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Properties

Label 640.2.j.c.607.3
Level $640$
Weight $2$
Character 640.607
Analytic conductor $5.110$
Analytic rank $0$
Dimension $18$
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] = [640,2,Mod(543,640)] 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("640.543"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(640, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.j (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,0,0,0,4,0,-2] 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: no
Analytic conductor: \(5.11042572936\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + \cdots + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 607.3
Root \(1.41323 + 0.0526497i\) of defining polynomial
Character \(\chi\) \(=\) 640.607
Dual form 640.2.j.c.543.7

$q$-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)
 
\(f(q)\) \(=\) \(q-1.28110i q^{3} +(0.841703 + 2.07160i) q^{5} +(1.13975 + 1.13975i) q^{7} +1.35879 q^{9} +(-2.32204 - 2.32204i) q^{11} -1.36502 q^{13} +(2.65392 - 1.07830i) q^{15} +(5.25380 + 5.25380i) q^{17} +(3.69752 + 3.69752i) q^{19} +(1.46013 - 1.46013i) q^{21} +(0.911118 - 0.911118i) q^{23} +(-3.58307 + 3.48735i) q^{25} -5.58403i q^{27} +(-2.37343 + 2.37343i) q^{29} +0.242577i q^{31} +(-2.97475 + 2.97475i) q^{33} +(-1.40178 + 3.32044i) q^{35} +3.34494 q^{37} +1.74872i q^{39} -2.66956i q^{41} +9.04874 q^{43} +(1.14370 + 2.81488i) q^{45} +(7.87820 - 7.87820i) q^{47} -4.40194i q^{49} +(6.73063 - 6.73063i) q^{51} -5.80113i q^{53} +(2.85587 - 6.76480i) q^{55} +(4.73688 - 4.73688i) q^{57} +(-5.91474 + 5.91474i) q^{59} +(6.67404 + 6.67404i) q^{61} +(1.54868 + 1.54868i) q^{63} +(-1.14894 - 2.82778i) q^{65} -4.54673 q^{67} +(-1.16723 - 1.16723i) q^{69} -15.4389 q^{71} +(1.49307 + 1.49307i) q^{73} +(4.46763 + 4.59026i) q^{75} -5.29308i q^{77} -10.3024 q^{79} -3.07731 q^{81} -3.26589i q^{83} +(-6.46165 + 15.3059i) q^{85} +(3.04060 + 3.04060i) q^{87} -9.77206 q^{89} +(-1.55578 - 1.55578i) q^{91} +0.310765 q^{93} +(-4.54758 + 10.7720i) q^{95} +(-1.63587 - 1.63587i) q^{97} +(-3.15516 - 3.15516i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 4 q^{5} - 2 q^{7} - 10 q^{9} - 2 q^{11} - 20 q^{15} - 6 q^{17} + 2 q^{19} + 16 q^{21} + 2 q^{23} + 6 q^{25} + 14 q^{29} - 8 q^{33} - 6 q^{35} - 8 q^{37} - 44 q^{43} + 4 q^{45} + 38 q^{47} + 8 q^{51}+ \cdots - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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
\(3\) − 1.28110i − 0.739642i −0.929103 − 0.369821i \(-0.879419\pi\)
0.929103 − 0.369821i \(-0.120581\pi\)
\(4\) 0 0
\(5\) 0.841703 + 2.07160i 0.376421 + 0.926449i
\(6\) 0 0
\(7\) 1.13975 + 1.13975i 0.430785 + 0.430785i 0.888895 − 0.458111i \(-0.151474\pi\)
−0.458111 + 0.888895i \(0.651474\pi\)
\(8\) 0 0
\(9\) 1.35879 0.452930
\(10\) 0 0
