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Properties

Label 444.2.bb.a.337.3
Level $444$
Weight $2$
Character 444.337
Analytic conductor $3.545$
Analytic rank $0$
Dimension $18$
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 = [18] 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: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 48 x^{16} + 954 x^{14} + 10172 x^{12} + 63093 x^{10} + 231255 x^{8} + 486910 x^{6} + \cdots + 36963 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 337.3
Root \(3.29901i\) of defining polynomial
Character \(\chi\) \(=\) 444.337
Dual form 444.2.bb.a.361.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.173648 + 0.984808i) q^{3} +(2.12056 + 2.52719i) q^{5} +(-1.00275 + 0.841410i) q^{7} +(-0.939693 + 0.342020i) q^{9} +(-2.81105 + 4.86888i) q^{11} +(1.10815 - 3.04461i) q^{13} +(-2.12056 + 2.52719i) q^{15} +(-0.873207 - 2.39912i) q^{17} +(-1.88838 + 0.332972i) q^{19} +(-1.00275 - 0.841410i) q^{21} +(5.61113 - 3.23959i) q^{23} +(-1.02166 + 5.79410i) q^{25} +(-0.500000 - 0.866025i) q^{27} +(-0.694918 - 0.401211i) q^{29} +9.58977i q^{31} +(-5.28305 - 1.92287i) q^{33} +(-4.25281 - 0.749885i) q^{35} +(-3.69170 - 4.83440i) q^{37} +(3.19078 + 0.562621i) q^{39} +(8.65603 + 3.15054i) q^{41} -4.57741i q^{43} +(-2.85703 - 1.64951i) q^{45} +(6.26284 + 10.8476i) q^{47} +(-0.917993 + 5.20620i) q^{49} +(2.21104 - 1.27654i) q^{51} +(6.53540 + 5.48385i) q^{53} +(-18.2656 + 3.22072i) q^{55} +(-0.655826 - 1.80187i) q^{57} +(5.94819 - 7.08877i) q^{59} +(3.55243 - 9.76023i) q^{61} +(0.654501 - 1.13363i) q^{63} +(10.0442 - 3.65579i) q^{65} +(-1.86400 + 1.56408i) q^{67} +(4.16473 + 4.96334i) q^{69} +(-1.86869 - 10.5979i) q^{71} +6.74190 q^{73} -5.88348 q^{75} +(-1.27794 - 7.24754i) q^{77} +(-7.46536 - 8.89687i) q^{79} +(0.766044 - 0.642788i) q^{81} +(-1.36439 + 0.496597i) q^{83} +(4.21133 - 7.29424i) q^{85} +(0.274445 - 0.754031i) q^{87} +(3.45108 - 4.11284i) q^{89} +(1.45057 + 3.98540i) q^{91} +(-9.44408 + 1.66525i) q^{93} +(-4.84591 - 4.06620i) q^{95} +(1.95274 - 1.12742i) q^{97} +(0.976267 - 5.53669i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 6 q^{11} + 18 q^{13} + 15 q^{17} - 12 q^{19} + 6 q^{25} - 9 q^{27} + 27 q^{29} - 6 q^{33} - 12 q^{35} - 9 q^{37} + 18 q^{39} + 33 q^{41} + 21 q^{47} + 30 q^{49} - 9 q^{51} - 18 q^{53} - 51 q^{55}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 2.12056 + 2.52719i 0.948345 + 1.13019i 0.991367 + 0.131120i \(0.0418572\pi\)
−0.0430214 + 0.999074i \(0.513698\pi\)
\(6\) 0 0
\(7\) −1.00275 + 0.841410i −0.379005 + 0.318023i −0.812312 − 0.583223i \(-0.801791\pi\)
0.433307 + 0.901247i \(0.357347\pi\)
\(8\) 0 0
\(9\) −0.939693 + 0.342020i −0.313231 + 0.114007i
\(10\) 0 0
\(11\) −2.81105 + 4.86888i −0.847563 + 1.46802i 0.0358127 + 0.999359i \(0.488598\pi\)
−0.883376 + 0.468665i \(0.844735\pi\)
\(12\) 0 0
\(13\) 1.10815 − 3.04461i 0.307345 − 0.844423i −0.685827 − 0.727764i \(-0.740559\pi\)
0.993172 − 0.116658i \(-0.0372183\pi\)
\(14\) 0 0
\(15\) −2.12056 + 2.52719i −0.547527 + 0.652518i
\(16\) 0 0
\(17\) −0.873207 − 2.39912i −0.211784 − 0.581871i 0.787628 − 0.616151i \(-0.211309\pi\)
−0.999412 + 0.0342791i \(0.989086\pi\)
\(18\) 0 0
\(19\) −1.88838 + 0.332972i −0.433223 + 0.0763889i −0.386007 − 0.922496i \(-0.626146\pi\)
−0.0472160 + 0.998885i \(0.515035\pi\)
\(20\) 0 0
\(21\) −1.00275 − 0.841410i −0.218819 − 0.183611i
\(22\) 0 0
\(23\) 5.61113 − 3.23959i 1.17000 − 0.675501i 0.216321 − 0.976322i \(-0.430594\pi\)
0.953681 + 0.300821i \(0.0972608\pi\)
\(24\) 0 0
\(25\) −1.02166 + 5.79410i −0.204331 + 1.15882i
\(26\) 0 0
\(27\) −0.500000 − 0.866025i −0.0962250 − 0.166667i
