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Properties

Label 80.11.i.a.13.19
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 13.19
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.19

$q$-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+(-28.4618 - 14.6261i) q^{2} -233.425i q^{3} +(596.154 + 832.572i) q^{4} +(-2188.11 + 2231.09i) q^{5} +(-3414.10 + 6643.70i) q^{6} +(4255.80 + 4255.80i) q^{7} +(-4790.33 - 32416.0i) q^{8} +4561.92 q^{9} +(94910.0 - 31497.4i) q^{10} +(-211820. + 211820. i) q^{11} +(194343. - 139157. i) q^{12} +398877. i q^{13} +(-58882.1 - 183374. i) q^{14} +(520792. + 510760. i) q^{15} +(-337778. + 992682. i) q^{16} +(-1.05042e6 + 1.05042e6i) q^{17} +(-129841. - 66723.2i) q^{18} +(1.79456e6 - 1.79456e6i) q^{19} +(-3.16200e6 - 491689. i) q^{20} +(993408. - 993408. i) q^{21} +(9.12689e6 - 2.93069e6i) q^{22} +(-2.99248e6 + 2.99248e6i) q^{23} +(-7.56669e6 + 1.11818e6i) q^{24} +(-189935. - 9.76378e6i) q^{25} +(5.83403e6 - 1.13528e7i) q^{26} -1.48484e7i q^{27} +(-1.00615e6 + 6.08037e6i) q^{28} +(1.55958e7 - 1.55958e7i) q^{29} +(-7.35228e6 - 2.21543e7i) q^{30} -1.02510e7 q^{31} +(2.41329e7 - 2.33132e7i) q^{32} +(4.94440e7 + 4.94440e7i) q^{33} +(4.52603e7 - 1.45333e7i) q^{34} +(-1.88072e7 + 182911. i) q^{35} +(2.71961e6 + 3.79813e6i) q^{36} +9.48654e7i q^{37} +(-7.73241e7 + 2.48291e7i) q^{38} +9.31078e7 q^{39} +(8.28049e7 + 6.02421e7i) q^{40} -1.88109e8i q^{41} +(-4.28039e7 + 1.37445e7i) q^{42} +1.96396e8 q^{43} +(-3.02633e8 - 5.00782e7i) q^{44} +(-9.98201e6 + 1.01781e7i) q^{45} +(1.28940e8 - 4.14032e7i) q^{46} +(-5.83419e7 + 5.83419e7i) q^{47} +(2.31717e8 + 7.88457e7i) q^{48} -2.46252e8i q^{49} +(-1.37400e8 + 2.80673e8i) q^{50} +(2.45193e8 + 2.45193e8i) q^{51} +(-3.32094e8 + 2.37792e8i) q^{52} -7.63638e8 q^{53} +(-2.17174e8 + 4.22612e8i) q^{54} +(-9.10389e6 - 9.36077e8i) q^{55} +(1.17569e8 - 1.58342e8i) q^{56} +(-4.18896e8 - 4.18896e8i) q^{57} +(-6.71990e8 + 2.15779e8i) q^{58} +(2.64858e8 + 2.64858e8i) q^{59} +(-1.14772e8 + 7.38089e8i) q^{60} +(-3.97559e8 - 3.97559e8i) q^{61} +(2.91763e8 + 1.49933e8i) q^{62} +(1.94146e7 + 1.94146e7i) q^{63} +(-1.02785e9 + 3.10567e8i) q^{64} +(-8.89933e8 - 8.72790e8i) q^{65} +(-6.84094e8 - 2.13044e9i) q^{66} -9.93553e8 q^{67} +(-1.50076e9 - 2.48338e8i) q^{68} +(6.98519e8 + 6.98519e8i) q^{69} +(5.37964e8 + 2.69871e8i) q^{70} +1.63307e9i q^{71} +(-2.18531e7 - 1.47879e8i) q^{72} +(1.99207e9 - 1.99207e9i) q^{73} +(1.38751e9 - 2.70004e9i) q^{74} +(-2.27911e9 + 4.43355e7i) q^{75} +(2.56394e9 + 4.24269e8i) q^{76} -1.80293e9 q^{77} +(-2.65002e9 - 1.36181e9i) q^{78} -4.43310e8i q^{79} +(-1.47567e9 - 2.92572e9i) q^{80} -3.19660e9 q^{81} +(-2.75130e9 + 5.35393e9i) q^{82} +2.39127e9i q^{83} +(1.41931e9 + 2.34860e8i) q^{84} +(-4.51463e7 - 4.64201e9i) q^{85} +(-5.58980e9 - 2.87251e9i) q^{86} +(-3.64044e9 - 3.64044e9i) q^{87} +(7.88104e9 + 5.85166e9i) q^{88} -7.16002e9 q^{89} +(4.32972e8 - 1.43689e8i) q^{90} +(-1.69754e9 + 1.69754e9i) q^{91} +(-4.27543e9 - 7.07479e8i) q^{92} +2.39284e9i q^{93} +(2.51383e9 - 8.07202e8i) q^{94} +(7.71292e7 + 7.93056e9i) q^{95} +(-5.44187e9 - 5.63321e9i) q^{96} +(-6.52877e9 + 6.52877e9i) q^{97} +(-3.60170e9 + 7.00878e9i) q^{98} +(-9.66307e8 + 9.66307e8i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −28.4618 − 14.6261i −0.889433 − 0.457066i
\(3\) − 233.425i − 0.960595i −0.877106 − 0.480298i \(-0.840529\pi\)
0.877106 − 0.480298i \(-0.159471\pi\)
\(4\) 596.154 + 832.572i 0.582181 + 0.813059i
\(5\) −2188.11 + 2231.09i −0.700197 + 0.713950i
\(6\) −3414.10 + 6643.70i −0.439056 + 0.854385i
\(7\) 4255.80 + 4255.80i 0.253216 + 0.253216i 0.822288 − 0.569072i \(-0.192697\pi\)
−0.569072 + 0.822288i \(0.692697\pi\)
\(8\) −4790.33 − 32416.0i −0.146189 − 0.989257i
\(9\) 4561.92 0.0772566
\(10\) 94910.0 − 31497.4i 0.949100 − 0.314974i
\(11\) −211820. + 211820.i −1.31524 + 1.31524i −0.397736 + 0.917500i \(0.630204\pi\)
−0.917500 + 0.397736i \(0.869796\pi\)
\(12\) 194343. − 139157.i 0.781021 − 0.559241i
\(13\) 398877.i 1.07429i 0.843489 + 0.537147i \(0.180498\pi\)
−0.843489 + 0.537147i \(0.819502\pi\)
\(14\) −58882.1 − 183374.i −0.109482 − 0.340955i
\(15\) 520792. + 510760.i 0.685817 + 0.672606i
\(16\) −337778. + 992682.i −0.322130 + 0.946695i
\(17\) −1.05042e6 + 1.05042e6i −0.739805 + 0.739805i −0.972540 − 0.232735i \(-0.925232\pi\)
0.232735 + 0.972540i \(0.425232\pi\)
\(18\) −129841. − 66723.2i −0.0687145 − 0.0353113i
\(19\) 1.79456e6 − 1.79456e6i 0.724755 − 0.724755i −0.244815 − 0.969570i \(-0.578727\pi\)
0.969570 + 0.244815i \(0.0787272\pi\)
\(20\) −3.16200e6 − 491689.i −0.988125 − 0.153653i
\(21\) 993408. − 993408.i 0.243238 − 0.243238i
\(22\) 9.12689e6 − 2.93069e6i 1.77096 − 0.568664i
\(23\) −2.99248e6 + 2.99248e6i −0.464935 + 0.464935i −0.900269 − 0.435334i \(-0.856630\pi\)
0.435334 + 0.900269i \(0.356630\pi\)
\(24\) −7.56669e6 + 1.11818e6i −0.950275 + 0.140429i
\(25\) −189935. − 9.76378e6i −0.0194493 − 0.999811i
\(26\) 5.83403e6 − 1.13528e7i 0.491023 − 0.955511i
\(27\) − 1.48484e7i − 1.03481i
