# Properties

 Label 4650.2.a.bq 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + q^6 + q^7 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + q^{8} + q^{9} - 5 q^{11} + q^{12} - 4 q^{13} + q^{14} + q^{16} + q^{18} - 5 q^{19} + q^{21} - 5 q^{22} - 9 q^{23} + q^{24} - 4 q^{26} + q^{27} + q^{28} - 2 q^{29} + q^{31} + q^{32} - 5 q^{33} + q^{36} + 8 q^{37} - 5 q^{38} - 4 q^{39} + 6 q^{41} + q^{42} + q^{43} - 5 q^{44} - 9 q^{46} - 12 q^{47} + q^{48} - 6 q^{49} - 4 q^{52} - 13 q^{53} + q^{54} + q^{56} - 5 q^{57} - 2 q^{58} + 10 q^{59} - 14 q^{61} + q^{62} + q^{63} + q^{64} - 5 q^{66} + 14 q^{67} - 9 q^{69} - 9 q^{71} + q^{72} - 9 q^{73} + 8 q^{74} - 5 q^{76} - 5 q^{77} - 4 q^{78} + 5 q^{79} + q^{81} + 6 q^{82} + 6 q^{83} + q^{84} + q^{86} - 2 q^{87} - 5 q^{88} + 3 q^{89} - 4 q^{91} - 9 q^{92} + q^{93} - 12 q^{94} + q^{96} - 18 q^{97} - 6 q^{98} - 5 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^6 + q^7 + q^8 + q^9 - 5 * q^11 + q^12 - 4 * q^13 + q^14 + q^16 + q^18 - 5 * q^19 + q^21 - 5 * q^22 - 9 * q^23 + q^24 - 4 * q^26 + q^27 + q^28 - 2 * q^29 + q^31 + q^32 - 5 * q^33 + q^36 + 8 * q^37 - 5 * q^38 - 4 * q^39 + 6 * q^41 + q^42 + q^43 - 5 * q^44 - 9 * q^46 - 12 * q^47 + q^48 - 6 * q^49 - 4 * q^52 - 13 * q^53 + q^54 + q^56 - 5 * q^57 - 2 * q^58 + 10 * q^59 - 14 * q^61 + q^62 + q^63 + q^64 - 5 * q^66 + 14 * q^67 - 9 * q^69 - 9 * q^71 + q^72 - 9 * q^73 + 8 * q^74 - 5 * q^76 - 5 * q^77 - 4 * q^78 + 5 * q^79 + q^81 + 6 * q^82 + 6 * q^83 + q^84 + q^86 - 2 * q^87 - 5 * q^88 + 3 * q^89 - 4 * q^91 - 9 * q^92 + q^93 - 12 * q^94 + q^96 - 18 * q^97 - 6 * q^98 - 5 * 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 1.00000 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.bq 1
5.b even 2 1 4650.2.a.f 1
5.c odd 4 2 930.2.d.b 2
15.e even 4 2 2790.2.d.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.d.b 2 5.c odd 4 2
2790.2.d.g 2 15.e even 4 2
4650.2.a.f 1 5.b even 2 1
4650.2.a.bq 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4650))$$:

 $$T_{7} - 1$$ T7 - 1 $$T_{11} + 5$$ T11 + 5 $$T_{13} + 4$$ T13 + 4 $$T_{19} + 5$$ T19 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T + 5$$
$13$ $$T + 4$$
$17$ $$T$$
$19$ $$T + 5$$
$23$ $$T + 9$$
$29$ $$T + 2$$
$31$ $$T - 1$$
$37$ $$T - 8$$
$41$ $$T - 6$$
$43$ $$T - 1$$
$47$ $$T + 12$$
$53$ $$T + 13$$
$59$ $$T - 10$$
$61$ $$T + 14$$
$67$ $$T - 14$$
$71$ $$T + 9$$
$73$ $$T + 9$$
$79$ $$T - 5$$
$83$ $$T - 6$$
$89$ $$T - 3$$
$97$ $$T + 18$$