# Properties

 Label 1078.2.e.c 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + 2 \zeta_{6} q^{5} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + 2 \zeta_{6} q^{5} + q^{8} + 3 \zeta_{6} q^{9} + ( 2 - 2 \zeta_{6} ) q^{10} + ( 1 - \zeta_{6} ) q^{11} -2 q^{13} -\zeta_{6} q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} + ( 3 - 3 \zeta_{6} ) q^{18} -2 q^{20} - q^{22} + 8 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + 2 \zeta_{6} q^{26} -2 q^{29} + ( -8 + 8 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} -2 q^{34} -3 q^{36} + 2 \zeta_{6} q^{37} + 2 \zeta_{6} q^{40} -10 q^{41} + 4 q^{43} + \zeta_{6} q^{44} + ( -6 + 6 \zeta_{6} ) q^{45} + ( 8 - 8 \zeta_{6} ) q^{46} + 8 \zeta_{6} q^{47} - q^{50} + ( 2 - 2 \zeta_{6} ) q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} + 2 q^{55} + 2 \zeta_{6} q^{58} + 10 \zeta_{6} q^{61} + 8 q^{62} + q^{64} -4 \zeta_{6} q^{65} + ( 12 - 12 \zeta_{6} ) q^{67} + 2 \zeta_{6} q^{68} + 16 q^{71} + 3 \zeta_{6} q^{72} + ( -14 + 14 \zeta_{6} ) q^{73} + ( 2 - 2 \zeta_{6} ) q^{74} + ( 2 - 2 \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + 10 \zeta_{6} q^{82} + 4 q^{85} -4 \zeta_{6} q^{86} + ( 1 - \zeta_{6} ) q^{88} -6 \zeta_{6} q^{89} + 6 q^{90} -8 q^{92} + ( 8 - 8 \zeta_{6} ) q^{94} -10 q^{97} + 3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + 2 q^{5} + 2 q^{8} + 3 q^{9} + O(q^{10})$$ $$2 q - q^{2} - q^{4} + 2 q^{5} + 2 q^{8} + 3 q^{9} + 2 q^{10} + q^{11} - 4 q^{13} - q^{16} + 2 q^{17} + 3 q^{18} - 4 q^{20} - 2 q^{22} + 8 q^{23} + q^{25} + 2 q^{26} - 4 q^{29} - 8 q^{31} - q^{32} - 4 q^{34} - 6 q^{36} + 2 q^{37} + 2 q^{40} - 20 q^{41} + 8 q^{43} + q^{44} - 6 q^{45} + 8 q^{46} + 8 q^{47} - 2 q^{50} + 2 q^{52} - 6 q^{53} + 4 q^{55} + 2 q^{58} + 10 q^{61} + 16 q^{62} + 2 q^{64} - 4 q^{65} + 12 q^{67} + 2 q^{68} + 32 q^{71} + 3 q^{72} - 14 q^{73} + 2 q^{74} + 2 q^{80} - 9 q^{81} + 10 q^{82} + 8 q^{85} - 4 q^{86} + q^{88} - 6 q^{89} + 12 q^{90} - 16 q^{92} + 8 q^{94} - 20 q^{97} + 6 q^{99} + O(q^{100})$$

## 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 0 −0.500000 + 0.866025i 1.00000 + 1.73205i 0 0 1.00000 1.50000 + 2.59808i 1.00000 1.73205i
177.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.00000 1.73205i 0 0 1.00000 1.50000 2.59808i 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.c 2
7.b odd 2 1 1078.2.e.b 2
7.c even 3 1 1078.2.a.j 1
7.c even 3 1 inner 1078.2.e.c 2
7.d odd 6 1 154.2.a.c 1
7.d odd 6 1 1078.2.e.b 2
21.g even 6 1 1386.2.a.b 1
21.h odd 6 1 9702.2.a.v 1
28.f even 6 1 1232.2.a.h 1
28.g odd 6 1 8624.2.a.o 1
35.i odd 6 1 3850.2.a.f 1
35.k even 12 2 3850.2.c.l 2
56.j odd 6 1 4928.2.a.n 1
56.m even 6 1 4928.2.a.o 1
77.i even 6 1 1694.2.a.c 1

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 - 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$1 - T + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$64 - 8 T + T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$64 + 8 T + T^{2}$$
$37$ $$4 - 2 T + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$64 - 8 T + T^{2}$$
$53$ $$36 + 6 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$100 - 10 T + T^{2}$$
$67$ $$144 - 12 T + T^{2}$$
$71$ $$( -16 + T )^{2}$$
$73$ $$196 + 14 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$( 10 + T )^{2}$$