Properties

Label 8016.2.a.z
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 8 x^{6} + 15 x^{5} + 19 x^{4} - 31 x^{3} - 13 x^{2} + 14 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} + q^{9} + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{11} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{13} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{15} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{17} + ( -5 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{21} + ( -\beta_{2} - \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{23} + ( 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{25} + q^{27} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{29} + ( -1 - 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{31} + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{33} + ( -4 + 2 \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{35} + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{7} ) q^{37} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{39} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{41} + ( -4 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{43} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{45} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 4 \beta_{7} ) q^{47} + ( 2 - 2 \beta_{1} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{49} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{51} + ( -5 + 2 \beta_{1} + 3 \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{53} + ( -3 - 3 \beta_{1} - \beta_{3} + 3 \beta_{4} + 2 \beta_{6} ) q^{55} + ( -5 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{57} + ( -4 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{59} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{7} ) q^{61} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{63} + ( 2 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 8 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{65} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{5} + 3 \beta_{7} ) q^{67} + ( -\beta_{2} - \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{69} + ( -1 - \beta_{1} + 5 \beta_{2} - \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - \beta_{7} ) q^{71} + ( -2 + 5 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{73} + ( 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{75} + ( -2 - \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{77} + ( 4 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{79} + q^{81} + ( -1 + \beta_{2} + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{83} + ( 1 - 8 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + 6 \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{85} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{87} + ( 1 + 5 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{89} + ( -4 + 2 \beta_{3} - \beta_{4} - 5 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{91} + ( -1 - 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{93} + ( 2 - 5 \beta_{1} - 5 \beta_{2} - \beta_{3} - \beta_{4} + 8 \beta_{5} + \beta_{7} ) q^{95} + ( -6 + 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 7 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{97} + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} + q^{5} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{3} + q^{5} + 8q^{9} - 5q^{11} + q^{15} - 7q^{17} - 24q^{19} - q^{23} + 3q^{25} + 8q^{27} - 11q^{29} - 30q^{31} - 5q^{33} - 26q^{35} + 11q^{37} + 10q^{41} - 24q^{43} + q^{45} + 3q^{47} + 6q^{49} - 7q^{51} - 25q^{53} - 25q^{55} - 24q^{57} - 45q^{59} + 16q^{61} - 10q^{65} - 18q^{67} - q^{69} - 21q^{71} - 8q^{73} + 3q^{75} - 18q^{77} - 10q^{79} + 8q^{81} - 7q^{83} - 11q^{85} - 11q^{87} + 26q^{89} - 15q^{91} - 30q^{93} - q^{95} - 3q^{97} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 8 x^{6} + 15 x^{5} + 19 x^{4} - 31 x^{3} - 13 x^{2} + 14 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 5 \nu^{2} + 3 \)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 7 \nu^{5} + 6 \nu^{4} + 13 \nu^{3} - 8 \nu^{2} - 5 \nu - 1 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} + \nu^{6} - 9 \nu^{5} - 8 \nu^{4} + 23 \nu^{3} + 16 \nu^{2} - 13 \nu - 3 \)\()/2\)
