Properties

Label 16.11.c.b
Level 16
Weight 11
Character orbit 16.c
Analytic conductor 10.166
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 16.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.1657160428\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{505})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{25}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( 1050 + \beta_{2} ) q^{5} + ( -12 \beta_{1} - \beta_{3} ) q^{7} + ( -47319 - 14 \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( 1050 + \beta_{2} ) q^{5} + ( -12 \beta_{1} - \beta_{3} ) q^{7} + ( -47319 - 14 \beta_{2} ) q^{9} + ( 93 \beta_{1} - 30 \beta_{3} ) q^{11} + ( -260246 + 69 \beta_{2} ) q^{13} + ( -3420 \beta_{1} - 171 \beta_{3} ) q^{15} + ( -1097502 - 70 \beta_{2} ) q^{17} + ( 8343 \beta_{1} - 266 \beta_{3} ) q^{19} + ( -1326336 - 596 \beta_{2} ) q^{21} + ( -16956 \beta_{1} + 513 \beta_{3} ) q^{23} + ( 9953195 + 2100 \beta_{2} ) q^{25} + ( 21450 \beta_{1} + 2394 \beta_{3} ) q^{27} + ( 7784202 - 7 \beta_{2} ) q^{29} + ( 70872 \beta_{1} + 3192 \beta_{3} ) q^{31} + ( 8394624 - 11538 \beta_{2} ) q^{33} + ( -117060 \beta_{1} - 732 \beta_{3} ) q^{35} + ( -41541670 + 15309 \beta_{2} ) q^{37} + ( 96716 \beta_{1} - 11799 \beta_{3} ) q^{39} + ( 43193682 + 23508 \beta_{2} ) q^{41} + ( -462111 \beta_{1} - 20868 \beta_{3} ) q^{43} + ( -310313430 - 62019 \beta_{2} ) q^{45} + ( 135408 \beta_{1} - 10794 \beta_{3} ) q^{47} + ( 220056817 - 6648 \beta_{2} ) q^{49} + ( 1263402 \beta_{1} + 11970 \beta_{3} ) q^{51} + ( -88972038 + 88053 \beta_{2} ) q^{53} + ( -1962540 \beta_{1} + 55503 \beta_{3} ) q^{55} + ( 874149504 + 2954 \beta_{2} ) q^{57} + ( 574557 \beta_{1} + 111720 \beta_{3} ) q^{59} + ( -487287094 - 11019 \beta_{2} ) q^{61} + ( 2030268 \beta_{1} + 42867 \beta_{3} ) q^{63} + ( 1011267780 - 187796 \beta_{2} ) q^{65} + ( 2782131 \beta_{1} - 65018 \beta_{3} ) q^{67} + ( -1777966848 - 17820 \beta_{2} ) q^{69} + ( -2659956 \beta_{1} - 69141 \beta_{3} ) q^{71} + ( 376254802 + 293466 \beta_{2} ) q^{73} + ( -14930195 \beta_{1} - 359100 \beta_{3} ) q^{75} + ( -1271722752 + 70548 \beta_{2} ) q^{77} + ( 11416680 \beta_{1} - 419330 \beta_{3} ) q^{79} + ( -393037551 + 498246 \beta_{2} ) q^{81} + ( 7551267 \beta_{1} + 217056 \beta_{3} ) q^{83} + ( -2455519500 - 1171002 \beta_{2} ) q^{85} + ( -7767612 \beta_{1} + 1197 \beta_{3} ) q^{87} + ( 1510358514 - 1420886 \beta_{2} ) q^{89} + ( -4084788 \beta_{1} + 282188 \beta_{3} ) q^{91} + ( 7697857536 + 2358384 \beta_{2} ) q^{93} + ( 8311740 \beta_{1} + 1777773 \beta_{3} ) q^{95} + ( -2073061886 + 1011090 \beta_{2} ) q^{97} + ( 24441993 \beta_{1} + 201528 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4200q^{5} - 189276q^{9} + O(q^{10}) \) \( 4q + 4200q^{5} - 189276q^{9} - 1040984q^{13} - 4390008q^{17} - 5305344q^{21} + 39812780q^{25} + 31136808q^{29} + 33578496q^{33} - 166166680q^{37} + 172774728q^{41} - 1241253720q^{45} + 880227268q^{49} - 355888152q^{53} + 3496598016q^{57} - 1949148376q^{61} + 4045071120q^{65} - 7111867392q^{69} + 1505019208q^{73} - 5086891008q^{77} - 1572150204q^{81} - 9822078000q^{85} + 6041434056q^{89} + 30791430144q^{93} - 8292247544q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 127 x^{2} + 126 x + 15876\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 1072 \nu^{3} - 8128 \nu^{2} + 264160 \nu - 376992 \)\()/8001\)
\(\beta_{2}\)\(=\)\((\)\( 384 \nu^{3} + 72768 \)\()/127\)
\(\beta_{3}\)\(=\)\((\)\( -2336 \nu^{3} - 471424 \nu^{2} - 1064768 \nu - 29994048 \)\()/8001\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-3 \beta_{3} - 8 \beta_{2} + 174 \beta_{1} + 1536\)\()/6144\)
\(\nu^{2}\)\(=\)\((\)\(-195 \beta_{3} + 16 \beta_{2} - 786 \beta_{1} - 777216\)\()/12288\)
\(\nu^{3}\)\(=\)\((\)\(127 \beta_{2} - 72768\)\()/384\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
−5.36805 + 9.29774i
5.86805 + 10.1638i
5.86805 10.1638i
−5.36805 9.29774i
0 408.379i 0 5364.66 0 9544.75i 0 −107724. 0
15.2 0 214.389i 0 −3264.66 0 5808.15i 0 13086.3 0
15.3 0 214.389i 0 −3264.66 0 5808.15i 0 13086.3 0
15.4 0 408.379i 0 5364.66 0 9544.75i 0 −107724. 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.11.c.b 4
3.b odd 2 1 144.11.g.d 4
4.b odd 2 1 inner 16.11.c.b 4
8.b even 2 1 64.11.c.b 4
8.d odd 2 1 64.11.c.b 4
12.b even 2 1 144.11.g.d 4
16.e even 4 2 256.11.d.g 8
16.f odd 4 2 256.11.d.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.11.c.b 4 1.a even 1 1 trivial
16.11.c.b 4 4.b odd 2 1 inner
64.11.c.b 4 8.b even 2 1
64.11.c.b 4 8.d odd 2 1
144.11.g.d 4 3.b odd 2 1
144.11.g.d 4 12.b even 2 1
256.11.d.g 8 16.e even 4 2
256.11.d.g 8 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 212736 T_{3}^{2} + 7665352704 \) acting on \(S_{11}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 23460 T^{2} + 3462362982 T^{4} - 81799962047460 T^{6} + 12157665459056928801 T^{8} \)
$5$ \( ( 1 - 2100 T + 2017430 T^{2} - 20507812500 T^{3} + 95367431640625 T^{4} )^{2} \)
$7$ \( 1 - 1005064132 T^{2} + 411300244861899078 T^{4} - \)\(80\!\cdots\!32\)\( T^{6} + \)\(63\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 - 19840803940 T^{2} + \)\(44\!\cdots\!82\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{6} + \)\(45\!\cdots\!01\)\( T^{8} \)
