Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [74,2,Mod(47,74)] 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("74.47"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(74, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,2] 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: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 63.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 74.63
Dual form 74.2.c.b.47.1

$q$-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+(0.500000 + 0.866025i) q^{2} +(1.00000 - 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} +2.00000 q^{6} +(2.00000 - 3.46410i) q^{7} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} -3.00000 q^{10} -6.00000 q^{11} +(1.00000 + 1.73205i) q^{12} +(-1.00000 + 1.73205i) q^{13} +4.00000 q^{14} +(3.00000 + 5.19615i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(-1.50000 - 2.59808i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.73205i) q^{19} +(-1.50000 - 2.59808i) q^{20} +(-4.00000 - 6.92820i) q^{21} +(-3.00000 - 5.19615i) q^{22} +6.00000 q^{23} +(-1.00000 + 1.73205i) q^{24} +(-2.00000 - 3.46410i) q^{25} -2.00000 q^{26} +4.00000 q^{27} +(2.00000 + 3.46410i) q^{28} +3.00000 q^{29} +(-3.00000 + 5.19615i) q^{30} +2.00000 q^{31} +(0.500000 - 0.866025i) q^{32} +(-6.00000 + 10.3923i) q^{33} +(1.50000 - 2.59808i) q^{34} +(6.00000 + 10.3923i) q^{35} +1.00000 q^{36} +(5.50000 - 2.59808i) q^{37} -2.00000 q^{38} +(2.00000 + 3.46410i) q^{39} +(1.50000 - 2.59808i) q^{40} +(-1.50000 + 2.59808i) q^{41} +(4.00000 - 6.92820i) q^{42} -4.00000 q^{43} +(3.00000 - 5.19615i) q^{44} +3.00000 q^{45} +(3.00000 + 5.19615i) q^{46} -6.00000 q^{47} -2.00000 q^{48} +(-4.50000 - 7.79423i) q^{49} +(2.00000 - 3.46410i) q^{50} -6.00000 q^{51} +(-1.00000 - 1.73205i) q^{52} +(3.00000 + 5.19615i) q^{53} +(2.00000 + 3.46410i) q^{54} +(9.00000 - 15.5885i) q^{55} +(-2.00000 + 3.46410i) q^{56} +(2.00000 + 3.46410i) q^{57} +(1.50000 + 2.59808i) q^{58} -6.00000 q^{60} +(0.500000 - 0.866025i) q^{61} +(1.00000 + 1.73205i) q^{62} -4.00000 q^{63} +1.00000 q^{64} +(-3.00000 - 5.19615i) q^{65} -12.0000 q^{66} +(-1.00000 + 1.73205i) q^{67} +3.00000 q^{68} +(6.00000 - 10.3923i) q^{69} +(-6.00000 + 10.3923i) q^{70} +(6.00000 - 10.3923i) q^{71} +(0.500000 + 0.866025i) q^{72} -10.0000 q^{73} +(5.00000 + 3.46410i) q^{74} -8.00000 q^{75} +(-1.00000 - 1.73205i) q^{76} +(-12.0000 + 20.7846i) q^{77} +(-2.00000 + 3.46410i) q^{78} +(-7.00000 + 12.1244i) q^{79} +3.00000 q^{80} +(5.50000 - 9.52628i) q^{81} -3.00000 q^{82} +(-3.00000 - 5.19615i) q^{83} +8.00000 q^{84} +9.00000 q^{85} +(-2.00000 - 3.46410i) q^{86} +(3.00000 - 5.19615i) q^{87} +6.00000 q^{88} +(-1.50000 - 2.59808i) q^{89} +(1.50000 + 2.59808i) q^{90} +(4.00000 + 6.92820i) q^{91} +(-3.00000 + 5.19615i) q^{92} +(2.00000 - 3.46410i) q^{93} +(-3.00000 - 5.19615i) q^{94} +(-3.00000 - 5.19615i) q^{95} +(-1.00000 - 1.73205i) q^{96} -13.0000 q^{97} +(4.50000 - 7.79423i) q^{98} +(3.00000 + 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{3} - q^{4} - 3 q^{5} + 4 q^{6} + 4 q^{7} - 2 q^{8} - q^{9} - 6 q^{10} - 12 q^{11} + 2 q^{12} - 2 q^{13} + 8 q^{14} + 6 q^{15} - q^{16} - 3 q^{17} + q^{18} - 2 q^{19} - 3 q^{20} - 8 q^{21}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 0.866025i 0.353553 + 0.612372i
