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Properties

Label 1776.2.q.k.1009.1
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 1009.1
Root \(0.500000 + 0.218662i\) of defining polynomial
Character \(\chi\) \(=\) 1776.1009
Dual form 1776.2.q.k.433.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(-1.67555 - 2.90213i) q^{5} +(-1.50000 - 2.59808i) q^{7} +(-0.500000 + 0.866025i) q^{9} -5.22983 q^{11} +(-3.17555 - 5.50021i) q^{13} +(-1.67555 + 2.90213i) q^{15} +(-0.0606331 + 0.105020i) q^{17} +(1.61491 + 2.79711i) q^{19} +(-1.50000 + 2.59808i) q^{21} -3.35109 q^{23} +(-3.11491 + 5.39519i) q^{25} +1.00000 q^{27} +8.82345 q^{29} -0.878734 q^{31} +(2.61491 + 4.52916i) q^{33} +(-5.02664 + 8.70640i) q^{35} +(5.64155 - 2.27439i) q^{37} +(-3.17555 + 5.50021i) q^{39} +(0.614914 + 1.06506i) q^{41} +9.05328 q^{43} +3.35109 q^{45} -8.82345 q^{47} +(-1.00000 + 1.73205i) q^{49} +0.121266 q^{51} +(4.29046 - 7.43130i) q^{53} +(8.76282 + 15.1776i) q^{55} +(1.61491 - 2.79711i) q^{57} +(-1.61491 + 2.79711i) q^{59} +(-6.22983 - 10.7904i) q^{61} +3.00000 q^{63} +(-10.6416 + 18.4317i) q^{65} +(-5.23618 - 9.06933i) q^{67} +(1.67555 + 2.90213i) q^{69} +(5.70219 + 9.87648i) q^{71} +15.8235 q^{73} +6.22983 q^{75} +(7.84474 + 13.5875i) q^{77} +(-4.23618 - 7.33728i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(4.00000 - 6.92820i) q^{83} +0.406374 q^{85} +(-4.41173 - 7.64133i) q^{87} +(-2.35109 + 4.07221i) q^{89} +(-9.52664 + 16.5006i) q^{91} +(0.439367 + 0.761006i) q^{93} +(5.41173 - 9.37339i) q^{95} -10.4724 q^{97} +(2.61491 - 4.52916i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + q^{5} - 9 q^{7} - 3 q^{9} - 10 q^{11} - 8 q^{13} + q^{15} - q^{19} - 9 q^{21} + 2 q^{23} - 8 q^{25} + 6 q^{27} + 8 q^{29} - 6 q^{31} + 5 q^{33} + 3 q^{35} - 10 q^{37} - 8 q^{39} - 7 q^{41}+ \cdots + 5 q^{99}+O(q^{100}) \) 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
\(5\) −1.67555 − 2.90213i −0.749327 − 1.29787i −0.948146 − 0.317837i \(-0.897044\pi\)
0.198818 − 0.980036i \(-0.436290\pi\)
\(6\) 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
\(11\) −5.22983 −1.57685 −0.788426 − 0.615129i \(-0.789104\pi\)
−0.788426 + 0.615129i \(0.789104\pi\)
\(12\) 0 0
\(13\) −3.17555 − 5.50021i −0.880738 − 1.52548i −0.850522 − 0.525940i \(-0.823714\pi\)
−0.0302165 − 0.999543i \(-0.509620\pi\)
\(14\) 0 0
\(15\) −1.67555 + 2.90213i −0.432624 + 0.749327i
\(16\) 0 0
\(17\) −0.0606331 + 0.105020i −0.0147057 + 0.0254710i −0.873285 − 0.487210i \(-0.838014\pi\)
0.858579 + 0.512681i \(0.171348\pi\)
\(18\) 0 0
\(19\) 1.61491 + 2.79711i 0.370487 + 0.641702i 0.989640 − 0.143568i \(-0.0458576\pi\)
−0.619154 + 0.785270i \(0.712524\pi\)
\(20\) 0 0
\(21\) −1.50000 + 2.59808i −0.327327 + 0.566947i
\(22\) 0 0
\(23\) −3.35109 −0.698751 −0.349376 − 0.936983i \(-0.613606\pi\)
−0.349376 + 0.936983i \(0.613606\pi\)
\(24\) 0 0
\(25\) −3.11491 + 5.39519i −0.622983 + 1.07904i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 8.82345 1.63847 0.819237 − 0.573455i \(-0.194397\pi\)
0.819237 + 0.573455i \(0.194397\pi\)
\(30\) 0 0
\(31\) −0.878734 −0.157825 −0.0789126 − 0.996882i \(-0.525145\pi\)
−0.0789126 + 0.996882i \(0.525145\pi\)
