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Properties

Label 432.9.q.a.305.1
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 305.1
Root \(430.310 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 432.305
Dual form 432.9.q.a.17.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-676.216 + 390.413i) q^{5} +(-2168.61 + 3756.14i) q^{7} +(5607.04 + 3237.22i) q^{11} +(-8497.75 - 14718.5i) q^{13} -29881.7i q^{17} +91768.1 q^{19} +(-106388. + 61423.0i) q^{23} +(109532. - 189716. i) q^{25} +(-434568. - 250898. i) q^{29} +(-505931. - 876298. i) q^{31} -3.38661e6i q^{35} +190279. q^{37} +(1.95619e6 - 1.12941e6i) q^{41} +(1.64497e6 - 2.84917e6i) q^{43} +(-110171. - 63607.2i) q^{47} +(-6.52331e6 - 1.12987e7i) q^{49} -1.40856e6i q^{53} -5.05542e6 q^{55} +(-1.79370e7 + 1.03560e7i) q^{59} +(9.62157e6 - 1.66651e7i) q^{61} +(1.14926e7 + 6.63527e6i) q^{65} +(-1.17156e7 - 2.02920e7i) q^{67} +9.12361e6i q^{71} +1.43931e7 q^{73} +(-2.43189e7 + 1.40405e7i) q^{77} +(-2.69804e7 + 4.67314e7i) q^{79} +(2.66155e7 + 1.53664e7i) q^{83} +(1.16662e7 + 2.02064e7i) q^{85} +2.32435e7i q^{89} +7.37131e7 q^{91} +(-6.20550e7 + 3.58275e7i) q^{95} +(2.46595e7 - 4.27114e7i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 438 q^{5} - 922 q^{7} - 28677 q^{11} + 1684 q^{13} + 269630 q^{19} - 1000452 q^{23} + 65177 q^{25} - 3797682 q^{29} + 164132 q^{31} - 1671668 q^{37} + 10239447 q^{41} - 791815 q^{43} + 31148628 q^{47}+ \cdots + 127049161 q^{97}+O(q^{100}) \) 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
\(5\) −676.216 + 390.413i −1.08194 + 0.624661i −0.931420 − 0.363945i \(-0.881430\pi\)
−0.150524 + 0.988606i \(0.548096\pi\)
\(6\) 0 0
\(7\) −2168.61 + 3756.14i −0.903210 + 1.56441i −0.0799078 + 0.996802i \(0.525463\pi\)
−0.823302 + 0.567603i \(0.807871\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5607.04 + 3237.22i 0.382968 + 0.221107i 0.679109 − 0.734038i \(-0.262366\pi\)
−0.296141 + 0.955144i \(0.595700\pi\)
\(12\) 0 0
\(13\) −8497.75 − 14718.5i −0.297530 − 0.515337i 0.678040 − 0.735025i \(-0.262829\pi\)
−0.975570 + 0.219688i \(0.929496\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 29881.7i − 0.357774i −0.983870 − 0.178887i \(-0.942750\pi\)
0.983870 − 0.178887i \(-0.0572497\pi\)
\(18\) 0 0
\(19\) 91768.1 0.704170 0.352085 − 0.935968i \(-0.385473\pi\)
0.352085 + 0.935968i \(0.385473\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −106388. + 61423.0i −0.380172 + 0.219492i −0.677893 − 0.735160i \(-0.737107\pi\)
0.297721 + 0.954653i \(0.403773\pi\)
\(24\) 0 0
\(25\) 109532. − 189716.i 0.280403 − 0.485672i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −434568. − 250898.i −0.614420 − 0.354736i 0.160273 − 0.987073i \(-0.448762\pi\)
−0.774693 + 0.632337i \(0.782096\pi\)
\(30\) 0 0
\(31\) −505931. − 876298.i −0.547828 − 0.948866i −0.998423 − 0.0561377i \(-0.982121\pi\)
0.450595 − 0.892729i \(-0.351212\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 3.38661e6i − 2.25680i
\(36\) 0 0
\(37\) 190279. 0.101528 0.0507638 − 0.998711i \(-0.483834\pi\)
