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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 593.8
Character \(\chi\) \(=\) 672.593
Dual form 672.2.bi.c.17.8

$q$-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 \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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 0
\(3\) −0.618109 − 1.61801i −0.356866 − 0.934156i
\(4\) 0 0
\(5\) −2.46958 − 1.42581i −1.10443 − 0.637643i −0.167049 − 0.985949i \(-0.553424\pi\)
−0.937381 + 0.348306i \(0.886757\pi\)
\(6\) 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 −0.956230 − 0.292617i \(-0.905474\pi\)
0.224701 − 0.974428i \(-0.427860\pi\)
\(12\) 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 0.996451 − 0.0841773i \(-0.0268262\pi\)
−0.571125 + 0.820863i \(0.693493\pi\)
\(18\) 0 0
\(19\) −3.14493 + 5.44717i −0.721496 + 1.24967i 0.238905 + 0.971043i \(0.423212\pi\)
−0.960400 + 0.278624i \(0.910122\pi\)
\(20\) 0 0
\(21\) −4.58037 − 0.142013i −0.999520 − 0.0309899i
\(22\) 0 0
\(23\) −3.15865 − 1.82365i −0.658624 − 0.380257i 0.133128 − 0.991099i \(-0.457498\pi\)
−0.791753 + 0.610842i \(0.790831\pi\)
\(24\) 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 −0.420962 − 0.907078i \(-0.638307\pi\)
0.575072 + 0.818103i \(0.304974\pi\)
\(32\) 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 0.584994 − 0.811038i \(-0.301097\pi\)
−0.409883 + 0.912138i \(0.634430\pi\)
\(38\) 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 \(-0.872399\pi\)
0.920722 − 0.390220i \(-0.127601\pi\)
\(44\) 0 0
\(45\) 8.37361 − 1.75172i 1.24826 − 0.261132i
\(46\) 0 0
\(47\) −5.61268 + 9.72145i −0.818693 + 1.41802i 0.0879518 + 0.996125i \(0.471968\pi\)
−0.906645 + 0.421894i \(0.861365\pi\)
\(48\) 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 0.926714 − 0.375767i \(-0.122621\pi\)
−0.788781 + 0.614674i \(0.789288\pi\)
\(54\) 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 −0.556876 − 0.830595i \(-0.688000\pi\)
0.440879 + 0.897567i \(0.354667\pi\)
\(60\) 0 0
\(61\) 1.24347 − 2.15376i 0.159211 − 0.275761i −0.775374 − 0.631503i \(-0.782438\pi\)
0.934584 + 0.355742i \(0.115772\pi\)
\(62\) 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 −0.774623 − 0.632423i \(-0.782060\pi\)
0.160383 + 0.987055i \(0.448727\pi\)
\(68\) 0 0
\(69\) −0.998281 + 6.23793i −0.120179 + 0.750958i
\(70\) 0 0
\(71\) − 9.75277i − 1.15744i −0.815526 − 0.578720i \(-0.803552\pi\)
0.815526 − 0.578720i \(-0.196448\pi\)
\(72\) 0 0
\(73\) 0.291019 − 0.168020i 0.0340612 − 0.0196652i −0.482873 − 0.875691i \(-0.660407\pi\)
0.516934 + 0.856025i \(0.327073\pi\)
\(74\) 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 −0.664346 − 0.747426i \(-0.731290\pi\)
0.979462 + 0.201627i \(0.0646230\pi\)
\(80\) 0 0
\(81\) 0.998337 − 8.94446i 0.110926 − 0.993829i
\(82\) 0 0
\(83\) − 0.138115i − 0.0151600i −0.999971 − 0.00758002i \(-0.997587\pi\)
0.999971 − 0.00758002i \(-0.00241282\pi\)
\(84\) 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 −0.833589 − 0.552385i \(-0.813718\pi\)
0.895174 + 0.445717i \(0.147051\pi\)
\(90\) 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 \(-0.809539\pi\)
0.826266 − 0.563280i \(-0.190461\pi\)
\(98\) 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.593.8 48
3.2 odd 2 inner 672.2.bi.c.593.7 48
4.3 odd 2 168.2.ba.c.5.14 yes 48
7.3 odd 6 inner 672.2.bi.c.17.18 48
8.3 odd 2 168.2.ba.c.5.21 yes 48
8.5 even 2 inner 672.2.bi.c.593.17 48
12.11 even 2 168.2.ba.c.5.11 yes 48
21.17 even 6 inner 672.2.bi.c.17.17 48
24.5 odd 2 inner 672.2.bi.c.593.18 48
24.11 even 2 168.2.ba.c.5.4 ✓ 48
28.3 even 6 168.2.ba.c.101.4 yes 48
56.3 even 6 168.2.ba.c.101.11 yes 48
56.45 odd 6 inner 672.2.bi.c.17.7 48
84.59 odd 6 168.2.ba.c.101.21 yes 48
168.59 odd 6 168.2.ba.c.101.14 yes 48
168.101 even 6 inner 672.2.bi.c.17.8 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.ba.c.5.4 ✓ 48 24.11 even 2
168.2.ba.c.5.11 yes 48 12.11 even 2
168.2.ba.c.5.14 yes 48 4.3 odd 2
168.2.ba.c.5.21 yes 48 8.3 odd 2
168.2.ba.c.101.4 yes 48 28.3 even 6
168.2.ba.c.101.11 yes 48 56.3 even 6
168.2.ba.c.101.14 yes 48 168.59 odd 6
168.2.ba.c.101.21 yes 48 84.59 odd 6
672.2.bi.c.17.7 48 56.45 odd 6 inner
672.2.bi.c.17.8 48 168.101 even 6 inner
672.2.bi.c.17.17 48 21.17 even 6 inner
672.2.bi.c.17.18 48 7.3 odd 6 inner
672.2.bi.c.593.7 48 3.2 odd 2 inner
672.2.bi.c.593.8 48 1.1 even 1 trivial
672.2.bi.c.593.17 48 8.5 even 2 inner
672.2.bi.c.593.18 48 24.5 odd 2 inner