Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [121,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 121.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.13923111069\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q-2.46410 q^{2} -0.535898 q^{3} -1.92820 q^{4} -1.53590 q^{5} +1.32051 q^{6} +28.2487 q^{7} +24.4641 q^{8} -26.7128 q^{9} +3.78461 q^{10} +1.03332 q^{12} -68.4641 q^{13} -69.6077 q^{14} +0.823085 q^{15} -44.8564 q^{16} -55.3538 q^{17} +65.8231 q^{18} +55.1769 q^{19} +2.96152 q^{20} -15.1384 q^{21} -178.315 q^{23} -13.1103 q^{24} -122.641 q^{25} +168.703 q^{26} +28.7846 q^{27} -54.4693 q^{28} -113.172 q^{29} -2.02817 q^{30} +70.8128 q^{31} -85.1821 q^{32} +136.397 q^{34} -43.3872 q^{35} +51.5077 q^{36} -210.664 q^{37} -135.962 q^{38} +36.6898 q^{39} -37.5744 q^{40} -191.928 q^{41} +37.3027 q^{42} +208.210 q^{43} +41.0282 q^{45} +439.387 q^{46} +512.515 q^{47} +24.0385 q^{48} +454.990 q^{49} +302.200 q^{50} +29.6640 q^{51} +132.013 q^{52} -375.449 q^{53} -70.9282 q^{54} +691.079 q^{56} -29.5692 q^{57} +278.867 q^{58} -506.508 q^{59} -1.58708 q^{60} +468.697 q^{61} -174.490 q^{62} -754.603 q^{63} +568.749 q^{64} +105.154 q^{65} -289.895 q^{67} +106.733 q^{68} +95.5589 q^{69} +106.910 q^{70} -394.010 q^{71} -653.505 q^{72} -289.538 q^{73} +519.098 q^{74} +65.7231 q^{75} -106.392 q^{76} -90.4074 q^{78} +169.587 q^{79} +68.8949 q^{80} +705.820 q^{81} +472.931 q^{82} +303.331 q^{83} +29.1900 q^{84} +85.0179 q^{85} -513.051 q^{86} +60.6486 q^{87} -1146.68 q^{89} -101.098 q^{90} -1934.02 q^{91} +343.828 q^{92} -37.9485 q^{93} -1262.89 q^{94} -84.7461 q^{95} +45.6489 q^{96} +641.600 q^{97} -1121.14 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 8 q^{3} + 10 q^{4} - 10 q^{5} - 32 q^{6} + 8 q^{7} + 42 q^{8} + 2 q^{9} - 34 q^{10} - 88 q^{12} - 130 q^{13} - 160 q^{14} + 64 q^{15} - 62 q^{16} + 14 q^{17} + 194 q^{18} + 48 q^{19} - 98 q^{20}+ \cdots - 822 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.46410 −0.871191 −0.435596 − 0.900142i \(-0.643462\pi\)
−0.435596 + 0.900142i \(0.643462\pi\)
\(3\) −0.535898 −0.103134 −0.0515668 − 0.998670i \(-0.516422\pi\)
−0.0515668 + 0.998670i \(0.516422\pi\)
\(4\) −1.92820 −0.241025
\(5\) −1.53590 −0.137375 −0.0686875 − 0.997638i \(-0.521881\pi\)
−0.0686875 + 0.997638i \(0.521881\pi\)
\(6\) 1.32051 0.0898492
\(7\) 28.2487 1.52529 0.762644 − 0.646819i \(-0.223901\pi\)
0.762644 + 0.646819i \(0.223901\pi\)
\(8\) 24.4641 1.08117
\(9\) −26.7128 −0.989363
\(10\) 3.78461 0.119680
\(11\) 0 0
\(12\) 1.03332 0.0248578
\(13\) −68.4641 −1.46066 −0.730328 − 0.683097i \(-0.760633\pi\)
−0.730328 + 0.683097i \(0.760633\pi\)
\(14\) −69.6077 −1.32882
\(15\) 0.823085 0.0141680
\(16\) −44.8564 −0.700881
\(17\) −55.3538 −0.789722 −0.394861 − 0.918741i \(-0.629207\pi\)
−0.394861 + 0.918741i \(0.629207\pi\)
\(18\) 65.8231 0.861925
\(19\) 55.1769 0.666234 0.333117 − 0.942885i \(-0.391900\pi\)
0.333117 + 0.942885i \(0.391900\pi\)
\(20\) 2.96152 0.0331108
\(21\) −15.1384 −0.157308
\(22\) 0 0
\(23\) −178.315 −1.61658 −0.808290 − 0.588785i \(-0.799606\pi\)
−0.808290 + 0.588785i \(0.799606\pi\)
\(24\) −13.1103 −0.111505
\(25\) −122.641 −0.981128
\(26\) 168.703 1.27251
\(27\) 28.7846 0.205170
\(28\) −54.4693 −0.367633
\(29\) −113.172 −0.724671 −0.362336 − 0.932048i \(-0.618021\pi\)
−0.362336 + 0.932048i \(0.618021\pi\)
\(30\) −2.02817 −0.0123430
\(31\) 70.8128 0.410269 0.205135 − 0.978734i \(-0.434237\pi\)
0.205135 + 0.978734i \(0.434237\pi\)
\(32\) −85.1821 −0.470569
\(33\) 0 0
\(34\) 136.397 0.687999
