# Properties

 Label 1014.2.e.j Level $1014$ Weight $2$ Character orbit 1014.e Analytic conductor $8.097$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.09683076496$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ 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: $$3$$ Twist minimal: no (minimal twist has level 78) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_1 q^{3} + (\beta_1 - 1) q^{4} - \beta_{3} q^{5} + (\beta_1 - 1) q^{6} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{7} - q^{8} + (\beta_1 - 1) q^{9}+O(q^{10})$$ q + b1 * q^2 + b1 * q^3 + (b1 - 1) * q^4 - b3 * q^5 + (b1 - 1) * q^6 + (-b3 + b2 + 3*b1 - 3) * q^7 - q^8 + (b1 - 1) * q^9 $$q + \beta_1 q^{2} + \beta_1 q^{3} + (\beta_1 - 1) q^{4} - \beta_{3} q^{5} + (\beta_1 - 1) q^{6} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{7} - q^{8} + (\beta_1 - 1) q^{9} - \beta_{2} q^{10} + ( - \beta_{2} - 3 \beta_1) q^{11} - q^{12} + ( - \beta_{3} - 3) q^{14} - \beta_{2} q^{15} - \beta_1 q^{16} + (3 \beta_{3} - 3 \beta_{2}) q^{17} - q^{18} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{19} + (\beta_{3} - \beta_{2}) q^{20} + ( - \beta_{3} - 3) q^{21} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{22} + ( - 3 \beta_{2} + 3 \beta_1) q^{23} - \beta_1 q^{24} - 2 q^{25} - q^{27} + ( - \beta_{2} - 3 \beta_1) q^{28} + 3 \beta_1 q^{29} + (\beta_{3} - \beta_{2}) q^{30} + ( - 2 \beta_{3} + 6) q^{31} + ( - \beta_1 + 1) q^{32} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{33} + 3 \beta_{3} q^{34} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{35} - \beta_1 q^{36} - 3 \beta_1 q^{37} + (\beta_{3} - 3) q^{38} + \beta_{3} q^{40} + (2 \beta_{2} - 3 \beta_1) q^{41} + ( - \beta_{2} - 3 \beta_1) q^{42} + ( - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{43} + (\beta_{3} + 3) q^{44} + (\beta_{3} - \beta_{2}) q^{45} + (3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{46} + (\beta_{3} - 3) q^{47} + ( - \beta_1 + 1) q^{48} + ( - 6 \beta_{2} - 5 \beta_1) q^{49} - 2 \beta_1 q^{50} + 3 \beta_{3} q^{51} + 3 q^{53} - \beta_1 q^{54} + (3 \beta_{2} + 3 \beta_1) q^{55} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{56} + (\beta_{3} - 3) q^{57} + (3 \beta_1 - 3) q^{58} + ( - 8 \beta_{3} + 8 \beta_{2}) q^{59} + \beta_{3} q^{60} + (3 \beta_{3} - 3 \beta_{2} + 10 \beta_1 - 10) q^{61} + ( - 2 \beta_{2} + 6 \beta_1) q^{62} + ( - \beta_{2} - 3 \beta_1) q^{63} + q^{64} + (\beta_{3} + 3) q^{66} + ( - \beta_{2} - 9 \beta_1) q^{67} + 3 \beta_{2} q^{68} + (3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{69} + (3 \beta_{3} + 3) q^{70} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 