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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 43.19
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.19

$q$-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+(-1.20482 - 3.81424i) q^{2} +(-12.1580 - 12.1580i) q^{3} +(-13.0968 + 9.19093i) q^{4} +(7.84203 + 7.84203i) q^{5} +(-31.7254 + 61.0217i) q^{6} +18.5203 q^{7} +(50.8357 + 38.8810i) q^{8} +214.634i q^{9} +(20.4632 - 39.3596i) q^{10} +(54.7379 - 54.7379i) q^{11} +(270.975 + 47.4880i) q^{12} +(-124.399 + 124.399i) q^{13} +(-22.3135 - 70.6407i) q^{14} -190.687i q^{15} +(87.0537 - 240.744i) q^{16} -250.724 q^{17} +(818.667 - 258.595i) q^{18} +(344.113 + 344.113i) q^{19} +(-174.781 - 30.6302i) q^{20} +(-225.169 - 225.169i) q^{21} +(-274.732 - 142.834i) q^{22} -349.713 q^{23} +(-145.345 - 1090.78i) q^{24} -502.005i q^{25} +(624.364 + 324.608i) q^{26} +(1624.73 - 1624.73i) q^{27} +(-242.557 + 170.218i) q^{28} +(469.469 - 469.469i) q^{29} +(-727.325 + 229.743i) q^{30} +1382.79i q^{31} +(-1023.14 - 41.9911i) q^{32} -1331.01 q^{33} +(302.077 + 956.322i) q^{34} +(145.236 + 145.236i) q^{35} +(-1972.69 - 2811.03i) q^{36} +(1278.46 + 1278.46i) q^{37} +(897.936 - 1727.12i) q^{38} +3024.88 q^{39} +(93.7487 + 703.561i) q^{40} +1637.67i q^{41} +(-587.562 + 1130.14i) q^{42} +(1498.53 - 1498.53i) q^{43} +(-213.801 + 1219.98i) q^{44} +(-1683.17 + 1683.17i) q^{45} +(421.341 + 1333.89i) q^{46} -808.118i q^{47} +(-3985.37 + 1868.57i) q^{48} +343.000 q^{49} +(-1914.77 + 604.825i) q^{50} +(3048.31 + 3048.31i) q^{51} +(485.889 - 2772.57i) q^{52} +(3171.86 + 3171.86i) q^{53} +(-8154.60 - 4239.60i) q^{54} +858.512 q^{55} +(941.490 + 720.087i) q^{56} -8367.47i q^{57} +(-2356.29 - 1225.04i) q^{58} +(1831.24 - 1831.24i) q^{59} +(1752.59 + 2497.39i) q^{60} +(-2225.77 + 2225.77i) q^{61} +(5274.28 - 1666.01i) q^{62} +3975.08i q^{63} +(1072.53 + 3953.09i) q^{64} -1951.08 q^{65} +(1603.62 + 5076.78i) q^{66} +(-3652.26 - 3652.26i) q^{67} +(3283.69 - 2304.39i) q^{68} +(4251.81 + 4251.81i) q^{69} +(378.983 - 728.950i) q^{70} +479.239 q^{71} +(-8345.21 + 10911.1i) q^{72} +602.260i q^{73} +(3336.03 - 6416.64i) q^{74} +(-6103.38 + 6103.38i) q^{75} +(-7669.51 - 1344.07i) q^{76} +(1013.76 - 1013.76i) q^{77} +(-3644.43 - 11537.6i) q^{78} +9372.00i q^{79} +(2570.60 - 1205.24i) q^{80} -22121.5 q^{81} +(6246.45 - 1973.09i) q^{82} +(3240.24 + 3240.24i) q^{83} +(5018.52 + 879.490i) q^{84} +(-1966.19 - 1966.19i) q^{85} +(-7521.20 - 3910.29i) q^{86} -11415.6 q^{87} +(4910.90 - 654.372i) q^{88} -904.032i q^{89} +(8447.92 + 4392.10i) q^{90} +(-2303.90 + 2303.90i) q^{91} +(4580.13 - 3214.19i) q^{92} +(16811.9 - 16811.9i) q^{93} +(-3082.36 + 973.636i) q^{94} +5397.09i q^{95} +(11928.8 + 12949.9i) q^{96} -2492.47 q^{97} +(-413.253 - 1308.28i) q^{98} +(11748.6 + 11748.6i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.20482 − 3.81424i −0.301205 − 0.953560i
\(3\) −12.1580 − 12.1580i −1.35089 − 1.35089i −0.884668 − 0.466222i \(-0.845615\pi\)
−0.466222 − 0.884668i \(-0.654385\pi\)
\(4\) −13.0968 + 9.19093i −0.818552 + 0.574433i
