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 $2$

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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 26)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} - 3 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} - 3 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} + q^{8} + 3 q^{10} + ( - 6 \zeta_{6} + 6) q^{11} - \zeta_{6} q^{13} + \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} + 3 q^{17} + 2 q^{19} + (3 \zeta_{6} - 3) q^{20} + 6 \zeta_{6} q^{22} + (4 \zeta_{6} - 4) q^{25} + q^{26} - q^{28} + ( - 6 \zeta_{6} + 6) q^{29} + 4 \zeta_{6} q^{31} - \zeta_{6} q^{32} + (3 \zeta_{6} - 3) q^{34} - 3 q^{35} - 7 q^{37} + (2 \zeta_{6} - 2) q^{38} - 3 \zeta_{6} q^{40} + ( - \zeta_{6} + 1) q^{43} - 6 q^{44} + ( - 3 \zeta_{6} + 3) q^{47} + 6 \zeta_{6} q^{49} - 4 \zeta_{6} q^{50} + (\zeta_{6} - 1) q^{52} - 18 q^{55} + ( - \zeta_{6} + 1) q^{56} + 6 \zeta_{6} q^{58} - 6 \zeta_{6} q^{59} + (8 \zeta_{6} - 8) q^{61} - 4 q^{62} + q^{64} + (3 \zeta_{6} - 3) q^{65} - 14 \zeta_{6} q^{67} - 3 \zeta_{6} q^{68} + ( - 3 \zeta_{6} + 3) q^{70} + 3 q^{71} + 2 q^{73} + ( - 7 \zeta_{6} + 7) q^{74} - 2 \zeta_{6} q^{76} - 6 \zeta_{6} q^{77} + (8 \zeta_{6} - 8) q^{79} + 3 q^{80} + ( - 12 \zeta_{6} + 12) q^{83} - 9 \zeta_{6} q^{85} + \zeta_{6} q^{86} + ( - 6 \zeta_{6} + 6) q^{88} + 6 q^{89} - q^{91} + 3 \zeta_{6} q^{94} - 6 \zeta_{6} q^{95} + ( - 10 \zeta_{6} + 10) q^{97} - 6 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - 3 q^{5} + q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} - 3 q^{5} + q^{7} + 2 q^{8} + 6 q^{10} + 6 q^{11} - q^{13} + q^{14} - q^{16} + 6 q^{17} + 4 q^{19} - 3 q^{20} + 6 q^{22} - 4 q^{25} + 2 q^{26} - 2 q^{28} + 6 q^{29} + 4 q^{31} - q^{32} - 3 q^{34} - 6 q^{35} - 14 q^{37} - 2 q^{38} - 3 q^{40} + q^{43} - 12 q^{44} + 3 q^{47} + 6 q^{49} - 4 q^{50} - q^{52} - 36 q^{55} + q^{56} + 6 q^{58} - 6 q^{59} - 8 q^{61} - 8 q^{62} + 2 q^{64} - 3 q^{65} - 14 q^{67} - 3 q^{68} + 3 q^{70} + 6 q^{71} + 4 q^{73} + 7 q^{74} - 2 q^{76} - 6 q^{77} - 8 q^{79} + 6 q^{80} + 12 q^{83} - 9 q^{85} + q^{86} + 6 q^{88} + 12 q^{89} - 2 q^{91} + 3 q^{94} - 6 q^{95} + 10 q^{97} - 12 q^{98}+O(q^{100}) \) 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 −1.50000 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:

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

\( T_{5}^{2} + 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 36 \) Copy content Toggle raw display
\( T_{19} - 2 \) 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
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$17$ \( (T - 3)^{2} \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) 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
$37$ \( (T + 7)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$71$ \( (T - 3)^{2} \) 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
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
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