\(11\) −2.32204 − 2.32204i −0.700120 − 0.700120i 0.264316 − 0.964436i \(-0.414854\pi\)
−0.964436 + 0.264316i \(0.914854\pi\)
\(12\) 0 0
\(13\) −1.36502 −0.378589 −0.189294 − 0.981920i \(-0.560620\pi\)
−0.189294 + 0.981920i \(0.560620\pi\)
\(14\) 0 0
\(15\) 2.65392 − 1.07830i 0.685240 − 0.278416i
\(16\) 0 0
\(17\) 5.25380 + 5.25380i 1.27423 + 1.27423i 0.943845 + 0.330389i \(0.107180\pi\)
0.330389 + 0.943845i \(0.392820\pi\)
\(18\) 0 0
\(19\) 3.69752 + 3.69752i 0.848269 + 0.848269i 0.989917 − 0.141648i \(-0.0452403\pi\)
−0.141648 + 0.989917i \(0.545240\pi\)
\(20\) 0 0
\(21\) 1.46013 − 1.46013i 0.318626 − 0.318626i
\(22\) 0 0
\(23\) 0.911118 − 0.911118i 0.189981 − 0.189981i −0.605707 − 0.795688i \(-0.707110\pi\)
0.795688 + 0.605707i \(0.207110\pi\)
\(24\) 0 0
\(25\) −3.58307 + 3.48735i −0.716615 + 0.697469i
\(26\) 0 0
\(27\) − 5.58403i − 1.07465i
\(28\) 0 0
\(29\) −2.37343 + 2.37343i −0.440736 + 0.440736i −0.892259 − 0.451524i \(-0.850881\pi\)
0.451524 + 0.892259i \(0.350881\pi\)
\(30\) 0 0
\(31\) 0.242577i 0.0435681i 0.999763 + 0.0217841i \(0.00693463\pi\)
−0.999763 + 0.0217841i \(0.993065\pi\)
\(32\) 0 0
\(33\) −2.97475 + 2.97475i −0.517838 + 0.517838i
\(34\) 0 0
\(35\) −1.40178 + 3.32044i −0.236944 + 0.561256i
\(36\) 0 0
\(37\) 3.34494 0.549905 0.274953 − 0.961458i \(-0.411338\pi\)
0.274953 + 0.961458i \(0.411338\pi\)
\(38\) 0 0
\(39\) 1.74872i 0.280020i
\(40\) 0 0
\(41\) − 2.66956i − 0.416915i −0.978031 − 0.208457i \(-0.933156\pi\)
0.978031 − 0.208457i \(-0.0668442\pi\)
\(42\) 0 0
\(43\) 9.04874 1.37992 0.689960 − 0.723847i \(-0.257628\pi\)
0.689960 + 0.723847i \(0.257628\pi\)
\(44\) 0 0
\(45\) 1.14370 + 2.81488i 0.170492 + 0.419617i
\(46\) 0 0
\(47\) 7.87820 − 7.87820i 1.14915 − 1.14915i 0.162435 − 0.986719i \(-0.448065\pi\)
0.986719 − 0.162435i \(-0.0519348\pi\)
\(48\) 0 0
\(49\) − 4.40194i − 0.628849i
\(50\) 0 0
\(51\) 6.73063 − 6.73063i 0.942476 − 0.942476i
\(52\) 0 0
\(53\) − 5.80113i − 0.796846i −0.917202 − 0.398423i \(-0.869558\pi\)
0.917202 − 0.398423i \(-0.130442\pi\)
\(54\) 0 0
\(55\) 2.85587 − 6.76480i 0.385086 − 0.912165i
\(56\) 0 0
\(57\) 4.73688 − 4.73688i 0.627415 − 0.627415i
\(58\) 0 0
\(59\) −5.91474 + 5.91474i −0.770033 + 0.770033i −0.978112 − 0.208079i \(-0.933279\pi\)
0.208079 + 0.978112i \(0.433279\pi\)
\(60\) 0 0
\(61\) 6.67404 + 6.67404i 0.854523 + 0.854523i 0.990686 − 0.136163i \(-0.0434772\pi\)
−0.136163 + 0.990686i \(0.543477\pi\)
\(62\) 0 0
\(63\) 1.54868 + 1.54868i 0.195116 + 0.195116i
\(64\) 0 0
\(65\) −1.14894 − 2.82778i −0.142509 − 0.350743i
\(66\) 0 0
\(67\) −4.54673 −0.555471 −0.277736 − 0.960658i \(-0.589584\pi\)
−0.277736 + 0.960658i \(0.589584\pi\)
\(68\) 0 0
\(69\) −1.16723 − 1.16723i −0.140518 − 0.140518i
\(70\) 0 0
\(71\) −15.4389 −1.83226 −0.916128 − 0.400885i \(-0.868703\pi\)
−0.916128 + 0.400885i \(0.868703\pi\)
\(72\) 0 0
\(73\) 1.49307 + 1.49307i 0.174750 + 0.174750i 0.789063 − 0.614313i \(-0.210567\pi\)
