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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 289.3
Root \(0.500000 - 1.48508i\) of defining polynomial
Character \(\chi\) \(=\) 432.289
Dual form 432.8.i.c.145.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-47.9866 - 83.1153i) q^{5} +(189.000 - 327.358i) q^{7} +(3436.63 - 5952.41i) q^{11} +(4826.64 + 8359.99i) q^{13} -21431.3 q^{17} -5518.94 q^{19} +(-31486.4 - 54536.1i) q^{23} +(34457.1 - 59681.4i) q^{25} +(111113. - 192454. i) q^{29} +(57729.1 + 99989.7i) q^{31} -36277.9 q^{35} +81737.7 q^{37} +(298773. + 517491. i) q^{41} +(33874.2 - 58671.8i) q^{43} +(-151740. + 262822. i) q^{47} +(340329. + 589468. i) q^{49} -846755. q^{53} -659649. q^{55} +(-793119. - 1.37372e6i) q^{59} +(1.12706e6 - 1.95213e6i) q^{61} +(463228. - 802335. i) q^{65} +(1.51172e6 + 2.61838e6i) q^{67} -4.41675e6 q^{71} +2.21484e6 q^{73} +(-1.29905e6 - 2.25002e6i) q^{77} +(153821. - 266426. i) q^{79} +(1.57735e6 - 2.73204e6i) q^{83} +(1.02841e6 + 1.78127e6i) q^{85} -1.93441e6 q^{89} +3.64895e6 q^{91} +(264836. + 458709. i) q^{95} +(-4.94528e6 + 8.56548e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 180 q^{5} + 84 q^{7} - 8460 q^{11} - 1848 q^{13} - 30564 q^{17} - 24432 q^{19} - 51588 q^{23} + 4746 q^{25} + 414648 q^{29} - 8196 q^{31} + 2210616 q^{35} + 139344 q^{37} + 1731582 q^{41} - 408372 q^{43}+ \cdots + 9977226 q^{97}+O(q^{100}) \) 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
\(5\) −47.9866 − 83.1153i −0.171682 − 0.297362i 0.767326 − 0.641257i \(-0.221587\pi\)
−0.939008 + 0.343895i \(0.888254\pi\)
\(6\) 0 0
\(7\) 189.000 − 327.358i 0.208266 − 0.360728i −0.742902 − 0.669400i \(-0.766551\pi\)
0.951169 + 0.308672i \(0.0998846\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3436.63 − 5952.41i 0.778499 − 1.34840i −0.154308 − 0.988023i \(-0.549315\pi\)
0.932807 − 0.360377i \(-0.117352\pi\)
\(12\) 0 0
\(13\) 4826.64 + 8359.99i 0.609317 + 1.05537i 0.991353 + 0.131220i \(0.0418896\pi\)
−0.382036 + 0.924147i \(0.624777\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −21431.3 −1.05798 −0.528989 − 0.848629i \(-0.677429\pi\)
−0.528989 + 0.848629i \(0.677429\pi\)
\(18\) 0 0
\(19\) −5518.94 −0.184594 −0.0922972 − 0.995732i \(-0.529421\pi\)
−0.0922972 + 0.995732i \(0.529421\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −31486.4 − 54536.1i −0.539605 − 0.934623i −0.998925 − 0.0463526i \(-0.985240\pi\)
0.459320 − 0.888271i \(-0.348093\pi\)
\(24\) 0 0
\(25\) 34457.1 − 59681.4i 0.441050 − 0.763922i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 111113. − 192454.i 0.846004 − 1.46532i −0.0387428 − 0.999249i \(-0.512335\pi\)
0.884747 − 0.466072i \(-0.154331\pi\)
\(30\) 0 0
\(31\) 57729.1 + 99989.7i 0.348040 + 0.602822i 0.985901 − 0.167329i \(-0.0535141\pi\)
−0.637862 + 0.770151i \(0.720181\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −36277.9 −0.143023
\(36\) 0 0
\(37\) 81737.7 0.265287 0.132644 − 0.991164i \(-0.457653\pi\)
