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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 53.4
Character \(\chi\) \(=\) 80.53
Dual form 80.11.t.a.77.4

$q$-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+(-31.8649 + 2.93778i) q^{2} -74.0023 q^{3} +(1006.74 - 187.224i) q^{4} +(0.479946 - 3125.00i) q^{5} +(2358.07 - 217.402i) q^{6} +(-2919.78 - 2919.78i) q^{7} +(-31529.6 + 8923.43i) q^{8} -53572.7 q^{9} +(9165.26 + 99579.1i) q^{10} +(33391.3 + 33391.3i) q^{11} +(-74501.0 + 13855.0i) q^{12} +605319. q^{13} +(101616. + 84460.8i) q^{14} +(-35.5172 + 231257. i) q^{15} +(978471. - 376971. i) q^{16} +(-323679. + 323679. i) q^{17} +(1.70709e6 - 157384. i) q^{18} +(1.18060e6 + 1.18060e6i) q^{19} +(-584591. - 3.14615e6i) q^{20} +(216071. + 216071. i) q^{21} +(-1.16211e6 - 965914. i) q^{22} +(-5.01513e6 + 5.01513e6i) q^{23} +(2.33326e6 - 660355. i) q^{24} +(-9.76562e6 - 2999.67i) q^{25} +(-1.92884e7 + 1.77829e6i) q^{26} +8.33427e6 q^{27} +(-3.48611e6 - 2.39281e6i) q^{28} +(1.17307e7 + 1.17307e7i) q^{29} +(-678251. - 7.36909e6i) q^{30} -1.09769e7 q^{31} +(-3.00714e7 + 1.48866e7i) q^{32} +(-2.47104e6 - 2.47104e6i) q^{33} +(9.36309e6 - 1.12649e7i) q^{34} +(-9.12572e6 + 9.12292e6i) q^{35} +(-5.39337e7 + 1.00301e7i) q^{36} +3.49450e7 q^{37} +(-4.10880e7 - 3.41513e7i) q^{38} -4.47950e7 q^{39} +(2.78706e7 + 9.85342e7i) q^{40} +3.44094e7i q^{41} +(-7.51983e6 - 6.25030e6i) q^{42} -6.84840e7i q^{43} +(3.98680e7 + 2.73647e7i) q^{44} +(-25712.0 + 1.67415e8i) q^{45} +(1.45073e8 - 1.74540e8i) q^{46} +(-2.46194e8 + 2.46194e8i) q^{47} +(-7.24091e7 + 2.78967e7i) q^{48} -2.65425e8i q^{49} +(3.11189e8 - 2.85936e7i) q^{50} +(2.39530e7 - 2.39530e7i) q^{51} +(6.09398e8 - 1.13330e8i) q^{52} -2.62087e7i q^{53} +(-2.65570e8 + 2.44842e7i) q^{54} +(1.04364e8 - 1.04332e8i) q^{55} +(1.18114e8 + 6.60050e7i) q^{56} +(-8.73672e7 - 8.73672e7i) q^{57} +(-4.08260e8 - 3.39336e8i) q^{58} +(8.38309e8 - 8.38309e8i) q^{59} +(4.32611e7 + 2.32822e8i) q^{60} +(9.28232e8 - 9.28232e8i) q^{61} +(3.49778e8 - 3.22478e7i) q^{62} +(1.56420e8 + 1.56420e8i) q^{63} +(9.14487e8 - 5.62704e8i) q^{64} +(290520. - 1.89162e9i) q^{65} +(8.59986e7 + 7.14799e7i) q^{66} +1.52579e9i q^{67} +(-2.65260e8 + 3.86461e8i) q^{68} +(3.71131e8 - 3.71131e8i) q^{69} +(2.63989e8 - 3.17510e8i) q^{70} +1.89392e9i q^{71} +(1.68912e9 - 4.78052e8i) q^{72} +(-3.99963e8 + 3.99963e8i) q^{73} +(-1.11352e9 + 1.02661e8i) q^{74} +(7.22679e8 + 221982. i) q^{75} +(1.40959e9 + 9.67520e8i) q^{76} -1.94991e8i