Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [363,2,Mod(124,363)] 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("363.124"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.e (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-2,-1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 202.1
Root \(0.809017 - 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 363.202
Dual form 363.2.e.c.124.1

$q$-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+(-0.500000 - 0.363271i) q^{2} +(0.309017 - 0.951057i) q^{3} +(-0.500000 - 1.53884i) q^{4} +(2.11803 - 1.53884i) q^{5} +(-0.500000 + 0.363271i) q^{6} +(-0.927051 - 2.85317i) q^{7} +(-0.690983 + 2.12663i) q^{8} +(-0.809017 - 0.587785i) q^{9} -1.61803 q^{10} -1.61803 q^{12} +(1.42705 + 1.03681i) q^{13} +(-0.572949 + 1.76336i) q^{14} +(-0.809017 - 2.48990i) q^{15} +(-1.50000 + 1.08981i) q^{16} +(1.30902 - 0.951057i) q^{17} +(0.190983 + 0.587785i) q^{18} +(-1.80902 + 5.56758i) q^{19} +(-3.42705 - 2.48990i) q^{20} -3.00000 q^{21} +3.47214 q^{23} +(1.80902 + 1.31433i) q^{24} +(0.572949 - 1.76336i) q^{25} +(-0.336881 - 1.03681i) q^{26} +(-0.809017 + 0.587785i) q^{27} +(-3.92705 + 2.85317i) q^{28} +(-1.38197 - 4.25325i) q^{29} +(-0.500000 + 1.53884i) q^{30} +(-2.30902 - 1.67760i) q^{31} +5.61803 q^{32} -1.00000 q^{34} +(-6.35410 - 4.61653i) q^{35} +(-0.500000 + 1.53884i) q^{36} +(0.0729490 + 0.224514i) q^{37} +(2.92705 - 2.12663i) q^{38} +(1.42705 - 1.03681i) q^{39} +(1.80902 + 5.56758i) q^{40} +(3.69098 - 11.3597i) q^{41} +(1.50000 + 1.08981i) q^{42} -6.23607 q^{43} -2.61803 q^{45} +(-1.73607 - 1.26133i) q^{46} +(0.500000 - 1.53884i) q^{47} +(0.572949 + 1.76336i) q^{48} +(-1.61803 + 1.17557i) q^{49} +(-0.927051 + 0.673542i) q^{50} +(-0.500000 - 1.53884i) q^{51} +(0.881966 - 2.71441i) q^{52} +(7.78115 + 5.65334i) q^{53} +0.618034 q^{54} +6.70820 q^{56} +(4.73607 + 3.44095i) q^{57} +(-0.854102 + 2.62866i) q^{58} +(3.19098 + 9.82084i) q^{59} +(-3.42705 + 2.48990i) q^{60} +(6.35410 - 4.61653i) q^{61} +(0.545085 + 1.67760i) q^{62} +(-0.927051 + 2.85317i) q^{63} +(0.190983 + 0.138757i) q^{64} +4.61803 q^{65} -9.56231 q^{67} +(-2.11803 - 1.53884i) q^{68} +(1.07295 - 3.30220i) q^{69} +(1.50000 + 4.61653i) q^{70} +(4.50000 - 3.26944i) q^{71} +(1.80902 - 1.31433i) q^{72} +(1.00000 + 3.07768i) q^{73} +(0.0450850 - 0.138757i) q^{74} +(-1.50000 - 1.08981i) q^{75} +9.47214 q^{76} -1.09017 q^{78} +(7.66312 + 5.56758i) q^{79} +(-1.50000 + 4.61653i) q^{80} +(0.309017 + 0.951057i) q^{81} +(-5.97214 + 4.33901i) q^{82} +(0.572949 - 0.416272i) q^{83} +(1.50000 + 4.61653i) q^{84} +(1.30902 - 4.02874i) q^{85} +(3.11803 + 2.26538i) q^{86} -4.47214 q^{87} +0.527864 q^{89} +(1.30902 + 0.951057i) q^{90} +(1.63525 - 5.03280i) q^{91} +(-1.73607 - 5.34307i) q^{92} +(-2.30902 + 1.67760i) q^{93} +(-0.809017 + 0.587785i) q^{94} +(4.73607 + 14.5761i) q^{95} +(1.73607 - 5.34307i) q^{96} +(11.3541 + 8.24924i) q^{97} +1.