# Properties

 Label 9075.2.a.q Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} - q^{4} + q^{6} + 4 q^{7} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 - q^4 + q^6 + 4 * q^7 - 3 * q^8 + q^9 $$q + q^{2} + q^{3} - q^{4} + q^{6} + 4 q^{7} - 3 q^{8} + q^{9} - q^{12} - 2 q^{13} + 4 q^{14} - q^{16} - 2 q^{17} + q^{18} + 4 q^{21} - 8 q^{23} - 3 q^{24} - 2 q^{26} + q^{27} - 4 q^{28} + 6 q^{29} - 8 q^{31} + 5 q^{32} - 2 q^{34} - q^{36} - 6 q^{37} - 2 q^{39} + 2 q^{41} + 4 q^{42} - 8 q^{46} - 8 q^{47} - q^{48} + 9 q^{49} - 2 q^{51} + 2 q^{52} - 6 q^{53} + q^{54} - 12 q^{56} + 6 q^{58} - 4 q^{59} - 6 q^{61} - 8 q^{62} + 4 q^{63} + 7 q^{64} + 4 q^{67} + 2 q^{68} - 8 q^{69} - 3 q^{72} - 14 q^{73} - 6 q^{74} - 2 q^{78} + 4 q^{79} + q^{81} + 2 q^{82} + 12 q^{83} - 4 q^{84} + 6 q^{87} - 6 q^{89} - 8 q^{91} + 8 q^{92} - 8 q^{93} - 8 q^{94} + 5 q^{96} - 2 q^{97} + 9 q^{98}+O(q^{100})$$ q + q^2 + q^3 - q^4 + q^6 + 4 * q^7 - 3 * q^8 + q^9 - q^12 - 2 * q^13 + 4 * q^14 - q^16 - 2 * q^17 + q^18 + 4 * q^21 - 8 * q^23 - 3 * q^24 - 2 * q^26 + q^27 - 4 * q^28 + 6 * q^29 - 8 * q^31 + 5 * q^32 - 2 * q^34 - q^36 - 6 * q^37 - 2 * q^39 + 2 * q^41 + 4 * q^42 - 8 * q^46 - 8 * q^47 - q^48 + 9 * q^49 - 2 * q^51 + 2 * q^52 - 6 * q^53 + q^54 - 12 * q^56 + 6 * q^58 - 4 * q^59 - 6 * q^61 - 8 * q^62 + 4 * q^63 + 7 * q^64 + 4 * q^67 + 2 * q^68 - 8 * q^69 - 3 * q^72 - 14 * q^73 - 6 * q^74 - 2 * q^78 + 4 * q^79 + q^81 + 2 * q^82 + 12 * q^83 - 4 * q^84 + 6 * q^87 - 6 * q^89 - 8 * q^91 + 8 * q^92 - 8 * q^93 - 8 * q^94 + 5 * q^96 - 2 * q^97 + 9 * 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 4.00000 −3.00000 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
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$-1$$

## 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.q 1
5.b even 2 1 363.2.a.b 1
11.b odd 2 1 825.2.a.a 1
15.d odd 2 1 1089.2.a.j 1
20.d odd 2 1 5808.2.a.t 1
33.d even 2 1 2475.2.a.g 1
55.d odd 2 1 33.2.a.a 1
55.e even 4 2 825.2.c.a 2
55.h odd 10 4 363.2.e.e 4
55.j even 10 4 363.2.e.g 4
165.d even 2 1 99.2.a.b 1
165.l odd 4 2 2475.2.c.d 2
220.g even 2 1 528.2.a.g 1
385.h even 2 1 1617.2.a.j 1
440.c even 2 1 2112.2.a.j 1
440.o odd 2 1 2112.2.a.bb 1
495.o odd 6 2 891.2.e.e 2
495.r even 6 2 891.2.e.g 2
660.g odd 2 1 1584.2.a.o 1
715.c odd 2 1 5577.2.a.a 1
935.h odd 2 1 9537.2.a.m 1
1155.e odd 2 1 4851.2.a.b 1
1320.b odd 2 1 6336.2.a.n 1
1320.u even 2 1 6336.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 55.d odd 2 1
99.2.a.b 1 165.d even 2 1
363.2.a.b 1 5.b even 2 1
363.2.e.e 4 55.h odd 10 4
363.2.e.g 4 55.j even 10 4
528.2.a.g 1 220.g even 2 1
825.2.a.a 1 11.b odd 2 1
825.2.c.a 2 55.e even 4 2
891.2.e.e 2 495.o odd 6 2
891.2.e.g 2 495.r even 6 2
1089.2.a.j 1 15.d odd 2 1
1584.2.a.o 1 660.g odd 2 1
1617.2.a.j 1 385.h even 2 1
2112.2.a.j 1 440.c even 2 1
2112.2.a.bb 1 440.o odd 2 1
2475.2.a.g 1 33.d even 2 1
2475.2.c.d 2 165.l odd 4 2
4851.2.a.b 1 1155.e odd 2 1
5577.2.a.a 1 715.c odd 2 1
5808.2.a.t 1 20.d odd 2 1
6336.2.a.n 1 1320.b odd 2 1
6336.2.a.x 1 1320.u even 2 1
9075.2.a.q 1 1.a even 1 1 trivial
9537.2.a.m 1 935.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{7} - 4$$ T7 - 4 $$T_{13} + 2$$ T13 + 2 $$T_{17} + 2$$ T17 + 2 $$T_{19}$$ T19 $$T_{23} + 8$$ T23 + 8 $$T_{37} + 6$$ T37 + 6

## Hecke characteristic polynomials

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