Properties

Label 4020.2.a.h
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.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + \beta_{1} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} - q^{5} + \beta_{1} q^{7} + q^{9} + ( -1 + \beta_{2} ) q^{11} + ( 1 + \beta_{3} + \beta_{6} ) q^{13} + q^{15} + ( 1 - \beta_{1} - \beta_{4} + \beta_{6} ) q^{17} + ( \beta_{1} + \beta_{4} - \beta_{5} ) q^{19} -\beta_{1} q^{21} + ( -1 + \beta_{1} - \beta_{3} ) q^{23} + q^{25} - q^{27} + ( -1 + \beta_{3} ) q^{29} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} ) q^{31} + ( 1 - \beta_{2} ) q^{33} -\beta_{1} q^{35} + ( -1 + \beta_{1} + \beta_{3} ) q^{37} + ( -1 - \beta_{3} - \beta_{6} ) q^{39} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{41} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{43} - q^{45} + ( -1 + \beta_{5} ) q^{47} + ( 2 - \beta_{3} + \beta_{4} + \beta_{5} ) q^{49} + ( -1 + \beta_{1} + \beta_{4} - \beta_{6} ) q^{51} + ( -1 - \beta_{2} - \beta_{3} - \beta_{5} ) q^{53} + ( 1 - \beta_{2} ) q^{55} + ( -\beta_{1} - \beta_{4} + \beta_{5} ) q^{57} + ( -4 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{6} ) q^{59} + ( 3 - \beta_{2} - \beta_{3} - \beta_{5} ) q^{61} + \beta_{1} q^{63} + ( -1 - \beta_{3} - \beta_{6} ) q^{65} - q^{67} + ( 1 - \beta_{1} + \beta_{3} ) q^{69} + ( -\beta_{4} - \beta_{5} - \beta_{6} ) q^{71} + ( 4 - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{73} - q^{75} + ( 1 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{77} + ( 5 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{79} + q^{81} + ( -3 + \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{83} + ( -1 + \beta_{1} + \beta_{4} - \beta_{6} ) q^{85} + ( 1 - \beta_{3} ) q^{87} + ( -\beta_{3} + \beta_{4} ) q^{89} + ( 4 - \beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{91} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} ) q^{93} + ( -\beta_{1} - \beta_{4} + \beta_{5} ) q^{95} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{97} + ( -1 + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 7q^{3} - 7q^{5} + 3q^{7} + 7q^{9} + O(q^{10}) \) \( 7q - 7q^{3} - 7q^{5} + 3q^{7} + 7q^{9} - 5q^{11} + 5q^{13} + 7q^{15} + 3q^{17} + 2q^{19} - 3q^{21} - 3q^{23} + 7q^{25} - 7q^{27} - 8q^{29} + 7q^{31} + 5q^{33} - 3q^{35} - 5q^{37} - 5q^{39} + 7q^{41} + 3q^{43} - 7q^{45} - 6q^{47} + 16q^{49} - 3q^{51} - 9q^{53} + 5q^{55} - 2q^{57} - 22q^{59} + 19q^{61} + 3q^{63} - 5q^{65} - 7q^{67} + 3q^{69} + 23q^{73} - 7q^{75} + 9q^{77} + 25q^{79} + 7q^{81} - 20q^{83} - 3q^{85} + 8q^{87} + q^{89} + 23q^{91} - 7q^{93} - 2q^{95} + 3q^{97} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 3 x^{6} - 28 x^{5} + 90 x^{4} + 143 x^{3} - 418 x^{2} - 256 x + 160\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + \nu^{5} - 24 \nu^{4} - 10 \nu^{3} + 99 \nu^{2} + 86 \nu + 60 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + \nu^{5} + 26 \nu^{4} - 34 \nu^{3} - 115 \nu^{2} + 64 \nu + 60 \)\()/20\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + \nu^{5} - 28 \nu^{4} - 14 \nu^{3} + 187 \nu^{2} + 78 \nu - 180 \)\()/20\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 27 \nu^{4} - 10 \nu^{3} - 141 \nu^{2} - 7 \nu + 30 \)\()/10\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{6} + 3 \nu^{5} - 188 \nu^{4} - 18 \nu^{3} + 1017 \nu^{2} + 610 \nu - 360 \)\()/40\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4} - \beta_{3} + 9\)
