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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 13.62
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.62

$q$-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+(2.05133 + 31.9342i) q^{2} -192.036i q^{3} +(-1015.58 + 131.015i) q^{4} +(-193.255 - 3119.02i) q^{5} +(6132.51 - 393.928i) q^{6} +(-18555.5 - 18555.5i) q^{7} +(-6267.14 - 32163.1i) q^{8} +22171.2 q^{9} +(99206.9 - 12569.6i) q^{10} +(14188.4 - 14188.4i) q^{11} +(25159.6 + 195029. i) q^{12} -558513. i q^{13} +(554492. - 630619. i) q^{14} +(-598964. + 37111.9i) q^{15} +(1.01425e6 - 266113. i) q^{16} +(145210. - 145210. i) q^{17} +(45480.3 + 708018. i) q^{18} +(-115045. + 115045. i) q^{19} +(604904. + 3.14231e6i) q^{20} +(-3.56333e6 + 3.56333e6i) q^{21} +(482201. + 423990. i) q^{22} +(7.04161e6 - 7.04161e6i) q^{23} +(-6.17647e6 + 1.20352e6i) q^{24} +(-9.69093e6 + 1.20553e6i) q^{25} +(1.78357e7 - 1.14569e6i) q^{26} -1.55972e7i q^{27} +(2.12757e7 + 1.64136e7i) q^{28} +(-4.93278e6 + 4.93278e6i) q^{29} +(-2.41381e6 - 1.90513e7i) q^{30} -3.10514e7 q^{31} +(1.05787e7 + 3.18432e7i) q^{32} +(-2.72469e6 - 2.72469e6i) q^{33} +(4.93505e6 + 4.33930e6i) q^{34} +(-5.42891e7 + 6.14610e7i) q^{35} +(-2.25167e7 + 2.90475e6i) q^{36} -1.55549e7i q^{37} +(-3.90985e6 - 3.43786e6i) q^{38} -1.07255e8 q^{39} +(-9.91061e7 + 2.57630e7i) q^{40} +1.88678e8i q^{41} +(-1.21102e8 - 1.06482e8i) q^{42} -1.25422e6 q^{43} +(-1.25506e7 + 1.62684e7i) q^{44} +(-4.28469e6 - 6.91523e7i) q^{45} +(2.39313e8 + 2.10424e8i) q^{46} +(8.45693e7 - 8.45693e7i) q^{47} +(-5.11033e7 - 1.94772e8i) q^{48} +4.06140e8i q^{49} +(-5.83769e7 - 3.06999e8i) q^{50} +(-2.78856e7 - 2.78856e7i) q^{51} +(7.31735e7 + 5.67217e8i) q^{52} -351742. q^{53} +(4.98084e8 - 3.19949e7i) q^{54} +(-4.69959e7 - 4.15120e7i) q^{55} +(-4.80513e8 + 7.13093e8i) q^{56} +(2.20927e7 + 2.20927e7i) q^{57} +(-1.67643e8 - 1.47406e8i) q^{58} +(-1.96733e8 - 1.96733e8i) q^{59} +(6.03436e8 - 1.16163e8i) q^{60} +(-6.86172e8 - 6.86172e8i) q^{61} +(-6.36965e7 - 9.91600e8i) q^{62} +(-4.11398e8 - 4.11398e8i) q^{63} +(-9.95188e8 + 4.03141e8i) q^{64} +(-1.74201e9 + 1.07935e8i) q^{65} +(8.14214e7 - 9.25999e7i) q^{66} +7.27908e7 q^{67} +(-1.28449e8 + 1.66498e8i) q^{68} +(-1.35224e9 - 1.35224e9i) q^{69} +(-2.07407e9 - 1.60760e9i) q^{70} +1.31887e9i q^{71} +(-1.38950e8 - 7.13094e8i) q^{72} +(1.78479e9 - 1.78479e9i) q^{73} +(4.96734e8 - 3.19082e7i) q^{74} +(2.31505e8 + 1.86101e9i) q^{75} +(1.01765e8 - 1.31910e8i) q^{76} -5.26547e8 