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Properties

Label 578.6.a.r.1.3
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 1.3
Root \(-14.3387\) 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)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} -21.2324 q^{3} +16.0000 q^{4} -54.9040 q^{5} +84.9296 q^{6} -170.486 q^{7} -64.0000 q^{8} +207.815 q^{9} +219.616 q^{10} +446.613 q^{11} -339.719 q^{12} +1185.01 q^{13} +681.945 q^{14} +1165.75 q^{15} +256.000 q^{16} -831.260 q^{18} -1587.71 q^{19} -878.465 q^{20} +3619.83 q^{21} -1786.45 q^{22} -1797.03 q^{23} +1358.87 q^{24} -110.545 q^{25} -4740.05 q^{26} +747.060 q^{27} -2727.78 q^{28} -1158.92 q^{29} -4662.98 q^{30} -1031.48 q^{31} -1024.00 q^{32} -9482.67 q^{33} +9360.39 q^{35} +3325.04 q^{36} -12924.9 q^{37} +6350.84 q^{38} -25160.6 q^{39} +3513.86 q^{40} -9852.60 q^{41} -14479.3 q^{42} +1577.22 q^{43} +7145.81 q^{44} -11409.9 q^{45} +7188.12 q^{46} +10225.0 q^{47} -5435.50 q^{48} +12258.6 q^{49} +442.182 q^{50} +18960.2 q^{52} -12362.6 q^{53} -2988.24 q^{54} -24520.9 q^{55} +10911.1 q^{56} +33710.9 q^{57} +4635.68 q^{58} -14454.6 q^{59} +18651.9 q^{60} +4989.39 q^{61} +4125.90 q^{62} -35429.6 q^{63} +4096.00 q^{64} -65061.9 q^{65} +37930.7 q^{66} +42323.6 q^{67} +38155.3 q^{69} -37441.5 q^{70} +15454.8 q^{71} -13300.2 q^{72} -75882.9 q^{73} +51699.5 q^{74} +2347.14 q^{75} -25403.4 q^{76} -76141.4 q^{77} +100643. q^{78} -74542.7 q^{79} -14055.4 q^{80} -66361.0 q^{81} +39410.4 q^{82} +76364.4 q^{83} +57917.3 q^{84} -6308.90 q^{86} +24606.7 q^{87} -28583.2 q^{88} -127941. q^{89} +45639.6 q^{90} -202028. q^{91} -28752.5 q^{92} +21900.7 q^{93} -40900.2 q^{94} +87171.7 q^{95} +21742.0 q^{96} +92363.3 q^{97} -49034.2 q^{98} +92813.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 64 q^{2} + 256 q^{4} - 1024 q^{8} + 1048 q^{9} + 4072 q^{13} + 1008 q^{15} + 4096 q^{16} - 4192 q^{18} - 376 q^{19} - 816 q^{21} + 15816 q^{25} - 16288 q^{26} - 4032 q^{30} - 16384 q^{32} - 11376 q^{33}+ \cdots - 113984 q^{98}+O(q^{100}) \) 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
\(3\) −21.2324 −1.36206 −0.681030 − 0.732256i \(-0.738468\pi\)
−0.681030 + 0.732256i \(0.738468\pi\)
\(4\) 16.0000 0.500000
\(5\) −54.9040 −0.982153 −0.491077 − 0.871116i \(-0.663397\pi\)
−0.491077 + 0.871116i \(0.663397\pi\)
\(6\) 84.9296 0.963122
\(7\) −170.486 −1.31506 −0.657528 − 0.753430i \(-0.728398\pi\)
−0.657528 + 0.753430i \(0.728398\pi\)
\(8\) −64.0000 −0.353553
\(9\) 207.815 0.855206
\(10\) 219.616 0.694487
\(11\) 446.613 1.11288 0.556442 − 0.830886i \(-0.312166\pi\)
0.556442 + 0.830886i \(0.312166\pi\)
\(12\) −339.719 −0.681030
\(13\) 1185.01 1.94475 0.972376 − 0.233421i \(-0.0749921\pi\)
0.972376 + 0.233421i \(0.0749921\pi\)
\(14\) 681.945 0.929885
\(15\) 1165.75 1.33775
\(16\) 256.000 0.250000
\(17\) 0 0
\(18\) −831.260 −0.604722
\(19\) −1587.71 −1.00899 −0.504495 − 0.863414i \(-0.668321\pi\)
−0.504495 + 0.863414i \(0.668321\pi\)
\(20\) −878.465 −0.491077
\(21\) 3619.83 1.79118
\(22\) −1786.45 −0.786928
\(23\) −1797.03 −0.708330 −0.354165 − 0.935183i \(-0.615235\pi\)
−0.354165 + 0.935183i \(0.615235\pi\)
\(24\) 1358.87 0.481561
\(25\) −110.545 −0.0353745
\(26\) −4740.05 −1.37515
\(27\) 747.060 0.197218
\(28\) −2727.78 −0.657528
