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(43,200)] 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("200.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 200.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,2,4] 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: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{20})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.3
Root \(-0.587785 + 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 200.43
Dual form 200.2.k.h.107.3

$q$-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+(0.642040 + 1.26007i) q^{2} +(1.61803 + 1.61803i) q^{3} +(-1.17557 + 1.61803i) q^{4} +(-1.00000 + 3.07768i) q^{6} +(-1.17557 - 1.17557i) q^{7} +(-2.79360 - 0.442463i) q^{8} +2.23607i q^{9} +1.23607 q^{11} +(-4.52015 + 0.715921i) q^{12} +(3.07768 - 3.07768i) q^{13} +(0.726543 - 2.23607i) q^{14} +(-1.23607 - 3.80423i) q^{16} +(1.00000 - 1.00000i) q^{17} +(-2.81761 + 1.43564i) q^{18} +2.00000i q^{19} -3.80423i q^{21} +(0.793604 + 1.55754i) q^{22} +(-2.62866 + 2.62866i) q^{23} +(-3.80423 - 5.23607i) q^{24} +(5.85410 + 1.90211i) q^{26} +(1.23607 - 1.23607i) q^{27} +(3.28408 - 0.520147i) q^{28} -1.45309 q^{29} +5.25731i q^{31} +(4.00000 - 4.00000i) q^{32} +(2.00000 + 2.00000i) q^{33} +(1.90211 + 0.618034i) q^{34} +(-3.61803 - 2.62866i) q^{36} +(-3.07768 - 3.07768i) q^{37} +(-2.52015 + 1.28408i) q^{38} +9.95959 q^{39} -7.70820 q^{41} +(4.79360 - 2.44246i) q^{42} +(-2.38197 - 2.38197i) q^{43} +(-1.45309 + 2.00000i) q^{44} +(-5.00000 - 1.62460i) q^{46} +(7.33094 + 7.33094i) q^{47} +(4.15537 - 8.15537i) q^{48} -4.23607i q^{49} +3.23607 q^{51} +(1.36176 + 8.59783i) q^{52} +(0.726543 - 0.726543i) q^{53} +(2.35114 + 0.763932i) q^{54} +(2.76393 + 3.80423i) q^{56} +(-3.23607 + 3.23607i) q^{57} +(-0.932938 - 1.83099i) q^{58} -8.47214i q^{59} -9.95959i q^{61} +(-6.62460 + 3.37540i) q^{62} +(2.62866 - 2.62866i) q^{63} +(7.60845 + 2.47214i) q^{64} +(-1.23607 + 3.80423i) q^{66} +(2.38197 - 2.38197i) q^{67} +(0.442463 + 2.79360i) q^{68} -8.50651 q^{69} +7.05342i q^{71} +(0.989378 - 6.24669i) q^{72} +(-8.70820 - 8.70820i) q^{73} +(1.90211 - 5.85410i) q^{74} +(-3.23607 - 2.35114i) q^{76} +(-1.45309 - 1.45309i) q^{77} +(6.39445 + 12.5498i) q^{78} -12.3107 q^{79} +10.7082 q^{81} +(-4.94897 - 9.71290i) q^{82} +(4.38197 + 4.38197i) q^{83} +(6.15537 + 4.47214i) q^{84} +(1.47214 - 4.53077i) q^{86} +(-2.35114 - 2.35114i) q^{87} +(-3.45309 - 0.546915i) q^{88} +6.47214i q^{89} -7.23607 q^{91} +(-1.16308 - 7.34342i) q^{92} +(-8.50651 + 8.50651i) q^{93} +(-4.53077 + 13.9443i) q^{94} +12.9443 