Properties

Label 504.6.a.l.1.1
Level $504$
Weight $6$
Character 504.1
Self dual yes
Analytic conductor $80.833$
Analytic rank $1$
Dimension $2$
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] = [504,6,Mod(1,504)] 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("504.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 504.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-48,0,-98] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(80.8334451857\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{37}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.54138\) of defining polynomial
Character \(\chi\) \(=\) 504.1

$q$-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)
 
\(f(q)\) \(=\) \(q-84.8276 q^{5} -49.0000 q^{7} -634.124 q^{11} +895.242 q^{13} +2057.46 q^{17} +2451.95 q^{19} -569.186 q^{23} +4070.73 q^{25} +1471.78 q^{29} -2006.65 q^{31} +4156.55 q^{35} +4860.54 q^{37} -17228.4 q^{41} -15481.3 q^{43} +5006.40 q^{47} +2401.00 q^{49} -19560.3 q^{53} +53791.3 q^{55} +14515.6 q^{59} +3572.67 q^{61} -75941.3 q^{65} -41480.7 q^{67} +9247.05 q^{71} -41350.0 q^{73} +31072.1 q^{77} -37962.2 q^{79} +79211.1 q^{83} -174530. q^{85} -92538.5 q^{89} -43866.9 q^{91} -207993. q^{95} +175419. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 48 q^{5} - 98 q^{7} - 368 q^{11} - 156 q^{13} + 3312 q^{17} + 3736 q^{19} - 1552 q^{23} + 2302 q^{25} - 1728 q^{29} - 3624 q^{31} + 2352 q^{35} + 6996 q^{37} - 26160 q^{41} - 30184 q^{43} + 11424 q^{47}+ \cdots + 66652 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −84.8276 −1.51744 −0.758721 − 0.651415i \(-0.774176\pi\)
−0.758721 + 0.651415i \(0.774176\pi\)
\(6\) 0 0
\(7\) −49.0000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −634.124 −1.58013 −0.790065 − 0.613023i \(-0.789953\pi\)
−0.790065 + 0.613023i \(0.789953\pi\)
\(12\) 0 0
\(13\) 895.242 1.46920 0.734602 − 0.678498i \(-0.237369\pi\)
0.734602 + 0.678498i \(0.237369\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2057.46 1.72667 0.863335 − 0.504630i \(-0.168371\pi\)
0.863335 + 0.504630i \(0.168371\pi\)
\(18\) 0 0
\(19\) 2451.95 1.55821 0.779106 − 0.626892i \(-0.215673\pi\)
0.779106 + 0.626892i \(0.215673\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −569.186 −0.224354 −0.112177 − 0.993688i \(-0.535782\pi\)
−0.112177 + 0.993688i \(0.535782\pi\)
\(24\) 0 0
\(25\) 4070.73 1.30263
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1471.78 0.324974 0.162487 − 0.986711i \(-0.448049\pi\)
0.162487 + 0.986711i \(0.448049\pi\)
\(30\) 0 0
\(31\) −2006.65 −0.375031 −0.187515 − 0.982262i \(-0.560043\pi\)
−0.187515 + 0.982262i \(0.560043\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4156.55 0.573539
\(36\) 0 0
\(37\) 4860.54 0.583687 0.291844 − 0.956466i \(-0.405731\pi\)
0.291844 + 0.956466i \(0.405731\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −17228.4 −1.60061 −0.800307 − 0.599591i \(-0.795330\pi\)
−0.800307 + 0.599591i \(0.795330\pi\)
\(42\) 0 0
\(43\) −15481.3 −1.27684 −0.638420 − 0.769689i \(-0.720412\pi\)
−0.638420 + 0.769689i \(0.720412\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5006.40 0.330583 0.165292 − 0.986245i \(-0.447143\pi\)
0.165292 + 0.986245i \(0.447143\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −19560.3 −0.956504 −0.478252 − 0.878223i \(-0.658729\pi\)
−0.478252 + 0.878223i \(0.658729\pi\)
\(54\) 0 0
\(55\) 53791.3 2.39776
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14515.6 0.542881 0.271441 − 0.962455i \(-0.412500\pi\)
0.271441 + 0.962455i \(0.412500\pi\)
\(60\) 0 0
\(61\) 3572.67 0.122933 0.0614664 − 0.998109i \(-0.480422\pi\)
0.0614664 + 0.998109i \(0.480422\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −75941.3 −2.22943
\(66\) 0 0
\(67\) −41480.7 −1.12891 −0.564455 − 0.825464i \(-0.690914\pi\)
−0.564455 + 0.825464i \(0.690914\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9247.05 0.217700 0.108850 − 0.994058i \(-0.465283\pi\)
0.108850 + 0.994058i \(0.465283\pi\)
\(72\) 0 0
\(73\) −41350.0 −0.908172 −0.454086 − 0.890958i \(-0.650034\pi\)
−0.454086 + 0.890958i \(0.650034\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 31072.1 0.597233
\(78\) 0 0
\(79\) −37962.2 −0.684359 −0.342179 − 0.939635i \(-0.611165\pi\)
−0.342179 + 0.939635i \(0.611165\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 79211.1 1.26209 0.631046 − 0.775746i \(-0.282626\pi\)
0.631046 + 0.775746i \(0.282626\pi\)
\(84\) 0 0
\(85\) −174530. −2.62012
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −92538.5 −1.23836 −0.619181 − 0.785248i \(-0.712535\pi\)
−0.619181 + 0.785248i \(0.712535\pi\)
\(90\) 0 0
\(91\) −43866.9 −0.555307
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −207993. −2.36450
\(96\) 0 0
\(97\) 175419. 1.89299 0.946495 − 0.322720i \(-0.104597\pi\)
0.946495 + 0.322720i \(0.104597\pi\)
\(98\) 0 0
\(99\) 0 0
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 504.6.a.l.1.1 ✓ 2
3.2 odd 2 504.6.a.r.1.2 yes 2
4.3 odd 2 1008.6.a.bh.1.1 2
12.11 even 2 1008.6.a.bs.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.6.a.l.1.1 ✓ 2 1.1 even 1 trivial
504.6.a.r.1.2 yes 2 3.2 odd 2
1008.6.a.bh.1.1 2 4.3 odd 2
1008.6.a.bs.1.2 2 12.11 even 2