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Properties

Label 440.2.y.b.201.3
Level $440$
Weight $2$
Character 440.201
Analytic conductor $3.513$
Analytic rank $0$
Dimension $12$
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] = [440,2,Mod(81,440)] 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("440.81"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(440, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.y (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-1,0,3,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 15 x^{10} - 22 x^{9} + 89 x^{8} - 118 x^{7} + 205 x^{6} - 68 x^{5} + 1061 x^{4} + \cdots + 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 201.3
Root \(0.945349 - 2.90948i\) of defining polynomial
Character \(\chi\) \(=\) 440.201
Dual form 440.2.y.b.81.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.584258 + 1.79816i) q^{3} +(0.809017 + 0.587785i) q^{5} +(-0.846938 + 2.60661i) q^{7} +(-0.464972 + 0.337822i) q^{9} +(-2.57134 + 2.09480i) q^{11} +(-0.159232 + 0.115689i) q^{13} +(-0.584258 + 1.79816i) q^{15} +(1.18771 + 0.862919i) q^{17} +(-0.828640 - 2.55029i) q^{19} -5.18193 q^{21} +1.81736 q^{23} +(0.309017 + 0.951057i) q^{25} +(3.70970 + 2.69525i) q^{27} +(0.426278 - 1.31195i) q^{29} +(-4.77653 + 3.47035i) q^{31} +(-5.26912 - 3.39978i) q^{33} +(-2.21731 + 1.61097i) q^{35} +(1.41413 - 4.35226i) q^{37} +(-0.301059 - 0.218732i) q^{39} +(0.381414 + 1.17387i) q^{41} +2.96514 q^{43} -0.574737 q^{45} +(2.87419 + 8.84584i) q^{47} +(-0.413981 - 0.300774i) q^{49} +(-0.857741 + 2.63985i) q^{51} +(-2.17939 + 1.58342i) q^{53} +(-3.31155 + 0.183336i) q^{55} +(4.10170 - 2.98006i) q^{57} +(4.39089 - 13.5138i) q^{59} +(-6.10021 - 4.43206i) q^{61} +(-0.486767 - 1.49811i) q^{63} -0.196821 q^{65} +10.6500 q^{67} +(1.06181 + 3.26790i) q^{69} +(5.56338 + 4.04203i) q^{71} +(3.22348 - 9.92086i) q^{73} +(-1.52961 + 1.11132i) q^{75} +(-3.28257 - 8.47665i) q^{77} +(-1.51552 + 1.10109i) q^{79} +(-3.21189 + 9.88518i) q^{81} +(4.68742 + 3.40561i) q^{83} +(0.453664 + 1.39623i) q^{85} +2.60815 q^{87} +17.3060 q^{89} +(-0.166695 - 0.513036i) q^{91} +(-9.03097 - 6.56138i) q^{93} +(0.828640 - 2.55029i) q^{95} +(14.8203 - 10.7675i) q^{97} +(0.487931 - 1.84268i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{3} + 3 q^{5} - 8 q^{7} + 10 q^{9} - 4 q^{11} - 7 q^{13} + q^{15} + 7 q^{17} + 3 q^{19} + 4 q^{21} + 36 q^{23} - 3 q^{25} + 8 q^{27} + 13 q^{29} + 2 q^{31} - 19 q^{33} - 2 q^{35} - 22 q^{37}+ \cdots - 79 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(221\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{4}{5}\right)\)

