Properties

Label 133.2.a.d.1.3
Level $133$
Weight $2$
Character 133.1
Self dual yes
Analytic conductor $1.062$
Analytic rank $0$
Dimension $3$
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] = [133,2,Mod(1,133)] 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("133.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(133, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 133 = 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 133.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,2] 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: \(1.06201034688\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.11491\) of defining polynomial
Character \(\chi\) \(=\) 133.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.47283 q^{2} -1.11491 q^{3} +4.11491 q^{4} -0.357926 q^{5} -2.75698 q^{6} -1.00000 q^{7} +5.22982 q^{8} -1.75698 q^{9} -0.885092 q^{10} +0.527166 q^{11} -4.58774 q^{12} -0.357926 q^{13} -2.47283 q^{14} +0.399055 q^{15} +4.70265 q^{16} -3.70265 q^{17} -4.34472 q^{18} +1.00000 q^{19} -1.47283 q^{20} +1.11491 q^{21} +1.30359 q^{22} +8.58774 q^{23} -5.83076 q^{24} -4.87189 q^{25} -0.885092 q^{26} +5.30359 q^{27} -4.11491 q^{28} -4.04113 q^{29} +0.986796 q^{30} -1.24302 q^{31} +1.16924 q^{32} -0.587741 q^{33} -9.15604 q^{34} +0.357926 q^{35} -7.22982 q^{36} +3.30359 q^{37} +2.47283 q^{38} +0.399055 q^{39} -1.87189 q^{40} +12.1623 q^{41} +2.75698 q^{42} -0.715853 q^{43} +2.16924 q^{44} +0.628870 q^{45} +21.2361 q^{46} +10.8176 q^{47} -5.24302 q^{48} +1.00000 q^{49} -12.0474 q^{50} +4.12811 q^{51} -1.47283 q^{52} -6.06058 q^{53} +13.1149 q^{54} -0.188687 q^{55} -5.22982 q^{56} -1.11491 q^{57} -9.99304 q^{58} -7.87189 q^{59} +1.64207 q^{60} +5.53341 q^{61} -3.07378 q^{62} +1.75698 q^{63} -6.51396 q^{64} +0.128111 q^{65} -1.45339 q^{66} +8.64832 q^{67} -15.2361 q^{68} -9.57454 q^{69} +0.885092 q^{70} -7.53341 q^{71} -9.18869 q^{72} -9.00624 q^{73} +8.16924 q^{74} +5.43171 q^{75} +4.11491 q^{76} -0.527166 q^{77} +0.986796 q^{78} -9.17548 q^{79} -1.68320 q^{80} -0.642074 q^{81} +30.0753 q^{82} +16.0606 q^{83} +4.58774 q^{84} +1.32528 q^{85} -1.77018 q^{86} +4.50548 q^{87} +2.75698 q^{88} -14.7981 q^{89} +1.55509 q^{90} +0.357926 q^{91} +35.3378 q^{92} +1.38585 q^{93} +26.7500 q^{94} -0.357926 q^{95} -1.30359 q^{96} -9.07378 q^{97} +2.47283 q^{98} -0.926221 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 3 q^{3} + 6 q^{4} - 2 q^{5} - q^{6} - 3 q^{7} + 3 q^{8} + 2 q^{9} - 9 q^{10} + 7 q^{11} - 2 q^{12} - 2 q^{13} - 2 q^{14} - 7 q^{15} - 4 q^{16} + 7 q^{17} + 6 q^{18} + 3 q^{19} + q^{20}+ \cdots + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.47283 1.74856 0.874279 − 0.485424i \(-0.161335\pi\)
