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(1,666)] 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("666.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(666, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,2,-1,0,-2] 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: yes
Analytic conductor: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 666.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.85410 q^{5} -3.23607 q^{7} -1.00000 q^{8} +3.85410 q^{10} +1.38197 q^{11} -2.85410 q^{13} +3.23607 q^{14} +1.00000 q^{16} +4.47214 q^{17} +4.47214 q^{19} -3.85410 q^{20} -1.38197 q^{22} -2.85410 q^{23} +9.85410 q^{25} +2.85410 q^{26} -3.23607 q^{28} +9.32624 q^{29} +7.38197 q^{31} -1.00000 q^{32} -4.47214 q^{34} +12.4721 q^{35} -1.00000 q^{37} -4.47214 q^{38} +3.85410 q^{40} -9.61803 q^{41} -5.23607 q^{43} +1.38197 q^{44} +2.85410 q^{46} +1.23607 q^{47} +3.47214 q^{49} -9.85410 q^{50} -2.85410 q^{52} -0.472136 q^{53} -5.32624 q^{55} +3.23607 q^{56} -9.32624 q^{58} +4.76393 q^{59} +10.6180 q^{61} -7.38197 q^{62} +1.00000 q^{64} +11.0000 q^{65} +1.09017 q^{67} +4.47214 q^{68} -12.4721 q^{70} -2.94427 q^{71} +7.09017 q^{73} +1.00000 q^{74} +4.47214 q^{76} -4.47214 q^{77} -8.56231 q^{79} -3.85410 q^{80} +9.61803 q^{82} +14.4721 q^{83} -17.2361 q^{85} +5.23607 q^{86} -1.38197 q^{88} +1.52786 q^{89} +9.23607 q^{91} -2.85410 q^{92} -1.23607 q^{94} -17.2361 q^{95} -0.472136 q^{97} -3.47214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8} + q^{10} + 5 q^{11} + q^{13} + 2 q^{14} + 2 q^{16} - q^{20} - 5 q^{22} + q^{23} + 13 q^{25} - q^{26} - 2 q^{28} + 3 q^{29} + 17 q^{31} - 2 q^{32}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.85410 −1.72361 −0.861803 − 0.507242i \(-0.830665\pi\)
−0.861803 + 0.507242i \(0.830665\pi\)
\(6\) 0 0
\(7\) −3.23607 −1.22312 −0.611559 − 0.791199i \(-0.709457\pi\)
−0.611559 + 0.791199i \(0.709457\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.85410 1.21877
\(11\) 1.38197 0.416678 0.208339 − 0.978057i \(-0.433194\pi\)
0.208339 + 0.978057i \(0.433194\pi\)
\(12\) 0 0
\(13\) −2.85410 −0.791585 −0.395793 − 0.918340i \(-0.629530\pi\)
−0.395793 + 0.918340i \(0.629530\pi\)
\(14\) 3.23607 0.864876
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.47214 1.08465 0.542326 − 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 0 0
\(19\) 4.47214 1.02598 0.512989 − 0.858395i \(-0.328538\pi\)
0.512989 + 0.858395i \(0.328538\pi\)
\(20\) −3.85410 −0.861803
\(21\) 0 0
\(22\) −1.38197 −0.294636
\(23\) −2.85410 −0.595121 −0.297561 − 0.954703i \(-0.596173\pi\)
−0.297561 + 0.954703i \(0.596173\pi\)
\(24\) 0 0
\(25\) 9.85410 1.97082
\(26\) 2.85410 0.559735
\(27\) 0 0
\(28\) −3.23607 −0.611559
\(29\) 9.32624 1.73184 0.865919 − 0.500183i \(-0.166734\pi\)
0.865919 + 0.500183i \(0.166734\pi\)
\(30\) 0 0
\(31\) 7.38197 1.32584 0.662920 − 0.748690i \(-0.269317\pi\)
0.662920 + 0.748690i \(0.269317\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.47214 −0.766965
\(35\) 12.4721 2.10818
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −4.47214 −0.725476