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
\(3\) 0.584258 + 1.79816i 0.337321 + 1.03817i 0.965567 + 0.260153i \(0.0837732\pi\)
−0.628246 + 0.778015i \(0.716227\pi\)
\(4\) 0 0
\(5\) 0.809017 + 0.587785i 0.361803 + 0.262866i
\(6\) 0 0
\(7\) −0.846938 + 2.60661i −0.320113 + 0.985205i 0.653486 + 0.756939i \(0.273306\pi\)
−0.973599 + 0.228267i \(0.926694\pi\)
\(8\) 0 0
\(9\) −0.464972 + 0.337822i −0.154991 + 0.112607i
\(10\) 0 0
\(11\) −2.57134 + 2.09480i −0.775288 + 0.631607i
\(12\) 0 0
\(13\) −0.159232 + 0.115689i −0.0441629 + 0.0320862i −0.609648 − 0.792672i \(-0.708689\pi\)
0.565485 + 0.824759i \(0.308689\pi\)
\(14\) 0 0
\(15\) −0.584258 + 1.79816i −0.150855 + 0.464283i
\(16\) 0 0
\(17\) 1.18771 + 0.862919i 0.288061 + 0.209289i 0.722426 − 0.691448i \(-0.243027\pi\)
−0.434365 + 0.900737i \(0.643027\pi\)
\(18\) 0 0
\(19\) −0.828640 − 2.55029i −0.190103 − 0.585077i 0.809896 − 0.586574i \(-0.199524\pi\)
−0.999999 + 0.00149654i \(0.999524\pi\)
\(20\) 0 0
\(21\) −5.18193 −1.13079
\(22\) 0 0
\(23\) 1.81736 0.378945 0.189473 − 0.981886i \(-0.439322\pi\)
0.189473 + 0.981886i \(0.439322\pi\)
\(24\) 0 0
\(25\) 0.309017 + 0.951057i 0.0618034 + 0.190211i
\(26\) 0 0
\(27\) 3.70970 + 2.69525i 0.713932 + 0.518702i
\(28\) 0 0
\(29\) 0.426278 − 1.31195i 0.0791577 − 0.243622i −0.903645 − 0.428283i \(-0.859119\pi\)
0.982802 + 0.184661i \(0.0591186\pi\)
\(30\) 0 0
\(31\) −4.77653 + 3.47035i −0.857889 + 0.623293i −0.927310 − 0.374294i \(-0.877885\pi\)
0.0694206 + 0.997587i \(0.477885\pi\)
\(32\) 0 0
\(33\) −5.26912 − 3.39978i −0.917236 − 0.591825i
\(34\) 0 0
\(35\) −2.21731 + 1.61097i −0.374794 + 0.272304i
\(36\) 0 0
\(37\) 1.41413 − 4.35226i 0.232482 − 0.715507i −0.764963 − 0.644074i \(-0.777243\pi\)
0.997445 − 0.0714327i \(-0.0227571\pi\)
\(38\) 0 0
\(39\) −0.301059 − 0.218732i −0.0482080 − 0.0350252i
\(40\) 0 0
\(41\) 0.381414 + 1.17387i 0.0595668 + 0.183328i 0.976412 − 0.215914i \(-0.0692732\pi\)
−0.916845 + 0.399242i \(0.869273\pi\)
\(42\) 0 0
\(43\) 2.96514 0.452180 0.226090 − 0.974106i \(-0.427406\pi\)
0.226090 + 0.974106i \(0.427406\pi\)
\(44\) 0 0
\(45\) −0.574737 −0.0856768
\(46\) 0 0
\(47\) 2.87419 + 8.84584i 0.419243 + 1.29030i 0.908400 + 0.418102i \(0.137305\pi\)
−0.489156 + 0.872196i \(0.662695\pi\)
\(48\) 0 0
\(49\) −0.413981 − 0.300774i −0.0591401 − 0.0429678i
\(50\) 0 0
\(51\) −0.857741 + 2.63985i −0.120108 + 0.369653i
\(52\) 0 0
\(53\) −2.17939 + 1.58342i −0.299362 + 0.217500i −0.727319 − 0.686300i \(-0.759234\pi\)
0.427956 + 0.903799i \(0.359234\pi\)
