Properties

Label 11.10.a.a.1.2
Level $11$
Weight $10$
Character 11.1
Self dual yes
Analytic conductor $5.665$
Analytic rank $1$
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] = [11,10,Mod(1,11)] 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("11.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(11, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 11.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.66539419780\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.2659452.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 306x - 836 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.80408\) of defining polynomial
Character \(\chi\) \(=\) 11.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.60816 q^{2} +5.22371 q^{3} -480.549 q^{4} -529.708 q^{5} +29.2954 q^{6} -3708.24 q^{7} -5566.37 q^{8} -19655.7 q^{9} -2970.69 q^{10} -14641.0 q^{11} -2510.24 q^{12} +30987.3 q^{13} -20796.4 q^{14} -2767.04 q^{15} +214824. q^{16} +250862. q^{17} -110232. q^{18} +438057. q^{19} +254551. q^{20} -19370.8 q^{21} -82109.0 q^{22} -1.78151e6 q^{23} -29077.1 q^{24} -1.67253e6 q^{25} +173782. q^{26} -205494. q^{27} +1.78199e6 q^{28} -2.59808e6 q^{29} -15518.0 q^{30} -2.09835e6 q^{31} +4.05475e6 q^{32} -76480.3 q^{33} +1.40687e6 q^{34} +1.96429e6 q^{35} +9.44552e6 q^{36} +2.62219e6 q^{37} +2.45669e6 q^{38} +161869. q^{39} +2.94855e6 q^{40} -2.91565e6 q^{41} -108634. q^{42} -3.01262e7 q^{43} +7.03571e6 q^{44} +1.04118e7 q^{45} -9.99100e6 q^{46} +1.45140e6 q^{47} +1.12218e6 q^{48} -2.66025e7 q^{49} -9.37983e6 q^{50} +1.31043e6 q^{51} -1.48909e7 q^{52} -3.32876e7 q^{53} -1.15244e6 q^{54} +7.75546e6 q^{55} +2.06415e7 q^{56} +2.28828e6 q^{57} -1.45704e7 q^{58} +1.40814e8 q^{59} +1.32970e6 q^{60} +1.32753e8 q^{61} -1.17678e7 q^{62} +7.28882e7 q^{63} -8.72501e7 q^{64} -1.64142e7 q^{65} -428913. q^{66} -4.13586e7 q^{67} -1.20551e8 q^{68} -9.30609e6 q^{69} +1.10160e7 q^{70} +1.81824e8 q^{71} +1.09411e8 q^{72} -4.13968e8 q^{73} +1.47056e7 q^{74} -8.73683e6 q^{75} -2.10508e8 q^{76} +5.42924e7 q^{77} +907785. q^{78} -5.90824e8 q^{79} -1.13794e8 q^{80} +3.85810e8 q^{81} -1.63514e7 q^{82} +6.29014e8 q^{83} +9.30860e6 q^{84} -1.32884e8 q^{85} -1.68953e8 q^{86} -1.35716e7 q^{87} +8.14972e7 q^{88} +7.73395e8 q^{89} +5.83910e7 q^{90} -1.14909e8 q^{91} +8.56103e8 q^{92} -1.09611e7 q^{93} +8.13966e6 q^{94} -2.32042e8 q^{95} +2.11808e7 q^{96} -1.00362e9 q^{97} -1.49191e8 q^{98} +2.87779e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 186 q^{3} + 912 q^{4} - 1824 q^{5} - 10956 q^{6} - 7260 q^{7} - 20064 q^{8} + 14553 q^{9} - 86724 q^{10} - 43923 q^{11} - 71040 q^{12} - 93258 q^{13} + 324264 q^{14} + 540330 q^{15} + 22656 q^{16}+ \cdots - 213070473 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.60816 0.247848 0.123924 − 0.992292i \(-0.460452\pi\)
