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(3,200)] 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("200.3"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 10, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 200.v (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [208,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(208\)
Relative dimension: \(26\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 123.3
Character \(\chi\) \(=\) 200.123
Dual form 200.2.v.c.187.3

$q$-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+(-1.34013 - 0.451717i) q^{2} +(-1.14950 + 0.585698i) q^{3} +(1.59190 + 1.21072i) q^{4} +(-1.39421 - 1.74819i) q^{5} +(1.80505 - 0.265665i) q^{6} +(1.03289 + 1.03289i) q^{7} +(-1.58646 - 2.34161i) q^{8} +(-0.785054 + 1.08053i) q^{9} +(1.07874 + 2.97259i) q^{10} +(0.0335875 - 0.0244028i) q^{11} +(-2.53900 - 0.459345i) q^{12} +(-5.40771 + 0.856497i) q^{13} +(-0.917634 - 1.85078i) q^{14} +(2.62656 + 1.19295i) q^{15} +(1.06831 + 3.85470i) q^{16} +(-5.96997 - 3.04185i) q^{17} +(1.54017 - 1.09344i) q^{18} +(-3.70807 + 1.20483i) q^{19} +(-0.102881 - 4.47095i) q^{20} +(-1.79226 - 0.582342i) q^{21} +(-0.0560348 + 0.0175308i) q^{22} +(-1.15283 + 7.27865i) q^{23} +(3.19511 + 1.76250i) q^{24} +(-1.11234 + 4.87470i) q^{25} +(7.63394 + 1.29494i) q^{26} +(0.875004 - 5.52456i) q^{27} +(0.393719 + 2.89480i) q^{28} +(0.211867 - 0.652058i) q^{29} +(-2.98105 - 2.78517i) q^{30} +(2.81237 - 0.913793i) q^{31} +(0.309557 - 5.64838i) q^{32} +(-0.0243161 + 0.0477230i) q^{33} +(6.62648 + 6.77322i) q^{34} +(0.365620 - 3.24576i) q^{35} +(-2.55796 + 0.769625i) q^{36} +(0.184731 + 1.16634i) q^{37} +(5.51354 + 0.0603756i) q^{38} +(5.71450 - 4.15183i) q^{39} +(-1.88173 + 6.03814i) q^{40} +(-2.05099 - 1.49013i) q^{41} +(2.13882 + 1.59001i) q^{42} +(-2.01817 - 2.01817i) q^{43} +(0.0830130 + 0.00181827i) q^{44} +(2.98351 - 0.134070i) q^{45} +(4.83283 - 9.23360i) q^{46} +(8.25787 - 4.20759i) q^{47} +(-3.48571 - 3.80526i) q^{48} -4.86628i q^{49} +(3.69267 - 6.03027i) q^{50} +8.64407 q^{51} +(-9.64553 - 5.18377i) q^{52} +(1.85568 + 3.64198i) q^{53} +(-3.66816 + 7.00838i) q^{54} +(-0.0894888 - 0.0246948i) q^{55} +(0.779996 - 4.05726i) q^{56} +(3.55675 - 3.55675i) q^{57} +(-0.578475 + 0.778140i) q^{58} +(-4.20081 + 5.78193i) q^{59} +(2.73689 + 5.07909i) q^{60} +(-4.43754 - 6.10774i) q^{61} +(-4.18172 - 0.0457915i) q^{62} +(-1.92695 + 0.305199i) q^{63} +(-2.96632 + 7.42974i) q^{64} +(9.03682 + 8.25957i) q^{65} +(0.0541441 - 0.0529711i) q^{66} +(13.0654 + 6.65716i) q^{67} +(-5.82078 - 12.0703i) q^{68} +(-2.93792 - 9.04200i) q^{69} +(-1.95614 + 4.18458i) q^{70} +(-11.7350 - 3.81292i) q^{71} +(3.77565 + 0.124074i) q^{72} +(7.94343 + 1.25812i) q^{73} +(0.279294 - 1.64650i) q^{74} +(-1.57647 - 6.25495i) q^{75} +(-7.36160 - 2.57147i) q^{76} +(0.0598975 + 0.00948684i) q^{77} +(-9.53363 + 2.98266i) q^{78} +(-1.62338 + 4.99625i) q^{79} +(5.24930 - 7.24188i) q^{80} +(0.991726 + 3.05222i) q^{81} +(2.07548 + 