Properties

Label 4020.2.a.e
Level 4020
Weight 2
Character orbit 4020.a
Self dual yes
Analytic conductor 32.100
Analytic rank 1
Dimension 5
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: \(5\)
Coefficient field: 5.5.1257629.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 9 x^{3} + 9 x^{2} + 17 x - 18\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 8 \nu^{2} + 11 \)
\(\beta_{4}\)\(=\)\( \nu^{4} + \nu^{3} - 9 \nu^{2} - 6 \nu + 16 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{4} - \beta_{3} + \beta_{2} + 6 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{3} + 8 \beta_{2} + 21\)

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
1.14431
−2.49431
−1.67454
1.48010
2.54445
0 −1.00000 0 1.00000 0 −4.78538 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −1.20813 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 −0.300647 0 1.00000 0
1.4 0 −1.00000 0 1.00000 0 0.397192 0 1.00000 0
1.5 0 −1.00000 0 1.00000 0 2.89696 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.e 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.a.e 5 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 \( T_{7}^{5} + 3 T_{7}^{4} - 12 T_{7}^{3} - 16 T_{7}^{2} + 3 T_{7} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 + T )^{5} \)
$5$ \( ( 1 - T )^{5} \)
$7$ \( 1 + 3 T + 23 T^{2} + 68 T^{3} + 241 T^{4} + 660 T^{5} + 1687 T^{6} + 3332 T^{7} + 7889 T^{8} + 7203 T^{9} + 16807 T^{10} \)
$11$ \( 1 - 3 T + 29 T^{2} - 60 T^{3} + 385 T^{4} - 712 T^{5} + 4235 T^{6} - 7260 T^{7} + 38599 T^{8} - 43923 T^{9} + 161051 T^{10} \)
$13$ \( 1 + 11 T + 102 T^{2} + 599 T^{3} + 3098 T^{4} + 11862 T^{5} + 40274 T^{6} + 101231 T^{7} + 224094 T^{8} + 314171 T^{9} + 371293 T^{10} \)
$17$ \( 1 + 11 T + 115 T^{2} + 695 T^{3} + 4088 T^{4} + 16919 T^{5} + 69496 T^{6} + 200855 T^{7} + 564995 T^{8} + 918731 T^{9} + 1419857 T^{10} \)
$19$ \( 1 - 6 T + 43 T^{2} - 258 T^{3} + 1470 T^{4} - 5525 T^{5} + 27930 T^{6} - 93138 T^{7} + 294937 T^{8} - 781926 T^{9} + 2476099 T^{10} \)
$23$ \( 1 - 3 T + 37 T^{2} - 98 T^{3} + 417 T^{4} - 2453 T^{5} + 9591 T^{6} - 51842 T^{7} + 450179 T^{8} - 839523 T^{9} + 6436343 T^{10} \)
$29$ \( 1 + 2 T + 45 T^{2} - 66 T^{3} + 2142 T^{4} + 1785 T^{5} + 62118 T^{6} - 55506 T^{7} + 1097505 T^{8} + 1414562 T^{9} + 20511149 T^{10} \)
$31$ \( 1 - T + 75 T^{2} + 328 T^{3} + 1275 T^{4} + 22862 T^{5} + 39525 T^{6} + 315208 T^{7} + 2234325 T^{8} - 923521 T^{9} + 28629151 T^{10} \)
$37$ \( 1 + 13 T + 135 T^{2} + 922 T^{3} + 5933 T^{4} + 31335 T^{5} + 219521 T^{6} + 1262218 T^{7} + 6838155 T^{8} + 24364093 T^{9} + 69343957 T^{10} \)
$41$ \( 1 + 13 T + 207 T^{2} + 1558 T^{3} + 14601 T^{4} + 81272 T^{5} + 598641 T^{6} + 2618998 T^{7} + 14266647 T^{8} + 36734893 T^{9} + 115856201 T^{10} \)
$43$ \( 1 + 9 T + 53 T^{2} + 120 T^{3} + 2299 T^{4} + 15920 T^{5} + 98857 T^{6} + 221880 T^{7} + 4213871 T^{8} + 30769209 T^{9} + 147008443 T^{10} \)
$47$ \( 1 + 12 T + 221 T^{2} + 2144 T^{3} + 20300 T^{4} + 148487 T^{5} + 954100 T^{6} + 4736096 T^{7} + 22944883 T^{8} + 58556172 T^{9} + 229345007 T^{10} \)
$53$ \( 1 + 19 T + 228 T^{2} + 1171 T^{3} + 1956 T^{4} - 25864 T^{5} + 103668 T^{6} + 3289339 T^{7} + 33943956 T^{8} + 149919139 T^{9} + 418195493 T^{10} \)
$59$ \( 1 - 6 T + 127 T^{2} - 562 T^{3} + 9064 T^{4} - 21293 T^{5} + 534776 T^{6} - 1956322 T^{7} + 26083133 T^{8} - 72704166 T^{9} + 714924299 T^{10} \)
$61$ \( 1 + 3 T + 238 T^{2} + 663 T^{3} + 25704 T^{4} + 59788 T^{5} + 1567944 T^{6} + 2467023 T^{7} + 54021478 T^{8} + 41537523 T^{9} + 844596301 T^{10} \)
$67$ \( ( 1 + T )^{5} \)
$71$ \( 1 - 6 T + 311 T^{2} - 1489 T^{3} + 41597 T^{4} - 152470 T^{5} + 2953387 T^{6} - 7506049 T^{7} + 111310321 T^{8} - 152470086 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 + 21 T + 465 T^{2} + 5845 T^{3} + 72518 T^{4} + 625715 T^{5} + 5293814 T^{6} + 31148005 T^{7} + 180892905 T^{8} + 596363061 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 + T + 23 T^{2} - 362 T^{3} + 4893 T^{4} - 32514 T^{5} + 386547 T^{6} - 2259242 T^{7} + 11339897 T^{8} + 38950081 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 + 8 T + 111 T^{2} + 377 T^{3} + 14981 T^{4} + 112874 T^{5} + 1243423 T^{6} + 2597153 T^{7} + 63468357 T^{8} + 379666568 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 + 29 T + 747 T^{2} + 11591 T^{3} + 161040 T^{4} + 1600867 T^{5} + 14332560 T^{6} + 91812311 T^{7} + 526611843 T^{8} + 1819524989 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 + 13 T + 172 T^{2} + 1289 T^{3} + 26352 T^{4} + 245878 T^{5} + 2556144 T^{6} + 12128201 T^{7} + 156979756 T^{8} + 1150880653 T^{9} + 8587340257 T^{10} \)
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