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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 = [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 81.2
Root \(0.421568 + 1.29745i\) of defining polynomial
Character \(\chi\) \(=\) 440.81
Dual form 440.2.y.b.201.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.260543 - 0.801870i) q^{3} +(0.809017 - 0.587785i) q^{5} +(1.46997 + 4.52411i) q^{7} +(1.85194 + 1.34551i) q^{9} +(0.231864 + 3.30851i) q^{11} +(-3.06058 - 2.22364i) q^{13} +(-0.260543 - 0.801870i) q^{15} +(-4.03055 + 2.92836i) q^{17} +(0.964489 - 2.96839i) q^{19} +4.01074 q^{21} +9.39001 q^{23} +(0.309017 - 0.951057i) q^{25} +(3.60777 - 2.62120i) q^{27} +(-0.805585 - 2.47934i) q^{29} +(2.98421 + 2.16816i) q^{31} +(2.71341 + 0.676086i) q^{33} +(3.84844 + 2.79605i) q^{35} +(-1.06536 - 3.27883i) q^{37} +(-2.58048 + 1.87483i) q^{39} +(2.88263 - 8.87181i) q^{41} -11.7051 q^{43} +2.28912 q^{45} +(-0.513451 + 1.58024i) q^{47} +(-12.6436 + 9.18613i) q^{49} +(1.29804 + 3.99494i) q^{51} +(1.56945 + 1.14027i) q^{53} +(2.13228 + 2.54035i) q^{55} +(-2.12897 - 1.54679i) q^{57} +(0.0321114 + 0.0988288i) q^{59} +(8.21258 - 5.96679i) q^{61} +(-3.36494 + 10.3562i) q^{63} -3.78308 q^{65} -4.99277 q^{67} +(2.44651 - 7.52957i) q^{69} +(5.03960 - 3.66149i) q^{71} +(-2.25783 - 6.94889i) q^{73} +(-0.682112 - 0.495583i) q^{75} +(-14.6272 + 5.91239i) q^{77} +(-8.53069 - 6.19791i) q^{79} +(0.960251 + 2.95535i) q^{81} +(-4.60338 + 3.34455i) q^{83} +(-1.53953 + 4.73819i) q^{85} -2.19800 q^{87} +13.6371 q^{89} +(5.56102 - 17.1151i) q^{91} +(2.51610 - 1.82805i) q^{93} +(-0.964489 - 2.96839i) q^{95} +(-11.0809 - 8.05072i) q^{97} +(-4.02224 + 6.43913i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{3} + 3 q^{5} - 8 q^{7} + 10 q^{9} - 4 q^{11} - 7 q^{13} + q^{15} + 7 q^{17} + 3 q^{19} + 4 q^{21} + 36 q^{23} - 3 q^{25} + 8 q^{27} + 13 q^{29} + 2 q^{31} - 19 q^{33} - 2 q^{35} - 22 q^{37}+ \cdots - 79 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 0.260543 − 0.801870i 0.150425 − 0.462960i −0.847244 − 0.531204i \(-0.821740\pi\)
0.997669 + 0.0682442i \(0.0217397\pi\)
\(4\) 0 0
\(5\) 0.809017 − 0.587785i 0.361803 − 0.262866i
\(6\) 0 0
\(7\) 1.46997 + 4.52411i 0.555597 + 1.70995i 0.694361 + 0.719627i \(0.255687\pi\)
−0.138764 + 0.990325i \(0.544313\pi\)
\(8\) 0 0
\(9\) 1.85194 + 1.34551i 0.617313 + 0.448504i
\(10\) 0 0
\(11\) 0.231864 + 3.30851i 0.0699096 + 0.997553i
\(12\) 0 0
\(13\) −3.06058 − 2.22364i −0.848851 − 0.616726i 0.0759779 − 0.997109i \(-0.475792\pi\)
−0.924829 + 0.380383i \(0.875792\pi\)
\(14\) 0 0
\(15\) −0.260543 − 0.801870i −0.0672720 − 0.207042i
\(16\) 0 0
\(17\) −4.03055 + 2.92836i −0.977551 + 0.710233i −0.957160 − 0.289559i \(-0.906491\pi\)
−0.0203914 + 0.999792i \(0.506491\pi\)
\(18\) 0 0
\(19\) 0.964489 − 2.96839i 0.221269 − 0.680996i −0.777380 − 0.629031i \(-0.783452\pi\)
0.998649 − 0.0519645i \(-0.0165483\pi\)
\(20\) 0 0
\(21\) 4.01074 0.875215
\(22\) 0 0
\(23\) 9.39001 1.95795 0.978976 − 0.203975i \(-0.0653861\pi\)
0.978976 + 0.203975i \(0.0653861\pi\)
\(24\) 0 0
\(25\) 0.309017 − 0.951057i 0.0618034 − 0.190211i
\(26\) 0 0
\(27\) 3.60777 − 2.62120i 0.694316 − 0.504450i
\(28\) 0 0
\(29\) −0.805585 − 2.47934i −0.149593 − 0.460401i 0.847980 − 0.530029i \(-0.177819\pi\)
−0.997573 + 0.0696276i \(0.977819\pi\)
\(30\) 0 0
\(31\) 2.98421 + 2.16816i 0.535980 + 0.389413i 0.822590 − 0.568635i \(-0.192528\pi\)
−0.286610 + 0.958047i \(0.592528\pi\)
\(32\) 0 0
\(33\) 2.71341 + 0.676086i 0.472343 + 0.117691i
