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Properties

Label 1323.2.g.e.361.1
Level $1323$
Weight $2$
Character 1323.361
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 361.1
Root \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 1323.361
Dual form 1323.2.g.e.667.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.439693 + 0.761570i) q^{2} +(0.613341 + 1.06234i) q^{4} +1.34730 q^{5} -2.83750 q^{8} +(-0.592396 + 1.02606i) q^{10} -1.65270 q^{11} +(-1.68479 + 2.91815i) q^{13} +(0.0209445 - 0.0362770i) q^{16} +(-0.233956 + 0.405223i) q^{17} +(-1.61334 - 2.79439i) q^{19} +(0.826352 + 1.43128i) q^{20} +(0.726682 - 1.25865i) q^{22} -8.94356 q^{23} -3.18479 q^{25} +(-1.48158 - 2.56617i) q^{26} +(3.13429 + 5.42874i) q^{29} +(4.61721 + 7.99724i) q^{31} +(-2.81908 - 4.88279i) q^{32} +(-0.205737 - 0.356347i) q^{34} +(-4.61721 - 7.99724i) q^{37} +2.83750 q^{38} -3.82295 q^{40} +(-1.70574 + 2.95442i) q^{41} +(2.20574 + 3.82045i) q^{43} +(-1.01367 - 1.75573i) q^{44} +(3.93242 - 6.81115i) q^{46} +(-4.67752 + 8.10170i) q^{47} +(1.40033 - 2.42544i) q^{50} -4.13341 q^{52} +(-0.286989 + 0.497079i) q^{53} -2.22668 q^{55} -5.51249 q^{58} +(5.19846 + 9.00400i) q^{59} +(3.81908 - 6.61484i) q^{61} -8.12061 q^{62} +5.04189 q^{64} +(-2.26991 + 3.93161i) q^{65} +(-0.298133 - 0.516382i) q^{67} -0.573978 q^{68} +0.554378 q^{71} +(1.02481 - 1.77503i) q^{73} +8.12061 q^{74} +(1.97906 - 3.42782i) q^{76} +(1.20187 - 2.08169i) q^{79} +(0.0282185 - 0.0488759i) q^{80} +(-1.50000 - 2.59808i) q^{82} +(7.52481 + 13.0334i) q^{83} +(-0.315207 + 0.545955i) q^{85} -3.87939 q^{86} +4.68954 q^{88} +(-4.54323 - 7.86911i) q^{89} +(-5.48545 - 9.50108i) q^{92} +(-4.11334 - 7.12452i) q^{94} +(-2.17365 - 3.76487i) q^{95} +(-0.949493 - 1.64457i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 3 q^{4} + 6 q^{5} - 12 q^{8} - 12 q^{11} - 3 q^{13} - 3 q^{16} - 6 q^{17} - 3 q^{19} + 6 q^{20} - 9 q^{22} - 24 q^{23} - 12 q^{25} + 3 q^{26} + 9 q^{29} - 3 q^{31} + 9 q^{34} + 3 q^{37}+ \cdots - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.439693 + 0.761570i −0.310910 + 0.538511i −0.978560 − 0.205964i \(-0.933967\pi\)
0.667650 + 0.744475i \(0.267300\pi\)
\(3\) 0 0
\(4\) 0.613341 + 1.06234i 0.306670 + 0.531169i
\(5\) 1.34730 0.602529 0.301265 − 0.953541i \(-0.402591\pi\)
0.301265 + 0.953541i \(0.402591\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −2.83750 −1.00321
\(9\) 0 0
\(10\) −0.592396 + 1.02606i −0.187332 + 0.324469i
\(11\) −1.65270 −0.498309 −0.249154 − 0.968464i \(-0.580153\pi\)
−0.249154 + 0.968464i \(0.580153\pi\)
\(12\) 0 0
\(13\) −1.68479 + 2.91815i −0.467277 + 0.809348i −0.999301 − 0.0373813i \(-0.988098\pi\)
0.532024 + 0.846729i \(0.321432\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.0209445 − 0.0362770i 0.00523613 − 0.00906925i
\(17\) −0.233956 + 0.405223i −0.0567426 + 0.0982810i −0.893001 − 0.450054i \(-0.851405\pi\)
0.836259 + 0.548335i \(0.184738\pi\)
\(18\) 0 0
