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Properties

Label 322.2.k.b.97.2
Level $322$
Weight $2$
Character 322.97
Analytic conductor $2.571$
Analytic rank $0$
Dimension $80$
Inner twists $4$

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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] = [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 97.2
Character \(\chi\) \(=\) 322.97
Dual form 322.2.k.b.83.2

$q$-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+(0.959493 + 0.281733i) q^{2} +(-1.76160 - 1.52644i) q^{3} +(0.841254 + 0.540641i) q^{4} +(0.360367 - 2.50641i) q^{5} +(-1.26020 - 1.96091i) q^{6} +(1.49769 + 2.18104i) q^{7} +(0.654861 + 0.755750i) q^{8} +(0.346290 + 2.40850i) q^{9} +(1.05191 - 2.30335i) q^{10} +(-1.57169 - 5.35270i) q^{11} +(-0.656701 - 2.23652i) q^{12} +(-4.54531 - 2.07577i) q^{13} +(0.822556 + 2.51464i) q^{14} +(-4.46070 + 3.86522i) q^{15} +(0.415415 + 0.909632i) q^{16} +(4.42824 - 2.84586i) q^{17} +(-0.346290 + 2.40850i) q^{18} +(1.92116 + 1.23465i) q^{19} +(1.65823 - 1.91370i) q^{20} +(0.690879 - 6.12826i) q^{21} -5.57868i q^{22} +(-4.70476 - 0.930168i) q^{23} -2.33094i q^{24} +(-1.35475 - 0.397792i) q^{25} +(-3.77638 - 3.27225i) q^{26} +(-0.714198 + 1.11131i) q^{27} +(0.0807810 + 2.64452i) q^{28} +(3.43903 - 2.21013i) q^{29} +(-5.36897 + 2.45193i) q^{30} +(-0.173313 + 0.150176i) q^{31} +(0.142315 + 0.989821i) q^{32} +(-5.40187 + 11.8284i) q^{33} +(5.05064 - 1.48300i) q^{34} +(6.00629 - 2.96785i) q^{35} +(-1.01082 + 2.21338i) q^{36} +(-0.348039 + 0.0500404i) q^{37} +(1.49550 + 1.72589i) q^{38} +(4.83850 + 10.5948i) q^{39} +(2.13021 - 1.36900i) q^{40} +(6.64275 + 0.955083i) q^{41} +(2.38942 - 5.68538i) q^{42} +(9.43839 + 8.17841i) q^{43} +(1.57169 - 5.35270i) q^{44} +6.16147 q^{45} +(-4.25213 - 2.21797i) q^{46} +9.86212i q^{47} +(0.656701 - 2.23652i) q^{48} +(-2.51384 + 6.53304i) q^{49} +(-1.18781 - 0.763357i) q^{50} +(-12.1448 - 1.74617i) q^{51} +(-2.70151 - 4.20363i) q^{52} +(7.75780 - 3.54287i) q^{53} +(-0.998361 + 0.865085i) q^{54} +(-13.9824 + 2.01037i) q^{55} +(-0.667538 + 2.56015i) q^{56} +(-1.49970 - 5.10750i) q^{57} +(3.92239 - 1.15172i) q^{58} +(-0.0115630 - 0.00528067i) q^{59} +(-5.84228 + 0.839993i) q^{60} +(-1.67529 - 1.93339i) q^{61} +(-0.208602 + 0.0952654i) q^{62} +(-4.73439 + 4.36246i) q^{63} +(-0.142315 + 0.989821i) q^{64} +(-6.84072 + 10.6444i) q^{65} +(-8.51551 + 9.82742i) q^{66} +(-1.97993 + 6.74301i) q^{67} +5.26386 q^{68} +(6.86809 + 8.82012i) q^{69} +(6.59913 - 1.15547i) q^{70} +(-8.11338 - 2.38230i) q^{71} +(-1.59345 + 1.83894i) q^{72} +(5.31431 - 8.26923i) q^{73} +(-0.348039 - 0.0500404i) q^{74} +(1.77934 + 2.76870i) q^{75} +(0.948677 + 2.07731i) q^{76} +(9.32052 - 11.4446i) q^{77} +(1.65760 + 11.5288i) q^{78} +(2.33017 + 1.06415i) q^{79} +(2.42961 - 0.713398i) q^{80} +(9.95859 - 2.92411i) q^{81} +(6.10459 + 2.78787i) q^{82} +(0.201284 + 1.39996i) q^{83} +(3.89439 - 4.78190i) q^{84} +(-5.53709 - 12.1245i) q^{85} +(6.75194 + 10.5062i) q^{86} +(-9.43185 - 1.35609i) q^{87} +(3.01606 - 4.69308i) q^{88} +(-4.15559 + 4.79581i) q^{89} +(5.91189 + 1.73589i) q^{90} +(-2.28014 - 13.0224i) q^{91} +(-3.45501 - 3.32609i) q^{92} +0.534544 q^{93} +(-2.77848 + 9.46263i) q^{94} +(3.78687 - 4.37028i) q^{95} +(1.26020 - 1.96091i) q^{96} +(-0.148787 + 1.03483i) q^{97} +(-4.25258 + 5.56018i) q^{98} +(12.3477 - 5.63901i) 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 \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.959493 + 0.281733i 0.678464 + 0.199215i
\(3\) −1.76160 − 1.52644i −1.01706 − 0.881290i −0.0240992 − 0.999710i \(-0.507672\pi\)
−0.992964 + 0.118420i \(0.962217\pi\)
\(4\) 0.841254 + 0.540641i 0.420627 + 0.270320i
\(5\) 0.360367 − 2.50641i 0.161161 − 1.12090i −0.735288 − 0.677754i \(-0.762953\pi\)
0.896450 − 0.443146i \(-0.146138\pi\)
\(6\) −1.26020 − 1.96091i −0.514474 − 0.800538i
\(7\) 1.49769 + 2.18104i 0.566074 + 0.824354i
\(8\) 0.654861 + 0.755750i 0.231528 + 0.267198i
\(9\) 0.346290 + 2.40850i 0.115430 + 0.802833i
\(10\) 1.05191 − 2.30335i 0.332642 − 0.728385i
\(11\) −1.57169 − 5.35270i −0.473884 − 1.61390i −0.756036 − 0.654530i \(-0.772866\pi\)
0.282152 − 0.959370i \(-0.408952\pi\)
\(12\) −0.656701 − 2.23652i −0.189573 − 0.645627i
\(13\) −4.54531 − 2.07577i −1.26064 − 0.575716i −0.330809 − 0.943698i \(-0.607322\pi\)
−0.929833 + 0.367981i \(0.880049\pi\)
\(14\) 0.822556 + 2.51464i 0.219837 + 0.672065i
\(15\) −4.46070 + 3.86522i −1.15175 + 0.997996i
\(16\) 0.415415 + 0.909632i 0.103854 + 0.227408i
\(17\) 4.42824 − 2.84586i 1.07401 − 0.690222i 0.120842 − 0.992672i \(-0.461441\pi\)
0.953165 + 0.302449i \(0.0978043\pi\)
\(18\) −0.346290 + 2.40850i −0.0816213 + 0.567689i
\(19\) 1.92116 + 1.23465i 0.440744 + 0.283249i 0.742133 − 0.670252i \(-0.233814\pi\)
−0.301389 + 0.953501i \(0.597450\pi\)
\(20\) 1.65823 − 1.91370i 0.370791 − 0.427915i
\(21\) 0.690879 − 6.12826i 0.150762 − 1.33730i
\(22\) − 5.57868i − 1.18938i
\(23\) −4.70476 − 0.930168i −0.981011 − 0.193953i
\(24\) − 2.33094i − 0.475800i
\(25\) −1.35475 − 0.397792i −0.270951 − 0.0795584i
\(26\) −3.77638 − 3.27225i −0.740609 − 0.641742i
