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Properties

Label 220.3.w.a.63.2
Level $220$
Weight $3$
Character 220.63
Analytic conductor $5.995$
Analytic rank $0$
Dimension $544$
Inner twists $8$

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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] = [220,3,Mod(7,220)] 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("220.7"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(220, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 5, 14])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 220.w (of order \(20\), degree \(8\), 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: \(5.99456581593\)
Analytic rank: \(0\)
Dimension: \(544\)
Relative dimension: \(68\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 63.2
Character \(\chi\) \(=\) 220.63
Dual form 220.3.w.a.7.2

$q$-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+(-1.99886 + 0.0675795i) q^{2} +(-1.94413 - 0.990585i) q^{3} +(3.99087 - 0.270163i) q^{4} +(-3.70748 - 3.35479i) q^{5} +(3.95299 + 1.84865i) q^{6} +(-1.57918 - 3.09932i) q^{7} +(-7.95892 + 0.809719i) q^{8} +(-2.49168 - 3.42950i) q^{9} +(7.63743 + 6.45520i) q^{10} +(6.04402 + 9.19075i) q^{11} +(-8.02639 - 3.42806i) q^{12} +(0.557233 - 3.51823i) q^{13} +(3.36601 + 6.08838i) q^{14} +(3.88462 + 10.1947i) q^{15} +(15.8540 - 2.15637i) q^{16} +(-1.87144 - 11.8158i) q^{17} +(5.21227 + 6.68669i) q^{18} +(-3.77795 + 1.22753i) q^{19} +(-15.7024 - 12.3869i) q^{20} +7.58980i q^{21} +(-12.7022 - 17.9625i) q^{22} +(-27.4174 + 27.4174i) q^{23} +(16.2753 + 6.30978i) q^{24} +(2.49076 + 24.8756i) q^{25} +(-0.876069 + 7.07010i) q^{26} +(4.51893 + 28.5314i) q^{27} +(-7.13963 - 11.9423i) q^{28} +(-6.37004 + 19.6050i) q^{29} +(-8.45375 - 20.1153i) q^{30} +(33.6775 + 46.3530i) q^{31} +(-31.5442 + 5.38169i) q^{32} +(-2.64616 - 23.8551i) q^{33} +(4.53924 + 23.4916i) q^{34} +(-4.54279 + 16.7885i) q^{35} +(-10.8705 - 13.0135i) q^{36} +(-10.3741 - 20.3602i) q^{37} +(7.46863 - 2.70897i) q^{38} +(-4.56844 + 6.28792i) q^{39} +(32.2239 + 23.6985i) q^{40} +(-66.3703 + 21.5650i) q^{41} +(-0.512915 - 15.1709i) q^{42} +(-3.10057 - 3.10057i) q^{43} +(26.6039 + 35.0462i) q^{44} +(-2.26742 + 21.0738i) q^{45} +(52.9506 - 56.6563i) q^{46} +(-17.8070 + 34.9482i) q^{47} +(-32.9584 - 11.5125i) q^{48} +(21.6895 - 29.8530i) q^{49} +(-6.65975 - 49.5545i) q^{50} +(-8.06622 + 24.8253i) q^{51} +(1.27334 - 14.1913i) q^{52} +(30.8259 + 4.88234i) q^{53} +(-10.9608 - 56.7248i) q^{54} +(8.42498 - 54.3509i) q^{55} +(15.0782 + 23.3885i) q^{56} +(8.56081 + 1.35590i) q^{57} +(11.4079 - 39.6180i) q^{58} +(26.3599 - 81.1273i) q^{59} +(18.2572 + 39.6363i) q^{60} +(8.24903 - 11.3538i) q^{61} +(-70.4490 - 90.3772i) q^{62} +(-6.69431 + 13.1383i) q^{63} +(62.6887 - 12.8890i) q^{64} +(-13.8689 + 11.1744i) q^{65} +(6.90141 + 47.5042i) q^{66} +(-39.5762 - 