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Properties

Label 200.2.m.c.161.3
Level $200$
Weight $2$
Character 200.161
Analytic conductor $1.597$
Analytic rank $0$
Dimension $16$
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] = [200,2,Mod(41,200)] 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("200.41"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 200.m (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] 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: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 12 x^{14} - 18 x^{13} + 100 x^{12} + 23 x^{11} + 567 x^{10} + 556 x^{9} + 3841 x^{8} + \cdots + 6400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 161.3
Root \(0.372462 - 1.14632i\) of defining polynomial
Character \(\chi\) \(=\) 200.161
Dual form 200.2.m.c.41.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.372462 - 1.14632i) q^{3} +(0.388572 + 2.20205i) q^{5} +1.59935 q^{7} +(1.25173 + 0.909432i) q^{9} +(2.86042 - 2.07822i) q^{11} +(-2.38530 - 1.73302i) q^{13} +(2.66898 + 0.374751i) q^{15} +(-0.357732 - 1.10099i) q^{17} +(-0.866689 - 2.66739i) q^{19} +(0.595698 - 1.83337i) q^{21} +(-2.55625 + 1.85723i) q^{23} +(-4.69802 + 1.71131i) q^{25} +(4.43408 - 3.22155i) q^{27} +(-3.04006 + 9.35636i) q^{29} +(0.572270 + 1.76126i) q^{31} +(-1.31690 - 4.05302i) q^{33} +(0.621463 + 3.52185i) q^{35} +(-3.63802 - 2.64318i) q^{37} +(-2.87503 + 2.08883i) q^{39} +(6.49601 + 4.71963i) q^{41} -11.5345 q^{43} +(-1.51623 + 3.10974i) q^{45} +(0.728491 - 2.24207i) q^{47} -4.44208 q^{49} -1.39533 q^{51} +(2.15111 - 6.62042i) q^{53} +(5.68781 + 5.49124i) q^{55} -3.38050 q^{57} +(-11.1170 - 8.07696i) q^{59} +(1.96004 - 1.42405i) q^{61} +(2.00195 + 1.45450i) q^{63} +(2.88933 - 5.92594i) q^{65} +(-4.27388 - 13.1537i) q^{67} +(1.17687 + 3.62203i) q^{69} +(-1.07380 + 3.30482i) q^{71} +(2.63248 - 1.91261i) q^{73} +(0.211872 + 6.02284i) q^{75} +(4.57481 - 3.32380i) q^{77} +(-3.06650 + 9.43770i) q^{79} +(-0.607051 - 1.86831i) q^{81} +(3.97495 + 12.2336i) q^{83} +(2.28542 - 1.21556i) q^{85} +(9.59308 + 6.96978i) q^{87} +(8.85248 - 6.43171i) q^{89} +(-3.81493 - 2.77171i) q^{91} +2.23212 q^{93} +(5.53696 - 2.94497i) q^{95} +(-2.21345 + 6.81231i) q^{97} +5.47046 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{3} - q^{5} - 6 q^{7} - 11 q^{9} - 10 q^{11} + q^{13} - 10 q^{15} - 4 q^{17} - 3 q^{21} + 11 q^{23} + 9 q^{25} + 13 q^{27} + 5 q^{29} - 9 q^{31} + 16 q^{33} + 24 q^{35} + 30 q^{37} + 14 q^{39}+ \cdots + 122 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{4}{5}\right)\)

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
\(3\) 0.372462 − 1.14632i 0.215041 − 0.661829i −0.784109 − 0.620623i \(-0.786880\pi\)
