Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(190,567)] 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("567.190"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-6,3,0,-3,18,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1156923.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 190.2
Root \(0.500000 + 2.43956i\) of defining polynomial
Character \(\chi\) \(=\) 567.190
Dual form 567.2.f.m.379.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.261988 + 0.453777i) q^{2} +(0.862724 - 1.49428i) q^{4} +(-1.10074 + 1.90653i) q^{5} +(-0.500000 - 0.866025i) q^{7} +1.95205 q^{8} -1.15352 q^{10} +(2.60074 + 4.50461i) q^{11} +(-1.57676 + 2.73103i) q^{13} +(0.261988 - 0.453777i) q^{14} +(-1.21404 - 2.10277i) q^{16} +3.24943 q^{17} +7.45090 q^{19} +(1.89926 + 3.28962i) q^{20} +(-1.36272 + 2.36031i) q^{22} +(2.20147 - 3.81306i) q^{23} +(0.0767598 + 0.132952i) q^{25} -1.65237 q^{26} -1.72545 q^{28} +(-0.576760 - 0.998977i) q^{29} +(-1.00000 + 1.73205i) q^{31} +(2.58817 - 4.48285i) q^{32} +(0.851311 + 1.47451i) q^{34} +2.20147 q^{35} +5.00000 q^{37} +(1.95205 + 3.38104i) q^{38} +(-2.14869 + 3.72164i) q^{40} +(-5.72545 + 9.91677i) q^{41} +(-4.64869 - 8.05177i) q^{43} +8.97487 q^{44} +2.30704 q^{46} +(-0.523976 - 0.907554i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(-0.0402203 + 0.0696636i) q^{50} +(2.72062 + 4.71225i) q^{52} +0.249425 q^{53} -11.4509 q^{55} +(-0.976024 - 1.69052i) q^{56} +(0.302209 - 0.523440i) q^{58} +(-4.04795 + 7.01126i) q^{59} +(-4.30221 - 7.45164i) q^{61} -1.04795 q^{62} -2.14386 q^{64} +(-3.47119 - 6.01228i) q^{65} +(3.80221 - 6.58562i) q^{67} +(2.80336 - 4.85556i) q^{68} +(0.576760 + 0.998977i) q^{70} -9.60442 q^{71} +0.846480 q^{73} +(1.30994 + 2.26888i) q^{74} +(6.42807 - 11.1337i) q^{76} +(2.60074 - 4.50461i) q^{77} +(3.80221 + 6.58562i) q^{79} +5.34533 q^{80} -6.00000 q^{82} +(5.72545 + 9.91677i) q^{83} +(-3.57676 + 6.19513i) q^{85} +(2.43580 - 4.21894i) q^{86} +(5.07676 + 8.79321i) q^{88} -9.24943 q^{89} +3.15352 q^{91} +(-3.79853 - 6.57924i) q^{92} +(0.274551 - 0.475537i) q^{94} +(-8.20147 + 14.2054i) q^{95} +(1.72545 + 2.98856i) q^{97} -0.523976 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 3 q^{5} - 3 q^{7} + 18 q^{8} + 6 q^{10} + 6 q^{11} - 3 q^{13} - 12 q^{16} - 6 q^{17} + 21 q^{20} + 3 q^{22} - 6 q^{23} - 6 q^{25} + 54 q^{26} + 12 q^{28} + 3 q^{29} - 6 q^{31} - 18 q^{32}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.261988 + 0.453777i 0.185254 + 0.320869i 0.943662 − 0.330911i \(-0.107356\pi\)
−0.758408 + 0.651780i \(0.774023\pi\)
\(3\) 0 0
\(4\) 0.862724 − 1.49428i 0.431362 − 0.747141i
