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Properties

Label 729.2.e.u.406.2
Level $729$
Weight $2$
Character 729.406
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 406.2
Root \(-1.37340i\) of defining polynomial
Character \(\chi\) \(=\) 729.406
Dual form 729.2.e.u.325.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.07220 + 1.73878i) q^{2} +(0.923353 + 5.23659i) q^{4} +(1.57153 + 0.571989i) q^{5} +(0.0869267 - 0.492986i) q^{7} +(-4.48686 + 7.77147i) q^{8} +(2.26195 + 3.91782i) q^{10} +(1.80207 - 0.655898i) q^{11} +(2.38426 - 2.00063i) q^{13} +(1.03732 - 0.870419i) q^{14} +(-12.8172 + 4.66506i) q^{16} +(-1.33234 - 2.30767i) q^{17} +(-2.89832 + 5.02003i) q^{19} +(-1.54420 + 8.75760i) q^{20} +(4.87470 + 1.77425i) q^{22} +(-0.806747 - 4.57529i) q^{23} +(-1.68769 - 1.41614i) q^{25} +8.41934 q^{26} +2.66183 q^{28} +(-2.00326 - 1.68093i) q^{29} +(-0.801317 - 4.54450i) q^{31} +(-17.8062 - 6.48092i) q^{32} +(1.25168 - 7.09860i) q^{34} +(0.418591 - 0.725020i) q^{35} +(2.42934 + 4.20773i) q^{37} +(-14.7346 + 5.36297i) q^{38} +(-11.4964 + 9.64664i) q^{40} +(-8.84640 + 7.42301i) q^{41} +(8.46131 - 3.07966i) q^{43} +(5.09861 + 8.83106i) q^{44} +(6.28369 - 10.8837i) q^{46} +(1.18641 - 6.72844i) q^{47} +(6.34237 + 2.30843i) q^{49} +(-1.03487 - 5.86907i) q^{50} +(12.6780 + 10.6381i) q^{52} -5.43322 q^{53} +3.20716 q^{55} +(3.44120 + 2.88751i) q^{56} +(-1.22838 - 6.96647i) q^{58} +(2.05916 + 0.749473i) q^{59} +(1.18781 - 6.73642i) q^{61} +(6.24140 - 10.8104i) q^{62} +(-11.9893 - 20.7661i) q^{64} +(4.89128 - 1.78028i) q^{65} +(9.56299 - 8.02430i) q^{67} +(10.8541 - 9.10770i) q^{68} +(2.12806 - 0.774549i) q^{70} +(1.41784 + 2.45578i) q^{71} +(-4.96749 + 8.60394i) q^{73} +(-2.28226 + 12.9434i) q^{74} +(-28.9640 - 10.5421i) q^{76} +(-0.166701 - 0.945408i) q^{77} +(4.06862 + 3.41398i) q^{79} -22.8109 q^{80} -31.2385 q^{82} +(-2.08988 - 1.75362i) q^{83} +(-0.773838 - 4.38865i) q^{85} +(22.8884 + 8.33069i) q^{86} +(-2.98832 + 16.9476i) q^{88} +(-5.60945 + 9.71585i) q^{89} +(-0.779029 - 1.34932i) q^{91} +(23.2140 - 8.44921i) q^{92} +(14.1578 - 11.8798i) q^{94} +(-7.42619 + 6.23132i) q^{95} +(6.47368 - 2.35623i) q^{97} +(9.12879 + 15.8115i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + 6 q^{4} - 6 q^{5} - 3 q^{7} - 6 q^{8} - 6 q^{10} - 12 q^{11} - 3 q^{13} - 15 q^{14} - 36 q^{16} + 9 q^{17} - 12 q^{19} - 42 q^{20} + 6 q^{22} - 6 q^{23} + 6 q^{25} + 48 q^{26} + 6 q^{28}+ \cdots - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.07220 + 1.73878i 1.46527 + 1.22950i 0.920398 + 0.390983i \(0.127865\pi\)
0.544869 + 0.838521i \(0.316579\pi\)
\(3\) 0 0
\(4\) 0.923353 + 5.23659i 0.461676 + 2.61830i
\(5\) 1.57153 + 0.571989i 0.702809 + 0.255801i 0.668610 − 0.743614i \(-0.266890\pi\)
0.0341990 + 0.999415i \(0.489112\pi\)
\(6\) 0 0
\(7\) 0.0869267 − 0.492986i 0.0328552 − 0.186331i −0.963963 − 0.266035i \(-0.914286\pi\)
0.996819 + 0.0797038i \(0.0253974\pi\)
