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Properties

Label 729.2.c.a.487.1
Level $729$
Weight $2$
Character 729.487
Analytic conductor $5.821$
Analytic rank $0$
Dimension $12$
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(244,729)] 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("729.244"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-3,0,-9] 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: no
Analytic conductor: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{10} + 105x^{8} + 266x^{6} + 306x^{4} + 132x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 487.1
Root \(-0.0878222i\) of defining polynomial
Character \(\chi\) \(=\) 729.487
Dual form 729.2.c.a.244.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.35253 + 2.34265i) q^{2} +(-2.65869 - 4.60498i) q^{4} +(0.836192 + 1.44833i) q^{5} +(-0.250296 + 0.433525i) q^{7} +8.97372 q^{8} -4.52391 q^{10} +(0.958859 - 1.66079i) q^{11} +(-1.55622 - 2.69545i) q^{13} +(-0.677066 - 1.17271i) q^{14} +(-6.81987 + 11.8124i) q^{16} +2.66467 q^{17} +5.79664 q^{19} +(4.44635 - 7.70130i) q^{20} +(2.59378 + 4.49255i) q^{22} +(2.32293 + 4.02344i) q^{23} +(1.10156 - 1.90797i) q^{25} +8.41934 q^{26} +2.66183 q^{28} +(1.30754 - 2.26472i) q^{29} +(2.30730 + 3.99636i) q^{31} +(-9.47446 - 16.4103i) q^{32} +(-3.60406 + 6.24241i) q^{34} -0.837181 q^{35} -4.85867 q^{37} +(-7.84014 + 13.5795i) q^{38} +(7.50375 + 12.9969i) q^{40} +(5.77408 + 10.0010i) q^{41} +(4.50217 - 7.79798i) q^{43} -10.1972 q^{44} -12.5674 q^{46} +(-3.41612 + 5.91689i) q^{47} +(3.37470 + 5.84516i) q^{49} +(2.97980 + 5.16117i) q^{50} +(-8.27499 + 14.3327i) q^{52} -5.43322 q^{53} +3.20716 q^{55} +(-2.24608 + 3.89033i) q^{56} +(3.53697 + 6.12621i) q^{58} +(1.09566 + 1.89773i) q^{59} +(-3.42017 + 5.92391i) q^{61} -12.4828 q^{62} +23.9786 q^{64} +(2.60259 - 4.50783i) q^{65} +(-6.24180 - 10.8111i) q^{67} +(-7.08453 - 12.2708i) q^{68} +(1.13231 - 1.96123i) q^{70} -2.83568 q^{71} +9.93497 q^{73} +(6.57151 - 11.3822i) q^{74} +(-15.4114 - 26.6934i) q^{76} +(0.479996 + 0.831378i) q^{77} +(-2.65561 + 4.59964i) q^{79} -22.8109 q^{80} -31.2385 q^{82} +(1.36408 - 2.36265i) q^{83} +(2.22818 + 3.85932i) q^{85} +(12.1787 + 21.0941i) q^{86} +(8.60453 - 14.9035i) q^{88} +11.2189 q^{89} +1.55806 q^{91} +(12.3519 - 21.3941i) q^{92} +(-9.24082 - 16.0056i) q^{94} +(4.84710 + 8.39543i) q^{95} +(3.44457 - 5.96617i) q^{97} -18.2576 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} - 9 q^{4} + 3 q^{5} - 6 q^{7} + 12 q^{8} + 12 q^{10} + 6 q^{11} - 6 q^{13} - 24 q^{14} - 15 q^{16} - 18 q^{17} + 24 q^{19} + 21 q^{20} - 3 q^{22} + 12 q^{23} - 9 q^{25} + 48 q^{26} + 6 q^{28}+ \cdots + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/729\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.35253 + 2.34265i −0.956385 + 1.65651i −0.225218 + 0.974308i \(0.572309\pi\)
−0.731167 + 0.682199i \(0.761024\pi\)
\(3\) 0 0
\(4\) −2.65869 − 4.60498i −1.32934 − 2.30249i
