# Properties

 Label 1078.2.e.s Level $1078$ Weight $2$ Character orbit 1078.e Analytic conductor $8.608$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$199$$ $$981$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
0.500000 + 0.866025i −0.707107 + 1.22474i −0.500000 + 0.866025i 0.707107 + 1.22474i −1.41421 0 −1.00000 0.500000 + 0.866025i −0.707107 + 1.22474i
67.2 0.500000 + 0.866025i 0.707107 1.22474i −0.500000 + 0.866025i −0.707107 1.22474i 1.41421 0 −1.00000 0.500000 + 0.866025i 0.707107 1.22474i
177.1 0.500000 0.866025i −0.707107 1.22474i −0.500000 0.866025i 0.707107 1.22474i −1.41421 0 −1.00000 0.500000 0.866025i −0.707107 1.22474i
177.2 0.500000 0.866025i 0.707107 + 1.22474i −0.500000 0.866025i −0.707107 + 1.22474i 1.41421 0 −1.00000 0.500000 0.866025i 0.707107 + 1.22474i
 $$n$$: 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.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.e.s 4
7.b odd 2 1 inner 1078.2.e.s 4
7.c even 3 1 1078.2.a.q 2
7.c even 3 1 inner 1078.2.e.s 4
7.d odd 6 1 1078.2.a.q 2
7.d odd 6 1 inner 1078.2.e.s 4
21.g even 6 1 9702.2.a.dq 2
21.h odd 6 1 9702.2.a.dq 2
28.f even 6 1 8624.2.a.bw 2
28.g odd 6 1 8624.2.a.bw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.q 2 7.c even 3 1
1078.2.a.q 2 7.d odd 6 1
1078.2.e.s 4 1.a even 1 1 trivial
1078.2.e.s 4 7.b odd 2 1 inner
1078.2.e.s 4 7.c even 3 1 inner
1078.2.e.s 4 7.d odd 6 1 inner
8624.2.a.bw 2 28.f even 6 1
8624.2.a.bw 2 28.g odd 6 1
9702.2.a.dq 2 21.g even 6 1
9702.2.a.dq 2 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 $$S_{2}^{\mathrm{new}}(1078, [\chi])$$:

 $$T_{3}^{4} + 2 T_{3}^{2} + 4$$ $$T_{5}^{4} + 2 T_{5}^{2} + 4$$ $$T_{13}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$4 + 2 T^{2} + T^{4}$$
$5$ $$4 + 2 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 1 - T + T^{2} )^{2}$$
$13$ $$( -8 + T^{2} )^{2}$$
$17$ $$1024 + 32 T^{2} + T^{4}$$
$19$ $$64 + 8 T^{2} + T^{4}$$
$23$ $$( 4 - 2 T + T^{2} )^{2}$$
$29$ $$( 6 + T )^{4}$$
$31$ $$4 + 2 T^{2} + T^{4}$$
$37$ $$( 4 - 2 T + T^{2} )^{2}$$
$41$ $$( -32 + T^{2} )^{2}$$
$43$ $$( -4 + T )^{4}$$
$47$ $$324 + 18 T^{2} + T^{4}$$
$53$ $$( 144 - 12 T + T^{2} )^{2}$$
$59$ $$324 + 18 T^{2} + T^{4}$$
$61$ $$1024 + 32 T^{2} + T^{4}$$
$67$ $$( 4 - 2 T + T^{2} )^{2}$$
$71$ $$( 10 + T )^{4}$$
$73$ $$40000 + 200 T^{2} + T^{4}$$
$79$ $$( 64 - 8 T + T^{2} )^{2}$$
$83$ $$( -8 + T^{2} )^{2}$$
$89$ $$2500 + 50 T^{2} + T^{4}$$
$97$ $$( -242 + T^{2} )^{2}$$