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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 43.10
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.10

$q$-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+(-3.35720 + 2.17468i) q^{2} +(7.22622 + 7.22622i) q^{3} +(6.54156 - 14.6016i) q^{4} +(-12.7239 - 12.7239i) q^{5} +(-39.9746 - 8.54517i) q^{6} +18.5203 q^{7} +(9.79252 + 63.2464i) q^{8} +23.4366i q^{9} +(70.3872 + 15.0463i) q^{10} +(70.0054 - 70.0054i) q^{11} +(152.786 - 58.2439i) q^{12} +(189.279 - 189.279i) q^{13} +(-62.1762 + 40.2756i) q^{14} -183.892i q^{15} +(-170.416 - 191.035i) q^{16} -361.018 q^{17} +(-50.9671 - 78.6814i) q^{18} +(292.706 + 292.706i) q^{19} +(-269.025 + 102.556i) q^{20} +(133.832 + 133.832i) q^{21} +(-82.7830 + 387.261i) q^{22} +919.386 q^{23} +(-386.270 + 527.796i) q^{24} -301.203i q^{25} +(-223.826 + 1047.07i) q^{26} +(415.966 - 415.966i) q^{27} +(121.151 - 270.426i) q^{28} +(-450.347 + 450.347i) q^{29} +(399.906 + 617.362i) q^{30} +80.4530i q^{31} +(987.560 + 270.743i) q^{32} +1011.75 q^{33} +(1212.01 - 785.098i) q^{34} +(-235.651 - 235.651i) q^{35} +(342.213 + 153.312i) q^{36} +(1745.99 + 1745.99i) q^{37} +(-1619.21 - 346.131i) q^{38} +2735.54 q^{39} +(680.144 - 929.342i) q^{40} +224.231i q^{41} +(-740.339 - 158.259i) q^{42} +(-451.772 + 451.772i) q^{43} +(-564.249 - 1480.14i) q^{44} +(298.206 - 298.206i) q^{45} +(-3086.56 + 1999.37i) q^{46} +610.249i q^{47} +(148.999 - 2611.93i) q^{48} +343.000 q^{49} +(655.019 + 1011.20i) q^{50} +(-2608.80 - 2608.80i) q^{51} +(-1525.60 - 4001.96i) q^{52} +(-2273.45 - 2273.45i) q^{53} +(-491.889 + 2301.07i) q^{54} -1781.49 q^{55} +(181.360 + 1171.34i) q^{56} +4230.32i q^{57} +(532.545 - 2491.26i) q^{58} +(-819.419 + 819.419i) q^{59} +(-2685.13 - 1202.94i) q^{60} +(4692.97 - 4692.97i) q^{61} +(-174.959 - 270.097i) q^{62} +434.052i q^{63} +(-3904.21 + 1238.68i) q^{64} -4816.74 q^{65} +(-3396.65 + 2200.23i) q^{66} +(-4048.17 - 4048.17i) q^{67} +(-2361.62 + 5271.46i) q^{68} +(6643.69 + 6643.69i) q^{69} +(1303.59 + 278.662i) q^{70} +1726.08 q^{71} +(-1482.28 + 229.504i) q^{72} -6871.84i q^{73} +(-9658.62 - 2064.68i) q^{74} +(2176.56 - 2176.56i) q^{75} +(6188.74 - 2359.23i) q^{76} +(1296.52 - 1296.52i) q^{77} +(-9183.75 + 5948.91i) q^{78} -2151.16i q^{79} +(-262.358 + 4599.08i) q^{80} +7910.09 q^{81} +(-487.631 - 752.789i) q^{82} +(-7633.83 - 7633.83i) q^{83} +(2829.63 - 1078.69i) q^{84} +(4593.57 + 4593.57i) q^{85} +(534.231 - 2499.15i) q^{86} -6508.61 q^{87} +(5113.12 + 3742.06i) q^{88} +8191.07i q^{89} +(-352.635 + 1649.64i) q^{90} +(3505.49 - 3505.49i) q^{91} +(6014.22 - 13424.6i) q^{92} +(-581.372 + 581.372i) q^{93} +(-1327.09 - 2048.73i) q^{94} -7448.74i q^{95} +(5179.87 + 9092.78i) q^{96} +4498.80 q^{97} +(-1151.52 + 745.914i) q^{98} +(1640.69 + 1640.69i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.35720 + 2.17468i −0.839300 + 0.543669i
\(3\) 7.22622 + 7.22622i 0.802914 + 0.802914i 0.983550 − 0.180636i \(-0.0578157\pi\)
−0.180636 + 0.983550i \(0.557816\pi\)
\(4\) 6.54156 − 14.6016i 0.408848 − 0.912603i
