# Properties

 Label 1014.2.a.o Level $1014$ Weight $2$ Character orbit 1014.a Self dual yes Analytic conductor $8.097$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.09683076496$$ Analytic rank: $$0$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{5} + q^{6} + ( - 2 \beta_{2} + \beta_1) q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + (-b2 - b1 - 1) * q^5 + q^6 + (-2*b2 + b1) * q^7 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{5} + q^{6} + ( - 2 \beta_{2} + \beta_1) q^{7} + q^{8} + q^{9} + ( - \beta_{2} - \beta_1 - 1) q^{10} + (\beta_{2} + 2 \beta_1) q^{11} + q^{12} + ( - 2 \beta_{2} + \beta_1) q^{14} + ( - \beta_{2} - \beta_1 - 1) q^{15} + q^{16} + (4 \beta_{2} - 2 \beta_1 + 6) q^{17} + q^{18} + (4 \beta_{2} - 4 \beta_1 + 4) q^{19} + ( - \beta_{2} - \beta_1 - 1) q^{20} + ( - 2 \beta_{2} + \beta_1) q^{21} + (\beta_{2} + 2 \beta_1) q^{22} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{23} + q^{24} + (4 \beta_{2} + 3 \beta_1 + 1) q^{25} + q^{27} + ( - 2 \beta_{2} + \beta_1) q^{28} + ( - 3 \beta_{2} - 2 \beta_1 + 4) q^{29} + ( - \beta_{2} - \beta_1 - 1) q^{30} + (5 \beta_{2} - 4 \beta_1) q^{31} + q^{32} + (\beta_{2} + 2 \beta_1) q^{33} + (4 \beta_{2} - 2 \beta_1 + 6) q^{34} + (\beta_1 + 1) q^{35} + q^{36} + ( - 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} + \beta_1) q^{42} + (2 \beta_{2} - 6 \beta_1) q^{43} + (\beta_{2} + 2 \beta_1) q^{44} + ( - \beta_{2} - \beta_1 - 1) q^{45} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{46} + 4 \beta_1 q^{47} + q^{48} + ( - 7 \beta_{2} + 4 \beta_1 - 5) q^{49} + (4 \beta_{2} + 3 \beta_1 + 1) q^{50} + (4 \beta_{2} - 2 \beta_1 + 6) q^{51} + ( - 3 \beta_{2} + 9 \beta_1 + 1) q^{53} + q^{54} + ( - 5 \beta_{2} - 3 \beta_1 - 8) q^{55} + ( - 2 \beta_{2} + \beta_1) q^{56} + (4 \beta_{2} - 4 \beta_1 + 4) q^{57} + ( - 3 \beta_{2} - 2 \beta_1 + 4) q^{58} + (2 \beta_{2} - \beta_1 - 2) q^{59} + ( - \beta_{2} - \beta_1 - 1) q^{60} + (2 \beta_1 - 4) q^{61} + (5 \beta_{2} - 4 \beta_1) q^{62} + ( - 2 \beta_{2} + \beta_1) q^{63} + q^{64} + (\beta_{2} + 2 \beta_1) q^{66} + ( - 10 \beta_{2} + 8 \beta_1 - 8) q^{67} + (4 \beta_{2} - 2 \beta_1 + 6) q^{68} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{69} + (\beta_1 + 1) q^{70} + ( - 6 \beta_{2} + 6 \beta_1 - 2) q^{71} + q^{72} + (\beta_{2} + 3 \beta_1 + 1) q^{73} + ( - 2 \beta_{2} + 4 \beta_1 - 6) q^{74} + (4 \beta_{2} + 3 \beta_1 + 1) q^{75} + (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} + q^{81} + (2 \beta_{2} - 4) q^{82} + (2 \beta_{2} - 11 \beta_1 + 2) q^{83} + ( - 2 \beta_{2} + \beta_1) q^{84} + ( - 6 \beta_{2} - 8 \beta_1 - 8) q^{85} + (2 \beta_{2} - 6 \beta_1) q^{86} + ( - 3 \beta_{2} - 2 \beta_1 + 4) q^{87} + (\beta_{2} + 2 \beta_1) q^{88} + (2 \beta_{2} - 2 \beta_1 - 2) q^{89} + ( - \beta_{2} - \beta_1 - 1) q^{90} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{92} + (5 \beta_{2} - 4 \beta_1) q^{93} + 4 \beta_1 q^{94} - 4 \beta_1 q^{95} + q^{96} + (3 \beta_{2} + \beta_1 + 3) q^{97} + ( - 7 \beta_{2} + 4 \beta_1 - 5) q^{98} + (\beta_{2} + 2 \beta_1) q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + (-b2 - b1 - 1) * q^5 + q^6 + (-2*b2 + b1) * q^7 + q^8 + q^9 + (-b2 - b1 - 1) * q^10 + (b2 + 2*b1) * q^11 + q^12 + (-2*b2 + b1) * q^14 + (-b2 - b1 - 1) * q^15 + q^16 + (4*b2 - 2*b1 + 6) * q^17 + q^18 + (4*b2 - 4*b1 + 4) * q^19 + (-b2 - b1 - 1) * q^20 + (-2*b2 + b1) * q^21 + (b2 + 2*b1) * q^22 + (-2*b2 + 2*b1 + 4) * q^23 + q^24 + (4*b2 + 3*b1 + 1) * q^25 + q^27 + (-2*b2 + b1) * q^28 + (-3*b2 - 2*b1 + 4) * q^29 + (-b2 - b1 - 1) * q^30 + (5*b2 - 4*b1) * q^31 + q^32 + (b2 + 2*b1) * q^33 + (4*b2 - 2*b1 + 6) * q^34 + (b1 + 1) * q^35 + q^36 + (-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 + b1) * q^42 + (2*b2 - 6*b1) * q^43 + (b2 + 2*b1) * q^44 + (-b2 - b1 - 1) * q^45 + (-2*b2 + 2*b1 + 4) * q^46 + 4*b1 * q^47 + q^48 + (-7*b2 + 4*b1 - 5) * q^49 + (4*b2 + 3*b1 + 1) * q^50 + (4*b2 - 2*b1 + 6) * q^51 + (-3*b2 + 9*b1 + 1) * q^53 + q^54 + (-5*b2 - 3*b1 - 8) * q^55 + (-2*b2 + b1) * q^56 + (4*b2 - 4*b1 + 4) * q^57 + (-3*b2 - 2*b1 + 4) * q^58 + (2*b2 - b1 - 2) * q^59 + (-b2 - b1 - 1) * q^60 + (2*b1 - 4) * q^61 + (5*b2 - 4*b1) * q^62 + (-2*b2 + b1) * q^63 + q^64 + (b2 + 2*b1) * q^66 + (-10*b2 + 8*b1 - 8) * q^67 + (4*b2 - 2*b1 + 6) * q^68 + (-2*b2 + 2*b1 + 4) * q^69 + (b1 + 1) * q^70 + (-6*b2 + 6*b1 - 2) * q^71 + q^72 + (b2 + 3*b1 + 1) * q^73 + (-2*b2 + 4*b1 - 6) * q^74 + (4*b2 + 3*b1 + 1) * q^75 + (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 + q^81 + (2*b2 - 4) * q^82 + (2*b2 - 11*b1 + 2) * q^83 + (-2*b2 + b1) * q^84 + (-6*b2 - 8*b1 - 8) * q^85 + (2*b2 - 6*b1) * q^86 + (-3*b2 - 2*b1 + 4) * q^87 + (b2 + 2*b1) * q^88 + (2*b2 - 2*b1 - 2) * q^89 + (-b2 - b1 - 1) * q^90 + (-2*b2 + 2*b1 + 4) * q^92 + (5*b2 - 4*b1) * q^93 + 4*b1 * q^94 - 4*b1 * q^95 + q^96 + (3*b2 + b1 + 3) * q^97 + (-7*b2 + 4*b1 - 5) * q^98 + (b2 + 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 + 3 * q^6 + 3 * q^7 + 3 * q^8 + 3 * q^9 $$3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + q^{11} + 3 q^{12} + 3 q^{14} - 3 q^{15} + 3 q^{16} + 12 q^{17} + 3 q^{18} + 4 q^{19} - 3 q^{20} + 3 q^{21} + q^{22} + 16 q^{23} + 3 q^{24} + 2 q^{25} + 3 q^{27} + 3 q^{28} + 13 q^{29} - 3 q^{30} - 9 q^{31} + 3 q^{32} + q^{33} + 12 q^{34} + 4 q^{35} + 3 q^{36} - 12 q^{37} + 4 q^{38} - 3 q^{40} - 14 q^{41} + 3 q^{42} - 8 q^{43} + q^{44} - 3 q^{45} + 16 q^{46} + 4 q^{47} + 3 q^{48} - 4 q^{49} + 2 q^{50} + 12 q^{51} + 15 q^{53} + 3 q^{54} - 22 q^{55} + 3 q^{56} + 4 q^{57} + 13 q^{58} - 9 q^{59} - 3 q^{60} - 10 q^{61} - 9 q^{62} + 3 q^{63} + 3 q^{64} + q^{66} - 6 q^{67} + 12 q^{68} + 16 q^{69} + 4 q^{70} + 6 q^{71} + 3 q^{72} + 5 q^{73} - 12 q^{74} + 2 q^{75} + 4 q^{76} - 6 q^{77} - 5 q^{79} - 3 q^{80} + 3 q^{81} - 14 q^{82} - 7 q^{83} + 3 q^{84} - 26 q^{85} - 8 q^{86} + 13 q^{87} + q^{88} - 10 q^{89} - 3 q^{90} + 16 q^{92} - 9 q^{93} + 4 q^{94} - 4 q^{95} + 3 q^{96} + 7 q^{97} - 4 q^{98} + q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 + 3 * q^6 + 3 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 + q^11 + 3 * q^12 + 3 * q^14 - 3 * q^15 + 3 * q^16 + 12 * q^17 + 3 * q^18 + 4 * q^19 - 3 * q^20 + 3 * q^21 + q^22 + 16 * q^23 + 3 * q^24 + 2 * q^25 + 3 * q^27 + 3 * q^28 + 13 * q^29 - 3 * q^30 - 9 * q^31 + 3 * q^32 + q^33 + 12 * q^34 + 4 * q^35 + 3 * q^36 - 12 * q^37 + 4 * q^38 - 3 * q^40 - 14 * q^41 + 3 * q^42 - 8 * q^43 + q^44 - 3 * q^45 + 16 * q^46 + 4 * q^47 + 3 * q^48 - 4 * q^49 + 2 * q^50 + 12 * q^51 + 15 * q^53 + 3 * q^54 - 22 * q^55 + 3 * q^56 + 4 * q^57 + 13 * q^58 - 9 * q^59 - 3 * q^60 - 10 * q^61 - 9 * q^62 + 3 * q^63 + 3 * q^64 + q^66 - 6 * q^67 + 12 * q^68 + 16 * q^69 + 4 * q^70 + 6 * q^71 + 3 * q^72 + 5 * q^73 - 12 * q^74 + 2 * q^75 + 4 * q^76 - 6 * q^77 - 5 * q^79 - 3 * q^80 + 3 * q^81 - 14 * q^82 - 7 * q^83 + 3 * q^84 - 26 * q^85 - 8 * q^86 + 13 * q^87 + q^88 - 10 * q^89 - 3 * q^90 + 16 * q^92 - 9 * q^93 + 4 * q^94 - 4 * q^95 + 3 * q^96 + 7 * q^97 - 4 * q^98 + q^99

