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Properties

Label 294.6.e.s.79.2
Level $294$
Weight $6$
Character 294.79
Analytic conductor $47.153$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [294,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 294.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-8,-18,-32,-53,144,0,256,-162,-212,-191] 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: \(47.1528430250\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{9601})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2401x^{2} + 2400x + 5760000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 79.2
Root \(-24.2462 + 41.9956i\) of defining polynomial
Character \(\chi\) \(=\) 294.79
Dual form 294.6.e.s.67.2

$q$-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+(-2.00000 + 3.46410i) q^{2} +(-4.50000 - 7.79423i) q^{3} +(-8.00000 - 13.8564i) q^{4} +(11.2462 - 19.4789i) q^{5} +36.0000 q^{6} +64.0000 q^{8} +(-40.5000 + 70.1481i) q^{9} +(44.9847 + 77.9158i) q^{10} +(-170.231 - 294.849i) q^{11} +(-72.0000 + 124.708i) q^{12} +728.416 q^{13} -202.431 q^{15} +(-128.000 + 221.703i) q^{16} +(404.923 + 701.348i) q^{17} +(-162.000 - 280.592i) q^{18} +(513.101 - 888.717i) q^{19} -359.878 q^{20} +1361.85 q^{22} +(-711.015 + 1231.51i) q^{23} +(-288.000 - 498.831i) q^{24} +(1309.55 + 2268.20i) q^{25} +(-1456.83 + 2523.31i) q^{26} +729.000 q^{27} +5218.03 q^{29} +(404.862 - 701.242i) q^{30} +(-3518.87 - 6094.87i) q^{31} +(-512.000 - 886.810i) q^{32} +(-1532.08 + 2653.64i) q^{33} -3239.39 q^{34} +1296.00 q^{36} +(-6396.05 + 11078.3i) q^{37} +(2052.40 + 3554.87i) q^{38} +(-3277.87 - 5677.44i) q^{39} +(719.755 - 1246.65i) q^{40} -1173.51 q^{41} +3664.17 q^{43} +(-2723.69 + 4717.58i) q^{44} +(910.940 + 1577.79i) q^{45} +(-2844.06 - 4926.06i) q^{46} +(4656.60 - 8065.46i) q^{47} +2304.00 q^{48} -10476.4 q^{50} +(3644.31 - 6312.13i) q^{51} +(-5827.33 - 10093.2i) q^{52} +(-17821.4 - 30867.6i) q^{53} +(-1458.00 + 2525.33i) q^{54} -7657.78 q^{55} -9235.81 q^{57} +(-10436.1 + 18075.8i) q^{58} +(-15188.1 - 26306.5i) q^{59} +(1619.45 + 2804.97i) q^{60} +(16093.1 - 27874.1i) q^{61} +28151.0 q^{62} +4096.00 q^{64} +(8191.89 - 14188.8i) q^{65} +(-6128.31 - 10614.5i) q^{66} +(-10675.5 - 18490.6i) q^{67} +(6478.78 - 11221.6i) q^{68} +12798.3 q^{69} +61153.7 q^{71} +(-2592.00 + 4489.48i) q^{72} +(20633.9 + 35739.0i) q^{73} +(-25584.2 - 44313.2i) q^{74} +(11785.9 - 20413.8i) q^{75} -16419.2 q^{76} +26223.0 q^{78} +(17500.2 - 30311.3i) q^{79} +(2879.02 + 4986.61i) q^{80} +(-3280.50 - 5681.99i) q^{81} +(2347.01 - 4065.14i) q^{82} -86193.0 q^{83} +18215.4 q^{85} +(-7328.33 + 12693.0i) q^{86} +(-23481.1 - 40670.5i) q^{87} +(-10894.8 - 18870.3i) q^{88} +(38996.3 - 67543.6i) q^{89} -7287.52 q^{90} +22752.5 q^{92} +(-31669.9 + 54853.8i) q^{93} +(18626.4 + 32261.9i) q^{94} +(-11540.8 - 19989.3i) q^{95} +(-4608.00 + 7981.29i) q^{96} +161765. q^{97} +27577.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} - 18 q^{3} - 32 q^{4} - 53 q^{5} + 144 q^{6} + 256 q^{8} - 162 q^{9} - 212 q^{10} - 191 q^{11} - 288 q^{12} + 758 q^{13} + 954 q^{15} - 512 q^{16} - 340 q^{17} - 648 q^{18} - 1769 q^{19} + 1696 q^{20}+ \cdots + 30942 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 + 3.46410i −0.353553 + 0.612372i
