Properties

Label 8016.2.a.r
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

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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: 5.5.11256624.1
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1002)
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.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{2} q^{5} + ( -2 + \beta_{4} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta_{2} q^{5} + ( -2 + \beta_{4} ) q^{7} + q^{9} -\beta_{3} q^{11} + ( 1 + \beta_{1} ) q^{13} -\beta_{2} q^{15} + ( 1 - \beta_{1} ) q^{17} + ( -2 + \beta_{3} ) q^{19} + ( 2 - \beta_{4} ) q^{21} + \beta_{3} q^{23} + ( 3 - 2 \beta_{1} + \beta_{4} ) q^{25} - q^{27} + ( -2 + 2 \beta_{1} - \beta_{3} ) q^{29} + ( -2 - \beta_{3} - \beta_{4} ) q^{31} + \beta_{3} q^{33} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{35} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{37} + ( -1 - \beta_{1} ) q^{39} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{41} + ( -4 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{43} + \beta_{2} q^{45} + ( 2 - 2 \beta_{1} - \beta_{4} ) q^{47} + ( 3 - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{49} + ( -1 + \beta_{1} ) q^{51} + ( -3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{53} + ( 2 - 2 \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{55} + ( 2 - \beta_{3} ) q^{57} + ( -3 + 3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{59} + ( 4 - 2 \beta_{1} - 2 \beta_{4} ) q^{61} + ( -2 + \beta_{4} ) q^{63} + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{65} + ( -2 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{67} -\beta_{3} q^{69} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{71} + ( 2 + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{73} + ( -3 + 2 \beta_{1} - \beta_{4} ) q^{75} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{77} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{79} + q^{81} + ( -1 + \beta_{1} + \beta_{2} + 4 \beta_{4} ) q^{83} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{85} + ( 2 - 2 \beta_{1} + \beta_{3} ) q^{87} + ( -2 + 2 \beta_{1} + 3 \beta_{4} ) q^{89} + ( -2 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{91} + ( 2 + \beta_{3} + \beta_{4} ) q^{93} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{95} + ( -2 + 4 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{97} -\beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{3} - q^{5} - 9q^{7} + 5q^{9} + O(q^{10}) \) \( 5q - 5q^{3} - q^{5} - 9q^{7} + 5q^{9} + 2q^{11} + 6q^{13} + q^{15} + 4q^{17} - 12q^{19} + 9q^{21} - 2q^{23} + 14q^{25} - 5q^{27} - 6q^{29} - 9q^{31} - 2q^{33} - 3q^{35} + 5q^{37} - 6q^{39} + 4q^{41} - 18q^{43} - q^{45} + 7q^{47} + 16q^{49} - 4q^{51} + 3q^{53} + 4q^{55} + 12q^{57} - 13q^{59} + 16q^{61} - 9q^{63} - 10q^{65} - 9q^{67} + 2q^{69} + 10q^{71} + 8q^{73} - 14q^{75} + 6q^{77} - 6q^{79} + 5q^{81} - q^{83} + 8q^{85} + 6q^{87} - 5q^{89} - 12q^{91} + 9q^{93} - 2q^{95} - 7q^{97} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 16 x^{3} + 20 x^{2} + 31 x - 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - 7 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + \nu^{2} - 11 \nu - 3 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 14 \nu^{2} + 8 \nu + 17 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2} + 7\)
\(\nu^{3}\)\(=\)\(2 \beta_{3} - 2 \beta_{2} + 11 \beta_{1} - 4\)
\(\nu^{4}\)\(=\)\(4 \beta_{4} + 28 \beta_{2} - 8 \beta_{1} + 81\)

