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 = [4,3,-1,3] 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.j.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+(1.30902 + 0.951057i) q^{2} +(0.309017 - 0.951057i) q^{3} +(0.190983 + 0.587785i) q^{4} +(0.309017 - 0.224514i) q^{5} +(1.30902 - 0.951057i) q^{6} +(-0.927051 - 2.85317i) q^{7} +(0.690983 - 2.12663i) q^{8} +(-0.809017 - 0.587785i) q^{9} +0.618034 q^{10} +0.618034 q^{12} +(5.04508 + 3.66547i) q^{13} +(1.50000 - 4.61653i) q^{14} +(-0.118034 - 0.363271i) q^{15} +(3.92705 - 2.85317i) q^{16} +(-0.500000 + 0.363271i) q^{17} +(-0.500000 - 1.53884i) q^{18} +(0.263932 - 0.812299i) q^{19} +(0.190983 + 0.138757i) q^{20} -3.00000 q^{21} -5.47214 q^{23} +(-1.80902 - 1.31433i) q^{24} +(-1.50000 + 4.61653i) q^{25} +(3.11803 + 9.59632i) q^{26} +(-0.809017 + 0.587785i) q^{27} +(1.50000 - 1.08981i) q^{28} +(1.38197 + 4.25325i) q^{29} +(0.190983 - 0.587785i) q^{30} +(3.11803 + 2.26538i) q^{31} +3.38197 q^{32} -1.00000 q^{34} +(-0.927051 - 0.673542i) q^{35} +(0.190983 - 0.587785i) q^{36} +(-1.30902 - 4.02874i) q^{37} +(1.11803 - 0.812299i) q^{38} +(5.04508 - 3.66547i) q^{39} +(-0.263932 - 0.812299i) q^{40} +(-1.83688 + 5.65334i) q^{41} +(-3.92705 - 2.85317i) q^{42} -1.76393 q^{43} -0.381966 q^{45} +(-7.16312 - 5.20431i) q^{46} +(-0.190983 + 0.587785i) q^{47} +(-1.50000 - 4.61653i) q^{48} +(-1.61803 + 1.17557i) q^{49} +(-6.35410 + 4.61653i) q^{50} +(0.190983 + 0.587785i) q^{51} +(-1.19098 + 3.66547i) q^{52} +(5.97214 + 4.33901i) q^{53} -1.61803 q^{54} -6.70820 q^{56} +(-0.690983 - 0.502029i) q^{57} +(-2.23607 + 6.88191i) q^{58} +(-1.64590 - 5.06555i) q^{59} +(0.190983 - 0.138757i) q^{60} +(0.927051 - 0.673542i) q^{61} +(1.92705 + 5.93085i) q^{62} +(-0.927051 + 2.85317i) q^{63} +(-3.42705 - 2.48990i) q^{64} +2.38197 q^{65} +10.5623 q^{67} +(-0.309017 - 0.224514i) q^{68} +(-1.69098 + 5.20431i) q^{69} +(-0.572949 - 1.76336i) q^{70} +(-11.7812 + 8.55951i) q^{71} +(-1.80902 + 1.31433i) q^{72} +(-0.381966 - 1.17557i) q^{73} +(2.11803 - 6.51864i) q^{74} +(3.92705 + 2.85317i) q^{75} +0.527864 q^{76} +10.0902 q^{78} +(0.427051 + 0.310271i) q^{79} +(0.572949 - 1.76336i) q^{80} +(0.309017 + 0.951057i) q^{81} +(-7.78115 + 5.65334i) q^{82} +(-10.2812 + 7.46969i) q^{83} +(-0.572949 - 1.76336i) q^{84} +(-0.0729490 + 0.224514i) q^{85} +(-2.30902 - 1.67760i) q^{86} +4.47214 q^{87} +9.47214 q^{89} +(-0.500000 - 0.363271i) q^{90} +(5.78115 - 17.7926i) q^{91} +(-1.04508 - 3.21644i) q^{92} +(3.11803 - 2.26538i) q^{93} +(-0.809017 + 0.587785i) q^{94} +(-0.100813 - 0.310271i) q^{95} +(1.04508 - 3.21644i) q^{96} +(-12.1631 - 8.83702i) q^{97} -3.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} + 3 q^{6} + 3 q^{7} + 5 q^{8} - q^{9} - 2 q^{10} - 2 q^{12} + 9 q^{13} + 6 q^{14} + 4 q^{15} + 9 q^{16} - 2 q^{17} - 2 q^{18} + 10 q^{19} + 3 q^{20} - 12 q^{21}+ \cdots - 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.30902 + 0.951057i 0.925615 + 0.672499i 0.944915 − 0.327315i \(-0.106144\pi\)
