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Properties

Label 294.2.f.b.215.1
Level $294$
Weight $2$
Character 294.215
Analytic conductor $2.348$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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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] = [294,2,Mod(215,294)] 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("294.215"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 294.f (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.34760181943\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 215.1
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 294.215
Dual form 294.2.f.b.227.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{2} +(-1.67303 - 0.448288i) q^{3} +(0.500000 + 0.866025i) q^{4} +(1.22474 - 2.12132i) q^{5} +(1.22474 + 1.22474i) q^{6} -1.00000i q^{8} +(2.59808 + 1.50000i) q^{9} +(-2.12132 + 1.22474i) q^{10} +(-0.448288 - 1.67303i) q^{12} -2.44949i q^{13} +(-3.00000 + 3.00000i) q^{15} +(-0.500000 + 0.866025i) q^{16} +(-2.44949 - 4.24264i) q^{17} +(-1.50000 - 2.59808i) q^{18} +(-2.12132 - 1.22474i) q^{19} +2.44949 q^{20} +(-5.19615 - 3.00000i) q^{23} +(-0.448288 + 1.67303i) q^{24} +(-0.500000 - 0.866025i) q^{25} +(-1.22474 + 2.12132i) q^{26} +(-3.67423 - 3.67423i) q^{27} -6.00000i q^{29} +(4.09808 - 1.09808i) q^{30} +(0.866025 - 0.500000i) q^{32} +4.89898i q^{34} +3.00000i q^{36} +(1.00000 - 1.73205i) q^{37} +(1.22474 + 2.12132i) q^{38} +(-1.09808 + 4.09808i) q^{39} +(-2.12132 - 1.22474i) q^{40} -4.89898 q^{41} +4.00000 q^{43} +(6.36396 - 3.67423i) q^{45} +(3.00000 + 5.19615i) q^{46} +(-2.44949 + 4.24264i) q^{47} +(1.22474 - 1.22474i) q^{48} +1.00000i q^{50} +(2.19615 + 8.19615i) q^{51} +(2.12132 - 1.22474i) q^{52} +(5.19615 - 3.00000i) q^{53} +(1.34486 + 5.01910i) q^{54} +(3.00000 + 3.00000i) q^{57} +(-3.00000 + 5.19615i) q^{58} +(6.12372 + 10.6066i) q^{59} +(-4.09808 - 1.09808i) q^{60} +(10.6066 + 6.12372i) q^{61} -1.00000 q^{64} +(-5.19615 - 3.00000i) q^{65} +(-4.00000 - 6.92820i) q^{67} +(2.44949 - 4.24264i) q^{68} +(7.34847 + 7.34847i) q^{69} +(1.50000 - 2.59808i) q^{72} +(8.48528 - 4.89898i) q^{73} +(-1.73205 + 1.00000i) q^{74} +(0.448288 + 1.67303i) q^{75} -2.44949i q^{76} +(3.00000 - 3.00000i) q^{78} +(5.00000 - 8.66025i) q^{79} +(1.22474 + 2.12132i) q^{80} +(4.50000 + 7.79423i) q^{81} +(4.24264 + 2.44949i) q^{82} +2.44949 q^{83} -12.0000 q^{85} +(-3.46410 - 2.00000i) q^{86} +(-2.68973 + 10.0382i) q^{87} -7.34847 q^{90} -6.00000i q^{92} +(4.24264 - 2.44949i) q^{94} +(-5.19615 + 3.00000i) q^{95} +(-1.67303 + 0.448288i) q^{96} +4.89898i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} - 24 q^{15} - 4 q^{16} - 12 q^{18} - 4 q^{25} + 12 q^{30} + 8 q^{37} + 12 q^{39} + 32 q^{43} + 24 q^{46} - 24 q^{51} + 24 q^{57} - 24 q^{58} - 12 q^{60} - 8 q^{64} - 32 q^{67} + 12 q^{72}+ \cdots - 96 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 − 0.500000i −0.612372 − 0.353553i
