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Properties

Label 279.6.a.f.1.2
Level $279$
Weight $6$
Character 279.1
Self dual yes
Analytic conductor $44.747$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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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] = [279,6,Mod(1,279)] 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("279.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(279, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 279 = 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 279.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.7470857278\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 199x^{6} + 256x^{5} + 12633x^{4} - 18583x^{3} - 260319x^{2} + 410640x + 275908 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5\cdot 13 \)
Twist minimal: no (minimal twist has level 31)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-7.89102\) of defining polynomial
Character \(\chi\) \(=\) 279.1

$q$-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)
 
\(f(q)\) \(=\) \(q-8.89102 q^{2} +47.0502 q^{4} +33.8630 q^{5} -134.850 q^{7} -133.811 q^{8} -301.077 q^{10} +191.415 q^{11} +766.219 q^{13} +1198.95 q^{14} -315.888 q^{16} -1357.39 q^{17} -164.832 q^{19} +1593.26 q^{20} -1701.88 q^{22} -3313.68 q^{23} -1978.30 q^{25} -6812.46 q^{26} -6344.71 q^{28} +6675.50 q^{29} +961.000 q^{31} +7090.52 q^{32} +12068.6 q^{34} -4566.42 q^{35} +10334.6 q^{37} +1465.53 q^{38} -4531.25 q^{40} -13434.9 q^{41} +16907.4 q^{43} +9006.12 q^{44} +29462.0 q^{46} +18714.2 q^{47} +1377.48 q^{49} +17589.1 q^{50} +36050.7 q^{52} -11410.8 q^{53} +6481.90 q^{55} +18044.4 q^{56} -59351.9 q^{58} -7475.61 q^{59} -55634.9 q^{61} -8544.27 q^{62} -52933.5 q^{64} +25946.5 q^{65} +26388.2 q^{67} -63865.5 q^{68} +40600.1 q^{70} +14258.2 q^{71} +6721.72 q^{73} -91885.2 q^{74} -7755.39 q^{76} -25812.3 q^{77} -29059.9 q^{79} -10696.9 q^{80} +119450. q^{82} +37568.9 q^{83} -45965.3 q^{85} -150324. q^{86} -25613.5 q^{88} -91193.1 q^{89} -103325. q^{91} -155909. q^{92} -166388. q^{94} -5581.72 q^{95} +95596.4 q^{97} -12247.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 7 q^{2} + 149 q^{4} - 128 q^{5} + 88 q^{7} - 924 q^{8} + 1581 q^{10} - 574 q^{11} - 122 q^{13} + 309 q^{14} + 833 q^{16} - 1932 q^{17} - 1796 q^{19} + 37 q^{20} - 9000 q^{22} + 4136 q^{23} + 3308 q^{25}+ \cdots - 288768 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.89102 −1.57172 −0.785862 − 0.618402i \(-0.787780\pi\)
−0.785862 + 0.618402i \(0.787780\pi\)
\(3\) 0 0
\(4\) 47.0502 1.47032
\(5\) 33.8630 0.605760 0.302880 − 0.953029i \(-0.402052\pi\)
0.302880 + 0.953029i \(0.402052\pi\)
\(6\) 0 0
\(7\) −134.850 −1.04017 −0.520086 − 0.854114i \(-0.674100\pi\)
−0.520086 + 0.854114i \(0.674100\pi\)
\(8\) −133.811 −0.739209
\(9\) 0 0
\(10\) −301.077 −0.952088
\(11\) 191.415 0.476974 0.238487 − 0.971146i \(-0.423349\pi\)
0.238487 + 0.971146i \(0.423349\pi\)
\(12\) 0 0
\(13\) 766.219 1.25746 0.628730 − 0.777623i \(-0.283575\pi\)
0.628730 + 0.777623i \(0.283575\pi\)
\(14\) 1198.95 1.63486
\(15\) 0 0
\(16\) −315.888 −0.308484
\(17\) −1357.39 −1.13915 −0.569577 − 0.821938i \(-0.692893\pi\)
−0.569577 + 0.821938i \(0.692893\pi\)
\(18\) 0 0
\(19\) −164.832 −0.104751 −0.0523755 − 0.998627i \(-0.516679\pi\)