0.0507638 + 0.998711i \(0.483834\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.95619e6 − 1.12941e6i 0.692269 − 0.399682i −0.112192 − 0.993687i \(-0.535787\pi\)
0.804462 + 0.594005i \(0.202454\pi\)
\(42\) 0 0
\(43\) 1.64497e6 − 2.84917e6i 0.481154 − 0.833383i −0.518612 − 0.855010i \(-0.673551\pi\)
0.999766 + 0.0216266i \(0.00688451\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −110171. − 63607.2i −0.0225775 − 0.0130351i 0.488669 − 0.872469i \(-0.337483\pi\)
−0.511246 + 0.859434i \(0.670816\pi\)
\(48\) 0 0
\(49\) −6.52331e6 − 1.12987e7i −1.13158 − 1.95995i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 1.40856e6i − 0.178514i −0.996009 − 0.0892571i \(-0.971551\pi\)
0.996009 − 0.0892571i \(-0.0284493\pi\)
\(54\) 0 0
\(55\) −5.05542e6 −0.552467
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.79370e7 + 1.03560e7i −1.48028 + 0.854638i −0.999750 − 0.0223421i \(-0.992888\pi\)
−0.480526 + 0.876980i \(0.659554\pi\)
\(60\) 0 0
\(61\) 9.62157e6 − 1.66651e7i 0.694907 − 1.20361i −0.275305 − 0.961357i \(-0.588779\pi\)
0.970212 − 0.242258i \(-0.0778879\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.14926e7 + 6.63527e6i 0.643822 + 0.371711i
\(66\) 0 0
\(67\) −1.17156e7 − 2.02920e7i −0.581386 − 1.00699i −0.995315 − 0.0966815i \(-0.969177\pi\)
0.413929 − 0.910309i \(-0.364156\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.12361e6i 0.359032i 0.983755 + 0.179516i \(0.0574532\pi\)
−0.983755 + 0.179516i \(0.942547\pi\)
\(72\) 0 0
\(73\) 1.43931e7 0.506832 0.253416 − 0.967357i \(-0.418446\pi\)
0.253416 + 0.967357i \(0.418446\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.43189e7 + 1.40405e7i −0.691801 + 0.399412i
\(78\) 0 0
\(79\) −2.69804e7 + 4.67314e7i −0.692691 + 1.19978i 0.278262 + 0.960505i \(0.410242\pi\)
−0.970953 + 0.239271i \(0.923092\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.66155e7 + 1.53664e7i 0.560818 + 0.323788i 0.753474 − 0.657478i \(-0.228377\pi\)
−0.192656 + 0.981266i \(0.561710\pi\)
\(84\) 0 0
\(85\) 1.16662e7 + 2.02064e7i 0.223488 + 0.387092i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.32435e7i 0.370460i 0.982695 + 0.185230i \(0.0593030\pi\)
−0.982695 + 0.185230i \(0.940697\pi\)
\(90\) 0 0
\(91\) 7.37131e7 1.07493
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.20550e7 + 3.58275e7i −0.761873 + 0.439867i
\(96\) 0 0
\(97\) 2.46595e7 − 4.27114e7i 0.278546 − 0.482455i −0.692478 − 0.721439i \(-0.743481\pi\)
0.971024 + 0.238984i \(0.0768143\pi\)
\(98\) 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.305.1 14
3.2 odd 2 144.9.q.a.65.2 14
4.3 odd 2 27.9.d.a.8.5 14
9.4 even 3 144.9.q.a.113.2 14
9.5 odd 6 inner 432.9.q.a.17.1 14
12.11 even 2 9.9.d.a.2.3 ✓ 14
36.7 odd 6 81.9.b.a.80.5 14
36.11 even 6 81.9.b.a.80.10 14
36.23 even 6 27.9.d.a.17.5 14
36.31 odd 6 9.9.d.a.5.3 yes 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.9.d.a.2.3 ✓ 14 12.11 even 2
9.9.d.a.5.3 yes 14 36.31 odd 6
27.9.d.a.8.5 14 4.3 odd 2
27.9.d.a.17.5 14 36.23 even 6
81.9.b.a.80.5 14 36.7 odd 6
81.9.b.a.80.10 14 36.11 even 6
144.9.q.a.65.2 14 3.2 odd 2
144.9.q.a.113.2 14 9.4 even 3
432.9.q.a.17.1 14 9.5 odd 6 inner
432.9.q.a.305.1 14 1.1 even 1 trivial