Properties

Label 8016.2.a.ba
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
Analytic rank $0$
Dimension $9$
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 16 x^{7} + 45 x^{6} + 67 x^{5} - 166 x^{4} - 83 x^{3} + 152 x^{2} + 51 x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( -1 + \beta_{1} ) q^{5} + ( 2 - \beta_{5} - \beta_{7} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -1 + \beta_{1} ) q^{5} + ( 2 - \beta_{5} - \beta_{7} ) q^{7} + q^{9} + ( -2 + \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} ) q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} ) q^{13} + ( 1 - \beta_{1} ) q^{15} + ( -1 - \beta_{2} - \beta_{6} - \beta_{7} ) q^{17} + ( 1 - \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{19} + ( -2 + \beta_{5} + \beta_{7} ) q^{21} + ( \beta_{1} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{23} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{25} - q^{27} + ( -5 + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{29} + ( 3 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{31} + ( 2 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{33} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{35} + ( 1 - \beta_{2} - \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{8} ) q^{37} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} ) q^{39} + ( -3 + 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{41} + ( 4 + 2 \beta_{3} - \beta_{7} ) q^{43} + ( -1 + \beta_{1} ) q^{45} + ( 1 + \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{47} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{49} + ( 1 + \beta_{2} + \beta_{6} + \beta_{7} ) q^{51} + ( -3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{53} + ( 3 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{55} + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{57} + ( -2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{8} ) q^{59} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{61} + ( 2 - \beta_{5} - \beta_{7} ) q^{63} + ( 2 + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{8} ) q^{65} + ( -1 + 2 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{8} ) q^{67} + ( -\beta_{1} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{69} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{8} ) q^{71} + ( 2 + \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{73} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{75} + ( -3 + \beta_{1} - \beta_{2} - \beta_{7} + 2 \beta_{8} ) q^{77} + ( 4 - 2 \beta_{2} - 2 \beta_{4} - \beta_{7} + 2 \beta_{8} ) q^{79} + q^{81} + ( 2 + 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - 3 \beta_{8} ) q^{83} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{85} + ( 5 - \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{87} + ( -6 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{7} + \beta_{8} ) q^{89} + ( 3 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{91} + ( -3 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{93} + ( 5 + 3 \beta_{1} + 4 \beta_{2} + \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - 3 \beta_{7} - 3 \beta_{8} ) q^{95} + ( 4 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{97} + ( -2 + \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} - 6q^{5} + 11q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} - 6q^{5} + 11q^{7} + 9q^{9} - q^{11} - 