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Properties

Label 144.9.q.a.113.2
Level $144$
Weight $9$
Character 144.113
Analytic conductor $58.663$
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] = [144,9,Mod(65,144)] 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("144.65"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 144.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,93] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(58.6625198488\)
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: \( x^{14} - x^{13} + 427 x^{12} - 1362 x^{11} + 135762 x^{10} - 371244 x^{9} + 18261508 x^{8} + \cdots + 872385888256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{21} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 113.2
Root \(4.05115 - 7.01679i\) of defining polynomial
Character \(\chi\) \(=\) 144.113
Dual form 144.9.q.a.65.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-54.3339 + 60.0735i) q^{3} +(676.216 + 390.413i) q^{5} +(-2168.61 - 3756.14i) q^{7} +(-656.650 - 6528.06i) q^{9} +(-5607.04 + 3237.22i) q^{11} +(-8497.75 + 14718.5i) q^{13} +(-60194.9 + 19410.0i) q^{15} -29881.7i q^{17} +91768.1 q^{19} +(343473. + 73809.9i) q^{21} +(106388. + 61423.0i) q^{23} +(109532. + 189716. i) q^{25} +(427842. + 315248. i) q^{27} +(434568. - 250898. i) q^{29} +(-505931. + 876298. i) q^{31} +(110181. - 512725. i) q^{33} -3.38661e6i q^{35} +190279. q^{37} +(-422478. - 1.31021e6i) q^{39} +(-1.95619e6 - 1.12941e6i) q^{41} +(1.64497e6 + 2.84917e6i) q^{43} +(2.10460e6 - 4.67074e6i) q^{45} +(110171. - 63607.2i) q^{47} +(-6.52331e6 + 1.12987e7i) q^{49} +(1.79510e6 + 1.62359e6i) q^{51} -1.40856e6i q^{53} -5.05542e6 q^{55} +(-4.98612e6 + 5.51283e6i) q^{57} +(1.79370e7 + 1.03560e7i) q^{59} +(9.62157e6 + 1.66651e7i) q^{61} +(-2.30963e7 + 1.66233e7i) q^{63} +(-1.14926e7 + 6.63527e6i) q^{65} +(-1.17156e7 + 2.02920e7i) q^{67} +(-9.47036e6 + 3.05373e6i) q^{69} +9.12361e6i q^{71} +1.43931e7 q^{73} +(-1.73482e7 - 3.72800e6i) q^{75} +(2.43189e7 + 1.40405e7i) q^{77} +(-2.69804e7 - 4.67314e7i) q^{79} +(-4.21843e7 + 8.57330e6i) q^{81} +(-2.66155e7 + 1.53664e7i) q^{83} +(1.16662e7 - 2.02064e7i) q^{85} +(-8.53946e6 + 3.97382e7i) q^{87} +2.32435e7i q^{89} +7.37131e7 q^{91} +(-2.51531e7 - 7.80057e7i) q^{93} +(6.20550e7 + 3.58275e7i) q^{95} +(2.46595e7 + 4.27114e7i) q^{97} +(2.48147e7 + 3.44773e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 93 q^{3} + 438 q^{5} - 922 q^{7} + 17973 q^{9} + 28677 q^{11} + 1684 q^{13} + 75276 q^{15} + 269630 q^{19} + 354054 q^{21} + 1000452 q^{23} + 65177 q^{25} + 524826 q^{27} + 3797682 q^{29} + 164132 q^{31}+ \cdots + 511060752 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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
\(3\) −54.3339 + 60.0735i −0.670789 + 0.741648i
\(4\) 0 0
\(5\) 676.216 + 390.413i 1.08194 + 0.624661i 0.931420 − 0.363945i \(-0.118570\pi\)
0.150524 + 0.988606i \(0.451904\pi\)
\(6\) 0 0
\(7\) −2168.61 − 3756.14i −0.903210 − 1.56441i −0.823302 − 0.567603i \(-0.807871\pi\)
−0.0799078 − 0.996802i \(-0.525463\pi\)
\(8\) 0 0
\(9\) −656.650 − 6528.06i −0.100084 − 0.994979i
\(10\) 0 0
\(11\) −5607.04 + 3237.22i −0.382968 + 0.221107i −0.679109 − 0.734038i \(-0.737634\pi\)
0.296141 + 0.955144i \(0.404300\pi\)
\(12\) 0 0
\(13\) −8497.75 + 14718.5i −0.297530 + 0.515337i −0.975570 − 0.219688i \(-0.929496\pi\)
0.678040 + 0.735025i \(0.262829\pi\)
\(14\) 0 0
\(15\) −60194.9 + 19410.0i −1.18904 + 0.383406i
\(16\) 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
\(21\) 343473. + 73809.9i 1.76610 + 0.379522i
\(22\) 0 0
\(23\) 106388. + 61423.0i 0.380172 + 0.219492i 0.677893 − 0.735160i \(-0.262893\pi\)
