Properties

Label 4020.2.a.g
Level 4020
Weight 2
Character orbit 4020.a
Self dual yes
Analytic conductor 32.100
Analytic rank 0
Dimension 6
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: \(6\)
Coefficient field: 6.6.195727752.1
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} - q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + q^{9} + ( -\beta_{2} + \beta_{4} ) q^{11} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} ) q^{13} - q^{15} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{17} + ( -1 - \beta_{1} - \beta_{5} ) q^{19} + ( \beta_{1} + \beta_{4} ) q^{21} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} + q^{25} + q^{27} + ( 3 + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{29} + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{5} ) q^{31} + ( -\beta_{2} + \beta_{4} ) q^{33} + ( -\beta_{1} - \beta_{4} ) q^{35} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{37} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} ) q^{39} + ( 2 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{41} + ( -1 + 2 \beta_{1} + 3 \beta_{2} - \beta_{5} ) q^{43} - q^{45} + ( 4 + \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{5} ) q^{47} + ( 1 + \beta_{2} + \beta_{3} + \beta_{5} ) q^{49} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{51} + ( 3 + \beta_{1} + 4 \beta_{3} + \beta_{5} ) q^{53} + ( \beta_{2} - \beta_{4} ) q^{55} + ( -1 - \beta_{1} - \beta_{5} ) q^{57} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{59} + ( 3 + \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{61} + ( \beta_{1} + \beta_{4} ) q^{63} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{65} + q^{67} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{69} + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{71} + ( -\beta_{1} + \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{73} + q^{75} + ( 5 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{77} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{5} ) q^{79} + q^{81} + ( 4 + \beta_{1} + \beta_{2} ) q^{83} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{85} + ( 3 + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{87} + ( 6 - 3 \beta_{1} + \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{89} + ( -5 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{91} + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{5} ) q^{93} + ( 1 + \beta_{1} + \beta_{5} ) q^{95} + ( 2 - \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{97} + ( -\beta_{2} + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{3} - 6q^{5} + 3q^{7} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{3} - 6q^{5} + 3q^{7} + 6q^{9} + 3q^{11} + 3q^{13} - 6q^{15} + 6q^{17} - 3q^{19} + 3q^{21} + 6q^{23} + 6q^{25} + 6q^{27} + 21q^{29} - 3q^{31} + 3q^{33} - 3q^{35} + 3q^{39} + 9q^{41} - 3q^{43} - 6q^{45} + 21q^{47} + 3q^{49} + 6q^{51} + 15q^{53} - 3q^{55} - 3q^{57} + 15q^{59} + 15q^{61} + 3q^{63} - 3q^{65} + 6q^{67} + 6q^{69} + 18q^{71} - 6q^{73} + 6q^{75} + 33q^{77} + 9q^{79} + 6q^{81} + 24q^{83} - 6q^{85} + 21q^{87} + 24q^{89} - 21q^{91} - 3q^{93} + 3q^{95} + 9q^{97} + 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 15 x^{4} - 25 x^{3} - 3 x^{2} + 9 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{5} - \nu^{4} - 14 \nu^{3} - 11 \nu^{2} + 8 \nu + 2 \)
\(\beta_{3}\)\(=\)\( -2 \nu^{5} + \nu^{4} + 30 \nu^{3} + 34 \nu^{2} - 16 \nu - 12 \)
\(\beta_{4}\)\(=\)\( -3 \nu^{5} + 2 \nu^{4} + 43 \nu^{3} + 48 \nu^{2} - 17 \nu - 16 \)
\(\beta_{5}\)\(=\)\( 4 \nu^{5} - 2 \nu^{4} - 59 \nu^{3} - 70 \nu^{2} + 22 \nu + 21 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(3 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 16 \beta_{1} + 13\)
\(\nu^{4}\)\(=\)\(18 \beta_{5} + 16 \beta_{4} + 19 \beta_{3} + 14 \beta_{2} + 68 \beta_{1} + 78\)
\(\nu^{5}\)\(=\)\(71 \beta_{5} + 55 \beta_{4} + 86 \beta_{3} + 54 \beta_{2} + 317 \beta_{1} + 313\)

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
−1.75666
−0.457832
0.600664
−0.587102
−2.33297
4.53390
0 1.00000 0 −1.00000 0 −3.63834 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −2.59172 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 1.05225 0 1.00000 0
1.4 0 1.00000 0 −1.00000 0 1.68378 0 1.00000 0
1.5 0 1.00000 0 −1.00000 0 3.15496 0 1.00000 0
1.6 0 1.00000 0 −1.00000 0 3.33907 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.a.g 6 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 \( T_{7}^{6} - 3 T_{7}^{5} - 18 T_{7}^{4} + 60 T_{7}^{3} + 51 T_{7}^{2} - 264 T_{7} + 176 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 - T )^{6} \)
