Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [320,5,Mod(193,320)] 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("320.193"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 320.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,10,0,6,0,110] 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: \(33.0783881868\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{241})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 121x^{2} + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 257.2
Root \(-8.26209i\) of defining polynomial
Character \(\chi\) \(=\) 320.257
Dual form 320.5.p.m.193.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(10.2621 - 10.2621i) q^{3} +(24.7863 - 3.26209i) q^{5} +(50.7863 + 50.7863i) q^{7} -129.621i q^{9} -2.62087 q^{11} +(43.4275 - 43.4275i) q^{13} +(220.883 - 287.835i) q^{15} +(131.331 + 131.331i) q^{17} -403.725i q^{19} +1042.35 q^{21} +(-334.455 + 334.455i) q^{23} +(603.718 - 161.710i) q^{25} +(-498.952 - 498.952i) q^{27} +1172.97i q^{29} +955.313 q^{31} +(-26.8956 + 26.8956i) q^{33} +(1424.47 + 1093.13i) q^{35} +(-673.305 - 673.305i) q^{37} -891.313i q^{39} +818.621 q^{41} +(-2.48083 + 2.48083i) q^{43} +(-422.835 - 3212.82i) q^{45} +(-1563.81 - 1563.81i) q^{47} +2757.49i q^{49} +2695.46 q^{51} +(-277.662 + 277.662i) q^{53} +(-64.9617 + 8.54952i) q^{55} +(-4143.06 - 4143.06i) q^{57} -6333.66i q^{59} -6518.49 q^{61} +(6582.96 - 6582.96i) q^{63} +(934.741 - 1218.07i) q^{65} +(713.488 + 713.488i) q^{67} +6864.42i q^{69} +288.280 q^{71} +(-5564.19 + 5564.19i) q^{73} +(4535.92 - 7854.88i) q^{75} +(-133.104 - 133.104i) q^{77} -4061.80i q^{79} +258.720 q^{81} +(-1284.98 + 1284.98i) q^{83} +(3683.61 + 2826.79i) q^{85} +(12037.1 + 12037.1i) q^{87} +4418.77i q^{89} +4411.04 q^{91} +(9803.51 - 9803.51i) q^{93} +(-1316.99 - 10006.8i) q^{95} +(-8805.47 - 8805.47i) q^{97} +339.720i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{3} + 6 q^{5} + 110 q^{7} + 300 q^{11} + 360 q^{13} + 542 q^{15} + 960 q^{17} + 1996 q^{21} - 810 q^{23} + 1856 q^{25} - 2120 q^{27} - 836 q^{31} - 1660 q^{33} + 2562 q^{35} + 660 q^{37} + 2964 q^{41}+ \cdots - 3180 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(1\)

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
\(3\) 10.2621 − 10.2621i 1.14023 − 1.14023i 0.151824 − 0.988407i \(-0.451485\pi\)
0.988407 − 0.151824i \(-0.0485148\pi\)
\(4\) 0 0
\(5\) 24.7863 − 3.26209i 0.991450 − 0.130483i
\(6\) 0 0
\(7\) 50.7863 + 50.7863i 1.03645 + 1.03645i 0.999310 + 0.0371444i \(0.0118262\pi\)
