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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 53.7
Character \(\chi\) \(=\) 80.53
Dual form 80.11.t.a.77.7

$q$-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+(-31.5860 - 5.13086i) q^{2} -69.2669 q^{3} +(971.348 + 324.127i) q^{4} +(-1327.58 + 2828.99i) q^{5} +(2187.86 + 355.399i) q^{6} +(-13184.8 - 13184.8i) q^{7} +(-29017.9 - 15221.7i) q^{8} -54251.1 q^{9} +(56448.0 - 82544.7i) q^{10} +(-38422.1 - 38422.1i) q^{11} +(-67282.3 - 22451.2i) q^{12} +87175.7 q^{13} +(348805. + 484103. i) q^{14} +(91957.2 - 195955. i) q^{15} +(838460. + 629680. i) q^{16} +(1.54495e6 - 1.54495e6i) q^{17} +(1.71357e6 + 278355. i) q^{18} +(-1.48189e6 - 1.48189e6i) q^{19} +(-2.20649e6 + 2.31763e6i) q^{20} +(913268. + 913268. i) q^{21} +(1.01646e6 + 1.41074e6i) q^{22} +(-1.71620e6 + 1.71620e6i) q^{23} +(2.00998e6 + 1.05436e6i) q^{24} +(-6.24070e6 - 7.51140e6i) q^{25} +(-2.75353e6 - 447287. i) q^{26} +7.84795e6 q^{27} +(-8.53347e6 - 1.70805e7i) q^{28} +(-1.44644e7 - 1.44644e7i) q^{29} +(-3.90998e6 + 5.71761e6i) q^{30} -1.29204e7 q^{31} +(-2.32528e7 - 2.41911e7i) q^{32} +(2.66138e6 + 2.66138e6i) q^{33} +(-5.67256e7 + 4.08717e7i) q^{34} +(5.48033e7 - 1.97957e7i) q^{35} +(-5.26967e7 - 1.75842e7i) q^{36} +1.54581e6 q^{37} +(3.92036e7 + 5.44103e7i) q^{38} -6.03839e6 q^{39} +(8.15856e7 - 6.18833e7i) q^{40} -1.52422e8i q^{41} +(-2.41606e7 - 3.35323e7i) q^{42} +1.14206e8i q^{43} +(-2.48676e7 - 4.97749e7i) q^{44} +(7.20226e7 - 1.53476e8i) q^{45} +(6.30135e7 - 4.54023e7i) q^{46} +(-8.86098e7 + 8.86098e7i) q^{47} +(-5.80775e7 - 4.36160e7i) q^{48} +6.52010e7i q^{49} +(1.58579e8 + 2.69275e8i) q^{50} +(-1.07014e8 + 1.07014e8i) q^{51} +(8.46780e7 + 2.82560e7i) q^{52} +1.73562e8i q^{53} +(-2.47885e8 - 4.02667e7i) q^{54} +(1.59704e8 - 5.76873e7i) q^{55} +(1.81900e8 + 5.83290e8i) q^{56} +(1.02646e8 + 1.02646e8i) q^{57} +(3.82657e8 + 5.31086e8i) q^{58} +(-3.88149e8 + 3.88149e8i) q^{59} +(1.52837e8 - 1.60535e8i) q^{60} +(-4.03435e8 + 4.03435e8i) q^{61} +(4.08103e8 + 6.62927e7i) q^{62} +(7.15288e8 + 7.15288e8i) q^{63} +(6.10341e8 + 8.83406e8i) q^{64} +(-1.15733e8 + 2.46619e8i) q^{65} +(-7.04072e7 - 9.77176e7i) q^{66} +2.56120e9i q^{67} +(2.00144e9 - 9.99923e8i) q^{68} +(1.18876e8 - 1.18876e8i) q^{69} +(-1.83259e9 + 3.44078e8i) q^{70} -1.85271e9i q^{71} +(1.57426e9 + 8.25795e8i) q^{72} +(1.03792e9 - 1.03792e9i) q^{73} +(-4.88258e7 - 7.93132e6i) q^{74} +(4.32274e8 + 5.20291e8i) q^{75} +(-9.59111e8 - 