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Properties

Label 444.2.i.a.121.1
Level $444$
Weight $2$
Character 444.121
Analytic conductor $3.545$
Analytic rank $0$
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] = [444,2,Mod(121,444)] 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("444.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.54535784974\)
Analytic rank: \(0\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 121.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 444.121
Dual form 444.2.i.a.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 \(\left(\mathbb{Z}/444\mathbb{Z}\right)^\times\).

\(n\) \(149\) \(223\) \(409\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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 \(0.0252088\pi\)
−0.429919 + 0.902867i \(0.641458\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 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 0.998899 − 0.0469020i \(-0.0149348\pi\)
−0.540068 + 0.841621i \(0.681602\pi\)
\(20\) 0 0
\(21\) −1.50000 + 2.59808i −0.327327 + 0.566947i
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 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 \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 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 \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 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 \(-0.607706\pi\)
0.982894 − 0.184172i \(-0.0589603\pi\)
\(60\) 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 0.759733 − 0.650236i \(-0.225330\pi\)
−0.942987 + 0.332830i \(0.891996\pi\)
\(68\) 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 \(-0.914425\pi\)
0.252010 − 0.967725i \(-0.418908\pi\)
\(72\) 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 \(-0.846971\pi\)
0.0428296 − 0.999082i \(-0.486363\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) 2.00000 − 3.46410i 0.219529 − 0.380235i −0.735135 − 0.677920i \(-0.762881\pi\)
0.954664 + 0.297686i \(0.0962148\pi\)
\(84\) 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 444.2.i.a.121.1 ✓ 2
3.2 odd 2 1332.2.j.c.1009.1 2
4.3 odd 2 1776.2.q.b.1009.1 2
37.26 even 3 inner 444.2.i.a.433.1 yes 2
111.26 odd 6 1332.2.j.c.433.1 2
148.63 odd 6 1776.2.q.b.433.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.i.a.121.1 ✓ 2 1.1 even 1 trivial
444.2.i.a.433.1 yes 2 37.26 even 3 inner
1332.2.j.c.433.1 2 111.26 odd 6
1332.2.j.c.1009.1 2 3.2 odd 2
1776.2.q.b.433.1 2 148.63 odd 6
1776.2.q.b.1009.1 2 4.3 odd 2