2.92344i) q^{82} +(-6.87163 + 13.4863i) q^{83} +(-2.14806 - 3.09696i) q^{84} +(3.00567 + 14.6776i) q^{85} +(1.79297 + 3.61625i) q^{86} +(0.138369 + 0.873629i) q^{87} +(-0.110427 - 0.0399351i) q^{88} +(3.26895 + 4.49933i) q^{89} +(-4.05886 - 1.16803i) q^{90} +(-6.47024 - 4.70090i) q^{91} +(-10.6476 + 10.1912i) q^{92} +(-2.69760 + 2.69760i) q^{93} +(-12.9673 + 1.90851i) q^{94} +(7.27611 + 4.80263i) q^{95} +(2.95241 + 6.67410i) q^{96} +(-2.62754 - 5.15685i) q^{97} +(-2.19818 + 6.52145i) q^{98} +0.0554500i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 208 q + 2 q^{2} - 16 q^{3} - 10 q^{4} - 26 q^{6} - 34 q^{8} - 20 q^{9} + 10 q^{10} - 4 q^{11} - 2 q^{12} - 30 q^{14} + 18 q^{16} - 32 q^{17} - 20 q^{18} - 10 q^{20} - 42 q^{22} - 40 q^{25} - 8 q^{26} - 28 q^{27}+ \cdots + 206 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{11}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.34013 − 0.451717i −0.947616 − 0.319412i
\(3\) −1.14950 + 0.585698i −0.663662 + 0.338153i −0.753163 − 0.657834i \(-0.771473\pi\)
0.0895006 + 0.995987i \(0.471473\pi\)
\(4\) 1.59190 + 1.21072i 0.795952 + 0.605360i
\(5\) −1.39421 − 1.74819i −0.623511 − 0.781815i
\(6\) 1.80505 − 0.265665i 0.736907 − 0.108457i
\(7\) 1.03289 + 1.03289i 0.390396 + 0.390396i 0.874828 − 0.484433i \(-0.160974\pi\)
−0.484433 + 0.874828i \(0.660974\pi\)
\(8\) −1.58646 − 2.34161i −0.560897 − 0.827886i
\(9\) −0.785054 + 1.08053i −0.261685 + 0.360178i
\(10\) 1.07874 + 2.97259i 0.341128 + 0.940017i
\(11\) 0.0335875 − 0.0244028i 0.0101270 − 0.00735771i −0.582710 − 0.812680i \(-0.698008\pi\)
0.592837 + 0.805322i \(0.298008\pi\)
\(12\) −2.53900 − 0.459345i −0.732948 − 0.132602i
\(13\) −5.40771 + 0.856497i −1.49983 + 0.237550i −0.851720 − 0.523997i \(-0.824440\pi\)
−0.648110 + 0.761547i \(0.724440\pi\)
\(14\) −0.917634 − 1.85078i −0.245248 − 0.494642i
\(15\) 2.62656 + 1.19295i 0.678174 + 0.308019i
\(16\) 1.06831 + 3.85470i 0.267078 + 0.963675i
\(17\) −5.96997 − 3.04185i −1.44793 − 0.737757i −0.459322 − 0.888270i \(-0.651908\pi\)
−0.988608 + 0.150513i \(0.951908\pi\)
\(18\) 1.54017 − 1.09344i 0.363022 − 0.257725i
\(19\) −3.70807 + 1.20483i −0.850690 + 0.276406i −0.701735 − 0.712438i \(-0.747591\pi\)
−0.148955 + 0.988844i \(0.547591\pi\)
\(20\) −0.102881 − 4.47095i −0.0230048 − 0.999735i
\(21\) −1.79226 − 0.582342i −0.391104 − 0.127077i
\(22\) −0.0560348 + 0.0175308i −0.0119467 + 0.00373759i
\(23\) −1.15283 + 7.27865i −0.240381 + 1.51770i 0.512006 + 0.858982i \(0.328903\pi\)
−0.752386 + 0.658722i \(0.771097\pi\)
\(24\) 3.19511 + 1.76250i 0.652198 + 0.359768i
\(25\) −1.11234 + 4.87470i −0.222468 + 0.974940i
\(26\) 7.63394 + 1.29494i 1.49714 + 0.253958i
\(27\) 0.875004 − 5.52456i 0.168395 − 1.06320i
\(28\) 0.393719 + 2.89480i 0.0744060 + 0.547066i
\(29\) 0.211867 − 0.652058i 0.0393426 − 0.121084i −0.929456 − 0.368932i \(-0.879723\pi\)
0.968799 + 0.247848i \(0.0797234\pi\)
\(30\) −2.98105 − 2.78517i −0.544263 − 0.508501i
\(31\) 2.81237 − 0.913793i 0.505116 − 0.164122i −0.0453640 − 0.998971i \(-0.514445\pi\)