0.0371444 + 0.999310i \(0.488174\pi\)
\(8\) 0 0
\(9\) − 129.621i − 1.60026i
\(10\) 0 0
\(11\) −2.62087 −0.0216601 −0.0108301 − 0.999941i \(-0.503447\pi\)
−0.0108301 + 0.999941i \(0.503447\pi\)
\(12\) 0 0
\(13\) 43.4275 − 43.4275i 0.256967 − 0.256967i −0.566852 − 0.823820i \(-0.691839\pi\)
0.823820 + 0.566852i \(0.191839\pi\)
\(14\) 0 0
\(15\) 220.883 − 287.835i 0.981702 − 1.27926i
\(16\) 0 0
\(17\) 131.331 + 131.331i 0.454432 + 0.454432i 0.896822 − 0.442391i \(-0.145870\pi\)
−0.442391 + 0.896822i \(0.645870\pi\)
\(18\) 0 0
\(19\) − 403.725i − 1.11835i −0.829049 − 0.559176i \(-0.811118\pi\)
0.829049 − 0.559176i \(-0.188882\pi\)
\(20\) 0 0
\(21\) 1042.35 2.36360
\(22\) 0 0
\(23\) −334.455 + 334.455i −0.632241 + 0.632241i −0.948630 − 0.316389i \(-0.897530\pi\)
0.316389 + 0.948630i \(0.397530\pi\)
\(24\) 0 0
\(25\) 603.718 − 161.710i 0.965948 − 0.258736i
\(26\) 0 0
\(27\) −498.952 − 498.952i −0.684433 − 0.684433i
\(28\) 0 0
\(29\) 1172.97i 1.39473i 0.716717 + 0.697364i \(0.245644\pi\)
−0.716717 + 0.697364i \(0.754356\pi\)
\(30\) 0 0
\(31\) 955.313 0.994082 0.497041 − 0.867727i \(-0.334420\pi\)
0.497041 + 0.867727i \(0.334420\pi\)
\(32\) 0 0
\(33\) −26.8956 + 26.8956i −0.0246976 + 0.0246976i
\(34\) 0 0
\(35\) 1424.47 + 1093.13i 1.16283 + 0.892353i
\(36\) 0 0
\(37\) −673.305 − 673.305i −0.491823 − 0.491823i 0.417057 − 0.908880i \(-0.363061\pi\)
−0.908880 + 0.417057i \(0.863061\pi\)
\(38\) 0 0
\(39\) − 891.313i − 0.586005i
\(40\) 0 0
\(41\) 818.621 0.486984 0.243492 − 0.969903i \(-0.421707\pi\)
0.243492 + 0.969903i \(0.421707\pi\)
\(42\) 0 0
\(43\) −2.48083 + 2.48083i −0.00134171 + 0.00134171i −0.707777 − 0.706436i \(-0.750302\pi\)
0.706436 + 0.707777i \(0.250302\pi\)
\(44\) 0 0
\(45\) −422.835 − 3212.82i −0.208807 − 1.58658i
\(46\) 0 0
\(47\) −1563.81 − 1563.81i −0.707926 − 0.707926i 0.258173 − 0.966099i \(-0.416880\pi\)
−0.966099 + 0.258173i \(0.916880\pi\)
\(48\) 0 0
\(49\) 2757.49i 1.14848i
\(50\) 0 0
\(51\) 2695.46 1.03632
\(52\) 0 0
\(53\) −277.662 + 277.662i −0.0988471 + 0.0988471i −0.754801 − 0.655954i \(-0.772267\pi\)
0.655954 + 0.754801i \(0.272267\pi\)
\(54\) 0 0
\(55\) −64.9617 + 8.54952i −0.0214749 + 0.00282629i
\(56\) 0 0
\(57\) −4143.06 − 4143.06i −1.27518 − 1.27518i
\(58\) 0 0
\(59\) − 6333.66i − 1.81949i −0.415163 − 0.909747i \(-0.636275\pi\)
0.415163 − 0.909747i \(-0.363725\pi\)
\(60\) 0 0
\(61\) −6518.49 −1.75181 −0.875906 − 0.482483i \(-0.839735\pi\)
−0.875906 + 0.482483i \(0.839735\pi\)
\(62\) 0 0
\(63\) 6582.96 − 6582.96i 1.65859 − 1.65859i
