# Properties

 Label 700.2.g.d Level $700$ Weight $2$ Character orbit 700.g Analytic conductor $5.590$ Analytic rank $0$ Dimension $4$ CM discriminant -20 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( 2 \beta_{1} - \beta_{2} ) q^{3} -2 q^{4} + 2 \beta_{3} q^{6} + ( \beta_{1} - 2 \beta_{2} ) q^{7} -2 \beta_{2} q^{8} + 7 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( 2 \beta_{1} - \beta_{2} ) q^{3} -2 q^{4} + 2 \beta_{3} q^{6} + ( \beta_{1} - 2 \beta_{2} ) q^{7} -2 \beta_{2} q^{8} + 7 q^{9} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{12} + ( 3 + \beta_{3} ) q^{14} + 4 q^{16} + 7 \beta_{2} q^{18} + ( 5 - 3 \beta_{3} ) q^{21} + \beta_{2} q^{23} -4 \beta_{3} q^{24} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{27} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{28} -6 q^{29} + 4 \beta_{2} q^{32} -14 q^{36} + 2 \beta_{3} q^{41} + ( 6 \beta_{1} + 2 \beta_{2} ) q^{42} + 9 \beta_{2} q^{43} -2 q^{46} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{47} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{48} + ( -2 - 3 \beta_{3} ) q^{49} + 8 \beta_{3} q^{54} + ( -6 - 2 \beta_{3} ) q^{56} -6 \beta_{2} q^{58} -6 \beta_{3} q^{61} + ( 7 \beta_{1} - 14 \beta_{2} ) q^{63} -8 q^{64} + 3 \beta_{2} q^{67} + 2 \beta_{3} q^{69} -14 \beta_{2} q^{72} + 19 q^{81} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{82} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -10 + 6 \beta_{3} ) q^{84} -18 q^{86} + ( -12 \beta_{1} + 6 \beta_{2} ) q^{87} -8 \beta_{3} q^{89} -2 \beta_{2} q^{92} -6 \beta_{3} q^{94} + 8 \beta_{3} q^{96} + ( 6 \beta_{1} - 5 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} + 28q^{9} + O(q^{10})$$ $$4q - 8q^{4} + 28q^{9} + 12q^{14} + 16q^{16} + 20q^{21} - 24q^{29} - 56q^{36} - 8q^{46} - 8q^{49} - 24q^{56} - 32q^{64} + 76q^{81} - 40q^{84} - 72q^{86} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 2$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{2} + \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 −1.58114 − 0.707107i 1.58114 − 0.707107i −1.58114 + 0.707107i 1.58114 + 0.707107i
1.41421i −3.16228 −2.00000 0 4.47214i −1.58114 + 2.12132i 2.82843i 7.00000 0
251.2 1.41421i 3.16228 −2.00000 0 4.47214i 1.58114 + 2.12132i 2.82843i 7.00000 0
251.3 1.41421i −3.16228 −2.00000 0 4.47214i −1.58114 2.12132i 2.82843i 7.00000 0
251.4 1.41421i 3.16228 −2.00000 0 4.47214i 1.58114 2.12132i 2.82843i 7.00000 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c 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.g.d 4
4.b odd 2 1 inner 700.2.g.d 4
5.b even 2 1 inner 700.2.g.d 4
5.c odd 4 2 140.2.c.a 4
7.b odd 2 1 inner 700.2.g.d 4
20.d odd 2 1 CM 700.2.g.d 4
20.e even 4 2 140.2.c.a 4
28.d even 2 1 inner 700.2.g.d 4
35.c odd 2 1 inner 700.2.g.d 4
35.f even 4 2 140.2.c.a 4
35.k even 12 4 980.2.s.b 8
35.l odd 12 4 980.2.s.b 8
40.i odd 4 2 2240.2.e.a 4
40.k even 4 2 2240.2.e.a 4
140.c even 2 1 inner 700.2.g.d 4
140.j odd 4 2 140.2.c.a 4
140.w even 12 4 980.2.s.b 8
140.x odd 12 4 980.2.s.b 8
280.s even 4 2 2240.2.e.a 4
280.y odd 4 2 2240.2.e.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.c.a 4 5.c odd 4 2
140.2.c.a 4 20.e even 4 2
140.2.c.a 4 35.f even 4 2
140.2.c.a 4 140.j odd 4 2
700.2.g.d 4 1.a even 1 1 trivial
700.2.g.d 4 4.b odd 2 1 inner
700.2.g.d 4 5.b even 2 1 inner
700.2.g.d 4 7.b odd 2 1 inner
700.2.g.d 4 20.d odd 2 1 CM
700.2.g.d 4 28.d even 2 1 inner
700.2.g.d 4 35.c odd 2 1 inner
700.2.g.d 4 140.c even 2 1 inner
980.2.s.b 8 35.k even 12 4
980.2.s.b 8 35.l odd 12 4
980.2.s.b 8 140.w even 12 4
980.2.s.b 8 140.x odd 12 4
2240.2.e.a 4 40.i odd 4 2
2240.2.e.a 4 40.k even 4 2
2240.2.e.a 4 280.s even 4 2
2240.2.e.a 4 280.y odd 4 2

## Hecke kernels

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

 $$T_{3}^{2} - 10$$ $$T_{11}$$ $$T_{19}$$ $$T_{37}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$( -10 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$49 + 4 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$( 2 + T^{2} )^{2}$$
$29$ $$( 6 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( 20 + T^{2} )^{2}$$
$43$ $$( 162 + T^{2} )^{2}$$
$47$ $$( -90 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 180 + T^{2} )^{2}$$
$67$ $$( 18 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$( -90 + T^{2} )^{2}$$
$89$ $$( 320 + T^{2} )^{2}$$
$97$ $$T^{4}$$