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Properties

Label 363.2.e.k.202.1
Level $363$
Weight $2$
Character 363.202
Analytic conductor $2.899$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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 = [4,4,1,6] 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: \(\Z[a_1, a_2, a_3]\)
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.k.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+(2.11803 + 1.53884i) q^{2} +(-0.309017 + 0.951057i) q^{3} +(1.50000 + 4.61653i) q^{4} +(0.500000 - 0.363271i) q^{5} +(-2.11803 + 1.53884i) q^{6} +(-0.309017 - 0.951057i) q^{7} +(-2.30902 + 7.10642i) q^{8} +(-0.809017 - 0.587785i) q^{9} +1.61803 q^{10} -4.85410 q^{12} +(-0.190983 - 0.138757i) q^{13} +(0.809017 - 2.48990i) q^{14} +(0.190983 + 0.587785i) q^{15} +(-7.97214 + 5.79210i) q^{16} +(0.927051 - 0.673542i) q^{17} +(-0.809017 - 2.48990i) q^{18} +(1.80902 - 5.56758i) q^{19} +(2.42705 + 1.76336i) q^{20} +1.00000 q^{21} +0.236068 q^{23} +(-6.04508 - 4.39201i) q^{24} +(-1.42705 + 4.39201i) q^{25} +(-0.190983 - 0.587785i) q^{26} +(0.809017 - 0.587785i) q^{27} +(3.92705 - 2.85317i) q^{28} +(-1.85410 - 5.70634i) q^{29} +(-0.500000 + 1.53884i) q^{30} +(4.92705 + 3.57971i) q^{31} -10.8541 q^{32} +3.00000 q^{34} +(-0.500000 - 0.363271i) q^{35} +(1.50000 - 4.61653i) q^{36} +(-1.92705 - 5.93085i) q^{37} +(12.3992 - 9.00854i) q^{38} +(0.190983 - 0.138757i) q^{39} +(1.42705 + 4.39201i) q^{40} +(0.0729490 - 0.224514i) q^{41} +(2.11803 + 1.53884i) q^{42} -6.70820 q^{43} -0.618034 q^{45} +(0.500000 + 0.363271i) q^{46} +(-3.11803 + 9.59632i) q^{47} +(-3.04508 - 9.37181i) q^{48} +(4.85410 - 3.52671i) q^{49} +(-9.78115 + 7.10642i) q^{50} +(0.354102 + 1.08981i) q^{51} +(0.354102 - 1.08981i) q^{52} +(0.309017 + 0.224514i) q^{53} +2.61803 q^{54} +7.47214 q^{56} +(4.73607 + 3.44095i) q^{57} +(4.85410 - 14.9394i) q^{58} +(2.28115 + 7.02067i) q^{59} +(-2.42705 + 1.76336i) q^{60} +(9.35410 - 6.79615i) q^{61} +(4.92705 + 15.1639i) q^{62} +(-0.309017 + 0.951057i) q^{63} +(-7.04508 - 5.11855i) q^{64} -0.145898 q^{65} +1.85410 q^{67} +(4.50000 + 3.26944i) q^{68} +(-0.0729490 + 0.224514i) q^{69} +(-0.500000 - 1.53884i) q^{70} +(-8.35410 + 6.06961i) q^{71} +(6.04508 - 4.39201i) q^{72} +(-1.76393 - 5.42882i) q^{73} +(5.04508 - 15.5272i) q^{74} +(-3.73607 - 2.71441i) q^{75} +28.4164 q^{76} +0.618034 q^{78} +(-8.89919 - 6.46564i) q^{79} +(-1.88197 + 5.79210i) q^{80} +(0.309017 + 0.951057i) q^{81} +(0.500000 - 0.363271i) q^{82} +(-1.19098 + 0.865300i) q^{83} +(1.50000 + 4.61653i) q^{84} +(0.218847 - 0.673542i) q^{85} +(-14.2082 - 10.3229i) q^{86} +6.00000 q^{87} -8.23607 q^{89} +(-1.30902 - 0.951057i) q^{90} +(-0.0729490 + 0.224514i) q^{91} +(0.354102 + 1.08981i) q^{92} +(-4.92705 + 3.57971i) q^{93} +(-21.3713 + 15.5272i) q^{94} +(-1.11803 - 3.44095i) q^{95} +(3.35410 - 10.3229i) q^{96} +(-6.35410 - 4.61653i) q^{97} +15.7082 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + q^{3} + 6 q^{4} + 2 q^{5} - 4 q^{6} + q^{7} - 7 q^{8} - q^{9} + 2 q^{10} - 6 q^{12} - 3 q^{13} + q^{14} + 3 q^{15} - 14 q^{16} - 3 q^{17} - q^{18} + 5 q^{19} + 3 q^{20} + 4 q^{21} - 8 q^{23}+ \cdots + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.11803 + 1.53884i 1.49768 + 1.08813i 0.971295 + 0.237877i \(0.0764514\pi\)
0.526381 + 0.850249i \(0.323549\pi\)
\(3\) −0.309017 + 0.951057i −0.178411 + 0.549093i
\(4\) 1.50000 + 4.61653i 0.750000 + 2.30826i
\(5\) 0.500000 − 0.363271i 0.223607 − 0.162460i −0.470342 − 0.882484i \(-0.655869\pi\)
0.693949 + 0.720024i \(0.255869\pi\)
\(6\) −2.11803 + 1.53884i −0.864684 + 0.628230i
\(7\) −0.309017 − 0.951057i −0.116797 − 0.359466i 0.875520 − 0.483181i \(-0.160519\pi\)
−0.992318 + 0.123716i \(0.960519\pi\)
\(8\) −2.30902 + 7.10642i −0.816361 + 2.51250i
\(9\) −0.809017 − 0.587785i −0.269672 − 0.195928i
\(10\) 1.61803 0.511667
\(11\) 0 0
\(12\) −4.85410 −1.40126
\(13\) −0.190983 − 0.138757i −0.0529692 − 0.0384843i 0.560986 − 0.827826i \(-0.310422\pi\)
−0.613955 + 0.789341i \(0.710422\pi\)
\(14\) 0.809017 − 2.48990i 0.216219 − 0.665453i
\(15\) 0.190983 + 0.587785i 0.0493116 + 0.151765i
\(16\) −7.97214 + 5.79210i −1.99303 + 1.44802i
\(17\) 0.927051 − 0.673542i 0.224843 − 0.163358i −0.469661 − 0.882847i \(-0.655624\pi\)
0.694504 + 0.719489i \(0.255624\pi\)
\(18\) −0.809017 − 2.48990i −0.190687 − 0.586875i
\(19\) 1.80902 − 5.56758i 0.415017 − 1.27729i −0.497219 − 0.867625i \(-0.665645\pi\)
0.912236 − 0.409666i \(-0.134355\pi\)
\(20\) 2.42705 + 1.76336i 0.542705 + 0.394298i
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 0.236068 0.0492236 0.0246118 − 0.999697i \(-0.492165\pi\)
0.0246118 + 0.999697i \(0.492165\pi\)
\(24\) −6.04508 − 4.39201i −1.23395 − 0.896516i
\(25\) −1.42705 + 4.39201i −0.285410 + 0.878402i
\(26\) −0.190983 − 0.587785i −0.0374548 − 0.115274i
\(27\) 0.809017 − 0.587785i 0.155695 − 0.113119i
\(28\) 3.92705 − 2.85317i 0.742143 − 0.539198i
\(29\) −1.85410 − 5.70634i −0.344298 − 1.05964i −0.961958 − 0.273196i \(-0.911919\pi\)
0.617660 − 0.786445i \(-0.288081\pi\)
\(30\) −0.500000 + 1.53884i −0.0912871 + 0.280953i
\(31\) 4.92705 + 3.57971i 0.884924 + 0.642935i 0.934550 − 0.355833i \(-0.115803\pi\)
−0.0496252 + 0.998768i \(0.515803\pi\)
\(32\) −10.8541 −1.91875
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) −0.500000 − 0.363271i −0.0845154 − 0.0614041i
\(36\) 1.50000 − 4.61653i 0.250000 − 0.769421i
\(37\) −1.92705 − 5.93085i −0.316805 − 0.975026i −0.975005 − 0.222183i \(-0.928682\pi\)
0.658200 − 0.752843i \(-0.271318\pi\)
\(38\) 12.3992 − 9.00854i 2.01141 − 1.46138i
\(39\) 0.190983 − 0.138757i 0.0305818 − 0.0222189i
