Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [207,2,Mod(68,207)] 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("207.68"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(207, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 207.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] 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: \(1.65290332184\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.57352136505929721.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3x^{9} + x^{6} - 24x^{3} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 137.1
Root \(-1.32313 - 0.499333i\) of defining polynomial
Character \(\chi\) \(=\) 207.137
Dual form 207.2.g.a.68.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.41713 - 1.39553i) q^{2} +(-1.73046 - 0.0741663i) q^{3} +(2.89500 + 5.01429i) q^{4} +(4.07924 + 2.59418i) q^{6} -10.5781i q^{8} +(2.98900 + 0.256684i) q^{9} +(-4.63780 - 8.89175i) q^{12} +(-2.18872 - 3.79097i) q^{13} +(-8.97205 + 15.5400i) q^{16} +(-6.86658 - 4.79167i) q^{18} +(-4.15331 + 2.39792i) q^{23} +(-0.784539 + 18.3050i) q^{24} +(2.50000 - 4.33013i) q^{25} +12.2177i q^{26} +(-5.15331 - 0.665865i) q^{27} +(-9.32413 - 5.38329i) q^{29} +(-4.36305 - 7.55702i) q^{31} +(25.0513 - 14.4634i) q^{32} +(7.36606 + 15.7308i) q^{36} +(3.50633 + 6.72247i) q^{39} +(11.0843 - 6.39951i) q^{41} +13.3854 q^{46} +(-1.77074 - 1.02234i) q^{47} +(16.6783 - 26.2260i) q^{48} +(-3.50000 - 6.06218i) q^{49} +(-12.0856 + 6.97764i) q^{50} +(12.6727 - 21.9497i) q^{52} +(11.5270 + 8.80107i) q^{54} +(15.0251 + 26.0242i) q^{58} +(-4.84669 + 2.79824i) q^{59} +24.3550i q^{62} -44.8481 q^{64} +(7.36499 - 3.84147i) q^{69} +4.77458i q^{71} +(2.71523 - 31.6179i) q^{72} +4.00210 q^{73} +(-4.64731 + 7.30771i) q^{75} +(0.906141 - 21.1422i) q^{78} +(8.86823 + 1.53446i) q^{81} -35.7228 q^{82} +(15.7358 + 10.0071i) q^{87} +(-24.0477 - 13.8839i) q^{92} +(6.98962 + 13.4007i) q^{93} +(2.85341 + 4.94225i) q^{94} +(-44.4231 + 23.1704i) q^{96} +19.5374i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{4} - 3 q^{6} - 30 q^{12} - 24 q^{16} - 21 q^{18} - 12 q^{24} + 30 q^{25} - 12 q^{27} + 54 q^{32} + 33 q^{36} + 24 q^{39} + 48 q^{48} - 42 q^{49} - 3 q^{52} + 15 q^{58} - 108 q^{59} - 54 q^{64}+ \cdots - 69 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41713 − 1.39553i −1.70917 − 0.986788i −0.935593 − 0.353082i \(-0.885134\pi\)
−0.773574 − 0.633706i \(-0.781533\pi\)
\(3\) −1.73046 − 0.0741663i −0.999083 − 0.0428199i
\(4\) 2.89500 + 5.01429i 1.44750 + 2.50714i
\(5\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 4.07924 + 2.59418i 1.66534 + 1.05907i
\(7\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) − 10.5781i − 3.73993i
\(9\) 2.98900 + 0.256684i 0.996333 + 0.0855613i
\(10\) 0 0
\(11\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) −4.63780 − 8.89175i −1.33882 − 2.56683i
\(13\) −2.18872 − 3.79097i −0.607042 − 1.05143i −0.991725 − 0.128379i \(-0.959023\pi\)
0.384684 − 0.923049i \(-0.374311\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −8.97205 + 15.5400i −2.24301 + 3.88501i
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −6.86658 − 4.79167i −1.61847 − 1.12941i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.15331 + 2.39792i −0.866025 + 0.500000i
\(24\) −0.784539 + 18.3050i −0.160143 + 3.73650i
\(25\) 2.50000 − 4.33013i 0.500000 − 0.866025i
\(26\) 12.2177i 2.39609i
\(27\) −5.15331 − 0.665865i −0.991755 − 0.128146i