q^{77} +(1.42739e9 - 1.31598e8i) q^{78} -3.93749e9i q^{79} +(-1.17756e9 - 3.05790e9i) q^{80} +2.54666e9 q^{81} +(-1.01087e8 - 1.09645e9i) q^{82} +4.36246e9 q^{83} +(2.57980e8 + 1.77073e8i) q^{84} +(1.01134e9 + 1.01165e9i) q^{85} +(2.01191e8 + 2.18223e9i) q^{86} +(-8.68101e8 - 8.68101e8i) q^{87} +(-1.35078e9 - 7.54849e8i) q^{88} +1.08001e10 q^{89} +(-4.91007e8 - 5.33472e9i) q^{90} +(-1.76740e9 - 1.76740e9i) q^{91} +(-4.10997e9 + 5.98787e9i) q^{92} +8.12319e8 q^{93} +(7.12168e9 - 8.56821e9i) q^{94} +(3.68994e9 - 3.68881e9i) q^{95} +(2.22535e9 - 1.10165e9i) q^{96} +(-1.02515e9 + 1.02515e9i) q^{97} +(7.79759e8 + 8.45773e9i) q^{98} +(-1.78886e9 - 1.78886e9i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.8649 + 2.93778i −0.995777 + 0.0918055i
\(3\) −74.0023 −0.304536 −0.152268 − 0.988339i \(-0.548658\pi\)
−0.152268 + 0.988339i \(0.548658\pi\)
\(4\) 1006.74 − 187.224i 0.983143 − 0.182836i
\(5\) 0.479946 − 3125.00i 0.000153583 − 1.00000i
\(6\) 2358.07 − 217.402i 0.303250 − 0.0279581i
\(7\) −2919.78 − 2919.78i −0.173724 − 0.173724i 0.614889 − 0.788613i \(-0.289201\pi\)
−0.788613 + 0.614889i \(0.789201\pi\)
\(8\) −31529.6 + 8923.43i −0.962206 + 0.272321i
\(9\) −53572.7 −0.907258
\(10\) 9165.26 + 99579.1i 0.0916526 + 0.995791i
\(11\) 33391.3 + 33391.3i 0.207334 + 0.207334i 0.803133 − 0.595799i \(-0.203165\pi\)
−0.595799 + 0.803133i \(0.703165\pi\)
\(12\) −74501.0 + 13855.0i −0.299403 + 0.0556801i
\(13\) 605319. 1.63030 0.815149 − 0.579251i \(-0.196655\pi\)
0.815149 + 0.579251i \(0.196655\pi\)
\(14\) 101616. + 84460.8i 0.188939 + 0.157042i
\(15\) −35.5172 + 231257.i −4.67716e−5 + 0.304536i
\(16\) 978471. − 376971.i 0.933142 − 0.359507i
\(17\) −323679. + 323679.i −0.227966 + 0.227966i −0.811842 − 0.583877i \(-0.801535\pi\)
0.583877 + 0.811842i \(0.301535\pi\)
\(18\) 1.70709e6 − 157384.i 0.903426 − 0.0832912i
\(19\) 1.18060e6 + 1.18060e6i 0.476799 + 0.476799i 0.904106 − 0.427308i \(-0.140538\pi\)
−0.427308 + 0.904106i \(0.640538\pi\)
\(20\) −584591. − 3.14615e6i −0.182685 − 0.983172i
\(21\) 216071. + 216071.i 0.0529053 + 0.0529053i
\(22\) −1.16211e6 − 965914.i −0.225493 − 0.187424i
\(23\) −5.01513e6 + 5.01513e6i −0.779189 + 0.779189i −0.979693 − 0.200504i \(-0.935742\pi\)
0.200504 + 0.979693i \(0.435742\pi\)
\(24\) 2.33326e6 − 660355.i 0.293027 − 0.0829318i
\(25\) −9.76562e6 − 2999.67i −1.00000 − 0.000307166i
\(26\) −1.92884e7 + 1.77829e6i −1.62341 + 0.149670i
\(27\) 8.33427e6 0.580829
\(28\) −3.48611e6 − 2.39281e6i −0.202559 − 0.139033i
