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Properties

Label 966.2.i.c.277.1
Level $966$
Weight $2$
Character 966.277
Analytic conductor $7.714$
Analytic rank $0$
Dimension $2$
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] = [966,2,Mod(277,966)] 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("966.277"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(966, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,1,-1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.71354883526\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 277.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 966.277
Dual form 966.2.i.c.415.1

$q$-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+(-0.500000 - 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} -1.00000 q^{6} +(-0.500000 - 2.59808i) q^{7} +1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +(-0.500000 + 0.866025i) q^{10} +(1.50000 - 2.59808i) q^{11} +(0.500000 + 0.866025i) q^{12} +2.00000 q^{13} +(-2.00000 + 1.73205i) q^{14} -1.00000 q^{15} +(-0.500000 - 0.866025i) q^{16} +(1.00000 - 1.73205i) q^{17} +(-0.500000 + 0.866025i) q^{18} +(1.00000 + 1.73205i) q^{19} +1.00000 q^{20} +(-2.50000 - 0.866025i) q^{21} -3.00000 q^{22} +(-0.500000 - 0.866025i) q^{23} +(0.500000 - 0.866025i) q^{24} +(2.00000 - 3.46410i) q^{25} +(-1.00000 - 1.73205i) q^{26} -1.00000 q^{27} +(2.50000 + 0.866025i) q^{28} -7.00000 q^{29} +(0.500000 + 0.866025i) q^{30} +(-3.50000 + 6.06218i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(-1.50000 - 2.59808i) q^{33} -2.00000 q^{34} +(-2.00000 + 1.73205i) q^{35} +1.00000 q^{36} +(1.00000 + 1.73205i) q^{37} +(1.00000 - 1.73205i) q^{38} +(1.00000 - 1.73205i) q^{39} +(-0.500000 - 0.866025i) q^{40} -2.00000 q^{41} +(0.500000 + 2.59808i) q^{42} -6.00000 q^{43} +(1.50000 + 2.59808i) q^{44} +(-0.500000 + 0.866025i) q^{45} +(-0.500000 + 0.866025i) q^{46} +(-3.00000 - 5.19615i) q^{47} -1.00000 q^{48} +(-6.50000 + 2.59808i) q^{49} -4.00000 q^{50} +(-1.00000 - 1.73205i) q^{51} +(-1.00000 + 1.73205i) q^{52} +(4.50000 - 7.79423i) q^{53} +(0.500000 + 0.866025i) q^{54} -3.00000 q^{55} +(-0.500000 - 2.59808i) q^{56} +2.00000 q^{57} +(3.50000 + 6.06218i) q^{58} +(4.50000 - 7.79423i) q^{59} +(0.500000 - 0.866025i) q^{60} +(-3.00000 - 5.19615i) q^{61} +7.00000 q^{62} +(-2.00000 + 1.73205i) q^{63} +1.00000 q^{64} +(-1.00000 - 1.73205i) q^{65} +(-1.50000 + 2.59808i) q^{66} +(-5.00000 + 8.66025i) q^{67} +(1.00000 + 1.73205i) q^{68} -1.00000 q^{69} +(2.50000 + 0.866025i) q^{70} -8.00000 q^{71} +(-0.500000 - 0.866025i) q^{72} +(5.00000 - 8.66025i) q^{73} +(1.00000 - 1.73205i) q^{74} +(-2.00000 - 3.46410i) q^{75} -2.00000 q^{76} +(-7.50000 - 2.59808i) q^{77} -2.00000 q^{78} +(7.50000 + 12.9904i) q^{79} +(-0.500000 + 0.866025i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(1.00000 + 1.73205i) q^{82} +1.00000 q^{83} +(2.00000 - 1.73205i) q^{84} -2.00000 q^{85} +(3.00000 + 5.19615i) q^{86} +(-3.50000 + 6.06218i) q^{87} +(1.50000 - 2.59808i) q^{88} +1.00000 q^{90} +(-1.00000 - 5.19615i) q^{91} +1.00000 q^{92} +(3.50000 + 6.06218i) q^{93} +(-3.00000 + 5.19615i) q^{94} +(1.00000 - 1.73205i) q^{95} +(0.500000 + 0.866025i) q^{96} +5.00000 q^{97} +(5.50000 + 4.33013i) q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} - q^{4} - q^{5} - 2 q^{6} - q^{7} + 2 q^{8} - q^{9} - q^{10} + 3 q^{11} + q^{12} + 4 q^{13} - 4 q^{14} - 2 q^{15} - q^{16} + 2 q^{17} - q^{18} + 2 q^{19} + 2 q^{20} - 5 q^{21}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/966\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.353553 − 0.612372i
\(3\) 0.500000 − 0.866025i 0.288675 − 0.500000i
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −0.500000 − 0.866025i −0.223607 − 0.387298i 0.732294 − 0.680989i \(-0.238450\pi\)
−0.955901 + 0.293691i \(0.905116\pi\)
\(6\) −1.00000 −0.408248
\(7\) −0.500000 − 2.59808i −0.188982 − 0.981981i
\(8\) 1.00000 0.353553
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) −0.500000 + 0.866025i −0.158114 + 0.273861i
\(11\) 1.50000 − 2.59808i 0.452267 − 0.783349i −0.546259 − 0.837616i \(-0.683949\pi\)
0.998526 + 0.0542666i \(0.0172821\pi\)
\(12\) 0.500000 + 0.866025i 0.144338 + 0.250000i
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −2.00000 + 1.73205i −0.534522 + 0.462910i
\(15\) −1.00000 −0.258199
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 1.00000 − 1.73205i 0.242536 − 0.420084i −0.718900 − 0.695113i \(-0.755354\pi\)
0.961436 + 0.275029i \(0.0886875\pi\)
\(18\) −0.500000 + 0.866025i −0.117851 + 0.204124i
\(19\) 1.00000 + 1.73205i 0.229416 + 0.397360i 0.957635 − 0.287984i \(-0.0929851\pi\)
−0.728219 + 0.685344i \(0.759652\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.50000 − 0.866025i −0.545545 − 0.188982i
\(22\) −3.00000 −0.639602
\(23\) −0.500000 − 0.866025i −0.104257 − 0.180579i
\(24\) 0.500000 − 0.866025i 0.102062 − 0.176777i
\(25\) 2.00000 − 3.46410i 0.400000 − 0.692820i
\(26\) −1.00000 − 1.73205i −0.196116 − 0.339683i
\(27\) −1.00000 −0.192450
\(28\) 2.50000 + 0.866025i 0.472456 + 0.163663i
\(29\) −7.00000 −1.29987 −0.649934 − 0.759991i \(-0.725203\pi\)
−0.649934 + 0.759991i \(0.725203\pi\)
\(30\) 0.500000 + 0.866025i 0.0912871 + 0.158114i
\(31\) −3.50000 + 6.06218i −0.628619 + 1.08880i 0.359211 + 0.933257i \(0.383046\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) −1.50000 − 2.59808i −0.261116 − 0.452267i
\(34\) −2.00000 −0.342997
\(35\) −2.00000 + 1.73205i −0.338062 + 0.292770i
\(36\) 1.00000 0.166667
\(37\) 1.00000 + 1.73205i 0.164399 + 0.284747i 0.936442 − 0.350823i \(-0.114098\pi\)
−0.772043 + 0.635571i \(0.780765\pi\)
\(38\) 1.00000 − 1.73205i 0.162221 − 0.280976i
\(39\) 1.00000 − 1.73205i 0.160128 − 0.277350i
\(40\) −0.500000 − 0.866025i −0.0790569 − 0.136931i
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0.500000 + 2.59808i 0.0771517 + 0.400892i
\(43\) −6.00000 −0.914991 −0.457496 − 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 1.50000 + 2.59808i 0.226134 + 0.391675i
\(45\) −0.500000 + 0.866025i −0.0745356 + 0.129099i
\(46\) −0.500000 + 0.866025i −0.0737210 + 0.127688i
\(47\) −3.00000 − 5.19615i −0.437595 − 0.757937i 0.559908 − 0.828554i \(-0.310836\pi\)
