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Properties

Label 220.3.w.a.107.24
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 107.24
Character \(\chi\) \(=\) 220.107
Dual form 220.3.w.a.183.24

$q$-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+(-0.954750 - 1.75740i) q^{2} +(1.02872 - 2.01897i) q^{3} +(-2.17690 + 3.35575i) q^{4} +(3.88835 + 3.14336i) q^{5} +(-4.53030 + 0.119746i) q^{6} +(-5.16446 + 2.63142i) q^{7} +(7.97580 + 0.621782i) q^{8} +(2.27209 + 3.12726i) q^{9} +(1.81173 - 9.83451i) q^{10} +(9.84371 - 4.90931i) q^{11} +(4.53575 + 7.84722i) q^{12} +(15.1080 + 2.39288i) q^{13} +(9.55523 + 6.56366i) q^{14} +(10.3464 - 4.61684i) q^{15} +(-6.52218 - 14.6103i) q^{16} +(12.0882 - 1.91459i) q^{17} +(3.32657 - 6.97872i) q^{18} +(-24.2978 + 7.89483i) q^{19} +(-19.0129 + 6.20556i) q^{20} +13.1339i q^{21} +(-18.0259 - 12.6122i) q^{22} +(11.4273 + 11.4273i) q^{23} +(9.46019 - 15.4633i) q^{24} +(5.23857 + 24.4450i) q^{25} +(-10.2192 - 28.8355i) q^{26} +(28.7936 - 4.56045i) q^{27} +(2.41212 - 23.0590i) q^{28} +(16.0454 - 49.3826i) q^{29} +(-17.9918 - 13.7747i) q^{30} +(-7.84453 - 10.7971i) q^{31} +(-19.4491 + 25.4113i) q^{32} +(0.214638 - 24.9244i) q^{33} +(-14.9059 - 19.4159i) q^{34} +(-28.3527 - 6.00185i) q^{35} +(-15.4404 + 0.816824i) q^{36} +(-7.85195 + 4.00077i) q^{37} +(37.0727 + 35.1633i) q^{38} +(20.3730 - 28.0411i) q^{39} +(29.0582 + 27.4885i) q^{40} +(-75.7259 + 24.6048i) q^{41} +(23.0814 - 12.5396i) q^{42} +(54.5989 - 54.5989i) q^{43} +(-4.95437 + 43.7202i) q^{44} +(-0.995428 + 19.3019i) q^{45} +(9.17211 - 30.9925i) q^{46} +(44.0767 + 22.4582i) q^{47} +(-36.2072 - 1.86179i) q^{48} +(-9.05424 + 12.4621i) q^{49} +(37.9581 - 32.5451i) q^{50} +(8.56986 - 26.3753i) q^{51} +(-40.9187 + 45.4898i) q^{52} +(-3.46642 + 21.8861i) q^{53} +(-35.5052 - 46.2477i) q^{54} +(53.7076 + 11.8532i) q^{55} +(-42.8269 + 17.7765i) q^{56} +(-9.05611 + 57.1780i) q^{57} +(-102.104 + 18.9499i) q^{58} +(-15.0901 + 46.4424i) q^{59} +(-7.03005 + 44.7702i) q^{60} +(55.4322 - 76.2959i) q^{61} +(-11.4852 + 24.0945i) q^{62} +(-19.9633 - 10.1718i) q^{63} +(63.2268 + 9.91841i) q^{64} +(51.2237 + 56.7944i) q^{65} +(-44.0071 + 23.4194i) q^{66} +(-5.11791 + 5.11791i) q^{67} +(-19.8900 + 44.7330i) q^{68} +(34.8268 - 11.3159i) q^{69} +(16.5221 + 55.5574i) q^{70} +(-10.7606 + 14.8107i) q^{71} +(16.1773 + 26.3552i) q^{72} +(-91.4110 + 46.5762i) q^{73} +(14.5276 + 9.97927i) q^{74} +(54.7427 + 14.5704i) q^{75} +(26.4008 - 98.7237i) q^{76} +(-37.9190 + 51.2569i) q^{77} +(-68.7305 - 9.03133i) q^{78} +(47.7369 + 65.7043i) q^{79} +(20.5649 - 