# Properties

 Label 5070.2.b.ba Level $5070$ Weight $2$ Character orbit 5070.b Analytic conductor $40.484$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: 8.0.17284886784.1 Defining polynomial: $$x^{8} - 2x^{7} + 2x^{6} + 30x^{5} + 185x^{4} + 36x^{3} + 8x^{2} + 208x + 2704$$ x^8 - 2*x^7 + 2*x^6 + 30*x^5 + 185*x^4 + 36*x^3 + 8*x^2 + 208*x + 2704 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + q^{3} - q^{4} + \beta_1 q^{5} - \beta_1 q^{6} + ( - \beta_{7} + \beta_1) q^{7} + \beta_1 q^{8} + q^{9}+O(q^{10})$$ q - b1 * q^2 + q^3 - q^4 + b1 * q^5 - b1 * q^6 + (-b7 + b1) * q^7 + b1 * q^8 + q^9 $$q - \beta_1 q^{2} + q^{3} - q^{4} + \beta_1 q^{5} - \beta_1 q^{6} + ( - \beta_{7} + \beta_1) q^{7} + \beta_1 q^{8} + q^{9} + q^{10} + ( - \beta_{3} - \beta_1) q^{11} - q^{12} + \beta_{6} q^{14} + \beta_1 q^{15} + q^{16} - 4 q^{17} - \beta_1 q^{18} + (\beta_{7} - \beta_{3} - \beta_1) q^{19} - \beta_1 q^{20} + ( - \beta_{7} + \beta_1) q^{21} + (\beta_{2} - 1) q^{22} + (\beta_{6} + \beta_{4} + \beta_{2} - 2) q^{23} + \beta_1 q^{24} - q^{25} + q^{27} + (\beta_{7} - \beta_1) q^{28} + ( - \beta_{6} + \beta_{4} + \beta_{2} + 2) q^{29} + q^{30} + (\beta_{7} + 3 \beta_{5} + \beta_{3} - \beta_1) q^{31} - \beta_1 q^{32} + ( - \beta_{3} - \beta_1) q^{33} + 4 \beta_1 q^{34} - \beta_{6} q^{35} - q^{36} + (3 \beta_{5} + \beta_{3} - 2 \beta_1) q^{37} + ( - \beta_{6} + \beta_{2}) q^{38} - q^{40} - 2 \beta_{7} q^{41} + \beta_{6} q^{42} + ( - \beta_{6} + 4 \beta_{4} - 3) q^{43} + (\beta_{3} + \beta_1) q^{44} + \beta_1 q^{45} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{46} + (2 \beta_{7} + 2 \beta_{5} + \beta_1) q^{47} + q^{48} + ( - \beta_{6} - 4 \beta_{4} - 3) q^{49} + \beta_1 q^{50} - 4 q^{51} + (\beta_{6} + 2 \beta_{4} - \beta_{2} + 2) q^{53} - \beta_1 q^{54} + ( - \beta_{2} + 1) q^{55} - \beta_{6} q^{56} + (\beta_{7} - \beta_{3} - \beta_1) q^{57} + ( - \beta_{7} + \beta_{5} + \beta_{3} - \beta_1) q^{58} + (\beta_{7} - 2 \beta_{5} - 2 \beta_1) q^{59} - \beta_1 q^{60} + (2 \beta_{4} + 4) q^{61} + ( - \beta_{6} - 3 \beta_{4} - \beta_{2}) q^{62} + ( - \beta_{7} + \beta_1) q^{63} - q^{64} + (\beta_{2} - 1) q^{66} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_1) q^{67} + 4 q^{68} + (\beta_{6} + \beta_{4} + \beta_{2} - 2) q^{69} + ( - \beta_{7} + \beta_1) q^{70} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} - 4 \beta_1) q^{71} + \beta_1 q^{72} + ( - 2 \beta_{7} + 4 \beta_{5} + 4 \beta_1) q^{73} + ( - 3 \beta_{4} - \beta_{2} - 