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Properties

Label 80.11.p.a.17.1
Level $80$
Weight $11$
Character 80.17
Analytic conductor $50.829$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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(17,80)] 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("80.17"); 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, 0, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-114] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(50.8285802139\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 80.17
Dual form 80.11.p.a.33.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-57.0000 + 57.0000i) q^{3} +(2925.00 + 1100.00i) q^{5} +(-6953.00 - 6953.00i) q^{7} +52551.0i q^{9} -75242.0 q^{11} +(109857. - 109857. i) q^{13} +(-229425. + 104025. i) q^{15} +(-1.52893e6 - 1.52893e6i) q^{17} +4.03868e6i q^{19} +792642. q^{21} +(712423. - 712423. i) q^{23} +(7.34562e6 + 6.43500e6i) q^{25} +(-6.36120e6 - 6.36120e6i) q^{27} +446120. i q^{29} +2.90807e7 q^{31} +(4.28879e6 - 4.28879e6i) q^{33} +(-1.26892e7 - 2.79858e7i) q^{35} +(-911847. - 911847. i) q^{37} +1.25237e7i q^{39} -1.63946e8 q^{41} +(-1.18423e8 + 1.18423e8i) q^{43} +(-5.78061e7 + 1.53712e8i) q^{45} +(-2.76320e8 - 2.76320e8i) q^{47} -1.85787e8i q^{49} +1.74298e8 q^{51} +(3.08460e8 - 3.08460e8i) q^{53} +(-2.20083e8 - 8.27662e7i) q^{55} +(-2.30205e8 - 2.30205e8i) q^{57} +9.40888e8i q^{59} -1.35361e9 q^{61} +(3.65387e8 - 3.65387e8i) q^{63} +(4.42174e8 - 2.00489e8i) q^{65} +(-8.53571e8 - 8.53571e8i) q^{67} +8.12162e7i q^{69} -2.82701e9 q^{71} +(-2.75330e9 + 2.75330e9i) q^{73} +(-7.85496e8 + 5.19056e7i) q^{75} +(5.23158e8 + 5.23158e8i) q^{77} -3.32450e9i q^{79} -2.37791e9 q^{81} +(-1.34634e9 + 1.34634e9i) q^{83} +(-2.79029e9 - 6.15393e9i) q^{85} +(-2.54288e7 - 2.54288e7i) q^{87} -2.66745e9i q^{89} -1.52767e9 q^{91} +(-1.65760e9 + 1.65760e9i) q^{93} +(-4.44255e9 + 1.18131e10i) q^{95} +(-5.26563e8 - 5.26563e8i) q^{97} -3.95404e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 114 q^{3} + 5850 q^{5} - 13906 q^{7} - 150484 q^{11} + 219714 q^{13} - 458850 q^{15} - 3057854 q^{17} + 1585284 q^{21} + 1424846 q^{23} + 14691250 q^{25} - 12722400 q^{27} + 58161436 q^{31} + 8577588 q^{33}+ \cdots - 1053125694 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −57.0000 + 57.0000i −0.234568 + 0.234568i −0.814596 − 0.580028i \(-0.803041\pi\)
0.580028 + 0.814596i \(0.303041\pi\)
\(4\) 0 0
\(5\) 2925.00 + 1100.00i 0.936000 + 0.352000i
\(6\) 0 0
\(7\) −6953.00 − 6953.00i −0.413697 − 0.413697i 0.469328 − 0.883024i \(-0.344496\pi\)
−0.883024 + 0.469328i \(0.844496\pi\)
\(8\) 0 0
\(9\) 52551.0i 0.889956i
\(10\) 0 0
\(11\) −75242.0 −0.467194 −0.233597 − 0.972334i \(-0.575050\pi\)
−0.233597 + 0.972334i \(0.575050\pi\)
\(12\) 0 0
\(13\) 109857. − 109857.i 0.295877 − 0.295877i −0.543520 − 0.839396i \(-0.682909\pi\)
0.839396 + 0.543520i \(0.182909\pi\)
\(14\) 0 0
\(15\) −229425. + 104025.i −0.302123 + 0.136988i
\(16\) 0 0
\(17\) −1.52893e6 − 1.52893e6i −1.07682 − 1.07682i −0.996793 − 0.0800247i \(-0.974500\pi\)
−0.0800247 − 0.996793i \(-0.525500\pi\)
\(18\) 0 0
\(19\) 4.03868e6i 1.63107i 0.578711 + 0.815533i \(0.303556\pi\)
−0.578711 + 0.815533i \(0.696444\pi\)
\(20\) 0 0
\(21\) 792642. 0.194080
\(22\) 0 0
\(23\) 712423. − 712423.i 0.110688 − 0.110688i −0.649594 − 0.760281i \(-0.725061\pi\)
0.760281 + 0.649594i \(0.225061\pi\)
\(24\) 0 0
\(25\) 7.34562e6 + 6.43500e6i 0.752192 + 0.658944i
\(26\) 0 0
\(27\) −6.36120e6 − 6.36120e6i −0.443323 − 0.443323i
\(28\) 0 0
\(29\) 446120.i 0.0217501i 0.999941 + 0.0108751i \(0.00346171\pi\)
−0.999941 + 0.0108751i \(0.996538\pi\)
