# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 700.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{3} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} -2 \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{12} q^{3} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} -2 \zeta_{12}^{2} q^{9} + ( 3 - 3 \zeta_{12}^{2} ) q^{11} -2 \zeta_{12}^{3} q^{13} -3 \zeta_{12} q^{17} -\zeta_{12}^{2} q^{19} + ( 1 - 3 \zeta_{12}^{2} ) q^{21} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{23} -5 \zeta_{12}^{3} q^{27} + 6 q^{29} + ( 7 - 7 \zeta_{12}^{2} ) q^{31} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{33} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{37} + ( 2 - 2 \zeta_{12}^{2} ) q^{39} + 6 q^{41} + 4 \zeta_{12}^{3} q^{43} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{47} + ( -5 + 8 \zeta_{12}^{2} ) q^{49} -3 \zeta_{12}^{2} q^{51} + 3 \zeta_{12} q^{53} -\zeta_{12}^{3} q^{57} + ( 9 - 9 \zeta_{12}^{2} ) q^{59} + \zeta_{12}^{2} q^{61} + ( -2 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{63} + 7 \zeta_{12} q^{67} -3 q^{69} -\zeta_{12} q^{73} + ( -9 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{77} -13 \zeta_{12}^{2} q^{79} + ( -1 + \zeta_{12}^{2} ) q^{81} -12 \zeta_{12}^{3} q^{83} + 6 \zeta_{12} q^{87} + 15 \zeta_{12}^{2} q^{89} + ( -6 + 4 \zeta_{12}^{2} ) q^{91} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{93} -10 \zeta_{12}^{3} q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} + 6q^{11} - 2q^{19} - 2q^{21} + 24q^{29} + 14q^{31} + 4q^{39} + 24q^{41} - 4q^{49} - 6q^{51} + 18q^{59} + 2q^{61} - 12q^{69} - 26q^{79} - 2q^{81} + 30q^{89} - 16q^{91} - 24q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$-\zeta_{12}^{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
0 −0.866025 + 0.500000i 0 0 0 1.73205 2.00000i 0 −1.00000 + 1.73205i 0
149.2 0 0.866025 0.500000i 0 0 0 −1.73205 + 2.00000i 0 −1.00000 + 1.73205i 0
249.1 0 −0.866025 0.500000i 0 0 0 1.73205 + 2.00000i 0 −1.00000 1.73205i 0
249.2 0 0.866025 + 0.500000i 0 0 0 −1.73205 2.00000i 0 −1.00000 1.73205i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.r.b 4
5.b even 2 1 inner 700.2.r.b 4
5.c odd 4 1 28.2.e.a 2
5.c odd 4 1 700.2.i.c 2
7.c even 3 1 inner 700.2.r.b 4
7.c even 3 1 4900.2.e.i 2
7.d odd 6 1 4900.2.e.h 2
15.e even 4 1 252.2.k.c 2
20.e even 4 1 112.2.i.b 2
35.f even 4 1 196.2.e.a 2
35.i odd 6 1 4900.2.e.h 2
35.j even 6 1 inner 700.2.r.b 4
35.j even 6 1 4900.2.e.i 2
35.k even 12 1 196.2.a.a 1
35.k even 12 1 196.2.e.a 2
35.k even 12 1 4900.2.a.n 1
35.l odd 12 1 28.2.e.a 2
35.l odd 12 1 196.2.a.b 1
35.l odd 12 1 700.2.i.c 2
35.l odd 12 1 4900.2.a.g 1
40.i odd 4 1 448.2.i.e 2
40.k even 4 1 448.2.i.c 2
45.k odd 12 1 2268.2.i.a 2
45.k odd 12 1 2268.2.l.h 2
45.l even 12 1 2268.2.i.h 2
45.l even 12 1 2268.2.l.a 2
60.l odd 4 1 1008.2.s.p 2
105.k odd 4 1 1764.2.k.b 2
105.w odd 12 1 1764.2.a.j 1
105.w odd 12 1 1764.2.k.b 2
105.x even 12 1 252.2.k.c 2
105.x even 12 1 1764.2.a.a 1
140.j odd 4 1 784.2.i.d 2
140.w even 12 1 112.2.i.b 2
140.w even 12 1 784.2.a.d 1
140.x odd 12 1 784.2.a.g 1
140.x odd 12 1 784.2.i.d 2
280.bp odd 12 1 3136.2.a.k 1
280.br even 12 1 448.2.i.c 2
280.br even 12 1 3136.2.a.s 1
280.bt odd 12 1 448.2.i.e 2
280.bt odd 12 1 3136.2.a.h 1
280.bv even 12 1 3136.2.a.v 1
315.bt odd 12 1 2268.2.l.h 2
315.bv even 12 1 2268.2.l.a 2
315.bx even 12 1 2268.2.i.h 2
315.ch odd 12 1 2268.2.i.a 2
420.bp odd 12 1 1008.2.s.p 2
420.bp odd 12 1 7056.2.a.f 1
420.br even 12 1 7056.2.a.bw 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.e.a 2 5.c odd 4 1
28.2.e.a 2 35.l odd 12 1
112.2.i.b 2 20.e even 4 1
112.2.i.b 2 140.w even 12 1
196.2.a.a 1 35.k even 12 1
196.2.a.b 1 35.l odd 12 1
196.2.e.a 2 35.f even 4 1
196.2.e.a 2 35.k even 12 1
252.2.k.c 2 15.e even 4 1
252.2.k.c 2 105.x even 12 1
448.2.i.c 2 40.k even 4 1
448.2.i.c 2 280.br even 12 1
448.2.i.e 2 40.i odd 4 1
448.2.i.e 2 280.bt odd 12 1
700.2.i.c 2 5.c odd 4 1
700.2.i.c 2 35.l odd 12 1
700.2.r.b 4 1.a even 1 1 trivial
700.2.r.b 4 5.b even 2 1 inner
700.2.r.b 4 7.c even 3 1 inner
700.2.r.b 4 35.j even 6 1 inner
784.2.a.d 1 140.w even 12 1
784.2.a.g 1 140.x odd 12 1
784.2.i.d 2 140.j odd 4 1
784.2.i.d 2 140.x odd 12 1
1008.2.s.p 2 60.l odd 4 1
1008.2.s.p 2 420.bp odd 12 1
1764.2.a.a 1 105.x even 12 1
1764.2.a.j 1 105.w odd 12 1
1764.2.k.b 2 105.k odd 4 1
1764.2.k.b 2 105.w odd 12 1
2268.2.i.a 2 45.k odd 12 1
2268.2.i.a 2 315.ch odd 12 1
2268.2.i.h 2 45.l even 12 1
2268.2.i.h 2 315.bx even 12 1
2268.2.l.a 2 45.l even 12 1
2268.2.l.a 2 315.bv even 12 1
2268.2.l.h 2 45.k odd 12 1
2268.2.l.h 2 315.bt odd 12 1
3136.2.a.h 1 280.bt odd 12 1
3136.2.a.k 1 280.bp odd 12 1
3136.2.a.s 1 280.br even 12 1
3136.2.a.v 1 280.bv even 12 1
4900.2.a.g 1 35.l odd 12 1
4900.2.a.n 1 35.k even 12 1
4900.2.e.h 2 7.d odd 6 1
4900.2.e.h 2 35.i odd 6 1
4900.2.e.i 2 7.c even 3 1
4900.2.e.i 2 35.j even 6 1
7056.2.a.f 1 420.bp odd 12 1
7056.2.a.bw 1 420.br even 12 1

