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Properties

Label 444.2.w.b.29.1
Level $444$
Weight $2$
Character 444.29
Analytic conductor $3.545$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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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(29,444)] 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("444.29"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 6, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label 29.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 444.29
Dual form 444.2.w.b.245.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.50000 + 0.866025i) q^{3} +(2.59808 - 4.50000i) q^{7} +(1.50000 + 2.59808i) q^{9} +(-0.0358984 - 0.133975i) q^{13} +(-0.830127 - 3.09808i) q^{19} +(7.79423 - 4.50000i) q^{21} +(-4.33013 + 2.50000i) q^{25} +5.19615i q^{27} +(6.36603 + 6.36603i) q^{31} +(5.00000 + 3.46410i) q^{37} +(0.0621778 - 0.232051i) q^{39} +(-8.56218 + 8.56218i) q^{43} +(-10.0000 - 17.3205i) q^{49} +(1.43782 - 5.36603i) q^{57} +(-14.2942 + 3.83013i) q^{61} +15.5885 q^{63} +(4.33013 + 2.50000i) q^{67} -17.0000i q^{73} -8.66025 q^{75} +(-2.16025 - 8.06218i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(-0.696152 - 0.186533i) q^{91} +(4.03590 + 15.0622i) q^{93} +(-6.90192 + 6.90192i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 6 q^{9} - 14 q^{13} + 14 q^{19} + 22 q^{31} + 20 q^{37} - 24 q^{39} - 10 q^{43} - 40 q^{49} + 30 q^{57} - 26 q^{61} + 26 q^{79} - 18 q^{81} + 18 q^{91} + 30 q^{93} - 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/444\mathbb{Z}\right)^\times\).

\(n\) \(149\) \(223\) \(409\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\)

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
\(3\) 1.50000 + 0.866025i 0.866025 + 0.500000i
\(4\) 0 0
\(5\) 0 0 −0.258819 − 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(6\) 0 0
\(7\) 2.59808 − 4.50000i 0.981981 − 1.70084i 0.327327 − 0.944911i \(-0.393852\pi\)
0.654654 − 0.755929i \(-0.272814\pi\)
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −0.0358984 − 0.133975i −0.00995642 − 0.0371579i 0.960769 − 0.277350i \(-0.0894562\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.965926 − 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(18\) 0 0
\(19\) −0.830127 − 3.09808i −0.190444 − 0.710747i −0.993399 − 0.114708i \(-0.963407\pi\)
0.802955 − 0.596040i \(-0.203260\pi\)
\(20\) 0 0
\(21\) 7.79423 − 4.50000i 1.70084 − 0.981981i
\(22\) 0 0
\(23\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 0 0
\(25\) −4.33013 + 2.50000i −0.866025 + 0.500000i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(30\) 0 0
\(31\) 6.36603 + 6.36603i 1.14337 + 1.14337i 0.987829 + 0.155543i \(0.0497126\pi\)
0.155543 + 0.987829i \(0.450287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00000 + 3.46410i 0.821995 + 0.569495i
\(38\) 0 0
\(39\) 0.0621778 − 0.232051i 0.00995642 − 0.0371579i
\(40\) 0 0
\(41\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) −8.56218 + 8.56218i −1.30572 + 1.30572i −0.381246 + 0.924473i \(0.624505\pi\)
−0.924473 + 0.381246i \(0.875495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −10.0000 − 17.3205i −1.42857 − 2.47436i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.43782 − 5.36603i 0.190444 − 0.710747i
\(58\) 0 0
\(59\) 0 0 −0.965926 − 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(60\) 0 0
\(61\) −14.2942 + 3.83013i −1.83019 + 0.490398i −0.997949 − 0.0640184i \(-0.979608\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 15.5885 1.96396
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.33013 + 2.50000i 0.529009 + 0.305424i 0.740613 − 0.671932i \(-0.234535\pi\)
−0.211604 + 0.977356i \(0.567869\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(72\) 0 0
\(73\) − 17.0000i − 1.98970i −0.101361 − 0.994850i \(-0.532320\pi\)
0.101361 − 0.994850i \(-0.467680\pi\)
\(74\) 0 0
\(75\) −8.66025 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.16025 − 8.06218i −0.243048 − 0.907066i −0.974355 − 0.225018i \(-0.927756\pi\)
0.731307 − 0.682048i \(-0.238911\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.258819 − 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(90\) 0 0
\(91\) −0.696152 − 0.186533i −0.0729766 − 0.0195540i
\(92\) 0 0
\(93\) 4.03590 + 15.0622i 0.418503 + 1.56188i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.90192 + 6.90192i −0.700784 + 0.700784i −0.964579 − 0.263795i \(-0.915026\pi\)
0.263795 + 0.964579i \(0.415026\pi\)
\(98\) 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 444.2.w.b.29.1 ✓ 4
3.2 odd 2 CM 444.2.w.b.29.1 ✓ 4
37.23 odd 12 inner 444.2.w.b.245.1 yes 4
111.23 even 12 inner 444.2.w.b.245.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.w.b.29.1 ✓ 4 1.1 even 1 trivial
444.2.w.b.29.1 ✓ 4 3.2 odd 2 CM
444.2.w.b.245.1 yes 4 37.23 odd 12 inner
444.2.w.b.245.1 yes 4 111.23 even 12 inner