\(5\) −1.10074 + 1.90653i −0.492264 + 0.852627i −0.999960 − 0.00890964i \(-0.997164\pi\)
0.507696 + 0.861536i \(0.330497\pi\)
\(6\) 0 0
\(7\) −0.500000 − 0.866025i −0.188982 − 0.327327i
\(8\) 1.95205 0.690153
\(9\) 0 0
\(10\) −1.15352 −0.364775
\(11\) 2.60074 + 4.50461i 0.784151 + 1.35819i 0.929505 + 0.368810i \(0.120235\pi\)
−0.145353 + 0.989380i \(0.546432\pi\)
\(12\) 0 0
\(13\) −1.57676 + 2.73103i −0.437314 + 0.757451i −0.997481 − 0.0709289i \(-0.977404\pi\)
0.560167 + 0.828380i \(0.310737\pi\)
\(14\) 0.261988 − 0.453777i 0.0700193 − 0.121277i
\(15\) 0 0
\(16\) −1.21404 − 2.10277i −0.303509 − 0.525693i
\(17\) 3.24943 0.788101 0.394051 − 0.919089i \(-0.371073\pi\)
0.394051 + 0.919089i \(0.371073\pi\)
\(18\) 0 0
\(19\) 7.45090 1.70935 0.854677 − 0.519161i \(-0.173755\pi\)
0.854677 + 0.519161i \(0.173755\pi\)
\(20\) 1.89926 + 3.28962i 0.424688 + 0.735582i
\(21\) 0 0
\(22\) −1.36272 + 2.36031i −0.290534 + 0.503219i
\(23\) 2.20147 − 3.81306i 0.459039 − 0.795078i −0.539872 − 0.841747i \(-0.681527\pi\)
0.998910 + 0.0466689i \(0.0148606\pi\)
\(24\) 0 0
\(25\) 0.0767598 + 0.132952i 0.0153520 + 0.0265904i
\(26\) −1.65237 −0.324056
\(27\) 0 0
\(28\) −1.72545 −0.326079
\(29\) −0.576760 − 0.998977i −0.107102 − 0.185505i 0.807493 − 0.589877i \(-0.200824\pi\)
−0.914595 + 0.404371i \(0.867490\pi\)
\(30\) 0 0
\(31\) −1.00000 + 1.73205i −0.179605 + 0.311086i −0.941745 − 0.336327i \(-0.890815\pi\)
0.762140 + 0.647412i \(0.224149\pi\)
\(32\) 2.58817 − 4.48285i 0.457529 − 0.792463i
\(33\) 0 0
\(34\) 0.851311 + 1.47451i 0.145999 + 0.252877i
\(35\) 2.20147 0.372117
\(36\) 0 0
\(37\) 5.00000 0.821995 0.410997 − 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 1.95205 + 3.38104i 0.316664 + 0.548478i
\(39\) 0 0
\(40\) −2.14869 + 3.72164i −0.339738 + 0.588443i
\(41\) −5.72545 + 9.91677i −0.894165 + 1.54874i −0.0593301 + 0.998238i \(0.518896\pi\)
−0.834835 + 0.550501i \(0.814437\pi\)
\(42\) 0 0
\(43\) −4.64869 − 8.05177i −0.708918 − 1.22788i −0.965259 − 0.261296i \(-0.915850\pi\)
0.256340 − 0.966587i \(-0.417483\pi\)
\(44\) 8.97487 1.35301
\(45\) 0 0
\(46\) 2.30704 0.340154
\(47\) −0.523976 − 0.907554i −0.0764298 − 0.132380i 0.825277 − 0.564728i \(-0.191019\pi\)
−0.901707 + 0.432347i \(0.857685\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) −0.0402203 + 0.0696636i −0.00568801 + 0.00985192i
\(51\) 0 0
\(52\) 2.72062 + 4.71225i 0.377282 + 0.653471i
\(53\) 0.249425 0.0342612 0.0171306 − 0.999853i \(-0.494547\pi\)
0.0171306 + 0.999853i \(0.494547\pi\)
\(54\) 0 0
\(55\) −11.4509 −1.54404
\(56\) −0.976024 − 1.69052i −0.130427 − 0.225906i
\(57\) 0 0
\(58\) 0.302209 − 0.523440i 0.0396819 − 0.0687311i
