Properties

Label 4020.2.a.f
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

Related objects

Downloads

Learn more about

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: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
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.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} + \beta_{1} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{5} + \beta_{1} q^{7} + q^{9} + ( -1 - \beta_{5} ) q^{11} + ( 1 - \beta_{3} ) q^{13} - q^{15} + ( 2 + \beta_{3} - \beta_{5} ) q^{17} + ( -\beta_{2} - \beta_{5} ) q^{19} -\beta_{1} q^{21} + ( -1 + \beta_{2} + \beta_{4} ) q^{23} + q^{25} - q^{27} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{29} + ( \beta_{1} - \beta_{3} + \beta_{5} ) q^{31} + ( 1 + \beta_{5} ) q^{33} + \beta_{1} q^{35} + ( 3 - \beta_{2} - \beta_{4} ) q^{37} + ( -1 + \beta_{3} ) q^{39} + ( 1 + \beta_{4} + \beta_{5} ) q^{41} + ( 1 - \beta_{1} - \beta_{2} ) q^{43} + q^{45} + ( \beta_{1} - \beta_{4} ) q^{47} + ( 2 + 2 \beta_{1} + \beta_{4} + \beta_{5} ) q^{49} + ( -2 - \beta_{3} + \beta_{5} ) q^{51} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{5} ) q^{53} + ( -1 - \beta_{5} ) q^{55} + ( \beta_{2} + \beta_{5} ) q^{57} + ( 1 + \beta_{3} ) q^{59} + ( 2 + \beta_{2} + \beta_{5} ) q^{61} + \beta_{1} q^{63} + ( 1 - \beta_{3} ) q^{65} + q^{67} + ( 1 - \beta_{2} - \beta_{4} ) q^{69} + ( -2 + \beta_{3} + \beta_{4} + \beta_{5} ) q^{71} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{73} - q^{75} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{5} ) q^{77} + ( 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{79} + q^{81} + ( 1 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{83} + ( 2 + \beta_{3} - \beta_{5} ) q^{85} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{87} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{5} ) q^{89} + ( -3 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{91} + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{93} + ( -\beta_{2} - \beta_{5} ) q^{95} + ( 7 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{97} + ( -1 - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} + 6q^{5} + q^{7} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} + 6q^{5} + q^{7} + 6q^{9} - 7q^{11} + 7q^{13} - 6q^{15} + 10q^{17} - 3q^{19} - q^{21} - 4q^{23} + 6q^{25} - 6q^{27} + 9q^{29} + 3q^{31} + 7q^{33} + q^{35} + 16q^{37} - 7q^{39} + 7q^{41} + 3q^{43} + 6q^{45} + q^{47} + 15q^{49} - 10q^{51} + 7q^{53} - 7q^{55} + 3q^{57} + 5q^{59} + 15q^{61} + q^{63} + 7q^{65} + 6q^{67} + 4q^{69} - 12q^{71} + 12q^{73} - 6q^{75} + 9q^{77} + 9q^{79} + 6q^{81} - 2q^{83} + 10q^{85} - 9q^{87} + 10q^{89} - 5q^{91} - 3q^{93} - 3q^{95} + 37q^{97} - 7q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 28 x^{4} - 12 x^{3} + 209 x^{2} + 360 x + 144\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{5} - 17 \nu^{4} - 104 \nu^{3} + 180 \nu^{2} + 685 \nu + 348 \)\()/24\)
\(\beta_{3}\)\(=\)\((\)\( -7 \nu^{5} + 19 \nu^{4} + 184 \nu^{3} - 204 \nu^{2} - 1415 \nu - 924 \)\()/48\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{5} - 19 \nu^{4} - 160 \nu^{3} + 204 \nu^{2} + 1055 \nu + 492 \)\()/24\)
\(\beta_{5}\)\(=\)\((\)\( -7 \nu^{5} + 19 \nu^{4} + 160 \nu^{3} - 180 \nu^{2} - 1103 \nu - 708 \)\()/24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4} + 2 \beta_{1} + 9\)
\(\nu^{3}\)\(=\)\(\beta_{4} + 2 \beta_{3} + 15 \beta_{1} + 18\)
\(\nu^{4}\)\(=\)\(10 \beta_{5} + 18 \beta_{4} + 6 \beta_{3} - 7 \beta_{2} + 45 \beta_{1} + 143\)
\(\nu^{5}\)\(=\)\(-2 \beta_{5} + 46 \beta_{4} + 62 \beta_{3} - 19 \beta_{2} + 256 \beta_{1} + 467\)