0.123924 + 0.992292i \(0.460452\pi\)
\(3\) 5.22371 0.0372334 0.0186167 − 0.999827i \(-0.494074\pi\)
0.0186167 + 0.999827i \(0.494074\pi\)
\(4\) −480.549 −0.938571
\(5\) −529.708 −0.379028 −0.189514 − 0.981878i \(-0.560691\pi\)
−0.189514 + 0.981878i \(0.560691\pi\)
\(6\) 29.2954 0.00922823
\(7\) −3708.24 −0.583750 −0.291875 − 0.956456i \(-0.594279\pi\)
−0.291875 + 0.956456i \(0.594279\pi\)
\(8\) −5566.37 −0.480471
\(9\) −19655.7 −0.998614
\(10\) −2970.69 −0.0939414
\(11\) −14641.0 −0.301511
\(12\) −2510.24 −0.0349462
\(13\) 30987.3 0.300912 0.150456 − 0.988617i \(-0.451926\pi\)
0.150456 + 0.988617i \(0.451926\pi\)
\(14\) −20796.4 −0.144681
\(15\) −2767.04 −0.0141125
\(16\) 214824. 0.819488
\(17\) 250862. 0.728476 0.364238 − 0.931306i \(-0.381330\pi\)
0.364238 + 0.931306i \(0.381330\pi\)
\(18\) −110232. −0.247504
\(19\) 438057. 0.771150 0.385575 − 0.922676i \(-0.374003\pi\)
0.385575 + 0.922676i \(0.374003\pi\)
\(20\) 254551. 0.355745
\(21\) −19370.8 −0.0217350
\(22\) −82109.0 −0.0747289
\(23\) −1.78151e6 −1.32744 −0.663718 − 0.747983i \(-0.731022\pi\)
−0.663718 + 0.747983i \(0.731022\pi\)
\(24\) −29077.1 −0.0178896
\(25\) −1.67253e6 −0.856337
\(26\) 173782. 0.0745803
\(27\) −205494. −0.0744153
\(28\) 1.78199e6 0.547891
\(29\) −2.59808e6 −0.682120 −0.341060 − 0.940041i \(-0.610786\pi\)
−0.341060 + 0.940041i \(0.610786\pi\)
\(30\) −15518.0 −0.00349776
\(31\) −2.09835e6 −0.408084 −0.204042 − 0.978962i \(-0.565408\pi\)
−0.204042 + 0.978962i \(0.565408\pi\)
\(32\) 4.05475e6 0.683579
\(33\) −76480.3 −0.0112263
\(34\) 1.40687e6 0.180551
\(35\) 1.96429e6 0.221258
\(36\) 9.44552e6 0.937270
\(37\) 2.62219e6 0.230015 0.115008 − 0.993365i \(-0.463311\pi\)
0.115008 + 0.993365i \(0.463311\pi\)
\(38\) 2.45669e6 0.191128
\(39\) 161869. 0.0112040
\(40\) 2.94855e6 0.182112
\(41\) −2.91565e6 −0.161142 −0.0805708 − 0.996749i \(-0.525674\pi\)
−0.0805708 + 0.996749i \(0.525674\pi\)
\(42\) −108634. −0.00538698
\(43\) −3.01262e7 −1.34381 −0.671903 − 0.740639i \(-0.734523\pi\)
−0.671903 + 0.740639i \(0.734523\pi\)
\(44\) 7.03571e6 0.282990
\(45\) 1.04118e7 0.378503
\(46\) −9.99100e6 −0.329002
\(47\) 1.45140e6 0.0433856 0.0216928 − 0.999765i \(-0.493094\pi\)
0.0216928 + 0.999765i \(0.493094\pi\)
\(48\) 1.12218e6 0.0305123
