Properties

Label 8016.2.a.s
Level 8016
Weight 2
Character orbit 8016.a
Self dual yes
Analytic conductor 64.008
Analytic rank 1
Dimension 5
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.284897.1
Defining polynomial: \(x^{5} - 2 x^{4} - 6 x^{3} + 5 x^{2} + 10 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4008)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( -\beta_{3} + \beta_{4} ) q^{5} + ( 1 + \beta_{2} - \beta_{3} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -\beta_{3} + \beta_{4} ) q^{5} + ( 1 + \beta_{2} - \beta_{3} ) q^{7} + q^{9} + ( \beta_{3} + \beta_{4} ) q^{11} + ( -2 - \beta_{1} + \beta_{2} - \beta_{4} ) q^{13} + ( \beta_{3} - \beta_{4} ) q^{15} + ( -3 + 2 \beta_{3} ) q^{17} + ( \beta_{1} - \beta_{2} ) q^{19} + ( -1 - \beta_{2} + \beta_{3} ) q^{21} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{23} + ( -1 + 3 \beta_{1} + \beta_{3} ) q^{25} - q^{27} + ( 4 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{29} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{31} + ( -\beta_{3} - \beta_{4} ) q^{33} + ( 2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{35} + ( -1 - 3 \beta_{2} ) q^{37} + ( 2 + \beta_{1} - \beta_{2} + \beta_{4} ) q^{39} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{41} + ( 4 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{43} + ( -\beta_{3} + \beta_{4} ) q^{45} + ( 1 - \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{47} + ( -2 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{49} + ( 3 - 2 \beta_{3} ) q^{51} + ( -2 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{53} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{55} + ( -\beta_{1} + \beta_{2} ) q^{57} + ( -2 + 5 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{59} + ( -4 - 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{61} + ( 1 + \beta_{2} - \beta_{3} ) q^{63} + ( -3 - \beta_{1} + 3 \beta_{3} - 4 \beta_{4} ) q^{65} + ( 4 - \beta_{1} + \beta_{2} + 4 \beta_{3} - 3 \beta_{4} ) q^{67} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{69} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} ) q^{71} + ( -5 + 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{73} + ( 1 - 3 \beta_{1} - \beta_{3} ) q^{75} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{77} + ( 2 + \beta_{1} + 4 \beta_{3} + 2 \beta_{4} ) q^{79} + q^{81} + ( 2 + 2 \beta_{1} + 5 \beta_{2} - \beta_{3} + \beta_{4} ) q^{83} + ( -4 - 4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{85} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{87} + ( -8 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{89} + ( -1 - 3 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{91} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{93} + ( 1 + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{95} + ( -6 + 2 \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{97} + ( \beta_{3} + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{3} + q^{5} + 4q^{7} + 5q^{9} + O(q^{10}) \) \( 5q - 5q^{3} + q^{5} + 4q^{7} + 5q^{9} + 3q^{11} - 14q^{13} - q^{15} - 13q^{17} + 2q^{19} - 4q^{21} + 5q^{23} + 2q^{25} - 5q^{27} + 13q^{29} - 2q^{31} - 3q^{33} + 12q^{35} - 5q^{37} + 14q^{39} - 20q^{41} + 20q^{43} + q^{45} - q^{47} - 9q^{49} + 13q^{51} - 3q^{53} + 3q^{55} - 2q^{57} + q^{59} - 34q^{61} + 4q^{63} - 22q^{65} + 16q^{67} - 5q^{69} + 5q^{71} - 12q^{73} - 2q^{75} - 8q^{77} + 20q^{79} + 5q^{81} + 15q^{83} - 27q^{85} - 13q^{87} - 48q^{89} - 7q^{91} + 2q^{93} + 5q^{95} - 21q^{97} + 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 6 x^{3} + 5 x^{2} + 10 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - 3 \nu^{3} - 3 \nu^{2} + 8 \nu + 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 3 \nu^{3} - 4 \nu^{2} + 9 \nu + 6 \)
\(\beta_{4}\)\(=\)\( 2 \nu^{4} - 5 \nu^{3} - 9 \nu^{2} + 14 \nu + 11 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{4} - 3 \beta_{3} + \beta_{2} + 5 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(3 \beta_{4} - 12 \beta_{3} + 7 \beta_{2} + 10 \beta_{1} + 18\)

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
1.80442
−0.759623
−1.65065
−0.435916
3.04177
0 −1.00000 0 −3.40664 0 −0.548479 0 1.00000 0
1.2 0 −1.00000 0 −0.473647 0 −0.663349 0 1.00000 0
1.3 0 −1.00000 0 −0.457884 0 2.37531 0 1.00000 0
1.4 0 −1.00000 0 2.07208 0 −1.37406 0 1.00000 0
