Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1900,2,Mod(1,1900)] 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("1900.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,2,0,0,0,2,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.19869\) of defining polynomial
Character \(\chi\) \(=\) 1900.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.19869 q^{3} -1.19869 q^{7} -1.56314 q^{9} -5.86718 q^{11} +0.364448 q^{13} -1.19869 q^{17} -1.00000 q^{19} +1.43686 q^{21} +8.23163 q^{23} +5.46980 q^{27} -7.86718 q^{29} +7.30404 q^{31} +7.03293 q^{33} -7.13828 q^{37} -0.436861 q^{39} +2.43686 q^{41} +7.39738 q^{43} +13.7014 q^{47} -5.56314 q^{49} +1.43686 q^{51} +7.39738 q^{53} +1.19869 q^{57} +12.8606 q^{59} -1.30404 q^{61} +1.87372 q^{63} -11.9330 q^{67} -9.86718 q^{69} +2.12628 q^{71} -2.50273 q^{73} +7.03293 q^{77} -7.74090 q^{79} -1.86718 q^{81} +3.02093 q^{83} +9.43032 q^{87} -5.68942 q^{89} -0.436861 q^{91} -8.75529 q^{93} -1.09334 q^{97} +9.17122 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} + 2 q^{7} + q^{9} - q^{11} + q^{13} + 2 q^{17} - 3 q^{19} + 10 q^{21} + 8 q^{23} + 11 q^{27} - 7 q^{29} + 11 q^{31} + 10 q^{33} - 5 q^{37} - 7 q^{39} + 13 q^{41} + 11 q^{43} + 19 q^{47}+ \cdots - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.19869 −0.692065 −0.346032 − 0.938223i \(-0.612471\pi\)
−0.346032 + 0.938223i \(0.612471\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.19869 −0.453063 −0.226531 − 0.974004i \(-0.572739\pi\)
−0.226531 + 0.974004i \(0.572739\pi\)
\(8\) 0 0
\(9\) −1.56314 −0.521046
\(10\) 0 0
\(11\) −5.86718 −1.76902 −0.884510 − 0.466521i \(-0.845507\pi\)
−0.884510 + 0.466521i \(0.845507\pi\)
\(12\) 0 0
\(13\) 0.364448 0.101080 0.0505399 − 0.998722i \(-0.483906\pi\)
0.0505399 + 0.998722i \(0.483906\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.19869 −0.290725 −0.145363 − 0.989378i \(-0.546435\pi\)
−0.145363 + 0.989378i \(0.546435\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 1.43686 0.313549
\(22\) 0 0
\(23\) 8.23163 1.71641 0.858206 − 0.513305i \(-0.171579\pi\)
0.858206 + 0.513305i \(0.171579\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.46980 1.05266
\(28\) 0 0
\(29\) −7.86718 −1.46090 −0.730449 − 0.682967i \(-0.760689\pi\)
−0.730449 + 0.682967i \(0.760689\pi\)
\(30\) 0 0
\(31\) 7.30404 1.31184 0.655922 − 0.754829i \(-0.272280\pi\)
0.655922 + 0.754829i \(0.272280\pi\)
\(32\) 0 0
\(33\) 7.03293 1.22428
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.13828 −1.17353 −0.586763 − 0.809759i \(-0.699598\pi\)
−0.586763 + 0.809759i \(0.699598\pi\)
\(38\) 0 0
\(39\) −0.436861 −0.0699537
\(40\) 0 0
\(41\) 2.43686 0.380574 0.190287 − 0.981729i \(-0.439058\pi\)
0.190287 + 0.981729i \(0.439058\pi\)
\(42\) 0 0
\(43\) 7.39738 1.12809 0.564045 − 0.825744i \(-0.309244\pi\)
0.564045 + 0.825744i \(0.309244\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 13.7014 1.99856 0.999279 − 0.0379717i \(-0.0120897\pi\)
0.999279 + 0.0379717i \(0.0120897\pi\)
\(48\) 0 0
\(49\) −5.56314 −0.794734
\(50\) 0 0
\(51\) 1.43686 0.201201
\(52\) 0 0
\(53\) 7.39738 1.01611 0.508054 − 0.861325i \(-0.330365\pi\)
0.508054 + 0.861325i \(0.330365\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.19869 0.158771
\(58\) 0 0
\(59\) 12.8606 1.67431 0.837156 − 0.546964i \(-0.184217\pi\)
0.837156 + 0.546964i \(0.184217\pi\)
\(60\) 0 0
\(61\) −1.30404 −0.166965 −0.0834825 − 0.996509i \(-0.526604\pi\)
−0.0834825 + 0.996509i \(0.526604\pi\)
\(62\) 0 0
\(63\) 1.87372 0.236067
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −11.9330 −1.45785 −0.728927 − 0.684592i \(-0.759981\pi\)
−0.728927 + 0.684592i \(0.759981\pi\)
\(68\) 0 0
\(69\) −9.86718 −1.18787
\(70\) 0 0
\(71\) 2.12628 0.252343 0.126171 − 0.992008i \(-0.459731\pi\)
0.126171 + 0.992008i \(0.459731\pi\)
\(72\) 0 0
\(73\) −2.50273 −0.292922 −0.146461 − 0.989216i \(-0.546788\pi\)
−0.146461 + 0.989216i \(0.546788\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.03293 0.801477
\(78\) 0 0
\(79\) −7.74090 −0.870919 −0.435460 − 0.900208i \(-0.643414\pi\)
−0.435460 + 0.900208i \(0.643414\pi\)
\(80\) 0 0
\(81\) −1.86718 −0.207464
\(82\) 0 0
\(83\) 3.02093 0.331590 0.165795 − 0.986160i \(-0.446981\pi\)
0.165795 + 0.986160i \(0.446981\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.43032 1.01104
\(88\) 0 0
\(89\) −5.68942 −0.603077 −0.301539 − 0.953454i \(-0.597500\pi\)
−0.301539 + 0.953454i \(0.597500\pi\)
\(90\) 0 0
\(91\) −0.436861 −0.0457954
\(92\) 0 0
\(93\) −8.75529 −0.907881
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.09334 −0.111012 −0.0555061 − 0.998458i \(-0.517677\pi\)
−0.0555061 + 0.998458i \(0.517677\pi\)
\(98\) 0 0
\(99\) 9.17122 0.921742
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 1900.2.a.h.1.1 yes 3
4.3 odd 2 7600.2.a.bj.1.3 3
5.2 odd 4 1900.2.c.g.1749.5 6
5.3 odd 4 1900.2.c.g.1749.2 6
5.4 even 2 1900.2.a.f.1.3 ✓ 3
20.19 odd 2 7600.2.a.by.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1900.2.a.f.1.3 ✓ 3 5.4 even 2
1900.2.a.h.1.1 yes 3 1.1 even 1 trivial
1900.2.c.g.1749.2 6 5.3 odd 4
1900.2.c.g.1749.5 6 5.2 odd 4
7600.2.a.bj.1.3 3 4.3 odd 2
7600.2.a.by.1.1 3 20.19 odd 2