Properties

Label 8016.2.a.be
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
Analytic rank $0$
Dimension $11$
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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + 292 x^{2} - 5687 x + 242\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( 1 - \beta_{1} ) q^{5} -\beta_{3} q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( 1 - \beta_{1} ) q^{5} -\beta_{3} q^{7} + q^{9} + \beta_{10} q^{11} + ( 1 - \beta_{4} ) q^{13} + ( 1 - \beta_{1} ) q^{15} + ( 1 - \beta_{4} + \beta_{5} + \beta_{7} ) q^{17} + ( 1 - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{19} -\beta_{3} q^{21} + ( 1 - \beta_{6} - \beta_{8} + \beta_{9} ) q^{23} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{25} + q^{27} + ( 2 - \beta_{7} ) q^{29} + ( \beta_{2} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{31} + \beta_{10} q^{33} + ( -2 - 3 \beta_{3} + 2 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{35} + ( 1 - 2 \beta_{2} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{37} + ( 1 - \beta_{4} ) q^{39} + ( 2 + \beta_{3} - \beta_{5} + \beta_{9} ) q^{41} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{6} + \beta_{8} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( 1 - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{8} ) q^{47} + ( 2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{8} - \beta_{9} ) q^{49} + ( 1 - \beta_{4} + \beta_{5} + \beta_{7} ) q^{51} + ( 4 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{53} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{55} + ( 1 - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{57} + ( -\beta_{3} + \beta_{4} + \beta_{6} - \beta_{10} ) q^{59} + ( 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{10} ) q^{61} -\beta_{3} q^{63} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{65} + ( 1 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{67} + ( 1 - \beta_{6} - \beta_{8} + \beta_{9} ) q^{69} + ( 2 + \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{71} + ( 2 - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{73} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{75} + ( 4 + 2 \beta_{1} - \beta_{2} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{77} + ( 1 + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{79} + q^{81} + ( -1 - \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{7} - \beta_{8} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{85} + ( 2 - \beta_{7} ) q^{87} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{89} + ( 2 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{91} + ( \beta_{2} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{93} + ( 1 - \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{7} + 2 \beta_{9} - \beta_{10} ) q^{95} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{6} - \beta_{10} ) q^{97} + \beta_{10} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q + 11q^{3} + 