\(35\) −43.3872 −0.209536
\(36\) 51.5077 0.238462
\(37\) −210.664 −0.936026 −0.468013 − 0.883722i \(-0.655030\pi\)
−0.468013 + 0.883722i \(0.655030\pi\)
\(38\) −135.962 −0.580418
\(39\) 36.6898 0.150643
\(40\) −37.5744 −0.148526
\(41\) −191.928 −0.731077 −0.365538 − 0.930796i \(-0.619115\pi\)
−0.365538 + 0.930796i \(0.619115\pi\)
\(42\) 37.3027 0.137046
\(43\) 208.210 0.738413 0.369207 − 0.929347i \(-0.379629\pi\)
0.369207 + 0.929347i \(0.379629\pi\)
\(44\) 0 0
\(45\) 41.0282 0.135914
\(46\) 439.387 1.40835
\(47\) 512.515 1.59060 0.795298 − 0.606218i \(-0.207314\pi\)
0.795298 + 0.606218i \(0.207314\pi\)
\(48\) 24.0385 0.0722845
\(49\) 454.990 1.32650
\(50\) 302.200 0.854750
\(51\) 29.6640 0.0814470
\(52\) 132.013 0.352055
\(53\) −375.449 −0.973054 −0.486527 − 0.873666i \(-0.661736\pi\)
−0.486527 + 0.873666i \(0.661736\pi\)
\(54\) −70.9282 −0.178743
\(55\) 0 0
\(56\) 691.079 1.64910
\(57\) −29.5692 −0.0687112
\(58\) 278.867 0.631327
\(59\) −506.508 −1.11766 −0.558828 − 0.829284i \(-0.688749\pi\)
−0.558828 + 0.829284i \(0.688749\pi\)
\(60\) −1.58708 −0.00341484
\(61\) 468.697 0.983779 0.491890 − 0.870658i \(-0.336306\pi\)
0.491890 + 0.870658i \(0.336306\pi\)
\(62\) −174.490 −0.357423
\(63\) −754.603 −1.50906
\(64\) 568.749 1.11084
\(65\) 105.154 0.200657
\(66\) 0 0
\(67\) −289.895 −0.528601 −0.264301 − 0.964440i \(-0.585141\pi\)
−0.264301 + 0.964440i \(0.585141\pi\)
\(68\) 106.733 0.190343
\(69\) 95.5589 0.166724
\(70\) 106.910 0.182546
\(71\) −394.010 −0.658597 −0.329299 − 0.944226i \(-0.606812\pi\)
−0.329299 + 0.944226i \(0.606812\pi\)
\(72\) −653.505 −1.06967
\(73\) −289.538 −0.464218 −0.232109 − 0.972690i \(-0.574563\pi\)
−0.232109 + 0.972690i \(0.574563\pi\)
\(74\) 519.098 0.815458
\(75\) 65.7231 0.101187
\(76\) −106.392 −0.160579
\(77\) 0 0
\(78\) −90.4074 −0.131239
\(79\) 169.587 0.241519 0.120760 − 0.992682i \(-0.461467\pi\)
0.120760 + 0.992682i \(0.461467\pi\)
\(80\) 68.8949 0.0962835
\(81\) 705.820 0.968203
\(82\) 472.931 0.636908
\(83\) 303.331 0.401143 0.200572 − 0.979679i \(-0.435720\pi\)
0.200572 + 0.979679i \(0.435720\pi\)
\(84\) 29.1900 0.0379153
\(85\) 85.0179 0.108488
\(86\) −513.051 −0.643299
\(87\) 60.6486 0.0747380
\(88\) 0 0
\(89\) −1146.68 −1.36571 −0.682854 − 0.730555i \(-0.739262\pi\)
−0.682854 + 0.730555i \(0.739262\pi\)
\(90\) −101.098 −0.118407
\(91\) −1934.02 −2.22792
\(92\) 343.828 0.389637
\(93\) −37.9485 −0.0423126
\(94\) −1262.89 −1.38571
\(95\) −84.7461 −0.0915239
\(96\) 45.6489 0.0485315
\(97\) 641.600 0.671594 0.335797 − 0.941934i \(-0.390994\pi\)
0.335797 + 0.941934i \(0.390994\pi\)
\(98\) −1121.14 −1.15564
\(99\) 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 121.4.a.e.1.1 yes 2
3.2 odd 2 1089.4.a.k.1.2 2
4.3 odd 2 1936.4.a.y.1.1 2
11.2 odd 10 121.4.c.g.81.1 8
11.3 even 5 121.4.c.d.9.1 8
11.4 even 5 121.4.c.d.27.1 8
11.5 even 5 121.4.c.d.3.2 8
11.6 odd 10 121.4.c.g.3.1 8
11.7 odd 10 121.4.c.g.27.2 8
11.8 odd 10 121.4.c.g.9.2 8
11.9 even 5 121.4.c.d.81.2 8
11.10 odd 2 121.4.a.b.1.2 ✓ 2
33.32 even 2 1089.4.a.x.1.1 2
44.43 even 2 1936.4.a.z.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.a.b.1.2 ✓ 2 11.10 odd 2
121.4.a.e.1.1 yes 2 1.1 even 1 trivial
121.4.c.d.3.2 8 11.5 even 5
121.4.c.d.9.1 8 11.3 even 5
121.4.c.d.27.1 8 11.4 even 5
121.4.c.d.81.2 8 11.9 even 5
121.4.c.g.3.1 8 11.6 odd 10
121.4.c.g.9.2 8 11.8 odd 10
121.4.c.g.27.2 8 11.7 odd 10
121.4.c.g.81.1 8 11.2 odd 10
1089.4.a.k.1.2 2 3.2 odd 2
1089.4.a.x.1.1 2 33.32 even 2
1936.4.a.y.1.1 2 4.3 odd 2
1936.4.a.z.1.1 2 44.43 even 2