−0.0193004 + 0.999814i \(0.506144\pi\)
\(3\) 0.309017 − 0.951057i 0.178411 − 0.549093i
\(4\) 0.190983 + 0.587785i 0.0954915 + 0.293893i
\(5\) 0.309017 − 0.224514i 0.138197 − 0.100406i −0.516539 − 0.856264i \(-0.672780\pi\)
0.654736 + 0.755858i \(0.272780\pi\)
\(6\) 1.30902 − 0.951057i 0.534404 − 0.388267i
\(7\) −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
\(10\) 0.618034 0.195440
\(11\) 0 0
\(12\) 0.618034 0.178411
\(13\) 5.04508 + 3.66547i 1.39925 + 1.01662i 0.994777 + 0.102070i \(0.0325466\pi\)
0.404478 + 0.914548i \(0.367453\pi\)
\(14\) 1.50000 − 4.61653i 0.400892 − 1.23382i
\(15\) −0.118034 − 0.363271i −0.0304762 − 0.0937962i
\(16\) 3.92705 − 2.85317i 0.981763 − 0.713292i
\(17\) −0.500000 + 0.363271i −0.121268 + 0.0881062i −0.646766 − 0.762688i \(-0.723879\pi\)
0.525498 + 0.850795i \(0.323879\pi\)
\(18\) −0.500000 − 1.53884i −0.117851 − 0.362708i
\(19\) 0.263932 − 0.812299i 0.0605502 − 0.186354i −0.916206 − 0.400707i \(-0.868764\pi\)
0.976756 + 0.214353i \(0.0687644\pi\)
\(20\) 0.190983 + 0.138757i 0.0427051 + 0.0310271i
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) −5.47214 −1.14102 −0.570510 − 0.821291i \(-0.693254\pi\)
−0.570510 + 0.821291i \(0.693254\pi\)
\(24\) −1.80902 − 1.31433i −0.369264 − 0.268286i
\(25\) −1.50000 + 4.61653i −0.300000 + 0.923305i
\(26\) 3.11803 + 9.59632i 0.611497 + 1.88199i
\(27\) −0.809017 + 0.587785i −0.155695 + 0.113119i
\(28\) 1.50000 − 1.08981i 0.283473 − 0.205955i
\(29\) 1.38197 + 4.25325i 0.256625 + 0.789809i 0.993505 + 0.113787i \(0.0362980\pi\)
−0.736881 + 0.676023i \(0.763702\pi\)
\(30\) 0.190983 − 0.587785i 0.0348686 − 0.107314i
\(31\) 3.11803 + 2.26538i 0.560015 + 0.406875i 0.831465 − 0.555578i \(-0.187503\pi\)
−0.271449 + 0.962453i \(0.587503\pi\)
\(32\) 3.38197 0.597853
\(33\) 0 0
\(34\) −1.00000 −0.171499
\(35\) −0.927051 − 0.673542i −0.156700 − 0.113849i
\(36\) 0.190983 − 0.587785i 0.0318305 − 0.0979642i
\(37\) −1.30902 − 4.02874i −0.215201 − 0.662321i −0.999139 − 0.0414819i \(-0.986792\pi\)
0.783938 − 0.620839i \(-0.213208\pi\)
\(38\) 1.11803 − 0.812299i 0.181369 − 0.131772i
\(39\) 5.04508 − 3.66547i 0.807860 − 0.586945i
\(40\) −0.263932 − 0.812299i −0.0417313 − 0.128436i
\(41\) −1.83688 + 5.65334i −0.286873 + 0.882903i 0.698958 + 0.715162i \(0.253647\pi\)
−0.985831 + 0.167741i \(0.946353\pi\)
\(42\) −3.92705 − 2.85317i −0.605957 − 0.440254i
\(43\) −1.76393 −0.268997 −0.134499 − 0.990914i \(-0.542942\pi\)
