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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [56,10,Mod(3,56)] 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("56.3"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(56, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 56.m (of order \(6\), 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: \(28.8420068252\)
Analytic rank: \(0\)
Dimension: \(140\)
Relative dimension: \(70\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 19.3
Character \(\chi\) \(=\) 56.19
Dual form 56.10.m.a.3.3

$q$-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+(-22.4274 + 3.00193i) q^{2} +(30.2929 - 17.4896i) q^{3} +(493.977 - 134.651i) q^{4} +(733.902 - 1271.16i) q^{5} +(-626.888 + 483.183i) q^{6} +(-4833.52 - 4121.98i) q^{7} +(-10674.4 + 4502.75i) q^{8} +(-9229.73 + 15986.4i) q^{9} +(-12643.6 + 30711.9i) q^{10} +(10934.3 + 18938.7i) q^{11} +(12609.0 - 12718.4i) q^{12} -85110.7 q^{13} +(120777. + 77935.4i) q^{14} -51342.6i q^{15} +(225882. - 133029. i) q^{16} +(-3510.53 + 2026.80i) q^{17} +(159009. - 386239. i) q^{18} +(-524035. - 302552. i) q^{19} +(191369. - 726742. i) q^{20} +(-218513. - 40330.3i) q^{21} +(-302080. - 391922. i) q^{22} +(1.34205e6 + 774832. i) q^{23} +(-244607. + 323092. i) q^{24} +(-100663. - 174353. i) q^{25} +(1.90881e6 - 255496. i) q^{26} +1.33419e6i q^{27} +(-2.94267e6 - 1.38533e6i) q^{28} -2.69795e6i q^{29} +(154127. + 1.15148e6i) q^{30} +(2.87323e6 + 4.97657e6i) q^{31} +(-4.66661e6 + 3.66157e6i) q^{32} +(662460. + 382471. i) q^{33} +(72647.7 - 55994.3i) q^{34} +(-8.78701e6 + 3.11902e6i) q^{35} +(-2.40670e6 + 9.13968e6i) q^{36} +(6.39979e6 + 3.69492e6i) q^{37} +(1.26610e7 + 5.21233e6i) q^{38} +(-2.57825e6 + 1.48855e6i) q^{39} +(-2.11027e6 + 1.68734e7i) q^{40} +6.03060e6i q^{41} +(5.02174e6 + 248545. i) q^{42} -4.80871e6 q^{43} +(7.95138e6 + 7.88297e6i) q^{44} +(1.35474e7 + 2.34649e7i) q^{45} +(-3.24246e7 - 1.33487e7i) q^{46} +(-2.80573e7 + 4.85966e7i) q^{47} +(4.51600e6 - 7.98041e6i) q^{48} +(6.37217e6 + 3.98473e7i) q^{49} +(2.78100e6 + 3.60810e6i) q^{50} +(-70895.9 + 122795. i) q^{51} +(-4.20427e7 + 1.14602e7i) q^{52} +(-2.28667e7 + 1.32021e7i) q^{53} +(-4.00515e6 - 2.99225e7i) q^{54} +3.20987e7 q^{55} +(7.01552e7 + 2.22356e7i) q^{56} -2.11660e7 q^{57} +(8.09905e6 + 6.05080e7i) q^{58} +(-1.12611e7 + 6.50159e6i) q^{59} +(-6.91333e6 - 2.53621e7i) q^{60} +(-4.14309e6 + 7.17604e6i) q^{61} +(-7.93783e7 - 1.02986e8i) q^{62} +(1.10507e8 - 3.92256e7i) q^{63} +(9.36682e7 - 9.61284e7i) q^{64} +(-6.24630e7 + 1.08189e8i) q^{65} +(-1.60054e7 - 6.58918e6i) q^{66} +(6.38170e7 + 1.10534e8i) q^{67} +(-1.46121e6 + 1.47389e6i) q^{68} +5.42060e7 q^{69} +(1.87707e8 - 