# Properties

 Label 700.2.i.b Level $700$ Weight $2$ Character orbit 700.i 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( -3 + \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( -3 + \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} + ( -6 + 6 \zeta_{6} ) q^{11} -2 q^{13} + ( -6 + 6 \zeta_{6} ) q^{17} -8 \zeta_{6} q^{19} + ( -2 + 3 \zeta_{6} ) q^{21} + 3 \zeta_{6} q^{23} + 5 q^{27} + 3 q^{29} + ( -2 + 2 \zeta_{6} ) q^{31} + 6 \zeta_{6} q^{33} + 8 \zeta_{6} q^{37} + ( -2 + 2 \zeta_{6} ) q^{39} -3 q^{41} -5 q^{43} + ( 8 - 5 \zeta_{6} ) q^{49} + 6 \zeta_{6} q^{51} + ( 12 - 12 \zeta_{6} ) q^{53} -8 q^{57} + \zeta_{6} q^{61} + ( -2 - 4 \zeta_{6} ) q^{63} + ( -7 + 7 \zeta_{6} ) q^{67} + 3 q^{69} + ( -10 + 10 \zeta_{6} ) q^{73} + ( 12 - 18 \zeta_{6} ) q^{77} + 4 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -3 q^{83} + ( 3 - 3 \zeta_{6} ) q^{87} + 3 \zeta_{6} q^{89} + ( 6 - 2 \zeta_{6} ) q^{91} + 2 \zeta_{6} q^{93} + 10 q^{97} -12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 5 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q + q^{3} - 5 q^{7} + 2 q^{9} - 6 q^{11} - 4 q^{13} - 6 q^{17} - 8 q^{19} - q^{21} + 3 q^{23} + 10 q^{27} + 6 q^{29} - 2 q^{31} + 6 q^{33} + 8 q^{37} - 2 q^{39} - 6 q^{41} - 10 q^{43} + 11 q^{49} + 6 q^{51} + 12 q^{53} - 16 q^{57} + q^{61} - 8 q^{63} - 7 q^{67} + 6 q^{69} - 10 q^{73} + 6 q^{77} + 4 q^{79} - q^{81} - 6 q^{83} + 3 q^{87} + 3 q^{89} + 10 q^{91} + 2 q^{93} + 20 q^{97} - 24 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 0 0 −2.50000 0.866025i 0 1.00000 1.73205i 0
501.1 0 0.500000 0.866025i 0 0 0 −2.50000 + 0.866025i 0 1.00000 + 1.73205i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.i.b 2
5.b even 2 1 140.2.i.a 2
5.c odd 4 2 700.2.r.a 4
7.c even 3 1 inner 700.2.i.b 2
7.c even 3 1 4900.2.a.i 1
7.d odd 6 1 4900.2.a.q 1
15.d odd 2 1 1260.2.s.c 2
20.d odd 2 1 560.2.q.f 2
35.c odd 2 1 980.2.i.f 2
35.i odd 6 1 980.2.a.e 1
35.i odd 6 1 980.2.i.f 2
35.j even 6 1 140.2.i.a 2
35.j even 6 1 980.2.a.g 1
35.k even 12 2 4900.2.e.n 2
35.l odd 12 2 700.2.r.a 4
35.l odd 12 2 4900.2.e.m 2
105.o odd 6 1 1260.2.s.c 2
105.o odd 6 1 8820.2.a.p 1
105.p even 6 1 8820.2.a.a 1
140.p odd 6 1 560.2.q.f 2
140.p odd 6 1 3920.2.a.k 1
140.s even 6 1 3920.2.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.a 2 5.b even 2 1
140.2.i.a 2 35.j even 6 1
560.2.q.f 2 20.d odd 2 1
560.2.q.f 2 140.p odd 6 1
700.2.i.b 2 1.a even 1 1 trivial
700.2.i.b 2 7.c even 3 1 inner
700.2.r.a 4 5.c odd 4 2
700.2.r.a 4 35.l odd 12 2
980.2.a.e 1 35.i odd 6 1
980.2.a.g 1 35.j even 6 1
980.2.i.f 2 35.c odd 2 1
980.2.i.f 2 35.i odd 6 1
1260.2.s.c 2 15.d odd 2 1
1260.2.s.c 2 105.o odd 6 1
3920.2.a.k 1 140.p odd 6 1
3920.2.a.w 1 140.s even 6 1
4900.2.a.i 1 7.c even 3 1
4900.2.a.q 1 7.d odd 6 1
4900.2.e.m 2 35.l odd 12 2
4900.2.e.n 2 35.k even 12 2
8820.2.a.a 1 105.p even 6 1
8820.2.a.p 1 105.o odd 6 1

## Hecke kernels

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

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{11}^{2} + 6 T_{11} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$36 + 6 T + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$64 + 8 T + T^{2}$$
$23$ $$9 - 3 T + T^{2}$$
$29$ $$( -3 + T )^{2}$$
$31$ $$4 + 2 T + T^{2}$$
$37$ $$64 - 8 T + T^{2}$$
$41$ $$( 3 + T )^{2}$$
$43$ $$( 5 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$144 - 12 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$49 + 7 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$100 + 10 T + T^{2}$$
$79$ $$16 - 4 T + T^{2}$$
$83$ $$( 3 + T )^{2}$$
$89$ $$9 - 3 T + T^{2}$$
$97$ $$( -10 + T )^{2}$$