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 397.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 666.397
Dual form 666.2.s.a.307.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-1.50000 + 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{7} -1.00000i q^{8} +1.73205 q^{10} +4.73205 q^{11} +(-3.00000 + 1.73205i) q^{13} -2.00000i q^{14} +(-0.500000 + 0.866025i) q^{16} +(-6.69615 - 3.86603i) q^{17} +(1.09808 - 0.633975i) q^{19} +(-1.50000 - 0.866025i) q^{20} +(-4.09808 - 2.36603i) q^{22} +4.73205i q^{23} +(-1.00000 + 1.73205i) q^{25} +3.46410 q^{26} +(-1.00000 + 1.73205i) q^{28} +8.66025i q^{29} +1.26795i q^{31} +(0.866025 - 0.500000i) q^{32} +(3.86603 + 6.69615i) q^{34} +(-3.00000 - 1.73205i) q^{35} +(-5.69615 + 2.13397i) q^{37} -1.26795 q^{38} +(0.866025 + 1.50000i) q^{40} +(4.96410 + 8.59808i) q^{41} +0.928203i q^{43} +(2.36603 + 4.09808i) q^{44} +(2.36603 - 4.09808i) q^{46} -4.73205 q^{47} +(1.50000 - 2.59808i) q^{49} +(1.73205 - 1.00000i) q^{50} +(-3.00000 - 1.73205i) q^{52} +(1.26795 - 2.19615i) q^{53} +(-7.09808 + 4.09808i) q^{55} +(1.73205 - 1.00000i) q^{56} +(4.33013 - 7.50000i) q^{58} +(-2.19615 - 1.26795i) q^{59} +(1.50000 - 0.866025i) q^{61} +(0.633975 - 1.09808i) q^{62} -1.00000 q^{64} +(3.00000 - 5.19615i) q^{65} +(5.09808 + 8.83013i) q^{67} -7.73205i q^{68} +(1.73205 + 3.00000i) q^{70} +(1.73205 + 3.00000i) q^{71} +4.00000 q^{73} +(6.00000 + 1.00000i) q^{74} +(1.09808 + 0.633975i) q^{76} +(4.73205 + 8.19615i) q^{77} +(-11.4904 + 6.63397i) q^{79} -1.73205i q^{80} -9.92820i q^{82} +(-2.83013 + 4.90192i) q^{83} +13.3923 q^{85} +(0.464102 - 0.803848i) q^{86} -4.73205i q^{88} +(5.89230 + 3.40192i) q^{89} +(-6.00000 - 3.46410i) q^{91} +(-4.09808 + 2.36603i) q^{92} +(4.09808 + 2.36603i) q^{94} +(-1.09808 + 1.90192i) q^{95} -7.73205i q^{97} +(-2.59808 + 1.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{11} - 12 q^{13} - 2 q^{16} - 6 q^{17} - 6 q^{19} - 6 q^{20} - 6 q^{22} - 4 q^{25} - 4 q^{28} + 12 q^{34} - 12 q^{35} - 2 q^{37} - 12 q^{38} + 6 q^{41} + 6 q^{44}+ \cdots + 6 q^{95}+O(q^{100}) \) 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
\(11\) 4.73205 1.42677 0.713384 − 0.700774i \(-0.247162\pi\)
0.713384 + 0.700774i \(0.247162\pi\)
\(12\) 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
\(17\) −6.69615 − 3.86603i −1.62406 − 0.937649i −0.985820 − 0.167808i \(-0.946331\pi\)
−0.638236 − 0.769841i \(-0.720336\pi\)
\(18\) 0 0
\(19\) 1.09808 − 0.633975i 0.251916 − 0.145444i −0.368725 − 0.929538i \(-0.620206\pi\)
0.620641 + 0.784095i \(0.286872\pi\)
\(20\) −1.50000 − 0.866025i −0.335410 − 0.193649i
\(21\) 0 0
\(22\) −4.09808 − 2.36603i −0.873713 − 0.504438i
\(23\) 4.73205i 0.986701i 0.869831 + 0.493350i \(0.164228\pi\)
−0.869831 + 0.493350i \(0.835772\pi\)
\(24\) 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
\(31\) 1.26795i 0.227730i 0.993496 + 0.113865i \(0.0363232\pi\)
−0.993496 + 0.113865i \(0.963677\pi\)
\(32\) 0.866025 − 0.500000i 0.153093 − 0.0883883i
\(33\) 0 0
\(34\) 3.86603 + 6.69615i 0.663018 + 1.14838i
\(35\) −3.00000 − 1.73205i −0.507093 − 0.292770i
\(36\) 0 0
\(37\) −5.69615 + 2.13397i −0.936442 + 0.350823i
\(38\) −1.26795 −0.205689
\(39\) 0 0
\(40\) 0.866025 + 1.50000i 0.136931 + 0.237171i
