Properties

Label 4012.2.a.f
Level 4012
Weight 2
Character orbit 4012.a
Self dual yes
Analytic conductor 32.036
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4012.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0359812909\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{3} -\beta_{1} q^{5} + q^{7} -2 q^{9} +O(q^{10})\) \( q - q^{3} -\beta_{1} q^{5} + q^{7} -2 q^{9} -2 q^{11} + ( 1 + \beta_{1} ) q^{13} + \beta_{1} q^{15} + q^{17} + ( -2 + \beta_{1} ) q^{19} - q^{21} + ( -\beta_{1} - \beta_{2} ) q^{23} + ( 6 + 2 \beta_{2} ) q^{25} + 5 q^{27} + ( -2 - \beta_{1} ) q^{29} + ( 2 + \beta_{1} - \beta_{2} ) q^{31} + 2 q^{33} -\beta_{1} q^{35} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{37} + ( -1 - \beta_{1} ) q^{39} + ( -2 + \beta_{1} ) q^{41} + ( -2 + \beta_{1} - \beta_{2} ) q^{43} + 2 \beta_{1} q^{45} + ( -4 + \beta_{1} + \beta_{2} ) q^{47} -6 q^{49} - q^{51} + ( 5 + 2 \beta_{2} ) q^{53} + 2 \beta_{1} q^{55} + ( 2 - \beta_{1} ) q^{57} - q^{59} + ( -3 - \beta_{1} ) q^{61} -2 q^{63} + ( -11 - \beta_{1} - 2 \beta_{2} ) q^{65} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{67} + ( \beta_{1} + \beta_{2} ) q^{69} + ( -2 + 2 \beta_{2} ) q^{71} + 2 q^{73} + ( -6 - 2 \beta_{2} ) q^{75} -2 q^{77} -5 q^{79} + q^{81} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{83} -\beta_{1} q^{85} + ( 2 + \beta_{1} ) q^{87} + ( 4 + \beta_{1} - \beta_{2} ) q^{89} + ( 1 + \beta_{1} ) q^{91} + ( -2 - \beta_{1} + \beta_{2} ) q^{93} + ( -11 + 2 \beta_{1} - 2 \beta_{2} ) q^{95} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{97} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + q^{5} + 3q^{7} - 6q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + q^{5} + 3q^{7} - 6q^{9} - 6q^{11} + 2q^{13} - q^{15} + 3q^{17} - 7q^{19} - 3q^{21} + 20q^{25} + 15q^{27} - 5q^{29} + 4q^{31} + 6q^{33} + q^{35} + 6q^{37} - 2q^{39} - 7q^{41} - 8q^{43} - 2q^{45} - 12q^{47} - 18q^{49} - 3q^{51} + 17q^{53} - 2q^{55} + 7q^{57} - 3q^{59} - 8q^{61} - 6q^{63} - 34q^{65} - 6q^{67} - 4q^{71} + 6q^{73} - 20q^{75} - 6q^{77} - 15q^{79} + 3q^{81} - 6q^{83} + q^{85} + 5q^{87} + 10q^{89} + 2q^{91} - 4q^{93} - 37q^{95} - 12q^{97} + 12q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 2 \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 6\)\()/2\)

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
2.46050
0.239123
−1.69963
0 −1.00000 0 −3.92101 0 1.00000 0 −2.00000 0
1.2 0 −1.00000 0 0.521753 0 1.00000 0 −2.00000 0
1.3 0 −1.00000 0 4.39926 0 1.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 4012.2.a.f 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4012.2.a.f 3 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(-1\)
\(59\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4012))\):

\( T_{3} + 1 \)
\( T_{5}^{3} - T_{5}^{2} - 17 T_{5} + 9 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 + T + 3 T^{2} )^{3} \)
$5$ \( 1 - T - 2 T^{2} - T^{3} - 10 T^{4} - 25 T^{5} + 125 T^{6} \)
$7$ \( ( 1 - T + 7 T^{2} )^{3} \)
$11$ \( ( 1 + 2 T + 11 T^{2} )^{3} \)
$13$ \( 1 - 2 T + 23 T^{2} - 44 T^{3} + 299 T^{4} - 338 T^{5} + 2197 T^{6} \)
$17$ \( ( 1 - T )^{3} \)
$19$ \( 1 + 7 T + 56 T^{2} + 235 T^{3} + 1064 T^{4} + 2527 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 33 T^{2} + 8 T^{3} + 759 T^{4} + 12167 T^{6} \)
$29$ \( 1 + 5 T + 78 T^{2} + 269 T^{3} + 2262 T^{4} + 4205 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 4 T + 49 T^{2} - 80 T^{3} + 1519 T^{4} - 3844 T^{5} + 29791 T^{6} \)
$37$ \( 1 - 6 T + 15 T^{2} + 188 T^{3} + 555 T^{4} - 8214 T^{5} + 50653 T^{6} \)
$41$ \( 1 + 7 T + 122 T^{2} + 543 T^{3} + 5002 T^{4} + 11767 T^{5} + 68921 T^{6} \)
$43$ \( 1 + 8 T + 101 T^{2} + 680 T^{3} + 4343 T^{4} + 14792 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 12 T + 153 T^{2} + 1040 T^{3} + 7191 T^{4} + 26508 T^{5} + 103823 T^{6} \)
$53$ \( 1 - 17 T + 154 T^{2} - 1085 T^{3} + 8162 T^{4} - 47753 T^{5} + 148877 T^{6} \)
$59$ \( ( 1 + T )^{3} \)
$61$ \( 1 + 8 T + 187 T^{2} + 952 T^{3} + 11407 T^{4} + 29768 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 6 T + 69 T^{2} + 460 T^{3} + 4623 T^{4} + 26934 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 4 T + 117 T^{2} + 760 T^{3} + 8307 T^{4} + 20164 T^{5} + 357911 T^{6} \)
$73$ \( ( 1 - 2 T + 73 T^{2} )^{3} \)
$79$ \( ( 1 + 5 T + 79 T^{2} )^{3} \)
$83$ \( 1 + 6 T + 153 T^{2} + 1212 T^{3} + 12699 T^{4} + 41334 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 10 T + 251 T^{2} - 1548 T^{3} + 22339 T^{4} - 79210 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 12 T + 207 T^{2} + 1936 T^{3} + 20079 T^{4} + 112908 T^{5} + 912673 T^{6} \)
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