\(5\) 0.836192 + 1.44833i 0.373957 + 0.647712i 0.990170 − 0.139867i \(-0.0446676\pi\)
−0.616214 + 0.787579i \(0.711334\pi\)
\(6\) 0 0
\(7\) −0.250296 + 0.433525i −0.0946028 + 0.163857i −0.909443 − 0.415829i \(-0.863491\pi\)
0.814840 + 0.579686i \(0.196825\pi\)
\(8\) 8.97372 3.17269
\(9\) 0 0
\(10\) −4.52391 −1.43059
\(11\) 0.958859 − 1.66079i 0.289107 − 0.500748i −0.684490 − 0.729022i \(-0.739975\pi\)
0.973597 + 0.228275i \(0.0733084\pi\)
\(12\) 0 0
\(13\) −1.55622 − 2.69545i −0.431617 − 0.747583i 0.565396 − 0.824820i \(-0.308723\pi\)
−0.997013 + 0.0772371i \(0.975390\pi\)
\(14\) −0.677066 − 1.17271i −0.180953 − 0.313421i
\(15\) 0 0
\(16\) −6.81987 + 11.8124i −1.70497 + 2.95309i
\(17\) 2.66467 0.646278 0.323139 − 0.946352i \(-0.395262\pi\)
0.323139 + 0.946352i \(0.395262\pi\)
\(18\) 0 0
\(19\) 5.79664 1.32984 0.664920 − 0.746915i \(-0.268466\pi\)
0.664920 + 0.746915i \(0.268466\pi\)
\(20\) 4.44635 − 7.70130i 0.994234 − 1.72206i
\(21\) 0 0
\(22\) 2.59378 + 4.49255i 0.552995 + 0.957815i
\(23\) 2.32293 + 4.02344i 0.484365 + 0.838945i 0.999839 − 0.0179603i \(-0.00571725\pi\)
−0.515473 + 0.856906i \(0.672384\pi\)
\(24\) 0 0
\(25\) 1.10156 − 1.90797i 0.220313 − 0.381593i
\(26\) 8.41934 1.65117
\(27\) 0 0
\(28\) 2.66183 0.503039
\(29\) 1.30754 − 2.26472i 0.242803 − 0.420547i −0.718709 − 0.695312i \(-0.755266\pi\)
0.961512 + 0.274764i \(0.0885997\pi\)
\(30\) 0 0
\(31\) 2.30730 + 3.99636i 0.414404 + 0.717768i 0.995366 − 0.0961626i \(-0.0306569\pi\)
−0.580962 + 0.813931i \(0.697324\pi\)
\(32\) −9.47446 − 16.4103i −1.67486 − 2.90095i
\(33\) 0 0
\(34\) −3.60406 + 6.24241i −0.618090 + 1.07056i
\(35\) −0.837181 −0.141509
\(36\) 0 0
\(37\) −4.85867 −0.798761 −0.399381 − 0.916785i \(-0.630775\pi\)
−0.399381 + 0.916785i \(0.630775\pi\)
\(38\) −7.84014 + 13.5795i −1.27184 + 2.20289i
\(39\) 0 0
\(40\) 7.50375 + 12.9969i 1.18645 + 2.05499i
\(41\) 5.77408 + 10.0010i 0.901760 + 1.56189i 0.825208 + 0.564828i \(0.191058\pi\)
0.0765514 + 0.997066i \(0.475609\pi\)
\(42\) 0 0
\(43\) 4.50217 − 7.79798i 0.686574 − 1.18918i −0.286365 − 0.958121i \(-0.592447\pi\)
0.972939 − 0.231061i \(-0.0742196\pi\)
\(44\) −10.1972 −1.53729
\(45\) 0 0
\(46\) −12.5674 −1.85296
\(47\) −3.41612 + 5.91689i −0.498292 + 0.863067i −0.999998 − 0.00197091i \(-0.999373\pi\)
0.501706 + 0.865038i \(0.332706\pi\)
\(48\) 0 0
\(49\) 3.37470 + 5.84516i 0.482101 + 0.835023i
\(50\) 2.97980 + 5.16117i 0.421408 + 0.729900i
\(51\) 0 0
\(52\) −8.27499 + 14.3327i −1.14754 + 1.98759i
\(53\) −5.43322 −0.746309 −0.373155 − 0.927769i \(-0.621724\pi\)
−0.373155 + 0.927769i \(0.621724\pi\)
\(54\) 0 0
\(55\) 3.20716 0.432454
\(56\) −2.24608 + 3.89033i −0.300145 + 0.519867i
\(57\) 0 0
\(58\) 3.53697 + 6.12621i 0.464427 + 0.804410i
\(59\) 1.09566 + 1.89773i 0.142642 + 0.247064i 0.928491 − 0.371355i \(-0.121107\pi\)
−0.785849 + 0.618419i \(0.787773\pi\)
\(60\) 0 0
\(61\) −3.42017 + 5.92391i −0.437908 + 0.758478i −0.997528 − 0.0702708i \(-0.977614\pi\)
