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,-5,-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: \(1\)
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: yes
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.a.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+(-1.80902 + 1.31433i) q^{2} +(0.309017 + 0.951057i) q^{3} +(0.927051 - 2.85317i) q^{4} +(-1.61803 - 1.17557i) q^{5} +(-1.80902 - 1.31433i) q^{6} +(-1.38197 + 4.25325i) q^{7} +(0.690983 + 2.12663i) q^{8} +(-0.809017 + 0.587785i) q^{9} +4.47214 q^{10} +3.00000 q^{12} +(-3.09017 - 9.51057i) q^{14} +(0.618034 - 1.90211i) q^{15} +(0.809017 + 0.587785i) q^{16} +(-3.61803 - 2.62866i) q^{17} +(0.690983 - 2.12663i) q^{18} +(-1.38197 - 4.25325i) q^{19} +(-4.85410 + 3.52671i) q^{20} -4.47214 q^{21} -4.00000 q^{23} +(-1.80902 + 1.31433i) q^{24} +(-0.309017 - 0.951057i) q^{25} +(-0.809017 - 0.587785i) q^{27} +(10.8541 + 7.88597i) q^{28} +(1.38197 - 4.25325i) q^{29} +(1.38197 + 4.25325i) q^{30} -6.70820 q^{32} +10.0000 q^{34} +(7.23607 - 5.25731i) q^{35} +(0.927051 + 2.85317i) q^{36} +(0.618034 - 1.90211i) q^{37} +(8.09017 + 5.87785i) q^{38} +(1.38197 - 4.25325i) q^{40} +(-1.38197 - 4.25325i) q^{41} +(8.09017 - 5.87785i) q^{42} +4.47214 q^{43} +2.00000 q^{45} +(7.23607 - 5.25731i) q^{46} +(2.47214 + 7.60845i) q^{47} +(-0.309017 + 0.951057i) q^{48} +(-10.5172 - 7.64121i) q^{49} +(1.80902 + 1.31433i) q^{50} +(1.38197 - 4.25325i) q^{51} +(-4.85410 + 3.52671i) q^{53} +2.23607 q^{54} -10.0000 q^{56} +(3.61803 - 2.62866i) q^{57} +(3.09017 + 9.51057i) q^{58} +(-4.85410 - 3.52671i) q^{60} +(-7.23607 - 5.25731i) q^{61} +(-1.38197 - 4.25325i) q^{63} +(10.5172 - 7.64121i) q^{64} -12.0000 q^{67} +(-10.8541 + 7.88597i) q^{68} +(-1.23607 - 3.80423i) q^{69} +(-6.18034 + 19.0211i) q^{70} +(6.47214 + 4.70228i) q^{71} +(-1.80902 - 1.31433i) q^{72} +(-2.76393 + 8.50651i) q^{73} +(1.38197 + 4.25325i) q^{74} +(0.809017 - 0.587785i) q^{75} -13.4164 q^{76} +(-10.8541 + 7.88597i) q^{79} +(-0.618034 - 1.90211i) q^{80} +(0.309017 - 0.951057i) q^{81} +(8.09017 + 5.87785i) q^{82} +(7.23607 + 5.25731i) q^{83} +(-4.14590 + 12.7598i) q^{84} +(2.76393 + 8.50651i) q^{85} +(-8.09017 + 5.87785i) q^{86} +4.47214 q^{87} -14.0000 q^{89} +(-3.61803 + 2.62866i) q^{90} +(-3.70820 + 11.4127i) q^{92} +(-14.4721 - 10.5146i) q^{94} +(-2.76393 + 8.50651i) q^{95} +(-2.07295 - 6.37988i) q^{96} +(-1.61803 + 1.17557i) q^{97} +29.0689 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 5 q^{2} - q^{3} - 3 q^{4} - 2 q^{5} - 5 q^{6} - 10 q^{7} + 5 q^{8} - q^{9} + 12 q^{12} + 10 q^{14} - 2 q^{15} + q^{16} - 10 q^{17} + 5 q^{18} - 10 q^{19} - 6 q^{20} - 16 q^{23} - 5 q^{24} + q^{25}+ \cdots - 2 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.80902 + 1.31433i −1.27917 + 0.929370i −0.999528 − 0.0307347i \(-0.990215\pi\)
−0.279641 + 0.960105i \(0.590215\pi\)
\(3\) 0.309017 + 0.951057i 0.178411 + 0.549093i
\(4\) 0.927051 − 2.85317i 0.463525 − 1.42658i
