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Properties

Label 1664.2.a.ba.1.2
Level $1664$
Weight $2$
Character 1664.1
Self dual yes
Analytic conductor $13.287$
Analytic rank $0$
Dimension $5$
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] = [1664,2,Mod(1,1664)] 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("1664.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1664 = 2^{7} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1664.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,1,0,-1,0,5,0,8,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(13.2871068963\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.592456.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 6x^{3} + 9x^{2} + 8x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.49707\) of defining polynomial
Character \(\chi\) \(=\) 1664.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.738305 q^{3} -3.70891 q^{5} -1.43136 q^{7} -2.45491 q^{9} -2.99415 q^{11} -1.00000 q^{13} +2.73830 q^{15} +2.23230 q^{17} -1.24815 q^{19} +1.05678 q^{21} -8.24230 q^{23} +8.75600 q^{25} +4.02738 q^{27} -10.2423 q^{29} +10.4708 q^{31} +2.21059 q^{33} +5.30879 q^{35} -6.93152 q^{37} +0.738305 q^{39} +1.17552 q^{41} +9.90212 q^{43} +9.10502 q^{45} +9.11856 q^{47} -4.95121 q^{49} -1.64812 q^{51} -2.82448 q^{53} +11.1050 q^{55} +0.921515 q^{57} -2.99415 q^{59} +1.77770 q^{61} +3.51386 q^{63} +3.70891 q^{65} +13.5875 q^{67} +6.08533 q^{69} +3.08033 q^{71} +1.90297 q^{73} -6.46460 q^{75} +4.28571 q^{77} -1.98830 q^{79} +4.39128 q^{81} -2.21676 q^{83} -8.27939 q^{85} +7.56194 q^{87} +6.33933 q^{89} +1.43136 q^{91} -7.73061 q^{93} +4.62927 q^{95} -14.5816 q^{97} +7.35035 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{3} - q^{5} + 5 q^{7} + 8 q^{9} + 6 q^{11} - 5 q^{13} + 9 q^{15} + 3 q^{17} + 12 q^{19} - 7 q^{21} - 2 q^{23} + 10 q^{25} - 5 q^{27} - 12 q^{29} + 22 q^{31} - 8 q^{33} + 15 q^{35} - 5 q^{37} - q^{39}+ \cdots + 66 q^{99}+O(q^{100}) \) 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
\(3\) −0.738305 −0.426261 −0.213130 − 0.977024i \(-0.568366\pi\)
−0.213130 + 0.977024i \(0.568366\pi\)
\(4\) 0 0
\(5\) −3.70891 −1.65867 −0.829337 − 0.558749i \(-0.811282\pi\)
−0.829337 + 0.558749i \(0.811282\pi\)
\(6\) 0 0
\(7\) −1.43136 −0.541004 −0.270502 − 0.962719i \(-0.587190\pi\)
−0.270502 + 0.962719i \(0.587190\pi\)
\(8\) 0 0
\(9\) −2.45491 −0.818302
\(10\) 0 0
\(11\) −2.99415 −0.902770 −0.451385 − 0.892329i \(-0.649070\pi\)
−0.451385 + 0.892329i \(0.649070\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 2.73830 0.707027
\(16\) 0 0
\(17\) 2.23230 0.541412 0.270706 − 0.962662i \(-0.412743\pi\)
0.270706 + 0.962662i \(0.412743\pi\)
\(18\) 0 0
\(19\) −1.24815 −0.286345 −0.143173 − 0.989698i \(-0.545730\pi\)
−0.143173 + 0.989698i \(0.545730\pi\)
\(20\) 0 0
\(21\) 1.05678 0.230608
\(22\) 0 0
\(23\) −8.24230 −1.71864 −0.859319 − 0.511440i \(-0.829112\pi\)
−0.859319 + 0.511440i \(0.829112\pi\)
\(24\) 0 0
\(25\) 8.75600 1.75120
\(26\) 0 0
\(27\) 4.02738 0.775070
\(28\) 0 0
\(29\) −10.2423 −1.90195 −0.950974 − 0.309272i \(-0.899915\pi\)
