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 discriminant -3
Inner twists 2

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Newspace parameters

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

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

$q$-expansion

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.37.b.a 1
3.b odd 2 1 CM 3.37.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.37.b.a 1 1.a even 1 1 trivial
3.37.b.a 1 3.b odd 2 1 CM

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 262144 T )( 1 + 262144 T ) \)
$3$ \( 1 - 387420489 T \)
$5$ \( ( 1 - 3814697265625 T )( 1 + 3814697265625 T ) \)
$7$ \( 1 - 2757049053441698 T + \)\(26\!\cdots\!01\)\( T^{2} \)
$11$ \( ( 1 - 5559917313492231481 T )( 1 + 5559917313492231481 T ) \)
$13$ \( 1 + \)\(17\!\cdots\!42\)\( T + \)\(12\!\cdots\!41\)\( T^{2} \)
$17$ \( ( 1 - \)\(14\!\cdots\!09\)\( T )( 1 + \)\(14\!\cdots\!09\)\( T ) \)
$19$ \( 1 + \)\(11\!\cdots\!18\)\( T + \)\(10\!\cdots\!81\)\( T^{2} \)
$23$ \( ( 1 - \)\(32\!\cdots\!69\)\( T )( 1 + \)\(32\!\cdots\!69\)\( T ) \)
$29$ \( ( 1 - \)\(21\!\cdots\!61\)\( T )( 1 + \)\(21\!\cdots\!61\)\( T ) \)
$31$ \( 1 - \)\(11\!\cdots\!82\)\( T + \)\(48\!\cdots\!81\)\( T^{2} \)
$37$ \( 1 + \)\(33\!\cdots\!42\)\( T + \)\(28\!\cdots\!41\)\( T^{2} \)
$41$ \( ( 1 - \)\(10\!\cdots\!21\)\( T )( 1 + \)\(10\!\cdots\!21\)\( T ) \)
$43$ \( 1 - \)\(28\!\cdots\!98\)\( T + \)\(63\!\cdots\!01\)\( T^{2} \)
$47$ \( ( 1 - \)\(12\!\cdots\!89\)\( T )( 1 + \)\(12\!\cdots\!89\)\( T ) \)
$53$ \( ( 1 - \)\(10\!\cdots\!89\)\( T )( 1 + \)\(10\!\cdots\!89\)\( T ) \)
$59$ \( ( 1 - \)\(75\!\cdots\!21\)\( T )( 1 + \)\(75\!\cdots\!21\)\( T ) \)
$61$ \( 1 + \)\(18\!\cdots\!38\)\( T + \)\(18\!\cdots\!61\)\( T^{2} \)
$67$ \( 1 - \)\(26\!\cdots\!18\)\( T + \)\(54\!\cdots\!81\)\( T^{2} \)
$71$ \( ( 1 - \)\(21\!\cdots\!61\)\( T )( 1 + \)\(21\!\cdots\!61\)\( T ) \)
$73$ \( 1 + \)\(60\!\cdots\!62\)\( T + \)\(12\!\cdots\!61\)\( T^{2} \)
$79$ \( 1 + \)\(89\!\cdots\!78\)\( T + \)\(20\!\cdots\!21\)\( T^{2} \)
$83$ \( ( 1 - \)\(34\!\cdots\!09\)\( T )( 1 + \)\(34\!\cdots\!09\)\( T ) \)
$89$ \( ( 1 - \)\(12\!\cdots\!81\)\( T )( 1 + \)\(12\!\cdots\!81\)\( T ) \)
$97$ \( 1 + \)\(11\!\cdots\!22\)\( T + \)\(33\!\cdots\!21\)\( T^{2} \)
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