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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [666,2,Mod(73,666)] 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("666.73"); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,0,-8] 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: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 73.3
Root \(-1.79129i\) of defining polynomial
Character \(\chi\) \(=\) 666.73
Dual form 666.2.c.b.73.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} -0.791288i q^{5} -2.00000 q^{7} -1.00000i q^{8} +0.791288 q^{10} +0.791288 q^{11} -3.79129i q^{13} -2.00000i q^{14} +1.00000 q^{16} -7.58258i q^{17} +1.58258i q^{19} +0.791288i q^{20} +0.791288i q^{22} +3.79129i q^{23} +4.37386 q^{25} +3.79129 q^{26} +2.00000 q^{28} -3.79129i q^{29} -8.37386i q^{31} +1.00000i q^{32} +7.58258 q^{34} +1.58258i q^{35} +(4.00000 - 4.58258i) q^{37} -1.58258 q^{38} -0.791288 q^{40} -9.79129 q^{41} -6.00000i q^{43} -0.791288 q^{44} -3.79129 q^{46} +7.58258 q^{47} -3.00000 q^{49} +4.37386i q^{50} +3.79129i q^{52} +1.58258 q^{53} -0.626136i q^{55} +2.00000i q^{56} +3.79129 q^{58} -1.58258i q^{59} +12.7913i q^{61} +8.37386 q^{62} -1.00000 q^{64} -3.00000 q^{65} +6.37386 q^{67} +7.58258i q^{68} -1.58258 q^{70} +9.16515 q^{71} -4.37386 q^{73} +(4.58258 + 4.00000i) q^{74} -1.58258i q^{76} -1.58258 q^{77} +8.20871i q^{79} -0.791288i q^{80} -9.79129i q^{82} -15.1652 q^{83} -6.00000 q^{85} +6.00000 q^{86} -0.791288i q^{88} -6.00000i q^{89} +7.58258i q^{91} -3.79129i q^{92} +7.58258i q^{94} +1.25227 q^{95} -13.5826i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 8 q^{7} - 6 q^{10} - 6 q^{11} + 4 q^{16} - 10 q^{25} + 6 q^{26} + 8 q^{28} + 12 q^{34} + 16 q^{37} + 12 q^{38} + 6 q^{40} - 30 q^{41} + 6 q^{44} - 6 q^{46} + 12 q^{47} - 12 q^{49} - 12 q^{53}+ \cdots + 60 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) − 0.791288i − 0.353875i −0.984222 − 0.176937i \(-0.943381\pi\)
0.984222 − 0.176937i \(-0.0566190\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 0.791288 0.250227
\(11\) 0.791288 0.238582 0.119291 − 0.992859i \(-0.461938\pi\)
0.119291 + 0.992859i \(0.461938\pi\)
\(12\) 0 0
\(13\) − 3.79129i − 1.05151i −0.850635 − 0.525757i \(-0.823782\pi\)
0.850635 − 0.525757i \(-0.176218\pi\)
\(14\) − 2.00000i − 0.534522i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 7.58258i − 1.83904i −0.393038 − 0.919522i \(-0.628576\pi\)
0.393038 − 0.919522i \(-0.371424\pi\)
\(18\) 0 0
\(19\) 1.58258i 0.363068i 0.983385 + 0.181534i \(0.0581062\pi\)
−0.983385 + 0.181534i \(0.941894\pi\)
\(20\) 0.791288i 0.176937i
\(21\) 0 0
\(22\) 0.791288i 0.168703i
\(23\) 3.79129i 0.790538i 0.918565 + 0.395269i \(0.129349\pi\)
−0.918565 + 0.395269i \(0.870651\pi\)
\(24\) 0 0
\(25\) 4.37386 0.874773
\(26\) 3.79129 0.743533
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) − 3.79129i − 0.704024i −0.935995 − 0.352012i \(-0.885498\pi\)
0.935995 − 0.352012i \(-0.114502\pi\)
\(30\) 0 0
\(31\) − 8.37386i − 1.50399i −0.659169 − 0.751995i \(-0.729092\pi\)
0.659169 − 0.751995i \(-0.270908\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 7.58258 1.30040
\(35\) 1.58258i 0.267504i
\(36\) 0 0
\(37\) 4.00000 − 4.58258i 0.657596 − 0.753371i
\(38\) −1.58258 −0.256728
\(39\) 0 0
\(40\) −0.791288 −0.125114
\(41\) −9.79129 −1.52914 −0.764571 − 0.644539i \(-0.777049\pi\)
−0.764571 + 0.644539i \(0.777049\pi\)
\(42\) 0 0
\(43\) − 6.00000i − 0.914991i −0.889212 − 0.457496i \(-0.848747\pi\)
