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Properties

Label 220.3.w.a.63.7
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 63.7
Character \(\chi\) \(=\) 220.63
Dual form 220.3.w.a.7.7

$q$-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+(-1.89633 + 0.635566i) q^{2} +(-4.39098 - 2.23732i) q^{3} +(3.19211 - 2.41048i) q^{4} +(4.88411 + 1.07027i) q^{5} +(9.74869 + 1.45192i) q^{6} +(-1.29034 - 2.53243i) q^{7} +(-4.52126 + 6.59986i) q^{8} +(8.98505 + 12.3669i) q^{9} +(-9.94210 + 1.07459i) q^{10} +(-2.28240 + 10.7606i) q^{11} +(-19.4095 + 3.44262i) q^{12} +(0.417679 - 2.63712i) q^{13} +(4.05642 + 3.98221i) q^{14} +(-19.0515 - 15.6268i) q^{15} +(4.37914 - 15.3891i) q^{16} +(0.0707521 + 0.446711i) q^{17} +(-24.8985 - 17.7410i) q^{18} +(22.1514 - 7.19742i) q^{19} +(18.1705 - 8.35664i) q^{20} +14.0067i q^{21} +(-2.51090 - 21.8562i) q^{22} +(16.7927 - 16.7927i) q^{23} +(34.6187 - 18.8644i) q^{24} +(22.7090 + 10.4547i) q^{25} +(0.884011 + 5.26631i) q^{26} +(-4.84625 - 30.5980i) q^{27} +(-10.2233 - 4.97345i) q^{28} +(7.96480 - 24.5131i) q^{29} +(46.0597 + 17.5251i) q^{30} +(-14.9598 - 20.5904i) q^{31} +(1.47649 + 31.9659i) q^{32} +(34.0968 - 42.1431i) q^{33} +(-0.418084 - 0.802143i) q^{34} +(-3.59175 - 13.7496i) q^{35} +(58.4914 + 17.8181i) q^{36} +(-32.1050 - 63.0096i) q^{37} +(-37.4318 + 27.7273i) q^{38} +(-7.73410 + 10.6451i) q^{39} +(-29.1460 + 27.3955i) q^{40} +(9.74917 - 3.16770i) q^{41} +(-8.90220 - 26.5613i) q^{42} +(1.75161 + 1.75161i) q^{43} +(18.6526 + 39.8507i) q^{44} +(30.6480 + 70.0175i) q^{45} +(-21.1715 + 42.5172i) q^{46} +(2.71100 - 5.32065i) q^{47} +(-53.6589 + 57.7755i) q^{48} +(24.0533 - 33.1065i) q^{49} +(-49.7084 - 5.39233i) q^{50} +(0.688763 - 2.11980i) q^{51} +(-5.02346 - 9.42480i) q^{52} +(69.4310 + 10.9968i) q^{53} +(28.6372 + 54.9438i) q^{54} +(-22.6643 + 50.1132i) q^{55} +(22.5476 + 2.93373i) q^{56} +(-113.369 - 17.9559i) q^{57} +(0.475861 + 51.5471i) q^{58} +(16.7213 - 51.4628i) q^{59} +(-98.4827 - 3.95930i) q^{60} +(-64.3142 + 88.5209i) q^{61} +(41.4552 + 29.5382i) q^{62} +(19.7244 - 38.7113i) q^{63} +(-23.1164 - 59.6794i) q^{64} +(4.86243 - 12.4330i) q^{65} +(-37.8740 + 101.588i) q^{66} +(70.1944 + 70.1944i) q^{67} +(1.30264 + 1.25541i) q^{68} +(-111.307 + 36.1657i) q^{69} +(15.5500 + 23.7910i) q^{70} +(59.0181 - 81.2315i) q^{71} +(-122.243 + 3.38627i) q^{72} +(-43.4809 - 85.3361i) q^{73} +(100.928 + 99.0819i) q^{74} +(-76.3245 - 96.7134i) q^{75} +(53.3604 - 76.3705i) q^{76} +(30.1955 - 8.10477i) q^{77} +(7.90073 - 25.1021i) q^{78} +(-42.4238 - 58.3914i) q^{79} +(37.8587 - 