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Properties

Label 1323.2.f.d.883.2
Level $1323$
Weight $2$
Character 1323.883
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(442,1323)] 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("1323.442"); 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, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,3,0,-3,-3] 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 883.2
Root \(-0.173648 + 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 1323.883
Dual form 1323.2.f.d.442.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.673648 - 1.16679i) q^{2} +(0.0923963 + 0.160035i) q^{4} +(-1.26604 - 2.19285i) q^{5} +2.94356 q^{8} -3.41147 q^{10} +(0.233956 - 0.405223i) q^{11} +(2.91147 + 5.04282i) q^{13} +(1.79813 - 3.11446i) q^{16} +3.87939 q^{17} +2.18479 q^{19} +(0.233956 - 0.405223i) q^{20} +(-0.315207 - 0.545955i) q^{22} +(-0.0530334 - 0.0918566i) q^{23} +(-0.705737 + 1.22237i) q^{25} +7.84524 q^{26} +(4.39053 - 7.60462i) q^{29} +(-3.84002 - 6.65111i) q^{31} +(0.520945 + 0.902302i) q^{32} +(2.61334 - 4.52644i) q^{34} -7.68004 q^{37} +(1.47178 - 2.54920i) q^{38} +(-3.72668 - 6.45480i) q^{40} +(1.11334 + 1.92836i) q^{41} +(-0.613341 + 1.06234i) q^{43} +0.0864665 q^{44} -0.142903 q^{46} +(2.66637 - 4.61830i) q^{47} +(0.950837 + 1.64690i) q^{50} +(-0.538019 + 0.931876i) q^{52} +0.716881 q^{53} -1.18479 q^{55} +(-5.91534 - 10.2457i) q^{58} +(-0.368241 - 0.637812i) q^{59} +(0.479055 - 0.829748i) q^{61} -10.3473 q^{62} +8.59627 q^{64} +(7.37211 - 12.7689i) q^{65} +(4.81908 + 8.34689i) q^{67} +(0.358441 + 0.620838i) q^{68} -13.2344 q^{71} +10.2686 q^{73} +(-5.17365 + 8.96102i) q^{74} +(0.201867 + 0.349643i) q^{76} +(6.31908 - 10.9450i) q^{79} -9.10607 q^{80} +3.00000 q^{82} +(1.36571 - 2.36549i) q^{83} +(-4.91147 - 8.50692i) q^{85} +(0.826352 + 1.43128i) q^{86} +(0.688663 - 1.19280i) q^{88} -8.11381 q^{89} +(0.00980018 - 0.0169744i) q^{92} +(-3.59240 - 6.22221i) q^{94} +(-2.76604 - 4.79093i) q^{95} +(-6.80200 + 11.7814i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 3 q^{4} - 3 q^{5} - 12 q^{8} + 6 q^{11} - 3 q^{13} - 3 q^{16} + 12 q^{17} + 6 q^{19} + 6 q^{20} - 9 q^{22} + 12 q^{23} + 6 q^{25} - 6 q^{26} + 9 q^{29} - 3 q^{31} + 9 q^{34} - 6 q^{37} - 6 q^{38}+ \cdots - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 −0.523291 − 0.852154i \(-0.675296\pi\)
0.999633 + 0.0271067i \(0.00862938\pi\)
\(3\) 0 0
\(4\) 0.0923963 + 0.160035i 0.0461981 + 0.0800175i
\(5\) −1.26604 − 2.19285i −0.566192 − 0.980674i −0.996938 − 0.0782003i \(-0.975083\pi\)
0.430745 − 0.902473i \(-0.358251\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 2.94356 1.04071
\(9\) 0 0
\(10\) −3.41147 −1.07880
\(11\) 0.233956 − 0.405223i 0.0705403 − 0.122179i −0.828598 − 0.559844i \(-0.810861\pi\)