\(29\) 1.17307e7 + 1.17307e7i 0.571919 + 0.571919i 0.932664 − 0.360745i \(-0.117478\pi\)
−0.360745 + 0.932664i \(0.617478\pi\)
\(30\) −678251. − 7.36909e6i −0.0279115 − 0.303255i
\(31\) −1.09769e7 −0.383418 −0.191709 − 0.981452i \(-0.561403\pi\)
−0.191709 + 0.981452i \(0.561403\pi\)
\(32\) −3.00714e7 + 1.48866e7i −0.896197 + 0.443657i
\(33\) −2.47104e6 − 2.47104e6i −0.0631407 − 0.0631407i
\(34\) 9.36309e6 − 1.12649e7i 0.206075 − 0.247932i
\(35\) −9.12572e6 + 9.12292e6i −0.173751 + 0.173697i
\(36\) −5.39337e7 + 1.00301e7i −0.891964 + 0.165879i
\(37\) 3.49450e7 0.503937 0.251969 − 0.967735i \(-0.418922\pi\)
0.251969 + 0.967735i \(0.418922\pi\)
\(38\) −4.10880e7 − 3.41513e7i −0.518558 − 0.431012i
\(39\) −4.47950e7 −0.496485
\(40\) 2.78706e7 + 9.85342e7i 0.272174 + 0.962248i
\(41\) 3.44094e7i 0.297001i 0.988912 + 0.148501i \(0.0474447\pi\)
−0.988912 + 0.148501i \(0.952555\pi\)
\(42\) −7.51983e6 − 6.25030e6i −0.0575389 − 0.0478249i
\(43\) − 6.84840e7i − 0.465850i −0.972495 − 0.232925i \(-0.925170\pi\)
0.972495 − 0.232925i \(-0.0748297\pi\)
\(44\) 3.98680e7 + 2.73647e7i 0.241747 + 0.165931i
\(45\) −25712.0 + 1.67415e8i −0.000139339 + 0.907258i
\(46\) 1.45073e8 − 1.74540e8i 0.704364 − 0.847432i
\(47\) −2.46194e8 + 2.46194e8i −1.07347 + 1.07347i −0.0763881 + 0.997078i \(0.524339\pi\)
−0.997078 + 0.0763881i \(0.975661\pi\)
\(48\) −7.24091e7 + 2.78967e7i −0.284176 + 0.109483i
\(49\) − 2.65425e8i − 0.939640i
\(50\) 3.11189e8 − 2.85936e7i 0.995805 − 0.0914996i
\(51\) 2.39530e7 − 2.39530e7i 0.0694239 − 0.0694239i
\(52\) 6.09398e8 − 1.13330e8i 1.60282 − 0.298077i
\(53\) − 2.62087e7i − 0.0626710i −0.999509 − 0.0313355i \(-0.990024\pi\)
0.999509 − 0.0313355i \(-0.00997604\pi\)
\(54\) −2.65570e8 + 2.44842e7i −0.578377 + 0.0533233i
\(55\) 1.04364e8 − 1.04332e8i 0.207366 − 0.207302i
\(56\) 1.18114e8 + 6.60050e7i 0.214467 + 0.119850i
\(57\) −8.73672e7 − 8.73672e7i −0.145203 − 0.145203i
\(58\) −4.08260e8 − 3.39336e8i −0.622009 − 0.516999i
\(59\) 8.38309e8 − 8.38309e8i 1.17258 − 1.17258i 0.190992 − 0.981592i \(-0.438829\pi\)
0.981592 − 0.190992i \(-0.0611705\pi\)
\(60\) 4.32611e7 + 2.32822e8i 0.0556341 + 0.299412i
\(61\) 9.28232e8 − 9.28232e8i 1.09902 − 1.09902i 0.104499 − 0.994525i \(-0.466676\pi\)
0.994525 − 0.104499i \(-0.0333239\pi\)
\(62\) 3.49778e8 − 3.22478e7i 0.381799 − 0.0351999i
\(63\) 1.56420e8 + 1.56420e8i 0.157613 + 0.157613i
\(64\) 9.14487e8 − 5.62704e8i 0.851682 − 0.524059i
\(65\) 290520. − 1.89162e9i 0.000250386 − 1.63030i
