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Properties

Label 1776.2.q.m.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,-3,0,3,0,3,0,-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 1009.2
Root \(0.330560 + 0.572547i\) of defining polynomial
Character \(\chi\) \(=\) 1776.1009
Dual form 1776.2.q.m.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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(0.169440 + 0.293478i) q^{5} +(0.500000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} -4.56292 q^{11} +(0.669440 + 1.15950i) q^{13} +(0.169440 - 0.293478i) q^{15} +(-0.450900 + 0.780981i) q^{17} +(-2.62034 - 4.53856i) q^{19} +(0.500000 - 0.866025i) q^{21} +8.14248 q^{23} +(2.44258 - 4.23067i) q^{25} +1.00000 q^{27} +10.2240 q^{29} -3.90180 q^{31} +(2.28146 + 3.95160i) q^{33} +(-0.169440 + 0.293478i) q^{35} +(1.12866 + 5.97713i) q^{37} +(0.669440 - 1.15950i) q^{39} +(6.18326 + 10.7097i) q^{41} -0.661120 q^{43} -0.338880 q^{45} +8.22404 q^{47} +(3.00000 - 5.19615i) q^{49} +0.901800 q^{51} +(-3.78978 + 6.56409i) q^{53} +(-0.773140 - 1.33912i) q^{55} +(-2.62034 + 4.53856i) q^{57} +(1.94258 - 3.36465i) q^{59} +(-5.56292 - 9.63526i) q^{61} -1.00000 q^{63} +(-0.226860 + 0.392932i) q^{65} +(3.44258 + 5.96272i) q^{67} +(-4.07124 - 7.05159i) q^{69} +(-4.32224 - 7.48634i) q^{71} +13.0276 q^{73} -4.88516 q^{75} +(-2.28146 - 3.95160i) q^{77} +(5.02214 + 8.69860i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(4.57956 - 7.93203i) q^{83} -0.305602 q^{85} +(-5.11202 - 8.85428i) q^{87} +(-3.24068 + 5.61302i) q^{89} +(-0.669440 + 1.15950i) q^{91} +(1.95090 + 3.37906i) q^{93} +(0.887980 - 1.53803i) q^{95} -6.20740 q^{97} +(2.28146 - 3.95160i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 3 q^{5} + 3 q^{7} - 3 q^{9} + 2 q^{11} + 6 q^{13} + 3 q^{15} + 10 q^{17} - 5 q^{19} + 3 q^{21} + 2 q^{23} - 4 q^{25} + 6 q^{27} + 28 q^{29} + 2 q^{31} - q^{33} - 3 q^{35} + 2 q^{37} + 6 q^{39}+ \cdots - q^{99}+O(q^{100}) \) 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
\(5\) 0.169440 + 0.293478i 0.0757758 + 0.131248i 0.901423 − 0.432939i \(-0.142523\pi\)
−0.825648 + 0.564186i \(0.809190\pi\)
\(6\) 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
\(11\) −4.56292 −1.37577 −0.687886 − 0.725819i \(-0.741461\pi\)
−0.687886 + 0.725819i \(0.741461\pi\)
\(12\) 0 0
\(13\) 0.669440 + 1.15950i 0.185669 + 0.321588i 0.943802 − 0.330512i \(-0.107221\pi\)
−0.758133 + 0.652100i \(0.773888\pi\)
\(14\) 0 0
\(15\) 0.169440 − 0.293478i 0.0437492 − 0.0757758i
\(16\) 0 0
\(17\) −0.450900 + 0.780981i −0.109359 + 0.189416i −0.915511 − 0.402293i \(-0.868213\pi\)
0.806152 + 0.591709i \(0.201547\pi\)
\(18\) 0 0
\(19\) −2.62034 − 4.53856i −0.601147 − 1.04122i −0.992648 − 0.121040i \(-0.961377\pi\)
0.391501 − 0.920178i \(-0.371956\pi\)
\(20\) 0 0
\(21\) 0.500000 − 0.866025i 0.109109 − 0.188982i
\(22\) 0 0
\(23\) 8.14248 1.69782 0.848912 − 0.528534i \(-0.177258\pi\)
0.848912 + 0.528534i \(0.177258\pi\)
\(24\) 0 0
\(25\) 2.44258 − 4.23067i 0.488516 − 0.846135i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 10.2240 1.89856 0.949278 − 0.314437i \(-0.101816\pi\)
0.949278 + 0.314437i \(0.101816\pi\)
\(30\) 0 0
\(31\) −3.90180 −0.700784 −0.350392 − 0.936603i \(-0.613952\pi\)
−0.350392 + 0.936603i \(0.613952\pi\)
