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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 49.6
Character \(\chi\) \(=\) 72.49
Dual form 72.8.i.b.25.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.287929 + 46.7645i) q^{3} +(193.063 - 334.394i) q^{5} +(-111.981 - 193.956i) q^{7} +(-2186.83 - 26.9297i) q^{9} +(-582.408 - 1008.76i) q^{11} +(-5469.58 + 9473.59i) q^{13} +(15582.2 + 9124.76i) q^{15} -33425.1 q^{17} -17982.9 q^{19} +(9102.52 - 5180.88i) q^{21} +(41313.1 - 71556.4i) q^{23} +(-35483.9 - 61460.0i) q^{25} +(1889.01 - 102258. i) q^{27} +(-86040.1 - 149026. i) q^{29} +(-69198.0 + 119854. i) q^{31} +(47341.8 - 26945.5i) q^{33} -86477.3 q^{35} -156788. q^{37} +(-441453. - 258510. i) q^{39} +(-194146. + 336270. i) q^{41} +(-389199. - 674112. i) q^{43} +(-431201. + 726066. i) q^{45} +(389309. + 674302. i) q^{47} +(386692. - 669770. i) q^{49} +(9624.04 - 1.56311e6i) q^{51} -912802. q^{53} -449765. q^{55} +(5177.79 - 840960. i) q^{57} +(935536. - 1.62040e6i) q^{59} +(637475. + 1.10414e6i) q^{61} +(239660. + 427166. i) q^{63} +(2.11194e6 + 3.65799e6i) q^{65} +(-211211. + 365828. i) q^{67} +(3.33441e6 + 1.95259e6i) q^{69} -410404. q^{71} -3.48224e6 q^{73} +(2.88436e6 - 1.64169e6i) q^{75} +(-130437. + 225923. i) q^{77} +(-1.35887e6 - 2.35364e6i) q^{79} +(4.78152e6 + 117782. i) q^{81} +(2.51869e6 + 4.36251e6i) q^{83} +(-6.45314e6 + 1.11772e7i) q^{85} +(6.99389e6 - 3.98071e6i) q^{87} +2.84043e6 q^{89} +2.44995e6 q^{91} +(-5.58500e6 - 3.27052e6i) q^{93} +(-3.47182e6 + 6.01337e6i) q^{95} +(-670541. - 1.16141e6i) q^{97} +(1.24646e6 + 2.22167e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 44 q^{3} - 125 q^{5} - 1245 q^{7} - 3766 q^{9} + 1699 q^{11} - 4937 q^{13} - 19349 q^{15} + 26540 q^{17} + 28976 q^{19} - 75315 q^{21} - 18239 q^{23} - 109168 q^{25} - 87680 q^{27} - 3525 q^{29}+ \cdots + 24043955 q^{99}+O(q^{100}) \) 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
\(3\) −0.287929 + 46.7645i −0.00615688 + 0.999981i
\(4\) 0 0
\(5\) 193.063 − 334.394i 0.690722 − 1.19637i −0.280879 − 0.959743i \(-0.590626\pi\)
0.971602 − 0.236623i \(-0.0760406\pi\)
\(6\) 0 0
\(7\) −111.981 − 193.956i −0.123396 − 0.213728i 0.797709 − 0.603043i \(-0.206045\pi\)
−0.921105 + 0.389315i \(0.872712\pi\)
\(8\) 0 0
\(9\) −2186.83 − 26.9297i −0.999924 − 0.0123135i
\(10\) 0 0
\(11\) −582.408 − 1008.76i −0.131933 − 0.228514i 0.792489 − 0.609886i \(-0.208785\pi\)
−0.924422 + 0.381372i \(0.875452\pi\)
\(12\) 0 0
\(13\) −5469.58 + 9473.59i −0.690482 + 1.19595i 0.281198 + 0.959650i \(0.409268\pi\)
−0.971680 + 0.236300i \(0.924065\pi\)
\(14\) 0 0
\(15\) 15582.2 + 9124.76i 1.19209 + 0.698075i
\(16\) 0 0
\(17\) −33425.1 −1.65007 −0.825033 − 0.565085i \(-0.808843\pi\)
−0.825033 + 0.565085i \(0.808843\pi\)
\(18\) 0 0
\(19\) −17982.9 −0.601480 −0.300740 − 0.953706i \(-0.597234\pi\)
−0.300740 + 0.953706i \(0.597234\pi\)
\(20\) 0 0
\(21\) 9102.52 − 5180.88i 0.214484 − 0.122078i
\(22\) 0 0
\(23\) 41313.1 − 71556.4i 0.708012 − 1.22631i −0.257581 − 0.966257i \(-0.582925\pi\)
0.965593 − 0.260056i \(-0.0837412\pi\)
\(24\) 0 0
\(25\) −35483.9 − 61460.0i −0.454194 − 0.786688i
\(26\) 0 0
\(27\) 1889.01 − 102258.i 0.0184697 − 0.999829i
\(28\) 0 0
\(29\) −86040.1 − 149026.i −0.655101 − 1.13467i −0.981869 − 0.189563i \(-0.939293\pi\)
0.326768 − 0.945105i \(-0.394041\pi\)
\(30\) 0 0
\(31\) −69198.0 + 119854.i −0.417184 + 0.722583i −0.995655 − 0.0931194i \(-0.970316\pi\)
0.578471 + 0.815703i \(0.303650\pi\)
\(32\) 0 0
\(33\) 47341.8 − 26945.5i 0.229322 − 0.130523i
