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Properties

Label 121.2.e.a.12.4
Level $121$
Weight $2$
Character 121.12
Analytic conductor $0.966$
Analytic rank $0$
Dimension $100$
Inner twists $2$

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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] = [121,2,Mod(12,121)] 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("121.12"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 121.e (of order \(11\), degree \(10\), minimal)

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: no
Analytic conductor: \(0.966189864457\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(10\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 12.4
Character \(\chi\) \(=\) 121.12
Dual form 121.2.e.a.111.4

$q$-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+(-0.800834 + 0.514665i) q^{2} -2.23328 q^{3} +(-0.454375 + 0.994941i) q^{4} +(0.373665 - 2.59890i) q^{5} +(1.78848 - 1.14939i) q^{6} +(1.22249 - 1.41083i) q^{7} +(-0.419137 - 2.91516i) q^{8} +1.98753 q^{9} +(1.03832 + 2.27360i) q^{10} +(-1.13514 - 3.11632i) q^{11} +(1.01474 - 2.22198i) q^{12} +(0.980538 - 2.14708i) q^{13} +(-0.252907 + 1.75901i) q^{14} +(-0.834498 + 5.80406i) q^{15} +(0.403438 + 0.465592i) q^{16} +(1.54007 + 0.452204i) q^{17} +(-1.59168 + 1.02291i) q^{18} +(-7.56971 + 2.22267i) q^{19} +(2.41597 + 1.55265i) q^{20} +(-2.73015 + 3.15077i) q^{21} +(2.51292 + 1.91144i) q^{22} +(-3.47230 - 4.00725i) q^{23} +(0.936049 + 6.51036i) q^{24} +(-1.81719 - 0.533574i) q^{25} +(0.319778 + 2.22410i) q^{26} +2.26113 q^{27} +(0.848222 + 1.85735i) q^{28} +(1.90926 - 0.560610i) q^{29} +(-2.31885 - 5.07758i) q^{30} +(1.29379 + 2.83300i) q^{31} +(5.08897 + 1.49426i) q^{32} +(2.53509 + 6.95961i) q^{33} +(-1.46607 + 0.430477i) q^{34} +(-3.20979 - 3.70430i) q^{35} +(-0.903081 + 1.97747i) q^{36} +(-4.74993 - 10.4009i) q^{37} +(4.91815 - 5.67585i) q^{38} +(-2.18981 + 4.79502i) q^{39} -7.73283 q^{40} +(-0.791510 + 0.508673i) q^{41} +(0.564812 - 3.92836i) q^{42} +(0.791045 + 5.50184i) q^{43} +(3.61634 + 0.286577i) q^{44} +(0.742670 - 5.16538i) q^{45} +(4.84313 + 1.42207i) q^{46} +(-0.488014 - 0.313627i) q^{47} +(-0.900988 - 1.03980i) q^{48} +(0.500250 + 3.47931i) q^{49} +(1.72988 - 0.507938i) q^{50} +(-3.43939 - 1.00990i) q^{51} +(1.69069 + 1.95116i) q^{52} +(7.29818 - 8.42255i) q^{53} +(-1.81079 + 1.16373i) q^{54} +(-8.52317 + 1.78566i) q^{55} +(-4.62518 - 2.97242i) q^{56} +(16.9053 - 4.96383i) q^{57} +(-1.24048 + 1.43159i) q^{58} +(-6.04650 - 3.88585i) q^{59} +(-5.39553 - 3.46749i) q^{60} +(6.69743 + 4.30418i) q^{61} +(-2.49416 - 1.60290i) q^{62} +(2.42973 - 2.80405i) q^{63} +(-6.02669 + 1.76959i) q^{64} +(-5.21365 - 3.35061i) q^{65} +(-5.61205 - 4.26877i) q^{66} +(4.73683 - 3.04417i) q^{67} +(-1.14968 + 1.32681i) q^{68} +(7.75461 + 8.94929i) q^{69} +(4.47699 + 1.31456i) q^{70} +(12.5864 - 3.69569i) q^{71} +(-0.833046 - 5.79396i) q^{72} +(5.54649 + 6.40099i) q^{73} +(9.15687 + 5.88477i) q^{74} +(4.05828 + 1.19162i) q^{75} +(1.22806 - 8.54134i) q^{76} +(-5.78428 - 2.20818i) q^{77} +(-0.714152 - 4.96704i) q^{78} +(-0.523964 + 3.64425i) q^{79} +(1.36078 - 0.874518i) q^{80} -11.0123 q^{81} +(0.372072 - 0.814725i) q^{82} +(-10.5043 + 12.1227i) q^{83} +(-1.89432 - 4.14797i) q^{84} +(1.75070 - 3.83350i) q^{85} +(-3.46510 - 3.99894i) q^{86} +(-4.26391 + 1.25200i) q^{87} +(-8.60880 + 4.61529i) q^{88} +(-2.25220 - 0.661305i) q^{89} +(2.06368 + 4.51884i) q^{90} +(-1.83046 - 4.00815i) q^{91} +(5.56470 - 1.63394i) q^{92} +(-2.88939 - 6.32688i) q^{93} +0.552231 q^{94} +(2.94795 + 20.5034i) q^{95} +(-11.3651 - 3.33709i) q^{96} +(-0.589344 - 4.09898i) q^{97} +(-2.19130 - 2.52889i) q^{98} +(-2.25612 - 6.19377i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 100 q - 6 q^{2} - 18 q^{3} - 16 q^{4} - 7 q^{5} - 23 q^{6} - q^{7} + 4 q^{8} + 70 q^{9} - 13 q^{10} - 12 q^{11} - 51 q^{12} - 34 q^{13} - 17 q^{14} - 46 q^{15} + 10 q^{16} + 9 q^{17} - 31 q^{18} + 9 q^{19}+ \cdots - 53 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{9}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.800834 + 0.514665i −0.566275 + 0.363923i −0.792237 − 0.610213i \(-0.791084\pi\)
0.225962 + 0.974136i \(0.427447\pi\)
\(3\) −2.23328 −1.28938 −0.644692 − 0.764443i \(-0.723014\pi\)
−0.644692 + 0.764443i \(0.723014\pi\)
\(4\) −0.454375 + 0.994941i −0.227187 + 0.497471i
\(5\) 0.373665 − 2.59890i 0.167108 − 1.16226i −0.717715 − 0.696337i \(-0.754812\pi\)
0.884823 − 0.465926i \(-0.154279\pi\)
\(6\) 1.78848 − 1.14939i 0.730146 − 0.469236i
\(7\) 1.22249 − 1.41083i 0.462057 − 0.533242i −0.476128 − 0.879376i \(-0.657960\pi\)
0.938185 + 0.346134i \(0.112506\pi\)
\(8\) −0.419137 − 2.91516i −0.148187 − 1.03067i
\(9\) 1.98753 0.662509
\(10\) 1.03832 + 2.27360i 0.328345 + 0.718975i
\(11\) −1.13514 − 3.11632i −0.342258 − 0.939606i
\(12\) 1.01474 − 2.22198i 0.292931 − 0.641430i
\(13\) 0.980538 − 2.14708i 0.271952 − 0.595493i −0.723545 − 0.690277i \(-0.757489\pi\)
0.995498 + 0.0947841i \(0.0302161\pi\)
\(14\) −0.252907 + 1.75901i −0.0675923 + 0.470115i
\(15\) −0.834498 + 5.80406i −0.215466 + 1.49860i
\(16\) 0.403438 + 0.465592i 0.100859 + 0.116398i
\(17\) 1.54007 + 0.452204i 0.373521 + 0.109676i 0.463105 − 0.886304i \(-0.346735\pi\)
−0.0895836 + 0.995979i \(0.528554\pi\)
\(18\) −1.59168 + 1.02291i −0.375162 + 0.241102i
\(19\) −7.56971 + 2.22267i −1.73661 + 0.509915i −0.988180 − 0.153299i \(-0.951010\pi\)
