Properties

Label 8016.2.a.u
Level 8016
Weight 2
Character orbit 8016.a
Self dual yes
Analytic conductor 64.008
Analytic rank 0
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.38569.1
Defining polynomial: \(x^{5} - 5 x^{3} + 4 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 501)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( \beta_{1} + \beta_{4} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( \beta_{1} + \beta_{4} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{7} + q^{9} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{11} + ( -1 + \beta_{2} + 2 \beta_{4} ) q^{13} + ( \beta_{1} + \beta_{4} ) q^{15} + ( 1 + 2 \beta_{2} - 2 \beta_{4} ) q^{17} + ( 4 - \beta_{2} - \beta_{3} ) q^{19} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{21} + ( -3 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{23} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{25} + q^{27} + ( -4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{29} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{31} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{33} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{35} + ( -4 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{37} + ( -1 + \beta_{2} + 2 \beta_{4} ) q^{39} + ( 1 - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{41} + ( 3 - \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{43} + ( \beta_{1} + \beta_{4} ) q^{45} + ( 1 + 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{47} + ( -3 + 2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{49} + ( 1 + 2 \beta_{2} - 2 \beta_{4} ) q^{51} + ( 2 + \beta_{1} - \beta_{2} + 5 \beta_{3} + \beta_{4} ) q^{53} + ( 2 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{55} + ( 4 - \beta_{2} - \beta_{3} ) q^{57} + ( 2 + 5 \beta_{1} - \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{59} + ( -3 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{61} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{63} + ( 5 - 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} ) q^{65} + ( 3 + 6 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} ) q^{67} + ( -3 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{69} + ( -1 + 5 \beta_{1} + \beta_{2} + 8 \beta_{3} + 3 \beta_{4} ) q^{71} + ( 4 \beta_{1} - \beta_{2} + 4 \beta_{3} + 3 \beta_{4} ) q^{73} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{75} + ( 5 + \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{77} + ( 1 + 3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{79} + q^{81} + ( 1 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - 5 \beta_{4} ) q^{83} + ( -2 + 5 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} ) q^{85} + ( -4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{87} + ( 4 \beta_{1} + \beta_{2} + 5 \beta_{3} + 4 \beta_{4} ) q^{89} + ( 2 + 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} ) q^{91} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} ) q^{95} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 5 \beta_{4} ) q^{97} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{3} - q^{5} + 4q^{7} + 5q^{9} + O(q^{10}) \) \( 5q + 5q^{3} - q^{5} + 4q^{7} + 5q^{9} + 7q^{11} - 8q^{13} - q^{15} + 5q^{17} + 20q^{19} + 4q^{21} - q^{23} - 6q^{25} + 5q^{27} - 5q^{29} + 18q^{31} + 7q^{33} + 6q^{35} - 17q^{37} - 8q^{39} + 6q^{41} + 10q^{43} - q^{45} + 3q^{47} - 13q^{49} + 5q^{51} + 15q^{53} + 9q^{55} + 20q^{57} + 17q^{59} - 2q^{61} + 4q^{63} + 26q^{65} + 24q^{67} - q^{69} - q^{71} + 2q^{73} - 6q^{75} + 26q^{77} + 6q^{79} + 5q^{81} + 7q^{83} - 11q^{85} - 5q^{87} + 15q^{91} + 18q^{93} - q^{95} - 13q^{97} + 7q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 5 x^{3} + 4 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - 4 \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( -\nu^{4} + 5 \nu^{2} - 3 \)
\(\beta_{4}\)\(=\)\( \nu^{4} + \nu^{3} - 5 \nu^{2} - 4 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{4} + \beta_{3} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(4 \beta_{3} + 5 \beta_{2} + 7\)

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
−1.95408
0.790734
−1.15098
0.275834
2.03850
0 1.00000 0 −3.11102 0 1.66149 0 1.00000 0
1.2 0 1.00000 0 −1.61314 0 −2.77861 0 1.00000 0
1.3 0 1.00000 0 0.0593478 0 1.53510 0 1.00000 0
1.4 0 1.00000 0 1.81885 0 0.619101 0 1.00000 0
1.5 0 1.00000 0 1.84596 0 2.96293 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

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

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.u 5
4.b odd 2 1 501.2.a.c 5
12.b even 2 1 1503.2.a.c 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
501.2.a.c 5 4.b odd 2 1
1503.2.a.c 5 12.b even 2 1
8016.2.a.u 5 1.a even 1 1 trivial

