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Properties

Label 1323.2.g.d.667.2
Level $1323$
Weight $2$
Character 1323.667
Analytic conductor $10.564$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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 = [6,3,0,-3,-6] 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: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) 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 667.2
Root \(-0.173648 - 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 1323.667
Dual form 1323.2.g.d.361.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.673648 + 1.16679i) q^{2} +(0.0923963 - 0.160035i) q^{4} -2.53209 q^{5} +2.94356 q^{8} +(-1.70574 - 2.95442i) q^{10} -0.467911 q^{11} +(-2.91147 - 5.04282i) q^{13} +(1.79813 + 3.11446i) q^{16} +(1.93969 + 3.35965i) q^{17} +(1.09240 - 1.89209i) q^{19} +(-0.233956 + 0.405223i) q^{20} +(-0.315207 - 0.545955i) q^{22} +0.106067 q^{23} +1.41147 q^{25} +(3.92262 - 6.79417i) q^{26} +(4.39053 - 7.60462i) q^{29} +(3.84002 - 6.65111i) q^{31} +(0.520945 - 0.902302i) q^{32} +(-2.61334 + 4.52644i) q^{34} +(3.84002 - 6.65111i) q^{37} +2.94356 q^{38} -7.45336 q^{40} +(-1.11334 - 1.92836i) q^{41} +(-0.613341 + 1.06234i) q^{43} +(-0.0432332 + 0.0748822i) q^{44} +(0.0714517 + 0.123758i) q^{46} +(-2.66637 - 4.61830i) q^{47} +(0.950837 + 1.64690i) q^{50} -1.07604 q^{52} +(-0.358441 - 0.620838i) q^{53} +1.18479 q^{55} +11.8307 q^{58} +(0.368241 - 0.637812i) q^{59} +(-0.479055 - 0.829748i) q^{61} +10.3473 q^{62} +8.59627 q^{64} +(7.37211 + 12.7689i) q^{65} +(4.81908 - 8.34689i) q^{67} +0.716881 q^{68} -13.2344 q^{71} +(5.13429 + 8.89284i) q^{73} +10.3473 q^{74} +(-0.201867 - 0.349643i) q^{76} +(6.31908 + 10.9450i) q^{79} +(-4.55303 - 7.88609i) q^{80} +(1.50000 - 2.59808i) q^{82} +(-1.36571 + 2.36549i) q^{83} +(-4.91147 - 8.50692i) q^{85} -1.65270 q^{86} -1.37733 q^{88} +(-4.05690 + 7.02676i) q^{89} +(0.00980018 - 0.0169744i) q^{92} +(3.59240 - 6.22221i) q^{94} +(-2.76604 + 4.79093i) q^{95} +(6.80200 - 11.7814i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 3 q^{4} - 6 q^{5} - 12 q^{8} - 12 q^{11} + 3 q^{13} - 3 q^{16} + 6 q^{17} + 3 q^{19} - 6 q^{20} - 9 q^{22} - 24 q^{23} - 12 q^{25} - 3 q^{26} + 9 q^{29} + 3 q^{31} - 9 q^{34} + 3 q^{37}+ \cdots + 3 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.673648 + 1.16679i 0.476341 + 0.825047i 0.999633 − 0.0271067i \(-0.00862938\pi\)
−0.523291 + 0.852154i \(0.675296\pi\)
\(3\) 0 0
\(4\) 0.0923963 − 0.160035i 0.0461981 − 0.0800175i
\(5\) −2.53209 −1.13238 −0.566192 − 0.824273i \(-0.691584\pi\)
−0.566192 + 0.824273i \(0.691584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 2.94356 1.04071
\(9\) 0 0
\(10\) −1.70574 − 2.95442i −0.539401 − 0.934271i
\(11\) −0.467911 −0.141081 −0.0705403 − 0.997509i \(-0.522472\pi\)
−0.0705403 + 0.997509i \(0.522472\pi\)
\(12\) 0 0
\(13\) −2.91147 − 5.04282i −0.807498 − 1.39863i −0.914592 − 0.404378i \(-0.867488\pi\)
0.107094 − 0.994249i \(-0.465845\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.79813 + 3.11446i 0.449533 + 0.778615i
