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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 43.13
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.13

$q$-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+(-2.74539 + 2.90910i) q^{2} +(1.91051 + 1.91051i) q^{3} +(-0.925711 - 15.9732i) q^{4} +(33.8528 + 33.8528i) q^{5} +(-10.8030 + 0.312775i) q^{6} +18.5203 q^{7} +(49.0090 + 41.1596i) q^{8} -73.6999i q^{9} +(-191.420 + 5.54214i) q^{10} +(100.769 - 100.769i) q^{11} +(28.7484 - 32.2856i) q^{12} +(28.6391 - 28.6391i) q^{13} +(-50.8453 + 53.8773i) q^{14} +129.353i q^{15} +(-254.286 + 29.5731i) q^{16} +549.495 q^{17} +(214.400 + 202.335i) q^{18} +(58.0004 + 58.0004i) q^{19} +(509.400 - 572.076i) q^{20} +(35.3832 + 35.3832i) q^{21} +(16.4971 + 569.795i) q^{22} -874.669 q^{23} +(14.9965 + 172.268i) q^{24} +1667.03i q^{25} +(4.68858 + 161.939i) q^{26} +(295.556 - 295.556i) q^{27} +(-17.1444 - 295.828i) q^{28} +(-307.151 + 307.151i) q^{29} +(-376.299 - 355.123i) q^{30} +763.779i q^{31} +(612.082 - 820.933i) q^{32} +385.040 q^{33} +(-1508.58 + 1598.53i) q^{34} +(626.963 + 626.963i) q^{35} +(-1177.22 + 68.2248i) q^{36} +(-292.192 - 292.192i) q^{37} +(-327.962 + 9.49541i) q^{38} +109.431 q^{39} +(265.726 + 3052.46i) q^{40} +1393.66i q^{41} +(-200.074 + 5.79268i) q^{42} +(757.492 - 757.492i) q^{43} +(-1702.88 - 1516.32i) q^{44} +(2494.95 - 2494.95i) q^{45} +(2401.30 - 2544.50i) q^{46} +494.734i q^{47} +(-542.317 - 429.317i) q^{48} +343.000 q^{49} +(-4849.55 - 4576.63i) q^{50} +(1049.82 + 1049.82i) q^{51} +(-483.969 - 430.946i) q^{52} +(203.756 + 203.756i) q^{53} +(48.3863 + 1671.22i) q^{54} +6822.61 q^{55} +(907.660 + 762.287i) q^{56} +221.621i q^{57} +(-50.2845 - 1736.78i) q^{58} +(3355.95 - 3355.95i) q^{59} +(2066.17 - 119.743i) q^{60} +(-4499.31 + 4499.31i) q^{61} +(-2221.91 - 2096.87i) q^{62} -1364.94i q^{63} +(707.773 + 4034.39i) q^{64} +1939.03 q^{65} +(-1057.08 + 1120.12i) q^{66} +(-5209.57 - 5209.57i) q^{67} +(-508.674 - 8777.19i) q^{68} +(-1671.07 - 1671.07i) q^{69} +(-3545.15 + 102.642i) q^{70} -6529.16 q^{71} +(3033.46 - 3611.96i) q^{72} -1866.03i q^{73} +(1652.20 - 47.8356i) q^{74} +(-3184.88 + 3184.88i) q^{75} +(872.760 - 980.144i) q^{76} +(1866.26 - 1866.26i) q^{77} +(-300.429 + 318.344i) q^{78} -7145.38i q^{79} +(-9609.44 - 7607.17i) q^{80} -4840.36 q^{81} +(-4054.28 - 3826.12i) q^{82} +(-1759.48 - 1759.48i) q^{83} +(532.428 - 597.937i) q^{84} +(18601.9 + 18601.9i) q^{85} +(124.011 + 4283.23i) q^{86} -1173.63 q^{87} +(9086.18 - 790.977i) q^{88} -1374.55i q^{89} +(408.455 + 14107.6i) q^{90} +(530.403 - 530.403i) q^{91} +(809.691 + 13971.3i) q^{92} +(-1459.21 + 1459.21i) q^{93} +(-1439.23 - 1358.23i) q^{94} +3926.95i q^{95} +(2737.79 - 399.012i) q^{96} +5954.34 q^{97} +(-941.667 + 997.821i) q^{98} +(-7426.64 - 7426.64i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.74539 + 2.90910i −0.686347 + 0.727275i
\(3\) 1.91051 + 1.91051i 0.212279 + 0.212279i 0.805235 − 0.592956i \(-0.202039\pi\)
−0.592956 + 0.805235i \(0.702039\pi\)
\(4\) −0.925711 − 15.9732i −0.0578570 − 0.998325i