0.999151 − 0.0412064i \(-0.0131201\pi\)
\(4\) 0 0
\(5\) 0.388572 + 2.20205i 0.173775 + 0.984785i
\(6\) 0 0
\(7\) 1.59935 0.604498 0.302249 − 0.953229i \(-0.402263\pi\)
0.302249 + 0.953229i \(0.402263\pi\)
\(8\) 0 0
\(9\) 1.25173 + 0.909432i 0.417242 + 0.303144i
\(10\) 0 0
\(11\) 2.86042 − 2.07822i 0.862449 − 0.626606i −0.0661014 − 0.997813i \(-0.521056\pi\)
0.928550 + 0.371207i \(0.121056\pi\)
\(12\) 0 0
\(13\) −2.38530 − 1.73302i −0.661563 − 0.480654i 0.205627 − 0.978630i \(-0.434076\pi\)
−0.867190 + 0.497977i \(0.834076\pi\)
\(14\) 0 0
\(15\) 2.66898 + 0.374751i 0.689128 + 0.0967604i
\(16\) 0 0
\(17\) −0.357732 − 1.10099i −0.0867628 − 0.267028i 0.898257 − 0.439471i \(-0.144834\pi\)
−0.985020 + 0.172443i \(0.944834\pi\)
\(18\) 0 0
\(19\) −0.866689 − 2.66739i −0.198832 − 0.611942i −0.999910 − 0.0133828i \(-0.995740\pi\)
0.801078 − 0.598559i \(-0.204260\pi\)
\(20\) 0 0
\(21\) 0.595698 − 1.83337i 0.129992 − 0.400074i
\(22\) 0 0
\(23\) −2.55625 + 1.85723i −0.533015 + 0.387258i −0.821484 − 0.570231i \(-0.806854\pi\)
0.288469 + 0.957489i \(0.406854\pi\)
\(24\) 0 0
\(25\) −4.69802 + 1.71131i −0.939605 + 0.342262i
\(26\) 0 0
\(27\) 4.43408 − 3.22155i 0.853339 − 0.619987i
\(28\) 0 0
\(29\) −3.04006 + 9.35636i −0.564526 + 1.73743i 0.104830 + 0.994490i \(0.466570\pi\)
−0.669356 + 0.742942i \(0.733430\pi\)
\(30\) 0 0
\(31\) 0.572270 + 1.76126i 0.102783 + 0.316332i 0.989204 − 0.146547i \(-0.0468161\pi\)
−0.886421 + 0.462880i \(0.846816\pi\)
\(32\) 0 0
\(33\) −1.31690 − 4.05302i −0.229244 − 0.705540i
\(34\) 0 0
\(35\) 0.621463 + 3.52185i 0.105046 + 0.595301i
\(36\) 0 0
\(37\) −3.63802 − 2.64318i −0.598087 − 0.434535i 0.247112 − 0.968987i \(-0.420518\pi\)
−0.845199 + 0.534451i \(0.820518\pi\)
\(38\) 0 0
\(39\) −2.87503 + 2.08883i −0.460374 + 0.334481i
\(40\) 0 0
\(41\) 6.49601 + 4.71963i 1.01451 + 0.737082i 0.965149 − 0.261699i \(-0.0842829\pi\)
0.0493568 + 0.998781i \(0.484283\pi\)
\(42\) 0 0
\(43\) −11.5345 −1.75899 −0.879496 − 0.475906i \(-0.842120\pi\)
−0.879496 + 0.475906i \(0.842120\pi\)
\(44\) 0 0
\(45\) −1.51623 + 3.10974i −0.226026 + 0.463573i
\(46\) 0 0
\(47\) 0.728491 − 2.24207i 0.106261 − 0.327039i −0.883763 − 0.467935i \(-0.844998\pi\)
0.990024 + 0.140896i \(0.0449982\pi\)
\(48\) 0 0
\(49\) −4.44208 −0.634583
\(50\) 0 0
\(51\) −1.39533 −0.195385
\(52\) 0 0
\(53\) 2.15111 − 6.62042i 0.295477 − 0.909385i −0.687584 − 0.726105i \(-0.741329\pi\)
0.983061 − 0.183280i \(-0.0586714\pi\)
\(54\) 0 0
\(55\) 5.68781 + 5.49124i 0.766944 + 0.740439i
\(56\) 0 0
\(57\) −3.38050 −0.447758
\(58\) 0 0
\(59\) −11.1170 − 8.07696i −1.44731 − 1.05153i −0.986451 − 0.164055i \(-0.947543\pi\)
−0.460856 − 0.887475i \(-0.652457\pi\)
\(60\) 0 0
