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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [200,5,Mod(57,200)] 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("200.57"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 200.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.6739926168\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.313431616.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 27x^{4} + 145x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 193.3
Root \(2.35597i\) of defining polynomial
Character \(\chi\) \(=\) 200.193
Dual form 200.5.l.e.57.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(11.3304 + 11.3304i) q^{3} +(42.8544 - 42.8544i) q^{7} +175.757i q^{9} -20.3872 q^{11} +(151.697 + 151.697i) q^{13} +(150.269 - 150.269i) q^{17} +277.202i q^{19} +971.118 q^{21} +(363.891 + 363.891i) q^{23} +(-1073.64 + 1073.64i) q^{27} -413.744i q^{29} -659.526 q^{31} +(-230.996 - 230.996i) q^{33} +(-1058.50 + 1058.50i) q^{37} +3437.59i q^{39} -801.206 q^{41} +(-2008.78 - 2008.78i) q^{43} +(1471.26 - 1471.26i) q^{47} -1272.01i q^{49} +3405.23 q^{51} +(1321.08 + 1321.08i) q^{53} +(-3140.82 + 3140.82i) q^{57} +937.237i q^{59} +2543.20 q^{61} +(7531.97 + 7531.97i) q^{63} +(-140.942 + 140.942i) q^{67} +8246.08i q^{69} +2532.44 q^{71} +(-5900.02 - 5900.02i) q^{73} +(-873.682 + 873.682i) q^{77} +196.937i q^{79} -10093.2 q^{81} +(9130.34 + 9130.34i) q^{83} +(4687.89 - 4687.89i) q^{87} -10678.4i q^{89} +13001.8 q^{91} +(-7472.70 - 7472.70i) q^{93} +(11052.2 - 11052.2i) q^{97} -3583.19i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{3} + 40 q^{7} + 72 q^{11} + 62 q^{13} - 418 q^{17} + 1736 q^{21} + 1760 q^{23} - 2248 q^{27} + 2864 q^{31} + 8 q^{33} - 5366 q^{37} + 1656 q^{41} + 2760 q^{43} + 432 q^{47} + 15840 q^{51} - 534 q^{53}+ \cdots + 9418 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) 11.3304 + 11.3304i 1.25894 + 1.25894i 0.951602 + 0.307334i \(0.0994369\pi\)
0.307334 + 0.951602i \(0.400563\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 42.8544 − 42.8544i 0.874581 − 0.874581i −0.118387 − 0.992968i \(-0.537772\pi\)
0.992968 + 0.118387i \(0.0377723\pi\)
\(8\) 0 0
\(9\) 175.757i 2.16984i
\(10\) 0 0
\(11\) −20.3872 −0.168489 −0.0842447 − 0.996445i \(-0.526848\pi\)
−0.0842447 + 0.996445i \(0.526848\pi\)
\(12\) 0 0
\(13\) 151.697 + 151.697i 0.897617 + 0.897617i 0.995225 − 0.0976082i \(-0.0311192\pi\)
−0.0976082 + 0.995225i \(0.531119\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 150.269 − 150.269i 0.519963 − 0.519963i −0.397597 − 0.917560i \(-0.630156\pi\)
0.917560 + 0.397597i \(0.130156\pi\)
\(18\) 0 0
\(19\) 277.202i 0.767873i 0.923359 + 0.383937i \(0.125432\pi\)
−0.923359 + 0.383937i \(0.874568\pi\)
\(20\) 0 0
\(21\) 971.118 2.20208
\(22\) 0 0
\(23\) 363.891 + 363.891i 0.687885 + 0.687885i 0.961764 − 0.273879i \(-0.0883069\pi\)
−0.273879 + 0.961764i \(0.588307\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1073.64 + 1073.64i −1.47275 + 1.47275i
\(28\) 0 0
\(29\) − 413.744i − 0.491967i −0.969274 − 0.245983i \(-0.920889\pi\)
0.969274 − 0.245983i \(-0.0791108\pi\)
\(30\) 0 0
\(31\) −659.526 −0.686291 −0.343145 − 0.939282i \(-0.611492\pi\)
−0.343145 + 0.939282i \(0.611492\pi\)
\(32\) 0 0
\(33\) −230.996 − 230.996i −0.212117 − 0.212117i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1058.50 + 1058.50i −0.773193 + 0.773193i −0.978663 − 0.205471i \(-0.934127\pi\)
