# 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: $$x^{8} - 2 x^{7} + 2 x^{6} + 30 x^{5} + 185 x^{4} + 36 x^{3} + 8 x^{2} + 208 x + 2704$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2225 \nu^{7} + 27304 \nu^{6} - 47714 \nu^{5} - 21770 \nu^{4} + 204731 \nu^{3} + 5488658 \nu^{2} + 258164 \nu - 320632$$$$)/20561424$$ $$\beta_{2}$$ $$=$$ $$($$$$-879 \nu^{7} + 1664 \nu^{6} - 1730 \nu^{5} - 23706 \nu^{4} - 214183 \nu^{3} - 10614 \nu^{2} + 785356 \nu - 231400$$$$)/790824$$ $$\beta_{3}$$ $$=$$ $$($$$$-879 \nu^{7} + 1664 \nu^{6} - 1730 \nu^{5} - 23706 \nu^{4} - 214183 \nu^{3} - 10614 \nu^{2} - 796292 \nu - 231400$$$$)/790824$$ $$\beta_{4}$$ $$=$$ $$($$$$79 \nu^{7} - 108 \nu^{6} - 634 \nu^{5} + 6190 \nu^{4} + 7075 \nu^{3} + 206 \nu^{2} - 33852 \nu + 174712$$$$)/19056$$ $$\beta_{5}$$ $$=$$ $$($$$$151 \nu^{7} - 822 \nu^{6} + 2018 \nu^{5} + 2554 \nu^{4} + 7135 \nu^{3} - 37828 \nu^{2} + 46500 \nu + 29224$$$$)/25896$$ $$\beta_{6}$$ $$=$$ $$($$$$3671 \nu^{7} - 5300 \nu^{6} - 24114 \nu^{5} + 242954 \nu^{4} + 411219 \nu^{3} + 14638 \nu^{2} - 1821820 \nu + 5391360$$$$)/395412$$ $$\beta_{7}$$ $$=$$ $$($$$$242617 \nu^{7} - 1102864 \nu^{6} + 2820138 \nu^{5} + 4476178 \nu^{4} + 18741717 \nu^{3} - 30958402 \nu^{2} + 96869068 \nu + 49538424$$$$)/20561424$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} + 2 \beta_{5} - \beta_{3} + 9 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-6 \beta_{7} - 6 \beta_{6} + 10 \beta_{5} + 10 \beta_{4} - 13 \beta_{3} - 13 \beta_{2} + 20 \beta_{1} - 14$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-23 \beta_{6} + 44 \beta_{4} - 28 \beta_{2} - 98$$ $$\nu^{5}$$ $$=$$ $$($$$$144 \beta_{7} - 144 \beta_{6} - 250 \beta_{5} + 250 \beta_{4} + 221 \beta_{3} - 221 \beta_{2} - 546 \beta_{1} - 402$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$471 \beta_{7} - 874 \beta_{5} + 619 \beta_{3} - 2095 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$($$$$2986 \beta_{7} + 2986 \beta_{6} - 5302 \beta_{5} - 5302 \beta_{4} + 4207 \beta_{3} + 4207 \beta_{2} - 11948 \beta_{1} + 8962$$$$)/2$$

## 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 $$S_{2}^{\mathrm{new}}(5070, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ $$( -1 + T )^{8}$$
$5$ $$( 1 + T^{2} )^{4}$$
$7$ $$2704 + 2376 T^{2} + 545 T^{4} + 42 T^{6} + T^{8}$$
$11$ $$32761 + 16152 T^{2} + 1766 T^{4} + 72 T^{6} + T^{8}$$
$13$ $$T^{8}$$
$17$ $$( 4 + T )^{8}$$
$19$ $$219024 + 64152 T^{2} + 4905 T^{4} + 126 T^{6} + T^{8}$$
$23$ $$( 52 - 16 T - 43 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$29$ $$( 376 + 148 T - 39 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$31$ $$913936 + 158088 T^{2} + 8777 T^{4} + 174 T^{6} + T^{8}$$
$37$ $$644809 + 130296 T^{2} + 8150 T^{4} + 168 T^{6} + T^{8}$$
$41$ $$692224 + 152064 T^{2} + 8720 T^{4} + 168 T^{6} + T^{8}$$
$43$ $$( -572 - 836 T - 43 T^{2} + 14 T^{3} + T^{4} )^{2}$$
$47$ $$1216609 + 421284 T^{2} + 16934 T^{4} + 228 T^{6} + T^{8}$$
$53$ $$( 52 + 4 T - 75 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$59$ $$80656 + 29048 T^{2} + 2985 T^{4} + 98 T^{6} + T^{8}$$
$61$ $$( 4 - 8 T + T^{2} )^{4}$$
$67$ $$123904 + 57344 T^{2} + 6864 T^{4} + 248 T^{6} + T^{8}$$
$71$ $$25240576 + 2743296 T^{2} + 61904 T^{4} + 456 T^{6} + T^{8}$$
$73$ $$20647936 + 1859072 T^{2} + 47760 T^{4} + 392 T^{6} + T^{8}$$
$79$ $$( 3508 - 860 T - 147 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$83$ $$25240576 + 2743296 T^{2} + 61904 T^{4} + 456 T^{6} + T^{8}$$
$89$ $$1008016 + 383736 T^{2} + 17489 T^{4} + 246 T^{6} + T^{8}$$
$97$ $$29246464 + 4499456 T^{2} + 88656 T^{4} + 536 T^{6} + T^{8}$$