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,-18,-12,0,16,-9,-36,72] 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.c.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} +(-9.00000 + 15.5885i) q^{5} -6.00000 q^{6} +8.00000 q^{8} +(-4.50000 + 7.79423i) q^{9} +(-18.0000 - 31.1769i) q^{10} +(36.0000 + 62.3538i) q^{11} +(6.00000 - 10.3923i) q^{12} -34.0000 q^{13} -54.0000 q^{15} +(-8.00000 + 13.8564i) q^{16} +(-3.00000 - 5.19615i) q^{17} +(-9.00000 - 15.5885i) q^{18} +(-46.0000 + 79.6743i) q^{19} +72.0000 q^{20} -144.000 q^{22} +(90.0000 - 155.885i) q^{23} +(12.0000 + 20.7846i) q^{24} +(-99.5000 - 172.339i) q^{25} +(34.0000 - 58.8897i) q^{26} -27.0000 q^{27} -114.000 q^{29} +(54.0000 - 93.5307i) q^{30} +(-28.0000 - 48.4974i) q^{31} +(-16.0000 - 27.7128i) q^{32} +(-108.000 + 187.061i) q^{33} +12.0000 q^{34} +36.0000 q^{36} +(17.0000 - 29.4449i) q^{37} +(-92.0000 - 159.349i) q^{38} +(-51.0000 - 88.3346i) q^{39} +(-72.0000 + 124.708i) q^{40} +6.00000 q^{41} +164.000 q^{43} +(144.000 - 249.415i) q^{44} +(-81.0000 - 140.296i) q^{45} +(180.000 + 311.769i) q^{46} +(-84.0000 + 145.492i) q^{47} -48.0000 q^{48} +398.000 q^{50} +(9.00000 - 15.5885i) q^{51} +(68.0000 + 117.779i) q^{52} +(-327.000 - 566.381i) q^{53} +(27.0000 - 46.7654i) q^{54} -1296.00 q^{55} -276.000 q^{57} +(114.000 - 197.454i) q^{58} +(246.000 + 426.084i) q^{59} +(108.000 + 187.061i) q^{60} +(125.000 - 216.506i) q^{61} +112.000 q^{62} +64.0000 q^{64} +(306.000 - 530.008i) q^{65} +(-216.000 - 374.123i) q^{66} +(62.0000 + 107.387i) q^{67} +(-12.0000 + 20.7846i) q^{68} +540.000 q^{69} +36.0000 q^{71} +(-36.0000 + 62.3538i) q^{72} +(-505.000 - 874.686i) q^{73} +(34.0000 + 58.8897i) q^{74} +(298.500 - 517.017i) q^{75} +368.000 q^{76} +204.000 q^{78} +(-28.0000 + 48.4974i) q^{79} +(-144.000 - 249.415i) q^{80} +(-40.5000 - 70.1481i) q^{81} +(-6.00000 + 10.3923i) q^{82} +228.000 q^{83} +108.000 q^{85} +(-164.000 + 284.056i) q^{86} +(-171.000 - 296.181i) q^{87} +(288.000 + 498.831i) q^{88} +(-195.000 + 337.750i) q^{89} +324.000 q^{90} -720.000 q^{92} +(84.0000 - 145.492i) q^{93} +(-168.000 - 290.985i) q^{94} +(-828.000 - 1434.14i) q^{95} +(48.0000 - 83.1384i) q^{96} -70.0000 q^{97} -648.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 3 q^{3} - 4 q^{4} - 18 q^{5} - 12 q^{6} + 16 q^{8} - 9 q^{9} - 36 q^{10} + 72 q^{11} + 12 q^{12} - 68 q^{13} - 108 q^{15} - 16 q^{16} - 6 q^{17} - 18 q^{18} - 92 q^{19} + 144 q^{20} - 288 q^{22}+ \cdots - 1296 q^{99}+O(q^{100}) \) 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
\(5\) −9.00000 + 15.5885i −0.804984 + 1.39427i 0.111317 + 0.993785i \(0.464493\pi\)
−0.916302 + 0.400489i \(0.868840\pi\)
\(6\) −6.00000 −0.408248
\(7\) 0 0
\(8\) 8.00000 0.353553
\(9\) −4.50000 + 7.79423i −0.166667 + 0.288675i
\(10\) −18.0000 − 31.1769i −0.569210 − 0.985901i
\(11\) 36.0000 + 62.3538i 0.986764 + 1.70913i 0.633817 + 0.773483i \(0.281487\pi\)
0.352947 + 0.935643i \(0.385180\pi\)
\(12\) 6.00000 − 10.3923i 0.144338 − 0.250000i