3) q^{71} + ( - \beta_1 + 1) q^{72} - 7 \beta_{3} q^{73} + ( - 3 \beta_1 + 3) q^{74} - 2 \beta_1 q^{75} + (\beta_{2} - 3 \beta_1) q^{76} + (6 \beta_{3} + 12) q^{77} + ( - 6 \beta_{3} - 2) q^{79} + \beta_{2} q^{80} - \beta_1 q^{81} + ( - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{82} + ( - 5 \beta_{3} - 3) q^{83} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{84} + (9 \beta_1 - 9) q^{85} + ( - 3 \beta_{3} - 1) q^{86} + (3 \beta_1 - 3) q^{87} + (\beta_{2} + 3 \beta_1) q^{88} + (2 \beta_{2} - 6 \beta_1) q^{89} + \beta_{3} q^{90} + (3 \beta_{3} - 3) q^{92} + ( - 2 \beta_{2} + 6 \beta_1) q^{93} + (\beta_{2} - 3 \beta_1) q^{94} + (3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{95} + q^{96} + (6 \beta_1 - 6) q^{97} + (6 \beta_{3} - 6 \beta_{2} - 5 \beta_1 + 5) q^{98} + (\beta_{3} + 3) q^{99}+O(q^{100})$$ q + b1 * q^2 + b1 * q^3 + (b1 - 1) * q^4 - b3 * q^5 + (b1 - 1) * q^6 + (-b3 + b2 + 3*b1 - 3) * q^7 - q^8 + (b1 - 1) * q^9 - b2 * q^10 + (-b2 - 3*b1) * q^11 - q^12 + (-b3 - 3) * q^14 - b2 * q^15 - b1 * q^16 + (3*b3 - 3*b2) * q^17 - q^18 + (b3 - b2 + 3*b1 - 3) * q^19 + (b3 - b2) * q^20 + (-b3 - 3) * q^21 + (b3 - b2 - 3*b1 + 3) * q^22 + (-3*b2 + 3*b1) * q^23 - b1 * q^24 - 2 * q^25 - q^27 + (-b2 - 3*b1) * q^28 + 3*b1 * q^29 + (b3 - b2) * q^30 + (-2*b3 + 6) * q^31 + (-b1 + 1) * q^32 + (b3 - b2 - 3*b1 + 3) * q^33 + 3*b3 * q^34 + (3*b3 - 3*b2 - 3*b1 + 3) * q^35 - b1 * q^36 - 3*b1 * q^37 + (b3 - 3) * q^38 + b3 * q^40 + (2*b2 - 3*b1) * q^41 + (-b2 - 3*b1) * q^42 + (-3*b3 + 3*b2 + b1 - 1) * q^43 + (b3 + 3) * q^44 + (b3 - b2) * q^45 + (3*b3 - 3*b2 + 3*b1 - 3) * q^46 + (b3 - 3) * q^47 + (-b1 + 1) * q^48 + (-6*b2 - 5*b1) * q^49 - 2*b1 * q^50 + 3*b3 * q^51 + 3 * q^53 - b1 * q^54 + (3*b2 + 3*b1) * q^55 + (b3 - b2 - 3*b1 + 3) * q^56 + (b3 - 3) * q^57 + (3*b1 - 3) * q^58 + (-8*b3 + 8*b2) * q^59 + b3 * q^60 + (3*b3 - 3*b2 + 10*b1 - 10) * q^61 + (-2*b2 + 6*b1) * q^62 + (-b2 - 3*b1) * q^63 + q^64 + (b3 + 3) * q^66 + (-b2 - 9*b1) * q^67 + 3*b2 * q^68 + (3*b3 - 3*b2 + 3*b1 - 3) * q^69 + (3*b3 + 3) * q^70 + (-3*b3 + 3*b2 + 3*b1 - 3) * q^71 + (-b1 + 1) * q^72 - 7*b3 * q^73 + (-3*b1 + 3) * q^74 - 2*b1 * q^75 + (b2 - 3*b1) * q^76 + (6*b3 + 12) * q^77 + (-6*b3 - 2) * q^79 + b2 * q^80 - b1 * q^81 + (-2*b3 + 2*b2 - 3*b1 + 3) * q^82 + (-5*b3 - 3) * q^83 + (b3 - b2 - 3*b1 + 3) * q^84 + (9*b1 - 9) * q^85 + (-3*b3 - 1) * q^86 + (3*b1 - 3) * q^87 + (b2 + 3*b1) * q^88 + (2*b2 - 6*b1) * q^89 + b3 * q^90 + (3*b3 - 3) * q^92 + (-2*b2 + 6*b1) * q^93 + (b2 - 3*b1) * q^94 + (3*b3 - 3*b2 + 3*b1 - 3) * q^95 + q^96 + (6*b1 - 6) * q^97 + (6*b3 - 6*b2 - 5*b1 + 5) * q^98 + (b3 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{6} - 