↑

Properties

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

Related objects

Downloads

Learn more

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,-3,0,1,0,-9,0,-3,0,-10] 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: 6.0.47545083.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 26x^{4} - 47x^{3} + 154x^{2} - 131x + 37 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
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 \(0.500000 + 3.63975i\) of defining polynomial
Character \(\chi\) \(=\) 1776.1009
Dual form 1776.2.q.k.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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(1.62445 + 2.81363i) q^{5} +(-1.50000 - 2.59808i) q^{7} +(-0.500000 + 0.866025i) q^{9} -4.55534 q^{11} +(0.124449 + 0.215552i) q^{13} +(1.62445 - 2.81363i) q^{15} +(2.90212 - 5.02662i) q^{17} +(1.27767 + 2.21299i) q^{19} +(-1.50000 + 2.59808i) q^{21} +3.24890 q^{23} +(-2.77767 + 4.81106i) q^{25} +1.00000 q^{27} -10.3020 q^{29} -6.80424 q^{31} +(2.27767 + 3.94504i) q^{33} +(4.87335 - 8.44088i) q^{35} +(-4.59568 - 3.98494i) q^{37} +(0.124449 - 0.215552i) q^{39} +(0.277669 + 0.480937i) q^{41} -10.7467 q^{43} -3.24890 q^{45} +10.3020 q^{47} +(-1.00000 + 1.73205i) q^{49} -5.80424 q^{51} +(0.653220 - 1.13141i) q^{53} +(-7.39991 - 12.8170i) q^{55} +(1.27767 - 2.21299i) q^{57} +(-1.27767 + 2.21299i) q^{59} +(-5.55534 - 9.62213i) q^{61} +3.00000 q^{63} +(-0.404322 + 0.700306i) q^{65} +(1.02657 + 1.77807i) q^{67} +(-1.62445 - 2.81363i) q^{69} +(-7.49780 - 12.9866i) q^{71} -3.30203 q^{73} +5.55534 q^{75} +(6.83301 + 11.8351i) q^{77} +(2.02657 + 3.51012i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(4.00000 - 6.92820i) q^{83} +18.8574 q^{85} +(5.15102 + 8.92182i) q^{87} +(4.24890 - 7.35931i) q^{89} +(0.373347 - 0.646656i) q^{91} +(3.40212 + 5.89264i) q^{93} +(-4.15102 + 7.18977i) q^{95} +2.05313 q^{97} +(2.27767 - 3.94504i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + q^{5} - 9 q^{7} - 3 q^{9} - 10 q^{11} - 8 q^{13} + q^{15} - q^{19} - 9 q^{21} + 2 q^{23} - 8 q^{25} + 6 q^{27} + 8 q^{29} - 6 q^{31} + 5 q^{33} + 3 q^{35} - 10 q^{37} - 8 q^{39} - 7 q^{41}+ \cdots + 5 q^{99}+O(q^{100}) \) 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
\(5\) 1.62445 + 2.81363i 0.726476 + 1.25829i 0.958364 + 0.285550i \(0.0921764\pi\)
−0.231888 + 0.972742i \(0.574490\pi\)
\(6\) 0 0
\(7\) −1.50000 − 2.59808i −0.566947 − 0.981981i −0.996866 − 0.0791130i \(-0.974791\pi\)
0.429919 − 0.902867i \(-0.358542\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −4.55534 −1.37349 −0.686743 − 0.726900i \(-0.740960\pi\)
−0.686743 + 0.726900i \(0.740960\pi\)
\(12\) 0 0
\(13\) 0.124449 + 0.215552i 0.0345159 + 0.0597833i 0.882767 − 0.469810i \(-0.155678\pi\)
−0.848251 + 0.529594i \(0.822344\pi\)
\(14\) 0 0
\(15\) 1.62445 − 2.81363i 0.419431 − 0.726476i
\(16\) 0 0
\(17\) 2.90212 − 5.02662i 0.703867 − 1.21913i −0.263232 − 0.964733i \(-0.584788\pi\)
0.967099 − 0.254401i \(-0.0818782\pi\)
\(18\) 0 0
\(19\) 1.27767 + 2.21299i 0.293117 + 0.507694i 0.974545 − 0.224191i \(-0.0719740\pi\)
−0.681428 + 0.731885i \(0.738641\pi\)
\(20\) 0 0
\(21\) −1.50000 + 2.59808i −0.327327 + 0.566947i
\(22\) 0 0
\(23\) 3.24890 0.677442 0.338721 − 0.940887i \(-0.390006\pi\)
0.338721 + 0.940887i \(0.390006\pi\)
\(24\) 0 0
\(25\) −2.77767 + 4.81106i −0.555534 + 0.962213i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −10.3020 −1.91304 −0.956520 − 0.291668i \(-0.905790\pi\)
