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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [672,2,Mod(17,672)] 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("672.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.bi (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0] 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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 17.8
Character \(\chi\) \(=\) 672.17
Dual form 672.2.bi.c.593.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.618109 + 1.61801i) q^{3} +(-2.46958 + 1.42581i) q^{5} +(1.02032 + 2.44110i) q^{7} +(-2.23588 - 2.00021i) q^{9} +(-2.42621 + 4.20231i) q^{11} +2.75221 q^{13} +(-0.780502 - 4.87710i) q^{15} +(1.75366 - 3.03743i) q^{17} +(-3.14493 - 5.44717i) q^{19} +(-4.58037 + 0.142013i) q^{21} +(-3.15865 + 1.82365i) q^{23} +(1.56588 - 2.71219i) q^{25} +(4.61837 - 2.38132i) q^{27} -3.90427 q^{29} +(0.858051 + 0.495396i) q^{31} +(-5.29970 - 6.52310i) q^{33} +(-6.00030 - 4.57370i) q^{35} +(1.06516 - 0.614970i) q^{37} +(-1.70117 + 4.45309i) q^{39} -2.10659 q^{41} +5.11768i q^{43} +(8.37361 + 1.75172i) q^{45} +(-5.61268 - 9.72145i) q^{47} +(-4.91791 + 4.98138i) q^{49} +(3.83063 + 4.71490i) q^{51} +(1.00417 - 1.73927i) q^{53} -13.8373i q^{55} +(10.7575 - 1.72156i) q^{57} +(-0.890996 - 0.514417i) q^{59} +(1.24347 + 2.15376i) q^{61} +(2.60139 - 7.49885i) q^{63} +(-6.79681 + 3.92414i) q^{65} +(-5.02777 - 2.90279i) q^{67} +(-0.998281 - 6.23793i) q^{69} +9.75277i q^{71} +(0.291019 + 0.168020i) q^{73} +(3.42045 + 4.21004i) q^{75} +(-12.7338 - 1.63491i) q^{77} +(2.80082 + 4.85116i) q^{79} +(0.998337 + 8.94446i) q^{81} +0.138115i q^{83} +10.0016i q^{85} +(2.41327 - 6.31714i) q^{87} +(0.580993 + 1.00631i) q^{89} +(2.80813 + 6.71841i) q^{91} +(-1.33192 + 1.08212i) q^{93} +(15.5333 + 8.96815i) q^{95} +11.0953i q^{97} +(13.8302 - 4.54296i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 4 q^{7} - 14 q^{9} - 4 q^{15} - 8 q^{25} - 48 q^{31} - 42 q^{33} + 8 q^{39} - 36 q^{49} + 4 q^{57} + 6 q^{63} - 36 q^{73} + 56 q^{79} + 42 q^{81} + 132 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.618109 + 1.61801i −0.356866 + 0.934156i
\(4\) 0 0
\(5\) −2.46958 + 1.42581i −1.10443 + 0.637643i −0.937381 − 0.348306i \(-0.886757\pi\)
−0.167049 + 0.985949i \(0.553424\pi\)
\(6\) 0 0
\(7\) 1.02032 + 2.44110i 0.385643 + 0.922648i
\(8\) 0 0
\(9\) −2.23588 − 2.00021i −0.745294 − 0.666736i
\(10\) 0 0
\(11\) −2.42621 + 4.20231i −0.731529 + 1.26705i 0.224701 + 0.974428i \(0.427860\pi\)
−0.956230 + 0.292617i \(0.905474\pi\)
\(12\) 0 0
