Properties

Label 4.45.b.a
Level $4$
Weight $45$
Character orbit 4.b
Self dual yes
Analytic conductor $49.048$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 45 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(49.0478939917\)
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

\(f(q)\) \(=\) \( q - 4194304q^{2} + 17592186044416q^{4} + 94681488501586q^{5} - 73786976294838206464q^{8} + 984770902183611232881q^{9} + O(q^{10}) \) \( q - 4194304q^{2} + 17592186044416q^{4} + 94681488501586q^{5} - 73786976294838206464q^{8} + 984770902183611232881q^{9} - 397122945948156166144q^{10} + 4746397497491127617277362q^{13} + 309485009821345068724781056q^{16} - 504652011666161847903676478q^{17} - 4130428534112329328517709824q^{18} + 1665654360682135200282443776q^{20} - 5675377301815925525037469625229q^{25} - 19907834009317026529656908546048q^{26} - 152160779312124788381478442879118q^{29} - 1298074214633706907132624082305024q^{32} + 2116663951139429303309781866381312q^{34} + 17324272922341479351919144385642496q^{36} + 35088063909469944088702337296151762q^{37} - 6986260747626522399085455059451904q^{40} - 94426535081528617629501241894433438q^{41} + 93239574851794058485356430783849266q^{45} + 15286700631942576193765185769276826401q^{49} + 23804257718515743693366758998976495616q^{50} + 83499507816614441641446090142323310592q^{52} + 163042973619137390933827733009903899282q^{53} + 638208565311962248407588558881656143872q^{58} + 2506573069383192859663671581243711923442q^{61} + 5444517870735015415413993718908291383296q^{64} + 449395980082662764778299879914338896132q^{65} - 8877932076919912884589431321290602446848q^{68} - 72663267215268556211671874973277863542784q^{72} - 196251180328857016590296490866486306864158q^{73} - 147170006807745424371020568130598519963648q^{74} + 29302501398812913404573720497679358754816q^{80} + 969773729787523602876821942164080815560161q^{81} + 396053593798595807037887576882789746737152q^{82} - 47781203639871946931933226250252733894108q^{85} + 13096261849231128405538165979936005417300642q^{89} - 391075121759179226681364419062422111780864q^{90} + 92246626358267604714915915792541346844458882q^{97} - 64117069607359275099814093732820870081019904q^{98} + O(q^{100}) \)

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
−4.19430e6 0 1.75922e13 9.46815e13 0 0 −7.37870e19 9.84771e20 −3.97123e20
\(n\): 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 4.45.b.a 1
4.b odd 2 1 CM 4.45.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.45.b.a 1 1.a even 1 1 trivial
4.45.b.a 1 4.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{45}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4194304 + T \)
$3$ \( T \)
$5$ \( -94681488501586 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -\)\(47\!\cdots\!62\)\( + T \)
$17$ \( \)\(50\!\cdots\!78\)\( + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( \)\(15\!\cdots\!18\)\( + T \)
$31$ \( T \)
$37$ \( -\)\(35\!\cdots\!62\)\( + T \)
$41$ \( \)\(94\!\cdots\!38\)\( + T \)
$43$ \( T \)
$47$ \( T \)
$53$ \( -\)\(16\!\cdots\!82\)\( + T \)
$59$ \( T \)
$61$ \( -\)\(25\!\cdots\!42\)\( + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( \)\(19\!\cdots\!58\)\( + T \)
$79$ \( T \)
$83$ \( T \)
$89$ \( -\)\(13\!\cdots\!42\)\( + T \)
$97$ \( -\)\(92\!\cdots\!82\)\( + T \)
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