\(3\) −4.50000 − 7.79423i −0.288675 − 0.500000i
\(4\) −8.00000 − 13.8564i −0.250000 − 0.433013i
\(5\) 11.2462 − 19.4789i 0.201178 − 0.348450i −0.747730 − 0.664002i \(-0.768856\pi\)
0.948908 + 0.315552i \(0.102190\pi\)
\(6\) 36.0000 0.408248
\(7\) 0 0
\(8\) 64.0000 0.353553
\(9\) −40.5000 + 70.1481i −0.166667 + 0.288675i
\(10\) 44.9847 + 77.9158i 0.142254 + 0.246391i
\(11\) −170.231 − 294.849i −0.424186 − 0.734712i 0.572158 − 0.820144i \(-0.306106\pi\)
−0.996344 + 0.0854314i \(0.972773\pi\)
\(12\) −72.0000 + 124.708i −0.144338 + 0.250000i
\(13\) 728.416 1.19542 0.597711 − 0.801712i \(-0.296077\pi\)
0.597711 + 0.801712i \(0.296077\pi\)
\(14\) 0 0
\(15\) −202.431 −0.232300
\(16\) −128.000 + 221.703i −0.125000 + 0.216506i
\(17\) 404.923 + 701.348i 0.339821 + 0.588588i 0.984399 − 0.175951i \(-0.0563001\pi\)
−0.644578 + 0.764539i \(0.722967\pi\)
\(18\) −162.000 − 280.592i −0.117851 − 0.204124i
\(19\) 513.101 − 888.717i 0.326076 − 0.564780i −0.655654 − 0.755062i \(-0.727607\pi\)
0.981730 + 0.190282i \(0.0609402\pi\)
\(20\) −359.878 −0.201178
\(21\) 0 0
\(22\) 1361.85 0.599890
\(23\) −711.015 + 1231.51i −0.280259 + 0.485423i −0.971448 − 0.237251i \(-0.923754\pi\)
0.691190 + 0.722674i \(0.257087\pi\)
\(24\) −288.000 − 498.831i −0.102062 − 0.176777i
\(25\) 1309.55 + 2268.20i 0.419055 + 0.725825i
\(26\) −1456.83 + 2523.31i −0.422645 + 0.732043i
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) 5218.03 1.15216 0.576079 − 0.817394i \(-0.304582\pi\)
0.576079 + 0.817394i \(0.304582\pi\)
\(30\) 404.862 − 701.242i 0.0821304 − 0.142254i
\(31\) −3518.87 − 6094.87i −0.657657 − 1.13909i −0.981221 − 0.192889i \(-0.938214\pi\)
0.323564 − 0.946206i \(-0.395119\pi\)
\(32\) −512.000 − 886.810i −0.0883883 − 0.153093i
\(33\) −1532.08 + 2653.64i −0.244904 + 0.424186i
\(34\) −3239.39 −0.480580
\(35\) 0 0
\(36\) 1296.00 0.166667
\(37\) −6396.05 + 11078.3i −0.768082 + 1.33036i 0.170519 + 0.985354i \(0.445456\pi\)
−0.938601 + 0.345004i \(0.887878\pi\)
\(38\) 2052.40 + 3554.87i 0.230570 + 0.399360i
\(39\) −3277.87 − 5677.44i −0.345088 − 0.597711i
\(40\) 719.755 − 1246.65i 0.0711270 − 0.123196i
\(41\) −1173.51 −0.109025 −0.0545124 − 0.998513i \(-0.517360\pi\)
−0.0545124 + 0.998513i \(0.517360\pi\)
\(42\) 0 0
\(43\) 3664.17 0.302207 0.151103 − 0.988518i \(-0.451717\pi\)
0.151103 + 0.988518i \(0.451717\pi\)
\(44\) −2723.69 + 4717.58i −0.212093 + 0.367356i
\(45\) 910.940 + 1577.79i 0.0670592 + 0.116150i
\(46\) −2844.06 − 4926.06i −0.198173 − 0.343246i
\(47\) 4656.60 − 8065.46i 0.307485 − 0.532580i −0.670326 − 0.742066i \(-0.733846\pi\)
0.977812 + 0.209487i \(0.0671793\pi\)
\(48\) 2304.00 0.144338
\(49\) 0 0
\(50\) −10476.4 −0.592633
\(51\) 3644.31 − 6312.13i 0.196196 − 0.339821i
\(52\) −5827.33 − 10093.2i −0.298855 − 0.517633i
\(53\) −17821.4 − 30867.6i −0.871469 − 1.50943i −0.860477 − 0.509489i \(-0.829835\pi\)
−0.0109916 − 0.999940i \(-0.503499\pi\)
\(54\) −1458.00 + 2525.33i −0.0680414 + 0.117851i
