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 47.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 74.47
Dual form 74.2.c.a.63.1

$q$-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} +(0.500000 + 0.866025i) q^{5} -2.00000 q^{6} -1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +1.00000 q^{10} +2.00000 q^{11} +(-1.00000 + 1.73205i) q^{12} +(3.00000 + 5.19615i) q^{13} +(1.00000 - 1.73205i) 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} +(0.500000 - 0.866025i) q^{20} +(1.00000 - 1.73205i) q^{22} -6.00000 q^{23} +(1.00000 + 1.73205i) q^{24} +(2.00000 - 3.46410i) q^{25} +6.00000 q^{26} -4.00000 q^{27} -9.00000 q^{29} +(-1.00000 - 1.73205i) q^{30} +10.0000 q^{31} +(0.500000 + 0.866025i) q^{32} +(-2.00000 - 3.46410i) q^{33} +(1.50000 + 2.59808i) q^{34} +1.00000 q^{36} +(-0.500000 + 6.06218i) q^{37} -2.00000 q^{38} +(6.00000 - 10.3923i) q^{39} +(-0.500000 - 0.866025i) q^{40} +(-1.50000 - 2.59808i) q^{41} -8.00000 q^{43} +(-1.00000 - 1.73205i) q^{44} -1.00000 q^{45} +(-3.00000 + 5.19615i) q^{46} +2.00000 q^{47} +2.00000 q^{48} +(3.50000 - 6.06218i) q^{49} +(-2.00000 - 3.46410i) q^{50} +6.00000 q^{51} +(3.00000 - 5.19615i) q^{52} +(3.00000 - 5.19615i) q^{53} +(-2.00000 + 3.46410i) q^{54} +(1.00000 + 1.73205i) q^{55} +(-2.00000 + 3.46410i) q^{57} +(-4.50000 + 7.79423i) q^{58} +(-4.00000 + 6.92820i) q^{59} -2.00000 q^{60} +(2.50000 + 4.33013i) q^{61} +(5.00000 - 8.66025i) q^{62} +1.00000 q^{64} +(-3.00000 + 5.19615i) q^{65} -4.00000 q^{66} +(3.00000 + 5.19615i) q^{67} +3.00000 q^{68} +(6.00000 + 10.3923i) q^{69} +(0.500000 - 0.866025i) q^{72} -2.00000 q^{73} +(5.00000 + 3.46410i) q^{74} -8.00000 q^{75} +(-1.00000 + 1.73205i) q^{76} +(-6.00000 - 10.3923i) q^{78} +(-3.00000 - 5.19615i) q^{79} -1.00000 q^{80} +(5.50000 + 9.52628i) q^{81} -3.00000 q^{82} +(-1.00000 + 1.73205i) q^{83} -3.00000 q^{85} +(-4.00000 + 6.92820i) q^{86} +(9.00000 + 15.5885i) q^{87} -2.00000 q^{88} +(6.50000 - 11.2583i) q^{89} +(-0.500000 + 0.866025i) q^{90} +(3.00000 + 5.19615i) q^{92} +(-10.0000 - 17.3205i) q^{93} +(1.00000 - 1.73205i) q^{94} +(1.00000 - 1.73205i) q^{95} +(1.00000 - 1.73205i) q^{96} +3.00000 q^{97} +(-3.50000 - 6.06218i) q^{98} +(-1.00000 + 1.73205i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} - q^{4} + q^{5} - 4 q^{6} - 2 q^{8} - q^{9} + 2 q^{10} + 4 q^{11} - 2 q^{12} + 6 q^{13} + 2 q^{15} - q^{16} - 3 q^{17} + q^{18} - 2 q^{19} + q^{20} + 2 q^{22} - 12 q^{23} + 2 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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 −0.995782 − 0.0917517i \(-0.970753\pi\)
0.418432 − 0.908248i \(-0.362580\pi\)
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i 0.955901 − 0.293691i \(-0.0948835\pi\)
−0.732294 + 0.680989i \(0.761550\pi\)
\(6\) −2.00000 −0.816497
\(7\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 1.00000 0.316228
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.00000 + 1.73205i −0.288675 + 0.500000i
\(13\) 3.00000 + 5.19615i 0.832050 + 1.44115i 0.896410 + 0.443227i \(0.146166\pi\)
−0.0643593 + 0.997927i \(0.520500\pi\)
\(14\) 0 0
\(15\) 1.00000 − 1.73205i 0.258199 − 0.447214i
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −1.50000 + 2.59808i −0.363803 + 0.630126i −0.988583 − 0.150675i \(-0.951855\pi\)
0.624780 + 0.780801i \(0.285189\pi\)
\(18\) 0.500000 + 0.866025i 0.117851 + 0.204124i
\(19\) −1.00000 − 1.73205i −0.229416 − 0.397360i 0.728219 − 0.685344i \(-0.240348\pi\)
−0.957635 + 0.287984i \(0.907015\pi\)
\(20\) 0.500000 − 0.866025i 0.111803 − 0.193649i
\(21\) 0 0
\(22\) 1.00000 − 1.73205i 0.213201 − 0.369274i
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 1.00000 + 1.73205i 0.204124 + 0.353553i
\(25\) 2.00000 − 3.46410i 0.400000 − 0.692820i
\(26\) 6.00000 1.17670
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) −1.00000 − 1.73205i −0.182574 − 0.316228i
\(31\) 10.0000 1.79605 0.898027 − 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0.500000 + 0.866025i 0.0883883 + 0.153093i
