Properties

Label 144.2.x.b
Level $144$
Weight $2$
Character orbit 144.x
Analytic conductor $1.150$
Analytic rank $1$
Dimension $4$
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.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.14984578911\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{2} + (\zeta_{12}^{2} - 2) q^{3} - 2 \zeta_{12} q^{4} + (2 \zeta_{12}^{3} - 2 \zeta_{12} - 2) q^{5} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} + 1) q^{6} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 2) q^{7} + (2 \zeta_{12}^{3} - 2) q^{8} + ( - 3 \zeta_{12}^{2} + 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{2} + (\zeta_{12}^{2} - 2) q^{3} - 2 \zeta_{12} q^{4} + (2 \zeta_{12}^{3} - 2 \zeta_{12} - 2) q^{5} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} + 1) q^{6} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 2) q^{7} + (2 \zeta_{12}^{3} - 2) q^{8} + ( - 3 \zeta_{12}^{2} + 3) q^{9} + (2 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{10} + (\zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12} - 3) q^{11} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{12} + ( - 4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12} - 1) q^{13} + (2 \zeta_{12}^{3} - 2 \zeta_{12} + 2) q^{14} + ( - 4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 4) q^{15} + 4 \zeta_{12}^{2} q^{16} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 4) q^{17} + ( - 3 \zeta_{12}^{3} - 3) q^{18} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{19} + (4 \zeta_{12} + 4) q^{20} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - \zeta_{12} + 3) q^{21} + (3 \zeta_{12}^{2} - \zeta_{12} + 3) q^{22} + (\zeta_{12}^{2} - 3 \zeta_{12} + 1) q^{23} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{24} + ( - 3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 3 \zeta_{12} + 8) q^{25} + (4 \zeta_{12}^{3} - 8 \zeta_{12} - 2) q^{26} + (6 \zeta_{12}^{2} - 3) q^{27} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 2) q^{28} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} - 2) q^{29} + ( - 2 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 2 \zeta_{12}) q^{30} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{31} + ( - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{32} + (\zeta_{12}^{3} - 4 \zeta_{12}^{2} - 5 \zeta_{12} + 5) q^{33} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 5 \zeta_{12} + 2) q^{34} + ( - 2 \zeta_{12}^{3} + 2) q^{35} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{36} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{37} + ( - \zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12}) q^{38} + (5 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} + 5) q^{39} + ( - 8 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{40} - 3 \zeta_{12} q^{41} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{42} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 5 \zeta_{12} - 3) q^{43} + ( - 2 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 6 \zeta_{12} + 2) q^{44} + (6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6) q^{45} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{46} + ( - 5 \zeta_{12}^{3} - \zeta_{12}^{2} - 5 \zeta_{12}) q^{47} + ( - 4 \zeta_{12}^{2} - 4) q^{48} + (4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 2 \zeta_{12} - 3) q^{49} + ( - 8 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + \zeta_{12} - 1) q^{50} + (3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 3 \zeta_{12} + 8) q^{51} + (6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 2 \zeta_{12} - 8) q^{52} + ( - 6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12} - 6) q^{53} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 6) q^{54} + ( - 8 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{55} + ( - 4 \zeta_{12} + 4) q^{56} + ( - 3 \zeta_{12} + 3) q^{57} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{58} + (\zeta_{12}^{3} - 5 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{59} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 8 \zeta_{12} - 8) q^{60} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6) q^{61} + (2 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12} + 2) q^{62} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{63} - 8 \zeta_{12}^{3} q^{64} + (6 \zeta_{12}^{3} + 14 \zeta_{12}^{2} + 6 \zeta_{12}) q^{65} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 9) q^{66} + (5 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{67} + ( - 2 \zeta_{12}^{2} + 8 \zeta_{12} - 2) q^{68} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12} - 3) q^{69} - 4 \zeta_{12}^{2} q^{70} + (4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{71} + (6 \zeta_{12}^{2} + 6 \zeta_{12} - 6) q^{72} + (8 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{73} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 6 \zeta_{12} - 4) q^{74} + (6 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 3 \zeta_{12} - 12) q^{75} + (2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{76} + (4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 7 \zeta_{12} + 7) q^{77} + ( - 12 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 12 \zeta_{12} + 4) q^{78} - 12 \zeta_{12}^{2} q^{79} + ( - 8 \zeta_{12}^{2} - 8 \zeta_{12}) q^{80} - 9 \zeta_{12}^{2} q^{81} + (3 \zeta_{12}^{3} - 3) q^{82} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{83} + (6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 6 \zeta_{12} + 4) q^{84} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{85} + (3 \zeta_{12}^{2} - 