\(8\) −4.48686 + 7.77147i −1.58634 + 2.74763i
\(9\) 0 0
\(10\) 2.26195 + 3.91782i 0.715293 + 1.23892i
\(11\) 1.80207 − 0.655898i 0.543343 − 0.197761i −0.0557432 − 0.998445i \(-0.517753\pi\)
0.599086 + 0.800684i \(0.295531\pi\)
\(12\) 0 0
\(13\) 2.38426 − 2.00063i 0.661276 − 0.554876i −0.249193 − 0.968454i \(-0.580165\pi\)
0.910469 + 0.413578i \(0.135721\pi\)
\(14\) 1.03732 − 0.870419i 0.277237 − 0.232629i
\(15\) 0 0
\(16\) −12.8172 + 4.66506i −3.20429 + 1.16627i
\(17\) −1.33234 − 2.30767i −0.323139 − 0.559693i 0.657995 − 0.753022i \(-0.271405\pi\)
−0.981134 + 0.193329i \(0.938071\pi\)
\(18\) 0 0
\(19\) −2.89832 + 5.02003i −0.664920 + 1.15167i 0.314387 + 0.949295i \(0.398201\pi\)
−0.979307 + 0.202380i \(0.935132\pi\)
\(20\) −1.54420 + 8.75760i −0.345294 + 1.95826i
\(21\) 0 0
\(22\) 4.87470 + 1.77425i 1.03929 + 0.378271i
\(23\) −0.806747 − 4.57529i −0.168218 − 0.954013i −0.945684 − 0.325088i \(-0.894606\pi\)
0.777466 − 0.628926i \(-0.216505\pi\)
\(24\) 0 0
\(25\) −1.68769 − 1.41614i −0.337539 − 0.283229i
\(26\) 8.41934 1.65117
\(27\) 0 0
\(28\) 2.66183 0.503039
\(29\) −2.00326 − 1.68093i −0.371996 − 0.312142i 0.437555 − 0.899192i \(-0.355845\pi\)
−0.809551 + 0.587050i \(0.800289\pi\)
\(30\) 0 0
\(31\) −0.801317 − 4.54450i −0.143921 − 0.816216i −0.968227 − 0.250072i \(-0.919546\pi\)
0.824306 − 0.566144i \(-0.191565\pi\)
\(32\) −17.8062 − 6.48092i −3.14772 − 1.14567i
\(33\) 0 0
\(34\) 1.25168 − 7.09860i 0.214661 − 1.21740i
\(35\) 0.418591 − 0.725020i 0.0707547 − 0.122551i
\(36\) 0 0
\(37\) 2.42934 + 4.20773i 0.399381 + 0.691747i 0.993650 − 0.112519i \(-0.0358919\pi\)
−0.594269 + 0.804266i \(0.702559\pi\)
\(38\) −14.7346 + 5.36297i −2.39027 + 0.869989i
\(39\) 0 0
\(40\) −11.4964 + 9.64664i −1.81774 + 1.52527i
\(41\) −8.84640 + 7.42301i −1.38158 + 1.15928i −0.412951 + 0.910753i \(0.635502\pi\)
−0.968625 + 0.248527i \(0.920054\pi\)
\(42\) 0 0
\(43\) 8.46131 − 3.07966i 1.29034 − 0.469644i 0.396500 − 0.918035i \(-0.370225\pi\)
0.893838 + 0.448390i \(0.148003\pi\)
\(44\) 5.09861 + 8.83106i 0.768645 + 1.33133i
\(45\) 0 0
\(46\) 6.28369 − 10.8837i 0.926479 − 1.60471i
\(47\) 1.18641 − 6.72844i 0.173055 − 0.981444i −0.767310 − 0.641277i \(-0.778405\pi\)
0.940365 − 0.340167i \(-0.110484\pi\)
\(48\) 0 0
\(49\) 6.34237 + 2.30843i 0.906053 + 0.329776i
\(50\) −1.03487 − 5.86907i −0.146353 − 0.830011i
\(51\) 0 0
\(52\) 12.6780 + 10.6381i 1.75813 + 1.47524i
\(53\) −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
\(56\) 3.44120 + 2.88751i 0.459849 + 0.385859i
\(57\) 0 0
\(58\) −1.22838 − 6.96647i −0.161294 − 0.914742i
\(59\) 2.05916 + 0.749473i 0.268080 + 0.0975731i 0.472563 − 0.881297i \(-0.343329\pi\)
−0.204483 + 0.978870i \(0.565551\pi\)
\(60\) 0 0
\(61\) 1.18781 − 6.73642i 0.152084 − 0.862510i −0.809320 − 0.587368i \(-0.800164\pi\)
0.961404 − 0.275142i \(-0.0887248\pi\)
\(62\) 6.24140 − 10.8104i 0.792659 − 1.37293i
\(63\) 0 0
\(64\) −11.9893 − 20.7661i −1.49866 − 2.59576i
