# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,2,Mod(10,37)]

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([4]))

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

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

chi := DirichletCharacter("37.10");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 37.c (of order $$3$$, 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + \zeta_{6} q^{4} - \zeta_{6} q^{5} - 2 \zeta_{6} q^{7} - 3 q^{8} + ( - 3 \zeta_{6} + 3) q^{9} +O(q^{10})$$ q + (z - 1) * q^2 + z * q^4 - z * q^5 - 2*z * q^7 - 3 * q^8 + (-3*z + 3) * q^9 $$q + (\zeta_{6} - 1) q^{2} + \zeta_{6} q^{4} - \zeta_{6} q^{5} - 2 \zeta_{6} q^{7} - 3 q^{8} + ( - 3 \zeta_{6} + 3) q^{9} + q^{10} - 2 q^{11} + 2 \zeta_{6} q^{13} + 2 q^{14} + ( - \zeta_{6} + 1) q^{16} + (3 \zeta_{6} - 3) q^{17} + 3 \zeta_{6} q^{18} + 6 \zeta_{6} q^{19} + ( - \zeta_{6} + 1) q^{20} + ( - 2 \zeta_{6} + 2) q^{22} - 4 q^{23} + ( - 4 \zeta_{6} + 4) q^{25} - 2 q^{26} + ( - 2 \zeta_{6} + 2) q^{28} + 9 q^{29} - 10 q^{31} - 5 \zeta_{6} q^{32} - 3 \zeta_{6} q^{34} + (2 \zeta_{6} - 2) q^{35} + 3 q^{36} + ( - 3 \zeta_{6} - 4) q^{37} - 6 q^{38} + 3 \zeta_{6} q^{40} + 9 \zeta_{6} q^{41} + 2 q^{43} - 2 \zeta_{6} q^{44} - 3 q^{45} + ( - 4 \zeta_{6} + 4) q^{46} + 6 q^{47} + ( - 3 \zeta_{6} + 3) q^{49} + 4 \zeta_{6} q^{50} + (2 \zeta_{6} - 2) q^{52} + ( - 2 \zeta_{6} + 2) q^{53} + 2 \zeta_{6} q^{55} + 6 \zeta_{6} q^{56} + (9 \zeta_{6} - 9) q^{58} + ( - 4 \zeta_{6} + 4) q^{59} - \zeta_{6} q^{61} + ( - 10 \zeta_{6} + 10) q^{62} - 6 q^{63} + 7 q^{64} + ( - 2 \zeta_{6} + 2) q^{65} + 10 \zeta_{6} q^{67} - 3 q^{68} - 2 \zeta_{6} q^{70} - 6 \zeta_{6} q^{71} + (9 \zeta_{6} - 9) q^{72} - 10 q^{73} + ( - 4 \zeta_{6} + 7) q^{74} + (6 \zeta_{6} - 6) q^{76} + 4 \zeta_{6} q^{77} - 10 \zeta_{6} q^{79} - q^{80} - 9 \zeta_{6} q^{81} - 9 q^{82} + ( - 12 \zeta_{6} + 12) q^{83} + 3 q^{85} + (2 \zeta_{6} - 2) q^{86} + 6 q^{88} + (7 \zeta_{6} - 7) q^{89} + ( - 3 \zeta_{6} + 3) q^{90} + ( - 4 \zeta_{6} + 4) q^{91} - 4 \zeta_{6} q^{92} + (6 \zeta_{6} - 6) q^{94} + ( - 6 \zeta_{6} + 6) q^{95} + 7 q^{97} + 3 \zeta_{6} q^{98} + (6 \zeta_{6} - 6) q^{99} +O(q^{100})$$ q + (z - 1) * q^2 + z * q^4 - z * q^5 - 2*z * q^7 - 3 * q^8 + (-3*z + 3) * q^9 + q^10 - 2 * q^11 + 2*z * q^13 + 2 * q^14 + (-z + 1) * q^16 + (3*z - 3) * q^17 + 3*z * q^18 + 6*z * q^19 + (-z + 1) * q^20 + (-2*z + 2) * q^22 - 4 * q^23 + (-4*z + 4) * q^25 - 2 * q^26 + (-2*z + 2) * q^28 + 9 * q^29 - 10 * q^31 - 5*z * q^32 - 3*z * q^34 + (2*z - 2) * q^35 + 3 * q^36 + (-3*z - 4) * q^37 - 6 * q^38 + 3*z * q^40 + 9*z * q^41 + 2 * q^43 - 2*z * q^44 - 3 * q^45 + (-4*z + 4) * q^46 + 6 * q^47 + (-3*z + 3) * q^49 + 4*z * q^50 + (2*z - 2) * q^52 + (-2*z + 2) * q^53 + 2*z * q^55 + 6*z * q^56 + (9*z - 9) * q^58 + (-4*z + 4) * q^59 - z * q^61 + (-10*z + 10) * q^62 - 6 * q^63 + 7 * q^64 + (-2*z + 2) * q^65 + 10*z * q^67 - 3 * q^68 - 2*z * q^70 - 6*z * q^71 + (9*z - 9) * q^72 - 10 * q^73 + (-4*z + 7) * q^74 + (6*z - 