2) q^{74} - q^{75} + ( - \beta_{7} + \beta_{3} + \beta_1) q^{76} + (\beta_{6} + 6 \beta_{4} - \beta_{2} + 2) q^{77} + ( - 3 \beta_{6} - 1) q^{79} + \beta_1 q^{80} + q^{81} + (2 \beta_{6} - 2) q^{82} + ( - 2 \beta_{7} - 2 \beta_{3} - 4 \beta_1) q^{83} + (\beta_{7} - \beta_1) q^{84} - 4 \beta_1 q^{85} + ( - \beta_{7} + 4 \beta_{5} + 4 \beta_1) q^{86} + ( - \beta_{6} + \beta_{4} + \beta_{2} + 2) q^{87} + ( - \beta_{2} + 1) q^{88} + (\beta_{7} + 4 \beta_{5} + \beta_{3} - \beta_1) q^{89} + q^{90} + ( - \beta_{6} - \beta_{4} - \beta_{2} + 2) q^{92} + (\beta_{7} + 3 \beta_{5} + \beta_{3} - \beta_1) q^{93} + ( - 2 \beta_{6} - 2 \beta_{4} + 3) q^{94} + (\beta_{6} - \beta_{2}) q^{95} - \beta_1 q^{96} + ( - 2 \beta_{7} + 2 \beta_{5} + 10 \beta_1) q^{97} + ( - \beta_{7} - 4 \beta_{5} + 4 \beta_1) q^{98} + ( - \beta_{3} - \beta_1) q^{99}+O(q^{100})$$ q - b1 * q^2 + q^3 - q^4 + b1 * q^5 - b1 * q^6 + (-b7 + b1) * q^7 + b1 * q^8 + q^9 + q^10 + (-b3 - b1) * q^11 - q^12 + b6 * q^14 + b1 * q^15 + q^16 - 4 * q^17 - b1 * q^18 + (b7 - b3 - b1) * q^19 - b1 * q^20 + (-b7 + b1) * q^21 + (b2 - 1) * q^22 + (b6 + b4 + b2 - 2) * q^23 + b1 * q^24 - q^25 + q^27 + (b7 - b1) * q^28 + (-b6 + b4 + b2 + 2) * q^29 + q^30 + (b7 + 3*b5 + b3 - b1) * q^31 - b1 * q^32 + (-b3 - b1) * q^33 + 4*b1 * q^34 - b6 * q^35 - q^36 + (3*b5 + b3 - 2*b1) * q^37 + (-b6 + b2) * q^38 - q^40 - 2*b7 * q^41 + b6 * q^42 + (-b6 + 4*b4 - 3) * q^43 + (b3 + b1) * q^44 + b1 * q^45 + (b7 + b5 + b3 + b1) * q^46 + (2*b7 + 2*b5 + b1) * q^47 + q^48 + (-b6 - 4*b4 - 3) * q^49 + b1 * q^50 - 4 * q^51 + (b6 + 2*b4 - b2 + 2) * q^53 - b1 * q^54 + (-b2 + 1) * q^55 - b6 * q^56 + (b7 - b3 - b1) * q^57 + (-b7 + b5 + b3 - b1) * q^58 + (b7 - 2*b5 - 2*b1) * q^59 - b1 * q^60 + (2*b4 + 4) * q^61 + (-b6 - 3*b4 - b2) * q^62 + (-b7 + b1) * q^63 - q^64 + (b2 - 1) * q^66 + (-2*b7 + 2*b5 - 2*b1) * q^67 + 4 * q^68 + (b6 + b4 + b2 - 2) * q^69 + (-b7 + b1) * q^70 + (2*b7 + 2*b5 + 2*b3 - 4*b1) * q^71 + b1 * q^72 + (-2*b7 + 4*b5 + 4*b1) * q^73 + (-3*b4 - b2 - 2) * q^74 - q^75 + (-b7 + b3 + b1) * q^76 + (b6 + 6*b4 - b2 + 2) * q^77 + (-3*b6 - 1) * q^79 + b1 * q^80 + q^81 + (2*b6 - 2) * q^82 + (-2*b7 - 2*b3 - 4*b1) * q^83 + (b7 - b1) * q^84 - 4*b1 * q^85 + (-b7 + 4*b5 + 4*b1) * q^86 + (-b6 + b4 + b2 + 2) * q^87 + (-b2 + 1) * q^88 + (b7 + 4*b5 + b3 - b1) * q^89 + q^90 + (-b6 - b4 - b2 + 2) * q^92 + (b7 + 3*b5 + b3 - b1) * q^93 + (-2*b6 - 2*b4 + 3) * q^94 + (b6 - b2) * q^95 - b1 * q^96 + (-2*b7 + 2*b5 + 10*b1) * q^97 + (-b7 - 4*b5 + 4*b1) * q^98 + (-b3 