# Properties

 Label 700.2.c.i Level $700$ Weight $2$ Character orbit 700.c 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.c (of order $$2$$, degree $$1$$, not 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: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} + ( \beta_{3} + \beta_{4} ) q^{3} + ( \beta_{3} + \beta_{4} + \beta_{7} ) q^{4} + ( \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{6} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{7} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{8} + ( -3 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{5} q^{2} + ( \beta_{3} + \beta_{4} ) q^{3} + ( \beta_{3} + \beta_{4} + \beta_{7} ) q^{4} + ( \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{6} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{7} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{8} + ( -3 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{9} + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{11} + ( -4 - 2 \beta_{7} ) q^{12} -2 q^{13} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{14} + ( -2 + \beta_{2} + \beta_{6} - 3 \beta_{7} ) q^{16} + ( -2 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{17} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{18} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{19} + ( -5 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{21} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{22} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{23} + ( 2 \beta_{2} - 2 \beta_{5} - 2 \beta_{6} ) q^{24} -2 \beta_{5} q^{26} + ( -2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 4 \beta_{7} ) q^{27} + ( -4 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{28} + 2 q^{29} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{31} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{32} + ( 8 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 4 \beta_{7} ) q^{33} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} ) q^{34} + ( -4 - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} + 3 \beta_{7} ) q^{36} + ( \beta_{1} - \beta_{4} + \beta_{5} ) q^{37} + ( 4 + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{38} + ( -2 \beta_{3} - 2 \beta_{4} ) q^{39} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{5} + \beta_{6} ) q^{41} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{42} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{43} + ( 4 + 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{7} ) q^{44} + ( 6 - \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - \beta_{6} + 4 \beta_{7} ) q^{46} + ( -3 \beta_{2} - 3 \beta_{6} + 6 \beta_{7} ) q^{47} + ( -4 - 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{48} + ( -4 - 2 \beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{5} + \beta_{7} ) q^{49} + ( 2 \beta_{2} + 2 \beta_{6} - 4 \beta_{7} ) q^{51} + ( -2 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{52} + ( -\beta_{1} + \beta_{4} - \beta_{5} ) q^{53} + ( 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{4} + 4 \beta_{6} ) q^{54} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{7} ) q^{56} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{57} + 2 \beta_{5} q^{58} + ( -3 \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{5} - \beta_{6} ) q^{59} + ( -\beta_{1} + \beta_{4} - \beta_{5} ) q^{61} + ( -4 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 4 \beta_{7} ) q^{62} + ( \beta_{1} - 4 \beta_{3} - 3 \beta_{4} - \beta_{5} - 5 \beta_{6} + 5 \beta_{7} ) q^{63} + ( -2 - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{64} + ( -4 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} + 4 \beta_{6} ) q^{66} + ( -4 \beta_{1} + \beta_{2} + 4 \beta_{3} + 4 \beta_{5} - \beta_{6} ) q^{67} + ( 4 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 