\(13\) 2.75221 0.763326 0.381663 − 0.924302i \(-0.375351\pi\)
0.381663 + 0.924302i \(0.375351\pi\)
\(14\) 0 0
\(15\) −0.780502 − 4.87710i −0.201525 − 1.25926i
\(16\) 0 0
\(17\) 1.75366 − 3.03743i 0.425326 − 0.736686i −0.571125 − 0.820863i \(-0.693493\pi\)
0.996451 + 0.0841773i \(0.0268262\pi\)
\(18\) 0 0
\(19\) −3.14493 − 5.44717i −0.721496 − 1.24967i −0.960400 − 0.278624i \(-0.910122\pi\)
0.238905 − 0.971043i \(-0.423212\pi\)
\(20\) 0 0
\(21\) −4.58037 + 0.142013i −0.999520 + 0.0309899i
\(22\) 0 0
\(23\) −3.15865 + 1.82365i −0.658624 + 0.380257i −0.791753 − 0.610842i \(-0.790831\pi\)
0.133128 + 0.991099i \(0.457498\pi\)
\(24\) 0 0
\(25\) 1.56588 − 2.71219i 0.313177 − 0.542438i
\(26\) 0 0
\(27\) 4.61837 − 2.38132i 0.888805 − 0.458286i
\(28\) 0 0
\(29\) −3.90427 −0.725005 −0.362503 − 0.931983i \(-0.618078\pi\)
−0.362503 + 0.931983i \(0.618078\pi\)
\(30\) 0 0
\(31\) 0.858051 + 0.495396i 0.154111 + 0.0889758i 0.575072 − 0.818103i \(-0.304974\pi\)
−0.420962 + 0.907078i \(0.638307\pi\)
\(32\) 0 0
\(33\) −5.29970 − 6.52310i −0.922560 − 1.13553i
\(34\) 0 0
\(35\) −6.00030 − 4.57370i −1.01424 − 0.773097i
\(36\) 0 0
\(37\) 1.06516 − 0.614970i 0.175111 − 0.101100i −0.409883 − 0.912138i \(-0.634430\pi\)
0.584994 + 0.811038i \(0.301097\pi\)
\(38\) 0 0
\(39\) −1.70117 + 4.45309i −0.272405 + 0.713065i
\(40\) 0 0
\(41\) −2.10659 −0.328995 −0.164497 − 0.986378i \(-0.552600\pi\)
−0.164497 + 0.986378i \(0.552600\pi\)
\(42\) 0 0
\(43\) 5.11768i 0.780439i 0.920722 + 0.390220i \(0.127601\pi\)
−0.920722 + 0.390220i \(0.872399\pi\)
\(44\) 0 0
\(45\) 8.37361 + 1.75172i 1.24826 + 0.261132i
\(46\) 0 0
\(47\) −5.61268 − 9.72145i −0.818693 − 1.41802i −0.906645 − 0.421894i \(-0.861365\pi\)
0.0879518 − 0.996125i \(-0.471968\pi\)
\(48\) 0 0
\(49\) −4.91791 + 4.98138i −0.702558 + 0.711626i
\(50\) 0 0
\(51\) 3.83063 + 4.71490i 0.536395 + 0.660218i
\(52\) 0 0
\(53\) 1.00417 − 1.73927i 0.137933 − 0.238907i −0.788781 − 0.614674i \(-0.789288\pi\)
0.926714 + 0.375767i \(0.122621\pi\)
\(54\) 0 0
\(55\) − 13.8373i − 1.86582i
\(56\) 0 0
\(57\) 10.7575 − 1.72156i 1.42486 − 0.228026i
\(58\) 0 0
\(59\) −0.890996 − 0.514417i −0.115998 − 0.0669714i 0.440879 − 0.897567i \(-0.354667\pi\)
−0.556876 + 0.830595i \(0.688000\pi\)
\(60\) 0 0
\(61\) 1.24347 + 2.15376i 0.159211 + 0.275761i 0.934584 − 0.355742i \(-0.115772\pi\)
−0.775374 + 0.631503i \(0.782438\pi\)
\(62\) 0 0
\(63\) 2.60139 − 7.49885i 0.327745 − 0.944766i
\(64\) 0 0
\(65\) −6.79681 + 3.92414i −0.843040 + 0.486729i
\(66\) 0 0
\(67\) −5.02777 − 2.90279i −0.614240 − 0.354632i 0.160383 − 0.987055i \(-0.448727\pi\)
−0.774623 + 0.632423i \(0.782060\pi\)
