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 107.2
Root \(-0.951057 + 0.309017i\) of defining polynomial
Character \(\chi\) \(=\) 200.107
Dual form 200.2.k.h.43.2

$q$-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+(0.221232 + 1.39680i) q^{2} +(-0.618034 + 0.618034i) q^{3} +(-1.90211 + 0.618034i) q^{4} +(-1.00000 - 0.726543i) q^{6} +(-1.90211 + 1.90211i) q^{7} +(-1.28408 - 2.52015i) q^{8} +2.23607i q^{9} -3.23607 q^{11} +(0.793604 - 1.55754i) q^{12} +(0.726543 + 0.726543i) q^{13} +(-3.07768 - 2.23607i) q^{14} +(3.23607 - 2.35114i) q^{16} +(1.00000 + 1.00000i) q^{17} +(-3.12334 + 0.494689i) q^{18} -2.00000i q^{19} -2.35114i q^{21} +(-0.715921 - 4.52015i) q^{22} +(4.25325 + 4.25325i) q^{23} +(2.35114 + 0.763932i) q^{24} +(-0.854102 + 1.17557i) q^{26} +(-3.23607 - 3.23607i) q^{27} +(2.44246 - 4.79360i) q^{28} +6.15537 q^{29} +8.50651i q^{31} +(4.00000 + 4.00000i) q^{32} +(2.00000 - 2.00000i) q^{33} +(-1.17557 + 1.61803i) q^{34} +(-1.38197 - 4.25325i) q^{36} +(-0.726543 + 0.726543i) q^{37} +(2.79360 - 0.442463i) q^{38} -0.898056 q^{39} +5.70820 q^{41} +(3.28408 - 0.520147i) q^{42} +(-4.61803 + 4.61803i) q^{43} +(6.15537 - 2.00000i) q^{44} +(-5.00000 + 6.88191i) q^{46} +(3.35520 - 3.35520i) q^{47} +(-0.546915 + 3.45309i) q^{48} -0.236068i q^{49} -1.23607 q^{51} +(-1.83099 - 0.932938i) q^{52} +(-3.07768 - 3.07768i) q^{53} +(3.80423 - 5.23607i) q^{54} +(7.23607 + 2.35114i) q^{56} +(1.23607 + 1.23607i) q^{57} +(1.36176 + 8.59783i) q^{58} -0.472136i q^{59} -0.898056i q^{61} +(-11.8819 + 1.88191i) q^{62} +(-4.25325 - 4.25325i) q^{63} +(-4.70228 + 6.47214i) q^{64} +(3.23607 + 2.35114i) q^{66} +(4.61803 + 4.61803i) q^{67} +(-2.52015 - 1.28408i) q^{68} -5.25731 q^{69} -11.4127i q^{71} +(5.63522 - 2.87129i) q^{72} +(4.70820 - 4.70820i) q^{73} +(-1.17557 - 0.854102i) q^{74} +(1.23607 + 3.80423i) q^{76} +(6.15537 - 6.15537i) q^{77} +(-0.198678 - 1.25441i) q^{78} -2.90617 q^{79} -2.70820 q^{81} +(1.26284 + 7.97323i) q^{82} +(6.61803 - 6.61803i) q^{83} +(1.45309 + 4.47214i) q^{84} +(-7.47214 - 5.42882i) q^{86} +(-3.80423 + 3.80423i) q^{87} +(4.15537 + 8.15537i) q^{88} +2.47214i q^{89} -2.76393 q^{91} +(-10.7188 - 5.46151i) q^{92} +(-5.25731 - 5.25731i) q^{93} +(5.42882 + 3.94427i) q^{94} -4.94427 q^{96} +(-4.23607 - 4.23607i) q^{97} +(0.329740 - 0.0522257i) q^{98} -7.23607i 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 \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.221232 + 1.39680i 0.156434 + 0.987688i
