Properties

Label 4014.2.a.s
Level 4014
Weight 2
Character orbit 4014.a
Self dual yes
Analytic conductor 32.052
Analytic rank 1
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4014.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0519513713\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.232773917.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1338)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( -1 + \beta_{1} ) q^{5} + ( 1 + \beta_{2} ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + ( -1 + \beta_{1} ) q^{5} + ( 1 + \beta_{2} ) q^{7} - q^{8} + ( 1 - \beta_{1} ) q^{10} + ( -\beta_{1} - \beta_{5} ) q^{11} + ( 1 - \beta_{2} - \beta_{5} ) q^{13} + ( -1 - \beta_{2} ) q^{14} + q^{16} + ( -2 - \beta_{3} + \beta_{4} + \beta_{5} ) q^{17} + ( -1 + \beta_{1} - \beta_{3} + \beta_{5} ) q^{19} + ( -1 + \beta_{1} ) q^{20} + ( \beta_{1} + \beta_{5} ) q^{22} + ( -\beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{23} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{25} + ( -1 + \beta_{2} + \beta_{5} ) q^{26} + ( 1 + \beta_{2} ) q^{28} + ( -1 + \beta_{3} + \beta_{5} ) q^{29} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{31} - q^{32} + ( 2 + \beta_{3} - \beta_{4} - \beta_{5} ) q^{34} + ( -2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{35} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{37} + ( 1 - \beta_{1} + \beta_{3} - \beta_{5} ) q^{38} + ( 1 - \beta_{1} ) q^{40} + ( -2 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{41} + ( -2 + \beta_{1} - \beta_{3} - \beta_{4} ) q^{43} + ( -\beta_{1} - \beta_{5} ) q^{44} + ( \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{46} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{47} + ( -\beta_{1} + \beta_{4} - \beta_{5} ) q^{49} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{50} + ( 1 - \beta_{2} - \beta_{5} ) q^{52} + ( -1 - \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{53} + ( -3 + \beta_{1} - \beta_{3} + 2 \beta_{5} ) q^{55} + ( -1 - \beta_{2} ) q^{56} + ( 1 - \beta_{3} - \beta_{5} ) q^{58} + ( -3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{59} + ( -1 + 2 \beta_{2} - \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{61} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{62} + q^{64} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{65} + ( -1 - \beta_{1} + 2 \beta_{4} ) q^{67} + ( -2 - \beta_{3} + \beta_{4} + \beta_{5} ) q^{68} + ( 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{70} + ( -1 + \beta_{1} - 4 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{71} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{73} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{74} + ( -1 + \beta_{1} - \beta_{3} + \beta_{5} ) q^{76} + ( 2 - 3 \beta_{1} + \beta_{2} - \beta_{4} - 3 \beta_{5} ) q^{77} + ( -2 + 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{79} + ( -1 + \beta_{1} ) q^{80} + ( 2 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{82} + ( 1 - 3 \beta_{2} + 3 \beta_{3} ) q^{83} + ( -3 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{85} + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} ) q^{86} + ( \beta_{1} + \beta_{5} ) q^{88} + ( 1 + 2 \beta_{1} + \beta_{2} - 4 \beta_{4} - 2 \beta_{5} ) q^{89} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{91} + ( -\beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{92} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{94} + ( 2 - 3 \beta_{1} - \beta_{2} + \beta_{4} - 3 \beta_{5} ) q^{95} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{97} + ( \beta_{1} - \beta_{4} + \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} + 