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Properties

Label 220.3.w.a.63.6
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 63.6
Character \(\chi\) \(=\) 220.63
Dual form 220.3.w.a.7.6

$q$-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+(-1.94429 + 0.468777i) q^{2} +(4.04592 + 2.06150i) q^{3} +(3.56050 - 1.82287i) q^{4} +(2.33189 - 4.42293i) q^{5} +(-8.83280 - 2.11151i) q^{6} +(3.03067 + 5.94803i) q^{7} +(-6.06810 + 5.21327i) q^{8} +(6.82960 + 9.40014i) q^{9} +(-2.46049 + 9.69257i) q^{10} +(2.74293 + 10.6525i) q^{11} +(18.1633 - 0.0352426i) q^{12} +(3.87209 - 24.4474i) q^{13} +(-8.68079 - 10.1440i) q^{14} +(18.5525 - 13.0876i) q^{15} +(9.35426 - 12.9807i) q^{16} +(2.85350 + 18.0163i) q^{17} +(-17.6853 - 15.0750i) q^{18} +(-10.7855 + 3.50442i) q^{19} +(0.240247 - 19.9986i) q^{20} +30.3129i q^{21} +(-10.3267 - 19.4257i) q^{22} +(20.8240 - 20.8240i) q^{23} +(-35.2982 + 8.58307i) q^{24} +(-14.1246 - 20.6276i) q^{25} +(3.93194 + 49.3479i) q^{26} +(1.86054 + 11.7470i) q^{27} +(21.6332 + 15.6534i) q^{28} +(-9.88305 + 30.4169i) q^{29} +(-29.9362 + 34.1430i) q^{30} +(8.80617 + 12.1207i) q^{31} +(-12.1023 + 29.6232i) q^{32} +(-10.8625 + 48.7538i) q^{33} +(-13.9936 - 33.6911i) q^{34} +(33.3749 + 0.465708i) q^{35} +(41.4520 + 21.0197i) q^{36} +(6.94136 + 13.6232i) q^{37} +(19.3273 - 11.8696i) q^{38} +(66.0644 - 90.9299i) q^{39} +(8.90776 + 38.9955i) q^{40} +(9.85319 - 3.20150i) q^{41} +(-14.2100 - 58.9370i) q^{42} +(-14.2190 - 14.2190i) q^{43} +(29.1844 + 32.9283i) q^{44} +(57.5020 - 8.28674i) q^{45} +(-30.7260 + 50.2496i) q^{46} +(-8.10427 + 15.9055i) q^{47} +(64.6062 - 33.2349i) q^{48} +(2.60743 - 3.58882i) q^{49} +(37.1319 + 33.4846i) q^{50} +(-25.5955 + 78.7748i) q^{51} +(-30.7780 - 94.1033i) 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} +(4.95673 - 63.7721i) q^{58} +(-24.7682 + 76.2288i) q^{59} +(42.1990 - 80.4172i) q^{60} +(-22.3830 + 30.8075i) q^{61} +(-22.8036 - 19.4379i) q^{62} +(-35.2140 + 69.1114i) q^{63} +(9.64368 - 63.2693i) q^{64} +(-99.0999 - 74.1347i) q^{65} +(-1.73484 - 99.8834i) q^{66} +(-15.3020 - 15.3020i) q^{67} +(43.0012 + 58.9453i) q^{68} +(127.181 - 41.3235i) q^{69} +(-65.1086 + 14.7399i) q^{70} +(12.2736 - 16.8932i) q^{71} +(-90.4481 - 21.4365i) q^{72} +(-36.2433 - 71.1314i) q^{73} +(-19.8822 - 23.2334i) q^{74} +(-14.6232 - 112.575i) q^{75} +(-32.0136 + 32.1380i) q^{76} +(-55.0486 + 48.5993i) q^{77} +(-85.8223 + 