10q^{5} + q^{7} + 11q^{9} + O(q^{10}) \) \( 11q + 11q^{3} + 10q^{5} + q^{7} + 11q^{9} + q^{11} + 10q^{13} + 10q^{15} + 17q^{17} - 2q^{19} + q^{21} + 3q^{23} + 21q^{25} + 11q^{27} + 17q^{29} + 15q^{31} + q^{33} - 11q^{35} + 4q^{37} + 10q^{39} + 16q^{41} - 10q^{43} + 10q^{45} + 16q^{47} + 22q^{49} + 17q^{51} + 42q^{53} + 5q^{55} - 2q^{57} + 2q^{59} + 12q^{61} + q^{63} + 10q^{65} + q^{67} + 3q^{69} + 9q^{71} + 24q^{73} + 21q^{75} + 22q^{77} + 30q^{79} + 11q^{81} - 16q^{83} + 25q^{85} + 17q^{87} + 37q^{89} - q^{91} + 15q^{93} - 5q^{95} + 4q^{97} + q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + 292 x^{2} - 5687 x + 242\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 6 \)
\(\beta_{3}\)\(=\)\((\)\(2623 \nu^{10} - 70944 \nu^{9} - 193391 \nu^{8} + 2378035 \nu^{7} + 2748114 \nu^{6} - 23458607 \nu^{5} - 15128561 \nu^{4} + 74577279 \nu^{3} + 34805559 \nu^{2} - 38238885 \nu - 7710406\)\()/10323192\)
\(\beta_{4}\)\(=\)\((\)\( -1379 \nu^{10} + 55664 \nu^{9} - 65533 \nu^{8} - 1226719 \nu^{7} + 1417390 \nu^{6} + 9091923 \nu^{5} - 7692731 \nu^{4} - 25269067 \nu^{3} + 11595133 \nu^{2} + 19499297 \nu - 2478234 \)\()/1876944\)
\(\beta_{5}\)\(=\)\((\)\(64985 \nu^{10} - 195192 \nu^{9} - 1989097 \nu^{8} + 5477813 \nu^{7} + 21384366 \nu^{6} - 52505737 \nu^{5} - 92700271 \nu^{4} + 195304833 \nu^{3} + 124569801 \nu^{2} - 220910235 \nu + 34175614\)\()/10323192\)
\(\beta_{6}\)\(=\)\((\)\(101765 \nu^{10} - 492056 \nu^{9} - 2608397 \nu^{8} + 12499993 \nu^{7} + 25094846 \nu^{6} - 110447061 \nu^{5} - 109201699 \nu^{4} + 387400045 \nu^{3} + 176633261 \nu^{2} - 427193567 \nu + 39160638\)\()/10323192\)
\(\beta_{7}\)\(=\)\((\)\(116141 \nu^{10} - 242652 \nu^{9} - 2909401 \nu^{8} + 5468837 \nu^{7} + 25357890 \nu^{6} - 43356601 \nu^{5} - 87108655 \nu^{4} + 142378425 \nu^{3} + 86939145 \nu^{2} - 163661007 \nu + 20257270\)\()/5161596\)
\(\beta_{8}\)\(=\)\((\)\(-522481 \nu^{10} + 1421984 \nu^{9} + 13457169 \nu^{8} - 32908293 \nu^{7} - 124595270 \nu^{6} + 263695489 \nu^{5} + 481600775 \nu^{4} - 850380841 \nu^{3} - 618583841 \nu^{2} + 928868747 \nu - 65756350\)\()/20646384\)
\(\beta_{9}\)\(=\)\((\)\(-596041 \nu^{10} + 2015712 \nu^{9} + 14695769 \nu^{8} - 46952653 \nu^{7} - 132016230 \nu^{6} + 379578137 \nu^{5} + 514603631 \nu^{4} - 1213924881 \nu^{3} - 743357145 \nu^{2} + 1176264339 \nu - 13787246\)\()/20646384\)
\(\beta_{10}\)\(=\)\((\)\(303385 \nu^{10} - 1032256 \nu^{9} - 7769641 \nu^{8} + 25611773 \nu^{7} + 71466478 \nu^{6} - 223268481 \nu^{5} - 278511527 \nu^{4} + 789881825 \nu^{3} + 385046857 \nu^{2} - 908813755 \nu + 47745918\)\()/10323192\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 6\)
\(\nu^{3}\)\(=\)\(\beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + \beta_{2} + 8 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-2 \beta_{10} + \beta_{9} + 2 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} + 11 \beta_{2} + 4 \beta_{1} + 48\)