1.91975e9i) q^{76} +1.01317e9i q^{77} +(1.90729e8 + 3.09822e7i) q^{78} +1.65699e9i q^{79} +(-2.89448e9 + 1.53604e9i) q^{80} +2.65987e9 q^{81} +(-7.82054e8 + 4.81438e9i) q^{82} +4.05643e9 q^{83} +(5.91087e8 + 1.18312e9i) q^{84} +(2.31959e9 + 6.42167e9i) q^{85} +(5.85976e8 - 3.60731e9i) q^{86} +(1.00190e9 + 1.00190e9i) q^{87} +(5.30081e8 + 1.69978e9i) q^{88} -2.67676e9 q^{89} +(-3.06237e9 + 4.47814e9i) q^{90} +(-1.14939e9 - 1.14939e9i) q^{91} +(-2.22329e9 + 1.11076e9i) q^{92} +8.94955e8 q^{93} +(3.25347e9 - 2.34418e9i) q^{94} +(6.15957e9 - 2.22492e9i) q^{95} +(1.61065e9 + 1.67564e9i) q^{96} +(-4.54638e9 + 4.54638e9i) q^{97} +(3.34538e8 - 2.05944e9i) q^{98} +(2.08444e9 + 2.08444e9i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.5860 − 5.13086i −0.987062 − 0.160339i
\(3\) −69.2669 −0.285049 −0.142524 − 0.989791i \(-0.545522\pi\)
−0.142524 + 0.989791i \(0.545522\pi\)
\(4\) 971.348 + 324.127i 0.948583 + 0.316530i
\(5\) −1327.58 + 2828.99i −0.424825 + 0.905275i
\(6\) 2187.86 + 355.399i 0.281361 + 0.0457046i
\(7\) −13184.8 − 13184.8i −0.784481 − 0.784481i 0.196103 − 0.980583i \(-0.437171\pi\)
−0.980583 + 0.196103i \(0.937171\pi\)
\(8\) −29017.9 − 15221.7i −0.885557 − 0.464530i
\(9\) −54251.1 −0.918747
\(10\) 56448.0 − 82544.7i 0.564480 − 0.825447i
\(11\) −38422.1 − 38422.1i −0.238571 − 0.238571i 0.577687 − 0.816258i \(-0.303955\pi\)
−0.816258 + 0.577687i \(0.803955\pi\)
\(12\) −67282.3 − 22451.2i −0.270392 − 0.0902265i
\(13\) 87175.7 0.234790 0.117395 − 0.993085i \(-0.462546\pi\)
0.117395 + 0.993085i \(0.462546\pi\)
\(14\) 348805. + 484103.i 0.648548 + 0.900114i
\(15\) 91957.2 − 195955.i 0.121096 − 0.258048i
\(16\) 838460. + 629680.i 0.799618 + 0.600510i
\(17\) 1.54495e6 − 1.54495e6i 1.08810 − 1.08810i 0.0923758 − 0.995724i \(-0.470554\pi\)
0.995724 − 0.0923758i \(-0.0294461\pi\)
\(18\) 1.71357e6 + 278355.i 0.906860 + 0.147311i
\(19\) −1.48189e6 − 1.48189e6i −0.598477 − 0.598477i 0.341430 − 0.939907i \(-0.389089\pi\)
−0.939907 + 0.341430i \(0.889089\pi\)
\(20\) −2.20649e6 + 2.31763e6i −0.689528 + 0.724259i
\(21\) 913268. + 913268.i 0.223615 + 0.223615i
\(22\) 1.01646e6 + 1.41074e6i 0.197232 + 0.273737i
\(23\) −1.71620e6 + 1.71620e6i −0.266642 + 0.266642i −0.827746 − 0.561104i \(-0.810377\pi\)
0.561104 + 0.827746i \(0.310377\pi\)
\(24\) 2.00998e6 + 1.05436e6i 0.252427 + 0.132414i
\(25\) −6.24070e6 − 7.51140e6i −0.639047 − 0.769168i
\(26\) −2.75353e6 − 447287.i −0.231752 − 0.0376460i
\(27\) 7.84795e6 0.546937
\(28\) −8.53347e6 − 1.70805e7i −0.495833 − 0.992456i
