# Properties

 Label 700.2.g.j Level $700$ Weight $2$ Character orbit 700.g Analytic conductor $5.590$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: 8.0.342102016.5 Defining polynomial: $$x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( \beta_{5} - \beta_{7} ) q^{3} + ( -\beta_{1} + \beta_{2} - \beta_{4} ) q^{4} + ( \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{6} + ( \beta_{2} - \beta_{5} + \beta_{7} ) q^{7} + ( -2 + \beta_{1} + \beta_{5} - \beta_{6} ) q^{8} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( \beta_{5} - \beta_{7} ) q^{3} + ( -\beta_{1} + \beta_{2} - \beta_{4} ) q^{4} + ( \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{6} + ( \beta_{2} - \beta_{5} + \beta_{7} ) q^{7} + ( -2 + \beta_{1} + \beta_{5} - \beta_{6} ) q^{8} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{9} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{11} + ( \beta_{3} - \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{12} + ( -\beta_{6} - \beta_{7} ) q^{13} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{14} + ( -2 - 3 \beta_{1} - \beta_{5} + \beta_{6} ) q^{16} + ( -\beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{17} + ( 4 + \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{18} + ( \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{19} + ( -5 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{21} + ( -4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{22} + ( -4 \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{23} + ( \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{24} + ( -\beta_{3} + \beta_{4} ) q^{26} + ( 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{27} + ( -2 - \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{28} -2 q^{29} + ( 2 \beta_{2} + 2 \beta_{4} ) q^{31} + ( 2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{32} + ( 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{33} + ( 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{34} + ( -4 + 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{36} -2 q^{37} + ( \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{38} + ( -2 \beta_{2} + 2 \beta_{4} ) q^{39} + ( -\beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{41} + ( -2 - 4 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{42} + ( 4 \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{43} + ( -4 - 2 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} ) q^{44} + ( 6 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{46} + ( 3 \beta_{2} + 3 \beta_{4} ) q^{47} + ( -2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{48} + ( 4 - \beta_{1} - \beta_{3} - 3 \beta_{6} - 2 \beta_{7} ) q^{49} + ( -4 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} ) q^{51} + ( \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{52} -2 q^{53} + ( 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} ) q^{54} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{56} + ( 8 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{57} -2 \beta_{1} q^{58} + ( 3 \beta_{2} + 3 \beta_{4} - \beta_{5} + \beta_{7} ) q^{59} + ( \beta_{6} + \beta_{7} ) q^{61} + ( -2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{62} + ( 2 \beta_{1} - 5 \beta_{4} - 2 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{63} + ( 2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{64} + ( 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 8 \beta_{7} ) q^{66} + ( 8 \beta_{1} - \beta_{2} + \beta_{4} + 4 \beta_{5} - 4 \beta_{6} ) q^{67} + ( 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{68} + ( -3 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} ) q^{69} + ( 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{71} + ( -10 - 7 \beta_{1} + 4 \beta_{2} - 4 \beta_{4} + \beta_{5} - \beta_{6} ) q^{72} + ( -3 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{73} -2 \beta_{1} q^{74} + ( 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{76} + ( -4 - 2 \beta_{1} - \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{77} + ( 4 - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{78} + ( -2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{79} + ( 7 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{81} + ( 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{82} + ( \beta_{5} - \beta_{7} ) q^{83} + ( 2 + 2 \beta_{1} - 6 \beta_{2} - \beta_{3} + 5 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} ) q^{84} + ( -2 - 4 \beta_{1} + \beta_{2} - \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{86} + ( -2 \beta_{5} + 2 \beta_{7} ) q^{87} + ( -4 - 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{88} + ( 6 \beta_{6} + 6 \beta_{7} ) q^{89} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{91} + ( 4 + 2 \beta_{1} - 4 \beta_{5} + 4 \beta_{6} ) q^{92} + ( 4 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{93} + ( -3 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} ) q^{94} + ( -2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 6 \beta_{6} - 6 \beta_{7} ) q^{96} + ( \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{97} + ( -2 + 5 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{98} + ( 2 \beta_{1} + 9 \beta_{2} - 9 \beta_{4} + \beta_{5} - \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{2} + 2q^{4} - 14q^{8} + 16q^{9} + O(q^{10})$$ $$8q - 2q^{2} + 2q^{4} - 14q^{8} + 16q^{9} - 6q^{14} - 14q^{16} + 30q^{18} - 36q^{21} - 28q^{22} - 14q^{28} - 16q^{29} + 18q^{32} - 30q^{36} - 16q^{37} - 8q^{42} - 20q^{44} + 44q^{46} + 36q^{49} - 16q^{53} + 2q^{56} + 48q^{57} + 4q^{58} + 2q^{64} - 62q^{72} + 4q^{74} - 24q^{77} + 24q^{78} + 48q^{81} + 8q^{84} - 20q^{86} - 36q^{88} + 12q^{92} + 16q^{93} - 26q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} - 3 \nu^{4} - 2 \nu^{2} - 8$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - \nu^{5} - 3 \nu^{4} + 6 \nu^{3} - 10 \nu^{2} - 8 \nu - 8$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} - \nu^{5} + 3 \nu^{4} + 6 \nu^{3} + 10 \nu^{2} + 24 \nu + 8$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} - \nu^{5} + 3 \nu^{4} + 6 \nu^{3} + 10 \nu^{2} - 8 \nu + 8$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{6} + 4 \nu^{5} + \nu^{4} + 4 \nu^{3} + 6 \nu^{2} + 8$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{6} + 4 \nu^{5} - \nu^{4} + 4 \nu^{3} - 6 \nu^{2} - 8$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-2 \nu^{7} + 3 \nu^{6} - 2 \nu^{5} + \nu^{4} + 6 \nu^{2} + 8$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} + \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} - \beta_{5} - 3 \beta_{1} - 2$$ $$\nu^{5}$$ $$=$$ $$($$$$-2 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-3 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} - \beta_{1} - 2$$ $$\nu^{7}$$ $$=$$ $$($$$$-14 \beta_{7} - 10 \beta_{6} + 4 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2}$$$$)/2$$

