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Properties

Label 1776.2.q.k.1009.2
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,-3,0,1,0,-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 1009.2
Root \(0.500000 - 2.99239i\) of defining polynomial
Character \(\chi\) \(=\) 1776.1009
Dual form 1776.2.q.k.433.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(0.551098 + 0.954529i) q^{5} +(-1.50000 - 2.59808i) q^{7} +(-0.500000 + 0.866025i) q^{9} +4.78517 q^{11} +(-0.948902 - 1.64355i) q^{13} +(0.551098 - 0.954529i) q^{15} +(-2.84148 + 4.92160i) q^{17} +(-3.39258 - 5.87613i) q^{19} +(-1.50000 + 2.59808i) q^{21} +1.10220 q^{23} +(1.89258 - 3.27805i) q^{25} +1.00000 q^{27} +5.47858 q^{29} +4.68297 q^{31} +(-2.39258 - 4.14407i) q^{33} +(1.65329 - 2.86359i) q^{35} +(-6.04588 - 0.668869i) q^{37} +(-0.948902 + 1.64355i) q^{39} +(-4.39258 - 7.60818i) q^{41} -4.30659 q^{43} -1.10220 q^{45} -5.47858 q^{47} +(-1.00000 + 1.73205i) q^{49} +5.68297 q^{51} +(-2.94368 + 5.09860i) q^{53} +(2.63709 + 4.56758i) q^{55} +(-3.39258 + 5.87613i) q^{57} +(3.39258 - 5.87613i) q^{59} +(3.78517 + 6.55610i) q^{61} +3.00000 q^{63} +(1.04588 - 1.81151i) q^{65} +(-5.79039 - 10.0292i) q^{67} +(-0.551098 - 0.954529i) q^{69} +(-3.20439 - 5.55017i) q^{71} +12.4786 q^{73} -3.78517 q^{75} +(-7.17775 - 12.4322i) q^{77} +(-4.79039 - 8.29719i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(4.00000 - 6.92820i) q^{83} -6.26374 q^{85} +(-2.73929 - 4.74459i) q^{87} +(2.10220 - 3.64111i) q^{89} +(-2.84671 + 4.93064i) q^{91} +(-2.34148 - 4.05557i) q^{93} +(3.73929 - 6.47664i) q^{95} -11.5808 q^{97} +(-2.39258 + 4.14407i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + q^{5} - 9 q^{7} - 3 q^{9} - 10 q^{11} - 8 q^{13} + q^{15} - q^{19} - 9 q^{21} + 2 q^{23} - 8 q^{25} + 6 q^{27} + 8 q^{29} - 6 q^{31} + 5 q^{33} + 3 q^{35} - 10 q^{37} - 8 q^{39} - 7 q^{41}+ \cdots + 5 q^{99}+O(q^{100}) \) 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
\(5\) 0.551098 + 0.954529i 0.246458 + 0.426878i 0.962541 − 0.271137i \(-0.0873998\pi\)
−0.716082 + 0.698016i \(0.754066\pi\)
\(6\) 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
\(11\) 4.78517 1.44278 0.721391 − 0.692528i \(-0.243503\pi\)
0.721391 + 0.692528i \(0.243503\pi\)
\(12\) 0 0
\(13\) −0.948902 − 1.64355i −0.263178 − 0.455838i 0.703907 − 0.710293i \(-0.251437\pi\)
−0.967085 + 0.254455i \(0.918104\pi\)
\(14\) 0 0
\(15\) 0.551098 − 0.954529i 0.142293 − 0.246458i
\(16\) 0 0
\(17\) −2.84148 + 4.92160i −0.689161 + 1.19366i 0.282948 + 0.959135i \(0.408688\pi\)
−0.972110 + 0.234527i \(0.924646\pi\)
\(18\) 0 0
\(19\) −3.39258 − 5.87613i −0.778312 − 1.34808i −0.932914 − 0.360099i \(-0.882743\pi\)
0.154602 − 0.987977i \(-0.450590\pi\)
\(20\) 0 0
\(21\) −1.50000 + 2.59808i −0.327327 + 0.566947i
\(22\) 0 0
\(23\) 1.10220 0.229824 0.114912 − 0.993376i \(-0.463341\pi\)
0.114912 + 0.993376i \(0.463341\pi\)
\(24\) 0 0
\(25\) 1.89258 − 3.27805i 0.378517 − 0.655610i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.47858 1.01735 0.508673 − 0.860960i \(-0.330136\pi\)
0.508673 + 0.860960i \(0.330136\pi\)
\(30\) 0 0
\(31\) 4.68297 0.841086 0.420543 − 0.907273i \(-0.361840\pi\)
0.420543 + 0.907273i \(0.361840\pi\)
