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Properties

Label 144.2.c.a.143.2
Level $144$
Weight $2$
Character 144.143
Analytic conductor $1.150$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,2,Mod(143,144)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("144.143"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 144.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 143.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 144.143
Dual form 144.2.c.a.143.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.24264i q^{5} +4.00000 q^{13} -4.24264i q^{17} -13.0000 q^{25} -4.24264i q^{29} +2.00000 q^{37} -12.7279i q^{41} +7.00000 q^{49} +12.7279i q^{53} -10.0000 q^{61} +16.9706i q^{65} +16.0000 q^{73} +18.0000 q^{85} +4.24264i q^{89} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{13} - 26 q^{25} + 4 q^{37} + 14 q^{49} - 20 q^{61} + 32 q^{73} + 36 q^{85} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 4.24264i 1.89737i 0.316228 + 0.948683i \(0.397584\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.24264i − 1.02899i −0.857493 − 0.514496i \(-0.827979\pi\)
0.857493 − 0.514496i \(-0.172021\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −13.0000 −2.60000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.24264i − 0.787839i −0.919145 − 0.393919i \(-0.871119\pi\)
0.919145 − 0.393919i \(-0.128881\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 12.7279i − 1.98777i −0.110432 − 0.993884i \(-0.535223\pi\)
0.110432 − 0.993884i \(-0.464777\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.7279i 1.74831i 0.485643 + 0.874157i \(0.338586\pi\)
−0.485643 + 0.874157i \(0.661414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 16.9706i 2.10494i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 16.0000 1.87266 0.936329 − 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 18.0000 1.95237
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.24264i 0.449719i 0.974391 + 0.224860i \(0.0721923\pi\)
−0.974391 + 0.224860i \(0.927808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 144.2.c.a.143.2 yes 2
3.2 odd 2 inner 144.2.c.a.143.1 ✓ 2
4.3 odd 2 CM 144.2.c.a.143.2 yes 2
5.2 odd 4 3600.2.o.a.3599.2 4
5.3 odd 4 3600.2.o.a.3599.3 4
5.4 even 2 3600.2.h.b.1151.2 2
7.6 odd 2 7056.2.h.b.4607.1 2
8.3 odd 2 576.2.c.a.575.1 2
8.5 even 2 576.2.c.a.575.1 2
9.2 odd 6 1296.2.s.h.863.2 4
9.4 even 3 1296.2.s.h.431.2 4
9.5 odd 6 1296.2.s.h.431.1 4
9.7 even 3 1296.2.s.h.863.1 4
12.11 even 2 inner 144.2.c.a.143.1 ✓ 2
15.2 even 4 3600.2.o.a.3599.1 4
15.8 even 4 3600.2.o.a.3599.4 4
15.14 odd 2 3600.2.h.b.1151.1 2
16.3 odd 4 2304.2.f.f.1151.4 4
16.5 even 4 2304.2.f.f.1151.1 4
16.11 odd 4 2304.2.f.f.1151.1 4
16.13 even 4 2304.2.f.f.1151.4 4
20.3 even 4 3600.2.o.a.3599.3 4
20.7 even 4 3600.2.o.a.3599.2 4
20.19 odd 2 3600.2.h.b.1151.2 2
21.20 even 2 7056.2.h.b.4607.2 2
24.5 odd 2 576.2.c.a.575.2 2
24.11 even 2 576.2.c.a.575.2 2
28.27 even 2 7056.2.h.b.4607.1 2
36.7 odd 6 1296.2.s.h.863.1 4
36.11 even 6 1296.2.s.h.863.2 4
36.23 even 6 1296.2.s.h.431.1 4
36.31 odd 6 1296.2.s.h.431.2 4
48.5 odd 4 2304.2.f.f.1151.3 4
48.11 even 4 2304.2.f.f.1151.3 4
48.29 odd 4 2304.2.f.f.1151.2 4
48.35 even 4 2304.2.f.f.1151.2 4
60.23 odd 4 3600.2.o.a.3599.4 4
60.47 odd 4 3600.2.o.a.3599.1 4
60.59 even 2 3600.2.h.b.1151.1 2
84.83 odd 2 7056.2.h.b.4607.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.c.a.143.1 ✓ 2 3.2 odd 2 inner
144.2.c.a.143.1 ✓ 2 12.11 even 2 inner
144.2.c.a.143.2 yes 2 1.1 even 1 trivial
144.2.c.a.143.2 yes 2 4.3 odd 2 CM
576.2.c.a.575.1 2 8.3 odd 2
576.2.c.a.575.1 2 8.5 even 2
576.2.c.a.575.2 2 24.5 odd 2
576.2.c.a.575.2 2 24.11 even 2
1296.2.s.h.431.1 4 9.5 odd 6
1296.2.s.h.431.1 4 36.23 even 6
1296.2.s.h.431.2 4 9.4 even 3
1296.2.s.h.431.2 4 36.31 odd 6
1296.2.s.h.863.1 4 9.7 even 3
1296.2.s.h.863.1 4 36.7 odd 6
1296.2.s.h.863.2 4 9.2 odd 6
1296.2.s.h.863.2 4 36.11 even 6
2304.2.f.f.1151.1 4 16.5 even 4
2304.2.f.f.1151.1 4 16.11 odd 4
2304.2.f.f.1151.2 4 48.29 odd 4
2304.2.f.f.1151.2 4 48.35 even 4
2304.2.f.f.1151.3 4 48.5 odd 4
2304.2.f.f.1151.3 4 48.11 even 4
2304.2.f.f.1151.4 4 16.3 odd 4
2304.2.f.f.1151.4 4 16.13 even 4
3600.2.h.b.1151.1 2 15.14 odd 2
3600.2.h.b.1151.1 2 60.59 even 2
3600.2.h.b.1151.2 2 5.4 even 2
3600.2.h.b.1151.2 2 20.19 odd 2
3600.2.o.a.3599.1 4 15.2 even 4
3600.2.o.a.3599.1 4 60.47 odd 4
3600.2.o.a.3599.2 4 5.2 odd 4
3600.2.o.a.3599.2 4 20.7 even 4
3600.2.o.a.3599.3 4 5.3 odd 4
3600.2.o.a.3599.3 4 20.3 even 4
3600.2.o.a.3599.4 4 15.8 even 4
3600.2.o.a.3599.4 4 60.23 odd 4
7056.2.h.b.4607.1 2 7.6 odd 2
7056.2.h.b.4607.1 2 28.27 even 2
7056.2.h.b.4607.2 2 21.20 even 2
7056.2.h.b.4607.2 2 84.83 odd 2