Properties

 Label 700.1.d.a Level $700$ Weight $1$ Character orbit 700.d Analytic conductor $0.349$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -35 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 700.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.349345508843$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.140.1 Artin image: $C_4\times S_3$ Artin field: Galois closure of 12.0.1200500000000.1

$q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -i q^{3} -i q^{7} +O(q^{10})$$ $$q -i q^{3} -i q^{7} - q^{11} -i q^{13} + i q^{17} - q^{21} -i q^{27} + q^{29} + i q^{33} - q^{39} + i q^{47} - q^{49} + q^{51} + 2 q^{71} + 2 i q^{73} + i q^{77} + q^{79} - q^{81} + 2 i q^{83} -i q^{87} - q^{91} + i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 2q^{11} - 2q^{21} + 2q^{29} - 2q^{39} - 2q^{49} + 2q^{51} + 4q^{71} + 2q^{79} - 2q^{81} - 2q^{91} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
601.1
 1.00000i − 1.00000i
0 1.00000i 0 0 0 1.00000i 0 0 0
601.2 0 1.00000i 0 0 0 1.00000i 0 0 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
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
5.b even 2 1 inner
7.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.1.d.a 2
4.b odd 2 1 2800.1.f.c 2
5.b even 2 1 inner 700.1.d.a 2
5.c odd 4 1 140.1.h.a 1
5.c odd 4 1 140.1.h.b yes 1
7.b odd 2 1 inner 700.1.d.a 2
15.e even 4 1 1260.1.p.a 1
15.e even 4 1 1260.1.p.b 1
20.d odd 2 1 2800.1.f.c 2
20.e even 4 1 560.1.p.a 1
20.e even 4 1 560.1.p.b 1
28.d even 2 1 2800.1.f.c 2
35.c odd 2 1 CM 700.1.d.a 2
35.f even 4 1 140.1.h.a 1
35.f even 4 1 140.1.h.b yes 1
35.k even 12 2 980.1.n.a 2
35.k even 12 2 980.1.n.b 2
35.l odd 12 2 980.1.n.a 2
35.l odd 12 2 980.1.n.b 2
40.i odd 4 1 2240.1.p.b 1
40.i odd 4 1 2240.1.p.c 1
40.k even 4 1 2240.1.p.a 1
40.k even 4 1 2240.1.p.d 1
105.k odd 4 1 1260.1.p.a 1
105.k odd 4 1 1260.1.p.b 1
140.c even 2 1 2800.1.f.c 2
140.j odd 4 1 560.1.p.a 1
140.j odd 4 1 560.1.p.b 1
140.w even 12 2 3920.1.br.a 2
140.w even 12 2 3920.1.br.b 2
140.x odd 12 2 3920.1.br.a 2
140.x odd 12 2 3920.1.br.b 2
280.s even 4 1 2240.1.p.b 1
280.s even 4 1 2240.1.p.c 1
280.y odd 4 1 2240.1.p.a 1
280.y odd 4 1 2240.1.p.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.h.a 1 5.c odd 4 1
140.1.h.a 1 35.f even 4 1
140.1.h.b yes 1 5.c odd 4 1
140.1.h.b yes 1 35.f even 4 1
560.1.p.a 1 20.e even 4 1
560.1.p.a 1 140.j odd 4 1
560.1.p.b 1 20.e even 4 1
560.1.p.b 1 140.j odd 4 1
700.1.d.a 2 1.a even 1 1 trivial
700.1.d.a 2 5.b even 2 1 inner
700.1.d.a 2 7.b odd 2 1 inner
700.1.d.a 2 35.c odd 2 1 CM
980.1.n.a 2 35.k even 12 2
980.1.n.a 2 35.l odd 12 2
980.1.n.b 2 35.k even 12 2
980.1.n.b 2 35.l odd 12 2
1260.1.p.a 1 15.e even 4 1
1260.1.p.a 1 105.k odd 4 1
1260.1.p.b 1 15.e even 4 1
1260.1.p.b 1 105.k odd 4 1
2240.1.p.a 1 40.k even 4 1
2240.1.p.a 1 280.y odd 4 1
2240.1.p.b 1 40.i odd 4 1
2240.1.p.b 1 280.s even 4 1
2240.1.p.c 1 40.i odd 4 1
2240.1.p.c 1 280.s even 4 1
2240.1.p.d 1 40.k even 4 1
2240.1.p.d 1 280.y odd 4 1
2800.1.f.c 2 4.b odd 2 1
2800.1.f.c 2 20.d odd 2 1
2800.1.f.c 2 28.d even 2 1
2800.1.f.c 2 140.c even 2 1
3920.1.br.a 2 140.w even 12 2
3920.1.br.a 2 140.x odd 12 2
3920.1.br.b 2 140.w even 12 2
3920.1.br.b 2 140.x odd 12 2

Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(700, [\chi])$$.

Hecke characteristic polynomials

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