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Properties

Label 11.6.a.b.1.2
Level $11$
Weight $6$
Character 11.1
Self dual yes
Analytic conductor $1.764$
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] = [11,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(1.76422201794\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.54492.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 52x - 38 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(8.04796\) 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)
 
\(f(q)\) \(=\) \(q+2.20859 q^{2} +16.8394 q^{3} -27.1221 q^{4} +75.2230 q^{5} +37.1913 q^{6} -225.525 q^{7} -130.577 q^{8} +40.5643 q^{9} +166.137 q^{10} +121.000 q^{11} -456.719 q^{12} +455.465 q^{13} -498.092 q^{14} +1266.71 q^{15} +579.518 q^{16} +190.657 q^{17} +89.5900 q^{18} -135.393 q^{19} -2040.21 q^{20} -3797.69 q^{21} +267.240 q^{22} +2796.65 q^{23} -2198.83 q^{24} +2533.51 q^{25} +1005.94 q^{26} -3408.89 q^{27} +6116.71 q^{28} -2608.58 q^{29} +2797.64 q^{30} -1056.76 q^{31} +5458.37 q^{32} +2037.56 q^{33} +421.082 q^{34} -16964.7 q^{35} -1100.19 q^{36} +12536.8 q^{37} -299.028 q^{38} +7669.74 q^{39} -9822.37 q^{40} +1130.09 q^{41} -8387.55 q^{42} -14671.0 q^{43} -3281.78 q^{44} +3051.37 q^{45} +6176.65 q^{46} -16882.2 q^{47} +9758.71 q^{48} +34054.4 q^{49} +5595.48 q^{50} +3210.54 q^{51} -12353.2 q^{52} +3313.02 q^{53} -7528.84 q^{54} +9101.99 q^{55} +29448.3 q^{56} -2279.93 q^{57} -5761.29 q^{58} +11454.0 q^{59} -34355.8 q^{60} -28227.5 q^{61} -2333.95 q^{62} -9148.26 q^{63} -6489.25 q^{64} +34261.4 q^{65} +4500.15 q^{66} -51431.0 q^{67} -5171.01 q^{68} +47093.8 q^{69} -37468.0 q^{70} -16218.0 q^{71} -5296.75 q^{72} -10168.8 q^{73} +27688.7 q^{74} +42662.6 q^{75} +3672.15 q^{76} -27288.5 q^{77} +16939.3 q^{78} +60841.2 q^{79} +43593.1 q^{80} -67260.7 q^{81} +2495.90 q^{82} +45770.6 q^{83} +103002. q^{84} +14341.8 q^{85} -32402.3 q^{86} -43926.9 q^{87} -15799.8 q^{88} -82267.9 q^{89} +6739.23 q^{90} -102719. q^{91} -75851.0 q^{92} -17795.1 q^{93} -37285.8 q^{94} -10184.7 q^{95} +91915.5 q^{96} +53097.0 q^{97} +75212.2 q^{98} +4908.28 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 34 q^{3} + 84 q^{4} + 24 q^{5} - 206 q^{6} + 84 q^{7} - 564 q^{8} - 7 q^{9} - 414 q^{10} + 363 q^{11} + 992 q^{12} + 486 q^{13} - 1020 q^{14} + 1654 q^{15} + 1992 q^{16} + 1086 q^{17} - 3706 q^{18}+ \cdots - 847 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.20859 0.390428 0.195214 − 0.980761i \(-0.437460\pi\)
