# Properties

 Label 9310.2.a.bc Level $9310$ Weight $2$ Character orbit 9310.a Self dual yes Analytic conductor $74.341$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta q^{3} + q^{4} - q^{5} - \beta q^{6} - q^{8} + (\beta + 1) q^{9} +O(q^{10})$$ q - q^2 + b * q^3 + q^4 - q^5 - b * q^6 - q^8 + (b + 1) * q^9 $$q - q^{2} + \beta q^{3} + q^{4} - q^{5} - \beta q^{6} - q^{8} + (\beta + 1) q^{9} + q^{10} + 4 q^{11} + \beta q^{12} + ( - 3 \beta + 2) q^{13} - \beta q^{15} + q^{16} + (\beta - 6) q^{17} + ( - \beta - 1) q^{18} + q^{19} - q^{20} - 4 q^{22} + 3 \beta q^{23} - \beta q^{24} + q^{25} + (3 \beta - 2) q^{26} + ( - \beta + 4) q^{27} + ( - 3 \beta + 2) q^{29} + \beta q^{30} + 2 \beta q^{31} - q^{32} + 4 \beta q^{33} + ( - \beta + 6) q^{34} + (\beta + 1) q^{36} - 6 q^{37} - q^{38} + ( - \beta - 12) q^{39} + q^{40} + ( - 4 \beta - 2) q^{41} + (2 \beta - 8) q^{43} + 4 q^{44} + ( - \beta - 1) q^{45} - 3 \beta q^{46} + ( - 4 \beta + 4) q^{47} + \beta q^{48} - q^{50} + ( - 5 \beta + 4) q^{51} + ( - 3 \beta + 2) q^{52} + ( - \beta - 2) q^{53} + (\beta - 4) q^{54} - 4 q^{55} + \beta q^{57} + (3 \beta - 2) q^{58} - \beta q^{59} - \beta q^{60} + ( - 2 \beta - 6) q^{61} - 2 \beta q^{62} + q^{64} + (3 \beta - 2) q^{65} - 4 \beta q^{66} - \beta q^{67} + (\beta - 6) q^{68} + (3 \beta + 12) q^{69} + 4 \beta q^{71} + ( - \beta - 1) q^{72} + (3 \beta - 6) q^{73} + 6 q^{74} + \beta q^{75} + q^{76} + (\beta + 12) q^{78} - 2 \beta q^{79} - q^{80} - 7 q^{81} + (4 \beta + 2) q^{82} + (2 \beta - 8) q^{83} + ( - \beta + 6) q^{85} + ( - 2 \beta + 8) q^{86} + ( - \beta - 12) q^{87} - 4 q^{88} - 2 q^{89} + (\beta + 1) q^{90} + 3 \beta q^{92} + (2 \beta + 8) q^{93} + (4 \beta - 4) q^{94} - q^{95} - \beta q^{96} - 6 q^{97} + (4 \beta + 4) q^{99} +O(q^{100})$$ q - q^2 + b * q^3 + q^4 - q^5 - b * q^6 - q^8 + (b + 1) * q^9 + q^10 + 4 * q^11 + b * q^12 + (-3*b + 2) * q^13 - b * q^15 + q^16 + (b - 6) * q^17 + (-b - 1) * q^18 + q^19 - q^20 - 4 * q^22 + 3*b * q^23 - b * q^24 + q^25 + (3*b - 2) * q^26 + (-b + 4) * q^27 + (-3*b + 2) * q^29 + b * q^30 + 2*b * q^31 - q^32 + 4*b * q^33 + (-b + 6) * q^34 + (b + 1) * q^36 - 6 * q^37 - q^38 + (-b - 12) * q^39 + q^40 + (-4*b - 2) * q^41 + (2*b - 8) * q^43 + 4 * q^44 + (-b - 1) * q^45 - 3*b * q^46 + (-4*b + 4) * q^47 + b * q^48 - q^50 + (-5*b + 4) * q^51 + (-3*b + 2) * q^52 + (-b - 2) * q^53 + (b - 4) * q^54 - 4 * q^55 + b * q^57 + (3*b - 2) * q^58 - b * q^59 - b * q^60 + (-2*b - 6) * q^61 - 2*b * q^62 + q^64 + (3*b - 2) * q^65 - 4*b * q^66 - b * q^67 + (b - 6) * q^68 + (3*b + 12) * q^69 + 4*b * q^71 + (-b - 1) * q^72 + (3*b - 6) * q^73 + 6 * q^74 + b * q^75 + q^76 + (b + 12) * q^78 - 2*b * q^79 - q^80 - 7 * q^81 + (4*b + 2) * q^82 + (2*b - 8) * q^83 + (-b + 6) * q^85 + (-2*b + 8) * q^86 + (-b - 12) * q^87 - 4 * q^88 - 2 * q^89 + (b + 1) * q^90 + 3*b * q^92 + (2*b + 8) * q^93 + (4*b - 4) * q^94 - q^95 - b * q^96 - 6 * q^97 + (4*b + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - q^{6} - 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 - q^6 - 2 * q^8 + 3 * q^9 $$2 q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - q^{6} - 