\(28\) 0 0
\(29\) −0.694918 − 0.401211i −0.129043 − 0.0745031i 0.434089 − 0.900870i \(-0.357070\pi\)
−0.563132 + 0.826367i \(0.690404\pi\)
\(30\) 0 0
\(31\) 9.58977i 1.72237i 0.508288 + 0.861187i \(0.330278\pi\)
−0.508288 + 0.861187i \(0.669722\pi\)
\(32\) 0 0
\(33\) −5.28305 − 1.92287i −0.919660 − 0.334729i
\(34\) 0 0
\(35\) −4.25281 − 0.749885i −0.718856 − 0.126754i
\(36\) 0 0
\(37\) −3.69170 − 4.83440i −0.606911 − 0.794770i
\(38\) 0 0
\(39\) 3.19078 + 0.562621i 0.510934 + 0.0900915i
\(40\) 0 0
\(41\) 8.65603 + 3.15054i 1.35184 + 0.492031i 0.913523 − 0.406787i \(-0.133351\pi\)
0.438321 + 0.898818i \(0.355573\pi\)
\(42\) 0 0
\(43\) − 4.57741i − 0.698048i −0.937114 − 0.349024i \(-0.886513\pi\)
0.937114 − 0.349024i \(-0.113487\pi\)
\(44\) 0 0
\(45\) −2.85703 − 1.64951i −0.425901 − 0.245894i
\(46\) 0 0
\(47\) 6.26284 + 10.8476i 0.913529 + 1.58228i 0.809041 + 0.587752i \(0.199987\pi\)
0.104487 + 0.994526i \(0.466680\pi\)
\(48\) 0 0
\(49\) −0.917993 + 5.20620i −0.131142 + 0.743743i
\(50\) 0 0
\(51\) 2.21104 − 1.27654i 0.309607 − 0.178752i
\(52\) 0 0
\(53\) 6.53540 + 5.48385i 0.897706 + 0.753265i 0.969741 − 0.244137i \(-0.0785048\pi\)
−0.0720347 + 0.997402i \(0.522949\pi\)
\(54\) 0 0
\(55\) −18.2656 + 3.22072i −2.46293 + 0.434282i
\(56\) 0 0
\(57\) −0.655826 − 1.80187i −0.0868663 − 0.238663i
\(58\) 0 0
\(59\) 5.94819 − 7.08877i 0.774388 − 0.922880i −0.224278 − 0.974525i \(-0.572002\pi\)
0.998665 + 0.0516457i \(0.0164467\pi\)
\(60\) 0 0
\(61\) 3.55243 − 9.76023i 0.454842 − 1.24967i −0.474436 − 0.880290i \(-0.657348\pi\)
0.929279 − 0.369379i \(-0.120430\pi\)
\(62\) 0 0
\(63\) 0.654501 − 1.13363i 0.0824594 − 0.142824i
\(64\) 0 0
\(65\) 10.0442 − 3.65579i 1.24583 − 0.453445i
\(66\) 0 0
\(67\) −1.86400 + 1.56408i −0.227724 + 0.191083i −0.749510 − 0.661993i \(-0.769711\pi\)
0.521785 + 0.853077i \(0.325266\pi\)
\(68\) 0 0
\(69\) 4.16473 + 4.96334i 0.501375 + 0.597516i
\(70\) 0 0
\(71\) −1.86869 − 10.5979i −0.221773 − 1.25773i −0.868760 − 0.495234i \(-0.835082\pi\)
0.646987 − 0.762501i \(-0.276029\pi\)
\(72\) 0 0
\(73\) 6.74190 0.789079 0.394540 − 0.918879i \(-0.370904\pi\)
0.394540 + 0.918879i \(0.370904\pi\)
\(74\) 0 0
\(75\) −5.88348 −0.679366
\(76\) 0 0
\(77\) −1.27794 − 7.24754i −0.145634 − 0.825933i
\(78\) 0 0
\(79\) −7.46536 − 8.89687i −0.839919 − 1.00098i −0.999904 − 0.0138566i \(-0.995589\pi\)
0.159985 − 0.987119i \(-0.448855\pi\)
\(80\) 0 0
\(81\) 0.766044 − 0.642788i 0.0851160 − 0.0714208i
\(82\) 0 0
\(83\) −1.36439 + 0.496597i −0.149761 + 0.0545086i −0.415813 − 0.909450i \(-0.636503\pi\)
0.266052 + 0.963959i \(0.414281\pi\)
\(84\) 0 0
\(85\) 4.21133 − 7.29424i 0.456783 − 0.791172i
\(86\) 0 0
\(87\) 0.274445 − 0.754031i 0.0294236 − 0.0808406i
\(88\) 0 0
\(89\) 3.45108 − 4.11284i 0.365814 − 0.435960i −0.551469 − 0.834195i \(-0.685933\pi\)
0.917283 + 0.398235i \(0.130377\pi\)
\(90\) 0 0
\(91\) 1.45057 + 3.98540i 0.152061 + 0.417783i
\(92\) 0 0
\(93\) −9.44408 + 1.66525i −0.979306 + 0.172678i
\(94\) 0 0
\(95\) −4.84591 − 4.06620i −0.497179 − 0.417183i
\(96\) 0 0
\(97\) 1.95274 − 1.12742i 0.198271 − 0.114472i −0.397578 − 0.917568i \(-0.630149\pi\)
0.595849 + 0.803097i \(0.296816\pi\)
\(98\) 0 0
\(99\) 0.976267 − 5.53669i 0.0981186 − 0.556458i
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.a.337.3 ✓ 18
3.2 odd 2 1332.2.ct.d.1225.1 18
37.28 even 18 inner 444.2.bb.a.361.3 yes 18
111.65 odd 18 1332.2.ct.d.361.1 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.bb.a.337.3 ✓ 18 1.1 even 1 trivial
444.2.bb.a.361.3 yes 18 37.28 even 18 inner
1332.2.ct.d.361.1 18 111.65 odd 18
1332.2.ct.d.1225.1 18 3.2 odd 2