Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [666,2,Mod(307,666)] 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("666.307"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(666, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,-6,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 397.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 666.397
Dual form 666.2.s.a.307.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-1.50000 + 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{7} +1.00000i q^{8} -1.73205 q^{10} +1.26795 q^{11} +(-3.00000 + 1.73205i) q^{13} +2.00000i q^{14} +(-0.500000 + 0.866025i) q^{16} +(3.69615 + 2.13397i) q^{17} +(-4.09808 + 2.36603i) q^{19} +(-1.50000 - 0.866025i) q^{20} +(1.09808 + 0.633975i) q^{22} -1.26795i q^{23} +(-1.00000 + 1.73205i) q^{25} -3.46410 q^{26} +(-1.00000 + 1.73205i) q^{28} +8.66025i q^{29} -4.73205i q^{31} +(-0.866025 + 0.500000i) q^{32} +(2.13397 + 3.69615i) q^{34} +(-3.00000 - 1.73205i) q^{35} +(4.69615 - 3.86603i) q^{37} -4.73205 q^{38} +(-0.866025 - 1.50000i) q^{40} +(-1.96410 - 3.40192i) q^{41} +12.9282i q^{43} +(0.633975 + 1.09808i) q^{44} +(0.633975 - 1.09808i) q^{46} -1.26795 q^{47} +(1.50000 - 2.59808i) q^{49} +(-1.73205 + 1.00000i) q^{50} +(-3.00000 - 1.73205i) q^{52} +(4.73205 - 8.19615i) q^{53} +(-1.90192 + 1.09808i) q^{55} +(-1.73205 + 1.00000i) q^{56} +(-4.33013 + 7.50000i) q^{58} +(8.19615 + 4.73205i) q^{59} +(1.50000 - 0.866025i) q^{61} +(2.36603 - 4.09808i) q^{62} -1.00000 q^{64} +(3.00000 - 5.19615i) q^{65} +(-0.0980762 - 0.169873i) q^{67} +4.26795i q^{68} +(-1.73205 - 3.00000i) q^{70} +(-1.73205 - 3.00000i) q^{71} +4.00000 q^{73} +(6.00000 - 1.00000i) q^{74} +(-4.09808 - 2.36603i) q^{76} +(1.26795 + 2.19615i) q^{77} +(14.4904 - 8.36603i) q^{79} -1.73205i q^{80} -3.92820i q^{82} +(5.83013 - 10.0981i) q^{83} -7.39230 q^{85} +(-6.46410 + 11.1962i) q^{86} +1.26795i q^{88} +(-14.8923 - 8.59808i) q^{89} +(-6.00000 - 3.46410i) q^{91} +(1.09808 - 0.633975i) q^{92} +(-1.09808 - 0.633975i) q^{94} +(4.09808 - 7.09808i) q^{95} +4.26795i q^{97} +(2.59808 - 1.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{11} - 12 q^{13} - 2 q^{16} - 6 q^{17} - 6 q^{19} - 6 q^{20} - 6 q^{22} - 4 q^{25} - 4 q^{28} + 12 q^{34} - 12 q^{35} - 2 q^{37} - 12 q^{38} + 6 q^{41} + 6 q^{44}+ \cdots + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 + 0.500000i 0.612372 + 0.353553i
\(3\) 0 0
\(4\) 0.500000 + 0.866025i 0.250000 + 0.433013i
\(5\) −1.50000 + 0.866025i −0.670820 + 0.387298i −0.796387 − 0.604787i \(-0.793258\pi\)
