# Properties

 Label 700.2.t.a Level $700$ Weight $2$ Character orbit 700.t Analytic conductor $5.590$ Analytic rank $1$ 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.t (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$1$$ 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.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−1.36603 0.366025i −1.50000 0.866025i 1.73205 + 1.00000i 0 1.73205 + 1.73205i −2.00000 1.73205i −2.00000 2.00000i 0 0
199.2 0.366025 1.36603i −1.50000 0.866025i −1.73205 1.00000i 0 −1.73205 + 1.73205i −2.00000 1.73205i −2.00000 + 2.00000i 0 0
299.1 −1.36603 + 0.366025i −1.50000 + 0.866025i 1.73205 1.00000i 0 1.73205 1.73205i −2.00000 + 1.73205i −2.00000 + 2.00000i 0 0
299.2 0.366025 + 1.36603i −1.50000 + 0.866025i −1.73205 + 1.00000i 0 −1.73205 1.73205i −2.00000 + 1.73205i −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
7.d odd 6 1 inner
20.d odd 2 1 inner
140.s 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.t.a 4
4.b odd 2 1 700.2.t.b 4
5.b even 2 1 700.2.t.b 4
5.c odd 4 1 28.2.f.a 4
5.c odd 4 1 700.2.p.a 4
7.d odd 6 1 inner 700.2.t.a 4
15.e even 4 1 252.2.bf.e 4
20.d odd 2 1 inner 700.2.t.a 4
20.e even 4 1 28.2.f.a 4
20.e even 4 1 700.2.p.a 4
28.f even 6 1 700.2.t.b 4
35.f even 4 1 196.2.f.a 4
35.i odd 6 1 700.2.t.b 4
35.k even 12 1 28.2.f.a 4
35.k even 12 1 196.2.d.b 4
35.k even 12 1 700.2.p.a 4
35.l odd 12 1 196.2.d.b 4
35.l odd 12 1 196.2.f.a 4
40.i odd 4 1 448.2.p.d 4
40.k even 4 1 448.2.p.d 4
60.l odd 4 1 252.2.bf.e 4
105.w odd 12 1 252.2.bf.e 4
105.w odd 12 1 1764.2.b.a 4
105.x even 12 1 1764.2.b.a 4
140.j odd 4 1 196.2.f.a 4
140.s even 6 1 inner 700.2.t.a 4
140.w even 12 1 196.2.d.b 4
140.w even 12 1 196.2.f.a 4
140.x odd 12 1 28.2.f.a 4
140.x odd 12 1 196.2.d.b 4
140.x odd 12 1 700.2.p.a 4
280.bp odd 12 1 448.2.p.d 4
280.bp odd 12 1 3136.2.f.e 4
280.br even 12 1 3136.2.f.e 4
280.bt odd 12 1 3136.2.f.e 4
280.bv even 12 1 448.2.p.d 4
280.bv even 12 1 3136.2.f.e 4
420.bp odd 12 1 1764.2.b.a 4
420.br even 12 1 252.2.bf.e 4
420.br even 12 1 1764.2.b.a 4

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

## Hecke kernels

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

 $$T_{3}^{2} + 3 T_{3} + 3$$ $$T_{13}^{2} - 12$$

## Hecke characteristic polynomials

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