↑

Properties

Label 80.11.t.a.53.14
Level $80$
Weight $11$
Character 80.53
Analytic conductor $50.829$
Analytic rank $0$
Dimension $236$
Inner twists $2$

Related objects

Downloads

Learn more

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 53.14
Character \(\chi\) \(=\) 80.53
Dual form 80.11.t.a.77.14

$q$-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+(-30.1354 + 10.7636i) q^{2} -201.343 q^{3} +(792.289 - 648.732i) q^{4} +(-2152.30 + 2265.67i) q^{5} +(6067.56 - 2167.18i) q^{6} +(3246.23 + 3246.23i) q^{7} +(-16893.3 + 28077.7i) q^{8} -18510.0 q^{9} +(40473.6 - 91443.4i) q^{10} +(165342. + 165342. i) q^{11} +(-159522. + 130618. i) q^{12} -331081. q^{13} +(-132768. - 62885.4i) q^{14} +(433349. - 456176. i) q^{15} +(206869. - 1.02797e6i) q^{16} +(-67881.6 + 67881.6i) q^{17} +(557808. - 199235. i) q^{18} +(2.25380e6 + 2.25380e6i) q^{19} +(-235428. + 3.19133e6i) q^{20} +(-653605. - 653605. i) q^{21} +(-6.76231e6 - 3.20297e6i) q^{22} +(2.29572e6 - 2.29572e6i) q^{23} +(3.40134e6 - 5.65325e6i) q^{24} +(-500877. - 9.75277e6i) q^{25} +(9.97726e6 - 3.56362e6i) q^{26} +1.56160e7 q^{27} +(4.67789e6 + 466018. i) q^{28} +(1.49119e7 + 1.49119e7i) q^{29} +(-8.14907e6 + 1.84115e7i) q^{30} +3.95648e7 q^{31} +(4.83057e6 + 3.32049e7i) q^{32} +(-3.32903e7 - 3.32903e7i) q^{33} +(1.31499e6 - 2.77629e6i) q^{34} +(-1.43417e7 + 368034. i) q^{35} +(-1.46653e7 + 1.20081e7i) q^{36} +9.28856e7 q^{37} +(-9.21781e7 - 4.36601e7i) q^{38} +6.66607e7 q^{39} +(-2.72555e7 - 9.87061e7i) q^{40} +1.35674e8i q^{41} +(2.67318e7 + 1.26615e7i) q^{42} +3.88020e7i q^{43} +(2.38261e8 + 2.37359e7i) q^{44} +(3.98391e7 - 4.19376e7i) q^{45} +(-4.44724e7 + 9.38930e7i) q^{46} +(2.56760e7 - 2.56760e7i) q^{47} +(-4.16515e7 + 2.06974e8i) q^{48} -2.61399e8i q^{49} +(1.20069e8 + 2.88513e8i) q^{50} +(1.36675e7 - 1.36675e7i) q^{51} +(-2.62312e8 + 2.14783e8i) q^{52} -1.99504e8i q^{53} +(-4.70594e8 + 1.68084e8i) q^{54} +(-7.30473e8 + 1.87452e7i) q^{55} +(-1.45986e8 + 3.63073e7i) q^{56} +(-4.53786e8 - 4.53786e8i) q^{57} +(-6.09882e8 - 2.88870e8i) q^{58} +(-7.46132e8 + 7.46132e8i) q^{59} +(4.74018e7 - 6.42551e8i) q^{60} +(-4.03505e8 + 4.03505e8i) q^{61} +(-1.19230e9 + 4.25860e8i) q^{62} +(-6.00878e7 - 6.00878e7i) q^{63} +(-5.02976e8 - 9.48650e8i) q^{64} +(7.12583e8 - 7.50119e8i) q^{65} +(1.36154e9 + 6.44895e8i) q^{66} -1.39301e9i q^{67} +(-9.74487e6 + 9.78189e7i) q^{68} +(-4.62228e8 + 4.62228e8i) q^{69} +(4.28233e8 - 1.65460e8i) q^{70} +2.21783e9i q^{71} +(3.12695e8 - 5.19720e8i) q^{72} +(-1.73750e9 + 