# Properties

 Label 2106.2.e.t Level $2106$ Weight $2$ Character orbit 2106.e Analytic conductor $16.816$ 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] = [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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} - q^{8} +O(q^{10})$$ q + (-z + 1) * q^2 - z * q^4 - z * q^5 + (z - 1) * q^7 - q^8 $$q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} - q^{8} - q^{10} + (2 \zeta_{6} - 2) q^{11} + \zeta_{6} q^{13} + \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} + 3 q^{17} + 6 q^{19} + (\zeta_{6} - 1) q^{20} + 2 \zeta_{6} q^{22} - 4 \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} + q^{26} + q^{28} + ( - 2 \zeta_{6} + 2) q^{29} - 4 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( - 3 \zeta_{6} + 3) q^{34} + q^{35} + 3 q^{37} + ( - 6 \zeta_{6} + 6) q^{38} + \zeta_{6} q^{40} + ( - 5 \zeta_{6} + 5) q^{43} + 2 q^{44} - 4 q^{46} + ( - 13 \zeta_{6} + 13) q^{47} + 6 \zeta_{6} q^{49} - 4 \zeta_{6} q^{50} + ( - \zeta_{6} + 1) q^{52} - 12 q^{53} + 2 q^{55} + ( - \zeta_{6} + 1) q^{56} - 2 \zeta_{6} q^{58} - 10 \zeta_{6} q^{59} + ( - 8 \zeta_{6} + 8) q^{61} - 4 q^{62} + q^{64} + ( - \zeta_{6} + 1) q^{65} + 2 \zeta_{6} q^{67} - 3 \zeta_{6} q^{68} + ( - \zeta_{6} + 1) q^{70} + 5 q^{71} - 10 q^{73} + ( - 3 \zeta_{6} + 3) q^{74} - 6 \zeta_{6} q^{76} - 2 \zeta_{6} q^{77} + ( - 4 \zeta_{6} + 4) q^{79} + q^{80} - 3 \zeta_{6} q^{85} - 5 \zeta_{6} q^{86} + ( - 2 \zeta_{6} + 2) q^{88} - 6 q^{89} - q^{91} + (4 \zeta_{6} - 4) q^{92} - 13 \zeta_{6} q^{94} - 6 \zeta_{6} q^{95} + (14 \zeta_{6} - 14) q^{97} + 6 q^{98} +O(q^{100})$$ q + (-z + 1) * q^2 - z * q^4 - z * q^5 + (z - 1) * q^7 - q^8 - q^10 + (2*z - 2) * q^11 + z * q^13 + z * q^14 + (z - 1) * q^16 + 3 * q^17 + 6 * q^19 + (z - 1) * q^20 + 2*z * q^22 - 4*z * q^23 + (-4*z + 4) * q^25 + q^26 + q^28 + (-2*z + 2) * q^29 - 4*z * q^31 + z * q^32 + (-3*z + 3) * q^34 + q^35 + 3 * q^37 + (-6*z + 6) * q^38 + z * q^40 + (-5*z + 5) * q^43 + 2 * q^44 - 4 * q^46 + (-13*z + 13) * q^47 + 6*z * q^49 - 4*z * q^50 + (-z + 1) * q^52 - 12 * q^53 + 2 * q^55 + (-z + 1) * q^56 - 2*z * q^58 - 10*z * q^59 + (-8*z + 8) * q^61 - 4 * q^62 + q^64 + (-z + 1) * q^65 + 2*z * q^67 - 3*z * q^68 + (-z + 1) * q^70 + 5 * q^71 - 10 * q^73 + (-3*z + 3) * q^74 - 6*z * q^76 - 2*z * q^77 + (-4*z + 4) * q^79 + q^80 - 3*z * q^85 - 5*z * q^86 + (-2*z + 2) * q^88 - 6 * q^89 - q^91 + (4*z - 