↑

Properties

Label 578.6.a.h.1.4
Level $578$
Weight $6$
Character 578.1
Self dual yes
Analytic conductor $92.702$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.4
Root \(-19.6131\) 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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} +14.6131 q^{3} +16.0000 q^{4} -93.3752 q^{5} +58.4523 q^{6} -50.0680 q^{7} +64.0000 q^{8} -29.4577 q^{9} -373.501 q^{10} +424.943 q^{11} +233.809 q^{12} +1183.75 q^{13} -200.272 q^{14} -1364.50 q^{15} +256.000 q^{16} -117.831 q^{18} -1722.85 q^{19} -1494.00 q^{20} -731.648 q^{21} +1699.77 q^{22} -3252.36 q^{23} +935.238 q^{24} +5593.92 q^{25} +4734.99 q^{26} -3981.45 q^{27} -801.088 q^{28} -5627.85 q^{29} -5458.00 q^{30} +4533.75 q^{31} +1024.00 q^{32} +6209.73 q^{33} +4675.11 q^{35} -471.323 q^{36} -2689.30 q^{37} -6891.42 q^{38} +17298.2 q^{39} -5976.01 q^{40} -1487.45 q^{41} -2926.59 q^{42} -5398.05 q^{43} +6799.09 q^{44} +2750.62 q^{45} -13009.4 q^{46} +7222.41 q^{47} +3740.95 q^{48} -14300.2 q^{49} +22375.7 q^{50} +18940.0 q^{52} -30980.1 q^{53} -15925.8 q^{54} -39679.1 q^{55} -3204.35 q^{56} -25176.2 q^{57} -22511.4 q^{58} +30559.3 q^{59} -21832.0 q^{60} -14510.0 q^{61} +18135.0 q^{62} +1474.89 q^{63} +4096.00 q^{64} -110533. q^{65} +24838.9 q^{66} -47607.9 q^{67} -47527.0 q^{69} +18700.4 q^{70} +26507.1 q^{71} -1885.29 q^{72} -72954.3 q^{73} -10757.2 q^{74} +81744.5 q^{75} -27565.7 q^{76} -21276.0 q^{77} +69192.9 q^{78} -57146.0 q^{79} -23904.0 q^{80} -51023.0 q^{81} -5949.79 q^{82} +23280.8 q^{83} -11706.4 q^{84} -21592.2 q^{86} -82240.2 q^{87} +27196.3 q^{88} +71847.6 q^{89} +11002.5 q^{90} -59267.9 q^{91} -52037.7 q^{92} +66252.1 q^{93} +28889.6 q^{94} +160872. q^{95} +14963.8 q^{96} -54239.8 q^{97} -57200.8 q^{98} -12517.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{2} - 22 q^{3} + 64 q^{4} - 110 q^{5} - 88 q^{6} - 42 q^{7} + 256 q^{8} + 218 q^{9} - 440 q^{10} - 148 q^{11} - 352 q^{12} + 10 q^{13} - 168 q^{14} + 796 q^{15} + 1024 q^{16} + 872 q^{18} + 706 q^{19}+ \cdots - 361042 q^{99}+O(q^{100}) \) 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
\(3\) 14.6131 0.937430 0.468715 − 0.883349i \(-0.344717\pi\)
0.468715 + 0.883349i \(0.344717\pi\)
\(4\) 16.0000 0.500000
\(5\) −93.3752 −1.67035 −0.835173 − 0.549988i \(-0.814632\pi\)
−0.835173 + 0.549988i \(0.814632\pi\)
\(6\) 58.4523 0.662863
\(7\) −50.0680 −0.386203 −0.193101 − 0.981179i \(-0.561855\pi\)
−0.193101 + 0.981179i \(0.561855\pi\)
\(8\) 64.0000 0.353553
\(9\) −29.4577 −0.121225
\(10\) −373.501 −1.18111
\(11\) 424.943 1.05888 0.529442 − 0.848346i \(-0.322401\pi\)
0.529442 + 0.848346i \(0.322401\pi\)
\(12\) 233.809 0.468715
\(13\) 1183.75 1.94268 0.971339 − 0.237697i \(-0.0763925\pi\)
0.971339 + 0.237697i \(0.0763925\pi\)
\(14\) −200.272 −0.273086
\(15\) −1364.50 −1.56583
\(16\) 256.000 0.250000
\(17\) 0 0
\(18\) −117.831 −0.0857191
\(19\) −1722.85 −1.09488 −0.547438 − 0.836847i \(-0.684397\pi\)
−0.547438 + 0.836847i \(0.684397\pi\)
\(20\) −1494.00 −0.835173
\(21\) −731.648 −0.362038
\(22\) 1699.77 0.748745
\(23\) −3252.36 −1.28197 −0.640986 − 0.767552i \(-0.721474\pi\)
−0.640986 + 0.767552i \(0.721474\pi\)
\(24\) 935.238 0.331432
\(25\) 5593.92 1.79005
\(26\) 4734.99 1.37368
