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Properties

Label 121.2.e.a.23.7
Level $121$
Weight $2$
Character 121.23
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 23.7
Character \(\chi\) \(=\) 121.23
Dual form 121.2.e.a.100.7

$q$-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+(0.333827 - 0.730979i) q^{2} +0.0294246 q^{3} +(0.886832 + 1.02346i) q^{4} +(0.779896 + 0.228998i) q^{5} +(0.00982271 - 0.0215087i) q^{6} +(0.0964221 + 0.670630i) q^{7} +(2.58627 - 0.759397i) q^{8} -2.99913 q^{9} +(0.427743 - 0.493642i) q^{10} +(0.219276 - 3.30937i) q^{11} +(0.0260947 + 0.0301148i) q^{12} +(0.295853 + 0.341432i) q^{13} +(0.522405 + 0.153392i) q^{14} +(0.0229481 + 0.00673818i) q^{15} +(-0.0771914 + 0.536878i) q^{16} +(-4.29650 - 2.76119i) q^{17} +(-1.00119 + 2.19230i) q^{18} +(-0.330107 + 0.212147i) q^{19} +(0.457267 + 1.00127i) q^{20} +(0.00283718 + 0.0197330i) q^{21} +(-2.34588 - 1.26504i) q^{22} +(-0.883277 + 6.14333i) q^{23} +(0.0760999 - 0.0223449i) q^{24} +(-3.65047 - 2.34601i) q^{25} +(0.348343 - 0.102283i) q^{26} -0.176522 q^{27} +(-0.600852 + 0.693421i) q^{28} +(3.92166 - 2.52030i) q^{29} +(0.0125862 - 0.0145252i) q^{30} +(-5.66785 + 6.54105i) q^{31} +(4.90180 + 3.15019i) q^{32} +(0.00645211 - 0.0973768i) q^{33} +(-3.45266 + 2.21889i) q^{34} +(-0.0783739 + 0.545103i) q^{35} +(-2.65973 - 3.06949i) q^{36} +(-2.06499 + 2.38312i) q^{37} +(0.0448763 + 0.312121i) q^{38} +(0.00870535 + 0.0100465i) q^{39} +2.19092 q^{40} +(3.53512 - 7.74082i) q^{41} +(0.0153715 + 0.00451349i) q^{42} +(0.845214 - 0.248177i) q^{43} +(3.58146 - 2.71043i) q^{44} +(-2.33901 - 0.686796i) q^{45} +(4.19578 + 2.69646i) q^{46} +(-1.67980 - 3.67826i) q^{47} +(-0.00227133 + 0.0157974i) q^{48} +(6.27600 - 1.84280i) q^{49} +(-2.93351 + 1.88525i) q^{50} +(-0.126423 - 0.0812470i) q^{51} +(-0.0870702 + 0.605586i) q^{52} +(-0.438597 - 3.05051i) q^{53} +(-0.0589278 + 0.129034i) q^{54} +(0.928852 - 2.53075i) q^{55} +(0.758648 + 1.66121i) q^{56} +(-0.00971326 + 0.00624233i) q^{57} +(-0.533129 - 3.70799i) q^{58} +(3.61444 + 7.91452i) q^{59} +(0.0134549 + 0.0294621i) q^{60} +(-2.93172 - 6.41957i) q^{61} +(2.88929 + 6.32665i) q^{62} +(-0.289183 - 2.01131i) q^{63} +(3.02648 - 1.94500i) q^{64} +(0.152547 + 0.334032i) q^{65} +(-0.0690264 - 0.0372233i) q^{66} +(-1.05158 + 2.30263i) q^{67} +(-0.984307 - 6.84601i) q^{68} +(-0.0259901 + 0.180765i) q^{69} +(0.372295 + 0.239260i) q^{70} +(10.6095 - 6.81833i) q^{71} +(-7.75657 + 2.27753i) q^{72} +(-0.780694 + 5.42985i) q^{73} +(1.05266 + 2.30501i) q^{74} +(-0.107414 - 0.0690305i) q^{75} +(-0.509873 - 0.149712i) q^{76} +(2.24051 - 0.172043i) q^{77} +(0.0102499 - 0.00300963i) q^{78} +(13.2963 + 3.90413i) q^{79} +(-0.183145 + 0.401033i) q^{80} +8.99221 q^{81} +(-4.47826 - 5.16819i) q^{82} +(1.16440 + 8.09855i) q^{83} +(-0.0176798 + 0.0204036i) q^{84} +(-2.71852 - 3.13734i) q^{85} +(0.100743 - 0.700682i) q^{86} +(0.115393 - 0.0741587i) q^{87} +(-1.94602 - 8.72543i) q^{88} +(0.399545 + 0.256772i) q^{89} +(-1.28286 + 1.48050i) q^{90} +(-0.200448 + 0.231330i) q^{91} +(-7.07076 + 4.54410i) q^{92} +(-0.166774 + 0.192468i) q^{93} -3.24949 q^{94} +(-0.306030 + 0.0898586i) q^{95} +(0.144233 + 0.0926931i) q^{96} +(-11.6531 + 3.42164i) q^{97} +(0.748050 - 5.20280i) q^{98} +(-0.657639 + 9.92524i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.333827 − 0.730979i 0.236051 − 0.516880i −0.754121 − 0.656736i \(-0.771937\pi\)
0.990172 + 0.139856i \(0.0446639\pi\)
\(3\) 0.0294246 0.0169883 0.00849414 − 0.999964i \(-0.497296\pi\)
0.00849414 + 0.999964i \(0.497296\pi\)
\(4\) 0.886832 + 1.02346i 0.443416 + 0.511729i
\(5\) 0.779896 + 0.228998i 0.348780 + 0.102411i 0.451430 − 0.892306i \(-0.350914\pi\)
−0.102650 + 0.994718i \(0.532732\pi\)
\(6\) 0.00982271 − 0.0215087i 0.00401011 − 0.00878091i
\(7\) 0.0964221 + 0.670630i 0.0364441 + 0.253474i 0.999896 − 0.0144062i \(-0.00458580\pi\)
−0.963452 + 0.267881i \(0.913677\pi\)
\(8\) 2.58627 − 0.759397i 0.914384 − 0.268487i
\(9\) −2.99913 −0.999711
\(10\) 0.427743 − 0.493642i 0.135264 − 0.156103i
\(11\) 0.219276 − 3.30937i 0.0661143 − 0.997812i
\(12\) 0.0260947 + 0.0301148i 0.00753288 + 0.00869341i
\(13\) 0.295853 + 0.341432i 0.0820548 + 0.0946963i 0.795292 − 0.606226i \(-0.207317\pi\)
−0.713238 + 0.700922i \(0.752772\pi\)
\(14\) 0.522405 + 0.153392i 0.139619 + 0.0409957i
\(15\) 0.0229481 + 0.00673818i 0.00592518 + 0.00173979i
\(16\) −0.0771914 + 0.536878i −0.0192979 + 0.134220i
\(17\) −4.29650 − 2.76119i −1.04205 − 0.669688i −0.0965606 − 0.995327i \(-0.530784\pi\)
−0.945494 + 0.325639i \(0.894421\pi\)
\(18\) −1.00119 + 2.19230i −0.235983 + 0.516731i
\(19\) −0.330107 + 0.212147i −0.0757317 + 0.0486698i −0.577959 − 0.816066i \(-0.696151\pi\)
0.502227 + 0.864736i \(0.332514\pi\)
\(20\) 0.457267 + 1.00127i 0.102248 + 0.223892i
\(21\) 0.00283718 + 0.0197330i 0.000619123 + 0.00430610i
\(22\) −2.34588 − 1.26504i −0.500143 − 0.269708i
\(23\) −0.883277 + 6.14333i −0.184176 + 1.28097i 0.662580 + 0.748991i \(0.269461\pi\)
−0.846756 + 0.531981i \(0.821448\pi\)
\(24\) 0.0760999 − 0.0223449i 0.0155338 − 0.00456114i
\(25\) −3.65047 − 2.34601i −0.730094 − 0.469203i
\(26\) 0.348343 − 0.102283i 0.0683158 − 0.0200593i