0.195214 + 0.980761i \(0.437460\pi\)
\(3\) 16.8394 1.08025 0.540123 − 0.841586i \(-0.318378\pi\)
0.540123 + 0.841586i \(0.318378\pi\)
\(4\) −27.1221 −0.847566
\(5\) 75.2230 1.34563 0.672815 − 0.739810i \(-0.265085\pi\)
0.672815 + 0.739810i \(0.265085\pi\)
\(6\) 37.1913 0.421758
\(7\) −225.525 −1.73960 −0.869799 − 0.493406i \(-0.835752\pi\)
−0.869799 + 0.493406i \(0.835752\pi\)
\(8\) −130.577 −0.721341
\(9\) 40.5643 0.166931
\(10\) 166.137 0.525371
\(11\) 121.000 0.301511
\(12\) −456.719 −0.915580
\(13\) 455.465 0.747474 0.373737 − 0.927535i \(-0.378076\pi\)
0.373737 + 0.927535i \(0.378076\pi\)
\(14\) −498.092 −0.679187
\(15\) 1266.71 1.45361
\(16\) 579.518 0.565935
\(17\) 190.657 0.160003 0.0800017 − 0.996795i \(-0.474507\pi\)
0.0800017 + 0.996795i \(0.474507\pi\)
\(18\) 89.5900 0.0651746
\(19\) −135.393 −0.0860424 −0.0430212 − 0.999074i \(-0.513698\pi\)
−0.0430212 + 0.999074i \(0.513698\pi\)
\(20\) −2040.21 −1.14051
\(21\) −3797.69 −1.87919
\(22\) 267.240 0.117718
\(23\) 2796.65 1.10235 0.551173 − 0.834391i \(-0.314180\pi\)
0.551173 + 0.834391i \(0.314180\pi\)
\(24\) −2198.83 −0.779225
\(25\) 2533.51 0.810722
\(26\) 1005.94 0.291835
\(27\) −3408.89 −0.899919
\(28\) 6116.71 1.47443
\(29\) −2608.58 −0.575983 −0.287991 − 0.957633i \(-0.592987\pi\)
−0.287991 + 0.957633i \(0.592987\pi\)
\(30\) 2797.64 0.567530
\(31\) −1056.76 −0.197502 −0.0987510 − 0.995112i \(-0.531485\pi\)
−0.0987510 + 0.995112i \(0.531485\pi\)
\(32\) 5458.37 0.942297
\(33\) 2037.56 0.325706
\(34\) 421.082 0.0624697
\(35\) −16964.7 −2.34086
\(36\) −1100.19 −0.141485
\(37\) 12536.8 1.50550 0.752752 − 0.658304i \(-0.228726\pi\)
0.752752 + 0.658304i \(0.228726\pi\)
\(38\) −299.028 −0.0335933
\(39\) 7669.74 0.807456
\(40\) −9822.37 −0.970658
\(41\) 1130.09 0.104991 0.0524954 − 0.998621i \(-0.483282\pi\)
0.0524954 + 0.998621i \(0.483282\pi\)
\(42\) −8387.55 −0.733689
\(43\) −14671.0 −1.21001 −0.605005 − 0.796222i \(-0.706829\pi\)
−0.605005 + 0.796222i \(0.706829\pi\)
\(44\) −3281.78 −0.255551
\(45\) 3051.37 0.224628
\(46\) 6176.65 0.430386
\(47\) −16882.2 −1.11477 −0.557383 − 0.830256i \(-0.688194\pi\)
−0.557383 + 0.830256i \(0.688194\pi\)
\(48\) 9758.71 0.611349
\(49\) 34054.4 2.02620
\(50\) 5595.48 0.316528
\(51\) 3210.54 0.172843
\(52\) −12353.2 −0.633534
\(53\) 3313.02 0.162007 0.0810035 − 0.996714i \(-0.474187\pi\)
0.0810035 + 0.996714i \(0.474187\pi\)
\(54\) −7528.84 −0.351353
\(55\) 9101.99 0.405723
\(56\) 29448.3 1.25484