$5$ \( ( 1 + T )^{6} \)
$7$ \( 1 - 3 T + 24 T^{2} - 45 T^{3} + 282 T^{4} - 474 T^{5} + 2458 T^{6} - 3318 T^{7} + 13818 T^{8} - 15435 T^{9} + 57624 T^{10} - 50421 T^{11} + 117649 T^{12} \)
$11$ \( 1 - 3 T + 30 T^{2} - 75 T^{3} + 348 T^{4} - 876 T^{5} + 3130 T^{6} - 9636 T^{7} + 42108 T^{8} - 99825 T^{9} + 439230 T^{10} - 483153 T^{11} + 1771561 T^{12} \)
$13$ \( 1 - 3 T + 39 T^{2} - 76 T^{3} + 648 T^{4} - 639 T^{5} + 8106 T^{6} - 8307 T^{7} + 109512 T^{8} - 166972 T^{9} + 1113879 T^{10} - 1113879 T^{11} + 4826809 T^{12} \)
$17$ \( 1 - 6 T + 78 T^{2} - 385 T^{3} + 2724 T^{4} - 11127 T^{5} + 57328 T^{6} - 189159 T^{7} + 787236 T^{8} - 1891505 T^{9} + 6514638 T^{10} - 8519142 T^{11} + 24137569 T^{12} \)
$19$ \( 1 + 3 T + 81 T^{2} + 258 T^{3} + 3060 T^{4} + 9339 T^{5} + 71452 T^{6} + 177441 T^{7} + 1104660 T^{8} + 1769622 T^{9} + 10556001 T^{10} + 7428297 T^{11} + 47045881 T^{12} \)
$23$ \( 1 - 6 T + 78 T^{2} - 436 T^{3} + 3354 T^{4} - 16590 T^{5} + 93454 T^{6} - 381570 T^{7} + 1774266 T^{8} - 5304812 T^{9} + 21827598 T^{10} - 38618058 T^{11} + 148035889 T^{12} \)
$29$ \( 1 - 21 T + 249 T^{2} - 2108 T^{3} + 14766 T^{4} - 90339 T^{5} + 507904 T^{6} - 2619831 T^{7} + 12418206 T^{8} - 51412012 T^{9} + 176112969 T^{10} - 430734129 T^{11} + 594823321 T^{12} \)
$31$ \( 1 + 3 T + 114 T^{2} + 209 T^{3} + 6642 T^{4} + 10098 T^{5} + 255648 T^{6} + 313038 T^{7} + 6382962 T^{8} + 6226319 T^{9} + 105281394 T^{10} + 85887453 T^{11} + 887503681 T^{12} \)
$37$ \( 1 + 120 T^{2} - 180 T^{3} + 7944 T^{4} - 10224 T^{5} + 370078 T^{6} - 378288 T^{7} + 10875336 T^{8} - 9117540 T^{9} + 224899320 T^{10} + 2565726409 T^{12} \)
$41$ \( 1 - 9 T + 186 T^{2} - 1421 T^{3} + 16128 T^{4} - 102162 T^{5} + 834046 T^{6} - 4188642 T^{7} + 27111168 T^{8} - 97936741 T^{9} + 525591546 T^{10} - 1042705809 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 + 3 T + 90 T^{2} + 609 T^{3} + 6102 T^{4} + 34602 T^{5} + 344926 T^{6} + 1487886 T^{7} + 11282598 T^{8} + 48419763 T^{9} + 307692090 T^{10} + 441025329 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 - 21 T + 303 T^{2} - 3044 T^{3} + 27474 T^{4} - 213243 T^{5} + 1570804 T^{6} - 10022421 T^{7} + 60690066 T^{8} - 316037212 T^{9} + 1478543343 T^{10} - 4816245147 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 - 15 T + 147 T^{2} - 1020 T^{3} + 8466 T^{4} - 77433 T^{5} + 656404 T^{6} - 4103949 T^{7} + 23780994 T^{8} - 151854540 T^{9} + 1159900707 T^{10} - 6272932395 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 - 15 T + 345 T^{2} - 3492 T^{3} + 46986 T^{4} - 360735 T^{5} + 3571414 T^{6} - 21283365 T^{7} + 163558266 T^{8} - 717183468 T^{9} + 4180489545 T^{10} - 10723864485 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 - 15 T + 333 T^{2} - 3566 T^{3} + 46368 T^{4} - 386895 T^{5} + 3658196 T^{6} - 23600595 T^{7} + 172535328 T^{8} - 809414246 T^{9} + 4610665053 T^{10} - 12668944515 T^{11} + 51520374361 T^{12} \)
$67$ \( ( 1 - T )^{6} \)
$71$ \( 1 - 18 T + 300 T^{2} - 3057 T^{3} + 34512 T^{4} - 321813 T^{5} + 3065020 T^{6} - 22848723 T^{7} + 173974992 T^{8} - 1094133927 T^{9} + 7623504300 T^{10} - 32476128318 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 + 6 T + 228 T^{2} + 1709 T^{3} + 28116 T^{4} + 226017 T^{5} + 2344662 T^{6} + 16499241 T^{7} + 149830164 T^{8} + 664830053 T^{9} + 6474798948 T^{10} + 12438429558 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 - 9 T + 84 T^{2} - 1001 T^{3} + 2028 T^{4} + 27336 T^{5} + 181484 T^{6} + 2159544 T^{7} + 12656748 T^{8} - 493532039 T^{9} + 3271806804 T^{10} - 27693507591 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 - 24 T + 708 T^{2} - 10769 T^{3} + 174270 T^{4} - 1854945 T^{5} + 20317210 T^{6} - 153960435 T^{7} + 1200546030 T^{8} - 6157574203 T^{9} + 33600491268 T^{10} - 94536975432 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 - 24 T + 276 T^{2} - 1213 T^{3} - 5016 T^{4} + 173625 T^{5} - 1927802 T^{6} + 15452625 T^{7} - 39731736 T^{8} - 855127397 T^{9} + 17316858516 T^{10} - 134017426776 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 - 9 T + 453 T^{2} - 2388 T^{3} + 81744 T^{4} - 252837 T^{5} + 9145678 T^{6} - 24525189 T^{7} + 769129296 T^{8} - 2179463124 T^{9} + 40103764293 T^{10} - 77286062313 T^{11} + 832972004929 T^{12} \)
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