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Properties

Label 1776.2.q.m.1009.3
Level $1776$
Weight $2$
Character 1776.1009
Analytic conductor $14.181$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1776,2,Mod(433,1776)] 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("1776.433"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1776, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1776.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-3,0,3,0,3,0,-3,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.1814313990\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.50898483.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 8x^{4} - 10x^{3} + 64x^{2} - 40x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 888)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1009.3
Root \(-1.55022 - 2.68505i\) of defining polynomial
Character \(\chi\) \(=\) 1776.1009
Dual form 1776.2.q.m.433.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(2.05022 + 3.55108i) q^{5} +(0.500000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +4.61268 q^{11} +(2.55022 + 4.41710i) q^{13} +(2.05022 - 3.55108i) q^{15} +(2.25612 - 3.90772i) q^{17} +(-1.79409 - 3.10746i) q^{19} +(0.500000 - 0.866025i) q^{21} +1.07594 q^{23} +(-5.90677 + 10.2308i) q^{25} +1.00000 q^{27} -2.71311 q^{29} +1.51225 q^{31} +(-2.30634 - 3.99470i) q^{33} +(-2.05022 + 3.55108i) q^{35} +(5.94474 + 1.28843i) q^{37} +(2.55022 - 4.41710i) q^{39} +(-3.81859 - 6.61398i) q^{41} +3.10043 q^{43} -4.10043 q^{45} -4.71311 q^{47} +(3.00000 - 5.19615i) q^{49} -4.51225 q^{51} +(-4.84431 + 8.39059i) q^{53} +(9.45699 + 16.3800i) q^{55} +(-1.79409 + 3.10746i) q^{57} +(-6.40677 + 11.0969i) q^{59} +(3.61268 + 6.25734i) q^{61} -1.00000 q^{63} +(-10.4570 + 18.1120i) q^{65} +(-4.90677 - 8.49878i) q^{67} +(-0.537969 - 0.931790i) q^{69} +(3.20086 + 5.54406i) q^{71} -10.7376 q^{73} +11.8135 q^{75} +(2.30634 + 3.99470i) q^{77} +(-1.21815 - 2.10990i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(6.68862 - 11.5850i) q^{83} +18.5022 q^{85} +(1.35656 + 2.34962i) q^{87} +(-1.58819 + 2.75082i) q^{89} +(-2.55022 + 4.41710i) q^{91} +(-0.756123 - 1.30964i) q^{93} +(7.35656 - 12.7419i) q^{95} +18.0144 q^{97} +(-2.30634 + 3.99470i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 3 q^{5} + 3 q^{7} - 3 q^{9} + 2 q^{11} + 6 q^{13} + 3 q^{15} + 10 q^{17} - 5 q^{19} + 3 q^{21} + 2 q^{23} - 4 q^{25} + 6 q^{27} + 28 q^{29} + 2 q^{31} - q^{33} - 3 q^{35} + 2 q^{37} + 6 q^{39}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1776\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(593\) \(1297\) \(1333\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.500000 − 0.866025i −0.288675 − 0.500000i
\(4\) 0 0
\(5\) 2.05022 + 3.55108i 0.916884 + 1.58809i 0.804119 + 0.594469i \(0.202638\pi\)
0.112766 + 0.993622i \(0.464029\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i 0.944911 − 0.327327i \(-0.106148\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 4.61268 1.39077 0.695387 − 0.718635i \(-0.255233\pi\)
0.695387 + 0.718635i \(0.255233\pi\)
\(12\) 0 0
\(13\) 2.55022 + 4.41710i 0.707303 + 1.22508i 0.965854 + 0.259087i \(0.0834215\pi\)
−0.258551 + 0.965997i \(0.583245\pi\)
\(14\) 0 0
\(15\) 2.05022 − 3.55108i 0.529363 − 0.916884i
\(16\) 0 0
\(17\) 2.25612 − 3.90772i 0.547190 − 0.947761i −0.451275 − 0.892385i \(-0.649031\pi\)
0.998466 − 0.0553765i \(-0.0176359\pi\)
\(18\) 0 0
\(19\) −1.79409 − 3.10746i −0.411593 − 0.712900i 0.583471 − 0.812134i \(-0.301694\pi\)
−0.995064 + 0.0992338i \(0.968361\pi\)
\(20\) 0 0
\(21\) 0.500000 − 0.866025i 0.109109 − 0.188982i
\(22\) 0 0
\(23\) 1.07594 0.224349 0.112174 − 0.993689i \(-0.464218\pi\)