\(34\) 0 0
\(35\) −86477.3 −0.340929
\(36\) 0 0
\(37\) −156788. −0.508870 −0.254435 − 0.967090i \(-0.581889\pi\)
−0.254435 + 0.967090i \(0.581889\pi\)
\(38\) 0 0
\(39\) −441453. − 258510.i −1.19168 − 0.697832i
\(40\) 0 0
\(41\) −194146. + 336270.i −0.439931 + 0.761982i −0.997684 − 0.0680247i \(-0.978330\pi\)
0.557753 + 0.830007i \(0.311664\pi\)
\(42\) 0 0
\(43\) −389199. − 674112.i −0.746504 − 1.29298i −0.949489 − 0.313800i \(-0.898398\pi\)
0.202985 − 0.979182i \(-0.434936\pi\)
\(44\) 0 0
\(45\) −431201. + 726066.i −0.705401 + 1.18777i
\(46\) 0 0
\(47\) 389309. + 674302.i 0.546955 + 0.947354i 0.998481 + 0.0550960i \(0.0175465\pi\)
−0.451526 + 0.892258i \(0.649120\pi\)
\(48\) 0 0
\(49\) 386692. − 669770.i 0.469547 − 0.813279i
\(50\) 0 0
\(51\) 9624.04 − 1.56311e6i 0.0101593 − 1.65003i
\(52\) 0 0
\(53\) −912802. −0.842191 −0.421096 − 0.907016i \(-0.638354\pi\)
−0.421096 + 0.907016i \(0.638354\pi\)
\(54\) 0 0
\(55\) −449765. −0.364515
\(56\) 0 0
\(57\) 5177.79 − 840960.i 0.00370324 − 0.601469i
\(58\) 0 0
\(59\) 935536. − 1.62040e6i 0.593032 − 1.02716i −0.400789 − 0.916170i \(-0.631264\pi\)
0.993821 − 0.110992i \(-0.0354026\pi\)
\(60\) 0 0
\(61\) 637475. + 1.10414e6i 0.359591 + 0.622829i 0.987892 − 0.155140i \(-0.0495830\pi\)
−0.628302 + 0.777970i \(0.716250\pi\)
\(62\) 0 0
\(63\) 239660. + 427166.i 0.120755 + 0.215231i
\(64\) 0 0
\(65\) 2.11194e6 + 3.65799e6i 0.953862 + 1.65214i
\(66\) 0 0
\(67\) −211211. + 365828.i −0.0857935 + 0.148599i −0.905729 − 0.423857i \(-0.860676\pi\)
0.819936 + 0.572456i \(0.194009\pi\)
\(68\) 0 0
\(69\) 3.33441e6 + 1.95259e6i 1.22193 + 0.715549i
\(70\) 0 0
\(71\) −410404. −0.136084 −0.0680421 − 0.997682i \(-0.521675\pi\)
−0.0680421 + 0.997682i \(0.521675\pi\)
\(72\) 0 0
\(73\) −3.48224e6 −1.04768 −0.523840 − 0.851817i \(-0.675501\pi\)
−0.523840 + 0.851817i \(0.675501\pi\)
\(74\) 0 0
\(75\) 2.88436e6 − 1.64169e6i 0.789469 − 0.449342i
\(76\) 0 0
\(77\) −130437. + 225923.i −0.0325599 + 0.0563954i
\(78\) 0 0
\(79\) −1.35887e6 − 2.35364e6i −0.310087 − 0.537087i 0.668294 − 0.743897i \(-0.267025\pi\)
−0.978381 + 0.206811i \(0.933692\pi\)
\(80\) 0 0
\(81\) 4.78152e6 + 117782.i 0.999697 + 0.0246252i
\(82\) 0 0
\(83\) 2.51869e6 + 4.36251e6i 0.483506 + 0.837457i 0.999821 − 0.0189418i \(-0.00602972\pi\)
−0.516314 + 0.856399i \(0.672696\pi\)
\(84\) 0 0
\(85\) −6.45314e6 + 1.11772e7i −1.13974 + 1.97408i
\(86\) 0 0
\(87\) 6.99389e6 − 3.98071e6i 1.13868 − 0.648102i
\(88\) 0 0
\(89\) 2.84043e6 0.427090 0.213545 − 0.976933i \(-0.431499\pi\)
0.213545 + 0.976933i \(0.431499\pi\)
\(90\) 0 0
\(91\) 2.44995e6 0.340810
\(92\) 0 0
\(93\) −5.58500e6 − 3.27052e6i −0.720001 − 0.421625i
\(94\) 0 0
\(95\) −3.47182e6 + 6.01337e6i −0.415456 + 0.719590i
\(96\) 0 0
\(97\) −670541. − 1.16141e6i −0.0745975 − 0.129207i 0.826314 − 0.563210i \(-0.190434\pi\)
−0.900911 + 0.434004i \(0.857101\pi\)
\(98\) 0 0
\(99\) 1.24646e6 + 2.22167e6i 0.129109 + 0.230121i
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.49.6 yes 22
3.2 odd 2 216.8.i.b.145.2 22
4.3 odd 2 144.8.i.f.49.6 22
9.2 odd 6 216.8.i.b.73.2 22
9.7 even 3 inner 72.8.i.b.25.6 ✓ 22
12.11 even 2 432.8.i.f.145.2 22
36.7 odd 6 144.8.i.f.97.6 22
36.11 even 6 432.8.i.f.289.2 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.8.i.b.25.6 ✓ 22 9.7 even 3 inner
72.8.i.b.49.6 yes 22 1.1 even 1 trivial
144.8.i.f.49.6 22 4.3 odd 2
144.8.i.f.97.6 22 36.7 odd 6
216.8.i.b.73.2 22 9.2 odd 6
216.8.i.b.145.2 22 3.2 odd 2
432.8.i.f.145.2 22 12.11 even 2
432.8.i.f.289.2 22 36.11 even 6