# Properties

 Label 1400.1.m.c Level $1400$ Weight $1$ Character orbit 1400.m Analytic conductor $0.699$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1400.m (of order $$2$$, degree $$1$$, 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 6.0.109760000.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} + q^{7} + q^{8} - q^{9} +O(q^{10})$$ $$q + \zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} + q^{7} + q^{8} - q^{9} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{11} + \zeta_{6}^{2} q^{14} + \zeta_{6}^{2} q^{16} -\zeta_{6}^{2} q^{18} + ( 1 + \zeta_{6} ) q^{22} + q^{23} -\zeta_{6} q^{28} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{29} -\zeta_{6} q^{32} + \zeta_{6} q^{36} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{37} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{43} + ( -1 + \zeta_{6}^{2} ) q^{44} + \zeta_{6}^{2} q^{46} + q^{49} + q^{56} + ( 1 + \zeta_{6} ) q^{58} - q^{63} + q^{64} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{67} + q^{71} - q^{72} + ( 1 + \zeta_{6} ) q^{74} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{77} + q^{79} + q^{81} + ( -1 - \zeta_{6} ) q^{86} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{88} -\zeta_{6} q^{92} + \zeta_{6}^{2} q^{98} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + 2q^{7} + 2q^{8} - 2q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{4} + 2q^{7} + 2q^{8} - 2q^{9} - q^{14} - q^{16} + q^{18} + 3q^{22} + 2q^{23} - q^{28} - q^{32} + q^{36} - 3q^{44} - q^{46} + 2q^{49} + 2q^{56} + 3q^{58} - 2q^{63} + 2q^{64} + 2q^{71} - 2q^{72} + 3q^{74} + 2q^{79} + 2q^{81} - 3q^{86} - q^{92} - q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1301.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 1.00000 1.00000 −1.00000 0
1301.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 1.00000 1.00000 −1.00000 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
8.b even 2 1 inner
56.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.1.m.c 2
5.b even 2 1 1400.1.m.d yes 2
5.c odd 4 2 1400.1.c.b 4
7.b odd 2 1 CM 1400.1.m.c 2
8.b even 2 1 inner 1400.1.m.c 2
35.c odd 2 1 1400.1.m.d yes 2
35.f even 4 2 1400.1.c.b 4
40.f even 2 1 1400.1.m.d yes 2
40.i odd 4 2 1400.1.c.b 4
56.h odd 2 1 inner 1400.1.m.c 2
280.c odd 2 1 1400.1.m.d yes 2
280.s even 4 2 1400.1.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.1.c.b 4 5.c odd 4 2
1400.1.c.b 4 35.f even 4 2
1400.1.c.b 4 40.i odd 4 2
1400.1.c.b 4 280.s even 4 2
1400.1.m.c 2 1.a even 1 1 trivial
1400.1.m.c 2 7.b odd 2 1 CM
1400.1.m.c 2 8.b even 2 1 inner
1400.1.m.c 2 56.h odd 2 1 inner
1400.1.m.d yes 2 5.b even 2 1
1400.1.m.d yes 2 35.c odd 2 1
1400.1.m.d yes 2 40.f even 2 1
1400.1.m.d yes 2 280.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1400, [\chi])$$:

 $$T_{3}$$ $$T_{11}^{2} + 3$$ $$T_{23} - 1$$ $$T_{113} + 1$$

## Hecke characteristic polynomials

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