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
 1.80194 0.445042 −1.24698
1.00000 1.00000 1.00000 −4.04892 1.00000 −0.692021 1.00000 1.00000 −4.04892
1.2 1.00000 1.00000 1.00000 0.356896 1.00000 4.04892 1.00000 1.00000 0.356896
1.3 1.00000 1.00000 1.00000 0.692021 1.00000 −0.356896 1.00000 1.00000 0.692021
 $$n$$: 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 1014.2.a.o yes 3
3.b odd 2 1 3042.2.a.bd 3
4.b odd 2 1 8112.2.a.bz 3
13.b even 2 1 1014.2.a.m 3
13.c even 3 2 1014.2.e.k 6
13.d odd 4 2 1014.2.b.g 6
13.e even 6 2 1014.2.e.m 6
13.f odd 12 4 1014.2.i.g 12
39.d odd 2 1 3042.2.a.be 3
39.f even 4 2 3042.2.b.r 6
52.b odd 2 1 8112.2.a.ce 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1014.2.a.m 3 13.b even 2 1
1014.2.a.o yes 3 1.a even 1 1 trivial
1014.2.b.g 6 13.d odd 4 2
1014.2.e.k 6 13.c even 3 2
1014.2.e.m 6 13.e even 6 2
1014.2.i.g 12 13.f odd 12 4
3042.2.a.bd 3 3.b odd 2 1
3042.2.a.be 3 39.d odd 2 1
3042.2.b.r 6 39.f even 4 2
8112.2.a.bz 3 4.b odd 2 1
8112.2.a.ce 3 52.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1014))$$:

 $$T_{5}^{3} + 3T_{5}^{2} - 4T_{5} + 1$$ 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

## Hecke characteristic polynomials

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