−0.134499 + 0.990914i \(0.542942\pi\)
\(44\) 0 0
\(45\) −0.381966 −0.0569401
\(46\) −7.16312 − 5.20431i −1.05614 − 0.767334i
\(47\) −0.190983 + 0.587785i −0.0278577 + 0.0857373i −0.964019 − 0.265834i \(-0.914353\pi\)
0.936161 + 0.351572i \(0.114353\pi\)
\(48\) −1.50000 − 4.61653i −0.216506 − 0.666338i
\(49\) −1.61803 + 1.17557i −0.231148 + 0.167939i
\(50\) −6.35410 + 4.61653i −0.898606 + 0.652875i
\(51\) 0.190983 + 0.587785i 0.0267430 + 0.0823064i
\(52\) −1.19098 + 3.66547i −0.165160 + 0.508309i
\(53\) 5.97214 + 4.33901i 0.820336 + 0.596009i 0.916809 − 0.399327i \(-0.130756\pi\)
−0.0964728 + 0.995336i \(0.530756\pi\)
\(54\) −1.61803 −0.220187
\(55\) 0 0
\(56\) −6.70820 −0.896421
\(57\) −0.690983 − 0.502029i −0.0915229 − 0.0664953i
\(58\) −2.23607 + 6.88191i −0.293610 + 0.903639i
\(59\) −1.64590 − 5.06555i −0.214278 − 0.659479i −0.999204 − 0.0398899i \(-0.987299\pi\)
0.784926 − 0.619589i \(-0.212701\pi\)
\(60\) 0.190983 − 0.138757i 0.0246558 − 0.0179135i
\(61\) 0.927051 − 0.673542i 0.118697 − 0.0862382i −0.526853 − 0.849956i \(-0.676628\pi\)
0.645550 + 0.763718i \(0.276628\pi\)
\(62\) 1.92705 + 5.93085i 0.244736 + 0.753219i
\(63\) −0.927051 + 2.85317i −0.116797 + 0.359466i
\(64\) −3.42705 − 2.48990i −0.428381 − 0.311237i
\(65\) 2.38197 0.295447
\(66\) 0 0
\(67\) 10.5623 1.29039 0.645196 − 0.764017i \(-0.276776\pi\)
0.645196 + 0.764017i \(0.276776\pi\)
\(68\) −0.309017 − 0.224514i −0.0374738 − 0.0272263i
\(69\) −1.69098 + 5.20431i −0.203570 + 0.626525i
\(70\) −0.572949 − 1.76336i −0.0684805 − 0.210761i
\(71\) −11.7812 + 8.55951i −1.39817 + 1.01583i −0.403253 + 0.915089i \(0.632120\pi\)
−0.994913 + 0.100738i \(0.967880\pi\)
\(72\) −1.80902 + 1.31433i −0.213195 + 0.154895i
\(73\) −0.381966 − 1.17557i −0.0447057 − 0.137590i 0.926212 − 0.377003i \(-0.123045\pi\)
−0.970918 + 0.239412i \(0.923045\pi\)
\(74\) 2.11803 − 6.51864i 0.246216 − 0.757776i
\(75\) 3.92705 + 2.85317i 0.453457 + 0.329456i
\(76\) 0.527864 0.0605502
\(77\) 0 0
\(78\) 10.0902 1.14249
\(79\) 0.427051 + 0.310271i 0.0480470 + 0.0349082i 0.611550 − 0.791206i \(-0.290547\pi\)
−0.563503 + 0.826114i \(0.690547\pi\)
\(80\) 0.572949 − 1.76336i 0.0640576 − 0.197149i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) −7.78115 + 5.65334i −0.859285 + 0.624307i
\(83\) −10.2812 + 7.46969i −1.12850 + 0.819906i −0.985476 − 0.169813i \(-0.945684\pi\)
−0.143027 + 0.989719i \(0.545684\pi\)
\(84\) −0.572949 − 1.76336i −0.0625139 − 0.192398i
\(85\) −0.0729490 + 0.224514i −0.00791243 + 0.0243520i
\(86\) −2.30902 − 1.67760i −0.248988 − 0.180900i
\(87\) 4.47214 0.479463
\(88\) 0 0
\(89\) 9.47214 1.00404 0.502022 − 0.864855i \(-0.332590\pi\)
0.502022 + 0.864855i \(0.332590\pi\)
