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Properties

Label 220.3.w.a.107.28
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 107.28
Character \(\chi\) \(=\) 220.107
Dual form 220.3.w.a.183.28

$q$-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+(-0.635566 - 1.89633i) q^{2} +(-2.23732 + 4.39098i) q^{3} +(-3.19211 + 2.41048i) q^{4} +(2.52716 + 4.31433i) q^{5} +(9.74869 + 1.45192i) q^{6} +(2.53243 - 1.29034i) q^{7} +(6.59986 + 4.52126i) q^{8} +(-8.98505 - 12.3669i) q^{9} +(6.57520 - 7.53437i) q^{10} +(-2.28240 + 10.7606i) q^{11} +(-3.44262 - 19.4095i) q^{12} +(-2.63712 - 0.417679i) q^{13} +(-4.05642 - 3.98221i) q^{14} +(-24.5982 + 1.44419i) q^{15} +(4.37914 - 15.3891i) q^{16} +(-0.446711 + 0.0707521i) q^{17} +(-17.7410 + 24.8985i) q^{18} +(-22.1514 + 7.19742i) q^{19} +(-18.4666 - 7.68014i) q^{20} +14.0067i q^{21} +(21.8562 - 2.51090i) q^{22} +(-16.7927 - 16.7927i) q^{23} +(-34.6187 + 18.8644i) q^{24} +(-12.2269 + 21.8060i) q^{25} +(0.884011 + 5.26631i) q^{26} +(30.5980 - 4.84625i) q^{27} +(-4.97345 + 10.2233i) q^{28} +(-7.96480 + 24.5131i) q^{29} +(18.3725 + 45.7283i) q^{30} +(-14.9598 - 20.5904i) q^{31} +(-31.9659 + 1.47649i) q^{32} +(-42.1431 - 34.0968i) q^{33} +(0.418084 + 0.802143i) q^{34} +(11.9668 + 7.66483i) q^{35} +(58.4914 + 17.8181i) q^{36} +(63.0096 - 32.1050i) q^{37} +(27.7273 + 37.4318i) q^{38} +(7.73410 - 10.6451i) q^{39} +(-2.82730 + 39.9000i) q^{40} +(9.74917 - 3.16770i) q^{41} +(26.5613 - 8.90220i) 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} +(5.32065 + 2.71100i) q^{47} +(57.7755 + 53.6589i) q^{48} +(-24.0533 + 33.1065i) q^{49} +(49.1224 + 9.32703i) q^{50} +(0.688763 - 2.11980i) q^{51} +(9.42480 - 5.02346i) q^{52} +(10.9968 - 69.4310i) q^{53} +(-28.6372 - 54.9438i) q^{54} +(-52.1928 + 17.3468i) q^{55} +(22.5476 + 2.93373i) q^{56} +(17.9559 - 113.369i) q^{57} +(51.5471 - 0.475861i) q^{58} +(-16.7213 + 51.4628i) q^{59} +(75.0390 - 63.9036i) q^{60} +(-64.3142 + 88.5209i) q^{61} +(-29.5382 + 41.4552i) q^{62} +(-38.7113 - 19.7244i) 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} +(1.25541 - 1.30264i) q^{68} +(111.307 - 36.1657i) q^{69} +(6.92935 - 27.5644i) q^{70} +(59.0181 - 81.2315i) q^{71} +(-3.38627 - 122.243i) q^{72} +(-85.3361 + 43.4809i) q^{73} +(-100.928 - 99.0819i) q^{74} +(-68.3944 - 102.475i) q^{75} +(53.3604 - 76.3705i) q^{76} +(8.10477 + 30.1955i) q^{77} +(-25.1021 - 7.90073i) q^{78} +(42.4238 + 58.3914i) q^{79} +(77.4603 - 19.9976i) q^{80} +(-4.66421 + 14.3550i) q^{81} +(-12.2032 - 16.4743i) q^{82} +(143.448 - 22.7200i) q^{83} +(-33.7629 - 44.7110i) q^{84} +(-1.43416 - 1.74846i) q^{85} +(-4.43489 - 2.20836i) q^{86} +(-89.8169 - 89.8169i) q^{87} +(-63.7151 + 60.6992i) q^{88} +73.3129i q^{89} +(-152.255 - 13.6179i) q^{90} +(-7.21727 + 2.34503i) q^{91} +(94.0824 + 13.1256i) q^{92} +(123.882 - 19.6209i) q^{93} +(1.75933 - 11.8127i) q^{94} +(-87.0322 - 77.3794i) q^{95} +(65.0346 - 143.665i) q^{96} +(50.9617 + 8.07153i) q^{97} +(78.0682 + 24.5715i) 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 \(\left(\mathbb{Z}/220\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.635566 − 1.89633i −0.317783 − 0.948163i
\(3\) −2.23732 + 4.39098i −0.745772 + 1.46366i 0.135364 + 0.990796i \(0.456780\pi\)
−0.881136 + 0.472864i \(0.843220\pi\)
\(4\) −3.19211 + 2.41048i −0.798028 + 0.602621i
\(5\) 2.52716 + 4.31433i 0.505432 + 0.862866i
\(6\) 9.74869 + 1.45192i 1.62478 + 0.241987i
\(7\) 2.53243 − 1.29034i 0.361775 − 0.184334i −0.263649 − 0.964619i \(-0.584926\pi\)
0.625424 + 0.780285i \(0.284926\pi\)
\(8\) 6.59986 + 4.52126i 0.824983 + 0.565158i
\(9\) −8.98505 − 12.3669i −0.998339 − 1.37410i
\(10\) 6.57520 − 7.53437i 0.657520 − 0.753437i
\(11\) −2.28240 + 10.7606i −0.207491 + 0.978237i
\(12\) −3.44262 − 19.4095i −0.286885 − 1.61746i
\(13\) −2.63712 − 0.417679i −0.202856 − 0.0321292i 0.0541796 − 0.998531i \(-0.482746\pi\)
−0.257035 + 0.966402i \(0.582746\pi\)
\(14\) −4.05642 − 3.98221i −0.289744 − 0.284444i
\(15\) −24.5982 + 1.44419i −1.63988 + 0.0962796i
\(16\) 4.37914 − 15.3891i 0.273696 − 0.961816i
\(17\) −0.446711 + 0.0707521i −0.0262771 + 0.00416189i −0.169559 − 0.985520i \(-0.554234\pi\)
0.143282 + 0.989682i \(0.454234\pi\)
\(18\) −17.7410 + 24.8985i −0.985611 + 1.38325i
\(19\) −22.1514 + 7.19742i −1.16586 + 0.378812i −0.827097 − 0.562059i \(-0.810009\pi\)
−0.338766 + 0.940871i \(0.610009\pi\)
\(20\) −18.4666 − 7.68014i −0.923330 − 0.384007i
\(21\) 14.0067i 0.666986i
\(22\) 21.8562 − 2.51090i 0.993466 − 0.114132i
\(23\) −16.7927 − 16.7927i −0.730115 − 0.730115i 0.240527 − 0.970642i \(-0.422680\pi\)
−0.970642 + 0.240527i \(0.922680\pi\)
\(24\) −34.6187 + 18.8644i −1.44245 + 0.786015i
\(25\) −12.2269 + 21.8060i −0.489076 + 0.872241i
\(26\) 0.884011 + 5.26631i 0.0340004 + 0.202550i
\(27\) 30.5980 − 4.84625i 1.13326 − 0.179491i
\(28\) −4.97345 + 10.2233i −0.177623 + 0.365116i
\(29\) −7.96480 + 24.5131i −0.274648 + 0.845281i 0.714664 + 0.699468i \(0.246580\pi\)
−0.989312 + 0.145813i \(0.953420\pi\)
