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Properties

Label 80.11.t.a.53.2
Level $80$
Weight $11$
Character 80.53
Analytic conductor $50.829$
Analytic rank $0$
Dimension $236$
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(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 53.2
Character \(\chi\) \(=\) 80.53
Dual form 80.11.t.a.77.2

$q$-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+(-31.9342 - 2.05133i) q^{2} -192.036 q^{3} +(1015.58 + 131.015i) q^{4} +(3119.02 - 193.255i) q^{5} +(6132.51 + 393.928i) q^{6} +(18555.5 + 18555.5i) q^{7} +(-32163.1 - 6267.14i) q^{8} -22171.2 q^{9} +(-99999.7 - 226.682i) q^{10} +(14188.4 + 14188.4i) q^{11} +(-195029. - 25159.6i) q^{12} -558513. q^{13} +(-554492. - 630619. i) q^{14} +(-598964. + 37111.9i) q^{15} +(1.01425e6 + 266113. i) q^{16} +(145210. - 145210. i) q^{17} +(708018. + 45480.3i) q^{18} +(115045. + 115045. i) q^{19} +(3.19295e6 + 212371. i) q^{20} +(-3.56333e6 - 3.56333e6i) q^{21} +(-423990. - 482201. i) q^{22} +(-7.04161e6 + 7.04161e6i) q^{23} +(6.17647e6 + 1.20352e6i) q^{24} +(9.69093e6 - 1.20553e6i) q^{25} +(1.78357e7 + 1.14569e6i) q^{26} +1.55972e7 q^{27} +(1.64136e7 + 2.12757e7i) q^{28} +(4.93278e6 + 4.93278e6i) q^{29} +(1.92036e7 + 43531.1i) q^{30} -3.10514e7 q^{31} +(-3.18432e7 - 1.05787e7i) q^{32} +(-2.72469e6 - 2.72469e6i) q^{33} +(-4.93505e6 + 4.33930e6i) q^{34} +(6.14610e7 + 5.42891e7i) q^{35} +(-2.25167e7 - 2.90475e6i) q^{36} +1.55549e7 q^{37} +(-3.43786e6 - 3.90985e6i) q^{38} +1.07255e8 q^{39} +(-1.01528e8 - 1.33317e7i) q^{40} -1.88678e8i q^{41} +(1.06482e8 + 1.21102e8i) q^{42} -1.25422e6i q^{43} +(1.25506e7 + 1.62684e7i) q^{44} +(-6.91523e7 + 4.28469e6i) q^{45} +(2.39313e8 - 2.10424e8i) q^{46} +(8.45693e7 - 8.45693e7i) q^{47} +(-1.94772e8 - 5.11033e7i) q^{48} +4.06140e8i q^{49} +(-3.11945e8 + 1.86184e7i) q^{50} +(-2.78856e7 + 2.78856e7i) q^{51} +(-5.67217e8 - 7.31735e7i) q^{52} -351742. i q^{53} +(-4.98084e8 - 3.19949e7i) q^{54} +(4.69959e7 + 4.15120e7i) q^{55} +(-4.80513e8 - 7.13093e8i) q^{56} +(-2.20927e7 - 2.20927e7i) q^{57} +(-1.47406e8 - 1.67643e8i) q^{58} +(1.96733e8 - 1.96733e8i) q^{59} +(-6.13160e8 - 4.07829e7i) q^{60} +(-6.86172e8 + 6.86172e8i) q^{61} +(9.91600e8 + 6.36965e7i) q^{62} +(-4.11398e8 - 4.11398e8i) q^{63} +(9.95188e8 + 4.03141e8i) q^{64} +(-1.74201e9 + 1.07935e8i) q^{65} +(8.14214e7 + 9.25999e7i) q^{66} -7.27908e7i q^{67} +(1.66498e8 - 1.28449e8i) q^{68} +(1.35224e9 - 1.35224e9i) q^{69} +(-1.85134e9 - 1.85975e9i) q^{70} -1.31887e9i q^{71} +(7.13094e8 + 1.38950e8i) q^{72} +(-1.78479e9 + 1.78479e9i) q^{73} +(-4.96734e8 - 3.19082e7i) q^{74} +(-1.86101e9 + 2.31505e8i) q^{75} +(1.01765e8 + 1.31910e8i) q^{76} +5.26547e8i q^{77} +(-3.42509e9 - 2.20014e8i) q^{78} -4.94128e9i q^{79} +(3.21488e9 + 6.34004e8i) q^{80} -1.68604e9 q^{81} +(-3.87040e8 + 6.02528e9i) q^{82} +6.37951e8 q^{83} +(-3.15201e9 - 4.08571e9i) q^{84} +(4.24851e8 - 4.80976e8i) q^{85} +(-2.57281e6 + 4.00525e7i) q^{86} +(-9.47272e8 - 9.47272e8i) q^{87} +(-3.67423e8 - 5.45264e8i) q^{88} -7.78731e9 q^{89} +(2.21711e9 + 5.02580e6i) q^{90} +(-1.03635e10 - 1.03635e10i) q^{91} +(-8.07391e9 + 6.22880e9i) q^{92} +5.96298e9 q^{93} +(-2.87413e9 + 2.52717e9i) q^{94} +(3.81059e8 + 3.36593e8i) q^{95} +(6.11505e9 + 2.03148e9i) q^{96} +(-1.05111e10 + 1.05111e10i) q^{97} +(8.33124e8 - 1.29697e10i) q^{98} +(-3.14574e8 - 3.14574e8i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.9342 − 2.05133i −0.997943 − 0.0641039i
\(3\) −192.036 −0.790272 −0.395136 − 0.918623i \(-0.629302\pi\)
−0.395136 + 0.918623i \(0.629302\pi\)
\(4\) 1015.58 + 131.015i 0.991781 + 0.127944i
\(5\) 3119.02 − 193.255i 0.998086 − 0.0618416i
\(6\) 6132.51 + 393.928i 0.788646 + 0.0506595i
\(7\) 18555.5 + 18555.5i 1.10404 + 1.10404i 0.993919 + 0.110117i \(0.0351224\pi\)
0.110117 + 0.993919i \(0.464878\pi\)
\(8\) −32163.1 − 6267.14i −0.981540 − 0.191258i
\(9\) −22171.2 −0.375471
\(10\) −99999.7 − 226.682i −0.999997 − 0.00226682i
\(11\) 14188.4 + 14188.4i 0.0880989 + 0.0880989i 0.749783 − 0.661684i \(-0.230158\pi\)
−0.661684 + 0.749783i \(0.730158\pi\)
\(12\) −195029. − 25159.6i −0.783777 − 0.101111i
\(13\) −558513. −1.50424 −0.752119 − 0.659027i \(-0.770968\pi\)
−0.752119 + 0.659027i \(0.770968\pi\)
\(14\) −554492. − 630619.i −1.03099 − 1.17254i
\(15\) −598964. + 37111.9i −0.788759 + 0.0488717i
\(16\) 1.01425e6 + 266113.i 0.967261 + 0.253785i
\(17\) 145210. − 145210.i 0.102271 − 0.102271i −0.654120 − 0.756391i \(-0.726961\pi\)
0.756391 + 0.654120i \(0.226961\pi\)
\(18\) 708018. + 45480.3i 0.374699 + 0.0240691i
\(19\) 115045. + 115045.i 0.0464620 + 0.0464620i 0.729956 − 0.683494i \(-0.239540\pi\)
−0.683494 + 0.729956i \(0.739540\pi\)
\(20\) 3.19295e6 + 212371.i 0.997795 + 0.0663659i
\(21\) −3.56333e6 − 3.56333e6i −0.872488 − 0.872488i
\(22\) −423990. − 482201.i −0.0822702 − 0.0935652i
\(23\) −7.04161e6 + 7.04161e6i −1.09404 + 1.09404i −0.0989468 + 0.995093i \(0.531547\pi\)
−0.995093 + 0.0989468i \(0.968453\pi\)
\(24\) 6.17647e6 + 1.20352e6i 0.775683 + 0.151146i
\(25\) 9.69093e6 − 1.20553e6i 0.992351 − 0.123446i
\(26\) 1.78357e7 + 1.14569e6i 1.50114 + 0.0964276i