207.763i) q^{78} +(-58.2496 - 80.1738i) q^{79} +(-35.5995 - 71.6427i) q^{80} +(15.6260 - 48.0919i) q^{81} +(-17.6566 + 10.8436i) q^{82} +(-16.4491 - 103.856i) q^{83} +(55.2567 + 107.929i) q^{84} +(86.3386 + 29.3911i) q^{85} +(34.3112 + 20.9802i) q^{86} +(-102.690 + 102.690i) q^{87} +(-72.1788 - 50.3410i) q^{88} +69.0839i q^{89} +(-107.916 + 43.0674i) q^{90} +(157.149 - 51.0608i) q^{91} +(36.1842 - 112.103i) q^{92} +(10.6423 + 67.1931i) q^{93} +(8.30087 - 34.7240i) q^{94} +(-9.65079 + 55.8753i) q^{95} +(-110.033 + 94.9041i) q^{96} +(22.7249 - 143.479i) q^{97} +(-3.38724 + 8.20000i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.94429 + 0.468777i −0.972143 + 0.234389i
\(3\) 4.04592 + 2.06150i 1.34864 + 0.687166i 0.971065 − 0.238814i \(-0.0767586\pi\)
0.377573 + 0.925980i \(0.376759\pi\)
\(4\) 3.56050 − 1.82287i 0.890124 − 0.455718i
\(5\) 2.33189 − 4.42293i 0.466378 − 0.884586i
\(6\) −8.83280 − 2.11151i −1.47213 − 0.351918i
\(7\) 3.03067 + 5.94803i 0.432953 + 0.849718i 0.999667 + 0.0258034i \(0.00821440\pi\)
−0.566714 + 0.823914i \(0.691786\pi\)
\(8\) −6.06810 + 5.21327i −0.758513 + 0.651658i
\(9\) 6.82960 + 9.40014i 0.758845 + 1.04446i
\(10\) −2.46049 + 9.69257i −0.246049 + 0.969257i
\(11\) 2.74293 + 10.6525i 0.249357 + 0.968412i
\(12\) 18.1633 − 0.0352426i 1.51361 − 0.00293689i
\(13\) 3.87209 − 24.4474i 0.297853 − 1.88057i −0.153384 − 0.988167i \(-0.549017\pi\)
0.451237 − 0.892404i \(-0.350983\pi\)
\(14\) −8.68079 − 10.1440i −0.620056 − 0.724568i
\(15\) 18.5525 − 13.0876i 1.23683 − 0.872507i
\(16\) 9.35426 − 12.9807i 0.584641 − 0.811292i
\(17\) 2.85350 + 18.0163i 0.167853 + 1.05978i 0.917440 + 0.397874i \(0.130252\pi\)
−0.749587 + 0.661906i \(0.769748\pi\)
\(18\) −17.6853 − 15.0750i −0.982515 − 0.837500i
\(19\) −10.7855 + 3.50442i −0.567657 + 0.184443i −0.578764 − 0.815495i \(-0.696465\pi\)
0.0111065 + 0.999938i \(0.496465\pi\)
\(20\) 0.240247 − 19.9986i 0.0120123 − 0.999928i
\(21\) 30.3129i 1.44347i
\(22\) −10.3267 − 19.4257i −0.469395 − 0.882988i
\(23\) 20.8240 − 20.8240i 0.905390 − 0.905390i −0.0905057 − 0.995896i \(-0.528848\pi\)
0.995896 + 0.0905057i \(0.0288483\pi\)
\(24\) −35.2982 + 8.58307i −1.47076 + 0.357628i
\(25\) −14.1246 − 20.6276i −0.564983 − 0.825103i
\(26\) 3.93194 + 49.3479i 0.151228 + 1.89800i
\(27\) 1.86054 + 11.7470i 0.0689089 + 0.435073i
\(28\) 21.6332 + 15.6534i 0.772614 + 0.559050i
\(29\) −9.88305 + 30.4169i −0.340795 + 1.04886i 0.623002 + 0.782220i \(0.285913\pi\)