\(5\) −1.61803 − 1.17557i −0.723607 − 0.525731i 0.163928 − 0.986472i \(-0.447584\pi\)
−0.887535 + 0.460741i \(0.847584\pi\)
\(6\) −1.80902 − 1.31433i −0.738528 − 0.536572i
\(7\) −1.38197 + 4.25325i −0.522334 + 1.60758i 0.247194 + 0.968966i \(0.420491\pi\)
−0.769528 + 0.638613i \(0.779509\pi\)
\(8\) 0.690983 + 2.12663i 0.244299 + 0.751876i
\(9\) −0.809017 + 0.587785i −0.269672 + 0.195928i
\(10\) 4.47214 1.41421
\(11\) 0 0
\(12\) 3.00000 0.866025
\(13\) 0 0 −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(14\) −3.09017 − 9.51057i −0.825883 − 2.54181i
\(15\) 0.618034 − 1.90211i 0.159576 − 0.491123i
\(16\) 0.809017 + 0.587785i 0.202254 + 0.146946i
\(17\) −3.61803 − 2.62866i −0.877502 − 0.637543i 0.0550873 − 0.998482i \(-0.482456\pi\)
−0.932589 + 0.360939i \(0.882456\pi\)
\(18\) 0.690983 − 2.12663i 0.162866 − 0.501251i
\(19\) −1.38197 − 4.25325i −0.317045 − 0.975763i −0.974905 − 0.222623i \(-0.928538\pi\)
0.657860 − 0.753140i \(-0.271462\pi\)
\(20\) −4.85410 + 3.52671i −1.08541 + 0.788597i
\(21\) −4.47214 −0.975900
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.80902 + 1.31433i −0.369264 + 0.268286i
\(25\) −0.309017 − 0.951057i −0.0618034 − 0.190211i
\(26\) 0 0
\(27\) −0.809017 − 0.587785i −0.155695 − 0.113119i
\(28\) 10.8541 + 7.88597i 2.05123 + 1.49031i
\(29\) 1.38197 − 4.25325i 0.256625 − 0.789809i −0.736881 − 0.676023i \(-0.763702\pi\)
0.993505 − 0.113787i \(-0.0362980\pi\)
\(30\) 1.38197 + 4.25325i 0.252311 + 0.776534i
\(31\) 0 0 −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(32\) −6.70820 −1.18585
\(33\) 0 0
\(34\) 10.0000 1.71499
\(35\) 7.23607 − 5.25731i 1.22312 − 0.888648i
\(36\) 0.927051 + 2.85317i 0.154508 + 0.475528i
\(37\) 0.618034 − 1.90211i 0.101604 − 0.312705i −0.887314 − 0.461165i \(-0.847432\pi\)
0.988918 + 0.148460i \(0.0474315\pi\)
\(38\) 8.09017 + 5.87785i 1.31240 + 0.953514i
\(39\) 0 0
\(40\) 1.38197 − 4.25325i 0.218508 − 0.672499i
\(41\) −1.38197 − 4.25325i −0.215827 − 0.664247i −0.999094 − 0.0425613i \(-0.986448\pi\)
0.783267 − 0.621685i \(-0.213552\pi\)
\(42\) 8.09017 − 5.87785i 1.24834 − 0.906972i
\(43\) 4.47214 0.681994 0.340997 − 0.940064i \(-0.389235\pi\)
0.340997 + 0.940064i \(0.389235\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 7.23607 − 5.25731i 1.06690 − 0.775148i
\(47\) 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
\(49\) −10.5172 − 7.64121i −1.50246 − 1.09160i
\(50\) 1.80902 + 1.31433i 0.255834 + 0.185874i
\(51\) 1.38197 − 4.25325i 0.193514 − 0.595575i
\(52\) 0 0
\(53\) −4.85410 + 3.52671i −0.666762 + 0.484431i −0.868940 − 0.494918i \(-0.835198\pi\)
0.202178 + 0.979349i \(0.435198\pi\)
\(54\) 2.23607 0.304290
\(55\) 0 0
\(56\) −10.0000 −1.33631
\(57\) 3.61803 − 2.62866i 0.479220 − 0.348174i
\(58\) 3.09017 + 9.51057i 0.405759 + 1.24880i
\(59\) 0 0 −0.951057 − 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(60\) −4.85410 − 3.52671i −0.626662 − 0.455296i
\(61\) −7.23607 − 5.25731i −0.926484 − 0.673130i 0.0186458 − 0.999826i \(-0.494065\pi\)
