Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [121,2,Mod(1,121)] 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("121.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 121.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.966189864457\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 121.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +2.00000 q^{7} -2.00000 q^{9} +2.00000 q^{10} -2.00000 q^{12} -4.00000 q^{13} +4.00000 q^{14} -1.00000 q^{15} -4.00000 q^{16} +2.00000 q^{17} -4.00000 q^{18} +2.00000 q^{20} -2.00000 q^{21} -1.00000 q^{23} -4.00000 q^{25} -8.00000 q^{26} +5.00000 q^{27} +4.00000 q^{28} -2.00000 q^{30} +7.00000 q^{31} -8.00000 q^{32} +4.00000 q^{34} +2.00000 q^{35} -4.00000 q^{36} +3.00000 q^{37} +4.00000 q^{39} +8.00000 q^{41} -4.00000 q^{42} +6.00000 q^{43} -2.00000 q^{45} -2.00000 q^{46} +8.00000 q^{47} +4.00000 q^{48} -3.00000 q^{49} -8.00000 q^{50} -2.00000 q^{51} -8.00000 q^{52} -6.00000 q^{53} +10.0000 q^{54} +5.00000 q^{59} -2.00000 q^{60} -12.0000 q^{61} +14.0000 q^{62} -4.00000 q^{63} -8.00000 q^{64} -4.00000 q^{65} -7.00000 q^{67} +4.00000 q^{68} +1.00000 q^{69} +4.00000 q^{70} -3.00000 q^{71} -4.00000 q^{73} +6.00000 q^{74} +4.00000 q^{75} +8.00000 q^{78} +10.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +16.0000 q^{82} +6.00000 q^{83} -4.00000 q^{84} +2.00000 q^{85} +12.0000 q^{86} +15.0000 q^{89} -4.00000 q^{90} -8.00000 q^{91} -2.00000 q^{92} -7.00000 q^{93} +16.0000 q^{94} +8.00000 q^{96} -7.00000 q^{97} -6.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 2.00000 1.00000
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) −2.00000 −0.816497
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 2.00000 0.632456
\(11\) 0 0
\(12\) −2.00000 −0.577350
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 4.00000 1.06904
\(15\) −1.00000 −0.258199
\(16\) −4.00000 −1.00000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −4.00000 −0.942809
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 0.447214
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 − 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) −8.00000 −1.56893
\(27\) 5.00000 0.962250
\(28\) 4.00000 0.755929
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −2.00000 −0.365148
\(31\) 7.00000 1.25724 0.628619 − 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) −8.00000 −1.41421
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) 2.00000 0.338062
\(36\) −4.00000 −0.666667
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 − 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) −4.00000 −0.617213
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) −2.00000 −0.294884
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 4.00000 0.577350
\(49\) −3.00000 −0.428571
\(50\) −8.00000 −1.13137
\(51\) −2.00000 −0.280056
\(52\) −8.00000 −1.10940
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 10.0000 1.36083
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.00000 0.650945 0.325472 − 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) −2.00000 −0.258199
\(61\) −12.0000 −1.53644 −0.768221 − 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 14.0000 1.77800
\(63\) −4.00000 −0.503953
\(64\) −8.00000 −1.00000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 − 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 4.00000 0.485071
\(69\) 1.00000 0.120386
\(70\) 4.00000 0.478091
\(71\) −3.00000 −0.356034 −0.178017 − 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 6.00000 0.697486
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 8.00000 0.905822
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 16.0000 1.76690
