# 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: $$x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{6} + \beta_{4}) q^{3} + ( - \beta_{4} + \beta_{3}) q^{7}+O(q^{10})$$ q + (-b6 + b4) * q^3 + (-b4 + b3) * q^7 $$q + ( - \beta_{6} + \beta_{4}) q^{3} + ( - \beta_{4} + \beta_{3}) q^{7} + ( - \beta_{2} + \beta_1 + 1) q^{11} + ( - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{13} + ( - 2 \beta_{6} - \beta_{5} + 2 \beta_{4}) q^{17} + ( - \beta_{7} + \beta_{2} + \beta_1) q^{19} + ( - \beta_{7} + 3 \beta_{2} - 1) q^{21} + ( - 3 \beta_{6} + 2 \beta_{3}) q^{23} + 3 \beta_{4} q^{27} + \beta_{7} q^{29} + ( - \beta_{2} - \beta_1 + 1) q^{31} + ( - 3 \beta_{6} + 3 \beta_{3}) q^{33} + ( - 5 \beta_{6} + \beta_{3}) q^{37} + ( - 4 \beta_{2} - 2 \beta_1 + 4) q^{39} + (\beta_{7} - 8) q^{41} + ( - \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3}) q^{43} + (3 \beta_{6} - \beta_{3}) q^{47} + (2 \beta_{7} + 2 \beta_{2} - \beta_1 - 3) q^{49} + (\beta_{7} - 5 \beta_{2} - \beta_1) q^{51} + ( - 2 \beta_{6} - 3 \beta_{5} + 2 \beta_{4}) q^{53} + ( - 3 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} + 3 \beta_{3}) q^{57} + (\beta_{2} + \beta_1 - 1) q^{59} + (2 \beta_{7} + 5 \beta_{2} - 2 \beta_1) q^{61} + (5 \beta_{6} + 4 \beta_{5} - 5 \beta_{4}) q^{67} + ( - 2 \beta_{7} - 5) q^{69} + ( - 2 \beta_{7} + 4) q^{71} + ( - 2 \beta_{6} + \beta_{5} + 2 \beta_{4}) q^{73} + ( - 3 \beta_{5} - 4 \beta_{4}) q^{77} + ( - \beta_{7} - 3 \beta_{2} + \beta_1) q^{79} + ( - 9 \beta_{2} + 9) q^{81} + (3 \beta_{6} + 3 \beta_{5} - \beta_{4} - 3 \beta_{3}) q^{83} + (\beta_{6} + 3 \beta_{5} - \beta_{4}) q^{87} + 7 \beta_{2} q^{89} + (12 \beta_{2} + 2 \beta_1 - 4) q^{91} + (\beta_{6} - 3 \beta_{3}) q^{93} + 4 \beta_{4} q^{97}+O(q^{100})$$ q + (-b6 + b4) * q^3 + (-b4 + b3) * q^7 + (-b2 + b1 + 1) * q^11 + (-2*b6 - 2*b5 + 2*b4 + 2*b3) * q^13 + (-2*b6 - b5 + 2*b4) * q^17 + (-b7 + b2 + b1) * q^19 + (-b7 + 3*b2 - 1) * q^21 + (-3*b6 + 2*b3) * q^23 + 3*b4 * q^27 + b7 * q^29 + (-b2 - b1 + 1) * q^31 + (-3*b6 + 3*b3) * q^33 + (-5*b6 + b3) * q^37 + (-4*b2 - 2*b1 + 4) * q^39 + (b7 - 8) * q^41 + (-b6 - b5 + 3*b4 + b3) * q^43 + (3*b6 - b3) * q^47 + (2*b7 + 2*b2 - b1 - 3) * q^49 + (b7 - 5*b2 - b1) * q^51 + (-2*b6 - 3*b5 + 2*b4) * q^53 + (-3*b6 - 3*b5 + 2*b4 + 3*b3) * q^57 + (b2 + b1 - 1) * q^59 + (2*b7 + 5*b2 - 2*b1) * q^61 + (5*b6 + 4*b5 - 5*b4) * q^67 + (-2*b7 - 5) * q^69 + (-2*b7 + 