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Properties

Label 729.2.e.j.325.2
Level $729$
Weight $2$
Character 729.325
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(82,729)] 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("729.82"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-6,0,6,6,0,-3,6,0,-6,12,0,-3,15,0,-36,-9,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == 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: \(2\) over \(\Q(\zeta_{9})\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 325.2
Root \(0.0878222i\) of defining polynomial
Character \(\chi\) \(=\) 729.325
Dual form 729.2.e.j.406.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.132507 - 0.111187i) q^{2} +(-0.342101 + 1.94015i) q^{4} +(3.51122 - 1.27798i) q^{5} +(0.526414 + 2.98544i) q^{7} +(0.343364 + 0.594724i) q^{8} +(0.323168 - 0.559743i) q^{10} +(2.34143 + 0.852210i) q^{11} +(-0.586130 - 0.491822i) q^{13} +(0.401695 + 0.337062i) q^{14} +(-3.59091 - 1.30699i) q^{16} +(-2.31139 + 4.00345i) q^{17} +(0.305922 + 0.529872i) q^{19} +(1.27828 + 7.24949i) q^{20} +(0.405011 - 0.147412i) q^{22} +(1.13295 - 6.42526i) q^{23} +(6.86521 - 5.76060i) q^{25} -0.132351 q^{26} -5.97229 q^{28} +(-5.01827 + 4.21083i) q^{29} +(1.13747 - 6.45091i) q^{31} +(-1.91177 + 0.695827i) q^{32} +(0.138854 + 0.787482i) q^{34} +(5.66369 + 9.80980i) q^{35} +(-2.47984 + 4.29522i) q^{37} +(0.0994517 + 0.0361975i) q^{38} +(1.96567 + 1.64939i) q^{40} +(4.02958 + 3.38122i) q^{41} +(-5.23463 - 1.90525i) q^{43} +(-2.45442 + 4.25118i) q^{44} +(-0.564280 - 0.977362i) q^{46} +(0.192335 + 1.09079i) q^{47} +(-2.05791 + 0.749017i) q^{49} +(0.269188 - 1.52664i) q^{50} +(1.15472 - 0.968928i) q^{52} +8.84310 q^{53} +9.31038 q^{55} +(-1.59476 + 1.33816i) q^{56} +(-0.196769 + 1.11593i) q^{58} +(11.1370 - 4.05354i) q^{59} +(1.42166 + 8.06263i) q^{61} +(-0.566533 - 0.981264i) q^{62} +(3.64541 - 6.31404i) q^{64} +(-2.68657 - 0.977832i) q^{65} +(-0.928705 - 0.779276i) q^{67} +(-6.97656 - 5.85403i) q^{68} +(1.84120 + 0.670142i) q^{70} +(2.45973 - 4.26038i) q^{71} +(-2.14972 - 3.72343i) q^{73} +(0.148974 + 0.844873i) q^{74} +(-1.13269 + 0.412265i) q^{76} +(-1.31166 + 7.43882i) q^{77} +(-9.03519 + 7.58143i) q^{79} -14.2788 q^{80} +0.909895 q^{82} +(6.90671 - 5.79542i) q^{83} +(-2.99948 + 17.0109i) q^{85} +(-0.905464 + 0.329562i) q^{86} +(0.297133 + 1.68512i) q^{88} +(-3.76943 - 6.52884i) q^{89} +(1.15976 - 2.00876i) q^{91} +(12.0784 + 4.39617i) q^{92} +(0.146767 + 0.123152i) q^{94} +(1.75133 + 1.46954i) q^{95} +(0.891161 + 0.324356i) q^{97} +(-0.189407 + 0.328062i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} + 6 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8} - 6 q^{10} + 12 q^{11} - 3 q^{13} + 15 q^{14} - 36 q^{16} - 9 q^{17} - 12 q^{19} + 42 q^{20} + 6 q^{22} + 6 q^{23} + 6 q^{25} - 48 q^{26} + 6 q^{28}+ \cdots + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{7}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.132507 − 0.111187i 0.0936968 − 0.0786209i −0.594736 − 0.803921i \(-0.702743\pi\)
0.688433 + 0.725300i \(0.258299\pi\)
\(3\) 0 0
\(4\) −0.342101 + 1.94015i −0.171050 + 0.970075i
