Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [294,4,Mod(67,294)] 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("294.67"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 294.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,-3,-4,-15,-12,0,-16,-9,30,9] 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: no
Analytic conductor: \(17.3465615417\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 79.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 294.79
Dual form 294.4.e.e.67.1

$q$-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 - 1.73205i) q^{2} +(-1.50000 - 2.59808i) q^{3} +(-2.00000 - 3.46410i) q^{4} +(-7.50000 + 12.9904i) q^{5} -6.00000 q^{6} -8.00000 q^{8} +(-4.50000 + 7.79423i) q^{9} +(15.0000 + 25.9808i) q^{10} +(4.50000 + 7.79423i) q^{11} +(-6.00000 + 10.3923i) q^{12} +88.0000 q^{13} +45.0000 q^{15} +(-8.00000 + 13.8564i) q^{16} +(-42.0000 - 72.7461i) q^{17} +(9.00000 + 15.5885i) q^{18} +(52.0000 - 90.0666i) q^{19} +60.0000 q^{20} +18.0000 q^{22} +(42.0000 - 72.7461i) q^{23} +(12.0000 + 20.7846i) q^{24} +(-50.0000 - 86.6025i) q^{25} +(88.0000 - 152.420i) q^{26} +27.0000 q^{27} +51.0000 q^{29} +(45.0000 - 77.9423i) q^{30} +(92.5000 + 160.215i) q^{31} +(16.0000 + 27.7128i) q^{32} +(13.5000 - 23.3827i) q^{33} -168.000 q^{34} +36.0000 q^{36} +(-22.0000 + 38.1051i) q^{37} +(-104.000 - 180.133i) q^{38} +(-132.000 - 228.631i) q^{39} +(60.0000 - 103.923i) q^{40} +168.000 q^{41} +326.000 q^{43} +(18.0000 - 31.1769i) q^{44} +(-67.5000 - 116.913i) q^{45} +(-84.0000 - 145.492i) q^{46} +(-69.0000 + 119.512i) q^{47} +48.0000 q^{48} -200.000 q^{50} +(-126.000 + 218.238i) q^{51} +(-176.000 - 304.841i) q^{52} +(-319.500 - 553.390i) q^{53} +(27.0000 - 46.7654i) q^{54} -135.000 q^{55} -312.000 q^{57} +(51.0000 - 88.3346i) q^{58} +(79.5000 + 137.698i) q^{59} +(-90.0000 - 155.885i) q^{60} +(361.000 - 625.270i) q^{61} +370.000 q^{62} +64.0000 q^{64} +(-660.000 + 1143.15i) q^{65} +(-27.0000 - 46.7654i) q^{66} +(83.0000 + 143.760i) q^{67} +(-168.000 + 290.985i) q^{68} -252.000 q^{69} +1086.00 q^{71} +(36.0000 - 62.3538i) q^{72} +(109.000 + 188.794i) q^{73} +(44.0000 + 76.2102i) q^{74} +(-150.000 + 259.808i) q^{75} -416.000 q^{76} -528.000 q^{78} +(291.500 - 504.893i) q^{79} +(-120.000 - 207.846i) q^{80} +(-40.5000 - 70.1481i) q^{81} +(168.000 - 290.985i) q^{82} +597.000 q^{83} +1260.00 q^{85} +(326.000 - 564.649i) q^{86} +(-76.5000 - 132.502i) q^{87} +(-36.0000 - 62.3538i) q^{88} +(-519.000 + 898.934i) q^{89} -270.000 q^{90} -336.000 q^{92} +(277.500 - 480.644i) q^{93} +(138.000 + 239.023i) q^{94} +(780.000 + 1351.00i) q^{95} +(48.0000 - 83.1384i) q^{96} +169.000 q^{97} -81.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} - 15 q^{5} - 12 q^{6} - 16 q^{8} - 9 q^{9} + 30 q^{10} + 9 q^{11} - 12 q^{12} + 176 q^{13} + 90 q^{15} - 16 q^{16} - 84 q^{17} + 18 q^{18} + 104 q^{19} + 120 q^{20} + 36 q^{22}+ \cdots - 162 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(199\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 − 1.73205i 0.353553 − 0.612372i
\(3\) −1.50000 − 2.59808i −0.288675 − 0.500000i
