# Properties

 Label 1078.2.e.g 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -4 \zeta_{6} q^{5} -3 q^{6} - q^{8} -6 \zeta_{6} q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -4 \zeta_{6} q^{5} -3 q^{6} - q^{8} -6 \zeta_{6} q^{9} + ( 4 - 4 \zeta_{6} ) q^{10} + ( 1 - \zeta_{6} ) q^{11} -3 \zeta_{6} q^{12} + q^{13} + 12 q^{15} -\zeta_{6} q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} + ( 6 - 6 \zeta_{6} ) q^{18} + 6 \zeta_{6} q^{19} + 4 q^{20} + q^{22} + 2 \zeta_{6} q^{23} + ( 3 - 3 \zeta_{6} ) q^{24} + ( -11 + 11 \zeta_{6} ) q^{25} + \zeta_{6} q^{26} + 9 q^{27} + q^{29} + 12 \zeta_{6} q^{30} + ( 4 - 4 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} + 3 \zeta_{6} q^{33} + 2 q^{34} + 6 q^{36} + 2 \zeta_{6} q^{37} + ( -6 + 6 \zeta_{6} ) q^{38} + ( -3 + 3 \zeta_{6} ) q^{39} + 4 \zeta_{6} q^{40} + 2 q^{41} + 4 q^{43} + \zeta_{6} q^{44} + ( -24 + 24 \zeta_{6} ) q^{45} + ( -2 + 2 \zeta_{6} ) q^{46} + 2 \zeta_{6} q^{47} + 3 q^{48} -11 q^{50} + 6 \zeta_{6} q^{51} + ( -1 + \zeta_{6} ) q^{52} + ( 12 - 12 \zeta_{6} ) q^{53} + 9 \zeta_{6} q^{54} -4 q^{55} -18 q^{57} + \zeta_{6} q^{58} + ( 9 - 9 \zeta_{6} ) q^{59} + ( -12 + 12 \zeta_{6} ) q^{60} -5 \zeta_{6} q^{61} + 4 q^{62} + q^{64} -4 \zeta_{6} q^{65} + ( -3 + 3 \zeta_{6} ) q^{66} + ( 9 - 9 \zeta_{6} ) q^{67} + 2 \zeta_{6} q^{68} -6 q^{69} + 4 q^{71} + 6 \zeta_{6} q^{72} + ( -2 + 2 \zeta_{6} ) q^{73} + ( -2 + 2 \zeta_{6} ) q^{74} -33 \zeta_{6} q^{75} -6 q^{76} -3 q^{78} + 15 \zeta_{6} q^{79} + ( -4 + 4 \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + 2 \zeta_{6} q^{82} + 6 q^{83} -8 q^{85} + 4 \zeta_{6} q^{86} + ( -3 + 3 \zeta_{6} ) q^{87} + ( -1 + \zeta_{6} ) q^{88} + 6 \zeta_{6} q^{89} -24 q^{90} -2 q^{92} + 12 \zeta_{6} q^{93} + ( -2 + 2 \zeta_{6} ) q^{94} + ( 24 - 24 \zeta_{6} ) q^{95} + 3 \zeta_{6} q^{96} + 5 q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 3 q^{3} - q^{4} - 4 q^{5} - 6 q^{6} - 2 q^{8} - 6 q^{9} + O(q^{10})$$ $$2 q + q^{2} - 3 q^{3} - q^{4} - 4 q^{5} - 6 q^{6} - 2 q^{8} - 6 q^{9} + 4 q^{10} + q^{11} - 3 q^{12} + 2 q^{13} + 24 q^{15} - q^{16} + 2 q^{17} + 6 q^{18} + 6 q^{19} + 8 q^{20} + 2 q^{22} + 2 q^{23} + 3 q^{24} - 11 q^{25} + q^{26} + 18 q^{27} + 2 q^{29} + 12 q^{30} + 4 q^{31} + q^{32} + 3 q^{33} + 4 q^{34} + 12 q^{36} + 2 q^{37} - 6 q^{38} - 3 q^{39} + 4 q^{40} + 4 q^{41} + 8 q^{43} + q^{44} - 24 q^{45} - 2 q^{46} + 2 q^{47} + 6 q^{48} - 22 q^{50} + 6 q^{51} - q^{52} + 12 q^{53} + 9 q^{54} - 8 q^{55} - 36 q^{57} + q^{58} + 9 q^{59} - 12 q^{60} - 5 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{65} - 3 q^{66} + 9 q^{67} + 2 q^{68} - 12 q^{69} + 8 q^{71} + 6 q^{72} - 2 q^{73} - 2 q^{74} - 33 q^{75} - 12 q^{76} - 6 q^{78} + 15 q^{79} - 4 q^{80} - 9 q^{81} + 2 q^{82} + 12 q^{83} - 16 q^{85} + 4 q^{86} - 3 q^{87} - q^{88} + 6 q^{89} - 48 q^{90} - 4 q^{92} + 12 q^{93} - 2 q^{94} + 24 q^{95} + 3 q^{96} + 10 q^{97} - 12 q^{99} + O(q^{100})$$

## 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 −2.00000 3.46410i −3.00000 0 −1.00000 −3.00000 5.19615i 2.00000 3.46410i
177.1 0.500000 0.866025i −1.50000 2.59808i −0.500000 0.866025i −2.00000 + 3.46410i −3.00000 0 −1.00000 −3.00000 + 5.19615i 2.00000 + 3.46410i
 $$n$$: 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.g 2
7.b odd 2 1 154.2.e.d 2
7.c even 3 1 1078.2.a.f 1
7.c even 3 1 inner 1078.2.e.g 2
7.d odd 6 1 154.2.e.d 2
7.d odd 6 1 1078.2.a.a 1
21.c even 2 1 1386.2.k.a 2
21.g even 6 1 1386.2.k.a 2
21.g even 6 1 9702.2.a.cg 1
21.h odd 6 1 9702.2.a.bb 1
28.d even 2 1 1232.2.q.a 2
28.f even 6 1 1232.2.q.a 2
28.f even 6 1 8624.2.a.bd 1
28.g odd 6 1 8624.2.a.d 1

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

## Hecke kernels

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

 $$T_{3}^{2} + 3 T_{3} + 9$$ $$T_{5}^{2} + 4 T_{5} + 16$$ $$T_{13} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$16 + 4 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$1 - T + T^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$36 - 6 T + T^{2}$$
$23$ $$4 - 2 T + T^{2}$$
$29$ $$( -1 + T )^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$4 - 2 T + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$4 - 2 T + T^{2}$$
$53$ $$144 - 12 T + T^{2}$$
$59$ $$81 - 9 T + T^{2}$$
$61$ $$25 + 5 T + T^{2}$$
$67$ $$81 - 9 T + T^{2}$$
$71$ $$( -4 + T )^{2}$$
$73$ $$4 + 2 T + T^{2}$$
$79$ $$225 - 15 T + T^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$( -5 + T )^{2}$$