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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [444,2,Mod(73,444)] 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("444.73"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 73.4
Root \(2.28825i\) of defining polynomial
Character \(\chi\) \(=\) 444.73
Dual form 444.2.e.a.73.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.70246i q^{5} +3.23607 q^{7} +1.00000 q^{9} -6.47214 q^{11} -1.74806i q^{13} -3.70246i q^{15} +6.53089i q^{17} +4.57649i q^{19} -3.23607 q^{21} -7.19859i q^{23} -8.70820 q^{25} -1.00000 q^{27} +4.78282i q^{29} +6.32456i q^{31} +6.47214 q^{33} +11.9814i q^{35} +(2.23607 + 5.65685i) q^{37} +1.74806i q^{39} +2.00000 q^{41} -2.82843i q^{43} +3.70246i q^{45} +4.00000 q^{47} +3.47214 q^{49} -6.53089i q^{51} -6.94427 q^{53} -23.9628i q^{55} -4.57649i q^{57} -0.874032i q^{59} -5.65685i q^{61} +3.23607 q^{63} +6.47214 q^{65} +5.70820 q^{67} +7.19859i q^{69} +8.94427 q^{71} -1.23607 q^{73} +8.70820 q^{75} -20.9443 q^{77} -15.8902i q^{79} +1.00000 q^{81} +10.4721 q^{83} -24.1803 q^{85} -4.78282i q^{87} +7.19859i q^{89} -5.65685i q^{91} -6.32456i q^{93} -16.9443 q^{95} +5.24419i q^{97} -6.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{7} + 4 q^{9} - 8 q^{11} - 4 q^{21} - 8 q^{25} - 4 q^{27} + 8 q^{33} + 8 q^{41} + 16 q^{47} - 4 q^{49} + 8 q^{53} + 4 q^{63} + 8 q^{65} - 4 q^{67} + 4 q^{73} + 8 q^{75} - 48 q^{77} + 4 q^{81}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/444\mathbb{Z}\right)^\times\).

\(n\) \(149\) \(223\) \(409\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 3.70246i 1.65579i 0.560883 + 0.827895i \(0.310462\pi\)
−0.560883 + 0.827895i \(0.689538\pi\)
\(6\) 0 0
\(7\) 3.23607 1.22312 0.611559 − 0.791199i \(-0.290543\pi\)
0.611559 + 0.791199i \(0.290543\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.47214 −1.95142 −0.975711 − 0.219061i \(-0.929701\pi\)
−0.975711 + 0.219061i \(0.929701\pi\)
\(12\) 0 0
\(13\) − 1.74806i − 0.484826i −0.970173 − 0.242413i \(-0.922061\pi\)
0.970173 − 0.242413i \(-0.0779389\pi\)
\(14\) 0 0
\(15\) − 3.70246i − 0.955971i
\(16\) 0 0
\(17\) 6.53089i 1.58397i 0.610539 + 0.791986i \(0.290953\pi\)
−0.610539 + 0.791986i \(0.709047\pi\)
\(18\) 0 0
\(19\) 4.57649i 1.04992i 0.851127 + 0.524960i \(0.175920\pi\)
−0.851127 + 0.524960i \(0.824080\pi\)
\(20\) 0 0
\(21\) −3.23607 −0.706168
\(22\) 0 0
\(23\) − 7.19859i − 1.50101i −0.660865 − 0.750505i \(-0.729811\pi\)
0.660865 − 0.750505i \(-0.270189\pi\)
\(24\) 0 0
\(25\) −8.70820 −1.74164
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.78282i 0.888148i 0.895990 + 0.444074i \(0.146467\pi\)
−0.895990 + 0.444074i \(0.853533\pi\)
\(30\) 0 0
\(31\) 6.32456i 1.13592i 0.823055 + 0.567962i \(0.192268\pi\)
−0.823055 + 0.567962i \(0.807732\pi\)
