Properties

Label 16.15.c.a
Level 16
Weight 15
Character orbit 16.c
Self dual yes
Analytic conductor 19.893
Analytic rank 0
Dimension 1
CM discriminant -4
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 16.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(19.8926349043\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 152886q^{5} + 4782969q^{9} + O(q^{10}) \) \( q - 152886q^{5} + 4782969q^{9} - 46322630q^{13} + 786851490q^{17} + 17270613371q^{25} + 19896480282q^{29} + 128202918410q^{37} - 389417726862q^{41} - 731248998534q^{45} + 678223072849q^{49} - 1711657125270q^{53} + 6001852736858q^{61} + 7082081610180q^{65} - 6748633251470q^{73} + 22876792454961q^{81} - 120298576900140q^{85} - 2971010122158q^{89} + 147017184612610q^{97} + O(q^{100}) \)

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
0 0 0 −152886. 0 0 0 4.78297e6 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 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.15.c.a 1
3.b odd 2 1 144.15.g.a 1
4.b odd 2 1 CM 16.15.c.a 1
8.b even 2 1 64.15.c.a 1
8.d odd 2 1 64.15.c.a 1
12.b even 2 1 144.15.g.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.15.c.a 1 1.a even 1 1 trivial
16.15.c.a 1 4.b odd 2 1 CM
64.15.c.a 1 8.b even 2 1
64.15.c.a 1 8.d odd 2 1
144.15.g.a 1 3.b odd 2 1
144.15.g.a 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{15}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 2187 T )( 1 + 2187 T ) \)
$5$ \( 1 + 152886 T + 6103515625 T^{2} \)
$7$ \( ( 1 - 823543 T )( 1 + 823543 T ) \)
$11$ \( ( 1 - 19487171 T )( 1 + 19487171 T ) \)
$13$ \( 1 + 46322630 T + 3937376385699289 T^{2} \)
$17$ \( 1 - 786851490 T + 168377826559400929 T^{2} \)
$19$ \( ( 1 - 893871739 T )( 1 + 893871739 T ) \)
$23$ \( ( 1 - 3404825447 T )( 1 + 3404825447 T ) \)
$29$ \( 1 - 19896480282 T + \)\(29\!\cdots\!81\)\( T^{2} \)
$31$ \( ( 1 - 27512614111 T )( 1 + 27512614111 T ) \)
$37$ \( 1 - 128202918410 T + \)\(90\!\cdots\!89\)\( T^{2} \)
$41$ \( 1 + 389417726862 T + \)\(37\!\cdots\!61\)\( T^{2} \)
$43$ \( ( 1 - 271818611107 T )( 1 + 271818611107 T ) \)
$47$ \( ( 1 - 506623120463 T )( 1 + 506623120463 T ) \)
$53$ \( 1 + 1711657125270 T + \)\(13\!\cdots\!69\)\( T^{2} \)
$59$ \( ( 1 - 2488651484819 T )( 1 + 2488651484819 T ) \)
$61$ \( 1 - 6001852736858 T + \)\(98\!\cdots\!41\)\( T^{2} \)
$67$ \( ( 1 - 6060711605323 T )( 1 + 6060711605323 T ) \)
$71$ \( ( 1 - 9095120158391 T )( 1 + 9095120158391 T ) \)
$73$ \( 1 + 6748633251470 T + \)\(12\!\cdots\!09\)\( T^{2} \)
$79$ \( ( 1 - 19203908986159 T )( 1 + 19203908986159 T ) \)
$83$ \( ( 1 - 27136050989627 T )( 1 + 27136050989627 T ) \)
$89$ \( 1 + 2971010122158 T + \)\(19\!\cdots\!41\)\( T^{2} \)
$97$ \( 1 - 147017184612610 T + \)\(65\!\cdots\!69\)\( T^{2} \)
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