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Properties

Label 220.3.w.a.63.13
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 63.13
Character \(\chi\) \(=\) 220.63
Dual form 220.3.w.a.7.13

$q$-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+(-1.70427 - 1.04665i) q^{2} +(-4.04592 - 2.06150i) q^{3} +(1.80904 + 3.56754i) q^{4} +(2.33189 - 4.42293i) q^{5} +(4.73765 + 7.74800i) q^{6} +(-3.03067 - 5.94803i) q^{7} +(0.650882 - 7.97348i) q^{8} +(6.82960 + 9.40014i) q^{9} +(-8.60342 + 5.09717i) q^{10} +(-2.74293 - 10.6525i) q^{11} +(0.0352426 - 18.1633i) q^{12} +(3.87209 - 24.4474i) q^{13} +(-1.06044 + 13.3091i) q^{14} +(-18.5525 + 13.0876i) q^{15} +(-9.45473 + 12.9077i) q^{16} +(2.85350 + 18.0163i) q^{17} +(-1.80079 - 23.1685i) q^{18} +(10.7855 - 3.50442i) q^{19} +(19.9975 + 0.317852i) q^{20} +30.3129i q^{21} +(-6.47480 + 21.0256i) q^{22} +(-20.8240 + 20.8240i) q^{23} +(-19.0707 + 30.9182i) q^{24} +(-14.1246 - 20.6276i) q^{25} +(-32.1870 + 37.6122i) q^{26} +(-1.86054 - 11.7470i) q^{27} +(15.7372 - 21.5723i) q^{28} +(-9.88305 + 30.4169i) q^{29} +(45.3165 - 2.88678i) q^{30} +(-8.80617 - 12.1207i) q^{31} +(29.6232 - 12.1023i) q^{32} +(-10.8625 + 48.7538i) q^{33} +(13.9936 - 33.6911i) q^{34} +(-33.3749 - 0.465708i) q^{35} +(-21.1804 + 41.3702i) q^{36} +(6.94136 + 13.6232i) q^{37} +(-22.0492 - 5.31618i) q^{38} +(-66.0644 + 90.9299i) q^{39} +(-33.7483 - 21.4721i) q^{40} +(9.85319 - 3.20150i) q^{41} +(31.7271 - 51.6613i) q^{42} +(14.2190 + 14.2190i) q^{43} +(33.0413 - 29.0564i) q^{44} +(57.5020 - 8.28674i) q^{45} +(57.2850 - 13.6942i) q^{46} +(8.10427 - 15.9055i) q^{47} +(64.8622 - 32.7325i) q^{48} +(2.60743 - 3.58882i) q^{49} +(2.48217 + 49.9384i) q^{50} +(25.5955 - 78.7748i) q^{51} +(94.2220 - 30.4126i) q^{52} +(-69.7703 - 11.0505i) q^{53} +(-9.12414 + 21.9673i) q^{54} +(-53.5116 - 12.7088i) q^{55} +(-49.3991 + 20.2935i) q^{56} +(-50.8615 - 8.05567i) q^{57} +(48.6792 - 41.4944i) q^{58} +(24.7682 - 76.2288i) q^{59} +(-80.2529 - 42.5107i) q^{60} +(-22.3830 + 30.8075i) q^{61} +(2.32196 + 29.8738i) q^{62} +(35.2140 - 69.1114i) q^{63} +(-63.1527 - 10.3796i) q^{64} +(-99.0999 - 74.1347i) q^{65} +(69.5408 - 71.7201i) q^{66} +(15.3020 + 15.3020i) q^{67} +(-59.1117 + 42.7722i) q^{68} +(127.181 - 41.3235i) q^{69} +(56.3922 + 35.7255i) q^{70} +(-12.2736 + 16.8932i) q^{71} +(79.3971 - 48.3373i) q^{72} +(-36.2433 - 71.1314i) q^{73} +(2.42880 - 30.4827i) q^{74} +(14.6232 + 112.575i) q^{75} +(32.0136 + 32.1380i) q^{76} +(-55.0486 + 48.5993i) q^{77} +(207.763 - 85.8223i) q^{78} +(58.2496 + 80.1738i) q^{79} +(35.0423 + 71.9169i) q^{80} +(15.6260 - 48.0919i) q^{81} +(-20.1433 - 4.85665i) q^{82} +(16.4491 + 103.856i) q^{83} +(-108.143 + 54.8374i) q^{84} +(86.3386 + 29.3911i) q^{85} +(-9.35060 - 39.1152i) q^{86} +(102.690 - 102.690i) q^{87} +(-86.7230 + 14.9371i) q^{88} +69.0839i q^{89} +(-106.672 - 46.0617i) q^{90} +(-157.149 + 51.0608i) q^{91} +(-111.962 - 36.6190i) q^{92} +(10.6423 + 67.1931i) q^{93} +(-30.4594 + 18.6249i) q^{94} +(9.65079 - 55.8753i) q^{95} +(-144.802 - 12.1032i) q^{96} +(22.7249 - 143.479i) q^{97} +(-8.20000 + 3.38724i) q^{98} +(81.4022 - 98.5364i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.70427 − 1.04665i −0.852133 − 0.523326i
\(3\) −4.04592 − 2.06150i −1.34864 − 0.687166i −0.377573 − 0.925980i \(-0.623241\pi\)
−0.971065 + 0.238814i \(0.923241\pi\)
\(4\) 1.80904 + 3.56754i 0.452261 + 0.891886i
\(5\) 2.33189 − 4.42293i 0.466378 − 0.884586i
\(6\) 4.73765 + 7.74800i 0.789608 + 1.29133i
\(7\) −3.03067 − 5.94803i −0.432953 − 0.849718i −0.999667 − 0.0258034i \(-0.991786\pi\)
0.566714 − 0.823914i \(-0.308214\pi\)
\(8\) 0.650882 − 7.97348i 0.0813603 − 0.996685i
\(9\) 6.82960 + 9.40014i 0.758845 + 1.04446i
\(10\) −8.60342 + 5.09717i −0.860342 + 0.509717i
\(11\) −2.74293 − 10.6525i −0.249357 − 0.968412i
\(12\) 0.0352426 − 18.1633i 0.00293689 − 1.51361i
\(13\) 3.87209 − 24.4474i 0.297853 − 1.88057i −0.153384 − 0.988167i \(-0.549017\pi\)
0.451237 − 0.892404i \(-0.350983\pi\)
\(14\) −1.06044 + 13.3091i −0.0757457 + 0.950648i
\(15\) −18.5525 + 13.0876i −1.23683 + 0.872507i
\(16\) −9.45473 + 12.9077i −0.590920 + 0.806730i
\(17\) 2.85350 + 18.0163i 0.167853 + 1.05978i 0.917440 + 0.397874i \(0.130252\pi\)
−0.749587 + 0.661906i \(0.769748\pi\)
\(18\) −1.80079 − 23.1685i −0.100044 − 1.28714i
\(19\) 10.7855 − 3.50442i 0.567657 − 0.184443i −0.0111065 − 0.999938i \(-0.503535\pi\)
0.578764 + 0.815495i \(0.303535\pi\)
\(20\) 19.9975 + 0.317852i 0.999874 + 0.0158926i
\(21\) 30.3129i 1.44347i
\(22\) −6.47480 + 21.0256i −0.294309 + 0.955710i
\(23\) −20.8240 + 20.8240i −0.905390 + 0.905390i −0.995896 − 0.0905057i \(-0.971152\pi\)
0.0905057 + 0.995896i \(0.471152\pi\)
\(24\) −19.0707 + 30.9182i −0.794613 + 1.28826i
\(25\) −14.1246 − 20.6276i −0.564983 − 0.825103i
\(26\) −32.1870 + 37.6122i −1.23796 + 1.44662i
\(27\) −1.86054 − 11.7470i −0.0689089 − 0.435073i
\(28\) 15.7372 − 21.5723i 0.562044 − 0.770439i
\(29\) −9.88305 + 30.4169i −0.340795 + 1.04886i 0.623002 + 0.782220i \(0.285913\pi\)
−0.963797 + 0.266638i \(0.914087\pi\)
\(30\) 45.3165 − 2.88678i 1.51055 − 0.0962262i