\(49\) −2.66025e7 −0.659236
\(50\) −9.37983e6 −0.212241
\(51\) 1.31043e6 0.0271236
\(52\) −1.48909e7 −0.282427
\(53\) −3.32876e7 −0.579483 −0.289742 − 0.957105i \(-0.593569\pi\)
−0.289742 + 0.957105i \(0.593569\pi\)
\(54\) −1.15244e6 −0.0184437
\(55\) 7.75546e6 0.114281
\(56\) 2.06415e7 0.280475
\(57\) 2.28828e6 0.0287126
\(58\) −1.45704e7 −0.169062
\(59\) 1.40814e8 1.51291 0.756455 − 0.654045i \(-0.226929\pi\)
0.756455 + 0.654045i \(0.226929\pi\)
\(60\) 1.32970e6 0.0132456
\(61\) 1.32753e8 1.22761 0.613806 − 0.789457i \(-0.289638\pi\)
0.613806 + 0.789457i \(0.289638\pi\)
\(62\) −1.17678e7 −0.101143
\(63\) 7.28882e7 0.582941
\(64\) −8.72501e7 −0.650064
\(65\) −1.64142e7 −0.114054
\(66\) −428913. −0.00278242
\(67\) −4.13586e7 −0.250743 −0.125372 − 0.992110i \(-0.540012\pi\)
−0.125372 + 0.992110i \(0.540012\pi\)
\(68\) −1.20551e8 −0.683726
\(69\) −9.30609e6 −0.0494250
\(70\) 1.10160e7 0.0548383
\(71\) 1.81824e8 0.849161 0.424580 − 0.905390i \(-0.360422\pi\)
0.424580 + 0.905390i \(0.360422\pi\)
\(72\) 1.09411e8 0.479805
\(73\) −4.13968e8 −1.70614 −0.853068 − 0.521800i \(-0.825261\pi\)
−0.853068 + 0.521800i \(0.825261\pi\)
\(74\) 1.47056e7 0.0570087
\(75\) −8.73683e6 −0.0318844
\(76\) −2.10508e8 −0.723780
\(77\) 5.42924e7 0.176007
\(78\) 907785. 0.00277688
\(79\) −5.90824e8 −1.70662 −0.853308 − 0.521407i \(-0.825407\pi\)
−0.853308 + 0.521407i \(0.825407\pi\)
\(80\) −1.13794e8 −0.310609
\(81\) 3.85810e8 0.995843
\(82\) −1.63514e7 −0.0399386
\(83\) 6.29014e8 1.45482 0.727409 − 0.686204i \(-0.240724\pi\)
0.727409 + 0.686204i \(0.240724\pi\)
\(84\) 9.30860e6 0.0203999
\(85\) −1.32884e8 −0.276113
\(86\) −1.68953e8 −0.333059
\(87\) −1.35716e7 −0.0253977
\(88\) 8.14972e7 0.144867
\(89\) 7.73395e8 1.30661 0.653305 − 0.757095i \(-0.273382\pi\)
0.653305 + 0.757095i \(0.273382\pi\)
\(90\) 5.83910e7 0.0938112
\(91\) −1.14909e8 −0.175657
\(92\) 8.56103e8 1.24589
\(93\) −1.09611e7 −0.0151944
\(94\) 8.13966e6 0.0107530
\(95\) −2.32042e8 −0.292288
\(96\) 2.11808e7 0.0254520
\(97\) −1.00362e9 −1.15105 −0.575527 − 0.817783i \(-0.695203\pi\)
−0.575527 + 0.817783i \(0.695203\pi\)
\(98\) −1.49191e8 −0.163390
\(99\) 2.87779e8 0.301093
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 11.10.a.a.1.2 ✓ 3
3.2 odd 2 99.10.a.b.1.2 3
4.3 odd 2 176.10.a.g.1.2 3
5.4 even 2 275.10.a.a.1.2 3
11.10 odd 2 121.10.a.b.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.10.a.a.1.2 ✓ 3 1.1 even 1 trivial
99.10.a.b.1.2 3 3.2 odd 2
121.10.a.b.1.2 3 11.10 odd 2
176.10.a.g.1.2 3 4.3 odd 2
275.10.a.a.1.2 3 5.4 even 2