0.874279 + 0.485424i \(0.161335\pi\)
\(3\) −1.11491 −0.643692 −0.321846 − 0.946792i \(-0.604303\pi\)
−0.321846 + 0.946792i \(0.604303\pi\)
\(4\) 4.11491 2.05745
\(5\) −0.357926 −0.160070 −0.0800348 − 0.996792i \(-0.525503\pi\)
−0.0800348 + 0.996792i \(0.525503\pi\)
\(6\) −2.75698 −1.12553
\(7\) −1.00000 −0.377964
\(8\) 5.22982 1.84902
\(9\) −1.75698 −0.585660
\(10\) −0.885092 −0.279891
\(11\) 0.527166 0.158947 0.0794733 − 0.996837i \(-0.474676\pi\)
0.0794733 + 0.996837i \(0.474676\pi\)
\(12\) −4.58774 −1.32437
\(13\) −0.357926 −0.0992709 −0.0496355 − 0.998767i \(-0.515806\pi\)
−0.0496355 + 0.998767i \(0.515806\pi\)
\(14\) −2.47283 −0.660893
\(15\) 0.399055 0.103036
\(16\) 4.70265 1.17566
\(17\) −3.70265 −0.898024 −0.449012 − 0.893526i \(-0.648224\pi\)
−0.449012 + 0.893526i \(0.648224\pi\)
\(18\) −4.34472 −1.02406
\(19\) 1.00000 0.229416
\(20\) −1.47283 −0.329336
\(21\) 1.11491 0.243293
\(22\) 1.30359 0.277927
\(23\) 8.58774 1.79067 0.895334 − 0.445395i \(-0.146937\pi\)
0.895334 + 0.445395i \(0.146937\pi\)
\(24\) −5.83076 −1.19020
\(25\) −4.87189 −0.974378
\(26\) −0.885092 −0.173581
\(27\) 5.30359 1.02068
\(28\) −4.11491 −0.777644
\(29\) −4.04113 −0.750419 −0.375209 − 0.926940i \(-0.622429\pi\)
−0.375209 + 0.926940i \(0.622429\pi\)
\(30\) 0.986796 0.180164
\(31\) −1.24302 −0.223253 −0.111626 − 0.993750i \(-0.535606\pi\)
−0.111626 + 0.993750i \(0.535606\pi\)
\(32\) 1.16924 0.206694
\(33\) −0.587741 −0.102313
\(34\) −9.15604 −1.57025
\(35\) 0.357926 0.0605006
\(36\) −7.22982 −1.20497
\(37\) 3.30359 0.543108 0.271554 − 0.962423i \(-0.412463\pi\)
0.271554 + 0.962423i \(0.412463\pi\)
\(38\) 2.47283 0.401147
\(39\) 0.399055 0.0638999
\(40\) −1.87189 −0.295972
\(41\) 12.1623 1.89943 0.949714 − 0.313117i \(-0.101373\pi\)
0.949714 + 0.313117i \(0.101373\pi\)
\(42\) 2.75698 0.425411
\(43\) −0.715853 −0.109167 −0.0545833 − 0.998509i \(-0.517383\pi\)
−0.0545833 + 0.998509i \(0.517383\pi\)
\(44\) 2.16924 0.327025
\(45\) 0.628870 0.0937464
\(46\) 21.2361 3.13109
\(47\) 10.8176 1.57790 0.788951 − 0.614456i \(-0.210624\pi\)
0.788951 + 0.614456i \(0.210624\pi\)
\(48\) −5.24302 −0.756765
\(49\) 1.00000 0.142857
\(50\) −12.0474 −1.70376
\(51\) 4.12811 0.578051
\(52\) −1.47283 −0.204245
\(53\) −6.06058 −0.832484 −0.416242 − 0.909254i \(-0.636653\pi\)
−0.416242 + 0.909254i \(0.636653\pi\)
\(54\) 13.1149 1.78471
\(55\) −0.188687 −0.0254425
\(56\) −5.22982 −0.698863
\(57\) −1.11491 −0.147673
\(58\) −9.99304 −1.31215