\(27\) −0.714198 + 1.11131i −0.137447 + 0.213872i
\(28\) 0.0807810 + 2.64452i 0.0152662 + 0.499767i
\(29\) 3.43903 − 2.21013i 0.638612 − 0.410411i −0.180877 − 0.983506i \(-0.557894\pi\)
0.819489 + 0.573095i \(0.194257\pi\)
\(30\) −5.36897 + 2.45193i −0.980236 + 0.447659i
\(31\) −0.173313 + 0.150176i −0.0311279 + 0.0269725i −0.670286 − 0.742103i \(-0.733828\pi\)
0.639158 + 0.769076i \(0.279283\pi\)
\(32\) 0.142315 + 0.989821i 0.0251579 + 0.174977i
\(33\) −5.40187 + 11.8284i −0.940344 + 2.05907i
\(34\) 5.05064 − 1.48300i 0.866178 − 0.254333i
\(35\) 6.00629 − 2.96785i 1.01525 − 0.501659i
\(36\) −1.01082 + 2.21338i −0.168469 + 0.368896i
\(37\) −0.348039 + 0.0500404i −0.0572172 + 0.00822659i −0.170864 − 0.985295i \(-0.554656\pi\)
0.113647 + 0.993521i \(0.463747\pi\)
\(38\) 1.49550 + 1.72589i 0.242602 + 0.279977i
\(39\) 4.83850 + 10.5948i 0.774780 + 1.69653i
\(40\) 2.13021 − 1.36900i 0.336815 − 0.216458i
\(41\) 6.64275 + 0.955083i 1.03742 + 0.149159i 0.639919 − 0.768442i \(-0.278968\pi\)
0.397503 + 0.917601i \(0.369877\pi\)
\(42\) 2.38942 − 5.68538i 0.368696 − 0.877273i
\(43\) 9.43839 + 8.17841i 1.43934 + 1.24720i 0.919591 + 0.392877i \(0.128520\pi\)
0.519750 + 0.854319i \(0.326025\pi\)
\(44\) 1.57169 − 5.35270i 0.236942 − 0.806950i
\(45\) 6.16147 0.918498
\(46\) −4.25213 − 2.21797i −0.626942 − 0.327022i
\(47\) 9.86212i 1.43854i 0.694732 + 0.719269i \(0.255523\pi\)
−0.694732 + 0.719269i \(0.744477\pi\)
\(48\) 0.656701 − 2.23652i 0.0947866 − 0.322813i
\(49\) −2.51384 + 6.53304i −0.359120 + 0.933291i
\(50\) −1.18781 − 0.763357i −0.167981 − 0.107955i
\(51\) −12.1448 − 1.74617i −1.70062 − 0.244512i
\(52\) −2.70151 − 4.20363i −0.374632 − 0.582939i
\(53\) 7.75780 − 3.54287i 1.06562 − 0.486650i 0.196114 − 0.980581i \(-0.437168\pi\)
0.869501 + 0.493931i \(0.164440\pi\)
\(54\) −0.998361 + 0.865085i −0.135860 + 0.117723i
\(55\) −13.9824 + 2.01037i −1.88539 + 0.271079i
\(56\) −0.667538 + 2.56015i −0.0892035 + 0.342115i
\(57\) −1.49970 − 5.10750i −0.198640 − 0.676505i
\(58\) 3.92239 − 1.15172i 0.515036 − 0.151228i
\(59\) −0.0115630 − 0.00528067i −0.00150538 − 0.000687484i 0.414662 − 0.909975i \(-0.363900\pi\)
−0.416168 + 0.909288i \(0.636627\pi\)
\(60\) −5.84228 + 0.839993i −0.754235 + 0.108443i
\(61\) −1.67529 − 1.93339i −0.214499 − 0.247545i 0.638296 − 0.769791i \(-0.279640\pi\)
−0.852795 + 0.522246i \(0.825094\pi\)
\(62\) −0.208602 + 0.0952654i −0.0264925 + 0.0120987i
\(63\) −4.73439 + 4.36246i −0.596477 + 0.549618i
\(64\) −0.142315 + 0.989821i −0.0177894 + 0.123728i
\(65\) −6.84072 + 10.6444i −0.848487 + 1.32027i
