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Properties

Label 729.2.a.c.1.6
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 1.6
Root \(1.96962\) 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)
 
\(f(q)\) \(=\) \(q+1.96962 q^{2} +1.87939 q^{4} -3.70167 q^{5} -2.34730 q^{7} -0.237565 q^{8} -7.29086 q^{10} +2.17853 q^{11} -4.71688 q^{13} -4.62327 q^{14} -4.22668 q^{16} +2.93512 q^{17} -6.22668 q^{19} -6.95686 q^{20} +4.29086 q^{22} -0.519030 q^{23} +8.70233 q^{25} -9.29044 q^{26} -4.41147 q^{28} +3.49276 q^{29} -4.30541 q^{31} -7.84981 q^{32} +5.78106 q^{34} +8.68891 q^{35} +2.41147 q^{37} -12.2642 q^{38} +0.879385 q^{40} -2.49860 q^{41} +1.06418 q^{43} +4.09429 q^{44} -1.02229 q^{46} +0.237565 q^{47} -1.49020 q^{49} +17.1403 q^{50} -8.86484 q^{52} -4.66717 q^{53} -8.06418 q^{55} +0.557635 q^{56} +6.87939 q^{58} +13.3122 q^{59} -3.67499 q^{61} -8.48000 q^{62} -7.00774 q^{64} +17.4603 q^{65} +14.2986 q^{67} +5.51622 q^{68} +17.1138 q^{70} -1.20307 q^{71} -4.68004 q^{73} +4.74968 q^{74} -11.7023 q^{76} -5.11365 q^{77} -12.8007 q^{79} +15.6458 q^{80} -4.92127 q^{82} +11.3040 q^{83} -10.8648 q^{85} +2.09602 q^{86} -0.517541 q^{88} -0.699287 q^{89} +11.0719 q^{91} -0.975457 q^{92} +0.467911 q^{94} +23.0491 q^{95} -7.08647 q^{97} -2.93512 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64}+ \cdots - 12 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.96962 1.39273 0.696364 − 0.717689i \(-0.254800\pi\)
0.696364 + 0.717689i \(0.254800\pi\)
\(3\) 0 0
\(4\) 1.87939 0.939693
\(5\) −3.70167 −1.65544 −0.827718 − 0.561145i \(-0.810361\pi\)
−0.827718 + 0.561145i \(0.810361\pi\)
\(6\) 0 0
\(7\) −2.34730 −0.887195 −0.443597 − 0.896226i \(-0.646298\pi\)
−0.443597 + 0.896226i \(0.646298\pi\)
\(8\) −0.237565 −0.0839918
\(9\) 0 0
\(10\) −7.29086 −2.30557
\(11\) 2.17853 0.656850 0.328425 − 0.944530i \(-0.393482\pi\)
0.328425 + 0.944530i \(0.393482\pi\)
\(12\) 0 0
\(13\) −4.71688 −1.30823 −0.654114 − 0.756396i \(-0.726958\pi\)
−0.654114 + 0.756396i \(0.726958\pi\)
\(14\) −4.62327 −1.23562
\(15\) 0 0
\(16\) −4.22668 −1.05667
\(17\) 2.93512 0.711871 0.355936 − 0.934510i \(-0.384162\pi\)
0.355936 + 0.934510i \(0.384162\pi\)
\(18\) 0 0
\(19\) −6.22668 −1.42850 −0.714249 − 0.699891i \(-0.753232\pi\)
−0.714249 + 0.699891i \(0.753232\pi\)
\(20\) −6.95686 −1.55560
\(21\) 0 0
\(22\) 4.29086 0.914814
\(23\) −0.519030 −0.108225 −0.0541126 − 0.998535i \(-0.517233\pi\)
−0.0541126 + 0.998535i \(0.517233\pi\)
\(24\) 0 0
\(25\) 8.70233 1.74047
\(26\) −9.29044 −1.82201
\(27\) 0 0
\(28\) −4.41147 −0.833690
\(29\) 3.49276 0.648588 0.324294 − 0.945956i \(-0.394873\pi\)
0.324294 + 0.945956i \(0.394873\pi\)
\(30\) 0 0
\(31\) −4.30541 −0.773274 −0.386637 − 0.922232i \(-0.626363\pi\)
−0.386637 + 0.922232i \(0.626363\pi\)
\(32\) −7.84981 −1.38766
\(33\) 0 0
\(34\) 5.78106 0.991443
\(35\) 8.68891 1.46869
\(36\) 0 0
\(37\) 2.41147 0.396444 0.198222 − 0.980157i \(-0.436483\pi\)
0.198222 + 0.980157i \(0.436483\pi\)
\(38\) −12.2642 −1.98951
\(39\) 0 0
\(40\) 0.879385 0.139043
\(41\) −2.49860 −0.390215 −0.195108 − 0.980782i \(-0.562506\pi\)
−0.195108 + 0.980782i \(0.562506\pi\)
\(42\) 0 0
\(43\) 1.06418 0.162286 0.0811428 − 0.996702i \(-0.474143\pi\)
