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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(99,925)] 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("925.99"); 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([9, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.ba (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,0,0,6,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 324.6
Character \(\chi\) \(=\) 925.324
Dual form 925.2.ba.b.374.6

$q$-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+(0.0186364 + 0.105693i) q^{2} +(-2.79245 - 0.492384i) q^{3} +(1.86856 - 0.680101i) q^{4} -0.304317i q^{6} +(2.52717 - 3.01176i) q^{7} +(0.214028 + 0.370707i) q^{8} +(4.73626 + 1.72386i) q^{9} +(2.81064 + 4.86818i) q^{11} +(-5.55274 + 0.979097i) q^{12} +(-2.10187 + 0.765019i) q^{13} +(0.365418 + 0.210974i) q^{14} +(3.01134 - 2.52681i) q^{16} +(5.73090 + 2.08588i) q^{17} +(-0.0939318 + 0.532714i) q^{18} +(-3.23982 - 0.571267i) q^{19} +(-8.53993 + 7.16586i) q^{21} +(-0.462149 + 0.387789i) q^{22} +(-1.10822 + 1.91949i) q^{23} +(-0.415132 - 1.14057i) q^{24} +(-0.120028 - 0.207895i) q^{26} +(-5.01005 - 2.89255i) q^{27} +(2.67387 - 7.34639i) q^{28} +(-1.38356 + 0.798797i) q^{29} -2.45822i q^{31} +(0.979005 + 0.821483i) q^{32} +(-5.45157 - 14.9781i) q^{33} +(-0.113658 + 0.644587i) q^{34} +10.0224 q^{36} +(-2.28710 + 5.63641i) q^{37} -0.353071i q^{38} +(6.24606 - 1.10135i) q^{39} +(8.92828 - 3.24963i) q^{41} +(-0.916531 - 0.769061i) q^{42} +12.5847 q^{43} +(8.56271 + 7.18497i) q^{44} +(-0.223528 - 0.0813577i) q^{46} +(-6.37482 - 3.68051i) q^{47} +(-9.65318 + 5.57326i) q^{48} +(-1.46859 - 8.32880i) q^{49} +(-14.9762 - 8.64652i) q^{51} +(-3.40719 + 2.85897i) q^{52} +(1.73609 + 2.06899i) q^{53} +(0.212352 - 0.583431i) q^{54} +(1.65737 + 0.292238i) q^{56} +(8.76575 + 3.19047i) q^{57} +(-0.110211 - 0.131345i) q^{58} +(0.823360 + 0.981242i) q^{59} +(-0.313577 - 0.861547i) q^{61} +(0.259815 - 0.0458124i) q^{62} +(17.1612 - 9.90800i) q^{63} +(3.86244 - 6.68995i) q^{64} +(1.48147 - 0.855328i) q^{66} +(1.73285 - 2.06513i) q^{67} +12.1272 q^{68} +(4.03976 - 4.81440i) q^{69} +(1.38565 - 7.85840i) q^{71} +(0.374645 + 2.12472i) q^{72} +5.34759i q^{73} +(-0.638350 - 0.136687i) q^{74} +(-6.44232 + 1.13595i) q^{76} +(21.7648 + 3.83771i) q^{77} +(0.232809 + 0.639636i) q^{78} +(5.87032 - 6.99598i) q^{79} +(0.982962 + 0.824803i) q^{81} +(0.509853 + 0.883091i) q^{82} +(0.771346 - 2.11925i) q^{83} +(-11.0839 + 19.1979i) q^{84} +(0.234535 + 1.33011i) q^{86} +(4.25683 - 1.54936i) q^{87} +(-1.20311 + 2.08385i) q^{88} +(0.660110 + 0.786689i) q^{89} +(-3.00773 + 8.26367i) q^{91} +(-0.765325 + 4.34037i) q^{92} +(-1.21039 + 6.86445i) q^{93} +(0.270198 - 0.742363i) q^{94} +(-2.32934 - 2.77600i) q^{96} +(-4.70063 + 8.14173i) q^{97} +(0.852923 - 0.310438i) q^{98} +(4.91989 + 27.9021i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 6 q^{4} - 6 q^{7} - 18 q^{8} - 42 q^{12} - 18 q^{13} + 18 q^{14} - 18 q^{16} + 12 q^{17} + 24 q^{18} + 18 q^{19} + 24 q^{22} - 24 q^{23} + 30 q^{24} - 48 q^{28} - 18 q^{29} + 24 q^{33} - 66 q^{34}+ \cdots + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0186364 + 0.105693i 0.0131780 + 0.0747359i 0.990688 − 0.136155i \(-0.0434746\pi\)
