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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 13.12
Character \(\chi\) \(=\) 36.13
Dual form 36.16.e.a.25.12

$q$-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)
 
\(f(q)\) \(=\) \(q+(2136.40 + 3128.05i) q^{3} +(115558. - 200153. i) q^{5} +(436872. + 756685. i) q^{7} +(-5.22049e6 + 1.33655e7i) q^{9} +(-5.47782e7 - 9.48786e7i) q^{11} +(1.41583e8 - 2.45229e8i) q^{13} +(8.72967e8 - 6.61344e7i) q^{15} -2.76394e9 q^{17} -2.81565e9 q^{19} +(-1.43361e9 + 2.98314e9i) q^{21} +(-1.11017e10 + 1.92287e10i) q^{23} +(-1.14486e10 - 1.98296e10i) q^{25} +(-5.29611e10 + 1.22242e10i) q^{27} +(1.47196e10 + 2.54951e10i) q^{29} +(1.04672e10 - 1.81297e10i) q^{31} +(1.79757e11 - 3.74048e11i) q^{33} +2.01937e11 q^{35} -2.83142e11 q^{37} +(1.06957e12 - 8.10284e10i) q^{39} +(-1.15380e12 + 1.99845e12i) q^{41} +(-1.48758e12 - 2.57656e12i) q^{43} +(2.07188e12 + 2.58939e12i) q^{45} +(-1.16731e12 - 2.02185e12i) q^{47} +(1.99207e12 - 3.45036e12i) q^{49} +(-5.90488e12 - 8.64575e12i) q^{51} -3.17055e12 q^{53} -2.53203e13 q^{55} +(-6.01536e12 - 8.80750e12i) q^{57} +(6.06788e12 - 1.05099e13i) q^{59} +(1.45086e13 + 2.51296e13i) q^{61} +(-1.23942e13 + 1.88876e12i) q^{63} +(-3.27222e13 - 5.66764e13i) q^{65} +(3.53741e13 - 6.12697e13i) q^{67} +(-8.38659e13 + 6.35353e12i) q^{69} -2.85483e13 q^{71} +5.01111e13 q^{73} +(3.75692e13 - 7.81759e13i) q^{75} +(4.78621e13 - 8.28996e13i) q^{77} +(-1.47142e14 - 2.54857e14i) q^{79} +(-1.51384e14 - 1.39549e14i) q^{81} +(-6.27413e13 - 1.08671e14i) q^{83} +(-3.19396e14 + 5.53210e14i) q^{85} +(-4.83029e13 + 1.00511e14i) q^{87} -2.39621e14 q^{89} +2.47415e14 q^{91} +(7.90727e13 - 5.99041e12i) q^{93} +(-3.25372e14 + 5.63560e14i) q^{95} +(-5.83535e14 - 1.01071e15i) q^{97} +(1.55407e15 - 2.36827e14i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 1911 q^{3} + 159846 q^{5} + 803706 q^{7} - 9247029 q^{9} + 85327287 q^{11} - 41509092 q^{13} + 1011307500 q^{15} - 4181504694 q^{17} - 4752917850 q^{19} - 38851541184 q^{21} - 2587803774 q^{23} - 90297075903 q^{25}+ \cdots + 18\!\cdots\!18 q^{99}+O(q^{100}) \) 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
\(3\) 2136.40 + 3128.05i 0.563992 + 0.825780i
\(4\) 0 0
\(5\) 115558. − 200153.i 0.661494 − 1.14574i −0.318729 − 0.947846i \(-0.603256\pi\)
0.980223 − 0.197895i \(-0.0634106\pi\)
\(6\) 0 0
