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 53.9
Character \(\chi\) \(=\) 80.53
Dual form 80.11.t.a.77.9

$q$-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+(-31.4526 + 5.89374i) q^{2} +259.387 q^{3} +(954.528 - 370.746i) q^{4} +(2667.14 - 1628.49i) q^{5} +(-8158.40 + 1528.76i) q^{6} +(820.915 + 820.915i) q^{7} +(-27837.3 + 17286.7i) q^{8} +8232.78 q^{9} +(-74290.5 + 66939.7i) q^{10} +(-16405.1 - 16405.1i) q^{11} +(247592. - 96166.9i) q^{12} -45239.7 q^{13} +(-30658.1 - 20981.6i) q^{14} +(691822. - 422410. i) q^{15} +(773670. - 707775. i) q^{16} +(934994. - 934994. i) q^{17} +(-258942. + 48521.9i) q^{18} +(-918795. - 918795. i) q^{19} +(1.94210e6 - 2.54327e6i) q^{20} +(212935. + 212935. i) q^{21} +(612672. + 419296. i) q^{22} +(5.46666e6 - 5.46666e6i) q^{23} +(-7.22063e6 + 4.48394e6i) q^{24} +(4.46165e6 - 8.68684e6i) q^{25} +(1.42290e6 - 266631. i) q^{26} -1.31811e7 q^{27} +(1.08794e6 + 479235. i) q^{28} +(-1.51640e6 - 1.51640e6i) q^{29} +(-1.92700e7 + 1.73633e7i) q^{30} +2.14191e7 q^{31} +(-2.01625e7 + 2.68212e7i) q^{32} +(-4.25529e6 - 4.25529e6i) q^{33} +(-2.38974e7 + 3.49186e7i) q^{34} +(3.52635e6 + 852641. i) q^{35} +(7.85842e6 - 3.05227e6i) q^{36} -4.85011e7 q^{37} +(3.43136e7 + 2.34833e7i) q^{38} -1.17346e7 q^{39} +(-4.60947e7 + 9.14387e7i) q^{40} +1.61349e7i q^{41} +(-7.95234e6 - 5.44237e6i) q^{42} +5.95418e7i q^{43} +(-2.17413e7 - 9.57702e6i) q^{44} +(2.19580e7 - 1.34070e7i) q^{45} +(-1.39721e8 + 2.04160e8i) q^{46} +(6.83461e7 - 6.83461e7i) q^{47} +(2.00680e8 - 1.83588e8i) q^{48} -2.81127e8i q^{49} +(-8.91322e7 + 2.99519e8i) q^{50} +(2.42526e8 - 2.42526e8i) q^{51} +(-4.31825e7 + 1.67724e7i) q^{52} -2.69339e8i q^{53} +(4.14579e8 - 7.76858e7i) q^{54} +(-7.04705e7 - 1.70392e7i) q^{55} +(-3.70429e7 - 8.66115e6i) q^{56} +(-2.38324e8 - 2.38324e8i) q^{57} +(5.66320e7 + 3.87575e7i) q^{58} +(-3.27283e8 + 3.27283e8i) q^{59} +(5.03756e8 - 6.59693e8i) q^{60} +(-4.12949e7 + 4.12949e7i) q^{61} +(-6.73686e8 + 1.26239e8i) q^{62} +(6.75842e6 + 6.75842e6i) q^{63} +(4.76085e8 - 9.62427e8i) q^{64} +(-1.20661e8 + 7.36725e7i) q^{65} +(1.58919e8 + 1.08760e8i) q^{66} -1.16917e9i q^{67} +(5.45832e8 - 1.23912e9i) q^{68} +(1.41798e9 - 1.41798e9i) q^{69} +(-1.15938e8 - 6.03436e6i) q^{70} -1.24695e9i q^{71} +(-2.29178e8 + 1.42317e8i) q^{72} +(2.46286e9 - 2.46286e9i) q^{73} +(1.52548e9 - 2.85853e8i) q^{74} +(1.15729e9 - 2.25326e9i) q^{75} +(-1.21765e9 - 5.36375e8i) q^{76} -2.69345e7i q^{77} +(3.69083e8 - 6.91607e7i) q^{78} +2.32462e8i