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(13,24)] 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("24.13"); 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([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 24.d (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: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 13.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 24.13
Dual form 24.2.d.a.13.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.00000i) q^{2} -1.00000i q^{3} +2.00000i q^{4} +2.00000i q^{5} +(-1.00000 + 1.00000i) q^{6} -2.00000 q^{7} +(2.00000 - 2.00000i) q^{8} -1.00000 q^{9} +(2.00000 - 2.00000i) q^{10} +2.00000 q^{12} -4.00000i q^{13} +(2.00000 + 2.00000i) q^{14} +2.00000 q^{15} -4.00000 q^{16} -2.00000 q^{17} +(1.00000 + 1.00000i) q^{18} +4.00000i q^{19} -4.00000 q^{20} +2.00000i q^{21} +4.00000 q^{23} +(-2.00000 - 2.00000i) q^{24} +1.00000 q^{25} +(-4.00000 + 4.00000i) q^{26} +1.00000i q^{27} -4.00000i q^{28} -6.00000i q^{29} +(-2.00000 - 2.00000i) q^{30} +2.00000 q^{31} +(4.00000 + 4.00000i) q^{32} +(2.00000 + 2.00000i) q^{34} -4.00000i q^{35} -2.00000i q^{36} +8.00000i q^{37} +(4.00000 - 4.00000i) q^{38} -4.00000 q^{39} +(4.00000 + 4.00000i) q^{40} +2.00000 q^{41} +(2.00000 - 2.00000i) q^{42} -4.00000i q^{43} -2.00000i q^{45} +(-4.00000 - 4.00000i) q^{46} -12.0000 q^{47} +4.00000i q^{48} -3.00000 q^{49} +(-1.00000 - 1.00000i) q^{50} +2.00000i q^{51} +8.00000 q^{52} +6.00000i q^{53} +(1.00000 - 1.00000i) q^{54} +(-4.00000 + 4.00000i) q^{56} +4.00000 q^{57} +(-6.00000 + 6.00000i) q^{58} +4.00000i q^{59} +4.00000i q^{60} +(-2.00000 - 2.00000i) q^{62} +2.00000 q^{63} -8.00000i q^{64} +8.00000 q^{65} -12.0000i q^{67} -4.00000i q^{68} -4.00000i q^{69} +(-4.00000 + 4.00000i) q^{70} +12.0000 q^{71} +(-2.00000 + 2.00000i) q^{72} -6.00000 q^{73} +(8.00000 - 8.00000i) q^{74} -1.00000i q^{75} -8.00000 q^{76} +(4.00000 + 4.00000i) q^{78} +10.0000 q^{79} -8.00000i q^{80} +1.00000 q^{81} +(-2.00000 - 2.00000i) q^{82} +16.0000i q^{83} -4.00000 q^{84} -4.00000i q^{85} +(-4.00000 + 4.00000i) q^{86} -6.00000 q^{87} -10.0000 q^{89} +(-2.00000 + 2.00000i) q^{90} +8.00000i q^{91} +8.00000i q^{92} -2.00000i q^{93} +(12.0000 + 12.0000i) q^{94} -8.00000 q^{95} +(4.00000 - 4.00000i) q^{96} -2.00000 q^{97} +(3.00000 + 3.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{6} - 4 q^{7} + 4 q^{8} - 2 q^{9} + 4 q^{10} + 4 q^{12} + 4 q^{14} + 4 q^{15} - 8 q^{16} - 4 q^{17} + 2 q^{18} - 8 q^{20} + 8 q^{23} - 4 q^{24} + 2 q^{25} - 8 q^{26} - 4 q^{30} + 4 q^{31}+ \cdots + 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 − 1.00000i −0.707107 − 0.707107i
\(3\) − 1.00000i − 0.577350i
\(4\) 2.00000i 1.00000i
\(5\) 2.00000i 0.894427i 0.894427 + 0.447214i \(0.147584\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) −1.00000 + 1.00000i −0.408248 + 0.408248i
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 2.00000 − 2.00000i 0.707107 − 0.707107i
\(9\) −1.00000 −0.333333
\(10\) 2.00000 − 2.00000i 0.632456 − 0.632456i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 2.00000 0.577350
\(13\) − 4.00000i − 1.10940i −0.832050 − 0.554700i \(-0.812833\pi\)
0.832050 − 0.554700i \(-0.187167\pi\)
\(14\) 2.00000 + 2.00000i 0.534522 + 0.534522i
\(15\) 2.00000 0.516398
\(16\) −4.00000 −1.00000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 + 1.00000i 0.235702 + 0.235702i