\(30\) 18.3725 + 45.7283i 0.612415 + 1.52428i
\(31\) −14.9598 − 20.5904i −0.482574 − 0.664206i 0.496423 − 0.868081i \(-0.334646\pi\)
−0.978997 + 0.203875i \(0.934646\pi\)
\(32\) −31.9659 + 1.47649i −0.998935 + 0.0461403i
\(33\) −42.1431 − 34.0968i −1.27706 − 1.03324i
\(34\) 0.418084 + 0.802143i 0.0122966 + 0.0235924i
\(35\) 11.9668 + 7.66483i 0.341908 + 0.218995i
\(36\) 58.4914 + 17.8181i 1.62476 + 0.494946i
\(37\) 63.0096 − 32.1050i 1.70296 − 0.867702i 0.717762 − 0.696288i \(-0.245166\pi\)
0.985199 − 0.171414i \(-0.0548336\pi\)
\(38\) 27.7273 + 37.4318i 0.729667 + 0.985048i
\(39\) 7.73410 − 10.6451i 0.198310 − 0.272951i
\(40\) −2.82730 + 39.9000i −0.0706826 + 0.997499i
\(41\) 9.74917 − 3.16770i 0.237785 − 0.0772609i −0.187701 − 0.982226i \(-0.560103\pi\)
0.425485 + 0.904965i \(0.360103\pi\)
\(42\) 26.5613 − 8.90220i 0.632412 − 0.211957i
\(43\) 1.75161 − 1.75161i 0.0407351 − 0.0407351i −0.686446 − 0.727181i \(-0.740830\pi\)
0.727181 + 0.686446i \(0.240830\pi\)
\(44\) −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
\(47\) 5.32065 + 2.71100i 0.113205 + 0.0576809i 0.509676 − 0.860366i \(-0.329765\pi\)
−0.396471 + 0.918047i \(0.629765\pi\)
\(48\) 57.7755 + 53.6589i 1.20366 + 1.11789i
\(49\) −24.0533 + 33.1065i −0.490883 + 0.675642i
\(50\) 49.1224 + 9.32703i 0.982447 + 0.186541i
\(51\) 0.688763 − 2.11980i 0.0135052 − 0.0415646i
\(52\) 9.42480 − 5.02346i 0.181246 − 0.0966051i
\(53\) 10.9968 − 69.4310i 0.207487 − 1.31002i −0.635507 − 0.772095i \(-0.719209\pi\)
0.842993 − 0.537924i \(-0.180791\pi\)
\(54\) −28.6372 − 54.9438i −0.530318 − 1.01748i
\(55\) −52.1928 + 17.3468i −0.948960 + 0.315396i
\(56\) 22.5476 + 2.93373i 0.402636 + 0.0523879i
\(57\) 17.9559 − 113.369i 0.315016 − 1.98893i
\(58\) 51.5471 − 0.475861i 0.888743 − 0.00820451i
\(59\) −16.7213 + 51.4628i −0.283412 + 0.872251i 0.703459 + 0.710736i \(0.251638\pi\)
−0.986870 + 0.161515i \(0.948362\pi\)
\(60\) 75.0390 − 63.9036i 1.25065 − 1.06506i
\(61\) −64.3142 + 88.5209i −1.05433 + 1.45116i −0.169338 + 0.985558i \(0.554163\pi\)
−0.884993 + 0.465604i \(0.845837\pi\)
\(62\) −29.5382 + 41.4552i −0.476422 + 0.668633i
\(63\) −38.7113 − 19.7244i −0.614466 − 0.313086i
\(64\) 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 −0.0488728 + 0.998805i \(0.515563\pi\)
−0.998805 + 0.0488728i \(0.984437\pi\)
\(68\) 1.25541 − 1.30264i 0.0184618 − 0.0191565i
\(69\) 111.307 − 36.1657i 1.61314 − 0.524141i
\(70\) 6.92935 − 27.5644i 0.0989907 − 0.393778i
\(71\) 59.0181 − 81.2315i 0.831241 − 1.14411i −0.156450 − 0.987686i \(-0.550005\pi\)
