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,-1] 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: \( x^{12} - x^{11} + 5 x^{10} + 4 x^{9} + 28 x^{8} - 81 x^{7} + 335 x^{6} - 235 x^{5} + 782 x^{4} + \cdots + 1 \) 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 81.1
Root \(1.85498 - 1.34772i\) of defining polynomial
Character \(\chi\) \(=\) 440.81
Dual form 440.2.y.c.201.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.01756 + 3.13172i) q^{3} +(0.809017 - 0.587785i) q^{5} +(1.08622 + 3.34304i) q^{7} +(-6.34518 - 4.61004i) q^{9} +(-1.91497 + 2.70793i) q^{11} +(3.45546 + 2.51054i) q^{13} +(1.01756 + 3.13172i) q^{15} +(-1.28632 + 0.934565i) q^{17} +(1.71533 - 5.27925i) q^{19} -11.5747 q^{21} -6.39042 q^{23} +(0.309017 - 0.951057i) q^{25} +(12.9019 - 9.37380i) q^{27} +(-0.117804 - 0.362562i) q^{29} +(0.615229 + 0.446990i) q^{31} +(-6.53187 - 8.75262i) q^{33} +(2.84376 + 2.06611i) q^{35} +(-0.448664 - 1.38085i) q^{37} +(-11.3784 + 8.26690i) q^{39} +(-1.89183 + 5.82245i) q^{41} -7.19067 q^{43} -7.84307 q^{45} +(1.33546 - 4.11012i) q^{47} +(-4.33291 + 3.14804i) q^{49} +(-1.61789 - 4.97936i) q^{51} +(5.62189 + 4.08454i) q^{53} +(0.0424355 + 3.31635i) q^{55} +(14.7877 + 10.7439i) q^{57} +(3.92793 + 12.0889i) q^{59} +(7.19700 - 5.22893i) q^{61} +(8.51929 - 26.2197i) q^{63} +4.27118 q^{65} +12.5135 q^{67} +(6.50261 - 20.0130i) q^{69} +(-5.95347 + 4.32545i) q^{71} +(2.17304 + 6.68793i) q^{73} +(2.66400 + 1.93551i) q^{75} +(-11.1328 - 3.46042i) q^{77} +(5.65811 + 4.11086i) q^{79} +(8.95672 + 27.5660i) q^{81} +(8.69768 - 6.31923i) q^{83} +(-0.491330 + 1.51216i) q^{85} +1.25531 q^{87} +0.451594 q^{89} +(-4.63944 + 14.2787i) q^{91} +(-2.02588 + 1.47189i) q^{93} +(-1.71533 - 5.27925i) q^{95} +(2.24871 + 1.63379i) q^{97} +(24.6345 - 8.35419i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{3} + 3 q^{5} - q^{7} - 10 q^{9} + 4 q^{11} + 18 q^{13} + q^{15} + 3 q^{17} + 4 q^{19} - 28 q^{21} - 18 q^{23} - 3 q^{25} + 23 q^{27} + 15 q^{29} - 8 q^{31} + 4 q^{33} + 6 q^{35} + 6 q^{37}+ \cdots + 79 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −1.01756 + 3.13172i −0.587487 + 1.80810i 0.00156040 + 0.999999i \(0.499503\pi\)
−0.589047 + 0.808099i \(0.700497\pi\)
\(4\) 0 0
\(5\) 0.809017 − 0.587785i 0.361803 − 0.262866i
\(6\) 0 0
\(7\) 1.08622 + 3.34304i 0.410552 + 1.26355i 0.916169 + 0.400791i \(0.131265\pi\)