\(31\) −8.80617 − 12.1207i −0.284070 − 0.390989i 0.643007 − 0.765861i \(-0.277687\pi\)
−0.927077 + 0.374872i \(0.877687\pi\)
\(32\) 29.6232 − 12.1023i 0.925725 − 0.378197i
\(33\) −10.8625 + 48.7538i −0.329167 + 1.47739i
\(34\) 13.9936 − 33.6911i 0.411577 − 0.990915i
\(35\) −33.3749 − 0.465708i −0.953568 − 0.0133059i
\(36\) −21.1804 + 41.3702i −0.588343 + 1.14917i
\(37\) 6.94136 + 13.6232i 0.187604 + 0.368194i 0.965582 − 0.260097i \(-0.0837545\pi\)
−0.777978 + 0.628291i \(0.783755\pi\)
\(38\) −22.0492 − 5.31618i −0.580243 − 0.139900i
\(39\) −66.0644 + 90.9299i −1.69396 + 2.33154i
\(40\) −33.7483 − 21.4721i −0.843708 − 0.536802i
\(41\) 9.85319 − 3.20150i 0.240322 − 0.0780853i −0.186380 − 0.982478i \(-0.559675\pi\)
0.426702 + 0.904392i \(0.359675\pi\)
\(42\) 31.7271 − 51.6613i 0.755406 − 1.23003i
\(43\) 14.2190 + 14.2190i 0.330673 + 0.330673i 0.852842 − 0.522169i \(-0.174877\pi\)
−0.522169 + 0.852842i \(0.674877\pi\)
\(44\) 33.0413 − 29.0564i 0.750938 − 0.660373i
\(45\) 57.5020 − 8.28674i 1.27782 − 0.184150i
\(46\) 57.2850 − 13.6942i 1.24533 − 0.297699i
\(47\) 8.10427 − 15.9055i 0.172431 − 0.338415i −0.788577 − 0.614937i \(-0.789182\pi\)
0.961008 + 0.276521i \(0.0891816\pi\)
\(48\) 64.8622 − 32.7325i 1.35130 − 0.681927i
\(49\) 2.60743 − 3.58882i 0.0532129 − 0.0732413i
\(50\) 2.48217 + 49.9384i 0.0496434 + 0.998767i
\(51\) 25.5955 − 78.7748i 0.501872 − 1.54460i
\(52\) 94.2220 − 30.4126i 1.81196 − 0.584857i
\(53\) −69.7703 − 11.0505i −1.31642 − 0.208501i −0.541576 − 0.840652i \(-0.682172\pi\)
−0.774845 + 0.632151i \(0.782172\pi\)
\(54\) −9.12414 + 21.9673i −0.168966 + 0.406802i
\(55\) −53.5116 − 12.7088i −0.972938 − 0.231068i
\(56\) −49.3991 + 20.2935i −0.882126 + 0.362384i
\(57\) −50.8615 − 8.05567i −0.892307 − 0.141328i
\(58\) 48.6792 − 41.4944i 0.839297 − 0.715420i
\(59\) 24.7682 − 76.2288i 0.419801 − 1.29201i −0.488085 − 0.872796i \(-0.662304\pi\)
0.907886 − 0.419218i \(-0.137696\pi\)
\(60\) −80.2529 − 42.5107i −1.33755 − 0.708512i
\(61\) −22.3830 + 30.8075i −0.366934 + 0.505042i −0.952064 − 0.305898i \(-0.901043\pi\)
0.585130 + 0.810939i \(0.301043\pi\)
\(62\) 2.32196 + 29.8738i 0.0374510 + 0.481836i
\(63\) 35.2140 − 69.1114i 0.558952 − 1.09701i
\(64\) −63.1527 − 10.3796i −0.986761 − 0.162181i
\(65\) −99.0999 − 74.1347i −1.52461 − 1.14053i
\(66\) 69.5408 − 71.7201i 1.05365 − 1.08667i
\(67\) 15.3020 + 15.3020i 0.228388 + 0.228388i 0.812019 − 0.583631i \(-0.198369\pi\)
−0.583631 + 0.812019i \(0.698369\pi\)
\(68\) −59.1117 + 42.7722i −0.869289 + 0.629002i
\(69\) 127.181 − 41.3235i 1.84320 − 0.598891i
\(70\) 56.3922 + 35.7255i 0.805603 + 0.510365i