4) * q^92 - 13*z * q^94 - 6*z * q^95 + (14*z - 14) * q^97 + 6 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - q^{5} - q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q + q^2 - q^4 - q^5 - q^7 - 2 * q^8 $$2 q + q^{2} - q^{4} - q^{5} - q^{7} - 2 q^{8} - 2 q^{10} - 2 q^{11} + q^{13} + q^{14} - q^{16} + 6 q^{17} + 12 q^{19} - q^{20} + 2 q^{22} - 4 q^{23} + 4 q^{25} + 2 q^{26} + 2 q^{28} + 2 q^{29} - 4 q^{31} + q^{32} + 3 q^{34} + 2 q^{35} + 6 q^{37} + 6 q^{38} + q^{40} + 5 q^{43} + 4 q^{44} - 8 q^{46} + 13 q^{47} + 6 q^{49} - 4 q^{50} + q^{52} - 24 q^{53} + 4 q^{55} + q^{56} - 2 q^{58} - 10 q^{59} + 8 q^{61} - 8 q^{62} + 2 q^{64} + q^{65} + 2 q^{67} - 3 q^{68} + q^{70} + 10 q^{71} - 20 q^{73} + 3 q^{74} - 6 q^{76} - 2 q^{77} + 4 q^{79} + 2 q^{80} - 3 q^{85} - 5 q^{86} + 2 q^{88} - 12 q^{89} - 2 q^{91} - 4 q^{92} - 13 q^{94} - 6 q^{95} - 14 q^{97} + 12 q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 - q^5 - q^7 - 2 * q^8 - 2 * q^10 - 2 * q^11 + q^13 + q^14 - q^16 + 6 * q^17 + 12 * q^19 - q^20 + 2 * q^22 - 4 * q^23 + 4 * q^25 + 2 * q^26 + 2 * q^28 + 2 * q^29 - 4 * q^31 + q^32 + 3 * q^34 + 2 * q^35 + 6 * q^37 + 6 * q^38 + q^40 + 5 * q^43 + 4 * q^44 - 8 * q^46 + 13 * q^47 + 6 * q^49 - 4 * q^50 + q^52 - 24 * q^53 + 4 * q^55 + q^56 - 2 * q^58 - 10 * q^59 + 8 * q^61 - 8 * q^62 + 2 * q^64 + q^65 + 2 * q^67 - 3 * q^68 + q^70 + 10 * q^71 - 20 * q^73 + 3 * q^74 - 6 * q^76 - 2 * q^77 + 4 * q^79 + 2 * q^80 - 3 * q^85 - 5 * q^86 + 2 * q^88 - 12 * q^89 - 2 * q^91 - 4 * q^92 - 13 * q^94 - 6 * q^95 - 14 * 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 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.00000 0 −1.00000
1405.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.00000 0 −1.00000
 $$n$$: 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.t 2
3.b odd 2 1 2106.2.e.h 2
9.c even 3 1 234.2.a.b 1
9.c even 3 1 inner 2106.2.e.t 2
9.d odd 6 1 26.2.a.b 1
9.d odd 6 1 2106.2.e.h 2
36.f odd 6 1 1872.2.a.m 1
36.h even 6 1 208.2.a.d 1
45.h odd 6 1 650.2.a.g 1
45.j even 6 1 5850.2.a.bn 1
45.k odd 12 2 5850.2.e.v 2
45.l even 12 2 650.2.b.a 2
63.i even 6 1 1274.2.f.a 2
63.j odd 6 1 1274.2.f.l 2
63.n odd 6 1 1274.2.f.l 2
63.o even 6 1 1274.2.a.o 1
63.s even 6 1 1274.2.f.a 2
72.j odd 6 1 832.2.a.j 1
72.l even 6 1 832.2.a.a 1
72.n even 6 1 7488.2.a.w 1
72.p odd 6 1 7488.2.a.v 1
99.g even 6 1 3146.2.a.a 1
117.k odd 6 1 338.2.c.c 2
117.m odd 6 1 338.2.c.g 2
