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Properties

Label 363.2.e.f.130.1
Level $363$
Weight $2$
Character 363.130
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 = [4,1,1,-9] 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 130.1
Root \(-0.309017 - 0.951057i\) of defining polynomial
Character \(\chi\) \(=\) 363.130
Dual form 363.2.e.f.148.1

$q$-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 - 2.48990i) q^{2} +(0.809017 - 0.587785i) q^{3} +(-3.92705 - 2.85317i) q^{4} +(-0.190983 - 0.587785i) q^{5} +(-0.809017 - 2.48990i) q^{6} +(-0.809017 - 0.587785i) q^{7} +(-6.04508 + 4.39201i) q^{8} +(0.309017 - 0.951057i) q^{9} -1.61803 q^{10} -4.85410 q^{12} +(-0.0729490 + 0.224514i) q^{13} +(-2.11803 + 1.53884i) q^{14} +(-0.500000 - 0.363271i) q^{15} +(3.04508 + 9.37181i) q^{16} +(0.354102 + 1.08981i) q^{17} +(-2.11803 - 1.53884i) q^{18} +(4.73607 - 3.44095i) q^{19} +(-0.927051 + 2.85317i) q^{20} -1.00000 q^{21} +0.236068 q^{23} +(-2.30902 + 7.10642i) q^{24} +(3.73607 - 2.71441i) q^{25} +(0.500000 + 0.363271i) q^{26} +(-0.309017 - 0.951057i) q^{27} +(1.50000 + 4.61653i) q^{28} +(-4.85410 - 3.52671i) q^{29} +(-1.30902 + 0.951057i) q^{30} +(-1.88197 + 5.79210i) q^{31} +10.8541 q^{32} +3.00000 q^{34} +(-0.190983 + 0.587785i) q^{35} +(-3.92705 + 2.85317i) q^{36} +(5.04508 + 3.66547i) q^{37} +(-4.73607 - 14.5761i) q^{38} +(0.0729490 + 0.224514i) q^{39} +(3.73607 + 2.71441i) q^{40} +(0.190983 - 0.138757i) q^{41} +(-0.809017 + 2.48990i) q^{42} +6.70820 q^{43} -0.618034 q^{45} +(0.190983 - 0.587785i) q^{46} +(8.16312 - 5.93085i) q^{47} +(7.97214 + 5.79210i) q^{48} +(-1.85410 - 5.70634i) q^{49} +(-3.73607 - 11.4984i) q^{50} +(0.927051 + 0.673542i) q^{51} +(0.927051 - 0.673542i) q^{52} +(-0.118034 + 0.363271i) q^{53} -2.61803 q^{54} +7.47214 q^{56} +(1.80902 - 5.56758i) q^{57} +(-12.7082 + 9.23305i) q^{58} +(-5.97214 - 4.33901i) q^{59} +(0.927051 + 2.85317i) q^{60} +(3.57295 + 10.9964i) q^{61} +(12.8992 + 9.37181i) q^{62} +(-0.809017 + 0.587785i) q^{63} +(2.69098 - 8.28199i) q^{64} +0.145898 q^{65} +1.85410 q^{67} +(1.71885 - 5.29007i) q^{68} +(0.190983 - 0.138757i) q^{69} +(1.30902 + 0.951057i) q^{70} +(3.19098 + 9.82084i) q^{71} +(2.30902 + 7.10642i) q^{72} +(-4.61803 - 3.35520i) q^{73} +(13.2082 - 9.59632i) q^{74} +(1.42705 - 4.39201i) q^{75} -28.4164 q^{76} +0.618034 q^{78} +(-3.39919 + 10.4616i) q^{79} +(4.92705 - 3.57971i) q^{80} +(-0.809017 - 0.587785i) q^{81} +(-0.190983 - 0.587785i) q^{82} +(-0.454915 - 1.40008i) q^{83} +(3.92705 + 2.85317i) q^{84} +(0.572949 - 0.416272i) q^{85} +(5.42705 - 16.7027i) q^{86} -6.00000 q^{87} -8.23607 q^{89} +(-0.500000 + 1.53884i) q^{90} +(0.190983 - 0.138757i) q^{91} +(-0.927051 - 0.673542i) q^{92} +(1.88197 + 5.79210i) q^{93} +(-8.16312 - 25.1235i) q^{94} +(-2.92705 - 2.12663i) q^{95} +(8.78115 - 6.37988i) q^{96} +(2.42705 - 7.46969i) q^{97} -15.7082 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + q^{3} - 9 q^{4} - 3 q^{5} - q^{6} - q^{7} - 13 q^{8} - q^{9} - 2 q^{10} - 6 q^{12} - 7 q^{13} - 4 q^{14} - 2 q^{15} + q^{16} - 12 q^{17} - 4 q^{18} + 10 q^{19} + 3 q^{20} - 4 q^{21}+ \cdots - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{5}\right)\)