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
−3.20623
−2.21014
−1.71821
−0.620094
4.08027
4.67440
0 −1.00000 0 1.00000 0 −3.20623 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −2.21014 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 −1.71821 0 1.00000 0
1.4 0 −1.00000 0 1.00000 0 −0.620094 0 1.00000 0
1.5 0 −1.00000 0 1.00000 0 4.08027 0 1.00000 0
1.6 0 −1.00000 0 1.00000 0 4.67440 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.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.a.f 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 \( T_{7}^{6} - T_{7}^{5} - 28 T_{7}^{4} - 12 T_{7}^{3} + 209 T_{7}^{2} + 360 T_{7} + 144 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 + T )^{6} \)
$5$ \( ( 1 - T )^{6} \)
$7$ \( 1 - T + 14 T^{2} - 47 T^{3} + 160 T^{4} - 382 T^{5} + 1698 T^{6} - 2674 T^{7} + 7840 T^{8} - 16121 T^{9} + 33614 T^{10} - 16807 T^{11} + 117649 T^{12} \)
$11$ \( 1 + 7 T + 52 T^{2} + 241 T^{3} + 1104 T^{4} + 3942 T^{5} + 14534 T^{6} + 43362 T^{7} + 133584 T^{8} + 320771 T^{9} + 761332 T^{10} + 1127357 T^{11} + 1771561 T^{12} \)
$13$ \( 1 - 7 T + 47 T^{2} - 214 T^{3} + 1146 T^{4} - 4513 T^{5} + 18522 T^{6} - 58669 T^{7} + 193674 T^{8} - 470158 T^{9} + 1342367 T^{10} - 2599051 T^{11} + 4826809 T^{12} \)
$17$ \( 1 - 10 T + 82 T^{2} - 461 T^{3} + 2598 T^{4} - 12491 T^{5} + 56936 T^{6} - 212347 T^{7} + 750822 T^{8} - 2264893 T^{9} + 6848722 T^{10} - 14198570 T^{11} + 24137569 T^{12} \)
$19$ \( 1 + 3 T + 73 T^{2} + 170 T^{3} + 2688 T^{4} + 5347 T^{5} + 63764 T^{6} + 101593 T^{7} + 970368 T^{8} + 1166030 T^{9} + 9513433 T^{10} + 7428297 T^{11} + 47045881 T^{12} \)
$23$ \( 1 + 4 T + 72 T^{2} + 152 T^{3} + 1528 T^{4} - 12 T^{5} + 18638 T^{6} - 276 T^{7} + 808312 T^{8} + 1849384 T^{9} + 20148552 T^{10} + 25745372 T^{11} + 148035889 T^{12} \)
$29$ \( 1 - 9 T + 115 T^{2} - 532 T^{3} + 4192 T^{4} - 11455 T^{5} + 107112 T^{6} - 332195 T^{7} + 3525472 T^{8} - 12974948 T^{9} + 81337315 T^{10} - 184600341 T^{11} + 594823321 T^{12} \)
$31$ \( 1 - 3 T + 116 T^{2} - 513 T^{3} + 6456 T^{4} - 31790 T^{5} + 237604 T^{6} - 985490 T^{7} + 6204216 T^{8} - 15282783 T^{9} + 107128436 T^{10} - 85887453 T^{11} + 887503681 T^{12} \)
$37$ \( 1 - 16 T + 256 T^{2} - 2444 T^{3} + 22360 T^{4} - 155208 T^{5} + 1050702 T^{6} - 5742696 T^{7} + 30610840 T^{8} - 123795932 T^{9} + 479785216 T^{10} - 1109503312 T^{11} + 2565726409 T^{12} \)
