Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [24,2,Mod(11,24)] 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("24.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(24, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 24.f (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: \(0.191640964851\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 11.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 24.11
Dual form 24.2.f.a.11.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} +(-1.00000 + 1.41421i) q^{3} -2.00000 q^{4} +(2.00000 + 1.41421i) q^{6} +2.82843i q^{8} +(-1.00000 - 2.82843i) q^{9} -2.82843i q^{11} +(2.00000 - 2.82843i) q^{12} +4.00000 q^{16} +5.65685i q^{17} +(-4.00000 + 1.41421i) q^{18} +2.00000 q^{19} -4.00000 q^{22} +(-4.00000 - 2.82843i) q^{24} -5.00000 q^{25} +(5.00000 + 1.41421i) q^{27} -5.65685i q^{32} +(4.00000 + 2.82843i) q^{33} +8.00000 q^{34} +(2.00000 + 5.65685i) q^{36} -2.82843i q^{38} -11.3137i q^{41} -10.0000 q^{43} +5.65685i q^{44} +(-4.00000 + 5.65685i) q^{48} +7.00000 q^{49} +7.07107i q^{50} +(-8.00000 - 5.65685i) q^{51} +(2.00000 - 7.07107i) q^{54} +(-2.00000 + 2.82843i) q^{57} +14.1421i q^{59} -8.00000 q^{64} +(4.00000 - 5.65685i) q^{66} +14.0000 q^{67} -11.3137i q^{68} +(8.00000 - 2.82843i) q^{72} +2.00000 q^{73} +(5.00000 - 7.07107i) q^{75} -4.00000 q^{76} +(-7.00000 + 5.65685i) q^{81} -16.0000 q^{82} -2.82843i q^{83} +14.1421i q^{86} +8.00000 q^{88} +5.65685i q^{89} +(8.00000 + 5.65685i) q^{96} -10.0000 q^{97} -9.89949i q^{98} +(-8.00000 + 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{4} + 4 q^{6} - 2 q^{9} + 4 q^{12} + 8 q^{16} - 8 q^{18} + 4 q^{19} - 8 q^{22} - 8 q^{24} - 10 q^{25} + 10 q^{27} + 8 q^{33} + 16 q^{34} + 4 q^{36} - 20 q^{43} - 8 q^{48} + 14 q^{49}+ \cdots - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(13\) \(17\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 1.00000i
\(3\) −1.00000 + 1.41421i −0.577350 + 0.816497i
\(4\) −2.00000 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 2.00000 + 1.41421i 0.816497 + 0.577350i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 2.82843i 1.00000i
\(9\) −1.00000 − 2.82843i −0.333333 − 0.942809i
\(10\) 0 0
\(11\) − 2.82843i − 0.852803i −0.904534 − 0.426401i \(-0.859781\pi\)
0.904534 − 0.426401i \(-0.140219\pi\)
\(12\) 2.00000 − 2.82843i 0.577350 − 0.816497i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 5.65685i 1.37199i 0.727607 + 0.685994i \(0.240633\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −4.00000 + 1.41421i −0.942809 + 0.333333i
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −4.00000 − 2.82843i −0.816497 − 0.577350i
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 5.00000 + 1.41421i 0.962250 + 0.272166i