Hecke kernels

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

\( T_{5}^{5} + T_{5}^{4} - 9 T_{5}^{3} - 2 T_{5}^{2} + 17 T_{5} - 1 \)
\( T_{7}^{5} - 4 T_{7}^{4} - 3 T_{7}^{3} + 29 T_{7}^{2} - 37 T_{7} + 13 \)
\( T_{11}^{5} - 7 T_{11}^{4} - 3 T_{11}^{3} + 92 T_{11}^{2} - 101 T_{11} - 121 \)
\( T_{13}^{5} + 8 T_{13}^{4} - T_{13}^{3} - 66 T_{13}^{2} - 48 T_{13} + 17 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{5} \)
$5$ \( 1 + T + 16 T^{2} + 18 T^{3} + 132 T^{4} + 129 T^{5} + 660 T^{6} + 450 T^{7} + 2000 T^{8} + 625 T^{9} + 3125 T^{10} \)
$7$ \( 1 - 4 T + 32 T^{2} - 83 T^{3} + 390 T^{4} - 757 T^{5} + 2730 T^{6} - 4067 T^{7} + 10976 T^{8} - 9604 T^{9} + 16807 T^{10} \)
$11$ \( 1 - 7 T + 52 T^{2} - 216 T^{3} + 1010 T^{4} - 3179 T^{5} + 11110 T^{6} - 26136 T^{7} + 69212 T^{8} - 102487 T^{9} + 161051 T^{10} \)
$13$ \( 1 + 8 T + 64 T^{2} + 350 T^{3} + 1603 T^{4} + 6413 T^{5} + 20839 T^{6} + 59150 T^{7} + 140608 T^{8} + 228488 T^{9} + 371293 T^{10} \)
$17$ \( 1 - 5 T + 51 T^{2} - 106 T^{3} + 901 T^{4} - 1007 T^{5} + 15317 T^{6} - 30634 T^{7} + 250563 T^{8} - 417605 T^{9} + 1419857 T^{10} \)
$19$ \( 1 - 20 T + 248 T^{2} - 2074 T^{3} + 13271 T^{4} - 64971 T^{5} + 252149 T^{6} - 748714 T^{7} + 1701032 T^{8} - 2606420 T^{9} + 2476099 T^{10} \)
$23$ \( 1 + T + 38 T^{2} + 66 T^{3} + 1278 T^{4} + 185 T^{5} + 29394 T^{6} + 34914 T^{7} + 462346 T^{8} + 279841 T^{9} + 6436343 T^{10} \)
$29$ \( 1 + 5 T + 74 T^{2} + 250 T^{3} + 3266 T^{4} + 10091 T^{5} + 94714 T^{6} + 210250 T^{7} + 1804786 T^{8} + 3536405 T^{9} + 20511149 T^{10} \)
$31$ \( 1 - 18 T + 273 T^{2} - 2579 T^{3} + 21028 T^{4} - 125483 T^{5} + 651868 T^{6} - 2478419 T^{7} + 8132943 T^{8} - 16623378 T^{9} + 28629151 T^{10} \)
$37$ \( 1 + 17 T + 208 T^{2} + 1432 T^{3} + 8968 T^{4} + 45509 T^{5} + 331816 T^{6} + 1960408 T^{7} + 10535824 T^{8} + 31860737 T^{9} + 69343957 T^{10} \)
$41$ \( 1 - 6 T + 165 T^{2} - 1039 T^{3} + 11876 T^{4} - 65025 T^{5} + 486916 T^{6} - 1746559 T^{7} + 11371965 T^{8} - 16954566 T^{9} + 115856201 T^{10} \)
$43$ \( 1 - 10 T + 163 T^{2} - 1031 T^{3} + 11132 T^{4} - 55845 T^{5} + 478676 T^{6} - 1906319 T^{7} + 12959641 T^{8} - 34188010 T^{9} + 147008443 T^{10} \)
$47$ \( 1 - 3 T + 172 T^{2} - 341 T^{3} + 13415 T^{4} - 19347 T^{5} + 630505 T^{6} - 753269 T^{7} + 17857556 T^{8} - 14639043 T^{9} + 229345007 T^{10} \)
$53$ \( 1 - 15 T + 200 T^{2} - 1763 T^{3} + 13441 T^{4} - 101901 T^{5} + 712373 T^{6} - 4952267 T^{7} + 29775400 T^{8} - 118357215 T^{9} + 418195493 T^{10} \)
$59$ \( 1 - 17 T + 222 T^{2} - 1631 T^{3} + 11419 T^{4} - 68367 T^{5} + 673721 T^{6} - 5677511 T^{7} + 45594138 T^{8} - 205995137 T^{9} + 714924299 T^{10} \)
$61$ \( 1 + 2 T + 158 T^{2} + 962 T^{3} + 12461 T^{4} + 93871 T^{5} + 760121 T^{6} + 3579602 T^{7} + 35862998 T^{8} + 27691682 T^{9} + 844596301 T^{10} \)
$67$ \( 1 - 24 T + 332 T^{2} - 3300 T^{3} + 29991 T^{4} - 250177 T^{5} + 2009397 T^{6} - 14813700 T^{7} + 99853316 T^{8} - 483626904 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 + T + 63 T^{2} + 77 T^{3} + 4051 T^{4} + 28891 T^{5} + 287621 T^{6} + 388157 T^{7} + 22548393 T^{8} + 25411681 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 - 2 T + 256 T^{2} - 646 T^{3} + 32717 T^{4} - 64257 T^{5} + 2388341 T^{6} - 3442534 T^{7} + 99588352 T^{8} - 56796482 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 - 6 T + 209 T^{2} - 425 T^{3} + 21440 T^{4} - 23747 T^{5} + 1693760 T^{6} - 2652425 T^{7} + 103045151 T^{8} - 233700486 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 - 7 T + 286 T^{2} - 1971 T^{3} + 41947 T^{4} - 223037 T^{5} + 3481601 T^{6} - 13578219 T^{7} + 163531082 T^{8} - 332208247 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 + 290 T^{2} - 322 T^{3} + 42889 T^{4} - 40245 T^{5} + 3817121 T^{6} - 2550562 T^{7} + 204441010 T^{8} + 5584059449 T^{10} \)
$97$ \( 1 + 13 T + 373 T^{2} + 3843 T^{3} + 59509 T^{4} + 500039 T^{5} + 5772373 T^{6} + 36158787 T^{7} + 340427029 T^{8} + 1150880653 T^{9} + 8587340257 T^{10} \)
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