Properties

Label 4.5.b.a
Level 4
Weight 5
Character orbit 4.b
Self dual yes
Analytic conductor 0.413
Analytic rank 0
Dimension 1
CM discriminant -4
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.413479852335\)
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

\(f(q)\) \(=\) \( q - 4q^{2} + 16q^{4} - 14q^{5} - 64q^{8} + 81q^{9} + O(q^{10}) \) \( q - 4q^{2} + 16q^{4} - 14q^{5} - 64q^{8} + 81q^{9} + 56q^{10} - 238q^{13} + 256q^{16} + 322q^{17} - 324q^{18} - 224q^{20} - 429q^{25} + 952q^{26} + 82q^{29} - 1024q^{32} - 1288q^{34} + 1296q^{36} + 2162q^{37} + 896q^{40} - 3038q^{41} - 1134q^{45} + 2401q^{49} + 1716q^{50} - 3808q^{52} + 2482q^{53} - 328q^{58} - 6958q^{61} + 4096q^{64} + 3332q^{65} + 5152q^{68} - 5184q^{72} + 1442q^{73} - 8648q^{74} - 3584q^{80} + 6561q^{81} + 12152q^{82} - 4508q^{85} - 9758q^{89} + 4536q^{90} - 1918q^{97} - 9604q^{98} + O(q^{100}) \)

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
−4.00000 0 16.0000 −14.0000 0 0 −64.0000 81.0000 56.0000
\(n\): 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 4.5.b.a 1
3.b odd 2 1 36.5.d.a 1
4.b odd 2 1 CM 4.5.b.a 1
5.b even 2 1 100.5.b.a 1
5.c odd 4 2 100.5.d.a 2
7.b odd 2 1 196.5.c.a 1
8.b even 2 1 64.5.c.a 1
8.d odd 2 1 64.5.c.a 1
12.b even 2 1 36.5.d.a 1
16.e even 4 2 256.5.d.c 2
16.f odd 4 2 256.5.d.c 2
20.d odd 2 1 100.5.b.a 1
20.e even 4 2 100.5.d.a 2
24.f even 2 1 576.5.g.b 1
24.h odd 2 1 576.5.g.b 1
28.d even 2 1 196.5.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.5.b.a 1 1.a even 1 1 trivial
4.5.b.a 1 4.b odd 2 1 CM
36.5.d.a 1 3.b odd 2 1
36.5.d.a 1 12.b even 2 1
64.5.c.a 1 8.b even 2 1
64.5.c.a 1 8.d odd 2 1
100.5.b.a 1 5.b even 2 1
100.5.b.a 1 20.d odd 2 1
100.5.d.a 2 5.c odd 4 2
100.5.d.a 2 20.e even 4 2
196.5.c.a 1 7.b odd 2 1
196.5.c.a 1 28.d even 2 1
256.5.d.c 2 16.e even 4 2
256.5.d.c 2 16.f odd 4 2
576.5.g.b 1 24.f even 2 1
576.5.g.b 1 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T \)
$3$ \( ( 1 - 9 T )( 1 + 9 T ) \)
$5$ \( 1 + 14 T + 625 T^{2} \)
$7$ \( ( 1 - 49 T )( 1 + 49 T ) \)
$11$ \( ( 1 - 121 T )( 1 + 121 T ) \)
$13$ \( 1 + 238 T + 28561 T^{2} \)
$17$ \( 1 - 322 T + 83521 T^{2} \)
$19$ \( ( 1 - 361 T )( 1 + 361 T ) \)
$23$ \( ( 1 - 529 T )( 1 + 529 T ) \)
$29$ \( 1 - 82 T + 707281 T^{2} \)
$31$ \( ( 1 - 961 T )( 1 + 961 T ) \)
$37$ \( 1 - 2162 T + 1874161 T^{2} \)
$41$ \( 1 + 3038 T + 2825761 T^{2} \)
$43$ \( ( 1 - 1849 T )( 1 + 1849 T ) \)
$47$ \( ( 1 - 2209 T )( 1 + 2209 T ) \)
$53$ \( 1 - 2482 T + 7890481 T^{2} \)
$59$ \( ( 1 - 3481 T )( 1 + 3481 T ) \)
$61$ \( 1 + 6958 T + 13845841 T^{2} \)
$67$ \( ( 1 - 4489 T )( 1 + 4489 T ) \)
$71$ \( ( 1 - 5041 T )( 1 + 5041 T ) \)
$73$ \( 1 - 1442 T + 28398241 T^{2} \)
$79$ \( ( 1 - 6241 T )( 1 + 6241 T ) \)
$83$ \( ( 1 - 6889 T )( 1 + 6889 T ) \)
$89$ \( 1 + 9758 T + 62742241 T^{2} \)
$97$ \( 1 + 1918 T + 88529281 T^{2} \)
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Additional information

This cusp form has an eta product $\eta(z)^4\eta(2z)^2\eta(4z)^4=q\prod_{n=1}^\infty (1-q^n)^4(1-q^{2n})^2(1-q^{4n})^4$ where $q=\exp(2\pi i z)$.