\(27\) −0.176522 −0.0339717
\(28\) −0.600852 + 0.693421i −0.113550 + 0.131044i
\(29\) 3.92166 − 2.52030i 0.728234 − 0.468008i −0.123258 − 0.992375i \(-0.539334\pi\)
0.851492 + 0.524367i \(0.175698\pi\)
\(30\) 0.0125862 − 0.0145252i 0.00229791 − 0.00265193i
\(31\) −5.66785 + 6.54105i −1.01798 + 1.17481i −0.0334724 + 0.999440i \(0.510657\pi\)
−0.984503 + 0.175367i \(0.943889\pi\)
\(32\) 4.90180 + 3.15019i 0.866523 + 0.556881i
\(33\) 0.00645211 − 0.0973768i 0.00112317 − 0.0169511i
\(34\) −3.45266 + 2.21889i −0.592126 + 0.380537i
\(35\) −0.0783739 + 0.545103i −0.0132476 + 0.0921392i
\(36\) −2.65973 − 3.06949i −0.443288 − 0.511582i
\(37\) −2.06499 + 2.38312i −0.339482 + 0.391783i −0.899661 − 0.436588i \(-0.856187\pi\)
0.560180 + 0.828371i \(0.310732\pi\)
\(38\) 0.0448763 + 0.312121i 0.00727989 + 0.0506328i
\(39\) 0.00870535 + 0.0100465i 0.00139397 + 0.00160873i
\(40\) 2.19092 0.346415
\(41\) 3.53512 − 7.74082i 0.552092 − 1.20891i −0.403706 − 0.914889i \(-0.632278\pi\)
0.955798 − 0.294025i \(-0.0949948\pi\)
\(42\) 0.0153715 + 0.00451349i 0.00237188 + 0.000696447i
\(43\) 0.845214 − 0.248177i 0.128894 − 0.0378467i −0.216649 − 0.976249i \(-0.569513\pi\)
0.345543 + 0.938403i \(0.387695\pi\)
\(44\) 3.58146 − 2.71043i 0.539926 − 0.408613i
\(45\) −2.33901 − 0.686796i −0.348680 − 0.102382i
\(46\) 4.19578 + 2.69646i 0.618634 + 0.397572i
\(47\) −1.67980 − 3.67826i −0.245024 − 0.536529i 0.746662 − 0.665203i \(-0.231655\pi\)
−0.991687 + 0.128674i \(0.958928\pi\)
\(48\) −0.00227133 + 0.0157974i −0.000327838 + 0.00228016i
\(49\) 6.27600 − 1.84280i 0.896572 − 0.263257i
\(50\) −2.93351 + 1.88525i −0.414861 + 0.266615i
\(51\) −0.126423 − 0.0812470i −0.0177027 − 0.0113769i
\(52\) −0.0870702 + 0.605586i −0.0120745 + 0.0839797i
\(53\) −0.438597 − 3.05051i −0.0602460 − 0.419020i −0.997517 − 0.0704197i \(-0.977566\pi\)
0.937271 − 0.348600i \(-0.113343\pi\)
\(54\) −0.0589278 + 0.129034i −0.00801905 + 0.0175593i
\(55\) 0.928852 − 2.53075i 0.125246 − 0.341246i
\(56\) 0.758648 + 1.66121i 0.101379 + 0.221988i
\(57\) −0.00971326 + 0.00624233i −0.00128655 + 0.000826817i
\(58\) −0.533129 − 3.70799i −0.0700033 − 0.486883i
\(59\) 3.61444 + 7.91452i 0.470560 + 1.03038i 0.984952 + 0.172828i \(0.0552904\pi\)
−0.514392 + 0.857555i \(0.671982\pi\)
\(60\) 0.0134549 + 0.0294621i 0.00173702 + 0.00380354i
\(61\) −2.93172 − 6.41957i −0.375368 − 0.821942i −0.999185 − 0.0403710i \(-0.987146\pi\)
0.623816 − 0.781571i \(-0.285581\pi\)
\(62\) 2.88929 + 6.32665i 0.366940 + 0.803486i
\(63\) −0.289183 − 2.01131i −0.0364336 − 0.253401i