\(65\) 4.89128 − 1.78028i 0.606688 − 0.220816i
\(66\) 0 0
\(67\) 9.56299 − 8.02430i 1.16831 − 0.980324i 0.168320 − 0.985732i \(-0.446166\pi\)
0.999985 + 0.00540797i \(0.00172142\pi\)
\(68\) 10.8541 − 9.10770i 1.31626 − 1.10447i
\(69\) 0 0
\(70\) 2.12806 − 0.774549i 0.254351 − 0.0925763i
\(71\) 1.41784 + 2.45578i 0.168267 + 0.291447i 0.937811 − 0.347147i \(-0.112850\pi\)
−0.769544 + 0.638594i \(0.779516\pi\)
\(72\) 0 0
\(73\) −4.96749 + 8.60394i −0.581400 + 1.00701i 0.413913 + 0.910316i \(0.364162\pi\)
−0.995314 + 0.0966986i \(0.969172\pi\)
\(74\) −2.28226 + 12.9434i −0.265308 + 1.50463i
\(75\) 0 0
\(76\) −28.9640 − 10.5421i −3.32240 − 1.20926i
\(77\) −0.166701 − 0.945408i −0.0189973 − 0.107739i
\(78\) 0 0
\(79\) 4.06862 + 3.41398i 0.457756 + 0.384103i 0.842305 − 0.539002i \(-0.181198\pi\)
−0.384549 + 0.923105i \(0.625643\pi\)
\(80\) −22.8109 −2.55033
\(81\) 0 0
\(82\) −31.2385 −3.44972
\(83\) −2.08988 − 1.75362i −0.229395 − 0.192485i 0.520844 − 0.853652i \(-0.325617\pi\)
−0.750239 + 0.661167i \(0.770062\pi\)
\(84\) 0 0
\(85\) −0.773838 − 4.38865i −0.0839345 − 0.476016i
\(86\) 22.8884 + 8.33069i 2.46812 + 0.898322i
\(87\) 0 0
\(88\) −2.98832 + 16.9476i −0.318556 + 1.80662i
\(89\) −5.60945 + 9.71585i −0.594600 + 1.02988i 0.399003 + 0.916950i \(0.369356\pi\)
−0.993603 + 0.112928i \(0.963977\pi\)
\(90\) 0 0
\(91\) −0.779029 − 1.34932i −0.0816644 − 0.141447i
\(92\) 23.2140 − 8.44921i 2.42023 − 0.880891i
\(93\) 0 0
\(94\) 14.1578 − 11.8798i 1.46026 − 1.22531i
\(95\) −7.42619 + 6.23132i −0.761911 + 0.639320i
\(96\) 0 0
\(97\) 6.47368 − 2.35623i 0.657302 − 0.239238i 0.00823103 − 0.999966i \(-0.497380\pi\)
0.649071 + 0.760728i \(0.275158\pi\)
\(98\) 9.12879 + 15.8115i 0.922147 + 1.59721i
\(99\) 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.u.406.2 12
3.2 odd 2 729.2.e.j.406.1 12
9.2 odd 6 729.2.e.s.649.1 12
9.4 even 3 729.2.e.k.163.1 12
9.5 odd 6 729.2.e.t.163.2 12
9.7 even 3 729.2.e.l.649.2 12
27.2 odd 18 729.2.c.d.244.6 12
27.4 even 9 729.2.e.l.82.2 12
27.5 odd 18 729.2.e.j.325.1 12
27.7 even 9 729.2.a.e.1.6 yes 6
27.11 odd 18 729.2.c.d.487.6 12
27.13 even 9 729.2.e.k.568.1 12
27.14 odd 18 729.2.e.t.568.2 12
27.16 even 9 729.2.c.a.487.1 12
27.20 odd 18 729.2.a.b.1.1 ✓ 6
27.22 even 9 inner 729.2.e.u.325.2 12
27.23 odd 18 729.2.e.s.82.1 12
27.25 even 9 729.2.c.a.244.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.1 ✓ 6 27.20 odd 18
729.2.a.e.1.6 yes 6 27.7 even 9
729.2.c.a.244.1 12 27.25 even 9
729.2.c.a.487.1 12 27.16 even 9
729.2.c.d.244.6 12 27.2 odd 18
729.2.c.d.487.6 12 27.11 odd 18
729.2.e.j.325.1 12 27.5 odd 18
729.2.e.j.406.1 12 3.2 odd 2
729.2.e.k.163.1 12 9.4 even 3
729.2.e.k.568.1 12 27.13 even 9
729.2.e.l.82.2 12 27.4 even 9
729.2.e.l.649.2 12 9.7 even 3
729.2.e.s.82.1 12 27.23 odd 18
729.2.e.s.649.1 12 9.2 odd 6
729.2.e.t.163.2 12 9.5 odd 6
729.2.e.t.568.2 12 27.14 odd 18
729.2.e.u.325.2 12 27.22 even 9 inner
729.2.e.u.406.2 12 1.1 even 1 trivial