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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(43,112)] 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("112.43"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.k (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: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 99.20
Character \(\chi\) \(=\) 112.99
Dual form 112.5.k.a.43.20

$q$-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+(-1.19668 - 3.81680i) q^{2} +(10.3892 - 10.3892i) q^{3} +(-13.1359 + 9.13497i) q^{4} +(9.65329 - 9.65329i) q^{5} +(-52.0860 - 27.2209i) q^{6} +18.5203 q^{7} +(50.5858 + 39.2055i) q^{8} -134.871i q^{9} +(-48.3966 - 25.2928i) q^{10} +(-135.330 - 135.330i) q^{11} +(-41.5666 + 231.376i) q^{12} +(-45.5832 - 45.5832i) q^{13} +(-22.1628 - 70.6881i) q^{14} -200.580i q^{15} +(89.1047 - 239.992i) q^{16} -50.1092 q^{17} +(-514.774 + 161.397i) q^{18} +(-10.6215 + 10.6215i) q^{19} +(-38.6223 + 214.987i) q^{20} +(192.410 - 192.410i) q^{21} +(-354.581 + 678.474i) q^{22} +434.248 q^{23} +(932.860 - 118.232i) q^{24} +438.628i q^{25} +(-119.434 + 228.531i) q^{26} +(-559.671 - 559.671i) q^{27} +(-243.281 + 169.182i) q^{28} +(467.158 + 467.158i) q^{29} +(-765.573 + 240.030i) q^{30} -830.849i q^{31} +(-1022.63 - 52.9009i) q^{32} -2811.94 q^{33} +(59.9646 + 191.257i) q^{34} +(178.781 - 178.781i) q^{35} +(1232.04 + 1771.65i) q^{36} +(-1421.60 + 1421.60i) q^{37} +(53.2506 + 27.8296i) q^{38} -947.146 q^{39} +(866.782 - 109.857i) q^{40} -985.146i q^{41} +(-964.646 - 504.139i) q^{42} +(1429.78 + 1429.78i) q^{43} +(3013.92 + 541.449i) q^{44} +(-1301.94 - 1301.94i) q^{45} +(-519.656 - 1657.44i) q^{46} -3399.25i q^{47} +(-1567.60 - 3419.05i) q^{48} +343.000 q^{49} +(1674.16 - 524.897i) q^{50} +(-520.593 + 520.593i) q^{51} +(1015.18 + 182.376i) q^{52} +(3183.56 - 3183.56i) q^{53} +(-1466.41 + 2805.90i) q^{54} -2612.76 q^{55} +(936.863 + 726.097i) q^{56} +220.697i q^{57} +(1224.01 - 2342.09i) q^{58} +(4144.96 + 4144.96i) q^{59} +(1832.29 + 2634.80i) q^{60} +(1014.38 + 1014.38i) q^{61} +(-3171.19 + 994.260i) q^{62} -2497.84i q^{63} +(1021.85 + 3966.49i) q^{64} -880.056 q^{65} +(3364.99 + 10732.6i) q^{66} +(1042.33 - 1042.33i) q^{67} +(658.230 - 457.746i) q^{68} +(4511.49 - 4511.49i) q^{69} +(-896.317 - 468.429i) q^{70} +3782.81 q^{71} +(5287.67 - 6822.54i) q^{72} -7864.12i q^{73} +(7127.15 + 3724.76i) q^{74} +(4556.99 + 4556.99i) q^{75} +(42.4961 - 236.550i) q^{76} +(-2506.35 - 2506.35i) q^{77} +(1133.43 + 3615.07i) q^{78} -3923.58i q^{79} +(-1456.56 - 3176.87i) q^{80} -704.549 q^{81} +(-3760.10 + 1178.90i) q^{82} +(-7983.39 + 7983.39i) q^{83} +(-769.825 + 4285.15i) q^{84} +(-483.718 + 483.718i) q^{85} +(3746.20 - 7168.18i) q^{86} +9706.79 q^{87} +(-1540.09 - 12151.5i) q^{88} -4273.58i q^{89} +(-3411.25 + 6527.27i) q^{90} +(-844.214 - 844.214i) q^{91} +(-5704.25 + 3966.84i) q^{92} +(-8631.85 - 8631.85i) q^{93} +(-12974.3 + 4067.81i) q^{94} +205.065i q^{95} +(-11173.9 + 10074.7i) q^{96} -10930.6 q^{97} +(-410.461 - 1309.16i) q^{98} +(-18252.0 + 18252.0i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 6 q^{4} - 132 q^{6} - 200 q^{10} - 96 q^{11} + 660 q^{12} - 294 q^{14} + 642 q^{16} - 810 q^{18} + 1408 q^{19} - 2604 q^{20} - 106 q^{22} - 1152 q^{23} + 3880 q^{24} + 6828 q^{26} + 3648 q^{27} - 864 q^{29}+ \cdots - 59552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.19668 − 3.81680i −0.299170 − 0.954200i
