# Properties

 Label 1014.2.e.g Level $1014$ Weight $2$ Character orbit 1014.e Analytic conductor $8.097$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-\zeta_{12}$$

## 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 0.267949 −0.500000 0.866025i −0.366025 0.633975i 1.00000 −0.500000 0.866025i −0.133975 + 0.232051i
529.2 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 3.73205 −0.500000 0.866025i 1.36603 + 2.36603i 1.00000 −0.500000 0.866025i −1.86603 + 3.23205i
991.1 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0.267949 −0.500000 + 0.866025i −0.366025 + 0.633975i 1.00000 −0.500000 + 0.866025i −0.133975 0.232051i
991.2 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 3.73205 −0.500000 + 0.866025i 1.36603 2.36603i 1.00000 −0.500000 + 0.866025i −1.86603 3.23205i
 $$n$$: 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.g 4
13.b even 2 1 1014.2.e.i 4
13.c even 3 1 1014.2.a.k 2
13.c even 3 1 inner 1014.2.e.g 4
13.d odd 4 1 78.2.i.a 4
13.d odd 4 1 1014.2.i.a 4
13.e even 6 1 1014.2.a.i 2
13.e even 6 1 1014.2.e.i 4
13.f odd 12 1 78.2.i.a 4
13.f odd 12 2 1014.2.b.e 4
13.f odd 12 1 1014.2.i.a 4
39.f even 4 1 234.2.l.c 4
39.h odd 6 1 3042.2.a.y 2
39.i odd 6 1 3042.2.a.p 2
39.k even 12 1 234.2.l.c 4
39.k even 12 2 3042.2.b.i 4
52.f even 4 1 624.2.bv.e 4
52.i odd 6 1 8112.2.a.bj 2
52.j odd 6 1 8112.2.a.bp 2
52.l even 12 1 624.2.bv.e 4
65.f even 4 1 1950.2.y.g 4
65.g odd 4 1 1950.2.bc.d 4
65.k even 4 1 1950.2.y.b 4
65.o even 12 1 1950.2.y.b 4
65.s odd 12 1 1950.2.bc.d 4
65.t even 12 1 1950.2.y.g 4
156.l odd 4 1 1872.2.by.h 4
156.v odd 12 1 1872.2.by.h 4

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$( 1 - 4 T + T^{2} )^{2}$$
$7$ $$4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4}$$
$11$ $$36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$169 - 104 T + 51 T^{2} - 8 T^{3} + T^{4}$$
$19$ $$36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$676 + 52 T + 30 T^{2} - 2 T^{3} + T^{4}$$
$29$ $$121 + 22 T + 15 T^{2} - 2 T^{3} + T^{4}$$
$31$ $$( -8 + 4 T + T^{2} )^{2}$$
$37$ $$1369 + 518 T + 159 T^{2} + 14 T^{3} + T^{4}$$
$41$ $$11449 - 214 T + 111 T^{2} + 2 T^{3} + T^{4}$$
$43$ $$5476 + 148 T + 78 T^{2} - 2 T^{3} + T^{4}$$
$47$ $$( -18 + 6 T + T^{2} )^{2}$$
$53$ $$( -3 + 6 T + T^{2} )^{2}$$
$59$ $$( 64 + 8 T + T^{2} )^{2}$$
$61$ $$121 + 88 T + 75 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$21316 - 292 T + 150 T^{2} + 2 T^{3} + T^{4}$$
$71$ $$36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4}$$
$73$ $$( 61 + 16 T + T^{2} )^{2}$$
$79$ $$( 24 + 12 T + T^{2} )^{2}$$
$83$ $$( -2 - 10 T + T^{2} )^{2}$$
$89$ $$576 - 288 T + 120 T^{2} - 12 T^{3} + T^{4}$$
$97$ $$( 36 - 6 T + T^{2} )^{2}$$
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