\(7\) 436872. + 756685.i 0.200502 + 0.347280i 0.948690 − 0.316207i \(-0.102409\pi\)
−0.748188 + 0.663487i \(0.769076\pi\)
\(8\) 0 0
\(9\) −5.22049e6 + 1.33655e7i −0.363825 + 0.931467i
\(10\) 0 0
\(11\) −5.47782e7 − 9.48786e7i −0.847544 − 1.46799i −0.883393 − 0.468633i \(-0.844747\pi\)
0.0358486 − 0.999357i \(-0.488587\pi\)
\(12\) 0 0
\(13\) 1.41583e8 − 2.45229e8i 0.625800 − 1.08392i −0.362585 − 0.931951i \(-0.618106\pi\)
0.988386 − 0.151967i \(-0.0485609\pi\)
\(14\) 0 0
\(15\) 8.72967e8 − 6.61344e7i 1.31921 − 0.0999408i
\(16\) 0 0
\(17\) −2.76394e9 −1.63366 −0.816830 − 0.576878i \(-0.804271\pi\)
−0.816830 + 0.576878i \(0.804271\pi\)
\(18\) 0 0
\(19\) −2.81565e9 −0.722648 −0.361324 − 0.932440i \(-0.617675\pi\)
−0.361324 + 0.932440i \(0.617675\pi\)
\(20\) 0 0
\(21\) −1.43361e9 + 2.98314e9i −0.173695 + 0.361434i
\(22\) 0 0
\(23\) −1.11017e10 + 1.92287e10i −0.679876 + 1.17758i 0.295141 + 0.955454i \(0.404633\pi\)
−0.975018 + 0.222127i \(0.928700\pi\)
\(24\) 0 0
\(25\) −1.14486e10 − 1.98296e10i −0.375149 − 0.649777i
\(26\) 0 0
\(27\) −5.29611e10 + 1.22242e10i −0.974382 + 0.224901i
\(28\) 0 0
\(29\) 1.47196e10 + 2.54951e10i 0.158457 + 0.274455i 0.934312 − 0.356456i \(-0.116015\pi\)
−0.775856 + 0.630910i \(0.782682\pi\)
\(30\) 0 0
\(31\) 1.04672e10 − 1.81297e10i 0.0683309 − 0.118353i −0.829836 − 0.558008i \(-0.811566\pi\)
0.898167 + 0.439655i \(0.144899\pi\)
\(32\) 0 0
\(33\) 1.79757e11 − 3.74048e11i 0.734228 − 1.52782i
\(34\) 0 0
\(35\) 2.01937e11 0.530524
\(36\) 0 0
\(37\) −2.83142e11 −0.490333 −0.245167 − 0.969481i \(-0.578843\pi\)
−0.245167 + 0.969481i \(0.578843\pi\)
\(38\) 0 0
\(39\) 1.06957e12 − 8.10284e10i 1.24802 − 0.0945481i
\(40\) 0 0
\(41\) −1.15380e12 + 1.99845e12i −0.925238 + 1.60256i −0.134060 + 0.990973i \(0.542801\pi\)
−0.791178 + 0.611586i \(0.790532\pi\)
\(42\) 0 0
\(43\) −1.48758e12 − 2.57656e12i −0.834577 − 1.44553i −0.894375 − 0.447319i \(-0.852379\pi\)
0.0597978 − 0.998211i \(-0.480954\pi\)
\(44\) 0 0
\(45\) 2.07188e12 + 2.58939e12i 0.826552 + 1.03301i
\(46\) 0 0
\(47\) −1.16731e12 − 2.02185e12i −0.336089 − 0.582123i 0.647605 − 0.761977i \(-0.275771\pi\)
−0.983693 + 0.179854i \(0.942438\pi\)
\(48\) 0 0
\(49\) 1.99207e12 − 3.45036e12i 0.419598 − 0.726765i
\(50\) 0 0
\(51\) −5.90488e12 − 8.64575e12i −0.921372 − 1.34904i
\(52\) 0 0
\(53\) −3.17055e12 −0.370737 −0.185369 − 0.982669i \(-0.559348\pi\)
−0.185369 + 0.982669i \(0.559348\pi\)
\(54\) 0 0
