# Properties

 Label 700.2.p.a Level $700$ Weight $2$ Character orbit 700.p Analytic conductor $5.590$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 700.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{4} + ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{6} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} +O(q^{10})$$ $$q + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{4} + ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{6} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + \zeta_{12} q^{11} + ( -4 + 2 \zeta_{12}^{2} ) q^{12} + ( -2 + 4 \zeta_{12}^{2} ) q^{13} + ( 2 + 3 \zeta_{12} - 3 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{14} + 4 \zeta_{12}^{2} q^{16} + ( 1 + \zeta_{12}^{2} ) q^{17} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{19} + ( -1 + 5 \zeta_{12}^{2} ) q^{21} + ( 1 + \zeta_{12}^{3} ) q^{22} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{23} + ( -2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{24} + ( -4 + 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{26} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( 6 - 2 \zeta_{12}^{2} ) q^{28} + 4 q^{29} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{31} + ( -4 + 4 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{32} + ( -2 + \zeta_{12}^{2} ) q^{33} + ( -1 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{34} + 3 \zeta_{12}^{2} q^{37} + ( -3 + 3 \zeta_{12} - 3 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{38} -6 \zeta_{12} q^{39} + ( -2 + 4 \zeta_{12}^{2} ) q^{41} + ( -5 + 4 \zeta_{12} + 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{42} -2 \zeta_{12}^{3} q^{43} + 2 \zeta_{12}^{2} q^{44} + ( -1 - \zeta_{12} + \zeta_{12}^{2} ) q^{46} + ( -5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{47} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{48} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{51} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{52} + ( -1 + \zeta_{12}^{2} ) q^{53} + ( -6 - 3 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{54} + ( 2 + 4 \zeta_{12} + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{56} + 9 q^{57} + ( 4 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{58} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{59} + ( -6 + 3 \zeta_{12}^{2} ) q^{61} + ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{62} + 8 \zeta_{12}^{3} q^{64} + ( -1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{66} -3 \zeta_{12} q^{67} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{68} + ( 1 - 2 \zeta_{12}^{2} ) q^{69} -14 \zeta_{12}^{3} q^{71} + ( -5 - 5 \zeta_{12}^{2} ) q^{73} + ( -3 + 3 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{74} + ( 6 - 12 \zeta_{12}^{2} ) q^{76} + ( 3 - \zeta_{12}^{2} ) q^{77} + ( -6 - 6 \zeta_{12}^{3} ) q^{78} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{79} + ( 9 - 9 \zeta_{12}^{2} ) q^{81} + ( -4 + 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{82} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{83} + ( -2 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{84} + ( 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{86} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{87} + ( -2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{88} + ( 18 - 9 \zeta_{12}^{2} ) q^{89} + ( 8 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{91} -2 q^{92} -3 \zeta_{12}^{2} q^{93} + ( -5 + 5 \zeta_{12} - 5 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{94} + ( -8 - 4 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{96} + ( 10 - 20 \zeta_{12}^{2} ) q^{97} + ( 8 - 5 \zeta_{12} - 5 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 8 q^{8} + O(q^{10})$$ $$4 q + 2 q^{2} + 8 q^{8} - 12 q^{12} + 2 q^{14} + 8 q^{16} + 6 q^{17} + 6 q^{21} + 4 q^{22} - 12 q^{24} - 12 q^{26} + 20 q^{28} + 16 q^{29} - 8 q^{32} - 6 q^{33} + 6 q^{37} - 18 q^{38} - 12 q^{42} + 4 q^{44} - 2 q^{46} - 4 q^{49} - 2 q^{53} - 18 q^{54} + 16 q^{56} + 36 q^{57} + 8 q^{58} - 18 q^{61} - 6 q^{66} - 30 q^{73} - 6 q^{74} + 10 q^{77} - 24 q^{78} + 18 q^{81} - 12 q^{82} - 4 q^{86} - 4 q^{88} + 54 q^{89} - 8 q^{92} - 6 q^{93} - 30 q^{94} - 24 q^{96} + 22 q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.366025 + 1.36603i 0.866025 1.50000i −1.73205 1.00000i 0 1.73205 + 1.73205i −1.73205 + 2.00000i 2.00000 2.00000i 0 0
