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Properties

Label 432.2.i.d.289.2
Level $432$
Weight $2$
Character 432.289
Analytic conductor $3.450$
Analytic rank $0$
Dimension $4$
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] = [432,2,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, 2); 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, 2, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 432.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-1] 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: \(3.44953736732\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.2
Root \(1.68614 + 0.396143i\) of defining polynomial
Character \(\chi\) \(=\) 432.289
Dual form 432.2.i.d.145.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.18614 + 2.05446i) q^{5} +(2.18614 - 3.78651i) q^{7} +(-0.500000 + 0.866025i) q^{11} +(0.186141 + 0.322405i) q^{13} +5.37228 q^{17} -0.627719 q^{19} +(0.186141 + 0.322405i) q^{23} +(-0.313859 + 0.543620i) q^{25} +(-2.18614 + 3.78651i) q^{29} +(3.18614 + 5.51856i) q^{31} +10.3723 q^{35} +8.74456 q^{37} +(-5.87228 - 10.1711i) q^{41} +(-0.872281 + 1.51084i) q^{43} +(-2.18614 + 3.78651i) q^{47} +(-6.05842 - 10.4935i) q^{49} -0.744563 q^{53} -2.37228 q^{55} +(-3.50000 - 6.06218i) q^{59} +(-1.18614 + 2.05446i) q^{61} +(-0.441578 + 0.764836i) q^{65} +(-1.87228 - 3.24289i) q^{67} -4.00000 q^{71} -12.1168 q^{73} +(2.18614 + 3.78651i) q^{77} +(-3.18614 + 5.51856i) q^{79} +(-4.81386 + 8.33785i) q^{83} +(6.37228 + 11.0371i) q^{85} -6.00000 q^{89} +1.62772 q^{91} +(-0.744563 - 1.28962i) q^{95} +(-0.872281 + 1.51084i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{5} + 3 q^{7} - 2 q^{11} - 5 q^{13} + 10 q^{17} - 14 q^{19} - 5 q^{23} - 7 q^{25} - 3 q^{29} + 7 q^{31} + 30 q^{35} + 12 q^{37} - 12 q^{41} + 8 q^{43} - 3 q^{47} - 7 q^{49} + 20 q^{53} + 2 q^{55}+ \cdots + 8 q^{97}+O(q^{100}) \) 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
\(5\) 1.18614 + 2.05446i 0.530458 + 0.918781i 0.999368 + 0.0355348i \(0.0113134\pi\)
−0.468910 + 0.883246i \(0.655353\pi\)
\(6\) 0 0
\(7\) 2.18614 − 3.78651i 0.826284 − 1.43117i −0.0746509 − 0.997210i \(-0.523784\pi\)
0.900934 − 0.433955i \(-0.142882\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.500000 + 0.866025i −0.150756 + 0.261116i −0.931505 − 0.363727i \(-0.881504\pi\)
0.780750 + 0.624844i \(0.214837\pi\)
\(12\) 0 0
\(13\) 0.186141 + 0.322405i 0.0516261 + 0.0894191i 0.890684 − 0.454624i \(-0.150226\pi\)
−0.839057 + 0.544043i \(0.816893\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.37228 1.30297 0.651485 − 0.758662i \(-0.274146\pi\)
0.651485 + 0.758662i \(0.274146\pi\)
\(18\) 0 0
\(19\) −0.627719 −0.144009 −0.0720043 − 0.997404i \(-0.522940\pi\)
−0.0720043 + 0.997404i \(0.522940\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.186141 + 0.322405i 0.0388130 + 0.0672261i 0.884779 − 0.466010i \(-0.154309\pi\)
−0.845966 + 0.533236i \(0.820976\pi\)
\(24\) 0 0
\(25\) −0.313859 + 0.543620i −0.0627719 + 0.108724i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.18614 + 3.78651i −0.405956 + 0.703137i −0.994432 − 0.105378i \(-0.966395\pi\)
0.588476 + 0.808515i \(0.299728\pi\)
\(30\) 0 0
\(31\) 3.18614 + 5.51856i 0.572248 + 0.991162i 0.996335 + 0.0855407i \(0.0272618\pi\)
−0.424087 + 0.905621i \(0.639405\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 10.3723 1.75324
\(36\) 0 0
\(37\) 8.74456 1.43760 0.718799 − 0.695218i \(-0.244692\pi\)
0.718799 + 0.695218i \(0.244692\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.87228 − 10.1711i −0.917096 − 1.58846i −0.803803 − 0.594896i \(-0.797193\pi\)