\(32\) 0 0
\(33\) 6.47214 1.12665
\(34\) 0 0
\(35\) 11.9814i 2.02523i
\(36\) 0 0
\(37\) 2.23607 + 5.65685i 0.367607 + 0.929981i
\(38\) 0 0
\(39\) 1.74806i 0.279914i
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) − 2.82843i − 0.431331i −0.976467 − 0.215666i \(-0.930808\pi\)
0.976467 − 0.215666i \(-0.0691921\pi\)
\(44\) 0 0
\(45\) 3.70246i 0.551930i
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 3.47214 0.496019
\(50\) 0 0
\(51\) − 6.53089i − 0.914507i
\(52\) 0 0
\(53\) −6.94427 −0.953869 −0.476935 − 0.878939i \(-0.658252\pi\)
−0.476935 + 0.878939i \(0.658252\pi\)
\(54\) 0 0
\(55\) − 23.9628i − 3.23115i
\(56\) 0 0
\(57\) − 4.57649i − 0.606171i
\(58\) 0 0
\(59\) − 0.874032i − 0.113789i −0.998380 − 0.0568946i \(-0.981880\pi\)
0.998380 − 0.0568946i \(-0.0181199\pi\)
\(60\) 0 0
\(61\) − 5.65685i − 0.724286i −0.932123 − 0.362143i \(-0.882045\pi\)
0.932123 − 0.362143i \(-0.117955\pi\)
\(62\) 0 0
\(63\) 3.23607 0.407706
\(64\) 0 0
\(65\) 6.47214 0.802770
\(66\) 0 0
\(67\) 5.70820 0.697368 0.348684 − 0.937240i \(-0.386629\pi\)
0.348684 + 0.937240i \(0.386629\pi\)
\(68\) 0 0
\(69\) 7.19859i 0.866608i
\(70\) 0 0
\(71\) 8.94427 1.06149 0.530745 − 0.847532i \(-0.321912\pi\)
0.530745 + 0.847532i \(0.321912\pi\)
\(72\) 0 0
\(73\) −1.23607 −0.144671 −0.0723354 − 0.997380i \(-0.523045\pi\)
−0.0723354 + 0.997380i \(0.523045\pi\)
\(74\) 0 0
\(75\) 8.70820 1.00554
\(76\) 0 0
\(77\) −20.9443 −2.38682
\(78\) 0 0
\(79\) − 15.8902i − 1.78779i −0.448279 − 0.893894i \(-0.647963\pi\)
0.448279 − 0.893894i \(-0.352037\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.4721 1.14947 0.574733 − 0.818341i \(-0.305106\pi\)
0.574733 + 0.818341i \(0.305106\pi\)
\(84\) 0 0
\(85\) −24.1803 −2.62273
\(86\) 0 0
\(87\) − 4.78282i − 0.512772i
\(88\) 0 0
\(89\) 7.19859i 0.763049i 0.924359 + 0.381524i \(0.124601\pi\)
−0.924359 + 0.381524i \(0.875399\pi\)
\(90\) 0 0
\(91\) − 5.65685i − 0.592999i
\(92\) 0 0
\(93\) − 6.32456i − 0.655826i
\(94\) 0 0
\(95\) −16.9443 −1.73845
\(96\) 0 0
\(97\) 5.24419i 0.532467i 0.963909 + 0.266234i \(0.0857792\pi\)
−0.963909 + 0.266234i \(0.914221\pi\)
\(98\) 0 0
\(99\) −6.47214 −0.650474
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 444.2.e.a.73.4 yes 4
3.2 odd 2 1332.2.e.e.73.1 4
4.3 odd 2 1776.2.h.f.961.4 4
12.11 even 2 5328.2.h.g.2737.1 4
37.36 even 2 inner 444.2.e.a.73.1 ✓ 4
111.110 odd 2 1332.2.e.e.73.4 4
148.147 odd 2 1776.2.h.f.961.1 4
444.443 even 2 5328.2.h.g.2737.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.e.a.73.1 ✓ 4 37.36 even 2 inner
444.2.e.a.73.4 yes 4 1.1 even 1 trivial
1332.2.e.e.73.1 4 3.2 odd 2
1332.2.e.e.73.4 4 111.110 odd 2
1776.2.h.f.961.1 4 148.147 odd 2
1776.2.h.f.961.4 4 4.3 odd 2
5328.2.h.g.2737.1 4 12.11 even 2
5328.2.h.g.2737.4 4 444.443 even 2