Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [180,5,Mod(37,180)] 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("180.37"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(180, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 180.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,6] 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: no
Analytic conductor: \(18.6065933551\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 37.2
Root \(8.26209i\) of defining polynomial
Character \(\chi\) \(=\) 180.37
Dual form 180.5.l.a.73.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(24.7863 - 3.26209i) q^{5} +(50.7863 + 50.7863i) q^{7} -2.62087 q^{11} +(-43.4275 + 43.4275i) q^{13} +(-131.331 - 131.331i) q^{17} +403.725i q^{19} +(334.455 - 334.455i) q^{23} +(603.718 - 161.710i) q^{25} +1172.97i q^{29} +955.313 q^{31} +(1424.47 + 1093.13i) q^{35} +(673.305 + 673.305i) q^{37} -818.621 q^{41} +(2.48083 - 2.48083i) q^{43} +(1563.81 + 1563.81i) q^{47} +2757.49i q^{49} +(-277.662 + 277.662i) q^{53} +(-64.9617 + 8.54952i) q^{55} -6333.66i q^{59} +6518.49 q^{61} +(-934.741 + 1218.07i) q^{65} +(-713.488 - 713.488i) q^{67} -288.280 q^{71} +(-5564.19 + 5564.19i) q^{73} +(-133.104 - 133.104i) q^{77} -4061.80i q^{79} +(-1284.98 + 1284.98i) q^{83} +(-3683.61 - 2826.79i) q^{85} -4418.77i q^{89} -4411.04 q^{91} +(1316.99 + 10006.8i) q^{95} +(-8805.47 - 8805.47i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{5} + 110 q^{7} + 300 q^{11} - 360 q^{13} - 960 q^{17} + 810 q^{23} + 1856 q^{25} - 836 q^{31} + 2562 q^{35} - 660 q^{37} - 2964 q^{41} + 3270 q^{43} + 2250 q^{47} - 1980 q^{53} - 6780 q^{55}+ \cdots - 3180 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 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
\(9\) 0 0
\(10\) 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 −0.823820 − 0.566852i \(-0.808161\pi\)
0.566852 + 0.823820i \(0.308161\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −131.331 − 131.331i −0.454432 − 0.454432i 0.442391 − 0.896822i \(-0.354130\pi\)
−0.896822 + 0.442391i \(0.854130\pi\)
\(18\) 0 0
\(19\) 403.725i 1.11835i 0.829049 + 0.559176i \(0.188882\pi\)
−0.829049 + 0.559176i \(0.811118\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 334.455 − 334.455i 0.632241 − 0.632241i −0.316389 − 0.948630i \(-0.602470\pi\)
0.948630 + 0.316389i \(0.102470\pi\)
\(24\) 0 0
\(25\) 603.718 − 161.710i 0.965948 − 0.258736i
\(26\) 0 0
\(27\) 0 0
\(28\) 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
\(33\) 0 0
\(34\) 0 0
\(35\) 1424.47 + 1093.13i 1.16283 + 0.892353i
\(36\) 0 0
\(37\) 673.305 + 673.305i 0.491823 + 0.491823i 0.908880 − 0.417057i \(-0.136939\pi\)
−0.417057 + 0.908880i \(0.636939\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −818.621 −0.486984 −0.243492 − 0.969903i \(-0.578293\pi\)
−0.243492 + 0.969903i \(0.578293\pi\)
\(42\) 0 0
\(43\) 2.48083 − 2.48083i 0.00134171 − 0.00134171i −0.706436 − 0.707777i \(-0.749698\pi\)
0.707777 + 0.706436i \(0.249698\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1563.81 + 1563.81i 0.707926 + 0.707926i 0.966099 − 0.258173i \(-0.0831203\pi\)
−0.258173 + 0.966099i \(0.583120\pi\)
\(48\) 0 0
\(49\) 2757.49i 1.14848i
\(50\) 0 0
\(51\) 0 0
\(52\) 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
\(57\) 0 0
\(58\) 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 \(-0.160265\pi\)
0.875906 + 0.482483i \(0.160265\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −934.741 + 1218.07i −0.221240 + 0.288300i
\(66\) 0 0
\(67\) −713.488 − 713.488i −0.158942 − 0.158942i 0.623156 − 0.782098i \(-0.285850\pi\)
−0.782098 + 0.623156i \(0.785850\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −288.280 −0.0571871 −0.0285935 − 0.999591i \(-0.509103\pi\)
−0.0285935 + 0.999591i \(0.509103\pi\)
\(72\) 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
\(75\) 0 0
\(76\) 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
\(81\) 0 0
\(82\) 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
\(87\) 0 0
\(88\) 0 0
\(89\) − 4418.77i − 0.557855i −0.960312 − 0.278927i \(-0.910021\pi\)
0.960312 − 0.278927i \(-0.0899789\pi\)
\(90\) 0 0
\(91\) −4411.04 −0.532670
\(92\) 0 0
\(93\) 0 0
\(94\) 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
\(99\) 0 0
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 180.5.l.a.37.2 4
3.2 odd 2 20.5.f.a.17.1 yes 4
5.2 odd 4 900.5.l.a.793.1 4
5.3 odd 4 inner 180.5.l.a.73.2 4
5.4 even 2 900.5.l.a.757.1 4
12.11 even 2 80.5.p.e.17.2 4
15.2 even 4 100.5.f.c.93.2 4
15.8 even 4 20.5.f.a.13.1 ✓ 4
15.14 odd 2 100.5.f.c.57.2 4
24.5 odd 2 320.5.p.m.257.2 4
24.11 even 2 320.5.p.l.257.1 4
60.23 odd 4 80.5.p.e.33.2 4
60.47 odd 4 400.5.p.h.193.1 4
60.59 even 2 400.5.p.h.257.1 4
120.53 even 4 320.5.p.m.193.2 4
120.83 odd 4 320.5.p.l.193.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.5.f.a.13.1 ✓ 4 15.8 even 4
20.5.f.a.17.1 yes 4 3.2 odd 2
80.5.p.e.17.2 4 12.11 even 2
80.5.p.e.33.2 4 60.23 odd 4
100.5.f.c.57.2 4 15.14 odd 2
100.5.f.c.93.2 4 15.2 even 4
180.5.l.a.37.2 4 1.1 even 1 trivial
180.5.l.a.73.2 4 5.3 odd 4 inner
320.5.p.l.193.1 4 120.83 odd 4
320.5.p.l.257.1 4 24.11 even 2
320.5.p.m.193.2 4 120.53 even 4
320.5.p.m.257.2 4 24.5 odd 2
400.5.p.h.193.1 4 60.47 odd 4
400.5.p.h.257.1 4 60.59 even 2
900.5.l.a.757.1 4 5.4 even 2
900.5.l.a.793.1 4 5.2 odd 4