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Properties

Label 200.2.m.c.161.4
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 161.4
Root \(0.772523 - 2.37758i\) of defining polynomial
Character \(\chi\) \(=\) 200.161
Dual form 200.2.m.c.41.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.772523 - 2.37758i) q^{3} +(1.16821 - 1.90664i) q^{5} -1.96923 q^{7} +(-2.62905 - 1.91012i) q^{9} +(-2.80465 + 2.03770i) q^{11} +(4.32714 + 3.14385i) q^{13} +(-3.63074 - 4.25043i) q^{15} +(-0.204922 - 0.630686i) q^{17} +(-0.163997 - 0.504731i) q^{19} +(-1.52128 + 4.68201i) q^{21} +(7.20196 - 5.23253i) q^{23} +(-2.27059 - 4.45471i) q^{25} +(-0.504994 + 0.366899i) q^{27} +(-0.939405 + 2.89119i) q^{29} +(0.921788 + 2.83697i) q^{31} +(2.67814 + 8.24246i) q^{33} +(-2.30047 + 3.75463i) q^{35} +(9.46267 + 6.87504i) q^{37} +(10.8176 - 7.85944i) q^{39} +(-0.662310 - 0.481196i) q^{41} -9.68524 q^{43} +(-6.71320 + 2.78126i) q^{45} +(-2.52063 + 7.75769i) q^{47} -3.12212 q^{49} -1.65782 q^{51} +(2.88802 - 8.88842i) q^{53} +(0.608757 + 7.72793i) q^{55} -1.32673 q^{57} +(2.82290 + 2.05096i) q^{59} +(-9.24453 + 6.71654i) q^{61} +(5.17722 + 3.76147i) q^{63} +(11.0492 - 4.57766i) q^{65} +(2.91032 + 8.95703i) q^{67} +(-6.87709 - 21.1655i) q^{69} +(1.27411 - 3.92130i) q^{71} +(-8.53720 + 6.20264i) q^{73} +(-12.3455 + 1.95715i) q^{75} +(5.52301 - 4.01270i) q^{77} +(-2.31326 + 7.11949i) q^{79} +(-2.53041 - 7.78781i) q^{81} +(-1.17148 - 3.60543i) q^{83} +(-1.44189 - 0.346057i) q^{85} +(6.14833 + 4.46702i) q^{87} +(11.3462 - 8.24351i) q^{89} +(-8.52115 - 6.19098i) q^{91} +7.45723 q^{93} +(-1.15392 - 0.276946i) q^{95} +(0.295759 - 0.910252i) q^{97} +11.2658 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{3} - q^{5} - 6 q^{7} - 11 q^{9} - 10 q^{11} + q^{13} - 10 q^{15} - 4 q^{17} - 3 q^{21} + 11 q^{23} + 9 q^{25} + 13 q^{27} + 5 q^{29} - 9 q^{31} + 16 q^{33} + 24 q^{35} + 30 q^{37} + 14 q^{39}+ \cdots + 122 q^{99}+O(q^{100}) \) 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
\(3\) 0.772523 − 2.37758i 0.446017 − 1.37270i −0.435349 − 0.900262i \(-0.643375\pi\)
0.881365 − 0.472436i \(-0.156625\pi\)
\(4\) 0 0
\(5\) 1.16821 − 1.90664i 0.522438 − 0.852677i
\(6\) 0 0
\(7\) −1.96923 −0.744300 −0.372150 − 0.928173i \(-0.621379\pi\)
−0.372150 + 0.928173i \(0.621379\pi\)
\(8\) 0 0
\(9\) −2.62905 − 1.91012i −0.876351 − 0.636706i
\(10\) 0 0
\(11\) −2.80465 + 2.03770i −0.845635 + 0.614389i −0.923939 − 0.382540i \(-0.875049\pi\)
0.0783043 + 0.996930i \(0.475049\pi\)
