# Properties

 Label 144.2.k.a Level $144$ Weight $2$ Character orbit 144.k Analytic conductor $1.150$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,2,Mod(37,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("144.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 144.k (of order $$4$$, 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: $$1.14984578911$$ 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: $$1$$ Twist minimal: no (minimal twist has level 16) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (i + 1) q^{2} + 2 i q^{4} + (i + 1) q^{5} - 2 i q^{7} + (2 i - 2) q^{8} +O(q^{10})$$ q + (i + 1) * q^2 + 2*i * q^4 + (i + 1) * q^5 - 2*i * q^7 + (2*i - 2) * q^8 $$q + (i + 1) q^{2} + 2 i q^{4} + (i + 1) q^{5} - 2 i q^{7} + (2 i - 2) q^{8} + 2 i q^{10} + ( - i - 1) q^{11} + (i - 1) q^{13} + ( - 2 i + 2) q^{14} - 4 q^{16} + 2 q^{17} + ( - 3 i + 3) q^{19} + (2 i - 2) q^{20} - 2 i q^{22} - 6 i q^{23} - 3 i q^{25} - 2 q^{26} + 4 q^{28} + (3 i - 3) q^{29} - 8 q^{31} + ( - 4 i - 4) q^{32} + (2 i + 2) q^{34} + ( - 2 i + 2) q^{35} + (3 i + 3) q^{37} + 6 q^{38} - 4 q^{40} + (5 i + 5) q^{43} + ( - 2 i + 2) q^{44} + ( - 6 i + 6) q^{46} - 8 q^{47} + 3 q^{49} + ( - 3 i + 3) q^{50} + ( - 2 i - 2) q^{52} + (5 i + 5) q^{53} - 2 i q^{55} + (4 i + 4) q^{56} - 6 q^{58} + (3 i + 3) q^{59} + (9 i - 9) q^{61} + ( - 8 i - 8) q^{62} - 8 i q^{64} - 2 q^{65} + (5 i - 5) q^{67} + 4 i q^{68} + 4 q^{70} + 10 i q^{71} - 4 i q^{73} + 6 i q^{74} + (6 i + 6) q^{76} + (2 i - 2) q^{77} + ( - 4 i - 4) q^{80} + ( - i + 1) q^{83} + (2 i + 2) q^{85} + 10 i q^{86} + 4 q^{88} - 4 i q^{89} + (2 i + 2) q^{91} + 12 q^{92} + ( - 8 i - 8) q^{94} + 6 q^{95} - 2 q^{97} + (3 i + 3) q^{98} +O(q^{100})$$ q + (i + 1) * q^2 + 2*i * q^4 + (i + 1) * q^5 - 2*i * q^7 + (2*i - 2) * q^8 + 2*i * q^10 + (-i - 1) * q^11 + (i - 1) * q^13 + (-2*i + 2) * q^14 - 4 * q^16 + 2 * q^17 + (-3*i + 3) * q^19 + (2*i - 2) * q^20 - 2*i * q^22 - 6*i * q^23 - 3*i * q^25 - 2 * q^26 + 4 * q^28 + (3*i - 3) * q^29 - 8 * q^31 + (-4*i - 4) * q^32 + (2*i + 2) * q^34 + (-2*i + 2) * q^35 + (3*i + 3) * q^37 + 6 * q^38 - 4 * q^40 + (5*i + 5) * q^43 + (-2*i + 2) * q^44 + (-6*i + 6) * q^46 - 8 * q^47 + 3 * q^49 + (-3*i + 3) * q^50 + (-2*i - 2) * q^52 + (5*i + 5) * q^53 - 2*i * q^55 + (4*i + 4) * q^56 - 6 * q^58 + (3*i + 3) * q^59 + (9*i - 9) * q^61 + (-8*i - 8) * q^62 - 8*i * q^64 - 2 * q^65 + (5*i - 5) * q^67 + 4*i * q^68 + 4 * q^70 + 10*i * q^71 - 4*i * q^73 + 6*i * q^74 + (6*i + 6) * q^76 + (2*i - 2) * q^77 + (-4*i - 4) * q^80 + (-i + 1) * q^83 + (2*i + 2) * q^85 + 10*i * q^86 + 4 * q^88 - 4*i * q^89 + (2*i + 2) * q^91 + 12 * q^92 + (-8*i - 8) * q^94 + 6 * q^95 - 2 * q^97 + (3*i + 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{5} - 4 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^5 - 4 * q^8 $$2 q + 2 q^{2} + 2 q^{5} - 4 q^{8} - 2 q^{11} - 2 q^{13} + 4 q^{14} - 8 q^{16} + 4 q^{17} + 6 q^{19} - 4 q^{20} - 4 q^{26} + 8 q^{28} - 6 q^{29} - 16 q^{31} - 8 q^{32} + 4 q^{34} + 4 q^{35} + 6 q^{37} + 12 q^{38} - 8 q^{40} + 10 q^{43} + 4 q^{44} + 12 q^{46} - 16 q^{47} + 6 q^{49} + 6 q^{50} - 4 q^{52} + 10 q^{53} + 8 q^{56} - 12 q^{58} + 6 q^{59} - 18 q^{61} - 16 q^{62} - 4 q^{65} - 10 q^{67} + 8 q^{70} + 12 q^{76} - 4 q^{77} - 8 q^{80} + 2 q^{83} + 4 q^{85} + 8 q^{88} + 4 q^{91} + 24 q^{92} - 16 q^{94} + 12 q^{95} - 4 q^{97} + 6 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^5 - 4 * q^8 - 2 * q^11 - 2 * q^13 + 4 * q^14 - 8 * q^16 + 4 * q^17 + 6 * q^19 - 4 * q^20 - 4 * q^26 + 8 * q^28 - 6 * q^29 - 16 * q^31 - 8 * q^32 + 4 * q^34 + 4 * q^35 + 6 * q^37 + 12 * q^38 - 8 * q^40 + 10 * q^43 + 4 * q^44 + 12 * q^46 - 16 * q^47 + 6 * q^49 + 6 * q^50 - 4 * q^52 + 10 * q^53 + 8 * q^56 - 12 * q^58 + 6 * q^59 - 18 * q^61 - 16 * q^62 - 4 * q^65 - 10 * q^67 + 8 * q^70 + 12 * q^76 - 4 * q^77 - 8 * q^80 + 2 * q^83 + 4 * q^85 + 8 * q^88 + 4 * q^91 + 24 * q^92 - 16 * q^94 + 12 * q^95 - 4 * q^97 + 6 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$i$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 1.00000i − 1.00000i
1.00000 + 1.00000i 0 2.00000i 1.00000 + 1.00000i 0 2.00000i −2.00000 + 2.00000i 0 2.00000i
109.1 1.00000 1.00000i 0 2.00000i 1.00000 1.00000i 0 2.00000i −2.00000 2.00000i 0 2.00000i
 $$n$$: 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
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.k.a 2
3.b odd 2 1 16.2.e.a 2
4.b odd 2 1 576.2.k.a 2
8.b even 2 1 1152.2.k.b 2
8.d odd 2 1 1152.2.k.a 2
12.b even 2 1 64.2.e.a 2
15.d odd 2 1 400.2.l.c 2
15.e even 4 1 400.2.q.a 2
15.e even 4 1 400.2.q.b 2
16.e even 4 1 inner 144.2.k.a 2
16.e even 4 1 1152.2.k.b 2
16.f odd 4 1 576.2.k.a 2
16.f odd 4 1 1152.2.k.a 2
21.c even 2 1 784.2.m.b 2
21.g even 6 2 784.2.x.c 4
21.h odd 6 2 784.2.x.f 4
24.f even 2 1 128.2.e.a 2
24.h odd 2 1 128.2.e.b 2
32.g even 8 2 9216.2.a.d 2
