# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1400.bl (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.698691017686$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{5} q^{2} -\zeta_{12}^{4} q^{4} + \zeta_{12}^{5} q^{7} + \zeta_{12}^{3} q^{8} -\zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{5} q^{2} -\zeta_{12}^{4} q^{4} + \zeta_{12}^{5} q^{7} + \zeta_{12}^{3} q^{8} -\zeta_{12}^{2} q^{9} -\zeta_{12}^{4} q^{14} -\zeta_{12}^{2} q^{16} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{17} + \zeta_{12} q^{18} -\zeta_{12}^{5} q^{23} + \zeta_{12}^{3} q^{28} + ( 1 + \zeta_{12}^{2} ) q^{31} + \zeta_{12} q^{32} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{34} - q^{36} + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{41} + \zeta_{12}^{4} q^{46} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{47} -\zeta_{12}^{4} q^{49} -\zeta_{12}^{2} q^{56} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{62} + \zeta_{12} q^{63} - q^{64} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{68} + q^{71} -\zeta_{12}^{5} q^{72} + \zeta_{12}^{2} q^{79} + \zeta_{12}^{4} q^{81} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{82} + ( -1 + \zeta_{12}^{4} ) q^{89} -\zeta_{12}^{3} q^{92} + ( 1 + \zeta_{12}^{2} ) q^{94} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{97} + \zeta_{12}^{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} - 2q^{9} + O(q^{10})$$ $$4q + 2q^{4} - 2q^{9} + 2q^{14} - 2q^{16} + 6q^{31} - 4q^{36} - 2q^{46} + 2q^{49} - 2q^{56} - 4q^{64} + 4q^{71} + 2q^{79} - 2q^{81} - 6q^{89} + 6q^{94} + O(q^{100})$$

## Character values

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

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
549.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 −0.866025 + 0.500000i 1.00000i −0.500000 0.866025i 0
549.2 0.866025 0.500000i 0 0.500000 0.866025i 0 0 0.866025 0.500000i 1.00000i −0.500000 0.866025i 0
1349.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 −0.866025 0.500000i 1.00000i −0.500000 + 0.866025i 0
1349.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 0.866025 + 0.500000i 1.00000i −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 $$\Q(\sqrt{2})$$
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner
40.f even 2 1 inner
56.j odd 6 1 inner
280.bk 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.bl.a 4
5.b even 2 1 inner 1400.1.bl.a 4
5.c odd 4 1 1400.1.bf.a 2
5.c odd 4 1 1400.1.bf.b yes 2
7.d odd 6 1 inner 1400.1.bl.a 4
8.b even 2 1 RM 1400.1.bl.a 4
35.i odd 6 1 inner 1400.1.bl.a 4
35.k even 12 1 1400.1.bf.a 2
35.k even 12 1 1400.1.bf.b yes 2
40.f even 2 1 inner 1400.1.bl.a 4
40.i odd 4 1 1400.1.bf.a 2
40.i odd 4 1 1400.1.bf.b yes 2
56.j odd 6 1 inner 1400.1.bl.a 4
280.bk odd 6 1 inner 1400.1.bl.a 4
280.bv even 12 1 1400.1.bf.a 2
280.bv even 12 1 1400.1.bf.b yes 2

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1400, [\chi])$$.

## Hecke characteristic polynomials

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