Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1323,2,Mod(361,1323)] 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("1323.361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1323, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-1,0,-3,10] 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: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 667.3
Root \(0.500000 - 0.224437i\) of defining polynomial
Character \(\chi\) \(=\) 1323.667
Dual form 1323.2.g.c.361.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.849814 + 1.47192i) q^{2} +(-0.444368 + 0.769668i) q^{4} +3.58836 q^{5} +1.88874 q^{8} +(3.04944 + 5.28179i) q^{10} +2.81089 q^{11} +(-0.500000 - 0.866025i) q^{13} +(2.49381 + 4.31941i) q^{16} +(-2.05563 - 3.56046i) q^{17} +(0.444368 - 0.769668i) q^{19} +(-1.59455 + 2.76185i) q^{20} +(2.38874 + 4.13741i) q^{22} -5.87636 q^{23} +7.87636 q^{25} +(0.849814 - 1.47192i) q^{26} +(-0.849814 + 1.47192i) q^{29} +(3.49381 - 6.05146i) q^{31} +(-2.34981 + 4.07000i) q^{32} +(3.49381 - 6.05146i) q^{34} +(-2.38255 + 4.12669i) q^{37} +1.51052 q^{38} +6.77747 q^{40} +(-2.70582 - 4.68661i) q^{41} +(-2.60507 + 4.51212i) q^{43} +(-1.24907 + 2.16345i) q^{44} +(-4.99381 - 8.64953i) q^{46} +(-1.33310 - 2.30900i) q^{47} +(6.69344 + 11.5934i) q^{50} +0.888736 q^{52} +(-0.0618219 - 0.107079i) q^{53} +10.0865 q^{55} -2.88874 q^{58} +(-4.43818 + 7.68715i) q^{59} +(-1.93818 - 3.35702i) q^{61} +11.8764 q^{62} +1.98762 q^{64} +(-1.79418 - 3.10761i) q^{65} +(-6.15452 + 10.6599i) q^{67} +3.65383 q^{68} +2.87636 q^{71} +(5.32072 + 9.21576i) q^{73} -8.09888 q^{74} +(0.394926 + 0.684031i) q^{76} +(3.54325 + 6.13709i) q^{79} +(8.94870 + 15.4996i) q^{80} +(4.59888 - 7.96550i) q^{82} +(-2.05563 + 3.56046i) q^{83} +(-7.37636 - 12.7762i) q^{85} -8.85532 q^{86} +5.30903 q^{88} +(4.80470 - 8.32199i) q^{89} +(2.61126 - 4.52284i) q^{92} +(2.26578 - 3.92445i) q^{94} +(1.59455 - 2.76185i) q^{95} +(-3.66071 + 6.34053i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - 3 q^{4} + 10 q^{5} + 12 q^{8} + 4 q^{11} - 3 q^{13} - 3 q^{16} - 12 q^{17} + 3 q^{19} - 16 q^{20} + 15 q^{22} + 12 q^{25} - q^{26} + q^{29} + 3 q^{31} - 8 q^{32} + 3 q^{34} + 3 q^{37} - 16 q^{38}+ \cdots - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.849814 + 1.47192i 0.600909 + 1.04081i 0.992684 + 0.120744i \(0.0385280\pi\)
−0.391774 + 0.920061i \(0.628139\pi\)
\(3\) 0 0
\(4\) −0.444368 + 0.769668i −0.222184 + 0.384834i
\(5\) 3.58836 1.60477 0.802383 − 0.596810i \(-0.203565\pi\)
0.802383 + 0.596810i \(0.203565\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.88874 0.667769
\(9\) 0 0
\(10\) 3.04944 + 5.28179i 0.964318 + 1.67025i
\(11\) 2.81089 0.847516 0.423758 − 0.905775i \(-0.360711\pi\)
0.423758 + 0.905775i \(0.360711\pi\)
\(12\) 0 0
\(13\) −0.500000 − 0.866025i −0.138675 − 0.240192i 0.788320 − 0.615265i \(-0.210951\pi\)
−0.926995 + 0.375073i \(0.877618\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 2.49381 + 4.31941i 0.623453 + 1.07985i
