Properties

Label 3.37.b.a
Level 3
Weight 37
Character orbit 3.b
Self dual Yes
Analytic conductor 24.627
Analytic rank 0
Dimension 1
CM disc. -3
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 37 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(24.6273775978\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 387420489q^{3} \) \(\mathstrut +\mathstrut 68719476736q^{4} \) \(\mathstrut +\mathstrut 2757049053441698q^{7} \) \(\mathstrut +\mathstrut 150094635296999121q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 387420489q^{3} \) \(\mathstrut +\mathstrut 68719476736q^{4} \) \(\mathstrut +\mathstrut 2757049053441698q^{7} \) \(\mathstrut +\mathstrut 150094635296999121q^{9} \) \(\mathstrut +\mathstrut 26623333280885243904q^{12} \) \(\mathstrut -\mathstrut 173106205278993698542q^{13} \) \(\mathstrut +\mathstrut 4722366482869645213696q^{16} \) \(\mathstrut -\mathstrut 113045623944631671449518q^{19} \) \(\mathstrut +\mathstrut 1068137292481369772150322q^{21} \) \(\mathstrut +\mathstrut 14551915228366851806640625q^{25} \) \(\mathstrut +\mathstrut 58149737003040059690390169q^{27} \) \(\mathstrut +\mathstrut 189462968287997586443337728q^{28} \) \(\mathstrut +\mathstrut 1103792338040358783246655682q^{31} \) \(\mathstrut +\mathstrut 10314424798490535546171949056q^{36} \) \(\mathstrut -\mathstrut 33239966002272825656727628942q^{37} \) \(\mathstrut -\mathstrut 67064890698122120117060227038q^{39} \) \(\mathstrut +\mathstrut 28264519692215715611104352498q^{43} \) \(\mathstrut +\mathstrut 1829541532030568071946613817344q^{48} \) \(\mathstrut +\mathstrut 4949588637224109442640061741603q^{49} \) \(\mathstrut -\mathstrut 11895767846527047856447566118912q^{52} \) \(\mathstrut -\mathstrut 43796190907939311077859602374302q^{57} \) \(\mathstrut -\mathstrut 183499879945319226427712967797038q^{61} \) \(\mathstrut +\mathstrut 413818272172268300516296330747458q^{63} \) \(\mathstrut +\mathstrut 324518553658426726783156020576256q^{64} \) \(\mathstrut +\mathstrut 262445612237681940025882001494418q^{67} \) \(\mathstrut -\mathstrut 6037179267121886675745264636116062q^{73} \) \(\mathstrut +\mathstrut 5637710113660432398319244384765625q^{75} \) \(\mathstrut -\mathstrut 7768436124769720698263947599413248q^{76} \) \(\mathstrut -\mathstrut 8944821450075693224847661238753278q^{79} \) \(\mathstrut +\mathstrut 22528399544939174411840147874772641q^{81} \) \(\mathstrut +\mathstrut 73401835821527517770697673373908992q^{84} \) \(\mathstrut -\mathstrut 477262299409333842017509539384604316q^{91} \) \(\mathstrut +\mathstrut 427631767358049101540864351935068498q^{93} \) \(\mathstrut -\mathstrut 1136057050255301120323176403638513022q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0
0 3.87420e8 6.87195e10 0 0 2.75705e15 0 1.50095e17 0
\(n\): 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
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) acting on \(S_{37}^{\mathrm{new}}(3, [\chi])\).