q^{79} +(9.10880e8 - 3.14765e9i) q^{80} -3.90514e9 q^{81} +(-9.50951e7 - 5.07485e8i) q^{82} +2.24207e9 q^{83} +(2.82197e8 + 1.24307e8i) q^{84} +(9.71129e8 - 4.01639e9i) q^{85} +(-3.50924e8 - 1.87274e9i) q^{86} +(-3.93336e8 - 3.93336e8i) q^{87} +(7.40265e8 + 1.73084e8i) q^{88} +3.67125e9 q^{89} +(-6.11617e8 + 5.51100e8i) q^{90} +(-3.71379e7 - 3.71379e7i) q^{91} +(3.19133e9 - 7.24482e9i) q^{92} +5.55584e9 q^{93} +(-1.74685e9 + 2.55247e9i) q^{94} +(-3.94680e9 - 9.54303e8i) q^{95} +(-5.22989e9 + 6.95707e9i) q^{96} +(-1.72048e9 + 1.72048e9i) q^{97} +(1.65689e9 + 8.84218e9i) q^{98} +(-1.35060e8 - 1.35060e8i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.4526 + 5.89374i −0.982893 + 0.184179i
\(3\) 259.387 1.06744 0.533719 − 0.845662i \(-0.320794\pi\)
0.533719 + 0.845662i \(0.320794\pi\)
\(4\) 954.528 − 370.746i 0.932156 − 0.362057i
\(5\) 2667.14 − 1628.49i 0.853485 − 0.521118i
\(6\) −8158.40 + 1528.76i −1.04918 + 0.196600i
\(7\) 820.915 + 820.915i 0.0488437 + 0.0488437i 0.731107 − 0.682263i \(-0.239004\pi\)
−0.682263 + 0.731107i \(0.739004\pi\)
\(8\) −27837.3 + 17286.7i −0.849526 + 0.527547i
\(9\) 8232.78 0.139423
\(10\) −74290.5 + 66939.7i −0.742905 + 0.669397i
\(11\) −16405.1 − 16405.1i −0.101863 − 0.101863i 0.654339 − 0.756202i \(-0.272947\pi\)
−0.756202 + 0.654339i \(0.772947\pi\)
\(12\) 247592. − 96166.9i 0.995018 − 0.386473i
\(13\) −45239.7 −0.121844 −0.0609218 − 0.998143i \(-0.519404\pi\)
−0.0609218 + 0.998143i \(0.519404\pi\)
\(14\) −30658.1 − 20981.6i −0.0570041 − 0.0390121i
\(15\) 691822. − 422410.i 0.911042 − 0.556261i
\(16\) 773670. − 707775.i 0.737829 − 0.674987i
\(17\) 934994. − 934994.i 0.658513 − 0.658513i −0.296515 − 0.955028i \(-0.595825\pi\)
0.955028 + 0.296515i \(0.0958246\pi\)
\(18\) −258942. + 48521.9i −0.137038 + 0.0256788i
\(19\) −918795. − 918795.i −0.371065 − 0.371065i 0.496800 − 0.867865i \(-0.334508\pi\)
−0.867865 + 0.496800i \(0.834508\pi\)
\(20\) 1.94210e6 − 2.54327e6i 0.606907 − 0.794773i
\(21\) 212935. + 212935.i 0.0521375 + 0.0521375i
\(22\) 612672. + 419296.i 0.118882 + 0.0813594i
\(23\) 5.46666e6 − 5.46666e6i 0.849343 − 0.849343i −0.140708 − 0.990051i \(-0.544938\pi\)
0.990051 + 0.140708i \(0.0449380\pi\)
\(24\) −7.22063e6 + 4.48394e6i −0.906816 + 0.563123i
\(25\) 4.46165e6 − 8.68684e6i 0.456872 − 0.889532i
\(26\) 1.42290e6 − 266631.i 0.119759 − 0.0224411i
\(27\) −1.31811e7 −0.918612
\(28\) 1.08794e6 + 479235.i 0.0632141 + 0.0278457i