6 q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 - 2 * q^6 - 6 * q^7 - 4 * q^8 - 2 * q^9 $$4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{6} - 6 q^{7} - 4 q^{8} - 2 q^{9} - 6 q^{11} - 4 q^{12} - 12 q^{14} - 2 q^{16} - 4 q^{18} - 6 q^{19} - 12 q^{21} + 6 q^{22} + 6 q^{23} - 2 q^{24} - 8 q^{25} - 4 q^{27} - 6 q^{28} + 6 q^{29} + 24 q^{31} + 2 q^{32} + 6 q^{33} + 6 q^{35} - 2 q^{36} - 6 q^{37} - 12 q^{38} - 6 q^{41} - 6 q^{42} - 2 q^{43} + 12 q^{44} - 6 q^{46} - 12 q^{47} + 2 q^{48} - 10 q^{49} - 4 q^{50} + 12 q^{53} - 2 q^{54} + 6 q^{55} + 6 q^{56} - 12 q^{57} - 6 q^{58} - 20 q^{61} + 12 q^{62} - 6 q^{63} + 4 q^{64} + 12 q^{66} - 18 q^{67} - 6 q^{69} + 12 q^{70} - 6 q^{71} + 2 q^{72} + 6 q^{74} - 4 q^{75} - 6 q^{76} + 48 q^{77} - 8 q^{79} - 2 q^{81} + 6 q^{82} - 12 q^{83} + 6 q^{84} - 18 q^{85} - 4 q^{86} - 6 q^{87} + 6 q^{88} - 12 q^{89} - 12 q^{92} + 12 q^{93} - 6 q^{94} - 6 q^{95} + 4 q^{96} - 12 q^{97} + 10 q^{98} + 12 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 - 2 * q^6 - 6 * q^7 - 4 * q^8 - 2 * q^9 - 6 * q^11 - 4 * q^12 - 12 * q^14 - 2 * q^16 - 4 * q^18 - 6 * q^19 - 12 * q^21 + 6 * q^22 + 6 * q^23 - 2 * q^24 - 8 * q^25 - 4 * q^27 - 6 * q^28 + 6 * q^29 + 24 * q^31 + 2 * q^32 + 6 * q^33 + 6 * q^35 - 2 * q^36 - 6 * q^37 - 12 * q^38 - 6 * q^41 - 6 * q^42 - 2 * q^43 + 12 * q^44 - 6 * q^46 - 12 * q^47 + 2 * q^48 - 10 * q^49 - 4 * q^50 + 12 * q^53 - 2 * q^54 + 6 * q^55 + 6 * q^56 - 12 * q^57 - 6 * q^58 - 20 * q^61 + 12 * q^62 - 6 * q^63 + 4 * q^64 + 12 * q^66 - 18 * q^67 - 6 * q^69 + 12 * q^70 - 6 * q^71 + 2 * q^72 + 6 * q^74 - 4 * q^75 - 6 * q^76 + 48 * q^77 - 8 * q^79 - 2 * q^81 + 6 * q^82 - 12 * q^83 + 6 * q^84 - 18 * q^85 - 4 * q^86 - 6 * q^87 + 6 * q^88 - 12 * q^89 - 12 * q^92 + 12 * q^93 - 6 * q^94 - 6 * q^95 + 4 * q^96 - 12 * q^97 + 10 * q^98 + 12 * q^99

Basis of coefficient ring

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
529.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i −1.73205 −0.500000 0.866025i −2.36603 4.09808i −1.00000 −0.500000 0.866025i −0.866025 + 1.50000i
529.2 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i 1.73205 −0.500000 0.866025i −0.633975 1.09808i −1.00000 −0.500000 0.866025i 0.866025 1.50000i
991.1 0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −1.73205 −0.500000 + 0.866025i −2.36603 + 4.09808i −1.00000 −0.500000 + 0.866025i −0.866025 1.50000i
991.2 0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.73205 −0.500000 + 0.866025i −0.633975 + 1.09808i −1.00000 −0.500000 + 0.866025i 0.866025 + 1.50000i
 $$n$$: 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 1014.2.e.j 4
