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 disc. -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 \(2\) and degree \(1\))

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: \( 1 \)
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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 CM by \(\Q(\sqrt{-1}) \) yes

Hecke kernels

There are no other newforms in \(S_{5}^{\mathrm{new}}(4, [\chi])\).