\(29\) −1.51640e6 − 1.51640e6i −0.0739307 − 0.0739307i 0.669175 − 0.743105i \(-0.266648\pi\)
−0.743105 + 0.669175i \(0.766648\pi\)
\(30\) −1.92700e7 + 1.73633e7i −0.793004 + 0.714540i
\(31\) 2.14191e7 0.748157 0.374078 − 0.927397i \(-0.377959\pi\)
0.374078 + 0.927397i \(0.377959\pi\)
\(32\) −2.01625e7 + 2.68212e7i −0.600888 + 0.799333i
\(33\) −4.25529e6 − 4.25529e6i −0.108732 − 0.108732i
\(34\) −2.38974e7 + 3.49186e7i −0.525963 + 0.768532i
\(35\) 3.52635e6 + 852641.i 0.0671406 + 0.0162340i
\(36\) 7.85842e6 − 3.05227e6i 0.129964 − 0.0504790i
\(37\) −4.85011e7 −0.699427 −0.349714 − 0.936857i \(-0.613721\pi\)
−0.349714 + 0.936857i \(0.613721\pi\)
\(38\) 3.43136e7 + 2.34833e7i 0.433060 + 0.296375i
\(39\) −1.17346e7 −0.130060
\(40\) −4.60947e7 + 9.14387e7i −0.450143 + 0.892956i
\(41\) 1.61349e7i 0.139267i 0.997573 + 0.0696334i \(0.0221830\pi\)
−0.997573 + 0.0696334i \(0.977817\pi\)
\(42\) −7.95234e6 − 5.44237e6i −0.0608483 − 0.0416430i
\(43\) 5.95418e7i 0.405023i 0.979280 + 0.202512i \(0.0649104\pi\)
−0.979280 + 0.202512i \(0.935090\pi\)
\(44\) −2.17413e7 − 9.57702e6i −0.131832 − 0.0580720i
\(45\) 2.19580e7 − 1.34070e7i 0.118995 − 0.0726558i
\(46\) −1.39721e8 + 2.04160e8i −0.678381 + 0.991244i
\(47\) 6.83461e7 − 6.83461e7i 0.298006 − 0.298006i −0.542227 − 0.840232i \(-0.682419\pi\)
0.840232 + 0.542227i \(0.182419\pi\)
\(48\) 2.00680e8 − 1.83588e8i 0.787587 − 0.720507i
\(49\) − 2.81127e8i − 0.995229i
\(50\) −8.91322e7 + 2.99519e8i −0.285223 + 0.958461i
\(51\) 2.42526e8 − 2.42526e8i 0.702921 − 0.702921i
\(52\) −4.31825e7 + 1.67724e7i −0.113577 + 0.0441143i
\(53\) − 2.69339e8i − 0.644050i −0.946731 − 0.322025i \(-0.895636\pi\)
0.946731 − 0.322025i \(-0.104364\pi\)
\(54\) 4.14579e8 − 7.76858e7i 0.902897 − 0.169189i
\(55\) −7.04705e7 − 1.70392e7i −0.140021 − 0.0338559i
\(56\) −3.70429e7 − 8.66115e6i −0.0672613 − 0.0157266i
\(57\) −2.38324e8 − 2.38324e8i −0.396089 − 0.396089i
\(58\) 5.66320e7 + 3.87575e7i 0.0862824 + 0.0590494i
\(59\) −3.27283e8 + 3.27283e8i −0.457787 + 0.457787i −0.897928 − 0.440142i \(-0.854928\pi\)
0.440142 + 0.897928i \(0.354928\pi\)
\(60\) 5.03756e8 − 6.59693e8i 0.647835 − 0.848371i
\(61\) −4.12949e7 + 4.12949e7i −0.0488930 + 0.0488930i −0.731131 − 0.682238i \(-0.761007\pi\)
0.682238 + 0.731131i \(0.261007\pi\)
\(62\) −6.73686e8 + 1.26239e8i −0.735358 + 0.137795i
\(63\) 6.75842e6 + 6.75842e6i 0.00680992 + 0.00680992i
\(64\) 4.76085e8 − 9.62427e8i 0.443388 − 0.896330i
\(65\) −1.20661e8 + 7.36725e7i −0.103992 + 0.0634949i
\(66\) 1.58919e8 + 1.08760e8i 0.126899 + 0.0868461i