## 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
 −0.599676 + 1.28078i 0.599676 − 1.28078i −0.599676 − 1.28078i 0.599676 + 1.28078i 1.17915 + 0.780776i −1.17915 − 0.780776i 1.17915 − 0.780776i −1.17915 + 0.780776i
−1.28078 0.599676i −0.936426 1.28078 + 1.53610i 0 1.19935 + 0.561553i 2.60399 + 0.468213i −0.719224 2.73546i −2.12311 0
251.2 −1.28078 0.599676i 0.936426 1.28078 + 1.53610i 0 −1.19935 0.561553i −2.60399 + 0.468213i −0.719224 2.73546i −2.12311 0
251.3 −1.28078 + 0.599676i −0.936426 1.28078 1.53610i 0 1.19935 0.561553i 2.60399 0.468213i −0.719224 + 2.73546i −2.12311 0
251.4 −1.28078 + 0.599676i 0.936426 1.28078 1.53610i 0 −1.19935 + 0.561553i −2.60399 0.468213i −0.719224 + 2.73546i −2.12311 0
251.5 0.780776 1.17915i −3.02045 −0.780776 1.84130i 0 −2.35829 + 3.56155i 2.17238 1.51022i −2.78078 0.516994i 6.12311 0
251.6 0.780776 1.17915i 3.02045 −0.780776 1.84130i 0 2.35829 3.56155i −2.17238 1.51022i −2.78078 0.516994i 6.12311 0
251.7 0.780776 + 1.17915i −3.02045 −0.780776 + 1.84130i 0 −2.35829 3.56155i 2.17238 + 1.51022i −2.78078 + 0.516994i 6.12311 0
251.8 0.780776 + 1.17915i 3.02045 −0.780776 + 1.84130i 0 2.35829 + 3.56155i −2.17238 + 1.51022i −2.78078 + 0.516994i 6.12311 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.8 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.j 8
4.b odd 2 1 inner 700.2.g.j 8
5.b even 2 1 140.2.g.c 8
5.c odd 4 1 700.2.c.i 8
5.c odd 4 1 700.2.c.j 8
7.b odd 2 1 inner 700.2.g.j 8
15.d odd 2 1 1260.2.c.c 8
20.d odd 2 1 140.2.g.c 8
20.e even 4 1 700.2.c.i 8
20.e even 4 1 700.2.c.j 8
28.d even 2 1 inner 700.2.g.j 8
35.c odd 2 1 140.2.g.c 8
35.f even 4 1 700.2.c.i 8
35.f even 4 1 700.2.c.j 8
35.i odd 6 2 980.2.o.e 16
35.j even 6 2 980.2.o.e 16
40.e odd 2 1 2240.2.k.e 8
40.f even 2 1 2240.2.k.e 8
60.h even 2 1 1260.2.c.c 8
105.g even 2 1 1260.2.c.c 8
140.c even 2 1 140.2.g.c 8
140.j odd 4 1 700.2.c.i 8
140.j odd 4 1 700.2.c.j 8
140.p odd 6 2 980.2.o.e 16
140.s even 6 2 980.2.o.e 16
280.c odd 2 1 2240.2.k.e 8
280.n even 2 1 2240.2.k.e 8
420.o odd 2 1 1260.2.c.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.c 8 5.b even 2 1
140.2.g.c 8 20.d odd 2 1
140.2.g.c 8 35.c odd 2 1
140.2.g.c 8 140.c even 2 1
700.2.c.i 8 5.c odd 4 1
700.2.c.i 8 20.e even 4 1
700.2.c.i 8 35.f even 4 1
700.2.c.i 8 140.j odd 4 1
700.2.c.j 8 5.c odd 4 1
700.2.c.j 8 20.e even 4 1
700.2.c.j 8 35.f even 4 1
700.2.c.j 8 140.j odd 4 1
700.2.g.j 8 1.a even 1 1 trivial
700.2.g.j 8 4.b odd 2 1 inner
700.2.g.j 8 7.b odd 2 1 inner
700.2.g.j 8 28.d even 2 1 inner
980.2.o.e 16 35.i odd 6 2
980.2.o.e 16 35.j even 6 2
980.2.o.e 16 140.p odd 6 2
980.2.o.e 16 140.s even 6 2
1260.2.c.c 8 15.d odd 2 1
1260.2.c.c 8 60.h even 2 1
1260.2.c.c 8 105.g even 2 1
1260.2.c.c 8 420.o odd 2 1
2240.2.k.e 8 40.e odd 2 1
2240.2.k.e 8 40.f even 2 1
2240.2.k.e 8 280.c odd 2 1
2240.2.k.e 8 280.n even 2 1

