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(11,74)] 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("74.11"); 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([5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2,2,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 27.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 74.27
Dual form 74.2.e.b.11.2

$q$-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+(0.866025 + 0.500000i) q^{2} +(-1.36603 - 2.36603i) q^{3} +(0.500000 + 0.866025i) q^{4} +(1.50000 - 0.866025i) q^{5} -2.73205i q^{6} +(1.00000 + 1.73205i) q^{7} +1.00000i q^{8} +(-2.23205 + 3.86603i) q^{9} +1.73205 q^{10} -4.73205 q^{11} +(1.36603 - 2.36603i) q^{12} +(-3.00000 + 1.73205i) q^{13} +2.00000i q^{14} +(-4.09808 - 2.36603i) q^{15} +(-0.500000 + 0.866025i) q^{16} +(6.69615 + 3.86603i) q^{17} +(-3.86603 + 2.23205i) q^{18} +(1.09808 - 0.633975i) q^{19} +(1.50000 + 0.866025i) q^{20} +(2.73205 - 4.73205i) q^{21} +(-4.09808 - 2.36603i) q^{22} -4.73205i q^{23} +(2.36603 - 1.36603i) q^{24} +(-1.00000 + 1.73205i) q^{25} -3.46410 q^{26} +4.00000 q^{27} +(-1.00000 + 1.73205i) q^{28} -8.66025i q^{29} +(-2.36603 - 4.09808i) q^{30} +1.26795i q^{31} +(-0.866025 + 0.500000i) q^{32} +(6.46410 + 11.1962i) q^{33} +(3.86603 + 6.69615i) q^{34} +(3.00000 + 1.73205i) q^{35} -4.46410 q^{36} +(-5.69615 + 2.13397i) q^{37} +1.26795 q^{38} +(8.19615 + 4.73205i) q^{39} +(0.866025 + 1.50000i) q^{40} +(-4.96410 - 8.59808i) q^{41} +(4.73205 - 2.73205i) q^{42} +0.928203i q^{43} +(-2.36603 - 4.09808i) q^{44} +7.73205i q^{45} +(2.36603 - 4.09808i) q^{46} +4.73205 q^{47} +2.73205 q^{48} +(1.50000 - 2.59808i) q^{49} +(-1.73205 + 1.00000i) q^{50} -21.1244i q^{51} +(-3.00000 - 1.73205i) q^{52} +(-1.26795 + 2.19615i) q^{53} +(3.46410 + 2.00000i) q^{54} +(-7.09808 + 4.09808i) q^{55} +(-1.73205 + 1.00000i) q^{56} +(-3.00000 - 1.73205i) q^{57} +(4.33013 - 7.50000i) q^{58} +(2.19615 + 1.26795i) q^{59} -4.73205i q^{60} +(1.50000 - 0.866025i) q^{61} +(-0.633975 + 1.09808i) q^{62} -8.92820 q^{63} -1.00000 q^{64} +(-3.00000 + 5.19615i) q^{65} +12.9282i q^{66} +(5.09808 + 8.83013i) q^{67} +7.73205i q^{68} +(-11.1962 + 6.46410i) q^{69} +(1.73205 + 3.00000i) q^{70} +(-1.73205 - 3.00000i) q^{71} +(-3.86603 - 2.23205i) q^{72} +4.00000 q^{73} +(-6.00000 - 1.00000i) q^{74} +5.46410 q^{75} +(1.09808 + 0.633975i) q^{76} +(-4.73205 - 8.19615i) q^{77} +(4.73205 + 8.19615i) q^{78} +(-11.4904 + 6.63397i) q^{79} +1.73205i q^{80} +(1.23205 + 2.13397i) q^{81} -9.92820i q^{82} +(2.83013 - 4.90192i) q^{83} +5.46410 q^{84} +13.3923 q^{85} +(-0.464102 + 0.803848i) q^{86} +(-20.4904 + 11.8301i) q^{87} -4.73205i q^{88} +(-5.89230 - 3.40192i) q^{89} +(-3.86603 + 6.69615i) q^{90} +(-6.00000 - 3.46410i) q^{91} +(4.09808 - 2.36603i) q^{92} +(3.00000 - 1.73205i) q^{93} +(4.09808 + 2.36603i) q^{94} +(1.09808 - 1.90192i) q^{95} +(2.36603 + 1.36603i) q^{96} -7.73205i q^{97} +(2.59808 - 1.50000i) q^{98} +(10.5622 - 18.2942i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 2 q^{4} + 6 q^{5} + 4 q^{7} - 2 q^{9} - 12 q^{11} + 2 q^{12} - 12 q^{13} - 6 q^{15} - 2 q^{16} + 6 q^{17} - 12 q^{18} - 6 q^{19} + 6 q^{20} + 4 q^{21} - 6 q^{22} + 6 q^{24} - 4 q^{25} + 16 q^{27}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 + 0.500000i 0.612372 + 0.353553i