\(59\) −7.87189 −1.02483 −0.512416 − 0.858737i \(-0.671250\pi\)
−0.512416 + 0.858737i \(0.671250\pi\)
\(60\) 1.64207 0.211991
\(61\) 5.53341 0.708480 0.354240 − 0.935154i \(-0.384740\pi\)
0.354240 + 0.935154i \(0.384740\pi\)
\(62\) −3.07378 −0.390370
\(63\) 1.75698 0.221359
\(64\) −6.51396 −0.814245
\(65\) 0.128111 0.0158902
\(66\) −1.45339 −0.178900
\(67\) 8.64832 1.05656 0.528280 − 0.849070i \(-0.322837\pi\)
0.528280 + 0.849070i \(0.322837\pi\)
\(68\) −15.2361 −1.84764
\(69\) −9.57454 −1.15264
\(70\) 0.885092 0.105789
\(71\) −7.53341 −0.894051 −0.447026 − 0.894521i \(-0.647517\pi\)
−0.447026 + 0.894521i \(0.647517\pi\)
\(72\) −9.18869 −1.08290
\(73\) −9.00624 −1.05410 −0.527051 − 0.849834i \(-0.676702\pi\)
−0.527051 + 0.849834i \(0.676702\pi\)
\(74\) 8.16924 0.949655
\(75\) 5.43171 0.627199
\(76\) 4.11491 0.472012
\(77\) −0.527166 −0.0600762
\(78\) 0.986796 0.111733
\(79\) −9.17548 −1.03232 −0.516161 − 0.856491i \(-0.672639\pi\)
−0.516161 + 0.856491i \(0.672639\pi\)
\(80\) −1.68320 −0.188188
\(81\) −0.642074 −0.0713415
\(82\) 30.0753 3.32126
\(83\) 16.0606 1.76288 0.881439 − 0.472299i \(-0.156576\pi\)
0.881439 + 0.472299i \(0.156576\pi\)
\(84\) 4.58774 0.500564
\(85\) 1.32528 0.143746
\(86\) −1.77018 −0.190884
\(87\) 4.50548 0.483039
\(88\) 2.75698 0.293895
\(89\) −14.7981 −1.56860 −0.784298 − 0.620384i \(-0.786977\pi\)
−0.784298 + 0.620384i \(0.786977\pi\)
\(90\) 1.55509 0.163921
\(91\) 0.357926 0.0375209
\(92\) 35.3378 3.68422
\(93\) 1.38585 0.143706
\(94\) 26.7500 2.75905
\(95\) −0.357926 −0.0367225
\(96\) −1.30359 −0.133048
\(97\) −9.07378 −0.921303 −0.460651 − 0.887581i \(-0.652384\pi\)
−0.460651 + 0.887581i \(0.652384\pi\)
\(98\) 2.47283 0.249794
\(99\) −0.926221 −0.0930887
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 133.2.a.d.1.3 ✓ 3
3.2 odd 2 1197.2.a.k.1.1 3
4.3 odd 2 2128.2.a.p.1.3 3
5.4 even 2 3325.2.a.r.1.1 3
7.2 even 3 931.2.f.l.704.1 6
7.3 odd 6 931.2.f.m.324.1 6
7.4 even 3 931.2.f.l.324.1 6
7.5 odd 6 931.2.f.m.704.1 6
7.6 odd 2 931.2.a.k.1.3 3
8.3 odd 2 8512.2.a.bp.1.1 3
8.5 even 2 8512.2.a.bi.1.3 3
19.18 odd 2 2527.2.a.f.1.1 3
21.20 even 2 8379.2.a.bo.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
133.2.a.d.1.3 ✓ 3 1.1 even 1 trivial
931.2.a.k.1.3 3 7.6 odd 2
931.2.f.l.324.1 6 7.4 even 3
931.2.f.l.704.1 6 7.2 even 3
931.2.f.m.324.1 6 7.3 odd 6
931.2.f.m.704.1 6 7.5 odd 6
1197.2.a.k.1.1 3 3.2 odd 2
2128.2.a.p.1.3 3 4.3 odd 2
2527.2.a.f.1.1 3 19.18 odd 2
3325.2.a.r.1.1 3 5.4 even 2
8379.2.a.bo.1.1 3 21.20 even 2
8512.2.a.bi.1.3 3 8.5 even 2
8512.2.a.bp.1.1 3 8.3 odd 2