# Properties

 Label 700.6.e.g Level $700$ Weight $6$ Character orbit 700.e Analytic conductor $112.269$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 700.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$112.268673869$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ Defining polynomial: $$x^{6} + 998x^{4} + 249001x^{2} + 44100$$ x^6 + 998*x^4 + 249001*x^2 + 44100 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{5}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{2} + \beta_1) q^{3} + 49 \beta_{2} q^{7} + (\beta_{4} - 5 \beta_{3} - 94) q^{9}+O(q^{10})$$ q + (-2*b2 + b1) * q^3 + 49*b2 * q^7 + (b4 - 5*b3 - 94) * q^9 $$q + ( - 2 \beta_{2} + \beta_1) q^{3} + 49 \beta_{2} q^{7} + (\beta_{4} - 5 \beta_{3} - 94) q^{9} + (\beta_{4} + 11 \beta_{3} - 5) q^{11} + ( - 2 \beta_{5} + 2 \beta_{2} + 13 \beta_1) q^{13} + (2 \beta_{5} + 14 \beta_{2} - 31 \beta_1) q^{17} + (9 \beta_{4} - 6 \beta_{3} - 779) q^{19} + (49 \beta_{3} + 98) q^{21} + ( - \beta_{5} - 1225 \beta_{2} - 70 \beta_1) q^{23} + ( - 6 \beta_{5} + 1244 \beta_{2} - 31 \beta_1) q^{27} + ( - 15 \beta_{4} + 99 \beta_{3} - 1359) q^{29} + (17 \beta_{4} - 248 \beta_{3} + 1957) q^{31} + (10 \beta_{5} - 3776 \beta_{2} - 137 \beta_1) q^{33} + ( - \beta_{5} + 3793 \beta_{2} - 64 \beta_1) q^{37} + (11 \beta_{4} + 293 \beta_{3} - 4571) q^{39} + ( - 31 \beta_{4} + 142 \beta_{3} + 3827) q^{41} + ( - 31 \beta_{5} - 6171 \beta_{2} - 322 \beta_1) q^{43} + (16 \beta_{5} + 7246 \beta_{2} - 113 \beta_1) q^{47} - 2401 q^{49} + ( - 29 \beta_{4} - 223 \beta_{3} + 10597) q^{51} + ( - 6 \beta_{5} - 2496 \beta_{2} + 594 \beta_1) q^{53} + ( - 15 \beta_{5} + 2449 \beta_{2} - 2282 \beta_1) q^{57} + ( - 64 \beta_{4} + 1840 \beta_{3} - 4108) q^{59} + (71 \beta_{4} - 110 \beta_{3} + 9037) q^{61} + (49 \beta_{5} - 4606 \beta_{2} + 245 \beta_1) q^{63} + ( - 8 \beta_{5} - 2556 \beta_{2} - 2228 \beta_1) q^{67} + ( - 71 \beta_{4} - 850 \beta_{3} + 20737) q^{69} + ( - 184 \beta_{4} + 136 \beta_{3} + 27392) q^{71} + (250 \beta_{5} - 160 \beta_{2} - 1772 \beta_1) q^{73} + (49 \beta_{5} - 245 \beta_{2} - 539 \beta_1) q^{77} + (95 \beta_{4} + 3919 \beta_{3} + 5277) q^{79} + (206 \beta_{4} + 1112 \beta_{3} - 10769) q^{81} + ( - 138 \beta_{5} + 28746 \beta_{2} - 960 \beta_1) q^{83} + (114 \beta_{5} - 28404 \beta_{2} + 1413 \beta_1) q^{87} + (127 \beta_{4} + 2870 \beta_{3} + 31861) q^{89} + (98 \beta_{4} + 637 \beta_{3} - 98) q^{91} + ( - 265 \beta_{5} + 76579 \beta_{2} - 1592 \beta_1) q^{93} + ( - 322 \beta_{5} - 58622 \beta_{2} - 931 \beta_1) q^{97} + (116 \beta_{4} - 