Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,3,Mod(17,72)] 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("72.17"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 72.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.96185790339\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 17.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 72.17
Dual form 72.3.e.a.17.1

$q$-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)
 
\(f(q)\) \(=\) \(q+7.07107i q^{5} +12.0000 q^{7} -5.65685i q^{11} -8.00000 q^{13} +9.89949i q^{17} -16.0000 q^{19} -39.5980i q^{23} -25.0000 q^{25} -29.6985i q^{29} -4.00000 q^{31} +84.8528i q^{35} +30.0000 q^{37} -21.2132i q^{41} -8.00000 q^{43} -16.9706i q^{47} +95.0000 q^{49} +49.4975i q^{53} +40.0000 q^{55} +79.1960i q^{59} -14.0000 q^{61} -56.5685i q^{65} -88.0000 q^{67} -28.2843i q^{71} -80.0000 q^{73} -67.8823i q^{77} +100.000 q^{79} -130.108i q^{83} -70.0000 q^{85} +148.492i q^{89} -96.0000 q^{91} -113.137i q^{95} -112.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 24 q^{7} - 16 q^{13} - 32 q^{19} - 50 q^{25} - 8 q^{31} + 60 q^{37} - 16 q^{43} + 190 q^{49} + 80 q^{55} - 28 q^{61} - 176 q^{67} - 160 q^{73} + 200 q^{79} - 140 q^{85} - 192 q^{91} - 224 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(5\) 7.07107i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) 0 0
\(7\) 12.0000 1.71429 0.857143 − 0.515079i \(-0.172237\pi\)
0.857143 + 0.515079i \(0.172237\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 5.65685i − 0.514259i −0.966377 − 0.257130i \(-0.917223\pi\)
0.966377 − 0.257130i \(-0.0827768\pi\)
\(12\) 0 0
\(13\) −8.00000 −0.615385 −0.307692 − 0.951486i \(-0.599557\pi\)
−0.307692 + 0.951486i \(0.599557\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 9.89949i 0.582323i 0.956674 + 0.291162i \(0.0940417\pi\)
−0.956674 + 0.291162i \(0.905958\pi\)
\(18\) 0 0
\(19\) −16.0000 −0.842105 −0.421053 − 0.907036i \(-0.638339\pi\)
−0.421053 + 0.907036i \(0.638339\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 39.5980i − 1.72165i −0.508900 − 0.860826i \(-0.669948\pi\)
0.508900 − 0.860826i \(-0.330052\pi\)
\(24\) 0 0
\(25\) −25.0000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 29.6985i − 1.02409i −0.858960 − 0.512043i \(-0.828889\pi\)
0.858960 − 0.512043i \(-0.171111\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.129032 −0.0645161 − 0.997917i \(-0.520550\pi\)
−0.0645161 + 0.997917i \(0.520550\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 84.8528i 2.42437i
\(36\) 0 0
\(37\) 30.0000 0.810811 0.405405 − 0.914137i \(-0.367130\pi\)
0.405405 + 0.914137i \(0.367130\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 21.2132i − 0.517395i −0.965958 − 0.258698i \(-0.916707\pi\)
0.965958 − 0.258698i \(-0.0832933\pi\)
\(42\) 0 0
\(43\) −8.00000 −0.186047 −0.0930233 − 0.995664i \(-0.529653\pi\)
−0.0930233 + 0.995664i \(0.529653\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 16.9706i − 0.361076i −0.983568 − 0.180538i \(-0.942216\pi\)
0.983568 − 0.180538i \(-0.0577838\pi\)
\(48\) 0 0
\(49\) 95.0000 1.93878
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 49.4975i 0.933915i 0.884280 + 0.466957i \(0.154650\pi\)
−0.884280 + 0.466957i \(0.845350\pi\)
\(54\) 0 0
\(55\) 40.0000 0.727273
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 79.1960i 1.34230i 0.741320 + 0.671152i \(0.234200\pi\)
−0.741320 + 0.671152i \(0.765800\pi\)
\(60\) 0 0
\(61\) −14.0000 −0.229508 −0.114754 − 0.993394i \(-0.536608\pi\)
−0.114754 + 0.993394i \(0.536608\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 56.5685i − 0.870285i
