Properties

Label 4020.2.a.b
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: 4.4.9301.1
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.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + ( \beta_{1} + \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} - q^{5} + ( \beta_{1} + \beta_{2} ) q^{7} + q^{9} + ( 1 + \beta_{1} ) q^{11} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{13} + q^{15} + ( \beta_{2} + \beta_{3} ) q^{17} + ( -1 - \beta_{1} + \beta_{3} ) q^{19} + ( -\beta_{1} - \beta_{2} ) q^{21} + ( 1 - 2 \beta_{1} - 2 \beta_{2} ) q^{23} + q^{25} - q^{27} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{29} + ( -4 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{31} + ( -1 - \beta_{1} ) q^{33} + ( -\beta_{1} - \beta_{2} ) q^{35} + ( 1 + 2 \beta_{3} ) q^{37} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{39} + ( -2 - 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{41} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} - q^{45} + ( 1 - \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{47} + ( -1 + 3 \beta_{1} + \beta_{3} ) q^{49} + ( -\beta_{2} - \beta_{3} ) q^{51} + ( 1 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{53} + ( -1 - \beta_{1} ) q^{55} + ( 1 + \beta_{1} - \beta_{3} ) q^{57} + ( 3 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{59} + ( -3 - \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{61} + ( \beta_{1} + \beta_{2} ) q^{63} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{65} + q^{67} + ( -1 + 2 \beta_{1} + 2 \beta_{2} ) q^{69} + ( -3 - 4 \beta_{1} + \beta_{2} - 6 \beta_{3} ) q^{71} + ( 4 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{73} - q^{75} + ( 3 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{77} + ( 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{79} + q^{81} + ( 5 - \beta_{2} + 2 \beta_{3} ) q^{83} + ( -\beta_{2} - \beta_{3} ) q^{85} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{87} + ( -7 + 2 \beta_{1} - \beta_{2} ) q^{89} + ( -6 - 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{91} + ( 4 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{93} + ( 1 + \beta_{1} - \beta_{3} ) q^{95} + ( -6 - 3 \beta_{1} - 2 \beta_{2} ) q^{97} + ( 1 + \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} - 4q^{5} - q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} - 4q^{5} - q^{7} + 4q^{9} + 5q^{11} - q^{13} + 4q^{15} - 4q^{17} - 7q^{19} + q^{21} + 6q^{23} + 4q^{25} - 4q^{27} - q^{29} - 11q^{31} - 5q^{33} + q^{35} + q^{39} - 13q^{41} + 15q^{43} - 4q^{45} + 7q^{47} - 3q^{49} + 4q^{51} + 13q^{53} - 5q^{55} + 7q^{57} + q^{59} - 15q^{61} - q^{63} + q^{65} + 4q^{67} - 6q^{69} - 6q^{71} + 8q^{73} - 4q^{75} + 9q^{77} - q^{79} + 4q^{81} + 18q^{83} + 4q^{85} + q^{87} - 24q^{89} - 29q^{91} + 11q^{93} + 7q^{95} - 23q^{97} + 5q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 3\)

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
−0.784476
0.869986
−1.68863
2.60312
0 −1.00000 0 −1.00000 0 −2.38460 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 −2.24312 0 1.00000 0
1.3 0 −1.00000 0 −1.00000 0 −0.148523 0 1.00000 0
1.4 0 −1.00000 0 −1.00000 0 3.77625 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.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.a.b 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 \( T_{7}^{4} + T_{7}^{3} - 12 T_{7}^{2} - 22 T_{7} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( 1 + T + 16 T^{2} - T^{3} + 123 T^{4} - 7 T^{5} + 784 T^{6} + 343 T^{7} + 2401 T^{8} \)
$11$ \( 1 - 5 T + 48 T^{2} - 161 T^{3} + 813 T^{4} - 1771 T^{5} + 5808 T^{6} - 6655 T^{7} + 14641 T^{8} \)
$13$ \( 1 + T + 27 T^{2} + 24 T^{3} + 365 T^{4} + 312 T^{5} + 4563 T^{6} + 2197 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 4 T + 52 T^{2} + 217 T^{3} + 1187 T^{4} + 3689 T^{5} + 15028 T^{6} + 19652 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 7 T + 79 T^{2} + 368 T^{3} + 2273 T^{4} + 6992 T^{5} + 28519 T^{6} + 48013 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 6 T + 56 T^{2} - 152 T^{3} + 1249 T^{4} - 3496 T^{5} + 29624 T^{6} - 73002 T^{7} + 279841 T^{8} \)
$29$ \( 1 + T + 79 T^{2} + 166 T^{3} + 2863 T^{4} + 4814 T^{5} + 66439 T^{6} + 24389 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 11 T + 116 T^{2} + 761 T^{3} + 4843 T^{4} + 23591 T^{5} + 111476 T^{6} + 327701 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 114 T^{2} + 32 T^{3} + 5747 T^{4} + 1184 T^{5} + 156066 T^{6} + 1874161 T^{8} \)
$41$ \( 1 + 13 T + 98 T^{2} + 387 T^{3} + 2053 T^{4} + 15867 T^{5} + 164738 T^{6} + 895973 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 15 T + 240 T^{2} - 2047 T^{3} + 16991 T^{4} - 88021 T^{5} + 443760 T^{6} - 1192605 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 7 T + 139 T^{2} - 726 T^{3} + 9219 T^{4} - 34122 T^{5} + 307051 T^{6} - 726761 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 13 T + 169 T^{2} - 1572 T^{3} + 12393 T^{4} - 83316 T^{5} + 474721 T^{6} - 1935401 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - T + 85 T^{2} + 80 T^{3} + 3389 T^{4} + 4720 T^{5} + 295885 T^{6} - 205379 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 15 T + 201 T^{2} + 1958 T^{3} + 15667 T^{4} + 119438 T^{5} + 747921 T^{6} + 3404715 T^{7} + 13845841 T^{8} \)
$67$ \( ( 1 - T )^{4} \)
$71$ \( 1 + 6 T - 32 T^{2} + 305 T^{3} + 11555 T^{4} + 21655 T^{5} - 161312 T^{6} + 2147466 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 8 T + 172 T^{2} - 415 T^{3} + 11367 T^{4} - 30295 T^{5} + 916588 T^{6} - 3112136 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + T + 214 T^{2} + 227 T^{3} + 23337 T^{4} + 17933 T^{5} + 1335574 T^{6} + 493039 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 18 T + 424 T^{2} - 4637 T^{3} + 56685 T^{4} - 384871 T^{5} + 2920936 T^{6} - 10292166 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 24 T + 542 T^{2} + 6953 T^{3} + 81163 T^{4} + 618817 T^{5} + 4293182 T^{6} + 16919256 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + 23 T + 511 T^{2} + 6898 T^{3} + 80385 T^{4} + 669106 T^{5} + 4807999 T^{6} + 20991479 T^{7} + 88529281 T^{8} \)
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