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Properties

Label 33.2.a.a.1.1
Level $33$
Weight $2$
Character 33.1
Self dual yes
Analytic conductor $0.264$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [33,2,Mod(1,33)] 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("33.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(33, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

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: yes
Analytic conductor: \(0.263506326670\)
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 \(\chi\) \(=\) 33.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +4.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +2.00000 q^{20} -4.00000 q^{21} +1.00000 q^{22} +8.00000 q^{23} +3.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} -6.00000 q^{29} +2.00000 q^{30} -8.00000 q^{31} +5.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} -8.00000 q^{35} -1.00000 q^{36} +6.00000 q^{37} +2.00000 q^{39} +6.00000 q^{40} -2.00000 q^{41} -4.00000 q^{42} -1.00000 q^{44} -2.00000 q^{45} +8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -2.00000 q^{55} -12.0000 q^{56} -6.00000 q^{58} -4.00000 q^{59} -2.00000 q^{60} +6.00000 q^{61} -8.00000 q^{62} +4.00000 q^{63} +7.00000 q^{64} +4.00000 q^{65} -1.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} -8.00000 q^{70} -3.00000 q^{72} -14.0000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{77} +2.00000 q^{78} -4.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +12.0000 q^{83} +4.00000 q^{84} +4.00000 q^{85} +6.00000 q^{87} -3.00000 q^{88} -6.00000 q^{89} -2.00000 q^{90} -8.00000 q^{91} -8.00000 q^{92} +8.00000 q^{93} +8.00000 q^{94} -5.00000 q^{96} +2.00000 q^{97} +9.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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 \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −1.00000 −0.408248
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 4.00000 1.06904
\(15\) 2.00000 0.516398
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 0.447214
\(21\) −4.00000 −0.872872
\(22\) 1.00000 0.213201
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 3.00000 0.612372
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) −4.00000 −0.755929
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 2.00000 0.365148
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 5.00000 0.883883
\(33\) −1.00000 −0.174078
\(34\) −2.00000 −0.342997
\(35\) −8.00000 −1.35225
\(36\) −1.00000 −0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 6.00000 0.948683
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −4.00000 −0.617213
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −1.00000 −0.150756
\(45\) −2.00000 −0.298142
\(46\) 8.00000 1.17954
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 2.00000 0.280056
\(52\) 2.00000 0.277350
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) −2.00000 −0.269680
\(56\) −12.0000 −1.60357
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) −2.00000 −0.258199
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −8.00000 −1.01600
\(63\) 4.00000 0.503953
\(64\) 7.00000 0.875000
\(65\) 4.00000 0.496139
\(66\) −1.00000 −0.123091
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.00000 0.242536
\(69\) −8.00000 −0.963087
\(70\) −8.00000 −0.956183
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −3.00000 −0.353553
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 2.00000 0.226455
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 4.00000 0.436436
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) −3.00000 −0.319801
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −2.00000 −0.210819
\(91\) −8.00000 −0.838628
\(92\) −8.00000 −0.834058
\(93\) 8.00000 0.829561
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 9.00000 0.909137
\(99\) 1.00000 0.100504
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 33.2.a.a.1.1 ✓ 1
3.2 odd 2 99.2.a.b.1.1 1
4.3 odd 2 528.2.a.g.1.1 1
5.2 odd 4 825.2.c.a.199.2 2
5.3 odd 4 825.2.c.a.199.1 2
5.4 even 2 825.2.a.a.1.1 1
7.6 odd 2 1617.2.a.j.1.1 1
8.3 odd 2 2112.2.a.j.1.1 1
8.5 even 2 2112.2.a.bb.1.1 1
9.2 odd 6 891.2.e.g.595.1 2
9.4 even 3 891.2.e.e.298.1 2
9.5 odd 6 891.2.e.g.298.1 2
9.7 even 3 891.2.e.e.595.1 2
11.2 odd 10 363.2.e.g.202.1 4
11.3 even 5 363.2.e.e.130.1 4
11.4 even 5 363.2.e.e.148.1 4
11.5 even 5 363.2.e.e.124.1 4
11.6 odd 10 363.2.e.g.124.1 4
11.7 odd 10 363.2.e.g.148.1 4
11.8 odd 10 363.2.e.g.130.1 4
11.9 even 5 363.2.e.e.202.1 4
11.10 odd 2 363.2.a.b.1.1 1
12.11 even 2 1584.2.a.o.1.1 1
13.12 even 2 5577.2.a.a.1.1 1
15.2 even 4 2475.2.c.d.199.1 2
15.8 even 4 2475.2.c.d.199.2 2
15.14 odd 2 2475.2.a.g.1.1 1
17.16 even 2 9537.2.a.m.1.1 1
21.20 even 2 4851.2.a.b.1.1 1
24.5 odd 2 6336.2.a.x.1.1 1
24.11 even 2 6336.2.a.n.1.1 1
33.32 even 2 1089.2.a.j.1.1 1
44.43 even 2 5808.2.a.t.1.1 1
55.54 odd 2 9075.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
99.2.a.b.1.1 1 3.2 odd 2
363.2.a.b.1.1 1 11.10 odd 2
363.2.e.e.124.1 4 11.5 even 5
363.2.e.e.130.1 4 11.3 even 5
363.2.e.e.148.1 4 11.4 even 5
363.2.e.e.202.1 4 11.9 even 5
363.2.e.g.124.1 4 11.6 odd 10
363.2.e.g.130.1 4 11.8 odd 10
363.2.e.g.148.1 4 11.7 odd 10
363.2.e.g.202.1 4 11.2 odd 10
528.2.a.g.1.1 1 4.3 odd 2
825.2.a.a.1.1 1 5.4 even 2
825.2.c.a.199.1 2 5.3 odd 4
825.2.c.a.199.2 2 5.2 odd 4
891.2.e.e.298.1 2 9.4 even 3
891.2.e.e.595.1 2 9.7 even 3
891.2.e.g.298.1 2 9.5 odd 6
891.2.e.g.595.1 2 9.2 odd 6
1089.2.a.j.1.1 1 33.32 even 2
1584.2.a.o.1.1 1 12.11 even 2
1617.2.a.j.1.1 1 7.6 odd 2
2112.2.a.j.1.1 1 8.3 odd 2
2112.2.a.bb.1.1 1 8.5 even 2
2475.2.a.g.1.1 1 15.14 odd 2
2475.2.c.d.199.1 2 15.2 even 4
2475.2.c.d.199.2 2 15.8 even 4
4851.2.a.b.1.1 1 21.20 even 2
5577.2.a.a.1.1 1 13.12 even 2
5808.2.a.t.1.1 1 44.43 even 2
6336.2.a.n.1.1 1 24.11 even 2
6336.2.a.x.1.1 1 24.5 odd 2
9075.2.a.q.1.1 1 55.54 odd 2
9537.2.a.m.1.1 1 17.16 even 2