# Properties

 Label 270.4.a.b Level $270$ Weight $4$ Character orbit 270.a Self dual yes Analytic conductor $15.931$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$270 = 2 \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 270.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.9305157015$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 4 q^{4} - 5 q^{5} + 8 q^{7} - 8 q^{8}+O(q^{10})$$ q - 2 * q^2 + 4 * q^4 - 5 * q^5 + 8 * q^7 - 8 * q^8 $$q - 2 q^{2} + 4 q^{4} - 5 q^{5} + 8 q^{7} - 8 q^{8} + 10 q^{10} - 18 q^{11} + 8 q^{13} - 16 q^{14} + 16 q^{16} - 15 q^{17} + 23 q^{19} - 20 q^{20} + 36 q^{22} - 63 q^{23} + 25 q^{25} - 16 q^{26} + 32 q^{28} - 156 q^{29} - 85 q^{31} - 32 q^{32} + 30 q^{34} - 40 q^{35} + 74 q^{37} - 46 q^{38} + 40 q^{40} - 246 q^{41} - 190 q^{43} - 72 q^{44} + 126 q^{46} - 288 q^{47} - 279 q^{49} - 50 q^{50} + 32 q^{52} + 177 q^{53} + 90 q^{55} - 64 q^{56} + 312 q^{58} - 792 q^{59} - 907 q^{61} + 170 q^{62} + 64 q^{64} - 40 q^{65} - 322 q^{67} - 60 q^{68} + 80 q^{70} + 270 q^{71} + 254 q^{73} - 148 q^{74} + 92 q^{76} - 144 q^{77} - 1123 q^{79} - 80 q^{80} + 492 q^{82} + 771 q^{83} + 75 q^{85} + 380 q^{86} + 144 q^{88} + 198 q^{89} + 64 q^{91} - 252 q^{92} + 576 q^{94} - 115 q^{95} - 1192 q^{97} + 558 q^{98}+O(q^{100})$$ q - 2 * q^2 + 4 * q^4 - 5 * q^5 + 8 * q^7 - 8 * q^8 + 10 * q^10 - 18 * q^11 + 8 * q^13 - 16 * q^14 + 16 * q^16 - 15 * q^17 + 23 * q^19 - 20 * q^20 + 36 * q^22 - 63 * q^23 + 25 * q^25 - 16 * q^26 + 32 * q^28 - 156 * q^29 - 85 * q^31 - 32 * q^32 + 30 * q^34 - 40 * q^35 + 74 * q^37 - 46 * q^38 + 40 * q^40 - 246 * q^41 - 190 * q^43 - 72 * q^44 + 126 * q^46 - 288 * q^47 - 279 * q^49 - 50 * q^50 + 32 * q^52 + 177 * q^53 + 90 * q^55 - 64 * q^56 + 312 * q^58 - 792 * q^59 - 907 * q^61 + 170 * q^62 + 64 * q^64 - 40 * q^65 - 322 * q^67 - 60 * q^68 + 80 * q^70 + 270 * q^71 + 254 * q^73 - 148 * q^74 + 92 * q^76 - 144 * q^77 - 1123 * q^79 - 80 * q^80 + 492 * q^82 + 771 * q^83 + 75 * q^85 + 380 * q^86 + 144 * q^88 + 198 * q^89 + 64 * q^91 - 252 * q^92 + 576 * q^94 - 115 * q^95 - 1192 * q^97 + 558 * 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 8.00000 −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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$

## 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 270.4.a.b 1
3.b odd 2 1 270.4.a.l yes 1
4.b odd 2 1 2160.4.a.c 1
5.b even 2 1 1350.4.a.t 1
5.c odd 4 2 1350.4.c.g 2
9.c even 3 2 810.4.e.v 2
9.d odd 6 2 810.4.e.b 2
12.b even 2 1 2160.4.a.m 1
15.d odd 2 1 1350.4.a.f 1
15.e even 4 2 1350.4.c.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.b 1 1.a even 1 1 trivial
270.4.a.l yes 1 3.b odd 2 1
810.4.e.b 2 9.d odd 6 2
810.4.e.v 2 9.c even 3 2
1350.4.a.f 1 15.d odd 2 1
1350.4.a.t 1 5.b even 2 1
1350.4.c.g 2 5.c odd 4 2
1350.4.c.n 2 15.e even 4 2
2160.4.a.c 1 4.b odd 2 1
2160.4.a.m 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(270))$$:

 $$T_{7} - 8$$ T7 - 8 $$T_{11} + 18$$ T11 + 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T - 8$$
$11$ $$T + 18$$
$13$ $$T - 8$$
$17$ $$T + 15$$
$19$ $$T - 23$$
$23$ $$T + 63$$
$29$ $$T + 156$$
$31$ $$T + 85$$
$37$ $$T - 74$$
$41$ $$T + 246$$
$43$ $$T + 190$$
$47$ $$T + 288$$
$53$ $$T - 177$$
$59$ $$T + 792$$
$61$ $$T + 907$$
$67$ $$T + 322$$
$71$ $$T - 270$$
$73$ $$T - 254$$
$79$ $$T + 1123$$
$83$ $$T - 771$$
$89$ $$T - 198$$
$97$ $$T + 1192$$
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