39.5762i) q^{67} +(-10.6609 - 46.6497i) q^{68} +(80.4622 - 26.1438i) q^{69} +(7.94584 - 33.8648i) q^{70} +(-38.5603 + 53.0737i) q^{71} +(22.6080 + 25.2775i) q^{72} +(54.8494 + 107.648i) q^{73} +(22.1122 + 39.9962i) q^{74} +(19.7990 - 50.8288i) q^{75} +(-14.7457 + 5.91958i) q^{76} +(18.9405 - 33.2462i) q^{77} +(8.70672 - 12.8774i) q^{78} +(-32.9819 - 45.3957i) q^{79} +(-66.0126 - 45.1922i) q^{80} +(7.68780 - 23.6606i) q^{81} +(131.207 - 47.5907i) q^{82} +(-14.8404 - 93.6988i) q^{83} +(2.05049 + 30.2899i) q^{84} +(-32.7012 + 50.0851i) q^{85} +(6.40713 + 5.98806i) q^{86} +(31.8046 - 31.8046i) q^{87} +(-55.5458 - 68.2544i) q^{88} +6.98261i q^{89} +(3.10809 - 42.2768i) q^{90} +(-11.7841 + 3.82889i) q^{91} +(-102.012 + 116.826i) q^{92} +(-19.5568 - 123.477i) q^{93} +(33.2318 - 71.0598i) q^{94} +(18.1248 + 8.12320i) q^{95} +(66.6571 + 20.7845i) q^{96} +(7.72602 - 48.7802i) q^{97} +(-41.3368 + 61.1378i) q^{98} +(16.4599 - 43.6283i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 544 q - 10 q^{2} - 12 q^{5} - 20 q^{6} - 10 q^{8} - 28 q^{12} - 20 q^{13} - 36 q^{16} - 20 q^{17} - 10 q^{18} - 40 q^{20} + 86 q^{22} - 12 q^{25} + 140 q^{26} - 10 q^{28} - 370 q^{30} - 100 q^{33} - 476 q^{36}+ \cdots + 148 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/220\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(111\) \(177\)
\(\chi(n)\) \(e\left(\frac{3}{10}\right)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99886 + 0.0675795i −0.999429 + 0.0337897i
\(3\) −1.94413 − 0.990585i −0.648044 − 0.330195i 0.0988849 − 0.995099i \(-0.468472\pi\)
−0.746929 + 0.664904i \(0.768472\pi\)
\(4\) 3.99087 − 0.270163i 0.997717 − 0.0675409i
\(5\) −3.70748 − 3.35479i −0.741495 − 0.670958i
\(6\) 3.95299 + 1.84865i 0.658831 + 0.308109i
\(7\) −1.57918 − 3.09932i −0.225598 − 0.442760i 0.750268 − 0.661134i \(-0.229925\pi\)
−0.975865 + 0.218374i \(0.929925\pi\)
\(8\) −7.95892 + 0.809719i −0.994865 + 0.101215i
\(9\) −2.49168 − 3.42950i −0.276853 − 0.381055i
\(10\) 7.63743 + 6.45520i 0.763743 + 0.645520i
\(11\) 6.04402 + 9.19075i 0.549456 + 0.835523i
\(12\) −8.02639 − 3.42806i −0.668866 − 0.285671i
\(13\) 0.557233 − 3.51823i 0.0428641 − 0.270633i −0.956943 − 0.290276i \(-0.906253\pi\)
0.999807 + 0.0196429i \(0.00625292\pi\)
\(14\) 3.36601 + 6.08838i 0.240430 + 0.434884i
\(15\) 3.88462 + 10.1947i 0.258974 + 0.679648i
\(16\) 15.8540 − 2.15637i 0.990876 − 0.134773i
\(17\) −1.87144 − 11.8158i −0.110085 − 0.695047i −0.979572 − 0.201092i \(-0.935551\pi\)
0.869488 − 0.493955i \(-0.164449\pi\)
\(18\) 5.21227 + 6.68669i 0.289571 + 0.371483i
\(19\) −3.77795 + 1.22753i −0.198840 + 0.0646069i −0.406744 − 0.913542i \(-0.633336\pi\)
0.207904 + 0.978149i \(0.433336\pi\)
\(20\) −15.7024 − 12.3869i −0.785119 − 0.619345i
\(21\) 7.58980i 0.361419i
\(22\) −12.7022 − 17.9625i −0.577375 − 0.816479i