8 \beta_{7} ) q^{68} + ( -2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 5 \beta_{4} + 4 \beta_{5} - 3 \beta_{6} ) q^{69} + ( -\beta_{2} + 3 \beta_{3} + 3 \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{71} + ( 4 \beta_{1} - 4 \beta_{2} + \beta_{3} - 5 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} ) q^{72} + ( -6 - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{6} - 6 \beta_{7} ) q^{73} + 2 \beta_{7} q^{74} + ( -2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{76} + ( 4 + \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{5} + 3 \beta_{6} + 4 \beta_{7} ) q^{77} + ( -2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{78} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{79} + ( 7 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{81} + ( 4 - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{7} ) q^{82} + ( \beta_{3} + \beta_{4} ) q^{83} + ( -2 + 2 \beta_{2} - 6 \beta_{3} - 4 \beta_{4} - 4 \beta_{6} - 2 \beta_{7} ) q^{84} + ( -2 + 3 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{6} - 4 \beta_{7} ) q^{86} + ( 2 \beta_{3} + 2 \beta_{4} ) q^{87} + ( -4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} ) q^{88} + ( 6 \beta_{1} - 6 \beta_{4} + 6 \beta_{5} ) q^{89} + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{91} + ( 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{92} + ( -2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} ) q^{93} + ( 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{6} ) q^{94} + ( 4 \beta_{1} + 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} ) q^{96} + ( 2 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{97} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 6 \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{98} + ( \beta_{2} + 9 \beta_{3} + 9 \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{4} - 16q^{9} + O(q^{10})$$ $$8q - 2q^{4} - 16q^{9} - 28q^{12} - 16q^{13} + 6q^{14} - 14q^{16} - 24q^{17} - 36q^{21} - 32q^{28} + 16q^{29} + 48q^{33} - 30q^{36} + 28q^{38} - 12q^{42} + 20q^{44} + 44q^{46} - 20q^{48} - 36q^{49} + 4q^{52} + 2q^{56} - 32q^{62} - 2q^{64} + 40q^{68} - 24q^{73} - 4q^{74} + 16q^{77} + 48q^{81} + 40q^{82} - 8q^{84} - 20q^{86} + 24q^{97} - 8q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} - 3 \nu^{4} + 8 \nu^{3} - 10 \nu^{2} - 8 \nu - 8$$$$)/16$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + 3 \nu^{5} - 3 \nu^{4} + 2 \nu^{3} + 6 \nu^{2} - 8$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} + 4 \nu^{5} - 3 \nu^{4} + 4 \nu^{3} - 10 \nu^{2} + 16 \nu - 8$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} - 4 \nu^{5} - 3 \nu^{4} - 4 \nu^{3} - 10 \nu^{2} - 16 \nu - 8$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{5} - 4 \nu^{3} - 4 \nu$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} - 3 \nu^{5} - 3 \nu^{4} - 2 \nu^{3} + 6 \nu^{2} - 8$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{6} + \nu^{4} + 2 \nu^{2}$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - \beta_{4} - \beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + 3 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$4 \beta_{7} - 3 \beta_{6} - \beta_{4} - \beta_{3} - 3 \beta_{2} - 4$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} + \beta_{3} + 5 \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-12 \beta_{7} - \beta_{6} - 3 \beta_{4} - 3 \beta_{3} - \beta_{2} - 4$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-3 \beta_{6} - 13 \beta_{5} + 10 \beta_{4} - \beta_{3} + 3 \beta_{2} - 9 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
699.1
 −1.17915 + 0.780776i −1.17915 − 0.780776i −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 3.02045i 0.780776 + 1.84130i 0 2.35829 3.56155i −1.51022 + 2.17238i 0.516994 2.78078i −6.12311 0