4) * q^71 + (-2*b6 + b5 + 2*b4) * q^73 + (-3*b5 - 4*b4) * q^77 + (-b7 - 3*b2 + b1) * q^79 + (-9*b2 + 9) * q^81 + (3*b6 + 3*b5 - b4 - 3*b3) * q^83 + (b6 + 3*b5 - b4) * q^87 + 7*b2 * q^89 + (12*b2 + 2*b1 - 4) * q^91 + (b6 - 3*b3) * q^93 + 4*b4 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 6 q^{11} + 2 q^{19} + 4 q^{29} + 2 q^{31} + 12 q^{39} - 60 q^{41} - 10 q^{49} - 18 q^{51} - 2 q^{59} + 24 q^{61} - 48 q^{69} + 24 q^{71} - 14 q^{79} + 36 q^{81} + 28 q^{89} + 20 q^{91}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{6} - 56\nu^{4} - 224\nu^{2} - 895 ) / 280$$ (v^6 - 56*v^4 - 224*v^2 - 895) / 280 $$\beta_{2}$$ $$=$$ $$( 9\nu^{6} + 56\nu^{4} + 504\nu^{2} + 2025 ) / 1400$$ (9*v^6 + 56*v^4 + 504*v^2 + 2025) / 1400 $$\beta_{3}$$ $$=$$ $$( 11\nu^{7} + 224\nu^{5} + 616\nu^{3} + 9475\nu ) / 7000$$ (11*v^7 + 224*v^5 + 616*v^3 + 9475*v) / 7000 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} + 16\nu^{5} + 44\nu^{3} + 175\nu ) / 500$$ (-v^7 + 16*v^5 + 44*v^3 + 175*v) / 500 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + 14\nu^{5} + 126\nu^{3} + 505\nu ) / 350$$ (v^7 + 14*v^5 + 126*v^3 + 505*v) / 350 $$\beta_{6}$$ $$=$$ $$( 47\nu^{7} + 448\nu^{5} + 1232\nu^{3} + 4975\nu ) / 7000$$ (47*v^7 + 448*v^5 + 1232*v^3 + 4975*v) / 7000 $$\beta_{7}$$ $$=$$ $$( -\nu^{6} - 9\nu^{4} - 31\nu^{2} - 100 ) / 25$$ (-v^6 - 9*v^4 - 31*v^2 - 100) / 25
 $$\nu$$ $$=$$ $$( -\beta_{6} - \beta_{4} + 3\beta_{3} ) / 3$$ (-b6 - b4 + 3*b3) / 3 $$\nu^{2}$$ $$=$$ $$( 2\beta_{7} + 13\beta_{2} - \beta _1 - 14 ) / 3$$ (2*b7 + 13*b2 - b1 - 14) / 3 $$\nu^{3}$$ $$=$$ $$( -2\beta_{6} + 12\beta_{5} + \beta_{4} - 12\beta_{3} ) / 3$$ (-2*b6 + 12*b5 + b4 - 12*b3) / 3 $$\nu^{4}$$ $$=$$ $$-3\beta_{7} - 17\beta_{2} - 3\beta _1 + 3$$ -3*b7 - 17*b2 - 3*b1 + 3 $$\nu^{5}$$ $$=$$ $$( 34\beta_{6} - 33\beta_{5} + 67\beta_{4} ) / 3$$ (34*b6 - 33*b5 + 67*b4) / 3 $$\nu^{6}$$ $$=$$ $$( -56\beta_{7} + 56\beta_{2} + 112\beta _1 + 53 ) / 3$$ (-56*b7 + 56*b2 + 112*b1 + 53) / 3 $$\nu^{7}$$ $$=$$ $$( 281\beta_{6} - 559\beta_{4} - 3\beta_{3} ) / 3$$ (281*b6 - 559*b4 - 3*b3) / 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 $$S_{2}^{\mathrm{new}}(700, [\chi])$$:

 $$T_{3}^{4} + 3T_{3}^{2} + 9$$ T3^4 + 3*T3^2 + 9 $$T_{11}^{4} - 3T_{11}^{3} + 21T_{11}^{2} + 36T_{11} + 144$$ T11^4 - 3*T11^3 + 21*T11^2 + 36*T11 + 144

## Hecke characteristic polynomials

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