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(226,1323)] 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("1323.226"); 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, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,2,0,6,-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: \(10.5642081874\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 802.1
Root \(0.500000 + 0.224437i\) of defining polynomial
Character \(\chi\) \(=\) 1323.802
Dual form 1323.2.h.d.226.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.69963 q^{2} +0.888736 q^{4} +(-1.79418 + 3.10761i) q^{5} +1.88874 q^{8} +(3.04944 - 5.28179i) q^{10} +(-1.40545 - 2.43430i) q^{11} +(-0.500000 - 0.866025i) q^{13} -4.98762 q^{16} +(-2.05563 + 3.56046i) q^{17} +(0.444368 + 0.769668i) q^{19} +(-1.59455 + 2.76185i) q^{20} +(2.38874 + 4.13741i) q^{22} +(2.93818 - 5.08907i) q^{23} +(-3.93818 - 6.82112i) q^{25} +(0.849814 + 1.47192i) q^{26} +(-0.849814 + 1.47192i) q^{29} -6.98762 q^{31} +4.69963 q^{32} +(3.49381 - 6.05146i) q^{34} +(-2.38255 - 4.12669i) q^{37} +(-0.755260 - 1.30815i) q^{38} +(-3.38874 + 5.86946i) q^{40} +(-2.70582 - 4.68661i) q^{41} +(-2.60507 + 4.51212i) q^{43} +(-1.24907 - 2.16345i) q^{44} +(-4.99381 + 8.64953i) q^{46} +2.66621 q^{47} +(6.69344 + 11.5934i) q^{50} +(-0.444368 - 0.769668i) q^{52} +(-0.0618219 + 0.107079i) q^{53} +10.0865 q^{55} +(1.44437 - 2.50172i) q^{58} +8.87636 q^{59} +3.87636 q^{61} +11.8764 q^{62} +1.98762 q^{64} +3.58836 q^{65} +12.3090 q^{67} +(-1.82691 + 3.16431i) q^{68} +2.87636 q^{71} +(5.32072 - 9.21576i) q^{73} +(4.04944 + 7.01384i) q^{74} +(0.394926 + 0.684031i) q^{76} -7.08650 q^{79} +(8.94870 - 15.4996i) q^{80} +(4.59888 + 7.96550i) q^{82} +(-2.05563 + 3.56046i) q^{83} +(-7.37636 - 12.7762i) q^{85} +(4.42766 - 7.66893i) q^{86} +(-2.65452 - 4.59776i) q^{88} +(4.80470 + 8.32199i) q^{89} +(2.61126 - 4.52284i) q^{92} -4.53156 q^{94} -3.18911 q^{95} +(-3.66071 + 6.34053i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 6 q^{4} - 5 q^{5} + 12 q^{8} - 2 q^{11} - 3 q^{13} + 6 q^{16} - 12 q^{17} + 3 q^{19} - 16 q^{20} + 15 q^{22} - 6 q^{25} - q^{26} + q^{29} - 6 q^{31} + 16 q^{32} + 3 q^{34} + 3 q^{37} + 8 q^{38}+ \cdots - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.69963 −1.20182 −0.600909 − 0.799317i \(-0.705195\pi\)
−0.600909 + 0.799317i \(0.705195\pi\)
\(3\) 0 0
\(4\) 0.888736 0.444368
\(5\) −1.79418 + 3.10761i −0.802383 + 1.38977i 0.115661 + 0.993289i \(0.463101\pi\)
−0.918044 + 0.396479i \(0.870232\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.88874 0.667769
\(9\) 0 0
\(10\) 3.04944 − 5.28179i 0.964318 − 1.67025i