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - q^{3} - 2 q^{4} + 4 q^{5} - 2 q^{6} + 3 q^{7} - 5 q^{8} - q^{9} - 2 q^{10} - 2 q^{12} - q^{13} - 9 q^{14} - q^{15} - 6 q^{16} + 3 q^{17} + 3 q^{18} - 5 q^{19} - 7 q^{20} - 12 q^{21}+ \cdots - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.363271i −0.353553 − 0.256872i 0.396805 − 0.917903i \(-0.370119\pi\)
−0.750358 + 0.661031i \(0.770119\pi\)
\(3\) 0.309017 − 0.951057i 0.178411 − 0.549093i
\(4\) −0.500000 − 1.53884i −0.250000 − 0.769421i
\(5\) 2.11803 − 1.53884i 0.947214 − 0.688191i −0.00293261 − 0.999996i \(-0.500933\pi\)
0.950146 + 0.311805i \(0.100933\pi\)
\(6\) −0.500000 + 0.363271i −0.204124 + 0.148305i
\(7\) −0.927051 − 2.85317i −0.350392 − 1.07840i −0.958633 − 0.284644i \(-0.908125\pi\)
0.608241 − 0.793752i \(-0.291875\pi\)
\(8\) −0.690983 + 2.12663i −0.244299 + 0.751876i
\(9\) −0.809017 − 0.587785i −0.269672 − 0.195928i
\(10\) −1.61803 −0.511667
\(11\) 0 0
\(12\) −1.61803 −0.467086
\(13\) 1.42705 + 1.03681i 0.395793 + 0.287560i 0.767825 − 0.640660i \(-0.221339\pi\)
−0.372032 + 0.928220i \(0.621339\pi\)
\(14\) −0.572949 + 1.76336i −0.153127 + 0.471277i
\(15\) −0.809017 − 2.48990i −0.208887 − 0.642889i
\(16\) −1.50000 + 1.08981i −0.375000 + 0.272453i
\(17\) 1.30902 − 0.951057i 0.317483 − 0.230665i −0.417618 − 0.908623i \(-0.637135\pi\)
0.735101 + 0.677958i \(0.237135\pi\)
\(18\) 0.190983 + 0.587785i 0.0450151 + 0.138542i
\(19\) −1.80902 + 5.56758i −0.415017 + 1.27729i 0.497219 + 0.867625i \(0.334355\pi\)
−0.912236 + 0.409666i \(0.865645\pi\)
\(20\) −3.42705 − 2.48990i −0.766312 − 0.556758i
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 3.47214 0.723990 0.361995 − 0.932180i \(-0.382096\pi\)
0.361995 + 0.932180i \(0.382096\pi\)
\(24\) 1.80902 + 1.31433i 0.369264 + 0.268286i
\(25\) 0.572949 − 1.76336i 0.114590 − 0.352671i
\(26\) −0.336881 − 1.03681i −0.0660678 − 0.203336i
\(27\) −0.809017 + 0.587785i −0.155695 + 0.113119i
\(28\) −3.92705 + 2.85317i −0.742143 + 0.539198i
\(29\) −1.38197 − 4.25325i −0.256625 − 0.789809i −0.993505 − 0.113787i \(-0.963702\pi\)
0.736881 − 0.676023i \(-0.236298\pi\)
\(30\) −0.500000 + 1.53884i −0.0912871 + 0.280953i
\(31\) −2.30902 − 1.67760i −0.414712 − 0.301306i 0.360795 − 0.932645i \(-0.382505\pi\)
−0.775507 + 0.631340i \(0.782505\pi\)
\(32\) 5.61803 0.993137
\(33\) 0 0
\(34\) −1.00000 −0.171499
\(35\) −6.35410 − 4.61653i −1.07404 − 0.780335i
\(36\) −0.500000 + 1.53884i −0.0833333 + 0.256474i
\(37\) 0.0729490 + 0.224514i 0.0119927 + 0.0369099i 0.956874 − 0.290504i \(-0.0938229\pi\)
−0.944881 + 0.327414i \(0.893823\pi\)
\(38\) 2.92705 − 2.12663i 0.474830 − 0.344984i
\(39\) 1.42705 − 1.03681i 0.228511 − 0.166023i
\(40\) 1.80902 + 5.56758i 0.286031 + 0.880312i
