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Properties

Label 578.6.a.r.1.4
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 1.4
Root \(-25.5434\) 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)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} -18.1990 q^{3} +16.0000 q^{4} +107.153 q^{5} +72.7958 q^{6} +12.5445 q^{7} -64.0000 q^{8} +88.2020 q^{9} -428.614 q^{10} +504.662 q^{11} -291.183 q^{12} +455.168 q^{13} -50.1780 q^{14} -1950.08 q^{15} +256.000 q^{16} -352.808 q^{18} -108.596 q^{19} +1714.46 q^{20} -228.297 q^{21} -2018.65 q^{22} +1607.65 q^{23} +1164.73 q^{24} +8356.86 q^{25} -1820.67 q^{26} +2817.16 q^{27} +200.712 q^{28} +1306.49 q^{29} +7800.32 q^{30} +7921.73 q^{31} -1024.00 q^{32} -9184.32 q^{33} +1344.19 q^{35} +1411.23 q^{36} +6011.77 q^{37} +434.386 q^{38} -8283.59 q^{39} -6857.82 q^{40} +9966.06 q^{41} +913.187 q^{42} -15299.3 q^{43} +8074.59 q^{44} +9451.15 q^{45} -6430.60 q^{46} +20750.9 q^{47} -4658.93 q^{48} -16649.6 q^{49} -33427.5 q^{50} +7282.70 q^{52} -23353.4 q^{53} -11268.7 q^{54} +54076.3 q^{55} -802.848 q^{56} +1976.34 q^{57} -5225.96 q^{58} +18408.8 q^{59} -31201.3 q^{60} -43948.4 q^{61} -31686.9 q^{62} +1106.45 q^{63} +4096.00 q^{64} +48772.9 q^{65} +36737.3 q^{66} +71284.9 q^{67} -29257.6 q^{69} -5376.74 q^{70} -50270.7 q^{71} -5644.93 q^{72} -61904.7 q^{73} -24047.1 q^{74} -152086. q^{75} -1737.54 q^{76} +6330.73 q^{77} +33134.4 q^{78} -3278.88 q^{79} +27431.3 q^{80} -72702.5 q^{81} -39864.3 q^{82} -25996.5 q^{83} -3652.75 q^{84} +61197.1 q^{86} -23776.8 q^{87} -32298.4 q^{88} +11963.2 q^{89} -37804.6 q^{90} +5709.86 q^{91} +25722.4 q^{92} -144167. q^{93} -83003.7 q^{94} -11636.5 q^{95} +18635.7 q^{96} -67490.4 q^{97} +66598.5 q^{98} +44512.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 64 q^{2} + 256 q^{4} - 1024 q^{8} + 1048 q^{9} + 4072 q^{13} + 1008 q^{15} + 4096 q^{16} - 4192 q^{18} - 376 q^{19} - 816 q^{21} + 15816 q^{25} - 16288 q^{26} - 4032 q^{30} - 16384 q^{32} - 11376 q^{33}+ \cdots - 113984 q^{98}+O(q^{100}) \) 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
\(3\) −18.1990 −1.16746 −0.583732 − 0.811947i \(-0.698408\pi\)
−0.583732 + 0.811947i \(0.698408\pi\)
\(4\) 16.0000 0.500000
\(5\) 107.153 1.91682 0.958410 − 0.285396i \(-0.0921251\pi\)
0.958410 + 0.285396i \(0.0921251\pi\)
\(6\) 72.7958 0.825521
\(7\) 12.5445 0.0967627 0.0483814 − 0.998829i \(-0.484594\pi\)
0.0483814 + 0.998829i \(0.484594\pi\)
\(8\) −64.0000 −0.353553
\(9\) 88.2020 0.362971
\(10\) −428.614 −1.35540
\(11\) 504.662 1.25753 0.628766 − 0.777595i \(-0.283560\pi\)
0.628766 + 0.777595i \(0.283560\pi\)
\(12\) −291.183 −0.583732
\(13\) 455.168 0.746988 0.373494 − 0.927633i \(-0.378160\pi\)
0.373494 + 0.927633i \(0.378160\pi\)
\(14\) −50.1780 −0.0684216
\(15\) −1950.08 −2.23782
\(16\) 256.000 0.250000
\(17\) 0 0
\(18\) −352.808 −0.256659
\(19\) −108.596 −0.0690131 −0.0345065 − 0.999404i \(-0.510986\pi\)
−0.0345065 + 0.999404i \(0.510986\pi\)
\(20\) 1714.46 0.958410
\(21\) −228.297 −0.112967
\(22\) −2018.65 −0.889209
\(23\) 1607.65 0.633683 0.316841 − 0.948478i \(-0.397378\pi\)
0.316841 + 0.948478i \(0.397378\pi\)
\(24\) 1164.73 0.412761
\(25\) 8356.86 2.67420
\(26\) −1820.67 −0.528200
\(27\) 2817.16 0.743708
\(28\) 200.712 0.0483814
