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Properties

Label 444.2.bb.b.361.3
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 361.3
Character \(\chi\) \(=\) 444.361
Dual form 444.2.bb.b.337.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.173648 + 0.984808i) q^{3} +(0.603708 - 0.719471i) q^{5} +(-0.0528059 - 0.0443094i) q^{7} +(-0.939693 - 0.342020i) q^{9} +(2.63394 + 4.56211i) q^{11} +(1.17307 + 3.22298i) q^{13} +(0.603708 + 0.719471i) q^{15} +(2.03377 - 5.58773i) q^{17} +(-0.627206 - 0.110593i) q^{19} +(0.0528059 - 0.0443094i) q^{21} +(6.09811 + 3.52075i) q^{23} +(0.715066 + 4.05534i) q^{25} +(0.500000 - 0.866025i) q^{27} +(-1.52560 + 0.880803i) q^{29} +0.412500i q^{31} +(-4.95018 + 1.80172i) q^{33} +(-0.0637586 + 0.0112424i) q^{35} +(-5.41331 + 2.77419i) q^{37} +(-3.37772 + 0.595583i) q^{39} +(-0.531995 + 0.193630i) q^{41} +10.3073i q^{43} +(-0.813373 + 0.469601i) q^{45} +(2.61827 - 4.53498i) q^{47} +(-1.21471 - 6.88897i) q^{49} +(5.14968 + 2.97317i) q^{51} +(8.07673 - 6.77718i) q^{53} +(4.87243 + 0.859141i) q^{55} +(0.217826 - 0.598473i) q^{57} +(2.47038 + 2.94408i) q^{59} +(-3.70229 - 10.1720i) q^{61} +(0.0344666 + 0.0596979i) q^{63} +(3.02703 + 1.10175i) q^{65} +(-6.53356 - 5.48231i) q^{67} +(-4.52618 + 5.39410i) q^{69} +(1.52777 - 8.66444i) q^{71} -14.5762 q^{73} -4.11790 q^{75} +(0.0630570 - 0.357614i) q^{77} +(4.65972 - 5.55324i) q^{79} +(0.766044 + 0.642788i) q^{81} +(-4.27337 - 1.55538i) q^{83} +(-2.79241 - 4.83659i) q^{85} +(-0.602505 - 1.65537i) q^{87} +(-3.11540 - 3.71279i) q^{89} +(0.0808634 - 0.222170i) q^{91} +(-0.406233 - 0.0716298i) q^{93} +(-0.458218 + 0.384490i) q^{95} +(3.69155 + 2.13132i) q^{97} +(-0.914756 - 5.18784i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 3 q^{7} + 6 q^{11} - 12 q^{13} - 15 q^{17} + 12 q^{19} + 3 q^{21} + 18 q^{25} + 12 q^{27} - 9 q^{29} + 18 q^{33} + 12 q^{35} + 12 q^{37} + 15 q^{39} + 15 q^{41} - 15 q^{47} + 15 q^{49} + 9 q^{51}+ \cdots + 9 q^{99}+O(q^{100}) \) 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
\(5\) 0.603708 − 0.719471i 0.269986 − 0.321757i −0.613968 − 0.789331i \(-0.710427\pi\)
0.883954 + 0.467574i \(0.154872\pi\)
\(6\) 0 0
\(7\) −0.0528059 − 0.0443094i −0.0199587 − 0.0167474i 0.632754 − 0.774353i \(-0.281925\pi\)
−0.652712 + 0.757606i \(0.726369\pi\)
\(8\) 0 0
\(9\) −0.939693 − 0.342020i −0.313231 − 0.114007i
\(10\) 0 0
\(11\) 2.63394 + 4.56211i 0.794161 + 1.37553i 0.923370 + 0.383910i \(0.125423\pi\)
−0.129209 + 0.991617i \(0.541244\pi\)
\(12\) 0 0
\(13\) 1.17307 + 3.22298i 0.325351 + 0.893894i 0.989271 + 0.146091i \(0.0466692\pi\)
−0.663920 + 0.747803i \(0.731109\pi\)
\(14\) 0 0
\(15\) 0.603708 + 0.719471i 0.155877 + 0.185767i
\(16\) 0 0
\(17\) 2.03377 − 5.58773i 0.493261 − 1.35522i −0.404419 − 0.914574i \(-0.632526\pi\)
0.897680 − 0.440649i \(-0.145252\pi\)
\(18\) 0 0
\(19\) −0.627206 − 0.110593i −0.143891 − 0.0253719i 0.101239 − 0.994862i \(-0.467719\pi\)
−0.245130 + 0.969490i \(0.578831\pi\)
\(20\) 0 0
\(21\) 0.0528059 − 0.0443094i 0.0115232 − 0.00966910i
\(22\) 0 0
\(23\) 6.09811 + 3.52075i 1.27154 + 0.734126i 0.975278 − 0.220979i \(-0.0709253\pi\)
0.296265 + 0.955106i \(0.404259\pi\)
\(24\) 0 0
\(25\) 0.715066 + 4.05534i 0.143013 + 0.811068i
\(26\) 0 0
\(27\) 0.500000 − 0.866025i 0.0962250 − 0.166667i
\(28\) 0 0
\(29\) −1.52560 + 0.880803i −0.283296 + 0.163561i −0.634915 − 0.772582i \(-0.718965\pi\)
0.351619 + 0.936143i \(0.385631\pi\)
\(30\) 0 0
