# Properties

 Label 9386.2.a.f 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + 3 q^{3} + q^{4} - q^{5} - 3 q^{6} + q^{7} - q^{8} + 6 q^{9}+O(q^{10})$$ q - q^2 + 3 * q^3 + q^4 - q^5 - 3 * q^6 + q^7 - q^8 + 6 * q^9 $$q - q^{2} + 3 q^{3} + q^{4} - q^{5} - 3 q^{6} + q^{7} - q^{8} + 6 q^{9} + q^{10} - 2 q^{11} + 3 q^{12} + q^{13} - q^{14} - 3 q^{15} + q^{16} - 3 q^{17} - 6 q^{18} - q^{20} + 3 q^{21} + 2 q^{22} - 4 q^{23} - 3 q^{24} - 4 q^{25} - q^{26} + 9 q^{27} + q^{28} - 2 q^{29} + 3 q^{30} - 4 q^{31} - q^{32} - 6 q^{33} + 3 q^{34} - q^{35} + 6 q^{36} - 3 q^{37} + 3 q^{39} + q^{40} - 3 q^{42} - 5 q^{43} - 2 q^{44} - 6 q^{45} + 4 q^{46} + 13 q^{47} + 3 q^{48} - 6 q^{49} + 4 q^{50} - 9 q^{51} + q^{52} - 12 q^{53} - 9 q^{54} + 2 q^{55} - q^{56} + 2 q^{58} + 10 q^{59} - 3 q^{60} - 8 q^{61} + 4 q^{62} + 6 q^{63} + q^{64} - q^{65} + 6 q^{66} + 2 q^{67} - 3 q^{68} - 12 q^{69} + q^{70} + 5 q^{71} - 6 q^{72} - 10 q^{73} + 3 q^{74} - 12 q^{75} - 2 q^{77} - 3 q^{78} + 4 q^{79} - q^{80} + 9 q^{81} + 3 q^{84} + 3 q^{85} + 5 q^{86} - 6 q^{87} + 2 q^{88} - 6 q^{89} + 6 q^{90} + q^{91} - 4 q^{92} - 12 q^{93} - 13 q^{94} - 3 q^{96} - 14 q^{97} + 6 q^{98} - 12 q^{99}+O(q^{100})$$ q - q^2 + 3 * q^3 + q^4 - q^5 - 3 * q^6 + q^7 - q^8 + 6 * q^9 + q^10 - 2 * q^11 + 3 * q^12 + q^13 - q^14 - 3 * q^15 + q^16 - 3 * q^17 - 6 * q^18 - q^20 + 3 * q^21 + 2 * q^22 - 4 * q^23 - 3 * q^24 - 4 * q^25 - q^26 + 9 * q^27 + q^28 - 2 * q^29 + 3 * q^30 - 4 * q^31 - q^32 - 6 * q^33 + 3 * q^34 - q^35 + 6 * q^36 - 3 * q^37 + 3 * q^39 + q^40 - 3 * q^42 - 5 * q^43 - 2 * q^44 - 6 * q^45 + 4 * q^46 + 13 * q^47 + 3 * q^48 - 6 * q^49 + 4 * q^50 - 9 * q^51 + q^52 - 12 * q^53 - 9 * q^54 + 2 * q^55 - q^56 + 2 * q^58 + 10 * q^59 - 3 * q^60 - 8 * q^61 + 4 * q^62 + 6 * q^63 + q^64 - q^65 + 6 * q^66 + 2 * q^67 - 3 * q^68 - 12 * q^69 + q^70 + 5 * q^71 - 6 * q^72 - 10 * q^73 + 3 * q^74 - 12 * q^75 - 2 * q^77 - 3 * q^78 + 4 * q^79 - q^80 + 9 * q^81 + 3 * q^84 + 3 * q^85 + 5 * q^86 - 6 * q^87 + 2 * q^88 - 6 * q^89 + 6 * q^90 + q^91 - 4 * q^92 - 12 * q^93 - 13 * q^94 - 3 * q^96 - 14 * 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 3.00000 1.00000 −1.00000 −3.00000 1.00000 −1.00000 6.00000 1.00000
 $$n$$: 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.f 1
19.b odd 2 1 26.2.a.b 1
57.d even 2 1 234.2.a.b 1
76.d even 2 1 208.2.a.d 1
95.d odd 2 1 650.2.a.g 1
95.g even 4 2 650.2.b.a 2
133.c even 2 1 1274.2.a.o 1
133.o even 6 2 1274.2.f.a 2
133.r odd 6 2 1274.2.f.l 2
152.b even 2 1 832.2.a.a 1
152.g odd 2 1 832.2.a.j 1
171.l even 6 2 2106.2.e.t 2
171.o odd 6 2 2106.2.e.h 2
209.d even 2 1 3146.2.a.a 1
228.b odd 2 1 1872.2.a.m 1
247.d odd 2 1 338.2.a.a 1
247.i even 4 2 338.2.b.a 2
247.m odd 6 2 338.2.c.g 2
247.u odd 6 2 338.2.c.c 2
247.bd even 12 4 338.2.e.d 4
285.b even 2 1 5850.2.a.bn 1
285.j odd 4 2 5850.2.e.v 2
304.j odd 4 2 3328.2.b.g 2
304.m even 4 2 3328.2.b.k 2
323.c odd 2 1 7514.2.a.i 1
380.d even 2 1 5200.2.a.c 1
456.l odd 2 1 7488.2.a.v 1
456.p even 2 1 7488.2.a.w 1
741.d even 2 1 3042.2.a.l 1
741.p odd 4 2 3042.2.b.f 2
988.g even 2 1 2704.2.a.n 1
988.p odd 4 2 2704.2.f.j 2
1235.e odd 2 1 8450.2.a.y 1

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

 $$T_{3} - 3$$ T3 - 3 $$T_{5} + 1$$ T5 + 1 $$T_{7} - 1$$ T7 - 1 $$T_{29} + 2$$ T29 + 2

## Hecke characteristic polynomials

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