0.125567 + 0.992085i \(0.459925\pi\)
\(6\) 0 0
\(7\) 1.00000 + 1.73205i 0.377964 + 0.654654i 0.990766 − 0.135583i \(-0.0432908\pi\)
−0.612801 + 0.790237i \(0.709957\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) −1.73205 −0.547723
\(11\) 1.26795 0.382301 0.191151 − 0.981561i \(-0.438778\pi\)
0.191151 + 0.981561i \(0.438778\pi\)
\(12\) 0 0
\(13\) −3.00000 + 1.73205i −0.832050 + 0.480384i −0.854554 − 0.519362i \(-0.826170\pi\)
0.0225039 + 0.999747i \(0.492836\pi\)
\(14\) 2.00000i 0.534522i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 3.69615 + 2.13397i 0.896449 + 0.517565i 0.876046 − 0.482227i \(-0.160172\pi\)
0.0204023 + 0.999792i \(0.493505\pi\)
\(18\) 0 0
\(19\) −4.09808 + 2.36603i −0.940163 + 0.542803i −0.890011 − 0.455938i \(-0.849304\pi\)
−0.0501517 + 0.998742i \(0.515970\pi\)
\(20\) −1.50000 − 0.866025i −0.335410 − 0.193649i
\(21\) 0 0
\(22\) 1.09808 + 0.633975i 0.234111 + 0.135164i
\(23\) − 1.26795i − 0.264386i −0.991224 − 0.132193i \(-0.957798\pi\)
0.991224 − 0.132193i \(-0.0422018\pi\)
\(24\) 0 0
\(25\) −1.00000 + 1.73205i −0.200000 + 0.346410i
\(26\) −3.46410 −0.679366
\(27\) 0 0
\(28\) −1.00000 + 1.73205i −0.188982 + 0.327327i
\(29\) 8.66025i 1.60817i 0.594515 + 0.804084i \(0.297344\pi\)
−0.594515 + 0.804084i \(0.702656\pi\)
\(30\) 0 0
\(31\) − 4.73205i − 0.849901i −0.905216 − 0.424951i \(-0.860291\pi\)
0.905216 − 0.424951i \(-0.139709\pi\)
\(32\) −0.866025 + 0.500000i −0.153093 + 0.0883883i
\(33\) 0 0
\(34\) 2.13397 + 3.69615i 0.365974 + 0.633885i
\(35\) −3.00000 − 1.73205i −0.507093 − 0.292770i
\(36\) 0 0
\(37\) 4.69615 − 3.86603i 0.772043 − 0.635571i
\(38\) −4.73205 −0.767640
\(39\) 0 0
\(40\) −0.866025 − 1.50000i −0.136931 − 0.237171i
\(41\) −1.96410 − 3.40192i −0.306741 − 0.531291i 0.670906 − 0.741542i \(-0.265905\pi\)
−0.977647 + 0.210251i \(0.932572\pi\)
\(42\) 0 0
\(43\) 12.9282i 1.97153i 0.168122 + 0.985766i \(0.446230\pi\)
−0.168122 + 0.985766i \(0.553770\pi\)
\(44\) 0.633975 + 1.09808i 0.0955753 + 0.165541i
\(45\) 0 0
\(46\) 0.633975 − 1.09808i 0.0934745 − 0.161903i
\(47\) −1.26795 −0.184949 −0.0924747 − 0.995715i \(-0.529478\pi\)
−0.0924747 + 0.995715i \(0.529478\pi\)
\(48\) 0 0
\(49\) 1.50000 − 2.59808i 0.214286 − 0.371154i
\(50\) −1.73205 + 1.00000i −0.244949 + 0.141421i
\(51\) 0 0
\(52\) −3.00000 − 1.73205i −0.416025 − 0.240192i
\(53\) 4.73205 − 8.19615i 0.649997 − 1.12583i −0.333126 − 0.942882i \(-0.608103\pi\)
0.983123 − 0.182946i \(-0.0585633\pi\)
\(54\) 0 0
\(55\) −1.90192 + 1.09808i −0.256455 + 0.148065i
\(56\) −1.73205 + 1.00000i −0.231455 + 0.133631i
\(57\) 0 0