\(39\) 0 0
\(40\) 3.85410 0.609387
\(41\) −9.61803 −1.50208 −0.751042 − 0.660254i \(-0.770449\pi\)
−0.751042 + 0.660254i \(0.770449\pi\)
\(42\) 0 0
\(43\) −5.23607 −0.798493 −0.399246 − 0.916844i \(-0.630728\pi\)
−0.399246 + 0.916844i \(0.630728\pi\)
\(44\) 1.38197 0.208339
\(45\) 0 0
\(46\) 2.85410 0.420814
\(47\) 1.23607 0.180299 0.0901495 − 0.995928i \(-0.471266\pi\)
0.0901495 + 0.995928i \(0.471266\pi\)
\(48\) 0 0
\(49\) 3.47214 0.496019
\(50\) −9.85410 −1.39358
\(51\) 0 0
\(52\) −2.85410 −0.395793
\(53\) −0.472136 −0.0648529 −0.0324264 − 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) 0 0
\(55\) −5.32624 −0.718190
\(56\) 3.23607 0.432438
\(57\) 0 0
\(58\) −9.32624 −1.22460
\(59\) 4.76393 0.620211 0.310106 − 0.950702i \(-0.399636\pi\)
0.310106 + 0.950702i \(0.399636\pi\)
\(60\) 0 0
\(61\) 10.6180 1.35950 0.679750 − 0.733444i \(-0.262088\pi\)
0.679750 + 0.733444i \(0.262088\pi\)
\(62\) −7.38197 −0.937511
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 11.0000 1.36438
\(66\) 0 0
\(67\) 1.09017 0.133185 0.0665927 − 0.997780i \(-0.478787\pi\)
0.0665927 + 0.997780i \(0.478787\pi\)
\(68\) 4.47214 0.542326
\(69\) 0 0
\(70\) −12.4721 −1.49071
\(71\) −2.94427 −0.349421 −0.174710 − 0.984620i \(-0.555899\pi\)
−0.174710 + 0.984620i \(0.555899\pi\)
\(72\) 0 0
\(73\) 7.09017 0.829842 0.414921 − 0.909858i \(-0.363809\pi\)
0.414921 + 0.909858i \(0.363809\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 4.47214 0.512989
\(77\) −4.47214 −0.509647
\(78\) 0 0
\(79\) −8.56231 −0.963335 −0.481667 − 0.876354i \(-0.659969\pi\)
−0.481667 + 0.876354i \(0.659969\pi\)
\(80\) −3.85410 −0.430902
\(81\) 0 0
\(82\) 9.61803 1.06213
\(83\) 14.4721 1.58852 0.794262 − 0.607576i \(-0.207858\pi\)
0.794262 + 0.607576i \(0.207858\pi\)
\(84\) 0 0
\(85\) −17.2361 −1.86951
\(86\) 5.23607 0.564620
\(87\) 0 0
\(88\) −1.38197 −0.147318
\(89\) 1.52786 0.161953 0.0809766 − 0.996716i \(-0.474196\pi\)
0.0809766 + 0.996716i \(0.474196\pi\)
\(90\) 0 0
\(91\) 9.23607 0.968203
\(92\) −2.85410 −0.297561
\(93\) 0 0
\(94\) −1.23607 −0.127491
\(95\) −17.2361 −1.76838
\(96\) 0 0
\(97\) −0.472136 −0.0479381 −0.0239691 − 0.999713i \(-0.507630\pi\)
−0.0239691 + 0.999713i \(0.507630\pi\)
\(98\) −3.47214 −0.350739
\(99\) 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.a.i.1.1 2
3.2 odd 2 74.2.a.b.1.1 ✓ 2
4.3 odd 2 5328.2.a.bc.1.1 2
12.11 even 2 592.2.a.g.1.2 2
15.2 even 4 1850.2.b.j.149.4 4
15.8 even 4 1850.2.b.j.149.1 4
15.14 odd 2 1850.2.a.t.1.2 2
21.20 even 2 3626.2.a.s.1.2 2
24.5 odd 2 2368.2.a.y.1.2 2
24.11 even 2 2368.2.a.u.1.1 2
33.32 even 2 8954.2.a.j.1.1 2
111.110 odd 2 2738.2.a.g.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.b.1.1 ✓ 2 3.2 odd 2
592.2.a.g.1.2 2 12.11 even 2
666.2.a.i.1.1 2 1.1 even 1 trivial
1850.2.a.t.1.2 2 15.14 odd 2
1850.2.b.j.149.1 4 15.8 even 4
1850.2.b.j.149.4 4 15.2 even 4
2368.2.a.u.1.1 2 24.11 even 2
2368.2.a.y.1.2 2 24.5 odd 2
2738.2.a.g.1.1 2 111.110 odd 2
3626.2.a.s.1.2 2 21.20 even 2
5328.2.a.bc.1.1 2 4.3 odd 2
8954.2.a.j.1.1 2 33.32 even 2