−0.0523755 + 0.998627i \(0.516679\pi\)
\(20\) 1593.26 0.890659
\(21\) 0 0
\(22\) −1701.88 −0.749672
\(23\) −3313.68 −1.30615 −0.653073 − 0.757295i \(-0.726520\pi\)
−0.653073 + 0.757295i \(0.726520\pi\)
\(24\) 0 0
\(25\) −1978.30 −0.633055
\(26\) −6812.46 −1.97638
\(27\) 0 0
\(28\) −6344.71 −1.52938
\(29\) 6675.50 1.47397 0.736985 − 0.675909i \(-0.236249\pi\)
0.736985 + 0.675909i \(0.236249\pi\)
\(30\) 0 0
\(31\) 961.000 0.179605
\(32\) 7090.52 1.22406
\(33\) 0 0
\(34\) 12068.6 1.79044
\(35\) −4566.42 −0.630095
\(36\) 0 0
\(37\) 10334.6 1.24105 0.620526 − 0.784186i \(-0.286919\pi\)
0.620526 + 0.784186i \(0.286919\pi\)
\(38\) 1465.53 0.164640
\(39\) 0 0
\(40\) −4531.25 −0.447783
\(41\) −13434.9 −1.24817 −0.624085 − 0.781356i \(-0.714528\pi\)
−0.624085 + 0.781356i \(0.714528\pi\)
\(42\) 0 0
\(43\) 16907.4 1.39446 0.697228 − 0.716850i \(-0.254417\pi\)
0.697228 + 0.716850i \(0.254417\pi\)
\(44\) 9006.12 0.701303
\(45\) 0 0
\(46\) 29462.0 2.05290
\(47\) 18714.2 1.23574 0.617869 − 0.786281i \(-0.287996\pi\)
0.617869 + 0.786281i \(0.287996\pi\)
\(48\) 0 0
\(49\) 1377.48 0.0819589
\(50\) 17589.1 0.994988
\(51\) 0 0
\(52\) 36050.7 1.84887
\(53\) −11410.8 −0.557990 −0.278995 − 0.960293i \(-0.590001\pi\)
−0.278995 + 0.960293i \(0.590001\pi\)
\(54\) 0 0
\(55\) 6481.90 0.288932
\(56\) 18044.4 0.768905
\(57\) 0 0
\(58\) −59351.9 −2.31667
\(59\) −7475.61 −0.279587 −0.139793 − 0.990181i \(-0.544644\pi\)
−0.139793 + 0.990181i \(0.544644\pi\)
\(60\) 0 0
\(61\) −55634.9 −1.91436 −0.957179 − 0.289498i \(-0.906512\pi\)
−0.957179 + 0.289498i \(0.906512\pi\)
\(62\) −8544.27 −0.282290
\(63\) 0 0
\(64\) −52933.5 −1.61540
\(65\) 25946.5 0.761719
\(66\) 0 0
\(67\) 26388.2 0.718161 0.359081 − 0.933307i \(-0.383090\pi\)
0.359081 + 0.933307i \(0.383090\pi\)
\(68\) −63865.5 −1.67492
\(69\) 0 0
\(70\) 40600.1 0.990335
\(71\) 14258.2 0.335674 0.167837 − 0.985815i \(-0.446322\pi\)
0.167837 + 0.985815i \(0.446322\pi\)
\(72\) 0 0
\(73\) 6721.72 0.147630 0.0738148 − 0.997272i \(-0.476483\pi\)
0.0738148 + 0.997272i \(0.476483\pi\)
\(74\) −91885.2 −1.95059
\(75\) 0 0
\(76\) −7755.39 −0.154017
\(77\) −25812.3 −0.496135
\(78\) 0 0
\(79\) −29059.9 −0.523874 −0.261937 − 0.965085i \(-0.584361\pi\)
−0.261937 + 0.965085i \(0.584361\pi\)
\(80\) −10696.9 −0.186867
\(81\) 0 0
\(82\) 119450. 1.96178
\(83\) 37568.9 0.598595 0.299298 − 0.954160i \(-0.403248\pi\)
0.299298 + 0.954160i \(0.403248\pi\)
\(84\) 0 0
\(85\) −45965.3 −0.690054
\(86\) −150324. −2.19170
\(87\) 0 0
\(88\) −25613.5 −0.352584
\(89\) −91193.1 −1.22036 −0.610179 − 0.792264i \(-0.708902\pi\)
−0.610179 + 0.792264i \(0.708902\pi\)
\(90\) 0 0
\(91\) −103325. −1.30798
\(92\) −155909. −1.92045
\(93\) 0 0
\(94\) −166388. −1.94224
\(95\) −5581.72 −0.0634540
\(96\) 0 0
\(97\) 95596.4 1.03160 0.515801 − 0.856708i \(-0.327494\pi\)
0.515801 + 0.856708i \(0.327494\pi\)
\(98\) −12247.2 −0.128817
\(99\) 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 279.6.a.f.1.2 8
3.2 odd 2 31.6.a.b.1.7 ✓ 8
12.11 even 2 496.6.a.h.1.3 8
15.14 odd 2 775.6.a.b.1.2 8
93.92 even 2 961.6.a.c.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
31.6.a.b.1.7 ✓ 8 3.2 odd 2
279.6.a.f.1.2 8 1.1 even 1 trivial
496.6.a.h.1.3 8 12.11 even 2
775.6.a.b.1.2 8 15.14 odd 2
961.6.a.c.1.7 8 93.92 even 2