\(19\) −1.61334 − 2.79439i −0.370126 − 0.641077i 0.619459 − 0.785029i \(-0.287352\pi\)
−0.989585 + 0.143953i \(0.954019\pi\)
\(20\) 0.826352 + 1.43128i 0.184778 + 0.320045i
\(21\) 0 0
\(22\) 0.726682 − 1.25865i 0.154929 − 0.268345i
\(23\) −8.94356 −1.86486 −0.932431 − 0.361348i \(-0.882317\pi\)
−0.932431 + 0.361348i \(0.882317\pi\)
\(24\) 0 0
\(25\) −3.18479 −0.636959
\(26\) −1.48158 − 2.56617i −0.290562 − 0.503268i
\(27\) 0 0
\(28\) 0 0
\(29\) 3.13429 + 5.42874i 0.582022 + 1.00809i 0.995239 + 0.0974595i \(0.0310717\pi\)
−0.413217 + 0.910632i \(0.635595\pi\)
\(30\) 0 0
\(31\) 4.61721 + 7.99724i 0.829276 + 1.43635i 0.898607 + 0.438754i \(0.144580\pi\)
−0.0693317 + 0.997594i \(0.522087\pi\)
\(32\) −2.81908 − 4.88279i −0.498347 − 0.863163i
\(33\) 0 0
\(34\) −0.205737 − 0.356347i −0.0352836 − 0.0611130i
\(35\) 0 0
\(36\) 0 0
\(37\) −4.61721 − 7.99724i −0.759065 − 1.31474i −0.943328 − 0.331862i \(-0.892323\pi\)
0.184263 − 0.982877i \(-0.441010\pi\)
\(38\) 2.83750 0.460303
\(39\) 0 0
\(40\) −3.82295 −0.604461
\(41\) −1.70574 + 2.95442i −0.266391 + 0.461403i −0.967927 − 0.251231i \(-0.919165\pi\)
0.701536 + 0.712634i \(0.252498\pi\)
\(42\) 0 0
\(43\) 2.20574 + 3.82045i 0.336372 + 0.582613i 0.983747 − 0.179558i \(-0.0574668\pi\)
−0.647376 + 0.762171i \(0.724133\pi\)
\(44\) −1.01367 − 1.75573i −0.152817 − 0.264686i
\(45\) 0 0
\(46\) 3.93242 − 6.81115i 0.579803 − 1.00425i
\(47\) −4.67752 + 8.10170i −0.682286 + 1.18175i 0.291995 + 0.956420i \(0.405681\pi\)
−0.974281 + 0.225335i \(0.927652\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.40033 − 2.42544i 0.198037 − 0.343009i
\(51\) 0 0
\(52\) −4.13341 −0.573201
\(53\) −0.286989 + 0.497079i −0.0394210 + 0.0682791i −0.885063 − 0.465472i \(-0.845885\pi\)
0.845642 + 0.533751i \(0.179218\pi\)
\(54\) 0 0
\(55\) −2.22668 −0.300246
\(56\) 0 0
\(57\) 0 0
\(58\) −5.51249 −0.723825
\(59\) 5.19846 + 9.00400i 0.676782 + 1.17222i 0.975945 + 0.218019i \(0.0699595\pi\)
−0.299162 + 0.954202i \(0.596707\pi\)
\(60\) 0 0
\(61\) 3.81908 − 6.61484i 0.488983 − 0.846943i −0.510937 − 0.859618i \(-0.670701\pi\)
0.999920 + 0.0126752i \(0.00403474\pi\)
\(62\) −8.12061 −1.03132
\(63\) 0 0
\(64\) 5.04189 0.630236
\(65\) −2.26991 + 3.93161i −0.281548 + 0.487656i
\(66\) 0 0
\(67\) −0.298133 − 0.516382i −0.0364228 − 0.0630861i 0.847239 − 0.531211i \(-0.178263\pi\)
−0.883662 + 0.468125i \(0.844930\pi\)
\(68\) −0.573978 −0.0696051
\(69\) 0 0
\(70\) 0 0
\(71\) 0.554378 0.0657925 0.0328963 − 0.999459i \(-0.489527\pi\)
0.0328963 + 0.999459i \(0.489527\pi\)
\(72\) 0 0
\(73\) 1.02481 − 1.77503i 0.119946 − 0.207752i −0.799800 − 0.600266i \(-0.795061\pi\)
0.919746 + 0.392514i \(0.128395\pi\)
\(74\) 8.12061 0.944002
\(75\) 0 0
\(76\) 1.97906 − 3.42782i 0.227013 − 0.393198i
\(77\) 0 0
\(78\) 0 0
\(79\) 1.20187 − 2.08169i 0.135221 − 0.234209i −0.790461 − 0.612512i \(-0.790159\pi\)
0.925682 + 0.378303i \(0.123492\pi\)
\(80\) 0.0282185 − 0.0488759i 0.00315492 − 0.00546449i
