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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 = [16,0,-3,0,-4,0,8] 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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 141 x^{12} - 220 x^{11} + 1105 x^{10} - 1935 x^{9} + \cdots + 10000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 81.3
Root \(-0.220438 + 0.678438i\) of defining polynomial
Character \(\chi\) \(=\) 440.81
Dual form 440.2.y.d.201.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.220438 - 0.678438i) q^{3} +(-0.809017 + 0.587785i) q^{5} +(0.116244 + 0.357761i) q^{7} +(2.01537 + 1.46425i) q^{9} +(-0.107091 + 3.31490i) q^{11} +(2.28815 + 1.66244i) q^{13} +(0.220438 + 0.678438i) q^{15} +(3.91377 - 2.84352i) q^{17} +(0.905388 - 2.78650i) q^{19} +0.268343 q^{21} +3.77226 q^{23} +(0.309017 - 0.951057i) q^{25} +(3.16901 - 2.30242i) q^{27} +(2.60933 + 8.03068i) q^{29} +(-6.50458 - 4.72586i) q^{31} +(2.22534 + 0.803383i) q^{33} +(-0.304330 - 0.221108i) q^{35} +(0.877578 + 2.70091i) q^{37} +(1.63226 - 1.18590i) q^{39} +(-1.14965 + 3.53825i) q^{41} -6.48484 q^{43} -2.49113 q^{45} +(0.800034 - 2.46225i) q^{47} +(5.54864 - 4.03132i) q^{49} +(-1.06641 - 3.28207i) q^{51} +(0.0394497 + 0.0286619i) q^{53} +(-1.86181 - 2.74475i) q^{55} +(-1.69089 - 1.22850i) q^{57} +(-0.509660 - 1.56857i) q^{59} +(-7.03606 + 5.11200i) q^{61} +(-0.289578 + 0.891229i) q^{63} -2.82831 q^{65} +11.4395 q^{67} +(0.831550 - 2.55925i) q^{69} +(-11.4246 + 8.30046i) q^{71} +(0.158595 + 0.488106i) q^{73} +(-0.577114 - 0.419298i) q^{75} +(-1.19839 + 0.347022i) q^{77} +(-10.5029 - 7.63082i) q^{79} +(1.44592 + 4.45010i) q^{81} +(-2.21418 + 1.60869i) q^{83} +(-1.49493 + 4.60091i) q^{85} +6.02351 q^{87} +12.0195 q^{89} +(-0.328773 + 1.01186i) q^{91} +(-4.64006 + 3.37120i) q^{93} +(0.905388 + 2.78650i) q^{95} +(-13.0046 - 9.44836i) q^{97} +(-5.06966 + 6.52392i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{3} - 4 q^{5} + 8 q^{7} - 7 q^{9} - 7 q^{11} - 11 q^{13} - 3 q^{15} + 9 q^{17} - 2 q^{19} + 12 q^{21} + 20 q^{23} - 4 q^{25} - 9 q^{27} + q^{29} - 2 q^{31} - 32 q^{33} - 2 q^{35} - 16 q^{37}+ \cdots - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 0.220438 − 0.678438i 0.127270 − 0.391696i −0.867038 − 0.498242i \(-0.833979\pi\)
0.994308 + 0.106546i \(0.0339791\pi\)
\(4\) 0 0
\(5\) −0.809017 + 0.587785i −0.361803 + 0.262866i
\(6\) 0 0
