# Properties

 Label 310.2.a.a Level $310$ Weight $2$ Character orbit 310.a Self dual yes Analytic conductor $2.475$ 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] = [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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - 2 q^{3} + q^{4} - q^{5} - 2 q^{6} - 4 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - 2 * q^3 + q^4 - q^5 - 2 * q^6 - 4 * q^7 + q^8 + q^9 $$q + q^{2} - 2 q^{3} + q^{4} - q^{5} - 2 q^{6} - 4 q^{7} + q^{8} + q^{9} - q^{10} - 2 q^{12} - 4 q^{13} - 4 q^{14} + 2 q^{15} + q^{16} + q^{18} - 4 q^{19} - q^{20} + 8 q^{21} - 6 q^{23} - 2 q^{24} + q^{25} - 4 q^{26} + 4 q^{27} - 4 q^{28} + 6 q^{29} + 2 q^{30} + q^{31} + q^{32} + 4 q^{35} + q^{36} + 8 q^{37} - 4 q^{38} + 8 q^{39} - q^{40} - 6 q^{41} + 8 q^{42} - 10 q^{43} - q^{45} - 6 q^{46} - 2 q^{48} + 9 q^{49} + q^{50} - 4 q^{52} + 4 q^{54} - 4 q^{56} + 8 q^{57} + 6 q^{58} - 12 q^{59} + 2 q^{60} + 14 q^{61} + q^{62} - 4 q^{63} + q^{64} + 4 q^{65} + 8 q^{67} + 12 q^{69} + 4 q^{70} + q^{72} - 4 q^{73} + 8 q^{74} - 2 q^{75} - 4 q^{76} + 8 q^{78} + 8 q^{79} - q^{80} - 11 q^{81} - 6 q^{82} + 6 q^{83} + 8 q^{84} - 10 q^{86} - 12 q^{87} - 18 q^{89} - q^{90} + 16 q^{91} - 6 q^{92} - 2 q^{93} + 4 q^{95} - 2 q^{96} - 10 q^{97} + 9 q^{98}+O(q^{100})$$ q + q^2 - 2 * q^3 + q^4 - q^5 - 2 * q^6 - 4 * q^7 + q^8 + q^9 - q^10 - 2 * q^12 - 4 * q^13 - 4 * q^14 + 2 * q^15 + q^16 + q^18 - 4 * q^19 - q^20 + 8 * q^21 - 6 * q^23 - 2 * q^24 + q^25 - 4 * q^26 + 4 * q^27 - 4 * q^28 + 6 * q^29 + 2 * q^30 + q^31 + q^32 + 4 * q^35 + q^36 + 8 * q^37 - 4 * q^38 + 8 * q^39 - q^40 - 6 * q^41 + 8 * q^42 - 10 * q^43 - q^45 - 6 * q^46 - 2 * q^48 + 9 * q^49 + q^50 - 4 * q^52 + 4 * q^54 - 4 * q^56 + 8 * q^57 + 6 * q^58 - 12 * q^59 + 2 * q^60 + 14 * q^61 + q^62 - 4 * q^63 + q^64 + 4 * q^65 + 8 * q^67 + 12 * q^69 + 4 * q^70 + q^72 - 4 * q^73 + 8 * q^74 - 2 * q^75 - 4 * q^76 + 8 * q^78 + 8 * q^79 - q^80 - 11 * q^81 - 6 * q^82 + 6 * q^83 + 8 * q^84 - 10 * q^86 - 12 * q^87 - 18 * q^89 - q^90 + 16 * q^91 - 6 * q^92 - 2 * q^93 + 4 * q^95 - 2 * q^96 - 10 * q^97 + 9 * q^98

## 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 −4.00000 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.a 1
3.b odd 2 1 2790.2.a.e 1
4.b odd 2 1 2480.2.a.n 1
5.b even 2 1 1550.2.a.e 1
5.c odd 4 2 1550.2.b.a 2
8.b even 2 1 9920.2.a.bc 1
8.d odd 2 1 9920.2.a.h 1
31.b odd 2 1 9610.2.a.b 1

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

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 2$$
$5$ $$T + 1$$
$7$ $$T + 4$$
$11$ $$T$$
$13$ $$T + 4$$
$17$ $$T$$
$19$ $$T + 4$$
$23$ $$T + 6$$
$29$ $$T - 6$$
$31$ $$T - 1$$
$37$ $$T - 8$$
$41$ $$T + 6$$
$43$ $$T + 10$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T + 12$$
$61$ $$T - 14$$
$67$ $$T - 8$$
$71$ $$T$$
$73$ $$T + 4$$
$79$ $$T - 8$$
$83$ $$T - 6$$
$89$ $$T + 18$$
$97$ $$T + 10$$