# Properties

 Label 37.2.f.b Level $37$ Weight $2$ Character orbit 37.f Analytic conductor $0.295$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,2,Mod(7,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([16]))

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

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

chi := DirichletCharacter("37.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 37.f (of order $$9$$, degree $$6$$, 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: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-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 $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \cdots + 1) q^{2}+ \cdots + (\zeta_{18}^{5} - \zeta_{18}^{4} + \cdots + 1) q^{9}+O(q^{10})$$ q + (-z^5 + z^4 - z^3 + z^2 - z + 1) * q^2 + (-z^4 - 1) * q^3 + (-z^5 + z^4 + z^3 + z - 2) * q^4 + (z^5 - z^4 - z^2 + z + 1) * q^5 + (z^5 - z^4 - 1) * q^6 + (2*z^4 - 2*z - 2) * q^7 + (3*z^5 - 2*z^4 + 2*z^3 - 2*z^2 + 3*z) * q^8 + (z^5 - z^4 - z^2 + 1) * q^9 $$q + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \cdots + 1) q^{2}+ \cdots + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + \cdots + 1) q^{99}+O(q^{100})$$ q + (-z^5 + z^4 - z^3 + z^2 - z + 1) * q^2 + (-z^4 - 1) * q^3 + (-z^5 + z^4 + z^3 + z - 2) * q^4 + (z^5 - z^4 - z^2 + z + 1) * q^5 + (z^5 - z^4 - 1) * q^6 + (2*z^4 - 2*z - 2) * q^7 + (3*z^5 - 2*z^4 + 2*z^3 - 2*z^2 + 3*z) * q^8 + (z^5 - z^4 - z^2 + 1) * q^9 + (z^5 + z^4 - 3*z^3 + z^2 - 2*z + 3) * q^10 + (-2*z^5 + z^4 + 3*z^3 + z^2 - 2*z) * q^11 + (-z^5 - z^3 + z^2 + 1) * q^12 + (-3*z^5 - 2*z^4 - 2*z^3 + 3*z - 1) * q^13 + (4*z^3 - 2*z^2 + 2*z - 4) * q^14 + (-z^5 + z^3 - z - 1) * q^15 + (3*z^5 - 3*z^3 - z + 3) * q^16 + (2*z^4 - z^3 - z^2 + 1) * q^17 + (2*z^4 - 3*z^3 + 2*z^2 - 3*z + 2) * q^18 + (-2*z^4 + z^3 + 2*z^2 + z - 2) * q^19 + (-4*z^5 + 4*z^4 - z^3 + 3*z^2 - 3) * q^20 + (2*z^2 + 2*z + 2) * q^21 + (-3*z^5 + 3*z^3 + 3*z^2 - 3*z) * q^22 + (-6*z^3 + 6) * q^23 + (-4*z^5 - z + 1) * q^24 + (z^5 + 2*z^4 + 2*z^3 - 2*z^2 - 2*z - 1) * q^25 + (4*z^5 - 7*z^4 + 2*z^3 - 7*z^2 + 4*z) * q^26 + (3*z^5 + 3*z^4 + z^3 - 3*z^2 - 1) * q^27 + (4*z^5 - 6*z^4 - 4*z^2 + 4) * q^28 + (3*z^4 + 3*z^3 + 3*z^2) * q^29 + (-z^4 + 2*z^3 - z - 1) * q^30 + (-z^5 - 2*z^4 + 3*z^2 + 3*z - 3) * q^31 + (3*z^4 - 3*z - 3) * q^32 + (3*z^5 - 4*z^4 - 4*z^3 + 5*z - 1) * q^33 + (z^4 + z^3 - 3*z^2 + z + 1) * q^34 + (4*z^4 - 2*z^3 + 2*z^2 - 4*z) * q^35 + (-5*z^5 + 3*z^4 + 2*z^2 + 2*z - 4) * q^36 + (5*z^5 - 2*z^4 - z^2 - 2*z - 2) * q^37 + (z^5 - 3*z^4 + 2*z^2 + 2*z - 2) * q^38 + (2*z^5 + 5*z^4 + 2*z^3 - 2*z^2 - 5*z - 2) * q^39 + (-5*z^4 + 7*z^3 - 3*z^2 + 7*z - 5) * q^40 + (2*z^5 - z^4 - z^3 - 2*z + 3) * q^41 + (-2*z^5 - 2*z^3 + 2*z^2 + 2*z + 2) * q^42 + (z^5 - z^2 - z - 3) * q^43 + (2*z^4 - z^3 - z - 1) * q^44 + (2*z^5 - 3*z^4 + 2*z^3 - 3*z^2 + 2*z) * q^45 + (-6*z^5 + 6*z^4 - 6*z^3) * q^46 + (-5*z^5 - 5*z^4 - 3*z^3 + z^2 + 4*z + 3) * q^47 + (-2*z^5 + 3*z^3 - 2*z) * q^48 + (-4*z^5 - z^4 + z + 4) * q^49 + (z^5 + 2*z^4 + 2*z^3 - 3*z + 1) * q^50 + (-2*z^5 - 2*z^4 + 2*z^3 + 3*z^2 - z - 2) * q^51 + (5*z^5 - 5*z^3 + 6*z^2 - 6*z + 11) * q^52 + (-3*z^5 + 3*z^3 + 5*z^2 + z + 2) * q^53 + (4*z^5 + 2*z^4 + z^3 - 3*z^2 + 3) * q^54 + (3*z^4 - 3*z^2 + 3) * q^55 + (4*z^4 - 10*z^3 + 2*z^2 - 10*z + 4) * q^56 + (z^5 + 3*z^4 - 3*z^3 - 4*z^2 + 4) * q^57 + (-3*z^5 + 3*z^3 + 3*z^2 + 3*z) * q^58 + (-2*z^5 + 2*z^3 + z - 2) * q^59 + (-2*z^3 + z^2 - z + 2) * q^60 + (z^5 + z^4 + z^3 + z - 2) * q^61 + (3*z^5 - 7*z^4 + 2*z^3 - 2*z^2 + 7*z - 3) * q^62 + (-2*z^5 + 4*z^4 - 2*z^3 + 4*z^2 - 2*z) * q^63 + (4*z^3 + 3*z^2 - 3*z - 4) * q^64 + (-7*z^5 + 2*z^4 - 3*z^3 + 4*z^2 - 4) * q^65 + (6*z^5 - 3*z^4 - 6*z^3 - 3*z^2 + 6*z) * q^66 + (3*z^5 - 5*z^4 + 2*z^3 - 3*z^2 + 3*z + 3) * q^67 + (3*z^5 - 2*z^4 - z^2 - z + 2) * q^68 + (6*z^3 - 6*z - 6) * q^69 + (-6*z^5 + 4*z^4 + 4*z^3 + 4*z - 8) * q^70 + (-2*z^4 - 11*z^2 - 2) * q^71 + (2*z^5 - 7*z^4 + 6*z^3 - 6*z^2 + 7*z - 2) * q^72 + (-z^5 + 4*z^4 - 3*z^2 - 3*z) * q^73 + (z^5 + 5*z^4 - 5*z^3 + 6*z^2 - 3*z - 3) * q^74 + (-z^5 - 3*z^4 + 4*z^2 + 4*z) * q^75 + (z^4 - z) * q^76 + (-2*z^4 - 2*z^3 - 2*z - 2) * q^77 + (-z^5 + 5*z^4 + 5*z^3 - 2*z - 3) * q^78 + (4*z^4 - 7*z^3 + 3*z + 3) * q^79 + (7*z^5 - 6*z^4 - z^2 - z + 4) * q^80 + (-3*z^5 - 6*z^4 + 5*z^3 + 3*z^2 + z + 1) * q^81 + (-5*z^5 + 7*z^4 - 6*z^3 + 7*z^2 - 5*z) * q^82 + (4*z^5 - 7*z^4 + 7*z^3 + 3*z^2 - 3) * q^83 + (2*z^5 + 2*z^4 + 4*z^3 - 2*z^2 - 4) * q^84 + (z^5 + z^4 - 3*z^3 + z^2 + z) * q^85 + (3*z^5 - z^4 + 2*z^3 - 2*z^2 + z - 3) * q^86 + (-3*z^5 - 6*z^4 - 6*z^3 + 3*z + 3) * q^87 + (6*z^5 + 6*z^4 - 3*z^3 - 9*z^2 + 3*z + 3) * q^88 + (2*z^5 - 2*z^3 - 5*z^2 - 2*z - 3) * q^89 + (5*z^5 - 5*z^3 + 2*z^2 - 6*z + 7) * q^90 + (12*z^5 + 2*z^4 + 10*z^3 - 2*z^2 + 2) * q^91 + (-6*z^4 + 12*z^3 - 6*z^2 + 12*z - 6) * q^92 + (5*z^4 - 3*z^3 - 5*z^2 - 3*z + 5) * q^93 + (-6*z^4 - 3*z^3 - 3*z^2 + 3) * q^94 + (-3*z^5 + 3*z^3 + 2*z^2 - 2*z - 1) * q^95 + (3*z^2 + 3*z + 3) * q^96 + (z^5 + z^4 + 9*z^3 - 4*z^2 + 3*z - 9) * q^97 + (-3*z^5 - z^4 - z^3 - 4*z + 5) * q^98 + (-z^5 + z^3 - z^2 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{2} - 6 q^{3} - 9 q^{4} + 6 q^{5} - 6 q^{6} - 12 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10})$$ 6 * q + 3 * q^2 - 6 * q^3 - 9 * q^4 + 6 * q^5 - 6 * q^6 - 12 * q^7 + 6 * q^8 + 6 * q^9 $$6 q + 3 q^{2} - 6 q^{3} - 9 q^{4} + 6 q^{5} - 6 q^{6} - 12 q^{7} + 6 q^{8} + 6 