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Properties

Label 220.3.w.a.63.19
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 63.19
Character \(\chi\) \(=\) 220.63
Dual form 220.3.w.a.7.19

$q$-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+(-1.37635 - 1.45109i) q^{2} +(-2.01897 - 1.02872i) q^{3} +(-0.211311 + 3.99441i) q^{4} +(4.19108 + 2.72669i) q^{5} +(1.28605 + 4.34558i) q^{6} +(-2.63142 - 5.16446i) q^{7} +(6.08708 - 5.19109i) 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} +(-2.39288 + 15.1080i) q^{13} +(-3.87232 + 10.9265i) q^{14} +(-5.65667 - 9.81653i) q^{15} +(-15.9107 - 1.68813i) q^{16} +(-1.91459 - 12.0882i) q^{17} +(-1.41074 + 7.60122i) q^{18} +(-24.2978 + 7.89483i) q^{19} +(-11.7772 + 16.1647i) q^{20} +13.1339i q^{21} +(20.6723 + 7.52715i) q^{22} +(11.4273 - 11.4273i) q^{23} +(-17.6298 + 4.21877i) q^{24} +(10.1303 + 22.8556i) q^{25} +(25.2165 - 17.3217i) q^{26} +(4.56045 + 28.7936i) q^{27} +(21.1850 - 9.41969i) q^{28} +(-16.0454 + 49.3826i) q^{29} +(-6.45909 + 21.7193i) q^{30} +(7.84453 + 10.7971i) q^{31} +(19.4491 + 25.4113i) q^{32} +(24.9244 + 0.214638i) q^{33} +(-14.9059 + 19.4159i) q^{34} +(3.05338 - 28.8197i) q^{35} +(12.9717 - 8.41484i) q^{36} +(4.00077 + 7.85195i) q^{37} +(44.8984 + 24.3922i) q^{38} +(20.3730 - 28.0411i) q^{39} +(39.6660 - 5.15867i) q^{40} +(-75.7259 + 24.6048i) q^{41} +(19.0584 - 18.0768i) q^{42} +(-54.5989 - 54.5989i) q^{43} +(-17.5297 - 40.3573i) q^{44} +(-0.995428 - 19.3019i) q^{45} +(-32.3100 - 0.854028i) q^{46} +(-22.4582 + 44.0767i) q^{47} +(30.3866 + 19.7759i) q^{48} +(9.05424 - 12.4621i) q^{49} +(19.2226 - 46.1573i) q^{50} +(-8.56986 + 26.3753i) q^{51} +(-59.8422 - 12.7506i) q^{52} +(-21.8861 - 3.46642i) q^{53} +(35.5052 - 46.2477i) q^{54} +(-54.6420 - 6.26545i) q^{55} +(-42.8269 - 17.7765i) q^{56} +(57.1780 + 9.05611i) q^{57} +(93.7425 - 44.6846i) q^{58} +(-15.0901 + 46.4424i) q^{59} +(40.4066 - 20.5207i) q^{60} +(55.4322 - 76.2959i) q^{61} +(4.87066 - 26.2437i) q^{62} +(-10.1718 + 19.9633i) q^{63} +(10.1052 - 63.1972i) q^{64} +(-51.2237 + 56.7944i) q^{65} +(-33.9933 - 36.4630i) q^{66} +(-5.11791 - 5.11791i) q^{67} +(48.6899 - 5.09328i) q^{68} +(-34.8268 + 11.3159i) q^{69} +(-46.0225 + 35.2354i) q^{70} +(10.7606 - 14.8107i) q^{71} +(-30.0643 - 7.24129i) q^{72} +(-46.5762 - 91.4110i) q^{73} +(5.88740 - 16.6125i) q^{74} +(3.05911 - 56.5659i) q^{75} +(-26.4008 - 98.7237i) q^{76} +(51.2569 + 37.9190i) q^{77} +(-68.7305 + 9.03133i) q^{78} +(47.7369 + 