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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 13.11
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.11

$q$-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+(-31.0905 + 7.57519i) q^{2} -417.831i q^{3} +(909.233 - 471.032i) q^{4} +(-2306.53 + 2108.44i) q^{5} +(3165.15 + 12990.6i) q^{6} +(-15187.1 - 15187.1i) q^{7} +(-24700.3 + 21532.2i) q^{8} -115534. q^{9} +(55739.2 - 83024.9i) q^{10} +(-77997.1 + 77997.1i) q^{11} +(-196812. - 379906. i) q^{12} -585273. i q^{13} +(587219. + 357129. i) q^{14} +(880974. + 963741. i) q^{15} +(604833. - 856556. i) q^{16} +(-1.38742e6 + 1.38742e6i) q^{17} +(3.59200e6 - 875192. i) q^{18} +(-1.95661e6 + 1.95661e6i) q^{19} +(-1.10403e6 + 3.00352e6i) q^{20} +(-6.34564e6 + 6.34564e6i) q^{21} +(1.83412e6 - 3.01581e6i) q^{22} +(2.56436e6 - 2.56436e6i) q^{23} +(8.99684e6 + 1.03206e7i) q^{24} +(874551. - 9.72639e6i) q^{25} +(4.43356e6 + 1.81964e7i) q^{26} +2.36011e7i q^{27} +(-2.09622e7 - 6.65499e6i) q^{28} +(-1.86703e7 + 1.86703e7i) q^{29} +(-3.46904e7 - 2.32896e7i) q^{30} +4.89331e6 q^{31} +(-1.23159e7 + 3.12125e7i) q^{32} +(3.25896e7 + 3.25896e7i) q^{33} +(3.26255e7 - 5.36454e7i) q^{34} +(6.70507e7 + 3.00837e6i) q^{35} +(-1.05047e8 + 5.44202e7i) q^{36} -7.57859e7i q^{37} +(4.60103e7 - 7.56538e7i) q^{38} -2.44545e8 q^{39} +(1.15725e7 - 1.01744e8i) q^{40} -9.68659e7i q^{41} +(1.49219e8 - 2.45358e8i) q^{42} -1.12030e8 q^{43} +(-3.41784e7 + 1.07657e8i) q^{44} +(2.66483e8 - 2.43597e8i) q^{45} +(-6.03017e7 + 9.91528e7i) q^{46} +(-2.81237e8 + 2.81237e8i) q^{47} +(-3.57896e8 - 2.52718e8i) q^{48} +1.78821e8i q^{49} +(4.64891e7 + 3.09023e8i) q^{50} +(5.79706e8 + 5.79706e8i) q^{51} +(-2.75683e8 - 5.32149e8i) q^{52} +2.69972e8 q^{53} +(-1.78783e8 - 7.33770e8i) q^{54} +(1.54502e7 - 3.44355e8i) q^{55} +(7.02138e8 + 4.81138e7i) q^{56} +(8.17535e8 + 8.17535e8i) q^{57} +(4.39036e8 - 7.21897e8i) q^{58} +(-1.52518e8 - 1.52518e8i) q^{59} +(1.25496e9 + 4.61298e8i) q^{60} +(1.91449e8 + 1.91449e8i) q^{61} +(-1.52135e8 + 3.70678e7i) q^{62} +(1.75462e9 + 1.75462e9i) q^{63} +(1.46468e8 - 1.06371e9i) q^{64} +(1.23402e9 + 1.34995e9i) q^{65} +(-1.26010e9 - 7.66354e8i) q^{66} +5.34124e8 q^{67} +(-6.07967e8 + 1.91500e9i) q^{68} +(-1.07147e9 - 1.07147e9i) q^{69} +(-2.10743e9 + 4.14390e8i) q^{70} +1.38988e8i q^{71} +(2.85372e9 - 2.48770e9i) q^{72} +(2.43030e9 - 2.43030e9i) q^{73} +(5.74093e8 + 2.35622e9i) q^{74} +(-4.06399e9 - 3.65415e8i) q^{75} +(-8.57389e8 + 