\(5\) −12.7239 − 12.7239i −0.508957 − 0.508957i 0.405249 − 0.914206i \(-0.367185\pi\)
−0.914206 + 0.405249i \(0.867185\pi\)
\(6\) −39.9746 − 8.54517i −1.11040 − 0.237366i
\(7\) 18.5203 0.377964
\(8\) 9.79252 + 63.2464i 0.153008 + 0.988225i
\(9\) 23.4366i 0.289341i
\(10\) 70.3872 + 15.0463i 0.703872 + 0.150463i
\(11\) 70.0054 − 70.0054i 0.578557 − 0.578557i −0.355948 − 0.934506i \(-0.615842\pi\)
0.934506 + 0.355948i \(0.115842\pi\)
\(12\) 152.786 − 58.2439i 1.06101 − 0.404472i
\(13\) 189.279 − 189.279i 1.11999 − 1.11999i 0.128250 − 0.991742i \(-0.459064\pi\)
0.991742 − 0.128250i \(-0.0409359\pi\)
\(14\) −62.1762 + 40.2756i −0.317225 + 0.205488i
\(15\) − 183.892i − 0.817298i
\(16\) −170.416 − 191.035i −0.665687 − 0.746231i
\(17\) −361.018 −1.24920 −0.624599 − 0.780946i \(-0.714737\pi\)
−0.624599 + 0.780946i \(0.714737\pi\)
\(18\) −50.9671 − 78.6814i −0.157306 − 0.242844i
\(19\) 292.706 + 292.706i 0.810820 + 0.810820i 0.984757 − 0.173937i \(-0.0556489\pi\)
−0.173937 + 0.984757i \(0.555649\pi\)
\(20\) −269.025 + 102.556i −0.672562 + 0.256390i
\(21\) 133.832 + 133.832i 0.303473 + 0.303473i
\(22\) −82.7830 + 387.261i −0.171039 + 0.800127i
\(23\) 919.386 1.73797 0.868985 − 0.494838i \(-0.164773\pi\)
0.868985 + 0.494838i \(0.164773\pi\)
\(24\) −386.270 + 527.796i −0.670607 + 0.916312i
\(25\) − 301.203i − 0.481925i
\(26\) −223.826 + 1047.07i −0.331104 + 1.54891i
\(27\) 415.966 − 415.966i 0.570598 − 0.570598i
\(28\) 121.151 − 270.426i 0.154530 − 0.344931i
\(29\) −450.347 + 450.347i −0.535489 + 0.535489i −0.922201 − 0.386711i \(-0.873611\pi\)
0.386711 + 0.922201i \(0.373611\pi\)
\(30\) 399.906 + 617.362i 0.444340 + 0.685958i
\(31\) 80.4530i 0.0837180i 0.999124 + 0.0418590i \(0.0133280\pi\)
−0.999124 + 0.0418590i \(0.986672\pi\)
\(32\) 987.560 + 270.743i 0.964414 + 0.264398i
\(33\) 1011.75 0.929063
\(34\) 1212.01 − 785.098i 1.04845 − 0.679150i
\(35\) −235.651 − 235.651i −0.192368 − 0.192368i
\(36\) 342.213 + 153.312i 0.264053 + 0.118296i
\(37\) 1745.99 + 1745.99i 1.27538 + 1.27538i 0.943225 + 0.332155i \(0.107776\pi\)
0.332155 + 0.943225i \(0.392224\pi\)
\(38\) −1619.21 − 346.131i −1.12134 − 0.239703i
\(39\) 2735.54 1.79851
\(40\) 680.144 − 929.342i 0.425090 − 0.580839i
\(41\) 224.231i 0.133392i 0.997773 + 0.0666958i \(0.0212457\pi\)
−0.997773 + 0.0666958i \(0.978754\pi\)
\(42\) −740.339 − 158.259i −0.419694 − 0.0897158i
\(43\) −451.772 + 451.772i −0.244333 + 0.244333i −0.818640 − 0.574307i \(-0.805272\pi\)
0.574307 + 0.818640i \(0.305272\pi\)
\(44\) −564.249 − 1480.14i −0.291451 − 0.764535i
\(45\) 298.206 − 298.206i 0.147262 − 0.147262i
\(46\) −3086.56 + 1999.37i −1.45868 + 0.944881i
\(47\) 610.249i 0.276256i 0.990414 + 0.138128i \(0.0441085\pi\)
−0.990414 + 0.138128i \(0.955891\pi\)
\(48\) 148.999 − 2611.93i 0.0646698 − 1.13365i
\(49\) 343.000 0.142857
\(50\) 655.019 + 1011.20i 0.262008 + 0.404479i
\(51\) −2608.80 − 2608.80i −1.00300 − 1.00300i
\(52\) −1525.60 − 4001.96i −0.564201 − 1.48001i
\(53\) −2273.45 − 2273.45i −0.809344 − 0.809344i 0.175191 − 0.984535i \(-0.443946\pi\)