\(29\) −1.44644e7 − 1.44644e7i −0.705195 − 0.705195i 0.260326 − 0.965521i \(-0.416170\pi\)
−0.965521 + 0.260326i \(0.916170\pi\)
\(30\) −3.90998e6 + 5.71761e6i −0.160904 + 0.235293i
\(31\) −1.29204e7 −0.451302 −0.225651 − 0.974208i \(-0.572451\pi\)
−0.225651 + 0.974208i \(0.572451\pi\)
\(32\) −2.32528e7 − 2.41911e7i −0.692987 − 0.720950i
\(33\) 2.66138e6 + 2.66138e6i 0.0680045 + 0.0680045i
\(34\) −5.67256e7 + 4.08717e7i −1.24849 + 0.899557i
\(35\) 5.48033e7 − 1.97957e7i 1.04344 − 0.376904i
\(36\) −5.26967e7 − 1.75842e7i −0.871507 − 0.290811i
\(37\) 1.54581e6 0.0222919 0.0111459 − 0.999938i \(-0.496452\pi\)
0.0111459 + 0.999938i \(0.496452\pi\)
\(38\) 3.92036e7 + 5.44103e7i 0.494775 + 0.686694i
\(39\) −6.03839e6 −0.0669265
\(40\) 8.15856e7 − 6.18833e7i 0.796735 − 0.604329i
\(41\) − 1.52422e8i − 1.31561i −0.753188 − 0.657805i \(-0.771485\pi\)
0.753188 − 0.657805i \(-0.228515\pi\)
\(42\) −2.41606e7 − 3.35323e7i −0.184868 − 0.256577i
\(43\) 1.14206e8i 0.776868i 0.921476 + 0.388434i \(0.126984\pi\)
−0.921476 + 0.388434i \(0.873016\pi\)
\(44\) −2.48676e7 − 4.97749e7i −0.150790 − 0.301819i
\(45\) 7.20226e7 − 1.53476e8i 0.390307 − 0.831719i
\(46\) 6.30135e7 − 4.54023e7i 0.305946 − 0.220439i
\(47\) −8.86098e7 + 8.86098e7i −0.386360 + 0.386360i −0.873387 − 0.487027i \(-0.838081\pi\)
0.487027 + 0.873387i \(0.338081\pi\)
\(48\) −5.80775e7 − 4.36160e7i −0.227930 − 0.171175i
\(49\) 6.52010e7i 0.230820i
\(50\) 1.58579e8 + 2.69275e8i 0.507451 + 0.861680i
\(51\) −1.07014e8 + 1.07014e8i −0.310162 + 0.310162i
\(52\) 8.46780e7 + 2.82560e7i 0.222717 + 0.0743179i
\(53\) 1.73562e8i 0.415027i 0.978232 + 0.207513i \(0.0665371\pi\)
−0.978232 + 0.207513i \(0.933463\pi\)
\(54\) −2.47885e8 − 4.02667e7i −0.539861 − 0.0876956i
\(55\) 1.59704e8 − 5.76873e7i 0.317324 − 0.114622i
\(56\) 1.81900e8 + 5.83290e8i 0.330288 + 1.05912i
\(57\) 1.02646e8 + 1.02646e8i 0.170595 + 0.170595i
\(58\) 3.82657e8 + 5.31086e8i 0.583001 + 0.809142i
\(59\) −3.88149e8 + 3.88149e8i −0.542923 + 0.542923i −0.924385 − 0.381462i \(-0.875421\pi\)
0.381462 + 0.924385i \(0.375421\pi\)
\(60\) 1.52837e8 − 1.60535e8i 0.196549 − 0.206449i
\(61\) −4.03435e8 + 4.03435e8i −0.477666 + 0.477666i −0.904385 − 0.426718i \(-0.859670\pi\)
0.426718 + 0.904385i \(0.359670\pi\)
\(62\) 4.08103e8 + 6.62927e7i 0.445463 + 0.0723615i
\(63\) 7.15288e8 + 7.15288e8i 0.720739 + 0.720739i
\(64\) 6.10341e8 + 8.83406e8i 0.568424 + 0.822736i
\(65\) −1.15733e8 + 2.46619e8i −0.0997445 + 0.212549i