q^{96} +(0.236068 - 0.236068i) q^{97} +(5.33776 - 2.71972i) q^{98} +2.76393i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 4 q^{3} - 8 q^{6} - 4 q^{8} - 8 q^{11} - 12 q^{12} + 8 q^{16} + 8 q^{17} - 10 q^{18} - 12 q^{22} + 20 q^{26} - 8 q^{27} + 20 q^{28} + 32 q^{32} + 16 q^{33} - 20 q^{36} + 4 q^{38} - 8 q^{41}+ \cdots + 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.642040 + 1.26007i 0.453990 + 0.891007i
\(3\) 1.61803 + 1.61803i 0.934172 + 0.934172i 0.997963 − 0.0637909i \(-0.0203191\pi\)
−0.0637909 + 0.997963i \(0.520319\pi\)
\(4\) −1.17557 + 1.61803i −0.587785 + 0.809017i
\(5\) 0 0
\(6\) −1.00000 + 3.07768i −0.408248 + 1.25646i
\(7\) −1.17557 − 1.17557i −0.444324 − 0.444324i 0.449138 − 0.893462i \(-0.351731\pi\)
−0.893462 + 0.449138i \(0.851731\pi\)
\(8\) −2.79360 − 0.442463i −0.987688 − 0.156434i
\(9\) 2.23607i 0.745356i
\(10\) 0 0
\(11\) 1.23607 0.372689 0.186344 − 0.982485i \(-0.440336\pi\)
0.186344 + 0.982485i \(0.440336\pi\)
\(12\) −4.52015 + 0.715921i −1.30485 + 0.206669i
\(13\) 3.07768 − 3.07768i 0.853596 − 0.853596i −0.136978 − 0.990574i \(-0.543739\pi\)
0.990574 + 0.136978i \(0.0437390\pi\)
\(14\) 0.726543 − 2.23607i 0.194177 − 0.597614i
\(15\) 0 0
\(16\) −1.23607 − 3.80423i −0.309017 − 0.951057i
\(17\) 1.00000 − 1.00000i 0.242536 − 0.242536i −0.575363 − 0.817898i \(-0.695139\pi\)
0.817898 + 0.575363i \(0.195139\pi\)
\(18\) −2.81761 + 1.43564i −0.664117 + 0.338385i
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 0 0
\(21\) − 3.80423i − 0.830150i
\(22\) 0.793604 + 1.55754i 0.169197 + 0.332068i
\(23\) −2.62866 + 2.62866i −0.548113 + 0.548113i −0.925895 − 0.377782i \(-0.876687\pi\)
0.377782 + 0.925895i \(0.376687\pi\)
\(24\) −3.80423 − 5.23607i −0.776534 − 1.06881i
\(25\) 0 0
\(26\) 5.85410 + 1.90211i 1.14808 + 0.373035i
\(27\) 1.23607 − 1.23607i 0.237881 − 0.237881i
\(28\) 3.28408 − 0.520147i 0.620633 − 0.0982985i
\(29\) −1.45309 −0.269831 −0.134916 − 0.990857i \(-0.543076\pi\)
−0.134916 + 0.990857i \(0.543076\pi\)
\(30\) 0 0
\(31\) 5.25731i 0.944241i 0.881534 + 0.472120i \(0.156511\pi\)
−0.881534 + 0.472120i \(0.843489\pi\)
\(32\) 4.00000 − 4.00000i 0.707107 − 0.707107i
\(33\) 2.00000 + 2.00000i 0.348155 + 0.348155i
\(34\) 1.90211 + 0.618034i 0.326210 + 0.105992i
\(35\) 0 0
\(36\) −3.61803 − 2.62866i −0.603006 − 0.438109i
\(37\) −3.07768 − 3.07768i −0.505968 − 0.505968i 0.407318 − 0.913286i \(-0.366464\pi\)
−0.913286 + 0.407318i \(0.866464\pi\)
\(38\) −2.52015 + 1.28408i −0.408822 + 0.208305i
\(39\) 9.95959 1.59481
\(40\) 0 0
\(41\) −7.70820 −1.20382 −0.601910 − 0.798564i \(-0.705593\pi\)
