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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 13.9
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.9

$q$-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+(-31.1873 + 7.16609i) q^{2} +113.551i q^{3} +(921.294 - 446.982i) q^{4} +(-3033.64 - 750.089i) q^{5} +(-813.717 - 3541.35i) q^{6} +(-9621.52 - 9621.52i) q^{7} +(-25529.6 + 20542.2i) q^{8} +46155.2 q^{9} +(99986.3 + 1653.87i) q^{10} +(-150277. + 150277. i) q^{11} +(50755.2 + 104614. i) q^{12} +409750. i q^{13} +(369018. + 231121. i) q^{14} +(85173.3 - 344473. i) q^{15} +(648990. - 823604. i) q^{16} +(442906. - 442906. i) q^{17} +(-1.43945e6 + 330752. i) q^{18} +(-1.68175e6 + 1.68175e6i) q^{19} +(-3.13015e6 + 664931. i) q^{20} +(1.09253e6 - 1.09253e6i) q^{21} +(3.60983e6 - 5.76362e6i) q^{22} +(2.37512e6 - 2.37512e6i) q^{23} +(-2.33259e6 - 2.89891e6i) q^{24} +(8.64036e6 + 4.55100e6i) q^{25} +(-2.93630e6 - 1.27790e7i) q^{26} +1.19460e7i q^{27} +(-1.31649e7 - 4.56361e6i) q^{28} +(-5.12889e6 + 5.12889e6i) q^{29} +(-187799. + 1.13535e7i) q^{30} -5.58774e7 q^{31} +(-1.43382e7 + 3.03367e7i) q^{32} +(-1.70641e7 - 1.70641e7i) q^{33} +(-1.06391e7 + 1.69870e7i) q^{34} +(2.19713e7 + 3.64053e7i) q^{35} +(4.25225e7 - 2.06305e7i) q^{36} +1.30084e8i q^{37} +(4.03976e7 - 6.45008e7i) q^{38} -4.65275e7 q^{39} +(9.28561e7 - 4.31684e7i) q^{40} -8.20307e7i q^{41} +(-2.62440e7 + 4.19023e7i) q^{42} -1.49294e8 q^{43} +(-7.12782e7 + 2.05620e8i) q^{44} +(-1.40018e8 - 3.46205e7i) q^{45} +(-5.70532e7 + 9.10938e7i) q^{46} +(2.18815e8 - 2.18815e8i) q^{47} +(9.35210e7 + 7.36935e7i) q^{48} -9.73279e7i q^{49} +(-3.02082e8 - 8.00159e7i) q^{50} +(5.02925e7 + 5.02925e7i) q^{51} +(1.83151e8 + 3.77500e8i) q^{52} +3.71825e8 q^{53} +(-8.56064e7 - 3.72565e8i) q^{54} +(5.68607e8 - 3.43165e8i) q^{55} +(4.43281e8 + 4.79857e7i) q^{56} +(-1.90964e8 - 1.90964e8i) q^{57} +(1.23202e8 - 1.96710e8i) q^{58} +(2.45572e8 + 2.45572e8i) q^{59} +(-7.55036e7 - 3.55432e8i) q^{60} +(-1.13359e9 - 1.13359e9i) q^{61} +(1.74266e9 - 4.00422e8i) q^{62} +(-4.44083e8 - 4.44083e8i) q^{63} +(2.29775e8 - 1.04887e9i) q^{64} +(3.07349e8 - 1.24303e9i) q^{65} +(6.54465e8 + 4.09900e8i) q^{66} +2.22350e9 q^{67} +(2.10076e8 - 6.06018e8i) q^{68} +(2.69697e8 + 2.69697e8i) q^{69} +(-9.46108e8 - 9.77933e8i) q^{70} -1.35353e9i q^{71} +(-1.17832e9 + 9.48130e8i) q^{72} +(-2.76638e9 + 2.76638e9i) q^{73} +(-9.32191e8 - 4.05695e9i) q^{74} +(-5.16771e8 + 9.81121e8i) q^{75} +(-7.97675e8 + 