\(3\) −0.618034 + 0.618034i −0.356822 + 0.356822i −0.862640 − 0.505818i \(-0.831191\pi\)
0.505818 + 0.862640i \(0.331191\pi\)
\(4\) −1.90211 + 0.618034i −0.951057 + 0.309017i
\(5\) 0 0
\(6\) −1.00000 − 0.726543i −0.408248 − 0.296610i
\(7\) −1.90211 + 1.90211i −0.718931 + 0.718931i −0.968386 − 0.249455i \(-0.919748\pi\)
0.249455 + 0.968386i \(0.419748\pi\)
\(8\) −1.28408 − 2.52015i −0.453990 − 0.891007i
\(9\) 2.23607i 0.745356i
\(10\) 0 0
\(11\) −3.23607 −0.975711 −0.487856 − 0.872924i \(-0.662221\pi\)
−0.487856 + 0.872924i \(0.662221\pi\)
\(12\) 0.793604 − 1.55754i 0.229094 − 0.449622i
\(13\) 0.726543 + 0.726543i 0.201507 + 0.201507i 0.800645 − 0.599139i \(-0.204490\pi\)
−0.599139 + 0.800645i \(0.704490\pi\)
\(14\) −3.07768 − 2.23607i −0.822546 − 0.597614i
\(15\) 0 0
\(16\) 3.23607 − 2.35114i 0.809017 − 0.587785i
\(17\) 1.00000 + 1.00000i 0.242536 + 0.242536i 0.817898 − 0.575363i \(-0.195139\pi\)
−0.575363 + 0.817898i \(0.695139\pi\)
\(18\) −3.12334 + 0.494689i −0.736179 + 0.116599i
\(19\) − 2.00000i − 0.458831i −0.973329 − 0.229416i \(-0.926318\pi\)
0.973329 − 0.229416i \(-0.0736815\pi\)
\(20\) 0 0
\(21\) − 2.35114i − 0.513061i
\(22\) −0.715921 − 4.52015i −0.152635 − 0.963699i
\(23\) 4.25325 + 4.25325i 0.886865 + 0.886865i 0.994221 − 0.107356i \(-0.0342384\pi\)
−0.107356 + 0.994221i \(0.534238\pi\)
\(24\) 2.35114 + 0.763932i 0.479925 + 0.155937i
\(25\) 0 0
\(26\) −0.854102 + 1.17557i −0.167503 + 0.230548i
\(27\) −3.23607 − 3.23607i −0.622782 − 0.622782i
\(28\) 2.44246 − 4.79360i 0.461582 − 0.905906i
\(29\) 6.15537 1.14302 0.571511 − 0.820594i \(-0.306357\pi\)
0.571511 + 0.820594i \(0.306357\pi\)
\(30\) 0 0
\(31\) 8.50651i 1.52781i 0.645326 + 0.763907i \(0.276721\pi\)
−0.645326 + 0.763907i \(0.723279\pi\)
\(32\) 4.00000 + 4.00000i 0.707107 + 0.707107i
\(33\) 2.00000 − 2.00000i 0.348155 − 0.348155i
\(34\) −1.17557 + 1.61803i −0.201609 + 0.277491i
\(35\) 0 0
\(36\) −1.38197 − 4.25325i −0.230328 − 0.708876i
\(37\) −0.726543 + 0.726543i −0.119443 + 0.119443i −0.764302 − 0.644859i \(-0.776916\pi\)
0.644859 + 0.764302i \(0.276916\pi\)
\(38\) 2.79360 − 0.442463i 0.453182 − 0.0717771i
\(39\) −0.898056 −0.143804
\(40\) 0 0
\(41\) 5.70820 0.891472 0.445736 − 0.895165i \(-0.352942\pi\)
0.445736 + 0.895165i \(0.352942\pi\)
\(42\) 3.28408 − 0.520147i 0.506744 − 0.0802604i
\(43\) −4.61803 + 4.61803i −0.704244 + 0.704244i −0.965319 − 0.261075i \(-0.915923\pi\)
0.261075 + 0.965319i \(0.415923\pi\)
\(44\) 6.15537 − 2.00000i 0.927957 − 0.301511i
