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Properties

Label 200.3.u.b.33.6
Level $200$
Weight $3$
Character 200.33
Analytic conductor $5.450$
Analytic rank $0$
Dimension $64$
Inner twists $2$

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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] = [200,3,Mod(17,200)] 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("200.17"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([0, 0, 13])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 200.u (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.44960528721\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 33.6
Character \(\chi\) \(=\) 200.33
Dual form 200.3.u.b.97.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.33419 + 0.211315i) q^{3} +(-4.90670 + 0.961386i) q^{5} +(-5.78356 + 5.78356i) q^{7} +(-6.82409 - 2.21728i) q^{9} +(-5.64771 - 17.3819i) q^{11} +(-8.75031 + 4.45850i) q^{13} +(-6.74965 + 0.245813i) q^{15} +(13.2695 - 2.10169i) q^{17} +(-13.9237 + 19.1644i) q^{19} +(-8.93854 + 6.49423i) q^{21} +(-2.96303 + 5.81527i) q^{23} +(23.1515 - 9.43447i) q^{25} +(-19.4684 - 9.91967i) q^{27} +(17.0950 + 23.5293i) q^{29} +(-25.1552 - 18.2763i) q^{31} +(-3.86208 - 24.3842i) q^{33} +(22.8180 - 33.9384i) q^{35} +(9.97352 + 19.5741i) q^{37} +(-12.6168 + 4.09943i) q^{39} +(-18.3883 + 56.5935i) q^{41} +(-51.6910 - 51.6910i) q^{43} +(35.6155 + 4.31896i) q^{45} +(-5.40431 + 34.1215i) q^{47} -17.8991i q^{49} +18.1482 q^{51} +(87.5908 + 13.8730i) q^{53} +(44.4223 + 79.8581i) q^{55} +(-22.6267 + 22.6267i) q^{57} +(-20.7741 - 6.74992i) q^{59} +(-36.8620 - 113.449i) q^{61} +(52.2913 - 26.6438i) q^{63} +(38.6488 - 30.2890i) q^{65} +(47.9275 - 7.59097i) q^{67} +(-5.18210 + 7.13255i) q^{69} +(-7.81903 + 5.68086i) q^{71} +(16.0773 - 31.5535i) q^{73} +(32.8822 - 7.69514i) q^{75} +(133.193 + 67.8652i) q^{77} +(51.6285 + 71.0605i) q^{79} +(28.3658 + 20.6089i) q^{81} +(-8.66416 - 54.7034i) q^{83} +(-63.0891 + 23.0695i) q^{85} +(17.8359 + 35.0050i) q^{87} +(-70.5667 + 22.9285i) q^{89} +(24.8219 - 76.3939i) q^{91} +(-29.6998 - 29.6998i) q^{93} +(49.8953 - 107.420i) q^{95} +(-8.90175 + 56.2034i) q^{97} +131.138i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 64 q + 6 q^{5} - 4 q^{7} - 40 q^{9} + 16 q^{11} + 24 q^{13} + 82 q^{15} - 8 q^{17} - 50 q^{19} - 100 q^{21} - 48 q^{23} - 200 q^{25} + 90 q^{27} + 108 q^{31} + 260 q^{33} + 2 q^{35} - 94 q^{37} + 320 q^{39}+ \cdots - 544 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{20}\right)\)

