Properties

Label 4.21.b.a
Level 4
Weight 21
Character orbit 4.b
Self dual Yes
Analytic conductor 10.141
Analytic rank 0
Dimension 1
CM disc. -4
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 21 \)
Character orbit: \([\chi]\) = 4.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(10.1405506041\)
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

\(f(q)\) \(=\) \( q - 1024q^{2} + 1048576q^{4} - 19306574q^{5} - 1073741824q^{8} + 3486784401q^{9} + O(q^{10}) \) \( q - 1024q^{2} + 1048576q^{4} - 19306574q^{5} - 1073741824q^{8} + 3486784401q^{9} + 19769931776q^{10} + 190840318802q^{13} + 1099511627776q^{16} + 750325121602q^{17} - 3570467226624q^{18} - 20244410138624q^{20} + 277376367976851q^{25} - 195420486453248q^{26} + 203154876160402q^{29} - 1125899906842624q^{32} - 768332924520448q^{34} + 3656158440062976q^{36} - 9492206529013198q^{37} + 20730275981950976q^{40} + 16082418088944802q^{41} - 67317861059952174q^{45} + 79792266297612001q^{49} - 284033400808295424q^{50} + 200110578128125952q^{52} + 263609364120076402q^{53} - 208030593188251648q^{58} + 342453856112605202q^{61} + 1152921504606846976q^{64} - 3684472737134404348q^{65} + 786772914708938752q^{68} - 3743906242624487424q^{72} + 5395059597962887202q^{73} + 9720019485709514752q^{74} - 21227802605517799424q^{80} + 12157665459056928801q^{81} - 16468396123079477248q^{82} - 14486207484268011548q^{85} + 10944684939688527202q^{89} + 68933489725391026176q^{90} - 72063723240789129598q^{97} - 81707280688754689024q^{98} + O(q^{100}) \)

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
−1024.00 0 1.04858e6 −1.93066e7 0 0 −1.07374e9 3.48678e9 1.97699e10
\(n\): 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

This newform can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{21}^{\mathrm{new}}(4, [\chi])\).