\(34\) 0 0
\(35\) 3.84844 + 2.79605i 0.650504 + 0.472619i
\(36\) 0 0
\(37\) −1.06536 − 3.27883i −0.175143 − 0.539036i 0.824497 − 0.565867i \(-0.191458\pi\)
−0.999640 + 0.0268308i \(0.991458\pi\)
\(38\) 0 0
\(39\) −2.58048 + 1.87483i −0.413208 + 0.300213i
\(40\) 0 0
\(41\) 2.88263 − 8.87181i 0.450190 − 1.38554i −0.426499 − 0.904488i \(-0.640253\pi\)
0.876690 − 0.481056i \(-0.159747\pi\)
\(42\) 0 0
\(43\) −11.7051 −1.78500 −0.892502 − 0.451043i \(-0.851052\pi\)
−0.892502 + 0.451043i \(0.851052\pi\)
\(44\) 0 0
\(45\) 2.28912 0.341242
\(46\) 0 0
\(47\) −0.513451 + 1.58024i −0.0748945 + 0.230502i −0.981495 − 0.191489i \(-0.938668\pi\)
0.906600 + 0.421990i \(0.138668\pi\)
\(48\) 0 0
\(49\) −12.6436 + 9.18613i −1.80623 + 1.31230i
\(50\) 0 0
\(51\) 1.29804 + 3.99494i 0.181761 + 0.559404i
\(52\) 0 0
\(53\) 1.56945 + 1.14027i 0.215580 + 0.156628i 0.690335 − 0.723490i \(-0.257463\pi\)
−0.474755 + 0.880118i \(0.657463\pi\)
\(54\) 0 0
\(55\) 2.13228 + 2.54035i 0.287516 + 0.342541i
\(56\) 0 0
\(57\) −2.12897 − 1.54679i −0.281990 − 0.204877i
\(58\) 0 0
\(59\) 0.0321114 + 0.0988288i 0.00418055 + 0.0128664i 0.953125 − 0.302577i \(-0.0978468\pi\)
−0.948944 + 0.315443i \(0.897847\pi\)
\(60\) 0 0
\(61\) 8.21258 − 5.96679i 1.05151 − 0.763969i 0.0790130 − 0.996874i \(-0.474823\pi\)
0.972499 + 0.232905i \(0.0748231\pi\)
\(62\) 0 0
\(63\) −3.36494 + 10.3562i −0.423943 + 1.30476i
\(64\) 0 0
\(65\) −3.78308 −0.469233
\(66\) 0 0
\(67\) −4.99277 −0.609964 −0.304982 − 0.952358i \(-0.598650\pi\)
−0.304982 + 0.952358i \(0.598650\pi\)
\(68\) 0 0
\(69\) 2.44651 − 7.52957i 0.294525 − 0.906454i
\(70\) 0 0
\(71\) 5.03960 − 3.66149i 0.598091 − 0.434538i −0.247110 − 0.968987i \(-0.579481\pi\)
0.845201 + 0.534449i \(0.179481\pi\)
\(72\) 0 0
\(73\) −2.25783 − 6.94889i −0.264259 − 0.813306i −0.991863 − 0.127309i \(-0.959366\pi\)
0.727604 − 0.685998i \(-0.240634\pi\)
\(74\) 0 0
\(75\) −0.682112 − 0.495583i −0.0787635 − 0.0572250i
\(76\) 0 0
\(77\) −14.6272 + 5.91239i −1.66693 + 0.673780i
\(78\) 0 0
\(79\) −8.53069 − 6.19791i −0.959778 − 0.697319i −0.00667849 − 0.999978i \(-0.502126\pi\)
−0.953099 + 0.302658i \(0.902126\pi\)
\(80\) 0 0
\(81\) 0.960251 + 2.95535i 0.106695 + 0.328372i
\(82\) 0 0
\(83\) −4.60338 + 3.34455i −0.505287 + 0.367113i −0.811033 − 0.585001i \(-0.801094\pi\)
0.305746 + 0.952113i \(0.401094\pi\)
\(84\) 0 0
\(85\) −1.53953 + 4.73819i −0.166986 + 0.513929i
\(86\) 0 0
\(87\) −2.19800 −0.235650
\(88\) 0 0
\(89\) 13.6371 1.44553 0.722767 − 0.691091i \(-0.242870\pi\)
0.722767 + 0.691091i \(0.242870\pi\)
\(90\) 0 0
\(91\) 5.56102 − 17.1151i 0.582953 − 1.79415i
\(92\) 0 0
\(93\) 2.51610 − 1.82805i 0.260907 − 0.189560i
\(94\) 0 0
\(95\) −0.964489 − 2.96839i −0.0989545 − 0.304551i
\(96\) 0 0
\(97\) −11.0809 − 8.05072i −1.12509 − 0.817427i −0.140118 − 0.990135i \(-0.544748\pi\)
−0.984973 + 0.172708i \(0.944748\pi\)
\(98\) 0 0
\(99\) −4.02224 + 6.43913i −0.404250 + 0.647157i
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.81.2 ✓ 12
4.3 odd 2 880.2.bo.j.81.2 12
11.3 even 5 inner 440.2.y.b.201.2 yes 12
11.5 even 5 4840.2.a.bf.1.4 6
11.6 odd 10 4840.2.a.be.1.4 6
44.3 odd 10 880.2.bo.j.641.2 12
44.27 odd 10 9680.2.a.cx.1.3 6
44.39 even 10 9680.2.a.cy.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.b.81.2 ✓ 12 1.1 even 1 trivial
440.2.y.b.201.2 yes 12 11.3 even 5 inner
880.2.bo.j.81.2 12 4.3 odd 2
880.2.bo.j.641.2 12 44.3 odd 10
4840.2.a.be.1.4 6 11.6 odd 10
4840.2.a.bf.1.4 6 11.5 even 5
9680.2.a.cx.1.3 6 44.27 odd 10
9680.2.a.cy.1.3 6 44.39 even 10