\(17\) 1.93969 + 3.35965i 0.470445 + 0.814834i 0.999429 − 0.0337978i \(-0.0107602\pi\)
−0.528984 + 0.848632i \(0.677427\pi\)
\(18\) 0 0
\(19\) 1.09240 − 1.89209i 0.250613 − 0.434074i −0.713082 − 0.701081i \(-0.752701\pi\)
0.963695 + 0.267007i \(0.0860345\pi\)
\(20\) −0.233956 + 0.405223i −0.0523141 + 0.0906106i
\(21\) 0 0
\(22\) −0.315207 − 0.545955i −0.0672025 − 0.116398i
\(23\) 0.106067 0.0221165 0.0110582 − 0.999939i \(-0.496480\pi\)
0.0110582 + 0.999939i \(0.496480\pi\)
\(24\) 0 0
\(25\) 1.41147 0.282295
\(26\) 3.92262 − 6.79417i 0.769289 − 1.33245i
\(27\) 0 0
\(28\) 0 0
\(29\) 4.39053 − 7.60462i 0.815301 − 1.41214i −0.0938108 − 0.995590i \(-0.529905\pi\)
0.909112 − 0.416552i \(-0.136762\pi\)
\(30\) 0 0
\(31\) 3.84002 − 6.65111i 0.689688 − 1.19458i −0.282250 − 0.959341i \(-0.591081\pi\)
0.971939 − 0.235235i \(-0.0755858\pi\)
\(32\) 0.520945 − 0.902302i 0.0920909 − 0.159506i
\(33\) 0 0
\(34\) −2.61334 + 4.52644i −0.448184 + 0.776278i
\(35\) 0 0
\(36\) 0 0
\(37\) 3.84002 − 6.65111i 0.631296 − 1.09344i −0.355991 − 0.934489i \(-0.615857\pi\)
0.987287 − 0.158947i \(-0.0508099\pi\)
\(38\) 2.94356 0.477509
\(39\) 0 0
\(40\) −7.45336 −1.17848
\(41\) −1.11334 − 1.92836i −0.173875 − 0.301160i 0.765897 − 0.642964i \(-0.222295\pi\)
−0.939771 + 0.341804i \(0.888962\pi\)
\(42\) 0 0
\(43\) −0.613341 + 1.06234i −0.0935336 + 0.162005i −0.908996 − 0.416806i \(-0.863150\pi\)
0.815462 + 0.578811i \(0.196483\pi\)
\(44\) −0.0432332 + 0.0748822i −0.00651766 + 0.0112889i
\(45\) 0 0
\(46\) 0.0714517 + 0.123758i 0.0105350 + 0.0182471i
\(47\) −2.66637 − 4.61830i −0.388931 − 0.673648i 0.603375 − 0.797457i \(-0.293822\pi\)
−0.992306 + 0.123810i \(0.960489\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0.950837 + 1.64690i 0.134469 + 0.232907i
\(51\) 0 0
\(52\) −1.07604 −0.149220
\(53\) −0.358441 − 0.620838i −0.0492356 − 0.0852786i 0.840357 − 0.542033i \(-0.182345\pi\)
−0.889593 + 0.456754i \(0.849012\pi\)
\(54\) 0 0
\(55\) 1.18479 0.159757
\(56\) 0 0
\(57\) 0 0
\(58\) 11.8307 1.55345
\(59\) 0.368241 − 0.637812i 0.0479409 − 0.0830360i −0.841059 − 0.540943i \(-0.818067\pi\)
0.889000 + 0.457907i \(0.151401\pi\)
\(60\) 0 0
\(61\) −0.479055 − 0.829748i −0.0613368 − 0.106238i 0.833726 − 0.552178i \(-0.186203\pi\)
−0.895063 + 0.445939i \(0.852870\pi\)
\(62\) 10.3473 1.31411
\(63\) 0 0
\(64\) 8.59627 1.07453
\(65\) 7.37211 + 12.7689i 0.914398 + 1.58378i
\(66\) 0 0
\(67\) 4.81908 − 8.34689i 0.588744 − 1.01973i −0.405653 − 0.914027i \(-0.632956\pi\)
0.994397 − 0.105708i \(-0.0337107\pi\)
\(68\) 0.716881 0.0869346
\(69\) 0 0
\(70\) 0 0
\(71\) −13.2344 −1.57064 −0.785318 − 0.619092i \(-0.787501\pi\)
−0.785318 + 0.619092i \(0.787501\pi\)
\(72\) 0 0
\(73\) 5.13429 + 8.89284i 0.600923 + 1.04083i 0.992682 + 0.120761i \(0.0385334\pi\)
−0.391759 + 0.920068i \(0.628133\pi\)
\(74\) 10.3473 1.20285
\(75\) 0 0
\(76\) −0.201867 − 0.349643i −0.0231557 − 0.0401068i
\(77\) 0 0
\(78\) 0 0
\(79\) 6.31908 + 10.9450i 0.710952 + 1.23140i 0.964500 + 0.264082i \(0.0850689\pi\)