\(\nu^{3}\)\(=\)\(-2 \beta_{6} - 3 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 17 \beta_{1} - 6\)
\(\nu^{4}\)\(=\)\(2 \beta_{6} + 25 \beta_{5} + 16 \beta_{4} - 23 \beta_{3} + 4 \beta_{2} - 19 \beta_{1} + 144\)
\(\nu^{5}\)\(=\)\(-46 \beta_{6} - 83 \beta_{5} + 14 \beta_{4} + 47 \beta_{3} + 28 \beta_{2} + 318 \beta_{1} - 264\)
\(\nu^{6}\)\(=\)\(74 \beta_{6} + 554 \beta_{5} + 281 \beta_{4} - 490 \beta_{3} + 98 \beta_{2} - 690 \beta_{1} + 2709\)

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
−4.78907
−2.11010
−0.941196
0.403734
3.15736
3.41543
3.86385
0 −1.00000 0 −1.00000 0 −4.78907 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 −2.11010 0 1.00000 0
1.3 0 −1.00000 0 −1.00000 0 −0.941196 0 1.00000 0
1.4 0 −1.00000 0 −1.00000 0 0.403734 0 1.00000 0
1.5 0 −1.00000 0 −1.00000 0 3.15736 0 1.00000 0
1.6 0 −1.00000 0 −1.00000 0 3.41543 0 1.00000 0
1.7 0 −1.00000 0 −1.00000 0 3.86385 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.h 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.a.h 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 \( T_{7}^{7} - 3 T_{7}^{6} - 28 T_{7}^{5} + 90 T_{7}^{4} + 143 T_{7}^{3} - 418 T_{7}^{2} - 256 T_{7} + 160 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 + T )^{7} \)
$5$ \( ( 1 + T )^{7} \)
$7$ \( 1 - 3 T + 21 T^{2} - 36 T^{3} + 192 T^{4} - 103 T^{5} + 1032 T^{6} + 188 T^{7} + 7224 T^{8} - 5047 T^{9} + 65856 T^{10} - 86436 T^{11} + 352947 T^{12} - 352947 T^{13} + 823543 T^{14} \)
$11$ \( 1 + 5 T + 21 T^{2} + 58 T^{3} + 292 T^{4} + 1049 T^{5} + 4080 T^{6} + 12192 T^{7} + 44880 T^{8} + 126929 T^{9} + 388652 T^{10} + 849178 T^{11} + 3382071 T^{12} + 8857805 T^{13} + 19487171 T^{14} \)
$13$ \( 1 - 5 T + 34 T^{2} - 139 T^{3} + 635 T^{4} - 2607 T^{5} + 10740 T^{6} - 37974 T^{7} + 139620 T^{8} - 440583 T^{9} + 1395095 T^{10} - 3969979 T^{11} + 12623962 T^{12} - 24134045 T^{13} + 62748517 T^{14} \)
$17$ \( 1 - 3 T + 31 T^{2} - 83 T^{3} + 651 T^{4} - 2330 T^{5} + 12209 T^{6} - 39446 T^{7} + 207553 T^{8} - 673370 T^{9} + 3198363 T^{10} - 6932243 T^{11} + 44015567 T^{12} - 72412707 T^{13} + 410338673 T^{14} \)
$19$ \( 1 - 2 T + 25 T^{2} - 70 T^{3} + 977 T^{4} - 2063 T^{5} + 20081 T^{6} - 51322 T^{7} + 381539 T^{8} - 744743 T^{9} + 6701243 T^{10} - 9122470 T^{11} + 61902475 T^{12} - 94091762 T^{13} + 893871739 T^{14} \)
$23$ \( 1 + 3 T + 87 T^{2} + 376 T^{3} + 3928 T^{4} + 19960 T^{5} + 119694 T^{6} + 598042 T^{7} + 2752962 T^{8} + 10558840 T^{9} + 47791976 T^{10} + 105220216 T^{11} + 559961841 T^{12} + 444107667 T^{13} + 3404825447 T^{14} \)
$29$ \( 1 + 8 T + 195 T^{2} + 1174 T^{3} + 16017 T^{4} + 76183 T^{5} + 746831 T^{6} + 2836022 T^{7} + 21658099 T^{8} + 64069903 T^{9} + 390638613 T^{10} + 830347894 T^{11} + 3999674055 T^{12} + 4758586568 T^{13} + 17249876309 T^{14} \)
$31$ \( 1 - 7 T + 103 T^{2} - 752 T^{3} + 6848 T^{4} - 42865 T^{5} + 308936 T^{6} - 1562360 T^{7} + 9577016 T^{8} - 41193265 T^{9} + 204008768 T^{10} - 694487792 T^{11} + 2948802553 T^{12} - 6212525767 T^{13} + 27512614111 T^{14} \)
