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Properties

Label 1323.2.h.c.226.2
Level $1323$
Weight $2$
Character 1323.226
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(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,-6,0,6,3] 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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 226.2
Root \(-0.173648 - 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 1323.226
Dual form 1323.2.h.c.802.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.34730 q^{2} -0.184793 q^{4} +(1.26604 + 2.19285i) q^{5} +2.94356 q^{8} +(-1.70574 - 2.95442i) q^{10} +(0.233956 - 0.405223i) q^{11} +(-2.91147 + 5.04282i) q^{13} -3.59627 q^{16} +(1.93969 + 3.35965i) q^{17} +(1.09240 - 1.89209i) q^{19} +(-0.233956 - 0.405223i) q^{20} +(-0.315207 + 0.545955i) q^{22} +(-0.0530334 - 0.0918566i) q^{23} +(-0.705737 + 1.22237i) q^{25} +(3.92262 - 6.79417i) q^{26} +(4.39053 + 7.60462i) q^{29} -7.68004 q^{31} -1.04189 q^{32} +(-2.61334 - 4.52644i) q^{34} +(3.84002 - 6.65111i) q^{37} +(-1.47178 + 2.54920i) q^{38} +(3.72668 + 6.45480i) q^{40} +(-1.11334 + 1.92836i) q^{41} +(-0.613341 - 1.06234i) q^{43} +(-0.0432332 + 0.0748822i) q^{44} +(0.0714517 + 0.123758i) q^{46} +5.33275 q^{47} +(0.950837 - 1.64690i) q^{50} +(0.538019 - 0.931876i) q^{52} +(-0.358441 - 0.620838i) q^{53} +1.18479 q^{55} +(-5.91534 - 10.2457i) q^{58} -0.736482 q^{59} +0.958111 q^{61} +10.3473 q^{62} +8.59627 q^{64} -14.7442 q^{65} -9.63816 q^{67} +(-0.358441 - 0.620838i) q^{68} -13.2344 q^{71} +(5.13429 + 8.89284i) q^{73} +(-5.17365 + 8.96102i) q^{74} +(-0.201867 + 0.349643i) q^{76} -12.6382 q^{79} +(-4.55303 - 7.88609i) q^{80} +(1.50000 - 2.59808i) q^{82} +(-1.36571 - 2.36549i) q^{83} +(-4.91147 + 8.50692i) q^{85} +(0.826352 + 1.43128i) q^{86} +(0.688663 - 1.19280i) q^{88} +(-4.05690 + 7.02676i) q^{89} +(0.00980018 + 0.0169744i) q^{92} -7.18479 q^{94} +5.53209 q^{95} +(6.80200 + 11.7814i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} + 3 q^{5} - 12 q^{8} + 6 q^{11} + 3 q^{13} + 6 q^{16} + 6 q^{17} + 3 q^{19} - 6 q^{20} - 9 q^{22} + 12 q^{23} + 6 q^{25} - 3 q^{26} + 9 q^{29} - 6 q^{31} - 9 q^{34} + 3 q^{37} + 6 q^{38}+ \cdots + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.34730 −0.952682 −0.476341 − 0.879261i \(-0.658037\pi\)
−0.476341 + 0.879261i \(0.658037\pi\)
\(3\) 0 0
\(4\) −0.184793 −0.0923963
\(5\) 1.26604 + 2.19285i 0.566192 + 0.980674i 0.996938 + 0.0782003i \(0.0249174\pi\)
−0.430745 + 0.902473i \(0.641749\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 2.94356 1.04071
\(9\) 0 0
\(10\) −1.70574 − 2.95442i −0.539401 − 0.934271i
\(11\) 0.233956 − 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 0.107094 + 0.994249i \(0.465845\pi\)
−0.914592 + 0.404378i \(0.867488\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.59627 −0.899067
\(17\) 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
\(23\) −0.0530334 − 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
\(26\) 3.92262 − 6.79417i 0.769289 − 1.33245i
\(27\) 0 0
\(28\) 0 0
\(29\) 4.39053 + 7.60462i 0.815301 + 1.41214i 0.909112 + 0.416552i \(0.136762\pi\)
−0.0938108 + 0.995590i \(0.529905\pi\)
\(30\) 0 0
\(31\) −7.68004 −1.37938 −0.689688 − 0.724106i \(-0.742252\pi\)
−0.689688 + 0.724106i \(0.742252\pi\)
\(32\) −1.04189 −0.184182
\(33\) 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 \(-0.0508099\pi\)
\(38\) −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 \(-0.888962\pi\)
0.765897 + 0.642964i \(0.222295\pi\)
\(42\) 0 0
\(43\) −0.613341 − 1.06234i −0.0935336 − 0.162005i 0.815462 − 0.578811i \(-0.196483\pi\)
−0.908996 + 0.416806i \(0.863150\pi\)
\(44\) −0.0432332 + 0.0748822i −0.00651766 + 0.0112889i