\(3\) 10.3892 − 10.3892i 1.15435 − 1.15435i 0.168684 − 0.985670i \(-0.446048\pi\)
0.985670 − 0.168684i \(-0.0539518\pi\)
\(4\) −13.1359 + 9.13497i −0.820995 + 0.570936i
\(5\) 9.65329 − 9.65329i 0.386132 − 0.386132i −0.487174 − 0.873305i \(-0.661972\pi\)
0.873305 + 0.487174i \(0.161972\pi\)
\(6\) −52.0860 − 27.2209i −1.44683 − 0.756137i
\(7\) 18.5203 0.377964
\(8\) 50.5858 + 39.2055i 0.790403 + 0.612587i
\(9\) − 134.871i − 1.66507i
\(10\) −48.3966 − 25.2928i −0.483966 − 0.252928i
\(11\) −135.330 − 135.330i −1.11843 − 1.11843i −0.991972 − 0.126458i \(-0.959639\pi\)
−0.126458 − 0.991972i \(-0.540361\pi\)
\(12\) −41.5666 + 231.376i −0.288657 + 1.60678i
\(13\) −45.5832 − 45.5832i −0.269723 − 0.269723i 0.559265 − 0.828989i \(-0.311083\pi\)
−0.828989 + 0.559265i \(0.811083\pi\)
\(14\) −22.1628 − 70.6881i −0.113076 − 0.360654i
\(15\) − 200.580i − 0.891465i
\(16\) 89.1047 − 239.992i 0.348065 − 0.937470i
\(17\) −50.1092 −0.173388 −0.0866940 − 0.996235i \(-0.527630\pi\)
−0.0866940 + 0.996235i \(0.527630\pi\)
\(18\) −514.774 + 161.397i −1.58881 + 0.498138i
\(19\) −10.6215 + 10.6215i −0.0294224 + 0.0294224i −0.721665 − 0.692243i \(-0.756623\pi\)
0.692243 + 0.721665i \(0.256623\pi\)
\(20\) −38.6223 + 214.987i −0.0965558 + 0.537468i
\(21\) 192.410 − 192.410i 0.436305 − 0.436305i
\(22\) −354.581 + 678.474i −0.732605 + 1.40181i
\(23\) 434.248 0.820885 0.410442 − 0.911887i \(-0.365374\pi\)
0.410442 + 0.911887i \(0.365374\pi\)
\(24\) 932.860 − 118.232i 1.61955 − 0.205264i
\(25\) 438.628i 0.701805i
\(26\) −119.434 + 228.531i −0.176677 + 0.338063i
\(27\) −559.671 − 559.671i −0.767725 − 0.767725i
\(28\) −243.281 + 169.182i −0.310307 + 0.215793i
\(29\) 467.158 + 467.158i 0.555479 + 0.555479i 0.928017 − 0.372538i \(-0.121512\pi\)
−0.372538 + 0.928017i \(0.621512\pi\)
\(30\) −765.573 + 240.030i −0.850636 + 0.266699i
\(31\) − 830.849i − 0.864567i −0.901738 − 0.432284i \(-0.857708\pi\)
0.901738 − 0.432284i \(-0.142292\pi\)
\(32\) −1022.63 − 52.9009i −0.998665 − 0.0516610i
\(33\) −2811.94 −2.58213
\(34\) 59.9646 + 191.257i 0.0518725 + 0.165447i
\(35\) 178.781 − 178.781i 0.145944 − 0.145944i
\(36\) 1232.04 + 1771.65i 0.950647 + 1.36701i
\(37\) −1421.60 + 1421.60i −1.03842 + 1.03842i −0.0391894 + 0.999232i \(0.512478\pi\)
−0.999232 + 0.0391894i \(0.987522\pi\)
\(38\) 53.2506 + 27.8296i 0.0368772 + 0.0192726i
\(39\) −947.146 −0.622713
\(40\) 866.782 − 109.857i 0.541739 − 0.0686607i
\(41\) − 985.146i − 0.586048i −0.956105 − 0.293024i \(-0.905338\pi\)
0.956105 − 0.293024i \(-0.0946615\pi\)
\(42\) −964.646 − 504.139i −0.546851 − 0.285793i
\(43\) 1429.78 + 1429.78i 0.773273 + 0.773273i 0.978677 − 0.205405i \(-0.0658510\pi\)
−0.205405 + 0.978677i \(0.565851\pi\)
\(44\) 3013.92 + 541.449i 1.55678 + 0.279674i
\(45\) −1301.94 − 1301.94i −0.642935 − 0.642935i
\(46\) −519.656 − 1657.44i −0.245584 − 0.783288i
\(47\) − 3399.25i − 1.53882i −0.638756 − 0.769409i \(-0.720551\pi\)
0.638756 − 0.769409i \(-0.279449\pi\)
\(48\) −1567.60 − 3419.05i −0.680382 − 1.48396i
\(49\) 343.000 0.142857
\(50\) 1674.16 − 524.897i 0.669662 − 0.209959i