\(81\) 0 0
\(82\) −1.50000 − 2.59808i −0.165647 − 0.286910i
\(83\) 7.52481 + 13.0334i 0.825956 + 1.43060i 0.901187 + 0.433431i \(0.142697\pi\)
−0.0752309 + 0.997166i \(0.523969\pi\)
\(84\) 0 0
\(85\) −0.315207 + 0.545955i −0.0341891 + 0.0592172i
\(86\) −3.87939 −0.418325
\(87\) 0 0
\(88\) 4.68954 0.499907
\(89\) −4.54323 − 7.86911i −0.481582 − 0.834124i 0.518195 − 0.855263i \(-0.326604\pi\)
−0.999777 + 0.0211385i \(0.993271\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −5.48545 − 9.50108i −0.571898 − 0.990556i
\(93\) 0 0
\(94\) −4.11334 − 7.12452i −0.424259 − 0.734838i
\(95\) −2.17365 − 3.76487i −0.223012 − 0.386267i
\(96\) 0 0
\(97\) −0.949493 − 1.64457i −0.0964064 − 0.166981i 0.813788 − 0.581161i \(-0.197402\pi\)
−0.910195 + 0.414181i \(0.864068\pi\)
\(98\) 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.e.361.1 6
3.2 odd 2 441.2.g.b.67.3 6
7.2 even 3 1323.2.h.b.226.3 6
7.3 odd 6 189.2.f.b.64.1 6
7.4 even 3 1323.2.f.d.442.1 6
7.5 odd 6 1323.2.h.c.226.3 6
7.6 odd 2 1323.2.g.d.361.1 6
9.2 odd 6 441.2.h.e.214.1 6
9.7 even 3 1323.2.h.b.802.3 6
21.2 odd 6 441.2.h.e.373.1 6
21.5 even 6 441.2.h.d.373.1 6
21.11 odd 6 441.2.f.c.148.3 6
21.17 even 6 63.2.f.a.22.3 ✓ 6
21.20 even 2 441.2.g.c.67.3 6
28.3 even 6 3024.2.r.k.1009.2 6
63.2 odd 6 441.2.g.b.79.3 6
63.4 even 3 3969.2.a.l.1.3 3
63.11 odd 6 441.2.f.c.295.3 6
63.16 even 3 inner 1323.2.g.e.667.1 6
63.20 even 6 441.2.h.d.214.1 6
63.25 even 3 1323.2.f.d.883.1 6
63.31 odd 6 567.2.a.c.1.3 3
63.32 odd 6 3969.2.a.q.1.1 3
63.34 odd 6 1323.2.h.c.802.3 6
63.38 even 6 63.2.f.a.43.3 yes 6
63.47 even 6 441.2.g.c.79.3 6
63.52 odd 6 189.2.f.b.127.1 6
63.59 even 6 567.2.a.h.1.1 3
63.61 odd 6 1323.2.g.d.667.1 6
84.59 odd 6 1008.2.r.h.337.2 6
252.31 even 6 9072.2.a.bs.1.2 3
252.59 odd 6 9072.2.a.ca.1.2 3
252.115 even 6 3024.2.r.k.2017.2 6
252.227 odd 6 1008.2.r.h.673.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.3 ✓ 6 21.17 even 6
63.2.f.a.43.3 yes 6 63.38 even 6
189.2.f.b.64.1 6 7.3 odd 6
189.2.f.b.127.1 6 63.52 odd 6
441.2.f.c.148.3 6 21.11 odd 6
441.2.f.c.295.3 6 63.11 odd 6
441.2.g.b.67.3 6 3.2 odd 2
441.2.g.b.79.3 6 63.2 odd 6
441.2.g.c.67.3 6 21.20 even 2
441.2.g.c.79.3 6 63.47 even 6
441.2.h.d.214.1 6 63.20 even 6
441.2.h.d.373.1 6 21.5 even 6
441.2.h.e.214.1 6 9.2 odd 6
441.2.h.e.373.1 6 21.2 odd 6
567.2.a.c.1.3 3 63.31 odd 6
567.2.a.h.1.1 3 63.59 even 6
1008.2.r.h.337.2 6 84.59 odd 6
1008.2.r.h.673.2 6 252.227 odd 6
1323.2.f.d.442.1 6 7.4 even 3
1323.2.f.d.883.1 6 63.25 even 3
1323.2.g.d.361.1 6 7.6 odd 2
1323.2.g.d.667.1 6 63.61 odd 6
1323.2.g.e.361.1 6 1.1 even 1 trivial
1323.2.g.e.667.1 6 63.16 even 3 inner
1323.2.h.b.226.3 6 7.2 even 3
1323.2.h.b.802.3 6 9.7 even 3
1323.2.h.c.226.3 6 7.5 odd 6
1323.2.h.c.802.3 6 63.34 odd 6
3024.2.r.k.1009.2 6 28.3 even 6
3024.2.r.k.2017.2 6 252.115 even 6
3969.2.a.l.1.3 3 63.4 even 3
3969.2.a.q.1.1 3 63.32 odd 6
9072.2.a.bs.1.2 3 252.31 even 6
9072.2.a.ca.1.2 3 252.59 odd 6