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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [36,16,Mod(13,36)] 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("36.13"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(36, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 36.e (of order \(3\), degree \(2\), 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: \(51.3696618360\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 13.15
Character \(\chi\) \(=\) 36.13
Dual form 36.16.e.a.25.15

$q$-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)
 
\(f(q)\) \(=\) \(q+(3752.00 + 520.947i) q^{3} +(121730. - 210843. i) q^{5} +(-734922. - 1.27292e6i) q^{7} +(1.38061e7 + 3.90919e6i) q^{9} +(5.77921e7 + 1.00099e8i) q^{11} +(-2.45388e7 + 4.25025e7i) q^{13} +(5.66570e8 - 7.27668e8i) q^{15} +2.40665e9 q^{17} -1.29424e9 q^{19} +(-2.09430e9 - 5.15886e9i) q^{21} +(4.29568e9 - 7.44034e9i) q^{23} +(-1.43777e10 - 2.49029e10i) q^{25} +(4.97642e10 + 2.18596e10i) q^{27} +(-3.67211e9 - 6.36028e9i) q^{29} +(1.10713e11 - 1.91760e11i) q^{31} +(1.64690e11 + 4.05678e11i) q^{33} -3.57849e11 q^{35} -9.47409e10 q^{37} +(-1.14211e11 + 1.46686e11i) q^{39} +(-6.83010e11 + 1.18301e12i) q^{41} +(-1.08181e12 - 1.87375e12i) q^{43} +(2.50485e12 - 2.43506e12i) q^{45} +(8.27197e10 + 1.43275e11i) q^{47} +(1.29356e12 - 2.24051e12i) q^{49} +(9.02977e12 + 1.25374e12i) q^{51} -1.41012e12 q^{53} +2.81402e13 q^{55} +(-4.85599e12 - 6.74230e11i) q^{57} +(1.81167e13 - 3.13790e13i) q^{59} +(-1.43654e13 - 2.48816e13i) q^{61} +(-5.17034e12 - 2.04471e13i) q^{63} +(5.97423e12 + 1.03477e13i) q^{65} +(-3.36650e13 + 5.83095e13i) q^{67} +(1.99934e13 - 2.56783e13i) q^{69} +1.51946e14 q^{71} +5.87225e13 q^{73} +(-4.09721e13 - 1.00926e14i) q^{75} +(8.49453e13 - 1.47130e14i) q^{77} +(6.03838e13 + 1.04588e14i) q^{79} +(1.75328e14 + 1.07942e14i) q^{81} +(-1.79017e13 - 3.10067e13i) q^{83} +(2.92963e14 - 5.07426e14i) q^{85} +(-1.04644e13 - 2.57768e13i) q^{87} -6.43264e14 q^{89} +7.21364e13 q^{91} +(5.15291e14 - 6.61808e14i) q^{93} +(-1.57548e14 + 2.72881e14i) q^{95} +(-2.69924e14 - 4.67522e14i) q^{97} +(4.06580e14 + 1.60790e15i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 1911 q^{3} + 159846 q^{5} + 803706 q^{7} - 9247029 q^{9} + 85327287 q^{11} - 41509092 q^{13} + 1011307500 q^{15} - 4181504694 q^{17} - 4752917850 q^{19} - 38851541184 q^{21} - 2587803774 q^{23} - 90297075903 q^{25}+ \cdots + 18\!\cdots\!18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 3752.00 + 520.947i 0.990498 + 0.137526i
