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 1.4
Root \(-0.857815\) 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)
 
\(f(q)\) \(=\) \(q+1.40634 q^{3} +0.422133 q^{5} -4.72205 q^{7} -1.02221 q^{9} +3.71563 q^{11} -1.00000 q^{13} +0.593662 q^{15} +2.39054 q^{17} +8.15998 q^{19} -6.64080 q^{21} +7.87561 q^{23} -4.82180 q^{25} -5.65659 q^{27} +5.87561 q^{29} -0.528307 q^{31} +5.22543 q^{33} -1.99333 q^{35} -1.20954 q^{37} -1.40634 q^{39} +9.03134 q^{41} +2.19374 q^{43} -0.431510 q^{45} +11.1348 q^{47} +15.2977 q^{49} +3.36191 q^{51} +5.03134 q^{53} +1.56849 q^{55} +11.4757 q^{57} +3.71563 q^{59} -14.6567 q^{61} +4.82694 q^{63} -0.422133 q^{65} +6.47144 q^{67} +11.0758 q^{69} -9.34063 q^{71} -16.5070 q^{73} -6.78109 q^{75} -17.5454 q^{77} +11.4313 q^{79} -4.88844 q^{81} +6.08396 q^{83} +1.00913 q^{85} +8.26309 q^{87} +8.63142 q^{89} +4.72205 q^{91} -0.742978 q^{93} +3.44460 q^{95} -0.755814 q^{97} -3.79816 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{3} - q^{5} + 5 q^{7} + 8 q^{9} + 6 q^{11} - 5 q^{13} + 9 q^{15} + 3 q^{17} + 12 q^{19} - 7 q^{21} - 2 q^{23} + 10 q^{25} - 5 q^{27} - 12 q^{29} + 22 q^{31} - 8 q^{33} + 15 q^{35} - 5 q^{37} - q^{39}+ \cdots + 66 q^{99}+O(q^{100}) \) 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
\(3\) 1.40634 0.811950 0.405975 − 0.913884i \(-0.366932\pi\)
0.405975 + 0.913884i \(0.366932\pi\)
\(4\) 0 0
\(5\) 0.422133 0.188784 0.0943918 − 0.995535i \(-0.469909\pi\)
0.0943918 + 0.995535i \(0.469909\pi\)
\(6\) 0 0
\(7\) −4.72205 −1.78477 −0.892383 − 0.451278i \(-0.850968\pi\)
−0.892383 + 0.451278i \(0.850968\pi\)
\(8\) 0 0
\(9\) −1.02221 −0.340738
\(10\) 0 0
\(11\) 3.71563 1.12030 0.560152 − 0.828390i \(-0.310743\pi\)
0.560152 + 0.828390i \(0.310743\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0.593662 0.153283
\(16\) 0 0
\(17\) 2.39054 0.579792 0.289896 − 0.957058i \(-0.406379\pi\)
0.289896 + 0.957058i \(0.406379\pi\)
\(18\) 0 0
\(19\) 8.15998 1.87203 0.936013 − 0.351964i \(-0.114486\pi\)
0.936013 + 0.351964i \(0.114486\pi\)
\(20\) 0 0
\(21\) −6.64080 −1.44914
\(22\) 0 0
\(23\) 7.87561 1.64218 0.821089 − 0.570801i \(-0.193367\pi\)
0.821089 + 0.570801i \(0.193367\pi\)
\(24\) 0 0
\(25\) −4.82180 −0.964361
\(26\) 0 0
\(27\) −5.65659 −1.08861
\(28\) 0 0
\(29\) 5.87561 1.09107 0.545536 − 0.838087i \(-0.316326\pi\)
0.545536 + 0.838087i \(0.316326\pi\)
\(30\) 0 0
\(31\) −0.528307 −0.0948867 −0.0474433 − 0.998874i \(-0.515107\pi\)
−0.0474433 + 0.998874i \(0.515107\pi\)
\(32\) 0 0
\(33\) 5.22543 0.909631
\(34\) 0 0
\(35\) −1.99333 −0.336935
\(36\) 0 0
\(37\) −1.20954 −0.198847 −0.0994233 − 0.995045i \(-0.531700\pi\)
−0.0994233 + 0.995045i \(0.531700\pi\)
\(38\) 0 0
\(39\) −1.40634 −0.225194