\(13\) −34.0000 −0.725377 −0.362689 − 0.931910i \(-0.618141\pi\)
−0.362689 + 0.931910i \(0.618141\pi\)
\(14\) 0 0
\(15\) −54.0000 −0.929516
\(16\) −8.00000 + 13.8564i −0.125000 + 0.216506i
\(17\) −3.00000 − 5.19615i −0.0428004 − 0.0741325i 0.843832 − 0.536608i \(-0.180295\pi\)
−0.886632 + 0.462476i \(0.846961\pi\)
\(18\) −9.00000 − 15.5885i −0.117851 − 0.204124i
\(19\) −46.0000 + 79.6743i −0.555428 + 0.962029i 0.442443 + 0.896797i \(0.354112\pi\)
−0.997870 + 0.0652319i \(0.979221\pi\)
\(20\) 72.0000 0.804984
\(21\) 0 0
\(22\) −144.000 −1.39550
\(23\) 90.0000 − 155.885i 0.815926 − 1.41323i −0.0927351 − 0.995691i \(-0.529561\pi\)
0.908661 − 0.417534i \(-0.137106\pi\)
\(24\) 12.0000 + 20.7846i 0.102062 + 0.176777i
\(25\) −99.5000 − 172.339i −0.796000 − 1.37871i
\(26\) 34.0000 − 58.8897i 0.256460 − 0.444201i
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −114.000 −0.729975 −0.364987 − 0.931012i \(-0.618927\pi\)
−0.364987 + 0.931012i \(0.618927\pi\)
\(30\) 54.0000 − 93.5307i 0.328634 − 0.569210i
\(31\) −28.0000 − 48.4974i −0.162224 − 0.280980i 0.773442 − 0.633867i \(-0.218533\pi\)
−0.935666 + 0.352887i \(0.885200\pi\)
\(32\) −16.0000 − 27.7128i −0.0883883 − 0.153093i
\(33\) −108.000 + 187.061i −0.569709 + 0.986764i
\(34\) 12.0000 0.0605289
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) 17.0000 − 29.4449i 0.0755347 − 0.130830i −0.825784 − 0.563987i \(-0.809267\pi\)
0.901319 + 0.433157i \(0.142600\pi\)
\(38\) −92.0000 − 159.349i −0.392747 − 0.680257i
\(39\) −51.0000 − 88.3346i −0.209398 − 0.362689i
\(40\) −72.0000 + 124.708i −0.284605 + 0.492950i
\(41\) 6.00000 0.0228547 0.0114273 − 0.999935i \(-0.496362\pi\)
0.0114273 + 0.999935i \(0.496362\pi\)
\(42\) 0 0
\(43\) 164.000 0.581622 0.290811 − 0.956780i \(-0.406075\pi\)
0.290811 + 0.956780i \(0.406075\pi\)
\(44\) 144.000 − 249.415i 0.493382 − 0.854563i
\(45\) −81.0000 − 140.296i −0.268328 − 0.464758i
\(46\) 180.000 + 311.769i 0.576947 + 0.999301i
\(47\) −84.0000 + 145.492i −0.260695 + 0.451537i −0.966427 − 0.256942i \(-0.917285\pi\)
0.705732 + 0.708479i \(0.250618\pi\)
\(48\) −48.0000 −0.144338
\(49\) 0 0
\(50\) 398.000 1.12571
\(51\) 9.00000 − 15.5885i 0.0247108 − 0.0428004i
\(52\) 68.0000 + 117.779i 0.181344 + 0.314098i
\(53\) −327.000 − 566.381i −0.847489 − 1.46789i −0.883442 − 0.468540i \(-0.844780\pi\)
0.0359535 − 0.999353i \(-0.488553\pi\)
\(54\) 27.0000 − 46.7654i 0.0680414 − 0.117851i
\(55\) −1296.00 −3.17732
\(56\) 0 0
\(57\) −276.000 −0.641353
\(58\) 114.000 − 197.454i 0.258085 − 0.447016i
\(59\) 246.000 + 426.084i 0.542822 + 0.940195i 0.998741 + 0.0501732i \(0.0159773\pi\)
−0.455919 + 0.890021i \(0.650689\pi\)
\(60\) 108.000 + 187.061i 0.232379 + 0.402492i
\(61\) 125.000 − 216.506i 0.262371 − 0.454439i −0.704501 − 0.709703i \(-0.748829\pi\)
0.966871 + 0.255264i \(0.0821624\pi\)
\(62\) 112.000 0.229420
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 306.000 − 530.008i 0.583917 − 1.01137i
\(66\) −216.000 − 374.123i −0.402845 − 0.697748i