\(23\) −27.4174 + 27.4174i −1.19206 + 1.19206i −0.215571 + 0.976488i \(0.569161\pi\)
−0.976488 + 0.215571i \(0.930839\pi\)
\(24\) 16.2753 + 6.30978i 0.678137 + 0.262908i
\(25\) 2.49076 + 24.8756i 0.0996302 + 0.995025i
\(26\) −0.876069 + 7.07010i −0.0336950 + 0.271927i
\(27\) 4.51893 + 28.5314i 0.167368 + 1.05672i
\(28\) −7.13963 − 11.9423i −0.254987 − 0.426512i
\(29\) −6.37004 + 19.6050i −0.219656 + 0.676033i 0.779134 + 0.626858i \(0.215659\pi\)
−0.998790 + 0.0491753i \(0.984341\pi\)
\(30\) −8.45375 − 20.1153i −0.281792 − 0.670510i
\(31\) 33.6775 + 46.3530i 1.08637 + 1.49526i 0.852310 + 0.523037i \(0.175201\pi\)
0.234059 + 0.972222i \(0.424799\pi\)
\(32\) −31.5442 + 5.38169i −0.985757 + 0.168178i
\(33\) −2.64616 − 23.8551i −0.0801865 − 0.722883i
\(34\) 4.53924 + 23.4916i 0.133507 + 0.690930i
\(35\) −4.54279 + 16.7885i −0.129794 + 0.479671i
\(36\) −10.8705 − 13.0135i −0.301958 − 0.361486i
\(37\) −10.3741 − 20.3602i −0.280380 − 0.550277i 0.707272 − 0.706942i \(-0.249926\pi\)
−0.987652 + 0.156665i \(0.949926\pi\)
\(38\) 7.46863 − 2.70897i 0.196543 − 0.0712887i
\(39\) −4.56844 + 6.28792i −0.117139 + 0.161229i
\(40\) 32.2239 + 23.6985i 0.805598 + 0.592462i
\(41\) −66.3703 + 21.5650i −1.61879 + 0.525976i −0.971655 − 0.236402i \(-0.924032\pi\)
−0.647132 + 0.762378i \(0.724032\pi\)
\(42\) −0.512915 − 15.1709i −0.0122123 − 0.361213i
\(43\) −3.10057 − 3.10057i −0.0721063 − 0.0721063i 0.670134 − 0.742240i \(-0.266237\pi\)
−0.742240 + 0.670134i \(0.766237\pi\)
\(44\) 26.6039 + 35.0462i 0.604633 + 0.796504i
\(45\) −2.26742 + 21.0738i −0.0503871 + 0.468308i
\(46\) 52.9506 − 56.6563i 1.15110 − 1.23166i
\(47\) −17.8070 + 34.9482i −0.378872 + 0.743578i −0.999168 − 0.0407899i \(-0.987013\pi\)
0.620296 + 0.784368i \(0.287013\pi\)
\(48\) −32.9584 − 11.5125i −0.686633 − 0.239843i
\(49\) 21.6895 − 29.8530i 0.442643 − 0.609246i
\(50\) −6.65975 − 49.5545i −0.133195 − 0.991090i
\(51\) −8.06622 + 24.8253i −0.158161 + 0.486770i
\(52\) 1.27334 − 14.1913i 0.0244874 − 0.272910i
\(53\) 30.8259 + 4.88234i 0.581620 + 0.0921196i 0.440310 − 0.897846i \(-0.354868\pi\)
0.141310 + 0.989965i \(0.454868\pi\)
\(54\) −10.9608 − 56.7248i −0.202978 − 1.05046i
\(55\) 8.42498 − 54.3509i 0.153182 − 0.988198i
\(56\) 15.0782 + 23.3885i 0.269253 + 0.417653i
\(57\) 8.56081 + 1.35590i 0.150190 + 0.0237877i
\(58\) 11.4079 − 39.6180i 0.196688 − 0.683069i
\(59\) 26.3599 − 81.1273i 0.446777 − 1.37504i −0.433745 − 0.901036i \(-0.642808\pi\)
0.880523 − 0.474004i \(-0.157192\pi\)
\(60\) 18.2572 + 39.6363i 0.304287 + 0.660605i
\(61\) 8.24903 − 11.3538i 0.135230 − 0.186128i −0.736031 − 0.676947i \(-0.763302\pi\)
0.871261 + 0.490819i \(0.163302\pi\)
\(62\) −70.4490 − 90.3772i −1.13627 − 1.45770i
\(63\) −6.69431 + 13.1383i −0.106259 + 0.208545i
\(64\) 62.6887 − 12.8890i 0.979511 − 0.201390i
\(65\) −13.8689 + 11.1744i −0.213367 + 0.171913i
\(66\) 6.90141 + 47.5042i 0.104567 + 0.719761i