−0.601910 + 0.798564i \(0.705593\pi\)
\(42\) 4.79360 − 2.44246i 0.739669 − 0.376880i
\(43\) −2.38197 − 2.38197i −0.363246 − 0.363246i 0.501760 − 0.865007i \(-0.332686\pi\)
−0.865007 + 0.501760i \(0.832686\pi\)
\(44\) −1.45309 + 2.00000i −0.219061 + 0.301511i
\(45\) 0 0
\(46\) −5.00000 − 1.62460i −0.737210 − 0.239534i
\(47\) 7.33094 + 7.33094i 1.06933 + 1.06933i 0.997411 + 0.0719165i \(0.0229115\pi\)
0.0719165 + 0.997411i \(0.477088\pi\)
\(48\) 4.15537 − 8.15537i 0.599776 − 1.17713i
\(49\) − 4.23607i − 0.605153i
\(50\) 0 0
\(51\) 3.23607 0.453140
\(52\) 1.36176 + 8.59783i 0.188842 + 1.19230i
\(53\) 0.726543 − 0.726543i 0.0997983 − 0.0997983i −0.655445 − 0.755243i \(-0.727519\pi\)
0.755243 + 0.655445i \(0.227519\pi\)
\(54\) 2.35114 + 0.763932i 0.319950 + 0.103958i
\(55\) 0 0
\(56\) 2.76393 + 3.80423i 0.369346 + 0.508361i
\(57\) −3.23607 + 3.23607i −0.428628 + 0.428628i
\(58\) −0.932938 − 1.83099i −0.122501 − 0.240421i
\(59\) − 8.47214i − 1.10298i −0.834182 − 0.551489i \(-0.814060\pi\)
0.834182 − 0.551489i \(-0.185940\pi\)
\(60\) 0 0
\(61\) − 9.95959i − 1.27520i −0.770370 − 0.637598i \(-0.779928\pi\)
0.770370 − 0.637598i \(-0.220072\pi\)
\(62\) −6.62460 + 3.37540i −0.841325 + 0.428676i
\(63\) 2.62866 − 2.62866i 0.331179 − 0.331179i
\(64\) 7.60845 + 2.47214i 0.951057 + 0.309017i
\(65\) 0 0
\(66\) −1.23607 + 3.80423i −0.152149 + 0.468268i
\(67\) 2.38197 − 2.38197i 0.291003 − 0.291003i −0.546473 − 0.837477i \(-0.684030\pi\)
0.837477 + 0.546473i \(0.184030\pi\)
\(68\) 0.442463 + 2.79360i 0.0536566 + 0.338774i
\(69\) −8.50651 −1.02406
\(70\) 0 0
\(71\) 7.05342i 0.837087i 0.908197 + 0.418544i \(0.137459\pi\)
−0.908197 + 0.418544i \(0.862541\pi\)
\(72\) 0.989378 − 6.24669i 0.116599 − 0.736179i
\(73\) −8.70820 − 8.70820i −1.01922 − 1.01922i −0.999812 − 0.0194065i \(-0.993822\pi\)
−0.0194065 − 0.999812i \(-0.506178\pi\)
\(74\) 1.90211 − 5.85410i 0.221116 − 0.680526i
\(75\) 0 0
\(76\) −3.23607 − 2.35114i −0.371202 − 0.269694i
\(77\) −1.45309 − 1.45309i −0.165594 − 0.165594i
\(78\) 6.39445 + 12.5498i 0.724029 + 1.42099i
\(79\) −12.3107 −1.38507 −0.692533 − 0.721386i \(-0.743505\pi\)
−0.692533 + 0.721386i \(0.743505\pi\)
\(80\) 0 0
\(81\) 10.7082 1.18980
\(82\) −4.94897 − 9.71290i −0.546522 − 1.07261i
\(83\) 4.38197 + 4.38197i 0.480983 + 0.480983i 0.905446 − 0.424462i \(-0.139537\pi\)
−0.424462 + 0.905446i \(0.639537\pi\)
\(84\) 6.15537 + 4.47214i 0.671606 + 0.487950i
\(85\) 0 0
\(86\) 1.47214 − 4.53077i 0.158745 − 0.488565i
\(87\) −2.35114 − 2.35114i −0.252069 − 0.252069i
\(88\) −3.45309 − 0.546915i −0.368100 − 0.0583013i
\(89\) 6.47214i 0.686045i 0.939327 + 0.343023i \(0.111451\pi\)