\(66\) −8.51551 + 9.82742i −1.04819 + 1.20967i
\(67\) −1.97993 + 6.74301i −0.241886 + 0.823789i 0.745642 + 0.666347i \(0.232143\pi\)
−0.987528 + 0.157442i \(0.949675\pi\)
\(68\) 5.26386 0.638337
\(69\) 6.86809 + 8.82012i 0.826820 + 1.06182i
\(70\) 6.59913 − 1.15547i 0.788747 − 0.138105i
\(71\) −8.11338 − 2.38230i −0.962881 − 0.282727i −0.237741 − 0.971329i \(-0.576407\pi\)
−0.725140 + 0.688601i \(0.758225\pi\)
\(72\) −1.59345 + 1.83894i −0.187790 + 0.216721i
\(73\) 5.31431 − 8.26923i 0.621993 − 0.967840i −0.377138 − 0.926157i \(-0.623092\pi\)
0.999131 − 0.0416828i \(-0.0132719\pi\)
\(74\) −0.348039 − 0.0500404i −0.0404587 − 0.00581708i
\(75\) 1.77934 + 2.76870i 0.205460 + 0.319702i
\(76\) 0.948677 + 2.07731i 0.108821 + 0.238284i
\(77\) 9.32052 − 11.4446i 1.06217 − 1.30424i
\(78\) 1.65760 + 11.5288i 0.187686 + 1.30538i
\(79\) 2.33017 + 1.06415i 0.262165 + 0.119727i 0.542161 − 0.840274i \(-0.317606\pi\)
−0.279997 + 0.960001i \(0.590333\pi\)
\(80\) 2.42961 − 0.713398i 0.271639 − 0.0797603i
\(81\) 9.95859 − 2.92411i 1.10651 − 0.324901i
\(82\) 6.10459 + 2.78787i 0.674139 + 0.307869i
\(83\) 0.201284 + 1.39996i 0.0220938 + 0.153666i 0.997882 − 0.0650526i \(-0.0207215\pi\)
−0.975788 + 0.218718i \(0.929812\pi\)
\(84\) 3.89439 − 4.78190i 0.424913 − 0.521748i
\(85\) −5.53709 − 12.1245i −0.600582 − 1.31509i
\(86\) 6.75194 + 10.5062i 0.728081 + 1.13292i
\(87\) −9.43185 − 1.35609i −1.01120 − 0.145389i
\(88\) 3.01606 − 4.69308i 0.321513 − 0.500284i
\(89\) −4.15559 + 4.79581i −0.440492 + 0.508354i −0.931970 − 0.362535i \(-0.881911\pi\)
0.491478 + 0.870890i \(0.336457\pi\)
\(90\) 5.91189 + 1.73589i 0.623168 + 0.182979i
\(91\) −2.28014 − 13.0224i −0.239023 − 1.36511i
\(92\) −3.45501 − 3.32609i −0.360210 − 0.346769i
\(93\) 0.534544 0.0554296
\(94\) −2.77848 + 9.46263i −0.286578 + 0.975996i
\(95\) 3.78687 − 4.37028i 0.388525 − 0.448381i
\(96\) 1.26020 − 1.96091i 0.128619 − 0.200134i
\(97\) −0.148787 + 1.03483i −0.0151070 + 0.105072i −0.995980 − 0.0895785i \(-0.971448\pi\)
0.980873 + 0.194650i \(0.0623571\pi\)
\(98\) −4.25258 + 5.56018i −0.429576 + 0.561663i
\(99\) 12.3477 − 5.63901i 1.24099 − 0.566742i
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.97.2 yes 80
7.6 odd 2 inner 322.2.k.b.97.7 yes 80
23.14 odd 22 inner 322.2.k.b.83.7 yes 80
161.83 even 22 inner 322.2.k.b.83.2 ✓ 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.k.b.83.2 ✓ 80 161.83 even 22 inner
322.2.k.b.83.7 yes 80 23.14 odd 22 inner
322.2.k.b.97.2 yes 80 1.1 even 1 trivial
322.2.k.b.97.7 yes 80 7.6 odd 2 inner