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Properties

Label 220.3.w.a.107.29
Level $220$
Weight $3$
Character 220.107
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 107.29
Character \(\chi\) \(=\) 220.107
Dual form 220.3.w.a.183.29

$q$-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+(-0.468777 - 1.94429i) q^{2} +(2.06150 - 4.04592i) q^{3} +(-3.56050 + 1.82287i) q^{4} +(-3.48586 + 3.58452i) q^{5} +(-8.83280 - 2.11151i) q^{6} +(-5.94803 + 3.03067i) q^{7} +(5.21327 + 6.06810i) 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} +(-24.4474 - 3.87209i) q^{13} +(8.68079 + 10.1440i) q^{14} +(7.31657 + 21.4930i) q^{15} +(9.35426 - 12.9807i) q^{16} +(-18.0163 + 2.85350i) q^{17} +(-15.0750 + 17.6853i) q^{18} +(10.7855 - 3.50442i) q^{19} +(5.87726 - 19.1169i) q^{20} +30.3129i q^{21} +(19.4257 - 10.3267i) q^{22} +(-20.8240 - 20.8240i) q^{23} +(35.2982 - 8.58307i) q^{24} +(-0.697556 - 24.9903i) q^{25} +(3.93194 + 49.3479i) q^{26} +(-11.7470 + 1.86054i) q^{27} +(15.6534 - 21.6332i) q^{28} +(9.88305 - 30.4169i) q^{29} +(38.3586 - 24.3009i) q^{30} +(8.80617 + 12.1207i) q^{31} +(-29.6232 - 12.1023i) q^{32} +(48.7538 + 10.8625i) q^{33} +(13.9936 + 33.6911i) q^{34} +(9.87049 - 31.8853i) q^{35} +(41.4520 + 21.0197i) q^{36} +(-13.6232 + 6.94136i) q^{37} +(-11.8696 - 19.3273i) q^{38} +(-66.0644 + 90.9299i) q^{39} +(-39.9239 - 2.46549i) q^{40} +(9.85319 - 3.20150i) q^{41} +(58.9370 - 14.2100i) 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} +(-15.9055 - 8.10427i) q^{47} +(-33.2349 - 64.6062i) q^{48} +(-2.60743 + 3.58882i) q^{49} +(-48.2612 + 13.0711i) q^{50} +(-25.5955 + 78.7748i) q^{51} +(94.1033 - 30.7780i) q^{52} +(-11.0505 + 69.7703i) q^{53} +(9.12414 + 21.9673i) q^{54} +(-47.7456 - 27.3012i) q^{55} +(-49.3991 - 20.2935i) q^{56} +(8.05567 - 50.8615i) q^{57} +(-63.7721 - 4.95673i) q^{58} +(24.7682 - 76.2288i) q^{59} +(-65.2296 - 63.1885i) q^{60} +(-22.3830 + 30.8075i) q^{61} +(19.4379 - 22.8036i) q^{62} +(69.1114 + 35.2140i) 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} +(58.9453 - 43.0012i) q^{68} +(-127.181 + 41.3235i) q^{69} +(-66.6212 - 4.24395i) q^{70} +(12.2736 - 16.8932i) q^{71} +(21.4365 - 90.4481i) q^{72} +(-71.1314 + 36.2433i) q^{73} +(19.8822 + 23.2334i) q^{74} +(-102.547 - 48.6951i) q^{75} +(-32.0136 + 32.1380i) q^{76} +(-48.5993 - 55.0486i) q^{77} +(207.763 + 85.8223i) q^{78} +(58.2496 + 80.1738i) q^{79} +(13.9218 + 78.7793i) q^{80} +(15.6260 - 48.0919i) q^{81} +(-10.8436 - 17.6566i) q^{82} +(-103.856 + 16.4491i) q^{83} +(-55.2567 - 107.929i) q^{84} +(52.5738 - 74.5265i) q^{85} +(34.3112 + 20.9802i) q^{86} +(-102.690 - 102.690i) q^{87} +(-50.3410 + 72.1788i) q^{88} -69.0839i q^{89} +(-10.8438 - 115.685i) q^{90} +(157.149 - 51.0608i) q^{91} +(112.103 + 36.1842i) q^{92} +(67.1931 - 10.6423i) q^{93} +(-8.30087 + 34.7240i) q^{94} +(-25.0350 + 50.8767i) q^{95} +(-110.033 + 94.9041i) q^{96} +(143.479 + 22.7249i) 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 \(\left(\mathbb{Z}/220\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.468777 − 1.94429i −0.234389 − 0.972143i
\(3\) 2.06150 − 4.04592i 0.687166 − 1.34864i −0.238814 − 0.971065i \(-0.576759\pi\)
0.925980 − 0.377573i \(-0.123241\pi\)
