Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(79,112)] 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("112.79"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 2])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5 x^{9} + 485 x^{8} - 1910 x^{7} + 81837 x^{6} - 238847 x^{5} + 5758115 x^{4} + \cdots + 1406445775 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 95.1
Root \(0.500000 - 13.8476i\) of defining polynomial
Character \(\chi\) \(=\) 112.95
Dual form 112.5.r.b.79.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-11.2423 - 6.49076i) q^{3} +(6.51387 + 11.2823i) q^{5} +(-44.8395 + 19.7590i) q^{7} +(43.7600 + 75.7946i) q^{9} +(-13.1019 - 7.56440i) q^{11} +144.884 q^{13} -169.120i q^{15} +(-41.9421 + 72.6458i) q^{17} +(265.488 - 153.280i) q^{19} +(632.352 + 68.9052i) q^{21} +(763.017 - 440.528i) q^{23} +(227.639 - 394.282i) q^{25} -84.6402i q^{27} +898.827 q^{29} +(-7.68271 - 4.43562i) q^{31} +(98.1974 + 170.083i) q^{33} +(-515.007 - 377.187i) q^{35} +(1294.62 + 2242.34i) q^{37} +(-1628.83 - 940.407i) q^{39} -2609.52 q^{41} +1788.64i q^{43} +(-570.094 + 987.432i) q^{45} +(-1398.45 + 807.394i) q^{47} +(1620.16 - 1771.97i) q^{49} +(943.054 - 544.473i) q^{51} +(1404.94 - 2433.43i) q^{53} -197.094i q^{55} -3979.61 q^{57} +(5606.24 + 3236.76i) q^{59} +(-2932.57 - 5079.35i) q^{61} +(-3459.80 - 2533.94i) q^{63} +(943.755 + 1634.63i) q^{65} +(3431.51 + 1981.18i) q^{67} -11437.4 q^{69} +2014.42i q^{71} +(-666.804 + 1154.94i) q^{73} +(-5118.39 + 2955.10i) q^{75} +(736.949 + 80.3028i) q^{77} +(4111.63 - 2373.85i) q^{79} +(2995.18 - 5187.81i) q^{81} +5679.48i q^{83} -1092.82 q^{85} +(-10104.9 - 5834.07i) q^{87} +(4633.28 + 8025.07i) q^{89} +(-6496.52 + 2862.76i) q^{91} +(57.5811 + 99.7334i) q^{93} +(3458.71 + 1996.89i) q^{95} +2823.42 q^{97} -1324.07i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 9 q^{3} + 9 q^{5} + 10 q^{7} + 74 q^{9} + 189 q^{11} + 84 q^{13} - 435 q^{17} + 357 q^{19} - 61 q^{21} + 1269 q^{23} - 776 q^{25} + 660 q^{29} + 969 q^{31} + 1759 q^{33} - 1521 q^{35} - 583 q^{37}+ \cdots - 43356 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −11.2423 − 6.49076i −1.24915 − 0.721196i −0.278208 − 0.960521i \(-0.589741\pi\)
−0.970940 + 0.239325i \(0.923074\pi\)
\(4\) 0 0
\(5\) 6.51387 + 11.2823i 0.260555 + 0.451294i 0.966389 − 0.257083i \(-0.0827612\pi\)
−0.705835 + 0.708377i \(0.749428\pi\)
\(6\) 0 0
\(7\) −44.8395 + 19.7590i −0.915092 + 0.403245i
\(8\) 0 0
\(9\) 43.7600 + 75.7946i 0.540247 + 0.935736i
\(10\) 0 0
\(11\) −13.1019 − 7.56440i −0.108280 − 0.0625157i 0.444882 − 0.895589i \(-0.353246\pi\)
−0.553162 + 0.833074i \(0.686579\pi\)
\(12\) 0 0
\(13\) 144.884 0.857301 0.428651 − 0.903470i \(-0.358989\pi\)
0.428651 + 0.903470i \(0.358989\pi\)
\(14\) 0 0
\(15\) − 169.120i − 0.751644i
\(16\) 0 0
\(17\) −41.9421 + 72.6458i −0.145128 + 0.251370i −0.929421 − 0.369022i \(-0.879693\pi\)
0.784292 + 0.620391i \(0.213026\pi\)
\(18\) 0 0
\(19\) 265.488 − 153.280i 0.735425 − 0.424598i −0.0849786 − 0.996383i \(-0.527082\pi\)
0.820403 + 0.571785i \(0.193749\pi\)
\(20\) 0 0
\(21\) 632.352 + 68.9052i 1.43390 + 0.156248i
\(22\) 0 0
\(23\) 763.017 − 440.528i 1.44238 − 0.832756i 0.444367 − 0.895845i \(-0.353428\pi\)
0.998008 + 0.0630888i \(0.0200951\pi\)
\(24\) 0 0
\(25\) 227.639 − 394.282i 0.364222 − 0.630852i
\(26\) 0 0
\(27\) − 84.6402i − 0.116105i
\(28\) 0 0
\(29\) 898.827 1.06876 0.534380 − 0.845244i \(-0.320545\pi\)
0.534380 + 0.845244i \(0.320545\pi\)
\(30\) 0 0
\(31\) −7.68271 − 4.43562i −0.00799450 − 0.00461563i 0.495997 − 0.868324i \(-0.334803\pi\)
