# Properties

 Label 5070.2.b.z Level $5070$ Weight $2$ Character orbit 5070.b Analytic conductor $40.484$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.153664.1 Defining polynomial: $$x^{6} + 5x^{4} + 6x^{2} + 1$$ x^6 + 5*x^4 + 6*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} + q^{3} - q^{4} - \beta_{5} q^{5} + \beta_{5} q^{6} + ( - 3 \beta_{5} - \beta_{3} + 2 \beta_1) q^{7} - \beta_{5} q^{8} + q^{9}+O(q^{10})$$ q + b5 * q^2 + q^3 - q^4 - b5 * q^5 + b5 * q^6 + (-3*b5 - b3 + 2*b1) * q^7 - b5 * q^8 + q^9 $$q + \beta_{5} q^{2} + q^{3} - q^{4} - \beta_{5} q^{5} + \beta_{5} q^{6} + ( - 3 \beta_{5} - \beta_{3} + 2 \beta_1) q^{7} - \beta_{5} q^{8} + q^{9} + q^{10} + ( - \beta_{3} - \beta_1) q^{11} - q^{12} + ( - \beta_{4} + \beta_{2} + 2) q^{14} - \beta_{5} q^{15} + q^{16} + ( - 3 \beta_{4} - 2 \beta_{2} + 4) q^{17} + \beta_{5} q^{18} + ( - 2 \beta_{5} - 3 \beta_{3} + 2 \beta_1) q^{19} + \beta_{5} q^{20} + ( - 3 \beta_{5} - \beta_{3} + 2 \beta_1) q^{21} + (2 \beta_{4} + \beta_{2} - 1) q^{22} + ( - \beta_{4} - 2 \beta_{2}) q^{23} - \beta_{5} q^{24} - q^{25} + q^{27} + (3 \beta_{5} + \beta_{3} - 2 \beta_1) q^{28} + ( - 7 \beta_{4} - 5 \beta_{2} + 4) q^{29} + q^{30} + (4 \beta_{5} - \beta_{3} + 3 \beta_1) q^{31} + \beta_{5} q^{32} + ( - \beta_{3} - \beta_1) q^{33} + (2 \beta_{5} - 2 \beta_{3} - \beta_1) q^{34} + (\beta_{4} - \beta_{2} - 2) q^{35} - q^{36} + ( - 7 \beta_{5} - 3 \beta_{3} - 2 \beta_1) q^{37} + (\beta_{4} + 3 \beta_{2} - 1) q^{38} - q^{40} + (6 \beta_{5} - \beta_{3} - 2 \beta_1) q^{41} + ( - \beta_{4} + \beta_{2} + 2) q^{42} + (4 \beta_{4} - \beta_{2} - 4) q^{43} + (\beta_{3} + \beta_1) q^{44} - \beta_{5} q^{45} + ( - 2 \beta_{5} - 2 \beta_{3} + \beta_1) q^{46} + ( - 3 \beta_{5} + 2 \beta_{3} + 4 \beta_1) q^{47} + q^{48} + (3 \beta_{4} - 2 \beta_{2} - 2) q^{49} - \beta_{5} q^{50} + ( - 3 \beta_{4} - 2 \beta_{2} + 4) q^{51} + (3 \beta_{4} + 3 \beta_{2} - 1) q^{53} + \beta_{5} q^{54} + ( - 2 \beta_{4} - \beta_{2} + 1) q^{55} + (\beta_{4} - \beta_{2} - 2) q^{56} + ( - 2 \beta_{5} - 3 \beta_{3} + 2 \beta_1) q^{57} + ( - \beta_{5} - 5 \beta_{3} - 2 \beta_1) q^{58} + ( - 3 \beta_{5} + \beta_{3} + 3 \beta_1) q^{59} + \beta_{5} q^{60} + ( - \beta_{2} + 2) q^{61} + ( - 2 \beta_{4} + \beta_{2} - 5) q^{62} + ( - 3 \beta_{5} - \beta_{3} + 2 \beta_1) q^{63} - q^{64} + (2 \beta_{4} + \beta_{2} - 1) q^{66} + (6 \beta_{5} + 2 \beta_{3} + \beta_1) q^{67} + (3 \beta_{4} + 2 \beta_{2} - 4) q^{68} + ( - \beta_{4} - 2 \beta_{2}) q^{69} + ( - 3 \beta_{5} - \beta_{3} + 2 \beta_1) q^{70} + ( - 4 \beta_{5} - 2 \beta_{3} + 3 \beta_1) q^{71} - \beta_{5} q^{72} + (2 \beta_{5} + 8 \beta_{3} - \beta_1) q^{73} + (5 \beta_{4} + 3 \beta_{2} + 4) q^{74} - q^{75} + (2 \beta_{5} + 3 \beta_{3} - 2 \beta_1) q^{76} + ( - 6 \beta_{4} - 5 \beta_{2} + 6) q^{77} + ( - 3 \beta_{4} - 7 \beta_{2} + 9) q^{79} - \beta_{5} q^{80} + q^{81} + (3 \beta_{4} + \beta_{2} - 7) q^{82} + (4 \beta_{5} + 7 \beta_{3} + \beta_1) q^{83} + (3 \beta_{5} + \beta_{3} - 2 \beta_1) q^{84} + ( - 2 \beta_{5} + 2 \beta_{3} + \beta_1) q^{85} + ( - 5 \beta_{5} - \beta_{3} + 5 \beta_1) q^{86} + ( - 7 \beta_{4} - 5 \beta_{2} + 4) q^{87} + ( - 2 \beta_{4} - \beta_{2} + 1) q^{88} + (6 \beta_{5} + 6 \beta_{3} - 9 \beta_1) q^{89} + q^{90} + (\beta_{4} + 2 \beta_{2}) q^{92} + (4 \beta_{5} - \beta_{3} + 3 \beta_1) q^{93} + ( - 6 \beta_{4} - 2 \beta_{2} + 5) q^{94} + ( - \beta_{4} - 3 \beta_{2} + 1) q^{95} + \beta_{5} q^{96} + (11 \beta_{5} + 5 \beta_{3} - 4 \beta_1) q^{97} + ( - 4 \beta_{5} - 2 \beta_{3} + 5 \beta_1) q^{98} + ( - \beta_{3} - \beta_1) q^{99}+O(q^{100})$$ q + b5 * q^2 + q^3 - q^4 - b5 * q^5 + b5 * q^6 + (-3*b5 - b3 + 2*b1) * q^7 - b5 * q^8 + q^9 + q^10 + (-b3 - b1) * q^11 - q^12 + (-b4 + b2 + 2) * q^14 - b5 * q^15 + q^16 + (-3*b4 - 2*b2 + 4) * q^17 + b5 * q^18 + (-2*b5 - 3*b3 + 2*b1) * q^19 + b5 * q^20 + (-3*b5 - b3 + 2*b1) * q^21 + (2*b4 + b2 - 1) * q^22 + (-b4 - 2*b2) * q^23 - b5 * q^24 - q^25 + q^27 + (3*b5 + b3 - 2*b1) * q^28 + (-7*b4 - 5*b2 + 4) * q^29 + q^30 + (4*b5 - b3 + 3*b1) * q^31 + b5 * q^32 + (-b3 - b1) * q^33 + (2*b5 - 2*b3 - b1) * q^34 + (b4 - b2 - 2) * q^35 - q^36 + (-7*b5 - 3*b3 - 2*b1) * q^37 + (b4 + 3*b2 - 1) * q^38 - q^40 + (6*b5 - b3 - 2*b1) * q^41 + (-b4 + b2 + 2) * q^42 + (4*b4 - b2 - 4) * q^43 + (b3 + b1) * q^44 - b5 * q^45 + (-2*b5 - 2*b3 + b1) * q^46 + (-3*b5 + 2*b3 + 4*b1) * q^47 + q^48 + (3*b4 - 2*b2 - 2) * q^49 - b5 * q^50 + (-3*b4 - 2*b2 + 4) * q^51 + (3*b4 + 3*b2 - 1) * q^53 + b5 * q^54 + (-2*b4 - b2 + 1) * q^55 + (b4 - b2 - 2) * q^56 + (-2*b5 - 3*b3 + 2*b1) * q^57 + (-b5 - 5*b3 - 2*b1) * q^58 + (-3*b5 + b3 + 3*b1) * q^59 + b5 * q^60 + (-b2 + 2) * q^61 + (-2*b4 + b2 - 5) * q^62 + (-3*b5 - b3 + 2*b1) * q^63 - q^64 + (2*b4 + b2 - 1) * q^66 + (6*b5 + 2*b3 + b1) * q^67 + (3*b4 + 2*b2 - 4) * q^68 + (-b4 - 2*b2) * q^69 + (-3*b5 - b3 + 2*b1) * q^70 + (-4*b5 - 2*b3 + 3*b1) * q^71 - b5 * q^72 + (2*b5 + 8*b3 - b1) * q^73 + (5*b4 + 3*b2 + 4) * q^74 - q^75 + (2*b5 + 3*b3 - 2*b1) * q^76 + (-6*b4 - 5*b2 + 6) * q^77 + (-3*b4 - 7*b2 + 9) * q^79 - b5 * q^80 + q^81 + (3*b4 + b2 - 7) * q^82 + (4*b5 + 7*b3 + b1) * q^83 + (3*b5 + b3 - 2*b1) * q^84 + (-2*b5 + 2*b3 + b1) * q^85 + (-5*b5 - b3 + 5*b1) * q^86 + (-7*b4 - 5*b2 + 4) * q^87 + (-2*b4 - b2 + 1) * q^88 + (6*b5 + 6*b3 - 9*b1) * q^89 + q^90 + (b4 + 2*b2) * q^92 + (4*b5 - b3 + 3*b1) * q^93 + (-6*b4 - 2*b2 + 5) * q^94 + (-b4 - 3*b2 + 1) * q^95 + b5 * q^96 + (11*b5 + 5*b3 - 4*b1) * q^97 + (-4*b5 - 2*b3 + 5*b1) * q^98 + (-b3 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10})$$ 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 $$6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} - 6 q^{12} + 12 q^{14} + 6 q^{16} + 14 q^{17} - 6 q^{23} - 6 q^{25} + 6 q^{27} + 6 q^{30} - 12 q^{35} - 6 q^{36} + 2 q^{38} - 6 q^{40} + 12 q^{42} - 18 q^{43} + 6 q^{48} - 10 q^{49} + 14 q^{51} + 6 q^{53} - 12 q^{56} + 10 q^{61} - 32 q^{62} - 6 q^{64} - 14 q^{68} - 6 q^{69} + 40 q^{74} - 6 q^{75} + 14 q^{77} + 34 q^{79} + 6 q^{81} - 34 q^{82} + 6 q^{90} + 6 q^{92} + 14 q^{94} - 2 q^{95}+O(q^{100})$$ 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 - 6 * q^12 + 12 * q^14 + 6 * q^16 + 14 * q^17 - 6 * q^23 - 6 * q^25 + 6 * q^27 + 6 * q^30 - 12 * q^35 - 6 * q^36 + 2 * q^38 - 6 * q^40 + 12 * q^42 - 18 * q^43 + 6 * q^48 - 10 * q^49 + 14 * q^51 + 6 * q^53 - 12 * q^56 + 10 * q^61 - 32 * q^62 - 6 * q^64 - 14 * q^68 - 6 * q^69 + 40 * q^74 - 6 * q^75 + 14 * q^77 + 34 * q^79 + 6 * q^81 - 34 * q^82 + 6 * q^90 + 6 * q^92 + 14 * q^94 - 2 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 5x^{4} + 6x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3\nu$$ v^3 + 3*v $$\beta_{4}$$ $$=$$ $$\nu^{4} + 3\nu^{2} + 1$$ v^4 + 3*v^2 + 1 $$\beta_{5}$$ $$=$$ $$\nu^{5} + 4\nu^{3} + 3\nu$$ v^5 + 4*v^3 + 3*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ b2 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3\beta_1$$ b3 - 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} - 3\beta_{2} + 5$$ b4 - 3*b2 + 5 $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4\beta_{3} + 9\beta_1$$ b5 - 4*b3 + 9*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1351.1
 − 1.80194i − 0.445042i 1.24698i − 1.24698i 0.445042i 1.80194i
1.00000i 1.00000 −1.00000 1.00000i 1.00000i 1.04892i 1.00000i 1.00000 1.00000