−0.963797 + 0.266638i \(0.914087\pi\)
\(30\) −29.9362 + 34.1430i −0.997872 + 1.13810i
\(31\) 8.80617 + 12.1207i 0.284070 + 0.390989i 0.927077 − 0.374872i \(-0.122313\pi\)
−0.643007 + 0.765861i \(0.722313\pi\)
\(32\) −12.1023 + 29.6232i −0.378197 + 0.925725i
\(33\) −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
\(36\) 41.4520 + 21.0197i 1.15145 + 0.583879i
\(37\) 6.94136 + 13.6232i 0.187604 + 0.368194i 0.965582 − 0.260097i \(-0.0837545\pi\)
−0.777978 + 0.628291i \(0.783755\pi\)
\(38\) 19.3273 − 11.8696i 0.508613 − 0.312357i
\(39\) 66.0644 − 90.9299i 1.69396 − 2.33154i
\(40\) 8.90776 + 38.9955i 0.222694 + 0.974888i
\(41\) 9.85319 − 3.20150i 0.240322 − 0.0780853i −0.186380 − 0.982478i \(-0.559675\pi\)
0.426702 + 0.904392i \(0.359675\pi\)
\(42\) −14.2100 − 58.9370i −0.338334 − 1.40326i
\(43\) −14.2190 − 14.2190i −0.330673 − 0.330673i 0.522169 − 0.852842i \(-0.325123\pi\)
−0.852842 + 0.522169i \(0.825123\pi\)
\(44\) 29.1844 + 32.9283i 0.663282 + 0.748370i
\(45\) 57.5020 − 8.28674i 1.27782 − 0.184150i
\(46\) −30.7260 + 50.2496i −0.667956 + 1.09238i
\(47\) −8.10427 + 15.9055i −0.172431 + 0.338415i −0.961008 − 0.276521i \(-0.910818\pi\)
0.788577 + 0.614937i \(0.210818\pi\)
\(48\) 64.6062 − 33.2349i 1.34596 − 0.692394i
\(49\) 2.60743 − 3.58882i 0.0532129 − 0.0732413i
\(50\) 37.1319 + 33.4846i 0.742639 + 0.669692i
\(51\) −25.5955 + 78.7748i −0.501872 + 1.54460i
\(52\) −30.7780 − 94.1033i −0.591885 − 1.80968i
\(53\) −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
\(58\) 4.95673 − 63.7721i 0.0854608 − 1.09952i
\(59\) −24.7682 + 76.2288i −0.419801 + 1.29201i 0.488085 + 0.872796i \(0.337696\pi\)
−0.907886 + 0.419218i \(0.862304\pi\)
\(60\) 42.1990 − 80.4172i 0.703316 − 1.34029i
\(61\) −22.3830 + 30.8075i −0.366934 + 0.505042i −0.952064 − 0.305898i \(-0.901043\pi\)
0.585130 + 0.810939i \(0.301043\pi\)
\(62\) −22.8036 − 19.4379i −0.367800 − 0.313514i
\(63\) −35.2140 + 69.1114i −0.558952 + 1.09701i
\(64\) 9.64368 − 63.2693i 0.150683 − 0.988582i
\(65\) −99.0999 − 74.1347i −1.52461 − 1.14053i
\(66\) −1.73484 − 99.8834i −0.0262854 − 1.51338i
\(67\) −15.3020 − 15.3020i −0.228388 − 0.228388i 0.583631 − 0.812019i \(-0.301631\pi\)
−0.812019 + 0.583631i \(0.801631\pi\)
\(68\) 43.0012 + 58.9453i 0.632371 + 0.866842i
\(69\) 127.181 − 41.3235i 1.84320 − 0.598891i
\(70\) −65.1086 + 14.7399i −0.930123 + 0.210570i