\(55\) −2.53203e13 −2.24258
\(56\) 0 0
\(57\) −6.01536e12 − 8.80750e12i −0.407568 − 0.596748i
\(58\) 0 0
\(59\) 6.06788e12 − 1.05099e13i 0.317429 − 0.549804i −0.662522 − 0.749043i \(-0.730514\pi\)
0.979951 + 0.199239i \(0.0638470\pi\)
\(60\) 0 0
\(61\) 1.45086e13 + 2.51296e13i 0.591087 + 1.02379i 0.994086 + 0.108592i \(0.0346342\pi\)
−0.403000 + 0.915200i \(0.632032\pi\)
\(62\) 0 0
\(63\) −1.23942e13 + 1.88876e12i −0.396428 + 0.0604120i
\(64\) 0 0
\(65\) −3.27222e13 − 5.66764e13i −0.827926 − 1.43401i
\(66\) 0 0
\(67\) 3.53741e13 − 6.12697e13i 0.713056 − 1.23505i −0.250648 − 0.968078i \(-0.580644\pi\)
0.963704 − 0.266972i \(-0.0860230\pi\)
\(68\) 0 0
\(69\) −8.38659e13 + 6.35353e12i −1.35587 + 0.102718i
\(70\) 0 0
\(71\) −2.85483e13 −0.372514 −0.186257 − 0.982501i \(-0.559636\pi\)
−0.186257 + 0.982501i \(0.559636\pi\)
\(72\) 0 0
\(73\) 5.01111e13 0.530900 0.265450 − 0.964125i \(-0.414479\pi\)
0.265450 + 0.964125i \(0.414479\pi\)
\(74\) 0 0
\(75\) 3.75692e13 − 7.81759e13i 0.324991 − 0.676259i
\(76\) 0 0
\(77\) 4.78621e13 − 8.28996e13i 0.339869 − 0.588670i
\(78\) 0 0
\(79\) −1.47142e14 − 2.54857e14i −0.862052 − 1.49312i −0.869945 − 0.493149i \(-0.835846\pi\)
0.00789293 − 0.999969i \(-0.497488\pi\)
\(80\) 0 0
\(81\) −1.51384e14 − 1.39549e14i −0.735262 − 0.677783i
\(82\) 0 0
\(83\) −6.27413e13 − 1.08671e14i −0.253786 − 0.439570i 0.710779 − 0.703415i \(-0.248343\pi\)
−0.964565 + 0.263845i \(0.915009\pi\)
\(84\) 0 0
\(85\) −3.19396e14 + 5.53210e14i −1.08066 + 1.87175i
\(86\) 0 0
\(87\) −4.83029e13 + 1.00511e14i −0.137271 + 0.285641i
\(88\) 0 0
\(89\) −2.39621e14 −0.574247 −0.287124 − 0.957894i \(-0.592699\pi\)
−0.287124 + 0.957894i \(0.592699\pi\)
\(90\) 0 0
\(91\) 2.47415e14 0.501897
\(92\) 0 0
\(93\) 7.90727e13 − 5.99041e12i 0.136271 − 0.0103237i
\(94\) 0 0
\(95\) −3.25372e14 + 5.63560e14i −0.478027 + 0.827968i
\(96\) 0 0
\(97\) −5.83535e14 − 1.01071e15i −0.733295 − 1.27010i −0.955467 − 0.295097i \(-0.904648\pi\)
0.222173 − 0.975007i \(-0.428685\pi\)
\(98\) 0 0
\(99\) 1.55407e15 − 2.36827e14i 1.67574 − 0.255368i
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.13.12 ✓ 30
3.2 odd 2 108.16.e.a.37.4 30
9.2 odd 6 108.16.e.a.73.4 30
9.7 even 3 inner 36.16.e.a.25.12 yes 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.16.e.a.13.12 ✓ 30 1.1 even 1 trivial
36.16.e.a.25.12 yes 30 9.7 even 3 inner
108.16.e.a.37.4 30 3.2 odd 2
108.16.e.a.73.4 30 9.2 odd 6