# Properties

 Label 1815.2.a.e Level $1815$ Weight $2$ Character orbit 1815.a Self dual yes Analytic conductor $14.493$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.4928479669$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} + 2 q^{7} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 - q^4 - q^5 + q^6 + 2 * q^7 - 3 * q^8 + q^9 $$q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} + 2 q^{7} - 3 q^{8} + q^{9} - q^{10} - q^{12} - 4 q^{13} + 2 q^{14} - q^{15} - q^{16} + 6 q^{17} + q^{18} + 6 q^{19} + q^{20} + 2 q^{21} + 4 q^{23} - 3 q^{24} + q^{25} - 4 q^{26} + q^{27} - 2 q^{28} - 6 q^{29} - q^{30} + 8 q^{31} + 5 q^{32} + 6 q^{34} - 2 q^{35} - q^{36} - 6 q^{37} + 6 q^{38} - 4 q^{39} + 3 q^{40} + 6 q^{41} + 2 q^{42} + 6 q^{43} - q^{45} + 4 q^{46} + 8 q^{47} - q^{48} - 3 q^{49} + q^{50} + 6 q^{51} + 4 q^{52} + 6 q^{53} + q^{54} - 6 q^{56} + 6 q^{57} - 6 q^{58} + q^{60} - 4 q^{61} + 8 q^{62} + 2 q^{63} + 7 q^{64} + 4 q^{65} + 12 q^{67} - 6 q^{68} + 4 q^{69} - 2 q^{70} + 8 q^{71} - 3 q^{72} - 16 q^{73} - 6 q^{74} + q^{75} - 6 q^{76} - 4 q^{78} - 2 q^{79} + q^{80} + q^{81} + 6 q^{82} - 2 q^{84} - 6 q^{85} + 6 q^{86} - 6 q^{87} + 10 q^{89} - q^{90} - 8 q^{91} - 4 q^{92} + 8 q^{93} + 8 q^{94} - 6 q^{95} + 5 q^{96} - 6 q^{97} - 3 q^{98}+O(q^{100})$$ q + q^2 + q^3 - q^4 - q^5 + q^6 + 2 * q^7 - 3 * q^8 + q^9 - q^10 - q^12 - 4 * q^13 + 2 * q^14 - q^15 - q^16 + 6 * q^17 + q^18 + 6 * q^19 + q^20 + 2 * q^21 + 4 * q^23 - 3 * q^24 + q^25 - 4 * q^26 + q^27 - 2 * q^28 - 6 * q^29 - q^30 + 8 * q^31 + 5 * q^32 + 6 * q^34 - 2 * q^35 - q^36 - 6 * q^37 + 6 * q^38 - 4 * q^39 + 3 * q^40 + 6 * q^41 + 2 * q^42 + 6 * q^43 - q^45 + 4 * q^46 + 8 * q^47 - q^48 - 3 * q^49 + q^50 + 6 * q^51 + 4 * q^52 + 6 * q^53 + q^54 - 6 * q^56 + 6 * q^57 - 6 * q^58 + q^60 - 4 * q^61 + 8 * q^62 + 2 * q^63 + 7 * q^64 + 4 * q^65 + 12 * q^67 - 6 * q^68 + 4 * q^69 - 2 * q^70 + 8 * q^71 - 3 * q^72 - 16 * q^73 - 6 * q^74 + q^75 - 6 * q^76 - 4 * q^78 - 2 * q^79 + q^80 + q^81 + 6 * q^82 - 2 * q^84 - 6 * q^85 + 6 * q^86 - 6 * q^87 + 10 * q^89 - q^90 - 8 * q^91 - 4 * q^92 + 8 * q^93 + 8 * q^94 - 6 * q^95 + 5 * q^96 - 6 * 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 −1.00000 1.00000 2.00000 −3.00000 1.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$1$$

## 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 1815.2.a.e yes 1
3.b odd 2 1 5445.2.a.e 1
5.b even 2 1 9075.2.a.d 1
11.b odd 2 1 1815.2.a.a 1
33.d even 2 1 5445.2.a.j 1
55.d odd 2 1 9075.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.a 1 11.b odd 2 1
1815.2.a.e yes 1 1.a even 1 1 trivial
5445.2.a.e 1 3.b odd 2 1
5445.2.a.j 1 33.d even 2 1
9075.2.a.d 1 5.b even 2 1
9075.2.a.n 1 55.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1815))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{7} - 2$$ T7 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T - 2$$
$11$ $$T$$
$13$ $$T + 4$$
$17$ $$T - 6$$
$19$ $$T - 6$$
$23$ $$T - 4$$
$29$ $$T + 6$$
$31$ $$T - 8$$
$37$ $$T + 6$$
$41$ $$T - 6$$
$43$ $$T - 6$$
$47$ $$T - 8$$
$53$ $$T - 6$$
$59$ $$T$$
$61$ $$T + 4$$
$67$ $$T - 12$$
$71$ $$T - 8$$
$73$ $$T + 16$$
$79$ $$T + 2$$
$83$ $$T$$
$89$ $$T - 10$$
$97$ $$T + 6$$
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