q^{9} + 9 q^{10} + 9 q^{11} + 3 q^{12} - 12 q^{13} - 12 q^{14} - 3 q^{15} + 9 q^{16} + 3 q^{17} + 3 q^{18} - 9 q^{19} - 21 q^{20} + 12 q^{21} + 9 q^{22} + 18 q^{23} + 6 q^{24} + 6 q^{26} - 3 q^{27} + 24 q^{28} + 9 q^{29} - 18 q^{31} - 18 q^{32} - 18 q^{33} + 9 q^{34} - 6 q^{35} - 24 q^{36} - 12 q^{37} - 12 q^{38} - 6 q^{39} - 9 q^{40} + 15 q^{41} + 6 q^{42} - 18 q^{43} - 9 q^{44} + 6 q^{45} - 18 q^{46} + 9 q^{47} + 9 q^{48} + 24 q^{49} + 12 q^{50} - 6 q^{51} + 51 q^{52} + 21 q^{53} + 21 q^{54} + 18 q^{55} - 6 q^{56} + 15 q^{57} + 9 q^{58} - 6 q^{59} + 6 q^{60} - 9 q^{61} - 12 q^{62} - 6 q^{63} - 12 q^{64} - 33 q^{65} - 18 q^{66} + 24 q^{67} + 12 q^{68} - 18 q^{69} - 36 q^{70} - 12 q^{71} + 6 q^{72} - 33 q^{74} - 18 q^{77} - 3 q^{78} - 3 q^{79} + 24 q^{80} + 21 q^{81} - 18 q^{82} + 3 q^{83} - 12 q^{84} - 9 q^{85} - 12 q^{86} + 9 q^{88} - 24 q^{89} + 27 q^{90} + 42 q^{91} + 21 q^{93} + 9 q^{94} + 3 q^{95} + 18 q^{96} - 27 q^{97} + 27 q^{98} + 9 q^{99}+O(q^{100})$$ 6 * q + 3 * q^2 - 6 * q^3 - 9 * q^4 + 6 * q^5 - 6 * q^6 - 12 * q^7 + 6 * q^8 + 6 * q^9 + 9 * q^10 + 9 * q^11 + 3 * q^12 - 12 * q^13 - 12 * q^14 - 3 * q^15 + 9 * q^16 + 3 * q^17 + 3 * q^18 - 9 * q^19 - 21 * q^20 + 12 * q^21 + 9 * q^22 + 18 * q^23 + 6 * q^24 + 6 * q^26 - 3 * q^27 + 24 * q^28 + 9 * q^29 - 18 * q^31 - 18 * q^32 - 18 * q^33 + 9 * q^34 - 6 * q^35 - 24 * q^36 - 12 * q^37 - 12 * q^38 - 6 * q^39 - 9 * q^40 + 15 * q^41 + 6 * q^42 - 18 * q^43 - 9 * q^44 + 6 * q^45 - 18 * q^46 + 9 * q^47 + 9 * q^48 + 24 * q^49 + 12 * q^50 - 6 * q^51 + 51 * q^52 + 21 * q^53 + 21 * q^54 + 18 * q^55 - 6 * q^56 + 15 * q^57 + 9 * q^58 - 6 * q^59 + 6 * q^60 - 9 * q^61 - 12 * q^62 - 6 * q^63 - 12 * q^64 - 33 * q^65 - 18 * q^66 + 24 * q^67 + 12 * q^68 - 18 * q^69 - 36 * q^70 - 12 * q^71 + 6 * q^72 - 33 * q^74 - 18 * q^77 - 3 * q^78 - 3 * q^79 + 24 * q^80 + 21 * q^81 - 18 * q^82 + 3 * q^83 - 12 * q^84 - 9 * q^85 - 12 * q^86 + 9 * q^88 - 24 * q^89 + 27 * q^90 + 42 * q^91 + 21 * q^93 + 9 * q^94 + 3 * q^95 + 18 * q^96 - 27 * q^97 + 27 * q^98 + 9 * q^99

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{18}^{5}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −0.173648 + 0.984808i −0.766044 + 0.642788i 0.939693 + 0.342020i −0.173648 − 0.984808i −0.766044 − 0.642788i 0.939693 − 0.342020i
1.26604 0.460802i −1.76604 0.642788i −0.141559 + 0.118782i 0.233956 + 1.32683i −2.53209 −0.120615 0.684040i −1.47178 + 2.54920i 0.407604 + 0.342020i 0.907604 + 1.57202i
9.1 −0.439693 2.49362i −0.0603074 + 0.342020i −4.14543 + 1.50881i 1.93969 + 1.62760i 0.879385 −2.34730 1.96962i 3.05303 + 5.28801i 2.70574 + 0.984808i 3.20574 5.55250i