\(32\) 0 0
\(33\) 2.61491 + 4.52916i 0.455198 + 0.788426i
\(34\) 0 0
\(35\) −5.02664 + 8.70640i −0.849657 + 1.47165i
\(36\) 0 0
\(37\) 5.64155 − 2.27439i 0.927466 − 0.373908i
\(38\) 0 0
\(39\) −3.17555 + 5.50021i −0.508494 + 0.880738i
\(40\) 0 0
\(41\) 0.614914 + 1.06506i 0.0960334 + 0.166335i 0.910039 − 0.414522i \(-0.136051\pi\)
−0.814006 + 0.580856i \(0.802718\pi\)
\(42\) 0 0
\(43\) 9.05328 1.38061 0.690306 − 0.723517i \(-0.257476\pi\)
0.690306 + 0.723517i \(0.257476\pi\)
\(44\) 0 0
\(45\) 3.35109 0.499552
\(46\) 0 0
\(47\) −8.82345 −1.28703 −0.643516 − 0.765432i \(-0.722525\pi\)
−0.643516 + 0.765432i \(0.722525\pi\)
\(48\) 0 0
\(49\) −1.00000 + 1.73205i −0.142857 + 0.247436i
\(50\) 0 0
\(51\) 0.121266 0.0169807
\(52\) 0 0
\(53\) 4.29046 − 7.43130i 0.589340 − 1.02077i −0.404979 − 0.914326i \(-0.632721\pi\)
0.994319 − 0.106441i \(-0.0339455\pi\)
\(54\) 0 0
\(55\) 8.76282 + 15.1776i 1.18158 + 2.04655i
\(56\) 0 0
\(57\) 1.61491 − 2.79711i 0.213901 − 0.370487i
\(58\) 0 0
\(59\) −1.61491 + 2.79711i −0.210244 + 0.364153i −0.951791 − 0.306748i \(-0.900759\pi\)
0.741547 + 0.670901i \(0.234092\pi\)
\(60\) 0 0
\(61\) −6.22983 − 10.7904i −0.797648 − 1.38157i −0.921144 − 0.389221i \(-0.872744\pi\)
0.123497 − 0.992345i \(-0.460589\pi\)
\(62\) 0 0
\(63\) 3.00000 0.377964
\(64\) 0 0
\(65\) −10.6416 + 18.4317i −1.31992 + 2.28617i
\(66\) 0 0
\(67\) −5.23618 − 9.06933i −0.639701 − 1.10799i −0.985498 − 0.169686i \(-0.945725\pi\)
0.345797 − 0.938309i \(-0.387609\pi\)
\(68\) 0 0
\(69\) 1.67555 + 2.90213i 0.201712 + 0.349376i
\(70\) 0 0
\(71\) 5.70219 + 9.87648i 0.676725 + 1.17212i 0.975961 + 0.217943i \(0.0699348\pi\)
−0.299236 + 0.954179i \(0.596732\pi\)
\(72\) 0 0
\(73\) 15.8235 1.85200 0.925998 − 0.377530i \(-0.123226\pi\)
0.925998 + 0.377530i \(0.123226\pi\)
\(74\) 0 0
\(75\) 6.22983 0.719358
\(76\) 0 0
\(77\) 7.84474 + 13.5875i 0.893991 + 1.54844i
\(78\) 0 0
\(79\) −4.23618 − 7.33728i −0.476607 − 0.825508i 0.523033 − 0.852312i \(-0.324800\pi\)
−0.999641 + 0.0268039i \(0.991467\pi\)
\(80\) 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
\(85\) 0.406374 0.0440775
\(86\) 0 0
\(87\) −4.41173 − 7.64133i −0.472987 − 0.819237i
\(88\) 0 0
\(89\) −2.35109 + 4.07221i −0.249215 + 0.431654i −0.963308 − 0.268397i \(-0.913506\pi\)
0.714093 + 0.700051i \(0.246839\pi\)
\(90\) 0 0
\(91\) −9.52664 + 16.5006i −0.998663 + 1.72974i
\(92\) 0 0
\(93\) 0.439367 + 0.761006i 0.0455602 + 0.0789126i
\(94\) 0 0
\(95\) 5.41173 − 9.37339i 0.555231 − 0.961689i
\(96\) 0 0
\(97\) −10.4724 −1.06331 −0.531654 − 0.846962i \(-0.678429\pi\)
−0.531654 + 0.846962i \(0.678429\pi\)
\(98\) 0 0
\(99\) 2.61491 − 4.52916i 0.262809 − 0.455198i
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.1009.1 6
4.3 odd 2 888.2.q.g.121.1 ✓ 6
12.11 even 2 2664.2.r.j.1009.3 6
37.26 even 3 inner 1776.2.q.k.433.1 6
148.63 odd 6 888.2.q.g.433.1 yes 6
444.359 even 6 2664.2.r.j.433.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.q.g.121.1 ✓ 6 4.3 odd 2
888.2.q.g.433.1 yes 6 148.63 odd 6
1776.2.q.k.433.1 6 37.26 even 3 inner
1776.2.q.k.1009.1 6 1.1 even 1 trivial
2664.2.r.j.433.3 6 444.359 even 6
2664.2.r.j.1009.3 6 12.11 even 2