\(32\) 0 0
\(33\) −2.39258 − 4.14407i −0.416495 − 0.721391i
\(34\) 0 0
\(35\) 1.65329 − 2.86359i 0.279458 − 0.484035i
\(36\) 0 0
\(37\) −6.04588 − 0.668869i −0.993936 − 0.109961i
\(38\) 0 0
\(39\) −0.948902 + 1.64355i −0.151946 + 0.263178i
\(40\) 0 0
\(41\) −4.39258 − 7.60818i −0.686006 − 1.18820i −0.973119 − 0.230301i \(-0.926029\pi\)
0.287113 − 0.957897i \(-0.407304\pi\)
\(42\) 0 0
\(43\) −4.30659 −0.656748 −0.328374 − 0.944548i \(-0.606501\pi\)
−0.328374 + 0.944548i \(0.606501\pi\)
\(44\) 0 0
\(45\) −1.10220 −0.164306
\(46\) 0 0
\(47\) −5.47858 −0.799133 −0.399566 − 0.916704i \(-0.630839\pi\)
−0.399566 + 0.916704i \(0.630839\pi\)
\(48\) 0 0
\(49\) −1.00000 + 1.73205i −0.142857 + 0.247436i
\(50\) 0 0
\(51\) 5.68297 0.795775
\(52\) 0 0
\(53\) −2.94368 + 5.09860i −0.404346 + 0.700347i −0.994245 − 0.107130i \(-0.965834\pi\)
0.589899 + 0.807477i \(0.299167\pi\)
\(54\) 0 0
\(55\) 2.63709 + 4.56758i 0.355586 + 0.615892i
\(56\) 0 0
\(57\) −3.39258 + 5.87613i −0.449359 + 0.778312i
\(58\) 0 0
\(59\) 3.39258 − 5.87613i 0.441677 − 0.765006i −0.556137 − 0.831090i \(-0.687717\pi\)
0.997814 + 0.0660840i \(0.0210505\pi\)
\(60\) 0 0
\(61\) 3.78517 + 6.55610i 0.484641 + 0.839422i 0.999844 − 0.0176455i \(-0.00561702\pi\)
−0.515204 + 0.857068i \(0.672284\pi\)
\(62\) 0 0
\(63\) 3.00000 0.377964
\(64\) 0 0
\(65\) 1.04588 − 1.81151i 0.129725 − 0.224690i
\(66\) 0 0
\(67\) −5.79039 − 10.0292i −0.707408 − 1.22527i −0.965815 − 0.259231i \(-0.916531\pi\)
0.258407 − 0.966036i \(-0.416802\pi\)
\(68\) 0 0
\(69\) −0.551098 − 0.954529i −0.0663444 − 0.114912i
\(70\) 0 0
\(71\) −3.20439 − 5.55017i −0.380291 − 0.658684i 0.610813 − 0.791775i \(-0.290843\pi\)
−0.991104 + 0.133092i \(0.957510\pi\)
\(72\) 0 0
\(73\) 12.4786 1.46051 0.730254 − 0.683176i \(-0.239402\pi\)
0.730254 + 0.683176i \(0.239402\pi\)
\(74\) 0 0
\(75\) −3.78517 −0.437073
\(76\) 0 0
\(77\) −7.17775 − 12.4322i −0.817980 − 1.41678i
\(78\) 0 0
\(79\) −4.79039 − 8.29719i −0.538961 − 0.933507i −0.998960 − 0.0455881i \(-0.985484\pi\)
0.460000 − 0.887919i \(-0.347850\pi\)
\(80\) 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
\(85\) −6.26374 −0.679398
\(86\) 0 0
\(87\) −2.73929 − 4.74459i −0.293683 − 0.508673i
\(88\) 0 0
\(89\) 2.10220 − 3.64111i 0.222832 − 0.385957i −0.732835 − 0.680407i \(-0.761803\pi\)
0.955667 + 0.294450i \(0.0951364\pi\)
\(90\) 0 0
\(91\) −2.84671 + 4.93064i −0.298416 + 0.516872i
\(92\) 0 0
\(93\) −2.34148 − 4.05557i −0.242801 − 0.420543i
\(94\) 0 0
\(95\) 3.73929 − 6.47664i 0.383643 − 0.664489i
\(96\) 0 0
\(97\) −11.5808 −1.17585 −0.587925 − 0.808916i \(-0.700055\pi\)
−0.587925 + 0.808916i \(0.700055\pi\)
\(98\) 0 0
\(99\) −2.39258 + 4.14407i −0.240464 + 0.416495i
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.1009.2 6
4.3 odd 2 888.2.q.g.121.2 ✓ 6
12.11 even 2 2664.2.r.j.1009.2 6
37.26 even 3 inner 1776.2.q.k.433.2 6
148.63 odd 6 888.2.q.g.433.2 yes 6
444.359 even 6 2664.2.r.j.433.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.q.g.121.2 ✓ 6 4.3 odd 2
888.2.q.g.433.2 yes 6 148.63 odd 6
1776.2.q.k.433.2 6 37.26 even 3 inner
1776.2.q.k.1009.2 6 1.1 even 1 trivial
2664.2.r.j.433.2 6 444.359 even 6
2664.2.r.j.1009.2 6 12.11 even 2