Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [285,2,Mod(106,285)] 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("285.106"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(285, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.27573645761\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 9x^{8} - 2x^{7} + 56x^{6} - 18x^{5} + 125x^{4} + x^{3} + 189x^{2} - 52x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 106.2
Root \(-0.690702 - 1.19633i\) of defining polynomial
Character \(\chi\) \(=\) 285.106
Dual form 285.2.i.f.121.2

$q$-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+(-0.690702 + 1.19633i) q^{2} +(0.500000 - 0.866025i) q^{3} +(0.0458624 + 0.0794360i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(0.690702 + 1.19633i) q^{6} -4.36264 q^{7} -2.88952 q^{8} +(-0.500000 - 0.866025i) q^{9} +(-0.690702 - 1.19633i) q^{10} -4.31625 q^{11} +0.0917248 q^{12} +(3.18132 + 5.51021i) q^{13} +(3.01329 - 5.21916i) q^{14} +(0.500000 + 0.866025i) q^{15} +(1.90407 - 3.29794i) q^{16} +(-2.85821 + 4.95056i) q^{17} +1.38140 q^{18} +(2.97100 - 3.18954i) q^{19} -0.0917248 q^{20} +(-2.18132 + 3.77816i) q^{21} +(2.98124 - 5.16366i) q^{22} +(0.289678 + 0.501738i) q^{23} +(-1.44476 + 2.50239i) q^{24} +(-0.500000 - 0.866025i) q^{25} -8.78938 q^{26} -1.00000 q^{27} +(-0.200081 - 0.346551i) q^{28} +(1.77672 + 3.07737i) q^{29} -1.38140 q^{30} +1.18345 q^{31} +(-0.259229 - 0.448998i) q^{32} +(-2.15812 + 3.73798i) q^{33} +(-3.94834 - 6.83872i) q^{34} +(2.18132 - 3.77816i) q^{35} +(0.0458624 - 0.0794360i) q^{36} -6.54609 q^{37} +(1.76367 + 5.75732i) q^{38} +6.36264 q^{39} +(1.44476 - 2.50239i) q^{40} +(0.381403 - 0.660610i) q^{41} +(-3.01329 - 5.21916i) q^{42} +(3.18132 - 5.51021i) q^{43} +(-0.197954 - 0.342866i) q^{44} +1.00000 q^{45} -0.800326 q^{46} +(1.36632 + 2.36653i) q^{47} +(-1.90407 - 3.29794i) q^{48} +12.0327 q^{49} +1.38140 q^{50} +(2.85821 + 4.95056i) q^{51} +(-0.291806 + 0.505423i) q^{52} +(-2.56853 - 4.44882i) q^{53} +(0.690702 - 1.19633i) q^{54} +(2.15812 - 3.73798i) q^{55} +12.6059 q^{56} +(-1.27672 - 4.16773i) q^{57} -4.90874 q^{58} +(-1.91484 + 3.31660i) q^{59} +(-0.0458624 + 0.0794360i) q^{60} +(6.01053 + 10.4105i) q^{61} +(-0.817411 + 1.41580i) q^{62} +(2.18132 + 3.77816i) q^{63} +8.33247 q^{64} -6.36264 q^{65} +(-2.98124 - 5.16366i) q^{66} +(2.00213 + 3.46779i) q^{67} -0.524337 q^{68} +0.579357 q^{69} +(3.01329 + 5.21916i) q^{70} +(-1.53953 + 2.66654i) q^{71} +(1.44476 + 2.50239i) q^{72} +(-4.04320 + 7.00303i) q^{73} +(4.52140 - 7.83129i) q^{74} -1.00000 q^{75} +(0.389622 + 0.0897246i) q^{76} +18.8303 q^{77} +(-4.39469 + 7.61182i) q^{78} +(-5.66836 + 9.81790i) q^{79} +(1.90407 + 3.29794i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(0.526872 + 0.912569i) q^{82} +7.67889 q^{83} -0.400163 q^{84} +(-2.85821 - 4.95056i) q^{85} +(4.39469 + 7.61182i) q^{86} +3.55344 q^{87} +12.4719 q^{88} +(-4.92093 - 8.52330i) q^{89} +(-0.690702 + 1.19633i) q^{90} +(-13.8790 - 24.0391i) q^{91} +(-0.0265707 + 0.0460218i) q^{92} +(0.591725 - 1.02490i) q^{93} -3.77487 q^{94} +(1.27672 + 4.16773i) q^{95} -0.518458 q^{96} +(-1.00000 + 1.73205i) q^{97} +(-8.31098 + 14.3950i) q^{98} +(2.15812 + 3.73798i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} + 5 q^{3} - 7 q^{4} - 5 q^{5} - q^{6} + 4 q^{7} - 12 q^{8} - 5 q^{9} + q^{10} + 10 q^{11} - 14 q^{12} + 8 q^{13} + 4 q^{14} + 5 q^{15} - 7 q^{16} - 10 q^{17} - 2 q^{18} + 5 q^{19} + 14 q^{20}+ \cdots - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/285\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(211\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.690702 + 1.19633i −0.488400 + 0.845933i −0.999911 − 0.0133433i \(-0.995753\pi\)
