Properties

Label 16.7.c.a
Level 16
Weight 7
Character orbit 16.c
Self dual yes
Analytic conductor 3.681
Analytic rank 0
Dimension 1
CM discriminant -4
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 16 = 2^{4} \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 16.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(3.68086533792\)
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

\(f(q)\) \(=\) \( q + 234q^{5} + 729q^{9} + O(q^{10}) \) \( q + 234q^{5} + 729q^{9} - 4070q^{13} - 990q^{17} + 39131q^{25} - 31878q^{29} - 55510q^{37} - 84942q^{41} + 170586q^{45} + 117649q^{49} - 29430q^{53} + 234938q^{61} - 952380q^{65} + 427570q^{73} + 531441q^{81} - 231660q^{85} + 1378962q^{89} - 1472510q^{97} + O(q^{100}) \)

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 0 234.000 0 0 0 729.000 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 16.7.c.a 1
3.b odd 2 1 144.7.g.a 1
4.b odd 2 1 CM 16.7.c.a 1
5.b even 2 1 400.7.b.a 1
5.c odd 4 2 400.7.h.a 2
8.b even 2 1 64.7.c.a 1
8.d odd 2 1 64.7.c.a 1
12.b even 2 1 144.7.g.a 1
16.e even 4 2 256.7.d.b 2
16.f odd 4 2 256.7.d.b 2
20.d odd 2 1 400.7.b.a 1
20.e even 4 2 400.7.h.a 2
24.f even 2 1 576.7.g.c 1
24.h odd 2 1 576.7.g.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.7.c.a 1 1.a even 1 1 trivial
16.7.c.a 1 4.b odd 2 1 CM
64.7.c.a 1 8.b even 2 1
64.7.c.a 1 8.d odd 2 1
144.7.g.a 1 3.b odd 2 1
144.7.g.a 1 12.b even 2 1
256.7.d.b 2 16.e even 4 2
256.7.d.b 2 16.f odd 4 2
400.7.b.a 1 5.b even 2 1
400.7.b.a 1 20.d odd 2 1
400.7.h.a 2 5.c odd 4 2
400.7.h.a 2 20.e even 4 2
576.7.g.c 1 24.f even 2 1
576.7.g.c 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{7}^{\mathrm{new}}(16, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 - 27 T )( 1 + 27 T ) \)
$5$ \( 1 - 234 T + 15625 T^{2} \)
$7$ \( ( 1 - 343 T )( 1 + 343 T ) \)
$11$ \( ( 1 - 1331 T )( 1 + 1331 T ) \)
$13$ \( 1 + 4070 T + 4826809 T^{2} \)
$17$ \( 1 + 990 T + 24137569 T^{2} \)
$19$ \( ( 1 - 6859 T )( 1 + 6859 T ) \)
$23$ \( ( 1 - 12167 T )( 1 + 12167 T ) \)
$29$ \( 1 + 31878 T + 594823321 T^{2} \)
$31$ \( ( 1 - 29791 T )( 1 + 29791 T ) \)
$37$ \( 1 + 55510 T + 2565726409 T^{2} \)
$41$ \( 1 + 84942 T + 4750104241 T^{2} \)
$43$ \( ( 1 - 79507 T )( 1 + 79507 T ) \)
$47$ \( ( 1 - 103823 T )( 1 + 103823 T ) \)
$53$ \( 1 + 29430 T + 22164361129 T^{2} \)
$59$ \( ( 1 - 205379 T )( 1 + 205379 T ) \)
$61$ \( 1 - 234938 T + 51520374361 T^{2} \)
$67$ \( ( 1 - 300763 T )( 1 + 300763 T ) \)
$71$ \( ( 1 - 357911 T )( 1 + 357911 T ) \)
$73$ \( 1 - 427570 T + 151334226289 T^{2} \)
$79$ \( ( 1 - 493039 T )( 1 + 493039 T ) \)
$83$ \( ( 1 - 571787 T )( 1 + 571787 T ) \)
$89$ \( 1 - 1378962 T + 496981290961 T^{2} \)
$97$ \( 1 + 1472510 T + 832972004929 T^{2} \)
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