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 − 2.48990i 0.572061 − 1.76062i −0.0739128 − 0.997265i \(-0.523549\pi\)
0.645974 − 0.763359i \(-0.276451\pi\)
\(3\) 0.809017 − 0.587785i 0.467086 − 0.339358i
\(4\) −3.92705 − 2.85317i −1.96353 − 1.42658i
\(5\) −0.190983 − 0.587785i −0.0854102 − 0.262866i 0.899226 − 0.437485i \(-0.144131\pi\)
−0.984636 + 0.174619i \(0.944131\pi\)
\(6\) −0.809017 − 2.48990i −0.330280 − 1.01650i
\(7\) −0.809017 − 0.587785i −0.305780 − 0.222162i 0.424304 − 0.905520i \(-0.360519\pi\)
−0.730084 + 0.683358i \(0.760519\pi\)
\(8\) −6.04508 + 4.39201i −2.13726 + 1.55281i
\(9\) 0.309017 − 0.951057i 0.103006 − 0.317019i
\(10\) −1.61803 −0.511667
\(11\) 0 0
\(12\) −4.85410 −1.40126
\(13\) −0.0729490 + 0.224514i −0.0202324 + 0.0622690i −0.960663 − 0.277717i \(-0.910422\pi\)
0.940431 + 0.339986i \(0.110422\pi\)
\(14\) −2.11803 + 1.53884i −0.566068 + 0.411273i
\(15\) −0.500000 − 0.363271i −0.129099 − 0.0937962i
\(16\) 3.04508 + 9.37181i 0.761271 + 2.34295i
\(17\) 0.354102 + 1.08981i 0.0858823 + 0.264319i 0.984770 − 0.173860i \(-0.0556239\pi\)
−0.898888 + 0.438178i \(0.855624\pi\)
\(18\) −2.11803 − 1.53884i −0.499225 − 0.362708i
\(19\) 4.73607 − 3.44095i 1.08653 − 0.789409i 0.107719 − 0.994181i \(-0.465645\pi\)
0.978810 + 0.204772i \(0.0656454\pi\)
\(20\) −0.927051 + 2.85317i −0.207295 + 0.637988i
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 0.236068 0.0492236 0.0246118 − 0.999697i \(-0.492165\pi\)
0.0246118 + 0.999697i \(0.492165\pi\)
\(24\) −2.30902 + 7.10642i −0.471326 + 1.45059i
\(25\) 3.73607 − 2.71441i 0.747214 − 0.542882i
\(26\) 0.500000 + 0.363271i 0.0980581 + 0.0712434i
\(27\) −0.309017 − 0.951057i −0.0594703 − 0.183031i
\(28\) 1.50000 + 4.61653i 0.283473 + 0.872441i
\(29\) −4.85410 − 3.52671i −0.901384 − 0.654894i 0.0374370 − 0.999299i \(-0.488081\pi\)
−0.938821 + 0.344405i \(0.888081\pi\)
\(30\) −1.30902 + 0.951057i −0.238993 + 0.173638i
\(31\) −1.88197 + 5.79210i −0.338011 + 1.04029i 0.627209 + 0.778851i \(0.284197\pi\)
−0.965220 + 0.261440i \(0.915803\pi\)
\(32\) 10.8541 1.91875
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) −0.190983 + 0.587785i −0.0322820 + 0.0993538i
\(36\) −3.92705 + 2.85317i −0.654508 + 0.475528i
\(37\) 5.04508 + 3.66547i 0.829407 + 0.602599i 0.919391 − 0.393344i \(-0.128682\pi\)
−0.0899846 + 0.995943i \(0.528682\pi\)
\(38\) −4.73607 − 14.5761i −0.768292 − 2.36456i
\(39\) 0.0729490 + 0.224514i 0.0116812 + 0.0359510i