0.0811428 + 0.996702i \(0.474143\pi\)
\(44\) 4.09429 0.617237
\(45\) 0 0
\(46\) −1.02229 −0.150728
\(47\) 0.237565 0.0346524 0.0173262 − 0.999850i \(-0.494485\pi\)
0.0173262 + 0.999850i \(0.494485\pi\)
\(48\) 0 0
\(49\) −1.49020 −0.212886
\(50\) 17.1403 2.42400
\(51\) 0 0
\(52\) −8.86484 −1.22933
\(53\) −4.66717 −0.641085 −0.320543 − 0.947234i \(-0.603865\pi\)
−0.320543 + 0.947234i \(0.603865\pi\)
\(54\) 0 0
\(55\) −8.06418 −1.08737
\(56\) 0.557635 0.0745171
\(57\) 0 0
\(58\) 6.87939 0.903308
\(59\) 13.3122 1.73310 0.866549 − 0.499092i \(-0.166333\pi\)
0.866549 + 0.499092i \(0.166333\pi\)
\(60\) 0 0
\(61\) −3.67499 −0.470535 −0.235267 − 0.971931i \(-0.575597\pi\)
−0.235267 + 0.971931i \(0.575597\pi\)
\(62\) −8.48000 −1.07696
\(63\) 0 0
\(64\) −7.00774 −0.875968
\(65\) 17.4603 2.16569
\(66\) 0 0
\(67\) 14.2986 1.74685 0.873426 − 0.486957i \(-0.161893\pi\)
0.873426 + 0.486957i \(0.161893\pi\)
\(68\) 5.51622 0.668940
\(69\) 0 0
\(70\) 17.1138 2.04549
\(71\) −1.20307 −0.142778 −0.0713891 − 0.997449i \(-0.522743\pi\)
−0.0713891 + 0.997449i \(0.522743\pi\)
\(72\) 0 0
\(73\) −4.68004 −0.547758 −0.273879 − 0.961764i \(-0.588307\pi\)
−0.273879 + 0.961764i \(0.588307\pi\)
\(74\) 4.74968 0.552139
\(75\) 0 0
\(76\) −11.7023 −1.34235
\(77\) −5.11365 −0.582754
\(78\) 0 0
\(79\) −12.8007 −1.44019 −0.720093 − 0.693877i \(-0.755901\pi\)
−0.720093 + 0.693877i \(0.755901\pi\)
\(80\) 15.6458 1.74925
\(81\) 0 0
\(82\) −4.92127 −0.543464
\(83\) 11.3040 1.24077 0.620385 − 0.784297i \(-0.286976\pi\)
0.620385 + 0.784297i \(0.286976\pi\)
\(84\) 0 0
\(85\) −10.8648 −1.17846
\(86\) 2.09602 0.226020
\(87\) 0 0
\(88\) −0.517541 −0.0551701
\(89\) −0.699287 −0.0741242 −0.0370621 − 0.999313i \(-0.511800\pi\)
−0.0370621 + 0.999313i \(0.511800\pi\)
\(90\) 0 0
\(91\) 11.0719 1.16065
\(92\) −0.975457 −0.101698
\(93\) 0 0
\(94\) 0.467911 0.0482613
\(95\) 23.0491 2.36479
\(96\) 0 0
\(97\) −7.08647 −0.719522 −0.359761 − 0.933045i \(-0.617142\pi\)
−0.359761 + 0.933045i \(0.617142\pi\)
\(98\) −2.93512 −0.296492
\(99\) 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.1.6 yes 6
3.2 odd 2 inner 729.2.a.c.1.1 ✓ 6
9.2 odd 6 729.2.c.c.244.6 12
9.4 even 3 729.2.c.c.487.1 12
9.5 odd 6 729.2.c.c.487.6 12
9.7 even 3 729.2.c.c.244.1 12
27.2 odd 18 729.2.e.r.568.2 12
27.4 even 9 729.2.e.q.406.2 12
27.5 odd 18 729.2.e.m.649.1 12
27.7 even 9 729.2.e.q.325.2 12
27.11 odd 18 729.2.e.m.82.1 12
27.13 even 9 729.2.e.r.163.1 12
27.14 odd 18 729.2.e.r.163.2 12
27.16 even 9 729.2.e.m.82.2 12
27.20 odd 18 729.2.e.q.325.1 12
27.22 even 9 729.2.e.m.649.2 12
27.23 odd 18 729.2.e.q.406.1 12
27.25 even 9 729.2.e.r.568.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.c.1.1 ✓ 6 3.2 odd 2 inner
729.2.a.c.1.6 yes 6 1.1 even 1 trivial
729.2.c.c.244.1 12 9.7 even 3
729.2.c.c.244.6 12 9.2 odd 6
729.2.c.c.487.1 12 9.4 even 3
729.2.c.c.487.6 12 9.5 odd 6
729.2.e.m.82.1 12 27.11 odd 18
729.2.e.m.82.2 12 27.16 even 9
729.2.e.m.649.1 12 27.5 odd 18
729.2.e.m.649.2 12 27.22 even 9
729.2.e.q.325.1 12 27.20 odd 18
729.2.e.q.325.2 12 27.7 even 9
729.2.e.q.406.1 12 27.23 odd 18
729.2.e.q.406.2 12 27.4 even 9
729.2.e.r.163.1 12 27.13 even 9
729.2.e.r.163.2 12 27.14 odd 18
729.2.e.r.568.1 12 27.25 even 9
729.2.e.r.568.2 12 27.2 odd 18