2.30110e9i) q^{76} +2.89178e9 q^{77} +(1.45107e9 - 3.33420e8i) q^{78} -4.90956e9i q^{79} +(-2.58658e9 + 2.01172e9i) q^{80} +1.36893e9 q^{81} +(5.87839e8 + 2.55832e9i) q^{82} -4.40182e9i q^{83} +(5.18202e8 - 1.49489e9i) q^{84} +(-1.67584e9 + 1.01140e9i) q^{85} +(4.65608e9 - 1.06985e9i) q^{86} +(-5.82390e8 - 5.82390e8i) q^{87} +(7.49481e8 - 6.92352e9i) q^{88} +9.02820e9 q^{89} +(4.61489e9 + 7.63348e7i) q^{90} +(3.94242e9 - 3.94242e9i) q^{91} +(1.12655e9 - 3.24982e9i) q^{92} -6.34493e9i q^{93} +(-5.25620e9 + 8.39229e9i) q^{94} +(6.36329e9 - 3.84037e9i) q^{95} +(-3.44476e9 - 1.62812e9i) q^{96} +(-5.94577e9 + 5.94577e9i) q^{97} +(6.97461e8 + 3.03539e9i) q^{98} +(-6.93605e9 + 6.93605e9i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.1873 + 7.16609i −0.974603 + 0.223940i
\(3\) 113.551i 0.467288i 0.972322 + 0.233644i \(0.0750650\pi\)
−0.972322 + 0.233644i \(0.924935\pi\)
\(4\) 921.294 − 446.982i 0.899701 − 0.436506i
\(5\) −3033.64 − 750.089i −0.970766 − 0.240028i
\(6\) −813.717 − 3541.35i −0.104645 − 0.455420i
\(7\) −9621.52 − 9621.52i −0.572471 − 0.572471i 0.360347 − 0.932818i \(-0.382658\pi\)
−0.932818 + 0.360347i \(0.882658\pi\)
\(8\) −25529.6 + 20542.2i −0.779100 + 0.626899i
\(9\) 46155.2 0.781642
\(10\) 99986.3 + 1653.87i 0.999863 + 0.0165387i
\(11\) −150277. + 150277.i −0.933101 + 0.933101i −0.997898 − 0.0647976i \(-0.979360\pi\)
0.0647976 + 0.997898i \(0.479360\pi\)
\(12\) 50755.2 + 104614.i 0.203974 + 0.420420i
\(13\) 409750.i 1.10358i 0.833985 + 0.551788i \(0.186054\pi\)
−0.833985 + 0.551788i \(0.813946\pi\)
\(14\) 369018. + 231121.i 0.686131 + 0.429733i
\(15\) 85173.3 − 344473.i 0.112162 − 0.453627i
\(16\) 648990. − 823604.i 0.618926 − 0.785450i
\(17\) 442906. − 442906.i 0.311937 − 0.311937i −0.533722 − 0.845660i \(-0.679207\pi\)
0.845660 + 0.533722i \(0.179207\pi\)
\(18\) −1.43945e6 + 330752.i −0.761790 + 0.175041i
\(19\) −1.68175e6 + 1.68175e6i −0.679193 + 0.679193i −0.959818 − 0.280624i \(-0.909458\pi\)
0.280624 + 0.959818i \(0.409458\pi\)
\(20\) −3.13015e6 + 664931.i −0.978173 + 0.207791i
\(21\) 1.09253e6 − 1.09253e6i 0.267509 − 0.267509i
\(22\) 3.60983e6 − 5.76362e6i 0.700444 − 1.11836i
\(23\) 2.37512e6 − 2.37512e6i 0.369017 − 0.369017i −0.498102 − 0.867118i \(-0.665969\pi\)
0.867118 + 0.498102i \(0.165969\pi\)
\(24\) −2.33259e6 − 2.89891e6i −0.292942 − 0.364064i
\(25\) 8.64036e6 + 4.55100e6i 0.884773 + 0.466023i
\(26\) −2.93630e6 − 1.27790e7i −0.247135 − 1.07555i
\(27\) 1.19460e7i 0.832540i
\(28\) −1.31649e7 − 4.56361e6i −0.764940 − 0.265166i