\(67\) −39.5762 − 39.5762i −0.590690 − 0.590690i 0.347128 − 0.937818i \(-0.387157\pi\)
−0.937818 + 0.347128i \(0.887157\pi\)
\(68\) −10.6609 − 46.6497i −0.156777 − 0.686024i
\(69\) 80.4622 − 26.1438i 1.16612 − 0.378895i
\(70\) 7.94584 − 33.8648i 0.113512 − 0.483783i
\(71\) −38.5603 + 53.0737i −0.543102 + 0.747516i −0.989056 − 0.147540i \(-0.952865\pi\)
0.445954 + 0.895056i \(0.352865\pi\)
\(72\) 22.6080 + 25.2775i 0.314000 + 0.351077i
\(73\) 54.8494 + 107.648i 0.751362 + 1.47463i 0.875940 + 0.482421i \(0.160242\pi\)
−0.124578 + 0.992210i \(0.539758\pi\)
\(74\) 22.1122 + 39.9962i 0.298814 + 0.540489i
\(75\) 19.7990 − 50.8288i 0.263987 − 0.677717i
\(76\) −14.7457 + 5.91958i −0.194022 + 0.0778892i
\(77\) 18.9405 − 33.2462i 0.245980 − 0.431769i
\(78\) 8.70672 − 12.8774i 0.111625 − 0.165095i
\(79\) −32.9819 − 45.3957i −0.417492 − 0.574629i 0.547533 − 0.836784i \(-0.315567\pi\)
−0.965026 + 0.262155i \(0.915567\pi\)
\(80\) −66.0126 − 45.1922i −0.825157 − 0.564903i
\(81\) 7.68780 − 23.6606i 0.0949111 − 0.292106i
\(82\) 131.207 − 47.5907i 1.60009 − 0.580374i
\(83\) −14.8404 − 93.6988i −0.178800 − 1.12890i −0.899909 − 0.436079i \(-0.856367\pi\)
0.721108 − 0.692822i \(-0.243633\pi\)
\(84\) 2.05049 + 30.2899i 0.0244106 + 0.360594i
\(85\) −32.7012 + 50.0851i −0.384720 + 0.589236i
\(86\) 6.40713 + 5.98806i 0.0745015 + 0.0696286i
\(87\) 31.8046 − 31.8046i 0.365570 − 0.365570i
\(88\) −55.5458 − 68.2544i −0.631202 − 0.775619i
\(89\) 6.98261i 0.0784563i 0.999230 + 0.0392281i \(0.0124899\pi\)
−0.999230 + 0.0392281i \(0.987510\pi\)
\(90\) 3.10809 − 42.2768i 0.0345344 − 0.469743i
\(91\) −11.7841 + 3.82889i −0.129496 + 0.0420757i
\(92\) −102.012 + 116.826i −1.10882 + 1.26985i
\(93\) −19.5568 − 123.477i −0.210288 − 1.32771i
\(94\) 33.2318 − 71.0598i 0.353530 − 0.755955i
\(95\) 18.1248 + 8.12320i 0.190787 + 0.0855073i
\(96\) 66.6571 + 20.7845i 0.694345 + 0.216505i
\(97\) 7.72602 − 48.7802i 0.0796497 − 0.502889i −0.915321 − 0.402725i \(-0.868063\pi\)
0.994971 − 0.100164i \(-0.0319368\pi\)
\(98\) −41.3368 + 61.1378i −0.421804 + 0.623855i
\(99\) 16.4599 − 43.6283i 0.166262 − 0.440690i
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 220.3.w.a.63.2 yes 544
4.3 odd 2 inner 220.3.w.a.63.9 yes 544
5.2 odd 4 inner 220.3.w.a.107.33 yes 544
11.7 odd 10 inner 220.3.w.a.183.43 yes 544
20.7 even 4 inner 220.3.w.a.107.43 yes 544
44.7 even 10 inner 220.3.w.a.183.33 yes 544
55.7 even 20 inner 220.3.w.a.7.9 yes 544
220.7 odd 20 inner 220.3.w.a.7.2 ✓ 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
220.3.w.a.7.2 ✓ 544 220.7 odd 20 inner
220.3.w.a.7.9 yes 544 55.7 even 20 inner
220.3.w.a.63.2 yes 544 1.1 even 1 trivial
220.3.w.a.63.9 yes 544 4.3 odd 2 inner
220.3.w.a.107.33 yes 544 5.2 odd 4 inner
220.3.w.a.107.43 yes 544 20.7 even 4 inner
220.3.w.a.183.33 yes 544 44.7 even 10 inner
220.3.w.a.183.43 yes 544 11.7 odd 10 inner