↑

Properties

Label 121.2.e.a.23.4
Level $121$
Weight $2$
Character 121.23
Analytic conductor $0.966$
Analytic rank $0$
Dimension $100$
Inner twists $2$

Related objects

Downloads

Learn more

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 23.4
Character \(\chi\) \(=\) 121.23
Dual form 121.2.e.a.100.4

$q$-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+(-0.597130 + 1.30753i) q^{2} -0.640035 q^{3} +(-0.0433545 - 0.0500337i) q^{4} +(-3.97756 - 1.16792i) q^{5} +(0.382184 - 0.836867i) q^{6} +(0.623172 + 4.33426i) q^{7} +(-2.66710 + 0.783131i) q^{8} -2.59035 q^{9} +(3.90221 - 4.50339i) q^{10} +(3.31624 - 0.0507878i) q^{11} +(0.0277484 + 0.0320233i) q^{12} +(-0.608526 - 0.702277i) q^{13} +(-6.03929 - 1.77330i) q^{14} +(2.54578 + 0.747508i) q^{15} +(0.587479 - 4.08601i) q^{16} +(2.51116 + 1.61382i) q^{17} +(1.54678 - 3.38697i) q^{18} +(-0.736041 + 0.473025i) q^{19} +(0.114010 + 0.249647i) q^{20} +(-0.398852 - 2.77408i) q^{21} +(-1.91382 + 4.36641i) q^{22} +(-0.753600 + 5.24141i) q^{23} +(1.70704 - 0.501231i) q^{24} +(10.2507 + 6.58772i) q^{25} +(1.28162 - 0.376317i) q^{26} +3.57802 q^{27} +(0.189842 - 0.219089i) q^{28} +(2.03939 - 1.31063i) q^{29} +(-2.49755 + 2.88233i) q^{30} +(-1.04055 + 1.20086i) q^{31} +(0.314927 + 0.202391i) q^{32} +(-2.12251 + 0.0325060i) q^{33} +(-3.60962 + 2.31976i) q^{34} +(2.58335 - 17.9676i) q^{35} +(0.112303 + 0.129605i) q^{36} +(-3.15579 + 3.64197i) q^{37} +(-0.178983 - 1.24486i) q^{38} +(0.389478 + 0.449482i) q^{39} +11.5232 q^{40} +(-2.06701 + 4.52611i) q^{41} +(3.86536 + 1.13497i) q^{42} +(5.10405 - 1.49868i) q^{43} +(-0.146315 - 0.163722i) q^{44} +(10.3033 + 3.02532i) q^{45} +(-6.40331 - 4.11516i) q^{46} +(-2.84451 - 6.22861i) q^{47} +(-0.376007 + 2.61519i) q^{48} +(-11.6810 + 3.42985i) q^{49} +(-14.7347 + 9.46938i) q^{50} +(-1.60723 - 1.03290i) q^{51} +(-0.00875518 + 0.0608936i) q^{52} +(1.28928 + 8.96711i) q^{53} +(-2.13654 + 4.67838i) q^{54} +(-13.2498 - 3.67108i) q^{55} +(-5.05635 - 11.0719i) q^{56} +(0.471092 - 0.302753i) q^{57} +(0.495918 + 3.44918i) q^{58} +(-2.63232 - 5.76399i) q^{59} +(-0.0729703 - 0.159783i) q^{60} +(-3.69025 - 8.08052i) q^{61} +(-0.948821 - 2.07763i) q^{62} +(-1.61424 - 11.2273i) q^{63} +(6.49275 - 4.17263i) q^{64} +(1.60025 + 3.50406i) q^{65} +(1.22491 - 2.79466i) q^{66} +(-2.62345 + 5.74455i) q^{67} +(-0.0281243 - 0.195609i) q^{68} +(0.482331 - 3.35468i) q^{69} +(21.9506 + 14.1068i) q^{70} +(-3.45927 + 2.22314i) q^{71} +(6.90873 - 2.02859i) q^{72} +(0.847223 - 5.89257i) q^{73} +(-2.87758 - 6.30102i) q^{74} +(-6.56080 - 