\(41\) 3.69098 − 11.3597i 0.576435 − 1.77408i −0.0548065 − 0.998497i \(-0.517454\pi\)
0.631241 − 0.775587i \(-0.282546\pi\)
\(42\) 1.50000 + 1.08981i 0.231455 + 0.168162i
\(43\) −6.23607 −0.950991 −0.475496 − 0.879718i \(-0.657731\pi\)
−0.475496 + 0.879718i \(0.657731\pi\)
\(44\) 0 0
\(45\) −2.61803 −0.390273
\(46\) −1.73607 − 1.26133i −0.255969 − 0.185973i
\(47\) 0.500000 − 1.53884i 0.0729325 − 0.224463i −0.907945 − 0.419089i \(-0.862349\pi\)
0.980877 + 0.194626i \(0.0623494\pi\)
\(48\) 0.572949 + 1.76336i 0.0826981 + 0.254518i
\(49\) −1.61803 + 1.17557i −0.231148 + 0.167939i
\(50\) −0.927051 + 0.673542i −0.131105 + 0.0952532i
\(51\) −0.500000 − 1.53884i −0.0700140 − 0.215481i
\(52\) 0.881966 − 2.71441i 0.122307 − 0.376421i
\(53\) 7.78115 + 5.65334i 1.06882 + 0.776546i 0.975700 − 0.219109i \(-0.0703150\pi\)
0.0931231 + 0.995655i \(0.470315\pi\)
\(54\) 0.618034 0.0841038
\(55\) 0 0
\(56\) 6.70820 0.896421
\(57\) 4.73607 + 3.44095i 0.627308 + 0.455766i
\(58\) −0.854102 + 2.62866i −0.112149 + 0.345159i
\(59\) 3.19098 + 9.82084i 0.415431 + 1.27856i 0.911865 + 0.410490i \(0.134642\pi\)
−0.496435 + 0.868074i \(0.665358\pi\)
\(60\) −3.42705 + 2.48990i −0.442430 + 0.321444i
\(61\) 6.35410 − 4.61653i 0.813559 − 0.591085i −0.101301 − 0.994856i \(-0.532301\pi\)
0.914860 + 0.403770i \(0.132301\pi\)
\(62\) 0.545085 + 1.67760i 0.0692259 + 0.213055i
\(63\) −0.927051 + 2.85317i −0.116797 + 0.359466i
\(64\) 0.190983 + 0.138757i 0.0238729 + 0.0173447i
\(65\) 4.61803 0.572797
\(66\) 0 0
\(67\) −9.56231 −1.16822 −0.584111 − 0.811674i \(-0.698557\pi\)
−0.584111 + 0.811674i \(0.698557\pi\)
\(68\) −2.11803 − 1.53884i −0.256849 − 0.186612i
\(69\) 1.07295 − 3.30220i 0.129168 − 0.397538i
\(70\) 1.50000 + 4.61653i 0.179284 + 0.551780i
\(71\) 4.50000 − 3.26944i 0.534052 − 0.388011i −0.287819 − 0.957685i \(-0.592930\pi\)
0.821871 + 0.569673i \(0.192930\pi\)
\(72\) 1.80902 − 1.31433i 0.213195 − 0.154895i
\(73\) 1.00000 + 3.07768i 0.117041 + 0.360216i 0.992367 − 0.123317i \(-0.0393533\pi\)
−0.875326 + 0.483533i \(0.839353\pi\)
\(74\) 0.0450850 − 0.138757i 0.00524102 − 0.0161302i
\(75\) −1.50000 − 1.08981i −0.173205 − 0.125841i
\(76\) 9.47214 1.08653
\(77\) 0 0
\(78\) −1.09017 −0.123437
\(79\) 7.66312 + 5.56758i 0.862168 + 0.626402i 0.928474 − 0.371397i \(-0.121121\pi\)
−0.0663057 + 0.997799i \(0.521121\pi\)
\(80\) −1.50000 + 4.61653i −0.167705 + 0.516143i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) −5.97214 + 4.33901i −0.659512 + 0.479164i
\(83\) 0.572949 − 0.416272i 0.0628893 − 0.0456918i −0.555896 − 0.831252i \(-0.687625\pi\)
0.618786 + 0.785560i \(0.287625\pi\)
\(84\) 1.50000 + 4.61653i 0.163663 + 0.503704i
\(85\) 1.30902 − 4.02874i 0.141983 − 0.436978i
\(86\) 3.11803 + 2.26538i 0.336226 + 0.244283i
\(87\) −4.47214 −0.479463
\(88\) 0 0