0.550480 + 0.834848i \(0.314445\pi\)
\(32\) 0.309557 − 5.64838i 0.0547225 − 0.998502i
\(33\) −0.0243161 + 0.0477230i −0.00423289 + 0.00830751i
\(34\) 6.62648 + 6.77322i 1.13643 + 1.16160i
\(35\) 0.365620 − 3.24576i 0.0618011 − 0.548633i
\(36\) −2.55796 + 0.769625i −0.426326 + 0.128271i
\(37\) 0.184731 + 1.16634i 0.0303695 + 0.191746i 0.998209 − 0.0598229i \(-0.0190536\pi\)
−0.967839 + 0.251569i \(0.919054\pi\)
\(38\) 5.51354 + 0.0603756i 0.894415 + 0.00979421i
\(39\) 5.71450 − 4.15183i 0.915052 − 0.664825i
\(40\) −1.88173 + 6.03814i −0.297528 + 0.954713i
\(41\) −2.05099 − 1.49013i −0.320311 − 0.232719i 0.415997 − 0.909366i \(-0.363433\pi\)
−0.736308 + 0.676646i \(0.763433\pi\)
\(42\) 2.13882 + 1.59001i 0.330026 + 0.245344i
\(43\) −2.01817 − 2.01817i −0.307768 − 0.307768i 0.536275 − 0.844043i \(-0.319831\pi\)
−0.844043 + 0.536275i \(0.819831\pi\)
\(44\) 0.0830130 + 0.00181827i 0.0125147 + 0.000274115i
\(45\) 2.98351 − 0.134070i 0.444756 − 0.0199860i
\(46\) 4.83283 − 9.23360i 0.712562 − 1.36142i
\(47\) 8.25787 − 4.20759i 1.20453 − 0.613741i 0.267696 − 0.963504i \(-0.413738\pi\)
0.936838 + 0.349763i \(0.113738\pi\)
\(48\) −3.48571 − 3.80526i −0.503119 − 0.549242i
\(49\) − 4.86628i − 0.695183i
\(50\) 3.69267 − 6.03027i 0.522222 − 0.852809i
\(51\) 8.64407 1.21041
\(52\) −9.64553 − 5.18377i −1.33759 − 0.718859i
\(53\) 1.85568 + 3.64198i 0.254897 + 0.500264i 0.982625 − 0.185602i \(-0.0594236\pi\)
−0.727728 + 0.685866i \(0.759424\pi\)
\(54\) −3.66816 + 7.00838i −0.499173 + 0.953719i
\(55\) −0.0894888 − 0.0246948i −0.0120667 − 0.00332984i
\(56\) 0.779996 − 4.05726i 0.104231 − 0.542175i
\(57\) 3.55675 − 3.55675i 0.471104 − 0.471104i
\(58\) −0.578475 + 0.778140i −0.0759575 + 0.102175i
\(59\) −4.20081 + 5.78193i −0.546900 + 0.752743i −0.989587 − 0.143933i \(-0.954025\pi\)
0.442688 + 0.896676i \(0.354025\pi\)
\(60\) 2.73689 + 5.07909i 0.353331 + 0.655708i
\(61\) −4.43754 − 6.10774i −0.568168 − 0.782016i 0.424168 − 0.905584i \(-0.360567\pi\)
−0.992336 + 0.123567i \(0.960567\pi\)
\(62\) −4.18172 − 0.0457915i −0.531078 − 0.00581553i
\(63\) −1.92695 + 0.305199i −0.242773 + 0.0384514i
\(64\) −2.96632 + 7.42974i −0.370790 + 0.928717i
\(65\) 9.03682 + 8.25957i 1.12088 + 1.02447i
\(66\) 0.0541441 − 0.0529711i 0.00666468 − 0.00652030i
\(67\) 13.0654 + 6.65716i 1.59619 + 0.813301i 0.999945 + 0.0105016i \(0.00334284\pi\)
0.596249 + 0.802800i \(0.296657\pi\)
\(68\) −5.82078 − 12.0703i −0.705873 − 1.46374i
\(69\) −2.93792 − 9.04200i −0.353684 − 1.08853i
\(70\) −1.95614 + 4.18458i −0.233804 + 0.500153i
\(71\) −11.7350 − 3.81292i −1.39268 − 0.452510i −0.485865 − 0.874034i \(-0.661495\pi\)
−0.906817 + 0.421524i \(0.861495\pi\)
\(72\) 3.77565 + 0.124074i 0.444965 + 0.0146223i
\(73\) 7.94343 + 1.25812i 0.929708 + 0.147251i 0.602880 − 0.797832i \(-0.294020\pi\)
0.326829 + 0.945084i \(0.394020\pi\)
\(74\) 0.279294 − 1.64650i 0.0324673 − 0.191402i
\(75\) −1.57647 − 6.25495i −0.182035 − 0.722259i