65.7043i) q^{79} +(-62.0800 - 50.4586i) q^{80} +(9.66242 - 29.7379i) q^{81} +(139.929 + 76.0200i) q^{82} +(9.70086 + 61.2488i) q^{83} +(-52.4621 - 2.77533i) q^{84} +(24.9367 - 55.8832i) q^{85} +(-4.08049 + 154.375i) q^{86} +(83.1957 - 83.1957i) q^{87} +(-34.4348 + 80.9830i) q^{88} +11.2250i q^{89} +(-26.6387 + 28.0107i) q^{90} +(84.3215 - 27.3977i) q^{91} +(43.2307 + 48.0601i) q^{92} +(-4.73074 - 29.8687i) q^{93} +(94.8695 - 28.0762i) q^{94} +(-123.361 - 33.1647i) q^{95} +(-13.1261 - 71.3121i) q^{96} +(7.98092 - 50.3895i) 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 \(\left(\mathbb{Z}/220\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.37635 − 1.45109i −0.688176 − 0.725544i
\(3\) −2.01897 − 1.02872i −0.672990 − 0.342905i 0.0838758 − 0.996476i \(-0.473270\pi\)
−0.756865 + 0.653571i \(0.773270\pi\)
\(4\) −0.211311 + 3.99441i −0.0528277 + 0.998604i
\(5\) 4.19108 + 2.72669i 0.838216 + 0.545338i
\(6\) 1.28605 + 4.34558i 0.214342 + 0.724263i
\(7\) −2.63142 − 5.16446i −0.375918 − 0.737780i 0.623098 − 0.782144i \(-0.285874\pi\)
−0.999015 + 0.0443641i \(0.985874\pi\)
\(8\) 6.08708 − 5.19109i 0.760885 − 0.648886i
\(9\) −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
\(13\) −2.39288 + 15.1080i −0.184068 + 1.16216i 0.706637 + 0.707576i \(0.250211\pi\)
−0.890705 + 0.454582i \(0.849789\pi\)
\(14\) −3.87232 + 10.9265i −0.276594 + 0.780467i
\(15\) −5.65667 − 9.81653i −0.377111 − 0.654436i
\(16\) −15.9107 − 1.68813i −0.994418 − 0.105508i
\(17\) −1.91459 − 12.0882i −0.112623 − 0.711072i −0.977790 − 0.209588i \(-0.932788\pi\)
0.865167 − 0.501484i \(-0.167212\pi\)
\(18\) −1.41074 + 7.60122i −0.0783743 + 0.422290i
\(19\) −24.2978 + 7.89483i −1.27883 + 0.415517i −0.868168 − 0.496270i \(-0.834703\pi\)
−0.410663 + 0.911787i \(0.634703\pi\)
\(20\) −11.7772 + 16.1647i −0.588858 + 0.808237i
\(21\) 13.1339i 0.625422i
\(22\) 20.6723 + 7.52715i 0.939648 + 0.342143i
\(23\) 11.4273 − 11.4273i 0.496839 − 0.496839i −0.413613 − 0.910453i \(-0.635733\pi\)
0.910453 + 0.413613i \(0.135733\pi\)
\(24\) −17.6298 + 4.21877i −0.734574 + 0.175782i
\(25\) 10.1303 + 22.8556i 0.405212 + 0.914223i
\(26\) 25.2165 − 17.3217i 0.969867 − 0.666220i
\(27\) 4.56045 + 28.7936i 0.168906 + 1.06643i
\(28\) 21.1850 − 9.41969i 0.756608 − 0.336417i
\(29\) −16.0454 + 49.3826i −0.553289 + 1.70285i 0.147132 + 0.989117i \(0.452996\pi\)
−0.700421 + 0.713730i \(0.747004\pi\)
\(30\) −6.45909 + 21.7193i −0.215303 + 0.723978i
\(31\) 7.84453 + 10.7971i 0.253049 + 0.348292i 0.916576 − 0.399860i \(-0.130941\pi\)