−0.748430 + 0.663214i \(0.769192\pi\)
\(20\) 2.41597 + 1.55265i 0.540227 + 0.347183i
\(21\) −2.73015 + 3.15077i −0.595768 + 0.687553i
\(22\) 2.51292 + 1.91144i 0.535757 + 0.407520i
\(23\) −3.47230 − 4.00725i −0.724025 − 0.835569i 0.267760 − 0.963486i \(-0.413716\pi\)
−0.991785 + 0.127917i \(0.959171\pi\)
\(24\) 0.936049 + 6.51036i 0.191070 + 1.32892i
\(25\) −1.81719 − 0.533574i −0.363437 − 0.106715i
\(26\) 0.319778 + 2.22410i 0.0627136 + 0.436183i
\(27\) 2.26113 0.435156
\(28\) 0.848222 + 1.85735i 0.160299 + 0.351006i
\(29\) 1.90926 − 0.560610i 0.354541 − 0.104103i −0.0996118 − 0.995026i \(-0.531760\pi\)
0.454153 + 0.890924i \(0.349942\pi\)
\(30\) −2.31885 − 5.07758i −0.423363 − 0.927035i
\(31\) 1.29379 + 2.83300i 0.232371 + 0.508823i 0.989516 − 0.144425i \(-0.0461333\pi\)
−0.757144 + 0.653248i \(0.773406\pi\)
\(32\) 5.08897 + 1.49426i 0.899611 + 0.264150i
\(33\) 2.53509 + 6.95961i 0.441302 + 1.21151i
\(34\) −1.46607 + 0.430477i −0.251429 + 0.0738263i
\(35\) −3.20979 − 3.70430i −0.542554 − 0.626141i
\(36\) −0.903081 + 1.97747i −0.150514 + 0.329579i
\(37\) −4.74993 − 10.4009i −0.780883 − 1.70990i −0.701078 − 0.713085i \(-0.747298\pi\)
−0.0798050 − 0.996810i \(-0.525430\pi\)
\(38\) 4.91815 − 5.67585i 0.797830 − 0.920744i
\(39\) −2.18981 + 4.79502i −0.350651 + 0.767818i
\(40\) −7.73283 −1.22267
\(41\) −0.791510 + 0.508673i −0.123613 + 0.0794413i −0.600986 − 0.799259i \(-0.705225\pi\)
0.477373 + 0.878701i \(0.341589\pi\)
\(42\) 0.564812 − 3.92836i 0.0871524 − 0.606158i
\(43\) 0.791045 + 5.50184i 0.120633 + 0.839022i 0.956842 + 0.290610i \(0.0938583\pi\)
−0.836208 + 0.548412i \(0.815233\pi\)
\(44\) 3.61634 + 0.286577i 0.545183 + 0.0432030i
\(45\) 0.742670 − 5.16538i 0.110711 − 0.770009i
\(46\) 4.84313 + 1.42207i 0.714080 + 0.209673i
\(47\) −0.488014 − 0.313627i −0.0711841 − 0.0457472i 0.504565 − 0.863374i \(-0.331653\pi\)
−0.575749 + 0.817627i \(0.695289\pi\)
\(48\) −0.900988 − 1.03980i −0.130046 − 0.150082i
\(49\) 0.500250 + 3.47931i 0.0714643 + 0.497045i
\(50\) 1.72988 − 0.507938i 0.244642 − 0.0718333i
\(51\) −3.43939 − 1.00990i −0.481612 − 0.141414i
\(52\) 1.69069 + 1.95116i 0.234456 + 0.270577i
\(53\) 7.29818 − 8.42255i 1.00248 − 1.15693i 0.0148897 − 0.999889i \(-0.495260\pi\)
0.987593 − 0.157037i \(-0.0501943\pi\)
\(54\) −1.81079 + 1.16373i −0.246418 + 0.158363i
\(55\) −8.52317 + 1.78566i −1.14926 + 0.240778i
\(56\) −4.62518 − 2.97242i −0.618065 − 0.397206i
\(57\) 16.9053 − 4.96383i 2.23916 − 0.657475i
\(58\) −1.24048 + 1.43159i −0.162883 + 0.187977i
\(59\) −6.04650 − 3.88585i −0.787187 − 0.505894i 0.0842246 − 0.996447i \(-0.473159\pi\)
−0.871412 + 0.490552i \(0.836795\pi\)
\(60\) −5.39553 − 3.46749i −0.696560 − 0.447652i