\(29\) −1158.92 −0.255893 −0.127947 − 0.991781i \(-0.540839\pi\)
−0.127947 + 0.991781i \(0.540839\pi\)
\(30\) −4662.98 −0.945933
\(31\) −1031.48 −0.192777 −0.0963884 − 0.995344i \(-0.530729\pi\)
−0.0963884 + 0.995344i \(0.530729\pi\)
\(32\) −1024.00 −0.176777
\(33\) −9482.67 −1.51581
\(34\) 0 0
\(35\) 9360.39 1.29159
\(36\) 3325.04 0.427603
\(37\) −12924.9 −1.55211 −0.776054 − 0.630666i \(-0.782782\pi\)
−0.776054 + 0.630666i \(0.782782\pi\)
\(38\) 6350.84 0.713464
\(39\) −25160.6 −2.64887
\(40\) 3513.86 0.347244
\(41\) −9852.60 −0.915358 −0.457679 − 0.889117i \(-0.651319\pi\)
−0.457679 + 0.889117i \(0.651319\pi\)
\(42\) −14479.3 −1.26656
\(43\) 1577.22 0.130083 0.0650417 − 0.997883i \(-0.479282\pi\)
0.0650417 + 0.997883i \(0.479282\pi\)
\(44\) 7145.81 0.556442
\(45\) −11409.9 −0.839944
\(46\) 7188.12 0.500865
\(47\) 10225.0 0.675182 0.337591 − 0.941293i \(-0.390388\pi\)
0.337591 + 0.941293i \(0.390388\pi\)
\(48\) −5435.50 −0.340515
\(49\) 12258.6 0.729372
\(50\) 442.182 0.0250136
\(51\) 0 0
\(52\) 18960.2 0.972376
\(53\) −12362.6 −0.604534 −0.302267 − 0.953223i \(-0.597743\pi\)
−0.302267 + 0.953223i \(0.597743\pi\)
\(54\) −2988.24 −0.139454
\(55\) −24520.9 −1.09302
\(56\) 10911.1 0.464943
\(57\) 33710.9 1.37431
\(58\) 4635.68 0.180944
\(59\) −14454.6 −0.540601 −0.270300 − 0.962776i \(-0.587123\pi\)
−0.270300 + 0.962776i \(0.587123\pi\)
\(60\) 18651.9 0.668876
\(61\) 4989.39 0.171681 0.0858406 − 0.996309i \(-0.472642\pi\)
0.0858406 + 0.996309i \(0.472642\pi\)
\(62\) 4125.90 0.136314
\(63\) −35429.6 −1.12464
\(64\) 4096.00 0.125000
\(65\) −65061.9 −1.91004
\(66\) 37930.7 1.07184
\(67\) 42323.6 1.15185 0.575924 − 0.817503i \(-0.304642\pi\)
0.575924 + 0.817503i \(0.304642\pi\)
\(68\) 0 0
\(69\) 38155.3 0.964788
\(70\) −37441.5 −0.913290
\(71\) 15454.8 0.363846 0.181923 − 0.983313i \(-0.441768\pi\)
0.181923 + 0.983313i \(0.441768\pi\)
\(72\) −13300.2 −0.302361
\(73\) −75882.9 −1.66662 −0.833311 − 0.552805i \(-0.813558\pi\)
−0.833311 + 0.552805i \(0.813558\pi\)
\(74\) 51699.5 1.09751
\(75\) 2347.14 0.0481822
\(76\) −25403.4 −0.504495
\(77\) −76141.4 −1.46350
\(78\) 100643. 1.87303
\(79\) −74542.7 −1.34381 −0.671904 − 0.740638i \(-0.734523\pi\)
−0.671904 + 0.740638i \(0.734523\pi\)
\(80\) −14055.4 −0.245538
\(81\) −66361.0 −1.12383
\(82\) 39410.4 0.647256
\(83\) 76364.4 1.21673 0.608367 − 0.793656i \(-0.291825\pi\)
0.608367 + 0.793656i \(0.291825\pi\)
\(84\) 57917.3 0.895592
\(85\) 0 0
\(86\) −6308.90 −0.0919829
\(87\) 24606.7 0.348542
\(88\) −28583.2 −0.393464
\(89\) −127941. −1.71212 −0.856059 − 0.516877i \(-0.827094\pi\)
−0.856059 + 0.516877i \(0.827094\pi\)
\(90\) 45639.6 0.593930
\(91\) −202028. −2.55746
\(92\) −28752.5 −0.354165
\(93\) 21900.7 0.262573
\(94\) −40900.2 −0.477426
\(95\) 87171.7 0.990984
\(96\) 21742.0 0.240780
\(97\) 92363.3 0.996713 0.498357 − 0.866972i \(-0.333937\pi\)
0.498357 + 0.866972i \(0.333937\pi\)
\(98\) −49034.2 −0.515744
\(99\) 92813.0 0.951745
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.1.3 16
17.10 odd 16 34.6.d.b.15.1 ✓ 16
17.12 odd 16 34.6.d.b.25.1 yes 16
17.16 even 2 inner 578.6.a.r.1.14 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
34.6.d.b.15.1 ✓ 16 17.10 odd 16
34.6.d.b.25.1 yes 16 17.12 odd 16
578.6.a.r.1.3 16 1.1 even 1 trivial
578.6.a.r.1.14 16 17.16 even 2 inner