4q^{13} + 6q^{15} - 9q^{17} + 8q^{19} - 11q^{21} + 7q^{23} - q^{25} - 9q^{27} - 9q^{29} + 25q^{31} + q^{33} - 5q^{35} - 6q^{37} + 4q^{39} - 4q^{41} + 24q^{43} - 6q^{45} + 16q^{47} + 4q^{49} + 9q^{51} - 26q^{53} + 29q^{55} - 8q^{57} + 4q^{59} - 20q^{61} + 11q^{63} - 8q^{65} + 25q^{67} - 7q^{69} + 15q^{71} - 10q^{73} + q^{75} - 20q^{77} + 34q^{79} + 9q^{81} + 4q^{83} - 13q^{85} + 9q^{87} + 13q^{89} + 21q^{91} - 25q^{93} + 7q^{95} - 4q^{97} - q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 16 x^{7} + 45 x^{6} + 67 x^{5} - 166 x^{4} - 83 x^{3} + 152 x^{2} + 51 x - 10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -253 \nu^{8} + 10903 \nu^{7} - 17171 \nu^{6} - 175482 \nu^{5} + 247955 \nu^{4} + 716568 \nu^{3} - 563660 \nu^{2} - 635601 \nu + 68945 \)\()/102665\)
\(\beta_{3}\)\(=\)\((\)\( -428 \nu^{8} + 9923 \nu^{7} - 2266 \nu^{6} - 178372 \nu^{5} + 67645 \nu^{4} + 878658 \nu^{3} - 38080 \nu^{2} - 988001 \nu - 301330 \)\()/102665\)
\(\beta_{4}\)\(=\)\((\)\( -3031 \nu^{8} + 7666 \nu^{7} + 48718 \nu^{6} - 99334 \nu^{5} - 199070 \nu^{4} + 207921 \nu^{3} + 208150 \nu^{2} + 200063 \nu - 130065 \)\()/102665\)
\(\beta_{5}\)\(=\)\((\)\( 3811 \nu^{8} - 23831 \nu^{7} - 35953 \nu^{6} + 387944 \nu^{5} - 79645 \nu^{4} - 1646101 \nu^{3} + 520415 \nu^{2} + 1605297 \nu + 255125 \)\()/102665\)
\(\beta_{6}\)\(=\)\((\)\( 5684 \nu^{8} - 5129 \nu^{7} - 118627 \nu^{6} + 65121 \nu^{5} + 751965 \nu^{4} - 184819 \nu^{3} - 1469865 \nu^{2} + 296533 \nu + 445920 \)\()/102665\)
\(\beta_{7}\)\(=\)\((\)\( -6589 \nu^{8} + 20594 \nu^{7} + 101842 \nu^{6} - 311796 \nu^{5} - 367380 \nu^{4} + 1137454 \nu^{3} + 148730 \nu^{2} - 769633 \nu + 59190 \)\()/102665\)
\(\beta_{8}\)\(=\)\((\)\( 16128 \nu^{8} - 24668 \nu^{7} - 293609 \nu^{6} + 290982 \nu^{5} + 1500975 \nu^{4} - 446023 \nu^{3} - 2000020 \nu^{2} - 396149 \nu + 286160 \)\()/102665\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} - \beta_{5} + \beta_{4} - \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(-2 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 8 \beta_{1} + 7\)
\(\nu^{4}\)\(=\)\(-3 \beta_{8} - 12 \beta_{7} + 6 \beta_{6} - 12 \beta_{5} + 7 \beta_{4} + 5 \beta_{3} - 17 \beta_{2} - 2 \beta_{1} + 54\)
\(\nu^{5}\)\(=\)\(-30 \beta_{8} - 45 \beta_{7} + 33 \beta_{6} - 27 \beta_{5} - 36 \beta_{4} + 34 \beta_{3} - 32 \beta_{2} + 73 \beta_{1} + 109\)
\(\nu^{6}\)\(=\)\(-56 \beta_{8} - 145 \beta_{7} + 106 \beta_{6} - 132 \beta_{5} + 54 \beta_{4} + 105 \beta_{3} - 225 \beta_{2} - 33 \beta_{1} + 635\)
\(\nu^{7}\)\(=\)\(-391 \beta_{8} - 576 \beta_{7} + 466 \beta_{6} - 324 \beta_{5} - 395 \beta_{4} + 506 \beta_{3} - 459 \beta_{2} + 691 \beta_{1} + 1496\)
\(\nu^{8}\)\(=\)\(-846 \beta_{8} - 1799 \beta_{7} + 1544 \beta_{6} - 1474 \beta_{5} + 418 \beta_{4} + 1662 \beta_{3} - 2818 \beta_{2} - 429 \beta_{1} + 7652\)

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
−3.17025
−1.99382
−0.985991
−0.482381
0.141931
1.41388
1.58856
2.94484
3.54323
0 −1.00000 0 −4.17025 0 1.78655 0 1.00000 0
1.2 0 −1.00000 0 −2.99382 0 4.65862 0 1.00000 0
1.3 0 −1.00000 0 −1.98599 0 0.0909950 0 1.00000 0
1.4 0 −1.00000 0 −1.48238 0 1.03908 0 1.00000 0
1.5 0 −1.00000 0 −0.858069 0 −2.33225 0 1.00000 0
1.6 0 −1.00000 0 0.413878 0 −2.72537 0 1.00000 0
1.7 0 −1.00000 0 0.588564 0 1.23518 0 1.00000 0
1.8 0 −1.00000 0 1.94484 0 3.19705 0 1.00000 0
