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Properties

Label 432.2.v.a.179.8
Level $432$
Weight $2$
Character 432.179
Analytic conductor $3.450$
Analytic rank $0$
Dimension $88$
Inner twists $4$

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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(35,432)] 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("432.35"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 9, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 432.v (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.44953736732\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 179.8
Character \(\chi\) \(=\) 432.179
Dual form 432.2.v.a.251.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.688374 - 1.23537i) q^{2} +(-1.05228 + 1.70079i) q^{4} +(-0.664471 + 0.178044i) q^{5} +(0.645693 + 1.11837i) q^{7} +(2.82548 + 0.129179i) q^{8} +(0.677355 + 0.698307i) q^{10} +(-3.21069 - 0.860301i) q^{11} +(-4.74727 + 1.27203i) q^{13} +(0.937128 - 1.56753i) q^{14} +(-1.78540 - 3.57943i) q^{16} +5.58523i q^{17} +(-2.49649 + 2.49649i) q^{19} +(0.396395 - 1.31748i) q^{20} +(1.14736 + 4.55860i) q^{22} +(2.36529 + 1.36560i) q^{23} +(-3.92031 + 2.26339i) q^{25} +(4.83933 + 4.98901i) q^{26} +(-2.58157 - 0.0786549i) q^{28} +(2.95682 + 0.792277i) q^{29} +(5.28160 + 3.04933i) q^{31} +(-3.19291 + 4.66962i) q^{32} +(6.89983 - 3.84472i) q^{34} +(-0.628165 - 0.628165i) q^{35} +(0.507420 - 0.507420i) q^{37} +(4.80260 + 1.36557i) q^{38} +(-1.90045 + 0.417225i) q^{40} +(-4.89892 + 8.48518i) q^{41} +(0.254540 - 0.949956i) q^{43} +(4.84175 - 4.55544i) q^{44} +(0.0588204 - 3.86205i) q^{46} +(-6.13774 - 10.6309i) q^{47} +(2.66616 - 4.61793i) q^{49} +(5.49476 + 3.28497i) q^{50} +(2.83202 - 9.41267i) q^{52} +(0.601793 + 0.601793i) q^{53} +2.28658 q^{55} +(1.67992 + 3.24335i) q^{56} +(-1.05664 - 4.19815i) q^{58} +(-1.28003 - 4.77715i) q^{59} +(-2.90167 + 10.8292i) q^{61} +(0.131344 - 8.62382i) q^{62} +(7.96663 + 0.729984i) q^{64} +(2.92795 - 1.69045i) q^{65} +(-0.0295686 - 0.110351i) q^{67} +(-9.49932 - 5.87724i) q^{68} +(-0.343604 + 1.20843i) q^{70} -0.0447904i q^{71} +13.2931i q^{73} +(-0.976146 - 0.277557i) q^{74} +(-1.61900 - 6.87302i) q^{76} +(-1.11098 - 4.14624i) q^{77} +(2.50052 - 1.44368i) q^{79} +(1.82364 + 2.06055i) q^{80} +(13.8546 + 0.211011i) q^{82} +(-1.01705 + 3.79568i) q^{83} +(-0.994419 - 3.71122i) q^{85} +(-1.34877 + 0.339473i) q^{86} +(-8.96058 - 2.84551i) q^{88} -12.7362 q^{89} +(-4.48788 - 4.48788i) q^{91} +(-4.81156 + 2.58587i) q^{92} +(-8.90802 + 14.9004i) q^{94} +(1.21436 - 2.10333i) q^{95} +(4.41066 + 7.63949i) q^{97} +(-7.54017 - 0.114840i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 88 q + 6 q^{2} - 2 q^{4} + 6 q^{5} - 4 q^{7} - 8 q^{10} + 6 q^{11} - 2 q^{13} + 6 q^{14} - 2 q^{16} - 8 q^{19} + 48 q^{20} - 2 q^{22} + 12 q^{23} + 8 q^{28} + 6 q^{29} + 6 q^{32} + 2 q^{34} - 8 q^{37} + 6 q^{38}+ \cdots - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.688374 − 1.23537i −0.486754 − 0.873539i
\(3\) 0 0
\(4\) −1.05228 + 1.70079i −0.526142 + 0.850397i
\(5\) −0.664471 + 0.178044i −0.297161 + 0.0796239i −0.404319 − 0.914618i \(-0.632491\pi\)
