Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7104,2,Mod(1,7104)] 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("7104.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7104 = 2^{6} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7104.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,0,-2,0,5,0,3,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(56.7257255959\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 888)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 7104.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.30777 q^{5} +4.71982 q^{7} +1.00000 q^{9} +1.77846 q^{11} -0.221543 q^{13} +3.30777 q^{15} +5.08623 q^{17} -1.77846 q^{19} -4.71982 q^{21} -8.96896 q^{23} +5.94137 q^{25} -1.00000 q^{27} -0.692226 q^{29} -3.55691 q^{31} -1.77846 q^{33} -15.6121 q^{35} +1.00000 q^{37} +0.221543 q^{39} -10.9966 q^{41} +4.00000 q^{43} -3.30777 q^{45} -9.43965 q^{47} +15.2767 q^{49} -5.08623 q^{51} -9.66119 q^{53} -5.88273 q^{55} +1.77846 q^{57} +4.86469 q^{59} +6.99656 q^{61} +4.71982 q^{63} +0.732814 q^{65} +4.11727 q^{67} +8.96896 q^{69} +5.88273 q^{71} -5.33537 q^{73} -5.94137 q^{75} +8.39400 q^{77} -12.0552 q^{79} +1.00000 q^{81} +7.66119 q^{83} -16.8241 q^{85} +0.692226 q^{87} -2.91377 q^{89} -1.04564 q^{91} +3.55691 q^{93} +5.88273 q^{95} +12.6155 q^{97} +1.77846 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 2 q^{5} + 5 q^{7} + 3 q^{9} - 3 q^{11} - 9 q^{13} + 2 q^{15} - q^{17} + 3 q^{19} - 5 q^{21} - 9 q^{23} + 17 q^{25} - 3 q^{27} - 10 q^{29} + 6 q^{31} + 3 q^{33} + 4 q^{35} + 3 q^{37} + 9 q^{39}+ \cdots - 3 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −3.30777 −1.47928 −0.739641 − 0.673002i \(-0.765005\pi\)
−0.739641 + 0.673002i \(0.765005\pi\)
\(6\) 0 0
\(7\) 4.71982 1.78393 0.891963 − 0.452109i \(-0.149328\pi\)
0.891963 + 0.452109i \(0.149328\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.77846 0.536225 0.268112 − 0.963388i \(-0.413600\pi\)
0.268112 + 0.963388i \(0.413600\pi\)
\(12\) 0 0
\(13\) −0.221543 −0.0614449 −0.0307225 − 0.999528i \(-0.509781\pi\)
−0.0307225 + 0.999528i \(0.509781\pi\)
\(14\) 0 0
\(15\) 3.30777 0.854063
\(16\) 0 0
\(17\) 5.08623 1.23359 0.616796 − 0.787123i \(-0.288430\pi\)
0.616796 + 0.787123i \(0.288430\pi\)
\(18\) 0 0
\(19\) −1.77846 −0.408006 −0.204003 − 0.978970i \(-0.565395\pi\)
−0.204003 + 0.978970i \(0.565395\pi\)
\(20\) 0 0
\(21\) −4.71982 −1.02995
\(22\) 0 0
\(23\) −8.96896 −1.87016 −0.935079 − 0.354439i \(-0.884672\pi\)
−0.935079 + 0.354439i \(0.884672\pi\)
\(24\) 0 0
\(25\) 5.94137 1.18827
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −0.692226 −0.128543 −0.0642716 − 0.997932i \(-0.520472\pi\)
−0.0642716 + 0.997932i \(0.520472\pi\)
