Properties

 Label 1650.2.a.h Level $1650$ Weight $2$ Character orbit 1650.a Self dual yes Analytic conductor $13.175$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1650.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$13.1753163335$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 330) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} + q^{11} + q^{12} - 6q^{13} + q^{16} - 2q^{17} - q^{18} - 4q^{19} - q^{22} - q^{24} + 6q^{26} + q^{27} - 10q^{29} - q^{32} + q^{33} + 2q^{34} + q^{36} - 6q^{37} + 4q^{38} - 6q^{39} + 2q^{41} - 4q^{43} + q^{44} + 8q^{47} + q^{48} - 7q^{49} - 2q^{51} - 6q^{52} + 10q^{53} - q^{54} - 4q^{57} + 10q^{58} - 4q^{59} - 2q^{61} + q^{64} - q^{66} + 4q^{67} - 2q^{68} - 8q^{71} - q^{72} - 2q^{73} + 6q^{74} - 4q^{76} + 6q^{78} - 8q^{79} + q^{81} - 2q^{82} + 12q^{83} + 4q^{86} - 10q^{87} - q^{88} - 6q^{89} - 8q^{94} - q^{96} - 18q^{97} + 7q^{98} + q^{99} + O(q^{100})$$

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
−1.00000 1.00000 1.00000 0 −1.00000 0 −1.00000 1.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$-1$$

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 1650.2.a.h 1
3.b odd 2 1 4950.2.a.bg 1
5.b even 2 1 330.2.a.d 1
5.c odd 4 2 1650.2.c.g 2
15.d odd 2 1 990.2.a.b 1
15.e even 4 2 4950.2.c.j 2
20.d odd 2 1 2640.2.a.t 1
55.d odd 2 1 3630.2.a.f 1
60.h even 2 1 7920.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.a.d 1 5.b even 2 1
990.2.a.b 1 15.d odd 2 1
1650.2.a.h 1 1.a even 1 1 trivial
1650.2.c.g 2 5.c odd 4 2
2640.2.a.t 1 20.d odd 2 1
3630.2.a.f 1 55.d odd 2 1
4950.2.a.bg 1 3.b odd 2 1
4950.2.c.j 2 15.e even 4 2
7920.2.a.m 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1650))$$:

 $$T_{7}$$ $$T_{13} + 6$$ $$T_{17} + 2$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$-1 + T$$
$13$ $$6 + T$$
$17$ $$2 + T$$
$19$ $$4 + T$$
$23$ $$T$$
$29$ $$10 + T$$
$31$ $$T$$
$37$ $$6 + T$$
$41$ $$-2 + T$$
$43$ $$4 + T$$
$47$ $$-8 + T$$
$53$ $$-10 + T$$
$59$ $$4 + T$$
$61$ $$2 + T$$
$67$ $$-4 + T$$
$71$ $$8 + T$$
$73$ $$2 + T$$
$79$ $$8 + T$$
$83$ $$-12 + T$$
$89$ $$6 + T$$
$97$ $$18 + T$$