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Properties

Label 555.2.bp.a.4.16
Level $555$
Weight $2$
Character 555.4
Analytic conductor $4.432$
Analytic rank $0$
Dimension $240$
Inner twists $4$

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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] = [555,2,Mod(4,555)] 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("555.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(555, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 9, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.bp (of order \(18\), degree \(6\), 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: \(4.43169731218\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(40\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 4.16
Character \(\chi\) \(=\) 555.4
Dual form 555.2.bp.a.139.16

$q$-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+(-0.169137 + 0.959222i) q^{2} +(0.984808 - 0.173648i) q^{3} +(0.987886 + 0.359561i) q^{4} +(-1.41362 + 1.73254i) q^{5} +0.974019i q^{6} +(0.930024 + 1.10836i) q^{7} +(-1.48601 + 2.57384i) q^{8} +(0.939693 - 0.342020i) q^{9} +(-1.42279 - 1.64901i) q^{10} +(0.000868333 - 0.00150400i) q^{11} +(1.03532 + 0.182554i) q^{12} +(1.40244 + 0.510445i) q^{13} +(-1.22046 + 0.704635i) q^{14} +(-1.09129 + 1.95169i) q^{15} +(-0.606879 - 0.509232i) q^{16} +(-2.64947 + 0.964327i) q^{17} +(0.169137 + 0.959222i) q^{18} +(-0.546064 + 0.0962858i) q^{19} +(-2.01945 + 1.20327i) q^{20} +(1.10836 + 0.930024i) q^{21} +(0.00129580 + 0.00108730i) q^{22} +(-0.661629 - 1.14597i) q^{23} +(-1.01649 + 2.79278i) q^{24} +(-1.00337 - 4.89829i) q^{25} +(-0.726833 + 1.25891i) q^{26} +(0.866025 - 0.500000i) q^{27} +(0.520235 + 1.42933i) q^{28} +(0.621640 + 0.358904i) q^{29} +(-1.68752 - 1.37689i) q^{30} -0.00328641i q^{31} +(-3.96227 + 3.32474i) q^{32} +(0.000593975 - 0.00163193i) q^{33} +(-0.476881 - 2.70453i) q^{34} +(-3.23497 + 0.0445054i) q^{35} +1.05129 q^{36} +(1.22843 + 5.95743i) q^{37} -0.540082i q^{38} +(1.46977 + 0.259160i) q^{39} +(-2.35863 - 6.21298i) q^{40} +(6.58428 + 2.39648i) q^{41} +(-1.07956 + 0.905861i) q^{42} +6.09566 q^{43} +(0.00139859 - 0.00117356i) q^{44} +(-0.735803 + 2.11154i) q^{45} +(1.21115 - 0.440822i) q^{46} +(-0.266099 + 0.153632i) q^{47} +(-0.686086 - 0.396112i) q^{48} +(0.852021 - 4.83205i) q^{49} +(4.86825 - 0.133976i) q^{50} +(-2.44176 + 1.40975i) q^{51} +(1.20191 + 1.00852i) q^{52} +(4.42328 - 5.27146i) q^{53} +(0.333134 + 0.915279i) q^{54} +(0.00137824 + 0.00363050i) q^{55} +(-4.23476 + 0.746702i) q^{56} +(-0.521048 + 0.189646i) q^{57} +(-0.449410 + 0.535586i) q^{58} +(-0.616300 + 0.734477i) q^{59} +(-1.77982 + 1.53566i) q^{60} +(-1.53451 + 4.21603i) q^{61} +(0.00315240 + 0.000555852i) q^{62} +(1.25302 + 0.723430i) q^{63} +(-3.31122 - 5.73521i) q^{64} +(-2.86687 + 1.70820i) q^{65} +(0.00146492 + 0.000845773i) q^{66} +(-0.160337 - 0.191082i) q^{67} -2.96411 q^{68} +(-0.850574 - 1.01367i) q^{69} +(0.504462 - 3.11058i) q^{70} +(0.802621 + 4.55189i) q^{71} +(-0.516084 + 2.92686i) q^{72} -3.16867i q^{73} +(-5.92227 + 0.170715i) q^{74} +(-1.83871 - 4.64964i) q^{75} +(-0.574069 - 0.101224i) q^{76} +(0.00247454 - 0.000436328i) q^{77} +(-0.497183 + 1.36600i) q^{78} +(-0.649181 - 0.773664i) q^{79} +(1.74016 - 0.331581i) q^{80} +(0.766044 - 0.642788i) q^{81} +(-3.41240 + 5.91045i) q^{82} +(-5.22929 - 14.3674i) q^{83} +(0.760533 + 1.31728i) q^{84} +(2.07460 - 5.95349i) q^{85} +(-1.03100 + 5.84709i) q^{86} +(0.674518 + 0.245505i) q^{87} +(0.00258070 + 0.00446990i) q^{88} +(5.07035 - 6.04261i) q^{89} +(-1.90098 - 1.06294i) q^{90} +(0.738543 + 2.02913i) q^{91} +(-0.241566 - 1.36999i) q^{92} +(-0.000570679 - 0.00323648i) q^{93} +(-0.102360 - 0.281233i) q^{94} +(0.605106 - 1.08219i) q^{95} +(-3.32474 + 3.96227i) q^{96} +(-1.57793 - 2.73305i) q^{97} +(4.49090 + 1.63455i) q^{98} +(0.000301569 - 0.00171028i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 12 q^{5} + 36 q^{14} + 12 q^{19} - 30 q^{20} + 12 q^{21} - 48 q^{25} - 36 q^{26} + 24 q^{30} + 48 q^{34} + 12 q^{35} - 264 q^{36} + 36 q^{39} + 66 q^{40} - 144 q^{41} + 192 q^{44} + 24 q^{46} - 12 q^{49}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/555\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\) \(371\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.169137 + 0.959222i −0.119598 + 0.678272i 0.864773 + 0.502163i \(0.167462\pi\)
−0.984371 + 0.176109i \(0.943649\pi\)
\(3\) 0.984808 − 0.173648i 0.568579 − 0.100256i
\(4\) 0.987886 + 0.359561i 0.493943 + 0.179781i
\(5\) −1.41362 + 1.73254i −0.632189 + 0.774814i
\(6\) 0.974019i 0.397642i
\(7\) 0.930024 + 1.10836i 0.351516 + 0.418921i 0.912610 − 0.408832i \(-0.134064\pi\)
−0.561094 + 0.827752i \(0.689619\pi\)
\(8\) −1.48601 + 2.57384i −0.525382 + 0.909989i
\(9\) 0.939693 − 0.342020i 0.313231 − 0.114007i
\(10\) −1.42279 − 1.64901i −0.449927 − 0.521462i
\(11\) 0.000868333 − 0.00150400i 0.000261812 − 0.000453472i −0.865894 − 0.500227i \(-0.833250\pi\)
0.866156 + 0.499773i \(0.166583\pi\)
\(12\) 1.03532 + 0.182554i 0.298870 + 0.0526988i
\(13\) 1.40244 + 0.510445i 0.388966 + 0.141572i 0.529098 − 0.848561i \(-0.322530\pi\)
−0.140132 + 0.990133i \(0.544753\pi\)
\(14\) −1.22046 + 0.704635i −0.326183 + 0.188322i
\(15\) −1.09129 + 1.95169i −0.281770 + 0.503924i
\(16\) −0.606879 − 0.509232i −0.151720 − 0.127308i
\(17\) −2.64947 + 0.964327i −0.642590 + 0.233884i −0.642702 − 0.766116i \(-0.722187\pi\)
0.000111815 1.00000i \(0.499964\pi\)
\(18\) 0.169137 + 0.959222i 0.0398659 + 0.226091i
\(19\) −0.546064 + 0.0962858i −0.125276 + 0.0220895i −0.235934 − 0.971769i \(-0.575815\pi\)
0.110659 + 0.993858i \(0.464704\pi\)