0.107158 + 0.994242i \(0.465825\pi\)
\(6\) 0 0
\(7\) 0.645693 + 1.11837i 0.244049 + 0.422705i 0.961864 − 0.273529i \(-0.0881909\pi\)
−0.717815 + 0.696234i \(0.754858\pi\)
\(8\) 2.82548 + 0.129179i 0.998957 + 0.0456717i
\(9\) 0 0
\(10\) 0.677355 + 0.698307i 0.214199 + 0.220824i
\(11\) −3.21069 − 0.860301i −0.968058 − 0.259390i −0.260051 − 0.965595i \(-0.583739\pi\)
−0.708008 + 0.706205i \(0.750406\pi\)
\(12\) 0 0
\(13\) −4.74727 + 1.27203i −1.31666 + 0.352797i −0.847725 − 0.530437i \(-0.822028\pi\)
−0.468933 + 0.883234i \(0.655361\pi\)
\(14\) 0.937128 − 1.56753i 0.250458 − 0.418940i
\(15\) 0 0
\(16\) −1.78540 − 3.57943i −0.446350 − 0.894859i
\(17\) 5.58523i 1.35462i 0.735699 + 0.677308i \(0.236854\pi\)
−0.735699 + 0.677308i \(0.763146\pi\)
\(18\) 0 0
\(19\) −2.49649 + 2.49649i −0.572733 + 0.572733i −0.932891 − 0.360158i \(-0.882723\pi\)
0.360158 + 0.932891i \(0.382723\pi\)
\(20\) 0.396395 − 1.31748i 0.0886366 − 0.294598i
\(21\) 0 0
\(22\) 1.14736 + 4.55860i 0.244618 + 0.971896i
\(23\) 2.36529 + 1.36560i 0.493197 + 0.284747i 0.725900 − 0.687801i \(-0.241424\pi\)
−0.232703 + 0.972548i \(0.574757\pi\)
\(24\) 0 0
\(25\) −3.92031 + 2.26339i −0.784061 + 0.452678i
\(26\) 4.83933 + 4.98901i 0.949070 + 0.978426i
\(27\) 0 0
\(28\) −2.58157 − 0.0786549i −0.487872 − 0.0148644i
\(29\) 2.95682 + 0.792277i 0.549067 + 0.147122i 0.522679 − 0.852530i \(-0.324933\pi\)
0.0263884 + 0.999652i \(0.491599\pi\)
\(30\) 0 0
\(31\) 5.28160 + 3.04933i 0.948604 + 0.547677i 0.892647 − 0.450757i \(-0.148846\pi\)
0.0559568 + 0.998433i \(0.482179\pi\)
\(32\) −3.19291 + 4.66962i −0.564432 + 0.825480i
\(33\) 0 0
\(34\) 6.89983 − 3.84472i 1.18331 − 0.659365i
\(35\) −0.628165 − 0.628165i −0.106179 − 0.106179i
\(36\) 0 0
\(37\) 0.507420 − 0.507420i 0.0834193 − 0.0834193i −0.664166 − 0.747585i \(-0.731213\pi\)
0.747585 + 0.664166i \(0.231213\pi\)
\(38\) 4.80260 + 1.36557i 0.779085 + 0.221525i
\(39\) 0 0
\(40\) −1.90045 + 0.417225i −0.300487 + 0.0659690i
\(41\) −4.89892 + 8.48518i −0.765083 + 1.32516i 0.175120 + 0.984547i \(0.443969\pi\)
−0.940203 + 0.340616i \(0.889365\pi\)
\(42\) 0 0
\(43\) 0.254540 − 0.949956i 0.0388170 − 0.144867i −0.943798 − 0.330524i \(-0.892775\pi\)
0.982615 + 0.185657i \(0.0594413\pi\)
\(44\) 4.84175 − 4.55544i 0.729921 − 0.686758i
\(45\) 0 0
\(46\) 0.0588204 − 3.86205i 0.00867260 − 0.569428i
\(47\) −6.13774 − 10.6309i −0.895281 − 1.55067i −0.833456 − 0.552586i \(-0.813641\pi\)
−0.0618250 − 0.998087i \(-0.519692\pi\)
\(48\) 0 0
\(49\) 2.66616 − 4.61793i 0.380880 − 0.659704i
\(50\) 5.49476 + 3.28497i 0.777076 + 0.464565i
\(51\) 0 0
\(52\) 2.83202 − 9.41267i 0.392731 − 1.30530i
\(53\) 0.601793 + 0.601793i 0.0826626 + 0.0826626i 0.747229 − 0.664567i \(-0.231384\pi\)
−0.664567 + 0.747229i \(0.731384\pi\)
\(54\) 0 0
\(55\) 2.28658 0.308322
\(56\) 1.67992 + 3.24335i 0.224489 + 0.433410i
\(57\) 0 0
\(58\) −1.05664 − 4.19815i −0.138744 − 0.551244i
\(59\) −1.28003 − 4.77715i −0.166646 − 0.621932i −0.997824 − 0.0659263i \(-0.979000\pi\)
0.831178 − 0.556006i \(-0.187667\pi\)
\(60\) 0 0
\(61\) −2.90167 + 10.8292i −0.371520 + 1.38653i 0.486842 + 0.873490i \(0.338149\pi\)