2342 \beta_{3} + 38084) q^{99}+O(q^{100})$$ q + (-2*b2 + b1) * q^3 + 49*b2 * q^7 + (b4 - 5*b3 - 94) * q^9 + (b4 + 11*b3 - 5) * q^11 + (-2*b5 + 2*b2 + 13*b1) * q^13 + (2*b5 + 14*b2 - 31*b1) * q^17 + (9*b4 - 6*b3 - 779) * q^19 + (49*b3 + 98) * q^21 + (-b5 - 1225*b2 - 70*b1) * q^23 + (-6*b5 + 1244*b2 - 31*b1) * q^27 + (-15*b4 + 99*b3 - 1359) * q^29 + (17*b4 - 248*b3 + 1957) * q^31 + (10*b5 - 3776*b2 - 137*b1) * q^33 + (-b5 + 3793*b2 - 64*b1) * q^37 + (11*b4 + 293*b3 - 4571) * q^39 + (-31*b4 + 142*b3 + 3827) * q^41 + (-31*b5 - 6171*b2 - 322*b1) * q^43 + (16*b5 + 7246*b2 - 113*b1) * q^47 - 2401 * q^49 + (-29*b4 - 223*b3 + 10597) * q^51 + (-6*b5 - 2496*b2 + 594*b1) * q^53 + (-15*b5 + 2449*b2 - 2282*b1) * q^57 + (-64*b4 + 1840*b3 - 4108) * q^59 + (71*b4 - 110*b3 + 9037) * q^61 + (49*b5 - 4606*b2 + 245*b1) * q^63 + (-8*b5 - 2556*b2 - 2228*b1) * q^67 + (-71*b4 - 850*b3 + 20737) * q^69 + (-184*b4 + 136*b3 + 27392) * q^71 + (250*b5 - 160*b2 - 1772*b1) * q^73 + (49*b5 - 245*b2 - 539*b1) * q^77 + (95*b4 + 3919*b3 + 5277) * q^79 + (206*b4 + 1112*b3 - 10769) * q^81 + (-138*b5 + 28746*b2 - 960*b1) * q^83 + (114*b5 - 28404*b2 + 1413*b1) * q^87 + (127*b4 + 2870*b3 + 31861) * q^89 + (98*b4 + 637*b3 - 98) * q^91 + (-265*b5 + 76579*b2 - 1592*b1) * q^93 + (-322*b5 - 58622*b2 - 931*b1) * q^97 + (116*b4 - 2342*b3 + 38084) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 562 q^{9}+O(q^{10})$$ 6 * q - 562 * q^9 $$6 q - 562 q^{9} - 28 q^{11} - 4656 q^{19} + 588 q^{21} - 8184 q^{29} + 11776 q^{31} - 27404 q^{39} + 22900 q^{41} - 14406 q^{49} + 63524 q^{51} - 24776 q^{59} + 54364 q^{61} + 124280 q^{69} + 163984 q^{71} + 31852 q^{79} - 64202 q^{81} + 191420 q^{89} - 392 q^{91} + 228736 q^{99}+O(q^{100})$$ 6 * q - 562 * q^9 - 28 * q^11 - 4656 * q^19 + 588 * q^21 - 8184 * q^29 + 11776 * q^31 - 27404 * q^39 + 22900 * q^41 - 14406 * q^49 + 63524 * q^51 - 24776 * q^59 + 54364 * q^61 + 124280 * q^69 + 163984 * q^71 + 31852 * q^79 - 64202 * q^81 + 191420 * q^89 - 392 * q^91 + 228736 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 998x^{4} + 249001x^{2} + 44100$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 499\nu ) / 210$$ (v^3 + 499*v) / 210 $$\beta_{3}$$ $$=$$ $$( \nu^{4} + 499\nu^{2} ) / 210$$ (v^4 + 499*v^2) / 210 $$\beta_{4}$$ $$=$$ $$( \nu^{4} + 709\nu^{2} + 69930 ) / 210$$ (v^4 + 709*v^2 + 69930) / 210 $$\beta_{5}$$ $$=$$ $$( \nu^{5} + 832\nu^{3} + 165957\nu ) / 210$$ (v^5 + 832*v^3 + 165957*v) / 210
