Properties

Label 4014.2.a.w
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 446)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_{3} q^{5} + ( 1 - \beta_{2} + \beta_{6} ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + \beta_{3} q^{5} + ( 1 - \beta_{2} + \beta_{6} ) q^{7} - q^{8} -\beta_{3} q^{10} + ( -2 + \beta_{1} + \beta_{4} + \beta_{6} ) q^{11} + ( \beta_{3} + \beta_{4} - \beta_{6} ) q^{13} + ( -1 + \beta_{2} - \beta_{6} ) q^{14} + q^{16} + ( \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{17} -2 \beta_{1} q^{19} + \beta_{3} q^{20} + ( 2 - \beta_{1} - \beta_{4} - \beta_{6} ) q^{22} + ( -2 - \beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} ) q^{23} + ( 1 - \beta_{1} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{25} + ( -\beta_{3} - \beta_{4} + \beta_{6} ) q^{26} + ( 1 - \beta_{2} + \beta_{6} ) q^{28} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{29} + ( -\beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{31} - q^{32} + ( -\beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{34} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{35} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{37} + 2 \beta_{1} q^{38} -\beta_{3} q^{40} + ( 6 + \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{41} + ( 2 + 2 \beta_{1} + 2 \beta_{4} ) q^{43} + ( -2 + \beta_{1} + \beta_{4} + \beta_{6} ) q^{44} + ( 2 + \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} ) q^{46} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} - 4 \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{47} + ( 1 + 2 \beta_{1} - 2 \beta_{2} ) q^{49} + ( -1 + \beta_{1} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{50} + ( \beta_{3} + \beta_{4} - \beta_{6} ) q^{52} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{53} + ( -1 + \beta_{1} - \beta_{2} - 3 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{55} + ( -1 + \beta_{2} - \beta_{6} ) q^{56} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{58} + ( \beta_{1} - \beta_{2} - \beta_{3} - 4 \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{59} + ( 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{61} + ( \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{62} + q^{64} + ( 5 - 3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} ) q^{65} + ( 4 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{67} + ( \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{68} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{70} + ( -4 \beta_{2} - 2 \beta_{3} + 4 \beta_{6} ) q^{71} + ( -5 + \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{73} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{74} -2 \beta_{1} q^{76} + ( 2 + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{77} + ( 4 - 2 \beta_{1} + 2 \beta_{5} + 2 \beta_{6} ) q^{79} + \beta_{3} q^{80} + ( -6 - \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{82} + ( 2 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{83} + ( 2 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{85} + ( -2 - 2 \beta_{1} - 2 \beta_{4} ) q^{86} + ( 2 - \beta_{1} - \beta_{4} - \beta_{6} ) q^{88} + ( 4 - 3 \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{6} ) q^{89} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{5} + \beta_{6} ) q^{91} + ( -2 - \beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} ) q^{92} + ( -2 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 4 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{94} + ( -2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{95} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{97} + ( -1 - 2 \beta_{1} + 2 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 7q^{2} + 7q^{4} - 2q^{5} + 6q^{7} - 7q^{8} + O(q^{10}) \) \( 7q - 7q^{2} + 7q^{4} - 2q^{5} + 6q^{7} - 7q^{8} + 2q^{10} - 9q^{11} - 2q^{13} - 6q^{14} + 7q^{16} + 7q^{17} - 2q^{19} - 2q^{20} + 9q^{22} - 15q^{23} + 13q^{25} + 2q^{26} + 6q^{28} - 9q^{29} - 2q^{31} - 7q^{32} - 7q^{34} + 4q^{35} + 5q^{37} + 2q^{38} + 2q^{40} + 33q^{41} + 20q^{43} - 9q^{44} + 15q^{46} + 2q^{47} + 3q^{49} - 13q^{50} - 2q^{52} + 13q^{53} - 18q^{55} - 6q^{56} + 9q^{58} - 9q^{59} + 8q^{61} + 2q^{62} + 7q^{64} + 44q^{65} + 29q^{67} + 7q^{68} - 4q^{70} - 37q^{73} - 5q^{74} - 2q^{76} + 18q^{77} + 32q^{79} - 2q^{80} - 33q^{82} + 6q^{83} - 4q^{85} - 20q^{86} + 9q^{88} + 17q^{89} - 4q^{91} - 15q^{92} - 2q^{94} + 12q^{95} + 12q^{97} - 3q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 14 x^{5} + 12 x^{4} + 50 x^{3} - 36 x^{2} - 38 x + 18\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -6 \nu^{6} + 37 \nu^{5} + 92 \nu^{4} - 388 \nu^{3} - 287 \nu^{2} + 703 \nu - 98 \)\()/239\)
\(\beta_{3}\)\(=\)\((\)\( 6 \nu^{6} - 37 \nu^{5} - 92 \nu^{4} + 388 \nu^{3} + 526 \nu^{2} - 703 \nu - 858 \)\()/239\)
\(\beta_{4}\)\(=\)\((\)\( 11 \nu^{6} - 28 \nu^{5} - 89 \nu^{4} + 313 \nu^{3} - 151 \nu^{2} - 771 \nu + 817 \)\()/239\)
\(\beta_{5}\)\(=\)\((\)\( -32 \nu^{6} + 38 \nu^{5} + 411 \nu^{4} - 476 \nu^{3} - 1212 \nu^{2} + 1439 \nu + 513 \)\()/239\)
\(\beta_{6}\)\(=\)\((\)\( -47 \nu^{6} + 11 \nu^{5} + 641 \nu^{4} - 12 \nu^{3} - 2049 \nu^{2} - 269 \nu + 985 \)\()/239\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} + 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{5} + 4 \beta_{4} + 10 \beta_{3} + 12 \beta_{2} + \beta_{1} + 25\)
\(\nu^{5}\)\(=\)\(10 \beta_{6} - 22 \beta_{5} - 12 \beta_{4} - 13 \beta_{3} + 4 \beta_{2} + 55 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(-3 \beta_{6} + 9 \beta_{5} + 52 \beta_{4} + 90 \beta_{3} + 121 \beta_{2} + 19 \beta_{1} + 188\)

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
0.409721
1.26499
−2.01250
3.08762
−0.922829
2.08198
−2.90898
−1.00000 0 1.00000 −4.32652 0 2.79833 −1.00000 0 4.32652
1.2 −1.00000 0 1.00000 −1.88711 0 −3.39932 −1.00000 0 1.88711
1.3 −1.00000 0 1.00000 −1.52490 0 0.908231 −1.00000 0 1.52490
1.4 −1.00000 0 1.00000 −0.580657 0 −1.39541 −1.00000 0 0.580657
1.5 −1.00000 0 1.00000 −0.437167 0 3.40247 −1.00000 0 0.437167
1.6 −1.00000 0 1.00000 3.23274 0 4.23794 −1.00000 0 −3.23274
1.7 −1.00000 0 1.00000 3.52361 0 −0.552246 −1.00000 0 −3.52361
\(n\): 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.w 7