0.511511 + 0.859277i \(0.329086\pi\)
\(3\) 0.500000 − 0.866025i 0.288675 − 0.500000i
\(4\) 0.0458624 + 0.0794360i 0.0229312 + 0.0397180i
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0.690702 + 1.19633i 0.281978 + 0.488400i
\(7\) −4.36264 −1.64892 −0.824462 − 0.565917i \(-0.808522\pi\)
−0.824462 + 0.565917i \(0.808522\pi\)
\(8\) −2.88952 −1.02160
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) −0.690702 − 1.19633i −0.218419 − 0.378313i
\(11\) −4.31625 −1.30140 −0.650699 − 0.759336i \(-0.725524\pi\)
−0.650699 + 0.759336i \(0.725524\pi\)
\(12\) 0.0917248 0.0264787
\(13\) 3.18132 + 5.51021i 0.882340 + 1.52826i 0.848732 + 0.528823i \(0.177366\pi\)
0.0336075 + 0.999435i \(0.489300\pi\)
\(14\) 3.01329 − 5.21916i 0.805334 − 1.39488i
\(15\) 0.500000 + 0.866025i 0.129099 + 0.223607i
\(16\) 1.90407 − 3.29794i 0.476017 − 0.824486i
\(17\) −2.85821 + 4.95056i −0.693217 + 1.20069i 0.277561 + 0.960708i \(0.410474\pi\)
−0.970778 + 0.239979i \(0.922860\pi\)
\(18\) 1.38140 0.325600
\(19\) 2.97100 − 3.18954i 0.681594 − 0.731730i
\(20\) −0.0917248 −0.0205103
\(21\) −2.18132 + 3.77816i −0.476003 + 0.824462i
\(22\) 2.98124 − 5.16366i 0.635603 − 1.10090i
\(23\) 0.289678 + 0.501738i 0.0604021 + 0.104620i 0.894645 − 0.446777i \(-0.147428\pi\)
−0.834243 + 0.551397i \(0.814095\pi\)
\(24\) −1.44476 + 2.50239i −0.294910 + 0.510799i
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) −8.78938 −1.72374
\(27\) −1.00000 −0.192450
\(28\) −0.200081 − 0.346551i −0.0378118 − 0.0654920i
\(29\) 1.77672 + 3.07737i 0.329929 + 0.571454i 0.982497 − 0.186276i \(-0.0596419\pi\)
−0.652569 + 0.757730i \(0.726309\pi\)
\(30\) −1.38140 −0.252209
\(31\) 1.18345 0.212554 0.106277 − 0.994337i \(-0.466107\pi\)
0.106277 + 0.994337i \(0.466107\pi\)
\(32\) −0.259229 − 0.448998i −0.0458257 − 0.0793724i
\(33\) −2.15812 + 3.73798i −0.375681 + 0.650699i
\(34\) −3.94834 − 6.83872i −0.677134 − 1.17283i
\(35\) 2.18132 − 3.77816i 0.368711 − 0.638626i
\(36\) 0.0458624 − 0.0794360i 0.00764374 − 0.0132393i
\(37\) −6.54609 −1.07617 −0.538086 − 0.842890i \(-0.680852\pi\)
−0.538086 + 0.842890i \(0.680852\pi\)
\(38\) 1.76367 + 5.75732i 0.286105 + 0.933960i
\(39\) 6.36264 1.01884
\(40\) 1.44476 − 2.50239i 0.228436 − 0.395663i
\(41\) 0.381403 − 0.660610i 0.0595652 − 0.103170i −0.834705 − 0.550697i \(-0.814362\pi\)
0.894270 + 0.447527i \(0.147695\pi\)
\(42\) −3.01329 − 5.21916i −0.464960 − 0.805334i
\(43\) 3.18132 − 5.51021i 0.485147 − 0.840299i −0.514707 − 0.857366i \(-0.672099\pi\)
0.999854 + 0.0170666i \(0.00543274\pi\)
\(44\) −0.197954 − 0.342866i −0.0298426 − 0.0516890i
\(45\) 1.00000 0.149071
\(46\) −0.800326 −0.118002
\(47\) 1.36632 + 2.36653i 0.199298 + 0.345194i 0.948301 − 0.317372i \(-0.102800\pi\)
−0.749003 + 0.662567i \(0.769467\pi\)
\(48\) −1.90407 − 3.29794i −0.274829 − 0.476017i
\(49\) 12.0327 1.71895
\(50\) 1.38140 0.195360
\(51\) 2.85821 + 4.95056i 0.400229 + 0.693217i