451.2 1.36603 + 0.366025i −0.866025 + 1.50000i 1.73205 + 1.00000i 0 −1.73205 + 1.73205i 1.73205 2.00000i 2.00000 + 2.00000i 0 0
551.1 −0.366025 1.36603i 0.866025 + 1.50000i −1.73205 + 1.00000i 0 1.73205 1.73205i −1.73205 2.00000i 2.00000 + 2.00000i 0 0
551.2 1.36603 0.366025i −0.866025 1.50000i 1.73205 1.00000i 0 −1.73205 1.73205i 1.73205 + 2.00000i 2.00000 2.00000i 0 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
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.p.a 4
4.b odd 2 1 inner 700.2.p.a 4
5.b even 2 1 28.2.f.a 4
5.c odd 4 1 700.2.t.a 4
5.c odd 4 1 700.2.t.b 4
7.d odd 6 1 inner 700.2.p.a 4
15.d odd 2 1 252.2.bf.e 4
20.d odd 2 1 28.2.f.a 4
20.e even 4 1 700.2.t.a 4
20.e even 4 1 700.2.t.b 4
28.f even 6 1 inner 700.2.p.a 4
35.c odd 2 1 196.2.f.a 4
35.i odd 6 1 28.2.f.a 4
35.i odd 6 1 196.2.d.b 4
35.j even 6 1 196.2.d.b 4
35.j even 6 1 196.2.f.a 4
35.k even 12 1 700.2.t.a 4
35.k even 12 1 700.2.t.b 4
40.e odd 2 1 448.2.p.d 4
40.f even 2 1 448.2.p.d 4
60.h even 2 1 252.2.bf.e 4
105.o odd 6 1 1764.2.b.a 4
105.p even 6 1 252.2.bf.e 4
105.p even 6 1 1764.2.b.a 4
140.c even 2 1 196.2.f.a 4
140.p odd 6 1 196.2.d.b 4
140.p odd 6 1 196.2.f.a 4
140.s even 6 1 28.2.f.a 4
140.s even 6 1 196.2.d.b 4
140.x odd 12 1 700.2.t.a 4
140.x odd 12 1 700.2.t.b 4
280.ba even 6 1 448.2.p.d 4
280.ba even 6 1 3136.2.f.e 4
280.bf even 6 1 3136.2.f.e 4
280.bi odd 6 1 3136.2.f.e 4
280.bk odd 6 1 448.2.p.d 4
280.bk odd 6 1 3136.2.f.e 4
420.ba even 6 1 1764.2.b.a 4
420.be odd 6 1 252.2.bf.e 4
420.be odd 6 1 1764.2.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 5.b even 2 1
28.2.f.a 4 20.d odd 2 1
28.2.f.a 4 35.i odd 6 1
28.2.f.a 4 140.s even 6 1
196.2.d.b 4 35.i odd 6 1
196.2.d.b 4 35.j even 6 1
196.2.d.b 4 140.p odd 6 1
196.2.d.b 4 140.s even 6 1
196.2.f.a 4 35.c odd 2 1
196.2.f.a 4 35.j even 6 1
196.2.f.a 4 140.c even 2 1
196.2.f.a 4 140.p odd 6 1
252.2.bf.e 4 15.d odd 2 1
252.2.bf.e 4 60.h even 2 1
252.2.bf.e 4 105.p even 6 1
252.2.bf.e 4 420.be odd 6 1
448.2.p.d 4 40.e odd 2 1
448.2.p.d 4 40.f even 2 1
448.2.p.d 4 280.ba even 6 1
448.2.p.d 4 280.bk odd 6 1
700.2.p.a 4 1.a even 1 1 trivial
700.2.p.a 4 4.b odd 2 1 inner
700.2.p.a 4 7.d odd 6 1 inner
700.2.p.a 4 28.f even 6 1 inner
700.2.t.a 4 5.c odd 4 1
700.2.t.a 4 20.e even 4 1
700.2.t.a 4 35.k even 12 1
700.2.t.a 4 140.x odd 12 1
700.2.t.b 4 5.c odd 4 1
700.2.t.b 4 20.e even 4 1
700.2.t.b 4 35.k even 12 1
700.2.t.b 4 140.x odd 12 1
1764.2.b.a 4 105.o odd 6 1
1764.2.b.a 4 105.p even 6 1
1764.2.b.a 4 420.ba even 6 1
1764.2.b.a 4 420.be odd 6 1
3136.2.f.e 4 280.ba even 6 1
3136.2.f.e 4 280.bf even 6 1
3136.2.f.e 4 280.bi odd 6 1
3136.2.f.e 4 280.bk odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(700, [\chi])$$:

 $$T_{3}^{4} + 3 T_{3}^{2} + 9$$ $$T_{17}^{2} - 3 T_{17} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$9 + 3 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$49 + 2 T^{2} + T^{4}$$
$11$ $$1 - T^{2} + T^{4}$$
$13$ $$( 12 + T^{2} )^{2}$$
$17$ $$( 3 - 3 T + T^{2} )^{2}$$
$19$ $$729 + 27 T^{2} + T^{4}$$
$23$ $$1 - T^{2} + T^{4}$$
$29$ $$( -4 + T )^{4}$$
$31$ $$9 + 3 T^{2} + T^{4}$$
$37$ $$( 9 - 3 T + T^{2} )^{2}$$
$41$ $$( 12 + T^{2} )^{2}$$
$43$ $$( 4 + T^{2} )^{2}$$
$47$ $$5625 + 75 T^{2} + T^{4}$$
$53$ $$( 1 + T + T^{2} )^{2}$$
$59$ $$729 + 27 T^{2} + T^{4}$$
$61$ $$( 27 + 9 T + T^{2} )^{2}$$
$67$ $$81 - 9 T^{2} + T^{4}$$
$71$ $$( 196 + T^{2} )^{2}$$
$73$ $$( 75 + 15 T + T^{2} )^{2}$$
$79$ $$6561 - 81 T^{2} + T^{4}$$
$83$ $$( -192 + T^{2} )^{2}$$
$89$ $$( 243 - 27 T + T^{2} )^{2}$$
$97$ $$( 300 + T^{2} )^{2}$$