\(17\) −2.05563 − 3.56046i −0.498564 − 0.863538i 0.501435 − 0.865196i \(-0.332806\pi\)
−0.999999 + 0.00165734i \(0.999472\pi\)
\(18\) 0 0
\(19\) 0.444368 − 0.769668i 0.101945 − 0.176574i −0.810541 − 0.585682i \(-0.800827\pi\)
0.912486 + 0.409108i \(0.134160\pi\)
\(20\) −1.59455 + 2.76185i −0.356553 + 0.617568i
\(21\) 0 0
\(22\) 2.38874 + 4.13741i 0.509280 + 0.882099i
\(23\) −5.87636 −1.22530 −0.612652 − 0.790352i \(-0.709897\pi\)
−0.612652 + 0.790352i \(0.709897\pi\)
\(24\) 0 0
\(25\) 7.87636 1.57527
\(26\) 0.849814 − 1.47192i 0.166662 − 0.288667i
\(27\) 0 0
\(28\) 0 0
\(29\) −0.849814 + 1.47192i −0.157807 + 0.273329i −0.934077 − 0.357071i \(-0.883776\pi\)
0.776271 + 0.630399i \(0.217109\pi\)
\(30\) 0 0
\(31\) 3.49381 − 6.05146i 0.627507 − 1.08687i −0.360544 − 0.932742i \(-0.617409\pi\)
0.988050 − 0.154131i \(-0.0492579\pi\)
\(32\) −2.34981 + 4.07000i −0.415392 + 0.719481i
\(33\) 0 0
\(34\) 3.49381 − 6.05146i 0.599183 − 1.03782i
\(35\) 0 0
\(36\) 0 0
\(37\) −2.38255 + 4.12669i −0.391688 + 0.678424i −0.992672 − 0.120837i \(-0.961442\pi\)
0.600984 + 0.799261i \(0.294775\pi\)
\(38\) 1.51052 0.245039
\(39\) 0 0
\(40\) 6.77747 1.07161
\(41\) −2.70582 − 4.68661i −0.422578 − 0.731926i 0.573613 − 0.819126i \(-0.305541\pi\)
−0.996191 + 0.0872002i \(0.972208\pi\)
\(42\) 0 0
\(43\) −2.60507 + 4.51212i −0.397270 + 0.688092i −0.993388 − 0.114805i \(-0.963376\pi\)
0.596118 + 0.802897i \(0.296709\pi\)
\(44\) −1.24907 + 2.16345i −0.188304 + 0.326153i
\(45\) 0 0
\(46\) −4.99381 − 8.64953i −0.736297 − 1.27530i
\(47\) −1.33310 − 2.30900i −0.194453 − 0.336803i 0.752268 − 0.658857i \(-0.228960\pi\)
−0.946721 + 0.322055i \(0.895627\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 6.69344 + 11.5934i 0.946595 + 1.63955i
\(51\) 0 0
\(52\) 0.888736 0.123245
\(53\) −0.0618219 − 0.107079i −0.00849190 − 0.0147084i 0.861748 − 0.507336i \(-0.169370\pi\)
−0.870240 + 0.492628i \(0.836036\pi\)
\(54\) 0 0
\(55\) 10.0865 1.36006
\(56\) 0 0
\(57\) 0 0
\(58\) −2.88874 −0.379310
\(59\) −4.43818 + 7.68715i −0.577802 + 1.00078i 0.417929 + 0.908479i \(0.362756\pi\)
−0.995731 + 0.0923022i \(0.970577\pi\)
\(60\) 0 0
\(61\) −1.93818 − 3.35702i −0.248158 − 0.429823i 0.714857 − 0.699271i \(-0.246492\pi\)
−0.963015 + 0.269448i \(0.913159\pi\)
\(62\) 11.8764 1.50830
\(63\) 0 0
\(64\) 1.98762 0.248453
\(65\) −1.79418 − 3.10761i −0.222541 − 0.385452i
\(66\) 0 0
\(67\) −6.15452 + 10.6599i −0.751894 + 1.30232i 0.195010 + 0.980801i \(0.437526\pi\)
−0.946904 + 0.321517i \(0.895807\pi\)
\(68\) 3.65383 0.443092
\(69\) 0 0
\(70\) 0 0
\(71\) 2.87636 0.341361 0.170680 − 0.985326i \(-0.445403\pi\)
0.170680 + 0.985326i \(0.445403\pi\)
\(72\) 0 0
\(73\) 5.32072 + 9.21576i 0.622744 + 1.07862i 0.988973 + 0.148099i \(0.0473154\pi\)
−0.366229 + 0.930525i \(0.619351\pi\)
\(74\) −8.09888 −0.941476
\(75\) 0 0
\(76\) 0.394926 + 0.684031i 0.0453011 + 0.0784638i
\(77\) 0 0
\(78\) 0 0
\(79\) 3.54325 + 6.13709i 0.398647 + 0.690477i 0.993559 − 0.113314i \(-0.0361465\pi\)
−0.594912 + 0.803791i \(0.702813\pi\)