\(5\) 3.51122 − 1.27798i 1.57027 − 0.571530i 0.597207 − 0.802087i \(-0.296277\pi\)
0.973059 + 0.230557i \(0.0740550\pi\)
\(6\) 0 0
\(7\) 0.526414 + 2.98544i 0.198966 + 1.12839i 0.906657 + 0.421869i \(0.138626\pi\)
−0.707691 + 0.706522i \(0.750263\pi\)
\(8\) 0.343364 + 0.594724i 0.121398 + 0.210267i
\(9\) 0 0
\(10\) 0.323168 − 0.559743i 0.102195 − 0.177006i
\(11\) 2.34143 + 0.852210i 0.705967 + 0.256951i 0.669957 − 0.742400i \(-0.266313\pi\)
0.0360107 + 0.999351i \(0.488535\pi\)
\(12\) 0 0
\(13\) −0.586130 − 0.491822i −0.162563 − 0.136407i 0.557877 − 0.829923i \(-0.311616\pi\)
−0.720441 + 0.693517i \(0.756060\pi\)
\(14\) 0.401695 + 0.337062i 0.107358 + 0.0900837i
\(15\) 0 0
\(16\) −3.59091 − 1.30699i −0.897729 − 0.326746i
\(17\) −2.31139 + 4.00345i −0.560595 + 0.970979i 0.436850 + 0.899534i \(0.356094\pi\)
−0.997445 + 0.0714442i \(0.977239\pi\)
\(18\) 0 0
\(19\) 0.305922 + 0.529872i 0.0701833 + 0.121561i 0.898982 − 0.437987i \(-0.144308\pi\)
−0.828798 + 0.559548i \(0.810975\pi\)
\(20\) 1.27828 + 7.24949i 0.285832 + 1.62104i
\(21\) 0 0
\(22\) 0.405011 − 0.147412i 0.0863486 − 0.0314283i
\(23\) 1.13295 − 6.42526i 0.236236 − 1.33976i −0.603760 − 0.797166i \(-0.706332\pi\)
0.839996 − 0.542593i \(-0.182557\pi\)
\(24\) 0 0
\(25\) 6.86521 − 5.76060i 1.37304 − 1.15212i
\(26\) −0.132351 −0.0259561
\(27\) 0 0
\(28\) −5.97229 −1.12866
\(29\) −5.01827 + 4.21083i −0.931870 + 0.781932i −0.976152 − 0.217087i \(-0.930344\pi\)
0.0442820 + 0.999019i \(0.485900\pi\)
\(30\) 0 0
\(31\) 1.13747 − 6.45091i 0.204296 − 1.15862i −0.694249 − 0.719735i \(-0.744263\pi\)
0.898544 − 0.438883i \(-0.144626\pi\)
\(32\) −1.91177 + 0.695827i −0.337956 + 0.123006i
\(33\) 0 0
\(34\) 0.138854 + 0.787482i 0.0238133 + 0.135052i
\(35\) 5.66369 + 9.80980i 0.957338 + 1.65816i
\(36\) 0 0
\(37\) −2.47984 + 4.29522i −0.407684 + 0.706129i −0.994630 − 0.103497i \(-0.966997\pi\)
0.586946 + 0.809626i \(0.300330\pi\)
\(38\) 0.0994517 + 0.0361975i 0.0161332 + 0.00587200i
\(39\) 0 0
\(40\) 1.96567 + 1.64939i 0.310800 + 0.260792i
\(41\) 4.02958 + 3.38122i 0.629314 + 0.528057i 0.900716 − 0.434409i \(-0.143043\pi\)
−0.271402 + 0.962466i \(0.587487\pi\)
\(42\) 0 0
\(43\) −5.23463 − 1.90525i −0.798273 − 0.290548i −0.0895024 − 0.995987i \(-0.528528\pi\)
−0.708771 + 0.705439i \(0.750750\pi\)
\(44\) −2.45442 + 4.25118i −0.370018 + 0.640889i
\(45\) 0 0
\(46\) −0.564280 − 0.977362i −0.0831986 − 0.144104i
\(47\) 0.192335 + 1.09079i 0.0280550 + 0.159108i 0.995617 − 0.0935267i \(-0.0298141\pi\)
−0.967562 + 0.252635i \(0.918703\pi\)
\(48\) 0 0
\(49\) −2.05791 + 0.749017i −0.293987 + 0.107002i
\(50\) 0.269188 − 1.52664i 0.0380690 − 0.215900i
\(51\) 0 0
\(52\) 1.15472 − 0.968928i 0.160131 − 0.134366i
\(53\) 8.84310 1.21469 0.607346 − 0.794437i \(-0.292234\pi\)
0.607346 + 0.794437i \(0.292234\pi\)
\(54\) 0 0
\(55\) 9.31038 1.25541
\(56\) −1.59476 + 1.33816i −0.213109 + 0.178820i
\(57\) 0 0
\(58\) −0.196769 + 1.11593i −0.0258370 + 0.146529i
\(59\) 11.1370 − 4.05354i 1.44992 − 0.527726i 0.507347 − 0.861742i \(-0.330626\pi\)
0.942568 + 0.334016i \(0.108404\pi\)
\(60\) 0 0
\(61\) 1.42166 + 8.06263i 0.182025 + 1.03231i 0.929719 + 0.368270i \(0.120050\pi\)