## Hecke kernels

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

 $$T_{3}^{4} - 10 T_{3}^{2} + 8$$ $$T_{11}^{4} + 28 T_{11}^{2} + 128$$ $$T_{19}^{4} - 28 T_{19}^{2} + 128$$ $$T_{37} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 2 T + T^{3} + T^{4} )^{2}$$
$3$ $$( 8 - 10 T^{2} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$2401 - 882 T^{2} + 162 T^{4} - 18 T^{6} + T^{8}$$
$11$ $$( 128 + 28 T^{2} + T^{4} )^{2}$$
$13$ $$( 4 + T^{2} )^{4}$$
$17$ $$( 64 + 52 T^{2} + T^{4} )^{2}$$
$19$ $$( 128 - 28 T^{2} + T^{4} )^{2}$$
$23$ $$( 1352 + 74 T^{2} + T^{4} )^{2}$$
$29$ $$( 2 + T )^{8}$$
$31$ $$( 512 - 56 T^{2} + T^{4} )^{2}$$
$37$ $$( 2 + T )^{8}$$
$41$ $$( 64 + 52 T^{2} + T^{4} )^{2}$$
$43$ $$( 8 + 58 T^{2} + T^{4} )^{2}$$
$47$ $$( 2592 - 126 T^{2} + T^{4} )^{2}$$
$53$ $$( 2 + T )^{8}$$
$59$ $$( 512 - 124 T^{2} + T^{4} )^{2}$$
$61$ $$( 4 + T^{2} )^{4}$$
$67$ $$( 2888 + 218 T^{2} + T^{4} )^{2}$$
$71$ $$( 2048 + 92 T^{2} + T^{4} )^{2}$$
$73$ $$( 20736 + 324 T^{2} + T^{4} )^{2}$$
$79$ $$( 32 + 20 T^{2} + T^{4} )^{2}$$
$83$ $$( 8 - 10 T^{2} + T^{4} )^{2}$$
$89$ $$( 144 + T^{2} )^{4}$$
$97$ $$( 64 + 52 T^{2} + T^{4} )^{2}$$