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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 81.4
Root \(-0.952275 + 2.93080i\) of defining polynomial
Character \(\chi\) \(=\) 440.81
Dual form 440.2.y.d.201.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.952275 - 2.93080i) q^{3} +(-0.809017 + 0.587785i) q^{5} +(0.0119064 + 0.0366440i) q^{7} +(-5.25571 - 3.81850i) q^{9} +(-1.11209 - 3.12462i) q^{11} +(-2.79380 - 2.02981i) q^{13} +(0.952275 + 2.93080i) q^{15} +(-0.726200 + 0.527615i) q^{17} +(0.373056 - 1.14815i) q^{19} +0.118734 q^{21} +7.60421 q^{23} +(0.309017 - 0.951057i) q^{25} +(-8.71688 + 6.33319i) q^{27} +(-2.33675 - 7.19177i) q^{29} +(4.81829 + 3.50069i) q^{31} +(-10.2167 + 0.283805i) q^{33} +(-0.0311712 - 0.0226472i) q^{35} +(3.31034 + 10.1882i) q^{37} +(-8.60944 + 6.25512i) q^{39} +(0.954637 - 2.93807i) q^{41} -9.68676 q^{43} +6.49642 q^{45} +(-0.403593 + 1.24213i) q^{47} +(5.66192 - 4.11362i) q^{49} +(0.854793 + 2.63078i) q^{51} +(-1.64407 - 1.19449i) q^{53} +(2.73630 + 1.87420i) q^{55} +(-3.00974 - 2.18671i) q^{57} +(-0.0573598 - 0.176535i) q^{59} +(7.91147 - 5.74802i) q^{61} +(0.0773487 - 0.238055i) q^{63} +3.45332 q^{65} +8.10586 q^{67} +(7.24130 - 22.2864i) q^{69} +(5.81639 - 4.22586i) q^{71} +(3.92691 + 12.0858i) q^{73} +(-2.49309 - 1.81133i) q^{75} +(0.101258 - 0.0779541i) q^{77} +(10.2417 + 7.44102i) q^{79} +(4.23793 + 13.0430i) q^{81} +(8.69073 - 6.31418i) q^{83} +(0.277384 - 0.853699i) q^{85} -23.3029 q^{87} -2.78161 q^{89} +(0.0411165 - 0.126544i) q^{91} +(14.8482 - 10.7878i) q^{93} +(0.373056 + 1.14815i) q^{95} +(-14.3301 - 10.4114i) q^{97} +(-6.08656 + 20.6686i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{3} - 4 q^{5} + 8 q^{7} - 7 q^{9} - 7 q^{11} - 11 q^{13} - 3 q^{15} + 9 q^{17} - 2 q^{19} + 12 q^{21} + 20 q^{23} - 4 q^{25} - 9 q^{27} + q^{29} - 2 q^{31} - 32 q^{33} - 2 q^{35} - 16 q^{37}+ \cdots - 10 q^{99}+O(q^{100}) \) 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
\(3\) 0.952275 − 2.93080i 0.549796 − 1.69210i −0.159509 − 0.987197i \(-0.550991\pi\)
0.709305 − 0.704902i \(-0.249009\pi\)
\(4\) 0 0
\(5\) −0.809017 + 0.587785i −0.361803 + 0.262866i
\(6\) 0 0
\(7\) 0.0119064 + 0.0366440i 0.00450018 + 0.0138501i 0.953281 − 0.302084i \(-0.0976822\pi\)
−0.948781 + 0.315934i \(0.897682\pi\)
\(8\) 0 0
\(9\) −5.25571 − 3.81850i −1.75190 − 1.27283i
\(10\) 0 0
\(11\) −1.11209 − 3.12462i −0.335307 − 0.942109i
\(12\) 0 0
\(13\) −2.79380 − 2.02981i −0.774860 − 0.562969i 0.128572 − 0.991700i \(-0.458961\pi\)
−0.903432 + 0.428731i \(0.858961\pi\)
\(14\) 0 0
\(15\) 0.952275 + 2.93080i 0.245876 + 0.756729i
\(16\) 0 0
\(17\) −0.726200 + 0.527615i −0.176129 + 0.127965i −0.672357 − 0.740227i \(-0.734718\pi\)
0.496228 + 0.868192i \(0.334718\pi\)
\(18\) 0 0
\(19\) 0.373056 − 1.14815i 0.0855850 − 0.263403i −0.899101 − 0.437741i \(-0.855779\pi\)
0.984686 + 0.174338i \(0.0557785\pi\)
\(20\) 0 0
\(21\) 0.118734 0.0259100
\(22\) 0 0
\(23\) 7.60421 1.58559 0.792794 − 0.609490i \(-0.208626\pi\)
0.792794 + 0.609490i \(0.208626\pi\)
\(24\) 0 0
\(25\) 0.309017 − 0.951057i 0.0618034 − 0.190211i
\(26\) 0 0
\(27\) −8.71688 + 6.33319i −1.67757 + 1.21882i
\(28\) 0 0
\(29\) −2.33675 − 7.19177i −0.433923 − 1.33548i −0.894186 − 0.447696i \(-0.852245\pi\)
0.460263 − 0.887783i \(-0.347755\pi\)
\(30\) 0 0
\(31\) 4.81829 + 3.50069i 0.865391 + 0.628743i 0.929346 − 0.369209i \(-0.120372\pi\)
−0.0639553 + 0.997953i \(0.520372\pi\)
\(32\) 0 0
\(33\) −10.2167 + 0.283805i −1.77849 + 0.0494041i
\(34\) 0 0
