# Properties

 Label 9386.2.a.j Level $9386$ Weight $2$ Character orbit 9386.a Self dual yes Analytic conductor $74.948$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9386,2,Mod(1,9386)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9386, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9386.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9386 = 2 \cdot 13 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9386.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$74.9475873372$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - 3 q^{5} - q^{6} - q^{7} + q^{8} - 2 q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 - 3 * q^5 - q^6 - q^7 + q^8 - 2 * q^9 $$q + q^{2} - q^{3} + q^{4} - 3 q^{5} - q^{6} - q^{7} + q^{8} - 2 q^{9} - 3 q^{10} + 6 q^{11} - q^{12} - q^{13} - q^{14} + 3 q^{15} + q^{16} - 3 q^{17} - 2 q^{18} - 3 q^{20} + q^{21} + 6 q^{22} - q^{24} + 4 q^{25} - q^{26} + 5 q^{27} - q^{28} - 6 q^{29} + 3 q^{30} + 4 q^{31} + q^{32} - 6 q^{33} - 3 q^{34} + 3 q^{35} - 2 q^{36} + 7 q^{37} + q^{39} - 3 q^{40} + q^{42} - q^{43} + 6 q^{44} + 6 q^{45} + 3 q^{47} - q^{48} - 6 q^{49} + 4 q^{50} + 3 q^{51} - q^{52} + 5 q^{54} - 18 q^{55} - q^{56} - 6 q^{58} + 6 q^{59} + 3 q^{60} + 8 q^{61} + 4 q^{62} + 2 q^{63} + q^{64} + 3 q^{65} - 6 q^{66} - 14 q^{67} - 3 q^{68} + 3 q^{70} + 3 q^{71} - 2 q^{72} + 2 q^{73} + 7 q^{74} - 4 q^{75} - 6 q^{77} + q^{78} - 8 q^{79} - 3 q^{80} + q^{81} + 12 q^{83} + q^{84} + 9 q^{85} - q^{86} + 6 q^{87} + 6 q^{88} + 6 q^{89} + 6 q^{90} + q^{91} - 4 q^{93} + 3 q^{94} - q^{96} + 10 q^{97} - 6 q^{98} - 12 q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 - 3 * q^5 - q^6 - q^7 + q^8 - 2 * q^9 - 3 * q^10 + 6 * q^11 - q^12 - q^13 - q^14 + 3 * q^15 + q^16 - 3 * q^17 - 2 * q^18 - 3 * q^20 + q^21 + 6 * q^22 - q^24 + 4 * q^25 - q^26 + 5 * q^27 - q^28 - 6 * q^29 + 3 * q^30 + 4 * q^31 + q^32 - 6 * q^33 - 3 * q^34 + 3 * q^35 - 2 * q^36 + 7 * q^37 + q^39 - 3 * q^40 + q^42 - q^43 + 6 * q^44 + 6 * q^45 + 3 * q^47 - q^48 - 6 * q^49 + 4 * q^50 + 3 * q^51 - q^52 + 5 * q^54 - 18 * q^55 - q^56 - 6 * q^58 + 6 * q^59 + 3 * q^60 + 8 * q^61 + 4 * q^62 + 2 * q^63 + q^64 + 3 * q^65 - 6 * q^66 - 14 * q^67 - 3 * q^68 + 3 * q^70 + 3 * q^71 - 2 * q^72 + 2 * q^73 + 7 * q^74 - 4 * q^75 - 6 * q^77 + q^78 - 8 * q^79 - 3 * q^80 + q^81 + 12 * q^83 + q^84 + 9 * q^85 - q^86 + 6 * q^87 + 6 * q^88 + 6 * q^89 + 6 * q^90 + q^91 - 4 * q^93 + 3 * q^94 - q^96 + 10 * q^97 - 6 * q^98 - 12 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 −3.00000 −1.00000 −1.00000 1.00000 −2.00000 −3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$13$$ $$1$$
$$19$$ $$-1$$

## 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 9386.2.a.j 1
19.b odd 2 1 26.2.a.a 1
57.d even 2 1 234.2.a.e 1
76.d even 2 1 208.2.a.a 1
95.d odd 2 1 650.2.a.j 1
95.g even 4 2 650.2.b.d 2
133.c even 2 1 1274.2.a.d 1
133.o even 6 2 1274.2.f.r 2
133.r odd 6 2 1274.2.f.p 2
152.b even 2 1 832.2.a.i 1
152.g odd 2 1 832.2.a.d 1
171.l even 6 2 2106.2.e.b 2
171.o odd 6 2 2106.2.e.ba 2
209.d even 2 1 3146.2.a.n 1
228.b odd 2 1 1872.2.a.q 1
247.d odd 2 1 338.2.a.f 1
247.i even 4 2 338.2.b.c 2
247.m odd 6 2 338.2.c.a 2
247.u odd 6 2 338.2.c.d 2
247.bd even 12 4 338.2.e.a 4
285.b even 2 1 5850.2.a.p 1
285.j odd 4 2 5850.2.e.a 2
304.j odd 4 2 3328.2.b.m 2
304.m even 4 2 3328.2.b.j 2
323.c odd 2 1 7514.2.a.c 1
380.d even 2 1 5200.2.a.x 1
456.l odd 2 1 7488.2.a.h 1
456.p even 2 1 7488.2.a.g 1
741.d even 2 1 3042.2.a.a 1
741.p odd 4 2 3042.2.b.a 2
988.g even 2 1 2704.2.a.f 1
988.p odd 4 2 2704.2.f.d 2
1235.e odd 2 1 8450.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 19.b odd 2 1
208.2.a.a 1 76.d even 2 1
234.2.a.e 1 57.d even 2 1
338.2.a.f 1 247.d odd 2 1
338.2.b.c 2 247.i even 4 2
338.2.c.a 2 247.m odd 6 2
338.2.c.d 2 247.u odd 6 2
338.2.e.a 4 247.bd even 12 4
650.2.a.j 1 95.d odd 2 1
650.2.b.d 2 95.g even 4 2
832.2.a.d 1 152.g odd 2 1
832.2.a.i 1 152.b even 2 1
1274.2.a.d 1 133.c even 2 1
1274.2.f.p 2 133.r odd 6 2
1274.2.f.r 2 133.o even 6 2
1872.2.a.q 1 228.b odd 2 1
2106.2.e.b 2 171.l even 6 2
2106.2.e.ba 2 171.o odd 6 2
2704.2.a.f 1 988.g even 2 1
2704.2.f.d 2 988.p odd 4 2
3042.2.a.a 1 741.d even 2 1
3042.2.b.a 2 741.p odd 4 2
3146.2.a.n 1 209.d even 2 1
3328.2.b.j 2 304.m even 4 2
3328.2.b.m 2 304.j odd 4 2
5200.2.a.x 1 380.d even 2 1
5850.2.a.p 1 285.b even 2 1
5850.2.e.a 2 285.j odd 4 2
7488.2.a.g 1 456.p even 2 1
7488.2.a.h 1 456.l odd 2 1
7514.2.a.c 1 323.c odd 2 1
8450.2.a.c 1 1235.e odd 2 1
9386.2.a.j 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9386))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{5} + 3$$ T5 + 3 $$T_{7} + 1$$ T7 + 1 $$T_{29} + 6$$ T29 + 6

## Hecke characteristic polynomials

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