\(40\) 3.73607 + 2.71441i 0.590724 + 0.429186i
\(41\) 0.190983 − 0.138757i 0.0298265 − 0.0216702i −0.572772 − 0.819715i \(-0.694132\pi\)
0.602599 + 0.798044i \(0.294132\pi\)
\(42\) −0.809017 + 2.48990i −0.124834 + 0.384200i
\(43\) 6.70820 1.02299 0.511496 − 0.859286i \(-0.329092\pi\)
0.511496 + 0.859286i \(0.329092\pi\)
\(44\) 0 0
\(45\) −0.618034 −0.0921311
\(46\) 0.190983 − 0.587785i 0.0281589 − 0.0866642i
\(47\) 8.16312 − 5.93085i 1.19071 − 0.865104i 0.197374 − 0.980328i \(-0.436759\pi\)
0.993339 + 0.115224i \(0.0367587\pi\)
\(48\) 7.97214 + 5.79210i 1.15068 + 0.836017i
\(49\) −1.85410 − 5.70634i −0.264872 − 0.815191i
\(50\) −3.73607 − 11.4984i −0.528360 − 1.62612i
\(51\) 0.927051 + 0.673542i 0.129813 + 0.0943147i
\(52\) 0.927051 − 0.673542i 0.128559 − 0.0934035i
\(53\) −0.118034 + 0.363271i −0.0162132 + 0.0498991i −0.958836 − 0.283961i \(-0.908351\pi\)
0.942623 + 0.333860i \(0.108351\pi\)
\(54\) −2.61803 −0.356269
\(55\) 0 0
\(56\) 7.47214 0.998506
\(57\) 1.80902 − 5.56758i 0.239610 − 0.737444i
\(58\) −12.7082 + 9.23305i −1.66867 + 1.21236i
\(59\) −5.97214 − 4.33901i −0.777506 − 0.564891i 0.126724 − 0.991938i \(-0.459554\pi\)
−0.904229 + 0.427047i \(0.859554\pi\)
\(60\) 0.927051 + 2.85317i 0.119682 + 0.368343i
\(61\) 3.57295 + 10.9964i 0.457469 + 1.40795i 0.868212 + 0.496194i \(0.165270\pi\)
−0.410742 + 0.911751i \(0.634730\pi\)
\(62\) 12.8992 + 9.37181i 1.63820 + 1.19022i
\(63\) −0.809017 + 0.587785i −0.101927 + 0.0740540i
\(64\) 2.69098 − 8.28199i 0.336373 − 1.03525i
\(65\) 0.145898 0.0180964
\(66\) 0 0
\(67\) 1.85410 0.226515 0.113257 − 0.993566i \(-0.463872\pi\)
0.113257 + 0.993566i \(0.463872\pi\)
\(68\) 1.71885 − 5.29007i 0.208441 − 0.641515i
\(69\) 0.190983 − 0.138757i 0.0229917 − 0.0167044i
\(70\) 1.30902 + 0.951057i 0.156457 + 0.113673i
\(71\) 3.19098 + 9.82084i 0.378700 + 1.16552i 0.940948 + 0.338550i \(0.109937\pi\)
−0.562248 + 0.826968i \(0.690063\pi\)
\(72\) 2.30902 + 7.10642i 0.272120 + 0.837500i
\(73\) −4.61803 − 3.35520i −0.540500 − 0.392696i 0.283771 − 0.958892i \(-0.408415\pi\)
−0.824271 + 0.566196i \(0.808415\pi\)
\(74\) 13.2082 − 9.59632i 1.53542 − 1.11555i
\(75\) 1.42705 − 4.39201i 0.164782 − 0.507146i
\(76\) −28.4164 −3.25959
\(77\) 0 0
\(78\) 0.618034 0.0699786
\(79\) −3.39919 + 10.4616i −0.382438 + 1.17702i 0.555883 + 0.831260i \(0.312380\pi\)
−0.938322 + 0.345764i \(0.887620\pi\)
\(80\) 4.92705 − 3.57971i 0.550861 − 0.400224i
\(81\) −0.809017 − 0.587785i −0.0898908 − 0.0653095i
\(82\) −0.190983 − 0.587785i −0.0210905 − 0.0649100i
\(83\) −0.454915 − 1.40008i −0.0499334 − 0.153679i 0.922981 − 0.384846i \(-0.125746\pi\)
−0.972914 + 0.231167i \(0.925746\pi\)
\(84\) 3.92705 + 2.85317i 0.428476 + 0.311306i
\(85\) 0.572949 − 0.416272i 0.0621450 − 0.0451510i
\(86\) 5.42705 − 16.7027i 0.585214 − 1.80110i