\(51\) −520.593 + 520.593i −0.200151 + 0.200151i
\(52\) 1015.18 + 182.376i 0.375436 + 0.0674468i
\(53\) 3183.56 − 3183.56i 1.13334 − 1.13334i 0.143726 − 0.989618i \(-0.454092\pi\)
0.989618 − 0.143726i \(-0.0459084\pi\)
\(54\) −1466.41 + 2805.90i −0.502883 + 0.962243i
\(55\) −2612.76 −0.863722
\(56\) 936.863 + 726.097i 0.298744 + 0.231536i
\(57\) 220.697i 0.0679278i
\(58\) 1224.01 − 2342.09i 0.363856 − 0.696221i
\(59\) 4144.96 + 4144.96i 1.19074 + 1.19074i 0.976860 + 0.213879i \(0.0686098\pi\)
0.213879 + 0.976860i \(0.431390\pi\)
\(60\) 1832.29 + 2634.80i 0.508969 + 0.731889i
\(61\) 1014.38 + 1014.38i 0.272610 + 0.272610i 0.830150 − 0.557540i \(-0.188255\pi\)
−0.557540 + 0.830150i \(0.688255\pi\)
\(62\) −3171.19 + 994.260i −0.824970 + 0.258652i
\(63\) − 2497.84i − 0.629337i
\(64\) 1021.85 + 3966.49i 0.249475 + 0.968381i
\(65\) −880.056 −0.208297
\(66\) 3364.99 + 10732.6i 0.772495 + 2.46387i
\(67\) 1042.33 − 1042.33i 0.232196 − 0.232196i −0.581413 − 0.813609i \(-0.697500\pi\)
0.813609 + 0.581413i \(0.197500\pi\)
\(68\) 658.230 − 457.746i 0.142351 − 0.0989934i
\(69\) 4511.49 − 4511.49i 0.947592 − 0.947592i
\(70\) −896.317 − 468.429i −0.182922 − 0.0955977i
\(71\) 3782.81 0.750408 0.375204 − 0.926942i \(-0.377573\pi\)
0.375204 + 0.926942i \(0.377573\pi\)
\(72\) 5287.67 − 6822.54i 1.02000 − 1.31608i
\(73\) − 7864.12i − 1.47572i −0.674953 − 0.737860i \(-0.735836\pi\)
0.674953 − 0.737860i \(-0.264164\pi\)
\(74\) 7127.15 + 3724.76i 1.30153 + 0.680197i
\(75\) 4556.99 + 4556.99i 0.810132 + 0.810132i
\(76\) 42.4961 − 236.550i 0.00735735 − 0.0409540i
\(77\) −2506.35 − 2506.35i −0.422727 − 0.422727i
\(78\) 1133.43 + 3615.07i 0.186297 + 0.594192i
\(79\) − 3923.58i − 0.628678i −0.949311 − 0.314339i \(-0.898217\pi\)
0.949311 − 0.314339i \(-0.101783\pi\)
\(80\) −1456.56 − 3176.87i −0.227588 − 0.496386i
\(81\) −704.549 −0.107384
\(82\) −3760.10 + 1178.90i −0.559207 + 0.175328i
\(83\) −7983.39 + 7983.39i −1.15886 + 1.15886i −0.174140 + 0.984721i \(0.555714\pi\)
−0.984721 + 0.174140i \(0.944286\pi\)
\(84\) −769.825 + 4285.15i −0.109102 + 0.607306i
\(85\) −483.718 + 483.718i −0.0669506 + 0.0669506i
\(86\) 3746.20 − 7168.18i 0.506517 − 0.969196i
\(87\) 9706.79 1.28244
\(88\) −1540.09 − 12151.5i −0.198876 − 1.56915i
\(89\) − 4273.58i − 0.539525i −0.962927 − 0.269763i \(-0.913055\pi\)
0.962927 − 0.269763i \(-0.0869452\pi\)
\(90\) −3411.25 + 6527.27i −0.421142 + 0.805836i
\(91\) −844.214 − 844.214i −0.101946 − 0.101946i
\(92\) −5704.25 + 3966.84i −0.673942 + 0.468672i
\(93\) −8631.85 − 8631.85i −0.998017 − 0.998017i
\(94\) −12974.3 + 4067.81i −1.46834 + 0.460368i
\(95\) 205.065i 0.0227219i
\(96\) −11173.9 + 10074.7i −1.21245 + 1.09318i
\(97\) −10930.6 −1.16172 −0.580858 − 0.814005i \(-0.697283\pi\)
−0.580858 + 0.814005i \(0.697283\pi\)
\(98\) −410.461 − 1309.16i −0.0427385 − 0.136314i
\(99\) −18252.0 + 18252.0i −1.86226 + 1.86226i
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.k.a.99.20 yes 96
4.3 odd 2 448.5.k.a.239.5 96
16.5 even 4 448.5.k.a.15.5 96
16.11 odd 4 inner 112.5.k.a.43.20 ✓ 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.20 ✓ 96 16.11 odd 4 inner
112.5.k.a.99.20 yes 96 1.1 even 1 trivial
448.5.k.a.15.5 96 16.5 even 4
448.5.k.a.239.5 96 4.3 odd 2