\(20\) −2.01945 + 1.20327i −0.451562 + 0.269059i
\(21\) 1.10836 + 0.930024i 0.241864 + 0.202948i
\(22\) 0.00129580 + 0.00108730i 0.000276265 + 0.000231814i
\(23\) −0.661629 − 1.14597i −0.137959 − 0.238952i 0.788765 − 0.614695i \(-0.210721\pi\)
−0.926724 + 0.375743i \(0.877388\pi\)
\(24\) −1.01649 + 2.79278i −0.207490 + 0.570073i
\(25\) −1.00337 − 4.89829i −0.200674 − 0.979658i
\(26\) −0.726833 + 1.25891i −0.142544 + 0.246893i
\(27\) 0.866025 − 0.500000i 0.166667 − 0.0962250i
\(28\) 0.520235 + 1.42933i 0.0983151 + 0.270119i
\(29\) 0.621640 + 0.358904i 0.115436 + 0.0666468i 0.556606 − 0.830777i \(-0.312103\pi\)
−0.441171 + 0.897423i \(0.645437\pi\)
\(30\) −1.68752 − 1.37689i −0.308098 − 0.251385i
\(31\) − 0.00328641i − 0.000590257i −1.00000 0.000295128i \(-0.999906\pi\)
1.00000 0.000295128i \(-9.39423e-5\pi\)
\(32\) −3.96227 + 3.32474i −0.700438 + 0.587737i
\(33\) 0.000593975 − 0.00163193i 0.000103398 − 0.000284083i
\(34\) −0.476881 − 2.70453i −0.0817845 − 0.463823i
\(35\) −3.23497 + 0.0445054i −0.546810 + 0.00752278i
\(36\) 1.05129 0.175214
\(37\) 1.22843 + 5.95743i 0.201952 + 0.979395i
\(38\) − 0.540082i − 0.0876128i
\(39\) 1.46977 + 0.259160i 0.235351 + 0.0414988i
\(40\) −2.35863 − 6.21298i −0.372932 − 0.982359i
\(41\) 6.58428 + 2.39648i 1.02829 + 0.374268i 0.800430 − 0.599426i \(-0.204604\pi\)
0.227862 + 0.973694i \(0.426827\pi\)
\(42\) −1.07956 + 0.905861i −0.166580 + 0.139777i
\(43\) 6.09566 0.929580 0.464790 − 0.885421i \(-0.346130\pi\)
0.464790 + 0.885421i \(0.346130\pi\)
\(44\) 0.00139859 − 0.00117356i 0.000210846 − 0.000176921i
\(45\) −0.735803 + 2.11154i −0.109687 + 0.314770i
\(46\) 1.21115 − 0.440822i 0.178574 − 0.0649957i
\(47\) −0.266099 + 0.153632i −0.0388146 + 0.0224096i −0.519282 − 0.854603i \(-0.673800\pi\)
0.480467 + 0.877013i \(0.340467\pi\)
\(48\) −0.686086 − 0.396112i −0.0990280 − 0.0571738i
\(49\) 0.852021 − 4.83205i 0.121717 − 0.690293i
\(50\) 4.86825 − 0.133976i 0.688475 − 0.0189471i
\(51\) −2.44176 + 1.40975i −0.341915 + 0.197405i
\(52\) 1.20191 + 1.00852i 0.166675 + 0.139857i
\(53\) 4.42328 − 5.27146i 0.607584 − 0.724091i −0.371298 − 0.928514i \(-0.621087\pi\)
0.978883 + 0.204423i \(0.0655317\pi\)
\(54\) 0.333134 + 0.915279i 0.0453338 + 0.124554i
\(55\) 0.00137824 + 0.00363050i 0.000185842 + 0.000489536i
\(56\) −4.23476 + 0.746702i −0.565893 + 0.0997823i
\(57\) −0.521048 + 0.189646i −0.0690145 + 0.0251192i
\(58\) −0.449410 + 0.535586i −0.0590105 + 0.0703259i
\(59\) −0.616300 + 0.734477i −0.0802354 + 0.0956208i −0.804669 − 0.593724i \(-0.797657\pi\)
0.724433 + 0.689345i \(0.242102\pi\)
\(60\) −1.77982 + 1.53566i −0.229774 + 0.198253i
\(61\) −1.53451 + 4.21603i −0.196474 + 0.539808i −0.998334 − 0.0577045i \(-0.981622\pi\)
0.801860 + 0.597512i \(0.203844\pi\)
\(62\) 0.00315240 0.000555852i 0.000400355 7.05933e-5i
