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Properties

Label 80.11.i.a.13.15
Level $80$
Weight $11$
Character 80.13
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(13,80)] 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("80.13"); 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, 3, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.i (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 13.15
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.15

$q$-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+(-30.3669 - 10.0922i) q^{2} +245.343i q^{3} +(820.295 + 612.938i) q^{4} +(-2236.79 + 2182.29i) q^{5} +(2476.05 - 7450.30i) q^{6} +(5366.20 + 5366.20i) q^{7} +(-18723.9 - 26891.6i) q^{8} -1144.21 q^{9} +(89948.5 - 43695.1i) q^{10} +(15881.4 - 15881.4i) q^{11} +(-150380. + 201254. i) q^{12} +220963. i q^{13} +(-108798. - 217111. i) q^{14} +(-535409. - 548782. i) q^{15} +(297190. + 1.00558e6i) q^{16} +(-21745.6 + 21745.6i) q^{17} +(34746.2 + 11547.6i) q^{18} +(-3.17239e6 + 3.17239e6i) q^{19} +(-3.17244e6 + 419103. i) q^{20} +(-1.31656e6 + 1.31656e6i) q^{21} +(-642549. + 321991. i) q^{22} +(-2.24101e6 + 2.24101e6i) q^{23} +(6.59767e6 - 4.59378e6i) q^{24} +(240866. - 9.76265e6i) q^{25} +(2.23000e6 - 6.70995e6i) q^{26} +1.42065e7i q^{27} +(1.11272e6 + 7.69101e6i) q^{28} +(-2.65851e7 + 2.65851e7i) q^{29} +(1.07203e7 + 2.20683e7i) q^{30} +4.44934e7 q^{31} +(1.12378e6 - 3.35356e7i) q^{32} +(3.89640e6 + 3.89640e6i) q^{33} +(879807. - 440885. i) q^{34} +(-2.37137e7 - 292489. i) q^{35} +(-938592. - 701332. i) q^{36} -4.17767e7i q^{37} +(1.28352e8 - 6.43192e7i) q^{38} -5.42117e7 q^{39} +(1.00567e8 + 1.92900e7i) q^{40} +2.16518e8i q^{41} +(5.32668e7 - 2.66928e7i) q^{42} +1.02882e8 q^{43} +(2.27618e7 - 3.29312e6i) q^{44} +(2.55937e6 - 2.49700e6i) q^{45} +(9.06691e7 - 4.54357e7i) q^{46} +(6.53374e7 - 6.53374e7i) q^{47} +(-2.46712e8 + 7.29136e7i) q^{48} -2.24883e8i q^{49} +(-1.05841e8 + 2.94030e8i) q^{50} +(-5.33513e6 - 5.33513e6i) q^{51} +(-1.35437e8 + 1.81255e8i) q^{52} -3.39788e8 q^{53} +(1.43375e8 - 4.31408e8i) q^{54} +(-865633. + 7.01814e7i) q^{55} +(4.38296e7 - 2.44782e8i) q^{56} +(-7.78325e8 - 7.78325e8i) q^{57} +(1.07561e9 - 5.39004e8i) q^{58} +(-6.10059e8 - 6.10059e8i) q^{59} +(-1.02824e8 - 7.78335e8i) q^{60} +(4.36247e7 + 4.36247e7i) q^{61} +(-1.35112e9 - 4.49037e8i) q^{62} +(-6.14007e6 - 6.14007e6i) q^{63} +(-3.72574e8 + 1.00703e9i) q^{64} +(-4.82205e8 - 4.94248e8i) q^{65} +(-7.89983e7 - 1.57645e8i) q^{66} -1.40324e9 q^{67} +(-3.11665e7 + 4.50910e6i) q^{68} +(-5.49816e8 - 5.49816e8i) q^{69} +(7.17158e8 + 2.48205e8i) q^{70} -1.87326e9i q^{71} +(2.14241e7 + 