2.70065e9i) q^{76} +2.36910e9 q^{77} +(7.60302e9 - 1.85248e9i) q^{78} -3.40748e8i q^{79} +(4.10936e8 + 3.25093e9i) q^{80} +3.03913e9 q^{81} +(7.33778e8 + 3.01161e9i) q^{82} -5.49935e9i q^{83} +(-2.78066e9 + 8.75867e9i) q^{84} +(2.74830e8 - 6.12542e9i) q^{85} +(3.48307e9 - 8.48650e8i) q^{86} +(7.80101e9 + 7.80101e9i) q^{87} +(2.47100e8 - 3.60600e9i) q^{88} -4.83960e9 q^{89} +(-6.43977e9 + 9.59219e9i) q^{90} +(-8.88860e9 + 8.88860e9i) q^{91} +(1.12371e9 - 3.53950e9i) q^{92} -2.04458e9i q^{93} +(6.61335e9 - 1.08742e10i) q^{94} +(3.87581e8 - 8.63841e9i) q^{95} +(1.30415e10 + 5.14599e9i) q^{96} +(6.88867e9 - 6.88867e9i) q^{97} +(-1.35460e9 - 5.55962e9i) q^{98} +(9.01131e9 - 9.01131e9i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.0905 + 7.57519i −0.971577 + 0.236725i
\(3\) − 417.831i − 1.71947i −0.510741 − 0.859735i \(-0.670629\pi\)
0.510741 − 0.859735i \(-0.329371\pi\)
\(4\) 909.233 − 471.032i 0.887923 − 0.459993i
\(5\) −2306.53 + 2108.44i −0.738090 + 0.674702i
\(6\) 3165.15 + 12990.6i 0.407041 + 1.67060i
\(7\) −15187.1 − 15187.1i −0.903618 − 0.903618i 0.0921294 − 0.995747i \(-0.470633\pi\)
−0.995747 + 0.0921294i \(0.970633\pi\)
\(8\) −24700.3 + 21532.2i −0.753793 + 0.657112i
\(9\) −115534. −1.95658
\(10\) 55739.2 − 83024.9i 0.557392 − 0.830249i
\(11\) −77997.1 + 77997.1i −0.484301 + 0.484301i −0.906502 − 0.422201i \(-0.861258\pi\)
0.422201 + 0.906502i \(0.361258\pi\)
\(12\) −196812. − 379906.i −0.790944 − 1.52676i
\(13\) − 585273.i − 1.57631i −0.615477 − 0.788155i \(-0.711037\pi\)
0.615477 − 0.788155i \(-0.288963\pi\)
\(14\) 587219. + 357129.i 1.09184 + 0.664025i
\(15\) 880974. + 963741.i 1.16013 + 1.26912i
\(16\) 604833. − 856556.i 0.576813 − 0.816876i
\(17\) −1.38742e6 + 1.38742e6i −0.977153 + 0.977153i −0.999745 − 0.0225915i \(-0.992808\pi\)
0.0225915 + 0.999745i \(0.492808\pi\)
\(18\) 3.59200e6 − 875192.i 1.90096 − 0.463170i
\(19\) −1.95661e6 + 1.95661e6i −0.790201 + 0.790201i −0.981527 − 0.191326i \(-0.938721\pi\)
0.191326 + 0.981527i \(0.438721\pi\)
\(20\) −1.10403e6 + 3.00352e6i −0.345009 + 0.938599i
\(21\) −6.34564e6 + 6.34564e6i −1.55374 + 1.55374i
\(22\) 1.83412e6 − 3.01581e6i 0.355889 − 0.585181i
\(23\) 2.56436e6 − 2.56436e6i 0.398420 − 0.398420i −0.479256 − 0.877675i \(-0.659093\pi\)
0.877675 + 0.479256i \(0.159093\pi\)
\(24\) 8.99684e6 + 1.03206e7i 1.12988 + 1.29612i
\(25\) 874551. − 9.72639e6i 0.0895540 − 0.995982i
\(26\) 4.43356e6 + 1.81964e7i 0.373152 + 1.53151i
\(27\) 2.36011e7i 1.64480i
