Properties

Label 4.17.b.a
Level $4$
Weight $17$
Character orbit 4.b
Self dual yes
Analytic conductor $6.493$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(6.49298175427\)
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

\(f(q)\) \(=\) \( q + 256q^{2} + 65536q^{4} + 329666q^{5} + 16777216q^{8} + 43046721q^{9} + O(q^{10}) \) \( q + 256q^{2} + 65536q^{4} + 329666q^{5} + 16777216q^{8} + 43046721q^{9} + 84394496q^{10} - 1631232958q^{13} + 4294967296q^{16} - 9937278718q^{17} + 11019960576q^{18} + 21604990976q^{20} - 43908219069q^{25} - 417595637248q^{26} + 981515008322q^{29} + 1099511627776q^{32} - 2543943351808q^{34} + 2821109907456q^{36} - 6167627357758q^{37} + 5530877689856q^{40} - 3168324620158q^{41} + 14191040325186q^{45} + 33232930569601q^{49} - 11240504081664q^{50} - 106904483135488q^{52} - 31962705295678q^{53} + 251267842130432q^{58} + 45990056420162q^{61} + 281474976710656q^{64} - 537762044332028q^{65} - 651249498062848q^{68} + 722204136308736q^{72} + 1381042818437762q^{73} - 1578912603586048q^{74} + 1415904688603136q^{80} + 1853020188851841q^{81} - 811091102760448q^{82} - 3275982925848188q^{85} - 6957151819021438q^{89} + 3632906323247616q^{90} + 14385701036152322q^{97} + 8507630225817856q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0
256.000 0 65536.0 329666. 0 0 1.67772e7 4.30467e7 8.43945e7
\(n\): 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 4.17.b.a 1
3.b odd 2 1 36.17.d.a 1
4.b odd 2 1 CM 4.17.b.a 1
8.b even 2 1 64.17.c.a 1
8.d odd 2 1 64.17.c.a 1
12.b even 2 1 36.17.d.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.17.b.a 1 1.a even 1 1 trivial
4.17.b.a 1 4.b odd 2 1 CM
36.17.d.a 1 3.b odd 2 1
36.17.d.a 1 12.b even 2 1
64.17.c.a 1 8.b even 2 1
64.17.c.a 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{17}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -256 + T \)
$3$ \( T \)
$5$ \( -329666 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( 1631232958 + T \)
$17$ \( 9937278718 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( -981515008322 + T \)
$31$ \( T \)
$37$ \( 6167627357758 + T \)
$41$ \( 3168324620158 + T \)
$43$ \( T \)
$47$ \( T \)
$53$ \( 31962705295678 + T \)
$59$ \( T \)
$61$ \( -45990056420162 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( -1381042818437762 + T \)
$79$ \( T \)
$83$ \( T \)
$89$ \( 6957151819021438 + T \)
$97$ \( -14385701036152322 + T \)
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