# Properties

 Label 2070.4.a.e Level $2070$ Weight $4$ Character orbit 2070.a Self dual yes Analytic conductor $122.134$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$122.133953712$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 4 q^{4} + 5 q^{5} - 18 q^{7} - 8 q^{8}+O(q^{10})$$ q - 2 * q^2 + 4 * q^4 + 5 * q^5 - 18 * q^7 - 8 * q^8 $$q - 2 q^{2} + 4 q^{4} + 5 q^{5} - 18 q^{7} - 8 q^{8} - 10 q^{10} + 32 q^{11} - 47 q^{13} + 36 q^{14} + 16 q^{16} - 20 q^{17} + 36 q^{19} + 20 q^{20} - 64 q^{22} + 23 q^{23} + 25 q^{25} + 94 q^{26} - 72 q^{28} + 27 q^{29} - 33 q^{31} - 32 q^{32} + 40 q^{34} - 90 q^{35} + 56 q^{37} - 72 q^{38} - 40 q^{40} + 157 q^{41} + 18 q^{43} + 128 q^{44} - 46 q^{46} - 65 q^{47} - 19 q^{49} - 50 q^{50} - 188 q^{52} + 14 q^{53} + 160 q^{55} + 144 q^{56} - 54 q^{58} + 744 q^{59} + 552 q^{61} + 66 q^{62} + 64 q^{64} - 235 q^{65} - 156 q^{67} - 80 q^{68} + 180 q^{70} - 699 q^{71} - 609 q^{73} - 112 q^{74} + 144 q^{76} - 576 q^{77} - 644 q^{79} + 80 q^{80} - 314 q^{82} - 512 q^{83} - 100 q^{85} - 36 q^{86} - 256 q^{88} + 102 q^{89} + 846 q^{91} + 92 q^{92} + 130 q^{94} + 180 q^{95} + 578 q^{97} + 38 q^{98}+O(q^{100})$$ q - 2 * q^2 + 4 * q^4 + 5 * q^5 - 18 * q^7 - 8 * q^8 - 10 * q^10 + 32 * q^11 - 47 * q^13 + 36 * q^14 + 16 * q^16 - 20 * q^17 + 36 * q^19 + 20 * q^20 - 64 * q^22 + 23 * q^23 + 25 * q^25 + 94 * q^26 - 72 * q^28 + 27 * q^29 - 33 * q^31 - 32 * q^32 + 40 * q^34 - 90 * q^35 + 56 * q^37 - 72 * q^38 - 40 * q^40 + 157 * q^41 + 18 * q^43 + 128 * q^44 - 46 * q^46 - 65 * q^47 - 19 * q^49 - 50 * q^50 - 188 * q^52 + 14 * q^53 + 160 * q^55 + 144 * q^56 - 54 * q^58 + 744 * q^59 + 552 * q^61 + 66 * q^62 + 64 * q^64 - 235 * q^65 - 156 * q^67 - 80 * q^68 + 180 * q^70 - 699 * q^71 - 609 * q^73 - 112 * q^74 + 144 * q^76 - 576 * q^77 - 644 * q^79 + 80 * q^80 - 314 * q^82 - 512 * q^83 - 100 * q^85 - 36 * q^86 - 256 * q^88 + 102 * q^89 + 846 * q^91 + 92 * q^92 + 130 * q^94 + 180 * q^95 + 578 * q^97 + 38 * q^98

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−2.00000 0 4.00000 5.00000 0 −18.0000 −8.00000 0 −10.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2070.4.a.e 1
3.b odd 2 1 230.4.a.e 1
12.b even 2 1 1840.4.a.d 1
15.d odd 2 1 1150.4.a.b 1
15.e even 4 2 1150.4.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.e 1 3.b odd 2 1
1150.4.a.b 1 15.d odd 2 1
1150.4.b.f 2 15.e even 4 2
1840.4.a.d 1 12.b even 2 1
2070.4.a.e 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2070))$$:

 $$T_{7} + 18$$ T7 + 18 $$T_{11} - 32$$ T11 - 32 $$T_{17} + 20$$ T17 + 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T + 18$$
$11$ $$T - 32$$
$13$ $$T + 47$$
$17$ $$T + 20$$
$19$ $$T - 36$$
$23$ $$T - 23$$
$29$ $$T - 27$$
$31$ $$T + 33$$
$37$ $$T - 56$$
$41$ $$T - 157$$
$43$ $$T - 18$$
$47$ $$T + 65$$
$53$ $$T - 14$$
$59$ $$T - 744$$
$61$ $$T - 552$$
$67$ $$T + 156$$
$71$ $$T + 699$$
$73$ $$T + 609$$
$79$ $$T + 644$$
$83$ $$T + 512$$
$89$ $$T - 102$$
$97$ $$T - 578$$