Properties

Label 2156.2.i.b
Level $2156$
Weight $2$
Character orbit 2156.i
Analytic conductor $17.216$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2156.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.2157466758\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} + 3 \zeta_{6} q^{5} + 2 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{3} + 3 \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{11} - 4 q^{13} - 3 q^{15} + (6 \zeta_{6} - 6) q^{17} - 8 \zeta_{6} q^{19} + 3 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} - 5 q^{27} + (5 \zeta_{6} - 5) q^{31} + \zeta_{6} q^{33} + \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{39} - 10 q^{43} + (6 \zeta_{6} - 6) q^{45} - 6 \zeta_{6} q^{51} + ( - 6 \zeta_{6} + 6) q^{53} + 3 q^{55} + 8 q^{57} + (3 \zeta_{6} - 3) q^{59} + 4 \zeta_{6} q^{61} - 12 \zeta_{6} q^{65} + ( - \zeta_{6} + 1) q^{67} - 3 q^{69} + 15 q^{71} + ( - 4 \zeta_{6} + 4) q^{73} - 4 \zeta_{6} q^{75} - 2 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 6 q^{83} - 18 q^{85} + 9 \zeta_{6} q^{89} - 5 \zeta_{6} q^{93} + ( - 24 \zeta_{6} + 24) q^{95} - 7 q^{97} + 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 3 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 3 q^{5} + 2 q^{9} + q^{11} - 8 q^{13} - 6 q^{15} - 6 q^{17} - 8 q^{19} + 3 q^{23} - 4 q^{25} - 10 q^{27} - 5 q^{31} + q^{33} + q^{37} + 4 q^{39} - 20 q^{43} - 6 q^{45} - 6 q^{51} + 6 q^{53} + 6 q^{55} + 16 q^{57} - 3 q^{59} + 4 q^{61} - 12 q^{65} + q^{67} - 6 q^{69} + 30 q^{71} + 4 q^{73} - 4 q^{75} - 2 q^{79} - q^{81} + 12 q^{83} - 36 q^{85} + 9 q^{89} - 5 q^{93} + 24 q^{95} - 14 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1079\) \(1277\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 0.866025i 0 1.50000 2.59808i 0 0 0 1.00000 1.73205i 0
1145.1 0 −0.500000 + 0.866025i 0 1.50000 + 2.59808i 0 0 0 1.00000 + 1.73205i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2156.2.i.b 2
7.b odd 2 1 2156.2.i.c 2
7.c even 3 1 44.2.a.a 1
7.c even 3 1 inner 2156.2.i.b 2
7.d odd 6 1 2156.2.a.a 1
7.d odd 6 1 2156.2.i.c 2
21.h odd 6 1 396.2.a.c 1
28.f even 6 1 8624.2.a.w 1
28.g odd 6 1 176.2.a.a 1
35.j even 6 1 1100.2.a.b 1
35.l odd 12 2 1100.2.b.c 2
56.k odd 6 1 704.2.a.i 1
56.p even 6 1 704.2.a.f 1
63.g even 3 1 3564.2.i.j 2
63.h even 3 1 3564.2.i.j 2
63.j odd 6 1 3564.2.i.a 2
63.n odd 6 1 3564.2.i.a 2
77.h odd 6 1 484.2.a.a 1
77.m even 15 4 484.2.e.a 4
77.o odd 30 4 484.2.e.b 4
84.n even 6 1 1584.2.a.p 1
91.r even 6 1 7436.2.a.d 1
105.o odd 6 1 9900.2.a.h 1
105.x even 12 2 9900.2.c.g 2
112.u odd 12 2 2816.2.c.k 2
112.w even 12 2 2816.2.c.e 2
140.p odd 6 1 4400.2.a.v 1
140.w even 12 2 4400.2.b.k 2
168.s odd 6 1 6336.2.a.j 1
168.v even 6 1 6336.2.a.i 1
231.l even 6 1 4356.2.a.j 1
308.n even 6 1 1936.2.a.c 1
616.y even 6 1 7744.2.a.bc 1
616.bg odd 6 1 7744.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 7.c even 3 1
176.2.a.a 1 28.g odd 6 1
396.2.a.c 1 21.h odd 6 1
484.2.a.a 1 77.h odd 6 1
484.2.e.a 4 77.m even 15 4
484.2.e.b 4 77.o odd 30 4
704.2.a.f 1 56.p even 6 1
704.2.a.i 1 56.k odd 6 1
1100.2.a.b 1 35.j even 6 1
1100.2.b.c 2 35.l odd 12 2
1584.2.a.p 1 84.n even 6 1
1936.2.a.c 1 308.n even 6 1
2156.2.a.a 1 7.d odd 6 1
2156.2.i.b 2 1.a even 1 1 trivial
2156.2.i.b 2 7.c even 3 1 inner
2156.2.i.c 2 7.b odd 2 1
2156.2.i.c 2 7.d odd 6 1
2816.2.c.e 2 112.w even 12 2
2816.2.c.k 2 112.u odd 12 2
3564.2.i.a 2 63.j odd 6 1
3564.2.i.a 2 63.n odd 6 1
3564.2.i.j 2 63.g even 3 1
3564.2.i.j 2 63.h even 3 1
4356.2.a.j 1 231.l even 6 1
4400.2.a.v 1 140.p odd 6 1
4400.2.b.k 2 140.w even 12 2
6336.2.a.i 1 168.v even 6 1
6336.2.a.j 1 168.s odd 6 1
7436.2.a.d 1 91.r even 6 1
7744.2.a.m 1 616.bg odd 6 1
7744.2.a.bc 1 616.y even 6 1
8624.2.a.w 1 28.f even 6 1
9900.2.a.h 1 105.o odd 6 1
9900.2.c.g 2 105.x even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2156, [\chi])\):

\( T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 3T_{5} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$37$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$71$ \( (T - 15)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$79$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$97$ \( (T + 7)^{2} \) Copy content Toggle raw display
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