\(27\) 1.55972e7 1.08700
\(28\) 1.64136e7 + 2.12757e7i 0.953707 + 1.23622i
\(29\) 4.93278e6 + 4.93278e6i 0.240493 + 0.240493i 0.817054 − 0.576561i \(-0.195606\pi\)
−0.576561 + 0.817054i \(0.695606\pi\)
\(30\) 1.92036e7 + 43531.1i 0.790270 + 0.00179140i
\(31\) −3.10514e7 −1.08461 −0.542304 − 0.840183i \(-0.682448\pi\)
−0.542304 + 0.840183i \(0.682448\pi\)
\(32\) −3.18432e7 − 1.05787e7i −0.949003 − 0.315268i
\(33\) −2.72469e6 − 2.72469e6i −0.0696221 − 0.0696221i
\(34\) −4.93505e6 + 4.33930e6i −0.108617 + 0.0955048i
\(35\) 6.14610e7 + 5.42891e7i 1.17020 + 1.03365i
\(36\) −2.25167e7 − 2.90475e6i −0.372385 − 0.0480393i
\(37\) 1.55549e7 0.224316 0.112158 − 0.993690i \(-0.464224\pi\)
0.112158 + 0.993690i \(0.464224\pi\)
\(38\) −3.43786e6 − 3.90985e6i −0.0433881 − 0.0493449i
\(39\) 1.07255e8 1.18876
\(40\) −1.01528e8 − 1.33317e7i −0.991489 − 0.130192i
\(41\) − 1.88678e8i − 1.62855i −0.580477 − 0.814277i \(-0.697134\pi\)
0.580477 − 0.814277i \(-0.302866\pi\)
\(42\) 1.06482e8 + 1.21102e8i 0.814763 + 0.926623i
\(43\) − 1.25422e6i − 0.00853161i −0.999991 − 0.00426581i \(-0.998642\pi\)
0.999991 − 0.00426581i \(-0.00135785\pi\)
\(44\) 1.25506e7 + 1.62684e7i 0.0761031 + 0.0986466i
\(45\) −6.91523e7 + 4.28469e6i −0.374752 + 0.0232197i
\(46\) 2.39313e8 − 2.10424e8i 1.16192 − 1.02166i
\(47\) 8.45693e7 − 8.45693e7i 0.368743 − 0.368743i −0.498276 − 0.867019i \(-0.666033\pi\)
0.867019 + 0.498276i \(0.166033\pi\)
\(48\) −1.94772e8 − 5.11033e7i −0.764399 − 0.200559i
\(49\) 4.06140e8i 1.43779i
\(50\) −3.11945e8 + 1.86184e7i −0.998224 + 0.0595789i
\(51\) −2.78856e7 + 2.78856e7i −0.0808220 + 0.0808220i
\(52\) −5.67217e8 − 7.31735e7i −1.49188 − 0.192459i
\(53\) − 351742.i − 0.000841096i −1.00000 0.000420548i \(-0.999866\pi\)
1.00000 0.000420548i \(-0.000133865\pi\)
\(54\) −4.98084e8 − 3.19949e7i −1.08476 − 0.0696807i
\(55\) 4.69959e7 + 4.15120e7i 0.0933785 + 0.0824821i
\(56\) −4.80513e8 − 7.13093e8i −0.872499 − 1.29481i
\(57\) −2.20927e7 − 2.20927e7i −0.0367176 − 0.0367176i
\(58\) −1.47406e8 − 1.67643e8i −0.224582 − 0.255415i
\(59\) 1.96733e8 − 1.96733e8i 0.275180 − 0.275180i −0.556001 − 0.831182i \(-0.687665\pi\)
0.831182 + 0.556001i \(0.187665\pi\)
\(60\) −6.13160e8 − 4.07829e7i −0.788529 − 0.0524471i
\(61\) −6.86172e8 + 6.86172e8i −0.812426 + 0.812426i −0.984997 − 0.172571i \(-0.944793\pi\)
0.172571 + 0.984997i \(0.444793\pi\)
\(62\) 9.91600e8 + 6.36965e7i 1.08238 + 0.0695276i
\(63\) −4.11398e8 − 4.11398e8i −0.414533 − 0.414533i
\(64\) 9.95188e8 + 4.03141e8i 0.926841 + 0.375455i
\(65\) −1.74201e9 + 1.07935e8i −1.50136 + 0.0930245i