$37$ \( 1 + 5 T + 213 T^{2} + 1068 T^{3} + 20860 T^{4} + 95468 T^{5} + 1212318 T^{6} + 4647094 T^{7} + 44855766 T^{8} + 130695692 T^{9} + 1056621580 T^{10} + 2001603948 T^{11} + 14770262841 T^{12} + 12828632045 T^{13} + 94931877133 T^{14} \)
$41$ \( 1 - 7 T + 201 T^{2} - 1558 T^{3} + 20510 T^{4} - 142227 T^{5} + 1320572 T^{6} - 7391784 T^{7} + 54143452 T^{8} - 239083587 T^{9} + 1413569710 T^{10} - 4402535638 T^{11} + 23287096401 T^{12} - 33250729687 T^{13} + 194754273881 T^{14} \)
$43$ \( 1 - 3 T + 211 T^{2} - 416 T^{3} + 19878 T^{4} - 22931 T^{5} + 1168364 T^{6} - 911236 T^{7} + 50239652 T^{8} - 42399419 T^{9} + 1580440146 T^{10} - 1422221216 T^{11} + 31018781473 T^{12} - 18964089147 T^{13} + 271818611107 T^{14} \)
$47$ \( 1 + 6 T + 281 T^{2} + 1420 T^{3} + 35583 T^{4} + 150355 T^{5} + 2641311 T^{6} + 9105486 T^{7} + 124141617 T^{8} + 332134195 T^{9} + 3694333809 T^{10} + 6929147020 T^{11} + 64445946967 T^{12} + 64675291974 T^{13} + 506623120463 T^{14} \)
$53$ \( 1 + 9 T + 296 T^{2} + 2235 T^{3} + 39989 T^{4} + 254943 T^{5} + 3255898 T^{6} + 17152618 T^{7} + 172562594 T^{8} + 716134887 T^{9} + 5953442353 T^{10} + 17635225035 T^{11} + 123785865928 T^{12} + 199479250161 T^{13} + 1174711139837 T^{14} \)
$59$ \( 1 + 22 T + 499 T^{2} + 6902 T^{3} + 92099 T^{4} + 937655 T^{5} + 9110405 T^{6} + 71721796 T^{7} + 537513895 T^{8} + 3263977055 T^{9} + 18915200521 T^{10} + 83634025622 T^{11} + 356747225201 T^{12} + 927971740102 T^{13} + 2488651484819 T^{14} \)
$61$ \( 1 - 19 T + 472 T^{2} - 6161 T^{3} + 88213 T^{4} - 867977 T^{5} + 8973786 T^{6} - 68695318 T^{7} + 547400946 T^{8} - 3229742417 T^{9} + 20022674953 T^{10} - 85304226401 T^{11} + 398649454072 T^{12} - 978887112859 T^{13} + 3142742836021 T^{14} \)
$67$ \( ( 1 + T )^{7} \)
$71$ \( 1 + 253 T^{2} - 419 T^{3} + 30766 T^{4} - 116730 T^{5} + 2499824 T^{6} - 12048814 T^{7} + 177487504 T^{8} - 588435930 T^{9} + 11011489826 T^{10} - 10647494339 T^{11} + 456470025803 T^{12} + 9095120158391 T^{14} \)
$73$ \( 1 - 23 T + 289 T^{2} - 1759 T^{3} + 6547 T^{4} - 96526 T^{5} + 2159079 T^{6} - 25271472 T^{7} + 157612767 T^{8} - 514387054 T^{9} + 2546894299 T^{10} - 49952505919 T^{11} + 599117690377 T^{12} - 3480687204647 T^{13} + 11047398519097 T^{14} \)
$79$ \( 1 - 25 T + 651 T^{2} - 9530 T^{3} + 142430 T^{4} - 1499035 T^{5} + 16738302 T^{6} - 142494620 T^{7} + 1322325858 T^{8} - 9355477435 T^{9} + 70223544770 T^{10} - 371194271930 T^{11} + 2003163715749 T^{12} - 6077186388025 T^{13} + 19203908986159 T^{14} \)
$83$ \( 1 + 20 T + 403 T^{2} + 4687 T^{3} + 61584 T^{4} + 626492 T^{5} + 7349918 T^{6} + 65413042 T^{7} + 610043194 T^{8} + 4315903388 T^{9} + 35212930608 T^{10} + 222437150527 T^{11} + 1587433379129 T^{12} + 6538807467380 T^{13} + 27136050989627 T^{14} \)
$89$ \( 1 - T + 533 T^{2} - 347 T^{3} + 127527 T^{4} - 53386 T^{5} + 17869735 T^{6} - 5417092 T^{7} + 1590406415 T^{8} - 422870506 T^{9} + 89902581663 T^{10} - 21771557627 T^{11} + 2976303686317 T^{12} - 496981290961 T^{13} + 44231334895529 T^{14} \)
$97$ \( 1 - 3 T + 454 T^{2} - 1093 T^{3} + 99495 T^{4} - 175093 T^{5} + 13812188 T^{6} - 19147754 T^{7} + 1339782236 T^{8} - 1647450037 T^{9} + 90806400135 T^{10} - 96762504133 T^{11} + 3898652476678 T^{12} - 2498916014787 T^{13} + 80798284478113 T^{14} \)
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