# Properties

 Label 700.2.a.c Level $700$ Weight $2$ Character orbit 700.a Self dual yes Analytic conductor $5.590$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 700.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + q^{7} + q^{9} + O(q^{10})$$ $$q - 2 q^{3} + q^{7} + q^{9} + 3 q^{11} - 4 q^{13} + 2 q^{19} - 2 q^{21} - 3 q^{23} + 4 q^{27} + 9 q^{29} + 8 q^{31} - 6 q^{33} + 5 q^{37} + 8 q^{39} - 6 q^{41} + 11 q^{43} + 6 q^{47} + q^{49} + 6 q^{53} - 4 q^{57} - 10 q^{61} + q^{63} + 5 q^{67} + 6 q^{69} + 15 q^{71} - 10 q^{73} + 3 q^{77} - 7 q^{79} - 11 q^{81} + 12 q^{83} - 18 q^{87} - 12 q^{89} - 4 q^{91} - 16 q^{93} + 8 q^{97} + 3 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −2.00000 0 0 0 1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.a.c 1
3.b odd 2 1 6300.2.a.s 1
4.b odd 2 1 2800.2.a.ba 1
5.b even 2 1 700.2.a.i yes 1
5.c odd 4 2 700.2.e.b 2
7.b odd 2 1 4900.2.a.t 1
15.d odd 2 1 6300.2.a.e 1
15.e even 4 2 6300.2.k.d 2
20.d odd 2 1 2800.2.a.e 1
20.e even 4 2 2800.2.g.e 2
35.c odd 2 1 4900.2.a.f 1
35.f even 4 2 4900.2.e.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.a.c 1 1.a even 1 1 trivial
700.2.a.i yes 1 5.b even 2 1
700.2.e.b 2 5.c odd 4 2
2800.2.a.e 1 20.d odd 2 1
2800.2.a.ba 1 4.b odd 2 1
2800.2.g.e 2 20.e even 4 2
4900.2.a.f 1 35.c odd 2 1
4900.2.a.t 1 7.b odd 2 1
4900.2.e.g 2 35.f even 4 2
6300.2.a.e 1 15.d odd 2 1
6300.2.a.s 1 3.b odd 2 1
6300.2.k.d 2 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(700))$$:

 $$T_{3} + 2$$ $$T_{11} - 3$$ $$T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$2 + T$$
$5$ $$T$$
$7$ $$-1 + T$$
$11$ $$-3 + T$$
$13$ $$4 + T$$
$17$ $$T$$
$19$ $$-2 + T$$
$23$ $$3 + T$$
$29$ $$-9 + T$$
$31$ $$-8 + T$$
$37$ $$-5 + T$$
$41$ $$6 + T$$
$43$ $$-11 + T$$
$47$ $$-6 + T$$
$53$ $$-6 + T$$
$59$ $$T$$
$61$ $$10 + T$$
$67$ $$-5 + T$$
$71$ $$-15 + T$$
$73$ $$10 + T$$
$79$ $$7 + T$$
$83$ $$-12 + T$$
$89$ $$12 + T$$
$97$ $$-8 + T$$