# Properties

 Label 700.2.c.d Level $700$ Weight $2$ Character orbit 700.c Analytic conductor $5.590$ Analytic rank $0$ Dimension $4$ CM discriminant -7 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{7})$$ Defining polynomial: $$x^{4} - 3 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -2 \beta_{1} + \beta_{3} ) q^{7} + ( \beta_{1} + 2 \beta_{3} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -2 \beta_{1} + \beta_{3} ) q^{7} + ( \beta_{1} + 2 \beta_{3} ) q^{8} + 3 q^{9} + ( -2 + 4 \beta_{2} ) q^{11} + ( -3 - \beta_{2} ) q^{14} + ( -1 + 3 \beta_{2} ) q^{16} + 3 \beta_{1} q^{18} + ( -2 \beta_{1} + 8 \beta_{3} ) q^{22} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{23} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{28} + 2 q^{29} + ( -\beta_{1} + 6 \beta_{3} ) q^{32} + ( 3 + 3 \beta_{2} ) q^{36} -6 \beta_{3} q^{37} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{43} + ( -10 + 6 \beta_{2} ) q^{44} + ( 6 + 2 \beta_{2} ) q^{46} + 7 q^{49} -10 \beta_{3} q^{53} + ( -1 - 5 \beta_{2} ) q^{56} + 2 \beta_{1} q^{58} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{63} + ( -7 + 5 \beta_{2} ) q^{64} + ( -12 \beta_{1} + 6 \beta_{3} ) q^{67} + ( 2 - 4 \beta_{2} ) q^{71} + ( 3 \beta_{1} + 6 \beta_{3} ) q^{72} + ( 6 - 6 \beta_{2} ) q^{74} -14 \beta_{3} q^{77} + ( 6 - 12 \beta_{2} ) q^{79} + 9 q^{81} + ( -6 - 2 \beta_{2} ) q^{86} + ( -10 \beta_{1} + 12 \beta_{3} ) q^{88} + ( 6 \beta_{1} + 4 \beta_{3} ) q^{92} + 7 \beta_{1} q^{98} + ( -6 + 12 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{4} + 12q^{9} + O(q^{10})$$ $$4q + 6q^{4} + 12q^{9} - 14q^{14} + 2q^{16} + 8q^{29} + 18q^{36} - 28q^{44} + 28q^{46} + 28q^{49} - 14q^{56} - 18q^{64} + 12q^{74} + 36q^{81} - 28q^{86} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 3 x^{2} + 4$$:

