2 - 3*b1 - 8) * q^55 + (2*b2 - b1) * q^56 + (-3*b2 - 2*b1 + 4) * q^58 + (-2*b2 + b1 + 2) * q^59 + (2*b1 - 4) * q^61 + (-5*b2 + 4*b1) * q^62 + q^64 + (-10*b2 + 8*b1 - 8) * q^67 + (-4*b2 + 2*b1 - 6) * q^68 + (b1 + 1) * q^70 + (6*b2 - 6*b1 + 2) * q^71 + (b2 + 3*b1 + 1) * q^73 + (2*b2 - 4*b1 + 6) * q^74 + (4*b2 - 4*b1 + 4) * q^76 + (-b2 + 2*b1 + 1) * q^77 + (6*b2 - 11*b1 + 4) * q^79 + (b2 + b1 + 1) * q^80 + (2*b2 - 4) * q^82 + (-2*b2 + 11*b1 - 2) * q^83 + (-6*b2 - 8*b1 - 8) * q^85 + (-2*b2 + 6*b1) * q^86 + (b2 + 2*b1) * q^88 + (-2*b2 + 2*b1 + 2) * q^89 + (2*b2 - 2*b1 - 4) * q^92 + 4*b1 * q^94 + 4*b1 * q^95 + (3*b2 + b1 + 3) * q^97 + (7*b2 - 4*b1 + 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 + 3 * q^7 - 3 * q^8 $$3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{10} - q^{11} - 3 q^{14} + 3 q^{16} - 12 q^{17} + 4 q^{19} + 3 q^{20} + q^{22} - 16 q^{23} + 2 q^{25} + 3 q^{28} - 13 q^{29} - 9 q^{31} - 3 q^{32} + 12 q^{34} - 4 q^{35} - 12 q^{37} - 4 q^{38} - 3 q^{40} + 14 q^{41} - 8 q^{43} - q^{44} + 16 q^{46} - 4 q^{47} - 4 q^{49} - 2 q^{50} - 15 q^{53} - 22 q^{55} - 3 q^{56} + 13 q^{58} + 9 q^{59} - 10 q^{61} + 9 q^{62} + 3 q^{64} - 6 q^{67} - 12 q^{68} + 4 q^{70} - 6 q^{71} + 5 q^{73} + 12 q^{74} + 4 q^{76} + 6 q^{77} - 5 q^{79} + 3 q^{80} - 14 q^{82} + 7 q^{83} - 26 q^{85} + 8 q^{86} + q^{88} + 10 q^{89} - 16 q^{92} + 4 q^{94} + 4 q^{95} + 7 q^{97} + 4 q^{98}+O(q^{100})$$ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 + 3 * q^7 - 3 * q^8 - 3 * q^10 - q^11 - 3 * q^14 + 3 * q^16 - 12 * q^17 + 4 * q^19 + 3 * q^20 + q^22 - 16 * q^23 + 2 * q^25 + 3 * q^28 - 13 * q^29 - 9 * q^31 - 3 * q^32 + 12 * q^34 - 4 * q^35 - 12 * q^37 - 4 * q^38 - 3 * q^40 + 14 * q^41 - 8 * q^43 - q^44 + 16 * q^46 - 4 * q^47 - 4 * q^49 - 2 * q^50 - 15 * q^53 - 22 * q^55 - 3 * q^56 + 13 * q^58 + 9 * q^59 - 10 * q^61 + 9 * q^62 + 3 * q^64 - 6 * q^67 - 12 * q^68 + 4 * q^70 - 6 * q^71 + 5 * q^73 + 12 * q^74 + 4 * q^76 + 6 * q^77 - 5 * q^79 + 3 * q^80 - 14 * q^82 + 7 * q^83 - 26 * q^85 + 8 * q^86 + q^88 + 10 * q^89 - 16 * q^92 + 4 * q^94 + 4 * q^95 + 7 * q^97 + 4 * q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## 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
 −1.24698 0.445042 1.80194
−1.00000 0 1.00000 −0.692021 0 −0.356896 −1.00000 0 0.692021
1.2 −1.00000 0 1.00000 −0.356896 0 4.04892 −1.00000 0 0.356896
1.3 −1.00000 0 1.00000 4.04892 0 −0.692021 −1.00000 0 −4.04892
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$13$$ $$-1$$

