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 $2$

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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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} - q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} - q^{8} - q^{10} + (2 \zeta_{6} - 2) q^{11} + \zeta_{6} q^{13} + \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} + 3 q^{17} + 6 q^{19} + (\zeta_{6} - 1) q^{20} + 2 \zeta_{6} q^{22} - 4 \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} + q^{26} + q^{28} + ( - 2 \zeta_{6} + 2) q^{29} - 4 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( - 3 \zeta_{6} + 3) q^{34} + q^{35} + 3 q^{37} + ( - 6 \zeta_{6} + 6) q^{38} + \zeta_{6} q^{40} + ( - 5 \zeta_{6} + 5) q^{43} + 2 q^{44} - 4 q^{46} + ( - 13 \zeta_{6} + 13) q^{47} + 6 \zeta_{6} q^{49} - 4 \zeta_{6} q^{50} + ( - \zeta_{6} + 1) q^{52} - 12 q^{53} + 2 q^{55} + ( - \zeta_{6} + 1) q^{56} - 2 \zeta_{6} q^{58} - 10 \zeta_{6} q^{59} + ( - 8 \zeta_{6} + 8) q^{61} - 4 q^{62} + q^{64} + ( - \zeta_{6} + 1) q^{65} + 2 \zeta_{6} q^{67} - 3 \zeta_{6} q^{68} + ( - \zeta_{6} + 1) q^{70} + 5 q^{71} - 10 q^{73} + ( - 3 \zeta_{6} + 3) q^{74} - 6 \zeta_{6} q^{76} - 2 \zeta_{6} q^{77} + ( - 4 \zeta_{6} + 4) q^{79} + q^{80} - 3 \zeta_{6} q^{85} - 5 \zeta_{6} q^{86} + ( - 2 \zeta_{6} + 2) q^{88} - 6 q^{89} - q^{91} + (4 \zeta_{6} - 4) q^{92} - 13 \zeta_{6} q^{94} - 6 \zeta_{6} q^{95} + (14 \zeta_{6} - 14) q^{97} + 6 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} - q^{5} - q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} - q^{5} - q^{7} - 2 q^{8} - 2 q^{10} - 2 q^{11} + q^{13} + q^{14} - q^{16} + 6 q^{17} + 12 q^{19} - q^{20} + 2 q^{22} - 4 q^{23} + 4 q^{25} + 2 q^{26} + 2 q^{28} + 2 q^{29} - 4 q^{31} + q^{32} + 3 q^{34} + 2 q^{35} + 6 q^{37} + 6 q^{38} + q^{40} + 5 q^{43} + 4 q^{44} - 8 q^{46} + 13 q^{47} + 6 q^{49} - 4 q^{50} + q^{52} - 24 q^{53} + 4 q^{55} + q^{56} - 2 q^{58} - 10 q^{59} + 8 q^{61} - 8 q^{62} + 2 q^{64} + q^{65} + 2 q^{67} - 3 q^{68} + q^{70} + 10 q^{71} - 20 q^{73} + 3 q^{74} - 6 q^{76} - 2 q^{77} + 4 q^{79} + 2 q^{80} - 3 q^{85} - 5 q^{86} + 2 q^{88} - 12 q^{89} - 2 q^{91} - 4 q^{92} - 13 q^{94} - 6 q^{95} - 14 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
0.500000 + 0.866025i
0.500000 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:

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 \(S_{2}^{\mathrm{new}}(2106, [\chi])\):

\( T_{5}^{2} + T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} + 4 \) Copy content Toggle raw display
\( T_{19} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

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