\(66\) 8.14214e7 + 9.25999e7i 0.0650158 + 0.0739419i
\(67\) − 7.27908e7i − 0.0539141i −0.999637 − 0.0269570i \(-0.991418\pi\)
0.999637 − 0.0269570i \(-0.00858173\pi\)
\(68\) 1.66498e8 − 1.28449e8i 0.114516 − 0.0883456i
\(69\) 1.35224e9 − 1.35224e9i 0.864588 − 0.864588i
\(70\) −1.85134e9 − 1.85975e9i −1.10153 − 1.10654i
\(71\) − 1.31887e9i − 0.730988i −0.930814 − 0.365494i \(-0.880900\pi\)
0.930814 − 0.365494i \(-0.119100\pi\)
\(72\) 7.13094e8 + 1.38950e8i 0.368539 + 0.0718118i
\(73\) −1.78479e9 + 1.78479e9i −0.860941 + 0.860941i −0.991447 − 0.130507i \(-0.958340\pi\)
0.130507 + 0.991447i \(0.458340\pi\)
\(74\) −4.96734e8 − 3.19082e7i −0.223854 − 0.0143795i
\(75\) −1.86101e9 + 2.31505e8i −0.784227 + 0.0975562i
\(76\) 1.01765e8 + 1.31910e8i 0.0401356 + 0.0520247i
\(77\) 5.26547e8i 0.194529i
\(78\) −3.42509e9 − 2.20014e8i −1.18631 − 0.0762040i
\(79\) − 4.94128e9i − 1.60585i −0.596083 − 0.802923i \(-0.703277\pi\)
0.596083 − 0.802923i \(-0.296723\pi\)
\(80\) 3.21488e9 + 6.34004e8i 0.981104 + 0.193483i
\(81\) −1.68604e9 −0.483551
\(82\) −3.87040e8 + 6.02528e9i −0.104397 + 1.62520i
\(83\) 6.37951e8 0.161956 0.0809780 − 0.996716i \(-0.474196\pi\)
0.0809780 + 0.996716i \(0.474196\pi\)
\(84\) −3.15201e9 − 4.08571e9i −0.753687 − 0.976947i
\(85\) 4.24851e8 − 4.80976e8i 0.0957508 − 0.108400i
\(86\) −2.57281e6 + 4.00525e7i −0.000546910 + 0.00851406i
\(87\) −9.47272e8 − 9.47272e8i −0.190055 − 0.190055i
\(88\) −3.67423e8 − 5.45264e8i −0.0696230 − 0.103322i
\(89\) −7.78731e9 −1.39456 −0.697281 − 0.716798i \(-0.745607\pi\)
−0.697281 + 0.716798i \(0.745607\pi\)
\(90\) 2.21711e9 + 5.02580e6i 0.375470 + 0.000851124i
\(91\) −1.03635e10 − 1.03635e10i −1.66073 − 1.66073i
\(92\) −8.07391e9 + 6.22880e9i −1.22502 + 0.945072i
\(93\) 5.96298e9 0.857134
\(94\) −2.87413e9 + 2.52717e9i −0.391622 + 0.344347i
\(95\) 3.81059e8 + 3.36593e8i 0.0492464 + 0.0434998i
\(96\) 6.11505e9 + 2.03148e9i 0.749970 + 0.249148i
\(97\) −1.05111e10 + 1.05111e10i −1.22403 + 1.22403i −0.257841 + 0.966187i \(0.583011\pi\)
−0.966187 + 0.257841i \(0.916989\pi\)
\(98\) 8.33124e8 − 1.29697e10i 0.0921678 − 1.43483i
\(99\) −3.14574e8 − 3.14574e8i −0.0330786 − 0.0330786i
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.53.2 yes 236
5.2 odd 4 80.11.i.a.37.62 yes 236
16.13 even 4 80.11.i.a.13.62 ✓ 236
80.77 odd 4 inner 80.11.t.a.77.2 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.62 ✓ 236 16.13 even 4
80.11.i.a.37.62 yes 236 5.2 odd 4
80.11.t.a.53.2 yes 236 1.1 even 1 trivial
80.11.t.a.77.2 yes 236 80.77 odd 4 inner