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Properties

Label 578.6.a.r.1.7
Level $578$
Weight $6$
Character 578.1
Self dual yes
Analytic conductor $92.702$
Analytic rank $0$
Dimension $16$
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] = [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 = [16,-64,0,256,0] 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: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 2234 x^{14} - 5644 x^{13} + 1696673 x^{12} + 12813520 x^{11} - 472386300 x^{10} + \cdots + 29\!\cdots\!28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 17^{10} \)
Twist minimal: no (minimal twist has level 34)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-9.94357\) 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)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} -6.25932 q^{3} +16.0000 q^{4} -36.2693 q^{5} +25.0373 q^{6} +146.885 q^{7} -64.0000 q^{8} -203.821 q^{9} +145.077 q^{10} -745.570 q^{11} -100.149 q^{12} +921.230 q^{13} -587.539 q^{14} +227.021 q^{15} +256.000 q^{16} +815.284 q^{18} -826.088 q^{19} -580.309 q^{20} -919.398 q^{21} +2982.28 q^{22} -3579.61 q^{23} +400.597 q^{24} -1809.54 q^{25} -3684.92 q^{26} +2796.80 q^{27} +2350.15 q^{28} +1921.81 q^{29} -908.085 q^{30} +6236.17 q^{31} -1024.00 q^{32} +4666.76 q^{33} -5327.40 q^{35} -3261.13 q^{36} -215.131 q^{37} +3304.35 q^{38} -5766.28 q^{39} +2321.24 q^{40} -18637.8 q^{41} +3677.59 q^{42} +3793.28 q^{43} -11929.1 q^{44} +7392.44 q^{45} +14318.4 q^{46} +13185.2 q^{47} -1602.39 q^{48} +4768.10 q^{49} +7238.15 q^{50} +14739.7 q^{52} -13887.2 q^{53} -11187.2 q^{54} +27041.3 q^{55} -9400.62 q^{56} +5170.75 q^{57} -7687.26 q^{58} +5505.23 q^{59} +3632.34 q^{60} +13113.1 q^{61} -24944.7 q^{62} -29938.2 q^{63} +4096.00 q^{64} -33412.4 q^{65} -18667.1 q^{66} -66928.4 q^{67} +22405.9 q^{69} +21309.6 q^{70} -53820.5 q^{71} +13044.5 q^{72} -56454.6 q^{73} +860.524 q^{74} +11326.5 q^{75} -13217.4 q^{76} -109513. q^{77} +23065.1 q^{78} +33390.7 q^{79} -9284.94 q^{80} +32022.4 q^{81} +74551.1 q^{82} +21536.5 q^{83} -14710.4 q^{84} -15173.1 q^{86} -12029.3 q^{87} +47716.5 q^{88} -33217.0 q^{89} -29569.8 q^{90} +135315. q^{91} -57273.7 q^{92} -39034.2 q^{93} -52740.9 q^{94} +29961.6 q^{95} +6409.55 q^{96} +48833.6 q^{97} -19072.4 q^{98} +151963. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 64 q^{2} + 256 q^{4} - 1024 q^{8} + 1048 q^{9} + 4072 q^{13} + 1008 q^{15} + 4096 q^{16} - 4192 q^{18} - 376 q^{19} - 816 q^{21} + 15816 q^{25} - 16288 q^{26} - 4032 q^{30} - 16384 q^{32} - 11376 q^{33}+ \cdots - 113984 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) −6.25932 −0.401536 −0.200768 − 0.979639i \(-0.564344\pi\)
−0.200768 + 0.979639i \(0.564344\pi\)
\(4\) 16.0000 0.500000
\(5\) −36.2693 −0.648805 −0.324403 − 0.945919i \(-0.605163\pi\)
−0.324403 + 0.945919i \(0.605163\pi\)
\(6\) 25.0373 0.283929
\(7\) 146.885 1.13300 0.566502 − 0.824061i \(-0.308296\pi\)
0.566502 + 0.824061i \(0.308296\pi\)
\(8\) −64.0000 −0.353553
\(9\) −203.821 −0.838769
\(10\) 145.077 0.458774
\(11\) −745.570 −1.85783 −0.928917 − 0.370289i \(-0.879259\pi\)
−0.928917 + 0.370289i \(0.879259\pi\)
\(12\) −100.149 −0.200768
\(13\) 921.230 1.51185 0.755927 − 0.654656i \(-0.227186\pi\)
0.755927 + 0.654656i \(0.227186\pi\)
\(14\) −587.539 −0.801155
\(15\) 227.021 0.260518
\(16\) 256.000 0.250000
\(17\) 0 0
\(18\) 815.284 0.593099
\(19\) −826.088 −0.524979 −0.262490 − 0.964935i \(-0.584544\pi\)
−0.262490 + 0.964935i \(0.584544\pi\)
\(20\) −580.309 −0.324403
\(21\) −919.398 −0.454941