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −4.00000 −0.436436
\(85\) 2.00000 0.216930
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 − 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) −4.00000 −0.421637
\(91\) −8.00000 −0.838628
\(92\) −2.00000 −0.208514
\(93\) −7.00000 −0.725866
\(94\) 16.0000 1.65027
\(95\) 0 0
\(96\) 8.00000 0.816497
\(97\) −7.00000 −0.710742 −0.355371 − 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) −6.00000 −0.606092
\(99\) 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 121.2.a.d.1.1 1
3.2 odd 2 1089.2.a.b.1.1 1
4.3 odd 2 1936.2.a.i.1.1 1
5.4 even 2 3025.2.a.a.1.1 1
7.6 odd 2 5929.2.a.h.1.1 1
8.3 odd 2 7744.2.a.k.1.1 1
8.5 even 2 7744.2.a.x.1.1 1
11.2 odd 10 121.2.c.e.81.1 4
11.3 even 5 121.2.c.a.9.1 4
11.4 even 5 121.2.c.a.27.1 4
11.5 even 5 121.2.c.a.3.1 4
11.6 odd 10 121.2.c.e.3.1 4
11.7 odd 10 121.2.c.e.27.1 4
11.8 odd 10 121.2.c.e.9.1 4
11.9 even 5 121.2.c.a.81.1 4
11.10 odd 2 11.2.a.a.1.1 ✓ 1
33.32 even 2 99.2.a.d.1.1 1
44.43 even 2 176.2.a.b.1.1 1
55.32 even 4 275.2.b.a.199.1 2
55.43 even 4 275.2.b.a.199.2 2
55.54 odd 2 275.2.a.b.1.1 1
77.10 even 6 539.2.e.g.177.1 2
77.32 odd 6 539.2.e.h.177.1 2
77.54 even 6 539.2.e.g.67.1 2
77.65 odd 6 539.2.e.h.67.1 2
77.76 even 2 539.2.a.a.1.1 1
88.21 odd 2 704.2.a.h.1.1 1
88.43 even 2 704.2.a.c.1.1 1
99.32 even 6 891.2.e.b.298.1 2
99.43 odd 6 891.2.e.k.595.1 2
99.65 even 6 891.2.e.b.595.1 2
99.76 odd 6 891.2.e.k.298.1 2
132.131 odd 2 1584.2.a.g.1.1 1
143.142 odd 2 1859.2.a.b.1.1 1
165.32 odd 4 2475.2.c.a.199.2 2
165.98 odd 4 2475.2.c.a.199.1 2
165.164 even 2 2475.2.a.a.1.1 1
176.21 odd 4 2816.2.c.j.1409.2 2
176.43 even 4 2816.2.c.f.1409.1 2
176.109 odd 4 2816.2.c.j.1409.1 2
176.131 even 4 2816.2.c.f.1409.2 2
187.186 odd 2 3179.2.a.a.1.1 1
209.208 even 2 3971.2.a.b.1.1 1
220.43 odd 4 4400.2.b.h.4049.2 2
220.87 odd 4 4400.2.b.h.4049.1 2
220.219 even 2 4400.2.a.i.1.1 1
231.230 odd 2 4851.2.a.t.1.1 1
253.252 even 2 5819.2.a.a.1.1 1
264.131 odd 2 6336.2.a.bu.1.1 1
264.197 even 2 6336.2.a.br.1.1 1
308.307 odd 2 8624.2.a.j.1.1 1
319.318 odd 2 9251.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.2.a.a.1.1 ✓ 1 11.10 odd 2
99.2.a.d.1.1 1 33.32 even 2
121.2.a.d.1.1 1 1.1 even 1 trivial
121.2.c.a.3.1 4 11.5 even 5
121.2.c.a.9.1 4 11.3 even 5
121.2.c.a.27.1 4 11.4 even 5
121.2.c.a.81.1 4 11.9 even 5
121.2.c.e.3.1 4 11.6 odd 10
121.2.c.e.9.1 4 11.8 odd 10
121.2.c.e.27.1 4 11.7 odd 10
121.2.c.e.81.1 4 11.2 odd 10
176.2.a.b.1.1 1 44.43 even 2
275.2.a.b.1.1 1 55.54 odd 2
275.2.b.a.199.1 2 55.32 even 4
275.2.b.a.199.2 2 55.43 even 4
539.2.a.a.1.1 1 77.76 even 2
539.2.e.g.67.1 2 77.54 even 6
539.2.e.g.177.1 2 77.10 even 6
539.2.e.h.67.1 2 77.65 odd 6
539.2.e.h.177.1 2 77.32 odd 6
704.2.a.c.1.1 1 88.43 even 2
704.2.a.h.1.1 1 88.21 odd 2
891.2.e.b.298.1 2 99.32 even 6
891.2.e.b.595.1 2 99.65 even 6
891.2.e.k.298.1 2 99.76 odd 6
891.2.e.k.595.1 2 99.43 odd 6
1089.2.a.b.1.1 1 3.2 odd 2
1584.2.a.g.1.1 1 132.131 odd 2
1859.2.a.b.1.1 1 143.142 odd 2
1936.2.a.i.1.1 1 4.3 odd 2
2475.2.a.a.1.1 1 165.164 even 2
2475.2.c.a.199.1 2 165.98 odd 4
2475.2.c.a.199.2 2 165.32 odd 4
2816.2.c.f.1409.1 2 176.43 even 4
2816.2.c.f.1409.2 2 176.131 even 4
2816.2.c.j.1409.1 2 176.109 odd 4
2816.2.c.j.1409.2 2 176.21 odd 4
3025.2.a.a.1.1 1 5.4 even 2
3179.2.a.a.1.1 1 187.186 odd 2
3971.2.a.b.1.1 1 209.208 even 2
4400.2.a.i.1.1 1 220.219 even 2
4400.2.b.h.4049.1 2 220.87 odd 4
4400.2.b.h.4049.2 2 220.43 odd 4
4851.2.a.t.1.1 1 231.230 odd 2
5819.2.a.a.1.1 1 253.252 even 2
5929.2.a.h.1.1 1 7.6 odd 2
6336.2.a.br.1.1 1 264.197 even 2
6336.2.a.bu.1.1 1 264.131 odd 2
7744.2.a.k.1.1 1 8.3 odd 2
7744.2.a.x.1.1 1 8.5 even 2
8624.2.a.j.1.1 1 308.307 odd 2
9251.2.a.d.1.1 1 319.318 odd 2