\(28\) −1.00615e6 + 6.08037e6i −0.0584619 + 0.353297i
\(29\) 1.55958e7 − 1.55958e7i 0.760355 − 0.760355i −0.216031 − 0.976386i \(-0.569311\pi\)
0.976386 + 0.216031i \(0.0693113\pi\)
\(30\) −7.35228e6 − 2.21543e7i −0.302563 − 0.911701i
\(31\) −1.02510e7 −0.358063 −0.179031 − 0.983843i \(-0.557296\pi\)
−0.179031 + 0.983843i \(0.557296\pi\)
\(32\) 2.41329e7 − 2.33132e7i 0.719215 − 0.694787i
\(33\) 4.94440e7 + 4.94440e7i 1.26341 + 1.26341i
\(34\) 4.52603e7 − 1.45333e7i 0.996146 − 0.319867i
\(35\) −1.88072e7 + 182911.i −0.358084 + 0.00348257i
\(36\) 2.71961e6 + 3.79813e6i 0.0449773 + 0.0628141i
\(37\) 9.48654e7i 1.36804i 0.729463 + 0.684020i \(0.239770\pi\)
−0.729463 + 0.684020i \(0.760230\pi\)
\(38\) −7.73241e7 + 2.48291e7i −0.975882 + 0.313360i
\(39\) 9.31078e7 1.03196
\(40\) 8.28049e7 + 6.02421e7i 0.808641 + 0.588302i
\(41\) − 1.88109e8i − 1.62364i −0.583906 − 0.811821i \(-0.698477\pi\)
0.583906 − 0.811821i \(-0.301523\pi\)
\(42\) −4.28039e7 + 1.37445e7i −0.327519 + 0.105168i
\(43\) 1.96396e8 1.33595 0.667976 − 0.744183i \(-0.267161\pi\)
0.667976 + 0.744183i \(0.267161\pi\)
\(44\) −3.02633e8 − 5.00782e7i −1.83507 − 0.303659i
\(45\) −9.98201e6 + 1.01781e7i −0.0540948 + 0.0551573i
\(46\) 1.28940e8 − 4.14032e7i 0.626034 − 0.201022i
\(47\) −5.83419e7 + 5.83419e7i −0.254385 + 0.254385i −0.822766 − 0.568381i \(-0.807570\pi\)
0.568381 + 0.822766i \(0.307570\pi\)
\(48\) 2.31717e8 + 7.88457e7i 0.909391 + 0.309437i
\(49\) − 2.46252e8i − 0.871764i
\(50\) −1.37400e8 + 2.80673e8i −0.439681 + 0.898154i
\(51\) 2.45193e8 + 2.45193e8i 0.710653 + 0.710653i
\(52\) −3.32094e8 + 2.37792e8i −0.873464 + 0.625433i
\(53\) −7.63638e8 −1.82603 −0.913015 − 0.407925i \(-0.866252\pi\)
−0.913015 + 0.407925i \(0.866252\pi\)
\(54\) −2.17174e8 + 4.22612e8i −0.472975 + 0.920392i
\(55\) −9.10389e6 − 9.36077e8i −0.0180890 − 1.85994i
\(56\) 1.17569e8 − 1.58342e8i 0.213478 − 0.287513i
\(57\) −4.18896e8 − 4.18896e8i −0.696196 − 0.696196i
\(58\) −6.71990e8 + 2.15779e8i −1.02382 + 0.328752i
\(59\) 2.64858e8 + 2.64858e8i 0.370470 + 0.370470i 0.867648 − 0.497179i \(-0.165631\pi\)
−0.497179 + 0.867648i \(0.665631\pi\)
\(60\) −1.14772e8 + 7.38089e8i −0.147598 + 0.949188i
\(61\) −3.97559e8 − 3.97559e8i −0.470708 − 0.470708i 0.431435 − 0.902144i \(-0.358007\pi\)
−0.902144 + 0.431435i \(0.858007\pi\)
\(62\) 2.91763e8 + 1.49933e8i 0.318473 + 0.163658i
\(63\) 1.94146e7 + 1.94146e7i 0.0195626 + 0.0195626i
\(64\) −1.02785e9 + 3.10567e8i −0.957257 + 0.289238i
\(65\) −8.89933e8 − 8.72790e8i −0.766992 − 0.752216i
