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,-1,1,1] 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 124.1
Root \(0.809017 + 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 363.124
Dual form 363.2.e.e.202.1

$q$-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+(-0.809017 + 0.587785i) q^{2} +(-0.309017 - 0.951057i) q^{3} +(-0.309017 + 0.951057i) q^{4} +(1.61803 + 1.17557i) q^{5} +(0.809017 + 0.587785i) q^{6} +(1.23607 - 3.80423i) q^{7} +(-0.927051 - 2.85317i) q^{8} +(-0.809017 + 0.587785i) q^{9} -2.00000 q^{10} +1.00000 q^{12} +(1.61803 - 1.17557i) q^{13} +(1.23607 + 3.80423i) q^{14} +(0.618034 - 1.90211i) q^{15} +(0.809017 + 0.587785i) q^{16} +(1.61803 + 1.17557i) q^{17} +(0.309017 - 0.951057i) q^{18} +(-1.61803 + 1.17557i) q^{20} -4.00000 q^{21} +8.00000 q^{23} +(-2.42705 + 1.76336i) q^{24} +(-0.309017 - 0.951057i) q^{25} +(-0.618034 + 1.90211i) q^{26} +(0.809017 + 0.587785i) q^{27} +(3.23607 + 2.35114i) q^{28} +(-1.85410 + 5.70634i) q^{29} +(0.618034 + 1.90211i) q^{30} +(6.47214 - 4.70228i) q^{31} +5.00000 q^{32} -2.00000 q^{34} +(6.47214 - 4.70228i) q^{35} +(-0.309017 - 0.951057i) q^{36} +(1.85410 - 5.70634i) q^{37} +(-1.61803 - 1.17557i) q^{39} +(1.85410 - 5.70634i) q^{40} +(-0.618034 - 1.90211i) q^{41} +(3.23607 - 2.35114i) q^{42} -2.00000 q^{45} +(-6.47214 + 4.70228i) q^{46} +(2.47214 + 7.60845i) q^{47} +(0.309017 - 0.951057i) q^{48} +(-7.28115 - 5.29007i) q^{49} +(0.809017 + 0.587785i) q^{50} +(0.618034 - 1.90211i) q^{51} +(0.618034 + 1.90211i) q^{52} +(-4.85410 + 3.52671i) q^{53} -1.00000 q^{54} -12.0000 q^{56} +(-1.85410 - 5.70634i) q^{58} +(-1.23607 + 3.80423i) q^{59} +(1.61803 + 1.17557i) q^{60} +(-4.85410 - 3.52671i) q^{61} +(-2.47214 + 7.60845i) q^{62} +(1.23607 + 3.80423i) q^{63} +(-5.66312 + 4.11450i) q^{64} +4.00000 q^{65} -4.00000 q^{67} +(-1.61803 + 1.17557i) q^{68} +(-2.47214 - 7.60845i) q^{69} +(-2.47214 + 7.60845i) q^{70} +(2.42705 + 1.76336i) q^{72} +(-4.32624 + 13.3148i) q^{73} +(1.85410 + 5.70634i) q^{74} +(-0.809017 + 0.587785i) q^{75} +2.00000 q^{78} +(3.23607 - 2.35114i) q^{79} +(0.618034 + 1.90211i) q^{80} +(0.309017 - 0.951057i) q^{81} +(1.61803 + 1.17557i) q^{82} +(-9.70820 - 7.05342i) q^{83} +(1.23607 - 3.80423i) q^{84} +(1.23607 + 3.80423i) q^{85} +6.00000 q^{87} -6.00000 q^{89} +(1.61803 - 1.17557i) q^{90} +(-2.47214 - 7.60845i) q^{91} +(-2.47214 + 7.60845i) q^{92} +(-6.47214 - 4.70228i) q^{93} +(-6.47214 - 4.70228i) q^{94} +(-1.54508 - 4.75528i) q^{96} +(-1.61803 + 1.17557i) q^{97} +9.