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Properties

Label 121.2.e.a.23.3
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 23.3
Character \(\chi\) \(=\) 121.23
Dual form 121.2.e.a.100.3

$q$-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+(-0.671026 + 1.46934i) q^{2} -1.09234 q^{3} +(-0.398970 - 0.460436i) q^{4} +(3.18729 + 0.935873i) q^{5} +(0.732990 - 1.60502i) q^{6} +(0.233950 + 1.62716i) q^{7} +(-2.15551 + 0.632915i) q^{8} -1.80679 q^{9} +(-3.51387 + 4.05523i) q^{10} +(-2.98590 - 1.44375i) q^{11} +(0.435811 + 0.502953i) q^{12} +(2.91034 + 3.35871i) q^{13} +(-2.54784 - 0.748114i) q^{14} +(-3.48161 - 1.02229i) q^{15} +(0.689844 - 4.79797i) q^{16} +(3.02629 + 1.94488i) q^{17} +(1.21240 - 2.65480i) q^{18} +(3.46328 - 2.22571i) q^{19} +(-0.840724 - 1.84093i) q^{20} +(-0.255554 - 1.77741i) q^{21} +(4.12498 - 3.41852i) q^{22} +(0.851457 - 5.92201i) q^{23} +(2.35455 - 0.691359i) q^{24} +(5.07670 + 3.26260i) q^{25} +(-6.88800 + 2.02250i) q^{26} +5.25066 q^{27} +(0.655864 - 0.756907i) q^{28} +(-7.13343 + 4.58438i) q^{29} +(3.83835 - 4.42969i) q^{30} +(1.32995 - 1.53484i) q^{31} +(2.80719 + 1.80407i) q^{32} +(3.26162 + 1.57706i) q^{33} +(-4.88841 + 3.14159i) q^{34} +(-0.777148 + 5.40518i) q^{35} +(0.720856 + 0.831912i) q^{36} +(7.45128 - 8.59924i) q^{37} +(0.946386 + 6.58226i) q^{38} +(-3.17908 - 3.66885i) q^{39} -7.46257 q^{40} +(2.48758 - 5.44703i) q^{41} +(2.78311 + 0.817196i) q^{42} +(-2.46608 + 0.724106i) q^{43} +(0.526532 + 1.95083i) q^{44} +(-5.75877 - 1.69093i) q^{45} +(8.13011 + 5.22491i) q^{46} +(-0.837901 - 1.83475i) q^{47} +(-0.753545 + 5.24102i) q^{48} +(4.12353 - 1.21078i) q^{49} +(-8.20047 + 5.27012i) q^{50} +(-3.30574 - 2.12447i) q^{51} +(0.385333 - 2.68005i) q^{52} +(-0.0209417 - 0.145653i) q^{53} +(-3.52333 + 7.71501i) q^{54} +(-8.16577 - 7.39606i) q^{55} +(-1.53414 - 3.35929i) q^{56} +(-3.78308 + 2.43124i) q^{57} +(-1.94930 - 13.5577i) q^{58} +(2.96026 + 6.48208i) q^{59} +(0.918358 + 2.01092i) q^{60} +(-2.06529 - 4.52234i) q^{61} +(1.36278 + 2.98407i) q^{62} +(-0.422699 - 2.93994i) q^{63} +(3.62113 - 2.32716i) q^{64} +(6.13276 + 13.4289i) q^{65} +(-4.50588 + 3.73419i) q^{66} +(-2.27836 + 4.98892i) q^{67} +(-0.311907 - 2.16936i) q^{68} +(-0.930081 + 6.46886i) q^{69} +(-7.42058 - 4.76891i) q^{70} +(-9.46687 + 6.08399i) q^{71} +(3.89456 - 1.14354i) q^{72} +(-1.33753 + 9.30273i) q^{73} +(7.63522 + 16.7188i) q^{74} +(-5.54548 - 3.56387i) q^{75} +(-2.40654 - 0.706625i) q^{76} +(1.65065 - 5.19630i) q^{77} +(7.52405 - 2.20926i) q^{78} +(-4.70734 - 1.38220i) q^{79} +(6.68903 - 14.6469i) q^{80} -0.315134 q^{81} +(6.33433 + 7.31021i) q^{82} +(-0.730562 - 5.08117i) q^{83} +(-0.716427 + 0.826801i) q^{84} +(7.82551 + 9.03112i) q^{85} +(0.590843 - 4.10941i) q^{86} +(7.79214 - 5.00771i) q^{87} +(7.34990 + 1.22219i) q^{88} +(-9.92034 - 6.37542i) q^{89} +(6.34884 - 7.32695i) q^{90} +(-4.78428 + 5.52135i) q^{91} +(-3.06641 + 1.97066i) q^{92} +(-1.45276 + 1.67657i) q^{93} +3.25812 q^{94} +(13.1215 - 3.85281i) q^{95} +(-3.06641 - 1.97066i) q^{96} +(1.80750 - 0.530730i) q^{97} +(-0.987951 + 6.87135i) q^{98} +(5.39490 + 2.60855i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.671026 + 1.46934i −0.474487 + 1.03898i 0.509455 + 0.860497i \(0.329847\pi\)
−0.983943 + 0.178485i \(0.942880\pi\)
\(3\) −1.09234 −0.630663 −0.315332 − 0.948982i \(-0.602116\pi\)
−0.315332 + 0.948982i \(0.602116\pi\)
\(4\) −0.398970 − 0.460436i −0.199485 − 0.230218i
\(5\) 3.18729 + 0.935873i 1.42540 + 0.418535i 0.901327 − 0.433139i \(-0.142594\pi\)
0.524072 + 0.851674i \(0.324412\pi\)
\(6\) 0.732990 − 1.60502i 0.299242 − 0.655248i
\(7\) 0.233950 + 1.62716i 0.0884249 + 0.615009i 0.985057 + 0.172231i \(0.0550977\pi\)
−0.896632 + 0.442777i \(0.853993\pi\)
\(8\) −2.15551 + 0.632915i −0.762088 + 0.223769i
\(9\) −1.80679 −0.602264
\(10\) −3.51387 + 4.05523i −1.11118 + 1.28238i
\(11\) −2.98590 − 1.44375i −0.900283 − 0.435306i
\(12\) 0.435811 + 0.502953i 0.125808 + 0.145190i
\(13\) 2.91034 + 3.35871i 0.807182 + 0.931538i 0.998752 − 0.0499413i \(-0.0159034\pi\)
−0.191570 + 0.981479i \(0.561358\pi\)
\(14\) −2.54784 − 0.748114i −0.680940 − 0.199942i
\(15\) −3.48161 − 1.02229i −0.898948 − 0.263955i
\(16\) 0.689844 − 4.79797i 0.172461 − 1.19949i
\(17\) 3.02629 + 1.94488i 0.733983 + 0.471702i 0.853476 − 0.521133i \(-0.174490\pi\)
−0.119492 + 0.992835i \(0.538127\pi\)
\(18\) 1.21240 − 2.65480i 0.285766 − 0.625741i
\(19\) 3.46328 − 2.22571i 0.794531 − 0.510614i −0.0792961 − 0.996851i \(-0.525267\pi\)
0.873827 + 0.486237i \(0.161631\pi\)
\(20\) −0.840724 − 1.84093i −0.187992 − 0.411644i
\(21\) −0.255554 − 1.77741i −0.0557663 − 0.387863i
\(22\) 4.12498 − 3.41852i 0.879448 − 0.728831i
\(23\) 0.851457 − 5.92201i 0.177541 − 1.23482i −0.684889 − 0.728648i \(-0.740149\pi\)
0.862430 − 0.506177i \(-0.168942\pi\)
\(24\) 2.35455 − 0.691359i 0.480621 − 0.141123i
\(25\) 5.07670 + 3.26260i 1.01534 + 0.652519i
\(26\) −6.88800 + 2.02250i −1.35085 + 0.396645i