−0.113293 − 0.993562i \(-0.536140\pi\)
\(42\) 0 0
\(43\) −0.872281 + 1.51084i −0.133022 + 0.230400i −0.924840 − 0.380356i \(-0.875801\pi\)
0.791818 + 0.610757i \(0.209135\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.18614 + 3.78651i −0.318881 + 0.552319i −0.980255 − 0.197738i \(-0.936640\pi\)
0.661374 + 0.750057i \(0.269974\pi\)
\(48\) 0 0
\(49\) −6.05842 − 10.4935i −0.865489 − 1.49907i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.744563 −0.102274 −0.0511368 − 0.998692i \(-0.516284\pi\)
−0.0511368 + 0.998692i \(0.516284\pi\)
\(54\) 0 0
\(55\) −2.37228 −0.319878
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.50000 − 6.06218i −0.455661 − 0.789228i 0.543065 − 0.839691i \(-0.317264\pi\)
−0.998726 + 0.0504625i \(0.983930\pi\)
\(60\) 0 0
\(61\) −1.18614 + 2.05446i −0.151870 + 0.263046i −0.931915 − 0.362677i \(-0.881863\pi\)
0.780045 + 0.625723i \(0.215196\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.441578 + 0.764836i −0.0547710 + 0.0948662i
\(66\) 0 0
\(67\) −1.87228 − 3.24289i −0.228736 − 0.396182i 0.728698 − 0.684835i \(-0.240126\pi\)
−0.957434 + 0.288653i \(0.906792\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) −12.1168 −1.41817 −0.709085 − 0.705123i \(-0.750892\pi\)
−0.709085 + 0.705123i \(0.750892\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.18614 + 3.78651i 0.249134 + 0.431512i
\(78\) 0 0
\(79\) −3.18614 + 5.51856i −0.358469 + 0.620886i −0.987705 − 0.156328i \(-0.950034\pi\)
0.629236 + 0.777214i \(0.283368\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.81386 + 8.33785i −0.528390 + 0.915198i 0.471062 + 0.882100i \(0.343871\pi\)
−0.999452 + 0.0330979i \(0.989463\pi\)
\(84\) 0 0
\(85\) 6.37228 + 11.0371i 0.691171 + 1.19714i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 1.62772 0.170631
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.744563 − 1.28962i −0.0763905 − 0.132312i
\(96\) 0 0
\(97\) −0.872281 + 1.51084i −0.0885667 + 0.153402i −0.906906 − 0.421334i \(-0.861562\pi\)
0.818339 + 0.574736i \(0.194895\pi\)
\(98\) 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 432.2.i.d.289.2 4
3.2 odd 2 144.2.i.d.97.1 4
4.3 odd 2 216.2.i.b.73.2 4
8.3 odd 2 1728.2.i.i.1153.1 4
8.5 even 2 1728.2.i.j.1153.1 4
9.2 odd 6 1296.2.a.n.1.2 2
9.4 even 3 inner 432.2.i.d.145.2 4
9.5 odd 6 144.2.i.d.49.1 4
9.7 even 3 1296.2.a.p.1.1 2
12.11 even 2 72.2.i.b.25.2 ✓ 4
24.5 odd 2 576.2.i.l.385.2 4
24.11 even 2 576.2.i.j.385.1 4
36.7 odd 6 648.2.a.g.1.1 2
36.11 even 6 648.2.a.f.1.2 2
36.23 even 6 72.2.i.b.49.2 yes 4
36.31 odd 6 216.2.i.b.145.2 4
72.5 odd 6 576.2.i.l.193.2 4
72.11 even 6 5184.2.a.bt.1.1 2
72.13 even 6 1728.2.i.j.577.1 4
72.29 odd 6 5184.2.a.bs.1.1 2
72.43 odd 6 5184.2.a.bp.1.2 2
72.59 even 6 576.2.i.j.193.1 4
72.61 even 6 5184.2.a.bo.1.2 2
72.67 odd 6 1728.2.i.i.577.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.i.b.25.2 ✓ 4 12.11 even 2
72.2.i.b.49.2 yes 4 36.23 even 6
144.2.i.d.49.1 4 9.5 odd 6
144.2.i.d.97.1 4 3.2 odd 2
216.2.i.b.73.2 4 4.3 odd 2
216.2.i.b.145.2 4 36.31 odd 6
432.2.i.d.145.2 4 9.4 even 3 inner
432.2.i.d.289.2 4 1.1 even 1 trivial
576.2.i.j.193.1 4 72.59 even 6
576.2.i.j.385.1 4 24.11 even 2
576.2.i.l.193.2 4 72.5 odd 6
576.2.i.l.385.2 4 24.5 odd 2
648.2.a.f.1.2 2 36.11 even 6
648.2.a.g.1.1 2 36.7 odd 6
1296.2.a.n.1.2 2 9.2 odd 6
1296.2.a.p.1.1 2 9.7 even 3
1728.2.i.i.577.1 4 72.67 odd 6
1728.2.i.i.1153.1 4 8.3 odd 2
1728.2.i.j.577.1 4 72.13 even 6
1728.2.i.j.1153.1 4 8.5 even 2
5184.2.a.bo.1.2 2 72.61 even 6
5184.2.a.bp.1.2 2 72.43 odd 6
5184.2.a.bs.1.1 2 72.29 odd 6
5184.2.a.bt.1.1 2 72.11 even 6