↑

Properties

Label 322.2.k.b.111.3
Level $322$
Weight $2$
Character 322.111
Analytic conductor $2.571$
Analytic rank $0$
Dimension $80$
Inner twists $4$

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] = [322,2,Mod(83,322)] 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("322.83"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(322, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 21])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.k (of order \(22\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [80,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(8\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 111.3
Character \(\chi\) \(=\) 322.111
Dual form 322.2.k.b.293.3

$q$-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+(-0.415415 - 0.909632i) q^{2} +(-0.521769 - 1.77698i) q^{3} +(-0.654861 + 0.755750i) q^{4} +(1.91138 + 1.22837i) q^{5} +(-1.39965 + 1.21280i) q^{6} +(2.44569 + 1.00927i) q^{7} +(0.959493 + 0.281733i) q^{8} +(-0.361658 + 0.232423i) q^{9} +(0.323348 - 2.24893i) q^{10} +(4.10456 + 1.87449i) q^{11} +(1.68464 + 0.769349i) q^{12} +(0.635676 + 0.0913964i) q^{13} +(-0.0979123 - 2.64394i) q^{14} +(1.18549 - 4.03741i) q^{15} +(-0.142315 - 0.989821i) q^{16} +(-3.91779 - 4.52137i) q^{17} +(0.361658 + 0.232423i) q^{18} +(-2.06468 + 2.38276i) q^{19} +(-2.18003 + 0.640113i) q^{20} +(0.517367 - 4.87254i) q^{21} -4.51233i q^{22} +(1.40318 + 4.58597i) q^{23} -1.85200i q^{24} +(0.0674053 + 0.147597i) q^{25} +(-0.180932 - 0.616198i) q^{26} +(-3.59723 - 3.11702i) q^{27} +(-2.36434 + 1.18740i) q^{28} +(-1.06343 - 1.22727i) q^{29} +(-4.16503 + 0.598840i) q^{30} +(-0.502822 + 1.71245i) q^{31} +(-0.841254 + 0.540641i) q^{32} +(1.18930 - 8.27178i) q^{33} +(-2.48527 + 5.44199i) q^{34} +(3.43488 + 4.93330i) q^{35} +(0.0611816 - 0.425527i) q^{36} +(-1.60073 - 2.49078i) q^{37} +(3.02514 + 0.888260i) q^{38} +(-0.169266 - 1.17727i) q^{39} +(1.48788 + 1.71711i) q^{40} +(3.85494 - 5.99840i) q^{41} +(-4.64714 + 1.55351i) q^{42} +(-2.69128 - 9.16567i) q^{43} +(-4.10456 + 1.87449i) q^{44} -0.976766 q^{45} +(3.58864 - 3.18146i) q^{46} +9.26338i q^{47} +(-1.68464 + 0.769349i) q^{48} +(4.96276 + 4.93670i) q^{49} +(0.106258 - 0.122628i) q^{50} +(-5.99020 + 9.32094i) q^{51} +(-0.485352 + 0.420560i) q^{52} +(0.370846 - 0.0533196i) q^{53} +(-1.34100 + 4.56701i) q^{54} +(5.54281 + 8.62478i) q^{55} +(2.06227 + 1.65741i) q^{56} +(5.31141 + 2.42564i) q^{57} +(-0.674596 + 1.47716i) q^{58} +(6.21434 + 0.893488i) q^{59} +(2.27494 + 3.53987i) q^{60} +(1.11958 + 0.328739i) q^{61} +(1.76658 - 0.253996i) q^{62} +(-1.11908 + 0.203425i) q^{63} +(0.841254 + 0.540641i) q^{64} +(1.10275 + 0.955537i) q^{65} +(-8.01833 + 2.35439i) q^{66} +(-9.35102 + 4.27047i) q^{67} +5.98263 q^{68} +(7.41704 - 4.88624i) q^{69} +(3.06058 - 5.17384i) q^{70} +(2.81918 + 6.17315i) q^{71} +(-0.412489 + 0.121118i) q^{72} +(-5.94098 - 5.14789i) q^{73} +(-1.60073 + 2.49078i) q^{74} +(0.227107 - 0.196789i) q^{75} +(-0.448697 - 3.12076i) q^{76} +(8.14661 + 8.72702i) q^{77} +(-1.00057 + 0.643026i) q^{78} +(-9.46660 - 1.36109i) q^{79} +(0.943848 - 2.06674i) q^{80} +(-4.19772 + 9.19173i) q^{81} +(-7.05774 - 1.01475i) q^{82} +(7.12112 - 4.57646i) q^{83} +(3.34362 + 3.58184i) q^{84} +(-1.93447 - 13.4545i) q^{85} +(-7.21939 + 6.25563i) q^{86} +(-1.62597 + 2.53005i) q^{87} +(3.41020 + 2.95495i) q^{88} +(-11.9349 + 3.50441i) q^{89} +(0.405763 + 0.888498i) q^{90} +(1.46242 + 0.865094i) q^{91} +(-4.38473 - 1.94272i) q^{92} +3.30536 q^{93} +(8.42626 - 3.84815i) q^{94} +(-6.87329 + 2.01818i) q^{95} +(1.39965 + 1.21280i) q^{96} +(-11.3857 - 7.31712i) q^{97} +(2.42898 - 6.56506i) q^{98} +(-1.92012 + 0.276072i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 8 q^{2} - 8 q^{4} + 11 q^{7} + 8 q^{8} + 14 q^{9} - 11 q^{14} - 8 q^{16} - 14 q^{18} - 33 q^{21} + 18 q^{23} + 6 q^{25} - 11 q^{28} + 8 q^{29} + 22 q^{30} + 8 q^{32} - 33 q^{35} - 8 q^{36} - 22 q^{37}+ \cdots + 132 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(-1\) \(e\left(\frac{15}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.415415 − 0.909632i −0.293743 − 0.643207i
\(3\) −0.521769 − 1.77698i −0.301243 − 1.02594i −0.961478 − 0.274881i \(-0.911361\pi\)
0.660235 − 0.751059i \(-0.270457\pi\)
\(4\) −0.654861 + 0.755750i −0.327430 + 0.377875i
\(5\) 1.91138 + 1.22837i 0.854795 + 0.549343i 0.893067 − 0.449924i \(-0.148549\pi\)
−0.0382720 + 0.999267i \(0.512185\pi\)
\(6\) −1.39965 + 1.21280i −0.571404 + 0.495124i
\(7\) 2.44569 + 1.00927i 0.924382 + 0.381467i
\(8\) 0.959493 + 0.281733i 0.339232 + 0.0996075i
\(9\) −0.361658 + 0.232423i −0.120553 + 0.0774744i
\(10\) 0.323348 − 2.24893i 0.102252 − 0.711176i
\(11\) 4.10456 + 1.87449i 1.23757 + 0.565180i 0.923273 − 0.384143i \(-0.125503\pi\)
0.314299 + 0.949324i \(0.398230\pi\)
\(12\) 1.68464 + 0.769349i 0.486313 + 0.222092i
\(13\) 0.635676 + 0.0913964i 0.176305 + 0.0253488i 0.229902 − 0.973214i \(-0.426160\pi\)
−0.0535970 + 0.998563i \(0.517069\pi\)
\(14\) −0.0979123 − 2.64394i −0.0261682 − 0.706622i
\(15\) 1.18549 − 4.03741i 0.306092 − 1.04245i
\(16\) −0.142315 − 0.989821i −0.0355787 − 0.247455i
\(17\) −3.91779 − 4.52137i −0.950203 − 1.09659i −0.995225 − 0.0976065i \(-0.968881\pi\)
0.0450223 − 0.998986i \(-0.485664\pi\)
\(18\) 0.361658 + 0.232423i 0.0852435 + 0.0547827i
\(19\) −2.06468 + 2.38276i −0.473669 + 0.546644i −0.941429 − 0.337212i \(-0.890516\pi\)
0.467759 + 0.883856i \(0.345061\pi\)