9.63296e7i) q^{70} -2.75148e8i q^{71} +(2.65393e7 - 2.12204e8i) q^{72} +(-1.34835e8 + 7.78472e7i) q^{73} +(-1.54623e8 - 6.36558e7i) q^{74} +(-6.09873e6 - 3.52110e6i) q^{75} +(-2.99600e8 - 7.88918e7i) q^{76} +(2.52140e7 - 1.36611e8i) q^{77} +(5.33549e7 - 4.11241e7i) q^{78} +(1.48500e8 + 8.57362e7i) q^{79} +(-3.32481e6 - 3.84762e8i) q^{80} +(-1.58334e8 - 2.74243e8i) q^{81} +(-1.81034e7 - 1.35251e8i) q^{82} +7.08642e8i q^{83} +(-1.13371e8 + 9.50069e6i) q^{84} +5.94990e6i q^{85} +(1.07847e8 - 1.44354e7i) q^{86} +(-4.71861e7 - 8.17287e7i) q^{87} +(-2.01993e8 - 1.52925e8i) q^{88} +(5.24187e8 + 3.02639e8i) q^{89} +(-3.74274e8 - 4.85587e8i) q^{90} +(4.11384e8 + 3.50825e8i) q^{91} +(7.67273e8 + 2.02041e8i) q^{92} +(1.74077e8 + 1.00503e8i) q^{93} +(4.83368e8 - 1.17412e9i) q^{94} +(-7.69180e8 + 4.44087e8i) q^{95} +(-7.73256e7 + 1.92537e8i) q^{96} +1.15562e9i q^{97} +(-2.62530e8 - 8.74543e8i) q^{98} -4.03681e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 140 q - 18 q^{2} - 6 q^{3} + 84 q^{4} + 17268 q^{8} + 433024 q^{9} - 26724 q^{10} - 65862 q^{11} - 82998 q^{12} - 214260 q^{14} + 410128 q^{16} - 6 q^{17} + 1110146 q^{18} - 6 q^{19} + 3455012 q^{22} - 5971242 q^{24}+ \cdots - 2771471384 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −22.4274 + 3.00193i −0.991161 + 0.132668i
\(3\) 30.2929 − 17.4896i 0.215921 − 0.124662i −0.388139 − 0.921601i \(-0.626882\pi\)
0.604060 + 0.796939i \(0.293549\pi\)
\(4\) 493.977 − 134.651i 0.964799 − 0.262990i
\(5\) 733.902 − 1271.16i 0.525138 − 0.909565i −0.474434 − 0.880291i \(-0.657347\pi\)
0.999571 − 0.0292740i \(-0.00931954\pi\)
\(6\) −626.888 + 483.183i −0.197474 + 0.152206i
\(7\) −4833.52 − 4121.98i −0.760890 − 0.648880i
\(8\) −10674.4 + 4502.75i −0.921380 + 0.388663i
\(9\) −9229.73 + 15986.4i −0.468919 + 0.812191i
\(10\) −12643.6 + 30711.9i −0.399826 + 0.971194i
\(11\) 10934.3 + 18938.7i 0.225176 + 0.390016i 0.956372 − 0.292151i \(-0.0943709\pi\)
−0.731196 + 0.682167i \(0.761038\pi\)
\(12\) 12609.0 − 12718.4i 0.175535 − 0.177059i
\(13\) −85110.7 −0.826493 −0.413246 − 0.910619i \(-0.635605\pi\)
−0.413246 + 0.910619i \(0.635605\pi\)
\(14\) 120777. + 77935.4i 0.840250 + 0.542199i
\(15\) − 51342.6i − 0.261859i
\(16\) 225882. − 133029.i 0.861673 − 0.507465i
\(17\) −3510.53 + 2026.80i −0.0101942 + 0.00588561i −0.505088 − 0.863068i \(-0.668540\pi\)
0.494894 + 0.868953i \(0.335207\pi\)
\(18\) 159009. − 386239.i 0.357022 − 0.867222i
\(19\) −524035. − 302552.i −0.922505 − 0.532609i −0.0380717 − 0.999275i \(-0.512122\pi\)
−0.884433 + 0.466666i \(0.845455\pi\)
\(20\) 191369. − 726742.i 0.267446 − 1.01565i
\(21\) −218513. − 40330.3i −0.245183 − 0.0452527i
\(22\) −302080. − 391922.i −0.274928 − 0.356695i
