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

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

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:

Atkin-Lehner signs

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

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 252 \) Copy content Toggle raw display
$5$ \( T - 4830 \) Copy content Toggle raw display
$7$ \( T - 16744 \) Copy content Toggle raw display
$11$ \( T + 534612 \) Copy content Toggle raw display
$13$ \( T + 577738 \) Copy content Toggle raw display
$17$ \( T + 6905934 \) Copy content Toggle raw display
$19$ \( T + 10661420 \) Copy content Toggle raw display
$23$ \( T + 18643272 \) Copy content Toggle raw display
$29$ \( T - 128406630 \) Copy content Toggle raw display
$31$ \( T - 52843168 \) Copy content Toggle raw display
$37$ \( T + 182213314 \) Copy content Toggle raw display
$41$ \( T - 308120442 \) Copy content Toggle raw display
$43$ \( T - 17125708 \) Copy content Toggle raw display
$47$ \( T + 2687348496 \) Copy content Toggle raw display
$53$ \( T + 1596055698 \) Copy content Toggle raw display
$59$ \( T - 5189203740 \) Copy content Toggle raw display
$61$ \( T - 6956478662 \) Copy content Toggle raw display
$67$ \( T - 15481826884 \) Copy content Toggle raw display
$71$ \( T + 9791485272 \) Copy content Toggle raw display
$73$ \( T - 1463791322 \) Copy content Toggle raw display
$79$ \( T + 38116845680 \) Copy content Toggle raw display
$83$ \( T - 29335099668 \) Copy content Toggle raw display
$89$ \( T + 24992917110 \) Copy content Toggle raw display
$97$ \( T - 75013568546 \) Copy content Toggle raw display
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