\(41\) 4.96410 + 8.59808i 0.775262 + 1.34279i 0.934647 + 0.355577i \(0.115716\pi\)
−0.159384 + 0.987217i \(0.550951\pi\)
\(42\) 0 0
\(43\) 0.928203i 0.141550i 0.997492 + 0.0707748i \(0.0225472\pi\)
−0.997492 + 0.0707748i \(0.977453\pi\)
\(44\) 2.36603 + 4.09808i 0.356692 + 0.617808i
\(45\) 0 0
\(46\) 2.36603 − 4.09808i 0.348851 − 0.604228i
\(47\) −4.73205 −0.690241 −0.345120 − 0.938558i \(-0.612162\pi\)
−0.345120 + 0.938558i \(0.612162\pi\)
\(48\) 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
\(53\) 1.26795 − 2.19615i 0.174166 − 0.301665i −0.765706 − 0.643191i \(-0.777610\pi\)
0.939872 + 0.341526i \(0.110944\pi\)
\(54\) 0 0
\(55\) −7.09808 + 4.09808i −0.957104 + 0.552584i
\(56\) 1.73205 − 1.00000i 0.231455 − 0.133631i
\(57\) 0 0
\(58\) 4.33013 − 7.50000i 0.568574 − 0.984798i
\(59\) −2.19615 − 1.26795i −0.285915 − 0.165073i 0.350183 − 0.936681i \(-0.386119\pi\)
−0.636098 + 0.771608i \(0.719453\pi\)
\(60\) 0 0
\(61\) 1.50000 − 0.866025i 0.192055 − 0.110883i −0.400889 − 0.916127i \(-0.631299\pi\)
0.592944 + 0.805243i \(0.297965\pi\)
\(62\) 0.633975 − 1.09808i 0.0805149 − 0.139456i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 3.00000 − 5.19615i 0.372104 − 0.644503i
\(66\) 0 0
\(67\) 5.09808 + 8.83013i 0.622829 + 1.07877i 0.988956 + 0.148207i \(0.0473502\pi\)
−0.366127 + 0.930565i \(0.619317\pi\)
\(68\) − 7.73205i − 0.937649i
\(69\) 0 0
\(70\) 1.73205 + 3.00000i 0.207020 + 0.358569i
\(71\) 1.73205 + 3.00000i 0.205557 + 0.356034i 0.950310 − 0.311305i \(-0.100766\pi\)
−0.744753 + 0.667340i \(0.767433\pi\)
\(72\) 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
\(76\) 1.09808 + 0.633975i 0.125958 + 0.0727219i
\(77\) 4.73205 + 8.19615i 0.539267 + 0.934038i
\(78\) 0 0
\(79\) −11.4904 + 6.63397i −1.29277 + 0.746380i −0.979144 − 0.203167i \(-0.934877\pi\)
−0.313625 + 0.949547i \(0.601543\pi\)
\(80\) − 1.73205i − 0.193649i
\(81\) 0 0
\(82\) − 9.92820i − 1.09639i
\(83\) −2.83013 + 4.90192i −0.310647 + 0.538056i −0.978503 − 0.206235i \(-0.933879\pi\)
0.667856 + 0.744291i \(0.267212\pi\)
\(84\) 0 0
\(85\) 13.3923 1.45260
\(86\) 0.464102 − 0.803848i 0.0500454 − 0.0866811i
\(87\) 0 0
\(88\) − 4.73205i − 0.504438i
\(89\) 5.89230 + 3.40192i 0.624583 + 0.360603i 0.778651 − 0.627457i \(-0.215904\pi\)
−0.154068 + 0.988060i \(0.549238\pi\)
\(90\) 0 0
\(91\) −6.00000 − 3.46410i −0.628971 − 0.363137i
\(92\) −4.09808 + 2.36603i −0.427254 + 0.246675i
\(93\) 0 0
\(94\) 4.09808 + 2.36603i 0.422684 + 0.244037i
\(95\) −1.09808 + 1.90192i −0.112660 + 0.195133i
\(96\) 0 0
\(97\) − 7.73205i − 0.785071i −0.919737 − 0.392535i \(-0.871598\pi\)
0.919737 − 0.392535i \(-0.128402\pi\)
\(98\) −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.397.1 4
3.2 odd 2 74.2.e.b.27.2 yes 4
12.11 even 2 592.2.w.e.545.2 4
37.11 even 6 inner 666.2.s.a.307.1 4
111.11 odd 6 74.2.e.b.11.2 ✓ 4
111.14 even 12 2738.2.a.i.1.1 2
111.23 even 12 2738.2.a.e.1.1 2
444.11 even 6 592.2.w.e.529.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.b.11.2 ✓ 4 111.11 odd 6
74.2.e.b.27.2 yes 4 3.2 odd 2
592.2.w.e.529.2 4 444.11 even 6
592.2.w.e.545.2 4 12.11 even 2
666.2.s.a.307.1 4 37.11 even 6 inner
666.2.s.a.397.1 4 1.1 even 1 trivial
2738.2.a.e.1.1 2 111.23 even 12
2738.2.a.i.1.1 2 111.14 even 12