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Properties

Label 444.2.bb.b.361.4
Level $444$
Weight $2$
Character 444.361
Analytic conductor $3.545$
Analytic rank $0$
Dimension $24$
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(25,444)] 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("444.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] 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: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 361.4
Character \(\chi\) \(=\) 444.361
Dual form 444.2.bb.b.337.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.173648 + 0.984808i) q^{3} +(2.03222 - 2.42190i) q^{5} +(3.93016 + 3.29779i) q^{7} +(-0.939693 - 0.342020i) q^{9} +(-1.96017 - 3.39511i) q^{11} +(-1.04562 - 2.87281i) q^{13} +(2.03222 + 2.42190i) q^{15} +(-0.996911 + 2.73899i) q^{17} +(5.48044 + 0.966349i) q^{19} +(-3.93016 + 3.29779i) q^{21} +(-0.453012 - 0.261546i) q^{23} +(-0.867465 - 4.91964i) q^{25} +(0.500000 - 0.866025i) q^{27} +(3.47913 - 2.00868i) q^{29} +9.47080i q^{31} +(3.68391 - 1.34083i) q^{33} +(15.9739 - 2.81663i) q^{35} +(-5.83530 - 1.71734i) q^{37} +(3.01074 - 0.530875i) q^{39} +(6.42436 - 2.33828i) q^{41} -5.76480i q^{43} +(-2.73800 + 1.58079i) q^{45} +(-0.791404 + 1.37075i) q^{47} +(3.35516 + 19.0280i) q^{49} +(-2.52427 - 1.45739i) q^{51} +(-10.9221 + 9.16475i) q^{53} +(-12.2061 - 2.15227i) q^{55} +(-1.90334 + 5.22937i) q^{57} +(2.95864 + 3.52597i) q^{59} +(-1.43171 - 3.93358i) q^{61} +(-2.56523 - 4.44310i) q^{63} +(-9.08261 - 3.30580i) q^{65} +(-0.984705 - 0.826266i) q^{67} +(0.336238 - 0.400712i) q^{69} +(2.47042 - 14.0105i) q^{71} -7.86813 q^{73} +4.99553 q^{75} +(3.49260 - 19.8075i) q^{77} +(-5.54444 + 6.60761i) q^{79} +(0.766044 + 0.642788i) q^{81} +(-13.3707 - 4.86654i) q^{83} +(4.60763 + 7.98065i) q^{85} +(1.37402 + 3.77508i) q^{87} +(-5.67034 - 6.75765i) q^{89} +(5.36450 - 14.7388i) q^{91} +(-9.32692 - 1.64459i) q^{93} +(13.4778 - 11.3093i) q^{95} +(-5.06169 - 2.92237i) q^{97} +(0.680759 + 3.86078i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 3 q^{7} + 6 q^{11} - 12 q^{13} - 15 q^{17} + 12 q^{19} + 3 q^{21} + 18 q^{25} + 12 q^{27} - 9 q^{29} + 18 q^{33} + 12 q^{35} + 12 q^{37} + 15 q^{39} + 15 q^{41} - 15 q^{47} + 15 q^{49} + 9 q^{51}+ \cdots + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(149\) \(223\) \(409\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{17}{18}\right)\)

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
\(3\) −0.173648 + 0.984808i −0.100256 + 0.568579i
\(4\) 0 0
\(5\) 2.03222 − 2.42190i 0.908836 − 1.08311i −0.0873784 − 0.996175i \(-0.527849\pi\)
0.996214 − 0.0869331i \(-0.0277066\pi\)
\(6\) 0 0
\(7\) 3.93016 + 3.29779i 1.48546 + 1.24645i 0.900106 + 0.435670i \(0.143489\pi\)
0.585353 + 0.810778i \(0.300956\pi\)
\(8\) 0 0
\(9\) −0.939693 − 0.342020i −0.313231 − 0.114007i
\(10\) 0 0
\(11\) −1.96017 − 3.39511i −0.591013 − 1.02366i −0.994096 − 0.108501i \(-0.965395\pi\)
0.403084 − 0.915163i \(-0.367938\pi\)
\(12\) 0 0
\(13\) −1.04562 − 2.87281i −0.290003 − 0.796775i −0.996065 − 0.0886259i \(-0.971752\pi\)
0.706062 − 0.708150i \(-0.250470\pi\)
\(14\) 0 0
\(15\) 2.03222 + 2.42190i 0.524717 + 0.625333i
\(16\) 0 0
\(17\) −0.996911 + 2.73899i −0.241786 + 0.664302i 0.758139 + 0.652093i \(0.226109\pi\)
−0.999925 + 0.0122096i \(0.996113\pi\)
\(18\) 0 0
\(19\) 5.48044 + 0.966349i 1.25730 + 0.221696i 0.762314 − 0.647207i \(-0.224063\pi\)
0.494984 + 0.868902i \(0.335174\pi\)
\(20\) 0 0
\(21\) −3.93016 + 3.29779i −0.857631 + 0.719637i
\(22\) 0 0
\(23\) −0.453012 − 0.261546i −0.0944595 − 0.0545362i 0.452026 − 0.892005i \(-0.350701\pi\)
−0.546486 + 0.837468i \(0.684035\pi\)
\(24\) 0 0
