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Properties

Label 288.2.r.b.49.4
Level $288$
Weight $2$
Character 288.49
Analytic conductor $2.300$
Analytic rank $0$
Dimension $16$
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] = [288,2,Mod(49,288)] 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("288.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 288.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] 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: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 49.4
Root \(1.41411 + 0.0174668i\) of defining polynomial
Character \(\chi\) \(=\) 288.49
Dual form 288.2.r.b.241.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.294546 + 1.70682i) q^{3} +(3.17262 + 1.83171i) q^{5} +(0.191926 + 0.332426i) q^{7} +(-2.82649 - 1.00547i) q^{9} +(1.73849 - 1.00372i) q^{11} +(-0.397799 - 0.229669i) q^{13} +(-4.06089 + 4.87557i) q^{15} -4.08495 q^{17} +4.72398i q^{19} +(-0.623923 + 0.229669i) q^{21} +(2.97594 - 5.15447i) q^{23} +(4.21034 + 7.29252i) q^{25} +(2.54870 - 4.52815i) q^{27} +(-2.03783 + 1.17654i) q^{29} +(-0.592083 + 1.02552i) q^{31} +(1.20110 + 3.26293i) q^{33} +1.40621i q^{35} +5.74432i q^{37} +(0.509175 - 0.611324i) q^{39} +(4.75281 - 8.23212i) q^{41} +(-1.03633 + 0.598327i) q^{43} +(-7.12562 - 8.36729i) q^{45} +(-3.27688 - 5.67572i) q^{47} +(3.42633 - 5.93458i) q^{49} +(1.20321 - 6.97229i) q^{51} -7.63807i q^{53} +7.35407 q^{55} +(-8.06300 - 1.39143i) q^{57} +(-0.603703 - 0.348548i) q^{59} +(4.23774 - 2.44666i) q^{61} +(-0.208231 - 1.13257i) q^{63} +(-0.841376 - 1.45731i) q^{65} +(-8.87932 - 5.12648i) q^{67} +(7.92122 + 6.59762i) q^{69} +3.73792 q^{71} -2.68275 q^{73} +(-13.6872 + 5.03832i) q^{75} +(0.667322 + 0.385279i) q^{77} +(5.35979 + 9.28342i) q^{79} +(6.97804 + 5.68392i) q^{81} +(5.49039 - 3.16988i) q^{83} +(-12.9600 - 7.48246i) q^{85} +(-1.40792 - 3.82477i) q^{87} +7.56802 q^{89} -0.176318i q^{91} +(-1.57598 - 1.31264i) q^{93} +(-8.65297 + 14.9874i) q^{95} +(-2.98511 - 5.17036i) q^{97} +(-5.92302 + 1.08898i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{7} + 2 q^{9} + 10 q^{15} - 28 q^{17} + 10 q^{23} + 2 q^{25} + 10 q^{31} - 2 q^{39} - 8 q^{41} - 6 q^{47} + 18 q^{49} + 4 q^{55} + 10 q^{57} - 2 q^{63} - 14 q^{65} - 72 q^{71} - 44 q^{73} + 30 q^{79}+ \cdots - 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(3\) −0.294546 + 1.70682i −0.170056 + 0.985434i
\(4\) 0 0
\(5\) 3.17262 + 1.83171i 1.41884 + 0.819167i 0.996197 − 0.0871306i \(-0.0277697\pi\)
0.422641 + 0.906297i \(0.361103\pi\)
\(6\) 0 0
\(7\) 0.191926 + 0.332426i 0.0725413 + 0.125645i 0.900014 − 0.435860i \(-0.143556\pi\)
−0.827473 + 0.561505i \(0.810222\pi\)
\(8\) 0 0
\(9\) −2.82649 − 1.00547i −0.942162 − 0.335158i
\(10\) 0 0
\(11\) 1.73849 − 1.00372i 0.524173 − 0.302632i −0.214467 − 0.976731i \(-0.568801\pi\)