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
0.308735
−1.14410
2.50259
3.20970
−3.87693
0 −1.00000 0 −3.45234 0 2.53613 0 1.00000 0
1.2 0 −1.00000 0 −2.84552 0 −4.19124 0 1.00000 0
1.3 0 −1.00000 0 −0.368511 0 −4.85901 0 1.00000 0
1.4 0 −1.00000 0 1.65109 0 −0.854515 0 1.00000 0
1.5 0 −1.00000 0 4.01528 0 −1.63136 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.r 5
4.b odd 2 1 1002.2.a.j 5
12.b even 2 1 3006.2.a.t 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.j 5 4.b odd 2 1
3006.2.a.t 5 12.b even 2 1
8016.2.a.r 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{5} + T_{5}^{4} - 19 T_{5}^{3} - 21 T_{5}^{2} + 60 T_{5} + 24 \)
\( T_{7}^{5} + 9 T_{7}^{4} + 15 T_{7}^{3} - 49 T_{7}^{2} - 132 T_{7} - 72 \)
\( T_{11}^{5} - 2 T_{11}^{4} - 28 T_{11}^{3} + 60 T_{11}^{2} + 144 T_{11} - 288 \)
\( T_{13}^{5} - 6 T_{13}^{4} - 2 T_{13}^{3} + 52 T_{13}^{2} - 48 T_{13} - 8 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{5} \)
$5$ \( 1 + T + 6 T^{2} - T^{3} + 25 T^{4} - 36 T^{5} + 125 T^{6} - 25 T^{7} + 750 T^{8} + 625 T^{9} + 3125 T^{10} \)
$7$ \( 1 + 9 T + 50 T^{2} + 203 T^{3} + 673 T^{4} + 1888 T^{5} + 4711 T^{6} + 9947 T^{7} + 17150 T^{8} + 21609 T^{9} + 16807 T^{10} \)
$11$ \( 1 - 2 T + 27 T^{2} - 28 T^{3} + 430 T^{4} - 420 T^{5} + 4730 T^{6} - 3388 T^{7} + 35937 T^{8} - 29282 T^{9} + 161051 T^{10} \)
$13$ \( 1 - 6 T + 63 T^{2} - 260 T^{3} + 1564 T^{4} - 4740 T^{5} + 20332 T^{6} - 43940 T^{7} + 138411 T^{8} - 171366 T^{9} + 371293 T^{10} \)
$17$ \( 1 - 4 T + 75 T^{2} - 248 T^{3} + 2404 T^{4} - 6144 T^{5} + 40868 T^{6} - 71672 T^{7} + 368475 T^{8} - 334084 T^{9} + 1419857 T^{10} \)
$19$ \( 1 + 12 T + 123 T^{2} + 812 T^{3} + 4918 T^{4} + 22368 T^{5} + 93442 T^{6} + 293132 T^{7} + 843657 T^{8} + 1563852 T^{9} + 2476099 T^{10} \)
$23$ \( 1 + 2 T + 87 T^{2} + 124 T^{3} + 3502 T^{4} + 3876 T^{5} + 80546 T^{6} + 65596 T^{7} + 1058529 T^{8} + 559682 T^{9} + 6436343 T^{10} \)
$29$ \( 1 + 6 T + 61 T^{2} + 396 T^{3} + 2830 T^{4} + 11148 T^{5} + 82070 T^{6} + 333036 T^{7} + 1487729 T^{8} + 4243686 T^{9} + 20511149 T^{10} \)
$31$ \( 1 + 9 T + 150 T^{2} + 887 T^{3} + 8569 T^{4} + 37440 T^{5} + 265639 T^{6} + 852407 T^{7} + 4468650 T^{8} + 8311689 T^{9} + 28629151 T^{10} \)
$37$ \( 1 - 5 T + 118 T^{2} - 325 T^{3} + 5909 T^{4} - 10586 T^{5} + 218633 T^{6} - 444925 T^{7} + 5977054 T^{8} - 9370805 T^{9} + 69343957 T^{10} \)
$41$ \( 1 - 4 T + 95 T^{2} - 788 T^{3} + 5080 T^{4} - 46992 T^{5} + 208280 T^{6} - 1324628 T^{7} + 6547495 T^{8} - 11303044 T^{9} + 115856201 T^{10} \)
$43$ \( 1 + 18 T + 191 T^{2} + 1052 T^{3} + 3586 T^{4} + 6484 T^{5} + 154198 T^{6} + 1945148 T^{7} + 15185837 T^{8} + 61538418 T^{9} + 147008443 T^{10} \)
$47$ \( 1 - 7 T + 174 T^{2} - 1073 T^{3} + 14593 T^{4} - 69780 T^{5} + 685871 T^{6} - 2370257 T^{7} + 18065202 T^{8} - 34157767 T^{9} + 229345007 T^{10} \)
$53$ \( 1 - 3 T + 56 T^{2} + 81 T^{3} + 5467 T^{4} - 17952 T^{5} + 289751 T^{6} + 227529 T^{7} + 8337112 T^{8} - 23671443 T^{9} + 418195493 T^{10} \)
$59$ \( 1 + 13 T + 168 T^{2} + 773 T^{3} + 5239 T^{4} + 4014 T^{5} + 309101 T^{6} + 2690813 T^{7} + 34503672 T^{8} + 157525693 T^{9} + 714924299 T^{10} \)
$61$ \( 1 - 16 T + 277 T^{2} - 2880 T^{3} + 32214 T^{4} - 247136 T^{5} + 1965054 T^{6} - 10716480 T^{7} + 62873737 T^{8} - 221533456 T^{9} + 844596301 T^{10} \)
$67$ \( 1 + 9 T + 180 T^{2} + 827 T^{3} + 16051 T^{4} + 66492 T^{5} + 1075417 T^{6} + 3712403 T^{7} + 54137340 T^{8} + 181360089 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 - 10 T + 171 T^{2} - 1364 T^{3} + 15538 T^{4} - 96324 T^{5} + 1103198 T^{6} - 6875924 T^{7} + 61202781 T^{8} - 254116810 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 - 8 T + 261 T^{2} - 1684 T^{3} + 32642 T^{4} - 163368 T^{5} + 2382866 T^{6} - 8974036 T^{7} + 101533437 T^{8} - 227185928 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 + 6 T + 293 T^{2} + 1088 T^{3} + 38140 T^{4} + 103348 T^{5} + 3013060 T^{6} + 6790208 T^{7} + 144460427 T^{8} + 233700486 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 + T + 134 T^{2} + 167 T^{3} + 15877 T^{4} + 3678 T^{5} + 1317791 T^{6} + 1150463 T^{7} + 76619458 T^{8} + 47458321 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 + 5 T + 240 T^{2} + 643 T^{3} + 33427 T^{4} + 85512 T^{5} + 2975003 T^{6} + 5093203 T^{7} + 169192560 T^{8} + 313711205 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 + 7 T + 232 T^{2} + 881 T^{3} + 26207 T^{4} + 67036 T^{5} + 2542079 T^{6} + 8289329 T^{7} + 211740136 T^{8} + 619704967 T^{9} + 8587340257 T^{10} \)
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