# Properties

 Label 1400.1.y.a Level $1400$ Weight $1$ Character orbit 1400.y Analytic conductor $0.699$ Analytic rank $0$ Dimension $4$ Projective image $D_{2}$ CM/RM discs -7, -40, 280 Inner twists $16$

# Related objects

## 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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{-7}, \sqrt{-10})$$ Artin image $OD_{16}:C_2$ Artin field Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-expansion

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

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

## Character values

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

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$\zeta_{8}^{2}$$

## 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
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
−0.707107 + 0.707107i 0 1.00000i 0 0 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0
307.2 0.707107 0.707107i 0 1.00000i 0 0 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0
643.1 −0.707107 0.707107i 0 1.00000i 0 0 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0
643.2 0.707107 + 0.707107i 0 1.00000i 0 0 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 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 $$\Q(\sqrt{-7})$$
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$
280.n even 2 1 RM by $$\Q(\sqrt{70})$$
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.k even 4 2 inner
56.e 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.a 4
5.b even 2 1 inner 1400.1.y.a 4
5.c odd 4 2 inner 1400.1.y.a 4
7.b odd 2 1 CM 1400.1.y.a 4
8.d odd 2 1 inner 1400.1.y.a 4
35.c odd 2 1 inner 1400.1.y.a 4
35.f even 4 2 inner 1400.1.y.a 4
40.e odd 2 1 CM 1400.1.y.a 4
40.k even 4 2 inner 1400.1.y.a 4
56.e even 2 1 inner 1400.1.y.a 4
280.n even 2 1 RM 1400.1.y.a 4
280.y odd 4 2 inner 1400.1.y.a 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11} + 2$$ acting on $$S_{1}^{\mathrm{new}}(1400, [\chi])$$.

## Hecke characteristic polynomials

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