Properties

Label 4029.2.a.d
Level 4029
Weight 2
Character orbit 4029.a
Self dual yes
Analytic conductor 32.172
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4029.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1717269744\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - q^{3} + ( 1 - \beta_{1} - \beta_{2} ) q^{4} + ( -2 + \beta_{1} ) q^{5} -\beta_{2} q^{6} + q^{7} -2 q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} - q^{3} + ( 1 - \beta_{1} - \beta_{2} ) q^{4} + ( -2 + \beta_{1} ) q^{5} -\beta_{2} q^{6} + q^{7} -2 q^{8} + q^{9} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{10} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} ) q^{12} + ( 3 + \beta_{1} + \beta_{2} ) q^{13} + \beta_{2} q^{14} + ( 2 - \beta_{1} ) q^{15} + ( -2 + 2 \beta_{1} ) q^{16} + q^{17} + \beta_{2} q^{18} + ( -\beta_{1} - \beta_{2} ) q^{19} + ( -3 + \beta_{1} + \beta_{2} ) q^{20} - q^{21} + ( -4 + 4 \beta_{2} ) q^{22} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{23} + 2 q^{24} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{25} + ( 2 + 2 \beta_{2} ) q^{26} - q^{27} + ( 1 - \beta_{1} - \beta_{2} ) q^{28} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{30} + ( -6 + 2 \beta_{1} + 2 \beta_{2} ) q^{31} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{32} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{33} + \beta_{2} q^{34} + ( -2 + \beta_{1} ) q^{35} + ( 1 - \beta_{1} - \beta_{2} ) q^{36} + ( -7 + 2 \beta_{1} ) q^{37} + ( -2 + \beta_{2} ) q^{38} + ( -3 - \beta_{1} - \beta_{2} ) q^{39} + ( 4 - 2 \beta_{1} ) q^{40} + ( -2 + 4 \beta_{1} + 4 \beta_{2} ) q^{41} -\beta_{2} q^{42} + ( 4 + 2 \beta_{1} + 4 \beta_{2} ) q^{43} + ( 8 - 4 \beta_{2} ) q^{44} + ( -2 + \beta_{1} ) q^{45} + ( 5 - 3 \beta_{1} ) q^{46} + ( 3 + \beta_{2} ) q^{47} + ( 2 - 2 \beta_{1} ) q^{48} -6 q^{49} + ( 6 - 4 \beta_{1} ) q^{50} - q^{51} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{52} + ( 4 + 5 \beta_{1} - 4 \beta_{2} ) q^{53} -\beta_{2} q^{54} + ( -6 + 2 \beta_{1} + 2 \beta_{2} ) q^{55} -2 q^{56} + ( \beta_{1} + \beta_{2} ) q^{57} + ( -4 + 4 \beta_{2} ) q^{58} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{59} + ( 3 - \beta_{1} - \beta_{2} ) q^{60} + ( -5 - 2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( 4 - 8 \beta_{2} ) q^{62} + q^{63} + ( -4 + 4 \beta_{2} ) q^{64} + ( -5 + 3 \beta_{1} - \beta_{2} ) q^{65} + ( 4 - 4 \beta_{2} ) q^{66} + ( -3 - 3 \beta_{1} - 5 \beta_{2} ) q^{67} + ( 1 - \beta_{1} - \beta_{2} ) q^{68} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{69} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{70} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{71} -2 q^{72} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{73} + ( -2 + 2 \beta_{1} - 7 \beta_{2} ) q^{74} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{75} + ( 3 + \beta_{1} - \beta_{2} ) q^{76} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{77} + ( -2 - 2 \beta_{2} ) q^{78} + q^{79} + ( 8 - 4 \beta_{1} + 2 \beta_{2} ) q^{80} + q^{81} + ( 8 - 6 \beta_{2} ) q^{82} + ( 4 + 4 \beta_{1} + \beta_{2} ) q^{83} + ( -1 + \beta_{1} + \beta_{2} ) q^{84} + ( -2 + \beta_{1} ) q^{85} + ( 10 - 2 \beta_{1} ) q^{86} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{87} + ( -4 + 4 \beta_{1} + 4 \beta_{2} ) q^{88} + ( -2 + 2 \beta_{1} + 6 \beta_{2} ) q^{89} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{90} + ( 3 + \beta_{1} + \beta_{2} ) q^{91} + ( 1 + \beta_{1} + 3 \beta_{2} ) q^{92} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{93} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{94} + ( -1 + \beta_{2} ) q^{95} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{96} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{97} -6 \beta_{2} q^{98} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 2q^{4} - 5q^{5} + 3q^{7} - 6q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 2q^{4} - 5q^{5} + 3q^{7} - 6q^{8} + 3q^{9} - 2q^{10} + 4q^{11} - 2q^{12} + 10q^{13} + 5q^{15} - 4q^{16} + 3q^{17} - q^{19} - 8q^{20} - 3q^{21} - 12q^{22} + q^{23} + 6q^{24} + 6q^{26} - 3q^{27} + 2q^{28} + 4q^{29} + 2q^{30} - 16q^{31} + 8q^{32} - 4q^{33} - 5q^{35} + 2q^{36} - 19q^{37} - 6q^{38} - 10q^{39} + 10q^{40} - 2q^{41} + 14q^{43} + 24q^{44} - 5q^{45} + 12q^{46} + 9q^{47} + 4q^{48} - 18q^{49} + 14q^{50} - 3q^{51} - 4q^{52} + 17q^{53} - 16q^{55} - 6q^{56} + q^{57} - 12q^{58} - q^{59} + 8q^{60} - 17q^{61} + 12q^{62} + 3q^{63} - 12q^{64} - 12q^{65} + 12q^{66} - 12q^{67} + 2q^{68} - q^{69} - 2q^{70} - 8q^{71} - 6q^{72} - 14q^{73} - 4q^{74} + 10q^{76} + 4q^{77} - 6q^{78} + 3q^{79} + 20q^{80} + 3q^{81} + 24q^{82} + 16q^{83} - 2q^{84} - 5q^{85} + 28q^{86} - 4q^{87} - 8q^{88} - 4q^{89} - 2q^{90} + 10q^{91} + 4q^{92} + 16q^{93} + 8q^{94} - 3q^{95} - 8q^{96} - 4q^{97} + 4q^{99} + O(q^{100}) \)