## 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 3042.2.a.bd 3
3.b odd 2 1 1014.2.a.o yes 3
12.b even 2 1 8112.2.a.bz 3
13.b even 2 1 3042.2.a.be 3
13.d odd 4 2 3042.2.b.r 6
39.d odd 2 1 1014.2.a.m 3
39.f even 4 2 1014.2.b.g 6
39.h odd 6 2 1014.2.e.m 6
39.i odd 6 2 1014.2.e.k 6
39.k even 12 4 1014.2.i.g 12
156.h even 2 1 8112.2.a.ce 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1014.2.a.m 3 39.d odd 2 1
1014.2.a.o yes 3 3.b odd 2 1
1014.2.b.g 6 39.f even 4 2
1014.2.e.k 6 39.i odd 6 2
1014.2.e.m 6 39.h odd 6 2
1014.2.i.g 12 39.k even 12 4
3042.2.a.bd 3 1.a even 1 1 trivial
3042.2.a.be 3 13.b even 2 1
3042.2.b.r 6 13.d odd 4 2
8112.2.a.bz 3 12.b even 2 1
8112.2.a.ce 3 156.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3042))$$:

 $$T_{5}^{3} - 3T_{5}^{2} - 4T_{5} - 1$$ T5^3 - 3*T5^2 - 4*T5 - 1 $$T_{7}^{3} - 3T_{7}^{2} - 4T_{7} - 1$$ T7^3 - 3*T7^2 - 4*T7 - 1 $$T_{11}^{3} + T_{11}^{2} - 16T_{11} + 13$$ T11^3 + T11^2 - 16*T11 + 13 $$T_{17}^{3} + 12T_{17}^{2} + 20T_{17} - 104$$ T17^3 + 12*T17^2 + 20*T17 - 104

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3} - 3 T^{2} - 4 T - 1$$
$7$ $$T^{3} - 3 T^{2} - 4 T - 1$$
$11$ $$T^{3} + T^{2} - 16 T + 13$$
$13$ $$T^{3}$$
$17$ $$T^{3} + 12 T^{2} + 20 T - 104$$
$19$ $$T^{3} - 4 T^{2} - 32 T + 64$$
$23$ $$T^{3} + 16 T^{2} + 76 T + 104$$
$29$ $$T^{3} + 13 T^{2} + 12 T - 223$$
$31$ $$T^{3} + 9 T^{2} - 22 T - 29$$
$37$ $$T^{3} + 12 T^{2} + 20 T + 8$$
$41$ $$T^{3} - 14 T^{2} + 56 T - 56$$
$43$ $$T^{3} + 8 T^{2} - 44 T - 344$$
$47$ $$T^{3} + 4 T^{2} - 32 T - 64$$
$53$ $$T^{3} + 15 T^{2} - 72 T - 1247$$
$59$ $$T^{3} - 9 T^{2} + 20 T - 13$$
$61$ $$T^{3} + 10 T^{2} + 24 T + 8$$
$67$ $$T^{3} + 6 T^{2} - 184 T - 1112$$
$71$ $$T^{3} + 6 T^{2} - 72 T - 104$$
$73$ $$T^{3} - 5 T^{2} - 22 T + 13$$
$79$ $$T^{3} + 5 T^{2} - 204 T - 1469$$
$83$ $$T^{3} - 7 T^{2} - 224 T + 1477$$
$89$ $$T^{3} - 10 T^{2} + 24 T - 8$$
$97$ $$T^{3} - 7 T^{2} - 14 T + 7$$