# Properties

 Label 1400.1.bf.a Level $1400$ Weight $1$ Character orbit 1400.bf Analytic conductor $0.699$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ RM discriminant 8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1400.bf (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.698691017686$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{6}$$ Projective field Galois closure of 6.0.5378240000.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} + \zeta_{6}^{2} q^{7} + q^{8} -\zeta_{6}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} + \zeta_{6}^{2} q^{7} + q^{8} -\zeta_{6}^{2} q^{9} -\zeta_{6} q^{14} + \zeta_{6}^{2} q^{16} + ( 1 - \zeta_{6}^{2} ) q^{17} + \zeta_{6} q^{18} + \zeta_{6}^{2} q^{23} + q^{28} + ( 1 - \zeta_{6}^{2} ) q^{31} -\zeta_{6} q^{32} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{34} - q^{36} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{41} -\zeta_{6} q^{46} + ( 1 + \zeta_{6} ) q^{47} -\zeta_{6} q^{49} + \zeta_{6}^{2} q^{56} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{62} + \zeta_{6} q^{63} + q^{64} + ( -1 - \zeta_{6} ) q^{68} + q^{71} -\zeta_{6}^{2} q^{72} + \zeta_{6}^{2} q^{79} -\zeta_{6} q^{81} + ( -1 - \zeta_{6} ) q^{82} + ( 1 + \zeta_{6} ) q^{89} + q^{92} + ( -1 + \zeta_{6}^{2} ) q^{94} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{97} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} - q^{7} + 2q^{8} + q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{4} - q^{7} + 2q^{8} + q^{9} - q^{14} - q^{16} + 3q^{17} + q^{18} - q^{23} + 2q^{28} + 3q^{31} - q^{32} - 2q^{36} - q^{46} + 3q^{47} - q^{49} - q^{56} + q^{63} + 2q^{64} - 3q^{68} + 2q^{71} + q^{72} - q^{79} - q^{81} - 3q^{82} + 3q^{89} + 2q^{92} - 3q^{94} + 2q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{6}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −0.500000 0.866025i 1.00000 0.500000 + 0.866025i 0
901.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −0.500000 + 0.866025i 1.00000 0.500000 0.866025i 0
 $$n$$: 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
8.b even 2 1 RM by $$\Q(\sqrt{2})$$
7.d odd 6 1 inner
56.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.1.bf.a 2
5.b even 2 1 1400.1.bf.b yes 2
5.c odd 4 2 1400.1.bl.a 4
7.d odd 6 1 inner 1400.1.bf.a 2
8.b even 2 1 RM 1400.1.bf.a 2
35.i odd 6 1 1400.1.bf.b yes 2
35.k even 12 2 1400.1.bl.a 4
40.f even 2 1 1400.1.bf.b yes 2
40.i odd 4 2 1400.1.bl.a 4
56.j odd 6 1 inner 1400.1.bf.a 2
280.bk odd 6 1 1400.1.bf.b yes 2
280.bv even 12 2 1400.1.bl.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.1.bf.a 2 1.a even 1 1 trivial
1400.1.bf.a 2 7.d odd 6 1 inner
1400.1.bf.a 2 8.b even 2 1 RM
1400.1.bf.a 2 56.j odd 6 1 inner
1400.1.bf.b yes 2 5.b even 2 1
1400.1.bf.b yes 2 35.i odd 6 1
1400.1.bf.b yes 2 40.f even 2 1
1400.1.bf.b yes 2 280.bk odd 6 1
1400.1.bl.a 4 5.c odd 4 2
1400.1.bl.a 4 35.k even 12 2
1400.1.bl.a 4 40.i odd 4 2
1400.1.bl.a 4 280.bv even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17}^{2} - 3 T_{17} + 3$$ acting on $$S_{1}^{\mathrm{new}}(1400, [\chi])$$.

## Hecke characteristic polynomials

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