\(4\) −3.56050 + 1.82287i −0.890124 + 0.455718i
\(5\) −3.48586 + 3.58452i −0.697172 + 0.716904i
\(6\) −8.83280 − 2.11151i −1.47213 − 0.351918i
\(7\) −5.94803 + 3.03067i −0.849718 + 0.432953i −0.823914 − 0.566714i \(-0.808214\pi\)
−0.0258034 + 0.999667i \(0.508214\pi\)
\(8\) 5.21327 + 6.06810i 0.651658 + 0.758513i
\(9\) −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
\(13\) −24.4474 − 3.87209i −1.88057 − 0.297853i −0.892404 − 0.451237i \(-0.850983\pi\)
−0.988167 + 0.153384i \(0.950983\pi\)
\(14\) 8.68079 + 10.1440i 0.620056 + 0.724568i
\(15\) 7.31657 + 21.4930i 0.487772 + 1.43287i
\(16\) 9.35426 − 12.9807i 0.584641 − 0.811292i
\(17\) −18.0163 + 2.85350i −1.05978 + 0.167853i −0.661906 − 0.749587i \(-0.730252\pi\)
−0.397874 + 0.917440i \(0.630252\pi\)
\(18\) −15.0750 + 17.6853i −0.837500 + 0.982515i
\(19\) 10.7855 − 3.50442i 0.567657 − 0.184443i −0.0111065 − 0.999938i \(-0.503535\pi\)
0.578764 + 0.815495i \(0.303535\pi\)
\(20\) 5.87726 − 19.1169i 0.293863 − 0.955847i
\(21\) 30.3129i 1.44347i
\(22\) 19.4257 − 10.3267i 0.882988 − 0.469395i
\(23\) −20.8240 − 20.8240i −0.905390 − 0.905390i 0.0905057 − 0.995896i \(-0.471152\pi\)
−0.995896 + 0.0905057i \(0.971152\pi\)
\(24\) 35.2982 − 8.58307i 1.47076 − 0.357628i
\(25\) −0.697556 − 24.9903i −0.0279022 − 0.999611i
\(26\) 3.93194 + 49.3479i 0.151228 + 1.89800i
\(27\) −11.7470 + 1.86054i −0.435073 + 0.0689089i
\(28\) 15.6534 − 21.6332i 0.559050 − 0.772614i
\(29\) 9.88305 − 30.4169i 0.340795 − 1.04886i −0.623002 − 0.782220i \(-0.714087\pi\)
0.963797 − 0.266638i \(-0.0859129\pi\)
\(30\) 38.3586 − 24.3009i 1.27862 − 0.810031i
\(31\) 8.80617 + 12.1207i 0.284070 + 0.390989i 0.927077 − 0.374872i \(-0.122313\pi\)
−0.643007 + 0.765861i \(0.722313\pi\)
\(32\) −29.6232 − 12.1023i −0.925725 − 0.378197i
\(33\) 48.7538 + 10.8625i 1.47739 + 0.329167i
\(34\) 13.9936 + 33.6911i 0.411577 + 0.990915i
\(35\) 9.87049 − 31.8853i 0.282014 − 0.911009i
\(36\) 41.4520 + 21.0197i 1.15145 + 0.583879i
\(37\) −13.6232 + 6.94136i −0.368194 + 0.187604i −0.628291 − 0.777978i \(-0.716245\pi\)
0.260097 + 0.965582i \(0.416245\pi\)
\(38\) −11.8696 − 19.3273i −0.312357 − 0.508613i
\(39\) −66.0644 + 90.9299i −1.69396 + 2.33154i
\(40\) −39.9239 − 2.46549i −0.998099 − 0.0616373i
\(41\) 9.85319 − 3.20150i 0.240322 − 0.0780853i −0.186380 − 0.982478i \(-0.559675\pi\)
0.426702 + 0.904392i \(0.359675\pi\)
\(42\) 58.9370 − 14.2100i 1.40326 − 0.338334i
\(43\) −14.2190 + 14.2190i −0.330673 + 0.330673i −0.852842 − 0.522169i \(-0.825123\pi\)
0.522169 + 0.852842i \(0.325123\pi\)
\(44\) −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
\(47\) −15.9055 − 8.10427i −0.338415 − 0.172431i 0.276521 − 0.961008i \(-0.410818\pi\)
−0.614937 + 0.788577i \(0.710818\pi\)
\(48\) −33.2349 − 64.6062i −0.692394 − 1.34596i
\(49\) −2.60743 + 3.58882i −0.0532129 + 0.0732413i
\(50\) −48.2612 + 13.0711i −0.965225 + 0.261422i
\(51\) −25.5955 + 78.7748i −0.501872 + 1.54460i
\(52\) 94.1033 − 30.7780i 1.80968 − 0.591885i
\(53\) −11.0505 + 69.7703i −0.208501 + 1.31642i 0.632151 + 0.774845i \(0.282172\pi\)
−0.840652 + 0.541576i \(0.817828\pi\)
\(54\) 9.12414 + 21.9673i 0.168966 + 0.406802i
\(55\) −47.7456 − 27.3012i −0.868103 − 0.496385i