77.3116i) q^{80} +(9.66242 - 29.7379i) q^{81} +(115.540 + 109.589i) q^{82} +(-61.2488 + 9.70086i) q^{83} +(-44.0740 - 28.5912i) q^{84} +(53.0215 + 30.5531i) q^{85} +(-148.080 - 43.8237i) q^{86} +(-83.1957 - 83.1957i) q^{87} +(81.5640 - 33.0350i) q^{88} -11.2250i q^{89} +(34.8715 - 16.6791i) q^{90} +(-84.3215 + 27.3977i) q^{91} +(-63.2234 + 13.4711i) q^{92} +(-29.8687 + 4.73074i) q^{93} +(-2.61421 - 98.9023i) q^{94} +(-119.295 - 45.6788i) q^{95} +(31.2970 + 65.4081i) q^{96} +(50.3895 + 7.98092i) q^{97} +(30.5454 + 4.01373i) q^{98} +(37.7185 + 19.6295i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.954750 − 1.75740i −0.477375 − 0.878700i
\(3\) 1.02872 − 2.01897i 0.342905 − 0.672990i −0.653571 − 0.756865i \(-0.726730\pi\)
0.996476 + 0.0838758i \(0.0267299\pi\)
\(4\) −2.17690 + 3.35575i −0.544226 + 0.838939i
\(5\) 3.88835 + 3.14336i 0.777671 + 0.628672i
\(6\) −4.53030 + 0.119746i −0.755050 + 0.0199577i
\(7\) −5.16446 + 2.63142i −0.737780 + 0.375918i −0.782144 − 0.623098i \(-0.785874\pi\)
0.0443641 + 0.999015i \(0.485874\pi\)
\(8\) 7.97580 + 0.621782i 0.996975 + 0.0777227i
\(9\) 2.27209 + 3.12726i 0.252454 + 0.347474i
\(10\) 1.81173 − 9.83451i 0.181173 − 0.983451i
\(11\) 9.84371 − 4.90931i 0.894883 − 0.446301i
\(12\) 4.53575 + 7.84722i 0.377979 + 0.653935i
\(13\) 15.1080 + 2.39288i 1.16216 + 0.184068i 0.707576 − 0.706637i \(-0.249789\pi\)
0.454582 + 0.890705i \(0.349789\pi\)
\(14\) 9.55523 + 6.56366i 0.682516 + 0.468833i
\(15\) 10.3464 − 4.61684i 0.689757 − 0.307789i
\(16\) −6.52218 − 14.6103i −0.407636 − 0.913144i
\(17\) 12.0882 − 1.91459i 0.711072 − 0.112623i 0.209588 − 0.977790i \(-0.432788\pi\)
0.501484 + 0.865167i \(0.332788\pi\)
\(18\) 3.32657 − 6.97872i 0.184810 − 0.387707i
\(19\) −24.2978 + 7.89483i −1.27883 + 0.415517i −0.868168 − 0.496270i \(-0.834703\pi\)
−0.410663 + 0.911787i \(0.634703\pi\)
\(20\) −19.0129 + 6.20556i −0.950646 + 0.310278i
\(21\) 13.1339i 0.625422i
\(22\) −18.0259 − 12.6122i −0.819359 − 0.573280i
\(23\) 11.4273 + 11.4273i 0.496839 + 0.496839i 0.910453 − 0.413613i \(-0.135733\pi\)
−0.413613 + 0.910453i \(0.635733\pi\)
\(24\) 9.46019 − 15.4633i 0.394175 − 0.644302i
\(25\) 5.23857 + 24.4450i 0.209543 + 0.977799i
\(26\) −10.2192 − 28.8355i −0.393045 − 1.10906i
\(27\) 28.7936 − 4.56045i 1.06643 − 0.168906i
\(28\) 2.41212 − 23.0590i 0.0861471 − 0.823536i
\(29\) 16.0454 − 49.3826i 0.553289 − 1.70285i −0.147132 − 0.989117i \(-0.547004\pi\)
0.700421 − 0.713730i \(-0.252996\pi\)
\(30\) −17.9918 − 13.7747i −0.599727 − 0.459158i
\(31\) −7.84453 − 10.7971i −0.253049 − 0.348292i 0.663527 − 0.748152i \(-0.269059\pi\)