\(66\) 8.59986e7 + 7.14799e7i 0.0686707 + 0.0570774i
\(67\) 1.52579e9i 1.13011i 0.825053 + 0.565056i \(0.191145\pi\)
−0.825053 + 0.565056i \(0.808855\pi\)
\(68\) −2.65260e8 + 3.86461e8i −0.182443 + 0.265804i
\(69\) 3.71131e8 − 3.71131e8i 0.237291 − 0.237291i
\(70\) 2.63989e8 − 3.17510e8i 0.157071 − 0.188915i
\(71\) 1.89392e9i 1.04971i 0.851192 + 0.524855i \(0.175880\pi\)
−0.851192 + 0.524855i \(0.824120\pi\)
\(72\) 1.68912e9 − 4.78052e8i 0.872969 − 0.247066i
\(73\) −3.99963e8 + 3.99963e8i −0.192932 + 0.192932i −0.796962 − 0.604029i \(-0.793561\pi\)
0.604029 + 0.796962i \(0.293561\pi\)
\(74\) −1.11352e9 + 1.02661e8i −0.501809 + 0.0462642i
\(75\) 7.22679e8 + 221982.i 0.304536 + 9.35431e-5i
\(76\) 1.40959e9 + 9.67520e8i 0.555937 + 0.381586i
\(77\) − 1.94991e8i − 0.0720378i
\(78\) 1.42739e9 − 1.31598e8i 0.494389 − 0.0455801i
\(79\) − 3.93749e9i − 1.27963i −0.768529 − 0.639815i \(-0.779011\pi\)
0.768529 − 0.639815i \(-0.220989\pi\)
\(80\) −1.17756e9 − 3.05790e9i −0.359364 − 0.933197i
\(81\) 2.54666e9 0.730374
\(82\) −1.01087e8 − 1.09645e9i −0.0272663 − 0.295747i
\(83\) 4.36246e9 1.10749 0.553747 − 0.832685i \(-0.313198\pi\)
0.553747 + 0.832685i \(0.313198\pi\)
\(84\) 2.57980e8 + 1.77073e8i 0.0616865 + 0.0423405i
\(85\) 1.01134e9 + 1.01165e9i 0.227931 + 0.228001i
\(86\) 2.01191e8 + 2.18223e9i 0.0427676 + 0.463883i
\(87\) −8.68101e8 − 8.68101e8i −0.174170 − 0.174170i
\(88\) −1.35078e9 − 7.54849e8i −0.255959 − 0.143036i
\(89\) 1.08001e10 1.93409 0.967043 − 0.254611i \(-0.0819475\pi\)
0.967043 + 0.254611i \(0.0819475\pi\)
\(90\) −4.91007e8 − 5.33472e9i −0.0831525 − 0.903439i
\(91\) −1.76740e9 − 1.76740e9i −0.283222 − 0.283222i
\(92\) −4.10997e9 + 5.98787e9i −0.623591 + 0.908518i
\(93\) 8.12319e8 0.116765
\(94\) 7.12168e9 − 8.56821e9i 0.970383 − 1.16748i
\(95\) 3.68994e9 − 3.68881e9i 0.476872 − 0.476725i
\(96\) 2.22535e9 − 1.10165e9i 0.272925 − 0.135110i
\(97\) −1.02515e9 + 1.02515e9i −0.119380 + 0.119380i −0.764273 − 0.644893i \(-0.776902\pi\)
0.644893 + 0.764273i \(0.276902\pi\)
\(98\) 7.79759e8 + 8.45773e9i 0.0862641 + 0.935672i
\(99\) −1.78886e9 − 1.78886e9i −0.188105 − 0.188105i
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.53.4 yes 236
5.2 odd 4 80.11.i.a.37.56 yes 236
16.13 even 4 80.11.i.a.13.56 ✓ 236
80.77 odd 4 inner 80.11.t.a.77.4 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.56 ✓ 236 16.13 even 4
80.11.i.a.37.56 yes 236 5.2 odd 4
80.11.t.a.53.4 yes 236 1.1 even 1 trivial
80.11.t.a.77.4 yes 236 80.77 odd 4 inner