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
699.1
 −1.32288 − 0.500000i −1.32288 + 0.500000i 1.32288 − 0.500000i 1.32288 + 0.500000i
−1.32288 0.500000i 0 1.50000 + 1.32288i 0 0 2.64575 −1.32288 2.50000i 3.00000 0
699.2 −1.32288 + 0.500000i 0 1.50000 1.32288i 0 0 2.64575 −1.32288 + 2.50000i 3.00000 0
699.3 1.32288 0.500000i 0 1.50000 1.32288i 0 0 −2.64575 1.32288 2.50000i 3.00000 0
699.4 1.32288 + 0.500000i 0 1.50000 + 1.32288i 0 0 −2.64575 1.32288 + 2.50000i 3.00000 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
4.b odd 2 1 inner
5.b even 2 1 inner
20.d 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.c.d 4
4.b odd 2 1 inner 700.2.c.d 4
5.b even 2 1 inner 700.2.c.d 4
5.c odd 4 1 28.2.d.a 2
5.c odd 4 1 700.2.g.a 2
7.b odd 2 1 CM 700.2.c.d 4
15.e even 4 1 252.2.b.a 2
20.d odd 2 1 inner 700.2.c.d 4
20.e even 4 1 28.2.d.a 2
20.e even 4 1 700.2.g.a 2
28.d even 2 1 inner 700.2.c.d 4
35.c odd 2 1 inner 700.2.c.d 4
35.f even 4 1 28.2.d.a 2
35.f even 4 1 700.2.g.a 2
35.k even 12 2 196.2.f.b 4
35.l odd 12 2 196.2.f.b 4
40.i odd 4 1 448.2.f.b 2
40.k even 4 1 448.2.f.b 2
60.l odd 4 1 252.2.b.a 2
80.i odd 4 1 1792.2.e.b 4
80.j even 4 1 1792.2.e.b 4
80.s even 4 1 1792.2.e.b 4
80.t odd 4 1 1792.2.e.b 4
105.k odd 4 1 252.2.b.a 2
120.q odd 4 1 4032.2.b.e 2
120.w even 4 1 4032.2.b.e 2
140.c even 2 1 inner 700.2.c.d 4
140.j odd 4 1 28.2.d.a 2
140.j odd 4 1 700.2.g.a 2
140.w even 12 2 196.2.f.b 4
140.x odd 12 2 196.2.f.b 4
280.s even 4 1 448.2.f.b 2
280.y odd 4 1 448.2.f.b 2
420.w even 4 1 252.2.b.a 2
560.r even 4 1 1792.2.e.b 4
560.u odd 4 1 1792.2.e.b 4
560.bm odd 4 1 1792.2.e.b 4
560.bn even 4 1 1792.2.e.b 4
840.bm even 4 1 4032.2.b.e 2
840.bp odd 4 1 4032.2.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.d.a 2 5.c odd 4 1
28.2.d.a 2 20.e even 4 1
28.2.d.a 2 35.f even 4 1
28.2.d.a 2 140.j odd 4 1
196.2.f.b 4 35.k even 12 2
196.2.f.b 4 35.l odd 12 2
196.2.f.b 4 140.w even 12 2
196.2.f.b 4 140.x odd 12 2
252.2.b.a 2 15.e even 4 1
252.2.b.a 2 60.l odd 4 1
252.2.b.a 2 105.k odd 4 1
252.2.b.a 2 420.w even 4 1
448.2.f.b 2 40.i odd 4 1
448.2.f.b 2 40.k even 4 1
448.2.f.b 2 280.s even 4 1
448.2.f.b 2 280.y odd 4 1
700.2.c.d 4 1.a even 1 1 trivial
700.2.c.d 4 4.b odd 2 1 inner
700.2.c.d 4 5.b even 2 1 inner
700.2.c.d 4 7.b odd 2 1 CM
700.2.c.d 4 20.d odd 2 1 inner
700.2.c.d 4 28.d even 2 1 inner
700.2.c.d 4 35.c odd 2 1 inner
700.2.c.d 4 140.c even 2 1 inner
700.2.g.a 2 5.c odd 4 1
700.2.g.a 2 20.e even 4 1
700.2.g.a 2 35.f even 4 1
700.2.g.a 2 140.j odd 4 1
1792.2.e.b 4 80.i odd 4 1
1792.2.e.b 4 80.j even 4 1
1792.2.e.b 4 80.s even 4 1
1792.2.e.b 4 80.t odd 4 1
1792.2.e.b 4 560.r even 4 1
1792.2.e.b 4 560.u odd 4 1
1792.2.e.b 4 560.bm odd 4 1
1792.2.e.b 4 560.bn even 4 1
4032.2.b.e 2 120.q odd 4 1
4032.2.b.e 2 120.w even 4 1
4032.2.b.e 2 840.bm even 4 1
4032.2.b.e 2 840.bp odd 4 1

## Hecke kernels

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

 $$T_{3}$$ $$T_{11}^{2} + 28$$ $$T_{13}$$ $$T_{19}$$ $$T_{23}^{2} - 28$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 3 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( -7 + T^{2} )^{2}$$
$11$ $$( 28 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$( -28 + T^{2} )^{2}$$
$29$ $$( -2 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$( 36 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -28 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$( 100 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$( -252 + T^{2} )^{2}$$
$71$ $$( 28 + T^{2} )^{2}$$
$73$ $$T^{4}$$
$79$ $$( 252 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$