\(76\) −7.36160 − 2.57147i −0.844433 − 0.294968i
\(77\) 0.0598975 + 0.00948684i 0.00682596 + 0.00108113i
\(78\) −9.53363 + 2.98266i −1.07947 + 0.337719i
\(79\) −1.62338 + 4.99625i −0.182644 + 0.562122i −0.999900 − 0.0141538i \(-0.995495\pi\)
0.817255 + 0.576276i \(0.195495\pi\)
\(80\) 5.24930 − 7.24188i 0.586889 − 0.809667i
\(81\) 0.991726 + 3.05222i 0.110192 + 0.339135i
\(82\) 2.07548 + 2.92344i 0.229198 + 0.322840i
\(83\) −6.87163 + 13.4863i −0.754259 + 1.48032i 0.118913 + 0.992905i \(0.462059\pi\)
−0.873172 + 0.487412i \(0.837941\pi\)
\(84\) −2.14806 − 3.09696i −0.234372 − 0.337907i
\(85\) 3.00567 + 14.6776i 0.326011 + 1.59201i
\(86\) 1.79297 + 3.61625i 0.193341 + 0.389951i
\(87\) 0.138369 + 0.873629i 0.0148347 + 0.0936628i
\(88\) −0.110427 − 0.0399351i −0.0117716 − 0.00425710i
\(89\) 3.26895 + 4.49933i 0.346509 + 0.476928i 0.946328 − 0.323207i \(-0.104761\pi\)
−0.599820 + 0.800135i \(0.704761\pi\)
\(90\) −4.05886 − 1.16803i −0.427842 − 0.123121i
\(91\) −6.47024 − 4.70090i −0.678265 − 0.492788i
\(92\) −10.6476 + 10.1912i −1.11009 + 1.06250i
\(93\) −2.69760 + 2.69760i −0.279728 + 0.279728i
\(94\) −12.9673 + 1.90851i −1.33747 + 0.196847i
\(95\) 7.27611 + 4.80263i 0.746513 + 0.492740i
\(96\) 2.95241 + 6.67410i 0.301329 + 0.681173i
\(97\) −2.62754 − 5.15685i −0.266787 − 0.523598i 0.718284 − 0.695750i \(-0.244928\pi\)
−0.985070 + 0.172152i \(0.944928\pi\)
\(98\) −2.19818 + 6.52145i −0.222050 + 0.658766i
\(99\) 0.0554500i 0.00557293i
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.v.c.123.3 ✓ 208
4.3 odd 2 800.2.bp.c.623.17 208
5.2 odd 4 1000.2.v.g.907.17 208
5.3 odd 4 1000.2.v.i.907.10 208
5.4 even 2 1000.2.v.h.843.24 208
8.3 odd 2 inner 200.2.v.c.123.26 yes 208
8.5 even 2 800.2.bp.c.623.18 208
25.9 even 10 1000.2.v.i.43.12 208
25.12 odd 20 inner 200.2.v.c.187.26 yes 208
25.13 odd 20 1000.2.v.h.707.1 208
25.16 even 5 1000.2.v.g.43.15 208
40.3 even 4 1000.2.v.i.907.12 208
40.19 odd 2 1000.2.v.h.843.1 208
40.27 even 4 1000.2.v.g.907.15 208
100.87 even 20 800.2.bp.c.687.18 208
200.37 odd 20 800.2.bp.c.687.17 208
200.59 odd 10 1000.2.v.i.43.10 208
200.91 odd 10 1000.2.v.g.43.17 208
200.163 even 20 1000.2.v.h.707.24 208
200.187 even 20 inner 200.2.v.c.187.3 yes 208
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.2.v.c.123.3 ✓ 208 1.1 even 1 trivial
200.2.v.c.123.26 yes 208 8.3 odd 2 inner
200.2.v.c.187.3 yes 208 200.187 even 20 inner
200.2.v.c.187.26 yes 208 25.12 odd 20 inner
800.2.bp.c.623.17 208 4.3 odd 2
800.2.bp.c.623.18 208 8.5 even 2
800.2.bp.c.687.17 208 200.37 odd 20
800.2.bp.c.687.18 208 100.87 even 20
1000.2.v.g.43.15 208 25.16 even 5
1000.2.v.g.43.17 208 200.91 odd 10
1000.2.v.g.907.15 208 40.27 even 4
1000.2.v.g.907.17 208 5.2 odd 4
1000.2.v.h.707.1 208 25.13 odd 20
1000.2.v.h.707.24 208 200.163 even 20
1000.2.v.h.843.1 208 40.19 odd 2
1000.2.v.h.843.24 208 5.4 even 2
1000.2.v.i.43.10 208 200.59 odd 10
1000.2.v.i.43.12 208 25.9 even 10
1000.2.v.i.907.10 208 5.3 odd 4
1000.2.v.i.907.12 208 40.3 even 4