\(33\) −2.00000 − 3.46410i −0.348155 − 0.603023i
\(34\) 1.50000 + 2.59808i 0.257248 + 0.445566i
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −0.500000 + 6.06218i −0.0821995 + 0.996616i
\(38\) −2.00000 −0.324443
\(39\) 6.00000 − 10.3923i 0.960769 − 1.66410i
\(40\) −0.500000 − 0.866025i −0.0790569 − 0.136931i
\(41\) −1.50000 − 2.59808i −0.234261 − 0.405751i 0.724797 − 0.688963i \(-0.241934\pi\)
−0.959058 + 0.283211i \(0.908600\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −1.00000 − 1.73205i −0.150756 − 0.261116i
\(45\) −1.00000 −0.149071
\(46\) −3.00000 + 5.19615i −0.442326 + 0.766131i
\(47\) 2.00000 0.291730 0.145865 − 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 2.00000 0.288675
\(49\) 3.50000 − 6.06218i 0.500000 − 0.866025i
\(50\) −2.00000 − 3.46410i −0.282843 − 0.489898i
\(51\) 6.00000 0.840168
\(52\) 3.00000 − 5.19615i 0.416025 − 0.720577i
\(53\) 3.00000 − 5.19615i 0.412082 − 0.713746i −0.583036 − 0.812447i \(-0.698135\pi\)
0.995117 + 0.0987002i \(0.0314685\pi\)
\(54\) −2.00000 + 3.46410i −0.272166 + 0.471405i
\(55\) 1.00000 + 1.73205i 0.134840 + 0.233550i
\(56\) 0 0
\(57\) −2.00000 + 3.46410i −0.264906 + 0.458831i
\(58\) −4.50000 + 7.79423i −0.590879 + 1.02343i
\(59\) −4.00000 + 6.92820i −0.520756 + 0.901975i 0.478953 + 0.877841i \(0.341016\pi\)
−0.999709 + 0.0241347i \(0.992317\pi\)
\(60\) −2.00000 −0.258199
\(61\) 2.50000 + 4.33013i 0.320092 + 0.554416i 0.980507 − 0.196485i \(-0.0629528\pi\)
−0.660415 + 0.750901i \(0.729619\pi\)
\(62\) 5.00000 − 8.66025i 0.635001 − 1.09985i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.00000 + 5.19615i −0.372104 + 0.644503i
\(66\) −4.00000 −0.492366
\(67\) 3.00000 + 5.19615i 0.366508 + 0.634811i 0.989017 − 0.147802i \(-0.0472198\pi\)
−0.622509 + 0.782613i \(0.713886\pi\)
\(68\) 3.00000 0.363803
\(69\) 6.00000 + 10.3923i 0.722315 + 1.25109i
\(70\) 0 0
\(71\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(72\) 0.500000 − 0.866025i 0.0589256 − 0.102062i
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 5.00000 + 3.46410i 0.581238 + 0.402694i
\(75\) −8.00000 −0.923760
\(76\) −1.00000 + 1.73205i −0.114708 + 0.198680i
\(77\) 0 0
\(78\) −6.00000 − 10.3923i −0.679366 − 1.17670i
\(79\) −3.00000 − 5.19615i −0.337526 − 0.584613i 0.646440 − 0.762964i \(-0.276257\pi\)
−0.983967 + 0.178352i \(0.942924\pi\)
\(80\) −1.00000 −0.111803
\(81\) 5.50000 + 9.52628i 0.611111 + 1.05848i
\(82\) −3.00000 −0.331295
\(83\) −1.00000 + 1.73205i −0.109764 + 0.190117i −0.915675 − 0.401920i \(-0.868343\pi\)
0.805910 + 0.592037i \(0.201676\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) −4.00000 + 6.92820i −0.431331 + 0.747087i
\(87\) 9.00000 + 15.5885i 0.964901 + 1.67126i
\(88\) −2.00000 −0.213201
\(89\) 6.50000 − 11.2583i 0.688999 − 1.19338i −0.283164 − 0.959072i \(-0.591384\pi\)
0.972162 − 0.234309i \(-0.0752827\pi\)
\(90\) −0.500000 + 0.866025i −0.0527046 + 0.0912871i
\(91\) 0 0
\(92\) 3.00000 + 5.19615i 0.312772 + 0.541736i
\(93\) −10.0000 − 17.3205i −1.03695 − 1.79605i
\(94\) 1.00000 − 1.73205i 0.103142 − 0.178647i
\(95\) 1.00000 − 1.73205i 0.102598 − 0.177705i
\(96\) 1.00000 − 1.73205i 0.102062 − 0.176777i
\(97\) 3.00000 0.304604 0.152302 − 0.988334i \(-0.451331\pi\)
0.152302 + 0.988334i \(0.451331\pi\)
\(98\) −3.50000 − 6.06218i −0.353553 − 0.612372i
\(99\) −1.00000 + 1.73205i −0.100504 + 0.174078i
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.a.47.1 ✓ 2
3.2 odd 2 666.2.f.b.343.1 2
4.3 odd 2 592.2.i.d.417.1 2
37.10 even 3 2738.2.a.b.1.1 1
37.26 even 3 inner 74.2.c.a.63.1 yes 2
37.27 even 6 2738.2.a.d.1.1 1
111.26 odd 6 666.2.f.b.433.1 2
148.63 odd 6 592.2.i.d.433.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.c.a.47.1 ✓ 2 1.1 even 1 trivial
74.2.c.a.63.1 yes 2 37.26 even 3 inner
592.2.i.d.417.1 2 4.3 odd 2
592.2.i.d.433.1 2 148.63 odd 6
666.2.f.b.343.1 2 3.2 odd 2
666.2.f.b.433.1 2 111.26 odd 6
2738.2.a.b.1.1 1 37.10 even 3
2738.2.a.d.1.1 1 37.27 even 6