Properties

Label 16.11.c.a
Level 16
Weight 11
Character orbit 16.c
Self dual yes
Analytic conductor 10.166
Analytic rank 0
Dimension 1
CM discriminant -4
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 16.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(10.1657160428\)
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

\(f(q)\) \(=\) \( q + 474q^{5} + 59049q^{9} + O(q^{10}) \) \( q + 474q^{5} + 59049q^{9} + 683050q^{13} + 2186850q^{17} - 9540949q^{25} - 32319798q^{29} + 11178650q^{37} + 207190098q^{41} + 27989226q^{45} + 282475249q^{49} - 783188550q^{53} - 1330090102q^{61} + 323765700q^{65} + 3740362450q^{73} + 3486784401q^{81} + 1036566900q^{85} - 8562016398q^{89} - 8684532350q^{97} + O(q^{100}) \)

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 0 474.000 0 0 0 59049.0 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 16.11.c.a 1
3.b odd 2 1 144.11.g.a 1
4.b odd 2 1 CM 16.11.c.a 1
8.b even 2 1 64.11.c.a 1
8.d odd 2 1 64.11.c.a 1
12.b even 2 1 144.11.g.a 1
16.e even 4 2 256.11.d.a 2
16.f odd 4 2 256.11.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.11.c.a 1 1.a even 1 1 trivial
16.11.c.a 1 4.b odd 2 1 CM
64.11.c.a 1 8.b even 2 1
64.11.c.a 1 8.d odd 2 1
144.11.g.a 1 3.b odd 2 1
144.11.g.a 1 12.b even 2 1
256.11.d.a 2 16.e even 4 2
256.11.d.a 2 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{11}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 243 T )( 1 + 243 T ) \)
$5$ \( 1 - 474 T + 9765625 T^{2} \)
$7$ \( ( 1 - 16807 T )( 1 + 16807 T ) \)
$11$ \( ( 1 - 161051 T )( 1 + 161051 T ) \)
$13$ \( 1 - 683050 T + 137858491849 T^{2} \)
$17$ \( 1 - 2186850 T + 2015993900449 T^{2} \)
$19$ \( ( 1 - 2476099 T )( 1 + 2476099 T ) \)
$23$ \( ( 1 - 6436343 T )( 1 + 6436343 T ) \)
$29$ \( 1 + 32319798 T + 420707233300201 T^{2} \)
$31$ \( ( 1 - 28629151 T )( 1 + 28629151 T ) \)
$37$ \( 1 - 11178650 T + 4808584372417849 T^{2} \)
$41$ \( 1 - 207190098 T + 13422659310152401 T^{2} \)
$43$ \( ( 1 - 147008443 T )( 1 + 147008443 T ) \)
$47$ \( ( 1 - 229345007 T )( 1 + 229345007 T ) \)
$53$ \( 1 + 783188550 T + 174887470365513049 T^{2} \)
$59$ \( ( 1 - 714924299 T )( 1 + 714924299 T ) \)
$61$ \( 1 + 1330090102 T + 713342911662882601 T^{2} \)
$67$ \( ( 1 - 1350125107 T )( 1 + 1350125107 T ) \)
$71$ \( ( 1 - 1804229351 T )( 1 + 1804229351 T ) \)
$73$ \( 1 - 3740362450 T + 4297625829703557649 T^{2} \)
$79$ \( ( 1 - 3077056399 T )( 1 + 3077056399 T ) \)
$83$ \( ( 1 - 3939040643 T )( 1 + 3939040643 T ) \)
$89$ \( 1 + 8562016398 T + 31181719929966183601 T^{2} \)
$97$ \( 1 + 8684532350 T + 73742412689492826049 T^{2} \)
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