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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 43.11
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.11

$q$-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+(-2.91721 + 2.73677i) q^{2} +(1.88359 + 1.88359i) q^{3} +(1.02022 - 15.9674i) q^{4} +(6.01207 + 6.01207i) q^{5} +(-10.6498 - 0.339881i) q^{6} -18.5203 q^{7} +(40.7230 + 49.3725i) q^{8} -73.9042i q^{9} +(-33.9921 - 1.08484i) q^{10} +(-99.3381 + 99.3381i) q^{11} +(31.9978 - 28.1545i) q^{12} +(-32.1684 + 32.1684i) q^{13} +(54.0275 - 50.6856i) q^{14} +22.6486i q^{15} +(-253.918 - 32.5806i) q^{16} -339.803 q^{17} +(202.258 + 215.594i) q^{18} +(-44.6062 - 44.6062i) q^{19} +(102.131 - 89.8637i) q^{20} +(-34.8846 - 34.8846i) q^{21} +(17.9249 - 561.655i) q^{22} -336.946 q^{23} +(-16.2922 + 169.703i) q^{24} -552.710i q^{25} +(5.80456 - 181.879i) q^{26} +(291.776 - 291.776i) q^{27} +(-18.8947 + 295.721i) q^{28} +(-43.1938 + 43.1938i) q^{29} +(-61.9839 - 66.0706i) q^{30} -1870.79i q^{31} +(829.898 - 599.871i) q^{32} -374.225 q^{33} +(991.277 - 929.962i) q^{34} +(-111.345 - 111.345i) q^{35} +(-1180.06 - 75.3985i) q^{36} +(-1082.73 - 1082.73i) q^{37} +(252.202 + 8.04887i) q^{38} -121.184 q^{39} +(-52.0014 + 541.660i) q^{40} -187.923i q^{41} +(197.237 + 6.29468i) q^{42} +(-2301.95 + 2301.95i) q^{43} +(1484.83 + 1687.52i) q^{44} +(444.317 - 444.317i) q^{45} +(982.941 - 922.142i) q^{46} +2754.62i q^{47} +(-416.910 - 539.647i) q^{48} +343.000 q^{49} +(1512.64 + 1612.37i) q^{50} +(-640.050 - 640.050i) q^{51} +(480.828 + 546.466i) q^{52} +(-1516.38 - 1516.38i) q^{53} +(-52.6490 + 1649.70i) q^{54} -1194.45 q^{55} +(-754.200 - 914.391i) q^{56} -168.040i q^{57} +(7.79402 - 244.217i) q^{58} +(756.421 - 756.421i) q^{59} +(361.640 + 23.1065i) q^{60} +(-1961.98 + 1961.98i) q^{61} +(5119.92 + 5457.49i) q^{62} +1368.72i q^{63} +(-779.281 + 4021.19i) q^{64} -386.797 q^{65} +(1091.69 - 1024.17i) q^{66} +(2923.15 + 2923.15i) q^{67} +(-346.674 + 5425.79i) q^{68} +(-634.668 - 634.668i) q^{69} +(629.542 + 20.0914i) q^{70} -774.916 q^{71} +(3648.83 - 3009.60i) q^{72} -46.7770i q^{73} +(6121.72 + 195.371i) q^{74} +(1041.08 - 1041.08i) q^{75} +(-757.754 + 666.738i) q^{76} +(1839.77 - 1839.77i) q^{77} +(353.520 - 331.653i) q^{78} +3673.93i q^{79} +(-1330.70 - 1722.45i) q^{80} -4887.06 q^{81} +(514.302 + 548.212i) q^{82} +(8170.35 + 8170.35i) q^{83} +(-592.608 + 521.428i) q^{84} +(-2042.92 - 2042.92i) q^{85} +(415.370 - 13015.1i) q^{86} -162.719 q^{87} +(-8949.91 - 859.226i) q^{88} -3939.40i q^{89} +(-80.1738 + 2512.16i) q^{90} +(595.767 - 595.767i) q^{91} +(-343.759 + 5380.16i) q^{92} +(3523.81 - 3523.81i) q^{93} +(-7538.75 - 8035.80i) q^{94} -536.351i q^{95} +(2693.10 + 433.278i) q^{96} -14945.6 q^{97} +(-1000.60 + 938.711i) q^{98} +(7341.50 + 7341.50i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.91721 + 2.73677i −0.729302 + 0.684192i
\(3\) 1.88359 + 1.88359i 0.209288 + 0.209288i 0.803965 − 0.594677i \(-0.202720\pi\)
−0.594677 + 0.803965i \(0.702720\pi\)
\(4\) 1.02022 − 15.9674i 0.0637637 − 0.997965i