\(3\) −1.36603 − 2.36603i −0.788675 − 1.36603i −0.926779 − 0.375608i \(-0.877434\pi\)
0.138104 − 0.990418i \(-0.455899\pi\)
\(4\) 0.500000 + 0.866025i 0.250000 + 0.433013i
\(5\) 1.50000 − 0.866025i 0.670820 − 0.387298i −0.125567 − 0.992085i \(-0.540075\pi\)
0.796387 + 0.604787i \(0.206742\pi\)
\(6\) − 2.73205i − 1.11536i
\(7\) 1.00000 + 1.73205i 0.377964 + 0.654654i 0.990766 − 0.135583i \(-0.0432908\pi\)
−0.612801 + 0.790237i \(0.709957\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −2.23205 + 3.86603i −0.744017 + 1.28868i
\(10\) 1.73205 0.547723
\(11\) −4.73205 −1.42677 −0.713384 − 0.700774i \(-0.752838\pi\)
−0.713384 + 0.700774i \(0.752838\pi\)
\(12\) 1.36603 − 2.36603i 0.394338 − 0.683013i
\(13\) −3.00000 + 1.73205i −0.832050 + 0.480384i −0.854554 − 0.519362i \(-0.826170\pi\)
0.0225039 + 0.999747i \(0.492836\pi\)
\(14\) 2.00000i 0.534522i
\(15\) −4.09808 − 2.36603i −1.05812 − 0.610905i
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 6.69615 + 3.86603i 1.62406 + 0.937649i 0.985820 + 0.167808i \(0.0536689\pi\)
0.638236 + 0.769841i \(0.279664\pi\)
\(18\) −3.86603 + 2.23205i −0.911231 + 0.526099i
\(19\) 1.09808 − 0.633975i 0.251916 − 0.145444i −0.368725 − 0.929538i \(-0.620206\pi\)
0.620641 + 0.784095i \(0.286872\pi\)
\(20\) 1.50000 + 0.866025i 0.335410 + 0.193649i
\(21\) 2.73205 − 4.73205i 0.596182 − 1.03262i
\(22\) −4.09808 − 2.36603i −0.873713 − 0.504438i
\(23\) − 4.73205i − 0.986701i −0.869831 − 0.493350i \(-0.835772\pi\)
0.869831 − 0.493350i \(-0.164228\pi\)
\(24\) 2.36603 − 1.36603i 0.482963 − 0.278839i
\(25\) −1.00000 + 1.73205i −0.200000 + 0.346410i
\(26\) −3.46410 −0.679366
\(27\) 4.00000 0.769800
\(28\) −1.00000 + 1.73205i −0.188982 + 0.327327i
\(29\) − 8.66025i − 1.60817i −0.594515 − 0.804084i \(-0.702656\pi\)
0.594515 − 0.804084i \(-0.297344\pi\)
\(30\) −2.36603 − 4.09808i −0.431975 − 0.748203i
\(31\) 1.26795i 0.227730i 0.993496 + 0.113865i \(0.0363232\pi\)
−0.993496 + 0.113865i \(0.963677\pi\)
\(32\) −0.866025 + 0.500000i −0.153093 + 0.0883883i
\(33\) 6.46410 + 11.1962i 1.12526 + 1.94900i
\(34\) 3.86603 + 6.69615i 0.663018 + 1.14838i
\(35\) 3.00000 + 1.73205i 0.507093 + 0.292770i
\(36\) −4.46410 −0.744017
\(37\) −5.69615 + 2.13397i −0.936442 + 0.350823i
\(38\) 1.26795 0.205689
\(39\) 8.19615 + 4.73205i 1.31243 + 0.757735i
\(40\) 0.866025 + 1.50000i 0.136931 + 0.237171i
\(41\) −4.96410 − 8.59808i −0.775262 − 1.34279i −0.934647 − 0.355577i \(-0.884284\pi\)
0.159384 − 0.987217i \(-0.449049\pi\)
\(42\) 4.73205 − 2.73205i 0.730171 − 0.421565i
\(43\) 0.928203i 0.141550i 0.997492 + 0.0707748i \(0.0225472\pi\)
−0.997492 + 0.0707748i \(0.977453\pi\)
\(44\) −2.36603 − 4.09808i −0.356692 − 0.617808i
\(45\) 7.73205i 1.15263i
\(46\) 2.36603 − 4.09808i 0.348851 − 0.604228i
\(47\) 4.73205 0.690241 0.345120 − 0.938558i \(-0.387838\pi\)
0.345120 + 0.938558i \(0.387838\pi\)
\(48\) 2.73205 0.394338
\(49\) 1.50000 − 2.59808i 0.214286 − 0.371154i
\(50\) −1.73205 + 1.00000i −0.244949 + 0.141421i
\(51\) − 21.1244i − 2.95800i
\(52\) −3.00000 − 1.73205i −0.416025 − 0.240192i
\(53\) −1.26795 + 2.19615i −0.174166 + 0.301665i −0.939872 − 0.341526i \(-0.889056\pi\)