32.h odd 8 2 9216.2.a.s 2
48.i odd 4 1 16.2.e.a 2
48.i odd 4 1 128.2.e.b 2
48.k even 4 1 64.2.e.a 2
48.k even 4 1 128.2.e.a 2
60.h even 2 1 1600.2.l.a 2
60.l odd 4 1 1600.2.q.a 2
60.l odd 4 1 1600.2.q.b 2
96.o even 8 2 1024.2.a.e 2
96.o even 8 2 1024.2.b.b 2
96.p odd 8 2 1024.2.a.b 2
96.p odd 8 2 1024.2.b.e 2
240.t even 4 1 1600.2.l.a 2
240.z odd 4 1 1600.2.q.a 2
240.bb even 4 1 400.2.q.b 2
240.bd odd 4 1 1600.2.q.b 2
240.bf even 4 1 400.2.q.a 2
240.bm odd 4 1 400.2.l.c 2
336.y even 4 1 784.2.m.b 2
336.bo even 12 2 784.2.x.c 4
336.bt odd 12 2 784.2.x.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 3.b odd 2 1
16.2.e.a 2 48.i odd 4 1
64.2.e.a 2 12.b even 2 1
64.2.e.a 2 48.k even 4 1
128.2.e.a 2 24.f even 2 1
128.2.e.a 2 48.k even 4 1
128.2.e.b 2 24.h odd 2 1
128.2.e.b 2 48.i odd 4 1
144.2.k.a 2 1.a even 1 1 trivial
144.2.k.a 2 16.e even 4 1 inner
400.2.l.c 2 15.d odd 2 1
400.2.l.c 2 240.bm odd 4 1
400.2.q.a 2 15.e even 4 1
400.2.q.a 2 240.bf even 4 1
400.2.q.b 2 15.e even 4 1
400.2.q.b 2 240.bb even 4 1
576.2.k.a 2 4.b odd 2 1
576.2.k.a 2 16.f odd 4 1
784.2.m.b 2 21.c even 2 1
784.2.m.b 2 336.y even 4 1
784.2.x.c 4 21.g even 6 2
784.2.x.c 4 336.bo even 12 2
784.2.x.f 4 21.h odd 6 2
784.2.x.f 4 336.bt odd 12 2
1024.2.a.b 2 96.p odd 8 2
1024.2.a.e 2 96.o even 8 2
1024.2.b.b 2 96.o even 8 2
1024.2.b.e 2 96.p odd 8 2
1152.2.k.a 2 8.d odd 2 1
1152.2.k.a 2 16.f odd 4 1
1152.2.k.b 2 8.b even 2 1
1152.2.k.b 2 16.e even 4 1
1600.2.l.a 2 60.h even 2 1
1600.2.l.a 2 240.t even 4 1
1600.2.q.a 2 60.l odd 4 1
1600.2.q.a 2 240.z odd 4 1
1600.2.q.b 2 60.l odd 4 1
1600.2.q.b 2 240.bd odd 4 1
9216.2.a.d 2 32.g even 8 2
9216.2.a.s 2 32.h odd 8 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 2T_{5} + 2$$ acting on $$S_{2}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 2$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2} + 2T + 2$$
$13$ $$T^{2} + 2T + 2$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} - 6T + 18$$
$23$ $$T^{2} + 36$$
$29$ $$T^{2} + 6T + 18$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} - 6T + 18$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 10T + 50$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} - 10T + 50$$
$59$ $$T^{2} - 6T + 18$$
$61$ $$T^{2} + 18T + 162$$
$67$ $$T^{2} + 10T + 50$$
$71$ $$T^{2} + 100$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 2T + 2$$
$89$ $$T^{2} + 16$$
$97$ $$(T + 2)^{2}$$