\(58\) −4.33013 + 7.50000i −0.568574 + 0.984798i
\(59\) 8.19615 + 4.73205i 1.06705 + 0.616061i 0.927373 − 0.374137i \(-0.122061\pi\)
0.139675 + 0.990197i \(0.455394\pi\)
\(60\) 0 0
\(61\) 1.50000 − 0.866025i 0.192055 − 0.110883i −0.400889 − 0.916127i \(-0.631299\pi\)
0.592944 + 0.805243i \(0.297965\pi\)
\(62\) 2.36603 − 4.09808i 0.300486 − 0.520456i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 3.00000 − 5.19615i 0.372104 − 0.644503i
\(66\) 0 0
\(67\) −0.0980762 − 0.169873i −0.0119819 − 0.0207533i 0.859972 − 0.510341i \(-0.170481\pi\)
−0.871954 + 0.489587i \(0.837147\pi\)
\(68\) 4.26795i 0.517565i
\(69\) 0 0
\(70\) −1.73205 − 3.00000i −0.207020 − 0.358569i
\(71\) −1.73205 − 3.00000i −0.205557 − 0.356034i 0.744753 − 0.667340i \(-0.232567\pi\)
−0.950310 + 0.311305i \(0.899234\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 6.00000 − 1.00000i 0.697486 − 0.116248i
\(75\) 0 0
\(76\) −4.09808 − 2.36603i −0.470082 − 0.271402i
\(77\) 1.26795 + 2.19615i 0.144496 + 0.250275i
\(78\) 0 0
\(79\) 14.4904 − 8.36603i 1.63030 − 0.941251i 0.646294 − 0.763088i \(-0.276318\pi\)
0.984001 − 0.178163i \(-0.0570155\pi\)
\(80\) − 1.73205i − 0.193649i
\(81\) 0 0
\(82\) − 3.92820i − 0.433797i
\(83\) 5.83013 − 10.0981i 0.639940 − 1.10841i −0.345506 − 0.938417i \(-0.612293\pi\)
0.985446 − 0.169991i \(-0.0543740\pi\)
\(84\) 0 0
\(85\) −7.39230 −0.801808
\(86\) −6.46410 + 11.1962i −0.697042 + 1.20731i
\(87\) 0 0
\(88\) 1.26795i 0.135164i
\(89\) −14.8923 − 8.59808i −1.57858 − 0.911394i −0.995058 − 0.0992979i \(-0.968340\pi\)
−0.583523 − 0.812096i \(-0.698326\pi\)
\(90\) 0 0
\(91\) −6.00000 − 3.46410i −0.628971 − 0.363137i
\(92\) 1.09808 − 0.633975i 0.114482 − 0.0660964i
\(93\) 0 0
\(94\) −1.09808 − 0.633975i −0.113258 − 0.0653895i
\(95\) 4.09808 − 7.09808i 0.420454 − 0.728247i
\(96\) 0 0
\(97\) 4.26795i 0.433345i 0.976244 + 0.216672i \(0.0695203\pi\)
−0.976244 + 0.216672i \(0.930480\pi\)
\(98\) 2.59808 − 1.50000i 0.262445 − 0.151523i
\(99\) 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 666.2.s.a.397.2 4
3.2 odd 2 74.2.e.b.27.1 yes 4
12.11 even 2 592.2.w.e.545.1 4
37.11 even 6 inner 666.2.s.a.307.2 4
111.11 odd 6 74.2.e.b.11.1 ✓ 4
111.14 even 12 2738.2.a.e.1.2 2
111.23 even 12 2738.2.a.i.1.2 2
444.11 even 6 592.2.w.e.529.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.b.11.1 ✓ 4 111.11 odd 6
74.2.e.b.27.1 yes 4 3.2 odd 2
592.2.w.e.529.1 4 444.11 even 6
592.2.w.e.545.1 4 12.11 even 2
666.2.s.a.307.2 4 37.11 even 6 inner
666.2.s.a.397.2 4 1.1 even 1 trivial
2738.2.a.e.1.2 2 111.14 even 12
2738.2.a.i.1.2 2 111.23 even 12