1351.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.35690i 1.00000i 1.00000 1.00000
1351.3 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.69202i 1.00000i 1.00000 1.00000
1351.4 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.69202i 1.00000i 1.00000 1.00000
1351.5 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.35690i 1.00000i 1.00000 1.00000
1351.6 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 1.04892i 1.00000i 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1351.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5070.2.b.z 6
13.b even 2 1 inner 5070.2.b.z 6
13.d odd 4 1 5070.2.a.bp 3
13.d odd 4 1 5070.2.a.bw yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5070.2.a.bp 3 13.d odd 4 1
5070.2.a.bw yes 3 13.d odd 4 1
5070.2.b.z 6 1.a even 1 1 trivial
5070.2.b.z 6 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(5070, [\chi])$$:

 $$T_{7}^{6} + 26T_{7}^{4} + 181T_{7}^{2} + 169$$ T7^6 + 26*T7^4 + 181*T7^2 + 169 $$T_{11}^{6} + 14T_{11}^{4} + 49T_{11}^{2} + 49$$ T11^6 + 14*T11^4 + 49*T11^2 + 49 $$T_{17}^{3} - 7T_{17}^{2} + 7$$ T17^3 - 7*T17^2 + 7 $$T_{31}^{6} + 118T_{31}^{4} + 2105T_{31}^{2} + 6889$$ T31^6 + 118*T31^4 + 2105*T31^2 + 6889

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{3}$$
$3$ $$(T - 1)^{6}$$
$5$ $$(T^{2} + 1)^{3}$$
$7$ $$T^{6} + 26 T^{4} + 181 T^{2} + \cdots + 169$$
$11$ $$T^{6} + 14 T^{4} + 49 T^{2} + 49$$
$13$ $$T^{6}$$
$17$ $$(T^{3} - 7 T^{2} + 7)^{2}$$
$19$ $$T^{6} + 33 T^{4} + 230 T^{2} + \cdots + 169$$
$23$ $$(T^{3} + 3 T^{2} - 4 T + 1)^{2}$$
$29$ $$(T^{3} - 91 T - 203)^{2}$$
$31$ $$T^{6} + 118 T^{4} + 2105 T^{2} + \cdots + 6889$$
$37$ $$T^{6} + 222 T^{4} + 11801 T^{2} + \cdots + 9409$$
$41$ $$T^{6} + 129 T^{4} + 2558 T^{2} + \cdots + 12769$$
$43$ $$(T^{3} + 9 T^{2} - 22 T - 169)^{2}$$
$47$ $$T^{6} + 147 T^{4} + 1715 T^{2} + \cdots + 2401$$
$53$ $$(T^{3} - 3 T^{2} - 18 T + 13)^{2}$$
$59$ $$T^{6} + 77 T^{4} + 294 T^{2} + \cdots + 49$$
$61$ $$(T^{3} - 5 T^{2} + 6 T - 1)^{2}$$
$67$ $$T^{6} + 129 T^{4} + 3986 T^{2} + \cdots + 5041$$
$71$ $$T^{6} + 49 T^{4} + 686 T^{2} + \cdots + 2401$$
$73$ $$T^{6} + 269 T^{4} + 19498 T^{2} + \cdots + 187489$$
$79$ $$(T^{3} - 17 T^{2} + 10 T + 587)^{2}$$
$83$ $$T^{6} + 278 T^{4} + 18829 T^{2} + \cdots + 121801$$
$89$ $$T^{6} + 297 T^{4} + 25434 T^{2} + \cdots + 613089$$
$97$ $$T^{6} + 290 T^{4} + 12337 T^{2} + \cdots + 28561$$