−0.505617 + 0.862758i \(0.668735\pi\)
\(8\) 0 0
\(9\) −6.34518 − 4.61004i −2.11506 − 1.53668i
\(10\) 0 0
\(11\) −1.91497 + 2.70793i −0.577386 + 0.816471i
\(12\) 0 0
\(13\) 3.45546 + 2.51054i 0.958372 + 0.696298i 0.952772 − 0.303687i \(-0.0982176\pi\)
0.00559963 + 0.999984i \(0.498218\pi\)
\(14\) 0 0
\(15\) 1.01756 + 3.13172i 0.262732 + 0.808606i
\(16\) 0 0
\(17\) −1.28632 + 0.934565i −0.311978 + 0.226665i −0.732745 − 0.680504i \(-0.761761\pi\)
0.420767 + 0.907169i \(0.361761\pi\)
\(18\) 0 0
\(19\) 1.71533 − 5.27925i 0.393525 − 1.21114i −0.536580 − 0.843849i \(-0.680284\pi\)
0.930105 − 0.367295i \(-0.119716\pi\)
\(20\) 0 0
\(21\) −11.5747 −2.52581
\(22\) 0 0
\(23\) −6.39042 −1.33249 −0.666247 − 0.745731i \(-0.732100\pi\)
−0.666247 + 0.745731i \(0.732100\pi\)
\(24\) 0 0
\(25\) 0.309017 − 0.951057i 0.0618034 − 0.190211i
\(26\) 0 0
\(27\) 12.9019 − 9.37380i 2.48298 − 1.80399i
\(28\) 0 0
\(29\) −0.117804 − 0.362562i −0.0218756 − 0.0673261i 0.939523 − 0.342486i \(-0.111269\pi\)
−0.961398 + 0.275160i \(0.911269\pi\)
\(30\) 0 0
\(31\) 0.615229 + 0.446990i 0.110498 + 0.0802818i 0.641662 − 0.766987i \(-0.278245\pi\)
−0.531164 + 0.847269i \(0.678245\pi\)
\(32\) 0 0
\(33\) −6.53187 − 8.75262i −1.13705 − 1.52364i
\(34\) 0 0
\(35\) 2.84376 + 2.06611i 0.480683 + 0.349236i
\(36\) 0 0
\(37\) −0.448664 − 1.38085i −0.0737599 − 0.227010i 0.907379 − 0.420313i \(-0.138080\pi\)
−0.981139 + 0.193304i \(0.938080\pi\)
\(38\) 0 0
\(39\) −11.3784 + 8.26690i −1.82200 + 1.32376i
\(40\) 0 0
\(41\) −1.89183 + 5.82245i −0.295454 + 0.909313i 0.687615 + 0.726076i \(0.258658\pi\)
−0.983069 + 0.183238i \(0.941342\pi\)
\(42\) 0 0
\(43\) −7.19067 −1.09657 −0.548284 − 0.836293i \(-0.684719\pi\)
−0.548284 + 0.836293i \(0.684719\pi\)
\(44\) 0 0
\(45\) −7.84307 −1.16918
\(46\) 0 0
\(47\) 1.33546 − 4.11012i 0.194797 − 0.599523i −0.805182 − 0.593028i \(-0.797932\pi\)
0.999979 − 0.00649536i \(-0.00206755\pi\)
\(48\) 0 0
\(49\) −4.33291 + 3.14804i −0.618987 + 0.449720i
\(50\) 0 0
\(51\) −1.61789 − 4.97936i −0.226550 − 0.697250i
\(52\) 0 0
\(53\) 5.62189 + 4.08454i 0.772226 + 0.561055i 0.902636 − 0.430405i \(-0.141629\pi\)
−0.130410 + 0.991460i \(0.541629\pi\)
\(54\) 0 0
\(55\) 0.0424355 + 3.31635i 0.00572200 + 0.447177i
\(56\) 0 0
\(57\) 14.7877 + 10.7439i 1.95868 + 1.42306i
\(58\) 0 0
\(59\) 3.92793 + 12.0889i 0.511373 + 1.57384i 0.789786 + 0.613382i \(0.210192\pi\)