4.21637i) q^{75} +(0.0555779 + 0.0163191i) q^{76} +(2.28671 + 14.3418i) q^{77} +(-0.820281 + 0.240856i) q^{78} +(2.78807 + 0.818652i) q^{79} +(-7.10885 + 15.5662i) q^{80} +5.48100 q^{81} +(-4.68376 - 5.40535i) q^{82} +(0.586961 + 4.08241i) q^{83} +(-0.121505 + 0.140225i) q^{84} +(-8.10348 - 9.35191i) q^{85} +(-1.08820 + 7.56862i) q^{86} +(-1.30528 + 0.838852i) q^{87} +(-8.80495 + 2.73250i) q^{88} +(6.74142 + 4.33245i) q^{89} +(-10.1081 + 11.6654i) q^{90} +(2.66463 - 3.07515i) q^{91} +(0.294919 - 0.189533i) q^{92} +(0.665991 - 0.768594i) q^{93} +9.84265 q^{94} +(3.48010 - 1.02185i) q^{95} +(-0.201564 - 0.129538i) q^{96} +(8.12344 - 2.38526i) q^{97} +(2.49043 - 17.3213i) q^{98} +(-8.59023 + 0.131558i) 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 \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{7}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.597130 + 1.30753i −0.422235 + 0.924565i 0.572289 + 0.820052i \(0.306056\pi\)
−0.994524 + 0.104513i \(0.966672\pi\)
\(3\) −0.640035 −0.369524 −0.184762 − 0.982783i \(-0.559152\pi\)
−0.184762 + 0.982783i \(0.559152\pi\)
\(4\) −0.0433545 − 0.0500337i −0.0216772 − 0.0250169i
\(5\) −3.97756 − 1.16792i −1.77882 − 0.522308i −0.783713 − 0.621123i \(-0.786677\pi\)
−0.995106 + 0.0988141i \(0.968495\pi\)
\(6\) 0.382184 − 0.836867i 0.156026 − 0.341649i
\(7\) 0.623172 + 4.33426i 0.235537 + 1.63819i 0.673490 + 0.739197i \(0.264795\pi\)
−0.437953 + 0.898998i \(0.644296\pi\)
\(8\) −2.66710 + 0.783131i −0.942962 + 0.276879i
\(9\) −2.59035 −0.863452
\(10\) 3.90221 − 4.50339i 1.23399 − 1.42410i
\(11\) 3.31624 − 0.0507878i 0.999883 − 0.0153131i
\(12\) 0.0277484 + 0.0320233i 0.00801027 + 0.00924434i
\(13\) −0.608526 − 0.702277i −0.168775 − 0.194776i 0.665061 − 0.746789i \(-0.268406\pi\)
−0.833836 + 0.552013i \(0.813860\pi\)
\(14\) −6.03929 − 1.77330i −1.61407 − 0.473933i
\(15\) 2.54578 + 0.747508i 0.657317 + 0.193006i
\(16\) 0.587479 − 4.08601i 0.146870 − 1.02150i
\(17\) 2.51116 + 1.61382i 0.609046 + 0.391410i 0.808499 − 0.588497i \(-0.200280\pi\)
−0.199453 + 0.979907i \(0.563917\pi\)
\(18\) 1.54678 − 3.38697i 0.364579 − 0.798317i
\(19\) −0.736041 + 0.473025i −0.168859 + 0.108519i −0.622341 − 0.782746i \(-0.713818\pi\)
0.453481 + 0.891266i \(0.350182\pi\)
\(20\) 0.114010 + 0.249647i 0.0254934 + 0.0558227i
\(21\) −0.398852 − 2.77408i −0.0870366 − 0.605353i
\(22\) −1.91382 + 4.36641i −0.408027 + 0.930922i
\(23\) −0.753600 + 5.24141i −0.157137 + 1.09291i 0.746740 + 0.665116i \(0.231618\pi\)
−0.903877 + 0.427793i \(0.859291\pi\)
\(24\) 1.70704 − 0.501231i 0.348447 − 0.102313i