12.1 0.673648 0.565258i −1.17365 0.984808i −0.213011 + 1.20805i 0.826352 0.300767i −1.34730 −3.53209 + 1.28558i 1.41875 + 2.45734i −0.113341 0.642788i 0.386659 0.669713i
16.1 1.26604 + 0.460802i −1.76604 + 0.642788i −0.141559 0.118782i 0.233956 1.32683i −2.53209 −0.120615 + 0.684040i −1.47178 2.54920i 0.407604 0.342020i 0.907604 1.57202i
33.1 −0.439693 + 2.49362i −0.0603074 0.342020i −4.14543 1.50881i 1.93969 1.62760i 0.879385 −2.34730 + 1.96962i 3.05303 5.28801i 2.70574 0.984808i 3.20574 + 5.55250i
34.1 0.673648 + 0.565258i −1.17365 + 0.984808i −0.213011 1.20805i 0.826352 + 0.300767i −1.34730 −3.53209 1.28558i 1.41875 2.45734i −0.113341 + 0.642788i 0.386659 + 0.669713i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.1 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.f even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.2.f.b 6
3.b odd 2 1 333.2.x.a 6
4.b odd 2 1 592.2.bc.c 6
5.b even 2 1 925.2.p.a 6
5.c odd 4 2 925.2.bc.b 12
37.f even 9 1 inner 37.2.f.b 6
37.f even 9 1 1369.2.a.i 3
37.h even 18 1 1369.2.a.l 3
37.i odd 36 2 1369.2.b.e 6
111.p odd 18 1 333.2.x.a 6
148.p odd 18 1 592.2.bc.c 6
185.x even 18 1 925.2.p.a 6
185.bd odd 36 2 925.2.bc.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.f.b 6 1.a even 1 1 trivial
37.2.f.b 6 37.f even 9 1 inner
333.2.x.a 6 3.b odd 2 1
333.2.x.a 6 111.p odd 18 1
592.2.bc.c 6 4.b odd 2 1
592.2.bc.c 6 148.p odd 18 1
925.2.p.a 6 5.b even 2 1
925.2.p.a 6 185.x even 18 1
925.2.bc.b 12 5.c odd 4 2
925.2.bc.b 12 185.bd odd 36 2
1369.2.a.i 3 37.f even 9 1
1369.2.a.l 3 37.h even 18 1
1369.2.b.e 6 37.i odd 36 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 3T_{2}^{5} + 9T_{2}^{4} - 24T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(37, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 3 T^{5} + \cdots + 9$$
$3$ $$T^{6} + 6 T^{5} + \cdots + 1$$
$5$ $$T^{6} - 6 T^{5} + \cdots + 9$$
$7$ $$T^{6} + 12 T^{5} + \cdots + 64$$
$11$ $$T^{6} - 9 T^{5} + \cdots + 81$$
$13$ $$T^{6} + 12 T^{5} + \cdots + 5329$$
$17$ $$T^{6} - 3 T^{5} + \cdots + 9$$
$19$ $$T^{6} + 9 T^{5} + \cdots + 1$$
$23$ $$(T^{2} - 6 T + 36)^{3}$$
$29$ $$T^{6} - 9 T^{5} + \cdots + 729$$
$31$ $$(T^{3} + 9 T^{2} + 6 T - 53)^{2}$$
$37$ $$T^{6} + 12 T^{5} + \cdots + 50653$$
$41$ $$T^{6} - 15 T^{5} + \cdots + 9$$
$43$ $$(T^{3} + 9 T^{2} + 24 T + 19)^{2}$$
$47$ $$T^{6} - 9 T^{5} + \cdots + 81$$
$53$ $$T^{6} - 21 T^{5} + \cdots + 2601$$
$59$ $$T^{6} + 6 T^{5} + \cdots + 9$$
$61$ $$T^{6} + 9 T^{5} + \cdots + 289$$
$67$ $$T^{6} - 24 T^{5} + \cdots + 1369$$
$71$ $$T^{6} + 12 T^{5} + \cdots + 1418481$$
$73$ $$(T^{3} - 39 T - 89)^{2}$$
$79$ $$T^{6} + 3 T^{5} + \cdots + 104329$$
$83$ $$T^{6} - 3 T^{5} + \cdots + 751689$$
$89$ $$T^{6} + 24 T^{5} + \cdots + 3249$$
$97$ $$T^{6} + 27 T^{5} + \cdots + 128881$$