\(66\) −6.84094e8 − 2.13044e9i −0.546256 − 1.70118i
\(67\) −9.93553e8 −0.735897 −0.367949 − 0.929846i \(-0.619940\pi\)
−0.367949 + 0.929846i \(0.619940\pi\)
\(68\) −1.50076e9 − 2.48338e8i −1.03221 − 0.170804i
\(69\) 6.98519e8 + 6.98519e8i 0.446614 + 0.446614i
\(70\) 5.37964e8 + 2.69871e8i 0.320083 + 0.160571i
\(71\) 1.63307e9i 0.905133i 0.891731 + 0.452567i \(0.149492\pi\)
−0.891731 + 0.452567i \(0.850508\pi\)
\(72\) −2.18531e7 − 1.47879e8i −0.0112941 − 0.0764266i
\(73\) 1.99207e9 − 1.99207e9i 0.960925 − 0.960925i −0.0383402 − 0.999265i \(-0.512207\pi\)
0.999265 + 0.0383402i \(0.0122071\pi\)
\(74\) 1.38751e9 − 2.70004e9i 0.625285 − 1.21678i
\(75\) −2.27911e9 + 4.43355e7i −0.960414 + 0.0186829i
\(76\) 2.56394e9 + 4.24269e8i 1.01121 + 0.167330i
\(77\) −1.80293e9 −0.666077
\(78\) −2.65002e9 − 1.36181e9i −0.917860 − 0.471674i
\(79\) − 4.43310e8i − 0.144070i −0.997402 − 0.0720348i \(-0.977051\pi\)
0.997402 − 0.0720348i \(-0.0229493\pi\)
\(80\) −1.47567e9 − 2.92572e9i −0.450339 − 0.892858i
\(81\) −3.19660e9 −0.916775
\(82\) −2.75130e9 + 5.35393e9i −0.742112 + 1.44412i
\(83\) 2.39127e9i 0.607069i 0.952820 + 0.303535i \(0.0981668\pi\)
−0.952820 + 0.303535i \(0.901833\pi\)
\(84\) 1.41931e9 + 2.34860e8i 0.339375 + 0.0561582i
\(85\) −4.51463e7 − 4.64201e9i −0.0101748 − 1.04619i
\(86\) −5.58980e9 − 2.87251e9i −1.18824 − 0.610618i
\(87\) −3.64044e9 − 3.64044e9i −0.730394 − 0.730394i
\(88\) 7.88104e9 + 5.85166e9i 1.49338 + 1.10883i
\(89\) −7.16002e9 −1.28223 −0.641113 − 0.767447i \(-0.721527\pi\)
−0.641113 + 0.767447i \(0.721527\pi\)
\(90\) 4.32972e8 − 1.43689e8i 0.0733242 − 0.0243338i
\(91\) −1.69754e9 + 1.69754e9i −0.272028 + 0.272028i
\(92\) −4.27543e9 − 7.07479e8i −0.648696 − 0.107343i
\(93\) 2.39284e9i 0.343953i
\(94\) 2.51383e9 − 8.07202e8i 0.342529 − 0.109987i
\(95\) 7.71292e7 + 7.93056e9i 0.00996784 + 1.02491i
\(96\) −5.44187e9 − 5.63321e9i −0.667409 − 0.690875i
\(97\) −6.52877e9 + 6.52877e9i −0.760278 + 0.760278i −0.976372 − 0.216094i \(-0.930668\pi\)
0.216094 + 0.976372i \(0.430668\pi\)
\(98\) −3.60170e9 + 7.00878e9i −0.398454 + 0.775375i
\(99\) −9.66307e8 + 9.66307e8i −0.101611 + 0.101611i
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.13.19 ✓ 236
5.2 odd 4 80.11.t.a.77.78 yes 236
16.5 even 4 80.11.t.a.53.78 yes 236
80.37 odd 4 inner 80.11.i.a.37.19 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.19 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.19 yes 236 80.37 odd 4 inner
80.11.t.a.53.78 yes 236 16.5 even 4
80.11.t.a.77.78 yes 236 5.2 odd 4