1.73750e9i) q^{73} +(-2.79915e9 + 9.99784e8i) q^{74} +(1.00848e8 + 1.96365e9i) q^{75} +(3.24777e9 + 3.23548e8i) q^{76} +1.07347e9i q^{77} +(-2.00885e9 + 7.17510e8i) q^{78} -1.14657e9i q^{79} +(1.88379e9 + 2.68118e9i) q^{80} -2.05116e9 q^{81} +(-1.46034e9 - 4.08860e9i) q^{82} -1.86695e9 q^{83} +(-9.41859e8 - 9.38295e7i) q^{84} +(-7.69593e6 - 2.99898e8i) q^{85} +(-4.17649e8 - 1.16931e9i) q^{86} +(-3.00240e9 - 3.00240e9i) q^{87} +(-7.43558e9 + 1.84925e9i) q^{88} +9.15816e9 q^{89} +(-7.49168e8 + 1.69262e9i) q^{90} +(-1.07476e9 - 1.07476e9i) q^{91} +(3.29567e8 - 3.30819e9i) q^{92} -7.96608e9 q^{93} +(-4.97391e8 + 1.05012e9i) q^{94} +(-9.95719e9 + 2.55519e8i) q^{95} +(-9.72601e8 - 6.68557e9i) q^{96} +(-9.49531e9 + 9.49531e9i) q^{97} +(2.81360e9 + 7.87738e9i) q^{98} +(-3.06048e9 - 3.06048e9i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −30.1354 + 10.7636i −0.941732 + 0.336363i
\(3\) −201.343 −0.828572 −0.414286 − 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(4\) 792.289 − 648.732i 0.773720 − 0.633528i
\(5\) −2152.30 + 2265.67i −0.688734 + 0.725014i
\(6\) 6067.56 − 2167.18i 0.780293 − 0.278701i
\(7\) 3246.23 + 3246.23i 0.193147 + 0.193147i 0.797055 − 0.603907i \(-0.206390\pi\)
−0.603907 + 0.797055i \(0.706390\pi\)
\(8\) −16893.3 + 28077.7i −0.515542 + 0.856864i
\(9\) −18510.0 −0.313469
\(10\) 40473.6 − 91443.4i 0.404736 − 0.914434i
\(11\) 165342. + 165342.i 1.02664 + 1.02664i 0.999635 + 0.0270057i \(0.00859724\pi\)
0.0270057 + 0.999635i \(0.491403\pi\)
\(12\) −159522. + 130618.i −0.641082 + 0.524923i
\(13\) −331081. −0.891696 −0.445848 − 0.895109i \(-0.647098\pi\)
−0.445848 + 0.895109i \(0.647098\pi\)
\(14\) −132768. − 62885.4i −0.246861 − 0.116926i
\(15\) 433349. − 456176.i 0.570666 − 0.600726i
\(16\) 206869. − 1.02797e6i 0.197285 − 0.980346i
\(17\) −67881.6 + 67881.6i −0.0478088 + 0.0478088i −0.730607 − 0.682798i \(-0.760763\pi\)
0.682798 + 0.730607i \(0.260763\pi\)
\(18\) 557808. − 199235.i 0.295204 − 0.105439i
\(19\) 2.25380e6 + 2.25380e6i 0.910221 + 0.910221i 0.996289 − 0.0860687i \(-0.0274305\pi\)
−0.0860687 + 0.996289i \(0.527430\pi\)
\(20\) −235428. + 3.19133e6i −0.0735713 + 0.997290i
\(21\) −653605. − 653605.i −0.160036 − 0.160036i
\(22\) −6.76231e6 − 3.20297e6i −1.31215 − 0.621497i
\(23\) 2.29572e6 − 2.29572e6i 0.356682 − 0.356682i −0.505907 − 0.862588i \(-0.668842\pi\)
0.862588 + 0.505907i \(0.168842\pi\)
\(24\) 3.40134e6 − 5.65325e6i 0.427163 − 0.709973i
\(25\) −500877. − 9.75277e6i −0.0512898 − 0.998684i
\(26\) 9.97726e6 − 3.56362e6i 0.839739 − 0.299934i