\(61\) 6.69743 + 4.30418i 0.857518 + 0.551093i 0.893911 − 0.448244i \(-0.147951\pi\)
−0.0363933 + 0.999338i \(0.511587\pi\)
\(62\) −2.49416 − 1.60290i −0.316759 − 0.203568i
\(63\) 2.42973 − 2.80405i 0.306117 − 0.353278i
\(64\) −6.02669 + 1.76959i −0.753336 + 0.221199i
\(65\) −5.21365 − 3.35061i −0.646674 − 0.415592i
\(66\) −5.61205 − 4.26877i −0.690796 − 0.525449i
\(67\) 4.73683 − 3.04417i 0.578696 − 0.371905i −0.218308 − 0.975880i \(-0.570054\pi\)
0.797003 + 0.603975i \(0.206417\pi\)
\(68\) −1.14968 + 1.32681i −0.139420 + 0.160899i
\(69\) 7.75461 + 8.94929i 0.933545 + 1.07737i
\(70\) 4.47699 + 1.31456i 0.535102 + 0.157120i
\(71\) 12.5864 − 3.69569i 1.49373 − 0.438598i 0.569999 − 0.821645i \(-0.306944\pi\)
0.923729 + 0.383047i \(0.125125\pi\)
\(72\) −0.833046 − 5.79396i −0.0981754 − 0.682825i
\(73\) 5.54649 + 6.40099i 0.649168 + 0.749179i 0.980968 − 0.194170i \(-0.0622014\pi\)
−0.331800 + 0.943350i \(0.607656\pi\)
\(74\) 9.15687 + 5.88477i 1.06447 + 0.684090i
\(75\) 4.05828 + 1.19162i 0.468610 + 0.137596i
\(76\) 1.22806 − 8.54134i 0.140868 − 0.979759i
\(77\) −5.78428 − 2.20818i −0.659180 − 0.251645i
\(78\) −0.714152 − 4.96704i −0.0808618 − 0.562406i
\(79\) −0.523964 + 3.64425i −0.0589506 + 0.410010i 0.938884 + 0.344234i \(0.111861\pi\)
−0.997834 + 0.0657761i \(0.979048\pi\)
\(80\) 1.36078 − 0.874518i 0.152139 − 0.0977741i
\(81\) −11.0123 −1.22359
\(82\) 0.372072 − 0.814725i 0.0410885 − 0.0899713i
\(83\) −10.5043 + 12.1227i −1.15300 + 1.33063i −0.218018 + 0.975945i \(0.569959\pi\)
−0.934984 + 0.354690i \(0.884586\pi\)
\(84\) −1.89432 − 4.14797i −0.206687 − 0.452581i
\(85\) 1.75070 − 3.83350i 0.189890 − 0.415802i
\(86\) −3.46510 − 3.99894i −0.373651 − 0.431216i
\(87\) −4.26391 + 1.25200i −0.457139 + 0.134228i
\(88\) −8.60880 + 4.61529i −0.917701 + 0.491991i
\(89\) −2.25220 − 0.661305i −0.238732 − 0.0700981i 0.160178 − 0.987088i \(-0.448793\pi\)
−0.398910 + 0.916990i \(0.630611\pi\)
\(90\) 2.06368 + 4.51884i 0.217531 + 0.476327i
\(91\) −1.83046 − 4.00815i −0.191884 − 0.420168i
\(92\) 5.56470 − 1.63394i 0.580160 − 0.170350i
\(93\) −2.88939 − 6.32688i −0.299616 − 0.656067i
\(94\) 0.552231 0.0569583
\(95\) 2.94795 + 20.5034i 0.302453 + 2.10361i
\(96\) −11.3651 − 3.33709i −1.15994 − 0.340590i
\(97\) −0.589344 − 4.09898i −0.0598388 − 0.416188i −0.997619 − 0.0689597i \(-0.978032\pi\)
0.937781 − 0.347228i \(-0.112877\pi\)
\(98\) −2.19130 − 2.52889i −0.221355 − 0.255457i
\(99\) −2.25612 − 6.19377i −0.226749 − 0.622497i
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 121.2.e.a.12.4 ✓ 100
121.111 even 11 inner 121.2.e.a.111.4 yes 100
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.2.e.a.12.4 ✓ 100 1.1 even 1 trivial
121.2.e.a.111.4 yes 100 121.111 even 11 inner