# Properties

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

# Learn more

## 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

 $$f(q)$$ $$=$$ $$q - q^{7} - 3q^{9} + O(q^{10})$$ $$q - q^{7} - 3q^{9} - 4q^{13} - 4q^{17} + 4q^{19} - 8q^{23} + 2q^{29} - 8q^{31} + 8q^{37} + 6q^{41} - 8q^{43} - 8q^{47} + q^{49} - 4q^{59} - 6q^{61} + 3q^{63} - 8q^{67} + 12q^{71} + 4q^{73} - 4q^{79} + 9q^{81} - 10q^{89} + 4q^{91} + 12q^{97} + O(q^{100})$$

## 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 0 0 0 0 −1.00000 0 −3.00000 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.f 1
3.b odd 2 1 6300.2.a.g 1
4.b odd 2 1 2800.2.a.s 1
5.b even 2 1 700.2.a.h 1
5.c odd 4 2 140.2.e.b 2
7.b odd 2 1 4900.2.a.m 1
15.d odd 2 1 6300.2.a.y 1
15.e even 4 2 1260.2.k.b 2
20.d odd 2 1 2800.2.a.o 1
20.e even 4 2 560.2.g.c 2
35.c odd 2 1 4900.2.a.l 1
35.f even 4 2 980.2.e.a 2
35.k even 12 4 980.2.q.e 4
35.l odd 12 4 980.2.q.d 4
40.i odd 4 2 2240.2.g.d 2
40.k even 4 2 2240.2.g.c 2
60.l odd 4 2 5040.2.t.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.b 2 5.c odd 4 2
560.2.g.c 2 20.e even 4 2
700.2.a.f 1 1.a even 1 1 trivial
700.2.a.h 1 5.b even 2 1
980.2.e.a 2 35.f even 4 2
980.2.q.d 4 35.l odd 12 4
980.2.q.e 4 35.k even 12 4
1260.2.k.b 2 15.e even 4 2
2240.2.g.c 2 40.k even 4 2
2240.2.g.d 2 40.i odd 4 2
2800.2.a.o 1 20.d odd 2 1
2800.2.a.s 1 4.b odd 2 1
4900.2.a.l 1 35.c odd 2 1
4900.2.a.m 1 7.b odd 2 1
5040.2.t.g 2 60.l odd 4 2
6300.2.a.g 1 3.b odd 2 1
6300.2.a.y 1 15.d odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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