\(\beta_{7}\)\(=\)\( -\nu^{7} + \nu^{6} + 8 \nu^{5} - 7 \nu^{4} - 18 \nu^{3} + 13 \nu^{2} + 9 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 5 \beta_{2} + 12\)
\(\nu^{5}\)\(=\)\(\beta_{7} + 2 \beta_{5} + \beta_{4} + 5 \beta_{3} + 16 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{5} + 8 \beta_{4} + 23 \beta_{2} + 51\)
\(\nu^{7}\)\(=\)\(8 \beta_{7} + \beta_{6} + 17 \beta_{5} + 9 \beta_{4} + 22 \beta_{3} + \beta_{2} + 65 \beta_{1} + 18\)

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
−1.63639
−0.0678707
−0.853788
0.688556
−2.05189
1.60046
2.22210
2.09883
0 1.00000 0 −3.62511 0 1.70865 0 1.00000 0
1.2 0 1.00000 0 −2.71255 0 5.14356 0 1.00000 0
1.3 0 1.00000 0 −0.621044 0 0.794861 0 1.00000 0
1.4 0 1.00000 0 0.0425261 0 −1.43576 0 1.00000 0
1.5 0 1.00000 0 0.925650 0 −2.76498 0 1.00000 0
1.6 0 1.00000 0 1.28584 0 0.120874 0 1.00000 0
1.7 0 1.00000 0 1.56175 0 −4.60397 0 1.00000 0
1.8 0 1.00000 0 4.14293 0 1.03675 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.z 8
4.b odd 2 1 501.2.a.d 8
12.b even 2 1 1503.2.a.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
501.2.a.d 8 4.b odd 2 1
1503.2.a.f 8 12.b even 2 1
8016.2.a.z 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\(T_{5}^{8} - \cdots\)
\( T_{7}^{8} - 31 T_{7}^{6} - 7 T_{7}^{5} + 175 T_{7}^{4} - 59 T_{7}^{3} - 224 T_{7}^{2} + 160 T_{7} - 16 \)
\(T_{11}^{8} + \cdots\)
\( T_{13}^{8} - 61 T_{13}^{6} - 10 T_{13}^{5} + 1088 T_{13}^{4} - 237 T_{13}^{3} - 6922 T_{13}^{2} + 3828 T_{13} + 9224 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( -1 + T )^{8} \)
$5$ \( -2 + 48 T - 18 T^{2} - 117 T^{3} + 91 T^{4} + 18 T^{5} - 21 T^{6} - T^{7} + T^{8} \)
$7$ \( -16 + 160 T - 224 T^{2} - 59 T^{3} + 175 T^{4} - 7 T^{5} - 31 T^{6} + T^{8} \)
$11$ \( 500 - 700 T - 460 T^{2} + 547 T^{3} + 151 T^{4} - 104 T^{5} - 23 T^{6} + 5 T^{7} + T^{8} \)
$13$ \( 9224 + 3828 T - 6922 T^{2} - 237 T^{3} + 1088 T^{4} - 10 T^{5} - 61 T^{6} + T^{8} \)
$17$ \( -1058 + 5532 T^{2} + 3735 T^{3} + 175 T^{4} - 318 T^{5} - 42 T^{6} + 7 T^{7} + T^{8} \)
$19$ \( -19872 - 47520 T - 41784 T^{2} - 16319 T^{3} - 2000 T^{4} + 518 T^{5} + 201 T^{6} + 24 T^{7} + T^{8} \)
$23$ \( -202112 - 204576 T - 43488 T^{2} + 12649 T^{3} + 4223 T^{4} - 226 T^{5} - 115 T^{6} + T^{7} + T^{8} \)
$29$ \( 5992 - 3388 T - 12574 T^{2} + 5515 T^{3} + 1545 T^{4} - 548 T^{5} - 53 T^{6} + 11 T^{7} + T^{8} \)
$31$ \( -115552 + 301520 T - 18776 T^{2} - 81029 T^{3} - 19628 T^{4} - 743 T^{5} + 246 T^{6} + 30 T^{7} + T^{8} \)
$37$ \( -464216 + 434236 T - 1118 T^{2} - 41165 T^{3} + 2973 T^{4} + 1200 T^{5} - 107 T^{6} - 11 T^{7} + T^{8} \)
$41$ \( 389482 + 217914 T - 37652 T^{2} - 26127 T^{3} + 2406 T^{4} + 1001 T^{5} - 96 T^{6} - 10 T^{7} + T^{8} \)
$43$ \( 322 + 77042 T + 64788 T^{2} - 5167 T^{3} - 7218 T^{4} - 651 T^{5} + 136 T^{6} + 24 T^{7} + T^{8} \)
$47$ \( 355556 - 2590656 T - 970912 T^{2} + 21773 T^{3} + 30172 T^{4} + 359 T^{5} - 303 T^{6} - 3 T^{7} + T^{8} \)
$53$ \( 191494 + 219978 T + 61412 T^{2} - 12003 T^{3} - 7332 T^{4} - 501 T^{5} + 155 T^{6} + 25 T^{7} + T^{8} \)
$59$ \( 288064 + 30352 T - 152444 T^{2} - 39529 T^{3} + 11618 T^{4} + 5413 T^{5} + 751 T^{6} + 45 T^{7} + T^{8} \)
$61$ \( -64516 + 73528 T - 4418 T^{2} - 14593 T^{3} + 1468 T^{4} + 1024 T^{5} - 39 T^{6} - 16 T^{7} + T^{8} \)
$67$ \( -209558 + 344414 T - 194308 T^{2} + 34503 T^{3} + 4602 T^{4} - 1572 T^{5} - 61 T^{6} + 18 T^{7} + T^{8} \)
$71$ \( -12220528 - 1253480 T + 1330380 T^{2} + 232649 T^{3} - 19051 T^{4} - 5595 T^{5} - 160 T^{6} + 21 T^{7} + T^{8} \)
$73$ \( -14504 + 23660 T + 135414 T^{2} + 87553 T^{3} + 11324 T^{4} - 1690 T^{5} - 223 T^{6} + 8 T^{7} + T^{8} \)
$79$ \( 2662174 - 8440882 T - 706442 T^{2} + 381923 T^{3} + 36438 T^{4} - 3705 T^{5} - 366 T^{6} + 10 T^{7} + T^{8} \)
$83$ \( -55504 + 101360 T + 33136 T^{2} - 66219 T^{3} + 18984 T^{4} - 567 T^{5} - 267 T^{6} + 7 T^{7} + T^{8} \)
$89$ \( 99128 + 99044 T - 298878 T^{2} + 183291 T^{3} - 44316 T^{4} + 4016 T^{5} + 49 T^{6} - 26 T^{7} + T^{8} \)
$97$ \( 4584952 - 457348 T - 704726 T^{2} + 9497 T^{3} + 28149 T^{4} - 13 T^{5} - 374 T^{6} + 3 T^{7} + T^{8} \)
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