Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,11,Mod(53,80)] 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("80.53"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.t (of order \(4\), 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: \(50.8285802139\)
Analytic rank: \(0\)
Dimension: \(236\)
Relative dimension: \(118\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 53.15
Character \(\chi\) \(=\) 80.53
Dual form 80.11.t.a.77.15

$q$-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+(-29.3849 - 12.6700i) q^{2} +223.074 q^{3} +(702.944 + 744.611i) q^{4} +(2920.49 + 1111.93i) q^{5} +(-6554.99 - 2826.33i) q^{6} +(-14947.8 - 14947.8i) q^{7} +(-11221.7 - 30786.6i) q^{8} -9287.21 q^{9} +(-71730.0 - 69676.4i) q^{10} +(-130332. - 130332. i) q^{11} +(156808. + 166103. i) q^{12} -18443.5 q^{13} +(249851. + 628626. i) q^{14} +(651483. + 248042. i) q^{15} +(-60316.1 + 1.04684e6i) q^{16} +(-261846. + 261846. i) q^{17} +(272904. + 117669. i) q^{18} +(1.88767e6 + 1.88767e6i) q^{19} +(1.22498e6 + 2.95625e6i) q^{20} +(-3.33445e6 - 3.33445e6i) q^{21} +(2.17849e6 + 5.48110e6i) q^{22} +(-3.07833e6 + 3.07833e6i) q^{23} +(-2.50327e6 - 6.86767e6i) q^{24} +(7.29284e6 + 6.49476e6i) q^{25} +(541959. + 233678. i) q^{26} -1.52440e7 q^{27} +(622837. - 2.16377e7i) q^{28} +(2.80470e7 + 2.80470e7i) q^{29} +(-1.60011e7 - 1.55430e7i) q^{30} -4.48688e7 q^{31} +(1.50358e7 - 2.99971e7i) q^{32} +(-2.90736e7 - 2.90736e7i) q^{33} +(1.10119e7 - 4.37674e6i) q^{34} +(-2.70338e7 - 6.02756e7i) q^{35} +(-6.52839e6 - 6.91536e6i) q^{36} +3.76279e7 q^{37} +(-3.15523e7 - 7.93858e7i) q^{38} -4.11425e6 q^{39} +(1.45971e6 - 1.02390e8i) q^{40} +1.02894e8i q^{41} +(5.57351e7 + 1.40230e8i) q^{42} +1.69725e8i q^{43} +(5.43062e6 - 1.88663e8i) q^{44} +(-2.71232e7 - 1.03267e7i) q^{45} +(1.29459e8 - 5.14541e7i) q^{46} +(-3.45536e7 + 3.45536e7i) q^{47} +(-1.34549e7 + 2.33522e8i) q^{48} +1.64396e8i q^{49} +(-1.32011e8 - 2.83248e8i) q^{50} +(-5.84109e7 + 5.84109e7i) q^{51} +(-1.29647e7 - 1.37332e7i) q^{52} -1.85113e8i q^{53} +(4.47943e8 + 1.93141e8i) q^{54} +(-2.35713e8 - 5.25553e8i) q^{55} +(-2.92451e8 + 6.27931e8i) q^{56} +(4.21090e8 + 4.21090e8i) q^{57} +(-4.68803e8 - 1.17951e9i) q^{58} +(-6.42866e8 + 6.42866e8i) q^{59} +(2.73261e8 + 6.59461e8i) q^{60} +(-3.14740e7 + 3.14740e7i) q^{61} +(1.31847e9 + 5.68487e8i) q^{62} +(1.38823e8 + 1.38823e8i) q^{63} +(-8.21888e8 + 6.90957e8i) q^{64} +(-5.38638e7 - 2.05079e7i) q^{65} +(4.85964e8 + 1.22269e9i) q^{66} -6.44840e8i q^{67} +(-3.79037e8 - 1.09105e7i) q^{68} +(-6.86695e8 + 6.86695e8i) q^{69} +(3.06961e7 + 2.11371e9i) q^{70} +2.39136e9i q^{71} +(1.04219e8 + 2.85922e8i) q^{72} +(-6.69025e8 + 6.69025e8i) q^{73} +(-1.10569e9 - 4.76745e8i) q^{74} +(1.62684e9 + 1.44881e9i) q^{75} +(-7.86547e7 + 2.73251e9i) q^{76} +3.89635e9i q^{77} +(1.20897e8 + 5.21274e7i) q^{78} +1.97302e9i q^{79} +(-1.34017e9 + 2.99021e9i) q^{80} -2.85213e9 q^{81} +(1.30367e9 - 3.02354e9i) q^{82} -6.30727e9 q^{83} +(1.38938e8 - 4.82680e9i) q^{84} +(-1.05587e9 + 4.73563e8i) q^{85} +(2.15042e9 - 4.98736e9i) q^{86} +(6.25653e9 + 6.25653e9i) q^{87} +(-2.54993e9 + 5.47503e9i) q^{88} +1.04063e10 q^{89} +(6.66172e8 + 6.47100e8i) q^{90} +(2.75688e8 + 2.75688e8i) q^{91} +(-4.45606e9 - 1.28267e8i) q^{92} -1.00090e10 q^{93} +(1.45315e9 - 5.77561e8i) q^{94} +(3.41396e9 + 7.61188e9i) q^{95} +(3.35409e9 - 6.69155e9i) q^{96} +(1.82158e9 - 1.82158e9i) q^{97} +(2.08289e9 - 4.83076e9i) q^{98} +(1.21042e9 + 1.21042e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 236 q - 2 q^{2} - 4 q^{3} - 1220 q^{4} - 2 q^{5} - 4 q^{6} - 69128 q^{8} + 4487724 q^{9} - 2050 q^{10} - 4 q^{11} + 502036 q^{12} - 4 q^{13} - 4 q^{15} - 2041064 q^{16} - 4 q^{17} - 5928762 q^{18} + 5107040 q^{19}+ \cdots + 28071956444 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{1}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −29.3849 − 12.6700i −0.918278 − 0.395937i
\(3\) 223.074 0.917998 0.458999 − 0.888437i \(-0.348208\pi\)
0.458999 + 0.888437i \(0.348208\pi\)
\(4\) 702.944 + 744.611i 0.686469 + 0.727160i
\(5\) 2920.49 + 1111.93i 0.934555 + 0.355818i
\(6\) −6554.99 − 2826.33i −0.842977 − 0.363469i
\(7\) −14947.8 − 14947.8i −0.889377 − 0.889377i 0.105086 − 0.994463i \(-0.466488\pi\)
−0.994463 + 0.105086i \(0.966488\pi\)
\(8\) −11221.7 − 30786.6i −0.342460 − 0.939532i
\(9\) −9287.21 −0.157280
\(10\) −71730.0 − 69676.4i −0.717300 − 0.696764i
\(11\) −130332. − 130332.i −0.809260 − 0.809260i 0.175262 − 0.984522i \(-0.443923\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(12\) 156808. + 166103.i 0.630177 + 0.667531i
\(13\) −18443.5 −0.0496736 −0.0248368 − 0.999692i \(-0.507907\pi\)
−0.0248368 + 0.999692i \(0.507907\pi\)
\(14\) 249851. + 628626.i 0.464559 + 1.16883i
\(15\) 651483. + 248042.i 0.857920 + 0.326640i
\(16\) −60316.1 + 1.04684e6i −0.0575219 + 0.998344i
\(17\) −261846. + 261846.i −0.184417 + 0.184417i −0.793277 − 0.608860i \(-0.791627\pi\)
0.608860 + 0.793277i \(0.291627\pi\)
\(18\) 272904. + 117669.i 0.144427 + 0.0622728i
\(19\) 1.88767e6 + 1.88767e6i 0.762357 + 0.762357i 0.976748 − 0.214391i \(-0.0687766\pi\)
−0.214391 + 0.976748i \(0.568777\pi\)
\(20\) 1.22498e6 + 2.95625e6i 0.382806 + 0.923829i
\(21\) −3.33445e6 − 3.33445e6i −0.816447 − 0.816447i
\(22\) 2.17849e6 + 5.48110e6i 0.422710 + 1.06354i
