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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [880,2,Mod(81,880)] 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("880.81"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.bo (of order \(5\), degree \(4\), not 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: \(7.02683537787\)
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: no (minimal twist has level 440)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 641.2
Root \(-0.220438 - 0.678438i\) of defining polynomial
Character \(\chi\) \(=\) 880.641
Dual form 880.2.bo.k.81.2

$q$-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 \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{4}{5}\right)\) \(1\)

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 \(-0.166021\pi\)
−0.994308 + 0.106546i \(0.966021\pi\)
\(4\) 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 \(-0.922648\pi\)
0.926682 + 0.375846i \(0.122648\pi\)
\(8\) 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 −0.223379 − 0.974732i \(-0.571709\pi\)
0.857997 + 0.513654i \(0.171709\pi\)
\(14\) 0 0
\(15\) −0.220438 + 0.678438i −0.0569168 + 0.175172i
\(16\) 0 0
\(17\) 3.91377 + 2.84352i 0.949228 + 0.689655i 0.950624 − 0.310344i \(-0.100444\pi\)
−0.00139602 + 0.999999i \(0.500444\pi\)
\(18\) 0 0
\(19\) −0.905388 − 2.78650i −0.207710 − 0.639267i −0.999591 − 0.0285910i \(-0.990898\pi\)
0.791881 − 0.610676i \(-0.209102\pi\)
\(20\) 0 0
\(21\) 0.268343 0.0585573
\(22\) 0 0
\(23\) −3.77226 −0.786571 −0.393285 − 0.919416i \(-0.628662\pi\)
−0.393285 + 0.919416i \(0.628662\pi\)
\(24\) 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 −0.348107 − 0.937455i \(-0.613175\pi\)
0.832646 − 0.553805i \(-0.186825\pi\)
\(30\) 0 0
\(31\) 6.50458 − 4.72586i 1.16826 − 0.848789i 0.177458 − 0.984128i \(-0.443213\pi\)
0.990799 + 0.135340i \(0.0432126\pi\)
\(32\) 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 −0.852644 − 0.522492i \(-0.825002\pi\)
0.996917 + 0.0784662i \(0.0250023\pi\)
\(38\) 0 0
\(39\) −1.63226 − 1.18590i −0.261370 − 0.189897i
\(40\) 0 0
\(41\) −1.14965 − 3.53825i −0.179545 − 0.552583i 0.820267 − 0.571981i \(-0.193825\pi\)
−0.999812 + 0.0193984i \(0.993825\pi\)
\(42\) 0 0
\(43\) 6.48484 0.988929 0.494465 − 0.869198i \(-0.335364\pi\)
0.494465 + 0.869198i \(0.335364\pi\)
\(44\) 0 0
\(45\) −2.49113 −0.371356
\(46\) 0 0
\(47\) −0.800034 − 2.46225i −0.116697 − 0.359157i 0.875600 − 0.483037i \(-0.160466\pi\)
−0.992297 + 0.123880i \(0.960466\pi\)
\(48\) 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 −0.585073 − 0.810981i \(-0.698934\pi\)
0.590491 + 0.807044i \(0.298934\pi\)
\(54\) 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 \(-0.865760\pi\)
0.978736 + 0.205126i \(0.0657603\pi\)
\(60\) 0 0
\(61\) −7.03606 − 5.11200i −0.900875 − 0.654524i 0.0378153 − 0.999285i \(-0.487960\pi\)
−0.938691 + 0.344760i \(0.887960\pi\)
\(62\) 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 \(-0.746273\pi\)
−0.698779 + 0.715338i \(0.746273\pi\)
\(68\) 0 0
\(69\) 0.831550 + 2.55925i 0.100107 + 0.308097i
\(70\) 0 0
\(71\) 11.4246 + 8.30046i 1.35585 + 0.985084i 0.998697 + 0.0510383i \(0.0162530\pi\)
0.357155 + 0.934045i \(0.383747\pi\)
\(72\) 0 0
\(73\) 0.158595 − 0.488106i 0.0185622 − 0.0571284i −0.941346 − 0.337442i \(-0.890438\pi\)
0.959909 + 0.280313i \(0.0904384\pi\)
\(74\) 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 \(-0.439374\pi\)
0.992359 + 0.123383i \(0.0393743\pi\)
\(80\) 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 \(-0.252007\pi\)
−0.459598 + 0.888127i \(0.652007\pi\)
\(84\) 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 −0.320486 + 0.947253i \(0.603846\pi\)
−0.999927 + 0.0120825i \(0.996154\pi\)
\(98\) 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 880.2.bo.k.641.2 16
4.3 odd 2 440.2.y.d.201.3 yes 16
11.2 odd 10 9680.2.a.de.1.4 8
11.4 even 5 inner 880.2.bo.k.81.2 16
11.9 even 5 9680.2.a.df.1.4 8
44.15 odd 10 440.2.y.d.81.3 ✓ 16
44.31 odd 10 4840.2.a.bg.1.5 8
44.35 even 10 4840.2.a.bh.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.d.81.3 ✓ 16 44.15 odd 10
440.2.y.d.201.3 yes 16 4.3 odd 2
880.2.bo.k.81.2 16 11.4 even 5 inner
880.2.bo.k.641.2 16 1.1 even 1 trivial
4840.2.a.bg.1.5 8 44.31 odd 10
4840.2.a.bh.1.5 8 44.35 even 10
9680.2.a.de.1.4 8 11.2 odd 10
9680.2.a.df.1.4 8 11.9 even 5