# Properties

 Label 310.2.a.e Level $310$ Weight $2$ Character orbit 310.a Self dual yes Analytic conductor $2.475$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [310,2,Mod(1,310)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(310, 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("310.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$310 = 2 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 310.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: $$2.47536246266$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} + q^{5} + ( - \beta_1 + 1) q^{6} + ( - \beta_{2} + \beta_1) q^{7} + q^{8} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10})$$ q + q^2 + (-b1 + 1) * q^3 + q^4 + q^5 + (-b1 + 1) * q^6 + (-b2 + b1) * q^7 + q^8 + (b2 - b1 + 1) * q^9 $$q + q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} + q^{5} + ( - \beta_1 + 1) q^{6} + ( - \beta_{2} + \beta_1) q^{7} + q^{8} + (\beta_{2} - \beta_1 + 1) q^{9} + q^{10} + (\beta_{2} + 2 \beta_1 - 1) q^{11} + ( - \beta_1 + 1) q^{12} + (\beta_1 - 3) q^{13} + ( - \beta_{2} + \beta_1) q^{14} + ( - \beta_1 + 1) q^{15} + q^{16} + ( - \beta_{2} + \beta_1) q^{17} + (\beta_{2} - \beta_1 + 1) q^{18} + (2 \beta_{2} + 2) q^{19} + q^{20} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{21} + (\beta_{2} + 2 \beta_1 - 1) q^{22} + ( - \beta_{2} - \beta_1) q^{23} + ( - \beta_1 + 1) q^{24} + q^{25} + (\beta_1 - 3) q^{26} + (2 \beta_{2} + 2) q^{27} + ( - \beta_{2} + \beta_1) q^{28} + ( - 3 \beta_{2} - 2 \beta_1 + 1) q^{29} + ( - \beta_1 + 1) q^{30} + q^{31} + q^{32} + ( - \beta_{2} - \beta_1 - 6) q^{33} + ( - \beta_{2} + \beta_1) q^{34} + ( - \beta_{2} + \beta_1) q^{35} + (\beta_{2} - \beta_1 + 1) q^{36} + (2 \beta_{2} - \beta_1 - 3) q^{37} + (2 \beta_{2} + 2) q^{38} + ( - \beta_{2} + 3 \beta_1 - 6) q^{39} + q^{40} + (3 \beta_{2} + \beta_1 - 2) q^{41} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{42} + (2 \beta_{2} - 3 \beta_1 - 3) q^{43} + (\beta_{2} + 2 \beta_1 - 1) q^{44} + (\beta_{2} - \beta_1 + 1) q^{45} + ( - \beta_{2} - \beta_1) q^{46} + ( - \beta_{2} - 3 \beta_1 + 8) q^{47} + ( - \beta_1 + 1) q^{48} + ( - 4 \beta_1 + 5) q^{49} + q^{50} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{51} + (\beta_1 - 3) q^{52} + (2 \beta_{2} - \beta_1 - 7) q^{53} + (2 \beta_{2} + 2) q^{54} + (\beta_{2} + 2 \beta_1 - 1) q^{55} + ( - \beta_{2} + \beta_1) q^{56} + (2 \beta_{2} - 6 \beta_1 + 4) q^{57} + ( - 3 \beta_{2} - 2 \beta_1 + 1) q^{58} + (2 \beta_1 + 