3.07697e7i) q^{72} +(-3.01032e8 + 3.01032e8i) q^{73} +(-4.21619e8 + 1.26863e9i) q^{74} +(2.39520e9 + 5.90948e7i) q^{75} +(-4.54678e9 + 6.57817e8i) q^{76} +1.70446e8 q^{77} +(1.64624e9 + 5.47116e8i) q^{78} +4.57796e9i q^{79} +(-2.85922e9 - 1.60072e9i) q^{80} -3.55304e9 q^{81} +(2.18514e9 - 6.57496e9i) q^{82} -4.34278e9i q^{83} +(-1.88694e9 + 2.72997e8i) q^{84} +(1.18526e6 - 9.60956e7i) q^{85} +(-3.12421e9 - 1.03831e9i) q^{86} +(-6.52247e9 - 6.52247e9i) q^{87} +(-7.24440e8 - 1.29715e8i) q^{88} +2.20491e9 q^{89} +(-1.02920e8 + 4.99965e7i) q^{90} +(-1.18573e9 + 1.18573e9i) q^{91} +(-3.21188e9 + 4.64688e8i) q^{92} +1.09161e10i q^{93} +(-2.64349e9 + 1.32469e9i) q^{94} +(1.72914e8 - 1.40191e10i) q^{95} +(8.22773e9 + 2.75711e8i) q^{96} +(7.96934e9 - 7.96934e9i) q^{97} +(-2.26957e9 + 6.82900e9i) q^{98} +(-1.81718e7 + 1.81718e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 236 q - 2 q^{2} + 1220 q^{4} - 2 q^{5} - 4 q^{6} - 69128 q^{8} - 4487724 q^{9} + 2046 q^{10} - 4 q^{11} + 738232 q^{12} - 4 q^{15} - 2041064 q^{16} - 4 q^{17} + 5924662 q^{18} - 5107040 q^{19} + 2913160 q^{20}+ \cdots - 28071956444 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −30.3669 − 10.0922i −0.948965 − 0.315382i
\(3\) 245.343i 1.00964i 0.863224 + 0.504821i \(0.168442\pi\)
−0.863224 + 0.504821i \(0.831558\pi\)
\(4\) 820.295 + 612.938i 0.801069 + 0.598572i
\(5\) −2236.79 + 2182.29i −0.715774 + 0.698332i
\(6\) 2476.05 − 7450.30i 0.318423 − 0.958115i
\(7\) 5366.20 + 5366.20i 0.319284 + 0.319284i 0.848492 − 0.529208i \(-0.177511\pi\)
−0.529208 + 0.848492i \(0.677511\pi\)
\(8\) −18723.9 − 26891.6i −0.571408 − 0.820666i
\(9\) −1144.21 −0.0193774
\(10\) 89948.5 − 43695.1i 0.899485 − 0.436951i
\(11\) 15881.4 − 15881.4i 0.0986113 − 0.0986113i −0.656080 − 0.754691i \(-0.727787\pi\)
0.754691 + 0.656080i \(0.227787\pi\)
\(12\) −150380. + 201254.i −0.604344 + 0.808793i
\(13\) 220963.i 0.595117i 0.954704 + 0.297559i \(0.0961724\pi\)
−0.954704 + 0.297559i \(0.903828\pi\)
\(14\) −108798. − 217111.i −0.202293 − 0.403685i
\(15\) −535409. − 548782.i −0.705066 − 0.722676i
\(16\) 297190. + 1.00558e6i 0.283423 + 0.958995i
\(17\) −21745.6 + 21745.6i −0.0153153 + 0.0153153i −0.714723 − 0.699408i \(-0.753447\pi\)
0.699408 + 0.714723i \(0.253447\pi\)
\(18\) 34746.2 + 11547.6i 0.0183884 + 0.00611126i
\(19\) −3.17239e6 + 3.17239e6i −1.28121 + 1.28121i −0.341224 + 0.939982i \(0.610841\pi\)
−0.939982 + 0.341224i \(0.889159\pi\)
\(20\) −3.17244e6 + 419103.i −0.991386 + 0.130970i
\(21\) −1.31656e6 + 1.31656e6i −0.322362 + 0.322362i
\(22\) −642549. + 321991.i −0.124679 + 0.0624785i
\(23\) −2.24101e6 + 2.24101e6i −0.348180 + 0.348180i −0.859431 − 0.511251i \(-0.829182\pi\)