\(4\) −2.00000 − 3.46410i −0.250000 − 0.433013i
\(5\) −7.50000 + 12.9904i −0.670820 + 1.16190i 0.306851 + 0.951757i \(0.400725\pi\)
−0.977672 + 0.210138i \(0.932609\pi\)
\(6\) −6.00000 −0.408248
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) −4.50000 + 7.79423i −0.166667 + 0.288675i
\(10\) 15.0000 + 25.9808i 0.474342 + 0.821584i
\(11\) 4.50000 + 7.79423i 0.123346 + 0.213641i 0.921085 − 0.389362i \(-0.127304\pi\)
−0.797739 + 0.603002i \(0.793971\pi\)
\(12\) −6.00000 + 10.3923i −0.144338 + 0.250000i
\(13\) 88.0000 1.87745 0.938723 − 0.344671i \(-0.112010\pi\)
0.938723 + 0.344671i \(0.112010\pi\)
\(14\) 0 0
\(15\) 45.0000 0.774597
\(16\) −8.00000 + 13.8564i −0.125000 + 0.216506i
\(17\) −42.0000 − 72.7461i −0.599206 − 1.03785i −0.992939 − 0.118630i \(-0.962150\pi\)
0.393733 − 0.919225i \(-0.371183\pi\)
\(18\) 9.00000 + 15.5885i 0.117851 + 0.204124i
\(19\) 52.0000 − 90.0666i 0.627875 − 1.08751i −0.360103 − 0.932913i \(-0.617258\pi\)
0.987977 − 0.154598i \(-0.0494083\pi\)
\(20\) 60.0000 0.670820
\(21\) 0 0
\(22\) 18.0000 0.174437
\(23\) 42.0000 − 72.7461i 0.380765 − 0.659505i −0.610406 − 0.792088i \(-0.708994\pi\)
0.991172 + 0.132583i \(0.0423272\pi\)
\(24\) 12.0000 + 20.7846i 0.102062 + 0.176777i
\(25\) −50.0000 − 86.6025i −0.400000 − 0.692820i
\(26\) 88.0000 − 152.420i 0.663778 − 1.14970i
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 51.0000 0.326568 0.163284 − 0.986579i \(-0.447791\pi\)
0.163284 + 0.986579i \(0.447791\pi\)
\(30\) 45.0000 − 77.9423i 0.273861 − 0.474342i
\(31\) 92.5000 + 160.215i 0.535919 + 0.928239i 0.999118 + 0.0419848i \(0.0133681\pi\)
−0.463199 + 0.886254i \(0.653299\pi\)
\(32\) 16.0000 + 27.7128i 0.0883883 + 0.153093i
\(33\) 13.5000 − 23.3827i 0.0712136 − 0.123346i
\(34\) −168.000 −0.847405
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) −22.0000 + 38.1051i −0.0977507 + 0.169309i −0.910753 − 0.412951i \(-0.864498\pi\)
0.813003 + 0.582260i \(0.197831\pi\)
\(38\) −104.000 − 180.133i −0.443974 − 0.768986i
\(39\) −132.000 − 228.631i −0.541972 − 0.938723i
\(40\) 60.0000 − 103.923i 0.237171 − 0.410792i
\(41\) 168.000 0.639932 0.319966 − 0.947429i \(-0.396329\pi\)
0.319966 + 0.947429i \(0.396329\pi\)
\(42\) 0 0
\(43\) 326.000 1.15615 0.578076 − 0.815983i \(-0.303804\pi\)
0.578076 + 0.815983i \(0.303804\pi\)
\(44\) 18.0000 − 31.1769i 0.0616728 − 0.106820i
\(45\) −67.5000 − 116.913i −0.223607 − 0.387298i
\(46\) −84.0000 − 145.492i −0.269242 − 0.466341i
\(47\) −69.0000 + 119.512i −0.214142 + 0.370905i −0.953007 − 0.302949i \(-0.902029\pi\)
0.738865 + 0.673854i \(0.235362\pi\)
\(48\) 48.0000 0.144338
\(49\) 0 0
\(50\) −200.000 −0.565685
\(51\) −126.000 + 218.238i −0.345952 + 0.599206i
\(52\) −176.000 − 304.841i −0.469362 − 0.812958i
\(53\) −319.500 − 553.390i −0.828051 − 1.43423i −0.899565 − 0.436787i \(-0.856116\pi\)
0.0715141 − 0.997440i \(-0.477217\pi\)
\(54\) 27.0000 − 46.7654i 0.0680414 − 0.117851i
\(55\) −135.000 −0.330971
\(56\) 0 0
\(57\) −312.000 −0.725007
\(58\) 51.0000 − 88.3346i 0.115459 − 0.199981i
\(59\) 79.5000 + 137.698i 0.175424 + 0.303843i 0.940308 − 0.340325i \(-0.110537\pi\)
−0.764884 + 0.644168i \(0.777204\pi\)
