↑

Properties

Label 121.2.a.a.1.1
Level $121$
Weight $2$
Character 121.1
Self dual yes
Analytic conductor $0.966$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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] = [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,-1] 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: yes
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{12} -1.00000 q^{13} -2.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} -1.00000 q^{20} +4.00000 q^{21} +2.00000 q^{23} +6.00000 q^{24} -4.00000 q^{25} +1.00000 q^{26} -4.00000 q^{27} -2.00000 q^{28} -9.00000 q^{29} -2.00000 q^{30} -2.00000 q^{31} -5.00000 q^{32} -5.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} -3.00000 q^{37} +6.00000 q^{38} -2.00000 q^{39} +3.00000 q^{40} +5.00000 q^{41} -4.00000 q^{42} +1.00000 q^{45} -2.00000 q^{46} +2.00000 q^{47} -2.00000 q^{48} -3.00000 q^{49} +4.00000 q^{50} +10.0000 q^{51} +1.00000 q^{52} +9.00000 q^{53} +4.00000 q^{54} +6.00000 q^{56} -12.0000 q^{57} +9.00000 q^{58} +8.00000 q^{59} -2.00000 q^{60} -6.00000 q^{61} +2.00000 q^{62} +2.00000 q^{63} +7.00000 q^{64} -1.00000 q^{65} +2.00000 q^{67} -5.00000 q^{68} +4.00000 q^{69} -2.00000 q^{70} +12.0000 q^{71} +3.00000 q^{72} +2.00000 q^{73} +3.00000 q^{74} -8.00000 q^{75} +6.00000 q^{76} +2.00000 q^{78} +10.0000 q^{79} -1.00000 q^{80} -11.0000 q^{81} -5.00000 q^{82} -6.00000 q^{83} -4.00000 q^{84} +5.00000 q^{85} -18.0000 q^{87} -9.00000 q^{89} -1.00000 q^{90} -2.00000 q^{91} -2.00000 q^{92} -4.00000 q^{93} -2.00000 q^{94} -6.00000 q^{95} -10.0000 q^{96} -13.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 2.00000 1.15470 0.577350 − 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) −1.00000 −0.500000
\(5\) 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 \(0.376624\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −2.00000 −0.577350
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) −2.00000 −0.534522
\(15\) 2.00000 0.516398
\(16\) −1.00000 −0.250000
\(17\) 5.00000 1.21268 0.606339 − 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) −1.00000 −0.235702
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) −1.00000 −0.223607
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 6.00000 1.22474
\(25\) −4.00000 −0.800000
\(26\) 1.00000 0.196116
\(27\) −4.00000 −0.769800
\(28\) −2.00000 −0.377964
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) −2.00000 −0.365148
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) −5.00000 −0.857493
\(35\) 2.00000 0.338062
\(36\) −1.00000 −0.166667
\(37\) −3.00000 −0.493197 −0.246598 − 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 6.00000 0.973329
\(39\) −2.00000 −0.320256
\(40\) 3.00000 0.474342
\(41\) 5.00000 0.780869 0.390434 − 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) −4.00000 −0.617213
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) −2.00000 −0.294884
\(47\) 2.00000 0.291730 0.145865 − 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) −2.00000 −0.288675
\(49\) −3.00000 −0.428571
\(50\) 4.00000 0.565685
\(51\) 10.0000 1.40028
\(52\) 1.00000 0.138675
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 6.00000 0.801784
\(57\) −12.0000 −1.58944
\(58\) 9.00000 1.18176
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) −2.00000 −0.258199
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 2.00000 0.254000
\(63\) 2.00000 0.251976
\(64\) 7.00000 0.875000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −5.00000 −0.606339
\(69\) 4.00000 0.481543
\(70\) −2.00000 −0.239046
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 3.00000 0.353553
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 3.00000 0.348743
\(75\) −8.00000 −0.923760
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.0000 −1.22222
\(82\) −5.00000 −0.552158
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) −4.00000 −0.436436
\(85\) 5.00000 0.542326
\(86\) 0 0
\(87\) −18.0000 −1.92980
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 − 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) −1.00000 −0.105409
\(91\) −2.00000 −0.209657
\(92\) −2.00000 −0.208514
\(93\) −4.00000 −0.414781
\(94\) −2.00000 −0.206284
\(95\) −6.00000 −0.615587
\(96\) −10.0000 −1.02062
\(97\) −13.0000 −1.31995 −0.659975 − 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 3.00000 0.303046
\(99\) 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.a.1.1 ✓ 1
3.2 odd 2 1089.2.a.i.1.1 1
4.3 odd 2 1936.2.a.a.1.1 1
5.4 even 2 3025.2.a.e.1.1 1
7.6 odd 2 5929.2.a.a.1.1 1
8.3 odd 2 7744.2.a.be.1.1 1
8.5 even 2 7744.2.a.f.1.1 1
11.2 odd 10 121.2.c.b.81.1 4
11.3 even 5 121.2.c.d.9.1 4
11.4 even 5 121.2.c.d.27.1 4
11.5 even 5 121.2.c.d.3.1 4
11.6 odd 10 121.2.c.b.3.1 4
11.7 odd 10 121.2.c.b.27.1 4
11.8 odd 10 121.2.c.b.9.1 4
11.9 even 5 121.2.c.d.81.1 4
11.10 odd 2 121.2.a.c.1.1 yes 1
33.32 even 2 1089.2.a.c.1.1 1
44.43 even 2 1936.2.a.b.1.1 1
55.54 odd 2 3025.2.a.b.1.1 1
77.76 even 2 5929.2.a.g.1.1 1
88.21 odd 2 7744.2.a.c.1.1 1
88.43 even 2 7744.2.a.bf.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
121.2.a.c.1.1 yes 1 11.10 odd 2
121.2.c.b.3.1 4 11.6 odd 10
121.2.c.b.9.1 4 11.8 odd 10
121.2.c.b.27.1 4 11.7 odd 10
121.2.c.b.81.1 4 11.2 odd 10
121.2.c.d.3.1 4 11.5 even 5
121.2.c.d.9.1 4 11.3 even 5
121.2.c.d.27.1 4 11.4 even 5
121.2.c.d.81.1 4 11.9 even 5
1089.2.a.c.1.1 1 33.32 even 2
1089.2.a.i.1.1 1 3.2 odd 2
1936.2.a.a.1.1 1 4.3 odd 2
1936.2.a.b.1.1 1 44.43 even 2
3025.2.a.b.1.1 1 55.54 odd 2
3025.2.a.e.1.1 1 5.4 even 2
5929.2.a.a.1.1 1 7.6 odd 2
5929.2.a.g.1.1 1 77.76 even 2
7744.2.a.c.1.1 1 88.21 odd 2
7744.2.a.f.1.1 1 8.5 even 2
7744.2.a.be.1.1 1 8.3 odd 2
7744.2.a.bf.1.1 1 88.43 even 2