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Properties

Label 578.6.a.h.1.3
Level $578$
Weight $6$
Character 578.1
Self dual yes
Analytic conductor $92.702$
Analytic rank $1$
Dimension $4$
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] = [578,6,Mod(1,578)] 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("578.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(578, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 578 = 2 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 578.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,16,-22,64,-110] 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: yes
Analytic conductor: \(92.7018478519\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 533x^{2} - 726x + 27729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-8.83700\) of defining polynomial
Character \(\chi\) \(=\) 578.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} +3.83700 q^{3} +16.0000 q^{4} +23.1865 q^{5} +15.3480 q^{6} +159.302 q^{7} +64.0000 q^{8} -228.277 q^{9} +92.7460 q^{10} -10.7249 q^{11} +61.3919 q^{12} -1147.03 q^{13} +637.207 q^{14} +88.9665 q^{15} +256.000 q^{16} -913.110 q^{18} -482.700 q^{19} +370.984 q^{20} +611.241 q^{21} -42.8997 q^{22} -3382.61 q^{23} +245.568 q^{24} -2587.39 q^{25} -4588.11 q^{26} -1808.29 q^{27} +2548.83 q^{28} +4304.02 q^{29} +355.866 q^{30} -7162.23 q^{31} +1024.00 q^{32} -41.1515 q^{33} +3693.65 q^{35} -3652.44 q^{36} -5717.32 q^{37} -1930.80 q^{38} -4401.14 q^{39} +1483.94 q^{40} -13417.8 q^{41} +2444.96 q^{42} +1760.00 q^{43} -171.599 q^{44} -5292.96 q^{45} -13530.4 q^{46} -2616.00 q^{47} +982.271 q^{48} +8570.08 q^{49} -10349.5 q^{50} -18352.5 q^{52} -17326.5 q^{53} -7233.16 q^{54} -248.674 q^{55} +10195.3 q^{56} -1852.12 q^{57} +17216.1 q^{58} +35686.5 q^{59} +1423.46 q^{60} +44856.5 q^{61} -28648.9 q^{62} -36365.0 q^{63} +4096.00 q^{64} -26595.6 q^{65} -164.606 q^{66} +55308.9 q^{67} -12979.0 q^{69} +14774.6 q^{70} +5819.00 q^{71} -14609.8 q^{72} +5413.20 q^{73} -22869.3 q^{74} -9927.79 q^{75} -7723.20 q^{76} -1708.50 q^{77} -17604.6 q^{78} +7016.76 q^{79} +5935.74 q^{80} +48533.0 q^{81} -53671.1 q^{82} -8886.27 q^{83} +9779.85 q^{84} +7040.01 q^{86} +16514.5 q^{87} -686.395 q^{88} -46280.7 q^{89} -21171.8 q^{90} -182724. q^{91} -54121.7 q^{92} -27481.5 q^{93} -10464.0 q^{94} -11192.1 q^{95} +3929.08 q^{96} +41530.9 q^{97} +34280.3 q^{98} +2448.26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{2} - 22 q^{3} + 64 q^{4} - 110 q^{5} - 88 q^{6} - 42 q^{7} + 256 q^{8} + 218 q^{9} - 440 q^{10} - 148 q^{11} - 352 q^{12} + 10 q^{13} - 168 q^{14} + 796 q^{15} + 1024 q^{16} + 872 q^{18} + 706 q^{19}+ \cdots - 361042 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) 3.83700 0.246143 0.123072 − 0.992398i \(-0.460725\pi\)
0.123072 + 0.992398i \(0.460725\pi\)
\(4\) 16.0000 0.500000
\(5\) 23.1865 0.414773 0.207386 − 0.978259i \(-0.433504\pi\)
0.207386 + 0.978259i \(0.433504\pi\)
\(6\) 15.3480 0.174050
\(7\) 159.302 1.22878 0.614392 − 0.789001i \(-0.289401\pi\)
0.614392 + 0.789001i \(0.289401\pi\)
\(8\) 64.0000 0.353553
\(9\) −228.277 −0.939413
\(10\) 92.7460 0.293289
\(11\) −10.7249 −0.0267247 −0.0133623 − 0.999911i \(-0.504253\pi\)
−0.0133623 + 0.999911i \(0.504253\pi\)
\(12\) 61.3919 0.123072
\(13\) −1147.03 −1.88242 −0.941209 − 0.337826i \(-0.890309\pi\)
−0.941209 + 0.337826i \(0.890309\pi\)
\(14\) 637.207 0.868882
\(15\) 88.9665 0.102094
\(16\) 256.000 0.250000
\(17\) 0 0
\(18\) −913.110 −0.664266
\(19\) −482.700 −0.306756 −0.153378 − 0.988168i \(-0.549015\pi\)