0.205471 + 0.978663i \(0.434127\pi\)
\(38\) 0 0
\(39\) 3437.59i 2.26008i
\(40\) 0 0
\(41\) −801.206 −0.476624 −0.238312 − 0.971189i \(-0.576594\pi\)
−0.238312 + 0.971189i \(0.576594\pi\)
\(42\) 0 0
\(43\) −2008.78 − 2008.78i −1.08642 − 1.08642i −0.995894 − 0.0905217i \(-0.971147\pi\)
−0.0905217 − 0.995894i \(-0.528853\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1471.26 − 1471.26i 0.666030 − 0.666030i −0.290765 − 0.956795i \(-0.593910\pi\)
0.956795 + 0.290765i \(0.0939097\pi\)
\(48\) 0 0
\(49\) − 1272.01i − 0.529782i
\(50\) 0 0
\(51\) 3405.23 1.30920
\(52\) 0 0
\(53\) 1321.08 + 1321.08i 0.470304 + 0.470304i 0.902013 − 0.431709i \(-0.142089\pi\)
−0.431709 + 0.902013i \(0.642089\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3140.82 + 3140.82i −0.966703 + 0.966703i
\(58\) 0 0
\(59\) 937.237i 0.269243i 0.990897 + 0.134622i \(0.0429819\pi\)
−0.990897 + 0.134622i \(0.957018\pi\)
\(60\) 0 0
\(61\) 2543.20 0.683473 0.341736 − 0.939796i \(-0.388985\pi\)
0.341736 + 0.939796i \(0.388985\pi\)
\(62\) 0 0
\(63\) 7531.97 + 7531.97i 1.89770 + 1.89770i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −140.942 + 140.942i −0.0313971 + 0.0313971i −0.722631 − 0.691234i \(-0.757067\pi\)
0.691234 + 0.722631i \(0.257067\pi\)
\(68\) 0 0
\(69\) 8246.08i 1.73200i
\(70\) 0 0
\(71\) 2532.44 0.502368 0.251184 − 0.967939i \(-0.419180\pi\)
0.251184 + 0.967939i \(0.419180\pi\)
\(72\) 0 0
\(73\) −5900.02 − 5900.02i −1.10715 − 1.10715i −0.993523 − 0.113630i \(-0.963752\pi\)
−0.113630 − 0.993523i \(-0.536248\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −873.682 + 873.682i −0.147357 + 0.147357i
\(78\) 0 0
\(79\) 196.937i 0.0315554i 0.999876 + 0.0157777i \(0.00502240\pi\)
−0.999876 + 0.0157777i \(0.994978\pi\)
\(80\) 0 0
\(81\) −10093.2 −1.53836
\(82\) 0 0
\(83\) 9130.34 + 9130.34i 1.32535 + 1.32535i 0.909377 + 0.415973i \(0.136559\pi\)
0.415973 + 0.909377i \(0.363441\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4687.89 − 4687.89i 0.619354 − 0.619354i
\(88\) 0 0
\(89\) − 10678.4i − 1.34811i −0.738680 − 0.674056i \(-0.764551\pi\)
0.738680 − 0.674056i \(-0.235449\pi\)
\(90\) 0 0
\(91\) 13001.8 1.57008
\(92\) 0 0
\(93\) −7472.70 − 7472.70i −0.863996 − 0.863996i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11052.2 − 11052.2i 1.17464 − 1.17464i 0.193550 − 0.981090i \(-0.438000\pi\)
0.981090 − 0.193550i \(-0.0620001\pi\)
\(98\) 0 0
\(99\) − 3583.19i − 0.365595i
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 200.5.l.e.193.3 6
4.3 odd 2 400.5.p.o.193.1 6
5.2 odd 4 inner 200.5.l.e.57.3 6
5.3 odd 4 40.5.l.c.17.1 ✓ 6
5.4 even 2 40.5.l.c.33.1 yes 6
15.8 even 4 360.5.v.c.217.1 6
15.14 odd 2 360.5.v.c.73.1 6
20.3 even 4 80.5.p.g.17.3 6
20.7 even 4 400.5.p.o.257.1 6
20.19 odd 2 80.5.p.g.33.3 6
40.3 even 4 320.5.p.o.257.1 6
40.13 odd 4 320.5.p.p.257.3 6
40.19 odd 2 320.5.p.o.193.1 6
40.29 even 2 320.5.p.p.193.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.5.l.c.17.1 ✓ 6 5.3 odd 4
40.5.l.c.33.1 yes 6 5.4 even 2
80.5.p.g.17.3 6 20.3 even 4
80.5.p.g.33.3 6 20.19 odd 2
200.5.l.e.57.3 6 5.2 odd 4 inner
200.5.l.e.193.3 6 1.1 even 1 trivial
320.5.p.o.193.1 6 40.19 odd 2
320.5.p.o.257.1 6 40.3 even 4
320.5.p.p.193.3 6 40.29 even 2
320.5.p.p.257.3 6 40.13 odd 4
360.5.v.c.73.1 6 15.14 odd 2
360.5.v.c.217.1 6 15.8 even 4
400.5.p.o.193.1 6 4.3 odd 2
400.5.p.o.257.1 6 20.7 even 4