\(5\) 6.01207 + 6.01207i 0.240483 + 0.240483i 0.817050 − 0.576567i \(-0.195608\pi\)
−0.576567 + 0.817050i \(0.695608\pi\)
\(6\) −10.6498 − 0.339881i −0.295827 − 0.00944114i
\(7\) −18.5203 −0.377964
\(8\) 40.7230 + 49.3725i 0.636296 + 0.771445i
\(9\) − 73.9042i − 0.912397i
\(10\) −33.9921 − 1.08484i −0.339921 − 0.0108484i
\(11\) −99.3381 + 99.3381i −0.820976 + 0.820976i −0.986248 − 0.165272i \(-0.947150\pi\)
0.165272 + 0.986248i \(0.447150\pi\)
\(12\) 31.9978 − 28.1545i 0.222207 − 0.195517i
\(13\) −32.1684 + 32.1684i −0.190346 + 0.190346i −0.795845 − 0.605500i \(-0.792973\pi\)
0.605500 + 0.795845i \(0.292973\pi\)
\(14\) 54.0275 − 50.6856i 0.275650 − 0.258600i
\(15\) 22.6486i 0.100660i
\(16\) −253.918 − 32.5806i −0.991868 − 0.127268i
\(17\) −339.803 −1.17579 −0.587895 − 0.808938i \(-0.700043\pi\)
−0.587895 + 0.808938i \(0.700043\pi\)
\(18\) 202.258 + 215.594i 0.624254 + 0.665413i
\(19\) −44.6062 − 44.6062i −0.123563 − 0.123563i 0.642621 − 0.766184i \(-0.277847\pi\)
−0.766184 + 0.642621i \(0.777847\pi\)
\(20\) 102.131 − 89.8637i 0.255327 − 0.224659i
\(21\) −34.8846 − 34.8846i −0.0791034 − 0.0791034i
\(22\) 17.9249 − 561.655i 0.0370348 − 1.16044i
\(23\) −336.946 −0.636948 −0.318474 − 0.947932i \(-0.603170\pi\)
−0.318474 + 0.947932i \(0.603170\pi\)
\(24\) −16.2922 + 169.703i −0.0282850 + 0.294623i
\(25\) − 552.710i − 0.884336i
\(26\) 5.80456 − 181.879i 0.00858663 − 0.269052i
\(27\) 291.776 − 291.776i 0.400242 − 0.400242i
\(28\) −18.8947 + 295.721i −0.0241004 + 0.377195i
\(29\) −43.1938 + 43.1938i −0.0513600 + 0.0513600i −0.732320 − 0.680960i \(-0.761563\pi\)
0.680960 + 0.732320i \(0.261563\pi\)
\(30\) −61.9839 − 66.0706i −0.0688709 − 0.0734118i
\(31\) − 1870.79i − 1.94671i −0.229297 − 0.973356i \(-0.573643\pi\)
0.229297 − 0.973356i \(-0.426357\pi\)
\(32\) 829.898 − 599.871i 0.810448 − 0.585811i
\(33\) −374.225 −0.343641
\(34\) 991.277 − 929.962i 0.857506 − 0.804465i
\(35\) −111.345 − 111.345i −0.0908939 − 0.0908939i
\(36\) −1180.06 − 75.3985i −0.910540 − 0.0581779i
\(37\) −1082.73 − 1082.73i −0.790890 − 0.790890i 0.190749 − 0.981639i \(-0.438908\pi\)
−0.981639 + 0.190749i \(0.938908\pi\)
\(38\) 252.202 + 8.04887i 0.174655 + 0.00557401i
\(39\) −121.184 −0.0796741
\(40\) −52.0014 + 541.660i −0.0325009 + 0.338537i
\(41\) − 187.923i − 0.111793i −0.998437 − 0.0558963i \(-0.982198\pi\)
0.998437 − 0.0558963i \(-0.0178016\pi\)
\(42\) 197.237 + 6.29468i 0.111812 + 0.00356841i
\(43\) −2301.95 + 2301.95i −1.24497 + 1.24497i −0.287054 + 0.957914i \(0.592676\pi\)
−0.957914 + 0.287054i \(0.907324\pi\)
\(44\) 1484.83 + 1687.52i 0.766957 + 0.871654i
\(45\) 444.317 − 444.317i 0.219416 − 0.219416i
\(46\) 982.941 − 922.142i 0.464528 − 0.435795i
\(47\) 2754.62i 1.24700i 0.781824 + 0.623499i \(0.214290\pi\)
−0.781824 + 0.623499i \(0.785710\pi\)
\(48\) −416.910 − 539.647i −0.180951 − 0.234222i
\(49\) 343.000 0.142857
\(50\) 1512.64 + 1612.37i 0.605055 + 0.644948i
\(51\) −640.050 − 640.050i −0.246079 − 0.246079i
\(52\) 480.828 + 546.466i 0.177821 + 0.202095i
\(53\) −1516.38 − 1516.38i −0.539830 − 0.539830i 0.383649 − 0.923479i \(-0.374667\pi\)
