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 67.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 294.67
Dual form 294.4.e.e.79.1

$q$-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 \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(199\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 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 −0.977672 − 0.210138i \(-0.932609\pi\)
0.306851 − 0.951757i \(-0.400725\pi\)
\(6\) −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 −0.797739 − 0.603002i \(-0.793971\pi\)
0.921085 + 0.389362i \(0.127304\pi\)
\(12\) −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 0.393733 + 0.919225i \(0.371183\pi\)
−0.992939 + 0.118630i \(0.962150\pi\)
\(18\) 9.00000 − 15.5885i 0.117851 − 0.204124i
\(19\) 52.0000 + 90.0666i 0.627875 + 1.08751i 0.987977 + 0.154598i \(0.0494083\pi\)
−0.360103 + 0.932913i \(0.617258\pi\)
\(20\) 60.0000 0.670820
\(21\) 0 0
\(22\) 18.0000 0.174437
\(23\) 42.0000 + 72.7461i 0.380765 + 0.659505i 0.991172 − 0.132583i \(-0.0423272\pi\)
−0.610406 + 0.792088i \(0.708994\pi\)
\(24\) 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 −0.463199 − 0.886254i \(-0.653299\pi\)
0.999118 − 0.0419848i \(-0.0133681\pi\)
\(32\) 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 0.813003 − 0.582260i \(-0.197831\pi\)
−0.910753 + 0.412951i \(0.864498\pi\)
\(38\) −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 0.738865 − 0.673854i \(-0.235362\pi\)
−0.953007 + 0.302949i \(0.902029\pi\)
\(48\) 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 0.0715141 + 0.997440i \(0.477217\pi\)
−0.899565 + 0.436787i \(0.856116\pi\)
\(54\) 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 −0.764884 − 0.644168i \(-0.777204\pi\)
0.940308 + 0.340325i \(0.110537\pi\)
\(60\) −90.0000 + 155.885i −0.193649 + 0.335410i
\(61\) 361.000 + 625.270i 0.757726 + 1.31242i 0.944007 + 0.329924i \(0.107023\pi\)
−0.186281 + 0.982497i \(0.559643\pi\)
\(62\) 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 −0.780378 − 0.625309i \(-0.784973\pi\)
0.931722 + 0.363173i \(0.118306\pi\)
\(68\) −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 −0.765318 − 0.643652i \(-0.777418\pi\)
0.940078 + 0.340959i \(0.110752\pi\)
\(74\) 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 0.995443 − 0.0953535i \(-0.0303981\pi\)
−0.580300 + 0.814403i \(0.697065\pi\)
\(80\) −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 −0.989826 − 0.142283i \(-0.954556\pi\)
0.371692 − 0.928356i \(-0.378778\pi\)
\(90\) −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.67.1 2
3.2 odd 2 882.4.g.l.361.1 2
7.2 even 3 inner 294.4.e.e.79.1 2
7.3 odd 6 294.4.a.a.1.1 1
7.4 even 3 294.4.a.g.1.1 1
7.5 odd 6 42.4.e.b.37.1 yes 2
7.6 odd 2 42.4.e.b.25.1 ✓ 2
21.2 odd 6 882.4.g.l.667.1 2
21.5 even 6 126.4.g.a.37.1 2
21.11 odd 6 882.4.a.h.1.1 1
21.17 even 6 882.4.a.r.1.1 1
21.20 even 2 126.4.g.a.109.1 2
28.3 even 6 2352.4.a.u.1.1 1
28.11 odd 6 2352.4.a.q.1.1 1
28.19 even 6 336.4.q.d.289.1 2
28.27 even 2 336.4.q.d.193.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.4.e.b.25.1 ✓ 2 7.6 odd 2
42.4.e.b.37.1 yes 2 7.5 odd 6
126.4.g.a.37.1 2 21.5 even 6
126.4.g.a.109.1 2 21.20 even 2
294.4.a.a.1.1 1 7.3 odd 6
294.4.a.g.1.1 1 7.4 even 3
294.4.e.e.67.1 2 1.1 even 1 trivial
294.4.e.e.79.1 2 7.2 even 3 inner
336.4.q.d.193.1 2 28.27 even 2
336.4.q.d.289.1 2 28.19 even 6
882.4.a.h.1.1 1 21.11 odd 6
882.4.a.r.1.1 1 21.17 even 6
882.4.g.l.361.1 2 3.2 odd 2
882.4.g.l.667.1 2 21.2 odd 6
2352.4.a.q.1.1 1 28.11 odd 6
2352.4.a.u.1.1 1 28.3 even 6