\(4\) 0 0
\(5\) 121730. − 210843.i 0.696825 − 1.20694i −0.272737 − 0.962089i \(-0.587929\pi\)
0.969562 − 0.244847i \(-0.0787377\pi\)
\(6\) 0 0
\(7\) −734922. − 1.27292e6i −0.337292 − 0.584206i 0.646631 − 0.762803i \(-0.276178\pi\)
−0.983922 + 0.178597i \(0.942844\pi\)
\(8\) 0 0
\(9\) 1.38061e7 + 3.90919e6i 0.962173 + 0.272438i
\(10\) 0 0
\(11\) 5.77921e7 + 1.00099e8i 0.894176 + 1.54876i 0.834821 + 0.550522i \(0.185571\pi\)
0.0593552 + 0.998237i \(0.481096\pi\)
\(12\) 0 0
\(13\) −2.45388e7 + 4.25025e7i −0.108462 + 0.187862i −0.915147 − 0.403119i \(-0.867926\pi\)
0.806685 + 0.590981i \(0.201259\pi\)
\(14\) 0 0
\(15\) 5.66570e8 − 7.27668e8i 0.856188 − 1.09964i
\(16\) 0 0
\(17\) 2.40665e9 1.42248 0.711241 − 0.702948i \(-0.248134\pi\)
0.711241 + 0.702948i \(0.248134\pi\)
\(18\) 0 0
\(19\) −1.29424e9 −0.332172 −0.166086 − 0.986111i \(-0.553113\pi\)
−0.166086 + 0.986111i \(0.553113\pi\)
\(20\) 0 0
\(21\) −2.09430e9 − 5.15886e9i −0.253743 − 0.625042i
\(22\) 0 0
\(23\) 4.29568e9 − 7.44034e9i 0.263071 − 0.455653i −0.703985 − 0.710215i \(-0.748598\pi\)
0.967056 + 0.254562i \(0.0819312\pi\)
\(24\) 0 0
\(25\) −1.43777e10 − 2.49029e10i −0.471129 − 0.816019i
\(26\) 0 0
\(27\) 4.97642e10 + 2.18596e10i 0.915564 + 0.402173i
\(28\) 0 0
\(29\) −3.67211e9 − 6.36028e9i −0.0395303 − 0.0684685i 0.845583 − 0.533843i \(-0.179253\pi\)
−0.885114 + 0.465375i \(0.845920\pi\)
\(30\) 0 0
\(31\) 1.10713e11 − 1.91760e11i 0.722744 − 1.25183i −0.237152 − 0.971473i \(-0.576214\pi\)
0.959896 − 0.280357i \(-0.0904528\pi\)
\(32\) 0 0
\(33\) 1.64690e11 + 4.05678e11i 0.672686 + 1.65701i
\(34\) 0 0
\(35\) −3.57849e11 −0.940133
\(36\) 0 0
\(37\) −9.47409e10 −0.164068 −0.0820341 − 0.996630i \(-0.526142\pi\)
−0.0820341 + 0.996630i \(0.526142\pi\)
\(38\) 0 0
\(39\) −1.14211e11 + 1.46686e11i −0.133267 + 0.171161i
\(40\) 0 0
\(41\) −6.83010e11 + 1.18301e12i −0.547707 + 0.948656i 0.450724 + 0.892663i \(0.351166\pi\)
−0.998431 + 0.0559931i \(0.982168\pi\)
\(42\) 0 0
\(43\) −1.08181e12 − 1.87375e12i −0.606928 − 1.05123i −0.991744 − 0.128236i \(-0.959068\pi\)
0.384816 − 0.922993i \(-0.374265\pi\)
\(44\) 0 0
\(45\) 2.50485e12 − 2.43506e12i 0.999281 − 0.971440i
\(46\) 0 0
\(47\) 8.27197e10 + 1.43275e11i 0.0238163 + 0.0412511i 0.877688 − 0.479233i \(-0.159085\pi\)
−0.853872 + 0.520484i \(0.825752\pi\)
\(48\) 0 0
\(49\) 1.29356e12 − 2.24051e12i 0.272469 − 0.471929i
\(50\) 0 0
\(51\) 9.02977e12 + 1.25374e12i 1.40897 + 0.195628i
\(52\) 0 0
