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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(151,925)] 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("925.151"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 176.7
Character \(\chi\) \(=\) 925.176
Dual form 925.2.bb.b.226.7

$q$-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+(-0.105693 + 0.0186364i) q^{2} +(-0.492384 + 2.79245i) q^{3} +(-1.86856 + 0.680101i) q^{4} -0.304317i q^{6} +(3.01176 + 2.52717i) q^{7} +(0.370707 - 0.214028i) q^{8} +(-4.73626 - 1.72386i) q^{9} +(2.81064 + 4.86818i) q^{11} +(-0.979097 - 5.55274i) q^{12} +(0.765019 + 2.10187i) q^{13} +(-0.365418 - 0.210974i) q^{14} +(3.01134 - 2.52681i) q^{16} +(-2.08588 + 5.73090i) q^{17} +(0.532714 + 0.0939318i) q^{18} +(3.23982 + 0.571267i) q^{19} +(-8.53993 + 7.16586i) q^{21} +(-0.387789 - 0.462149i) q^{22} +(1.91949 + 1.10822i) q^{23} +(0.415132 + 1.14057i) q^{24} +(-0.120028 - 0.207895i) q^{26} +(2.89255 - 5.01005i) q^{27} +(-7.34639 - 2.67387i) q^{28} +(1.38356 - 0.798797i) q^{29} -2.45822i q^{31} +(-0.821483 + 0.979005i) q^{32} +(-14.9781 + 5.45157i) q^{33} +(0.113658 - 0.644587i) q^{34} +10.0224 q^{36} +(-5.63641 - 2.28710i) q^{37} -0.353071 q^{38} +(-6.24606 + 1.10135i) q^{39} +(8.92828 - 3.24963i) q^{41} +(0.769061 - 0.916531i) q^{42} -12.5847i q^{43} +(-8.56271 - 7.18497i) q^{44} +(-0.223528 - 0.0813577i) q^{46} +(3.68051 - 6.37482i) q^{47} +(5.57326 + 9.65318i) q^{48} +(1.46859 + 8.32880i) q^{49} +(-14.9762 - 8.64652i) q^{51} +(-2.85897 - 3.40719i) q^{52} +(2.06899 - 1.73609i) q^{53} +(-0.212352 + 0.583431i) q^{54} +(1.65737 + 0.292238i) q^{56} +(-3.19047 + 8.76575i) q^{57} +(-0.131345 + 0.110211i) q^{58} +(-0.823360 - 0.981242i) q^{59} +(-0.313577 - 0.861547i) q^{61} +(0.0458124 + 0.259815i) q^{62} +(-9.90800 - 17.1612i) q^{63} +(-3.86244 + 6.68995i) q^{64} +(1.48147 - 0.855328i) q^{66} +(2.06513 + 1.73285i) q^{67} -12.1272i q^{68} +(-4.03976 + 4.81440i) q^{69} +(1.38565 - 7.85840i) q^{71} +(-2.12472 + 0.374645i) q^{72} +5.34759 q^{73} +(0.638350 + 0.136687i) q^{74} +(-6.44232 + 1.13595i) q^{76} +(-3.83771 + 21.7648i) q^{77} +(0.639636 - 0.232809i) q^{78} +(-5.87032 + 6.99598i) q^{79} +(0.982962 + 0.824803i) q^{81} +(-0.883091 + 0.509853i) q^{82} +(-2.11925 - 0.771346i) q^{83} +(11.0839 - 19.1979i) q^{84} +(0.234535 + 1.33011i) q^{86} +(1.54936 + 4.25683i) q^{87} +(2.08385 + 1.20311i) q^{88} +(-0.660110 - 0.786689i) q^{89} +(-3.00773 + 8.26367i) q^{91} +(-4.34037 - 0.765325i) q^{92} +(6.86445 + 1.21039i) q^{93} +(-0.270198 + 0.742363i) q^{94} +(-2.32934 - 2.77600i) q^{96} +(-8.14173 - 4.70063i) q^{97} +(-0.310438 - 0.852923i) q^{98} +(-4.91989 - 27.9021i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 6 q^{2} - 6 q^{4} + 6 q^{7} + 18 q^{8} - 30 q^{12} + 30 q^{13} - 18 q^{14} - 18 q^{16} + 30 q^{18} - 18 q^{19} - 30 q^{22} - 30 q^{24} + 24 q^{27} - 30 q^{28} + 18 q^{29} + 36 q^{32} + 12 q^{33}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.105693 + 0.0186364i −0.0747359 + 0.0131780i −0.210891 − 0.977510i \(-0.567637\pi\)
0.136155 + 0.990688i \(0.456525\pi\)
\(3\) −0.492384 + 2.79245i −0.284278 + 1.61222i 0.423574 + 0.905861i \(0.360775\pi\)
