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Properties

Label 2664.2.r.h.433.2
Level $2664$
Weight $2$
Character 2664.433
Analytic conductor $21.272$
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] = [2664,2,Mod(433,2664)] 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("2664.433"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2664, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2664 = 2^{3} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2664.r (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-3,0,-3,0,0,0,2] 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: \(21.2721470985\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 888)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 433.2
Root \(0.330560 - 0.572547i\) of defining polynomial
Character \(\chi\) \(=\) 2664.433
Dual form 2664.2.r.h.1009.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.169440 + 0.293478i) q^{5} +(-0.500000 + 0.866025i) q^{7} -4.56292 q^{11} +(0.669440 - 1.15950i) q^{13} +(0.450900 + 0.780981i) q^{17} +(2.62034 - 4.53856i) q^{19} +8.14248 q^{23} +(2.44258 + 4.23067i) q^{25} -10.2240 q^{29} +3.90180 q^{31} +(-0.169440 - 0.293478i) q^{35} +(1.12866 - 5.97713i) q^{37} +(-6.18326 + 10.7097i) q^{41} +0.661120 q^{43} +8.22404 q^{47} +(3.00000 + 5.19615i) q^{49} +(3.78978 + 6.56409i) q^{53} +(0.773140 - 1.33912i) q^{55} +(1.94258 + 3.36465i) q^{59} +(-5.56292 + 9.63526i) q^{61} +(0.226860 + 0.392932i) q^{65} +(-3.44258 + 5.96272i) q^{67} +(-4.32224 + 7.48634i) q^{71} +13.0276 q^{73} +(2.28146 - 3.95160i) q^{77} +(-5.02214 + 8.69860i) q^{79} +(4.57956 + 7.93203i) q^{83} -0.305602 q^{85} +(3.24068 + 5.61302i) q^{89} +(0.669440 + 1.15950i) q^{91} +(0.887980 + 1.53803i) q^{95} -6.20740 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} - 3 q^{7} + 2 q^{11} + 6 q^{13} - 10 q^{17} + 5 q^{19} + 2 q^{23} - 4 q^{25} - 28 q^{29} - 2 q^{31} - 3 q^{35} + 2 q^{37} + 3 q^{41} + 16 q^{47} + 18 q^{49} + 14 q^{53} - 16 q^{55} - 7 q^{59}+ \cdots + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2664\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1333\) \(1999\) \(2369\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) −0.169440 + 0.293478i −0.0757758 + 0.131248i −0.901423 − 0.432939i \(-0.857477\pi\)
0.825648 + 0.564186i \(0.190810\pi\)
\(6\) 0 0
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i −0.944911 − 0.327327i \(-0.893852\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 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 −0.758133 − 0.652100i \(-0.773888\pi\)
0.943802 + 0.330512i \(0.107221\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.450900 + 0.780981i 0.109359 + 0.189416i 0.915511 − 0.402293i \(-0.131787\pi\)
−0.806152 + 0.591709i \(0.798453\pi\)
\(18\) 0 0
\(19\) 2.62034 − 4.53856i 0.601147 − 1.04122i −0.391501 − 0.920178i \(-0.628044\pi\)
0.992648 − 0.121040i \(-0.0386228\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 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
\(27\) 0 0
\(28\) 0 0
\(29\) −10.2240 −1.89856 −0.949278 − 0.314437i \(-0.898184\pi\)
−0.949278 + 0.314437i \(0.898184\pi\)
\(30\) 0 0
\(31\) 3.90180 0.700784 0.350392 − 0.936603i \(-0.386048\pi\)
0.350392 + 0.936603i \(0.386048\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 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 0
\(40\) 0 0
\(41\) −6.18326 + 10.7097i −0.965663 + 1.67258i −0.257840 + 0.966188i \(0.583011\pi\)
−0.707823 + 0.706389i \(0.750323\pi\)
\(42\) 0 0
\(43\) 0.661120 0.100820 0.0504100 − 0.998729i \(-0.483947\pi\)
0.0504100 + 0.998729i \(0.483947\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 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 0
\(52\) 0 0
\(53\) 3.78978 + 6.56409i 0.520566 + 0.901647i 0.999714 + 0.0239130i \(0.00761246\pi\)
−0.479148 + 0.877734i \(0.659054\pi\)
\(54\) 0 0
\(55\) 0.773140 − 1.33912i 0.104250 − 0.180567i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.94258 + 3.36465i 0.252902 + 0.438040i 0.964324 − 0.264726i \(-0.0852815\pi\)
−0.711421 + 0.702766i \(0.751948\pi\)
\(60\) 0 0
\(61\) −5.56292 + 9.63526i −0.712259 + 1.23367i 0.251748 + 0.967793i \(0.418994\pi\)
−0.964007 + 0.265876i \(0.914339\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 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 \(-0.971506\pi\)
0.575418 + 0.817860i \(0.304839\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.32224 + 7.48634i −0.512956 + 0.888465i 0.486932 + 0.873440i \(0.338116\pi\)
−0.999887 + 0.0150250i \(0.995217\pi\)
\(72\) 0 0
\(73\) 13.0276 1.52477 0.762385 − 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.28146 − 3.95160i 0.259996 − 0.450327i
\(78\) 0 0
\(79\) −5.02214 + 8.69860i −0.565035 + 0.978669i 0.432012 + 0.901868i \(0.357804\pi\)
−0.997046 + 0.0768010i \(0.975529\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.57956 + 7.93203i 0.502672 + 0.870653i 0.999995 + 0.00308800i \(0.000982943\pi\)
−0.497323 + 0.867565i \(0.665684\pi\)
\(84\) 0 0
\(85\) −0.305602 −0.0331471
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.24068 + 5.61302i 0.343511 + 0.594979i 0.985082 − 0.172085i \(-0.0550503\pi\)
−0.641571 + 0.767064i \(0.721717\pi\)
\(90\) 0 0
\(91\) 0.669440 + 1.15950i 0.0701764 + 0.121549i
\(92\) 0 0
\(93\) 0 0
\(94\) 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
\(99\) 0 0
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 2664.2.r.h.433.2 6
3.2 odd 2 888.2.q.h.433.2 yes 6
12.11 even 2 1776.2.q.m.433.2 6
37.10 even 3 inner 2664.2.r.h.1009.2 6
111.47 odd 6 888.2.q.h.121.2 ✓ 6
444.47 even 6 1776.2.q.m.1009.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.q.h.121.2 ✓ 6 111.47 odd 6
888.2.q.h.433.2 yes 6 3.2 odd 2
1776.2.q.m.433.2 6 12.11 even 2
1776.2.q.m.1009.2 6 444.47 even 6
2664.2.r.h.433.2 6 1.1 even 1 trivial
2664.2.r.h.1009.2 6 37.10 even 3 inner