Properties

Label 16.12.a.a
Level 16
Weight 12
Character orbit 16.a
Self dual yes
Analytic conductor 12.293
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.2934908890\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 252q^{3} + 4830q^{5} + 16744q^{7} - 113643q^{9} + O(q^{10}) \) \( q - 252q^{3} + 4830q^{5} + 16744q^{7} - 113643q^{9} - 534612q^{11} - 577738q^{13} - 1217160q^{15} - 6905934q^{17} - 10661420q^{19} - 4219488q^{21} - 18643272q^{23} - 25499225q^{25} + 73279080q^{27} + 128406630q^{29} + 52843168q^{31} + 134722224q^{33} + 80873520q^{35} - 182213314q^{37} + 145589976q^{39} + 308120442q^{41} + 17125708q^{43} - 548895690q^{45} - 2687348496q^{47} - 1696965207q^{49} + 1740295368q^{51} - 1596055698q^{53} - 2582175960q^{55} + 2686677840q^{57} + 5189203740q^{59} + 6956478662q^{61} - 1902838392q^{63} - 2790474540q^{65} + 15481826884q^{67} + 4698104544q^{69} - 9791485272q^{71} + 1463791322q^{73} + 6425804700q^{75} - 8951543328q^{77} - 38116845680q^{79} + 1665188361q^{81} + 29335099668q^{83} - 33355661220q^{85} - 32358470760q^{87} - 24992917110q^{89} - 9673645072q^{91} - 13316478336q^{93} - 51494658600q^{95} + 75013568546q^{97} + 60754911516q^{99} + O(q^{100}) \)

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 −252.000 0 4830.00 0 16744.0 0 −113643. 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 16.12.a.a 1
3.b odd 2 1 144.12.a.d 1
4.b odd 2 1 1.12.a.a 1
8.b even 2 1 64.12.a.f 1
8.d odd 2 1 64.12.a.b 1
12.b even 2 1 9.12.a.b 1
16.e even 4 2 256.12.b.c 2
16.f odd 4 2 256.12.b.e 2
20.d odd 2 1 25.12.a.b 1
20.e even 4 2 25.12.b.b 2
28.d even 2 1 49.12.a.a 1
28.f even 6 2 49.12.c.c 2
28.g odd 6 2 49.12.c.b 2
36.f odd 6 2 81.12.c.d 2
36.h even 6 2 81.12.c.b 2
44.c even 2 1 121.12.a.b 1
52.b odd 2 1 169.12.a.a 1
60.h even 2 1 225.12.a.b 1
60.l odd 4 2 225.12.b.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a 1 4.b odd 2 1
9.12.a.b 1 12.b even 2 1
16.12.a.a 1 1.a even 1 1 trivial
25.12.a.b 1 20.d odd 2 1
25.12.b.b 2 20.e even 4 2
49.12.a.a 1 28.d even 2 1
49.12.c.b 2 28.g odd 6 2
49.12.c.c 2 28.f even 6 2
64.12.a.b 1 8.d odd 2 1
64.12.a.f 1 8.b even 2 1
81.12.c.b 2 36.h even 6 2
81.12.c.d 2 36.f odd 6 2
121.12.a.b 1 44.c even 2 1
144.12.a.d 1 3.b odd 2 1
169.12.a.a 1 52.b odd 2 1
225.12.a.b 1 60.h even 2 1
225.12.b.d 2 60.l odd 4 2
256.12.b.c 2 16.e even 4 2
256.12.b.e 2 16.f odd 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 252 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 252 T + 177147 T^{2} \)
$5$ \( 1 - 4830 T + 48828125 T^{2} \)
$7$ \( 1 - 16744 T + 1977326743 T^{2} \)
$11$ \( 1 + 534612 T + 285311670611 T^{2} \)
$13$ \( 1 + 577738 T + 1792160394037 T^{2} \)
$17$ \( 1 + 6905934 T + 34271896307633 T^{2} \)
$19$ \( 1 + 10661420 T + 116490258898219 T^{2} \)
$23$ \( 1 + 18643272 T + 952809757913927 T^{2} \)
$29$ \( 1 - 128406630 T + 12200509765705829 T^{2} \)
$31$ \( 1 - 52843168 T + 25408476896404831 T^{2} \)
$37$ \( 1 + 182213314 T + 177917621779460413 T^{2} \)
$41$ \( 1 - 308120442 T + 550329031716248441 T^{2} \)
$43$ \( 1 - 17125708 T + 929293739471222707 T^{2} \)
$47$ \( 1 + 2687348496 T + 2472159215084012303 T^{2} \)
$53$ \( 1 + 1596055698 T + 9269035929372191597 T^{2} \)
$59$ \( 1 - 5189203740 T + 30155888444737842659 T^{2} \)
$61$ \( 1 - 6956478662 T + 43513917611435838661 T^{2} \)
$67$ \( 1 - 15481826884 T + \)\(12\!\cdots\!83\)\( T^{2} \)
$71$ \( 1 + 9791485272 T + \)\(23\!\cdots\!71\)\( T^{2} \)
$73$ \( 1 - 1463791322 T + \)\(31\!\cdots\!77\)\( T^{2} \)
$79$ \( 1 + 38116845680 T + \)\(74\!\cdots\!79\)\( T^{2} \)
$83$ \( 1 - 29335099668 T + \)\(12\!\cdots\!67\)\( T^{2} \)
$89$ \( 1 + 24992917110 T + \)\(27\!\cdots\!89\)\( T^{2} \)
$97$ \( 1 - 75013568546 T + \)\(71\!\cdots\!53\)\( T^{2} \)
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