\(27\) −3981.45 −1.05107
\(28\) −801.088 −0.193101
\(29\) −5627.85 −1.24265 −0.621323 − 0.783555i \(-0.713404\pi\)
−0.621323 + 0.783555i \(0.713404\pi\)
\(30\) −5458.00 −1.10721
\(31\) 4533.75 0.847332 0.423666 − 0.905818i \(-0.360743\pi\)
0.423666 + 0.905818i \(0.360743\pi\)
\(32\) 1024.00 0.176777
\(33\) 6209.73 0.992630
\(34\) 0 0
\(35\) 4675.11 0.645092
\(36\) −471.323 −0.0606126
\(37\) −2689.30 −0.322950 −0.161475 − 0.986877i \(-0.551625\pi\)
−0.161475 + 0.986877i \(0.551625\pi\)
\(38\) −6891.42 −0.774194
\(39\) 17298.2 1.82113
\(40\) −5976.01 −0.590556
\(41\) −1487.45 −0.138192 −0.0690959 − 0.997610i \(-0.522011\pi\)
−0.0690959 + 0.997610i \(0.522011\pi\)
\(42\) −2926.59 −0.255999
\(43\) −5398.05 −0.445211 −0.222605 − 0.974909i \(-0.571456\pi\)
−0.222605 + 0.974909i \(0.571456\pi\)
\(44\) 6799.09 0.529442
\(45\) 2750.62 0.202488
\(46\) −13009.4 −0.906491
\(47\) 7222.41 0.476911 0.238456 − 0.971153i \(-0.423359\pi\)
0.238456 + 0.971153i \(0.423359\pi\)
\(48\) 3740.95 0.234357
\(49\) −14300.2 −0.850848
\(50\) 22375.7 1.26576
\(51\) 0 0
\(52\) 18940.0 0.971339
\(53\) −30980.1 −1.51493 −0.757467 − 0.652874i \(-0.773563\pi\)
−0.757467 + 0.652874i \(0.773563\pi\)
\(54\) −15925.8 −0.743219
\(55\) −39679.1 −1.76870
\(56\) −3204.35 −0.136543
\(57\) −25176.2 −1.02637
\(58\) −22511.4 −0.878683
\(59\) 30559.3 1.14292 0.571458 − 0.820632i \(-0.306378\pi\)
0.571458 + 0.820632i \(0.306378\pi\)
\(60\) −21832.0 −0.782916
\(61\) −14510.0 −0.499279 −0.249639 − 0.968339i \(-0.580312\pi\)
−0.249639 + 0.968339i \(0.580312\pi\)
\(62\) 18135.0 0.599154
\(63\) 1474.89 0.0468175
\(64\) 4096.00 0.125000
\(65\) −110533. −3.24495
\(66\) 24838.9 0.701896
\(67\) −47607.9 −1.29566 −0.647832 − 0.761783i \(-0.724324\pi\)
−0.647832 + 0.761783i \(0.724324\pi\)
\(68\) 0 0
\(69\) −47527.0 −1.20176
\(70\) 18700.4 0.456149
\(71\) 26507.1 0.624045 0.312023 − 0.950075i \(-0.398994\pi\)
0.312023 + 0.950075i \(0.398994\pi\)
\(72\) −1885.29 −0.0428596
\(73\) −72954.3 −1.60230 −0.801150 − 0.598464i \(-0.795778\pi\)
−0.801150 + 0.598464i \(0.795778\pi\)
\(74\) −10757.2 −0.228360
\(75\) 81744.5 1.67805
\(76\) −27565.7 −0.547438
\(77\) −21276.0 −0.408944
\(78\) 69192.9 1.28773
\(79\) −57146.0 −1.03019 −0.515096 − 0.857132i \(-0.672244\pi\)
−0.515096 + 0.857132i \(0.672244\pi\)
\(80\) −23904.0 −0.417586
\(81\) −51023.0 −0.864079
\(82\) −5949.79 −0.0977163
\(83\) 23280.8 0.370939 0.185469 − 0.982650i \(-0.440619\pi\)
0.185469 + 0.982650i \(0.440619\pi\)
\(84\) −11706.4 −0.181019
\(85\) 0 0
\(86\) −21592.2 −0.314812
\(87\) −82240.2 −1.16489
\(88\) 27196.3 0.374372
\(89\) 71847.6 0.961474 0.480737 − 0.876865i \(-0.340369\pi\)
0.480737 + 0.876865i \(0.340369\pi\)
\(90\) 11002.5 0.143181
\(91\) −59267.9 −0.750268
\(92\) −52037.7 −0.640986
\(93\) 66252.1 0.794314
\(94\) 28889.6 0.337227
\(95\) 160872. 1.82882
\(96\) 14963.8 0.165716
\(97\) −54239.8 −0.585313 −0.292657 − 0.956218i \(-0.594539\pi\)
−0.292657 + 0.956218i \(0.594539\pi\)
\(98\) −57200.8 −0.601640
\(99\) −12517.8 −0.128363
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.1.4 ✓ 4
17.16 even 2 578.6.a.j.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
578.6.a.h.1.4 ✓ 4 1.1 even 1 trivial
578.6.a.j.1.1 yes 4 17.16 even 2