\(35\) −0.0311712 − 0.0226472i −0.00526890 − 0.00382808i
\(36\) 0 0
\(37\) 3.31034 + 10.1882i 0.544216 + 1.67492i 0.722847 + 0.691008i \(0.242833\pi\)
−0.178631 + 0.983916i \(0.557167\pi\)
\(38\) 0 0
\(39\) −8.60944 + 6.25512i −1.37861 + 1.00162i
\(40\) 0 0
\(41\) 0.954637 − 2.93807i 0.149089 − 0.458849i −0.848425 − 0.529316i \(-0.822449\pi\)
0.997514 + 0.0704663i \(0.0224487\pi\)
\(42\) 0 0
\(43\) −9.68676 −1.47722 −0.738609 − 0.674135i \(-0.764517\pi\)
−0.738609 + 0.674135i \(0.764517\pi\)
\(44\) 0 0
\(45\) 6.49642 0.968429
\(46\) 0 0
\(47\) −0.403593 + 1.24213i −0.0588700 + 0.181183i −0.976167 − 0.217020i \(-0.930366\pi\)
0.917297 + 0.398204i \(0.130366\pi\)
\(48\) 0 0
\(49\) 5.66192 − 4.11362i 0.808845 − 0.587661i
\(50\) 0 0
\(51\) 0.854793 + 2.63078i 0.119695 + 0.368383i
\(52\) 0 0
\(53\) −1.64407 − 1.19449i −0.225831 − 0.164075i 0.469117 − 0.883136i \(-0.344572\pi\)
−0.694947 + 0.719061i \(0.744572\pi\)
\(54\) 0 0
\(55\) 2.73630 + 1.87420i 0.368963 + 0.252718i
\(56\) 0 0
\(57\) −3.00974 − 2.18671i −0.398650 − 0.289636i
\(58\) 0 0
\(59\) −0.0573598 − 0.176535i −0.00746761 − 0.0229829i 0.947253 − 0.320486i \(-0.103846\pi\)
−0.954721 + 0.297503i \(0.903846\pi\)
\(60\) 0 0
\(61\) 7.91147 − 5.74802i 1.01296 − 0.735958i 0.0481316 − 0.998841i \(-0.484673\pi\)
0.964828 + 0.262883i \(0.0846733\pi\)
\(62\) 0 0
\(63\) 0.0773487 − 0.238055i 0.00974502 − 0.0299921i
\(64\) 0 0
\(65\) 3.45332 0.428332
\(66\) 0 0
\(67\) 8.10586 0.990288 0.495144 − 0.868811i \(-0.335115\pi\)
0.495144 + 0.868811i \(0.335115\pi\)
\(68\) 0 0
\(69\) 7.24130 − 22.2864i 0.871750 − 2.68297i
\(70\) 0 0
\(71\) 5.81639 − 4.22586i 0.690279 − 0.501517i −0.186473 − 0.982460i \(-0.559706\pi\)
0.876752 + 0.480943i \(0.159706\pi\)
\(72\) 0 0
\(73\) 3.92691 + 12.0858i 0.459610 + 1.41453i 0.865636 + 0.500673i \(0.166914\pi\)
−0.406026 + 0.913861i \(0.633086\pi\)
\(74\) 0 0
\(75\) −2.49309 − 1.81133i −0.287877 − 0.209155i
\(76\) 0 0
\(77\) 0.101258 − 0.0779541i 0.0115394 − 0.00888370i
\(78\) 0 0
\(79\) 10.2417 + 7.44102i 1.15228 + 0.837181i 0.988782 − 0.149363i \(-0.0477224\pi\)
0.163498 + 0.986544i \(0.447722\pi\)
\(80\) 0 0
\(81\) 4.23793 + 13.0430i 0.470881 + 1.44922i
\(82\) 0 0
\(83\) 8.69073 − 6.31418i 0.953931 − 0.693072i 0.00219777 − 0.999998i \(-0.499300\pi\)
0.951733 + 0.306926i \(0.0993004\pi\)
\(84\) 0 0
\(85\) 0.277384 − 0.853699i 0.0300865 − 0.0925967i
\(86\) 0 0
\(87\) −23.3029 −2.49833
\(88\) 0 0
\(89\) −2.78161 −0.294850 −0.147425 − 0.989073i \(-0.547099\pi\)
−0.147425 + 0.989073i \(0.547099\pi\)
\(90\) 0 0
\(91\) 0.0411165 − 0.126544i 0.00431018 − 0.0132654i
\(92\) 0 0
\(93\) 14.8482 − 10.7878i 1.53968 − 1.11865i
\(94\) 0 0
\(95\) 0.373056 + 1.14815i 0.0382748 + 0.117798i
\(96\) 0 0
\(97\) −14.3301 − 10.4114i −1.45500 − 1.05712i −0.984630 − 0.174655i \(-0.944119\pi\)
−0.470374 − 0.882467i \(-0.655881\pi\)
\(98\) 0 0
\(99\) −6.08656 + 20.6686i −0.611722 + 2.07727i
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.81.4 ✓ 16
4.3 odd 2 880.2.bo.k.81.1 16
11.3 even 5 inner 440.2.y.d.201.4 yes 16
11.5 even 5 4840.2.a.bg.1.7 8
11.6 odd 10 4840.2.a.bh.1.7 8
44.3 odd 10 880.2.bo.k.641.1 16
44.27 odd 10 9680.2.a.df.1.2 8
44.39 even 10 9680.2.a.de.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.d.81.4 ✓ 16 1.1 even 1 trivial
440.2.y.d.201.4 yes 16 11.3 even 5 inner
880.2.bo.k.81.1 16 4.3 odd 2
880.2.bo.k.641.1 16 44.3 odd 10
4840.2.a.bg.1.7 8 11.5 even 5
4840.2.a.bh.1.7 8 11.6 odd 10
9680.2.a.de.1.2 8 44.39 even 10
9680.2.a.df.1.2 8 44.27 odd 10