\(27\) 1.56160e7 1.08830
\(28\) 4.67789e6 + 466018.i 0.271806 + 0.0270778i
\(29\) 1.49119e7 + 1.49119e7i 0.727013 + 0.727013i 0.970024 − 0.243010i \(-0.0781349\pi\)
−0.243010 + 0.970024i \(0.578135\pi\)
\(30\) −8.14907e6 + 1.84115e7i −0.335353 + 0.757674i
\(31\) 3.95648e7 1.38197 0.690987 − 0.722867i \(-0.257176\pi\)
0.690987 + 0.722867i \(0.257176\pi\)
\(32\) 4.83057e6 + 3.32049e7i 0.143962 + 0.989583i
\(33\) −3.32903e7 − 3.32903e7i −0.850646 − 0.850646i
\(34\) 1.31499e6 − 2.77629e6i 0.0289420 − 0.0611041i
\(35\) −1.43417e7 + 368034.i −0.273062 + 0.00700725i
\(36\) −1.46653e7 + 1.20081e7i −0.242537 + 0.198591i
\(37\) 9.28856e7 1.33949 0.669745 − 0.742591i \(-0.266403\pi\)
0.669745 + 0.742591i \(0.266403\pi\)
\(38\) −9.21781e7 − 4.36601e7i −1.16335 − 0.551020i
\(39\) 6.66607e7 0.738834
\(40\) −2.72555e7 − 9.87061e7i −0.266167 − 0.963927i
\(41\) 1.35674e8i 1.17106i 0.810652 + 0.585528i \(0.199113\pi\)
−0.810652 + 0.585528i \(0.800887\pi\)
\(42\) 2.67318e7 + 1.26615e7i 0.204542 + 0.0968812i
\(43\) 3.88020e7i 0.263944i 0.991253 + 0.131972i \(0.0421309\pi\)
−0.991253 + 0.131972i \(0.957869\pi\)
\(44\) 2.38261e8 + 2.37359e7i 1.44474 + 0.143927i
\(45\) 3.98391e7 − 4.19376e7i 0.215897 − 0.227269i
\(46\) −4.44724e7 + 9.38930e7i −0.215924 + 0.455873i
\(47\) 2.56760e7 − 2.56760e7i 0.111954 − 0.111954i −0.648911 − 0.760864i \(-0.724775\pi\)
0.760864 + 0.648911i \(0.224775\pi\)
\(48\) −4.16515e7 + 2.06974e8i −0.163465 + 0.812287i
\(49\) − 2.61399e8i − 0.925388i
\(50\) 1.20069e8 + 2.88513e8i 0.384221 + 0.923241i
\(51\) 1.36675e7 − 1.36675e7i 0.0396130 − 0.0396130i
\(52\) −2.62312e8 + 2.14783e8i −0.689923 + 0.564914i
\(53\) − 1.99504e8i − 0.477058i −0.971135 − 0.238529i \(-0.923335\pi\)
0.971135 − 0.238529i \(-0.0766652\pi\)
\(54\) −4.70594e8 + 1.68084e8i −1.02489 + 0.366065i
\(55\) −7.30473e8 + 1.87452e7i −1.45141 + 0.0372458i
\(56\) −1.45986e8 + 3.63073e7i −0.265077 + 0.0659255i
\(57\) −4.53786e8 − 4.53786e8i −0.754183 − 0.754183i
\(58\) −6.09882e8 − 2.88870e8i −0.929192 − 0.440112i
\(59\) −7.46132e8 + 7.46132e8i −1.04365 + 1.04365i −0.0446491 + 0.999003i \(0.514217\pi\)
−0.999003 + 0.0446491i \(0.985783\pi\)
\(60\) 4.74018e7 − 6.42551e8i 0.0609591 − 0.826326i
\(61\) −4.03505e8 + 4.03505e8i −0.477748 + 0.477748i −0.904411 − 0.426662i \(-0.859689\pi\)
0.426662 + 0.904411i \(0.359689\pi\)
\(62\) −1.19230e9 + 4.25860e8i −1.30145 + 0.464845i
\(63\) −6.00878e7 − 6.00878e7i −0.0605458 − 0.0605458i
\(64\) −5.02976e8 − 9.48650e8i −0.468433 − 0.883499i
\(65\) 7.12583e8 − 7.50119e8i 0.614142 − 0.646492i
