Properties

Label 16.14.a.d
Level 16
Weight 14
Character orbit 16.a
Self dual yes
Analytic conductor 17.157
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1569486323\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 1836q^{3} + 3990q^{5} + 433432q^{7} + 1776573q^{9} + O(q^{10}) \) \( q + 1836q^{3} + 3990q^{5} + 433432q^{7} + 1776573q^{9} - 1619772q^{11} - 10878466q^{13} + 7325640q^{15} + 60569298q^{17} + 243131740q^{19} + 795781152q^{21} + 606096456q^{23} - 1204783025q^{25} + 334611000q^{27} + 5258639310q^{29} + 1824312928q^{31} - 2973901392q^{33} + 1729393680q^{35} - 3005875402q^{37} - 19972863576q^{39} - 49704880758q^{41} - 58766693084q^{43} + 7088526270q^{45} + 42095878032q^{47} + 90974288217q^{49} + 111205231128q^{51} - 181140755706q^{53} - 6462890280q^{55} + 446389874640q^{57} - 206730587820q^{59} - 124479015058q^{61} + 770023588536q^{63} - 43405079340q^{65} - 95665133588q^{67} + 1112793093216q^{69} + 371436487128q^{71} - 1800576064726q^{73} - 2211981633900q^{75} - 702061017504q^{77} - 1557932091920q^{79} - 2218085399079q^{81} - 2492790917604q^{83} + 241671499020q^{85} + 9654861773160q^{87} + 2994235754490q^{89} - 4715075275312q^{91} + 3349438535808q^{93} + 970095642600q^{95} + 4382492665058q^{97} - 2877643201356q^{99} + O(q^{100}) \)

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
0
0 1836.00 0 3990.00 0 433432. 0 1.77657e6 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 16.14.a.d 1
3.b odd 2 1 144.14.a.d 1
4.b odd 2 1 2.14.a.a 1
8.b even 2 1 64.14.a.a 1
8.d odd 2 1 64.14.a.i 1
12.b even 2 1 18.14.a.d 1
20.d odd 2 1 50.14.a.e 1
20.e even 4 2 50.14.b.e 2
28.d even 2 1 98.14.a.b 1
28.f even 6 2 98.14.c.e 2
28.g odd 6 2 98.14.c.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.14.a.a 1 4.b odd 2 1
16.14.a.d 1 1.a even 1 1 trivial
18.14.a.d 1 12.b even 2 1
50.14.a.e 1 20.d odd 2 1
50.14.b.e 2 20.e even 4 2
64.14.a.a 1 8.b even 2 1
64.14.a.i 1 8.d odd 2 1
98.14.a.b 1 28.d even 2 1
98.14.c.e 2 28.f even 6 2
98.14.c.h 2 28.g odd 6 2
144.14.a.d 1 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1836 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 1836 T + 1594323 T^{2} \)
$5$ \( 1 - 3990 T + 1220703125 T^{2} \)
$7$ \( 1 - 433432 T + 96889010407 T^{2} \)
$11$ \( 1 + 1619772 T + 34522712143931 T^{2} \)
$13$ \( 1 + 10878466 T + 302875106592253 T^{2} \)
$17$ \( 1 - 60569298 T + 9904578032905937 T^{2} \)
$19$ \( 1 - 243131740 T + 42052983462257059 T^{2} \)
$23$ \( 1 - 606096456 T + 504036361936467383 T^{2} \)
$29$ \( 1 - 5258639310 T + 10260628712958602189 T^{2} \)
$31$ \( 1 - 1824312928 T + 24417546297445042591 T^{2} \)
$37$ \( 1 + 3005875402 T + \)\(24\!\cdots\!97\)\( T^{2} \)
$41$ \( 1 + 49704880758 T + \)\(92\!\cdots\!21\)\( T^{2} \)
$43$ \( 1 + 58766693084 T + \)\(17\!\cdots\!43\)\( T^{2} \)
$47$ \( 1 - 42095878032 T + \)\(54\!\cdots\!27\)\( T^{2} \)
$53$ \( 1 + 181140755706 T + \)\(26\!\cdots\!73\)\( T^{2} \)
$59$ \( 1 + 206730587820 T + \)\(10\!\cdots\!79\)\( T^{2} \)
$61$ \( 1 + 124479015058 T + \)\(16\!\cdots\!81\)\( T^{2} \)
$67$ \( 1 + 95665133588 T + \)\(54\!\cdots\!87\)\( T^{2} \)
$71$ \( 1 - 371436487128 T + \)\(11\!\cdots\!11\)\( T^{2} \)
$73$ \( 1 + 1800576064726 T + \)\(16\!\cdots\!33\)\( T^{2} \)
$79$ \( 1 + 1557932091920 T + \)\(46\!\cdots\!39\)\( T^{2} \)
$83$ \( 1 + 2492790917604 T + \)\(88\!\cdots\!63\)\( T^{2} \)
$89$ \( 1 - 2994235754490 T + \)\(21\!\cdots\!69\)\( T^{2} \)
$97$ \( 1 - 4382492665058 T + \)\(67\!\cdots\!77\)\( T^{2} \)
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