\(64\) 3.02648 − 1.94500i 0.378311 − 0.243125i
\(65\) 0.152547 + 0.334032i 0.0189211 + 0.0414315i
\(66\) −0.0690264 − 0.0372233i −0.00849657 − 0.00458187i
\(67\) −1.05158 + 2.30263i −0.128471 + 0.281311i −0.962927 − 0.269763i \(-0.913055\pi\)
0.834456 + 0.551074i \(0.185782\pi\)
\(68\) −0.984307 − 6.84601i −0.119365 − 0.830200i
\(69\) −0.0259901 + 0.180765i −0.00312883 + 0.0217615i
\(70\) 0.372295 + 0.239260i 0.0444978 + 0.0285970i
\(71\) 10.6095 − 6.81833i 1.25912 − 0.809187i 0.270957 − 0.962592i \(-0.412660\pi\)
0.988164 + 0.153404i \(0.0490236\pi\)
\(72\) −7.75657 + 2.27753i −0.914120 + 0.268410i
\(73\) −0.780694 + 5.42985i −0.0913733 + 0.635515i 0.891744 + 0.452540i \(0.149482\pi\)
−0.983117 + 0.182975i \(0.941427\pi\)
\(74\) 1.05266 + 2.30501i 0.122370 + 0.267952i
\(75\) −0.107414 − 0.0690305i −0.0124030 − 0.00797095i
\(76\) −0.509873 − 0.149712i −0.0584864 − 0.0171732i
\(77\) 2.24051 − 0.172043i 0.255329 − 0.0196061i
\(78\) 0.0102499 − 0.00300963i 0.00116057 − 0.000340773i
\(79\) 13.2963 + 3.90413i 1.49595 + 0.439249i 0.924433 − 0.381344i \(-0.124539\pi\)
0.571513 + 0.820593i \(0.306357\pi\)
\(80\) −0.183145 + 0.401033i −0.0204763 + 0.0448368i
\(81\) 8.99221 0.999134
\(82\) −4.47826 − 5.16819i −0.494541 − 0.570731i
\(83\) 1.16440 + 8.09855i 0.127809 + 0.888931i 0.948323 + 0.317305i \(0.102778\pi\)
−0.820514 + 0.571626i \(0.806313\pi\)
\(84\) −0.0176798 + 0.0204036i −0.00192903 + 0.00222622i
\(85\) −2.71852 − 3.13734i −0.294865 − 0.340292i
\(86\) 0.100743 − 0.700682i 0.0108634 − 0.0755564i
\(87\) 0.115393 − 0.0741587i 0.0123715 − 0.00795065i
\(88\) −1.94602 − 8.72543i −0.207446 − 0.930134i
\(89\) 0.399545 + 0.256772i 0.0423517 + 0.0272178i 0.561645 − 0.827378i \(-0.310169\pi\)
−0.519294 + 0.854596i \(0.673805\pi\)
\(90\) −1.28286 + 1.48050i −0.135225 + 0.156058i
\(91\) −0.200448 + 0.231330i −0.0210127 + 0.0242499i
\(92\) −7.07076 + 4.54410i −0.737178 + 0.473755i
\(93\) −0.166774 + 0.192468i −0.0172937 + 0.0199580i
\(94\) −3.24949 −0.335159
\(95\) −0.306030 + 0.0898586i −0.0313981 + 0.00921930i
\(96\) 0.144233 + 0.0926931i 0.0147208 + 0.00946045i
\(97\) −11.6531 + 3.42164i −1.18319 + 0.347415i −0.813402 − 0.581702i \(-0.802387\pi\)
−0.369786 + 0.929117i \(0.620569\pi\)
\(98\) 0.748050 − 5.20280i 0.0755644 − 0.525562i
\(99\) −0.657639 + 9.92524i −0.0660952 + 0.997524i
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.23.7 ✓ 100
121.100 even 11 inner 121.2.e.a.100.7 yes 100
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.2.e.a.23.7 ✓ 100 1.1 even 1 trivial
121.2.e.a.100.7 yes 100 121.100 even 11 inner