\(61\) 1.96004 − 1.42405i 0.250958 − 0.182331i −0.455194 − 0.890393i \(-0.650430\pi\)
0.706151 + 0.708061i \(0.250430\pi\)
\(62\) 0 0
\(63\) 2.00195 + 1.45450i 0.252222 + 0.183250i
\(64\) 0 0
\(65\) 2.88933 − 5.92594i 0.358378 − 0.735023i
\(66\) 0 0
\(67\) −4.27388 − 13.1537i −0.522138 − 1.60698i −0.769907 − 0.638156i \(-0.779697\pi\)
0.247769 − 0.968819i \(-0.420303\pi\)
\(68\) 0 0
\(69\) 1.17687 + 3.62203i 0.141678 + 0.436042i
\(70\) 0 0
\(71\) −1.07380 + 3.30482i −0.127437 + 0.392210i −0.994337 − 0.106271i \(-0.966109\pi\)
0.866900 + 0.498482i \(0.166109\pi\)
\(72\) 0 0
\(73\) 2.63248 − 1.91261i 0.308108 − 0.223854i −0.422976 − 0.906141i \(-0.639015\pi\)
0.731084 + 0.682287i \(0.239015\pi\)
\(74\) 0 0
\(75\) 0.211872 + 6.02284i 0.0244649 + 0.695458i
\(76\) 0 0
\(77\) 4.57481 − 3.32380i 0.521348 − 0.378782i
\(78\) 0 0
\(79\) −3.06650 + 9.43770i −0.345008 + 1.06182i 0.616572 + 0.787299i \(0.288521\pi\)
−0.961580 + 0.274526i \(0.911479\pi\)
\(80\) 0 0
\(81\) −0.607051 − 1.86831i −0.0674501 − 0.207590i
\(82\) 0 0
\(83\) 3.97495 + 12.2336i 0.436307 + 1.34282i 0.891741 + 0.452546i \(0.149484\pi\)
−0.455434 + 0.890270i \(0.650516\pi\)
\(84\) 0 0
\(85\) 2.28542 − 1.21556i 0.247888 − 0.131845i
\(86\) 0 0
\(87\) 9.59308 + 6.96978i 1.02849 + 0.747239i
\(88\) 0 0
\(89\) 8.85248 − 6.43171i 0.938361 − 0.681759i −0.00966436 − 0.999953i \(-0.503076\pi\)
0.948026 + 0.318194i \(0.103076\pi\)
\(90\) 0 0
\(91\) −3.81493 − 2.77171i −0.399913 − 0.290554i
\(92\) 0 0
\(93\) 2.23212 0.231461
\(94\) 0 0
\(95\) 5.53696 − 2.94497i 0.568080 − 0.302147i
\(96\) 0 0
\(97\) −2.21345 + 6.81231i −0.224742 + 0.691685i 0.773576 + 0.633704i \(0.218466\pi\)
−0.998318 + 0.0579809i \(0.981534\pi\)
\(98\) 0 0
\(99\) 5.47046 0.549802
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 200.2.m.c.161.3 yes 16
4.3 odd 2 400.2.u.g.161.2 16
5.2 odd 4 1000.2.q.d.449.4 32
5.3 odd 4 1000.2.q.d.449.5 32
5.4 even 2 1000.2.m.c.801.2 16
25.4 even 10 5000.2.a.m.1.3 8
25.9 even 10 1000.2.m.c.201.2 16
25.12 odd 20 1000.2.q.d.49.5 32
25.13 odd 20 1000.2.q.d.49.4 32
25.16 even 5 inner 200.2.m.c.41.3 ✓ 16
25.21 even 5 5000.2.a.l.1.6 8
100.71 odd 10 10000.2.a.bk.1.3 8
100.79 odd 10 10000.2.a.bh.1.6 8
100.91 odd 10 400.2.u.g.241.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.2.m.c.41.3 ✓ 16 25.16 even 5 inner
200.2.m.c.161.3 yes 16 1.1 even 1 trivial
400.2.u.g.161.2 16 4.3 odd 2
400.2.u.g.241.2 16 100.91 odd 10
1000.2.m.c.201.2 16 25.9 even 10
1000.2.m.c.801.2 16 5.4 even 2
1000.2.q.d.49.4 32 25.13 odd 20
1000.2.q.d.49.5 32 25.12 odd 20
1000.2.q.d.449.4 32 5.2 odd 4
1000.2.q.d.449.5 32 5.3 odd 4
5000.2.a.l.1.6 8 25.21 even 5
5000.2.a.m.1.3 8 25.4 even 10
10000.2.a.bh.1.6 8 100.79 odd 10
10000.2.a.bk.1.3 8 100.71 odd 10