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) −4.00000 −0.894427
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −2.00000 − 2.00000i −0.408248 − 0.408248i
\(25\) 1.00000 0.200000
\(26\) −4.00000 + 4.00000i −0.784465 + 0.784465i
\(27\) 1.00000i 0.192450i
\(28\) − 4.00000i − 0.755929i
\(29\) − 6.00000i − 1.11417i −0.830455 − 0.557086i \(-0.811919\pi\)
0.830455 − 0.557086i \(-0.188081\pi\)
\(30\) −2.00000 − 2.00000i −0.365148 − 0.365148i
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 4.00000 + 4.00000i 0.707107 + 0.707107i
\(33\) 0 0
\(34\) 2.00000 + 2.00000i 0.342997 + 0.342997i
\(35\) − 4.00000i − 0.676123i
\(36\) − 2.00000i − 0.333333i
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 4.00000 − 4.00000i 0.648886 − 0.648886i
\(39\) −4.00000 −0.640513
\(40\) 4.00000 + 4.00000i 0.632456 + 0.632456i
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 2.00000 − 2.00000i 0.308607 − 0.308607i
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) − 2.00000i − 0.298142i
\(46\) −4.00000 − 4.00000i −0.589768 − 0.589768i
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 4.00000i 0.577350i
\(49\) −3.00000 −0.428571
\(50\) −1.00000 − 1.00000i −0.141421 − 0.141421i
\(51\) 2.00000i 0.280056i
\(52\) 8.00000 1.10940
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 1.00000 − 1.00000i 0.136083 − 0.136083i
\(55\) 0 0
\(56\) −4.00000 + 4.00000i −0.534522 + 0.534522i
\(57\) 4.00000 0.529813
\(58\) −6.00000 + 6.00000i −0.787839 + 0.787839i
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 4.00000i 0.516398i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) −2.00000 − 2.00000i −0.254000 − 0.254000i
\(63\) 2.00000 0.251976
\(64\) − 8.00000i − 1.00000i
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) − 12.0000i − 1.46603i −0.680211 − 0.733017i \(-0.738112\pi\)
0.680211 − 0.733017i \(-0.261888\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) − 4.00000i − 0.481543i
\(70\) −4.00000 + 4.00000i −0.478091 + 0.478091i
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −2.00000 + 2.00000i −0.235702 + 0.235702i
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 8.00000 − 8.00000i 0.929981 − 0.929981i
\(75\) − 1.00000i − 0.115470i
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) 4.00000 + 4.00000i 0.452911 + 0.452911i
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) − 8.00000i − 0.894427i
\(81\) 1.00000 0.111111
\(82\) −2.00000 − 2.00000i −0.220863 − 0.220863i
\(83\) 16.0000i 1.75623i 0.478451 + 0.878114i \(0.341198\pi\)
−0.478451 + 0.878114i \(0.658802\pi\)
\(84\) −4.00000 −0.436436
\(85\) − 4.00000i − 0.433861i
\(86\) −4.00000 + 4.00000i −0.431331 + 0.431331i
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −2.00000 + 2.00000i −0.210819 + 0.210819i
\(91\) 8.00000i 0.838628i
\(92\) 8.00000i 0.834058i
\(93\) − 2.00000i − 0.207390i
\(94\) 12.0000 + 12.0000i 1.23771 + 1.23771i
\(95\) −8.00000 −0.820783
\(96\) 4.00000 − 4.00000i 0.408248 − 0.408248i
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 3.00000 + 3.00000i 0.303046 + 0.303046i
\(99\) 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 24.2.d.a.13.1 ✓ 2
3.2 odd 2 72.2.d.b.37.2 2
4.3 odd 2 96.2.d.a.49.2 2
5.2 odd 4 600.2.d.c.349.1 2
5.3 odd 4 600.2.d.b.349.2 2
5.4 even 2 600.2.k.b.301.2 2
7.6 odd 2 1176.2.c.a.589.1 2
8.3 odd 2 96.2.d.a.49.1 2
8.5 even 2 inner 24.2.d.a.13.2 yes 2
9.2 odd 6 648.2.n.c.109.2 4
9.4 even 3 648.2.n.k.541.2 4
9.5 odd 6 648.2.n.c.541.1 4