\(23\) 1.34205e6 + 774832.i 0.999984 + 0.577341i 0.908243 − 0.418442i \(-0.137424\pi\)
0.0917402 + 0.995783i \(0.470757\pi\)
\(24\) −244607. + 323092.i −0.150494 + 0.198781i
\(25\) −100663. − 174353.i −0.0515393 − 0.0892687i
\(26\) 1.90881e6 − 255496.i 0.819187 − 0.109649i
\(27\) 1.33419e6i 0.483149i
\(28\) −2.94267e6 − 1.38533e6i −0.904755 − 0.425932i
\(29\) − 2.69795e6i − 0.708342i −0.935181 − 0.354171i \(-0.884763\pi\)
0.935181 − 0.354171i \(-0.115237\pi\)
\(30\) 154127. + 1.15148e6i 0.0347402 + 0.259544i
\(31\) 2.87323e6 + 4.97657e6i 0.558782 + 0.967839i 0.997599 + 0.0692619i \(0.0220644\pi\)
−0.438817 + 0.898577i \(0.644602\pi\)
\(32\) −4.66661e6 + 3.66157e6i −0.786732 + 0.617295i
\(33\) 662460. + 382471.i 0.0972404 + 0.0561418i
\(34\) 72647.7 − 55994.3i 0.00932323 − 0.00718602i
\(35\) −8.78701e6 + 3.11902e6i −0.989771 + 0.351328i
\(36\) −2.40670e6 + 9.13968e6i −0.238814 + 0.906922i
\(37\) 6.39979e6 + 3.69492e6i 0.561382 + 0.324114i 0.753700 − 0.657219i \(-0.228267\pi\)
−0.192318 + 0.981333i \(0.561601\pi\)
\(38\) 1.26610e7 + 5.21233e6i 0.985011 + 0.405514i
\(39\) −2.57825e6 + 1.48855e6i −0.178457 + 0.103032i
\(40\) −2.11027e6 + 1.68734e7i −0.130337 + 1.04216i
\(41\) 6.03060e6i 0.333298i 0.986016 + 0.166649i \(0.0532947\pi\)
−0.986016 + 0.166649i \(0.946705\pi\)
\(42\) 5.02174e6 + 248545.i 0.249019 + 0.0123249i
\(43\) −4.80871e6 −0.214496 −0.107248 − 0.994232i \(-0.534204\pi\)
−0.107248 + 0.994232i \(0.534204\pi\)
\(44\) 7.95138e6 + 7.88297e6i 0.319820 + 0.317068i
\(45\) 1.35474e7 + 2.34649e7i 0.492494 + 0.853024i
\(46\) −3.24246e7 − 1.33487e7i −1.06774 − 0.439572i
\(47\) −2.80573e7 + 4.85966e7i −0.838697 + 1.45267i 0.0522882 + 0.998632i \(0.483349\pi\)
−0.890985 + 0.454033i \(0.849985\pi\)
\(48\) 4.51600e6 − 7.98041e6i 0.122792 − 0.216990i
\(49\) 6.37217e6 + 3.98473e7i 0.157908 + 0.987454i
\(50\) 2.78100e6 + 3.60810e6i 0.0629268 + 0.0816420i
\(51\) −70895.9 + 122795.i −0.00146742 + 0.00254165i
\(52\) −4.20427e7 + 1.14602e7i −0.797399 + 0.217359i
\(53\) −2.28667e7 + 1.32021e7i −0.398072 + 0.229827i −0.685652 − 0.727930i \(-0.740483\pi\)
0.287580 + 0.957757i \(0.407149\pi\)
\(54\) −4.00515e6 − 2.99225e7i −0.0640983 − 0.478879i
\(55\) 3.20987e7 0.472994
\(56\) 7.01552e7 + 2.22356e7i 0.953265 + 0.302136i
\(57\) −2.11660e7 −0.265584
\(58\) 8.09905e6 + 6.05080e7i 0.0939741 + 0.702081i
\(59\) −1.12611e7 + 6.50159e6i −0.120989 + 0.0698531i −0.559273 − 0.828983i \(-0.688920\pi\)
0.438284 + 0.898836i \(0.355586\pi\)
\(60\) −6.91333e6 − 2.53621e7i −0.0688662 − 0.252641i
\(61\) −4.14309e6 + 7.17604e6i −0.0383125 + 0.0663591i −0.884546 − 0.466453i \(-0.845532\pi\)
0.846233 + 0.532812i \(0.178865\pi\)
\(62\) −7.93783e7 − 1.02986e8i −0.682243 − 0.885151i
\(63\) 1.10507e8 − 3.92256e7i 0.883811 − 0.313716i
