# Properties

 Label 272.2.a.b Level $272$ Weight $2$ Character orbit 272.a Self dual yes Analytic conductor $2.172$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$272 = 2^{4} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 272.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.17193093498$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 17) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{5} - 4 q^{7} - 3 q^{9}+O(q^{10})$$ q - 2 * q^5 - 4 * q^7 - 3 * q^9 $$q - 2 q^{5} - 4 q^{7} - 3 q^{9} - 2 q^{13} + q^{17} + 4 q^{19} - 4 q^{23} - q^{25} + 6 q^{29} - 4 q^{31} + 8 q^{35} - 2 q^{37} - 6 q^{41} - 4 q^{43} + 6 q^{45} + 9 q^{49} + 6 q^{53} + 12 q^{59} - 10 q^{61} + 12 q^{63} + 4 q^{65} - 4 q^{67} + 4 q^{71} - 6 q^{73} - 12 q^{79} + 9 q^{81} + 4 q^{83} - 2 q^{85} + 10 q^{89} + 8 q^{91} - 8 q^{95} + 2 q^{97}+O(q^{100})$$ q - 2 * q^5 - 4 * q^7 - 3 * q^9 - 2 * q^13 + q^17 + 4 * q^19 - 4 * q^23 - q^25 + 6 * q^29 - 4 * q^31 + 8 * q^35 - 2 * q^37 - 6 * q^41 - 4 * q^43 + 6 * q^45 + 9 * q^49 + 6 * q^53 + 12 * q^59 - 10 * q^61 + 12 * q^63 + 4 * q^65 - 4 * q^67 + 4 * q^71 - 6 * q^73 - 12 * q^79 + 9 * q^81 + 4 * q^83 - 2 * q^85 + 10 * q^89 + 8 * q^91 - 8 * q^95 + 2 * q^97

## 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
0 0 0 −2.00000 0 −4.00000 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$17$$ $$-1$$

## 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 272.2.a.b 1
3.b odd 2 1 2448.2.a.o 1
4.b odd 2 1 17.2.a.a 1
5.b even 2 1 6800.2.a.n 1
8.b even 2 1 1088.2.a.h 1
8.d odd 2 1 1088.2.a.i 1
12.b even 2 1 153.2.a.c 1
17.b even 2 1 4624.2.a.d 1
20.d odd 2 1 425.2.a.d 1
20.e even 4 2 425.2.b.b 2
24.f even 2 1 9792.2.a.n 1
24.h odd 2 1 9792.2.a.i 1
28.d even 2 1 833.2.a.a 1
28.f even 6 2 833.2.e.a 2
28.g odd 6 2 833.2.e.b 2
44.c even 2 1 2057.2.a.e 1
52.b odd 2 1 2873.2.a.c 1
60.h even 2 1 3825.2.a.d 1
68.d odd 2 1 289.2.a.a 1
68.f odd 4 2 289.2.b.a 2
68.g odd 8 4 289.2.c.a 4
68.i even 16 8 289.2.d.d 8
76.d even 2 1 6137.2.a.b 1
84.h odd 2 1 7497.2.a.l 1
92.b even 2 1 8993.2.a.a 1
204.h even 2 1 2601.2.a.g 1
340.d odd 2 1 7225.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 4.b odd 2 1
153.2.a.c 1 12.b even 2 1
272.2.a.b 1 1.a even 1 1 trivial
289.2.a.a 1 68.d odd 2 1
289.2.b.a 2 68.f odd 4 2
289.2.c.a 4 68.g odd 8 4
289.2.d.d 8 68.i even 16 8
425.2.a.d 1 20.d odd 2 1
425.2.b.b 2 20.e even 4 2
833.2.a.a 1 28.d even 2 1
833.2.e.a 2 28.f even 6 2
833.2.e.b 2 28.g odd 6 2
1088.2.a.h 1 8.b even 2 1
1088.2.a.i 1 8.d odd 2 1
2057.2.a.e 1 44.c even 2 1
2448.2.a.o 1 3.b odd 2 1
2601.2.a.g 1 204.h even 2 1
2873.2.a.c 1 52.b odd 2 1
3825.2.a.d 1 60.h even 2 1
4624.2.a.d 1 17.b even 2 1
6137.2.a.b 1 76.d even 2 1
6800.2.a.n 1 5.b even 2 1
7225.2.a.g 1 340.d odd 2 1
7497.2.a.l 1 84.h odd 2 1
8993.2.a.a 1 92.b even 2 1
9792.2.a.i 1 24.h odd 2 1
9792.2.a.n 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(272))$$:

 $$T_{3}$$ T3 $$T_{5} + 2$$ T5 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 2$$
$7$ $$T + 4$$
$11$ $$T$$
$13$ $$T + 2$$
$17$ $$T - 1$$
$19$ $$T - 4$$
$23$ $$T + 4$$
$29$ $$T - 6$$
$31$ $$T + 4$$
$37$ $$T + 2$$
$41$ $$T + 6$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T - 12$$
$61$ $$T + 10$$
$67$ $$T + 4$$
$71$ $$T - 4$$
$73$ $$T + 6$$
$79$ $$T + 12$$
$83$ $$T - 4$$
$89$ $$T - 10$$
$97$ $$T - 2$$