\(64\) 0 0
\(65\) 934.741 − 1218.07i 0.221240 − 0.288300i
\(66\) 0 0
\(67\) 713.488 + 713.488i 0.158942 + 0.158942i 0.782098 − 0.623156i \(-0.214150\pi\)
−0.623156 + 0.782098i \(0.714150\pi\)
\(68\) 0 0
\(69\) 6864.42i 1.44180i
\(70\) 0 0
\(71\) 288.280 0.0571871 0.0285935 − 0.999591i \(-0.490897\pi\)
0.0285935 + 0.999591i \(0.490897\pi\)
\(72\) 0 0
\(73\) −5564.19 + 5564.19i −1.04413 + 1.04413i −0.0451542 + 0.998980i \(0.514378\pi\)
−0.998980 + 0.0451542i \(0.985622\pi\)
\(74\) 0 0
\(75\) 4535.92 − 7854.88i 0.806386 − 1.39642i
\(76\) 0 0
\(77\) −133.104 − 133.104i −0.0224497 − 0.0224497i
\(78\) 0 0
\(79\) − 4061.80i − 0.650825i −0.945572 − 0.325413i \(-0.894497\pi\)
0.945572 − 0.325413i \(-0.105503\pi\)
\(80\) 0 0
\(81\) 258.720 0.0394330
\(82\) 0 0
\(83\) −1284.98 + 1284.98i −0.186527 + 0.186527i −0.794193 − 0.607666i \(-0.792106\pi\)
0.607666 + 0.794193i \(0.292106\pi\)
\(84\) 0 0
\(85\) 3683.61 + 2826.79i 0.509842 + 0.391251i
\(86\) 0 0
\(87\) 12037.1 + 12037.1i 1.59031 + 1.59031i
\(88\) 0 0
\(89\) 4418.77i 0.557855i 0.960312 + 0.278927i \(0.0899789\pi\)
−0.960312 + 0.278927i \(0.910021\pi\)
\(90\) 0 0
\(91\) 4411.04 0.532670
\(92\) 0 0
\(93\) 9803.51 − 9803.51i 1.13348 − 1.13348i
\(94\) 0 0
\(95\) −1316.99 − 10006.8i −0.145927 − 1.10879i
\(96\) 0 0
\(97\) −8805.47 − 8805.47i −0.935857 − 0.935857i 0.0622067 − 0.998063i \(-0.480186\pi\)
−0.998063 + 0.0622067i \(0.980186\pi\)
\(98\) 0 0
\(99\) 339.720i 0.0346618i
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 320.5.p.m.257.2 4
4.3 odd 2 320.5.p.l.257.1 4
5.3 odd 4 inner 320.5.p.m.193.2 4
8.3 odd 2 80.5.p.e.17.2 4
8.5 even 2 20.5.f.a.17.1 yes 4
20.3 even 4 320.5.p.l.193.1 4
24.5 odd 2 180.5.l.a.37.2 4
40.3 even 4 80.5.p.e.33.2 4
40.13 odd 4 20.5.f.a.13.1 ✓ 4
40.19 odd 2 400.5.p.h.257.1 4
40.27 even 4 400.5.p.h.193.1 4
40.29 even 2 100.5.f.c.57.2 4
40.37 odd 4 100.5.f.c.93.2 4
120.29 odd 2 900.5.l.a.757.1 4
120.53 even 4 180.5.l.a.73.2 4
120.77 even 4 900.5.l.a.793.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.5.f.a.13.1 ✓ 4 40.13 odd 4
20.5.f.a.17.1 yes 4 8.5 even 2
80.5.p.e.17.2 4 8.3 odd 2
80.5.p.e.33.2 4 40.3 even 4
100.5.f.c.57.2 4 40.29 even 2
100.5.f.c.93.2 4 40.37 odd 4
180.5.l.a.37.2 4 24.5 odd 2
180.5.l.a.73.2 4 120.53 even 4
320.5.p.l.193.1 4 20.3 even 4
320.5.p.l.257.1 4 4.3 odd 2
320.5.p.m.193.2 4 5.3 odd 4 inner
320.5.p.m.257.2 4 1.1 even 1 trivial
400.5.p.h.193.1 4 40.27 even 4
400.5.p.h.257.1 4 40.19 odd 2
900.5.l.a.757.1 4 120.29 odd 2
900.5.l.a.793.1 4 120.77 even 4