# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 700.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} + i q^{7} - q^{9} +O(q^{10})$$ $$q + 2 i q^{3} + i q^{7} - q^{9} + 3 q^{11} + 4 i q^{13} -2 q^{19} -2 q^{21} + 3 i q^{23} + 4 i q^{27} -9 q^{29} + 8 q^{31} + 6 i q^{33} + 5 i q^{37} -8 q^{39} -6 q^{41} -11 i q^{43} + 6 i q^{47} - q^{49} -6 i q^{53} -4 i q^{57} -10 q^{61} -i q^{63} + 5 i q^{67} -6 q^{69} + 15 q^{71} + 10 i q^{73} + 3 i q^{77} + 7 q^{79} -11 q^{81} -12 i q^{83} -18 i q^{87} + 12 q^{89} -4 q^{91} + 16 i q^{93} + 8 i q^{97} -3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{9} + 6 q^{11} - 4 q^{19} - 4 q^{21} - 18 q^{29} + 16 q^{31} - 16 q^{39} - 12 q^{41} - 2 q^{49} - 20 q^{61} - 12 q^{69} + 30 q^{71} + 14 q^{79} - 22 q^{81} + 24 q^{89} - 8 q^{91} - 6 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 1.00000i 0 −1.00000 0
449.2 0 2.00000i 0 0 0 1.00000i 0 −1.00000 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
5.b even 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(700, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$9 + T^{2}$$
$29$ $$( 9 + T )^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$25 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$121 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$25 + T^{2}$$
$71$ $$( -15 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( -7 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( -12 + T )^{2}$$
$97$ $$64 + T^{2}$$