# Properties

 Label 4650.2.a.bk 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} - 3 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + q^6 - 3 * q^7 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + q^{6} - 3 q^{7} + q^{8} + q^{9} + 3 q^{11} + q^{12} + 2 q^{13} - 3 q^{14} + q^{16} - 8 q^{17} + q^{18} - 7 q^{19} - 3 q^{21} + 3 q^{22} - 7 q^{23} + q^{24} + 2 q^{26} + q^{27} - 3 q^{28} - 8 q^{29} - q^{31} + q^{32} + 3 q^{33} - 8 q^{34} + q^{36} + 4 q^{37} - 7 q^{38} + 2 q^{39} - 3 q^{42} - q^{43} + 3 q^{44} - 7 q^{46} - 6 q^{47} + q^{48} + 2 q^{49} - 8 q^{51} + 2 q^{52} - 5 q^{53} + q^{54} - 3 q^{56} - 7 q^{57} - 8 q^{58} + 6 q^{59} + 2 q^{61} - q^{62} - 3 q^{63} + q^{64} + 3 q^{66} - 10 q^{67} - 8 q^{68} - 7 q^{69} + 9 q^{71} + q^{72} - q^{73} + 4 q^{74} - 7 q^{76} - 9 q^{77} + 2 q^{78} + 13 q^{79} + q^{81} + 16 q^{83} - 3 q^{84} - q^{86} - 8 q^{87} + 3 q^{88} - 3 q^{89} - 6 q^{91} - 7 q^{92} - q^{93} - 6 q^{94} + q^{96} - 6 q^{97} + 2 q^{98} + 3 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^6 - 3 * q^7 + q^8 + q^9 + 3 * q^11 + q^12 + 2 * q^13 - 3 * q^14 + q^16 - 8 * q^17 + q^18 - 7 * q^19 - 3 * q^21 + 3 * q^22 - 7 * q^23 + q^24 + 2 * q^26 + q^27 - 3 * q^28 - 8 * q^29 - q^31 + q^32 + 3 * q^33 - 8 * q^34 + q^36 + 4 * q^37 - 7 * q^38 + 2 * q^39 - 3 * q^42 - q^43 + 3 * q^44 - 7 * q^46 - 6 * q^47 + q^48 + 2 * q^49 - 8 * q^51 + 2 * q^52 - 5 * q^53 + q^54 - 3 * q^56 - 7 * q^57 - 8 * q^58 + 6 * q^59 + 2 * q^61 - q^62 - 3 * q^63 + q^64 + 3 * q^66 - 10 * q^67 - 8 * q^68 - 7 * q^69 + 9 * q^71 + q^72 - q^73 + 4 * q^74 - 7 * q^76 - 9 * q^77 + 2 * q^78 + 13 * q^79 + q^81 + 16 * q^83 - 3 * q^84 - q^86 - 8 * q^87 + 3 * q^88 - 3 * q^89 - 6 * q^91 - 7 * q^92 - q^93 - 6 * q^94 + q^96 - 6 * q^97 + 2 * q^98 + 3 * 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 −3.00000 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.bk 1
5.b even 2 1 930.2.a.e 1
5.c odd 4 2 4650.2.d.ba 2
15.d odd 2 1 2790.2.a.u 1
20.d odd 2 1 7440.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.e 1 5.b even 2 1
2790.2.a.u 1 15.d odd 2 1
4650.2.a.bk 1 1.a even 1 1 trivial
4650.2.d.ba 2 5.c odd 4 2
7440.2.a.x 1 20.d odd 2 1

## Hecke kernels

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

 $$T_{7} + 3$$ T7 + 3 $$T_{11} - 3$$ T11 - 3 $$T_{13} - 2$$ T13 - 2 $$T_{19} + 7$$ T19 + 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 3$$
$11$ $$T - 3$$
$13$ $$T - 2$$
$17$ $$T + 8$$
$19$ $$T + 7$$
$23$ $$T + 7$$
$29$ $$T + 8$$
$31$ $$T + 1$$
$37$ $$T - 4$$
$41$ $$T$$
$43$ $$T + 1$$
$47$ $$T + 6$$
$53$ $$T + 5$$
$59$ $$T - 6$$
$61$ $$T - 2$$
$67$ $$T + 10$$
$71$ $$T - 9$$
$73$ $$T + 1$$
$79$ $$T - 13$$
$83$ $$T - 16$$
$89$ $$T + 3$$
$97$ $$T + 6$$