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 0
\(3\) 1.33419 + 0.211315i 0.444731 + 0.0704385i 0.374784 − 0.927112i \(-0.377717\pi\)
0.0699472 + 0.997551i \(0.477717\pi\)
\(4\) 0 0
\(5\) −4.90670 + 0.961386i −0.981341 + 0.192277i
\(6\) 0 0
\(7\) −5.78356 + 5.78356i −0.826223 + 0.826223i −0.986992 − 0.160769i \(-0.948602\pi\)
0.160769 + 0.986992i \(0.448602\pi\)
\(8\) 0 0
\(9\) −6.82409 − 2.21728i −0.758232 − 0.246365i
\(10\) 0 0
\(11\) −5.64771 − 17.3819i −0.513428 − 1.58017i −0.786123 − 0.618070i \(-0.787915\pi\)
0.272695 − 0.962101i \(-0.412085\pi\)
\(12\) 0 0
\(13\) −8.75031 + 4.45850i −0.673101 + 0.342962i −0.756909 − 0.653520i \(-0.773292\pi\)
0.0838087 + 0.996482i \(0.473292\pi\)
\(14\) 0 0
\(15\) −6.74965 + 0.245813i −0.449976 + 0.0163875i
\(16\) 0 0
\(17\) 13.2695 − 2.10169i 0.780561 − 0.123629i 0.246577 − 0.969123i \(-0.420694\pi\)
0.533984 + 0.845495i \(0.320694\pi\)
\(18\) 0 0
\(19\) −13.9237 + 19.1644i −0.732829 + 1.00865i 0.266171 + 0.963926i \(0.414242\pi\)
−0.998999 + 0.0447261i \(0.985758\pi\)
\(20\) 0 0
\(21\) −8.93854 + 6.49423i −0.425645 + 0.309249i
\(22\) 0 0
\(23\) −2.96303 + 5.81527i −0.128827 + 0.252838i −0.946406 − 0.322979i \(-0.895316\pi\)
0.817579 + 0.575816i \(0.195316\pi\)
\(24\) 0 0
\(25\) 23.1515 − 9.43447i 0.926059 − 0.377379i
\(26\) 0 0
\(27\) −19.4684 − 9.91967i −0.721054 − 0.367395i
\(28\) 0 0
\(29\) 17.0950 + 23.5293i 0.589483 + 0.811354i 0.994695 − 0.102869i \(-0.0328023\pi\)
−0.405212 + 0.914223i \(0.632802\pi\)
\(30\) 0 0
\(31\) −25.1552 − 18.2763i −0.811456 − 0.589558i 0.102796 − 0.994702i \(-0.467221\pi\)
−0.914253 + 0.405145i \(0.867221\pi\)
\(32\) 0 0
\(33\) −3.86208 − 24.3842i −0.117033 − 0.738916i
\(34\) 0 0
\(35\) 22.8180 − 33.9384i 0.651942 − 0.969670i
\(36\) 0 0
\(37\) 9.97352 + 19.5741i 0.269555 + 0.529031i 0.985615 − 0.169008i \(-0.0540564\pi\)
−0.716060 + 0.698039i \(0.754056\pi\)
\(38\) 0 0
\(39\) −12.6168 + 4.09943i −0.323506 + 0.105114i
\(40\) 0 0
\(41\) −18.3883 + 56.5935i −0.448496 + 1.38033i 0.430108 + 0.902777i \(0.358475\pi\)
−0.878604 + 0.477551i \(0.841525\pi\)
\(42\) 0 0
\(43\) −51.6910 − 51.6910i −1.20212 − 1.20212i −0.973522 − 0.228594i \(-0.926587\pi\)
−0.228594 − 0.973522i \(-0.573413\pi\)
\(44\) 0 0
\(45\) 35.6155 + 4.31896i 0.791455 + 0.0959768i
\(46\) 0 0
\(47\) −5.40431 + 34.1215i −0.114985 + 0.725989i 0.861073 + 0.508481i \(0.169793\pi\)
−0.976059 + 0.217508i \(0.930207\pi\)
\(48\) 0 0
\(49\) − 17.8991i − 0.365288i
\(50\) 0 0
\(51\) 18.1482 0.355848
\(52\) 0 0