\(80\) 8.94870 + 15.4996i 1.00049 + 1.73291i
\(81\) 0 0
\(82\) 4.59888 − 7.96550i 0.507862 − 0.879642i
\(83\) −2.05563 + 3.56046i −0.225635 + 0.390811i −0.956510 − 0.291700i \(-0.905779\pi\)
0.730875 + 0.682512i \(0.239112\pi\)
\(84\) 0 0
\(85\) −7.37636 − 12.7762i −0.800078 − 1.38578i
\(86\) −8.85532 −0.954893
\(87\) 0 0
\(88\) 5.30903 0.565945
\(89\) 4.80470 − 8.32199i 0.509297 − 0.882129i −0.490645 − 0.871360i \(-0.663239\pi\)
0.999942 − 0.0107692i \(-0.00342802\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.61126 − 4.52284i 0.272243 − 0.471539i
\(93\) 0 0
\(94\) 2.26578 − 3.92445i 0.233697 − 0.404776i
\(95\) 1.59455 − 2.76185i 0.163598 − 0.283360i
\(96\) 0 0
\(97\) −3.66071 + 6.34053i −0.371688 + 0.643783i −0.989825 − 0.142287i \(-0.954554\pi\)
0.618137 + 0.786070i \(0.287888\pi\)
\(98\) 0 0
\(99\) 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 1323.2.g.c.667.3 6
3.2 odd 2 441.2.g.e.79.1 6
7.2 even 3 189.2.f.a.127.3 6
7.3 odd 6 1323.2.h.e.802.1 6
7.4 even 3 1323.2.h.d.802.1 6
7.5 odd 6 1323.2.f.c.883.3 6
7.6 odd 2 1323.2.g.b.667.3 6
9.4 even 3 1323.2.h.d.226.1 6
9.5 odd 6 441.2.h.c.373.3 6
21.2 odd 6 63.2.f.b.43.1 yes 6
21.5 even 6 441.2.f.d.295.1 6
21.11 odd 6 441.2.h.c.214.3 6
21.17 even 6 441.2.h.b.214.3 6
21.20 even 2 441.2.g.d.79.1 6
28.23 odd 6 3024.2.r.g.2017.1 6
63.2 odd 6 567.2.a.d.1.3 3
63.4 even 3 inner 1323.2.g.c.361.3 6
63.5 even 6 441.2.f.d.148.1 6
63.13 odd 6 1323.2.h.e.226.1 6
63.16 even 3 567.2.a.g.1.1 3
63.23 odd 6 63.2.f.b.22.1 ✓ 6
63.31 odd 6 1323.2.g.b.361.3 6
63.32 odd 6 441.2.g.e.67.1 6
63.40 odd 6 1323.2.f.c.442.3 6
63.41 even 6 441.2.h.b.373.3 6
63.47 even 6 3969.2.a.m.1.3 3
63.58 even 3 189.2.f.a.64.3 6
63.59 even 6 441.2.g.d.67.1 6
63.61 odd 6 3969.2.a.p.1.1 3
84.23 even 6 1008.2.r.k.673.1 6
252.23 even 6 1008.2.r.k.337.1 6
252.79 odd 6 9072.2.a.cd.1.3 3
252.191 even 6 9072.2.a.bq.1.1 3
252.247 odd 6 3024.2.r.g.1009.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.1 ✓ 6 63.23 odd 6
63.2.f.b.43.1 yes 6 21.2 odd 6
189.2.f.a.64.3 6 63.58 even 3
189.2.f.a.127.3 6 7.2 even 3
441.2.f.d.148.1 6 63.5 even 6
441.2.f.d.295.1 6 21.5 even 6
441.2.g.d.67.1 6 63.59 even 6
441.2.g.d.79.1 6 21.20 even 2
441.2.g.e.67.1 6 63.32 odd 6
441.2.g.e.79.1 6 3.2 odd 2
441.2.h.b.214.3 6 21.17 even 6
441.2.h.b.373.3 6 63.41 even 6
441.2.h.c.214.3 6 21.11 odd 6
441.2.h.c.373.3 6 9.5 odd 6
567.2.a.d.1.3 3 63.2 odd 6
567.2.a.g.1.1 3 63.16 even 3
1008.2.r.k.337.1 6 252.23 even 6
1008.2.r.k.673.1 6 84.23 even 6
1323.2.f.c.442.3 6 63.40 odd 6
1323.2.f.c.883.3 6 7.5 odd 6
1323.2.g.b.361.3 6 63.31 odd 6
1323.2.g.b.667.3 6 7.6 odd 2
1323.2.g.c.361.3 6 63.4 even 3 inner
1323.2.g.c.667.3 6 1.1 even 1 trivial
1323.2.h.d.226.1 6 9.4 even 3
1323.2.h.d.802.1 6 7.4 even 3
1323.2.h.e.226.1 6 63.13 odd 6
1323.2.h.e.802.1 6 7.3 odd 6
3024.2.r.g.1009.1 6 252.247 odd 6
3024.2.r.g.2017.1 6 28.23 odd 6
3969.2.a.m.1.3 3 63.47 even 6
3969.2.a.p.1.1 3 63.61 odd 6
9072.2.a.bq.1.1 3 252.191 even 6
9072.2.a.cd.1.3 3 252.79 odd 6