Properties

Label 144.2.k.a
Level 144
Weight 2
Character orbit 144.k
Analytic conductor 1.150
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 144.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + i ) q^{2} + 2 i q^{4} + ( 1 + i ) q^{5} -2 i q^{7} + ( -2 + 2 i ) q^{8} +O(q^{10})\) \( q + ( 1 + i ) q^{2} + 2 i q^{4} + ( 1 + i ) q^{5} -2 i q^{7} + ( -2 + 2 i ) q^{8} + 2 i q^{10} + ( -1 - i ) q^{11} + ( -1 + i ) q^{13} + ( 2 - 2 i ) q^{14} -4 q^{16} + 2 q^{17} + ( 3 - 3 i ) q^{19} + ( -2 + 2 i ) q^{20} -2 i q^{22} -6 i q^{23} -3 i q^{25} -2 q^{26} + 4 q^{28} + ( -3 + 3 i ) q^{29} -8 q^{31} + ( -4 - 4 i ) q^{32} + ( 2 + 2 i ) q^{34} + ( 2 - 2 i ) q^{35} + ( 3 + 3 i ) q^{37} + 6 q^{38} -4 q^{40} + ( 5 + 5 i ) q^{43} + ( 2 - 2 i ) q^{44} + ( 6 - 6 i ) q^{46} -8 q^{47} + 3 q^{49} + ( 3 - 3 i ) q^{50} + ( -2 - 2 i ) q^{52} + ( 5 + 5 i ) q^{53} -2 i q^{55} + ( 4 + 4 i ) q^{56} -6 q^{58} + ( 3 + 3 i ) q^{59} + ( -9 + 9 i ) q^{61} + ( -8 - 8 i ) q^{62} -8 i q^{64} -2 q^{65} + ( -5 + 5 i ) q^{67} + 4 i q^{68} + 4 q^{70} + 10 i q^{71} -4 i q^{73} + 6 i q^{74} + ( 6 + 6 i ) q^{76} + ( -2 + 2 i ) q^{77} + ( -4 - 4 i ) q^{80} + ( 1 - i ) q^{83} + ( 2 + 2 i ) q^{85} + 10 i q^{86} + 4 q^{88} -4 i q^{89} + ( 2 + 2 i ) q^{91} + 12 q^{92} + ( -8 - 8 i ) q^{94} + 6 q^{95} -2 q^{97} + ( 3 + 3 i ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{5} - 4q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{5} - 4q^{8} - 2q^{11} - 2q^{13} + 4q^{14} - 8q^{16} + 4q^{17} + 6q^{19} - 4q^{20} - 4q^{26} + 8q^{28} - 6q^{29} - 16q^{31} - 8q^{32} + 4q^{34} + 4q^{35} + 6q^{37} + 12q^{38} - 8q^{40} + 10q^{43} + 4q^{44} + 12q^{46} - 16q^{47} + 6q^{49} + 6q^{50} - 4q^{52} + 10q^{53} + 8q^{56} - 12q^{58} + 6q^{59} - 18q^{61} - 16q^{62} - 4q^{65} - 10q^{67} + 8q^{70} + 12q^{76} - 4q^{77} - 8q^{80} + 2q^{83} + 4q^{85} + 8q^{88} + 4q^{91} + 24q^{92} - 16q^{94} + 12q^{95} - 4q^{97} + 6q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(i\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
1.00000i
1.00000i
1.00000 + 1.00000i 0 2.00000i 1.00000 + 1.00000i 0 2.00000i −2.00000 + 2.00000i 0 2.00000i
109.1 1.00000 1.00000i 0 2.00000i 1.00000 1.00000i 0 2.00000i −2.00000 2.00000i 0 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.k.a 2
3.b odd 2 1 16.2.e.a 2
4.b odd 2 1 576.2.k.a 2
8.b even 2 1 1152.2.k.b 2
8.d odd 2 1 1152.2.k.a 2
12.b even 2 1 64.2.e.a 2
15.d odd 2 1 400.2.l.c 2
15.e even 4 1 400.2.q.a 2
15.e even 4 1 400.2.q.b 2
16.e even 4 1 inner 144.2.k.a 2
16.e even 4 1 1152.2.k.b 2
16.f odd 4 1 576.2.k.a 2
16.f odd 4 1 1152.2.k.a 2
21.c even 2 1 784.2.m.b 2
21.g even 6 2 784.2.x.c 4
21.h odd 6 2 784.2.x.f 4
24.f even 2 1 128.2.e.a 2
24.h odd 2 1 128.2.e.b 2
32.g even 8 2 9216.2.a.d 2
32.h odd 8 2 9216.2.a.s 2
48.i odd 4 1 16.2.e.a 2
48.i odd 4 1 128.2.e.b 2
48.k even 4 1 64.2.e.a 2
48.k even 4 1 128.2.e.a 2
60.h even 2 1 1600.2.l.a 2
60.l odd 4 1 1600.2.q.a 2