2 q^{8} + 3 q^{9} + 2 q^{10} + 8 q^{11} + q^{12} + q^{13} - q^{15} + 2 q^{16} - 11 q^{17} - 3 q^{18} + 2 q^{19} - 2 q^{20} - 8 q^{22} + 3 q^{23} - q^{24} + 2 q^{25} - q^{26} + 7 q^{27} + q^{29} + q^{30} + 2 q^{31} - 2 q^{32} + 4 q^{33} + 11 q^{34} + 3 q^{36} - 12 q^{37} - 2 q^{38} - 25 q^{39} + 2 q^{40} - 8 q^{41} - 14 q^{43} + 8 q^{44} - 3 q^{45} - 3 q^{46} + 4 q^{47} + q^{48} - 2 q^{50} + 3 q^{51} + q^{52} - 5 q^{53} - 7 q^{54} - 8 q^{55} + q^{57} - q^{58} - q^{59} - q^{60} - 14 q^{61} - 2 q^{62} + 2 q^{64} - q^{65} - 4 q^{66} - q^{67} - 11 q^{68} + 27 q^{69} + 4 q^{71} - 3 q^{72} - 9 q^{73} + 12 q^{74} + q^{75} + 2 q^{76} + 25 q^{78} - 2 q^{79} - 2 q^{80} - 14 q^{81} + 8 q^{82} - 14 q^{83} + 11 q^{85} + 14 q^{86} - 25 q^{87} - 8 q^{88} - 4 q^{89} + 3 q^{90} + 3 q^{92} + 18 q^{93} - 4 q^{94} - 2 q^{95} - q^{96} - 12 q^{97} + 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 - q^6 - 2 * q^8 + 3 * q^9 + 2 * q^10 + 8 * q^11 + q^12 + q^13 - q^15 + 2 * q^16 - 11 * q^17 - 3 * q^18 + 2 * q^19 - 2 * q^20 - 8 * q^22 + 3 * q^23 - q^24 + 2 * q^25 - q^26 + 7 * q^27 + q^29 + q^30 + 2 * q^31 - 2 * q^32 + 4 * q^33 + 11 * q^34 + 3 * q^36 - 12 * q^37 - 2 * q^38 - 25 * q^39 + 2 * q^40 - 8 * q^41 - 14 * q^43 + 8 * q^44 - 3 * q^45 - 3 * q^46 + 4 * q^47 + q^48 - 2 * q^50 + 3 * q^51 + q^52 - 5 * q^53 - 7 * q^54 - 8 * q^55 + q^57 - q^58 - q^59 - q^60 - 14 * q^61 - 2 * q^62 + 2 * q^64 - q^65 - 4 * q^66 - q^67 - 11 * q^68 + 27 * q^69 + 4 * q^71 - 3 * q^72 - 9 * q^73 + 12 * q^74 + q^75 + 2 * q^76 + 25 * q^78 - 2 * q^79 - 2 * q^80 - 14 * q^81 + 8 * q^82 - 14 * q^83 + 11 * q^85 + 14 * q^86 - 25 * q^87 - 8 * q^88 - 4 * q^89 + 3 * q^90 + 3 * q^92 + 18 * q^93 - 4 * q^94 - 2 * q^95 - q^96 - 12 * q^97 + 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
 −1.56155 2.56155
−1.00000 −1.56155 1.00000 −1.00000 1.56155 0 −1.00000 −0.561553 1.00000
1.2 −1.00000 2.56155 1.00000 −1.00000 −2.56155 0 −1.00000 3.56155 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.bc 2
7.b odd 2 1 190.2.a.d 2
21.c even 2 1 1710.2.a.w 2
28.d even 2 1 1520.2.a.n 2
35.c odd 2 1 950.2.a.h 2
35.f even 4 2 950.2.b.f 4
56.e even 2 1 6080.2.a.bb 2
56.h odd 2 1 6080.2.a.bh 2
105.g even 2 1 8550.2.a.br 2
133.c even 2 1 3610.2.a.t 2
140.c even 2 1 7600.2.a.y 2

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

 $$T_{3}^{2} - T_{3} - 4$$ T3^2 - T3 - 4 $$T_{11} - 4$$ T11 - 4 $$T_{13}^{2} - T_{13} - 38$$ T13^2 - T13 - 38 $$T_{17}^{2} + 11T_{17} + 26$$ T17^2 + 11*T17 + 26

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} - T - 4$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} - T - 38$$
$17$ $$T^{2} + 11T + 26$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} - 3T - 36$$
$29$ $$T^{2} - T - 38$$
$31$ $$T^{2} - 2T - 16$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} + 8T - 52$$
$43$ $$T^{2} + 14T + 32$$
$47$ $$T^{2} - 4T - 64$$
$53$ $$T^{2} + 5T + 2$$
$59$ $$T^{2} + T - 4$$
$61$ $$T^{2} + 14T + 32$$
$67$ $$T^{2} + T - 4$$
$71$ $$T^{2} - 4T - 64$$
$73$ $$T^{2} + 9T - 18$$
$79$ $$T^{2} + 2T - 16$$
$83$ $$T^{2} + 14T + 32$$
$89$ $$(T + 2)^{2}$$
$97$ $$(T + 6)^{2}$$