Properties

Label 16.13.c.b
Level 16
Weight 13
Character orbit 16.c
Analytic conductor 14.624
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 16.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.6239010764\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{2521})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{24}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -3726 - \beta_{3} ) q^{5} + ( -162 \beta_{1} + 5 \beta_{2} ) q^{7} + ( 28209 - 10 \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -3726 - \beta_{3} ) q^{5} + ( -162 \beta_{1} + 5 \beta_{2} ) q^{7} + ( 28209 - 10 \beta_{3} ) q^{9} + ( 1815 \beta_{1} + 462 \beta_{2} ) q^{11} + ( 3138130 + 243 \beta_{3} ) q^{13} + ( -7566 \beta_{1} + 7263 \beta_{2} ) q^{15} + ( 14357250 - 1346 \beta_{3} ) q^{17} + ( 16605 \beta_{1} + 41098 \beta_{2} ) q^{19} + ( 82752384 + 1940 \beta_{3} ) q^{21} + ( -79254 \beta_{1} + 104715 \beta_{2} ) q^{23} + ( 141479027 + 7452 \beta_{3} ) q^{25} + ( 521250 \beta_{1} + 72630 \beta_{2} ) q^{27} + ( -149362542 - 29105 \beta_{3} ) q^{29} + ( -1636200 \beta_{1} - 326232 \beta_{2} ) q^{31} + ( -799824960 + 11418 \beta_{3} ) q^{33} + ( 1471452 \beta_{1} - 1176036 \beta_{2} ) q^{35} + ( -948166030 + 80595 \beta_{3} ) q^{37} + ( 4071250 \beta_{1} - 1764909 \beta_{2} ) q^{39} + ( -2870147358 - 84420 \beta_{3} ) q^{41} + ( -10650609 \beta_{1} - 562140 \beta_{2} ) q^{43} + ( 3612259026 + 9051 \beta_{3} ) q^{45} + ( 3260868 \beta_{1} + 3530370 \beta_{2} ) q^{47} + ( 157692193 - 366120 \beta_{3} ) q^{49} + ( 9188610 \beta_{1} + 9775998 \beta_{2} ) q^{51} + ( 23639445810 + 224859 \beta_{3} ) q^{53} + ( 8975934 \beta_{1} + 13235013 \beta_{2} ) q^{55} + ( 1744077120 + 2464222 \beta_{3} ) q^{57} + ( -35415285 \beta_{1} + 6170712 \beta_{2} ) q^{59} + ( 21860355986 - 3237165 \beta_{3} ) q^{61} + ( 4108542 \beta_{1} - 11433015 \beta_{2} ) q^{63} + ( -102024660348 - 4043548 \beta_{3} ) q^{65} + ( 72148401 \beta_{1} - 39156710 \beta_{2} ) q^{67} + ( 65617907328 + 7494300 \beta_{3} ) q^{69} + ( -162491250 \beta_{1} - 69550359 \beta_{2} ) q^{71} + ( -189854974430 + 1789614 \beta_{3} ) q^{73} + ( 170094707 \beta_{1} - 54123876 \beta_{2} ) q^{75} + ( 124535283840 - 1268916 \beta_{3} ) q^{77} + ( 59348700 \beta_{1} - 2006150 \beta_{2} ) q^{79} + ( -229468712031 - 5878590 \beta_{3} ) q^{81} + ( -60014067 \beta_{1} + 51125280 \beta_{2} ) q^{83} + ( 446862317796 - 9342054 \beta_{3} ) q^{85} + ( -261125742 \beta_{1} + 211389615 \beta_{2} ) q^{87} + ( -170459453214 + 12752510 \beta_{3} ) q^{89} + ( -719262180 \beta_{1} + 296940308 \beta_{2} ) q^{91} + ( 743213422080 - 4516848 \beta_{3} ) q^{93} + ( 1894415466 \beta_{1} + 125287287 \beta_{2} ) q^{95} + ( -101950322750 + 32213862 \beta_{3} ) q^{97} + ( 208585575 \beta_{1} + 162596808 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 14904q^{5} + 112836q^{9} + O(q^{10}) \) \( 4q - 14904q^{5} + 112836q^{9} + 12552520q^{13} + 57429000q^{17} + 331009536q^{21} + 565916108q^{25} - 597450168q^{29} - 3199299840q^{33} - 3792664120q^{37} - 11480589432q^{41} + 14449036104q^{45} + 630768772q^{49} + 94557783240q^{53} + 6976308480q^{57} + 87441423944q^{61} - 408098641392q^{65} + 262471629312q^{69} - 759419897720q^{73} + 498141135360q^{77} - 917874848124q^{81} + 1787449271184q^{85} - 681837812856q^{89} + 2972853688320q^{93} - 407801291000q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 631 x^{2} + 630 x + 396900\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 5128 \nu^{3} - 55528 \nu^{2} + 6416008 \nu - 14260680 \)\()/198765\)
\(\beta_{2}\)\(=\)\((\)\( -1024 \nu^{3} + 646144 \nu^{2} - 646144 \nu + 202890240 \)\()/198765\)
\(\beta_{3}\)\(=\)\((\)\( 768 \nu^{3} + 726144 \)\()/631\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-8 \beta_{3} + 33 \beta_{2} + 384 \beta_{1} + 3072\)\()/12288\)
\(\nu^{2}\)\(=\)\((\)\(8 \beta_{3} + 3813 \beta_{2} + 384 \beta_{1} - 3873792\)\()/12288\)
\(\nu^{3}\)\(=\)\((\)\(631 \beta_{3} - 726144\)\()/768\)

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
−12.3024 21.3084i
12.8024 22.1744i
12.8024 + 22.1744i
−12.3024 + 21.3084i
0 834.288i 0 −23006.5 0 144023.i 0 −164596. 0
15.2 0 557.160i 0 15554.5 0 81391.8i 0 221014. 0
15.3 0 557.160i 0 15554.5 0 81391.8i 0 221014. 0
15.4 0 834.288i 0 −23006.5 0 144023.i 0 −164596. 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 16.13.c.b 4
3.b odd 2 1 144.13.g.g 4
4.b odd 2 1 inner 16.13.c.b 4
8.b even 2 1 64.13.c.e 4
8.d odd 2 1 64.13.c.e 4
12.b even 2 1 144.13.g.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.13.c.b 4 1.a even 1 1 trivial
16.13.c.b 4 4.b odd 2 1 inner
64.13.c.e 4 8.b even 2 1
64.13.c.e 4 8.d odd 2 1
144.13.g.g 4 3.b odd 2 1
144.13.g.g 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 1006464 T_{3}^{2} + 216068788224 \) acting on \(S_{13}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 1119300 T^{2} + 840893537862 T^{4} - 316123380183183300 T^{6} + \)\(79\!\cdots\!61\)\( T^{8} \)
$5$ \( ( 1 + 7452 T + 130427750 T^{2} + 1819335937500 T^{3} + 59604644775390625 T^{4} )^{2} \)
$7$ \( 1 - 27997958788 T^{2} + \)\(52\!\cdots\!38\)\( T^{4} - \)\(53\!\cdots\!88\)\( T^{6} + \)\(36\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 - 8719629633220 T^{2} + \)\(36\!\cdots\!82\)\( T^{4} - \)\(85\!\cdots\!20\)\( T^{6} + \)\(97\!\cdots\!81\)\( T^{8} \)
$13$ \( ( 1 - 6276260 T + 34493357065638 T^{2} - \)\(14\!\cdots\!60\)\( T^{3} + \)\(54\!\cdots\!61\)\( T^{4} )^{2} \)
