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Properties

Label 729.2.a.c.1.4
Level $729$
Weight $2$
Character 729.1
Self dual yes
Analytic conductor $5.821$
Analytic rank $1$
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] = [729,2,Mod(1,729)] 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("729.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.82109430735\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{36})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 6x^{4} + 9x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.684040\) of defining polynomial
Character \(\chi\) \(=\) 729.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.684040 q^{2} -1.53209 q^{4} +1.04801 q^{5} -0.120615 q^{7} -2.41609 q^{8} +0.716881 q^{10} -5.43372 q^{11} -4.57398 q^{13} -0.0825054 q^{14} +1.41147 q^{16} +4.77833 q^{17} -0.588526 q^{19} -1.60565 q^{20} -3.71688 q^{22} -7.79596 q^{23} -3.90167 q^{25} -3.12879 q^{26} +0.184793 q^{28} +5.06975 q^{29} -8.75877 q^{31} +5.79769 q^{32} +3.26857 q^{34} -0.126406 q^{35} -2.18479 q^{37} -0.402575 q^{38} -2.53209 q^{40} +7.55839 q^{41} -1.30541 q^{43} +8.32494 q^{44} -5.33275 q^{46} +2.41609 q^{47} -6.98545 q^{49} -2.66890 q^{50} +7.00774 q^{52} -3.04628 q^{53} -5.69459 q^{55} +0.291416 q^{56} +3.46791 q^{58} -0.0439002 q^{59} -10.2121 q^{61} -5.99135 q^{62} +1.14290 q^{64} -4.79358 q^{65} -1.85978 q^{67} -7.32083 q^{68} -0.0864665 q^{70} -6.51038 q^{71} +12.2344 q^{73} -1.49449 q^{74} +0.901674 q^{76} +0.655386 q^{77} +0.702333 q^{79} +1.47924 q^{80} +5.17024 q^{82} +6.77660 q^{83} +5.00774 q^{85} -0.892951 q^{86} +13.1284 q^{88} +6.85565 q^{89} +0.551689 q^{91} +11.9441 q^{92} +1.65270 q^{94} -0.616781 q^{95} -9.02734 q^{97} -4.77833 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.684040 0.483690 0.241845 − 0.970315i \(-0.422248\pi\)
0.241845 + 0.970315i \(0.422248\pi\)
\(3\) 0 0
\(4\) −1.53209 −0.766044
\(5\) 1.04801 0.468685 0.234342 − 0.972154i \(-0.424706\pi\)
0.234342 + 0.972154i \(0.424706\pi\)
\(6\) 0 0
\(7\) −0.120615 −0.0455881 −0.0227940 − 0.999740i \(-0.507256\pi\)
−0.0227940 + 0.999740i \(0.507256\pi\)
\(8\) −2.41609 −0.854217
\(9\) 0 0
\(10\) 0.716881 0.226698
\(11\) −5.43372 −1.63833 −0.819164 − 0.573560i \(-0.805562\pi\)
−0.819164 + 0.573560i \(0.805562\pi\)
\(12\) 0 0
\(13\) −4.57398 −1.26859 −0.634297 − 0.773090i \(-0.718710\pi\)
−0.634297 + 0.773090i \(0.718710\pi\)
\(14\) −0.0825054 −0.0220505
\(15\) 0 0
\(16\) 1.41147 0.352869
\(17\) 4.77833 1.15892 0.579458 − 0.815002i \(-0.303264\pi\)
0.579458 + 0.815002i \(0.303264\pi\)
\(18\) 0 0
\(19\) −0.588526 −0.135017 −0.0675085 − 0.997719i \(-0.521505\pi\)
−0.0675085 + 0.997719i \(0.521505\pi\)
\(20\) −1.60565 −0.359033
\(21\) 0 0
\(22\) −3.71688 −0.792442
\(23\) −7.79596 −1.62557 −0.812785 − 0.582564i \(-0.802049\pi\)
−0.812785 + 0.582564i \(0.802049\pi\)
\(24\) 0 0
\(25\) −3.90167 −0.780335
\(26\) −3.12879 −0.613605
\(27\) 0 0
\(28\) 0.184793 0.0349225
\(29\) 5.06975 0.941428 0.470714 − 0.882286i \(-0.343996\pi\)
0.470714 + 0.882286i \(0.343996\pi\)
\(30\) 0 0
\(31\) −8.75877 −1.57312 −0.786561 − 0.617513i \(-0.788140\pi\)
−0.786561 + 0.617513i \(0.788140\pi\)
\(32\) 5.79769 1.02490
\(33\) 0 0
\(34\) 3.26857 0.560555
\(35\) −0.126406 −0.0213664
\(36\) 0 0
\(37\) −2.18479 −0.359178 −0.179589 − 0.983742i \(-0.557477\pi\)
−0.179589 + 0.983742i \(0.557477\pi\)
\(38\) −0.402575 −0.0653064
\(39\) 0 0
\(40\) −2.53209 −0.400358