−0.253548 + 0.967323i \(0.581598\pi\)
\(80\) −4.55303 − 7.88609i −0.509045 − 0.881691i
\(81\) 0 0
\(82\) 1.50000 − 2.59808i 0.165647 − 0.286910i
\(83\) −1.36571 + 2.36549i −0.149907 + 0.259646i −0.931193 − 0.364527i \(-0.881231\pi\)
0.781286 + 0.624173i \(0.214564\pi\)
\(84\) 0 0
\(85\) −4.91147 − 8.50692i −0.532724 − 0.922705i
\(86\) −1.65270 −0.178216
\(87\) 0 0
\(88\) −1.37733 −0.146823
\(89\) −4.05690 + 7.02676i −0.430031 + 0.744835i −0.996875 − 0.0789894i \(-0.974831\pi\)
0.566845 + 0.823825i \(0.308164\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.00980018 − 0.0169744i 0.00102174 − 0.00176970i
\(93\) 0 0
\(94\) 3.59240 − 6.22221i 0.370527 − 0.641772i
\(95\) −2.76604 + 4.79093i −0.283790 + 0.491539i
\(96\) 0 0
\(97\) 6.80200 − 11.7814i 0.690639 − 1.19622i −0.280990 − 0.959711i \(-0.590663\pi\)
0.971629 − 0.236511i \(-0.0760039\pi\)
\(98\) 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.d.667.2 6
3.2 odd 2 441.2.g.c.79.2 6
7.2 even 3 189.2.f.b.127.2 6
7.3 odd 6 1323.2.h.b.802.2 6
7.4 even 3 1323.2.h.c.802.2 6
7.5 odd 6 1323.2.f.d.883.2 6
7.6 odd 2 1323.2.g.e.667.2 6
9.4 even 3 1323.2.h.c.226.2 6
9.5 odd 6 441.2.h.d.373.2 6
21.2 odd 6 63.2.f.a.43.2 yes 6
21.5 even 6 441.2.f.c.295.2 6
21.11 odd 6 441.2.h.d.214.2 6
21.17 even 6 441.2.h.e.214.2 6
21.20 even 2 441.2.g.b.79.2 6
28.23 odd 6 3024.2.r.k.2017.3 6
63.2 odd 6 567.2.a.h.1.2 3
63.4 even 3 inner 1323.2.g.d.361.2 6
63.5 even 6 441.2.f.c.148.2 6
63.13 odd 6 1323.2.h.b.226.2 6
63.16 even 3 567.2.a.c.1.2 3
63.23 odd 6 63.2.f.a.22.2 ✓ 6
63.31 odd 6 1323.2.g.e.361.2 6
63.32 odd 6 441.2.g.c.67.2 6
63.40 odd 6 1323.2.f.d.442.2 6
63.41 even 6 441.2.h.e.373.2 6
63.47 even 6 3969.2.a.q.1.2 3
63.58 even 3 189.2.f.b.64.2 6
63.59 even 6 441.2.g.b.67.2 6
63.61 odd 6 3969.2.a.l.1.2 3
84.23 even 6 1008.2.r.h.673.1 6
252.23 even 6 1008.2.r.h.337.1 6
252.79 odd 6 9072.2.a.bs.1.1 3
252.191 even 6 9072.2.a.ca.1.3 3
252.247 odd 6 3024.2.r.k.1009.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.2 ✓ 6 63.23 odd 6
63.2.f.a.43.2 yes 6 21.2 odd 6
189.2.f.b.64.2 6 63.58 even 3
189.2.f.b.127.2 6 7.2 even 3
441.2.f.c.148.2 6 63.5 even 6
441.2.f.c.295.2 6 21.5 even 6
441.2.g.b.67.2 6 63.59 even 6
441.2.g.b.79.2 6 21.20 even 2
441.2.g.c.67.2 6 63.32 odd 6
441.2.g.c.79.2 6 3.2 odd 2
441.2.h.d.214.2 6 21.11 odd 6
441.2.h.d.373.2 6 9.5 odd 6
441.2.h.e.214.2 6 21.17 even 6
441.2.h.e.373.2 6 63.41 even 6
567.2.a.c.1.2 3 63.16 even 3
567.2.a.h.1.2 3 63.2 odd 6
1008.2.r.h.337.1 6 252.23 even 6
1008.2.r.h.673.1 6 84.23 even 6
1323.2.f.d.442.2 6 63.40 odd 6
1323.2.f.d.883.2 6 7.5 odd 6
1323.2.g.d.361.2 6 63.4 even 3 inner
1323.2.g.d.667.2 6 1.1 even 1 trivial
1323.2.g.e.361.2 6 63.31 odd 6
1323.2.g.e.667.2 6 7.6 odd 2
1323.2.h.b.226.2 6 63.13 odd 6
1323.2.h.b.802.2 6 7.3 odd 6
1323.2.h.c.226.2 6 9.4 even 3
1323.2.h.c.802.2 6 7.4 even 3
3024.2.r.k.1009.3 6 252.247 odd 6
3024.2.r.k.2017.3 6 28.23 odd 6
3969.2.a.l.1.2 3 63.61 odd 6
3969.2.a.q.1.2 3 63.47 even 6
9072.2.a.bs.1.1 3 252.79 odd 6
9072.2.a.ca.1.3 3 252.191 even 6