Properties

 Label 9310.2.a.u Level $9310$ Weight $2$ Character orbit 9310.a Self dual yes Analytic conductor $74.341$ Analytic rank $0$ 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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + 3 q^{3} + q^{4} + q^{5} + 3 q^{6} + q^{8} + 6 q^{9}+O(q^{10})$$ q + q^2 + 3 * q^3 + q^4 + q^5 + 3 * q^6 + q^8 + 6 * q^9 $$q + q^{2} + 3 q^{3} + q^{4} + q^{5} + 3 q^{6} + q^{8} + 6 q^{9} + q^{10} - 4 q^{11} + 3 q^{12} + q^{13} + 3 q^{15} + q^{16} + 3 q^{17} + 6 q^{18} - q^{19} + q^{20} - 4 q^{22} + 7 q^{23} + 3 q^{24} + q^{25} + q^{26} + 9 q^{27} - 3 q^{29} + 3 q^{30} + 2 q^{31} + q^{32} - 12 q^{33} + 3 q^{34} + 6 q^{36} - 2 q^{37} - q^{38} + 3 q^{39} + q^{40} + 6 q^{41} + 6 q^{43} - 4 q^{44} + 6 q^{45} + 7 q^{46} + 3 q^{48} + q^{50} + 9 q^{51} + q^{52} - 13 q^{53} + 9 q^{54} - 4 q^{55} - 3 q^{57} - 3 q^{58} + 9 q^{59} + 3 q^{60} + 12 q^{61} + 2 q^{62} + q^{64} + q^{65} - 12 q^{66} - 3 q^{67} + 3 q^{68} + 21 q^{69} + 6 q^{72} - 11 q^{73} - 2 q^{74} + 3 q^{75} - q^{76} + 3 q^{78} - 2 q^{79} + q^{80} + 9 q^{81} + 6 q^{82} + 10 q^{83} + 3 q^{85} + 6 q^{86} - 9 q^{87} - 4 q^{88} - 2 q^{89} + 6 q^{90} + 7 q^{92} + 6 q^{93} - q^{95} + 3 q^{96} + 2 q^{97} - 24 q^{99}+O(q^{100})$$ q + q^2 + 3 * q^3 + q^4 + q^5 + 3 * q^6 + q^8 + 6 * q^9 + q^10 - 4 * q^11 + 3 * q^12 + q^13 + 3 * q^15 + q^16 + 3 * q^17 + 6 * q^18 - q^19 + q^20 - 4 * q^22 + 7 * q^23 + 3 * q^24 + q^25 + q^26 + 9 * q^27 - 3 * q^29 + 3 * q^30 + 2 * q^31 + q^32 - 12 * q^33 + 3 * q^34 + 6 * q^36 - 2 * q^37 - q^38 + 3 * q^39 + q^40 + 6 * q^41 + 6 * q^43 - 4 * q^44 + 6 * q^45 + 7 * q^46 + 3 * q^48 + q^50 + 9 * q^51 + q^52 - 13 * q^53 + 9 * q^54 - 4 * q^55 - 3 * q^57 - 3 * q^58 + 9 * q^59 + 3 * q^60 + 12 * q^61 + 2 * q^62 + q^64 + q^65 - 12 * q^66 - 3 * q^67 + 3 * q^68 + 21 * q^69 + 6 * q^72 - 11 * q^73 - 2 * q^74 + 3 * q^75 - q^76 + 3 * q^78 - 2 * q^79 + q^80 + 9 * q^81 + 6 * q^82 + 10 * q^83 + 3 * q^85 + 6 * q^86 - 9 * q^87 - 4 * q^88 - 2 * q^89 + 6 * q^90 + 7 * q^92 + 6 * q^93 - q^95 + 3 * q^96 + 2 * q^97 - 24 * 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 0 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$
$$19$$ $$1$$

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.u 1
7.b odd 2 1 190.2.a.b 1
21.c even 2 1 1710.2.a.g 1
28.d even 2 1 1520.2.a.j 1
35.c odd 2 1 950.2.a.c 1
35.f even 4 2 950.2.b.a 2
56.e even 2 1 6080.2.a.b 1
56.h odd 2 1 6080.2.a.x 1
105.g even 2 1 8550.2.a.bm 1
133.c even 2 1 3610.2.a.e 1
140.c even 2 1 7600.2.a.a 1

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

 $$T_{3} - 3$$ T3 - 3 $$T_{11} + 4$$ T11 + 4 $$T_{13} - 1$$ T13 - 1 $$T_{17} - 3$$ T17 - 3

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 3$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T - 1$$
$17$ $$T - 3$$
$19$ $$T + 1$$
$23$ $$T - 7$$
$29$ $$T + 3$$
$31$ $$T - 2$$
$37$ $$T + 2$$
$41$ $$T - 6$$
$43$ $$T - 6$$
$47$ $$T$$
$53$ $$T + 13$$
$59$ $$T - 9$$
$61$ $$T - 12$$
$67$ $$T + 3$$
$71$ $$T$$
$73$ $$T + 11$$
$79$ $$T + 2$$
$83$ $$T - 10$$
$89$ $$T + 2$$
$97$ $$T - 2$$