−0.950974 + 0.309272i \(0.899915\pi\)
\(30\) 0 0
\(31\) 10.4708 1.88060 0.940302 − 0.340342i \(-0.110543\pi\)
0.940302 + 0.340342i \(0.110543\pi\)
\(32\) 0 0
\(33\) 2.21059 0.384815
\(34\) 0 0
\(35\) 5.30879 0.897349
\(36\) 0 0
\(37\) −6.93152 −1.13953 −0.569767 − 0.821806i \(-0.692967\pi\)
−0.569767 + 0.821806i \(0.692967\pi\)
\(38\) 0 0
\(39\) 0.738305 0.118223
\(40\) 0 0
\(41\) 1.17552 0.183585 0.0917925 − 0.995778i \(-0.470740\pi\)
0.0917925 + 0.995778i \(0.470740\pi\)
\(42\) 0 0
\(43\) 9.90212 1.51006 0.755029 − 0.655691i \(-0.227623\pi\)
0.755029 + 0.655691i \(0.227623\pi\)
\(44\) 0 0
\(45\) 9.10502 1.35730
\(46\) 0 0
\(47\) 9.11856 1.33008 0.665040 − 0.746808i \(-0.268415\pi\)
0.665040 + 0.746808i \(0.268415\pi\)
\(48\) 0 0
\(49\) −4.95121 −0.707315
\(50\) 0 0
\(51\) −1.64812 −0.230782
\(52\) 0 0
\(53\) −2.82448 −0.387973 −0.193986 − 0.981004i \(-0.562142\pi\)
−0.193986 + 0.981004i \(0.562142\pi\)
\(54\) 0 0
\(55\) 11.1050 1.49740
\(56\) 0 0
\(57\) 0.921515 0.122058
\(58\) 0 0
\(59\) −2.99415 −0.389805 −0.194902 − 0.980823i \(-0.562439\pi\)
−0.194902 + 0.980823i \(0.562439\pi\)
\(60\) 0 0
\(61\) 1.77770 0.227611 0.113806 − 0.993503i \(-0.463696\pi\)
0.113806 + 0.993503i \(0.463696\pi\)
\(62\) 0 0
\(63\) 3.51386 0.442704
\(64\) 0 0
\(65\) 3.70891 0.460033
\(66\) 0 0
\(67\) 13.5875 1.65998 0.829988 − 0.557782i \(-0.188347\pi\)
0.829988 + 0.557782i \(0.188347\pi\)
\(68\) 0 0
\(69\) 6.08533 0.732588
\(70\) 0 0
\(71\) 3.08033 0.365567 0.182784 − 0.983153i \(-0.441489\pi\)
0.182784 + 0.983153i \(0.441489\pi\)
\(72\) 0 0
\(73\) 1.90297 0.222725 0.111363 − 0.993780i \(-0.464478\pi\)
0.111363 + 0.993780i \(0.464478\pi\)
\(74\) 0 0
\(75\) −6.46460 −0.746467
\(76\) 0 0
\(77\) 4.28571 0.488402
\(78\) 0 0
\(79\) −1.98830 −0.223701 −0.111850 − 0.993725i \(-0.535678\pi\)
−0.111850 + 0.993725i \(0.535678\pi\)
\(80\) 0 0
\(81\) 4.39128 0.487920
\(82\) 0 0
\(83\) −2.21676 −0.243321 −0.121660 − 0.992572i \(-0.538822\pi\)
−0.121660 + 0.992572i \(0.538822\pi\)
\(84\) 0 0
\(85\) −8.27939 −0.898026
\(86\) 0 0
\(87\) 7.56194 0.810725
\(88\) 0 0
\(89\) 6.33933 0.671968 0.335984 − 0.941868i \(-0.390931\pi\)
0.335984 + 0.941868i \(0.390931\pi\)
\(90\) 0 0
\(91\) 1.43136 0.150047
\(92\) 0 0
\(93\) −7.73061 −0.801627
\(94\) 0 0
\(95\) 4.62927 0.474954
\(96\) 0 0
\(97\) −14.5816 −1.48054 −0.740270 − 0.672310i \(-0.765302\pi\)
−0.740270 + 0.672310i \(0.765302\pi\)
\(98\) 0 0
\(99\) 7.35035 0.738738
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 1664.2.a.ba.1.2 yes 5
4.3 odd 2 1664.2.a.y.1.4 ✓ 5
8.3 odd 2 1664.2.a.bb.1.2 yes 5
8.5 even 2 1664.2.a.z.1.4 yes 5
16.3 odd 4 3328.2.b.bd.1665.7 10
16.5 even 4 3328.2.b.bc.1665.7 10
16.11 odd 4 3328.2.b.bd.1665.4 10
16.13 even 4 3328.2.b.bc.1665.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1664.2.a.y.1.4 ✓ 5 4.3 odd 2
1664.2.a.z.1.4 yes 5 8.5 even 2
1664.2.a.ba.1.2 yes 5 1.1 even 1 trivial
1664.2.a.bb.1.2 yes 5 8.3 odd 2
3328.2.b.bc.1665.4 10 16.13 even 4
3328.2.b.bc.1665.7 10 16.5 even 4
3328.2.b.bd.1665.4 10 16.11 odd 4
3328.2.b.bd.1665.7 10 16.3 odd 4