Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [855,2,Mod(406,855)] 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("855.406"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(855, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,-1,0,-7,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.82720937282\)
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: no (minimal twist has level 285)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 676.4
Root \(-0.690702 - 1.19633i\) of defining polynomial
Character \(\chi\) \(=\) 855.676
Dual form 855.2.k.i.406.4

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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 −0.511511 − 0.859277i \(-0.670914\pi\)
0.999911 + 0.0133433i \(0.00424744\pi\)
\(3\) 0 0
\(4\) 0.0458624 + 0.0794360i 0.0229312 + 0.0397180i
\(5\) 0.500000 − 0.866025i 0.223607 − 0.387298i
\(6\) 0 0
\(7\) −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 0
\(10\) −0.690702 − 1.19633i −0.218419 − 0.378313i
\(11\) 4.31625 1.30140 0.650699 − 0.759336i \(-0.274476\pi\)
0.650699 + 0.759336i \(0.274476\pi\)
\(12\) 0 0
\(13\) 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 0
\(16\) 1.90407 − 3.29794i 0.476017 − 0.824486i
\(17\) 2.85821 − 4.95056i 0.693217 − 1.20069i −0.277561 − 0.960708i \(-0.589526\pi\)
0.970778 − 0.239979i \(-0.0771405\pi\)
\(18\) 0 0
\(19\) 2.97100 − 3.18954i 0.681594 − 0.731730i
\(20\) 0.0917248 0.0205103
\(21\) 0 0
\(22\) 2.98124 − 5.16366i 0.635603 − 1.10090i
\(23\) −0.289678 − 0.501738i −0.0604021 − 0.104620i 0.834243 − 0.551397i \(-0.185905\pi\)
−0.894645 + 0.446777i \(0.852572\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 8.78938 1.72374
\(27\) 0 0
\(28\) −0.200081 − 0.346551i −0.0378118 − 0.0654920i
\(29\) −1.77672 − 3.07737i −0.329929 − 0.571454i 0.652569 − 0.757730i \(-0.273691\pi\)
−0.982497 + 0.186276i \(0.940358\pi\)
\(30\) 0 0
\(31\) 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
\(33\) 0 0
\(34\) −3.94834 − 6.83872i −0.677134 − 1.17283i
\(35\) −2.18132 + 3.77816i −0.368711 + 0.638626i
\(36\) 0 0
\(37\) −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
\(39\) 0 0
\(40\) 1.44476 − 2.50239i 0.228436 − 0.395663i
\(41\) −0.381403 + 0.660610i −0.0595652 + 0.103170i −0.894270 − 0.447527i \(-0.852305\pi\)
0.834705 + 0.550697i \(0.185638\pi\)
\(42\) 0 0
\(43\) 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
\(45\) 0 0
\(46\) −0.800326 −0.118002
\(47\) −1.36632 − 2.36653i −0.199298 − 0.345194i 0.749003 − 0.662567i \(-0.230533\pi\)
−0.948301 + 0.317372i \(0.897200\pi\)
\(48\) 0 0
\(49\) 12.0327 1.71895
\(50\) −1.38140 −0.195360
\(51\) 0 0
\(52\) −0.291806 + 0.505423i −0.0404662 + 0.0700896i
\(53\) 2.56853 + 4.44882i 0.352814 + 0.611092i 0.986741 − 0.162300i \(-0.0518914\pi\)
−0.633927 + 0.773393i \(0.718558\pi\)
\(54\) 0 0
\(55\) 2.15812 − 3.73798i 0.291001 − 0.504029i
\(56\) −12.6059 −1.68454
\(57\) 0 0
\(58\) −4.90874 −0.644549
\(59\) 1.91484 − 3.31660i 0.249291 − 0.431785i −0.714038 − 0.700107i \(-0.753136\pi\)
0.963329 + 0.268322i \(0.0864691\pi\)
\(60\) 0 0
\(61\) 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
\(63\) 0 0
\(64\) 8.33247 1.04156
\(65\) 6.36264 0.789189
\(66\) 0 0
\(67\) 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 0
\(70\) 3.01329 + 5.21916i 0.360156 + 0.623809i
\(71\) 1.53953 − 2.66654i 0.182708 − 0.316460i −0.760094 − 0.649814i \(-0.774847\pi\)
0.942802 + 0.333354i \(0.108180\pi\)
\(72\) 0 0
\(73\) −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
\(75\) 0 0
\(76\) 0.389622 + 0.0897246i 0.0446927 + 0.0102921i
\(77\) −18.8303 −2.14591
\(78\) 0 0
\(79\) −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 0
\(82\) 0.526872 + 0.912569i 0.0581833 + 0.100776i
\(83\) −7.67889 −0.842868 −0.421434 − 0.906859i \(-0.638473\pi\)
−0.421434 + 0.906859i \(0.638473\pi\)
\(84\) 0 0
\(85\) −2.85821 − 4.95056i −0.310016 − 0.536963i
\(86\) −4.39469 − 7.61182i −0.473891 − 0.820804i
\(87\) 0 0
\(88\) 12.4719 1.32951
\(89\) 4.92093 + 8.52330i 0.521618 + 0.903468i 0.999684 + 0.0251445i \(0.00800458\pi\)
−0.478066 + 0.878324i \(0.658662\pi\)
\(90\) 0 0
\(91\) −13.8790 − 24.0391i −1.45491 − 2.51998i
\(92\) 0.0265707 − 0.0460218i 0.00277019 − 0.00479811i
\(93\) 0 0
\(94\) −3.77487 −0.389348
\(95\) −1.27672 − 4.16773i −0.130989 − 0.427600i
\(96\) 0 0
\(97\) −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
\(99\) 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 855.2.k.i.676.4 10
3.2 odd 2 285.2.i.f.106.2 ✓ 10
19.7 even 3 inner 855.2.k.i.406.4 10
57.8 even 6 5415.2.a.z.1.2 5
57.11 odd 6 5415.2.a.y.1.4 5
57.26 odd 6 285.2.i.f.121.2 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.i.f.106.2 ✓ 10 3.2 odd 2
285.2.i.f.121.2 yes 10 57.26 odd 6
855.2.k.i.406.4 10 19.7 even 3 inner
855.2.k.i.676.4 10 1.1 even 1 trivial
5415.2.a.y.1.4 5 57.11 odd 6
5415.2.a.z.1.2 5 57.8 even 6