9.7 even 3 648.2.n.k.109.1 4
12.11 even 2 288.2.d.b.145.1 2
15.2 even 4 1800.2.d.b.1549.2 2
15.8 even 4 1800.2.d.i.1549.1 2
15.14 odd 2 1800.2.k.a.901.1 2
16.3 odd 4 768.2.a.d.1.1 1
16.5 even 4 768.2.a.a.1.1 1
16.11 odd 4 768.2.a.e.1.1 1
16.13 even 4 768.2.a.h.1.1 1
20.3 even 4 2400.2.d.b.49.1 2
20.7 even 4 2400.2.d.c.49.2 2
20.19 odd 2 2400.2.k.a.1201.1 2
24.5 odd 2 72.2.d.b.37.1 2
24.11 even 2 288.2.d.b.145.2 2
28.27 even 2 4704.2.c.a.2353.1 2
36.7 odd 6 2592.2.r.f.433.1 4
36.11 even 6 2592.2.r.g.433.2 4
36.23 even 6 2592.2.r.g.2161.1 4
36.31 odd 6 2592.2.r.f.2161.2 4
40.3 even 4 2400.2.d.c.49.1 2
40.13 odd 4 600.2.d.c.349.2 2
40.19 odd 2 2400.2.k.a.1201.2 2
40.27 even 4 2400.2.d.b.49.2 2
40.29 even 2 600.2.k.b.301.1 2
40.37 odd 4 600.2.d.b.349.1 2
48.5 odd 4 2304.2.a.o.1.1 1
48.11 even 4 2304.2.a.l.1.1 1
48.29 odd 4 2304.2.a.e.1.1 1
48.35 even 4 2304.2.a.b.1.1 1
56.13 odd 2 1176.2.c.a.589.2 2
56.27 even 2 4704.2.c.a.2353.2 2
60.23 odd 4 7200.2.d.g.2449.1 2
60.47 odd 4 7200.2.d.d.2449.2 2
60.59 even 2 7200.2.k.d.3601.2 2
72.5 odd 6 648.2.n.c.541.2 4
72.11 even 6 2592.2.r.g.433.1 4
72.13 even 6 648.2.n.k.541.1 4
72.29 odd 6 648.2.n.c.109.1 4
72.43 odd 6 2592.2.r.f.433.2 4
72.59 even 6 2592.2.r.g.2161.2 4
72.61 even 6 648.2.n.k.109.2 4
72.67 odd 6 2592.2.r.f.2161.1 4
120.29 odd 2 1800.2.k.a.901.2 2
120.53 even 4 1800.2.d.b.1549.1 2
120.59 even 2 7200.2.k.d.3601.1 2
120.77 even 4 1800.2.d.i.1549.2 2
120.83 odd 4 7200.2.d.d.2449.1 2
120.107 odd 4 7200.2.d.g.2449.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.2.d.a.13.1 ✓ 2 1.1 even 1 trivial
24.2.d.a.13.2 yes 2 8.5 even 2 inner
72.2.d.b.37.1 2 24.5 odd 2
72.2.d.b.37.2 2 3.2 odd 2
96.2.d.a.49.1 2 8.3 odd 2
96.2.d.a.49.2 2 4.3 odd 2
288.2.d.b.145.1 2 12.11 even 2
288.2.d.b.145.2 2 24.11 even 2
600.2.d.b.349.1 2 40.37 odd 4
600.2.d.b.349.2 2 5.3 odd 4
600.2.d.c.349.1 2 5.2 odd 4
600.2.d.c.349.2 2 40.13 odd 4
600.2.k.b.301.1 2 40.29 even 2
600.2.k.b.301.2 2 5.4 even 2
648.2.n.c.109.1 4 72.29 odd 6
648.2.n.c.109.2 4 9.2 odd 6
648.2.n.c.541.1 4 9.5 odd 6
648.2.n.c.541.2 4 72.5 odd 6
648.2.n.k.109.1 4 9.7 even 3
648.2.n.k.109.2 4 72.61 even 6
648.2.n.k.541.1 4 72.13 even 6
648.2.n.k.541.2 4 9.4 even 3
768.2.a.a.1.1 1 16.5 even 4
768.2.a.d.1.1 1 16.3 odd 4
768.2.a.e.1.1 1 16.11 odd 4
768.2.a.h.1.1 1 16.13 even 4
1176.2.c.a.589.1 2 7.6 odd 2
1176.2.c.a.589.2 2 56.13 odd 2
1800.2.d.b.1549.1 2 120.53 even 4
1800.2.d.b.1549.2 2 15.2 even 4
1800.2.d.i.1549.1 2 15.8 even 4
1800.2.d.i.1549.2 2 120.77 even 4
1800.2.k.a.901.1 2 15.14 odd 2
1800.2.k.a.901.2 2 120.29 odd 2
2304.2.a.b.1.1 1 48.35 even 4
2304.2.a.e.1.1 1 48.29 odd 4
2304.2.a.l.1.1 1 48.11 even 4
2304.2.a.o.1.1 1 48.5 odd 4
2400.2.d.b.49.1 2 20.3 even 4
2400.2.d.b.49.2 2 40.27 even 4
2400.2.d.c.49.1 2 40.3 even 4
2400.2.d.c.49.2 2 20.7 even 4
2400.2.k.a.1201.1 2 20.19 odd 2
2400.2.k.a.1201.2 2 40.19 odd 2
2592.2.r.f.433.1 4 36.7 odd 6
2592.2.r.f.433.2 4 72.43 odd 6
2592.2.r.f.2161.1 4 72.67 odd 6
2592.2.r.f.2161.2 4 36.31 odd 6
2592.2.r.g.433.1 4 72.11 even 6
2592.2.r.g.433.2 4 36.11 even 6
2592.2.r.g.2161.1 4 36.23 even 6
2592.2.r.g.2161.2 4 72.59 even 6
4704.2.c.a.2353.1 2 28.27 even 2
4704.2.c.a.2353.2 2 56.27 even 2
7200.2.d.d.2449.1 2 120.83 odd 4
7200.2.d.d.2449.2 2 60.47 odd 4
7200.2.d.g.2449.1 2 60.23 odd 4
7200.2.d.g.2449.2 2 120.107 odd 4
7200.2.k.d.3601.1 2 120.59 even 2
7200.2.k.d.3601.2 2 60.59 even 2