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Properties

Label 220.3.w.a.107.45
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 107.45
Character \(\chi\) \(=\) 220.107
Dual form 220.3.w.a.183.45

$q$-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+(1.04665 - 1.70427i) q^{2} +(-2.06150 + 4.04592i) q^{3} +(-1.80904 - 3.56754i) q^{4} +(-3.48586 + 3.58452i) q^{5} +(4.73765 + 7.74800i) q^{6} +(5.94803 - 3.03067i) q^{7} +(-7.97348 - 0.650882i) 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} +(-24.4474 - 3.87209i) q^{13} +(1.06044 - 13.3091i) q^{14} +(-7.31657 - 21.4930i) q^{15} +(-9.45473 + 12.9077i) q^{16} +(-18.0163 + 2.85350i) q^{17} +(-23.1685 + 1.80079i) q^{18} +(-10.7855 + 3.50442i) q^{19} +(19.0940 + 5.95141i) q^{20} +30.3129i q^{21} +(-21.0256 - 6.47480i) q^{22} +(20.8240 + 20.8240i) q^{23} +(19.0707 - 30.9182i) q^{24} +(-0.697556 - 24.9903i) q^{25} +(-32.1870 + 37.6122i) q^{26} +(11.7470 - 1.86054i) q^{27} +(-21.5723 - 15.7372i) q^{28} +(9.88305 - 30.4169i) q^{29} +(-44.2876 - 10.0263i) q^{30} +(-8.80617 - 12.1207i) q^{31} +(12.1023 + 29.6232i) q^{32} +(48.7538 + 10.8625i) q^{33} +(-13.9936 + 33.6911i) q^{34} +(-9.87049 + 31.8853i) q^{35} +(-21.1804 + 41.3702i) q^{36} +(-13.6232 + 6.94136i) q^{37} +(-5.31618 + 22.0492i) q^{38} +(66.0644 - 90.9299i) q^{39} +(30.1275 - 26.3122i) q^{40} +(9.85319 - 3.20150i) q^{41} +(51.6613 + 31.7271i) 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} +(15.9055 + 8.10427i) q^{47} +(-32.7325 - 64.8622i) q^{48} +(-2.60743 + 3.58882i) q^{49} +(-43.3202 - 24.9673i) q^{50} +(25.5955 - 78.7748i) q^{51} +(30.4126 + 94.2220i) 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} +(-41.4944 - 48.6792i) q^{58} +(-24.7682 + 76.2288i) q^{59} +(-63.4411 + 64.9839i) q^{60} +(-22.3830 + 30.8075i) q^{61} +(-29.8738 + 2.32196i) q^{62} +(-69.1114 - 35.2140i) 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} +(42.7722 + 59.1117i) q^{68} +(-127.181 + 41.3235i) q^{69} +(44.0101 + 50.1947i) q^{70} +(-12.2736 + 16.8932i) q^{71} +(48.3373 + 79.3971i) q^{72} +(-71.1314 + 36.2433i) q^{73} +(-2.42880 + 30.4827i) q^{74} +(102.547 + 48.6951i) q^{75} +(32.0136 + 32.1380i) q^{76} +(-48.5993 - 55.0486i) q^{77} +(-85.8223 - 207.763i) q^{78} +(-58.2496 - 80.1738i) q^{79} +(-13.3100 - 78.8850i) q^{80} +(15.6260 - 48.0919i) q^{81} +(4.85665 - 20.1433i) q^{82} +(103.856 - 16.4491i) q^{83} +(108.143 - 54.8374i) q^{84} +(52.5738 - 74.5265i) q^{85} +(-9.35060 - 39.1152i) q^{86} +(102.690 + 102.690i) q^{87} +(14.9371 + 86.7230i) q^{88} -69.0839i q^{89} +(74.3073 - 89.3254i) q^{90} +(-157.149 + 51.0608i) q^{91} +(36.6190 - 111.962i) q^{92} +(67.1931 - 10.6423i) q^{93} +(30.4594 - 18.6249i) q^{94} +(25.0350 - 50.8767i) q^{95} +(-144.802 - 12.1032i) q^{96} +(143.479 + 22.7249i) 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 \(\left(\mathbb{Z}/220\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.04665 − 1.70427i 0.523326 − 0.852133i
\(3\) −2.06150 + 4.04592i −0.687166 + 1.34864i 0.238814 + 0.971065i \(0.423241\pi\)