0.899138 + 0.437665i \(0.144194\pi\)
\(12\) 0 0
\(13\) 2.91147 + 5.04282i 0.807498 + 1.39863i 0.914592 + 0.404378i \(0.132512\pi\)
−0.107094 + 0.994249i \(0.534155\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.79813 − 3.11446i 0.449533 − 0.778615i
\(17\) 3.87939 0.940889 0.470445 − 0.882430i \(-0.344094\pi\)
0.470445 + 0.882430i \(0.344094\pi\)
\(18\) 0 0
\(19\) 2.18479 0.501226 0.250613 − 0.968087i \(-0.419368\pi\)
0.250613 + 0.968087i \(0.419368\pi\)
\(20\) 0.233956 − 0.405223i 0.0523141 − 0.0906106i
\(21\) 0 0
\(22\) −0.315207 − 0.545955i −0.0672025 − 0.116398i
\(23\) −0.0530334 − 0.0918566i −0.0110582 − 0.0191534i 0.860443 − 0.509546i \(-0.170187\pi\)
−0.871502 + 0.490393i \(0.836853\pi\)
\(24\) 0 0
\(25\) −0.705737 + 1.22237i −0.141147 + 0.244474i
\(26\) 7.84524 1.53858
\(27\) 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 −0.971939 − 0.235235i \(-0.924414\pi\)
0.282250 − 0.959341i \(-0.408919\pi\)
\(32\) 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
\(37\) −7.68004 −1.26259 −0.631296 − 0.775542i \(-0.717477\pi\)
−0.631296 + 0.775542i \(0.717477\pi\)
\(38\) 1.47178 − 2.54920i 0.238754 − 0.413535i
\(39\) 0 0
\(40\) −3.72668 − 6.45480i −0.589240 − 1.02059i
\(41\) 1.11334 + 1.92836i 0.173875 + 0.301160i 0.939771 − 0.341804i \(-0.111038\pi\)
−0.765897 + 0.642964i \(0.777705\pi\)
\(42\) 0 0
\(43\) −0.613341 + 1.06234i −0.0935336 + 0.162005i −0.908996 − 0.416806i \(-0.863150\pi\)
0.815462 + 0.578811i \(0.196483\pi\)
\(44\) 0.0864665 0.0130353
\(45\) 0 0
\(46\) −0.142903 −0.0210700
\(47\) 2.66637 − 4.61830i 0.388931 − 0.673648i −0.603375 − 0.797457i \(-0.706178\pi\)
0.992306 + 0.123810i \(0.0395112\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0.950837 + 1.64690i 0.134469 + 0.232907i
\(51\) 0 0
\(52\) −0.538019 + 0.931876i −0.0746098 + 0.129228i
\(53\) 0.716881 0.0984712 0.0492356 − 0.998787i \(-0.484321\pi\)
0.0492356 + 0.998787i \(0.484321\pi\)
\(54\) 0 0
\(55\) −1.18479 −0.159757
\(56\) 0 0
\(57\) 0 0
\(58\) −5.91534 − 10.2457i −0.776723 − 1.34532i
\(59\) −0.368241 − 0.637812i −0.0479409 − 0.0830360i 0.841059 − 0.540943i \(-0.181933\pi\)
−0.889000 + 0.457907i \(0.848599\pi\)
\(60\) 0 0
\(61\) 0.479055 − 0.829748i 0.0613368 − 0.106238i −0.833726 − 0.552178i \(-0.813797\pi\)
0.895063 + 0.445939i \(0.147130\pi\)
\(62\) −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 0.994397 + 0.105708i \(0.0337107\pi\)
−0.405653 + 0.914027i \(0.632956\pi\)
\(68\) 0.358441 + 0.620838i 0.0434673 + 0.0752876i
\(69\) 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
\(73\) 10.2686 1.20185 0.600923 − 0.799307i \(-0.294800\pi\)
0.600923 + 0.799307i \(0.294800\pi\)
\(74\) −5.17365 + 8.96102i −0.601424 + 1.04170i
\(75\) 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 −0.253548 − 0.967323i \(-0.581598\pi\)
