# Properties

 Label 37.2.a.b Level $37$ Weight $2$ Character orbit 37.a Self dual yes Analytic conductor $0.295$ 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] = [37,2,Mod(1,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("37.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 37.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: $$0.295446487479$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 q^{4} - q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 - 2 * q^4 - q^7 - 2 * q^9 $$q + q^{3} - 2 q^{4} - q^{7} - 2 q^{9} + 3 q^{11} - 2 q^{12} - 4 q^{13} + 4 q^{16} + 6 q^{17} + 2 q^{19} - q^{21} + 6 q^{23} - 5 q^{25} - 5 q^{27} + 2 q^{28} - 6 q^{29} - 4 q^{31} + 3 q^{33} + 4 q^{36} + q^{37} - 4 q^{39} - 9 q^{41} + 8 q^{43} - 6 q^{44} + 3 q^{47} + 4 q^{48} - 6 q^{49} + 6 q^{51} + 8 q^{52} - 3 q^{53} + 2 q^{57} + 12 q^{59} + 8 q^{61} + 2 q^{63} - 8 q^{64} - 4 q^{67} - 12 q^{68} + 6 q^{69} - 15 q^{71} + 11 q^{73} - 5 q^{75} - 4 q^{76} - 3 q^{77} - 10 q^{79} + q^{81} + 9 q^{83} + 2 q^{84} - 6 q^{87} + 6 q^{89} + 4 q^{91} - 12 q^{92} - 4 q^{93} + 8 q^{97} - 6 q^{99}+O(q^{100})$$ q + q^3 - 2 * q^4 - q^7 - 2 * q^9 + 3 * q^11 - 2 * q^12 - 4 * q^13 + 4 * q^16 + 6 * q^17 + 2 * q^19 - q^21 + 6 * q^23 - 5 * q^25 - 5 * q^27 + 2 * q^28 - 6 * q^29 - 4 * q^31 + 3 * q^33 + 4 * q^36 + q^37 - 4 * q^39 - 9 * q^41 + 8 * q^43 - 6 * q^44 + 3 * q^47 + 4 * q^48 - 6 * q^49 + 6 * q^51 + 8 * q^52 - 3 * q^53 + 2 * q^57 + 12 * q^59 + 8 * q^61 + 2 * q^63 - 8 * q^64 - 4 * q^67 - 12 * q^68 + 6 * q^69 - 15 * q^71 + 11 * q^73 - 5 * q^75 - 4 * q^76 - 3 * q^77 - 10 * q^79 + q^81 + 9 * q^83 + 2 * q^84 - 6 * q^87 + 6 * q^89 + 4 * q^91 - 12 * q^92 - 4 * q^93 + 8 * q^97 - 6 * 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
0 1.00000 −2.00000 0 0 −1.00000 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$37$$ $$-1$$

## 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 37.2.a.b 1
3.b odd 2 1 333.2.a.b 1
4.b odd 2 1 592.2.a.a 1
5.b even 2 1 925.2.a.b 1
5.c odd 4 2 925.2.b.e 2
7.b odd 2 1 1813.2.a.b 1
8.b even 2 1 2368.2.a.d 1
8.d odd 2 1 2368.2.a.m 1
11.b odd 2 1 4477.2.a.a 1
12.b even 2 1 5328.2.a.k 1
13.b even 2 1 6253.2.a.b 1
15.d odd 2 1 8325.2.a.p 1
37.b even 2 1 1369.2.a.c 1
37.d odd 4 2 1369.2.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.a.b 1 1.a even 1 1 trivial
333.2.a.b 1 3.b odd 2 1
592.2.a.a 1 4.b odd 2 1
925.2.a.b 1 5.b even 2 1
925.2.b.e 2 5.c odd 4 2
1369.2.a.c 1 37.b even 2 1
1369.2.b.a 2 37.d odd 4 2
1813.2.a.b 1 7.b odd 2 1
2368.2.a.d 1 8.b even 2 1
2368.2.a.m 1 8.d odd 2 1
4477.2.a.a 1 11.b odd 2 1
5328.2.a.k 1 12.b even 2 1
6253.2.a.b 1 13.b even 2 1
8325.2.a.p 1 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(37))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T - 3$$
$13$ $$T + 4$$
$17$ $$T - 6$$
$19$ $$T - 2$$
$23$ $$T - 6$$
$29$ $$T + 6$$
$31$ $$T + 4$$
$37$ $$T - 1$$
$41$ $$T + 9$$
$43$ $$T - 8$$
$47$ $$T - 3$$
$53$ $$T + 3$$
$59$ $$T - 12$$
$61$ $$T - 8$$
$67$ $$T + 4$$
$71$ $$T + 15$$
$73$ $$T - 11$$
$79$ $$T + 10$$
$83$ $$T - 9$$
$89$ $$T - 6$$
$97$ $$T - 8$$