−0.916576 + 0.399860i \(0.869059\pi\)
\(32\) −19.4491 + 25.4113i −0.607784 + 0.794102i
\(33\) 0.214638 − 24.9244i 0.00650418 − 0.755286i
\(34\) −14.9059 − 19.4159i −0.438410 − 0.571055i
\(35\) −28.3527 − 6.00185i −0.810078 − 0.171481i
\(36\) −15.4404 + 0.816824i −0.428901 + 0.0226896i
\(37\) −7.85195 + 4.00077i −0.212215 + 0.108129i −0.556867 − 0.830602i \(-0.687997\pi\)
0.344652 + 0.938730i \(0.387997\pi\)
\(38\) 37.0727 + 35.1633i 0.975597 + 0.925351i
\(39\) 20.3730 − 28.0411i 0.522385 − 0.719002i
\(40\) 29.0582 + 27.4885i 0.726456 + 0.687213i
\(41\) −75.7259 + 24.6048i −1.84697 + 0.600118i −0.849621 + 0.527395i \(0.823169\pi\)
−0.997352 + 0.0727233i \(0.976831\pi\)
\(42\) 23.0814 − 12.5396i 0.549558 − 0.298561i
\(43\) 54.5989 − 54.5989i 1.26974 − 1.26974i 0.323519 − 0.946222i \(-0.395134\pi\)
0.946222 − 0.323519i \(-0.104866\pi\)
\(44\) −4.95437 + 43.7202i −0.112599 + 0.993640i
\(45\) −0.995428 + 19.3019i −0.0221206 + 0.428931i
\(46\) 9.17211 − 30.9925i 0.199394 − 0.673751i
\(47\) 44.0767 + 22.4582i 0.937802 + 0.477834i 0.854940 − 0.518727i \(-0.173594\pi\)
0.0828618 + 0.996561i \(0.473594\pi\)
\(48\) −36.2072 − 1.86179i −0.754317 − 0.0387872i
\(49\) −9.05424 + 12.4621i −0.184780 + 0.254328i
\(50\) 37.9581 − 32.5451i 0.759161 − 0.650902i
\(51\) 8.56986 − 26.3753i 0.168036 − 0.517163i
\(52\) −40.9187 + 45.4898i −0.786898 + 0.874804i
\(53\) −3.46642 + 21.8861i −0.0654041 + 0.412945i 0.933164 + 0.359451i \(0.117036\pi\)
−0.998568 + 0.0534944i \(0.982964\pi\)
\(54\) −35.5052 − 46.2477i −0.657504 − 0.856439i
\(55\) 53.7076 + 11.8532i 0.976501 + 0.215513i
\(56\) −42.8269 + 17.7765i −0.764765 + 0.317438i
\(57\) −9.05611 + 57.1780i −0.158879 + 1.00312i
\(58\) −102.104 + 18.9499i −1.76042 + 0.326722i
\(59\) −15.0901 + 46.4424i −0.255764 + 0.787159i 0.737915 + 0.674894i \(0.235811\pi\)
−0.993678 + 0.112265i \(0.964189\pi\)
\(60\) −7.03005 + 44.7702i −0.117167 + 0.746171i
\(61\) 55.4322 − 76.2959i 0.908725 − 1.25075i −0.0588751 − 0.998265i \(-0.518751\pi\)
0.967600 − 0.252487i \(-0.0812486\pi\)
\(62\) −11.4852 + 24.0945i −0.185245 + 0.388620i
\(63\) −19.9633 − 10.1718i −0.316877 − 0.161457i
\(64\) 63.2268 + 9.91841i 0.987918 + 0.154975i
\(65\) 51.2237 + 56.7944i 0.788057 + 0.873760i
\(66\) −44.0071 + 23.4194i −0.666774 + 0.354839i
\(67\) −5.11791 + 5.11791i −0.0763867 + 0.0763867i −0.744268 − 0.667881i \(-0.767201\pi\)
0.667881 + 0.744268i \(0.267201\pi\)
\(68\) −19.8900 + 44.7330i −0.292500 + 0.657838i
\(69\) 34.8268 − 11.3159i 0.504736 − 0.163999i
\(70\) 16.5221 + 55.5574i 0.236031 + 0.793677i
