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Properties

Label 121.2.e.a.12.3
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 12.3
Character \(\chi\) \(=\) 121.12
Dual form 121.2.e.a.111.3

$q$-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+(-1.29895 + 0.834784i) q^{2} -0.564640 q^{3} +(0.159574 - 0.349419i) q^{4} +(-0.144022 + 1.00170i) q^{5} +(0.733439 - 0.471353i) q^{6} +(-1.26378 + 1.45848i) q^{7} +(-0.355076 - 2.46961i) q^{8} -2.68118 q^{9} +(-0.649122 - 1.42138i) q^{10} +(-1.99988 + 2.64584i) q^{11} +(-0.0901021 + 0.197296i) q^{12} +(-0.862426 + 1.88845i) q^{13} +(0.424070 - 2.94947i) q^{14} +(0.0813207 - 0.565598i) q^{15} +(3.02592 + 3.49210i) q^{16} +(-6.07927 - 1.78503i) q^{17} +(3.48272 - 2.23821i) q^{18} +(5.06741 - 1.48793i) q^{19} +(0.327029 + 0.210169i) q^{20} +(0.713580 - 0.823515i) q^{21} +(0.389045 - 5.10628i) q^{22} +(3.23470 + 3.73304i) q^{23} +(0.200490 + 1.39444i) q^{24} +(3.81481 + 1.12013i) q^{25} +(-0.456199 - 3.17294i) q^{26} +3.20782 q^{27} +(0.307953 + 0.674324i) q^{28} +(-1.32154 + 0.388040i) q^{29} +(0.366520 + 0.802568i) q^{30} +(4.40341 + 9.64212i) q^{31} +(-2.05779 - 0.604221i) q^{32} +(1.12922 - 1.49395i) q^{33} +(9.38678 - 2.75621i) q^{34} +(-1.27894 - 1.47597i) q^{35} +(-0.427848 + 0.936856i) q^{36} +(-1.56921 - 3.43609i) q^{37} +(-5.34021 + 6.16293i) q^{38} +(0.486960 - 1.06629i) q^{39} +2.52493 q^{40} +(1.04899 - 0.674148i) q^{41} +(-0.239447 + 1.66539i) q^{42} +(-0.685810 - 4.76992i) q^{43} +(0.605376 + 1.12101i) q^{44} +(0.386149 - 2.68573i) q^{45} +(-7.31798 - 2.14875i) q^{46} +(-4.61114 - 2.96340i) q^{47} +(-1.70856 - 1.97178i) q^{48} +(0.466182 + 3.24237i) q^{49} +(-5.89031 + 1.72955i) q^{50} +(3.43260 + 1.00790i) q^{51} +(0.522239 + 0.602696i) q^{52} +(-4.10562 + 4.73813i) q^{53} +(-4.16680 + 2.67784i) q^{54} +(-2.36230 - 2.38433i) q^{55} +(4.05060 + 2.60316i) q^{56} +(-2.86126 + 0.840143i) q^{57} +(1.39269 - 1.60725i) q^{58} +(3.57627 + 2.29833i) q^{59} +(-0.184654 - 0.118670i) q^{60} +(-5.45435 - 3.50530i) q^{61} +(-13.7689 - 8.84872i) q^{62} +(3.38842 - 3.91044i) q^{63} +(-5.68972 + 1.67065i) q^{64} +(-1.76744 - 1.13587i) q^{65} +(-0.219670 + 2.88321i) q^{66} +(-7.69397 + 4.94461i) q^{67} +(-1.59382 + 1.83937i) q^{68} +(-1.82644 - 2.10782i) q^{69} +(2.89339 + 0.849577i) q^{70} +(12.7112 - 3.73235i) q^{71} +(0.952023 + 6.62146i) q^{72} +(0.0733885 + 0.0846949i) q^{73} +(4.90672 + 3.15336i) q^{74} +(-2.15400 - 0.632471i) q^{75} +(0.288719 - 2.00809i) q^{76} +(-1.33149 - 6.26054i) q^{77} +(0.257589 + 1.79157i) q^{78} +(-0.784248 + 5.45457i) q^{79} +(-3.93382 + 2.52811i) q^{80} +6.23228 q^{81} +(-0.799823 + 1.75137i) q^{82} +(-2.40179 + 2.77181i) q^{83} +(-0.173883 - 0.380750i) q^{84} +(2.66361 - 5.83249i) q^{85} +(4.87268 + 5.62337i) q^{86} +(0.746197 - 0.219103i) q^{87} +(7.24429 + 3.99945i) q^{88} +(-5.27235 - 1.54810i) q^{89} +(1.74041 + 3.81097i) q^{90} +(-1.66435 - 3.64441i) q^{91} +(1.82057 - 0.534567i) q^{92} +(-2.48634 - 5.44433i) q^{93} +8.46344 q^{94} +(0.760630 + 5.29029i) q^{95} +(1.16191 + 0.341167i) q^{96} +(-1.23092 - 8.56126i) q^{97} +(-3.31222 - 3.82251i) q^{98} +(5.36205 - 7.09397i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.29895 + 0.834784i −0.918496 + 0.590281i −0.912221 − 0.409699i \(-0.865634\pi\)
−0.00627487 + 0.999980i \(0.501997\pi\)
\(3\) −0.564640 −0.325995 −0.162998 − 0.986626i \(-0.552116\pi\)
−0.162998 + 0.986626i \(0.552116\pi\)
\(4\) 0.159574 − 0.349419i 0.0797872 − 0.174710i
\(5\) −0.144022 + 1.00170i −0.0644086 + 0.447972i 0.931941 + 0.362609i \(0.118114\pi\)
−0.996350 + 0.0853627i \(0.972795\pi\)
\(6\) 0.733439 − 0.471353i 0.299425 − 0.192429i
\(7\) −1.26378 + 1.45848i −0.477663 + 0.551253i −0.942527 − 0.334129i \(-0.891558\pi\)
0.464864 + 0.885382i \(0.346103\pi\)
\(8\) −0.355076 − 2.46961i −0.125538 − 0.873138i
\(9\) −2.68118 −0.893727
\(10\) −0.649122 − 1.42138i −0.205270 − 0.449479i
\(11\) −1.99988 + 2.64584i −0.602988 + 0.797750i
\(12\) −0.0901021 + 0.197296i −0.0260102 + 0.0569545i
\(13\) −0.862426 + 1.88845i −0.239194 + 0.523761i −0.990716 − 0.135946i \(-0.956593\pi\)
0.751522 + 0.659708i \(0.229320\pi\)
\(14\) 0.424070 − 2.94947i 0.113337 − 0.788279i
\(15\) 0.0813207 − 0.565598i 0.0209969 − 0.146037i
\(16\) 3.02592 + 3.49210i 0.756480 + 0.873024i
\(17\) −6.07927 − 1.78503i −1.47444 − 0.432934i −0.556900 − 0.830580i \(-0.688009\pi\)
−0.917539 + 0.397645i \(0.869827\pi\)
\(18\) 3.48272 − 2.23821i 0.820884 − 0.527550i
\(19\) 5.06741 − 1.48793i 1.16254 − 0.341354i 0.357123 − 0.934057i \(-0.383758\pi\)
0.805421 + 0.592704i \(0.201940\pi\)
\(20\) 0.327029 + 0.210169i 0.0731260 + 0.0469952i
\(21\) 0.713580 − 0.823515i 0.155716 − 0.179706i
\(22\) 0.389045 − 5.10628i 0.0829446 − 1.08866i
\(23\) 3.23470 + 3.73304i 0.674481 + 0.778392i 0.985070 − 0.172152i \(-0.0550720\pi\)
−0.310590 + 0.950544i \(0.600527\pi\)
\(24\) 0.200490 + 1.39444i 0.0409249 + 0.284639i
\(25\) 3.81481 + 1.12013i 0.762963 + 0.224026i
\(26\) −0.456199 − 3.17294i −0.0894681 − 0.622264i
