# Properties

 Label 1014.2.e.h 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: $$x^{4} - x^{2} + 1$$ 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.

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

## Character values

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

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

## 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 −1.73205 0.500000 + 0.866025i 0.633975 + 1.09808i 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 2.36603 + 4.09808i 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 0.633975 1.09808i 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 2.36603 4.09808i 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.h 4
13.b even 2 1 1014.2.e.j 4
13.c even 3 1 1014.2.a.j 2
13.c even 3 1 inner 1014.2.e.h 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.h 2
13.e even 6 1 1014.2.e.j 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.v 2
39.i odd 6 1 3042.2.a.s 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.bq 2
52.j odd 6 1 8112.2.a.bx 2
52.l even 12 1 624.2.bv.d 4
65.f even 4 1 1950.2.y.h 4
65.g odd 4 1 1950.2.bc.c 4
65.k even 4 1 1950.2.y.a 4
65.o even 12 1 1950.2.y.a 4
65.s odd 12 1 1950.2.bc.c 4
65.t even 12 1 1950.2.y.h 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.e even 6 1
1014.2.a.j 2 13.c even 3 1
1014.2.b.d 4 13.f odd 12 2
1014.2.e.h 4 1.a even 1 1 trivial
1014.2.e.h 4 13.c even 3 1 inner
1014.2.e.j 4 13.b even 2 1
1014.2.e.j 4 13.e even 6 1
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.k even 4 1
1950.2.y.a 4 65.o even 12 1
1950.2.y.h 4 65.f even 4 1
1950.2.y.h 4 65.t 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.i odd 6 1
3042.2.a.v 2 39.h odd 6 1
3042.2.b.l 4 39.k even 12 2
8112.2.a.bq 2 52.i odd 6 1
8112.2.a.bx 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 $$S_{2}^{\mathrm{new}}(1014, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ $$( -3 + T^{2} )^{2}$$
$7$ $$36 - 36 T + 30 T^{2} - 6 T^{3} + T^{4}$$
$11$ $$36 - 36 T + 30 T^{2} - 6 T^{3} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$729 + 27 T^{2} + T^{4}$$
$19$ $$36 - 36 T + 30 T^{2} - 6 T^{3} + T^{4}$$
$23$ $$324 + 108 T + 54 T^{2} - 6 T^{3} + T^{4}$$
$29$ $$( 9 - 3 T + T^{2} )^{2}$$
$31$ $$( 24 + 12 T + T^{2} )^{2}$$
$37$ $$( 9 - 3 T + T^{2} )^{2}$$
$41$ $$9 + 18 T + 39 T^{2} - 6 T^{3} + T^{4}$$
$43$ $$676 - 52 T + 30 T^{2} + 2 T^{3} + T^{4}$$
$47$ $$( 6 - 6 T + T^{2} )^{2}$$
$53$ $$( -3 + T )^{4}$$
$59$ $$36864 + 192 T^{2} + T^{4}$$
$61$ $$5329 + 1460 T + 327 T^{2} + 20 T^{3} + T^{4}$$
$67$ $$6084 - 1404 T + 246 T^{2} - 18 T^{3} + T^{4}$$
$71$ $$324 + 108 T + 54 T^{2} - 6 T^{3} + T^{4}$$
$73$ $$( -147 + T^{2} )^{2}$$
$79$ $$( -104 + 4 T + T^{2} )^{2}$$
$83$ $$( -66 - 6 T + T^{2} )^{2}$$
$89$ $$576 - 288 T + 120 T^{2} - 12 T^{3} + T^{4}$$
$97$ $$( 36 - 6 T + T^{2} )^{2}$$