0.987691 − 0.156419i \(-0.0499952\pi\)
\(72\) −3.38627 − 122.243i −0.0470315 − 1.69782i
\(73\) −85.3361 + 43.4809i −1.16899 + 0.595629i −0.927151 − 0.374688i \(-0.877750\pi\)
−0.241837 + 0.970317i \(0.577750\pi\)
\(74\) −100.928 − 99.0819i −1.36390 − 1.33894i
\(75\) −68.3944 − 102.475i −0.911925 − 1.36633i
\(76\) 53.3604 − 76.3705i 0.702111 − 1.00488i
\(77\) 8.10477 + 30.1955i 0.105257 + 0.392149i
\(78\) −25.1021 − 7.90073i −0.321821 − 0.101291i
\(79\) 42.4238 + 58.3914i 0.537010 + 0.739131i 0.988179 − 0.153308i \(-0.0489926\pi\)
−0.451168 + 0.892439i \(0.648993\pi\)
\(80\) 77.4603 − 19.9976i 0.968254 − 0.249970i
\(81\) −4.66421 + 14.3550i −0.0575829 + 0.177222i
\(82\) −12.2032 − 16.4743i −0.148820 − 0.200906i
\(83\) 143.448 − 22.7200i 1.72829 − 0.273734i 0.788389 − 0.615178i \(-0.210916\pi\)
0.939902 + 0.341443i \(0.110916\pi\)
\(84\) −33.7629 − 44.7110i −0.401940 − 0.532273i
\(85\) −1.43416 − 1.74846i −0.0168725 − 0.0205701i
\(86\) −4.43489 − 2.20836i −0.0515685 − 0.0256786i
\(87\) −89.8169 − 89.8169i −1.03238 − 1.03238i
\(88\) −63.7151 + 60.6992i −0.724035 + 0.689763i
\(89\) 73.3129i 0.823741i 0.911242 + 0.411870i \(0.135124\pi\)
−0.911242 + 0.411870i \(0.864876\pi\)
\(90\) −152.255 − 13.6179i −1.69172 − 0.151310i
\(91\) −7.21727 + 2.34503i −0.0793106 + 0.0257696i
\(92\) 94.0824 + 13.1256i 1.02264 + 0.142670i
\(93\) 123.882 − 19.6209i 1.33206 − 0.210978i
\(94\) 1.75933 − 11.8127i 0.0187162 − 0.125667i
\(95\) −87.0322 − 77.3794i −0.916129 − 0.814520i
\(96\) 65.0346 − 143.665i 0.677444 − 1.49651i
\(97\) 50.9617 + 8.07153i 0.525378 + 0.0832117i 0.413489 − 0.910509i \(-0.364310\pi\)
0.111889 + 0.993721i \(0.464310\pi\)
\(98\) 78.0682 + 24.5715i 0.796614 + 0.250729i
\(99\) 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.107.28 yes 544
4.3 odd 2 inner 220.3.w.a.107.48 yes 544
5.3 odd 4 inner 220.3.w.a.63.7 yes 544
11.7 odd 10 inner 220.3.w.a.7.14 yes 544
20.3 even 4 inner 220.3.w.a.63.14 yes 544
44.7 even 10 inner 220.3.w.a.7.7 ✓ 544
55.18 even 20 inner 220.3.w.a.183.48 yes 544
220.183 odd 20 inner 220.3.w.a.183.28 yes 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
220.3.w.a.7.7 ✓ 544 44.7 even 10 inner
220.3.w.a.7.14 yes 544 11.7 odd 10 inner
220.3.w.a.63.7 yes 544 5.3 odd 4 inner
220.3.w.a.63.14 yes 544 20.3 even 4 inner
220.3.w.a.107.28 yes 544 1.1 even 1 trivial
220.3.w.a.107.48 yes 544 4.3 odd 2 inner
220.3.w.a.183.28 yes 544 220.183 odd 20 inner
220.3.w.a.183.48 yes 544 55.18 even 20 inner