Properties

Label 16.15.c.c
Level 16
Weight 15
Character orbit 16.c
Analytic conductor 19.893
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 16.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.8926349043\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -21750 + \beta_{3} ) q^{5} + ( 245 \beta_{1} - \beta_{2} ) q^{7} + ( -5000199 + 226 \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -21750 + \beta_{3} ) q^{5} + ( 245 \beta_{1} - \beta_{2} ) q^{7} + ( -5000199 + 226 \beta_{3} ) q^{9} + ( 3057 \beta_{1} - 270 \beta_{2} ) q^{11} + ( 16716154 - 1131 \beta_{3} ) q^{13} + ( -52977 \beta_{1} - 9531 \beta_{2} ) q^{15} + ( 3279138 - 7030 \beta_{3} ) q^{17} + ( -80045 \beta_{1} - 112826 \beta_{2} ) q^{19} + ( -2395550976 + 55084 \beta_{3} ) q^{21} + ( 454707 \beta_{1} - 651807 \beta_{2} ) q^{23} + ( -4222801285 - 43500 \beta_{3} ) q^{25} + ( -7274532 \beta_{1} - 2154006 \beta_{2} ) q^{27} + ( -4375446438 - 407767 \beta_{3} ) q^{29} + ( 4798512 \beta_{1} - 4065288 \beta_{2} ) q^{31} + ( -29549344896 + 613662 \beta_{3} ) q^{33} + ( -12933984 \beta_{1} - 2344572 \beta_{2} ) q^{35} + ( -34996433590 + 1942029 \beta_{3} ) q^{37} + ( 52033891 \beta_{1} + 10779561 \beta_{2} ) q^{39} + ( 165522102642 - 2823372 \beta_{3} ) q^{41} + ( -62935245 \beta_{1} + 39873372 \beta_{2} ) q^{43} + ( 426883644090 - 9915699 \beta_{3} ) q^{45} + ( 107341674 \beta_{1} + 72228726 \beta_{2} ) q^{47} + ( 91599855697 + 13424712 \beta_{3} ) q^{49} + ( 222804948 \beta_{1} + 67002930 \beta_{2} ) q^{51} + ( -743793406038 + 33160533 \beta_{3} ) q^{53} + ( -149697819 \beta_{1} - 31695057 \beta_{2} ) q^{55} + ( 932608892544 - 50358406 \beta_{3} ) q^{57} + ( 339074283 \beta_{1} - 249381720 \beta_{2} ) q^{59} + ( -1656242712934 - 32995419 \beta_{3} ) q^{61} + ( -2943831639 \beta_{1} - 529788573 \beta_{2} ) q^{63} + ( -1955630580540 + 41315404 \beta_{3} ) q^{65} + ( 1229905543 \beta_{1} - 679535978 \beta_{2} ) q^{67} + ( -3584710764288 - 83653020 \beta_{3} ) q^{69} + ( 2286480849 \beta_{1} - 411388821 \beta_{2} ) q^{71} + ( -3008283147278 + 355525386 \beta_{3} ) q^{73} + ( -2864426785 \beta_{1} + 414598500 \beta_{2} ) q^{75} + ( -7245770902272 + 149257428 \beta_{3} ) q^{77} + ( 3951839050 \beta_{1} + 1805219230 \beta_{2} ) q^{79} + ( 50106626153649 - 1179138954 \beta_{3} ) q^{81} + ( -5754348195 \beta_{1} + 3410578656 \beta_{2} ) q^{83} + ( -9967113686700 + 156181638 \beta_{3} ) q^{85} + ( 8357893671 \beta_{1} + 3886427277 \beta_{2} ) q^{87} + ( 26576753624274 + 1035147034 \beta_{3} ) q^{89} + ( 12696977384 \beta_{1} + 2659594028 \beta_{2} ) q^{91} + ( -41557394433024 - 78208656 \beta_{3} ) q^{93} + ( 9360700671 \beta_{1} - 306343107 \beta_{2} ) q^{95} + ( 23948291484034 + 1255201890 \beta_{3} ) q^{97} + ( -34090631937 \beta_{1} - 7140214152 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 87000q^{5} - 20000796q^{9} + O(q^{10}) \) \( 4q - 87000q^{5} - 20000796q^{9} + 66864616q^{13} + 13116552q^{17} - 9582203904q^{21} - 16891205140q^{25} - 17501785752q^{29} - 118197379584q^{33} - 139985734360q^{37} + 662088410568q^{41} + 1707534576360q^{45} + 366399422788q^{49} - 2975173624152q^{53} + 3730435570176q^{57} - 6624970851736q^{61} - 7822522322160q^{65} - 14338843057152q^{69} - 12033132589112q^{73} - 28983083609088q^{77} + 200426504614596q^{81} - 39868454746800q^{85} + 106307014497096q^{89} - 166229577732096q^{93} + 95793165936136q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 9547 x^{2} + 9546 x + 91126116\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -25760 \nu^{3} + 2902288 \nu^{2} - 488959152 \nu + 13606715664 \)\()/15189277\)
\(\beta_{2}\)\(=\)\((\)\( 73184 \nu^{3} + 30397648 \nu^{2} + 1427772944 \nu + 145786588368 \)\()/45567831\)
\(\beta_{3}\)\(=\)\((\)\( -384 \nu^{3} - 5498688 \)\()/9547\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(32 \beta_{3} + 171 \beta_{2} - 597 \beta_{1} + 6144\)\()/24576\)
\(\nu^{2}\)\(=\)\((\)\(-32 \beta_{3} + 28809 \beta_{2} + 28041 \beta_{1} - 117307392\)\()/24576\)
\(\nu^{3}\)\(=\)\((\)\(-9547 \beta_{3} - 5498688\)\()/384\)

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
−48.6025 + 84.1819i
49.1025 + 85.0479i
49.1025 85.0479i
−48.6025 84.1819i
0 4273.45i 0 −59268.7 0 1.04417e6i 0 −1.34794e7 0
15.2 0 1141.90i 0 15768.7 0 288003.i 0 3.47902e6 0
15.3 0 1141.90i 0 15768.7 0 288003.i 0 3.47902e6 0
15.4 0 4273.45i 0 −59268.7 0 1.04417e6i 0 −1.34794e7 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 16.15.c.c 4
3.b odd 2 1 144.15.g.g 4
4.b odd 2 1 inner 16.15.c.c 4
8.b even 2 1 64.15.c.c 4
8.d odd 2 1 64.15.c.c 4
12.b even 2 1 144.15.g.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.15.c.c 4 1.a even 1 1 trivial
16.15.c.c 4 4.b odd 2 1 inner
64.15.c.c 4 8.b even 2 1
64.15.c.c 4 8.d odd 2 1
144.15.g.g 4 3.b odd 2 1
144.15.g.g 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 19566336 T_{3}^{2} + \)\(23\!\cdots\!84\)\( \) acting on \(S_{15}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 434460 T^{2} - 26096451597018 T^{4} + 9939051249982356060 T^{6} + \)\(52\!\cdots\!21\)\( T^{8} \)
$5$ \( ( 1 + 43500 T + 11272441910 T^{2} + 265502929687500 T^{3} + 37252902984619140625 T^{4} )^{2} \)
$7$ \( 1 - 1539645857092 T^{2} + \)\(12\!\cdots\!58\)\( T^{4} - \)\(70\!\cdots\!92\)\( T^{6} + \)\(21\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 - 1334996682152740 T^{2} + \)\(73\!\cdots\!22\)\( T^{4} - \)\(19\!\cdots\!40\)\( T^{6} + \)\(20\!\cdots\!61\)\( T^{8} \)
