# Properties

 Label 9310.2.a.o Level $9310$ Weight $2$ Character orbit 9310.a Self dual yes Analytic conductor $74.341$ 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] = [9310,2,Mod(1,9310)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9310, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9310.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9310 = 2 \cdot 5 \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9310.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: $$74.3407242818$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} - 2 q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 - q^5 - q^6 + q^8 - 2 * q^9 $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} - 2 q^{9} - q^{10} - q^{12} + q^{13} + q^{15} + q^{16} + 3 q^{17} - 2 q^{18} - q^{19} - q^{20} + 3 q^{23} - q^{24} + q^{25} + q^{26} + 5 q^{27} - 3 q^{29} + q^{30} - 2 q^{31} + q^{32} + 3 q^{34} - 2 q^{36} - 10 q^{37} - q^{38} - q^{39} - q^{40} - 6 q^{41} + 2 q^{43} + 2 q^{45} + 3 q^{46} - q^{48} + q^{50} - 3 q^{51} + q^{52} + 3 q^{53} + 5 q^{54} + q^{57} - 3 q^{58} - 3 q^{59} + q^{60} - 8 q^{61} - 2 q^{62} + q^{64} - q^{65} - 7 q^{67} + 3 q^{68} - 3 q^{69} + 12 q^{71} - 2 q^{72} + 13 q^{73} - 10 q^{74} - q^{75} - q^{76} - q^{78} + 14 q^{79} - q^{80} + q^{81} - 6 q^{82} - 6 q^{83} - 3 q^{85} + 2 q^{86} + 3 q^{87} - 6 q^{89} + 2 q^{90} + 3 q^{92} + 2 q^{93} + q^{95} - q^{96} + 10 q^{97}+O(q^{100})$$ q + q^2 - q^3 + q^4 - q^5 - q^6 + q^8 - 2 * q^9 - q^10 - q^12 + q^13 + q^15 + q^16 + 3 * q^17 - 2 * q^18 - q^19 - q^20 + 3 * q^23 - q^24 + q^25 + q^26 + 5 * q^27 - 3 * q^29 + q^30 - 2 * q^31 + q^32 + 3 * q^34 - 2 * q^36 - 10 * q^37 - q^38 - q^39 - q^40 - 6 * q^41 + 2 * q^43 + 2 * q^45 + 3 * q^46 - q^48 + q^50 - 3 * q^51 + q^52 + 3 * q^53 + 5 * q^54 + q^57 - 3 * q^58 - 3 * q^59 + q^60 - 8 * q^61 - 2 * q^62 + q^64 - q^65 - 7 * q^67 + 3 * q^68 - 3 * q^69 + 12 * q^71 - 2 * q^72 + 13 * q^73 - 10 * q^74 - q^75 - q^76 - q^78 + 14 * q^79 - q^80 + q^81 - 6 * q^82 - 6 * q^83 - 3 * q^85 + 2 * q^86 + 3 * q^87 - 6 * q^89 + 2 * q^90 + 3 * q^92 + 2 * q^93 + q^95 - q^96 + 10 * q^97

## 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 −1.00000 −1.00000 0 1.00000 −2.00000 −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
$$2$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$
$$19$$ $$1$$

## 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 9310.2.a.o 1
7.b odd 2 1 190.2.a.c 1
21.c even 2 1 1710.2.a.d 1
28.d even 2 1 1520.2.a.d 1
35.c odd 2 1 950.2.a.a 1
35.f even 4 2 950.2.b.e 2
56.e even 2 1 6080.2.a.p 1
56.h odd 2 1 6080.2.a.h 1
105.g even 2 1 8550.2.a.bd 1
133.c even 2 1 3610.2.a.b 1
140.c even 2 1 7600.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.c 1 7.b odd 2 1
950.2.a.a 1 35.c odd 2 1
950.2.b.e 2 35.f even 4 2
1520.2.a.d 1 28.d even 2 1
1710.2.a.d 1 21.c even 2 1
3610.2.a.b 1 133.c even 2 1
6080.2.a.h 1 56.h odd 2 1
6080.2.a.p 1 56.e even 2 1
7600.2.a.m 1 140.c even 2 1
8550.2.a.bd 1 105.g even 2 1
9310.2.a.o 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9310))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{11}$$ T11 $$T_{13} - 1$$ T13 - 1 $$T_{17} - 3$$ T17 - 3

## Hecke characteristic polynomials

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