6) * q^76 + 4*z * q^77 - 10*z * q^79 - q^80 - 9*z * q^81 - 9 * q^82 + (-12*z + 12) * q^83 + 3 * q^85 + (2*z - 2) * q^86 + 6 * q^88 + (7*z - 7) * q^89 + (-3*z + 3) * q^90 + (-4*z + 4) * q^91 - 4*z * q^92 + (6*z - 6) * q^94 + (-6*z + 6) * q^95 + 7 * q^97 + 3*z * q^98 + (6*z - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{4} - q^{5} - 2 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q - q^2 + q^4 - q^5 - 2 * q^7 - 6 * q^8 + 3 * q^9 $$2 q - q^{2} + q^{4} - q^{5} - 2 q^{7} - 6 q^{8} + 3 q^{9} + 2 q^{10} - 4 q^{11} + 2 q^{13} + 4 q^{14} + q^{16} - 3 q^{17} + 3 q^{18} + 6 q^{19} + q^{20} + 2 q^{22} - 8 q^{23} + 4 q^{25} - 4 q^{26} + 2 q^{28} + 18 q^{29} - 20 q^{31} - 5 q^{32} - 3 q^{34} - 2 q^{35} + 6 q^{36} - 11 q^{37} - 12 q^{38} + 3 q^{40} + 9 q^{41} + 4 q^{43} - 2 q^{44} - 6 q^{45} + 4 q^{46} + 12 q^{47} + 3 q^{49} + 4 q^{50} - 2 q^{52} + 2 q^{53} + 2 q^{55} + 6 q^{56} - 9 q^{58} + 4 q^{59} - q^{61} + 10 q^{62} - 12 q^{63} + 14 q^{64} + 2 q^{65} + 10 q^{67} - 6 q^{68} - 2 q^{70} - 6 q^{71} - 9 q^{72} - 20 q^{73} + 10 q^{74} - 6 q^{76} + 4 q^{77} - 10 q^{79} - 2 q^{80} - 9 q^{81} - 18 q^{82} + 12 q^{83} + 6 q^{85} - 2 q^{86} + 12 q^{88} - 7 q^{89} + 3 q^{90} + 4 q^{91} - 4 q^{92} - 6 q^{94} + 6 q^{95} + 14 q^{97} + 3 q^{98} - 6 q^{99}+O(q^{100})$$ 2 * q - q^2 + q^4 - q^5 - 2 * q^7 - 6 * q^8 + 3 * q^9 + 2 * q^10 - 4 * q^11 + 2 * q^13 + 4 * q^14 + q^16 - 3 * q^17 + 3 * q^18 + 6 * q^19 + q^20 + 2 * q^22 - 8 * q^23 + 4 * q^25 - 4 * q^26 + 2 * q^28 + 18 * q^29 - 20 * q^31 - 5 * q^32 - 3 * q^34 - 2 * q^35 + 6 * q^36 - 11 * q^37 - 12 * q^38 + 3 * q^40 + 9 * q^41 + 4 * q^43 - 2 * q^44 - 6 * q^45 + 4 * q^46 + 12 * q^47 + 3 * q^49 + 4 * q^50 - 2 * q^52 + 2 * q^53 + 2 * q^55 + 6 * q^56 - 9 * q^58 + 4 * q^59 - q^61 + 10 * q^62 - 12 * q^63 + 14 * q^64 + 2 * q^65 + 10 * q^67 - 6 * q^68 - 2 * q^70 - 6 * q^71 - 9 * q^72 - 20 * q^73 + 10 * q^74 - 6 * q^76 + 4 * q^77 - 10 * q^79 - 2 * q^80 - 9 * q^81 - 18 * q^82 + 12 * q^83 + 6 * q^85 - 2 * q^86 + 12 * q^88 - 7 * q^89 + 3 * q^90 + 4 * q^91 - 4 * q^92 - 6 * q^94 + 6 * q^95 + 14 * q^97 + 3 * q^98 - 6 * q^99

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 1.73205i −3.00000 1.50000 2.59808i 1.00000
26.1 −0.500000 0.866025i 0 0.500000 0.866025i −0.500000 + 0.866025i 0 −1.00000 + 1.73205i −3.00000 1.50000 + 2.59808i 1.00000
 $$n$$: 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.c even 3 1 inner

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} - 2T + 4$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} - 6T + 36$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T - 9)^{2}$$
$31$ $$(T + 10)^{2}$$
$37$ $$T^{2} + 11T + 37$$
$41$ $$T^{2} - 9T + 81$$
$43$ $$(T - 2)^{2}$$
$47$ $$(T - 6)^{2}$$
$53$ $$T^{2} - 2T + 4$$
$59$ $$T^{2} - 4T + 16$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} - 10T + 100$$
$71$ $$T^{2} + 6T + 36$$
$73$ $$(T + 10)^{2}$$
$79$ $$T^{2} + 10T + 100$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$T^{2} + 7T + 49$$
$97$ $$(T - 7)^{2}$$