\(11\) −1.40545 − 2.43430i −0.423758 − 0.733970i 0.572546 − 0.819873i \(-0.305956\pi\)
−0.996304 + 0.0859026i \(0.972623\pi\)
\(12\) 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
\(16\) −4.98762 −1.24691
\(17\) −2.05563 + 3.56046i −0.498564 + 0.863538i −0.999999 − 0.00165734i \(-0.999472\pi\)
0.501435 + 0.865196i \(0.332806\pi\)
\(18\) 0 0
\(19\) 0.444368 + 0.769668i 0.101945 + 0.176574i 0.912486 − 0.409108i \(-0.134160\pi\)
−0.810541 + 0.585682i \(0.800827\pi\)
\(20\) −1.59455 + 2.76185i −0.356553 + 0.617568i
\(21\) 0 0
\(22\) 2.38874 + 4.13741i 0.509280 + 0.882099i
\(23\) 2.93818 − 5.08907i 0.612652 − 1.06115i −0.378139 − 0.925749i \(-0.623436\pi\)
0.990792 − 0.135396i \(-0.0432308\pi\)
\(24\) 0 0
\(25\) −3.93818 − 6.82112i −0.787636 − 1.36422i
\(26\) 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
\(31\) −6.98762 −1.25501 −0.627507 − 0.778611i \(-0.715925\pi\)
−0.627507 + 0.778611i \(0.715925\pi\)
\(32\) 4.69963 0.830785
\(33\) 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 0.600984 − 0.799261i \(-0.294775\pi\)
−0.992672 + 0.120837i \(0.961442\pi\)
\(38\) −0.755260 − 1.30815i −0.122519 − 0.212210i
\(39\) 0 0
\(40\) −3.38874 + 5.86946i −0.535806 + 0.928044i
\(41\) −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
\(47\) 2.66621 0.388906 0.194453 − 0.980912i \(-0.437707\pi\)
0.194453 + 0.980912i \(0.437707\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 6.69344 + 11.5934i 0.946595 + 1.63955i
\(51\) 0 0
\(52\) −0.444368 − 0.769668i −0.0616227 − 0.106734i
\(53\) −0.0618219 + 0.107079i −0.00849190 + 0.0147084i −0.870240 − 0.492628i \(-0.836036\pi\)
0.861748 + 0.507336i \(0.169370\pi\)
\(54\) 0 0
\(55\) 10.0865 1.36006
\(56\) 0 0
\(57\) 0 0
\(58\) 1.44437 − 2.50172i 0.189655 − 0.328492i
\(59\) 8.87636 1.15560 0.577802 − 0.816177i \(-0.303911\pi\)
0.577802 + 0.816177i \(0.303911\pi\)
\(60\) 0 0
\(61\) 3.87636 0.496317 0.248158 − 0.968720i \(-0.420175\pi\)
0.248158 + 0.968720i \(0.420175\pi\)
\(62\) 11.8764 1.50830
\(63\) 0 0
\(64\) 1.98762 0.248453
\(65\) 3.58836 0.445082
\(66\) 0 0
\(67\) 12.3090 1.50379 0.751894 − 0.659284i \(-0.229141\pi\)
0.751894 + 0.659284i \(0.229141\pi\)
\(68\) −1.82691 + 3.16431i −0.221546 + 0.383729i
\(69\) 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 −0.366229 − 0.930525i \(-0.619351\pi\)
0.988973 − 0.148099i \(-0.0473154\pi\)
\(74\) 4.04944 + 7.01384i 0.470738 + 0.815342i
\(75\) 0 0
\(76\) 0.394926 + 0.684031i 0.0453011 + 0.0784638i
\(77\) 0 0
\(78\) 0 0
\(79\) −7.08650 −0.797294 −0.398647 − 0.917104i \(-0.630520\pi\)
−0.398647 + 0.917104i \(0.630520\pi\)