\(29\) 1306.49 0.288477 0.144239 − 0.989543i \(-0.453927\pi\)
0.144239 + 0.989543i \(0.453927\pi\)
\(30\) 7800.32 1.58238
\(31\) 7921.73 1.48052 0.740262 − 0.672318i \(-0.234701\pi\)
0.740262 + 0.672318i \(0.234701\pi\)
\(32\) −1024.00 −0.176777
\(33\) −9184.32 −1.46812
\(34\) 0 0
\(35\) 1344.19 0.185477
\(36\) 1411.23 0.181486
\(37\) 6011.77 0.721935 0.360967 − 0.932578i \(-0.382447\pi\)
0.360967 + 0.932578i \(0.382447\pi\)
\(38\) 434.386 0.0487996
\(39\) −8283.59 −0.872081
\(40\) −6857.82 −0.677698
\(41\) 9966.06 0.925900 0.462950 − 0.886384i \(-0.346791\pi\)
0.462950 + 0.886384i \(0.346791\pi\)
\(42\) 913.187 0.0798797
\(43\) −15299.3 −1.26183 −0.630913 − 0.775854i \(-0.717320\pi\)
−0.630913 + 0.775854i \(0.717320\pi\)
\(44\) 8074.59 0.628766
\(45\) 9451.15 0.695750
\(46\) −6430.60 −0.448082
\(47\) 20750.9 1.37023 0.685114 − 0.728436i \(-0.259752\pi\)
0.685114 + 0.728436i \(0.259752\pi\)
\(48\) −4658.93 −0.291866
\(49\) −16649.6 −0.990637
\(50\) −33427.5 −1.89094
\(51\) 0 0
\(52\) 7282.70 0.373494
\(53\) −23353.4 −1.14199 −0.570993 − 0.820955i \(-0.693442\pi\)
−0.570993 + 0.820955i \(0.693442\pi\)
\(54\) −11268.7 −0.525881
\(55\) 54076.3 2.41046
\(56\) −802.848 −0.0342108
\(57\) 1976.34 0.0805703
\(58\) −5225.96 −0.203984
\(59\) 18408.8 0.688485 0.344242 − 0.938881i \(-0.388136\pi\)
0.344242 + 0.938881i \(0.388136\pi\)
\(60\) −31201.3 −1.11891
\(61\) −43948.4 −1.51223 −0.756116 − 0.654438i \(-0.772905\pi\)
−0.756116 + 0.654438i \(0.772905\pi\)
\(62\) −31686.9 −1.04689
\(63\) 1106.45 0.0351221
\(64\) 4096.00 0.125000
\(65\) 48772.9 1.43184
\(66\) 36737.3 1.03812
\(67\) 71284.9 1.94004 0.970019 − 0.243029i \(-0.0781410\pi\)
0.970019 + 0.243029i \(0.0781410\pi\)
\(68\) 0 0
\(69\) −29257.6 −0.739802
\(70\) −5376.74 −0.131152
\(71\) −50270.7 −1.18350 −0.591751 − 0.806121i \(-0.701563\pi\)
−0.591751 + 0.806121i \(0.701563\pi\)
\(72\) −5644.93 −0.128330
\(73\) −61904.7 −1.35962 −0.679809 − 0.733390i \(-0.737937\pi\)
−0.679809 + 0.733390i \(0.737937\pi\)
\(74\) −24047.1 −0.510485
\(75\) −152086. −3.12203
\(76\) −1737.54 −0.0345065
\(77\) 6330.73 0.121682
\(78\) 33134.4 0.616655
\(79\) −3278.88 −0.0591095 −0.0295547 − 0.999563i \(-0.509409\pi\)
−0.0295547 + 0.999563i \(0.509409\pi\)
\(80\) 27431.3 0.479205
\(81\) −72702.5 −1.23122
\(82\) −39864.3 −0.654710
\(83\) −25996.5 −0.414208 −0.207104 − 0.978319i \(-0.566404\pi\)
−0.207104 + 0.978319i \(0.566404\pi\)
\(84\) −3652.75 −0.0564835
\(85\) 0 0
\(86\) 61197.1 0.892246
\(87\) −23776.8 −0.336786
\(88\) −32298.4 −0.444605
\(89\) 11963.2 0.160093 0.0800465 − 0.996791i \(-0.474493\pi\)
0.0800465 + 0.996791i \(0.474493\pi\)
\(90\) −37804.6 −0.491969
\(91\) 5709.86 0.0722806
\(92\) 25722.4 0.316841
\(93\) −144167. −1.72846
\(94\) −83003.7 −0.968898
\(95\) −11636.5 −0.132286
\(96\) 18635.7 0.206380
\(97\) −67490.4 −0.728304 −0.364152 − 0.931340i \(-0.618641\pi\)
−0.364152 + 0.931340i \(0.618641\pi\)
\(98\) 66598.5 0.700486
\(99\) 44512.2 0.456448
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.1.4 16
17.11 odd 16 34.6.d.b.19.4 yes 16
17.14 odd 16 34.6.d.b.9.4 ✓ 16
17.16 even 2 inner 578.6.a.r.1.13 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
34.6.d.b.9.4 ✓ 16 17.14 odd 16
34.6.d.b.19.4 yes 16 17.11 odd 16
578.6.a.r.1.4 16 1.1 even 1 trivial
578.6.a.r.1.13 16 17.16 even 2 inner