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 67.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 294.67
Dual form 294.4.e.c.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} +(-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 \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

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

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
\(5\) −9.00000 − 15.5885i −0.804984 − 1.39427i −0.916302 − 0.400489i \(-0.868840\pi\)
0.111317 − 0.993785i \(-0.464493\pi\)
\(6\) −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 0.352947 − 0.935643i \(-0.385180\pi\)
0.633817 − 0.773483i \(-0.281487\pi\)
\(12\) 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 −0.886632 − 0.462476i \(-0.846961\pi\)
0.843832 + 0.536608i \(0.180295\pi\)
\(18\) −9.00000 + 15.5885i −0.117851 + 0.204124i
\(19\) −46.0000 − 79.6743i −0.555428 − 0.962029i −0.997870 − 0.0652319i \(-0.979221\pi\)
0.442443 − 0.896797i \(-0.354112\pi\)
\(20\) 72.0000 0.804984
\(21\) 0 0
\(22\) −144.000 −1.39550
\(23\) 90.0000 + 155.885i 0.815926 + 1.41323i 0.908661 + 0.417534i \(0.137106\pi\)
−0.0927351 + 0.995691i \(0.529561\pi\)
\(24\) 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 −0.935666 − 0.352887i \(-0.885200\pi\)
0.773442 + 0.633867i \(0.218533\pi\)
\(32\) −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 0.901319 − 0.433157i \(-0.142600\pi\)
−0.825784 + 0.563987i \(0.809267\pi\)
\(38\) −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 0.705732 − 0.708479i \(-0.250618\pi\)
−0.966427 + 0.256942i \(0.917285\pi\)
\(48\) −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 0.0359535 + 0.999353i \(0.488553\pi\)
−0.883442 + 0.468540i \(0.844780\pi\)
\(54\) 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 −0.455919 − 0.890021i \(-0.650689\pi\)
0.998741 − 0.0501732i \(-0.0159773\pi\)
\(60\) 108.000 − 187.061i 0.232379 − 0.402492i
\(61\) 125.000 + 216.506i 0.262371 + 0.454439i 0.966871 − 0.255264i \(-0.0821624\pi\)
−0.704501 + 0.709703i \(0.748829\pi\)
\(62\) 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 −0.803947 − 0.594701i \(-0.797271\pi\)
0.917000 + 0.398888i \(0.130604\pi\)
\(68\) −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 0.103426 + 0.994637i \(0.467020\pi\)
−0.913094 + 0.407749i \(0.866314\pi\)
\(74\) 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 0.845398 − 0.534136i \(-0.179363\pi\)
−0.885275 + 0.465068i \(0.846030\pi\)
\(80\) −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 0.726222 − 0.687460i \(-0.241274\pi\)
−0.958469 + 0.285197i \(0.907941\pi\)
\(90\) 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.67.1 2
3.2 odd 2 882.4.g.w.361.1 2
7.2 even 3 inner 294.4.e.c.79.1 2
7.3 odd 6 294.4.a.i.1.1 1
7.4 even 3 42.4.a.a.1.1 ✓ 1
7.5 odd 6 294.4.e.b.79.1 2
7.6 odd 2 294.4.e.b.67.1 2
21.2 odd 6 882.4.g.w.667.1 2
21.5 even 6 882.4.g.o.667.1 2
21.11 odd 6 126.4.a.a.1.1 1
21.17 even 6 882.4.a.g.1.1 1
21.20 even 2 882.4.g.o.361.1 2
28.3 even 6 2352.4.a.a.1.1 1
28.11 odd 6 336.4.a.l.1.1 1
35.4 even 6 1050.4.a.g.1.1 1
35.18 odd 12 1050.4.g.a.799.1 2
35.32 odd 12 1050.4.g.a.799.2 2
56.11 odd 6 1344.4.a.a.1.1 1
56.53 even 6 1344.4.a.o.1.1 1
84.11 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 7.4 even 3
126.4.a.a.1.1 1 21.11 odd 6
294.4.a.i.1.1 1 7.3 odd 6
294.4.e.b.67.1 2 7.6 odd 2
294.4.e.b.79.1 2 7.5 odd 6
294.4.e.c.67.1 2 1.1 even 1 trivial
294.4.e.c.79.1 2 7.2 even 3 inner
336.4.a.l.1.1 1 28.11 odd 6
882.4.a.g.1.1 1 21.17 even 6
882.4.g.o.361.1 2 21.20 even 2
882.4.g.o.667.1 2 21.5 even 6
882.4.g.w.361.1 2 3.2 odd 2
882.4.g.w.667.1 2 21.2 odd 6
1008.4.a.b.1.1 1 84.11 even 6
1050.4.a.g.1.1 1 35.4 even 6
1050.4.g.a.799.1 2 35.18 odd 12
1050.4.g.a.799.2 2 35.32 odd 12
1344.4.a.a.1.1 1 56.11 odd 6
1344.4.a.o.1.1 1 56.53 even 6
2352.4.a.a.1.1 1 28.3 even 6