Properties

Label 4.37.b.a
Level 4
Weight 37
Character orbit 4.b
Self dual Yes
Analytic conductor 32.837
Analytic rank 0
Dimension 1
CM disc. -4
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 37 \)
Character orbit: \([\chi]\) = 4.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(32.8365034637\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 262144q^{2} \) \(\mathstrut +\mathstrut 68719476736q^{4} \) \(\mathstrut -\mathstrut 4228490555534q^{5} \) \(\mathstrut -\mathstrut 18014398509481984q^{8} \) \(\mathstrut +\mathstrut 150094635296999121q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 262144q^{2} \) \(\mathstrut +\mathstrut 68719476736q^{4} \) \(\mathstrut -\mathstrut 4228490555534q^{5} \) \(\mathstrut -\mathstrut 18014398509481984q^{8} \) \(\mathstrut +\mathstrut 150094635296999121q^{9} \) \(\mathstrut +\mathstrut 1108473428189904896q^{10} \) \(\mathstrut -\mathstrut 152935864759452674158q^{13} \) \(\mathstrut +\mathstrut 4722366482869645213696q^{16} \) \(\mathstrut -\mathstrut 23125093646441048317118q^{17} \) \(\mathstrut -\mathstrut 39346408075296537575424q^{18} \) \(\mathstrut -\mathstrut 290579658359414429057024q^{20} \) \(\mathstrut +\mathstrut 3328217149873384131384531q^{25} \) \(\mathstrut +\mathstrut 40091219331501961814474752q^{26} \) \(\mathstrut +\mathstrut 178879349123443365836288722q^{29} \) \(\mathstrut -\mathstrut 1237940039285380274899124224q^{32} \) \(\mathstrut +\mathstrut 6062104548852642170042580992q^{34} \) \(\mathstrut +\mathstrut 10314424798490535546171949056q^{36} \) \(\mathstrut +\mathstrut 31870691915293393072612781042q^{37} \) \(\mathstrut +\mathstrut 76173713960970336090724499456q^{40} \) \(\mathstrut +\mathstrut 142745614735837791048783844642q^{41} \) \(\mathstrut -\mathstrut 634673747789680938240399685614q^{45} \) \(\mathstrut +\mathstrut 2651730845859653471779023381601q^{49} \) \(\mathstrut -\mathstrut 872472156536408409737666494464q^{50} \) \(\mathstrut -\mathstrut 10509672600437250277893669388288q^{52} \) \(\mathstrut -\mathstrut 18046568200945234989160879491278q^{53} \) \(\mathstrut -\mathstrut 46892148096615937693788070739968q^{58} \) \(\mathstrut +\mathstrut 271467758018404231546709127105362q^{61} \) \(\mathstrut +\mathstrut 324518553658426726783156020576256q^{64} \) \(\mathstrut +\mathstrut 646687859737770731428143305690372q^{65} \) \(\mathstrut -\mathstrut 1589144334854427029023642351566848q^{68} \) \(\mathstrut -\mathstrut 2703864574375502950215699413336064q^{72} \) \(\mathstrut +\mathstrut 6512963410113083479146903535884962q^{73} \) \(\mathstrut -\mathstrut 8354710661442671233627004873474048q^{74} \) \(\mathstrut -\mathstrut 19968482072584607784166883185393664q^{80} \) \(\mathstrut +\mathstrut 22528399544939174411840147874772641q^{81} \) \(\mathstrut -\mathstrut 37419906429311461896692392169832448q^{82} \) \(\mathstrut +\mathstrut 97784240079815282180431627621831012q^{85} \) \(\mathstrut +\mathstrut 75073270510125487261136916910657762q^{89} \) \(\mathstrut +\mathstrut 166375914940578119874091335185596416q^{90} \) \(\mathstrut -\mathstrut 918873327359871231451163027078417278q^{97} \) \(\mathstrut -\mathstrut 695135330857032999706040305346412544q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−262144. 0 6.87195e10 −4.22849e12 0 0 −1.80144e16 1.50095e17 1.10847e18
\(n\): 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 \(S_{37}^{\mathrm{new}}(4, [\chi])\).