# Properties

 Label 1400.1.bs.b Level $1400$ Weight $1$ Character orbit 1400.bs Analytic conductor $0.699$ Analytic rank $0$ Dimension $4$ Projective image $D_{5}$ 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.bs (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.698691017686$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{5}$$ Projective field Galois closure of 5.1.1225000000.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{10}^{4} q^{2} + ( 1 - \zeta_{10} ) q^{3} -\zeta_{10}^{3} q^{4} + \zeta_{10}^{4} q^{5} + ( 1 + \zeta_{10}^{4} ) q^{6} + q^{7} + \zeta_{10}^{2} q^{8} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{9} +O(q^{10})$$ $$q + \zeta_{10}^{4} q^{2} + ( 1 - \zeta_{10} ) q^{3} -\zeta_{10}^{3} q^{4} + \zeta_{10}^{4} q^{5} + ( 1 + \zeta_{10}^{4} ) q^{6} + q^{7} + \zeta_{10}^{2} q^{8} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{9} -\zeta_{10}^{3} q^{10} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{12} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{13} + \zeta_{10}^{4} q^{14} + ( 1 + \zeta_{10}^{4} ) q^{15} -\zeta_{10} q^{16} + ( 1 - \zeta_{10} + \zeta_{10}^{4} ) q^{18} + 2 \zeta_{10}^{2} q^{19} + \zeta_{10}^{2} q^{20} + ( 1 - \zeta_{10} ) q^{21} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{23} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{24} -\zeta_{10}^{3} q^{25} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{26} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{27} -\zeta_{10}^{3} q^{28} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{30} + q^{32} + \zeta_{10}^{4} q^{35} + ( 1 - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{36} -2 \zeta_{10} q^{38} + ( 1 - \zeta_{10}^{3} + 2 \zeta_{10}^{4} ) q^{39} -\zeta_{10} q^{40} + ( 1 + \zeta_{10}^{4} ) q^{42} + ( 1 - \zeta_{10} + \zeta_{10}^{4} ) q^{45} + ( 1 - \zeta_{10} ) q^{46} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{48} + q^{49} + \zeta_{10}^{2} q^{50} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{52} + ( 1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{54} + \zeta_{10}^{2} q^{56} + ( 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{57} + ( 1 + \zeta_{10}^{2} ) q^{59} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{60} + ( 1 - \zeta_{10}^{3} ) q^{61} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{63} + \zeta_{10}^{4} q^{64} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{65} + ( -\zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{69} -\zeta_{10}^{3} q^{70} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{71} + ( \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{72} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{75} + 2 q^{76} + ( \zeta_{10}^{2} - 2 \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{78} + ( 1 - \zeta_{10} ) q^{79} + q^{80} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{81} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{83} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{84} + ( 1 - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{90} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{91} + ( 1 + \zeta_{10}^{4} ) q^{92} -2 \zeta_{10} q^{95} + ( 1 - \zeta_{10} ) q^{96} + \zeta_{10}^{4} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{2} + 3q^{3} - q^{4} - q^{5} + 3q^{6} + 4q^{7} - q^{8} + 2q^{9} + O(q^{10})$$ $$4q - q^{2} + 3q^{3} - q^{4} - q^{5} + 3q^{6} + 4q^{7} - q^{8} + 2q^{9} - q^{10} - 2q^{12} - 2q^{13} - q^{14} + 3q^{15} - q^{16} + 2q^{18} - 2q^{19} - q^{20} + 3q^{21} - 2q^{23} - 2q^{24} - q^{25} - 2q^{26} + q^{27} - q^{28} - 2q^{30} + 4q^{32} - q^{35} + 2q^{36} - 2q^{38} + q^{39} - q^{40} + 3q^{42} + 2q^{45} + 3q^{46} - 2q^{48} + 4q^{49} - q^{50} - 2q^{52} + q^{54} - q^{56} - 4q^{57} + 3q^{59} - 2q^{60} + 3q^{61} + 2q^{63} - q^{64} - 2q^{65} - 4q^{69} - q^{70} - 2q^{71} - 3q^{72} - 2q^{75} + 8q^{76} - 4q^{78} + 3q^{79} + 4q^{80} - 2q^{83} - 2q^{84} + 2q^{90} - 2q^{91} + 3q^{92} - 2q^{95} + 3q^{96} - q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 −0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 − 0.951057i
0.309017 + 0.951057i 1.30902 0.951057i −0.809017 + 0.587785i 0.309017 + 0.951057i 1.30902 + 0.951057i 1.00000 −0.809017 0.587785i 0.500000 1.53884i −0.809017 + 0.587785i
461.1 −0.809017 + 0.587785i 0.190983 0.587785i 0.309017 0.951057i −0.809017 + 0.587785i 0.190983 + 0.587785i 1.00000 0.309017 + 0.951057i 0.500000 + 0.363271i 0.309017 0.951057i
741.1 −0.809017 0.587785i 0.190983 + 0.587785i 0.309017 + 0.951057i −0.809017 0.587785i 0.190983 0.587785i 1.00000 0.309017 0.951057i 0.500000 0.363271i 0.309017 + 0.951057i
1021.1 0.309017 0.951057i 1.30902 + 0.951057i −0.809017 0.587785i 0.309017 0.951057i 1.30902 0.951057i 1.00000 −0.809017 + 0.587785i 0.500000 + 1.53884i −0.809017 0.587785i
 $$n$$: 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 $$\Q(\sqrt{-14})$$
25.d even 5 1 inner
1400.bs odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.1.bs.b yes 4
7.b odd 2 1 1400.1.bs.a 4
8.b even 2 1 1400.1.bs.a 4
25.d even 5 1 inner 1400.1.bs.b yes 4
56.h odd 2 1 CM 1400.1.bs.b yes 4
175.l odd 10 1 1400.1.bs.a 4
200.t even 10 1 1400.1.bs.a 4
1400.bs odd 10 1 inner 1400.1.bs.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.1.bs.a 4 7.b odd 2 1
1400.1.bs.a 4 8.b even 2 1
1400.1.bs.a 4 175.l odd 10 1
1400.1.bs.a 4 200.t even 10 1
1400.1.bs.b yes 4 1.a even 1 1 trivial
1400.1.bs.b yes 4 25.d even 5 1 inner
1400.1.bs.b yes 4 56.h odd 2 1 CM
1400.1.bs.b yes 4 1400.bs odd 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 3 T_{3}^{3} + 4 T_{3}^{2} - 2 T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1400, [\chi])$$.

## Hecke characteristic polynomials

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