70.4750i) q^{80} +(-4.66421 + 14.3550i) q^{81} +(-16.4743 + 12.2032i) q^{82} +(22.7200 + 143.448i) q^{83} +(33.7629 + 44.7110i) q^{84} +(-0.132542 + 2.25751i) q^{85} +(-4.43489 - 2.20836i) q^{86} +(-89.8169 + 89.8169i) q^{87} +(-60.6992 - 63.7151i) q^{88} -73.3129i q^{89} +(-102.619 - 113.297i) q^{90} +(-7.21727 + 2.34503i) q^{91} +(13.1256 - 94.0824i) q^{92} +(19.6209 + 123.882i) q^{93} +(-1.75933 + 11.8127i) q^{94} +(115.893 - 11.4450i) q^{95} +(65.0346 - 143.665i) q^{96} +(8.07153 - 50.9617i) q^{97} +(-24.5715 + 78.0682i) q^{98} +(-153.582 + 68.4584i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.89633 + 0.635566i −0.948163 + 0.317783i
\(3\) −4.39098 − 2.23732i −1.46366 − 0.745772i −0.472864 − 0.881136i \(-0.656780\pi\)
−0.990796 + 0.135364i \(0.956780\pi\)
\(4\) 3.19211 − 2.41048i 0.798028 − 0.602621i
\(5\) 4.88411 + 1.07027i 0.976822 + 0.214054i
\(6\) 9.74869 + 1.45192i 1.62478 + 0.241987i
\(7\) −1.29034 − 2.53243i −0.184334 − 0.361775i 0.780285 − 0.625424i \(-0.215074\pi\)
−0.964619 + 0.263649i \(0.915074\pi\)
\(8\) −4.52126 + 6.59986i −0.565158 + 0.824983i
\(9\) 8.98505 + 12.3669i 0.998339 + 1.37410i
\(10\) −9.94210 + 1.07459i −0.994210 + 0.107459i
\(11\) −2.28240 + 10.7606i −0.207491 + 0.978237i
\(12\) −19.4095 + 3.44262i −1.61746 + 0.286885i
\(13\) 0.417679 − 2.63712i 0.0321292 − 0.202856i −0.966402 − 0.257035i \(-0.917254\pi\)
0.998531 + 0.0541796i \(0.0172543\pi\)
\(14\) 4.05642 + 3.98221i 0.289744 + 0.284444i
\(15\) −19.0515 − 15.6268i −1.27010 − 1.04179i
\(16\) 4.37914 − 15.3891i 0.273696 − 0.961816i
\(17\) 0.0707521 + 0.446711i 0.00416189 + 0.0262771i 0.989682 − 0.143282i \(-0.0457656\pi\)
−0.985520 + 0.169559i \(0.945766\pi\)
\(18\) −24.8985 − 17.7410i −1.38325 − 0.985611i
\(19\) 22.1514 − 7.19742i 1.16586 − 0.378812i 0.338766 − 0.940871i \(-0.389991\pi\)
0.827097 + 0.562059i \(0.189991\pi\)
\(20\) 18.1705 − 8.35664i 0.908524 − 0.417832i
\(21\) 14.0067i 0.666986i
\(22\) −2.51090 − 21.8562i −0.114132 − 0.993466i
\(23\) 16.7927 − 16.7927i 0.730115 − 0.730115i −0.240527 − 0.970642i \(-0.577320\pi\)
0.970642 + 0.240527i \(0.0773203\pi\)
\(24\) 34.6187 − 18.8644i 1.44245 − 0.786015i
\(25\) 22.7090 + 10.4547i 0.908361 + 0.418186i
\(26\) 0.884011 + 5.26631i 0.0340004 + 0.202550i
\(27\) −4.84625 − 30.5980i −0.179491 − 1.13326i
\(28\) −10.2233 − 4.97345i −0.365116 − 0.177623i
\(29\) 7.96480 − 24.5131i 0.274648 − 0.845281i −0.714664 − 0.699468i \(-0.753420\pi\)
0.989312 − 0.145813i \(-0.0465796\pi\)
\(30\) 46.0597 + 17.5251i 1.53532 + 0.584170i
\(31\) −14.9598 − 20.5904i −0.482574 − 0.664206i 0.496423 − 0.868081i \(-0.334646\pi\)