60.l odd 4 1 1600.2.q.b 2
96.o even 8 2 1024.2.a.e 2
96.o even 8 2 1024.2.b.b 2
96.p odd 8 2 1024.2.a.b 2
96.p odd 8 2 1024.2.b.e 2
240.t even 4 1 1600.2.l.a 2
240.z odd 4 1 1600.2.q.a 2
240.bb even 4 1 400.2.q.b 2
240.bd odd 4 1 1600.2.q.b 2
240.bf even 4 1 400.2.q.a 2
240.bm odd 4 1 400.2.l.c 2
336.y even 4 1 784.2.m.b 2
336.bo even 12 2 784.2.x.c 4
336.bt odd 12 2 784.2.x.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 3.b odd 2 1
16.2.e.a 2 48.i odd 4 1
64.2.e.a 2 12.b even 2 1
64.2.e.a 2 48.k even 4 1
128.2.e.a 2 24.f even 2 1
128.2.e.a 2 48.k even 4 1
128.2.e.b 2 24.h odd 2 1
128.2.e.b 2 48.i odd 4 1
144.2.k.a 2 1.a even 1 1 trivial
144.2.k.a 2 16.e even 4 1 inner
400.2.l.c 2 15.d odd 2 1
400.2.l.c 2 240.bm odd 4 1
400.2.q.a 2 15.e even 4 1
400.2.q.a 2 240.bf even 4 1
400.2.q.b 2 15.e even 4 1
400.2.q.b 2 240.bb even 4 1
576.2.k.a 2 4.b odd 2 1
576.2.k.a 2 16.f odd 4 1
784.2.m.b 2 21.c even 2 1
784.2.m.b 2 336.y even 4 1
784.2.x.c 4 21.g even 6 2
784.2.x.c 4 336.bo even 12 2
784.2.x.f 4 21.h odd 6 2
784.2.x.f 4 336.bt odd 12 2
1024.2.a.b 2 96.p odd 8 2
1024.2.a.e 2 96.o even 8 2
1024.2.b.b 2 96.o even 8 2
1024.2.b.e 2 96.p odd 8 2
1152.2.k.a 2 8.d odd 2 1
1152.2.k.a 2 16.f odd 4 1
1152.2.k.b 2 8.b even 2 1
1152.2.k.b 2 16.e even 4 1
1600.2.l.a 2 60.h even 2 1
1600.2.l.a 2 240.t even 4 1
1600.2.q.a 2 60.l odd 4 1
1600.2.q.a 2 240.z odd 4 1
1600.2.q.b 2 60.l odd 4 1
1600.2.q.b 2 240.bd odd 4 1
9216.2.a.d 2 32.g even 8 2
9216.2.a.s 2 32.h odd 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 2 T_{5} + 2 \) acting on \(S_{2}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 2 T^{2} \)
$3$ 1
$5$ \( ( 1 - 4 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} ) \)
$7$ \( 1 - 10 T^{2} + 49 T^{4} \)
$11$ \( 1 + 2 T + 2 T^{2} + 22 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \)
$17$ \( ( 1 - 2 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 6 T + 18 T^{2} - 114 T^{3} + 361 T^{4} \)
$23$ \( 1 - 10 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 4 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} ) \)
$31$ \( ( 1 + 8 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 41 T^{2} )^{2} \)
$43$ \( 1 - 10 T + 50 T^{2} - 430 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 + 8 T + 47 T^{2} )^{2} \)
$53$ \( ( 1 - 14 T + 53 T^{2} )( 1 + 4 T + 53 T^{2} ) \)
$59$ \( 1 - 6 T + 18 T^{2} - 354 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 18 T + 162 T^{2} + 1098 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 10 T + 50 T^{2} + 670 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 42 T^{2} + 5041 T^{4} \)
$73$ \( 1 - 130 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 79 T^{2} )^{2} \)
$83$ \( 1 - 2 T + 2 T^{2} - 166 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 162 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 + 2 T + 97 T^{2} )^{2} \)
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