Properties

Label 144.6.a.j
Level 144
Weight 6
Character orbit 144.a
Self dual yes
Analytic conductor 23.095
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 144.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.0952700531\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 66q^{5} - 176q^{7} + O(q^{10}) \) \( q + 66q^{5} - 176q^{7} - 60q^{11} - 658q^{13} + 414q^{17} - 956q^{19} + 600q^{23} + 1231q^{25} - 5574q^{29} + 3592q^{31} - 11616q^{35} - 8458q^{37} - 19194q^{41} - 13316q^{43} - 19680q^{47} + 14169q^{49} + 31266q^{53} - 3960q^{55} + 26340q^{59} - 31090q^{61} - 43428q^{65} + 16804q^{67} + 6120q^{71} - 25558q^{73} + 10560q^{77} - 74408q^{79} - 6468q^{83} + 27324q^{85} + 32742q^{89} + 115808q^{91} - 63096q^{95} + 166082q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 66.0000 0 −176.000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.6.a.j 1
3.b odd 2 1 48.6.a.c 1
4.b odd 2 1 18.6.a.b 1
8.b even 2 1 576.6.a.i 1
8.d odd 2 1 576.6.a.j 1
12.b even 2 1 6.6.a.a 1
20.d odd 2 1 450.6.a.m 1
20.e even 4 2 450.6.c.j 2
24.f even 2 1 192.6.a.o 1
24.h odd 2 1 192.6.a.g 1
28.d even 2 1 882.6.a.a 1
36.f odd 6 2 162.6.c.h 2
36.h even 6 2 162.6.c.e 2
48.i odd 4 2 768.6.d.p 2
48.k even 4 2 768.6.d.c 2
60.h even 2 1 150.6.a.d 1
60.l odd 4 2 150.6.c.b 2
84.h odd 2 1 294.6.a.m 1
84.j odd 6 2 294.6.e.a 2
84.n even 6 2 294.6.e.g 2
132.d odd 2 1 726.6.a.a 1
156.h even 2 1 1014.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.6.a.a 1 12.b even 2 1
18.6.a.b 1 4.b odd 2 1
48.6.a.c 1 3.b odd 2 1
144.6.a.j 1 1.a even 1 1 trivial
150.6.a.d 1 60.h even 2 1
150.6.c.b 2 60.l odd 4 2
162.6.c.e 2 36.h even 6 2
162.6.c.h 2 36.f odd 6 2
192.6.a.g 1 24.h odd 2 1
192.6.a.o 1 24.f even 2 1
294.6.a.m 1 84.h odd 2 1
294.6.e.a 2 84.j odd 6 2
294.6.e.g 2 84.n even 6 2
450.6.a.m 1 20.d odd 2 1
450.6.c.j 2 20.e even 4 2
576.6.a.i 1 8.b even 2 1
576.6.a.j 1 8.d odd 2 1
726.6.a.a 1 132.d odd 2 1
768.6.d.c 2 48.k even 4 2
768.6.d.p 2 48.i odd 4 2
882.6.a.a 1 28.d even 2 1
1014.6.a.c 1 156.h even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 66 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(144))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 66 T + 3125 T^{2} \)
$7$ \( 1 + 176 T + 16807 T^{2} \)
$11$ \( 1 + 60 T + 161051 T^{2} \)
$13$ \( 1 + 658 T + 371293 T^{2} \)
$17$ \( 1 - 414 T + 1419857 T^{2} \)
$19$ \( 1 + 956 T + 2476099 T^{2} \)
$23$ \( 1 - 600 T + 6436343 T^{2} \)
$29$ \( 1 + 5574 T + 20511149 T^{2} \)
$31$ \( 1 - 3592 T + 28629151 T^{2} \)
$37$ \( 1 + 8458 T + 69343957 T^{2} \)
$41$ \( 1 + 19194 T + 115856201 T^{2} \)
$43$ \( 1 + 13316 T + 147008443 T^{2} \)
$47$ \( 1 + 19680 T + 229345007 T^{2} \)
$53$ \( 1 - 31266 T + 418195493 T^{2} \)
$59$ \( 1 - 26340 T + 714924299 T^{2} \)
$61$ \( 1 + 31090 T + 844596301 T^{2} \)
$67$ \( 1 - 16804 T + 1350125107 T^{2} \)
$71$ \( 1 - 6120 T + 1804229351 T^{2} \)
$73$ \( 1 + 25558 T + 2073071593 T^{2} \)
$79$ \( 1 + 74408 T + 3077056399 T^{2} \)
$83$ \( 1 + 6468 T + 3939040643 T^{2} \)
$89$ \( 1 - 32742 T + 5584059449 T^{2} \)
$97$ \( 1 - 166082 T + 8587340257 T^{2} \)
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