\(59\) −4.04795 + 7.01126i −0.526999 + 0.912788i 0.472506 + 0.881327i \(0.343349\pi\)
−0.999505 + 0.0314611i \(0.989984\pi\)
\(60\) 0 0
\(61\) −4.30221 − 7.45164i −0.550841 − 0.954085i −0.998214 − 0.0597376i \(-0.980974\pi\)
0.447373 − 0.894348i \(-0.352360\pi\)
\(62\) −1.04795 −0.133090
\(63\) 0 0
\(64\) −2.14386 −0.267982
\(65\) −3.47119 − 6.01228i −0.430549 − 0.745732i
\(66\) 0 0
\(67\) 3.80221 − 6.58562i 0.464514 − 0.804561i −0.534666 − 0.845064i \(-0.679562\pi\)
0.999179 + 0.0405023i \(0.0128958\pi\)
\(68\) 2.80336 − 4.85556i 0.339957 − 0.588823i
\(69\) 0 0
\(70\) 0.576760 + 0.998977i 0.0689360 + 0.119401i
\(71\) −9.60442 −1.13983 −0.569917 − 0.821702i \(-0.693025\pi\)
−0.569917 + 0.821702i \(0.693025\pi\)
\(72\) 0 0
\(73\) 0.846480 0.0990730 0.0495365 − 0.998772i \(-0.484226\pi\)
0.0495365 + 0.998772i \(0.484226\pi\)
\(74\) 1.30994 + 2.26888i 0.152278 + 0.263752i
\(75\) 0 0
\(76\) 6.42807 − 11.1337i 0.737350 − 1.27713i
\(77\) 2.60074 − 4.50461i 0.296381 − 0.513348i
\(78\) 0 0
\(79\) 3.80221 + 6.58562i 0.427782 + 0.740940i 0.996676 − 0.0814710i \(-0.0259618\pi\)
−0.568894 + 0.822411i \(0.692628\pi\)
\(80\) 5.34533 0.597626
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 5.72545 + 9.91677i 0.628450 + 1.08851i 0.987863 + 0.155328i \(0.0496435\pi\)
−0.359413 + 0.933178i \(0.617023\pi\)
\(84\) 0 0
\(85\) −3.57676 + 6.19513i −0.387954 + 0.671956i
\(86\) 2.43580 − 4.21894i 0.262659 − 0.454939i
\(87\) 0 0
\(88\) 5.07676 + 8.79321i 0.541184 + 0.937359i
\(89\) −9.24943 −0.980437 −0.490219 − 0.871600i \(-0.663083\pi\)
−0.490219 + 0.871600i \(0.663083\pi\)
\(90\) 0 0
\(91\) 3.15352 0.330579
\(92\) −3.79853 − 6.57924i −0.396024 − 0.685933i
\(93\) 0 0
\(94\) 0.274551 − 0.475537i 0.0283178 − 0.0490479i
\(95\) −8.20147 + 14.2054i −0.841453 + 1.45744i
\(96\) 0 0
\(97\) 1.72545 + 2.98856i 0.175193 + 0.303443i 0.940228 − 0.340546i \(-0.110612\pi\)
−0.765035 + 0.643988i \(0.777278\pi\)
\(98\) −0.523976 −0.0529296
\(99\) 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 567.2.f.m.190.2 6
3.2 odd 2 567.2.f.l.190.2 6
9.2 odd 6 567.2.f.l.379.2 6
9.4 even 3 567.2.a.e.1.2 ✓ 3
9.5 odd 6 567.2.a.f.1.2 yes 3
9.7 even 3 inner 567.2.f.m.379.2 6
36.23 even 6 9072.2.a.cb.1.1 3
36.31 odd 6 9072.2.a.bu.1.3 3
63.13 odd 6 3969.2.a.o.1.2 3
63.41 even 6 3969.2.a.n.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.a.e.1.2 ✓ 3 9.4 even 3
567.2.a.f.1.2 yes 3 9.5 odd 6
567.2.f.l.190.2 6 3.2 odd 2
567.2.f.l.379.2 6 9.2 odd 6
567.2.f.m.190.2 6 1.1 even 1 trivial
567.2.f.m.379.2 6 9.7 even 3 inner
3969.2.a.n.1.2 3 63.41 even 6
3969.2.a.o.1.2 3 63.13 odd 6
9072.2.a.bu.1.3 3 36.31 odd 6
9072.2.a.cb.1.1 3 36.23 even 6