# Properties

 Label 1400.2.bh.g Level $1400$ Weight $2$ Character orbit 1400.bh Analytic conductor $11.179$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1400.bh (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1790562830$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 280) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{3} + ( \zeta_{24} - \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{7} + ( -2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{3} + ( \zeta_{24} - \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{7} + ( -2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{9} + ( -2 \zeta_{24} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{7} ) q^{11} + 2 \zeta_{24}^{6} q^{13} + ( 4 \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{17} + ( -4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{19} + ( -1 + 2 \zeta_{24} - 3 \zeta_{24}^{4} + 2 \zeta_{24}^{7} ) q^{21} + ( 7 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{23} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{27} + ( 5 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{29} + ( -2 \zeta_{24} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{7} ) q^{31} + ( -6 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{33} + ( 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{37} + ( 2 \zeta_{24} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{7} ) q^{39} + ( 3 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{41} + ( 7 \zeta_{24} - 7 \zeta_{24}^{3} - 7 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{43} + ( 6 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{47} + ( -5 + 4 \zeta_{24} + 2 \zeta_{24}^{3} + 5 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{49} + ( 6 - 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{51} + ( -4 \zeta_{24} + 4 \zeta_{24}^{7} ) q^{53} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 8 \zeta_{24}^{6} ) q^{57} -4 \zeta_{24}^{4} q^{59} + ( -1 - 4 \zeta_{24}^{3} + \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{61} + ( 2 \zeta_{24} + 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{63} + ( 7 \zeta_{24} + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} - 7 \zeta_{24}^{7} ) q^{67} + ( -9 + 8 \zeta_{24} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} ) q^{69} -12 q^{71} + ( 4 \zeta_{24} - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{73} + ( 4 \zeta_{24} - 6 \zeta_{24}^{2} + 8 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{77} + ( -4 + 4 \zeta_{24}^{4} ) q^{79} + ( -6 \zeta_{24} + \zeta_{24}^{4} - 6 \zeta_{24}^{7} ) q^{81} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 9 \zeta_{24}^{6} ) q^{83} + ( 3 \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{87} + ( -11 + 4 \zeta_{24}^{3} + 11 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{89} + ( 2 \zeta_{24} + 4 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{91} + ( -6 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{93} -6 \zeta_{24}^{6} q^{97} + ( 8 - 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 8q^{11} - 20q^{21} + 40q^{29} + 8q^{31} - 8q^{39} + 24q^{41} - 20q^{49} + 24q^{51} - 16q^{59} - 4q^{61} - 72q^{69} - 96q^{71} - 16q^{79} + 4q^{81} - 44q^{89} - 8q^{91} + 64q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
249.1
