Properties

Label 4026.2.a.q
Level 4026
Weight 2
Character orbit 4026.a
Self dual yes
Analytic conductor 32.148
Analytic rank 0
Dimension 4
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.6809.1
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} - x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1} + 1\)

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
−2.06963
−0.582772
0.361989
2.29041
−1.00000 −1.00000 1.00000 −2.06963 1.00000 −4.35299 −1.00000 1.00000 2.06963
1.2 −1.00000 −1.00000 1.00000 −0.582772 1.00000 1.07760 −1.00000 1.00000 0.582772
1.3 −1.00000 −1.00000 1.00000 0.361989 1.00000 2.23095 −1.00000 1.00000 −0.361989
1.4 −1.00000 −1.00000 1.00000 2.29041 1.00000 −0.955570 −1.00000 1.00000 −2.29041
\(n\): 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.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.q 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{4} - 5 T_{5}^{2} - T_{5} + 1 \)
\( T_{7}^{4} + 2 T_{7}^{3} - 11 T_{7}^{2} - T_{7} + 10 \)
\( T_{13}^{4} + 4 T_{13}^{3} - 21 T_{13}^{2} + 9 T_{13} + 17 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( 1 + 15 T^{2} - T^{3} + 101 T^{4} - 5 T^{5} + 375 T^{6} + 625 T^{8} \)
$7$ \( 1 + 2 T + 17 T^{2} + 41 T^{3} + 150 T^{4} + 287 T^{5} + 833 T^{6} + 686 T^{7} + 2401 T^{8} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( 1 + 4 T + 31 T^{2} + 165 T^{3} + 485 T^{4} + 2145 T^{5} + 5239 T^{6} + 8788 T^{7} + 28561 T^{8} \)
$17$ \( 1 + T + 43 T^{2} + 26 T^{3} + 892 T^{4} + 442 T^{5} + 12427 T^{6} + 4913 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 4 T + 35 T^{2} - 151 T^{3} + 600 T^{4} - 2869 T^{5} + 12635 T^{6} - 27436 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 16 T + 161 T^{2} - 1085 T^{3} + 5920 T^{4} - 24955 T^{5} + 85169 T^{6} - 194672 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 5 T + 83 T^{2} + 319 T^{3} + 3271 T^{4} + 9251 T^{5} + 69803 T^{6} + 121945 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 10 T + 97 T^{2} + 423 T^{3} + 2897 T^{4} + 13113 T^{5} + 93217 T^{6} + 297910 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 10 T + 91 T^{2} + 727 T^{3} + 5542 T^{4} + 26899 T^{5} + 124579 T^{6} + 506530 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 16 T + 162 T^{2} - 984 T^{3} + 6251 T^{4} - 40344 T^{5} + 272322 T^{6} - 1102736 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + 8 T + 121 T^{2} + 591 T^{3} + 5984 T^{4} + 25413 T^{5} + 223729 T^{6} + 636056 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 7 T + 120 T^{2} + 770 T^{3} + 8022 T^{4} + 36190 T^{5} + 265080 T^{6} + 726761 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 12 T + 143 T^{2} - 1135 T^{3} + 9944 T^{4} - 60155 T^{5} + 401687 T^{6} - 1786524 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 2 T + 191 T^{2} - 175 T^{3} + 15451 T^{4} - 10325 T^{5} + 664871 T^{6} - 410758 T^{7} + 12117361 T^{8} \)
$61$ \( ( 1 - T )^{4} \)
$67$ \( 1 + 5 T + 116 T^{2} + 312 T^{3} + 5862 T^{4} + 20904 T^{5} + 520724 T^{6} + 1503815 T^{7} + 20151121 T^{8} \)
$71$ \( 1 - 15 T + 288 T^{2} - 2726 T^{3} + 30728 T^{4} - 193546 T^{5} + 1451808 T^{6} - 5368665 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 28 T + 397 T^{2} + 3589 T^{3} + 29536 T^{4} + 261997 T^{5} + 2115613 T^{6} + 10892476 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 3 T + 200 T^{2} - 270 T^{3} + 20254 T^{4} - 21330 T^{5} + 1248200 T^{6} - 1479117 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 32 T + 649 T^{2} - 8773 T^{3} + 92400 T^{4} - 728159 T^{5} + 4470961 T^{6} - 18297184 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 16 T + 447 T^{2} + 4489 T^{3} + 63905 T^{4} + 399521 T^{5} + 3540687 T^{6} + 11279504 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 2 T + 199 T^{2} - 105 T^{3} + 24953 T^{4} - 10185 T^{5} + 1872391 T^{6} - 1825346 T^{7} + 88529281 T^{8} \)
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