Properties

Label 4014.2.a.v
Level 4014
Weight 2
Character orbit 4014.a
Self dual yes
Analytic conductor 32.052
Analytic rank 0
Dimension 7
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: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( -1 - \beta_{4} ) q^{5} -\beta_{3} q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + ( -1 - \beta_{4} ) q^{5} -\beta_{3} q^{7} - q^{8} + ( 1 + \beta_{4} ) q^{10} + ( \beta_{2} - \beta_{6} ) q^{11} + ( 1 + \beta_{2} - \beta_{5} ) q^{13} + \beta_{3} q^{14} + q^{16} + ( -3 - \beta_{3} - \beta_{5} ) q^{17} + ( \beta_{1} - \beta_{6} ) q^{19} + ( -1 - \beta_{4} ) q^{20} + ( -\beta_{2} + \beta_{6} ) q^{22} + ( -1 + \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} ) q^{23} + ( 3 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{25} + ( -1 - \beta_{2} + \beta_{5} ) q^{26} -\beta_{3} q^{28} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{29} + ( 1 - 2 \beta_{4} - \beta_{5} ) q^{31} - q^{32} + ( 3 + \beta_{3} + \beta_{5} ) q^{34} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{35} + ( 2 - 2 \beta_{1} - \beta_{4} - \beta_{5} ) q^{37} + ( -\beta_{1} + \beta_{6} ) q^{38} + ( 1 + \beta_{4} ) q^{40} + ( -4 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{41} + ( 1 + 4 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{43} + ( \beta_{2} - \beta_{6} ) q^{44} + ( 1 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} ) q^{46} + ( -2 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{47} + ( 3 + \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + \beta_{6} ) q^{49} + ( -3 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{50} + ( 1 + \beta_{2} - \beta_{5} ) q^{52} + ( -\beta_{1} + 2 \beta_{5} + \beta_{6} ) q^{53} + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{55} + \beta_{3} q^{56} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{58} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{59} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{61} + ( -1 + 2 \beta_{4} + \beta_{5} ) q^{62} + q^{64} + ( -1 - 2 \beta_{2} + 3 \beta_{5} ) q^{65} + ( -1 + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{67} + ( -3 - \beta_{3} - \beta_{5} ) q^{68} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{70} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{71} + ( 3 - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{73} + ( -2 + 2 \beta_{1} + \beta_{4} + \beta_{5} ) q^{74} + ( \beta_{1} - \beta_{6} ) q^{76} + ( -\beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{77} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{79} + ( -1 - \beta_{4} ) q^{80} + ( 4 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{82} + ( 3 \beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{83} + ( 3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{85} + ( -1 - 4 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{86} + ( -\beta_{2} + \beta_{6} ) q^{88} + ( 2 \beta_{1} + \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{89} + ( -1 - \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{91} + ( -1 + \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} ) q^{92} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{94} + ( 2 - 3 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 3 \beta_{6} ) q^{95} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{97} + ( -3 - \beta_{1} + 3 \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 7q^{2} + 7q^{4} - 6q^{5} + 3q^{7} - 7q^{8} + O(q^{10}) \) \( 7q - 7q^{2} + 7q^{4} - 6q^{5} + 3q^{7} - 7q^{8} + 6q^{10} + q^{11} + 8q^{13} - 3q^{14} + 7q^{16} - 16q^{17} + 2q^{19} - 6q^{20} - q^{22} - 8q^{23} + 19q^{25} - 8q^{26} + 3q^{28} - 4q^{29} + 11q^{31} - 7q^{32} + 16q^{34} + 17q^{37} - 2q^{38} + 6q^{40} - 18q^{41} - q^{43} + q^{44} + 8q^{46} - 5q^{47} + 24q^{49} - 19q^{50} + 8q^{52} - 6q^{53} + 21q^{55} - 3q^{56} + 4q^{58} + 7q^{59} + 24q^{61} - 11q^{62} + 7q^{64} - 11q^{65} - 4q^{67} - 16q^{68} + 7q^{71} + 28q^{73} - 17q^{74} + 2q^{76} + 2q^{77} - 6q^{80} + 18q^{82} + 2q^{83} + 4q^{85} + q^{86} - q^{88} - 5q^{89} + 2q^{91} - 8q^{92} + 5q^{94} + 14q^{95} + 23q^{97} - 24q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 18 x^{5} - 8 x^{4} + 51 x^{3} + 47 x^{2} - 2 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{6} - \nu^{5} - 17 \nu^{4} + 9 \nu^{3} + 42 \nu^{2} + 5 \nu - 8 \)
\(\beta_{3}\)\(=\)\((\)\( -9 \nu^{6} + 7 \nu^{5} + 156 \nu^{4} - 49 \nu^{3} - 413 \nu^{2} - 104 \nu + 88 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -21 \nu^{6} + 16 \nu^{5} + 366 \nu^{4} - 112 \nu^{3} - 989 \nu^{2} - 221 \nu + 226 \)\()/6\)
\(\beta_{5}\)\(=\)\((\)\( 11 \nu^{6} - 9 \nu^{5} - 191 \nu^{4} + 68 \nu^{3} + 512 \nu^{2} + 104 \nu - 123 \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( -14 \nu^{6} + 10 \nu^{5} + 245 \nu^{4} - 63 \nu^{3} - 670 \nu^{2} - 181 \nu + 148 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} - \beta_{3} - 2 \beta_{2} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(2 \beta_{6} - \beta_{5} - 4 \beta_{4} - \beta_{2} + 13 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(17 \beta_{6} + 11 \beta_{5} - 4 \beta_{4} - 18 \beta_{3} - 29 \beta_{2} + 18 \beta_{1} + 59\)
\(\nu^{5}\)\(=\)\(40 \beta_{6} - 17 \beta_{5} - 70 \beta_{4} - 11 \beta_{3} - 29 \beta_{2} + 188 \beta_{1} + 57\)
\(\nu^{6}\)\(=\)\(269 \beta_{6} + 137 \beta_{5} - 102 \beta_{4} - 275 \beta_{3} - 428 \beta_{2} + 330 \beta_{1} + 831\)

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
2.01737
−3.57857
0.369925
−0.718922
4.01465
−1.31546
−0.788998
−1.00000 0 1.00000 −3.79108 0 −2.04653 −1.00000 0 3.79108
1.2 −1.00000 0 1.00000 −3.27125 0 3.11004 −1.00000 0 3.27125
1.3 −1.00000 0 1.00000 −2.69148 0 2.17419 −1.00000 0 2.69148
1.4 −1.00000 0 1.00000 −2.18886 0 −2.20076 −1.00000 0 2.18886
1.5 −1.00000 0 1.00000 0.603045 0 4.55050 −1.00000 0 −0.603045
1.6 −1.00000 0 1.00000 1.60399 0 −4.86544 −1.00000 0 −1.60399
1.7 −1.00000 0 1.00000 3.73564 0 2.27800 −1.00000 0 −3.73564
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.v 7