−0.978997 + 0.203875i \(0.934646\pi\)
\(32\) 1.47649 + 31.9659i 0.0461403 + 0.998935i
\(33\) 34.0968 − 42.1431i 1.03324 − 1.27706i
\(34\) −0.418084 − 0.802143i −0.0122966 − 0.0235924i
\(35\) −3.59175 − 13.7496i −0.102622 − 0.392847i
\(36\) 58.4914 + 17.8181i 1.62476 + 0.494946i
\(37\) −32.1050 − 63.0096i −0.867702 − 1.70296i −0.696288 − 0.717762i \(-0.745166\pi\)
−0.171414 − 0.985199i \(-0.554834\pi\)
\(38\) −37.4318 + 27.7273i −0.985048 + 0.729667i
\(39\) −7.73410 + 10.6451i −0.198310 + 0.272951i
\(40\) −29.1460 + 27.3955i −0.728650 + 0.684887i
\(41\) 9.74917 − 3.16770i 0.237785 − 0.0772609i −0.187701 − 0.982226i \(-0.560103\pi\)
0.425485 + 0.904965i \(0.360103\pi\)
\(42\) −8.90220 − 26.5613i −0.211957 − 0.632412i
\(43\) 1.75161 + 1.75161i 0.0407351 + 0.0407351i 0.727181 − 0.686446i \(-0.240830\pi\)
−0.686446 + 0.727181i \(0.740830\pi\)
\(44\) 18.6526 + 39.8507i 0.423922 + 0.905699i
\(45\) 30.6480 + 70.0175i 0.681068 + 1.55594i
\(46\) −21.1715 + 42.5172i −0.460250 + 0.924287i
\(47\) 2.71100 − 5.32065i 0.0576809 − 0.113205i −0.860366 − 0.509676i \(-0.829765\pi\)
0.918047 + 0.396471i \(0.129765\pi\)
\(48\) −53.6589 + 57.7755i −1.11789 + 1.20366i
\(49\) 24.0533 − 33.1065i 0.490883 − 0.675642i
\(50\) −49.7084 − 5.39233i −0.994168 − 0.107847i
\(51\) 0.688763 − 2.11980i 0.0135052 − 0.0415646i
\(52\) −5.02346 − 9.42480i −0.0966051 − 0.181246i
\(53\) 69.4310 + 10.9968i 1.31002 + 0.207487i 0.772095 − 0.635507i \(-0.219209\pi\)
0.537924 + 0.842993i \(0.319209\pi\)
\(54\) 28.6372 + 54.9438i 0.530318 + 1.01748i
\(55\) −22.6643 + 50.1132i −0.412078 + 0.911149i
\(56\) 22.5476 + 2.93373i 0.402636 + 0.0523879i
\(57\) −113.369 − 17.9559i −1.98893 − 0.315016i
\(58\) 0.475861 + 51.5471i 0.00820451 + 0.888743i
\(59\) 16.7213 − 51.4628i 0.283412 − 0.872251i −0.703459 − 0.710736i \(-0.748362\pi\)
0.986870 − 0.161515i \(-0.0516380\pi\)
\(60\) −98.4827 − 3.95930i −1.64138 − 0.0659884i
\(61\) −64.3142 + 88.5209i −1.05433 + 1.45116i −0.169338 + 0.985558i \(0.554163\pi\)
−0.884993 + 0.465604i \(0.845837\pi\)
\(62\) 41.4552 + 29.5382i 0.668633 + 0.476422i
\(63\) 19.7244 − 38.7113i 0.313086 − 0.614466i
\(64\) −23.1164 − 59.6794i −0.361193 − 0.932491i
\(65\) 4.86243 − 12.4330i 0.0748066 − 0.191276i
\(66\) −37.8740 + 101.588i −0.573849 + 1.53921i
\(67\) 70.1944 + 70.1944i 1.04768 + 1.04768i 0.998805 + 0.0488728i \(0.0155629\pi\)
0.0488728 + 0.998805i \(0.484437\pi\)
\(68\) 1.30264 + 1.25541i 0.0191565 + 0.0184618i
\(69\) −111.307 + 36.1657i −1.61314 + 0.524141i
\(70\) 15.5500 + 23.7910i 0.222142 + 0.339872i