\(66\) 0 0
\(67\) −88.0000 −1.31343 −0.656716 − 0.754138i \(-0.728055\pi\)
−0.656716 + 0.754138i \(0.728055\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 28.2843i − 0.398370i −0.979962 − 0.199185i \(-0.936171\pi\)
0.979962 − 0.199185i \(-0.0638295\pi\)
\(72\) 0 0
\(73\) −80.0000 −1.09589 −0.547945 − 0.836514i \(-0.684590\pi\)
−0.547945 + 0.836514i \(0.684590\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 67.8823i − 0.881588i
\(78\) 0 0
\(79\) 100.000 1.26582 0.632911 − 0.774224i \(-0.281860\pi\)
0.632911 + 0.774224i \(0.281860\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 130.108i − 1.56756i −0.621037 − 0.783781i \(-0.713288\pi\)
0.621037 − 0.783781i \(-0.286712\pi\)
\(84\) 0 0
\(85\) −70.0000 −0.823529
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 148.492i 1.66845i 0.551421 + 0.834227i \(0.314086\pi\)
−0.551421 + 0.834227i \(0.685914\pi\)
\(90\) 0 0
\(91\) −96.0000 −1.05495
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 113.137i − 1.19092i
\(96\) 0 0
\(97\) −112.000 −1.15464 −0.577320 − 0.816518i \(-0.695901\pi\)
−0.577320 + 0.816518i \(0.695901\pi\)
\(98\) 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 72.3.e.a.17.2 yes 2
3.2 odd 2 inner 72.3.e.a.17.1 ✓ 2
4.3 odd 2 144.3.e.a.17.2 2
5.2 odd 4 1800.3.c.a.449.3 4
5.3 odd 4 1800.3.c.a.449.1 4
5.4 even 2 1800.3.l.a.1601.1 2
7.6 odd 2 3528.3.d.a.1961.1 2
8.3 odd 2 576.3.e.a.449.1 2
8.5 even 2 576.3.e.h.449.1 2
9.2 odd 6 648.3.m.a.377.2 4
9.4 even 3 648.3.m.a.593.2 4
9.5 odd 6 648.3.m.a.593.1 4
9.7 even 3 648.3.m.a.377.1 4
12.11 even 2 144.3.e.a.17.1 2
15.2 even 4 1800.3.c.a.449.4 4
15.8 even 4 1800.3.c.a.449.2 4
15.14 odd 2 1800.3.l.a.1601.2 2
16.3 odd 4 2304.3.h.h.2177.3 4
16.5 even 4 2304.3.h.a.2177.2 4
16.11 odd 4 2304.3.h.h.2177.2 4
16.13 even 4 2304.3.h.a.2177.3 4
20.3 even 4 3600.3.c.c.449.4 4
20.7 even 4 3600.3.c.c.449.2 4
20.19 odd 2 3600.3.l.l.1601.2 2
21.20 even 2 3528.3.d.a.1961.2 2
24.5 odd 2 576.3.e.h.449.2 2
24.11 even 2 576.3.e.a.449.2 2
36.7 odd 6 1296.3.q.k.1025.1 4
36.11 even 6 1296.3.q.k.1025.2 4
36.23 even 6 1296.3.q.k.593.1 4
36.31 odd 6 1296.3.q.k.593.2 4
48.5 odd 4 2304.3.h.a.2177.4 4
48.11 even 4 2304.3.h.h.2177.4 4
48.29 odd 4 2304.3.h.a.2177.1 4
48.35 even 4 2304.3.h.h.2177.1 4
60.23 odd 4 3600.3.c.c.449.3 4
60.47 odd 4 3600.3.c.c.449.1 4
60.59 even 2 3600.3.l.l.1601.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.3.e.a.17.1 ✓ 2 3.2 odd 2 inner
72.3.e.a.17.2 yes 2 1.1 even 1 trivial
144.3.e.a.17.1 2 12.11 even 2
144.3.e.a.17.2 2 4.3 odd 2
576.3.e.a.449.1 2 8.3 odd 2
576.3.e.a.449.2 2 24.11 even 2
576.3.e.h.449.1 2 8.5 even 2
576.3.e.h.449.2 2 24.5 odd 2
648.3.m.a.377.1 4 9.7 even 3
648.3.m.a.377.2 4 9.2 odd 6
648.3.m.a.593.1 4 9.5 odd 6
648.3.m.a.593.2 4 9.4 even 3
1296.3.q.k.593.1 4 36.23 even 6
1296.3.q.k.593.2 4 36.31 odd 6
1296.3.q.k.1025.1 4 36.7 odd 6
1296.3.q.k.1025.2 4 36.11 even 6
1800.3.c.a.449.1 4 5.3 odd 4
1800.3.c.a.449.2 4 15.8 even 4
1800.3.c.a.449.3 4 5.2 odd 4
1800.3.c.a.449.4 4 15.2 even 4
1800.3.l.a.1601.1 2 5.4 even 2
1800.3.l.a.1601.2 2 15.14 odd 2
2304.3.h.a.2177.1 4 48.29 odd 4
2304.3.h.a.2177.2 4 16.5 even 4
2304.3.h.a.2177.3 4 16.13 even 4
2304.3.h.a.2177.4 4 48.5 odd 4
2304.3.h.h.2177.1 4 48.35 even 4
2304.3.h.h.2177.2 4 16.11 odd 4
2304.3.h.h.2177.3 4 16.3 odd 4
2304.3.h.h.2177.4 4 48.11 even 4
3528.3.d.a.1961.1 2 7.6 odd 2
3528.3.d.a.1961.2 2 21.20 even 2
3600.3.c.c.449.1 4 60.47 odd 4
3600.3.c.c.449.2 4 20.7 even 4
3600.3.c.c.449.3 4 60.23 odd 4
3600.3.c.c.449.4 4 20.3 even 4
3600.3.l.l.1601.1 2 60.59 even 2
3600.3.l.l.1601.2 2 20.19 odd 2