\(56\) −49.3991 − 20.2935i −0.882126 − 0.362384i
\(57\) 8.05567 − 50.8615i 0.141328 − 0.892307i
\(58\) −63.7721 − 4.95673i −1.09952 − 0.0854608i
\(59\) 24.7682 − 76.2288i 0.419801 − 1.29201i −0.488085 − 0.872796i \(-0.662304\pi\)
0.907886 − 0.419218i \(-0.137696\pi\)
\(60\) −65.2296 − 63.1885i −1.08716 − 1.05314i
\(61\) −22.3830 + 30.8075i −0.366934 + 0.505042i −0.952064 − 0.305898i \(-0.901043\pi\)
0.585130 + 0.810939i \(0.301043\pi\)
\(62\) 19.4379 − 22.8036i 0.313514 − 0.367800i
\(63\) 69.1114 + 35.2140i 1.09701 + 0.558952i
\(64\) −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 \(-0.698369\pi\)
0.812019 + 0.583631i \(0.198369\pi\)
\(68\) 58.9453 − 43.0012i 0.866842 − 0.632371i
\(69\) −127.181 + 41.3235i −1.84320 + 0.598891i
\(70\) −66.6212 − 4.24395i −0.951732 − 0.0606279i
\(71\) 12.2736 − 16.8932i 0.172868 − 0.237932i −0.713788 − 0.700362i \(-0.753022\pi\)
0.886656 + 0.462429i \(0.153022\pi\)
\(72\) 21.4365 − 90.4481i 0.297728 − 1.25622i
\(73\) −71.1314 + 36.2433i −0.974403 + 0.496483i −0.867311 − 0.497766i \(-0.834154\pi\)
−0.107092 + 0.994249i \(0.534154\pi\)
\(74\) 19.8822 + 23.2334i 0.268679 + 0.313965i
\(75\) −102.547 − 48.6951i −1.36729 − 0.649268i
\(76\) −32.0136 + 32.1380i −0.421231 + 0.422869i
\(77\) −48.5993 − 55.0486i −0.631160 − 0.714917i
\(78\) 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 \(0.584194\pi\)
\(80\) 13.9218 + 78.7793i 0.174023 + 0.984742i
\(81\) 15.6260 − 48.0919i 0.192914 − 0.593727i
\(82\) −10.8436 − 17.6566i −0.132239 − 0.215325i
\(83\) −103.856 + 16.4491i −1.25127 + 0.198182i −0.746684 − 0.665179i \(-0.768355\pi\)
−0.504587 + 0.863361i \(0.668355\pi\)
\(84\) −55.2567 − 107.929i −0.657817 − 1.28487i
\(85\) 52.5738 − 74.5265i 0.618515 − 0.876783i
\(86\) 34.3112 + 20.9802i 0.398968 + 0.243956i
\(87\) −102.690 − 102.690i −1.18035 − 1.18035i
\(88\) −50.3410 + 72.1788i −0.572057 + 0.820214i
\(89\) − 69.0839i − 0.776223i −0.921612 − 0.388112i \(-0.873128\pi\)
0.921612 − 0.388112i \(-0.126872\pi\)
\(90\) −10.8438 − 115.685i −0.120487 − 1.28539i
\(91\) 157.149 − 51.0608i 1.72691 − 0.561107i
\(92\) 112.103 + 36.1842i 1.21851 + 0.393306i
\(93\) 67.1931 − 10.6423i 0.722506 − 0.114434i
\(94\) −8.30087 + 34.7240i −0.0883071 + 0.369404i
\(95\) −25.0350 + 50.8767i −0.263527 + 0.535544i
\(96\) −110.033 + 94.9041i −1.14618 + 0.988584i
\(97\) 143.479 + 22.7249i 1.47917 + 0.234277i 0.843268 − 0.537494i \(-0.180629\pi\)
0.635901 + 0.771771i \(0.280629\pi\)
\(98\) 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.107.29 yes 544
4.3 odd 2 inner 220.3.w.a.107.45 yes 544
5.3 odd 4 inner 220.3.w.a.63.6 yes 544
11.7 odd 10 inner 220.3.w.a.7.13 yes 544
20.3 even 4 inner 220.3.w.a.63.13 yes 544
44.7 even 10 inner 220.3.w.a.7.6 ✓ 544
55.18 even 20 inner 220.3.w.a.183.45 yes 544
220.183 odd 20 inner 220.3.w.a.183.29 yes 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
220.3.w.a.7.6 ✓ 544 44.7 even 10 inner
220.3.w.a.7.13 yes 544 11.7 odd 10 inner
220.3.w.a.63.6 yes 544 5.3 odd 4 inner
220.3.w.a.63.13 yes 544 20.3 even 4 inner
220.3.w.a.107.29 yes 544 1.1 even 1 trivial
220.3.w.a.107.45 yes 544 4.3 odd 2 inner
220.3.w.a.183.29 yes 544 220.183 odd 20 inner
220.3.w.a.183.45 yes 544 55.18 even 20 inner