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Properties

Label 200.3.u.b.73.1
Level $200$
Weight $3$
Character 200.73
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 73.1
Character \(\chi\) \(=\) 200.73
Dual form 200.3.u.b.137.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.33681 + 2.20972i) q^{3} +(0.913560 - 4.91583i) q^{5} +(-4.23023 + 4.23023i) q^{7} +(8.63502 - 11.8851i) q^{9} +(5.29425 - 3.84650i) q^{11} +(-0.370366 - 2.33840i) q^{13} +(6.90065 + 23.3377i) q^{15} +(22.0546 + 11.2374i) q^{17} +(25.7489 - 8.36632i) q^{19} +(8.99811 - 27.6933i) q^{21} +(15.6106 + 2.47248i) q^{23} +(-23.3308 - 8.98182i) q^{25} +(-4.33305 + 27.3578i) q^{27} +(38.4796 + 12.5028i) q^{29} +(-1.53159 - 4.71375i) q^{31} +(-14.4605 + 28.3803i) q^{33} +(16.9305 + 24.6597i) q^{35} +(32.3795 - 5.12841i) q^{37} +(6.77341 + 9.32279i) q^{39} +(-65.8117 - 47.8150i) q^{41} +(-43.9478 - 43.9478i) q^{43} +(-50.5365 - 53.3061i) q^{45} +(-1.81688 - 3.56582i) q^{47} +13.2103i q^{49} -120.478 q^{51} +(-7.55410 + 3.84901i) q^{53} +(-14.0721 - 29.5397i) q^{55} +(-93.1809 + 93.1809i) q^{57} +(53.8981 - 74.1844i) q^{59} +(74.4356 - 54.0806i) q^{61} +(13.7485 + 86.8048i) q^{63} +(-11.8335 - 0.315612i) q^{65} +(-21.9187 - 11.1681i) q^{67} +(-73.1639 + 23.7724i) q^{69} +(-37.5374 + 115.528i) q^{71} +(115.118 + 18.2329i) q^{73} +(121.029 - 12.6020i) q^{75} +(-6.12433 + 38.6675i) q^{77} +(70.2432 + 22.8234i) q^{79} +(-0.804034 - 2.47456i) q^{81} +(-50.1266 + 98.3789i) q^{83} +(75.3893 - 98.1507i) q^{85} +(-194.506 + 30.8068i) q^{87} +(-9.28274 - 12.7766i) q^{89} +(11.4587 + 8.32524i) q^{91} +(17.0583 + 17.0583i) q^{93} +(-17.6043 - 134.220i) q^{95} +(-17.2696 - 33.8934i) q^{97} -96.1373i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 64 q + 6 q^{5} - 4 q^{7} - 40 q^{9} + 16 q^{11} + 24 q^{13} + 82 q^{15} - 8 q^{17} - 50 q^{19} - 100 q^{21} - 48 q^{23} - 200 q^{25} + 90 q^{27} + 108 q^{31} + 260 q^{33} + 2 q^{35} - 94 q^{37} + 320 q^{39}+ \cdots - 544 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{11}{20}\right)\)

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
\(3\) −4.33681 + 2.20972i −1.44560 + 0.736572i −0.988269 − 0.152723i \(-0.951196\pi\)
−0.457335 + 0.889295i \(0.651196\pi\)
\(4\) 0 0
\(5\) 0.913560 − 4.91583i 0.182712 − 0.983166i
\(6\) 0 0
\(7\) −4.23023 + 4.23023i −0.604319 + 0.604319i −0.941456 − 0.337137i \(-0.890542\pi\)
0.337137 + 0.941456i \(0.390542\pi\)
\(8\) 0 0
\(9\) 8.63502 − 11.8851i 0.959447 − 1.32057i
\(10\) 0 0
\(11\) 5.29425 − 3.84650i 0.481296 − 0.349682i −0.320531 − 0.947238i \(-0.603861\pi\)
0.801827 + 0.597556i \(0.203861\pi\)
\(12\) 0 0
\(13\) −0.370366 − 2.33840i −0.0284897 − 0.179877i 0.969340 − 0.245723i \(-0.0790254\pi\)
−0.997830 + 0.0658461i \(0.979025\pi\)
\(14\) 0 0
\(15\) 6.90065 + 23.3377i 0.460044 + 1.55585i
\(16\) 0 0
\(17\) 22.0546 + 11.2374i 1.29733 + 0.661022i 0.959904 − 0.280330i \(-0.0904438\pi\)
0.337426 + 0.941352i \(0.390444\pi\)
\(18\) 0 0
\(19\) 25.7489 − 8.36632i 1.35521 − 0.440333i 0.460766 − 0.887522i \(-0.347575\pi\)
0.894439 + 0.447189i \(0.147575\pi\)
\(20\) 0 0
\(21\) 8.99811 − 27.6933i 0.428481 − 1.31873i
\(22\) 0 0
\(23\) 15.6106 + 2.47248i 0.678724 + 0.107499i 0.486272 − 0.873807i \(-0.338356\pi\)
0.192451 + 0.981307i \(0.438356\pi\)
\(24\) 0 0
\(25\) −23.3308 − 8.98182i −0.933233 − 0.359273i
\(26\) 0 0
\(27\) −4.33305 + 27.3578i −0.160483 + 1.01325i
\(28\) 0 0
\(29\) 38.4796 + 12.5028i 1.32688 + 0.431130i 0.884853 − 0.465871i \(-0.154259\pi\)
0.442029 + 0.897001i \(0.354259\pi\)
\(30\) 0 0