0.889212 − 0.457496i \(-0.151253\pi\)
\(44\) −0.791288 −0.119291
\(45\) 0 0
\(46\) −3.79129 −0.558995
\(47\) 7.58258 1.10603 0.553016 − 0.833171i \(-0.313477\pi\)
0.553016 + 0.833171i \(0.313477\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 4.37386i 0.618558i
\(51\) 0 0
\(52\) 3.79129i 0.525757i
\(53\) 1.58258 0.217383 0.108692 − 0.994076i \(-0.465334\pi\)
0.108692 + 0.994076i \(0.465334\pi\)
\(54\) 0 0
\(55\) − 0.626136i − 0.0844282i
\(56\) 2.00000i 0.267261i
\(57\) 0 0
\(58\) 3.79129 0.497820
\(59\) − 1.58258i − 0.206034i −0.994680 − 0.103017i \(-0.967150\pi\)
0.994680 − 0.103017i \(-0.0328496\pi\)
\(60\) 0 0
\(61\) 12.7913i 1.63776i 0.573967 + 0.818878i \(0.305404\pi\)
−0.573967 + 0.818878i \(0.694596\pi\)
\(62\) 8.37386 1.06348
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) 6.37386 0.778691 0.389346 − 0.921092i \(-0.372701\pi\)
0.389346 + 0.921092i \(0.372701\pi\)
\(68\) 7.58258i 0.919522i
\(69\) 0 0
\(70\) −1.58258 −0.189154
\(71\) 9.16515 1.08770 0.543852 − 0.839181i \(-0.316965\pi\)
0.543852 + 0.839181i \(0.316965\pi\)
\(72\) 0 0
\(73\) −4.37386 −0.511922 −0.255961 − 0.966687i \(-0.582392\pi\)
−0.255961 + 0.966687i \(0.582392\pi\)
\(74\) 4.58258 + 4.00000i 0.532714 + 0.464991i
\(75\) 0 0
\(76\) − 1.58258i − 0.181534i
\(77\) −1.58258 −0.180351
\(78\) 0 0
\(79\) 8.20871i 0.923552i 0.886997 + 0.461776i \(0.152788\pi\)
−0.886997 + 0.461776i \(0.847212\pi\)
\(80\) − 0.791288i − 0.0884687i
\(81\) 0 0
\(82\) − 9.79129i − 1.08127i
\(83\) −15.1652 −1.66459 −0.832296 − 0.554332i \(-0.812974\pi\)
−0.832296 + 0.554332i \(0.812974\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 6.00000 0.646997
\(87\) 0 0
\(88\) − 0.791288i − 0.0843516i
\(89\) − 6.00000i − 0.635999i −0.948091 − 0.317999i \(-0.896989\pi\)
0.948091 − 0.317999i \(-0.103011\pi\)
\(90\) 0 0
\(91\) 7.58258i 0.794870i
\(92\) − 3.79129i − 0.395269i
\(93\) 0 0
\(94\) 7.58258i 0.782083i
\(95\) 1.25227 0.128480
\(96\) 0 0
\(97\) − 13.5826i − 1.37910i −0.724237 − 0.689551i \(-0.757808\pi\)
0.724237 − 0.689551i \(-0.242192\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) 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.c.b.73.3 4
3.2 odd 2 74.2.b.a.73.2 ✓ 4
4.3 odd 2 5328.2.h.m.2737.2 4
12.11 even 2 592.2.g.c.369.2 4
15.2 even 4 1850.2.c.h.1849.2 4
15.8 even 4 1850.2.c.g.1849.3 4
15.14 odd 2 1850.2.d.e.1701.3 4
24.5 odd 2 2368.2.g.j.961.1 4
24.11 even 2 2368.2.g.h.961.3 4
37.36 even 2 inner 666.2.c.b.73.2 4
111.68 even 4 2738.2.a.k.1.1 2
111.80 even 4 2738.2.a.h.1.1 2
111.110 odd 2 74.2.b.a.73.4 yes 4
148.147 odd 2 5328.2.h.m.2737.3 4
444.443 even 2 592.2.g.c.369.1 4
555.332 even 4 1850.2.c.g.1849.2 4
555.443 even 4 1850.2.c.h.1849.3 4
555.554 odd 2 1850.2.d.e.1701.1 4
888.221 odd 2 2368.2.g.j.961.2 4
888.443 even 2 2368.2.g.h.961.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.b.a.73.2 ✓ 4 3.2 odd 2
74.2.b.a.73.4 yes 4 111.110 odd 2
592.2.g.c.369.1 4 444.443 even 2
592.2.g.c.369.2 4 12.11 even 2
666.2.c.b.73.2 4 37.36 even 2 inner
666.2.c.b.73.3 4 1.1 even 1 trivial
1850.2.c.g.1849.2 4 555.332 even 4
1850.2.c.g.1849.3 4 15.8 even 4
1850.2.c.h.1849.2 4 15.2 even 4
1850.2.c.h.1849.3 4 555.443 even 4
1850.2.d.e.1701.1 4 555.554 odd 2
1850.2.d.e.1701.3 4 15.14 odd 2
2368.2.g.h.961.3 4 24.11 even 2
2368.2.g.h.961.4 4 888.443 even 2
2368.2.g.j.961.1 4 24.5 odd 2
2368.2.g.j.961.2 4 888.221 odd 2
2738.2.a.h.1.1 2 111.80 even 4
2738.2.a.k.1.1 2 111.68 even 4
5328.2.h.m.2737.2 4 4.3 odd 2
5328.2.h.m.2737.3 4 148.147 odd 2