3.b odd 2 1 446.2.a.e 7
12.b even 2 1 3568.2.a.l 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
446.2.a.e 7 3.b odd 2 1
3568.2.a.l 7 12.b even 2 1
4014.2.a.w 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\( T_{5}^{7} + 2 T_{5}^{6} - 22 T_{5}^{5} - 42 T_{5}^{4} + 92 T_{5}^{3} + 256 T_{5}^{2} + 174 T_{5} + 36 \)
\( T_{7}^{7} - 6 T_{7}^{6} - 8 T_{7}^{5} + 88 T_{7}^{4} - 48 T_{7}^{3} - 224 T_{7}^{2} + 80 T_{7} + 96 \)
\( T_{11}^{7} + 9 T_{11}^{6} - 14 T_{11}^{5} - 254 T_{11}^{4} - 58 T_{11}^{3} + 2028 T_{11}^{2} + 394 T_{11} - 3494 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{7} \)
$3$ 1
$5$ \( 1 + 2 T + 13 T^{2} + 18 T^{3} + 67 T^{4} + 166 T^{5} + 429 T^{6} + 1296 T^{7} + 2145 T^{8} + 4150 T^{9} + 8375 T^{10} + 11250 T^{11} + 40625 T^{12} + 31250 T^{13} + 78125 T^{14} \)
$7$ \( 1 - 6 T + 41 T^{2} - 164 T^{3} + 701 T^{4} - 2170 T^{5} + 7157 T^{6} - 18328 T^{7} + 50099 T^{8} - 106330 T^{9} + 240443 T^{10} - 393764 T^{11} + 689087 T^{12} - 705894 T^{13} + 823543 T^{14} \)
$11$ \( 1 + 9 T + 63 T^{2} + 340 T^{3} + 1713 T^{4} + 7187 T^{5} + 28125 T^{6} + 96298 T^{7} + 309375 T^{8} + 869627 T^{9} + 2280003 T^{10} + 4977940 T^{11} + 10146213 T^{12} + 15944049 T^{13} + 19487171 T^{14} \)
$13$ \( 1 + 2 T + 27 T^{2} + 78 T^{3} + 749 T^{4} + 1430 T^{5} + 12173 T^{6} + 26660 T^{7} + 158249 T^{8} + 241670 T^{9} + 1645553 T^{10} + 2227758 T^{11} + 10024911 T^{12} + 9653618 T^{13} + 62748517 T^{14} \)
$17$ \( 1 - 7 T + 75 T^{2} - 422 T^{3} + 2657 T^{4} - 13017 T^{5} + 64695 T^{6} - 268656 T^{7} + 1099815 T^{8} - 3761913 T^{9} + 13053841 T^{10} - 35245862 T^{11} + 106489275 T^{12} - 168962983 T^{13} + 410338673 T^{14} \)
$19$ \( 1 + 2 T + 77 T^{2} + 132 T^{3} + 3061 T^{4} + 4686 T^{5} + 81073 T^{6} + 107896 T^{7} + 1540387 T^{8} + 1691646 T^{9} + 20995399 T^{10} + 17202372 T^{11} + 190659623 T^{12} + 94091762 T^{13} + 893871739 T^{14} \)
$23$ \( 1 + 15 T + 205 T^{2} + 1902 T^{3} + 15529 T^{4} + 103889 T^{5} + 615469 T^{6} + 3131844 T^{7} + 14155787 T^{8} + 54957281 T^{9} + 188941343 T^{10} + 532257582 T^{11} + 1319450315 T^{12} + 2220538335 T^{13} + 3404825447 T^{14} \)
$29$ \( 1 + 9 T + 107 T^{2} + 722 T^{3} + 4829 T^{4} + 23967 T^{5} + 139055 T^{6} + 602588 T^{7} + 4032595 T^{8} + 20156247 T^{9} + 117774481 T^{10} + 510656882 T^{11} + 2194692943 T^{12} + 5353409889 T^{13} + 17249876309 T^{14} \)
$31$ \( 1 + 2 T + 125 T^{2} - 100 T^{3} + 6369 T^{4} - 22362 T^{5} + 216297 T^{6} - 1064192 T^{7} + 6705207 T^{8} - 21489882 T^{9} + 189738879 T^{10} - 92352100 T^{11} + 3578643875 T^{12} + 1775007362 T^{13} + 27512614111 T^{14} \)
$37$ \( 1 - 5 T + 51 T^{2} - 402 T^{3} + 3485 T^{4} - 16691 T^{5} + 132407 T^{6} - 755612 T^{7} + 4899059 T^{8} - 22849979 T^{9} + 176525705 T^{10} - 753412722 T^{11} + 3536541807 T^{12} - 12828632045 T^{13} + 94931877133 T^{14} \)
$41$ \( 1 - 33 T + 655 T^{2} - 9442 T^{3} + 107781 T^{4} - 1018431 T^{5} + 8167163 T^{6} - 56270876 T^{7} + 334853683 T^{8} - 1711982511 T^{9} + 7428374301 T^{10} - 26680835362 T^{11} + 75885811655 T^{12} - 156753439953 T^{13} + 194754273881 T^{14} \)