\(67\) 62.0000 + 107.387i 0.113052 + 0.195812i 0.917000 − 0.398888i \(-0.130604\pi\)
−0.803947 + 0.594701i \(0.797271\pi\)
\(68\) −12.0000 + 20.7846i −0.0214002 + 0.0370662i
\(69\) 540.000 0.942150
\(70\) 0 0
\(71\) 36.0000 0.0601748 0.0300874 − 0.999547i \(-0.490421\pi\)
0.0300874 + 0.999547i \(0.490421\pi\)
\(72\) −36.0000 + 62.3538i −0.0589256 + 0.102062i
\(73\) −505.000 − 874.686i −0.809668 − 1.40239i −0.913094 − 0.407749i \(-0.866314\pi\)
0.103426 − 0.994637i \(-0.467020\pi\)
\(74\) 34.0000 + 58.8897i 0.0534111 + 0.0925107i
\(75\) 298.500 − 517.017i 0.459571 − 0.796000i
\(76\) 368.000 0.555428
\(77\) 0 0
\(78\) 204.000 0.296134
\(79\) −28.0000 + 48.4974i −0.0398765 + 0.0690682i −0.885275 − 0.465068i \(-0.846030\pi\)
0.845398 + 0.534136i \(0.179363\pi\)
\(80\) −144.000 − 249.415i −0.201246 − 0.348569i
\(81\) −40.5000 − 70.1481i −0.0555556 − 0.0962250i
\(82\) −6.00000 + 10.3923i −0.00808036 + 0.0139956i
\(83\) 228.000 0.301521 0.150761 − 0.988570i \(-0.451828\pi\)
0.150761 + 0.988570i \(0.451828\pi\)
\(84\) 0 0
\(85\) 108.000 0.137815
\(86\) −164.000 + 284.056i −0.205635 + 0.356170i
\(87\) −171.000 − 296.181i −0.210726 − 0.364987i
\(88\) 288.000 + 498.831i 0.348874 + 0.604267i
\(89\) −195.000 + 337.750i −0.232247 + 0.402263i −0.958469 − 0.285197i \(-0.907941\pi\)
0.726222 + 0.687460i \(0.241274\pi\)
\(90\) 324.000 0.379473
\(91\) 0 0
\(92\) −720.000 −0.815926
\(93\) 84.0000 − 145.492i 0.0936602 − 0.162224i
\(94\) −168.000 − 290.985i −0.184339 − 0.319285i
\(95\) −828.000 − 1434.14i −0.894221 − 1.54884i
\(96\) 48.0000 − 83.1384i 0.0510310 − 0.0883883i
\(97\) −70.0000 −0.0732724 −0.0366362 − 0.999329i \(-0.511664\pi\)
−0.0366362 + 0.999329i \(0.511664\pi\)
\(98\) 0 0
\(99\) −648.000 −0.657843
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.c.79.1 2
3.2 odd 2 882.4.g.w.667.1 2
7.2 even 3 42.4.a.a.1.1 ✓ 1
7.3 odd 6 294.4.e.b.67.1 2
7.4 even 3 inner 294.4.e.c.67.1 2
7.5 odd 6 294.4.a.i.1.1 1
7.6 odd 2 294.4.e.b.79.1 2
21.2 odd 6 126.4.a.a.1.1 1
21.5 even 6 882.4.a.g.1.1 1
21.11 odd 6 882.4.g.w.361.1 2
21.17 even 6 882.4.g.o.361.1 2
21.20 even 2 882.4.g.o.667.1 2
28.19 even 6 2352.4.a.a.1.1 1
28.23 odd 6 336.4.a.l.1.1 1
35.2 odd 12 1050.4.g.a.799.2 2
35.9 even 6 1050.4.a.g.1.1 1
35.23 odd 12 1050.4.g.a.799.1 2
56.37 even 6 1344.4.a.o.1.1 1
56.51 odd 6 1344.4.a.a.1.1 1
84.23 even 6 1008.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.4.a.a.1.1 ✓ 1 7.2 even 3
126.4.a.a.1.1 1 21.2 odd 6
294.4.a.i.1.1 1 7.5 odd 6
294.4.e.b.67.1 2 7.3 odd 6
294.4.e.b.79.1 2 7.6 odd 2
294.4.e.c.67.1 2 7.4 even 3 inner
294.4.e.c.79.1 2 1.1 even 1 trivial
336.4.a.l.1.1 1 28.23 odd 6
882.4.a.g.1.1 1 21.5 even 6
882.4.g.o.361.1 2 21.17 even 6
882.4.g.o.667.1 2 21.20 even 2
882.4.g.w.361.1 2 21.11 odd 6
882.4.g.w.667.1 2 3.2 odd 2
1008.4.a.b.1.1 1 84.23 even 6
1050.4.a.g.1.1 1 35.9 even 6
1050.4.g.a.799.1 2 35.23 odd 12
1050.4.g.a.799.2 2 35.2 odd 12
1344.4.a.a.1.1 1 56.51 odd 6
1344.4.a.o.1.1 1 56.37 even 6
2352.4.a.a.1.1 1 28.19 even 6