# Properties

 Label 4650.2.a.z Level $4650$ Weight $2$ Character orbit 4650.a Self dual yes Analytic conductor $37.130$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$37.1304369399$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) 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 $$q + q^{2} - q^{3} + q^{4} - q^{6} - 2 q^{7} + q^{8} + q^{9} - q^{12} + 2 q^{13} - 2 q^{14} + q^{16} + q^{18} - 4 q^{19} + 2 q^{21} + 2 q^{23} - q^{24} + 2 q^{26} - q^{27} - 2 q^{28} - 10 q^{29} - q^{31} + q^{32} + q^{36} - 2 q^{37} - 4 q^{38} - 2 q^{39} + 2 q^{41} + 2 q^{42} + 4 q^{43} + 2 q^{46} + 4 q^{47} - q^{48} - 3 q^{49} + 2 q^{52} + 2 q^{53} - q^{54} - 2 q^{56} + 4 q^{57} - 10 q^{58} - 6 q^{59} - 8 q^{61} - q^{62} - 2 q^{63} + q^{64} + 8 q^{67} - 2 q^{69} - 8 q^{71} + q^{72} - 10 q^{73} - 2 q^{74} - 4 q^{76} - 2 q^{78} + 12 q^{79} + q^{81} + 2 q^{82} + 4 q^{83} + 2 q^{84} + 4 q^{86} + 10 q^{87} - 14 q^{89} - 4 q^{91} + 2 q^{92} + q^{93} + 4 q^{94} - q^{96} - 4 q^{97} - 3 q^{98}+O(q^{100})$$ q + q^2 - q^3 + q^4 - q^6 - 2 * q^7 + q^8 + q^9 - q^12 + 2 * q^13 - 2 * q^14 + q^16 + q^18 - 4 * q^19 + 2 * q^21 + 2 * q^23 - q^24 + 2 * q^26 - q^27 - 2 * q^28 - 10 * q^29 - q^31 + q^32 + q^36 - 2 * q^37 - 4 * q^38 - 2 * q^39 + 2 * q^41 + 2 * q^42 + 4 * q^43 + 2 * q^46 + 4 * q^47 - q^48 - 3 * q^49 + 2 * q^52 + 2 * q^53 - q^54 - 2 * q^56 + 4 * q^57 - 10 * q^58 - 6 * q^59 - 8 * q^61 - q^62 - 2 * q^63 + q^64 + 8 * q^67 - 2 * q^69 - 8 * q^71 + q^72 - 10 * q^73 - 2 * q^74 - 4 * q^76 - 2 * q^78 + 12 * q^79 + q^81 + 2 * q^82 + 4 * q^83 + 2 * q^84 + 4 * q^86 + 10 * q^87 - 14 * q^89 - 4 * q^91 + 2 * q^92 + q^93 + 4 * q^94 - q^96 - 4 * q^97 - 3 * 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
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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$31$$ $$1$$

## 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 4650.2.a.z 1
5.b even 2 1 4650.2.a.u 1
5.c odd 4 2 930.2.d.d 2
15.e even 4 2 2790.2.d.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.d.d 2 5.c odd 4 2
2790.2.d.d 2 15.e even 4 2
4650.2.a.u 1 5.b even 2 1
4650.2.a.z 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4650))$$:

 $$T_{7} + 2$$ T7 + 2 $$T_{11}$$ T11 $$T_{13} - 2$$ T13 - 2 $$T_{19} + 4$$ T19 + 4

## Hecke characteristic polynomials

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