\(20\) −2.18003 + 0.640113i −0.487469 + 0.143134i
\(21\) 0.517367 − 4.87254i 0.112899 − 1.06328i
\(22\) − 4.51233i − 0.962033i
\(23\) 1.40318 + 4.58597i 0.292583 + 0.956240i
\(24\) − 1.85200i − 0.378038i
\(25\) 0.0674053 + 0.147597i 0.0134811 + 0.0295194i
\(26\) −0.180932 − 0.616198i −0.0354837 − 0.120846i
\(27\) −3.59723 − 3.11702i −0.692287 − 0.599870i
\(28\) −2.36434 + 1.18740i −0.446818 + 0.224397i
\(29\) −1.06343 − 1.22727i −0.197475 − 0.227898i 0.648372 − 0.761323i \(-0.275450\pi\)
−0.845847 + 0.533425i \(0.820905\pi\)
\(30\) −4.16503 + 0.598840i −0.760426 + 0.109333i
\(31\) −0.502822 + 1.71245i −0.0903095 + 0.307566i −0.992244 − 0.124307i \(-0.960329\pi\)
0.901934 + 0.431873i \(0.142147\pi\)
\(32\) −0.841254 + 0.540641i −0.148714 + 0.0955727i
\(33\) 1.18930 − 8.27178i 0.207031 − 1.43993i
\(34\) −2.48527 + 5.44199i −0.426221 + 0.933293i
\(35\) 3.43488 + 4.93330i 0.580601 + 0.833879i
\(36\) 0.0611816 − 0.425527i 0.0101969 − 0.0709212i
\(37\) −1.60073 − 2.49078i −0.263158 − 0.409482i 0.684380 − 0.729126i \(-0.260073\pi\)
−0.947538 + 0.319644i \(0.896437\pi\)
\(38\) 3.02514 + 0.888260i 0.490742 + 0.144095i
\(39\) −0.169266 − 1.17727i −0.0271043 − 0.188514i
\(40\) 1.48788 + 1.71711i 0.235255 + 0.271499i
\(41\) 3.85494 − 5.99840i 0.602040 − 0.936793i −0.397773 − 0.917484i \(-0.630217\pi\)
0.999814 − 0.0193094i \(-0.00614677\pi\)
\(42\) −4.64714 + 1.55351i −0.717069 + 0.239712i
\(43\) −2.69128 − 9.16567i −0.410417 − 1.39775i −0.862627 − 0.505840i \(-0.831183\pi\)
0.452210 − 0.891911i \(-0.350636\pi\)
\(44\) −4.10456 + 1.87449i −0.618786 + 0.282590i
\(45\) −0.976766 −0.145608
\(46\) 3.58864 − 3.18146i 0.529116 − 0.469080i
\(47\) 9.26338i 1.35120i 0.737267 + 0.675601i \(0.236116\pi\)
−0.737267 + 0.675601i \(0.763884\pi\)
\(48\) −1.68464 + 0.769349i −0.243157 + 0.111046i
\(49\) 4.96276 + 4.93670i 0.708965 + 0.705243i
\(50\) 0.106258 − 0.122628i 0.0150271 − 0.0173422i
\(51\) −5.99020 + 9.32094i −0.838796 + 1.30519i
\(52\) −0.485352 + 0.420560i −0.0673062 + 0.0583211i
\(53\) 0.370846 − 0.0533196i 0.0509396 − 0.00732401i −0.116798 − 0.993156i \(-0.537263\pi\)
0.167738 + 0.985832i \(0.446354\pi\)
\(54\) −1.34100 + 4.56701i −0.182486 + 0.621492i
\(55\) 5.54281 + 8.62478i 0.747393 + 1.16297i
\(56\) 2.06227 + 1.65741i 0.275583 + 0.221481i
\(57\) 5.31141 + 2.42564i 0.703513 + 0.321284i
\(58\) −0.674596 + 1.47716i −0.0885788 + 0.193961i
\(59\) 6.21434 + 0.893488i 0.809038 + 0.116322i 0.534407 − 0.845227i \(-0.320535\pi\)
0.274632 + 0.961550i \(0.411444\pi\)
\(60\) 2.27494 + 3.53987i 0.293693 + 0.456996i
\(61\) 1.11958 + 0.328739i 0.143348 + 0.0420908i 0.352619 − 0.935767i \(-0.385291\pi\)
−0.209271 + 0.977858i \(0.567109\pi\)
\(62\) 1.76658 − 0.253996i 0.224356 − 0.0322576i