−0.956520 + 0.291668i \(0.905790\pi\)
\(30\) 0 0
\(31\) −6.80424 −1.22208 −0.611038 − 0.791601i \(-0.709248\pi\)
−0.611038 + 0.791601i \(0.709248\pi\)
\(32\) 0 0
\(33\) 2.27767 + 3.94504i 0.396491 + 0.686743i
\(34\) 0 0
\(35\) 4.87335 − 8.44088i 0.823746 − 1.42677i
\(36\) 0 0
\(37\) −4.59568 − 3.98494i −0.755525 − 0.655120i
\(38\) 0 0
\(39\) 0.124449 − 0.215552i 0.0199278 − 0.0345159i
\(40\) 0 0
\(41\) 0.277669 + 0.480937i 0.0433646 + 0.0751097i 0.886893 − 0.461975i \(-0.152859\pi\)
−0.843528 + 0.537084i \(0.819526\pi\)
\(42\) 0 0
\(43\) −10.7467 −1.63886 −0.819428 − 0.573183i \(-0.805709\pi\)
−0.819428 + 0.573183i \(0.805709\pi\)
\(44\) 0 0
\(45\) −3.24890 −0.484317
\(46\) 0 0
\(47\) 10.3020 1.50271 0.751353 − 0.659901i \(-0.229402\pi\)
0.751353 + 0.659901i \(0.229402\pi\)
\(48\) 0 0
\(49\) −1.00000 + 1.73205i −0.142857 + 0.247436i
\(50\) 0 0
\(51\) −5.80424 −0.812756
\(52\) 0 0
\(53\) 0.653220 − 1.13141i 0.0897267 − 0.155411i −0.817669 − 0.575689i \(-0.804734\pi\)
0.907396 + 0.420278i \(0.138067\pi\)
\(54\) 0 0
\(55\) −7.39991 − 12.8170i −0.997804 − 1.72825i
\(56\) 0 0
\(57\) 1.27767 − 2.21299i 0.169231 − 0.293117i
\(58\) 0 0
\(59\) −1.27767 + 2.21299i −0.166338 + 0.288106i −0.937130 − 0.348981i \(-0.886528\pi\)
0.770791 + 0.637088i \(0.219861\pi\)
\(60\) 0 0
\(61\) −5.55534 − 9.62213i −0.711288 − 1.23199i −0.964374 − 0.264543i \(-0.914779\pi\)
0.253086 − 0.967444i \(-0.418555\pi\)
\(62\) 0 0
\(63\) 3.00000 0.377964
\(64\) 0 0
\(65\) −0.404322 + 0.700306i −0.0501500 + 0.0868623i
\(66\) 0 0
\(67\) 1.02657 + 1.77807i 0.125415 + 0.217225i 0.921895 − 0.387440i \(-0.126640\pi\)
−0.796480 + 0.604665i \(0.793307\pi\)
\(68\) 0 0
\(69\) −1.62445 − 2.81363i −0.195561 − 0.338721i
\(70\) 0 0
\(71\) −7.49780 − 12.9866i −0.889825 − 1.54122i −0.840082 − 0.542459i \(-0.817493\pi\)
−0.0497427 − 0.998762i \(-0.515840\pi\)
\(72\) 0 0
\(73\) −3.30203 −0.386474 −0.193237 − 0.981152i \(-0.561899\pi\)
−0.193237 + 0.981152i \(0.561899\pi\)
\(74\) 0 0
\(75\) 5.55534 0.641475
\(76\) 0 0
\(77\) 6.83301 + 11.8351i 0.778693 + 1.34874i
\(78\) 0 0
\(79\) 2.02657 + 3.51012i 0.228007 + 0.394919i 0.957217 − 0.289370i \(-0.0934459\pi\)
−0.729211 + 0.684289i \(0.760113\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) 4.00000 − 6.92820i 0.439057 − 0.760469i −0.558560 − 0.829464i \(-0.688646\pi\)
0.997617 + 0.0689950i \(0.0219793\pi\)
\(84\) 0 0
\(85\) 18.8574 2.04537
\(86\) 0 0
\(87\) 5.15102 + 8.92182i 0.552247 + 0.956520i
\(88\) 0 0
\(89\) 4.24890 − 7.35931i 0.450382 − 0.780085i −0.548027 − 0.836460i \(-0.684621\pi\)
0.998410 + 0.0563754i \(0.0179544\pi\)
\(90\) 0 0
\(91\) 0.373347 − 0.646656i 0.0391374 − 0.0677879i
\(92\) 0 0
\(93\) 3.40212 + 5.89264i 0.352783 + 0.611038i
\(94\) 0 0
\(95\) −4.15102 + 7.18977i −0.425885 + 0.737655i
\(96\) 0 0
\(97\) 2.05313 0.208464 0.104232 − 0.994553i \(-0.466762\pi\)
0.104232 + 0.994553i \(0.466762\pi\)
\(98\) 0 0
\(99\) 2.27767 − 3.94504i 0.228914 − 0.396491i
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.k.1009.3 6
4.3 odd 2 888.2.q.g.121.3 ✓ 6
12.11 even 2 2664.2.r.j.1009.1 6
37.26 even 3 inner 1776.2.q.k.433.3 6
148.63 odd 6 888.2.q.g.433.3 yes 6
444.359 even 6 2664.2.r.j.433.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.q.g.121.3 ✓ 6 4.3 odd 2
888.2.q.g.433.3 yes 6 148.63 odd 6
1776.2.q.k.433.3 6 37.26 even 3 inner
1776.2.q.k.1009.3 6 1.1 even 1 trivial
2664.2.r.j.433.1 6 444.359 even 6
2664.2.r.j.1009.1 6 12.11 even 2