\(71\) 12.2736 − 16.8932i 0.172868 − 0.237932i −0.713788 − 0.700362i \(-0.753022\pi\)
0.886656 + 0.462429i \(0.153022\pi\)
\(72\) −90.4481 − 21.4365i −1.25622 − 0.297728i
\(73\) −36.2433 − 71.1314i −0.496483 − 0.974403i −0.994249 − 0.107092i \(-0.965846\pi\)
0.497766 − 0.867311i \(-0.334154\pi\)
\(74\) −19.8822 − 23.2334i −0.268679 − 0.313965i
\(75\) −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
\(78\) −85.8223 + 207.763i −1.10029 + 2.66363i
\(79\) −58.2496 − 80.1738i −0.737337 − 1.01486i −0.998767 − 0.0496354i \(-0.984194\pi\)
0.261430 − 0.965222i \(-0.415806\pi\)
\(80\) −35.5995 − 71.6427i −0.444993 − 0.895534i
\(81\) 15.6260 − 48.0919i 0.192914 − 0.593727i
\(82\) −17.6566 + 10.8436i −0.215325 + 0.132239i
\(83\) −16.4491 − 103.856i −0.198182 − 1.25127i −0.863361 − 0.504587i \(-0.831645\pi\)
0.665179 − 0.746684i \(-0.268355\pi\)
\(84\) 55.2567 + 107.929i 0.657817 + 1.28487i
\(85\) 86.3386 + 29.3911i 1.01575 + 0.345778i
\(86\) 34.3112 + 20.9802i 0.398968 + 0.243956i
\(87\) −102.690 + 102.690i −1.18035 + 1.18035i
\(88\) −72.1788 − 50.3410i −0.820214 − 0.572057i
\(89\) 69.0839i 0.776223i 0.921612 + 0.388112i \(0.126872\pi\)
−0.921612 + 0.388112i \(0.873128\pi\)
\(90\) −107.916 + 43.0674i −1.19906 + 0.478527i
\(91\) 157.149 − 51.0608i 1.72691 − 0.561107i
\(92\) 36.1842 − 112.103i 0.393306 − 1.21851i
\(93\) 10.6423 + 67.1931i 0.114434 + 0.722506i
\(94\) 8.30087 − 34.7240i 0.0883071 − 0.369404i
\(95\) −9.65079 + 55.8753i −0.101587 + 0.588161i
\(96\) −110.033 + 94.9041i −1.14618 + 0.988584i
\(97\) 22.7249 − 143.479i 0.234277 − 1.47917i −0.537494 − 0.843268i \(-0.680629\pi\)
0.771771 − 0.635901i \(-0.219371\pi\)
\(98\) −3.38724 + 8.20000i −0.0345636 + 0.0836735i
\(99\) −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.63.6 yes 544
4.3 odd 2 inner 220.3.w.a.63.13 yes 544
5.2 odd 4 inner 220.3.w.a.107.29 yes 544
11.7 odd 10 inner 220.3.w.a.183.45 yes 544
20.7 even 4 inner 220.3.w.a.107.45 yes 544
44.7 even 10 inner 220.3.w.a.183.29 yes 544
55.7 even 20 inner 220.3.w.a.7.13 yes 544
220.7 odd 20 inner 220.3.w.a.7.6 ✓ 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
220.3.w.a.7.6 ✓ 544 220.7 odd 20 inner
220.3.w.a.7.13 yes 544 55.7 even 20 inner
220.3.w.a.63.6 yes 544 1.1 even 1 trivial
220.3.w.a.63.13 yes 544 4.3 odd 2 inner
220.3.w.a.107.29 yes 544 5.2 odd 4 inner
220.3.w.a.107.45 yes 544 20.7 even 4 inner
220.3.w.a.183.29 yes 544 44.7 even 10 inner
220.3.w.a.183.45 yes 544 11.7 odd 10 inner