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Properties

Label 432.9.q.a.305.3
Level $432$
Weight $9$
Character 432.305
Analytic conductor $175.988$
Analytic rank $0$
Dimension $14$
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] = [432,9,Mod(17,432)] 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("432.17"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 432.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,0,0,-438] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(175.987559546\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5 x^{13} - 930122 x^{12} + 122593669 x^{11} + 316468329343 x^{10} - 78164131766942 x^{9} + \cdots + 19\!\cdots\!59 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{30} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 305.3
Root \(200.431 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 432.305
Dual form 432.9.q.a.17.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-331.396 + 191.331i) q^{5} +(-467.516 + 809.762i) q^{7} +(4890.10 + 2823.30i) q^{11} +(17663.1 + 30593.3i) q^{13} +152914. i q^{17} -191248. q^{19} +(132313. - 76390.7i) q^{23} +(-122097. + 211478. i) q^{25} +(-401612. - 231871. i) q^{29} +(393956. + 682351. i) q^{31} -357802. i q^{35} -1.10561e6 q^{37} +(3.14915e6 - 1.81816e6i) q^{41} +(1.51318e6 - 2.62091e6i) q^{43} +(4.75308e6 + 2.74419e6i) q^{47} +(2.44526e6 + 4.23531e6i) q^{49} +1.41381e7i q^{53} -2.16075e6 q^{55} +(7.58082e6 - 4.37679e6i) q^{59} +(-3.47102e6 + 6.01198e6i) q^{61} +(-1.17069e7 - 6.75900e6i) q^{65} +(-7.08899e6 - 1.22785e7i) q^{67} +7.97888e6i q^{71} -4.61414e6 q^{73} +(-4.57241e6 + 2.63988e6i) q^{77} +(1.37621e6 - 2.38366e6i) q^{79} +(3.52291e7 + 2.03395e7i) q^{83} +(-2.92573e7 - 5.06751e7i) q^{85} -2.90621e7i q^{89} -3.30311e7 q^{91} +(6.33787e7 - 3.65917e7i) q^{95} +(-4.58061e7 + 7.93386e7i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 438 q^{5} - 922 q^{7} - 28677 q^{11} + 1684 q^{13} + 269630 q^{19} - 1000452 q^{23} + 65177 q^{25} - 3797682 q^{29} + 164132 q^{31} - 1671668 q^{37} + 10239447 q^{41} - 791815 q^{43} + 31148628 q^{47}+ \cdots + 127049161 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) −331.396 + 191.331i −0.530233 + 0.306130i −0.741111 − 0.671382i \(-0.765701\pi\)
0.210878 + 0.977512i \(0.432368\pi\)
\(6\) 0 0
\(7\) −467.516 + 809.762i −0.194717 + 0.337260i −0.946808 − 0.321800i \(-0.895712\pi\)
0.752091 + 0.659060i \(0.229046\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4890.10 + 2823.30i 0.334001 + 0.192835i 0.657616 − 0.753353i \(-0.271565\pi\)
−0.323615 + 0.946189i \(0.604898\pi\)
\(12\) 0 0
\(13\) 17663.1 + 30593.3i 0.618433 + 1.07116i 0.989772 + 0.142660i \(0.0455654\pi\)
−0.371339 + 0.928497i \(0.621101\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 152914.i 1.83085i 0.402491 + 0.915424i \(0.368144\pi\)
−0.402491 + 0.915424i \(0.631856\pi\)
\(18\) 0 0
\(19\) −191248. −1.46751 −0.733756 − 0.679413i \(-0.762234\pi\)
−0.733756 + 0.679413i \(0.762234\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 132313. − 76390.7i 0.472814 − 0.272979i −0.244603 − 0.969623i \(-0.578658\pi\)
0.717417 + 0.696644i \(0.245324\pi\)
\(24\) 0 0
\(25\) −122097. + 211478.i −0.312568 + 0.541384i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −401612. − 231871.i −0.567825 − 0.327834i 0.188455 − 0.982082i \(-0.439652\pi\)
−0.756280 + 0.654248i \(0.772985\pi\)
\(30\) 0 0