\(3\) −1.67303 − 0.448288i −0.965926 − 0.258819i
\(4\) 0.500000 + 0.866025i 0.250000 + 0.433013i
\(5\) 1.22474 − 2.12132i 0.547723 − 0.948683i −0.450708 − 0.892672i \(-0.648828\pi\)
0.998430 − 0.0560116i \(-0.0178384\pi\)
\(6\) 1.22474 + 1.22474i 0.500000 + 0.500000i
\(7\) 0 0
\(8\) − 1.00000i − 0.353553i
\(9\) 2.59808 + 1.50000i 0.866025 + 0.500000i
\(10\) −2.12132 + 1.22474i −0.670820 + 0.387298i
\(11\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) −0.448288 − 1.67303i −0.129410 − 0.482963i
\(13\) − 2.44949i − 0.679366i −0.940540 − 0.339683i \(-0.889680\pi\)
0.940540 − 0.339683i \(-0.110320\pi\)
\(14\) 0 0
\(15\) −3.00000 + 3.00000i −0.774597 + 0.774597i
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −2.44949 − 4.24264i −0.594089 − 1.02899i −0.993675 − 0.112296i \(-0.964180\pi\)
0.399586 − 0.916696i \(-0.369154\pi\)
\(18\) −1.50000 − 2.59808i −0.353553 − 0.612372i
\(19\) −2.12132 − 1.22474i −0.486664 − 0.280976i 0.236525 − 0.971625i \(-0.423991\pi\)
−0.723190 + 0.690650i \(0.757325\pi\)
\(20\) 2.44949 0.547723
\(21\) 0 0
\(22\) 0 0
\(23\) −5.19615 − 3.00000i −1.08347 − 0.625543i −0.151642 − 0.988436i \(-0.548456\pi\)
−0.931831 + 0.362892i \(0.881789\pi\)
\(24\) −0.448288 + 1.67303i −0.0915064 + 0.341506i
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) −1.22474 + 2.12132i −0.240192 + 0.416025i
\(27\) −3.67423 − 3.67423i −0.707107 − 0.707107i
\(28\) 0 0
\(29\) − 6.00000i − 1.11417i −0.830455 − 0.557086i \(-0.811919\pi\)
0.830455 − 0.557086i \(-0.188081\pi\)
\(30\) 4.09808 − 1.09808i 0.748203 − 0.200480i
\(31\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 0.866025 − 0.500000i 0.153093 − 0.0883883i
\(33\) 0 0
\(34\) 4.89898i 0.840168i
\(35\) 0 0
\(36\) 3.00000i 0.500000i
\(37\) 1.00000 − 1.73205i 0.164399 − 0.284747i −0.772043 − 0.635571i \(-0.780765\pi\)
0.936442 + 0.350823i \(0.114098\pi\)
\(38\) 1.22474 + 2.12132i 0.198680 + 0.344124i
\(39\) −1.09808 + 4.09808i −0.175833 + 0.656217i
\(40\) −2.12132 − 1.22474i −0.335410 − 0.193649i
\(41\) −4.89898 −0.765092 −0.382546 − 0.923936i \(-0.624953\pi\)
−0.382546 + 0.923936i \(0.624953\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 6.36396 − 3.67423i 0.948683 − 0.547723i
\(46\) 3.00000 + 5.19615i 0.442326 + 0.766131i
\(47\) −2.44949 + 4.24264i −0.357295 + 0.618853i −0.987508 − 0.157569i \(-0.949634\pi\)
0.630213 + 0.776422i \(0.282968\pi\)
\(48\) 1.22474 − 1.22474i 0.176777 − 0.176777i
