# Properties

 Label 2106.2.e.b Level $2106$ Weight $2$ Character orbit 2106.e Analytic conductor $16.816$ 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] = [2106,2,Mod(703,2106)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2106, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("2106.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2106 = 2 \cdot 3^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2106.e (of order $$3$$, degree $$2$$, not 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: $$16.8164946657$$ 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: no (minimal twist has level 26) 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2106\mathbb{Z}\right)^\times$$.

 $$n$$ $$1379$$ $$1783$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
703.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −1.50000 2.59808i 0 0.500000 0.866025i 1.00000 0 3.00000
1405.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.50000 + 2.59808i 0 0.500000 + 0.866025i 1.00000 0 3.00000
 $$n$$: 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2106.2.e.b 2
3.b odd 2 1 2106.2.e.ba 2
9.c even 3 1 234.2.a.e 1
9.c even 3 1 inner 2106.2.e.b 2
9.d odd 6 1 26.2.a.a 1
9.d odd 6 1 2106.2.e.ba 2
36.f odd 6 1 1872.2.a.q 1
36.h even 6 1 208.2.a.a 1
45.h odd 6 1 650.2.a.j 1
45.j even 6 1 5850.2.a.p 1
45.k odd 12 2 5850.2.e.a 2
45.l even 12 2 650.2.b.d 2
63.i even 6 1 1274.2.f.r 2
63.j odd 6 1 1274.2.f.p 2
63.n odd 6 1 1274.2.f.p 2
63.o even 6 1 1274.2.a.d 1
63.s even 6 1 1274.2.f.r 2
72.j odd 6 1 832.2.a.d 1
72.l even 6 1 832.2.a.i 1
72.n even 6 1 7488.2.a.g 1
72.p odd 6 1 7488.2.a.h 1
99.g even 6 1 3146.2.a.n 1
117.k odd 6 1 338.2.c.d 2
117.m odd 6 1 338.2.c.a 2
117.n odd 6 1 338.2.a.f 1
117.t even 6 1 3042.2.a.a 1
117.u odd 6 1 338.2.c.d 2
117.v odd 6 1 338.2.c.a 2
117.x even 12 2 338.2.e.a 4
117.y odd 12 2 3042.2.b.a 2
117.z even 12 2 338.2.b.c 2
117.bc even 12 2 338.2.e.a 4
144.u even 12 2 3328.2.b.j 2
144.w odd 12 2 3328.2.b.m 2
153.i odd 6 1 7514.2.a.c 1
171.l even 6 1 9386.2.a.j 1
180.n even 6 1 5200.2.a.x 1
468.x even 6 1 2704.2.a.f 1
468.ch odd 12 2 2704.2.f.d 2
585.bo odd 6 1 8450.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 9.d odd 6 1
208.2.a.a 1 36.h even 6 1
234.2.a.e 1 9.c even 3 1
338.2.a.f 1 117.n odd 6 1
338.2.b.c 2 117.z even 12 2
338.2.c.a 2 117.m odd 6 1
338.2.c.a 2 117.v odd 6 1
338.2.c.d 2 117.k odd 6 1
338.2.c.d 2 117.u odd 6 1
338.2.e.a 4 117.x even 12 2
338.2.e.a 4 117.bc even 12 2
650.2.a.j 1 45.h odd 6 1
650.2.b.d 2 45.l even 12 2
832.2.a.d 1 72.j odd 6 1
832.2.a.i 1 72.l even 6 1
1274.2.a.d 1 63.o even 6 1
1274.2.f.p 2 63.j odd 6 1
1274.2.f.p 2 63.n odd 6 1
1274.2.f.r 2 63.i even 6 1
1274.2.f.r 2 63.s even 6 1
1872.2.a.q 1 36.f odd 6 1
2106.2.e.b 2 1.a even 1 1 trivial
2106.2.e.b 2 9.c even 3 1 inner
2106.2.e.ba 2 3.b odd 2 1
2106.2.e.ba 2 9.d odd 6 1
2704.2.a.f 1 468.x even 6 1
2704.2.f.d 2 468.ch odd 12 2
3042.2.a.a 1 117.t even 6 1
3042.2.b.a 2 117.y odd 12 2
3146.2.a.n 1 99.g even 6 1
3328.2.b.j 2 144.u even 12 2
3328.2.b.m 2 144.w odd 12 2
5200.2.a.x 1 180.n even 6 1
5850.2.a.p 1 45.j even 6 1
5850.2.e.a 2 45.k odd 12 2
7488.2.a.g 1 72.n even 6 1
7488.2.a.h 1 72.p odd 6 1
7514.2.a.c 1 153.i odd 6 1
8450.2.a.c 1 585.bo odd 6 1
9386.2.a.j 1 171.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2106, [\chi])$$:

 $$T_{5}^{2} + 3T_{5} + 9$$ T5^2 + 3*T5 + 9 $$T_{7}^{2} - T_{7} + 1$$ T7^2 - T7 + 1 $$T_{11}^{2} - 6T_{11} + 36$$ T11^2 - 6*T11 + 36 $$T_{19} - 2$$ T19 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2} - 6T + 36$$
$13$ $$T^{2} + T + 1$$
$17$ $$(T - 3)^{2}$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 6T + 36$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$(T + 7)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2} - 3T + 9$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 6T + 36$$
$61$ $$T^{2} + 8T + 64$$
$67$ $$T^{2} + 14T + 196$$
$71$ $$(T - 3)^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2} + 8T + 64$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} - 10T + 100$$