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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 341.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 444.341
Dual form 444.2.w.b.125.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.50000 + 0.866025i) q^{3} +(-2.59808 + 4.50000i) q^{7} +(1.50000 + 2.59808i) q^{9} +(-6.96410 + 1.86603i) q^{13} +(7.83013 - 2.09808i) q^{19} +(-7.79423 + 4.50000i) q^{21} +(4.33013 - 2.50000i) q^{25} +5.19615i q^{27} +(4.63397 - 4.63397i) q^{31} +(5.00000 + 3.46410i) q^{37} +(-12.0622 - 3.23205i) q^{39} +(3.56218 + 3.56218i) q^{43} +(-10.0000 - 17.3205i) q^{49} +(13.5622 + 3.63397i) q^{57} +(1.29423 + 4.83013i) q^{61} -15.5885 q^{63} +(-4.33013 - 2.50000i) q^{67} +17.0000i q^{73} +8.66025 q^{75} +(15.1603 - 4.06218i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(9.69615 - 36.1865i) q^{91} +(10.9641 - 2.93782i) q^{93} +(-12.0981 - 12.0981i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 6 q^{9} - 14 q^{13} + 14 q^{19} + 22 q^{31} + 20 q^{37} - 24 q^{39} - 10 q^{43} - 40 q^{49} + 30 q^{57} - 26 q^{61} + 26 q^{79} - 18 q^{81} + 18 q^{91} + 30 q^{93} - 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 0.965926 − 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(6\) 0 0
\(7\) −2.59808 + 4.50000i −0.981981 + 1.70084i −0.327327 + 0.944911i \(0.606148\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 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
\(13\) −6.96410 + 1.86603i −1.93149 + 0.517542i −0.960769 + 0.277350i \(0.910544\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.258819 − 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(18\) 0 0
\(19\) 7.83013 − 2.09808i 1.79635 − 0.481332i 0.802955 − 0.596040i \(-0.203260\pi\)
0.993399 + 0.114708i \(0.0365932\pi\)
\(20\) 0 0
\(21\) −7.79423 + 4.50000i −1.70084 + 0.981981i
\(22\) 0 0
\(23\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 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 \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(30\) 0 0
\(31\) 4.63397 − 4.63397i 0.832286 − 0.832286i −0.155543 − 0.987829i \(-0.549713\pi\)
0.987829 + 0.155543i \(0.0497126\pi\)
\(32\) 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
\(39\) −12.0622 − 3.23205i −1.93149 − 0.517542i
\(40\) 0 0
\(41\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) 3.56218 + 3.56218i 0.543227 + 0.543227i 0.924473 − 0.381246i \(-0.124505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 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
\(57\) 13.5622 + 3.63397i 1.79635 + 0.481332i
\(58\) 0 0
\(59\) 0 0 0.258819 − 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(60\) 0 0
\(61\) 1.29423 + 4.83013i 0.165709 + 0.618434i 0.997949 + 0.0640184i \(0.0203916\pi\)
−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 0.211604 − 0.977356i \(-0.432131\pi\)
−0.740613 + 0.671932i \(0.765465\pi\)
\(68\) 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 \(0.467680\pi\)
−0.101361 + 0.994850i \(0.532320\pi\)
\(74\) 0 0
\(75\) 8.66025 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 15.1603 − 4.06218i 1.70566 − 0.457031i 0.731307 − 0.682048i \(-0.238911\pi\)
0.974355 + 0.225018i \(0.0722440\pi\)
\(80\) 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 −0.965926 − 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(90\) 0 0
\(91\) 9.69615 − 36.1865i 1.01643 − 3.79338i
\(92\) 0 0
\(93\) 10.9641 − 2.93782i 1.13692 − 0.304638i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.0981 − 12.0981i −1.22837 − 1.22837i −0.964579 − 0.263795i \(-0.915026\pi\)
−0.263795 − 0.964579i \(-0.584974\pi\)
\(98\) 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.341.1 yes 4
3.2 odd 2 CM 444.2.w.b.341.1 yes 4
37.14 odd 12 inner 444.2.w.b.125.1 ✓ 4
111.14 even 12 inner 444.2.w.b.125.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.w.b.125.1 ✓ 4 37.14 odd 12 inner
444.2.w.b.125.1 ✓ 4 111.14 even 12 inner
444.2.w.b.341.1 yes 4 1.1 even 1 trivial
444.2.w.b.341.1 yes 4 3.2 odd 2 CM