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 discriminant -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 \(2\), degree \(1\), minimal)

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: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

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
−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 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 4.37.b.a 1
4.b odd 2 1 CM 4.37.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.37.b.a 1 1.a even 1 1 trivial
4.37.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 \(S_{37}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 262144 T \)
$3$ \( ( 1 - 387420489 T )( 1 + 387420489 T ) \)
$5$ \( 1 + 4228490555534 T + \)\(14\!\cdots\!25\)\( T^{2} \)
$7$ \( ( 1 - 1628413597910449 T )( 1 + 1628413597910449 T ) \)
$11$ \( ( 1 - 5559917313492231481 T )( 1 + 5559917313492231481 T ) \)
$13$ \( 1 + \)\(15\!\cdots\!58\)\( T + \)\(12\!\cdots\!41\)\( T^{2} \)
$17$ \( 1 + \)\(23\!\cdots\!18\)\( T + \)\(19\!\cdots\!81\)\( T^{2} \)
$19$ \( ( 1 - \)\(10\!\cdots\!41\)\( T )( 1 + \)\(10\!\cdots\!41\)\( T ) \)
$23$ \( ( 1 - \)\(32\!\cdots\!69\)\( T )( 1 + \)\(32\!\cdots\!69\)\( T ) \)
$29$ \( 1 - \)\(17\!\cdots\!22\)\( T + \)\(44\!\cdots\!21\)\( T^{2} \)
$31$ \( ( 1 - \)\(69\!\cdots\!41\)\( T )( 1 + \)\(69\!\cdots\!41\)\( T ) \)
$37$ \( 1 - \)\(31\!\cdots\!42\)\( T + \)\(28\!\cdots\!41\)\( T^{2} \)
$41$ \( 1 - \)\(14\!\cdots\!42\)\( T + \)\(11\!\cdots\!41\)\( T^{2} \)
$43$ \( ( 1 - \)\(25\!\cdots\!49\)\( T )( 1 + \)\(25\!\cdots\!49\)\( T ) \)
$47$ \( ( 1 - \)\(12\!\cdots\!89\)\( T )( 1 + \)\(12\!\cdots\!89\)\( T ) \)
$53$ \( 1 + \)\(18\!\cdots\!78\)\( T + \)\(11\!\cdots\!21\)\( T^{2} \)
$59$ \( ( 1 - \)\(75\!\cdots\!21\)\( T )( 1 + \)\(75\!\cdots\!21\)\( T ) \)
$61$ \( 1 - \)\(27\!\cdots\!62\)\( T + \)\(18\!\cdots\!61\)\( T^{2} \)
$67$ \( ( 1 - \)\(74\!\cdots\!09\)\( T )( 1 + \)\(74\!\cdots\!09\)\( T ) \)
$71$ \( ( 1 - \)\(21\!\cdots\!61\)\( T )( 1 + \)\(21\!\cdots\!61\)\( T ) \)
$73$ \( 1 - \)\(65\!\cdots\!62\)\( T + \)\(12\!\cdots\!61\)\( T^{2} \)
$79$ \( ( 1 - \)\(14\!\cdots\!61\)\( T )( 1 + \)\(14\!\cdots\!61\)\( T ) \)
$83$ \( ( 1 - \)\(34\!\cdots\!09\)\( T )( 1 + \)\(34\!\cdots\!09\)\( T ) \)
$89$ \( 1 - \)\(75\!\cdots\!62\)\( T + \)\(15\!\cdots\!61\)\( T^{2} \)
$97$ \( 1 + \)\(91\!\cdots\!78\)\( T + \)\(33\!\cdots\!21\)\( T^{2} \)
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