−0.939327 + 0.343023i \(0.888549\pi\)
\(90\) 0 0
\(91\) −7.23607 −0.758546
\(92\) −1.16308 − 7.34342i −0.121260 − 0.765605i
\(93\) −8.50651 + 8.50651i −0.882084 + 0.882084i
\(94\) −4.53077 + 13.9443i −0.467313 + 1.43824i
\(95\) 0 0
\(96\) 12.9443 1.32112
\(97\) 0.236068 − 0.236068i 0.0239691 − 0.0239691i −0.695021 − 0.718990i \(-0.744605\pi\)
0.718990 + 0.695021i \(0.244605\pi\)
\(98\) 5.33776 − 2.71972i 0.539195 − 0.274734i
\(99\) 2.76393i 0.277786i
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.k.h.43.3 8
4.3 odd 2 800.2.o.g.143.2 8
5.2 odd 4 inner 200.2.k.h.107.1 8
5.3 odd 4 40.2.k.a.27.4 yes 8
5.4 even 2 40.2.k.a.3.2 ✓ 8
8.3 odd 2 inner 200.2.k.h.43.1 8
8.5 even 2 800.2.o.g.143.1 8
15.8 even 4 360.2.w.c.307.1 8
15.14 odd 2 360.2.w.c.163.3 8
20.3 even 4 160.2.o.a.47.3 8
20.7 even 4 800.2.o.g.207.1 8
20.19 odd 2 160.2.o.a.143.4 8
40.3 even 4 40.2.k.a.27.2 yes 8
40.13 odd 4 160.2.o.a.47.4 8
40.19 odd 2 40.2.k.a.3.4 yes 8
40.27 even 4 inner 200.2.k.h.107.3 8
40.29 even 2 160.2.o.a.143.3 8
40.37 odd 4 800.2.o.g.207.2 8
60.23 odd 4 1440.2.bi.c.847.4 8
60.59 even 2 1440.2.bi.c.1423.1 8
80.3 even 4 1280.2.n.q.767.3 8
80.13 odd 4 1280.2.n.m.767.1 8
80.19 odd 4 1280.2.n.m.1023.1 8
80.29 even 4 1280.2.n.q.1023.3 8
80.43 even 4 1280.2.n.m.767.2 8
80.53 odd 4 1280.2.n.q.767.4 8
80.59 odd 4 1280.2.n.q.1023.4 8
80.69 even 4 1280.2.n.m.1023.2 8
120.29 odd 2 1440.2.bi.c.1423.4 8
120.53 even 4 1440.2.bi.c.847.1 8
120.59 even 2 360.2.w.c.163.1 8
120.83 odd 4 360.2.w.c.307.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.k.a.3.2 ✓ 8 5.4 even 2
40.2.k.a.3.4 yes 8 40.19 odd 2
40.2.k.a.27.2 yes 8 40.3 even 4
40.2.k.a.27.4 yes 8 5.3 odd 4
160.2.o.a.47.3 8 20.3 even 4
160.2.o.a.47.4 8 40.13 odd 4
160.2.o.a.143.3 8 40.29 even 2
160.2.o.a.143.4 8 20.19 odd 2
200.2.k.h.43.1 8 8.3 odd 2 inner
200.2.k.h.43.3 8 1.1 even 1 trivial
200.2.k.h.107.1 8 5.2 odd 4 inner
200.2.k.h.107.3 8 40.27 even 4 inner
360.2.w.c.163.1 8 120.59 even 2
360.2.w.c.163.3 8 15.14 odd 2
360.2.w.c.307.1 8 15.8 even 4
360.2.w.c.307.3 8 120.83 odd 4
800.2.o.g.143.1 8 8.5 even 2
800.2.o.g.143.2 8 4.3 odd 2
800.2.o.g.207.1 8 20.7 even 4
800.2.o.g.207.2 8 40.37 odd 4
1280.2.n.m.767.1 8 80.13 odd 4
1280.2.n.m.767.2 8 80.43 even 4
1280.2.n.m.1023.1 8 80.19 odd 4
1280.2.n.m.1023.2 8 80.69 even 4
1280.2.n.q.767.3 8 80.3 even 4
1280.2.n.q.767.4 8 80.53 odd 4
1280.2.n.q.1023.3 8 80.29 even 4
1280.2.n.q.1023.4 8 80.59 odd 4
1440.2.bi.c.847.1 8 120.53 even 4
1440.2.bi.c.847.4 8 60.23 odd 4
1440.2.bi.c.1423.1 8 60.59 even 2
1440.2.bi.c.1423.4 8 120.29 odd 2