\(63\) 1.25302 + 0.723430i 0.157865 + 0.0911436i
\(64\) −3.31122 − 5.73521i −0.413903 − 0.716901i
\(65\) −2.86687 + 1.70820i −0.355592 + 0.211876i
\(66\) 0.00146492 0.000845773i 0.000180319 0.000104107i
\(67\) −0.160337 − 0.191082i −0.0195883 − 0.0233444i 0.756161 − 0.654385i \(-0.227072\pi\)
−0.775750 + 0.631041i \(0.782628\pi\)
\(68\) −2.96411 −0.359451
\(69\) −0.850574 − 1.01367i −0.102397 − 0.122032i
\(70\) 0.504462 − 3.11058i 0.0602947 − 0.371786i
\(71\) 0.802621 + 4.55189i 0.0952536 + 0.540210i 0.994669 + 0.103116i \(0.0328813\pi\)
−0.899416 + 0.437094i \(0.856008\pi\)
\(72\) −0.516084 + 2.92686i −0.0608211 + 0.344934i
\(73\) − 3.16867i − 0.370865i −0.982657 − 0.185433i \(-0.940631\pi\)
0.982657 − 0.185433i \(-0.0593687\pi\)
\(74\) −5.92227 + 0.170715i −0.688450 + 0.0198453i
\(75\) −1.83871 − 4.64964i −0.212316 − 0.536894i
\(76\) −0.574069 − 0.101224i −0.0658503 − 0.0116112i
\(77\) 0.00247454 0.000436328i 0.000282000 4.97242e-5i
\(78\) −0.497183 + 1.36600i −0.0562949 + 0.154669i
\(79\) −0.649181 − 0.773664i −0.0730386 − 0.0870440i 0.728288 − 0.685271i \(-0.240316\pi\)
−0.801326 + 0.598227i \(0.795872\pi\)
\(80\) 1.74016 − 0.331581i 0.194555 − 0.0370719i
\(81\) 0.766044 − 0.642788i 0.0851160 − 0.0714208i
\(82\) −3.41240 + 5.91045i −0.376837 + 0.652700i
\(83\) −5.22929 − 14.3674i −0.573989 − 1.57702i −0.798142 − 0.602469i \(-0.794184\pi\)
0.224153 − 0.974554i \(-0.428038\pi\)
\(84\) 0.760533 + 1.31728i 0.0829809 + 0.143727i
\(85\) 2.07460 − 5.95349i 0.225022 − 0.645747i
\(86\) −1.03100 + 5.84709i −0.111176 + 0.630508i
\(87\) 0.674518 + 0.245505i 0.0723160 + 0.0263209i
\(88\) 0.00258070 + 0.00446990i 0.000275103 + 0.000476493i
\(89\) 5.07035 − 6.04261i 0.537456 − 0.640515i −0.427159 − 0.904176i \(-0.640486\pi\)
0.964615 + 0.263661i \(0.0849301\pi\)
\(90\) −1.90098 − 1.06294i −0.200381 − 0.112043i
\(91\) 0.738543 + 2.02913i 0.0774203 + 0.212711i
\(92\) −0.241566 − 1.36999i −0.0251850 − 0.142831i
\(93\) −0.000570679 − 0.00323648i −5.91767e−5 − 0.000335608i
\(94\) −0.102360 − 0.281233i −0.0105577 − 0.0290070i
\(95\) 0.605106 − 1.08219i 0.0620826 − 0.111030i
\(96\) −3.32474 + 3.96227i −0.339330 + 0.404398i
\(97\) −1.57793 − 2.73305i −0.160214 − 0.277500i 0.774731 − 0.632291i \(-0.217885\pi\)
−0.934946 + 0.354791i \(0.884552\pi\)
\(98\) 4.49090 + 1.63455i 0.453650 + 0.165115i
\(99\) 0.000301569 − 0.00171028i 3.03088e−5 − 0.000171890i
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 555.2.bp.a.4.16 ✓ 240
5.4 even 2 inner 555.2.bp.a.4.25 yes 240
37.28 even 18 inner 555.2.bp.a.139.25 yes 240
185.139 even 18 inner 555.2.bp.a.139.16 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.bp.a.4.16 ✓ 240 1.1 even 1 trivial
555.2.bp.a.4.25 yes 240 5.4 even 2 inner
555.2.bp.a.139.16 yes 240 185.139 even 18 inner
555.2.bp.a.139.25 yes 240 37.28 even 18 inner