\(12\) 0 0
\(13\) 4.32714 + 3.14385i 1.20013 + 0.871948i 0.994298 − 0.106638i \(-0.0340086\pi\)
0.205836 + 0.978586i \(0.434009\pi\)
\(14\) 0 0
\(15\) −3.63074 − 4.25043i −0.937453 − 1.09746i
\(16\) 0 0
\(17\) −0.204922 − 0.630686i −0.0497010 − 0.152964i 0.923126 − 0.384498i \(-0.125625\pi\)
−0.972827 + 0.231534i \(0.925625\pi\)
\(18\) 0 0
\(19\) −0.163997 − 0.504731i −0.0376235 − 0.115793i 0.930481 − 0.366340i \(-0.119389\pi\)
−0.968104 + 0.250547i \(0.919389\pi\)
\(20\) 0 0
\(21\) −1.52128 + 4.68201i −0.331970 + 1.02170i
\(22\) 0 0
\(23\) 7.20196 − 5.23253i 1.50171 − 1.09106i 0.532019 − 0.846733i \(-0.321434\pi\)
0.969693 − 0.244325i \(-0.0785664\pi\)
\(24\) 0 0
\(25\) −2.27059 − 4.45471i −0.454118 − 0.890942i
\(26\) 0 0
\(27\) −0.504994 + 0.366899i −0.0971861 + 0.0706098i
\(28\) 0 0
\(29\) −0.939405 + 2.89119i −0.174443 + 0.536881i −0.999608 − 0.0280119i \(-0.991082\pi\)
0.825165 + 0.564892i \(0.191082\pi\)
\(30\) 0 0
\(31\) 0.921788 + 2.83697i 0.165558 + 0.509535i 0.999077 − 0.0429557i \(-0.0136774\pi\)
−0.833519 + 0.552491i \(0.813677\pi\)
\(32\) 0 0
\(33\) 2.67814 + 8.24246i 0.466204 + 1.43483i
\(34\) 0 0
\(35\) −2.30047 + 3.75463i −0.388850 + 0.634648i
\(36\) 0 0
\(37\) 9.46267 + 6.87504i 1.55565 + 1.13025i 0.939462 + 0.342653i \(0.111326\pi\)
0.616192 + 0.787596i \(0.288674\pi\)
\(38\) 0 0
\(39\) 10.8176 − 7.85944i 1.73220 − 1.25852i
\(40\) 0 0
\(41\) −0.662310 − 0.481196i −0.103435 − 0.0751502i 0.534865 − 0.844937i \(-0.320362\pi\)
−0.638301 + 0.769787i \(0.720362\pi\)
\(42\) 0 0
\(43\) −9.68524 −1.47699 −0.738493 − 0.674261i \(-0.764462\pi\)
−0.738493 + 0.674261i \(0.764462\pi\)
\(44\) 0 0
\(45\) −6.71320 + 2.78126i −1.00074 + 0.414606i
\(46\) 0 0
\(47\) −2.52063 + 7.75769i −0.367671 + 1.13158i 0.580620 + 0.814174i \(0.302810\pi\)
−0.948291 + 0.317401i \(0.897190\pi\)
\(48\) 0 0
\(49\) −3.12212 −0.446018
\(50\) 0 0
\(51\) −1.65782 −0.232141
\(52\) 0 0
\(53\) 2.88802 − 8.88842i 0.396700 − 1.22092i −0.530929 − 0.847416i \(-0.678157\pi\)
0.927630 − 0.373502i \(-0.121843\pi\)
\(54\) 0 0
\(55\) 0.608757 + 7.72793i 0.0820848 + 1.04203i
\(56\) 0 0
\(57\) −1.32673 −0.175730
\(58\) 0 0
\(59\) 2.82290 + 2.05096i 0.367511 + 0.267012i 0.756178 − 0.654366i \(-0.227064\pi\)
−0.388667 + 0.921378i \(0.627064\pi\)
\(60\) 0 0
\(61\) −9.24453 + 6.71654i −1.18364 + 0.859965i −0.992578 − 0.121613i \(-0.961193\pi\)
−0.191063 + 0.981578i \(0.561193\pi\)
\(62\) 0 0
\(63\) 5.17722 + 3.76147i 0.652268 + 0.473900i
\(64\) 0 0
\(65\) 11.0492 − 4.57766i 1.37049 − 0.567789i
