Properties

Label 4026.2.a.n
Level 4026
Weight 2
Character orbit 4026.a
Self dual yes
Analytic conductor 32.148
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4026.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1477718538\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 root \(\beta\) of the polynomial \(x^{3} - 3 x - 1\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.87939
−0.347296
−1.53209
−1.00000 1.00000 1.00000 −2.87939 −1.00000 −0.347296 −1.00000 1.00000 2.87939
1.2 −1.00000 1.00000 1.00000 −0.652704 −1.00000 −1.53209 −1.00000 1.00000 0.652704
1.3 −1.00000 1.00000 1.00000 0.532089 −1.00000 1.87939 −1.00000 1.00000 −0.532089
\(n\): 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 4026.2.a.n 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.n 3 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(11\) \(-1\)
\(61\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{3} + 3 T_{5}^{2} - 1 \)
\( T_{7}^{3} - 3 T_{7} - 1 \)
\( T_{13}^{3} + 3 T_{13}^{2} - 3 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( ( 1 - T )^{3} \)
$5$ \( 1 + 3 T + 15 T^{2} + 29 T^{3} + 75 T^{4} + 75 T^{5} + 125 T^{6} \)
$7$ \( 1 + 18 T^{2} - T^{3} + 126 T^{4} + 343 T^{6} \)
$11$ \( ( 1 - T )^{3} \)
$13$ \( 1 + 3 T + 39 T^{2} + 75 T^{3} + 507 T^{4} + 507 T^{5} + 2197 T^{6} \)
$17$ \( 1 - 3 T + 42 T^{2} - 99 T^{3} + 714 T^{4} - 867 T^{5} + 4913 T^{6} \)
$19$ \( 1 + 6 T + 48 T^{2} + 177 T^{3} + 912 T^{4} + 2166 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 6 T + 78 T^{2} + 277 T^{3} + 1794 T^{4} + 3174 T^{5} + 12167 T^{6} \)
$29$ \( 1 - 6 T + 60 T^{2} - 297 T^{3} + 1740 T^{4} - 5046 T^{5} + 24389 T^{6} \)
$31$ \( 1 + 15 T + 129 T^{2} + 771 T^{3} + 3999 T^{4} + 14415 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 6 T + 66 T^{2} + 501 T^{3} + 2442 T^{4} + 8214 T^{5} + 50653 T^{6} \)
$41$ \( 1 + 15 T + 186 T^{2} + 1287 T^{3} + 7626 T^{4} + 25215 T^{5} + 68921 T^{6} \)
$43$ \( 1 + 6 T + 78 T^{2} + 407 T^{3} + 3354 T^{4} + 11094 T^{5} + 79507 T^{6} \)
$47$ \( 1 - 9 T + 51 T^{2} - 369 T^{3} + 2397 T^{4} - 19881 T^{5} + 103823 T^{6} \)
$53$ \( 1 - 6 T + 78 T^{2} - 439 T^{3} + 4134 T^{4} - 16854 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 9 T + 141 T^{2} + 891 T^{3} + 8319 T^{4} + 31329 T^{5} + 205379 T^{6} \)
$61$ \( ( 1 + T )^{3} \)
$67$ \( 1 + 27 T + 333 T^{2} + 2915 T^{3} + 22311 T^{4} + 121203 T^{5} + 300763 T^{6} \)
$71$ \( 1 - 15 T + 225 T^{2} - 2111 T^{3} + 15975 T^{4} - 75615 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 6 T + 102 T^{2} - 177 T^{3} + 7446 T^{4} - 31974 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 3 T + 237 T^{2} + 473 T^{3} + 18723 T^{4} + 18723 T^{5} + 493039 T^{6} \)
$83$ \( 1 - 6 T + 132 T^{2} - 675 T^{3} + 10956 T^{4} - 41334 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 3 T + 159 T^{2} - 747 T^{3} + 14151 T^{4} - 23763 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 9 T + 261 T^{2} - 1439 T^{3} + 25317 T^{4} - 84681 T^{5} + 912673 T^{6} \)
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