\(30\) 0 0
\(31\) −3.55691 −0.638841 −0.319420 − 0.947613i \(-0.603488\pi\)
−0.319420 + 0.947613i \(0.603488\pi\)
\(32\) 0 0
\(33\) −1.77846 −0.309590
\(34\) 0 0
\(35\) −15.6121 −2.63893
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 0.221543 0.0354753
\(40\) 0 0
\(41\) −10.9966 −1.71737 −0.858687 − 0.512500i \(-0.828719\pi\)
−0.858687 + 0.512500i \(0.828719\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −3.30777 −0.493094
\(46\) 0 0
\(47\) −9.43965 −1.37691 −0.688457 − 0.725277i \(-0.741712\pi\)
−0.688457 + 0.725277i \(0.741712\pi\)
\(48\) 0 0
\(49\) 15.2767 2.18239
\(50\) 0 0
\(51\) −5.08623 −0.712215
\(52\) 0 0
\(53\) −9.66119 −1.32707 −0.663533 − 0.748147i \(-0.730944\pi\)
−0.663533 + 0.748147i \(0.730944\pi\)
\(54\) 0 0
\(55\) −5.88273 −0.793228
\(56\) 0 0
\(57\) 1.77846 0.235562
\(58\) 0 0
\(59\) 4.86469 0.633328 0.316664 − 0.948538i \(-0.397437\pi\)
0.316664 + 0.948538i \(0.397437\pi\)
\(60\) 0 0
\(61\) 6.99656 0.895818 0.447909 − 0.894079i \(-0.352169\pi\)
0.447909 + 0.894079i \(0.352169\pi\)
\(62\) 0 0
\(63\) 4.71982 0.594642
\(64\) 0 0
\(65\) 0.732814 0.0908944
\(66\) 0 0
\(67\) 4.11727 0.503004 0.251502 − 0.967857i \(-0.419075\pi\)
0.251502 + 0.967857i \(0.419075\pi\)
\(68\) 0 0
\(69\) 8.96896 1.07974
\(70\) 0 0
\(71\) 5.88273 0.698152 0.349076 − 0.937094i \(-0.386495\pi\)
0.349076 + 0.937094i \(0.386495\pi\)
\(72\) 0 0
\(73\) −5.33537 −0.624458 −0.312229 − 0.950007i \(-0.601076\pi\)
−0.312229 + 0.950007i \(0.601076\pi\)
\(74\) 0 0
\(75\) −5.94137 −0.686050
\(76\) 0 0
\(77\) 8.39400 0.956586
\(78\) 0 0
\(79\) −12.0552 −1.35632 −0.678158 − 0.734916i \(-0.737221\pi\)
−0.678158 + 0.734916i \(0.737221\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.66119 0.840925 0.420462 − 0.907310i \(-0.361868\pi\)
0.420462 + 0.907310i \(0.361868\pi\)
\(84\) 0 0
\(85\) −16.8241 −1.82483
\(86\) 0 0
\(87\) 0.692226 0.0742144
\(88\) 0 0
\(89\) −2.91377 −0.308859 −0.154429 − 0.988004i \(-0.549354\pi\)
−0.154429 + 0.988004i \(0.549354\pi\)
\(90\) 0 0
\(91\) −1.04564 −0.109613
\(92\) 0 0
\(93\) 3.55691 0.368835
\(94\) 0 0
\(95\) 5.88273 0.603556
\(96\) 0 0
\(97\) 12.6155 1.28091 0.640457 − 0.767994i \(-0.278745\pi\)
0.640457 + 0.767994i \(0.278745\pi\)
\(98\) 0 0
\(99\) 1.77846 0.178742
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 7104.2.a.bq.1.1 3
4.3 odd 2 7104.2.a.bw.1.1 3
8.3 odd 2 888.2.a.i.1.3 ✓ 3
8.5 even 2 1776.2.a.s.1.3 3
24.5 odd 2 5328.2.a.bl.1.1 3
24.11 even 2 2664.2.a.m.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.a.i.1.3 ✓ 3 8.3 odd 2
1776.2.a.s.1.3 3 8.5 even 2
2664.2.a.m.1.1 3 24.11 even 2
5328.2.a.bl.1.1 3 24.5 odd 2
7104.2.a.bq.1.1 3 1.1 even 1 trivial
7104.2.a.bw.1.1 3 4.3 odd 2