\(\nu^{5}\)\(=\)\(-4 \beta_{10} + 14 \beta_{9} - 14 \beta_{8} + 3 \beta_{7} + 19 \beta_{6} - 14 \beta_{5} + 3 \beta_{3} + 17 \beta_{2} + 73 \beta_{1} + 50\)
\(\nu^{6}\)\(=\)\(-39 \beta_{10} + 17 \beta_{9} - 2 \beta_{8} + 35 \beta_{7} + 68 \beta_{6} + 20 \beta_{5} - 9 \beta_{4} - 16 \beta_{3} + 122 \beta_{2} + 74 \beta_{1} + 432\)
\(\nu^{7}\)\(=\)\(-89 \beta_{10} + 167 \beta_{9} - 151 \beta_{8} + 77 \beta_{7} + 294 \beta_{6} - 163 \beta_{5} + 4 \beta_{4} + 54 \beta_{3} + 241 \beta_{2} + 722 \beta_{1} + 656\)
\(\nu^{8}\)\(=\)\(-576 \beta_{10} + 244 \beta_{9} - 31 \beta_{8} + 516 \beta_{7} + 1101 \beta_{6} + 116 \beta_{5} - 55 \beta_{4} - 186 \beta_{3} + 1378 \beta_{2} + 1072 \beta_{1} + 4199\)
\(\nu^{9}\)\(=\)\(-1462 \beta_{10} + 1922 \beta_{9} - 1489 \beta_{8} + 1378 \beta_{7} + 4200 \beta_{6} - 1860 \beta_{5} + 147 \beta_{4} + 704 \beta_{3} + 3209 \beta_{2} + 7555 \beta_{1} + 8094\)
\(\nu^{10}\)\(=\)\(-7771 \beta_{10} + 3303 \beta_{9} - 341 \beta_{8} + 7202 \beta_{7} + 15782 \beta_{6} - 172 \beta_{5} - 44 \beta_{4} - 1868 \beta_{3} + 15861 \beta_{2} + 14338 \beta_{1} + 43210\)

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
3.54432
3.11942
2.80306
2.38119
1.09328
0.0427374
−1.60189
−2.17907
−2.34947
−2.83671
−3.01685
0 1.00000 0 −2.54432 0 −0.525270 0 1.00000 0
1.2 0 1.00000 0 −2.11942 0 0.802640 0 1.00000 0
1.3 0 1.00000 0 −1.80306 0 4.10861 0 1.00000 0
1.4 0 1.00000 0 −1.38119 0 0.260099 0 1.00000 0
1.5 0 1.00000 0 −0.0932775 0 −3.86231 0 1.00000 0
1.6 0 1.00000 0 0.957263 0 0.898491 0 1.00000 0
1.7 0 1.00000 0 2.60189 0 3.58131 0 1.00000 0
1.8 0 1.00000 0 3.17907 0 −0.651548 0 1.00000 0
1.9 0 1.00000 0 3.34947 0 −3.54581 0 1.00000 0
1.10 0 1.00000 0 3.83671 0 −4.48179 0 1.00000 0
1.11 0 1.00000 0 4.01685 0 4.41557 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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.be 11
4.b odd 2 1 4008.2.a.k 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.k 11 4.b odd 2 1
8016.2.a.be 11 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} \)
$3$ \( ( -1 + T )^{11} \)
$5$ \( 512 + 5280 T - 2906 T^{2} - 6948 T^{3} + 2246 T^{4} + 2665 T^{5} - 864 T^{6} - 385 T^{7} + 155 T^{8} + 12 T^{9} - 10 T^{10} + T^{11} \)
$7$ \( 256 - 704 T - 2096 T^{2} + 3304 T^{3} + 3572 T^{4} - 4561 T^{5} - 692 T^{6} + 800 T^{7} + 46 T^{8} - 49 T^{9} - T^{10} + T^{11} \)
$11$ \( 86400 - 69984 T - 101976 T^{2} + 76492 T^{3} + 26456 T^{4} - 19694 T^{5} - 2359 T^{6} + 1865 T^{7} + 84 T^{8} - 73 T^{9} - T^{10} + T^{11} \)
$13$ \( 10688 + 16160 T - 19504 T^{2} - 22496 T^{3} + 16124 T^{4} + 8438 T^{5} - 5699 T^{6} - 408 T^{7} + 484 T^{8} - 27 T^{9} - 10 T^{10} + T^{11} \)
$17$ \( 20128 + 60120 T + 18268 T^{2} - 63798 T^{3} - 17386 T^{4} + 28492 T^{5} - 1937 T^{6} - 2693 T^{7} + 464 T^{8} + 54 T^{9} - 17 T^{10} + T^{11} \)
$19$ \( -858112 - 1081344 T + 666112 T^{2} + 371936 T^{3} - 130272 T^{4} - 52404 T^{5} + 9917 T^{6} + 3544 T^{7} - 284 T^{8} - 105 T^{9} + 2 T^{10} + T^{11} \)