\(45\) 0 0
\(46\) −5.00000 + 6.88191i −0.737210 + 1.01468i
\(47\) 3.35520 − 3.35520i 0.489406 − 0.489406i −0.418713 − 0.908119i \(-0.637519\pi\)
0.908119 + 0.418713i \(0.137519\pi\)
\(48\) −0.546915 + 3.45309i −0.0789404 + 0.498410i
\(49\) − 0.236068i − 0.0337240i
\(50\) 0 0
\(51\) −1.23607 −0.173084
\(52\) −1.83099 − 0.932938i −0.253913 − 0.129375i
\(53\) −3.07768 − 3.07768i −0.422752 − 0.422752i 0.463398 − 0.886150i \(-0.346630\pi\)
−0.886150 + 0.463398i \(0.846630\pi\)
\(54\) 3.80423 − 5.23607i 0.517690 − 0.712539i
\(55\) 0 0
\(56\) 7.23607 + 2.35114i 0.966960 + 0.314184i
\(57\) 1.23607 + 1.23607i 0.163721 + 0.163721i
\(58\) 1.36176 + 8.59783i 0.178808 + 1.12895i
\(59\) − 0.472136i − 0.0614669i −0.999528 − 0.0307334i \(-0.990216\pi\)
0.999528 − 0.0307334i \(-0.00978430\pi\)
\(60\) 0 0
\(61\) − 0.898056i − 0.114984i −0.998346 − 0.0574921i \(-0.981690\pi\)
0.998346 − 0.0574921i \(-0.0183104\pi\)
\(62\) −11.8819 + 1.88191i −1.50900 + 0.239003i
\(63\) −4.25325 − 4.25325i −0.535860 − 0.535860i
\(64\) −4.70228 + 6.47214i −0.587785 + 0.809017i
\(65\) 0 0
\(66\) 3.23607 + 2.35114i 0.398332 + 0.289405i
\(67\) 4.61803 + 4.61803i 0.564183 + 0.564183i 0.930493 − 0.366310i \(-0.119379\pi\)
−0.366310 + 0.930493i \(0.619379\pi\)
\(68\) −2.52015 − 1.28408i −0.305613 − 0.155717i
\(69\) −5.25731 −0.632906
\(70\) 0 0
\(71\) − 11.4127i − 1.35444i −0.735783 − 0.677218i \(-0.763185\pi\)
0.735783 − 0.677218i \(-0.236815\pi\)
\(72\) 5.63522 − 2.87129i 0.664117 − 0.338385i
\(73\) 4.70820 − 4.70820i 0.551054 − 0.551054i −0.375691 − 0.926745i \(-0.622595\pi\)
0.926745 + 0.375691i \(0.122595\pi\)
\(74\) −1.17557 − 0.854102i −0.136657 − 0.0992873i
\(75\) 0 0
\(76\) 1.23607 + 3.80423i 0.141787 + 0.436375i
\(77\) 6.15537 − 6.15537i 0.701469 − 0.701469i
\(78\) −0.198678 − 1.25441i −0.0224959 − 0.142034i
\(79\) −2.90617 −0.326970 −0.163485 − 0.986546i \(-0.552273\pi\)
−0.163485 + 0.986546i \(0.552273\pi\)
\(80\) 0 0
\(81\) −2.70820 −0.300912
\(82\) 1.26284 + 7.97323i 0.139457 + 0.880496i
\(83\) 6.61803 − 6.61803i 0.726424 − 0.726424i −0.243482 − 0.969905i \(-0.578290\pi\)
0.969905 + 0.243482i \(0.0782896\pi\)
\(84\) 1.45309 + 4.47214i 0.158545 + 0.487950i
\(85\) 0 0
\(86\) −7.47214 − 5.42882i −0.805741 − 0.585405i
\(87\) −3.80423 + 3.80423i −0.407856 + 0.407856i
\(88\) 4.15537 + 8.15537i 0.442964 + 0.869365i
\(89\) 2.47214i 0.262046i 0.991379 + 0.131023i \(0.0418262\pi\)
−0.991379 + 0.131023i \(0.958174\pi\)
\(90\) 0 0
\(91\) −2.76393 −0.289739