6q^{4} - 6q^{5} + 5q^{7} - 6q^{8} + O(q^{10}) \) \( 6q - 6q^{2} + 6q^{4} - 6q^{5} + 5q^{7} - 6q^{8} + 6q^{10} - q^{11} + 6q^{13} - 5q^{14} + 6q^{16} - 10q^{17} - 4q^{19} - 6q^{20} + q^{22} - 2q^{23} - 6q^{26} + 5q^{28} - 6q^{29} - 5q^{31} - 6q^{32} + 10q^{34} + 2q^{35} + q^{37} + 4q^{38} + 6q^{40} - 12q^{41} - 11q^{43} - q^{44} + 2q^{46} - 5q^{47} - q^{49} + 6q^{52} - 4q^{53} - 15q^{55} - 5q^{56} + 6q^{58} - q^{59} - 8q^{61} + 5q^{62} + 6q^{64} - 5q^{65} - 6q^{67} - 10q^{68} - 2q^{70} - q^{71} - 10q^{73} - q^{74} - 4q^{76} + 8q^{77} - 10q^{79} - 6q^{80} + 12q^{82} + 6q^{83} - 2q^{85} + 11q^{86} + q^{88} + 3q^{89} - 16q^{91} - 2q^{92} + 5q^{94} + 10q^{95} + 3q^{97} + q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 12 x^{4} - x^{3} + 33 x^{2} + 5 x - 22\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{5} - \nu^{4} - 10 \nu^{3} + 9 \nu^{2} + 15 \nu - 11 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - \nu^{4} - 10 \nu^{3} + 10 \nu^{2} + 15 \nu - 15 \)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{5} - 3 \nu^{4} - 31 \nu^{3} + 28 \nu^{2} + 51 \nu - 36 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{5} + 7 \nu^{4} + 51 \nu^{3} - 66 \nu^{2} - 81 \nu + 86 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 6 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{5} + \beta_{4} + 10 \beta_{3} - 9 \beta_{2} + 26\)
\(\nu^{5}\)\(=\)\(\beta_{5} - 19 \beta_{4} + 11 \beta_{3} + 21 \beta_{2} + 45 \beta_{1} + 11\)

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
−2.85387
−1.36937
−1.29309
0.862793
1.72012
2.93341
−1.00000 0 1.00000 −3.85387 0 −2.71491 −1.00000 0 3.85387
1.2 −1.00000 0 1.00000 −2.36937 0 3.68274 −1.00000 0 2.36937
1.3 −1.00000 0 1.00000 −2.29309 0 0.862607 −1.00000 0 2.29309
1.4 −1.00000 0 1.00000 −0.137207 0 3.14283 −1.00000 0 0.137207
1.5 −1.00000 0 1.00000 0.720124 0 −2.15975 −1.00000 0 −0.720124
1.6 −1.00000 0 1.00000 1.93341 0 2.18648 −1.00000 0 −1.93341
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 4014.2.a.s 6
3.b odd 2 1 1338.2.a.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1338.2.a.i 6 3.b odd 2 1
4014.2.a.s 6 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(223\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\( T_{5}^{6} + 6 T_{5}^{5} + 3 T_{5}^{4} - 29 T_{5}^{3} - 27 T_{5}^{2} + 26 T_{5} + 4 \)
\( T_{7}^{6} - 5 T_{7}^{5} - 8 T_{7}^{4} + 61 T_{7}^{3} - 12 T_{7}^{2} - 176 T_{7} + 128 \)
\( T_{11}^{6} + T_{11}^{5} - 26 T_{11}^{4} - 80 T_{11}^{3} - 56 T_{11}^{2} + 13 T_{11} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{6} \)
$3$ \( \)
$5$ \( 1 + 6 T + 33 T^{2} + 121 T^{3} + 408 T^{4} + 1091 T^{5} + 2684 T^{6} + 5455 T^{7} + 10200 T^{8} + 15125 T^{9} + 20625 T^{10} + 18750 T^{11} + 15625 T^{12} \)
$7$ \( 1 - 5 T + 34 T^{2} - 114 T^{3} + 499 T^{4} - 1345 T^{5} + 4468 T^{6} - 9415 T^{7} + 24451 T^{8} - 39102 T^{9} + 81634 T^{10} - 84035 T^{11} + 117649 T^{12} \)
$11$ \( 1 + T + 40 T^{2} - 25 T^{3} + 615 T^{4} - 1417 T^{5} + 6516 T^{6} - 15587 T^{7} + 74415 T^{8} - 33275 T^{9} + 585640 T^{10} + 161051 T^{11} + 1771561 T^{12} \)
$13$ \( 1 - 6 T + 68 T^{2} - 322 T^{3} + 2040 T^{4} - 7657 T^{5} + 34410 T^{6} - 99541 T^{7} + 344760 T^{8} - 707434 T^{9} + 1942148 T^{10} - 2227758 T^{11} + 4826809 T^{12} \)
$17$ \( 1 + 10 T + 97 T^{2} + 649 T^{3} + 3893 T^{4} + 18857 T^{5} + 86090 T^{6} + 320569 T^{7} + 1125077 T^{8} + 3188537 T^{9} + 8101537 T^{10} + 14198570 T^{11} + 24137569 T^{12} \)
$19$ \( 1 + 4 T + 70 T^{2} + 306 T^{3} + 2570 T^{4} + 10439 T^{5} + 59754 T^{6} + 198341 T^{7} + 927770 T^{8} + 2098854 T^{9} + 9122470 T^{10} + 9904396 T^{11} + 47045881 T^{12} \)
