Properties

Label 4020.2.a.c
Level 4020
Weight 2
Character orbit 4020.a
Self dual yes
Analytic conductor 32.100
Analytic rank 1
Dimension 4
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.98117.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + \beta_{1} q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} - q^{5} + \beta_{1} q^{7} + q^{9} + ( -1 - \beta_{1} + \beta_{3} ) q^{11} + ( -1 - \beta_{2} ) q^{13} - q^{15} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} + ( 1 + \beta_{2} - \beta_{3} ) q^{19} + \beta_{1} q^{21} -3 q^{23} + q^{25} + q^{27} + ( -1 - \beta_{1} ) q^{29} + ( -\beta_{2} - \beta_{3} ) q^{31} + ( -1 - \beta_{1} + \beta_{3} ) q^{33} -\beta_{1} q^{35} + ( 3 + 2 \beta_{1} ) q^{37} + ( -1 - \beta_{2} ) q^{39} + ( -4 - \beta_{2} + \beta_{3} ) q^{41} + ( -2 - \beta_{2} + 2 \beta_{3} ) q^{43} - q^{45} + ( -1 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{47} + ( -3 + \beta_{2} - \beta_{3} ) q^{49} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{51} + ( -1 + \beta_{2} + \beta_{3} ) q^{53} + ( 1 + \beta_{1} - \beta_{3} ) q^{55} + ( 1 + \beta_{2} - \beta_{3} ) q^{57} + ( -7 + 4 \beta_{1} + \beta_{2} ) q^{59} + ( -7 + \beta_{2} - \beta_{3} ) q^{61} + \beta_{1} q^{63} + ( 1 + \beta_{2} ) q^{65} - q^{67} -3 q^{69} + ( -5 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{71} + ( -3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{73} + q^{75} + ( -5 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{77} + ( -4 - \beta_{2} ) q^{79} + q^{81} + ( 3 - 2 \beta_{1} - 3 \beta_{3} ) q^{83} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{85} + ( -1 - \beta_{1} ) q^{87} + ( -5 + 2 \beta_{1} + 5 \beta_{3} ) q^{89} + ( -4 \beta_{1} - \beta_{2} ) q^{91} + ( -\beta_{2} - \beta_{3} ) q^{93} + ( -1 - \beta_{2} + \beta_{3} ) q^{95} + ( -4 + 5 \beta_{1} + 3 \beta_{3} ) q^{97} + ( -1 - \beta_{1} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 4q^{5} + q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 4q^{5} + q^{7} + 4q^{9} - 5q^{11} - 5q^{13} - 4q^{15} - 8q^{17} + 5q^{19} + q^{21} - 12q^{23} + 4q^{25} + 4q^{27} - 5q^{29} - q^{31} - 5q^{33} - q^{35} + 14q^{37} - 5q^{39} - 17q^{41} - 9q^{43} - 4q^{45} - 7q^{47} - 11q^{49} - 8q^{51} - 3q^{53} + 5q^{55} + 5q^{57} - 23q^{59} - 27q^{61} + q^{63} + 5q^{65} - 4q^{67} - 12q^{69} - 20q^{71} - 2q^{73} + 4q^{75} - 23q^{77} - 17q^{79} + 4q^{81} + 10q^{83} + 8q^{85} - 5q^{87} - 18q^{89} - 5q^{91} - q^{93} - 5q^{95} - 11q^{97} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 8 x^{2} + 4 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 8 \nu - 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 8 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 8 \beta_{1} + 1\)

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
−2.60457
−0.184077
0.668475
3.12017
0 1.00000 0 −1.00000 0 −2.60457 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −0.184077 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 0.668475 0 1.00000 0
1.4 0 1.00000 0 −1.00000 0 3.12017 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
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.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.a.c 4 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 \( T_{7}^{4} - T_{7}^{3} - 8 T_{7}^{2} + 4 T_{7} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( 1 - T + 20 T^{2} - 17 T^{3} + 183 T^{4} - 119 T^{5} + 980 T^{6} - 343 T^{7} + 2401 T^{8} \)
$11$ \( 1 + 5 T + 26 T^{2} + 95 T^{3} + 411 T^{4} + 1045 T^{5} + 3146 T^{6} + 6655 T^{7} + 14641 T^{8} \)
$13$ \( 1 + 5 T + 31 T^{2} + 70 T^{3} + 341 T^{4} + 910 T^{5} + 5239 T^{6} + 10985 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 8 T + 58 T^{2} + 289 T^{3} + 1469 T^{4} + 4913 T^{5} + 16762 T^{6} + 39304 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 5 T + 51 T^{2} - 224 T^{3} + 1409 T^{4} - 4256 T^{5} + 18411 T^{6} - 34295 T^{7} + 130321 T^{8} \)
$23$ \( ( 1 + 3 T + 23 T^{2} )^{4} \)
$29$ \( 1 + 5 T + 117 T^{2} + 422 T^{3} + 5095 T^{4} + 12238 T^{5} + 98397 T^{6} + 121945 T^{7} + 707281 T^{8} \)
$31$ \( 1 + T + 70 T^{2} - 119 T^{3} + 2217 T^{4} - 3689 T^{5} + 67270 T^{6} + 29791 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 14 T + 188 T^{2} - 1492 T^{3} + 10941 T^{4} - 55204 T^{5} + 257372 T^{6} - 709142 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 17 T + 238 T^{2} + 2123 T^{3} + 16155 T^{4} + 87043 T^{5} + 400078 T^{6} + 1171657 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + 9 T + 136 T^{2} + 807 T^{3} + 7703 T^{4} + 34701 T^{5} + 251464 T^{6} + 715563 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 7 T + 61 T^{2} - 378 T^{3} - 2059 T^{4} - 17766 T^{5} + 134749 T^{6} + 726761 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 3 T + 161 T^{2} + 582 T^{3} + 11405 T^{4} + 30846 T^{5} + 452249 T^{6} + 446631 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 23 T + 255 T^{2} + 1812 T^{3} + 11745 T^{4} + 106908 T^{5} + 887655 T^{6} + 4723717 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 27 T + 483 T^{2} + 5690 T^{3} + 52101 T^{4} + 347090 T^{5} + 1797243 T^{6} + 6128487 T^{7} + 13845841 T^{8} \)
$67$ \( ( 1 + T )^{4} \)
$71$ \( 1 + 20 T + 310 T^{2} + 3443 T^{3} + 32439 T^{4} + 244453 T^{5} + 1562710 T^{6} + 7158220 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 2 T + 160 T^{2} + 317 T^{3} + 16077 T^{4} + 23141 T^{5} + 852640 T^{6} + 778034 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 17 T + 394 T^{2} + 4021 T^{3} + 49295 T^{4} + 317659 T^{5} + 2458954 T^{6} + 8381663 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 10 T + 240 T^{2} - 1505 T^{3} + 25403 T^{4} - 124915 T^{5} + 1653360 T^{6} - 5717870 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 18 T + 144 T^{2} - 297 T^{3} - 9529 T^{4} - 26433 T^{5} + 1140624 T^{6} + 12689442 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + 11 T + 173 T^{2} + 2468 T^{3} + 24581 T^{4} + 239396 T^{5} + 1627757 T^{6} + 10039403 T^{7} + 88529281 T^{8} \)
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