\(7\) 0.116244 + 0.357761i 0.0439359 + 0.135221i 0.970618 − 0.240625i \(-0.0773523\pi\)
−0.926682 + 0.375846i \(0.877352\pi\)
\(8\) 0 0
\(9\) 2.01537 + 1.46425i 0.671789 + 0.488083i
\(10\) 0 0
\(11\) −0.107091 + 3.31490i −0.0322892 + 0.999479i
\(12\) 0 0
\(13\) 2.28815 + 1.66244i 0.634619 + 0.461077i 0.857997 − 0.513654i \(-0.171709\pi\)
−0.223379 + 0.974732i \(0.571709\pi\)
\(14\) 0 0
\(15\) 0.220438 + 0.678438i 0.0569168 + 0.175172i
\(16\) 0 0
\(17\) 3.91377 − 2.84352i 0.949228 − 0.689655i −0.00139602 − 0.999999i \(-0.500444\pi\)
0.950624 + 0.310344i \(0.100444\pi\)
\(18\) 0 0
\(19\) 0.905388 − 2.78650i 0.207710 − 0.639267i −0.791881 − 0.610676i \(-0.790898\pi\)
0.999591 − 0.0285910i \(-0.00910205\pi\)
\(20\) 0 0
\(21\) 0.268343 0.0585573
\(22\) 0 0
\(23\) 3.77226 0.786571 0.393285 − 0.919416i \(-0.371338\pi\)
0.393285 + 0.919416i \(0.371338\pi\)
\(24\) 0 0
\(25\) 0.309017 − 0.951057i 0.0618034 − 0.190211i
\(26\) 0 0
\(27\) 3.16901 − 2.30242i 0.609876 − 0.443101i
\(28\) 0 0
\(29\) 2.60933 + 8.03068i 0.484540 + 1.49126i 0.832646 + 0.553805i \(0.186825\pi\)
−0.348107 + 0.937455i \(0.613175\pi\)
\(30\) 0 0
\(31\) −6.50458 − 4.72586i −1.16826 − 0.848789i −0.177458 − 0.984128i \(-0.556787\pi\)
−0.990799 + 0.135340i \(0.956787\pi\)
\(32\) 0 0
\(33\) 2.22534 + 0.803383i 0.387383 + 0.139851i
\(34\) 0 0
\(35\) −0.304330 − 0.221108i −0.0514411 − 0.0373742i
\(36\) 0 0
\(37\) 0.877578 + 2.70091i 0.144273 + 0.444026i 0.996917 − 0.0784662i \(-0.0250023\pi\)
−0.852644 + 0.522492i \(0.825002\pi\)
\(38\) 0 0
\(39\) 1.63226 − 1.18590i 0.261370 − 0.189897i
\(40\) 0 0
\(41\) −1.14965 + 3.53825i −0.179545 + 0.552583i −0.999812 − 0.0193984i \(-0.993825\pi\)
0.820267 + 0.571981i \(0.193825\pi\)
\(42\) 0 0
\(43\) −6.48484 −0.988929 −0.494465 − 0.869198i \(-0.664636\pi\)
−0.494465 + 0.869198i \(0.664636\pi\)
\(44\) 0 0
\(45\) −2.49113 −0.371356
\(46\) 0 0
\(47\) 0.800034 − 2.46225i 0.116697 − 0.359157i −0.875600 − 0.483037i \(-0.839534\pi\)
0.992297 + 0.123880i \(0.0395338\pi\)
\(48\) 0 0
\(49\) 5.54864 − 4.03132i 0.792663 − 0.575903i
\(50\) 0 0
\(51\) −1.06641 − 3.28207i −0.149327 − 0.459582i
\(52\) 0 0
\(53\) 0.0394497 + 0.0286619i 0.00541884 + 0.00393702i 0.590491 − 0.807044i \(-0.298934\pi\)
−0.585073 + 0.810981i \(0.698934\pi\)
\(54\) 0 0
\(55\) −1.86181 − 2.74475i −0.251046 − 0.370102i
\(56\) 0 0
\(57\) −1.69089 − 1.22850i −0.223963 − 0.162719i
\(58\) 0 0