q^{77} +(-2.20014e8 - 3.42509e9i) q^{78} -4.94128e9i q^{79} +(-1.02602e9 - 3.11203e9i) q^{80} -1.68604e9 q^{81} +(-6.02528e9 + 3.87040e8i) q^{82} +6.37951e8i q^{83} +(3.15201e9 - 4.08571e9i) q^{84} +(-4.80976e8 - 4.24851e8i) q^{85} +(-2.57281e6 - 4.00525e7i) q^{86} +(9.47272e8 + 9.47272e8i) q^{87} +(-5.45264e8 - 3.67423e8i) q^{88} +7.78731e9 q^{89} +(2.19953e9 - 2.78682e8i) q^{90} +(-1.03635e10 + 1.03635e10i) q^{91} +(-6.22880e9 + 8.07391e9i) q^{92} +5.96298e9i q^{93} +(2.87413e9 + 2.52717e9i) q^{94} +(3.81059e8 + 3.36593e8i) q^{95} +(6.11505e9 - 2.03148e9i) q^{96} +(-1.05111e10 + 1.05111e10i) q^{97} +(-1.29697e10 + 8.33124e8i) q^{98} +(3.14574e8 - 3.14574e8i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.05133 + 31.9342i 0.0641039 + 0.997943i
\(3\) − 192.036i − 0.790272i −0.918623 − 0.395136i \(-0.870698\pi\)
0.918623 − 0.395136i \(-0.129302\pi\)
\(4\) −1015.58 + 131.015i −0.991781 + 0.127944i
\(5\) −193.255 − 3119.02i −0.0618416 − 0.998086i
\(6\) 6132.51 − 393.928i 0.788646 − 0.0506595i
\(7\) −18555.5 − 18555.5i −1.10404 − 1.10404i −0.993919 − 0.110117i \(-0.964878\pi\)
−0.110117 − 0.993919i \(-0.535122\pi\)
\(8\) −6267.14 − 32163.1i −0.191258 − 0.981540i
\(9\) 22171.2 0.375471
\(10\) 99206.9 − 12569.6i 0.992069 − 0.125696i
\(11\) 14188.4 − 14188.4i 0.0880989 − 0.0880989i −0.661684 − 0.749783i \(-0.730158\pi\)
0.749783 + 0.661684i \(0.230158\pi\)
\(12\) 25159.6 + 195029.i 0.101111 + 0.783777i
\(13\) − 558513.i − 1.50424i −0.659027 − 0.752119i \(-0.729032\pi\)
0.659027 − 0.752119i \(-0.270968\pi\)
\(14\) 554492. − 630619.i 1.03099 − 1.17254i
\(15\) −598964. + 37111.9i −0.788759 + 0.0488717i
\(16\) 1.01425e6 − 266113.i 0.967261 − 0.253785i
\(17\) 145210. − 145210.i 0.102271 − 0.102271i −0.654120 − 0.756391i \(-0.726961\pi\)
0.756391 + 0.654120i \(0.226961\pi\)
\(18\) 45480.3 + 708018.i 0.0240691 + 0.374699i
\(19\) −115045. + 115045.i −0.0464620 + 0.0464620i −0.729956 − 0.683494i \(-0.760460\pi\)
0.683494 + 0.729956i \(0.260460\pi\)
\(20\) 604904. + 3.14231e6i 0.189033 + 0.981971i
\(21\) −3.56333e6 + 3.56333e6i −0.872488 + 0.872488i
\(22\) 482201. + 423990.i 0.0935652 + 0.0822702i
\(23\) 7.04161e6 − 7.04161e6i 1.09404 − 1.09404i 0.0989468 − 0.995093i \(-0.468453\pi\)
0.995093 − 0.0989468i \(-0.0315474\pi\)
\(24\) −6.17647e6 + 1.20352e6i −0.775683 + 0.151146i
\(25\) −9.69093e6 + 1.20553e6i −0.992351 + 0.123446i
\(26\) 1.78357e7 − 1.14569e6i 1.50114 − 0.0964276i
\(27\) − 1.55972e7i − 1.08700i
\(28\) 2.12757e7 + 1.64136e7i 1.23622 + 0.953707i