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) − 5.65685i − 1.00000i
\(33\) 4.00000 + 2.82843i 0.696311 + 0.492366i
\(34\) 8.00000 1.37199
\(35\) 0 0
\(36\) 2.00000 + 5.65685i 0.333333 + 0.942809i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) − 2.82843i − 0.458831i
\(39\) 0 0
\(40\) 0 0
\(41\) − 11.3137i − 1.76690i −0.468521 − 0.883452i \(-0.655213\pi\)
0.468521 − 0.883452i \(-0.344787\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 5.65685i 0.852803i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −4.00000 + 5.65685i −0.577350 + 0.816497i
\(49\) 7.00000 1.00000
\(50\) 7.07107i 1.00000i
\(51\) −8.00000 − 5.65685i −1.12022 − 0.792118i
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 2.00000 − 7.07107i 0.272166 − 0.962250i
\(55\) 0 0
\(56\) 0 0
\(57\) −2.00000 + 2.82843i −0.264906 + 0.374634i
\(58\) 0 0
\(59\) 14.1421i 1.84115i 0.390567 + 0.920575i \(0.372279\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 4.00000 − 5.65685i 0.492366 − 0.696311i
\(67\) 14.0000 1.71037 0.855186 − 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) − 11.3137i − 1.37199i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 8.00000 − 2.82843i 0.942809 − 0.333333i
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 5.00000 − 7.07107i 0.577350 − 0.816497i
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −7.00000 + 5.65685i −0.777778 + 0.628539i
\(82\) −16.0000 −1.76690
\(83\) − 2.82843i − 0.310460i −0.987878 − 0.155230i \(-0.950388\pi\)
0.987878 − 0.155230i \(-0.0496119\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 14.1421i 1.52499i
\(87\) 0 0
\(88\) 8.00000 0.852803
\(89\) 5.65685i 0.599625i 0.953998 + 0.299813i \(0.0969242\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 8.00000 + 5.65685i 0.816497 + 0.577350i
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) − 9.89949i − 1.00000i
\(99\) −8.00000 + 2.82843i −0.804030 + 0.284268i
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 24.2.f.a.11.1 ✓ 2
3.2 odd 2 inner 24.2.f.a.11.2 yes 2
4.3 odd 2 96.2.f.a.47.1 2
5.2 odd 4 600.2.m.a.299.4 4
5.3 odd 4 600.2.m.a.299.1 4
5.4 even 2 600.2.b.a.251.2 2
8.3 odd 2 CM 24.2.f.a.11.1 ✓ 2
8.5 even 2 96.2.f.a.47.1 2
9.2 odd 6 648.2.l.b.539.2 4
9.4 even 3 648.2.l.b.107.2 4
9.5 odd 6 648.2.l.b.107.1 4
9.7 even 3 648.2.l.b.539.1 4
12.11 even 2 96.2.f.a.47.2 2
15.2 even 4 600.2.m.a.299.2 4
15.8 even 4 600.2.m.a.299.3 4
15.14 odd 2 600.2.b.a.251.1 2
16.3 odd 4 768.2.c.h.767.4 4
16.5 even 4 768.2.c.h.767.4 4
16.11 odd 4 768.2.c.h.767.1 4
16.13 even 4 768.2.c.h.767.1 4
20.3 even 4 2400.2.m.a.1199.4 4
20.7 even 4 2400.2.m.a.1199.1 4
20.19 odd 2 2400.2.b.a.2351.2 2
24.5 odd 2 96.2.f.a.47.2 2
24.11 even 2 inner 24.2.f.a.11.2 yes 2
36.7 odd 6 2592.2.p.b.2159.2 4