\(32\) 0 0
\(33\) 2.28146 + 3.95160i 0.397151 + 0.687886i
\(34\) 0 0
\(35\) −0.169440 + 0.293478i −0.0286406 + 0.0496069i
\(36\) 0 0
\(37\) 1.12866 + 5.97713i 0.185550 + 0.982635i
\(38\) 0 0
\(39\) 0.669440 − 1.15950i 0.107196 − 0.185669i
\(40\) 0 0
\(41\) 6.18326 + 10.7097i 0.965663 + 1.67258i 0.707823 + 0.706389i \(0.249677\pi\)
0.257840 + 0.966188i \(0.416989\pi\)
\(42\) 0 0
\(43\) −0.661120 −0.100820 −0.0504100 − 0.998729i \(-0.516053\pi\)
−0.0504100 + 0.998729i \(0.516053\pi\)
\(44\) 0 0
\(45\) −0.338880 −0.0505172
\(46\) 0 0
\(47\) 8.22404 1.19960 0.599800 − 0.800150i \(-0.295247\pi\)
0.599800 + 0.800150i \(0.295247\pi\)
\(48\) 0 0
\(49\) 3.00000 − 5.19615i 0.428571 − 0.742307i
\(50\) 0 0
\(51\) 0.901800 0.126277
\(52\) 0 0
\(53\) −3.78978 + 6.56409i −0.520566 + 0.901647i 0.479148 + 0.877734i \(0.340946\pi\)
−0.999714 + 0.0239130i \(0.992388\pi\)
\(54\) 0 0
\(55\) −0.773140 − 1.33912i −0.104250 − 0.180567i
\(56\) 0 0
\(57\) −2.62034 + 4.53856i −0.347072 + 0.601147i
\(58\) 0 0
\(59\) 1.94258 − 3.36465i 0.252902 − 0.438040i −0.711421 − 0.702766i \(-0.751948\pi\)
0.964324 + 0.264726i \(0.0852815\pi\)
\(60\) 0 0
\(61\) −5.56292 − 9.63526i −0.712259 − 1.23367i −0.964007 − 0.265876i \(-0.914339\pi\)
0.251748 − 0.967793i \(-0.418994\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −0.226860 + 0.392932i −0.0281385 + 0.0487373i
\(66\) 0 0
\(67\) 3.44258 + 5.96272i 0.420578 + 0.728463i 0.995996 − 0.0893968i \(-0.0284939\pi\)
−0.575418 + 0.817860i \(0.695161\pi\)
\(68\) 0 0
\(69\) −4.07124 − 7.05159i −0.490120 − 0.848912i
\(70\) 0 0
\(71\) −4.32224 − 7.48634i −0.512956 − 0.888465i −0.999887 − 0.0150250i \(-0.995217\pi\)
0.486932 − 0.873440i \(-0.338116\pi\)
\(72\) 0 0
\(73\) 13.0276 1.52477 0.762385 − 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(74\) 0 0
\(75\) −4.88516 −0.564090
\(76\) 0 0
\(77\) −2.28146 − 3.95160i −0.259996 − 0.450327i
\(78\) 0 0
\(79\) 5.02214 + 8.69860i 0.565035 + 0.978669i 0.997046 + 0.0768010i \(0.0244706\pi\)
−0.432012 + 0.901868i \(0.642196\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) 4.57956 − 7.93203i 0.502672 − 0.870653i −0.497323 − 0.867565i \(-0.665684\pi\)
0.999995 − 0.00308800i \(-0.000982943\pi\)
\(84\) 0 0
\(85\) −0.305602 −0.0331471
\(86\) 0 0
\(87\) −5.11202 − 8.85428i −0.548066 − 0.949278i
\(88\) 0 0
\(89\) −3.24068 + 5.61302i −0.343511 + 0.594979i −0.985082 − 0.172085i \(-0.944950\pi\)
0.641571 + 0.767064i \(0.278283\pi\)
\(90\) 0 0
\(91\) −0.669440 + 1.15950i −0.0701764 + 0.121549i
\(92\) 0 0
\(93\) 1.95090 + 3.37906i 0.202299 + 0.350392i
\(94\) 0 0
\(95\) 0.887980 − 1.53803i 0.0911048 − 0.157798i
\(96\) 0 0
\(97\) −6.20740 −0.630266 −0.315133 − 0.949048i \(-0.602049\pi\)
−0.315133 + 0.949048i \(0.602049\pi\)
\(98\) 0 0
\(99\) 2.28146 − 3.95160i 0.229295 − 0.397151i
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.1009.2 6
4.3 odd 2 888.2.q.h.121.2 ✓ 6
12.11 even 2 2664.2.r.h.1009.2 6
37.26 even 3 inner 1776.2.q.m.433.2 6
148.63 odd 6 888.2.q.h.433.2 yes 6
444.359 even 6 2664.2.r.h.433.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.q.h.121.2 ✓ 6 4.3 odd 2
888.2.q.h.433.2 yes 6 148.63 odd 6
1776.2.q.m.433.2 6 37.26 even 3 inner
1776.2.q.m.1009.2 6 1.1 even 1 trivial
2664.2.r.h.433.2 6 444.359 even 6
2664.2.r.h.1009.2 6 12.11 even 2