\(66\) −7.04072e7 − 9.77176e7i −0.0562208 − 0.0780284i
\(67\) 2.56120e9i 1.89701i 0.316763 + 0.948505i \(0.397404\pi\)
−0.316763 + 0.948505i \(0.602596\pi\)
\(68\) 2.00144e9 − 9.99923e8i 1.37657 − 0.687736i
\(69\) 1.18876e8 − 1.18876e8i 0.0760061 − 0.0760061i
\(70\) −1.83259e9 + 3.44078e8i −1.09037 + 0.204723i
\(71\) − 1.85271e9i − 1.02687i −0.858128 − 0.513436i \(-0.828372\pi\)
0.858128 − 0.513436i \(-0.171628\pi\)
\(72\) 1.57426e9 + 8.25795e8i 0.813603 + 0.426785i
\(73\) 1.03792e9 − 1.03792e9i 0.500669 − 0.500669i −0.410977 − 0.911646i \(-0.634812\pi\)
0.911646 + 0.410977i \(0.134812\pi\)
\(74\) −4.88258e7 − 7.93132e6i −0.0220034 − 0.00357427i
\(75\) 4.32274e8 + 5.20291e8i 0.182160 + 0.219250i
\(76\) −9.59111e8 − 1.91975e9i −0.378269 − 0.757141i
\(77\) 1.01317e9i 0.374309i
\(78\) 1.90729e8 + 3.09822e7i 0.0660606 + 0.0107310i
\(79\) 1.65699e9i 0.538500i 0.963070 + 0.269250i \(0.0867758\pi\)
−0.963070 + 0.269250i \(0.913224\pi\)
\(80\) −2.89448e9 + 1.53604e9i −0.883324 + 0.468763i
\(81\) 2.65987e9 0.762843
\(82\) −7.82054e8 + 4.81438e9i −0.210944 + 1.29859i
\(83\) 4.05643e9 1.02980 0.514901 − 0.857249i \(-0.327829\pi\)
0.514901 + 0.857249i \(0.327829\pi\)
\(84\) 5.91087e8 + 1.18312e9i 0.141337 + 0.282899i
\(85\) 2.31959e9 + 6.42167e9i 0.522778 + 1.44728i
\(86\) 5.85976e8 − 3.60731e9i 0.124563 − 0.766817i
\(87\) 1.00190e9 + 1.00190e9i 0.201015 + 0.201015i
\(88\) 5.30081e8 + 1.69978e9i 0.100445 + 0.322092i
\(89\) −2.67676e9 −0.479357 −0.239678 − 0.970852i \(-0.577042\pi\)
−0.239678 + 0.970852i \(0.577042\pi\)
\(90\) −3.06237e9 + 4.47814e9i −0.518614 + 0.758377i
\(91\) −1.14939e9 − 1.14939e9i −0.184188 − 0.184188i
\(92\) −2.22329e9 + 1.11076e9i −0.337332 + 0.168532i
\(93\) 8.94955e8 0.128643
\(94\) 3.25347e9 − 2.34418e9i 0.443310 − 0.319413i
\(95\) 6.15957e9 − 2.22492e9i 0.796035 − 0.287539i
\(96\) 1.61065e9 + 1.67564e9i 0.197535 + 0.205506i
\(97\) −4.54638e9 + 4.54638e9i −0.529428 + 0.529428i −0.920402 − 0.390974i \(-0.872138\pi\)
0.390974 + 0.920402i \(0.372138\pi\)
\(98\) 3.34538e8 − 2.05944e9i 0.0370096 − 0.227834i
\(99\) 2.08444e9 + 2.08444e9i 0.219187 + 0.219187i
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.53.7 yes 236
5.2 odd 4 80.11.i.a.37.67 yes 236
16.13 even 4 80.11.i.a.13.67 ✓ 236
80.77 odd 4 inner 80.11.t.a.77.7 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.67 ✓ 236 16.13 even 4
80.11.i.a.37.67 yes 236 5.2 odd 4
80.11.t.a.53.7 yes 236 1.1 even 1 trivial
80.11.t.a.77.7 yes 236 80.77 odd 4 inner