\(31\) −1.53159 − 4.71375i −0.0494062 − 0.152057i 0.923310 − 0.384056i \(-0.125473\pi\)
−0.972716 + 0.232000i \(0.925473\pi\)
\(32\) 0 0
\(33\) −14.4605 + 28.3803i −0.438197 + 0.860010i
\(34\) 0 0
\(35\) 16.9305 + 24.6597i 0.483730 + 0.704563i
\(36\) 0 0
\(37\) 32.3795 − 5.12841i 0.875122 − 0.138606i 0.297315 − 0.954779i \(-0.403909\pi\)
0.577807 + 0.816174i \(0.303909\pi\)
\(38\) 0 0
\(39\) 6.77341 + 9.32279i 0.173677 + 0.239046i
\(40\) 0 0
\(41\) −65.8117 − 47.8150i −1.60516 − 1.16622i −0.876571 − 0.481272i \(-0.840175\pi\)
−0.728592 − 0.684948i \(-0.759825\pi\)
\(42\) 0 0
\(43\) −43.9478 − 43.9478i −1.02204 − 1.02204i −0.999752 − 0.0222914i \(-0.992904\pi\)
−0.0222914 − 0.999752i \(-0.507096\pi\)
\(44\) 0 0
\(45\) −50.5365 − 53.3061i −1.12303 − 1.18458i
\(46\) 0 0
\(47\) −1.81688 − 3.56582i −0.0386570 − 0.0758686i 0.870870 − 0.491513i \(-0.163556\pi\)
−0.909527 + 0.415645i \(0.863556\pi\)
\(48\) 0 0
\(49\) 13.2103i 0.269597i
\(50\) 0 0
\(51\) −120.478 −2.36231
\(52\) 0 0
\(53\) −7.55410 + 3.84901i −0.142530 + 0.0726228i −0.523801 − 0.851841i \(-0.675487\pi\)
0.381271 + 0.924463i \(0.375487\pi\)
\(54\) 0 0
\(55\) −14.0721 − 29.5397i −0.255857 − 0.537085i
\(56\) 0 0
\(57\) −93.1809 + 93.1809i −1.63475 + 1.63475i
\(58\) 0 0
\(59\) 53.8981 − 74.1844i 0.913527 − 1.25736i −0.0524204 − 0.998625i \(-0.516694\pi\)
0.965948 − 0.258737i \(-0.0833064\pi\)
\(60\) 0 0
\(61\) 74.4356 − 54.0806i 1.22026 − 0.886567i 0.224134 − 0.974558i \(-0.428045\pi\)
0.996121 + 0.0879911i \(0.0280447\pi\)
\(62\) 0 0
\(63\) 13.7485 + 86.8048i 0.218231 + 1.37785i
\(64\) 0 0
\(65\) −11.8335 − 0.315612i −0.182054 − 0.00485556i
\(66\) 0 0
\(67\) −21.9187 − 11.1681i −0.327145 − 0.166689i 0.282702 − 0.959208i \(-0.408769\pi\)
−0.609847 + 0.792519i \(0.708769\pi\)
\(68\) 0 0
\(69\) −73.1639 + 23.7724i −1.06035 + 0.344527i
\(70\) 0 0
\(71\) −37.5374 + 115.528i −0.528695 + 1.62716i 0.228195 + 0.973615i \(0.426718\pi\)
−0.756891 + 0.653542i \(0.773282\pi\)
\(72\) 0 0
\(73\) 115.118 + 18.2329i 1.57696 + 0.249765i 0.882691 − 0.469954i \(-0.155730\pi\)
0.694265 + 0.719720i \(0.255730\pi\)
\(74\) 0 0
\(75\) 121.029 − 12.6020i 1.61371 − 0.168027i
\(76\) 0 0
\(77\) −6.12433 + 38.6675i −0.0795368 + 0.502176i
\(78\) 0 0
\(79\) 70.2432 + 22.8234i 0.889154 + 0.288904i 0.717753 − 0.696298i \(-0.245171\pi\)
0.171401 + 0.985201i \(0.445171\pi\)
\(80\) 0 0
\(81\) −0.804034 − 2.47456i −0.00992635 − 0.0305502i
\(82\) 0 0
\(83\) −50.1266 + 98.3789i −0.603934 + 1.18529i 0.363364 + 0.931647i \(0.381628\pi\)
−0.967298 + 0.253641i \(0.918372\pi\)
\(84\) 0 0
\(85\) 75.3893 − 98.1507i 0.886933 − 1.15471i
\(86\) 0 0
\(87\) −194.506 + 30.8068i −2.23570 + 0.354101i
\(88\) 0 0
\(89\) −9.28274 − 12.7766i −0.104300 − 0.143557i 0.753676 − 0.657246i \(-0.228279\pi\)
−0.857977 + 0.513689i \(0.828279\pi\)
\(90\) 0 0
\(91\) 11.4587 + 8.32524i 0.125920 + 0.0914861i
\(92\) 0 0
\(93\) 17.0583 + 17.0583i 0.183422 + 0.183422i
\(94\) 0 0
\(95\) −17.6043 − 134.220i −0.185308 − 1.41285i
\(96\) 0 0
\(97\) −17.2696 − 33.8934i −0.178037 − 0.349417i 0.784692 − 0.619886i \(-0.212821\pi\)
−0.962729 + 0.270469i \(0.912821\pi\)
\(98\) 0 0
\(99\) − 96.1373i − 0.971084i
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.73.1 ✓ 64
4.3 odd 2 400.3.bg.f.273.8 64
25.12 odd 20 inner 200.3.u.b.137.1 yes 64
100.87 even 20 400.3.bg.f.337.8 64
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.3.u.b.73.1 ✓ 64 1.1 even 1 trivial
200.3.u.b.137.1 yes 64 25.12 odd 20 inner
400.3.bg.f.273.8 64 4.3 odd 2
400.3.bg.f.337.8 64 100.87 even 20