−0.278413 + 0.960461i \(0.589808\pi\)
\(60\) 0 0
\(61\) 7.19700 − 5.22893i 0.921482 − 0.669496i −0.0224105 − 0.999749i \(-0.507134\pi\)
0.943892 + 0.330253i \(0.107134\pi\)
\(62\) 0 0
\(63\) 8.51929 − 26.2197i 1.07333 − 3.30337i
\(64\) 0 0
\(65\) 4.27118 0.529775
\(66\) 0 0
\(67\) 12.5135 1.52877 0.764383 − 0.644763i \(-0.223044\pi\)
0.764383 + 0.644763i \(0.223044\pi\)
\(68\) 0 0
\(69\) 6.50261 − 20.0130i 0.782823 − 2.40928i
\(70\) 0 0
\(71\) −5.95347 + 4.32545i −0.706547 + 0.513336i −0.882058 − 0.471141i \(-0.843842\pi\)
0.175511 + 0.984477i \(0.443842\pi\)
\(72\) 0 0
\(73\) 2.17304 + 6.68793i 0.254335 + 0.782763i 0.993960 + 0.109743i \(0.0350029\pi\)
−0.739625 + 0.673019i \(0.764997\pi\)
\(74\) 0 0
\(75\) 2.66400 + 1.93551i 0.307612 + 0.223493i
\(76\) 0 0
\(77\) −11.1328 − 3.46042i −1.26870 − 0.394352i
\(78\) 0 0
\(79\) 5.65811 + 4.11086i 0.636587 + 0.462508i 0.858676 − 0.512519i \(-0.171288\pi\)
−0.222089 + 0.975026i \(0.571288\pi\)
\(80\) 0 0
\(81\) 8.95672 + 27.5660i 0.995191 + 3.06288i
\(82\) 0 0
\(83\) 8.69768 − 6.31923i 0.954694 − 0.693626i 0.00278190 − 0.999996i \(-0.499114\pi\)
0.951912 + 0.306370i \(0.0991145\pi\)
\(84\) 0 0
\(85\) −0.491330 + 1.51216i −0.0532922 + 0.164017i
\(86\) 0 0
\(87\) 1.25531 0.134584
\(88\) 0 0
\(89\) 0.451594 0.0478689 0.0239344 − 0.999714i \(-0.492381\pi\)
0.0239344 + 0.999714i \(0.492381\pi\)
\(90\) 0 0
\(91\) −4.63944 + 14.2787i −0.486345 + 1.49682i
\(92\) 0 0
\(93\) −2.02588 + 1.47189i −0.210074 + 0.152627i
\(94\) 0 0
\(95\) −1.71533 − 5.27925i −0.175990 − 0.541640i
\(96\) 0 0
\(97\) 2.24871 + 1.63379i 0.228322 + 0.165886i 0.696065 − 0.717979i \(-0.254933\pi\)
−0.467743 + 0.883865i \(0.654933\pi\)
\(98\) 0 0
\(99\) 24.6345 − 8.35419i 2.47586 − 0.839628i
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.c.81.1 ✓ 12
4.3 odd 2 880.2.bo.i.81.3 12
11.3 even 5 inner 440.2.y.c.201.1 yes 12
11.5 even 5 4840.2.a.bb.1.1 6
11.6 odd 10 4840.2.a.ba.1.1 6
44.3 odd 10 880.2.bo.i.641.3 12
44.27 odd 10 9680.2.a.dc.1.6 6
44.39 even 10 9680.2.a.dd.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.c.81.1 ✓ 12 1.1 even 1 trivial
440.2.y.c.201.1 yes 12 11.3 even 5 inner
880.2.bo.i.81.3 12 4.3 odd 2
880.2.bo.i.641.3 12 44.3 odd 10
4840.2.a.ba.1.1 6 11.6 odd 10
4840.2.a.bb.1.1 6 11.5 even 5
9680.2.a.dc.1.6 6 44.27 odd 10
9680.2.a.dd.1.6 6 44.39 even 10