−0.858363 + 0.513043i \(0.828518\pi\)
\(62\) 0.131344 − 8.62382i 0.0166807 − 1.09523i
\(63\) 0 0
\(64\) 7.96663 + 0.729984i 0.995828 + 0.0912480i
\(65\) 2.92795 − 1.69045i 0.363167 − 0.209675i
\(66\) 0 0
\(67\) −0.0295686 − 0.110351i −0.00361238 − 0.0134816i 0.964096 − 0.265554i \(-0.0855547\pi\)
−0.967708 + 0.252072i \(0.918888\pi\)
\(68\) −9.49932 − 5.87724i −1.15196 − 0.712720i
\(69\) 0 0
\(70\) −0.343604 + 1.20843i −0.0410686 + 0.144435i
\(71\) − 0.0447904i − 0.00531565i −0.999996 − 0.00265782i \(-0.999154\pi\)
0.999996 − 0.00265782i \(-0.000846013\pi\)
\(72\) 0 0
\(73\) 13.2931i 1.55585i 0.628360 + 0.777923i \(0.283726\pi\)
−0.628360 + 0.777923i \(0.716274\pi\)
\(74\) −0.976146 − 0.277557i −0.113475 − 0.0322654i
\(75\) 0 0
\(76\) −1.61900 − 6.87302i −0.185712 − 0.788389i
\(77\) −1.11098 − 4.14624i −0.126608 − 0.472507i
\(78\) 0 0
\(79\) 2.50052 − 1.44368i 0.281331 − 0.162426i −0.352695 − 0.935738i \(-0.614735\pi\)
0.634026 + 0.773312i \(0.281401\pi\)
\(80\) 1.82364 + 2.06055i 0.203890 + 0.230377i
\(81\) 0 0
\(82\) 13.8546 + 0.211011i 1.52999 + 0.0233023i
\(83\) −1.01705 + 3.79568i −0.111636 + 0.416630i −0.999013 − 0.0444135i \(-0.985858\pi\)
0.887378 + 0.461043i \(0.152525\pi\)
\(84\) 0 0
\(85\) −0.994419 − 3.71122i −0.107860 − 0.402539i
\(86\) −1.34877 + 0.339473i −0.145441 + 0.0366064i
\(87\) 0 0
\(88\) −8.96058 − 2.84551i −0.955201 − 0.303333i
\(89\) −12.7362 −1.35003 −0.675017 − 0.737802i \(-0.735864\pi\)
−0.675017 + 0.737802i \(0.735864\pi\)
\(90\) 0 0
\(91\) −4.48788 − 4.48788i −0.470458 − 0.470458i
\(92\) −4.81156 + 2.58587i −0.501639 + 0.269595i
\(93\) 0 0
\(94\) −8.90802 + 14.9004i −0.918792 + 1.53686i
\(95\) 1.21436 − 2.10333i 0.124590 − 0.215797i
\(96\) 0 0
\(97\) 4.41066 + 7.63949i 0.447835 + 0.775673i 0.998245 − 0.0592215i \(-0.0188618\pi\)
−0.550410 + 0.834895i \(0.685529\pi\)
\(98\) −7.54017 − 0.114840i −0.761672 − 0.0116005i
\(99\) 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.v.a.179.8 88
3.2 odd 2 144.2.u.a.131.15 yes 88
4.3 odd 2 1728.2.z.a.1583.10 88
9.2 odd 6 inner 432.2.v.a.35.15 88
9.7 even 3 144.2.u.a.83.8 yes 88
12.11 even 2 576.2.y.a.239.7 88
16.5 even 4 1728.2.z.a.719.10 88
16.11 odd 4 inner 432.2.v.a.395.15 88
36.7 odd 6 576.2.y.a.47.5 88
36.11 even 6 1728.2.z.a.1007.10 88
48.5 odd 4 576.2.y.a.527.5 88
48.11 even 4 144.2.u.a.59.8 yes 88
144.11 even 12 inner 432.2.v.a.251.8 88
144.43 odd 12 144.2.u.a.11.15 ✓ 88
144.101 odd 12 1728.2.z.a.143.10 88
144.133 even 12 576.2.y.a.335.7 88
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.u.a.11.15 ✓ 88 144.43 odd 12
144.2.u.a.59.8 yes 88 48.11 even 4
144.2.u.a.83.8 yes 88 9.7 even 3
144.2.u.a.131.15 yes 88 3.2 odd 2
432.2.v.a.35.15 88 9.2 odd 6 inner
432.2.v.a.179.8 88 1.1 even 1 trivial
432.2.v.a.251.8 88 144.11 even 12 inner
432.2.v.a.395.15 88 16.11 odd 4 inner
576.2.y.a.47.5 88 36.7 odd 6
576.2.y.a.239.7 88 12.11 even 2
576.2.y.a.335.7 88 144.133 even 12
576.2.y.a.527.5 88 48.5 odd 4
1728.2.z.a.143.10 88 144.101 odd 12
1728.2.z.a.719.10 88 16.5 even 4
1728.2.z.a.1007.10 88 36.11 even 6
1728.2.z.a.1583.10 88 4.3 odd 2