−0.977510 + 0.210891i \(0.932363\pi\)
\(3\) −2.79245 − 0.492384i −1.61222 − 0.284278i −0.706360 − 0.707852i \(-0.749664\pi\)
−0.905861 + 0.423574i \(0.860775\pi\)
\(4\) 1.86856 − 0.680101i 0.934281 − 0.340050i
\(5\) 0 0
\(6\) − 0.304317i − 0.124237i
\(7\) 2.52717 − 3.01176i 0.955180 − 1.13834i −0.0351189 − 0.999383i \(-0.511181\pi\)
0.990299 − 0.138956i \(-0.0443746\pi\)
\(8\) 0.214028 + 0.370707i 0.0756703 + 0.131065i
\(9\) 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 \(0.511476\pi\)
\(12\) −5.55274 + 0.979097i −1.60294 + 0.282641i
\(13\) −2.10187 + 0.765019i −0.582954 + 0.212178i −0.616628 − 0.787255i \(-0.711502\pi\)
0.0336733 + 0.999433i \(0.489279\pi\)
\(14\) 0.365418 + 0.210974i 0.0976621 + 0.0563852i
\(15\) 0 0
\(16\) 3.01134 − 2.52681i 0.752835 − 0.631703i
\(17\) 5.73090 + 2.08588i 1.38995 + 0.505900i 0.925179 − 0.379532i \(-0.123915\pi\)
0.464769 + 0.885432i \(0.346137\pi\)
\(18\) −0.0939318 + 0.532714i −0.0221399 + 0.125562i
\(19\) −3.23982 − 0.571267i −0.743265 − 0.131058i −0.210823 − 0.977524i \(-0.567614\pi\)
−0.532442 + 0.846467i \(0.678726\pi\)
\(20\) 0 0
\(21\) −8.53993 + 7.16586i −1.86357 + 1.56372i
\(22\) −0.462149 + 0.387789i −0.0985306 + 0.0826770i
\(23\) −1.10822 + 1.91949i −0.231079 + 0.400240i −0.958126 − 0.286347i \(-0.907559\pi\)
0.727047 + 0.686588i \(0.240892\pi\)
\(24\) −0.415132 − 1.14057i −0.0847384 − 0.232817i
\(25\) 0 0
\(26\) −0.120028 − 0.207895i −0.0235395 − 0.0407716i
\(27\) −5.01005 − 2.89255i −0.964184 − 0.556672i
\(28\) 2.67387 − 7.34639i 0.505313 − 1.38834i
\(29\) −1.38356 + 0.798797i −0.256920 + 0.148333i −0.622929 − 0.782279i \(-0.714057\pi\)
0.366009 + 0.930611i \(0.380724\pi\)
\(30\) 0 0
\(31\) − 2.45822i − 0.441509i −0.975329 − 0.220755i \(-0.929148\pi\)
0.975329 − 0.220755i \(-0.0708520\pi\)
\(32\) 0.979005 + 0.821483i 0.173065 + 0.145219i
\(33\) −5.45157 − 14.9781i −0.948996 − 2.60735i
\(34\) −0.113658 + 0.644587i −0.0194922 + 0.110546i
\(35\) 0 0
\(36\) 10.0224 1.67040
\(37\) −2.28710 + 5.63641i −0.375997 + 0.926621i
\(38\) − 0.353071i − 0.0572757i
\(39\) 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 \(0.122802\pi\)
\(42\) −0.916531 − 0.769061i −0.141424 − 0.118669i
\(43\) 12.5847 1.91916 0.959578 − 0.281444i \(-0.0908134\pi\)
0.959578 + 0.281444i \(0.0908134\pi\)
\(44\) 8.56271 + 7.18497i 1.29088 + 1.08317i
\(45\) 0 0
\(46\) −0.223528 − 0.0813577i −0.0329575 − 0.0119955i
\(47\) −6.37482 − 3.68051i −0.929864 − 0.536857i −0.0430951 − 0.999071i \(-0.513722\pi\)
−0.886768 + 0.462214i \(0.847055\pi\)
\(48\) −9.65318 + 5.57326i −1.39332 + 0.804431i
\(49\) −1.46859 − 8.32880i −0.209799 − 1.18983i
\(50\) 0 0
\(51\) −14.9762 − 8.64652i −2.09709 − 1.21075i
\(52\) −3.40719 + 2.85897i −0.472492 + 0.396468i
\(53\) 1.73609 + 2.06899i 0.238470 + 0.284197i 0.871985 − 0.489533i \(-0.162833\pi\)
−0.633515 + 0.773731i \(0.718388\pi\)
\(54\) 0.212352 − 0.583431i 0.0288974 − 0.0793950i
\(55\) 0 0
\(56\) 1.65737 + 0.292238i 0.221475 + 0.0390520i