−0.925980 + 0.377573i \(0.876759\pi\)
\(4\) −1.80904 − 3.56754i −0.452261 − 0.891886i
\(5\) −3.48586 + 3.58452i −0.697172 + 0.716904i
\(6\) 4.73765 + 7.74800i 0.789608 + 1.29133i
\(7\) 5.94803 − 3.03067i 0.849718 − 0.432953i 0.0258034 − 0.999667i \(-0.491786\pi\)
0.823914 + 0.566714i \(0.191786\pi\)
\(8\) −7.97348 − 0.650882i −0.996685 − 0.0813603i
\(9\) −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
\(13\) −24.4474 − 3.87209i −1.88057 − 0.297853i −0.892404 − 0.451237i \(-0.850983\pi\)
−0.988167 + 0.153384i \(0.950983\pi\)
\(14\) 1.06044 − 13.3091i 0.0757457 − 0.950648i
\(15\) −7.31657 − 21.4930i −0.487772 − 1.43287i
\(16\) −9.45473 + 12.9077i −0.590920 + 0.806730i
\(17\) −18.0163 + 2.85350i −1.05978 + 0.167853i −0.661906 − 0.749587i \(-0.730252\pi\)
−0.397874 + 0.917440i \(0.630252\pi\)
\(18\) −23.1685 + 1.80079i −1.28714 + 0.100044i
\(19\) −10.7855 + 3.50442i −0.567657 + 0.184443i −0.578764 − 0.815495i \(-0.696465\pi\)
0.0111065 + 0.999938i \(0.496465\pi\)
\(20\) 19.0940 + 5.95141i 0.954700 + 0.297570i
\(21\) 30.3129i 1.44347i
\(22\) −21.0256 − 6.47480i −0.955710 − 0.294309i
\(23\) 20.8240 + 20.8240i 0.905390 + 0.905390i 0.995896 − 0.0905057i \(-0.0288483\pi\)
−0.0905057 + 0.995896i \(0.528848\pi\)
\(24\) 19.0707 − 30.9182i 0.794613 − 1.28826i
\(25\) −0.697556 − 24.9903i −0.0279022 − 0.999611i
\(26\) −32.1870 + 37.6122i −1.23796 + 1.44662i
\(27\) 11.7470 − 1.86054i 0.435073 − 0.0689089i
\(28\) −21.5723 − 15.7372i −0.770439 − 0.562044i
\(29\) 9.88305 − 30.4169i 0.340795 − 1.04886i −0.623002 − 0.782220i \(-0.714087\pi\)
0.963797 − 0.266638i \(-0.0859129\pi\)
\(30\) −44.2876 − 10.0263i −1.47625 − 0.334209i
\(31\) −8.80617 − 12.1207i −0.284070 − 0.390989i 0.643007 − 0.765861i \(-0.277687\pi\)
−0.927077 + 0.374872i \(0.877687\pi\)
\(32\) 12.1023 + 29.6232i 0.378197 + 0.925725i
\(33\) 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
\(36\) −21.1804 + 41.3702i −0.588343 + 1.14917i
\(37\) −13.6232 + 6.94136i −0.368194 + 0.187604i −0.628291 − 0.777978i \(-0.716245\pi\)
0.260097 + 0.965582i \(0.416245\pi\)
\(38\) −5.31618 + 22.0492i −0.139900 + 0.580243i
\(39\) 66.0644 − 90.9299i 1.69396 − 2.33154i
\(40\) 30.1275 − 26.3122i 0.753188 − 0.657805i
\(41\) 9.85319 − 3.20150i 0.240322 − 0.0780853i −0.186380 − 0.982478i \(-0.559675\pi\)
0.426702 + 0.904392i \(0.359675\pi\)
\(42\) 51.6613 + 31.7271i 1.23003 + 0.755406i
\(43\) 14.2190 − 14.2190i 0.330673 − 0.330673i −0.522169 − 0.852842i \(-0.674877\pi\)
0.852842 + 0.522169i \(0.174877\pi\)
\(44\) −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
\(47\) 15.9055 + 8.10427i 0.338415 + 0.172431i 0.614937 − 0.788577i \(-0.289182\pi\)
−0.276521 + 0.961008i \(0.589182\pi\)
\(48\) −32.7325 − 64.8622i −0.681927 − 1.35130i
\(49\) −2.60743 + 3.58882i −0.0532129 + 0.0732413i
\(50\) −43.3202 − 24.9673i −0.866403 − 0.499345i
\(51\) 25.5955 − 78.7748i 0.501872 − 1.54460i
\(52\) 30.4126 + 94.2220i 0.584857 + 1.81196i
\(53\) −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