\(89\) 0.527864 0.0559535 0.0279767 − 0.999609i \(-0.491094\pi\)
0.0279767 + 0.999609i \(0.491094\pi\)
\(90\) 1.30902 + 0.951057i 0.137983 + 0.100250i
\(91\) 1.63525 − 5.03280i 0.171421 − 0.527580i
\(92\) −1.73607 − 5.34307i −0.180998 − 0.557053i
\(93\) −2.30902 + 1.67760i −0.239434 + 0.173959i
\(94\) −0.809017 + 0.587785i −0.0834437 + 0.0606254i
\(95\) 4.73607 + 14.5761i 0.485910 + 1.49548i
\(96\) 1.73607 − 5.34307i 0.177187 − 0.545325i
\(97\) 11.3541 + 8.24924i 1.15283 + 0.837583i 0.988855 − 0.148881i \(-0.0475670\pi\)
0.163979 + 0.986464i \(0.447567\pi\)
\(98\) 1.23607 0.124862
\(99\) 0 0
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 363.2.e.c.202.1 4
11.2 odd 10 33.2.e.a.16.1 ✓ 4
11.3 even 5 inner 363.2.e.c.124.1 4
11.4 even 5 363.2.e.j.130.1 4
11.5 even 5 363.2.a.e.1.2 2
11.6 odd 10 363.2.a.h.1.1 2
11.7 odd 10 33.2.e.a.31.1 yes 4
11.8 odd 10 363.2.e.h.124.1 4
11.9 even 5 363.2.e.j.148.1 4
11.10 odd 2 363.2.e.h.202.1 4
33.2 even 10 99.2.f.b.82.1 4
33.5 odd 10 1089.2.a.s.1.1 2
33.17 even 10 1089.2.a.m.1.2 2
33.29 even 10 99.2.f.b.64.1 4
44.7 even 10 528.2.y.f.97.1 4
44.27 odd 10 5808.2.a.bm.1.1 2
44.35 even 10 528.2.y.f.49.1 4
44.39 even 10 5808.2.a.bl.1.1 2
55.2 even 20 825.2.bx.b.49.2 8
55.7 even 20 825.2.bx.b.724.1 8
55.13 even 20 825.2.bx.b.49.1 8
55.18 even 20 825.2.bx.b.724.2 8
55.24 odd 10 825.2.n.f.676.1 4
55.29 odd 10 825.2.n.f.526.1 4
55.39 odd 10 9075.2.a.x.1.2 2
55.49 even 10 9075.2.a.bv.1.1 2
99.2 even 30 891.2.n.a.676.1 8
99.7 odd 30 891.2.n.d.757.1 8
99.13 odd 30 891.2.n.d.379.1 8
99.29 even 30 891.2.n.a.757.1 8
99.40 odd 30 891.2.n.d.460.1 8
99.68 even 30 891.2.n.a.379.1 8
99.79 odd 30 891.2.n.d.676.1 8
99.95 even 30 891.2.n.a.460.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.a.16.1 ✓ 4 11.2 odd 10
33.2.e.a.31.1 yes 4 11.7 odd 10
99.2.f.b.64.1 4 33.29 even 10
99.2.f.b.82.1 4 33.2 even 10
363.2.a.e.1.2 2 11.5 even 5
363.2.a.h.1.1 2 11.6 odd 10
363.2.e.c.124.1 4 11.3 even 5 inner
363.2.e.c.202.1 4 1.1 even 1 trivial
363.2.e.h.124.1 4 11.8 odd 10
363.2.e.h.202.1 4 11.10 odd 2
363.2.e.j.130.1 4 11.4 even 5
363.2.e.j.148.1 4 11.9 even 5
528.2.y.f.49.1 4 44.35 even 10
528.2.y.f.97.1 4 44.7 even 10
825.2.n.f.526.1 4 55.29 odd 10
825.2.n.f.676.1 4 55.24 odd 10
825.2.bx.b.49.1 8 55.13 even 20
825.2.bx.b.49.2 8 55.2 even 20
825.2.bx.b.724.1 8 55.7 even 20
825.2.bx.b.724.2 8 55.18 even 20
891.2.n.a.379.1 8 99.68 even 30
891.2.n.a.460.1 8 99.95 even 30
891.2.n.a.676.1 8 99.2 even 30
891.2.n.a.757.1 8 99.29 even 30
891.2.n.d.379.1 8 99.13 odd 30
891.2.n.d.460.1 8 99.40 odd 30
891.2.n.d.676.1 8 99.79 odd 30
891.2.n.d.757.1 8 99.7 odd 30
1089.2.a.m.1.2 2 33.17 even 10
1089.2.a.s.1.1 2 33.5 odd 10
5808.2.a.bl.1.1 2 44.39 even 10
5808.2.a.bm.1.1 2 44.27 odd 10
9075.2.a.x.1.2 2 55.39 odd 10
9075.2.a.bv.1.1 2 55.49 even 10