# Properties

 Label 1014.2.e.k Level $1014$ Weight $2$ Character orbit 1014.e Analytic conductor $8.097$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

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## 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

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

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 $$\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$$.

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

## 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 −4.04892 −0.500000 0.866025i 0.346011 + 0.599308i 1.00000 −0.500000 0.866025i 2.02446 3.50647i
529.2 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0.356896 −0.500000 0.866025i −2.02446 3.50647i 1.00000 −0.500000 0.866025i −0.178448 + 0.309081i
529.3 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0.692021 −0.500000 0.866025i 0.178448 + 0.309081i 1.00000 −0.500000 0.866025i −0.346011 + 0.599308i
991.1 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −4.04892 −0.500000 + 0.866025i 0.346011 0.599308i 1.00000 −0.500000 + 0.866025i 2.02446 + 3.50647i
991.2 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0.356896 −0.500000 + 0.866025i −2.02446 + 3.50647i 1.00000 −0.500000 + 0.866025i −0.178448 0.309081i
991.3 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0.692021 −0.500000 + 0.866025i 0.178448 0.309081i 1.00000 −0.500000 + 0.866025i −0.346011 0.599308i
 $$n$$: 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.k 6
13.b even 2 1 1014.2.e.m 6
13.c even 3 1 1014.2.a.o yes 3
13.c even 3 1 inner 1014.2.e.k 6
13.d odd 4 2 1014.2.i.g 12
13.e even 6 1 1014.2.a.m 3
13.e even 6 1 1014.2.e.m 6
13.f odd 12 2 1014.2.b.g 6
13.f odd 12 2 1014.2.i.g 12
39.h odd 6 1 3042.2.a.be 3
39.i odd 6 1 3042.2.a.bd 3
39.k even 12 2 3042.2.b.r 6
52.i odd 6 1 8112.2.a.ce 3
52.j odd 6 1 8112.2.a.bz 3

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

 $$T_{5}^{3} + 3 T_{5}^{2} - 4 T_{5} + 1$$ $$T_{7}^{6} + 3 T_{7}^{5} + 13 T_{7}^{4} - 14 T_{7}^{3} + 13 T_{7}^{2} - 4 T_{7} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{3}$$
$3$ $$( 1 + T + T^{2} )^{3}$$
$5$ $$( 1 - 4 T + 3 T^{2} + T^{3} )^{2}$$
$7$ $$1 - 4 T + 13 T^{2} - 14 T^{3} + 13 T^{4} + 3 T^{5} + T^{6}$$
$11$ $$169 - 208 T + 243 T^{2} - 42 T^{3} + 17 T^{4} + T^{5} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$10816 - 2080 T + 1648 T^{2} + 448 T^{3} + 124 T^{4} + 12 T^{5} + T^{6}$$
$19$ $$4096 + 2048 T + 1280 T^{2} + 48 T^{4} + 4 T^{5} + T^{6}$$
$23$ $$10816 + 7904 T + 4112 T^{2} + 1008 T^{3} + 180 T^{4} + 16 T^{5} + T^{6}$$
$29$ $$49729 - 2676 T + 3043 T^{2} + 602 T^{3} + 157 T^{4} + 13 T^{5} + T^{6}$$
$31$ $$( -29 - 22 T + 9 T^{2} + T^{3} )^{2}$$
$37$ $$64 - 160 T + 304 T^{2} - 224 T^{3} + 124 T^{4} - 12 T^{5} + T^{6}$$
$41$ $$3136 - 3136 T + 2352 T^{2} - 672 T^{3} + 140 T^{4} - 14 T^{5} + T^{6}$$
$43$ $$118336 - 15136 T + 4688 T^{2} - 336 T^{3} + 108 T^{4} - 8 T^{5} + T^{6}$$
$47$ $$( 64 - 32 T - 4 T^{2} + T^{3} )^{2}$$
$53$ $$( 1247 - 72 T - 15 T^{2} + T^{3} )^{2}$$
$59$ $$169 - 260 T + 283 T^{2} - 154 T^{3} + 61 T^{4} - 9 T^{5} + T^{6}$$
$61$ $$64 - 192 T + 496 T^{2} - 224 T^{3} + 76 T^{4} - 10 T^{5} + T^{6}$$
$67$ $$1236544 - 204608 T + 40528 T^{2} - 1120 T^{3} + 220 T^{4} - 6 T^{5} + T^{6}$$
$71$ $$10816 + 7488 T + 5808 T^{2} - 224 T^{3} + 108 T^{4} + 6 T^{5} + T^{6}$$
$73$ $$( 13 - 22 T - 5 T^{2} + T^{3} )^{2}$$
$79$ $$( -1469 - 204 T + 5 T^{2} + T^{3} )^{2}$$
$83$ $$( -1477 - 224 T + 7 T^{2} + T^{3} )^{2}$$
$89$ $$64 - 192 T + 496 T^{2} - 224 T^{3} + 76 T^{4} - 10 T^{5} + T^{6}$$
$97$ $$49 + 98 T + 245 T^{2} - 84 T^{3} + 63 T^{4} + 7 T^{5} + T^{6}$$
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