−0.153378 + 0.988168i \(0.549015\pi\)
\(20\) 370.984 0.207386
\(21\) 611.241 0.302457
\(22\) −42.8997 −0.0188972
\(23\) −3382.61 −1.33331 −0.666656 − 0.745365i \(-0.732275\pi\)
−0.666656 + 0.745365i \(0.732275\pi\)
\(24\) 245.568 0.0870248
\(25\) −2587.39 −0.827964
\(26\) −4588.11 −1.33107
\(27\) −1808.29 −0.477374
\(28\) 2548.83 0.614392
\(29\) 4304.02 0.950340 0.475170 − 0.879894i \(-0.342387\pi\)
0.475170 + 0.879894i \(0.342387\pi\)
\(30\) 355.866 0.0721911
\(31\) −7162.23 −1.33858 −0.669290 − 0.743001i \(-0.733402\pi\)
−0.669290 + 0.743001i \(0.733402\pi\)
\(32\) 1024.00 0.176777
\(33\) −41.1515 −0.00657811
\(34\) 0 0
\(35\) 3693.65 0.509666
\(36\) −3652.44 −0.469707
\(37\) −5717.32 −0.686575 −0.343287 − 0.939230i \(-0.611540\pi\)
−0.343287 + 0.939230i \(0.611540\pi\)
\(38\) −1930.80 −0.216909
\(39\) −4401.14 −0.463345
\(40\) 1483.94 0.146644
\(41\) −13417.8 −1.24658 −0.623290 − 0.781990i \(-0.714205\pi\)
−0.623290 + 0.781990i \(0.714205\pi\)
\(42\) 2444.96 0.213870
\(43\) 1760.00 0.145158 0.0725792 − 0.997363i \(-0.476877\pi\)
0.0725792 + 0.997363i \(0.476877\pi\)
\(44\) −171.599 −0.0133623
\(45\) −5292.96 −0.389643
\(46\) −13530.4 −0.942794
\(47\) −2616.00 −0.172740 −0.0863702 − 0.996263i \(-0.527527\pi\)
−0.0863702 + 0.996263i \(0.527527\pi\)
\(48\) 982.271 0.0615359
\(49\) 8570.08 0.509911
\(50\) −10349.5 −0.585459
\(51\) 0 0
\(52\) −18352.5 −0.941209
\(53\) −17326.5 −0.847271 −0.423635 − 0.905833i \(-0.639246\pi\)
−0.423635 + 0.905833i \(0.639246\pi\)
\(54\) −7233.16 −0.337554
\(55\) −248.674 −0.0110847
\(56\) 10195.3 0.434441
\(57\) −1852.12 −0.0755060
\(58\) 17216.1 0.671992
\(59\) 35686.5 1.33467 0.667335 − 0.744758i \(-0.267435\pi\)
0.667335 + 0.744758i \(0.267435\pi\)
\(60\) 1423.46 0.0510468
\(61\) 44856.5 1.54348 0.771739 − 0.635939i \(-0.219387\pi\)
0.771739 + 0.635939i \(0.219387\pi\)
\(62\) −28648.9 −0.946519
\(63\) −36365.0 −1.15434
\(64\) 4096.00 0.125000
\(65\) −26595.6 −0.780775
\(66\) −164.606 −0.00465142
\(67\) 55308.9 1.50525 0.752623 − 0.658451i \(-0.228788\pi\)
0.752623 + 0.658451i \(0.228788\pi\)
\(68\) 0 0
\(69\) −12979.0 −0.328186
\(70\) 14774.6 0.360389
\(71\) 5819.00 0.136994 0.0684972 − 0.997651i \(-0.478180\pi\)
0.0684972 + 0.997651i \(0.478180\pi\)
\(72\) −14609.8 −0.332133
\(73\) 5413.20 0.118890 0.0594452 − 0.998232i \(-0.481067\pi\)
0.0594452 + 0.998232i \(0.481067\pi\)
\(74\) −22869.3 −0.485482
\(75\) −9927.79 −0.203798
\(76\) −7723.20 −0.153378
\(77\) −1708.50 −0.0328389
\(78\) −17604.6 −0.327634
\(79\) 7016.76 0.126494 0.0632468 − 0.997998i \(-0.479854\pi\)
0.0632468 + 0.997998i \(0.479854\pi\)
\(80\) 5935.74 0.103693
\(81\) 48533.0 0.821911
\(82\) −53671.1 −0.881466
\(83\) −8886.27 −0.141587 −0.0707936 − 0.997491i \(-0.522553\pi\)
−0.0707936 + 0.997491i \(0.522553\pi\)
\(84\) 9779.85 0.151229
\(85\) 0 0
\(86\) 7040.01 0.102642
\(87\) 16514.5 0.233920
\(88\) −686.395 −0.00944860
\(89\) −46280.7 −0.619335 −0.309667 − 0.950845i \(-0.600218\pi\)
−0.309667 + 0.950845i \(0.600218\pi\)
\(90\) −21171.8 −0.275519
\(91\) −182724. −2.31309
\(92\) −54121.7 −0.666656
\(93\) −27481.5 −0.329483
\(94\) −10464.0 −0.122146
\(95\) −11192.1 −0.127234
\(96\) 3929.08 0.0435124
\(97\) 41530.9 0.448170 0.224085 − 0.974570i \(-0.428061\pi\)
0.224085 + 0.974570i \(0.428061\pi\)
\(98\) 34280.3 0.360562
\(99\) 2448.26 0.0251055
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 578.6.a.h.1.3 ✓ 4
17.16 even 2 578.6.a.j.1.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
578.6.a.h.1.3 ✓ 4 1.1 even 1 trivial
578.6.a.j.1.2 yes 4 17.16 even 2