Properties

Label 2112.2.a.be
Level $2112$
Weight $2$
Character orbit 2112.a
Self dual yes
Analytic conductor $16.864$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
0 −1.00000 0 −3.23607 0 −1.23607 0 1.00000 0
1.2 0 −1.00000 0 1.23607 0 3.23607 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.be 2
3.b odd 2 1 6336.2.a.ct 2
4.b odd 2 1 2112.2.a.bf 2
8.b even 2 1 1056.2.a.l yes 2
8.d odd 2 1 1056.2.a.k 2
12.b even 2 1 6336.2.a.cs 2
24.f even 2 1 3168.2.a.be 2
24.h odd 2 1 3168.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1056.2.a.k 2 8.d odd 2 1
1056.2.a.l yes 2 8.b even 2 1
2112.2.a.be 2 1.a even 1 1 trivial
2112.2.a.bf 2 4.b odd 2 1
3168.2.a.be 2 24.f even 2 1
3168.2.a.bf 2 24.h odd 2 1
6336.2.a.cs 2 12.b even 2 1
6336.2.a.ct 2 3.b odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 2 T_{5} - 4 \)
\( T_{7}^{2} - 2 T_{7} - 4 \)
\( T_{13}^{2} + 6 T_{13} + 4 \)
\( T_{17}^{2} - 20 \)
\( T_{19}^{2} + 4 T_{19} - 16 \)
\( T_{23}^{2} - 2 T_{23} - 44 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{2} \)
$5$ \( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ \( 1 - 2 T + 10 T^{2} - 14 T^{3} + 49 T^{4} \)
$11$ \( ( 1 - T )^{2} \)
$13$ \( 1 + 6 T + 30 T^{2} + 78 T^{3} + 169 T^{4} \)
$17$ \( 1 + 14 T^{2} + 289 T^{4} \)
$19$ \( 1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( 1 - 2 T + 2 T^{2} - 46 T^{3} + 529 T^{4} \)
$29$ \( 1 + 8 T + 54 T^{2} + 232 T^{3} + 841 T^{4} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( ( 1 + 6 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 4 T + 6 T^{2} - 164 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 8 T + 22 T^{2} - 344 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 2 T + 50 T^{2} + 94 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 18 T + 182 T^{2} + 954 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 20 T + 198 T^{2} - 1180 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 10 T + 102 T^{2} + 610 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 + 8 T + 67 T^{2} )^{2} \)
$71$ \( 1 - 2 T + 98 T^{2} - 142 T^{3} + 5041 T^{4} \)
$73$ \( ( 1 + 6 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 10 T + 138 T^{2} + 790 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 8 T + 102 T^{2} - 664 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 2 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 8 T + 190 T^{2} + 776 T^{3} + 9409 T^{4} \)
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