\(29\) −5.12889e6 + 5.12889e6i −0.250054 + 0.250054i −0.820993 − 0.570939i \(-0.806579\pi\)
0.570939 + 0.820993i \(0.306579\pi\)
\(30\) −187799. + 1.13535e7i −0.00772836 + 0.467224i
\(31\) −5.58774e7 −1.95176 −0.975882 − 0.218298i \(-0.929949\pi\)
−0.975882 + 0.218298i \(0.929949\pi\)
\(32\) −1.43382e7 + 3.03367e7i −0.427313 + 0.904104i
\(33\) −1.70641e7 − 1.70641e7i −0.436027 − 0.436027i
\(34\) −1.06391e7 + 1.69870e7i −0.234160 + 0.373870i
\(35\) 2.19713e7 + 3.64053e7i 0.418326 + 0.693145i
\(36\) 4.25225e7 − 2.06305e7i 0.703244 − 0.341191i
\(37\) 1.30084e8i 1.87592i 0.346746 + 0.937959i \(0.387287\pi\)
−0.346746 + 0.937959i \(0.612713\pi\)
\(38\) 4.03976e7 − 6.45008e7i 0.509845 − 0.814042i
\(39\) −4.65275e7 −0.515687
\(40\) 9.28561e7 − 4.31684e7i 0.906798 − 0.421566i
\(41\) − 8.20307e7i − 0.708039i −0.935238 − 0.354019i \(-0.884815\pi\)
0.935238 − 0.354019i \(-0.115185\pi\)
\(42\) −2.62440e7 + 4.19023e7i −0.200809 + 0.320621i
\(43\) −1.49294e8 −1.01555 −0.507774 − 0.861490i \(-0.669531\pi\)
−0.507774 + 0.861490i \(0.669531\pi\)
\(44\) −7.12782e7 + 2.05620e8i −0.432208 + 1.24682i
\(45\) −1.40018e8 − 3.46205e7i −0.758791 − 0.187616i
\(46\) −5.70532e7 + 9.10938e7i −0.277007 + 0.442282i
\(47\) 2.18815e8 − 2.18815e8i 0.954086 − 0.954086i −0.0449050 − 0.998991i \(-0.514299\pi\)
0.998991 + 0.0449050i \(0.0142985\pi\)
\(48\) 9.35210e7 + 7.36935e7i 0.367031 + 0.289217i
\(49\) − 9.73279e7i − 0.344554i
\(50\) −3.02082e8 − 8.00159e7i −0.966663 − 0.256051i
\(51\) 5.02925e7 + 5.02925e7i 0.145765 + 0.145765i
\(52\) 1.83151e8 + 3.77500e8i 0.481717 + 0.992888i
\(53\) 3.71825e8 0.889118 0.444559 − 0.895749i \(-0.353360\pi\)
0.444559 + 0.895749i \(0.353360\pi\)
\(54\) −8.56064e7 − 3.72565e8i −0.186439 − 0.811396i
\(55\) 5.68607e8 − 3.43165e8i 1.12979 − 0.681852i
\(56\) 4.43281e8 + 4.79857e7i 0.804894 + 0.0871309i
\(57\) −1.90964e8 − 1.90964e8i −0.317379 − 0.317379i
\(58\) 1.23202e8 − 1.96710e8i 0.187706 − 0.299700i
\(59\) 2.45572e8 + 2.45572e8i 0.343494 + 0.343494i 0.857679 − 0.514185i \(-0.171906\pi\)
−0.514185 + 0.857679i \(0.671906\pi\)
\(60\) −7.55036e7 − 3.55432e8i −0.0970982 − 0.457089i
\(61\) −1.13359e9 − 1.13359e9i −1.34217 − 1.34217i −0.893898 − 0.448271i \(-0.852040\pi\)
−0.448271 − 0.893898i \(-0.647960\pi\)
\(62\) 1.74266e9 − 4.00422e8i 1.90220 − 0.437079i
\(63\) −4.44083e8 − 4.44083e8i −0.447467 − 0.447467i
\(64\) 2.29775e8 − 1.04887e9i 0.213995 − 0.976835i
\(65\) 3.07349e8 − 1.24303e9i 0.264889 − 1.07131i