\(5\) 7.84203 + 7.84203i 0.313681 + 0.313681i 0.846334 − 0.532653i \(-0.178805\pi\)
−0.532653 + 0.846334i \(0.678805\pi\)
\(6\) −31.7254 + 61.0217i −0.881260 + 1.69505i
\(7\) 18.5203 0.377964
\(8\) 50.8357 + 38.8810i 0.794307 + 0.607516i
\(9\) 214.634i 2.64981i
\(10\) 20.4632 − 39.3596i 0.204632 − 0.393596i
\(11\) 54.7379 − 54.7379i 0.452379 − 0.452379i −0.443764 − 0.896144i \(-0.646357\pi\)
0.896144 + 0.443764i \(0.146357\pi\)
\(12\) 270.975 + 47.4880i 1.88177 + 0.329778i
\(13\) −124.399 + 124.399i −0.736087 + 0.736087i −0.971818 − 0.235731i \(-0.924251\pi\)
0.235731 + 0.971818i \(0.424251\pi\)
\(14\) −22.3135 − 70.6407i −0.113845 − 0.360412i
\(15\) − 190.687i − 0.847498i
\(16\) 87.0537 − 240.744i 0.340054 − 0.940406i
\(17\) −250.724 −0.867558 −0.433779 − 0.901019i \(-0.642820\pi\)
−0.433779 + 0.901019i \(0.642820\pi\)
\(18\) 818.667 − 258.595i 2.52675 − 0.798134i
\(19\) 344.113 + 344.113i 0.953223 + 0.953223i 0.998954 − 0.0457312i \(-0.0145618\pi\)
−0.0457312 + 0.998954i \(0.514562\pi\)
\(20\) −174.781 − 30.6302i −0.436953 − 0.0765755i
\(21\) −225.169 − 225.169i −0.510588 − 0.510588i
\(22\) −274.732 − 142.834i −0.567629 − 0.295112i
\(23\) −349.713 −0.661083 −0.330542 − 0.943791i \(-0.607231\pi\)
−0.330542 + 0.943791i \(0.607231\pi\)
\(24\) −145.345 − 1090.78i −0.252335 − 1.89371i
\(25\) − 502.005i − 0.803208i
\(26\) 624.364 + 324.608i 0.923615 + 0.480190i
\(27\) 1624.73 − 1624.73i 2.22871 − 2.22871i
\(28\) −242.557 + 170.218i −0.309383 + 0.217115i
\(29\) 469.469 − 469.469i 0.558227 − 0.558227i −0.370575 − 0.928802i \(-0.620840\pi\)
0.928802 + 0.370575i \(0.120840\pi\)
\(30\) −727.325 + 229.743i −0.808139 + 0.255270i
\(31\) 1382.79i 1.43890i 0.694543 + 0.719452i \(0.255607\pi\)
−0.694543 + 0.719452i \(0.744393\pi\)
\(32\) −1023.14 − 41.9911i −0.999159 − 0.0410069i
\(33\) −1331.01 −1.22223
\(34\) 302.077 + 956.322i 0.261312 + 0.827268i
\(35\) 145.236 + 145.236i 0.118560 + 0.118560i
\(36\) −1972.69 − 2811.03i −1.52214 − 2.16900i
\(37\) 1278.46 + 1278.46i 0.933861 + 0.933861i 0.997945 − 0.0640831i \(-0.0204123\pi\)
−0.0640831 + 0.997945i \(0.520412\pi\)
\(38\) 897.936 − 1727.12i 0.621840 − 1.19607i
\(39\) 3024.88 1.98874
\(40\) 93.7487 + 703.561i 0.0585930 + 0.439726i
\(41\) 1637.67i 0.974222i 0.873340 + 0.487111i \(0.161949\pi\)
−0.873340 + 0.487111i \(0.838051\pi\)
\(42\) −587.562 + 1130.14i −0.333085 + 0.640668i
\(43\) 1498.53 − 1498.53i 0.810453 − 0.810453i −0.174249 − 0.984702i \(-0.555750\pi\)
0.984702 + 0.174249i \(0.0557496\pi\)
\(44\) −213.801 + 1219.98i −0.110434 + 0.630157i
\(45\) −1683.17 + 1683.17i −0.831195 + 0.831195i
\(46\) 421.341 + 1333.89i 0.199121 + 0.630382i
\(47\) − 808.118i − 0.365830i −0.983129 − 0.182915i \(-0.941447\pi\)
0.983129 − 0.182915i \(-0.0585533\pi\)
\(48\) −3985.37 + 1868.57i −1.72976 + 0.811010i
\(49\) 343.000 0.142857
\(50\) −1914.77 + 604.825i −0.765907 + 0.241930i
\(51\) 3048.31 + 3048.31i 1.17198 + 1.17198i
\(52\) 485.889 − 2772.57i 0.179693 − 1.02536i
\(53\) 3171.86 + 3171.86i 1.12918 + 1.12918i 0.990311 + 0.138867i \(0.0443460\pi\)
