Properties

Label 2112.2.a.bh
Level $2112$
Weight $2$
Character orbit 2112.a
Self dual yes
Analytic conductor $16.864$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2112.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.8644049069\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1056)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{1} q^{5} + ( -1 - \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{1} q^{5} + ( -1 - \beta_{2} ) q^{7} + q^{9} + q^{11} + \beta_{1} q^{13} + \beta_{1} q^{15} + ( 3 + \beta_{1} - \beta_{2} ) q^{17} + ( 1 + \beta_{1} - \beta_{2} ) q^{19} + ( -1 - \beta_{2} ) q^{21} + ( 2 - \beta_{1} ) q^{23} + ( 5 + 2 \beta_{2} ) q^{25} + q^{27} + ( -3 - \beta_{1} + \beta_{2} ) q^{29} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{31} + q^{33} + ( -4 - 4 \beta_{1} ) q^{35} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{37} + \beta_{1} q^{39} + ( 3 - \beta_{1} - \beta_{2} ) q^{41} + ( 3 + \beta_{1} + \beta_{2} ) q^{43} + \beta_{1} q^{45} + ( 6 + \beta_{1} ) q^{47} + ( 9 + 2 \beta_{1} ) q^{49} + ( 3 + \beta_{1} - \beta_{2} ) q^{51} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{53} + \beta_{1} q^{55} + ( 1 + \beta_{1} - \beta_{2} ) q^{57} + 2 \beta_{1} q^{59} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{61} + ( -1 - \beta_{2} ) q^{63} + ( 10 + 2 \beta_{2} ) q^{65} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 2 - \beta_{1} ) q^{69} + ( 10 - \beta_{1} ) q^{71} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{73} + ( 5 + 2 \beta_{2} ) q^{75} + ( -1 - \beta_{2} ) q^{77} + ( -5 + 2 \beta_{1} - \beta_{2} ) q^{79} + q^{81} -8 q^{83} + ( 6 + 2 \beta_{2} ) q^{85} + ( -3 - \beta_{1} + \beta_{2} ) q^{87} + 2 q^{89} + ( -4 - 4 \beta_{1} ) q^{91} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{93} + ( 6 - 2 \beta_{1} + 2 \beta_{2} ) q^{95} + ( 2 - 2 \beta_{1} ) q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} - 4q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} - 4q^{7} + 3q^{9} + 3q^{11} + 8q^{17} + 2q^{19} - 4q^{21} + 6q^{23} + 17q^{25} + 3q^{27} - 8q^{29} - 4q^{31} + 3q^{33} - 12q^{35} + 2q^{37} + 8q^{41} + 10q^{43} + 18q^{47} + 27q^{49} + 8q^{51} + 4q^{53} + 2q^{57} - 8q^{61} - 4q^{63} + 32q^{65} - 4q^{67} + 6q^{69} + 30q^{71} + 10q^{73} + 17q^{75} - 4q^{77} - 16q^{79} + 3q^{81} - 24q^{83} + 20q^{85} - 8q^{87} + 6q^{89} - 12q^{91} - 4q^{93} + 20q^{95} + 6q^{97} + 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 4 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 5\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.86081
−0.254102
2.11491
0 1.00000 0 −3.72161 0 −2.92520 0 1.00000 0
1.2 0 1.00000 0 −0.508203 0 3.87086 0 1.00000 0
1.3 0 1.00000 0 4.22982 0 −4.94567 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2112.2.a.bh 3
3.b odd 2 1 6336.2.a.cy 3
4.b odd 2 1 2112.2.a.bg 3
8.b even 2 1 1056.2.a.m 3
8.d odd 2 1 1056.2.a.n yes 3
12.b even 2 1 6336.2.a.cz 3
24.f even 2 1 3168.2.a.bh 3
24.h odd 2 1 3168.2.a.bg 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1056.2.a.m 3 8.b even 2 1
1056.2.a.n yes 3 8.d odd 2 1
2112.2.a.bg 3 4.b odd 2 1
2112.2.a.bh 3 1.a even 1 1 trivial
3168.2.a.bg 3 24.h odd 2 1
3168.2.a.bh 3 24.f even 2 1
6336.2.a.cy 3 3.b odd 2 1
6336.2.a.cz 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2112))\):

\( T_{5}^{3} - 16 T_{5} - 8 \)
\( T_{7}^{3} + 4 T_{7}^{2} - 16 T_{7} - 56 \)
\( T_{13}^{3} - 16 T_{13} - 8 \)
\( T_{17}^{3} - 8 T_{17}^{2} - 4 T_{17} + 64 \)
\( T_{19}^{3} - 2 T_{19}^{2} - 24 T_{19} + 32 \)
\( T_{23}^{3} - 6 T_{23}^{2} - 4 T_{23} + 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{3} \)
$5$ \( 1 - T^{2} - 8 T^{3} - 5 T^{4} + 125 T^{6} \)
$7$ \( 1 + 4 T + 5 T^{2} + 35 T^{4} + 196 T^{5} + 343 T^{6} \)
$11$ \( ( 1 - T )^{3} \)
$13$ \( 1 + 23 T^{2} - 8 T^{3} + 299 T^{4} + 2197 T^{6} \)
$17$ \( 1 - 8 T + 47 T^{2} - 208 T^{3} + 799 T^{4} - 2312 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 2 T + 33 T^{2} - 44 T^{3} + 627 T^{4} - 722 T^{5} + 6859 T^{6} \)
$23$ \( 1 - 6 T + 65 T^{2} - 244 T^{3} + 1495 T^{4} - 3174 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 8 T + 83 T^{2} + 400 T^{3} + 2407 T^{4} + 6728 T^{5} + 24389 T^{6} \)
$31$ \( 1 + 4 T - 3 T^{2} - 8 T^{3} - 93 T^{4} + 3844 T^{5} + 29791 T^{6} \)
$37$ \( 1 - 2 T + 11 T^{2} - 204 T^{3} + 407 T^{4} - 2738 T^{5} + 50653 T^{6} \)
$41$ \( 1 - 8 T + 95 T^{2} - 448 T^{3} + 3895 T^{4} - 13448 T^{5} + 68921 T^{6} \)
$43$ \( 1 - 10 T + 113 T^{2} - 828 T^{3} + 4859 T^{4} - 18490 T^{5} + 79507 T^{6} \)
$47$ \( 1 - 18 T + 233 T^{2} - 1820 T^{3} + 10951 T^{4} - 39762 T^{5} + 103823 T^{6} \)
$53$ \( 1 - 4 T + 39 T^{2} - 192 T^{3} + 2067 T^{4} - 11236 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 113 T^{2} - 64 T^{3} + 6667 T^{4} + 205379 T^{6} \)
$61$ \( 1 + 8 T + 127 T^{2} + 584 T^{3} + 7747 T^{4} + 29768 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 4 T + 9 T^{2} - 488 T^{3} + 603 T^{4} + 17956 T^{5} + 300763 T^{6} \)
$71$ \( 1 - 30 T + 497 T^{2} - 5092 T^{3} + 35287 T^{4} - 151230 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 10 T + 55 T^{2} - 76 T^{3} + 4015 T^{4} - 53290 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 16 T + 261 T^{2} + 2536 T^{3} + 20619 T^{4} + 99856 T^{5} + 493039 T^{6} \)
$83$ \( ( 1 + 8 T + 83 T^{2} )^{3} \)
$89$ \( ( 1 - 2 T + 89 T^{2} )^{3} \)
$97$ \( 1 - 6 T + 239 T^{2} - 980 T^{3} + 23183 T^{4} - 56454 T^{5} + 912673 T^{6} \)
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