\(54\) 0 0
\(55\) −3.31155 + 0.183336i −0.446530 + 0.0247211i
\(56\) 0 0
\(57\) 4.10170 − 2.98006i 0.543283 − 0.394718i
\(58\) 0 0
\(59\) 4.39089 − 13.5138i 0.571646 − 1.75934i −0.0756817 − 0.997132i \(-0.524113\pi\)
0.647327 − 0.762212i \(-0.275887\pi\)
\(60\) 0 0
\(61\) −6.10021 − 4.43206i −0.781052 − 0.567467i 0.124243 − 0.992252i \(-0.460350\pi\)
−0.905294 + 0.424785i \(0.860350\pi\)
\(62\) 0 0
\(63\) −0.486767 − 1.49811i −0.0613269 − 0.188745i
\(64\) 0 0
\(65\) −0.196821 −0.0244127
\(66\) 0 0
\(67\) 10.6500 1.30111 0.650553 − 0.759461i \(-0.274537\pi\)
0.650553 + 0.759461i \(0.274537\pi\)
\(68\) 0 0
\(69\) 1.06181 + 3.26790i 0.127826 + 0.393409i
\(70\) 0 0
\(71\) 5.56338 + 4.04203i 0.660252 + 0.479701i 0.866748 − 0.498746i \(-0.166206\pi\)
−0.206496 + 0.978447i \(0.566206\pi\)
\(72\) 0 0
\(73\) 3.22348 − 9.92086i 0.377280 − 1.16115i −0.564647 − 0.825332i \(-0.690988\pi\)
0.941927 − 0.335817i \(-0.109012\pi\)
\(74\) 0 0
\(75\) −1.52961 + 1.11132i −0.176624 + 0.128325i
\(76\) 0 0
\(77\) −3.28257 − 8.47665i −0.374083 − 0.966004i
\(78\) 0 0
\(79\) −1.51552 + 1.10109i −0.170509 + 0.123882i −0.669767 − 0.742571i \(-0.733606\pi\)
0.499258 + 0.866453i \(0.333606\pi\)
\(80\) 0 0
\(81\) −3.21189 + 9.88518i −0.356876 + 1.09835i
\(82\) 0 0
\(83\) 4.68742 + 3.40561i 0.514512 + 0.373815i 0.814532 − 0.580118i \(-0.196994\pi\)
−0.300021 + 0.953933i \(0.596994\pi\)
\(84\) 0 0
\(85\) 0.453664 + 1.39623i 0.0492067 + 0.151443i
\(86\) 0 0
\(87\) 2.60815 0.279623
\(88\) 0 0
\(89\) 17.3060 1.83443 0.917214 − 0.398396i \(-0.130433\pi\)
0.917214 + 0.398396i \(0.130433\pi\)
\(90\) 0 0
\(91\) −0.166695 − 0.513036i −0.0174744 − 0.0537807i
\(92\) 0 0
\(93\) −9.03097 − 6.56138i −0.936468 − 0.680383i
\(94\) 0 0
\(95\) 0.828640 − 2.55029i 0.0850167 − 0.261655i
\(96\) 0 0
\(97\) 14.8203 − 10.7675i 1.50477 − 1.09328i 0.536336 − 0.844005i \(-0.319808\pi\)
0.968433 − 0.249274i \(-0.0801919\pi\)
\(98\) 0 0
\(99\) 0.487931 − 1.84268i 0.0490389 − 0.185196i
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 440.2.y.b.201.3 yes 12
4.3 odd 2 880.2.bo.j.641.1 12
11.2 odd 10 4840.2.a.be.1.5 6
11.4 even 5 inner 440.2.y.b.81.3 ✓ 12
11.9 even 5 4840.2.a.bf.1.5 6
44.15 odd 10 880.2.bo.j.81.1 12
44.31 odd 10 9680.2.a.cx.1.2 6
44.35 even 10 9680.2.a.cy.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.b.81.3 ✓ 12 11.4 even 5 inner
440.2.y.b.201.3 yes 12 1.1 even 1 trivial
880.2.bo.j.81.1 12 44.15 odd 10
880.2.bo.j.641.1 12 4.3 odd 2
4840.2.a.be.1.5 6 11.2 odd 10
4840.2.a.bf.1.5 6 11.9 even 5
9680.2.a.cx.1.2 6 44.31 odd 10
9680.2.a.cy.1.2 6 44.35 even 10