\(25\) 10.2507 + 6.58772i 2.05014 + 1.31754i
\(26\) 1.28162 − 0.376317i 0.251346 − 0.0738019i
\(27\) 3.57802 0.688591
\(28\) 0.189842 − 0.219089i 0.0358767 − 0.0414039i
\(29\) 2.03939 − 1.31063i 0.378705 − 0.243379i −0.337415 − 0.941356i \(-0.609553\pi\)
0.716120 + 0.697977i \(0.245916\pi\)
\(30\) −2.49755 + 2.88233i −0.455988 + 0.526239i
\(31\) −1.04055 + 1.20086i −0.186889 + 0.215681i −0.841460 − 0.540319i \(-0.818304\pi\)
0.654572 + 0.756000i \(0.272849\pi\)
\(32\) 0.314927 + 0.202391i 0.0556718 + 0.0357781i
\(33\) −2.12251 + 0.0325060i −0.369481 + 0.00565857i
\(34\) −3.60962 + 2.31976i −0.619044 + 0.397835i
\(35\) 2.58335 − 17.9676i 0.436666 − 3.03708i
\(36\) 0.112303 + 0.129605i 0.0187172 + 0.0216008i
\(37\) −3.15579 + 3.64197i −0.518808 + 0.598737i −0.953332 − 0.301924i \(-0.902371\pi\)
0.434524 + 0.900660i \(0.356917\pi\)
\(38\) −0.178983 − 1.24486i −0.0290349 − 0.201942i
\(39\) 0.389478 + 0.449482i 0.0623664 + 0.0719747i
\(40\) 11.5232 1.82197
\(41\) −2.06701 + 4.52611i −0.322812 + 0.706860i −0.999569 − 0.0293464i \(-0.990657\pi\)
0.676757 + 0.736206i \(0.263385\pi\)
\(42\) 3.86536 + 1.13497i 0.596438 + 0.175130i
\(43\) 5.10405 − 1.49868i 0.778360 − 0.228547i 0.131663 − 0.991294i \(-0.457968\pi\)
0.646697 + 0.762747i \(0.276150\pi\)
\(44\) −0.146315 − 0.163722i −0.0220578 − 0.0246820i
\(45\) 10.3033 + 3.02532i 1.53592 + 0.450988i
\(46\) −6.40331 − 4.11516i −0.944116 − 0.606747i
\(47\) −2.84451 − 6.22861i −0.414915 − 0.908537i −0.995538 − 0.0943641i \(-0.969918\pi\)
0.580623 − 0.814173i \(-0.302809\pi\)
\(48\) −0.376007 + 2.61519i −0.0542720 + 0.377470i
\(49\) −11.6810 + 3.42985i −1.66871 + 0.489978i
\(50\) −14.7347 + 9.46938i −2.08379 + 1.33917i
\(51\) −1.60723 − 1.03290i −0.225057 − 0.144636i
\(52\) −0.00875518 + 0.0608936i −0.00121413 + 0.00844443i
\(53\) 1.28928 + 8.96711i 0.177096 + 1.23173i 0.863440 + 0.504451i \(0.168305\pi\)
−0.686345 + 0.727276i \(0.740786\pi\)
\(54\) −2.13654 + 4.67838i −0.290747 + 0.636647i
\(55\) −13.2498 − 3.67108i −1.78661 − 0.495008i
\(56\) −5.05635 − 11.0719i −0.675683 − 1.47954i
\(57\) 0.471092 − 0.302753i 0.0623977 − 0.0401006i
\(58\) 0.495918 + 3.44918i 0.0651172 + 0.452900i
\(59\) −2.63232 − 5.76399i −0.342700 − 0.750407i 0.657295 − 0.753633i \(-0.271700\pi\)
−0.999995 + 0.00322587i \(0.998973\pi\)
\(60\) −0.0729703 − 0.159783i −0.00942042 − 0.0206278i
\(61\) −3.69025 − 8.08052i −0.472488 − 1.03460i −0.984461 − 0.175603i \(-0.943813\pi\)
0.511973 − 0.859001i \(-0.328915\pi\)
\(62\) −0.948821 − 2.07763i −0.120500 − 0.263859i