\(41\) 7.55839 1.18042 0.590211 − 0.807249i \(-0.299044\pi\)
0.590211 + 0.807249i \(0.299044\pi\)
\(42\) 0 0
\(43\) −1.30541 −0.199073 −0.0995364 − 0.995034i \(-0.531736\pi\)
−0.0995364 + 0.995034i \(0.531736\pi\)
\(44\) 8.32494 1.25503
\(45\) 0 0
\(46\) −5.33275 −0.786271
\(47\) 2.41609 0.352423 0.176212 − 0.984352i \(-0.443616\pi\)
0.176212 + 0.984352i \(0.443616\pi\)
\(48\) 0 0
\(49\) −6.98545 −0.997922
\(50\) −2.66890 −0.377440
\(51\) 0 0
\(52\) 7.00774 0.971799
\(53\) −3.04628 −0.418439 −0.209219 − 0.977869i \(-0.567092\pi\)
−0.209219 + 0.977869i \(0.567092\pi\)
\(54\) 0 0
\(55\) −5.69459 −0.767859
\(56\) 0.291416 0.0389421
\(57\) 0 0
\(58\) 3.46791 0.455359
\(59\) −0.0439002 −0.00571532 −0.00285766 − 0.999996i \(-0.500910\pi\)
−0.00285766 + 0.999996i \(0.500910\pi\)
\(60\) 0 0
\(61\) −10.2121 −1.30753 −0.653765 − 0.756698i \(-0.726811\pi\)
−0.653765 + 0.756698i \(0.726811\pi\)
\(62\) −5.99135 −0.760902
\(63\) 0 0
\(64\) 1.14290 0.142863
\(65\) −4.79358 −0.594570
\(66\) 0 0
\(67\) −1.85978 −0.227209 −0.113604 − 0.993526i \(-0.536240\pi\)
−0.113604 + 0.993526i \(0.536240\pi\)
\(68\) −7.32083 −0.887781
\(69\) 0 0
\(70\) −0.0864665 −0.0103347
\(71\) −6.51038 −0.772640 −0.386320 − 0.922365i \(-0.626254\pi\)
−0.386320 + 0.922365i \(0.626254\pi\)
\(72\) 0 0
\(73\) 12.2344 1.43193 0.715965 − 0.698136i \(-0.245987\pi\)
0.715965 + 0.698136i \(0.245987\pi\)
\(74\) −1.49449 −0.173730
\(75\) 0 0
\(76\) 0.901674 0.103429
\(77\) 0.655386 0.0746882
\(78\) 0 0
\(79\) 0.702333 0.0790187 0.0395093 − 0.999219i \(-0.487421\pi\)
0.0395093 + 0.999219i \(0.487421\pi\)
\(80\) 1.47924 0.165384
\(81\) 0 0
\(82\) 5.17024 0.570958
\(83\) 6.77660 0.743828 0.371914 − 0.928267i \(-0.378702\pi\)
0.371914 + 0.928267i \(0.378702\pi\)
\(84\) 0 0
\(85\) 5.00774 0.543166
\(86\) −0.892951 −0.0962894
\(87\) 0 0
\(88\) 13.1284 1.39949
\(89\) 6.85565 0.726697 0.363349 − 0.931653i \(-0.381633\pi\)
0.363349 + 0.931653i \(0.381633\pi\)
\(90\) 0 0
\(91\) 0.551689 0.0578327
\(92\) 11.9441 1.24526
\(93\) 0 0
\(94\) 1.65270 0.170463
\(95\) −0.616781 −0.0632804
\(96\) 0 0
\(97\) −9.02734 −0.916588 −0.458294 − 0.888801i \(-0.651539\pi\)
−0.458294 + 0.888801i \(0.651539\pi\)
\(98\) −4.77833 −0.482684
\(99\) 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 729.2.a.c.1.4 yes 6
3.2 odd 2 inner 729.2.a.c.1.3 ✓ 6
9.2 odd 6 729.2.c.c.244.4 12
9.4 even 3 729.2.c.c.487.3 12
9.5 odd 6 729.2.c.c.487.4 12
9.7 even 3 729.2.c.c.244.3 12
27.2 odd 18 729.2.e.m.568.2 12
27.4 even 9 729.2.e.r.406.2 12
27.5 odd 18 729.2.e.q.649.1 12
27.7 even 9 729.2.e.r.325.2 12
27.11 odd 18 729.2.e.q.82.1 12
27.13 even 9 729.2.e.m.163.1 12
27.14 odd 18 729.2.e.m.163.2 12
27.16 even 9 729.2.e.q.82.2 12
27.20 odd 18 729.2.e.r.325.1 12
27.22 even 9 729.2.e.q.649.2 12
27.23 odd 18 729.2.e.r.406.1 12
27.25 even 9 729.2.e.m.568.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.c.1.3 ✓ 6 3.2 odd 2 inner
729.2.a.c.1.4 yes 6 1.1 even 1 trivial
729.2.c.c.244.3 12 9.7 even 3
729.2.c.c.244.4 12 9.2 odd 6
729.2.c.c.487.3 12 9.4 even 3
729.2.c.c.487.4 12 9.5 odd 6
729.2.e.m.163.1 12 27.13 even 9
729.2.e.m.163.2 12 27.14 odd 18
729.2.e.m.568.1 12 27.25 even 9
729.2.e.m.568.2 12 27.2 odd 18
729.2.e.q.82.1 12 27.11 odd 18
729.2.e.q.82.2 12 27.16 even 9
729.2.e.q.649.1 12 27.5 odd 18
729.2.e.q.649.2 12 27.22 even 9
729.2.e.r.325.1 12 27.20 odd 18
729.2.e.r.325.2 12 27.7 even 9
729.2.e.r.406.1 12 27.23 odd 18
729.2.e.r.406.2 12 27.4 even 9