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

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

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
0.311108
2.17009
−1.48119
−2.21432 −1.00000 2.90321 −1.68889 2.21432 1.00000 −2.00000 1.00000 3.73975
1.2 0.539189 −1.00000 −1.70928 0.170086 −0.539189 1.00000 −2.00000 1.00000 0.0917087
1.3 1.67513 −1.00000 0.806063 −3.48119 −1.67513 1.00000 −2.00000 1.00000 −5.83146
\(n\): 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 4029.2.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4029.2.a.d 3 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(17\) \(-1\)
\(79\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4029))\):

\( T_{2}^{3} - 4 T_{2} + 2 \)
\( T_{5}^{3} + 5 T_{5}^{2} + 5 T_{5} - 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + 2 T^{3} + 4 T^{4} + 8 T^{6} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 1 + 5 T + 20 T^{2} + 49 T^{3} + 100 T^{4} + 125 T^{5} + 125 T^{6} \)
$7$ \( ( 1 - T + 7 T^{2} )^{3} \)
$11$ \( 1 - 4 T + 17 T^{2} - 56 T^{3} + 187 T^{4} - 484 T^{5} + 1331 T^{6} \)
$13$ \( 1 - 10 T + 67 T^{2} - 280 T^{3} + 871 T^{4} - 1690 T^{5} + 2197 T^{6} \)
$17$ \( ( 1 - T )^{3} \)
$19$ \( 1 + T + 52 T^{2} + 37 T^{3} + 988 T^{4} + 361 T^{5} + 6859 T^{6} \)
$23$ \( 1 - T + 48 T^{2} - 75 T^{3} + 1104 T^{4} - 529 T^{5} + 12167 T^{6} \)
$29$ \( 1 - 4 T + 71 T^{2} - 200 T^{3} + 2059 T^{4} - 3364 T^{5} + 24389 T^{6} \)
$31$ \( 1 + 16 T + 157 T^{2} + 1024 T^{3} + 4867 T^{4} + 15376 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 19 T + 218 T^{2} + 1575 T^{3} + 8066 T^{4} + 26011 T^{5} + 50653 T^{6} \)
$41$ \( 1 + 2 T + 39 T^{2} + 60 T^{3} + 1599 T^{4} + 3362 T^{5} + 68921 T^{6} \)
$43$ \( 1 - 14 T + 133 T^{2} - 860 T^{3} + 5719 T^{4} - 25886 T^{5} + 79507 T^{6} \)
$47$ \( 1 - 9 T + 164 T^{2} - 859 T^{3} + 7708 T^{4} - 19881 T^{5} + 103823 T^{6} \)
$53$ \( 1 - 17 T + 68 T^{2} + 47 T^{3} + 3604 T^{4} - 47753 T^{5} + 148877 T^{6} \)
$59$ \( 1 + T + 164 T^{2} + 95 T^{3} + 9676 T^{4} + 3481 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 17 T + 242 T^{2} + 1965 T^{3} + 14762 T^{4} + 63257 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 12 T + 149 T^{2} + 1004 T^{3} + 9983 T^{4} + 53868 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 8 T + 221 T^{2} + 1138 T^{3} + 15691 T^{4} + 40328 T^{5} + 357911 T^{6} \)
$73$ \( 1 + 14 T + 261 T^{2} + 2054 T^{3} + 19053 T^{4} + 74606 T^{5} + 389017 T^{6} \)
$79$ \( ( 1 - T )^{3} \)
$83$ \( 1 - 16 T + 285 T^{2} - 2646 T^{3} + 23655 T^{4} - 110224 T^{5} + 571787 T^{6} \)
$89$ \( 1 + 4 T + 139 T^{2} + 1128 T^{3} + 12371 T^{4} + 31684 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 4 T + 281 T^{2} + 738 T^{3} + 27257 T^{4} + 37636 T^{5} + 912673 T^{6} \)
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