0.738640 + 0.674100i \(0.235468\pi\)
\(12\) 0 0
\(13\) −0.397799 − 0.229669i −0.110330 − 0.0636988i 0.443820 − 0.896116i \(-0.353623\pi\)
−0.554149 + 0.832417i \(0.686956\pi\)
\(14\) 0 0
\(15\) −4.06089 + 4.87557i −1.04852 + 1.25887i
\(16\) 0 0
\(17\) −4.08495 −0.990747 −0.495373 − 0.868680i \(-0.664969\pi\)
−0.495373 + 0.868680i \(0.664969\pi\)
\(18\) 0 0
\(19\) 4.72398i 1.08376i 0.840457 + 0.541878i \(0.182286\pi\)
−0.840457 + 0.541878i \(0.817714\pi\)
\(20\) 0 0
\(21\) −0.623923 + 0.229669i −0.136151 + 0.0501179i
\(22\) 0 0
\(23\) 2.97594 − 5.15447i 0.620525 − 1.07478i −0.368863 − 0.929484i \(-0.620253\pi\)
0.989388 − 0.145298i \(-0.0464140\pi\)
\(24\) 0 0
\(25\) 4.21034 + 7.29252i 0.842068 + 1.45850i
\(26\) 0 0
\(27\) 2.54870 − 4.52815i 0.490497 − 0.871443i
\(28\) 0 0
\(29\) −2.03783 + 1.17654i −0.378416 + 0.218479i −0.677129 − 0.735864i \(-0.736776\pi\)
0.298713 + 0.954343i \(0.403443\pi\)
\(30\) 0 0
\(31\) −0.592083 + 1.02552i −0.106341 + 0.184188i −0.914285 − 0.405071i \(-0.867247\pi\)
0.807944 + 0.589259i \(0.200580\pi\)
\(32\) 0 0
\(33\) 1.20110 + 3.26293i 0.209085 + 0.568003i
\(34\) 0 0
\(35\) 1.40621i 0.237693i
\(36\) 0 0
\(37\) 5.74432i 0.944360i 0.881502 + 0.472180i \(0.156533\pi\)
−0.881502 + 0.472180i \(0.843467\pi\)
\(38\) 0 0
\(39\) 0.509175 − 0.611324i 0.0815332 − 0.0978902i
\(40\) 0 0
\(41\) 4.75281 − 8.23212i 0.742265 − 1.28564i −0.209197 − 0.977874i \(-0.567085\pi\)
0.951462 − 0.307767i \(-0.0995817\pi\)
\(42\) 0 0
\(43\) −1.03633 + 0.598327i −0.158039 + 0.0912440i −0.576934 − 0.816791i \(-0.695751\pi\)
0.418895 + 0.908035i \(0.362418\pi\)
\(44\) 0 0
\(45\) −7.12562 − 8.36729i −1.06222 − 1.24732i
\(46\) 0 0
\(47\) −3.27688 − 5.67572i −0.477982 − 0.827889i 0.521699 − 0.853129i \(-0.325298\pi\)
−0.999681 + 0.0252403i \(0.991965\pi\)
\(48\) 0 0
\(49\) 3.42633 − 5.93458i 0.489476 − 0.847796i
\(50\) 0 0
\(51\) 1.20321 − 6.97229i 0.168482 − 0.976316i
\(52\) 0 0
\(53\) − 7.63807i − 1.04917i −0.851358 − 0.524585i \(-0.824221\pi\)
0.851358 − 0.524585i \(-0.175779\pi\)
\(54\) 0 0
\(55\) 7.35407 0.991623
\(56\) 0 0
\(57\) −8.06300 − 1.39143i −1.06797 − 0.184299i
\(58\) 0 0
\(59\) −0.603703 − 0.348548i −0.0785954 − 0.0453771i 0.460187 − 0.887822i \(-0.347782\pi\)
−0.538783 + 0.842445i \(0.681116\pi\)
\(60\) 0 0
\(61\) 4.23774 − 2.44666i 0.542587 − 0.313263i −0.203540 − 0.979067i \(-0.565245\pi\)
0.746127 + 0.665804i \(0.231911\pi\)
\(62\) 0 0
\(63\) −0.208231 − 1.13257i −0.0262346 − 0.142691i
\(64\) 0 0
\(65\) −0.841376 − 1.45731i −0.104360 − 0.180757i
\(66\) 0 0
\(67\) −8.87932 − 5.12648i −1.08478 − 0.626299i −0.152599 − 0.988288i \(-0.548764\pi\)
−0.932182 + 0.361989i \(0.882098\pi\)
\(68\) 0 0
\(69\) 7.92122 + 6.59762i 0.953603 + 0.794260i
\(70\) 0 0
\(71\) 3.73792 0.443610 0.221805 − 0.975091i \(-0.428805\pi\)