\(92\) −10.7188 − 5.46151i −1.11751 − 0.569402i
\(93\) −5.25731 − 5.25731i −0.545158 − 0.545158i
\(94\) 5.42882 + 3.94427i 0.559940 + 0.406821i
\(95\) 0 0
\(96\) −4.94427 −0.504623
\(97\) −4.23607 − 4.23607i −0.430108 − 0.430108i 0.458557 − 0.888665i \(-0.348366\pi\)
−0.888665 + 0.458557i \(0.848366\pi\)
\(98\) 0.329740 − 0.0522257i 0.0333088 − 0.00527560i
\(99\) − 7.23607i − 0.727252i
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.107.2 8
4.3 odd 2 800.2.o.g.207.4 8
5.2 odd 4 40.2.k.a.3.1 ✓ 8
5.3 odd 4 inner 200.2.k.h.43.4 8
5.4 even 2 40.2.k.a.27.3 yes 8
8.3 odd 2 inner 200.2.k.h.107.4 8
8.5 even 2 800.2.o.g.207.3 8
15.2 even 4 360.2.w.c.163.4 8
15.14 odd 2 360.2.w.c.307.2 8
20.3 even 4 800.2.o.g.143.3 8
20.7 even 4 160.2.o.a.143.2 8
20.19 odd 2 160.2.o.a.47.1 8
40.3 even 4 inner 200.2.k.h.43.2 8
40.13 odd 4 800.2.o.g.143.4 8
40.19 odd 2 40.2.k.a.27.1 yes 8
40.27 even 4 40.2.k.a.3.3 yes 8
40.29 even 2 160.2.o.a.47.2 8
40.37 odd 4 160.2.o.a.143.1 8
60.47 odd 4 1440.2.bi.c.1423.2 8
60.59 even 2 1440.2.bi.c.847.3 8
80.19 odd 4 1280.2.n.q.767.2 8
80.27 even 4 1280.2.n.q.1023.1 8
80.29 even 4 1280.2.n.m.767.4 8
80.37 odd 4 1280.2.n.m.1023.3 8
80.59 odd 4 1280.2.n.m.767.3 8
80.67 even 4 1280.2.n.m.1023.4 8
80.69 even 4 1280.2.n.q.767.1 8
80.77 odd 4 1280.2.n.q.1023.2 8
120.29 odd 2 1440.2.bi.c.847.2 8
120.59 even 2 360.2.w.c.307.4 8
120.77 even 4 1440.2.bi.c.1423.3 8
120.107 odd 4 360.2.w.c.163.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.k.a.3.1 ✓ 8 5.2 odd 4
40.2.k.a.3.3 yes 8 40.27 even 4
40.2.k.a.27.1 yes 8 40.19 odd 2
40.2.k.a.27.3 yes 8 5.4 even 2
160.2.o.a.47.1 8 20.19 odd 2
160.2.o.a.47.2 8 40.29 even 2
160.2.o.a.143.1 8 40.37 odd 4
160.2.o.a.143.2 8 20.7 even 4
200.2.k.h.43.2 8 40.3 even 4 inner
200.2.k.h.43.4 8 5.3 odd 4 inner
200.2.k.h.107.2 8 1.1 even 1 trivial
200.2.k.h.107.4 8 8.3 odd 2 inner
360.2.w.c.163.2 8 120.107 odd 4
360.2.w.c.163.4 8 15.2 even 4
360.2.w.c.307.2 8 15.14 odd 2
360.2.w.c.307.4 8 120.59 even 2
800.2.o.g.143.3 8 20.3 even 4
800.2.o.g.143.4 8 40.13 odd 4
800.2.o.g.207.3 8 8.5 even 2
800.2.o.g.207.4 8 4.3 odd 2
1280.2.n.m.767.3 8 80.59 odd 4
1280.2.n.m.767.4 8 80.29 even 4
1280.2.n.m.1023.3 8 80.37 odd 4
1280.2.n.m.1023.4 8 80.67 even 4
1280.2.n.q.767.1 8 80.69 even 4
1280.2.n.q.767.2 8 80.19 odd 4
1280.2.n.q.1023.1 8 80.27 even 4
1280.2.n.q.1023.2 8 80.77 odd 4
1440.2.bi.c.847.2 8 120.29 odd 2
1440.2.bi.c.847.3 8 60.59 even 2
1440.2.bi.c.1423.2 8 60.47 odd 4
1440.2.bi.c.1423.3 8 120.77 even 4