\(22\) 2982.28 1.31369
\(23\) −3579.61 −1.41096 −0.705482 − 0.708728i \(-0.749269\pi\)
−0.705482 + 0.708728i \(0.749269\pi\)
\(24\) 400.597 0.141964
\(25\) −1809.54 −0.579052
\(26\) −3684.92 −1.06904
\(27\) 2796.80 0.738332
\(28\) 2350.15 0.566502
\(29\) 1921.81 0.424342 0.212171 − 0.977233i \(-0.431947\pi\)
0.212171 + 0.977233i \(0.431947\pi\)
\(30\) −908.085 −0.184214
\(31\) 6236.17 1.16550 0.582752 − 0.812650i \(-0.301976\pi\)
0.582752 + 0.812650i \(0.301976\pi\)
\(32\) −1024.00 −0.176777
\(33\) 4666.76 0.745986
\(34\) 0 0
\(35\) −5327.40 −0.735098
\(36\) −3261.13 −0.419385
\(37\) −215.131 −0.0258344 −0.0129172 − 0.999917i \(-0.504112\pi\)
−0.0129172 + 0.999917i \(0.504112\pi\)
\(38\) 3304.35 0.371216
\(39\) −5766.28 −0.607063
\(40\) 2321.24 0.229387
\(41\) −18637.8 −1.73155 −0.865774 − 0.500435i \(-0.833173\pi\)
−0.865774 + 0.500435i \(0.833173\pi\)
\(42\) 3677.59 0.321692
\(43\) 3793.28 0.312855 0.156428 − 0.987689i \(-0.450002\pi\)
0.156428 + 0.987689i \(0.450002\pi\)
\(44\) −11929.1 −0.928917
\(45\) 7392.44 0.544198
\(46\) 14318.4 0.997702
\(47\) 13185.2 0.870648 0.435324 − 0.900274i \(-0.356634\pi\)
0.435324 + 0.900274i \(0.356634\pi\)
\(48\) −1602.39 −0.100384
\(49\) 4768.10 0.283697
\(50\) 7238.15 0.409452
\(51\) 0 0
\(52\) 14739.7 0.755927
\(53\) −13887.2 −0.679087 −0.339544 − 0.940590i \(-0.610273\pi\)
−0.339544 + 0.940590i \(0.610273\pi\)
\(54\) −11187.2 −0.522079
\(55\) 27041.3 1.20537
\(56\) −9400.62 −0.400577
\(57\) 5170.75 0.210798
\(58\) −7687.26 −0.300055
\(59\) 5505.23 0.205895 0.102947 − 0.994687i \(-0.467173\pi\)
0.102947 + 0.994687i \(0.467173\pi\)
\(60\) 3632.34 0.130259
\(61\) 13113.1 0.451211 0.225606 − 0.974219i \(-0.427564\pi\)
0.225606 + 0.974219i \(0.427564\pi\)
\(62\) −24944.7 −0.824136
\(63\) −29938.2 −0.950328
\(64\) 4096.00 0.125000
\(65\) −33412.4 −0.980898
\(66\) −18667.1 −0.527492
\(67\) −66928.4 −1.82148 −0.910738 − 0.412984i \(-0.864486\pi\)
−0.910738 + 0.412984i \(0.864486\pi\)
\(68\) 0 0
\(69\) 22405.9 0.566552
\(70\) 21309.6 0.519793
\(71\) −53820.5 −1.26707 −0.633537 − 0.773713i \(-0.718397\pi\)
−0.633537 + 0.773713i \(0.718397\pi\)
\(72\) 13044.5 0.296550
\(73\) −56454.6 −1.23992 −0.619958 − 0.784635i \(-0.712850\pi\)
−0.619958 + 0.784635i \(0.712850\pi\)
\(74\) 860.524 0.0182677
\(75\) 11326.5 0.232510
\(76\) −13217.4 −0.262490
\(77\) −109513. −2.10493
\(78\) 23065.1 0.429259
\(79\) 33390.7 0.601947 0.300973 − 0.953633i \(-0.402688\pi\)
0.300973 + 0.953633i \(0.402688\pi\)
\(80\) −9284.94 −0.162201
\(81\) 32022.4 0.542303
\(82\) 74551.1 1.22439
\(83\) 21536.5 0.343146 0.171573 − 0.985171i \(-0.445115\pi\)
0.171573 + 0.985171i \(0.445115\pi\)
\(84\) −14710.4 −0.227471
\(85\) 0 0
\(86\) −15173.1 −0.221222
\(87\) −12029.3 −0.170389
\(88\) 47716.5 0.656843
\(89\) −33217.0 −0.444514 −0.222257 − 0.974988i \(-0.571342\pi\)
−0.222257 + 0.974988i \(0.571342\pi\)
\(90\) −29569.8 −0.384806
\(91\) 135315. 1.71294
\(92\) −57273.7 −0.705482
\(93\) −39034.2 −0.467992
\(94\) −52740.9 −0.615641
\(95\) 29961.6 0.340609
\(96\) 6409.55 0.0709822
\(97\) 48833.6 0.526975 0.263487 − 0.964663i \(-0.415127\pi\)
0.263487 + 0.964663i \(0.415127\pi\)
\(98\) −19072.4 −0.200604
\(99\) 151963. 1.55829
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.r.1.7 16
17.11 odd 16 34.6.d.b.19.3 yes 16
17.14 odd 16 34.6.d.b.9.3 ✓ 16
17.16 even 2 inner 578.6.a.r.1.10 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
34.6.d.b.9.3 ✓ 16 17.14 odd 16
34.6.d.b.19.3 yes 16 17.11 odd 16
578.6.a.r.1.7 16 1.1 even 1 trivial
578.6.a.r.1.10 16 17.16 even 2 inner