\(3\) 1.00000 − 1.73205i 0.577350 − 1.00000i −0.418432 − 0.908248i \(-0.637420\pi\)
0.995782 − 0.0917517i \(-0.0292466\pi\)
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −1.50000 + 2.59808i −0.670820 + 1.16190i 0.306851 + 0.951757i \(0.400725\pi\)
−0.977672 + 0.210138i \(0.932609\pi\)
\(6\) 2.00000 0.816497
\(7\) 2.00000 − 3.46410i 0.755929 − 1.30931i −0.188982 − 0.981981i \(-0.560519\pi\)
0.944911 − 0.327327i \(-0.106148\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) −3.00000 −0.948683
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 1.00000 + 1.73205i 0.288675 + 0.500000i
\(13\) −1.00000 + 1.73205i −0.277350 + 0.480384i −0.970725 − 0.240192i \(-0.922790\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 4.00000 1.06904
\(15\) 3.00000 + 5.19615i 0.774597 + 1.34164i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −1.50000 − 2.59808i −0.363803 − 0.630126i 0.624780 − 0.780801i \(-0.285189\pi\)
−0.988583 + 0.150675i \(0.951855\pi\)
\(18\) 0.500000 − 0.866025i 0.117851 − 0.204124i
\(19\) −1.00000 + 1.73205i −0.229416 + 0.397360i −0.957635 − 0.287984i \(-0.907015\pi\)
0.728219 + 0.685344i \(0.240348\pi\)
\(20\) −1.50000 − 2.59808i −0.335410 − 0.580948i
\(21\) −4.00000 − 6.92820i −0.872872 − 1.51186i
\(22\) −3.00000 − 5.19615i −0.639602 − 1.10782i
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −1.00000 + 1.73205i −0.204124 + 0.353553i
\(25\) −2.00000 − 3.46410i −0.400000 − 0.692820i
\(26\) −2.00000 −0.392232
\(27\) 4.00000 0.769800
\(28\) 2.00000 + 3.46410i 0.377964 + 0.654654i
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) −3.00000 + 5.19615i −0.547723 + 0.948683i
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0.500000 − 0.866025i 0.0883883 − 0.153093i
\(33\) −6.00000 + 10.3923i −1.04447 + 1.80907i
\(34\) 1.50000 − 2.59808i 0.257248 − 0.445566i
\(35\) 6.00000 + 10.3923i 1.01419 + 1.75662i
\(36\) 1.00000 0.166667
\(37\) 5.50000 − 2.59808i 0.904194 − 0.427121i
\(38\) −2.00000 −0.324443
\(39\) 2.00000 + 3.46410i 0.320256 + 0.554700i
\(40\) 1.50000 − 2.59808i 0.237171 − 0.410792i
\(41\) −1.50000 + 2.59808i −0.234261 + 0.405751i −0.959058 − 0.283211i \(-0.908600\pi\)
0.724797 + 0.688963i \(0.241934\pi\)
\(42\) 4.00000 − 6.92820i 0.617213 − 1.06904i
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 3.00000 − 5.19615i 0.452267 − 0.783349i
\(45\) 3.00000 0.447214
\(46\) 3.00000 + 5.19615i 0.442326 + 0.766131i
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −2.00000 −0.288675
\(49\) −4.50000 − 7.79423i −0.642857 − 1.11346i
\(50\) 2.00000 − 3.46410i 0.282843 − 0.489898i
\(51\) −6.00000 −0.840168
\(52\) −1.00000 − 1.73205i −0.138675 − 0.240192i
\(53\) 3.00000 + 5.19615i 0.412082 + 0.713746i 0.995117 − 0.0987002i \(-0.0314685\pi\)
−0.583036 + 0.812447i \(0.698135\pi\)