3.b odd 2 1 1338.2.a.j 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1338.2.a.j 7 3.b odd 2 1
4014.2.a.v 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\( T_{5}^{7} + 6 T_{5}^{6} - 9 T_{5}^{5} - 105 T_{5}^{4} - 91 T_{5}^{3} + 316 T_{5}^{2} + 304 T_{5} - 264 \)
\( T_{7}^{7} - 3 T_{7}^{6} - 32 T_{7}^{5} + 101 T_{7}^{4} + 224 T_{7}^{3} - 736 T_{7}^{2} - 448 T_{7} + 1536 \)
\( T_{11}^{7} - T_{11}^{6} - 48 T_{11}^{5} + 8 T_{11}^{4} + 624 T_{11}^{3} + 305 T_{11}^{2} - 2384 T_{11} - 2512 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{7} \)
$3$ 1
$5$ \( 1 + 6 T + 26 T^{2} + 75 T^{3} + 209 T^{4} + 466 T^{5} + 1064 T^{6} + 2146 T^{7} + 5320 T^{8} + 11650 T^{9} + 26125 T^{10} + 46875 T^{11} + 81250 T^{12} + 93750 T^{13} + 78125 T^{14} \)
$7$ \( 1 - 3 T + 17 T^{2} - 25 T^{3} + 133 T^{4} - 113 T^{5} + 581 T^{6} + 346 T^{7} + 4067 T^{8} - 5537 T^{9} + 45619 T^{10} - 60025 T^{11} + 285719 T^{12} - 352947 T^{13} + 823543 T^{14} \)
$11$ \( 1 - T + 29 T^{2} - 58 T^{3} + 525 T^{4} - 1158 T^{5} + 6713 T^{6} - 16614 T^{7} + 73843 T^{8} - 140118 T^{9} + 698775 T^{10} - 849178 T^{11} + 4670479 T^{12} - 1771561 T^{13} + 19487171 T^{14} \)
$13$ \( 1 - 8 T + 87 T^{2} - 534 T^{3} + 3352 T^{4} - 15709 T^{5} + 72570 T^{6} - 263078 T^{7} + 943410 T^{8} - 2654821 T^{9} + 7364344 T^{10} - 15251574 T^{11} + 32302491 T^{12} - 38614472 T^{13} + 62748517 T^{14} \)
$17$ \( 1 + 16 T + 182 T^{2} + 1459 T^{3} + 9888 T^{4} + 55744 T^{5} + 279481 T^{6} + 1215546 T^{7} + 4751177 T^{8} + 16110016 T^{9} + 48579744 T^{10} + 121857139 T^{11} + 258413974 T^{12} + 386201104 T^{13} + 410338673 T^{14} \)
$19$ \( 1 - 2 T + 57 T^{2} + 30 T^{3} + 1522 T^{4} + 1831 T^{5} + 43250 T^{6} + 14418 T^{7} + 821750 T^{8} + 660991 T^{9} + 10439398 T^{10} + 3909630 T^{11} + 141137643 T^{12} - 94091762 T^{13} + 893871739 T^{14} \)
$23$ \( 1 + 8 T + 94 T^{2} + 389 T^{3} + 2978 T^{4} + 6480 T^{5} + 54941 T^{6} + 63078 T^{7} + 1263643 T^{8} + 3427920 T^{9} + 36233326 T^{10} + 108858149 T^{11} + 605016242 T^{12} + 1184287112 T^{13} + 3404825447 T^{14} \)
$29$ \( 1 + 4 T + 101 T^{2} + 216 T^{3} + 2780 T^{4} - 3931 T^{5} - 11736 T^{6} - 397206 T^{7} - 340344 T^{8} - 3305971 T^{9} + 67801420 T^{10} + 152772696 T^{11} + 2071626049 T^{12} + 2379293284 T^{13} + 17249876309 T^{14} \)
$31$ \( 1 - 11 T + 167 T^{2} - 1267 T^{3} + 11993 T^{4} - 74685 T^{5} + 552771 T^{6} - 2866186 T^{7} + 17135901 T^{8} - 71772285 T^{9} + 357283463 T^{10} - 1170101107 T^{11} + 4781068217 T^{12} - 9762540491 T^{13} + 27512614111 T^{14} \)
$37$ \( 1 - 17 T + 280 T^{2} - 2814 T^{3} + 27841 T^{4} - 212197 T^{5} + 1603258 T^{6} - 9844176 T^{7} + 59320546 T^{8} - 290497693 T^{9} + 1410230173 T^{10} - 5273889054 T^{11} + 19416307960 T^{12} - 43617348953 T^{13} + 94931877133 T^{14} \)
$41$ \( 1 + 18 T + 244 T^{2} + 2031 T^{3} + 15320 T^{4} + 87734 T^{5} + 576219 T^{6} + 3259314 T^{7} + 23624979 T^{8} + 147480854 T^{9} + 1055869720 T^{10} + 5739120591 T^{11} + 28268913044 T^{12} + 85501876338 T^{13} + 194754273881 T^{14} \)
