Properties

Label 16.13.c.a
Level 16
Weight 13
Character orbit 16.c
Analytic conductor 14.624
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1155}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 24\sqrt{-1155}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} -270 q^{5} -222 \beta q^{7} -133839 q^{9} +O(q^{10})\) \( q -\beta q^{3} -270 q^{5} -222 \beta q^{7} -133839 q^{9} + 2025 \beta q^{11} -4386350 q^{13} + 270 \beta q^{15} -39383550 q^{17} -66525 \beta q^{19} -147692160 q^{21} + 294678 \beta q^{23} -244067725 q^{25} -397602 \beta q^{27} + 163560978 q^{29} -283800 \beta q^{31} + 1347192000 q^{33} + 59940 \beta q^{35} + 3600024050 q^{37} + 4386350 \beta q^{39} + 2124864738 q^{41} -3193359 \beta q^{43} + 36136530 q^{45} -21555396 \beta q^{47} -18946372319 q^{49} + 39383550 \beta q^{51} -13585251150 q^{53} -546750 \beta q^{55} -44257752000 q^{57} -30494475 \beta q^{59} + 35496554258 q^{61} + 29712258 \beta q^{63} + 1184314500 q^{65} -149495985 \beta q^{67} + 196043379840 q^{69} + 152996850 \beta q^{71} -5982269150 q^{73} + 244067725 \beta q^{75} + 299076624000 q^{77} -149070300 \beta q^{79} -335644190559 q^{81} -487109133 \beta q^{83} + 10633558500 q^{85} -163560978 \beta q^{87} -753989286942 q^{89} + 973769700 \beta q^{91} -188806464000 q^{93} + 17961750 \beta q^{95} -970603845950 q^{97} -271023975 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 540q^{5} - 267678q^{9} + O(q^{10}) \) \( 2q - 540q^{5} - 267678q^{9} - 8772700q^{13} - 78767100q^{17} - 295384320q^{21} - 488135450q^{25} + 327121956q^{29} + 2694384000q^{33} + 7200048100q^{37} + 4249729476q^{41} + 72273060q^{45} - 37892744638q^{49} - 27170502300q^{53} - 88515504000q^{57} + 70993108516q^{61} + 2368629000q^{65} + 392086759680q^{69} - 11964538300q^{73} + 598153248000q^{77} - 671288381118q^{81} + 21267117000q^{85} - 1507978573884q^{89} - 377612928000q^{93} - 1941207691900q^{97} + O(q^{100}) \)

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
0.500000 + 16.9926i
0.500000 16.9926i
0 815.647i 0 −270.000 0 181074.i 0 −133839. 0
15.2 0 815.647i 0 −270.000 0 181074.i 0 −133839. 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.a 2
3.b odd 2 1 144.13.g.d 2
4.b odd 2 1 inner 16.13.c.a 2
8.b even 2 1 64.13.c.b 2
8.d odd 2 1 64.13.c.b 2
12.b even 2 1 144.13.g.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.13.c.a 2 1.a even 1 1 trivial
16.13.c.a 2 4.b odd 2 1 inner
64.13.c.b 2 8.b even 2 1
64.13.c.b 2 8.d odd 2 1
144.13.g.d 2 3.b odd 2 1
144.13.g.d 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 665280 \) acting on \(S_{13}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 397602 T^{2} + 282429536481 T^{4} \)
$5$ \( ( 1 + 270 T + 244140625 T^{2} )^{2} \)
$7$ \( 1 + 5105085118 T^{2} + \)\(19\!\cdots\!01\)\( T^{4} \)
$11$ \( 1 - 3548792953442 T^{2} + \)\(98\!\cdots\!41\)\( T^{4} \)
$13$ \( ( 1 + 4386350 T + 23298085122481 T^{2} )^{2} \)
$17$ \( ( 1 + 39383550 T + 582622237229761 T^{2} )^{2} \)
$19$ \( 1 - 1482382886332322 T^{2} + \)\(48\!\cdots\!21\)\( T^{4} \)
$23$ \( 1 + 13940422220450878 T^{2} + \)\(48\!\cdots\!41\)\( T^{4} \)
$29$ \( ( 1 - 163560978 T + 353814783205469041 T^{2} )^{2} \)
$31$ \( 1 - 1521742293093899522 T^{2} + \)\(62\!\cdots\!21\)\( T^{4} \)
$37$ \( ( 1 - 3600024050 T + 6582952005840035281 T^{2} )^{2} \)
$41$ \( ( 1 - 2124864738 T + 22563490300366186081 T^{2} )^{2} \)
$43$ \( 1 - 73135041050432481122 T^{2} + \)\(15\!\cdots\!01\)\( T^{4} \)
$47$ \( 1 + 76729470925866191998 T^{2} + \)\(13\!\cdots\!81\)\( T^{4} \)
$53$ \( ( 1 + 13585251150 T + \)\(49\!\cdots\!41\)\( T^{2} )^{2} \)
$59$ \( 1 - \)\(29\!\cdots\!62\)\( T^{2} + \)\(31\!\cdots\!61\)\( T^{4} \)
$61$ \( ( 1 - 35496554258 T + \)\(26\!\cdots\!21\)\( T^{2} )^{2} \)
$67$ \( 1 - \)\(14\!\cdots\!22\)\( T^{2} + \)\(66\!\cdots\!21\)\( T^{4} \)
$71$ \( 1 - \)\(17\!\cdots\!82\)\( T^{2} + \)\(26\!\cdots\!81\)\( T^{4} \)
$73$ \( ( 1 + 5982269150 T + \)\(22\!\cdots\!21\)\( T^{2} )^{2} \)
$79$ \( 1 - \)\(10\!\cdots\!82\)\( T^{2} + \)\(34\!\cdots\!81\)\( T^{4} \)
$83$ \( 1 - \)\(55\!\cdots\!02\)\( T^{2} + \)\(11\!\cdots\!21\)\( T^{4} \)
$89$ \( ( 1 + 753989286942 T + \)\(24\!\cdots\!21\)\( T^{2} )^{2} \)
$97$ \( ( 1 + 970603845950 T + \)\(69\!\cdots\!41\)\( T^{2} )^{2} \)
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