# Properties

 Label 700.1.bf.a Level $700$ Weight $1$ Character orbit 700.bf Analytic conductor $0.349$ Analytic rank $0$ Dimension $8$ Projective image $D_{6}$ CM discriminant -20 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 700.bf (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.349345508843$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ 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 6.0.6722800.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{11} q^{2} + ( \zeta_{24}^{5} + \zeta_{24}^{9} ) q^{3} -\zeta_{24}^{10} q^{4} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{6} + \zeta_{24}^{7} q^{7} -\zeta_{24}^{9} q^{8} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{9} +O(q^{10})$$ $$q -\zeta_{24}^{11} q^{2} + ( \zeta_{24}^{5} + \zeta_{24}^{9} ) q^{3} -\zeta_{24}^{10} q^{4} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{6} + \zeta_{24}^{7} q^{7} -\zeta_{24}^{9} q^{8} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{9} + ( \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{12} + \zeta_{24}^{6} q^{14} -\zeta_{24}^{8} q^{16} + ( -\zeta_{24} - \zeta_{24}^{5} + \zeta_{24}^{9} ) q^{18} + ( -1 - \zeta_{24}^{4} ) q^{21} -\zeta_{24}^{5} q^{23} + ( \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{24} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{27} + \zeta_{24}^{5} q^{28} + \zeta_{24}^{6} q^{29} -\zeta_{24}^{7} q^{32} + ( -1 - \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{36} + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{41} + ( -\zeta_{24}^{3} + \zeta_{24}^{11} ) q^{42} -\zeta_{24}^{9} q^{43} -\zeta_{24}^{4} q^{46} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{48} -\zeta_{24}^{2} q^{49} + ( -\zeta_{24}^{6} - \zeta_{24}^{10} ) q^{54} + \zeta_{24}^{4} q^{56} + \zeta_{24}^{5} q^{58} + ( 1 + \zeta_{24}^{4} ) q^{61} + ( \zeta_{24} - \zeta_{24}^{5} - \zeta_{24}^{9} ) q^{63} -\zeta_{24}^{6} q^{64} + \zeta_{24}^{7} q^{67} + ( \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{69} + ( -\zeta_{24}^{3} + \zeta_{24}^{7} + \zeta_{24}^{11} ) q^{72} + \zeta_{24}^{4} q^{81} + ( -\zeta_{24}^{3} - \zeta_{24}^{7} ) q^{82} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{83} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{84} -\zeta_{24}^{8} q^{86} + ( -\zeta_{24}^{3} + \zeta_{24}^{11} ) q^{87} + ( -\zeta_{24}^{6} - \zeta_{24}^{10} ) q^{89} -\zeta_{24}^{3} q^{92} + ( 1 + \zeta_{24}^{4} ) q^{96} -\zeta_{24} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 4q^{16} - 12q^{21} - 16q^{36} - 4q^{46} + 4q^{56} + 12q^{61} + 4q^{81} + 4q^{86} + 12q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$-\zeta_{24}^{8}$$ $$-1$$ $$-\zeta_{24}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 −0.258819 − 0.965926i 0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i
−0.258819 + 0.965926i −1.67303 + 0.448288i −0.866025 0.500000i 0 1.73205i 0.965926 0.258819i 0.707107 0.707107i 1.73205 1.00000i 0
143.2 0.258819 0.965926i 1.67303 0.448288i −0.866025 0.500000i 0 1.73205i −0.965926 + 0.258819i −0.707107 + 0.707107i 1.73205 1.00000i 0
243.1 −0.965926 + 0.258819i 0.448288 1.67303i 0.866025 0.500000i 0 1.73205i 0.258819 0.965926i −0.707107 + 0.707107i −1.73205 1.00000i 0
243.2 0.965926 0.258819i −0.448288 + 1.67303i 0.866025 0.500000i 0 1.73205i −0.258819 + 0.965926i 0.707107 0.707107i −1.73205 1.00000i 0
507.1 −0.965926 0.258819i 0.448288 + 1.67303i 0.866025 + 0.500000i 0 1.73205i 0.258819 + 0.965926i −0.707107 0.707107i −1.73205 + 1.00000i 0
507.2 0.965926 + 0.258819i −0.448288 1.67303i 0.866025 + 0.500000i 0 1.73205i −0.258819 0.965926i 0.707107 + 0.707107i −1.73205 + 1.00000i 0
607.1 −0.258819 0.965926i −1.67303 0.448288i −0.866025 + 0.500000i 0 1.73205i 0.965926 + 0.258819i 0.707107 + 0.707107i 1.73205 + 1.00000i 0
607.2 0.258819 + 0.965926i 1.67303 + 0.448288i −0.866025 + 0.500000i 0 1.73205i −0.965926 0.258819i −0.707107 0.707107i 1.73205 + 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 607.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
7.d odd 6 1 inner
20.e even 4 2 inner
28.f even 6 1 inner
35.i odd 6 1 inner
35.k even 12 2 inner
140.s even 6 1 inner
140.x odd 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.1.bf.a 8
4.b odd 2 1 inner 700.1.bf.a 8
5.b even 2 1 inner 700.1.bf.a 8
5.c odd 4 2 inner 700.1.bf.a 8
7.d odd 6 1 inner 700.1.bf.a 8
20.d odd 2 1 CM 700.1.bf.a 8
20.e even 4 2 inner 700.1.bf.a 8
28.f even 6 1 inner 700.1.bf.a 8
35.i odd 6 1 inner 700.1.bf.a 8
35.k even 12 2 inner 700.1.bf.a 8
140.s even 6 1 inner 700.1.bf.a 8
140.x odd 12 2 inner 700.1.bf.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.1.bf.a 8 1.a even 1 1 trivial
700.1.bf.a 8 4.b odd 2 1 inner
700.1.bf.a 8 5.b even 2 1 inner
700.1.bf.a 8 5.c odd 4 2 inner
700.1.bf.a 8 7.d odd 6 1 inner
700.1.bf.a 8 20.d odd 2 1 CM
700.1.bf.a 8 20.e even 4 2 inner
700.1.bf.a 8 28.f even 6 1 inner
700.1.bf.a 8 35.i odd 6 1 inner
700.1.bf.a 8 35.k even 12 2 inner
700.1.bf.a 8 140.s even 6 1 inner
700.1.bf.a 8 140.x odd 12 2 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(700, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ $$81 - 9 T^{4} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$1 - T^{4} + T^{8}$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$1 - T^{4} + T^{8}$$
$29$ $$( 1 + T^{2} )^{4}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$( 3 + T^{2} )^{4}$$
$43$ $$( 1 + T^{4} )^{2}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$( 3 - 3 T + T^{2} )^{4}$$
$67$ $$1 - T^{4} + T^{8}$$
$71$ $$T^{8}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$( 9 + T^{4} )^{2}$$
$89$ $$( 9 + 3 T^{2} + T^{4} )^{2}$$
$97$ $$T^{8}$$