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} - 333$$ b4 - b3 - 333 $$\nu^{3}$$ $$=$$ $$210\beta_{2} - 499\beta_1$$ 210*b2 - 499*b1 $$\nu^{4}$$ $$=$$ $$-499\beta_{4} + 709\beta_{3} + 166167$$ -499*b4 + 709*b3 + 166167 $$\nu^{5}$$ $$=$$ $$210\beta_{5} - 174720\beta_{2} + 249211\beta_1$$ 210*b5 - 174720*b2 + 249211*b1

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 22.5458i − 22.1248i 0.420991i − 0.420991i 22.1248i 22.5458i
0 24.5458i 0 0 0 49.0000i 0 −359.498 0
449.2 0 20.1248i 0 0 0 49.0000i 0 −162.009 0
449.3 0 1.57901i 0 0 0 49.0000i 0 240.507 0
449.4 0 1.57901i 0 0 0 49.0000i 0 240.507 0
449.5 0 20.1248i 0 0 0 49.0000i 0 −162.009 0
449.6 0 24.5458i 0 0 0 49.0000i 0 −359.498 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.6 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.6.e.g 6
5.b even 2 1 inner 700.6.e.g 6
5.c odd 4 1 140.6.a.d 3
5.c odd 4 1 700.6.a.i 3
20.e even 4 1 560.6.a.t 3
35.f even 4 1 980.6.a.h 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.6.a.d 3 5.c odd 4 1
560.6.a.t 3 20.e even 4 1
700.6.a.i 3 5.c odd 4 1
700.6.e.g 6 1.a even 1 1 trivial
700.6.e.g 6 5.b even 2 1 inner
980.6.a.h 3 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 1010T_{3}^{4} + 246529T_{3}^{2} + 608400$$ acting on $$S_{6}^{\mathrm{new}}(700, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 1010 T^{4} + 246529 T^{2} + \cdots + 608400$$
$5$ $$T^{6}$$
$7$ $$(T^{2} + 2401)^{3}$$
$11$ $$(T^{3} + 14 T^{2} - 147231 T + 12436740)^{2}$$
$13$ $$T^{6} + 850758 T^{4} + \cdots + 37708614058756$$
$17$ $$T^{6} + 1668310 T^{4} + \cdots + 77\!\cdots\!00$$
$19$ $$(T^{3} + 2328 T^{2} + \cdots - 11429521264)^{2}$$
$23$ $$T^{6} + 9508744 T^{4} + \cdots + 43\!\cdots\!16$$
$29$ $$(T^{3} + 4092 T^{2} + \cdots - 17580868722)^{2}$$
$31$ $$(T^{3} - 5888 T^{2} + \cdots + 211802104832)^{2}$$
$37$ $$T^{6} + 47359404 T^{4} + \cdots + 22\!\cdots\!00$$
$41$ $$(T^{3} - 11450 T^{2} + \cdots + 478579953600)^{2}$$
$43$ $$T^{6} + 370031384 T^{4} + \cdots + 56\!\cdots\!76$$
$47$ $$T^{6} + 214248754 T^{4} + \cdots + 30\!\cdots\!36$$
$53$ $$T^{6} + 379440036 T^{4} + \cdots + 55\!\cdots\!00$$
$59$ $$(T^{3} + 12388 T^{2} + \cdots - 41176040028480)^{2}$$
$61$ $$(T^{3} - 27182 T^{2} + \cdots - 169955356480)^{2}$$
$67$ $$T^{6} + 4971185168 T^{4} + \cdots + 23\!\cdots\!00$$
$71$ $$(T^{3} - 81992 T^{2} + \cdots + 113425504819200)^{2}$$
$73$ $$T^{6} + 13818436332 T^{4} + \cdots + 12\!\cdots\!00$$
$79$ $$(T^{3} - 15926 T^{2} + \cdots + 274092525845520)^{2}$$
$83$ $$T^{6} + 6449648688 T^{4} + \cdots + 16\!\cdots\!00$$
$89$ $$(T^{3} - 95710 T^{2} + \cdots + 305457269205600)^{2}$$
$97$ $$T^{6} + 28176239622 T^{4} + \cdots + 17\!\cdots\!24$$