$17$ \( ( 1 - 28714500 T + 697893999497606 T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!21\)\( T^{4} )^{2} \)
$19$ \( 1 + 1379946502958780 T^{2} + \)\(76\!\cdots\!42\)\( T^{4} + \)\(67\!\cdots\!80\)\( T^{6} + \)\(23\!\cdots\!41\)\( T^{8} \)
$23$ \( 1 - 4191099606953860 T^{2} + \)\(49\!\cdots\!82\)\( T^{4} - \)\(20\!\cdots\!60\)\( T^{6} + \)\(23\!\cdots\!81\)\( T^{8} \)
$29$ \( ( 1 + 298725084 T + 415040300804049446 T^{2} + \)\(10\!\cdots\!44\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} )^{2} \)
$31$ \( 1 - 311340683481463300 T^{2} + \)\(62\!\cdots\!42\)\( T^{4} - \)\(19\!\cdots\!00\)\( T^{6} + \)\(38\!\cdots\!41\)\( T^{8} \)
$37$ \( ( 1 + 1896332060 T + 11650287819649513062 T^{2} + \)\(12\!\cdots\!60\)\( T^{3} + \)\(43\!\cdots\!61\)\( T^{4} )^{2} \)
$41$ \( ( 1 + 5740294716 T + 50715457869976185926 T^{2} + \)\(12\!\cdots\!96\)\( T^{3} + \)\(50\!\cdots\!61\)\( T^{4} )^{2} \)
$43$ \( 1 - 49567287012768743620 T^{2} + \)\(37\!\cdots\!02\)\( T^{4} - \)\(79\!\cdots\!20\)\( T^{6} + \)\(25\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - \)\(38\!\cdots\!28\)\( T^{2} + \)\(63\!\cdots\!58\)\( T^{4} - \)\(52\!\cdots\!68\)\( T^{6} + \)\(18\!\cdots\!61\)\( T^{8} \)
$53$ \( ( 1 - 47278891620 T + \)\(15\!\cdots\!26\)\( T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + \)\(24\!\cdots\!81\)\( T^{4} )^{2} \)
$59$ \( 1 - \)\(54\!\cdots\!60\)\( T^{2} + \)\(13\!\cdots\!22\)\( T^{4} - \)\(17\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!21\)\( T^{8} \)
$61$ \( ( 1 - 43720711972 T + \)\(18\!\cdots\!38\)\( T^{2} - \)\(11\!\cdots\!12\)\( T^{3} + \)\(70\!\cdots\!41\)\( T^{4} )^{2} \)
$67$ \( 1 - \)\(15\!\cdots\!80\)\( T^{2} + \)\(12\!\cdots\!42\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{6} + \)\(44\!\cdots\!41\)\( T^{8} \)
$71$ \( 1 - \)\(19\!\cdots\!28\)\( T^{2} + \)\(11\!\cdots\!58\)\( T^{4} - \)\(53\!\cdots\!68\)\( T^{6} + \)\(72\!\cdots\!61\)\( T^{8} \)
$73$ \( ( 1 + 379709948860 T + \)\(80\!\cdots\!46\)\( T^{2} + \)\(86\!\cdots\!60\)\( T^{3} + \)\(52\!\cdots\!41\)\( T^{4} )^{2} \)
$79$ \( 1 - \)\(23\!\cdots\!64\)\( T^{2} + \)\(20\!\cdots\!86\)\( T^{4} - \)\(81\!\cdots\!84\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{8} \)
$83$ \( 1 - \)\(40\!\cdots\!48\)\( T^{2} + \)\(63\!\cdots\!18\)\( T^{4} - \)\(46\!\cdots\!08\)\( T^{6} + \)\(13\!\cdots\!41\)\( T^{8} \)
$89$ \( ( 1 + 340918906428 T + \)\(46\!\cdots\!38\)\( T^{2} + \)\(84\!\cdots\!88\)\( T^{3} + \)\(61\!\cdots\!41\)\( T^{4} )^{2} \)
$97$ \( ( 1 + 203900645500 T + \)\(10\!\cdots\!38\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(48\!\cdots\!81\)\( T^{4} )^{2} \)
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