−0.663527 + 0.748152i \(0.730941\pi\)
\(32\) 19.4491 + 25.4113i 0.607784 + 0.794102i
\(33\) 24.9244 + 0.214638i 0.755286 + 0.00650418i
\(34\) −14.9059 + 19.4159i −0.438410 + 0.571055i
\(35\) 3.05338 − 28.8197i 0.0872394 − 0.823421i
\(36\) 12.9717 − 8.41484i 0.360325 − 0.233746i
\(37\) 4.00077 + 7.85195i 0.108129 + 0.212215i 0.938730 − 0.344652i \(-0.112003\pi\)
−0.830602 + 0.556867i \(0.812003\pi\)
\(38\) 44.8984 + 24.3922i 1.18154 + 0.641899i
\(39\) 20.3730 − 28.0411i 0.522385 − 0.719002i
\(40\) 39.6660 − 5.15867i 0.991649 − 0.128967i
\(41\) −75.7259 + 24.6048i −1.84697 + 0.600118i −0.849621 + 0.527395i \(0.823169\pi\)
−0.997352 + 0.0727233i \(0.976831\pi\)
\(42\) 19.0584 − 18.0768i 0.453771 − 0.430400i
\(43\) −54.5989 − 54.5989i −1.26974 − 1.26974i −0.946222 − 0.323519i \(-0.895134\pi\)
−0.323519 − 0.946222i \(-0.604866\pi\)
\(44\) −17.5297 − 40.3573i −0.398403 − 0.917210i
\(45\) −0.995428 − 19.3019i −0.0221206 − 0.428931i
\(46\) −32.3100 − 0.854028i −0.702391 − 0.0185658i
\(47\) −22.4582 + 44.0767i −0.477834 + 0.937802i 0.518727 + 0.854940i \(0.326406\pi\)
−0.996561 + 0.0828618i \(0.973594\pi\)
\(48\) 30.3866 + 19.7759i 0.633054 + 0.411997i
\(49\) 9.05424 − 12.4621i 0.184780 − 0.254328i
\(50\) 19.2226 − 46.1573i 0.384451 − 0.923145i
\(51\) −8.56986 + 26.3753i −0.168036 + 0.517163i
\(52\) −59.8422 − 12.7506i −1.15081 − 0.245205i
\(53\) −21.8861 − 3.46642i −0.412945 − 0.0654041i −0.0534944 − 0.998568i \(-0.517036\pi\)
−0.359451 + 0.933164i \(0.617036\pi\)
\(54\) 35.5052 − 46.2477i 0.657504 − 0.856439i
\(55\) −54.6420 − 6.26545i −0.993490 − 0.113917i
\(56\) −42.8269 − 17.7765i −0.764765 − 0.317438i
\(57\) 57.1780 + 9.05611i 1.00312 + 0.158879i
\(58\) 93.7425 − 44.6846i 1.61625 − 0.770423i
\(59\) −15.0901 + 46.4424i −0.255764 + 0.787159i 0.737915 + 0.674894i \(0.235811\pi\)
−0.993678 + 0.112265i \(0.964189\pi\)
\(60\) 40.4066 − 20.5207i 0.673444 − 0.342012i
\(61\) 55.4322 − 76.2959i 0.908725 − 1.25075i −0.0588751 − 0.998265i \(-0.518751\pi\)
0.967600 − 0.252487i \(-0.0812486\pi\)
\(62\) 4.87066 − 26.2437i 0.0785590 − 0.423285i
\(63\) −10.1718 + 19.9633i −0.161457 + 0.316877i
\(64\) 10.1052 − 63.1972i 0.157893 − 0.987456i
\(65\) −51.2237 + 56.7944i −0.788057 + 0.873760i
\(66\) −33.9933 − 36.4630i −0.515050 − 0.552469i
\(67\) −5.11791 − 5.11791i −0.0763867 − 0.0763867i 0.667881 − 0.744268i \(-0.267201\pi\)
−0.744268 + 0.667881i \(0.767201\pi\)
\(68\) 48.6899 − 5.09328i 0.716029 − 0.0749012i