\(55\) −7657.78 −0.341347
\(56\) 0 0
\(57\) −9235.81 −0.376520
\(58\) −10436.1 + 18075.8i −0.407349 + 0.705549i
\(59\) −15188.1 − 26306.5i −0.568033 − 0.983861i −0.996760 − 0.0804276i \(-0.974371\pi\)
0.428728 − 0.903434i \(-0.358962\pi\)
\(60\) 1619.45 + 2804.97i 0.0580750 + 0.100589i
\(61\) 16093.1 − 27874.1i 0.553753 − 0.959128i −0.444247 − 0.895904i \(-0.646529\pi\)
0.997999 − 0.0632231i \(-0.0201380\pi\)
\(62\) 28151.0 0.930067
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 8191.89 − 14188.8i 0.240492 − 0.416544i
\(66\) −6128.31 − 10614.5i −0.173173 − 0.299945i
\(67\) −10675.5 − 18490.6i −0.290538 − 0.503226i 0.683399 − 0.730045i \(-0.260501\pi\)
−0.973937 + 0.226819i \(0.927168\pi\)
\(68\) 6478.78 − 11221.6i 0.169911 − 0.294294i
\(69\) 12798.3 0.323615
\(70\) 0 0
\(71\) 61153.7 1.43972 0.719859 − 0.694121i \(-0.244207\pi\)
0.719859 + 0.694121i \(0.244207\pi\)
\(72\) −2592.00 + 4489.48i −0.0589256 + 0.102062i
\(73\) 20633.9 + 35739.0i 0.453184 + 0.784937i 0.998582 − 0.0532401i \(-0.0169549\pi\)
−0.545398 + 0.838177i \(0.683622\pi\)
\(74\) −25584.2 − 44313.2i −0.543116 − 0.940705i
\(75\) 11785.9 − 20413.8i 0.241942 − 0.419055i
\(76\) −16419.2 −0.326076
\(77\) 0 0
\(78\) 26223.0 0.488029
\(79\) 17500.2 − 30311.3i 0.315483 − 0.546433i −0.664057 − 0.747682i \(-0.731167\pi\)
0.979540 + 0.201249i \(0.0645001\pi\)
\(80\) 2879.02 + 4986.61i 0.0502944 + 0.0871125i
\(81\) −3280.50 − 5681.99i −0.0555556 − 0.0962250i
\(82\) 2347.01 − 4065.14i 0.0385461 − 0.0667638i
\(83\) −86193.0 −1.37334 −0.686668 − 0.726972i \(-0.740927\pi\)
−0.686668 + 0.726972i \(0.740927\pi\)
\(84\) 0 0
\(85\) 18215.4 0.273458
\(86\) −7328.33 + 12693.0i −0.106846 + 0.185063i
\(87\) −23481.1 − 40670.5i −0.332599 − 0.576079i
\(88\) −10894.8 − 18870.3i −0.149972 − 0.259760i
\(89\) 38996.3 − 67543.6i 0.521853 − 0.903876i −0.477824 − 0.878456i \(-0.658574\pi\)
0.999677 − 0.0254206i \(-0.00809249\pi\)
\(90\) −7287.52 −0.0948361
\(91\) 0 0
\(92\) 22752.5 0.280259
\(93\) −31669.9 + 54853.8i −0.379698 + 0.657657i
\(94\) 18626.4 + 32261.9i 0.217425 + 0.376591i
\(95\) −11540.8 − 19989.3i −0.131198 − 0.227242i
\(96\) −4608.00 + 7981.29i −0.0510310 + 0.0883883i
\(97\) 161765. 1.74565 0.872823 − 0.488037i \(-0.162287\pi\)
0.872823 + 0.488037i \(0.162287\pi\)
\(98\) 0 0
\(99\) 27577.4 0.282791
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 294.6.e.s.79.2 4
7.2 even 3 294.6.a.w.1.1 2
7.3 odd 6 42.6.e.c.25.1 ✓ 4
7.4 even 3 inner 294.6.e.s.67.2 4
7.5 odd 6 294.6.a.r.1.2 2
7.6 odd 2 42.6.e.c.37.1 yes 4
21.2 odd 6 882.6.a.bb.1.2 2
21.5 even 6 882.6.a.bh.1.1 2
21.17 even 6 126.6.g.h.109.2 4
21.20 even 2 126.6.g.h.37.2 4
28.3 even 6 336.6.q.f.193.1 4
28.27 even 2 336.6.q.f.289.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.6.e.c.25.1 ✓ 4 7.3 odd 6
42.6.e.c.37.1 yes 4 7.6 odd 2
126.6.g.h.37.2 4 21.20 even 2
126.6.g.h.109.2 4 21.17 even 6
294.6.a.r.1.2 2 7.5 odd 6
294.6.a.w.1.1 2 7.2 even 3
294.6.e.s.67.2 4 7.4 even 3 inner
294.6.e.s.79.2 4 1.1 even 1 trivial
336.6.q.f.193.1 4 28.3 even 6
336.6.q.f.289.1 4 28.27 even 2
882.6.a.bb.1.2 2 21.2 odd 6
882.6.a.bh.1.1 2 21.5 even 6