Properties

Label 2106.2.e.q
Level $2106$
Weight $2$
Character orbit 2106.e
Analytic conductor $16.816$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Self dual: no
Analytic conductor: \(16.8164946657\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
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 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - 2 \zeta_{6} q^{5} + (4 \zeta_{6} - 4) q^{7} - q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - 2 \zeta_{6} q^{5} + (4 \zeta_{6} - 4) q^{7} - q^{8} - 2 q^{10} + ( - 4 \zeta_{6} + 4) q^{11} - \zeta_{6} q^{13} + 4 \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} + 2 q^{17} - 8 q^{19} + (2 \zeta_{6} - 2) q^{20} - 4 \zeta_{6} q^{22} + ( - \zeta_{6} + 1) q^{25} - q^{26} + 4 q^{28} + (6 \zeta_{6} - 6) q^{29} + 4 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( - 2 \zeta_{6} + 2) q^{34} + 8 q^{35} - 2 q^{37} + (8 \zeta_{6} - 8) q^{38} + 2 \zeta_{6} q^{40} + 10 \zeta_{6} q^{41} + (4 \zeta_{6} - 4) q^{43} - 4 q^{44} + (8 \zeta_{6} - 8) q^{47} - 9 \zeta_{6} q^{49} - \zeta_{6} q^{50} + (\zeta_{6} - 1) q^{52} - 10 q^{53} - 8 q^{55} + ( - 4 \zeta_{6} + 4) q^{56} + 6 \zeta_{6} q^{58} - 4 \zeta_{6} q^{59} + ( - 2 \zeta_{6} + 2) q^{61} + 4 q^{62} + q^{64} + (2 \zeta_{6} - 2) q^{65} + 16 \zeta_{6} q^{67} - 2 \zeta_{6} q^{68} + ( - 8 \zeta_{6} + 8) q^{70} - 8 q^{71} + 2 q^{73} + (2 \zeta_{6} - 2) q^{74} + 8 \zeta_{6} q^{76} + 16 \zeta_{6} q^{77} + (8 \zeta_{6} - 8) q^{79} + 2 q^{80} + 10 q^{82} + (12 \zeta_{6} - 12) q^{83} - 4 \zeta_{6} q^{85} + 4 \zeta_{6} q^{86} + (4 \zeta_{6} - 4) q^{88} + 14 q^{89} + 4 q^{91} + 8 \zeta_{6} q^{94} + 16 \zeta_{6} q^{95} + (10 \zeta_{6} - 10) q^{97} - 9 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} - 2 q^{5} - 4 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} - 2 q^{5} - 4 q^{7} - 2 q^{8} - 4 q^{10} + 4 q^{11} - q^{13} + 4 q^{14} - q^{16} + 4 q^{17} - 16 q^{19} - 2 q^{20} - 4 q^{22} + q^{25} - 2 q^{26} + 8 q^{28} - 6 q^{29} + 4 q^{31} + q^{32} + 2 q^{34} + 16 q^{35} - 4 q^{37} - 8 q^{38} + 2 q^{40} + 10 q^{41} - 4 q^{43} - 8 q^{44} - 8 q^{47} - 9 q^{49} - q^{50} - q^{52} - 20 q^{53} - 16 q^{55} + 4 q^{56} + 6 q^{58} - 4 q^{59} + 2 q^{61} + 8 q^{62} + 2 q^{64} - 2 q^{65} + 16 q^{67} - 2 q^{68} + 8 q^{70} - 16 q^{71} + 4 q^{73} - 2 q^{74} + 8 q^{76} + 16 q^{77} - 8 q^{79} + 4 q^{80} + 20 q^{82} - 12 q^{83} - 4 q^{85} + 4 q^{86} - 4 q^{88} + 28 q^{89} + 8 q^{91} + 8 q^{94} + 16 q^{95} - 10 q^{97} - 18 q^{98}+O(q^{100}) \) 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.

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 −1.00000 1.73205i 0 −2.00000 + 3.46410i −1.00000 0 −2.00000
1405.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.00000 + 1.73205i 0 −2.00000 3.46410i −1.00000 0 −2.00000
\(n\): 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.q 2
3.b odd 2 1 2106.2.e.j 2
9.c even 3 1 78.2.a.a 1
9.c even 3 1 inner 2106.2.e.q 2
9.d odd 6 1 234.2.a.c 1
9.d odd 6 1 2106.2.e.j 2
36.f odd 6 1 624.2.a.h 1
36.h even 6 1 1872.2.a.c 1
45.h odd 6 1 5850.2.a.d 1
45.j even 6 1 1950.2.a.w 1
45.k odd 12 2 1950.2.e.i 2
45.l even 12 2 5850.2.e.bb 2
63.l odd 6 1 3822.2.a.j 1
72.j odd 6 1 7488.2.a.bz 1
72.l even 6 1 7488.2.a.bk 1
72.n even 6 1 2496.2.a.t 1
72.p odd 6 1 2496.2.a.b 1
99.h odd 6 1 9438.2.a.t 1
117.f even 3 1 1014.2.e.f 2
117.h even 3 1 1014.2.e.f 2
117.l even 6 1 1014.2.e.c 2
117.n odd 6 1 3042.2.a.f 1
117.r even 6 1 1014.2.e.c 2
117.t even 6 1 1014.2.a.d 1
117.w odd 12 2 1014.2.i.d 4
117.y odd 12 2 1014.2.b.b 2
117.z even 12 2 3042.2.b.g 2
117.bb odd 12 2 1014.2.i.d 4
468.bg odd 6 1 8112.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 9.c even 3 1
234.2.a.c 1 9.d odd 6 1
624.2.a.h 1 36.f odd 6 1
1014.2.a.d 1 117.t even 6 1
1014.2.b.b 2 117.y odd 12 2
1014.2.e.c 2 117.l even 6 1
1014.2.e.c 2 117.r even 6 1
1014.2.e.f 2 117.f even 3 1
1014.2.e.f 2 117.h even 3 1
1014.2.i.d 4 117.w odd 12 2
1014.2.i.d 4 117.bb odd 12 2
1872.2.a.c 1 36.h even 6 1
1950.2.a.w 1 45.j even 6 1
1950.2.e.i 2 45.k odd 12 2
2106.2.e.j 2 3.b odd 2 1
2106.2.e.j 2 9.d odd 6 1
2106.2.e.q 2 1.a even 1 1 trivial
2106.2.e.q 2 9.c even 3 1 inner
2496.2.a.b 1 72.p odd 6 1
2496.2.a.t 1 72.n even 6 1
3042.2.a.f 1 117.n odd 6 1
3042.2.b.g 2 117.z even 12 2
3822.2.a.j 1 63.l odd 6 1
5850.2.a.d 1 45.h odd 6 1
5850.2.e.bb 2 45.l even 12 2
7488.2.a.bk 1 72.l even 6 1
7488.2.a.bz 1 72.j odd 6 1
8112.2.a.v 1 468.bg odd 6 1
9438.2.a.t 1 99.h odd 6 1

Hecke kernels

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

\( T_{5}^{2} + 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 4T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} + 16 \) Copy content Toggle raw display
\( T_{19} + 8 \) 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
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$13$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( (T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$53$ \( (T + 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$89$ \( (T - 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
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