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + q^{3} + q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 3 q^{8} - q^{9} - 8 q^{10} + 4 q^{12} + 2 q^{13} - 4 q^{14} - 2 q^{15} + q^{16} + 2 q^{17} - q^{18} - 2 q^{20} - 16 q^{21} + 32 q^{23} - 3 q^{24}+ \cdots + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(1\) \(e\left(\frac{4}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.809017 + 0.587785i −0.572061 + 0.415627i −0.835853 − 0.548953i \(-0.815027\pi\)
0.263792 + 0.964580i \(0.415027\pi\)
\(3\) −0.309017 − 0.951057i −0.178411 − 0.549093i
\(4\) −0.309017 + 0.951057i −0.154508 + 0.475528i
\(5\) 1.61803 + 1.17557i 0.723607 + 0.525731i 0.887535 − 0.460741i \(-0.152416\pi\)
−0.163928 + 0.986472i \(0.552416\pi\)
\(6\) 0.809017 + 0.587785i 0.330280 + 0.239962i
\(7\) 1.23607 − 3.80423i 0.467190 − 1.43786i −0.389018 − 0.921230i \(-0.627186\pi\)
0.856208 − 0.516632i \(-0.172814\pi\)
\(8\) −0.927051 − 2.85317i −0.327762 − 1.00875i
\(9\) −0.809017 + 0.587785i −0.269672 + 0.195928i
\(10\) −2.00000 −0.632456
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) 1.61803 − 1.17557i 0.448762 − 0.326045i −0.340345 − 0.940301i \(-0.610544\pi\)
0.789107 + 0.614256i \(0.210544\pi\)
\(14\) 1.23607 + 3.80423i 0.330353 + 1.01672i
\(15\) 0.618034 − 1.90211i 0.159576 − 0.491123i
\(16\) 0.809017 + 0.587785i 0.202254 + 0.146946i
\(17\) 1.61803 + 1.17557i 0.392431 + 0.285118i 0.766451 − 0.642303i \(-0.222021\pi\)
−0.374020 + 0.927421i \(0.622021\pi\)
\(18\) 0.309017 − 0.951057i 0.0728360 − 0.224166i
\(19\) 0 0 0.951057 − 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(20\) −1.61803 + 1.17557i −0.361803 + 0.262866i
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) −2.42705 + 1.76336i −0.495420 + 0.359943i
\(25\) −0.309017 − 0.951057i −0.0618034 − 0.190211i
\(26\) −0.618034 + 1.90211i −0.121206 + 0.373035i
\(27\) 0.809017 + 0.587785i 0.155695 + 0.113119i
\(28\) 3.23607 + 2.35114i 0.611559 + 0.444324i
\(29\) −1.85410 + 5.70634i −0.344298 + 1.05964i 0.617660 + 0.786445i \(0.288081\pi\)
−0.961958 + 0.273196i \(0.911919\pi\)
\(30\) 0.618034 + 1.90211i 0.112837 + 0.347277i
\(31\) 6.47214 − 4.70228i 1.16243 − 0.844555i 0.172347 − 0.985036i \(-0.444865\pi\)
0.990083 + 0.140482i \(0.0448651\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 6.47214 − 4.70228i 1.09399 − 0.794831i
\(36\) −0.309017 − 0.951057i −0.0515028 − 0.158509i
\(37\) 1.85410 − 5.70634i 0.304812 − 0.938116i −0.674935 − 0.737878i \(-0.735828\pi\)
0.979747 − 0.200239i \(-0.0641718\pi\)
\(38\) 0 0
\(39\) −1.61803 − 1.17557i −0.259093 − 0.188242i
\(40\) 1.85410 − 5.70634i 0.293159 − 0.902251i
