# Properties

 Label 37.2.b.a Level $37$ Weight $2$ Character orbit 37.b 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(36,37)]

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([1]))

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

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

chi := DirichletCharacter("37.36");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## $q$-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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
36.1
 − 1.00000i 1.00000i
2.00000i −1.00000 −2.00000 2.00000i 2.00000i 3.00000 0 −2.00000 4.00000
36.2 2.00000i −1.00000 −2.00000 2.00000i 2.00000i 3.00000 0 −2.00000 4.00000
 $$n$$: 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.2.b.a 2
3.b odd 2 1 333.2.c.a 2
4.b odd 2 1 592.2.g.b 2
5.b even 2 1 925.2.c.b 2
5.c odd 4 1 925.2.d.a 2
5.c odd 4 1 925.2.d.d 2
8.b even 2 1 2368.2.g.f 2
8.d odd 2 1 2368.2.g.b 2
12.b even 2 1 5328.2.h.c 2
37.b even 2 1 inner 37.2.b.a 2
37.d odd 4 1 1369.2.a.a 1
37.d odd 4 1 1369.2.a.f 1
111.d odd 2 1 333.2.c.a 2
148.b odd 2 1 592.2.g.b 2
185.d even 2 1 925.2.c.b 2
185.h odd 4 1 925.2.d.a 2
185.h odd 4 1 925.2.d.d 2
296.e even 2 1 2368.2.g.f 2
296.h odd 2 1 2368.2.g.b 2
444.g even 2 1 5328.2.h.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.b.a 2 1.a even 1 1 trivial
37.2.b.a 2 37.b even 2 1 inner
333.2.c.a 2 3.b odd 2 1
333.2.c.a 2 111.d odd 2 1
592.2.g.b 2 4.b odd 2 1
592.2.g.b 2 148.b odd 2 1
925.2.c.b 2 5.b even 2 1
925.2.c.b 2 185.d even 2 1
925.2.d.a 2 5.c odd 4 1
925.2.d.a 2 185.h odd 4 1
925.2.d.d 2 5.c odd 4 1
925.2.d.d 2 185.h odd 4 1
1369.2.a.a 1 37.d odd 4 1
1369.2.a.f 1 37.d odd 4 1
2368.2.g.b 2 8.d odd 2 1
2368.2.g.b 2 296.h odd 2 1
2368.2.g.f 2 8.b even 2 1
2368.2.g.f 2 296.e even 2 1
5328.2.h.c 2 12.b even 2 1
5328.2.h.c 2 444.g even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$(T - 3)^{2}$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2} + 36$$
$23$ $$T^{2} + 16$$
$29$ $$T^{2} + 16$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 2T + 37$$
$41$ $$(T + 3)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$(T - 3)^{2}$$
$53$ $$(T - 9)^{2}$$
$59$ $$T^{2} + 16$$
$61$ $$T^{2}$$
$67$ $$(T + 12)^{2}$$
$71$ $$(T + 3)^{2}$$
$73$ $$(T - 9)^{2}$$
$79$ $$T^{2} + 36$$
$83$ $$(T - 9)^{2}$$
$89$ $$T^{2} + 196$$
$97$ $$T^{2} + 144$$