\(49\) 0 0
\(50\) 1.00000i 0.141421i
\(51\) 2.19615 + 8.19615i 0.307523 + 1.14769i
\(52\) 2.12132 − 1.22474i 0.294174 − 0.169842i
\(53\) 5.19615 − 3.00000i 0.713746 − 0.412082i −0.0987002 − 0.995117i \(-0.531468\pi\)
0.812447 + 0.583036i \(0.198135\pi\)
\(54\) 1.34486 + 5.01910i 0.183013 + 0.683013i
\(55\) 0 0
\(56\) 0 0
\(57\) 3.00000 + 3.00000i 0.397360 + 0.397360i
\(58\) −3.00000 + 5.19615i −0.393919 + 0.682288i
\(59\) 6.12372 + 10.6066i 0.797241 + 1.38086i 0.921406 + 0.388600i \(0.127041\pi\)
−0.124165 + 0.992262i \(0.539625\pi\)
\(60\) −4.09808 − 1.09808i −0.529059 − 0.141761i
\(61\) 10.6066 + 6.12372i 1.35804 + 0.784063i 0.989359 − 0.145495i \(-0.0464774\pi\)
0.368677 + 0.929557i \(0.379811\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −5.19615 − 3.00000i −0.644503 − 0.372104i
\(66\) 0 0
\(67\) −4.00000 − 6.92820i −0.488678 − 0.846415i 0.511237 − 0.859440i \(-0.329187\pi\)
−0.999915 + 0.0130248i \(0.995854\pi\)
\(68\) 2.44949 − 4.24264i 0.297044 − 0.514496i
\(69\) 7.34847 + 7.34847i 0.884652 + 0.884652i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 1.50000 − 2.59808i 0.176777 − 0.306186i
\(73\) 8.48528 − 4.89898i 0.993127 − 0.573382i 0.0869195 − 0.996215i \(-0.472298\pi\)
0.906208 + 0.422833i \(0.138964\pi\)
\(74\) −1.73205 + 1.00000i −0.201347 + 0.116248i
\(75\) 0.448288 + 1.67303i 0.0517638 + 0.193185i
\(76\) − 2.44949i − 0.280976i
\(77\) 0 0
\(78\) 3.00000 − 3.00000i 0.339683 − 0.339683i
\(79\) 5.00000 − 8.66025i 0.562544 − 0.974355i −0.434730 − 0.900561i \(-0.643156\pi\)
0.997274 − 0.0737937i \(-0.0235106\pi\)
\(80\) 1.22474 + 2.12132i 0.136931 + 0.237171i
\(81\) 4.50000 + 7.79423i 0.500000 + 0.866025i
\(82\) 4.24264 + 2.44949i 0.468521 + 0.270501i
\(83\) 2.44949 0.268866 0.134433 − 0.990923i \(-0.457079\pi\)
0.134433 + 0.990923i \(0.457079\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) −3.46410 − 2.00000i −0.373544 − 0.215666i
\(87\) −2.68973 + 10.0382i −0.288369 + 1.07621i
\(88\) 0 0
\(89\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) −7.34847 −0.774597
\(91\) 0 0
\(92\) − 6.00000i − 0.625543i
\(93\) 0 0
\(94\) 4.24264 − 2.44949i 0.437595 − 0.252646i
\(95\) −5.19615 + 3.00000i −0.533114 + 0.307794i
\(96\) −1.67303 + 0.448288i −0.170753 + 0.0457532i
\(97\) 4.89898i 0.497416i 0.968579 + 0.248708i \(0.0800060\pi\)
−0.968579 + 0.248708i \(0.919994\pi\)
\(98\) 0 0
\(99\) 0 0
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 294.2.f.b.215.1 8
3.2 odd 2 inner 294.2.f.b.215.3 8
7.2 even 3 42.2.d.a.41.4 yes 4
7.3 odd 6 inner 294.2.f.b.227.3 8
7.4 even 3 inner 294.2.f.b.227.4 8
7.5 odd 6 42.2.d.a.41.3 yes 4
7.6 odd 2 inner 294.2.f.b.215.2 8
21.2 odd 6 42.2.d.a.41.1 ✓ 4
