Properties

Label 4020.2.a.i
Level 4020
Weight 2
Character orbit 4020.a
Self dual yes
Analytic conductor 32.100
Analytic rank 0
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4020.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} -\beta_{4} q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + q^{5} -\beta_{4} q^{7} + q^{9} + ( \beta_{3} + \beta_{6} ) q^{11} + ( 1 - \beta_{2} + \beta_{6} ) q^{13} + q^{15} + ( 2 - \beta_{3} ) q^{17} + ( -\beta_{4} + \beta_{5} - \beta_{6} ) q^{19} -\beta_{4} q^{21} + ( -\beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{23} + q^{25} + q^{27} + ( 2 + \beta_{1} - \beta_{3} - \beta_{5} ) q^{29} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{31} + ( \beta_{3} + \beta_{6} ) q^{33} -\beta_{4} q^{35} + ( 2 + 3 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{37} + ( 1 - \beta_{2} + \beta_{6} ) q^{39} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{6} ) q^{41} + ( 1 + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{43} + q^{45} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{6} ) q^{47} + ( 3 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{49} + ( 2 - \beta_{3} ) q^{51} + ( 3 + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{53} + ( \beta_{3} + \beta_{6} ) q^{55} + ( -\beta_{4} + \beta_{5} - \beta_{6} ) q^{57} + ( 2 \beta_{1} - 2 \beta_{4} - \beta_{6} ) q^{59} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{61} -\beta_{4} q^{63} + ( 1 - \beta_{2} + \beta_{6} ) q^{65} - q^{67} + ( -\beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{69} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{71} + ( 3 - 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} ) q^{73} + q^{75} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{77} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{79} + q^{81} + ( 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{83} + ( 2 - \beta_{3} ) q^{85} + ( 2 + \beta_{1} - \beta_{3} - \beta_{5} ) q^{87} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} ) q^{89} + ( -1 - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{91} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{93} + ( -\beta_{4} + \beta_{5} - \beta_{6} ) q^{95} + ( -2 - 2 \beta_{1} + 2 \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{97} + ( \beta_{3} + \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 7q^{3} + 7q^{5} + q^{7} + 7q^{9} + O(q^{10}) \) \( 7q + 7q^{3} + 7q^{5} + q^{7} + 7q^{9} + 5q^{11} + 9q^{13} + 7q^{15} + 11q^{17} + 2q^{19} + q^{21} + 7q^{23} + 7q^{25} + 7q^{27} + 8q^{29} + 9q^{31} + 5q^{33} + q^{35} + 7q^{37} + 9q^{39} + 9q^{41} + 5q^{43} + 7q^{45} + 2q^{47} + 12q^{49} + 11q^{51} + 15q^{53} + 5q^{55} + 2q^{57} + 11q^{61} + q^{63} + 9q^{65} - 7q^{67} + 7q^{69} + 18q^{71} + 21q^{73} + 7q^{75} + 5q^{77} + 5q^{79} + 7q^{81} + 6q^{83} + 11q^{85} + 8q^{87} + 7q^{89} - 9q^{91} + 9q^{93} + 2q^{95} - 9q^{97} + 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 21 x^{5} - 12 x^{4} + 93 x^{3} + 18 x^{2} - 120 x + 48\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - 19 \nu^{4} - 16 \nu^{3} + 65 \nu^{2} + 44 \nu - 60 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + 20 \nu^{4} + 15 \nu^{3} - 78 \nu^{2} - 46 \nu + 72 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + \nu^{5} - 21 \nu^{4} - 30 \nu^{3} + 74 \nu^{2} + 72 \nu - 64 