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Properties

Label 1776.2.q.b.1009.1
Level $1776$
Weight $2$
Character 1776.1009
Analytic conductor $14.181$
Analytic rank $1$
Dimension $2$
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 = [2,0,-1,0,-2,0,-3,0,-1,0,-4] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 444)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1009.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1776.1009
Dual form 1776.2.q.b.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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{5} +(-1.50000 - 2.59808i) q^{7} +(-0.500000 + 0.866025i) q^{9} -2.00000 q^{11} +(1.50000 + 2.59808i) q^{13} +(-1.00000 + 1.73205i) q^{15} +(-2.00000 - 3.46410i) q^{19} +(-1.50000 + 2.59808i) q^{21} -6.00000 q^{23} +(0.500000 - 0.866025i) q^{25} +1.00000 q^{27} -1.00000 q^{31} +(1.00000 + 1.73205i) q^{33} +(-3.00000 + 5.19615i) q^{35} +(-5.00000 + 3.46410i) q^{37} +(1.50000 - 2.59808i) q^{39} +5.00000 q^{43} +2.00000 q^{45} +4.00000 q^{47} +(-1.00000 + 1.73205i) q^{49} +(-3.00000 + 5.19615i) q^{53} +(2.00000 + 3.46410i) q^{55} +(-2.00000 + 3.46410i) q^{57} +(-5.00000 + 8.66025i) q^{59} +(1.00000 + 1.73205i) q^{61} +3.00000 q^{63} +(3.00000 - 5.19615i) q^{65} +(1.50000 + 2.59808i) q^{67} +(3.00000 + 5.19615i) q^{69} +(6.00000 + 10.3923i) q^{71} +1.00000 q^{73} -1.00000 q^{75} +(3.00000 + 5.19615i) q^{77} +(7.50000 + 12.9904i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(-2.00000 + 3.46410i) q^{83} +(5.00000 - 8.66025i) q^{89} +(4.50000 - 7.79423i) q^{91} +(0.500000 + 0.866025i) q^{93} +(-4.00000 + 6.92820i) q^{95} +9.00000 q^{97} +(1.00000 - 1.73205i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 2 q^{5} - 3 q^{7} - q^{9} - 4 q^{11} + 3 q^{13} - 2 q^{15} - 4 q^{19} - 3 q^{21} - 12 q^{23} + q^{25} + 2 q^{27} - 2 q^{31} + 2 q^{33} - 6 q^{35} - 10 q^{37} + 3 q^{39} + 10 q^{43} + 4 q^{45}+ \cdots + 2 q^{99}+O(q^{100}) \) 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
\(5\) −1.00000 − 1.73205i −0.447214 − 0.774597i 0.550990 − 0.834512i \(-0.314250\pi\)
−0.998203 + 0.0599153i \(0.980917\pi\)
\(6\) 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
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 1.50000 + 2.59808i 0.416025 + 0.720577i 0.995535 − 0.0943882i \(-0.0300895\pi\)
−0.579510 + 0.814965i \(0.696756\pi\)
\(14\) 0 0
\(15\) −1.00000 + 1.73205i −0.258199 + 0.447214i
\(16\) 0 0
\(17\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) −2.00000 − 3.46410i −0.458831 − 0.794719i 0.540068 − 0.841621i \(-0.318398\pi\)
−0.998899 + 0.0469020i \(0.985065\pi\)
\(20\) 0 0
\(21\) −1.50000 + 2.59808i −0.327327 + 0.566947i
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 0.500000 − 0.866025i 0.100000 − 0.173205i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 − 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 0 0
\(33\) 1.00000 + 1.73205i 0.174078 + 0.301511i
\(34\) 0 0
\(35\) −3.00000 + 5.19615i −0.507093 + 0.878310i
\(36\) 0 0
\(37\) −5.00000 + 3.46410i −0.821995 + 0.569495i
\(38\) 0 0
\(39\) 1.50000 − 2.59808i 0.240192 − 0.416025i
\(40\) 0 0
\(41\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 − 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) −1.00000 + 1.73205i −0.142857 + 0.247436i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.00000 + 5.19615i −0.412082 + 0.713746i −0.995117 − 0.0987002i \(-0.968532\pi\)
0.583036 + 0.812447i \(0.301865\pi\)
\(54\) 0 0
\(55\) 2.00000 + 3.46410i 0.269680 + 0.467099i
\(56\) 0 0
\(57\) −2.00000 + 3.46410i −0.264906 + 0.458831i
\(58\) 0 0
\(59\) −5.00000 + 8.66025i −0.650945 + 1.12747i 0.331949 + 0.943297i \(0.392294\pi\)
−0.982894 + 0.184172i \(0.941040\pi\)
\(60\) 0 0
\(61\) 1.00000 + 1.73205i 0.128037 + 0.221766i 0.922916 − 0.385002i \(-0.125799\pi\)
−0.794879 + 0.606768i \(0.792466\pi\)
\(62\) 0 0
\(63\) 3.00000 0.377964
\(64\) 0 0
\(65\) 3.00000 − 5.19615i 0.372104 − 0.644503i
\(66\) 0 0
\(67\) 1.50000 + 2.59808i 0.183254 + 0.317406i 0.942987 − 0.332830i \(-0.108004\pi\)
−0.759733 + 0.650236i \(0.774670\pi\)
\(68\) 0 0
\(69\) 3.00000 + 5.19615i 0.361158 + 0.625543i
\(70\) 0 0
\(71\) 6.00000 + 10.3923i 0.712069 + 1.23334i 0.964079 + 0.265615i \(0.0855750\pi\)
−0.252010 + 0.967725i \(0.581092\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 − 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 3.00000 + 5.19615i 0.341882 + 0.592157i
\(78\) 0 0
\(79\) 7.50000 + 12.9904i 0.843816 + 1.46153i 0.886646 + 0.462450i \(0.153029\pi\)
−0.0428296 + 0.999082i \(0.513637\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −2.00000 + 3.46410i −0.219529 + 0.380235i −0.954664 − 0.297686i \(-0.903785\pi\)
0.735135 + 0.677920i \(0.237119\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.00000 − 8.66025i 0.529999 − 0.917985i −0.469389 − 0.882992i \(-0.655526\pi\)
0.999388 − 0.0349934i \(-0.0111410\pi\)
\(90\) 0 0
\(91\) 4.50000 − 7.79423i 0.471728 − 0.817057i
\(92\) 0 0
\(93\) 0.500000 + 0.866025i 0.0518476 + 0.0898027i
\(94\) 0 0
\(95\) −4.00000 + 6.92820i −0.410391 + 0.710819i
\(96\) 0 0
\(97\) 9.00000 0.913812 0.456906 − 0.889515i \(-0.348958\pi\)
0.456906 + 0.889515i \(0.348958\pi\)
\(98\) 0 0
\(99\) 1.00000 − 1.73205i 0.100504 − 0.174078i
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.b.1009.1 2
4.3 odd 2 444.2.i.a.121.1 ✓ 2
12.11 even 2 1332.2.j.c.1009.1 2
37.26 even 3 inner 1776.2.q.b.433.1 2
148.63 odd 6 444.2.i.a.433.1 yes 2
444.359 even 6 1332.2.j.c.433.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.i.a.121.1 ✓ 2 4.3 odd 2
444.2.i.a.433.1 yes 2 148.63 odd 6
1332.2.j.c.433.1 2 444.359 even 6
1332.2.j.c.1009.1 2 12.11 even 2
1776.2.q.b.433.1 2 37.26 even 3 inner
1776.2.q.b.1009.1 2 1.1 even 1 trivial