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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [432,8,Mod(145,432)] 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("432.145"); S:= CuspForms(chi, 8); 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, 2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 432.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(134.950331009\)
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: \( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{21} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.5
Root \(0.500000 - 9.08282i\) of defining polynomial
Character \(\chi\) \(=\) 432.289
Dual form 432.8.i.c.145.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(145.304 + 251.673i) q^{5} +(555.940 - 962.916i) q^{7} +(-2245.36 + 3889.07i) q^{11} +(-1218.29 - 2110.14i) q^{13} -15905.4 q^{17} +49949.6 q^{19} +(-34692.5 - 60089.2i) q^{23} +(-3163.73 + 5479.74i) q^{25} +(-47035.8 + 81468.4i) q^{29} +(-9963.58 - 17257.4i) q^{31} +323120. q^{35} +331750. q^{37} +(121133. + 209809. i) q^{41} +(415713. - 720036. i) q^{43} +(80005.3 - 138573. i) q^{47} +(-206366. - 357437. i) q^{49} -311589. q^{53} -1.30503e6 q^{55} +(156177. + 270506. i) q^{59} +(28723.9 - 49751.3i) q^{61} +(354044. - 613221. i) q^{65} +(-2.05100e6 - 3.55243e6i) q^{67} -403110. q^{71} -823496. q^{73} +(2.49656e6 + 4.32418e6i) q^{77} +(489414. - 847689. i) q^{79} +(1.85204e6 - 3.20782e6i) q^{83} +(-2.31111e6 - 4.00296e6i) q^{85} -2.09023e6 q^{89} -2.70918e6 q^{91} +(7.25786e6 + 1.25710e7i) q^{95} +(-1.75125e6 + 3.03325e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 180 q^{5} + 84 q^{7} - 8460 q^{11} - 1848 q^{13} - 30564 q^{17} - 24432 q^{19} - 51588 q^{23} + 4746 q^{25} + 414648 q^{29} - 8196 q^{31} + 2210616 q^{35} + 139344 q^{37} + 1731582 q^{41} - 408372 q^{43}+ \cdots + 9977226 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 145.304 + 251.673i 0.519854 + 0.900413i 0.999734 + 0.0230788i \(0.00734688\pi\)
−0.479880 + 0.877334i \(0.659320\pi\)
\(6\) 0 0
\(7\) 555.940 − 962.916i 0.612611 − 1.06107i −0.378188 − 0.925729i \(-0.623453\pi\)
0.990799 − 0.135344i \(-0.0432139\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2245.36 + 3889.07i −0.508640 + 0.880991i 0.491310 + 0.870985i \(0.336518\pi\)
−0.999950 + 0.0100060i \(0.996815\pi\)
\(12\) 0 0
\(13\) −1218.29 − 2110.14i −0.153797 − 0.266385i 0.778823 − 0.627244i \(-0.215817\pi\)
−0.932620 + 0.360859i \(0.882484\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −15905.4 −0.785187 −0.392593 − 0.919712i \(-0.628422\pi\)
−0.392593 + 0.919712i \(0.628422\pi\)
\(18\) 0 0
\(19\) 49949.6 1.67069 0.835343 − 0.549730i \(-0.185269\pi\)
0.835343 + 0.549730i \(0.185269\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −34692.5 − 60089.2i −0.594550 − 1.02979i −0.993610 − 0.112867i \(-0.963997\pi\)
0.399060 − 0.916925i \(-0.369337\pi\)
\(24\) 0 0
\(25\) −3163.73 + 5479.74i −0.0404957 + 0.0701406i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −47035.8 + 81468.4i −0.358126 + 0.620292i −0.987648 − 0.156691i \(-0.949917\pi\)
0.629522 + 0.776983i \(0.283251\pi\)
\(30\) 0 0
\(31\) −9963.58 − 17257.4i −0.0600689 − 0.104042i 0.834427 − 0.551118i \(-0.185799\pi\)