\(41\) −0.618034 − 1.90211i −0.0965207 − 0.297060i 0.891126 − 0.453755i \(-0.149916\pi\)
−0.987647 + 0.156695i \(0.949916\pi\)
\(42\) 3.23607 − 2.35114i 0.499336 − 0.362789i
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) −6.47214 + 4.70228i −0.954264 + 0.693314i
\(47\) 2.47214 + 7.60845i 0.360598 + 1.10981i 0.952692 + 0.303938i \(0.0983015\pi\)
−0.592094 + 0.805869i \(0.701699\pi\)
\(48\) 0.309017 − 0.951057i 0.0446028 − 0.137273i
\(49\) −7.28115 − 5.29007i −1.04016 − 0.755724i
\(50\) 0.809017 + 0.587785i 0.114412 + 0.0831254i
\(51\) 0.618034 − 1.90211i 0.0865421 − 0.266349i
\(52\) 0.618034 + 1.90211i 0.0857059 + 0.263776i
\(53\) −4.85410 + 3.52671i −0.666762 + 0.484431i −0.868940 − 0.494918i \(-0.835198\pi\)
0.202178 + 0.979349i \(0.435198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −12.0000 −1.60357
\(57\) 0 0
\(58\) −1.85410 − 5.70634i −0.243456 − 0.749279i
\(59\) −1.23607 + 3.80423i −0.160922 + 0.495268i −0.998713 − 0.0507240i \(-0.983847\pi\)
0.837790 + 0.545992i \(0.183847\pi\)
\(60\) 1.61803 + 1.17557i 0.208887 + 0.151765i
\(61\) −4.85410 − 3.52671i −0.621504 − 0.451549i 0.231942 − 0.972730i \(-0.425492\pi\)
−0.853447 + 0.521180i \(0.825492\pi\)
\(62\) −2.47214 + 7.60845i −0.313962 + 0.966274i
\(63\) 1.23607 + 3.80423i 0.155730 + 0.479287i
\(64\) −5.66312 + 4.11450i −0.707890 + 0.514312i
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −1.61803 + 1.17557i −0.196215 + 0.142559i
\(69\) −2.47214 − 7.60845i −0.297610 − 0.915950i
\(70\) −2.47214 + 7.60845i −0.295477 + 0.909384i
\(71\) 0 0 0.587785 − 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(72\) 2.42705 + 1.76336i 0.286031 + 0.207813i
\(73\) −4.32624 + 13.3148i −0.506348 + 1.55838i 0.292145 + 0.956374i \(0.405631\pi\)
−0.798493 + 0.602004i \(0.794369\pi\)
\(74\) 1.85410 + 5.70634i 0.215535 + 0.663348i
\(75\) −0.809017 + 0.587785i −0.0934172 + 0.0678716i
\(76\) 0 0
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 3.23607 − 2.35114i 0.364086 − 0.264524i −0.390668 − 0.920532i \(-0.627756\pi\)
0.754754 + 0.656007i \(0.227756\pi\)
\(80\) 0.618034 + 1.90211i 0.0690983 + 0.212663i
\(81\) 0.309017 − 0.951057i 0.0343352 − 0.105673i
\(82\) 1.61803 + 1.17557i 0.178682 + 0.129820i
\(83\) −9.70820 − 7.05342i −1.06561 − 0.774214i −0.0904951 − 0.995897i \(-0.528845\pi\)
−0.975119 + 0.221683i \(0.928845\pi\)
\(84\) 1.23607 − 3.80423i 0.134866 − 0.415075i
\(85\) 1.23607 + 3.80423i 0.134070 + 0.412626i
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 1.61803 − 1.17557i 0.170556 − 0.123916i
\(91\) −2.47214 − 7.60845i −0.259150 − 0.797582i