\(27\) 3.20782 0.617346
\(28\) 0.307953 + 0.674324i 0.0581977 + 0.127435i
\(29\) −1.32154 + 0.388040i −0.245405 + 0.0720573i −0.402122 − 0.915586i \(-0.631727\pi\)
0.156717 + 0.987644i \(0.449909\pi\)
\(30\) 0.366520 + 0.802568i 0.0669171 + 0.146528i
\(31\) 4.40341 + 9.64212i 0.790875 + 1.73178i 0.674129 + 0.738614i \(0.264519\pi\)
0.116746 + 0.993162i \(0.462754\pi\)
\(32\) −2.05779 − 0.604221i −0.363769 − 0.106812i
\(33\) 1.12922 − 1.49395i 0.196571 − 0.260063i
\(34\) 9.38678 − 2.75621i 1.60982 − 0.472685i
\(35\) −1.27894 − 1.47597i −0.216180 − 0.249485i
\(36\) −0.427848 + 0.936856i −0.0713080 + 0.156143i
\(37\) −1.56921 − 3.43609i −0.257977 − 0.564891i 0.735682 − 0.677327i \(-0.236862\pi\)
−0.993659 + 0.112437i \(0.964135\pi\)
\(38\) −5.34021 + 6.16293i −0.866296 + 0.999759i
\(39\) 0.486960 − 1.06629i 0.0779760 − 0.170744i
\(40\) 2.52493 0.399227
\(41\) 1.04899 − 0.674148i 0.163825 − 0.105284i −0.456159 − 0.889898i \(-0.650775\pi\)
0.619985 + 0.784614i \(0.287139\pi\)
\(42\) −0.239447 + 1.66539i −0.0369474 + 0.256975i
\(43\) −0.685810 − 4.76992i −0.104585 − 0.727405i −0.972872 − 0.231343i \(-0.925688\pi\)
0.868287 − 0.496062i \(-0.165221\pi\)
\(44\) 0.605376 + 1.12101i 0.0912639 + 0.168998i
\(45\) 0.386149 − 2.68573i 0.0575637 − 0.400365i
\(46\) −7.31798 − 2.14875i −1.07898 − 0.316816i
\(47\) −4.61114 − 2.96340i −0.672604 − 0.432256i 0.159259 − 0.987237i \(-0.449089\pi\)
−0.831863 + 0.554980i \(0.812726\pi\)
\(48\) −1.70856 − 1.97178i −0.246609 − 0.284602i
\(49\) 0.466182 + 3.24237i 0.0665974 + 0.463195i
\(50\) −5.89031 + 1.72955i −0.833016 + 0.244596i
\(51\) 3.43260 + 1.00790i 0.480660 + 0.141135i
\(52\) 0.522239 + 0.602696i 0.0724215 + 0.0835789i
\(53\) −4.10562 + 4.73813i −0.563950 + 0.650833i −0.964076 − 0.265627i \(-0.914421\pi\)
0.400126 + 0.916460i \(0.368966\pi\)
\(54\) −4.16680 + 2.67784i −0.567030 + 0.364408i
\(55\) −2.36230 − 2.38433i −0.318532 − 0.321504i
\(56\) 4.05060 + 2.60316i 0.541285 + 0.347862i
\(57\) −2.86126 + 0.840143i −0.378984 + 0.111280i
\(58\) 1.39269 − 1.60725i 0.182869 − 0.211042i
\(59\) 3.57627 + 2.29833i 0.465591 + 0.299217i 0.752324 − 0.658793i \(-0.228933\pi\)
−0.286733 + 0.958011i \(0.592569\pi\)
\(60\) −0.184654 − 0.118670i −0.0238387 − 0.0153202i
\(61\) −5.45435 − 3.50530i −0.698357 − 0.448807i 0.142691 − 0.989767i \(-0.454425\pi\)
−0.841048 + 0.540960i \(0.818061\pi\)
\(62\) −13.7689 − 8.84872i −1.74865 − 1.12379i
\(63\) 3.38842 − 3.91044i 0.426901 − 0.492669i
\(64\) −5.68972 + 1.67065i −0.711215 + 0.208831i