−0.945129 + 0.326696i \(0.894065\pi\)
\(62\) 0 0
\(63\) −1.38197 − 4.25325i −0.174111 − 0.535860i
\(64\) 10.5172 − 7.64121i 1.31465 − 0.955151i
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −10.8541 + 7.88597i −1.31625 + 0.956314i
\(69\) −1.23607 − 3.80423i −0.148805 − 0.457975i
\(70\) −6.18034 + 19.0211i −0.738692 + 2.27346i
\(71\) 6.47214 + 4.70228i 0.768101 + 0.558058i 0.901384 − 0.433020i \(-0.142552\pi\)
−0.133283 + 0.991078i \(0.542552\pi\)
\(72\) −1.80902 − 1.31433i −0.213195 − 0.154895i
\(73\) −2.76393 + 8.50651i −0.323494 + 0.995611i 0.648622 + 0.761111i \(0.275346\pi\)
−0.972116 + 0.234501i \(0.924654\pi\)
\(74\) 1.38197 + 4.25325i 0.160650 + 0.494431i
\(75\) 0.809017 − 0.587785i 0.0934172 − 0.0678716i
\(76\) −13.4164 −1.53897
\(77\) 0 0
\(78\) 0 0
\(79\) −10.8541 + 7.88597i −1.22118 + 0.887241i −0.996198 − 0.0871218i \(-0.972233\pi\)
−0.224984 + 0.974362i \(0.572233\pi\)
\(80\) −0.618034 − 1.90211i −0.0690983 − 0.212663i
\(81\) 0.309017 − 0.951057i 0.0343352 − 0.105673i
\(82\) 8.09017 + 5.87785i 0.893410 + 0.649100i
\(83\) 7.23607 + 5.25731i 0.794262 + 0.577065i 0.909225 − 0.416305i \(-0.136675\pi\)
−0.114963 + 0.993370i \(0.536675\pi\)
\(84\) −4.14590 + 12.7598i −0.452355 + 1.39220i
\(85\) 2.76393 + 8.50651i 0.299791 + 0.922660i
\(86\) −8.09017 + 5.87785i −0.872385 + 0.633825i
\(87\) 4.47214 0.479463
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) −3.61803 + 2.62866i −0.381374 + 0.277085i
\(91\) 0 0
\(92\) −3.70820 + 11.4127i −0.386607 + 1.18985i
\(93\) 0 0
\(94\) −14.4721 − 10.5146i −1.49269 − 1.08450i
\(95\) −2.76393 + 8.50651i −0.283573 + 0.872749i
\(96\) −2.07295 − 6.37988i −0.211569 − 0.651144i
\(97\) −1.61803 + 1.17557i −0.164286 + 0.119361i −0.666891 − 0.745155i \(-0.732375\pi\)
0.502604 + 0.864517i \(0.332375\pi\)
\(98\) 29.0689 2.93640
\(99\) 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.a.124.1 4
11.2 odd 10 363.2.a.g.1.1 ✓ 2
11.3 even 5 363.2.e.l.148.1 4
11.4 even 5 inner 363.2.e.a.202.1 4
11.5 even 5 363.2.e.l.130.1 4
11.6 odd 10 inner 363.2.e.a.130.1 4
11.7 odd 10 363.2.e.l.202.1 4
11.8 odd 10 inner 363.2.e.a.148.1 4
11.9 even 5 363.2.a.g.1.2 yes 2
11.10 odd 2 363.2.e.l.124.1 4
33.2 even 10 1089.2.a.p.1.2 2
33.20 odd 10 1089.2.a.p.1.1 2
44.31 odd 10 5808.2.a.bx.1.2 2
44.35 even 10 5808.2.a.bx.1.1 2
55.9 even 10 9075.2.a.bi.1.1 2
55.24 odd 10 9075.2.a.bi.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.2.a.g.1.1 ✓ 2 11.2 odd 10
363.2.a.g.1.2 yes 2 11.9 even 5
363.2.e.a.124.1 4 1.1 even 1 trivial
363.2.e.a.130.1 4 11.6 odd 10 inner
363.2.e.a.148.1 4 11.8 odd 10 inner
363.2.e.a.202.1 4 11.4 even 5 inner
363.2.e.l.124.1 4 11.10 odd 2
363.2.e.l.130.1 4 11.5 even 5
363.2.e.l.148.1 4 11.3 even 5
363.2.e.l.202.1 4 11.7 odd 10
1089.2.a.p.1.1 2 33.20 odd 10
1089.2.a.p.1.2 2 33.2 even 10
5808.2.a.bx.1.1 2 44.35 even 10
5808.2.a.bx.1.2 2 44.31 odd 10
9075.2.a.bi.1.1 2 55.9 even 10
9075.2.a.bi.1.2 2 55.24 odd 10