\(58\) −41.4944 − 48.6792i −0.715420 − 0.839297i
\(59\) −24.7682 + 76.2288i −0.419801 + 1.29201i 0.488085 + 0.872796i \(0.337696\pi\)
−0.907886 + 0.419218i \(0.862304\pi\)
\(60\) −63.4411 + 64.9839i −1.05735 + 1.08307i
\(61\) −22.3830 + 30.8075i −0.366934 + 0.505042i −0.952064 − 0.305898i \(-0.901043\pi\)
0.585130 + 0.810939i \(0.301043\pi\)
\(62\) −29.8738 + 2.32196i −0.481836 + 0.0374510i
\(63\) −69.1114 − 35.2140i −1.09701 − 0.558952i
\(64\) 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 \(-0.801631\pi\)
0.583631 + 0.812019i \(0.301631\pi\)
\(68\) 42.7722 + 59.1117i 0.629002 + 0.869289i
\(69\) −127.181 + 41.3235i −1.84320 + 0.598891i
\(70\) 44.0101 + 50.1947i 0.628715 + 0.717067i
\(71\) −12.2736 + 16.8932i −0.172868 + 0.237932i −0.886656 − 0.462429i \(-0.846978\pi\)
0.713788 + 0.700362i \(0.246978\pi\)
\(72\) 48.3373 + 79.3971i 0.671351 + 1.10274i
\(73\) −71.1314 + 36.2433i −0.974403 + 0.496483i −0.867311 − 0.497766i \(-0.834154\pi\)
−0.107092 + 0.994249i \(0.534154\pi\)
\(74\) −2.42880 + 30.4827i −0.0328216 + 0.411929i
\(75\) 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
\(78\) −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 \(-0.415806\pi\)
\(80\) −13.3100 − 78.8850i −0.166375 − 0.986063i
\(81\) 15.6260 − 48.0919i 0.192914 − 0.593727i
\(82\) 4.85665 − 20.1433i 0.0592275 − 0.245650i
\(83\) 103.856 − 16.4491i 1.25127 − 0.198182i 0.504587 − 0.863361i \(-0.331645\pi\)
0.746684 + 0.665179i \(0.231645\pi\)
\(84\) 108.143 − 54.8374i 1.28741 − 0.652826i
\(85\) 52.5738 − 74.5265i 0.618515 − 0.876783i
\(86\) −9.35060 − 39.1152i −0.108728 − 0.454828i
\(87\) 102.690 + 102.690i 1.18035 + 1.18035i
\(88\) 14.9371 + 86.7230i 0.169740 + 0.985489i
\(89\) − 69.0839i − 0.776223i −0.921612 − 0.388112i \(-0.873128\pi\)
0.921612 − 0.388112i \(-0.126872\pi\)
\(90\) 74.3073 − 89.3254i 0.825637 − 0.992504i
\(91\) −157.149 + 51.0608i −1.72691 + 0.561107i
\(92\) 36.6190 − 111.962i 0.398032 − 1.21698i
\(93\) 67.1931 − 10.6423i 0.722506 − 0.114434i
\(94\) 30.4594 − 18.6249i 0.324036 − 0.198137i
\(95\) 25.0350 − 50.8767i 0.263527 − 0.535544i
\(96\) −144.802 − 12.1032i −1.50835 − 0.126075i
\(97\) 143.479 + 22.7249i 1.47917 + 0.234277i 0.843268 − 0.537494i \(-0.180629\pi\)
0.635901 + 0.771771i \(0.280629\pi\)
\(98\) 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.107.45 yes 544
4.3 odd 2 inner 220.3.w.a.107.29 yes 544
5.3 odd 4 inner 220.3.w.a.63.13 yes 544
11.7 odd 10 inner 220.3.w.a.7.6 ✓ 544
20.3 even 4 inner 220.3.w.a.63.6 yes 544
44.7 even 10 inner 220.3.w.a.7.13 yes 544
55.18 even 20 inner 220.3.w.a.183.29 yes 544
220.183 odd 20 inner 220.3.w.a.183.45 yes 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
220.3.w.a.7.6 ✓ 544 11.7 odd 10 inner
220.3.w.a.7.13 yes 544 44.7 even 10 inner
220.3.w.a.63.6 yes 544 20.3 even 4 inner
220.3.w.a.63.13 yes 544 5.3 odd 4 inner
220.3.w.a.107.29 yes 544 4.3 odd 2 inner
220.3.w.a.107.45 yes 544 1.1 even 1 trivial
220.3.w.a.183.29 yes 544 55.18 even 20 inner
220.3.w.a.183.45 yes 544 220.183 odd 20 inner