$13$ \( ( 1 + 520492 T + 254812664694 T^{2} + 71754242139469708 T^{3} + \)\(19\!\cdots\!01\)\( T^{4} )^{2} \)
$17$ \( ( 1 + 2195004 T + 5145278472902 T^{2} + 4425114675461156796 T^{3} + \)\(40\!\cdots\!01\)\( T^{4} )^{2} \)
$19$ \( 1 - 3663898849060 T^{2} + \)\(45\!\cdots\!22\)\( T^{4} - \)\(13\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!01\)\( T^{8} \)
$23$ \( 1 - 82119401461060 T^{2} + \)\(46\!\cdots\!82\)\( T^{4} - \)\(14\!\cdots\!60\)\( T^{6} + \)\(29\!\cdots\!01\)\( T^{8} \)
$29$ \( ( 1 - 15568404 T + 902007355177526 T^{2} - \)\(65\!\cdots\!04\)\( T^{3} + \)\(17\!\cdots\!01\)\( T^{4} )^{2} \)
$31$ \( 1 - 1229395698322180 T^{2} + \)\(92\!\cdots\!02\)\( T^{4} - \)\(82\!\cdots\!80\)\( T^{6} + \)\(45\!\cdots\!01\)\( T^{8} \)
$37$ \( ( 1 + 83083340 T + 6979856299974678 T^{2} + \)\(39\!\cdots\!60\)\( T^{3} + \)\(23\!\cdots\!01\)\( T^{4} )^{2} \)
$41$ \( ( 1 - 86387364 T + 18423149137257446 T^{2} - \)\(11\!\cdots\!64\)\( T^{3} + \)\(18\!\cdots\!01\)\( T^{4} )^{2} \)
$43$ \( 1 + 888082514945180 T^{2} - \)\(50\!\cdots\!78\)\( T^{4} + \)\(41\!\cdots\!80\)\( T^{6} + \)\(21\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - 196091355038080132 T^{2} + \)\(15\!\cdots\!38\)\( T^{4} - \)\(54\!\cdots\!32\)\( T^{6} + \)\(76\!\cdots\!01\)\( T^{8} \)
$53$ \( ( 1 + 177944076 T + 213352476870696662 T^{2} + \)\(31\!\cdots\!24\)\( T^{3} + \)\(30\!\cdots\!01\)\( T^{4} )^{2} \)
$59$ \( 1 - 815549572994663140 T^{2} + \)\(68\!\cdots\!82\)\( T^{4} - \)\(21\!\cdots\!40\)\( T^{6} + \)\(68\!\cdots\!01\)\( T^{8} \)
$61$ \( ( 1 + 974574188 T + 1661874172242478518 T^{2} + \)\(69\!\cdots\!88\)\( T^{3} + \)\(50\!\cdots\!01\)\( T^{4} )^{2} \)
$67$ \( 1 - 5292742361877207460 T^{2} + \)\(13\!\cdots\!82\)\( T^{4} - \)\(17\!\cdots\!60\)\( T^{6} + \)\(11\!\cdots\!01\)\( T^{8} \)
$71$ \( 1 - 11040185613269882308 T^{2} + \)\(50\!\cdots\!18\)\( T^{4} - \)\(11\!\cdots\!08\)\( T^{6} + \)\(11\!\cdots\!01\)\( T^{8} \)
$73$ \( ( 1 - 752509604 T + 7133539166909268582 T^{2} - \)\(32\!\cdots\!96\)\( T^{3} + \)\(18\!\cdots\!01\)\( T^{4} )^{2} \)
$79$ \( 1 + 5042168236353513596 T^{2} - \)\(12\!\cdots\!94\)\( T^{4} + \)\(45\!\cdots\!96\)\( T^{6} + \)\(80\!\cdots\!01\)\( T^{8} \)
$83$ \( 1 - 45281007302789483812 T^{2} + \)\(92\!\cdots\!58\)\( T^{4} - \)\(10\!\cdots\!12\)\( T^{6} + \)\(57\!\cdots\!01\)\( T^{8} \)
$89$ \( ( 1 - 3020717028 T + 27059817309971120678 T^{2} - \)\(94\!\cdots\!28\)\( T^{3} + \)\(97\!\cdots\!01\)\( T^{4} )^{2} \)
$97$ \( ( 1 + 4146123772 T + \)\(13\!\cdots\!94\)\( T^{2} + \)\(30\!\cdots\!28\)\( T^{3} + \)\(54\!\cdots\!01\)\( T^{4} )^{2} \)
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