\(59\) −0.509660 − 1.56857i −0.0663521 − 0.204211i 0.912384 − 0.409336i \(-0.134240\pi\)
−0.978736 + 0.205126i \(0.934240\pi\)
\(60\) 0 0
\(61\) −7.03606 + 5.11200i −0.900875 + 0.654524i −0.938691 − 0.344760i \(-0.887960\pi\)
0.0378153 + 0.999285i \(0.487960\pi\)
\(62\) 0 0
\(63\) −0.289578 + 0.891229i −0.0364834 + 0.112284i
\(64\) 0 0
\(65\) −2.82831 −0.350809
\(66\) 0 0
\(67\) 11.4395 1.39756 0.698779 − 0.715338i \(-0.253727\pi\)
0.698779 + 0.715338i \(0.253727\pi\)
\(68\) 0 0
\(69\) 0.831550 − 2.55925i 0.100107 − 0.308097i
\(70\) 0 0
\(71\) −11.4246 + 8.30046i −1.35585 + 0.985084i −0.357155 + 0.934045i \(0.616253\pi\)
−0.998697 + 0.0510383i \(0.983747\pi\)
\(72\) 0 0
\(73\) 0.158595 + 0.488106i 0.0185622 + 0.0571284i 0.959909 − 0.280313i \(-0.0904384\pi\)
−0.941346 + 0.337442i \(0.890438\pi\)
\(74\) 0 0
\(75\) −0.577114 − 0.419298i −0.0666394 − 0.0484163i
\(76\) 0 0
\(77\) −1.19839 + 0.347022i −0.136569 + 0.0395469i
\(78\) 0 0
\(79\) −10.5029 − 7.63082i −1.18167 − 0.858534i −0.189312 − 0.981917i \(-0.560626\pi\)
−0.992359 + 0.123383i \(0.960626\pi\)
\(80\) 0 0
\(81\) 1.44592 + 4.45010i 0.160658 + 0.494455i
\(82\) 0 0
\(83\) −2.21418 + 1.60869i −0.243037 + 0.176577i −0.702635 − 0.711550i \(-0.747993\pi\)
0.459598 + 0.888127i \(0.347993\pi\)
\(84\) 0 0
\(85\) −1.49493 + 4.60091i −0.162148 + 0.499039i
\(86\) 0 0
\(87\) 6.02351 0.645789
\(88\) 0 0
\(89\) 12.0195 1.27407 0.637034 − 0.770835i \(-0.280161\pi\)
0.637034 + 0.770835i \(0.280161\pi\)
\(90\) 0 0
\(91\) −0.328773 + 1.01186i −0.0344648 + 0.106072i
\(92\) 0 0
\(93\) −4.64006 + 3.37120i −0.481152 + 0.349577i
\(94\) 0 0
\(95\) 0.905388 + 2.78650i 0.0928909 + 0.285889i
\(96\) 0 0
\(97\) −13.0046 − 9.44836i −1.32041 − 0.959336i −0.999927 − 0.0120825i \(-0.996154\pi\)
−0.320486 − 0.947253i \(-0.603846\pi\)
\(98\) 0 0
\(99\) −5.06966 + 6.52392i −0.509520 + 0.655678i
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.d.81.3 ✓ 16
4.3 odd 2 880.2.bo.k.81.2 16
11.3 even 5 inner 440.2.y.d.201.3 yes 16
11.5 even 5 4840.2.a.bg.1.5 8
11.6 odd 10 4840.2.a.bh.1.5 8
44.3 odd 10 880.2.bo.k.641.2 16
44.27 odd 10 9680.2.a.df.1.4 8
44.39 even 10 9680.2.a.de.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.d.81.3 ✓ 16 1.1 even 1 trivial
440.2.y.d.201.3 yes 16 11.3 even 5 inner
880.2.bo.k.81.2 16 4.3 odd 2
880.2.bo.k.641.2 16 44.3 odd 10
4840.2.a.bg.1.5 8 11.5 even 5
4840.2.a.bh.1.5 8 11.6 odd 10
9680.2.a.de.1.4 8 44.39 even 10
9680.2.a.df.1.4 8 44.27 odd 10