0.964500 − 0.264082i \(-0.0850689\pi\)
\(80\) −9.10607 −1.01809
\(81\) 0 0
\(82\) 3.00000 0.331295
\(83\) 1.36571 − 2.36549i 0.149907 − 0.259646i −0.781286 − 0.624173i \(-0.785436\pi\)
0.931193 + 0.364527i \(0.118769\pi\)
\(84\) 0 0
\(85\) −4.91147 − 8.50692i −0.532724 − 0.922705i
\(86\) 0.826352 + 1.43128i 0.0891078 + 0.154339i
\(87\) 0 0
\(88\) 0.688663 − 1.19280i 0.0734117 − 0.127153i
\(89\) −8.11381 −0.860062 −0.430031 − 0.902814i \(-0.641497\pi\)
−0.430031 + 0.902814i \(0.641497\pi\)
\(90\) 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 \(0.409337\pi\)
−0.971629 + 0.236511i \(0.923996\pi\)
\(98\) 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.f.d.883.2 6
3.2 odd 2 441.2.f.c.295.2 6
7.2 even 3 1323.2.h.b.802.2 6
7.3 odd 6 1323.2.g.d.667.2 6
7.4 even 3 1323.2.g.e.667.2 6
7.5 odd 6 1323.2.h.c.802.2 6
7.6 odd 2 189.2.f.b.127.2 6
9.2 odd 6 3969.2.a.q.1.2 3
9.4 even 3 inner 1323.2.f.d.442.2 6
9.5 odd 6 441.2.f.c.148.2 6
9.7 even 3 3969.2.a.l.1.2 3
21.2 odd 6 441.2.h.e.214.2 6
21.5 even 6 441.2.h.d.214.2 6
21.11 odd 6 441.2.g.b.79.2 6
21.17 even 6 441.2.g.c.79.2 6
21.20 even 2 63.2.f.a.43.2 yes 6
28.27 even 2 3024.2.r.k.2017.3 6
63.4 even 3 1323.2.h.b.226.2 6
63.5 even 6 441.2.g.c.67.2 6
63.13 odd 6 189.2.f.b.64.2 6
63.20 even 6 567.2.a.h.1.2 3
63.23 odd 6 441.2.g.b.67.2 6
63.31 odd 6 1323.2.h.c.226.2 6
63.32 odd 6 441.2.h.e.373.2 6
63.34 odd 6 567.2.a.c.1.2 3
63.40 odd 6 1323.2.g.d.361.2 6
63.41 even 6 63.2.f.a.22.2 ✓ 6
63.58 even 3 1323.2.g.e.361.2 6
63.59 even 6 441.2.h.d.373.2 6
84.83 odd 2 1008.2.r.h.673.1 6
252.83 odd 6 9072.2.a.ca.1.3 3
252.139 even 6 3024.2.r.k.1009.3 6
252.167 odd 6 1008.2.r.h.337.1 6
252.223 even 6 9072.2.a.bs.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.2 ✓ 6 63.41 even 6
63.2.f.a.43.2 yes 6 21.20 even 2
189.2.f.b.64.2 6 63.13 odd 6
189.2.f.b.127.2 6 7.6 odd 2
441.2.f.c.148.2 6 9.5 odd 6
441.2.f.c.295.2 6 3.2 odd 2
441.2.g.b.67.2 6 63.23 odd 6
441.2.g.b.79.2 6 21.11 odd 6
441.2.g.c.67.2 6 63.5 even 6
441.2.g.c.79.2 6 21.17 even 6
441.2.h.d.214.2 6 21.5 even 6
441.2.h.d.373.2 6 63.59 even 6
441.2.h.e.214.2 6 21.2 odd 6
441.2.h.e.373.2 6 63.32 odd 6
567.2.a.c.1.2 3 63.34 odd 6
567.2.a.h.1.2 3 63.20 even 6
1008.2.r.h.337.1 6 252.167 odd 6
1008.2.r.h.673.1 6 84.83 odd 2
1323.2.f.d.442.2 6 9.4 even 3 inner
1323.2.f.d.883.2 6 1.1 even 1 trivial
1323.2.g.d.361.2 6 63.40 odd 6
1323.2.g.d.667.2 6 7.3 odd 6
1323.2.g.e.361.2 6 63.58 even 3
1323.2.g.e.667.2 6 7.4 even 3
1323.2.h.b.226.2 6 63.4 even 3
1323.2.h.b.802.2 6 7.2 even 3
1323.2.h.c.226.2 6 63.31 odd 6
1323.2.h.c.802.2 6 7.5 odd 6
3024.2.r.k.1009.3 6 252.139 even 6
3024.2.r.k.2017.3 6 28.27 even 2
3969.2.a.l.1.2 3 9.7 even 3
3969.2.a.q.1.2 3 9.2 odd 6
9072.2.a.bs.1.1 3 252.223 even 6
9072.2.a.ca.1.3 3 252.83 odd 6