\(57\) −2279.93 −0.0929469
\(58\) −5761.29 −0.224880
\(59\) 11454.0 0.428378 0.214189 − 0.976792i \(-0.431289\pi\)
0.214189 + 0.976792i \(0.431289\pi\)
\(60\) −34355.8 −1.23203
\(61\) −28227.5 −0.971286 −0.485643 − 0.874157i \(-0.661415\pi\)
−0.485643 + 0.874157i \(0.661415\pi\)
\(62\) −2333.95 −0.0771102
\(63\) −9148.26 −0.290394
\(64\) −6489.25 −0.198036
\(65\) 34261.4 1.00582
\(66\) 4500.15 0.127165
\(67\) −51431.0 −1.39971 −0.699855 − 0.714285i \(-0.746752\pi\)
−0.699855 + 0.714285i \(0.746752\pi\)
\(68\) −5171.01 −0.135614
\(69\) 47093.8 1.19081
\(70\) −37468.0 −0.913935
\(71\) −16218.0 −0.381814 −0.190907 − 0.981608i \(-0.561143\pi\)
−0.190907 + 0.981608i \(0.561143\pi\)
\(72\) −5296.75 −0.120414
\(73\) −10168.8 −0.223337 −0.111669 − 0.993745i \(-0.535620\pi\)
−0.111669 + 0.993745i \(0.535620\pi\)
\(74\) 27688.7 0.587791
\(75\) 42662.6 0.875779
\(76\) 3672.15 0.0729266
\(77\) −27288.5 −0.524509
\(78\) 16939.3 0.315253
\(79\) 60841.2 1.09681 0.548404 − 0.836214i \(-0.315236\pi\)
0.548404 + 0.836214i \(0.315236\pi\)
\(80\) 43593.1 0.761540
\(81\) −67260.7 −1.13907
\(82\) 2495.90 0.0409913
\(83\) 45770.6 0.729275 0.364638 − 0.931150i \(-0.381193\pi\)
0.364638 + 0.931150i \(0.381193\pi\)
\(84\) 103002. 1.59274
\(85\) 14341.8 0.215306
\(86\) −32402.3 −0.472421
\(87\) −43926.9 −0.622203
\(88\) −15799.8 −0.217492
\(89\) −82267.9 −1.10092 −0.550460 − 0.834862i \(-0.685548\pi\)
−0.550460 + 0.834862i \(0.685548\pi\)
\(90\) 6739.23 0.0877009
\(91\) −102719. −1.30031
\(92\) −75851.0 −0.934312
\(93\) −17795.1 −0.213351
\(94\) −37285.8 −0.435235
\(95\) −10184.7 −0.115781
\(96\) 91915.5 1.01791
\(97\) 53097.0 0.572981 0.286491 − 0.958083i \(-0.407511\pi\)
0.286491 + 0.958083i \(0.407511\pi\)
\(98\) 75212.2 0.791085
\(99\) 4908.28 0.0503317
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 11.6.a.b.1.2 ✓ 3
3.2 odd 2 99.6.a.g.1.2 3
4.3 odd 2 176.6.a.i.1.2 3
5.2 odd 4 275.6.b.b.199.4 6
5.3 odd 4 275.6.b.b.199.3 6
5.4 even 2 275.6.a.b.1.2 3
7.6 odd 2 539.6.a.e.1.2 3
8.3 odd 2 704.6.a.t.1.2 3
8.5 even 2 704.6.a.q.1.2 3
11.10 odd 2 121.6.a.d.1.2 3
33.32 even 2 1089.6.a.r.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.2 ✓ 3 1.1 even 1 trivial
99.6.a.g.1.2 3 3.2 odd 2
121.6.a.d.1.2 3 11.10 odd 2
176.6.a.i.1.2 3 4.3 odd 2
275.6.a.b.1.2 3 5.4 even 2
275.6.b.b.199.3 6 5.3 odd 4
275.6.b.b.199.4 6 5.2 odd 4
539.6.a.e.1.2 3 7.6 odd 2
704.6.a.q.1.2 3 8.5 even 2
704.6.a.t.1.2 3 8.3 odd 2
1089.6.a.r.1.2 3 33.32 even 2