# Properties

 Label 1400.1.m.b Level $1400$ Weight $1$ Character orbit 1400.m Self dual yes Analytic conductor $0.699$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -56 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: yes Analytic conductor: $$0.698691017686$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Projective image $$D_{4}$$ Projective field Galois closure of 4.2.11200.2 Artin image $D_8$ Artin field Galois closure of 8.0.19208000000.2

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} -\beta q^{3} + q^{4} + \beta q^{6} + q^{7} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} -\beta q^{3} + q^{4} + \beta q^{6} + q^{7} - q^{8} + q^{9} -\beta q^{12} + \beta q^{13} - q^{14} + q^{16} - q^{18} -\beta q^{19} -\beta q^{21} + \beta q^{24} -\beta q^{26} + q^{28} - q^{32} + q^{36} + \beta q^{38} -2 q^{39} + \beta q^{42} -\beta q^{48} + q^{49} + \beta q^{52} - q^{56} + 2 q^{57} + \beta q^{59} + \beta q^{61} + q^{63} + q^{64} - q^{72} -\beta q^{76} + 2 q^{78} - q^{81} + \beta q^{83} -\beta q^{84} + \beta q^{91} + \beta q^{96} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} + 2q^{7} - 2q^{8} + 2q^{9} - 2q^{14} + 2q^{16} - 2q^{18} + 2q^{28} - 2q^{32} + 2q^{36} - 4q^{39} + 2q^{49} - 2q^{56} + 4q^{57} + 2q^{63} + 2q^{64} - 2q^{72} + 4q^{78} - 2q^{81} - 2q^{98} + O(q^{100})$$

## Character values

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

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

## 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
 1.41421 −1.41421
−1.00000 −1.41421 1.00000 0 1.41421 1.00000 −1.00000 1.00000 0
1301.2 −1.00000 1.41421 1.00000 0 −1.41421 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
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
7.b odd 2 1 inner
8.b even 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.b 2
5.b even 2 1 1400.1.m.e 2
5.c odd 4 2 280.1.c.a 4
7.b odd 2 1 inner 1400.1.m.b 2
8.b even 2 1 inner 1400.1.m.b 2
15.e even 4 2 2520.1.h.e 4
20.e even 4 2 1120.1.c.a 4
35.c odd 2 1 1400.1.m.e 2
35.f even 4 2 280.1.c.a 4
35.k even 12 4 1960.1.bk.a 8
35.l odd 12 4 1960.1.bk.a 8
40.f even 2 1 1400.1.m.e 2
40.i odd 4 2 280.1.c.a 4
40.k even 4 2 1120.1.c.a 4
56.h odd 2 1 CM 1400.1.m.b 2
105.k odd 4 2 2520.1.h.e 4
120.w even 4 2 2520.1.h.e 4
140.j odd 4 2 1120.1.c.a 4
280.c odd 2 1 1400.1.m.e 2
280.s even 4 2 280.1.c.a 4
280.y odd 4 2 1120.1.c.a 4
280.bt odd 12 4 1960.1.bk.a 8
280.bv even 12 4 1960.1.bk.a 8
840.bp odd 4 2 2520.1.h.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.1.c.a 4 5.c odd 4 2
280.1.c.a 4 35.f even 4 2
280.1.c.a 4 40.i odd 4 2
280.1.c.a 4 280.s even 4 2
1120.1.c.a 4 20.e even 4 2
1120.1.c.a 4 40.k even 4 2
1120.1.c.a 4 140.j odd 4 2
1120.1.c.a 4 280.y odd 4 2
1400.1.m.b 2 1.a even 1 1 trivial
1400.1.m.b 2 7.b odd 2 1 inner
1400.1.m.b 2 8.b even 2 1 inner
1400.1.m.b 2 56.h odd 2 1 CM
1400.1.m.e 2 5.b even 2 1
1400.1.m.e 2 35.c odd 2 1
1400.1.m.e 2 40.f even 2 1
1400.1.m.e 2 280.c odd 2 1
1960.1.bk.a 8 35.k even 12 4
1960.1.bk.a 8 35.l odd 12 4
1960.1.bk.a 8 280.bt odd 12 4
1960.1.bk.a 8 280.bv even 12 4
2520.1.h.e 4 15.e even 4 2
2520.1.h.e 4 105.k odd 4 2
2520.1.h.e 4 120.w even 4 2
2520.1.h.e 4 840.bp odd 4 2

## Hecke kernels

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

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

## Hecke characteristic polynomials

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