# Properties

 Label 1078.2.e.l 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( 2 - 2 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + 2 \zeta_{6} q^{5} + 2 q^{6} - q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( 2 - 2 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + 2 \zeta_{6} q^{5} + 2 q^{6} - q^{8} -\zeta_{6} q^{9} + ( -2 + 2 \zeta_{6} ) q^{10} + ( -1 + \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{12} + 4 q^{13} + 4 q^{15} -\zeta_{6} q^{16} + ( 1 - \zeta_{6} ) q^{18} + 4 \zeta_{6} q^{19} -2 q^{20} - q^{22} -4 \zeta_{6} q^{23} + ( -2 + 2 \zeta_{6} ) q^{24} + ( 1 - \zeta_{6} ) q^{25} + 4 \zeta_{6} q^{26} + 4 q^{27} + 2 q^{29} + 4 \zeta_{6} q^{30} + ( -10 + 10 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} + 2 \zeta_{6} q^{33} + q^{36} + 6 \zeta_{6} q^{37} + ( -4 + 4 \zeta_{6} ) q^{38} + ( 8 - 8 \zeta_{6} ) q^{39} -2 \zeta_{6} q^{40} -4 q^{43} -\zeta_{6} q^{44} + ( 2 - 2 \zeta_{6} ) q^{45} + ( 4 - 4 \zeta_{6} ) q^{46} + 10 \zeta_{6} q^{47} -2 q^{48} + q^{50} + ( -4 + 4 \zeta_{6} ) q^{52} + ( 14 - 14 \zeta_{6} ) q^{53} + 4 \zeta_{6} q^{54} -2 q^{55} + 8 q^{57} + 2 \zeta_{6} q^{58} + ( 10 - 10 \zeta_{6} ) q^{59} + ( -4 + 4 \zeta_{6} ) q^{60} -8 \zeta_{6} q^{61} -10 q^{62} + q^{64} + 8 \zeta_{6} q^{65} + ( -2 + 2 \zeta_{6} ) q^{66} + ( -8 + 8 \zeta_{6} ) q^{67} -8 q^{69} -4 q^{71} + \zeta_{6} q^{72} + ( 4 - 4 \zeta_{6} ) q^{73} + ( -6 + 6 \zeta_{6} ) q^{74} -2 \zeta_{6} q^{75} -4 q^{76} + 8 q^{78} -16 \zeta_{6} q^{79} + ( 2 - 2 \zeta_{6} ) q^{80} + ( 11 - 11 \zeta_{6} ) q^{81} -4 q^{83} -4 \zeta_{6} q^{86} + ( 4 - 4 \zeta_{6} ) q^{87} + ( 1 - \zeta_{6} ) q^{88} + 10 \zeta_{6} q^{89} + 2 q^{90} + 4 q^{92} + 20 \zeta_{6} q^{93} + ( -10 + 10 \zeta_{6} ) q^{94} + ( -8 + 8 \zeta_{6} ) q^{95} -2 \zeta_{6} q^{96} -6 q^{97} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{8} - q^{9} + O(q^{10})$$ $$2 q + q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{8} - q^{9} - 2 q^{10} - q^{11} + 2 q^{12} + 8 q^{13} + 8 q^{15} - q^{16} + q^{18} + 4 q^{19} - 4 q^{20} - 2 q^{22} - 4 q^{23} - 2 q^{24} + q^{25} + 4 q^{26} + 8 q^{27} + 4 q^{29} + 4 q^{30} - 10 q^{31} + q^{32} + 2 q^{33} + 2 q^{36} + 6 q^{37} - 4 q^{38} + 8 q^{39} - 2 q^{40} - 8 q^{43} - q^{44} + 2 q^{45} + 4 q^{46} + 10 q^{47} - 4 q^{48} + 2 q^{50} - 4 q^{52} + 14 q^{53} + 4 q^{54} - 4 q^{55} + 16 q^{57} + 2 q^{58} + 10 q^{59} - 4 q^{60} - 8 q^{61} - 20 q^{62} + 2 q^{64} + 8 q^{65} - 2 q^{66} - 8 q^{67} - 16 q^{69} - 8 q^{71} + q^{72} + 4 q^{73} - 6 q^{74} - 2 q^{75} - 8 q^{76} + 16 q^{78} - 16 q^{79} + 2 q^{80} + 11 q^{81} - 8 q^{83} - 4 q^{86} + 4 q^{87} + q^{88} + 10 q^{89} + 4 q^{90} + 8 q^{92} + 20 q^{93} - 10 q^{94} - 8 q^{95} - 2 q^{96} - 12 q^{97} + 2 q^{99} + O(q^{100})$$

## 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 1.00000 1.73205i −0.500000 + 0.866025i 1.00000 + 1.73205i 2.00000 0 −1.00000 −0.500000 0.866025i −1.00000 + 1.73205i
177.1 0.500000 0.866025i 1.00000 + 1.73205i −0.500000 0.866025i 1.00000 1.73205i 2.00000 0 −1.00000 −0.500000 + 0.866025i −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.l 2
7.b odd 2 1 1078.2.e.h 2
7.c even 3 1 1078.2.a.b 1
7.c even 3 1 inner 1078.2.e.l 2
7.d odd 6 1 154.2.a.b 1
7.d odd 6 1 1078.2.e.h 2
21.g even 6 1 1386.2.a.f 1
21.h odd 6 1 9702.2.a.bz 1
28.f even 6 1 1232.2.a.c 1
28.g odd 6 1 8624.2.a.z 1
35.i odd 6 1 3850.2.a.o 1
35.k even 12 2 3850.2.c.d 2
56.j odd 6 1 4928.2.a.d 1
56.m even 6 1 4928.2.a.bf 1
77.i even 6 1 1694.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.b 1 7.d odd 6 1
1078.2.a.b 1 7.c even 3 1
1078.2.e.h 2 7.b odd 2 1
1078.2.e.h 2 7.d odd 6 1
1078.2.e.l 2 1.a even 1 1 trivial
1078.2.e.l 2 7.c even 3 1 inner
1232.2.a.c 1 28.f even 6 1
1386.2.a.f 1 21.g even 6 1
1694.2.a.i 1 77.i even 6 1
3850.2.a.o 1 35.i odd 6 1
3850.2.c.d 2 35.k even 12 2
4928.2.a.d 1 56.j odd 6 1
4928.2.a.bf 1 56.m even 6 1
8624.2.a.z 1 28.g odd 6 1
9702.2.a.bz 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 $$S_{2}^{\mathrm{new}}(1078, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$4 - 2 T + T^{2}$$
$5$ $$4 - 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$1 + T + T^{2}$$
$13$ $$( -4 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$16 - 4 T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$100 + 10 T + T^{2}$$
$37$ $$36 - 6 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$100 - 10 T + T^{2}$$
$53$ $$196 - 14 T + T^{2}$$
$59$ $$100 - 10 T + T^{2}$$
$61$ $$64 + 8 T + T^{2}$$
$67$ $$64 + 8 T + T^{2}$$
$71$ $$( 4 + T )^{2}$$
$73$ $$16 - 4 T + T^{2}$$
$79$ $$256 + 16 T + T^{2}$$
$83$ $$( 4 + T )^{2}$$
$89$ $$100 - 10 T + T^{2}$$
$97$ $$( 6 + T )^{2}$$