1.9 0 −1.00000 0 2.54323 0 4.05015 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.ba 9
4.b odd 2 1 4008.2.a.i 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.i 9 4.b odd 2 1
8016.2.a.ba 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\(T_{5}^{9} + \cdots\)
\(T_{7}^{9} - \cdots\)
\(T_{11}^{9} + \cdots\)
\(T_{13}^{9} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \)
$3$ \( ( 1 + T )^{9} \)
$5$ \( 38 - 80 T - 132 T^{2} + 179 T^{3} + 200 T^{4} - 41 T^{5} - 67 T^{6} - 4 T^{7} + 6 T^{8} + T^{9} \)
$7$ \( -80 + 1064 T - 2148 T^{2} + 1265 T^{3} + 242 T^{4} - 446 T^{5} + 88 T^{6} + 27 T^{7} - 11 T^{8} + T^{9} \)
$11$ \( 400 + 780 T - 600 T^{2} - 1612 T^{3} - 71 T^{4} + 595 T^{5} - 18 T^{6} - 47 T^{7} + T^{8} + T^{9} \)
$13$ \( -5360 + 11568 T - 4048 T^{2} - 4140 T^{3} + 1539 T^{4} + 612 T^{5} - 152 T^{6} - 43 T^{7} + 4 T^{8} + T^{9} \)
$17$ \( 2620 - 34842 T - 24818 T^{2} + 8558 T^{3} + 7289 T^{4} - 261 T^{5} - 588 T^{6} - 48 T^{7} + 9 T^{8} + T^{9} \)
$19$ \( -8000 - 70880 T + 58744 T^{2} + 21582 T^{3} - 22327 T^{4} + 1588 T^{5} + 874 T^{6} - 97 T^{7} - 8 T^{8} + T^{9} \)
$23$ \( 239104 - 374656 T + 152416 T^{2} + 15802 T^{3} - 19639 T^{4} + 1637 T^{5} + 672 T^{6} - 85 T^{7} - 7 T^{8} + T^{9} \)
$29$ \( -539584 - 876200 T - 281972 T^{2} + 59814 T^{3} + 35793 T^{4} + 1099 T^{5} - 1134 T^{6} - 101 T^{7} + 9 T^{8} + T^{9} \)
$31$ \( 2058080 - 842400 T - 316612 T^{2} + 162999 T^{3} + 6491 T^{4} - 9673 T^{5} + 757 T^{6} + 152 T^{7} - 25 T^{8} + T^{9} \)
$37$ \( -40760 + 166692 T - 50946 T^{2} - 90943 T^{3} + 18748 T^{4} + 6469 T^{5} - 809 T^{6} - 168 T^{7} + 6 T^{8} + T^{9} \)
$41$ \( -380 + 4430 T - 15112 T^{2} + 20832 T^{3} - 13003 T^{4} + 3242 T^{5} + 39 T^{6} - 112 T^{7} + 4 T^{8} + T^{9} \)
$43$ \( 1370216 - 776126 T - 166844 T^{2} + 141994 T^{3} - 2081 T^{4} - 8112 T^{5} + 833 T^{6} + 128 T^{7} - 24 T^{8} + T^{9} \)
$47$ \( -10204 - 12176 T + 21504 T^{2} + 6161 T^{3} - 7859 T^{4} - 1149 T^{5} + 602 T^{6} + 20 T^{7} - 16 T^{8} + T^{9} \)
$53$ \( 270098 + 1164046 T + 906042 T^{2} + 59895 T^{3} - 134771 T^{4} - 42239 T^{5} - 3998 T^{6} + 50 T^{7} + 26 T^{8} + T^{9} \)
$59$ \( 10720 + 3056 T - 1033304 T^{2} - 393129 T^{3} + 9543 T^{4} + 15137 T^{5} + 346 T^{6} - 206 T^{7} - 4 T^{8} + T^{9} \)
$61$ \( 160 - 1948 T - 2180 T^{2} + 4410 T^{3} + 3955 T^{4} - 1344 T^{5} - 512 T^{6} + 53 T^{7} + 20 T^{8} + T^{9} \)
$67$ \( 1990 + 1531510 T + 2059494 T^{2} + 242891 T^{3} - 162115 T^{4} - 12230 T^{5} + 4191 T^{6} - 35 T^{7} - 25 T^{8} + T^{9} \)
$71$ \( -5162848 - 800848 T + 1484608 T^{2} + 59766 T^{3} - 124523 T^{4} + 4003 T^{5} + 3021 T^{6} - 190 T^{7} - 15 T^{8} + T^{9} \)
$73$ \( -187912544 + 96764040 T - 998868 T^{2} - 3806018 T^{3} + 174741 T^{4} + 56530 T^{5} - 2460 T^{6} - 381 T^{7} + 10 T^{8} + T^{9} \)
$79$ \( 237200 + 3806574 T - 4365108 T^{2} + 1183316 T^{3} + 16389 T^{4} - 38648 T^{5} + 3015 T^{6} + 230 T^{7} - 34 T^{8} + T^{9} \)
$83$ \( 5840 + 423280 T + 674784 T^{2} - 102709 T^{3} - 77439 T^{4} + 10149 T^{5} + 2144 T^{6} - 296 T^{7} - 4 T^{8} + T^{9} \)
$89$ \( 69802664 + 33177572 T - 878550 T^{2} - 2143285 T^{3} - 130681 T^{4} + 41540 T^{5} + 2787 T^{6} - 321 T^{7} - 13 T^{8} + T^{9} \)
$97$ \( -10035400 + 33779140 T - 9361610 T^{2} - 6101369 T^{3} + 292502 T^{4} + 101080 T^{5} - 2163 T^{6} - 567 T^{7} + 4 T^{8} + T^{9} \)
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