$23$ \( -32768 + 135168 T - 138240 T^{2} + 7296 T^{3} + 49744 T^{4} - 15084 T^{5} - 5145 T^{6} + 2149 T^{7} + 178 T^{8} - 91 T^{9} - 3 T^{10} + T^{11} \)
$29$ \( -30400 - 1792 T + 105520 T^{2} - 13328 T^{3} - 70788 T^{4} + 33952 T^{5} + 697 T^{6} - 3159 T^{7} + 468 T^{8} + 55 T^{9} - 17 T^{10} + T^{11} \)
$31$ \( -1515520 - 4776448 T - 2200544 T^{2} + 932288 T^{3} + 558652 T^{4} - 58023 T^{5} - 43497 T^{6} + 2065 T^{7} + 1361 T^{8} - 64 T^{9} - 15 T^{10} + T^{11} \)
$37$ \( -62176 + 975008 T - 2658584 T^{2} + 1259260 T^{3} + 870936 T^{4} - 242493 T^{5} - 59172 T^{6} + 14349 T^{7} + 873 T^{8} - 224 T^{9} - 4 T^{10} + T^{11} \)
$41$ \( -337072 + 1246992 T - 714448 T^{2} - 738062 T^{3} + 324172 T^{4} + 101444 T^{5} - 40515 T^{6} - 3132 T^{7} + 1707 T^{8} - 52 T^{9} - 16 T^{10} + T^{11} \)
$43$ \( 48168832 + 79900744 T + 23528560 T^{2} - 7560314 T^{3} - 3635448 T^{4} + 45984 T^{5} + 154353 T^{6} + 9406 T^{7} - 2225 T^{8} - 194 T^{9} + 10 T^{10} + T^{11} \)
$47$ \( 151369024 - 24363936 T - 81596940 T^{2} - 4236232 T^{3} + 7626528 T^{4} + 176155 T^{5} - 283869 T^{6} + 5405 T^{7} + 4190 T^{8} - 210 T^{9} - 16 T^{10} + T^{11} \)
$53$ \( 64144928 - 42283208 T - 21033750 T^{2} + 17705982 T^{3} - 1060476 T^{4} - 1645021 T^{5} + 486583 T^{6} - 46071 T^{7} - 1360 T^{8} + 590 T^{9} - 42 T^{10} + T^{11} \)
$59$ \( 1956736 - 515200 T - 1808992 T^{2} + 405316 T^{3} + 352622 T^{4} - 78295 T^{5} - 23899 T^{6} + 5683 T^{7} + 514 T^{8} - 144 T^{9} - 2 T^{10} + T^{11} \)
$61$ \( 61760 - 6868688 T - 77126592 T^{2} + 36003900 T^{3} + 12382736 T^{4} - 1657414 T^{5} - 374529 T^{6} + 34552 T^{7} + 3936 T^{8} - 331 T^{9} - 12 T^{10} + T^{11} \)
$67$ \( -248382664 - 33896048 T + 88538278 T^{2} + 23729506 T^{3} - 4080316 T^{4} - 1446279 T^{5} + 59487 T^{6} + 32098 T^{7} - 195 T^{8} - 301 T^{9} - T^{10} + T^{11} \)
$71$ \( 382383104 - 782295296 T + 583838656 T^{2} - 197858096 T^{3} + 26457488 T^{4} + 1209392 T^{5} - 657717 T^{6} + 34413 T^{7} + 4371 T^{8} - 374 T^{9} - 9 T^{10} + T^{11} \)
$73$ \( -360896 + 2073280 T + 715056 T^{2} - 2079408 T^{3} + 65220 T^{4} + 422504 T^{5} - 56053 T^{6} - 15834 T^{7} + 2848 T^{8} + 29 T^{9} - 24 T^{10} + T^{11} \)
$79$ \( -6033532768 + 459622920 T + 2537522356 T^{2} - 878255166 T^{3} + 38714512 T^{4} + 17101188 T^{5} - 1853549 T^{6} - 60140 T^{7} + 14081 T^{8} - 224 T^{9} - 30 T^{10} + T^{11} \)
$83$ \( 93972800 + 75574752 T - 147011360 T^{2} + 48506548 T^{3} + 3605934 T^{4} - 3659211 T^{5} + 287289 T^{6} + 52377 T^{7} - 4390 T^{8} - 354 T^{9} + 16 T^{10} + T^{11} \)
$89$ \( 158510752 - 278944016 T + 41206496 T^{2} + 103099624 T^{3} - 51125558 T^{4} + 7852663 T^{5} + 4509 T^{6} - 108598 T^{7} + 7755 T^{8} + 197 T^{9} - 37 T^{10} + T^{11} \)
$97$ \( -20250080 + 313478704 T - 443880192 T^{2} + 64500792 T^{3} + 36202454 T^{4} - 6069457 T^{5} - 542422 T^{6} + 99412 T^{7} + 2673 T^{8} - 555 T^{9} - 4 T^{10} + T^{11} \)
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