# Properties

 Label 310.2.a.b Level $310$ Weight $2$ Character orbit 310.a Self dual yes Analytic conductor $2.475$ 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] = [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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + 2 q^{3} + q^{4} - q^{5} + 2 q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + 2 * q^3 + q^4 - q^5 + 2 * q^6 + q^8 + q^9 $$q + q^{2} + 2 q^{3} + q^{4} - q^{5} + 2 q^{6} + q^{8} + q^{9} - q^{10} + 2 q^{11} + 2 q^{12} - 2 q^{15} + q^{16} + 2 q^{17} + q^{18} - 4 q^{19} - q^{20} + 2 q^{22} - 4 q^{23} + 2 q^{24} + q^{25} - 4 q^{27} - 4 q^{29} - 2 q^{30} - q^{31} + q^{32} + 4 q^{33} + 2 q^{34} + q^{36} - 8 q^{37} - 4 q^{38} - q^{40} + 6 q^{41} + 2 q^{43} + 2 q^{44} - q^{45} - 4 q^{46} + 2 q^{48} - 7 q^{49} + q^{50} + 4 q^{51} + 8 q^{53} - 4 q^{54} - 2 q^{55} - 8 q^{57} - 4 q^{58} + 8 q^{59} - 2 q^{60} - q^{62} + q^{64} + 4 q^{66} + 4 q^{67} + 2 q^{68} - 8 q^{69} + q^{72} + 6 q^{73} - 8 q^{74} + 2 q^{75} - 4 q^{76} - 4 q^{79} - q^{80} - 11 q^{81} + 6 q^{82} + 6 q^{83} - 2 q^{85} + 2 q^{86} - 8 q^{87} + 2 q^{88} - 6 q^{89} - q^{90} - 4 q^{92} - 2 q^{93} + 4 q^{95} + 2 q^{96} - 2 q^{97} - 7 q^{98} + 2 q^{99}+O(q^{100})$$ q + q^2 + 2 * q^3 + q^4 - q^5 + 2 * q^6 + q^8 + q^9 - q^10 + 2 * q^11 + 2 * q^12 - 2 * q^15 + q^16 + 2 * q^17 + q^18 - 4 * q^19 - q^20 + 2 * q^22 - 4 * q^23 + 2 * q^24 + q^25 - 4 * q^27 - 4 * q^29 - 2 * q^30 - q^31 + q^32 + 4 * q^33 + 2 * q^34 + q^36 - 8 * q^37 - 4 * q^38 - q^40 + 6 * q^41 + 2 * q^43 + 2 * q^44 - q^45 - 4 * q^46 + 2 * q^48 - 7 * q^49 + q^50 + 4 * q^51 + 8 * q^53 - 4 * q^54 - 2 * q^55 - 8 * q^57 - 4 * q^58 + 8 * q^59 - 2 * q^60 - q^62 + q^64 + 4 * q^66 + 4 * q^67 + 2 * q^68 - 8 * q^69 + q^72 + 6 * q^73 - 8 * q^74 + 2 * q^75 - 4 * q^76 - 4 * q^79 - q^80 - 11 * q^81 + 6 * q^82 + 6 * q^83 - 2 * q^85 + 2 * q^86 - 8 * q^87 + 2 * q^88 - 6 * q^89 - q^90 - 4 * q^92 - 2 * q^93 + 4 * q^95 + 2 * q^96 - 2 * q^97 - 7 * q^98 + 2 * 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 2.00000 1.00000 −1.00000 2.00000 0 1.00000 1.00000 −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
$$2$$ $$-1$$
$$5$$ $$1$$
$$31$$ $$1$$

## 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.b 1
3.b odd 2 1 2790.2.a.h 1
4.b odd 2 1 2480.2.a.c 1
5.b even 2 1 1550.2.a.a 1
5.c odd 4 2 1550.2.b.e 2
8.b even 2 1 9920.2.a.d 1
8.d odd 2 1 9920.2.a.bg 1
31.b odd 2 1 9610.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.a.b 1 1.a even 1 1 trivial
1550.2.a.a 1 5.b even 2 1
1550.2.b.e 2 5.c odd 4 2
2480.2.a.c 1 4.b odd 2 1
2790.2.a.h 1 3.b odd 2 1
9610.2.a.a 1 31.b odd 2 1
9920.2.a.d 1 8.b even 2 1
9920.2.a.bg 1 8.d odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

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