Properties

 Label 370.2.a.b Level $370$ Weight $2$ Character orbit 370.a Self dual yes Analytic conductor $2.954$ 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] = [370,2,Mod(1,370)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(370, 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("370.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.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: $$2.95446487479$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{5} - q^{8} - 3 q^{9}+O(q^{10})$$ q - q^2 + q^4 - q^5 - q^8 - 3 * q^9 $$q - q^{2} + q^{4} - q^{5} - q^{8} - 3 q^{9} + q^{10} - 4 q^{11} + 2 q^{13} + q^{16} - 2 q^{17} + 3 q^{18} - 4 q^{19} - q^{20} + 4 q^{22} + q^{25} - 2 q^{26} - 6 q^{29} - 4 q^{31} - q^{32} + 2 q^{34} - 3 q^{36} - q^{37} + 4 q^{38} + q^{40} - 6 q^{41} + 4 q^{43} - 4 q^{44} + 3 q^{45} - 8 q^{47} - 7 q^{49} - q^{50} + 2 q^{52} + 10 q^{53} + 4 q^{55} + 6 q^{58} + 4 q^{59} + 10 q^{61} + 4 q^{62} + q^{64} - 2 q^{65} - 8 q^{67} - 2 q^{68} + 3 q^{72} + 10 q^{73} + q^{74} - 4 q^{76} - 4 q^{79} - q^{80} + 9 q^{81} + 6 q^{82} + 2 q^{85} - 4 q^{86} + 4 q^{88} + 2 q^{89} - 3 q^{90} + 8 q^{94} + 4 q^{95} + 6 q^{97} + 7 q^{98} + 12 q^{99}+O(q^{100})$$ q - q^2 + q^4 - q^5 - q^8 - 3 * q^9 + q^10 - 4 * q^11 + 2 * q^13 + q^16 - 2 * q^17 + 3 * q^18 - 4 * q^19 - q^20 + 4 * q^22 + q^25 - 2 * q^26 - 6 * q^29 - 4 * q^31 - q^32 + 2 * q^34 - 3 * q^36 - q^37 + 4 * q^38 + q^40 - 6 * q^41 + 4 * q^43 - 4 * q^44 + 3 * q^45 - 8 * q^47 - 7 * q^49 - q^50 + 2 * q^52 + 10 * q^53 + 4 * q^55 + 6 * q^58 + 4 * q^59 + 10 * q^61 + 4 * q^62 + q^64 - 2 * q^65 - 8 * q^67 - 2 * q^68 + 3 * q^72 + 10 * q^73 + q^74 - 4 * q^76 - 4 * q^79 - q^80 + 9 * q^81 + 6 * q^82 + 2 * q^85 - 4 * q^86 + 4 * q^88 + 2 * q^89 - 3 * q^90 + 8 * q^94 + 4 * q^95 + 6 * q^97 + 7 * q^98 + 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
 0
−1.00000 0 1.00000 −1.00000 0 0 −1.00000 −3.00000 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
$$2$$ $$+1$$
$$5$$ $$+1$$
$$37$$ $$+1$$

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 370.2.a.b 1
3.b odd 2 1 3330.2.a.w 1
4.b odd 2 1 2960.2.a.g 1
5.b even 2 1 1850.2.a.k 1
5.c odd 4 2 1850.2.b.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.b 1 1.a even 1 1 trivial
1850.2.a.k 1 5.b even 2 1
1850.2.b.d 2 5.c odd 4 2
2960.2.a.g 1 4.b odd 2 1
3330.2.a.w 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(370))$$:

 $$T_{3}$$ T3 $$T_{7}$$ T7

Hecke characteristic polynomials

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