Properties

Label 4014.2.a.r
Level 4014
Weight 2
Character orbit 4014.a
Self dual yes
Analytic conductor 32.052
Analytic rank 1
Dimension 5
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: \(5\)
Coefficient field: 5.5.356173.1
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( -1 + \beta_{1} - \beta_{2} ) q^{5} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( -1 + \beta_{1} - \beta_{2} ) q^{5} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{7} + q^{8} + ( -1 + \beta_{1} - \beta_{2} ) q^{10} + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{11} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{14} + q^{16} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{17} + ( -1 + \beta_{2} + \beta_{3} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} ) q^{20} + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{22} + ( -2 - \beta_{1} + 2 \beta_{4} ) q^{23} + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{25} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{26} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{28} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{29} + ( -1 + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{31} + q^{32} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{34} + ( -3 - \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{35} + ( -\beta_{1} + 4 \beta_{3} - 2 \beta_{4} ) q^{37} + ( -1 + \beta_{2} + \beta_{3} ) q^{38} + ( -1 + \beta_{1} - \beta_{2} ) q^{40} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{41} + ( 4 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{43} + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{44} + ( -2 - \beta_{1} + 2 \beta_{4} ) q^{46} + ( -4 - \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{47} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{49} + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{50} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{52} + ( -5 + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{53} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{55} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{56} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{58} + ( -5 + \beta_{1} + 4 \beta_{2} + 3 \beta_{4} ) q^{59} + ( -5 + 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{61} + ( -1 + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{62} + q^{64} + ( -1 + 3 \beta_{2} - 2 \beta_{3} ) q^{65} + ( -3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 4 \beta_{4} ) q^{67} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{68} + ( -3 - \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{70} + ( -2 - \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{71} + ( 3 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{73} + ( -\beta_{1} + 4 \beta_{3} - 2 \beta_{4} ) q^{74} + ( -1 + \beta_{2} + \beta_{3} ) q^{76} + ( -1 - 2 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} ) q^{77} + ( -3 - \beta_{1} + \beta_{2} - 4 \beta_{3} + 5 \beta_{4} ) q^{79} + ( -1 + \beta_{1} - \beta_{2} ) q^{80} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{82} + ( -3 + \beta_{2} - 6 \beta_{3} + 2 \beta_{4} ) q^{83} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{85} + ( 4 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{86} + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{88} + ( \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{89} + ( -3 + 3 \beta_{1} - \beta_{3} + \beta_{4} ) q^{91} + ( -2 - \beta_{1} + 2 \beta_{4} ) q^{92} + ( -4 - \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{94} + ( -1 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{95} + ( -3 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{97} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{2} + 5q^{4} - 5q^{5} - q^{7} + 5q^{8} + O(q^{10}) \) \( 5q + 5q^{2} + 5q^{4} - 5q^{5} - q^{7} + 5q^{8} - 5q^{10} - 9q^{11} - q^{14} + 5q^{16} - 6q^{17} - 4q^{19} - 5q^{20} - 9q^{22} - 16q^{23} + 8q^{25} - q^{28} - 8q^{29} - q^{31} + 5q^{32} - 6q^{34} - 22q^{35} - 2q^{37} - 4q^{38} - 5q^{40} - 4q^{41} + 3q^{43} - 9q^{44} - 16q^{46} - 18q^{47} + 2q^{49} + 8q^{50} - 26q^{53} + q^{55} - q^{56} - 8q^{58} - 21q^{59} - 20q^{61} - q^{62} + 5q^{64} + 3q^{65} - 5q^{67} - 6q^{68} - 22q^{70} - 17q^{71} + 5q^{73} - 2q^{74} - 4q^{76} - 2q^{77} - 21q^{79} - 5q^{80} - 4q^{82} - 11q^{83} - 12q^{85} + 3q^{86} - 9q^{88} + 5q^{89} - 10q^{91} - 16q^{92} - 18q^{94} - 10q^{95} - 11q^{97} + 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 7 x^{3} + 9 x^{2} + 14 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 1 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 3 \nu^{2} + 5 \nu - 2 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 7 \nu + 4 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{4} + \beta_{3} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{4} + \beta_{3} + \beta_{2} + 5 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-5 \beta_{4} + 7 \beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 15\)

