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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 43.20
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.20

$q$-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 \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

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

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 0.985670 + 0.168684i \(0.0539518\pi\)
0.168684 + 0.985670i \(0.446048\pi\)
\(4\) −13.1359 − 9.13497i −0.820995 − 0.570936i
\(5\) 9.65329 + 9.65329i 0.386132 + 0.386132i 0.873305 − 0.487174i \(-0.161972\pi\)
−0.487174 + 0.873305i \(0.661972\pi\)
\(6\) −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 −0.126458 + 0.991972i \(0.540361\pi\)
−0.991972 + 0.126458i \(0.959639\pi\)
\(12\) −41.5666 − 231.376i −0.288657 − 1.60678i
\(13\) −45.5832 + 45.5832i −0.269723 + 0.269723i −0.828989 − 0.559265i \(-0.811083\pi\)
0.559265 + 0.828989i \(0.311083\pi\)
\(14\) −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 0.692243 − 0.721665i \(-0.256623\pi\)
−0.721665 + 0.692243i \(0.756623\pi\)
\(20\) −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 −0.372538 − 0.928017i \(-0.621512\pi\)
0.928017 + 0.372538i \(0.121512\pi\)
\(30\) −765.573 − 240.030i −0.850636 − 0.266699i
\(31\) 830.849i 0.864567i 0.901738 + 0.432284i \(0.142292\pi\)
−0.901738 + 0.432284i \(0.857708\pi\)
\(32\) −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 −0.999232 − 0.0391894i \(-0.987522\pi\)
−0.0391894 − 0.999232i \(-0.512478\pi\)
\(38\) 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 \(0.0946615\pi\)
−0.956105 + 0.293024i \(0.905338\pi\)
\(42\) −964.646 + 504.139i −0.546851 + 0.285793i
\(43\) 1429.78 − 1429.78i 0.773273 − 0.773273i −0.205405 − 0.978677i \(-0.565851\pi\)
0.978677 + 0.205405i \(0.0658510\pi\)
\(44\) 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 \(0.279449\pi\)
−0.638756 + 0.769409i \(0.720551\pi\)
\(48\) −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 0.989618 + 0.143726i \(0.0459084\pi\)
0.143726 + 0.989618i \(0.454092\pi\)
\(54\) −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 0.213879 − 0.976860i \(-0.431390\pi\)
0.976860 − 0.213879i \(-0.0686098\pi\)
\(60\) 1832.29 − 2634.80i 0.508969 − 0.731889i
\(61\) 1014.38 − 1014.38i 0.272610 − 0.272610i −0.557540 − 0.830150i \(-0.688255\pi\)
0.830150 + 0.557540i \(0.188255\pi\)
\(62\) −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 0.813609 − 0.581413i \(-0.197500\pi\)
−0.581413 + 0.813609i \(0.697500\pi\)
\(68\) 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 \(0.264164\pi\)
−0.674953 + 0.737860i \(0.735836\pi\)
\(74\) 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 \(0.101783\pi\)
−0.949311 + 0.314339i \(0.898217\pi\)
\(80\) −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 −0.984721 − 0.174140i \(-0.944286\pi\)
−0.174140 − 0.984721i \(-0.555714\pi\)
\(84\) −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 \(0.0869452\pi\)
−0.962927 + 0.269763i \(0.913055\pi\)
\(90\) −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.43.20 ✓ 96
4.3 odd 2 448.5.k.a.15.5 96
16.3 odd 4 inner 112.5.k.a.99.20 yes 96
16.13 even 4 448.5.k.a.239.5 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.20 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.20 yes 96 16.3 odd 4 inner
448.5.k.a.15.5 96 4.3 odd 2
448.5.k.a.239.5 96 16.13 even 4