# Properties

 Label 9438.2.a.t Level $9438$ Weight $2$ Character orbit 9438.a Self dual yes Analytic conductor $75.363$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9438 = 2 \cdot 3 \cdot 11^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9438.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$75.3628094277$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + 2 q^{5} - q^{6} - 4 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 + 2 * q^5 - q^6 - 4 * q^7 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} + 2 q^{5} - q^{6} - 4 q^{7} + q^{8} + q^{9} + 2 q^{10} - q^{12} - q^{13} - 4 q^{14} - 2 q^{15} + q^{16} - 2 q^{17} + q^{18} + 8 q^{19} + 2 q^{20} + 4 q^{21} - q^{24} - q^{25} - q^{26} - q^{27} - 4 q^{28} - 6 q^{29} - 2 q^{30} - 4 q^{31} + q^{32} - 2 q^{34} - 8 q^{35} + q^{36} - 2 q^{37} + 8 q^{38} + q^{39} + 2 q^{40} + 10 q^{41} + 4 q^{42} - 4 q^{43} + 2 q^{45} + 8 q^{47} - q^{48} + 9 q^{49} - q^{50} + 2 q^{51} - q^{52} - 10 q^{53} - q^{54} - 4 q^{56} - 8 q^{57} - 6 q^{58} + 4 q^{59} - 2 q^{60} + 2 q^{61} - 4 q^{62} - 4 q^{63} + q^{64} - 2 q^{65} - 16 q^{67} - 2 q^{68} - 8 q^{70} - 8 q^{71} + q^{72} - 2 q^{73} - 2 q^{74} + q^{75} + 8 q^{76} + q^{78} - 8 q^{79} + 2 q^{80} + q^{81} + 10 q^{82} - 12 q^{83} + 4 q^{84} - 4 q^{85} - 4 q^{86} + 6 q^{87} + 14 q^{89} + 2 q^{90} + 4 q^{91} + 4 q^{93} + 8 q^{94} + 16 q^{95} - q^{96} + 10 q^{97} + 9 q^{98}+O(q^{100})$$ q + q^2 - q^3 + q^4 + 2 * q^5 - q^6 - 4 * q^7 + q^8 + q^9 + 2 * q^10 - q^12 - q^13 - 4 * q^14 - 2 * q^15 + q^16 - 2 * q^17 + q^18 + 8 * q^19 + 2 * q^20 + 4 * q^21 - q^24 - q^25 - q^26 - q^27 - 4 * q^28 - 6 * q^29 - 2 * q^30 - 4 * q^31 + q^32 - 2 * q^34 - 8 * q^35 + q^36 - 2 * q^37 + 8 * q^38 + q^39 + 2 * q^40 + 10 * q^41 + 4 * q^42 - 4 * q^43 + 2 * q^45 + 8 * q^47 - q^48 + 9 * q^49 - q^50 + 2 * q^51 - q^52 - 10 * q^53 - q^54 - 4 * q^56 - 8 * q^57 - 6 * q^58 + 4 * q^59 - 2 * q^60 + 2 * q^61 - 4 * q^62 - 4 * q^63 + q^64 - 2 * q^65 - 16 * q^67 - 2 * q^68 - 8 * q^70 - 8 * q^71 + q^72 - 2 * q^73 - 2 * q^74 + q^75 + 8 * q^76 + q^78 - 8 * q^79 + 2 * q^80 + q^81 + 10 * q^82 - 12 * q^83 + 4 * q^84 - 4 * q^85 - 4 * q^86 + 6 * q^87 + 14 * q^89 + 2 * q^90 + 4 * q^91 + 4 * q^93 + 8 * q^94 + 16 * q^95 - q^96 + 10 * 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 2.00000 −1.00000 −4.00000 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$11$$ $$-1$$
$$13$$ $$1$$

## 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 9438.2.a.t 1
11.b odd 2 1 78.2.a.a 1
33.d even 2 1 234.2.a.c 1
44.c even 2 1 624.2.a.h 1
55.d odd 2 1 1950.2.a.w 1
55.e even 4 2 1950.2.e.i 2
77.b even 2 1 3822.2.a.j 1
88.b odd 2 1 2496.2.a.t 1
88.g even 2 1 2496.2.a.b 1
99.g even 6 2 2106.2.e.j 2
99.h odd 6 2 2106.2.e.q 2
132.d odd 2 1 1872.2.a.c 1
143.d odd 2 1 1014.2.a.d 1
143.g even 4 2 1014.2.b.b 2
143.i odd 6 2 1014.2.e.c 2
143.k odd 6 2 1014.2.e.f 2
143.o even 12 4 1014.2.i.d 4
165.d even 2 1 5850.2.a.d 1
165.l odd 4 2 5850.2.e.bb 2
264.m even 2 1 7488.2.a.bz 1
264.p odd 2 1 7488.2.a.bk 1
429.e even 2 1 3042.2.a.f 1
429.l odd 4 2 3042.2.b.g 2
572.b even 2 1 8112.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 11.b odd 2 1
234.2.a.c 1 33.d even 2 1
624.2.a.h 1 44.c even 2 1
1014.2.a.d 1 143.d odd 2 1
1014.2.b.b 2 143.g even 4 2
1014.2.e.c 2 143.i odd 6 2
1014.2.e.f 2 143.k odd 6 2
1014.2.i.d 4 143.o even 12 4
1872.2.a.c 1 132.d odd 2 1
1950.2.a.w 1 55.d odd 2 1
1950.2.e.i 2 55.e even 4 2
2106.2.e.j 2 99.g even 6 2
2106.2.e.q 2 99.h odd 6 2
2496.2.a.b 1 88.g even 2 1
2496.2.a.t 1 88.b odd 2 1
3042.2.a.f 1 429.e even 2 1
3042.2.b.g 2 429.l odd 4 2
3822.2.a.j 1 77.b even 2 1
5850.2.a.d 1 165.d even 2 1
5850.2.e.bb 2 165.l odd 4 2
7488.2.a.bk 1 264.p odd 2 1
7488.2.a.bz 1 264.m even 2 1
8112.2.a.v 1 572.b even 2 1
9438.2.a.t 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9438))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{7} + 4$$ T7 + 4 $$T_{17} + 2$$ T17 + 2 $$T_{19} - 8$$ T19 - 8 $$T_{29} + 6$$ T29 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T - 2$$
$7$ $$T + 4$$
$11$ $$T$$
$13$ $$T + 1$$
$17$ $$T + 2$$
$19$ $$T - 8$$
$23$ $$T$$
$29$ $$T + 6$$
$31$ $$T + 4$$
$37$ $$T + 2$$
$41$ $$T - 10$$
$43$ $$T + 4$$
$47$ $$T - 8$$
$53$ $$T + 10$$
$59$ $$T - 4$$
$61$ $$T - 2$$
$67$ $$T + 16$$
$71$ $$T + 8$$
$73$ $$T + 2$$
$79$ $$T + 8$$
$83$ $$T + 12$$
$89$ $$T - 14$$
$97$ $$T - 10$$