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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,8,Mod(25,72)] 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("72.25"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 72.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.4917218349\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 49.5
Character \(\chi\) \(=\) 72.49
Dual form 72.8.i.b.25.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-9.85077 + 45.7161i) q^{3} +(-186.216 + 322.535i) q^{5} +(726.432 + 1258.22i) q^{7} +(-1992.92 - 900.678i) q^{9} +(3510.15 + 6079.76i) q^{11} +(-262.017 + 453.827i) q^{13} +(-12910.7 - 11690.3i) q^{15} +17080.7 q^{17} -21851.6 q^{19} +(-64676.7 + 20815.2i) q^{21} +(-1873.48 + 3244.97i) q^{23} +(-30290.1 - 52463.9i) q^{25} +(60807.3 - 82236.4i) q^{27} +(15522.1 + 26885.1i) q^{29} +(129142. - 223680. i) q^{31} +(-312521. + 100580. i) q^{33} -541092. q^{35} -438830. q^{37} +(-18166.1 - 16449.0i) q^{39} +(-99343.5 + 172068. i) q^{41} +(96178.1 + 166585. i) q^{43} +(661614. - 475068. i) q^{45} +(-441116. - 764035. i) q^{47} +(-643637. + 1.11481e6i) q^{49} +(-168258. + 780864. i) q^{51} +1.78412e6 q^{53} -2.61458e6 q^{55} +(215255. - 998970. i) q^{57} +(-464224. + 804060. i) q^{59} +(646828. + 1.12034e6i) q^{61} +(-314476. - 3.16181e6i) q^{63} +(-97583.4 - 169019. i) q^{65} +(2.05729e6 - 3.56333e6i) q^{67} +(-129892. - 117614. i) q^{69} +5.60557e6 q^{71} +2.84957e6 q^{73} +(2.69683e6 - 867934. i) q^{75} +(-5.09978e6 + 8.83308e6i) q^{77} +(-1.91864e6 - 3.32318e6i) q^{79} +(3.16053e6 + 3.58997e6i) q^{81} +(4.14708e6 + 7.18296e6i) q^{83} +(-3.18070e6 + 5.50913e6i) q^{85} +(-1.38199e6 + 444772. i) q^{87} +7.67933e6 q^{89} -761351. q^{91} +(8.95362e6 + 8.10727e6i) q^{93} +(4.06911e6 - 7.04791e6i) q^{95} +(-5.27913e6 - 9.14373e6i) q^{97} +(-1.51956e6 - 1.52780e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 44 q^{3} - 125 q^{5} - 1245 q^{7} - 3766 q^{9} + 1699 q^{11} - 4937 q^{13} - 19349 q^{15} + 26540 q^{17} + 28976 q^{19} - 75315 q^{21} - 18239 q^{23} - 109168 q^{25} - 87680 q^{27} - 3525 q^{29}+ \cdots + 24043955 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) −9.85077 + 45.7161i −0.210642 + 0.977563i
\(4\) 0 0
\(5\) −186.216 + 322.535i −0.666225 + 1.15394i 0.312726 + 0.949843i \(0.398758\pi\)
−0.978952 + 0.204093i \(0.934575\pi\)
\(6\) 0 0
\(7\) 726.432 + 1258.22i 0.800483 + 1.38648i 0.919299 + 0.393561i \(0.128757\pi\)
−0.118816 + 0.992916i \(0.537910\pi\)
\(8\) 0 0
\(9\) −1992.92 − 900.678i −0.911260 − 0.411833i
\(10\) 0 0
\(11\) 3510.15 + 6079.76i 0.795155 + 1.37725i 0.922741 + 0.385421i \(0.125944\pi\)
−0.127586 + 0.991828i \(0.540723\pi\)
\(12\) 0 0
\(13\) −262.017 + 453.827i −0.0330771 + 0.0572913i −0.882090 − 0.471081i \(-0.843864\pi\)
0.849013 + 0.528372i \(0.177197\pi\)
\(14\) 0 0
\(15\) −12910.7 − 11690.3i −0.987710 − 0.894345i
\(16\) 0 0
\(17\) 17080.7 0.843208 0.421604 − 0.906780i \(-0.361467\pi\)
0.421604 + 0.906780i \(0.361467\pi\)
\(18\) 0 0
\(19\) −21851.6 −0.730879 −0.365440 − 0.930835i \(-0.619081\pi\)
−0.365440 + 0.930835i \(0.619081\pi\)
\(20\) 0 0
\(21\) −64676.7 + 20815.2i −1.52398 + 0.490472i
\(22\) 0 0
\(23\) −1873.48 + 3244.97i −0.0321072 + 0.0556113i −0.881633 − 0.471937i \(-0.843555\pi\)
0.849525 + 0.527548i \(0.176888\pi\)
\(24\) 0 0
\(25\) −30290.1 − 52463.9i −0.387713 − 0.671538i
\(26\) 0 0
\(27\) 60807.3 − 82236.4i 0.594542 − 0.804064i
\(28\) 0 0
\(29\) 15522.1 + 26885.1i 0.118184 + 0.204700i 0.919048 − 0.394146i \(-0.128959\pi\)
−0.800864 + 0.598846i \(0.795626\pi\)
\(30\) 0 0
\(31\) 129142. − 223680.i 0.778574 − 1.34853i −0.154189 − 0.988041i \(-0.549277\pi\)