21.5 even 6 42.2.d.a.41.2 yes 4
21.11 odd 6 inner 294.2.f.b.227.2 8
21.17 even 6 inner 294.2.f.b.227.1 8
21.20 even 2 inner 294.2.f.b.215.4 8
28.19 even 6 336.2.k.b.209.3 4
28.23 odd 6 336.2.k.b.209.2 4
35.2 odd 12 1050.2.d.b.1049.1 4
35.9 even 6 1050.2.b.b.251.1 4
35.12 even 12 1050.2.d.b.1049.4 4
35.19 odd 6 1050.2.b.b.251.2 4
35.23 odd 12 1050.2.d.e.1049.4 4
35.33 even 12 1050.2.d.e.1049.1 4
56.5 odd 6 1344.2.k.c.1217.3 4
56.19 even 6 1344.2.k.d.1217.2 4
56.37 even 6 1344.2.k.c.1217.2 4
56.51 odd 6 1344.2.k.d.1217.3 4
63.2 odd 6 1134.2.m.g.377.1 8
63.5 even 6 1134.2.m.g.755.4 8
63.16 even 3 1134.2.m.g.377.4 8
63.23 odd 6 1134.2.m.g.755.3 8
63.40 odd 6 1134.2.m.g.755.1 8
63.47 even 6 1134.2.m.g.377.2 8
63.58 even 3 1134.2.m.g.755.2 8
63.61 odd 6 1134.2.m.g.377.3 8
84.23 even 6 336.2.k.b.209.4 4
84.47 odd 6 336.2.k.b.209.1 4
105.2 even 12 1050.2.d.e.1049.2 4
105.23 even 12 1050.2.d.b.1049.3 4
105.44 odd 6 1050.2.b.b.251.4 4
105.47 odd 12 1050.2.d.e.1049.3 4
105.68 odd 12 1050.2.d.b.1049.2 4
105.89 even 6 1050.2.b.b.251.3 4
168.5 even 6 1344.2.k.c.1217.1 4
168.107 even 6 1344.2.k.d.1217.1 4
168.131 odd 6 1344.2.k.d.1217.4 4
168.149 odd 6 1344.2.k.c.1217.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.d.a.41.1 ✓ 4 21.2 odd 6
42.2.d.a.41.2 yes 4 21.5 even 6
42.2.d.a.41.3 yes 4 7.5 odd 6
42.2.d.a.41.4 yes 4 7.2 even 3
294.2.f.b.215.1 8 1.1 even 1 trivial
294.2.f.b.215.2 8 7.6 odd 2 inner
294.2.f.b.215.3 8 3.2 odd 2 inner
294.2.f.b.215.4 8 21.20 even 2 inner
294.2.f.b.227.1 8 21.17 even 6 inner
294.2.f.b.227.2 8 21.11 odd 6 inner
294.2.f.b.227.3 8 7.3 odd 6 inner
294.2.f.b.227.4 8 7.4 even 3 inner
336.2.k.b.209.1 4 84.47 odd 6
336.2.k.b.209.2 4 28.23 odd 6
336.2.k.b.209.3 4 28.19 even 6
336.2.k.b.209.4 4 84.23 even 6
1050.2.b.b.251.1 4 35.9 even 6
1050.2.b.b.251.2 4 35.19 odd 6
1050.2.b.b.251.3 4 105.89 even 6
1050.2.b.b.251.4 4 105.44 odd 6
1050.2.d.b.1049.1 4 35.2 odd 12
1050.2.d.b.1049.2 4 105.68 odd 12
1050.2.d.b.1049.3 4 105.23 even 12
1050.2.d.b.1049.4 4 35.12 even 12
1050.2.d.e.1049.1 4 35.33 even 12
1050.2.d.e.1049.2 4 105.2 even 12
1050.2.d.e.1049.3 4 105.47 odd 12
1050.2.d.e.1049.4 4 35.23 odd 12
1134.2.m.g.377.1 8 63.2 odd 6
1134.2.m.g.377.2 8 63.47 even 6
1134.2.m.g.377.3 8 63.61 odd 6
1134.2.m.g.377.4 8 63.16 even 3
1134.2.m.g.755.1 8 63.40 odd 6
1134.2.m.g.755.2 8 63.58 even 3
1134.2.m.g.755.3 8 63.23 odd 6
1134.2.m.g.755.4 8 63.5 even 6
1344.2.k.c.1217.1 4 168.5 even 6
1344.2.k.c.1217.2 4 56.37 even 6
1344.2.k.c.1217.3 4 56.5 odd 6
1344.2.k.c.1217.4 4 168.149 odd 6
1344.2.k.d.1217.1 4 168.107 even 6
1344.2.k.d.1217.2 4 56.19 even 6
1344.2.k.d.1217.3 4 56.51 odd 6
1344.2.k.d.1217.4 4 168.131 odd 6