$13$ \( ( 1 - 33432308 T + 6353569240644054 T^{2} - \)\(13\!\cdots\!12\)\( T^{3} + \)\(15\!\cdots\!21\)\( T^{4} )^{2} \)
$17$ \( ( 1 - 6558276 T + 267198985045368902 T^{2} - \)\(11\!\cdots\!04\)\( T^{3} + \)\(28\!\cdots\!41\)\( T^{4} )^{2} \)
$19$ \( 1 - 2057925827207927140 T^{2} + \)\(23\!\cdots\!22\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{6} + \)\(40\!\cdots\!81\)\( T^{8} \)
$23$ \( 1 - 11694673362806264260 T^{2} + \)\(27\!\cdots\!22\)\( T^{4} - \)\(15\!\cdots\!60\)\( T^{6} + \)\(18\!\cdots\!61\)\( T^{8} \)
$29$ \( ( 1 + 8750892876 T + \)\(38\!\cdots\!46\)\( T^{2} + \)\(26\!\cdots\!56\)\( T^{3} + \)\(88\!\cdots\!61\)\( T^{4} )^{2} \)
$31$ \( 1 - \)\(14\!\cdots\!60\)\( T^{2} + \)\(11\!\cdots\!82\)\( T^{4} - \)\(81\!\cdots\!60\)\( T^{6} + \)\(32\!\cdots\!81\)\( T^{8} \)
$37$ \( ( 1 + 69992867180 T + \)\(13\!\cdots\!38\)\( T^{2} + \)\(63\!\cdots\!20\)\( T^{3} + \)\(81\!\cdots\!21\)\( T^{4} )^{2} \)
$41$ \( ( 1 - 331044205284 T + \)\(92\!\cdots\!26\)\( T^{2} - \)\(12\!\cdots\!24\)\( T^{3} + \)\(14\!\cdots\!21\)\( T^{4} )^{2} \)
$43$ \( 1 - \)\(11\!\cdots\!20\)\( T^{2} + \)\(93\!\cdots\!42\)\( T^{4} - \)\(60\!\cdots\!20\)\( T^{6} + \)\(29\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - \)\(36\!\cdots\!52\)\( T^{2} + \)\(15\!\cdots\!58\)\( T^{4} - \)\(24\!\cdots\!72\)\( T^{6} + \)\(43\!\cdots\!21\)\( T^{8} \)
$53$ \( ( 1 + 1487586812076 T + \)\(17\!\cdots\!22\)\( T^{2} + \)\(20\!\cdots\!44\)\( T^{3} + \)\(19\!\cdots\!61\)\( T^{4} )^{2} \)
$59$ \( 1 - \)\(18\!\cdots\!00\)\( T^{2} + \)\(15\!\cdots\!02\)\( T^{4} - \)\(70\!\cdots\!00\)\( T^{6} + \)\(14\!\cdots\!41\)\( T^{8} \)
$61$ \( ( 1 + 3312485425868 T + \)\(20\!\cdots\!98\)\( T^{2} + \)\(32\!\cdots\!88\)\( T^{3} + \)\(97\!\cdots\!81\)\( T^{4} )^{2} \)
$67$ \( 1 - \)\(86\!\cdots\!80\)\( T^{2} + \)\(42\!\cdots\!42\)\( T^{4} - \)\(11\!\cdots\!80\)\( T^{6} + \)\(18\!\cdots\!81\)\( T^{8} \)
$71$ \( 1 - \)\(22\!\cdots\!48\)\( T^{2} + \)\(25\!\cdots\!98\)\( T^{4} - \)\(15\!\cdots\!28\)\( T^{6} + \)\(46\!\cdots\!21\)\( T^{8} \)
$73$ \( ( 1 + 6016566294556 T + \)\(75\!\cdots\!62\)\( T^{2} + \)\(73\!\cdots\!04\)\( T^{3} + \)\(14\!\cdots\!81\)\( T^{4} )^{2} \)
$79$ \( 1 - \)\(88\!\cdots\!24\)\( T^{2} + \)\(43\!\cdots\!66\)\( T^{4} - \)\(12\!\cdots\!64\)\( T^{6} + \)\(18\!\cdots\!21\)\( T^{8} \)
$83$ \( 1 - \)\(15\!\cdots\!72\)\( T^{2} + \)\(14\!\cdots\!18\)\( T^{4} - \)\(82\!\cdots\!52\)\( T^{6} + \)\(29\!\cdots\!81\)\( T^{8} \)
$89$ \( ( 1 - 53153507248548 T + \)\(31\!\cdots\!18\)\( T^{2} - \)\(10\!\cdots\!68\)\( T^{3} + \)\(38\!\cdots\!81\)\( T^{4} )^{2} \)
$97$ \( ( 1 - 47896582968068 T + \)\(11\!\cdots\!94\)\( T^{2} - \)\(31\!\cdots\!92\)\( T^{3} + \)\(42\!\cdots\!61\)\( T^{4} )^{2} \)
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