# Properties

 Label 9537.2.a.m Level $9537$ Weight $2$ Character orbit 9537.a Self dual yes Analytic conductor $76.153$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9537 = 3 \cdot 11 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9537.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$76.1533284077$$ Analytic rank: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} - q^{4} + 2 q^{5} + q^{6} - 4 q^{7} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 - q^4 + 2 * q^5 + q^6 - 4 * q^7 - 3 * q^8 + q^9 $$q + q^{2} + q^{3} - q^{4} + 2 q^{5} + q^{6} - 4 q^{7} - 3 q^{8} + q^{9} + 2 q^{10} - q^{11} - q^{12} - 2 q^{13} - 4 q^{14} + 2 q^{15} - q^{16} + q^{18} - 2 q^{20} - 4 q^{21} - q^{22} - 8 q^{23} - 3 q^{24} - q^{25} - 2 q^{26} + q^{27} + 4 q^{28} + 6 q^{29} + 2 q^{30} + 8 q^{31} + 5 q^{32} - q^{33} - 8 q^{35} - q^{36} - 6 q^{37} - 2 q^{39} - 6 q^{40} + 2 q^{41} - 4 q^{42} + q^{44} + 2 q^{45} - 8 q^{46} + 8 q^{47} - q^{48} + 9 q^{49} - q^{50} + 2 q^{52} + 6 q^{53} + q^{54} - 2 q^{55} + 12 q^{56} + 6 q^{58} - 4 q^{59} - 2 q^{60} - 6 q^{61} + 8 q^{62} - 4 q^{63} + 7 q^{64} - 4 q^{65} - q^{66} - 4 q^{67} - 8 q^{69} - 8 q^{70} - 3 q^{72} + 14 q^{73} - 6 q^{74} - q^{75} + 4 q^{77} - 2 q^{78} + 4 q^{79} - 2 q^{80} + q^{81} + 2 q^{82} + 12 q^{83} + 4 q^{84} + 6 q^{87} + 3 q^{88} - 6 q^{89} + 2 q^{90} + 8 q^{91} + 8 q^{92} + 8 q^{93} + 8 q^{94} + 5 q^{96} - 2 q^{97} + 9 q^{98} - q^{99}+O(q^{100})$$ q + q^2 + q^3 - q^4 + 2 * q^5 + q^6 - 4 * q^7 - 3 * q^8 + q^9 + 2 * q^10 - q^11 - q^12 - 2 * q^13 - 4 * q^14 + 2 * q^15 - q^16 + q^18 - 2 * q^20 - 4 * q^21 - q^22 - 8 * q^23 - 3 * q^24 - q^25 - 2 * q^26 + q^27 + 4 * q^28 + 6 * q^29 + 2 * q^30 + 8 * q^31 + 5 * q^32 - q^33 - 8 * q^35 - q^36 - 6 * q^37 - 2 * q^39 - 6 * q^40 + 2 * q^41 - 4 * q^42 + q^44 + 2 * q^45 - 8 * q^46 + 8 * q^47 - q^48 + 9 * q^49 - q^50 + 2 * q^52 + 6 * q^53 + q^54 - 2 * q^55 + 12 * q^56 + 6 * q^58 - 4 * q^59 - 2 * q^60 - 6 * q^61 + 8 * q^62 - 4 * q^63 + 7 * q^64 - 4 * q^65 - q^66 - 4 * q^67 - 8 * q^69 - 8 * q^70 - 3 * q^72 + 14 * q^73 - 6 * q^74 - q^75 + 4 * q^77 - 2 * q^78 + 4 * q^79 - 2 * q^80 + q^81 + 2 * q^82 + 12 * q^83 + 4 * q^84 + 6 * q^87 + 3 * q^88 - 6 * q^89 + 2 * q^90 + 8 * q^91 + 8 * q^92 + 8 * q^93 + 8 * q^94 + 5 * q^96 - 2 * q^97 + 9 * q^98 - 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 2.00000 1.00000 −4.00000 −3.00000 1.00000 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$1$$
$$17$$ $$1$$

## 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 9537.2.a.m 1
17.b even 2 1 33.2.a.a 1
51.c odd 2 1 99.2.a.b 1
68.d odd 2 1 528.2.a.g 1
85.c even 2 1 825.2.a.a 1
85.g odd 4 2 825.2.c.a 2
119.d odd 2 1 1617.2.a.j 1
136.e odd 2 1 2112.2.a.j 1
136.h even 2 1 2112.2.a.bb 1
153.h even 6 2 891.2.e.e 2
153.i odd 6 2 891.2.e.g 2
187.b odd 2 1 363.2.a.b 1
187.j even 10 4 363.2.e.e 4
187.l odd 10 4 363.2.e.g 4
204.h even 2 1 1584.2.a.o 1
221.b even 2 1 5577.2.a.a 1
255.h odd 2 1 2475.2.a.g 1
255.o even 4 2 2475.2.c.d 2
357.c even 2 1 4851.2.a.b 1
408.b odd 2 1 6336.2.a.x 1
408.h even 2 1 6336.2.a.n 1
561.h even 2 1 1089.2.a.j 1
748.f even 2 1 5808.2.a.t 1
935.h odd 2 1 9075.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 17.b even 2 1
99.2.a.b 1 51.c odd 2 1
363.2.a.b 1 187.b odd 2 1
363.2.e.e 4 187.j even 10 4
363.2.e.g 4 187.l odd 10 4
528.2.a.g 1 68.d odd 2 1
825.2.a.a 1 85.c even 2 1
825.2.c.a 2 85.g odd 4 2
891.2.e.e 2 153.h even 6 2
891.2.e.g 2 153.i odd 6 2
1089.2.a.j 1 561.h even 2 1
1584.2.a.o 1 204.h even 2 1
1617.2.a.j 1 119.d odd 2 1
2112.2.a.j 1 136.e odd 2 1
2112.2.a.bb 1 136.h even 2 1
2475.2.a.g 1 255.h odd 2 1
2475.2.c.d 2 255.o even 4 2
4851.2.a.b 1 357.c even 2 1
5577.2.a.a 1 221.b even 2 1
5808.2.a.t 1 748.f even 2 1
6336.2.a.n 1 408.h even 2 1
6336.2.a.x 1 408.b odd 2 1
9075.2.a.q 1 935.h odd 2 1
9537.2.a.m 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9537))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{5} - 2$$ T5 - 2 $$T_{7} + 4$$ T7 + 4 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

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