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
−1.38363
2.75496
−1.85688
0.253142
2.23241
1.00000 0 1.00000 −4.35484 0 3.05676 1.00000 0 −4.35484
1.2 1.00000 0 1.00000 −1.54501 0 −1.28984 1.00000 0 −1.54501
1.3 1.00000 0 1.00000 −1.43386 0 −1.87103 1.00000 0 −1.43386
1.4 1.00000 0 1.00000 −0.686428 0 2.87549 1.00000 0 −0.686428
1.5 1.00000 0 1.00000 3.02013 0 −3.77138 1.00000 0 3.02013
\(n\): 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 4014.2.a.r 5
3.b odd 2 1 1338.2.a.h 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1338.2.a.h 5 3.b odd 2 1
4014.2.a.r 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\( T_{5}^{5} + 5 T_{5}^{4} - 4 T_{5}^{3} - 41 T_{5}^{2} - 54 T_{5} - 20 \)
\( T_{7}^{5} + T_{7}^{4} - 18 T_{7}^{3} - 15 T_{7}^{2} + 72 T_{7} + 80 \)
\( T_{11}^{5} + 9 T_{11}^{4} + 6 T_{11}^{3} - 82 T_{11}^{2} - 38 T_{11} + 67 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{5} \)
$3$ \( \)
$5$ \( 1 + 5 T + 21 T^{2} + 59 T^{3} + 136 T^{4} + 320 T^{5} + 680 T^{6} + 1475 T^{7} + 2625 T^{8} + 3125 T^{9} + 3125 T^{10} \)
$7$ \( 1 + T + 17 T^{2} + 13 T^{3} + 184 T^{4} + 164 T^{5} + 1288 T^{6} + 637 T^{7} + 5831 T^{8} + 2401 T^{9} + 16807 T^{10} \)
$11$ \( 1 + 9 T + 61 T^{2} + 314 T^{3} + 1370 T^{4} + 4797 T^{5} + 15070 T^{6} + 37994 T^{7} + 81191 T^{8} + 131769 T^{9} + 161051 T^{10} \)
$13$ \( 1 + 41 T^{2} - 54 T^{3} + 735 T^{4} - 1391 T^{5} + 9555 T^{6} - 9126 T^{7} + 90077 T^{8} + 371293 T^{10} \)
$17$ \( 1 + 6 T + 80 T^{2} + 341 T^{3} + 2587 T^{4} + 8190 T^{5} + 43979 T^{6} + 98549 T^{7} + 393040 T^{8} + 501126 T^{9} + 1419857 T^{10} \)
$19$ \( 1 + 4 T + 79 T^{2} + 216 T^{3} + 2575 T^{4} + 5267 T^{5} + 48925 T^{6} + 77976 T^{7} + 541861 T^{8} + 521284 T^{9} + 2476099 T^{10} \)
$23$ \( 1 + 16 T + 182 T^{2} + 1407 T^{3} + 9129 T^{4} + 46862 T^{5} + 209967 T^{6} + 744303 T^{7} + 2214394 T^{8} + 4477456 T^{9} + 6436343 T^{10} \)
$29$ \( 1 + 8 T + 87 T^{2} + 554 T^{3} + 3949 T^{4} + 18747 T^{5} + 114521 T^{6} + 465914 T^{7} + 2121843 T^{8} + 5658248 T^{9} + 20511149 T^{10} \)
$31$ \( 1 + T + 101 T^{2} + 143 T^{3} + 5000 T^{4} + 6444 T^{5} + 155000 T^{6} + 137423 T^{7} + 3008891 T^{8} + 923521 T^{9} + 28629151 T^{10} \)
$37$ \( 1 + 2 T + 72 T^{2} - 51 T^{3} + 3779 T^{4} - 146 T^{5} + 139823 T^{6} - 69819 T^{7} + 3647016 T^{8} + 3748322 T^{9} + 69343957 T^{10} \)
$41$ \( 1 + 4 T + 96 T^{2} - 57 T^{3} + 3551 T^{4} - 12502 T^{5} + 145591 T^{6} - 95817 T^{7} + 6616416 T^{8} + 11303044 T^{9} + 115856201 T^{10} \)
$43$ \( 1 - 3 T + 27 T^{2} + 246 T^{3} + 2882 T^{4} - 10727 T^{5} + 123926 T^{6} + 454854 T^{7} + 2146689 T^{8} - 10256403 T^{9} + 147008443 T^{10} \)
$47$ \( 1 + 18 T + 275 T^{2} + 2774 T^{3} + 24661 T^{4} + 177861 T^{5} + 1159067 T^{6} + 6127766 T^{7} + 28551325 T^{8} + 87834258 T^{9} + 229345007 T^{10} \)
$53$ \( 1 + 26 T + 501 T^{2} + 6338 T^{3} + 66181 T^{4} + 524547 T^{5} + 3507593 T^{6} + 17803442 T^{7} + 74587377 T^{8} + 205152506 T^{9} + 418195493 T^{10} \)
$59$ \( 1 + 21 T + 191 T^{2} + 1174 T^{3} + 7728 T^{4} + 57933 T^{5} + 455952 T^{6} + 4086694 T^{7} + 39227389 T^{8} + 254464581 T^{9} + 714924299 T^{10} \)
$61$ \( 1 + 20 T + 309 T^{2} + 3002 T^{3} + 27263 T^{4} + 201443 T^{5} + 1663043 T^{6} + 11170442 T^{7} + 70137129 T^{8} + 276916820 T^{9} + 844596301 T^{10} \)
$67$ \( 1 + 5 T + 207 T^{2} + 1391 T^{3} + 20426 T^{4} + 142580 T^{5} + 1368542 T^{6} + 6244199 T^{7} + 62257941 T^{8} + 100755605 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 + 17 T + 390 T^{2} + 4187 T^{3} + 54301 T^{4} + 418152 T^{5} + 3855371 T^{6} + 21106667 T^{7} + 139585290 T^{8} + 431998577 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 - 5 T + 247 T^{2} - 1122 T^{3} + 28636 T^{4} - 113547 T^{5} + 2090428 T^{6} - 5979138 T^{7} + 96087199 T^{8} - 141991205 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 + 21 T + 385 T^{2} + 3818 T^{3} + 39648 T^{4} + 299687 T^{5} + 3132192 T^{6} + 23828138 T^{7} + 189820015 T^{8} + 817951701 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 + 11 T + 213 T^{2} + 2511 T^{3} + 31820 T^{4} + 244036 T^{5} + 2641060 T^{6} + 17298279 T^{7} + 121790631 T^{8} + 522041531 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 - 5 T + 391 T^{2} - 1393 T^{3} + 64134 T^{4} - 168644 T^{5} + 5707926 T^{6} - 11033953 T^{7} + 275642879 T^{8} - 313711205 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 + 11 T + 347 T^{2} + 3389 T^{3} + 60520 T^{4} + 444664 T^{5} + 5870440 T^{6} + 31887101 T^{7} + 316697531 T^{8} + 973822091 T^{9} + 8587340257 T^{10} \)
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