# Properties

 Label 700.2.a.b Level $700$ Weight $2$ Character orbit 700.a Self dual yes Analytic conductor $5.590$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3q^{3} + q^{7} + 6q^{9} + O(q^{10})$$ $$q - 3q^{3} + q^{7} + 6q^{9} - 5q^{11} + 3q^{13} + q^{17} + 6q^{19} - 3q^{21} - 6q^{23} - 9q^{27} - 9q^{29} - 4q^{31} + 15q^{33} - 2q^{37} - 9q^{39} - 4q^{41} - 10q^{43} + q^{47} + q^{49} - 3q^{51} - 4q^{53} - 18q^{57} - 8q^{59} - 8q^{61} + 6q^{63} - 12q^{67} + 18q^{69} + 8q^{71} - 2q^{73} - 5q^{77} + 13q^{79} + 9q^{81} + 4q^{83} + 27q^{87} + 4q^{89} + 3q^{91} + 12q^{93} + 13q^{97} - 30q^{99} + O(q^{100})$$

## 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 −3.00000 0 0 0 1.00000 0 6.00000 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.b 1
3.b odd 2 1 6300.2.a.bf 1
4.b odd 2 1 2800.2.a.be 1
5.b even 2 1 140.2.a.b 1
5.c odd 4 2 700.2.e.a 2
7.b odd 2 1 4900.2.a.u 1
15.d odd 2 1 1260.2.a.h 1
15.e even 4 2 6300.2.k.p 2
20.d odd 2 1 560.2.a.a 1
20.e even 4 2 2800.2.g.c 2
35.c odd 2 1 980.2.a.b 1
35.f even 4 2 4900.2.e.a 2
35.i odd 6 2 980.2.i.j 2
35.j even 6 2 980.2.i.b 2
40.e odd 2 1 2240.2.a.bb 1
40.f even 2 1 2240.2.a.c 1
60.h even 2 1 5040.2.a.bd 1
105.g even 2 1 8820.2.a.n 1
140.c even 2 1 3920.2.a.bl 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.a.b 1 5.b even 2 1
560.2.a.a 1 20.d odd 2 1
700.2.a.b 1 1.a even 1 1 trivial
700.2.e.a 2 5.c odd 4 2
980.2.a.b 1 35.c odd 2 1
980.2.i.b 2 35.j even 6 2
980.2.i.j 2 35.i odd 6 2
1260.2.a.h 1 15.d odd 2 1
2240.2.a.c 1 40.f even 2 1
2240.2.a.bb 1 40.e odd 2 1
2800.2.a.be 1 4.b odd 2 1
2800.2.g.c 2 20.e even 4 2
3920.2.a.bl 1 140.c even 2 1
4900.2.a.u 1 7.b odd 2 1
4900.2.e.a 2 35.f even 4 2
5040.2.a.bd 1 60.h even 2 1
6300.2.a.bf 1 3.b odd 2 1
6300.2.k.p 2 15.e even 4 2
8820.2.a.n 1 105.g even 2 1

## Hecke kernels

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

 $$T_{3} + 3$$ $$T_{11} + 5$$ $$T_{13} - 3$$

## Hecke characteristic polynomials

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