−0.503992 + 0.863708i \(0.668136\pi\)
\(32\) 0 0
\(33\) 98.1974 + 170.083i 0.0901721 + 0.156183i
\(34\) 0 0
\(35\) −515.007 − 377.187i −0.420414 − 0.307908i
\(36\) 0 0
\(37\) 1294.62 + 2242.34i 0.945665 + 1.63794i 0.754413 + 0.656400i \(0.227921\pi\)
0.191252 + 0.981541i \(0.438745\pi\)
\(38\) 0 0
\(39\) −1628.83 − 940.407i −1.07090 − 0.618282i
\(40\) 0 0
\(41\) −2609.52 −1.55236 −0.776182 − 0.630509i \(-0.782846\pi\)
−0.776182 + 0.630509i \(0.782846\pi\)
\(42\) 0 0
\(43\) 1788.64i 0.967354i 0.875247 + 0.483677i \(0.160699\pi\)
−0.875247 + 0.483677i \(0.839301\pi\)
\(44\) 0 0
\(45\) −570.094 + 987.432i −0.281528 + 0.487621i
\(46\) 0 0
\(47\) −1398.45 + 807.394i −0.633068 + 0.365502i −0.781939 − 0.623355i \(-0.785769\pi\)
0.148871 + 0.988857i \(0.452436\pi\)
\(48\) 0 0
\(49\) 1620.16 − 1771.97i 0.674787 − 0.738013i
\(50\) 0 0
\(51\) 943.054 − 544.473i 0.362574 − 0.209332i
\(52\) 0 0
\(53\) 1404.94 − 2433.43i 0.500158 − 0.866299i −0.499842 − 0.866116i \(-0.666609\pi\)
1.00000 0.000182194i \(-5.79940e-5\pi\)
\(54\) 0 0
\(55\) − 197.094i − 0.0651550i
\(56\) 0 0
\(57\) −3979.61 −1.22487
\(58\) 0 0
\(59\) 5606.24 + 3236.76i 1.61053 + 0.929837i 0.989248 + 0.146247i \(0.0467194\pi\)
0.621278 + 0.783590i \(0.286614\pi\)
\(60\) 0 0
\(61\) −2932.57 − 5079.35i −0.788112 − 1.36505i −0.927122 − 0.374760i \(-0.877725\pi\)
0.139010 − 0.990291i \(-0.455608\pi\)
\(62\) 0 0
\(63\) −3459.80 − 2533.94i −0.871707 − 0.638432i
\(64\) 0 0
\(65\) 943.755 + 1634.63i 0.223374 + 0.386895i
\(66\) 0 0
\(67\) 3431.51 + 1981.18i 0.764427 + 0.441342i 0.830883 − 0.556447i \(-0.187836\pi\)
−0.0664561 + 0.997789i \(0.521169\pi\)
\(68\) 0 0
\(69\) −11437.4 −2.40232
\(70\) 0 0
\(71\) 2014.42i 0.399607i 0.979836 + 0.199804i \(0.0640304\pi\)
−0.979836 + 0.199804i \(0.935970\pi\)
\(72\) 0 0
\(73\) −666.804 + 1154.94i −0.125127 + 0.216727i −0.921783 − 0.387707i \(-0.873267\pi\)
0.796655 + 0.604434i \(0.206601\pi\)
\(74\) 0 0
\(75\) −5118.39 + 2955.10i −0.909936 + 0.525352i
\(76\) 0 0
\(77\) 736.949 + 80.3028i 0.124296 + 0.0135441i
\(78\) 0 0
\(79\) 4111.63 − 2373.85i 0.658809 − 0.380364i −0.133014 − 0.991114i \(-0.542465\pi\)
0.791823 + 0.610750i \(0.209132\pi\)
\(80\) 0 0
\(81\) 2995.18 − 5187.81i 0.456513 − 0.790704i
\(82\) 0 0
\(83\) 5679.48i 0.824427i 0.911087 + 0.412214i \(0.135244\pi\)
−0.911087 + 0.412214i \(0.864756\pi\)
\(84\) 0 0
\(85\) −1092.82 −0.151256
\(86\) 0 0
\(87\) −10104.9 − 5834.07i −1.33504 − 0.770785i
\(88\) 0 0
\(89\) 4633.28 + 8025.07i 0.584936 + 1.01314i 0.994883 + 0.101030i \(0.0322137\pi\)
−0.409948 + 0.912109i \(0.634453\pi\)
\(90\) 0 0
\(91\) −6496.52 + 2862.76i −0.784510 + 0.345703i
\(92\) 0 0
\(93\) 57.5811 + 99.7334i 0.00665754 + 0.0115312i
\(94\) 0 0
\(95\) 3458.71 + 1996.89i 0.383237 + 0.221262i
\(96\) 0 0
\(97\) 2823.42 0.300076 0.150038 − 0.988680i \(-0.452060\pi\)
0.150038 + 0.988680i \(0.452060\pi\)
\(98\) 0 0
\(99\) − 1324.07i − 0.135096i
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 112.5.r.b.95.1 yes 10
4.3 odd 2 112.5.r.a.95.5 yes 10
7.2 even 3 112.5.r.a.79.5 ✓ 10
7.3 odd 6 784.5.d.l.687.2 10
7.4 even 3 784.5.d.k.687.9 10
28.3 even 6 784.5.d.l.687.9 10
28.11 odd 6 784.5.d.k.687.2 10
28.23 odd 6 inner 112.5.r.b.79.1 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.r.a.79.5 ✓ 10 7.2 even 3
112.5.r.a.95.5 yes 10 4.3 odd 2
112.5.r.b.79.1 yes 10 28.23 odd 6 inner
112.5.r.b.95.1 yes 10 1.1 even 1 trivial
784.5.d.k.687.2 10 28.11 odd 6
784.5.d.k.687.9 10 7.4 even 3
784.5.d.l.687.2 10 7.3 odd 6
784.5.d.l.687.9 10 28.3 even 6