# Properties

 Label 700.2.i.f Level $700$ Weight $2$ Character orbit 700.i 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.2702336256.1 Defining polynomial: $$x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{4} - \beta_{6} ) q^{3} + ( \beta_{3} - \beta_{4} ) q^{7} +O(q^{10})$$ $$q + ( \beta_{4} - \beta_{6} ) q^{3} + ( \beta_{3} - \beta_{4} ) q^{7} + ( 1 + \beta_{1} - \beta_{2} ) q^{11} + ( 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{13} + ( 2 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{17} + ( \beta_{1} + \beta_{2} - \beta_{7} ) q^{19} + ( -1 + 3 \beta_{2} - \beta_{7} ) q^{21} + ( 2 \beta_{3} - 3 \beta_{6} ) q^{23} + 3 \beta_{4} q^{27} + \beta_{7} q^{29} + ( 1 - \beta_{1} - \beta_{2} ) q^{31} + ( 3 \beta_{3} - 3 \beta_{6} ) q^{33} + ( \beta_{3} - 5 \beta_{6} ) q^{37} + ( 4 - 2 \beta_{1} - 4 \beta_{2} ) q^{39} + ( -8 + \beta_{7} ) q^{41} + ( \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} ) q^{43} + ( -\beta_{3} + 3 \beta_{6} ) q^{47} + ( -3 - \beta_{1} + 2 \beta_{2} + 2 \beta_{7} ) q^{49} + ( -\beta_{1} - 5 \beta_{2} + \beta_{7} ) q^{51} + ( 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{53} + ( 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{57} + ( -1 + \beta_{1} + \beta_{2} ) q^{59} + ( -2 \beta_{1} + 5 \beta_{2} + 2 \beta_{7} ) q^{61} + ( -5 \beta_{4} + 4 \beta_{5} + 5 \beta_{6} ) q^{67} + ( -5 - 2 \beta_{7} ) q^{69} + ( 4 - 2 \beta_{7} ) q^{71} + ( 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{73} + ( -4 \beta_{4} - 3 \beta_{5} ) q^{77} + ( \beta_{1} - 3 \beta_{2} - \beta_{7} ) q^{79} + ( 9 - 9 \beta_{2} ) q^{81} + ( -3 \beta_{3} - \beta_{4} + 3 \beta_{5} + 3 \beta_{6} ) q^{83} + ( -\beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{87} + 7 \beta_{2} q^{89} + ( -4 + 2 \beta_{1} + 12 \beta_{2} ) q^{91} + ( -3 \beta_{3} + \beta_{6} ) q^{93} + 4 \beta_{4} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q + 6 q^{11} + 2 q^{19} + 4 q^{29} + 2 q^{31} + 12 q^{39} - 60 q^{41} - 10 q^{49} - 18 q^{51} - 2 q^{59} + 24 q^{61} - 48 q^{69} + 24 q^{71} - 14 q^{79} + 36 q^{81} + 28 q^{89} + 20 q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} - 56 \nu^{4} - 224 \nu^{2} - 895$$$$)/280$$ $$\beta_{2}$$ $$=$$ $$($$$$9 \nu^{6} + 56 \nu^{4} + 504 \nu^{2} + 2025$$$$)/1400$$ $$\beta_{3}$$ $$=$$ $$($$$$11 \nu^{7} + 224 \nu^{5} + 616 \nu^{3} + 9475 \nu$$$$)/7000$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 16 \nu^{5} + 44 \nu^{3} + 175 \nu$$$$)/500$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 14 \nu^{5} + 126 \nu^{3} + 505 \nu$$$$)/350$$ $$\beta_{6}$$ $$=$$ $$($$$$47 \nu^{7} + 448 \nu^{5} + 1232 \nu^{3} + 4975 \nu$$$$)/7000$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{6} - 9 \nu^{4} - 31 \nu^{2} - 100$$$$)/25$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{6} - \beta_{4} + 3 \beta_{3}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{7} + 13 \beta_{2} - \beta_{1} - 14$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{6} + 12 \beta_{5} + \beta_{4} - 12 \beta_{3}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{7} - 17 \beta_{2} - 3 \beta_{1} + 3$$ $$\nu^{5}$$ $$=$$ $$($$$$34 \beta_{6} - 33 \beta_{5} + 67 \beta_{4}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-56 \beta_{7} + 56 \beta_{2} + 112 \beta_{1} + 53$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$281 \beta_{6} - 559 \beta_{4} - 3 \beta_{3}$$$$)/3$$

