Properties

 Label 700.1.j.a Level $700$ Weight $1$ Character orbit 700.j Analytic conductor $0.349$ Analytic rank $0$ Dimension $4$ Projective image $D_{2}$ CM/RM discs -7, -20, 140 Inner twists $16$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 700.j (of order $$4$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.349345508843$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-5}, \sqrt{-7})$$ Artin image: $OD_{16}:C_2$ Artin field: Galois closure of 16.0.960400000000000000.1

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{8}^{3} q^{2} -\zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{7} -\zeta_{8} q^{8} + \zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{8}^{3} q^{2} -\zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{7} -\zeta_{8} q^{8} + \zeta_{8}^{2} q^{9} -\zeta_{8}^{2} q^{14} - q^{16} + \zeta_{8} q^{18} -2 \zeta_{8} q^{23} -\zeta_{8} q^{28} + 2 \zeta_{8}^{2} q^{29} + \zeta_{8}^{3} q^{32} + q^{36} + 2 \zeta_{8} q^{43} -2 q^{46} -\zeta_{8}^{2} q^{49} - q^{56} + 2 \zeta_{8} q^{58} + \zeta_{8} q^{63} + \zeta_{8}^{2} q^{64} + 2 \zeta_{8}^{3} q^{67} -\zeta_{8}^{3} q^{72} - q^{81} + 2 q^{86} + 2 \zeta_{8}^{3} q^{92} -\zeta_{8} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 4q^{16} + 4q^{36} - 8q^{46} - 4q^{56} - 4q^{81} + 8q^{86} + O(q^{100})$$

Character values

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

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{8}^{2}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
−0.707107 + 0.707107i 0 1.00000i 0 0 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0
307.2 0.707107 0.707107i 0 1.00000i 0 0 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0
643.1 −0.707107 0.707107i 0 1.00000i 0 0 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0
643.2 0.707107 + 0.707107i 0 1.00000i 0 0 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 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.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
140.c even 2 1 RM by $$\Q(\sqrt{35})$$
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.e even 4 2 inner
28.d even 2 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner
140.j odd 4 2 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.1.j.a 4
4.b odd 2 1 inner 700.1.j.a 4
5.b even 2 1 inner 700.1.j.a 4
5.c odd 4 2 inner 700.1.j.a 4
7.b odd 2 1 CM 700.1.j.a 4
20.d odd 2 1 CM 700.1.j.a 4
20.e even 4 2 inner 700.1.j.a 4
28.d even 2 1 inner 700.1.j.a 4
35.c odd 2 1 inner 700.1.j.a 4
35.f even 4 2 inner 700.1.j.a 4
140.c even 2 1 RM 700.1.j.a 4
140.j odd 4 2 inner 700.1.j.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.1.j.a 4 1.a even 1 1 trivial
700.1.j.a 4 4.b odd 2 1 inner
700.1.j.a 4 5.b even 2 1 inner
700.1.j.a 4 5.c odd 4 2 inner
700.1.j.a 4 7.b odd 2 1 CM
700.1.j.a 4 20.d odd 2 1 CM
700.1.j.a 4 20.e even 4 2 inner
700.1.j.a 4 28.d even 2 1 inner
700.1.j.a 4 35.c odd 2 1 inner
700.1.j.a 4 35.f even 4 2 inner
700.1.j.a 4 140.c even 2 1 RM
700.1.j.a 4 140.j odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}$$ acting on $$S_{1}^{\mathrm{new}}(700, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$1 + T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$16 + T^{4}$$
$29$ $$( 4 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$16 + T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$16 + T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$