\(53\) −1.41012e12 −0.164887 −0.0824436 − 0.996596i \(-0.526272\pi\)
−0.0824436 + 0.996596i \(0.526272\pi\)
\(54\) 0 0
\(55\) 2.81402e13 2.49234
\(56\) 0 0
\(57\) −4.85599e12 − 6.74230e11i −0.329015 − 0.0456822i
\(58\) 0 0
\(59\) 1.81167e13 − 3.13790e13i 0.947738 − 1.64153i 0.197563 − 0.980290i \(-0.436697\pi\)
0.750175 − 0.661240i \(-0.229969\pi\)
\(60\) 0 0
\(61\) −1.43654e13 − 2.48816e13i −0.585253 − 1.01369i −0.994844 − 0.101418i \(-0.967662\pi\)
0.409591 − 0.912269i \(-0.365671\pi\)
\(62\) 0 0
\(63\) −5.17034e12 − 2.04471e13i −0.165373 − 0.653999i
\(64\) 0 0
\(65\) 5.97423e12 + 1.03477e13i 0.151158 + 0.261814i
\(66\) 0 0
\(67\) −3.36650e13 + 5.83095e13i −0.678606 + 1.17538i 0.296795 + 0.954941i \(0.404082\pi\)
−0.975401 + 0.220439i \(0.929251\pi\)
\(68\) 0 0
\(69\) 1.99934e13 − 2.56783e13i 0.323236 − 0.415144i
\(70\) 0 0
\(71\) 1.51946e14 1.98268 0.991341 − 0.131316i \(-0.0419203\pi\)
0.991341 + 0.131316i \(0.0419203\pi\)
\(72\) 0 0
\(73\) 5.87225e13 0.622133 0.311067 − 0.950388i \(-0.399314\pi\)
0.311067 + 0.950388i \(0.399314\pi\)
\(74\) 0 0
\(75\) −4.09721e13 − 1.00926e14i −0.354429 − 0.873058i
\(76\) 0 0
\(77\) 8.49453e13 − 1.47130e14i 0.603196 − 1.04477i
\(78\) 0 0
\(79\) 6.03838e13 + 1.04588e14i 0.353767 + 0.612742i 0.986906 − 0.161296i \(-0.0515673\pi\)
−0.633139 + 0.774038i \(0.718234\pi\)
\(80\) 0 0
\(81\) 1.75328e14 + 1.07942e14i 0.851555 + 0.524265i
\(82\) 0 0
\(83\) −1.79017e13 − 3.10067e13i −0.0724118 − 0.125421i 0.827546 − 0.561398i \(-0.189736\pi\)
−0.899958 + 0.435977i \(0.856403\pi\)
\(84\) 0 0
\(85\) 2.92963e14 − 5.07426e14i 0.991220 − 1.71684i
\(86\) 0 0
\(87\) −1.04644e13 − 2.57768e13i −0.0297385 − 0.0732544i
\(88\) 0 0
\(89\) −6.43264e14 −1.54157 −0.770787 − 0.637093i \(-0.780137\pi\)
−0.770787 + 0.637093i \(0.780137\pi\)
\(90\) 0 0
\(91\) 7.21364e13 0.146334
\(92\) 0 0
\(93\) 5.15291e14 − 6.61808e14i 0.888036 − 1.14054i
\(94\) 0 0
\(95\) −1.57548e14 + 2.72881e14i −0.231465 + 0.400910i
\(96\) 0 0
\(97\) −2.69924e14 − 4.67522e14i −0.339198 − 0.587509i 0.645084 − 0.764112i \(-0.276822\pi\)
−0.984282 + 0.176603i \(0.943489\pi\)
\(98\) 0 0
\(99\) 4.06580e14 + 1.60790e15i 0.438412 + 1.73378i
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 36.16.e.a.13.15 ✓ 30
3.2 odd 2 108.16.e.a.37.3 30
9.2 odd 6 108.16.e.a.73.3 30
9.7 even 3 inner 36.16.e.a.25.15 yes 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.16.e.a.13.15 ✓ 30 1.1 even 1 trivial
36.16.e.a.25.15 yes 30 9.7 even 3 inner
108.16.e.a.37.3 30 3.2 odd 2
108.16.e.a.73.3 30 9.2 odd 6