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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 37.62
Character \(\chi\) \(=\) 80.37
Dual form 80.11.i.a.13.62

$q$-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 \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.05133 − 31.9342i 0.0641039 − 0.997943i
\(3\) 192.036i 0.790272i 0.918623 + 0.395136i \(0.129302\pi\)
−0.918623 + 0.395136i \(0.870698\pi\)
\(4\) −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 −0.110117 + 0.993919i \(0.535122\pi\)
−0.993919 + 0.110117i \(0.964878\pi\)
\(8\) −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 0.749783 − 0.661684i \(-0.230158\pi\)
−0.661684 + 0.749783i \(0.730158\pi\)
\(12\) 25159.6 − 195029.i 0.101111 − 0.783777i
\(13\) 558513.i 1.50424i 0.659027 + 0.752119i \(0.270968\pi\)
−0.659027 + 0.752119i \(0.729032\pi\)
\(14\) 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 0.756391 − 0.654120i \(-0.226961\pi\)
−0.654120 + 0.756391i \(0.726961\pi\)
\(18\) 45480.3 − 708018.i 0.0240691 − 0.374699i
\(19\) −115045. − 115045.i −0.0464620 − 0.0464620i 0.683494 − 0.729956i \(-0.260460\pi\)
−0.729956 + 0.683494i \(0.760460\pi\)
\(20\) 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 0.995093 + 0.0989468i \(0.0315474\pi\)
0.0989468 + 0.995093i \(0.468453\pi\)
\(24\) −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 0.576561 − 0.817054i \(-0.304394\pi\)
−0.817054 + 0.576561i \(0.804394\pi\)
\(30\) −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 \(0.0357762\pi\)
−0.993690 + 0.112158i \(0.964224\pi\)
\(38\) −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 \(-0.697134\pi\)
0.580477 − 0.814277i \(-0.302866\pi\)
\(42\) −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 0.867019 − 0.498276i \(-0.166033\pi\)
−0.498276 + 0.867019i \(0.666033\pi\)
\(48\) −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 −0.831182 − 0.556001i \(-0.812335\pi\)
0.556001 + 0.831182i \(0.312335\pi\)
\(60\) 6.03436e8 + 1.16163e8i 0.776024 + 0.149387i
\(61\) −6.86172e8 + 6.86172e8i −0.812426 + 0.812426i −0.984997 − 0.172571i \(-0.944793\pi\)
0.172571 + 0.984997i \(0.444793\pi\)
\(62\) −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 \(-0.880900\pi\)
0.930814 − 0.365494i \(-0.119100\pi\)
\(72\) −1.38950e8 + 7.13094e8i −0.0718118 + 0.368539i
\(73\) 1.78479e9 + 1.78479e9i 0.860941 + 0.860941i 0.991447 − 0.130507i \(-0.0416604\pi\)
−0.130507 + 0.991447i \(0.541660\pi\)
\(74\) 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 \(0.296723\pi\)
−0.596083 + 0.802923i \(0.703277\pi\)
\(80\) −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 \(-0.974196\pi\)
0.996716 − 0.0809780i \(-0.0258044\pi\)
\(84\) 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 −0.966187 − 0.257841i \(-0.916989\pi\)
−0.257841 − 0.966187i \(-0.583011\pi\)
\(98\) −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.37.62 yes 236
5.3 odd 4 80.11.t.a.53.2 yes 236
16.13 even 4 80.11.t.a.77.2 yes 236
80.13 odd 4 inner 80.11.i.a.13.62 ✓ 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.62 ✓ 236 80.13 odd 4 inner
80.11.i.a.37.62 yes 236 1.1 even 1 trivial
80.11.t.a.53.2 yes 236 5.3 odd 4
80.11.t.a.77.2 yes 236 16.13 even 4