Properties

 Label 16.2.e.a Level $16$ Weight $2$ Character orbit 16.e Analytic conductor $0.128$ 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] = [16,2,Mod(5,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("16.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 16.e (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: $$0.127760643234$$ 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: yes 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.

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

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/16\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$15$$ $$\chi(n)$$ $$i$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 1.00000i − 1.00000i
−1.00000 1.00000i −1.00000 + 1.00000i 2.00000i −1.00000 1.00000i 2.00000 2.00000i 2.00000 2.00000i 1.00000i 2.00000i
13.1 −1.00000 + 1.00000i −1.00000 1.00000i 2.00000i −1.00000 + 1.00000i 2.00000 2.00000i 2.00000 + 2.00000i 1.00000i 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 16.2.e.a 2
3.b odd 2 1 144.2.k.a 2
4.b odd 2 1 64.2.e.a 2
5.b even 2 1 400.2.l.c 2
5.c odd 4 1 400.2.q.a 2
5.c odd 4 1 400.2.q.b 2
7.b odd 2 1 784.2.m.b 2
7.c even 3 2 784.2.x.f 4
7.d odd 6 2 784.2.x.c 4
8.b even 2 1 128.2.e.b 2
8.d odd 2 1 128.2.e.a 2
12.b even 2 1 576.2.k.a 2
16.e even 4 1 inner 16.2.e.a 2
16.e even 4 1 128.2.e.b 2
16.f odd 4 1 64.2.e.a 2
16.f odd 4 1 128.2.e.a 2
20.d odd 2 1 1600.2.l.a 2
20.e even 4 1 1600.2.q.a 2
20.e even 4 1 1600.2.q.b 2
24.f even 2 1 1152.2.k.a 2
24.h odd 2 1 1152.2.k.b 2
32.g even 8 2 1024.2.a.b 2
32.g even 8 2 1024.2.b.e 2
32.h odd 8 2 1024.2.a.e 2
32.h odd 8 2 1024.2.b.b 2
48.i odd 4 1 144.2.k.a 2
48.i odd 4 1 1152.2.k.b 2
48.k even 4 1 576.2.k.a 2
48.k even 4 1 1152.2.k.a 2
80.i odd 4 1 400.2.q.b 2
80.j even 4 1 1600.2.q.b 2
80.k odd 4 1 1600.2.l.a 2
80.q even 4 1 400.2.l.c 2
80.s even 4 1 1600.2.q.a 2
80.t odd 4 1 400.2.q.a 2
96.o even 8 2 9216.2.a.s 2
96.p odd 8 2 9216.2.a.d 2
112.l odd 4 1 784.2.m.b 2
112.w even 12 2 784.2.x.f 4
112.x odd 12 2 784.2.x.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 1.a even 1 1 trivial
16.2.e.a 2 16.e even 4 1 inner
64.2.e.a 2 4.b odd 2 1
64.2.e.a 2 16.f odd 4 1
128.2.e.a 2 8.d odd 2 1
128.2.e.a 2 16.f odd 4 1
128.2.e.b 2 8.b even 2 1
128.2.e.b 2 16.e even 4 1
144.2.k.a 2 3.b odd 2 1
144.2.k.a 2 48.i odd 4 1
400.2.l.c 2 5.b even 2 1
400.2.l.c 2 80.q even 4 1
400.2.q.a 2 5.c odd 4 1
400.2.q.a 2 80.t odd 4 1
400.2.q.b 2 5.c odd 4 1
400.2.q.b 2 80.i odd 4 1
576.2.k.a 2 12.b even 2 1
576.2.k.a 2 48.k even 4 1
784.2.m.b 2 7.b odd 2 1
784.2.m.b 2 112.l odd 4 1
784.2.x.c 4 7.d odd 6 2
784.2.x.c 4 112.x odd 12 2
784.2.x.f 4 7.c even 3 2
784.2.x.f 4 112.w even 12 2
1024.2.a.b 2 32.g even 8 2
1024.2.a.e 2 32.h odd 8 2
1024.2.b.b 2 32.h odd 8 2
1024.2.b.e 2 32.g even 8 2
1152.2.k.a 2 24.f even 2 1
1152.2.k.a 2 48.k even 4 1
1152.2.k.b 2 24.h odd 2 1
1152.2.k.b 2 48.i odd 4 1
1600.2.l.a 2 20.d odd 2 1
1600.2.l.a 2 80.k odd 4 1
1600.2.q.a 2 20.e even 4 1
1600.2.q.a 2 80.s even 4 1
1600.2.q.b 2 20.e even 4 1
1600.2.q.b 2 80.j even 4 1
9216.2.a.d 2 96.p odd 8 2
9216.2.a.s 2 96.o even 8 2

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(16, [\chi])$$.

Hecke characteristic polynomials

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