0.765706 + 0.643191i \(0.222390\pi\)
\(54\) 3.46410 + 2.00000i 0.471405 + 0.272166i
\(55\) −7.09808 + 4.09808i −0.957104 + 0.552584i
\(56\) −1.73205 + 1.00000i −0.231455 + 0.133631i
\(57\) −3.00000 − 1.73205i −0.397360 − 0.229416i
\(58\) 4.33013 − 7.50000i 0.568574 − 0.984798i
\(59\) 2.19615 + 1.26795i 0.285915 + 0.165073i 0.636098 − 0.771608i \(-0.280547\pi\)
−0.350183 + 0.936681i \(0.613881\pi\)
\(60\) − 4.73205i − 0.610905i
\(61\) 1.50000 − 0.866025i 0.192055 − 0.110883i −0.400889 − 0.916127i \(-0.631299\pi\)
0.592944 + 0.805243i \(0.297965\pi\)
\(62\) −0.633975 + 1.09808i −0.0805149 + 0.139456i
\(63\) −8.92820 −1.12485
\(64\) −1.00000 −0.125000
\(65\) −3.00000 + 5.19615i −0.372104 + 0.644503i
\(66\) 12.9282i 1.59135i
\(67\) 5.09808 + 8.83013i 0.622829 + 1.07877i 0.988956 + 0.148207i \(0.0473502\pi\)
−0.366127 + 0.930565i \(0.619317\pi\)
\(68\) 7.73205i 0.937649i
\(69\) −11.1962 + 6.46410i −1.34786 + 0.778186i
\(70\) 1.73205 + 3.00000i 0.207020 + 0.358569i
\(71\) −1.73205 − 3.00000i −0.205557 − 0.356034i 0.744753 − 0.667340i \(-0.232567\pi\)
−0.950310 + 0.311305i \(0.899234\pi\)
\(72\) −3.86603 − 2.23205i −0.455615 − 0.263050i
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −6.00000 − 1.00000i −0.697486 − 0.116248i
\(75\) 5.46410 0.630940
\(76\) 1.09808 + 0.633975i 0.125958 + 0.0727219i
\(77\) −4.73205 − 8.19615i −0.539267 − 0.934038i
\(78\) 4.73205 + 8.19615i 0.535799 + 0.928032i
\(79\) −11.4904 + 6.63397i −1.29277 + 0.746380i −0.979144 − 0.203167i \(-0.934877\pi\)
−0.313625 + 0.949547i \(0.601543\pi\)
\(80\) 1.73205i 0.193649i
\(81\) 1.23205 + 2.13397i 0.136895 + 0.237108i
\(82\) − 9.92820i − 1.09639i
\(83\) 2.83013 − 4.90192i 0.310647 − 0.538056i −0.667856 − 0.744291i \(-0.732788\pi\)
0.978503 + 0.206235i \(0.0661210\pi\)
\(84\) 5.46410 0.596182
\(85\) 13.3923 1.45260
\(86\) −0.464102 + 0.803848i −0.0500454 + 0.0866811i
\(87\) −20.4904 + 11.8301i −2.19680 + 1.26832i
\(88\) − 4.73205i − 0.504438i
\(89\) −5.89230 − 3.40192i −0.624583 − 0.360603i 0.154068 − 0.988060i \(-0.450762\pi\)
−0.778651 + 0.627457i \(0.784096\pi\)
\(90\) −3.86603 + 6.69615i −0.407515 + 0.705836i
\(91\) −6.00000 − 3.46410i −0.628971 − 0.363137i
\(92\) 4.09808 − 2.36603i 0.427254 − 0.246675i
\(93\) 3.00000 − 1.73205i 0.311086 − 0.179605i
\(94\) 4.09808 + 2.36603i 0.422684 + 0.244037i
\(95\) 1.09808 − 1.90192i 0.112660 − 0.195133i
\(96\) 2.36603 + 1.36603i 0.241481 + 0.139419i
\(97\) − 7.73205i − 0.785071i −0.919737 − 0.392535i \(-0.871598\pi\)
0.919737 − 0.392535i \(-0.128402\pi\)
\(98\) 2.59808 − 1.50000i 0.262445 − 0.151523i
\(99\) 10.5622 − 18.2942i 1.06154 − 1.83864i
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.e.b.27.2 yes 4
3.2 odd 2 666.2.s.a.397.1 4
4.3 odd 2 592.2.w.e.545.2 4
37.11 even 6 inner 74.2.e.b.11.2 ✓ 4
37.14 odd 12 2738.2.a.i.1.1 2
37.23 odd 12 2738.2.a.e.1.1 2
111.11 odd 6 666.2.s.a.307.1 4
148.11 odd 6 592.2.w.e.529.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.b.11.2 ✓ 4 37.11 even 6 inner
74.2.e.b.27.2 yes 4 1.1 even 1 trivial
592.2.w.e.529.2 4 148.11 odd 6
592.2.w.e.545.2 4 4.3 odd 2
666.2.s.a.307.1 4 111.11 odd 6
666.2.s.a.397.1 4 3.2 odd 2
2738.2.a.e.1.1 2 37.23 odd 12
2738.2.a.i.1.1 2 37.14 odd 12