−0.707852 + 0.706360i \(0.750336\pi\)
\(4\) −1.86856 + 0.680101i −0.934281 + 0.340050i
\(5\) 0 0
\(6\) − 0.304317i − 0.124237i
\(7\) 3.01176 + 2.52717i 1.13834 + 0.955180i 0.999383 − 0.0351189i \(-0.0111810\pi\)
0.138956 + 0.990299i \(0.455625\pi\)
\(8\) 0.370707 − 0.214028i 0.131065 − 0.0756703i
\(9\) −4.73626 − 1.72386i −1.57875 − 0.574619i
\(10\) 0 0
\(11\) 2.81064 + 4.86818i 0.847441 + 1.46781i 0.883485 + 0.468460i \(0.155191\pi\)
−0.0360439 + 0.999350i \(0.511476\pi\)
\(12\) −0.979097 − 5.55274i −0.282641 − 1.60294i
\(13\) 0.765019 + 2.10187i 0.212178 + 0.582954i 0.999433 − 0.0336733i \(-0.0107206\pi\)
−0.787255 + 0.616628i \(0.788498\pi\)
\(14\) −0.365418 − 0.210974i −0.0976621 − 0.0563852i
\(15\) 0 0
\(16\) 3.01134 − 2.52681i 0.752835 − 0.631703i
\(17\) −2.08588 + 5.73090i −0.505900 + 1.38995i 0.379532 + 0.925179i \(0.376085\pi\)
−0.885432 + 0.464769i \(0.846137\pi\)
\(18\) 0.532714 + 0.0939318i 0.125562 + 0.0221399i
\(19\) 3.23982 + 0.571267i 0.743265 + 0.131058i 0.532442 − 0.846467i \(-0.321274\pi\)
0.210823 + 0.977524i \(0.432386\pi\)
\(20\) 0 0
\(21\) −8.53993 + 7.16586i −1.86357 + 1.56372i
\(22\) −0.387789 − 0.462149i −0.0826770 − 0.0985306i
\(23\) 1.91949 + 1.10822i 0.400240 + 0.231079i 0.686588 − 0.727047i \(-0.259108\pi\)
−0.286347 + 0.958126i \(0.592441\pi\)
\(24\) 0.415132 + 1.14057i 0.0847384 + 0.232817i
\(25\) 0 0
\(26\) −0.120028 − 0.207895i −0.0235395 − 0.0407716i
\(27\) 2.89255 − 5.01005i 0.556672 − 0.964184i
\(28\) −7.34639 − 2.67387i −1.38834 − 0.505313i
\(29\) 1.38356 − 0.798797i 0.256920 − 0.148333i −0.366009 − 0.930611i \(-0.619276\pi\)
0.622929 + 0.782279i \(0.285943\pi\)
\(30\) 0 0
\(31\) − 2.45822i − 0.441509i −0.975329 − 0.220755i \(-0.929148\pi\)
0.975329 − 0.220755i \(-0.0708520\pi\)
\(32\) −0.821483 + 0.979005i −0.145219 + 0.173065i
\(33\) −14.9781 + 5.45157i −2.60735 + 0.948996i
\(34\) 0.113658 − 0.644587i 0.0194922 − 0.110546i
\(35\) 0 0
\(36\) 10.0224 1.67040
\(37\) −5.63641 − 2.28710i −0.926621 − 0.375997i
\(38\) −0.353071 −0.0572757
\(39\) −6.24606 + 1.10135i −1.00017 + 0.176357i
\(40\) 0 0
\(41\) 8.92828 − 3.24963i 1.39436 − 0.507507i 0.467863 − 0.883801i \(-0.345024\pi\)
0.926500 + 0.376294i \(0.122802\pi\)
\(42\) 0.769061 − 0.916531i 0.118669 − 0.141424i
\(43\) − 12.5847i − 1.91916i −0.281444 − 0.959578i \(-0.590813\pi\)
0.281444 − 0.959578i \(-0.409187\pi\)
\(44\) −8.56271 − 7.18497i −1.29088 − 1.08317i
\(45\) 0 0
\(46\) −0.223528 − 0.0813577i −0.0329575 − 0.0119955i
\(47\) 3.68051 − 6.37482i 0.536857 − 0.929864i −0.462214 − 0.886768i \(-0.652945\pi\)
0.999071 − 0.0430951i \(-0.0137219\pi\)
\(48\) 5.57326 + 9.65318i 0.804431 + 1.39332i
\(49\) 1.46859 + 8.32880i 0.209799 + 1.18983i
\(50\) 0 0
\(51\) −14.9762 − 8.64652i −2.09709 − 1.21075i
\(52\) −2.85897 − 3.40719i −0.396468 − 0.472492i
\(53\) 2.06899 − 1.73609i 0.284197 − 0.238470i −0.489533 − 0.871985i \(-0.662833\pi\)
0.773731 + 0.633515i \(0.218388\pi\)
\(54\) −0.212352 + 0.583431i −0.0288974 + 0.0793950i
\(55\) 0 0
\(56\) 1.65737 + 0.292238i 0.221475 + 0.0390520i
\(57\) −3.19047 + 8.76575i −0.422588 + 1.16105i
\(58\) −0.131345 + 0.110211i −0.0172464 + 0.0144715i