\(71\) −12.2736 + 16.8932i −0.172868 + 0.237932i −0.886656 − 0.462429i \(-0.846978\pi\)
0.713788 + 0.700362i \(0.246978\pi\)
\(72\) 79.3971 − 48.3373i 1.10274 − 0.671351i
\(73\) −36.2433 − 71.1314i −0.496483 − 0.974403i −0.994249 − 0.107092i \(-0.965846\pi\)
0.497766 − 0.867311i \(-0.334154\pi\)
\(74\) 2.42880 − 30.4827i 0.0328216 − 0.411929i
\(75\) 14.6232 + 112.575i 0.194976 + 1.50100i
\(76\) 32.0136 + 32.1380i 0.421231 + 0.422869i
\(77\) −55.0486 + 48.5993i −0.714917 + 0.631160i
\(78\) 207.763 − 85.8223i 2.66363 − 1.10029i
\(79\) 58.2496 + 80.1738i 0.737337 + 1.01486i 0.998767 + 0.0496354i \(0.0158059\pi\)
−0.261430 + 0.965222i \(0.584194\pi\)
\(80\) 35.0423 + 71.9169i 0.438029 + 0.898961i
\(81\) 15.6260 − 48.0919i 0.192914 − 0.593727i
\(82\) −20.1433 − 4.85665i −0.245650 − 0.0592275i
\(83\) 16.4491 + 103.856i 0.198182 + 1.25127i 0.863361 + 0.504587i \(0.168355\pi\)
−0.665179 + 0.746684i \(0.731645\pi\)
\(84\) −108.143 + 54.8374i −1.28741 + 0.652826i
\(85\) 86.3386 + 29.3911i 1.01575 + 0.345778i
\(86\) −9.35060 − 39.1152i −0.108728 − 0.454828i
\(87\) 102.690 − 102.690i 1.18035 − 1.18035i
\(88\) −86.7230 + 14.9371i −0.985489 + 0.169740i
\(89\) 69.0839i 0.776223i 0.921612 + 0.388112i \(0.126872\pi\)
−0.921612 + 0.388112i \(0.873128\pi\)
\(90\) −106.672 − 46.0617i −1.18524 − 0.511797i
\(91\) −157.149 + 51.0608i −1.72691 + 0.561107i
\(92\) −111.962 − 36.6190i −1.21698 − 0.398032i
\(93\) 10.6423 + 67.1931i 0.114434 + 0.722506i
\(94\) −30.4594 + 18.6249i −0.324036 + 0.198137i
\(95\) 9.65079 − 55.8753i 0.101587 − 0.588161i
\(96\) −144.802 − 12.1032i −1.50835 − 0.126075i
\(97\) 22.7249 − 143.479i 0.234277 − 1.47917i −0.537494 − 0.843268i \(-0.680629\pi\)
0.771771 − 0.635901i \(-0.219371\pi\)
\(98\) −8.20000 + 3.38724i −0.0836735 + 0.0345636i
\(99\) 81.4022 − 98.5364i 0.822244 − 0.995317i
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.63.13 yes 544
4.3 odd 2 inner 220.3.w.a.63.6 yes 544
5.2 odd 4 inner 220.3.w.a.107.45 yes 544
11.7 odd 10 inner 220.3.w.a.183.29 yes 544
20.7 even 4 inner 220.3.w.a.107.29 yes 544
44.7 even 10 inner 220.3.w.a.183.45 yes 544
55.7 even 20 inner 220.3.w.a.7.6 ✓ 544
220.7 odd 20 inner 220.3.w.a.7.13 yes 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
220.3.w.a.7.6 ✓ 544 55.7 even 20 inner
220.3.w.a.7.13 yes 544 220.7 odd 20 inner
220.3.w.a.63.6 yes 544 4.3 odd 2 inner
220.3.w.a.63.13 yes 544 1.1 even 1 trivial
220.3.w.a.107.29 yes 544 20.7 even 4 inner
220.3.w.a.107.45 yes 544 5.2 odd 4 inner
220.3.w.a.183.29 yes 544 11.7 odd 10 inner
220.3.w.a.183.45 yes 544 44.7 even 10 inner