 −0.965926 + 0.258819i −0.258819 − 0.965926i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.965926 + 0.258819i 0.258819 − 0.965926i
0 −2.09077 1.20711i 0 0 0 0.358719 2.62132i 0 1.41421 + 2.44949i 0
249.2 0 −0.358719 0.207107i 0 0 0 2.09077 1.62132i 0 −1.41421 2.44949i 0
249.3 0 0.358719 + 0.207107i 0 0 0 −2.09077 + 1.62132i 0 −1.41421 2.44949i 0
249.4 0 2.09077 + 1.20711i 0 0 0 −0.358719 + 2.62132i 0 1.41421 + 2.44949i 0
849.1 0 −2.09077 + 1.20711i 0 0 0 0.358719 + 2.62132i 0 1.41421 2.44949i 0
849.2 0 −0.358719 + 0.207107i 0 0 0 2.09077 + 1.62132i 0 −1.41421 + 2.44949i 0
849.3 0 0.358719 0.207107i 0 0 0 −2.09077 1.62132i 0 −1.41421 + 2.44949i 0
849.4 0 2.09077 1.20711i 0 0 0 −0.358719 2.62132i 0 1.41421 2.44949i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 849.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.bh.g 8
5.b even 2 1 inner 1400.2.bh.g 8
5.c odd 4 1 280.2.q.d 4
5.c odd 4 1 1400.2.q.h 4
7.c even 3 1 inner 1400.2.bh.g 8
15.e even 4 1 2520.2.bi.k 4
20.e even 4 1 560.2.q.j 4
35.f even 4 1 1960.2.q.q 4
35.j even 6 1 inner 1400.2.bh.g 8
35.k even 12 1 1960.2.a.t 2
35.k even 12 1 1960.2.q.q 4
35.k even 12 1 9800.2.a.br 2
35.l odd 12 1 280.2.q.d 4
35.l odd 12 1 1400.2.q.h 4
35.l odd 12 1 1960.2.a.p 2
35.l odd 12 1 9800.2.a.bz 2
105.x even 12 1 2520.2.bi.k 4
140.w even 12 1 560.2.q.j 4
140.w even 12 1 3920.2.a.bz 2
140.x odd 12 1 3920.2.a.bp 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.d 4 5.c odd 4 1
280.2.q.d 4 35.l odd 12 1
560.2.q.j 4 20.e even 4 1
560.2.q.j 4 140.w even 12 1
1400.2.q.h 4 5.c odd 4 1
1400.2.q.h 4 35.l odd 12 1
1400.2.bh.g 8 1.a even 1 1 trivial
1400.2.bh.g 8 5.b even 2 1 inner
1400.2.bh.g 8 7.c even 3 1 inner
1400.2.bh.g 8 35.j even 6 1 inner
1960.2.a.p 2 35.l odd 12 1
1960.2.a.t 2 35.k even 12 1
1960.2.q.q 4 35.f even 4 1
1960.2.q.q 4 35.k even 12 1
2520.2.bi.k 4 15.e even 4 1
2520.2.bi.k 4 105.x even 12 1
3920.2.a.bp 2 140.x odd 12 1
3920.2.a.bz 2 140.w even 12 1
9800.2.a.br 2 35.k even 12 1
9800.2.a.bz 2 35.l odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1400, [\chi])$$:

 $$T_{3}^{8} - 6 T_{3}^{6} + 35 T_{3}^{4} - 6 T_{3}^{2} + 1$$ $$T_{11}^{4} - 4 T_{11}^{3} + 20 T_{11}^{2} + 16 T_{11} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$1 - 6 T^{2} + 35 T^{4} - 6 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$2401 + 490 T^{2} + 51 T^{4} + 10 T^{6} + T^{8}$$
$11$ $$( 16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$13$ $$( 4 + T^{2} )^{4}$$
$17$ $$614656 - 56448 T^{2} + 4400 T^{4} - 72 T^{6} + T^{8}$$
$19$ $$( 1024 + 32 T^{2} + T^{4} )^{2}$$
$23$ $$4879681 - 225318 T^{2} + 8195 T^{4} - 102 T^{6} + T^{8}$$
$29$ $$( 17 - 10 T + T^{2} )^{4}$$
$31$ $$( 16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$37$ $$( 1024 - 32 T^{2} + T^{4} )^{2}$$
$41$ $$( 1 - 6 T + T^{2} )^{4}$$
$43$ $$( 7921 + 214 T^{2} + T^{4} )^{2}$$
$47$ $$256 - 2176 T^{2} + 18480 T^{4} - 136 T^{6} + T^{8}$$
$53$ $$( 1024 - 32 T^{2} + T^{4} )^{2}$$
$59$ $$( 16 + 4 T + T^{2} )^{4}$$
$61$ $$( 961 - 62 T + 35 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$67$ $$62742241 - 1695094 T^{2} + 37875 T^{4} - 214 T^{6} + T^{8}$$
$71$ $$( 12 + T )^{8}$$
$73$ $$614656 - 56448 T^{2} + 4400 T^{4} - 72 T^{6} + T^{8}$$
$79$ $$( 16 + 4 T + T^{2} )^{4}$$
$83$ $$( 3969 + 198 T^{2} + T^{4} )^{2}$$
$89$ $$( 7921 + 1958 T + 395 T^{2} + 22 T^{3} + T^{4} )^{2}$$
$97$ $$( 36 + T^{2} )^{4}$$