−0.297721 + 0.954653i \(0.596227\pi\)
\(24\) 0 0
\(25\) 109532. + 189716.i 0.280403 + 0.485672i
\(26\) 0 0
\(27\) 427842. + 315248.i 0.805059 + 0.593194i
\(28\) 0 0
\(29\) 434568. − 250898.i 0.614420 − 0.354736i −0.160273 − 0.987073i \(-0.551238\pi\)
0.774693 + 0.632337i \(0.217904\pi\)
\(30\) 0 0
\(31\) −505931. + 876298.i −0.547828 + 0.948866i 0.450595 + 0.892729i \(0.351212\pi\)
−0.998423 + 0.0561377i \(0.982121\pi\)
\(32\) 0 0
\(33\) 110181. − 512725.i 0.0929075 − 0.432344i
\(34\) 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
\(39\) −422478. − 1.31021e6i −0.182619 − 0.566345i
\(40\) 0 0
\(41\) −1.95619e6 − 1.12941e6i −0.692269 − 0.399682i 0.112192 − 0.993687i \(-0.464213\pi\)
−0.804462 + 0.594005i \(0.797546\pi\)
\(42\) 0 0
\(43\) 1.64497e6 + 2.84917e6i 0.481154 + 0.833383i 0.999766 − 0.0216266i \(-0.00688451\pi\)
−0.518612 + 0.855010i \(0.673551\pi\)
\(44\) 0 0
\(45\) 2.10460e6 − 4.67074e6i 0.513240 − 1.13903i
\(46\) 0 0
\(47\) 110171. − 63607.2i 0.0225775 − 0.0130351i −0.488669 − 0.872469i \(-0.662517\pi\)
0.511246 + 0.859434i \(0.329184\pi\)
\(48\) 0 0
\(49\) −6.52331e6 + 1.12987e7i −1.13158 + 1.95995i
\(50\) 0 0
\(51\) 1.79510e6 + 1.62359e6i 0.265343 + 0.239991i
\(52\) 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
\(57\) −4.98612e6 + 5.51283e6i −0.472349 + 0.522246i
\(58\) 0 0
\(59\) 1.79370e7 + 1.03560e7i 1.48028 + 0.854638i 0.999750 − 0.0223421i \(-0.00711229\pi\)
0.480526 + 0.876980i \(0.340446\pi\)
\(60\) 0 0
\(61\) 9.62157e6 + 1.66651e7i 0.694907 + 1.20361i 0.970212 + 0.242258i \(0.0778879\pi\)
−0.275305 + 0.961357i \(0.588779\pi\)
\(62\) 0 0
\(63\) −2.30963e7 + 1.66233e7i −1.46615 + 1.05525i
\(64\) 0 0
\(65\) −1.14926e7 + 6.63527e6i −0.643822 + 0.371711i
\(66\) 0 0
\(67\) −1.17156e7 + 2.02920e7i −0.581386 + 1.00699i 0.413929 + 0.910309i \(0.364156\pi\)
−0.995315 + 0.0966815i \(0.969177\pi\)
\(68\) 0 0
\(69\) −9.47036e6 + 3.05373e6i −0.417801 + 0.134721i
\(70\) 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
\(75\) −1.73482e7 − 3.72800e6i −0.548289 − 0.117823i
\(76\) 0 0
\(77\) 2.43189e7 + 1.40405e7i 0.691801 + 0.399412i
\(78\) 0 0
\(79\) −2.69804e7 − 4.67314e7i −0.692691 − 1.19978i −0.970953 − 0.239271i \(-0.923092\pi\)
0.278262 − 0.960505i \(-0.410242\pi\)
\(80\) 0 0
\(81\) −4.21843e7 + 8.57330e6i −0.979966 + 0.199163i
\(82\) 0 0
\(83\) −2.66155e7 + 1.53664e7i −0.560818 + 0.323788i −0.753474 − 0.657478i \(-0.771623\pi\)
0.192656 + 0.981266i \(0.438290\pi\)
\(84\) 0 0
\(85\) 1.16662e7 − 2.02064e7i 0.223488 − 0.387092i
\(86\) 0 0
\(87\) −8.53946e6 + 3.97382e7i −0.149057 + 0.693636i
\(88\) 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
\(93\) −2.51531e7 − 7.80057e7i −0.336248 − 1.04278i
\(94\) 0 0
\(95\) 6.20550e7 + 3.58275e7i 0.761873 + 0.439867i
\(96\) 0 0
\(97\) 2.46595e7 + 4.27114e7i 0.278546 + 0.482455i 0.971024 − 0.238984i \(-0.0768143\pi\)
−0.692478 + 0.721439i \(0.743481\pi\)
\(98\) 0 0
\(99\) 2.48147e7 + 3.44773e7i 0.258326 + 0.358916i
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 144.9.q.a.113.2 14
3.2 odd 2 432.9.q.a.17.1 14
4.3 odd 2 9.9.d.a.5.3 yes 14
9.2 odd 6 inner 144.9.q.a.65.2 14
9.7 even 3 432.9.q.a.305.1 14
12.11 even 2 27.9.d.a.17.5 14
36.7 odd 6 27.9.d.a.8.5 14
36.11 even 6 9.9.d.a.2.3 ✓ 14
36.23 even 6 81.9.b.a.80.10 14
36.31 odd 6 81.9.b.a.80.5 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.9.d.a.2.3 ✓ 14 36.11 even 6
9.9.d.a.5.3 yes 14 4.3 odd 2
27.9.d.a.8.5 14 36.7 odd 6
27.9.d.a.17.5 14 12.11 even 2
81.9.b.a.80.5 14 36.31 odd 6
81.9.b.a.80.10 14 36.23 even 6
144.9.q.a.65.2 14 9.2 odd 6 inner
144.9.q.a.113.2 14 1.1 even 1 trivial
432.9.q.a.17.1 14 3.2 odd 2
432.9.q.a.305.1 14 9.7 even 3