0.932764 − 0.360489i \(-0.117390\pi\)
\(32\) 0 0
\(33\) −312521. + 100580.i −1.51384 + 0.487207i
\(34\) 0 0
\(35\) −541092. −2.13321
\(36\) 0 0
\(37\) −438830. −1.42426 −0.712132 − 0.702045i \(-0.752270\pi\)
−0.712132 + 0.702045i \(0.752270\pi\)
\(38\) 0 0
\(39\) −18166.1 − 16449.0i −0.0490384 − 0.0444030i
\(40\) 0 0
\(41\) −99343.5 + 172068.i −0.225111 + 0.389903i −0.956353 − 0.292215i \(-0.905608\pi\)
0.731242 + 0.682118i \(0.238941\pi\)
\(42\) 0 0
\(43\) 96178.1 + 166585.i 0.184475 + 0.319519i 0.943399 − 0.331659i \(-0.107608\pi\)
−0.758925 + 0.651178i \(0.774275\pi\)
\(44\) 0 0
\(45\) 661614. − 475068.i 1.08233 − 0.777162i
\(46\) 0 0
\(47\) −441116. − 764035.i −0.619741 − 1.07342i −0.989533 − 0.144308i \(-0.953904\pi\)
0.369792 − 0.929114i \(-0.379429\pi\)
\(48\) 0 0
\(49\) −643637. + 1.11481e6i −0.781546 + 1.35368i
\(50\) 0 0
\(51\) −168258. + 780864.i −0.177615 + 0.824289i
\(52\) 0 0
\(53\) 1.78412e6 1.64610 0.823052 − 0.567966i \(-0.192269\pi\)
0.823052 + 0.567966i \(0.192269\pi\)
\(54\) 0 0
\(55\) −2.61458e6 −2.11901
\(56\) 0 0
\(57\) 215255. − 998970.i 0.153954 − 0.714481i
\(58\) 0 0
\(59\) −464224. + 804060.i −0.294270 + 0.509690i −0.974815 − 0.223016i \(-0.928410\pi\)
0.680545 + 0.732706i \(0.261743\pi\)
\(60\) 0 0
\(61\) 646828. + 1.12034e6i 0.364867 + 0.631968i 0.988755 − 0.149546i \(-0.0477812\pi\)
−0.623888 + 0.781514i \(0.714448\pi\)
\(62\) 0 0
\(63\) −314476. − 3.16181e6i −0.158451 − 1.59311i
\(64\) 0 0
\(65\) −97583.4 − 169019.i −0.0440737 − 0.0763378i
\(66\) 0 0
\(67\) 2.05729e6 − 3.56333e6i 0.835667 − 1.44742i −0.0578186 − 0.998327i \(-0.518415\pi\)
0.893486 − 0.449091i \(-0.148252\pi\)
\(68\) 0 0
\(69\) −129892. − 117614.i −0.0476004 − 0.0431009i
\(70\) 0 0
\(71\) 5.60557e6 1.85873 0.929363 − 0.369167i \(-0.120357\pi\)
0.929363 + 0.369167i \(0.120357\pi\)
\(72\) 0 0
\(73\) 2.84957e6 0.857333 0.428667 − 0.903463i \(-0.358984\pi\)
0.428667 + 0.903463i \(0.358984\pi\)
\(74\) 0 0
\(75\) 2.69683e6 − 867934.i 0.738140 − 0.237559i
\(76\) 0 0
\(77\) −5.09978e6 + 8.83308e6i −1.27302 + 2.20493i
\(78\) 0 0
\(79\) −1.91864e6 − 3.32318e6i −0.437823 − 0.758331i 0.559699 − 0.828696i \(-0.310917\pi\)
−0.997521 + 0.0703650i \(0.977584\pi\)
\(80\) 0 0
\(81\) 3.16053e6 + 3.58997e6i 0.660788 + 0.750573i
\(82\) 0 0
\(83\) 4.14708e6 + 7.18296e6i 0.796104 + 1.37889i 0.922136 + 0.386866i \(0.126442\pi\)
−0.126032 + 0.992026i \(0.540224\pi\)
\(84\) 0 0
\(85\) −3.18070e6 + 5.50913e6i −0.561767 + 0.973009i
\(86\) 0 0
\(87\) −1.38199e6 + 444772.i −0.225002 + 0.0724136i
\(88\) 0 0
\(89\) 7.67933e6 1.15467 0.577335 − 0.816507i \(-0.304092\pi\)
0.577335 + 0.816507i \(0.304092\pi\)
\(90\) 0 0
\(91\) −761351. −0.105911
\(92\) 0 0
\(93\) 8.95362e6 + 8.10727e6i 1.15427 + 1.04516i
\(94\) 0 0
\(95\) 4.06911e6 − 7.04791e6i 0.486930 − 0.843388i
\(96\) 0 0
\(97\) −5.27913e6 − 9.14373e6i −0.587302 − 1.01724i −0.994584 − 0.103935i \(-0.966857\pi\)
0.407282 − 0.913303i \(-0.366477\pi\)
\(98\) 0 0
\(99\) −1.51956e6 − 1.52780e7i −0.157397 − 1.58250i
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 72.8.i.b.49.5 yes 22
3.2 odd 2 216.8.i.b.145.10 22
4.3 odd 2 144.8.i.f.49.7 22
9.2 odd 6 216.8.i.b.73.10 22
9.7 even 3 inner 72.8.i.b.25.5 ✓ 22
12.11 even 2 432.8.i.f.145.10 22
36.7 odd 6 144.8.i.f.97.7 22
36.11 even 6 432.8.i.f.289.10 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.8.i.b.25.5 ✓ 22 9.7 even 3 inner
72.8.i.b.49.5 yes 22 1.1 even 1 trivial
144.8.i.f.49.7 22 4.3 odd 2
144.8.i.f.97.7 22 36.7 odd 6
216.8.i.b.73.10 22 9.2 odd 6
216.8.i.b.145.10 22 3.2 odd 2
432.8.i.f.145.10 22 12.11 even 2
432.8.i.f.289.10 22 36.11 even 6