# Properties

 Label 270.4.a.e 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

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 4 q^{4} + 5 q^{5} - 4 q^{7} - 8 q^{8}+O(q^{10})$$ q - 2 * q^2 + 4 * q^4 + 5 * q^5 - 4 * q^7 - 8 * q^8 $$q - 2 q^{2} + 4 q^{4} + 5 q^{5} - 4 q^{7} - 8 q^{8} - 10 q^{10} - 42 q^{11} + 20 q^{13} + 8 q^{14} + 16 q^{16} - 93 q^{17} + 59 q^{19} + 20 q^{20} + 84 q^{22} - 9 q^{23} + 25 q^{25} - 40 q^{26} - 16 q^{28} - 120 q^{29} + 47 q^{31} - 32 q^{32} + 186 q^{34} - 20 q^{35} - 262 q^{37} - 118 q^{38} - 40 q^{40} - 126 q^{41} - 178 q^{43} - 168 q^{44} + 18 q^{46} - 144 q^{47} - 327 q^{49} - 50 q^{50} + 80 q^{52} - 741 q^{53} - 210 q^{55} + 32 q^{56} + 240 q^{58} + 444 q^{59} + 221 q^{61} - 94 q^{62} + 64 q^{64} + 100 q^{65} - 538 q^{67} - 372 q^{68} + 40 q^{70} - 690 q^{71} - 1126 q^{73} + 524 q^{74} + 236 q^{76} + 168 q^{77} + 665 q^{79} + 80 q^{80} + 252 q^{82} - 75 q^{83} - 465 q^{85} + 356 q^{86} + 336 q^{88} + 1086 q^{89} - 80 q^{91} - 36 q^{92} + 288 q^{94} + 295 q^{95} + 1544 q^{97} + 654 q^{98}+O(q^{100})$$ q - 2 * q^2 + 4 * q^4 + 5 * q^5 - 4 * q^7 - 8 * q^8 - 10 * q^10 - 42 * q^11 + 20 * q^13 + 8 * q^14 + 16 * q^16 - 93 * q^17 + 59 * q^19 + 20 * q^20 + 84 * q^22 - 9 * q^23 + 25 * q^25 - 40 * q^26 - 16 * q^28 - 120 * q^29 + 47 * q^31 - 32 * q^32 + 186 * q^34 - 20 * q^35 - 262 * q^37 - 118 * q^38 - 40 * q^40 - 126 * q^41 - 178 * q^43 - 168 * q^44 + 18 * q^46 - 144 * q^47 - 327 * q^49 - 50 * q^50 + 80 * q^52 - 741 * q^53 - 210 * q^55 + 32 * q^56 + 240 * q^58 + 444 * q^59 + 221 * q^61 - 94 * q^62 + 64 * q^64 + 100 * q^65 - 538 * q^67 - 372 * q^68 + 40 * q^70 - 690 * q^71 - 1126 * q^73 + 524 * q^74 + 236 * q^76 + 168 * q^77 + 665 * q^79 + 80 * q^80 + 252 * q^82 - 75 * q^83 - 465 * q^85 + 356 * q^86 + 336 * q^88 + 1086 * q^89 - 80 * q^91 - 36 * q^92 + 288 * q^94 + 295 * q^95 + 1544 * q^97 + 654 * 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 −4.00000 −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.e 1
3.b odd 2 1 270.4.a.i yes 1
4.b odd 2 1 2160.4.a.o 1
5.b even 2 1 1350.4.a.u 1
5.c odd 4 2 1350.4.c.d 2
9.c even 3 2 810.4.e.q 2
9.d odd 6 2 810.4.e.h 2
12.b even 2 1 2160.4.a.e 1
15.d odd 2 1 1350.4.a.g 1
15.e even 4 2 1350.4.c.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.e 1 1.a even 1 1 trivial
270.4.a.i yes 1 3.b odd 2 1
810.4.e.h 2 9.d odd 6 2
810.4.e.q 2 9.c even 3 2
1350.4.a.g 1 15.d odd 2 1
1350.4.a.u 1 5.b even 2 1
1350.4.c.d 2 5.c odd 4 2
1350.4.c.q 2 15.e even 4 2
2160.4.a.e 1 12.b even 2 1
2160.4.a.o 1 4.b odd 2 1

## Hecke kernels

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

 $$T_{7} + 4$$ T7 + 4 $$T_{11} + 42$$ T11 + 42

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T + 4$$
$11$ $$T + 42$$
$13$ $$T - 20$$
$17$ $$T + 93$$
$19$ $$T - 59$$
$23$ $$T + 9$$
$29$ $$T + 120$$
$31$ $$T - 47$$
$37$ $$T + 262$$
$41$ $$T + 126$$
$43$ $$T + 178$$
$47$ $$T + 144$$
$53$ $$T + 741$$
$59$ $$T - 444$$
$61$ $$T - 221$$
$67$ $$T + 538$$
$71$ $$T + 690$$
$73$ $$T + 1126$$
$79$ $$T - 665$$
$83$ $$T + 75$$
$89$ $$T - 1086$$
$97$ $$T - 1544$$
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