\(66\) 6.54465e8 + 4.09900e8i 0.522597 + 0.327309i
\(67\) 2.22350e9 1.64688 0.823441 − 0.567402i \(-0.192051\pi\)
0.823441 + 0.567402i \(0.192051\pi\)
\(68\) 2.10076e8 − 6.06018e8i 0.144488 − 0.416813i
\(69\) 2.69697e8 + 2.69697e8i 0.172437 + 0.172437i
\(70\) −9.46108e8 − 9.77933e8i −0.562925 − 0.581861i
\(71\) − 1.35353e9i − 0.750200i −0.926984 − 0.375100i \(-0.877608\pi\)
0.926984 − 0.375100i \(-0.122392\pi\)
\(72\) −1.17832e9 + 9.48130e8i −0.608977 + 0.490011i
\(73\) −2.76638e9 + 2.76638e9i −1.33444 + 1.33444i −0.433080 + 0.901356i \(0.642573\pi\)
−0.901356 + 0.433080i \(0.857427\pi\)
\(74\) −9.32191e8 − 4.05695e9i −0.420094 − 1.82828i
\(75\) −5.16771e8 + 9.81121e8i −0.217767 + 0.413444i
\(76\) −7.97675e8 + 2.30110e9i −0.314599 + 0.907543i
\(77\) 2.89178e9 1.06835
\(78\) 1.45107e9 − 3.33420e8i 0.502591 − 0.115483i
\(79\) − 4.90956e9i − 1.59554i −0.602964 − 0.797768i \(-0.706014\pi\)
0.602964 − 0.797768i \(-0.293986\pi\)
\(80\) −2.58658e9 + 2.01172e9i −0.789362 + 0.613928i
\(81\) 1.36893e9 0.392606
\(82\) 5.87839e8 + 2.55832e9i 0.158558 + 0.690057i
\(83\) − 4.40182e9i − 1.11748i −0.829341 − 0.558742i \(-0.811284\pi\)
0.829341 − 0.558742i \(-0.188716\pi\)
\(84\) 5.18202e8 − 1.49489e9i 0.123909 − 0.357447i
\(85\) −1.67584e9 + 1.01140e9i −0.377692 + 0.227944i
\(86\) 4.65608e9 − 1.06985e9i 0.989756 − 0.227422i
\(87\) −5.82390e8 − 5.82390e8i −0.116847 − 0.116847i
\(88\) 7.49481e8 − 6.92352e9i 0.142019 − 1.31194i
\(89\) 9.02820e9 1.61678 0.808390 − 0.588647i \(-0.200339\pi\)
0.808390 + 0.588647i \(0.200339\pi\)
\(90\) 4.61489e9 + 7.63348e7i 0.781535 + 0.0129274i
\(91\) 3.94242e9 − 3.94242e9i 0.631765 − 0.631765i
\(92\) 1.12655e9 − 3.24982e9i 0.170927 − 0.493083i
\(93\) − 6.34493e9i − 0.912036i
\(94\) −5.25620e9 + 8.39229e9i −0.716197 + 1.14351i
\(95\) 6.36329e9 − 3.84037e9i 0.822363 − 0.496312i
\(96\) −3.44476e9 − 1.62812e9i −0.422477 − 0.199678i
\(97\) −5.94577e9 + 5.94577e9i −0.692387 + 0.692387i −0.962757 − 0.270369i \(-0.912854\pi\)
0.270369 + 0.962757i \(0.412854\pi\)
\(98\) 6.97461e8 + 3.03539e9i 0.0771595 + 0.335803i
\(99\) −6.93605e9 + 6.93605e9i −0.729351 + 0.729351i
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.13.9 ✓ 236
5.2 odd 4 80.11.t.a.77.52 yes 236
16.5 even 4 80.11.t.a.53.52 yes 236
80.37 odd 4 inner 80.11.i.a.37.9 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.9 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.9 yes 236 80.37 odd 4 inner
80.11.t.a.53.52 yes 236 16.5 even 4
80.11.t.a.77.52 yes 236 5.2 odd 4