\(63\) −1.61424 − 11.2273i −0.203375 − 1.41450i
\(64\) 6.49275 − 4.17263i 0.811593 − 0.521579i
\(65\) 1.60025 + 3.50406i 0.198486 + 0.434625i
\(66\) 1.22491 − 2.79466i 0.150776 − 0.343999i
\(67\) −2.62345 + 5.74455i −0.320505 + 0.701808i −0.999476 − 0.0323564i \(-0.989699\pi\)
0.678971 + 0.734165i \(0.262426\pi\)
\(68\) −0.0281243 − 0.195609i −0.00341058 − 0.0237211i
\(69\) 0.482331 − 3.35468i 0.0580658 − 0.403857i
\(70\) 21.9506 + 14.1068i 2.62360 + 1.68608i
\(71\) −3.45927 + 2.22314i −0.410539 + 0.263838i −0.729569 − 0.683907i \(-0.760279\pi\)
0.319030 + 0.947745i \(0.396643\pi\)
\(72\) 6.90873 − 2.02859i 0.814202 − 0.239071i
\(73\) 0.847223 − 5.89257i 0.0991600 − 0.689673i −0.878232 − 0.478236i \(-0.841277\pi\)
0.977392 − 0.211437i \(-0.0678144\pi\)
\(74\) −2.87758 − 6.30102i −0.334512 − 0.732479i
\(75\) −6.56080 − 4.21637i −0.757576 − 0.486865i
\(76\) 0.0555779 + 0.0163191i 0.00637522 + 0.00187193i
\(77\) 2.28671 + 14.3418i 0.260595 + 1.63440i
\(78\) −0.820281 + 0.240856i −0.0928785 + 0.0272716i
\(79\) 2.78807 + 0.818652i 0.313683 + 0.0921056i 0.434785 − 0.900535i \(-0.356825\pi\)
−0.121102 + 0.992640i \(0.538643\pi\)
\(80\) −7.10885 + 15.5662i −0.794794 + 1.74036i
\(81\) 5.48100 0.609000
\(82\) −4.68376 − 5.40535i −0.517235 − 0.596921i
\(83\) 0.586961 + 4.08241i 0.0644274 + 0.448102i 0.996344 + 0.0854301i \(0.0272264\pi\)
−0.931917 + 0.362672i \(0.881864\pi\)
\(84\) −0.121505 + 0.140225i −0.0132573 + 0.0152998i
\(85\) −8.10348 − 9.35191i −0.878946 − 1.01436i
\(86\) −1.08820 + 7.56862i −0.117344 + 0.816145i
\(87\) −1.30528 + 0.838852i −0.139941 + 0.0899344i
\(88\) −8.80495 + 2.73250i −0.938611 + 0.291286i
\(89\) 6.74142 + 4.33245i 0.714589 + 0.459238i 0.846751 − 0.531990i \(-0.178556\pi\)
−0.132162 + 0.991228i \(0.542192\pi\)
\(90\) −10.1081 + 11.6654i −1.06549 + 1.22964i
\(91\) 2.66463 − 3.07515i 0.279329 − 0.322363i
\(92\) 0.294919 − 0.189533i 0.0307474 − 0.0197602i
\(93\) 0.665991 − 0.768594i 0.0690600 − 0.0796995i
\(94\) 9.84265 1.01519
\(95\) 3.48010 − 1.02185i 0.357051 − 0.104840i
\(96\) −0.201564 − 0.129538i −0.0205721 − 0.0132209i
\(97\) 8.12344 − 2.38526i 0.824810 − 0.242186i 0.158024 − 0.987435i \(-0.449488\pi\)
0.666786 + 0.745249i \(0.267669\pi\)
\(98\) 2.49043 − 17.3213i 0.251571 − 1.74972i
\(99\) −8.59023 + 0.131558i −0.863350 + 0.0132221i
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.23.4 ✓ 100
121.100 even 11 inner 121.2.e.a.100.4 yes 100
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.2.e.a.23.4 ✓ 100 1.1 even 1 trivial
121.2.e.a.100.4 yes 100 121.100 even 11 inner