# Properties

 Label 1014.2.e.n Level $1014$ Weight $2$ Character orbit 1014.e Analytic conductor $8.097$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.64827.1 Defining polynomial: $$x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 3 \nu^{4} - 9 \nu^{3} + 5 \nu^{2} - 2 \nu + 6$$$$)/13$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{5} + 9 \nu^{4} - 14 \nu^{3} + 15 \nu^{2} - 6 \nu + 18$$$$)/13$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{5} - \nu^{4} - 10 \nu^{3} - 6 \nu^{2} - 34 \nu - 2$$$$)/13$$ $$\beta_{5}$$ $$=$$ $$($$$$-6 \nu^{5} + 5 \nu^{4} - 15 \nu^{3} - 9 \nu^{2} - 25 \nu + 10$$$$)/13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} - 3 \beta_{4} - 4 \beta_{1} - 2$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 9 \beta_{2} - 9 \beta_{1}$$

## Character values

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

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{5}$$

## 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
 0.900969 + 1.56052i 0.222521 + 0.385418i −0.623490 − 1.07992i 0.900969 − 1.56052i 0.222521 − 0.385418i −0.623490 + 1.07992i
0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i −3.15883 −0.500000 0.866025i 2.34601 + 4.06341i −1.00000 −0.500000 0.866025i −1.57942 + 2.73563i
529.2 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i −2.13706 −0.500000 0.866025i −0.0244587 0.0423637i −1.00000 −0.500000 0.866025i −1.06853 + 1.85075i
529.3 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i 4.29590 −0.500000 0.866025i 2.17845 + 3.77318i −1.00000 −0.500000 0.866025i 2.14795 3.72036i
991.1 0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −3.15883 −0.500000 + 0.866025i 2.34601 4.06341i −1.00000 −0.500000 + 0.866025i −1.57942 2.73563i
991.2 0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −2.13706 −0.500000 + 0.866025i −0.0244587 + 0.0423637i −1.00000 −0.500000 + 0.866025i −1.06853 1.85075i
991.3 0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 4.29590 −0.500000 + 0.866025i 2.17845 3.77318i −1.00000 −0.500000 + 0.866025i 2.14795 + 3.72036i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 991.3 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.n 6
13.b even 2 1 1014.2.e.l 6
13.c even 3 1 1014.2.a.l 3
13.c even 3 1 inner 1014.2.e.n 6
13.d odd 4 2 1014.2.i.h 12
13.e even 6 1 1014.2.a.n yes 3
13.e even 6 1 1014.2.e.l 6
13.f odd 12 2 1014.2.b.f 6
13.f odd 12 2 1014.2.i.h 12
39.h odd 6 1 3042.2.a.ba 3
39.i odd 6 1 3042.2.a.bh 3
39.k even 12 2 3042.2.b.o 6
52.i odd 6 1 8112.2.a.cm 3
52.j odd 6 1 8112.2.a.cj 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1014.2.a.l 3 13.c even 3 1
1014.2.a.n yes 3 13.e even 6 1
1014.2.b.f 6 13.f odd 12 2
1014.2.e.l 6 13.b even 2 1
1014.2.e.l 6 13.e even 6 1
1014.2.e.n 6 1.a even 1 1 trivial
1014.2.e.n 6 13.c even 3 1 inner
1014.2.i.h 12 13.d odd 4 2
1014.2.i.h 12 13.f odd 12 2
3042.2.a.ba 3 39.h odd 6 1
3042.2.a.bh 3 39.i odd 6 1
3042.2.b.o 6 39.k even 12 2
8112.2.a.cj 3 52.j odd 6 1
8112.2.a.cm 3 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 $$S_{2}^{\mathrm{new}}(1014, [\chi])$$:

 $$T_{5}^{3} + T_{5}^{2} - 16 T_{5} - 29$$ $$T_{7}^{6} - 9 T_{7}^{5} + 61 T_{7}^{4} - 182 T_{7}^{3} + 409 T_{7}^{2} + 20 T_{7} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{3}$$
$3$ $$( 1 - T + T^{2} )^{3}$$
$5$ $$( -29 - 16 T + T^{2} + T^{3} )^{2}$$
$7$ $$1 + 20 T + 409 T^{2} - 182 T^{3} + 61 T^{4} - 9 T^{5} + T^{6}$$
$11$ $$1 + 8 T + 59 T^{2} + 42 T^{3} + 33 T^{4} - 5 T^{5} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$64 + 96 T + 208 T^{2} - 112 T^{3} + 52 T^{4} - 8 T^{5} + T^{6}$$
$19$ $$4096 + 2048 T + 1280 T^{2} + 48 T^{4} + 4 T^{5} + T^{6}$$
$23$ $$3136 + 1568 T + 784 T^{2} + 112 T^{3} + 28 T^{4} + T^{6}$$
$29$ $$841 - 696 T + 895 T^{2} + 322 T^{3} + 97 T^{4} + 11 T^{5} + T^{6}$$
$31$ $$( -125 - 50 T + 5 T^{2} + T^{3} )^{2}$$
$37$ $$64 + 352 T + 2000 T^{2} - 336 T^{3} + 108 T^{4} + 8 T^{5} + T^{6}$$
$41$ $$53824 - 14848 T + 4560 T^{2} - 336 T^{3} + 68 T^{4} - 2 T^{5} + T^{6}$$
$43$ $$10816 - 2080 T + 1648 T^{2} + 448 T^{3} + 124 T^{4} + 12 T^{5} + T^{6}$$
$47$ $$( -64 - 32 T + 4 T^{2} + T^{3} )^{2}$$
$53$ $$( -43 - 36 T - 5 T^{2} + T^{3} )^{2}$$
$59$ $$27889 + 6012 T + 2131 T^{2} + 154 T^{3} + 61 T^{4} + 5 T^{5} + T^{6}$$
$61$ $$107584 + 49856 T + 15888 T^{2} + 2688 T^{3} + 332 T^{4} + 22 T^{5} + T^{6}$$
$67$ $$10816 - 1664 T + 880 T^{2} - 112 T^{3} + 52 T^{4} - 6 T^{5} + T^{6}$$
$71$ $$64 - 640 T + 6544 T^{2} + 1456 T^{3} + 244 T^{4} + 18 T^{5} + T^{6}$$
$73$ $$( 13 - 30 T - 13 T^{2} + T^{3} )^{2}$$
$79$ $$( -533 + 276 T - 31 T^{2} + T^{3} )^{2}$$
$83$ $$( -1567 - 128 T + 13 T^{2} + T^{3} )^{2}$$
$89$ $$3136 + 3136 T + 3920 T^{2} - 672 T^{3} + 252 T^{4} + 14 T^{5} + T^{6}$$
$97$ $$9409 + 8730 T + 5869 T^{2} + 1876 T^{3} + 439 T^{4} + 23 T^{5} + T^{6}$$