# Properties

 Label 5070.2.b.p Level $5070$ Weight $2$ Character orbit 5070.b Analytic conductor $40.484$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.00000i 1.00000i 1.00000 1.00000
1351.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.00000i 1.00000i 1.00000 1.00000
1351.3 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.00000i 1.00000i 1.00000 1.00000
1351.4 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.00000i 1.00000i 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 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.p 4
13.b even 2 1 inner 5070.2.b.p 4
13.c even 3 1 390.2.bb.a 4
13.d odd 4 1 5070.2.a.ba 2
13.d odd 4 1 5070.2.a.be 2
13.e even 6 1 390.2.bb.a 4
39.h odd 6 1 1170.2.bs.d 4
39.i odd 6 1 1170.2.bs.d 4
65.l even 6 1 1950.2.bc.a 4
65.n even 6 1 1950.2.bc.a 4
65.q odd 12 1 1950.2.y.d 4
65.q odd 12 1 1950.2.y.e 4
65.r odd 12 1 1950.2.y.d 4
65.r odd 12 1 1950.2.y.e 4

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

 $$T_{7}^{2} + 9$$ $$T_{11}^{4} + 14 T_{11}^{2} + 1$$ $$T_{17} - 4$$ $$T_{31}^{4} + 104 T_{31}^{2} + 1936$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$( 9 + T^{2} )^{2}$$
$11$ $$1 + 14 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( -4 + T )^{4}$$
$19$ $$169 + 38 T^{2} + T^{4}$$
$23$ $$( -12 + T^{2} )^{2}$$
$29$ $$( -8 + 4 T + T^{2} )^{2}$$
$31$ $$1936 + 104 T^{2} + T^{4}$$
$37$ $$2209 + 98 T^{2} + T^{4}$$
$41$ $$( 16 + T^{2} )^{2}$$
$43$ $$( -6 + T )^{4}$$
$47$ $$9 + 42 T^{2} + T^{4}$$
$53$ $$( 1 + 4 T + T^{2} )^{2}$$
$59$ $$2704 + 152 T^{2} + T^{4}$$
$61$ $$( 4 + 8 T + T^{2} )^{2}$$
$67$ $$64 + 32 T^{2} + T^{4}$$
$71$ $$144 + 168 T^{2} + T^{4}$$
$73$ $$( 48 + T^{2} )^{2}$$
$79$ $$( 52 - 20 T + T^{2} )^{2}$$
$83$ $$576 + 96 T^{2} + T^{4}$$
$89$ $$20449 + 302 T^{2} + T^{4}$$
$97$ $$10816 + 224 T^{2} + T^{4}$$