\(28\) −2.09622e7 − 6.65499e6i −1.21800 − 0.386685i
\(29\) −1.86703e7 + 1.86703e7i −0.910249 + 0.910249i −0.996291 − 0.0860424i \(-0.972578\pi\)
0.0860424 + 0.996291i \(0.472578\pi\)
\(30\) −3.46904e7 − 2.32896e7i −1.42759 − 0.958419i
\(31\) 4.89331e6 0.170920 0.0854602 − 0.996342i \(-0.472764\pi\)
0.0854602 + 0.996342i \(0.472764\pi\)
\(32\) −1.23159e7 + 3.12125e7i −0.367044 + 0.930204i
\(33\) 3.25896e7 + 3.25896e7i 0.832741 + 0.832741i
\(34\) 3.26255e7 − 5.36454e7i 0.718063 − 1.18070i
\(35\) 6.70507e7 + 3.00837e6i 1.27662 + 0.0572785i
\(36\) −1.05047e8 + 5.44202e7i −1.73729 + 0.900011i
\(37\) − 7.57859e7i − 1.09290i −0.837492 − 0.546449i \(-0.815979\pi\)
0.837492 − 0.546449i \(-0.184021\pi\)
\(38\) 4.60103e7 − 7.56538e7i 0.580680 − 0.954801i
\(39\) −2.44545e8 −2.71042
\(40\) 1.15725e7 − 1.01744e8i 0.113013 − 0.993594i
\(41\) − 9.68659e7i − 0.836088i −0.908427 − 0.418044i \(-0.862716\pi\)
0.908427 − 0.418044i \(-0.137284\pi\)
\(42\) 1.49219e8 − 2.45358e8i 1.14177 − 1.87739i
\(43\) −1.12030e8 −0.762066 −0.381033 − 0.924561i \(-0.624432\pi\)
−0.381033 + 0.924561i \(0.624432\pi\)
\(44\) −3.41784e7 + 1.07657e8i −0.207247 + 0.652797i
\(45\) 2.66483e8 − 2.43597e8i 1.44413 − 1.32011i
\(46\) −6.03017e7 + 9.91528e7i −0.292779 + 0.481411i
\(47\) −2.81237e8 + 2.81237e8i −1.22626 + 1.22626i −0.260893 + 0.965368i \(0.584017\pi\)
−0.965368 + 0.260893i \(0.915983\pi\)
\(48\) −3.57896e8 − 2.52718e8i −1.40459 − 0.991813i
\(49\) 1.78821e8i 0.633050i
\(50\) 4.64891e7 + 3.09023e8i 0.148765 + 0.988873i
\(51\) 5.79706e8 + 5.79706e8i 1.68019 + 1.68019i
\(52\) −2.75683e8 − 5.32149e8i −0.725091 − 1.39964i
\(53\) 2.69972e8 0.645565 0.322783 − 0.946473i \(-0.395382\pi\)
0.322783 + 0.946473i \(0.395382\pi\)
\(54\) −1.78783e8 − 7.33770e8i −0.389366 − 1.59805i
\(55\) 1.54502e7 − 3.44355e8i 0.0306988 − 0.684216i
\(56\) 7.02138e8 + 4.81138e7i 1.27492 + 0.0873634i
\(57\) 8.17535e8 + 8.17535e8i 1.35873 + 1.35873i
\(58\) 4.39036e8 − 7.21897e8i 0.668898 − 1.09986i
\(59\) −1.52518e8 − 1.52518e8i −0.213334 − 0.213334i 0.592348 − 0.805682i \(-0.298201\pi\)
−0.805682 + 0.592348i \(0.798201\pi\)
\(60\) 1.25496e9 + 4.61298e8i 1.61389 + 0.593232i
\(61\) 1.91449e8 + 1.91449e8i 0.226675 + 0.226675i 0.811302 − 0.584627i \(-0.198759\pi\)
−0.584627 + 0.811302i \(0.698759\pi\)
\(62\) −1.52135e8 + 3.70678e7i −0.166062 + 0.0404611i
\(63\) 1.75462e9 + 1.75462e9i 1.76800 + 1.76800i
\(64\) 1.46468e8 − 1.06371e9i 0.136409 − 0.990653i
\(65\) 1.23402e9 + 1.34995e9i 1.06354 + 1.16346i