\(29\) −4.93278e6 + 4.93278e6i −0.240493 + 0.240493i −0.817054 − 0.576561i \(-0.804394\pi\)
0.576561 + 0.817054i \(0.304394\pi\)
\(30\) −2.41381e6 − 1.90513e7i −0.0993337 − 0.784004i
\(31\) −3.10514e7 −1.08461 −0.542304 − 0.840183i \(-0.682448\pi\)
−0.542304 + 0.840183i \(0.682448\pi\)
\(32\) 1.05787e7 + 3.18432e7i 0.315268 + 0.949003i
\(33\) −2.72469e6 − 2.72469e6i −0.0696221 − 0.0696221i
\(34\) 4.93505e6 + 4.33930e6i 0.108617 + 0.0955048i
\(35\) −5.42891e7 + 6.14610e7i −1.03365 + 1.17020i
\(36\) −2.25167e7 + 2.90475e6i −0.372385 + 0.0480393i
\(37\) − 1.55549e7i − 0.224316i −0.993690 − 0.112158i \(-0.964224\pi\)
0.993690 − 0.112158i \(-0.0357762\pi\)
\(38\) −3.90985e6 − 3.43786e6i −0.0493449 − 0.0433881i
\(39\) −1.07255e8 −1.18876
\(40\) −9.91061e7 + 2.57630e7i −0.967833 + 0.251592i
\(41\) 1.88678e8i 1.62855i 0.580477 + 0.814277i \(0.302866\pi\)
−0.580477 + 0.814277i \(0.697134\pi\)
\(42\) −1.21102e8 − 1.06482e8i −0.926623 − 0.814763i
\(43\) −1.25422e6 −0.00853161 −0.00426581 − 0.999991i \(-0.501358\pi\)
−0.00426581 + 0.999991i \(0.501358\pi\)
\(44\) −1.25506e7 + 1.62684e7i −0.0761031 + 0.0986466i
\(45\) −4.28469e6 − 6.91523e7i −0.0232197 − 0.374752i
\(46\) 2.39313e8 + 2.10424e8i 1.16192 + 1.02166i
\(47\) 8.45693e7 − 8.45693e7i 0.368743 − 0.368743i −0.498276 − 0.867019i \(-0.666033\pi\)
0.867019 + 0.498276i \(0.166033\pi\)
\(48\) −5.11033e7 − 1.94772e8i −0.200559 − 0.764399i
\(49\) 4.06140e8i 1.43779i
\(50\) −5.83769e7 − 3.06999e8i −0.186806 − 0.982397i
\(51\) −2.78856e7 − 2.78856e7i −0.0808220 − 0.0808220i
\(52\) 7.31735e7 + 5.67217e8i 0.192459 + 1.49188i
\(53\) −351742. −0.000841096 −0.000420548 − 1.00000i \(-0.500134\pi\)
−0.000420548 1.00000i \(0.500134\pi\)
\(54\) 4.98084e8 − 3.19949e7i 1.08476 − 0.0696807i
\(55\) −4.69959e7 − 4.15120e7i −0.0933785 − 0.0824821i
\(56\) −4.80513e8 + 7.13093e8i −0.872499 + 1.29481i
\(57\) 2.20927e7 + 2.20927e7i 0.0367176 + 0.0367176i
\(58\) −1.67643e8 − 1.47406e8i −0.255415 − 0.224582i
\(59\) −1.96733e8 − 1.96733e8i −0.275180 − 0.275180i 0.556001 − 0.831182i \(-0.312335\pi\)
−0.831182 + 0.556001i \(0.812335\pi\)
\(60\) 6.03436e8 − 1.16163e8i 0.776024 − 0.149387i
\(61\) −6.86172e8 − 6.86172e8i −0.812426 − 0.812426i 0.172571 − 0.984997i \(-0.444793\pi\)
−0.984997 + 0.172571i \(0.944793\pi\)
\(62\) −6.36965e7 − 9.91600e8i −0.0695276 − 1.08238i
\(63\) −4.11398e8 − 4.11398e8i −0.414533 − 0.414533i
\(64\) −9.95188e8 + 4.03141e8i −0.926841 + 0.375455i
\(65\) −1.74201e9 + 1.07935e8i −1.50136 + 0.0930245i