\(30\) 0 0
\(31\) 2.90807e7 1.01577 0.507886 − 0.861424i \(-0.330427\pi\)
0.507886 + 0.861424i \(0.330427\pi\)
\(32\) 0 0
\(33\) 4.28879e6 − 4.28879e6i 0.109589 − 0.109589i
\(34\) 0 0
\(35\) −1.26892e7 − 2.79858e7i −0.241599 − 0.532841i
\(36\) 0 0
\(37\) −911847. − 911847.i −0.0131496 − 0.0131496i 0.700501 − 0.713651i \(-0.252960\pi\)
−0.713651 + 0.700501i \(0.752960\pi\)
\(38\) 0 0
\(39\) 1.25237e7i 0.138806i
\(40\) 0 0
\(41\) −1.63946e8 −1.41508 −0.707540 − 0.706674i \(-0.750195\pi\)
−0.707540 + 0.706674i \(0.750195\pi\)
\(42\) 0 0
\(43\) −1.18423e8 + 1.18423e8i −0.805551 + 0.805551i −0.983957 − 0.178406i \(-0.942906\pi\)
0.178406 + 0.983957i \(0.442906\pi\)
\(44\) 0 0
\(45\) −5.78061e7 + 1.53712e8i −0.313264 + 0.832999i
\(46\) 0 0
\(47\) −2.76320e8 − 2.76320e8i −1.20482 − 1.20482i −0.972682 − 0.232142i \(-0.925427\pi\)
−0.232142 − 0.972682i \(-0.574573\pi\)
\(48\) 0 0
\(49\) − 1.85787e8i − 0.657710i
\(50\) 0 0
\(51\) 1.74298e8 0.505174
\(52\) 0 0
\(53\) 3.08460e8 − 3.08460e8i 0.737598 − 0.737598i −0.234515 − 0.972113i \(-0.575350\pi\)
0.972113 + 0.234515i \(0.0753501\pi\)
\(54\) 0 0
\(55\) −2.20083e8 − 8.27662e7i −0.437293 − 0.164452i
\(56\) 0 0
\(57\) −2.30205e8 − 2.30205e8i −0.382596 − 0.382596i
\(58\) 0 0
\(59\) 9.40888e8i 1.31607i 0.752989 + 0.658034i \(0.228612\pi\)
−0.752989 + 0.658034i \(0.771388\pi\)
\(60\) 0 0
\(61\) −1.35361e9 −1.60267 −0.801336 − 0.598215i \(-0.795877\pi\)
−0.801336 + 0.598215i \(0.795877\pi\)
\(62\) 0 0
\(63\) 3.65387e8 − 3.65387e8i 0.368172 − 0.368172i
\(64\) 0 0
\(65\) 4.42174e8 − 2.00489e8i 0.381089 − 0.172792i
\(66\) 0 0
\(67\) −8.53571e8 − 8.53571e8i −0.632216 − 0.632216i 0.316407 − 0.948623i \(-0.397523\pi\)
−0.948623 + 0.316407i \(0.897523\pi\)
\(68\) 0 0
\(69\) 8.12162e7i 0.0519275i
\(70\) 0 0
\(71\) −2.82701e9 −1.56688 −0.783441 − 0.621466i \(-0.786537\pi\)
−0.783441 + 0.621466i \(0.786537\pi\)
\(72\) 0 0
\(73\) −2.75330e9 + 2.75330e9i −1.32812 + 1.32812i −0.421119 + 0.907005i \(0.638363\pi\)
−0.907005 + 0.421119i \(0.861637\pi\)
\(74\) 0 0
\(75\) −7.85496e8 + 5.19056e7i −0.331007 + 0.0218730i
\(76\) 0 0
\(77\) 5.23158e8 + 5.23158e8i 0.193276 + 0.193276i
\(78\) 0 0
\(79\) − 3.32450e9i − 1.08042i −0.841532 − 0.540208i \(-0.818346\pi\)
0.841532 − 0.540208i \(-0.181654\pi\)
\(80\) 0 0
\(81\) −2.37791e9 −0.681977
\(82\) 0 0
\(83\) −1.34634e9 + 1.34634e9i −0.341794 + 0.341794i −0.857041 − 0.515248i \(-0.827700\pi\)
0.515248 + 0.857041i \(0.327700\pi\)
\(84\) 0 0
\(85\) −2.79029e9 − 6.15393e9i −0.628861 − 1.38694i
\(86\) 0 0
\(87\) −2.54288e7 − 2.54288e7i −0.00510188 − 0.00510188i
\(88\) 0 0
\(89\) − 2.66745e9i − 0.477690i −0.971058 − 0.238845i \(-0.923231\pi\)
0.971058 − 0.238845i \(-0.0767688\pi\)
\(90\) 0 0
\(91\) −1.52767e9 −0.244807
\(92\) 0 0
\(93\) −1.65760e9 + 1.65760e9i −0.238268 + 0.238268i
\(94\) 0 0
\(95\) −4.44255e9 + 1.18131e10i −0.574135 + 1.52668i
\(96\) 0 0
\(97\) −5.26563e8 − 5.26563e8i −0.0613185 − 0.0613185i 0.675783 − 0.737101i \(-0.263806\pi\)
−0.737101 + 0.675783i \(0.763806\pi\)
\(98\) 0 0
\(99\) − 3.95404e9i − 0.415782i
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.p.a.17.1 2
4.3 odd 2 10.11.c.b.7.1 yes 2
5.3 odd 4 inner 80.11.p.a.33.1 2
12.11 even 2 90.11.g.a.37.1 2
20.3 even 4 10.11.c.b.3.1 ✓ 2
20.7 even 4 50.11.c.b.43.1 2
20.19 odd 2 50.11.c.b.7.1 2
60.23 odd 4 90.11.g.a.73.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.11.c.b.3.1 ✓ 2 20.3 even 4
10.11.c.b.7.1 yes 2 4.3 odd 2
50.11.c.b.7.1 2 20.19 odd 2
50.11.c.b.43.1 2 20.7 even 4
80.11.p.a.17.1 2 1.1 even 1 trivial
80.11.p.a.33.1 2 5.3 odd 4 inner
90.11.g.a.37.1 2 12.11 even 2
90.11.g.a.73.1 2 60.23 odd 4