0.221805 + 0.975091i \(0.428805\pi\)
\(72\) 0 0
\(73\) −2.68275 −0.313992 −0.156996 − 0.987599i \(-0.550181\pi\)
−0.156996 + 0.987599i \(0.550181\pi\)
\(74\) 0 0
\(75\) −13.6872 + 5.03832i −1.58046 + 0.581775i
\(76\) 0 0
\(77\) 0.667322 + 0.385279i 0.0760484 + 0.0439066i
\(78\) 0 0
\(79\) 5.35979 + 9.28342i 0.603023 + 1.04447i 0.992361 + 0.123372i \(0.0393707\pi\)
−0.389337 + 0.921095i \(0.627296\pi\)
\(80\) 0 0
\(81\) 6.97804 + 5.68392i 0.775338 + 0.631546i
\(82\) 0 0
\(83\) 5.49039 − 3.16988i 0.602648 − 0.347939i −0.167434 − 0.985883i \(-0.553548\pi\)
0.770083 + 0.637944i \(0.220215\pi\)
\(84\) 0 0
\(85\) −12.9600 − 7.48246i −1.40571 − 0.811586i
\(86\) 0 0
\(87\) −1.40792 − 3.82477i −0.150945 − 0.410058i
\(88\) 0 0
\(89\) 7.56802 0.802208 0.401104 − 0.916032i \(-0.368627\pi\)
0.401104 + 0.916032i \(0.368627\pi\)
\(90\) 0 0
\(91\) − 0.176318i − 0.0184832i
\(92\) 0 0
\(93\) −1.57598 − 1.31264i −0.163422 − 0.136115i
\(94\) 0 0
\(95\) −8.65297 + 14.9874i −0.887776 + 1.53767i
\(96\) 0 0
\(97\) −2.98511 − 5.17036i −0.303092 − 0.524971i 0.673743 − 0.738966i \(-0.264686\pi\)
−0.976835 + 0.213995i \(0.931352\pi\)
\(98\) 0 0
\(99\) −5.92302 + 1.08898i −0.595286 + 0.109447i
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 288.2.r.b.49.4 16
3.2 odd 2 864.2.r.b.145.1 16
4.3 odd 2 72.2.n.b.13.3 ✓ 16
8.3 odd 2 72.2.n.b.13.8 yes 16
8.5 even 2 inner 288.2.r.b.49.5 16
9.2 odd 6 864.2.r.b.721.8 16
9.4 even 3 2592.2.d.j.1297.1 8
9.5 odd 6 2592.2.d.k.1297.8 8
9.7 even 3 inner 288.2.r.b.241.5 16
12.11 even 2 216.2.n.b.37.6 16
24.5 odd 2 864.2.r.b.145.8 16
24.11 even 2 216.2.n.b.37.1 16
36.7 odd 6 72.2.n.b.61.8 yes 16
36.11 even 6 216.2.n.b.181.1 16
36.23 even 6 648.2.d.k.325.6 8
36.31 odd 6 648.2.d.j.325.3 8
72.5 odd 6 2592.2.d.k.1297.1 8
72.11 even 6 216.2.n.b.181.6 16
72.13 even 6 2592.2.d.j.1297.8 8
72.29 odd 6 864.2.r.b.721.1 16
72.43 odd 6 72.2.n.b.61.3 yes 16
72.59 even 6 648.2.d.k.325.5 8
72.61 even 6 inner 288.2.r.b.241.4 16
72.67 odd 6 648.2.d.j.325.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.n.b.13.3 ✓ 16 4.3 odd 2
72.2.n.b.13.8 yes 16 8.3 odd 2
72.2.n.b.61.3 yes 16 72.43 odd 6
72.2.n.b.61.8 yes 16 36.7 odd 6
216.2.n.b.37.1 16 24.11 even 2
216.2.n.b.37.6 16 12.11 even 2
216.2.n.b.181.1 16 36.11 even 6
216.2.n.b.181.6 16 72.11 even 6
288.2.r.b.49.4 16 1.1 even 1 trivial
288.2.r.b.49.5 16 8.5 even 2 inner
288.2.r.b.241.4 16 72.61 even 6 inner
288.2.r.b.241.5 16 9.7 even 3 inner
648.2.d.j.325.3 8 36.31 odd 6
648.2.d.j.325.4 8 72.67 odd 6
648.2.d.k.325.5 8 72.59 even 6
648.2.d.k.325.6 8 36.23 even 6
864.2.r.b.145.1 16 3.2 odd 2
864.2.r.b.145.8 16 24.5 odd 2
864.2.r.b.721.1 16 72.29 odd 6
864.2.r.b.721.8 16 9.2 odd 6
2592.2.d.j.1297.1 8 9.4 even 3
2592.2.d.j.1297.8 8 72.13 even 6
2592.2.d.k.1297.1 8 72.5 odd 6
2592.2.d.k.1297.8 8 9.5 odd 6