\(66\) 8.14214e7 − 9.25999e7i 0.0650158 − 0.0739419i
\(67\) 7.27908e7 0.0539141 0.0269570 − 0.999637i \(-0.491418\pi\)
0.0269570 + 0.999637i \(0.491418\pi\)
\(68\) −1.28449e8 + 1.66498e8i −0.0883456 + 0.114516i
\(69\) −1.35224e9 − 1.35224e9i −0.864588 − 0.864588i
\(70\) −2.07407e9 − 1.60760e9i −1.23405 − 0.956507i
\(71\) 1.31887e9i 0.730988i 0.930814 + 0.365494i \(0.119100\pi\)
−0.930814 + 0.365494i \(0.880900\pi\)
\(72\) −1.38950e8 − 7.13094e8i −0.0718118 − 0.368539i
\(73\) 1.78479e9 − 1.78479e9i 0.860941 − 0.860941i −0.130507 − 0.991447i \(-0.541660\pi\)
0.991447 + 0.130507i \(0.0416604\pi\)
\(74\) 4.96734e8 − 3.19082e7i 0.223854 − 0.0143795i
\(75\) 2.31505e8 + 1.86101e9i 0.0975562 + 0.784227i
\(76\) 1.01765e8 − 1.31910e8i 0.0401356 − 0.0520247i
\(77\) −5.26547e8 −0.194529
\(78\) −2.20014e8 − 3.42509e9i −0.0762040 − 1.18631i
\(79\) − 4.94128e9i − 1.60585i −0.596083 − 0.802923i \(-0.703277\pi\)
0.596083 − 0.802923i \(-0.296723\pi\)
\(80\) −1.02602e9 − 3.11203e9i −0.313116 − 0.949715i
\(81\) −1.68604e9 −0.483551
\(82\) −6.02528e9 + 3.87040e8i −1.62520 + 0.104397i
\(83\) 6.37951e8i 0.161956i 0.996716 + 0.0809780i \(0.0258044\pi\)
−0.996716 + 0.0809780i \(0.974196\pi\)
\(84\) 3.15201e9 − 4.08571e9i 0.753687 − 0.976947i
\(85\) −4.80976e8 − 4.24851e8i −0.108400 − 0.0957508i
\(86\) −2.57281e6 − 4.00525e7i −0.000546910 − 0.00851406i
\(87\) 9.47272e8 + 9.47272e8i 0.190055 + 0.190055i
\(88\) −5.45264e8 − 3.67423e8i −0.103322 − 0.0696230i
\(89\) 7.78731e9 1.39456 0.697281 − 0.716798i \(-0.254393\pi\)
0.697281 + 0.716798i \(0.254393\pi\)
\(90\) 2.19953e9 − 2.78682e8i 0.372493 − 0.0471950i
\(91\) −1.03635e10 + 1.03635e10i −1.66073 + 1.66073i
\(92\) −6.22880e9 + 8.07391e9i −0.945072 + 1.22502i
\(93\) 5.96298e9i 0.857134i
\(94\) 2.87413e9 + 2.52717e9i 0.391622 + 0.344347i
\(95\) 3.81059e8 + 3.36593e8i 0.0492464 + 0.0434998i
\(96\) 6.11505e9 − 2.03148e9i 0.749970 − 0.249148i
\(97\) −1.05111e10 + 1.05111e10i −1.22403 + 1.22403i −0.257841 + 0.966187i \(0.583011\pi\)
−0.966187 + 0.257841i \(0.916989\pi\)
\(98\) −1.29697e10 + 8.33124e8i −1.43483 + 0.0921678i
\(99\) 3.14574e8 − 3.14574e8i 0.0330786 − 0.0330786i
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.13.62 ✓ 236
5.2 odd 4 80.11.t.a.77.2 yes 236
16.5 even 4 80.11.t.a.53.2 yes 236
80.37 odd 4 inner 80.11.i.a.37.62 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.62 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.62 yes 236 80.37 odd 4 inner
80.11.t.a.53.2 yes 236 16.5 even 4
80.11.t.a.77.2 yes 236 5.2 odd 4