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} + \nu^{5} - 22 \nu^{4} - 29 \nu^{3} + 91 \nu^{2} + 74 \nu - 96 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} + \nu^{5} - 21 \nu^{4} - 32 \nu^{3} + 80 \nu^{2} + 90 \nu - 88 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(-2 \beta_{6} + 3 \beta_{5} - \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 9 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-2 \beta_{6} + 16 \beta_{5} - 14 \beta_{4} + 20 \beta_{3} + 20 \beta_{2} + 11 \beta_{1} + 56\)
\(\nu^{5}\)\(=\)\(-32 \beta_{6} + 65 \beta_{5} - 29 \beta_{4} + 73 \beta_{3} + 69 \beta_{2} + 120 \beta_{1} + 113\)
\(\nu^{6}\)\(=\)\(-70 \beta_{6} + 287 \beta_{5} - 217 \beta_{4} + 363 \beta_{3} + 367 \beta_{2} + 309 \beta_{1} + 847\)

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
0.819259
−1.72664
−2.67981
4.35097
0.575750
1.67614
−3.01567
0 1.00000 0 1.00000 0 −4.84231 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 −2.80679 0 1.00000 0
1.3 0 1.00000 0 1.00000 0 −0.239317 0 1.00000 0
1.4 0 1.00000 0 1.00000 0 0.772018 0 1.00000 0
1.5 0 1.00000 0 1.00000 0 1.48734 0 1.00000 0
1.6 0 1.00000 0 1.00000 0 1.75949 0 1.00000 0
1.7 0 1.00000 0 1.00000 0 4.86957 0 1.00000 0
\(n\): 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 4020.2.a.i 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.a.i 7 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(67\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{7} - T_{7}^{6} - 30 T_{7}^{5} + 34 T_{7}^{4} + 149 T_{7}^{3} - 258 T_{7}^{2} + 64 T_{7} + 32 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 - T )^{7} \)
$5$ \( ( 1 - T )^{7} \)
$7$ \( 1 - T + 19 T^{2} - 8 T^{3} + 128 T^{4} - 41 T^{5} + 498 T^{6} - 444 T^{7} + 3486 T^{8} - 2009 T^{9} + 43904 T^{10} - 19208 T^{11} + 319333 T^{12} - 117649 T^{13} + 823543 T^{14} \)
$11$ \( 1 - 5 T + 47 T^{2} - 124 T^{3} + 782 T^{4} - 1205 T^{5} + 8984 T^{6} - 10948 T^{7} + 98824 T^{8} - 145805 T^{9} + 1040842 T^{10} - 1815484 T^{11} + 7569397 T^{12} - 8857805 T^{13} + 19487171 T^{14} \)
$13$ \( 1 - 9 T + 82 T^{2} - 501 T^{3} + 2853 T^{4} - 13419 T^{5} + 57830 T^{6} - 218514 T^{7} + 751790 T^{8} - 2267811 T^{9} + 6268041 T^{10} - 14309061 T^{11} + 30446026 T^{12} - 43441281 T^{13} + 62748517 T^{14} \)
$17$ \( 1 - 11 T + 131 T^{2} - 919 T^{3} + 6497 T^{4} - 34076 T^{5} + 177839 T^{6} - 735646 T^{7} + 3023263 T^{8} - 9847964 T^{9} + 31919761 T^{10} - 76755799 T^{11} + 186001267 T^{12} - 265513259 T^{13} + 410338673 T^{14} \)
$19$ \( 1 - 2 T + 73 T^{2} - 22 T^{3} + 2165 T^{4} + 3821 T^{5} + 39189 T^{6} + 134574 T^{7} + 744591 T^{8} + 1379381 T^{9} + 14849735 T^{10} - 2867062 T^{11} + 180755227 T^{12} - 94091762 T^{13} + 893871739 T^{14} \)
$23$ \( 1 - 7 T + 119 T^{2} - 704 T^{3} + 6512 T^{4} - 32704 T^{5} + 219182 T^{6} - 929666 T^{7} + 5041186 T^{8} - 17300416 T^{9} + 79231504 T^{10} - 197008064 T^{11} + 765924817 T^{12} - 1036251223 T^{13} + 3404825447 T^{14} \)
$29$ \( 1 - 8 T + 173 T^{2} - 1024 T^{3} + 12881 T^{4} - 60359 T^{5} + 566477 T^{6} - 2164546 T^{7} + 16427833 T^{8} - 50761919 T^{9} + 314154709 T^{10} - 724255744 T^{11} + 3548428777 T^{12} - 4758586568 T^{13} + 17249876309 T^{14} \)
$31$ \( 1 - 9 T + 157 T^{2} - 684 T^{3} + 6900 T^{4} - 2043 T^{5} + 96962 T^{6} + 651768 T^{7} + 3005822 T^{8} - 1963323 T^{9} + 205557900 T^{10} - 631688364 T^{11} + 4494776707 T^{12} - 7987533129 T^{13} + 27512614111 T^{14} \)
