# Properties

 Label 1650.2.a.m 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} - 2 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 - q^6 - 2 * q^7 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} - q^{6} - 2 q^{7} + q^{8} + q^{9} - q^{11} - q^{12} + 4 q^{13} - 2 q^{14} + q^{16} + 6 q^{17} + q^{18} - 4 q^{19} + 2 q^{21} - q^{22} - 6 q^{23} - q^{24} + 4 q^{26} - q^{27} - 2 q^{28} + 6 q^{29} + 8 q^{31} + q^{32} + q^{33} + 6 q^{34} + q^{36} + 10 q^{37} - 4 q^{38} - 4 q^{39} + 6 q^{41} + 2 q^{42} - 8 q^{43} - q^{44} - 6 q^{46} + 6 q^{47} - q^{48} - 3 q^{49} - 6 q^{51} + 4 q^{52} - q^{54} - 2 q^{56} + 4 q^{57} + 6 q^{58} + 8 q^{61} + 8 q^{62} - 2 q^{63} + q^{64} + q^{66} + 4 q^{67} + 6 q^{68} + 6 q^{69} + 6 q^{71} + q^{72} - 2 q^{73} + 10 q^{74} - 4 q^{76} + 2 q^{77} - 4 q^{78} + 14 q^{79} + q^{81} + 6 q^{82} + 12 q^{83} + 2 q^{84} - 8 q^{86} - 6 q^{87} - q^{88} - 6 q^{89} - 8 q^{91} - 6 q^{92} - 8 q^{93} + 6 q^{94} - q^{96} - 14 q^{97} - 3 q^{98} - q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 - q^6 - 2 * q^7 + q^8 + q^9 - q^11 - q^12 + 4 * q^13 - 2 * q^14 + q^16 + 6 * q^17 + q^18 - 4 * q^19 + 2 * q^21 - q^22 - 6 * q^23 - q^24 + 4 * q^26 - q^27 - 2 * q^28 + 6 * q^29 + 8 * q^31 + q^32 + q^33 + 6 * q^34 + q^36 + 10 * q^37 - 4 * q^38 - 4 * q^39 + 6 * q^41 + 2 * q^42 - 8 * q^43 - q^44 - 6 * q^46 + 6 * q^47 - q^48 - 3 * q^49 - 6 * q^51 + 4 * q^52 - q^54 - 2 * q^56 + 4 * q^57 + 6 * q^58 + 8 * q^61 + 8 * q^62 - 2 * q^63 + q^64 + q^66 + 4 * q^67 + 6 * q^68 + 6 * q^69 + 6 * q^71 + q^72 - 2 * q^73 + 10 * q^74 - 4 * q^76 + 2 * q^77 - 4 * q^78 + 14 * q^79 + q^81 + 6 * q^82 + 12 * q^83 + 2 * q^84 - 8 * q^86 - 6 * q^87 - q^88 - 6 * q^89 - 8 * q^91 - 6 * q^92 - 8 * q^93 + 6 * q^94 - q^96 - 14 * q^97 - 3 * q^98 - q^99

## 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 −2.00000 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.m 1
3.b odd 2 1 4950.2.a.g 1
5.b even 2 1 66.2.a.a 1
5.c odd 4 2 1650.2.c.d 2
15.d odd 2 1 198.2.a.e 1
15.e even 4 2 4950.2.c.r 2
20.d odd 2 1 528.2.a.d 1
35.c odd 2 1 3234.2.a.d 1
40.e odd 2 1 2112.2.a.v 1
40.f even 2 1 2112.2.a.i 1
45.h odd 6 2 1782.2.e.f 2
45.j even 6 2 1782.2.e.s 2
55.d odd 2 1 726.2.a.i 1
55.h odd 10 4 726.2.e.b 4
55.j even 10 4 726.2.e.k 4
60.h even 2 1 1584.2.a.h 1
105.g even 2 1 9702.2.a.bu 1
120.i odd 2 1 6336.2.a.bj 1
120.m even 2 1 6336.2.a.bf 1
165.d even 2 1 2178.2.a.b 1
220.g even 2 1 5808.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.a 1 5.b even 2 1
198.2.a.e 1 15.d odd 2 1
528.2.a.d 1 20.d odd 2 1
726.2.a.i 1 55.d odd 2 1
726.2.e.b 4 55.h odd 10 4
726.2.e.k 4 55.j even 10 4
1584.2.a.h 1 60.h even 2 1
1650.2.a.m 1 1.a even 1 1 trivial
1650.2.c.d 2 5.c odd 4 2
1782.2.e.f 2 45.h odd 6 2
1782.2.e.s 2 45.j even 6 2
2112.2.a.i 1 40.f even 2 1
2112.2.a.v 1 40.e odd 2 1
2178.2.a.b 1 165.d even 2 1
3234.2.a.d 1 35.c odd 2 1
4950.2.a.g 1 3.b odd 2 1
4950.2.c.r 2 15.e even 4 2
5808.2.a.l 1 220.g even 2 1
6336.2.a.bf 1 120.m even 2 1
6336.2.a.bj 1 120.i odd 2 1
9702.2.a.bu 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1650))$$:

 $$T_{7} + 2$$ T7 + 2 $$T_{13} - 4$$ T13 - 4 $$T_{17} - 6$$ T17 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T + 2$$
$11$ $$T + 1$$
$13$ $$T - 4$$
$17$ $$T - 6$$
$19$ $$T + 4$$
$23$ $$T + 6$$
$29$ $$T - 6$$
$31$ $$T - 8$$
$37$ $$T - 10$$
$41$ $$T - 6$$
$43$ $$T + 8$$
$47$ $$T - 6$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 8$$
$67$ $$T - 4$$
$71$ $$T - 6$$
$73$ $$T + 2$$
$79$ $$T - 14$$
$83$ $$T - 12$$
$89$ $$T + 6$$
$97$ $$T + 14$$