\(65\) −1.76744 − 1.13587i −0.219224 − 0.140887i
\(66\) −0.219670 + 2.88321i −0.0270395 + 0.354899i
\(67\) −7.69397 + 4.94461i −0.939968 + 0.604081i −0.918386 − 0.395686i \(-0.870507\pi\)
−0.0215823 + 0.999767i \(0.506870\pi\)
\(68\) −1.59382 + 1.83937i −0.193279 + 0.223056i
\(69\) −1.82644 − 2.10782i −0.219877 − 0.253752i
\(70\) 2.89339 + 0.849577i 0.345827 + 0.101544i
\(71\) 12.7112 − 3.73235i 1.50854 − 0.442948i 0.580138 − 0.814519i \(-0.302999\pi\)
0.928405 + 0.371571i \(0.121181\pi\)
\(72\) 0.952023 + 6.62146i 0.112197 + 0.780347i
\(73\) 0.0733885 + 0.0846949i 0.00858948 + 0.00991279i 0.760028 − 0.649890i \(-0.225185\pi\)
−0.751439 + 0.659803i \(0.770640\pi\)
\(74\) 4.90672 + 3.15336i 0.570395 + 0.366571i
\(75\) −2.15400 − 0.632471i −0.248722 − 0.0730314i
\(76\) 0.288719 − 2.00809i 0.0331183 − 0.230343i
\(77\) −1.33149 − 6.26054i −0.151737 − 0.713455i
\(78\) 0.257589 + 1.79157i 0.0291662 + 0.202855i
\(79\) −0.784248 + 5.45457i −0.0882348 + 0.613687i 0.896942 + 0.442147i \(0.145783\pi\)
−0.985177 + 0.171539i \(0.945126\pi\)
\(80\) −3.93382 + 2.52811i −0.439814 + 0.282651i
\(81\) 6.23228 0.692475
\(82\) −0.799823 + 1.75137i −0.0883256 + 0.193406i
\(83\) −2.40179 + 2.77181i −0.263630 + 0.304246i −0.872096 − 0.489334i \(-0.837240\pi\)
0.608466 + 0.793580i \(0.291785\pi\)
\(84\) −0.173883 − 0.380750i −0.0189722 − 0.0415433i
\(85\) 2.66361 − 5.83249i 0.288909 − 0.632622i
\(86\) 4.87268 + 5.62337i 0.525435 + 0.606384i
\(87\) 0.746197 − 0.219103i 0.0800007 − 0.0234903i
\(88\) 7.24429 + 3.99945i 0.772244 + 0.426343i
\(89\) −5.27235 − 1.54810i −0.558868 − 0.164098i −0.00991357 − 0.999951i \(-0.503156\pi\)
−0.548954 + 0.835853i \(0.684974\pi\)
\(90\) 1.74041 + 3.81097i 0.183456 + 0.401712i
\(91\) −1.66435 − 3.64441i −0.174471 − 0.382038i
\(92\) 1.82057 − 0.534567i 0.189807 − 0.0557325i
\(93\) −2.48634 − 5.44433i −0.257822 − 0.564550i
\(94\) 8.46344 0.872937
\(95\) 0.760630 + 5.29029i 0.0780389 + 0.542773i
\(96\) 1.16191 + 0.341167i 0.118587 + 0.0348203i
\(97\) −1.23092 − 8.56126i −0.124981 − 0.869264i −0.951783 − 0.306771i \(-0.900751\pi\)
0.826802 − 0.562493i \(-0.190158\pi\)
\(98\) −3.31222 − 3.82251i −0.334585 − 0.386132i
\(99\) 5.36205 − 7.09397i 0.538907 − 0.712971i
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.12.3 ✓ 100
121.111 even 11 inner 121.2.e.a.111.3 yes 100
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.2.e.a.12.3 ✓ 100 1.1 even 1 trivial
121.2.e.a.111.3 yes 100 121.111 even 11 inner