$41$ \( 1 - 7 T + 200 T^{2} - 1161 T^{3} + 17720 T^{4} - 84576 T^{5} + 918738 T^{6} - 3467616 T^{7} + 29787320 T^{8} - 80017281 T^{9} + 565152200 T^{10} - 810993407 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 - 3 T + 222 T^{2} - 599 T^{3} + 21772 T^{4} - 49864 T^{5} + 1210538 T^{6} - 2144152 T^{7} + 40256428 T^{8} - 47624693 T^{9} + 758973822 T^{10} - 441025329 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 - T + 213 T^{2} - 102 T^{3} + 21168 T^{4} - 5801 T^{5} + 1257772 T^{6} - 272647 T^{7} + 46760112 T^{8} - 10589946 T^{9} + 1039372053 T^{10} - 229345007 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 - 7 T + 249 T^{2} - 1418 T^{3} + 28238 T^{4} - 129827 T^{5} + 1892976 T^{6} - 6880831 T^{7} + 79320542 T^{8} - 211107586 T^{9} + 1964729769 T^{10} - 2927368451 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 - 5 T + 313 T^{2} - 1348 T^{3} + 43144 T^{4} - 151913 T^{5} + 3320862 T^{6} - 8962867 T^{7} + 150184264 T^{8} - 276850892 T^{9} + 3792733993 T^{10} - 3574621495 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 - 15 T + 415 T^{2} - 4412 T^{3} + 66966 T^{4} - 528689 T^{5} + 5538276 T^{6} - 32250029 T^{7} + 249180486 T^{8} - 1001440172 T^{9} + 5746024015 T^{10} - 12668944515 T^{11} + 51520374361 T^{12} \)
$67$ \( ( 1 - T )^{6} \)
$71$ \( 1 + 12 T + 362 T^{2} + 3665 T^{3} + 58658 T^{4} + 478899 T^{5} + 5395088 T^{6} + 34001829 T^{7} + 295694978 T^{8} + 1311743815 T^{9} + 9199028522 T^{10} + 21650752212 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 - 12 T + 320 T^{2} - 3055 T^{3} + 50012 T^{4} - 385121 T^{5} + 4601870 T^{6} - 28113833 T^{7} + 266513948 T^{8} - 1188446935 T^{9} + 9087437120 T^{10} - 24876859116 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 - 9 T + 214 T^{2} - 971 T^{3} + 14970 T^{4} - 38514 T^{5} + 890304 T^{6} - 3042606 T^{7} + 93427770 T^{8} - 478740869 T^{9} + 8335317334 T^{10} - 27693507591 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 + 2 T + 132 T^{2} - 319 T^{3} + 18328 T^{4} - 37297 T^{5} + 1553942 T^{6} - 3095651 T^{7} + 126261592 T^{8} - 182400053 T^{9} + 6264498372 T^{10} + 7878081286 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 - 10 T + 362 T^{2} - 3087 T^{3} + 60038 T^{4} - 436915 T^{5} + 6351790 T^{6} - 38885435 T^{7} + 475560998 T^{8} - 2176239303 T^{9} + 22712691242 T^{10} - 55840594490 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 - 37 T + 881 T^{2} - 15720 T^{3} + 230894 T^{4} - 2872481 T^{5} + 30509558 T^{6} - 278630657 T^{7} + 2172481646 T^{8} - 14347219560 T^{9} + 77994296561 T^{10} - 317731589509 T^{11} + 832972004929 T^{12} \)
show more
show less