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{3} - 8 q^{4} + 8 q^{9}+O(q^{10})$$ 8 * q + 8 * q^3 - 8 * q^4 + 8 * q^9 $$8 q + 8 q^{3} - 8 q^{4} + 8 q^{9} + 8 q^{10} - 8 q^{12} + 4 q^{14} + 8 q^{16} - 32 q^{17} - 4 q^{22} - 8 q^{23} - 8 q^{25} + 8 q^{27} + 16 q^{29} + 8 q^{30} - 4 q^{35} - 8 q^{36} - 8 q^{40} + 4 q^{42} - 28 q^{43} + 8 q^{48} - 28 q^{49} - 32 q^{51} + 16 q^{53} + 4 q^{55} - 4 q^{56} + 32 q^{61} - 8 q^{62} - 8 q^{64} - 4 q^{66} + 32 q^{68} - 8 q^{69} - 20 q^{74} - 8 q^{75} + 16 q^{77} - 20 q^{79} + 8 q^{81} - 8 q^{82} + 16 q^{87} + 4 q^{88} + 8 q^{90} + 8 q^{92} + 16 q^{94}+O(q^{100})$$ 8 * q + 8 * q^3 - 8 * q^4 + 8 * q^9 + 8 * q^10 - 8 * q^12 + 4 * q^14 + 8 * q^16 - 32 * q^17 - 4 * q^22 - 8 * q^23 - 8 * q^25 + 8 * q^27 + 16 * q^29 + 8 * q^30 - 4 * q^35 - 8 * q^36 - 8 * q^40 + 4 * q^42 - 28 * q^43 + 8 * q^48 - 28 * q^49 - 32 * q^51 + 16 * q^53 + 4 * q^55 - 4 * q^56 + 32 * q^61 - 8 * q^62 - 8 * q^64 - 4 * q^66 + 32 * q^68 - 8 * q^69 - 20 * q^74 - 8 * q^75 + 16 * q^77 - 20 * q^79 + 8 * q^81 - 8 * q^82 + 16 * q^87 + 4 * q^88 + 8 * q^90 + 8 * q^92 + 16 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 2x^{6} + 30x^{5} + 185x^{4} + 36x^{3} + 8x^{2} + 208x + 2704$$ :

 $$\beta_{1}$$ $$=$$ $$( - 2225 \nu^{7} + 27304 \nu^{6} - 47714 \nu^{5} - 21770 \nu^{4} + 204731 \nu^{3} + 5488658 \nu^{2} + 258164 \nu - 320632 ) / 20561424$$ (-2225*v^7 + 27304*v^6 - 47714*v^5 - 21770*v^4 + 204731*v^3 + 5488658*v^2 + 258164*v - 320632) / 20561424 $$\beta_{2}$$ $$=$$ $$( - 879 \nu^{7} + 1664 \nu^{6} - 1730 \nu^{5} - 23706 \nu^{4} - 214183 \nu^{3} - 10614 \nu^{2} + 785356 \nu - 231400 ) / 790824$$ (-879*v^7 + 1664*v^6 - 1730*v^5 - 23706*v^4 - 214183*v^3 - 10614*v^2 + 785356*v - 231400) / 790824 $$\beta_{3}$$ $$=$$ $$( - 879 \nu^{7} + 1664 \nu^{6} - 1730 \nu^{5} - 23706 \nu^{4} - 214183 \nu^{3} - 10614 \nu^{2} - 796292 \nu - 231400 ) / 790824$$ (-879*v^7 + 1664*v^6 - 1730*v^5 - 23706*v^4 - 214183*v^3 - 10614*v^2 - 796292*v - 231400) / 790824 $$\beta_{4}$$ $$=$$ $$( 79\nu^{7} - 108\nu^{6} - 634\nu^{5} + 6190\nu^{4} + 7075\nu^{3} + 206\nu^{2} - 33852\nu + 174712 ) / 19056$$ (79*v^7 - 108*v^6 - 634*v^5 + 6190*v^4 + 7075*v^3 + 206*v^2 - 33852*v + 174712) / 19056 $$\beta_{5}$$ $$=$$ $$( 151\nu^{7} - 822\nu^{6} + 2018\nu^{5} + 2554\nu^{4} + 7135\nu^{3} - 37828\nu^{2} + 46500\nu + 29224 ) / 25896$$ (151*v^7 - 822*v^6 + 2018*v^5 + 2554*v^4 + 7135*v^3 - 37828*v^2 + 46500*v + 29224) / 25896 $$\beta_{6}$$ $$=$$ $$( 3671 \nu^{7} - 5300 \nu^{6} - 24114 \nu^{5} + 242954 \nu^{4} + 