\(23\) −3.07833e6 + 3.07833e6i −0.478274 + 0.478274i −0.904579 − 0.426306i \(-0.859815\pi\)
0.426306 + 0.904579i \(0.359815\pi\)
\(24\) −2.50327e6 − 6.86767e6i −0.314377 − 0.862489i
\(25\) 7.29284e6 + 6.49476e6i 0.746787 + 0.665063i
\(26\) 541959. + 233678.i 0.0456141 + 0.0196676i
\(27\) −1.52440e7 −1.06238
\(28\) 622837. − 2.16377e7i 0.0361896 − 1.25725i
\(29\) 2.80470e7 + 2.80470e7i 1.36740 + 1.36740i 0.864128 + 0.503273i \(0.167871\pi\)
0.503273 + 0.864128i \(0.332129\pi\)
\(30\) −1.60011e7 − 1.55430e7i −0.658480 − 0.639628i
\(31\) −4.48688e7 −1.56724 −0.783622 − 0.621238i \(-0.786630\pi\)
−0.783622 + 0.621238i \(0.786630\pi\)
\(32\) 1.50358e7 − 2.99971e7i 0.448102 − 0.893982i
\(33\) −2.90736e7 − 2.90736e7i −0.742899 − 0.742899i
\(34\) 1.10119e7 − 4.37674e6i 0.242364 − 0.0963287i
\(35\) −2.70338e7 − 6.02756e7i −0.514716 − 1.14763i
\(36\) −6.52839e6 − 6.91536e6i −0.107968 − 0.114367i
\(37\) 3.76279e7 0.542627 0.271314 − 0.962491i \(-0.412542\pi\)
0.271314 + 0.962491i \(0.412542\pi\)
\(38\) −3.15523e7 − 7.93858e7i −0.398211 − 1.00190i
\(39\) −4.11425e6 −0.0456002
\(40\) 1.45971e6 − 1.02390e8i 0.0142550 − 0.999898i
\(41\) 1.02894e8i 0.888122i 0.895997 + 0.444061i \(0.146463\pi\)
−0.895997 + 0.444061i \(0.853537\pi\)
\(42\) 5.57351e7 + 1.40230e8i 0.426464 + 1.07299i
\(43\) 1.69725e8i 1.15453i 0.816557 + 0.577264i \(0.195880\pi\)
−0.816557 + 0.577264i \(0.804120\pi\)
\(44\) 5.43062e6 − 1.88663e8i 0.0329296 − 1.14399i
\(45\) −2.71232e7 − 1.03267e7i −0.146987 − 0.0559630i
\(46\) 1.29459e8 − 5.14541e7i 0.628554 − 0.249822i
\(47\) −3.45536e7 + 3.45536e7i −0.150662 + 0.150662i −0.778414 − 0.627752i \(-0.783975\pi\)
0.627752 + 0.778414i \(0.283975\pi\)
\(48\) −1.34549e7 + 2.33522e8i −0.0528050 + 0.916478i
\(49\) 1.64396e8i 0.581984i
\(50\) −1.32011e8 − 2.83248e8i −0.422435 − 0.906393i
\(51\) −5.84109e7 + 5.84109e7i −0.169295 + 0.169295i
\(52\) −1.29647e7 − 1.37332e7i −0.0340993 − 0.0361206i
\(53\) − 1.85113e8i − 0.442647i −0.975200 − 0.221324i \(-0.928962\pi\)
0.975200 − 0.221324i \(-0.0710377\pi\)
\(54\) 4.47943e8 + 1.93141e8i 0.975560 + 0.420635i
\(55\) −2.35713e8 − 5.25553e8i −0.468349 − 1.04425i
\(56\) −2.92451e8 + 6.27931e8i −0.531023 + 1.14017i
\(57\) 4.21090e8 + 4.21090e8i 0.699842 + 0.699842i
\(58\) −4.68803e8 − 1.17951e9i −0.714250 − 1.79706i
\(59\) −6.42866e8 + 6.42866e8i −0.899208 + 0.899208i −0.995366 − 0.0961582i \(-0.969345\pi\)
0.0961582 + 0.995366i \(0.469345\pi\)
\(60\) 2.73261e8 + 6.59461e8i 0.351415 + 0.848073i
\(61\) −3.14740e7 + 3.14740e7i −0.0372651 + 0.0372651i −0.725494 − 0.688229i \(-0.758389\pi\)
0.688229 + 0.725494i \(0.258389\pi\)
\(62\) 1.31847e9 + 5.68487e8i 1.43916 + 0.620529i