117.n odd 6 1 338.2.a.a 1
117.t even 6 1 3042.2.a.l 1
117.u odd 6 1 338.2.c.c 2
117.v odd 6 1 338.2.c.g 2
117.x even 12 2 338.2.e.d 4
117.y odd 12 2 3042.2.b.f 2
117.z even 12 2 338.2.b.a 2
117.bc even 12 2 338.2.e.d 4
144.u even 12 2 3328.2.b.k 2
144.w odd 12 2 3328.2.b.g 2
153.i odd 6 1 7514.2.a.i 1
171.l even 6 1 9386.2.a.f 1
180.n even 6 1 5200.2.a.c 1
468.x even 6 1 2704.2.a.n 1
468.ch odd 12 2 2704.2.f.j 2
585.bo odd 6 1 8450.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 9.d odd 6 1
208.2.a.d 1 36.h even 6 1
234.2.a.b 1 9.c even 3 1
338.2.a.a 1 117.n odd 6 1
338.2.b.a 2 117.z even 12 2
338.2.c.c 2 117.k odd 6 1
338.2.c.c 2 117.u odd 6 1
338.2.c.g 2 117.m odd 6 1
338.2.c.g 2 117.v odd 6 1
338.2.e.d 4 117.x even 12 2
338.2.e.d 4 117.bc even 12 2
650.2.a.g 1 45.h odd 6 1
650.2.b.a 2 45.l even 12 2
832.2.a.a 1 72.l even 6 1
832.2.a.j 1 72.j odd 6 1
1274.2.a.o 1 63.o even 6 1
1274.2.f.a 2 63.i even 6 1
1274.2.f.a 2 63.s even 6 1
1274.2.f.l 2 63.j odd 6 1
1274.2.f.l 2 63.n odd 6 1
1872.2.a.m 1 36.f odd 6 1
2106.2.e.h 2 3.b odd 2 1
2106.2.e.h 2 9.d odd 6 1
2106.2.e.t 2 1.a even 1 1 trivial
2106.2.e.t 2 9.c even 3 1 inner
2704.2.a.n 1 468.x even 6 1
2704.2.f.j 2 468.ch odd 12 2
3042.2.a.l 1 117.t even 6 1
3042.2.b.f 2 117.y odd 12 2
3146.2.a.a 1 99.g even 6 1
3328.2.b.g 2 144.w odd 12 2
3328.2.b.k 2 144.u even 12 2
5200.2.a.c 1 180.n even 6 1
5850.2.a.bn 1 45.j even 6 1
5850.2.e.v 2 45.k odd 12 2
7488.2.a.v 1 72.p odd 6 1
7488.2.a.w 1 72.n even 6 1
7514.2.a.i 1 153.i odd 6 1
8450.2.a.y 1 585.bo odd 6 1
9386.2.a.f 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 $$S_{2}^{\mathrm{new}}(2106, [\chi])$$:

 $$T_{5}^{2} + T_{5} + 1$$ T5^2 + T5 + 1 $$T_{7}^{2} + T_{7} + 1$$ T7^2 + T7 + 1 $$T_{11}^{2} + 2T_{11} + 4$$ T11^2 + 2*T11 + 4 $$T_{19} - 6$$ T19 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2} + T + 1$$
$11$ $$T^{2} + 2T + 4$$
$13$ $$T^{2} - T + 1$$
$17$ $$(T - 3)^{2}$$
$19$ $$(T - 6)^{2}$$
$23$ $$T^{2} + 4T + 16$$
$29$ $$T^{2} - 2T + 4$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$(T - 3)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 5T + 25$$
$47$ $$T^{2} - 13T + 169$$
$53$ $$(T + 12)^{2}$$
$59$ $$T^{2} + 10T + 100$$
$61$ $$T^{2} - 8T + 64$$
$67$ $$T^{2} - 2T + 4$$
$71$ $$(T - 5)^{2}$$
$73$ $$(T + 10)^{2}$$
$79$ $$T^{2} - 4T + 16$$
$83$ $$T^{2}$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 14T + 196$$