\(80\) 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
\(86\) 4.42766 − 7.66893i 0.477447 − 0.826962i
\(87\) 0 0
\(88\) −2.65452 − 4.59776i −0.282972 − 0.490123i
\(89\) 4.80470 + 8.32199i 0.509297 + 0.882129i 0.999942 + 0.0107692i \(0.00342802\pi\)
−0.490645 + 0.871360i \(0.663239\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.61126 − 4.52284i 0.272243 − 0.471539i
\(93\) 0 0
\(94\) −4.53156 −0.467395
\(95\) −3.18911 −0.327196
\(96\) 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.h.d.802.1 6
3.2 odd 2 441.2.h.c.214.3 6
7.2 even 3 1323.2.g.c.667.3 6
7.3 odd 6 1323.2.f.c.883.3 6
7.4 even 3 189.2.f.a.127.3 6
7.5 odd 6 1323.2.g.b.667.3 6
7.6 odd 2 1323.2.h.e.802.1 6
9.4 even 3 1323.2.g.c.361.3 6
9.5 odd 6 441.2.g.e.67.1 6
21.2 odd 6 441.2.g.e.79.1 6
21.5 even 6 441.2.g.d.79.1 6
21.11 odd 6 63.2.f.b.43.1 yes 6
21.17 even 6 441.2.f.d.295.1 6
21.20 even 2 441.2.h.b.214.3 6
28.11 odd 6 3024.2.r.g.2017.1 6
63.4 even 3 189.2.f.a.64.3 6
63.5 even 6 441.2.h.b.373.3 6
63.11 odd 6 567.2.a.d.1.3 3
63.13 odd 6 1323.2.g.b.361.3 6
63.23 odd 6 441.2.h.c.373.3 6
63.25 even 3 567.2.a.g.1.1 3
63.31 odd 6 1323.2.f.c.442.3 6
63.32 odd 6 63.2.f.b.22.1 ✓ 6
63.38 even 6 3969.2.a.m.1.3 3
63.40 odd 6 1323.2.h.e.226.1 6
63.41 even 6 441.2.g.d.67.1 6
63.52 odd 6 3969.2.a.p.1.1 3
63.58 even 3 inner 1323.2.h.d.226.1 6
63.59 even 6 441.2.f.d.148.1 6
84.11 even 6 1008.2.r.k.673.1 6
252.11 even 6 9072.2.a.bq.1.1 3
252.67 odd 6 3024.2.r.g.1009.1 6
252.95 even 6 1008.2.r.k.337.1 6
252.151 odd 6 9072.2.a.cd.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.1 ✓ 6 63.32 odd 6
63.2.f.b.43.1 yes 6 21.11 odd 6
189.2.f.a.64.3 6 63.4 even 3
189.2.f.a.127.3 6 7.4 even 3
441.2.f.d.148.1 6 63.59 even 6
441.2.f.d.295.1 6 21.17 even 6
441.2.g.d.67.1 6 63.41 even 6
441.2.g.d.79.1 6 21.5 even 6
441.2.g.e.67.1 6 9.5 odd 6
441.2.g.e.79.1 6 21.2 odd 6
441.2.h.b.214.3 6 21.20 even 2
441.2.h.b.373.3 6 63.5 even 6
441.2.h.c.214.3 6 3.2 odd 2
441.2.h.c.373.3 6 63.23 odd 6
567.2.a.d.1.3 3 63.11 odd 6
567.2.a.g.1.1 3 63.25 even 3
1008.2.r.k.337.1 6 252.95 even 6
1008.2.r.k.673.1 6 84.11 even 6
1323.2.f.c.442.3 6 63.31 odd 6
1323.2.f.c.883.3 6 7.3 odd 6
1323.2.g.b.361.3 6 63.13 odd 6
1323.2.g.b.667.3 6 7.5 odd 6
1323.2.g.c.361.3 6 9.4 even 3
1323.2.g.c.667.3 6 7.2 even 3
1323.2.h.d.226.1 6 63.58 even 3 inner
1323.2.h.d.802.1 6 1.1 even 1 trivial
1323.2.h.e.226.1 6 63.40 odd 6
1323.2.h.e.802.1 6 7.6 odd 2
3024.2.r.g.1009.1 6 252.67 odd 6
3024.2.r.g.2017.1 6 28.11 odd 6
3969.2.a.m.1.3 3 63.38 even 6
3969.2.a.p.1.1 3 63.52 odd 6
9072.2.a.bq.1.1 3 252.11 even 6
9072.2.a.cd.1.3 3 252.151 odd 6