\(90\) −0.500000 − 0.363271i −0.0527046 − 0.0382922i
\(91\) 5.78115 − 17.7926i 0.606029 − 1.86517i
\(92\) −1.04508 − 3.21644i −0.108958 − 0.335337i
\(93\) 3.11803 − 2.26538i 0.323325 − 0.234909i
\(94\) −0.809017 + 0.587785i −0.0834437 + 0.0606254i
\(95\) −0.100813 − 0.310271i −0.0103432 − 0.0318331i
\(96\) 1.04508 − 3.21644i 0.106664 − 0.328277i
\(97\) −12.1631 − 8.83702i −1.23498 − 0.897264i −0.237724 − 0.971333i \(-0.576402\pi\)
−0.997253 + 0.0740689i \(0.976402\pi\)
\(98\) −3.23607 −0.326892
\(99\) 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.j.202.1 4
11.2 odd 10 363.2.e.h.148.1 4
11.3 even 5 inner 363.2.e.j.124.1 4
11.4 even 5 363.2.e.c.130.1 4
11.5 even 5 363.2.a.e.1.1 2
11.6 odd 10 363.2.a.h.1.2 2
11.7 odd 10 363.2.e.h.130.1 4
11.8 odd 10 33.2.e.a.25.1 yes 4
11.9 even 5 363.2.e.c.148.1 4
11.10 odd 2 33.2.e.a.4.1 ✓ 4
33.5 odd 10 1089.2.a.s.1.2 2
33.8 even 10 99.2.f.b.91.1 4
33.17 even 10 1089.2.a.m.1.1 2
33.32 even 2 99.2.f.b.37.1 4
44.19 even 10 528.2.y.f.289.1 4
44.27 odd 10 5808.2.a.bm.1.2 2
44.39 even 10 5808.2.a.bl.1.2 2
44.43 even 2 528.2.y.f.433.1 4
55.8 even 20 825.2.bx.b.124.2 8
55.19 odd 10 825.2.n.f.751.1 4
55.32 even 4 825.2.bx.b.499.2 8
55.39 odd 10 9075.2.a.x.1.1 2
55.43 even 4 825.2.bx.b.499.1 8
55.49 even 10 9075.2.a.bv.1.2 2
55.52 even 20 825.2.bx.b.124.1 8
55.54 odd 2 825.2.n.f.301.1 4
99.32 even 6 891.2.n.a.136.1 8
99.41 even 30 891.2.n.a.784.1 8
99.43 odd 6 891.2.n.d.433.1 8
99.52 odd 30 891.2.n.d.190.1 8
99.65 even 6 891.2.n.a.433.1 8
99.74 even 30 891.2.n.a.190.1 8
99.76 odd 6 891.2.n.d.136.1 8
99.85 odd 30 891.2.n.d.784.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.a.4.1 ✓ 4 11.10 odd 2
33.2.e.a.25.1 yes 4 11.8 odd 10
99.2.f.b.37.1 4 33.32 even 2
99.2.f.b.91.1 4 33.8 even 10
363.2.a.e.1.1 2 11.5 even 5
363.2.a.h.1.2 2 11.6 odd 10
363.2.e.c.130.1 4 11.4 even 5
363.2.e.c.148.1 4 11.9 even 5
363.2.e.h.130.1 4 11.7 odd 10
363.2.e.h.148.1 4 11.2 odd 10
363.2.e.j.124.1 4 11.3 even 5 inner
363.2.e.j.202.1 4 1.1 even 1 trivial
528.2.y.f.289.1 4 44.19 even 10
528.2.y.f.433.1 4 44.43 even 2
825.2.n.f.301.1 4 55.54 odd 2
825.2.n.f.751.1 4 55.19 odd 10
825.2.bx.b.124.1 8 55.52 even 20
825.2.bx.b.124.2 8 55.8 even 20
825.2.bx.b.499.1 8 55.43 even 4
825.2.bx.b.499.2 8 55.32 even 4
891.2.n.a.136.1 8 99.32 even 6
891.2.n.a.190.1 8 99.74 even 30
891.2.n.a.433.1 8 99.65 even 6
891.2.n.a.784.1 8 99.41 even 30
891.2.n.d.136.1 8 99.76 odd 6
891.2.n.d.190.1 8 99.52 odd 30
891.2.n.d.433.1 8 99.43 odd 6
891.2.n.d.784.1 8 99.85 odd 30
1089.2.a.m.1.1 2 33.17 even 10
1089.2.a.s.1.2 2 33.5 odd 10
5808.2.a.bl.1.2 2 44.39 even 10
5808.2.a.bm.1.2 2 44.27 odd 10
9075.2.a.x.1.1 2 55.39 odd 10
9075.2.a.bv.1.2 2 55.49 even 10