# Properties

 Label 1650.2.a.k Level $1650$ Weight $2$ Character orbit 1650.a Self dual yes Analytic conductor $13.175$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 66) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} + 4q^{7} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} + 4q^{7} - q^{8} + q^{9} - q^{11} + q^{12} + 6q^{13} - 4q^{14} + q^{16} - 2q^{17} - q^{18} + 4q^{19} + 4q^{21} + q^{22} - 4q^{23} - q^{24} - 6q^{26} + q^{27} + 4q^{28} + 6q^{29} - q^{32} - q^{33} + 2q^{34} + q^{36} - 6q^{37} - 4q^{38} + 6q^{39} - 6q^{41} - 4q^{42} - 4q^{43} - q^{44} + 4q^{46} + 12q^{47} + q^{48} + 9q^{49} - 2q^{51} + 6q^{52} - 2q^{53} - q^{54} - 4q^{56} + 4q^{57} - 6q^{58} + 12q^{59} - 14q^{61} + 4q^{63} + q^{64} + q^{66} - 4q^{67} - 2q^{68} - 4q^{69} - 12q^{71} - q^{72} + 6q^{73} + 6q^{74} + 4q^{76} - 4q^{77} - 6q^{78} - 4q^{79} + q^{81} + 6q^{82} - 4q^{83} + 4q^{84} + 4q^{86} + 6q^{87} + q^{88} + 10q^{89} + 24q^{91} - 4q^{92} - 12q^{94} - q^{96} + 14q^{97} - 9q^{98} - q^{99} + O(q^{100})$$

## 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 4.00000 −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.k 1
3.b odd 2 1 4950.2.a.bu 1
5.b even 2 1 66.2.a.b 1
5.c odd 4 2 1650.2.c.e 2
15.d odd 2 1 198.2.a.a 1
15.e even 4 2 4950.2.c.p 2
20.d odd 2 1 528.2.a.j 1
35.c odd 2 1 3234.2.a.t 1
40.e odd 2 1 2112.2.a.e 1
40.f even 2 1 2112.2.a.r 1
45.h odd 6 2 1782.2.e.v 2
45.j even 6 2 1782.2.e.e 2
55.d odd 2 1 726.2.a.c 1
55.h odd 10 4 726.2.e.o 4
55.j even 10 4 726.2.e.g 4
60.h even 2 1 1584.2.a.f 1
105.g even 2 1 9702.2.a.x 1
120.i odd 2 1 6336.2.a.bw 1
120.m even 2 1 6336.2.a.cj 1
165.d even 2 1 2178.2.a.g 1
220.g even 2 1 5808.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.b 1 5.b even 2 1
198.2.a.a 1 15.d odd 2 1
528.2.a.j 1 20.d odd 2 1
726.2.a.c 1 55.d odd 2 1
726.2.e.g 4 55.j even 10 4
726.2.e.o 4 55.h odd 10 4
1584.2.a.f 1 60.h even 2 1
1650.2.a.k 1 1.a even 1 1 trivial
1650.2.c.e 2 5.c odd 4 2
1782.2.e.e 2 45.j even 6 2
1782.2.e.v 2 45.h odd 6 2
2112.2.a.e 1 40.e odd 2 1
2112.2.a.r 1 40.f even 2 1
2178.2.a.g 1 165.d even 2 1
3234.2.a.t 1 35.c odd 2 1
4950.2.a.bu 1 3.b odd 2 1
4950.2.c.p 2 15.e even 4 2
5808.2.a.bc 1 220.g even 2 1
6336.2.a.bw 1 120.i odd 2 1
6336.2.a.cj 1 120.m even 2 1
9702.2.a.x 1 105.g even 2 1

## Hecke kernels

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

 $$T_{7} - 4$$ $$T_{13} - 6$$ $$T_{17} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$-4 + T$$
$11$ $$1 + T$$
$13$ $$-6 + T$$
$17$ $$2 + T$$
$19$ $$-4 + T$$
$23$ $$4 + T$$
$29$ $$-6 + T$$
$31$ $$T$$
$37$ $$6 + T$$
$41$ $$6 + T$$
$43$ $$4 + T$$
$47$ $$-12 + T$$
$53$ $$2 + T$$
$59$ $$-12 + T$$
$61$ $$14 + T$$
$67$ $$4 + T$$
$71$ $$12 + T$$
$73$ $$-6 + T$$
$79$ $$4 + T$$
$83$ $$4 + T$$
$89$ $$-10 + T$$
$97$ $$-14 + T$$