\(92\) −2.47214 + 7.60845i −0.257738 + 0.793236i
\(93\) −6.47214 − 4.70228i −0.671129 − 0.487604i
\(94\) −6.47214 − 4.70228i −0.667550 − 0.485003i
\(95\) 0 0
\(96\) −1.54508 − 4.75528i −0.157695 − 0.485334i
\(97\) −1.61803 + 1.17557i −0.164286 + 0.119361i −0.666891 − 0.745155i \(-0.732375\pi\)
0.502604 + 0.864517i \(0.332375\pi\)
\(98\) 9.00000 0.909137
\(99\) 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.e.124.1 4
11.2 odd 10 363.2.a.b.1.1 1
11.3 even 5 inner 363.2.e.e.148.1 4
11.4 even 5 inner 363.2.e.e.202.1 4
11.5 even 5 inner 363.2.e.e.130.1 4
11.6 odd 10 363.2.e.g.130.1 4
11.7 odd 10 363.2.e.g.202.1 4
11.8 odd 10 363.2.e.g.148.1 4
11.9 even 5 33.2.a.a.1.1 ✓ 1
11.10 odd 2 363.2.e.g.124.1 4
33.2 even 10 1089.2.a.j.1.1 1
33.20 odd 10 99.2.a.b.1.1 1
44.31 odd 10 528.2.a.g.1.1 1
44.35 even 10 5808.2.a.t.1.1 1
55.9 even 10 825.2.a.a.1.1 1
55.24 odd 10 9075.2.a.q.1.1 1
55.42 odd 20 825.2.c.a.199.2 2
55.53 odd 20 825.2.c.a.199.1 2
77.20 odd 10 1617.2.a.j.1.1 1
88.53 even 10 2112.2.a.bb.1.1 1
88.75 odd 10 2112.2.a.j.1.1 1
99.20 odd 30 891.2.e.g.595.1 2
99.31 even 15 891.2.e.e.298.1 2
99.86 odd 30 891.2.e.g.298.1 2
99.97 even 15 891.2.e.e.595.1 2
132.119 even 10 1584.2.a.o.1.1 1
143.64 even 10 5577.2.a.a.1.1 1
165.53 even 20 2475.2.c.d.199.2 2
165.119 odd 10 2475.2.a.g.1.1 1
165.152 even 20 2475.2.c.d.199.1 2
187.152 even 10 9537.2.a.m.1.1 1
231.20 even 10 4851.2.a.b.1.1 1
264.53 odd 10 6336.2.a.x.1.1 1
264.251 even 10 6336.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.a.a.1.1 ✓ 1 11.9 even 5
99.2.a.b.1.1 1 33.20 odd 10
363.2.a.b.1.1 1 11.2 odd 10
363.2.e.e.124.1 4 1.1 even 1 trivial
363.2.e.e.130.1 4 11.5 even 5 inner
363.2.e.e.148.1 4 11.3 even 5 inner
363.2.e.e.202.1 4 11.4 even 5 inner
363.2.e.g.124.1 4 11.10 odd 2
363.2.e.g.130.1 4 11.6 odd 10
363.2.e.g.148.1 4 11.8 odd 10
363.2.e.g.202.1 4 11.7 odd 10
528.2.a.g.1.1 1 44.31 odd 10
825.2.a.a.1.1 1 55.9 even 10
825.2.c.a.199.1 2 55.53 odd 20
825.2.c.a.199.2 2 55.42 odd 20
891.2.e.e.298.1 2 99.31 even 15
891.2.e.e.595.1 2 99.97 even 15
891.2.e.g.298.1 2 99.86 odd 30
891.2.e.g.595.1 2 99.20 odd 30
1089.2.a.j.1.1 1 33.2 even 10
1584.2.a.o.1.1 1 132.119 even 10
1617.2.a.j.1.1 1 77.20 odd 10
2112.2.a.j.1.1 1 88.75 odd 10
2112.2.a.bb.1.1 1 88.53 even 10
2475.2.a.g.1.1 1 165.119 odd 10
2475.2.c.d.199.1 2 165.152 even 20
2475.2.c.d.199.2 2 165.53 even 20
4851.2.a.b.1.1 1 231.20 even 10
5577.2.a.a.1.1 1 143.64 even 10
5808.2.a.t.1.1 1 44.35 even 10
6336.2.a.n.1.1 1 264.251 even 10
6336.2.a.x.1.1 1 264.53 odd 10
9075.2.a.q.1.1 1 55.24 odd 10
9537.2.a.m.1.1 1 187.152 even 10