−0.614313 + 0.789063i \(0.710567\pi\)
\(74\) 0 0
\(75\) 4.46763 + 4.59026i 0.515877 + 0.530038i
\(76\) 0 0
\(77\) − 5.29308i − 0.603202i
\(78\) 0 0
\(79\) −10.3024 −1.15911 −0.579556 − 0.814932i \(-0.696774\pi\)
−0.579556 + 0.814932i \(0.696774\pi\)
\(80\) 0 0
\(81\) −3.07731 −0.341924
\(82\) 0 0
\(83\) − 3.26589i − 0.358478i −0.983806 − 0.179239i \(-0.942636\pi\)
0.983806 − 0.179239i \(-0.0573636\pi\)
\(84\) 0 0
\(85\) −6.46165 + 15.3059i −0.700864 + 1.66016i
\(86\) 0 0
\(87\) 3.04060 + 3.04060i 0.325986 + 0.325986i
\(88\) 0 0
\(89\) −9.77206 −1.03584 −0.517918 − 0.855430i \(-0.673293\pi\)
−0.517918 + 0.855430i \(0.673293\pi\)
\(90\) 0 0
\(91\) −1.55578 − 1.55578i −0.163090 − 0.163090i
\(92\) 0 0
\(93\) 0.310765 0.0322248
\(94\) 0 0
\(95\) −4.54758 + 10.7720i −0.466571 + 1.10518i
\(96\) 0 0
\(97\) −1.63587 − 1.63587i −0.166097 − 0.166097i 0.619164 − 0.785262i \(-0.287472\pi\)
−0.785262 + 0.619164i \(0.787472\pi\)
\(98\) 0 0
\(99\) −3.15516 − 3.15516i −0.317106 − 0.317106i
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 640.2.j.c.607.3 18
4.3 odd 2 640.2.j.d.607.7 18
5.3 odd 4 640.2.s.c.223.7 18
8.3 odd 2 80.2.j.b.67.2 yes 18
8.5 even 2 320.2.j.b.47.7 18
16.3 odd 4 320.2.s.b.207.3 18
16.5 even 4 640.2.s.d.287.3 18
16.11 odd 4 640.2.s.c.287.7 18
16.13 even 4 80.2.s.b.27.4 yes 18
20.3 even 4 640.2.s.d.223.3 18
24.11 even 2 720.2.bd.g.307.8 18
40.3 even 4 80.2.s.b.3.4 yes 18
40.13 odd 4 320.2.s.b.303.3 18
40.19 odd 2 400.2.j.d.307.8 18
40.27 even 4 400.2.s.d.243.6 18
40.29 even 2 1600.2.j.d.1007.3 18
40.37 odd 4 1600.2.s.d.943.7 18
48.29 odd 4 720.2.z.g.667.6 18
80.3 even 4 320.2.j.b.143.3 18
80.13 odd 4 80.2.j.b.43.2 ✓ 18
80.19 odd 4 1600.2.s.d.207.7 18
80.29 even 4 400.2.s.d.107.6 18
80.43 even 4 inner 640.2.j.c.543.7 18
80.53 odd 4 640.2.j.d.543.3 18
80.67 even 4 1600.2.j.d.143.7 18
80.77 odd 4 400.2.j.d.43.8 18
120.83 odd 4 720.2.z.g.163.6 18
240.173 even 4 720.2.bd.g.523.8 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.b.43.2 ✓ 18 80.13 odd 4
80.2.j.b.67.2 yes 18 8.3 odd 2
80.2.s.b.3.4 yes 18 40.3 even 4
80.2.s.b.27.4 yes 18 16.13 even 4
320.2.j.b.47.7 18 8.5 even 2
320.2.j.b.143.3 18 80.3 even 4
320.2.s.b.207.3 18 16.3 odd 4
320.2.s.b.303.3 18 40.13 odd 4
400.2.j.d.43.8 18 80.77 odd 4
400.2.j.d.307.8 18 40.19 odd 2
400.2.s.d.107.6 18 80.29 even 4
400.2.s.d.243.6 18 40.27 even 4
640.2.j.c.543.7 18 80.43 even 4 inner
640.2.j.c.607.3 18 1.1 even 1 trivial
640.2.j.d.543.3 18 80.53 odd 4
640.2.j.d.607.7 18 4.3 odd 2
640.2.s.c.223.7 18 5.3 odd 4
640.2.s.c.287.7 18 16.11 odd 4
640.2.s.d.223.3 18 20.3 even 4
640.2.s.d.287.3 18 16.5 even 4
720.2.z.g.163.6 18 120.83 odd 4
720.2.z.g.667.6 18 48.29 odd 4
720.2.bd.g.307.8 18 24.11 even 2
720.2.bd.g.523.8 18 240.173 even 4
1600.2.j.d.143.7 18 80.67 even 4
1600.2.j.d.1007.3 18 40.29 even 2
1600.2.s.d.207.7 18 80.19 odd 4
1600.2.s.d.943.7 18 40.37 odd 4