7 \zeta_{12} + 3) q^{86} + ( - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{87} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 6 \zeta_{12}) q^{88} + 2 \zeta_{12}^{3} q^{89} + (6 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{90} + (6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 6) q^{91} + ( - 2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 2 \zeta_{12}) q^{92} + ( - 6 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{93} + (10 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 6 \zeta_{12} - 6) q^{94} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{95} + (4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 8 \zeta_{12} - 4) q^{96} + (\zeta_{12}^{3} - 10 \zeta_{12}^{2} + \zeta_{12}) q^{97} + (\zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12} + 1) q^{98} + ( - 6 \zeta_{12}^{3} + 9 \zeta_{12}^{2} + 9 \zeta_{12} - 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 6 q^{3} - 8 q^{5} + 6 q^{6} - 6 q^{7} - 8 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 6 q^{3} - 8 q^{5} + 6 q^{6} - 6 q^{7} - 8 q^{8} + 6 q^{9} - 10 q^{11} - 10 q^{13} + 8 q^{14} + 12 q^{15} + 8 q^{16} - 16 q^{17} - 12 q^{18} - 6 q^{19} + 16 q^{20} + 6 q^{21} + 18 q^{22} + 6 q^{23} + 12 q^{24} + 24 q^{25} - 8 q^{26} - 8 q^{28} - 6 q^{29} - 12 q^{30} - 8 q^{31} + 8 q^{32} + 12 q^{33} + 14 q^{34} + 8 q^{35} + 12 q^{37} + 6 q^{38} + 24 q^{39} + 8 q^{40} - 12 q^{42} - 16 q^{43} - 4 q^{44} - 12 q^{45} - 12 q^{46} - 2 q^{47} - 24 q^{48} - 6 q^{49} - 18 q^{50} + 24 q^{51} - 20 q^{52} - 16 q^{53} + 18 q^{54} + 16 q^{56} + 12 q^{57} + 6 q^{59} - 24 q^{60} - 12 q^{61} + 16 q^{62} + 28 q^{65} - 36 q^{66} + 16 q^{67} - 12 q^{68} - 12 q^{69} - 8 q^{70} - 12 q^{72} - 12 q^{74} - 24 q^{75} + 22 q^{77} + 12 q^{78} - 24 q^{79} - 16 q^{80} - 18 q^{81} - 12 q^{82} - 2 q^{83} + 12 q^{84} + 20 q^{85} + 18 q^{86} + 6 q^{87} + 4 q^{88} + 36 q^{90} + 20 q^{91} + 12 q^{92} - 32 q^{94} - 20 q^{97} + 12 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(-1 + \zeta_{12}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
−1.36603 0.366025i −1.50000 + 0.866025i 1.73205 + 1.00000i −0.267949 1.00000i 2.36603 0.633975i −2.36603 + 1.36603i −2.00000 2.00000i 1.50000 2.59808i 1.46410i
61.1 0.366025 + 1.36603i −1.50000 0.866025i −1.73205 + 1.00000i −3.73205 1.00000i 0.633975 2.36603i −0.633975 0.366025i −2.00000 2.00000i 1.50000 + 2.59808i 5.46410i
85.1 0.366025 1.36603i −1.50000 + 0.866025i −1.73205 1.00000i −3.73205 + 1.00000i 0.633975 + 2.36603i −0.633975 + 0.366025i −2.00000 + 2.00000i 1.50000 2.59808i 5.46410i
133.1 −1.36603 + 0.366025i −1.50000 0.866025i 1.73205 1.00000i −0.267949 + 1.00000i 2.36603 + 0.633975i −2.36603 1.36603i −2.00000 + 2.00000i 1.50000 + 2.59808i 1.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
144.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.x.b 4
3.b odd 2 1 432.2.y.c 4
4.b odd 2 1 576.2.bb.d 4
9.c even 3 1 144.2.x.c yes 4
9.d odd 6 1 432.2.y.b 4
12.b even 2 1 1728.2.bc.d 4
16.e even 4 1 144.2.x.c yes 4
16.f odd 4 1 576.2.bb.c 4
36.f odd 6 1 576.2.bb.c 4
36.h even 6 1 1728.2.bc.a 4
48.i odd 4 1 432.2.y.b 4
48.k even 4 1 1728.2.bc.a 4
144.u even 12 1 1728.2.bc.d 4
144.v odd 12 1 576.2.bb.d 4
144.w odd 12 1 432.2.y.c 4
144.x even 12 1 inner 144.2.x.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.x.b 4 1.a even 1 1 trivial
144.2.x.b 4 144.x even 12 1 inner
144.2.x.c yes 4 9.c even 3 1
144.2.x.c yes 4 16.e even 4 1
432.2.y.b 4 9.d odd 6 1
432.2.y.b 4 48.i odd 4 1
432.2.y.c 4 3.b odd 2 1
432.2.y.c 4 144.w odd 12 1
576.2.bb.c 4 16.f odd 4 1
576.2.bb.c 4 36.f odd 6 1
576.2.bb.d 4 4.b odd 2 1
576.2.bb.d 4 144.v odd 12 1
1728.2.bc.a 4 36.h even 6 1
1728.2.bc.a 4 48.k even 4 1
1728.2.bc.d 4 12.b even 2 1
1728.2.bc.d 4 144.u even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 8T_{5}^{3} + 20T_{5}^{2} + 16T_{5} + 16 \) acting on \(S_{2}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 8 T^{3} + 20 T^{2} + 16 T + 16 \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} + 14 T^{2} + 12 T + 4 \) Copy content Toggle raw display
$11$ \( T^{4} + 10 T^{3} + 41 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{4} + 10 T^{3} + 74 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$17$ \( (T^{2} + 8 T + 13)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + 18 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} + 6 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + 18 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$31$ \( T^{4} + 8 T^{3} + 60 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$37$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$41$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$43$ \( T^{4} + 16 T^{3} + 65 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$47$ \( T^{4} + 2 T^{3} + 78 T^{2} + \cdots + 5476 \) Copy content Toggle raw display
$53$ \( T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} + 45 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$61$ \( T^{4} + 12 T^{3} + 180 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$67$ \( T^{4} - 16 T^{3} + 113 T^{2} + \cdots + 1369 \) Copy content Toggle raw display
$71$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$73$ \( T^{4} + 134T^{2} + 3721 \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T + 144)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$89$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 20 T^{3} + 303 T^{2} + \cdots + 9409 \) Copy content Toggle raw display
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