699.2 −1.17915 + 0.780776i 3.02045i 0.780776 1.84130i 0 2.35829 + 3.56155i −1.51022 2.17238i 0.516994 + 2.78078i −6.12311 0
699.3 −0.599676 1.28078i 0.936426i −1.28078 + 1.53610i 0 1.19935 0.561553i 0.468213 + 2.60399i 2.73546 + 0.719224i 2.12311 0
699.4 −0.599676 + 1.28078i 0.936426i −1.28078 1.53610i 0 1.19935 + 0.561553i 0.468213 2.60399i 2.73546 0.719224i 2.12311 0
699.5 0.599676 1.28078i 0.936426i −1.28078 1.53610i 0 −1.19935 0.561553i −0.468213 2.60399i −2.73546 + 0.719224i 2.12311 0
699.6 0.599676 + 1.28078i 0.936426i −1.28078 + 1.53610i 0 −1.19935 + 0.561553i −0.468213 + 2.60399i −2.73546 0.719224i 2.12311 0
699.7 1.17915 0.780776i 3.02045i 0.780776 1.84130i 0 −2.35829 3.56155i 1.51022 2.17238i −0.516994 2.78078i −6.12311 0
699.8 1.17915 + 0.780776i 3.02045i 0.780776 + 1.84130i 0 −2.35829 + 3.56155i 1.51022 + 2.17238i −0.516994 + 2.78078i −6.12311 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 699.8 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
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.i 8
4.b odd 2 1 inner 700.2.c.i 8
5.b even 2 1 700.2.c.j 8
5.c odd 4 1 140.2.g.c 8
5.c odd 4 1 700.2.g.j 8
7.b odd 2 1 700.2.c.j 8
15.e even 4 1 1260.2.c.c 8
20.d odd 2 1 700.2.c.j 8
20.e even 4 1 140.2.g.c 8
20.e even 4 1 700.2.g.j 8
28.d even 2 1 700.2.c.j 8
35.c odd 2 1 inner 700.2.c.i 8
35.f even 4 1 140.2.g.c 8
35.f even 4 1 700.2.g.j 8
35.k even 12 2 980.2.o.e 16
35.l odd 12 2 980.2.o.e 16
40.i odd 4 1 2240.2.k.e 8
40.k even 4 1 2240.2.k.e 8
60.l odd 4 1 1260.2.c.c 8
105.k odd 4 1 1260.2.c.c 8
140.c even 2 1 inner 700.2.c.i 8
140.j odd 4 1 140.2.g.c 8
140.j odd 4 1 700.2.g.j 8
140.w even 12 2 980.2.o.e 16
140.x odd 12 2 980.2.o.e 16
280.s even 4 1 2240.2.k.e 8
280.y odd 4 1 2240.2.k.e 8
420.w even 4 1 1260.2.c.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.c 8 5.c odd 4 1
140.2.g.c 8 20.e even 4 1
140.2.g.c 8 35.f even 4 1
140.2.g.c 8 140.j odd 4 1
700.2.c.i 8 1.a even 1 1 trivial
700.2.c.i 8 4.b odd 2 1 inner
700.2.c.i 8 35.c odd 2 1 inner
700.2.c.i 8 140.c even 2 1 inner
700.2.c.j 8 5.b even 2 1
700.2.c.j 8 7.b odd 2 1
700.2.c.j 8 20.d odd 2 1
700.2.c.j 8 28.d even 2 1
700.2.g.j 8 5.c odd 4 1
700.2.g.j 8 20.e even 4 1
700.2.g.j 8 35.f even 4 1
700.2.g.j 8 140.j odd 4 1
980.2.o.e 16 35.k even 12 2
980.2.o.e 16 35.l odd 12 2
980.2.o.e 16 140.w even 12 2
980.2.o.e 16 140.x odd 12 2
1260.2.c.c 8 15.e even 4 1
1260.2.c.c 8 60.l odd 4 1
1260.2.c.c 8 105.k odd 4 1
1260.2.c.c 8 420.w even 4 1
2240.2.k.e 8 40.i odd 4 1
2240.2.k.e 8 40.k even 4 1
2240.2.k.e 8 280.s even 4 1
2240.2.k.e 8 280.y odd 4 1

## Hecke kernels

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

 $$T_{3}^{4} + 10 T_{3}^{2} + 8$$ $$T_{11}^{4} + 28 T_{11}^{2} + 128$$ $$T_{13} + 2$$ $$T_{19}^{4} - 28 T_{19}^{2} + 128$$ $$T_{23}^{4} - 74 T_{23}^{2} + 1352$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 4 T^{2} + 4 T^{4} + T^{6} + T^{8}$$
$3$ $$( 8 + 10 T^{2} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$2401 + 882 T^{2} + 162 T^{4} + 18 T^{6} + T^{8}$$
$11$ $$( 128 + 28 T^{2} + T^{4} )^{2}$$
$13$ $$( 2 + T )^{8}$$
$17$ $$( -8 + 6 T + T^{2} )^{4}$$
$19$ $$( 128 - 28 T^{2} + T^{4} )^{2}$$
$23$ $$( 1352 - 74 T^{2} + T^{4} )^{2}$$
$29$ $$( -2 + T )^{8}$$
$31$ $$( 512 - 56 T^{2} + T^{4} )^{2}$$
$37$ $$( 4 + T^{2} )^{4}$$
$41$ $$( 64 + 52 T^{2} + T^{4} )^{2}$$
$43$ $$( 8 - 58 T^{2} + T^{4} )^{2}$$
$47$ $$( 2592 + 126 T^{2} + T^{4} )^{2}$$
$53$ $$( 4 + T^{2} )^{4}$$
$59$ $$( 512 - 124 T^{2} + T^{4} )^{2}$$
$61$ $$( 4 + T^{2} )^{4}$$
$67$ $$( 2888 - 218 T^{2} + T^{4} )^{2}$$
$71$ $$( 2048 + 92 T^{2} + T^{4} )^{2}$$
$73$ $$( -144 + 6 T + T^{2} )^{4}$$
$79$ $$( 32 + 20 T^{2} + T^{4} )^{2}$$
$83$ $$( 8 + 10 T^{2} + T^{4} )^{2}$$
$89$ $$( 144 + T^{2} )^{4}$$
$97$ $$( -8 - 6 T + T^{2} )^{4}$$