\(63\) −1.11908 + 0.203425i −0.140991 + 0.0256291i
\(64\) 0.841254 + 0.540641i 0.105157 + 0.0675801i
\(65\) 1.10275 + 0.955537i 0.136779 + 0.118520i
\(66\) −8.01833 + 2.35439i −0.986988 + 0.289806i
\(67\) −9.35102 + 4.27047i −1.14241 + 0.521721i −0.894502 − 0.447063i \(-0.852470\pi\)
−0.247906 + 0.968784i \(0.579743\pi\)
\(68\) 5.98263 0.725500
\(69\) 7.41704 − 4.88624i 0.892907 − 0.588234i
\(70\) 3.06058 − 5.17384i 0.365810 − 0.618392i
\(71\) 2.81918 + 6.17315i 0.334576 + 0.732618i 0.999903 − 0.0139333i \(-0.00443526\pi\)
−0.665327 + 0.746552i \(0.731708\pi\)
\(72\) −0.412489 + 0.121118i −0.0486123 + 0.0142739i
\(73\) −5.94098 − 5.14789i −0.695339 − 0.602514i 0.233783 − 0.972289i \(-0.424890\pi\)
−0.929122 + 0.369774i \(0.879435\pi\)
\(74\) −1.60073 + 2.49078i −0.186081 + 0.289547i
\(75\) 0.227107 − 0.196789i 0.0262241 − 0.0227233i
\(76\) −0.448697 − 3.12076i −0.0514691 − 0.357975i
\(77\) 8.14661 + 8.72702i 0.928392 + 0.994536i
\(78\) −1.00057 + 0.643026i −0.113292 + 0.0728083i
\(79\) −9.46660 − 1.36109i −1.06508 − 0.153135i −0.412575 − 0.910923i \(-0.635371\pi\)
−0.652500 + 0.757789i \(0.726280\pi\)
\(80\) 0.943848 − 2.06674i 0.105525 − 0.231068i
\(81\) −4.19772 + 9.19173i −0.466414 + 1.02130i
\(82\) −7.05774 − 1.01475i −0.779397 − 0.112060i
\(83\) 7.12112 − 4.57646i 0.781644 − 0.502332i −0.0879346 − 0.996126i \(-0.528027\pi\)
0.869579 + 0.493794i \(0.164390\pi\)
\(84\) 3.34362 + 3.58184i 0.364819 + 0.390810i
\(85\) −1.93447 − 13.4545i −0.209823 − 1.45935i
\(86\) −7.21939 + 6.25563i −0.778487 + 0.674562i
\(87\) −1.62597 + 2.53005i −0.174322 + 0.271250i
\(88\) 3.41020 + 2.95495i 0.363528 + 0.314999i
\(89\) −11.9349 + 3.50441i −1.26510 + 0.371467i −0.844390 − 0.535728i \(-0.820037\pi\)
−0.420710 + 0.907195i \(0.638219\pi\)
\(90\) 0.405763 + 0.888498i 0.0427712 + 0.0936559i
\(91\) 1.46242 + 0.865094i 0.153303 + 0.0906865i
\(92\) −4.38473 − 1.94272i −0.457140 − 0.202542i
\(93\) 3.30536 0.342749
\(94\) 8.42626 − 3.84815i 0.869103 − 0.396906i
\(95\) −6.87329 + 2.01818i −0.705185 + 0.207061i
\(96\) 1.39965 + 1.21280i 0.142851 + 0.123781i
\(97\) −11.3857 − 7.31712i −1.15604 − 0.742941i −0.185206 − 0.982700i \(-0.559295\pi\)
−0.970832 + 0.239759i \(0.922931\pi\)
\(98\) 2.42898 − 6.56506i 0.245364 − 0.663172i
\(99\) −1.92012 + 0.276072i −0.192980 + 0.0277463i
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 322.2.k.b.111.3 ✓ 80
7.6 odd 2 inner 322.2.k.b.111.6 yes 80
23.17 odd 22 inner 322.2.k.b.293.6 yes 80
161.132 even 22 inner 322.2.k.b.293.3 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.k.b.111.3 ✓ 80 1.1 even 1 trivial
322.2.k.b.111.6 yes 80 7.6 odd 2 inner
322.2.k.b.293.3 yes 80 161.132 even 22 inner
322.2.k.b.293.6 yes 80 23.17 odd 22 inner