0.112174 + 0.993689i \(0.464218\pi\)
\(24\) 0 0
\(25\) −5.90677 + 10.2308i −1.18135 + 2.04617i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.71311 −0.503812 −0.251906 − 0.967752i \(-0.581057\pi\)
−0.251906 + 0.967752i \(0.581057\pi\)
\(30\) 0 0
\(31\) 1.51225 0.271607 0.135804 − 0.990736i \(-0.456638\pi\)
0.135804 + 0.990736i \(0.456638\pi\)
\(32\) 0 0
\(33\) −2.30634 − 3.99470i −0.401482 − 0.695387i
\(34\) 0 0
\(35\) −2.05022 + 3.55108i −0.346550 + 0.600242i
\(36\) 0 0
\(37\) 5.94474 + 1.28843i 0.977309 + 0.211817i
\(38\) 0 0
\(39\) 2.55022 − 4.41710i 0.408361 − 0.707303i
\(40\) 0 0
\(41\) −3.81859 − 6.61398i −0.596363 − 1.03293i −0.993353 − 0.115108i \(-0.963279\pi\)
0.396990 − 0.917823i \(-0.370055\pi\)
\(42\) 0 0
\(43\) 3.10043 0.472811 0.236406 − 0.971654i \(-0.424031\pi\)
0.236406 + 0.971654i \(0.424031\pi\)
\(44\) 0 0
\(45\) −4.10043 −0.611256
\(46\) 0 0
\(47\) −4.71311 −0.687478 −0.343739 − 0.939065i \(-0.611694\pi\)
−0.343739 + 0.939065i \(0.611694\pi\)
\(48\) 0 0
\(49\) 3.00000 − 5.19615i 0.428571 − 0.742307i
\(50\) 0 0
\(51\) −4.51225 −0.631841
\(52\) 0 0
\(53\) −4.84431 + 8.39059i −0.665417 + 1.15254i 0.313755 + 0.949504i \(0.398413\pi\)
−0.979172 + 0.203032i \(0.934920\pi\)
\(54\) 0 0
\(55\) 9.45699 + 16.3800i 1.27518 + 2.20868i
\(56\) 0 0
\(57\) −1.79409 + 3.10746i −0.237633 + 0.411593i
\(58\) 0 0
\(59\) −6.40677 + 11.0969i −0.834091 + 1.44469i 0.0606782 + 0.998157i \(0.480674\pi\)
−0.894769 + 0.446530i \(0.852660\pi\)
\(60\) 0 0
\(61\) 3.61268 + 6.25734i 0.462556 + 0.801171i 0.999088 − 0.0427097i \(-0.0135991\pi\)
−0.536531 + 0.843880i \(0.680266\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −10.4570 + 18.1120i −1.29703 + 2.24652i
\(66\) 0 0
\(67\) −4.90677 − 8.49878i −0.599458 − 1.03829i −0.992901 − 0.118942i \(-0.962050\pi\)
0.393444 − 0.919349i \(-0.371284\pi\)
\(68\) 0 0
\(69\) −0.537969 − 0.931790i −0.0647639 − 0.112174i
\(70\) 0 0
\(71\) 3.20086 + 5.54406i 0.379873 + 0.657959i 0.991043 − 0.133540i \(-0.0426345\pi\)
−0.611171 + 0.791499i \(0.709301\pi\)
\(72\) 0 0
\(73\) −10.7376 −1.25674 −0.628371 − 0.777914i \(-0.716278\pi\)
−0.628371 + 0.777914i \(0.716278\pi\)
\(74\) 0 0
\(75\) 11.8135 1.36411
\(76\) 0 0
\(77\) 2.30634 + 3.99470i 0.262832 + 0.455238i
\(78\) 0 0
\(79\) −1.21815 − 2.10990i −0.137053 − 0.237383i 0.789327 − 0.613973i \(-0.210430\pi\)
−0.926380 + 0.376590i \(0.877096\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) 6.68862 − 11.5850i 0.734171 − 1.27162i −0.220915 − 0.975293i \(-0.570904\pi\)
0.955086 − 0.296329i \(-0.0957624\pi\)
\(84\) 0 0
\(85\) 18.5022 2.00684
\(86\) 0 0
\(87\) 1.35656 + 2.34962i 0.145438 + 0.251906i
\(88\) 0 0
\(89\) −1.58819 + 2.75082i −0.168347 + 0.291586i −0.937839 − 0.347071i \(-0.887176\pi\)
0.769492 + 0.638657i \(0.220510\pi\)
\(90\) 0 0
\(91\) −2.55022 + 4.41710i −0.267335 + 0.463038i
\(92\) 0 0
\(93\) −0.756123 − 1.30964i −0.0784063 − 0.135804i
\(94\) 0 0
\(95\) 7.35656 − 12.7419i 0.754767 − 1.30729i
\(96\) 0 0
\(97\) 18.0144 1.82909 0.914543 − 0.404489i \(-0.132551\pi\)
0.914543 + 0.404489i \(0.132551\pi\)
\(98\) 0 0
\(99\) −2.30634 + 3.99470i −0.231796 + 0.401482i
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 1776.2.q.m.1009.3 6
4.3 odd 2 888.2.q.h.121.3 ✓ 6
12.11 even 2 2664.2.r.h.1009.1 6
37.26 even 3 inner 1776.2.q.m.433.3 6
148.63 odd 6 888.2.q.h.433.3 yes 6
444.359 even 6 2664.2.r.h.433.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.q.h.121.3 ✓ 6 4.3 odd 2
888.2.q.h.433.3 yes 6 148.63 odd 6
1776.2.q.m.433.3 6 37.26 even 3 inner
1776.2.q.m.1009.3 6 1.1 even 1 trivial
2664.2.r.h.433.1 6 444.359 even 6
2664.2.r.h.1009.1 6 12.11 even 2