\(31\) 0.412500i 0.0740871i 0.999314 + 0.0370436i \(0.0117940\pi\)
−0.999314 + 0.0370436i \(0.988206\pi\)
\(32\) 0 0
\(33\) −4.95018 + 1.80172i −0.861716 + 0.313639i
\(34\) 0 0
\(35\) −0.0637586 + 0.0112424i −0.0107772 + 0.00190031i
\(36\) 0 0
\(37\) −5.41331 + 2.77419i −0.889942 + 0.456074i
\(38\) 0 0
\(39\) −3.37772 + 0.595583i −0.540868 + 0.0953696i
\(40\) 0 0
\(41\) −0.531995 + 0.193630i −0.0830836 + 0.0302400i −0.383228 − 0.923654i \(-0.625187\pi\)
0.300144 + 0.953894i \(0.402965\pi\)
\(42\) 0 0
\(43\) 10.3073i 1.57184i 0.618327 + 0.785921i \(0.287811\pi\)
−0.618327 + 0.785921i \(0.712189\pi\)
\(44\) 0 0
\(45\) −0.813373 + 0.469601i −0.121250 + 0.0700040i
\(46\) 0 0
\(47\) 2.61827 − 4.53498i 0.381914 − 0.661495i −0.609422 − 0.792846i \(-0.708598\pi\)
0.991336 + 0.131351i \(0.0419316\pi\)
\(48\) 0 0
\(49\) −1.21471 − 6.88897i −0.173530 − 0.984139i
\(50\) 0 0
\(51\) 5.14968 + 2.97317i 0.721099 + 0.416327i
\(52\) 0 0
\(53\) 8.07673 − 6.77718i 1.10942 − 0.930918i 0.111402 − 0.993775i \(-0.464466\pi\)
0.998023 + 0.0628575i \(0.0200213\pi\)
\(54\) 0 0
\(55\) 4.87243 + 0.859141i 0.656998 + 0.115847i
\(56\) 0 0
\(57\) 0.217826 − 0.598473i 0.0288518 − 0.0792697i
\(58\) 0 0
\(59\) 2.47038 + 2.94408i 0.321616 + 0.383287i 0.902493 − 0.430704i \(-0.141735\pi\)
−0.580877 + 0.813991i \(0.697290\pi\)
\(60\) 0 0
\(61\) −3.70229 − 10.1720i −0.474029 − 1.30239i −0.914488 − 0.404612i \(-0.867406\pi\)
0.440459 − 0.897773i \(-0.354816\pi\)
\(62\) 0 0
\(63\) 0.0344666 + 0.0596979i 0.00434238 + 0.00752123i
\(64\) 0 0
\(65\) 3.02703 + 1.10175i 0.375457 + 0.136655i
\(66\) 0 0
\(67\) −6.53356 − 5.48231i −0.798202 − 0.669771i 0.149559 − 0.988753i \(-0.452215\pi\)
−0.947761 + 0.318982i \(0.896659\pi\)
\(68\) 0 0
\(69\) −4.52618 + 5.39410i −0.544888 + 0.649373i
\(70\) 0 0
\(71\) 1.52777 − 8.66444i 0.181314 − 1.02828i −0.749288 − 0.662245i \(-0.769604\pi\)
0.930601 − 0.366035i \(-0.119285\pi\)
\(72\) 0 0
\(73\) −14.5762 −1.70601 −0.853005 − 0.521903i \(-0.825222\pi\)
−0.853005 + 0.521903i \(0.825222\pi\)
\(74\) 0 0
\(75\) −4.11790 −0.475494
\(76\) 0 0
\(77\) 0.0630570 − 0.357614i 0.00718602 − 0.0407539i
\(78\) 0 0
\(79\) 4.65972 − 5.55324i 0.524260 − 0.624789i −0.437323 − 0.899305i \(-0.644073\pi\)
0.961583 + 0.274516i \(0.0885176\pi\)
\(80\) 0 0
\(81\) 0.766044 + 0.642788i 0.0851160 + 0.0714208i
\(82\) 0 0
\(83\) −4.27337 − 1.55538i −0.469064 − 0.170725i 0.0966645 − 0.995317i \(-0.469183\pi\)
−0.565728 + 0.824592i \(0.691405\pi\)
\(84\) 0 0
\(85\) −2.79241 − 4.83659i −0.302879 − 0.524602i
\(86\) 0 0
\(87\) −0.602505 − 1.65537i −0.0645953 − 0.177474i
\(88\) 0 0
\(89\) −3.11540 − 3.71279i −0.330232 − 0.393555i 0.575224 − 0.817996i \(-0.304915\pi\)
−0.905456 + 0.424441i \(0.860471\pi\)
\(90\) 0 0
\(91\) 0.0808634 − 0.222170i 0.00847679 − 0.0232898i
\(92\) 0 0
\(93\) −0.406233 − 0.0716298i −0.0421244 − 0.00742766i
\(94\) 0 0
\(95\) −0.458218 + 0.384490i −0.0470122 + 0.0394479i
\(96\) 0 0
\(97\) 3.69155 + 2.13132i 0.374821 + 0.216403i 0.675562 − 0.737303i \(-0.263901\pi\)
−0.300742 + 0.953706i \(0.597234\pi\)
\(98\) 0 0
\(99\) −0.914756 − 5.18784i −0.0919364 − 0.521397i
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.361.3 yes 24
3.2 odd 2 1332.2.ct.e.361.2 24
37.4 even 18 inner 444.2.bb.b.337.3 ✓ 24
111.41 odd 18 1332.2.ct.e.1225.2 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.bb.b.337.3 ✓ 24 37.4 even 18 inner
444.2.bb.b.361.3 yes 24 1.1 even 1 trivial
1332.2.ct.e.361.2 24 3.2 odd 2
1332.2.ct.e.1225.2 24 111.41 odd 18