\(67\) − 1.16917e9i − 0.865974i −0.901400 − 0.432987i \(-0.857460\pi\)
0.901400 − 0.432987i \(-0.142540\pi\)
\(68\) 5.45832e8 − 1.23912e9i 0.375418 − 0.852256i
\(69\) 1.41798e9 − 1.41798e9i 0.906620 − 0.906620i
\(70\) −1.15938e8 − 6.03436e6i −0.0689820 − 0.00359038i
\(71\) − 1.24695e9i − 0.691126i −0.938396 − 0.345563i \(-0.887688\pi\)
0.938396 − 0.345563i \(-0.112312\pi\)
\(72\) −2.29178e8 + 1.42317e8i −0.118443 + 0.0735521i
\(73\) 2.46286e9 − 2.46286e9i 1.18802 − 1.18802i 0.210409 − 0.977613i \(-0.432520\pi\)
0.977613 − 0.210409i \(-0.0674796\pi\)
\(74\) 1.52548e9 − 2.85853e8i 0.687462 − 0.128820i
\(75\) 1.15729e9 − 2.25326e9i 0.487683 − 0.949520i
\(76\) −1.21765e9 − 5.36375e8i −0.480238 − 0.211544i
\(77\) − 2.69345e7i − 0.00995073i
\(78\) 3.69083e8 − 6.91607e7i 0.127835 − 0.0239544i
\(79\) 2.32462e8i 0.0755469i 0.999286 + 0.0377735i \(0.0120265\pi\)
−0.999286 + 0.0377735i \(0.987973\pi\)
\(80\) 9.10880e8 − 3.14765e9i 0.277978 − 0.960587i
\(81\) −3.90514e9 −1.11998
\(82\) −9.50951e7 − 5.07485e8i −0.0256501 − 0.136884i
\(83\) 2.24207e9 0.569193 0.284597 − 0.958647i \(-0.408140\pi\)
0.284597 + 0.958647i \(0.408140\pi\)
\(84\) 2.82197e8 + 1.24307e8i 0.0674771 + 0.0297236i
\(85\) 9.71129e8 − 4.01639e9i 0.218868 − 0.905193i
\(86\) −3.50924e8 − 1.87274e9i −0.0745969 − 0.398094i
\(87\) −3.93336e8 − 3.93336e8i −0.0789164 − 0.0789164i
\(88\) 7.40265e8 + 1.73084e8i 0.140273 + 0.0327977i
\(89\) 3.67125e9 0.657452 0.328726 − 0.944425i \(-0.393381\pi\)
0.328726 + 0.944425i \(0.393381\pi\)
\(90\) −6.11617e8 + 5.51100e8i −0.103578 + 0.0933293i
\(91\) −3.71379e7 − 3.71379e7i −0.00595129 − 0.00595129i
\(92\) 3.19133e9 − 7.24482e9i 0.484209 − 1.09923i
\(93\) 5.55584e9 0.798611
\(94\) −1.74685e9 + 2.55247e9i −0.238021 + 0.347794i
\(95\) −3.94680e9 − 9.54303e8i −0.510067 − 0.123330i
\(96\) −5.22989e9 + 6.95707e9i −0.641411 + 0.853238i
\(97\) −1.72048e9 + 1.72048e9i −0.200351 + 0.200351i −0.800150 − 0.599799i \(-0.795247\pi\)
0.599799 + 0.800150i \(0.295247\pi\)
\(98\) 1.65689e9 + 8.84218e9i 0.183301 + 0.978203i
\(99\) −1.35060e8 − 1.35060e8i −0.0142020 − 0.0142020i
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.53.9 yes 236
5.2 odd 4 80.11.i.a.37.51 yes 236
16.13 even 4 80.11.i.a.13.51 ✓ 236
80.77 odd 4 inner 80.11.t.a.77.9 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.51 ✓ 236 16.13 even 4
80.11.i.a.37.51 yes 236 5.2 odd 4
80.11.t.a.53.9 yes 236 1.1 even 1 trivial
80.11.t.a.77.9 yes 236 80.77 odd 4 inner