\(25\) −0.867465 − 4.91964i −0.173493 − 0.983927i
\(26\) 0 0
\(27\) 0.500000 − 0.866025i 0.0962250 − 0.166667i
\(28\) 0 0
\(29\) 3.47913 − 2.00868i 0.646058 − 0.373002i −0.140886 − 0.990026i \(-0.544995\pi\)
0.786944 + 0.617024i \(0.211662\pi\)
\(30\) 0 0
\(31\) 9.47080i 1.70101i 0.525971 + 0.850503i \(0.323702\pi\)
−0.525971 + 0.850503i \(0.676298\pi\)
\(32\) 0 0
\(33\) 3.68391 − 1.34083i 0.641286 − 0.233409i
\(34\) 0 0
\(35\) 15.9739 − 2.81663i 2.70008 − 0.476097i
\(36\) 0 0
\(37\) −5.83530 − 1.71734i −0.959317 − 0.282330i
\(38\) 0 0
\(39\) 3.01074 − 0.530875i 0.482104 − 0.0850080i
\(40\) 0 0
\(41\) 6.42436 − 2.33828i 1.00332 − 0.365177i 0.212454 − 0.977171i \(-0.431854\pi\)
0.790863 + 0.611994i \(0.209632\pi\)
\(42\) 0 0
\(43\) − 5.76480i − 0.879123i −0.898212 − 0.439562i \(-0.855134\pi\)
0.898212 − 0.439562i \(-0.144866\pi\)
\(44\) 0 0
\(45\) −2.73800 + 1.58079i −0.408157 + 0.235650i
\(46\) 0 0
\(47\) −0.791404 + 1.37075i −0.115438 + 0.199945i −0.917955 − 0.396685i \(-0.870161\pi\)
0.802517 + 0.596630i \(0.203494\pi\)
\(48\) 0 0
\(49\) 3.35516 + 19.0280i 0.479308 + 2.71829i
\(50\) 0 0
\(51\) −2.52427 − 1.45739i −0.353468 − 0.204075i
\(52\) 0 0
\(53\) −10.9221 + 9.16475i −1.50027 + 1.25888i −0.619811 + 0.784751i \(0.712791\pi\)
−0.880458 + 0.474125i \(0.842765\pi\)
\(54\) 0 0
\(55\) −12.2061 − 2.15227i −1.64587 − 0.290212i
\(56\) 0 0
\(57\) −1.90334 + 5.22937i −0.252103 + 0.692647i
\(58\) 0 0
\(59\) 2.95864 + 3.52597i 0.385182 + 0.459042i 0.923443 − 0.383737i \(-0.125363\pi\)
−0.538261 + 0.842778i \(0.680918\pi\)
\(60\) 0 0
\(61\) −1.43171 − 3.93358i −0.183311 − 0.503644i 0.813666 − 0.581332i \(-0.197468\pi\)
−0.996978 + 0.0776885i \(0.975246\pi\)
\(62\) 0 0
\(63\) −2.56523 − 4.44310i −0.323188 − 0.559779i
\(64\) 0 0
\(65\) −9.08261 − 3.30580i −1.12656 − 0.410034i
\(66\) 0 0
\(67\) −0.984705 − 0.826266i −0.120301 − 0.100944i 0.580653 − 0.814151i \(-0.302797\pi\)
−0.700953 + 0.713207i \(0.747242\pi\)
\(68\) 0 0
\(69\) 0.336238 − 0.400712i 0.0404783 − 0.0482401i
\(70\) 0 0
\(71\) 2.47042 − 14.0105i 0.293185 − 1.66274i −0.381302 − 0.924451i \(-0.624524\pi\)
0.674487 − 0.738287i \(-0.264365\pi\)
\(72\) 0 0
\(73\) −7.86813 −0.920895 −0.460447 − 0.887687i \(-0.652311\pi\)
−0.460447 + 0.887687i \(0.652311\pi\)
\(74\) 0 0
\(75\) 4.99553 0.576834
\(76\) 0 0
\(77\) 3.49260 − 19.8075i 0.398019 − 2.25728i
\(78\) 0 0
\(79\) −5.54444 + 6.60761i −0.623798 + 0.743414i −0.981719 − 0.190338i \(-0.939041\pi\)
0.357920 + 0.933752i \(0.383486\pi\)
\(80\) 0 0
\(81\) 0.766044 + 0.642788i 0.0851160 + 0.0714208i
\(82\) 0 0
\(83\) −13.3707 − 4.86654i −1.46762 − 0.534172i −0.520170 − 0.854063i \(-0.674131\pi\)
−0.947454 + 0.319891i \(0.896354\pi\)
\(84\) 0 0
\(85\) 4.60763 + 7.98065i 0.499767 + 0.865623i
\(86\) 0 0
\(87\) 1.37402 + 3.77508i 0.147310 + 0.404731i
\(88\) 0 0
\(89\) −5.67034 − 6.75765i −0.601055 − 0.716309i 0.376635 − 0.926362i \(-0.377081\pi\)
−0.977690 + 0.210052i \(0.932637\pi\)
\(90\) 0 0
\(91\) 5.36450 − 14.7388i 0.562353 − 1.54505i
\(92\) 0 0
\(93\) −9.32692 − 1.64459i −0.967156 − 0.170536i
\(94\) 0 0
\(95\) 13.4778 − 11.3093i 1.38280 − 1.16031i
\(96\) 0 0
\(97\) −5.06169 − 2.92237i −0.513936 − 0.296721i 0.220514 − 0.975384i \(-0.429227\pi\)
−0.734450 + 0.678663i \(0.762560\pi\)
\(98\) 0 0
\(99\) 0.680759 + 3.86078i 0.0684188 + 0.388023i
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.bb.b.361.4 yes 24
3.2 odd 2 1332.2.ct.e.361.1 24
37.4 even 18 inner 444.2.bb.b.337.4 ✓ 24
111.41 odd 18 1332.2.ct.e.1225.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.bb.b.337.4 ✓ 24 37.4 even 18 inner
444.2.bb.b.361.4 yes 24 1.1 even 1 trivial
1332.2.ct.e.361.1 24 3.2 odd 2
1332.2.ct.e.1225.1 24 111.41 odd 18