Properties

 Label 9075.2.a.l Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$9075 = 3 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9075.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$72.4642398343$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 825) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 q^{4} + q^{7} + q^{9}+O(q^{10})$$ q + q^3 - 2 * q^4 + q^7 + q^9 $$q + q^{3} - 2 q^{4} + q^{7} + q^{9} - 2 q^{12} + q^{13} + 4 q^{16} + 6 q^{17} + 7 q^{19} + q^{21} + 6 q^{23} + q^{27} - 2 q^{28} + 6 q^{29} - 7 q^{31} - 2 q^{36} + 2 q^{37} + q^{39} + 6 q^{41} + q^{43} + 4 q^{48} - 6 q^{49} + 6 q^{51} - 2 q^{52} - 6 q^{53} + 7 q^{57} - 5 q^{61} + q^{63} - 8 q^{64} + 5 q^{67} - 12 q^{68} + 6 q^{69} - 12 q^{71} - 14 q^{73} - 14 q^{76} + 4 q^{79} + q^{81} + 6 q^{83} - 2 q^{84} + 6 q^{87} + 6 q^{89} + q^{91} - 12 q^{92} - 7 q^{93} + 17 q^{97}+O(q^{100})$$ q + q^3 - 2 * q^4 + q^7 + q^9 - 2 * q^12 + q^13 + 4 * q^16 + 6 * q^17 + 7 * q^19 + q^21 + 6 * q^23 + q^27 - 2 * q^28 + 6 * q^29 - 7 * q^31 - 2 * q^36 + 2 * q^37 + q^39 + 6 * q^41 + q^43 + 4 * q^48 - 6 * q^49 + 6 * q^51 - 2 * q^52 - 6 * q^53 + 7 * q^57 - 5 * q^61 + q^63 - 8 * q^64 + 5 * q^67 - 12 * q^68 + 6 * q^69 - 12 * q^71 - 14 * q^73 - 14 * q^76 + 4 * q^79 + q^81 + 6 * q^83 - 2 * q^84 + 6 * q^87 + 6 * q^89 + q^91 - 12 * q^92 - 7 * q^93 + 17 * q^97

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
0 1.00000 −2.00000 0 0 1.00000 0 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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$-1$$

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 9075.2.a.l 1
5.b even 2 1 9075.2.a.i 1
11.b odd 2 1 825.2.a.c yes 1
33.d even 2 1 2475.2.a.e 1
55.d odd 2 1 825.2.a.b 1
55.e even 4 2 825.2.c.b 2
165.d even 2 1 2475.2.a.f 1
165.l odd 4 2 2475.2.c.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.b 1 55.d odd 2 1
825.2.a.c yes 1 11.b odd 2 1
825.2.c.b 2 55.e even 4 2
2475.2.a.e 1 33.d even 2 1
2475.2.a.f 1 165.d even 2 1
2475.2.c.h 2 165.l odd 4 2
9075.2.a.i 1 5.b even 2 1
9075.2.a.l 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2}$$ T2 $$T_{7} - 1$$ T7 - 1 $$T_{13} - 1$$ T13 - 1 $$T_{17} - 6$$ T17 - 6 $$T_{19} - 7$$ T19 - 7 $$T_{23} - 6$$ T23 - 6 $$T_{37} - 2$$ T37 - 2

Hecke characteristic polynomials

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