\(54\) 2.00000 + 3.46410i 0.272166 + 0.471405i
\(55\) 9.00000 − 15.5885i 1.21356 − 2.10195i
\(56\) −2.00000 + 3.46410i −0.267261 + 0.462910i
\(57\) 2.00000 + 3.46410i 0.264906 + 0.458831i
\(58\) 1.50000 + 2.59808i 0.196960 + 0.341144i
\(59\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) −6.00000 −0.774597
\(61\) 0.500000 − 0.866025i 0.0640184 − 0.110883i −0.832240 − 0.554416i \(-0.812942\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 1.00000 + 1.73205i 0.127000 + 0.219971i
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) −3.00000 − 5.19615i −0.372104 − 0.644503i
\(66\) −12.0000 −1.47710
\(67\) −1.00000 + 1.73205i −0.122169 + 0.211604i −0.920623 − 0.390453i \(-0.872318\pi\)
0.798454 + 0.602056i \(0.205652\pi\)
\(68\) 3.00000 0.363803
\(69\) 6.00000 − 10.3923i 0.722315 − 1.25109i
\(70\) −6.00000 + 10.3923i −0.717137 + 1.24212i
\(71\) 6.00000 − 10.3923i 0.712069 − 1.23334i −0.252010 − 0.967725i \(-0.581092\pi\)
0.964079 − 0.265615i \(-0.0855750\pi\)
\(72\) 0.500000 + 0.866025i 0.0589256 + 0.102062i
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 5.00000 + 3.46410i 0.581238 + 0.402694i
\(75\) −8.00000 −0.923760
\(76\) −1.00000 − 1.73205i −0.114708 − 0.198680i
\(77\) −12.0000 + 20.7846i −1.36753 + 2.36863i
\(78\) −2.00000 + 3.46410i −0.226455 + 0.392232i
\(79\) −7.00000 + 12.1244i −0.787562 + 1.36410i 0.139895 + 0.990166i \(0.455323\pi\)
−0.927457 + 0.373930i \(0.878010\pi\)
\(80\) 3.00000 0.335410
\(81\) 5.50000 − 9.52628i 0.611111 − 1.05848i
\(82\) −3.00000 −0.331295
\(83\) −3.00000 − 5.19615i −0.329293 − 0.570352i 0.653079 − 0.757290i \(-0.273477\pi\)
−0.982372 + 0.186938i \(0.940144\pi\)
\(84\) 8.00000 0.872872
\(85\) 9.00000 0.976187
\(86\) −2.00000 − 3.46410i −0.215666 − 0.373544i
\(87\) 3.00000 − 5.19615i 0.321634 − 0.557086i
\(88\) 6.00000 0.639602
\(89\) −1.50000 − 2.59808i −0.159000 − 0.275396i 0.775509 − 0.631337i \(-0.217494\pi\)
−0.934508 + 0.355942i \(0.884160\pi\)
\(90\) 1.50000 + 2.59808i 0.158114 + 0.273861i
\(91\) 4.00000 + 6.92820i 0.419314 + 0.726273i
\(92\) −3.00000 + 5.19615i −0.312772 + 0.541736i
\(93\) 2.00000 − 3.46410i 0.207390 − 0.359211i
\(94\) −3.00000 − 5.19615i −0.309426 − 0.535942i
\(95\) −3.00000 − 5.19615i −0.307794 − 0.533114i
\(96\) −1.00000 − 1.73205i −0.102062 − 0.176777i
\(97\) −13.0000 −1.31995 −0.659975 − 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 4.50000 − 7.79423i 0.454569 − 0.787336i
\(99\) 3.00000 + 5.19615i 0.301511 + 0.522233i
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 74.2.c.b.63.1 yes 2
3.2 odd 2 666.2.f.d.433.1 2
4.3 odd 2 592.2.i.a.433.1 2
37.10 even 3 inner 74.2.c.b.47.1 ✓ 2
37.11 even 6 2738.2.a.c.1.1 1
37.26 even 3 2738.2.a.a.1.1 1
111.47 odd 6 666.2.f.d.343.1 2
148.47 odd 6 592.2.i.a.417.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.c.b.47.1 ✓ 2 37.10 even 3 inner
74.2.c.b.63.1 yes 2 1.1 even 1 trivial
592.2.i.a.417.1 2 148.47 odd 6
592.2.i.a.433.1 2 4.3 odd 2
666.2.f.d.343.1 2 111.47 odd 6
666.2.f.d.433.1 2 3.2 odd 2
2738.2.a.a.1.1 1 37.26 even 3
2738.2.a.c.1.1 1 37.11 even 6