Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6664,2,Mod(1,6664)] 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("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6664.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,1,0,1] 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: \(53.2123079070\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.30091192.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 9x^{4} + 10x^{3} + 18x^{2} - 23x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 952)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(2.36506\) of defining polynomial
Character \(\chi\) \(=\) 6664.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.36506 q^{3} -1.89992 q^{5} +2.59349 q^{9} +3.76870 q^{11} -0.189847 q^{13} -4.49341 q^{15} -1.00000 q^{17} -0.880612 q^{19} -3.97038 q^{23} -1.39032 q^{25} -0.961410 q^{27} -6.30040 q^{29} -5.88017 q^{31} +8.91320 q^{33} -10.8935 q^{37} -0.448999 q^{39} +3.28277 q^{41} -0.477269 q^{43} -4.92742 q^{45} +2.44464 q^{47} -2.36506 q^{51} -7.16378 q^{53} -7.16022 q^{55} -2.08270 q^{57} -4.53281 q^{59} +11.3227 q^{61} +0.360693 q^{65} +8.76314 q^{67} -9.39016 q^{69} -6.72185 q^{71} +0.695543 q^{73} -3.28818 q^{75} -3.98071 q^{79} -10.0543 q^{81} -7.77110 q^{83} +1.89992 q^{85} -14.9008 q^{87} +0.237007 q^{89} -13.9069 q^{93} +1.67309 q^{95} -6.81746 q^{97} +9.77411 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + q^{5} + q^{9} - 6 q^{11} - 2 q^{13} - 6 q^{17} + 2 q^{19} - 5 q^{23} + q^{25} - 8 q^{27} - 15 q^{29} + 8 q^{31} + 11 q^{33} - 16 q^{37} - 2 q^{39} + 4 q^{41} + 24 q^{43} + 2 q^{45} - q^{47}+ \cdots + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.36506 1.36547 0.682733 − 0.730668i \(-0.260791\pi\)
0.682733 + 0.730668i \(0.260791\pi\)
\(4\) 0 0
\(5\) −1.89992 −0.849668 −0.424834 − 0.905271i \(-0.639668\pi\)
−0.424834 + 0.905271i \(0.639668\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.59349 0.864498
\(10\) 0 0
\(11\) 3.76870 1.13631 0.568154 − 0.822923i \(-0.307658\pi\)
0.568154 + 0.822923i \(0.307658\pi\)
\(12\) 0 0
\(13\) −0.189847 −0.0526541 −0.0263270 − 0.999653i \(-0.508381\pi\)
−0.0263270 + 0.999653i \(0.508381\pi\)
\(14\) 0 0
\(15\) −4.49341 −1.16019
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) −0.880612 −0.202026 −0.101013 − 0.994885i \(-0.532208\pi\)
−0.101013 + 0.994885i \(0.532208\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.97038 −0.827881 −0.413940 − 0.910304i \(-0.635848\pi\)
−0.413940 + 0.910304i \(0.635848\pi\)
\(24\) 0 0
\(25\) −1.39032 −0.278063
\(26\) 0 0
\(27\) −0.961410 −0.185023
\(28\) 0 0
\(29\) −6.30040 −1.16996 −0.584978 − 0.811049i \(-0.698897\pi\)
−0.584978 + 0.811049i \(0.698897\pi\)
\(30\) 0 0
\(31\) −5.88017 −1.05611 −0.528055 − 0.849210i \(-0.677078\pi\)
−0.528055 + 0.849210i \(0.677078\pi\)
\(32\) 0 0
\(33\) 8.91320 1.55159
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.8935 −1.79088 −0.895440 − 0.445182i \(-0.853139\pi\)
−0.895440 + 0.445182i \(0.853139\pi\)
\(38\) 0 0
\(39\) −0.448999 −0.0718973
\(40\) 0 0
\(41\) 3.28277 0.512683 0.256341 − 0.966586i \(-0.417483\pi\)
0.256341 + 0.966586i \(0.417483\pi\)
\(42\) 0 0
\(43\) −0.477269 −0.0727828 −0.0363914 − 0.999338i \(-0.511586\pi\)
−0.0363914 + 0.999338i \(0.511586\pi\)
\(44\) 0 0
\(45\) −4.92742 −0.734537
\(46\) 0 0
\(47\) 2.44464 0.356587 0.178293 − 0.983977i \(-0.442942\pi\)
0.178293 + 0.983977i \(0.442942\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −2.36506 −0.331174
\(52\) 0 0
\(53\) −7.16378 −0.984021 −0.492011 − 0.870589i \(-0.663738\pi\)
−0.492011 + 0.870589i \(0.663738\pi\)
\(54\) 0 0
\(55\) −7.16022 −0.965484
\(56\) 0 0
\(57\) −2.08270 −0.275860
\(58\) 0 0
\(59\) −4.53281 −0.590121 −0.295061 − 0.955479i \(-0.595340\pi\)
−0.295061 + 0.955479i \(0.595340\pi\)
\(60\) 0 0
\(61\) 11.3227 1.44973 0.724863 − 0.688894i \(-0.241903\pi\)
0.724863 + 0.688894i \(0.241903\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.360693 0.0447385
\(66\) 0 0
\(67\) 8.76314 1.07059 0.535294 − 0.844666i \(-0.320201\pi\)
0.535294 + 0.844666i \(0.320201\pi\)
\(68\) 0 0
\(69\) −9.39016 −1.13044
\(70\) 0 0
\(71\) −6.72185 −0.797737 −0.398868 − 0.917008i \(-0.630597\pi\)
−0.398868 + 0.917008i \(0.630597\pi\)
\(72\) 0 0
\(73\) 0.695543 0.0814072 0.0407036 − 0.999171i \(-0.487040\pi\)
0.0407036 + 0.999171i \(0.487040\pi\)
\(74\) 0 0
\(75\) −3.28818 −0.379686
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.98071 −0.447865 −0.223933 − 0.974605i \(-0.571890\pi\)
−0.223933 + 0.974605i \(0.571890\pi\)
\(80\) 0 0
\(81\) −10.0543 −1.11714
\(82\) 0 0
\(83\) −7.77110 −0.852989 −0.426495 − 0.904490i \(-0.640252\pi\)
−0.426495 + 0.904490i \(0.640252\pi\)
\(84\) 0 0
\(85\) 1.89992 0.206075
\(86\) 0 0
\(87\) −14.9008 −1.59753
\(88\) 0 0
\(89\) 0.237007 0.0251227 0.0125613 − 0.999921i \(-0.496001\pi\)
0.0125613 + 0.999921i \(0.496001\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −13.9069 −1.44208
\(94\) 0 0
\(95\) 1.67309 0.171655
\(96\) 0 0
\(97\) −6.81746 −0.692208 −0.346104 − 0.938196i \(-0.612496\pi\)
−0.346104 + 0.938196i \(0.612496\pi\)
\(98\) 0 0
\(99\) 9.77411 0.982335
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 6664.2.a.s.1.6 6
7.2 even 3 952.2.q.c.137.1 ✓ 12
7.4 even 3 952.2.q.c.681.1 yes 12
7.6 odd 2 6664.2.a.r.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.q.c.137.1 ✓ 12 7.2 even 3
952.2.q.c.681.1 yes 12 7.4 even 3
6664.2.a.r.1.1 6 7.6 odd 2
6664.2.a.s.1.6 6 1.1 even 1 trivial