\(5\) 33.8528 + 33.8528i 1.35411 + 1.35411i 0.881004 + 0.473109i \(0.156868\pi\)
0.473109 + 0.881004i \(0.343132\pi\)
\(6\) −10.8030 + 0.312775i −0.300082 + 0.00868821i
\(7\) 18.5203 0.377964
\(8\) 49.0090 + 41.1596i 0.765766 + 0.643119i
\(9\) − 73.6999i − 0.909875i
\(10\) −191.420 + 5.54214i −1.91420 + 0.0554214i
\(11\) 100.769 − 100.769i 0.832799 − 0.832799i −0.155100 − 0.987899i \(-0.549570\pi\)
0.987899 + 0.155100i \(0.0495699\pi\)
\(12\) 28.7484 − 32.2856i 0.199642 − 0.224205i
\(13\) 28.6391 − 28.6391i 0.169462 − 0.169462i −0.617281 − 0.786743i \(-0.711766\pi\)
0.786743 + 0.617281i \(0.211766\pi\)
\(14\) −50.8453 + 53.8773i −0.259415 + 0.274884i
\(15\) 129.353i 0.574900i
\(16\) −254.286 + 29.5731i −0.993305 + 0.115520i
\(17\) 549.495 1.90137 0.950683 − 0.310164i \(-0.100384\pi\)
0.950683 + 0.310164i \(0.100384\pi\)
\(18\) 214.400 + 202.335i 0.661729 + 0.624490i
\(19\) 58.0004 + 58.0004i 0.160666 + 0.160666i 0.782862 − 0.622196i \(-0.213759\pi\)
−0.622196 + 0.782862i \(0.713759\pi\)
\(20\) 509.400 − 572.076i 1.27350 − 1.43019i
\(21\) 35.3832 + 35.3832i 0.0802340 + 0.0802340i
\(22\) 16.4971 + 569.795i 0.0340850 + 1.17726i
\(23\) −874.669 −1.65344 −0.826719 − 0.562615i \(-0.809795\pi\)
−0.826719 + 0.562615i \(0.809795\pi\)
\(24\) 14.9965 + 172.268i 0.0260355 + 0.299077i
\(25\) 1667.03i 2.66724i
\(26\) 4.68858 + 161.939i 0.00693577 + 0.239555i
\(27\) 295.556 − 295.556i 0.405427 − 0.405427i
\(28\) −17.1444 − 295.828i −0.0218679 − 0.377331i
\(29\) −307.151 + 307.151i −0.365221 + 0.365221i −0.865731 − 0.500510i \(-0.833146\pi\)
0.500510 + 0.865731i \(0.333146\pi\)
\(30\) −376.299 − 355.123i −0.418110 − 0.394581i
\(31\) 763.779i 0.794775i 0.917651 + 0.397387i \(0.130083\pi\)
−0.917651 + 0.397387i \(0.869917\pi\)
\(32\) 612.082 − 820.933i 0.597737 − 0.801693i
\(33\) 385.040 0.353572
\(34\) −1508.58 + 1598.53i −1.30500 + 1.38282i
\(35\) 626.963 + 626.963i 0.511807 + 0.511807i
\(36\) −1177.22 + 68.2248i −0.908351 + 0.0526426i
\(37\) −292.192 − 292.192i −0.213435 − 0.213435i 0.592290 − 0.805725i \(-0.298224\pi\)
−0.805725 + 0.592290i \(0.798224\pi\)
\(38\) −327.962 + 9.49541i −0.227121 + 0.00657577i
\(39\) 109.431 0.0719465
\(40\) 265.726 + 3052.46i 0.166078 + 1.90779i
\(41\) 1393.66i 0.829063i 0.910035 + 0.414532i \(0.136055\pi\)
−0.910035 + 0.414532i \(0.863945\pi\)
\(42\) −200.074 + 5.79268i −0.113420 + 0.00328383i
\(43\) 757.492 − 757.492i 0.409677 − 0.409677i −0.471949 − 0.881626i \(-0.656449\pi\)
0.881626 + 0.471949i \(0.156449\pi\)
\(44\) −1702.88 − 1516.32i −0.879587 − 0.783221i
\(45\) 2494.95 − 2494.95i 1.23207 − 1.23207i
\(46\) 2401.30 − 2544.50i 1.13483 − 1.20250i
\(47\) 494.734i 0.223963i 0.993710 + 0.111981i \(0.0357197\pi\)
−0.993710 + 0.111981i \(0.964280\pi\)
\(48\) −542.317 − 429.317i −0.235381 − 0.186336i
\(49\) 343.000 0.142857
\(50\) −4849.55 − 4576.63i −1.93982 − 1.83065i
\(51\) 1049.82 + 1049.82i 0.403621 + 0.403621i
\(52\) −483.969 − 430.946i −0.178983 − 0.159374i
\(53\) 203.756 + 203.756i 0.0725369 + 0.0725369i 0.742445 − 0.669908i \(-0.233666\pi\)