## Hecke kernels

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

 $$T_{3}^{4} - T_{3}^{2} + 1$$ $$T_{11}^{2} - 3 T_{11} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$49 + 2 T^{2} + T^{4}$$
$11$ $$( 9 - 3 T + T^{2} )^{2}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$81 - 9 T^{2} + T^{4}$$
$19$ $$( 1 + T + T^{2} )^{2}$$
$23$ $$81 - 9 T^{2} + T^{4}$$
$29$ $$( -6 + T )^{4}$$
$31$ $$( 49 - 7 T + T^{2} )^{2}$$
$37$ $$1 - T^{2} + T^{4}$$
$41$ $$( -6 + T )^{4}$$
$43$ $$( 16 + T^{2} )^{2}$$
$47$ $$6561 - 81 T^{2} + T^{4}$$
$53$ $$81 - 9 T^{2} + T^{4}$$
$59$ $$( 81 - 9 T + T^{2} )^{2}$$
$61$ $$( 1 - T + T^{2} )^{2}$$
$67$ $$2401 - 49 T^{2} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$1 - T^{2} + T^{4}$$
$79$ $$( 169 + 13 T + T^{2} )^{2}$$
$83$ $$( 144 + T^{2} )^{2}$$
$89$ $$( 225 - 15 T + T^{2} )^{2}$$
$97$ $$( 100 + T^{2} )^{2}$$
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