\(52\) −0.291806 + 0.505423i −0.0404662 + 0.0700896i
\(53\) −2.56853 − 4.44882i −0.352814 − 0.611092i 0.633927 − 0.773393i \(-0.281442\pi\)
−0.986741 + 0.162300i \(0.948109\pi\)
\(54\) 0.690702 − 1.19633i 0.0939926 − 0.162800i
\(55\) 2.15812 − 3.73798i 0.291001 − 0.504029i
\(56\) 12.6059 1.68454
\(57\) −1.27672 − 4.16773i −0.169106 − 0.552029i
\(58\) −4.90874 −0.644549
\(59\) −1.91484 + 3.31660i −0.249291 + 0.431785i −0.963329 − 0.268322i \(-0.913531\pi\)
0.714038 + 0.700107i \(0.246864\pi\)
\(60\) −0.0458624 + 0.0794360i −0.00592081 + 0.0102551i
\(61\) 6.01053 + 10.4105i 0.769569 + 1.33293i 0.937797 + 0.347185i \(0.112862\pi\)
−0.168227 + 0.985748i \(0.553804\pi\)
\(62\) −0.817411 + 1.41580i −0.103811 + 0.179806i
\(63\) 2.18132 + 3.77816i 0.274821 + 0.476003i
\(64\) 8.33247 1.04156
\(65\) −6.36264 −0.789189
\(66\) −2.98124 − 5.16366i −0.366965 − 0.635603i
\(67\) 2.00213 + 3.46779i 0.244599 + 0.423658i 0.962019 − 0.272983i \(-0.0880104\pi\)
−0.717420 + 0.696641i \(0.754677\pi\)
\(68\) −0.524337 −0.0635852
\(69\) 0.579357 0.0697464
\(70\) 3.01329 + 5.21916i 0.360156 + 0.623809i
\(71\) −1.53953 + 2.66654i −0.182708 + 0.316460i −0.942802 − 0.333354i \(-0.891820\pi\)
0.760094 + 0.649814i \(0.225153\pi\)
\(72\) 1.44476 + 2.50239i 0.170266 + 0.294910i
\(73\) −4.04320 + 7.00303i −0.473221 + 0.819643i −0.999530 − 0.0306504i \(-0.990242\pi\)
0.526309 + 0.850293i \(0.323575\pi\)
\(74\) 4.52140 − 7.83129i 0.525602 − 0.910369i
\(75\) −1.00000 −0.115470
\(76\) 0.389622 + 0.0897246i 0.0446927 + 0.0102921i
\(77\) 18.8303 2.14591
\(78\) −4.39469 + 7.61182i −0.497601 + 0.861869i
\(79\) −5.66836 + 9.81790i −0.637741 + 1.10460i 0.348187 + 0.937425i \(0.386798\pi\)
−0.985927 + 0.167174i \(0.946536\pi\)
\(80\) 1.90407 + 3.29794i 0.212881 + 0.368721i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0.526872 + 0.912569i 0.0581833 + 0.100776i
\(83\) 7.67889 0.842868 0.421434 − 0.906859i \(-0.361527\pi\)
0.421434 + 0.906859i \(0.361527\pi\)
\(84\) −0.400163 −0.0436613
\(85\) −2.85821 − 4.95056i −0.310016 − 0.536963i
\(86\) 4.39469 + 7.61182i 0.473891 + 0.820804i
\(87\) 3.55344 0.380969
\(88\) 12.4719 1.32951
\(89\) −4.92093 − 8.52330i −0.521618 − 0.903468i −0.999684 − 0.0251445i \(-0.991995\pi\)
0.478066 − 0.878324i \(-0.341338\pi\)
\(90\) −0.690702 + 1.19633i −0.0728063 + 0.126104i
\(91\) −13.8790 − 24.0391i −1.45491 − 2.51998i
\(92\) −0.0265707 + 0.0460218i −0.00277019 + 0.00479811i
\(93\) 0.591725 − 1.02490i 0.0613590 − 0.106277i
\(94\) −3.77487 −0.389348
\(95\) 1.27672 + 4.16773i 0.130989 + 0.427600i
\(96\) −0.518458 −0.0529149
\(97\) −1.00000 + 1.73205i −0.101535 + 0.175863i −0.912317 − 0.409484i \(-0.865709\pi\)
0.810782 + 0.585348i \(0.199042\pi\)
\(98\) −8.31098 + 14.3950i −0.839536 + 1.45412i
\(99\) 2.15812 + 3.73798i 0.216900 + 0.375681i
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 285.2.i.f.106.2 ✓ 10
3.2 odd 2 855.2.k.i.676.4 10
19.7 even 3 inner 285.2.i.f.121.2 yes 10
19.8 odd 6 5415.2.a.z.1.2 5
19.11 even 3 5415.2.a.y.1.4 5
57.26 odd 6 855.2.k.i.406.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.i.f.106.2 ✓ 10 1.1 even 1 trivial
285.2.i.f.121.2 yes 10 19.7 even 3 inner
855.2.k.i.406.4 10 57.26 odd 6
855.2.k.i.676.4 10 3.2 odd 2
5415.2.a.y.1.4 5 19.11 even 3
5415.2.a.z.1.2 5 19.8 odd 6