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −8.23607 −0.873021 −0.436511 − 0.899699i \(-0.643786\pi\)
−0.436511 + 0.899699i \(0.643786\pi\)
\(90\) −0.500000 + 1.53884i −0.0527046 + 0.162208i
\(91\) 0.190983 − 0.138757i 0.0200205 − 0.0145457i
\(92\) −0.927051 − 0.673542i −0.0966517 − 0.0702216i
\(93\) 1.88197 + 5.79210i 0.195151 + 0.600612i
\(94\) −8.16312 − 25.1235i −0.841961 − 2.59129i
\(95\) −2.92705 − 2.12663i −0.300309 − 0.218187i
\(96\) 8.78115 − 6.37988i 0.896223 − 0.651144i
\(97\) 2.42705 − 7.46969i 0.246430 − 0.758433i −0.748968 − 0.662606i \(-0.769451\pi\)
0.995398 − 0.0958268i \(-0.0305495\pi\)
\(98\) −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.f.130.1 4
11.2 odd 10 363.2.e.k.124.1 4
11.3 even 5 363.2.e.b.202.1 4
11.4 even 5 363.2.a.i.1.2 2
11.5 even 5 inner 363.2.e.f.148.1 4
11.6 odd 10 33.2.e.b.16.1 ✓ 4
11.7 odd 10 363.2.a.d.1.1 2
11.8 odd 10 363.2.e.k.202.1 4
11.9 even 5 363.2.e.b.124.1 4
11.10 odd 2 33.2.e.b.31.1 yes 4
33.17 even 10 99.2.f.a.82.1 4
33.26 odd 10 1089.2.a.l.1.1 2
33.29 even 10 1089.2.a.t.1.2 2
33.32 even 2 99.2.f.a.64.1 4
44.7 even 10 5808.2.a.cj.1.1 2
44.15 odd 10 5808.2.a.ci.1.1 2
44.39 even 10 528.2.y.b.49.1 4
44.43 even 2 528.2.y.b.97.1 4
55.4 even 10 9075.2.a.u.1.1 2
55.17 even 20 825.2.bx.d.49.2 8
55.28 even 20 825.2.bx.d.49.1 8
55.29 odd 10 9075.2.a.cb.1.2 2
55.32 even 4 825.2.bx.d.724.1 8
55.39 odd 10 825.2.n.c.676.1 4
55.43 even 4 825.2.bx.d.724.2 8
55.54 odd 2 825.2.n.c.526.1 4
99.32 even 6 891.2.n.b.460.1 8
99.43 odd 6 891.2.n.c.757.1 8
99.50 even 30 891.2.n.b.379.1 8
99.61 odd 30 891.2.n.c.676.1 8
99.65 even 6 891.2.n.b.757.1 8
99.76 odd 6 891.2.n.c.460.1 8
99.83 even 30 891.2.n.b.676.1 8
99.94 odd 30 891.2.n.c.379.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.b.16.1 ✓ 4 11.6 odd 10
33.2.e.b.31.1 yes 4 11.10 odd 2
99.2.f.a.64.1 4 33.32 even 2
99.2.f.a.82.1 4 33.17 even 10
363.2.a.d.1.1 2 11.7 odd 10
363.2.a.i.1.2 2 11.4 even 5
363.2.e.b.124.1 4 11.9 even 5
363.2.e.b.202.1 4 11.3 even 5
363.2.e.f.130.1 4 1.1 even 1 trivial
363.2.e.f.148.1 4 11.5 even 5 inner
363.2.e.k.124.1 4 11.2 odd 10
363.2.e.k.202.1 4 11.8 odd 10
528.2.y.b.49.1 4 44.39 even 10
528.2.y.b.97.1 4 44.43 even 2
825.2.n.c.526.1 4 55.54 odd 2
825.2.n.c.676.1 4 55.39 odd 10
825.2.bx.d.49.1 8 55.28 even 20
825.2.bx.d.49.2 8 55.17 even 20
825.2.bx.d.724.1 8 55.32 even 4
825.2.bx.d.724.2 8 55.43 even 4
891.2.n.b.379.1 8 99.50 even 30
891.2.n.b.460.1 8 99.32 even 6
891.2.n.b.676.1 8 99.83 even 30
891.2.n.b.757.1 8 99.65 even 6
891.2.n.c.379.1 8 99.94 odd 30
891.2.n.c.460.1 8 99.76 odd 6
891.2.n.c.676.1 8 99.61 odd 30
891.2.n.c.757.1 8 99.43 odd 6
1089.2.a.l.1.1 2 33.26 odd 10
1089.2.a.t.1.2 2 33.29 even 10
5808.2.a.ci.1.1 2 44.15 odd 10
5808.2.a.cj.1.1 2 44.7 even 10
9075.2.a.u.1.1 2 55.4 even 10
9075.2.a.cb.1.2 2 55.29 odd 10