\(40\) 0 0
\(41\) 9.03134 1.41046 0.705229 − 0.708979i \(-0.250844\pi\)
0.705229 + 0.708979i \(0.250844\pi\)
\(42\) 0 0
\(43\) 2.19374 0.334542 0.167271 − 0.985911i \(-0.446504\pi\)
0.167271 + 0.985911i \(0.446504\pi\)
\(44\) 0 0
\(45\) −0.431510 −0.0643257
\(46\) 0 0
\(47\) 11.1348 1.62418 0.812089 − 0.583533i \(-0.198330\pi\)
0.812089 + 0.583533i \(0.198330\pi\)
\(48\) 0 0
\(49\) 15.2977 2.18539
\(50\) 0 0
\(51\) 3.36191 0.470762
\(52\) 0 0
\(53\) 5.03134 0.691108 0.345554 − 0.938399i \(-0.387691\pi\)
0.345554 + 0.938399i \(0.387691\pi\)
\(54\) 0 0
\(55\) 1.56849 0.211495
\(56\) 0 0
\(57\) 11.4757 1.51999
\(58\) 0 0
\(59\) 3.71563 0.483734 0.241867 − 0.970309i \(-0.422240\pi\)
0.241867 + 0.970309i \(0.422240\pi\)
\(60\) 0 0
\(61\) −14.6567 −1.87660 −0.938299 − 0.345826i \(-0.887599\pi\)
−0.938299 + 0.345826i \(0.887599\pi\)
\(62\) 0 0
\(63\) 4.82694 0.608137
\(64\) 0 0
\(65\) −0.422133 −0.0523592
\(66\) 0 0
\(67\) 6.47144 0.790613 0.395306 − 0.918549i \(-0.370638\pi\)
0.395306 + 0.918549i \(0.370638\pi\)
\(68\) 0 0
\(69\) 11.0758 1.33337
\(70\) 0 0
\(71\) −9.34063 −1.10853 −0.554265 − 0.832341i \(-0.687000\pi\)
−0.554265 + 0.832341i \(0.687000\pi\)
\(72\) 0 0
\(73\) −16.5070 −1.93200 −0.966001 − 0.258540i \(-0.916759\pi\)
−0.966001 + 0.258540i \(0.916759\pi\)
\(74\) 0 0
\(75\) −6.78109 −0.783012
\(76\) 0 0
\(77\) −17.5454 −1.99948
\(78\) 0 0
\(79\) 11.4313 1.28612 0.643059 − 0.765817i \(-0.277665\pi\)
0.643059 + 0.765817i \(0.277665\pi\)
\(80\) 0 0
\(81\) −4.88844 −0.543160
\(82\) 0 0
\(83\) 6.08396 0.667801 0.333901 − 0.942608i \(-0.391635\pi\)
0.333901 + 0.942608i \(0.391635\pi\)
\(84\) 0 0
\(85\) 1.00913 0.109455
\(86\) 0 0
\(87\) 8.26309 0.885896
\(88\) 0 0
\(89\) 8.63142 0.914929 0.457464 − 0.889228i \(-0.348758\pi\)
0.457464 + 0.889228i \(0.348758\pi\)
\(90\) 0 0
\(91\) 4.72205 0.495005
\(92\) 0 0
\(93\) −0.742978 −0.0770432
\(94\) 0 0
\(95\) 3.44460 0.353408
\(96\) 0 0
\(97\) −0.755814 −0.0767413 −0.0383706 − 0.999264i \(-0.512217\pi\)
−0.0383706 + 0.999264i \(0.512217\pi\)
\(98\) 0 0
\(99\) −3.79816 −0.381730
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.1.4 yes 5
4.3 odd 2 1664.2.a.y.1.2 ✓ 5
8.3 odd 2 1664.2.a.bb.1.4 yes 5
8.5 even 2 1664.2.a.z.1.2 yes 5
16.3 odd 4 3328.2.b.bd.1665.3 10
16.5 even 4 3328.2.b.bc.1665.3 10
16.11 odd 4 3328.2.b.bd.1665.8 10
16.13 even 4 3328.2.b.bc.1665.8 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1664.2.a.y.1.2 ✓ 5 4.3 odd 2
1664.2.a.z.1.2 yes 5 8.5 even 2
1664.2.a.ba.1.4 yes 5 1.1 even 1 trivial
1664.2.a.bb.1.4 yes 5 8.3 odd 2
3328.2.b.bc.1665.3 10 16.5 even 4
3328.2.b.bc.1665.8 10 16.13 even 4
3328.2.b.bd.1665.3 10 16.3 odd 4
3328.2.b.bd.1665.8 10 16.11 odd 4