\(66\) 1.36154e9 + 6.44895e8i 1.08721 + 0.514955i
\(67\) − 1.39301e9i − 1.03177i −0.856659 − 0.515883i \(-0.827464\pi\)
0.856659 − 0.515883i \(-0.172536\pi\)
\(68\) −9.74487e6 + 9.78189e7i −0.00670242 + 0.0672788i
\(69\) −4.62228e8 + 4.62228e8i −0.295536 + 0.295536i
\(70\) 4.28233e8 − 1.65460e8i 0.254794 − 0.0984468i
\(71\) 2.21783e9i 1.22924i 0.788824 + 0.614619i \(0.210690\pi\)
−0.788824 + 0.614619i \(0.789310\pi\)
\(72\) 3.12695e8 − 5.19720e8i 0.161606 − 0.268601i
\(73\) −1.73750e9 + 1.73750e9i −0.838126 + 0.838126i −0.988612 − 0.150486i \(-0.951916\pi\)
0.150486 + 0.988612i \(0.451916\pi\)
\(74\) −2.79915e9 + 9.99784e8i −1.26144 + 0.450555i
\(75\) 1.00848e8 + 1.96365e9i 0.0424973 + 0.827481i
\(76\) 3.24777e9 + 3.23548e8i 1.28091 + 0.127606i
\(77\) 1.07347e9i 0.396586i
\(78\) −2.00885e9 + 7.17510e8i −0.695784 + 0.248516i
\(79\) − 1.14657e9i − 0.372620i −0.982491 − 0.186310i \(-0.940347\pi\)
0.982491 − 0.186310i \(-0.0596528\pi\)
\(80\) 1.88379e9 + 2.68118e9i 0.574887 + 0.818233i
\(81\) −2.05116e9 −0.588268
\(82\) −1.46034e9 − 4.08860e9i −0.393900 − 1.10282i
\(83\) −1.86695e9 −0.473961 −0.236980 − 0.971514i \(-0.576158\pi\)
−0.236980 + 0.971514i \(0.576158\pi\)
\(84\) −9.41859e8 − 9.38295e7i −0.225211 − 0.0224359i
\(85\) −7.69593e6 − 2.99898e8i −0.00173447 − 0.0675895i
\(86\) −4.17649e8 − 1.16931e9i −0.0887809 − 0.248564i
\(87\) −3.00240e9 − 3.00240e9i −0.602382 − 0.602382i
\(88\) −7.43558e9 + 1.84925e9i −1.40897 + 0.350416i
\(89\) 9.15816e9 1.64005 0.820027 − 0.572325i \(-0.193959\pi\)
0.820027 + 0.572325i \(0.193959\pi\)
\(90\) −7.49168e8 + 1.69262e9i −0.126872 + 0.286647i
\(91\) −1.07476e9 − 1.07476e9i −0.172229 − 0.172229i
\(92\) 3.29567e8 − 3.30819e9i 0.0500040 − 0.501939i
\(93\) −7.96608e9 −1.14507
\(94\) −4.97391e8 + 1.05012e9i −0.0677733 + 0.143087i
\(95\) −9.95719e9 + 2.55519e8i −1.28682 + 0.0330222i
\(96\) −9.72601e8 − 6.68557e9i −0.119283 − 0.819941i
\(97\) −9.49531e9 + 9.49531e9i −1.10573 + 1.10573i −0.112029 + 0.993705i \(0.535735\pi\)
−0.993705 + 0.112029i \(0.964265\pi\)
\(98\) 2.81360e9 + 7.87738e9i 0.311266 + 0.871468i
\(99\) −3.06048e9 − 3.06048e9i −0.321820 − 0.321820i
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.53.14 yes 236
5.2 odd 4 80.11.i.a.37.46 yes 236
16.13 even 4 80.11.i.a.13.46 ✓ 236
80.77 odd 4 inner 80.11.t.a.77.14 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.46 ✓ 236 16.13 even 4
80.11.i.a.37.46 yes 236 5.2 odd 4
80.11.t.a.53.14 yes 236 1.1 even 1 trivial
80.11.t.a.77.14 yes 236 80.77 odd 4 inner