$37$ \( 1 - 7 T + 97 T^{2} - 224 T^{3} + 3260 T^{4} + 3352 T^{5} + 85122 T^{6} + 362766 T^{7} + 3149514 T^{8} + 4588888 T^{9} + 165128780 T^{10} - 419812064 T^{11} + 6726363829 T^{12} - 17960084863 T^{13} + 94931877133 T^{14} \)
$41$ \( 1 - 9 T + 119 T^{2} - 696 T^{3} + 6948 T^{4} - 31269 T^{5} + 313624 T^{6} - 1370892 T^{7} + 12858584 T^{8} - 52563189 T^{9} + 478863108 T^{10} - 1966729656 T^{11} + 13786887919 T^{12} - 42750938169 T^{13} + 194754273881 T^{14} \)
$43$ \( 1 - 5 T + 151 T^{2} - 622 T^{3} + 10436 T^{4} - 32797 T^{5} + 484086 T^{6} - 1283088 T^{7} + 20815698 T^{8} - 60641653 T^{9} + 829735052 T^{10} - 2126494222 T^{11} + 22198274893 T^{12} - 31606815245 T^{13} + 271818611107 T^{14} \)
$47$ \( 1 - 2 T + 203 T^{2} - 856 T^{3} + 18173 T^{4} - 117683 T^{5} + 1046795 T^{6} - 7697422 T^{7} + 49199365 T^{8} - 259961747 T^{9} + 1886775379 T^{10} - 4177006936 T^{11} + 46557036421 T^{12} - 21558430658 T^{13} + 506623120463 T^{14} \)
$53$ \( 1 - 15 T + 374 T^{2} - 3975 T^{3} + 57039 T^{4} - 470469 T^{5} + 4901962 T^{6} - 32054298 T^{7} + 259803986 T^{8} - 1321547421 T^{9} + 8491795203 T^{10} - 31364661975 T^{11} + 156405114382 T^{12} - 332465416935 T^{13} + 1174711139837 T^{14} \)
$59$ \( 1 + 197 T^{2} + 108 T^{3} + 22893 T^{4} + 15603 T^{5} + 1834513 T^{6} + 1268472 T^{7} + 108236267 T^{8} + 54314043 T^{9} + 4701741447 T^{10} + 1308674988 T^{11} + 140840086903 T^{12} + 2488651484819 T^{14} \)
$61$ \( 1 - 11 T + 214 T^{2} - 883 T^{3} + 12641 T^{4} + 24563 T^{5} + 347856 T^{6} + 4536294 T^{7} + 21219216 T^{8} + 91398923 T^{9} + 2869266821 T^{10} - 12225877603 T^{11} + 180743608414 T^{12} - 566724117971 T^{13} + 3142742836021 T^{14} \)
$67$ \( ( 1 + T )^{7} \)
$71$ \( 1 - 18 T + 527 T^{2} - 6351 T^{3} + 105570 T^{4} - 953004 T^{5} + 11665102 T^{6} - 84162318 T^{7} + 828222242 T^{8} - 4804093164 T^{9} + 37784664270 T^{10} - 161389586031 T^{11} + 950828867977 T^{12} - 2305805110578 T^{13} + 9095120158391 T^{14} \)
$73$ \( 1 - 21 T + 511 T^{2} - 6561 T^{3} + 92751 T^{4} - 882216 T^{5} + 9543917 T^{6} - 75489564 T^{7} + 696705941 T^{8} - 4701329064 T^{9} + 36081715767 T^{10} - 186320859201 T^{11} + 1059339584023 T^{12} - 3178018752069 T^{13} + 11047398519097 T^{14} \)
$79$ \( 1 - 5 T + 385 T^{2} - 628 T^{3} + 60698 T^{4} + 73721 T^{5} + 5848464 T^{6} + 14581032 T^{7} + 462028656 T^{8} + 460092761 T^{9} + 29926481222 T^{10} - 24460650868 T^{11} + 1184666713615 T^{12} - 1215437277605 T^{13} + 19203908986159 T^{14} \)
$83$ \( 1 - 6 T + 431 T^{2} - 2205 T^{3} + 82446 T^{4} - 361914 T^{5} + 9684928 T^{6} - 36558774 T^{7} + 803849024 T^{8} - 2493225546 T^{9} + 47141551002 T^{10} - 104645597805 T^{11} + 1697726517133 T^{12} - 1961642240214 T^{13} + 27136050989627 T^{14} \)
$89$ \( 1 - 7 T + 365 T^{2} - 1685 T^{3} + 61205 T^{4} - 158422 T^{5} + 6655073 T^{6} - 11367164 T^{7} + 592301497 T^{8} - 1254860662 T^{9} + 43147627645 T^{10} - 105720676085 T^{11} + 2038181698885 T^{12} - 3478869036727 T^{13} + 44231334895529 T^{14} \)
$97$ \( 1 + 9 T + 394 T^{2} + 3615 T^{3} + 86565 T^{4} + 702723 T^{5} + 12173822 T^{6} + 85552134 T^{7} + 1180860734 T^{8} + 6611920707 T^{9} + 79005538245 T^{10} + 320033350815 T^{11} + 3383412061258 T^{12} + 7496748044361 T^{13} + 80798284478113 T^{14} \)
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