411219 \nu^{3} + 14638 \nu^{2} - 1821820 \nu + 5391360 ) / 395412$$ (3671*v^7 - 5300*v^6 - 24114*v^5 + 242954*v^4 + 411219*v^3 + 14638*v^2 - 1821820*v + 5391360) / 395412 $$\beta_{7}$$ $$=$$ $$( 242617 \nu^{7} - 1102864 \nu^{6} + 2820138 \nu^{5} + 4476178 \nu^{4} + 18741717 \nu^{3} - 30958402 \nu^{2} + 96869068 \nu + 49538424 ) / 20561424$$ (242617*v^7 - 1102864*v^6 + 2820138*v^5 + 4476178*v^4 + 18741717*v^3 - 30958402*v^2 + 96869068*v + 49538424) / 20561424
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$-\beta_{7} + 2\beta_{5} - \beta_{3} + 9\beta_1$$ -b7 + 2*b5 - b3 + 9*b1 $$\nu^{3}$$ $$=$$ $$( -6\beta_{7} - 6\beta_{6} + 10\beta_{5} + 10\beta_{4} - 13\beta_{3} - 13\beta_{2} + 20\beta _1 - 14 ) / 2$$ (-6*b7 - 6*b6 + 10*b5 + 10*b4 - 13*b3 - 13*b2 + 20*b1 - 14) / 2 $$\nu^{4}$$ $$=$$ $$-23\beta_{6} + 44\beta_{4} - 28\beta_{2} - 98$$ -23*b6 + 44*b4 - 28*b2 - 98 $$\nu^{5}$$ $$=$$ $$( 144\beta_{7} - 144\beta_{6} - 250\beta_{5} + 250\beta_{4} + 221\beta_{3} - 221\beta_{2} - 546\beta _1 - 402 ) / 2$$ (144*b7 - 144*b6 - 250*b5 + 250*b4 + 221*b3 - 221*b2 - 546*b1 - 402) / 2 $$\nu^{6}$$ $$=$$ $$471\beta_{7} - 874\beta_{5} + 619\beta_{3} - 2095\beta_1$$ 471*b7 - 874*b5 + 619*b3 - 2095*b1 $$\nu^{7}$$ $$=$$ $$( 2986 \beta_{7} + 2986 \beta_{6} - 5302 \beta_{5} - 5302 \beta_{4} + 4207 \beta_{3} + 4207 \beta_{2} - 11948 \beta _1 + 8962 ) / 2$$ (2986*b7 + 2986*b6 - 5302*b5 - 5302*b4 + 4207*b3 + 4207*b2 - 11948*b1 + 8962) / 2

## 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
 1.33404 + 1.33404i −1.80668 − 1.80668i 3.17270 + 3.17270i −1.70006 − 1.70006i −1.70006 + 1.70006i 3.17270 − 3.17270i −1.80668 + 1.80668i 1.33404 − 1.33404i
1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.64466i 1.00000i 1.00000 1.00000
1351.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 1.32258i 1.00000i 1.00000 1.00000
1351.3 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 2.32258i 1.00000i 1.00000 1.00000
1351.4 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 4.64466i 1.00000i 1.00000 1.00000
1351.5 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 4.64466i 1.00000i 1.00000 1.00000
1351.6 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 2.32258i 1.00000i 1.00000 1.00000
1351.7 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 1.32258i 1.00000i 1.00000 1.00000
1351.8 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.64466i 1.00000i 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1351.8 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.ba 8