0.511251 + 0.859431i \(0.329182\pi\)
\(24\) 6.59767e6 − 4.59378e6i 0.828579 − 0.576917i
\(25\) 240866. − 9.76265e6i 0.0246647 − 0.999696i
\(26\) 2.23000e6 − 6.70995e6i 0.187689 − 0.564745i
\(27\) 1.42065e7i 0.990078i
\(28\) 1.11272e6 + 7.69101e6i 0.0646539 + 0.446882i
\(29\) −2.65851e7 + 2.65851e7i −1.29613 + 1.29613i −0.365202 + 0.930928i \(0.619000\pi\)
−0.930928 + 0.365202i \(0.881000\pi\)
\(30\) 1.07203e7 + 2.20683e7i 0.441164 + 0.908158i
\(31\) 4.44934e7 1.55413 0.777064 − 0.629421i \(-0.216708\pi\)
0.777064 + 0.629421i \(0.216708\pi\)
\(32\) 1.12378e6 − 3.35356e7i 0.0334912 − 0.999439i
\(33\) 3.89640e6 + 3.89640e6i 0.0995621 + 0.0995621i
\(34\) 879807. − 440885.i 0.0193639 − 0.00970355i
\(35\) −2.37137e7 − 292489.i −0.451501 − 0.00556891i
\(36\) −938592. − 701332.i −0.0155226 − 0.0115987i
\(37\) − 4.17767e7i − 0.602456i −0.953552 − 0.301228i \(-0.902603\pi\)
0.953552 − 0.301228i \(-0.0973966\pi\)
\(38\) 1.28352e8 − 6.43192e7i 1.61989 − 0.811751i
\(39\) −5.42117e7 −0.600856
\(40\) 1.00567e8 + 1.92900e7i 0.982096 + 0.188379i
\(41\) 2.16518e8i 1.86885i 0.356164 + 0.934424i \(0.384085\pi\)
−0.356164 + 0.934424i \(0.615915\pi\)
\(42\) 5.32668e7 − 2.66928e7i 0.407577 − 0.204243i
\(43\) 1.02882e8 0.699838 0.349919 − 0.936780i \(-0.386209\pi\)
0.349919 + 0.936780i \(0.386209\pi\)
\(44\) 2.27618e7 − 3.29312e6i 0.138020 − 0.0199685i
\(45\) 2.55937e6 − 2.49700e6i 0.0138698 − 0.0135318i
\(46\) 9.06691e7 − 4.54357e7i 0.440220 − 0.220601i
\(47\) 6.53374e7 − 6.53374e7i 0.284887 − 0.284887i −0.550167 − 0.835054i \(-0.685436\pi\)
0.835054 + 0.550167i \(0.185436\pi\)
\(48\) −2.46712e8 + 7.29136e7i −0.968242 + 0.286156i
\(49\) − 2.24883e8i − 0.796116i
\(50\) −1.05841e8 + 2.94030e8i −0.338692 + 0.940897i
\(51\) −5.33513e6 − 5.33513e6i −0.0154630 − 0.0154630i
\(52\) −1.35437e8 + 1.81255e8i −0.356221 + 0.476730i
\(53\) −3.39788e8 −0.812511 −0.406256 − 0.913759i \(-0.633166\pi\)
−0.406256 + 0.913759i \(0.633166\pi\)
\(54\) 1.43375e8 − 4.31408e8i 0.312252 − 0.939549i
\(55\) −865633. + 7.01814e7i −0.00171997 + 0.139447i
\(56\) 4.38296e7 − 2.44782e8i 0.0795842 − 0.444466i
\(57\) −7.78325e8 − 7.78325e8i −1.29356 − 1.29356i
\(58\) 1.07561e9 − 5.39004e8i 1.63876 − 0.821206i
\(59\) −6.10059e8 − 6.10059e8i −0.853320 − 0.853320i 0.137221 − 0.990541i \(-0.456183\pi\)
−0.990541 + 0.137221i \(0.956183\pi\)
\(60\) −1.02824e8 − 7.78335e8i −0.132233 − 1.00095i
\(61\) 4.36247e7 + 4.36247e7i 0.0516515 + 0.0516515i 0.732461 − 0.680809i \(-0.238372\pi\)
−0.680809 + 0.732461i \(0.738372\pi\)
\(62\) −1.35112e9 − 4.49037e8i −1.47481 − 0.490144i
\(63\) −6.14007e6 − 6.14007e6i −0.00618687 − 0.00618687i