$23$ \( 1 + 2 T + 65 T^{2} + 235 T^{3} + 2357 T^{4} + 9465 T^{5} + 62842 T^{6} + 217695 T^{7} + 1246853 T^{8} + 2859245 T^{9} + 18189665 T^{10} + 12872686 T^{11} + 148035889 T^{12} \)
$29$ \( 1 + 6 T + 142 T^{2} + 710 T^{3} + 9120 T^{4} + 36961 T^{5} + 338694 T^{6} + 1071869 T^{7} + 7669920 T^{8} + 17316190 T^{9} + 100433902 T^{10} + 123066894 T^{11} + 594823321 T^{12} \)
$31$ \( 1 + 5 T + 126 T^{2} + 530 T^{3} + 7933 T^{4} + 27909 T^{5} + 306992 T^{6} + 865179 T^{7} + 7623613 T^{8} + 15789230 T^{9} + 116363646 T^{10} + 143145755 T^{11} + 887503681 T^{12} \)
$37$ \( 1 - T + 127 T^{2} + 61 T^{3} + 8406 T^{4} + 5520 T^{5} + 382600 T^{6} + 204240 T^{7} + 11507814 T^{8} + 3089833 T^{9} + 238018447 T^{10} - 69343957 T^{11} + 2565726409 T^{12} \)
$41$ \( 1 + 12 T + 251 T^{2} + 1995 T^{3} + 23819 T^{4} + 141129 T^{5} + 1246146 T^{6} + 5786289 T^{7} + 40039739 T^{8} + 137497395 T^{9} + 709266011 T^{10} + 1390274412 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 + 11 T + 246 T^{2} + 1877 T^{3} + 24479 T^{4} + 143495 T^{5} + 1364840 T^{6} + 6170285 T^{7} + 45261671 T^{8} + 149234639 T^{9} + 841025046 T^{10} + 1617092873 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 + 5 T + 204 T^{2} + 1111 T^{3} + 19280 T^{4} + 101170 T^{5} + 1117673 T^{6} + 4754990 T^{7} + 42589520 T^{8} + 115347353 T^{9} + 995454924 T^{10} + 1146725035 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 + 4 T + 232 T^{2} + 684 T^{3} + 24888 T^{4} + 57715 T^{5} + 1635504 T^{6} + 3058895 T^{7} + 69910392 T^{8} + 101831868 T^{9} + 1830591592 T^{10} + 1672781972 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 + T + 198 T^{2} + 531 T^{3} + 16363 T^{4} + 74891 T^{5} + 963204 T^{6} + 4418569 T^{7} + 56959603 T^{8} + 109056249 T^{9} + 2399237478 T^{10} + 714924299 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 + 8 T + 96 T^{2} + 868 T^{3} + 12058 T^{4} + 71629 T^{5} + 630206 T^{6} + 4369369 T^{7} + 44867818 T^{8} + 197019508 T^{9} + 1329200736 T^{10} + 6756770408 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 6 T + 341 T^{2} + 1697 T^{3} + 51708 T^{4} + 207911 T^{5} + 4466256 T^{6} + 13930037 T^{7} + 232117212 T^{8} + 510394811 T^{9} + 6871532261 T^{10} + 8100750642 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 + T + 157 T^{2} - 254 T^{3} + 17047 T^{4} - 15675 T^{5} + 1462102 T^{6} - 1112925 T^{7} + 85933927 T^{8} - 90909394 T^{9} + 3989633917 T^{10} + 1804229351 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 + 10 T + 249 T^{2} + 2366 T^{3} + 32637 T^{4} + 293431 T^{5} + 2852527 T^{6} + 21420463 T^{7} + 173922573 T^{8} + 920414222 T^{9} + 7071162009 T^{10} + 20730715930 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 + 10 T + 165 T^{2} + 1548 T^{3} + 19319 T^{4} + 124055 T^{5} + 1493471 T^{6} + 9800345 T^{7} + 120569879 T^{8} + 763224372 T^{9} + 6426763365 T^{10} + 30770563990 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 - 6 T + 243 T^{2} - 2429 T^{3} + 38130 T^{4} - 303279 T^{5} + 4244252 T^{6} - 25172157 T^{7} + 262677570 T^{8} - 1388870623 T^{9} + 11532372003 T^{10} - 23634243858 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 - 3 T + 212 T^{2} + 432 T^{3} + 27629 T^{4} + 18147 T^{5} + 3437316 T^{6} + 1615083 T^{7} + 218849309 T^{8} + 304546608 T^{9} + 13301355092 T^{10} - 16752178347 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 - 3 T + 506 T^{2} - 1466 T^{3} + 113327 T^{4} - 280531 T^{5} + 14292972 T^{6} - 27211507 T^{7} + 1066293743 T^{8} - 1337978618 T^{9} + 44795816186 T^{10} - 25762020771 T^{11} + 832972004929 T^{12} \)
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