\(71\) −10.7606 + 14.8107i −0.151558 + 0.208602i −0.878044 − 0.478579i \(-0.841152\pi\)
0.726486 + 0.687181i \(0.241152\pi\)
\(72\) 16.1773 + 26.3552i 0.224684 + 0.366044i
\(73\) −91.4110 + 46.5762i −1.25220 + 0.638030i −0.949114 − 0.314932i \(-0.898018\pi\)
−0.303090 + 0.952962i \(0.598018\pi\)
\(74\) 14.5276 + 9.97927i 0.196319 + 0.134855i
\(75\) 54.7427 + 14.5704i 0.729902 + 0.194272i
\(76\) 26.4008 − 98.7237i 0.347380 − 1.29900i
\(77\) −37.9190 + 51.2569i −0.492454 + 0.665674i
\(78\) −68.7305 − 9.03133i −0.881161 − 0.115786i
\(79\) 47.7369 + 65.7043i 0.604265 + 0.831700i 0.996090 − 0.0883400i \(-0.0281562\pi\)
−0.391825 + 0.920040i \(0.628156\pi\)
\(80\) 20.5649 − 77.3116i 0.257062 − 0.966395i
\(81\) 9.66242 − 29.7379i 0.119289 − 0.367134i
\(82\) 115.540 + 109.589i 1.40902 + 1.33645i
\(83\) −61.2488 + 9.70086i −0.737937 + 0.116878i −0.514078 − 0.857744i \(-0.671866\pi\)
−0.223860 + 0.974621i \(0.571866\pi\)
\(84\) −44.0740 − 28.5912i −0.524691 − 0.340371i
\(85\) 53.0215 + 30.5531i 0.623782 + 0.359448i
\(86\) −148.080 − 43.8237i −1.72186 − 0.509578i
\(87\) −83.1957 − 83.1957i −0.956273 − 0.956273i
\(88\) 81.5640 − 33.0350i 0.926864 − 0.375398i
\(89\) − 11.2250i − 0.126124i −0.998010 − 0.0630619i \(-0.979913\pi\)
0.998010 − 0.0630619i \(-0.0200865\pi\)
\(90\) 34.8715 − 16.6791i 0.387461 − 0.185324i
\(91\) −84.3215 + 27.3977i −0.926610 + 0.301074i
\(92\) −63.2234 + 13.4711i −0.687210 + 0.146425i
\(93\) −29.8687 + 4.73074i −0.321169 + 0.0508682i
\(94\) −2.61421 − 98.9023i −0.0278108 − 1.05215i
\(95\) −119.295 − 45.6788i −1.25573 − 0.480830i
\(96\) 31.2970 + 65.4081i 0.326010 + 0.681334i
\(97\) 50.3895 + 7.98092i 0.519480 + 0.0822775i 0.410668 − 0.911785i \(-0.365295\pi\)
0.108812 + 0.994062i \(0.465295\pi\)
\(98\) 30.5454 + 4.01373i 0.311688 + 0.0409564i
\(99\) 37.7185 + 19.6295i 0.380995 + 0.198278i
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.107.24 yes 544
4.3 odd 2 inner 220.3.w.a.107.52 yes 544
5.3 odd 4 inner 220.3.w.a.63.11 yes 544
11.7 odd 10 inner 220.3.w.a.7.19 yes 544
20.3 even 4 inner 220.3.w.a.63.19 yes 544
44.7 even 10 inner 220.3.w.a.7.11 ✓ 544
55.18 even 20 inner 220.3.w.a.183.52 yes 544
220.183 odd 20 inner 220.3.w.a.183.24 yes 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
220.3.w.a.7.11 ✓ 544 44.7 even 10 inner
220.3.w.a.7.19 yes 544 11.7 odd 10 inner
220.3.w.a.63.11 yes 544 5.3 odd 4 inner
220.3.w.a.63.19 yes 544 20.3 even 4 inner
220.3.w.a.107.24 yes 544 1.1 even 1 trivial
220.3.w.a.107.52 yes 544 4.3 odd 2 inner
220.3.w.a.183.24 yes 544 220.183 odd 20 inner
220.3.w.a.183.52 yes 544 55.18 even 20 inner