\(27\) 5.25066 1.01049
\(28\) 0.655864 − 0.756907i 0.123947 − 0.143042i
\(29\) −7.13343 + 4.58438i −1.32465 + 0.851298i −0.995663 − 0.0930382i \(-0.970342\pi\)
−0.328983 + 0.944336i \(0.606706\pi\)
\(30\) 3.83835 − 4.42969i 0.700783 − 0.808747i
\(31\) 1.32995 − 1.53484i 0.238866 − 0.275666i −0.623641 − 0.781711i \(-0.714347\pi\)
0.862507 + 0.506044i \(0.168893\pi\)
\(32\) 2.80719 + 1.80407i 0.496246 + 0.318918i
\(33\) 3.26162 + 1.57706i 0.567775 + 0.274531i
\(34\) −4.88841 + 3.14159i −0.838356 + 0.538779i
\(35\) −0.777148 + 5.40518i −0.131362 + 0.913642i
\(36\) 0.720856 + 0.831912i 0.120143 + 0.138652i
\(37\) 7.45128 − 8.59924i 1.22498 − 1.41371i 0.345066 − 0.938578i \(-0.387857\pi\)
0.879917 − 0.475127i \(-0.157598\pi\)
\(38\) 0.946386 + 6.58226i 0.153524 + 1.06778i
\(39\) −3.17908 − 3.66885i −0.509060 − 0.587487i
\(40\) −7.46257 −1.17994
\(41\) 2.48758 − 5.44703i 0.388494 − 0.850684i −0.609814 − 0.792544i \(-0.708756\pi\)
0.998308 − 0.0581393i \(-0.0185168\pi\)
\(42\) 2.78311 + 0.817196i 0.429444 + 0.126096i
\(43\) −2.46608 + 0.724106i −0.376073 + 0.110425i −0.464306 − 0.885675i \(-0.653696\pi\)
0.0882328 + 0.996100i \(0.471878\pi\)
\(44\) 0.526532 + 1.95083i 0.0793777 + 0.294098i
\(45\) −5.75877 − 1.69093i −0.858466 − 0.252069i
\(46\) 8.13011 + 5.22491i 1.19872 + 0.770370i
\(47\) −0.837901 − 1.83475i −0.122220 − 0.267625i 0.838626 − 0.544708i \(-0.183360\pi\)
−0.960846 + 0.277083i \(0.910632\pi\)
\(48\) −0.753545 + 5.24102i −0.108765 + 0.756476i
\(49\) 4.12353 − 1.21078i 0.589076 − 0.172968i
\(50\) −8.20047 + 5.27012i −1.15972 + 0.745308i
\(51\) −3.30574 − 2.12447i −0.462896 − 0.297485i
\(52\) 0.385333 − 2.68005i 0.0534360 − 0.371656i
\(53\) −0.0209417 − 0.145653i −0.00287657 − 0.0200070i 0.988333 − 0.152311i \(-0.0486716\pi\)
−0.991209 + 0.132304i \(0.957762\pi\)
\(54\) −3.52333 + 7.71501i −0.479464 + 1.04988i
\(55\) −8.16577 − 7.39606i −1.10107 − 0.997285i
\(56\) −1.53414 − 3.35929i −0.205008 − 0.448904i
\(57\) −3.78308 + 2.43124i −0.501081 + 0.322026i
\(58\) −1.94930 − 13.5577i −0.255956 − 1.78021i
\(59\) 2.96026 + 6.48208i 0.385394 + 0.843894i 0.998545 + 0.0539287i \(0.0171744\pi\)
−0.613151 + 0.789966i \(0.710098\pi\)
\(60\) 0.918358 + 2.01092i 0.118559 + 0.259609i
\(61\) −2.06529 − 4.52234i −0.264433 − 0.579027i 0.730113 − 0.683326i \(-0.239467\pi\)
−0.994546 + 0.104299i \(0.966740\pi\)
\(62\) 1.36278 + 2.98407i 0.173073 + 0.378978i
\(63\) −0.422699 − 2.93994i −0.0532551 − 0.370397i
\(64\) 3.62113 − 2.32716i 0.452641 − 0.290895i