−0.984535 + 0.175191i \(0.943946\pi\)
\(54\) −491.889 + 2301.07i −0.168686 + 0.789119i
\(55\) −1781.49 −0.588922
\(56\) 181.360 + 1171.34i 0.0578317 + 0.373514i
\(57\) 4230.32i 1.30204i
\(58\) 532.545 − 2491.26i 0.158307 − 0.740565i
\(59\) −819.419 + 819.419i −0.235398 + 0.235398i −0.814941 − 0.579544i \(-0.803231\pi\)
0.579544 + 0.814941i \(0.303231\pi\)
\(60\) −2685.13 − 1202.94i −0.745868 − 0.334150i
\(61\) 4692.97 − 4692.97i 1.26121 − 1.26121i 0.310707 − 0.950506i \(-0.399434\pi\)
0.950506 − 0.310707i \(-0.100566\pi\)
\(62\) −174.959 − 270.097i −0.0455149 − 0.0702645i
\(63\) 434.052i 0.109361i
\(64\) −3904.21 + 1238.68i −0.953177 + 0.302413i
\(65\) −4816.74 −1.14006
\(66\) −3396.65 + 2200.23i −0.779763 + 0.505103i
\(67\) −4048.17 − 4048.17i −0.901798 − 0.901798i 0.0937940 − 0.995592i \(-0.470100\pi\)
−0.995592 + 0.0937940i \(0.970100\pi\)
\(68\) −2361.62 + 5271.46i −0.510732 + 1.14002i
\(69\) 6643.69 + 6643.69i 1.39544 + 1.39544i
\(70\) 1303.59 + 278.662i 0.266039 + 0.0568698i
\(71\) 1726.08 0.342407 0.171204 − 0.985236i \(-0.445234\pi\)
0.171204 + 0.985236i \(0.445234\pi\)
\(72\) −1482.28 + 229.504i −0.285934 + 0.0442715i
\(73\) − 6871.84i − 1.28952i −0.764386 − 0.644759i \(-0.776958\pi\)
0.764386 − 0.644759i \(-0.223042\pi\)
\(74\) −9658.62 − 2064.68i −1.76381 − 0.377041i
\(75\) 2176.56 − 2176.56i 0.386944 − 0.386944i
\(76\) 6188.74 − 2359.23i 1.07146 − 0.408454i
\(77\) 1296.52 − 1296.52i 0.218674 − 0.218674i
\(78\) −9183.75 + 5948.91i −1.50949 + 0.977796i
\(79\) − 2151.16i − 0.344682i −0.985037 − 0.172341i \(-0.944867\pi\)
0.985037 − 0.172341i \(-0.0551330\pi\)
\(80\) −262.358 + 4599.08i −0.0409934 + 0.718606i
\(81\) 7910.09 1.20562
\(82\) −487.631 − 752.789i −0.0725209 − 0.111956i
\(83\) −7633.83 − 7633.83i −1.10812 − 1.10812i −0.993398 − 0.114721i \(-0.963403\pi\)
−0.114721 − 0.993398i \(-0.536597\pi\)
\(84\) 2829.63 − 1078.69i 0.401024 − 0.152876i
\(85\) 4593.57 + 4593.57i 0.635788 + 0.635788i
\(86\) 534.231 − 2499.15i 0.0722324 − 0.337905i
\(87\) −6508.61 −0.859904
\(88\) 5113.12 + 3742.06i 0.660269 + 0.483221i
\(89\) 8191.07i 1.03410i 0.855957 + 0.517048i \(0.172969\pi\)
−0.855957 + 0.517048i \(0.827031\pi\)
\(90\) −352.635 + 1649.64i −0.0435352 + 0.203659i
\(91\) 3505.49 − 3505.49i 0.423317 − 0.423317i
\(92\) 6014.22 − 13424.6i 0.710565 − 1.58608i
\(93\) −581.372 + 581.372i −0.0672183 + 0.0672183i
\(94\) −1327.09 − 2048.73i −0.150192 − 0.231861i
\(95\) − 7448.74i − 0.825346i
\(96\) 5179.87 + 9092.78i 0.562052 + 0.986630i
\(97\) 4498.80 0.478138 0.239069 − 0.971003i \(-0.423158\pi\)
0.239069 + 0.971003i \(0.423158\pi\)
\(98\) −1151.52 + 745.914i −0.119900 + 0.0776670i
\(99\) 1640.69 + 1640.69i 0.167400 + 0.167400i
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.43.10 ✓ 96
4.3 odd 2 448.5.k.a.15.12 96
16.3 odd 4 inner 112.5.k.a.99.10 yes 96
16.13 even 4 448.5.k.a.239.12 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.10 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.10 yes 96 16.3 odd 4 inner
448.5.k.a.15.12 96 4.3 odd 2
448.5.k.a.239.12 96 16.13 even 4