6) q^{59} + ( - \beta_1 + 1) q^{60} + ( - \beta_{2} - 2 \beta_1 - 5) q^{61} + q^{62} + ( - \beta_{2} + 5 \beta_1 - 12) q^{63} + q^{64} + (\beta_1 - 3) q^{65} + ( - \beta_{2} - \beta_1 - 6) q^{66} + ( - 4 \beta_{2} - 2 \beta_1 - 2) q^{67} + ( - \beta_{2} + \beta_1) q^{68} + (2 \beta_1 + 2) q^{69} + ( - \beta_{2} + \beta_1) q^{70} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{71} + (\beta_{2} - \beta_1 + 1) q^{72} + ( - 3 \beta_{2} + \beta_1 - 6) q^{73} + (2 \beta_{2} - \beta_1 - 3) q^{74} + ( - \beta_1 + 1) q^{75} + (2 \beta_{2} + 2) q^{76} + 4 \beta_{2} q^{77} + ( - \beta_{2} + 3 \beta_1 - 6) q^{78} + (6 \beta_1 - 2) q^{79} + q^{80} + ( - \beta_{2} - 3 \beta_1 + 1) q^{81} + (3 \beta_{2} + \beta_1 - 2) q^{82} + (2 \beta_{2} - \beta_1 + 3) q^{83} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{84} + ( - \beta_{2} + \beta_1) q^{85} + (2 \beta_{2} - 3 \beta_1 - 3) q^{86} + ( - \beta_{2} + 5 \beta_1 + 4) q^{87} + (\beta_{2} + 2 \beta_1 - 1) q^{88} + (2 \beta_{2} - 2 \beta_1 + 6) q^{89} + (\beta_{2} - \beta_1 + 1) q^{90} + (4 \beta_{2} - 4 \beta_1 + 4) q^{91} + ( - \beta_{2} - \beta_1) q^{92} + ( - \beta_1 + 1) q^{93} + ( - \beta_{2} - 3 \beta_1 + 8) q^{94} + (2 \beta_{2} + 2) q^{95} + ( - \beta_1 + 1) q^{96} + ( - \beta_{2} + 3 \beta_1 - 4) q^{97} + ( - 4 \beta_1 + 5) q^{98} + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{99}+O(q^{100})$$ q + q^2 + (-b1 + 1) * q^3 + q^4 + q^5 + (-b1 + 1) * q^6 + (-b2 + b1) * q^7 + q^8 + (b2 - b1 + 1) * q^9 + q^10 + (b2 + 2*b1 - 1) * q^11 + (-b1 + 1) * q^12 + (b1 - 3) * q^13 + (-b2 + b1) * q^14 + (-b1 + 1) * q^15 + q^16 + (-b2 + b1) * q^17 + (b2 - b1 + 1) * q^18 + (2*b2 + 2) * q^19 + q^20 + (-2*b2 + 2*b1 - 4) * q^21 + (b2 + 2*b1 - 1) * q^22 + (-b2 - b1) * q^23 + (-b1 + 1) * q^24 + q^25 + (b1 - 3) * q^26 + (2*b2 + 2) * q^27 + (-b2 + b1) * q^28 + (-3*b2 - 2*b1 + 1) * q^29 + (-b1 + 1) * q^30 + q^31 + q^32 + (-b2 - b1 - 6) * q^33 + (-b2 + b1) * q^34 + (-b2 + b1) * q^35 + (b2 - b1 + 1) * q^36 + (2*b2 - b1 - 3) * q^37 + (2*b2 + 2) * q^38 + (-b2 + 3*b1 - 6) * q^39 + q^40 + (3*b2 + b1 - 2) * q^41 + (-2*b2 + 2*b1 - 4) * q^42 + (2*b2 - 3*b1 - 3) * q^43 + (b2 + 2*b1 - 1) * q^44 + (b2 - b1 + 1) * q^45 + (-b2 - b1) * q^46 + (-b2 - 3*b1 + 8) * q^47 + (-b1 + 1) * q^48 + (-4*b1 + 5) * q^49 + q^50 + (-2*b2 + 2*b1 - 4) * q^51 + (b1 - 3) * q^52 + (2*b2 - b1 - 7) * q^53 + (2*b2 + 2) * q^54 + (b2 + 2*b1 - 1) * q^55 + (-b2 + b1) * q^56 + (2*b2 - 6*b1 + 4) * q^57 + (-3*b2 - 2*b1 + 1) * q^58 + (2*b1 + 6) * q^59 + (-b1 + 1) * q^60 + (-b2 - 2*b1 - 5) * q^61 + q^62 + (-b2 + 5*b1 - 12) * q^63 + q^64 + (b1 - 3) * q^65 + (-b2 - b1 - 6) * q^66 + (-4*b2 - 2*b1 - 2) * q^67 + (-b2 + b1) * q^68 + (2*b1 + 2) * q^69 + (-b2 + b1) * q^70 + (-2*b2 - 2*b1 + 4) * q^71 + (b2 - b1 + 1) * q^72 + (-3*b2 + b1 - 6) * q^73 + (2*b2 - b1 - 3) * q^74 + (-b1 + 1) * q^75 + (2*b2 + 2) * q^76 + 4*b2 * q^77 + (-b2 + 3*b1 - 6) * q^78 + (6*b1 - 2) * q^79 + q^80 + (-b2 - 3*b1 + 1) * q^81 + (3*b2 + b1 - 2) * q^82 + (2*b2 - b1 + 3) * q^83 + (-2*b2 + 2*b1 - 4) * q^84 + (-b2 + b1) * q^85 + (2*b2 - 3*b1 - 3) * q^86 + (-b2 + 5*b1 + 4) * q^87 + (b2 + 2*b1 - 1) * q^88 + (2*b2 - 2*b1 + 6) * q^89 + (b2 - b1 + 1) * q^90 + (4*b2 - 4*b1 + 4) * q^91 + (-b2 - b1) * q^92 + (-b1 + 1) * q^93 + (-b2 - 3*b1 + 8) * q^94 + (2*b2 + 2) * q^95 + (-b1 + 1) * q^96 + (-b2 + 3*b1 - 4) * q^97 + (-4*b1 + 5) * q^98 + (-3*b2 + 2*b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} + 2 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^5 + 2 * q^6 + 3 * q^8 + 3 * q^9 $$3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} + 2 q^{6} + 3 q^{8} + 3 q^{9} + 3 q^{10} + 2 q^{12} - 8 q^{13} + 2 q^{15} + 3 q^{16} + 3 q^{18} + 8 q^{19} + 3 q^{20} - 12 q^{21} - 2 q^{23} + 2 q^{24} + 3 q^{25} - 8 q^{26} + 8 q^{27} - 2 q^{29} + 2 q^{30} + 3 q^{31} + 3 q^{32} - 20 q^{33} + 3 q^{36} - 8 q^{37} + 8 q^{38} - 16 q^{39} + 3 q^{40} - 2 q^{41} - 12 q^{42} - 10 q^{43} + 3 q^{45} - 2 q^{46} + 20 q^{47} + 2 q^{48} + 11 q^{49} + 3 q^{50} - 12 q^{51} - 8 q^{52} - 20 q^{53} + 8 q^{54} + 8 q^{57} - 2 q^{58} + 20 q^{59} + 2 q^{60} - 18 q^{61} + 3 q^{62} - 32 q^{63} + 3 q^{64} - 8 q^{65} - 20 q^{66} - 12 q^{67} + 8 q^{69} + 8 q^{71} + 3 q^{72} - 20 q^{73} - 8 q^{74} + 2 q^{75} + 8 q^{76} + 4 q^{77} - 16 q^{78} + 3 q^{80} - q^{81} - 2 q^{82} + 10 q^{83} - 12 q^{84} - 10 q^{86} + 16 q^{87} + 18 q^{89} + 3 q^{90} + 12 q^{91} - 2 q^{92} + 2 q^{93} + 20 q^{94} + 8 q^{95} + 2 q^{96} - 10 q^{97} + 11 q^{98} - 4 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^5 + 2 * q^6 + 3 * q^8 + 3 * q^9 + 3 * q^10 + 2 * q^12 - 8 * q^13 + 2 * q^15 + 3 * q^16 + 3 * q^18 + 8 * q^19 + 3 * q^20 - 12 * q^21 - 2 * q^23 + 2 * q^24 + 3 * q^25 - 8 * q^26 + 8 * q^27 - 2 * q^29 + 2 * q^30 + 3 * q^31 + 3 * q^32 - 20 * q^33 + 3 * q^36 - 8 * q^37 + 8 * q^38 - 16 * q^39 + 3 * q^40 - 2 * q^41 - 12 * q^42 - 10 * q^43 + 3 * q^45 - 2 * q^46 + 20 * q^47 + 2 * q^48 + 11 * q^49 + 3 * q^50 - 12 * q^51 - 8 * q^52 - 20 * q^53 + 8 * q^54 + 8 * q^57 - 2 * q^58 + 20 * q^59 + 2 * q^60 - 18 * q^61 + 3 * q^62 - 32 * q^63 + 3 * q^64 - 8 * q^65 - 20 * q^66 - 12 * q^67 + 8 * q^69 + 8 * q^71 + 3 * q^72 - 20 * q^73 - 8 * q^74 + 2 * q^75 + 8 * q^76 + 4 * q^77 - 16 * q^78 + 3 * q^80 - q^81 - 2 * q^82 + 10 * q^83 - 12 * q^84 - 10 * q^86 + 16 * q^87 + 18 * q^89 + 3 * q^90 + 12 * q^91 - 2 * q^92 + 2 * q^93 + 20 * q^94 + 8 * q^95 + 2 * q^96 - 10 * q^97 + 11 * q^98 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2\nu + 2$$ -v^2 + 2*v + 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 2$$ b1 + 2

## 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
 2.17009 −1.48119 0.311108
1.00000 −1.70928 1.00000 1.00000 −1.70928 1.07838 1.00000 −0.0783777 1.00000
1.2 1.00000 0.806063 1.00000 1.00000 0.806063 3.35026 1.00000 −2.35026 1.00000
1.3 1.00000 2.90321 1.00000 1.00000 2.90321 −4.42864 1.00000 5.42864 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$31$$ $$-1$$

## 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 310.2.a.e 3
3.b odd 2 1 2790.2.a.bi 3
4.b odd 2 1 2480.2.a.u 3
5.b even 2 1 1550.2.a.k 3
5.c odd 4 2 1550.2.b.j 6
8.b even 2 1 9920.2.a.bw 3
8.d odd 2 1 9920.2.a.bx 3
31.b odd 2 1 9610.2.a.u 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.a.e 3 1.a even 1 1 trivial
1550.2.a.k 3 5.b even 2 1
1550.2.b.j 6 5.c odd 4 2
2480.2.a.u 3 4.b odd 2 1
2790.2.a.bi 3 3.b odd 2 1
9610.2.a.u 3 31.b odd 2 1
9920.2.a.bw 3 8.b even 2 1
9920.2.a.bx 3 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} - 2T_{3}^{2} - 4T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(310))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{3}$$
$3$ $$T^{3} - 2 T^{2} + \cdots + 4$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - 16T + 16$$
$11$ $$T^{3} - 28T - 52$$
$13$ $$T^{3} + 8 T^{2} + \cdots + 4$$
$17$ $$T^{3} - 16T + 16$$
$19$ $$T^{3} - 8 T^{2} + \cdots + 160$$
$23$ $$T^{3} + 2 T^{2} + \cdots - 8$$
$29$ $$T^{3} + 2 T^{2} + \cdots - 260$$
$31$ $$(T - 1)^{3}$$
$37$ $$T^{3} + 8 T^{2} + \cdots - 92$$
$41$ $$T^{3} + 2 T^{2} + \cdots + 232$$
$43$ $$T^{3} + 10 T^{2} + \cdots - 604$$
$47$ $$T^{3} - 20 T^{2} + \cdots + 208$$
$53$ $$T^{3} + 20 T^{2} + \cdots + 4$$
$59$ $$T^{3} - 20 T^{2} + \cdots - 160$$
$61$ $$T^{3} + 18 T^{2} + \cdots + 100$$
$67$ $$T^{3} + 12 T^{2} + \cdots - 1184$$
$71$ $$T^{3} - 8 T^{2} + \cdots + 128$$
$73$ $$T^{3} + 20 T^{2} + \cdots - 464$$
$79$ $$T^{3} - 192T - 160$$
$83$ $$T^{3} - 10 T^{2} + \cdots + 124$$
$89$ $$T^{3} - 18 T^{2} + \cdots + 40$$
$97$ $$T^{3} + 10 T^{2} + \cdots - 8$$