\(65\) 6.13276 + 13.4289i 0.760676 + 1.66565i
\(66\) −4.50588 + 3.73419i −0.554635 + 0.459647i
\(67\) −2.27836 + 4.98892i −0.278346 + 0.609494i −0.996238 − 0.0866614i \(-0.972380\pi\)
0.717891 + 0.696155i \(0.245107\pi\)
\(68\) −0.311907 − 2.16936i −0.0378243 − 0.263074i
\(69\) −0.930081 + 6.46886i −0.111969 + 0.778759i
\(70\) −7.42058 − 4.76891i −0.886928 − 0.569994i
\(71\) −9.46687 + 6.08399i −1.12351 + 0.722037i −0.964196 − 0.265191i \(-0.914565\pi\)
−0.159315 + 0.987228i \(0.550929\pi\)
\(72\) 3.89456 − 1.14354i 0.458978 − 0.134768i
\(73\) −1.33753 + 9.30273i −0.156546 + 1.08880i 0.748391 + 0.663258i \(0.230827\pi\)
−0.904937 + 0.425545i \(0.860082\pi\)
\(74\) 7.63522 + 16.7188i 0.887576 + 1.94352i
\(75\) −5.54548 − 3.56387i −0.640337 − 0.411520i
\(76\) −2.40654 − 0.706625i −0.276050 − 0.0810555i
\(77\) 1.65065 − 5.19630i 0.188109 − 0.592174i
\(78\) 7.52405 − 2.20926i 0.851931 − 0.250149i
\(79\) −4.70734 − 1.38220i −0.529618 − 0.155510i 0.00598151 − 0.999982i \(-0.498096\pi\)
−0.535599 + 0.844472i \(0.679914\pi\)
\(80\) 6.68903 − 14.6469i 0.747856 − 1.63758i
\(81\) −0.315134 −0.0350148
\(82\) 6.33433 + 7.31021i 0.699510 + 0.807277i
\(83\) −0.730562 − 5.08117i −0.0801896 − 0.557731i −0.989821 − 0.142315i \(-0.954545\pi\)
0.909632 − 0.415415i \(-0.136364\pi\)
\(84\) −0.716427 + 0.826801i −0.0781686 + 0.0902114i
\(85\) 7.82551 + 9.03112i 0.848796 + 0.979562i
\(86\) 0.590843 − 4.10941i 0.0637123 − 0.443129i
\(87\) 7.79214 − 5.00771i 0.835405 − 0.536882i
\(88\) 7.34990 + 1.22219i 0.783502 + 0.130286i
\(89\) −9.92034 − 6.37542i −1.05155 − 0.675793i −0.103736 − 0.994605i \(-0.533080\pi\)
−0.947818 + 0.318812i \(0.896716\pi\)
\(90\) 6.34884 − 7.32695i 0.669226 − 0.772328i
\(91\) −4.78428 + 5.52135i −0.501529 + 0.578795i
\(92\) −3.06641 + 1.97066i −0.319696 + 0.205456i
\(93\) −1.45276 + 1.67657i −0.150644 + 0.173853i
\(94\) 3.25812 0.336050
\(95\) 13.1215 − 3.85281i 1.34623 − 0.395290i
\(96\) −3.06641 − 1.97066i −0.312964 − 0.201130i
\(97\) 1.80750 − 0.530730i 0.183524 − 0.0538875i −0.188679 − 0.982039i \(-0.560421\pi\)
0.372203 + 0.928151i \(0.378602\pi\)
\(98\) −0.987951 + 6.87135i −0.0997981 + 0.694111i
\(99\) 5.39490 + 2.60855i 0.542208 + 0.262169i
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.23.3 ✓ 100
121.100 even 11 inner 121.2.e.a.100.3 yes 100
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.2.e.a.23.3 ✓ 100 1.1 even 1 trivial
121.2.e.a.100.3 yes 100 121.100 even 11 inner