\(64\) 9.36682e7 − 9.61284e7i 0.697882 − 0.716212i
\(65\) −6.24630e7 + 1.08189e8i −0.434023 + 0.751749i
\(66\) −1.60054e7 − 6.58918e6i −0.103829 − 0.0427449i
\(67\) 6.38170e7 + 1.10534e8i 0.386901 + 0.670132i 0.992031 − 0.125995i \(-0.0402122\pi\)
−0.605130 + 0.796127i \(0.706879\pi\)
\(68\) −1.46121e6 + 1.47389e6i −0.00828747 + 0.00835939i
\(69\) 5.42060e7 0.287890
\(70\) 1.87707e8 − 9.63296e7i 0.934413 − 0.479533i
\(71\) − 2.75148e8i − 1.28500i −0.766285 − 0.642501i \(-0.777897\pi\)
0.766285 − 0.642501i \(-0.222103\pi\)
\(72\) 2.65393e7 − 2.12204e8i 0.116384 − 0.930588i
\(73\) −1.34835e8 + 7.78472e7i −0.555713 + 0.320841i −0.751423 − 0.659821i \(-0.770632\pi\)
0.195710 + 0.980662i \(0.437299\pi\)
\(74\) −1.54623e8 − 6.36558e7i −0.599419 − 0.246772i
\(75\) −6.09873e6 − 3.52110e6i −0.0222568 − 0.0128500i
\(76\) −2.99600e8 − 7.88918e7i −1.03010 − 0.271250i
\(77\) 2.52140e7 − 1.36611e8i 0.0817397 − 0.442872i
\(78\) 5.33549e7 − 4.11241e7i 0.163211 − 0.125797i
\(79\) 1.48500e8 + 8.57362e7i 0.428947 + 0.247652i 0.698898 − 0.715222i \(-0.253674\pi\)
−0.269951 + 0.962874i \(0.587008\pi\)
\(80\) −3.32481e6 − 3.84762e8i −0.00907533 − 1.05024i
\(81\) −1.58334e8 − 2.74243e8i −0.408688 − 0.707869i
\(82\) −1.81034e7 − 1.35251e8i −0.0442179 − 0.330352i
\(83\) 7.08642e8i 1.63899i 0.573088 + 0.819494i \(0.305745\pi\)
−0.573088 + 0.819494i \(0.694255\pi\)
\(84\) −1.13371e8 + 9.50069e6i −0.248453 + 0.0208208i
\(85\) 5.94990e6i 0.0123630i
\(86\) 1.07847e8 − 1.44354e7i 0.212600 − 0.0284567i
\(87\) −4.71861e7 − 8.17287e7i −0.0883033 − 0.152946i
\(88\) −2.01993e8 − 1.52925e8i −0.359058 − 0.271836i
\(89\) 5.24187e8 + 3.02639e8i 0.885587 + 0.511294i 0.872496 − 0.488621i \(-0.162500\pi\)
0.0130904 + 0.999914i \(0.495833\pi\)
\(90\) −3.74274e8 − 4.85587e8i −0.601309 − 0.780146i
\(91\) 4.11384e8 + 3.50825e8i 0.628871 + 0.536295i
\(92\) 7.67273e8 + 2.02041e8i 1.11662 + 0.294032i
\(93\) 1.74077e8 + 1.00503e8i 0.241305 + 0.139318i
\(94\) 4.83368e8 − 1.17412e9i 0.638561 − 1.55109i
\(95\) −7.69180e8 + 4.44087e8i −0.968885 + 0.559386i
\(96\) −7.73256e7 + 1.92537e8i −0.0929186 + 0.231362i
\(97\) 1.15562e9i 1.32539i 0.748891 + 0.662694i \(0.230587\pi\)
−0.748891 + 0.662694i \(0.769413\pi\)
\(98\) −2.62530e8 − 8.74543e8i −0.287516 − 0.957776i
\(99\) −4.03681e8 −0.422357
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 56.10.m.a.19.3 yes 140
7.3 odd 6 inner 56.10.m.a.3.44 yes 140
8.3 odd 2 inner 56.10.m.a.19.44 yes 140
56.3 even 6 inner 56.10.m.a.3.3 ✓ 140
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.10.m.a.3.3 ✓ 140 56.3 even 6 inner
56.10.m.a.3.44 yes 140 7.3 odd 6 inner
56.10.m.a.19.3 yes 140 1.1 even 1 trivial
56.10.m.a.19.44 yes 140 8.3 odd 2 inner