# Properties

 Label 1400.1.y.b Level $1400$ Weight $1$ Character orbit 1400.y Analytic conductor $0.699$ Analytic rank $0$ Dimension $8$ Projective image $D_{6}$ CM discriminant -7 Inner twists $16$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1400.y (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.698691017686$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{6}$$ Projective field Galois closure of 6.0.78400000.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{7} q^{2} -\zeta_{24}^{2} q^{4} + \zeta_{24}^{3} q^{7} + \zeta_{24}^{9} q^{8} + \zeta_{24}^{6} q^{9} +O(q^{10})$$ $$q -\zeta_{24}^{7} q^{2} -\zeta_{24}^{2} q^{4} + \zeta_{24}^{3} q^{7} + \zeta_{24}^{9} q^{8} + \zeta_{24}^{6} q^{9} + q^{11} -\zeta_{24}^{10} q^{14} + \zeta_{24}^{4} q^{16} + \zeta_{24} q^{18} -\zeta_{24}^{7} q^{22} + \zeta_{24}^{9} q^{23} -\zeta_{24}^{5} q^{28} + ( \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{29} -\zeta_{24}^{11} q^{32} -\zeta_{24}^{8} q^{36} -\zeta_{24}^{3} q^{37} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{43} -\zeta_{24}^{2} q^{44} + \zeta_{24}^{4} q^{46} + \zeta_{24}^{6} q^{49} -2 \zeta_{24}^{9} q^{53} - q^{56} + ( -\zeta_{24}^{5} - \zeta_{24}^{9} ) q^{58} + \zeta_{24}^{9} q^{63} -\zeta_{24}^{6} q^{64} + ( \zeta_{24}^{7} + \zeta_{24}^{11} ) q^{67} + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{71} -\zeta_{24}^{3} q^{72} + \zeta_{24}^{10} q^{74} + \zeta_{24}^{3} q^{77} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{79} - q^{81} + ( -1 + \zeta_{24}^{8} ) q^{86} + \zeta_{24}^{9} q^{88} -\zeta_{24}^{11} q^{92} + \zeta_{24} q^{98} + \zeta_{24}^{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 8q^{11} + 4q^{16} + 4q^{36} + 4q^{46} - 8q^{56} - 8q^{81} - 12q^{86} + O(q^{100})$$

## Character values

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

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$-\zeta_{24}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 −0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i 0.258819 − 0.965926i −0.258819 − 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i 0.258819 + 0.965926i
−0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0
307.2 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0
307.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0
307.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0
643.1 −0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0
643.2 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0
643.3 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0
643.4 0.965926 0.258819i 0 0.866025 0.500000i 0 0 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 643.4 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 $$\Q(\sqrt{-7})$$
5.b even 2 1 inner
5.c odd 4 2 inner
8.d odd 2 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner
40.e odd 2 1 inner
40.k even 4 2 inner
56.e even 2 1 inner
280.n even 2 1 inner
280.y odd 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.1.y.b 8
5.b even 2 1 inner 1400.1.y.b 8
5.c odd 4 2 inner 1400.1.y.b 8
7.b odd 2 1 CM 1400.1.y.b 8
8.d odd 2 1 inner 1400.1.y.b 8
35.c odd 2 1 inner 1400.1.y.b 8
35.f even 4 2 inner 1400.1.y.b 8
40.e odd 2 1 inner 1400.1.y.b 8
40.k even 4 2 inner 1400.1.y.b 8
56.e even 2 1 inner 1400.1.y.b 8
280.n even 2 1 inner 1400.1.y.b 8
280.y odd 4 2 inner 1400.1.y.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.1.y.b 8 1.a even 1 1 trivial
1400.1.y.b 8 5.b even 2 1 inner
1400.1.y.b 8 5.c odd 4 2 inner
1400.1.y.b 8 7.b odd 2 1 CM
1400.1.y.b 8 8.d odd 2 1 inner
1400.1.y.b 8 35.c odd 2 1 inner
1400.1.y.b 8 35.f even 4 2 inner
1400.1.y.b 8 40.e odd 2 1 inner
1400.1.y.b 8 40.k even 4 2 inner
1400.1.y.b 8 56.e even 2 1 inner
1400.1.y.b 8 280.n even 2 1 inner
1400.1.y.b 8 280.y odd 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11} - 1$$ acting on $$S_{1}^{\mathrm{new}}(1400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 1 + T^{4} )^{2}$$
$11$ $$( -1 + T )^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$( 1 + T^{4} )^{2}$$
$29$ $$( -3 + T^{2} )^{4}$$
$31$ $$T^{8}$$
$37$ $$( 1 + T^{4} )^{2}$$
$41$ $$T^{8}$$
$43$ $$( 9 + T^{4} )^{2}$$
$47$ $$T^{8}$$
$53$ $$( 16 + T^{4} )^{2}$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$( 9 + T^{4} )^{2}$$
$71$ $$( 3 + T^{2} )^{4}$$
$73$ $$T^{8}$$
$79$ $$( -3 + T^{2} )^{4}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$
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