−0.894496 + 0.447076i \(0.852465\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 323120. 1.27387
\(36\) 0 0
\(37\) 331750. 1.07673 0.538363 − 0.842713i \(-0.319043\pi\)
0.538363 + 0.842713i \(0.319043\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 121133. + 209809.i 0.274486 + 0.475423i 0.970005 − 0.243084i \(-0.0781591\pi\)
−0.695520 + 0.718507i \(0.744826\pi\)
\(42\) 0 0
\(43\) 415713. − 720036.i 0.797359 − 1.38107i −0.123971 − 0.992286i \(-0.539563\pi\)
0.921330 − 0.388781i \(-0.127104\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 80005.3 − 138573.i 0.112403 − 0.194687i −0.804336 − 0.594175i \(-0.797479\pi\)
0.916738 + 0.399488i \(0.130812\pi\)
\(48\) 0 0
\(49\) −206366. − 357437.i −0.250583 − 0.434023i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −311589. −0.287486 −0.143743 − 0.989615i \(-0.545914\pi\)
−0.143743 + 0.989615i \(0.545914\pi\)
\(54\) 0 0
\(55\) −1.30503e6 −1.05767
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 156177. + 270506.i 0.0989997 + 0.171473i 0.911271 − 0.411807i \(-0.135102\pi\)
−0.812271 + 0.583280i \(0.801769\pi\)
\(60\) 0 0
\(61\) 28723.9 − 49751.3i 0.0162028 − 0.0280640i −0.857810 − 0.513966i \(-0.828176\pi\)
0.874013 + 0.485902i \(0.161509\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 354044. − 613221.i 0.159904 − 0.276962i
\(66\) 0 0
\(67\) −2.05100e6 − 3.55243e6i −0.833111 − 1.44299i −0.895559 − 0.444943i \(-0.853224\pi\)
0.0624478 − 0.998048i \(-0.480109\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −403110. −0.133666 −0.0668328 − 0.997764i \(-0.521289\pi\)
−0.0668328 + 0.997764i \(0.521289\pi\)
\(72\) 0 0
\(73\) −823496. −0.247760 −0.123880 − 0.992297i \(-0.539534\pi\)
−0.123880 + 0.992297i \(0.539534\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.49656e6 + 4.32418e6i 0.623197 + 1.07941i
\(78\) 0 0
\(79\) 489414. − 847689.i 0.111682 − 0.193438i −0.804767 − 0.593591i \(-0.797710\pi\)
0.916448 + 0.400153i \(0.131043\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.85204e6 − 3.20782e6i 0.355530 − 0.615796i −0.631678 − 0.775231i \(-0.717634\pi\)
0.987209 + 0.159434i \(0.0509670\pi\)
\(84\) 0 0
\(85\) −2.31111e6 − 4.00296e6i −0.408182 − 0.706993i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.09023e6 −0.314289 −0.157145 − 0.987576i \(-0.550229\pi\)
−0.157145 + 0.987576i \(0.550229\pi\)
\(90\) 0 0
\(91\) −2.70918e6 −0.376872
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.25786e6 + 1.25710e7i 0.868512 + 1.50431i
\(96\) 0 0
\(97\) −1.75125e6 + 3.03325e6i −0.194826 + 0.337448i −0.946843 − 0.321695i \(-0.895747\pi\)
0.752018 + 0.659143i \(0.229081\pi\)
\(98\) 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 432.8.i.c.289.5 12
3.2 odd 2 144.8.i.c.97.1 12
4.3 odd 2 27.8.c.a.19.2 12
9.4 even 3 inner 432.8.i.c.145.5 12
9.5 odd 6 144.8.i.c.49.1 12
12.11 even 2 9.8.c.a.7.5 yes 12
36.7 odd 6 81.8.a.c.1.5 6
36.11 even 6 81.8.a.e.1.2 6
36.23 even 6 9.8.c.a.4.5 ✓ 12
36.31 odd 6 27.8.c.a.10.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.5 ✓ 12 36.23 even 6
9.8.c.a.7.5 yes 12 12.11 even 2
27.8.c.a.10.2 12 36.31 odd 6
27.8.c.a.19.2 12 4.3 odd 2
81.8.a.c.1.5 6 36.7 odd 6
81.8.a.e.1.2 6 36.11 even 6
144.8.i.c.49.1 12 9.5 odd 6
144.8.i.c.97.1 12 3.2 odd 2
432.8.i.c.145.5 12 9.4 even 3 inner
432.8.i.c.289.5 12 1.1 even 1 trivial