\(28\) 0 0
\(29\) −9.32413 − 5.38329i −1.73145 − 0.999652i −0.878920 − 0.476969i \(-0.841735\pi\)
−0.852527 − 0.522682i \(-0.824931\pi\)
\(30\) 0 0
\(31\) −4.36305 − 7.55702i −0.783627 − 1.35728i −0.929816 − 0.368025i \(-0.880034\pi\)
0.146189 − 0.989257i \(-0.453299\pi\)
\(32\) 25.0513 − 14.4634i 4.42849 − 2.55679i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 7.36606 + 15.7308i 1.22768 + 2.62180i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 3.50633 + 6.72247i 0.561463 + 1.07646i
\(40\) 0 0
\(41\) 11.0843 − 6.39951i 1.73107 − 0.999436i 0.848748 − 0.528798i \(-0.177357\pi\)
0.882326 − 0.470638i \(-0.155976\pi\)
\(42\) 0 0
\(43\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 13.3854 1.97358
\(47\) −1.77074 − 1.02234i −0.258290 − 0.149124i 0.365265 − 0.930904i \(-0.380979\pi\)
−0.623554 + 0.781780i \(0.714312\pi\)
\(48\) 16.6783 − 26.2260i 2.40731 − 3.78540i
\(49\) −3.50000 − 6.06218i −0.500000 − 0.866025i
\(50\) −12.0856 + 6.97764i −1.70917 + 0.986788i
\(51\) 0 0
\(52\) 12.6727 − 21.9497i 1.75739 − 3.04388i
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 11.5270 + 8.80107i 1.56862 + 1.19767i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 15.0251 + 26.0242i 1.97289 + 3.41714i
\(59\) −4.84669 + 2.79824i −0.630985 + 0.364299i −0.781133 − 0.624364i \(-0.785358\pi\)
0.150148 + 0.988663i \(0.452025\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 24.3550i 3.09309i
\(63\) 0 0
\(64\) −44.8481 −5.60602
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 0 0
\(69\) 7.36499 − 3.84147i 0.886641 − 0.462458i
\(70\) 0 0
\(71\) 4.77458i 0.566639i 0.959026 + 0.283319i \(0.0914356\pi\)
−0.959026 + 0.283319i \(0.908564\pi\)
\(72\) 2.71523 − 31.6179i 0.319993 − 3.72621i
\(73\) 4.00210 0.468410 0.234205 − 0.972187i \(-0.424751\pi\)
0.234205 + 0.972187i \(0.424751\pi\)
\(74\) 0 0
\(75\) −4.64731 + 7.30771i −0.536625 + 0.843821i
\(76\) 0 0
\(77\) 0 0
\(78\) 0.906141 − 21.1422i 0.102600 − 2.39389i
\(79\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) 8.86823 + 1.53446i 0.985359 + 0.170495i
\(82\) −35.7228 −3.94493
\(83\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 15.7358 + 10.0071i 1.68705 + 1.07288i
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −24.0477 − 13.8839i −2.50714 − 1.44750i
\(93\) 6.98962 + 13.4007i 0.724789 + 1.38959i
\(94\) 2.85341 + 4.94225i 0.294307 + 0.509754i
\(95\) 0 0
\(96\) −44.4231 + 23.1704i −4.53391 + 2.36482i
\(97\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 19.5374i 1.97358i
\(99\) 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 207.2.g.a.137.1 yes 12
3.2 odd 2 621.2.g.a.413.6 12
9.2 odd 6 1863.2.c.a.1862.1 12
9.4 even 3 621.2.g.a.206.6 12
9.5 odd 6 inner 207.2.g.a.68.1 ✓ 12
9.7 even 3 1863.2.c.a.1862.12 12
23.22 odd 2 CM 207.2.g.a.137.1 yes 12
69.68 even 2 621.2.g.a.413.6 12
207.22 odd 6 621.2.g.a.206.6 12
207.68 even 6 inner 207.2.g.a.68.1 ✓ 12
207.137 even 6 1863.2.c.a.1862.1 12
207.160 odd 6 1863.2.c.a.1862.12 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
207.2.g.a.68.1 ✓ 12 9.5 odd 6 inner
207.2.g.a.68.1 ✓ 12 207.68 even 6 inner
207.2.g.a.137.1 yes 12 1.1 even 1 trivial
207.2.g.a.137.1 yes 12 23.22 odd 2 CM
621.2.g.a.206.6 12 9.4 even 3
621.2.g.a.206.6 12 207.22 odd 6
621.2.g.a.413.6 12 3.2 odd 2
621.2.g.a.413.6 12 69.68 even 2
1863.2.c.a.1862.1 12 9.2 odd 6
1863.2.c.a.1862.1 12 207.137 even 6
1863.2.c.a.1862.12 12 9.7 even 3
1863.2.c.a.1862.12 12 207.160 odd 6