\(66\) 0 0
\(67\) 2.91032 + 8.95703i 0.355552 + 1.09428i 0.955689 + 0.294379i \(0.0951127\pi\)
−0.600137 + 0.799897i \(0.704887\pi\)
\(68\) 0 0
\(69\) −6.87709 − 21.1655i −0.827904 − 2.54803i
\(70\) 0 0
\(71\) 1.27411 − 3.92130i 0.151209 − 0.465373i −0.846548 − 0.532312i \(-0.821323\pi\)
0.997757 + 0.0669389i \(0.0213233\pi\)
\(72\) 0 0
\(73\) −8.53720 + 6.20264i −0.999203 + 0.725964i −0.961917 − 0.273341i \(-0.911871\pi\)
−0.0372862 + 0.999305i \(0.511871\pi\)
\(74\) 0 0
\(75\) −12.3455 + 1.95715i −1.42554 + 0.225992i
\(76\) 0 0
\(77\) 5.52301 − 4.01270i 0.629406 − 0.457290i
\(78\) 0 0
\(79\) −2.31326 + 7.11949i −0.260262 + 0.801005i 0.732485 + 0.680783i \(0.238360\pi\)
−0.992747 + 0.120222i \(0.961640\pi\)
\(80\) 0 0
\(81\) −2.53041 − 7.78781i −0.281157 − 0.865312i
\(82\) 0 0
\(83\) −1.17148 − 3.60543i −0.128586 − 0.395748i 0.865951 − 0.500129i \(-0.166714\pi\)
−0.994537 + 0.104381i \(0.966714\pi\)
\(84\) 0 0
\(85\) −1.44189 − 0.346057i −0.156395 − 0.0375352i
\(86\) 0 0
\(87\) 6.14833 + 4.46702i 0.659170 + 0.478915i
\(88\) 0 0
\(89\) 11.3462 − 8.24351i 1.20270 − 0.873810i 0.208149 − 0.978097i \(-0.433256\pi\)
0.994547 + 0.104287i \(0.0332561\pi\)
\(90\) 0 0
\(91\) −8.52115 − 6.19098i −0.893260 − 0.648991i
\(92\) 0 0
\(93\) 7.45723 0.773279
\(94\) 0 0
\(95\) −1.15392 − 0.276946i −0.118390 − 0.0284140i
\(96\) 0 0
\(97\) 0.295759 − 0.910252i 0.0300298 − 0.0924221i −0.934918 − 0.354863i \(-0.884527\pi\)
0.964948 + 0.262441i \(0.0845275\pi\)
\(98\) 0 0
\(99\) 11.2658 1.13226
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.161.4 yes 16
4.3 odd 2 400.2.u.g.161.1 16
5.2 odd 4 1000.2.q.d.449.2 32
5.3 odd 4 1000.2.q.d.449.7 32
5.4 even 2 1000.2.m.c.801.1 16
25.4 even 10 5000.2.a.m.1.1 8
25.9 even 10 1000.2.m.c.201.1 16
25.12 odd 20 1000.2.q.d.49.7 32
25.13 odd 20 1000.2.q.d.49.2 32
25.16 even 5 inner 200.2.m.c.41.4 ✓ 16
25.21 even 5 5000.2.a.l.1.8 8
100.71 odd 10 10000.2.a.bk.1.1 8
100.79 odd 10 10000.2.a.bh.1.8 8
100.91 odd 10 400.2.u.g.241.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.2.m.c.41.4 ✓ 16 25.16 even 5 inner
200.2.m.c.161.4 yes 16 1.1 even 1 trivial
400.2.u.g.161.1 16 4.3 odd 2
400.2.u.g.241.1 16 100.91 odd 10
1000.2.m.c.201.1 16 25.9 even 10
1000.2.m.c.801.1 16 5.4 even 2
1000.2.q.d.49.2 32 25.13 odd 20
1000.2.q.d.49.7 32 25.12 odd 20
1000.2.q.d.449.2 32 5.2 odd 4
1000.2.q.d.449.7 32 5.3 odd 4
5000.2.a.l.1.8 8 25.21 even 5
5000.2.a.m.1.1 8 25.4 even 10
10000.2.a.bh.1.8 8 100.79 odd 10
10000.2.a.bk.1.1 8 100.71 odd 10