# Properties

 Label 700.2.g.h Level $700$ Weight $2$ Character orbit 700.g Analytic conductor $5.590$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} -\beta_{3} q^{3} -2 \beta_{1} q^{4} + ( \beta_{2} - \beta_{3} ) q^{6} + ( -\beta_{1} - \beta_{3} ) q^{7} + ( -2 - 2 \beta_{1} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} -\beta_{3} q^{3} -2 \beta_{1} q^{4} + ( \beta_{2} - \beta_{3} ) q^{6} + ( -\beta_{1} - \beta_{3} ) q^{7} + ( -2 - 2 \beta_{1} ) q^{8} + 3 q^{9} + 5 \beta_{1} q^{11} + 2 \beta_{2} q^{12} -\beta_{2} q^{13} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{14} -4 q^{16} + 2 \beta_{2} q^{17} + ( 3 - 3 \beta_{1} ) q^{18} + ( 6 + \beta_{2} ) q^{21} + ( 5 + 5 \beta_{1} ) q^{22} + \beta_{1} q^{23} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{24} + ( -\beta_{2} - \beta_{3} ) q^{26} + ( -2 + 2 \beta_{2} ) q^{28} -5 q^{29} + 3 \beta_{3} q^{31} + ( -4 + 4 \beta_{1} ) q^{32} -5 \beta_{2} q^{33} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{34} -6 \beta_{1} q^{36} -3 q^{37} + 6 \beta_{1} q^{39} + 5 \beta_{2} q^{41} + ( 6 - 6 \beta_{1} + \beta_{2} + \beta_{3} ) q^{42} + 11 \beta_{1} q^{43} + 10 q^{44} + ( 1 + \beta_{1} ) q^{46} -2 \beta_{3} q^{47} + 4 \beta_{3} q^{48} + ( 5 + 2 \beta_{2} ) q^{49} -12 \beta_{1} q^{51} -2 \beta_{3} q^{52} -4 q^{53} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{56} + ( -5 + 5 \beta_{1} ) q^{58} -5 \beta_{3} q^{59} -5 \beta_{2} q^{61} + ( -3 \beta_{2} + 3 \beta_{3} ) q^{62} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{63} + 8 \beta_{1} q^{64} + ( -5 \beta_{2} - 5 \beta_{3} ) q^{66} + 3 \beta_{1} q^{67} + 4 \beta_{3} q^{68} -\beta_{2} q^{69} -5 \beta_{1} q^{71} + ( -6 - 6 \beta_{1} ) q^{72} -\beta_{2} q^{73} + ( -3 + 3 \beta_{1} ) q^{74} + ( 5 - 5 \beta_{2} ) q^{77} + ( 6 + 6 \beta_{1} ) q^{78} -9 \beta_{1} q^{79} -9 q^{81} + ( 5 \beta_{2} + 5 \beta_{3} ) q^{82} -\beta_{3} q^{83} + ( -12 \beta_{1} + 2 \beta_{3} ) q^{84} + ( 11 + 11 \beta_{1} ) q^{86} + 5 \beta_{3} q^{87} + ( 10 - 10 \beta_{1} ) q^{88} -\beta_{2} q^{89} + ( 6 \beta_{1} - \beta_{3} ) q^{91} + 2 q^{92} -18 q^{93} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{94} + ( -4 \beta_{2} + 4 \beta_{3} ) q^{96} -3 \beta_{2} q^{97} + ( 5 - 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{98} + 15 \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} - 8q^{8} + 12q^{9} + O(q^{10})$$ $$4q + 4q^{2} - 8q^{8} + 12q^{9} - 4q^{14} - 16q^{16} + 12q^{18} + 24q^{21} + 20q^{22} - 8q^{28} - 20q^{29} - 16q^{32} - 12q^{37} + 24q^{42} + 40q^{44} + 4q^{46} + 20q^{49} - 16q^{53} - 8q^{56} - 20q^{58} - 24q^{72} - 12q^{74} + 20q^{77} + 24q^{78} - 36q^{81} + 44q^{86} + 40q^{88} + 8q^{92} - 72q^{93} + 20q^{98} + O(q^{100})$$

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

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

## 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
 1.22474 + 1.22474i −1.22474 − 1.22474i 1.22474 − 1.22474i −1.22474 + 1.22474i
1.00000 1.00000i −2.44949 2.00000i 0 −2.44949 + 2.44949i −2.44949 1.00000i −2.00000 2.00000i 3.00000 0
251.2 1.00000 1.00000i 2.44949 2.00000i 0 2.44949 2.44949i 2.44949 1.00000i −2.00000 2.00000i 3.00000 0
251.3 1.00000 + 1.00000i −2.44949 2.00000i 0 −2.44949 2.44949i −2.44949 + 1.00000i −2.00000 + 2.00000i 3.00000 0
251.4 1.00000 + 1.00000i 2.44949 2.00000i 0 2.44949 + 2.44949i 2.44949 + 1.00000i −2.00000 + 2.00000i 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
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d 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.h yes 4
4.b odd 2 1 inner 700.2.g.h yes 4
5.b even 2 1 700.2.g.b 4
5.c odd 4 1 700.2.c.a 4
5.c odd 4 1 700.2.c.g 4
7.b odd 2 1 inner 700.2.g.h yes 4
20.d odd 2 1 700.2.g.b 4
20.e even 4 1 700.2.c.a 4
20.e even 4 1 700.2.c.g 4
28.d even 2 1 inner 700.2.g.h yes 4
35.c odd 2 1 700.2.g.b 4
35.f even 4 1 700.2.c.a 4
35.f even 4 1 700.2.c.g 4
140.c even 2 1 700.2.g.b 4
140.j odd 4 1 700.2.c.a 4
140.j odd 4 1 700.2.c.g 4

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

## Hecke kernels

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

 $$T_{3}^{2} - 6$$ $$T_{11}^{2} + 25$$ $$T_{19}$$ $$T_{37} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 - 2 T + T^{2} )^{2}$$
$3$ $$( -6 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$49 - 10 T^{2} + T^{4}$$
$11$ $$( 25 + T^{2} )^{2}$$
$13$ $$( 6 + T^{2} )^{2}$$
$17$ $$( 24 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$( 1 + T^{2} )^{2}$$
$29$ $$( 5 + T )^{4}$$
$31$ $$( -54 + T^{2} )^{2}$$
$37$ $$( 3 + T )^{4}$$
$41$ $$( 150 + T^{2} )^{2}$$
$43$ $$( 121 + T^{2} )^{2}$$
$47$ $$( -24 + T^{2} )^{2}$$
$53$ $$( 4 + T )^{4}$$
$59$ $$( -150 + T^{2} )^{2}$$
$61$ $$( 150 + T^{2} )^{2}$$
$67$ $$( 9 + T^{2} )^{2}$$
$71$ $$( 25 + T^{2} )^{2}$$
$73$ $$( 6 + T^{2} )^{2}$$
$79$ $$( 81 + T^{2} )^{2}$$
$83$ $$( -6 + T^{2} )^{2}$$
$89$ $$( 6 + T^{2} )^{2}$$
$97$ $$( 54 + T^{2} )^{2}$$