$43$ \( 1 + T + 19 T^{2} + 56 T^{3} + 1695 T^{4} - 6240 T^{5} - 6539 T^{6} - 47162 T^{7} - 281177 T^{8} - 11537760 T^{9} + 134764365 T^{10} + 191452856 T^{11} + 2793160417 T^{12} + 6321363049 T^{13} + 271818611107 T^{14} \)
$47$ \( 1 + 5 T + 61 T^{2} - 128 T^{3} + 2482 T^{4} - 14063 T^{5} + 41963 T^{6} - 1627436 T^{7} + 1972261 T^{8} - 31065167 T^{9} + 257688686 T^{10} - 624599168 T^{11} + 13990045427 T^{12} + 53896076645 T^{13} + 506623120463 T^{14} \)
$53$ \( 1 + 6 T + 189 T^{2} + 1106 T^{3} + 17250 T^{4} + 106541 T^{5} + 1103088 T^{6} + 6768406 T^{7} + 58463664 T^{8} + 299273669 T^{9} + 2568128250 T^{10} + 8726871986 T^{11} + 79038948177 T^{12} + 132986166774 T^{13} + 1174711139837 T^{14} \)
$59$ \( 1 - 7 T + 221 T^{2} - 1540 T^{3} + 25061 T^{4} - 148516 T^{5} + 1965993 T^{6} - 9737770 T^{7} + 115993587 T^{8} - 516984196 T^{9} + 5147003119 T^{10} - 18660735940 T^{11} + 157998270079 T^{12} - 295263735487 T^{13} + 2488651484819 T^{14} \)
$61$ \( 1 - 24 T + 565 T^{2} - 8504 T^{3} + 115340 T^{4} - 1255803 T^{5} + 12166384 T^{6} - 100810822 T^{7} + 742149424 T^{8} - 4672842963 T^{9} + 26179988540 T^{10} - 117745031864 T^{11} + 477196910065 T^{12} - 1236488984664 T^{13} + 3142742836021 T^{14} \)
$67$ \( 1 + 4 T + 334 T^{2} + 1393 T^{3} + 54365 T^{4} + 208604 T^{5} + 5511936 T^{6} + 17838814 T^{7} + 369299712 T^{8} + 936423356 T^{9} + 16350980495 T^{10} + 28070511553 T^{11} + 450941785738 T^{12} + 361833528676 T^{13} + 6060711605323 T^{14} \)
$71$ \( 1 - 7 T + 304 T^{2} - 2759 T^{3} + 42642 T^{4} - 454357 T^{5} + 3931755 T^{6} - 41713722 T^{7} + 279154605 T^{8} - 2290413637 T^{9} + 15262040862 T^{10} - 70110827879 T^{11} + 548485722704 T^{12} - 896701987447 T^{13} + 9095120158391 T^{14} \)
$73$ \( 1 - 28 T + 674 T^{2} - 10822 T^{3} + 155858 T^{4} - 1792357 T^{5} + 18944622 T^{6} - 168136784 T^{7} + 1382957406 T^{8} - 9551470453 T^{9} + 60631411586 T^{10} - 307325764102 T^{11} + 1397250253682 T^{12} - 4237358336092 T^{13} + 11047398519097 T^{14} \)
$79$ \( 1 + 278 T^{2} + 350 T^{3} + 42298 T^{4} + 68845 T^{5} + 4389542 T^{6} + 7723498 T^{7} + 346773818 T^{8} + 429661645 T^{9} + 20854563622 T^{10} + 13632528350 T^{11} + 855421678922 T^{12} + 19203908986159 T^{14} \)
$83$ \( 1 - 2 T + 124 T^{2} - 227 T^{3} + 9871 T^{4} - 46034 T^{5} + 748440 T^{6} - 4215522 T^{7} + 62120520 T^{8} - 317128226 T^{9} + 5644109477 T^{10} - 10773038867 T^{11} + 488441039732 T^{12} - 653880746738 T^{13} + 27136050989627 T^{14} \)
$89$ \( 1 + 5 T + 427 T^{2} + 1871 T^{3} + 90273 T^{4} + 340543 T^{5} + 11970235 T^{6} + 38144970 T^{7} + 1065350915 T^{8} + 2697441103 T^{9} + 63639666537 T^{10} + 117390732911 T^{11} + 2384393384723 T^{12} + 2484906454805 T^{13} + 44231334895529 T^{14} \)
$97$ \( 1 - 23 T + 477 T^{2} - 5871 T^{3} + 66981 T^{4} - 495073 T^{5} + 4100541 T^{6} - 25668914 T^{7} + 397752477 T^{8} - 4658141857 T^{9} + 61131750213 T^{10} - 519755408751 T^{11} + 4096161302589 T^{12} - 19158356113367 T^{13} + 80798284478113 T^{14} \)
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