Properties

Label 16.14.a.b
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 4)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 468q^{3} + 56214q^{5} - 333032q^{7} - 1375299q^{9} + O(q^{10}) \) \( q - 468q^{3} + 56214q^{5} - 333032q^{7} - 1375299q^{9} + 6397380q^{11} + 15199742q^{13} - 26308152q^{15} + 43114194q^{17} + 365115484q^{19} + 155858976q^{21} + 57226824q^{23} + 1939310671q^{25} + 1389783096q^{27} - 46418994q^{29} + 5682185824q^{31} - 2993973840q^{33} - 18721060848q^{35} - 1887185098q^{37} - 7113479256q^{39} - 7336802934q^{41} + 26886674980q^{43} - 77311057986q^{45} - 101839834224q^{47} + 14021302617q^{49} - 20177442792q^{51} + 278731884294q^{53} + 359622319320q^{55} - 170874046512q^{57} - 59573945772q^{59} - 27484470418q^{61} + 458018576568q^{63} + 854438296788q^{65} - 784410054932q^{67} - 26782153632q^{69} + 360365227992q^{71} - 1592635413718q^{73} - 907597394028q^{75} - 2130532256160q^{77} + 23161184752q^{79} + 1542252338649q^{81} - 2050158110436q^{83} + 2423621301516q^{85} + 21724089192q^{87} - 3485391237126q^{89} - 5062000477744q^{91} - 2659262965632q^{93} + 20524601817576q^{95} + 6706667416802q^{97} - 8798310316620q^{99} + O(q^{100}) \)

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 −468.000 0 56214.0 0 −333032. 0 −1.37530e6 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.b 1
3.b odd 2 1 144.14.a.a 1
4.b odd 2 1 4.14.a.a 1
8.b even 2 1 64.14.a.g 1
8.d odd 2 1 64.14.a.c 1
12.b even 2 1 36.14.a.a 1
20.d odd 2 1 100.14.a.a 1
20.e even 4 2 100.14.c.a 2
28.d even 2 1 196.14.a.a 1
28.f even 6 2 196.14.e.b 2
28.g odd 6 2 196.14.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.14.a.a 1 4.b odd 2 1
16.14.a.b 1 1.a even 1 1 trivial
36.14.a.a 1 12.b even 2 1
64.14.a.c 1 8.d odd 2 1
64.14.a.g 1 8.b even 2 1
100.14.a.a 1 20.d odd 2 1
100.14.c.a 2 20.e even 4 2
144.14.a.a 1 3.b odd 2 1
196.14.a.a 1 28.d even 2 1
196.14.e.a 2 28.g odd 6 2
196.14.e.b 2 28.f even 6 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 468 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 468 T + 1594323 T^{2} \)
$5$ \( 1 - 56214 T + 1220703125 T^{2} \)
$7$ \( 1 + 333032 T + 96889010407 T^{2} \)
$11$ \( 1 - 6397380 T + 34522712143931 T^{2} \)
$13$ \( 1 - 15199742 T + 302875106592253 T^{2} \)
$17$ \( 1 - 43114194 T + 9904578032905937 T^{2} \)
$19$ \( 1 - 365115484 T + 42052983462257059 T^{2} \)
$23$ \( 1 - 57226824 T + 504036361936467383 T^{2} \)
$29$ \( 1 + 46418994 T + 10260628712958602189 T^{2} \)
$31$ \( 1 - 5682185824 T + 24417546297445042591 T^{2} \)
$37$ \( 1 + 1887185098 T + \)\(24\!\cdots\!97\)\( T^{2} \)
$41$ \( 1 + 7336802934 T + \)\(92\!\cdots\!21\)\( T^{2} \)
$43$ \( 1 - 26886674980 T + \)\(17\!\cdots\!43\)\( T^{2} \)
$47$ \( 1 + 101839834224 T + \)\(54\!\cdots\!27\)\( T^{2} \)
$53$ \( 1 - 278731884294 T + \)\(26\!\cdots\!73\)\( T^{2} \)
$59$ \( 1 + 59573945772 T + \)\(10\!\cdots\!79\)\( T^{2} \)
$61$ \( 1 + 27484470418 T + \)\(16\!\cdots\!81\)\( T^{2} \)
$67$ \( 1 + 784410054932 T + \)\(54\!\cdots\!87\)\( T^{2} \)
$71$ \( 1 - 360365227992 T + \)\(11\!\cdots\!11\)\( T^{2} \)
$73$ \( 1 + 1592635413718 T + \)\(16\!\cdots\!33\)\( T^{2} \)
$79$ \( 1 - 23161184752 T + \)\(46\!\cdots\!39\)\( T^{2} \)
$83$ \( 1 + 2050158110436 T + \)\(88\!\cdots\!63\)\( T^{2} \)
$89$ \( 1 + 3485391237126 T + \)\(21\!\cdots\!69\)\( T^{2} \)
$97$ \( 1 - 6706667416802 T + \)\(67\!\cdots\!77\)\( T^{2} \)
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