36.11 even 6 2592.2.p.b.2159.1 4
36.23 even 6 2592.2.p.b.431.2 4
36.31 odd 6 2592.2.p.b.431.1 4
40.3 even 4 600.2.m.a.299.1 4
40.13 odd 4 2400.2.m.a.1199.4 4
40.19 odd 2 600.2.b.a.251.2 2
40.27 even 4 600.2.m.a.299.4 4
40.29 even 2 2400.2.b.a.2351.2 2
40.37 odd 4 2400.2.m.a.1199.1 4
48.5 odd 4 768.2.c.h.767.2 4
48.11 even 4 768.2.c.h.767.3 4
48.29 odd 4 768.2.c.h.767.3 4
48.35 even 4 768.2.c.h.767.2 4
60.23 odd 4 2400.2.m.a.1199.2 4
60.47 odd 4 2400.2.m.a.1199.3 4
60.59 even 2 2400.2.b.a.2351.1 2
72.5 odd 6 2592.2.p.b.431.2 4
72.11 even 6 648.2.l.b.539.2 4
72.13 even 6 2592.2.p.b.431.1 4
72.29 odd 6 2592.2.p.b.2159.1 4
72.43 odd 6 648.2.l.b.539.1 4
72.59 even 6 648.2.l.b.107.1 4
72.61 even 6 2592.2.p.b.2159.2 4
72.67 odd 6 648.2.l.b.107.2 4
120.29 odd 2 2400.2.b.a.2351.1 2
120.53 even 4 2400.2.m.a.1199.2 4
120.59 even 2 600.2.b.a.251.1 2
120.77 even 4 2400.2.m.a.1199.3 4
120.83 odd 4 600.2.m.a.299.3 4
120.107 odd 4 600.2.m.a.299.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.2.f.a.11.1 ✓ 2 1.1 even 1 trivial
24.2.f.a.11.1 ✓ 2 8.3 odd 2 CM
24.2.f.a.11.2 yes 2 3.2 odd 2 inner
24.2.f.a.11.2 yes 2 24.11 even 2 inner
96.2.f.a.47.1 2 4.3 odd 2
96.2.f.a.47.1 2 8.5 even 2
96.2.f.a.47.2 2 12.11 even 2
96.2.f.a.47.2 2 24.5 odd 2
600.2.b.a.251.1 2 15.14 odd 2
600.2.b.a.251.1 2 120.59 even 2
600.2.b.a.251.2 2 5.4 even 2
600.2.b.a.251.2 2 40.19 odd 2
600.2.m.a.299.1 4 5.3 odd 4
600.2.m.a.299.1 4 40.3 even 4
600.2.m.a.299.2 4 15.2 even 4
600.2.m.a.299.2 4 120.107 odd 4
600.2.m.a.299.3 4 15.8 even 4
600.2.m.a.299.3 4 120.83 odd 4
600.2.m.a.299.4 4 5.2 odd 4
600.2.m.a.299.4 4 40.27 even 4
648.2.l.b.107.1 4 9.5 odd 6
648.2.l.b.107.1 4 72.59 even 6
648.2.l.b.107.2 4 9.4 even 3
648.2.l.b.107.2 4 72.67 odd 6
648.2.l.b.539.1 4 9.7 even 3
648.2.l.b.539.1 4 72.43 odd 6
648.2.l.b.539.2 4 9.2 odd 6
648.2.l.b.539.2 4 72.11 even 6
768.2.c.h.767.1 4 16.11 odd 4
768.2.c.h.767.1 4 16.13 even 4
768.2.c.h.767.2 4 48.5 odd 4
768.2.c.h.767.2 4 48.35 even 4
768.2.c.h.767.3 4 48.11 even 4
768.2.c.h.767.3 4 48.29 odd 4
768.2.c.h.767.4 4 16.3 odd 4
768.2.c.h.767.4 4 16.5 even 4
2400.2.b.a.2351.1 2 60.59 even 2
2400.2.b.a.2351.1 2 120.29 odd 2
2400.2.b.a.2351.2 2 20.19 odd 2
2400.2.b.a.2351.2 2 40.29 even 2
2400.2.m.a.1199.1 4 20.7 even 4
2400.2.m.a.1199.1 4 40.37 odd 4
2400.2.m.a.1199.2 4 60.23 odd 4
2400.2.m.a.1199.2 4 120.53 even 4
2400.2.m.a.1199.3 4 60.47 odd 4
2400.2.m.a.1199.3 4 120.77 even 4
2400.2.m.a.1199.4 4 20.3 even 4
2400.2.m.a.1199.4 4 40.13 odd 4
2592.2.p.b.431.1 4 36.31 odd 6
2592.2.p.b.431.1 4 72.13 even 6
2592.2.p.b.431.2 4 36.23 even 6
2592.2.p.b.431.2 4 72.5 odd 6
2592.2.p.b.2159.1 4 36.11 even 6
2592.2.p.b.2159.1 4 72.29 odd 6
2592.2.p.b.2159.2 4 36.7 odd 6
2592.2.p.b.2159.2 4 72.61 even 6