\(68\) 0 0
\(69\) −0.998281 − 6.23793i −0.120179 − 0.750958i
\(70\) 0 0
\(71\) 9.75277i 1.15744i 0.815526 + 0.578720i \(0.196448\pi\)
−0.815526 + 0.578720i \(0.803552\pi\)
\(72\) 0 0
\(73\) 0.291019 + 0.168020i 0.0340612 + 0.0196652i 0.516934 − 0.856025i \(-0.327073\pi\)
−0.482873 + 0.875691i \(0.660407\pi\)
\(74\) 0 0
\(75\) 3.42045 + 4.21004i 0.394960 + 0.486133i
\(76\) 0 0
\(77\) −12.7338 − 1.63491i −1.45115 − 0.186316i
\(78\) 0 0
\(79\) 2.80082 + 4.85116i 0.315117 + 0.545798i 0.979462 − 0.201627i \(-0.0646230\pi\)
−0.664346 + 0.747426i \(0.731290\pi\)
\(80\) 0 0
\(81\) 0.998337 + 8.94446i 0.110926 + 0.993829i
\(82\) 0 0
\(83\) 0.138115i 0.0151600i 0.999971 + 0.00758002i \(0.00241282\pi\)
−0.999971 + 0.00758002i \(0.997587\pi\)
\(84\) 0 0
\(85\) 10.0016i 1.08482i
\(86\) 0 0
\(87\) 2.41327 − 6.31714i 0.258729 − 0.677268i
\(88\) 0 0
\(89\) 0.580993 + 1.00631i 0.0615852 + 0.106669i 0.895174 − 0.445717i \(-0.147051\pi\)
−0.833589 + 0.552385i \(0.813718\pi\)
\(90\) 0 0
\(91\) 2.80813 + 6.71841i 0.294372 + 0.704281i
\(92\) 0 0
\(93\) −1.33192 + 1.08212i −0.138114 + 0.112211i
\(94\) 0 0
\(95\) 15.5333 + 8.96815i 1.59368 + 0.920113i
\(96\) 0 0
\(97\) 11.0953i 1.12656i 0.826266 + 0.563280i \(0.190461\pi\)
−0.826266 + 0.563280i \(0.809539\pi\)
\(98\) 0 0
\(99\) 13.8302 − 4.54296i 1.38999 − 0.456585i
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 672.2.bi.c.17.8 48
3.2 odd 2 inner 672.2.bi.c.17.7 48
4.3 odd 2 168.2.ba.c.101.14 yes 48
7.5 odd 6 inner 672.2.bi.c.593.18 48
8.3 odd 2 168.2.ba.c.101.21 yes 48
8.5 even 2 inner 672.2.bi.c.17.17 48
12.11 even 2 168.2.ba.c.101.11 yes 48
21.5 even 6 inner 672.2.bi.c.593.17 48
24.5 odd 2 inner 672.2.bi.c.17.18 48
24.11 even 2 168.2.ba.c.101.4 yes 48
28.19 even 6 168.2.ba.c.5.4 ✓ 48
56.5 odd 6 inner 672.2.bi.c.593.7 48
56.19 even 6 168.2.ba.c.5.11 yes 48
84.47 odd 6 168.2.ba.c.5.21 yes 48
168.5 even 6 inner 672.2.bi.c.593.8 48
168.131 odd 6 168.2.ba.c.5.14 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.ba.c.5.4 ✓ 48 28.19 even 6
168.2.ba.c.5.11 yes 48 56.19 even 6
168.2.ba.c.5.14 yes 48 168.131 odd 6
168.2.ba.c.5.21 yes 48 84.47 odd 6
168.2.ba.c.101.4 yes 48 24.11 even 2
168.2.ba.c.101.11 yes 48 12.11 even 2
168.2.ba.c.101.14 yes 48 4.3 odd 2
168.2.ba.c.101.21 yes 48 8.3 odd 2
672.2.bi.c.17.7 48 3.2 odd 2 inner
672.2.bi.c.17.8 48 1.1 even 1 trivial
672.2.bi.c.17.17 48 8.5 even 2 inner
672.2.bi.c.17.18 48 24.5 odd 2 inner
672.2.bi.c.593.7 48 56.5 odd 6 inner
672.2.bi.c.593.8 48 168.5 even 6 inner
672.2.bi.c.593.17 48 21.5 even 6 inner
672.2.bi.c.593.18 48 7.5 odd 6 inner