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,-8,-12,0,-16,-9,16,-40] 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: yes
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.f.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} +(-4.00000 - 6.92820i) q^{5} -6.00000 q^{6} -8.00000 q^{8} +(-4.50000 - 7.79423i) q^{9} +(8.00000 - 13.8564i) q^{10} +(-20.0000 + 34.6410i) q^{11} +(-6.00000 - 10.3923i) q^{12} +4.00000 q^{13} +24.0000 q^{15} +(-8.00000 - 13.8564i) q^{16} +(42.0000 - 72.7461i) q^{17} +(9.00000 - 15.5885i) q^{18} +(-74.0000 - 128.172i) q^{19} +32.0000 q^{20} -80.0000 q^{22} +(-42.0000 - 72.7461i) q^{23} +(12.0000 - 20.7846i) q^{24} +(30.5000 - 52.8275i) q^{25} +(4.00000 + 6.92820i) q^{26} +27.0000 q^{27} +58.0000 q^{29} +(24.0000 + 41.5692i) q^{30} +(68.0000 - 117.779i) q^{31} +(16.0000 - 27.7128i) q^{32} +(-60.0000 - 103.923i) q^{33} +168.000 q^{34} +36.0000 q^{36} +(111.000 + 192.258i) q^{37} +(148.000 - 256.344i) q^{38} +(-6.00000 + 10.3923i) q^{39} +(32.0000 + 55.4256i) q^{40} +420.000 q^{41} -164.000 q^{43} +(-80.0000 - 138.564i) q^{44} +(-36.0000 + 62.3538i) q^{45} +(84.0000 - 145.492i) q^{46} +(-244.000 - 422.620i) q^{47} +48.0000 q^{48} +122.000 q^{50} +(126.000 + 218.238i) q^{51} +(-8.00000 + 13.8564i) q^{52} +(-239.000 + 413.960i) q^{53} +(27.0000 + 46.7654i) q^{54} +320.000 q^{55} +444.000 q^{57} +(58.0000 + 100.459i) q^{58} +(-274.000 + 474.582i) q^{59} +(-48.0000 + 83.1384i) q^{60} +(-346.000 - 599.290i) q^{61} +272.000 q^{62} +64.0000 q^{64} +(-16.0000 - 27.7128i) q^{65} +(120.000 - 207.846i) q^{66} +(454.000 - 786.351i) q^{67} +(168.000 + 290.985i) q^{68} +252.000 q^{69} -524.000 q^{71} +(36.0000 + 62.3538i) q^{72} +(-220.000 + 381.051i) q^{73} +(-222.000 + 384.515i) q^{74} +(91.5000 + 158.483i) q^{75} +592.000 q^{76} -24.0000 q^{78} +(-608.000 - 1053.09i) q^{79} +(-64.0000 + 110.851i) q^{80} +(-40.5000 + 70.1481i) q^{81} +(420.000 + 727.461i) q^{82} -684.000 q^{83} -672.000 q^{85} +(-164.000 - 284.056i) q^{86} +(-87.0000 + 150.688i) q^{87} +(160.000 - 277.128i) q^{88} +(-302.000 - 523.079i) q^{89} -144.000 q^{90} +336.000 q^{92} +(204.000 + 353.338i) q^{93} +(488.000 - 845.241i) q^{94} +(-592.000 + 1025.37i) q^{95} +(48.0000 + 83.1384i) q^{96} -832.000 q^{97} +360.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} - 8 q^{5} - 12 q^{6} - 16 q^{8} - 9 q^{9} + 16 q^{10} - 40 q^{11} - 12 q^{12} + 8 q^{13} + 48 q^{15} - 16 q^{16} + 84 q^{17} + 18 q^{18} - 148 q^{19} + 64 q^{20} - 160 q^{22}+ \cdots + 720 q^{99}+O(q^{100}) \) 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
\(5\) −4.00000 − 6.92820i −0.357771 − 0.619677i 0.629817 − 0.776743i \(-0.283130\pi\)
−0.987588 + 0.157066i \(0.949796\pi\)
\(6\) −6.00000 −0.408248
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) −4.50000 − 7.79423i −0.166667 − 0.288675i
\(10\) 8.00000 − 13.8564i 0.252982 − 0.438178i