\(59\) −0.823360 − 0.981242i −0.107192 − 0.127747i 0.709781 − 0.704423i \(-0.248794\pi\)
−0.816973 + 0.576676i \(0.804350\pi\)
\(60\) 0 0
\(61\) −0.313577 − 0.861547i −0.0401495 − 0.110310i 0.917998 − 0.396585i \(-0.129805\pi\)
−0.958147 + 0.286276i \(0.907583\pi\)
\(62\) 0.0458124 + 0.259815i 0.00581819 + 0.0329966i
\(63\) −9.90800 − 17.1612i −1.24829 − 2.16210i
\(64\) −3.86244 + 6.68995i −0.482805 + 0.836244i
\(65\) 0 0
\(66\) 1.48147 − 0.855328i 0.182356 − 0.105284i
\(67\) 2.06513 + 1.73285i 0.252296 + 0.211701i 0.760160 − 0.649736i \(-0.225120\pi\)
−0.507864 + 0.861437i \(0.669565\pi\)
\(68\) − 12.1272i − 1.47063i
\(69\) −4.03976 + 4.81440i −0.486330 + 0.579586i
\(70\) 0 0
\(71\) 1.38565 − 7.85840i 0.164446 − 0.932620i −0.785188 − 0.619258i \(-0.787434\pi\)
0.949634 − 0.313362i \(-0.101455\pi\)
\(72\) −2.12472 + 0.374645i −0.250401 + 0.0441524i
\(73\) 5.34759 0.625888 0.312944 − 0.949772i \(-0.398685\pi\)
0.312944 + 0.949772i \(0.398685\pi\)
\(74\) 0.638350 + 0.136687i 0.0742067 + 0.0158895i
\(75\) 0 0
\(76\) −6.44232 + 1.13595i −0.738985 + 0.130303i
\(77\) −3.83771 + 21.7648i −0.437348 + 2.48032i
\(78\) 0.639636 − 0.232809i 0.0724246 − 0.0263604i
\(79\) −5.87032 + 6.99598i −0.660463 + 0.787109i −0.987452 − 0.157919i \(-0.949521\pi\)
0.326989 + 0.945028i \(0.393966\pi\)
\(80\) 0 0
\(81\) 0.982962 + 0.824803i 0.109218 + 0.0916447i
\(82\) −0.883091 + 0.509853i −0.0975211 + 0.0563038i
\(83\) −2.11925 − 0.771346i −0.232618 − 0.0846662i 0.223081 − 0.974800i \(-0.428389\pi\)
−0.455699 + 0.890134i \(0.650611\pi\)
\(84\) 11.0839 − 19.1979i 1.20935 − 2.09466i
\(85\) 0 0
\(86\) 0.234535 + 1.33011i 0.0252905 + 0.143430i
\(87\) 1.54936 + 4.25683i 0.166109 + 0.456380i
\(88\) 2.08385 + 1.20311i 0.222139 + 0.128252i
\(89\) −0.660110 − 0.786689i −0.0699715 − 0.0833888i 0.729923 − 0.683529i \(-0.239556\pi\)
−0.799895 + 0.600140i \(0.795111\pi\)
\(90\) 0 0
\(91\) −3.00773 + 8.26367i −0.315296 + 0.866268i
\(92\) −4.34037 − 0.765325i −0.452515 − 0.0797907i
\(93\) 6.86445 + 1.21039i 0.711811 + 0.125511i
\(94\) −0.270198 + 0.742363i −0.0278688 + 0.0765689i
\(95\) 0 0
\(96\) −2.32934 − 2.77600i −0.237737 − 0.283324i
\(97\) −8.14173 − 4.70063i −0.826668 − 0.477277i 0.0260428 − 0.999661i \(-0.491709\pi\)
−0.852710 + 0.522384i \(0.825043\pi\)
\(98\) −0.310438 − 0.852923i −0.0313590 − 0.0861582i
\(99\) −4.91989 − 27.9021i −0.494468 − 2.80427i
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 925.2.bb.b.176.7 72
5.2 odd 4 925.2.ba.c.324.7 72
5.3 odd 4 925.2.ba.b.324.6 72
5.4 even 2 185.2.w.a.176.6 yes 72
37.4 even 18 inner 925.2.bb.b.226.7 72
185.4 even 18 185.2.w.a.41.6 ✓ 72
185.39 odd 36 6845.2.a.u.1.18 36
185.78 odd 36 925.2.ba.c.374.7 72
185.109 odd 36 6845.2.a.t.1.19 36
185.152 odd 36 925.2.ba.b.374.6 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.41.6 ✓ 72 185.4 even 18
185.2.w.a.176.6 yes 72 5.4 even 2
925.2.ba.b.324.6 72 5.3 odd 4
925.2.ba.b.374.6 72 185.152 odd 36
925.2.ba.c.324.7 72 5.2 odd 4
925.2.ba.c.374.7 72 185.78 odd 36
925.2.bb.b.176.7 72 1.1 even 1 trivial
925.2.bb.b.226.7 72 37.4 even 18 inner
6845.2.a.t.1.19 36 185.109 odd 36
6845.2.a.u.1.18 36 185.39 odd 36