1.5 0 −1.00000 0 3.26608 0 4.21058 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 8016.2.a.s 5
4.b odd 2 1 4008.2.a.f 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.f 5 4.b odd 2 1
8016.2.a.s 5 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(167\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{5} - T_{5}^{4} - 13 T_{5}^{3} + 12 T_{5}^{2} + 19 T_{5} + 5 \)
\( T_{7}^{5} - 4 T_{7}^{4} - 5 T_{7}^{3} + 13 T_{7}^{2} + 17 T_{7} + 5 \)
\( T_{11}^{5} - 3 T_{11}^{4} - 19 T_{11}^{3} + 30 T_{11}^{2} + 47 T_{11} - 55 \)
\( T_{13}^{5} + 14 T_{13}^{4} + 61 T_{13}^{3} + 62 T_{13}^{2} - 132 T_{13} - 215 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{5} \)
$5$ \( 1 - T + 12 T^{2} - 8 T^{3} + 74 T^{4} - 25 T^{5} + 370 T^{6} - 200 T^{7} + 1500 T^{8} - 625 T^{9} + 3125 T^{10} \)
$7$ \( 1 - 4 T + 30 T^{2} - 99 T^{3} + 402 T^{4} - 989 T^{5} + 2814 T^{6} - 4851 T^{7} + 10290 T^{8} - 9604 T^{9} + 16807 T^{10} \)
$11$ \( 1 - 3 T + 36 T^{2} - 102 T^{3} + 630 T^{4} - 1573 T^{5} + 6930 T^{6} - 12342 T^{7} + 47916 T^{8} - 43923 T^{9} + 161051 T^{10} \)
$13$ \( 1 + 14 T + 126 T^{2} + 790 T^{3} + 3937 T^{4} + 15593 T^{5} + 51181 T^{6} + 133510 T^{7} + 276822 T^{8} + 399854 T^{9} + 371293 T^{10} \)
$17$ \( 1 + 13 T + 119 T^{2} + 830 T^{3} + 4589 T^{4} + 20715 T^{5} + 78013 T^{6} + 239870 T^{7} + 584647 T^{8} + 1085773 T^{9} + 1419857 T^{10} \)
$19$ \( 1 - 2 T + 82 T^{2} - 108 T^{3} + 2829 T^{4} - 2651 T^{5} + 53751 T^{6} - 38988 T^{7} + 562438 T^{8} - 260642 T^{9} + 2476099 T^{10} \)
$23$ \( 1 - 5 T + 78 T^{2} - 386 T^{3} + 2904 T^{4} - 12749 T^{5} + 66792 T^{6} - 204194 T^{7} + 949026 T^{8} - 1399205 T^{9} + 6436343 T^{10} \)
$29$ \( 1 - 13 T + 158 T^{2} - 1100 T^{3} + 7688 T^{4} - 39579 T^{5} + 222952 T^{6} - 925100 T^{7} + 3853462 T^{8} - 9194653 T^{9} + 20511149 T^{10} \)
$31$ \( 1 + 2 T + 105 T^{2} + 215 T^{3} + 5580 T^{4} + 8789 T^{5} + 172980 T^{6} + 206615 T^{7} + 3128055 T^{8} + 1847042 T^{9} + 28629151 T^{10} \)
$37$ \( 1 + 5 T + 114 T^{2} + 372 T^{3} + 5706 T^{4} + 14757 T^{5} + 211122 T^{6} + 509268 T^{7} + 5774442 T^{8} + 9370805 T^{9} + 69343957 T^{10} \)
$41$ \( 1 + 20 T + 219 T^{2} + 1955 T^{3} + 16226 T^{4} + 114713 T^{5} + 665266 T^{6} + 3286355 T^{7} + 15093699 T^{8} + 56515220 T^{9} + 115856201 T^{10} \)
$43$ \( 1 - 20 T + 293 T^{2} - 3203 T^{3} + 28482 T^{4} - 201705 T^{5} + 1224726 T^{6} - 5922347 T^{7} + 23295551 T^{8} - 68376020 T^{9} + 147008443 T^{10} \)
$47$ \( 1 + T + 80 T^{2} - 105 T^{3} + 5053 T^{4} + 627 T^{5} + 237491 T^{6} - 231945 T^{7} + 8305840 T^{8} + 4879681 T^{9} + 229345007 T^{10} \)
$53$ \( 1 + 3 T + 174 T^{2} + 423 T^{3} + 15171 T^{4} + 27597 T^{5} + 804063 T^{6} + 1188207 T^{7} + 25904598 T^{8} + 23671443 T^{9} + 418195493 T^{10} \)
$59$ \( 1 - T + 94 T^{2} + 389 T^{3} - 829 T^{4} + 52407 T^{5} - 48911 T^{6} + 1354109 T^{7} + 19305626 T^{8} - 12117361 T^{9} + 714924299 T^{10} \)
$61$ \( 1 + 34 T + 606 T^{2} + 7646 T^{3} + 77597 T^{4} + 659411 T^{5} + 4733417 T^{6} + 28450766 T^{7} + 137550486 T^{8} + 470758594 T^{9} + 844596301 T^{10} \)
$67$ \( 1 - 16 T + 232 T^{2} - 2478 T^{3} + 23361 T^{4} - 190905 T^{5} + 1565187 T^{6} - 11123742 T^{7} + 69777016 T^{8} - 322417936 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 - 5 T + 215 T^{2} - 595 T^{3} + 19727 T^{4} - 33999 T^{5} + 1400617 T^{6} - 2999395 T^{7} + 76950865 T^{8} - 127058405 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 + 12 T + 262 T^{2} + 1656 T^{3} + 24261 T^{4} + 111631 T^{5} + 1771053 T^{6} + 8824824 T^{7} + 101922454 T^{8} + 340778892 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 - 20 T + 363 T^{2} - 4417 T^{3} + 51806 T^{4} - 475675 T^{5} + 4092674 T^{6} - 27566497 T^{7} + 178973157 T^{8} - 779001620 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 - 15 T + 214 T^{2} - 1165 T^{3} + 7325 T^{4} + 3901 T^{5} + 607975 T^{6} - 8025685 T^{7} + 122362418 T^{8} - 711874815 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 + 48 T + 1306 T^{2} + 24216 T^{3} + 335755 T^{4} + 3585017 T^{5} + 29882195 T^{6} + 191814936 T^{7} + 920689514 T^{8} + 3011627568 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 + 21 T + 593 T^{2} + 8159 T^{3} + 124997 T^{4} + 1187307 T^{5} + 12124709 T^{6} + 76768031 T^{7} + 541215089 T^{8} + 1859114901 T^{9} + 8587340257 T^{10} \)
show more
show less