0.138867 + 0.990311i \(0.455654\pi\)
\(54\) −8154.60 − 4239.60i −2.79650 − 1.45391i
\(55\) 858.512 0.283806
\(56\) 941.490 + 720.087i 0.300220 + 0.229619i
\(57\) − 8367.47i − 2.57540i
\(58\) −2356.29 − 1225.04i −0.700443 − 0.364162i
\(59\) 1831.24 − 1831.24i 0.526067 − 0.526067i −0.393330 − 0.919397i \(-0.628677\pi\)
0.919397 + 0.393330i \(0.128677\pi\)
\(60\) 1752.59 + 2497.39i 0.486831 + 0.693721i
\(61\) −2225.77 + 2225.77i −0.598165 + 0.598165i −0.939824 − 0.341659i \(-0.889011\pi\)
0.341659 + 0.939824i \(0.389011\pi\)
\(62\) 5274.28 − 1666.01i 1.37208 − 0.433404i
\(63\) 3975.08i 1.00153i
\(64\) 1072.53 + 3953.09i 0.261849 + 0.965109i
\(65\) −1951.08 −0.461793
\(66\) 1603.62 + 5076.78i 0.368141 + 1.16547i
\(67\) −3652.26 − 3652.26i −0.813603 − 0.813603i 0.171569 − 0.985172i \(-0.445116\pi\)
−0.985172 + 0.171569i \(0.945116\pi\)
\(68\) 3283.69 − 2304.39i 0.710141 − 0.498354i
\(69\) 4251.81 + 4251.81i 0.893051 + 0.893051i
\(70\) 378.983 − 728.950i 0.0773434 − 0.148765i
\(71\) 479.239 0.0950683 0.0475341 − 0.998870i \(-0.484864\pi\)
0.0475341 + 0.998870i \(0.484864\pi\)
\(72\) −8345.21 + 10911.1i −1.60980 + 2.10476i
\(73\) 602.260i 0.113016i 0.998402 + 0.0565078i \(0.0179966\pi\)
−0.998402 + 0.0565078i \(0.982003\pi\)
\(74\) 3336.03 − 6416.64i 0.609209 − 1.17178i
\(75\) −6103.38 + 6103.38i −1.08505 + 1.08505i
\(76\) −7669.51 − 1344.07i −1.32782 − 0.232700i
\(77\) 1013.76 − 1013.76i 0.170983 − 0.170983i
\(78\) −3644.43 − 11537.6i −0.599019 − 1.89639i
\(79\) 9372.00i 1.50168i 0.660482 + 0.750842i \(0.270352\pi\)
−0.660482 + 0.750842i \(0.729648\pi\)
\(80\) 2570.60 − 1205.24i 0.401656 − 0.188319i
\(81\) −22121.5 −3.37167
\(82\) 6246.45 − 1973.09i 0.928979 − 0.293440i
\(83\) 3240.24 + 3240.24i 0.470350 + 0.470350i 0.902028 − 0.431678i \(-0.142078\pi\)
−0.431678 + 0.902028i \(0.642078\pi\)
\(84\) 5018.52 + 879.490i 0.711242 + 0.124644i
\(85\) −1966.19 − 1966.19i −0.272137 − 0.272137i
\(86\) −7521.20 − 3910.29i −1.01693 − 0.528703i
\(87\) −11415.6 −1.50821
\(88\) 4910.90 − 654.372i 0.634156 − 0.0845006i
\(89\) − 904.032i − 0.114131i −0.998370 − 0.0570655i \(-0.981826\pi\)
0.998370 − 0.0570655i \(-0.0181744\pi\)
\(90\) 8447.92 + 4392.10i 1.04295 + 0.542234i
\(91\) −2303.90 + 2303.90i −0.278215 + 0.278215i
\(92\) 4580.13 − 3214.19i 0.541131 − 0.379748i
\(93\) 16811.9 − 16811.9i 1.94380 − 1.94380i
\(94\) −3082.36 + 973.636i −0.348841 + 0.110190i
\(95\) 5397.09i 0.598016i
\(96\) 11928.8 + 12949.9i 1.29436 + 1.40515i
\(97\) −2492.47 −0.264903 −0.132452 − 0.991189i \(-0.542285\pi\)
−0.132452 + 0.991189i \(0.542285\pi\)
\(98\) −413.253 − 1308.28i −0.0430292 − 0.136223i
\(99\) 11748.6 + 11748.6i 1.19872 + 1.19872i
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.43.19 ✓ 96
4.3 odd 2 448.5.k.a.15.47 96
16.3 odd 4 inner 112.5.k.a.99.19 yes 96
16.13 even 4 448.5.k.a.239.47 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.19 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.19 yes 96 16.3 odd 4 inner
448.5.k.a.15.47 96 4.3 odd 2
448.5.k.a.239.47 96 16.13 even 4