\(40\) 1.42705 + 4.39201i 0.225637 + 0.694438i
\(41\) 0.0729490 − 0.224514i 0.0113927 − 0.0350632i −0.945199 − 0.326496i \(-0.894132\pi\)
0.956591 + 0.291433i \(0.0941320\pi\)
\(42\) 2.11803 + 1.53884i 0.326820 + 0.237448i
\(43\) −6.70820 −1.02299 −0.511496 − 0.859286i \(-0.670908\pi\)
−0.511496 + 0.859286i \(0.670908\pi\)
\(44\) 0 0
\(45\) −0.618034 −0.0921311
\(46\) 0.500000 + 0.363271i 0.0737210 + 0.0535614i
\(47\) −3.11803 + 9.59632i −0.454812 + 1.39977i 0.416544 + 0.909116i \(0.363241\pi\)
−0.871356 + 0.490652i \(0.836759\pi\)
\(48\) −3.04508 − 9.37181i −0.439520 − 1.35270i
\(49\) 4.85410 − 3.52671i 0.693443 − 0.503816i
\(50\) −9.78115 + 7.10642i −1.38326 + 1.00500i
\(51\) 0.354102 + 1.08981i 0.0495842 + 0.152604i
\(52\) 0.354102 − 1.08981i 0.0491051 − 0.151130i
\(53\) 0.309017 + 0.224514i 0.0424467 + 0.0308394i 0.608806 − 0.793319i \(-0.291649\pi\)
−0.566360 + 0.824158i \(0.691649\pi\)
\(54\) 2.61803 0.356269
\(55\) 0 0
\(56\) 7.47214 0.998506
\(57\) 4.73607 + 3.44095i 0.627308 + 0.455766i
\(58\) 4.85410 − 14.9394i 0.637375 − 1.96164i
\(59\) 2.28115 + 7.02067i 0.296981 + 0.914013i 0.982549 + 0.186004i \(0.0595539\pi\)
−0.685568 + 0.728009i \(0.740446\pi\)
\(60\) −2.42705 + 1.76336i −0.313331 + 0.227648i
\(61\) 9.35410 − 6.79615i 1.19767 − 0.870158i 0.203617 − 0.979051i \(-0.434730\pi\)
0.994054 + 0.108893i \(0.0347304\pi\)
\(62\) 4.92705 + 15.1639i 0.625736 + 1.92582i
\(63\) −0.309017 + 0.951057i −0.0389325 + 0.119822i
\(64\) −7.04508 − 5.11855i −0.880636 − 0.639819i
\(65\) −0.145898 −0.0180964
\(66\) 0 0
\(67\) 1.85410 0.226515 0.113257 − 0.993566i \(-0.463872\pi\)
0.113257 + 0.993566i \(0.463872\pi\)
\(68\) 4.50000 + 3.26944i 0.545705 + 0.396478i
\(69\) −0.0729490 + 0.224514i −0.00878203 + 0.0270283i
\(70\) −0.500000 − 1.53884i −0.0597614 − 0.183927i
\(71\) −8.35410 + 6.06961i −0.991449 + 0.720330i −0.960238 − 0.279183i \(-0.909937\pi\)
−0.0312115 + 0.999513i \(0.509937\pi\)
\(72\) 6.04508 − 4.39201i 0.712420 − 0.517603i
\(73\) −1.76393 − 5.42882i −0.206453 − 0.635396i −0.999651 − 0.0264320i \(-0.991585\pi\)
0.793198 − 0.608964i \(-0.208415\pi\)
\(74\) 5.04508 − 15.5272i 0.586479 − 1.80500i
\(75\) −3.73607 − 2.71441i −0.431404 − 0.313433i
\(76\) 28.4164 3.25959
\(77\) 0 0
\(78\) 0.618034 0.0699786
\(79\) −8.89919 − 6.46564i −1.00124 − 0.727441i −0.0388837 − 0.999244i \(-0.512380\pi\)
−0.962353 + 0.271803i \(0.912380\pi\)
\(80\) −1.88197 + 5.79210i −0.210410 + 0.647576i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) 0.500000 − 0.363271i 0.0552158 − 0.0401166i
\(83\) −1.19098 + 0.865300i −0.130727 + 0.0949790i −0.651227 − 0.758883i \(-0.725746\pi\)
0.520500 + 0.853862i \(0.325746\pi\)
\(84\) 1.50000 + 4.61653i 0.163663 + 0.503704i
\(85\) 0.218847 − 0.673542i 0.0237373 − 0.0730559i
\(86\) −14.2082 − 10.3229i −1.53211 − 1.11314i