13.b even 2 1 inner 5070.2.b.ba 8
13.c even 3 1 390.2.bb.c 8
13.d odd 4 1 5070.2.a.bz 4
13.d odd 4 1 5070.2.a.ca 4
13.e even 6 1 390.2.bb.c 8
39.h odd 6 1 1170.2.bs.f 8
39.i odd 6 1 1170.2.bs.f 8
65.l even 6 1 1950.2.bc.g 8
65.n even 6 1 1950.2.bc.g 8
65.q odd 12 1 1950.2.y.j 8
65.q odd 12 1 1950.2.y.k 8
65.r odd 12 1 1950.2.y.j 8
65.r odd 12 1 1950.2.y.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.c 8 13.c even 3 1
390.2.bb.c 8 13.e even 6 1
1170.2.bs.f 8 39.h odd 6 1
1170.2.bs.f 8 39.i odd 6 1
1950.2.y.j 8 65.q odd 12 1
1950.2.y.j 8 65.r odd 12 1
1950.2.y.k 8 65.q odd 12 1
1950.2.y.k 8 65.r odd 12 1
1950.2.bc.g 8 65.l even 6 1
1950.2.bc.g 8 65.n even 6 1
5070.2.a.bz 4 13.d odd 4 1
5070.2.a.ca 4 13.d odd 4 1
5070.2.b.ba 8 1.a even 1 1 trivial
5070.2.b.ba 8 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 $$S_{2}^{\mathrm{new}}(5070, [\chi])$$:

 $$T_{7}^{8} + 42T_{7}^{6} + 545T_{7}^{4} + 2376T_{7}^{2} + 2704$$ T7^8 + 42*T7^6 + 545*T7^4 + 2376*T7^2 + 2704 $$T_{11}^{8} + 72T_{11}^{6} + 1766T_{11}^{4} + 16152T_{11}^{2} + 32761$$ T11^8 + 72*T11^6 + 1766*T11^4 + 16152*T11^2 + 32761 $$T_{17} + 4$$ T17 + 4 $$T_{31}^{8} + 174T_{31}^{6} + 8777T_{31}^{4} + 158088T_{31}^{2} + 913936$$ T31^8 + 174*T31^6 + 8777*T31^4 + 158088*T31^2 + 913936

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{4}$$
$3$ $$(T - 1)^{8}$$
$5$ $$(T^{2} + 1)^{4}$$
$7$ $$T^{8} + 42 T^{6} + 545 T^{4} + \cdots + 2704$$
$11$ $$T^{8} + 72 T^{6} + 1766 T^{4} + \cdots + 32761$$
$13$ $$T^{8}$$
$17$ $$(T + 4)^{8}$$
$19$ $$T^{8} + 126 T^{6} + 4905 T^{4} + \cdots + 219024$$
$23$ $$(T^{4} + 4 T^{3} - 43 T^{2} - 16 T + 52)^{2}$$
$29$ $$(T^{4} - 8 T^{3} - 39 T^{2} + 148 T + 376)^{2}$$
$31$ $$T^{8} + 174 T^{6} + 8777 T^{4} + \cdots + 913936$$
$37$ $$T^{8} + 168 T^{6} + 8150 T^{4} + \cdots + 644809$$
$41$ $$T^{8} + 168 T^{6} + 8720 T^{4} + \cdots + 692224$$
$43$ $$(T^{4} + 14 T^{3} - 43 T^{2} - 836 T - 572)^{2}$$
$47$ $$T^{8} + 228 T^{6} + 16934 T^{4} + \cdots + 1216609$$
$53$ $$(T^{4} - 8 T^{3} - 75 T^{2} + 4 T + 52)^{2}$$
$59$ $$T^{8} + 98 T^{6} + 2985 T^{4} + \cdots + 80656$$
$61$ $$(T^{2} - 8 T + 4)^{4}$$
$67$ $$T^{8} + 248 T^{6} + 6864 T^{4} + \cdots + 123904$$
$71$ $$T^{8} + 456 T^{6} + \cdots + 25240576$$
$73$ $$T^{8} + 392 T^{6} + \cdots + 20647936$$
$79$ $$(T^{4} + 10 T^{3} - 147 T^{2} - 860 T + 3508)^{2}$$
$83$ $$T^{8} + 456 T^{6} + \cdots + 25240576$$
$89$ $$T^{8} + 246 T^{6} + 17489 T^{4} + \cdots + 1008016$$
$97$ $$T^{8} + 536 T^{6} + \cdots + 29246464$$