# Properties

 Label 1078.2.e.f Level $1078$ Weight $2$ Character orbit 1078.e Analytic conductor $8.608$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1078 = 2 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1078.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.60787333789$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + 2 \zeta_{6} q^{5} -3 q^{6} + q^{8} -6 \zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + 2 \zeta_{6} q^{5} -3 q^{6} + q^{8} -6 \zeta_{6} q^{9} + ( 2 - 2 \zeta_{6} ) q^{10} + ( 1 - \zeta_{6} ) q^{11} + 3 \zeta_{6} q^{12} + 7 q^{13} + 6 q^{15} -\zeta_{6} q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} + ( -6 + 6 \zeta_{6} ) q^{18} -2 q^{20} - q^{22} + 8 \zeta_{6} q^{23} + ( 3 - 3 \zeta_{6} ) q^{24} + ( 1 - \zeta_{6} ) q^{25} -7 \zeta_{6} q^{26} -9 q^{27} -5 q^{29} -6 \zeta_{6} q^{30} + ( 4 - 4 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} -3 \zeta_{6} q^{33} -2 q^{34} + 6 q^{36} -4 \zeta_{6} q^{37} + ( 21 - 21 \zeta_{6} ) q^{39} + 2 \zeta_{6} q^{40} -4 q^{41} -8 q^{43} + \zeta_{6} q^{44} + ( 12 - 12 \zeta_{6} ) q^{45} + ( 8 - 8 \zeta_{6} ) q^{46} + 2 \zeta_{6} q^{47} -3 q^{48} - q^{50} -6 \zeta_{6} q^{51} + ( -7 + 7 \zeta_{6} ) q^{52} + ( 6 - 6 \zeta_{6} ) q^{53} + 9 \zeta_{6} q^{54} + 2 q^{55} + 5 \zeta_{6} q^{58} + ( 3 - 3 \zeta_{6} ) q^{59} + ( -6 + 6 \zeta_{6} ) q^{60} + \zeta_{6} q^{61} -4 q^{62} + q^{64} + 14 \zeta_{6} q^{65} + ( -3 + 3 \zeta_{6} ) q^{66} + ( -9 + 9 \zeta_{6} ) q^{67} + 2 \zeta_{6} q^{68} + 24 q^{69} -2 q^{71} -6 \zeta_{6} q^{72} + ( 4 - 4 \zeta_{6} ) q^{73} + ( -4 + 4 \zeta_{6} ) q^{74} -3 \zeta_{6} q^{75} -21 q^{78} -9 \zeta_{6} q^{79} + ( 2 - 2 \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + 4 \zeta_{6} q^{82} -6 q^{83} + 4 q^{85} + 8 \zeta_{6} q^{86} + ( -15 + 15 \zeta_{6} ) q^{87} + ( 1 - \zeta_{6} ) q^{88} + 6 \zeta_{6} q^{89} -12 q^{90} -8 q^{92} -12 \zeta_{6} q^{93} + ( 2 - 2 \zeta_{6} ) q^{94} + 3 \zeta_{6} q^{96} -7 q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + 3q^{3} - q^{4} + 2q^{5} - 6q^{6} + 2q^{8} - 6q^{9} + O(q^{10})$$ $$2q - q^{2} + 3q^{3} - q^{4} + 2q^{5} - 6q^{6} + 2q^{8} - 6q^{9} + 2q^{10} + q^{11} + 3q^{12} + 14q^{13} + 12q^{15} - q^{16} + 2q^{17} - 6q^{18} - 4q^{20} - 2q^{22} + 8q^{23} + 3q^{24} + q^{25} - 7q^{26} - 18q^{27} - 10q^{29} - 6q^{30} + 4q^{31} - q^{32} - 3q^{33} - 4q^{34} + 12q^{36} - 4q^{37} + 21q^{39} + 2q^{40} - 8q^{41} - 16q^{43} + q^{44} + 12q^{45} + 8q^{46} + 2q^{47} - 6q^{48} - 2q^{50} - 6q^{51} - 7q^{52} + 6q^{53} + 9q^{54} + 4q^{55} + 5q^{58} + 3q^{59} - 6q^{60} + q^{61} - 8q^{62} + 2q^{64} + 14q^{65} - 3q^{66} - 9q^{67} + 2q^{68} + 48q^{69} - 4q^{71} - 6q^{72} + 4q^{73} - 4q^{74} - 3q^{75} - 42q^{78} - 9q^{79} + 2q^{80} - 9q^{81} + 4q^{82} - 12q^{83} + 8q^{85} + 8q^{86} - 15q^{87} + q^{88} + 6q^{89} - 24q^{90} - 16q^{92} - 12q^{93} + 2q^{94} + 3q^{96} - 14q^{97} - 12q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$981$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 0.866025i 1.50000 2.59808i −0.500000 + 0.866025i 1.00000 + 1.73205i −3.00000 0 1.00000 −3.00000 5.19615i 1.00000 1.73205i
177.1 −0.500000 + 0.866025i 1.50000 + 2.59808i −0.500000 0.866025i 1.00000 1.73205i −3.00000 0 1.00000 −3.00000 + 5.19615i 1.00000 + 1.73205i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.e.f 2
7.b odd 2 1 154.2.e.a 2
7.c even 3 1 1078.2.a.g 1
7.c even 3 1 inner 1078.2.e.f 2
7.d odd 6 1 154.2.e.a 2
7.d odd 6 1 1078.2.a.m 1
21.c even 2 1 1386.2.k.o 2
21.g even 6 1 1386.2.k.o 2
21.g even 6 1 9702.2.a.i 1
21.h odd 6 1 9702.2.a.y 1
28.d even 2 1 1232.2.q.e 2
28.f even 6 1 1232.2.q.e 2
28.f even 6 1 8624.2.a.b 1
28.g odd 6 1 8624.2.a.be 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.a 2 7.b odd 2 1
154.2.e.a 2 7.d odd 6 1
1078.2.a.g 1 7.c even 3 1
1078.2.a.m 1 7.d odd 6 1
1078.2.e.f 2 1.a even 1 1 trivial
1078.2.e.f 2 7.c even 3 1 inner
1232.2.q.e 2 28.d even 2 1
1232.2.q.e 2 28.f even 6 1
1386.2.k.o 2 21.c even 2 1
1386.2.k.o 2 21.g even 6 1
8624.2.a.b 1 28.f even 6 1
8624.2.a.be 1 28.g odd 6 1
9702.2.a.i 1 21.g even 6 1
9702.2.a.y 1 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1078, [\chi])$$:

 $$T_{3}^{2} - 3 T_{3} + 9$$ $$T_{5}^{2} - 2 T_{5} + 4$$ $$T_{13} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$9 - 3 T + T^{2}$$
$5$ $$4 - 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$1 - T + T^{2}$$
$13$ $$( -7 + T )^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$64 - 8 T + T^{2}$$
$29$ $$( 5 + T )^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$16 + 4 T + T^{2}$$
$41$ $$( 4 + T )^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$4 - 2 T + T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$9 - 3 T + T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$81 + 9 T + T^{2}$$
$71$ $$( 2 + T )^{2}$$
$73$ $$16 - 4 T + T^{2}$$
$79$ $$81 + 9 T + T^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$( 7 + T )^{2}$$