Properties

Label 4026.2.a.u
Level 4026
Weight 2
Character orbit 4026.a
Self dual yes
Analytic conductor 32.148
Analytic rank 0
Dimension 5
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: \(5\)
Coefficient field: 5.5.9176805.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 12 x^{3} + 7 x^{2} + 30 x - 20\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 8 \nu^{2} + \nu + 6 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 10 \nu^{2} + 3 \nu + 14 \)\()/2\)
\(\beta_{4}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 7 \nu + 8 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 9 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} - 10 \beta_{3} + 12 \beta_{2} + 16 \beta_{1} + 26\)

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
3.32361
1.62906
0.689091
−2.15973
−2.48204
−1.00000 1.00000 1.00000 −3.32361 −1.00000 −3.91566 −1.00000 1.00000 3.32361
1.2 −1.00000 1.00000 1.00000 −1.62906 −1.00000 −4.09482 −1.00000 1.00000 1.62906
1.3 −1.00000 1.00000 1.00000 −0.689091 −1.00000 4.91945 −1.00000 1.00000 0.689091
1.4 −1.00000 1.00000 1.00000 2.15973 −1.00000 −1.48663 −1.00000 1.00000 −2.15973
1.5 −1.00000 1.00000 1.00000 2.48204 −1.00000 1.57766 −1.00000 1.00000 −2.48204
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.u 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.u 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{5} + T_{5}^{4} - 12 T_{5}^{3} - 7 T_{5}^{2} + 30 T_{5} + 20 \)
\( T_{7}^{5} + 3 T_{7}^{4} - 26 T_{7}^{3} - 84 T_{7}^{2} + 62 T_{7} + 185 \)
\( T_{13}^{5} - 2 T_{13}^{4} - 34 T_{13}^{3} + 56 T_{13}^{2} + 193 T_{13} - 339 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{5} \)
$3$ \( ( 1 - T )^{5} \)
$5$ \( 1 + T + 13 T^{2} + 13 T^{3} + 100 T^{4} + 100 T^{5} + 500 T^{6} + 325 T^{7} + 1625 T^{8} + 625 T^{9} + 3125 T^{10} \)
$7$ \( 1 + 3 T + 9 T^{2} + 6 T^{4} - 109 T^{5} + 42 T^{6} + 3087 T^{8} + 7203 T^{9} + 16807 T^{10} \)
$11$ \( ( 1 + T )^{5} \)
$13$ \( 1 - 2 T + 31 T^{2} - 48 T^{3} + 557 T^{4} - 911 T^{5} + 7241 T^{6} - 8112 T^{7} + 68107 T^{8} - 57122 T^{9} + 371293 T^{10} \)
$17$ \( 1 - 6 T + 54 T^{2} - 207 T^{3} + 995 T^{4} - 3429 T^{5} + 16915 T^{6} - 59823 T^{7} + 265302 T^{8} - 501126 T^{9} + 1419857 T^{10} \)
$19$ \( 1 - 7 T + 31 T^{2} - 114 T^{3} + 536 T^{4} - 3085 T^{5} + 10184 T^{6} - 41154 T^{7} + 212629 T^{8} - 912247 T^{9} + 2476099 T^{10} \)
$23$ \( 1 + 12 T + 128 T^{2} + 909 T^{3} + 5835 T^{4} + 29074 T^{5} + 134205 T^{6} + 480861 T^{7} + 1557376 T^{8} + 3358092 T^{9} + 6436343 T^{10} \)
$29$ \( 1 - 4 T + 96 T^{2} - 109 T^{3} + 3365 T^{4} + 970 T^{5} + 97585 T^{6} - 91669 T^{7} + 2341344 T^{8} - 2829124 T^{9} + 20511149 T^{10} \)
$31$ \( 1 + T + 133 T^{2} + 111 T^{3} + 7658 T^{4} + 5020 T^{5} + 237398 T^{6} + 106671 T^{7} + 3962203 T^{8} + 923521 T^{9} + 28629151 T^{10} \)
$37$ \( 1 - 3 T + 103 T^{2} - 174 T^{3} + 5588 T^{4} - 8037 T^{5} + 206756 T^{6} - 238206 T^{7} + 5217259 T^{8} - 5622483 T^{9} + 69343957 T^{10} \)
$41$ \( 1 + 3 T + 112 T^{2} + 213 T^{3} + 6631 T^{4} + 7488 T^{5} + 271871 T^{6} + 358053 T^{7} + 7719152 T^{8} + 8477283 T^{9} + 115856201 T^{10} \)
$43$ \( 1 - 24 T + 400 T^{2} - 4559 T^{3} + 41953 T^{4} - 301526 T^{5} + 1803979 T^{6} - 8429591 T^{7} + 31802800 T^{8} - 82051224 T^{9} + 147008443 T^{10} \)
$47$ \( 1 - 18 T + 261 T^{2} - 2562 T^{3} + 23795 T^{4} - 170127 T^{5} + 1118365 T^{6} - 5659458 T^{7} + 27097803 T^{8} - 87834258 T^{9} + 229345007 T^{10} \)
$53$ \( 1 + 13 T + 305 T^{2} + 2764 T^{3} + 34410 T^{4} + 219955 T^{5} + 1823730 T^{6} + 7764076 T^{7} + 45407485 T^{8} + 102576253 T^{9} + 418195493 T^{10} \)
$59$ \( 1 + 16 T + 187 T^{2} + 2226 T^{3} + 21689 T^{4} + 167401 T^{5} + 1279651 T^{6} + 7748706 T^{7} + 38405873 T^{8} + 193877776 T^{9} + 714924299 T^{10} \)
$61$ \( ( 1 + T )^{5} \)
$67$ \( 1 - 18 T + 279 T^{2} - 2904 T^{3} + 32349 T^{4} - 267409 T^{5} + 2167383 T^{6} - 13036056 T^{7} + 83912877 T^{8} - 362720178 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 + 31 T + 687 T^{2} + 10159 T^{3} + 122096 T^{4} + 1126272 T^{5} + 8668816 T^{6} + 51211519 T^{7} + 245884857 T^{8} + 787762111 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 - 8 T + 176 T^{2} - 691 T^{3} + 12623 T^{4} - 18442 T^{5} + 921479 T^{6} - 3682339 T^{7} + 68466992 T^{8} - 227185928 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 - 32 T + 521 T^{2} - 6042 T^{3} + 60211 T^{4} - 551709 T^{5} + 4756669 T^{6} - 37708122 T^{7} + 256873319 T^{8} - 1246402592 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 - 8 T + 222 T^{2} - 875 T^{3} + 16877 T^{4} - 33826 T^{5} + 1400791 T^{6} - 6027875 T^{7} + 126936714 T^{8} - 379666568 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 - T + 249 T^{2} + 215 T^{3} + 30584 T^{4} + 42508 T^{5} + 2721976 T^{6} + 1703015 T^{7} + 175537281 T^{8} - 62742241 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 + 4 T + 361 T^{2} + 1626 T^{3} + 61187 T^{4} + 232277 T^{5} + 5935139 T^{6} + 15299034 T^{7} + 329474953 T^{8} + 354117124 T^{9} + 8587340257 T^{10} \)
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