## Character values

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

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 −0.656712 + 2.13746i 1.52274 − 1.63746i −1.52274 + 1.63746i 0.656712 − 2.13746i −0.656712 − 2.13746i 1.52274 + 1.63746i −1.52274 − 1.63746i 0.656712 + 2.13746i
0 −0.866025 1.50000i 0 0 0 0.209313 + 2.63746i 0 0 0
401.2 0 −0.866025 1.50000i 0 0 0 2.38876 1.13746i 0 0 0
401.3 0 0.866025 + 1.50000i 0 0 0 −2.38876 + 1.13746i 0 0 0
401.4 0 0.866025 + 1.50000i 0 0 0 −0.209313 2.63746i 0 0 0
501.1 0 −0.866025 + 1.50000i 0 0 0 0.209313 2.63746i 0 0 0
501.2 0 −0.866025 + 1.50000i 0 0 0 2.38876 + 1.13746i 0 0 0
501.3 0 0.866025 1.50000i 0 0 0 −2.38876 1.13746i 0 0 0
501.4 0 0.866025 1.50000i 0 0 0 −0.209313 + 2.63746i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 501.4 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.i.f 8
5.b even 2 1 inner 700.2.i.f 8
5.c odd 4 1 140.2.q.a 4
5.c odd 4 1 140.2.q.b yes 4
7.c even 3 1 inner 700.2.i.f 8
7.c even 3 1 4900.2.a.be 4
7.d odd 6 1 4900.2.a.bf 4
15.e even 4 1 1260.2.bm.a 4
15.e even 4 1 1260.2.bm.b 4
20.e even 4 1 560.2.bw.a 4
20.e even 4 1 560.2.bw.e 4
35.f even 4 1 980.2.q.b 4
35.f even 4 1 980.2.q.g 4
35.i odd 6 1 4900.2.a.bf 4
35.j even 6 1 inner 700.2.i.f 8
35.j even 6 1 4900.2.a.be 4
35.k even 12 2 980.2.e.c 4
35.k even 12 1 980.2.q.b 4
35.k even 12 1 980.2.q.g 4
35.l odd 12 1 140.2.q.a 4
35.l odd 12 1 140.2.q.b yes 4
35.l odd 12 2 980.2.e.f 4
105.x even 12 1 1260.2.bm.a 4
105.x even 12 1 1260.2.bm.b 4
140.w even 12 1 560.2.bw.a 4
140.w even 12 1 560.2.bw.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.q.a 4 5.c odd 4 1
140.2.q.a 4 35.l odd 12 1
140.2.q.b yes 4 5.c odd 4 1
140.2.q.b yes 4 35.l odd 12 1
560.2.bw.a 4 20.e even 4 1
560.2.bw.a 4 140.w even 12 1
560.2.bw.e 4 20.e even 4 1
560.2.bw.e 4 140.w even 12 1
700.2.i.f 8 1.a even 1 1 trivial
700.2.i.f 8 5.b even 2 1 inner
700.2.i.f 8 7.c even 3 1 inner
700.2.i.f 8 35.j even 6 1 inner
980.2.e.c 4 35.k even 12 2
980.2.e.f 4 35.l odd 12 2
980.2.q.b 4 35.f even 4 1
980.2.q.b 4 35.k even 12 1
980.2.q.g 4 35.f even 4 1
980.2.q.g 4 35.k even 12 1
1260.2.bm.a 4 15.e even 4 1
1260.2.bm.a 4 105.x even 12 1
1260.2.bm.b 4 15.e even 4 1
1260.2.bm.b 4 105.x even 12 1
4900.2.a.be 4 7.c even 3 1
4900.2.a.be 4 35.j even 6 1
4900.2.a.bf 4 7.d odd 6 1
4900.2.a.bf 4 35.i odd 6 1

## Hecke kernels

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

 $$T_{3}^{4} + 3 T_{3}^{2} + 9$$ $$T_{11}^{4} - 3 T_{11}^{3} + 21 T_{11}^{2} + 36 T_{11} + 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 9 + 3 T^{2} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$2401 + 245 T^{2} - 24 T^{4} + 5 T^{6} + T^{8}$$
$11$ $$( 144 + 36 T + 21 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$13$ $$( 256 - 44 T^{2} + T^{4} )^{2}$$
$17$ $$16 + 92 T^{2} + 525 T^{4} + 23 T^{6} + T^{8}$$
$19$ $$( 196 + 14 T + 15 T^{2} - T^{3} + T^{4} )^{2}$$
$23$ $$2401 + 3038 T^{2} + 3795 T^{4} + 62 T^{6} + T^{8}$$
$29$ $$( -14 - T + T^{2} )^{4}$$
$31$ $$( 196 + 14 T + 15 T^{2} - T^{3} + T^{4} )^{2}$$
$37$ $$9834496 + 410816 T^{2} + 14025 T^{4} + 131 T^{6} + T^{8}$$
$41$ $$( 42 + 15 T + T^{2} )^{4}$$
$43$ $$( 196 - 47 T^{2} + T^{4} )^{2}$$
$47$ $$38416 + 9212 T^{2} + 2013 T^{4} + 47 T^{6} + T^{8}$$
$53$ $$3111696 + 153468 T^{2} + 5805 T^{4} + 87 T^{6} + T^{8}$$
$59$ $$( 196 - 14 T + 15 T^{2} + T^{3} + T^{4} )^{2}$$
$61$ $$( 441 + 252 T + 165 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$67$ $$5764801 + 494606 T^{2} + 40035 T^{4} + 206 T^{6} + T^{8}$$
$71$ $$( -48 - 6 T + T^{2} )^{4}$$
$73$ $$38416 + 9212 T^{2} + 2013 T^{4} + 47 T^{6} + T^{8}$$
$79$ $$( 4 - 14 T + 51 T^{2} + 7 T^{3} + T^{4} )^{2}$$
$83$ $$( 1764 - 87 T^{2} + T^{4} )^{2}$$
$89$ $$( 49 - 7 T + T^{2} )^{4}$$
$97$ $$( -48 + T^{2} )^{4}$$