\(11\) −20.0000 + 34.6410i −0.548202 + 0.949514i 0.450195 + 0.892930i \(0.351354\pi\)
−0.998398 + 0.0565844i \(0.981979\pi\)
\(12\) −6.00000 − 10.3923i −0.144338 − 0.250000i
\(13\) 4.00000 0.0853385 0.0426692 − 0.999089i \(-0.486414\pi\)
0.0426692 + 0.999089i \(0.486414\pi\)
\(14\) 0 0
\(15\) 24.0000 0.413118
\(16\) −8.00000 − 13.8564i −0.125000 − 0.216506i
\(17\) 42.0000 − 72.7461i 0.599206 − 1.03785i −0.393733 − 0.919225i \(-0.628817\pi\)
0.992939 − 0.118630i \(-0.0378502\pi\)
\(18\) 9.00000 − 15.5885i 0.117851 − 0.204124i
\(19\) −74.0000 − 128.172i −0.893514 − 1.54761i −0.835633 − 0.549288i \(-0.814899\pi\)
−0.0578808 − 0.998324i \(-0.518434\pi\)
\(20\) 32.0000 0.357771
\(21\) 0 0
\(22\) −80.0000 −0.775275
\(23\) −42.0000 − 72.7461i −0.380765 − 0.659505i 0.610406 − 0.792088i \(-0.291006\pi\)
−0.991172 + 0.132583i \(0.957673\pi\)
\(24\) 12.0000 − 20.7846i 0.102062 − 0.176777i
\(25\) 30.5000 − 52.8275i 0.244000 − 0.422620i
\(26\) 4.00000 + 6.92820i 0.0301717 + 0.0522589i
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 58.0000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 24.0000 + 41.5692i 0.146059 + 0.252982i
\(31\) 68.0000 − 117.779i 0.393973 − 0.682381i −0.598997 − 0.800752i \(-0.704434\pi\)
0.992970 + 0.118370i \(0.0377670\pi\)
\(32\) 16.0000 − 27.7128i 0.0883883 − 0.153093i
\(33\) −60.0000 − 103.923i −0.316505 − 0.548202i
\(34\) 168.000 0.847405
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) 111.000 + 192.258i 0.493197 + 0.854242i 0.999969 − 0.00783774i \(-0.00249486\pi\)
−0.506772 + 0.862080i \(0.669162\pi\)
\(38\) 148.000 − 256.344i 0.631810 − 1.09433i
\(39\) −6.00000 + 10.3923i −0.0246351 + 0.0426692i
\(40\) 32.0000 + 55.4256i 0.126491 + 0.219089i
\(41\) 420.000 1.59983 0.799914 − 0.600114i \(-0.204878\pi\)
0.799914 + 0.600114i \(0.204878\pi\)
\(42\) 0 0
\(43\) −164.000 −0.581622 −0.290811 − 0.956780i \(-0.593925\pi\)
−0.290811 + 0.956780i \(0.593925\pi\)
\(44\) −80.0000 − 138.564i −0.274101 − 0.474757i
\(45\) −36.0000 + 62.3538i −0.119257 + 0.206559i
\(46\) 84.0000 − 145.492i 0.269242 − 0.466341i
\(47\) −244.000 − 422.620i −0.757257 − 1.31161i −0.944245 − 0.329245i \(-0.893206\pi\)
0.186988 − 0.982362i \(-0.440127\pi\)
\(48\) 48.0000 0.144338
\(49\) 0 0
\(50\) 122.000 0.345068
\(51\) 126.000 + 218.238i 0.345952 + 0.599206i
\(52\) −8.00000 + 13.8564i −0.0213346 + 0.0369527i
\(53\) −239.000 + 413.960i −0.619418 + 1.07286i 0.370174 + 0.928963i \(0.379298\pi\)
−0.989592 + 0.143902i \(0.954035\pi\)
\(54\) 27.0000 + 46.7654i 0.0680414 + 0.117851i
\(55\) 320.000 0.784523
\(56\) 0 0
\(57\) 444.000 1.03174
\(58\) 58.0000 + 100.459i 0.131306 + 0.227429i
\(59\) −274.000 + 474.582i −0.604606 + 1.04721i 0.387507 + 0.921867i \(0.373336\pi\)
−0.992114 + 0.125342i \(0.959997\pi\)
\(60\) −48.0000 + 83.1384i −0.103280 + 0.178885i
\(61\) −346.000 − 599.290i −0.726242 − 1.25789i −0.958461 − 0.285224i \(-0.907932\pi\)