\(31\) 393956. + 682351.i 0.426580 + 0.738858i 0.996567 − 0.0827958i \(-0.0263849\pi\)
−0.569987 + 0.821654i \(0.693052\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 357802.i − 0.238435i
\(36\) 0 0
\(37\) −1.10561e6 −0.589921 −0.294960 − 0.955509i \(-0.595306\pi\)
−0.294960 + 0.955509i \(0.595306\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.14915e6 − 1.81816e6i 1.11444 − 0.643425i 0.174468 − 0.984663i \(-0.444180\pi\)
0.939977 + 0.341238i \(0.110846\pi\)
\(42\) 0 0
\(43\) 1.51318e6 − 2.62091e6i 0.442607 − 0.766617i −0.555275 − 0.831667i \(-0.687387\pi\)
0.997882 + 0.0650494i \(0.0207205\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.75308e6 + 2.74419e6i 0.974055 + 0.562371i 0.900470 − 0.434918i \(-0.143223\pi\)
0.0735846 + 0.997289i \(0.476556\pi\)
\(48\) 0 0
\(49\) 2.44526e6 + 4.23531e6i 0.424170 + 0.734685i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.41381e7i 1.79179i 0.444269 + 0.895894i \(0.353464\pi\)
−0.444269 + 0.895894i \(0.646536\pi\)
\(54\) 0 0
\(55\) −2.16075e6 −0.236131
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.58082e6 − 4.37679e6i 0.625617 − 0.361200i −0.153436 − 0.988159i \(-0.549034\pi\)
0.779052 + 0.626959i \(0.215701\pi\)
\(60\) 0 0
\(61\) −3.47102e6 + 6.01198e6i −0.250690 + 0.434209i −0.963716 − 0.266929i \(-0.913991\pi\)
0.713026 + 0.701138i \(0.247324\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.17069e7 − 6.75900e6i −0.655827 − 0.378642i
\(66\) 0 0
\(67\) −7.08899e6 − 1.22785e7i −0.351791 − 0.609321i 0.634772 − 0.772700i \(-0.281094\pi\)
−0.986563 + 0.163379i \(0.947761\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.97888e6i 0.313985i 0.987600 + 0.156992i \(0.0501798\pi\)
−0.987600 + 0.156992i \(0.949820\pi\)
\(72\) 0 0
\(73\) −4.61414e6 −0.162480 −0.0812399 − 0.996695i \(-0.525888\pi\)
−0.0812399 + 0.996695i \(0.525888\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.57241e6 + 2.63988e6i −0.130071 + 0.0750968i
\(78\) 0 0
\(79\) 1.37621e6 − 2.38366e6i 0.0353326 − 0.0611979i −0.847818 − 0.530287i \(-0.822084\pi\)
0.883151 + 0.469089i \(0.155418\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.52291e7 + 2.03395e7i 0.742316 + 0.428577i 0.822911 − 0.568170i \(-0.192349\pi\)
−0.0805945 + 0.996747i \(0.525682\pi\)
\(84\) 0 0
\(85\) −2.92573e7 − 5.06751e7i −0.560478 − 0.970776i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 2.90621e7i − 0.463199i −0.972811 − 0.231599i \(-0.925604\pi\)
0.972811 − 0.231599i \(-0.0743958\pi\)
\(90\) 0 0
\(91\) −3.30311e7 −0.481678
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.33787e7 − 3.65917e7i 0.778124 − 0.449250i
\(96\) 0 0
\(97\) −4.58061e7 + 7.93386e7i −0.517412 + 0.896185i 0.482383 + 0.875960i \(0.339771\pi\)
−0.999795 + 0.0202242i \(0.993562\pi\)
\(98\) 0 0
\(99\) 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 432.9.q.a.305.3 14
3.2 odd 2 144.9.q.a.65.4 14
4.3 odd 2 27.9.d.a.8.3 14
9.4 even 3 144.9.q.a.113.4 14
9.5 odd 6 inner 432.9.q.a.17.3 14
12.11 even 2 9.9.d.a.2.5 ✓ 14
36.7 odd 6 81.9.b.a.80.9 14
36.11 even 6 81.9.b.a.80.6 14
36.23 even 6 27.9.d.a.17.3 14
36.31 odd 6 9.9.d.a.5.5 yes 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.9.d.a.2.5 ✓ 14 12.11 even 2
9.9.d.a.5.5 yes 14 36.31 odd 6
27.9.d.a.8.3 14 4.3 odd 2
27.9.d.a.17.3 14 36.23 even 6
81.9.b.a.80.6 14 36.11 even 6
81.9.b.a.80.9 14 36.7 odd 6
144.9.q.a.65.4 14 3.2 odd 2
144.9.q.a.113.4 14 9.4 even 3
432.9.q.a.17.3 14 9.5 odd 6 inner
432.9.q.a.305.3 14 1.1 even 1 trivial