−0.747694 + 0.664043i \(0.768839\pi\)
\(62\) −0.566533 − 0.981264i −0.0719498 − 0.124621i
\(63\) 0 0
\(64\) 3.64541 − 6.31404i 0.455677 − 0.789255i
\(65\) −2.68657 − 0.977832i −0.333228 − 0.121285i
\(66\) 0 0
\(67\) −0.928705 − 0.779276i −0.113459 − 0.0952037i 0.584293 − 0.811543i \(-0.301372\pi\)
−0.697752 + 0.716339i \(0.745816\pi\)
\(68\) −6.97656 − 5.85403i −0.846032 − 0.709905i
\(69\) 0 0
\(70\) 1.84120 + 0.670142i 0.220066 + 0.0800973i
\(71\) 2.45973 − 4.26038i 0.291916 − 0.505614i −0.682346 − 0.731029i \(-0.739040\pi\)
0.974263 + 0.225415i \(0.0723738\pi\)
\(72\) 0 0
\(73\) −2.14972 − 3.72343i −0.251606 − 0.435795i 0.712362 − 0.701812i \(-0.247625\pi\)
−0.963968 + 0.266017i \(0.914292\pi\)
\(74\) 0.148974 + 0.844873i 0.0173179 + 0.0982145i
\(75\) 0 0
\(76\) −1.13269 + 0.412265i −0.129928 + 0.0472900i
\(77\) −1.31166 + 7.43882i −0.149478 + 0.847732i
\(78\) 0 0
\(79\) −9.03519 + 7.58143i −1.01654 + 0.852977i −0.989189 − 0.146649i \(-0.953151\pi\)
−0.0273498 + 0.999626i \(0.508707\pi\)
\(80\) −14.2788 −1.59642
\(81\) 0 0
\(82\) 0.909895 0.100481
\(83\) 6.90671 − 5.79542i 0.758110 − 0.636130i −0.179524 − 0.983754i \(-0.557456\pi\)
0.937634 + 0.347624i \(0.113011\pi\)
\(84\) 0 0
\(85\) −2.99948 + 17.0109i −0.325339 + 1.84509i
\(86\) −0.905464 + 0.329562i −0.0976387 + 0.0355376i
\(87\) 0 0
\(88\) 0.297133 + 1.68512i 0.0316744 + 0.179635i
\(89\) −3.76943 − 6.52884i −0.399558 − 0.692055i 0.594113 − 0.804382i \(-0.297503\pi\)
−0.993671 + 0.112326i \(0.964170\pi\)
\(90\) 0 0
\(91\) 1.15976 − 2.00876i 0.121576 − 0.210575i
\(92\) 12.0784 + 4.39617i 1.25926 + 0.458332i
\(93\) 0 0
\(94\) 0.146767 + 0.123152i 0.0151379 + 0.0127022i
\(95\) 1.75133 + 1.46954i 0.179682 + 0.150771i
\(96\) 0 0
\(97\) 0.891161 + 0.324356i 0.0904837 + 0.0329334i 0.386865 − 0.922136i \(-0.373558\pi\)
−0.296382 + 0.955070i \(0.595780\pi\)
\(98\) −0.189407 + 0.328062i −0.0191330 + 0.0331393i
\(99\) 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.e.j.325.2 12
3.2 odd 2 729.2.e.u.325.1 12
9.2 odd 6 729.2.e.k.568.2 12
9.4 even 3 729.2.e.s.82.2 12
9.5 odd 6 729.2.e.l.82.1 12
9.7 even 3 729.2.e.t.568.1 12
27.2 odd 18 729.2.e.k.163.2 12
27.4 even 9 729.2.a.b.1.4 ✓ 6
27.5 odd 18 729.2.c.a.244.4 12
27.7 even 9 729.2.e.s.649.2 12
27.11 odd 18 729.2.e.u.406.1 12
27.13 even 9 729.2.c.d.487.3 12
27.14 odd 18 729.2.c.a.487.4 12
27.16 even 9 inner 729.2.e.j.406.2 12
27.20 odd 18 729.2.e.l.649.1 12
27.22 even 9 729.2.c.d.244.3 12
27.23 odd 18 729.2.a.e.1.3 yes 6
27.25 even 9 729.2.e.t.163.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.4 ✓ 6 27.4 even 9
729.2.a.e.1.3 yes 6 27.23 odd 18
729.2.c.a.244.4 12 27.5 odd 18
729.2.c.a.487.4 12 27.14 odd 18
729.2.c.d.244.3 12 27.22 even 9
729.2.c.d.487.3 12 27.13 even 9
729.2.e.j.325.2 12 1.1 even 1 trivial
729.2.e.j.406.2 12 27.16 even 9 inner
729.2.e.k.163.2 12 27.2 odd 18
729.2.e.k.568.2 12 9.2 odd 6
729.2.e.l.82.1 12 9.5 odd 6
729.2.e.l.649.1 12 27.20 odd 18
729.2.e.s.82.2 12 9.4 even 3
729.2.e.s.649.2 12 27.7 even 9
729.2.e.t.163.1 12 27.25 even 9
729.2.e.t.568.1 12 9.7 even 3
729.2.e.u.325.1 12 3.2 odd 2
729.2.e.u.406.1 12 27.11 odd 18