0.232219 − 0.972664i \(-0.425401\pi\)
\(62\) 272.000 0.557162
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −16.0000 − 27.7128i −0.0305316 − 0.0528823i
\(66\) 120.000 − 207.846i 0.223803 − 0.387638i
\(67\) 454.000 − 786.351i 0.827835 − 1.43385i −0.0718987 − 0.997412i \(-0.522906\pi\)
0.899733 − 0.436440i \(-0.143761\pi\)
\(68\) 168.000 + 290.985i 0.299603 + 0.518927i
\(69\) 252.000 0.439670
\(70\) 0 0
\(71\) −524.000 −0.875878 −0.437939 − 0.899005i \(-0.644291\pi\)
−0.437939 + 0.899005i \(0.644291\pi\)
\(72\) 36.0000 + 62.3538i 0.0589256 + 0.102062i
\(73\) −220.000 + 381.051i −0.352727 + 0.610941i −0.986726 − 0.162393i \(-0.948079\pi\)
0.633999 + 0.773334i \(0.281412\pi\)
\(74\) −222.000 + 384.515i −0.348743 + 0.604040i
\(75\) 91.5000 + 158.483i 0.140873 + 0.244000i
\(76\) 592.000 0.893514
\(77\) 0 0
\(78\) −24.0000 −0.0348393
\(79\) −608.000 − 1053.09i −0.865890 − 1.49977i −0.866160 − 0.499766i \(-0.833419\pi\)
0.000269874 − 1.00000i \(-0.499914\pi\)
\(80\) −64.0000 + 110.851i −0.0894427 + 0.154919i
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) 420.000 + 727.461i 0.565625 + 0.979691i
\(83\) −684.000 −0.904563 −0.452282 − 0.891875i \(-0.649390\pi\)
−0.452282 + 0.891875i \(0.649390\pi\)
\(84\) 0 0
\(85\) −672.000 −0.857513
\(86\) −164.000 − 284.056i −0.205635 − 0.356170i
\(87\) −87.0000 + 150.688i −0.107211 + 0.185695i
\(88\) 160.000 − 277.128i 0.193819 − 0.335704i
\(89\) −302.000 − 523.079i −0.359685 − 0.622992i 0.628223 − 0.778033i \(-0.283782\pi\)
−0.987908 + 0.155041i \(0.950449\pi\)
\(90\) −144.000 −0.168655
\(91\) 0 0
\(92\) 336.000 0.380765
\(93\) 204.000 + 353.338i 0.227460 + 0.393973i
\(94\) 488.000 − 845.241i 0.535461 − 0.927446i
\(95\) −592.000 + 1025.37i −0.639347 + 1.10738i
\(96\) 48.0000 + 83.1384i 0.0510310 + 0.0883883i
\(97\) −832.000 −0.870895 −0.435447 − 0.900214i \(-0.643410\pi\)
−0.435447 + 0.900214i \(0.643410\pi\)
\(98\) 0 0
\(99\) 360.000 0.365468
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.f.67.1 2
3.2 odd 2 882.4.g.j.361.1 2
7.2 even 3 inner 294.4.e.f.79.1 2
7.3 odd 6 294.4.a.b.1.1 ✓ 1
7.4 even 3 294.4.a.f.1.1 yes 1
7.5 odd 6 294.4.e.j.79.1 2
7.6 odd 2 294.4.e.j.67.1 2
21.2 odd 6 882.4.g.j.667.1 2
21.5 even 6 882.4.g.c.667.1 2
21.11 odd 6 882.4.a.j.1.1 1
21.17 even 6 882.4.a.q.1.1 1
21.20 even 2 882.4.g.c.361.1 2
28.3 even 6 2352.4.a.z.1.1 1
28.11 odd 6 2352.4.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.4.a.b.1.1 ✓ 1 7.3 odd 6
294.4.a.f.1.1 yes 1 7.4 even 3
294.4.e.f.67.1 2 1.1 even 1 trivial
294.4.e.f.79.1 2 7.2 even 3 inner
294.4.e.j.67.1 2 7.6 odd 2
294.4.e.j.79.1 2 7.5 odd 6
882.4.a.j.1.1 1 21.11 odd 6
882.4.a.q.1.1 1 21.17 even 6
882.4.g.c.361.1 2 21.20 even 2
882.4.g.c.667.1 2 21.5 even 6
882.4.g.j.361.1 2 3.2 odd 2
882.4.g.j.667.1 2 21.2 odd 6
2352.4.a.m.1.1 1 28.11 odd 6
2352.4.a.z.1.1 1 28.3 even 6