$43$ \( 1 - 20 T + 373 T^{2} - 4328 T^{3} + 47717 T^{4} - 402636 T^{5} + 3292073 T^{6} - 21857328 T^{7} + 141559139 T^{8} - 744473964 T^{9} + 3793835519 T^{10} - 14796570728 T^{11} + 54834149239 T^{12} - 126427260980 T^{13} + 271818611107 T^{14} \)
$47$ \( 1 - 2 T + 113 T^{2} + 20 T^{3} + 8501 T^{4} + 906 T^{5} + 518921 T^{6} + 118368 T^{7} + 24389287 T^{8} + 2001354 T^{9} + 882599323 T^{10} + 97593620 T^{11} + 25915985791 T^{12} - 21558430658 T^{13} + 506623120463 T^{14} \)
$53$ \( 1 - 13 T + 259 T^{2} - 2626 T^{3} + 33677 T^{4} - 278539 T^{5} + 2685015 T^{6} - 18247356 T^{7} + 142305795 T^{8} - 782416051 T^{9} + 5013730729 T^{10} - 20720403106 T^{11} + 108312632687 T^{12} - 288136694677 T^{13} + 1174711139837 T^{14} \)
$59$ \( 1 + 9 T + 203 T^{2} + 1630 T^{3} + 23925 T^{4} + 170923 T^{5} + 1964329 T^{6} + 12171514 T^{7} + 115895411 T^{8} + 594982963 T^{9} + 4913692575 T^{10} + 19751298430 T^{11} + 145129632697 T^{12} + 379624802769 T^{13} + 2488651484819 T^{14} \)
$61$ \( 1 - 8 T + 239 T^{2} - 2446 T^{3} + 30153 T^{4} - 299532 T^{5} + 2695533 T^{6} - 21956832 T^{7} + 164427513 T^{8} - 1114558572 T^{9} + 6844158093 T^{10} - 33866927086 T^{11} + 201858515939 T^{12} - 412162994888 T^{13} + 3142742836021 T^{14} \)
$67$ \( 1 - 29 T + 547 T^{2} - 7824 T^{3} + 93989 T^{4} - 1001607 T^{5} + 9666465 T^{6} - 83193754 T^{7} + 647653155 T^{8} - 4496213823 T^{9} + 28268413607 T^{10} - 157662370704 T^{11} + 738518433529 T^{12} - 2623293082901 T^{13} + 6060711605323 T^{14} \)
$71$ \( 1 + 73 T^{2} + 48 T^{3} + 7885 T^{4} - 42944 T^{5} + 476501 T^{6} - 2121440 T^{7} + 33831571 T^{8} - 216480704 T^{9} + 2822128235 T^{10} + 1219760688 T^{11} + 131708742623 T^{12} + 9095120158391 T^{14} \)
$73$ \( 1 + 37 T + 983 T^{2} + 18018 T^{3} + 274469 T^{4} + 3384747 T^{5} + 36363747 T^{6} + 330718412 T^{7} + 2654553531 T^{8} + 18037316763 T^{9} + 106773106973 T^{10} + 511679506338 T^{11} + 2037829375919 T^{12} + 5599366372693 T^{13} + 11047398519097 T^{14} \)
$79$ \( 1 - 32 T + 825 T^{2} - 14640 T^{3} + 224645 T^{4} - 2807072 T^{5} + 31109933 T^{6} - 292707104 T^{7} + 2457684707 T^{8} - 17518936352 T^{9} + 110758746155 T^{10} - 570229185840 T^{11} + 2538571529175 T^{12} - 7778798576672 T^{13} + 19203908986159 T^{14} \)
$83$ \( 1 - 6 T + 381 T^{2} - 2108 T^{3} + 73989 T^{4} - 359722 T^{5} + 9067089 T^{6} - 37267848 T^{7} + 752568387 T^{8} - 2478124858 T^{9} + 42305948343 T^{10} - 100042140668 T^{11} + 1500774484983 T^{12} - 1961642240214 T^{13} + 27136050989627 T^{14} \)
$89$ \( 1 - 17 T + 447 T^{2} - 7166 T^{3} + 104309 T^{4} - 1338359 T^{5} + 14938191 T^{6} - 148859160 T^{7} + 1329498999 T^{8} - 10601141639 T^{9} + 73534611421 T^{10} - 449610899006 T^{11} + 2496074573703 T^{12} - 8448681946337 T^{13} + 44231334895529 T^{14} \)
$97$ \( 1 - 12 T + 251 T^{2} - 2024 T^{3} + 40393 T^{4} - 296116 T^{5} + 5234683 T^{6} - 36247216 T^{7} + 507764251 T^{8} - 2786155444 T^{9} + 36865600489 T^{10} - 179183264744 T^{11} + 2155422404507 T^{12} - 9995664059148 T^{13} + 80798284478113 T^{14} \)
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