\(53\) 87.5908 + 13.8730i 1.65266 + 0.261755i 0.912018 − 0.410149i \(-0.134523\pi\)
0.740638 + 0.671904i \(0.234523\pi\)
\(54\) 0 0
\(55\) 44.4223 + 79.8581i 0.807679 + 1.45196i
\(56\) 0 0
\(57\) −22.6267 + 22.6267i −0.396960 + 0.396960i
\(58\) 0 0
\(59\) −20.7741 − 6.74992i −0.352104 − 0.114405i 0.127625 − 0.991823i \(-0.459265\pi\)
−0.479728 + 0.877417i \(0.659265\pi\)
\(60\) 0 0
\(61\) −36.8620 − 113.449i −0.604294 − 1.85983i −0.501570 − 0.865117i \(-0.667244\pi\)
−0.102724 − 0.994710i \(-0.532756\pi\)
\(62\) 0 0
\(63\) 52.2913 − 26.6438i 0.830021 − 0.422917i
\(64\) 0 0
\(65\) 38.6488 − 30.2890i 0.594597 − 0.465984i
\(66\) 0 0
\(67\) 47.9275 − 7.59097i 0.715335 − 0.113298i 0.211851 − 0.977302i \(-0.432051\pi\)
0.503485 + 0.864004i \(0.332051\pi\)
\(68\) 0 0
\(69\) −5.18210 + 7.13255i −0.0751030 + 0.103370i
\(70\) 0 0
\(71\) −7.81903 + 5.68086i −0.110127 + 0.0800121i −0.641486 − 0.767135i \(-0.721682\pi\)
0.531359 + 0.847147i \(0.321682\pi\)
\(72\) 0 0
\(73\) 16.0773 − 31.5535i 0.220237 − 0.432240i −0.754280 − 0.656553i \(-0.772014\pi\)
0.974517 + 0.224313i \(0.0720138\pi\)
\(74\) 0 0
\(75\) 32.8822 − 7.69514i 0.438429 − 0.102602i
\(76\) 0 0
\(77\) 133.193 + 67.8652i 1.72978 + 0.881366i
\(78\) 0 0
\(79\) 51.6285 + 71.0605i 0.653525 + 0.899500i 0.999246 − 0.0388373i \(-0.0123654\pi\)
−0.345720 + 0.938338i \(0.612365\pi\)
\(80\) 0 0
\(81\) 28.3658 + 20.6089i 0.350195 + 0.254431i
\(82\) 0 0
\(83\) −8.66416 − 54.7034i −0.104388 − 0.659077i −0.983286 − 0.182068i \(-0.941721\pi\)
0.878898 − 0.477009i \(-0.158279\pi\)
\(84\) 0 0
\(85\) −63.0891 + 23.0695i −0.742225 + 0.271406i
\(86\) 0 0
\(87\) 17.8359 + 35.0050i 0.205011 + 0.402356i
\(88\) 0 0
\(89\) −70.5667 + 22.9285i −0.792885 + 0.257624i −0.677332 − 0.735678i \(-0.736864\pi\)
−0.115553 + 0.993301i \(0.536864\pi\)
\(90\) 0 0
\(91\) 24.8219 − 76.3939i 0.272768 − 0.839494i
\(92\) 0 0
\(93\) −29.6998 − 29.6998i −0.319352 − 0.319352i
\(94\) 0 0
\(95\) 49.8953 − 107.420i 0.525214 − 1.13074i
\(96\) 0 0
\(97\) −8.90175 + 56.2034i −0.0917706 + 0.579417i 0.898359 + 0.439262i \(0.144760\pi\)
−0.990130 + 0.140155i \(0.955240\pi\)
\(98\) 0 0
\(99\) 131.138i 1.32463i
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 200.3.u.b.33.6 ✓ 64
4.3 odd 2 400.3.bg.f.33.3 64
25.22 odd 20 inner 200.3.u.b.97.6 yes 64
100.47 even 20 400.3.bg.f.97.3 64
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.3.u.b.33.6 ✓ 64 1.1 even 1 trivial
200.3.u.b.97.6 yes 64 25.22 odd 20 inner
400.3.bg.f.33.3 64 4.3 odd 2
400.3.bg.f.97.3 64 100.47 even 20