−0.669908 + 0.742445i \(0.733666\pi\)
\(54\) 48.3863 + 1671.22i 0.0165934 + 0.573120i
\(55\) 6822.61 2.25541
\(56\) 907.660 + 762.287i 0.289432 + 0.243076i
\(57\) 221.621i 0.0682121i
\(58\) −50.2845 − 1736.78i −0.0149478 − 0.516284i
\(59\) 3355.95 − 3355.95i 0.964077 − 0.964077i −0.0352997 − 0.999377i \(-0.511239\pi\)
0.999377 + 0.0352997i \(0.0112386\pi\)
\(60\) 2066.17 − 119.743i 0.573937 − 0.0332620i
\(61\) −4499.31 + 4499.31i −1.20917 + 1.20917i −0.237871 + 0.971297i \(0.576450\pi\)
−0.971297 + 0.237871i \(0.923550\pi\)
\(62\) −2221.91 − 2096.87i −0.578020 − 0.545491i
\(63\) − 1364.94i − 0.343900i
\(64\) 707.773 + 4034.39i 0.172796 + 0.984958i
\(65\) 1939.03 0.458941
\(66\) −1057.08 + 1120.12i −0.242673 + 0.257144i
\(67\) −5209.57 − 5209.57i −1.16052 − 1.16052i −0.984362 − 0.176157i \(-0.943633\pi\)
−0.176157 − 0.984362i \(-0.556367\pi\)
\(68\) −508.674 − 8777.19i −0.110007 − 1.89818i
\(69\) −1671.07 − 1671.07i −0.350991 − 0.350991i
\(70\) −3545.15 + 102.642i −0.723501 + 0.0209473i
\(71\) −6529.16 −1.29521 −0.647606 − 0.761975i \(-0.724230\pi\)
−0.647606 + 0.761975i \(0.724230\pi\)
\(72\) 3033.46 − 3611.96i 0.585158 − 0.696752i
\(73\) − 1866.03i − 0.350165i −0.984554 − 0.175082i \(-0.943981\pi\)
0.984554 − 0.175082i \(-0.0560192\pi\)
\(74\) 1652.20 − 47.8356i 0.301716 − 0.00873549i
\(75\) −3184.88 + 3184.88i −0.566200 + 0.566200i
\(76\) 872.760 − 980.144i 0.151101 − 0.169692i
\(77\) 1866.26 − 1866.26i 0.314768 − 0.314768i
\(78\) −300.429 + 318.344i −0.0493802 + 0.0523249i
\(79\) − 7145.38i − 1.14491i −0.819937 − 0.572454i \(-0.805991\pi\)
0.819937 − 0.572454i \(-0.194009\pi\)
\(80\) −9609.44 − 7607.17i −1.50147 − 1.18862i
\(81\) −4840.36 −0.737748
\(82\) −4054.28 − 3826.12i −0.602957 − 0.569025i
\(83\) −1759.48 − 1759.48i −0.255404 − 0.255404i 0.567778 − 0.823182i \(-0.307803\pi\)
−0.823182 + 0.567778i \(0.807803\pi\)
\(84\) 532.428 − 597.937i 0.0754575 − 0.0847417i
\(85\) 18601.9 + 18601.9i 2.57466 + 2.57466i
\(86\) 124.011 + 4283.23i 0.0167673 + 0.579127i
\(87\) −1173.63 −0.155058
\(88\) 9086.18 − 790.977i 1.17332 − 0.102141i
\(89\) − 1374.55i − 0.173532i −0.996229 − 0.0867662i \(-0.972347\pi\)
0.996229 − 0.0867662i \(-0.0276533\pi\)
\(90\) 408.455 + 14107.6i 0.0504265 + 1.74169i
\(91\) 530.403 − 530.403i 0.0640506 − 0.0640506i
\(92\) 809.691 + 13971.3i 0.0956629 + 1.65067i
\(93\) −1459.21 + 1459.21i −0.168714 + 0.168714i
\(94\) −1439.23 − 1358.23i −0.162882 − 0.153716i
\(95\) 3926.95i 0.435120i
\(96\) 2737.79 − 399.012i 0.297070 − 0.0432956i
\(97\) 5954.34 0.632835 0.316417 − 0.948620i \(-0.397520\pi\)
0.316417 + 0.948620i \(0.397520\pi\)
\(98\) −941.667 + 997.821i −0.0980495 + 0.103896i
\(99\) −7426.64 − 7426.64i −0.757743 − 0.757743i
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.43.13 ✓ 96
4.3 odd 2 448.5.k.a.15.20 96
16.3 odd 4 inner 112.5.k.a.99.13 yes 96
16.13 even 4 448.5.k.a.239.20 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.13 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.13 yes 96 16.3 odd 4 inner
448.5.k.a.15.20 96 4.3 odd 2
448.5.k.a.239.20 96 16.13 even 4