−0.923479 + 0.383649i \(0.874667\pi\)
\(54\) −52.6490 + 1649.70i −0.0180552 + 0.565739i
\(55\) −1194.45 −0.394861
\(56\) −754.200 − 914.391i −0.240497 − 0.291579i
\(57\) − 168.040i − 0.0517204i
\(58\) 7.79402 − 244.217i 0.00231689 − 0.0725971i
\(59\) 756.421 − 756.421i 0.217300 − 0.217300i −0.590060 − 0.807360i \(-0.700896\pi\)
0.807360 + 0.590060i \(0.200896\pi\)
\(60\) 361.640 + 23.1065i 0.100455 + 0.00641848i
\(61\) −1961.98 + 1961.98i −0.527273 + 0.527273i −0.919758 − 0.392485i \(-0.871616\pi\)
0.392485 + 0.919758i \(0.371616\pi\)
\(62\) 5119.92 + 5457.49i 1.33192 + 1.41974i
\(63\) 1368.72i 0.344854i
\(64\) −779.281 + 4021.19i −0.190254 + 0.981735i
\(65\) −386.797 −0.0915497
\(66\) 1091.69 − 1024.17i 0.250618 − 0.235116i
\(67\) 2923.15 + 2923.15i 0.651180 + 0.651180i 0.953277 − 0.302097i \(-0.0976867\pi\)
−0.302097 + 0.953277i \(0.597687\pi\)
\(68\) −346.674 + 5425.79i −0.0749727 + 1.17340i
\(69\) −634.668 − 634.668i −0.133306 − 0.133306i
\(70\) 629.542 + 20.0914i 0.128478 + 0.00410029i
\(71\) −774.916 −0.153723 −0.0768613 − 0.997042i \(-0.524490\pi\)
−0.0768613 + 0.997042i \(0.524490\pi\)
\(72\) 3648.83 − 3009.60i 0.703864 − 0.580555i
\(73\) − 46.7770i − 0.00877781i −0.999990 − 0.00438891i \(-0.998603\pi\)
0.999990 − 0.00438891i \(-0.00139704\pi\)
\(74\) 6121.72 + 195.371i 1.11792 + 0.0356776i
\(75\) 1041.08 − 1041.08i 0.185081 − 0.185081i
\(76\) −757.754 + 666.738i −0.131190 + 0.115433i
\(77\) 1839.77 − 1839.77i 0.310300 − 0.310300i
\(78\) 353.520 − 331.653i 0.0581065 − 0.0545124i
\(79\) 3673.93i 0.588677i 0.955701 + 0.294339i \(0.0950993\pi\)
−0.955701 + 0.294339i \(0.904901\pi\)
\(80\) −1330.70 − 1722.45i −0.207921 − 0.269133i
\(81\) −4887.06 −0.744865
\(82\) 514.302 + 548.212i 0.0764876 + 0.0815306i
\(83\) 8170.35 + 8170.35i 1.18600 + 1.18600i 0.978164 + 0.207835i \(0.0666417\pi\)
0.207835 + 0.978164i \(0.433358\pi\)
\(84\) −592.608 + 521.428i −0.0839864 + 0.0738985i
\(85\) −2042.92 − 2042.92i −0.282757 − 0.282757i
\(86\) 415.370 − 13015.1i 0.0561614 − 1.75976i
\(87\) −162.719 −0.0214981
\(88\) −8949.91 − 859.226i −1.15572 − 0.110954i
\(89\) − 3939.40i − 0.497336i −0.968589 − 0.248668i \(-0.920007\pi\)
0.968589 − 0.248668i \(-0.0799927\pi\)
\(90\) −80.1738 + 2512.16i −0.00989800 + 0.310143i
\(91\) 595.767 − 595.767i 0.0719439 − 0.0719439i
\(92\) −343.759 + 5380.16i −0.0406142 + 0.635652i
\(93\) 3523.81 − 3523.81i 0.407424 − 0.407424i
\(94\) −7538.75 − 8035.80i −0.853186 − 0.909439i
\(95\) − 536.351i − 0.0594294i
\(96\) 2693.10 + 433.278i 0.292220 + 0.0470137i
\(97\) −14945.6 −1.58843 −0.794217 − 0.607634i \(-0.792119\pi\)
−0.794217 + 0.607634i \(0.792119\pi\)
\(98\) −1000.60 + 938.711i −0.104186 + 0.0977417i
\(99\) 7341.50 + 7341.50i 0.749056 + 0.749056i
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.43.11 ✓ 96
4.3 odd 2 448.5.k.a.15.21 96
16.3 odd 4 inner 112.5.k.a.99.11 yes 96
16.13 even 4 448.5.k.a.239.21 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.11 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.11 yes 96 16.3 odd 4 inner
448.5.k.a.15.21 96 4.3 odd 2
448.5.k.a.239.21 96 16.13 even 4