\(45\) 0 0
\(46\) 0.0714517 + 0.123758i 0.0105350 + 0.0182471i
\(47\) 5.33275 0.777861 0.388931 − 0.921267i \(-0.372845\pi\)
0.388931 + 0.921267i \(0.372845\pi\)
\(48\) 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
\(53\) −0.358441 − 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
\(58\) −5.91534 − 10.2457i −0.776723 − 1.34532i
\(59\) −0.736482 −0.0958818 −0.0479409 − 0.998850i \(-0.515266\pi\)
−0.0479409 + 0.998850i \(0.515266\pi\)
\(60\) 0 0
\(61\) 0.958111 0.122674 0.0613368 − 0.998117i \(-0.480464\pi\)
0.0613368 + 0.998117i \(0.480464\pi\)
\(62\) 10.3473 1.31411
\(63\) 0 0
\(64\) 8.59627 1.07453
\(65\) −14.7442 −1.82880
\(66\) 0 0
\(67\) −9.63816 −1.17749 −0.588744 − 0.808320i \(-0.700377\pi\)
−0.588744 + 0.808320i \(0.700377\pi\)
\(68\) −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
\(73\) 5.13429 + 8.89284i 0.600923 + 1.04083i 0.992682 + 0.120761i \(0.0385334\pi\)
−0.391759 + 0.920068i \(0.628133\pi\)
\(74\) −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
\(79\) −12.6382 −1.42190 −0.710952 − 0.703241i \(-0.751736\pi\)
−0.710952 + 0.703241i \(0.751736\pi\)
\(80\) −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 0.781286 − 0.624173i \(-0.214564\pi\)
−0.931193 + 0.364527i \(0.881231\pi\)
\(84\) 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
\(89\) −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
\(94\) −7.18479 −0.741055
\(95\) 5.53209 0.567580
\(96\) 0 0
\(97\) 6.80200 + 11.7814i 0.690639 + 1.19622i 0.971629 + 0.236511i \(0.0760039\pi\)
−0.280990 + 0.959711i \(0.590663\pi\)
\(98\) 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.c.226.2 6
3.2 odd 2 441.2.h.d.373.2 6
7.2 even 3 189.2.f.b.64.2 6
7.3 odd 6 1323.2.g.e.361.2 6
7.4 even 3 1323.2.g.d.361.2 6
7.5 odd 6 1323.2.f.d.442.2 6
7.6 odd 2 1323.2.h.b.226.2 6
9.2 odd 6 441.2.g.c.79.2 6
9.7 even 3 1323.2.g.d.667.2 6
21.2 odd 6 63.2.f.a.22.2 ✓ 6
21.5 even 6 441.2.f.c.148.2 6
21.11 odd 6 441.2.g.c.67.2 6
21.17 even 6 441.2.g.b.67.2 6
21.20 even 2 441.2.h.e.373.2 6
28.23 odd 6 3024.2.r.k.1009.3 6
63.2 odd 6 63.2.f.a.43.2 yes 6
63.5 even 6 3969.2.a.q.1.2 3
63.11 odd 6 441.2.h.d.214.2 6
63.16 even 3 189.2.f.b.127.2 6
63.20 even 6 441.2.g.b.79.2 6
63.23 odd 6 567.2.a.h.1.2 3
63.25 even 3 inner 1323.2.h.c.802.2 6
63.34 odd 6 1323.2.g.e.667.2 6
63.38 even 6 441.2.h.e.214.2 6
63.40 odd 6 3969.2.a.l.1.2 3
63.47 even 6 441.2.f.c.295.2 6
63.52 odd 6 1323.2.h.b.802.2 6
63.58 even 3 567.2.a.c.1.2 3
63.61 odd 6 1323.2.f.d.883.2 6
84.23 even 6 1008.2.r.h.337.1 6
252.23 even 6 9072.2.a.ca.1.3 3
252.79 odd 6 3024.2.r.k.2017.3 6
252.191 even 6 1008.2.r.h.673.1 6
252.247 odd 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 21.2 odd 6
63.2.f.a.43.2 yes 6 63.2 odd 6
189.2.f.b.64.2 6 7.2 even 3
189.2.f.b.127.2 6 63.16 even 3
441.2.f.c.148.2 6 21.5 even 6
441.2.f.c.295.2 6 63.47 even 6
441.2.g.b.67.2 6 21.17 even 6
441.2.g.b.79.2 6 63.20 even 6
441.2.g.c.67.2 6 21.11 odd 6
441.2.g.c.79.2 6 9.2 odd 6
441.2.h.d.214.2 6 63.11 odd 6
441.2.h.d.373.2 6 3.2 odd 2
441.2.h.e.214.2 6 63.38 even 6
441.2.h.e.373.2 6 21.20 even 2
567.2.a.c.1.2 3 63.58 even 3
567.2.a.h.1.2 3 63.23 odd 6
1008.2.r.h.337.1 6 84.23 even 6
1008.2.r.h.673.1 6 252.191 even 6
1323.2.f.d.442.2 6 7.5 odd 6
1323.2.f.d.883.2 6 63.61 odd 6
1323.2.g.d.361.2 6 7.4 even 3
1323.2.g.d.667.2 6 9.7 even 3
1323.2.g.e.361.2 6 7.3 odd 6
1323.2.g.e.667.2 6 63.34 odd 6
1323.2.h.b.226.2 6 7.6 odd 2
1323.2.h.b.802.2 6 63.52 odd 6
1323.2.h.c.226.2 6 1.1 even 1 trivial
1323.2.h.c.802.2 6 63.25 even 3 inner
3024.2.r.k.1009.3 6 28.23 odd 6
3024.2.r.k.2017.3 6 252.79 odd 6
3969.2.a.l.1.2 3 63.40 odd 6
3969.2.a.q.1.2 3 63.5 even 6
9072.2.a.bs.1.1 3 252.247 odd 6
9072.2.a.ca.1.3 3 252.23 even 6