\(71\) 59.0181 − 81.2315i 0.831241 − 1.14411i −0.156450 − 0.987686i \(-0.550005\pi\)
0.987691 − 0.156419i \(-0.0499952\pi\)
\(72\) −122.243 + 3.38627i −1.69782 + 0.0470315i
\(73\) −43.4809 − 85.3361i −0.595629 − 1.16899i −0.970317 − 0.241837i \(-0.922250\pi\)
0.374688 − 0.927151i \(-0.377750\pi\)
\(74\) 100.928 + 99.0819i 1.36390 + 1.33894i
\(75\) −76.3245 − 96.7134i −1.01766 − 1.28951i
\(76\) 53.3604 − 76.3705i 0.702111 − 1.00488i
\(77\) 30.1955 − 8.10477i 0.392149 − 0.105257i
\(78\) 7.90073 − 25.1021i 0.101291 − 0.321821i
\(79\) −42.4238 − 58.3914i −0.537010 − 0.739131i 0.451168 − 0.892439i \(-0.351007\pi\)
−0.988179 + 0.153308i \(0.951007\pi\)
\(80\) 37.8587 − 70.4750i 0.473233 − 0.880937i
\(81\) −4.66421 + 14.3550i −0.0575829 + 0.177222i
\(82\) −16.4743 + 12.2032i −0.200906 + 0.148820i
\(83\) 22.7200 + 143.448i 0.273734 + 1.72829i 0.615178 + 0.788389i \(0.289084\pi\)
−0.341443 + 0.939902i \(0.610916\pi\)
\(84\) 33.7629 + 44.7110i 0.401940 + 0.532273i
\(85\) −0.132542 + 2.25751i −0.00155931 + 0.0265590i
\(86\) −4.43489 − 2.20836i −0.0515685 − 0.0256786i
\(87\) −89.8169 + 89.8169i −1.03238 + 1.03238i
\(88\) −60.6992 − 63.7151i −0.689763 − 0.724035i
\(89\) − 73.3129i − 0.823741i −0.911242 − 0.411870i \(-0.864876\pi\)
0.911242 − 0.411870i \(-0.135124\pi\)
\(90\) −102.619 − 113.297i −1.14022 − 1.25886i
\(91\) −7.21727 + 2.34503i −0.0793106 + 0.0257696i
\(92\) 13.1256 − 94.0824i 0.142670 − 1.02264i
\(93\) 19.6209 + 123.882i 0.210978 + 1.33206i
\(94\) −1.75933 + 11.8127i −0.0187162 + 0.125667i
\(95\) 115.893 − 11.4450i 1.21993 − 0.120474i
\(96\) 65.0346 − 143.665i 0.677444 − 1.49651i
\(97\) 8.07153 − 50.9617i 0.0832117 − 0.525378i −0.910509 − 0.413489i \(-0.864310\pi\)
0.993721 − 0.111889i \(-0.0356900\pi\)
\(98\) −24.5715 + 78.0682i −0.250729 + 0.796614i
\(99\) −153.582 + 68.4584i −1.55134 + 0.691499i
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.63.7 yes 544
4.3 odd 2 inner 220.3.w.a.63.14 yes 544
5.2 odd 4 inner 220.3.w.a.107.28 yes 544
11.7 odd 10 inner 220.3.w.a.183.48 yes 544
20.7 even 4 inner 220.3.w.a.107.48 yes 544
44.7 even 10 inner 220.3.w.a.183.28 yes 544
55.7 even 20 inner 220.3.w.a.7.14 yes 544
220.7 odd 20 inner 220.3.w.a.7.7 ✓ 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
220.3.w.a.7.7 ✓ 544 220.7 odd 20 inner
220.3.w.a.7.14 yes 544 55.7 even 20 inner
220.3.w.a.63.7 yes 544 1.1 even 1 trivial
220.3.w.a.63.14 yes 544 4.3 odd 2 inner
220.3.w.a.107.28 yes 544 5.2 odd 4 inner
220.3.w.a.107.48 yes 544 20.7 even 4 inner
220.3.w.a.183.28 yes 544 44.7 even 10 inner
220.3.w.a.183.48 yes 544 11.7 odd 10 inner