\(69\) −34.8268 + 11.3159i −0.504736 + 0.163999i
\(70\) −46.0225 + 35.2354i −0.657464 + 0.503363i
\(71\) 10.7606 − 14.8107i 0.151558 − 0.208602i −0.726486 − 0.687181i \(-0.758848\pi\)
0.878044 + 0.478579i \(0.158848\pi\)
\(72\) −30.0643 − 7.24129i −0.417560 − 0.100573i
\(73\) −46.5762 − 91.4110i −0.638030 − 1.25220i −0.952962 − 0.303090i \(-0.901982\pi\)
0.314932 − 0.949114i \(-0.398018\pi\)
\(74\) 5.88740 − 16.6125i 0.0795595 − 0.224493i
\(75\) 3.05911 − 56.5659i 0.0407882 − 0.754212i
\(76\) −26.4008 − 98.7237i −0.347380 − 1.29900i
\(77\) 51.2569 + 37.9190i 0.665674 + 0.492454i
\(78\) −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 \(0.628156\pi\)
\(80\) −62.0800 − 50.4586i −0.776000 − 0.630733i
\(81\) 9.66242 − 29.7379i 0.119289 − 0.367134i
\(82\) 139.929 + 76.0200i 1.70645 + 0.927073i
\(83\) 9.70086 + 61.2488i 0.116878 + 0.737937i 0.974621 + 0.223860i \(0.0718657\pi\)
−0.857744 + 0.514078i \(0.828134\pi\)
\(84\) −52.4621 − 2.77533i −0.624549 − 0.0330396i
\(85\) 24.9367 − 55.8832i 0.293373 − 0.657449i
\(86\) −4.08049 + 154.375i −0.0474475 + 1.79506i
\(87\) 83.1957 − 83.1957i 0.956273 − 0.956273i
\(88\) −34.4348 + 80.9830i −0.391305 + 0.920261i
\(89\) 11.2250i 0.126124i 0.998010 + 0.0630619i \(0.0200865\pi\)
−0.998010 + 0.0630619i \(0.979913\pi\)
\(90\) −26.6387 + 28.0107i −0.295985 + 0.311230i
\(91\) 84.3215 − 27.3977i 0.926610 − 0.301074i
\(92\) 43.2307 + 48.0601i 0.469899 + 0.522392i
\(93\) −4.73074 − 29.8687i −0.0508682 − 0.321169i
\(94\) 94.8695 − 28.0762i 1.00925 − 0.298683i
\(95\) −123.361 − 33.1647i −1.29853 − 0.349102i
\(96\) −13.1261 − 71.3121i −0.136731 − 0.742835i
\(97\) 7.98092 − 50.3895i 0.0822775 − 0.519480i −0.911785 − 0.410668i \(-0.865295\pi\)
0.994062 − 0.108812i \(-0.0347045\pi\)
\(98\) −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.63.19 yes 544
4.3 odd 2 inner 220.3.w.a.63.11 yes 544
5.2 odd 4 inner 220.3.w.a.107.52 yes 544
11.7 odd 10 inner 220.3.w.a.183.24 yes 544
20.7 even 4 inner 220.3.w.a.107.24 yes 544
44.7 even 10 inner 220.3.w.a.183.52 yes 544
55.7 even 20 inner 220.3.w.a.7.11 ✓ 544
220.7 odd 20 inner 220.3.w.a.7.19 yes 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
220.3.w.a.7.11 ✓ 544 55.7 even 20 inner
220.3.w.a.7.19 yes 544 220.7 odd 20 inner
220.3.w.a.63.11 yes 544 4.3 odd 2 inner
220.3.w.a.63.19 yes 544 1.1 even 1 trivial
220.3.w.a.107.24 yes 544 20.7 even 4 inner
220.3.w.a.107.52 yes 544 5.2 odd 4 inner
220.3.w.a.183.24 yes 544 11.7 odd 10 inner
220.3.w.a.183.52 yes 544 44.7 even 10 inner