↑

Properties

Label 48.2.c.a.47.2
Level $48$
Weight $2$
Character 48.47
Analytic conductor $0.383$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [48,2,Mod(47,48)] 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("48.47"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(48, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 48.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: \(0.383281929702\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 47.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 48.47
Dual form 48.2.c.a.47.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} -3.46410i q^{7} -3.00000 q^{9} -2.00000 q^{13} +3.46410i q^{19} +6.00000 q^{21} +5.00000 q^{25} -5.19615i q^{27} +10.3923i q^{31} -10.0000 q^{37} -3.46410i q^{39} -10.3923i q^{43} -5.00000 q^{49} -6.00000 q^{57} +14.0000 q^{61} +10.3923i q^{63} +3.46410i q^{67} +10.0000 q^{73} +8.66025i q^{75} -17.3205i q^{79} +9.00000 q^{81} +6.92820i q^{91} -18.0000 q^{93} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{9} - 4 q^{13} + 12 q^{21} + 10 q^{25} - 20 q^{37} - 10 q^{49} - 12 q^{57} + 28 q^{61} + 20 q^{73} + 18 q^{81} - 36 q^{93} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 1.73205i 1.00000i
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) − 3.46410i − 1.30931i −0.755929 − 0.654654i \(-0.772814\pi\)
0.755929 − 0.654654i \(-0.227186\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 3.46410i 0.794719i 0.917663 + 0.397360i \(0.130073\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) − 5.19615i − 1.00000i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 10.3923i 1.86651i 0.359211 + 0.933257i \(0.383046\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) − 3.46410i − 0.554700i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) − 10.3923i − 1.58481i −0.609994 − 0.792406i \(-0.708828\pi\)
0.609994 − 0.792406i \(-0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 10.3923i 1.30931i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.46410i 0.423207i 0.977356 + 0.211604i \(0.0678686\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 8.66025i 1.00000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) − 17.3205i − 1.94871i −0.225018 − 0.974355i \(-0.572244\pi\)
0.225018 − 0.974355i \(-0.427756\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 6.92820i 0.726273i
\(92\) 0 0
\(93\) −18.0000 −1.86651
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 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 48.2.c.a.47.2 yes 2
3.2 odd 2 CM 48.2.c.a.47.2 yes 2
4.3 odd 2 inner 48.2.c.a.47.1 ✓ 2
5.2 odd 4 1200.2.o.i.1199.4 4
5.3 odd 4 1200.2.o.i.1199.1 4
5.4 even 2 1200.2.h.e.1151.1 2
7.6 odd 2 2352.2.h.c.2255.1 2
8.3 odd 2 192.2.c.a.191.2 2
8.5 even 2 192.2.c.a.191.1 2
9.2 odd 6 1296.2.s.b.863.1 2
9.4 even 3 1296.2.s.e.431.1 2
9.5 odd 6 1296.2.s.e.431.1 2
9.7 even 3 1296.2.s.b.863.1 2
12.11 even 2 inner 48.2.c.a.47.1 ✓ 2
15.2 even 4 1200.2.o.i.1199.4 4
15.8 even 4 1200.2.o.i.1199.1 4
15.14 odd 2 1200.2.h.e.1151.1 2
16.3 odd 4 768.2.f.d.383.3 4
16.5 even 4 768.2.f.d.383.4 4
16.11 odd 4 768.2.f.d.383.1 4
16.13 even 4 768.2.f.d.383.2 4
20.3 even 4 1200.2.o.i.1199.3 4
20.7 even 4 1200.2.o.i.1199.2 4
20.19 odd 2 1200.2.h.e.1151.2 2
21.20 even 2 2352.2.h.c.2255.1 2
24.5 odd 2 192.2.c.a.191.1 2
24.11 even 2 192.2.c.a.191.2 2
28.27 even 2 2352.2.h.c.2255.2 2
36.7 odd 6 1296.2.s.e.863.1 2
36.11 even 6 1296.2.s.e.863.1 2
36.23 even 6 1296.2.s.b.431.1 2
36.31 odd 6 1296.2.s.b.431.1 2
48.5 odd 4 768.2.f.d.383.4 4
48.11 even 4 768.2.f.d.383.1 4
48.29 odd 4 768.2.f.d.383.2 4
48.35 even 4 768.2.f.d.383.3 4
60.23 odd 4 1200.2.o.i.1199.3 4
60.47 odd 4 1200.2.o.i.1199.2 4
60.59 even 2 1200.2.h.e.1151.2 2
84.83 odd 2 2352.2.h.c.2255.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.c.a.47.1 ✓ 2 4.3 odd 2 inner
48.2.c.a.47.1 ✓ 2 12.11 even 2 inner
48.2.c.a.47.2 yes 2 1.1 even 1 trivial
48.2.c.a.47.2 yes 2 3.2 odd 2 CM
192.2.c.a.191.1 2 8.5 even 2
192.2.c.a.191.1 2 24.5 odd 2
192.2.c.a.191.2 2 8.3 odd 2
192.2.c.a.191.2 2 24.11 even 2
768.2.f.d.383.1 4 16.11 odd 4
768.2.f.d.383.1 4 48.11 even 4
768.2.f.d.383.2 4 16.13 even 4
768.2.f.d.383.2 4 48.29 odd 4
768.2.f.d.383.3 4 16.3 odd 4
768.2.f.d.383.3 4 48.35 even 4
768.2.f.d.383.4 4 16.5 even 4
768.2.f.d.383.4 4 48.5 odd 4
1200.2.h.e.1151.1 2 5.4 even 2
1200.2.h.e.1151.1 2 15.14 odd 2
1200.2.h.e.1151.2 2 20.19 odd 2
1200.2.h.e.1151.2 2 60.59 even 2
1200.2.o.i.1199.1 4 5.3 odd 4
1200.2.o.i.1199.1 4 15.8 even 4
1200.2.o.i.1199.2 4 20.7 even 4
1200.2.o.i.1199.2 4 60.47 odd 4
1200.2.o.i.1199.3 4 20.3 even 4
1200.2.o.i.1199.3 4 60.23 odd 4
1200.2.o.i.1199.4 4 5.2 odd 4
1200.2.o.i.1199.4 4 15.2 even 4
1296.2.s.b.431.1 2 36.23 even 6
1296.2.s.b.431.1 2 36.31 odd 6
1296.2.s.b.863.1 2 9.2 odd 6
1296.2.s.b.863.1 2 9.7 even 3
1296.2.s.e.431.1 2 9.4 even 3
1296.2.s.e.431.1 2 9.5 odd 6
1296.2.s.e.863.1 2 36.7 odd 6
1296.2.s.e.863.1 2 36.11 even 6
2352.2.h.c.2255.1 2 7.6 odd 2
2352.2.h.c.2255.1 2 21.20 even 2
2352.2.h.c.2255.2 2 28.27 even 2
2352.2.h.c.2255.2 2 84.83 odd 2