\(66\) −1.26010e9 − 7.66354e8i −1.00620 − 0.611941i
\(67\) 5.34124e8 0.395611 0.197805 − 0.980241i \(-0.436619\pi\)
0.197805 + 0.980241i \(0.436619\pi\)
\(68\) −6.07967e8 + 1.91500e9i −0.418153 + 1.31712i
\(69\) −1.07147e9 − 1.07147e9i −0.685070 − 0.685070i
\(70\) −2.10743e9 + 4.14390e8i −1.25390 + 0.246558i
\(71\) 1.38988e8i 0.0770344i 0.999258 + 0.0385172i \(0.0122634\pi\)
−0.999258 + 0.0385172i \(0.987737\pi\)
\(72\) 2.85372e9 − 2.48770e9i 1.47485 − 1.28569i
\(73\) 2.43030e9 − 2.43030e9i 1.17232 − 1.17232i 0.190666 − 0.981655i \(-0.438935\pi\)
0.981655 − 0.190666i \(-0.0610646\pi\)
\(74\) 5.74093e8 + 2.35622e9i 0.258716 + 1.06183i
\(75\) −4.06399e9 − 3.65415e8i −1.71256 − 0.153985i
\(76\) −8.57389e8 + 2.70065e9i −0.338151 + 1.06512i
\(77\) 2.36910e9 0.875246
\(78\) 7.60302e9 − 1.85248e9i 2.63338 − 0.641623i
\(79\) − 3.40748e8i − 0.110738i −0.998466 − 0.0553691i \(-0.982366\pi\)
0.998466 − 0.0553691i \(-0.0176335\pi\)
\(80\) 4.10936e8 + 3.25093e9i 0.125408 + 0.992105i
\(81\) 3.03913e9 0.871615
\(82\) 7.33778e8 + 3.01161e9i 0.197923 + 0.812323i
\(83\) − 5.49935e9i − 1.39611i −0.716042 − 0.698057i \(-0.754048\pi\)
0.716042 − 0.698057i \(-0.245952\pi\)
\(84\) −2.78066e9 + 8.75867e9i −0.664893 + 2.09431i
\(85\) 2.74830e8 − 6.12542e9i 0.0619398 − 1.38051i
\(86\) 3.48307e9 − 8.48650e8i 0.740406 − 0.180400i
\(87\) 7.80101e9 + 7.80101e9i 1.56515 + 1.56515i
\(88\) 2.47100e8 − 3.60600e9i 0.0468231 − 0.683302i
\(89\) −4.83960e9 −0.866681 −0.433341 − 0.901230i \(-0.642665\pi\)
−0.433341 + 0.901230i \(0.642665\pi\)
\(90\) −6.43977e9 + 9.59219e9i −1.09058 + 1.62445i
\(91\) −8.88860e9 + 8.88860e9i −1.42438 + 1.42438i
\(92\) 1.12371e9 − 3.53950e9i 0.170496 − 0.537036i
\(93\) − 2.04458e9i − 0.293893i
\(94\) 6.61335e9 − 1.08742e10i 0.901120 − 1.48169i
\(95\) 3.87581e8 − 8.63841e9i 0.0500892 − 1.11639i
\(96\) 1.30415e10 + 5.14599e9i 1.59946 + 0.631121i
\(97\) 6.88867e9 − 6.88867e9i 0.802189 − 0.802189i −0.181248 − 0.983437i \(-0.558014\pi\)
0.983437 + 0.181248i \(0.0580137\pi\)
\(98\) −1.35460e9 − 5.55962e9i −0.149859 − 0.615056i
\(99\) 9.01131e9 − 9.01131e9i 0.947572 − 0.947572i
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.13.11 ✓ 236
5.2 odd 4 80.11.t.a.77.50 yes 236
16.5 even 4 80.11.t.a.53.50 yes 236
80.37 odd 4 inner 80.11.i.a.37.11 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.11 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.11 yes 236 80.37 odd 4 inner
80.11.t.a.53.50 yes 236 16.5 even 4
80.11.t.a.77.50 yes 236 5.2 odd 4