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −8.23607 −0.873021 −0.436511 − 0.899699i \(-0.643786\pi\)
−0.436511 + 0.899699i \(0.643786\pi\)
\(90\) −1.30902 − 0.951057i −0.137983 − 0.100250i
\(91\) −0.0729490 + 0.224514i −0.00764713 + 0.0235355i
\(92\) 0.354102 + 1.08981i 0.0369177 + 0.113621i
\(93\) −4.92705 + 3.57971i −0.510911 + 0.371199i
\(94\) −21.3713 + 15.5272i −2.20428 + 1.60151i
\(95\) −1.11803 − 3.44095i −0.114708 − 0.353035i
\(96\) 3.35410 − 10.3229i 0.342327 − 1.05357i
\(97\) −6.35410 − 4.61653i −0.645161 − 0.468737i 0.216458 − 0.976292i \(-0.430549\pi\)
−0.861620 + 0.507555i \(0.830549\pi\)
\(98\) 15.7082 1.58677
\(99\) 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.k.202.1 4
11.2 odd 10 363.2.e.f.148.1 4
11.3 even 5 inner 363.2.e.k.124.1 4
11.4 even 5 33.2.e.b.31.1 yes 4
11.5 even 5 363.2.a.d.1.1 2
11.6 odd 10 363.2.a.i.1.2 2
11.7 odd 10 363.2.e.f.130.1 4
11.8 odd 10 363.2.e.b.124.1 4
11.9 even 5 33.2.e.b.16.1 ✓ 4
11.10 odd 2 363.2.e.b.202.1 4
33.5 odd 10 1089.2.a.t.1.2 2
33.17 even 10 1089.2.a.l.1.1 2
33.20 odd 10 99.2.f.a.82.1 4
33.26 odd 10 99.2.f.a.64.1 4
44.15 odd 10 528.2.y.b.97.1 4
44.27 odd 10 5808.2.a.cj.1.1 2
44.31 odd 10 528.2.y.b.49.1 4
44.39 even 10 5808.2.a.ci.1.1 2
55.4 even 10 825.2.n.c.526.1 4
55.9 even 10 825.2.n.c.676.1 4
55.37 odd 20 825.2.bx.d.724.1 8
55.39 odd 10 9075.2.a.u.1.1 2
55.42 odd 20 825.2.bx.d.49.2 8
55.48 odd 20 825.2.bx.d.724.2 8
55.49 even 10 9075.2.a.cb.1.2 2
55.53 odd 20 825.2.bx.d.49.1 8
99.4 even 15 891.2.n.c.460.1 8
99.20 odd 30 891.2.n.b.676.1 8
99.31 even 15 891.2.n.c.379.1 8
99.59 odd 30 891.2.n.b.460.1 8
99.70 even 15 891.2.n.c.757.1 8
99.86 odd 30 891.2.n.b.379.1 8
99.92 odd 30 891.2.n.b.757.1 8
99.97 even 15 891.2.n.c.676.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.b.16.1 ✓ 4 11.9 even 5
33.2.e.b.31.1 yes 4 11.4 even 5
99.2.f.a.64.1 4 33.26 odd 10
99.2.f.a.82.1 4 33.20 odd 10
363.2.a.d.1.1 2 11.5 even 5
363.2.a.i.1.2 2 11.6 odd 10
363.2.e.b.124.1 4 11.8 odd 10
363.2.e.b.202.1 4 11.10 odd 2
363.2.e.f.130.1 4 11.7 odd 10
363.2.e.f.148.1 4 11.2 odd 10
363.2.e.k.124.1 4 11.3 even 5 inner
363.2.e.k.202.1 4 1.1 even 1 trivial
528.2.y.b.49.1 4 44.31 odd 10
528.2.y.b.97.1 4 44.15 odd 10
825.2.n.c.526.1 4 55.4 even 10
825.2.n.c.676.1 4 55.9 even 10
825.2.bx.d.49.1 8 55.53 odd 20
825.2.bx.d.49.2 8 55.42 odd 20
825.2.bx.d.724.1 8 55.37 odd 20
825.2.bx.d.724.2 8 55.48 odd 20
891.2.n.b.379.1 8 99.86 odd 30
891.2.n.b.460.1 8 99.59 odd 30
891.2.n.b.676.1 8 99.20 odd 30
891.2.n.b.757.1 8 99.92 odd 30
891.2.n.c.379.1 8 99.31 even 15
891.2.n.c.460.1 8 99.4 even 15
891.2.n.c.676.1 8 99.97 even 15
891.2.n.c.757.1 8 99.70 even 15
1089.2.a.l.1.1 2 33.17 even 10
1089.2.a.t.1.2 2 33.5 odd 10
5808.2.a.ci.1.1 2 44.39 even 10
5808.2.a.cj.1.1 2 44.27 odd 10
9075.2.a.u.1.1 2 55.39 odd 10
9075.2.a.cb.1.2 2 55.49 even 10