# Properties

 Label 37.2.e.a Level $37$ Weight $2$ Character orbit 37.e Analytic conductor $0.295$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,2,Mod(11,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

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

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

chi := DirichletCharacter("37.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 37.e (of order $$6$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$0.295446487479$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/37\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 1.36603 2.36603i −0.500000 + 0.866025i 0.232051 + 0.133975i 2.73205i −1.73205 + 3.00000i 3.00000i −2.23205 3.86603i −0.267949
11.2 0.866025 0.500000i −0.366025 + 0.633975i −0.500000 + 0.866025i −3.23205 1.86603i 0.732051i 1.73205 3.00000i 3.00000i 1.23205 + 2.13397i −3.73205
27.1 −0.866025 0.500000i 1.36603 + 2.36603i −0.500000 0.866025i 0.232051 0.133975i 2.73205i −1.73205 3.00000i 3.00000i −2.23205 + 3.86603i −0.267949
27.2 0.866025 + 0.500000i −0.366025 0.633975i −0.500000 0.866025i −3.23205 + 1.86603i 0.732051i 1.73205 + 3.00000i 3.00000i 1.23205 2.13397i −3.73205
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.2.e.a 4
3.b odd 2 1 333.2.s.b 4
4.b odd 2 1 592.2.w.c 4
5.b even 2 1 925.2.n.a 4
5.c odd 4 1 925.2.m.a 4
5.c odd 4 1 925.2.m.b 4
37.c even 3 1 1369.2.b.d 4
37.e even 6 1 inner 37.2.e.a 4
37.e even 6 1 1369.2.b.d 4
37.g odd 12 1 1369.2.a.g 2
37.g odd 12 1 1369.2.a.h 2
111.h odd 6 1 333.2.s.b 4
148.j odd 6 1 592.2.w.c 4
185.l even 6 1 925.2.n.a 4
185.r odd 12 1 925.2.m.a 4
185.r odd 12 1 925.2.m.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.e.a 4 1.a even 1 1 trivial
37.2.e.a 4 37.e even 6 1 inner
333.2.s.b 4 3.b odd 2 1
333.2.s.b 4 111.h odd 6 1
592.2.w.c 4 4.b odd 2 1
592.2.w.c 4 148.j odd 6 1
925.2.m.a 4 5.c odd 4 1
925.2.m.a 4 185.r odd 12 1
925.2.m.b 4 5.c odd 4 1
925.2.m.b 4 185.r odd 12 1
925.2.n.a 4 5.b even 2 1
925.2.n.a 4 185.l even 6 1
1369.2.a.g 2 37.g odd 12 1
1369.2.a.h 2 37.g odd 12 1
1369.2.b.d 4 37.c even 3 1
1369.2.b.d 4 37.e even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(37, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4} - 2 T^{3} + \cdots + 4$$
$5$ $$T^{4} + 6 T^{3} + \cdots + 1$$
$7$ $$T^{4} + 12T^{2} + 144$$
$11$ $$(T^{2} + 6 T + 6)^{2}$$
$13$ $$(T^{2} - 6 T + 12)^{2}$$
$17$ $$T^{4} - 6 T^{3} + \cdots + 1$$
$19$ $$T^{4} + 6 T^{3} + \cdots + 36$$
$23$ $$T^{4} + 8T^{2} + 4$$
$29$ $$T^{4} + 14T^{2} + 1$$
$31$ $$T^{4} + 72T^{2} + 324$$
$37$ $$(T^{2} - T + 37)^{2}$$
$41$ $$(T^{2} + 3 T + 9)^{2}$$
$43$ $$(T^{2} + 12)^{2}$$
$47$ $$(T^{2} - 6 T - 18)^{2}$$
$53$ $$T^{4} - 12 T^{3} + \cdots + 576$$
$59$ $$T^{4} + 12 T^{3} + \cdots + 7744$$
$61$ $$T^{4} + 6 T^{3} + \cdots + 1089$$
$67$ $$T^{4} - 18 T^{3} + \cdots + 6084$$
$71$ $$(T^{2} + 6 T + 36)^{2}$$
$73$ $$T^{4}$$
$79$ $$T^{4} + 18 T^{3} + \cdots + 2916$$
$83$ $$T^{4} - 6 T^{3} + \cdots + 324$$
$89$ $$T^{4} - 18 T^{3} + \cdots + 121$$
$97$ $$T^{4} + 78T^{2} + 1089$$