13.b even 2 1 1014.2.e.h 4
13.c even 3 1 1014.2.a.h 2
13.c even 3 1 inner 1014.2.e.j 4
13.d odd 4 1 78.2.i.b 4
13.d odd 4 1 1014.2.i.f 4
13.e even 6 1 1014.2.a.j 2
13.e even 6 1 1014.2.e.h 4
13.f odd 12 1 78.2.i.b 4
13.f odd 12 2 1014.2.b.d 4
13.f odd 12 1 1014.2.i.f 4
39.f even 4 1 234.2.l.a 4
39.h odd 6 1 3042.2.a.s 2
39.i odd 6 1 3042.2.a.v 2
39.k even 12 1 234.2.l.a 4
39.k even 12 2 3042.2.b.l 4
52.f even 4 1 624.2.bv.d 4
52.i odd 6 1 8112.2.a.bx 2
52.j odd 6 1 8112.2.a.bq 2
52.l even 12 1 624.2.bv.d 4
65.f even 4 1 1950.2.y.a 4
65.g odd 4 1 1950.2.bc.c 4
65.k even 4 1 1950.2.y.h 4
65.o even 12 1 1950.2.y.h 4
65.s odd 12 1 1950.2.bc.c 4
65.t even 12 1 1950.2.y.a 4
156.l odd 4 1 1872.2.by.k 4
156.v odd 12 1 1872.2.by.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.b 4 13.d odd 4 1
78.2.i.b 4 13.f odd 12 1
234.2.l.a 4 39.f even 4 1
234.2.l.a 4 39.k even 12 1
624.2.bv.d 4 52.f even 4 1
624.2.bv.d 4 52.l even 12 1
1014.2.a.h 2 13.c even 3 1
1014.2.a.j 2 13.e even 6 1
1014.2.b.d 4 13.f odd 12 2
1014.2.e.h 4 13.b even 2 1
1014.2.e.h 4 13.e even 6 1
1014.2.e.j 4 1.a even 1 1 trivial
1014.2.e.j 4 13.c even 3 1 inner
1014.2.i.f 4 13.d odd 4 1
1014.2.i.f 4 13.f odd 12 1
1872.2.by.k 4 156.l odd 4 1
1872.2.by.k 4 156.v odd 12 1
1950.2.y.a 4 65.f even 4 1
1950.2.y.a 4 65.t even 12 1
1950.2.y.h 4 65.k even 4 1
1950.2.y.h 4 65.o even 12 1
1950.2.bc.c 4 65.g odd 4 1
1950.2.bc.c 4 65.s odd 12 1
3042.2.a.s 2 39.h odd 6 1
3042.2.a.v 2 39.i odd 6 1
3042.2.b.l 4 39.k even 12 2
8112.2.a.bq 2 52.j odd 6 1
8112.2.a.bx 2 52.i odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 3$$ T5^2 - 3 $$T_{7}^{4} + 6T_{7}^{3} + 30T_{7}^{2} + 36T_{7} + 36$$ T7^4 + 6*T7^3 + 30*T7^2 + 36*T7 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T^{2} - 3)^{2}$$
$7$ $$T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36$$
$11$ $$T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 27T^{2} + 729$$
$19$ $$T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36$$
$23$ $$T^{4} - 6 T^{3} + 54 T^{2} + 108 T + 324$$
$29$ $$(T^{2} - 3 T + 9)^{2}$$
$31$ $$(T^{2} - 12 T + 24)^{2}$$
$37$ $$(T^{2} + 3 T + 9)^{2}$$
$41$ $$T^{4} + 6 T^{3} + 39 T^{2} - 18 T + 9$$
$43$ $$T^{4} + 2 T^{3} + 30 T^{2} - 52 T + 676$$
$47$ $$(T^{2} + 6 T + 6)^{2}$$
$53$ $$(T - 3)^{4}$$
$59$ $$T^{4} + 192 T^{2} + 36864$$
$61$ $$T^{4} + 20 T^{3} + 327 T^{2} + \cdots + 5329$$
$67$ $$T^{4} + 18 T^{3} + 246 T^{2} + \cdots + 6084$$
$71$ $$T^{4} + 6 T^{3} + 54 T^{2} - 108 T + 324$$
$73$ $$(T^{2} - 147)^{2}$$
$79$ $$(T^{2} + 4 T - 104)^{2}$$
$83$ $$(T^{2} + 6 T - 66)^{2}$$
$89$ $$T^{4} + 12 T^{3} + 120 T^{2} + \cdots + 576$$
$97$ $$(T^{2} + 6 T + 36)^{2}$$