−0.997503 + 0.0706177i \(0.977503\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) −4.00000 −0.565685
\(51\) −1.00000 − 1.73205i −0.140028 − 0.242536i
\(52\) −1.00000 + 1.73205i −0.138675 + 0.240192i
\(53\) 4.50000 − 7.79423i 0.618123 − 1.07062i −0.371706 − 0.928351i \(-0.621227\pi\)
0.989828 − 0.142269i \(-0.0454398\pi\)
\(54\) 0.500000 + 0.866025i 0.0680414 + 0.117851i
\(55\) −3.00000 −0.404520
\(56\) −0.500000 − 2.59808i −0.0668153 − 0.347183i
\(57\) 2.00000 0.264906
\(58\) 3.50000 + 6.06218i 0.459573 + 0.796003i
\(59\) 4.50000 − 7.79423i 0.585850 − 1.01472i −0.408919 − 0.912571i \(-0.634094\pi\)
0.994769 − 0.102151i \(-0.0325726\pi\)
\(60\) 0.500000 − 0.866025i 0.0645497 − 0.111803i
\(61\) −3.00000 − 5.19615i −0.384111 − 0.665299i 0.607535 − 0.794293i \(-0.292159\pi\)
−0.991645 + 0.128994i \(0.958825\pi\)
\(62\) 7.00000 0.889001
\(63\) −2.00000 + 1.73205i −0.251976 + 0.218218i
\(64\) 1.00000 0.125000
\(65\) −1.00000 − 1.73205i −0.124035 − 0.214834i
\(66\) −1.50000 + 2.59808i −0.184637 + 0.319801i
\(67\) −5.00000 + 8.66025i −0.610847 + 1.05802i 0.380251 + 0.924883i \(0.375838\pi\)
−0.991098 + 0.133135i \(0.957496\pi\)
\(68\) 1.00000 + 1.73205i 0.121268 + 0.210042i
\(69\) −1.00000 −0.120386
\(70\) 2.50000 + 0.866025i 0.298807 + 0.103510i
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −0.500000 − 0.866025i −0.0589256 − 0.102062i
\(73\) 5.00000 − 8.66025i 0.585206 − 1.01361i −0.409644 − 0.912245i \(-0.634347\pi\)
0.994850 − 0.101361i \(-0.0323196\pi\)
\(74\) 1.00000 − 1.73205i 0.116248 − 0.201347i
\(75\) −2.00000 − 3.46410i −0.230940 − 0.400000i
\(76\) −2.00000 −0.229416
\(77\) −7.50000 − 2.59808i −0.854704 − 0.296078i
\(78\) −2.00000 −0.226455
\(79\) 7.50000 + 12.9904i 0.843816 + 1.46153i 0.886646 + 0.462450i \(0.153029\pi\)
−0.0428296 + 0.999082i \(0.513637\pi\)
\(80\) −0.500000 + 0.866025i −0.0559017 + 0.0968246i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 1.00000 + 1.73205i 0.110432 + 0.191273i
\(83\) 1.00000 0.109764 0.0548821 − 0.998493i \(-0.482522\pi\)
0.0548821 + 0.998493i \(0.482522\pi\)
\(84\) 2.00000 − 1.73205i 0.218218 − 0.188982i
\(85\) −2.00000 −0.216930
\(86\) 3.00000 + 5.19615i 0.323498 + 0.560316i
\(87\) −3.50000 + 6.06218i −0.375239 + 0.649934i
\(88\) 1.50000 − 2.59808i 0.159901 − 0.276956i
\(89\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 1.00000 0.105409
\(91\) −1.00000 − 5.19615i −0.104828 − 0.544705i
\(92\) 1.00000 0.104257
\(93\) 3.50000 + 6.06218i 0.362933 + 0.628619i
\(94\) −3.00000 + 5.19615i −0.309426 + 0.535942i
\(95\) 1.00000 − 1.73205i 0.102598 − 0.177705i
\(96\) 0.500000 + 0.866025i 0.0510310 + 0.0883883i
\(97\) 5.00000 0.507673 0.253837 − 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 5.50000 + 4.33013i 0.555584 + 0.437409i
\(99\) −3.00000 −0.301511
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 966.2.i.c.277.1 ✓ 2
7.2 even 3 inner 966.2.i.c.415.1 yes 2
7.3 odd 6 6762.2.a.bh.1.1 1
7.4 even 3 6762.2.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.i.c.277.1 ✓ 2 1.1 even 1 trivial
966.2.i.c.415.1 yes 2 7.2 even 3 inner
6762.2.a.bb.1.1 1 7.4 even 3
6762.2.a.bh.1.1 1 7.3 odd 6