0.559620 + 0.828749i \(0.310947\pi\)
\(62\) −12.4828 −1.58532
\(63\) 0 0
\(64\) 23.9786 2.99733
\(65\) 2.60259 − 4.50783i 0.322812 − 0.559127i
\(66\) 0 0
\(67\) −6.24180 − 10.8111i −0.762557 − 1.32079i −0.941529 − 0.336933i \(-0.890610\pi\)
0.178971 − 0.983854i \(-0.442723\pi\)
\(68\) −7.08453 − 12.2708i −0.859126 − 1.48805i
\(69\) 0 0
\(70\) 1.13231 − 1.96123i 0.135337 − 0.234411i
\(71\) −2.83568 −0.336534 −0.168267 − 0.985741i \(-0.553817\pi\)
−0.168267 + 0.985741i \(0.553817\pi\)
\(72\) 0 0
\(73\) 9.93497 1.16280 0.581400 − 0.813618i \(-0.302505\pi\)
0.581400 + 0.813618i \(0.302505\pi\)
\(74\) 6.57151 − 11.3822i 0.763923 − 1.32315i
\(75\) 0 0
\(76\) −15.4114 − 26.6934i −1.76781 − 3.06194i
\(77\) 0.479996 + 0.831378i 0.0547007 + 0.0947443i
\(78\) 0 0
\(79\) −2.65561 + 4.59964i −0.298779 + 0.517500i −0.975857 − 0.218411i \(-0.929913\pi\)
0.677078 + 0.735911i \(0.263246\pi\)
\(80\) −22.8109 −2.55033
\(81\) 0 0
\(82\) −31.2385 −3.44972
\(83\) 1.36408 − 2.36265i 0.149727 − 0.259334i −0.781400 − 0.624031i \(-0.785494\pi\)
0.931126 + 0.364697i \(0.118827\pi\)
\(84\) 0 0
\(85\) 2.22818 + 3.85932i 0.241680 + 0.418602i
\(86\) 12.1787 + 21.0941i 1.31326 + 2.27463i
\(87\) 0 0
\(88\) 8.60453 − 14.9035i 0.917246 − 1.58872i
\(89\) 11.2189 1.18920 0.594600 − 0.804021i \(-0.297310\pi\)
0.594600 + 0.804021i \(0.297310\pi\)
\(90\) 0 0
\(91\) 1.55806 0.163329
\(92\) 12.3519 − 21.3941i 1.28778 − 2.23049i
\(93\) 0 0
\(94\) −9.24082 − 16.0056i −0.953118 − 1.65085i
\(95\) 4.84710 + 8.39543i 0.497302 + 0.861353i
\(96\) 0 0
\(97\) 3.44457 − 5.96617i 0.349743 − 0.605773i −0.636461 − 0.771309i \(-0.719602\pi\)
0.986204 + 0.165536i \(0.0529355\pi\)
\(98\) −18.2576 −1.84429
\(99\) 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.c.a.487.1 12
3.2 odd 2 729.2.c.d.487.6 12
9.2 odd 6 729.2.a.b.1.1 ✓ 6
9.4 even 3 inner 729.2.c.a.244.1 12
9.5 odd 6 729.2.c.d.244.6 12
9.7 even 3 729.2.a.e.1.6 yes 6
27.2 odd 18 729.2.e.j.325.1 12
27.4 even 9 729.2.e.k.163.1 12
27.5 odd 18 729.2.e.j.406.1 12
27.7 even 9 729.2.e.l.82.2 12
27.11 odd 18 729.2.e.t.568.2 12
27.13 even 9 729.2.e.l.649.2 12
27.14 odd 18 729.2.e.s.649.1 12
27.16 even 9 729.2.e.k.568.1 12
27.20 odd 18 729.2.e.s.82.1 12
27.22 even 9 729.2.e.u.406.2 12
27.23 odd 18 729.2.e.t.163.2 12
27.25 even 9 729.2.e.u.325.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.1 ✓ 6 9.2 odd 6
729.2.a.e.1.6 yes 6 9.7 even 3
729.2.c.a.244.1 12 9.4 even 3 inner
729.2.c.a.487.1 12 1.1 even 1 trivial
729.2.c.d.244.6 12 9.5 odd 6
729.2.c.d.487.6 12 3.2 odd 2
729.2.e.j.325.1 12 27.2 odd 18
729.2.e.j.406.1 12 27.5 odd 18
729.2.e.k.163.1 12 27.4 even 9
729.2.e.k.568.1 12 27.16 even 9
729.2.e.l.82.2 12 27.7 even 9
729.2.e.l.649.2 12 27.13 even 9
729.2.e.s.82.1 12 27.20 odd 18
729.2.e.s.649.1 12 27.14 odd 18
729.2.e.t.163.2 12 27.23 odd 18
729.2.e.t.568.2 12 27.11 odd 18
729.2.e.u.325.2 12 27.25 even 9
729.2.e.u.406.2 12 27.22 even 9