\(57\) 8.76575 + 3.19047i 1.16105 + 0.422588i
\(58\) −0.110211 − 0.131345i −0.0144715 − 0.0172464i
\(59\) 0.823360 + 0.981242i 0.107192 + 0.127747i 0.816973 − 0.576676i \(-0.195650\pi\)
−0.709781 + 0.704423i \(0.751206\pi\)
\(60\) 0 0
\(61\) −0.313577 − 0.861547i −0.0401495 − 0.110310i 0.917998 − 0.396585i \(-0.129805\pi\)
−0.958147 + 0.286276i \(0.907583\pi\)
\(62\) 0.259815 − 0.0458124i 0.0329966 − 0.00581819i
\(63\) 17.1612 − 9.90800i 2.16210 − 1.24829i
\(64\) 3.86244 − 6.68995i 0.482805 − 0.836244i
\(65\) 0 0
\(66\) 1.48147 − 0.855328i 0.182356 − 0.105284i
\(67\) 1.73285 − 2.06513i 0.211701 − 0.252296i −0.649736 − 0.760160i \(-0.725120\pi\)
0.861437 + 0.507864i \(0.169565\pi\)
\(68\) 12.1272 1.47063
\(69\) 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 \(-0.101455\pi\)
\(72\) 0.374645 + 2.12472i 0.0441524 + 0.250401i
\(73\) 5.34759i 0.625888i 0.949772 + 0.312944i \(0.101315\pi\)
−0.949772 + 0.312944i \(0.898685\pi\)
\(74\) −0.638350 − 0.136687i −0.0742067 − 0.0158895i
\(75\) 0 0
\(76\) −6.44232 + 1.13595i −0.738985 + 0.130303i
\(77\) 21.7648 + 3.83771i 2.48032 + 0.437348i
\(78\) 0.232809 + 0.639636i 0.0263604 + 0.0724246i
\(79\) 5.87032 − 6.99598i 0.660463 − 0.787109i −0.326989 − 0.945028i \(-0.606034\pi\)
0.987452 + 0.157919i \(0.0504786\pi\)
\(80\) 0 0
\(81\) 0.982962 + 0.824803i 0.109218 + 0.0916447i
\(82\) 0.509853 + 0.883091i 0.0563038 + 0.0975211i
\(83\) 0.771346 − 2.11925i 0.0846662 − 0.232618i −0.890134 − 0.455699i \(-0.849389\pi\)
0.974800 + 0.223081i \(0.0716114\pi\)
\(84\) −11.0839 + 19.1979i −1.20935 + 2.09466i
\(85\) 0 0
\(86\) 0.234535 + 1.33011i 0.0252905 + 0.143430i
\(87\) 4.25683 − 1.54936i 0.456380 − 0.166109i
\(88\) −1.20311 + 2.08385i −0.128252 + 0.222139i
\(89\) 0.660110 + 0.786689i 0.0699715 + 0.0833888i 0.799895 − 0.600140i \(-0.204889\pi\)
−0.729923 + 0.683529i \(0.760444\pi\)
\(90\) 0 0
\(91\) −3.00773 + 8.26367i −0.315296 + 0.866268i
\(92\) −0.765325 + 4.34037i −0.0797907 + 0.452515i
\(93\) −1.21039 + 6.86445i −0.125511 + 0.711811i
\(94\) 0.270198 − 0.742363i 0.0278688 − 0.0765689i
\(95\) 0 0
\(96\) −2.32934 − 2.77600i −0.237737 − 0.283324i
\(97\) −4.70063 + 8.14173i −0.477277 + 0.826668i −0.999661 − 0.0260428i \(-0.991709\pi\)
0.522384 + 0.852710i \(0.325043\pi\)
\(98\) 0.852923 − 0.310438i 0.0861582 − 0.0313590i
\(99\) 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.ba.b.324.6 72
5.2 odd 4 925.2.bb.b.176.7 72
5.3 odd 4 185.2.w.a.176.6 yes 72
5.4 even 2 925.2.ba.c.324.7 72
37.4 even 18 925.2.ba.c.374.7 72
185.4 even 18 inner 925.2.ba.b.374.6 72
185.78 odd 36 185.2.w.a.41.6 ✓ 72
185.113 even 36 6845.2.a.u.1.18 36
185.152 odd 36 925.2.bb.b.226.7 72
185.183 even 36 6845.2.a.t.1.19 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.41.6 ✓ 72 185.78 odd 36
185.2.w.a.176.6 yes 72 5.3 odd 4
925.2.ba.b.324.6 72 1.1 even 1 trivial
925.2.ba.b.374.6 72 185.4 even 18 inner
925.2.ba.c.324.7 72 5.4 even 2
925.2.ba.c.374.7 72 37.4 even 18
925.2.bb.b.176.7 72 5.2 odd 4
925.2.bb.b.226.7 72 185.152 odd 36
6845.2.a.t.1.19 36 185.183 even 36
6845.2.a.u.1.18 36 185.113 even 36