\(63\) 1.38823e8 + 1.38823e8i 0.139881 + 0.139881i
\(64\) −8.21888e8 + 6.90957e8i −0.765443 + 0.643504i
\(65\) −5.38638e7 − 2.05079e7i −0.0464227 − 0.0176748i
\(66\) 4.85964e8 + 1.22269e9i 0.388047 + 0.976329i
\(67\) − 6.44840e8i − 0.477615i −0.971067 − 0.238807i \(-0.923244\pi\)
0.971067 − 0.238807i \(-0.0767564\pi\)
\(68\) −3.79037e8 − 1.09105e7i −0.260697 − 0.00750412i
\(69\) −6.86695e8 + 6.86695e8i −0.439054 + 0.439054i
\(70\) 3.06961e7 + 2.11371e9i 0.0182638 + 1.25764i
\(71\) 2.39136e9i 1.32542i 0.748877 + 0.662709i \(0.230593\pi\)
−0.748877 + 0.662709i \(0.769407\pi\)
\(72\) 1.04219e8 + 2.85922e8i 0.0538620 + 0.147769i
\(73\) −6.69025e8 + 6.69025e8i −0.322722 + 0.322722i −0.849810 − 0.527089i \(-0.823284\pi\)
0.527089 + 0.849810i \(0.323284\pi\)
\(74\) −1.10569e9 − 4.76745e8i −0.498283 − 0.214846i
\(75\) 1.62684e9 + 1.44881e9i 0.685549 + 0.610527i
\(76\) −7.86547e7 + 2.73251e9i −0.0310211 + 1.07769i
\(77\) 3.89635e9i 1.43947i
\(78\) 1.20897e8 + 5.21274e7i 0.0418737 + 0.0180548i
\(79\) 1.97302e9i 0.641205i 0.947214 + 0.320602i \(0.103885\pi\)
−0.947214 + 0.320602i \(0.896115\pi\)
\(80\) −1.34017e9 + 2.99021e9i −0.408986 + 0.912540i
\(81\) −2.85213e9 −0.817983
\(82\) 1.30367e9 − 3.02354e9i 0.351640 − 0.815543i
\(83\) −6.30727e9 −1.60122 −0.800610 − 0.599185i \(-0.795491\pi\)
−0.800610 + 0.599185i \(0.795491\pi\)
\(84\) 1.38938e8 − 4.82680e9i 0.0332220 − 1.15415i
\(85\) −1.05587e9 + 4.73563e8i −0.237967 + 0.106729i
\(86\) 2.15042e9 − 4.98736e9i 0.457120 − 1.06018i
\(87\) 6.25653e9 + 6.25653e9i 1.25527 + 1.25527i
\(88\) −2.54993e9 + 5.47503e9i −0.483187 + 1.03747i
\(89\) 1.04063e10 1.86358 0.931790 − 0.362997i \(-0.118246\pi\)
0.931790 + 0.362997i \(0.118246\pi\)
\(90\) 6.66172e8 + 6.47100e8i 0.112817 + 0.109587i
\(91\) 2.75688e8 + 2.75688e8i 0.0441786 + 0.0441786i
\(92\) −4.45606e9 − 1.28267e8i −0.676101 − 0.0194614i
\(93\) −1.00090e10 −1.43873
\(94\) 1.45315e9 − 5.77561e8i 0.198002 − 0.0786971i
\(95\) 3.41396e9 + 7.61188e9i 0.441204 + 0.983726i
\(96\) 3.35409e9 − 6.69155e9i 0.411357 − 0.820674i
\(97\) 1.82158e9 − 1.82158e9i 0.212124 − 0.212124i −0.593045 − 0.805169i \(-0.702074\pi\)
0.805169 + 0.593045i \(0.202074\pi\)
\(98\) 2.08289e9 − 4.83076e9i 0.230429 − 0.534423i
\(99\) 1.21042e9 + 1.21042e9i 0.127280 + 0.127280i
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 80.11.t.a.53.15 yes 236
5.2 odd 4 80.11.i.a.37.74 yes 236
16.13 even 4 80.11.i.a.13.74 ✓ 236
80.77 odd 4 inner 80.11.t.a.77.15 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.74 ✓ 236 16.13 even 4
80.11.i.a.37.74 yes 236 5.2 odd 4
80.11.t.a.53.15 yes 236 1.1 even 1 trivial
80.11.t.a.77.15 yes 236 80.77 odd 4 inner