\(64\) −3.72574e8 + 1.00703e9i −0.346987 + 0.937870i
\(65\) −4.82205e8 − 4.94248e8i −0.415590 − 0.425969i
\(66\) −7.89983e7 − 1.57645e8i −0.0630809 − 0.125881i
\(67\) −1.40324e9 −1.03934 −0.519669 − 0.854368i \(-0.673945\pi\)
−0.519669 + 0.854368i \(0.673945\pi\)
\(68\) −3.11665e7 + 4.50910e6i −0.0214360 + 0.00310131i
\(69\) −5.49816e8 − 5.49816e8i −0.351537 − 0.351537i
\(70\) 7.17158e8 + 2.48205e8i 0.426702 + 0.147680i
\(71\) − 1.87326e9i − 1.03826i −0.854694 − 0.519132i \(-0.826255\pi\)
0.854694 − 0.519132i \(-0.173745\pi\)
\(72\) 2.14241e7 + 3.07697e7i 0.0110724 + 0.0159023i
\(73\) −3.01032e8 + 3.01032e8i −0.145211 + 0.145211i −0.775975 − 0.630764i \(-0.782742\pi\)
0.630764 + 0.775975i \(0.282742\pi\)
\(74\) −4.21619e8 + 1.26863e9i −0.190004 + 0.571710i
\(75\) 2.39520e9 + 5.90948e7i 1.00934 + 0.0249025i
\(76\) −4.54678e9 + 6.57817e8i −1.79323 + 0.259440i
\(77\) 1.70446e8 0.0629699
\(78\) 1.64624e9 + 5.47116e8i 0.570191 + 0.189499i
\(79\) 4.57796e9i 1.48777i 0.668306 + 0.743886i \(0.267020\pi\)
−0.668306 + 0.743886i \(0.732980\pi\)
\(80\) −2.85922e9 − 1.60072e9i −0.872564 − 0.488501i
\(81\) −3.55304e9 −1.01900
\(82\) 2.18514e9 − 6.57496e9i 0.589400 − 1.77347i
\(83\) − 4.34278e9i − 1.10250i −0.834341 − 0.551249i \(-0.814152\pi\)
0.834341 − 0.551249i \(-0.185848\pi\)
\(84\) −1.88694e9 + 2.72997e8i −0.451191 + 0.0652773i
\(85\) 1.18526e6 − 9.60956e7i 0.000267129 − 0.0216575i
\(86\) −3.12421e9 − 1.03831e9i −0.664121 − 0.220716i
\(87\) −6.52247e9 − 6.52247e9i −1.30863 − 1.30863i
\(88\) −7.24440e8 − 1.29715e8i −0.137274 − 0.0245797i
\(89\) 2.20491e9 0.394859 0.197429 − 0.980317i \(-0.436741\pi\)
0.197429 + 0.980317i \(0.436741\pi\)
\(90\) −1.02920e8 + 4.99965e7i −0.0174296 + 0.00846695i
\(91\) −1.18573e9 + 1.18573e9i −0.190011 + 0.190011i
\(92\) −3.21188e9 + 4.64688e8i −0.487327 + 0.0705053i
\(93\) 1.09161e10i 1.56911i
\(94\) −2.64349e9 + 1.32469e9i −0.360196 + 0.180500i
\(95\) 1.72914e8 − 1.40191e10i 0.0223467 − 1.81176i
\(96\) 8.22773e9 + 2.75711e8i 1.00908 + 0.0338141i
\(97\) 7.96934e9 − 7.96934e9i 0.928034 − 0.928034i −0.0695449 − 0.997579i \(-0.522155\pi\)
0.997579 + 0.0695449i \(0.0221547\pi\)
\(98\) −2.26957e9 + 6.82900e9i −0.251080 + 0.755486i
\(99\) −1.81718e7 + 1.81718e7i −0.00191083 + 0.00191083i
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.i.a.13.15 ✓ 236
5.2 odd 4 80.11.t.a.77.73 yes 236
16.5 even 4 80.11.t.a.53.73 yes 236
80.37 odd 4 inner 80.11.i.a.37.15 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.15 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.15 yes 236 80.37 odd 4 inner
80.11.t.a.53.73 yes 236 16.5 even 4
80.11.t.a.77.73 yes 236 5.2 odd 4