\(60\) −90.0000 − 155.885i −0.193649 − 0.335410i
\(61\) 361.000 − 625.270i 0.757726 − 1.31242i −0.186281 − 0.982497i \(-0.559643\pi\)
0.944007 − 0.329924i \(-0.107023\pi\)
\(62\) 370.000 0.757904
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −660.000 + 1143.15i −1.25943 + 2.18140i
\(66\) −27.0000 − 46.7654i −0.0503556 − 0.0872185i
\(67\) 83.0000 + 143.760i 0.151344 + 0.262136i 0.931722 − 0.363173i \(-0.118306\pi\)
−0.780378 + 0.625309i \(0.784973\pi\)
\(68\) −168.000 + 290.985i −0.299603 + 0.518927i
\(69\) −252.000 −0.439670
\(70\) 0 0
\(71\) 1086.00 1.81527 0.907637 − 0.419755i \(-0.137884\pi\)
0.907637 + 0.419755i \(0.137884\pi\)
\(72\) 36.0000 − 62.3538i 0.0589256 − 0.102062i
\(73\) 109.000 + 188.794i 0.174760 + 0.302693i 0.940078 − 0.340959i \(-0.110752\pi\)
−0.765318 + 0.643652i \(0.777418\pi\)
\(74\) 44.0000 + 76.2102i 0.0691202 + 0.119720i
\(75\) −150.000 + 259.808i −0.230940 + 0.400000i
\(76\) −416.000 −0.627875
\(77\) 0 0
\(78\) −528.000 −0.766464
\(79\) 291.500 − 504.893i 0.415143 − 0.719049i −0.580300 − 0.814403i \(-0.697065\pi\)
0.995443 + 0.0953535i \(0.0303981\pi\)
\(80\) −120.000 − 207.846i −0.167705 − 0.290474i
\(81\) −40.5000 − 70.1481i −0.0555556 − 0.0962250i
\(82\) 168.000 − 290.985i 0.226250 − 0.391876i
\(83\) 597.000 0.789509 0.394755 − 0.918787i \(-0.370830\pi\)
0.394755 + 0.918787i \(0.370830\pi\)
\(84\) 0 0
\(85\) 1260.00 1.60784
\(86\) 326.000 − 564.649i 0.408761 − 0.707996i
\(87\) −76.5000 − 132.502i −0.0942720 − 0.163284i
\(88\) −36.0000 − 62.3538i −0.0436092 − 0.0755334i
\(89\) −519.000 + 898.934i −0.618134 + 1.07064i 0.371692 + 0.928356i \(0.378778\pi\)
−0.989826 + 0.142283i \(0.954556\pi\)
\(90\) −270.000 −0.316228
\(91\) 0 0
\(92\) −336.000 −0.380765
\(93\) 277.500 − 480.644i 0.309413 − 0.535919i
\(94\) 138.000 + 239.023i 0.151421 + 0.262270i
\(95\) 780.000 + 1351.00i 0.842382 + 1.45905i
\(96\) 48.0000 − 83.1384i 0.0510310 − 0.0883883i
\(97\) 169.000 0.176901 0.0884503 − 0.996081i \(-0.471809\pi\)
0.0884503 + 0.996081i \(0.471809\pi\)
\(98\) 0 0
\(99\) −81.0000 −0.0822304
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 294.4.e.e.79.1 2
3.2 odd 2 882.4.g.l.667.1 2
7.2 even 3 294.4.a.g.1.1 1
7.3 odd 6 42.4.e.b.25.1 ✓ 2
7.4 even 3 inner 294.4.e.e.67.1 2
7.5 odd 6 294.4.a.a.1.1 1
7.6 odd 2 42.4.e.b.37.1 yes 2
21.2 odd 6 882.4.a.h.1.1 1
21.5 even 6 882.4.a.r.1.1 1
21.11 odd 6 882.4.g.l.361.1 2
21.17 even 6 126.4.g.a.109.1 2
21.20 even 2 126.4.g.a.37.1 2
28.3 even 6 336.4.q.d.193.1 2
28.19 even 6 2352.4.a.u.1.1 1
28.23 odd 6 2352.4.a.q.1.1 1
28.27 even 2 336.4.q.d.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.4.e.b.25.1 ✓ 2 7.3 odd 6
42.4.e.b.37.1 yes 2 7.6 odd 2
126.4.g.a.37.1 2 21.20 even 2
126.4.g.a.109.1 2 21.17 even 6
294.4.a.a.1.1 1 7.5 odd 6
294.4.a.g.1.1 1 7.2 even 3
294.4.e.e.67.1 2 7.4 even 3 inner
294.4.e.e.79.1 2 1.1 even 1 trivial
336.4.q.d.193.1 2 28.3 even 6
336.4.q.d.289.1 2 28.27 even 2
882.4.a.h.1.1 1 21.2 odd 6
882.4.a.r.1.1 1 21.5 even 6
882.4.g.l.361.1 2 21.11 odd 6
882.4.g.l.667.1 2 3.2 odd 2
2352.4.a.q.1.1 1 28.23 odd 6
2352.4.a.u.1.1 1 28.19 even 6