0.132644 + 0.991164i \(0.457653\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 298773. + 517491.i 0.677015 + 1.17262i 0.975875 + 0.218328i \(0.0700602\pi\)
−0.298860 + 0.954297i \(0.596606\pi\)
\(42\) 0 0
\(43\) 33874.2 − 58671.8i 0.0649725 − 0.112536i −0.831709 − 0.555211i \(-0.812637\pi\)
0.896682 + 0.442676i \(0.145971\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −151740. + 262822.i −0.213186 + 0.369249i −0.952710 − 0.303881i \(-0.901717\pi\)
0.739524 + 0.673130i \(0.235051\pi\)
\(48\) 0 0
\(49\) 340329. + 589468.i 0.413250 + 0.715770i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −846755. −0.781254 −0.390627 − 0.920549i \(-0.627742\pi\)
−0.390627 + 0.920549i \(0.627742\pi\)
\(54\) 0 0
\(55\) −659649. −0.534618
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −793119. − 1.37372e6i −0.502755 − 0.870797i −0.999995 − 0.00318395i \(-0.998987\pi\)
0.497240 − 0.867613i \(-0.334347\pi\)
\(60\) 0 0
\(61\) 1.12706e6 − 1.95213e6i 0.635760 − 1.10117i −0.350593 − 0.936528i \(-0.614020\pi\)
0.986353 − 0.164641i \(-0.0526467\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 463228. − 802335.i 0.209218 − 0.362376i
\(66\) 0 0
\(67\) 1.51172e6 + 2.61838e6i 0.614060 + 1.06358i 0.990549 + 0.137162i \(0.0437980\pi\)
−0.376489 + 0.926421i \(0.622869\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.41675e6 −1.46453 −0.732266 − 0.681018i \(-0.761537\pi\)
−0.732266 + 0.681018i \(0.761537\pi\)
\(72\) 0 0
\(73\) 2.21484e6 0.666366 0.333183 − 0.942862i \(-0.391877\pi\)
0.333183 + 0.942862i \(0.391877\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.29905e6 − 2.25002e6i −0.324271 − 0.561653i
\(78\) 0 0
\(79\) 153821. − 266426.i 0.0351011 − 0.0607969i −0.847941 − 0.530090i \(-0.822158\pi\)
0.883042 + 0.469293i \(0.155491\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.57735e6 − 2.73204e6i 0.302798 − 0.524462i −0.673970 − 0.738758i \(-0.735412\pi\)
0.976769 + 0.214296i \(0.0687458\pi\)
\(84\) 0 0
\(85\) 1.02841e6 + 1.78127e6i 0.181636 + 0.314603i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.93441e6 −0.290859 −0.145430 − 0.989369i \(-0.546456\pi\)
−0.145430 + 0.989369i \(0.546456\pi\)
\(90\) 0 0
\(91\) 3.64895e6 0.507601
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 264836. + 458709.i 0.0316916 + 0.0548914i
\(96\) 0 0
\(97\) −4.94528e6 + 8.56548e6i −0.550161 + 0.952907i 0.448101 + 0.893983i \(0.352100\pi\)
−0.998262 + 0.0589243i \(0.981233\pi\)
\(98\) 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.289.3 12
3.2 odd 2 144.8.i.c.97.4 12
4.3 odd 2 27.8.c.a.19.3 12
9.4 even 3 inner 432.8.i.c.145.3 12
9.5 odd 6 144.8.i.c.49.4 12
12.11 even 2 9.8.c.a.7.4 yes 12
36.7 odd 6 81.8.a.c.1.4 6
36.11 even 6 81.8.a.e.1.3 6
36.23 even 6 9.8.c.a.4.4 ✓ 12
36.31 odd 6 27.8.c.a.10.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.4 ✓ 12 36.23 even 6
9.8.c.a.7.4 yes 12 12.11 even 2
27.8.c.a.10.3 12 36.31 odd 6
27.8.c.a.19.3 12 4.3 odd 2
81.8.a.c.1.4 6 36.7 odd 6
81.8.a.e.1.3 6 36.11 even 6
144.8.i.c.49.4 12 9.5 odd 6
144.8.i.c.97.4 12 3.2 odd 2
432.8.i.c.145.3 12 9.4 even 3 inner
432.8.i.c.289.3 12 1.1 even 1 trivial